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^1Dipartimento di Fisica, Università di Roma Sapienza, Italy ^2Dipartimento di Fisica, Universitá di Perugia and Istituto per l'Officina Materiali of CNR, Via A. Pascoli, I-06123 Perugia, Italy ^3Dipartimento di Fisica, Università di Perugia, Via Alessandro Pascoli, I-06123 Perugia, Italy The temperature dependence of the surface plasmon resonance in small metal spheres is calculated using an electron gas model within the Random Phase Approximation. The calculation is mainly devoted to the study of spheres with diameters up to at least 10 nm, where quantum effects can still be relevant and simple plasmon pole approximation for the dielectric function is no more appropriate. We find a possible blue shift of the plasmon resonance position when the temperature is increased while keeping the size of the sphere fixed. The blue shift is appreciable only when the temperature is a large fraction of the Fermi energy. These results provide a guide for pump and probe experiments with a high time resolution, and tailored to study the excited electron system before thermalisation with the lattice takes place. 71.10.Ca; 71.45.Gm; 78.67.Bf Temperature dependence of the surface plasmon resonance in small electron gas fragments, self consistent field approximation. C. Fasolato^1, F. Sacchetti^2, P. Tozzi^3, C. Petrillo^2December 30, 2023 ============================================================================================================================= § INTRODUCTION The electron gas is a proven prototype for interacting electrons in metals and it is a powerful reference model in the development of modern theories of many electron systems. The knowledge of the electron gas as a highly symmetric model containing, nonetheless, the physics of the real systems, has been instrumental to the development of the Density Functional Theory <cit.> (DFT). So, there exists an enormous body of studies <cit.> on the electron gas properties, even though a less touched subject concerns the properties of the electron gas when confined within a limited volume <cit.>. Small metal fragments have been extensively investigated over the last years because of their very exciting optical properties that are exploited for applications in different areas <cit.>. Small, that is nanometer scale, metal spheres are used in optical techniques, like SERS, SEIRA, etc., whose empirical development relies upon the classical Mie's theory <cit.> of light diffusion from a metal surface, which provides a reasonable working model for experiments in the visible region <cit.>. In practical applications rather small nanoparticles are employed, which might cause quantum mechanics effects to show up following the transition from the continuous energy spectrum of an extended system into the discrete energy spectrum of a size-limited system. Although quantum effects should be visible in optical spectra, no experimental evidence is available. Moereover, there are controversial indications on quantum effects as being quite visible in small spheres <cit.> against their conjectured suppression. At the core of the debate is the capability of a proper treatment of the dynamic screening[From the book of Pines and Nozieres <cit.>: Indeed, the central problem in developing a divergence-free theory of electron systems is that of introducing the concept of dynamic screening in consistent fashion.] in the extended versus confined electron system.The theoretical approach is nowadays challenged by the experimental possibilities opened by new sources like the Free Electron Lasers operating in a wide wavelength region <cit.>. Indeed, it is becoming possible to get experimental data on the behavior of materials in conditions of a highly excited electron system, which enhances the role of the dynamical screening.The seminal paper of Wood and Ashcroft <cit.> on optical response of small fragments of electron gas, initiated the theoretical research on specific fragments containing enough electrons using the most advanced techniques <cit.>. These modern techniques are very efficient with a solid theoretical foundation, based on the DFT, extended to the time dependent problems <cit.>, although two basic limitations affect such an approach. First, the size of the fragment is too small and second, and more important, the calculation is purely numerical so limiting the understanding of the physical mechanisms responsible for the observed behavior. Whereas the first condition might be relaxed by improving the numerical technique, the second one is intrinsic to fully numerical approaches. Further, the DFT is tailored to describe zero-temperature problems, whereas the investigation of quantum effects in small fragments finds its counterpart in the experiments on matter in extreme conditions. This possibility is offered by the high energy density available at the new coherent light sources, the Free Electron Lasers (FEL), over wide wavelength regions <cit.>. FELs experiments can be exploited to investigate the behavior of matter at extreme temperature conditions, as T> 10^4 K. The good time resolution achievable in pump and probe experiments allows first for strongly exciting the electron system, which emphasizes the role of the dynamic screening, and, second, for probing the optical response at these conditions. The finite temperature is still a challenging problem even for the electron gas <cit.>, i.e. a system where the most advanced theoretical and simulation techniques can be applied. In this paper we present a study of the longitudinal dielectric function of a fragment of electron gas using the self-consistent approximation <cit.> for electrons confined in a finite volume Ω. A spherical volume Ω is chosen because it corresponds to a more realistic shape and it is characterized by non-uniformly spaced energy levels, a condition that is relevant when searching for quantum effects evidence. This study goes beyond that presented in Ref. <cit.> on a fragment of electron gas treated within the Random Phase Approximation (RPA). The present approach enables to study a system containing several thousands of electrons to recover the surface plasmon response using the classical boundary conditions <cit.>. In the following sections the fundamental results of the self-consistent field approach <cit.> are recalled for the straightforward application to the confined electron gas, in order to identify the quantum effects related to the discrete distribution of the electron states arising from the finite volume boundary conditions. Our calculation is computationally fast enough to enable the investigation of the effects brought about by the change of relevant parameters like the electron density n_e = 3/4 π (r_s a_B)^3 (a_B, Bohr radius) and the size of the fragments. We focus on the temperature dependence of the surface plasmon resonance in a single gold nanoparticle up to T ≃ 10^5 K, that is k_B T = 1 Ryd. § CALCULATION OF THE DIELECTRIC FUNCTION IN A FINITE VOLUME We start from the general result of the self-consistent field approximation <cit.> and consider an electron gas confined in the finite volume Ω. In a finite system the boundaries break down the translational symmetry of the electron gas thus introducing several effects which are not present in the extended system <cit.>.Let ϕ_k( r) be the single particle wave function, k a generic quantum number, and ϵ_k the associated energy. The actual expression of the wave function depends on the shape of the system volume and analytical forms are available for the parallelepiped and the sphere cases. For a sphere, the wave function ϕ_k( r) is proportional to the product of a spherical Bessel function j_l(k_n r) times a spherical harmonic Y_l m(r̂). The corresponding energy ϵ_n l is given by the n-th zero z_n l of j_l(x), that is ϵ_n l = ħ^2 k_n^2 / 2 m = ħ^2 z_n l^2 / 2 m R^2, being R the sphereradius. Unfortunately, there is no explicit formula for the values of the zeros of the spherical Bessel functions when l > 0, therefore they must be calculated numerically. It is important to note that, apart from the case l = 0, the spacing of z_n l is not uniform and there is a continuous, almost linear, increase of the position of the first zero on increasing l.The diagonal part of the longitudinal dielectric function ϵ(q,ω), at thermodynamic equilibrium and in the frequency domain, can be obtained from the von Neumann-Liouville equation for the single particle density matrix linearized in order to derive the response function of the system to an external (weak and adiabatically switched) electric potential. We confine ourselves to systems with a paired spin distribution. The result is: ϵ(q,ω) = 1 - 2 v(q)lim_η→ 0^+ ∑_k k' f(E_k') - f(E_k)E_k' - E_k - ħω + i η M^2_k k'( q)where f(E) is the (Fermi) population factor, η is a vanishingly small positive number that governs the adiabatic switching of the external potential, v(q) is the Fourier transform of the bare Coulomb electron potential 4 π e^2 / q^2 and M^2_k k'( q) is the matrix element of the Fourier transform of the electron density. This expression reduces to the well known Lindhard dielectric function for the bulk electron gas and it does not contain further approximation other than those related to the RPA <cit.>. We observe that in the case of a small fragment, the broken translational symmetry allows also for intra-band transitions that are forbidden in the homogeneous gas.A reliable calculation of the dielectric function requires a highly accurate calculation of the matrix elements. This task becomes increasingly complex for larger sizes of the fragments; a situation that is relevant for the present calculation, as small fragments can be treated using atomistic approaches (e.g. TDDFT). In the case of a spherical fragment, the calculation of the matrix elements is more complex as no fully analytical formulation exists. The matrix elements are expressed in terms of integrals of three spherical Bessel functions and 3j symbols. Through a proper manipulation of these formulas, the calculation reduces to the sum of Bessel function integrals and 3j symbols that depend only on the three angular quantum numbers l_1, l_2, l_3 over a wide range of values, namely up to 100-200 depending on the size of the particle. Considering that the matrix elements depend on q only through the integrals of the Bessel functions, the calculation of the 3j symbols can be carried out just once and then stored. Similarly, the matrix elements are calculated at each q value and stored to perform the calculation of the dielectric function as a function of the energy. To determine the effect of the parameter η, we note that the limit as a function of η cannot be carried out numerically and the only possibility is to calculate ϵ(q,ω) at different values of η. In any case, η should be considered with some care. Indeed, 1 η is often identified as a relaxation time responsible forbroadening the quantum peaks that are expected to appear as a consequence of the discreteness of the electron energy spectrum in the confined system. This straightforward relaxation time approximation is, in our opinion, not correct since the continuity equation is locally violated as discussed in Ref. <cit.>. For a more detailed discussion on the problem of producing a conserving theoretical approach one can refer to the reference paper of Baym and Kadanoff <cit.> and to some applications like that of Refs. <cit.>. § NUMERICAL CALCULATION OF THE DIELECTRIC FUNCTION The numerical determination of ϵ (q, ω) requires, as said, the calculation of integrals of the spherical Bessel functions and 3j-symbols over a wide range of angular momentum with high accuracy. In the high temperature case, the maximum angular momentum to be included is fairly large but the calculation could be carried on in a rather accurate way. For a safely accurate calculation up to temperatures T = 10^5 K in the case of a 10 nm sphere, we decided to include all the contributions up to l = 200, so that energies up to 5 Ryd were considered. Accordingly, the integrals of the Bessel functions had to be calculated with a six digit accuracy. The calculation of the 3j symbols was performed following the proposal of Ref. <cit.> using a specifically written code.After several tests, we concluded that the calculation of the dielectric function was accurate to better than three digits. A description of the numerical techniques is given in a more extended paper that is in preparation.The calculation of the dielectric function was performed at r_s = 3 (the typical average electron density of a gold particle) for several q values and for an energy range up to 1 Ryd, with varying η. As a typical result, we show in Fig.<ref> the dielectric function of a sphere with radius R = 3.1 nm, at q = 0.2 a.u. andT = 0 K.The results obtained using η = 0.0001 Ryd and η = 0.001 Ryd are shown in the two panels of Fig.<ref>.The overall trend is similar to the RPA result in the homogeneous electron gas, although a complex structure is apparent in the low energy region, which is clearly related to the discrete level structure of the fragment. Also, while the imaginary part of the dielectric function of the electron gas is zero outside the particle-hole pair region, limited by the two parabolic functions ħω / (2 k_F)^2 = (q/2 k_F)^2 + q/2 k_F and ħω / (2 k_F)^2 = (q/2 k_F)^2 - q/2 k_F, the same function calculated for the sphere can extend beyond this region, due to the breaking of the translational symmetry for the reduced spherical volume. We note also that the η = 0.001 Ryd corresponds to a relaxation time τ≃ 5·10^-14 s^-1, a value fairly higher than that derived from the Drude model for metals Ref. <cit.>. Further insight on the temperature dependence of the plasmon resonance is provided by the study of the response function which is proportional to the extinction cross section, under different conditions <cit.>, i.e.A(q,ω) = - [1 ϵ(q,ω) + C ϵ_m ]where ϵ_m is the dielectric function of the external medium and C is a constant equal to 0 for a bulk system, to 1 for an infinite surface and to 2 for a sphere <cit.>. We assumed ϵ_m = 1 corresponding to the case of particles in the empty space, as done in Ref. <cit.> so that a direct comparison is possible for the case of the R = 1.34 nm sphere. In Fig. <ref> (a) we show the response A(q,ω) for a sphere (C = 2) with radius R = 3.1 nm, containing 7445 electrons at r_s = 3, in vacuum (ϵ_m = 1), as a function of temperature from T = 0 to T = 50000 K. Larger systems can be treated using the present approximation over the same temperature range. We fixed q = 0.01 a.u. to approximate the q → 0 limit. We also show the result at T = 0 obtained using ϵ_m = 4, so that the plasmon peak is red-shifted. It is quite interesting to observe that most part of the complex structure visible in the dielectric function (Fig.<ref>) disappears, and the response results in a rather smooth function with a single peak that can be attributed to the surface plasmon. Secondary structures are visible in a fashion similar to what reported in Ref. <cit.>. For comparison purposes, the response A(q,ω) calculated at R = 1.34 nm and T = 316 K is shown in Fig.<ref> (b) against the companion results from Ref. <cit.>. We observe that the present calculation of the response function provides results very similar to the advanced TDDFT, which represents a reasonable support for the validity of our high temperature calculation. On further increasing the temperature, further smoothing of the peak structure occurs, as expected. Considering that the function A(q,ω) is related to the quantity measured in a real transmission experiment, these results indicate that quantum effects can be difficult to observe in practical conditions.Fig. <ref> shows the trend of the surface plasmon resonance position as a function of temperature. A continuous increase of the energy of the resonance is readily observed in the range from 0 to 10^5 K, by a quantity that is well detectable with the current experimental capabilities. This result suggests that a pump and probe experiment with adequate time resolution is feasible. Therefore, by heating small metal particles by means of a short (say 50 fs) pump laser pulse, the electronic temperature could be determined by measuring the position of the surface plasmon resonance with a probe delayed by a time interval short enough to maintain a hot electron plasma and no much energy transferred to the lattice. § CONCLUSIONS The present results show that a free electron approximation provides good results for the description of the dielectric function of electron gas fragments, particularly when the fragment is rather large and contains thousands of electrons, at thermal energies comparable to or higher than the Fermi energy. The proposed approach is quite flexible and it is easy to include the exchange-energy contribution. Further improvements can also be considered. Considering that the dielectric function (Eq. <ref>) is the sum of several state-dependent contributions, the state populations can be changed by properly modeling the effect of an incoming photon beam that produces transitions from one state to the other. This also contributes to the increase of the sample electronic temperature or to the presence of hot carriers <cit.>. § ACKNOLEDGEMENTS The Sincrotrone Trieste SCpA is acknowledged for the support the PhD program in Physics. This research has been partially supported by the PRIN (2012Z3N9R9) project of MIUR, Italy.99kohn W. Kohn, Rev. Mod. Phys. 71, 1253 (1998). mahan G. D. Mahan, Many-Particle Physics, (Plenum Press, New York, 1990). simgas E. Townsend and G. W. Bryant, Nano Lett. 12, 429 (2012). sfere1 G. Friesecke, Commun. Math Phys. 184, 143 (1997). sfere2 Q. Sato, Y. Tanaka, M. Kobayashi and Akira Hasegawa Phys. Rev. B 48, 1947 (1993). sfere3 P.-F. Loos and P. M. W. Gill, J. Chem. Phys. 135, 214111 (2011). rev16 W. Zhu, R. 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Dionne, Nature 483, 421 (2012). qm6 E. B. Guidez and C. M. Aikens, Nanoscale 6, 11512 (2014). qm7 J. M. Pitarke, V. M. Silkin, E. V. Chulkov and P. M. Echenique, Rep. Prog. Phys. 70, 87 (2007). qm8 S. M. Morton, D. W. Silverstein, and L. Jensen Chem. Rev. 111, 3962 (2011). mcht S. Groth, T. Schoof, T. Dornheim and M. Bonitz, Phys. Rev. B 93, 085102 (2016). mcht2 E. W. Brown, B. K. Clark, J. L. DuBois and D. M. Ceperley, Phys. Rev. Lett. 110, 146405 (2013).cohen H. Eherenreich and M. H. Cohen, Phys. Rev. 115, 786 (1959). byka G. Baym and L. P. Kadanoff, Phys. Rev. 124, 287 (1961). ash2 G. S. Atwal and N. W. Ashcroft, Phys. Rev. B 65, 115109, (2002). merm N. D. Mermin, Phys. Rev. B 1, 2362 (1970). 3jcal A. Messiah Quantum Mechanics, Vol. 2. (Amsterdam, Netherlands: North-Holland), Appendix C.I pp. 1054-1060, (1962). merash N. D. Mermin and N. W. Ashcroft, Solid State Physics, Saunders College Publishing, (1976). | http://arxiv.org/abs/1704.08655v1 | {
"authors": [
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"Francesco Sacchetti",
"Pietro Tozzi",
"Caterina Petrillo"
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"published": "20170427165857",
"title": "Temperature dependence of the surface plasmon resonance in small electron gas fragments, self consistent field approximation"
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http://arxiv.org/abs/1704.08571v1 | {
"authors": [
"M. Hassanvand",
"Y. Akaishi",
"T. Yamazaki"
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"published": "20170426101513",
"title": "Clear indication of a strong I=0 Kbar-N attraction in the Lambda (1405) region from the CLAS photo-production data"
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Hybrid cell-centred/vertex model for multicellular systems with equilibrium-preserving remodelling P. Mosaffa, A. Rodríguez-Ferran, J.J. Muñoz ([email protected])Unviersitat Politècnica de Catalunya, Barcelona, SpainDecember 30, 2023 ==========================================================================================================================We present a hybrid vertex/cell-centred model for mechanically simulating planar cellular monolayers undergoing cell reorganisation.Cell centres are represented by a triangular nodal network, while the cell boundaries are formed by an associated vertex network. The two networks are coupled through a kinematic constraint which we allow to relax progressively. Special attention is paid to the change of cell-cell connectivity due to cell reorganisation or remodelling events. We handle thesesituations by using a variable resting length and applying an Equilibrium-Preserving Mapping (EPM) on the new connectivity, which computesa new set of resting lengths that preserve nodal and vertex equilibrium. We illustrate the properties of the model by simulating monolayers subjected to imposed extension and during a wound healing process. The evolution of forces and the EPM are analysed during the remodelling events. As a by-product, the proposed technique enables to recover fully vertex or fully cell-centred models in a seamlessly manner by modifying a numerical parameter of the model. keywords: cell-centred, vertex model, remodelling, tessellation, biomechanics, tissues. § INTRODUCTION Mechanical analysis of embryonic tissues has gained attention in recent years. Biologists and experimentalists have been able to accurately track the kinematic information of tissues and organs, but the mechanical forces that drive these shape changes have resulted far more elusive, despite evidence that genetic expression and mechanics are tightly coupled in cell migration <cit.>, wound healing <cit.> or embryo development <cit.>. The quantification of themechanical forces in morphogenesis has given rise to numerous and diverse numerical approaches <cit.>, which can be classified into two main types: continuum and cell-based models. The former allow to incorporate well-known constitutive behaviour of solids or fluids <cit.> and can be discretised with robust techniques such as finite elements <cit.>. The latter instead have the advantage of explicitly representing junctional mechanics and capturing the discrete and cellular nature of tissues <cit.>. Due to recent evidence on the role of contractile forces at cell-cell junctions in embryonic development <cit.> and wound healing <cit.>, we will here present a methodology based on the latter approach.Cell-based models can be described through cell-centred or off-latice models <cit.>, or vertex models (see for instance <cit.> and the review articles <cit.>). The first approach focuses on establishing forces between cell-centres and can easily include variations on the number of cells (cell proliferation or apoptosis). The second approach is instead driven by the mechanical forces at the cell-cell junctions <cit.>, which seem to determine the emergent properties of tissues and monolayers <cit.>.The model proposed here aims to gather the advantages of the two approaches: define cell-cell interactions between centres and at the cell-cell junctions, but include the cell as an essential unit in order to ease the transitions in the cell-cell contacts. We resort to Delaunay triangulation of the cell-centres, and a barycentric interpolation of the vertices on the cell-boundaries. Both nodes and vertices are kinematically coupled by this interpolation, which has effects on the resulting equilibrium equations. The use of Voronoi tessellations has been well studied for domain decomposition <cit.> or for discretising partial differential equations in elasticity, diffusion, fluid dynamics or electrostatics. Some examples are the Natural ElementMethod <cit.>, the Voronoi Cell Finite Element method <cit.>, the Voronoi Interface Element <cit.> or the particle-in-cell methodology <cit.>. In these methods, the tessellation is used for either constructing the interpolation functions, or describing the heterogeneities or interfaces. We resort here to the related barycentric tessellation, where the vertices of the network are built from the barycentres of each triangle instead of the bisectors, as it is the case in the Voronoi diagram. We choose this alternative tessellation to guarantee that the vertices are inside each triangle, even when the Delaunay triangulation is deformed, and thus may potentially violate the Delaunay condition. The use of automatic tessellation is also motivated in our case by the need to handle cell-cell connectivity changes in a robust and accurate manner, and thus avoid the design of specific algorithms during remodelling events, as it is customary in vertex models in two <cit.> and three dimensions <cit.>. The proposed model extends a previous cell-centred model <cit.> with a hybrid approach that incorporates mechanics at the cell boundaries in order to model morphogenetic events driven by contractile forces <cit.>, like for instance germ band extension <cit.> or wound healing <cit.>. Other recent hybrid techniques that couple cell-centred and continuum approaches may be found in <cit.>, but with no specific mechanics at the cell junctions.We point out that our aim is to be able to model multicellular systems, with hundreds of cells. We therefore focus our approach at the cell rather than at the subcellular scale. Other methods for modelling cell mechanics such as the Subcellular Element Model <cit.> or the Immersed Boundary Method <cit.> are more suitable at smaller scales and therefore can simulate cell-cell interaction more accurately. We will first define the model kinematics in Section <ref> andthe equations that describe the mechanical equilibrium of the multicellular system in Section <ref>. The particular viscoelastic rheological model is presented in Section <ref>; it allows to handle inter-cellular remodelling by using the equilibrium-preserving mapping described in Section <ref>. Representative results are presented in Section <ref> and some conclusions are highlighted in Section <ref>.§ TISSUE DISCRETISATION§.§ Nodal and vertex networks In the proposed model the tissue kinematics is defined by the cell-centres or nodes and the cell boundaries, which are formed by a set of vertices. We will denote by ^i the nodal positions (lower case superscript), and by^I the vertex positions (upper case superscript). In <ref> we give a complete list of the notation employed in the article. Figure <ref> shows an example of the nodal and vertex networks that define the domain of a tissue. The bar elements that define each one of the networks will be in turn employed to write the mechanical equilibrium equation. In the next subsections we detail the definitions of the nodal and vertex positions and their relation.§.§ Nodal geometry We will assume that a tissue forms a flat surface and has a constant number of nodes N_nodes. These are kinematically described by their cell-centres positions = {^1,...,^N_nodes} and connectivity , which define a triangulation of the domain into N_tri triangles 𝒯^I, I=1,…, N_tri and N_D edges. We will denote by _n and _n the set of nodal coordinates and connectivity at time t_n. Figure <ref> illustrates the connectivity of the nodal network.The position of the nodes is resolved using mechanical equilibrium, which will be explained in Section <ref>. The connectivities are found resorting to a trimmed Delaunay triangulation in order to obtain a not necessarily convex boundary. Triangles with an aspect ratio larger than a given tolerance are removed, and each pair of connected nodes ^i and ^j are connected with a bar element, with a rheology that will be detailed later. Figure <ref> illustrates this trimming process and the steps for obtaining configuration {_n+1, _n+1} from {_n, _ n}. §.§ Vertex geometryThe boundaries of the cells are defined by a set of connected vertices{^1,…,. .^N_tri}, which define a tessellation of the tissue domain into N̅_nodes cell domains Ω^i, i=1,…, N̅_nodes. Note that N̅_nodes< N_nodes because N̅_nodes does not include the external nodes. Each triangle 𝒯^I is associated to vertex ^I, and each interior node i is surrounded by a number of vertices which is not necessarily constant between time-steps and may vary from cell to cell (see Figure <ref>). The position of vertex ^I is given by a local parametric coordinate ^I in triangle 𝒯^I. The kinematic relation between the nodal positions ^i and the vertices is given by the interpolation ^I=∑_i∈𝒯^I p^i(^I)^i. The previous summation extends to the three nodes of triangle 𝒯^I where vertex I is located. Function p^i(^I) is the standard finite element interpolation function of node i in triangle 𝒯^I evaluated at coordinate ^I. We will initially consider that all parameters ^I have a constant value ^I=1/3{1 1}, whichcorresponds to a barycentric tessellation of the domain. We will eventually allow varying values of ^I in Section <ref>, where ξ-relaxation is introduced. Every two vertices ^I and ^J are connected with a bar element if their corresponding triangles 𝒯^I and 𝒯^J have a common edge. The positions and the connectivity of nodes and vertices in the tessellated network is uniquely defined by , , and all the local coordinates ={^1, …, ^N_tri} which define the vertex locations ^I, I=1,…, N_tri. The rheology ofthe N_V bar elements that join the vertices along the boundary of cells will be also described in Section <ref>. We remark that the Voronoi tessellation of the tissue may be obtained by computingspecificvalues of the parameter ^I for each vertex. However, we will not consider this tessellation in this article because our initial Delaunay triangulation deforms due to mechanical equilibrium, with a potential loss of its Delaunay character. In this case, Voronoi tessellation may become undefined, or lead to crossing bars or overlapping domains. § MECHANICAL EQUILIBRIUM Mechanical equilibrium of the bar elements that form the nodal and vertex networks is computed by minimising the total elastic energy of the two networks. This energy is decomposed as the sum of a nodal contribution W_D(), and a contribution of the vertex network, W_V(()). The minimisation of the total elastic energy W_D() + W_V(()) with respect to the nodal positions in , which are considered the principal kinematic variables, yields the equations ∂ W_D()/∂^i + ∂ W_V(())/∂^i=,i=1,…, N_nodes We will consider each one of the two terms on the left separately in the next subsections.§.§ Cell-centred mechanical equilibrium The cell-cell connectivity defined byincludes information on the set of N_D pairs ij between the N_nodes nodes. Each pair of connected nodes are joined with a bar element that represents the forces between the two cells. This force is derived here from an elastic strain function, W^ij_D() =1/2k_D(^ij)^2,W_D() =∑_ij=1^N_DW^ij_D(),where k_D is the material inter-cellular stiffness, ε^ij=l^ij-L^ij/L^ij is the scalar elastic strain, and l^ij=^i-^j and L^ij are the current and reference lengths. In Section <ref> we will introduce a rheological law where the reference length L^ij (stress-free length of the element) is allowed to vary along time, and thus we may have that L^ij≠ L_0^ij:=_0^i-_0^j. W_D is the total strain function of the network of nodes. In the absence of any other strain function, the minimisation of W_D leads to the equations _D^i:=∑_j∈S^i^ij_D=, i=1,…,N_nodes, where S^i denotes the set of nodes connected to node i and ^ij_D is the nodal traction at node i due to bar ij, which is derived from the elastic strain function W^ij_D as (no summation on i) ^ij_D=∂ W^ij_D/∂^i=-^ji_D=-∂ W^ij_D/∂^j. Figure <ref> shows the traction vectors between two nodes ^i and ^j. Since the system of equations (<ref>) is non-linear with respect to the nodal positions ^i, we resort to a full Newton-Raphson method, which requires linearisation of the set of equations. The expression of the resulting Jacobian is given in <ref>. §.§ Adding vertex mechanical equilibrium The force between any two vertices is also derived here from an elastic strain function, W^IJ_V()=1/2k_V(^IJ)^2W_V()=∑_IJ=1^N_VW_V^IJ() with k_V the cell boundary stretching stiffness. The total mechanical strain energy of the system is the sum of the contributions of the nodal and vertex networks, W_D()+W_V(()). The new nodal positions are found by solving the minimisation problem ^*=_(W_D() + W_V(())). which may be solved in two manners: as a constrained minimisation, where nodes ^i and vertices ^I are independent and coupled through the constraint in (<ref>), or by using this constraint in the expression of the objective function (total strain energy). We choose the latter approach in order to reduce the number of unknowns, and thus the size of the resulting system of equations. In order to deduce the expression of ∂ W_V/∂^i, we define first the vertex tractions as ^IJ_V=∂ W^IJ_V/∂^I=-^JI_V=-∂ W^IJ_V/∂^J. The nodal residuals due to contributions of the vertex network, denoted by _V^i, may be then computed by using the chain rule and the kinematic relation in (<ref>), _V^i :=∂ W_V/∂^i =∑_IJ(∂ W^IJ_V/∂^I∂^I/∂^i +∂ W^IJ_V/∂^J∂^J/∂^i) =∑_IJ(^IJ_V p^i(^I) +^JI_V p^i(^J))=∑_I∈ B^ip^i(^I)∑_J∈ S^I^IJ_V. In the last expression B^i denotes the set of vertices that form the boundary of cell i, centred on ^i, and S^I is the set of vertices connected to vertex I. Note also that the last equality follows from the fact that p^j(^K) vanishes if K∉ B^j. Total mechanical equilibrium is then found by solving the minimisation in (<ref>), which yields, ∑_j∈ S^i^ij_D + ∑_I∈ B^ip^i(^I)∑_J∈ S^I^IJ_V =, i=1,…,N_nodes, which in terms of the force contributions ^i_D and ^i_V reads ^i_D + ^i_V=,i=1,…,N_nodes.The summation in the second term of (<ref>) involves the vertex bars that have at least one vertex on the triangles that surround node ^i. Figure <ref> shows a schematic view of how the boundary of each cell is defined within the tissue, and the traction vectors ^IJ_V and ^JI_V.Mechanical equilibrium of the system is obtained at cell centres (nodes) by solving the set of equations in(<ref>). Since this equation is non-linear with respect to the positions of the nodes, we resort toNewton-Raphson method for linearisation of the equations and to obtain the solution. The linearisation of the terms in (<ref>) is givenin <ref>.Note that the second term in (<ref>) arises due to the kinematic interpolation in (<ref>). This term represents the nodal contribution of the vertex forces (reactions of the constraints in (<ref>)), which is proportional to the values of the shape functions p^i(^I). This equation shows the coupling between nodal and vertex equilibrium. When vertex forces exist (k_V≠ 0), nodal forces and vertex forces are not necessarily equilibrated at nodes and vertex, respectively, that is, we may have that _D^i≠ and ∑_J∈ S^I^IJ_V≠. The latter condition is the equilibrium equation usually imposed in purely vertex models <cit.>. We will analyse the evolution of these resultants in Section <ref> (Numerical results). §.§ Area constraint Cell volume invariance under tissue extension is relevant when the size and the number of cells within the tissue is considered as constant. A two-dimensional area constraint will be imposed here by adding the energy term, W_A=λ_A/2∑_m=1^N̅_nodes(A^m-A_0^m)^2, where λ_A is a penalisation coefficient and A_0^m and A^m are the initial and the current areas of cell m, respectively. The area of cell m can be expressed in terms of its vertices by using Gauss theorem A^m=∫_Ω^mdA=1/2∫_∂Ω^m· ds, whereis an arbitrary point on the boundary of cell m, ds is the differential segment of the cell boundary andis the outward normal. Since each cell boundary forms a polygon, we will break the integral over the whole cell boundary into N_m line integrals. Points between vertices I and J can be obtained by using a linear interpolation =q^I(α)^I + q^J(α)^J, with α∈[-1,1] a local coordinate along the cell boundary segment IJ, andq^I(α)=1/2(1-α) and q^J(α)=1/2(1+α) the interpolation functions. By inserting equation (<ref>) into (<ref>) and noting that ds=l^IJ dα /2, with l^IJ=||^I-^J||, we have A^m=1/2∑_IJ∈ P^m^N_m∫_-1^1∑_I q^I(α)^I·^IJl^IJ/2dα =1/2∑_IJ∈ P^m^N_ml^IJ/2(^I+^J)·^IJ, where P^m denotes the segments of the polygon that surrounds node ^m (see Figure <ref>). The expression above can be simplified as A^m =1/2∑_IJ∈ P^m^N_m(^I×^J)·_z =1/2∑_IJ∈ P^m^N_m^I·^J with =[[0 -1;10 ]]=-^T and such that (^I×^J)·_z=^I·^J. Finally, the total area of the whole set of N̅_nodes cells in the tissue, A_T, can be expressed as A_T=1/2∑_m=1^N̅_nodes∑_IJ∈ P^m^I·^J.The expression of the contribution in (<ref>) is inserted in the energy term in (<ref>), and appended to the total elastic energy,W=W_D()+W_V(())+W_A(()), which is minimised with respect to the nodal positions ^i. This gives riseto an additional nodal contribution,^i_A :=∂ W_A/∂^i=λ_A/2∑_m∈S̅^i(A^m-A^m_0)∑_IJ∈ P ^m(p^i(^I)^J-p^i(^J)^I). The set S̅^i in the first summation includes the nodes that surround node i and also node i itself. Since the force vector above is non-linear, the Jacobian must be complemented with additional terms arising from the linearisation of ^i_A. These terms are given in <ref>. §.§ ξ-RelaxationWhen the values of ^I are kept constant, vertices and cell-centred positions are coupled through the constraint in (<ref>). As pointed out in Section <ref>, this constraint has the effect of altering the usual equilibrium conditions in cell-centred and vertex networks (vanishing of the sum of forces at nodes and at vertices, respectively). In fact, in our equilibrium equations in (<ref>) and (<ref>), the additional force due to _V^i (which contains the tractions ^IJ_V) may be regarded as a reaction force stemming from the constraints in (<ref>).This modified equilibrium may furnish non-smooth and unrealistic deformations at the tissue boundaries, which can then exhibit a zig-zag shape.In order to avoid these effects, we will disregard the constraint (<ref>) for those vertices at the boundary, and relax the value of ^I, which can attain values different from {11}^T/3. Those vertices are then allowed to change their relative positions within the correspondingtriangle 𝒯^I, and may be not necessarily located at the barycentre. In this case, mechanical equilibrium is expressed as a vanishing sum of tractions at the vertex location, as it is customary in vertex models <cit.>. In our hybrid model, we interpret the parametric coordinatesof those vertices as additional unknowns. The energy terms including the vertices are now made dependent on these extra parametric coordinates, i.e. we write W_V((, )) and W_A((, )).When relaxing the constraint, we will further limit the increments ofbetween time-steps, so that their positions are kept not too far from their otherwiseinterpolated value in order to minimise large discontinuities between discrete time-points on the resulting force contributions. This is achieved by adding to the total energy of the system W and at each time t_n+1 a term that penalises the variations of , W_ξ() =λ_ξ/2∑_Irelaxed||_n+1^I-_n^I||^2. By interpreting the factor λ_ξ as a viscous coefficient ≈η/Δ t, this additional term is equivalent to a viscous-like effect, since it generates forces proportional to the incremental vertex positions (or vertex velocities).The extension of the system with additional variablesalso modifies the minimisation problem in (<ref>), which now takes the form {^*, ^*}=_,W(, ), with W(, )=W_D() + W_V((,)) + W_A((,))+W_ξ(). Equilibrium is now represented by two systems of equations, g:={[ _x; _y ]}=, with _x=∇_ W(, ) and _y=∇_ W(, ). Each residual contribution in the total residualis the sum of different energy contributions in (<ref>), so that =_D+_V+_A+_ξ, where each termcontains in turn nodal () and vertex () contributions,^i_x :=∂ W(, )/∂^i=^i_D+_V^i+_A^i+^i_ξ, ^I_y :=∂ W(, )/∂^I=^I_D+_V^I+_A^I+^I_ξ. Since the nodal strain energy W_D does not depend on ^I, and the penalty term W_ξ does not depend on the nodal positions ^i (see equations (<ref>) and (<ref>)), we have that_D^I= and _ξ^i=. The nodal contributions ^i_D, ^i_V and _A^i have been given respectively in (<ref>), (<ref>) and (<ref>). The vertexcontributions require the computations of ∇_ W= ∇_ W_V +∇_ W_A+λ_ξ(_n+1-_n) ∂^I/∂^I =∑_^i∈𝒯^I^i⊗∇ p^i(^I) so that we have, also from equations (<ref>) and (<ref>),^I_V:=∂ W_V/∂^I =∑_JK∂ W^JK_V/∂^J∂^J/∂^I +∂ W^JK_V/∂^K∂^K/∂^I =∑_K∈ S^I∑_^i∈𝒯^I(^IK_V·^i)∇ p^i(^I) ^I_A:=λ_A∑_m=1^N̅_nodes(A^m-A^m_0)∂ A^m/∂^I ^I_ξ :=∇_ W_ξ =λ_ξ(_n+1^I-_n^I) with ∂^I/∂^I given in (<ref>), and∂ A^m/∂^I=1/2∑_KL∈ P^m( δ_KI(∂^K/∂^I)^T^L -δ_LI(∂^L/∂^I)^T^K ).The symbol δ_KI above is the Kronecker delta, which is equal to 1 if K=I and 0 otherwise. We note that if we extended ξ-relaxation to the whole tissue, we could recover standard vertex models, that is, a model where the vertices positions are solely determined by their mechanical equilibrium: sum of forces at each vertex equal to zero. In our numerical simulations we have though just applied ξ-relaxation to specific boundaries of the domain.§ RHEOLOGICAL MODEL So far, the bar elements of the cell-centred and vertex networks have been considered as purely elastic, with a strain function given in equations (<ref>) and (<ref>) respectively. Since cells exhibit both elastic and viscous response <cit.>, we here extend the elastic strain energy function of the bars with the ability to vary their resting length L. The rate of changeof the resting length is given by the evolution law L̇/L=γ where γ is the remodelling rate, andis the elastic strain used either in (<ref>) or (<ref>). It has been previously shown that such a rheological model is equivalent to a Maxwell viscoelastic behaviour <cit.>, and that can be used to simulate tissue fluidisation<cit.> or cell cortex response <cit.>.In order to include the inherent contractility that cells exert <cit.>, the previous evolution law is modified as L̇/L=γ(-^c) with ^c a contractility parameter. This modification aims to attain a homoeostatic elastic strain equal to ^c,at which no further modifications of the resting length take place. The ordinary differential equation (ODE) in (<ref>) is employed for the bar elements of the nodal and vertex networks, and it is solved together with the non-linear equations in (<ref>). In fact, the evolution law is taken into account by first discretising in time the ODE in (<ref>) with a β-weighted scheme. By using the strain definition =(l-L)/L, the discretisation of (<ref>)yields L_n+1-L_n=Δ tγ(l_n+β-L_n+β-^cL_n+β), with (∙)_n+β=(1-β)(∙)_n+β(∙)_n+1. In our numerical tests we have used the value β=0.5. The discretisation in (<ref>) allows us to write ∂ L/∂ l=βΔ tγ/1+βΔ tγ(1+^c). This term is inserted in the traction definitions of ^ij_D ad _V^IJ in (<ref>) and (<ref>), which are then computed with the help of the following derivation, ∂^ij/∂^i=1/L(1-l/L∂ L/∂ l)^ij, ∂^IJ/∂^I=1/L(1-l/L∂ L/∂ l)^IJ, with ^ij=-^ji=^i-^j/||^i-^j|| , ^IJ=-^JI=^I-^J/||^i-^j||. The traction forces in (<ref>) and (<ref>) read then respectively, _D^ij =∂ W_D^ij/∂^i =^ij/L^ij(1-l^ij/L^ij∂ L^ij/∂ l^ij)^ij, _V^IJ =∂ W_V^IJ/∂^I =^IJ/L^IJ(1-l^IJ/L^IJ∂ L^IJ/∂ l^IJ)^IJ.§ REMODELLING: EQUILIBRIUM-PRESERVING MAP One of the key features of soft biological tissues is their ability to remodel, that is, to change their neighbouring cells during growth, mobility and morphogenesis. We aim to include this feature in our model by computing a new connectivity _n+1 after each time point t_n. In this work we resort to the Delaunay triangulation of the nodal network, which guarantees a minimum aspect-ratio of the resulting triangles. We also assume that these optimal aspect ratios will not be exceedingly spoiled during tissue deformation. The redefinition of the network topology from _n to _n+1 may involve drastic changes in the nodal and vertex equilibrium equations. Furthermore, the resting lengths L^ij and L^IJ are undefined for the newly created bar elements. In order to smooth mechanical transition between time-steps, we will here present an Equilibrium-Preserving Map that computes L^ij and L^IJ by minimising the error of the mechanical equilibrium for the new connectivity. We will consider two approaches: a map that preserves the nodal and vertex equilibrium in a coupled manner (full-network mapping), and a map that preserves nodal equilibrium and vertex equilibrium independently (split-network mapping). The computational process depicted in Figure <ref> is nowcompleted with the EPM as shown in Figure <ref>.§.§ Full-network mapping In this approach, we aim to compute a new set of resting lengths L^ij and L^IJ that minimises the functional π̂_F(L^ij,L^IJ)=∑_i^nodes_D^i+_V^i+_A^i-^i^2. This functional measures the error in the mechanical equilibrium considering all the residual contributions at node i due to the cell-centres (_D^i), the vertex network (_V^i) and area constraints (_A^i).The latter is the value obtained from the expression in (<ref>), while ^i is the total reaction for those nodes that have prescribed displacements. The residual contributions are computed as a function of nodal and vertex tractions as _D^i =∑_j∈ S^i_D^ij=∑_j∈ S^i k_D(l^ij/L^ij-1)^ij _V^i =∑_I∈ B^ip^i(^I)∑_J∈ S^I_V^IJ =∑_I∈ B^ip^i(^I)∑_J∈ S^I k_V(l^IJ/L^IJ-1)^IJ Note that _D^ij are _V^IJ are not defined as ∂ W_D^ij/∂^i or ∂ W_V^IJ/∂^I, but with a simpler purely elastic law, which disregards any rheologicalevolution of the resting lengths. We emphasise that while computing the new resting lengths and thus the variables L^ij and L^IJ, the nodal and vertex positions ^i and ^I, and also the current lengths l^ij and l^IJ, are all constant.The minimisation of π̂_F in (<ref>) gives rise to a non-linear system of equations in terms of the unknowns L^ij and L^IJ, but that is linear with respect to the inverse of these quantities. We will denote these inverses by θ^ij=1/L^ij and θ^IJ=1/L^IJ. The new functional, denoted by π_F(θ^ij, θ^IJ), is obtained by inserting this change of variables(θ^ij,θ^IJ)^*=π_F(θ^ij,θ^IJ).The optimal variables θ^ij^* and θ^IJ^* are found by solving the associated normal equations of this least-squares problem, which after making use of (<ref>) reads [[ _DD _DV; _DV^T _VV; ]]{[ _D; _V ]}={[ _D; _V ]} with _D and _V vectors containing all the inverses of the resting lengths for the nodal and vertex networks, 1/L^ij and 1/L^IJ respectively, and^mn,pq_DD= k_D^2l^mn^mn^T(∑_j∈ S^ml^mj^mjδ_mj^pq-∑_j∈ S^nl^nj^njδ_nj^pq)_DV^mn,PQ= k_D k_Vl^mn^mn^T( ∑_I∈ B^mp^m(^I)∑_J∈ S^I l^IJ^IJδ_IJ^PQ..-∑_I∈ B^np^n(^I)∑_J∈ S^I l^IJ^IJδ_IJ^PQ) _VV^MN,PQ= k_V^2∑_i^N_nodes(∑_I∈ B^i p^i(^I)∑_J∈ S^Il^IJ^IJδ_IJ^PQ)(∑_I ∈ B^ip^i(^I)∑_J∈ S^Il^IJ^IJδ^MN_IJ)_D^mn= k_Dl^mn(^m-^n)^T^mn_V^MN= ∑_i^N_nodesk_V^i^T (∑_I∈ B^ip^i(^I)∑_J∈ S^Il^IJ^IJδ^MN_IJ) In the equations above, we have defined ^i = k_D∑_j∈ S^il^ij^ij+k_V∑_I∈ B^i p^i(^I)∑_J∈ S^I^IJ-_A^i+^i δ_mj^pq ={[ 1, mj=pq,mj=qp,; 0,]. δ_IJ^PQ ={[ 1, IJ=PQ,IJ=QP,; 0,]. The uniqueness of the solution of system of equations in (<ref>), and thus the regularity of the system matrix, is in general not guaranteed, since more than one combination of tractions in equilibrium with the reaction field may be found in some cases. This is algebraically reflected by a large condition number of the system matrix. For this reason, the functional is regularised by adding an extra term, π_Fλ(θ^ij, θ^IJ)=π_F(θ^ij, θ^IJ) +λ_L(∑_ij ||θ^ij-1/l^ij||^2 +∑_IJ||θ^IJ-1/l^IJ||^2) with l^ij and l^IJ the current distances between connected nodes and vertices, respectively. This regularisation adds a factor λ_L on the diagonal components and factors λ_L/l^mn andλ_L/l^MN on _D^mn and _V^MN, which ensure that the system will have a unique solution for a sufficiently large value of the regularisation parameter λ_L. In our numerical examples we have used λ_L=10^-12. §.§ Split-network mapping The previous approach allows to find equilibrated tractions with a possible redistribution of forces between the vertex and nodal networks. In some cases though, it is desirable to keep the traction contributions of the two networks split. For this reason, we present an alternative Equilibrium-Preserving Map that aims to compute the resting lengths by considering equilibrium conditions for the nodal and vertex networks independently. This is achieved by minimising the functional π_S(θ^ij,θ^IJ)=π_D(θ^ij) + π_V(θ^IJ) with π_D(θ^ij)=∑_i^N_nodes||^i_D-_D^i||^2 π_V(θ^IJ)=∑_i^N_nodes||^i_V-_V^i||^2 where _D^i is the contribution from the nodal network on node i before remodelling, and _V^i is the contribution from the vertex network to node i before remodelling. This contributions are obtained from the residual contributions before remodelling takes place as _D^i =_D^i, _V^i =_V^i+_A^i. Applying the same approach as in Section <ref> to π_F, the minimisation of π_S yields two uncoupled systems of equations, _DD_D='_D_VV_V='_V. Matrices _DD and _VV are those written in equation (<ref>), while the right-hand-sides are now given by_D'^mn= k_Dl^mn(^m_D-^n_D)^T^mn_V'^MN= ∑_i^N_nodesk_V^i_V^T (∑_I∈ B^ip^i(^I)∑_J∈ S^Il^IJ^IJδ^MN_IJ) with ^i_D = k_D∑_j∈ S^il^ij^ij+^i_D, ^i_V = k_V∑_I∈ B^i p^i(^I)∑_J∈ S^I^IJ+^i_V. Like in the previous section, a regularisation term, equal to the one used in (<ref>)is added to the functional π_S in order to ensure the regularity and uniqueness of the solution, with the same value of the regularisation parameter λ_L=10^-12.The split-network approach is in fact relevant when the stresses in the nodal and vertex networks follow different patterns, and it is necessaryto maintain this difference between the networks, such as wound healing, where the stresses around the wound ring are significantly higher. Preserving stress residuals independently at each network guarantees the stress contrast. The full-network approach on the other hand, spoils this contrast and may transfer some of the stresses on the wound ring to the nodal network. The numerical example in Section <ref> illustrates this fact. § NUMERICAL RESULTS§.§ Extension of square tissue We test our methodology by extending a square domain obtained from a random perturbation of a 10×10 grid of nodes (see Figure <ref>a). The domain is formed by 81 cells, and subjected to a uniform 30% extension applied within 60 time-steps. We will test two situations: extension with constant topology (evolution from (a)-(b)), and with remodelling (evolution (a)-(c)). In the two situations we will apply the full and split approaches of the Equilibrium-Preserving Map (EPM). §.§.§ Validation of EPM: fixed topology To inquire the accuracy and effects of the EPM, we measure the total reaction at the right side and the elastic energy of the tissue during extension while keeping the topology constant. Figure <ref> shows the evolution of the two quantities whenk_D=0.1 k_V (Figures <ref>a-b) and whenk_D=10 k_V (Figures <ref>c-d). It can be observed that in all cases the full-network and the split-network mappings give the same values as the tests with no mapping. This fact shows that the EPM is able to recover the same traction values as the ones when no computation of the resting lengths is applied, and that the system regularisation is not altering these lengths or the elastic response of the tissue.§.§.§ Validation of the EPM: variable topology We now apply the same boundary conditions as in the previous tests, but allowing the tissue to remodel according to the Delaunay triangulation of the nodal positions. Figure <ref> shows the total reaction at the right end and the total elastic energy. We have monitored these quantities under three conditions: no remodelling/mapping, remodelling with full-network mapping and remodelling with split-network mapping. We have tested also two sets of material properties: k_D=10k_V (Figure <ref>a-b), and k_D=0.1k_V (Figure <ref>c-d). The totalnumber of remodelling events (elements that change their connectivity) is also plotted at each time-step, whenever this number is positive. From the plots in Figure <ref> it can be observed that the evolution of the total reaction is not substantially affected by the remodelling process. The elastic energy, however, suffers some deviations with respect to the case with no remodelling when the split-network EPM is used and the vertex network is stiffer than the nodal network. This drift is more severe when more remodelling events are encountered. Indeed, the split-network approach prevents the transfer of energy between the vertex and nodal networks, preventing in some cases the full preservation of the equilibrium conditions before the remodelling events. The total reaction of the tissue is in all cases not much affected by the mapping, which is in agreement with the fact that EPM aims to compute resting lengths distributions that match the nodal resultants before remodelling. For the two sets of material parameters, the total reaction, and thus the tissue response, is very much unaffected by the remodellingfor the two EPM approaches. This allows to keep the correct aspect ratio of the cells while keeping the elastic response. Although cells may use remodelling events to relax their stress state, we here aim to independently control the stress relaxation and the remodelling events. In our example, the stress relaxation is prevented by using a small value of the remodelling rate γ=10^-6. §.§.§ Analysis of ξ-relaxation Tissue stiffness against tissue total reaction and strain energy is investigated by assigning a range of values to {k_D k_V} at a constant total stiffness, k_D+k_V=1, under two conditions: 1) when vertices arerigidly anchored at barycentres (=1/3{11}), and 2) when vertices are allowed to change their relative positions with respect to the barycentres (-relaxation).Figure <ref> compares the vertex network shown in Figure <ref>b for the two situations. The red network displays vertices anchored at barycenters, while in the green network vertices are relaxed under a penalisation factor λ_=10^-4.In order to inspect the effect of ξ-relaxation we have analysed the reaction and energy of mainly nodal-driven or mainly vertex-driven tissues for different values of λ_ξ. Figure <ref> shows the tissue response for different values of k_V∈[0,1] while keeping k_V+k_D=1, and when the tissue is subjected to an 30% extension.Figure <ref>a shows that the total reaction decreases astractions concentrate on the vertex network. This reduction is steeper when vertices are relaxed (lower values of λ_ξ). Figure <ref>b shows a faster drop in tissue total energyand a lower growth in vertex network energy, while no significant effect on nodal network energy when -relaxation is allowed. We have also analysed the difference of our equilibrated tractions with respect to the purely nodal and vertex equilibrium conditions: null sum of tractions at nodes and at vertices. This difference is computed as the mean value of the following nodal and vertex measures, E_i =||∑_j∈ S^i t^ij_D||/∑_j∈ S^i|| t^ij_D||, i=1,…,N_nodesE_I =||∑_J∈ S^I t^IJ_V||/∑_J∈ S^I|| t^IJ_V||, I=1,…,N_triFigures <ref>c and <ref>d plot the means E̅_D=∑_i E_i/N_nodes and E̅_V=∑_I E_I/N_tri for the whole tissue.As expected, the nodal difference is zero when no stiffness is assigned to the vertex network (k_V=0). As k_V increases, pure nodal equilibrium is increasingly violated, due to the coupling between the two networks. In most cases, this difference is below 10%, except when vertices are fixed. Pure vertex equilibrium is more severely affected by the kinematic constraint, but the difference also decreases rapidly as λ_ξ decreases. It can be observed that while the positions of the vertices in the two networks is very similar, purely vertex equilibrium drastically improves for approximately λ_ξ<10^-2. §.§ Wound healing The model is testedto simulate a wound healing process in monolayers <cit.>. The evolution law in (<ref>) is applied to the nodal and vertex networks with the values given in Table <ref>, which also indicatesthat the area constraint is imposed in order to mimic mechanical properties of the tissue. Topological changes in the tissue are allowed to examine the role of cell motility and cell intercalation during wound healing.Wounding and wound healing processes are simulated during the consecutive steps below: * To resemble the initial condition of in-vivo tissue before wounding, the modelled tissue is let to reach a contractile state given by the values of ^c_D and ^c_V in Table <ref> and the evolution law affecting elements resting lengths, during 50 time-steps. This time is found to be sufficient to reach a steady asymptotic state. * Wounding by laser ablation of cells is analogised by a significant reduction of stiffness in nodal and vertex elements encircled by the wound edge,as well as removing the area constraint on wounded cells. In wounded areas we set k_D^wounded=0.1 k_Dand k_V^wounded=0.1 k_V. Also, vertices at the wound edge are allowed to relax by resorting to the -relaxation. This isdone to avoid unrealistic zig-zag effects on the profile of the wound edge. Figures <ref>a <ref>d and <ref>g show the tissue initially after wounding, without remodelling, and with full- and split-network remodelling, respectively.* To simulate tissue eventual response to wounding, after 12 time-steps, contractility on the elements of the vertex network surrounding the wound (wound ring) is multiplied by 5 in order to pattern actomyosin concentration, as it has been experimentally tested <cit.>. Figures <ref>b and <ref>e show how the extra contractility on the wound edge results in higher tractions on the wound ring, at both non-remodelling and remodelling tissues. * Additional tractions on the wound ring cause the wounded area being squeezed by the cells on the wound boundary. Figures <ref>c and <ref>f show the wound closure with and without remodelling. Including remodelling during the tissue evolution results in less cell elongation at the wound edge and allows cells to relocate during wound closure.In the full-network strategy (Figures <ref>d-f), since the total residual of nodal and vertex networks were preserved at the nodes, the interplay of stresses in nodal and vertex networks could not preserve the higher stress in the vertex elements at the wound ring. Instead, the split-network strategy could provide the expected higher stress in the elements at the wound ring. This is due to preserving nodal residual independently in each of the networks. § CONCLUSIONS We have presented a hybrid cell-centred and vertex discretisation for biological tissues. This approach allows to independently control the material properties of the cell-boundaries and the cytoplasm (cell interior). The methodologysolves the mechanical equilibrium of the two networks in a coupled manner, and it has been shown that can reproduce relevant phenomena such as tissue extension or wound healing. The method resorts to a rheological law that is based on an evolution law of the resting length <cit.>. This evolution is controlled through the remodelling rate γ. For high values of γ, the tissue relaxes and adapts its reference free configuration rapidly, while for very low values of γ, an purely elastic response is recovered. The variations of the resting lengths allow also to design an Equilibrium-Preserving Map (EPM) that computes a set of resting lengths and traction field that mimics the force distribution on the nodal and vertex network before remodelling. The numerical examples presented show that this recovery of tractions alters minimally the stress state. We have just presented two-dimensional examples, but a three dimensional extension does not involve substantial changes neither in the hybrid approach and in the EPM strategy, if the vertex mechanics is maintained along bar elements. In case that mechanics at the cell boundaries is carried by the vertex faces, some additional modifications should be applied to the tractions and functional in the EPM.The strategy described here opens also the possibility to energy decaying or methods where the actual reaction is relaxed in a controlled manner. This could be achieved by progressively reducing the nodal reaction used in the functional of the EPM. Also, the hybrid approach could be modified for handling cell proliferation or apoptosis (addition or removal of nodes). Current research is now being undertaken in this direction. § ACKNOWLEDGEMENTS The authors gratefully acknowledge the financial support of the Spanish Ministry of Economy, Science and Competitiveness (MINECO) under grants DPI2013-32727-R and DPI2016-74929-R, and the Generalitat de Catalunya under grant 2014-SGR-1471. PM is also supported by the European Molecular and Biology Organisation (EMBO) under grant ASTF 351-2016.§ NOTATION The notation used in this article is summarised in Tables <ref> and <ref>. § LINEARISATION§.§ General linearisation steps with ξ-relaxation When ξ-relaxation is included, the total residual vector ={_x^T _y^T}^T is split in a nodal _x and ξ contributions _y (see equation (<ref>)). Each nodal and vertex contribution is given by ^i_x =^i_D+^i_V+^i_A, ^I_y =^I_V+^i_A+^I_ξ. Vectors ^i_D, ^i_V and ^i_A are written in equations (<ref>), (<ref>) and (<ref>), and the vertex contributions^I_V, ^I_A and ^I_ξ given in equations (<ref>). The non-linear equations = are solved with a Newton-Raphson process that at each iteration k reads {[ δ; δ ]} =-[[_xx_xy; _y x_yy ]]_k^-1{[ _x; _y ]}_k and is updated as {[ ;]}_k+1 ={[ ;]}_k +{[ δ; δ ]}as long as the two following conditions are met,{[ √(δx^2+δ^2)>tol; g>tol ]. with tol a sufficiently small tolerance. In our numerical examples we used tol=1e-10.The block matrices in (<ref>) correspond to the following linearisation terms, _xx^ij =∂^i_D/∂^j +∂^i_V/∂^j + ∂^i_A/∂^j _xy^iJ =∂^i_D/∂^J+∂^i_V/∂^J + ∂^i_A/∂^J _y x^Ij =∂^I_V/∂^j +∂^I_A/∂^j _yy^IJ =∂^I_V/∂^J +∂^I_A/∂^J + ∂^I_ξ/∂^J where due to the expressions of _D^i and _ξ^I, we have used the fact that ∂^i_D/∂^J and ∂^I_ξ/∂^j vanish. Also note that since our equlibrium equations stem from the linearisation of an energy function W(, ), we have that _xy^iJ=∂^2 (W_V+W_A)/∂^i∂^J=[∂^2 (W_V+W_A)/∂^I∂^j]^T=_yx^Ij^T.In the next sections we will give the linearisation of the terms in (<ref>)-(<ref>). §.§ Linearisation of nodal and vertex tractions t_D and t_V Many of the derivations detailed below will involve the linearisation of the traction vectors given in (<ref>), _D^ij =∂ W_D^ij/∂^i =^ij/L^ij(1-l^ij/L^ij∂ L^ij/∂ l^ij)^ij _V^IJ =∂ W_V^IJ/∂^I =^IJ/L^IJ(1-l^IJ/L^IJ∂ L^IJ/∂ l^IJ)^IJ The factor ∂ L/∂ l is zero when the resting length is constant, but for the rheological law presented in Section <ref>, this factor is given in equation (<ref>). In the subsequent expressions we will need the derivatives of the traction vectors above. We define matrix^ii_t:=∂_D^ij/∂^i=-∂_D^ji/∂^i=-^ji_t=-^ij_t=^jj_t which after making use of (<ref>), it can be deduced that ^ij_t =(-1)^δ_ij+1((a^ija^ij-^ij/l^ij a^ij+^ijb^ij)^ij⊗^ij +^ija^ij/l^ij𝐈)a^ij =1/L^ij(1-l^ij/L^ij∂ L/∂ l)b^ij =1/L^ij∂ L/∂ l(-a^ij+1/L^ij(l^ij/L^ij^2-1))A similar derivation is obtained for ∂_V^IJ/∂^I, but replacing ij by IJ. In this case, we also note that from the interpolation in (<ref>) we have, ∂_V^IJ/∂^j =^IJ_t(∂^J/∂^j-∂^I/∂^j) =_t^IJ(p^j(^J)-p^j(^I))∂_V^IJ/∂^J =∂_V^IJ/∂^I∂^I/∂^J +∂_V^IJ/∂^J∂^J/∂^J =^IJ_t∑_^j∈𝒯^J^j⊗∇ p^j(^J) where p^i(^I)=0 if i∉𝒯^I. §.§ Linearisation terms in ^ij_xx By using the expressions of _D^i, _V^i and _A^i in (<ref>), (<ref>) and (<ref>), and the definition of _t^ij in (<ref>), it can be deduced that ∂^i_D/∂^j =∑_j∈ S^i_t^ij ∂^i_V/∂^j =∑_I∈ B^i∑_J∈ S^I_t^IJ(p^j(^J)-p^j(^I)) ∂^i_A/∂^j =λ_A/2∑_m∈S̅^i(A^m-A^m_0)∑_IJ∈ P ^m(p^i(^I)p^j(^J)-p^i(^J)p^j(^I))+ λ_A/4∑_m∈S̅^i∑_IJ∈ P^m(p^i(^I)^J-p^i(^J)^I)⊗∑_KL∈ P^m(p^j(^K)^L-p^j(^L)^K)§.§ Linearisation terms in ^iJ_xy From the expressions of _V^i and _A^i in (<ref>) and (<ref>), and from equation (<ref>),it can be also deduced that ∂^i_V/∂^J=(∑_K∈ S^J^JK_V)⊗∇ p^i(^J) +∑_I∈ B^ip^i(^I)∑_J∈ S^I^IJ_t∂^J/∂^J ∂^i_A/∂^J = λ_A/2∑_m∈S̅^i(A^m-A^m_0)∑_IJ∈ P ^m^N_m( p^i(^I)∂^J/∂^J-^I⊗∇ p^i(^J) )+λ_A/2∑_m∈S̅^i∑_IJ∈ P ^m^N_m( p^i(^I)^J-p^i(^J)^I)⊗∂ A^m/∂^J with ∂ A^m/∂^J give in (<ref>). §.§ Linearisation terms in ^IJ_yy The linearisation of _V^I, _A^I and _ξ^I in (<ref>) yields ∂^I_V/∂^J = ∑_K∈ S^I∑_i∈𝒯^I(∇ p^i(^I)⊗^i)(_t^IIδ_IJ∂^I/∂^J+_t^IKδ_KJ∂^K/∂^J) ∂^I_A/∂^J =λ_A∑_m=1^N̅_nodes∂ A^m/∂^I⊗∂ A^m/∂^J +λ_A ∑_m=1^N̅_nodes(A^m-A^m_0)∂^2 A^m/∂^I∂^J ∂^I_ξ/∂^J =λ_ξδ_IJ𝐈 where the expressions of ∂^I/∂^I and ∂ A^m/∂^I are given in (<ref>) and in (<ref>), respectively, and∂^2 A^m/∂^I∂^J=∑_KL∈ P^m( δ_KIδ_LJ(∂^K/∂^I )^T ∂^L/∂^J -δ_LIδ_KJ(∂^L/∂^I )^T ∂^K/∂^J).wileyj10 urlstylesunyer15 Sunyer R, Conte V, Escribano J, Elosegui-Artola A, Labernadie A, Valon L, Navajas D, García-Aznar J, Muñoz J, Roca-Cusachs P, et al.. 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"authors": [
"Payman Mosaffa",
"Antonio Rodríguez-Ferran",
"José J. Muñoz"
],
"categories": [
"q-bio.CB",
"74B20, 74D10, 92B05, 74S99"
],
"primary_category": "q-bio.CB",
"published": "20170426220455",
"title": "Hybrid cell-centred/vertex model for multicellular systems with equilibrium-preserving remodelling"
} |
MITP/17-023 April 26, 2017^aInstitut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany^bPRISMA Cluster of Excellence & MITP, Johannes Gutenberg University, 55099 Mainz, Germany^cDepartment of Physics & LEPP, Cornell University, Ithaca, NY 14853, U.S.A. We argue that a large region of so far unconstrained parameter space for axion-like particles (ALPs), where their couplings to the Standard Model are of order (0.01-1) ^-1, can be explored by searches for the exotic Higgs decays h→ Za and h→ aa in Run-2 of the LHC. Almost the complete region in which ALPs can explain the anomalous magnetic moment of the muon can be probed by searches for these decays with subsequent decay a→γγ, even if the relevant couplings are loop suppressed and the a→γγ branching ratio is less than 1. LHC as an Axion Factory: Probing an Axion Explanation for (𝐠-2)_μ with Exotic Higgs Decays Andrea Thamm^b December 30, 2023 ============================================================================================Axion-like particles (ALPs) appear in well motivated extensions of the Standard Model (SM), e.g. as a way to address the strong CP problem, as mediators between the SM and a hidden sector, or as pseudo Nambu-Goldstone bosons in extensions of the SM with a broken global symmetry. If ALP couplings to muons and photons are present, the 3.6σ deviation of the anomalous magnetic moment of the muon a_μ=(g-2)_μ/2 from its SM value can be explained by ALP exchange <cit.>. Collider experiments can be used to search directly and indirectly for ALPs. Besides ALP production in association with photons, jets and electroweak gauge bosons <cit.>, searches for the decay Z→γ a are sensitive to ALPs with up to weak-scale masses <cit.>. Utilizing the exotic Higgs decay h→ aa to search for light pseudoscalars was proposed in <cit.>. Several experimental searches for this mode have been performed, constraining various final states <cit.>. Surprisingly, the related decay h→ Za has not been studied experimentally, even though analogous searches for new heavy scalar bosons decaying into Za have been performed <cit.>. The reason is, perhaps, the suppression of the h→ Za decay in the decoupling limit in two-Higgs-doublet models in general and supersymmetric models in particular <cit.>. In models featuring a gauge-singlet ALP, there is no dimension-5 operator mediating h→ Za decay at tree level, and hence this mode has not received much theoretical attention either. Here we point out that fermion-loop graphs arising at dimension-5 order and tree-level contributions of dimension-7 operators can naturally induce a h→ Za decay rate of similar magnitude as the h→ Zγ decay rate in the SM, which is a prime target for Run-2 at the LHC. Furthermore, in certain classes of UV completions the h→ Zγ branching ratio can be enhanced parametrically to the level of O(10%) and higher. A search for this decay mode is therefore well motivated and can provide non-trivial information about the underlying UV theory. In this letter we show that searches for h→ Za and h→ aa decays in Run-2 at the LHC can probe a large region of so far unconstrained parameter space in the planes spanned by the ALP mass and its couplings to photons or leptons, covering in particular the difficult region above 30 MeV and probing ALP–photon couplings as small as 10^-10 ^-1. If the (g-2)_μ anomaly is explained by a light pseudoscalar, this particle will be copiously produced in Higgs decays and should be discovered at the LHC. A detailed discussion of the searches presented here, along with a comprehensive analysis of electroweak precision bounds, flavor constraints and the relevance of ALPs to other low-energy anomalies, will be presented elsewhere <cit.>.We consider a light, gauge-singlet CP-odd boson a, whose mass is protected by a (approximate) shift symmetry. Its interactions with SM fermions and gauge fields start at dimension-5 order and are described by the effective Lagrangian <cit.> L_ eff = g_s^2 C_GG a/Λ G_μν^A G̃^μν,A+ g^2 C_WW a/Λ W_μν^A W̃^μν,A+ g^' 2 C_BB a/Λ B_μν B̃^μν + ∑_f c_ff/2 ∂^μ a/Λ f̅γ_μγ_5 f.Here Λ is the characteristic scale of global symmetry breaking (often called f_a in the literature on ALPs), which we assume to be above the weak scale. We neglect flavor off-diagonal couplings, which will play no role for our analysis. After electroweak symmetry breaking, the effective ALP coupling to two photons is described by a term analogous to the hypercharge coupling, but with gauge coupling e^2 and coefficient C_γγ=C_WW+C_BB. Note that at this order there are no ALP couplings to the Higgs doublet ϕ. They appear first at dimension-6 and 7 and are given by L_ eff^D≥ 6 = C_ah/Λ^2( ∂_μ a)( ∂^μ a) ϕ^†ϕ+ C_Zh^(7)/Λ^3( ∂^μ a)( ϕ^† iD_μ ϕ + ) ϕ^†ϕ + … .The first term is the leading Higgs-portal interaction allowed by the shift symmetry, while the second term is the leading polynomial operator mediating the decay h→ Za at tree level <cit.>. If the electroweak symmetry is realized non-linearly, insertions of ϕ^†ϕ are accompanied by factors 1/f^2 rather than 1/Λ^2, where f is the analog of the pion decay constant <cit.>. As a result, the contribution of C_Zh^(7) can be enhanced by a factor ∼Λ^2/f^2 if f<Λ <cit.>. Importantly, in models featuring heavy particles which receive their mass from electroweak symmetry breaking, an additional non-polynomial dimension-5 operatorL_ eff^ non-pol = C_Zh^(5)/Λ( ∂^μ a) ( ϕ^† iD_μ ϕ + ) lnϕ^†ϕ/μ^2 + …can be generated <cit.>. It gives a contribution to the h→ Za amplitude that is parametrically enhanced compared with the h→ aa amplitude. The decay h→ Za is unique in the sense that a tree-level dimension-5 coupling can only arise from a non-polynomial operator.A search for this decay mode can thus provide complementary information to h→ aa searches and offer important clues about the underlying UV theory. In this letter we consider decays of the ALP into photons and charged leptons, with decay rates given byΓ(a→γγ)= 4πα^2 m_a^3/Λ^2 C_γγ^2,Γ(a→ℓ^+ℓ^-)= m_a m_ℓ^2/8πΛ^2 c_ℓℓ^2 √(1-4m_ℓ^2/m_a^2) .The same couplings enter the diagrams shown in Figure <ref>, which show the ALP-induced contributions to the anomalous magnetic moment a_μ of the muon, whose experimental value differs by more than 3σ from the SM prediction: a_μ^ exp-a_μ^ SM=(288± 63± 49)· 10^-11 <cit.>. It has been emphasized recently that this discrepancy can be explained by postulating the existence of an ALP with sizeable couplings to both photons and muons <cit.>. While the first graph in Figure <ref> gives a contribution of the wrong sign <cit.>, the second diagram can overcome this contribution if the Wilson coefficient C_γγ is sufficiently large <cit.>. At one-loop order, we find the new-physics contribution δ a_μ=m_μ^2/Λ^2 { - c_μμ^2/16π^2 h_1(x)- 2α/π c_μμ C_γγ[ lnΛ^2/m_μ^2- h_2(x) ] } ,where x=m_a^2/m_μ^2. The functions h_i(x) are positive and satisfy h_1(0)=h_2(0)=1 and h_1(x)≈ 0, h_2(x)≈(ln x+3/2) for x≫ 1. Their analytical expressions will be given in <cit.>. Our result for the logarithmically enhanced contribution proportional to C_γγ agrees with <cit.>. We omit the numerically subdominant contribution from Z exchange, which is suppressed by (1-4sin^2θ_w) and comes with a smaller logarithm ln(Λ^2/m_Z^2). A positive shift of a_μ can be obtained if c_μμ and C_γγ have opposite signs. Figure <ref> shows the parameter space in the c_μμ-C_γγ plane in which the muon anomaly can be explained in terms of an ALP with mass of 1 GeV (we use Λ=1 TeV in the argument of the logarithm). The contours are insensitive to m_a for lighter ALP masses and broaden slightly for m_a>1 GeV. A resolution of the anomaly is possible without much tuning as long as one of the two coefficients is of order Λ/, while the other can be of similar order or larger. Since c_μμ enters observables always in combination with m_μ, it is less constrained by perturbativity than C_γγ. We thus consider the region where |C_γγ|/Λ≲ 2 ^-1 and |c_μμ|≥|C_γγ| as the most plausible parameter space.We now turn our attention to the exotic Higgs decays h→ Za and h→ aa, arguing that over wide regions of parameter space – including the region motivated by (g-2)_μ – the high-luminosity LHC can serve as an ALP factory. At tree-level, the effective interactions in (<ref>) and (<ref>) yield the decay ratesΓ(h→ Za)= m_h^3/16πΛ^2 C_Zh^2 λ^3/2(m_Z^2/m_h^2,m_a^2/m_h^2),Γ(h→ aa)= v^2 m_h^3/32πΛ^4 C_ah^2 ( 1-2m_a^2/m_h^2)^2 √(1-4m_a^2/m_h^2) ,where λ(x,y)=(1-x-y)^2-4xy, and we have defined C_Zh≡ C_Zh^(5)+v^2/2Λ^2 C_Zh^(7). Integrating out the top-quark yields the one-loop contributions δ C_Zh≈ -0.016 c_tt and δ C_ah≈ 0.173 c_tt^2 <cit.>. For natural values of the Wilson coefficients the rates in (<ref>) can give rise to large branching ratios. For instance, one finds (h→ Za)=0.1 for |C_Zh|/Λ≈ 0.34 ^-1 and (h→ aa)=0.1 for |C_ah|/Λ^2≈ 0.62 ^-2. Even in the absence of large tree-level contributions, the loop-induced top-quark contribution yields (h→ aa)=0.01 for |c_tt|/Λ≈ 1.04 ^-1, while a combination of the top-quark contribution and the dimension-7 contribution from C_Zh^(7) can give (h→ Za)= O(10^-3) without tuning. With such rates, large samples of ALPs will be produced in Run-2 of the LHC. The model-independent bound Br(h→BSM)<0.34 derived from the global analysis of Higgs couplings <cit.> implies |C_Zh|/Λ≲ 0.72 ^-1 and |C_ah|/Λ^2<1.34 ^-2 at 95% CL. If the ALP is light or weakly coupled to SM fields, its decay length can become macroscopic, and hence only a small fraction of ALPs decay inside the detector. Since to good approximation Higgs bosons at the LHC are produced along the beam direction, the average decay length of the ALP perpendicular to the beam is L_a^⊥(θ)=sinθ β_aγ_a/Γ_a, where θ is the angle of the ALP with respect to the beam axis in the Higgs-boson rest frame, β_a and γ_a are the usual relativistic factors in that frame, and Γ_a is the total decay width of the ALP. If the ALP is observed in the decay mode a→ XX̅, we can express its total width in terms of the branching fraction and partial width for this decay, i.e. L_a^⊥(θ) = sinθ√(γ_a^2-1)(a→ XX̅)/Γ(a→ XX̅) .The boost factor is γ_a=(m_h^2-m_Z^2+m_a^2)/(2m_a m_h) for h→ Za and γ_a=m_h/(2m_a) for h→ aa. As a consequence, only a fraction of events given by f_ dec = 1 - ⟨ e^-L_ det/L_a^⊥(θ)⟩ ,where the brackets mean an average over solid angle, decays before the ALP has traveled a distance L_ det set by the relevant detector components. We define the effective branching ratios(h→ Za→ℓ^+ℓ^- XX̅) |_ eff = (h→ Za) ×(a→ XX̅) f_ dec (Z→ℓ^+ℓ^-), (h→ aa→ 4X) |_ eff= (h→ aa) (a→ XX̅)^2 f_ dec^2,where (Z→ℓ^+ℓ^-)=0.0673 for ℓ=e,μ. If the ALPs are observed in their decay into photons, we require L_det=1.5 m, such that the decay occurs before the electromagnetic calorimeter. For a given value of the Wilson coefficients C_Zh or C_ah, we can now present the reach of high-luminosity LHC searches for h→ Za→ℓ^+ℓ^- γγ and h→ aa→ 4γ decays in the m_a-|C_γγ| plane. We require at least 100 signal events in a dataset of 300 fb^-1 at √(s)=13 TeV (Run-2), considering gluon-fusion induced Higgs production with cross section σ(pp→ h+X)=48.52 pb <cit.> and the effective Higgs branching ratios defined above. Figure <ref> shows this parameter space in light green. In the upper panel we present the reach of Run-2 searches for h→ Za→ℓ^+ℓ^-γγ decays assuming |C_Zh|/Λ=0.72 ^-1 (solid contour), 0.1 ^-1 (dashed contour) and 0.015 ^-1 (dotted contour). Reaching sensitivity to smaller h→ Za branching ratios obtained with |C_Zh|/Λ<0.015 ^-1 would require larger luminosity. The lower panel shows the reach of searches for h→ aa→ 4γ decays assuming |C_ah|/Λ^2=1 ^-2 (solid), 0.1 ^-2 (dashed) and 0.01 ^-2 (dotted). These contours are essentially independent of the a→γγ branching ratio unless this quantity falls below certain threshold values. For h→ Za, one needs (a→γγ)>3· 10^-4 (solid), 0.011 (dashed) and 0.46 (dotted). For h→ aa, one needs instead (a→γγ)>0.006 (solid), 0.049 (dashed) and 0.49 (dotted). It is thus possible to probe the ALP–photon coupling even if the ALP predominantly decays into other final states. The insensitivity of the contours to (a→γγ) can be understood by considering the behavior of the quantity f_ dec in (<ref>). The contours limiting the green regions from the left arise from the region of large ALP decay length, L_a≫ L_ det, in which case f_ dec≈ (π/2) L_ det/L_a∝Γ(a→ XX̅)/(a→ XX̅). In this region the effective branching ratios in (<ref>) become independent of (a→γγ) and only depend on the partial rate Γ(a→ XX̅)∝ m_a^3 C_γγ^2. On the other hand, the number of signal events inside the probed contour regions is bounded by the yield computed with f_ dec=1 (prompt ALP decays), and this number becomes too small if (a→ XX̅) falls below a critical value.The red band in the panels shows the parameter region in which the (g-2)_μ anomaly can be explained. We consider only the theoretically preferred region |c_μμ|≥|C_γγ| and impose the constraint |c_μμ|/Λ≤ 10 ^-1. In principle, larger values of |C_γγ| can also explain the anomaly. Almost the entire parameter space where the red band is not excluded by existing experiments – the region between 30 MeV and 60 GeV – can be covered by searches for exotic Higgs decays. Even if the relevant couplings C_Zh and C_ah are loop suppressed, large event yields in this region can be expected in Run-2. Existing searches for h→ aa→ 4γ decay already imply interesting bounds on the ALP parameter space. ATLAS has performed dedicated searches for this signature at m_a=100 MeV, 200 MeV and 400 MeV <cit.>, as well as in the high-mass region m_a=10-62.5 GeV <cit.>. Lighter ALPs produced in Higgs decays would be highly boosted, and the final-state photon pairs would therefore be strongly collimated. For m_a≲ 100 MeV these pairs cannot be resolved in the calorimeter and would be reconstructed as single photons <cit.>. Hence, the existing measurements of the h→γγ rate <cit.> can also be used to derive constraints on the ALP couplings. At present, non-trivial exclusion regions can be derived for values |C_ah|/Λ^2≳ 0.1 ^-2 <cit.>. Currently, there exist no dedicated searches for the h→ Za→ℓ^+ℓ^-γγ decay channel. However, for m_a≲ 50 MeV the current upper bounds on the h→ Zγ rate <cit.> imply a weak constraint. Since the h→ Za signal does not interfere with the decay h→ Zγ, its contribution would lead to an enhancement of the h→ Zγ rate. This would provide a very interesting signal once the decay h→ Zγ becomes within reach of the LHC. The couplings of ALPs to other SM particles can be probed in an analogous way. In Figure <ref> we consider the decay a→ e^+ e^-. We use L_ det=2 cm, such that reconstructed events correspond to decays before the inner tracker, and require 100 signal events in a dataset of 300 fb^-1. The two panels show the reach of Run-2 searches for h→ Za→ℓ^+ℓ^-e^+e^- (top) and h→ aa→ e^+e^-e^+e^- decays (bottom), using the same values for the Wilson coefficients C_Zh and C_ah as in Figure <ref>. Once again, the contours are essentially independent of the a→ e^+e^- branching ratio unless this quantity falls below certain threshold values, which are the same as before. For h→ Za, one needs (a→ e^+e^-)>2· 10^-4 (solid), 0.011 (dashed) and 0.46 (dotted). For h→ aa, one needs instead (a→ e^+e^-)>0.006 (solid), 0.049 (dashed) and 0.49 (dotted). In summary, we have shown that LHC searches for the exotic Higgs decays h→ Za and h→ aa in Run-2 with an integrated luminosity of 300 fb^-1 can probe the ALP couplings to photons and electrons over a large region in parameter space, which almost perfectly complements the regions covered by existing searches. Importantly, the parameter space in which an ALP can provide an explanation of the (g-2)_μ anomaly is almost completely covered by these searches. The reach can be extended with more luminosity (the event yields increase by a factor ∼ 10.7 for 3000 fb^-1 luminosity at √(s)=14), and similar searches can be performed at a future lepton collider. Our yield estimates can be improved using dedicated analyses, including reconstruction efficiencies and exploiting displaced-vertex signatures. 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"Matthias Neubert",
"Andrea Thamm"
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"published": "20170426164328",
"title": "LHC as an Axion Factory: Probing an Axion Explanation for $(g-2)_μ$ with Exotic Higgs Decays"
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-1cm | http://arxiv.org/abs/1704.08052v2 | {
"authors": [
"Dipanjan Mandal",
"R. Rajesh"
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"published": "20170426104529",
"title": "The columnar-disorder phase boundary in a mixture of hard squares and dimers"
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^1 AWE plc, Aldermaston, Reading RG7 4PR, UK ^2 Department of Physics and Astronomy, University of Kentucky, Lexington, KY 40506, USA ^3 Department of Physics, University of Strathclyde, Glasgow G4 0NG, UK ^4 Royal Observatory of Belgium, Ringlaan 3, 1180 Brussels, BelgiumWe compare the results of the semi-classical (SC) and quantum-mechanical (QM) formalisms for angular-momentum changing transitions in Rydberg atom collisions given by Vrinceanu & Flannery, J. Phys. B 34, L1 (2001), and Vrinceanu, Onofrio & Sadeghpour, ApJ 747, 56 (2012), with those of the SC formalism using a modified Monte Carlo realization.We find that this revised SC formalism agrees well with the QM results.This provides further evidence that the rates derived from the QM treatment are appropriate to be used when modelling recombination through Rydberg cascades, an important process in understanding the state of material in the early universe.The rates for Δℓ=±1 derived from the QM formalism diverge when integrated to sufficiently large impact parameter, b.Further to the empirical limits to the b integration suggested by Pengelly & Seaton, MNRAS 127, 165 (1964), we suggest that the fundamental issue causing this divergence in the theory is that it does not fully cater for the finite time taken for such distant collisions to complete. 32.80.Ee, 34.10.+x, 34.50.FaThermodynamically-consistent semi-classical ℓ-changing rates R. J. R. Williams^1, F. Guzmán^2, N. R. Badnell^3, P. A. M. van Hoof^4, M. Chatzikos^2, G. J. Ferland^3December 30, 2023 ============================================================================================================== § INTRODUCTIONThere has been significant recent interest in the rates for ℓ-changing collisions in Rydberg atoms.While this may seem a somewhat obscure corner of atomic physics, the time to pass through the ladder of high-angular momentum levels in highly excited atoms proves to be a bottleneck in the process of atomic (re-)combination in the early universe.The details of the ℓ-changing rates therefore have a major impact on our understanding of this important stage in cosmic development <cit.>.The atomic physics of these rates can be calculated by a variety of approximations, as described in detail by <cit.> and <cit.> (hereafter VF01 and VOS12, respectively).In this paper, we will concentrate on the differences between the most detailed of these formulations, those based on quantum-mechanical perturbation theory (QM) and semi-classical trajectory theory (SC). As discussed elsewhere (<cit.>, and references therein), these theories result in predictions for integrated rates which differ by up to an order of magnitude, sufficient to have a major impact on the interpretation of observations.This difference is primarily the result of the different range in impact parameter over which |Δℓ| = 1 transitions are active in the different theories: the QM theory has a logarithmic divergence in transition rate, which requires a limit to be placed on the largest impact parameter for which collisions are active in causing transitions, while in the VF01/VOS12 SC approach, this limit arises as a result of the finite domain in impact parameter over which a single collision can cause a complete |Δℓ| = 1 transition.In this paper, we will present the results of calculations using the semi-classical theory of VF01, but using an alternative Monte Carlo formalism.We find that with this alternative approach, the classical results agree well with those of the quantum theory.The differences which there are, are of the kind that would usually be expected when relating a quantum theory to its classical approximation, and are consistent with the correspondence principle.When integrated over impact parameter, b, and a thermal spectrum of colliders, it is clear that the differences in rates computed using this modified SC approach and the original QM formalism will be minimal.In Section <ref> we compare the quantum mechanical transition probabilities with semi-classical probabilities as calculated by VF01 and VOS12, and with our revised approach.In Section <ref>, we briefly discuss how the discretely-sampled VF01 SC approach can be made somewhat more accurate, and consistent with the discrete detailed balance relations.Finally, in Section <ref>, we summarize our results, and discuss the processes which prevent the overall dipole transition rate from diverging in a plasma of finite density. § CALCULATIONS VF01 and VOS12 provide detailed formulae for the rates of transitions ℓ→ℓ' for precise values of ℓ, ℓ'.They derive their semi-classical formulae using approximations which correspond to the continuum limit n,ℓ,ℓ' →∞, with ℓ/n and ℓ'/n finite, and then apply them in the case of finite quantum numbers.They apply what they term as a microcanonical ensemble, sampling the fundamentally continuous classical-limit expressions at discrete values of incoming and outgoing angular momentum appropriate for the angular momentum quantum number.Using this procedure provides values for the overall collision rate which are finite for all Δℓ, as noted above.The process of returning from the continuum to the discrete limit is not, however, unique.In the present paper, we use a method similar to that discussed by <cit.>.To ensure thermodynamic consistency for the derived total rates, it is better to follow a finite-volume formalism (see, e.g., <cit.>), where each of the discrete quantum numbers is taken to correspond to a finite range of continuum values.The simplest assumption which will ensure results are consistent with the thermodynamical equilibrium is to assume that the probability density of states in the continuum band corresponding to each of the aggregate states is internally in thermodynamic equilibrium.As there is no energy difference between states in the case considered by Vrinceanu et al., this corresponds to assuming a uniform population.This procedure provides results consistent with the thermodynamic requirements of unitarity and detailed balance and with the quantum and classical limits, as well as with usual practice in Monte Carlo simulation of off-lattice systems <cit.>.However, it leaves us looking to statistical physics, rather than numerical convention, to resolve the divergence in dipole transition rates.For the standard quantum mechanical association of the radial quantum number n, the total angular momentum quantum number ℓ satisfies 0≤ℓ≤ n-1. If we assume that each value of ℓ maps to a shell with total angular momentum between ℓħ and (ℓ+1) ħ, with a classical density of states ∝ℓ (which may be visualized as a two-dimensional polar coordinate system), then the area of this shell is ∝ 2ℓ+1.This is consistent with the number of z-angular momentum eigenstates | m |≤ℓ corresponding to each total angular momentum eigenstate.It results in a mean-squared angular momentum in the shell of⟨ L^2⟩ = [ℓ(ℓ+1)+1 2]ħ^2,which is a constant ħ^2/2 greater than the value which enters in quantum mechanical calculations, ⟨ L^2⟩ = ℓ(ℓ+1)ħ^2.For comparison, assuming that the total angular momentum corresponding to a quantum number ℓ is exactly ℓħ underestimates ⟨ L^2⟩ by ℓħ^2, which is a significantly larger error for large ℓ. [Taking the angular momentum for the discrete state to be (ℓ+12)ħ is more accurate, with the classical and quantum density of states being equivalent, and the mean-square angular momentum ⟨ L^2⟩ over-estimated by ħ^2/4.]Transition probabilities in the finite volume regime can be derived from the semi-classical results of VF01/VOS12 by interpreting them as probability density functions in the continuum limit, so the transition probability from the state (n,ℓ) to (n,ℓ') becomes⟨ P^ SC⟩_nℓℓ' = ∫_ℓ/n^(ℓ+1)/n dλ∫_ℓ'/n^(ℓ'+1)/n dλ'P^ SC(λ,λ',χ)g(λ)g(λ') / ∫_ℓ/n^(ℓ+1)/n dλg(λ).,where g(λ) = 2λ is the classical density of states, normalized so that ∫_0^1 g(λ) dλ = 1, andP^ SC(λ,λ',χ)= 2λ'/nπħsinχ{[0, |sinχ| < |sin(η_1-η_2)|;K(B/A)√(A) |sinχ| > |sin(η_1+η_2)|;K(A/B)√(B) ,; ].is the SC transition probability given by VOS12.In this expression, K is the complete elliptic integral,A= sin^2χ - sin^2(η_1-η_2), B= sin^2(η_1+η_2)-sin^2(η_1-η_2),cosη_1= λ, cosη_2= λ',and χ is given in terms of n and other implicit parameters of the collision bycosχ =1+α^2cos(√(1+α^2)ΔΦ)1+α^2, α = 3Z_1 2(a_n v_n bv),where the swept angle ΔΦ is assumed to be π.This procedure replaces the closed-form expressions of VF01 and VOS12 with a double integral, so is not as suitable for numerical work. However, it seems worthwhile to compare the results with those of the discrete interpretation in order to inform possible modifications to the VF01/VOS12 formalism which might be made to ensure compatibility with the limit of thermodynamic equilibrium.There are a number of desirable properties for any set of approximate transition probabilities.These include unitarity, i.e. that the system must reside in one of the angular momentum states at the end of the transition∑_l' P_nℓℓ' = 1,and detailed balance, i.e.(2ℓ+1) P_nℓℓ' = (2ℓ'+1)P_nℓ'ℓ,for the quantum level degeneracy g(ℓ)=2ℓ+1.Note that the sum for the unitarity requirement includes the probability that the scattering leads to no transition, P_nℓℓ' with ℓ' = ℓ.It is possible to determine this rate using the same analytic forms as for the ℓ-changing interactions, and this rate is included in the plots shown below.The symmetry of the expressions for A and B in λ and λ', together with the overall factor of λ' in equation (<ref>), means that equation (<ref>) satisfies the detailed balance relations in the continuum limit,2λ P^ SC(n,λ,λ') = 2λ'P^ SC(n,λ',λ),given the classical density of states g(λ) = 2λ.As a result of this, it is simple to verify that the phase-space average, equation (<ref>), satisfies the discrete detailed balance relation(2ℓ+1)⟨ P^ SC⟩_nℓℓ' = (2ℓ'+1)⟨ P^ SC⟩_nℓ'ℓ. Beyond these absolute requirements, we also suggest that the rates should be subject to another statistical requirement for collisions at small impact parameter.For these scatterings, the output state of the interaction is dependent on complex interference phenomena, sensitive to many details of the atomic physics.However, the net effect of this complexity, when averaged over some small range of incoming particle properties, would be expected to be asymptotically close to the output states being in statistical equilibrium (cf., for the classical case, <cit.>).We therefore suggest that, in the limit of close scatterings b→ 0, the rates should be subject to an ergodicity property⟨ P⟩_nℓℓ'≃2 ℓ'+1 n^2,i.e. when the collider passes close enough to the core of the target atom, the effect of the collision is to randomize the output state, when the input state is coarse-grained over a suitable domain.Of course, in reality scatterings will cease to be purely ℓ-changing in this limit.Even so, it is to be expected that the output state angular momentum will become statistically independent within the shell.This requirement seems to be the best physical interpretation which can be put on the statement in <cit.>, hereafter PS64, that in the core the scattering probability becomes a rapidly-oscillating function with mean value 12.This is what would result from the core ergodicity principle in the case of a two-level system, so the core ergodicity principle seems like a reasonable generalization, agreeing with the work of PS64 at least in spirit.As we will see, it is also a reasonable description of what in fact happens when the quantum and shell-averaged classical transition probabilities are calculated in detail. In Figures <ref> and <ref>, we compare the quantum mechanical probability distributions with the classical transition probabilities sampled at specific ℓ, ℓ'.The QM dipole transition rates, Δℓ = ± 1 decay slowly as b increases, which is the origin of the divergence of the rate integral for these transitions.The classical transition probabilities show sharp edges where the transitions first become allowed, for all Δℓ: the transition rates for all Δℓ are similar, as there is nothing in the SC formulation which fundamentally distinguished a |Δℓ|=1 transition from one with a larger change in angular momentum.There are also internal peaks for many cases, corresponding to orbital resonances.These are used by VOS12 to limit the domain over which |Δℓ|=1 transitions are allowed, avoiding the divergence in the integrated transition rate found for the QM dipole transition rate.It is clear that the classical transition probabilities cannot satisfy unitarity, as where any transition ceases to be allowed, there is no corresponding increase in the others.Indeed, at sufficiently large radii, the probability of no transition, P_nℓℓ, increases above unity, which is inconsistent with usual definition of probability.The quantum transition probabilities do satisfy unitarity (note that the curves as plotted are divided by 2ℓ'+1 to make the ergodicity property at small b more obvious, but this means that this summation property for the probabilities is less obvious as shown). In Figures <ref> and <ref>, we compare the quantum mechanical probability distributions with the classical probability distributions averaged over angular momentum shells.These plots are significantly more alike than those for the comparison between the quantum mechanical probability and the discretely-sampled classical transition probability.While there are no longer any sharp edges in the shell-averaged classical probabilities, for |Δℓ| > 1 they will be non-zero only within some range of α values. (The integration over a quantized shell width means that there is now a genuine distinction in qualitative behaviour between transitions with different |Δℓ|.)The shell-averaged classical probabilities also satisfy unitarity and the quantum-weighted detailed balance constraint.Away from the region where the discretely-sampled transition probability is zero, the binning has a relatively minor effect, simply smoothing out the steepest peaks.The general form of the transition probability distributions shown in these figures is of interest.Working from large b inwards, it is initially most likely that no change in ℓ will result.At least in the QM case, all other values of Δℓ are possible, with probabilities reducing as |Δℓ| increases.As b becomes smaller, the probabilities of the higher |Δℓ| transitions increase, following a smooth power law dependency, until the statisical weight of the output state reaches a similar level to that of the Δℓ=0 transition. Thereafter, the probabilities are subject to significant oscillations, around an average level consistent with statistical balance, apart from a strong spike at the very smallest values of b.This suggests that the combination of the asymptotic behaviour at large b, the core ergodicity principle, and the fundamental requirements of unitarity and detailed balance, should be sufficient to provide thermodynamically-consistent estimates of the impact-parameter and thermal-averaged rates which would be acceptably accurate for many applications.§ USE OF SEMI-CLASSICAL PROBABILITIESThe shell-sampled probabilities presented in the previous section are determined using a computationally expensive double integral.It may be possible to perform one or both of these integrals analytically, but numerical results were sufficient for the present analysis.However, given that the use of a classical transition rate is already a significant approximation, using a point sample of the transition rate rather than an integral is likely to be an acceptable approximation, at least away from the case of Δℓ = ±1, b→∞.As these rates are being attributed to quantum mechanical rather than classical states, it makes sense to ensure the rates are chosen so as to satisfy quantum mechanical rather than classical statistics.The most consistent identification of quantum mechanical states with a continuum band has ℓħ as the angular momentum at the innermost edge of the band.Hence, if a single value based on the classical transition probability is to be used for this quantum number, it will be accurate to a higher order if the angular momentum used in this expression is somewhat higher than ℓħ. In particular, in order to be consistent with the quantum mechanical detailed balance condition, the value 2ℓ'+1 should be used in the numerator of the prefactor of equation (6) of VOS12.This means that the ℓ→ℓ'=0 rates will not be strictly zero, as required by the expressions given by VF01.Given that VF01 provide expressions for the transition rate out of an ℓ=0 state, a zero inward rate is in clear violation of the detailed balance requirement.The values used in the elliptic integral terms must be symmetric functions of ℓ,ℓ', as is true for the expressions given. Ideally they should also be chosen to satisfy unitarity, but in reality the correction to the overall transition rates as a result of violating this constraint will be small compared to the other approximations underlying this approach.Using cosη_1 = (ℓ+12)/n, etc., will be at least somewhat more accurate than without the 12, and also means that the cases ℓ=0, ℓ'=0 do not require a special treatment.§ CONCLUSIONSWe have shown that, by using an alternative form for the Monte Carlo realization, the results for the SC and QM formalisms described by VF01 and VOS12 can be brought closely into line.As this is consistent with what is expected as a result of the correspondence principle, it provides further evidence that the results of the QM formalism of VF01 and VOS12 should be preferred over their SC formalism.While finding agreement between the different forms of theory is satisfying, this does not take into account the major reason given by VF01 and VOS12 for preferring their semi-classical results, specifically the need for an outer limit to be imposed on the integration over impact parameters to prevent the total collision rate diverging for Δℓ = ± 1 when using the QM theory.This type of divergence is common in other areas of collision rate physics (in particular, the two-body relaxation time, <cit.>), so is not unexpected, and the empirical limits of PS64 are similar to those applied to the Coulomb logarithm in these contexts.However, as b increases, so does the time over which the collision takes place: the treatment of collisions as independent events must therefore eventually become inaccurate.Following the reasoning underlying the diffusion-based approach to modelling transition rates between degenerate levels discussed by <cit.> (which requires no empirical cut-offs), we note that during the extended period taken to complete the most distant encounters, there will potentially be time for many collisions at smaller impact parameters.While, at first order, the effects of these collisions will superpose linearly, for collisions at sufficiently large b, a limit will be reached where the smaller impact parameter collisions together are sufficient to randomize the angular momentum of the target orbital.Once b increases above the level where this occurs, b_ eq, the effect of collisions at larger impact parameter will be felt, in effect, as a superposition of N_coll∼τ_coll/τ_eq partial interactions adding in quadrature, rather than linearly (where τ_coll is the collision time at the large b of interest, and τ_eq is the collision time at the smallest radius leading to effective randomization).A reduction in contribution to the transition probability by ∼ N_coll^-1/2∝ b^-1/2 at the largest b will be sufficient to prevent the weak logarithmic divergence in the overall rate.This is a somewhat academic argument, as radiative lifetimes and plasma particle correlations of the type discussed by PS64 will often result in more stringent limits to the range of b over which collisions are effective.Nevertheless, given that the agreement we now find between the SC and QM gives greater confidence in SC results for the Rydberg scattering problem, it may be possible to investigate the corrections required for these multiple interactions in a believable manner using explicit classical trajectory calculations.§ ACKNOWLEDGEMENTS We thank D. Vrinceanu and H. Sadeghpour for helpful responses to several queries about their published work.Parts of this work have been supported by the NSF (1108928, 1109061, and 1412155), NASA (10-ATP10-0053, 10-ADAP10-0073, NNX12AH73G, and ATP13-0153), and STScI (HST-AR-13245, GO-12560, HST-GO-12309, GO-13310.002-A, and HST-AR-13914). MC has been supported by STScI (HST-AR-14286.001-A). PvH was funded by the Belgian Science Policy Office under contract no. BR/154/PI/MOLPLAN. | http://arxiv.org/abs/1704.08722v1 | {
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[(INPAC, Shanghai Key Laboratory for Particle Physics and Cosmology, School of Physics and Astronomy, Shanghai Jiao-Tong University, Shanghai, 200240, China We discuss the implications ofthe recently reported R_K and R_K^* anomalies, the lepton flavor non-universality in the B→ Kℓ^+ℓ^- and B→ K^*ℓ^+ℓ^-.Using two sets ofhadronic inputs of form factors, we perform a fit of the new physicsto theR_K and R_K^* data, and significantnew physics contributionsare found.We suggest to studythe lepton flavor universality ina number of related rare B, B_s, B_c and Λ_b decay channels,and in particular we give the predictions forthe μ-to-e ratios of decay widthswith different polarizations ofthe final state particles,and ofthe b→ dℓ^+ℓ^- processes which are presumablymore sensitive to the structure of the underlying new physics. With the new physics contributions embedded in Wilson coefficients, we present theoreticalpredictions for lepton flavor non-universality in these processes. Implications of the R_K and R_K^* anomalies Wei Wang and Shuai Zhao [[email protected]] December 30, 2023 ====================================================== § INTRODUCTION The standard model (SM) of particle physics is now completed by the discovery of Higgs boson. Thus thefocusin particle physics has been gradually switched to the search for new physics (NP) beyond the SM. Thiscan proceed in two distinctways. One is the direct search at the high energy frontier, in which new particles beyond the SM are produced and detected directly. The other is called indirect search, which is at the high intensity frontier. The new particles will presumably manifest themselves as intermediate loop effects, and might be detectableby low-energy experiments with high precision. In flavor physics, theb → s ℓ^+ℓ^- process is a flavor changing neutral current (FCNC) transition. This process isof special interest sinceit isinduced by loop effects in the SM, which leads totiny branching fractions.Many extensions of the SMcan generate sizableeffects that can be experimentally validated. In particular, the B → K^*(→ Kπ)μ^+ μ^- decayoffers a large number of observables to test the SM, ranging from the differential decay widths, polarizations, toa full analysis of angular distributions of the final state particles, for an incomplete list one can refer to Refs. <cit.> and many references therein. In the past few years,quite a few observables in the channels mediated byb→ sℓ^+ℓ^- transitionhave exhibited deviations from the SM expectations.The LHCb experiment has firstobserved the so-called P_5^' anomaly, a sizeable discrepancy at 3.7 σbetween the measurement and the SM prediction inone bin for the angular observable P_5^' <cit.>. This discrepancy was reproduced in a later LHCb analysis forthetwo adjacent bins at large K^* recoil <cit.>. To accommodate this discrepancy,considerableattentions have been paid to explorenew physics contributions (see Refs. <cit.> and references therein), whileat the same time, thishas also triggered the thoughts to revisit the hadronic uncertainties <cit.>.More strikingly, the LHCb measurementofthe ratio <cit.>:R_K[q_ min^2,q_ max^2]≡∫_q_ min^2^q_ max^2dq^2 dΓ(B^+ → K^+ μ^+ μ^-)/dq^2/∫_q_ min^2^q_ max^2dq^2 dΓ(B^+ → K^+ e^+ e^-)/dq^2,gives a hint forthelepton flavour universalityviolation (LFUV). A plausible speculation is that deviations from the SM arepresent in b→ sμ^+μ^- transitions insteadin b→ s e^+e^- ones. Very recently the LHCb collaboration has found sizable differences betweenB→ K^*e^+e^- andB→ K^*μ^+μ^-at both low q^2 region and central q^2 region <cit.>. Results for ratiosR_K^*[q_ min^2,q_ max^2]≡∫_q_ min^2^q_ max^2dq^2 dΓ(B → K^*μ^+ μ^-)/dq^2/∫_q_ min^2^q_ max^2dq^2 dΓ(B → K^* e^+ e^-)/dq^2,are given in Tab. <ref>, from which we can see the data showed significant deviations from unity. Theseinteresting results havesubsequentlyattracted many theoretical attentions <cit.>. The statistics significance in the datais low at this stage, about 3σ level. In order to obtain more conclusive results, one shouldmeasure the muon-versus-electron ratiosin the B→ Kℓ^+ℓ^- and B→ K^*ℓ^+ℓ^- more precisely, meanwhile one should also investigate more channels with better sensitivities to the structures of new physics contributions. In this paper, we will focus on the latter.To do so, we will first discuss the implications of the R_K and R_K^* anomalies in a model-independent way, where the new particle contributions are parameterizedin terms ofeffective operators. Since there is lack of enough data, we analyze their impact on the Wilson coefficients of SM operators O_9,10.Wethen propose to study thelepton flavor universality ina number ofrare B, B_s, B_c and Λ_b decay channels. Incorporatingthenew physics contributions,we will presentthe predictions for the muon-versus-electron ratios in thesechannels, making use of various updates of form factors <cit.>. We will demonstratethatthemeasurementsoflepton flavor non-universality with different polarizations ofthe final state hadron,and in the b→ dℓ^+ℓ^- processes are of great value to decode the structure of the underlying new physics. The rest of this paper is organized as follows. In the next section, we will use a model-independent approach and quantifythe new physics effects in terms of the short-distance Wilson coefficients. In SectionIII, we will study the LFUV in various FCNC channels. Our conclusionis given in the last section. § IMPLICATIONS FROMTHE R_K AND R_K^* In this section, we will firststudy the impact of theR_K and R_K^* data. In the SM, the effective Hamiltonian for the transitionb→ sℓ^+ℓ^- H_eff=-G_F/√(2)V_tbV^*_ts∑_i=1^10C_i(μ)O_i(μ)involves thefour-quark and the magnetic penguin operators O_i. HereC_i(μ) are theWilson coefficients for these local operators O_i. G_F is the Fermi constant, V_tb and V_ts are CKM matrix elements. The dominant contributions to b→ sℓ^+ℓ^-come from the following operators:O_7 = e m_b/8π^2s̅σ^μν(1+γ_5)bF_μν+e m_s/8π^2s̅σ^μν(1-γ_5)bF_μν, O_9 = α_em/2π(l̅γ_μl) s̅γ^μ(1-γ_5)b ,O_10=α_em/2π(l̅γ_μγ_5l) s̅γ^μ(1-γ_5)b. The above effective Hamiltonian givesthe B→ Kℓ^+ℓ^- decay widthas:dΓ(B→ Kℓ^+ℓ^-)/dq^2 =G_F^2√(λ)α_em^2β_l/1536m_B^3π^5|V_tbV_ts^*|^2 ×[ λ (1+2m̂_l^2) | C_9f_+(q^2)+C_72m_b f_T(q^2)/ m_B+m_K |^2..+λβ_l^2 |C_10|^2 f_+^2 (q^2)+6m̂_l^2 |C_10|^2(m_B^2-m_K^2)^2 f_0^2(q^2)],where m̂_l=m_l/√(q^2),β_l=√(1-m̂_l^2), λ= (m^2_B-m^2_K-q^2)^2-4m^2_Kq^2, and f_+,f_0 and f_T are the B→ K form factors. In the above expression, we have neglected the non-factorizable contributions which are expected to be negligible for R_K.Thedecay width for B→ K^*ℓ^+ℓ^- can bederivedin terms of the helicity amplitude <cit.>. The differential decay width is given asdΓ(B→ K^* ℓ^+ ℓ^-)/dq^2 = 3/4(I_1^c+ 2I_1^s)-1/4(I_2^c+2I_2^s),withI_1^c = (|A^1_L0|^2+|A^1_R0|^2)+8m̂_l^2Re[A^1_L0A^1*_R0 ]+4 m̂_l^2|A^1_t|^2,I_1^s = (3/4- m̂_l^2 )[|A^1_L⊥|^2+|A^1_L|||^2+|A^1_R⊥|^2+|A^1_R|||^2] + 4 m̂_l^2 Re[A^1_L⊥A^1*_R⊥+ A^1_L||A^1*_R||], I_2^c =-β_l^2(|A^1_L0|^2+ |A^1_R0|^2), I_2^s = 1/4β_l^2(|A^1_L⊥|^2+|A^1_L|||^2+|A^1_R⊥|^2+|A^1_R|||^2).The handednesslabel L or R corresponds tothe chirality of the di-lepton system. Functions A_L/Rican beexpressed in terms of B→ K^* form factorsA^1_t =2 √(N_K_J^*) N_1C_10√(λ)/√(q^2)A_0(q^2),A^1_L 0=N_1 √( N_K_J^*)/2m_K^*_J√(q^2)[ (C_9-C_10)[(m_B^2-m_K^*^2-q^2)(m_B+m_K^*)A_1-λ/m_B+m_K^*A_2]. . + 2m_b C_7[ (m_B^2+3m_K^*^2-q^2)T_2 -λ/m_B^2-m_K^*^2T_3]],A^1_L ⊥ =- √(2N_K_J^*) N_1[(C_9-C_10)√(λ)V/m_B+m_K^* +2m_b C_7/q^2√(λ)T_1],A^1_L || = √(2N_K_J^*) N_1[(C_9-C_10) (m_B+m_K^*)A_1 +2m_b C_7/q^2(m_B^2-m_K^*^2)T_2 ] , with N_1= iG_F/4√(2)α_ em/π V_tbV_ts^*, N_K_J^* = 8/3√(λ)q^2β_l/(256π^3 m_B^3) and λ≡ (m^2_B-m^2_K^*-q^2)^2-4m^2_K^*q^2.The right-handed decay amplitudes areobtained by reversing the sign of C_10:A_Ri =A_Li|_C_10→ -C_10.Within the SM, one can easilyfind thatresults for R_K and R_K^* are extremely close to 1 and thus deviate from the experimental data. If new physicsis indeed present, it can be in b→ sμ^+μ^- and/orb→ se^+e^-transitions. In order to explain theR_K and R_K^* data, one can enhance the partial width for the electronic mode or reduce the one for the muonic mode.It seems that the SM result for the B→ Ke^+e^- is consistent with the data, and thus here we will adopt the strategythat the muonic decay width is reduced by new physics. After integrating out the high scale intermediate states the new physics contributions can be incorporated into the effective operators. As there is lack of enough data that shows significant deviationswith SM, we will assume that NP contributions can be incorporated into Wilson coefficients C_9 and C_10. For this purpose, we defineδ C_9^μ =C_9^μ -C_9^ SM,δ C_10^μ =C_10^μ -C_10^ SM.The O_7 contribution to b→ sℓ^+ℓ^- arises from the coupling ofa photon with the lepton pair. On one hand, this couplingishighly constrained from the b→ sγ data. On the other hand, this coefficient is flavor blinded and thus even if NP affect C_7, the μ-to-e will not be affected.For the analysis, we adopt three scenarios,*Only C_9 is affectedwith δ C_9^μ≠0.*Only C_10 is affectedwith δ C_10^μ≠0.*BothC_9 and C_10 are affected in the form:δ C_9^μ=-δ C_10^μ≠0.Using the R_K and R_K^* data, we show our results in FIG. <ref>. The left panel corresponds to scenario 1, andthe middle panel corresponds to the constraint on δ C_10^μ, the last one corresponds to the scenario 3 with a nonzero δ C^μ_9- δ C^μ_10. In this analysis, we have usedtwo sets ofB→ K and B→ K^* form factors. One is from the light-cone sum rules (LCSR) <cit.>, corresponding to the dashed curves. The other is from Lattice QCD (LQCD) <cit.>, which gives the solid curves. As one can see clearly from the figure,the results are not sensitive to the form factors, and this alsopartly validate the neglect of other hadronic uncertainties like non-factorizable contributions.Using the LQCD set of form factors <cit.> and the data in Tab.I,we found the best-fittedcentral value and the 1σ range for δ C_9^μin scenario 1 asδ C_9^μ = -1.83 ,-2.63< δ C_9^μ< -1.25 .For scenario 2, we haveδ C_10^μ = 1.43,1.04< δ C_10^μ< 1.89,whilefor the δ C_9^μ =- δ C_10^μ, we obtainδ C_9^μ- δ C_10^μ= -1.47,-1.89< δ C_9^μ- δ C_10^μ< -1.08. A fewremarks are given in order.*Since the Wilson coefficient in the electron channel is unchanged,the δ C_9^μ and δ C_10^μ could be viewed as the difference between theWilson coefficients for the lepton and muon case. *We have found the largest deviation between the fitted results and the data comes fromthe low-q^2 region. Removing this data, we show the χ^2 in FIG. <ref> as dotted and dot-dashed curves, where the χ^2 has been greatly reduced.The reason is that in low-q^2 region, the dominant contribution to R_K^* arises from the transverse polarization of K^*. From Eq. (<ref>) and (<ref>), one can see this contribution is dominated byO_7 andless sensitive to O_9,10. A light mediator that only couples to the μ^+μ^- is explored for instance in Refs. <cit.>.*For theR_K and R_K^* predictionsin Refs. <cit.>, theoretical errors are typically less than one percent, while Ref. <cit.>gives the prediction with even smaller uncertainty R_K= 1.0003± 0.0001. However it is necessary to stress thattheseresults did not consider the electromagnetic corrections properly. We give the Feynman diagrams in Fig. <ref>. Fig. <ref>(a) is the typical Sudakov form factor, which usually introduces a double logarithm in terms of α/πln (q^2/m_ℓ^2).The difference between thedouble logarithms for the electron and muon mode is about 3%.A complete analysis requests the detailed calculation of all diagrams inFig. <ref> and analyses can be foundin Ref. <cit.>. The nonfactorizable corrections to the amplitude can be found in Ref. <cit.>. *It is necessary to point out that there are a number of observables in B→ K μ^+μ^- and B→ K^* μ^+μ^- that have been experimentally measured. These observables are of great values to provide very stringent constraints on the Wilson coefficients in the factorization approach. On the other hand, most of these observables in B→ K μ^+μ^- and B→ K^* μ^+μ^- are not sensitive to the flavor non-universality coupling since only the mu lepton is involved. The exploration of the μ-to-e ratios will be able to detect thedifference in the new physics couplings to fermions.It is always meaningful to conducta comprehensive global analysis andincorporate as many observables as possible. At this stage, the study of flavor non-universality in flavor physics is at the beginning, and we believe measuring more μ to e ratios (for instance the ones in Table II shown in the following section) will be helpful.*For a more comprehensive analysis, one may combine various experimental data on the flavor changing neutral current processes for instancein Refs. <cit.>. We quote the results in scenario I in Ref. <cit.>,δ C_9^μ =-1.58±0.28,δ C_9^e =-0.10±0.45,from which we can see that the results are close to our scenario 1.This implies that for the determination of flavor dependent Wilson coefficient, the R_K and R_K^* are dominant.From a practical viewpoint, since the main purpose of this paper is to explore the implications of the large lepton flavor non-universality, we will use our fitted resultsto predict the lepton flavor non-universality for a number of other channels.Explicit models which can realize these scenariosinclude the flavor non-universal Z' model, leptoquark model and vector-like models, see, e.g., Refs. <cit.> and many references therein. Their genericcontributions are shown in FIG. <ref>. Taking the Z' model as an example,the SM can be extended by including an additional U(1)^' symmetry, which can leads to the Lagrangian of Z^'b̅s couplings L_FCNC^Z^'=-g^'(B_sb^Ls̅_Lγ_μb_L + B_sb^Rs̅_Rγ_μb_R)Z^'μ +h.c..It contributes to the b→ sℓ^+ℓ^- decay at tree levelH_eff^Z^'= 8G_F/√(2) (ρ_sb^Ls̅_Lγ_μb_L + ρ_sb^Rs̅_Rγ_μb_R) (ρ_ll^Lℓ̅_Lγ^μℓ_L +ρ_ll^Rℓ̅_Rγ^μℓ_R) ,where the coupling isρ_ff'^L,R≡g'M_Z/gM_Z' B_ff'^L,Rwherethe g standard model SU(2)_L coupling.For simplicity, onecanassume thatthe FCNC couplings of the Z^' and quarks only occur in the left-handed sector:ρ_sb^R=0. Thus in this case the effects of the Z^'willmodify the Wilson coefficients C_9 and C_10:C_9^Z^' =C_9^ -4π/α_ emρ_sb^L (ρ_ll^L+ρ_ll^R)/V_tbV^*_ts,C_10^Z^' =C_10+4π/α_ emρ_sb^L(ρ_ll^L-ρ_ll^R) / V_tbV^*_ts.From this expression, we can see that the δ C_9^μ and δ C_10^μ are not entirely correlated. This corresponds to the scenario 1 and 2 in our previous analysis.The impact in a leptoquark model hasbeen discussed for instance in Ref. <cit.>, where the NP contribution satisfiesδ C_9^LQ,μ=-δ C_10^LQ,μ.This corresponds to the scenario 3. § LEPTON FLAVOR UNIVERSALITY IN FCNC CHANNELSIn this section, we will study the μ-to-e ratios of decay widthsin various FCNCchannels.Since the three scenarios considered in the last section describe the data equally well, we will choose the first one for illustration in the following. We follow a similar definitionR_B, M[q_ min^2,q_ max^2]≡∫_q_ min^2^q_ max^2dq^2 dΓ(B→ M μ^+ μ^-)/dq^2/∫_q_ min^2^q_ max^2dq^2 dΓ(B→ M e^+ e^-)/dq^2,where B denotes a heavy particle and M denotes a final state. The channels to be studied includeB→ K^*_0,2(1430)ℓ^+ℓ^-, B_s→ f_0(980)ℓ^+ℓ^-, B→ K_1(1270)ℓ^+ℓ^-, B_s→ f_2(1525)ℓ^+ℓ^-, B_s→ϕℓ^+ℓ^-, B_c→ D_sℓ^+ℓ^-, B_c→ D_s^*ℓ^+ℓ^-.The expressions for their decay widths have been given in the last section.In addition, we will also analyzeon the R ratio forthe baryonic decay Λ_b→Λℓ^+ℓ^-. The differential decay width for Λ_b→Λℓ^+ℓ^- is given as <cit.> Γ/ q^2[Λ_b→Λℓ^+ℓ^-]= 2 K_1ss + K_1cc ,whereK_1ss(q^2) = 1/4[|A_⊥_1^R|^2 + |A__1^R|^2 + 2 |A_⊥_0^R|^2 + 2 |A__0^R|^2 + (R ↔ L)] ,K_1cc(q^2) = 1/2[|A_⊥_1^R|^2 + |A__1^R|^2 + (R ↔ L)].The functions A are definedasA_⊥_1^L(R)= √(2) N [(C_9∓ C_10) H_+^V - 2 m_b C_7/q^2 H_+^T ], A__1^L(R)= -√(2) N [(C_9∓ C_10)H_+^A + 2 m_bC_7/q^2 H_+^T5],A_⊥_0^L(R)= √(2) N [(C_9∓ C_10) H_0^V - 2 m_bC_7/q^2 H_0^T ] ,A__0^L(R) = -√(2) N [(C_9∓ C_10) H_0^A + 2 m_bC_7 /q^2 H_0^T5],where the normalization factor N isN=G_F V_tbV^*_tsα_ em√(q^2√(λ(m^2_Λ_b,m^2_Λ,q^2))/3· 2^11 m^3_Λ_bπ^5). The helicity amplitudes are given byH_0^V =f_0^V(q^2) m_Λ_b + m_Λ/√(q^2) √(s_-),H_+^V = -f_⊥^V(q^2)√(2 s_-), H_0^A=f_0^A(q^2)m_Λ_b - m_Λ/√(q^2) √(s_+),H_+^A= - f_⊥^A(q^2)√(2 s_+) H_0^T =-f_0^T(q^2)√(q^2) √(s_-) ,H_+^T =f_⊥^T(q^2) ( + )√(2 s_-) , H_0^T5= f_0^T5(q^2)√(q^2) √(s_+) ,H_+^T5 = - f_⊥^T5(q^2) ( - )√(2 s_+),where s_±≡ (m_Λ_b± m_Λ)^2 - q^2. The f^i_0/⊥ with i=V, A, T, T5 are theΛ_b→Λ form factors. The B_s→ϕℓ^+ℓ^- and Λ_b→Λ form factors are used fromLQCD calculation in Refs. <cit.>, respectively. The B→ K_0^*(1430) and B_s→ f_0(980) form factors are taken from Ref. <cit.>. The B→ K_1(1270) form factors are calculated in the perturbative QCD approach <cit.>, and the mixing angle between K_1(1^++) and K_1(1^+-) is set to be approximately 45^∘. In this case the B→ K_1(1400)ℓ^+ℓ^- is greatly suppresed <cit.>. The B→ K_2 and B_s→ f_2(1525) form factors are taken from Ref. <cit.>.The B_c→ D_s/D_s^* form factors are provided in light-front quark model <cit.>, and in this work we have calculatedthe previously-missing tensor form factors. Using the Wilson coefficient δ C_9^μ in Eq. (<ref>), we presentour numerical results for R_B, M in TABLE <ref>. Three kinematics regions are chosen in the analysis: lowq^2 with [0.045, 1] GeV^2, central q^2with [1, 6] GeV^2 and high q^2 region with[14 GeV^2, q^2_ max=(m_B-m_M)^2]. For a vector final state, the longitudinal and transverse polarizations are separated and labeled as L and T, respectively.For Λ_b→Λℓ^+ℓ^-, a similar decomposition is used, in which the superscript0 means the Λ_b and Λ have the same polarizationwhile 1 corresponds to different polarizations. The SM predictions for these ratios are listed in Tab. <ref>.A few remarks are given in order.*From the decay widthsfor B→ K^*ℓ^+ℓ^-, we can see thatin the transverse polarization,the contribution from O_7is enhanced at low q^2, and thus the R_B,M^T is less sensitive to the NP in O_9,10. Measurements of the μ-to-eratiointhe transverse polarization of B→ Vℓ^+ℓ^-at low q^2 can tell whether the NP is fromthe q^2 independent contribution in C_9,10 or the q^2 dependent contribution in C_7. *In the central q^2region,the operators O_7 and O_9,10 will contribute destructively to the transverse polarization of B→ Vℓ^+ℓ^-. ReducingC_9 with δ C_9^μ<0will affect the cancellation, and as a result the decay width for the muonic decay mode will be enhanced. Thus instead of having a ratio smaller than 1, one will obtain a surplus for this ratio.*Results for Λ_b→Λ with different polarizations aresimilar, but it should be pointed out thatdifferential decay widths in Eq. (<ref>) have neglected the kinematic lepton mass corrections. Thus the results in thelow q^2 region are not accurate.*For the B→ K_0,2(1430)ℓ^+ℓ^- and B_c→ D_s^*,the high q^2 region has a limited kinematics, and thus the results are difficulttobe measured.*Among the decay processes involved in Table II, a few of them have been experimentally investigated: the branching fractions of B_s→ϕℓ^+ℓ^- <cit.>, Λ_b→Λℓ^+ℓ^- <cit.> and B_s→ f_0(980)ℓ^+ℓ^- <cit.> have been measured. So for these channels, the measurement of the μ-to-e ratio will be straightforward when enough statistical luminosity is accumulated. For the other channels, we believe most of them except the B_c decay might also be experimentally measurable, especially at the Belle-II with the designed 50 ab^-1 data and the high luminosity upgrade of LHC. *In FIG. <ref>, a new particle likeZ' or leptoquarkcan contribute to the R_K and R_K^*. The couplingstrength is unknown, and in principleit could be different from the CKM pattern.In the SM, theB→πℓ^+ℓ^- and B_s→ Kℓ^+ℓ^- have smaller CKM matrix elements. Thus if the NP contributions hadthe same magnitude as in b→ sℓ^+ℓ^-, their impact inB→πℓ^+ℓ^- and B_s→ Kℓ^+ℓ^-would be much larger. But in many frameworks,the new physics in b→ dℓ^+ℓ^- is suppressed compared to those in b→ sℓ^-ℓ^-, for recent discussions see Ref. <cit.>.This can be resolvedby experiments in the future. *The weak phases from Z' and leptoquark can be different from that in b→ sμ^+μ^- orb→ dμ^+μ^-, whichmayinduce direct CP violations. In the b→ dμ^+μ^- process,the current data on B→πμ^+μ^- contains a large uncertainty <cit.> A_CP(B^±→π^±μ^+μ^-) =(-0.12±0.12±0.01).This can be certainly refinedin the future. It should be noticedthat the SM contribution mayalso contain CP violation source <cit.> since the up-type quark loop contributions are sizable.§ CONCLUSIONS Due to the smallbranching fractions in the SM, rare decaysof heavy mesons canprovidea rich laboratory to search for effects of physics beyond the SM.Up to date, quite a fewquantities in B decayshave exhibited moderate deviations from the SM. This happens in both tree operator and penguin operator induced processes. The so-called R_D(D^*) anomaly gives a hint that the tau lepton might have a different interaction with the light leptons. TheV_ub and V_cb puzzles referto the difference for the CKM matrix elementsextracted from the exclusive and inclusive decay modes. In the b→ sℓ^+ℓ^- mode, the P_5'in B→ K^*ℓ^+ℓ^- has received considerable attentions on both the reliable estimates of hadronic uncertainties and new physics effects. In addition, LHCb also observed a systematic deficit with respect to SM predictions for the branching ratios of several decay modes, such as B_s →ϕμ^+μ^- <cit.>. Though the statistical significance is low, all these anomaliesindicate that the NP particles couldbe detectedin flavor physics. In this work, we have presented an analysis of the recently observedR_K and R_K^* anomalies. In terms ofthe effective operators,we have performeda model-independent fit to theR_K and R_K^* data. In the analysis, we have used two sets of form factors and found the results are rather stable against thesehadronic inputs. Since the statistical significance in R_K and R_K^* is rather low, weproposed to study a number of related rare B, B_s, B_c and Λ_b decay channels,and in particular wehave pointed outthat theμ-to-e ratios of decay widths with different polarizations ofthe final state particles,and in the b→ dℓ^+ℓ^- processes are likely more sensitive to the structure of the underlying new physics. After taking into account the new physics contributions, we made theoreticalpredictions onlepton flavor non-universality in these processes which canstringently examined by experiments in future. § ACKNOWLEDGEMENTSWe thank Yun Jiang and Yu-Ming Wang for useful discussions. This work is supportedin part by NationalNatural Science Foundation of China under GrantNo.11575110, 11655002, 11735010,NaturalScience Foundation of Shanghai under GrantNo. 15DZ2272100 and No. 15ZR1423100,by the Young Thousand Talents Plan, andbyKey Laboratory for Particle Physics, Astrophysics and Cosmology, Ministry of Education. § DEFINITIONS OF R^L,T AND R^0,1 For B decays to vector final state, we define the longitudinal and transverse ratios R^L and R^T asR^L,T_V[q_ min^2,q_ max^2]≡∫_q_ min^2^q_ max^2dq^2 dΓ^L,T(B → V μ^+ μ^-)/dq^2/∫_q_ min^2^q_ max^2dq^2 dΓ^L,T(B → V e^+ e^-)/dq^2,where the longitudinal and transverse differential widths are defined bydΓ^L(B → V μ^+ μ^-)/dq^2 = 3/4 I^c_1-1/4 I^c_2,dΓ^T(B → V μ^+ μ^-)/dq^2 = 3/2 I^s_1-1/2 I^s_2, V denotes a vector final state. The expressions for I^c,s_1 and I^c,s_2 are given by Eq. 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"authors": [
"Wei Wang",
"Shuai Zhao"
],
"categories": [
"hep-ph",
"hep-ex"
],
"primary_category": "hep-ph",
"published": "20170426154510",
"title": "Implications of the $R_K$ and $R_{K^*}$ anomalies"
} |
Department of Physics, Cornell University, Ithaca, New York 14853Department of Physics, Cornell University, Ithaca, New York 14853Department of Physics, Cornell University, Ithaca, New York 14853By combining confocal microscopy and Stress Assessment from Local Structural Anisotropy (SALSA), we directly measure stresses in 3D quiescent colloidal liquids. Our non-invasive and non-perturbative method allows us to measure forces ≲ 50 fN with a small and tunable probing volume, enabling us to resolve the stress fluctuations arising from particle thermal motions. We use the Green-Kubo relation to relate these measured stress fluctuations to the bulk Brownian viscosity at different volume fractions and comparing against simulations and conventional rheometry measurements. We demonstrate that the Green-Kubo analysis gives excellent agreement with these prior results. This agreement provides a strong demonstration of the applicability of the Green-Kubo relation in nearly hard-sphere suspensions and opens the door to investigations of local flow properties in many poorly understood far-from-equilibrium systems, including suspensions that are glassy, strongly-sheared, or highly-confined. 05.40.-a, 05.60.-k, 82.70.Dd, 83.85.CgExperimental Realization of the Green-Kubo Relation in ColloidalSuspensions Enabled by Image-based Stress Measurements Itai Cohen========================================================================================================================All quiescent thermal systems may seem static macroscopically, but microscopically they fluctuate strongly. By observing the system's response to these thermal fluctuations, a material's linear transport coefficients can be predicted using the Green-Kubo relation <cit.>. This foundational relation – a central achievement of nonequilibrium statistical mechanics – has enabled numerous diverse theoretical calculations ranging from electrical and magnetic susceptibilities in quantum systems <cit.> to thermal conductivities in nanotubes <cit.>. In particular, it has been widely used to theoretically determine the viscosities in bulk <cit.>, confined <cit.>, supercooled <cit.>, and quantum <cit.> liquids, where external load is problematic or heterogeneities play a crucial role. Unfortunately, these applications have remained strictly theoretical due to the difficulties in experimentally observing fluctuations in atomic systems, which are too rapid (∼ ps) and weak (∼ μN) to mechanically resolve in experiments.Here, by using high-speed confocal microscopy in conjunction with Stress Assessment from Local Structural Anisotropy (SALSA) <cit.>, we directly measure the stress fluctuations in nearly hard-sphere colloidal liquids. Colloidal suspensions are comprised of particles that are small enough to demonstrate Brownian motions, while large enough to be optically imaged, providing length- and time-scales that are associated with system relaxation <cit.>. To measure a suspension's stress fluctuations, we use a confocal microscope to image the 3D microstructure of the sample, then use SALSA to determine its Brownian stress arising from interparticle thermal collisions. Since SALSA is image-based, non-invasive, non-perturbative, and able to measure the suspension stress with a tunable probing volume, it can resolve the weak stress fluctuations that are usually averaged out in conventional bulk measurements due to the requisite large probing volume.The suspension samples are comprised of silica spheres with a radius a= 490 nm in a water-glycerine mixture that has a matched refractive index and viscosity η_0=60 mPa·s. We add 1.25 mg/ml of fluorescein sodium salt to the solvent to shorten screening length (≤10 nm) and obtain nearly hard sphere interactions. The added fluorescein also makes the solvent fluorescent, so the solvent appears bright and the particles appeardark. We then image the particle configuration using a high-speed confocal microscope with a hyper-fine scanner that maximizes the stability in the vertical (z-axis) scanning position (schematic in Fig. <ref>(a)). To ensure that the suspension structure remains homogenous throughout the experiment, we image the sample within a minute after the sample cell is made. We capture 216 frames per second and acquire stacks of 100 images within 0.5 s∼ 0.02τ_B, where τ_B=6 π a^3 η_0/k_B T is the self-diffusion time of the sample.By implementing the previously developed SALSA method <cit.>, we determine the stress in our 3D suspensions. SALSA uses the featured particle positions to calculate the local structural anisotropy or fabric tensor ψ_ij^α(Δ) = ∑_β∈ nnr̂_i^αβr̂^αβ_j of particle α, where nn is the set of colliding neighbors that lie within a distance 2a + Δ from particle α (Δ = 106 nm in the current work), i,j are spatial indices, and r̂_ij is the unit vector between particles (see Fig. <ref>(b) and SI). Scaling the ensemble-averaged ψ_ij^α(Δ) by Δ enables us to estimate the probability of thermal collisions between particles. Consequently, the instantaneous Brownian stress of the sample can be approximated: σ_ij (V, Δ) = k_B T/Va/Δ∑_α∈ Vψ_ij^α(Δ) + nk_B T δ_ij, where V is the averaging window volume, k_B T is thermal energy, n is number density, and δ_ij is Kronecker delta function. Here, nk_B T is simply the ideal gas term.The typical volume of our probed region V = 61μm×15μm×12μm ∼ 10 pL contains approximately 6,000 particles at a volume fraction ϕ∼ 0.27. This small volume ensures that the stress fluctuations are not suppressed by the volume averaging, ∝ 1/V, while preserving bulk behavior. We plot the instantaneous stress σ_xz and σ_xy in Fig. <ref>(b), where ẑ is the gravitational axis and x̂ and ŷ are horizontal. In contrast to a flat line at zero level anticipated in a macroscopic measurement, we find that both σ_xz and σ_xy fluctuate up to ± 0.5 mPa. We note that the force fluctuations corresponding to these stresses are less than 50 fN, difficult to resolve using mechanical methods. We calculate the time-time autocorrelation function ⟨σ_ij(t+Δ t)σ_ij(t)⟩ for the stress components σ_xz and σ_xy, and show the correlation decay in a log-linear plot, see Fig. <ref> (a). Despite the slight sedimentation due to the density mismatch between the particle and solvent, both autocorrelation functions decay in the same fashion indicating an isotropic viscosity of the sample (see SI). We further examine the cross-correlation ⟨σ_xz(t + Δ t) σ_xy(t)⟩ and find it negligibly small, which is consistent with the system symmetry. While the exact function form of the autocorrelation decay cannot be determined from the current data due to the limited measurement time span, we use an exponential decay (∼ e^-Δ t/τ) to quantify the correlation time. In doing this, we find that the correlation time τ varies weakly with the suspension volume fraction ϕ (see Fig. 2 (b)). We compare our observed trend with previous simulations of short-time diffusivity D_ss where a^2 / D_ss roughly sets the relaxation time-scale of the system [19,20]. In simulations, D_ss decays approximately as D_ss∼ D_0(1- b ϕ) (red line, Fig. 2 (b)) with b on the order of 1.5 at intermediate volume fractions. Here, we find a weaker trend b∼0.60±0.23 (blue dashed line) indicating either our measurements are not sufficiently precise to determine b accurately or that the functional form changes at volume fractions approaching close-packing. With the measured stress fluctuations, we can directly calculate the shear viscosity of our sample via the Green-Kubo formula η_B =⟨V/k_B T∫⟨σ_ij(t+Δ t) σ_ij(t)⟩ dΔ t ⟩_i≠ j, where η_B is the Brownian contribution to the total shear viscosity η_tot. Since our suspension systems are nearly hard-sphere, we anticipate that the stresses are weakly correlated in space, and thus the sample viscosity is roughly independent of probe window size. To verify this, we change our probing (averaging) volume V, and investigate how the stress fluctuations vary. In Fig. <ref>(c), we plot the mean variance of shear stress C_ij = ⟨σ_ij(t) σ_ij(t) ⟩_t, i ≠ j as a function of V/V_p where V_p is the particle volume (4/3) π a^3. We find that C_ij is inversely proportional to V/V_p when V/V_p ≥ 200 corresponding to a cubic volume that is approximately six particles across. This inverse proportionality and constant viscosity shown in the inset of Fig. <ref>(c) are consistent with the Green-Kubo formula. When V/V_p ≤ 200, we find that the viscosity slightly deviates from its bulk value. The viscosity reduction is around 20% of the mean for the smallest probing volume explored – a three-particle wide cube. While this reduction is reminiscent of the system size-dependent viscosity associated with long-ranged stress correlations in atomic simulations <cit.>, in our nearly hard-sphere liquid system we do not anticipate such long-ranged correlations that lead to nonlocal viscosities. Instead, at small volumes, the stress fluctuations are strongly influenced by changes in particle number as particles pass into and out of the constrained field of view.To compare our results with macroscopic flow measurements and simulations, we use the measured stress autocorrelation in conjunction with the Green-Kubo relation to determine the Brownian viscosity η_B of suspensions at eight different volume fractions 0.12 ≤ϕ≤ 0.45 (see Fig. <ref>). The resulting viscosities (red circles) show excellent agreement with previous hydrodynamic Stokesian simulations (blue squares) <cit.>. To further confirm the accuracy of our SALSA stress measurement, we also use Brownian Dynamics simulations to generate sets of particle configurations matching the experimental parameters (e.g. particle size, solvent viscosity, and temperature), and compare the stresses calculated from actual virials F_ij X_ij (purple diamonds) with those calculated on the same data set with SALSA (green diamonds) <cit.>. Both results again show a quantitative agreement with the experimental measurements. Finally, the measured Brownian viscosities are compared with conventional mechanical measurements by subtracting the hydrodynamic contribution η_H from the total viscosity η_tot determined using rheometry (purple crosses) <cit.>. The rheology data points (colloidal PMMA and silica systems) are obtained from previous experiments <cit.> and the hydrodynamic contribution is calculated from previous analytical approximation for the high frequency viscosity <cit.>. We find good agreement between the viscosities determined by our stress fluctuation measurements and conventional rheometry at all volume fractions explored. Collectively, the agreements between our results, simulations, and bulk measurements provide a clear demonstration of the Green-Kubo formula in hard-sphere systems. In contrast to conventional mechanical measurements, which can only measure the flow-gradient stress and the difference between normal stresses, SALSA measures all stress components simultaneously. In Fig. <ref>(a) we report the pressure of the suspension at the eight volume fractions explored in Fig. <ref>. We find that the measured osmotic pressure arising from Brownian collisions (red disks) is well described by the Carnahan-Starling equation of state (gray line) [We note that in previous confocal measurements where the pressure is determined by calculating the available volume to insert an additional sphere into a system, and its surface area <cit.>, the volume and corresponding surface area become exceedingly small and difficult to measure when suspensions are dense. This uncertainty results in a mismatch between data and theory. In our experiment, SALSA accurately captures the particle collision probabilities and correctly reports the pressure at all tested volume fractions.]. In addition, we find that both the shear (η_B, light red points) and bulk (η_B^bulk, dark red points) viscosities roughly exhibit Π^2 scaling (dashed black line), as shown in Fig. <ref>(b). While the underlying mechanism of such an empirical scaling remains an open question, we can qualitatively understand this scaling for the bulk viscosity using a dimensional analysis. Since the correlations in the Green-Kubo formula decay approximately exponentially in time, and the relaxation time τ does not increases significantly with increasing pressure over the range measured, we haveη_B^bulk ∼∫_0^∞⟨ (Π(t+Δ t)-Π̅) (Π(t) - Π̅) ⟩ dΔ t ∼∫_0^∞ C_Π e^-Δ t/τ dΔ t∼⟨Π^2 ⟩ - ⟨Π⟩^2where C_Π is the variance of pressure <cit.>.While many previous studies have made analogies between the transport phenomena of colloidal systems and simple liquids <cit.>, we find that the observed Π^2 scaling is actually absent in atomic systems. Specifically, the atomic viscosity (blue curve in Fig. <ref>(b)) exhibits a similar scaling behavior, but only at high pressures corresponding to large ϕ. At low pressures, the viscosity trend deviates from the Π^2 scaling. We conjecture that this deviation is associated with the kinetic contribution to the viscosity <cit.>, which is associated with atom velocity, insensitive to Π, and dominates in the dilute limit (see SI). Collectively, our findings, which are made possible by SALSA, suggest that even the Brownian contribution to the colloidal viscosity can have a distinct transport mechanism than that in simple liquids.In conclusion, we measure the stress fluctuation in colloidal liquids with SALSA, and experimentally demonstrate the well-known Green-Kubo relation <cit.>. Our measurements essentially show that “as far as linear responses are concerned, the admittance is reduced to the calculation of time-fluctuations in equilibrium” <cit.>. Previous pioneering experiments were able to combine the Green-Kubo relation with numerical simulations to extract the viscosity of a 2D dusty plasma <cit.>. These measurements, however, relied on assumptions for the interparticle potentials and ignored power-law decays in the stress correlation characteristic of 2D systems, which are known to lead to diverging integrals <cit.>. The analysis presented here avoids many of these complications and opens the door to further investigations of stress distributions in liquids under shear, confinement, and at high densities where the suspension becomes glassy <cit.>. In such situations SALSA is still applicable since the solvent remains in equilibrium. More importantly, since the SALSA measurement is non-invasive, it also allows for probing the mechanical heterogeneity in a 3D colloidal glass <cit.>, in which we can perform a time-average for particle-scale stress calculation. Measuring the temporal and spatial stress fluctuations in such a system will shed light on the generalization of the Green-Kubo relation in far-from-equilibrium systems and elucidate the mechanisms that underly the flow behaviors of disordered systems.The authors thank James Sethna, Brian Leahy, James Swan, John Brady, and Wilson Poon for helpful discussions. I.C. and N.Y.C.L. gratefully acknowledge the Poon Laboratory at School of Physics & Astronomy, University of Edinburgh for generous use of their PMMA suspensions. I.C. and N.Y.C.L. acknowledge funding from National Science Foundation (NSF) NSF CBET-PMP Award 1509308. M. B. was supported by Department of Energy DOE-DE-FG02-07ER46393 and continued support from NSF DMR-1507607. 53 natexlab#1#1bibnamefont#1#1bibfnamefont#1#1citenamefont#1#1url<#>1urlprefixURL[Kubo(1957)]kubo1957statistical authorR. Kubo, journalJournal of the Physical Society of Japan volume12, pages570 (year1957).[Green(1954)]green1954markoff authorM. S. 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"authors": [
"Neil Y. C. Lin",
"Matthew Bierbaum",
"Itai Cohen"
],
"categories": [
"cond-mat.soft",
"cond-mat.stat-mech",
"physics.flu-dyn"
],
"primary_category": "cond-mat.soft",
"published": "20170427011324",
"title": "Experimental Realization of the Green-Kubo Relation in Colloidal Suspensions Enabled by Image-based Stress Measurements"
} |
Connected Vehicular Transportation: Data Analytics and Traffic-dependent Networking Cailian Chen, Shanghai Jiao Tong University Tom Hao Luan, Deakin University Xinping Guan, Shanghai Jiao Tong University Ning Lu, Thompson Rivers UniversityYunshu Liu, Shanghai Jiao Tong UniversityReceived: date / Revised version: date =================================================================================================================================================================================================================With onboard operating systems becoming increasingly common in vehicles, the real-time broadband infotainment and Intelligent Transportation System (ITS) service applications in fast-motion vehicles become ever demanding, which are highly expected to significantly improve the efficiency and safety of our daily on-road lives. The emerging ITS and vehicular applications, e.g., trip planning, however, require substantial efforts on the real-time pervasive information collection and big data processing so as to provide quick decision making and feedbacks to the fast moving vehicles, which thus impose the significant challenges on the development of an efficient vehicular communication platform. In this article, we present TrasoNET, an integrated network framework to provide realtime intelligent transportation services to connected vehicles by exploring the data analytics and networking techniques. TrasoNET is built upon two key components. The first one guides vehicles to the appropriate access networks by exploring the information of realtime traffic status, specific user preferences, service applications and network conditions. The second component mainly involves a distributed automatic access engine, which enables individual vehicles to make distributed access decisions based on access recommender, local observation and historic information. We showcase the application of TrasoNET in a case study on real-time traffic sensing based on real traces of taxis. § INTRODUCTIONOur earth is facing the unstoppable increasing trend of vehicles. In United States, there are on average 812 cars for every 1,000 people. In China, the amount of vehicles is estimated to be 250 million in 2020. The massive increase in vehicles has brought a series of social and environmental issues to our cities and daily lives such as frequent traffic jams, vehicle crashes, throat-choking air pollution, etc. A sustainable, intelligent and green transportation system is thus of crucial importance. Towards this goal, one practical solution is to use the cutting-edge wireless information and communication technologies to provide real-time transport-related information services to road administrators and vehicles, namely Connected Vehicular Transportation System (CVTS) <cit.>. As a result, the transportation efficiency can be significantly improved with more smooth traffic flows and travellers can get informed for more wise route selections and enhanced travel experience. Furthermore, both Google and Apple released their mobile operating systems for autos in 2014. It is estimated that the global Connected Car Market will reach 30.2 billion in 2015, and 80% of all autos sold in 2016 will be connected. Therefore, it is foreseeable that in near future, connected vehicles would embody a pragmatic solution towards Intelligent Transportation System (ITS).CVTS aims to make safer and more coordinated use of transportation networks. Such a system lies on the timely collection of road traffic information, effective data analytics, and quick decision making and feedbacks to the traffic management facilities and vehicles. Besides the roadside sensors (e.g. GPS, cameras, inductive loops, RFID and in-road reflectors) deployed in the city-wide for traditional traffic sensing, connected vehicles provide a new efficient traffic monitoring method by the live data streams of large number of the off-the-shelf mobile terminals (e.g., on-board wireless communication facilities, smartphones, tachographs and wearable devices). It is envisioned that with the adoptions of the embedded, tethered or smartphone integrated vehicular sensing and communication facilities, the demand of on board infotainment services would become more demanding, which eventually would generate a large volume of data required for processing. Moreover, the large variety of data sources and applications require CVTS with the assets of fast response and processing rate, and more importantly, with high accuracy, reliability and security.With the increasingly growing data in CVTS <cit.>, there rise the fundamental engineering challenges from the following three aspects: (i) big data collection from ubiquitous roadside and in-vehicle sensors in the city; (ii) deep data analysis in traffic management center; and (iii) real-time decisions returned to traffic management facilities and vehicles. In this cycle, the first step is to timely, effectively and economically collect monitoring data from the ubiquitous sensors in the city. Compared to the dramatic improvement on technical tools for handling data <cit.>, vehicular networking for CVTS applications has, however, adapted much slower towards the low-cost and efficient data collection, which motives this article.In this article, we unfold our journey by first reviewing the impact of data analysis on some representative realtime traffic-related services. The basic requirements for vehicular networking architecture are then identified. It is followed by a traffic-dependent network architecture for traffic data collecting and efficient service provisioning. Lastly, a case study is presented to show how realtime traffic estimation and timely network access can be implemented under the proposed architecture. The main contributions of this paper are summarized as follows: * A novel Traffic-Social Network framework, called TrasoNET, is presented for the first time to build the connection of realtime traffic and networking. Under this framework, the data analytics of CVTS take effects on the macroscopic, midscopic and microscopic network resource allocation and network access. Networking information is an effective data source for traffic sensing. It makes data analytics and traffic-dependent networking mutually beneficial, which is the core idea of this work.* A new traffic-dependent network access scheme is developed with network access recommendation from higher layer (network) and distributed automatic access decision-making in lower layer (terminal). It enables individual vehicles to make access decisions based on access recommender and local observation on network conditions.* A case study is presented to show the real data analytics for traffic estimation in Shanghai, China. Extensive simulations demonstrate that TrasoNEt can effectively select optimum network to ensure QoS of vehicles/ mobile devices, and network resource is fully utilized without network congestions by data offloading.§ DATA ANALYTICS FOR CVTS APPLICATIONS The continuous monitoring on movements (e.g., safety services by short range v2x communications), mobility applications and vehicles condition monitoring would result in exponential growth of diverse source data which provides a wealth of information that is valuable to traffic management parties, drivers, repair shops and automakers.In the following, we list some representative CVTS applications to describe their dependance on data analytics.§.§.§ Real-time traffic estimationBy using the moving vehicles or smartphones on-board vehicles for data sensing, and uploading the sensing reports (such as time, location and heading direction of vehicles) to the data center, the realtime traffic conditions of the roads, such as average running speed and traffic density, can be achieved by data analysis.The taxi/bus management system in Shanghai, China, represents a practical deployment of the CVTS platform. Around 40,000 taxis and buses of Shanghai are now equipped with the on-board GPS and sensors which periodically report the vehicle information (GPS location, velocity, heading direction, passengers on/off) in cycles ranging from 30 seconds to 5 minutes. This results in 65 million records transmitted to the traffic management center everyday through cellular networks, and enables multiple management purposes. Road traffic conditions can be estimated efficiently by sparse sensing and advanced estimation methods. For example, compressive sensing and matrix completion based methods are reported in <cit.> based on the GPS dataset of Shanghai.§.§.§ Online navigation for connected vehiclesTraffic prediction is more difficult than traffic estimation. Fortunately, through correlation analysis of big data, traffic patterns can be gleaned more easily, faster and clearly than before. For example, a social proximity mobility pattern of vehicles is adopted in <cit.>, i.e., each vehicle has a restricted mobility region around a specific social spot such as a financial and sport center. By using data analysis, the social spots of vehicles in the real-world can be identified. The traffic peak probably appear at the rush hour around the social spot (SP), which makes traffics predictable. Five SPs has been shown in Fig. <ref>. In another example, the traffic can also be predicted from the message published in social networks, such as network group events (e.g., big show and football game) information including time, place, and number of attendees. The traffic can be predicted to influx to the social places before the event and outflow after the event.The traffic also has strong correlations with online navigation services: the more people in a particular geographic place search for the routes to a particular destination online, the more probably the traffic congestion happens on the route to the destination. In this case, the traffic can be predicted with the data from search engine. The online navigation server can provide more feasible path plans for vehicles.§.§.§ Remote vehicle diagnostics and road condition warningBased on the data collected from cars, the drivers can arrange better service interval by taking their own driving habits, and predicted wear and tear into account rather than conventional means based on defined number of kilometers. More importantly, valuable information on potential vehicle or road hazards can be delivered to drivers in realtime. For example, if a number of vehicles' traction control systems are activated at the same time and place, the cars in this area can be warned about “icy conditions", fuel-efficient driving in heavy traffic, and etc.§.§.§ Fuel up or chargeDuring travelling in the city, the electric vehicles/hybrid electric vehicles (EVs/HEVs) may make decision on what time and where to fuel up or load at location-specific charging piles by estimating the driving mileage according to real-time traffic conditions. The information provided by CVTS is valuable to design efficient ways of resource management in smart grid system in the city <cit.>.§.§.§ Dynamic urban planningSensory data from vehicles and mobile devices provide a pervasive way to understand how people use the city's infrastructure and affect the city, including urban dynamics, energy consumption and environment impacts such as noise and pollution. The big data related to real-time traffic can be used to improve city's services. For example, the planning and management parties can estimate how residential and working areas in cities are connected temporally, what the dynamic correlation of traffic density and pollution level appears, and how to reduce operational costs by optimizing planning. It also creates feedback loops with vehicles to reduce energy consumption and environmental impact.§ FEATURES OF DATA ANALYTICS IN CVTSFor the aforementioned applications, to explore the strong correlation among multi-source data is the key, which helps us capture the present road conditions and predict the future. The data collection for connected vehicles and related applications of CVTS distinguish with the traditional ones from the following three aspects: * From static sensing to dynamic sensing: As a large amount of traffic data can now be harnessed through ubiquitous roadside sensors, vehicles and mobile devices, static traffic sampling no longer makes as much sense. Moreover, due to the complicated traffic conditions, accurate traffic estimation and prediction can hardly be achieved on small random samples, and require as much data as possible. Connected vehicles make it possible.* From precise data to messy data <cit.>: In the applications, allowing for imprecision (for messiness) of data may be an advantage, rather than a shortcoming. We can infer the vehicles' direction, speed and position with messy GPS data thus traffic estimation can be improved to the level of predicting the traffic congestion in a particular road rather than a region in the city by using 65 millions of “dirty" (or “noisy") taxi GPS data rather than small precise samples from digital camera and loop detector <cit.>. The data could be messy if they cover as many streets as possible.* From parametric data to nonparametric data: Correlations are useful in a small-data world, but they really shine in the context of large volume data and/or big data.For traffic sensing and prediction, different (complicated) traffic models can be assumed to parameterize the relationship of traffic flows. However, multi-source data in CVTS allows us to pick a nonparametric model with simpler algorithms, and it results in more accurate than the sophisticated solution <cit.>. To summarize, for data analytics of connected vehicles, we can relax the standards of allowable errors and increase messiness by combining different types of data information from different sources. In dealing with even more comprehensive datasets, we no longer need to worry so much about individual data points, but biasing the overall analysis. Through them we can glean insights more easily, faster, and more clearly than before. Correlations of multi-source data let us analyze the city traffic not by shedding light on its inner working but by identifying a useful proxy (e.g., the online navigation requests implies the possible congestion) for it. It is foreseeable that big data enhances the data analytics in CVTS to change the way of services provisioning in and enables connected vehicles from multi-dimensions of traffic, vehicular network and ITS.§ TRAFFIC-DEPENDENT NETWORKING FOR CVTS To engineer an efficient and economic network architecture is the foremost issue to facilitate the data collection, decision feedbacks and traffic related services. §.§ Network FrameworkThe network framework is challenged by following issues: * From a traffic sensing perspective, even with the broad mobility of vehicles and the dense deployment of static sensors on the road, it cannot be guaranteed that the traffic information for all the roads in all the time could be sensed. It therefore calls for an efficient and economical sampling way for traffic sensing. Crowdsensing by more vehicles could be one of the solutions in the very near future.* From a social perspective, the spatial distribution of traffic may follow a specific social pattern such as the power-law distribution features for the traffic in Shanghai, China which is shown in Fig.<ref>. The traffic density decays from the business hot spots towards the boarder of the vehicles' mobility regions. One of the main challenges is finding a good guidance for wireless access to heterogeneous wireless networks (cellular network and vehicular ad hoc networks) such that the distributed traffic-dependent service requests can be satisfied with good Quality of Service (QoS)<cit.>. In this article, we describe a Traffic-Social Network (TrasoNET) framework for CVTS as in Fig. <ref> to support the crowdsensing and network access guidance according to realtime traffic and traffic pattern.TrasoNET consists of three layers: access network layer, data aggregation layer and application layer. In the access network layer, sensory nodes, including vehicles and the mobile devices, could connect to roadside communication infrastructures (e.g., cellular base stations and roadside units) and communicate through LTE/5G cellular networks and/or vehicular ad hoc networks (VANETs). The static sensors (e.g. cameras, inductive loops, RFID and in-road reflectors) transmit data through wired communication. In the data aggregation layer, the roadside communication infrastructure are connected to corresponding backbone routers. Data flows are combined through the so-called central controller sub-layer or so-called fog computing server, and further delivered to the cloud server through Internet. In the application layer, the traffic management center (TMC) aggregates the collected multi-source data from cloud and analyze the data to estimate and predict the road traffic. The cloud also connects to other service providers such that the traffic-related information can be fused out and provided in the application layer. Different traffic-related services are then delivered to vehicles through cellular core network and regional VANET.We elaborate on the four core components in the framework to highlight the characteristics in traffic-dependent networking.Infrastructure: The access infrastructures consists of the evolved NodeBs (eNBs) and RSUs. It is assumed eNBs cover the whole city, and the communication link between mobile device and eNB is more stable than that between mobile device with RSU. RSU is equipped with a wireless transceiver operating on DSRC and/or WiFi, and hence the transmission range is small compared with eNB. But it provides high-rate transmission for mobile devices. Due to the explosive growth of mobile data traffic, the cellular network nowadays is straining to meet the current mobile data demand and faces an increasingly severe overload problem. RSU is not only an alternative for V2I (vehicular to infrastructure) communications, but also enables an offloading for cellular networks <cit.>.Mobile Devices: We do not discriminate what kind of mobile devices they are, but care about what network they access to. Normally, smartphones can connect to cellular network through LTE/5G and VANET infrastructures through WiFi, while vehicles can additionally connect to VANET infrastructure and other vehicles through DSRC. Since WiFi and DSRC technologies can be applied to drive-thru connection when they are moving on the road <cit.>, we propose an automatic network access engine in the mobile devices to offload data originally targeted for cellular networks, which is referred to as the automatic offloading engine.Central Controller: The central controller is connected to base stations (e.g., eNBs for LTE), RSUs and Internet backbones. It allocates the network radio resources based on the realtime traffic estimated by TMC, and service demands requested by mobile devices. It acts as an interface between the physical network routers and the network operators to specify network services. The controller builds a logical control plane separated from data plane. Different from Internet Protocol (IP) based networks, such a frame enables mobile devices to move between different access interfaces without changing identities or violating specifications. The control function can be implemented by a protocol known as OpenFlow which enable controller to drive the access network edge hardware in order to create an easily programmable identity-based overlay on the traditional IP core.Cloud: As the data analytics center for TMC and other service providers, the cloud receives data from static traffic sensors and mobile devices, and analyzes them for traffic estimation and prediction. Other traffic-related services are then analyzed based on the realtime traffic and data from other service providers. One key feature provided by cloud is the access guidance for the mobile devices to facilitate the automatic offloading engine.As it is shown in Fig. <ref>, TrasoNET builds the connection of data collection, analytics and traffic-dependent networking from the following three aspects: * Firstly, the traffic big data are collected from static and mobile sensors through access network of TrasoNET. The aforementioned static sensors (e.g., cameras and inductive loops) transmit the traffic data to Regional Traffic Management Center through wired networks. Ubiquitous data from Mobile Devices (e.g., embedded, tethered or integrated on-board units, and smartphone) could be transmitted through wireless access network. For example, the probe vehicles (such as Taxis and buses) and floating cars (such as police cars from Public Security Bureau) in the city could provide sparse GPS data for preliminary traffic estimation. Then the traffic-dependent networking mechanism to be introduced in Subsection <ref> could facilitate big data collection from ubiquitous Mobile Devices. More data improves the traffic estimation and other traffic related services. * Secondly, on the aggregation layer and application layer of TrasoNET, the data analytics provides the real-time regional and global traffic conditions. It facilitates the Central Controller to allocate wide-area network radio resources (e.g., base stations, RSUs and Internet backbones) according to the estimated traffic density, speed, acceleration and other information of vehicles/users in the city. Another key feature dependent on real-time traffic condition is the regional network access guidance for Mobile Devices, which realizes locally network resource management. As for Mobile Devices, the decision-making of network selection and handover can be given locally by the guidance-based access mechanism for efficiency and offloading purposes. In this sense, the data analytics take effects on the macroscopic, midscopic and microscopic network resource allocation and network access. * Thirdly, the various data from different network components provide complimentary data for deep data analytics. For example, the number of vehicles connected to an access point of wireless communication can reflect vehicle density, which can reduce the cost for satisfactory traffic estimation accuracy compared to traditional sensing methods with digital cameras and loop detectors. Besides the realtime estimation, the TrasoNET facilitates online navigation, remote vehicle diagnostics, fuel up and charge and other emerging applications.§.§ Traffic-dependent Network Access Mechanism The access control of networks is one of the key mechanism to guarantee the real-time CVTS applications. In the framework of TrasoNET, we give a guidance-based automatic access mechanism for efficient and offloading purposes. From the perspective of network access, the aforementioned four components map the phases of guidance, information push and distributed decision-making into Access Recommender Console, Broadcasting and Automatic offloading Engine.Access recommender console: To recommend an “optimum network” to vehicles based on multiple criteria, the cloud could apply intelligent computation methods to set the priority of network access for a specific region under realtime traffic condition. The well-known Analytic Hierarchy Process (AHP) for multi-criteria decision <cit.> is one of good solutions. The logical flowchart of AHP algorithm is given in Fig. <ref>. The key steps are introduced in the following. * Model the network recommendation problem as a hierarchy which contains the goal, alternatives for reaching the goal, and criteria for evaluating alternatives. * Establish priorities among the elements of the hierarchy by making a series of judgments based on pair-wise comparisons of these elements. The compared results construct a pair-wise comparison matrix A=a_ij,i,j=1,2,⋯,n, where n is the number of criteria of second level,and every element a_ij is based on a standardized comparison scale from equal importance to dominance. * The pair-wise comparison matrix should satisfy transitive preference and strength relations, it is necessary to check its consistency. Calculate consistency indicators C.I., random consistency indicators RI, and get the consistency ratio CR=CI/RI. For example, consistency of judgement matrix is acceptable for the case of CR<0.1. * Synthesize these priority vectors to construct an overall priority vector and check the consistency again. The final priorities of alternative networks for the “Optimum Network” can be got through the above algorithm. Traffic density is the critical factor and reflects the feature of vehicles' mobility.Distributed Automatic Access Engine: The engine operators in an automatic process shown in Fig. <ref>. The QoS requirements (⟨data rate, delay, cost⟩) of various applications are registered with local observation of vehicle speed and the access recommender pushed through cellular network. The access option can be decided by analyzing the registered information, the received signal strength (RSS) of communication links and the statistical knowledge in the past. It is noted that the knowledge base is defined as𝒬⟨Speed, Application, Access option, QoS|Access Recommender⟩,which can be abbreviated as 𝒬⟨ S,A,O,Q|R ⟩. The knowledge base could be updated by the new achieved QoS periodically. The trustworthiness on access recommender can be adapted according to local observation and achieved QoS (access trials or QoS in a specific accessed network) for device's access decision-making (handover to another access network or not). The adapted process can be implemented by designing proper low-complexity algorithm in APP such as in Fig. <ref> through rule based inference <cit.> for decision-making. In this process, the aforementioned proximity traffic pattern, locations of infrastructure and RSS statistics are the preferences for consideration so that the rules could be logically given.In the following, fuzzy rules are powerful to represent the relation between the achieved QoS under accessed network and the criteria ⟨S,A,O,R⟩ for automatic access engine. In fuzzy theory, a rulebase is a function F that maps an input vector into outputs. Here, the premise variables are set as the four factors ⟨S,A,O,R⟩. The achievable QoS level is defined as the output. The membership function for each variable can be defined. It could be simplified into singleton fuzzified levels for each premise variable. For example, set Low and High for S, classify Voice, Text and Video for A, and let Cellular, WiFi and VANET for both of access recommender R and access option of the engine. An exemplary fuzzy rule with l levels of output could be as follows: Rule i:If S is Low, A is Voice, O is Cellular and R is Cellular, then the achievable QoS could be Level_l. Comparing the achievable QoS Level_l through fuzzy decision-making and the achieved QoS Level_c, we can decide whether or not to handover to the “optimum network". Only if achievable QoS Level_l is better in a certain degree than the achieved QoS Level_c, the handover happens.§ TRAFFIC SENSING AND TRAFFIC-DEPENDENT NETWORKING: A CASE STUDYIn this section, we describe a prototype of TrasoNET based data analytics for realtime traffic sensing and service provisioning. Based on the framework depicted in Fig. <ref>, the prototyping system consists in three basic components: probe vehicles (PVs), TMC, and a cloud server for traffic analysis and network access recommender. Fig. <ref> shows the system structure which is applied to estimate the traffic of Shanghai, China based on real GPS dataset of 32,122 taxies (as PVs) on Jan. 24, 2013. To offload the cellular data traffic, the city-wide WLANs have been developed in Shanghai and the number of AP is over 130,000 including i-Shanghai free WiFi in important social spots. Thus the cellular network and WLAN network form the access network layer. The TMC and cloud server are on the data aggregation layer. The application considered in this case is the on-demand network service provisioning for vehicles in a certain region.§.§ Data analytics for traffic sensing in CVTS The taxies in Shanghai generate sensing reports every 30 seconds and report the readings of GPS, i.e. location, time of report, current speed and headings, to TMC through cellular networks. The TMC collects all the traffic reports and constructs a huge traffic matrix X={x_ij}, in which each entry x_ij represents the traffic condition of the i-th road (e.g. average speed based on all the reports from PVs on the road) at the j-th duty cycle of a day. For example, the location of each report is matched to one road by map matching algorithm, data from different PVs are fused to get the traffic matrix X. Since the PVs cannot cover all the roads for all the time, TMC needs to estimate the traffic of un-sampled road in the traffic matrix. Matrix completion is applied in <cit.> with the low rank property of the traffic matrix. The matrix completion based estimation could be computed in the cloud. The estimation result is then sent to TMC for traffic management and message publishing to the vehicles in the city through the traffic bulletin board or information push through cellular network.The main idea of the real trace analytics is as follows. Firstly, estimate the values of average speed in the un-sampled roads <cit.> with the constraints of the temporal continuity and bound of the traffic data (the speed limit), respectively. Secondly, use the sampled data, together with the estimated data, to solve the optimization problem by minimizing the rank of the traffic matrix. The so-called HaTTEM algorithm is presented in <cit.>.However, the integrity analysis of sensing report about specific roads tells the fact that only 69% of over 35,000 roads in Shanghai have sensing reports for only 30% time of the day. There are not any GPS reports of taxies or buses in 17% roads within a whole day <cit.>. The coverage of taxies' traces in the city is quite uneven due to the aforementioned social proximity. So we need floating cars (FCs, e.g. police cars from Public Security Bureau and patrol cars from Traffic Management Department) to provide more data. These cars don't need to change the patrolling area, but just adjust the patrolling path for better traffic sensing. It is well-known that the disorder of samples can be expressed by entropy. The relation between the entropy and the estimation error is seen in Fig. <ref>. By planning the paths of only 260 controllable FCs for the whole city of Shanghai, China, even with the 15% of current PV samples, the average entropy is reduced from 0.233 nats (unity of entropy) to 0.05 nats. Thus, it is seen from Fig. <ref> that the estimation error could be reduced from 35% to 15% with the complimentary GPS data of FCs. §.§ Recommendation algorithm for Traffic-dependent networkingThe PVs and FCs connect not only to TMC for management, but also frequently to Internet for providing more emerging services, (e.g., the new free taxi calling services in Shanghai, China with mobile APPs called Diditaxi and Kuaidadi[URLs: www.xiaojukeji.com; www.kuaidadi.com]). Access to Internet would become a standard feature of future motor vehicles. However, simply using the cellular infrastructure for vehicle Internet access may result in an increasingly severe data overloading issue, which eventually would degrade the communication service performance of both traditional smartphone and vehicular mobile users. This advances of citywide free WLAN access in Shanghai make it possible to serve vehicular users in the near future. This article provides an access network recommendation mechanism for different network applications based on the estimated traffic which could be achieved by the method in Subsection <ref>.In order to demonstrates the feasibility of traffic-dependent networking, we provide in Fig. <ref> the intelligent network access system (INAS) for efficient and economical communication. INAS consists of network recommender by TMC and automatic access engine in mobile devices. It works in three phases, i.e. information gathering, network selection and access execution. The context repository module in Fig. <ref> is the knowledge base in Fig. <ref>.Model the urban traffic as scalable grids. In the simulation, consider 5 SPs and 20,000 vehicles in the area of 10KM×10KM with restricted mobility region for each vehicle. There are 20 vertical and horizontal streets, respectively. Assume that vehicles mobility region is partitioned into multiple tiers co-centered at their SPs. The distribution of mobility follows social proximity model and the vehicle dense obeys the power-law decaying from the center of SP to the border of the mobility region with the exponent γ=2. Without loss of any generality, consider two types of real-time applications, i.e. Voice Service and Video Service for individual vehicles. Assume the service requirements are 3 minutes of voice and 5 minutes of video on average. The data flow rate is 0.6Kbps and 5Mbps for voice and video service, respectively. The data rates are RMB1/Mb for cellular network and RMB10/2Gb per month, respectively. Based on the aforementioned AHP method, the network access recommendation can be given based on the comparison matrices in Table <ref>. It implies that Voice Service is sensitive to network access delay, while Video Service need more priority for bandwidth. Furthermore, we show the access network recommendation result in Fig. <ref> for different service types according to the traffic condition (vehicle density) demonstrated on the bottom X-Y layer of Fig. <ref>.For the traffic density, its second level pair-wise comparison matrix is formed as A_2×2. The simulation result about the values of density-tolerance for the two applications (voice and video services) shows that without the network selection algorithm, the successful transmission probability is nearly zero when traffic density is 0.04 for voice service, and 0.06 for video service. The result shows that without the algorithm, it's almost impossible to satisfy every cars' QoS need.It is noted that Fig. <ref> represents the average index values for the recommendation of cellular network and VANET, respectively. If there are only two accessible networks, the priority can be normalized. It is easily seen from Fig. <ref> that cellular network is recommended for voice service in a large region around SP where the vehicular traffic density is relatively high. Therefore, cellular network is still the first choice for voice service, especially at SPs. On the other hand, VANET is recommended to offload the cellular network for video service in the region close to SP (except SP due to QoS requirement). It indicates that the network selection/ handover is closely related to vehicular traffic condition, which is demonstrated the necessity of traffic-dependent networking.With the access network recommendation, the procedure of distributed automatic network access decision-making could be shown in Fig. <ref>.There are 4 premise variables ⟨S,A,O,R⟩, which represent Speed of vehicle, Application of network (i.e. voice or vedio), current Option of access network, and Recommendation of access network, respectively. The output is achievable QoS represented by 𝒬. The fuzzy sets and corresponding membership functions for each premise variable can be seen in Fig. <ref>. S is in the range of 0∼ 80km/h. The fuzzifier for A, O and R is singleton. Hence, we have the following 16 fuzzy rules: * Rule 1: If S is Low, A is Voice, O is Cellular and R is Cellular, then 𝒬 could be level_h;* Rule 2: If S is Low, A is Voice, O is Cellular and R is VANET, then 𝒬 could be level_h;* ⋯;* Rule 16: If S is High, A is Video, O is VANET and R is VANET, then 𝒬 could be level_l. With defuzzifier of fuzzy inference result, the distributed automatic network access engine determines the network selection and handover. In order to avoid ping-pong handover due to the mobility and perturbation of QoS, set two thresholds for QoS and delay, respectively. Calculate the QoS improvement by switching the current network to the other. Only if the improvement exceeds the QoS threshold for the time longer than the delay threshold, the handover happens.§ CONCLUSION AND FUTURE RESEARCH TOPICS This article describes an architecture called TransoNET for data analytics and networking in connected vehicles enabled transportation systems. To efficient manage network resources for CVTS applications, we describe the features of data analytics and subsequently introduce the traffic-dependent networking approach for data collecting. It shows how vehicular traffic can be estimated by matrix completion and how the recommendation-automatic integrated method provides efficient guidance to vehicles for network accessing. The data analysis based on real traces of taxies gives an exemplary study on traffic sensing. In particular, we study a case of multiple-network selection by the combination of network access recommendation from cloud and automatic access engine in vehicles. It has been demonstrated the necessity to explore the relationship between vehicular traffic and networking for providing real-time services in CVTS.Based on the proposed CVTS architecture, potential research directions can be envisioned to improve the data analytics and networking performance from both cloud and vehicle sides. On the cloud side, big data processing algorithm can be incorporated, e.g. crowdsourcing technologies, for ubiquitous traffic sensing such that more vehicles could take the roles of PV and FC. The social patterns of the vehicles may be considered to improve the traffic crowdsensing. On the vehicle side, automatic network access engine needs low-complexity decision-making algorithms for explosively increasing infotainment services through vehicles to Internet connection. As the terminals of crowdsensing, the vehicles could be more intelligent by automatically adapting the cycles of sensing and reporting according to local vehicular traffic. We believe CVTS will attract enormous attention from academia and industry in the near future.IEEEtran§ BIOGRAPHIESCailian Chen ([email protected]) is currently a Professor of Shanghai Jiao Tong University, China. Her research interests include vehicular ad hoc networks, wireless sensor and actuator network and computational intelligence. Dr. Chen was one of the First Prize Winners of University Natural Science Award from The Ministry of Education of China in 2007. She received the "IEEE Transactions on Fuzzy Systems Outstanding Paper Award" in 2008. She was honored "New Century Excellent Talents in University" by Ministry of Education of China, "Pujiang Scholar" and "Shanghai Rising-Star" by Science and Technology Commission of Shanghai Municipality, China.Tom Hao Luan ([email protected]) recieved the B.Eng. degree from Xi'an Jiao Tong University, China, in 2004, M.Phil. degree from Hong Kong University of Science and Technology in 2007, and PhD degree from University of Waterloo, Canada, in 2012. He is currently a Lecturer in the School of Information Technology at the Deakin University, Melbourne, Australia. From March 2013 to August 2013, he was a visiting research scientist in the Institute of Information Engineering, Chinese Academy of Sciences.Xinping Guan ([email protected]) is currently a Distinguished Professor of Shanghai Jiao Tong University, China. He is also the Professor of "Cheung Kong Scholar" Program, appointed by Ministry of Education of P. R. China, and the winner of "National Outstanding Youth Foundation", granted by NSF of China (NSFC). His current research interests include wireless sensor networks, cognitive radio and wireless technologies for smart grid and smart community. He received First Prize Winners of University Natural Science Award from The Ministry of Education of China in 2006, and the Second Prize of National Natural Science Award from The Ministry of Science and Technology of China in 2008. He received the "IEEE Transaction on Fuzzy Systems Outstanding Paper Award" in 2008.Ning Lu ([email protected]) received the B.Sc. and M.Sc. degrees from Tongji University, Shanghai, China, in 2007 and 2010, respectively, and PhD degree from University of Waterloo, Waterloo, Canada in 2015. He is currently an Assistant Professor in the Department of Computing Science at Thompson Rivers University, Canada. His research interests include capacity and delay analysis, media access control, and routing protocol design for vehicular networks.Yunshu Liu ([email protected]) is pursuing the M.Sc. degree at the Department of Automation, Shanghai Jiao Tong University, China. His research interests include vehicular network based traffic monitoring and application of compressive sensing. | http://arxiv.org/abs/1704.08125v1 | {
"authors": [
"Cailian Chen",
"Tom Hao Luan",
"Xinping Guan",
"Ning Lu",
"Yunshu Liu"
],
"categories": [
"cs.NI"
],
"primary_category": "cs.NI",
"published": "20170426135611",
"title": "Connected Vehicular Transportation: Data Analytics and Traffic-dependent Networking"
} |
[email protected] School of Physics and Astronomy, Monash University, Clayton, Victoria 3800, Australia School of Physics and Astronomy, Monash University, Clayton, Victoria 3800, Australia OzGrav: The ARC Centre of Excellence for Gravitational-wave Discovery, Hawthorn, Victoria 3122, Australia There are at least two formation scenarios consistent with the first gravitational-wave observations of binary black hole mergers. In field models, black hole binaries are formed from stellar binaries that may undergo common envelope evolution. In dynamic models, black hole binaries are formed through capture events in globular clusters. Both classes of models are subject to significant theoretical uncertainties. Nonetheless, the conventional wisdom holds that the distribution of spin orientations of dynamically merging black holes is nearly isotropic while field-model black holes prefer to spin in alignment with the orbital angular momentum. We present a framework in which observations of black hole mergers can be used to measure ensemble properties of black hole spin such as the typical black hole spin misalignment. We show how to obtain constraints on population hyperparameters using minimal assumptions so that the results are not strongly dependent on the uncertain physics of formation models. These data-driven constraints will facilitate tests of theoretical models and help determine the formation history of binary black holes using information encoded in their observed spins. We demonstrate that the ensemble properties of binary detections can be used to search for and characterize the properties of two distinct populations of black hole mergers. Determining the population properties of spinning black holes Eric Thrane December 30, 2023 =============================================================§ INTRODUCTIONAt present, merging black holes are the only directly detected source of gravitational waves <cit.>. A variety of mechanisms by which black hole binaries can form have been proposed. These mechanisms might yield significantly different distributions of the intrinsic parameters of binaries <cit.>. In this work we focus on the distribution of spin orientations to probe black hole binary formation mechanisms. We consider two mechanisms which are expected to dominate, the field and dynamical models (see, e.g., <cit.> for a detailed review).In dynamical models, the binary forms when two black holes become gravitationally bound in dense stellar environments such as globular clusters <cit.>. Due to mass segregation such clusters arrange themselves with more massive objects being found in the center and less massive objects on the outside. This means that binaries are expected to have mass ratios close to unity <cit.>. It is expected that the spins of the two companions will be isotropically oriented <cit.>.The distribution of spin orientations in field models is subject to more theoretical uncertainty (e.g., <cit.>). In field models, a stellar binary forms and the components of the binary then coevolve. Although such stars are expected to form with their angular momenta aligned with the total angular momentum of the binary, there are exceptions (e.g., <cit.>). If binaries are formed with misaligned spins, tidal interactions and mass transfer processes between the stars can align the angular momenta of the stars with the total angular momentum of the binary (e.g., <cit.>). When the first star explodes in a supernova and collapses to form a black hole, a natal kick may be imparted on the two companions due to asymmetry of the explosion (e.g., <cit.>), increasing misalignment between spin and angular momentum vectors. The subsequent evolution of the secondary, possibly involving a common envelope phase, can reverse this misalignment <cit.>. This is followed by the supernova of the secondary, which may give each black hole another kick and some additional degree of misalignment. The net effect is to leave the population of black hole spin orientations distributed about the angular momentum vector of the binary with some unknown typical misalignment angle <cit.>.Following the formation of the black hole binary (either through dynamical capture or common evolution) the spin orientation of nonaligned spinning black holes changes due toprecession. Isotropic spin orientation distributions are expected to remain isotropic throughout such evolution <cit.>. However, anisotropic distributions, such as those predicted by field models, may change significantly <cit.>. Here, we are interested in the distribution of spin orientations at the moment the binary enters LIGO's observing band. We therefore measure our spin orientations at f_ref=[20]Hz. Advanced LIGO's observing band will eventually extend down to [10]Hz, but we use [20]Hz here for the sake of convenience. One may use the spin orientation at f_ref to reverse engineer the spin alignment distribution at the moment of formation, but this is not our present goal.In this paper, we use Bayesian hierarchical modeling (e.g., <cit.>) and model selection to infer the parameters describing the distribution of spins of black hole binaries. We construct a mixture model, which treats the fraction of dynamical mergers, the fraction of isolated binary mergers, and the typical spin misalignment of the primary and secondary black holes as free parameters. We apply the model to simulated data (including noise) to show that we can both detect the presence of distinct populations, and also measure hyperparameters describing typical spin misalignment.Our method builds on a body of research using gravitational waves to study the ensemble properties of compact binaries. In <cit.>, it was shown that Bayesian model selection can be used to distinguish between formation channels using nonparametrized mass distributions. Clustering was used in <cit.> to show that model-independent statements about the existence of distinct mass subpopulations can be made with an ensemble of detections. In <cit.>, it was shown that the spin magnitude distribution can be used to determine whether observed merging black holes formed through hierarchical mergers of smaller black holes. Hierarchical merger models predict an isotropic distribution of black hole spin orientations since all binaries form through dynamical capture.Vitale et al. <cit.> showed that model selection can be used to distinguish between models which predict mutually exclusive spin orientations of merging compact binaries, both binary black holes and neutron star black hole binaries. In order to generate two distinct populations with different spin distributions, binaries were generated with random spin angles. Those with tilt angles (between the black hole spin and the Newtonian orbital angular momentum) <10^∘ were considered to be a fieldlike binary while those with tilt angles >10^∘ were considered to be dynamiclike. The authors showed that, after ∼100 detections, one can recover the proportion of binaries in each population to within ∼10% at 1σ.Stevenson et al. <cit.> used Bayesian hierarchical modeling to recover the proportion of binaries taken from a set of four populations distributed according to astrophysically motivated, spin orientation distributions with fixed spin magnitudes (a_i=0.7). Unlike <cit.>, the populations overlap so that even precise knowledge of a binary's spin parameters does not provide certain knowledge about its parent population. Of the four populations, three are different distributions predicted by population synthesis models of isolated binary evolution and the fourth is the isotropic distribution predicted for dynamic formation. They achieve a similar result to Vitale et al., measuring the relative proportion of different populations at the ∼10% level after 100 events. They also demonstrate that their two “extreme hypotheses" (perfect alignment and isotropy) can be ruled out at > 5σ after as few as five events if they are not good descriptions of nature. We build on these studies by employing a (hyper)parametrized model of the spin orientation distribution for the field model in order to measure not just the fraction of binaries from different populations, but also properties of the field model. In particular, we aim to measure the typical black hole misalignment for black hole binaries formed in the field. The advantage of this approach is that our modeling employs a broadly accepted idea from theoretical modeling (black holes in field binaries should be somewhat aligned) without assuming less certain details about the size of the misalignment. Since our model is agnostic with respect to the detailed physics of binary formation and subsequent evolution, the resulting methodology is robust against theoretical bias and provides a measurement of black hole spin misalignment for binaries formed in the field. The remainder of the paper is organized as follows. In the next section we review how the properties of merging binary black holes are recovered from observed data and briefly discuss the current observational results. We then introduce a useful parametrization to describe an admixture of field and dynamical black hole mergers. We follow this with a description hierarchical inference. We then present the results of a proof-of-principle study using simulated data. We introduce a new tool for visualizing spin orientations, spin maps. Finally, closing thoughts are provided.§ GRAVITATIONAL-WAVE PARAMETER ESTIMATION In order to determine the parameters describing the sources of gravitational waves Θ from gravitational-wave strain data h, we employ Bayesian inference. Merging binary black hole waveforms are described by 15 parameters: two masses {m_1,m_2}, two three-dimensional spin vectors {S_1,S_2},and seven additional parameters to specify the position and orientation of the source relative to Earth. It is possible that in both the field and dynamical formation models the presence of a third companion will induce eccentricity when the binary enters LIGO’s observing band through Lidov-Kozai cycles <cit.>. However, we consider only circular binaries. Most gravitational-wave parameter estimation results obtained to date have been obtained using the Bayesian parameter estimation code LALInference <cit.>. For our study we use the LALInference implementation of nested sampling <cit.>. We employ reduced order modeling and reduced order quadrature <cit.> tolimit the computational time of the analysis.Performing parameter estimation over this 15-dimensional space is computationally intensive. In order to maximize the efficiency sampling this high-dimensional space, the effect of the two spin vectors on the waveform is approximately represented using two spin parameters <cit.>,χ_eff =a_1cos(θ_1) + qa_2cos(θ_2)/1+q χ_p =max(a_1sin(θ_1), (4q+3/4+3q)qa_2sin(θ_2)).Here (a_1, a_2) are the dimensionless spin magnitudes, q=m_1 / m_2 <1 is the mass ratio and (θ_1, θ_2) are the angles between the spin angular momenta and the Newtonian orbital angular momentum of the binary.The variable χ_eff is “the effective spin parameter." When χ_eff >0, the binary merges at a higher frequency than for χ_eff=0 and hence spends more time in the observing band <cit.>. Similarly, binaries with χ_eff <0 spend less time in the observing band. The variable χ_p describes the precession of the binary, which is manifest as a long-period modulation of the signal <cit.>.Using numerical relativity to compute all of the waveforms necessary for parameter estimation is computationally prohibitive. Parameter estimation therefore relies on “approximants," which can be used for rapid waveform estimation. We use the IMRPhenomP approximant <cit.>, which has been used in many recent parameter estimation studies, including parameter estimation for recently observed binaries (e.g., <cit.>). IMRPhenomP approximates a generically precessing binary waveform using χ_eff and χ_p. Parameter estimation of the confirmed binary black hole detections, GW150914 <cit.>, GW151226 <cit.> and GW170104 <cit.>, yield (slightly) informative posterior distributions for χ_eff. However, the posterior distributions for χ_p show no significant deviation from the prior.The observed distribution of these two effective spin parameters will depend on the mass and spin magnitude distributions of black holes. The distributions are expected to differ for binaries formed through different mechanisms <cit.>. We do not consider these effects. Instead we work directly with the spin orientations of each black hole. For our purposes, it will be useful to define two additional variables:z_1 =cos(θ_1) z_2 =cos(θ_2) . Instead of working with χ_eff and χ_p, we work with distributions of z_1, z_2. We note that z_i≈1 corresponds to aligned spin while z_i≈ -1 corresponds to antialigned spin and z_i=0 corresponds to black holes spinning in the orbital plane. § MODELS For the purpose of this work we ignore the detailed formation history used in population synthesis studies. Instead, we introduce a simple parametrization designed to capture the salient features of the field and dynamic models. More sophisticated parametrizations are possible and will (eventually) be necessary to accurately describe realistic populations. However, we believe this is a suitable starting point given current theoretical uncertainty.We hypothesize that the distribution of {z_1, z_2} can be approximated as an admixture of two populations. The first population is described by a truncated Gaussian peaked at (z_1, z_2) = (1,1) with width (σ_1, σ_2). This is our proxy for the population formed in the field. The Gaussian shape mimics theform of distributions predicted by population synthesis models, which are clustered about z=1 with some unknown spread. The second population is uniform in (z_1, z_2), this represents the dynamically formed population. The relative abundances of each population are given by ξ (field) and 1 - ξ (dynamic). Thus, according to our parametrization, the true distribution of black hole mergers can be approximately described as follows:p_0(z_1, z_2) = 1/4 p_1(z_1, z_2) = 2/π1/σ_1e^-(z_1 - 1)^2/2σ_1^2/erf(√(2)/σ_1)1/σ_2e^-(z_2 - 1)^2/2σ_2^2/erf(√(2)/σ_2)p(z_1,z_2) = (1 - ξ) p_0 + ξ p_1Here, p_0(z_1, z_2) is the true dynamic-only distribution, p_1(z_1, z_2) is the true field-only distribution, and p(z_1, z_2) is the true distribution for all black hole binaries. These distributions depend on three hyperparameters: two widths (σ_1, σ_2) and one fraction ξ.For each of our population hyperparameters {σ_1, σ_2, ξ}, we choose uniform prior distributions between 0 and 1. For ξ this covers the full allowed range of values. For σ, this prior is chosen to be consistent with the most conservative estimates on spin misalignments predicted byfield models (isotropically distributed kicks with the same velocity distribution as neutron stars, isotropic full kicks in <cit.>). In Fig. <ref>, we plot p_1 for various values of σ.There are two interesting limiting cases. We note that p_1(z|σ)→δ(z-1) as σ→ 0. This corresponds to perfect alignment of black hole spins. We also note that p_1(z|σ) → p_0 as σ→∞. Thus, depending on the choice of prior, the dynamical model is degenerate with the field model evaluated at one point in hyperparameter space. A consequence of this limiting behavior is that it is far more difficult to distinguish samples drawn from a broad aligned distribution (σ=1), than an almost perfectly aligned distribution (σ=0.01). It is simple to extend this model to include more terms describing additional subpopulations or alter the form of the existing terms to better fit physically motivated distributions. § BAYESIAN HIERARCHICAL MODELING Bayesian hierarchical modeling involves splitting a Bayesian inference problem into multiple stages. In the case of merging compact binaries these steps are as follows:0em* Perform gravitational-wave parameter estimation as described above. We adopt priors that are uniform in spin magnitude and isotropic in spin orientations. * Assume the population from which events are drawn is described by hyperparameters Λ. Calculate a likelihood function for the data given Λ by marginalizing over the parameters for individual events Θ.* Combine multiple events to derive a joint likelihood for Λ.* Use the joint likelihood to derive posterior distributions for Λ, which, in turn, may be used to construct Bayes factors or odds ratios comparing different population models and confidence intervals on hyperparameters.Step (i) produces a set of n_k posterior samples {Θ_i}, sampled according to the likelihood of the binary having each set of parameters, p(Θ|h). This step is computationally expensive and requires the application of a specialized tool such as LALInference.In Step (ii), we estimate Λ using the posterior samples {z_i}. Our likelihood requires marginalization over z, for each event. Since LALInference approximates the posterior for Θ with a list of posterior sample points, the marginalization integral over (z_1, z_2) can be approximated by summing the probability of each sample in the LALInference posterior chain for our population model (see, e.g., <cit.> for details). Step (iii): To combine data from N events, we multiply the likelihoods:ℒ_k(h_k|Λ) = ∫ dz_1 dz_2 p ( z_1, z_2|h_k ) p ( z_1, z_2 | Λ)= 1/n_k∑_α = 1^n_k p ( z_α1, z_α2 | Λ)ℒ({h_k}|Λ) = ∏_k=1^N ℒ_k(h_k|Λ).Here, ℒ_k(h_k|Λ) is the likelihood function for the kth event with strain data h_k. The joint likelihood function ℒ({h_k}|Λ) combines data from all N measurements to arrive at the best possible constraints on Λ.Step (iv): At last, we arrive at the posterior distribution for Λ, p(Λ|{h_k}). Combining the joint likelihood ℒ({h_k}|Λ) with a prior distribution for the hyperparameters Λ, π(Λ|H), for a particular population model, H, we obtainp(Λ|{h_k}) = ℒ({h_k}|Λ)π(Λ|H)/Z({h_k}|H) = π(Λ|H)/Z({h_k}|H)∏_k=1^N 1/n_k∑_α = 1^n_k p ( z_α1, z_α2|Λ) ∝∏_k=1^N ∑_α = 1^n_k p(z_α1, z_α1 |Λ).Here, Z({h_k}|H) is the Bayesian evidence for the data from N observations {h_k}, for a model H, which is given by marginalizing over the hyperprior spaceZ({h_k}|H) = ∫ dΛ ℒ({h_k}|Λ,H)π(Λ|H) .From our (hyper)posterior distribution p(Λ|{h_k}), we construct confidence intervals for our hyperparameters.The odds ratio of two models is:𝒪^i_j = Z({h_k}|H_i)p(H_i)/Z({h_k}|H_j)p(H_j) .We use the odds ratio to select between different models. Here, the p(H_i) are the prior probabilities assigned to each model. In our study, we assign equal probabilities to each model. Thus, the odds ratio is equivalent to the Bayes factor:B^i_j=Z({h_k}|H_i)/Z({h_k}|H_j).We impose a somewhat arbitrary, but commonly used threshold of |ln(B)| > 8 (∼ 3.6 σ) to define the point at which one model is significantly preferred over another.Now that we have derived a number of statistical tools, it is worthwhile to pause and consider what astrophysical questions we can answer with them.* If p(σ_1, σ_2| {h_k}) excludes σ_1=σ_2=∞, then it necessarily follows that p(ξ|{h_k}) excludes ξ=0, and we may infer that at least some binaries merge through fieldlike models. * If p(ξ|{h_k}) excludes ξ=1, we may infer that not all binaries can be formed via fieldlike models. * If both ξ=0 and ξ=1 are excluded, then we may infer the existence of at least two distinct populations. * If the (σ_1,σ_2) posterior distributionp(σ_1, σ_2| {h_k}) excludes σ_1=σ_2=0, we may infer that not all binaries are perfectly aligned.In this way we can distinguish between different formation channels or specific models, i.e., perfect alignment in case (iv).We employ Bayes factors to compare our population models. We calculate evidences for three hypotheses:0em* Z_dyn – Dynamic formation only, ξ=0.* Z_field – Field formation only, ξ=1.* Z_mix – Mixture of field and dynamic, ξ∈[0,1].We then define three Bayes' factors to compare these three hypotheses:0em* B^mix_field = Z_mix/Z_field.* B^mix_dyn = Z_mix/Z_dyn.* B^field_dyn = Z_field/Z_dyn.In the next section, we apply these tools to a variety of simulated data sets in order to show under what circumstances we can measure various hyperparameters and carry out model selection.§ SIMULATED POPULATION STUDYWe use a simulated population to test our models. For the sake of simplicity, we construct a somewhat contrived population in which every binary shares some parameters corresponding to the best-fit parameters of GW150914: * (m_1, m_2)=(35 M_⊙,30 M_⊙).* d_L=[410]Mpc.* (a_1, a_2)=(0.6, 0.6).Here, d_L is luminosity distance and (a_1, a_2) are the black hole spin magnitudes. The remaining extrinsic parameters (sky position and source orientation) are sampled from isotropic distributions. We emphasize that the distance and mass and spin magnitude distributions are not representative of the full population of black hole binaries, which is poorly constrained. These distributions represent a subset of GW150914-like events, chosen for illustrative purposes. In reality, for every GW150914-like event, there are likely to be a large number of more distant (and possibly lower mass) events, which contribute relatively less information about spin.We inject 160 binary merger signals into simulated Gaussian noise corresponding to Advanced LIGO at design sensitivity <cit.>. Of these, we generate 80 distributed according to p_0 and 80 distributed according to p_1; see Eq. (<ref>). The injected values of (z_1, z_2) are shown in Fig. <ref>. The red diamonds correspond to the p_0 dynamical model and the blue circles to the p_1 fieldlike model. From these we construct “universes" summarized in Table <ref>. Each universe contains a different mixture of field and dynamical binaries. In every universe, (σ_1,σ_2)=(0.3, 0.5).For each universe, we present the results of the methods described above. In Fig. <ref>, we plot the 1σ (dark), 2σ (lighter), and 3σ (lightest) confidence regions as a function of the number of GW150914-like events. In Fig. <ref>, we plot the three Bayes factors defined in Eq. (<ref>) as a function of the number of GW150914-like events. Each row in Fig. <ref> and panel in Fig. <ref> represents a different universe.First we consider universe A, consisting of only dynamically formed binaries, ξ=0; see the top row of Fig. <ref>.Since all binaries form dynamically in this universe, σ is undefined. We see that after O(1) event we rule out ξ=1 at 3σ (the hypothesis that all binaries form in the field).Next we consider universe E in which all events are drawn from the aligned model, ξ=1; see the bottom row of Fig. <ref> and the bottom panel of Fig. <ref>. For this universe, σ_1=0.3, σ_2=0.5. We rule out ξ=0 (dynamical only) at 3σ after O(1) event. The Bayes factors also rule out all binaries forming dynamically after ≲10 events. The threshold |ln(B)|=8 is shown by the dashed line. After 80 events, the 1σ confidence intervals for σ_1 and σ_2 have shrunk to ∼30% and the 1σ confidence interval for ξ has shrunk to 3%. The Bayes factor comparing the two-population hypothesis to the purely field hypothesis B^mix_field (the blue line in the bottom panel of Fig. <ref>) does not strongly favor field-only formation.Universes B, C and D are mixtures of the field and dynamical populations. Of these, B and D have only 10% drawn from the subdominant population. We recover marginally weaker constraints than the corresponding single population universes. The hypothesis that all binaries form through the dominant mechanism is disfavored at 1σ after a few tens of events for universes B and D, establishing a weak preference for the presence of two distinct populations. For some realizations we can rule out both one component models after 80 events, however generally we see a subthreshold preference for the mixture model. This is unsurprising since each one-population model is a subset of our two-population model. For universe C, an equal mixture of events drawn from the field and dynamical populations. Both ξ=0 and ξ=1 are excluded at 3σ after tens of events establishing the presence of two distinct subpopulations.For all five universes, the presence of a perfectly aligned component (σ=0) is excluded after fewer than 20 events. For many realizations this number is <5. For universes B, C and D (consisting of a mixture of field and dynamical mergers), we can rule out the entire population forming from one of the two channels after 10–40 GW150914-like events. When there is a large contribution from the aligned model, we observe that the allowed region for σ_1 becomes small faster than the allowed region for σ_2. There are two effects, which explain this. First, the secondary black hole's spin has a less significant effect on the waveform <cit.>. The spin orientation of the secondary is therefore less well constrained for each event. This translates to a larger uncertainty for σ_2 compared to σ_1. Second, the width of the distribution of spin tilts is broader for the secondary black holes. This broader distribution is intrinsically more difficult to resolve. § SPIN MAPSIn addition to our hierarchical analysis, we present a visualization tool for the distribution of spin orientations. We introduce “spin maps": histograms of posterior spin orientation probability density, averaged over many events, and plotted using a Mollweide projection of the sphere defining the spin orientation, see Fig. <ref>. The maps use HEALPix <cit.>. For each posterior sample the latitude is the spin tilt of the primary black hole, θ_1, and the longitude the difference in azimuthal angles of the two black holes, ΔΦ. The difference in azimuthal angles may give information about the history of the binary, specifically by identifying spin-orbit resonances at ΔΦ=0,π <cit.>. These resonances, if detected, would appear as bands of constant longitude. We do not utilize azimuthal angle in this work and our injected distributions are isotropic in ΔΦ. In the future, it would also be interesting to produce ensemble spin disk plots (e.g., Fig. 5 of <cit.>), showing the spin magnitude and orientation for a population of binaries.The spin maps in Fig. <ref> include contributions from 80 events for universes A and C (see Table <ref>). This simple representation is useful because it provides qualitative insight into the distribution of spins and helps us to see trends and patterns that might not be obvious from our likelihood formalism. The north pole on these maps corresponds to spin aligned with the total angular momentum of the binary. We see the preference for the spin to be aligned with the angular momentum vector of the binary by the clustering in the northern hemisphere.§ DISCUSSIONThe physics underlying the formation of black hole binaries is poorly constrained both theoretically and observationally. We do not know which of the proposed mechanisms is the main source of binary mergers: preferentially aligned mergers formed in the field versus randomly aligned mergers formed dynamically. We are also not confident in the predicted characteristics of binaries formed through either channel. We therefore create a simple (hyper)parametrization, describing the ensemble properties of black hole binaries. We demonstrate that we can measure hyperparameters describing the spin properties of an ensemble of black hole mergers with multiple populations. Previous work by Vitale et al. <cit.> and Stevenson et al. <cit.> demonstrated that the fraction of binaries drawn from different populations can be inferred after O(10) events. We show that after a similar number of events, the shape of the spin-orientation distribution can be inferred using a simple hyperparametrization. We reproduce the finding from Stevenson et al., that O(1) event is required to distinguish an isotropically oriented distribution, ξ=0, from a perfectly aligned distribution, ξ=1, σ_1=σ_2=0. After fewer than 40 GW150914-like events we can determine the properties of the dominant formation mechanism for all of our considered scenarios. We also introduce the concept of spin maps, which provide a tool for visualizing the distribution of spin orientations from an ensemble of detections.One limitation of our study is that, for the sake of simplicity, we employ a population of binaries with masses, distance, and spins fixed to values consistent with GW150914. The advantage of this simple model is that we are able to isolate the effect of spin orientation by holding other parameters fixed. The disadvantage is that the GW150914-like population is not a realistic description of nature. By changing from a population of binaries at a fixed distance to a population distributed uniformly in comoving volume, more events will be required for measurement of population hyperparameters. This is because most events, coming from the edge of the visible volume, will contribute only marginally to our knowledge of these hyperparameters. We assume fixed spin magnitudes of a_1=a_2=0.6. For a binary with aligned spins, this would imply χ_eff=0.6. Based on recent LIGO detections, this might be optimistic. For GW151226, χ_eff=0.21^+0.20_ - 0.10. For all other observed events, χ_eff is consistent with 0. This implies either that the observed black holes are not spinning rapidly or that the merging black holes observed so far possess significantly misaligned spins <cit.>. If we have overestimated the typical black hole spin magnitude a, the number of events required to determine the distribution of spin orientation will increase. Implementing a theoretically motivated distribution of these parameters is left to future studies. Another area of future work is extending the method to other physically motivated spin orientation distributions. We thank Yuri Levin,Simon Stevenson and Richard O'Shaughnessy for helpful comments. This is LIGO Document No. DCC P1700077. E. T. is supported through ARC FT150100281 and CE170100004. | http://arxiv.org/abs/1704.08370v2 | {
"authors": [
"Colm Talbot",
"Eric Thrane"
],
"categories": [
"astro-ph.HE"
],
"primary_category": "astro-ph.HE",
"published": "20170426224317",
"title": "Determining the population properties of spinning black holes"
} |
School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA [][email protected] School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GAA long standing postulate in crystal plasticity of metals is that yielding commences once the resolved shear stress on a slip plane reaches a critical value. This assumption, known as Schmid law, implies that the onset of plasticity is independent of the normal stress acting on the slip plane. We examine the validity of this assumption in single crystal perfect lattices at zero temperature by subjecting them to a wide range of combined normal and shear stresses and identifying the onset of plasticity. We employ phonon stability analysis on four distinct single crystal metals and identify the onset of plasticity with the onset of an instability. Our results show significant dependence of yielding on the normal stress, thereby illustrating the necessity of considering non-Schmid effects in crystal plasticity. Finally, contrary to the common assumption that instabilities in single crystals are of long wavelength type, we show that short wavelength instabilities are abundant in the nucleation of defects for a wide range of loading conditions. Non-Schmid Effects In Perfect Single Crystal Metals At Zero Temperature Julian J. Rimoli December 30, 2023 =======================================================================Schmid law in crystal plasticity states that glide on a given slip system commences when its resolved shear stress reaches a critical value <cit.>. By assuming that this value is constant, it neglects any effect that the normal component of the traction acting on the slip plane could have on yield initiation. Inspired by the physics of friction, the present letter aims to scrutinize this assumption in single crystal metals at zero temperature and demonstrate how, at certain stress levels, the critical resolved shear stress depends on the normal traction component.Loading conditions leading to the onset of plasticity have been extensively investigated over the last century.In 1900, Guest <cit.> showed through various experiments that yielding starts when the maximum shear stress reaches a critical value. This idea was utilized by M.T. Huber (unpublished) and von Mises <cit.> to propose yield criteria. The following decade, crystal plasticity and the effect of material texture became a major research focus. G. I. Taylor, in his 1934 seminal paper, explained the basics of deformation mechanisms in crystal plasticity <cit.>. In the same year, Boas and Schmid proposed their celebrated Schmid law in crystal plasticity. Later, Bridgman conducted several experiments on a sample under various hydrostatic pressures and concluded that the yield stress is independent of the hydrostatic pressure <cit.>.However, in 1983, Christian <cit.> observed non-Schmid effects experimentally in iron and other body-centered cubic (BCC) metals. Similar behaviors were observed in certain alloys, like Ni_3Al <cit.>, and crystal plasticity models based on non-Schmid effects have been proposed <cit.>. These models attribute non-Schmid effects to the non-closed packedness present in BCC single crystals. Molecular dynamics simulations have also been employed to study non-Schmid effects in various materials <cit.>. The aforementioned models and numerical studies were either phenomenological, like most models in crystal plasticity or based on molecular dynamics simulations at finite temperature.The abundance of defects (e.g., dislocations) in bulk crystalline materials has resulted in research focusing primarily on defect evolution or nucleation from existing ones (e.g., Frank-Read source) as opposed to defect nucleation in a perfect lattice.However, with recent advances in nanoscale devices, investigating defect nucleation in perfect lattices is gaining attention <cit.>. We investigate the validity of Schmid law by looking at the onset of plasticity, i.e. defect nucleation, in perfect single crystal metals. It is worth mentioning that, since stress is a continuum notion and this paper focuses on discrete lattices, in the remainder of the paper the various loading configurations are attained by enforcing displacement boundary conditions and computing the equivalent stresses from an energetic perspective. Four common metals are chosen for this study: Fe (BCC), Cu (FCC), Ag (FCC) and Ni (FCC). We deliberately chose three common FCC metals in order to test the hypothesis of non-closed packedness leading to non-Schmid effects <cit.>. Defect nucleation is identified by lattice instability analysis using phonon calculations. We model inter-atomic interactions through Mishin potentials, which belong to the widely adopted class of Embedded-Atom Method (EAM) potentials <cit.>.Each lattice system is constructed using its standard primitive vectors written in the standard basis <cit.>. We consider a sufficiently large lattice such that finite size effects are not encountered and subject the atoms at the boundary of the lattice to an affine deformation with deformation gradient F. The deformed configuration of the boundary is obtained by x= F, where x andare the position vectors of the atoms in the deformed and the reference configurations, respectively.In our study, the applied deformation gradient consists of a hydrostatic component β and a simple shear part γ, as followsF=[ 1+β γ 0; 0 1+β 0; 0 0 1+β ].This particular choice of F is motivated by our objective to investigate the effect of hydrostatic deformation β on the shear value at the instability point, which we indicate as γ_d. Also, for the sake of consistency in both the lattice systems, F is written in the standard basis. An alternate approach to examine the Schmid law would consist in applying F in a coordinate system in which two of the basis vectors lie within one of the slip planes (for FCC). Note that this would result in the same hydrostatic strain as in our approach, as the hydrostatic component of the deformation gradient is invariant under a coordinate transformation. We believe this indirect way of subjecting the slip plane to shear loading allows us to compare the shear-normal coupling in FCC and BCC metals under identical deformation gradient.The deformation gradient F is applied in two stages: We first subject the lattice to a hydrostatic deformation (β) and then impose shear deformation (γ). At a fixed β, shear deformation is increased from 0 in small steps until the onset of instability, at γ=γ_d.This procedure is performed for 80 different values of β, spanning from -0.04 to 0.04. Note that under this procedure the lattice undergoes an affine deformation, and that an affine displacement for atoms is always a valid equilibrium solution for the lattice, owing to translational symmetry. As the deformation increases, this affine deformation solution becomes unstable and a defect nucleates. Indeed, since the affine deformation is stable to infinitesimal perturbations before the instability point, no defect nucleates under quasistatic loading. The nucleation of defects in a real single crystal beyond this critical deformation point will depend on finite size effects and on the nature of perturbations in the lattice. Our procedure thus seeks to identify a lower bound for the yield strength of the material under the considered boundary conditions. We seek to identify the instability point by analyzing the stability under infinitesimal perturbations in the Fourier space, corresponding to a phonon stability analysis <cit.>. Since instabilities are associated with the loss of positive definiteness of the Hessian of the energy functional, we use a second order approximation about the deformed configuration and write the equilibrium of an arbitrary atom `r' located in the interior, far from the boundary as: ∑_s=1^N K_rsδ u_s= δ_r,where N is the number of atoms in the lattice.and δ are, respectively, the second derivative of the potential energy with respect to atomic coordinates and the change in displacement with respect to the deformed configuration due to force perturbation, δ. Without loss of generality, the origin is placed at the atom `r' and the discrete Fourier transform of δ is written as: δ_s=∑_h=1^N e^-i_h ·_sδ_h,whereand δ are wave vector in the reciprocal basis and the Fourier transform of the displacement perturbation, respectively. Substituting Eq. <ref> into Eq. <ref> and employing periodicity and orthogonality of the Fourier basis leads to:∑_s=1^N_rs e^-𝑖._sδ_r = δ_r,where similarly δ is the force perturbation in the Fourier space. Eq. <ref> is written for a fixed wave vector =. Lattice instability is identified by the loss of positive definiteness of the stiffness matrix ∑_s=1^N_rs e^-𝑖._s. The analysis is performed by looking at the entire first Birillioun Zone (BZ) in the deformed configuration.The key message of this work is summarized in Fig. <ref>, illustrating the dependence of the shear at the onset of instability, γ_d, on the hydrostatic deformation β. Since Schmid law focuses on the stress values, it is worth emphasizing that this dependency in the deformation space is an indirect way of investigating dependency in the stress space. Indeed, an independencyof critical shear stress τ_c on hydrostatic pressure P implies an independency of the critical shear strain γ_d on thehydrostatic strain β.Note that the form of dependency τ_c(P) can be different, in general,from γ_d(β) due to geometric and material nonlinear effects. Schmid law implies a horizontal line, i.e. the critical shear strain γ_d is independent of the hydrostatic deformation β. As mentioned previously, there is experimental evidence of non-Schmid effects in BCC metals, while FCC metals are considered to follow the Schmid plasticity <cit.>. Indeed, a strong coupling exists in iron (BCC). However, Fig. <ref> illustrates that hydrostatic deformation significantly affects shear instabilities in FCC metals too. We observe that the trend and extentof shear normal coupling are considerably different in various materials. While silver does not exhibit Schmid type behavior, copper and nickel follow closely the Schmid assumption in a vast regime of deformations. To investigate the nature of instabilities, after the wave vector associated with the instability point (i.e. _d) is computed, we look at all the wave vectors along the line connecting Γ to the boundary of the first BZ, passing through _d. If the phonon softens along the entire line (eigenvalues become zero), it corresponds to a long wavelength instability. On the other hand, nonzero eigenvalues close to κ=0 and along that path imply short wavelength instability. We performed this calculation for all the loading conditions. Note that the square root of eigenvalues of the stiffness matrix, denoted by λ, is proportional to the frequency of the acoustic modes. Fig. <ref> shows λ values for the considered materials. For each material, four different β values are chosen and their associated λ values are plotted along the aforementioned path.We observe that for β<0, i.e. hydrostatic compression, instabilities of short wavelength happen in the Ag, Ni and Fe. In Ni, Cu and Ag, a transition from short to long wavelength instabilities occur. The transition point depends significantly on the material: it happens around β=-0.005 for Cu, while it only happens, at β>0.04 for Ag. To illustrate these transitions, we choose distinct β values for various metals in Fig. <ref>. We also observe that instabilities in iron are always of a short wavelength nature (in the chosen deformation regime). It is widely assumed that instabilities in single crystals are of long wavelength nature and if a lattice is stable at long wavelengths, it will be stable at short wavelengths <cit.>. While counterexamples have been shown previously <cit.>, our results demonstrate an abundance of such short wavelength instabilities. Since short wavelength instabilities do not have a simple homogenized continuum analogue,they might not be captured by a first order continuum model. Based on our results, the common practice of using an elastic stability analysis to study nanoindentation and uniaxial tension at the nanoscale may have led to erroneous results <cit.>. We observe that the type of instability is dependent on both the material and the loading conditions. To now relate the strains to the material yield stress, we consider the behavior of an equivalent continuum hyperelastic materialuntil the onset of instability. We also perform an elastic stability analysis to quantify the difference between the results obtained by the two approaches and to further illustrate the pitfalls of using an elastic stability analysis.For this purpose, a homogenized continuum model of the lattice is considered. Let W be the homogenized energy of the continuum under two assumptions: I) The Cauchy-Born hypothesis is satisfied, implying that the underlying lattice will deform under the same deformation gradient as the continuum, II) The strain energy density W( F) in the continuum model is equal to the energy of a single unit cell normalized by its volume, obtained from the interatomic potentials <cit.>. Within this framework, instability occurs following the violation of the strong ellipticity condition <cit.>:δ^T: ∂^2 W( F)/∂ F ∂ F : δ > 0,where δ is a perturbation of the displacement field about the deformed configuration. The continuum body is subjected to exactly the same affine deformation as the lattice. We seek instabilities which result from perturbations having a plane wave basis, i.e. δu = δu_0 e^i· x <cit.>. Fig. <ref> compares the elastic and phonon instability results. We observe that the difference in the results are significant, e.g. 50% for iron at β=-0.03. Evidently, instabilities that happen at short or finite wavelengths are not captured by an elastic stability analysis. Indeed, since elastic instabilities under affine deformations are a subset of phonon instabilities, the γ_d obtained by phonon calculations either coincide with elastic stability results or predict a smaller value.At the onset of instability, the Cauchy stress tensor is computed by <cit.>:= (F)^-1( ∂ W∂F) F^T.Fig. <ref> illustrates the shear stress component, τ=σ_12 for different hydrostatic pressure values (P), where P=Tr()/3. Evidently, by increasing the hydrostatic pressure, the shear stress at the onset of instability increases. These observations motivate the necessity of a pressure-dependent plasticity model for perfect single crystal metals. While hydrostatic deformation is chosen to understand non-Schmid effects, applying this type of deformation in the nano-scale experiments might be very challenging. Fig. <ref> is demonstrating phonon stability results for uniaxial deformation F', i.e. a combined uniaxial deformation and simple shear, which is anticipated to be useful for nano-scale experiments.F'=[ 1+β γ 0; 0 1 0; 0 0 1 ]. In summary, the pitfalls of Schmid law are investigated and a significant dependence of the critical shear strain γ_d on thehydrostatic strain β is shown. While certain metals like copper follow the Schmid law reasonably well, others such as iron and nickel demonstrate a strong non-Schmid behavior. It is verified that depending on the crystal and the loading conditions, short wavelength instabilities could be dominant. The method pursued in this research relies on rigorous mathematical formulations. The only possible source of error is the interatomic potentials, which have been extensivelydeveloped and used by the scientific community and are believed to be accurate enough for this study. 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Curtarolo, @noopjournal journal Computational Materials Science volume 49, pages 299 (year 2010)NoStop [Elliott et al.(2006)Elliott, Triantafyllidis, and Shaw]elliott2006stability author author R. S. Elliott, author N. Triantafyllidis,and author J. A. Shaw, @noopjournal journal Journal of the Mechanics and Physics of Solids volume 54, pages 161 (year 2006)NoStop [Tadmor et al.(1999)Tadmor, Smith, Bernstein, and Kaxiras]tadmor1999mixed author author E. Tadmor, author G. Smith, author N. Bernstein,and author E. Kaxiras, @noop journal journal Physical Review B volume 59, pages 235 (year 1999)NoStop [Born(1940)]born1940stability author author M. Born, in @noopbooktitle Mathematical Proceedings of the Cambridge Philosophical Society, Vol. volume 36 (organization Cambridge Univ Press, year 1940) pp. pages 160–172NoStop [Grimvall et al.(2012)Grimvall, Magyari-Köpe, Ozoliņš, and Persson]grimvall2012lattice author author G. Grimvall, author B. Magyari-Köpe, author V. Ozoliņš,and author K. A. Persson, @noopjournal journal Reviews of Modern Physics volume 84, pages 945 (year 2012)NoStop [Wallace and Patrick(1965)]wallace1965stability author author D. C. Wallace and author J. L. Patrick, @noopjournal journal Physical Review volume 137, pages A152 (year 1965)NoStop [Liu et al.(2016)Liu, Gu, Shen, and Li]liu2016crystal author author X. Liu, author J. Gu, author Y. Shen,and author J. Li, @noopjournal journal NPG Asia Materials volume 8, pages e320 (year 2016)NoStop [Liu et al.(2010)Liu, Gu, Shen, Li, and Chen]liu2010lattice author author X. Liu, author J. Gu, author Y. Shen, author J. Li,and author C. Chen, @noopjournal journal Acta Materialia volume 58, pages 510 (year 2010)NoStop [Pal et al.(2016)Pal, Ruzzene, and Rimoli]pal2016continuum author author R. K. Pal, author M. Ruzzene,and author J. J. Rimoli, @noopjournal journal International Journal of Solids and Structures volume 96, pages 300 (year 2016)NoStop [Geymonat et al.(1993)Geymonat, Müller, and Triantafyllidis]geymonat1993homogenization author author G. Geymonat, author S. Müller,and author N. Triantafyllidis, @noopjournal journal Archive for rational mechanics and analysis volume 122, pages 231 (year 1993)NoStop [Michel et al.(2007)Michel, Lopez-Pamies, Castañeda, and Triantafyllidis]michel2007microscopic author author J.-C. Michel, author O. Lopez-Pamies, author P. P. Castañeda,and author N. Triantafyllidis, @noopjournal journal Journal of the Mechanics and Physics of Solids volume 55, pages 900 (year 2007)NoStop [Gurtin(1982)]gurtin1982introduction author author M. E. Gurtin, @nooptitle An introduction to continuum mechanics, Vol. volume 158 (publisher Academic press, year 1982)NoStop | http://arxiv.org/abs/1704.08376v1 | {
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Tweeting AI: Perceptions of AI-Tweeters (AIT) vs Expert AI-Tweeters (EAIT)Lydia Manikonda, Cameron Dudley, Subbarao Kambhampati Arizona State University, Tempe, AZ {lmanikon, cjdudley, rao}@asu.edu====================================================================================================================================With the recent advancements in Artificial Intelligence (AI), various organizations and individuals started debating about the progress of AI as a blessing or a curse for the future of the society. This paper conducts an investigation on how the public perceives the progress of AI by utilizing the data shared on Twitter. Specifically, this paper performs a comparative analysis on the understanding of users from two categories – general AI-Tweeters (AIT) and the expert AI-Tweeters (EAIT) who share posts about AI on Twitter. Our analysis revealed that users from both the categories express distinct emotions and interests towards AI. Users from both the categories regard AI as positive and are optimistic about the progress of AI but the experts are more negative than the general AI-Tweeters. Characterization of users manifested that `London' is the popular location of users from where they tweet about AI. Tweets posted by AIT are highly retweeted than posts made by EAIT that reveals greater diffusion of information from AIT.§ INTRODUCTIONDue to the rapid progress of technology especially in the field of AI, there are various discussions and concerns about the threats and benefits of AI. Some of these discussions include – improving the everyday lives of individuals (https://goo.gl/ViLdgV), ethical issues associated with the intelligent systems (https://goo.gl/5KlmXk), etc. Social media platforms are ideal repositories of opinions and discussion threads. Twitter is one of the popular platformswhere individuals post their statuses, opinions and perceptions about the ongoing issues in the society <cit.>. This paper focuses on investigating the Twitter posts on AI to understand the perceptions of individuals. Specifically, we compare and contrast the perceptions of users from two categories – general AI-Tweeters (AIT) and expert AI-Tweeters (EAIT). We believe that the findings from this analysis can help research funding agencies, organizations, industries who are curious about the public perceptions of AI. Efforts to understand public perceptions of AI are not new. Recent work by Fast et. al <cit.> conducts a longitudinal study about articles published on AI from New York Times between January 1986 and May 2016 are transformed over the years. This study revealed that from 2009 the discussion on AI has sharply increased and is more optimistic than pessimistic. It also disclosed that fears about losing control over AI systems have been increasing in the recent years. Another recent survey <cit.> conducted by the Harvard Business Review on individuals who do not have any background in technology, stated the positive perceptions towards AI. In contrast, even though online social media platforms are the main channels of communication <cit.>, there is no existing work on how and what users share about AI on these platforms. This paper attempts to learn the perceptions of individuals manifested by their posts shared on Twitter. Towards this goal, we attempt to answer 5 important questions through a thorough quantitative and comparative investigation of the posts shared on Twitter. 1) What are the insights that could be learned by characterizing the individuals and their interests who are making AI-related posts? 2) What is the Twitter engagement rate for the AI-tweets? 3) Are the posts about AI optimistic or pessimistic? 4) What are the most interesting topics of discussion to the users? 5) What can we learn about the frequently co-occurring terms with AI vocabulary? We address each of these questions in the next few sections.Our analysis reveals intriguing differences between the posts shared by AIT and EAIT on Twitter. Specifically, this analysis reveals four interesting findings about the perceptions of individuals who are using Twitter to share their opinions about AI. Firstly, users from both the categories are emotionally positive (or optimistic) towards the progress of AI. Secondly, even though users are positive overall, users from EAIT are more negative than those from AIT. Thirdly, we found that the tweets shared by EAIT have lower diffusion rate than the tweets posted by AIT as measured by the magnitude of retweets. Fourth, and last, users from AIT are geographically distributed (mostly Europe and US) with London and New York City taking the top positions. In the next section, we describe the process of collecting data from the two categories of users that we use for the study presented in this paper. We compare and contrast each of the 5 research questions we posed for Twitter users from the two categories. Results of a similar analysis conducted on the AI posts shared on Reddit, is attached as an appendix.§ DATA COLLECTION §.§ AI-Tweeters (AIT)We employ the official API of Twitter [https://dev.twitter.com/overview/api] along with a frequency-based hashtag selection approach to crawl the data from common Twitter users who tweet about AI. We first identify an appropriate set of hashtags that focus on the artificial intelligence in social media. In our case, these are – #ai and #artificialintelligence. With these seed hashtags, we crawled 2 million unique tweets. Using the hashtags presented in these tweets, we iteratively obtained the co-occurring hashtags and sort them based on their frequency. We remove non-technical hashtags included in this sorted hashtag list for example: #trump, #politics, etc. The top-15 co-occurring hashtags after the pre-processing are shown in Table <ref>.We used the top-4 hashtags from this list: #ai, #artificialintelligence, #machinelearning and #bigdata as the final hashtag set to crawl a set of 2.3 million tweets. In our dataset we found that the set of tweets obtained using these 4 hashtags are the super set of all the tweets crawled by utilizing the remaining hashtags presented in Table <ref>. Each tweet in this dataset is public and contains the following post-related information: * tweet id * posting date* number of favorites received* number of times it is retweeted* the url links shared as a part of it* text of the tweet including the hashtags* geolocation if taggedA tweet may contain more than a single hashtag. From this set of tweets, we remove the redundant tweets that have more than one of these four hashtags we considered. This resulted in a dataset of 0.2 million tweets that are unique and are posted by a unique set of 33K users. Due to the download limit of the Twitter API, all the tweets in our dataset are from February 2017, and none of the tweets we crawled were tagged with a geolocation.§.§ Expert AI-Tweeters (EAIT)The team at IBM Watson has recently shared an article (https://goo.gl/PdRlHT) listing 30 people in AI as “AI Influencers 2017”. This article suggested that these are the top-30 people in AI to follow on Twitter to get updated information about AI. For the purposes of our investigation, we contacted the author of this article through a private email to learn the approach used to compile this list. According to the author, this list was compiled by using a combination of thought leaders that the author's team was aware of. This list includes – AI experts who are most active on Twitter, influencers who speak regularly and present research at AI events & conferences, and entrepreneurs in the AI space who are working on interesting products and technology. The IBM team also interviewed multiple experts to ask them who they follow on Twitter to stay updated on the trending news about AI. Table <ref> enlists the AI Influencers 2017 named by this article (sorted alphabetically). We then utilize the Twitter official API to crawl the timelines of these 30 AI influencers.§ RQ1: CHARACTERIZATION OF USERSBefore we delve deep in to investigating the research questions posed earlier, we present few details about the demographics of users from both the categories. We first focus on the influence attributes of users – #statuses shared, #followers, #friends, #favorites. To understand the differences between the two types of user categories based on their activity and influence of tweets, we plot the logarithmic frequencies of these attributes in Figure <ref>. Surprisingly, the influence attribute values of experts are significantly lower than the general Twitter users.Figure <ref> shows that both sets of users are highly active on Twitter by sharing statuses and favoriting tweets. When we consider the other influence attributes – followers and friends, on an average both sets of users have large number of followers than friends (users you are following). The statuses shared by EAIT are approximately equal to the number of favorites. However, AIT share more number of statuses than favoriting other tweets. This observation also suggests that the users in AIT and EAIT are active on Twitter as user activity is influential in attracting more number of followers <cit.>. Table <ref> compares the statistics about the length of posts made by AIT and EAIT. On average, EAIT's tweets are longer compared to the tweets posted by AIT. We then looked in to the particulars of the users' geographical location and professional background which we obtained from their biography profiles. Figure <ref> shows that the highest percentage of users in our dataset who tweet about AI are from London (6% of the total set of users) followed by New York City (4%) and Paris (3%). The histogram shows only the top-10 locations of users as we also have a fair percentage of users tweeting from locations such as Sydney (1%), Berlin (0.98%), etc. Where as from EAIT, large percentage (14%) of users in this set do not provide their geographical location. However, the top-2 locations of experts are San Francisco (10%) followed by Seattle (7%).We conducted a unigram-based analysis of the profiles of the users to decipher their professional background. Table <ref> shows that based on the frequencies of professions stated by users on Twitter, majority of the Twitter users contributing to AI-related tweets are pursuing careers in technology.§.§ User Interests To examine the interests of the users in our dataset, we crawled the recent 100 posts made by these users. We extract the topics from these posts made by users using the Twitter LDA package <cit.>. By utilizing these topics, we aim to measure the level of users' interest in technology which can quantify their perceptions about AI. §.§.§ AITAs mentioned earlier, there are a total of 33K unique set of users who contributed towards our dataset. We crawl their recent tweets to extract the topics. We empirically decided to extract 5 topics and their corresponding vocabulary are shown as below: * Topic-1 [Science & Technology]: data, learning, intelligence, business, machine, digital, analytics, cloud, trends, science* Topic-2 [Personal Status & Opinions]: people, time, day, good, love, today, life, work, happy, story* Topic-3 [General News]: latest, stories, daily, join, great, world, march, live, week, day, event, network, news* Topic-4 [Non-English Tweets]: pour, les, des, para, los, dans, qui, avec, une, se, las* Topic-5 [Daily Updates]: follow, daily, chapter, updates, translation, cardBy considering the topics, we obtain the percentage distribution of each individual's tweets to these 5 different topics. We then aggregate all the distributions of users across these topics and the percentage distributions are shown in Figure <ref>. This figure shows that majority of the tweets posted by the users are about technology and science.§.§.§ EAIT We conduct a similar investigation as above on the tweets posted by experts. The topics extracted and their corresponding vocabulary are shown here: * Topic-1 [Rise of AI]: robots, intelligence, jobs, future, human, tesla, cars, driving, learning, machine * Topic-2 [AI subscriptions and research]: papers, news, subscribing, events, latest, learning, deep, code, work, image, model, people* Topic-3 [AI News from industry]: zapchain, google, sharing, bitcoin, startup, slack, twitter, working, app, facebook, medium, chatbots* Topic-4 [Marketing & AI]: virtualreality, ibmwatson, team, marketing, virtual, work, channel, bit, time* Topic-5 [Opinions about AI]: essays, tweeting, enjoy, visit, brain, neuroscience, conspiracy, cleaning, fiction, theory, fun, hype, post These topics reveal that experts predominantly focus on posting about AI, its impacts, industry aspects of AI, their opinions and subscription recommendations which may help provide more information about AI. These topics are different from the topics focused by AIT. The pie chart shown in Figure <ref> reveals that more than 50% of the tweets posted by experts on Twitter are about the impacts of AI and research directions in this field. AIT share large percentage of personal opinions and statuses where as EAIT post the least percentage of tweets about their opinions. § RQ2: TWITTER ENGAGEMENTWe seek to study the attributes which might disclose the holistic picture of the overall engagement rate of AI-related tweets. This rate provides us with an information on patterns of public interests and perceptions in AI. We measure the engagement by considering the `favorites', `likes', `replies' and `mentions' of a Twitter post. We first compute Twitter engagement statistics that are shown in Table <ref>. Tweets made by AIT are more likely to be retweeted than favorited by the users on this platform. 63.3% of their tweets are retweeted atleast once. This is significantly higher than the general dataset which is 11.99% as shown by the existing literature <cit.>. Where as, the tweets posted by EAIT received more number of favorites than retweets as only 35% of the tweets are retweeted atleast once. Tweets from both the categories of users have higher probabilities of containing atleast one user handle. This may suggest that users are more likely to interact or engage in discussions with each other about AI on Twitter. On the other hand, 68.5% of the tweets posted by AIT in our dataset have atleast one url shared as part of the tweet where as 48% of the tweets made by EAIT have urls. Sharing large percentage of urls by AIT compared to EAIT could be one of the reasons for receiving more retweets than favorites. Literature <cit.> considers retweeting as one of the features to measure information diffusion. Based on these results, tweets posted by AIT diffuse faster (higher retweet rate) than the tweets posted by EAIT.§ RQ3: OPTIMISTIC OR PESSIMISTICOptimism is defined as being hopeful and confident about the future whose synonym is `positive' and pessimism is defined as tending to see or believing that the worst will happen whose synonym is `negative'. In this work we measure these two attributes – positive, negative emotions alongside assessing cognitive mechanisms by employing the popular psycho-linguistic tool LIWC <cit.>. Tausczik et. al in their work introducing LIWC mention that the way people express emotion and the degree to which they express it can tell us how people are experiencing the world. Existing literature <cit.> states that LIWC is powerful in accurately identifying emotion in the usage of language. This is the motivation for us to use LIWC to measure the emotionality of tweets shared by AIT and EAIT. §.§ AITFigure <ref> reveals that individuals are more postive (65% greater than negative) and optimistic towards AI and its related topics. This concurs with the recent analysis <cit.> and survey <cit.> on New York Times articles and interviews with individuals respectively. This is a useful finding because Twitter is known for users making posts that are emotionally negative <cit.>. In other words, despite the general negative emotional content on Twitter, thes subset of tweets focusing on artificial intelligence are more positive than being negative. §.§ EAITWe conduct the emotion analysis on tweets posted by experts and it reveals similar findings as earlier (shown in Figure <ref>) but with relatively higher cognitive mechanisms than AIT. Cognitive mechanisms or complexity can be considered as a rich way of reasoning <cit.>. The horizontal axis represents the gravity of a given emotion and the vertical axis represents the number of posts with a gravity value shown on horizontal axis. The distribution in each plot are normalized and the sum of all the values in different buckets of gravity will sum upto 100%.Compared to AIT, tweets made by EAIT have more negativity overall. However, positive emotion is thrice as dominating as the negative emotion. When we compare the positive and negative emotions of the two categories of users, the results reveal that expert users are 38% more negative than the general AI-Tweeters.§ RQ4: TOPICS HEAVILY DISCUSSED BY USERS ON TWITTERIn Section <ref>, we have presented the analysis on the interests of users by crawling their timelines and extracting topics from these timelines. In order to better understand the public perceptions about AI, we extract topics from the tweets that are exclusively about AI. To perform this, we first consider all the tweets posted by AIT using the hashtag approach. In Table <ref>, we present the topics extracted by considering the aggregated set of tweets. These topics display that the focus of the AI-tweets are on stories and statistics about AI followed by discussions about deep learning. These topics reveal the high-level interests of individuals about the emerging trends, impacts and career opportunities related to AI. These topics also display that individuals have continued interests in the similar topics over years due to partial alignment of these topics with the findings shown by Fast et. al <cit.> who consider NYT articles published before 2017. We perform this analysis on the AI-tagged tweets from Twitter as we assume that this set may comprise both experts and general Twitter users. § RQ5: CO-OCCURRING CONCEPTS DISCUSSEDThe questions we investigated until now provides valuable insights into whether and how individuals perceive the issues about AI advancements. However, we note that conceptual relationships could significantly quantify and measure the perceptions of individuals. Towards addressing this challenge, we employ the popular word2vec analysis to detect relationships between words that are frequently co-occurring. Word2Vec <cit.> is a popular two-layer neural network that is used to process text. It considers a text corpus as an input and generates feature vectors for words present in that corpus. Word2vec represents words in a higher-dimensional feature space and makes accurate predictions about the meaning of a word based on its past occurrences. These vectors can then be utilized to detect relationships between words which are highly accurate given enough data to learn these vectors. To detect the relationships, we train the Word2Vec model on the Text8 corpus (<http://mattmahoney.net/dc/>) which is created using the articles from Wikipedia. As a processing step, we first remove stop words from the tweets and consider each tweet independently. We utilized the pre-existing lists from academia[https://www.cs.utexas.edu/users/novak/aivocab.html] and industry[http://www.techrepublic.com/article/mini-glossary-ai-terms-you-should-know/] to manually compile the AI vocabulary of 61 words. Figure <ref> provides two pairs of co-occurring pattern comparisons. These pairs of co-occurring patterns tell us that AIT are in general fantasizing about the future where as EAIT are grounded and realistic. The entire list of co-occurrences for the 61 words in AI vocabulary can be found here: AIT (<https://goo.gl/5WPA9L>) and EAIT (<https://goo.gl/8WxBaP>). § CONCLUSIONSSocial media platforms are one of the primary channels of communication in the lives of individuals. These platforms are reshaping our ideas and the way we share those ideas. Given the increasing interest in AI from different communities, multiple debates are commencing to evaluate the benefits and drawbacks of AI to humans and society as a whole. This paper presents the findings from our investigation on public perceptions about AI using the AI-related posts shared on Twitter. Alongside, we performed a comparative analysis between how the posts made by AIT and EAIT are engaged. Some of the key findings from our analysis are:* Tweets about AI are overall more positive compared to the general tweets. * Tweets posted by experts are more negative than the general AI tweeters. * Tweets posted by experts have lower diffusion than the tweets posted by general AI tweeters. * General AI-tweeters are geographically distributed with London and New York City as the top locations.The co-occurring pattern mapping tells us that users belonging to EAIT are more grounded and realistic in their perceptions about AI. Additionally, analysis on user interests revealed that tweeters who are posting about AI are interested in technology. The most discussed topics on Twitter by the general AI tweeters are about the stories and statistics alongside the impact of deep learning, career and recruitment aspects. Our LIWC analysis revealed that discussions are cognitively loaded with a larger variance of gravities among Twitter users.We hope that our findings will benefit different organizations and communities who are debating about the benefits and threats of AI to our society. Some of the future directions include a longitudinal study across several years as well as multiple mediums of communication.§.§ AcknowledgementsThis research is supported in part by a Google research award, the ONR grants N00014-16-1-2892, N00014-13-1-0176, N00014-13-1-0519, N00014-15-1-2027 and the NASA grant NNX17AD06G. We thank Miles Brundage for his feedback on parts of this analysis.aaai § APPENDIX – ANALYSIS ON REDDITReddit is one of the popular online social media platforms that is considered as a social news and a discussion platform. Users on this platform can share posts that may contain either text or media. Other users on this platform can then vote or comment on these posts. There are different sub-communities organized by the areas of interest also called as Subreddits. Posts on any subreddit are of two types – self post or a link post. Self post is defined as a post that contains text and the content of this information is saved on the Reddit's server. Link post submission redirects the user to an external website. Similar to the Twitter analysis, we consider the subreddits that focus on AI to capture user perceptions about AI. To start the analysis, we match the hashtags used in Twitter analysis to the names of subreddits to conduct a comparative study between Reddit and Twitter. Also, please note that all the users considered for Reddit analysis are categorized as common Reddit users with no distinction between experts and non-experts. §.§ Data CollectionTo crawl the data from Reddit, we employed the Python Reddit API Wrapper [https://praw.readthedocs.io/en/latest/] (PRAW). We utilized the subreddits that focus on the same topics as the hashtags used to crawl the data from Twitter – r/artificial, r/machinelearning and r/datascience. Across the 3 subreddits, a total of 2,550 unique thread posts are crawled from 5 different categories of posts – hot, new, rising, controversial and top. The total number of posts aggregated across all the threads consists of 21,420 posts.For each post crawled using PRAW, the metadata includes the title of the post, content of the post, score of the post, up votes, down votes, posting date, username of the author. In our dataset, we have 5,382unique number of users. For understanding the users and their background, we separately crawl the recent 100 posts made by each individual user on Reddit. The posts we use for our analysis come from the time period between November 2016 until March 2017.§.§ RQ1: Reddit EngagementOur first research question explores different set of statistics to understand the engagement on the subreddits related to artificial intelligence. Towards this goal, we first understand whether the posts are questions or opinions. We utilize the `wh' questions along with the post that matches the '?' pattern. 52.8% of the posts we crawled from Reddit related to AI are questions. This suggests that the interest to seek information on Reddit is high. Each post made on the subreddit can be voted as up or down which also allows users to comment on this post. The score of a post is defined as the sum of up votes and down votes. In our dataset, there are no posts which have a down vote. This might be due to the inherent nature of subreddit that we are focusing on. On average, each post received around 20 votes with a median of 7 votes. The post made by Google Brain team encouraging the community to ask questions about machine learning received the maximum number of votes (4196).§.§ RQ2: Optimistic or PessimisticWe conduct a similar analysis on Reddit posts, that we have conducted on tweets, to measure the emotional gravity. To do this, we first pre-process the data by considering only the textual content attached to the post that doesn't consider the title of the post. Similar to the Twitter analysis, we do not remove any stopwords because in the LIWC <cit.> analysis, stopwords may count towards providing emotional insights. Figure <ref> shows that, Reddit posts are emotionally more positive than Twitter.§.§ RQ3: Topics heavily discussed and concerning to the publicAnalysis on emotions shows that there is a higher percentage of positive emotions alongside cognitive complexity and expressing insights. To explain this phenomenon, exploring the topics focused in the posts can be beneficial. To do this, we consider the Twitter LDA <cit.> package and the extracted topics are shown in Table <ref>. The posts about data science and careers are the most discussed topics. We define the unclassified text as the non-english text or a post that includes emoticons. However, we also notice that Reddit posts on Reddit focus on various topics of AI – comparison of intelligent systems with humans (Topic-6), properties of intelligent machines (Topic-7), how to learn or train these intelligent models (Topic-8), neural networks (Topic-4), applications like games (Topic-3), how industry is spreading its interest around AI (Topic-5), future applications that could be envisioned (Topic-2).In comparison to Twitter that focuses only on 7.2% of tweets on data science, Reddit has higher percentage of posts on this topic. However, 7 of the 9 topics (that constitute 72.1% of the Reddit posts) very tightly focus on the technical details about intelligent systems and their performance. For example – details about training the learning models, properties of intelligence machines, nuances of technicalities from the perspective of industry, emphasizes on technical aspects of AI systems. On the other hand, Twitter topics are relatively skewed in terms of mostly focusing on marketing, recruitment, stories, impact, etc and do not heavily focus on the technical details about the intelligent systems. This might be due to the fact that Reddit as a platform does not have any constraints on the length of the post. This could be one of the reasons why the posts on Reddit focus on in-depth technicalities about the AI systems.§.§ RQ4: User AnalysisAs mentioned earlier, understanding the conclusions we inferred from the Reddit posts could be more valuable if we recognize the interests of users. To understand the background of the 5,382 unique set of Reddit users from our dataset, we crawl their recent 100 Reddit posts.* Topic-1 [Topics about life]: people, good, pretty, money, games, things, years, lot, high, pay* Topic-2 [Political issues]: government, country, world, time, years, women, good, bad, person, life* Topic-3 [Mobile or Web Applications]: work, love, great, windows, app, phone, video, version, link, find* Topic-4 [Training neural networks]: model, code, neural, function, training, network, deep, learn, image* Topic-5 [Intelligence aspects]: ai, people, human, brain, work, learning, data, intelligence, understand* Topic-6 [Jobs and programs offered in Data Science]: science, python, job, math, experience, programming, courses, statistics, degree, ml* Topic-7 [External URL shares]: en.wikipedia.org, www.youtube.com, watch, i.imgur.com, delete, youtu.be, definition According to the topic distributions shown in Figure <ref>, these users are interested in technology – especially neural networks. Eventhough on an average, 44% of the reddit posts made by a user focuses on general topics about life and politics, 25% of their posts talk about training neural networks and intelligence. This may shed more light on the perceptions of AI that are mainly shared and discussed by users with interests in technology especially related to AI.The main findings from the Reddit analysis are that the users are more optimistic than the users on Twitter. The topic analysis shows the technical depth in to different aspects of AI. This might be due to the nature of Reddit platform that doesn't have restrictions on the length of a post. | http://arxiv.org/abs/1704.08389v2 | {
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"title": "Tweeting AI: Perceptions of AI-Tweeters (AIT) vs Expert AI-Tweeters (EAIT)"
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[pages=1-last]paper.pdf | http://arxiv.org/abs/1704.08738v1 | {
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"published": "20170427203314",
"title": "Portfolio-driven Resource Management for Transient Cloud Servers"
} |
Mass spectrum and decay constants of radially excited vector mesons Bruno El-Bennich December 30, 2023 =================================================================== Finding central nodes is a fundamental problem in network analysis. Betweenness centrality is a well-known measure which quantifies the importance of a node based on the fraction of shortest paths going though it. Due to the dynamic nature of many today's networks, algorithms that quickly update centrality scores have become a necessity.For betweenness, several dynamic algorithms have been proposed over the years, targeting different update types (incremental- and decremental-only, fully-dynamic). In this paper we introduce a new dynamic algorithm for updating betweenness centrality after an edge insertion or an edge weight decrease.Our method is a combination of two independent contributions: a faster algorithm for updating pairwise distances as well as number of shortest paths, and a faster algorithm for updating dependencies. Whereas the worst-case running time of our algorithm is the same as recomputation, our techniques considerably reduce the number of operations performed by existing dynamic betweenness algorithms.Our experimental evaluation on a variety of real-world networks reveals that our approach is significantly faster than the current state-of-the-art dynamic algorithms, approximately by one order of magnitude on average. § INTRODUCTIONOver the last years, increasing attention has been devoted to the analysis of complex networks.A common sub-problem for many graph based applications is to identify the most central nodes in a network.Examples include facility location <cit.>, marketing strategies <cit.> and identification of key infrastructure nodesas well as disease propagation control and crime prevention <cit.>.As the meaning of “central” heavily depends on the context, various centrality measures have been proposed (see <cit.> for an overview). Betweenness centrality is a well-known measure which ranks nodes according to their participation in the shortest paths of the network. Formally, the betweenness of a node v is defined as c_B(v) = ∑_s ≠ v ≠ tσ_st(v)/σ_st, where σ_st is the number of shortest paths between two nodes s and t and σ_st(v) is the number of these paths that go through node v. The fastest algorithm for computing betweenness centrality is due to Brandes <cit.>, which we refer to as , from Brandes's algorithm. This algorithm is composed of two parts: an augmented APSP (all-pairs shortest paths) step, where pairwise distances and shortest paths are computed, and a dependency accumulation step, where the actual betweenness scores are computed.The augmented APSP is computed by running a SSSP (single-source shortest paths) computation from each node s and the dependency accumulation is performed by traversing only once the edges that lie in shortest paths between s and the other nodes.Therefore,requires Θ(|V| |E|) time on unweighted and Θ(|V| |E| + |V|^2 log |V|) time on weighted graphs (i.e. the time of running n SSSPs).Networks such as the Web graph and social networks continuously undergo changes. Since an update in the graph might affect only a small fraction of nodes, recomputing betweenness withafter each update would be very inefficient. For this reason, several dynamic algorithms have been proposed over the last years <cit.>.As , these approaches usually solve two sub-tasks: the update of the augmented APSP data structures and the update of the betweenness scores. Although none of these algorithms is in general asymptotically faster than recomputation with , good speedups overhave been reported for some of them, in particular for <cit.> and <cit.>. Nonetheless, an exhaustive comparison of these methods is missing in the literature.In our paper, we only consider incremental updates, i.e. edge insertions or edge weight decreases (node insertions can be handled treating the new node as an isolated node and adding its neighboring edges one by one). Although it might seem reductive to only consider these kinds of updates, it is important to note that several real-world dynamic networks evolve only this way and do not shrink. For example, in a co-authorship network, a new author (node) or a new edge (coauthored publication) might be added to the network, but existing nodes or edges will not disappear. Another possible application is the centrality maximization problem, which consists in finding a set of edges that, if added to the graph, would maximize the centrality of a certain node. The problem can be approximated with a heuristic <cit.>, which requires to add several edges to the graph and to recompute distances after each edge insertion.*Our contribution We present a new algorithm for updating betweenness centrality after an edge insertion or an edge weight decrease. Our method is a combination of two contributions: a new dynamic algorithm for the augmented APSP, and a new approach for updating the betweenness scores. Based on properties of the newly-created shortest paths, our dynamic APSP algorithm efficiently identifies the node pairs affected by the edge update (i.e. those for which the distance and/or number of shortest paths change as a consequence of the update). The betweenness update method works by accumulating values in a fashion similar to that of . However, differently from , our method only processes nodes that lie in shortest paths between affected pairs.We compare our new approach with two of the dynamic algorithms for which the best speedups over recomputation have been reported in the literature, i.e. <cit.> and <cit.>. Compared to them, our algorithm for the augmented APSP update is asymptotically faster on dense graphs: O(|V|^2) in the worst case versus O(|V||E|). This is due to the fact that we iterate over the edges between affected nodes only once, whereasanddo it several times. Moreover, our dependency update works also for weighted graphs (whereasdoes not) and it is asymptotically faster than the dependency update offor sparse graphs (O(|V||E| + |V| log |V|) in the worst case versus O(|V|^3)).Our experimental evaluation on a variety of real-world networks reveals that our approach is significantly faster than bothand , on average by a factor 14.7 and 7.4, respectively.§ PRELIMINARIES §.§ NotationLet G = (V, E, ω) be a graph with node set V = V(G), edge set E = E(G)and edge weights ω: E →ℝ_> 0. In the following we will use n := |V| to denote the number of nodes and m := |E| for the number of edges. Let d(s, t) be the shortest-path distance between any two nodes s, t ∈ V. On a shortest path from s to t in G, we say w is a predecessor of t, or t is a successor of w, if (w,t) ∈ E and d(s,w) + ω(w,t) = d(s,t). We denote the set of predecessors of t as P_s(t). For a given source node s ∈ V, we call the graph composed of the nodes reachable from s and the edges that lie in at least one shortest path from s to any other node the SSSP DAG of s. We use σ_st to denote the number of shortest paths between s and t and we use σ_st(v) for the number of shortest paths between s and t that go through v. Then, the betweenness centrality c_B(v) of a node v is defined as: c_B(v) = ∑_s ≠ v ≠ tσ_st(v)/σ_st.Our goal is to keep track of the betweenness scores of all nodes after an update (u, v, ω'(u, v)) in the graph, which could either be an edge insertion or an edge weight decrease. We use G' = (V, E', ω') to denote the new graph after the edge update and d', σ' and P' to denote the new distances, numbers of shortest paths and sets of predecessors, respectively. Also, we define the set of affected sources S(t) of a node t∈ V as { s ∈ V : d(s, t) > d'(s, t) ∨σ_st≠σ'_st}. Analogously, we define the set of affected targets of s ∈ V as T(s) := { t ∈ V : d(s, t) > d'(s, t) ∨σ_st≠σ'_st}. In the following we will assume G to be directed. However, the algorithms can be easily extended to undirected graphs.§.§ Related WorkThe basic idea of dynamic betweenness algorithms is to keep track of the old betweenness scores (and additional data structures) and efficiently update the information after some modification in the graph.Based on the type of updates they can handle, dynamic algorithms are classified as incremental (only edge insertions and weight decreases), decremental (only edge deletions and weight increases) or fully-dynamic (all kinds of edge updates). However, one commonality of all these approaches is that they build on the techniques used by <cit.>, which we therefore describe in Section <ref> in more detail.The approach proposed by Green <cit.> for unweighted graphsmaintains all previously calculated betweenness values and additional information, such as pairwise distances, number of shortest paths and lists of predecessors of each node in the shortest paths from each source node s ∈ V. Using this information, the algorithm tries to limit the recomputation to the nodes whose betweenness has been affected by the edge insertion. Kourtellis <cit.> modify the approach by Green <cit.> in order to reduce the memory requirements from O(nm) to O(n^2).Instead of being stored, the predecessors are recomputed every time the algorithm requires them. The authors show that not only using less memory allows them to scale to larger graphs, but their approach (which we refer to as , from the authors's initials) turns out to be also faster than the one by Green <cit.> in practice (most likely because of the cost of maintaining the data structure of the algorithm by Green ).Kas <cit.> extend an existing algorithm for the dynamic all-pairs shortest paths (APSP) problem by Ramalingam and Reps <cit.> to also update betweenness scores. Differently from the previous two approaches, this algorithm can handle also weighted graphs. Although good speedups have been reported for this approach, no experimental evaluation compares its performance with that of the approaches by Green <cit.> andKourtellis <cit.>. We refer to this algorithm as , from the authors's initials.Nasre <cit.> compare the distances between each node pair before and after the update and then recompute the dependencies from scratch as in(see Section <ref>). Although this algorithm is faster than recomputation on some graph classes (i.e. when only edge insertions are allowed and the graph is sparse and weighted), it was shown in <cit.> that its practical performance is much worse than that of the algorithm proposed by Green <cit.>. This is quite intuitive, since recomputing all dependencies requires Ω(n^2) time independently of the number of nodes that are actually affected by the insertion.Pontecorvi and Ramachandran <cit.> extend existing fully-dynamic APSP algorithms with new data structures to update all shortest paths and then recompute dependencies as in . To our knowledge, this algorithm has never been implemented, probably because of the quite complicated data structures it requires. Also, since it recomputes dependencies from scratch as Nasre <cit.>, we expect its practical performance to be similar.Differently from the other algorithms, the approach by Lee <cit.> is not based on dynamic APSP algorithms. The idea is to decompose the graph into its biconnected components and then recompute the betweenness values from scratch only for the nodes in the component affected by the update. Although this allows for a smaller memory requirement (Θ(m) versus Ω(n^2) needed by the other approaches), the speedups on recomputation reported in <cit.> are significantly worse than those reported for example by Kourtellis <cit.>.To summarize, <cit.> and <cit.> are the most promising methods for a comparison with our new algorithm. For this reason, we will describe them in more detail in Section <ref> and Section <ref> and evaluate them in our experiments.Since computing betweenness exactly can be too expensive for large networks, several approximation algorithms and heuristics have been introduced in the literature <cit.> and, recently, also dynamic algorithms that update an approximation of betweenness centrality have been proposed <cit.>. However, we will not consider them in our experimental evaluation since our focus here is on exact methods.§ BRANDES'S ALGORITHM () Betweenness centrality can be easily computed in time Θ(n^3) by simply applying its definition. In 2001, Brandes proposed an algorithm () <cit.> which requires time Θ(nm) for unweighted and Θ(n(m + nlog n)) for weighted graphs, i.e. the time of computing n single-source shortest paths (SSSPs). The algorithm is composed of two parts: the augmented APSP computation phase based on n SSSPs and the dependency accumulation phase. As dynamic algorithms based onbuild on these two steps as well, we explain them now in more detail.*Augmented APSP In this first part,needs to perform an augmented APSP, meaning that instead of simply computing distances between all node pairs (s, t), it also finds the number of shortest paths σ_st and the set of predecessors P_s(t). This can be done while computing an SSSP from each node s (i.e. BFS for unweighted and Dijkstra for weighted graphs). When a node w is extracted from the SSSP (priority) queue,computes P_s(w) as {v : (v, w) ∈ E∧d(s,w) = d(s, v) + ω(v,w)} and σ_sw as ∑_v ∈ P_s(w)σ_sv. *Dependency accumulation Brandes defines the one-side dependency of a node s on a node v as δ_s ∙(v) := ∑_t ≠ vσ_st(v)/ σ_st. It can be proven <cit.> thatδ_s ∙(v) = ∑_w : v ∈ P_s(w)σ_sv/σ_sw (1 + δ_s ∙(w)), ∀ s, v ∈ VIntuitively, the term δ_s ∙(w) in Eq. (<ref>) represents the contribution of the sub-DAG (of the SSSP DAG of s) rooted in w to the betweenness of v, whereas the term 1 is the contribution of w itself. For all nodes v such that {w : v ∈ P_s(w)} = ∅ (i.e. the nodes that have no successors), we know that δ_s ∙(v) = 0. Starting from these nodes, we can compute δ_s ∙(v) ∀ v ∈ V by “walking up” the SSSP DAG rooted in s, using Eq. (<ref>). Notice that it is fundamental that we process the nodes in order of decreasing distance from s, because to correctly compute δ_s ∙(v), we need to know δ_s ∙(w) for all successors of v. This can be done by inserting the nodes into a stack as soon as they are extracted from the SSSP (priority) queue in the first step. The betweenness of v is then simply computed as ∑_s ≠ vδ_s ∙(v).§ DYNAMIC AUGMENTED APSP As mentioned in Section <ref>, also dynamic algorithms based onbuild on its two steps. In the following, we will see how <cit.> and <cit.> update the augmented APSP data structures (i.e. distances and number of shortest paths) after an edge insertion or a weight decrease. One difference between these two approaches is thatdoes not store the predecessors explicitly, whereasdoes. However, since in <cit.> it was shown that keeping track of the predecessors only introduces overhead, we report a slightly-modified version ofthat recomputes them “on the fly” when needed (we will also use this version in our experiments in Section <ref>).We will then introduce our new approach in Section <ref>.§.§ Algorithm by Kourtellis et al. () Let (u, v) be the new edge inserted into G (we recall thatworks only on unweighted graphs, so edge weight modifications are not supported). For each source node s ∈ V, there are three possibilities: (i) d(s, u) = d(s, v), (ii) |d(s, u) - d(s, v)| = 1 and (iii) |d(s, u) - d(s, v)| > 1 (in case (ii) and (iii), let us assume that d(s, u) < d(s, v) without loss of generality). We recall that d is the distance before the edge insertion.In the first case, it is easy to see that the insertion does not affect any shortest path rooted in s, and therefore nothing needs to be updated for s. In case (ii), the distance between s and the other nodes is not affected, since there already existed an alternative shortest-path from s to v. However, the insertion creates new shortest paths from s to to v and consequently to all the nodes t in the sub-DAG (of the SSSP DAG from s) rooted in v. To account for this, for each of these nodes t, we add σ_su·σ_vt to the old value of σ_st (where σ_su·σ_vt is the number of new shortest paths between s and t going through (u, v)).Finally, in case (iii), a part of the sub-DAG rooted in v might get closer to s. This case is handled with a BFS traversal rooted in v. In the traversal, all neighbors y of nodes x extracted from the BFS queue are examined and all the ones such that d(s, y) ≥ d'(s, x) are also enqueued. For each traversed node y, the new distance d'(s, y) is computed as min_z : (z, y) ∈ E d'(s, z) + 1 and the number of shortest paths σ'_sy as ∑_z ∈ P'_s(y)σ_sz. §.§ Algorithm by Kas et al. ()r0.31< g r a p h i c s >Insertion of (u,v). updates the augmented APSP based on a dynamic APSP algorithm by Ramalingam and Reps <cit.>. Instead of checking for each source s whether the new edge (or the weight decrease) changes the SSSP DAG rooted in s,first identifies the affected sources S = { s: d(s, v) ≥ d(s, u) + ω'(u, v)}. These are exactly the nodes for which there is some change in the SSSP DAG. The affected sources are identified by running a pruned BFS rooted in u on G transposed (i.e. the graph obtained by reversing the direction of edges in G). For each node s traversed in the BFS,checks whether the neighbors of s are also affected sources and, if not, it does not continue the traversal from them. Notice that even on weighted graphs, a (pruned) BFS is sufficient since we already know all distances to v and we can basically sidestep the use of a priority queue. Once all affected sources s are identified,starts a pruned BFS rooted in v for each of them. In the pruned BFS, only nodes t such that d(s,t) ≥ d(s, u) + ω'(u, v) + d(v, t) are traversed (the affected targets of s). The new distance d'(s, t) is set to d(s, u) + ω'(u, v) + d(v, t) and the new number of shortest paths σ'(s, t) is set to ∑_z ∈ P'_s(t)σ_sz as in . Compared to , the augmented APSP update ofrequires fewer operations. First, it efficiently identifies the affected sources instead of checking all nodes. Second, in case (iii),might traverse more nodes than . For example, assume (u, v) is a new edge and the resulting SSSP DAG of u is as in Figure <ref>. Then,will prune the BFS in t, since d(u, t) < d(u, v) + d(v, t), skipping all the SSSP DAGs rooted in t. On the contrary,will traverse the whole subtree rooted in t, although neither the distances nor the number of shortest paths from u to those nodes are affected. The reason for this will be made clearer in Section <ref>.§.§ Faster augmented APSP updater0.3< g r a p h i c s >Affected targets (in green) and affected sources (x_1, x_2, u).To explain our idea for improving the APSP update step, let us start with an example, shown in Figure <ref>.The insertion of (u, v) decreases the distance from nodes x_1, x_2, u toall the nodes shown in green. would first identify the affected sources S = {x_1, x_2, u} and, for each of them, run a pruned BFS rooted in v.This means we are repeating almost exactly the same procedure for each of the affected sources. We clearly have to update the distances and number of shortest paths between each affected source and the affected targets (and this cannot be avoided). However,also goes through the outgoing edges of each affected target multiple times, leading to a worst-case running time of O(mn).[Notice that this is true also for , with the difference thatstarts a BFS from each node instead of first identifying the affected sources and that it also visits additional nodes.]Our basic idea is to avoid this redundancy and is based on the following proposition (a similar result was proven also in <cit.>).Let t ∈ V and y ∈ P_v(t) be given. Then, S(t) ⊆ S(y).Let s be any node in S(t), i.e. either d'(s,t) = d(s,t) and σ'_st≠σ_st (case (i)), or d'(s,t) < d(s,t) (case (ii)).We want to show that s ∈ S(y). Before proving this, we show that y has to be in P'_s(t). In fact, if s ∈ S(t), there have to be shortest paths between s and t going through (u, v), i.e. d'(s, t) = d(s, u) + ω'(u, v) + d(v, t). On the other hand, we know y ∈ P_v(t) and thus d'(s, t) = d(s, u) + ω'(u, v) + d(v, y) + ω(y, t).Now, d(s, u) + ω'(u, v) + d(v, y) cannot be larger than d'(s, y), or this would mean that d'(s, t) > d'(s, v) + ω(y, t), which contradicts the triangle inequality. Also, d(s, u) + ω'(u, v) + d(v, y) cannot be smaller than d'(s, y) by definition of distance. Thus, d'(s, y) = d(s, u) + ω'(u, v) + d(v, y). If we substitute this in Eq. (<ref>), we obtain d'(s, t) = d'(s, y) + ω(y, t), which means y ∈ P'_s(t).Now, let us consider case (i). We have two options: either y was a predecessor of t from s also before the edge update, i.e. y ∈ P_s(t), or it was not. If it was not, it means d(s, y) + ω(y, t) > d(s, t) = d'(s, t) = d'(s, y) + ω(y, t), which implies d(s, y) > d'(s, y) and thus s ∈ S(y). If it was, we can similarly show that d(s, y) = d'(s, y). Since we have seen before that d'(s, y) = d(s, u) + ω'(u, v) + d(v, y), there has to be at least one new shortest path from s to y in G' going through (u, v), which means σ'_sy > σ_sy and therefore s ∈ S(y). Case (ii) can be easily proven by contradiction. We know d(s,t) ≤ d(s,y) + ω(y,t) (by the triangle inequality) and that ω'(y,t) = ω(y,t). Thus, if it were true that d(s,y) = d'(s,y) thend(s,t) ≤ d(s,y) + ω(y,t) = d'(s,y) + ω(y,t) = d'(s,t),which contradicts our hypothesis that d'(s,t) < d(s,t) (case (ii)). Thus, d(s,y) ≠ d'(s,y). Since pairwise distances in G' can only be equal to or shorter than pairwise distances in G, d(s,y) ≠ d'(s,y) implies d(s,y) > d'(s,y) and thus s ∈ S(y). In particular, this implies that S(t) ⊆ S(v) for each t ∈ T(u). Consequently, it is sufficient to compute S(v) and T(u) once via two pruned BFSs. Our approach is described in Algorithm <ref>. The pruned BFS to compute S(v) is performed in Line <ref>. Then, a pruned BFS from v is executed, whereby for each t ∈ T(u) we store one of its predecessors p(t) in the BFS (Line <ref>). Let d^⋆(s, t) be the length of a shortest path between s and t going through (u, v), i.e. d^⋆(s, t) := d(s, u) + ω'(u, v) + d(v, t). To finally compute S(t) all that is left to do is to test whether d^⋆(s, t) ≤ d(s, t) for each s ∈ S(p(t)) once we remove t from the queue (Lines <ref> - <ref>).Note that this implies that S(p(t)) was already computed.In case d^⋆(s,t) < d(s,t), the path from s to t via edge (u,v) is shorter than before and therefore we set d'(s,t) to d^⋆(s,t) and σ'_st to σ_su·σ_vt, since all new shortest paths now go through (u,v)).Also in case of equality (d^⋆(s,t) = d(s,t)), s is in S(t), since its number of shortest paths has changed. Consequently we set σ'_st to σ_st + σ_su·σ_vt (since in this case also old shortest paths are still valid).If d^⋆(s, t) > d(s, t), the edge (u, v) does not lie on any shortest path from s to t, hence s ∉ S(t) (and s is not added to S(t) in Lines <ref> - <ref>).1em1em § DYNAMIC DEPENDENCY ACCUMULATION After updating distances and number of shortest paths, dynamic algorithms need to update the betweenness scores. This means increasing the score of all nodes that lie in new shortest paths, but also decreasing that of nodes that used to be in old shortest paths between affected nodes. Again, we will first see howandupdate the dependencies and then we will present our new approach in Section <ref>. §.§ Algorithm by Kourtellis et al. () In addition to d and σ,keeps track of the old dependencies δ_s ∙(v) ∀ s, v ∈ V. The dependency update is done in a way similar to(see Section <ref>). Also in this case, nodes v are processed in decreasing order of their new distance d'(s, v) from s (otherwise it would not be possible to apply Eq. (<ref>)). However, in this case we would only like to process nodes for which the dependency has actually changed. To do this, while still making sure that the nodes are processed in the right order,replaces the stack used inwith a bucket list. Every node that is traversed during the APSP update is inserted into the bucket list in a position equal to its new distance from s. Then, nodes are extracted from the bucket list starting from the ones with maximum distance. Every time a node v is extracted, we compute its new dependency as δ'_s ∙(v) = ∑_w : v ∈ P'_s(w)σ'_sv/σ'_sw (1 + δ'_s ∙(w)). Since we are processing the nodes in order of decreasing new distance, we can be sure that δ'_s ∙(v) is computed correctly. The score of v is then updated by adding the new dependency δ'_s ∙(v) and subtracting the old δ_s ∙(v), which was previously stored. Also, all neighbors y ∈ P'_s(v) that are not in the bucket list yet are inserted at level d'(s, y) = d'(s, v) - 1. Notice that, in the example in Figure <ref>, all the nodes in the sub-DAG of t are necessary to compute the new dependency of t, although they have not been affected by the insertion. This is why they are traversed during the APSP update. §.§ Algorithm by Kas et al. ()does not store dependencies. On the contrary, for every node pair (s, t) for which either d(s, t) or σ_st has been affected by the insertion, all the nodes in the new shortest paths and the ones in the old shortest paths between s and t are processed.More specifically, starting from t, all the nodes y ∈ P'_s(t) are inserted into a queue. When a node y is extracted, we increase its betweenness by σ'(s, y) ·σ' (y, t) / σ'(s, t) (i.e. the fraction of shortest paths between s and t going through y). Then, also y enqueues all nodes in P'_s(y) and the process is repeated until we reach s. Decreasing the betweenness of nodes in the old paths is done in a similar fashion, with the only difference that nodes in P_s(y) are enqueued (instead of nodes in P'_s(y)) and that σ(s, y) ·σ (y, t) / σ(s, t) is subtracted from the scores of processed nodes. Notice that the worst-case complexity of this approach is O(n^3), whereas that ofis O(nm). This cubic running time is due to the fact that, for each affected node pair (s, t) (at most Θ(n^2)), there could be up to Θ(n) nodes lying in either one of the old or new shortest paths between s and t. (In the running time analysis of <cit.>, this is represented by the term |σ_old| I.) This means that, if many nodes are affected,can even be slower than recomputation with . On the other hand, we have seen in Section <ref> thatalso processes nodes for which the betweenness has not changed (see Figure <ref> and its explaination), which in some cases might result in a higher running time than .§.§ Faster betweenness update We propose a new approach for updating the betweenness scores. As , we do not store the old dependencies (resulting in a lower memory requirement) and we only process the nodes whose betweenness has actually been affected. However, we do this by accumulating contributions of nodes only once for each affected source, in a fashion similar to .For an affected source s ∈ S and for any node v ∈ V, let us define Δ_s, ∙(v) as ∑_t ∈ T(s)σ_st(v)/σ_st. This is the contribution of nodes whose old shortest paths from s went through v, but which have been affected by the edge insertion. Analogously, we can define Δ'_s, ∙(v) as ∑_t ∈ T(s)σ'_st(v)/σ'_st. Then, the new dependency δ'_s, ∙ (v) can be expressed as:δ'_s, ∙ (v) = δ_s, ∙ (v) - Δ_s, ∙(v) + Δ'_s, ∙(v)Notice that for all nodes t ∉ T(s), σ'_st = σ_st and σ'_st(v) = σ_st(v), therefore their contribution to δ_s, ∙ (v) is not affected by the edge update. The new betweenness c'_B(v) can then be computed as c_B(v) - ∑_s ∈ SΔ_s, ∙(v) +∑_s ∈ SΔ'_s, ∙(v). The following theorem allows us to compute Δ_s, ∙(v) and Δ'_s, ∙(v) efficiently.For any s ∈ T, v ∈ V:Δ_s, ∙(v)= ∑_w : v ∈ P_s(w) ∧ w ∈ T(s)σ_sv/σ_sw (1 + Δ_s, ∙(w)) + ∑_w : v ∈ P_s(w) ∧ w ∉ T(s)σ_sv/σ_sw·Δ_s, ∙(w) .Similarly:Δ'_s, ∙(v)= ∑_w : v ∈ P'_s(w) ∧ w ∈ T(s)σ'_sv/σ'_sw (1 + Δ'_s, ∙(w)) + ∑_w : v ∈ P'_s(w) ∧ w ∉ T(s)σ'_sv/σ'_sw·Δ'_s, ∙(w) .We prove only the equation for Δ_s, ∙(v), the one for Δ'_s, ∙(v) can be proven analogously. Let t be any node in T(s), t ≠ v. Then, σ_st(v)/σ_st can be rewritten as ∑_w : v ∈ P_s(w)σ_st(v, w)/σ_st, where σ_st(v, w) is the number of shortest paths between s and t going through both v and w. Then: Δ_s, ∙(v) = ∑_t ∈ T(s)σ_st(v)/σ_st = ∑_t ∈ T(s)∑_w : v ∈ P_s(w)σ_st(v, w)/σ_st = ∑_w : v ∈ P_s(w)∑_t ∈ T(s)σ_st(v, w)/σ_st . Now, of the σ_sw paths from s to w, there are σ_sv many that also go through v. Therefore, for t ≠ w, there are σ_sv/σ_sw·σ_st(w) shortest paths from s to t containing both v and w, i.e. σ_st(v, w) = σ_sv/σ_sw·σ_st(w). On the other hand, if t = w, σ_st(v, w) is simply σ_sv. Therefore, we can rewrite the equation above as: ∑_w : v ∈ P_s(w) ∧ w ∈ T(s){σ_sv/σ_sw + ∑_t ∈ T(s) - {w}σ_st(v, w)/σ_st}+ ∑_w : v ∈ P_s(w) ∧ w ∉ T(s)∑_t ∈ T(s)σ_st(v, w)/σ_st= ∑_w : v ∈ P_s(w) ∧ w ∈ T(s)σ_sv/σ_sw(1 + ∑_t ∈ T(s) - {w}σ_st(w)/σ_st)+ ∑_w : v ∈ P_s(w) ∧ w ∉ T(s)σ_sv/σ_sw∑_t ∈ T(s)σ_st(w)/σ_st= ∑_w : v ∈ P_s(w) ∧ w ∈ T(s)σ_sv/σ_sw (1 + Δ_s, ∙(w)) + ∑_w : v ∈ P_s(w) ∧ w ∉ T(s)σ_sv/σ_sw·Δ_s, ∙(w) .Theorem <ref> allows us to accumulate the dependency changes in a way similar to . To compute Δ_s, ∙, we need to process nodes in decreasing order of d(s, ·), whereas to compute Δ'_s, ∙ we need to process them in decreasing order of d'(s, ·). To do this, we use two priority queues PQ_s and PQ'_s (if the graph is unweighted, we can use bucket lists as the ones used in ). Notice that nodes w such that σ_st(w) = 0 ∧σ'_st(w) = 0∀ t ∈ T(s) do not need to be added to the queue. PQ_s and PQ'_s are filled with all nodes in T(s) during the APSP update in Algorithm <ref>. In PQ_s, nodes w are inserted with priority d(s, w) and PQ'_s with priority d'(s, w). Algorithm <ref> shows how we decrease betweenness of nodes that lied in old shortest paths from s (notice that this is repeated for each s ∈ S(v)). In Lines <ref> - <ref>, Theorem <ref> is applied to compute Δ_s, ∙(y) for each predecessor y of w. Then, y is also enqueued and this is repeated until PQ_s is empty (i.e. when we reach s). The betweenness update of nodes in the new shortest paths works in a very similar way. The only difference is that PQ'_s is used instead of PQ, that d' and σ ' are used instead of d and σ and that Δ'_s, ∙ is added to c_B and not subtracted in Line <ref>. At the end of the update, σ is set to σ' and d is set to d'.In undirected graphs, we can notice that ∑_s ∈ S(w)Δ_s, ∙(w) = ∑_t ∈ T(w)Δ_t, ∙(w). Thus, to account also for the changes in the shortest paths between w and the nodes in T(w), 2 Δ_s, ∙ is subtracted from c_B(w) in Line <ref> (and analogously 2 Δ'_s, ∙ is added in the update of nodes in the new shortest paths). § TIME COMPLEXITY Let us study the complexity of our two new algorithms for updating APSP and betweenness scores described in Section <ref> and Section <ref>, respectively.We define the extended size ||A|| of a set of nodes A as the sum of the number of nodes in A and the number of edges that have a node of A as their endpoint. Then, the following holds.The running time of Algorithm <ref> for updating the augmented APSP after an edge insertion (or weight decrease) (u, v, ω'(u,v)) is Θ(||S(v)|| + ||T(u)|| + ∑_y ∈ T(u) |S(p(y))|), where p(y) can be any node in P_u(y).The functionin Line <ref> identifies the set of affected sources starting a BFS in v and visiting only the nodes s ∈ S(v). This takes Θ(||S(v)||), since this pruned BFS visits all nodes in S(v) and their incident edges.Then, the while loop of Lines <ref> - <ref> identifies all the affected targets T(u) with a pruned BFS. This part (excluding Lines <ref> - <ref>) requires Θ(||T(u)||) operations, since all affected targets and their incident edges are visited.In Lines <ref> - <ref>, for each affected node t ∈ T(u), all the affected sources of the predecessor p(y) of y are scanned. This part requires in total Θ(∑_t ∈ T(u) |S(p(y))|) operations.Notice that, since |S(p(y))| is O(n) and both ||T(u)|| and ||S(v)|| are O(n+m), the worst-case complexity of Algorithm <ref> is O(n^2).To show the complexity of the dependency update described in Algorithm <ref>, let us introduce, for a given source node s, the set τ(s) := T(s) ∪{w ∈ V : Δ_s, ∙(w) > 0 }. Then, the following theorem holds. The running time of Algorithm <ref> is Θ(|| τ(s) || + |τ(s)| log |τ(s)|) for weighted graphs and Θ(|| τ(s) ||) for unweighted graphs. In the following, we assume a binary heap priority queue for weighted graphs and a bucket list priority queue for unweighted graphs.Then, theoperation in Line <ref> requires constant time for unweighted and logarithmic time for weighted graphs. Also, for each node extracted from PQ, all neighbors are visited in Lines <ref> - <ref>. Therefore, it is sufficient to prove that the set of nodes inserted into (and therefore extracted from) PQ is exactly τ(s). As we said in the description of Algorithm <ref>, PQ is initially populated with the nodes in T(s). Then, all nodes y inserted into PQ in Line <ref> are nodes that lied in at least one shortest path between s and a node in T(s) before the insertion. This means that there is at least one t ∈ T(s) such that σ_st(y) > 0, which implies that Δ_s, ∙(y) > 0, by definition of Δ_s, ∙(y). The running time necessary to increase the betweenness score of nodes such that Δ'_s, ∙ > 0 can be computed analogously, defining τ'(s) = T(s) ∪{w ∈ V : Δ'_s, ∙(w) > 0 }. Overall, the running time of the betweenness update score described in Section <ref> is Θ(∑_s ∈ S ||τ(s) || + ||τ'(s)||) for unweighted and Θ(∑_s ∈ S ||τ(s) || + ||τ'(s)|| +|τ(s)| log |τ(s)| +|τ '(s)| log |τ '(s)|) for weighted graphs. Consequently, in the worst case, this is O(nm) for unweighted and O(n(m + n log n)) for weighted graphs, which matches the running time of . For sparse graphs, this is asymptotically faster than , which requires Θ(n^3) operations in the worst case. § EXPERIMENTAL RESULTS *Implementation and settingsFor our experiments, we implemented , , , and our new approach, which we refer to as(from Incremental Betweenness). All the algorithms were implemented in C++, building on the open-source NetworKit framework <cit.>.All codes are sequential; they were executed on a 64bit machine with 2 x 8 Intel(R) Xeon(R) E5-2680 cores at 2.7 GHz with 256 GB RAM with a single thread on a single CPU.*Data sets and experimental design For our experiments, we consider a set of real-world networks belonging to different domains, taken from SNAP <cit.>, KONECT <cit.>, and LASAGNE (<piluc.dsi.unifi.it/lasagne >). Sincecannot handle weighted graphs and the pseudocode given in <cit.> is only for undirected graphs, all graphs used in the experiments are undirected and unweighted.The networks are reported in Table <ref>. Due to the time required by the static algorithm and the memory constraints of all dynamic algorithms (Θ(n^2)), we only considered networks with up to about 26000 nodes.To simulate real edge insertions, we remove an existing edge from the graph (chosen uniformly at random), compute betweenness on the graph without the edge and then re-insert the edge, updating betweenness with the incremental algorithms (and recomputing it with ).For all networks, we consider 100 edge insertions and report the average over these 100 runs. *Experimental results In Table <ref> the running times offor each graph and the speedups of the three incremental algorithms onare reported. The last line shows the geometric mean of the speedups onover all tested networks.Our new methodclearly outperforms the other two approaches and is always faster than both of them. On average,is faster thanby a factor 179.1, whereasby a factor 13.0 andby a factor 22.9.Figure <ref> compares the APSP update (on the left) and dependency update (on the right) steps for thegraph (a similar behavior was observed also for the other graphs of Table <ref>.On the left, the running time of the APSP update phase of the three incremental algorithms on 100 edge insertions are reported, sorted by the running time taken by . It is clear that the APSP update ofis always faster than the competitors. This is due to the fact thatprocesses the edges between the affected targets only once instead of doing it once for each affected source as bothand . Also, the running time of the APSP update ofvaries significantly. On about one third of the updates, it is basically as fast as . This means that in these cases,only visits a small amount of nodes in addition to the affected ones (see Figure <ref> and its explanation). However, in other casescan be much slower, as shown in the figure.On the right of Figure <ref>, the running times of the dependency update step are reported. Also for this step,is faster than bothand . However, for this part there is not a clear winner betweenand . In fact, in some casesneeds to process additional nodes in order to recompute dependencies, whereasonly processes nodes in the shortest paths between affected nodes. However,processes each node at most once for each source node s, whereas might process the same node several times if it lies in several shortest paths between s and other nodes (we recall that the worst-case running time ofis O(n^3), whereas that ofis O(n m)). Notice also that in some rare casesis slightly faster thanin the dependency update. This is probably due to the fact that our implementation ofis based on a priority queue, whereason a bucket list.Figure <ref> on the left reports the total running times of , ,andon . Although the running times vary significantly among the updates,is always the fastest among all algorithms. On the contrary, there is not always a clear winner betweenand . On the right, Figure <ref> shows the geometric mean of the speedups on recomputation for the three incremental algorithms, considering the complete update, the APSP update step only and the dependency update step only, respectively.is the method with the highest speedup both overall and on the APSP update and dependency update steps separately, meaning that each of the improvements described in Section <ref> and Section <ref> contribute to the final speedup. On average,is a factor 82.7 faster thanand a factor 28.5 faster thanon the APSP update step and it is a factor 9.4 faster thanand a factor 4.9 faster thanon the dependency update step. Overall, the speedup ofonranges from 6.6 to 29.7 and is on average (geometric mean of the speedups) 14.7 times faster. The average speedup onis 7.4, ranging from a factor 4.1 to a factor 16.0. § CONCLUSIONS AND FUTURE WORKComputing betweenness centrality is a problem of great practical relevance. In this paper we have proposed and evaluated new techniques for the betweenness update after the insertion (or weight decrease) of an edge. Compared to other approaches, our new algorithm is easy to implement and significantly reduces the number of operations of both the APSP update and the dependency update. Our experiments on real-world networks show that our approach outperforms existing methods, on average approximately by one order of magnitude.Future work might include parallelization for further acceleration. Furthermore, we plan to extend our techniques also to the decremental case (where an edge can be deleted from the graph or its weight can be increased) and to batch updates, where several edge updates might occur at the same time.Although dynamic betweenness algorithms can be much faster than recomputation, a major limitation for their scalability is their memory requirement of Θ(n^2). An interesting research direction is the design of scalable dynamic algorithms with a smaller memory footprint.Our implementations are based on NetworKit <cit.>, the open-source framework for network analysis, and we will publish our source code in upcoming releases of the package. abbrv | http://arxiv.org/abs/1704.08592v1 | {
"authors": [
"Elisabetta Bergamini",
"Henning Meyerhenke",
"Mark Ortmann",
"Arie Slobbe"
],
"categories": [
"cs.DS"
],
"primary_category": "cs.DS",
"published": "20170427142119",
"title": "Faster Betweenness Centrality Updates in Evolving Networks"
} |
In this article I will review a couple of recent results on the inclusion of higher-order QCD corrections to the Monte Carlo simulation of final states involving top-pair and heavy electroweak bosons. In section <ref> I will focus on the recent progress achieved in the matching of QCD NLO corrections with parton shower simulations (NLO+PS) for the process pp→ W^+ W^- bb̅, whereas in section <ref> I will discuss the NLO+PS merging of W^+W^- and W^+W^-+1 jet production using themethod.[Unless otherwise stated, throughout this document, we indicate with “W” the lepton-neutrino final-state pair arising from a W bosons, i.e. W bosons are treated as unstable, have a finite width and they decay leptonically.]§ TOP-PAIR PRODUCTIONIt is known that having an accurate simulation of the process pp→ W^+W^-bb̅ is important for several reasons at the LHC, for instance to measure the top quark mass, or to have an unified treatment of tt̅ and the so-called “single-top Wt” production.At fixed order in QCD, W^+W^-bb̅ hadronic production is very well known and much studied. Despite the fully differential cross section has been known for several years at NLO <cit.>, with the exception of a first study appeared in <cit.>, Monte Carlo NLO+PS event generators addressing all the issues related to the complete simulation of this final state started to be available only recently in theapproach <cit.>. In the rest of this section I'll focus on these issues (and solutions thereof) within theapproach, although substantial work in this direction is also pursued within thematching scheme, and complete results for single-top t-channel production in the 4-flavour scheme were published in ref. <cit.>.The problem with the simulation of W^+W^-bb̅ production and the inclusion of finite width effects can be stated as follows: unless special care is taken, the intermediate top-quark virtuality is not preserved among different parts of the computation, leading to the evaluation of matrix elements at phase space points which have different top virtualities. When this happens, three problems will in general occur: * at the level of computing NLO corrections with a subtraction method, the cancellation of collinear singularities associated to gluon emission off final-state b-quarks can become delicate, eventually failing when approaching the narrow-width limit.* when the hardest radiation is generated in , the phase-space region associated to final-state gluon emission off the b-quark is handled by a mapping that, in general, does not preserve the virtuality of the intermediate resonance. Unless m^2_bg≪Γ_t E_bg, real and Born matrix elements (R and B, respectively) will not be on the resonance peak at the same time, hence the ratio R/B in theSudakov can become large when R is on peak and B is not, yielding a spurious “Sudakov suppression”.* further problems can arise during the parton-showering stage: from the second emission onward, the shower should be instructed to preserve the mass of the resonances. This could be done easily if there was an unique mechanism to “assign” the radiation to a given resonance. For processes where interference(s) is(are) present, no obvious choice is possible. An intermediate solution to the previous issues was presented in ref. <cit.>, where a fully consistent NLO+PS simulation for W^+W^-bb̅ production was obtained in the narrow-width limit, and off-shellenss and interference effects were implemented in an approximate way. I refer to the original paper, or to the review <cit.>, for more details. Here it suffices to say that, by using the narrow-with approximation to compute NLO corrections, production and decay can be clearly separated (no interference arises), thereby allowing a non-ambiguous “resonance assignment” for final-state radiation, as well as the use of an improved (“resonance aware”) subtraction method, where radiation in the decay is generated by first boosting momenta in the resonance rest-frame. In this way, B and R are always evaluated with the same virtuality for the intermediate resonance, so that the subtraction can be safely performed, and no spurious Sudakov suppression can arise.More recently, a general solution to include off-shellness and interference effects in theapproach was proposed in ref. <cit.>, and later applied to the W^+W^-bb̅ process in ref. <cit.>, where matrix elements were obtained using OpenLoops <cit.>.The main new concept introduced in <cit.> is that one separates all contributions to the cross section into terms with definite resonance structure, i.e. each term should only have peaks associated to a given resonance structure (“resonance history”). For W^+W^-bb̅ production, for instance, one has two types of resonance histories: one where at least one s-channel top-propagator appears (this includes both doubly- and single-resonant contributions) and another associated to non-resonant production. By means of projectors Π_f_b built by combining Breit-Wigner like functions P^f_b, a partition of the unit can be constructed, so that a given Born (and virtual) partonic subprocess B can be separated into contributions B_f_b that are, individually, dominated by one and only one resonance history (labeled by f_b):[For simplicity we suppressed the labels F_b and F_r used in <cit.>, which represent the “bare” structure, i.e. the flavour of external partons of Born and real matrix elements. Moreover in <cit.> the symbols f_b and f_r represent a “full” structure, since they label a given resonance history related to a given set of external partons, i.e. they also contain the explicit information on the external partons, which we are suppressing in this document.] B() = ∑_f_b B_f_b() ≡∑_f_bΠ_f_b() B(),Π_f_b() = P^f_b()/∑_f'_bP^f'_b() . Since each Born matrix element is separated according to resonance histories, one needs to set-up a similar mechanism for real matrix elements, such that, eventually, each projected real matrix element can be associated to an unique resonance history, with a counterpart in the corresponding list of Born's ones. As usual, real matrix elements also need be separated according to their collinear singularities: to this end, one requires that a collinear region α_r is admitted only if the two collinear partons both arise either from the same resonance, or from the hard interaction. This separation is achieved schematically as R = ∑_α_r R_α_r,R_α_r = P^f_r d^-1(α_r)/∑_f'_r(P^f'_r∑_α'_rd^-1(α'_r)) R . In eq. (<ref>), f_r denotes a given resonance history assignment for R, d(α_r)→ 0 when the collinear region α_r is approached, and the sums in the denominator run only on the possible resonance histories f'_r present in R, and on the compatible singular regions α'_r associated to a given f'_r: hencea given R_α_r becomes dominant only if the collinear partons of region α_r have the smallest k_t and the corresponding resonance history f_r is the closest to its mass shell.The above prescriptions allowed to build agenerator able to simulate processes with intermediate resonances, keeping all finite-width effects and interferences. In fact, having separated each contribution as explained above, for the singular regions associated to a radiation in a resonance decay, it becomes now possible to safely use the “resonance-aware” subtraction method developed in <cit.>, thereby avoiding the mismatches mentioned at the beginning of this section. Similarly, because an index α_r is naturally associated to the hardest radiation generated by , it's possible to unambiguously assign the radiation to a given resonance, preventing the parton shower to distort the mass of the resonances.[I want to mention that another technical but crucial issue was addressed in <cit.>, related to the computation of soft-collinear contributions to be added to to the virtual terms. In <cit.>, these terms were computed independently for production and for each radiating resonance decay, and in different frames. This posed no problem, because in the narrow-width limit no interferences are present. When interferences are present, this is clearly no longer possible, and a substantial generalization of the subtraction scheme adopted in POWHEG was worked out in <cit.>, leading to the development of a new framework, dubbed POWHEG-BOX-RES.]The left panel of Fig. <ref>shows the differences in shape of the reconstructed top peak obtained using three tools, namely the new generator of ref. <cit.> (), the resonance-improved generator based on an approximate treatment of off-shell effects <cit.> (), and the original generator <cit.> based on on-shell NLO matrix elements for tt̅ production (). As expected, the “” and “” generators are fairly consistent, especially close to the resonance peak, whereas the “” generator shows larger deviations. The p_T spectrum of the hardest b-jet is instead shown in the right panel of fig. <ref>, without imposing any particular cuts. The shape difference at small p_T can be attributed to the fact that the “Wt” contribution is missing (approximate) in the “” (“”) generator, whereas it's fully taken into account in the “” one.§ W-BOSON PAIR PRODUCTION The study of vector boson pair-production is central for the LHC Physics program. Not only is W^+W^- production measured to access anomalous gauge couplings, but it's also an important background for several searches, notably for those where the H→ W^+W^- decay is present. For these and other similar reasons, it is important to have flexible and fully realistic theoretical predictions that allow to simultaneously model, with high accuracy, the production of W^+W^-, inclusively as well as in presence of jets. The methods aiming at this task are usually referred to as “NLO+PS merging”. NLO+PS merging for pp→ VV+jet(s) was achieved using the MEPS@NLO <cit.> and FxFx <cit.> methods. In this section, I'll review theformalism and show how it was used to merge at NLO the processes pp→ W^+W^- and pp→ W^+W^-+ jet <cit.>.The(Multi-scale Improved NLO) procedure <cit.> was originally introduced as a prescription to a-priori choose the renormalization (μ_R) and factorization (μ_F) scales in multileg NLO computations: since these computations can probe kinematic regimes involving several different scales, the choice of μ_R and μ_F is indeed ambiguous, and themethod addresses this issue by consistently including CKKW-like corrections <cit.> into a standard NLO computation. In practice this is achieved by associating a “most-probable” branching history to each kinematic configuration, through which it becomes possible to evaluate the couplings at the branching scales, as well as to include () Sudakov form factors (FF). This prescription regularizes the NLO computation also in the regions where jets become unresolved, hence theprocedure can be used within theformalism to regulate the B̅ function for processes involving jets at LO.In a single equation, for a qq̅-induced process as W^+W^- production, the -improvedB̅ function reads:B̅_WWJ-MiNLO =(q_T) Δ^2_q(q_T,M_X)[ B ( 1-2Δ^(1)_q(q_T,M_X) ) + V(μ̅_R) +∫ d R ] ,where X is the color-singlet system (WW in this case), q_T is its transverse momentum, μ̅_R is set to q_T, and Δ_q(q_T,Q)=exp{-∫_q^2_T^Q^2dq^2/q^2(q^2)/2π[ A_qlogQ^2/q^2 + B_q]} is theSudakov FF associated to the jet present at LO. Convolutions with PDFs are understood, B is the leading-order matrix element for the process pp→ X+1 jet (stripped off of the strong coupling), and Δ_q^(1)(q_T,Q) (the 𝒪 () expansion of Δ_q) is removed to avoid double counting. We also notice that B̅_WWJ-MiNLO is a function of Φ_X+j, i.e. the phase space to produce the X system and an extra parton, which can be arbitrarily soft and/or collinear.In ref. <cit.> it was also realized that, if X is a color singlet, upon integration over the full phase space for the leading jet, one can formally recover NLO+PS accuracy for the process pp→ X by properly applyingto NLO+PS simulations for processes of the type pp→ X+1 jet.[The idea has been generalized recently in ref. <cit.>.] Besides setting μ_F and μ_R equal to q_T in all their occurrences, the key point is to include at least part of the Next-to-Next-to-Leading Logarithmic (NNLL) corrections into theSudakov form factor, namely the B_2 term: by omitting it, the full integral of eq. (<ref>) over Φ_X+j, albeit finite, differs from σ_pp→ X^NLO by a relative amount (M_X)^3/2, thereby hampering a claim of NLO accuracy.The B_2 coefficient is process-dependent, and formally also a function of Φ_X, because part of it stems from the 1-loop correction to the pp→ X process. For Higgs, Drell-Yan, and VH production, these 1-loop corrections can be expressed as form factors: B_2 becomes just a number as its dependence upon Φ_X disappears, and the analogous of eq. (<ref>) can be easily implemented <cit.>. For diboson production, the situation is more delicate. First, extracting B_2 for the WW case is more subtle, as the virtual corrections to the pp→ WW process don't factorize on the Born squared amplitude, hence B_2=B_2(Φ_X). As a consequence, a mismatch between different phase spaces becomes apparent, because in eq. (<ref>) B̅_WWJ-MiNLO depends upon Φ_X+j, whereas B_2 needs to be computed as a function of Φ_X. In ref. <cit.> these two issues were handled as follows: * to compute B_2 we started from the relatively simple expression used for the Drell-Yan case, and replaced its process-dependent part [V/B]^ DY=C_F(π^2-8) with the corresponding term for WW production: [V/B]^WW(Φ_WW)=V^WW(Φ_WW)/B^WW(Φ_WW).* in order to evaluate B_2, we defined on an event-by-event basis a projection of the WW+1 jet state onto a WW one, using the FKS mapping relevant for initial-state radiation as implemented in the POWHEG BOX <cit.>. For real emission events, a similar mapping was used. In all cases, in the q_T→ 0 limit, the effect of these projections on the final state kinematics smoothly vanishes, making sure that the precise numerical determination of B_2 is affected only beyond the required accuracy. In ref. <cit.> we have built agenerator for the pp→ W^+W^-+1 jet process, and upgraded it with , according to the aforementioned procedure.We worked in the 4-flavour scheme, including exactly the vector bosons' decay products, as well as finite-width effects and single-resonant contributions. Tree-level matrix elements were obtained with an interface to MadGraph 4 <cit.>, whereas one loop corrections were computed with GoSam 2.0 <cit.>.The left panel of Fig. <ref>shows the transverse momentum spectrum of the WW system as obtained with the WWJ-MiNLO generator against the one obtained with the originalgenerator for pp→ W^+W^- <cit.>. The importance of NLO corrections is manifest in the high-p_T tail, whereas the differences at small p_T can be attributed to the differences among the POWHEG and the MiNLO Sudakovs.The right panel shows instead a comparison for the leading jet p_T spectrum between the parton-level computation pp→ WW+1 jet at NLO, and theresult. 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Cullen et al.,Eur. Phys. J. C 74, no. 8, 3001 (2014) Melia:2011tjT. Melia, P. Nason, R. Rontsch and G. Zanderighi,JHEP 1111, 078 (2011) Alioli:2016xabS. Alioli, F. Caola, G. Luisoni and R. Röntsch,Phys. Rev. D 95, no. 3, 034042 (2017) Grazzini:2016ctrM. Grazzini, S. Kallweit, S. Pozzorini, D. Rathlev and M. Wiesemann,JHEP 1608, 140 (2016) | http://arxiv.org/abs/1704.08577v1 | {
"authors": [
"Emanuele Re"
],
"categories": [
"hep-ph"
],
"primary_category": "hep-ph",
"published": "20170427140242",
"title": "Latest developments in the simulation of final states involving top-pair and heavy bosons"
} |
We prove the existence of global, smooth solutions to the 2D Muskat problem in the stable regime whenever the product of the maximal and minimal slope is less than 1.The curvature of these solutions solutions decays to 0 as t goes to infinity, and they are unique when the initial data is C^1,ϵ.We do this by getting a priori estimates using a nonlinear maximum principle first introduced in <cit.>, where the authors proved global well-posedness for the surface quasi-geostraphic equation. Black hole entropy emission property Shao-Wen Wei [[email protected]], Yu-Xiao Liu [[email protected]]============================================================================ § INTRODUCTION The Muskat problem was originally introduced by Muskat in <cit.> in order to model the interface between water and oil in tar sands.In general, it describes the interface between two incompressible, immiscible fluids of different constant densities in a porous media.The fluids evolve according to Darcy's law, giving an evolution of the interface (see <cit.> for derivation of equations), and in 2D is analogous to the two phase Hele-Shaw cell (see <cit.>).In the case that the two fluids are of equal viscosity and the interface is given by the graph y=f(t,x) with the denser fluid on bottom (i.e. the stable regime), the function f satisfiesf_t(t,x) = ∫_(f_x(t,y) - f_x(t,x))(y-x)/(f(t,y)-f(t,x))^2 + (y-x)^2 dy,after the appropriate renormalization.By making a change of variables, (see the proof of Lemma 5.1 of <cit.>) we get the equivalent systemf_t(t,x) = ∫_f(t,y)-f(t,x) - (y-x)f_x(t,x)/(f(t,y)-f(t,x))^2 + (y-x)^2 dy,which will be more useful for our purposes.Since the function f is Lipschitz, the above integral can be viewed as a nonlinear perturbation of the half Laplacian.In fact, it is easy to see that linearizing around a flat solution givesf_t(t,x) = -c(-Δ)^1/2f(t,x),demonstrating the natural parabolicity of the problem.The Muskat problem is known to be locally well-posed in H^k for k≥ 3 with solutions satisfying L^∞ and L^2 maximum principles, but neither imply any gain of derivatives (see <cit.>, <cit.>).Under the assumption ||f_0'||_L^∞<1, there have been a number of positive results.In <cit.> the authors prove an L^∞ maximal principle for the slope f_x along with the existence of global weak Lipschitz solutions using a regularized system. Recently, <cit.> improved the L^2 energy estimate of <cit.> (which holds for any solution) to one analogous with the energy estimate from the linear equation under this assumption on the slope.When the initial data f_0∈ H^2() with ||f_0||_1 = || |ξ| f̂_0(ξ)||_L_ξ^1 less than some explicit constant ≈ 1/3 (which implies slope less than 1), <cit.> proves that a unique global strong solution exists.In this case <cit.> proves optimal decay estimates on the norms ||f(t,·)||_s = || |ξ|^s f̂(t,ξ)||_L_ξ^1, matching the estimates for the linear equation.Recently, <cit.> was also able to prove the existence of global weak solutions for arbitrarily large monotonic initial data.They did this using the regularized system from <cit.> to prove that both f and f_x still obey the maximum principle under this monotonicity assumption. Because solutions to (<ref>) have the natural scaling 1/rf(rt,rx), we see that L^∞ or sign bounds on the slope f_x are scale invariant properties.We fit these two types of assumptions into the same framework by showing that the critical quantity is in fact the product of the maximal and minimal slopes,β(f_0'):= (sup_x f_0'(x))(sup_y -f_0'(y)).As we shall see in section 3, the derivative f_x obeys the equation(f_x)_t(t,x) = f_xx(t,x) ∫_-h/δ_hf(t,x)^2+h^2 dh + ∫_δ_hf_x(t,x) K(t,x,h) dh.where δ_hf(t,x):=f(t,x+h)-f(t,x) and the kernel K is uniformly elliptic of order 1 whenever β(f_0')<1.Thus we naturally get regularizing effects from the equation whenever the initial data satisfies this bound.It's clear that ||f_0'||_L^∞<1 implies β(f_0')<1, and for bounded monotonic data we get that β(f_0')=0 since either sup f_0'=0 or inf f_0' =0.Thus this β(f_0')<1 provides a natural interpolation between these two types of assumptions.In contrast to the positive results, <cit.> shows that there is an open subset of initial data in H^4 such that the Rayleigh-Taylor condition breaks down in finite time.That is, lim_t→ t_0- ||f_x(t,·)||_L^∞ = ∞ for some time t_0, after which the interface between the fluids can no longer be described by a graph. The authors of <cit.> made great progress towards proving global regularity.They proved that if the initial data f_0∈ H^k, then the solution f will exist and remain in H^k so long as the slope f_x(t,·) remains bounded and uniformly continuous.Thus the natural next step is to prove the generation of a modulus of continuity for f_x, henceLet f_0∈ W^1,∞() withβ(f_0'):=(sup_x f_0'(x))(sup_y -f_0'(y))<1.Then there exists a classical solutionf∈ C([0,∞)×)∩ C^1,α_loc((0 ∞)×)∩ L^∞_loc((0,∞);C^1,1),to (<ref>) with f_x satisfying both the maximum principle andf_x(t,x)-f_x(t,y) ≤ρ(|x-y|/t),t>0, x≠ y ∈,for some Lipschitz modulus of continuity ρ depending solely on β(f_0'),||f_0'||_L^∞.In the case that f_0∈ C^1,ϵ() for some ϵ>0, then the solution f is unique with f∈ L^∞([0,∞);C^1,ϵ).The uniqueness statement follows essentially from the uniqueness theorem of <cit.>.We note in the appendix the few small changes needed to their proof in order to apply it here. The most vital part of Theorem <ref> is the spontaneous generation of the modulus ρ(·/t), as everything else will follow from that.The spontaneous generation/propogation of a general modulus of continuity has old roots as classical Holder estimates, but its only recently that the idea to tailor make moduli for specific equations emerged.The technique first appeared in <cit.>, where the authors used it to prove global well-posedness for the surface quasi-geostraphic equation.It has had great success at proving regularity for a number of active scalar equations, that is equations of the formθ_t + (u·∇)θ + ℒθ=0,where u is a flow depending on θ and ℒ is some diffusive operator.See <cit.>, <cit.> for a good overview of results using this method.To date, these tailor made moduli have only been applied to cases where all the nonlinearity has been in the flow velocity u, and the diffusive term ℒ has been rather nice (typically (-Δ)^α, or at least a Fourier multiplier). We will be applying this method to f_x, which solves the active scalar equation (<ref>).Note that in this equation, the kernel K defined in (<ref>) is a highly nonlinear function of f, f_x.Thus this is the first time the method has been applied in a fully nonlinear equation. We prove Theorem <ref> by deriving a priori estimates for smooth solutions to (<ref>) with initial data f_0∈ C^∞_c() depending primarily on β(f_0'),||f_0'||_L^∞.We prove enough estimates that by approximating in W^1,∞_loc with smooth compactly supported initial data, we get solutions f^ϵ which will converge along subsequences in C^1_loc to a solution f solving (<ref>) for arbitrary initial data f_0∈ W^1,∞() with β(f_0')<1. The rest of the paper is organized as follows.We begin by repeating the breakthrough argument of <cit.> in Section 2.In Section 3, we differentiate (<ref>) to derive the equation for f_x, showing that it satisfies the maximum principle when β(f_0')<1.In Section 4, we state how a modulus of continuity ω interacts with the equation in our main technical lemma.In Sections 5 and 6 we then derive the bounds on the drift and diffusion terms necessary to prove that lemma.In Section 7, we apply our main technical lemma to a specific modulus of continuity, and finally in Section 8 we complete the proof of (<ref>) by choosing the correct modulus ρ.In Section 9, we then use (<ref>) to prove a few estimates on regularity in time, guaranteeing enough compactness to prove that there are classical solutions for rough initial data.Finally in the appendix, we give a quick outline for how to modify the uniqueness proof of <cit.> to work for initial data f_0∈ C^1,ϵ() with β(f_0')<1. § BREAKTHROUGH SCENARIO Assume that f_0∈ C^∞_c() with β(f_0')<1, so that there exists a solution f∈ C^1((0,T_+); H^k) for k arbitrarily large and some T_+>0 by <cit.>.Note that under the assumption that β(f_0')<1, we will show that the maximum principle holds (see Section 3 Proposition <ref>) and hence ||f_x||_L^∞([0,T_+)×)≤ ||f_0'||_L^∞ is uniformly bounded.Fix a Lipschitz modulus ρ which we will define later. For sufficiently small times, f_x(t,·) will have modulus ρ(·/t) since it is smooth and bounded.It then follows by the main theorem of <cit.> that as long as f_x(t,·) continues to have modulus ρ(·/t), the solution f will exist with T_+>t. So, we proceed as in <cit.>'s proof for quasi-geostraphic equation.Suppose that f_x(t,·) satisfies (<ref>) for all t<T.Then by continuity,f_x(T,x)-f_x(T,y) ≤ρ(|x-y|/T), ∀ x≠ y∈.We first prove that if we have the strict inequality f_x(T,x)-f_x(T,y) < ρ(|x-y|/T), then f_x(t,·) will have modulus ρ(·/ t) for t≤ T+ϵ. Let f∈ C([0,T_+); C^3_0()), and T∈ (0,T_+).Suppose that f(T,·) satisfiesf_x(T,x)-f_x(T,y) < ρ(|x-y|/T), ∀ x≠ y∈,for some Lipschitz modulus of continuity ρ with ρ”(0) = -∞.Thenf_x(T+ϵ, x) - f_x(T+ϵ, y) < ρ(|x-y| / (T+ϵ)), ∀ x≠ y∈,for all ϵ>0 sufficiently small.To begin, note that for any compact compact subset K⊂^2∖{(x,x)|x∈},f_x(T,x)-f_x(T,y) < ρ(|x-y|/T) ∀ (x,y)∈ K ⇒ f_x(T+ϵ, x)-f_x(T+ϵ,y) < ρ(|x-y|/(T+ϵ)) ∀ (x,y)∈ K,for ϵ >0 sufficiently small by uniform continuity.So, we only need to focus on pairs (x,y) that are either close to the diagonal, or that are large.To handle (x,y) near the diagonal, we start by noting that f(T,·)∈ C^3() and ρ”(0)=-∞.Thus for every x we get that|f_xx(T,x)| < ρ'(0)/T.Since f∈ C([0,T_+); C^3_0()), f_xx(T,x)→ 0 as x→∞.Thus we can take the point where max_x |f_xx(T,x)| is achieved to get that||f_xx(T,·)||_L^∞ < ρ'(0)/T.By continuity of f_xx, we thus have ||f_xx(T+ϵ,·)||_L^∞ < ρ'(0)/T+ϵ for ϵ>0 sufficiently small.Hence,f_x(T+ϵ,x)-f_x(T+ϵ,y) < ρ(|x-y|/T+ϵ),|x-y|<δ,for ϵ, δ sufficiently small.Now let R_1,R_2>0 be such thatρ(R_1/ (T+ϵ)) > _ f_x(T+ϵ,·),and that |x|>R_2 implies|f_x(T+ϵ,x)| < ρ(δ/(T+ϵ))/2,for ϵ>0 sufficiently small. Taking R = R_1+R_2, it's easy to check that |x|>R implies that|f_x(T+ϵ,x) - f_x(T+ϵ,y)| < ρ(|x-y|/(T+ϵ)), ∀ y≠ x. Finally, taking K= { (x,y)∈^2: |x-y|≥δ, x,y∈B_R}, we're done.Thus by the lemma, if f_x was to lose its modulus after time T, we must have that there exist x≠ y∈ withf_x(T,x) - f_x(T,y) = ρ(|x-y|/T).We will show for a smooth solution f of (<ref>) and the correct choice of ρ that in this cased/dt(f_x(t,x) - f_x(t,y))|_t=T < d/dt(ρ(|x-y|/t))|_t=T,contradicting the fact that f_x had modulus ρ(· /t) for time t<T.Thus we just need to prove (<ref>) to complete the proof of the generation of modulus of continuity (<ref>) of Theorem <ref>. § EQUATION FOR F_X So, we just need to prove (<ref>).To begin, we need to examine the equation that f_x solves. Since everything we will be doing is for some fixed time T>0, we will suppress the time variable from now on.Differentiating(<ref>), we see that f_x solves(f_x)_t(x) = f_xx(x)∫_x-y/(f(y)-f(x))^2 + (y-x)^2 dy + ∫_(f(y)-f(x) - (y-x)f_x(x))2((f(y)-f(x))f_x(x)+(y-x))/((f(y)-f(x))^2 + (y-x)^2)^2 dy. To simplify notation, we reparametrize (<ref>) by taking y=x+h, and lettingδ_hf(x):= f(x+h) -f(x),we get(f_x)_t(x) = f_xx(x)∫_-h/(δ_hf(x))^2 + h^2 dh + ∫_(δ_hf(x) - hf_x(x))2(δ_hf(x)f_x(x)+h)/(δ_hf(x)^2 + h^2)^2 dh. Note thatδ_hf(x) - hf_x(x) = ∫_0^h δ_s f_x(x) ds,for h>0, andδ_hf(x) - hf_x(x) = -∫_h^0 δ_s f_x(x) ds,for h<0.With that in mind, definek(x,s) = 2(δ_sf(x)f_x(x)+s)/(δ_sf(x)^2 + s^2)^2,andK(x,h) = {[∫_h^∞ k(x,s)ds,h>0; ∫_-∞^h -k(x,s) ds,h<0 ]. .Then integrating (<ref>) by parts, we have that f_x solves the equation(f_x)_t(x) = f_xx(x) ∫_-h/δ_hf(x)^2+h^2 dh + ∫_δ_hf_x(x) K(x,h) dh. As-β(f_x)/s≤f_x(x)δ_sf(x)/s≤||f_x||_L^∞^2/s,we see that2(1-β(f_x))/(1+||f_x||_L^∞^2)^21/|s|^3≤(s) k(x,s)≤2(1+||f_x||_L^∞^2)/|s|^3,and hence1-β(f_x)/(1+||f_x||_L^∞^2)^21/h^2≤ K(x,h)≤1+||f_x||_L^∞^2/h^2.Thus in the case that β(f_x)≤ 1, we then have that the kernel K is a nonnegative, from which we get immediately (Maximum Principle)Let f_x be a sufficiently smooth solution to (<ref>) with β(f_0')≤ 1. Then for any 0≤ s≤ t, we have thatinf_y f_x(s,y)≤inf_y f_x(t,y)≤sup_y f_x(t,y)≤sup_y f_x(s,y).In particular, since β(f_0')<1 the maximum principle tells us thatβ(f_x)≤β(f_0')<1,||f_x||_L^∞≤ ||f_0'||_L^∞<∞. Thus we get that0<λ/h^2≤ K(x,h) ≤Λ/h^2,whereλ = 1-β(f_0')/(1+||f_0'||_L^∞^2)^2, Λ = 1+||f_0'||_L^∞^2.Thus K is comparable to the kernel for (-Δ)^1/2, so f_x solves the uniformly elliptic equation (<ref>).Note that the sole reason we require β(f_0')<1 is to ensure this ellipticity of K.§ MODULI ESTIMATES Our goal is to show that if f_x(T,·) has modulus ρ(·/T) and equality is achieved at two points (<ref>), then (<ref>) must hold, contradicting the assumptions of the breakthrough argument (see section 2).To that end, we first need to understand how a modulus of continuity interacts with the equation for f_x (<ref>).Hence,Let f: [0,∞)×→ be a bounded smooth solution to (<ref>) with β(f_0')<1, and ω: [0,∞)→ [0,∞) be some fixed modulus of continuity.Assume that at some fixed time T thatδ_hf_x(T,x)≤ω(|h|), f_x(T,ξ/2)-f_x(T,-ξ/2) = ω(ξ),for all h∈, and for some ξ>0.Thend/dt(f_x(t,ξ/2)-f_x(t,-ξ/2))|_t=T≤Aω'(ξ)(∫_0^ξω(h)/hdh + ξ∫_ξ^∞ω(h)/h^2 dh+ ln(M+1)ω(ξ)) +Aω(ξ)∫_Mξ^∞ω(h)/h^2dh +2(Λ-λ)∫_ξ^Mξ(ω(h-ξ) - ω(ξ))_+/h^2dh +2λ∫_0^ξδ_hω(ξ) + δ_-hω(ξ)/h^2dh + 2λ∫_ξ^∞ω(h+ξ)-ω(h)-ω(ξ)/h^2dh,for any M≥ 1, where A depends only on ||f_0'||_L^∞ and λ, Λ are as in (<ref>). This is the main technical lemma that we need.Since solutions to (<ref>) are closed under translation and sign change, it suffices to consider the above situation for our proof of (<ref>).Note that (4.2) holds for any value of the parameter M≥ 1.Later in Lemma 6.1, we will essentially use two different values of M depending on the size of ξ.In the small ξ regime we can simply take M=1, but in the large ξ regime we will need to take M to be a sufficiently large constant depending only on initial data (but not on exact size of ξ) in order to control the size of the error term ω(ξ)∫_Mξ^∞ω(h)/h^2dh. The proof for Lemma <ref> is essentially a nondivergence form argument; our function f_x is touched from above at ξ/2 by our modulus ω, and its touched from below at -ξ/2 by -ω.Specifically,δ_h f_x(ξ/2)≤δ_hω(ξ), ∀ h>-ξ,δ_h f_x(-ξ/2) ≥ -δ_-hω(ξ), ∀ h<ξ.From (<ref>), we want to derive as much information as we can and bound d/dt(f_x(ξ/2)-f_x(-ξ/2)).To that end, by dividing (<ref>) through by h and taking the limit as h→ 0, we then get thatf_xx(ξ/2) = f_xx(-ξ/2) = ω'(ξ). Hence by our equation for f_x (<ref>), we have thatd/dt(f_x(ξ/2) -f_x(-ξ/2)) = ω'(ξ)∫_(-h/δ_hf(ξ/2)^2+h^2 - -h/δ_hf(-ξ/2)^2+h^2)dh+ ∫_δ_hf_x(ξ/2)K(ξ/2,h) - δ_hf_x(-ξ/2)K(-ξ/2,h)dh =ω'(ξ)∫_(-h/δ_hf(ξ/2)^2+h^2 - -h/δ_hf(-ξ/2)^2+h^2)dh +ω'(ξ)∫_-Mξ^Mξ(hK(ξ/2,h) - hK(-ξ/2,h))dh+ ∫_-Mξ^Mξ (δ_hf_x(ξ/2)-hω'(ξ))K(ξ/2,h) - (δ_hf_x(-ξ/2)-hω'(ξ))K(-ξ/2,h)dh+∫_|h|>Mξδ_hf_x(ξ/2)K(ξ/2,h) -δ_hf_x(-ξ/2)K(-ξ/2,h)dh,for any M≥ 1.The first two terms of the RHS of (<ref>) act as a drift, giving rise to the first two error terms of (<ref>).The latter two terms of (<ref>) act as a diffusion, giving rise to both the helpful (negative) terms in (<ref>), as well as additional error terms (the middle terms of (<ref>)) arising from the difference in the kernels, |K(ξ/2,h)-K(-ξ/2,h)|.§ BOUNDS ON DRIFT TERMS We begin proving Lemma <ref> by bounding the drift terms of (<ref>), starting withUnder the assumptions of Lemma <ref>, ω'(ξ)|∫_-h/δ_hf(ξ/2)^2+h^2 - -h/δ_hf(-ξ/2)^2+h^2dh| ≲ω'(ξ)(∫_0^ξω(h)/hdh + ξ∫_ξ^∞ω(h)/h^2dh).We want to bound (<ref>) by symmetrizing the kernels for |h|<ξ, and and then using the continuity in the first variable for |h|>ξ.To that end,ω'(ξ)∫_ (-h/δ_hf(ξ/2)^2+h^2 - -h/δ_hf(-ξ/2)^2+h^2)dh ≤ω'(ξ)∫_0^ξ h|δ_hf(ξ/2)^2 - δ_-hf(ξ/2)^2/(δ_hf(ξ/2)^2+h^2)(δ_-hf(ξ/2)^2+h^2) + δ_hf(-ξ/2)^2 - δ_-hf(-ξ/2)^2/(δ_hf(-ξ/2)^2+h^2)(δ_-hf(-ξ/2)^2+h^2)|dh+ ω'(ξ)∫_|h|>ξ |h||δ_hf(ξ/2)^2 - δ_hf(-ξ/2)^2/(δ_hf(ξ/2)^2+h^2)(δ_hf(-ξ/2)^2+h^2)|dh . We bound the first integral using|δ_h f(x)|≲ |h|, |δ_h f(x)+δ_-hf(x)| = |∫_0^h f_x(x+s)-f_x(x+s-h) ds|≤ω(h)h,Thus get that for 0≤ h<ξ,|δ_hf(x)^2-δ_-hf(x)^2/(δ_hf(x)^2+h^2)(δ_-hf(x)^2+h^2)| ≲ω(h)/h^2,and hence∫_0^ξ h|δ_hf(ξ/2)^2 - δ_-hf(ξ/2)^2/(δ_hf(ξ/2)^2+h^2)(δ_-hf(ξ/2)^2+h^2) dh |≲∫_0^ξω(h)/hdh.For |h|≥ξ, we bound |δ_h f(ξ/2)+δ_hf(-ξ/2)|≲ |h| and|δ_h f(ξ/2)- δ_h f(-ξ/2) |=| ∫_0^h f_x(ξ/2+s)-f_x(-ξ/2+s)ds | = |∫_0^ξf_x(h-ξ/2+s) - f_x(-ξ/2+s)ds|≤ξω(|h|),in order to get∫_|h|>ξ |h||δ_hf(ξ/2)^2-δ_hf(-ξ/2)^2/(δ_hf(ξ/2)^2+h^2)(δ_hf(-ξ/2)^2+h^2)|dh ≲ξ∫_ξ^∞ω(h)/h^2dh. Putting (<ref>) and (<ref>) together, we thus haveω'(ξ)∫_(-h/δ_hf(ξ/2)^2+h^2 - -h/δ_hf(-ξ/2)^2+h^2)dh ≲ω'(ξ)(∫_0^ξω(h)/hdh + ξ∫_ξ^∞ω(h)/h^2dh).That leaves us with the second drift term of (<ref>),Under the assumptions of Lemma <ref>, for any M≥ 1ω'(ξ)|∫_-Mξ^Mξ hK(ξ/2,h) - hK(-ξ/2,h)dh| ≲ω'(ξ)(∫_0^ξω(h)/hdh + ξ∫_ξ^∞ω(h)/h^2dh + ln(M+1)ω(ξ)). To begin, we noteω'(ξ)|∫_-Mξ^Mξ hK(ξ/2,h) - hK(-ξ/2,h)dh | ≤ω'(ξ)∫_0^Mξ h|K(ξ/2,h)-K(ξ/2, -h) - K(-ξ/2,h)+K(-ξ/2,-h)|dh. Recall the definition of K, (<ref>),K(x,h) = {[∫_h^∞ k(x,s)ds,h>0; ∫_-∞^h -k(x,s) ds,h<0 ]. , k(x,s) = 2(δ_sf(x)f_x(x)+s)/(δ_sf(x)^2 + s^2)^2.So, to control (<ref>) we first need to bound |k(x,s)+k(x,-s)| for 0≤ s<ξ, and |k(ξ/2,s)-k(-ξ/2,s)| for |s|>ξ. For the first, using the bounds (<ref>) we see that|k(x,s)+k(x,-s)|= |2(δ_sf(x)f_x(x)+s)/(δ_sf(x)^2 + s^2)^2 + 2(δ_-sf(x)f_x(x)-s)/(δ_-sf(x)^2 + s^2)^2| ≤2|δ_sf(x) + δ_-sf(x)|· |f_x(x)|/(δ_-sf(x)^2 + s^2)^2 + 2|δ_sf(x)f_x(x)+s||(δ_sf(x)^2 + s^2)^2 - (δ_-sf(x)^2 + s^2)^2/(δ_sf(x)^2 + s^2)^2(δ_-sf(x)^2 + s^2)^2| ≲ω(s)/s^3 + s|δ_sf(x)^4-δ_-sf(x)^4 + 2s^2(δ_sf(x)^2-δ_-sf(x)^2)/s^8| ≲ω(s)/s^3. For the second, using (<ref>), (<ref>), and (<ref>) we get that|k(ξ/2,s)-k(-ξ/2,s)|= |2(δ_sf(ξ/2)f_x(ξ/2)+s)/(δ_sf(ξ/2)^2 + s^2)^2 - 2(δ_sf(-ξ/2)f_x(-ξ/2)+s)/(δ_sf(-ξ/2)^2 + s^2)^2| ≤ 2|δ_sf(ξ/2)f_x(ξ/2)-δ_sf(-ξ/2)f_x(-ξ/2)|/(δ_sf(-ξ/2)^2 + s^2)^2 + 2|δ_sf(ξ/2)f_x(ξ/2)+s| |(δ_sf(ξ/2)^2 + s^2)^2 - (δ_sf(-ξ/2)^2 + s^2)^2/(δ_sf(ξ/2)^2 + s^2)^2(δ_sf(-ξ/2)^2 + s^2)^2| ≲|δ_sf(ξ/2)-δ_sf(-ξ/2)|· |f_x(ξ/2)|/ s^4 + |δ_sf(-ξ/2)|· |f_x(ξ/2)-f_x(-ξ/2)|/s^4 + |s| |δ_sf(ξ/2)^4-δ_sf(-ξ/2)^4 + s^2(δ_sf(ξ/2)^2-δ_sf(-ξ/2)^2)/s^8| ≲ξω(s)/s^4 + ω(ξ)/s^3. So using (<ref>) and (<ref>), we can first bound∫_0^ξ h|K(ξ/2,h)-K(ξ/2, -h) - K(-ξ/2,h)+K(-ξ/2,-h)|dh ≲∫_0^ξ h∫_h^ξω(s)/s^3 ds dh+ ∫_0^ξ h∫_ξ^∞ξω(s)/s^4 + ω(ξ)/s^3ds dh ≲∫_0^ξω(s)/s^3∫_0^s hdh ds+ ∫_ξ^∞ξ^3ω(s)/s^4 +ξ^2ω(ξ)/s^3ds ≲∫_0^ξω(s)/sds + ξ∫_ξ^∞ω(s)/s^2ds + ω(ξ). For the rest of (<ref>), we use (<ref>) again to also bound∫_Mξ>|h|>ξ |h||K(ξ/2, h) - K(-ξ/2,h)| dh≲∫_ξ^Mξ h∫_h^∞ω(ξ)/s^3 + ξω(s)/s^4 ds ≲ω(ξ)∫_ξ^Mξ1/h dh + ξ∫_ξ^Mξω(h)/h^2dh ≲ln(M)ω(ξ) + ξ∫_ξ^∞ω(h)/h^2dh.§ BOUNDS ON DIFFUSIVE TERMS Now we move on to proving an upper bound for the diffusive terms of (<ref>).We can rewrite them as∫_-Mξ^Mξ (δ_hf_x (ξ/2)-hω'(ξ))K(ξ/2,h) - (δ_hf_x(-ξ/2)-hω'(ξ))K(-ξ/2,h)dh +∫_|h|>Mξδ_hf_x(ξ/2)K(ξ/2,h) -δ_hf_x(-ξ/2)K(-ξ/2,h)dh =∫_-Mξ^Mξ (δ_hf_x(ξ/2)-hω'(ξ))K(ξ/2,h) - (δ_hf_x(-ξ/2)-hω'(ξ))K(-ξ/2,h)dh +∫_|h|>Mξ[δ_hf_x(ξ/2) - δ_h f_x(-ξ/2)]K(ξ/2, h)dh + ∫_|h|>Mξδ_hf_x(-ξ/2)[K(ξ/2, h)-K(-ξ/2,h)] dh. We begin by bounding the last term, which is an error term.Under the assumptions of Lemma <ref>,|∫_|h|>Mξδ_hf_x(-ξ/2)[K(ξ/2, h)-K(-ξ/2,h)]| dh ≲ω(ξ)∫_Mξ^∞ω(h)/h^2dh + ω'(ξ)ξ∫_ξ^∞ω(h)/h^2dh. Using the fact that f_x has modulus ω and the bounds <ref>, it follows that∫_|h|>Mξδ_hf_x(-ξ/2)[K(ξ/2, h)-K(-ξ/2,h)] dh≲∫_Mξ^∞ω(h)∫_h^∞ω(ξ)/s^3+ξω(s)/s^4 ds dh ≲ω(ξ)∫_Mξ^∞ω(h)/h^2dh + ∫_Mξ^∞ω(h)∫_h^∞ξω(ξ)+ξω'(ξ)(s-ξ)/s^4ds dh ≲ω(ξ)∫_Mξ^∞ω(h)/h^2dh + ω(ξ)∫_Mξ^∞ξω(h)/h^3dh + ω'(ξ)ξ∫_Mξ^∞ω(h)/h^2dh ≲ω(ξ)∫_Mξ^∞ω(h)/h^2dh.+ω'(ξ)ξ∫_ξ^∞ω(h)/h^2dh.For the other two terms in (<ref>), we bound them in two stages. Under the assumptions of Lemma <ref>,∫_-Mξ^Mξ (δ_hf_x(ξ/2)-hω'(ξ))K(ξ/2,h) - (δ_hf_x(-ξ/2)-hω'(ξ))K(-ξ/2,h)dh +∫_|h|>Mξ[δ_hf_x(ξ/2) - δ_h f_x(-ξ/2)]K(ξ/2, h)dh ≤λ∫_δ_hf_x(ξ/2) - δ_h f_x(-ξ/2)/h^2dh + 2(Λ-λ)∫_ξ^Mξ(ω(h-ξ) - ω(ξ))_+/h^2dh + ω'(ξ)∫_ξ<|h|<Mξ|h[K(ξ/2,h) - K(-ξ/2,h)]|dh. We can bound the second term of (<ref>) rather easily.Sinceδ_hf_x(ξ/2) - δ_h f_x(-ξ/2) = (f_x(h+ξ/2)-f_x(h-ξ/2)) - ω(ξ)≤ 0,by the uniform ellipticity of K,∫_|h|>Mξ[δ_hf_x(ξ/2) - δ_h f_x(-ξ/2)]K(ξ/2, h) dh ≤λ∫_|h|>Mξδ_hf_x(ξ/2) - δ_h f_x(-ξ/2)/h^2dh.To bound the first term, we first defineG(ξ, h) = (δ_hf_x(ξ/2)-hω'(ξ))K(ξ/2,h) - (δ_hf_x(-ξ/2)-hω'(ξ))K(-ξ/2,h).Note that since ω is concave and touches f_x from above (see (<ref>)), it follows thatδ_h f_x(ξ/2) - ω'(ξ)h ≤δ_hω(ξ)-ω'(ξ)h ≤ 0, h≥-ξ δ_h f_x(-ξ/2) - ω'(ξ)h ≥ -δ_-hω(ξ)-hω'(ξ)≥ 0,h≤ξThus for |h|≤ξ, by the uniform ellipticity of K we have the boundG(ξ,h)≤λδ_hf_x(ξ/2)-δ_hf_x(-ξ/2)/h^2. That just leaves us with the caseξ≤ |h| ≤ Mξ to analyze.Note that we can write G in two distinct ways:G(ξ,h)= (δ_hf_x(ξ/2)-δ_hf_x(-ξ/2)) K(ξ/2,h) + ( δ_hf_x(-ξ/2)-hω'(ξ))(K(ξ/2,h)-K(-ξ/2,h)) =(δ_hf_x(ξ/2)-δ_hf_x(-ξ/2)) K(-ξ/2,h) + ( δ_hf_x(ξ/2)-hω'(ξ))(K(ξ/2,h)-K(-ξ/2,h)).By (<ref>), δ_hf_x(ξ/2)-hω'(ξ)≤ 0 for all h>ξ.Thus if K(ξ/2,h)-K(-ξ/2,h)≥ 0, thenG(ξ,h)≤λδ_hf_x(ξ/2)-δ_hf_x(-ξ/2)/h^2, K(ξ/2,h)-K(-ξ/2,h)≥ 0On the other hand, sinceδ_hf_x(-ξ/2) = δ_h-ξf(ξ/2) + ω(ξ) ≥- ω(h-ξ) + ω(ξ)for h≥ξ, we see thatG(ξ,h)≤λδ_hf_x(ξ/2)-δ_hf_x(-ξ/2)/h^2 + ( δ_hf_x(-ξ/2)-hω'(ξ))(K(ξ/2,h)-K(-ξ/2,h)) ≤λδ_hf_x(ξ/2)-δ_hf_x(-ξ/2)/h^2 +(Λ-λ)(ω(h-ξ)-ω(ξ))_+/h^2 +hω'(ξ)|K(ξ/2,h)-K(-ξ/2,h)|, K(ξ/2,h)-K(-ξ/2,h)≤ 0. Putting these two together, we get thatG(ξ,h) ≤λδ_hf_x(ξ/2)-δ_hf_x(-ξ/2)/h^2 +(Λ-λ)(ω(h-ξ)-ω(ξ))_+/h^2 +hω'(ξ)|K(ξ/2,h)-K(-ξ/2,h)|.for h≥ξ.A similar argument can be made in the case that h≤ -ξ.Putting this all together,∫_-Mξ^Mξ G(ξ,h)dh+∫_|h|>Mξ[δ_hf_x(ξ/2) - δ_h f_x(-ξ/2)]K(ξ/2, h)dh ≤λ∫_δ_hf_x(ξ/2) - δ_h f_x(-ξ/2)/h^2dh + 2(Λ-λ)∫_ξ^Mξ(ω(h-ξ) - ω(ξ))_+/h^2dh + ω'(ξ)∫_ξ<|h|<Mξ| h[K(ξ/2,h) - K(-ξ/2,h)]|dh.It's clear that we can bound ∫_ξ<|h|<Mξ|h[K(ξ/2,h) - K(-ξ/2,h)]|dh as in (<ref>).Thus the only thing remaining to prove (<ref>) isUnder the assumptions of Lemma <ref>,λ∫_δ_hf_x(ξ/2) - δ_hf_x(-ξ/2)/h^2dh≤ 2λ∫_0^ξδ_hω(ξ) + δ_-hω(ξ)/h^2dh + 2λ∫_ξ^∞ω(ξ+h)-ω(h)-ω(ξ)/h^2dh.To see this, note that formally we should have∫_δ_hf_x(ξ/2) - δ_hf_x(-ξ/2)/h^2dh = ∫_ f_x(y)(1/(y-ξ/2)^2 - 1/(y+ξ/2)^2) - ω(ξ)/y^2 dy.Thus in order to get an upper bound on (<ref>), we should be taking an upper bound on f_x(y) when y>0 and a lower bound when y<0.Note by (<ref>) thatf_x(y)≤ f_x(ξ/2) + ω(y+ξ/2) - ω(ξ) = f_x(-ξ/2)+ω(y+ξ/2),y>-ξ/2, f_x(y)≥ f_x(-ξ/2) -ω(-y+ξ/2) + ω(ξ) = f_x(ξ/2)-ω(-y+ξ/2),y<ξ/2.In particular, using the upper bounds bounds on δ_hf_x(±ξ/2) for h>0 and the lower bounds for δ_hf_x(±ξ/2) for h<0 give the result.To rigorously justify this though, we will bound∫_ϵ^∞δ_hf_x(ξ/2) - δ_hf_x(-ξ/2)/h^2dhfrom above.Taking ϵ→ 0, we'll get ∫_0^∞δ_hf_x(ξ/2) - δ_hf_x(-ξ/2)/h^2dh≤∫_0^ξδ_hω(ξ) + δ_-hω(ξ)/h^2dh +∫_ξ^∞ω(ξ+h)-ω(h)-ω(ξ)/h^2dh.The bound for ∫_-∞^0 follows from identical arguments.So, fix some ϵ<<ξ.By splitting the integral into a several pieces and reparameterizing, we get that ∫_ϵ^∞δ_hf_x(ξ/2) - δ_hf_x(-ξ/2)/h^2dh = ∫_ϵ+ξ/2^∞f_x(y)/(y-ξ/2)^2 dy - ∫_ϵ - ξ/2^∞f_x(y)/(y+ξ/2)^2 dy - ∫_ϵ^∞ω(ξ)/y^2dy= ∫_ϵ+ξ/2^∞ f_x(y)(1/(y-ξ/2)^2 -1/(y+ξ/2)^2)dy - ∫_ϵ^∞ω(ξ)/y^2dy - ∫_ϵ - ξ/2^ϵ+ξ/2f_x(y)/(y+ξ/2)^2 dy .In the first integral of the second line, since y>ξ/2 we have that (y-ξ/2)^-2>(y+ξ/2)^-2.So applying the upper bound in (<ref>) gives an upper bound on the integral,∫_ϵ+ξ/2^∞ f_x(y)(1/(y-ξ/2)^2 -1/(y+ξ/2)^2)dy ≤∫_ϵ+ξ/2^∞(f_x(ξ/2) + ω(y+ξ/2) - ω(ξ))(1/(y-ξ/2)^2 -1/(y+ξ/2)^2)dy = ∫_ϵ+ξ/2^∞f_x(ξ/2) + ω(y+ξ/2) - ω(ξ)/(y-ξ/2)^2dy -∫_ϵ + ξ/2^∞f_x(ξ/2) + ω(y+ξ/2) - ω(ξ)/(y+ξ/2)^2dyBy reparametrizing back, we get that ∫_ϵ+3ξ/2^∞f_x(ξ/2) + ω(y+ξ/2) - ω(ξ)/(y-ξ/2)^2dy -∫_ϵ + ξ/2^∞f_x(ξ/2) + ω(y+ξ/2) - ω(ξ)/(y+ξ/2)^2dy -∫_ϵ+ξ^∞ω(ξ)/y^2dy = ∫_ϵ+ξ^∞ω(ξ+h)-ω(h)-ω(ξ)/h^2dh Hence combining (<ref>),(<ref>), and (<ref>)gives us ∫_ϵ^∞δ_hf_x(ξ/2) - δ_hf_x(-ξ/2)/h^2dh ≤∫_ϵ+ξ^∞ω(ξ+h)-ω(h)-ω(ξ)/h^2dh + ∫_ϵ^ϵ+ξf_x(ξ/2) + ω(ξ+h) - ω(ξ)/h^2dh- ∫_ϵ^ϵ+ξω(ξ)/h^2 dh- ∫_ϵ^ϵ+ξf_x(h-ξ/2)/h^2dh=∫_ϵ+ξ^∞ω(ξ+h)-ω(h)-ω(ξ)/h^2dh+∫_ϵ^ϵ+ξδ_h ω(ξ) + f_x(ξ/2) - f_x(h-ξ/2)-ω(ξ)/h^2dh.Now for h<ξ, we have that f_x(ξ/2) - f_x(h-ξ/2)≤ω(ξ-h), and thus ∫_ϵ^ξδ_h ω(ξ) + f_x(ξ/2) - f_x(h-ξ/2)-ω(ξ)/h^2dh≤∫_ϵ^ξδ_hω(ξ) + δ_-hω(ξ)/h^2dh.Taking the limit as ϵ→ 0, we then get∫_0^∞δ_hf_x(ξ/2) - δ_hf_x(-ξ/2)/h^2dh≤∫_0^ξδ_hω(ξ) + δ_-hω(ξ)/h^2dh +∫_ξ^∞ω(ξ+h)-ω(h)-ω(ξ)/h^2dh. § MODULUS INEQUALITYCombining all the estimates from the previous two sections, we get a proof of Lemma <ref>.Thus under the assumptions (<ref>), we havethatd/dt(f_x(ξ/2)-f_x(-ξ/2)) ≤Aω'(ξ)(∫_0^ξω(h)/hdh + ξ∫_ξ^∞ω(h)/h^2 dh+ ln(M+1)ω(ξ)) +Aω(ξ)∫_Mξ^∞ω(h)/h^2dh +2(Λ-λ)∫_ξ^Mξ(ω(h-ξ) - ω(ξ))_+/h^2dh +2λ∫_0^ξδ_hω(ξ) + δ_-hω(ξ)/h^2dh + 2λ∫_ξ^∞ω(h+ξ)-ω(h)-ω(ξ)/h^2dh,for any M≥ 1, where A is a constant depending only on ||f_0'||_L^∞.In <cit.>, the authors showed that the modulus{[ω(ξ) = ξ-ξ^3/2,0≤ξ≤δ; ω'(ξ) = γ/ξ(4+log(ξ/δ)),ξ≥δ ]. ,satisfiesAω'(ξ)(∫_0^ξω(h)/hdh + ξ∫_ξ^∞ω(h)/h^2 dh ) +λ∫_0^ξδ_hω(ξ) + δ_-hω(ξ)/h^2dh + λ∫_ξ^∞ω(h+ξ)-ω(h)-ω(ξ)/h^2dh < 0,for all ξ∈ so long as δ, γ are sufficiently small.With that in mind, we will show thatUnder the assumptions of Lemma <ref> for the modulus ω defined in (<ref>),d/dt(f_x(ξ/2)-f_x(-ξ/2)) < -ω'(ξ)ω(ξ),as long as δ, γ are taken sufficiently small depending on β(f_0'),||f_0'||_L^∞. By the Lemma <ref> and (<ref>) which was proven in <cit.>, it suffices to showAω'(ξ)ln(M+1)ω(ξ)+Aω(ξ)∫_Mξ^∞ω(h)/h^2dh +2(Λ-λ)∫_ξ^Mξ(ω(h-ξ) - ω(ξ))_+/h^2dh +λ∫_0^ξδ_hω(ξ) + δ_-hω(ξ)/h^2dh + λ∫_ξ^∞ω(h+ξ)-ω(h)-ω(ξ)/h^2dh≤ -ω'(ξ)ω(ξ)for the correct choices of M, and δ, γ sufficiently small.We proceed very similarly to <cit.>. To begin, for ξ≤δ we take M=1.Then we just need to show thatAω'(ξ)ω(ξ) + Aω(ξ)∫_ξ^∞ω(h)/h^2dh + λ∫_0^ξδ_hω(ξ) + δ_-hω(ξ)/h^2dh + λ∫_ξ^∞ω(h+ξ)-ω(h)-ω(ξ)/h^2dh≤ -ω'(ξ)ω(ξ) . In this regime, note that we have the bounds{[∫_ξ^δω(h)/h^2dh ≤log(δ/ξ),; ∫_δ^∞ω(h)/h^2dh = ω(δ)/δ + γ∫_δ^∞1/h^2(4+log(h/δ)) dh ≤ 1 + γ/4δ≤ 2 γ<4δ,; ω'(ξ)≤ 1,; ω(ξ)≤ξ,;∫_0^ξω(ξ+h)+ω(ξ-h)-2ω(ξ)/h^2≤ξω”(ξ) = -3/2ξξ^-1/2. ].Putting this all together, we get that(A +1)ω'(ξ)ω(ξ) + Aω(ξ)∫_ξ^∞ω(h)/h^2dh+ λ∫_0^ξω(ξ+h) + ω(ξ-h)-2ω(ξ)/h^2dh + λ∫_ξ^∞ω(ξ+h)-ω(h)-ω(ξ)/h^2dh ≤ξ( (A+1)(3+log(δ/ξ)) - 3/2λξ^-1/2)<0,assuming that δ is sufficiently small.Now assume that ξ≥δ.Then what we need to show isAω'(ξ) ln(M+1)ω(ξ) + Aω(ξ)∫_Mξ^∞ω(h)/h^2dh +2(Λ-λ)∫_ξ^Mξ(ω(h-ξ) - ω(ξ))_+/h^2dh + λ∫_0^ξδ_hω(ξ) + δ_-hω(ξ)/h^2dh + λ∫_ξ^∞ω(h+ξ)-ω(h)-ω(ξ)/h^2dh≤ -ω'(ξ)ω(ξ) . We first bound our new error terms.Using the definition of ω and integrating by parts, we see that2(Λ-λ)∫_2ξ^Mξω(h-ξ) - ω(ξ)/h^2dh≤ 2(Λ-λ)∫_ξ^∞ω(h)-ω(ξ)/h^2 dh ≤ 2(Λ-λ) ∫_ξ^∞γ/h^2(4+log(h/δ))dh≤2(Λ-λ)γ/ξ≤λ/4ω(δ)/ξ≤λ/4ω(ξ)/ξ,assuming γ≤λ/8(Λ-λ)ω(δ).In order to bound our other new error term, we will be taking M sufficiently large and then γ sufficiently small depending on M,δ.Noting that ω(ξ)≤ 2||f_0'||_L^∞, we can bound our other new error term by integrating by partsAω(ξ)∫_Mξ^∞ω(h)/h^2dh≤2A||f_0'||_L^∞/Mω(Mξ)/ξ +2A||f_0'||_L^∞∫_Mξ^∞γ/h^2(4+log(h/δ))dh ≤2A||f_0'||_L^∞/Mω(Mξ)/ξ +2A||f_0'||_L^∞/Mγ/ξ≤λ/16ω(Mξ)/ξ + λ/8ω(ξ)/ξ,assuming thatM ≥32A||f_0'||_L^∞/λ,and then γ is sufficiently small so that2||f_0'||_L^∞A/Mγ≤λ/8ω(δ)≤λ/8ω(ξ). Note that this is where we set a value for M, and that γ is taken sufficiently small depending on M.Now that the value for M is fixed, we can also control the value ω(Mξ) by taking γ sufficiently small thatω(Mξ)= ω(ξ) + ∫_ξ^Mξγ/h(4+log(h/δ))dh ≤ω(ξ) + γln(M)≤ω(ξ)+ω(δ) ≤ 2ω(ξ).Hence,Aω(ξ)∫_Mξ^∞ω(h)/h^2dh ≤λ/16ω(Mξ)/ξ + λ/8ω(ξ)/ξ≤λ/4ω(ξ)/ξ.Using the same integration by parts tricks, we can also showλ∫_ξ^∞ω(h+ξ)-ω(h)-ω(ξ)/h^2dh ≤ -3/4λω(ξ)/ξ.for γ sufficiently small. So combining these together, we get thatAω(ξ)∫_Mξ^∞ω(h)/h^2dh +2(Λ-λ)∫_2ξ^Mξω(h-ξ) - ω(ξ)/h^2dh + λ∫_ξ^∞ω(h+ξ)-ω(h)-ω(ξ)/h^2dh ≤-λ/4ω(ξ)/ξ. Since ω'(ξ)ω(ξ)≤γω(ξ)/ξ, we finally get that(Aln(M+1)+1)ω'(ξ)ω(ξ) - λ/4ω(ξ)/ξ≤ω(ξ)/ξ((Aln(M+1)+1)γ -λ/4)<0,if γ is taken sufficiently small.§ OUR CHOICE FOR THE MODULUS ΡWe've now shown that for the modulus defined in (<ref>) that if the assumptions (<ref>) hold thatd/dt(f_x(t,ξ/2)-f_x(t,-ξ/2))|_t=T < -ω'(ξ)ω(ξ).We claim that in fact (<ref>) will hold for any rescaling ω_r(h) = ω(rh) as well.To see this, fix some r>0, and suppose that f(t,x) satisfies the conditions of Lemma <ref> for ω_r at time T and distance ξ. Take f̃ (t,x) = rf(t/r, x/r), which is also a solution of (<ref>).Then f̃_x is a solution of (<ref>) with β(f̃_0') = β(f_0'), ||f̃_0'||_L^∞ = ||f_0'||_L^∞, and satisfying the conditions of Lemma <ref> for ω at time rT and distance rξ.Hence by Lemma <ref> d/dt(f_x(t,ξ/2)-f_x(t,-ξ/2))|_t=T = rd/dt(f̃_x(t,rξ/2)-f̃_x(t,-rξ/2))|_t=rT < -rω'(rξ)ω(rξ) = -ω_r'(ξ)ω_r(ξ).So, (<ref>) will hold for any rescaling ω_r.Also note that for f_x(T,ξ/2)-f_x(T,-ξ/2)= ω(ξ) to hold, we must necessarily have ω(ξ)≤ 2||f_x(T,·)||_L^∞<2||f_0'||_L^∞.Thus takingC = sup_0<h<ω^-1(2||f_0'||_L^∞)h/ω(h) = ω^-1(2||f_0'||_L^∞)/2||f_0'||_L^∞,we see thatω(h)≥h/C.for all relevant h. Defineρ(h):= ω(Ch),so thatρ(h)≥ h,for all h∈ [0,ρ^-1(2||f_0'||_L^∞)].Now, suppose that at time T, f satisfies the assumptions (<ref>) for ρ(·/T).Then since ρ(·/T) is a rescaling of ω, we have thatd/dt(f_x(T,ξ/2)-f_x(T,-ξ/2)) < -d/dhρ(h/T)|_h=ξρ(ξ/T) = -1/Tρ'(ξ/T)ρ(ξ/T)≤-ξ/T^2ρ'(ξ/T) = d/dtρ(ξ/t)|_t=T.Thus we've constructed a modulus ρ which satisfies (<ref>), completing the proof of the generation of a Lipschitz modulus of continuity (<ref>) in our main theorem. § REGULARITY IN TIME With the construction of the modulus ρ, we get universal Lipschitz bounds in space for f_x(t,·).By the structure of (<ref>), we also get regularity in space for f_t.Let f:(0,T)×→ be a classical solution to (<ref>) with ||f(t,·)||_W^1,∞ bounded and ||f_xx(t,·)||_L^∞≲ 1/t..Then f_t(t,·) is Log-Lipschitz in space with|f_t(t,·) | ≲max{-log(t), 1},|f_t(t,x)-f_t(t,y)| ≲ -log(|x-y|)|x-y|(1+1/t) 0<|x-y|<1/2. For t<1, we have that|f_t(t,x)|= |∫_δ_hf(t,x) - hf_x(t,x)/δ_hf(t,x)^2+h^2 dh|≤|∫_0^∞δ_hf(t,x) +δ_-hf(t,x)/δ_-hf(t,x)^2 +h^2dh| + |∫_0^∞(δ_hf(t,x)-hf_x(t,x))(δ_hf(t,x)^2-δ_-hf(t,x)^2)/(δ_hf(t,x)^2+h^2)(δ_-hf(t,x)^2 +h^2) dh| ≲∫_0^t 1/tdh+ ∫_t^11/hdh +∫_1^∞1/h^2+1/h^3dh ≲ -log(t) + 1.For t>1, you can similarly show |f_t(t,x)|≲ 1, proving the first bound.For regularity in space, we see thatf_t(t,x) -f_t(t,y)= ∫_δ_hf(t,x) - h f_x(t,x)/δ_hf(t,x)^2 + h^2 - δ_hf(t,y) - h f_x(t,y)/δ_hf(t,y)^2 + h^2 dh = ∫_δ_h f(t,x) - hf_x(t,x) - (δ_hf(t,y)-hf_x(t,y))/δ_h f(t,y)^2+h^2+ (δ_hf(t,x) - h f_x(t,x))(δ_hf(t,x)^2-δ_hf(t,y)^2)/( δ_hf(t,x)^2 + h^2)(δ_hf(t,y)^2 + h^2) dh ≤|∫_|h|<|x-y|| + |∫_|x-y|<|h|<1|+|∫_|h|>1| For |h|<|x-y|, we can bound similarly to before to get that|∫_|h|<|x-y||≲∫_0^|x-y|1/tdh = |x-y|/t.For midsize |x-y| < |h| < 1, we have that|δ_h f(t,x) - hf_x(t,x) - (δ_hf(t,y)-hf_x(t,y))| = |∫_0^h δ_sf_x(t,x)- δ_sf_x(t,y)ds |≲|x-y|h/t,| δ_hf(t,x) - δ_hf(t,y)| = | ∫_0^h f_x(t,x+s) - f_x(t,y+s)ds|≲|x-y|h/t.Thus|∫_|x-y|<|h|<1|≲|x-y|/t∫_|x-y|^1 1/hdh= -ln(|x-y|)|x-y|/t.Finally, we use L^∞ bounds on f to get that|∫_|h|>1|≤|∫_|h|>1δ_h f(t,x) - δ_hf(t,y)/δ_h f(t,y)^2+h^2 + (δ_hf(t,x) - h f_x(t,x))(δ_hf(t,x)^2-δ_hf(t,y)^2)/( δ_hf(t,x)^2 + h^2)(δ_hf(t,y)^2 + h^2) dh|+|f_x(t,x)-f_x(t,y)||∫_|h|>1-h/δ_hf(t,y)^2 +h^2 dh | ≲ |x-y|∫_1^∞1/h^2 + 1/h^3 dh + |x-y|/t∫_1^∞1/h^3dh ≲(1+1/t)|x-y|.Putting this all together, we thus have that|f_t(t,x)-f_t(t,y)|≲ -ln(|x-y|)|x-y|(1+1/t). Recall that in section 2, we assumed that our initial data f_0∈ C^∞_c() so that by the local existence results of <cit.>, there is a unique solution f∈ C^1((0,T_+); H^k) for k arbitrarily large and some T_+>0.We were then able to prove the existence of the modulus ρ as in Theorem <ref> depending only on β(f_0'),||f_0'||_L^∞, and hence with the solution f existing for all time by the main theorem of <cit.>.For an arbitrary f_0 ∈ W^1,∞() with β(f_0')<1, the same result holds true by compactness.Let η∈ C^∞_c() be a smooth mollifier, and ϕ∈ C^∞_c() be a smooth cutoff function.For f_0∈ W^1,∞() with β(f_0')<1, take f_0^(ϵ)(x):= (f_0*η_ϵ)(x)ϕ(ϵ x).Then f_0^(ϵ)→ f_0 in W^1,∞_loc, with β(f_0^(ϵ)'), ||f_0^(ϵ)||_W^1,∞()→β(f_0'),||f_0||_W^1,∞() respectively as ϵ→ 0.Thus for ϵ sufficiently small, β(f_0^(ϵ)' )<1 and the results of the previous section hold for the solution to the mollified problem f^(ϵ).The L^∞ bound on f_t^(ϵ) proven above along with the maximum principle for f_x^(ϵ) is enough to ensure that there a subsequence f^(ϵ_k) converging in C_loc([0,∞)×) to a Lipschitz (weak) solution f to the original problem.In order to get a classical C^1 solution, we need regularity estimates for f_x^(ϵ),f_t^(ϵ) in both time and space.The modulus ρ and Proposition <ref> give the regularity in space that we need for f_x,f_t.All that leaves is to prove regularity in time. Let f be a sufficiently smooth solution to (<ref>) with β(f_0')<1.Then f_x,f_t ∈ C^α_loc((0,∞)×) with||f_x||_C^α(Q_t/4(t,x)), ||f_t||_C^α(Q_t/4(t,x))≤ C(β(f_0'),||f||_L_t^∞((t/2,3t/2); W_x^2,∞())) max{t^-α,1},where Q_r(s,y) = (s-r,s]× B_r(y), and α>0 depends only on β(f_0'),||f_0'||_L^∞. We have that f_x solves(f_x)_t (t,x) =f_xx(t,x) ∫_-h/δ_hf(t,x)^2+h^2 dh + ∫_δ_hf_x(t,x) K(t,x,h) dh,where λ/h^2≤ K(t,x,h) ≤Λ/h^2 is uniformly elliptic with ellipticity constants λ, Λ depending on β(f_0'),||f_0'||_L^∞. Rewriting this, we have that f_x satisfies(f_x)_t - ∫_δ_h f_x(t,x) (K(t,x,h)+K(t,x,-h)/2) dh=f_xx(t,x) ∫_-h/δ_hf(t,x)^2+h^2 dh + ∫_δ_h f_x(t,x) (K(t,x,h)-K(t,x,-h)/2) dh. Let F(t,x) denote the righthand side of (<ref>).Then F(t,x) is locally bounded with |F(t,x)| controlled by ||f(t,·)||_W^2,∞.Then since (K(t,x,h)+K(t,x,-h))/2 is a symmetric uniformly elliptic kernel, it follows that we have local C^α bounds for α≤α_0 for some α_0 depending on ellipticity constants (see <cit.>).So, all we have to do is give bounds on F(t,x) depending only on ||f(t,·)||_W^2,∞.Similar to proof of Lemma <ref>,∫_-h/δ_hf(t,x)^2 +h^2 dh = ∫_0^∞ h δ_hf(t,x)^2 - δ_-hf(t,x)^2/(δ_hf(t,x)^2 +h^2)(δ_-hf(t,x)^2 +h^2) dh≲∫_0^1 1 dh + ∫_1^∞1/h^3 dh ≲ 1.Also similar to the proof of Lemma <ref> (specifically (<ref>)), we have that|K(t,x,h) - K(t,x,-h)| ≲min{1/h, 1/h^3},so|∫_δ_h f_x(t,x) (K(t,x,h)-K(t,x,-h)/2) dh | ≲∫_0^1 1 dh + ∫_1^∞1/h^3 dh ≲ 1. Thus since we've bounded the right hand side of (<ref>) depending only on ||f(t,·)||_W^2,∞, we have our local C^α bounds for f_x for all α sufficiently small.A C^α bound that is uniform in x for f_x then gives a log C^α estimate for f_t, similar to the proof for regularity in space in Proposition <ref>.Thus we have C^α estimates for both f_x,f_t. § UNIQUENESS We now prove that if our initial data f_0∈ C^1,ϵ() with β(f_0') <1, then the solution f given by Theorem <ref> is unique with f∈ L^∞([0,∞) ; C^1,ϵ).As mentioned before, this essentially follows from the uniqueness theorem given in <cit.>, which under our assumptions simplifies to (Constantin et al)Let f∈ L^∞ ([0,T]; W^1,∞) be a classical, C^1 solution to (<ref>) with initial data f(0,x)=f_0(x).Assume that lim_x→∞f(t,x) = 0, and that there is some modulus of continuity ρ̃ such thatf_x(t,x)-f_x(t,y)≤ρ̃(|x-y|), ∀ 0≤ t≤ T,x≠y∈.Then the solution f is unique. The authors of <cit.> note that the uniform continuity assumption should be the only real assumption; the decay is assumed for convenience in their proof.So, we start by proving that if f_0∈ C^1,ϵ(), then the solution f∈ L^∞ ([0,∞); C^1,ϵ).To begin, suppose that f_0∈ C^1,1(). Then necessarily f_0' has modulus ρ(·/δ) for some δ>0 sufficiently small.The same proof for the instantaneous generation of the modulus ρ will give that f_x(t,·) has modulus ρ(·/t+δ).Hence f_x(t,·) has modulus ρ(·/δ) for all t≥ 0.If f_0∈ C^1,ϵ(), we can make the same essential argument by changing the definition of ρ , ω.You can repeat the arguments of section 7 and 8 for the modulus{[ω^(ϵ)(ξ) = ξ^ϵ,0≤ξ≤δ; ω^(ϵ)'(ξ) = γ/ξ(4+log(ξ/δ)),ξ≥δ ]. .All the error terms for ξ≤δ are of order ξ^2ϵ-1, while the diffusion term is of the order ξ^ϵ -1, so there are no problems as long as δ is sufficiently small.The argument for ξ≥δ is identical to the original. Taking ρ^(ϵ) to be some suitable rescaling of ω^(ϵ), we then have that if f_0' has modulus ρ^(ϵ)(·/δ), then f_x(t,·) will have modulus ρ^(ϵ)(·/t+δ).Thus if f_0∈ C^1,ϵ(), then the solution f given by Theorem <ref> will satisfy the main uniform continuity assumption of Theorem <ref>.Our solution f will not decay as x→∞, but that assumption isn't truly necessary.Let f_1,f_2 be two uniformly continuous, classical solutions to (<ref>) with the same initial data, and let M(t) = ||f_1(t,·)-f_2(t,·)||_L^∞.With the decay assumption, the authors of <cit.> are able to assume that for almost every t, there is a point x(t)∈ such thatM(t) = |f_1(t,x(t)) - f_2(t,x(t))|, d/dtM(t) = (d/dt|f_1 - f_2|)(t,x(t)).They then bound d/dt|f_1(t,x(t)) - f_2(t,x(t))| using equation (<ref>), ρ̃, and W^1,∞ bounds.Without the decay assumption, you instead use thatd/dtM(t) ≤sup{d/dt|f_1(t,x) - f_2(t,x)| : |f_1(t,x)-f_2(t,x)| ≥ M(t)-δ},where δ>0 is arbitrary.When you go to bound d/dt|f_1(t,x) - f_2(t,x)|, you then get new error terms which can be bounded byC(ρ̃,max_i ||f_i(t,·)||_W^1,∞,M(t)) (δ + |f_1,x(t,x)-f_2,x(t,x)|).Since f_i,x(t,x) is bounded and has modulus ρ̃, it then follows that|f_1,x(t,x)-f_2,x(t,x)| = o_δ(1).Thus by taking δ sufficiently small depending on ρ̃,max_i ||f_i(t,·)||_W^1,∞,M(t), we can guarantee that the new error terms ≲ M(t).Then the original proof of <cit.> goes through. § ACKNOWLEDGEMENTSI would like to thank my advisor Luis Silvestre for suggesting the problem, pointing me towards good resources, and just giving good advice in general.abbrv | http://arxiv.org/abs/1704.08401v3 | {
"authors": [
"Stephen Cameron"
],
"categories": [
"math.AP"
],
"primary_category": "math.AP",
"published": "20170427011535",
"title": "Global well-posedness for the 2D Muskat problem with slope less than 1"
} |
Embedded Si patterns in grapheneNosraty Alamdary et al. e-mail , Phone: +43-1-427772855University of Vienna, Faculty of Physics, Boltzmanngasse 5, 1090 Vienna, AustriaXXXX, revised XXXX, accepted XXXXXXXXRecent experiments have revealed the possibility of precise electron beam manipulation of silicon impurities in graphene. Motivated by these findings and studies on metal surface quantum corrals, the question arises what kind of embedded Si structures are possible within the hexagonal lattice, and how these are limited by the distortion caused by the preference of Si for sp^3 hybridization. In this work, we study the geometry and stability of elementary Si patterns in graphene, including lines, hexagons, triangles, circles and squares. Due to the size of the required unit cells, to obtain the relaxed geometries we use an empirical bond-order potential as a starting point for density functional theory. Despite some interesting discrepancies, the classical geometries provide an effective route for the simulation of large structures.[]Abstract_figure.png A relaxed hexagonal structure of 30 Si embedded within the graphene lattice, containing in total of 1152 atoms in the unit cell. This pattern corrugates the graphene lattice in a symmetric way around a central plateau.Structure and energetics of embedded Si patterns in grapheneDaryoush Nosraty Alamdary, Jani Kotakoski, Toma Susi ===============================================================§ INTRODUCTIONSingle-layer graphene not only has remarkable electronic <cit.> and mechanical <cit.> properties, but it is also highly suitable for atomic-resolution transmission electron microscopy studies <cit.>. Due to its two-dimensional (2D) nature, each atom can be directly imaged, and the high conductivity reduces radiolysis and ionization, completely suppressing beam damage at electron acceleration voltages below 80 kV <cit.>. However, C atoms next to impurities such as Si heteroatoms embedded within the lattice <cit.> are less strongly bound than atoms of the bulk <cit.>. Scanning transmission electron microscopy (STEM) with 60 keV electrons cannot quite outright eject them, but instead induces out-of-plane dynamics <cit.> that allow the Si atoms to be non-destructively moved with atomic precision <cit.>. These findings have raised the question of what kinds of stable patterns could be possible within the bounds of lattice symmetry.Precisely designed Si structures could be of importance for at least two reasons. First, they raise the possibility of confinement of the graphene surface states, similarly to quantum corrals created by scanning tunnelling microscopy on metal surfaces <cit.> since the early 1990s. Although embedded Si impurities certainly differ from adatoms on a metal surface, it is possible that closed rings or similar structures could also confine graphene electronic states into standing wave patterns. The second reason is the possible enhancement of graphene surface plasmons <cit.> near the impurities. Electron energy loss spectroscopy near single Si impurities has provided evidence for localized enhancement of the plasmon resonances <cit.>. Arranging many impurities into patterns whose dimensions match the plasmon wavelength might result in stronger antenna enhancement <cit.>, and their shape might allow plasmons to be directed in interesting ways <cit.>.Before such experiments can be realized, we need to know what kinds of Si patterns are possible. The hexagonal symmetry of graphene restricts the possibilities for placing Si atoms within the lattice, and the two symmetry directions, zigzag (ZZ) and armchair (AC), further limit the number of inequivalent patterns. Within these limitations, at least five categories of elementary structures appear possible, namely lines (both ZZ and AC), hexagons, triangles, circles, and squares, with the latter two being generally impossible to perfectly realize (Fig. <ref>).Two further considerations are important. One is relative stability: C–C bonds are more stable than Si–C bonds, which in turn are more stable than Si–Si bonds <cit.>. Thus while 2D silicon carbide is stable <cit.> and indeed more stable than 2D silicene <cit.>, it is significantly less stable and more reactive than graphene <cit.>. Similarly for Si impurity patterns, Si–Si bonds increase the energy of the system (although it may still be stable <cit.>), as will bonds between Si and C. More importantly, though, since the beam manipulation method is based on the inversion of Si–C bonds <cit.>, neighbouring impurities are difficult to control. The second issue is computational: the size of the unit cell needs to be large enough so that the structures and the distortion they cause in the graphene lattice do not interact significantly with their periodic images. Even distortions caused by small vacancies in graphene become apparent only in simulations involving hundreds of atoms <cit.>. For this reason, computationally efficient empirical bond-order potentials are required to relax structures with up to 1000 atoms.Despite close similarities between the analytical potential and DFT, we do find some differences in the local geometries of the Si. These mostly subtle differences are not trivial, as they highlight the role of the electronic structure of materials in their local bonding and overall geometrical configuration. In few cases, this leads to surprisingly different overall shapes. In general, the analytical potential has a tendency to introduce stronger out-of-plane corrugation of the graphene sheet, whereas DFT consistently prefers flatter atomic arrangements.§ METHODS The Atomic Simulation Environment (ASE) <cit.> enables the efficient design and manual adjustment of atomic structures along with structure optimization (we used FIRE <cit.> for force minimization). To obtain potential energies and forces, this needs to be coupled to a calculator, either based on an analytical potential or density functional theory (DFT). For the classical calculations, we settled on the Erhart-Albe (EA) <cit.> Si–C potential as implemented in the Atomistica package <cit.>. For DFT, we used the grid-based projector-augmented wave code Gpaw <cit.> with the PBE <cit.> functional and k-point spacings of less than 0.2 Å^-1. For smaller systems we used a combination of plane wave (enabling a strain filter; cutoff energy 600 eV) and finite-difference (FD) modes (grid spacing 0.18 Å), and for large ones, the highly efficient atom-based-orbital (LCAO) implementation <cit.> with a polarized double-zeta basis.For each structure type, we studied using the EA potential the influence of both the structure size (e.g. how large an area is delimited by the Si atoms) as well as the unit cell size (amount of graphene between the structures). After designing a Si structure into the lattice, we found the optimal unit cell size by scaling the structure separately in the x and y directions while relaxing the atomic positions, and selecting the minimum energy size. After this, the structure was further relaxed by a successively stricter three-stage iteration of a strain filter (minimizing the stress; for smaller structures using DFT) and force minimization (maximum forces <0.001 eV/Å). We then took the converged size of each structure, scaled it by the difference between the DFT and EA equilibrium C–C bond length, relaxed each with LCAO-DFT, and finally converged the electron density and total energy using FD-DFT.To estimate the smallest cell where the periodic images of the structures do not interact and to compare stability between structures, we calculate the embedding energy per Si atom asε = E_tot-N_Cμ_C/N_Si-μ_Si,where μ_C is the chemical potential of C (energy per atom in pristine graphene), E_tot the total energy of the system, N_C the number of the C atoms, and μ_Si the chemical potential of Si calculated for a single Si atom in vacuum (zero for any classical potential). The value of μ_C was calculated in EA and DFT respectively to be -7.374 and -9.223 eV, while the value of μ_Si in DFT is -0.805 eV. This energy becomes constant for sufficiently large unit cells.§ RESULTS We first compared the geometry a single trivalent Si substitution relaxed using the EA potential or with DFT (Fig. <ref>a-b). The equilibrium graphene C–C bond length in EA is 3.6% larger at 1.475 Å (DFT: 1.424 Å), while the Si–C bond is 1.770 Å (DFT: 1.762 Å). Thus graphene is slightly underbound with EA, whereas the Si atoms are comparatively overbound. The main difference, however, is the local corrugation: with EA, the second-nearest C neighbors buckle almost 0.3 Å (8.3%) further from the plane than with DFT, resulting in a Si height of 1.815 Å (DFT: 1.676 Å). Despite these discrepancies, the overall agreement is good. For multiple Si atoms in a unit cell, alternating their placement above and below the plane results in a lower energy <cit.>. For EA, the symmetric and antisymmetric energies are equal for cell sizes above 6× (number of pristine graphene hexagons between the Si), where the DFT energy is converged within 20 meV.Next we turn to the different Si line structures (Fig. <ref>c-j). Due to its simplicity (Si atoms on the same sublattice), the ZZ line is an elementary building block of most of the larger structures. In addition to a dense ZZ line, where every other C atom is replaced by Si, sparser arrangements such as the dashed ZZ line can be envisaged. Finally, an armchair line can be embedded into the lattice in the other symmetry direction, superficially resembling an A-B ZZ line where Si atoms are placed on alternating sublattices.Most relaxed line structures are very similar between EA and DFT, with the exception of the AC line. Here EA predicts a significant out-of-plane corrugation (Fig. <ref>i), while DFT finds a nearly flat structure (Fig. <ref>j). For the ZZ line, EA yields a corrugated structure (Fig. <ref>e), which is also reproduced by DFT (Fig. <ref>f) when the EA configuration is used as the starting point for relaxation.However, when initialized from a flat geometry, DFT can also yield another, rather surprising ZZ line: one of the four Si is lifted from the graphene plane, remaining bound to just one C atom with its other two C neighbours bonding in a pentagon (Fig. <ref>). The structure is otherwise almost completely flat—presumably to minimize the energy of the C atoms—reducing the embedding energy by 0.6 eV compared to the corrugated ZZ line. It appears that the DFT energy penalty of C atoms bound to two Si is so high that breaking two Si–C bonds to create just one more C–C bond becomes energetically favourable when a periodicity of at least four Si atoms is available in the cell. This can also be seen in the embedding energies plotted in Fig. <ref>, which show how the introduction of new in-plane Si atoms by increasing unit cell size increases the energy until the number of Si atoms reaches eight, which allows a second Si atom to buckle out of the plane. Before this, the separation of the singly-bound Si atoms is too short, which would result in too high strain between the C pentagons.For the line structures, the embedding energy is converged to <10 meV when there are six rows of carbon between the Si lines. While a weak indirect interaction between the lines remains even at this distance, simulations with the system size doubled to introduce a second Si line into the unit cell show that the symmetry of the corrugation of the second line with respect to the first does not significantly influence the embedding energy. Since the second line neither leads to appreciable changes in the overall atomic structure, we limit the discussion here to the symmetric case possible in the smaller simulation cell.In Table <ref> we have listed the embedding energy per Si as given by Eq. <ref> for all of our structure classes using both the EA and DFT potentials. It is immediately obvious that bringing several Si atoms to close proximity reduces the embedding energy (thus stabilizing the structure), in some cases by more than 1 eV per Si. The embedding energies as calculated with the two methods show a trend that is similar to the atomic structures discussed above: the relaxed configurations that are alike between the two methods also show similar relative embedding energies. In fact, for most structures, the differences remain below 0.5 eV per silicon atom, the largest one (∼1 eV) arising for the AC line with its EA out-of-plane corrugation.For the other pattern classes, we were mainly interested in large closed structures that could be reasonably simulated, and also potentially fabricated using electron beam manipulation. The first obvious closed structures are hexagons delineated by dense ZZ lines. Based on their EA energy convergence in terms of feature size and super-cell size, we settled on a square 7×14 supercell of nominally 392 C atoms as a sufficient yet minimal graphene template. For the hexagon pattern, we considered both a dense 24 Si atom structure and a sparse version with only 12 Si, which we compare in Fig. <ref>. The six ZZ lines are each ∼10 Å in length, enclosing 27 full C hexagons within an area of ∼280 Å^2, nearly comparable to graphene quantum dots that have been studied on metal surfaces <cit.>.The main difference between the two structures is the overall height of the "mesa" delineated by the Si atoms. The preferred local bonding of the Si within the line leads to their periodic up-and-down oscillation, similar to the corrugated lines. In the dense pattern, this seems only possible by raising the entire enclosed area several Å above the lattice plane, whereas in the sparse hexagon, the Si atoms bond above and below the plane, greatly reducing the overall buckling. We also created sparser versions of the other structures, which exhibit the same kinds of differences and thus do not need to be discussed at length.Like hexagons, triangles fully respect the lattice symmetry, with the 24-Si one shown in Fig. <ref>a appearing qualitatively quite similar to the hexagon. Of structures that do not completely respect the symmetry, circles are particularly interesting. While it is not possible to make ones of arbitrary size that are fully circular, a 12-Si ring comes close and encloses exactly seven carbon hexagons (Fig. <ref>b), i.e., embedded coronene. For this small structure where the Si atoms on the two halves of the circle occupy different sublattices, the "mesa" rising from the graphene plane is highly symmetrical and flat (consistent with smaller structures of other types). For larger circles, the Si atoms at the circumference weave up and down similar to the hexagon. Lastly we made structures as similar to a square shape as possible. While these also do not respect the hexagonal lattice symmetry, it is possible to make them using ZZ lines in one direction and AC lines in the other (Fig. <ref>c), arguably making them an elementary shape.§ DISCUSSIONThe easy availability of carefully parametrized and extensively tested bond-order potentials for C and Si makes it straightforward to simulate Si patterns in graphene (which is not the case for another interesting heteroatom, phosphorus <cit.>). However, although we have shown that the classical structures mostly provide a good starting point for DFT, one has to be careful to avoid structurally and energetically distinct local minima. Starting from the corrugated EA-relaxed geometry of the AC line, DFT is able to find the correct flat structure. This is not the case for the flat ZZ line, which cannot be reached from the corrugated starting point.In terms of the general differences in the relaxed structures, DFT tends prefer more symmetric arrangements with flatter graphene areas, perhaps due to the short range of the bond-order potential. However, the overall agreement of both the geometries and the relative energetics is surprisingly good. Unfortunately, the same does not extend to the simulation of electron irradiation: we found that EA gives qualitatively wrong results for the ejection of C neighbors to the Si, failing to reproduce the Si–C bond inversion <cit.> underlying the mechanism of electron beam manipulation.Nonetheless, for obtaining relaxed geometries, starting with the EA potential offers a dramatic speedup. For example, the initial classical relaxation of the large hexagon takes only 10 min on single processor, whereas even with optimized LCAO calculator parameters, each of the subsequent relaxation steps consumes on average over 2000 CPU-min (well over 140 000 CPU-h in total).All of the above structures have been designed using the simple trivalent Si substitution. A planar tetravalent bonding configuration <cit.>, with the Si bonding to four C atoms in a graphene divancy, is also possible despite being slightly higher in energy <cit.>. However, while it would certainly reduce the corrugation of the lattice, that configuration is not possible to manipulate <cit.> and thus is of little practical interest to us.Finally, we should note that even larger structures can be designed and relaxed using the methodology of combining progressively more accurate simulations. For example, the unit cell with the 30-Si hexagon shown in the abstract figure contains 1152 atoms in total, yet it is even possible to obtain its high-quality wavefunctions due to the excellent parallelization of the Gpaw code. Unfortunately, despite modern computational resources, such system sizes are still prohibitively expensive for systematic studies, not the mention for simulating their electron beam stability and phonon modes using DFT <cit.>.§ CONCLUSIONSWe have presented here a multimodal approach for the efficient prediction of large embedded Si structures in graphene. After designing a Si pattern, its structure and unit cell size can be first roughly optimised using a classical bond-order potential. The structure can then be scaled to correct for the C–C bond length mismatch with respect to DFT, and then further relaxed using a computationally efficient atom-based orbital basis. 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KotakoskiSingle-atom spectroscopy of phosphorus dopants implanted into graphene,2D Materials 4(2), 021013 (2017). | http://arxiv.org/abs/1704.08019v1 | {
"authors": [
"Daryoush Nosraty-Alamdary",
"Jani Kotakoski",
"Toma Susi"
],
"categories": [
"cond-mat.mtrl-sci"
],
"primary_category": "cond-mat.mtrl-sci",
"published": "20170426090634",
"title": "Structure and energetics of embedded Si patterns in graphene"
} |
http://arxiv.org/abs/1704.08256v1 | {
"authors": [
"Asher Berlin"
],
"categories": [
"hep-ph",
"astro-ph.CO"
],
"primary_category": "hep-ph",
"published": "20170426180001",
"title": "WIMPs with GUTs: Dark Matter Coannihilation with a Lighter Species"
} |
|
[][email protected] Unidade de Educação a Distância e Tecnologia, Universidade Federal Rural de Pernambuco, Recife, Pernambuco 52171-900 Brazil In this work, we analyze the nonequilibrium thermodynamics of a class of neural networks known as Restricted Boltzmann Machines (RBMs) in the context of unsupervised learning. We show how the network is described as a discrete Markov process and how the detailed balance condition and the Maxwell-Boltzmann equilibrium distribution are sufficient conditions for a complete thermodynamics description, including nonequilibrium fluctuation theorems. Numerical simulations in a fully trained RBM are performed and the heat exchange fluctuation theorem is verified with excellent agreement to the theory. We observe how the contrastive divergence functional, mostly used in unsupervised learning of RBMs, is closely related to nonequilibrium thermodynamic quantities. We also use the framework to interpret the estimation of the partition function of RBMs with the Annealed Importance Sampling method from a thermodynamics standpoint. Finally, we argue that unsupervised learning of RBMs is equivalent to a work protocol in a system driven by the laws of thermodynamics in the absence of labeled data.07.05.Mh, 05.70.Ln Nonequilibrium Thermodynamics of Restricted Boltzmann Machines Domingos S. P. Salazar Received: date / Revised version: date ============================================================== § INTRODUCTION Neural networks learn from noisy environments by adjusting its internal configuration (or weights) in order to map input variables into known outputs (or labels). This type of learning is known as supervised and it requires a reasonable volume of labeled data. When this condition is met, supervised learning becomes extremely effective in a variety of applications, specially with deep architectures consisting on multiple layers of neurons <cit.>. More recently, a stochastic thermodynamic analysis of a supervised learning rule was successfully developed <cit.>, enhancing the understanding of supervised learning efficiency. Despite of the importance of supervised learning, most of the biologic systems learning tasks are likely to happen unsupervised, taking place in the absence of labeled data <cit.>. The general process of synaptic plasticity resulting in learning representations of the world from unlabeled sensory data seems essential in the quest for understanding intelligence <cit.>. Such important role in biologic systems poses a question whether unsupervised learning is also ruled by some fundamental laws such as thermodynamics. In addition to the desirable ability of learning representations, unsupervised learning also had major importance in the origins of deep learning <cit.>. Although shallow artificial neural networks (ANNs) have been used for a long time, the use of multiple layers of neurons in ANNs, the so called deep learning, had practical applications just recently <cit.>. Deep learning was put forward with the introduction of the contrastive divergence (CD) learning algorithm <cit.> to pre train a specific type of network, the Restricted Boltzmann Machine (RBM), as building blocks of deep architectures <cit.>. Those new ideas influenced older supervised successful algorithms <cit.> to be recast into speech <cit.> and image recognition <cit.> problems with new available data and computational power resulting in extraordinary performance. Deep learning applications are now used from high energy physics <cit.> and phase transitions <cit.> to genomics <cit.> and gaming <cit.>.In order to understand some fundamental laws of unsupervised learning, we study a type of stochastic neural network known as Restricted Boltzmann Machine (RBM) <cit.>. Beyond their important role in the development of deep learning, RBMs are closely related to physical systems <cit.>. The RBM has two layers (called visible and hidden layers) of binary units. There are connections between any neurons from different layers, but neurons from the same layer are not connected, therefore the network forms a bipartite graph. The RBM has a scalar energy function associated to each state of the network and the probability of finding a state is given by the Maxwell-Boltzmann (MB) distribution. The network is generative in the sense that it can be used to randomly create data (visible layer) from a given configuration of the hidden layer. The unsupervised training of the RBM requires a systematic adjustment of its weights until it is able to generate data from the training distribution with some accuracy. In this case, we say the network has learned the data distribution in an unsupervised way. The contrastive divergence (CD) algorithm <cit.> allows unsupervised learning of RBMs with a very simple learning rule for updating its weights. During the training process, the RBM learns internal (hidden) representations of the input (visible) data until it is able to generate random outputs that resembles the original data. The CD learning algorithm in RBMs has been used in image recognition <cit.> but also applied to learn other general data distributions, such as the Ising model <cit.>. Some adaptations to CD have been proposed <cit.>, but they remain based on the Gibbs sampling procedure, which is the rule used to randomly generate one layer of the network as a stochastic function of the other layer.In this paper, we show that unsupervised learning of RBMs is somehow driven by thermodynamics. We start by introducing a physical motivation for RBMs by considering the Gibbs sampling procedure as the systems discrete time dynamics. In this case, physical observables, such as the energy, fluctuate randomly in discrete time, akin to other systems from continuous stochastic thermodynamics <cit.>. As a consequence, the system behaves as a Markov chain and satisfies the detailed balance condition. Suitable definitions of thermodynamic work and heat are adapted from a framework <cit.> for discrete Markov chains. As the Gibbs sampling dynamics allows the RBM to exchange heat and perform work, the system behaves in accordance to the first law of thermodynamics. As expected from the framework used, we obtain the Crooks Fluctuation Theorem (CFT) <cit.> and the underlying second law of thermodynamics. We also show how the RBM, initially prepared in equilibrium at temperature T_1, obeys the heat exchange fluctuation theorem (XFT) <cit.> when placed in contact a reservoir of different temperature T_2, with excellent agreement to numerical simulations. Then, we analyze the contrastive divergence (CD) learning functional within the nonequilibrium thermodynamics framework and rewrite it in terms of physical quantities. Finally, we use the concepts presented in the paper to interpret Annealed Importance Sampling (AIS), a known method for estimating the partition function of RBMs, in the light of stochastic thermodynamics concepts.The paper is organized as follows. Section II introduces mathematical properties of RBMs and defines the thermodynamic observables resulting in the first law. Section III treats the derivation of nonequilibrium fluctuation theorems for the system compared to numerical simulations and obtains the second law. Section IV describes the unsupervised CD learning written in terms of nonequilibrium thermodynamic quantities. Section V uses the framework to interpret AIS method for estimating the partition function. Section VI contains conclusion and perspectives on unsupervised learning understood as a thermodynamic process. § FIRST LAW OF THERMODYNAMICS IN RBMS In this section, we review the formalism of Restricted Boltzmann Machines (RBMs) in the context of a discrete stochastic process. Then, we allow the weights (parameters of the RBM) to change in time, which in turn leads to a natural definition of thermodynamic observables (heat and work) and the underlying first law. The structure of RBMs <cit.> is composed by two layers of neurons (or units) with binary states. The visible layer (m units) is fully connected to the hidden layer (n units), however there are not connections between neurons in the same layer. The state s of the network is determined by a pair of vectors formed by the states of the visible (v) and hidden (h) neurons, s=(v,h), where v={v_i}, i=1,...,m, and h={h_j}, j=1,...,n. The neurons v_i and h_j assume values 0 or 1. In a given configuration λ, the energy of a given state s is defined:E(s,λ)=-∑_i=1^ma_iv_i-∑_j=1^nb_jh_j-∑_i,j=1^m,nv_iw_ijh_j,where s=(v,h) is a state and λ={a_i,b_j,w_ij} represents a configuration of weights (or parameters). The model also assigns a probability for each state of the network that depends only on its energy:p_λ(s)=1/Z(β,λ)e^-β E(s,λ),where the partition function, Z(β,λ)=∑_s e^-β E(s,λ), is the sum of the Boltzmann factor over all possible states of the network, assuring the probability adds up to 1. Notice that (<ref>) is the Maxwell-Boltzmann (MB) probability distribution from statistical mechanics. We have deliberately included the parameter β representing the inverse temperature (β=1/T, for k_B=1). Most original RBM formulations set β=1, because β is usually kept constant during simulations. Recently, temperature has been introduced as a parameter in the temperature based RBM for a variety of purposes <cit.>. The properties of the temperature based RBM remain unchanged with the introduction of a constant β, since the parameter could be rescaled in the original weights λ by a simple transformation, βλ→λ. However, the introduction of temperature allows the notion of different thermal reservoirs, which is a central motivation for the heat exchange fluctuation theorem (XFT) discussed in the next section.A very useful property of RBMs is the independence of the neurons from the same layer. Using (<ref>) and (<ref>) it can be deduced <cit.> the conditional probability of finding a hidden (visible) unit given a visible (hidden) vector:p_λ(h_j=1|v)=σ(β b_j+β∑_iv_iw_ij), p_λ(v_i=1|h)=σ(β a_i+β∑_jh_jw_ij),where σ(x)=1/(1+exp(-x)) is the sigmoid function. The subscript λ is explicitly written for clarity, since they will be adjusted during the learning process. This property above makes it possible to numerically estimate sample averages easily, which are used during training the parameters λ.Simulations on RMBs use (<ref>) and (<ref>) to generate a layer based on the opposite layer as a Markov chain. This is called the Gibbs sampling <cit.> and it works as if the dynamics of a RBM understood as a discrete time Markov chain. The conditional probabilities in the identity above can be written in terms of (<ref>) and (<ref>) asp_λ(v|h)=∏_i^m p_λ(v_i|h), p_λ(h|v)=∏_i^n p_λ(h_j|v).Although the probability (<ref>) is a conditional probability defined from the MB distribution (<ref>), the Gibbs sampling dynamics assigns it to the transition probability of a single step, p^(1)_λ(s→ s'), from a state s=(v,h) to a final state s=(v',h') in the discrete stochastic process: p^(1)_λ(s→ s')≡ p_λ(v'|h)p_λ(h'|v'),with p_λ(v'|h) and p_λ(h'|v') defined in (<ref>). The statistical dependence of the layers are depicted in Fig.1. This equivalence is the starting point of the thermodynamic analysis, since it defines a stochastic dynamics that encodes an arrow of time. In this case, it is clear that the dynamic process (<ref>) is a Markov chain, since by definition the probability of finding state s_K at time step K depends only on the previous state.In other words, the dynamics does not have a memory of previous states of the chain. It is also essential to notice that the Gibbs sampling dynamical process defined in (<ref>) satisfies detailed balance condition. For K=1 step, detailed balance reads for a constant λ:p_λ^(1)(s→ s')/p_λ^(1)(s'→ s)=p_λ(v'|h)p_λ(h'|v')/p_λ(v|h)p_λ(h|v')=p_λ(s')/p_λ(s),where the last identity was obtained using Bayes theorem. For multiple steps, K>1, notice that the transition probability may be written in terms of the one step transitions. For simplicity, we consider a constant λ, which is the relevant case for the heat exchange fluctuation theorem:p_λ^(K)(s→ s')=∑_s_1,...,s_K-2∏_i=0^K-1p^(1)_λ(s_i→ s_i+1),where p_λ^(K)(s→ s') is the transition probability of state s to state s' after K steps in the dynamics, s_0=s and s'=s_K. Upon using (<ref>) in (<ref>), one obtains the detailed balance condition also for K stepsp_λ^(K)(s→ s')/p_λ^(K)(s'→ s)=p_λ(s')/p_λ(s),where p_λ(s) and p_λ(s') are MB distributions (<ref>). As discussed in the next section, detailed balance plays a major role in the derivation of nonequilibrium fluctuation theorems (FTs).In general, during the discrete steps of the dynamics depicted in Fig.1, one could allow the weights λ to be adjusted as a function of time. Actually, the process of learning in RBMs (and other networks) is a type of weight adjustment and it can be done in many ways. In all type of learning rules, there will be an iterative change of parameters configuration, from λ={a_i,b_j,w_ij} to λ'={a_i',b_j',w_ij'}, where λ may be understood as an external set of controlled parameters. There is also noise from the stochastic (Gibbs sampling) dynamics itself. More precisely, defining the energy E(s_k,λ_k), of the configurations s_k generated with λ_k, given by (<ref>), the variation Δ E during a sequence of K steps, Σ=(s_0,...,s_K), is given by Δ E = E(s_K,λ_K)-E(s_0,λ_0). This variation can be conveniently written as a contribution of two factors as pointed in <cit.> for discrete Markov chains:Q=∑_k=0^K-1 E(s_k+1,λ_k+1)-E(s_k,λ_k+1), W=∑_k=0^K-1 E(s_k,λ_k+1)-E(s_k,λ_k),which can be understood as the heat and work for the trajectory Σ=(s_0,...,s_K). Definitions above result in the first law of thermodynamics for RBMs, since Δ E = W + Q. From the specific form the energy (<ref>) in RBMs one gets complete expressions for work and heat in terms of the neurons values (v,h) and the network configuration λ. Notice that during learning processes, there are changes in the weights from λ_k to λ_k+1, which allows the work to be different from zero in (<ref>). Alternatively, heat accounts for the energy variation due to stochastic change of states s_k→ s_k+1, even with the same configuration λ, in analogy with the thermodynamics observable also found in other stochastic systems <cit.>.§ FLUCTUATION THEOREMS AND THE SECOND LAW In this section, we use general properties of the dynamics of the RBMs as Markov chains to derive known fluctuation theorems and the second law of thermodynamics. There are complete reviews of fluctuation theorems (FTs) in Markov systems with continuous time dynamics <cit.>. Here we explore FTs in the discrete time dynamics observed in RBMs. §.§ Crooks Fluctuation Theorem and the Second Law In a discrete Markov chain, the system undergoes a given trajectory, Σ=(s_0,...,s_K), with the configuration being adjusted in a controllable protocol, Λ=(λ_0,...,λ_K). The work, W, defined in (<ref>) is a random variable that depends on the trajectory. In this case, the Crooks Fluctuation Theorem (CFT) <cit.> states a property for the probability density function of the random variable W as the identity: P_s_0→ s_K(W)/P_s_0→ s_0(-W)=e^β(W-Δ F),where P_s_0→ s_K(W) is the probability of finding the thermodynamic work over all trajectories going from state s_0 to state s_K. The variation of the free energy is defined in terms of the partition function, Δ F=-T log (Z(β,λ_K)/Z(β,λ_0)), for k_B=1. The theorem is valid when both the initial and final distributions are the equilibrium distribution. For RBMs, the derivation of CFT is easily adapted from the original formulation <cit.>, since the result was originally introduced for discrete Markov chains satisfying detailed balance, which is the case of this paper. The slight difference comes from the multidimensional control parameter λ. Therefore, we will keep the calculations brief and refer to the original when needed.Defining the backwards trajectory as γ'=(s_K,...,s_0), we can write from (<ref>) and the Markov property:P(γ)/P(γ')=p_eq(s_0)/p_eq(s_K)∏_k=0^K-1p_λ_k+1(s_k+1|s_k)/p_λ_k+1(s_k|s_k+1),where P(γ) is the probability of the trajectory γ. After rearranging the factors above and using detailed balance (<ref>), one gets:P(γ)/P(γ')= p_eq(s_0)/p_λ_0(s_0)p_λ_K(s_K)/p_eq(s_K)∏_k=0^K-1p_λ_k(s_k)/p_λ_k+1(s_k).Finally, using the explicit form of the equilibrium distributions in (<ref>) leads toP(γ)/P(γ')= p_eq(s_0)/p_λ_0(s_0)p_λ_K(s_K)/p_eq(s_K) e^β(W-Δ F),with W defined in (<ref>) as the work of the forward trajectory γ. Considering the initial and final distributions to be equilibrium distributions (<ref>) and summing over all possible trajectories with the same work leads to the identity (<ref>).A consequence of CFT (<ref>) is the Jarzynski equality (JE) <cit.>:⟨ e^-β W⟩ = e^-βΔ F.where the ensemble average above is taken over all possible trajectories also starting from configurations λ_0 to λ_K. Jensen's inequality, ⟨ exp(x) ⟩≥ exp⟨ x ⟩, applied in (<ref>) results in ⟨ W ⟩≥Δ F, which is the the second law of thermodynamics. Actually, by defining the Shannon entropy asS(β,λ)=-∑_sp_λ(s)logp_λ(s),and using (<ref>), one gets the expression for the entropy variation from configurations λ_0 to λ_K (with constant temperature):Δ S=β⟨Δ E ⟩ +log(Z(β,λ_K)/Z(β,λ_0)).Now using (<ref>) and (<ref>), the expression for the entropy finally gets the formΔ S=β⟨ Q ⟩ + β⟨ W ⟩ - βΔ F ≥β⟨ Q ⟩,where the inequality follows from ⟨ W ⟩ - Δ F ≥ 0, which represents the irreversible work in fine time processes <cit.>. Expression above is a common statement of the second law of thermodynamics. §.§ Heat Exchange Fluctuation Theorem In the absence of work, the energy variation in the RBM is totally due to heat exchange. When approaching equilibrium, the system's energy variation after a single discrete time step is expected to approach zero on average. However, when the RBM is prepared with a given temperature T_1 and then placed in thermal contact with a reservoir with a different temperature T_2, there will be a nonequilibrium fluctuation for the heat Q random variable (<ref>). The heat exchange fluctuation theorem (XFT) <cit.> states a identity for the nonequilibrium heat probability:P(Q)/P(-Q)=e^Q(β_1-β_2),for Q the heat transferred to the RBM, where β_1 and β_2 are the inverse temperatures of reservoirs 1 and 2, respectively. The identity holds for any number of steps K in (<ref>), where we have adapted the identity to our sign notation for the heat. The original derivation uses a small coupling between the two systems in consideration <cit.>, as well as continuous time dynamics. In the case of RBMs, as expected in other stochastic systems <cit.>, we show that XFT also follows from the discrete Markov dynamics with detailed balance as an exact result, without further assumptions on the magnitude of the coupling. We start by noticing that Δ E = Q, in the absence of work. The probability, P(Δ E), of finding the energy variation, Δ E, after any number K of steps is given in terms of the joint probability of states:P^(K)(Δ E)=∑_s,s' p_2^(K)(s→ s')p_1(s)δ(E'-E-Δ E),where s and s' are the initial and final states with energies E=E(s) and E'=E(s'). For clarity. the probabilities p_2 and p_1 are generated from (<ref>) and (<ref>), using T_2 and T_1, respectively, with the same constant λ in both cases (the λ subscript was omitted for simplicity). The function δ is defined as δ(x)=1, if x=0, and δ(x)=0 otherwise. Replacing the MB distribution p_1(s) using (<ref>) leads toP^(K)(Δ E)=∑_s,s' p_2^(K)(s→ s')e^-β_1 E/Z(β_1,λ)δ(E'-E-Δ E).The expression above can be rearranged easily after the introduction of the equilibrium distribution p_1(s')P^(K)=e^β_1 Δ E∑_s,s' p_2^(K)(s→ s')p_1(s')δ(E'-E-Δ E),and after exchanging the summation variables (s,s'), one getsP^(K)(Δ E)=e^β_1 Δ E∑_s',s p_2^(K)(s'→ s)p_1(s)δ(E'-E+Δ E).Applying the detailed balance (<ref>) in the transition probability p_2^(K)(s'→ s) above leads toP^(K)(Δ E)=e^(β_1-β_2)Δ E× ×∑_s',s p_2^(K)(s→ s')p_1(s)δ(E'-E+Δ E),where last double sum can be identified as P^K(-Δ E) from definition (<ref>). Finally, equation (<ref>) results in the identity P^(K)(Δ E)=e^(β_1-β_2)Δ EP^(K)(-Δ E),for any K, which is the original XFT identity (<ref>). In order to verity the XFT numerically, a simulation was implemented using a known RBM architecture (m=784,n=500) <cit.> for the task of image recognition of hand written digits (MNITS) <cit.>. Details on the unsupervised training of the RBM are given in Appendix (<ref>). After training the configuration λ, an ensemble of N=2·10^7 RBMs, with the same λ, was put in thermal equilibrium with the temperature T_1=1. The equilibrium was prepared after a Markov chain Monte Carlo simulation for several steps (K>100) from an initial random state and T_1=1. Finally, we take the final state s of each RBMs, supposedly in equilibrium, with energy E=E(s), and perform a single step in the dynamics (K=1) with a different temperature T_2 and same configuration λ. This procedure results in the state s' and energy E'=E(s'). The energy variation, Δ E = E'-E, is used to compute the numerical pdf from N=2·10^7 RMBs and the ratio P(Δ E)/P(-Δ E) is evaluated and displayed in FIG. <ref> for different final temperatures T_2={0.8,0.9,1.1,1.2}. In all cases, the XFTpredictions (<ref>) for the nonequilibrium case of K=1 step are remarkably consistent with the simulations. Notice that the ascending lines, T_2={1.2, 1.1}, suggest that the heating process (T_2>T_1) favors a positive variation of the energy (P(Δ E)/P(-Δ E)>1), as expected. Alternatively, the descending lines, T_2={0.8,0.9}, represent cooling processes (T_2<T_1) for which the energy of the RBM is expected to decrease (P(Δ E)/P(-Δ E)<1). It is important to notice that the results derived above (<ref>) are true for any parameter configuration λ, including a biased model fully trained over a data set as presented. The property of heat exchange has been observed experimentally for different physical systems <cit.>. The derivation presented above for a discrete Markov chain relying on detailed balance is general and it suits well the formalism of RBMs presented in this paper. The same approach and numerical simulation setup could possibly be applied in other learning systems with deep architectures <cit.>, where the RBMs are used as building blocks.§ UNSUPERVISED LEARNING AS THERMODYNAMIC PROCESSIn this section, we explore the unsupervised learning process of contrastive divergence (CD) in the context of nonequilibrium thermodynamics. First, we review the necessary notation of the CD. Then, we analyze the relation between the algorithm and thermodynamics. §.§ Contrastive Divergence (CD) The contrastive divergence (CD) algorithm <cit.> is one of the most successful unsupervised learning rules for RBMs. It works by updating the weights of a RBM iteratively so it better generates a given data distribution. Due to its simplicity and speed, several applications of RBM as generative models became possible <cit.>. In this section, we analyze CD in the framework of stochastic thermodynamics introduced in this paper. The algorithm is motivated by the optimization of the log likelihood function, L(λ,D), over a training data set D={v_i}^N_i=1 of m dimensional vectors, v_i, defined asL(λ,D)=∑_i=1^Nlog p_λ(v_i),where p_λ(v_i) the is the observed probability (<ref>) of v_i given by the model with configuration λ. A perfect model would reproduce training data exactly, thus p_λ(v_i)=1 for all i, resulting in L(λ,D)=0. But this ideal situation is not reachable in real data sets. Typically, one would adjust the weights, λ={a_i,b_i,h_ij}={θ}, of the generative model iteratively to maximize the log likelihood (<ref>), using a stochastic gradient ascent (SGA) approach <cit.>. In every iteration τ, the Maximum Likelihood (ML) learning increments each parameter, θ_τ, from the set λ_τ, asθ_τ+1 = θ_τ + η·∂_θ L(λ,D)|_λ_τ,where ∂_θ=∂/∂θ is a short notation for the partial derivative, the constant η is a positive learning rate and L(λ,D) is taken from (<ref>). Upon replacing (<ref>) in (<ref>), the RBM gets a simple form for the expression (<ref>), where the increments of the parameters may be easily represented as averages of the energy function (<ref>):∂_θ L(λ,D)= -β(⟨∂_θ E⟩_D-⟨∂_θE⟩_λ),where the partial derivatives are immediate due to the linear dependence of E(s,λ) taken from (<ref>) for each parameter {θ}={a_i,b_i,w_ij}. The expression⟨ f(v,h) ⟩_D represents the average of a function of the state s=(v,h) over the training data D:⟨ f(v,h) ⟩_D = ∑_v∈ D,hp_D(v)p_λ(h|v)f(v,h),with p_D(v) representing the relative frequency of v∈ D and p_λ(h|v) given in (<ref>). Similarly, the value ⟨ f(v,h) ⟩_λ represents the average of the function f(v,h) as evaluated by the RBM with parameters λ,⟨ f(v,h) ⟩_λ=∑_v,hp_λ(v,h)f(v,h),where p_λ(v,h)=p_λ(s=(v,h)), given by (<ref>). Inserting (<ref>) in (<ref>) leads to the increments for each parameter θ of the RBM:Δ a_i = -ηβ (⟨ v_i ⟩_D - ⟨ v_i ⟩_λ), Δ b_j = -ηβ (⟨ h_j ⟩_D - ⟨ h_j ⟩_λ), Δ w_ij = -ηβ (⟨ v_i h_j ⟩_D - ⟨ v_i h_j ⟩_λ),where the averages above are evaluated over a sample of the data (usually called a minibatch). Although the expressions for the learning rules (<ref>) are simple, the computation of (<ref>) is unfeasible in most architectures, since it would involve the knowledge of the partition function, Z(β, λ), which is a sum of 2^m· n Boltzmann terms. To avoid this problem, a Markov chain Monte Carlo (MCMC) method can be used to sample the equilibrium distribution by performing the Gibbs sampling dynamics (<ref>) for a very large number of iterations. However, the large number of iterations makes the algorithm very slow for practical use in big architectures.In this sense, contrastive divergence (CD) <cit.> is an idea that simplified the MCMC approach as it approximates the average in (<ref>) by n Gibbs steps drawn from the training data using the dynamics (<ref>), where the most simple case is n=1. Therefore, the learning rule (<ref>) for CD_n is given byθ_τ+1 = θ_τ + η·β(⟨∂_θ E⟩_D-⟨∂_θE⟩_n),In the data average, ⟨⟩_D, a sample of visible vectors, v ∈ D, is used to generate the hidden vectors, h, using (<ref>), and ⟨ f(v,h) ⟩_D is evaluated in the resulting ensemble of states {s=(v,h)}. Alternatively, the model average ⟨ f(v,h) ⟩_n represents the empirical average of the function f(v,h) after the application of a n steps Gibbs sampling from the dynamics (<ref>). In this case, a visible vector v' is generated from the hidden vector h using (<ref>) and a new hidden vector h' is generated from v' analogously using (<ref>). The process is repeated iteratively for n steps. Finally, ⟨ f(v',h') ⟩_n is evaluated as an average in the final ensemble of states of the type {s'=(v',h')}. §.§ Stochastic Thermodynamics of CD In the subsection above, it was argued that maximum likelihood (ML) learning deals with the maximization of a known functional (<ref>), but the gradient ascent steps (<ref>) require unfeasible computation. Contrastive divergence rule, CD_n, solves this issue by approximating the model average (<ref>), although this approximation does not ensure it maximizes the log likelihood (<ref>). Actually, CD_n learning <cit.> is equivalent to the minimization of the the following expression:CD_n=KL(p_Dp_λ)-KL(p_np_λ),where p_D is the data distribution, p_n is the resulting distribution after n Gibbs steps (<ref>) and p_λ is the model distribution (often written as p_∞). The functional KL(pq) is the Kullback-Leibler divergence, defined asKL(pq)=∑_sp(s)logp(s)/q(s),for probability distributions p and q, summed over all states s.Now we show that functional (<ref>) has sound physical interpretation based on the stochastic thermodynamics presented in Sec. <ref>. First, notice that the probability p_λ is known (<ref>), but there are not closed formulas for p_D and p_n. Upon replacing (<ref>) in (<ref>) and using definition (<ref>) one obtainsCD_n=-β∑_s p_n(s)E(s,λ)+β∑_s p_D(s) E(s,λ) -∑_s p_n(s)log p_n(s) + ∑_s p_D(s)log p_D(s),where the partition function Z(β,λ) has been conveniently canceled out. From the definition of averages (<ref>) and (<ref>) and using the definition of the Shannon entropy (<ref>), the expression above is rewritten as CD_n=-β(⟨ E(s,λ)⟩_n-⟨ E(s,λ)⟩_D)+ S_n - S_0.Notice that from the first term above can be written in terms of the stochastic heat defined in (<ref>) for a constant λ and n steps:Q_n=∑_k=1^n E(s_k+1,λ)-E(s_k,λ) =E(s_n,λ)-E(s_0,λ),which in turn allows one to write (<ref>) asCD_n= (S_n - S_0) -β⟨ Q_n⟩.The derivation above shows that the CD_n functional (<ref>) is composed of two terms. The first term is the variation of the Shannon entropy from the data distribution, p_D, to the nonequilibrium distribution, p_n. The second term is minus the average heat observed in the process of taking a data vector and subjecting it to n steps in the Gibbs sampling dynamics (<ref>). If the model distribution, p_λ, is close to the data distribution, p_D, their Shannon entropy difference is expected to be negligible, as well as the average heat observed in the process.Actually, the nonequilibrium expression (<ref>) is a measure of how irreversible is this process. The expression turns to a familiar form in the particular case of data being drawn from a MB distribution with configuration λ_D. By letting n→∞, the entropy difference is given by (<ref>), where λ_0=λ_D and λ_K=λ. In this case, (<ref>) becomes the irreversible workCD_∞=β⟨ W ⟩ -βΔ F,which is always positive. In other words, contrastive divergence (unsupervised learning) is approximately minimizing the difference between the entropy variation and the average heat (<ref>) in the process of taking a data vector and placing it in the RBM dynamics. The expression for the optimized functional, CD_n, is well defined in the nonequilibrium stochastic thermodynamics framework. When the number of steps is very large (n→∞) and data comes from a MB distribution, the stochastic thermodynamics expression turns to the familiar irreversible work (<ref>), where the partition functions (and the free energy) may be defined.§ APPLICATION IN ESTIMATION OF THE PARTITION FUNCTION In this section, we show that the Jarzynski Equality (JE) can be explored to estimate the partition function of RBMs with large architectures.Computing the partition function, Z=Z(β,λ), of a RBM model (<ref>) is necessary to find the probability of each state according to model. This is important for calculating the Log likelihood (<ref>) over a data set in order to estimate the performance of a trained model in the unsupervised learning task. However, most practical applications of RBMs uses architectures with large visible and hidden layers <cit.>, which makes the computation of Z(β,λ) unfeasible (as a sum of 2^m· n Boltzmann factors). Different methods have been proposed to estimate the partition function in RBMs in the recent years <cit.>. The Annealed Importance Sampling (AIS) <cit.> is a general method for estimating the expectation of some random variable x (or a function of it) drawn from some (intractable) distribution p(x). The idea is based on making a convenient sequence of intermediate distributions that converges to p(x). AIS found application in the estimation of the partition function of RBMs <cit.> with an excellent performance. In the original AIS formalism for RBMs, one defines p_λ(v)=p^*_λ(v)/Z(β,λ), where p_λ(v)=∑_hp_λ(s=(v,h)), obtained from (<ref>), so the estimate of Z(β,λ), for a configuration λ=(a_i,b_j,w_ij),can be written in terms of a known partition function Z(β,λ_0) as:Z(β,λ)/Z(β,λ_0)=∑_v p_λ^*(v)/p_λ_0^*(v)p_λ_0(v)=⟨p^*_λ(v)/p^*_λ_0(v)⟩_p_λ_0,where we used Z(β,λ)=∑_vp_λ^*(v) andZ(β,λ_0)^-1=p_λ_0(v)/p_λ_0^*(v) for any v. The known configuration λ_0 could be, for instance, the case λ_0=(a_i,b_j,0), for which the partition function can be computed analytically due to its separability (lack of interaction terms between the layers). In the last identity of (<ref>), we could slightly modify the original AIS formalism in order to sum over all possible states of the RBM, s=(v,h), for a clearer interpretation within thermodynamics. It results in an equivalent expressionZ(β,λ)/Z(β,λ_0)=∑_s p_λ_0(s)e^-β (E(s,λ)-E(s,λ_0)),where the definition Z(β,λ)=∑_s e^-β E(s,λ) was used, as well as the identity Z(β,λ_0)^-1=p_λ_0(s)e^β E(s,λ_0) that comes from (<ref>). One can easily notice that the exponents in (<ref>) are the stochastic work (<ref>) defined in the thermodynamics formalism for a system prepared at equilibrium (β,λ_0) after a single (k=1) Gibbs step (<ref>) with parameters (β,λ). So it is immediate that one could estimate Z(β,λ) by computing the average of a stochastic quantity e^-β W, a function of the stochastic work, W, over a ensemble of states starting from a known MB distribution with configuration (β,λ_0). However, the quality of such estimate depends on the size of the ensemble, and since λ_0 and λ may differ greatly, the variance of sampling (<ref>) (the single step protocol) may be large.Fortunately, a refinement of this single step estimate can be done by considering a slow protocol going from the configuration (β, λ_0) to (β, λ=λ_K) as sequence of intermediate steps λ_k. Notice that, using (<ref>), the ratio between partition functions Z(β, λ_K) and Z(β, λ_0) can be written as the productZ(β,λ_K)/Z(β,λ_0)=∏_k=0^K-1Z(β,λ_k+1)/Z(β,λ_k)=⟨ e^-β W⟩,since the intermediate factors cancel out in the first identity above. The last identity follows from using (<ref>) in each factor of the product. In this expression, the average is taken over all possible trajectories Σ=(s_0,...,s_K), with the weights being adjusted in a controllable protocol, Λ=(λ_0,...,λ_K). Notice that the ratio of the partition functions in (<ref>) may be written in terms of the free energy, βΔ F = -log(Z(β,λ)/Z(β,λ_0)), which makes the expression equivalent to the Jarzynski equality (<ref>), obtained independently <cit.> from AIS. This evidence supports the claim that AIS and JE provides essentially the same method for computing the partition function, as claimed originally in <cit.>. A close inspection in (<ref>) shows that it could only be used in the step k of the protocol Λ in (<ref>) if the system is approximately in equilibrium in the configuration (β,λ_k-1). Therefore, the protocol Λ should be slow enough (||λ_k-λ_k-1||≪1) to account for this condition. Physically, this protocol represents a quasi-static (reversible) isothermal “expansion” (or “compression”), where the configuration λ could be understood as an external set of controlled parameters, akin to the volume of the system in equilibrium thermodynamics. By performing a slow protocol, the system is managed to stay locally in thermal equilibrium, ie, its nonequilibrium distributions are approximately MB during the whole process.Actually, the physical condition of a quasi static process Λ is met in real AIS estimations of the partition function <cit.>. Typically, simulations regarding AIS uses a specific time protocol λ_t=(a_i,b_i,w_ij(t)), where the weights a_i and b_i are constant and the interaction terms w_ij(t) goes from 0 to w_ij in a linear or exponential behavior. Due to the functional form of the MB distribution (<ref>), this sort of transformation in the the configuration parameters resembles a fine tuning in the inverse temperature of the system. Therefore the process is often seen as a simulated annealing approach. However, we point out that the work protocol interpretation of the AIS approach presented in this section (<ref>), in which the temperature is held constant, corroborates with the stochastic thermodynamics framework of RBMs.§ SUMMARY AND CONCLUSIONSIn this paper, we have analyzed Restricted Boltzmann Machines (RBMs) in a stochastic thermodynamics approach. We start by presenting the RBM as discrete Markov chains satisfying detailed balance condition. This property allowed us toadapt the framework <cit.> to define the stochastic heat and work, leading to the first law of thermodynamics.We highlighted nonequilibrium fluctuation theorems arising from this approach. Notably, the Crooks Fluctuation Theorem (CFT) followed immediately, since it was originally derived in the context of Markov chains. The Jarzynski equality (JE) and the second law of thermodynamics followed from CFT as expected. Then, the heat exchange fluctuation theorem (XFT) was derived for any configuration of RBMs, which differs from its original presentation <cit.> based on a hamiltonian with a small thermal coupling. Our presentation uses the general facts that the equilibrium distribution is MB (<ref>) and the detailed balance condition. Numerical simulations in a fully trained RBM shows excellent agreement with XFT predictions in the nonequilibrium case of a single Gibbs step in the dynamics, K=1, for both heating and cooling situations.We also interpreted the known contrastive divergence (CD) unsupervised learning algorithm <cit.> in the context of stochastic thermodynamics. We showed that the CD_n functional of two distributions (p,q), defined in terms of the Kullback-Leibler divergence, can be written as thermodynamic observables. It turns out that CD_n is a measure of how irreversible is the process of propagating data vectors with the RBM dynamics. Namely, the CD_n functional to be optimized in the learning process is the difference between the entropy variation and the average stochastic heat of that process. In the particular case of MB distributions, for a infinite number of steps, the expression is reduced to the known irreversible work, which plays important role in stochastic thermodynamics <cit.>.Finally, we presented how the ratio of partition functions may be estimated by averaging a thermodynamic observable. The derivation, that is closely related to the Jarzynski equality (JE), is mathematically equivalent to the widely used Annealed Importance Sampling (AIS) algorithm, as claimed in the original derivation of AIS <cit.>. The difference of interpretations being that, in the stochastic thermodynamics framework, a work protocol is produced in the estimation of the partition function at constant temperature. A process that resembles a physical isothermal transformation, opposed to the original annealing interpretation in which the temperature is slowly changed during the process.We point that artificial neural networks (ANNs) have produced astonishing results over the years for image, text and speech recognition, specially when stacked in form of multiple layers, known as deep learning. Most of the applications, including the ones observed in physics <cit.> are trained supervised, a situation that requires a lot of labeled data. Actually, the vast majority of available data in the world is not labeled. Biologic systems also learn representations of the world from sensory data in a unsupervised manner. These observations make generative models to be speculated as the next frontier in artificial intelligence, for which the Restricted Boltzmann Machines are a type of building block. The results presented in this paper supports that unsupervised learning obey general rules observed in thermodynamics, mostly due to the fundamental properties of its dynamics, such as detailed balance. The results are also general enough and could possibly be extended to deep ANNs such as the Deep Belief Network (DBN).§ ACKNOWLEDGMENTS This work was supported by Fundação de Amparo à Ciência e Tecnologia de Pernambuco (FACEPE) under grant APQ 0073-1.05/15.§ EXPERIMENTSIn this section, the known benchmark MNIST data set is briefly described. We also provide details on the unsupervised learning experiment on MNIST that produced the heat exchange fluctuation theorem (XFT) of subsection <ref>.§.§ MNIST data setThe MNIST is a data set of images of handwritten digits of size 28×28 pixels <cit.>. It is widely used as a benchmark for machine learning algorithms. The set contains 60,000 images used for training and 10,000 images used for testing. The images were binarized, so pixel values are either 0 or 1. Since its creation, several algorithms have reached very low error rates for the supervised (or labeled) problem, aimed to classify an image in one of the 10 categories (digits). However, the goal of section (<ref>), as in previous unsupervised learning applications of RBMs <cit.>, is to learn the handwritten data distribution by trying to optimize the log likelihood (<ref>). It means the algorithm should try to generate the original distribution of MNIST images as close as possible to the original set, without using any information of the image labels. §.§ Training the RBMThe training procedure for the contrastive divergence algorithm is straightforward. We use the increments for the parameters λ=(a_i,b_j,w_ij) from (<ref>). It is worth to point out that the average in D is to be understood as taken from data (positive phase) and the average n=1 is taken from the reconstructed image (negative phase) <cit.>, after a single Gibbs sampling starting from the original image.The RBM has the same architecture with m=784 (representing 28×28 pixels of MNIST) and n=500 neurons in the visible and hidden layers, respectively. We initialize the biases (a_i,b_j) at zero and w_ij from a uniform distribution (from -0.1 to 0.1). The training set is split in 600 minibatches of 100 images. For each iteration, all the images of a minibatch are used to generate the positive and negative phases, used to compute the increments of the weights (<ref>) and the weights are updated. Passing through all minibatches is called an epoch. In our experiment, the learning rate was set η=0.004 for 300 epochs. A linear weight decay of α=10^-4 was used and momentum was set to 0. The inverse temperature is a constant β_1=1. 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"authors": [
"Domingos S. P. Salazar"
],
"categories": [
"cond-mat.stat-mech"
],
"primary_category": "cond-mat.stat-mech",
"published": "20170427192337",
"title": "Nonequilibrium Thermodynamics of Restricted Boltzmann Machines"
} |
]Tuning the collective decay of two entangled emitters by means of a nearby surface ^1 Dipartimento di Fisica e Chimica, Università degli Studi di Palermo, Via Archirafi 36, 90123 Palermo, Italy^2 INFN, Laboratori Nazionali del Sud, 95123 Catania, Italy ^3 Physikalisches Institut, Albert-Ludwigs-Universitä̈t Freiburg, Hermann-Herder-Str. 3, 79104 Freiburg, Germany^4 Freiburg Institute for Advanced Studies, Albert-Ludwigs-Universitä̈t Freiburg, Albertstr. 19, 79104 Freiburg, Germany We consider the radiative properties of a system of two identical correlated atoms interacting with the electromagnetic field in its vacuum state in the presence of a generic dielectric environment. We suppose that the two emitters are prepared in a symmetric or antisymmetric superposition of one ground state and one excited state and we evaluate the transition rate to the collective ground state, showing distinctive cooperative radiative features. Using a macroscopic quantum electrodynamics approach to describe the electromagnetic field, we first obtain an analytical expression for the decay rate of the two entangled two-level atoms in terms of the Green's tensor of the generic external environment. We then investigate the emission process when both atoms are in free space and subsequently when a perfectly reflecting mirror is present, showing how the boundary affects the physical features of the superradiant and subradiant emission by the two coupled emitters. The possibility to control and tailor radiative processes is also discussed.[ P. Barcellona^3 and S. Y. Buhmann^3,4 December 30, 2023 ========================================= § INTRODUCTION Spontaneous emission processes by multi-atom systems coherently coupled to the electromagnetic field have been extensively explored in the literature since the seminal work by Dicke in 1954 <cit.>, in which the collective emission of N identical atoms with interatomic separation much shorter than the wavelength of the atomic transition was considered. It was shown that the resulting radiation intensity was proportional to N^2, rather than N (as expected for the intensity radiated by independent atoms), with a decay time proportional to the inverse of the number of emitters <cit.>. This phenomenon is commonly known as Superradiance and it occurs when the sample is prepared in a symmetric superposition of atomic states. The underlying physical process responsible for such enhanced radiative behavior is the constructive interference between emitted waves and researchers have pursued and explored it by studying quantum emitters coupled to various environments including microcavities <cit.> or plasmonic waveguides <cit.> and left-handed metamaterials <cit.> as well as classical emitters near a metal interface <cit.>, withthe potential applications in quantum communication <cit.>. A closely related process is the collective Lamb Shift of an ensemble of many identical two-level atoms interacting cooperatively with a resonant radiation field. In this case, a virtual photon emitted by one atom may be reabsorbed by another one within the ensemble and, as for the superradiant emission, the process sensitively depends on the spatial arrangement of the system. Contrarily to superradiance, this process is difficult to observe at high atomic densities because of atom-atom interactions which tend to mask it. However, it becomes experimentally accessible in the case of an extended sample, for which R≫λ_0 (R being the interatomic separation and λ_0 the atomic transition wavelength) as it has been explored for an ensemble of Fe atoms embedded in a low-Q planar cavity <cit.>, for a confined vapor nanolayer of Rb atoms <cit.> and for a cold-atom ensemble <cit.>.The counterpart of superradiance is Subradiance <cit.>, occurring when the system is prepared in an antisymmetric state. In this situation, destructive interference between the neighboring radiators leads to a drastic suppression of the emission intensity, which is vanishing when the interatomic distance is much smaller than the atomic transition wavelength. Contrarily to superradiance, subradiance is more elusive, mainly because the corresponding states are weakly coupled to the environment. They are not affected by decoherence, therefore appealing for quantum information processing. Indeed in Ref. <cit.> the authors propose a combined optical/solid-state approach to realize a quantum processor based on "subradiant dimers" of quantum dots, resonantly coupled by dipole-dipole interaction and implanted in low-temperature solid host materials at controllable nanoscale separations. Other works focused on the controlled production of superradiant and subradiant states when the artificial atomic systems are tightly confined in optical lattices <cit.> or in quantum electrodynamics (QED) circuits <cit.>.In this framework, the main aim of the present paper is to investigate the influence of a structured environment on the collective spontaneous emission of a system of two identical correlated atoms. By using macroscopic quantum electrodynamics <cit.> to characterize the electromagnetic field in the presence of macroscopic magnetodielectric objects, the first result we have obtained is an analytical expression for the decay rate of the joint two atom-field system, prepared in their symmetric (antisymmetric) entangled state in terms of the Green's tensor of a general structured environment.We then use this result to analyze the radiative behavior of the two entangled quantum emitters in two specific cases: when they are located in free space for any interatomic distance, so recovering the known superradiant and subradiant features in particular in the subwavelength spatial region; when a perfectly reflecting planar surface is present, showing how its presence significantly affects the physical features of the collective spontaneous emission. We explicitly consider the cases when both atomic dipoles are parallel or perpendicular to the line connecting them. Our results suggest the possibility to tune and manipulate the system's superradiant and subradiant behavior by suitably placing the two radiators with respect to the reflecting boundary.The paper is organized as follows. Section 2 outlines the basic theory of macroscopic quantum electrodynamics, presenting the main properties of the Green's tensor we have exploited in our analysis. Section 3 is dedicated to the methodology and results. Through a suitably adapted version of the Wigner-Weisskopf theory <cit.> we find the decay probability of the initial state of our joint two atom-field system that we then apply in two specific situations: atoms in free space and in the presence of a perfectly reflecting mirror, illustrated in Sections 4 and 5, respectively. Our concluding remarks are given in Section 6.§ MEDIUM-ASSISTED FIELD FORMALISM As mentioned, the technique we exploit to describe the body-assisted electromagnetic field and to investigate the collective radiative behavior of our two-atom system embedded in a general macroscopic environment, is based on the Green's tensor formalism <cit.>. This method has been widely applied in various contexts, ranging from QED to quantum optics since the Green's tensor includes all the properties of the electromagnetic field with which the system interacts. It is thus a general method to study matter-field interaction in any external environment. As an example, this formalism has been recently applied to the study of the dynamical Casimir-Polder force between a chiral molecule and a plate <cit.> and the van der Waals interaction between a ground and an excited atom in a generic environment <cit.>. In addition, the macroscopic QED approach has been exploited to unveil cooperative effects in a two-atoms system in the presence of left-handed media <cit.> and in determining how the Casimir-Polder interaction energy of a polarizable atom is affected by a metallic surface <cit.>.In this section, we introduce some properties of the electromagnetic Green's tensor involved, starting with the quantized body-assisted electric field <cit.> 𝐄(𝐫) =∫_0^∞dω𝐄(𝐫,ω)+h.c.==∫_0^∞dω∑_λ=e,m∫ d^3r'𝐆_λ(𝐫,𝐫',ω)·𝐟_λ(𝐫',ω) +h.c. where 𝐆(𝐫,𝐫^',ω) is the Green's tensor, 𝐟_λ(𝐫,ω) and 𝐟_λ^†(𝐫,ω) are bosonic annihilation and creation operators representing the collective excitations of the body-field system, satisfying the following commutation rules [𝐟_λ(𝐫,ω),𝐟_λ'(𝐫',ω')]=0, [𝐟_λ(𝐫,ω),𝐟_λ'^†(𝐫',ω')]=δ_λλ'δ(𝐫-𝐫')δ(ω-ω'). where the subscript λ=e,m identifies the electric and magnetic parts. The 𝐆_λ(𝐫,𝐫',ω) obeys the following integral relation ∑_λ=e,m∫ d^3s𝐆_λ(𝐫,𝐬,ω) ·𝐆_λ^*T(𝐫^',𝐬,ω)==ħμ_0/πω^2Im𝐆(𝐫,𝐫^',ω). According to the geometry of the chosen environment, generally one needs to consider two contributions to the total tensor 𝐆(𝐫,𝐫^',ω): 𝐆^(0)(𝐫,𝐫^',ω), describing the electromagnetic field in the bulk region of the medium, and a scattering part 𝐆^(1)(𝐫,𝐫^',ω) accounting for additional contributions due to reflections at or transmission through the boundaries. For the purpose of this work, a perfectly reflecting planar surface will be considered as macroscopic environment for the two emitters. The analytical expressions for the bulk and scattering parts of the Green's tensor are, respectively, the free space Green's tensor <cit.>, <cit.> 𝐆^(0)(𝐫,𝐫^',ω)= -c^2/3ω^2δ(ϱ)-c^2e^iωϱ/c/4πω^2ϱ^3{[1-iωϱ/c....-(ωϱ/c)^2]𝐈-[3-3iωϱ/c-(ωϱ/c)^2]𝐞_ϱ𝐞_ϱ} where ϱ=𝐫-𝐫^', ϱ=|ϱ|, 𝐞_ϱ=ϱ/|ϱ| and <cit.> 𝐆^(1)(𝐫,𝐫^',ω)=𝐆^(0)(𝐫,𝐫^'*,ω)·𝐑, with 𝐫^'* being the position of the source's image behind the mirror and 𝐑 the reflection matrix defined by 𝐑=([ -100;0 -10;001 ]). The scattering component 𝐆^(1)(𝐫,𝐫^',ω) is related to the free-space Green's tensor at the distance ϱ_+, that is the distance that a reflected wave on the mirror needs to travel from one atom to reach the other. An alternative interpretation is that the interaction occurs between one atom, say A, located at 𝐫, and the mirror image of atom B, located at 𝐫^'*, behind the plate.Moreover, given two oriented dipoles 𝐝_A=(d_Ax,d_Ay,d_Az) and 𝐝_B=(d_Bx,d_By,d_Bz), their images are constructed by a spatial reflection at the z=0 plane, together with an interchange of positive and negative charges: 𝐝_A^*=𝐑·𝐝_A,𝐝_B^*=𝐑·𝐝_B. Two additional useful properties are the following: the free space Green's tensor is symmetric under exchange of its spatial arguments 𝐆^(0)(𝐫,𝐫^',ω)=𝐆^(0)T(𝐫^',𝐫,ω)=𝐆^(0)(𝐫^',𝐫,ω), a key feature in our computation of the system's collective decay rate in the following; its imaginary part is finite for 𝐫=𝐫^' and reads Im𝐆^(0)(𝐫,𝐫,ω)=ω/6π c𝐈. with 𝐈 being the unit tensor.The next sections will focus on the methodology we use to evaluate the collective decay rate of the two entangled atoms when they are located in an arbitrary macroscopic environment, first, and subsequently when a perfectly reflecting surface is present. § THE COLLECTIVE SPONTANEOUS DECAY RATE We use an approach based on second-order time-dependent perturbation theory holding for a general physical system in the presence of an external perturbation. It leads to the following result C_i(t)=C_i(0)e^-i(_i-i_i/2)t/ħ for the probability C_i(t) to find the system in its initial state |i⟩, of energy ħω_0, at some instant of time t <cit.>. The above expression contains an energy shift _i and the decay rate of the amplitude C_i(t), given by _i=P.V.(∑_I≠ i|⟨ I|Ĥ_1|i⟩|^2/ħ(ω_i-ω_I))_i=2π/ħ∑_I≠ i|⟨ I|Ĥ_1|i⟩|^2δ(ħ[ω_i-ω_I]). respectively, where Ĥ_1 is the perturbation, P.V. indicates the principal value and |I⟩ the intermediate states involved in the process. In the following we will focus on the decay rate.We thus first define the total Hamiltonian and the initial state vectors of our composite two-atoms and field system Ĥ=Ĥ_0+Ĥ_1=∑_ξ=A,BĤ_at(ξ)+Ĥ_F+∑_ξ=A,BĤ_int(ξ) where Ĥ_at(ξ) is the Hamiltonian of the atomic species ξ, Ĥ_F the field Hamiltonian and Ĥ_1 the interaction Hamiltonian of the two atoms with the electromagnetic field. We consider electric atoms and adopt the multipolar coupling scheme in the dipole approximation, so that Ĥ_1=-𝐝_A·𝐄(𝐫_A)-𝐝_B·𝐄(𝐫_B) with 𝐝_ξ the transition dipole moment operator of atom ξ and 𝐄(𝐫_ξ) the electric field operator evaluated at the atomic position 𝐫_ξ. In addition, we need to specify the initial state of the system. A generic eigenvector of Ĥ_0 is expressed as product of atomic and field states with eigenenergy given by the sum of the energy of the atomic species and of the photons in some field modes. Here, we consider two identical atoms prepared in the correlated symmetric (antisymmetric) state |i_±⟩ =1/√(2)(|e_A,g_B;{ 0}⟩±|g_A,e_B;{ 0}⟩) with the atomic excitation being delocalized over the two atoms, and the field in the vacuum state (e and g indicate ground and excited states, respectively). These are the superradiant (symmetric) and subradiant (antisymmetric) states in the Dicke model.Due to the expression of the interaction Hamiltonian, only two intermediate states contribute|I_1⟩ =|g_A,g_B;1_λ(𝐫,ω)⟩, |I_2⟩ =|e_A,e_B;1_λ(𝐫,ω)⟩so that, using the expression of the electric field in terms of the Green's tensor, given in equation (<ref>), the matrix elements for the intermediate states |I_1⟩ and |I_2⟩ can be now evaluated. By collecting equations (<ref>),(<ref>),(<ref>) and (<ref>), one finds the matrix element for the symmetric correlated state⟨ i_+|Ĥ_1|I_1⟩ = -1/√(2)[𝐝_A^eg·𝐆_λ(𝐫_A,𝐫,ω)+𝐝_B^eg·𝐆_λ(𝐫_B,𝐫,ω)]where we have assumed that the dipole operator 𝐝 has only off-diagonal matrix elements due to selection rules and 𝐝_ξ^eg are matrix elements between the excited and the ground state. To compute ⟨ I_1|Ĥ_1|i_+⟩ it is sufficient to take the hermitian conjugate of (<ref>). Similarly, the matrix element containing the second intermediate state |I_2⟩ is ⟨ i_+|Ĥ_1|I_2⟩ = -1/√(2)[𝐝_B^ge·𝐆_λ(𝐫_𝐁,𝐫,ω)+𝐝_A^ge·𝐆_λ(𝐫_𝐀,𝐫,ω)]and the hermitian conjugate can be computed. The matrix elements for the antisymmetric correlated state can be found in an analogous way.By exploiting the integral relation fulfilled by the Green's tensor equation (<ref>), from (<ref>) we can evaluate the decay rate of the initial state of the system, when the two atoms are placed in the environment described by the Green's tensor 𝐆(𝐫,𝐫',ω) <cit.>:_i±= μ_0ω_0^2/ħ[𝐝_A^eg·Im𝐆(𝐫_A,𝐫_A,ω_0)·𝐝_A^ge+𝐝_B^eg·Im𝐆(𝐫_B,𝐫_B,ω_0)·𝐝_B^ge±𝐝_A^eg·Im𝐆(𝐫_A,𝐫_B,ω_0)·𝐝_B^ge±𝐝_B^eg·Im𝐆(𝐫_B,𝐫_A,ω_0)·𝐝_A^ge] with ω_0 being the atomic transition frequency and the + or - signs refer to the symmetric or antisymmetric state.When the atoms satisfy time-reversal symmetry 𝐝^ge=𝐝^eg, the above expression (<ref>) can be written in the following form_i±=_A+_B/2±_AB.where:_A=2μ_0ω_0^2/ħ𝐝_A^eg·Im𝐆(𝐫_A,𝐫_A,ω_0)·𝐝_A^geis the single-atom decay rate and_AB=2μ_0ω_0^2/ħ𝐝_A^eg·Im𝐆(𝐫_A,𝐫_B,ω_0)·𝐝_B^geis an interference term. Equations (<ref>)describe the overall decay rates of the two atom-field system prepared in a symmetric and antisymmetric state, where the excitation energy is delocalized over the two atoms. These relations are given in terms of the Green's tensor of a generic structured environment surrounding the system and contain both terms evaluated at the position of the single atoms, corresponding to the decay rates of independent atoms, and terms describing interference effects, as we shall discuss in detail in the following sections. The reason for the non-vanishing interference term is that, in the transition from the correlated initial state to the intermediate state, it is not possible to know which atom emits the photon. § ATOMS IN FREE SPACE The method we followed in the previous paragraph has led to the expression (<ref>) which can be applied to any external environment, provided the form of the Green's tensor of the specific environment is known.We first analyze the behaviour of the decay rate in free space varying the interatomic separation and hence observing it in the non-retarded short-distance limit (ϱ=|𝐫_B-𝐫_A|≪ c/ω_0)and in the retarded long-distance limit (ϱ=|𝐫_B-𝐫_A|≫ c/ω_0). This allows us to recover the known superradiant and subradiant behaviours.We use the free-space Green's tensor 𝐆(𝐫,𝐫',ω)=𝐆^(0)(𝐫,𝐫',ω), choosing the axis in order to have 𝐫_𝐀=(0,0,𝐳_𝐀) and 𝐫_𝐁=(0,0,𝐳_𝐁), then ϱ=𝐫_A-𝐫_B=(0,0,z) with z=z_A-z_B. Furthermore, we consider dipole moments with arbitrary spatial orientations: 𝐝_A=(d_Ax,d_Ay,d_Az) and 𝐝_B=(d_Bx,d_By,d_Bz).Since the free space Green's tensor is symmetric with respect to exchange of its position arguments, the mixed terms in equation (<ref>) coincide, thus giving the collective decay rate in the following form_i+= μ_0ω_0^2/ħ[𝐝_A^eg·Im𝐆^0)(𝐫_A,𝐫_A,ω_0)·𝐝_A^ge+𝐝_B^eg·Im𝐆^0)(𝐫_B,𝐫_B,ω_0)·𝐝_B^ge+2𝐝_A^eg·Im𝐆^0)(𝐫_A,𝐫_B,ω_0)·𝐝_B^ge] Equivalently _i+=_A+_B/2+_AB, by identifying the single-atom contributions _A and _B (free-space spontaneous decay of an independent excited atom), and the "interference" term 2_AB by the equations: _A= | 𝐝_A^eg|^2ω _0^3/3πε _0ħ c^3, _AB = 1/2πε _0ħ z^3( ( 𝐝_A^eg·𝐝_B^ge - 3d_Az^eg d_Bz^ge)×( λcosλ - sinλ)+ ( 𝐝_A^eg·𝐝_B^ge - d_Az^eg d_Bz^ge)λ^2sinλ)where λ=z ω/c <cit.>.Firstly, we take the imaginary part of the free space Green's tensor (<ref>) <cit.> which will hence be applied to the first two terms on the right-hand side of equation (<ref>). Secondly, we consider the two asymptotic behaviors, starting with the nonretarded limit ϱ≪ c/ω_0.When 𝐫_A→𝐫_B (short-distance regime), Im𝐆(𝐫_A,𝐫_B,ω_0)→ω_0/6π c𝐈 and the interference term in equation (<ref>) yields _AB→_A+_B/2 (when 𝐝_A and 𝐝_B are parallel, otherwise it vanishes) so that _i+^nret=_A+_B. In the non-retarded limit of very small interatomic distances, the transition rate of the system to the collective ground state is the sum of the two individual atom's rates, showing the superradiant decay of the initial state (<ref>).In the opposite case, the retarded limit ϱ≫ c/ω_0, the interference term vanishes due to the Green's tensor boundary condition 𝐆^(0)(𝐫,𝐫',ω)→0 for |𝐫-𝐫'|→∞. Therefore, we find _i+^ret= _A+_B/2For intermediate distances ϱ, the decay rate displays an oscillatory behaviour due to the presence of periodic functions in the imaginary part of the free-space Green's tensor, scaling with the interatomic distance z as sin(zω_0/c)/z. The oscillation will be damped as the two atoms are further far apart, yielding the independent-atoms decay at large distances. In Figure <ref>, we display the ratio between the total decay rate and the sum of decay rates from independent atoms, so that the non-retarded and retarded limit behaviours are clearly visible. We considered the transition frequency for the 2p→1s transition of the Hydrogen atom.With regard to the blue continous curve, a peak is observed for the minimum atomic separation considered (see equation (<ref>)). This is a signature of the cooperative behaviour (superradiance) due to a constructive correlation between the two identical atoms. The emission rate rapidly decreases with increasing distances, leading to the ordinary spontaneous emission of independent atoms in the limit of large interatomic separations.Furthermore, whenever the distance between the two atoms is ϱ=(n+1/4)λ_0 with n∈ℕ^+ positive integer, the transition rate is increased, compared to that of independent atoms. On the contrary, when ϱ=(m+3/4)λ_0 with m∈ℕ, we observe a reduction of the transition probability rate. The points corresponding to ϱ=nλ_0/2 indicate a vanishing interference term. Analogue behaviours are observed in <cit.> where the authors study the emission rate of a bi-atomic system prepared in a correlated symmetric state and interacting with the massless scalar field.The transition rate from the antisymmetric state to the collective ground state in free space is given by_i-= μ_0ω_0^2/ħ[𝐝_A^eg·Im𝐆(𝐫_A,𝐫_A,ω_0)·𝐝_A^ge+𝐝_B^eg·Im𝐆(𝐫_B,𝐫_B,ω_0)·𝐝_B^ge-2𝐝_A^eg·Im𝐆(𝐫_A,𝐫_B,ω_0)·𝐝_B^ge] that we can write as_i-=_A+_B/2-_AB. In this case, the behaviour is completely different for the non-retarded limit ϱ≪ c/ω_0. In fact, since in this limit 2_AB→_A+_B, a complete inhibition of the spontaneous transition rate due to destructive interference of quantum correlations between the two atoms is observed, recovering the known subradiant behaviour. In contrast with the previous case, we get a lower transition rate for ϱ=(n+1/4)λ_0, while for ϱ=(m+3/4)λ_0 the interference effects lead to an enhancement of the output rate, as displayed in Figure <ref> (red dashed line). As before, the points corresponding to ϱ=nλ_0/2 indicate a vanishing interference term. Similar results for the antisymmetric state are also obtained in the massless scalar field case discussed in <cit.>.In both cases we find that the quantum interference term between the atoms produces vanishing contributions for large interatomic separations ϱ≫ c/ω_0. The influence of the quantum interference is stronger for short distances between the atoms, compared to their transition wavelength. In the free-space case, our results thus agree with those reported in Refs.<cit.> with different methods. Superradiance has been observed in experiments both with atoms <cit.> and quantum dots <cit.>.§ ATOMS NEAR A PERFECTLY REFLECTING PLATEHaving obtained in Section 3 the general expression for the decay rate of the two entangled atoms in terms of the Green's tensor, we can consider other environments by using the relative expression of 𝐆(𝐫,𝐫^',ω). In this section we will focus on how the cooperative emission rate is affected when a perfectly reflecting mirror is present.Let us consider both atoms aligned along the z-axis perpendicular to the surface as shown in Figure <ref>. In addition to the interatomic distance, the presence of the mirror introduces a new spatial parameter: ϱ_+=z_A+z_B, that is the distance between one atom and the mirror image of the other behind the plate. In this physical configuration, two terms contribute to the system's decay rate Γ=Γ^(0)+Γ^(1), the first one on the right hand side is due to the free-space interaction between the atoms and the second to the presence of the boundary, which importantly modifies the modes of the electromagnetic field. Each term in equation (<ref>) refers to the respective component of the total Green's tensor 𝐆(𝐫,𝐫^',ω)=𝐆^(0)(𝐫,𝐫^',ω)+𝐆^(1)(𝐫,𝐫^',ω), where the scattering component 𝐆^(1)(𝐫,𝐫^',ω) is given in equation (<ref>).Taking into account equation (<ref>) the modification of the decay due to the presence of the mirror can be thought of as an effective cooperative emission from the correlated state of one atom and the mirror image of the other atom.By recalling equation (<ref>) (the same procedure can be used also for the decay rate from the antisymmetric state), and taking the imaginary part of 𝐆(𝐫,𝐫^',ω), we consider the non-retarded (r_A,r_B,ϱ ,ϱ_+≪ c/ω_0) and the retarded regime (r_A,r_B,ϱ ,ϱ_+≫ c/ω_0), evaluating first the Green's tensor in these two limits.Let us start with the non-retarded case. By recalling the property (<ref>), holding for 𝐫=𝐫^'and observing that for 𝐫_A→𝐫_A^*, 𝐫_B→𝐫_B^* Im𝐆^(1)(𝐫_A,𝐫_A,ω_0)=Im𝐆^(1)(𝐫_B,𝐫_B,ω_0)=ω_0/6π c𝐑 and for 𝐫_A→𝐫_B we have also 𝐫_A→𝐫_B^* so that Im𝐆^(0)(𝐫_A,𝐫_B,ω_0)=ω_0/6π c𝐈,Im𝐆^(1)(𝐫_A,𝐫_B,ω_0)=ω_0/6π c𝐑, the total decay rate in the non-retarded limit reads_i+^nret =μ_0ω_0^2/ħ[𝐝_A^eg·(ω_0/6π c𝐈+ω_0/6π c𝐑)·𝐝_A^ge..+𝐝_B^eg·(ω_0/6π c𝐈+ω_0/6π c𝐑)·𝐝_B^ge..+2𝐝_A^eg·(ω_0/6π c𝐈+ω_0/6π c𝐑)·𝐝_B^ge]=μ_0ω_0^2/ħω_0/6π c[𝐝_A^eg·𝐝_A^ge+ 𝐝_A^eg·𝐝_A^ge*+𝐝_B^eg·𝐝_B^ge+ 𝐝_B^eg·𝐝_B^ge*+2𝐝_A^eg·𝐝_B^ge+2𝐝_A^eg·𝐝_B^ge*],where we have used equations (<ref>,<ref>).For what concerns the retarded limit, we consider the following configuration: one atom, say B, in a fixed position close to the surface and r_A,ϱ ,ϱ_+≫ c/ω_0. We thus estimate the decay rate to be _i+^ret =μ_0ω_0^2/ħω_0/6π c[𝐝_A^eg·𝐈·𝐝_A^ge+𝐝_B^eg·(𝐈+𝐑)·𝐝_B^ge]=μ_0ω_0^2/ħω_0/6π c[𝐝_A^eg·𝐝_A^ge+𝐝_B^eg·𝐝_B^ge+𝐝_B^eg·𝐝_B^ge*].Besides the self-interaction due to the free-space component of the Green's tensor, there is an additional term, due to the proximity of atom B to the plate, containing (𝐝_B^eg·𝐝_B^ge*), while atom A presents only the free-space contribution due to its very large distance from the plate.On the other hand, if atom B is very distant from the surface too, the free-space behaviour of the spontaneous emission rate, obtained in the previous section, is recovered, that is Γ→Γ^(0).Figure <ref> shows a plot of the decay rate for the symmetric state for increasing distance of one of the two atoms (A) with respect to the mirror, when the other atom (B) is at a fixed distance from the plate (z_B=10 A). The distances of bothatoms from the mirror are compatible with the dipole approximation and the assumption of a perfectly reflecting mirror. Note that in this plot and in the following ones, we assume an atomic transition angular frequency equal to that of the 2p→ 1s transition in the Hydrogen atom (ω_0 ≃ 1.55 · 10^16^-1, λ_0 =c/(2πω_0) ≃ 1.2 · 10^-7), a minimum interatomic separation z_A-z_B=10 A, and we consider perpendicular and parallel orientations of both (identical) atomic dipoles with respect to the line connecting them. Controlling distance and dipole orientation of a quantum emitter with respect to an external environment, for example a nanoparticle, is experimentally achievable and it has been used to obtain enhancement of the spontaneous emission rate <cit.>. As Figure <ref> shows, in the near zone (all distances much smaller than λ_0), the decay rate in the case of dipole moments perpendicular to the wall is essentially doubled with respect to the free-space case (red dashed line and blue continuous line, respectively). This can be explained in terms of additional constructive interference between the emitters and their mirror images, since the image dipole of 𝐝_⊥ coincides with 𝐝_⊥. If emitter A is far from the boundary while B remains close to it, the interference term Γ_AB≃ 0; however, the scaled decay rate is greater than the unity value of the empty-space case, as Figure <ref> shows. In fact, in addition to the free-space decay term Γ_A^(0) of atom A, we must consider the term Γ_B^(1)= Γ_B^(0), due to the proximity of atom B to the plate, therefore doubling the decay term related to the emitter B. A different result is obtained for a parallel alignment configuration, where the overall decay rate is suppressed in the near zone of both atoms (orange dotted line), since the image dipole of 𝐝_‖ is -𝐝_‖ and therefore their sum vanishes. With increasing distance of atom A from the mirror, the decay rate grows but it is always lower than the respective free-space result (green dot-dashed line) because Γ_B^(1)= -Γ_B^(0), and only the free-space decay term Γ_A^(0) of atom A survives. It is also worth discussing the decay rate for specific positions of the fixed atom B related to the transition wavelength of the two atoms. If we locate the emitter B in a node of the electromagnetic field mode resonant with the atomic transition and vary the position of atom A, we obtain the results shown in Figure <ref> which should be compared with Figure <ref>. For atomic dipoles oriented along z, we observe a better matching between the continuous blue curve (free space) and the red dashed curve (mirror at z=0); a similar result is obtained in the case of dipoles along x,given by the green dot-dashed line (free space) and the orange dotted line (mirror at z=0). This matching further improves if we locate atom B in an antinode, as shown in Figure <ref>. The physical reason for such behaviour might be interpreted in the following way: fixing one atom in a position which is multiple of the atomic transition wavelength λ_0 seems to reduce any eventual positive or negative influence of the boundary on the collective spontaneous emission, if compared to the atomic position effect of Figure <ref>. The mirror has a relevant impact on the free-space oscillatory profile which becomes more regular when atom B is placed in the antinode (comparing the parallel alignment configurations of Figures <ref> and <ref>).Analogous remarks can be made for the plots representing the scaled collective spontaneous emission rate of the two-atom system prepared in the antisymmetric state, as Figures <ref>, <ref> and <ref> show.All this clearly shows that the presence of the mirror can significantly affect the collective radiative behaviour of the two-atom system.§ CONCLUSIONS In this paper we have investigated the spontaneous emission rate of a system composed by two identical entangled atoms in a generic macroscopic environment and interacting with the electromagnetic field. We have considered both symmetric and antisymmetric states of the two emitters. The first result we have obtained is a general analytical expression of the collective transition rate of our system from the initial symmetric or antisymmetric state to the collective ground state, expressed in terms of the Green's tensor of the electromagnetic field. This expression can be applied to any external environment, whose magnetoelectric properties are contained in the electromagnetic Green's tensor.We have then considered two specific cases: both atoms in free space and near a perfectly reflecting plate. In free space we have analyzed the decay rate as a function of the interatomic distance, recovering the known results of superradiant and subradiant behaviours. In the non-retarded limit of small interatomic separations, the outcome shows an enhanced emission rate for the symmetric initial state with respect to the case of isolated atoms (superradiance), and the ordinary behaviour of independent atoms for increasing distances. Concerning with the transition rate from the antisymmetric state, it results completely inhibited in the non-retarded limit (subradiance). The enhancement or the inhibition are due to cooperative processes arising from constructive and destructive interference effects between the two correlated atoms, when they are very close to each other.Subsequently, we have considered the case of two correlated atoms in the presence of a perfectly reflecting planar surface. We have shown that the presence of the boundary significantly affects the superradiant and subradiant decay processes of our atomic system and discussed the results as a function of the interatomic distance as well as of the distance of the atoms from the plate. Our results show that the presence of the plate can enhance or weaken the superradiant/subradiant decay features (if compared to the free-space case) according to the specific orientation of the dipole moments and to atom-plate distances with respect to the atomic transition wavelength. This shows the possibility of controlling and manipulating the collective decay through the external environment. R. Palacino wishes to thank G. Baio and J. Hemmerich for stimulating discussions on the subject of the present paper. This work was supported by the German Research Foundation (DFG, Grants BU 1803/3-1 and GRK 2079/1). S.Y.B is grateful for support by the Freiburg Institute of Advanced Studies.§ REFERENCES99[1]dickeDicke R H 1954 Phys. Rev. 93 99[2]grossGross M and Haroche S 1982 Phys. Rep. 93 304[3]pengPeng J S and Li G X 1998 Introduction to Modern Quantum Optics (Singapore: World Scientific Publishing Co. Pte. Ltd.)[4]persicoLeonardi C, Persico F and Vetri G 1986 Riv. 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A 84 061805[23]buhmannBuhmann S Y 2012 Dispersion Forces I (Springer, Vol 247)[24]knollKnö̈ll L, Scheel S and Welsh D G 2001 in: Coherence and Statistics of Photons and Atoms (New York: Wiley)[25]scheelScheel S and Buhmann S Y 2008 Acta Physica Slovaca 58 675[26]santraSantra R and Cederbaum L S 2002 Phys. Rep. 368 10[27]pabloBarcellona P, Passante R, Rizzuto L and Buhmann S Y 2016 Phys. Rev. A 93 032508[28]pablo1Barcellona P, Passante R, Rizzuto L and Buhmann S Y 2016 Phys. Rev. A 94 012705[29]haakhHaakh H R, Henkel C, Spagnolo S, Rizzuto L and Passante R 2014 Phys. Rev. A 89 022509[30]tanasTanaś R and Ficek Z 2004 Journ. Opt. B 6 2[31]ariasArias E, Duenas J G, Menezes G and Svaiter N F 2016 Journ. of High Energy Phys. 5 1 [32]skribanowitzSkribanowitz N, Herman I P , MacGillivray J C and Feld M S 1973 Phys. Rev. Lett. 30 309[33]ScheibnerScheibner M, Schmidt T, Worschech L, Forchel A, Bacher G, Passow T and Hommel D 2007 Nature Phys. 3 106[34]Novotny11Novotny L and van Hulst N 2011 Nature Photon. 5 83 | http://arxiv.org/abs/1704.08594v2 | {
"authors": [
"R. Palacino",
"R. Passante",
"L. Rizzuto",
"P. Barcellona",
"S. Y. Buhmann"
],
"categories": [
"quant-ph"
],
"primary_category": "quant-ph",
"published": "20170427142146",
"title": "Tuning the collective decay of two entangled emitters by means of a nearby surface"
} |
[email protected] [email protected][label1]Department of Applied Mathematics, School of Sciences,Xi'an University of Technology, Xi'an, Shaanxi 710054, P.R.China In this article, we consider discrete schemes for a fractional diffusion equation involving a tempered fractional derivative in time. We present a semi-discrete scheme by usingthe local discontinuous Galerkin (LDG) discretization in the spatial variables. We prove that the semi-discrete scheme is unconditionally stable in L^2 norm and convergence with optimal convergence rate 𝒪(h^k+1). We develop a class of fully discrete LDG schemes by combining the weighted and shifted Lubich difference operators with respect to the time variable, and establish theerror estimates. Finally, numerical experiments are presented to verify the theoretical results. local discontinuous Galerkin methodstime tempered fractional diffusion equationstabilityconvergence.§ INTRODUCTION In this paper we discuss a local discontinuous Galerkin method to solve the followingtime tempered fractional subdiffusion equation <cit.>u_t(x,t)=κ_α _0D_t^1-α,λ(u_xx(x,t))-λ u(x,t),where u(x,t) represents the probability density of finding a particle on x at time t,κ_α>0 is the diffusion coefficient, and _0D_t^1-α,λ(0<α<1) denotes the Riemann-Liouville tempered fractional derivative operator. TheRiemann-Liouville tempered fractional derivativeof order γ (n-1<γ<n) is defined by <cit.>_0D_t^γ,λu(t) =_0D_t^n,λ_0I_t^n-γ,λu(t),where _0I_t^n-α,λ denotes the Riemann-Liouville fractional tempered integral <cit.>_0I_t^σ,λu(t)=1/Γ(σ)∫_0^te^-λ( t-s)(t-s)^σ-1u(s)ds,σ=n-α,and_0D_t^n,λ=(d/d t+λ)^n=(d/d t+λ)⋯(d/d t+λ)_n times.Tempered fractional calculus can be recognized as the generalization of fractional calculus. If we taking λ=0 in (<ref>), then the tempered fractional integral and derivative operators reduceto the Riemann-Liouville fractional integral _0I_t^σ and derivative _0D_t^γ operators, respectively. In recent years, many numerical methods such as finite difference methods <cit.>, finite element methods <cit.> and spectral methods <cit.> have been developed for the numerical solutions of fractional subdiffusion and superdiffusion equations. Limited works are reported for solving thetempered fractional differential equations, when compared with a large volume of literature on numerical solutions of fractional differential equations. In this literature, Baeumera and Meerschaert <cit.> provide finite difference and particle tracking methods for solving the tempered fractional diffusion equation with the second order accuracy. The stability and convergence of the provided schemes are discussed. Cartea and del-Castillo-Negrete <cit.> derive a finite difference scheme to numerically solve a Black-Merton-Scholes model with tempered fractional derivatives.Hanert and Piret <cit.> presented a Chebyshev pseudo-spectral scheme to solve the space-time tempered fractional diffusion equation, and proved that the method yields an exponential convergence rate. Zayernouri et al. <cit.>derived an efficient Petrov-Galerkin method for solving tempered fractional ODEs by using the eigenfunctions of tempered fractional Sturm-Liouville problems.By using the weighted and shifted Grünwald difference (WSGD) operators, Li and Deng <cit.> designed a series of high order numerical schemes for the space tempered fractional diffusion equations. This technique is used to solve the tempered fractional Black-Scholes equation for European double barrier option <cit.>. Using the properties of generalized Laguerre functions, Huang et al. <cit.> used Laguerre functions to approximate the substantial fractional ODEs on the half line. Li et al. <cit.>analysed the well-posedness and developed the Jacobi-predictor-corrector algorithm for the tempered fractional ordinary differential equation. Yu et al. <cit.> developed the third and fourth order quasi-compact approximations for one and two dimensional space tempered fractional diffusion equations. By using the weighted and shifted Grünwald-Letnikov formula suggested in <cit.>, Hao et al. <cit.> constructed a second-order approximation for the time tempered fractional diffusion equation. By introducing fractional integral spaces, Zhao et al. <cit.> discussed spectral Galerkin and Petrov-Galerkin methods for tempered fractional advection problems and tempered fractional diffusion problems. Recently, the high order and fast numerical methods for fractional differential equationsdraw the wide interests of the researchers <cit.>. The local discontinuous Galerkin (LDG) method is one of the most popular methods in this literature. TheLDG method was first introduced to solve a convection-diffusion problems by Cockburn and Shu <cit.>. These methodshave recently become increasingly popular due to their flexibility for adaptive simulations, suitability for parallel computations, applicability to problems with discontinuoussolutions, and compatibility with other numerical methods. Nowadays, the LDG method has been successfully used in solving linear and nonlinear elliptic, parabolic, hyperbolic equations, and some mixed schemes. For the recent development of discontinuous Galerkin methods, see the the monograph and review articles <cit.>and the references therein. More recently, some researchers pay attention to solving the fractional partial equations by the LDG method. For the time fractionaldifferential equations,Mustapha and McLean <cit.> employed a piecewise-linear, discontinuous Galerkin method for the time discretization of a sub-diffusion equation.ALDG method for space discretization of a time fractional diffusion equation is discussed inXu and Zheng's work <cit.>. By using L1 time discretization, Wei et al.<cit.> developed an implicit fully discrete LDG finite element method for solving the time-fractional Schröinger equation, Guo et al. <cit.> studied a LDG method for some time fractional fourth-order differential equations. Liu et al. <cit.> proposed LDG method combined with a third order weighted and shifted Grünwald difference operatorsfor a fractional subdiffusion equation. For the space fractionaldifferential equations, based on two (or four) auxiliary variables in one dimension (or two dimensions) and the Caputo derivative as the spatial derivative, Ji and Tang <cit.>developed the high-order accurate Runge-Kutta LDGmethods for one- and two-dimensional space-fractional diffusion equations with the variable diffusive coefficients. Deng and Hesthaven <cit.> have developed a LDG method forspace fractional diffusion equation and given a fundamental frame to combine the LDG methods with fractional operators.In this paper,we will develop and analyze a new class of LDG method for the model (<ref>). Our newmethod is based on a combination of the weighted and shifted Lubich difference approaches in the time direction and a LDG method in the space direction. Stability and convergence of semi-discrete and fully discrete LDG schemes are rigorously analyzed. Weshowthat the fully discrete scheme is unconditionally stable with convergence order of 𝒪(τ^q+h^k+1),q=1,2,3,4,5.The rest of the article is organized as follows. In section <ref>, we first consider the initial boundary value problem of the tempered fractional diffusion equation. Then in section <ref> we construct a semi-discrete LDG method for the considered equation. We perform the detailed theoretical analysis for the stability and error estimate of the semi-discrete numerical scheme in this section. In section <ref>, weapply the weighted and shifted Lubich difference approximation for the temporal discretization of the time fractional tempered equation. The error estimates are provided for the full-discrete LDG scheme. Finally, some numerical examples and physical simulations are presented in section <ref> which confirm the theoretical statement.Some concluding remarks are given in the final section. § INITIAL BOUNDARY VALUE PROBLEM OF THE TEMPERED FRACTIONAL DIFFUSION EQUATIONInstead ofdesigning the numerical scheme of the equation (<ref>) directly, we constructing the numerical scheme for its equivalent form. By simple calculation, the equation (<ref>) can be rewritten asd/dt( e^λ tu(x,t))=κ_α _0D_t^1-α(e^λ tu_xx(x,t)).Performing Riemann-Liouville fractional integral _0I_t^1-α on both side of (<ref>), we arrive at_0^CD_t^α(e^λ tu(x,t))=κ_α e^λ tu_xx(x,t),where _0^CD_t^α denotes the Caputofractional derivative <cit.>_0^CD_t^αu(t)=1/Γ(1-α)∫_0^t1/(t-s)^αd u(s)/d sds,0<α<1.In view of the Caputo tempered fractional derivative <cit.>_0^CD_t^α,λu(t)=e^-λ t _0^CD_t^α(e^λ tu(t))=e^-λ t/Γ(1-α)∫_0^t1/(t-s)^αd (e^λ su(s))/d sds,we get the following time tempered fractional diffusion equation_0^CD_t^α,λu(x,t)=κ_αu_xx(x,t).Let Ω= (0,L) be the space domain. We now consider the initial boundary value problem of the fractional tempered diffusion equation (<ref>) in the domain (0,L)×(0,T], subject to the initial conditionu(x,0)=u_0(x), x∈Ωand the boundary conditionsu(x,t)=u(x+L,t),L>0,t∈ (0,T]. <cit.>For function u(t)absolutely continuous on [0,t], holds the the followinginequality1/2_0^CD_t^α(u^2(t))≤u(t)_0^CD_t^α(u(t)),0<α<1. For the initial-boundary value problem (<ref>)-(<ref>), we have the following stability. The solution u(x, t) of the initial-boundary valueproblem (<ref>)-(<ref>) holds the prior estimateu(·,t)^2+2κ_α _0I_t^α,2λu_x(·,t)^2≤ e^-2λ tu_0^2,where _0I_t^α,2λ denotesthe Riemann-Liouville fractional tempered integral operator defined by (<ref>). Taking v(x,t)=e^λ tu(x,t) in (<ref>), we have_0^CD_t^γv(x,t)=κ_α v_xx(x,t),x∈Ω,t>0.Takinginner product in equation (<ref>) with the space variable, we get( v(·,t),_0^CD_t^γv(·,t))_Ω=-κ_α( v_x(x,t), v_x(x,t))_Ω +κ_α v_x(x,t) v(x,t)|^L_0,where (u(x),v(x))_Ω=∫_Ω u(x)v(x)dx. Denoting the L^2-norm as u=(u(x),u(x))_Ω^1/2, using the boundary conditions (<ref>) and using the inequality(<ref>), we get_0^CD_t^α v(·,t)^2+2κ_α v_x(·,t)^2≤ 0.By applying the fractional integraloperator _0I_t^α to both sides of inequality (<ref>), using the composite properties of fractional calculus <cit.>_0I_t^α _0D_t^α(w(t))=w(t)-w(0), we obtainv(·,t)^2+2κ_α _0 I_t^αv_x(·,t)^2≤v_0^2.Taking u(x,t)=e^-λ tv(x,t) in (<ref>) we getthe desired estimate (<ref>).§ THE SEMI-DISCRETE LDG SCHEME In this section, we present and analyze a local discontinuous Galerkin method for the equation (<ref>) subjects to the initial condition (<ref>) and the periodic boundary conditions (<ref>). For the interval Ω=[0,L], we divide it into N cells as follows0=x_1/2<x_3/2<⋯<x_N-1/2<x_N+1/2=L.We denote I_j=(x_j-1/2,x_j+1/2),x_j=(x_j-1/2+x_j+1/2)/2, and h_j=x_j+1/2-x_j-1/2,h= max_1≤ j≤ Nh_j. Furthermore, we define the mesh 𝒯={I_j=(x_j-1/2,x_j+1/2),j=1,2,...,N}. The finite element space is defined byV_h={v: v|_I_j∈ P^k(I_j),j=1,2,⋯,N},where P^k(I_j) denotes the set of allpolynomials ofdegree at most k on cell I_j. We define u^-_j+1/2,u^+_j+1/2 represent the values of u at x_j+1/2 from the left cell I_j and the right cell I_j+1, respectively.To define the local discontinuous Galerkin method, we rewrite (<ref>) as a first-order system[ _0^CD_t^α,λu(x,t)-κ_αp_x=0,;p-u_x=0. ]Now we can define the local discontinuous Galerkin method to the system (<ref>). Find u_h,p_h,for all v_h, w_h∈ V_h such that[ (_0^CD_t^α,λu_h, v_h)_I_j+κ_α((p_h,(v_h)_x)_I_j -(p_hv_h^-)_j+1/2 +(p_hv_h^+)_j-1/2)=0,; (p_h ,w_h)_I_j+(u_h ,(w_h)_x)_I_j - (u_h w_h^-)_j+1/2 +(u_h w_h^+)_j-1/2=0. ]The 'hat' terms in (<ref>) are the numerical probability density fluxes. We chose the alternating numerical fluxes <cit.>asu_h=u_h^-,p_h=p_h^+.§.§ Stabilityanalysis of the semi-discrete LDG schemeIn this section, we present the stability and convergence analysis for the semi-discrete scheme (<ref>) in L^2 sense. To do so, we follow the technique used by Cockburn and Shu <cit.>.For function u(t)absolutely continuous on [0,t], holds the the followinginequalitye^-λ t1/2_0^CD_t^α,λ(u(t))^2 ≤ u(t) _0^CD_t^α,λ(u(t)),0< α <1,λ>0.Taking v(t)=e^λ tu(t) in the inequality (<ref>), we have1/2_0^CD_t^α(e^λ tu(t))^2 ≤ e^λ tu(t) _0^CD_t^α(e^λ tu(t)) .which leadse^-2λ t1/2_0^CD_t^α(e^λ tu(t))^2≤ u(t) _0^CD_t^α,λ(u(t)).Let u_h,p_h to be the solution of semi-discrete LDG scheme (<ref>) with the flux u_h ,p_h defined in (<ref>), holds(u_h, _0^CD_t^α,λu_h)_Ω+κ_αp_h^2=0. For simplicity, we denoteB(u_h,p_h;v_h,w_h) := (_0^CD_t^α,λ(u_h),v_h)_Ω+ κ_α((p_h,(v_h)_x)_Ω- ∑_j=1^N[(p_h)_j+1/2v^-_j+1/2-( p_h)_j-1/2(v_h)^+_j-1/2]) + (u_h ,(w_h)_x)_Ω+(p_h, w_h)_Ω- ∑_j=1^N[(u_h)_j+1/2 (w_h)^-_j+1/2 -(p_h)_j-1/2(w_h)^+_j-1/2] .If we take v_h=u_h, w_h=κ_αp_h in (<ref>), we haveB(u_h,p_h;u_h,κ_α p_h) = (u_h, _0^CD_t^α,λu_h)_Ω+ κ_α((p_h,(u_h)_x)_Ω + (p_h,p_h)_Ω- ∑_j=1^N[(p_h)_j+1/2(u_h^-)_j+1/2 - ( p_h)_j-1/2(u_h^+)_j-1/2] + (u_h ,(p_h)_x)_Ω-∑_j=1^N[(u_h)_j+1/2 (p_h)^-_j+1/2 -(u_h)_j-1/2(p_h)^+_j-1/2])= 0.Combining the numerical flux defined by (<ref>) and periodic boundary conditions (<ref>) we arrive at (<ref>). (L^2-stability) The semi-discrete LDG scheme (<ref>) with the flux choice (<ref>) is L^2-stable, i.e.u_h(·,t)^2 +2κ_α _0I_t^α,λ( e^λ tp_h(·,t)^2) ≤ e^-λ tu_h(·,0)^2. Using the inequality (<ref>), we can obtaine^-λ t1/2_0^CD_t^α,λu_h(·,t)^2 +κ_αp_h(·,t)^2≤ 0.Recall the composite properties of the Caputo tempered fractional derivative and the Riemann-Liouville tempered fractional integral, we have_0I_t^α,λ[_0^CD_t^α,λ u(t)]=u(t)-e^-λ t u(0).Applying the operator _0I_t^α,λ on both sides of the inequality (<ref>) leads to (<ref>). §.§ Convergence analysis of the semi-discrete LDG schemeNow, we given the L^2 error estimate. In order to give more detailed error estimate, the following two special projections operators introduced in <cit.> will be used.(𝒫^-w(x)-w(x)),v(x))_I_j=0,∀ v∈ P^k-1(I_j),𝒫^-w(x^-_j+1/2)=w(x_j+1/2), (𝒫^+w(x)-w(x))v(x))_I_j=0,∀ v∈ P^k-1(I_j),𝒫^+w(x^+_j-1/2)=w(x_j-1/2). <cit.> For projection operators 𝒫^±, the following estimate holdsw^e + hw^e_∞ + h^1/2w^e_Γ_h≤ Ch^k + 1,where w^e=𝒫^±w-w,C is a positive constant depending u and its derivatives but independent of h.<cit.> Let u(t) be continuous and non-negative on [0,T]. Ifu(t)≤φ(t)+M∫_0^tu(s)(t-s)^βds,0≤ t≤ T,where 0≤α<1. φ(t) is nonnegative monotonic increasing continuous function on [0,T], and M is a positive constant, thenu(t)≤φ(t) E_1-α(MΓ(1-α)t^1-α), 0≤ t<T,where E_1-α(z) denotes the Mittag-Leffler function defined for all 0≤α<1 by <cit.>E_1-α(z)=∑_k=0^∞z^k/Γ((1-α) k+1).Let u_h,p_h to be the solution of semi-discrete LDG scheme (<ref>), and u,p be the exact solution of(<ref>) with initial condition (<ref>) and the periodic boundary (<ref>), the following error estimate holdsu(·,t)-u_h(·,t)≤ Ch^k+1. With the denote in (<ref>), we directly getB(u_h,p_h;v_h,w_h)=0, ∀ v_h,w_h∈ V_h, andB(u,p;v_h,w_h)=0, ∀ v_h,w_h∈ V_h,Subtracting(<ref>) from (<ref>), then we obtain the error equationB(e,e;v_h,w_h)=0, ∀ v_h,w_h∈ V_h,where we denote e=u-u_h,e=p-p_h. We divide the error both e and e into two parts[ e=u-u_h=(u-𝒫^-u)+(𝒫^-u-u_h)=ε_h+e_h,; e=p-p_h=(p-𝒫^+p)+(𝒫^+p-p_h)=ε_h+e_h. ]If we take v_h=e_h,w_h=κ_αe_h in (<ref>), we getB(e_h,e_h;e_h,κ_αe_h)=-B(ε_h,ε_h;e_h,κ_αe_h).For the left side of (<ref>), using the equation (<ref>) in Lemma <ref>, we haveB(e_h,e_h;e_h,κ_αe_h)= (e_h, _0^CD_t^α,λe_h)_Ω+κ_α(e,e)_Ω=0.Obviously, the right of (<ref>) can be written as-B(ε_h,ε_h;e_h,κ_αe_h) : = -(e_h_0^CD_t^α,λ,ε_h)_Ω- κ_α((ε_h,(e_h)_x)_Ω- ∑_j=1^N[(ε_h)^+_j+1/2e^-_j+1/2 -(ε_h)^+_j-1/2(v_h)^+_j-1/2] + (ε_h, (e_h)_x)_Ω+ (ε_h, e_h)_Ω- ∑_j=1^N[(ε_h)^-_j+1/2 (e_h)^-_j+1/2 -(ε_h)^-_j-1/2(e_h)^+_j-1/2].Since (e_h)_x and (e_h)_x are polynomials of degree at most k-1, applying the properties (<ref>) and (<ref>) of the projections 𝒫^±, we obtain(ε_h,(e_h)_x)_I_j=0 and(ε_h, (e_h)_x)_I_j=0.In other way,(ε_h)^-_j+1/2=u_j+1/2-(𝒫u)^-_j+1/2=0 and (ε_h)^+_j-1/2=p_j-1/2-(𝒫p)^+_j-1/2 =0.By the Cauchy's inequality, we get(e_h, _0^CD_t^α,λe_h)_Ω+κ_α(e,e)_Ω ≤ 1/2(_0^CD_t^α,λε_h ,_0^CD_t^α,λε_h)_Ω +1/2(e_h,e_h)_Ω+ 1/2κ_α(ε_h,ε_h)_Ω +1/2κ_α(e_h,e_h)_Ω.From the Lemma <ref>, we conclude that2(e_h _0^CD_t^α,λ,e_h)_Ω+κ_α(e,e)_Ω≤(e_h,e_h)_Ω+Ch^2k+2.Using the inequality (<ref>), we have_0^CD_t^α,λe_h(·,t)^2 +κ_αe^λ te(·,t)^2 ≤ e^λ te_h(·,t)^2+e^λ tCh^2k+2.Combining the composite properties (<ref>) and the definition of Riemann-Liouville tempered fractional integral, we arrive ate^2λ te_h(·,t)^2 ≤ e^λ t/Γ(α)∫_0^t(t-τ)^α-1(e^λτe_h(·,τ))^2dτ +Ch^2k+2e^λ t/Γ(α)∫_0^t(t-τ)^α-1e^2λτdτ, ≤ e^λ t/Γ(α)∫_0^t(t-τ)^α-1(e^λτe_h(·,τ))^2dτ + C e^3λ t h^2k+2,where we used the fact∫_0^t(t-τ)^α-1e^2λτdτ= e^2λ t∫_0^ts^α-1e^-2λ sds ≤e^2λ t/(2λ)^αΓ(α).Furthermore, using the fractional Gronwall's lemma <ref>, we havee_h(·,t)^2 ≤ Ce^λ TE_α(e^λ tt^α)h^2k+2,where E_α(·) denotes the Mittag-Leffler function is defined by (<ref>).§ FULLY DISCRETE LDG SCHEMESIn this section we discrete the time in the semi-discrete scheme by virtue of high orderapproximation. Let 0= t_0 < t_1 <⋯ <t_n < t_n+1 <⋯< t_M = T be the subdivision of the time interval [0,T ], with the time step τ= t_n+1-t_n. To achieve the high order accuracy, we employ the q-th order approximations given in <cit.> to approximate the Riemann-Liouville tempered derivative, i.e._0D_t^α,λ v(t)|_t_n=τ^-α∑_k=0^nd_k^q,αv(t_n-k)+R^n, q=1,2,3,4,5,whereR^n=𝒪(τ^q) andd_k^q,α=e^-λ k τl_k^q,α, q=1,2,3,4,5.More detailsof l_k^q,α, one can refer to <cit.>. Using (<ref>), we find_0D_t^α,λ u(x,t)|_(x_i,t_n)=τ^-α∑_k=0^nd_k^q,αu(x_i,t_n-k)+𝒪(τ^q), _0D_t^α,λ [e^-λ tu(x,0)]_(x_i,t_n)=τ^-α∑_k=0^nd_k^q,αe^-λ (n-k)τu(x_i,0)+𝒪(τ^q).Recalling the relation of Riemann-Liouville and Caputo tempered fractional derivatives <cit.>^C_0D_t^α,λv( t) =_0D_t^α,λ [v( t)-e^-λ tv( 0)],the weak form of the first order system (<ref>) at t_n can be rewritten as[ τ^-α∑_k=0^nd_k^α(u(x,t_n-k), v)_Ω -τ^-α∑_k=0^nd_k^αe^-λ (n-k)τ(u(x,t_0), v)_Ω +κ_α(p(x,t_n),v_x)_Ω; -κ_α∑_j=1^N[(p(x,t_n)v^-)_j+1/2- (p(x,t_n)v^+)_j-1/2] =(T^n,v)_Ω; (p(x,t_n), w)_Ω+(u(x,t_n), w_x)_Ω-∑_j=1^N[(u(x,t_n) w^-)_j+1/2 -(u(x,t_n) w^+)_j-1/2]=0. ]Let u_h^n,p_h^n ∈ V_h be the approximate solution of u(x,t_n),p(x,t_n), respectively. We propose the fully discrete LDG schemes as follows: Find u_h^n,p_h^n∈ V_h,[ (p_h^n, w)_Ω + (u_h^n, w_x)_Ω - ∑_j=1^N[(u^n_h w^-)_j+1/2 - (u_h^n w^+)_j-1/2]=0,; l_0^q,α( u_h^n, v)_Ω + κ_ατ^α((p_h^n, v_x)_Ω -∑_j=1^N[(p^n_h v^-)_j+1/2 - (p_h^n v^+)_j-1/2]);=e^-λ n τ∑_k=0^n-1l_k^q,α(u_h^0, v)_Ω -∑_k=1^n-1e^-λ k τl_k^q,α(u_h^n-k, v)_Ω, ]for all v,w∈ V_h, j=1,2,...,N. We take numerical flux to be u_h^n=(u_h^n)^-,q_h^n=(q_h^n)^+ with the same choice of (<ref>). In the following, we prove the stability and error estimate of the schemes (<ref>) with q=1 in L_2 norm. For convenience, we denote l_k^1,α by w_k, where the coefficientsw_k=(-1)^k ( αk ), w_0=1, w_k=(1-α+1/k)w_k-1, k ≥1. <cit.> The coefficients w_k defined in (<ref>) satisfyw_0=1; w_k<0, k≥ 1; ∑_k=0^n-1w_k>0; ∑_k=0^∞w_k=0;and1/n^αΓ(1-α)< ∑_k=0^n-1w_k=-∑_k=n^∞w_k ≤1/n^α, for n≥ 1.The fully discrete LDG scheme (<ref>) of initial-boundary problem (<ref>)-(<ref>) is unconditional stability and holdsu_h^n≤u_h^0, n≥1.Setting v = u_h^n,w=κ_ατ^α p_h^n in (<ref>) and summing over all elements, we obtain[ u_h^n^2+κ_ατ^αp^n_h^2+κ_ατ^α∑_j=1^N[F_j+1/2(u_h^n,p_h^n)-F_j-1/2(u_h^n,p_h^n)+Θ_j-1/2(u_h^n,p_h^n)];=e^-λ n τ∑_k=0^n-1w_k(u_h^0, u_h^n)_Ω -∑_k=1^n-1e^-λ k τw_k(u_h^n-k, u_h^n)_Ω, ]whereF(u_h^n,p_h^n)=(p_h^n)^-(u^n_h)^- - (u_h^n)(p_h^n)^- - (u_h^n)^-(p_h^n),andΘ(u_h^n,p_h^n) = (p_h^n)^-(u_h^n)^- + (u_h^n)(p_h^n)^+ + (u_h^n)^+(p_h^n) - (u_h^n)^+(p_h^n)^+ - (u_h^n)(p_h^n)^- - (u_h^n)^-(p_h^n).Recallingthe numerical flux in (<ref>), we have Θ(u_h^n,p_h^n)=0. On the other hand, in view of the periodic boundary conditions (<ref>), we get∑_j=1^N[F_j+1/2(u_h^n,p_h^n)-F_j-1/2(u_h^n,p_h^n)] =F_N+1/2(u_h^n,p_h^n)-F_1/2(u_h^n,p_h^n)=0.Using the Cauchy-Schwartz inequality, we arrive at[ u_h^n^2+κ_ατ^αp_h^n^2 ≤ e^-λ n τ∑_k=0^n-1w_ku_h^0u_h^n -∑_k=1^n-1w_ke^-λ k τu_h^n u_h^n-k. ]Therefore, we haveu_h^n ≤ e^-λ n τ∑_k=0^n-1w_ku_h^0 -∑_k=1^n-1e^-λ k τw_ku_h^n-k≤∑_k=0^n-1w_ku_h^0 - ∑_k=1^n-1w_ku_h^n-k.Next we need to prove the following estimate by mathematical inductionu_h^n≤u_h^0.From the inequality (<ref>), we can see the inequality (<ref>) holds obviously when n=1. Assumingu_h^m≤u_h^0, for m=1,2,…,n-1,then from the inequality (<ref>), we obtainu_h^n≤∑_k=0^n-1 w_ku_h^0- ∑_k=1^n-1w_ku_h^n-k≤∑_k=0^n-1 w_ku_h^0- ∑_k=1^n-1w_ku_h^0=u_h^0.The proof is complete.Let u(x,t_n) be the exact solution of the problem (<ref>)-(<ref>), which is sufficiently smooth such that u∈ H^m+1 with 0≤ m≤ k+1. Let u_h^n be the numerical solution of the fully discrete LDG scheme (<ref>), then there holds the following error estimateu(x,t_n)-u_h^n≤ C(τ+h^k+1), n=1,⋯,M,where C is a constant depending on u, T, α but independent of τ and h. To simplify the notation, we decompose the errors as follows:e_u^n=u(x,t_n)-𝒫^-u(x,t_n)+𝒫^-u(x,t_n)-u_h^n=𝒫^-e_u^n-𝒫^-ε_u^n, e_p^n=p(x,t_n)-𝒫^+p(x,t_n)+𝒫^+p(x,t_n)-p_h^n=𝒫^+e_p^n-𝒫^+ε_p^n.Combining (<ref>) and (<ref>), we have[ w_0(e_u^n, v)_Ω + κ_ατ^α((e_p^n, v_x)_Ω -∑_j=1^N[((e^n_p)^+ v^-)_j+1/2 - ((e_p^n)^+ v^+)_j-1/2]) +(e_p^n, w)_Ω + (e_u^n, w_x)_Ω - ∑_j=1^N[((e^n_u)^- w^-)_j+1/2 - ((e_u^n)^- w^+)_j-1/2] = e^-λ n τ∑_k=0^n-1w_k(e_u^0, v)_Ω -∑_k=1^n-1e^-λ k τw_k(e_u^n-k, v)_Ω -τ^α(R^n, v)_Ω. ]Substituting (<ref>) into (<ref>) and notice that e_u^0=0, we can get the error equation[ w_0(𝒫^-e_u^n, v)_Ω + κ_ατ^α((𝒫^+e_p^n, v_x)_Ω -∑_j=1^N[((𝒫^+e^n_p)^+ v^-)_j+1/2 - ((𝒫^+e_p^n)^+ v^+)_j-1/2]);+(𝒫^+e_p^n, w)_Ω + (𝒫^-e_u^n, w_x)_Ω - ∑_j=1^N[((𝒫^-e^n_u)^- w^-)_j+1/2 - ((𝒫^-e_u^n)^- w^+)_j-1/2];=-∑_k=1^n-1e^-λ k τw_k(𝒫^-e_u^n-k, v)_Ω -τ^α(R^n, v)_Ω +w_0(𝒫^-ε_p^n,v)_Ω;+ κ_ατ^α(𝒫^+ε_p^n, v_x)_Ω -κ_ατ^α∑_j=1^N[((𝒫^+ε_u^n)^+ v^-)_j+1/2 - ((𝒫^+ε_p^n)^+ v^+)_j-1/2]; +(𝒫^+ε_p^n, w)_Ω- ∑_j=1^N[((𝒫^-ε_u^n)^- w^-)_j+1/2 - ((𝒫^-ε_p^n)^- w^+)_j-1/2]; + ∑_k=1^n-1e^-λ k τw_k(𝒫^-ε_u^n-k, v)_Ω + (𝒫^-ε_u^n, w_x)_Ω. ]Taking v=𝒫^-e_u^n, w=κ_ατ^α𝒫^+e_p^n in (<ref>), we get[ w_0(𝒫^-e_u^n,𝒫^-e_u^n )_Ω + κ_ατ^α(𝒫^+e_p^n,𝒫^+e_p^n )_Ω; =-∑_k=1^n-1e^-λ k τw_k(𝒫^-e_u^n-k, 𝒫^-e_u^n)_Ω;-τ^α(R^n, 𝒫^-e_u^n)_Ω + w_0(𝒫^-ε_u^n,𝒫^-e_u^n)_Ω + κ_ατ^α(𝒫^+ε_p^m, (𝒫^-e_u^n)_x)_Ω;-κ_ατ^α∑_j=1^N[((𝒫^+ε_p^n)^+ (𝒫^-e_u^n)^-)_j+1/2 - ((𝒫^+ε_p^n)^+ (𝒫^-e_u^n)^+)_j-1/2]; +κ_ατ^α(𝒫^+ε_p^n, 𝒫^+e_p^n)_Ω + κ_ατ^α(𝒫^-ε_u^n, (𝒫^+e_p^n)_x)_Ω; - κ_ατ^α∑_j=1^N[((𝒫^-ε_u^n)^- (𝒫^+e_p^n)^-)_j+1/2 - ((𝒫^-ε_u^n)^- (𝒫^+e_p^n)^+)_j-1/2];+ ∑_k=1^n-1e^-λ k τw_k(𝒫^-ε_u^n-k, 𝒫^-e_u^n)_Ω. ]Using the properties of projections 𝒫^± and w_0=1, we can further get[(𝒫^-e_u^n)^2 + κ_ατ^α(𝒫^+e_p^n)^2; =-∑_k=1^n-1e^-λ k τw_k(𝒫^-e_u^n-k, 𝒫^-e_u^n)_Ω -τ^α(R^n, 𝒫^-e_u^n)_Ω; +(𝒫^-ε_u^n,𝒫^-e_u^n)_Ω +κ_ατ^α(𝒫^+ε_p^n, 𝒫^+e_p^n)_Ω;+ ∑_k=1^n-1e^-λ k τw_k(𝒫^-ε_u^n-k, 𝒫^-e_u^n)_Ω. ]Applying the Cauchy-Schwarz inequality, we have[ (𝒫^-e_u^n)^2+ κ_ατ^α(𝒫^+e_p^n)^2; ≤ -∑_k=1^n-1e^-λ k τw_k𝒫^-e_u^n-k ||𝒫^-e_u^n|| +τ^αR^n𝒫^-e_u^n;+𝒫^-ε_u^n𝒫^-e_u^n +κ_ατ^α𝒫^+ε_p^n𝒫^+e_p^n; - ∑_k=1^n-1e^-λ k τw_k 𝒫^-ε_u^n-k𝒫^-e_u^n. ]Combining the inequality 2ab≤ a^2 + b^2 and e^-λ k τ∈ (0,1], from the above inequality (<ref>), we can derive[(𝒫^-e_u^n)^2+ κ_ατ^α(𝒫^+e_p^n)^2; ≤1/2(-∑_k=1^n-1w_k𝒫^-e_u^n-k +τ^αR^n+𝒫^-ε_u^n; - ∑_k=1^n-1w_k 𝒫^-ε_u^n-k)^2 +1/2𝒫^-e_u^n^2;+1/2κ_ατ^α𝒫^+ε_p^n^2 +1/2κ_ατ^α𝒫^+e_p^n^2. ]Moreover, we have[(𝒫^-e_u^n)≤ -∑_k=1^n-1w_k𝒫^-e_u^n-k +τ^αR^n; +𝒫^-ε_u^n- ∑_k=1^n-1w_k 𝒫^-ε_u^n-k +√(κ_ατ^α)𝒫^+ε_p^n. ]Next, we prove the following estimate by mathematical introduction(𝒫^-e_u^n)≤ C(τ+h^k+1).For n=1, using the properties (<ref>) of the projections 𝒫^±, it can be seen that the inequality (<ref>) holds obviously. Assuming(𝒫^-e_u^m)≤ C(τ+h^k+1), for m=1,2⋯,n-1.Remembering -∑_k=1^n-1w_k<1 and the properties of the projections 𝒫^±, we have(𝒫^-e_u^n)≤ C(τ+h^k+1).Finally, combining the triangle inequality and lemma <ref> to haveu(x,t_n)-u_h^n = 𝒫^-e_u^n - 𝒫^-ε_u^n≤ 𝒫^-e_u^n + 𝒫^-ε_u^n≤C(τ+h^k+1).§ NUMERICAL EXPERIMENTSIn this section, we perform three examples to illustrate the effectiveness of our numerical schemes and confirmour theoretical results. Without loss of generality, we add a force term f(x,t) on the right hand side of the equation (<ref>), we consider_0^CD_t^α,λu(x,t)=κ_αu_xx(x,t)+f(x,t), (x,t)∈[0,1]×(0,T],with periodic boundary conditions u(x+π,t)=u(x,t) and initial condition u(x,0)=sin(2π x). If we take the force term f(x,t) as f(x,t)=e^-λ t(Γ(β+1)/Γ(β+1-α)t^β-α +4κ_απ^2(t^β+1))sin(2π x), then the exact solution of the problem (<ref>) with the corresponding initial-boundary condition givesu(x,t)=e^-λ t(t^β+1)sin(2π x).The L^2 errors and orders of the fully discrete LDG scheme (<ref>) on uniform meshes are present in Table <ref>-Table <ref>. Table <ref>-Table <ref> list the L^2 errors and orders of accuracy for schemes (<ref>) with different k and fixed α=0.5. In these tests we take τ=h^(k+1)/q. All the numerical results given in Table <ref>-Table <ref> are consistent with the theoretical analysis which presented in theorem <ref>. Table <ref> shows the errors and orders of scheme (<ref>) for solving the problem (<ref>) with the different parameters α and λ. As expected, we observe that our scheme can achieve higher order accuracy in space, as well as in time. To test the high order of scheme (<ref>) in time direction, we list the errors and orders of scheme (<ref>) in Table <ref>-Table <ref>. We can again clearly observe the desired orders of accuracy from these tables.In this example, we examine the followinghomogeneousequation_0^CD_t^α,λu(x,t)= u_xx(x,t),(x,t)∈(0,1)×(0,T],subjects to the boundary conditions u(0,t)=0,u(1,t)=0, and the initial value u(x,0)=sin(2π x). We can check that the exact solution of this initial-boundary value problem (<ref>) is u(x,t)=e^-λ tE_α(- 4π^2t^α)sin(2π x), where the generalized Mittag-Leffler function E_α(·) defined in(<ref>).In this test, the finite element space is piecewise linear and piecewise quadratic polynomials for the second and third order schemes, respectively. The numerical results are shown in Table <ref>-Table <ref>. The evolution of numerical solutions with different α, λ at different times are given in Fig. <ref>. In this example, we will test the dynamics behavior of the tempered fractional diffusion equation (<ref>) withhomogeneous Dirichlet boundary conditions on a finite domain [-4,4]. We take the Gaussian functionu(x,0)=1/σ√(2π)exp( -x^2/2σ^2),as the initial condition.The numerical results for this example are calculated by the fully discrete scheme (<ref>). In the computation, we set h=1/40, τ=h^2, σ=0.01. The probability density function of a diffusion particle for different values of α, λ at different times are given in Fig. <ref>. It can be seen that, the different parameters α, λ, t have different effect for the probability density of a particle, which is in agreement with theanalytic resultsgiven in <cit.>.The effectiveness of our numerical schemes is confirmed once again. § CONCLUSIONS We have presented a numerical method for a time fractional tempered diffusion equation. The proposed method is based on a combination of the weighted and shifted Lubich difference approaches in the time direction and a LDG method in the space direction. The convergence rate of the method is proven by providing a priori error estimate, and confirmed by a series of numerical tests. It has been proved that the proposed scheme is unconditionally stable and of q-order convergence in time and k+1-order convergence in space. Some numerical experiments have been carried out to support the theoretical results. § ACKNOWLEDGMENTS This research was partially supported by the National Natural Science Foundation of China under Grant No.11426174, the Starting Research Fund from the Xian university of Technology under Grant Nos. 108-211206, 2014CX022, the Natural Science Basic Research Plan in Shaanxi Province of China under Grant No.2015JQ1022, the Shaanxi science and technology research projects under Grant No.2015GY004. § REFERENCE 99Henry:06B.I. Henry, T.A.M. Langlands, S.L. 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Alikhanov, A priori estimates for solutions of boundary value problem for fractional-order equations, Diff.Eq. 46 (2010) 660-666.Dixo:86 J.Dixo, S. Mckee,Weakly singular discrete gronwall inequalities, Z. angew. Math. Mech. 66(1986) 535-544.Kirby:14 R. M. Kirby, G. E. Karniadakis,Selecting the numerical flux in discontinuous Galerkin methods for diffusion problems, J. Sci. Comput. 22 (2005) 385-411. | http://arxiv.org/abs/1704.07995v1 | {
"authors": [
"Xiaorui Sun",
"Fengfqun Zhao",
"Can Li"
],
"categories": [
"math.NA"
],
"primary_category": "math.NA",
"published": "20170426073213",
"title": "Local discontinuous Galerkin methods for the time tempered fractional diffusion equation"
} |
-20mm -25mm 240mm 190mm propositionPropositionlemma[proposition]Lemma theorem[proposition]Theorem definition[proposition]Definition corollary[proposition]Corollary remark[proposition]Remark claim[proposition]Claimexampleemph[proposition]Exampleexamplefoo[proposition]Remarks Remark Remarks Economic Neutral Position: How to best replicate not fully replicable liabilities? Andreas Kunz,Markus Popp Munich Re. Letters: Königinstrasse 107, 80802 München, Germany. Emails: {akunz,mpopp}@munichre.com.Version: December 30, 2023 =================================================================================================================================== Financial undertakings often have to deal with liabilities of the form “non-hedgeable claim size times value of a tradeable asset", e.g. foreign property insurance claims times fx rates. Which strategy to invest in the tradeable asset is risk minimal?We generalize the Gram-Charlier series for the sum of two dependent random variable, which allows us to expand the capital requirements based on value-at-risk and expected shortfall. We derive a stable and fairly model independent approximation of the risk minimal asset allocationin terms of the claim size distribution and the moments of asset return. The results enable a correct and easy-to-implement modularization of capital requirements into a market risk and a non-hedgeable risk component. Keywords: risk measure; risk minimal asset allocation; incomplete markets; modular capital requirements; perturbation theory; Gram-Chalier series; Cornish-Fisher quantile approximation; quantos; Solvency II; standard formula; SCR; market risk; internal model; replicating portfolio;JEL Classification: D81; G11; G22; G28; § INTRODUCTIONWe consider a liability of product structure ∑_i _i ·_i, where _i are hedgeable risk factors and _i represent stochastic notionals or claim sizes that are not replicable by financial instruments. It is well known that such liability is not perfectly replicable, since the number of risk drivers exceeds the number of involved hedgeable capital market factors.This liability structure is of high practical relevance. Prominent examples stem from insurance:_i denoting the claims from property insurance portfolios in foreign currencies and _i denoting the exchange rates, or, _i the benefit payments of pure endowment policies staggered by maturities (depending on realized mortality) and _i the risk-free discount factors.Also for the banking industry such liability structure is relevant, in particular for measuring the credit value adjustment (CVA) risk for non-collateralized derivatives with counterparties for whichno liquid credit default swaps exists: e.g. the CVA for a non-collateralized commodity forward contractcan be written in the above structure with _i denoting the default rate of the counterparty in the time interval t_i (multiplied by the loss-given-default ratio) and _i denoting a commodity call option expiring at t_i. The latter represents the loss potential due to counterparty default at t_i in case of increasing commodity prices.[Hereby we assumed independence of the default rates from the credit exposure against the counterparty due to an increase of the commodity forward rates beyond the pre-agreed strike, refer e.g. to <cit.> for details.]To which extent can the risk from the above liability structure be mitigated by trading in the capital market factors _i? The residual risk must be warehoused and backed with capital. The capital requirement for a financial institution is obtained in theory by applying a risk measure ρ on the distribution of its surplus (i.e. excess of the value of assets over liabilities) in one year, which is the typical time horizon for risk measurement. Hence we aim to find the optimal strategy to invest in the assets _i that minimizes the capital requirements.Intuition tells us that investing more than the expected claim size into the respective hedgeable asset _i makes sense, sincelarge liability losses are usually driven by events where both the claim sizes and the asset values develop adversely. As risk measures focus on tail events, the excess investments in X_i mitigate that part of the liability losses that stems from an increase in _i. The essential task now is to quantify this excess amount.Without loosing too much of generality we assume that _i and _j are pairwise independent for any combination of i and j and that there is no continuous increase in information concerning the states ofL_i during the risk measurement horizon. The latter assumption is almost tantamount to the assumption that claim sizes _i are not hedgeable. As a consequence there is no need to adjust the holdings in _i dynamically within the year. If _i and _j were not independent, then in most practical applications _i can be expressed by regression techniques as a functionof the capital market factors _j plus some residual _i' which then is independent of all _j by construction.Even if the _i and _j are normally or log-normally distributed, the derivation of the risk minimal asset allocation is not straight forward, since products of log-normal variables are again log-normal but sums are not and vice versa for normal variables. This paper is to be interpreted in the context of hedging in incomplete markets. The results relate to the approach of quantile hedging or efficient hedging initiated by Föllmer & Leukert <cit.> and <cit.> and extended in particular by Cvitanic & Spivac <cit.>, Cvitanic <cit.>,Cvitanic & Karazas <cit.> and Pham<cit.>, see also chapter 8 ofFöllmer & Schied <cit.> and the reference therein. For a given budget constraint on the hedge, the (static) quantile hedging strategy resultsfor a liability of product structure as described above in holding a certain amount of the tradeable asset, which corresponds to the distribution of the non-hedgeable claim size distribution truncated at a particular quantile. The efficient hedging framework provides some determining conditions for that truncation level. Similar conditions are derived also when the shortfall risk of failing to (over-)hedging the liability is minimized instead of the probability. The results of this paper allow to approximate this truncation level explicitly in terms of characteristics of the claim size and asset distributions.Another approach to hedging in incomplete markets is mean variance hedging or – more specifically – (local) quadratc risk-minimizing strategies initiated by Föllmer & Sondermann <cit.> and developed further by Föllmer & Schweizer <cit.> and Schweizer <cit.>. Applications of these techniques to insurance mathematics have been intensively studied in particular byMøller <cit.>, <cit.>, <cit.> and <cit.>. Here the insurance risk process (stochastic mortality) is time-continuous and hence reveals a dynamic hedging strategy that reacts immediately to insurance risk changes. As the variance of the hedging error is minimized instead of a down-side focussed risk measure, the replication is always based on the current expectation of the insurance risk factor (mortality), i.e. no overhedging of the best-estimate claim size by a specific fraction of the pure insurance risk occurs as in our approach. Moreover the hedging risk is minimized under the risk-neutral measure and not under the physical measure that is relevant for risk measurement.A further approach towards hedging of insurance claims in an incomplete market is the utility indifference pricing approach initiated by Schweizer <cit.> and Becherer <cit.>, refer also to Møller <cit.>,Henderson & Hobson <cit.> and alsoto the survey paper Dahl & Møller <cit.> that combines utility indifference pricing with quadratic risk-minimization. In this paper, we analyzethe risk measures value-at-risk and expected shortfall. Our first results concern the particular asset allocation, i.e. the initial holding in the asset X which makes the capital requirements independent of the asset distribution. We showin section <ref> that in the one-dimensional case this particular asset allocation equals for both risk measures the value-at-risk of the non-hedgeable claim size distribution, i.e. coincides with the capital requirement when the asset volatility tends to zero. Moreover, this particular asset allocation is risk minimal in the expected shortfall case; the value-at-risk basedcapital requirements on the other hand are still decreasing when less than this exceptional amount is invested in .In the second part of this paper we apply perturbation techniques to the capital requirements. Classical expansion techniques such as the Gram-Charlier series (refer to <cit.> for the seminal paper) approximate the distribution of a random variable in terms of its moments or cumulants. Typically the Gaussian density is used as base function resulting in an expansion in terms of Hermite polynomials. The Cornish-Fisher expansion (first published in <cit.>) uses a similar approach to expand the quantiles of random variables. Similar to the Gram-Charlier series, the Edgeworth expansion <cit.> approximates the distance of the sum of i.i.d. random variables (properly scaled) to the Gaussian density, which is closely linked to the bootstrap method, refer to Hall<cit.>. For detailson classical expansion techniques and further developments refer to the monographs Kolassa <cit.>, Johnson et al. <cit.>, Wallace <cit.>, and the references therein. These classical expansion techniques celebrate a revival in financial mathematics,refer e.g. to Ait-Sahalia et al. <cit.> and the references therein. A straight-forward application of the Cornish-Fisher approach to expand the value-at-risk of the surplus in terms of Hermite polynomials fails to reproduce the distribution-independent relationat the particular asset allocation, which we derive in the first part of this paper. The reason is that due to the product structure of the liability the distribution of the surplus becomes so irregular that thequantile cannot be well approximated by the third and forth excess moments compared to the Gaussian distribution. We prove in Proposition <ref>a Gram-Charlier-like expansion of the sum of two dependent random variables, where not the Gaussian density is used as base function but the distribution of one variable instead.Writing the surplus as sum of a non-hedgeable term and a perturbation term based on the hedgeable assets, Proposition <ref> yields an expansion of the surplus distribution in terms of moments of the hedgeable assets.Expandingin terms of the normal or log-normal asset volatility, we obtain an approximation of the capital requirement (value-at-risk and expected shortfall based) up to forth order in the asset volatility (refer to Theorem <ref> and Corollary <ref>), which also results in an expansion of the optimal asset allocation. The approach generalizes easily to the multivariate case where several assets and non-hedgeable claim sizes are involved; the second order expansion of the capital requirementsin terms of asset volatility is presented in Theorem <ref> (value-at-risk) and Corollary <ref> (expected shortfall). We show that the sum of the optimal investment amounts is given by the optimalamount in the associated univariate case; further, the allocation of the total optimal investment amount into the single asset dimensions follows the covariance principle as long as the non-hedgeable claim sizes are multi-variate Gaussian (refer to Theorems <ref> and <ref>). Numerical studies showthat the derived expansions are stable even for large log-normal asset volatility levels. Our results relate also to the replicating portfolio techniques, that have been recently studied with financial mathematical rigour, refer to the work of Natolski & Werner <cit.>, Pelsser & Schweizer <cit.> and Cambou & Filipović <cit.>. The main focus of these papers is to analyze how to best approximate complex not-perfectly hedgeable claims by investment strategies based on a specified investment universe (including derivatives); this best approximating replicating portfolio is then used for measuring market risk. Whereas the admissible financial claims are much more complex and general thanliabilities of product type (asanalyzed in our paper), the stochastic modelling ofinsurance risk factors and the interaction of the insurance and financial stochastics is not explicitly analyzed. To determinethe asset allocation that minimizes capital requirements in a rather generic and model independent way is important for its own sake.This objective is even more relevant for the modularization ofcapital requirements into a capital market and a non-hedgeable risk component. This has become market standard since deriving capital requirements via a joint stochastic modeling of all (hedgeable and non-hedgeable) risk factors turned out to be too complex. The financial benchmark (Economic Neutral Position) against which the actual investment portfolio is measured to obtain the capital market risk component must obviously coincide with the risk minimal asset allocation. Our results show that the Economic Neutral Position replicates the financial risk factors of the liabilities on the basis of the expected claim size plus some safety margin. Solvency II, the new capital regime for European insurers, does not recognize this safety marginin the modularized Standard Formula approach, which can result in significant distortions of the total risk compared with the (correct) fully stochastic approach, refer to <cit.> for details. The results of this paper provide a simple and stable approximation of the required safety margin in the Economic Neutral Position, such that the modularized capital requirement approach keeps its easy-to-implement property; e.g. for non-hedgeable risks with normal tails the safety margin amounts to 85% of the insurance risk component in the Solvency II context. § SETUP AND PRELIMINARY RESULTS Consider a financial undertaking whose capital requirement is determined by applying a risk measure ρ onits surplus S in one year.Thevalue of the liabilitiesat year one shall factorize in the form∑_i=1^n X_i · L_i, where the real-valued random variables X_i and L_i denote the value of a i-the tradeable asset and the claim size associated to this asset, respectively. These variables live on a probability space with measuretogether with a risk free numeraire investment (money market account).The X_i are assumed strictly positive and independent ofL_j, i,j=1, …, n. All financial quantities are expressed in units of the numeraire.The financial undertaking can invest its assets with initial value A_0≥ 0into the tradeable assets X_i with initial value x_i or into the numeraire.We assume thatadditional information concerning the claim sizes becomes known only at year one, i.e. there is no continuous increase in information concerning the state of L_i during the year.Hence there is no need to adjust the holdings in X_i dynamically within the year. We denote by ϕ_i ≥ 0 the number of units the financial undertaking invests statically into the asset X_i as of today; the remaining asset value A_0- ∑_i=1^n ϕ_i· x_i is invested into the numeraire.We denote in the sequel column vectors and matrices in bold face, e.g.is the column vector (ϕ_1, …, ϕ_n)', where the prime superscript denotes the transposed vector or matrix, respectively. By ··we denote the scalar product.The value of the surplusat year oneis a function of the asset allocation and reads expressed in units of the numeraireS() := ∑_i=1^nϕ_i· X_i + A_0 - ∑_i=1^nϕ_i· x_i - ∑_i=1^n X_i· L_i =-+ A_0 - . We analyze the risk measures value-at-risk _α and expected shortfall _α at tolerance level 1-α for some small α >0. Typically α=0.01 for banks and =0.005 for European insurance companies. Refer to <cit.> for details of the definition of _α and_α. We use the notation ρ if the expression is valid for both analyzed risk measures ρ∈{_α,_α}. We aim to find the optimal holdings ^* in the tradeable assets that minimize the risk of the surplus, i.e.ρ[S(^*)] = min_∈_+^n ρ[S()] . Note that we do not allow for leverage, i.e. ϕ_i < 0 is forbidden. We assume the following technical conditions: To simplify the minimization of ρ[S()] we assume without loss of generality[] = = ,[] = 0,A_0 = 0,whereand 0 denote the column vector with all entries equal to one and zero, respectively.The first assumption means in particular thatis fairly priced. Further these assumptions imply that S() has zero mean and hence readsS()=-- .These simplifying assumptions can be justified by centering and normalizing S(), i.e. subtracting its mean and dividing by [X_i], making use of thepositive homogeneity and cash invariance property of ρ.Ifhas non-zero excess return, i.e. [] ≠, then the additional linear term “ times excess return” arises, which enters the minimization of the risk of the surplus with respect toin a straight forward way. Similarly, ifhas non-zero mean (claim size distributions are typically positive, the centered variable - [ ] is regarded instead. The detailed justification of the simplifying assumption is transferred to the appendix.The following lemma shows that the α-quantile of the surplus S() is well defined and states further preliminary results. We denote by _A the indicator function of some set A; further F_Y, F̅_Y = 1-F_Y, and F_Y^-1 denotes the cumulative distribution function, the tail function, and the quantile function of some scalar random variable Y, respectively. Assume (<ref>) and (<ref>). Then for every ∈_+^n and α∈(0,1) * (S()≤ z) =α has a unique solution z = z_,α, i.e. the α-quantile of S() is well defined.* _α[S()] = -z_,α and _α[S()] = -α^-1·[S()·_S() ≤ z_,α ].* ↦ρ[S()] is differentiable for both risk measures ρ∈{_α, _α}. * ↦_α[S()] is convex.We denote the quantile of S() by z_omitting the subscript α when there is no confusion about the risk tolerance. Part (a) and (c) result basically from the implicit function theorem applied to (z,) ↦ F_S()(z); (b) is a consequence of the continuous distribution of S(), and (d) follows from the convexity of the expected shortfall. The details of the proofs are transferred to the appendix. * Ifhas atoms, i.e. does not admit a density, then the function ↦_α[S()] might not be continuousbut can have kinks at the singular values of .* Assumption (<ref>) can be relaxed; it suffices to assume thatadmits a strictly positive density in someopen set around {∈^n:= F^-1_(1-α)}.We introduce some further notation: for two scalar functions a(t) and b(t) we denotea(t) ∼ b(t), or a(t) = o(b(t)) as t → t_0, if lim sup_t→ t_0 |a(t) / b(t)| < ∞, or lim_t→ t_0 a(t) / b(t) = 1 or =0,respectively.We call a vectorof tradeable assets admissible if X_i is strictly positive with unit mean and satisfies condition (<ref>) for every i=1,…,n.Recalling the well-known link between expected shortfall and value-at-risk _α[·] = α^-1∫_0^α_β[·] dβ, we present a result concerning the integration with respect to the confidence level.Consider a real-valued random variable with strictly positive density f which enables a continuous quantile function F^-1. Further consider a differentiable function G:→ with G(x)→ 0 as x→∞. Then for every α∈ (0,1)∫_0^αG'∘ F^-1 (1-β)/f∘ F^-1 (1-β)dβ = - G∘ F^-1 (1-α) . This result follows directly from the change of variable β→ y:= F^-1(1-β), which impliesdβ = -f(y)dy.§ PARTICULAR VALUE OF Φ (ONE-DIMENSIONAL CASE) The results of this section only hold in the one-dimensional case, i.e. if n=1. We abandon in the sequel the subscript i equal to one and refrain from matrix notation.We identify a particular initial investment amount ϕ into the tradeable asset X such that ρ[S(ϕ)] becomes fairly independent of the distribution of X.To separate the distribution of the tradeable asset X from the claim size L, we analyze the event{ S(ϕ) ≤ -ϕ} for any ϕ≥ 0 and derive the following equivalent events: { S(ϕ) ≤ -ϕ} = {ϕ·(X-1)-X· L ≤ -ϕ} = { X·(ϕ-L) ≤ 0}={ϕ-L ≤ 0} = {L≥ϕ},wherethe last but one equality follows from the strict positivity of X. Hence we derive that ( S(ϕ) ≤ -ϕ) = 1 - F_L(ϕ). As we are interested in the α-quantile of S(ϕ), we need to choose ϕ = := F_L^-1(1-α),which is well defined due to assumption (<ref>). This impliesz_ = -or, equivalently, _α[ S(q)] = q.Also for the expected shortfall, ϕ = is a special case: since {S()≤ z_} = {L≥}, which follows directly from (<ref>), we conclude-α·_α[S()]= [S()·_S()≤ z_]=[ (·(X-1)-X· L) ·_L≥] = ·[ X-1] ·(L≥) - [X] ·[L·_L≥ ] =[-L·_-L≤ - =F_-L^-1(α)] = -α·_α[-L] ,where the third equality follows from the independence of X and L and the forth equality from the unit mean of X.Also the first derivative of the function ϕ↦ρ[S(ϕ)] shows special properties at ϕ =.We summarize the findings in the following theorem together with all other results concerning the particular value for ϕ.Assume (<ref>) and (<ref>). If := F_L^-1(1-α) = _α[-L] units are initially invested in X, i.e. if ϕ=,then* ρ[S()] = ρ[-L] for ρ∈{_α,_α}.* the differential of the risk of the surplus with respect to ϕ evaluated at ϕ = reads(∂_ϕ ρ[S(ϕ)])_|_ϕ= = {[ (-1)·([X^-1]^-1-1) ≥ 0 ρ = _α,;0 ρ = _α.;].and the above inequality becomes strict if X is not constant.[Since in the expression for the value-at-risk the figure (-1) appears four times in this formula, we propose the name ] * the function ϕ↦_α[S(ϕ)] attains its global minimum value _α[-L] at ϕ^* =. (ϕ^* is not necessarily unique.) Part (a) has already been shown above, the proof of (b) is transferred to the appendix, and (c) follows from (b) using the differentiability and convexity ofϕ↦_α[S(ϕ)], see Lemma <ref>.*The particular asset allocation q is model-independent in the following sense: the risk ρ[S(q)] becomes independent of the asset distribution for both risk measures value-at-risk and expected shortfall, as long as the asset is strictly positive.* The model-independent risk value at the particular asset value equals ρ[-L] which coincides with the risk of the surplus if the volatility of X collapse to zero and X becomes constant (with value one).* The initial amount ϕ^* invested in X that minimizes the risk ρ[S(ϕ)] is less than ρ[-L] for both risk measures ρ∈{_α,_α}.For _α this follows from part (b) of the theorem, for _α the minimum is attained at ϕ^* = _α[-L] < _α[-L]. This phenomenon is due to the diversificationbetween X and L. The probability of a synchronous realization of X and L beyond their respective (1-α)-quantiles amounts to α^2 ≪α. Hence it makes sense to immunize against shocks in X based on a claim size notional below ρ[-L]. * In the general multi-dimensional case we can not expect to find a particular asset allocation ^* such that the risk of the surplus ρ[S(^*)] becomes independent of the distribution of the asset vector. The reason is that the separation of claims sizes from the tradeable assetsdoes not work any more as in the univariate case. Similar to(<ref>) we deriveDue to thescalar product structure the positivity ofis not sufficient to deduce that - is positive in all dimensions as in the univariate case.§ EXPANSION RESULTS§.§ Gram-Charlier-like expansion The classical Cornish-Fisher method <cit.> yields an expansion of the quantile of the surplus based on its moments.These can be easily computed from(<ref>) in terms of the moments ofandusingtheir independence.Figure<ref> compares the forth order Cornish-Fisher expansion with the true value-at-risk profile of the surplus as a function of the asset allocation ϕ in the univariate case.This Cornish-Fisher expansion fails to reproduce the relation _α[S(q)] = q of Theorem <ref>.(a) which holds independently of the distributions of X and L. The reason is that due to the product structure of the liability the third and higher moments of S(ϕ) differ considerably from those of the normal distribution.We suggest an expansionthat preserves the relation of Theorem <ref>.(a).To this aim we prove an expansion similar to the Gram-Charlier series <cit.> for the sum of two not necessarily independent random variables. This expansion does not use the Gaussian distribution as base function but the distribution of one of the variables itself.Consider two scalar random variables Y_0 and Y_1 such that Y_0+Y_1 has a density which is differentiable for any order and the differentials are integrable.ThenF_Y_0 + Y_1(z) = (Y_0 + Y_1 ≤ z) = ∑_r=0^∞1/r!· (-D_z)^r [Y_1^r ·_Y_0≤ z] . This theorem is proved by means of the Fourier transform; the details are transferred to the appendix. If Y_0 and Y_1 are independent, the expansion reads F_Y_0 + Y_1 = ∑_r=0^∞1/r!· m_r(Y_1) · (-D_z)^r F_Y_0, wherem_r(Y_1) denotes the r-th moment of Y_1. This results is in line with classical Gram-Charlier series that are based on directly expanding the characteristic function instead of the cumulant generating function, refer to sec. 12 of <cit.>To apply Proposition <ref> tothe surplus S()=-- we rewrite it in the formS()= Y_0 + Y_1 with a purely non-hedgeable base function Y_0 := - perturbed by a noise term Y_1 := - - that depends linearly on the hedgeable asset. An application of Proposition <ref> leads (S() ≤ z)= (-≤ z) +∑_i ≥ 2(-1)^i/i!· D_z^i[ - - ^i ·_-≤ z] .The first order term vanishes since the terms involvingandare independent andhas unit mean. Noting that - - ^i = ∑_j_1, ⋯, j_i = 1^n ∏_k = 1^i (X_j_k-1)· (ϕ_j_k-L_j_k), we can again use this independenceto integrate the i-th order term with respect to the asset dimension to deduce (S() ≤ z)= F̅_[-](-z)+ ∑_i ≥ 2 1/i! ·∑_j_1, ⋯, j_i = 1^n m̅_j_1, ⋯, j_i· D^i K_j_1, ⋯, j_i[-](-z) ,whereK_j_1, ⋯, j_i[-]( y) := _ [∏_k = 1^i (ϕ_j_k-L_j_k)·_ > y]depends only on the claim size and m̅_j_1, ⋯, j_i:= _[∏_k = 1^i (X_j_k-1)] represents the i-th multidimensional central moment of the tradeable assets; furtherF̅_ is the tail function of the random variable . Note that the (-1)^i terms have vanished since the terms _-≤ z are now referenced in the function K_j_1, ⋯, j_i by the expression (_≥ y)_|_y=-z and i-times differentiation reproduces these (-1)^i terms.§.§ Second order expansionWe have derived an expansion of the cumulative distribution of the surplus S() in terms of the (multi-dimensional) moments of the tradeable assets . But what we need is an expansion of the α-quantile z = z() ofS() when the financial asset vectorbecomes more and more deterministic, i.e. approaches the constant vector .We denote bythe covariance matrix of the tradeable assets , i.e. Σ_ij= [(X_i - 1)· (X_j - 1)]. We consider convergence oftoin quadratic norm, i.e. -_2 := ( [ -- ])^1/2→ 0. Note that -_2^2 = () = _*, where (·) denotes the trace operator and ·_* the nuclear norm.Due to the equivalence of matrix norms there exists some constant C>0 such that for any vectors ,∈^n|·| ≤_2 ·_2·_2 ≤ C ·_*·· =C ·-_2^2 ··.This implies that for every , ∈^n· = O(-_2^2)-_2→ 0. * Relation (<ref>) holds true independently of the particular convergence of →: for any family (_σ)_σ>0 with _σ-_2 ∼σ as σ→ 0 and _σ admissible for every σ>0 we have _σ· = O(σ^2) as σ→ 0 where _σ denotes the covariance matrix of _σ. * The term _σ· can contain terms of higher order than σ^2 if some dimensions ofconverge faster to the constant than others, e.g. _σ = (1+σ· (X_1-1), 1+σ^2 · (X_2-1)) with some independent admissible X_i. We choose an expansion of the α-quantile z of the surplus as σ :=- _2 → 0 in the formWhen we insert the α-quantile z() into equation (<ref>), the left hand side equals α by definition of the quantile. We then expand all σ-dependent terms of the right hand side of (<ref>) in orders of σ^i. Note that only the moments ofin the expansion (<ref>) depend directly on σ; all other terms depend only via the quantile z on σ. This enables us to evaluate sequentially the terms z_i in increasing Let us start to expand the terms in equation (<ref>) in orders of σ^i as σ→ 0. The first term of the right hand side of equation (<ref>) reads as σ→ 0F̅_(-z)= F̅_(-z_0) - f_(-z_0)· (-z_1-z_2 - …) -f_'(-z_0)· (-z_1- …)^2+…. We start to evaluate the zero and first order terms z_0 and z_1 of the quantile expansion.Having (<ref>) in mind, relation (<ref>) reads for the α-quantilein first order approximationα = F̅_(-z_0 - z_1) + o(σ)=F̅_(-z_0) -f_(-z_0)· (-z_1) + o(σ) .Collecting the zero order terms we obtain 1-α = F_(-z_0). Denoting again := F_^-1(1-α) we deduce that -z_0 = q. Collecting the first order terms we obtain 0 = f_(q)· z_1. From the positivity of the density f_ we conclude that z_1 ≡ 0. Before we start the evaluation of the second order term z_2, we define some useful functionals: (y) :=_[ ·_ > y] ,K_[](y) :=_[··_ > y],for any ^n-valued random variable .This allows us to rewrite the second order term in the expansion (<ref>) as 1/2· K_[ - ]”(-z). By equation (<ref>) we know that K_[ - ](y) = O(σ^2) and hence also K_[ - ]”(y) = O(σ^2) as σ→ 0 for every y∈.To evaluate the second order term z_2 we collect in the relation (<ref>) combined with the expansion (<ref>) all terms ∼σ^2 as σ→ 0 and obtain0= -f_(-z_0)· (-z_2) + · K_[ - ]”(-z_0)+ o(σ^2) . The following theorem reformulates this second order expansion result for the value-at-risk of S() and derives the risk minimizing asset allocation. *Define := _α[-] = F_^-1(1-α) and denote the covariance matrix ofby . The expansion of _α[S()] up to second order in σ:=- _2 = √(())→ 0is given by _α[S(ϕ)] =+ · f_(q)^-1· K_[ - ]”(q) + o(σ^2)=- ·{·· f_' (q)+ 2 ·”(q) - K_[]” (q) } + o(σ^2). * If f_'() ≠ 0 andis invertible, the minimum of the second order expansion of _α[S()] is attained at ^* = -f_'(q)^-1·”(q) and equals_α[S(^*)] =+ ·{ f_' (q)^-1·”() ·”()+ K_[]” (q) }. part a) follows from solving (<ref>) for z_2 and expressing K_[- ] via the K-terms defined in (<ref>). Differentiating the second equation of part a) with respect to , setting it to zero, and multiplying from the left by f_(q) ·^-1 proves the first assertion of part b). Inserting this into the second equation of part a) yields the second assertion. The investment amount^* in the tradeable assets that minimizes the second order expansion of _α[S()] (when the asset volatility tends to zero) is completely independent of the asset distribution. Only the value-at-risk of the surplus at the optimal asset allocation ^* depends on the assets via . We now turn to the expected shortfall of the surplus which can be characterized in terms of the value-at risk by _α[S()] = α^-1∫_0^α_β[S()] dβ. Its expansionis an immediate consequence of Lemma <ref> when setting G := K_[-]'.* The expansion of _α[S()] up to second order in σ = -_2→ 0_α[S(ϕ)] = _α[-] - · K_[ - ]'(q) + o(σ^2) = _α[-] + {·· f_ (q)+ 2 ·'(q) - K_[]' (q)} + o(σ^2). * Ifis invertible, the minimum of the second order expansion of _α[S()] is attained atand equals_α[S(^*)] = _α[-] - { f_ (q)^-1·'() ·'()+ K_[]' (q) }. We analyze the total optimal investment amount Φ^*:= ∑_i ϕ^*_i = ^* in all tradeable assets defined asthe sum of the optimal investment amounts ϕ^*_i in the tradeable assets X_i that minimize the second order expansion of ρ[S()]. We establish a link to the associated single-asset case that is characterized as follows: there is only onetradeable asset X_0, i.e. X_i = X_0 for every i=1, …, n, and the surplusreadsS_0(ϕ_0) = ϕ_0 · (X_0-1) - X_0 ·, where ϕ_0 >0 is the investment amount into this single asset. We denote by ϕ_0^* the optimal investment amount that minimizes the second order expansion ofρ[S_0(ϕ_0)] in the associated single asset case.In second order approximation ofρ[S()] according to Theorem <ref> the total optimal investment amount Φ^* satisfies: * Φ^* =+ f_(q)/f_'(q) if ρ = _α, and Φ^*= if ρ = _α.* Φ^* = ϕ_0^* for ρ∈{_α, _α}, i.e. the total optimal investment amount coincides with the optimal investment amount in the associated single-asset case.we denote by K_(z) := [·_ >z]= ∫_q^∞t· f_ (t) dt. Observe that Φ^*=^*=- K_”(q) / f_'(q) if ρ=_α by Theorem <ref> and = - K_'(q) / f_(q) if ρ=_α by Corollary<ref>. Furthernote that K_'(q) = -q · f_ (q) and K_”(q) = -q · f_' (q)-f_(q), which proves part a). As a) also holds in the one-dimensional case, part b) follows by inspection of the formula in a) in the one-dimensional associated single-asset case. Hence ^* can be interpreted as an allocation of ϕ_0^* in the sense that ∑_i ϕ_i^* = ϕ_0^*. We investigate the impact of the multivariate claim size distribution on this allocation:if a particular claim size L_i is more volatile and only weakly correlatedto the other claim sizes L_j, j≠ i, then a material amount in the asset X_i should show up in the risk-minimal asset allocation ^*.If the claim sizes are multivariate normally distributed we obtain the following result, the proof of which is transferred to the appendix.Assume that the claim sizes ∼(, ) follow a multivariate normal distribution with covariance matrix . * Then for ρ∈{_α, _α} the investments ϕ_i^* in the tradeable assets X_i that minimizeρ[S()] expanded up to second order in the asset volatility σ = -_2→ 0 follow the covariance allocation principle with respect to, i.e.ϕ_i^* = Σ^_ii + ∑_j ≠ iΣ^_ij/··ϕ_0^*(i = 1, …, n) ,where ϕ_0^* is the risk-minimal investment in the associated single-asset case according toand ·is the total variance of ∑_i L_i.* The minimum of the risk of the surplus ρ[S(^*)] in second order approximation for ρ∈{_α, _α} equals_α[S(^*)]=q + (ln f_)'(q)/2·{( 1+ (ln f_)'(q)^-2/·)····/· - (·)}, _α[S(^*)]= _α[-] - f_(q)/2α·{···/· -(·) }.Theorem <ref> and Corollary<ref> describe the expansion results in terms of derivatives of the K-terms defined in (<ref>). In order to calculate these terms explicitly a rotation in the state space ofproofs useful: let ∈ SO(n) be a rotation matrix in the n-dimensional special orthogonal group[I.e. has unit determinate andpairwise orthogonal columns with unit l_2-norm], such that the first column ofis parallel to thevector. The rotation matrix can be written = ( n^-1/2·| ), whereis a n×(n-1) matrix oforthogonal coordinates that span the hyperplane orthogonal to the vector . In two and three dimensions the rotation matrixreads _(n=2) = ·[1 -1;11 ], _(n=3) = ·[√(2) 1 -√(3);√(2) 1√(3);√(2)-2 0 ].Rewriting (y) =∫_{∈^n:> y}· f_() d we applythe change in variable := ' (implying =), which yields(y) = ∫_{∈^n:> y}·f_() d = ∫_^n-1∫_y/√(n)^∞( λ_1/√(n)· + ) · g(λ_1, ) dλ_1 d,where g() : = f_() denotes the rotated density. The last equation follows from the observation that = n^-1/2·λ_1 · + · = √(n)·λ_1. A similar expression can be derived for K[](y).The following result reformulates the derivatives of the K-terms accordingly.Defining the expressions(y) := ∫_^n-1· g( , )d,h_2(y) := ∫_^n-1'···· g( , )d, the first and second derivative of the K-terms defined in (<ref>) reads * '(y) = - · f_(y) · - ·(y),* K_[]' (y) = - ··· f_(y) - ·' ··(y) - h_2(y),* ”(y) = - 1/n·( f_(y) + y· f_'(y)) · - ·'(y),* K_[]”(y) = - ···( 2 f_(y) + y · f_'(y) ) - ·⟨' ··,(y)+y ·'(y)⟩ - h_2'(y).The minimum values of part (b) of Theorem <ref> and Corollary <ref> read* _α[S(^*)] =+ ·{f_ (q)^2/n^2 f_' (q)·· + 1/f_'(q)·⟨'(q),' ···'(q) ⟩ + 2/n·⟨ln(f_)'(q) ·'(q) - (q),' ··⟩ -h_2'(y)}.* _α[S(^*)] = _α[-] - { f_ (q)^-1·⟨(q),' ···(q) ⟩ - h_2(y)}.the relation 1/√(n)∫_^n-1 g( , )d=D_y ∫_{∈^n:>y} d = f_(y)is derived analogously to (<ref>). Part a) follows from differentiating (<ref>) and applying this relation.Part b) follows analog to a); c) and d) is obtained by differentiating a) and b) again. Part e) and f) are obtained by inserting part a) to d) into the corresponding expressions of Theorem <ref> and Corollary <ref>, respectively.§.§ Higher order expansionDeriving the third and higher order expansion terms is in principle straight forward but tedious, since the higher order expansion results are not any more independent of the specific convergence of the asset vectorto the constant , refer toRemark <ref>(a). Let us choose a family (_σ)_σ>0 of admissible asset vectors with _σ-_2 ∼σ as σ→ 0 as in Remark <ref>(a). In order to expand the Gram-Charlier-like formula (<ref>) to third or higher order in σas σ→ 0 we need to expand the i-th central moments m̅_j_1, ⋯, j_i(σ):= _[∏_k = 1^i (X_σ,j_k-1)] in terms of σ as followsm̅_j_1, ⋯, j_i(σ) =m̅_j_1, ⋯, j_i^(0)·σ^i + m̅_j_1, ⋯, j_i^(1)·σ^i+1 + … + m̅_j_1, ⋯, j_i^(k)·σ^i+k + o(σ^i+k)σ→ 0.Recall that also for the second moments third and higher order terms can appear, refer to Remark <ref>(b).Extendingequation (<ref>), from which we derivedthe second order terms, up to third order, we derive from (<ref>) using (<ref>)0 =-f_(-z_0)· (-z_2- z_3)+ ·∑_i, j =1^n (m̅_i,j^(0)·σ^2 + m̅_i,j^(1)·σ^3 ) · K_i,j[-]”(-z_0)+ ·∑_i, j,k =1^n m̅_i,j,k^(0)·σ^3 · K_i,j,k[-]”'(-z_0) .Solving for the third order term z_3 we obtain the following result. Let us choose a family (_σ)_σ>0 of admissible asset vectors with _σ-_2 ∼σ as σ→ 0 and consider the expansion of the higher order moments as in (<ref>). Then the third order expansion of the value-at-risk of the surplus in σ reads _α[S(ϕ)] =+ σ^2/2 · f_(q)· K_^(0)[ - ]”(q) + σ^3/6 · f_(q)·{ 3 · K_^(1)[ - ]”(q) + ∑_i, j,k =1^n m̅_i,j,k^(0)· K_i,j,k[-]”'(q)}+ o(σ^3) ,where ^(k) :=(m̅_i,j^(k))_ij denotes the matrices of the expansion of the second order moments according to (<ref>) and the term K_i,j,k is defined in (<ref>). In the sequel we demonstrate the effects of particular converging families of asset distributions that are important in practice andderive the forth order terms.Due to the increased complexity, we restrict to the one-dimensional case, i.e. n=1.The expansion (<ref>) of the cumulative distribution of the surplus then reads in the one-dimensional case(S() ≤ z)= F̅_L(-z) + ∑_i ≥ 2m̅_i/i!· D^i _i(-z) , _i(y) := ∫_y^∞ (ϕ-ℓ)^i· f_L(ℓ) dℓ,and m̅_idenotes the i-th central moment of the tradeable asset X. A straight forward way to construct a family of admissible assets converging to the constant 1 is to scale a fixed asset variable X by its normal volatility. In financial application also the log-normal volatility is of high importance. Hence weintroduce for a fixed tradeable asset X the following two families of admissible assets (X_σ)_σ≥ 0 indexed by the normal as well as log-normal volatility:we set X_σ_N := 1 + σ_N Y in the normal case and X_σ_lN := e^σ_lNY / M(σ_lN) in the log-normal case, where Y denotes the centered and normalized version of X or ln X, respectively,[ Normal case: Y = (X-1)/√([X]), log-normal case: Y = (ln(Y) - [ln Y]) /√([ln X]). ] and M(σ) := [e^σ Y] is the moment generating function of Y. Note that the standard deviation of X_σ_N or ln X_σ_lN equals σ_N or σ_lN, respectively. Further X_σ^* coincides with the original tradeable asset X if σ^* = √([X]) in the normal and =√([ln X]) in the log-normal case. Moreover, X_σ_N and ln X_σ_lN keep the unit mean property due to the normalization. Hence X_σ is admissible for every σ >0 in the normal as well as in the log-normal case. The central moments m̅_i = m̅_i(σ):=[(X_σ-1)^i] of X_σ for σ∈{σ_N, σ_lN} show the following expansions in terms of the normal and log-normal asset volatility: denote by μ_i := [Y^i] the i-th moment of Y, which coincides with the i-th centered and normalized moment of X or ln X, respectively. In the normal case the expansion of m̅_i is trivially given by m̅_i = σ_N^i ·μ_i, whereas in the log-normal case the expansion of m̅_i up to forth order in σ_lN readsm̅_2= σ_lN^2 + μ_3 ·σ_lN^3 +(μ_4 - )·σ_lN^4 +o(σ_lN^4) ,m̅_3= μ_3 ·σ_lN^3+(μ_4-1) ·σ_lN^4 +o(σ_lN^4) , m̅_4= μ_4 ·σ_lN^4 +o(σ_lN^4) . We summarize the results for the fourth order expansion of the _α[S(ϕ)] in the following theorem. The proof is transferred to the appendix together with proof of (<ref>). We denote by id the identity function.Consider the one-dimensional case, i.e. n=1.* The expansion of_α[S(ϕ)] in the log-normal volatility σ_lN of the financial asset X up to fourth order as σ_lN→ 0 is given by _α[S(ϕ)]= -1/f_L(q)·{σ_lN^2/2·[(ϕ-id)^2· f_L]'(q) + σ_lN^3 ·μ_3/6·[(ϕ-id)^3 ·f_L']'(q) + σ_lN^4/24·[μ_4·[(ϕ-id)^4 ·f_L']' - 3·((ϕ-id)^2 ·f_L)' ^2/f_L +(2μ_4- 6) · (ϕ-id)^3· f_L' + (μ_4 +3) · (ϕ-id)^2 ·f_L ]'(q) } + o(σ_lN^4) , where μ_3 and μ_4 denote the third and forth centered normalized moments of ln X, respectively. * If μ_3· f_L”(q) ≠ 0, theexpansion of _α[S(ϕ)] in (a) up to third order attains its local minimum at ϕ^* =+ f_L”(q)^-1 ·( (1-δ) · f_L'(q) - √((1-δ)^2 · f_L'(q)^2 + 2·δ· f_L”(q)· f_L(q) )) , δ:=If μ_3· f_L”(q) = 0 but f_L'() ≠ 0, the minimumis attained at ϕ^* =+ f_L() / f_L'(). * The expansion of _α[S(ϕ)] only involves local properties of L around its (1-α)-quantile, i.e. (higher order)derivatives of f_L at . * If the skew of ln(X) vanishes and L is normally distributed with volatility σ_L, then = σ_L · u_1-α where u_1-α denotesthe (1-α)-quantile of the standard normal distribution. Hence f_L'() / f_L() = -/σ_L^2 = - u_1-α/σ_L. Part (b) of the theorem implies thatϕ^*/ = 1-u_1-α^-2, which amounts to 0.815 or 0.849 for the risk tolerance 1-α = 0.99 (Basel II) or = 0.995 (Solvency II), respectively. This means that the total Solvency II capital requirement of an insurance undertaking (when evaluated via a fully stochastic model) is minimized, if in addition to the expected claim size also 84.9% of the non-hedgeable risk component, i.e. the 99.5%-quantile of the centered claim size L, is initially invested in X. * The presence of a negative log-normal asset skew(the usual case in practical applications) shifts the optimal asset allocation ϕ^* nearer to the 1-α quantile q of L, refer to Figures <ref> and <ref>. The reason is that the diversification effect that reduces the risk minimal asset allocation ϕ^* to a value lower than q, refer to Remark <ref>(c), is less pronounced if ln X is negatively skewed. Vice versa for a positive log-normal skew of X. Repeating the proof of the expansion in the above theorem using the normal instead of the log-normal asset volatility gives the following results. In the one-dimensional case, the expansion of_α[S(ϕ)] in the normal asset volatility σ_Nup to forth order as σ_N → 0 is given by _α[S(ϕ)] = -1/f_L(q)·{σ_N^2/2·[(ϕ-id)^2 ·f_L]'(q) + σ_N^3·μ_3/6·[(ϕ-id)^3 ·f_L]”(q) +σ_N^4/24·[μ_4·((ϕ-id)^4 ·f_L)” - 3·((ϕ-id)^2· f_L)' ^2/f_L]'(q) } + o(σ_N^4) . The corresponding result for the expected shortfall is again a direct consequence of Lemma <ref>. In the one-dimensional case,the expansion of_α[S(ϕ)] in the asset volatility σ∈{σ_N,σ_lN}up to forth order as σ→ 0 is given by_α[S(ϕ)]= ES_α[-L] +σ^2/2α· (ϕ-)^2· f_L(q) +{[ [ σ^3 ·μ_3/6α·(ϕ-q)^3 · f_L'(q) + σ^4/24α·[μ_4·[(ϕ-id)^4· f_L']' - 3·((ϕ-id)^2· f_L)' ^2/f_L;+(2μ_4- 6) · (ϕ-id)^3 ·f_L' + (μ_4 +3) · (ϕ-id)^2 ·f_L ](q) + o(σ^4); ](σ = σ_lN) ,;; [ σ^3 ·μ_3/6α·((ϕ-id)^3· f_L)'(q)+σ^4/24α·[μ_4·((ϕ-id)^4· f_L)”;- 3·((ϕ-id)^2· f_L)' ^2/f_L](q) + o(σ^4); ] (σ = σ_N) .; ]. In contrast to the value-at-risk case, all correction terms of theexpansions of ϕ→_α[S(ϕ)] up to fourth order have ϕ^* = q as (local) minimum, refer also to Figure <ref>.This is consistent with Theorem <ref> stating that the risk-minimizing asset allocation equalsindependently of the distribution of X and L. § NUMERICAL ANALYSIS§.§ Univariate CaseWe now compare our perturbation results in the univariate casewith numerical analysis. To this end we use numerical integration and sample the cumulative distribution functionof the surplus around the α-quantile of the surplus S(ϕ) in order to obtain the inverse. Figure <ref> shows the function ϕ↦ρ[S(ϕ)] for the risk measures ρ∈{_α, _α} with the Solvency II risk tolerance 1-α = 99.5%. The claim size L is normally distributed such that q = 1. Log-normal volatility and skew of the asset X are calibrated to typical values of a 30 year discount factor. It can be seen that the analytical expansion results for the log-normal asset volatility(Theorem <ref> and Corollary <ref>) approximate the numerical behavior quite well. As predicted the risk minimal investment amount in X is around ϕ^* ≈ 0.85 for ρ = _α and ϕ^* = 1 for ρ = _α, respectively.Figure <ref> displays the same situation as Figure <ref>, but with a much more volatile asset (comparable to an emerging market single stock). For both risk measures the third and fourth order expansions based on normal asset volatility are less accurate than the expansions based on log-normal asset volatility.In the value-at-risk case the second order approximation still fits the overall shape quite well, whereas the third and fourth order expansion are more accurate for investment amounts ϕ not too far from q; the optimal investment ϕ^* ≈ 0.9 is higher than in the second order approximation due to the massive negative asset skew;in this setting ϕ^* is very close to the optimal investment in the third order approximation, whereas the fourth order correction of the optimal investment does not add precision if ϕ is away from q. In the expected shortfall case, the third order (log-normal volatility based) approximation produces the best fit for the risk profile, whereas the fourth-order approximation adds only little additional accuracy for ϕ not too far from q. These observations are consistent with the fact that the Gram-Charlier series are known to converge slowly, see e.g. <cit.>.Next we analyze for the value-at-riskthe location of the risk minimal investment amount ϕ^* in more detail, which depends on the characteristics of the hedgeable risk factor X. Figure <ref> shows the dependence of ϕ^* on the log-normal volatility σ for various log-normal skew values μ_3. In case of zero skew the third order expansion term vanishes. Higher order terms lead only to very small corrections to our theoretical prediction of ϕ^* ≈ 0.85. For realistic skew values of around μ_3 = -0.3 the third order expansion is a good approximation up to σ = 0.5. In case of very high skew μ_3 = -1.0 the approximation is only good up to σ = 0.3. To sum up, for realistic parametrizations of the hedgeable risk factor X our perturbation results up to third order reflect the behavior of the risk minimal investment amount ϕ^* very well.§.§ Bivariate CaseNext let us consider the case of two financial assets X_1 and X_2, which are used to hedge two different claim sizes L_1 and L_2. Based on Monte Carlo simulation we compare the numerical results for the risk minimal investment amounts ϕ^*_1 and ϕ^*_2 with the findings of our perturbation approach.Figure <ref> shows the numerical results for the value-at-risk VaR_α[S()]as a function of the units =(ϕ_1, ϕ_2) of the financial asset . As in the univariate case the risk tolerance is set to 1-α = 99.5% and the claim sizeis normally distributed such that q = 1. The financial assets X_1 and X_2 are chosen to be independent and log-normally distributed with log-normal volatility σ = 0.3. For the symmetric case (a) the analytical expansion results in second order (Theorem <ref>) predict risk minimal investment amounts of ϕ^*_1= ϕ^*_2=ϕ_0^*/2≈ 0.425. In the asymmetric case (b) we obtain ϕ^*_1 ≈ 0.79 and ϕ^*_2≈ 0.06. In both cases the numerical results coincide quite well with the theoretical prediction. § APPLICATION TO SOLVENCY II MARKET RISK MEASUREMENT In general, there are two ways of how to set up an internal model for calculating the Solvency Capital Requirement (SCR) under Solvency II: The integrated risk model calculates the surplus (= excess assets over liabilities) distribution of the economic balance sheet, by simulating simultaneously the stochastics of all risk drivers (hedgeable and non-hedgeable). Although it is the more adequate approach, it is rarely used in practice both for operational and steering reasons. Market standard is a modular approach similar to the one used in the Solvency II standard formula. In the modular risk model the profit and loss distribution for each risk category is computed in a separate module and the different risk modules are subsequently aggregated to the total SCR of the company. For risk categories which affect only one side of the economic balance sheet this approach works fine. The market risk module is more problematic, because risk drivers like foreign exchange rates or interest rates affect both sides of the balance sheet. Therefore so-called replicating portfolios are introduced, which translate the capital market sensitivities of the liability side into a portfolio of financial instruments (e.g. zero coupon bonds). The key question is, how the notional value of the liabilities should be chosen for the replicating portfolio? Market standard is to take the best-estimate value,which implies that the capital backing the surplus is attributed to the risk-free investment, e.g. EUR cash.We will show that this can lead to significant distortions of the measured market risk SCR as compared to an integrated risk model. To avoid this we have introduced at Munich Re the concept of the Economic Neutral Position (ENP) which is defined as (virtual) asset portfolio, which minimizes the total SCR of the integrated model. The ENP is the risk-neutral reference point for Solvency II market risk measurementin Munich Re's certified internal model.[Except for with-profit life insurance business which exhibits significant interaction between the asset and the liability side of the insurer's balance sheet.] This means that any mismatch between assets and ENP produces market risk by definition. For liabilities exhibiting the product structure ∑_i L_i· X_i defined in section <ref>, the ENP corresponds exactly to the solution of the optimization problem addressed in this paper. The ENP consists ofassets X_i (represented by zero coupon bonds of different maturity and currency), which back the claim cash flows of the liability side in a risk minimal way. The investment amounts of the assets in the ENPequal the best estimate values of L_i· X_i plus a safety margin corresponding to the risk minimal investment amount ϕ^*_i.If the L_i are normally distributed then the total safety margin equals 85% of the total insurance risk component SCR_L_i defined as the risk contribution for the non-hedgeable claim size L_i fully diversified within all non-hedgeable risks. This component is allocated to the single assets X_i (e.g. the different maturities of the zero bonds)according to the covariance principle (Theorem <ref>).Let us now analyze the total SCR of a modular risk model, which uses the ENP as risk-neutral reference portfolio for market risk measurement, and compare it with the outcome of an integrated risk model. We assume that the surplus S is of the form (<ref>) for the one-dimensional case. Let us consider the Solvency II risk measure _α[S] with risk tolerance 1-α = 99.5%. The non-hedgeable SCR_L of theinsurance liabilities is computed in the insurancerisk module (e.g. the P/C module). For our simple example SCR_L equals our definition of q and can be set to one without loss of generality (SCR_L = q = _α[L] = 1).The market risk SCR_M is measured by the value-at-risk of the mismatch portfolio of assets minus ENP, i.e. S_M(ϕ) = (ϕ - ϕ^*) · X- ϕ, and is a function of the units ϕ of the financial asset X. For the sake of simplicity the total SCR_T of the surplus is calculated by aggregating SCR_L and SCR_M based on the square root formula, which is also used in the Solvency II standard formula (remember that L and X are assumed to be independent): SCR_T= √( SCR_L^2 +SCR_M^2). This aggregation method is only valid for a sum of normally distributed stochastic variables. Therefore we assume that both risk drivers L and X follow a normal distribution,i.e. we violate here the positivity assumption on X for technical reasons.Otherwise the aggregation method needs to be adjusted accordingly.Figure <ref> compares the total SCR_T of the modular risk model with the total SCR_T of the integrated model, which is simply the value-at-risk of S(ϕ) at risk tolerance 1-α = 99.5% with joint stochastics of all risk drivers. The integrated and the ENP-based modular approachyield in good approximation the same total SCR, as desired. Only if the asset value ϕ differs strongly from the risk minimal value ϕ^*, deviations between the outcomes of the two models can be observed. This is due to the fact, that the square root formula used for aggregation only holds for a sum of normally distributed stochastic variables. Due to the product structure L · X the total distribution of the surplus is in general not normally distributed(even though both L and X are normally distributed). This effect can be healed to some extent by refining the aggregation method for the modular model.For comparison we show in Figure <ref> also the industry standard, which measures market risk versus the replicating portfolio (RP). This corresponds to setting the notional of the liability L equal to its best-estimate value, which is zero in our example. 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Annals of Mathematical Statistics. 29, 635–654.§ PROOFS Justification of the simplifying assumptions (<ref>):[S()] =[] -+ A_0 - [][], henceS() - [S()] = - [] - ( - [][]) =- [] -[]- -[]=- -=: S̃() ,where X̃_i := X_i / [X_i], L̃_i :=[X_i] ·(L_i - [L_i]), and ϕ̃_i :=[X_i] ·(ϕ_i - [L_i]). If [] =, the cash invariance property of the risk measure yields ρ[S(ϕ)] = ρ[S̃()] + A_0 - [][].If [] ≠, the additional linear term [] - appears.Proof of Lemma <ref>: Set G(,z):= (S()≤ z)= _[ ∫_{∈^n: ≥ --z} f_() d]. Changing to the rotated variable = (λ_1,)' defined by = as in Theorem <ref>, which implies = λ_1/√(n)+, we obtain G(,z) = _[ ∫_^n-1 d∫_√(n)/ v^∞ dλ_1 g(λ_1,) ], where v = v(, ,z,) := -1 - z - and g() := f_() is the rotated density. The differentials D_y of G with y ∈{z,ϕ_1, …, ϕ_n} read D_y G(,z) = -_[ ∫_^n-1 g(√(n)/ v,) ·√(n)/ D_y v d], where D_y v = -1 if y = z and = X_i -1 if y = ϕ_i. Differentiation and integration can be interchanged by dominated convergence as the (rotated) density g ofis bounded and 1/ is integrable by assumption. Note that the partial derivatives of G are continuous, which implies that the total differential of G exists.In particular, z↦ G(,z) is continuous and is an increasing function with G(,) = [0,1]. Hence for every ∈_+^n and α∈[0,1] there exists a unique z_ϕ,α∈ such that (S()≤ z_ϕ,α) = G(,z_ϕ,α) =α, which proves (a). The latter also implies that S(ϕ) hasno atoms, and hence upper and lower quantile of S() coincide; the representation for the expected shortfalls follows from Corollary 4.49 of <cit.>, hence (b) is proved.Ad (c): since G is continuously differentiable and D_z G >0 by the strict positivity of the density of , the implicit function theorem implies that ϕ↦ z_ϕ,α is differentiable. For the expected shortfall the differentiability with respect to ϕ_i follows fromthe representation_α[S()]=α^-1·∫_0^α_β[S(ϕ)] dβ, since the differential ∂_ϕ_i and the integral ∫_0^α can be interchanged. This proofs (c) Ad (d): for _1, _2 ∈_+^n and λ∈ [0,1],S(λ·_1 +(1-λ)·_2) = -λ·_1 +(1-λ)·_2 -=λ· [-_1 - ]+(1-λ)·[-_2 - ] =λ· S(_1) +(1-λ)· S(_2) .Hence the assertion follows from the convexity of the expected shortfall. Proof of part (b) of Theorem <ref>:In the one-dimensional case, the cumulative distribution of the surplus can be written F_S(ϕ)(z) = (ϕ·(X-1)-X· L ≤ z)= _X [ (L ≥ϕ-(z+ϕ)/X| X ) ] = _X [ F̅_L (w(ϕ,z,X) ) ] , w(ϕ,z,X) := ϕ-(z+ϕ)/X ,where the last two equations follow from the strict positivity of X and its independence fromL. Since the quantile z_ϕ is implicitly defined as the z solving α = F_S(ϕ)(z) = _X [ F̅_L (w(ϕ,z,X) ) ], we can determine ∂_ϕ z_ϕ at ϕ = q from the implicit function theorem (whose conditions are satisfied as shown in proof of Lemma <ref>).We denote by D_ϕ = ∂_ϕ + (∂_ϕ z_ϕ)·∂_z the total differential with respect to ϕ. Applying D_ϕ on the defining equation of z_ϕyields0 = D_ϕ _X[F̅_L(w(ϕ,z_ϕ,X)) ] = -_X[f_L(w(ϕ,z_ϕ,X)) · [∂_ϕ + ∂_ϕ z_ϕ·∂_z] w (ϕ,z_ϕ,X)] Since ∂_ϕ w = 1-1/X and ∂_z w = -1/X we deduce∂_ϕ z_ϕ = _X[f_L( w) ·(1-1/X)]/_X[f_L( w) ·(1/X)] = _X[f_L( w)]/_X[f_L( w) ·(1/X)] -1,provided the denominator is not zero. Since z_q = -q, the termw(q,z_q,X) = q-(q+z_q)/X = q becomes constant. Hence also f( w) becomes constant and the expression for ∂_ϕ z_ϕ above collapses to(∂_ϕ z_ϕ)_|_ϕ= = [X^-1]^-1 -1 ≤ 0 ,with < if X is non constant. The latter inequality follows from the strict convexity of the inverse function and Jensen's inequality, which implies [X^-1] > [X]^-1 = 1 for non-constant X. Multiplying (<ref>) with -1 yields the assertion of the theorem for the value-at-risk. For the expected shortfall, we can show that at ϕ = the derivative with respect to ϕ vanishes: from the second equation in (<ref>) we find that {S(ϕ) ≤ z_ϕ} = {L ≥ w(ϕ,z_ϕ,X)}. Similar to (<ref>) we calculate[S(ϕ)·_S(ϕ)≤ z_ϕ] = _X[ (ϕ·(X-1) -X· L) ·_L≥ w(ϕ,z_ϕ,X)] = ϕ·_X[ (X-1) ·F̅_L( w(ϕ,z_ϕ,X))] - _X[X ·∫_w(ϕ,z_ϕ,X)^∞ l· f_L(l) dl]. Differentiation with respect to ϕ yields∂_ϕ[S(ϕ)·_S(ϕ)≤ z_ϕ] = _X[ (X-1) ·F̅_L(w)] - ϕ·_X[ (X-1) · f_L(w)· D_ϕ w] + _X[X · w· f_L(w)· D_ϕ w] .Recall that at ϕ =, the term w(,z_,X) = becomes constant. Hence the above expression simplifies∂_ϕ[S(ϕ)·_S(ϕ)≤ z_ϕ]_|_ϕ= = F̅_L()·_X[ X-1] + · f_L()·_X[ (-(X-1)+X) · D_ϕ w] = · f_L()·_X[ (D_ϕ w)(,z_,X)] = 0 ,where the last equality follows from the unit-mean property of X and from (<ref>) evaluated at ϕ= together with the fact that f_L(w) becomes a positive constant. This proves the assertion of the theorem for the expected shortfall. Proof of Proposition <ref>:The characteristic function of Y_0 + Y_1can be written as ϕ_Y_0 + Y_1(t) := [e^it(Y_0 + Y_1)]= _Y_1[e^itY_1·ϕ_Y_0 | Y_1(t) ], where ϕ_Y_0 | Y_1(t) := [e^itY_0 | Y_1] denotes the conditional characteristic function of Y_0 conditioned on Y_1.We show that ϕ_Y_0 + Y_1 and ϕ_Y_0 | Y_1 are integrable: by assumption the differential of any order of the density f_Y_0+Y_1 exists and is integrable. Since f_Y_0+Y_1 is continuous and hence locally bounded, it is also L^2-integrable. We deduce from Parceval's theorem and the differentiation rules for the Fourier transformation that ∫_ |D^k f_Y_0+Y_1|^2 dx = 1/√(2π)∫_ |t^k ·ϕ_Y_0+Y_1(t)|^2 dt for every k ∈_0. As any characteristic function is bounded, ϕ_Y_0+Y_1 is integrable since the tails are integrable by Cauchy-Schwartz: ∫_T_0^∞ |ϕ_Y_0+Y_1| dt≤ (∫_T_0^∞ t^-2dt) · (∫_T_0^∞ t^2|ϕ_Y_0+Y_1| dt < ∞, and analogously for the negative tail.Since F_Y_0+Y_1(z) = _Y_1[F_Y_0| Y_1 (z-Y_1 )], the differentiability- and integrability-assumptions for F_Y_0+Y_1 also hold for the conditional cumulative distribution F_Y_0| Y_1. Repeating the above arguments, we deduce thatϕ_Y_0 | Y_1 is also integrable. By the inversion formula, the cumulative distribution of Y_0+Y_1 can be recovered forz_0<zF_Y_0 + Y_1(z) - F_Y_0 + Y_1(z_0)=(2π)^-1∫_ e^-itz_0-e^-itz/it·ϕ_Y_0 + Y_1(t) dt=(2π)^-1∫_ e^-itz_0-e^-itz/it·_Y_1[e^itY_1·ϕ_Y_0 | Y_1(t) ] dt=(2π)^-1_Y_1[∫_∑_r = 0^∞(itY_1)^r/r!· e^-itz_0-e^-itz/it·ϕ_Y_0 | Y_1(t) dt]=(2π)^-1∑_r = 0^∞(-1)^r/r!·_Y_1[ Y_1^r ∫_ (-it)^r · e^-itz_0-e^-itz/it·ϕ_Y_0 | Y_1(t) dt] = ∑_r = 0^∞(-1)^r/r!·_Y_1[ Y_1^r ·( D_z^r F_Y_0|Y_1(z) - D_z^r F_Y_0|Y_1(z_0) ) ],where the third equation follows from Fubini's theorem (since (t,y_1)↦ϕ_Y_0 | y_1(t) is integrable on the product measure) and from expanding e^itY_1; the fourth equation follows from the fact that the convergence of the exponential series is uniform on {w ∈ :w ≤ 1 } and the last equation follows from the differentiation rules for Fourier transforms. Letting z_0 tend to -∞ we obtainF_Y_0 + Y_1(z) = ∑_r = 0^∞(-1)^r/r!· D_z^r _Y_1[ Y_1^r · F_Y_0|Y_1(z) ] = ∑_r = 0^∞1/r!· (-D_z)^r_Y_1[ Y_1^r ·[_Y_0≤ z|Y_1]] ,which proves the assertion. Proof of Theorem <ref>:We start with some preparations. Since ∼(, ), also (, ) is distributed according to a centered (n+1)-dimensional normal distribution with covariance matrix Γ = ( [_11_12; _12' Γ_22 ]) , with _11 = Σ^, _12 = Σ^·, and Γ_22 = ·. From the theory of conditional normal distributions we derive thatconditioned on the event { = x } follows a n-dimensional normal distribution := | _ { = x }∼( x ·_12/Γ_22,_11 - _12·_12'/Γ_22) =( x ··/·,Σ^ - (Σ^·) · (Σ^·)'/·) .Hence[| ]= []= ·^-1···, [L_i· L_j| ]= [Y_i· Y_j] = [( - [])_i ·( - [])_j] + [Y_i]·[Y_j] = ^_ij -·^-1· (^·)_i · (^·)_j +^2··^-2·(^·)_i · (^·)_j .Denoting the K-terms of the associated single-asset case by _i(q) := [^i ·_>q] we deduce(q)= [·_>q] =[[ |] ·_>q] = _1(q)/·,_[](q)= [··_>q] =[[· |] ·_>q]= (·) ·F̅_(q) - ···/··(F̅_(q) - _2(q)/·) . Ad a): Value-at-risk case:combining Theorem <ref>.(b) with equation (<ref>) gives^* = - ”(q)/ f_'(q) = - _1”(q)/ f_'(q)··/· = ϕ_0^* ··/·,which proves the assertion. The expected shortfall case follows similarly.Ad b): Value-at-risk case: according to Theorem <ref>.(b) using (<ref>) and (<ref>)_α[S(^*)]=+ 1/2 f_(q)·{ f_' (q)^-1·”() ·”()+ K_[]” (q) }=+ 1/2 f_(q)·{····_1”(q)^2/·^2 · f_'(q) -(·) · f_'(q) +···/··( f_'(q) + _2”(q)/·)}=+ f_'(q)/2 f_(q)·{···/· -(·)+···/·^2· f_'(q)·( _1”(q)^2/f_'(q) + _2”(q) )} ,which proves the assertions using the fact that _1”(q)^2/f_'(q) + _2”(q) =f_(q)^2/f_'(q), refer also to (<ref>).Expected shortfall case: according to Corollary <ref>.(b) using (<ref>) and (<ref>)_α[S(^*)]= _α[-] - { f_ (q)^-1·'() ·'()+ K_[]' (q) }= _α[-] - {·· ,·_1'(q)^2/^2 · f_(q) -(·) · f_(q) +·· ,/·( f_(q) + _2'(q)/)},which proves the assertions recalling that -_2'(q)= _1'(q)^2/f_.Proof of Equation (<ref>):The non-centered i-th moment of X_σ_lN is given by m_i (σ_lN) := [X_σ_lN^i] = M(iσ_lN)/M(σ_lN)^i. The moment generating function of Y has the expansion M(σ) = 1+μ_2σ^2/2 + μ_3σ^3/6 + μ_3σ^4/24 +o(σ^4) as σ→ 0, where μ_i are the moments of Y. Further (1+x)^-i = 1 - ix + i(i+1) x^2/2 + o(x^2) as x→ 0. Hence we can write having in mind that μ_2=1 by construction of Ym_i(σ_lN)= [1+(iσ_lN)^2/2 + μ_3(iσ_lN)^3/6 + μ_4(iσ_lN)^4/24] · ·[1- i(σ_lN^2/2 + μ_3σ_lN^3/6+ μ_4σ_lN^4/24) + i(i+1)σ_lN^4/8] + o(σ_lN^4)=1 + i (i-1) σ_lN^2 / 2 + μ_3 i(i^2-1)σ_lN^3/6 + i( μ_4 (i^3-1) - 6i^2+ 3i+3) σ_lN^4/24+ o(σ_lN^4) .The assertion of (<ref>) follows by applying the rule to derive the centered moments m̅_i from the non-centered m_i via m̅_i = ∑_k=0^i ik (-1)^k- i m_k. Proof of Theorem <ref>: Expanding the relation (<ref>) up to fourth order in σ∈{σ_N,σ_lN} in a similar way as for the derivation of (<ref>) having relation (<ref>) in mind and omitting the zero and first order terms (which add up to zero by construction) yields0 = -f_L(-z_0) ·(-z_2 -z_3 - z_4) - 12· f_L'(-z_0) · z_2^2 + 12· (σ^2 +a_3σ^3 + a_4σ^4 ) · [_2”(-z_0)++ _2”'(-z_0)· (-z_2)] + 16· (σ^3 μ_3 + b_4 σ^4) ·_3”'(-z_0) + 124·σ^4 μ_4 ·_4””(-z_0) +o(σ^4),where a_3 = μ_3, a_4=(μ_4 - )and b_4=(μ_4-1), i.e. equal to the third and fourth order terms of the expansion (<ref>). (Note that if σ = σ_N then a_3=a_4=b_4 = 0.)We observe _j' = -(ϕ-id)^j f_L and_j” = j(ϕ-id)^j-1 f_L - (ϕ-id)^j f_L' = -j_j-1' - (ϕ-id)^j f_L' .Setting the second order terms in the above equation equal to zero we recover z_2 = -σ^2/2 f_L()·_2”(q) = σ^2/2 f_L()·((ϕ-id)^2 f_L)', which is the one-dimensional variant of Theorem <ref>. Setting the third order terms equal to zero leads z_3 = -σ^3/6 f_L()· (3 · a_3 ·_2”(q) + μ_3·_3”'(q)) = σ^3μ_3/6 f_L()·[(ϕ-id)^3 f_L']'(q), where the second equation follows from (<ref>). Setting the fourth order term equal to zero we obtain 0=f_L(q) z_4 +σ^4 [ - f_L'_2”^2/8 f_L^2 + a_4 _2”/2 + _2”' _2”/4f_L + b_4_3”'/6 + μ_4 _4””/24](q) .Observing that (_2”^2/f_L)' = -f_L'_2”^2/f_L^2 + 2_2”_2”'/f_L we derivez_4= -σ^4/24 f_L(q)·[ μ_4 _4”' +3_2”^2/f_L + 12a_4 _2' + 4b_4_3”]'(q)=-σ^4/24 f_L(q)·[-μ_4[(ϕ-id)^4 f_L']' +3_2”^2/f_L + (7μ_4-15) _2' + (6μ_4 -4μ_4-6)_3”]'(q)=-σ^4/24 f_L(q)·[- μ_4[(ϕ-id)^4 f_L']' + 3_2”^2/f_L + (7μ_4-15-3(2μ_4-6)) _2' - (2μ_4 -6) (ϕ-id)^3 f_L' ]'(q)= σ^4/24 f_L(q)·[ μ_4[(ϕ-id)^4 f_L']' -3_2”^2/f_L - (μ_4 +3) _2' + (2μ_4 -6) (ϕ-id)^3 f_L' ]'(q) ,where the second and third equality follow again from (<ref>), which proofs the fourth order expansion; hence part a) is proved.Ad b): Let's turn to the expression for ϕ^*: setting ψ = ϕ-, we can rewrite the value-at-risk in third order expansion of part a) whenperforming the differentiation _α[S(ϕ)]=-1/f_L(q)·{(ψ^2 f_L'(q)-2ψ f_L(q)) ·σ_lN^2/2 + (ψ^3 f_L”(q) -3ψ^2 f_L'(q) ) ·σ_lN^3 μ_3/6} + o(σ_lN^3)=(a/3)·ψ^3 + (b/2) ·ψ^2 + c·ψ + q + o(σ_lN^3),with a = -(μ_3σ_lN^3/2) · (f_L”/f_L)(q),b= (μ_3σ_lN-1)σ_lN^2· (f_L'/f_L)(q), and c = σ_lN^2.Setting the differential with respect to ψ equal to zero yields the quadratic formulawhich is solved by ψ_± = (-b ±√(b^2-4ac))/(2a). Only ψ_+ constitutes a (local) minimum of the third order polynomial in ψ, since its second order derivative evaluated at ψ_± reads 2aψ_± +b = ±√(b^2-4ac) which is only positive for ψ_+. Hence the locally minimal ϕ is given by ϕ^* = +ψ_+. Inserting the parameters a,b, and c and straight forward calculus leads the assertion. | http://arxiv.org/abs/1704.08523v3 | {
"authors": [
"Andreas Kunz",
"Markus Popp"
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"Primary: 91B30, Secondary: 60E05, 62P05"
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"published": "20170427120415",
"title": "Economic Neutral Position: How to best replicate not fully replicable liabilities"
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theoremTheorem lemmaLemma definitionDefinition corollaryCorollary | http://arxiv.org/abs/1704.08600v2 | {
"authors": [
"Karoly F Pal",
"Tamas Vertesi"
],
"categories": [
"quant-ph"
],
"primary_category": "quant-ph",
"published": "20170427143252",
"title": "Family of Bell inequalities violated by higher-dimensional bound entangled states"
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ϵ α_s†μν | http://arxiv.org/abs/1704.08127v1 | {
"authors": [
"Tai-ran Liang",
"Bin Zhu",
"Ran Ding",
"Tianjun Li"
],
"categories": [
"hep-ph"
],
"primary_category": "hep-ph",
"published": "20170426135908",
"title": "Natural Supersymmetry from the Yukawa Deflect Mediations"
} |
http://arxiv.org/abs/1704.08418v2 | {
"authors": [
"Zi-Wei Lin"
],
"categories": [
"nucl-th",
"nucl-ex"
],
"primary_category": "nucl-th",
"published": "20170427031841",
"title": "Extension of the Bjorken energy density formula of the initial state for relativistic heavy ion collisions"
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|
[@twocolumnfalseFluctuations in an established transmission in the presence of a complex environment Fabrice Mortessagne December 30, 2023 ==================================================================================== Thediscrete wavelet transform (DWT) can be found in the heart of many image-processing algorithms. Until recently, several studies have compared the performance of such transform on various shared-memory parallel architectures, especially on graphics processing units (GPUs). All these studies, however, considered only separable calculation schemes. We show that corresponding separable parts can be merged into non-separable units, which halves the number of steps. In addition, we introduce an optional optimization approach leading to a reduction in the number of arithmetic operations. The discussed schemes were adapted on the OpenCL framework and pixel shaders, and then evaluated using GPUs of two biggest vendors. We demonstrate the performance of the proposed non-separable methods by comparison with existing separable schemes. The non-separable schemes outperform their separable counterparts on numerous setups, especially considering the pixel shaders.§.§ KeywordsDiscrete wavelet transform, Image processing, Synchronization, Graphics processors]§ INTRODUCTIONThe discrete wavelet transform became a very popular image processing tool in last decades. A widespread use of this transform has resulted in a development of fast algorithms on all sorts of computer systems, including shared-memory parallel architectures. At present, the GPU is considered as a typical representative of such parallel architectures. In this regard, several studies have compared the performance of variousDWT computational approaches on GPUs. All of these studies are based on separable schemes, whose operations are oriented either horizontally or vertically. These schemes comprise the convolution and lifting. The lifting requires fewer arithmetic operations as compared with the convolution, at the cost of introducing some data dependencies. The number of operations should be proportional to a transform performance. However, also the data dependencies may form a bottleneck, especially on shared-memory parallel architectures.In this paper, we show that the fastest scheme for a given architecture can be obtained by fusing the corresponding parts of the separable schemes into new structures. Several new non-separable schemes are obtained in this way. More precisely, the underlying operations of these schemes can be associated with neither horizontal nor vertical axes. In addition, we present an approach where each scheme can be adapted to a particular platform in order to reduce the number of operations. This possibility was completely omitted in existing studies. Our reasoning is supported by extensive experiments on GPUs using OpenCL and pixel shaders (fragment shaders in OpenGL terminology). The presented schemes are general, and they are not limited to any specific type of DWT. To clarify the situation, they all compute the same values.The rest of this paper is organized as follows. Section sec:background formally introduces the problem definition. Section sec:related-work briefly presents the existing separable approaches. Section sec:proposed-schemes presents the proposed non-separable schemes. Section sec:improvements discusses the optimization approach that reduces the number of operations. Section sec:performance evaluates the performance on GPUs in the pixel shaders and OpenCL framework. Eventually, Section sec:conclusion closes the paper. This section is followed by Section sec:appendix for readers not familiar with signal-processing notations.§ BACKGROUNDSince the separable schemes are built on the one-dimensional transform, a widely-used z-transform is used for the description of underlying FIR filters. The transfer function of the filter ( g_k ) is the polynomialG(z) = ∑_kg_k z^-k,where the k refers to the time axis. Below in the text, the one-dimensional transforms are used in conjunction with two-dimensional signals. For this case, the transfer function of the filter ( g_k_m,k_n) is defined as the bivariate polynomialG(z_m,z_n) = ∑_k_m∑_k_ng_k_m,k_nz_m^-k_m z_n^-k_n,where the subscript m refers to the horizontal axis and n to the vertical one. The G^*(z_m,z_n) = G(z_n,z_m) is a polynomial transposed to a polynomial G(z_m,z_n). A shortened notation G is only written in order to keep the notation readable. A discrete wavelet transform is a signal-processing tool which is suitable for the decomposition of a signal into low-pass and high-pass components. In detail, the single-scale transform splits the input signal into two components, according to a parity of its samples. Therefore, the DWT is described by 2 × 2 matrices. As shown by Mallat <cit.>, the transform can be computed by a pair of filters followed by subsampling by a factor of 2. The filters are referred to as G_0, G_1. The transform can also be represented by the polyphase matrix[ G_1 G_1; G_0 G_0 ],where the polynomials G and G refer to the even and odd terms of G. This polyphase matrix defines the convolution scheme. To avoid misunderstandings, it is necessary to say that, in this paper, column vectors are transformed to become another columns. For example, y = Mx and y = M_2 M_1 x are transforms represented by one and two matrices, respectively.Following the algorithm by Sweldens <cit.>, the convolution scheme in (<ref>) can be factored into a sequence∏_k[1 ^(k);01 ][10; ^(k)1 ]of K pairs of short filterings, known as the lifting scheme. The filters employed in (<ref>) are referred to as the lifting steps. Usually, the first step ^(k) in the kth pair is referred to as the predict and the second one ^(k) as the update. The lifting scheme reduces the number of operations by up to half. Since this paper is mostly focused on a single pair of steps, the superscript (k) is omitted in the text below. Note that the number of operations is calculated as the number of distinct (in a column) terms of all polynomials in all matrices, excluding units on diagonals. Considering the shared-memory parallel architectures, the processing of single or several samples is mapped to independent processing units. In order to avoid race conditions during data exchange, the units must use some synchronization method (barrier). In the lifting scheme, the barriers are required before the lifting steps. In the convolution scheme, the barrier is only required before starting the calculation. In this paper, the barriers are indicated by the | symbol. For example, M_2 | M_1 are two adjacent lifting steps separated by the barrier. For simplicity, the number of barriers is also called the number of steps in the text below. Thetransform is defined as a tensor product oftransforms. Consequently, the transform splits the signal into a quadruple of wavelet coefficients. Therefore, theDWT is described by 4 × 4 matrices. See Section sec:appendix for details.Following the pioneering paper of Mallat <cit.>, thetransforms are applied in both directions sequentially. By its nature, this scheme can be referred to as the separable convolution. The calculations in a single direction are performed in a single step. This means two steps for the two dimensions. The scheme can formally be described as[]^V|[]^H| ,where ^H and ^V aretransforms in horizontal and in vertical direction.For the well-known Cohen-Daubechies-Feauveau (CDF) wavelet with 9/7 samples, such as used in the JPEG 2000 standard, these matrices are graphically illustrated in fig:dataflow-Separable-Convolution. Here, only the horizontal part is shown. Particularly, the filters in the figure are of sizes 9 and 7 taps. The 0 < g r a p h i c s > , 0 < g r a p h i c s > , 0 < g r a p h i c s > , and 0 < g r a p h i c s >circles represent the quadruple of wavelet coefficients. Figures shown are for illustration purpose only. Another scheme used fortransform is the separable lifting. Similarly to the previous case, the predict and update lifting steps can be applied in both directions sequentially. Moreover, horizontal and vertical steps can be arbitrarily interleaved thanks to the linear nature of the filters. Therefore, the scheme is defined as[]^V|[]^H|[]^V|[]^H| ,wherein the predict stepsalways precede the update steps . The above mapping corresponds to a singleandpair of lifting steps. For multiple pairs, the scheme is separately applied to each such pair. In order to describematrices, the lifting steps must be extended into two dimensions as[ G^*; G^* ] = [ G^*(z_m,z_n); G^*(z_m,z_n) ] = [ G(z_m); G(z_n) ].Then, the individual steps are defined as[]^H= [ 1 0 0 0; 1 0 0; 0 0 1 0; 0 0 1; ], []^V= [1000;0100; ^*010;0 ^*01;], []^H= [ 1 0 0; 0 1 0 0; 0 0 1; 0 0 0 1; ], []^V= [10 ^*0;010 ^*;0010;0001;].For the CDF wavelets, the matrices are also illustrated in fig:dataflow-Separable-Lifting, again showing the horizontal part only.§ RELATED WORKThis section briefly reviews papers that motivated our research.So far, several papers have compared the performance of the separable lifting and separable convolution schemes on GPUs. Especially, Tenllado et al. <cit.> compared these schemes on GPUs using pixel shaders. The authors mapped data totextures, constituted by four floating-point elements. They concluded that the separable convolution is more efficient than the separable lifting scheme in most cases. They further noted that fusing several consecutive kernels might significantly speed up the execution, even if the complexity of the resulting fused pixel program is higher. Kucis et al. <cit.> compared the performance of several recently published schedules for computing theDWT using the OpenCL framework. All of these schedules use separable schemes, either the convolution or lifting. In more detail, the work compares a convolution-based algorithm proposed in <cit.> against several lifting-based methods <cit.> in the horizontal part of the transform. The authors concluded that the lifting-based algorithm of Blazewicz et al. <cit.> is the fastest method. Furthermore, Laan et al. <cit.> compared the performance of their separable lifting-based method against a convolution-based method. They concluded that the lifting is the fastest method. The authors also compared the performance of implementations in CUDA and pixel shaders, based on the work of Tenllado <cit.>. The CUDA implementation proved to be the faster choice. In this regard, the authors noted that a speedup in CUDA occurs because the CUDA effectively makes use of on-chip memory. This use is not possible in pixel shaders, which exchange the data using off-chip memory. Other important separable approaches can be found in <cit.>. This paper is based on the previous works in <cit.>. In those works, we introduced several non-separable schemes for calculation ofDWT. However, we have not considered important structures, such as polyconvolutions. We contribute this consideration with this paper. Moreover, differences and similarities between the separable schemes and their non-separable counterparts are homogeneously discussed here. All these schemes are also thoroughly analyzed and evaluated.Considering the present papers, we see that a possible fusion of separable parts into new non-separable structures is not considered. Therefore, we investigate on this promising technique in the following sections.[@twocolumnfalse[b] < g r a p h i c s > < g r a p h i c s > < g r a p h i c s > < g r a p h i c s > figure Non-separable convolution scheme for the CDF 9/7 wavelet. The individual rows ofare depicted in separate subfigures. The sizes are from top to bottom and left to right: 9×9, 7×9, 9×7, 7×7. ]§ PROPOSED SCHEMESAs stated above, the existing approaches did not study the possibility of a partial fusion of lifting polyphase matrices. This section presents three alternative non-separable schemes for the calculation of thetransform. The contribution of this paper starts with this section. To avoid confusion, please note that the proposed schemes compute the same values as the original ones.The non-separable convolution scheme is a counterpart to the separable convolution. Unlike the separable scheme, all horizontal and vertical calculations are performed in a single step|,where = ^V ^H is a product oftransforms in horizonal and vertical directions. The drawback of this scheme is that it requires the highest number of arithmetic operations. For the CDF 9/7 wavelet, the matrix is graphically illustrated in fig:dataflow-Non-Separable-Convolution. Here, thefilters are of sizes 9×9, 7×9, 9×7, and 7×7. These sizes make the calculation computationally demanding. Aside from the GPUs, this approach was earlier discussed in Hsia et al. <cit.>.In order to reduce computational complexity, it would be a good idea to construct some smaller non-separable steps. Indeed, the non-separable convolution can be broken into smaller units, referred here to as the non-separable polyconvolutions. For a single pair of lifting steps, the scheme follows from the mapping[,]| ,where[,] = [ ^* ^* ^* ^*; ^* ^* ^* ^*; ^* ^*; ^* ^* 1;]and =+ 1. For the CDF wavelets, the scheme is graphically illustrated in fig:dataflow-Non-Separable-Polyconvolution. In this case, the employed filters are of sizes 5×5, 3×5, 5×3, and 3×3. Note that only half of the operations are required specifically for the CDF 9/7 wavelet, compared to the non-separable convolution. For the sake of completeness, it should be pointed out that it is also possible to formulate the separable polyconvolution scheme. In our experiments, this one was however not proven to be useful concerning the performance.By combining the corresponding horizontal and vertical steps of the separable lifting scheme, the non-separable lifting scheme is formed. The number of operations has slightly been increased. The scheme consists of a spatial predict and spatial update step, thus two steps in total for each pair of the original lifting steps. Formally, for each pair ofand , the scheme follows from[]|[]| ,where[]= [1000; 100; ^*010; ^* ^* 1;], []= [1^* ^*;010 ^*;001 ;0001;].Note that the spatial filters in ^* and ^* may be computationally demanding, depending on their sizes. However, the situation is always better than in the previous two cases. For the CDF wavelets, the scheme is graphically illustrated in fig:dataflow-Non-Separable-Lifting.§ OPTIMIZATION APPROACHThis section presents an optimization approach that reduces the number of operations, while the number of steps remains unaffected. Such an approach was not covered in existing studies.Regardless of the underlying platform, an important observation can be made. A very special form of the operations guarantees that the processing units never access the results belonging to their neighbors. These operations comprise only constants. Since the convolution is a linear operation, the polynomials can be pulled out of the original matrices, and calculated in a different step. Formally, the original polynomials are split as = [0] + [1] and = [0] + [1]. The [0] and [0] are constant. As a next step, the [0] and [0] are substituted into the separable lifting scheme. The separable lifting scheme was chosen because it has the lowest number of operations. This part is illustrated in fig:the-trick. In contrast, the [1] and [1] are kept in original schemes. These two steps are then computed without any barrier. The observation is further exploited to adapt schemes for a particular platform.Now, the improved schemes for the shaders and OpenCL are briefly described. These schemes exploit the above-described observation with the polynomials [0] and [0] . On recent GPUs, OpenCL schemes also omit memory barriers due to thearchitecture. Note that the non-separable polyconvolution scheme makes sense only when K>1, which is the case of the CDF 9/7 wavelet. Implementations in the pixel shaders map input and output data totextures. There is no possibility to retain some results in registers, and the results are exchanged through textures in off-chip memory. Considering the OpenCL implementations, a data format is not constrained. The image is divided into overlapping blocks and on-chip memory shared by all threads in a block is utilized to exchange the results. Additionally, some results are passed in registers.This paper explores the performance for three frequently used wavelets, namely, CDF 5/3, CDF 9/7 <cit.>, and DD 13/7 <cit.>. Their fundamental properties are listed in tab:parameters-baseline: number of steps and arithmetic operations. Note that the number of operations is commonly proportional to a transform performance. Additionally, the number of steps correspond to the number of synchronizations on parallel architectures, which also form a performance bottleneck.§ EVALUATIONThe experiments in this paper were performed on GPUs of the two biggest vendors NVIDIA and AMD using the OpenCL and pixel shaders. In these experiments, only a transform performance was measured, usually in gigabytes per second (GB/s). The host system does not help in the calculation, i.e. with respect to padding or pre/post-processing. Results for only two GPUs are shown for the sake of brevity: AMD Radeon HD 6970 and NVIDIA Titan X. Their technical parameters are summarized in tab:gpus.The implementations were created using the DirectX HLSL and OpenCL. The HLSL implementation is used on the NVIDIA Titan X, whereas the OpenCL implementation on the AMD 6970. The results of the performance comparison are shown in Figures <ref>, <ref>, and <ref>. The value on the x-axis is the image resolution in kilo/megapixels (kpel or Mpel). Except for the convolutions for the DD 13/7 wavelet, the non-separable schemes always outperform their separable counterparts. For CDF wavelets, having short lifting filters, the non-separable (poly)convolutions have a better performance than the non-separable lifting scheme. Unfortunately, for the DD 13/7 wavelet, which is characterized by a high number of operations in lifting filters, the results are not conclusive. Considering the implementation in pixel shaders, similar results were also achieved on other GPUs, including NVIDIA unified architectures and AMD GPUs based on Graphics Core Next (GCN) architecture. Whereas for the OpenCL implementation, the non-separable schemes are only proved to be useful for very long instruction word (VLIW) architectures.Looking at the experiments with the pixel-shader implementations, some transients can be seen at the beginning of the plots (in lower 2 Mpel region). We concluded that these transients are caused by a suboptimal use of cache system, or alternatively by some overhead made by the graphics API. It should be interesting to show some measures provided by an OpenCL profiler. Our profiling revealed that the implementations exhibit only an occupancy 95.24 %. This occupancy is caused by making use of 256 threads in OpenCL work groups and due to maximal number 1344 of threads in multiprocessors (256 times 5 work groups is 1280 out of 1344).[@twocolumnfalse[b] < g r a p h i c s > < g r a p h i c s > (a) OpenCL(b) pixel shader < g r a p h i c s > figure Performance for the CDF 5/3 wavelet. [b] < g r a p h i c s > < g r a p h i c s > (a) OpenCL(b) pixel shader < g r a p h i c s > figure Performance for the CDF 9/7 wavelet. [b] < g r a p h i c s > < g r a p h i c s > (a) OpenCL(b) pixel shader < g r a p h i c s > figure Performance for the DD 13/7 wavelet. ]§ CONCLUSIONSThis paper presented and discussed several non-separable schemes for the computation of thediscrete wavelet transform on parallel architectures, exemplarily on modern GPUs. As an option, an optimization approach leading to a reduction in the number of operations was presented. Using this approach, the schemes were adapted on the OpenCL framework and pixel shaders. The implementations were then evaluated using GPUs of the two biggest vendors. Considering OpenCL, the schemes exploit features of recent GPUs, such as warping.For CDF wavelets, the non-separable schemes exhibit a better performance than their separable counterparts on both the OpenCL and pixel shaders.In the evaluation, we reached the following conclusions. Fusing several consecutive steps of the schemes might significantly speed up the execution, irrespective of their higher complexity. The non-separable schemes outperform their separable counterparts on numerous setups, especially considering the pixel shaders. All of the schemes are general and they can be used on any discrete wavelet transform. In future work, we plan to focus on general-purpose processors and multi-scale transforms. Acknowledgements This work has been supported by the Ministry of Education, Youth and Sports from the National Programme of Sustainability (NPU II) project IT4Innovations excellence in science (no. LQ1602), and the Technology Agency of the Czech Republic (TA CR) Competence Centres project V3C – Visual Computing Competence Center (no. TE01020415).§ APPENDIXFor readers who are not familiar with signal-processing notations, a relationship between polyphase matrices and data-flow diagrams is explained here. Thediscrete wavelet transform divides the image into four polyphase components. Therefore, the 4×4 matrices of Laurent polynomials are used to describe thediscrete wavelet transform. These matrices are commonly referred to as the polyphase matrices. The Laurent polynomials correspond toFIR filters, that define the transform. In most cases, the transform is described using a sequence of such matrices. One particular matrix thus defines a step of calculation in this case. For example, the matrix[]^H= [ 1 0 0 0; 1 0 0; 0 0 1 0; 0 0 1; ]maps four polyphase components to another four components, while using twoFIR filters represented by the polynomials . Moreover, when we substitute a particular polynomial, say P(z) = -1/2( 1 + z^-1 ), into the matrix, the mapping gets a specific shape. Such a substitution illustrated by the data-flow diagram in fig:dataflow-appendix. The solid arrows correspond to multiplication by -1/2 along with subsequent summation.myabbrvnat | http://arxiv.org/abs/1704.08657v2 | {
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Institute of Theoretical Physics, Lanzhou University, Lanzhou 730000, People's Republic of China In this work, we examine the entropy emission property of black holes. When the greybody factor is considered, it is found that Schwarzschild black hole is a one-dimensional entropy emitter, which is independent of the spacetime dimension and the spin of the emitted quanta. However, when generalized to other black holes with two or more parameters, the result shows that the one-dimensional entropy emission property will be violated. Thus our result implies that not all black holes behave as one-dimensional entropy emitters.04.70.Dy, 04.50.-h, 11.10.Kk Black hole entropy emission property Shao-Wen Wei [[email protected]], Yu-Xiao Liu [[email protected]]============================================================================ § INTRODUCTIONCombining quantum mechanics and general relativity, black holes are found to radiate particles characterized by the well-known Hawking temperature <cit.>. There are several approaches proposed for obtaining the temperature, for examples the collapse geometry, tunneling process <cit.>, and the gauge and gravitational anomalies <cit.>.Exploration of the property corresponding to Hawking radiation is an important subject. In 2001, Bekenstein and Mayo <cit.> studied the black hole entropy flow related with the radiation particles at the Hawking temperature, and found that the entropy emission rate for a (3+1)-dimensional Schwarzschild black hole is Ṡ=(ν^2Γ̅π P/480 ħ)^1/2,where the values of ν and Γ̅ can be found in Page's work <cit.>. This result strongly recommends that black hole radiation is completely different from that of a hot body in three-dimensional space, but like a hot body in one-dimensional space. Thus they concluded that Schwarzschild black holes are one-dimensional entropy emitters, and the information flow accompanied by Hawking radiation from a (3+1)-dimensional Schwarzschild black hole is one-dimensional. Therefore, such property can be treated as a complementary statement to the famous “holographic principle" <cit.>.Very recently, the one-dimensional nature of the black hole entropy emission rate has been explored with a direct calculation of the entropy emission rate. The first interesting example is the d-dimensional Schwarzschild black hole. In Refs. <cit.>, it was shown that its entropy emission rate has a similar form as that of a one-dimensional hot body, which implies that a Schwarzschild black hole in arbitrary spacetime dimensions is a one-dimensional entropy emitter. This greatly confirms Bekenstein and Mayo's result, and supports the conjecture that all black holes are one-dimensional entropy emitters. However, when generalized other types of black holes, the result behaves very differently. For a (2+1)-dimensional non-rotating Banados-Teitelboim-Zanelli (BTZ) black hole, the result <cit.> shows that the entropy flow or information out of the black hole is three-dimensional. For Lovelock black holes, the result indicates that the channel of the entropy flow is equal to d for odd d-dimensional spacetime, and 1+ε(Λ) for even d-dimensional spacetime <cit.>. On the other hand, Hod <cit.> noted that the entropy emission rate of a Reissner-Nordström (RN) black hole characterized by the neutral sector of the Hawking radiation spectra can be studied analytically in the near-extremal regime. Unfortunately, the result indicates that such black hole is not one-dimensional entropy emitter.Nevertheless, we would like to reexamine the black hole entropy emission property. In the previous works, the study showed that the Schwarzschild black hole is a one-dimensional entropy emitter of boson quanta. However, how about the fermion radiation? The answer is worth searching. On the other hand, when some other parameters are included in, do the black holes still behave as one-dimensional entropy emitters? Motivated by these questions, we would like to explore the entropy emission properties of black holes, even the gravitational effect is considered. § BEKENSTEIN AND MAYO'S TREATMENT Here, we would like to give a brief review of the Bekenstein and Mayo's treatment given in Ref. <cit.>. In flat spacetime, the energy and entropy transmission out of a closed black object with temperature T and area A into3-dimensional space areP = π^2AT^4/120 h^3, Ṡ = 4P/3T.Combining the above two equations yieldṠ=2/3(2π^2AP^3/15ħ^3)^1/4.On the other side, if we replace the black object with a black hole, the entropy emitted rate is also in the form (<ref>). However, the area A and temperature T will not be independent of each other. For a Schwarzschild black hole, we always haveA=16π M^2, T=ħ/8π M,where M is the black hole mass. Thus, one will obtain <cit.>Ṡ=(ν^2Γ̅π/480ħ)^1/2× P^1/2.As claimed by Bekenstein and Mayo, this result implies that a d=4-dimensional Schwarzschild black hole is a one-dimensional entropy emitter.This result was soon generalized to the higher dimensional Schwarzschild black hole cases (see Refs. <cit.>). The starting point is the generalized Stefan-Boltzmann law, which in d-dimensional spacetime is given by <cit.>P=σ_d𝒜T^d,where σ_d is the generalized Stefan-Boltzmann constant. In addition, for a d-dimensional perfect black-body emitter, there exists a relation <cit.>Ṡ=d+1/d×P/T.For a d-dimensional Schwarzschild black hole, its temperature is given by <cit.>T=(d-3)ħ/4π r_h.The parameter r_h is the radius for the black hole event horizon. While the effective area in Eq. (<ref>) has different interpretations. Mirza, Oboudiat, and Zare evaluated it as the area of black hole horizon, i.e., 𝒜=A_h. Hod adopted another expression𝒜=Γ(d/2)/√(π)(d-1)Γ(d-1/2)(d/2)^d-1/d-2(d/d-2)^d-1/2 A_h.Nevertheless,𝒜∝ A_h.Making use of Eqs. (<ref>)-(<ref>), one will getṠ∝ P^1/2,which implies that a d-dimensional Schwarzschild black hole is also a one-dimensional entropy emitter.§ OUR ARGUMENT During the treatment, one important key is the relation (<ref>) between the entropy emitted rate and the energy power. However, when we replace the black body with a black hole, we should consider the gravitational effect of the black hole when it emits particles. It will lead to the deviation of black hole from a black body. Therefore, the grey-body factor should be included in. Moreover, the black hole energy will be taken away by the emitted quanta, which will lead to the continuous decrease of the black hole size. We show the sketch picture in Fig. <ref>.Considering these effect, the energy radiation power and entropy emission rate of spin s particles for a Schwarzschild black hole areP = ∑_jN_j^(s)∫_0^∞ω h_j^(s)(ω)dω, Ṡ = ∑_jN_j^(s)∫_0^∞ g_j^(s)(ω)dω,where j is the angular harmonic index. For scalar, vector, and tensor modes, we respectively have <cit.>N_j^(s) = (2j+d-3)(j+d-4)/j!(d-3)!, N_j^(s) = j(j+d-3)(2j+d-3)(j+d-5)!/(j+1)!(d-4)!, N_j^(s) = (d-4)(d-1)(j+d-2)(j-1)/2(j+1)!(d-3)! × (2j+d-3)(j+d-5)!.The integrands are given by <cit.>h_j^(s)(ω) = 1/2π|A_j^(s)|^2/e^ω/T-(-1)^2s, g_j^(s)(ω) = 1/2π[|A_j^(s)|^2/e^ω/T-(-1)^2s ×ln(e^ω/T-(-1)^2s/|A_j^(s)|^2+(-1)^2s) +(-1)^2sln(1+(-1)^2s×|A_j^(s)|^2/e^ω/T-(-1)^2s)].Here |A_j^(s)|^2 is the dimensionless greybody factor, which is a function of ω and the black hole horizon radius, i.e. |A_j^(s)|^2=|A_j^(s)|^2(ω, r_h). Or, more explicitly, we can express the greybody factor as |A_j^(s)|^2=|A_j^(s)|^2(ω r_h) with ω r_h being a dimensionless parameter. For the d-dimensional Schwarzschild black hole, we have T∝1/r_h. Then we can define a new dimensionless variablex=ω/T.Then the integrals (<ref>) and (<ref>) will beP = T^2×∑_jN_j^(s)∫_0^∞ x h_j^(s)(x)dx, Ṡ = T×∑_jN_j^(s)∫_0^∞ g_j^(s)(x)dx,The integral parts in the above equations give pure numbers. So, we haveP∝ T^2 and Ṡ∝ T. Combining these two relations yieldsṠ=C(d,s)× P^1/2.Here the number C(d,s) is only dependent of the dimension number d of the spacetime and the kind of the emitted particles, while independent of the black hole parameter. It can be obtained by performing the integrals,C(d,s)=∑_j√(N_j^(s))∫_0^∞ g_j^(s)(x)dx/√(∫_0^∞ x h_j^(s)(x)dx).During the radiation, the black hole horizon shrinks, which leads to the rise of the black hole temperature. However, the relation (<ref>) always holds during the process. Thus, a d-dimensional Schwarzschild black hole is a one-dimensional entropy emitter, even the gravitational effective is included in. It is also clear that such property is independent of the nature of the emitted quanta.§ GENERAL ARGUMENTIt was reported in Refs. <cit.> that, some black holes, such as charged RN black holes, Lovelock black holes, are not one-dimensional entropy emitters. Here we would like to examine the entropy emitter for a general case using the method of dimensional analysis. We adopt the units k_B=G=c=ħ=1. For a black hole, the radius of its event horizon is a characteristic length, so we can express all the parameters with the length dimension [L]. Taking an example, the black hole temperature T=T(r_h, α_i) with α_i being the black hole characteristic parameters (except the mass), such as the charge and angular momentum. Then we can construct a dimensionless temperature T̃=r_hT(α̃_̃ĩ)=T̃(α̃_̃ĩ), where α̃_̃ĩ has been non-dimensionalized with r_h. For a black hole, the temperature, energy radiation power, and entropy emission rate have the following dimensions[T]=[L]^-1, [P]=[T^2]=[L]^-2 , [̇Ṡ]̇ =[T]=[L]^-1.Then, according to the dimensions, we finally getṠ= C(d,s;α̃_i)× P^1/2.At a first glance, one will obtain the result that it is the same functional relation as the one-dimensional entropy emitter. However, we should note the expression of the parameters α̃_i:α̃_i=α_i/r_h^δ_i,where δ_i is the dimensional number of the black hole parameter α_i. We need clearly recall that, during the radiation of the quanta, the characteristic parameters α_i keeps constant, while the horizon shrinks. So the dimensionless parameters α̃_i varies in the emitted process. And the coefficient C(d,s;α̃_i) can not be a constant. Therefore, for a black hole with nonvanishing α_i, it will not a one-dimensional black hole entropy emitter. On the other hand, a Schwarzschild black hole behaves as a limit for other black holes. Thus in the limit α_i→0, C(d,s;α̃_i)→ C(d,s), then the property of one-dimensional entropy emitter will be recovered.In fact, if the emitted quanta can change the black hole parameters (like the charge and angular momentum) a specific manner such that α̃_i keep as fixed constants, then the black hole will be exactly a one-dimensional entropy emitter.§ SUMMARY At last, we would like to give a brief summary of this work. We considered the gravitational effect of black hole on the entropy emission properties.For a d-dimensional Schwarzschild black hole, our result confirms that of Refs. <cit.>. Moreover, we also showed that the one-dimensional entropy emission property of a d-dimensional Schwarzschild black hole is independent of the spin s of the emitted quanta. This result holds both for boson and fermion emitted quanta.When other parameters are included in, for example, the black hole charge and angular momentum, the black hole entropy emission will deviate from the one-dimensional property. This result also agrees with Refs. <cit.>.In summary, a Schwarzschild black hole is a one-dimensional entropy emitter for any spacetime dimension d and for any emitted quanta with spin s. While other black holes will not hold such intriguing property in general. § ACKNOWLEDGEMENTSThis work was supported by the National Natural Science Foundation of China (Grants No. 11675064, No. 11522541, No. 11375075, andNo 11205074), and the Fundamental Research Funds for the Central Universities (No. lzujbky-2016-121 and lzujbky-2016-k04).99HawkingS. Hawking, Particle creation by black holes, Commun. Math. Phys. 43, 199 (1975).HawkingbS. Hawking,Black Hole Evaporation,Nature (London) 248, 30 (1974).ParikhM. K. Parikh and F. 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Phys. 25, 2888 (1984); Symmetric-tensor eigenspectrum of the Laplacian on n-spheres26, 65 (1985).WaldR. M. Wald, On particle creation by black holes,Commun. Math. Phys. 45, 9 (1975). HawkingdS. W. Hawking,Breakdown of predictability in gravitational collapse,Phys. Rev. D 14, 2460 (1976). | http://arxiv.org/abs/1704.08402v1 | {
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"primary_category": "gr-qc",
"published": "20170427012415",
"title": "Black hole entropy emission property"
} |
N. D. Sandham, R. Johnstone, C. T. JacobsSurface-sampled simulations of turbulent flowFaculty of Engineering and the Environment, University of Southampton, University Road, Southampton, SO17 1BJ, United KingdomFaculty of Engineering and the Environment, University of Southampton, University Road, Southampton, SO17 1BJ, United Kingdom. A new approach to turbulence simulation, based on a combination of large-eddy simulation (LES) for the whole flow and an array of non-space-filling quasi-direct numerical simulations (QDNS), which sample the response of near-wall turbulence to large-scale forcing, is proposed and evaluated. The technique overcomes some of the cost limitations of turbulence simulation, since the main flow is treated with a coarse-grid LES, with the equivalent of wall functions supplied by the near-wall sampled QDNS. Two cases are tested, at friction Reynolds number _τ=4200 and 20 000. The total grid node count for the first case is less than half a million and less than two million for the second case, with the calculations only requiring a desktop computer. A good agreement with published DNS is found at _τ=4200, both in terms of the mean velocity profile and the streamwise velocity fluctuation statistics, which correctly show a substantial increase in near-wall turbulence levels due to a modulation of near-wall streaks by large-scale structures. The trend continues at _τ=20 000, in agreement with experiment, which represents one of the major achievements of the new approach. A number of detailed aspects of the model, including numerical resolution, LES-QDNS coupling strategy and sub-grid model are explored. A low level of grid sensitivity is demonstrated for both the QDNS and LES aspects. Since the method does not assume a law of the wall, it can in principle be applied to flows that are out of equilibrium. Surface-sampled simulations of turbulent flow at high Reynolds number Neil D. Sandham, Roderick Johnstone, Christian T. Jacobs December 30, 2023 ===================================================================== § INTRODUCTIONDespite advances in hardware and in particular the use of massively parallel supercomputers, applications of direct numerical simulation (DNS) are limited in terms of the Reynolds number () that can be reached, owing to the cost of the simulations. Measured in terms of number of grid points, the cost scales strongly with , for example the number of grid points required scales as _L^37/14 (where L is the distance from the leading edge) for boundary layer flow <cit.> and smaller timesteps are also required as the grid becomes finer. A cheaper approach is large-eddy simulation (LES) where only the larger scales are simulated, while smaller scales are modelled. However, near a wall the smaller scales play a predominant role and to obtain sufficient accuracy many LES in practice end up being `wall-resolved' LES, where grid node counts are significantly lower than DNS (typically of the order of 1%) but a strong scaling withremains, meaning that LES is also too expensive for routine application, for example to flow over a commercial aircraft wing. The alternative of wall-modelled LES has much more attractive scaling characteristics (fixed in terms of boundary layer thickness, for example), but relies very heavily on a wall treatment. Given that there is no accurate reduced-order model for turbulence near a wall (which would require some kind of breakthrough solution of the `turbulence problem'), a lot of reliance would be placed on the near-wall model, with little likelihood of significant improvements over second-moment closure approaches based on the Reynolds-averaged equations. In this paper we consider an alternative approach whereby small-domain simulations are used to represent the near-wall turbulence, in a non-space-filling manner, and linked to an LES away from the wall, where the sub-grid models might be expected to work with reasonable accuracy. To understand the new approach, an appreciation of recent progress in understanding the physics of near-wall turbulence is useful. The inner region, consisting of the viscous sublayer and the buffer layer, out to a wall-normal distance of z^+≈ 100 (where z is the wall normal distance and the dimensionless form is z^+=z u_τ/ν, where ν is the kinematic viscosity and u_τ=√(τ_w/ρ) is the friction velocity, with τ_w=μ( du/dz )_w the wall shear stress, μ=ρν being the viscosity and ρ the density) follows a known regeneration cycle <cit.>, whereby vortices develop streamwise streaks, which give rise to instabilities that create new vortices. The streamwise scales are up to 1000 in wall units (ν/u_τ), while the spanwise scale is 100 (sufficient to sustain near-wall turbulent cycles <cit.>), but one should note that the probability distributions are smooth over a range of scales, and the regeneration process doesn't involve single Fourier modes with these wavelengths. The outer region of a turbulent flow follows a different known scaling, where a defect velocity (relative to the centreline in internal flows, or the external velocity in boundary layers) scales with u_τ and the geometry of the flow (for example boundary layer thickness). As the Reynolds number is increased an overlap between these inner and outer-layers is found and, at very high Re, recent pipe flow experiments <cit.> provide good evidence for a logarithmic region in the mean velocity profile. Within the logarithmic region of turbulent boundary layers, pipes and channels very large scale motions (VLSMs) (sometimes referred to as `superstructures') have been observed, for example in <cit.>. These structures are in addition to the near-wall turbulence cycle and possible organised motions in the outer part of the flow. Interestingly these VLSM structures are longer than those of the outer layer <cit.>. The presence of both outer-layer motions and VLSMs means that the near-wall flow cannot be considered as a separate feature, but one that is modulated by larger-scale flow features. This leads to increases in the near-wall fluctuations asis increased, as has been shown experimentally. For example <cit.> shows a small increase in the near wall (z^+=12) peak in streamwise fluctuation level and a much larger increase for z^+>100, eventually leading to a separate peak in the fluctuation profile.Further insight into the near-wall structure of turbulent flow has been obtained recently from a resolvent-mode analysis of the mean flow <cit.>. The resolvent modes are obtained from a singular value decomposition of the linearised Navier-Stokes equations subject to forcing and shows the response of the flow. From this type of analysis, Moarref et al. <cit.> extracted near-wall, outer layer and mixed scalings. In particular, at very high Reynolds number three kinds of structures were shown to be present, including a near-wall structure whose scaling was in good agreement with the regeneration cycle discussed above. In the outer region the spanwise width of structures was shown to scale with the channel half height, whereas in the logarithmic region the width had a mixed scaling. Given these insights into the key structures in turbulent wall-bounded flow, it is interesting to consider a simulation approach based on resolving these classes of structures.There have been a small number of previous attempts to combine different simulations to resolve the various layers of flow near a wall. A multi-block approach was developed by Pascarelli et al. <cit.>. This method includes a multi-layer structure with a large block covering the channel central region and smaller blocks near the wall that were periodically-replicated. Simulations were only carried out at lowbut it was observed that the flow adjusted very quickly to the imposition of periodic spanwise boundary conditions at the block interfaces. The method envisioned more layers at high . The cost saving at thesimulated was found to be modest and the method would not capture the modulation of small scales by large scales, since the same near-wall box was used everywhere. Another approach has been proposed recently <cit.> in which a minimal flow unit for near-wall turbulence is coupled to a coarse-grid LES for the whole domain, with a rescaling of both simulation at each timestep. It is not clear from the description whether the minimal flow simulation feeds back the correct local shear stresses to the large structures, but results from this approach are shown to reproduce experimental correlations for skin friction <cit.> within 5% up to _τ=10,000.In the present contribution we consider an approach that uses multiple near-wall simulations that are able locally to respond to changes in the outer-layer environment, provided by an LES. In return the near-wall simulations provide the wall shear stress required by the LES as a boundary condition. The general arrangement is sketched on Figure <ref> for a simulation of turbulent channel flow. In effect the set of near-wall simulations (shown in red on the figure) are used as the near-wall model. However these simulations are only sampled (not continuous) in space, hence a large saving in computational cost is possible. As a shorthand notation we will refer to the near-wall simulations as quasi-DNS (QDNS) since no sub-grid model is used, but resolutions do not need to be fine enough for these to be fully-resolved DNS. The approach proposed here follows the style of heterogeneous multiscale methods (HMM), a general framework in which different modelling techniques/algorithms are applied to different scales and/or areas of the computational grid <cit.>. More specifically, the crux of HMM is the coupling of an overall macroscale model (i.e. the LES in this case) with several microscale models (i.e. the QDNS blocks); it is these microscale models that can provide missing/more accurate data (i.e. the shear stress boundary conditions) back to the macroscale model.A similar multiscale reduced-order approach was formulated independently by <cit.> and applied to a quasigeostrophic model of the Antarctic Cirumpolar Current. The velocity field and potential vorticity gradient were advanced in time using a coarse grid model; this model comprised small embedded subdomains at each `coarse' grid location (in contrast to the use of blocks encompassing multiple coarse grid points in this work), within which smaller-scale eddies evolved on a separate spatial and temporal scale. This is similar to the method proposed here in that the domain comprises smaller turbulence-resolving simulations that are coupled with a coarser grid simulation. Furthermore, the components of the eddy potential vorticity flux divergence were computed and averaged in the subdomains and fed back to the coarse grid model, much like the near-wall averaged shear stresses computed in the approach described here. However, unlike the present work, the state of the eddy-resolving embedded subdomains was not carried over between coarse grid time-steps and was reset each time to a given initial condition.In this paper we set out the method and present results from a proof-of-concept simulation of turbulent channel flow, also showing the sensitivity of the method to various numerical parameters. Section <ref> provides details of the numerical approach and its implementation as a Fortran code. Section <ref> presents the proof-of-concept results from the simulation of turbulent channel flow. The potential for extension of the method to very high Reynolds number is then discussed in Section <ref>. The paper closes with some conclusions in Section <ref>.§ NUMERICAL FORMULATION§.§ Numerical methodThe same numerical method is used for both the LES and the near-wall QDNS domains shown in Figure <ref>, all of which have periodic boundary conditions applied in the wall-parallel directions x and y. Within these domains the incompressible Navier-Stokes equations are solved on stretched (in z) grids, using staggered variables (with pressure p defined at the cell centre and velocity components u_i at the centres of the faces), by an Adams-Bashforth method. The governing equations are the continuity equation ∂ u_i/∂ x_i=0and the momentum equations ∂ u_i/∂ t+ ∂ u_i u_j/∂ x_j=δ_i1-∂ p/∂ x_i + 1/_τ∂^2 u_i/∂ x_j ∂ x_j,where all variable are dimensionless (normalised using the channel half height, friction velocity, density and kinematic viscosity) and the term δ_i1 provides the driving pressure gradient. Enforcing a constant pressure gradient or constant mass flow rate are the two main approaches to ensuring that the flow field evolves with a near-constant wall shear velocity <cit.>. In the present work, the constant pressure gradient δ_i1 frequently used in similar channel flow simulation setups (e.g. <cit.>) is used only in the LES, whereas for the QDNSs it is set to zero and a constant mass flow rate is employed for consistency reasons so that there is conservation of mass between the LES and QDNS. It was found that using only the LES stresses alone to drive the QDNS simulations resulted in too high a flow velocity. Any inaccuracies in the shear stresses would increase over time since there was no mechanism in place to keep the wall shear velocity (and therefore Re_τ) near the desired constant value.Grids are uniform in the wall-parallel directions x and y and stretched in the wall-normal (z) direction according to z=tanh(aζ)/tanh(a),where a is a stretching parameter and ζ is uniformly spaced on an appropriate interval (-1 ≤ζ≤1 in the LES for example).The Adams-Bashforth method advances the solution in time using two steps. In the first step a provisional update of the velocity field is made according tou_i^*=u_i^n+Δ t [ 3/2H_i^n -1/2H_i^n-1+1/2∂ p^n-1/∂ x_i +δ_i1],whereH_i=-∂ u_i u_j/∂ x_j+1/Re∂^2 u_i/∂ x_j ∂ x_j.A final correction is then made to give u_i^n+1=u_i^*-3/2Δ t ∂ p^n/∂ x_i,where the pressure is obtained by solution of ∂^2 p/∂ x_i ∂ x_i=2/3Δ t∂ u_i^*/∂ x_i.Application of a fast Fourier transform in horizontal planes leads to a tridiagonal matrix that is solved directly. §.§ Model implementationThe model code was written in Fortran 90, with conditional statements used to enable/disable the LES parameterisation depending on the flag set in the simulation setup/configuration file. Each iteration of the combined LES-QDNS approach entailed first running each QDNS simulation individually with its own setup file (containing the number of timesteps to perform, for example); the LES was then run immediately afterwards to complete the iteration (and thus a single LES timestep, as explained in the next subsection). The setup and execution of these simulations was performed using a Python script that ensured the simulations were run in the correct order, and also performed statistical averaging and postprocessing of the simulation results. Such postprocessing includes the averaging of the shear stresses from all the QDNS and writing out these results to a file in a format that the LES expects, as discussed in the next section. Note that, while the model itself was written in Fortran and could only be executed in serial, the Python script that handled the execution of the simulations was parallelised such that all of the QDNS were executed at the same time, with the results then being combined/postprocessed via MPI Send/Receive operations. The mpi4py library <cit.> was used for this purpose. For a setup involving N × N QDNS per wall, the LES-QDNS approach requires (N × N × 2) + 1 MPI processes (N × N × 2 processes for the total number of QDNS, and one process for the LES). §.§ Interconnection between LES and QDNSThe basic arrangement for the simulations is as shown on figure <ref>. To illustrate the details we consider a baseline case at _τ=4200, corresponding to the highest current _τ for DNS of channel flow <cit.>. The DNS used a domain of size 2π by π by 2 with a 2048 × 2048 × 1081 grid. The smallest resolved length scale in a DNS needs to be O(η), where η is the Kolmogorov length scale <cit.>. The choice of O(η) grid spacing in the DNS of <cit.> therefore satisfied this requirement, and is consistent with known guidelines for the choice of wall units in turbulent channel flow simulations (see e.g. <cit.>). Here we attempt the same configuration using an LES in a domain 6 × 3 × 2[Note that the domain size of 6 × 3 × 2 did not match exactly with the DNS domain size of 2π×π× 2 because such round numbers were convenient for wall unit measurements and choice of QDNS block size. The results were found not to be sensitive to this small inconsistency.] on a 24× 24 × 42 grid (with stretching parameter a set to 1.577) with a 4× 4 array of QDNS on each wall, each QDNS using a 24^3 grid (with stretching parameter a set to 1.4) covering a domain in wall units of 1000 × 500 × 200. The total number of grid points is less than half a million, or 0.01% of the DNS. In this baseline case the QDNS grid spacing in wall units is Δ x^+=41.7, Δ y^+=20 with the first cell centre at z^+=1.5. The choice of QDNS resolution follows guideline values in the literature (e.g. Δ x^+ typically less than 50 in the spanwise direction compared to 20 for DNS <cit.>) such that the cost of the QDNS is approximately an order of magnitude less than a full DNS near the wall <cit.>. It was found that refining this further had little impact on the accuracy of the results, as discussed in Section <ref>. Seen in plan view the entire QDNS occupies one LES cell (i.e. L_x,QDNS= Δ x_LES and L_y,QDNS = Δ y_LES. In the wall-normal direction the QDNS overlaps the LES, in this case by three cells, to avoid using the immediate near-wall points that are most susceptible to errors in the accuracy of the sub-grid modelling. These three cells cover the region out to z^+=200 with the centre of the first LES cell at z^+=30. The LES grid was deliberately kept very coarse in order to highlight the potential savings of the proposed method and how it takes advantage of the separation of scales, although it was found a posteriori that it needed refining to a 96 × 96 × 56 grid in order to yield a much better mean flow prediction (see Section <ref>).The required resolution for DNS and QDNS scales strongly with Reynolds number <cit.>, with the number of DNS grid points being proportional to ^9/4 <cit.> (or _L^37/14 in the more recent calculations of <cit.>). The resolution requirements for QDNS are likely to be similar to that of wall-resolving LES which scales proportional to ∼^2 <cit.>, while wall-modelled LES scales weakly with Reynolds number (^2/5 <cit.>). In terms of resolving the turbulence structures, small-scale eddies and streaks near the wall scale with wall units while the LSMs scale with domain size <cit.>.The time step for the QDNS is set to Δ t=0.0001 and 25 QDNS steps are run before one LES update (i.e. the LES operates on a timestep of 0.0025). The respective Courant number criteria need to be respected for both the LES and QDNS simulations, which determines the number of QDNS steps per LES step.The QDNS are driven by the LES. The QDNS are run in constant mass-flux mode with the mass fluxes in x and y provided by the LES. At the upper boundary conditions the QDNS use w=0 and apply a viscous stress corresponding to the shear stresses from the LES. This effectively sets du/dz and dv/dz at the upper boundary of the QDNS and, together with the enforced mass flux, drives the QDNS to match the LES in these aspects. Each QDNS is thus driven by the local LES conditions and simulates the response of wall turbulence to large-scales present in the LES. Figure <ref> shows a snapshot of the results from a simulation. The streamwise velocity is shown in a plan view. Part (a) of the figure shows the whole LES domain at z^+=335, with QDNS sub-domains visible as the dark areas. Part (b) of the figure zooms in on one of the QDNS domains, showing streamwise velocity contours near the wall (z^+=13). In this arrangement it can be seen how the 4 × 4 array of QDNS samples the large-scale structures from the LES. At the end of the 25 QDNS time steps the shear stresses (du/dz)_w and (dv/dz)_w are averaged over each QDNS and linearly interpolated back to the LES to provide the lower boundary condition. Such a boundary condition is considered a good first approximation, despite the QDNS blocks not resolving turbulence structures down to the Kolmagorov length scale, because the QDNSs are capable of resolving the near-wall streaks to reduce the empiricism required at the wall <cit.>. It may be more desirable to use more information from the QDNS (e.g. transferring all components of the Reynolds stress tensor back to the LES and computing a contribution to the eddy viscosity for use in the LES) to obtain a more accurate result. Nevertheless, the current sampling technique and the interpolation back to the full LES domain is advantageous since it exploits the emerging spectral gap that exists between the large and small scales at large Reynolds number <cit.>.Larger domains are handled by increasing the size of the LES domain and increasing the number of QDNS blocks. It should be noted that there is only a very small amount of communication between the LES and QDNS calculations (four floating point numbers into each QDNS and two returned per 25 steps of computational effort). Thus the introduction of the QDNS subdomains brings with it an additional level of parallelism, with parallel treatment also possible within the LES and QDNS blocks using conventional strategies.Once fully developed, the turbulent dynamics are homogeneous in the spanwise and streamwise directions <cit.> and thus the use of a regular grid on each wall is a justifiable initial choice. However, instead of keeping the QDNS blocks stationary, it may be more appropriate to move the blocks downstream with the flow speed in an attempt to track smaller-scale turbulent structures. It is possible that the effects of these turbulent small-scale structures are being dissipated by the averaging procedure or simply by the region of lower resolution outside the QDNS block, with downstream blocks becoming increasingly inaccurate as a result. It is unclear how many QDNS blocks will be required in general, but the number is likely to scale with _τ in order to obtain adequate sampling near the wall.§ PROOF OF CONCEPT AND SENSITIVITY TO NUMERICAL PARAMETERSThe mean streamwise velocity u^+ and root mean square (RMS) of the streamwise velocity fluctuations u^+_RMS were used as performance measures. These are defined, for each point k in the z-direction, by u^+ = 1/SN_xN_y∑_s=1^S∑_i=1^N_x∑_j=1^N_yu^+_i,j,k,and u^+_RMS = √((1/SN_xN_y∑_s=1^S∑_i=1^N_x∑_j=1^N_yu^+_i,j,k^2) - u^+^2).where u^+_i,j,k is the dimensionless velocity at grid point (i,j,k). The quantities N_x and N_y are the number of grid points in the x and y directions. The quantities were not accumulated over all time-steps, but were instead accumulated every S timesteps, where S was chosen to be sufficiently small to ensure a steady average. In addition, the mean velocity relative to the friction velocity was also considered. This quantity is defined as u^+ = 1/2∫_-1^1 u^+dz. The mean streamwise velocity for the baseline case is shown in figure <ref> in linear and semi-logarithmic co-ordinates in parts (a) and (b) respectively, showing a composite of the LES results (with squares, omitting the first 3 cells) and the near-wall QDNS (shown with triangles). Overall a reasonable match to the reference DNS is observed despite the very low grid node count. The QDNS simulations correctly capture the viscous sublayer and buffer layer, while the LES captures the outer layer. Both the QDNS and LES undershoot the reference DNS by about 5% at the LES/QDNS interface and the LES gives noticeably too low a centreline velocity (by 3%). The mean velocity relative to the friction velocity is 23.3 which is ∼0.9% lower than the DNS and 2.9% lower than Dean's correlation <cit.>, which together provide a useful measure of the overall accuracy of this approach. With all the data available from the QDNS, it would in principle be possible to improve the near-wall sub-grid modelling in the LES to address the undershoot at the interface (for example the eddy viscosity can be computed from the QDNS and used in the LES), however in the present contribution we use the same (Smagorinsky) sub-grid model for all cases.An interesting feature emerges when one considers the root mean square (RMS) of streamwise velocity fluctuations from the QDNS simulations, shown on figure <ref>. To assemble this figure, as with the QDNS shown in figure <ref>, all 16 QDNS on one wall were averaged in horizontal planes and over time. The result is generally in good agreement with the DNS. There is an overshoot in the peak at z^+=12, which is likely due to under-resolution within the QDNS blocks; similar over-shoots have been observed in the RMS streamwise velocity for large eddy simulations of turbulent channel flow (at lower _τ values of 180, 395 and 640) where the near-wall zone is not adequately resolved by the grid <cit.>. The RMS levels agree well with DNS further away from the wall, showing that the current methodology has correctly captured the modulation of near-wall turbulence by outer-layer motions that is seen experimentally <cit.>. For comparison, a separate QDNS was run with only the mean mass flow and velocity gradients imposed, giving an unmodulated result (shown on figure <ref> with the chain dotted line) for comparison. It can be seen that the effect of modulation of near-wall turbulence by outer-layer structures is to increase the RMS levels by a factor of ∼2.5 at this Reynolds number. The effect of increasing RMS with Reynolds number would only be properly obtained in conventional LES using the wall-resolved approach, which would however be significantly more expensive than the current method. A wall-resolved LES grid to do the same calculation as shown here (allowing for a factor of four under-resolution in all directions compared to the reference DNS) would need 71 million grid points, compared to less than half a million employed here. The nested LES approach of <cit.> also gives the modulation effect, but not the multi-block model of <cit.>, which uses the same replicated near-wall block everywhere on the wall.The extremely coarse-grid LES shows significant errors in the structure of the turbulence as the wall is approached. Figure <ref> shows RMS values of all velocity components compared to DNS. Here, only the resolved part of the LES is shown, but nevertheless there is a significant overshoot relative to the DNS. In particular the streamwise velocity fluctuations are significantly higher and the wall-normal velocity fluctuations are significantly lower than the DNS. In both cases the effect of the wall extends to much higher values of z than it should, due no doubt to the severe under-resolution of turbulence near the wall, with only larger structures resolved on the LES grid. It should be noted that the sub-grid model used here is the classical Smagorinsky model and no effort has been made to optimise the model formulation in the near-wall region. Other formulations such as dynamic Smagorinsky or WALE would be expected to do better, but the grid is so coarse in these cases that good agreement is not to be expected. A more limited expectation is that the LES resolve sufficient features of the turbulence to provide a reasonable model of the outer-flow, with the shear stress at the wall provided by the QDNS and not so dependent on the subgrid modelling (since only the local flow derivatives are passed to the QDNS as boundary conditions).Any simulation-based model of turbulence is only useful if it provides a suitable degree of grid independency. In the current case the resolution required for the QDNS is reasonably well known, based on previous DNS. Figure <ref> (a) shows a negligible effect on the mean flow of increasing the near-wall QDNS from 24^3 to 32^3, which is still well below the levels required for a resolved DNS (64^3 would give a resolution of Δ x^+=15.6, Δ y^+=7.8 and a first grid point at z^+<1). One effect of the increased resolution is the reduced near-wall peak of the RMS streamwise velocity, shown on figure <ref>(b), with the correct trend to agree with the DNS in the limit of very fine resolution. Additionally there is a slight improvement (<4%) in the RMS from around z^+=80 onwards.The effect of the grid resolution of the LES in all directions is tested in figure <ref>(a), where the LES grid is changed from 24 × 24 × 42 to 96 × 96 × 56 and the stretching parameter a is decreased from 1.577 to 1.28 (in order for the LES to overlap the QDNS blocks by three cells as before). This increases the LES grid point count by a factor of 21 and the timestep is reduced by a factor of 4 due to Courant number restrictions, but relatively little change is seen in the mean flow. The main effect is for the centreline velocity prediction to change from a 3% undershoot to a <1% undershoot. Similarly, the disagreement at the interface between the LES and QDNS blocks is reduced from about 5% to 2%. While the agreement at the near-wall peak in the streamwise velocity RMS results, shown in figure <ref>(b), is better when a refined LES grid is used, the same cannot be said for the results for z^+ > ∼25 which deviate away from the DNS data. Both RMS velocity curves from our simulations followed a trend similar to that of the DNS results (namely the initial peak in the near-wall region followed by a relatively gradual decrease further away from the wall). These RMS curves were found to be sensitive to the method of averaging the bulk velocity and velocity derivatives from the LES to enforce the mass flow rate in the QDNSs. For each block, a number of LES grid points were used for the averaging. It was observed that too small an averaging window caused the RMS velocity curve to be significantly higher than the DNS results, which was likely caused by small grid-to-grid point oscillations (in turn caused by under-resolution of the turbulence) being picked up near the wall. On the other hand, too large an averaging window can introduce turbulence smoothing, reducing the turbulent kinetic energy levels in the QDNS and therefore causing the curve to be lower than that of the DNS. The latter may have had an effect here since the number of grid points used in each averaging window (N_x/4 × N_y/4) was obviously greater in the refined case (with the length and width of the averaging window remaining the same). Note that the choice of averaging size did not significantly alter the mean streamwise velocity results which were consistently better than the results from the coarser LES grid. The ultimate convergence of the LES back to the DNS would require much finer grids and large parallel simulations, which is beyond the scope of the current investigation. Nevertheless, the limited sensitivity to the grid at these very low resolutions is promising.Finally in this section, we consider the effect of the basic arrangement of the QDNS blocks. The baseline configuration has 4 × 4 blocks, as sketched in figure <ref>. This configuration seems to be capable of resolving near-wall flow features, as illustrated by the velocity contours that were shown on figure <ref>. Figure <ref> shows the effect of reducing the number of near-wall blocks to 2 × 2 and 1 × 1, which clearly under-samples the flow features. The mean flow on figure <ref>(a) shows that the principal effect of reducing the near-wall block count is to slightly diminish the accuracy of the near-wall turbulence. This is possibly due to aliasing effects when trying to sample the very high-frequency turbulent structures. Whilst this result is not catastrophic, it does lead to the conclusion that 4 × 4 blocks is probably a minimum number of blocks for a reasonable prediction of the mean flow for the current domain size. On the other hand, increasing the number of near-wall blocks to 6 × 6 yields an improved mean flow prediction particularly near the LES-QDNS interface. While the RMS curve for the 6 × 6 case displays the correct shape, the values continue to overshoot the DNS data near the wall. As already noted, these RMS values are sensitive to the averaging procedure used to enforce the mass flow rate in the QDNS blocks.§ EXTENSION TO HIGHER REYNOLDS NUMBERSince the method has been proposed here as a way of simulating high Reynolds number flows, it is of interest to test the approach at even high Reynolds numbers. In this section we consider a simulation at _τ=20 000, which is a factor of nearly 5 higher than that used in the previous section. If we keep the same near-wall QDNS configuration, with 4 × 4 blocks, each of 32^3 points on the same domains in wall units, we end up with sub-domains that are 0.05 long, 0.025 in the spanwise direction, with z^+=200 reached at z=0.01. Maintaining the same link between the LES and QDNS (i.e. one Δ x_LES matching to the entire QDNS subdomain) as in the previous section, and retaining approximately the same stretching property of the grid (i.e. maximum to minimum Δ z) we end up with an LES grid of 120 × 120 × 90. Courant number considerations again lead to a choice of 25 iterations of the QDNS per LES step, with Δ t_LES=0.00035. Even at the higher _τ most of the cost (>90%) resides in the QDNS simulations and most of the additional cost is due to the increased number of time steps required at the higher _τ, which (if it works) represents a linear scaling of the total simulation cost with _τ in the channel flow example here. Figure <ref> shows a plan view of the simulation at _τ=20 000, for comparison with figure <ref> which showed the equivalent figure at _τ=4200. Part (a) of figure <ref> shows the streamwise velocity field from the LES at z^+=322, with the QDNS block superimposed, although these are too small to be clearly visible. Figure <ref>(b) shows the flow in one of the QDNS blocks at z^+=13, showing qualitatively the same near-wall streak structure as was seen in the lower Reynolds number case. Compared with figure <ref>(a), figure <ref>(a) shows a much wider range of scales. The imprint of very large structures can be seen in figure <ref>(a) as streamwise-elongated zones of higher- or lower-than-average streamwise velocity. Superimposed on this are smaller-scale structures down to the grid scale. On the one hand this increase in the range of scales is a more accurate picture of a turbulent flow than the picture shown in figure <ref>(a), since a wider range of the turbulent energy cascade is captured. On the other hand, this picture also illustrates a possible weakness of the current approach, since the linear interpolation method used to feedback the shear stress from the QDNS to the LES will clearly not be accurate, apart from very close to the QDNS locations. Statistical results for the simulation at _τ=20 000 are shown on figure <ref>, comparing the results with the logarithmic law of the wall u^+=1/κlog z^+ +b with κ=0.39 and b=4.5 (where these values have been chosen to agree with the DNS data from <cit.>). The solution overshoots the log law by about 6% near the LES-QDNS interface. It seems unlikely that sub-grid models can be blamed for the overshoot, although this is something that could be tested. Compared to <cit.> the mean flow prediction is approximately 9% too low, although we should note that the Reynolds number in this simulation is well above the highest Reynolds number used by Dean to make his correlations. In general the RMS streamwise velocity fluctuations shown on figure <ref>(b) follow the expected trend, with the RMS increasing as _τ increases. The near-wall peak clearly increases with the fivefold increase in _τ, although the RMS at z^+=100 decreases by 20% before once again rising slightly above the _τ = 4200 line. Therefore, the method proposed here clearly has limitations dependent on _τ, which could be mitigated through the use of finer LES resolution or more QDNS blocks which have been shown to improve the results in the _τ=4 200 case.In summary, the effect of increasing Reynolds number is partly captured by the method presented in this section, which is based on a computational cost (including the number of time steps) that scales approximately proportional to _τ. However the results at _τ=20,000 deviate from the logarithmic law, suggesting that the quality of the results will decrease with further increases in _τ. To improve on this probably requires more computational resource, and in this respect it is interesting that the trend to over-predict the logarithmic law of the wall was also seen when the number of QDNS blocks was reduced to 2 × 2 and 1 × 1, as shown in figure <ref>. This suggests that one method to increase the accuracy of the simulations is to increase the number of QDNS blocks, for example to 32 × 32 at _τ=20 000. Although this parallelises trivially, it formally represents a scaling of the computational grid with _τ^2 for channel flow, albeit with a much lower constant of proportionality than wall-resolved LES. Another method of increasing the accuracy at higher _τ would be to increase the domain size of the QDNS, also resulting in a higher scaling exponent. These estimates may be reduced if the resolution of structures associated with the mixed scaling of <cit.> is the limiting factor. Otherwise, for very high _τ one may need to apply the method recursively, with successively smaller domains as the wall is approached.§ CONCLUSIONSA new approach to simulating near wall flows at high Reynolds number has been presented and tested. The method relies on LES for the whole domain, but with the skin friction supplied from a set of quasi-DNS of the near-wall region (out to a wall normal distance of z^+=200). These near-wall simulations use periodic boundary conditions and are not space-filling, but provide an estimate of the two components of skin friction, given the instantaneous near-wall velocity gradients. The method has an extremely small communication overhead between the LES and quasi-DNS and is thus suitable for scaling to large core counts. The accuracy of the method was demonstrated for a turbulent channel flow at _τ=4200, for which less than half a million points were used, compared to the reference DNS that used over 4 billion points. Besides the low cost, a particular feature of the new simulation approach is that it is able to predict the effect of modulation of small-scale near-wall features by large structures, residing either in the logarithmic or outer regions of the flow. This makes it possible, for example, to study the effects of wall-based flow control schemes in a high-Reynolds number external environment. The method is found to be robust to changes in grid resolution. An O(_τ) total cost extrapolation to _τ=20 000 demonstrated some limitations, suggesting that accurate simulations at higher _τ probably have a higher total cost scaling (including an increase in grid points and in the number of timesteps), however at much lower cost relative to wall-resolved LES. For the particular case considered here, that of turbulent channel flow, wall functions for LES based on the logarithmic law of the wall would be expected to work well. The advantage of the current approach is that the log law is not assumed and it would be expected that the effects of a range of non-equilibrium flow conditions could be captured, so long as the surface sampling is sufficient relative to the dominant large-scale structure in the flow. Overall the new method offers the potential for engineering calculations at high Reynolds number at a substantially lower computational cost compared to current LES techniques. A significant part of this work was first presented at the 8th International Conference on Computational Fluid Dynamics (ICCFD); see <cit.>. CTJ was supported by a European Commission Horizon 2020 project grant entitled “ExaFLOW: Enabling Exascale Fluid Dynamics Simulations” (grant reference 671571). RJ was supported partially by the UK Turbulence Consortium (EPSRC grant EP/L000261/1). The authors would like to acknowledge the support of iSolutions at the University of Southampton and the use of the in-house Iridis 4 compute cluster. The data files generated as part of this work will be available through the University of Southampton's institutional repository service.wileyj.bst | http://arxiv.org/abs/1704.08368v1 | {
"authors": [
"Neil D. Sandham",
"Roderick Johnstone",
"Christian T. Jacobs"
],
"categories": [
"physics.flu-dyn",
"cs.CE",
"physics.comp-ph"
],
"primary_category": "physics.flu-dyn",
"published": "20170426223113",
"title": "Surface-sampled simulations of turbulent flow at high Reynolds number"
} |
=115212 LastPage May 18, 2018May 08, 2019arrows,patterns, decorations.pathmorphingtheorem lemma corollary proposition claim definitionproofsketchproofideaproofof[1] proofsketchof[1]arrows, shapes, calcmnode=[ circle, fill=,draw=, minimum size=6pt,inner sep=0pt ]mnodeinvisible=[ minimum size=6pt,inner sep=0pt ]invisible=[ minimum size=0pt,inner sep=0pt ]invisiblel=[ minimum size=10pt,inner sep=0pt ]invisibleEdge=[ transparent ] nameNode=[ font=] namingNode=[ font=]mEdge=[ -latex',thick,shorten >=3pt,shorten <=3pt, draw=black!80, ]dDashedEdge=[ -latex',thick,shorten >=3pt,shorten <=3pt, draw=black!80, dashed ] dEdge=[ -latex',thick,shorten >=1pt,shorten <=1pt, draw=black!80, ] dhEdge=[ -latex',thick,shorten >=3pt,shorten <=3pt, draw=black!80, ]uEdge=[thick,shorten >=3pt,shorten <=3pt, draw=black!80, ]uhEdge=[thick,shorten >=3pt,shorten <=3pt, draw=black!80, ]cEdge=[ultra thick,shorten >=3pt,shorten <=3pt, draw=black!80, ] dotsEdge=[ very thick,loosely dotted,shorten >=7pt,shorten <=7pt ] class rectangle=[ draw=black, inner sep=0.2cm, rounded corners=5pt, thick ]mline=[ draw=black, inner sep=0.2cm, rounded corners=5pt, thick ]mainclass rectangle=[draw=blue, inner sep=0.2cm, rounded corners=5pt, very thick ] background substructure edges foregroundbackground,substructure,edges,main,foreground background rectangle=[ fill=black!10, draw=black!20, line width = 5pt,inner sep=0.4cm, outer sep=0.4cm, rounded corners=5pt ]A Strategy for Dynamic Programs: Start over and Muddle through]A Strategy for Dynamic Programs: Start over and Muddle through This article is the full version of<cit.>. The authors acknowledge the financial support by the DAAD-DST grant “Exploration of New Frontiers in Dynamic Complexity”. The first and the second authors were partially funded by a grant from Infosys foundation. The second author was partially supported by a TCS PhD fellowship. The last three authors acknowledge the financial support by DFG grant SCHW 678/6-2 on “Dynamic Expressiveness of Logics”.S. Datta]Samir Dattaa aChennai Mathematical Institute & UMI ReLaX, Chennai, India [email protected]. Mukherjee]Anish Mukherjeeb bChennai Mathematical Institute, Chennai, India [email protected] T. Schwentick]Thomas Schwentickc cTU Dortmund University, Dortmund, Germany {thomas.schwentick,nils.vortmeier,thomas.zeume}@tu-dortmund.deN. Vortmeier]Nils Vortmeierc T. Zeume]Thomas Zeumec In the setting of , dynamic programs update the stored result of a query whenever the underlying data changes.This update is expressed in terms of first-order logic.We introduce a strategy for constructing dynamic programs that utilises periodic computation of auxiliary data from scratch and the ability to maintain a query for a limited number of change steps. We show that if some program can maintain a query for log n change steps after an ^1-computable initialisation, it can be maintained by a first-order dynamic program as well, i.e., in .As an application, it is shown that decision and optimisation problems defined by monadic second-order ()formulas are in , if only change sequences that produce graphs of bounded treewidth are allowed. To establish this result,a Feferman-Vaught-type composition theorem for is established that might be useful in its own right. [ [ =====§ INTRODUCTION Each time a database is changed, any previously computed and stored result of a fixed query might become outdated. However, when the change is small, it is plausible that the new query result is highly related to the old result.In that case it might be more efficient to use previously computed information for obtaining the new answer to the query instead of recomputing the query result from scratch. A theoretical framework for studying when the result of a query over relational databases can be updated in a declarative fashion was formalised by Patnaik and Immerman <cit.>, and Dong, Su, and Topor <cit.>. In their formalisation, a dynamic program consists of a set of logical formulas that update a query result after the insertion or deletion of a tuple. The formulas may use additional auxiliary relations that, of course, need to be updated as well. The queries maintainable in this way via first-order formulas constitute the dynamic complexity class .Recent work has confirmed that is a quite powerful class, since it captures, e.g., the reachability query for directed graphs <cit.>, and even allows for more complex change operations than single-tuple changes <cit.>. In this article we introduce a general strategy for dynamic programs that further underscores the expressive power of . All prior results for yield dynamic programs that are able to maintain a query for arbitrary long sequences of changes. Even if this is not (known to be) possible for a given query, in a practical scenario it might still be favourable to maintain the query result dynamically for a bounded number of changes, then to apply a more complex algorithm that recomputes certain auxiliary information from scratch, such that afterwards the query can again be maintained for some time, and so on.Here we formalise this approach. Letbe a complexity class and f → a function.A queryis called (,f)-maintainable, if there is a dynamic program (with first-order definable updates) that, starting from some input structureand auxiliary relations computed infrom , can answerfor f(||) many steps, where || denotes the size of the universe of . We feel that this notion might be interesting in its own right. However, in this article we concentrate on the case whereis (uniform) ^1 and f(n)=log n. The class ^1 contains all queries that can be computed by a (uniform) circuit of depth (log n) that uses polynomially many ∧-, ∨-, and -gates, where ∧- and ∨-gates may have unbounded fan-in. We show that (^1,log n)-maintainable queries are actually in , and thus can be maintained for arbitrary long change sequences. We apply this insight to show that all queries and optimisation problems definable in monadic second-order logic () are in for (classes of) structures of bounded treewidth, by proving that they are (^1,log n)-maintainable. The same can be said about guarded second-order logic (), simply because it is expressively equivalent to on such classes <cit.>. This implies that decision problems like 3-Colourability or HamiltonCycle as well as optimisation problems like VertexCover and DominatingSet are in , for such classes of structures. This result is therefore a dynamic version of Courcelle's Theorem which states that all problems definable in (certain extensions of) can statically be solved in linear time for graphs of bounded treewidth <cit.>.The proof that -definable queries are (^1,log n)-maintainable on structures of bounded treewidth makes use of a Feferman–Vaught-type composition theorem for which might be useful in its own right. The result that (^1,log n)-maintainable queries are in comes with a technical restriction: in a nutshell, it holds for queries that are invariant under insertion of (many) isolated elements. We call such queries almost domain independent and refer to Section <ref> for a precise definition.We emphasise that the main technical challenge in maintaining -queries on graphs of bounded treewidth is that tree decompositions might change drastically after an edge insertion, and can therefore not be maintained incrementally in any obvious way. In particular, the result does not simply follow from the -maintainability of regular tree languages shown in <cit.>. We circumvent this problem by periodically recomputing a new tree decomposition (this can be done in logarithmic space <cit.> and thus in ^1) and by showing that -queries can be maintained for (log n) many change operations, even if they make the tree decomposition invalid. §.§ ContributionsWe briefly summarise the contributions described above. In this article, we introduce the notion of (, f)-maintainability and show that, amongst others, (almost domain independent) (^1,log n)-maintainable queries are in . We show that -definable decision problems and optimisation problems are (^1,log n)-maintainable and therefore in , for structures of bounded treewidth. These proofs make use of a Feferman–Vaught-type composition theorem for logic.§.§ Related work The simulation-based technique for proving that (^1,log n)-maintainable queries are in is inspired by proof techniques from <cit.> and<cit.>. As mentioned above, in <cit.> it has been shown that tree languages, i.e. MSO on trees, can be maintained in . Independently, the maintenance of definable queries on graphs of bounded treewidth is also studied in <cit.>, though in the restricted setting where the tree decomposition stays the same for all changes. A static but parallel version of Courcelle's Theorem is given in <cit.>: every -definable problem for graphs of bounded treewidth can be solved with logarithmic space. §.§ OrganisationBasic terminology is recalled in Section <ref>, followed by a short introduction into dynamic complexity in Section <ref>. In Section <ref> we introduce the notion of(, f)-maintainability and show that (^1,log n)-maintainable queries are in . A glimpse on the proof techniques for proving that queries are in for graphs of bounded treewidth is given in Section <ref> via the example 3-Colourability. The proof of the general results is presented in Section <ref>. An extension to optimisation problems can be found in Section <ref>. This article is the full version of <cit.>. § PRELIMINARIES We now introduce some notation and notions regarding logics, graph theory and complexity theory. We assume familiarity with first-order logic and other notions from finite model theory <cit.>.§.§ Relational structuresIn this article we consider finite relational structures over relational signatures Σ = {R_1, …, R_ℓ,c_1,…,c_m}, where each R_i is a relation symbol with a corresponding arity (R_i), and each c_j is a constant symbol. A Σ-structureconsists of a finite domain A, also called the universe of , a relation R_i^⊆ A^(R_i), and a constant c_j^∈ A, for each i ∈{1, …, ℓ}, j ∈{1, …, m}.The active domain ()of a structurecontains all elements used in some tuple or as some constant of .For a set B ⊆ A that contains all constants and a relation R, the restriction RB of R to B is the relation R ∩ B^(R). The structure B induced by B is the structure obtained fromby restricting the domain and all relations to B. Sometimes, especially in Section <ref>, we consider relational structures as relational databases. This terminology is common in the context of dynamic complexity due to its original motivation from relational databases.Also for this reason, dynamic complexity classesdefined later will be defined as classes of queries of arbitrary arity, and not as a class of decision problems. However, we will mostly consider queries over structures with a single binary relation symbol E, that is, queries on graphs.We will often use structureswith a linear order ≤ on the universe A, and compatible ternary relations encoding arithmetical operations+ and × or, alternatively, a binary relation encoding the relation = {(i,j) | the j-th bit in the binary representation of i is 1}. The linear order in particular allows us to identify A with the first |A| natural numbers. We write (+,×) or () to emphasise that we allowfirst-order formulas to use such additional relations.[The question of <-invariance (c.f. <cit.>) will not be relevant in the context of this article as a specific relation ≤ will be available in the structure.] We also use that (+,×)=() <cit.>. §.§ Tree decompositions and treewidthA tree decomposition (T,B) of G consists of a (rooted, directed) tree T = (I,F,r), with (tree) nodes I, (tree) edges F, a distinguished root node r ∈ I, and a function BI → 2^V such that(1) the set {i ∈ I | v ∈ B(i)} is non-empty for each node v ∈ V, (2) there is an i ∈ I with {u,v}⊆ B(i) for each edge (u, v) ∈ E, and (3) the subgraph T[{i ∈ I | v ∈ B(i)}] is connected for each node v ∈ V. We refer to the number of children of a node i of T as its degree, and to the set B(i) as its bag. We denote the parent node of i by p(i). The width of a tree decomposition is defined as the maximal size of a bag minus 1. The treewidth of a graph G is the minimal width among all tree decompositions of G.A tree decomposition is d-nice, for some d ∈, if (1) T has depth at most d log n, (2) the degree of the nodes is at most 2, and (3) all bags are distinct.Often we do not make the constant d explicit and just speak of nice tree decompositions.Later we will use that tree decompositions can be transformed into nice tree decompositions with slightly increased width. This is formalized in the following lemma, whose proof is an adaption of <cit.>.For every k ∈ there is a constant d ∈ such that for every graph of treewidth k, a d-nice tree decomposition of width 4k+5 can be computed in logarithmic space. Let G be a graph of treewidth k. By <cit.> a tree decomposition (T,B) of width 4k+3 can be computed in logarithmic space, such that each non-leaf node has degree 2 and the depth is at most d log n, for a constant d that only depends on k.To obtain a tree decomposition with distinct bags, we compose this algorithm with three further algorithms, each reading a tree decomposition (T,B) and transforming it into a tree decomposition (T',B') with a particular property. Since each of the four algorithms requires only logarithmic space, the same holds for their composition. The first transformation algorithm produces a tree decomposition, in which for each leaf bag i it holds B(i)⊈B(p(i)). In particular, after this transformation, each bag of a leaf node i contains some graph node u(i) that does not appear in any other bag. This transformation inspects each node i separately in a bottom-up fashion, and removes it if (1) B(i)⊆ B(p(i)) and (2) every bag below i is a subset of B(i). Clearly, logarithmic space suffices for this.The second transformation (inductively) removes an inner node i of degree 1 with a child i' and inserts an edge between B(p(i)) and B(i') whenever B(i)⊆ B(p(i)) or B(i)⊆ B(i') holds. For this transformation, only one linear chain of nodes in T has to be considered at any time and therefore logarithmic space suffices again. Clearly, the connectivity property is not affected by these deletions.The third transformation adds to every bag of an inner node i the nodes u(i_1) and u(i_2), guaranteed to exist by the first transformation, of theleftmost and rightmost leaf nodes i_1 and i_2 of the subtree rooted at i, respectively. Here, we assume the children of every node to be ordered by the representation of T as input to the algorithm.After this transformation, each node of degree 2 has a different bag than its two children thanks to the addition ofu(i_1) and u(i_2). Each node of degree 1 has a different bag than its child, since this was already the case before (and to both of them the same two nodes might have been added). Altogether, all bags are pairwise distinct and the bag sizes have increased by at most 2.We emphasise that, whenever a leftmost graph nodeu(i_1) is added to B(i), it is also added to all bags of nodes on the path from i to i_1 and therefore the connectivity property is not corrupted. It is easy to see that the third transformation can also be carried out in logarithmic space. The three presented algorithms never increase the depth of a tree decomposition, so the final result is a d-nice tree decomposition for G of width 4k+5. In this paper we only consider nice tree decompositions, and due to property (3) of these decompositions we can identify bags with nodes from I. For two nodes i,i' of I, we write i≼ i' if i' is in the subtree of T rooted at i andi ≺ i' if, in addition, i'≠i. A triangle δ of T is a triple (i_0,i_1,i_2) of nodes from I such that i_0≼ i_1,i_0≼ i_2, and (1) i_1=i_2 or (2) neither i_1≼ i_2 nor i_2≼ i_1. In case of (2) we call the triangle proper, in case of (1) unary, unless i_0=i_1=i_2 in which we call it open(see Figure <ref> for an illustration). The subtree T(δ) induced by a triangle consists of all nodes jof T for which the following holds: (i)i_0 ≼ j, (ii) if i_0≺ i_1 theni_1⊀j, and (iii) if i_0≺ i_2 theni_2⊀j.That is, for a proper or unary triangle, T(δ) contains all nodes of the subtree rooted at i_0 which are not below i_1 or i_2. For an open triangle δ=(i_0,i_0,i_0),T(δ) is just the subtree rooted at i_0. Each triangle δ induces a subgraph G(δ) of Gas follows: V(δ) is the union of allbags of T(δ). By B(δ) we denote the set B(i_0)∪ B(i_1)∪ B(i_2) of interface nodes of V(δ). All other nodes in V(δ) are called inner nodes. The edge set of G(δ) consists of all edges of G that involve at least one inner node of V(δ). §.§ MSO-logic and MSO-types is the extension of first-order logic that allows existential and universal quantification over set variables X,X_1,…. The (quantifier) depth of an formula is the maximum nesting depth of (second-order and first-order) quantifiers in the syntax tree of the formula. For a signature Σ and a natural number d≥ 0, the depth-d type of a Σ-structure is defined as the set of all sentences φ over Σ of quantifier depth at most d, for which φ holds. We also define the notion of types for structures with additional constants and formulas with free variables. Let be a Σ-structure and v=(v_1,…,v_m) a tuple of elements from . We write (, v) for the structure over Σ∪{c_1,…,c_m} which interprets c_i as v_i, for every i∈{1,…,m}. For asetof first-order and second-order variables and an assignment α for the variables of , the depth-d type of(, v, α) is the set of formulas with free variables fromof depth d that hold in(, v, α).For every depth-d type τ, there is a depth-d formula α_τ that is true in exactly the structures and for those assignments of type τ. The logic guarded second-order logic () extends by guarded second-order quantification. Thus, it syntactically allows to quantify over non-unary relation variables. However, this quantification is semantically restricted: a tuple t = (a_1, …, a_m) can only occur in a quantified relation, if all elements from {a_1, …, a_m } occur together in some tuple of the structure, in which the formula is evaluated.For more background on logic and types, readers might consult, e.g., <cit.>. §.§ Complexity classes and descriptive characterisationsOur main result refers to the complexity class (uniform) ^1. It contains all queries that can be computed by (families of uniform) circuits of depth (log n), consisting of polynomially many “and”, “or” and “not” gates, where “and” and “or” gates may have unbounded fan-in. It contains the classes and , and it can be characterised as the class log nof problems that can be expressed by applying a first-order formula (log n) times <cit.>. Here, n denotes the size of the universe and the formulas can use built-in relations + and ×.More generally, this characterisation is also valid for the analogously defined classes [f(n)] and f(n), where the depth of the circuits and the number of applications of the first-order formula is f(n), respectively, for some function f →. Technically, the function f needs to be first-order constructible, that is, there has to be aformula ψ_f( x) such that ψ_f( a) if and only if a is a base-n representation of f(n), for any ordered structurewith domain {0, …, n-1}.Our proofs often assume that log n is a natural number, but they can be easily adapted to the general case. § DYNAMIC COMPLEXITYWe briefly repeat the essentials of dynamic complexity, closely following <cit.>. The goal of a dynamic program is to answer a given query on an input database subjected to changes that insert or delete single tuples. The program may use an auxiliary data structure represented by an auxiliary database over the same domain. Initially, both input and auxiliary database are empty; and the domain is fixed during each run of the program.A dynamic program has a set of update rules that specify how auxiliary relations are updated after a change of the input database. An update rule for updating an auxiliary relation T is basically a formula φ. As an example, if φ( x,y) is the update rule for auxiliary relation T under insertions into input relation R, then the new version of T after insertion of a tuple a to R is T { b | (, ) φ( a,b)} whereandare the current input and auxiliary databases. For a state = (, ) of the dynamic programwith input databaseand auxiliary databasewe denote the state of the program after applying a sequence α of changes by _α(). The dynamic programmaintains a k-ary queryif, for each non-empty sequence α of changes and each empty input structure _∅, a designated auxiliary relation Q in _α(_∅) and (α(_∅)) coincide. Here, _∅=(_∅, _∅), where _∅ denotes the empty auxiliary structure over the domain of _∅, and α(_∅) is the input database after applying α. In this article, we are particularly interested in maintaining queries for structures of bounded treewidth. There are several ways toadjust the dynamic setting to restricted classesof structures.Sometimes it is possible that a dynamic program itself detects that a change operation would yield a structure outside the class . However, here we simply disallow change sequences that construct structures outside . That is, in the above definition, only change sequencesα are considered, for which each prefix transforms aninitially empty structure into a structure from .We say that a program maintainsfor a classof structures, if Q contains its result after each change sequence α such that the application of each prefix of α to _∅ yields a structure from . The class of queries that can be maintained by a dynamic program with first-order update formulas is called . We say that a queryis in for a classof structures, if there is such a dynamic program thatmaintainsfor . Programs for queries in have three particular auxiliary relations ≤, +, × that are initialised as a linear order and the corresponding addition and multiplication relations. For a wide class of queries, membership in implies membership in <cit.>.Queries of this class have the property that the query result does not change considerably when elements are added to the domain but not to any relation. Informally, a queryis called almost domain independent if there is a constant c such that if a structure already has at least c “non-active” elements, adding more “non-active” elements does not change the query result with respect to the original elements. More formally, a queryis almost domain independent if there is a c ∈ such that, for every structureand every set B ⊆ A ∖() with |B| ≥ c it holds()(() ∪ B) = ((() ∪ B)). * The binary reachability query , that maps a directed input graph G=(V,E) to its transitive closure relation, is almost domain independent with c = 0: adding any set B ⊆ V ∖(G) of isolated nodes to a graph does not create or destroy paths in the remaining graph. Note that, for each node v ∈ V, the tuple (v,v) is part of the query result (G), so (G) ≠(G(G)) in general and thereforeis not domain independent in the sense of <cit.>.* The definable Boolean query _||=2, which is true if and only if exactly two elements are not in the active domain, is almost domain independent with c=3.* The Boolean query _even, which is true for domains of even size and false otherwise, is not almost domain independent. Furthermore, all properties definable in monadic second-order logic are almost domain dependent. All -definable queries are almost domain independent. This can be easily shown by an Ehrenfeucht game. Let φ be an -formula of quantifier depth d with e free (node) variables and let c=2^dd+e. Consider two graphs _1 and _2 that result from adding c_1≥ c and c_2≥ c isolated nodes to some graph , respectively. Since φ might have free variables, the game is played on (_1, a_1) and (_2, a_2), where a_1 and a_2 are tuples of elements of length e. Within d moves the spoiler can use at most d set moves and can therefore induce at most 2^d different “colours” on _1 and _2. However, the duplicator can easily guarantee that, for each such “colour”, the number of isolated nodes (which do not occur in the initial tuples on both structures) of that colour is the same in _1 and _2 or it is larger than d, in both of them. Since the spoiler can have at most d node moves, he can not make use of this difference in the two structures.The following proposition adapts Proposition 7 from <cit.>.Letbe an almost domain independent query. Ifthen also .The same statement is proved for weakly domain independent queries in <cit.>. A queryis weakly domain independent, if ()() = (()) for all structures , that is, if it is almost domain independent with c=0. The proof of <cit.> can easily be adapted for this more general statement, so we omit a full proof here. However, for readers who are familiar with the proof of <cit.>, we sketch the necessary changes. We assume familiarity with the proof of <cit.> and only repeat its main outline. That proof explains, for a given -program , how to construct a -program ' that is equivalent to .This program ' relies on the observation that a linear order, addition and multiplication can be maintained on the activated domain <cit.>, that is, on all elements that were part of the active domain at some time during the dynamic process. The linear order is determined by the order in which the elements are activated.The program ' can be regarded as the parallel composition of multiple copies, called threads, of the same dynamic program. Each thread simulatesfor a certain period of time: thread i starts when f(i-1) elements are activated and does some initialisation, and it is in charge of answering the query whenever more than f(i) but at most f(i+1) elements are activated, for some function f. Thanks to weak domain independence, thread i only needs to simulateon a domain with f(i+1) elements, for which it can define the arithmetic relations ≤, + and × based on the available arithmetic relations on the f(i-1) activated elements.For almost domain independent queries we extend this technique slightly.Let c be the constant from almost domain independence.The query result for a structure with domain size n coincides with the result of a simulation on a domain of size n', if n = n' or if there are at least c non-activated elements in both domains. To ensure that the simulation can in principle represent these c elements, phase i is generally in charge as long as the number of activated elements is at least f(i)-c+1 but at most f(i+1)-c, as in phase i one can simulateon a domain of size f(i+1).However, if in the original structure the number of non-activated elements becomes smaller than c, the simulation has to switch to a domain of the same size as the original domain. Therefore in phase i there is not only one simulation with domain size f(i+1), but one simulation (denoted by the pair (i,ℓ)) for each domain size ℓ∈{f(i)-c+1, …, f(i+1)}.During phase i the simulation denoted by (i,f(i+1)) is in charge unless there are fewer than c non-activated elements in the original domain, which can be easily detected by a first-order formula. As soon as that happens, the simulation denoted by (i,n) takes over, where n is the size of the domain. § ALGORITHMIC TECHNIQUEThe definition ofrequires that for the problem at hand each change can be handled by a first-order definable update operation.There are alternative definitions of , where the initial structure is non-empty and the initial auxiliary relations can be computed within some complexity class <cit.>. However, in a practical scenario of dynamic query answering it is conceivable that the quality of the auxiliary relations decreases over time and that they are therefore recomputed from scratch at times. We formalise this notion by a relaxed definition of maintainability in which the initial structure is non-empty, the dynamic program is allowed to apply some preprocessing, and query answers need only be given for a certain number of change steps. A queryis called (,f)-maintainable, for some complexity class[Strictly speakingshould be a complexity class of functions. In this paper, the implied class of functions will always be clear from the stated class of decision problems.]and some function , if there is a dynamic programand a -algorithmsuch that for each input databaseover a domain of size n, each linear order ≤ on the domain, and each change sequence α of length |α| ≤ f(n), the relation Q in _α() and (α()) coincide where = (, (,≤)).Although we feel that (,f)-maintainability deserves further investigation, in this paper we exclusively use it as a tool to prove that queries are actually maintainable in . To this end, we show next that every (^1,log n)-maintainable query is actually inand prove later that the queries in which we are interested are (^1,log n)-maintainable. Every(^1,log n)-maintainable, almost domain independent query is in . We do not prove this theorem directly, but instead give a more general result, strengthening the correspondence between depth of the initialising circuit families and number of change steps the query has to be maintained. Let f: → be a first-order constructible function with f ∈(n). Every ([f(n)],f(n))-maintainable, almost domain independent query is in . Let f: → be first-order constructible with f ∈(n) and assume that an [f(n)] algorithmand a dynamic programwitness that an almost domain independent queryis ([f(n)],f(n))-maintainable. Thanks to Proposition <ref> it suffices to construct a dynamic program ' that witnesses ∈. We restrict ourselves to graphs, for simplicity. The overall idea is to use a simulation technique similar to the ones used in Proposition <ref> and in <cit.>. We first present the computations performed by ' intuitively and later explain how they can be expressed in first-order logic.We consider each application of one change as a time step and refer to the graph after time step t as G_t=(V,E_t). After each time step t, the program ' starts a thread that is in charge of answering the query at time point t + f(n). Each thread works in two phases, each lasting f(n)/2 time steps. Roughly speaking, the first phase is in charge of simulatingand in the second phaseis used to apply all changes that occur from time step t+1 to time step t + f(n).Using f(n) many threads, ' is able to answer the query from time point f(n) onwards.We now give more details on the two phases and describe afterwards how to deal with time points earlier than f(n). For the first phase, we make use of the equality[f(n)]=f(n), see <cit.>. Let ψ be an inductive formula that is applied d f(n) times, for some d, to get the auxiliary relations (G,≤) for a given graph G and the given order ≤. The program ' applies ψ to G_t, 2d times during each time step, and thus the result ofon (G_t, ≤) is obtained after f(n)/2 steps. The change operations that occur during these steps are not applied to G_t directly but rather stored in some additional relation. If some edge e is changed multiple times, the stored change for e is adjusted accordingly.During the second phase the f(n)/2 stored change operations and the f(n)/2 change operations that happen during the next f(n)/2 steps are applied to the state after phase 1. To this end, it suffices for ' toapply two changes during each time step by simulating two update steps of . Observe that ' processes the changes in a different order than they actually occur. However, both change sequences result in the same graph. Sincecan maintainfor f(n) changes, the program ' can give the correct query answer forabout G_t+f(n) at the end of phase 2, that is, at time point t+f(n).The following auxiliary relations are used by thread i:* a binary relation Ê_i that contains the edges currently considered by the thread, * binary relations Δ^+_i and Δ^-_i that cache edges inserted and deleted during the first phase, respectively, * a relation R̂_i for each auxiliary relation R ofwith the same arity, * and a relation C_i that is used as a counter: it contains exactly one tuple which is interpreted as the counter value, according to its position in the lexicographic order induced by ≤. When thread i starts its first phase at time point t, it sets Ê_i to E_t and the counter C_i to 0; its other auxiliary relations are empty in the beginning.Whenever an edge (u,v) is inserted (or deleted), Ê_i is not changed, (u,v) is inserted into Δ^+_i (or Δ^-_i)[If an edge (u,v) with (u,v) ∈Δ^-_i is inserted, it is instead deleted from Δ^-_i, and accordingly for deletions of edges (u,v) ∈Δ^+_i.], and the counter C_i is incremented by one. The relations R̂_i are replaced by the result of applying their defining first-order formulas 2d times, as explained above.When the counter value is at least f(n)/2,the thread enters its second phase and proceeds as follows.When an edge (u,v) is inserted (or deleted), it applies this change and the change implied by the lexicographically smallest tuple in Δ^+_i and Δ^-_i, if these relations are not empty: it simulatesfor these changes using the edge set Ê_i and auxiliary relations R̂_i, replaces the auxiliary relations accordingly and adjusts Ê_i, Δ^+_i and Δ^-_i. Again, the counter C_i is incremented. If the counter value is f(n), the thread's query result is used as the query result of ', and the thread stops.All steps are easily seen to be first-order expressible.So far we have seen how ' can give the query answer from time step f(n) onwards. For time steps earlier than f(n) the approach needs to be slightly adapted as the program does not have enough time to simulate . The idea is that for time steps t < f(n) the active domain is small and, exploiting the almost domain independence of , it suffices to compute the query result with respect to this small domain extended by c isolated elements, where c is the constant from almost domain independence. The result on this restricted domain can afterwards be used to define the result for the whole domain. Towards making this idea more precise, let n_0, b be such that bn ≥ f(n) for all n ≥ n_0. We focus on explaining how ' handles structures with n ≥ n_0, as small graphs with less than n_0 nodes can be dealt with separately.The program ' starts a new thread at time t/2 for the graph G_t/2 with at most t/2 edges. Such a thread is responsible for providing the query result after t time steps, and works in two phases that are similar to the phases described above. It computes relative to a domain D_t of size min{2t+c,n}, where c is the constant from (almost) domain independence.The size of D_t is large enough to account for possible new nodes used in edge insertions in the following t/2 change steps. The domain D_t is chosen as the first |D_t| elements of the full domain (with respect to ≤). The program ' maintains a bijection π between the active domain D_G of the current graph G and the first |D_G| elements of the domain to allow a translation between D_G and D_t. The first phase of the thread for t starts at time point t/2+1 and applies ψ for 8bcd times during each of the next time steps. This simulation ofis finished after at most (2t+c)b/8bc≤t/4 time steps, and therefore the auxiliary relations are properly initialised at time point 3t/4.In the second phase, starting at time step 3t/4+1 and ending at time step t, the changes that occurred in the first phase are applied, two at a time. The thread is then ready to answerat time point t. Since at time t at most 2t elements are used by edges, the almost domain independence ofguarantees that the result computed by the thread relative to D_t coincides with the D_t-restriction of the query result for π(G_t). The query result for G_t is obtained by translating the obtained result according to π^-1, and extending it to the full domain. More precisely, a tuple t is included in the query result, if it can be generated from a tuple t' of the restricted query result by replacing elements from π^-1(D_t) ∖(G_t) by elements from V ∖(G_t) (under consideration of equality constraints among these elements). Again, all steps are easily seen to be first-order definable using the auxiliary relations from above.The above presentation assumes a separate thread for each time point and each thread uses its own relations. These threads can be combined into one dynamic program as follows. Since at each time point at most f(n) threads are active, we can number them in a round robin fashion with numbers 1,…,f(n) that we can encode by tuples of constant arity. The arity of all auxiliary relations is incremented accordingly and the additional dimensions are used to indicate the number of the thread to which a tuple belongs. § WARM-UP: 3-COLOURABILITYIn this section, we show that the 3-colourability problem 3Col for graphs of bounded treewidth can be maintained in . Given an undirected graph, 3Col asks whether its vertices can be coloured with three colours such that adjacent vertices have different colours. For every k, 3Col is in forgraphs with treewidth at most k. The remainder of this section is dedicated to aproof for this theorem. Thanks to Theorem <ref> and the fact that 3Col is almost domain independent, it suffices to show that 3Col is (^1,log n)-maintainable for graphs with treewidth at most k. In a nutshell, our approach can be summarised as follows. The ^1 initialisation computes a nice tree decomposition (T,B) of width at most 4k+5 and maximum bag size ℓ 4k+6, as well as information about the 3-colourability of induced subgraphs of G. More precisely, it computes, for each triangle δ of T and each 3-colouring C of the nodes of B(δ), whether there exists a colouring C'of the inner vertices of G(δ), such that all edges involving at least one inner vertex are consistent with C∪ C'.During the following log n change operations, the dynamic program does not need to do much.It only maintains a set S of special bags: for each affected graph node v that participates in any changed (i.e. deleted or inserted) edge, S contains one bag in which v occurs. Also, if two bags are special, their least common ancestor is considered special and is included in S.It will be guaranteed that there are at most 4 log n special bags. With the auxiliary information, a first-order formula φ can test whether G is 3-colourable as follows. By existentially quantifying 8ℓ variables, the formula can choose two bits of information for each of the at most 4ℓlog n nodes in special bags. For each such node, these two bits are interpreted as encoding of one of three colours and the formula φchecks that this colouring of the special bags can be extended to a colouring of G. This can be done with the help of the auxiliary relations computed during the initialisation which provide all necessary information about colourability of subgraphs induced by triangles consisting of special bags. Before we give a detailed proof, we need some more notation. Let G = (V,E) be a graph and (T,B) with T = (I,F,r)a nice tree decomposition with bags of size at most ℓ. A colouring of a set U of vertices is just a mapping from U to {1,2,3}. An edge (u, v) is properly coloured ifu and v are mapped to different colours. For a triangle δ, we say that a colouring C of B(δ) isconsistent, if there exists a colouring C' of the inner vertices of G(δ) such that all edges of G(δ) are properly coloured by C∪ C'. Recall that G(δ) only contains edges that involve at least one inner vertex. We say that a tuple v(i)=(v_1,…,v_ℓ) represents a tree node i∈ I (or, the bag B(i)) if B(i)={v_1,…,v_ℓ}. A tuple v(δ)=( v(i_0),v(i_1),v(i_2)) represents the triangle δ=(i_0,i_1,i_2). If v(δ) = (v_1, …, v_3ℓ) represents the triangle δ and c is a tuple from{1,2,3}^3ℓ such that c_j = c_j' whenever v_j = v_j' forj, j'∈{1,…,3ℓ}, we write C_ c, v for the colouringof B(δ) defined by C_ c, v (v_j)=c_j, for every j∈{1,…,3ℓ}.Theorem <ref> Let G = (V,E) be a graph of treewidth at most k. The ^1 initialisation first computes a d-nice tree decomposition (T,B) with bags of size at most ℓ= 4k+6, for the constant d guaranteed to exist by Lemma <ref>, and the predecessor relation ≼ of T. Also, it initialises the relations ≤ and .Next, it computes the following auxiliary relations in a bottom-up fashion with respect to T = (I, F, r).For each tuple c∈{1,2,3}^3ℓ the auxiliary relation R_ c contains all tuples v(δ) from V^3ℓ that represent some triangle δ such that C_ c, v is a consistent colouring of B(δ). The auxiliary relations are computed inductively and bottom-up, that is, the auxiliary information for a tuple representing a triangle (i_0, i_1, i_2) is computed by using the information for the triangles rooted at the two children of i_0. It will be easy to see that each inductive step can be defined by a first-order formula and, since T has depth d log n, the induction reaches a fixpoint after d log n iterations. Therefore the initialisation is in log n = [log n]. We recall that triangles can be open, unary or proper, depending on whether they are induced by a single bag, by two, or by three bags. For the base case, a tuple v representing an open triangle corresponding to a leaf of T is in R_ c if and only if, for each j∈{1,…ℓ}, c_j=c_ℓ+j=c_2ℓ+j, since there are no inner vertices to worry about. The inductive cases are straightforward. We only describein detail the case of a proper triangle δ=(i_0,i_1,i_2) where i_1 and i_2 are in different subtrees of i_0; the other cases are similar. Let i'_1 and i'_2 be the two children of i_0 such that i'_1≼ i_1 and i'_2≼ i_2. Figure <ref> illustrates this situation. By the induction hypothesis, the auxiliary information for all triangles rooted at i'_1 and i'_2 has already been computed.A tuple v=v(δ) is in some R_ c, if there are tuples d,e ∈{1,2,3}^3ℓ such that u =v((i'_1,i_1,i_1)) ∈ R_ d, w =v((i'_2,i_2,i_2)) ∈ R_ e and it holds that * C_ c, v and C_ d, u coincide on B(i_0)∩ B(i'_1) and on B(i_1),* C_ c, v and C_ e, w coincide on B(i_0)∩ B(i'_2) and on B(i_2), and* all edges over B(i_0)∪ B(i'_1)∪ B(i'_2) ∪ B(i_1) ∪ B(i_2) of which at least one node is not in B(i_0) ∪ B(i_1) ∪ B(i_2) are properly coloured byC_ c, v∪ C_ d, u∪ C_ e, w.We next describe how a dynamic program can maintain 3-colourability for log n change steps starting from the above initial auxiliary relations with the help of an additional ℓ-ary relation S and another binary relation N. Whenever an edge (u,v) is inserted into or deleted from E, we consider both u and v as affected. With each affected graph node v we associate a tree node i(v)∈ I such that v∈ B(i(v)). Tree nodes of the form i(v) for affected nodes v are called special. Furthermore, if node i is the least common ancestor of two special nodes i_1,i_2 it becomes special, as well. The dynamic program keeps track of all special nodes using the relation S which contains all tuples v that represent some special node. Furthermore, using the relation N it maintains a bijection between the first ℓ|S| nodes of V with respect to the linear order ≤ and the graph nodes in S. We call a triangle δ=(i_0,i_1,i_2) of T clean ifthere are no special nodes in T(δ) apart from i_0, i_1, i_2.It only remains to describe how a first-order formula can check 3-colourability of G given the relation S and the relations R_ c.We note first that within log n steps at most 2log n graph nodes can be affected resulting in at most 4log n special tree nodes altogether (since each new special node can contribute at most one new least common ancestor of special nodes). That is, the set Z of graph nodes occurring in some tuple of S contains at most 4ℓlog n nodes. A colouring of Z can be represented by 8ℓlog n bits and can thus be guessed by a first-order formula byquantifying over 8ℓ first-order variables x_1,…,x_4ℓ,y_1,…,y_4ℓ. More precisely, the j-th bits of x_r and y_r together represent the colour of the special node at position (r-1)log n+j with respect to the linear order represented by N. The first-order formula can easily check that the colouring C of S representedbyx_1,…,x_4ℓ,y_1,…,y_4ℓ is consistent for edges between special nodes and that for each clean triangle of T induced by special nodes it can be extended to a consistent colouring of the inner nodes. The latter information is available in the relations R_ c.§ MSO QUERIES In this section we prove a dynamic version of Courcelle's Theorem: all properties can be maintained in for graphs with bounded treewidth. More precisely, for a given sentence φ we consider the model checking problem MC_φ that asks whether a given graph G satisfies φ, that is, whether G φ holds. For every sentence φ and every k, MC_φ is in for graphs with treewidth at most k. Since, for every k,guarded second-order logic () has the same expressive power ason graphs with treewidth at most k <cit.>, we can immediately conclude the following corollary. For every sentence φ and every k, MC_φ is in for graphs with treewidth at most k.We first give a rough sketch of the proof. Let φ be a fixed formula of quantifier depth d and k a treewidth. We show that MC_φ is (^1,log n)-maintainable for graphs with treewidth at most k.The construction of a dynamic program for MC_φ is similar to the one in the proof for 3-Colourability (Theorem <ref>). At each point, the program needs to evaluate φ on a graph G'=(V,(E ∖ E^-) ∪ E^+) with n nodes, where |E^- ∪ E^+|=(log n), using a nice tree decomposition (T,B) of width 4k+5 for the initial graph G=(V,E) and auxiliary information on the type of depth d for each triangle of T (defined as in Section <ref>). The graph G' can be viewed as havinga center C⊆ V of logarithmic size, that contains the nodes with edges in E^- ∪ E^+ and, additionally, for each of these nodes v all nodes of one bag that contains v.Furthermore, there are node sets D_1,…,D_ℓ that, together with C, contain all nodes from V, such that the sets D_i-C are pairwise disjoint and disconnected, and each set D_i∩ C has size (1) (cf. Figure <ref>). From the type information for the triangles, the program can infer the depth-d types of each G'[D_i].The situation is similar as in the Composition Theorem of Elberfeld, Grohe and Tantau <cit.>. However, in their setting, the size of C is bounded by a constant, and they show, very roughly, that there is a first-order formula that can be evaluated on a suitable extension of G[C] by information on the types of the G[D_i] to yield the same result as φ on G.We show that in our setting one can construct an formula ψsuch that G'φ if and only if ψ, whereis a structure, which extends G'[C] by type information about the G'[D_i].This construction is detailed in Section <ref> below. It uses well-known techniques and, in particular, a composition theorem by Shelah (cf. Theorem <ref>). The dynamic program then uses a first-order formula (to be evaluated in a suitable extension of G' with auxiliary relations) that is obtained from ψ by replacing the second-order quantification over C by first-order quantification over V. This is possible, since sets of size (log n) over C can be encoded by (1) elements of V. In the remainder of this section we make these ideas more precise. In the next subsection we state and prove a composition theorem for graphs with a center of the form described above. Then, in Section <ref>, we show how this theorem is applied to dynamically evaluate an formula φ. §.§ A Feferman–Vaught-type composition theoremIn the following, we give an adaptation of the Feferman–Vaught-type composition theorem from <cit.> that will be useful for maintaining properties.Intuitively, the idea is very easy, but the formal presentation will come with some technicalities. For ease of presentation, we explain the basic idea for graphs first. We consider graphs G=(V,E) with a center C⊆ V, such that there are sets D_1,…,D_ℓ such that, for some w>0, the following conditions hold. * C∪⋃_i=1^ℓD_i= V.* For all i≠j, D_i∩ D_j ⊆ C.* All edges in E have both end nodes in C or in some D_i. * For every i, |D_i∩ C|≤ w.* For each i there is some element v_i∈ D_i∩ C that is not contained in any D_j, for j≠ i.In this case, we say that C has connection width w in G. See Figure <ref> for an illustration.We refer to the sets D_1,…,D_ℓ as petals and the nodes v_1,…,v_ℓ as identifiers of their respective petals. We emphasise that ℓ is bounded by |C|, but not assumed to be bounded by a constant. Readers who have read the proof of Theorem <ref> can roughly think of C as the set of vertices from special bags. Our goal is to show the following.If a graph G has a center C of connection width w, for come constant w, then GC can be extended by the information about the types of its petals in a suitable way, resulting in a structurewith universe C, such that formulas over G have equivalent formulas over . In the following, we work out the above plan in more detail. Although, for Theorem <ref>, we need the composition theorem only for coloured graphs with some constants, we deal in the following with arbitrary signatures. We fix some relational signature Σ and assume that it contains a unary relation symbol C. The definition of the connection width of sets C easily carries over to Σ-structures. In particular, tuples need to be entirely in C or in some petal D_i, andall constants of the structure need to be included in C. For every i, we call the set I_i D_i∩ C the interface of D_i and the nodes of D_i-Cinner elements of D_i.Let be a Σ-structure, C a center of connection width w with petals D_1, …, D_ℓ. For every i, let u^i=(u^i_1,…,u^i_w) be a tuple of elements from the interface I_i of D_i such that u^i_1 is an identifier of its petal D_i and every node from I_i occurs in u^i. By (_i,u^i) we denote the substructure ofinduced byD_iwith u^i_1,…,u^i_w as constants but without all tuples over C, i.e., (_i,u^i) only contains tuples with at least one inner element of D_i.Let d>0. The depth d, width w indicator structure ofrelative to C and tuples u^i is the unique structurewhich expandsC by the following relations: * a w-ary relation J that contains all tuples u^i, and* for every depth-d -type τ over Σ∪{c_1,…,c_w}, a unary relation R_τ containing those identifier nodes u^i_1 for which the depth-d -type of (_i, u^i) is τ. We note that different choices of the tuples u^i result in different indicator structures and we denote the set of all indicator structures ofrelative to C by (,C,w,d).We are now ready to formulate the desired composition theorem. For each d>0, every sentence φ with depth d, and each w, there is a number d' and a sentence ψ such that for every Σ-structure , every center C ofwith connection width w and every ∈(,C,w,d') it holds φif and only if ψ. The proof of Theorem <ref> uses Shelah's generalised sums <cit.>. We follow the exposition from Blumensath et al. <cit.>. In a nutshell, a generalised sum is a composition of several disjoint component structures along an index structure. Shelah's composition theorem states that sentences on a generalised sum can be translated to sentences on the index structure enriched by type information on the components.We apply the composition theorem on the basis of the following ideas, illustrated in Figure <ref>. From the center C and the petals D_i we define an index structureand component structures _i, respectively.In the generalised sum, these disjoint structures are again composed into a structure that is very similar to . More precisely,can be defined in the generalised sum by a first-order interpretation, and thus, thanks to Lemma <ref> below, we can translate the formula φ forinto a formula φ' on the generalised sum. Shelah's composition theorem then provides a translation of φ' into anformula ψ' on the structureenriched with type information on the structures _i.This enriched index structure is again very similar to an indicator structure : there is a first-order interpretation that defines the enriched index structure in . As a consequence, byLemma <ref> again, the formula ψ' can be translated into a formula φ on .Before we proceed to the proof of Theorem <ref>, we formally introduce the notions of a generalised sum and a first-order interpretation, and state the corresponding results on translations of formulas. We start with first-order interpretations. Let Σ, Γ be relational signatures. A first-order interpretation Υ from Σ to Γ consists of a first-order formula φ_U(x) and first-order formulas φ_R(x_1, …, x_r) for each r-ary relation symbol R ∈Γ, each over signature Σ.The first-order interpretation Υ interprets, in a Σ-structure , the Γ-structure Υ() with universe U^Υ(){a |φ_U(a)} and relationsR^Υ(){(a_1, …, a_r) |φ_R(a_1, …, a_r), a_1, …, a_r ∈ U^Υ()}for each R ∈Γ. A first-order interpretation from Σ to Γ not only interprets a Γ-structure in a Σ-structure, it also translates Γ-formulas to Σ-formulas.Let Υ be a first-order interpretation from Σ to Γ. For every () formula φ(x_1, …, x_ℓ) over Γ there is an () formula φ^Υ(x_1,…, x_ℓ) over Σ such that φ^Υ(a_1,…, a_ℓ) ⇔Υ() φ(a_1, …, a_ℓ) for all Σ-structuresand all elements a_i ∈ U^Υ(). We now turn to the definition of generalised sums. Let = (I,S_1,…, S_r) be a structure and (_i)_i ∈ I a sequence of structures _i = (D_i, R^i_1, …, R^i_t) indexed by elements i of . The generalised sum of (_i)_i ∈ I is the structure∑_i ∈ I_i(U, ∼, R'_1, …, R'_t, S'_1, …, S'_r)with universe U {i,a| i ∈ I, a ∈ D_i} and relations* i,a∼i',a' if and only if i = i' * R'_j {( i,a_1,…, i,a_ℓ ) | i ∈ I, (a_1, …, a_ℓ) ∈ R^i_j} * S'_j {( i_1,a_1,…,i_ℓ,a_ℓ ) | (i_1, …, i_ℓ) ∈ S_j, a_k ∈ D_i_k for all k ∈{1, …, ℓ}}The structuresand _i in this definition are also referred to as index structure and component structures, respectively. From every sentence φ, a finite sequence χ_1, …, χ_s of formulas and an formula ψ can be constructed such that ∑_i ∈ I_i φ ⇔(, χ_1 , …, χ_s ) ψfor all index structuresand component structures _i, where χ{i ∈ I |_i χ}.Intuitively, the formulas χ_i encode the type information on the component structures of the generalized sum.With the necessary notions in place, we can now prove Theorem <ref>.Theorem <ref> Suppose thatis a Σ-structure with center C of connection width w, tuples u^i collecting the interface nodes for each petal D_i as described above, and let φ be an formula over signature Σ. The proof is in three steps, depicted in Figure <ref>: * We present an index structureand component structures _i, and show that there is an FO-interpretation Υ that interprets the structurein the generalised sum ∑_i ∈ I_i.Thus there is an formula φ' such that φ if and only if ∑_i ∈ I_i φ' by Lemma <ref>.* From Theorem <ref> we obtain formulas ψ' and χ_1, …, χ_s such that ∑_i ∈ I_i φ' if and only if (, χ_1 , …, χ_s ) ψ'.* Then we show that there is an FO-interpretation Υ' that interprets (, χ_1 , …, χ_s ) in each indicator structure ∈(, C, w, d') with appropriate d'. Thus there is an MSO formula ψ that satisfies ψ if and only if (, χ_1 , …, χ_s ) ψ' by Lemma <ref>.Combining these three steps allows us to concludeφ(A)⟺∑_i ∈ I_i φ'(B)⟺ (, χ_1 , …, χ_s ) ψ' (C)⟺ψfor anyindicator structure ∈(, C, w, d'). It remains to prove (A) and (C), since (B) is a direct application ofTheorem <ref>. Towards proving (A) we construct structuresand _i which are closely related to the substructure C ofand the structures (_i,u^i), respectively. Let R_1,…,R_q be the relation symbols of Σ and letΓ={S_1,…,S_q}. The structurehas universe C, Γ-relations S_ i defined as the restriction of the respective Σ-relation R_i ofto C, and the relation J as described above. Each structure _v for v ∈ C is over signature Σ∪{U_1 …, U_w } and defined as follows. If v is an identifier u^i_1 of a petal, then _v = (_i, {u^i_1}, …, {u^i_w}), that is, the restriction ofto the elements from D_i, without any tuples consisting only of elements from C∩ D_i, and with additional unary, singleton relations U_1, …, U_w such that U_j includes only the j-th interface node u^i_j. If v is no identifier node then _v is the structure with universe {v} and empty relations. In the generalised sum ∑_v ∈ I_v= (U, ∼, R'_1, …, R'_q, U'_1,… U'_w,S'_1, …, S'_q,J'), the universe U consists of elements of the form u^i_1,w, where w∈ D_i, and of the form u,u where u∈ C is not an identifier of any petal. We emphasise that the formulas of the interpretation that defines (a copy of)incan not access the components u and v of an elementu,v∈ U.However, U can be partitioned intofour kinds of elements, each of which can easily be distinguished from the others in a first-order fashion: * Elements of the form v,v, for which D_v is not a petal. They can be identified since they constitute an equivalence class of size 1 with respect to ∼; * Elements of the form v,v, for which D_v is a petal. These are precisely the elements in U'_1;* Elements of the form v,u, where u∈ C is in the petal D_v. These elements occur in some U'_i, for i>1 (but not in U'_1);* Elements of the form v,u, where u∉C is in the petal D_v. They do not occur in any U'_i and are not of type (i). Elements of type (iv) are in one-to-one correspondence with theinner elements of petals D_i. Elements from C might have several copies in U, but only one of the types (i) or (ii). Thus, the formula that defines the universe for the first-order interpretation Υ ofin ∑_i ∈ I_i simply drops all elements of type (iii). Tuples ofof a relation R_i thatentirely consist of nodes from C (that is, elements of type (i) or (ii) in ) are directly induced by the corresponding relation S'_i.In order to define tuples with at least one node of type (iv) in , we first observe that it can be expressed in a first order fashion, whether for a type (iii) element v_1,u_1 and a type (i) element v_2,v_2 it holds u_1=v_2, i.e., that, intuitively, v_2,v_2 is the copy representing u_1 inthat survives in the universe of the interpretation. We claim that this condition holds, if and only if v_2,v_2 occurs as the i-th entry in some tuple of J' withfirst entry v_1,u_1, where i is the unique number such that v_1,u_1∈ U'_i. From this claim, first-order expressibility follows instantly. The “only if”-part of the claim is straightforward. For the “if”-part, it followsfrom the latter condition that there is a tuple with first entry v_1 and i-th entry v_2 in J, by the definition of J'. Since v_1,u_1∈ U'_i, there is also a tuple in J with v_1 as first entry and u_1 as i-th entry. However, since J has at most one tuple with any given value as first entry, u_1=v_2 follows, as claimed.A tuple with some element v,u of type (iv) is now in a relation R_i of the interpretation, if it can be transformed into a tuple of R'_i by replacing some elements w,w of type (i)with v,w.It follows from the construction that Υ() is isomorphic toand therefore φ if and only if Υ() φ. We obtain the formulas φ', χ_1, …, χ_s and ψ' as explained above.Let d' be the maximal quantifier depth of any formula χ_j.This concludes step (A) of the proof.Towards proving (C), recall that we need to show that there is a first-order interpretation Υ' which interprets (, χ_1 , …, χ_s ) in , for any ∈(, C, w, d'). Letbe such a structure. The universe ofis C, that is, the same as the universe of . Thus the formula of Υ' that defines the universe is trivial.For the definitions of the relations χ_j, the idea is as follows. If v is not an identifier u^i_1 of a petal, then v ∈χ_j if and only if χ_j holds in the structure consisting of only one element and with empty relations, which can be hard-coded in the defining formula. Otherwise, if v = u^i_1 for some i, we need to determine whether χ_j holds in the structure _v = (_i, {u^i_1}, …, {u^i_w}), which is a structure over signature Σ∪{U_1, …, U_w}. The structurecontains information about the types of the structures (_i,u^i), but (_i,u^i) is a structure over signature Σ∪{c_1, …, c_w}. Yet it is easy to see that for the formula χ'_j that is obtained from χ_j by replacing every atom U_k(x) by x = c_k it holds (_i,u^i) χ'_j if and only if _v χ_j. So, in this case v ∈χ_j if and only if v ∈ R_τ^ for a depth-d' type τ with χ'_j ∈τ. All these conditions can even be expressed by quantifier-free first-order formulas for fixed formulas χ_j.As a result, by Lemma <ref> we obtain from Υ' and ψ' a formula ψ with ψ⇔φ.§.§ The dynamic program We proceed to show that every -definable property can be maintained in , and thus prove Theorem <ref>. Thanks to Theorem <ref> and Proposition <ref> it suffices to showthat MC_φ is (^1,log n)-maintainable for graphs G with treewidth at most k.The idea for our dynamic program is similar to the idea for maintaining 3-colourability: during its initialisation the program constructs a tree decomposition and appropriate types for all triangles (instead of partial colourings as in the proof of Theorem <ref>). During the change sequence, a set C of nodes is defined that contains, for each affected graph node v, all nodes of at least one special bag containing v. The set C has connection width w for some constant w and the dynamic program basically maintains an indicator structurefor G relative to C. As there are only log n many change steps, the size of C is bounded by (log n).By Theorem <ref> there is an formula ψ with the property that G φ if and only if ψ. Although the dynamic program maintains , it cannot directly evaluate ψ, as it is restricted to use first-order formulas.For this reason we first show that second-order quantification over sets of size (log n) can be simulated in first-order logic, if a particular relation is present. Afterwards we present the details of the dynamic program. We call an -formula C-restricted, if all its quantified subformulas are of one of the following forms. * ∃ x(C(x) φ) or ∀ x(C(x) φ),* ∃ X(∀ x (X(x)C(x)) φ) or ∀ X(∀ x (X(x)C(x)) φ). Letbe a structure with a unary relation C and a (k+1)-ary relation Sub, for some k. We say that Sub encodes subsets of C if, for each subset C'⊆ C, there is a k-tuple t such that, for every element c∈ C it holds c∈ C' if and only if ( t,c)∈Sub. Clearly, such an encoding of subsets only exists if |V|^k≥ 2^|C| and thus if |C|≤ klog |V|.For each C-restricted -sentence ψ over a signature Σ (containing C) and every k there is a first-order sentence χ over Σ∪{S} where S is a (k+1)-ary relation symbol such that, for every Σ-structureand (k+1)-ary relation Sub that encodes subsets of C (of ), it holdsψ if and only if(, Sub) χ. The proof is straightforward. Formulas ∃ X(∀ x (X(x)C(x)) φ) are translated into formulas ∃ x φ', where x is a tuple of k variables and φ' results from φ by simply replacing every atomic formula X(y) by Sub( x,y). Universal set quantification is translated analogously.Theorem <ref> Thanks to Theorem <ref> and Proposition <ref>it suffices to show that MC_φ is (^1,log n)-maintainable in for graphs with treewidth at most k. Let d be the quantifier depth of φ and let d' and ψ be the number and the sentence guaranteed to exist by Theorem <ref>. Given a graph G = (V,E), the ^1 initialisation first ensures that relations ≤, +, × andare available. Then it computes a d_tree-nice tree decomposition (T,B) with T = (I,F,r) with bags of size at most ℓ 4k+6, for the constant d_tree guaranteed to exist by Lemma <ref>, together with the predecessor order ≼ on I.With each node i ∈ I, we associate a tuple v(i)=(v_1,…,v_m,v_1,…,v_1) of length ℓ, where B(i)={v_1,…,v_m} and v_1<⋯<v_m. That is, if the bag size of i is ℓ, this tuple just contains all graph nodes of the bag in increasing order. If the bag size is smaller, the smallest graph node is repeated.Similarly, with each triangle δ = (i_0,i_1,i_2) such that the subgraph G(δ) has at least one inner node, we associate a tuple v(δ) = (v(δ), v(i_0), v(i_1), v(i_2)), where v(δ) denotes the smallest inner node of G(δ) with respect to ≤.The dynamic program further uses auxiliary relations S, C, N, and D_τ, for each depth-d' type τ over the signature that consists of the binary relation symbol E and 3 ℓ + 1 constant symbols c_1, …, c_3 ℓ +1.The intended meaning is that C is a center of G with connection width 3 ℓ +1 and that from these relations an indicator structurerelative to C can be defined in first-order. The relation S stores tuples v(i) representing special bags, as in the proof of Theorem <ref>.The relations D_τ provide type information for all triangles. More precisely, for each triangle δ = (i_0, i_1, i_2) for which the subgraph G(δ) has at least one inner node, D_τ contains the tuple v(δ) ifand only if the depth-d' type of (G(δ), v(δ)) is τ.The set C always contains all graph nodes that occur in special bags (and thus in S), plus one inner node v(δ), for each maximal clean triangle[For such a triangle δ=(i_0,i_1,i_2), the nodes i_0, i_1, i_2 are exactly the special nodes in T(δ).] δ with at least one inner node.The relation N defines a bijection between C and an initial segment of ≤. We observe that C is a center of G and that the petals induced by C correspond to the maximal clean triangles, with respect to the special nodes stored in S, with at least two inner nodes.The interface I(δ) of a petal corresponding to a maximal clean triangle δ = (i_0, i_1, i_2) contains the nodes from B(i_0), B(i_1), and B(i_2) as well as the node v(δ), so C has connection width w 3ℓ+1. Figure <ref> givesan illustration. Now, an indicator structure ∈(G, C, w, d')can be first-order defined as follows.Clearly, maximal clean triangles can be easily first-order defined from the relation S. For each maximal clean triangle δ = (i_0, i_1, i_2) with at least two inner nodes, the relation J contains the tuple v(δ), and the relation R_τ contains v(δ) if and only if is v(δ) ∈ D_τ. We translate the formula ψ to a C-restricted formula χ' such that ψ⇔ (G,) χ', whereis the auxiliary database stored by the dynamic program. This translation is basically as described by Lemma <ref>.The formula χ' results from ψ by * C-restricting every quantified subformula, so, for example, replacing every quantified subformula ∃ Xθ by ∃ X(∀ x (X(x)C(x)) θ) and every quantified subformula ∀ Xθ by ∀ X(∀ x (X(x)C(x)) θ), and * replacing every atom A( x) by θ_A( x), where θ_A is the first-order formula that defines A in (G, ). It clearly holds that (G,) χ' ⇔ψ, and by Theorem <ref> also (G,) χ' ⇔ G φ.We now define a relation Sub that encodes subsets of C. We observe that C is of size at most b log n for some b ∈. Thus a subset C' of C can be represented by a tuple (a_1, …, a_b) of nodes, where an element c ∈ C is in C' if and only if c is the m-th element of C with respect to the mapping defined by N, m = (ℓ -1) log n + j and the j-th bit of a_ℓ is one. By Proposition <ref> we finally obtain a first-order formula χ such that (G,,Sub) χ⇔ (G,) χ' ⇔ G φ. That means that a dynamic program that maintains the auxiliary relations as intended can maintain the query MC_φ.It thus remains to describe how the auxiliary relations can be initialised and updated.The set C is initially the bag B(r) of the root of T plus one inner node v(r,r,r), and S contains the tuple v(r). The relations D_τ are computed in d_treelog n inductive steps, each of which can be defined in first-order logic, and therefore this computation can be carried out in ^1, thanks to log n = ^1.More precisely, the computation of the relations D_τ proceeds inductively in a bottom-up fashion. It starts with trianglesδ = (i_0, i_1, i_2) for which T(δ) has exactly one or two inner tree nodes (i,.e., nodes different fromi_0, i_1, i_2). Since such graphs G(δ) have at most 5ℓ nodes, their type can be determined by a first-order formula.[Basically, all isomorphism types of such graphs and their respective types can be directly encoded into first-order formulas.] For larger triangles, several cases need to be distinguished. Here we explain the case of a triangle δ = (i_0, i_1, i_2), for which i_0 has child nodes i'_1 and i'_2 such that i'_1≼ i_1 and i'_2≼ i_2 (cf., Figure <ref>).In this case, the type τ of (G(δ), v(δ)) can be determined from the types τ_1 of (G(δ_1), v(δ_1)) and τ_2 of (G(δ_2), v(δ_2)), where δ_1=(i'_1, i_1, i_1) and δ_2=(i'_2, i_2, i_2), and the type τ_0 of the graph G_0 that includes all edges of G(δ) that are not already in G(δ_1) or G(δ_2).More precisely, τ_0 is the type of (G_0,v(i_0), v(i_1), v(i_2), v(i'_1), v(i'_2)) and G_0 is the subgraph of G with node set V_0 = ⋃{B(i_0), B(i_1), B(i_2), B(i'_1), B(i'_2) } and all edges from G[V_0] that have at least one endpoint in B(i'_1) ∪ B(i'_2). These types are either already computed or the graphs are of size at most 5ℓ and their type can therefore be determined by a first-order formula as before. We make this more precise. We observe that (G(δ), v(δ)) can be composed from the graphs (G_0,v(i_0), v(i_1), v(i_2), v(i'_1), v(i'_2)), (G(δ_1), v(δ_1)) and (G(δ_2), v(δ_2)) by first taking the disjoint union of these graphs and afterwards fusing nodes according to the identities induced by the additional constants. For both operations, the depth-d type of the resulting structure only depends on the depth-d type(s) of the original structure(s) <cit.>. The type τ of (G(δ), v(δ)) is therefore determined by a finite function f as τ=f(τ_0,τ_1,τ_2), which can be directly encoded into first-order formulas.Finally, we describe how a dynamic program can maintain S, C, and N for log n many changes. The relation D_τ is not adapted during the changes. Whenever an edge (u,v) is inserted to or deleted from G, the nodes u and v are viewed as affected. For every affected node u that is not yet in a bag stored in S, a special tree node is selected(in some canonical way, e.g. always the smallest node with respect to ≤ is selected) such that u ∈ B(j). Furthermore, if node i is the least common ancestor of j and another special node it becomes special, as well. It is easy to see that when selecting j as a special node, at most one furthernode becomes special. The tuples v(j) and v(i) (if i exists) are added to S, their elements are added to C, the identifier nodes in C for maximal clean triangles are corrected, and N is updated accordingly.§ MSO OPTIMISATION PROBLEMSWith the techniques presented in the previous section, alsodefinable optimisation problems can be maintained in for graphs with bounded treewidth. An definable optimisation problem _φ is induced by an MSO formula φ(X_1, …, X_m) with free set variables X_1, …, X_m. Given a graph G with vertex set V, it asks for sets A_1, …, A_m ⊆ V of minimal[The adaptation to maximisation problems is straightforward.] size ∑_i=1^m |A_i| such that Gφ(A_1, …, A_m).Examples (with m=1) for such problems are the vertex cover problem and the dominating set problem.We require from a dynamic program for such a problem that it maintains unary query relations Q_1,…,Q_mthat store, at any time, an optimal solution for the current graph.For every formula φ(X_1, …, X_m) and every k, OPT_φ is in for graphs with treewidth at most k. As already mentioned in the previous section, for every k and everyformula φ there is an formula ψ that is equivalent on graphs with treewidth k <cit.>. Moreover, if φ = ∃ X_1 ⋯∃ X_mφ', then ψ is of the form ∃ X_1^1 ⋯∃ X_1^ℓ⋯∃ X_m^1 ⋯∃ X_m^ℓψ', for some natural number ℓ. So, we can conclude the following corollary.For every formula φ(X_1, …, X_m) and every k, OPT_φ is in for graphs with treewidth at most k. Given the machinery from the previous section, the plan for a dynamic program for an MSO-definable optimisation problem is relatively straightforward. Again, it suffices to show (^1,log n)-maintainability. The affected nodes of the graph after log n changes are again collected in a center C of the graph (with (log n) additional nodes as before). For each petal D_i and each relevant MSO-type τ we basically maintain a collection (B_1,…,B_m) of subsets of D_i-C that yields type τ in D_i and isminimal with respect to ∑_i=1^m |B_i|. Then, it is easy to compute in a first-order fashion, for every possible colouring of C, the minimum achievable overall sum for extensions of the colouring that make φ true.Theorem <ref> We only prove the special case of m=1, the extension to the general case is straightforward. Let φ(X) be an formula of quantifier depth d.The proof of Theorem <ref> shows how one can obtain a dynamic program that (^1, log n)-maintains the model checking problem MC_ψ for ψ∃ X φ. We adapt this proof, and reuse its notation, in order to obtain such a dynamic program for OPT_φ, using almost the same auxiliary relations.Together with Theorem <ref> and Proposition <ref>, the result follows. In the following, we sketch the proof idea.We consider φ to be an sentence over the signature {E,X}. Let G^+X = (V,E,X) be an arbitrary expansion of a graph G with a center C of connection width w, for some constant w. By Theorem <ref> there is a number d' and an sentence ψ such that for every ^+X∈(G^+X,C,w,d') it holds that G^+Xφ if and only if ^+Xψ. So, the formula ψ uses the type information on the petals provided by ^+X as well as G^+XC directly to check whether the relation X represents a feasible solution of the problem _φ. In the proof of Theorem <ref> we explained how to obtain a first-order formula χ from ψ such that (G, , Sub) χ⇔ψ, for the auxiliary databasemaintained by the dynamic program constructed in the proof of Theorem <ref> and a relation Sub encoding subsets of C. Let ∈(G,C,w,d'+1) be an indicator structure for G. Our goal is to maintain some relations that augment the type information provided bysuch that a formula χ' similar to χ can “guess” a relation X, check that it is a feasible solution, compute its size and verify that no feasible solution of smaller size exists. Of course, a relation X of unrestricted size cannot be quantified in first-order logic, even in the presence of , but we will see that the restriction of X to C and the type information on the petals can be quantified, which is sufficient for our purpose.We now give the details of the construction. The structurecontains relations R_τ such that u^i_1 ∈ R_τ if and only the depth-(d'+1) type of (_i,u^i) over signature Σ = {E, c_1, …, c_3ℓ+1} is τ, where the subgraph _i over universe D_i and the tuple u^i are defined as in Subsection <ref>. We say that a depth-d' type τ' over signature Σ^+XΣ∪{X} can be realised in (_i,u^i) by a set A_i ⊆ D_i, if τ' is the depth-d' type of (_i,u^i, A_i). If (_i,u^i) has depth-(d'+1) type τ, the existence of such a set is equivalent to the statement ∃ X α_τ'∈τ. We note that τ' already determines whether u_j^i ∈ A_i shall hold, for each constant u_j^i from the tupel u^i.The dynamic program maintains relations #R_τ' and Q_τ', for each depth-d' type over Σ^+X.The relations #R_τ' give the minimal size of a set that realises the type τ'. So, if τ' can be realised in (_i, u^i) by some set A, and s is the minimal size of such a set, then #R_τ' shall contain the tuple (u^i_1, v_s), where v_s is the (s+1)-th element[We ignore the case that the size could be as large as |V|, which can be handled by some additional encoding.] with respect to ≤. Furthermore, for the lexicographically minimal set A of this kind and size s, Q_τ' shall contain all tuples (u^i_1,a), where a ∈ A.We construct a first-order formula χ' from χ that is able to define an optimal solution X for _φ from (G, , ) expanded by the relations #R_τ' and Q_τ'. First, this formula quantifies for each depth-d' type τ' the set of identifiers u^i_1 such that X realises τ' in (_i,u^i) and checks consistency: as for each node v ∈ C that appears in u^i the type τ' already determines whether v ∈ X shall hold, the respective types need to agree for nodes that appear in multiple petals. For each u^i_1 the assigned type also needs to be realisable in the respective substructure (_i,u^i), which can be checked using the relations R_τ of . Using this information, χ' can apply χ to check that the implied set X is a feasible solution. With the help of #R_τ it can compute the size of X, as is able to add up logarithmically many numbers <cit.> and C is only of logarithmic size in |V|. Also χ' checks that no other assignment of types τ' to identifier nodes results in feasible solutions of smaller size. Finally, χ' uses the relations Q_τ' to actually return an optimal solution X.Building on the proof of Theorem <ref>, it remains to show that the additional auxiliary relations #R_τ' and Q_τ' can be initialised and maintained. Actually, we maintain similarly defined relations #D_τ' and F_τ', the relations #R_τ' and Q_τ' are then first-order definable by the dynamic program using these relations.Let δ be a triangle such that G(δ) has at least one inner node.Similar to the relations D_τ used in the proof of Theorem <ref>, here a relation #D_τ' contains the tuple ( v(δ), u) if and only if (1) the depth-d' type τ' is realisable in (G(δ), v(δ)), and (2)u is the (s+1)-th element with respect to ≤, where s is the minimal size of a set that realises this type. Furthermore, for the lexicographically minimal set A of this kind and size s, F_τ' contains all tuples ( v(δ),a), where a ∈ A. It is clear that these relations suffice to define the relations #R_τ' and Q_τ', given the other relations of the proof of Theorem <ref>.The proof of Theorem <ref> can be extended to show that the initial versions of these auxiliary relations can be computed in ^1. For the inductive step of this computation, a type τ' realisable in a structure (G(δ),v(δ)) might be achievable by a finite number of combinations of types of its substructures. Here, the overall size of the realising set for X needs to be computed and the minimal solution needs to be picked. This is possible by a -formula since the number of possible combinations is bounded by a constant depending only on d and k.The updates of the auxiliary relations are exactly as in the proof of Theorem <ref>. Since D_τ needs no updates there, neither #D_τ'nor F_τ'do, here. From the proof it is easy to see that a dynamic program can also maintain the size s of an optimal solution, either implicitly as ∑_j=1^m|Q_j| for distinguished relations Q_j, or as {v_s}.Additionally, it can easily be adapted for optimisation problems on weighted graphs, where nodes and edges have polynomial weights in n.§ CONCLUSION In this paper, we introduced a strategy for maintaining queries by periodically restarting its computation from scratch and limiting the number of change steps that have to be taken into account. This has been captured in the notion of (, f)-maintainable queries, and we proved in particular that all (^1, log n)-maintainable, almost domain independent queries are actually in . As a consequence, decision and optimisation queries definable in- and -logic are infor graphs of bounded treewidth.For this, we stated a Feferman-Vaught-type composition theorem for these logics, which might be interesting in its own right. Though we phrase our results for and for graphs only, their proofs translate swiftly to general relational structures. Apart from this paper, this strategy is already used in <cit.> to prove that the reachability query can be maintained dynamically under insertions and deletions of a non-constant number of edges per change step.We believe that our strategy will find further applications. For instance, it is conceivable that interesting queries on planar graphs, such as the shortest-path query, can be maintained for a bounded number of changes using auxiliary data computed by an ^1 algorithm (in particular since many important data structures for planar graphs can be constructed in logarithmic space and therefore in ^1). § ACKNOWLEDGMENTWe thank an anonymous referee for valuable suggestions that greatly simplified the proof of Theorem <ref>.alpha | http://arxiv.org/abs/1704.07998v5 | {
"authors": [
"Samir Datta",
"Anish Mukherjee",
"Thomas Schwentick",
"Nils Vortmeier",
"Thomas Zeume"
],
"categories": [
"cs.LO",
"F.1.2"
],
"primary_category": "cs.LO",
"published": "20170426075109",
"title": "A Strategy for Dynamic Programs: Start over and Muddle through"
} |
L[1]>p#1 C[1]>p#1 R[1]>p#1Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am Mühlenberg 1, Potsdam-Golm, 14476, Germany Department of Physics, University of Maryland, College Park, Maryland 20742, USAMax Planck Institute for Gravitational Physics (Albert Einstein Institute), Am Mühlenberg 1, Potsdam-Golm, 14476, GermanyMax Planck Institute for Gravitational Physics (Albert Einstein Institute), Am Mühlenberg 1, Potsdam-Golm, 14476, GermanyMax Planck Institute for Gravitational Physics (Albert Einstein Institute), Am Mühlenberg 1, Potsdam-Golm, 14476, Germany Department of Physics, University of Maryland, College Park, Maryland 20742, USAMax Planck Institute for Gravitational Physics (Albert Einstein Institute), Am Mühlenberg 1, Potsdam-Golm, 14476, Germany [email protected] systems containing boson stars—self-gravitating configurations of a complex scalar field— can potentially mimic black holes or neutron stars as gravitational-wave sources. We investigate the extent to which tidal effects in the gravitational-wave signal can be used to discriminate between these standard sources and boson stars. We consider spherically symmetric boson stars within two classes of scalar self-interactions: an effective-field-theoretically motivated quartic potential and a solitonic potential constructed to produce very compact stars. We compute the tidal deformability parameter characterizing the dominant tidal imprint in the gravitational-wave signals for a large span of the parameter space of each boson star model, covering the entire space in the quartic case, and an extensive portion of interest in the solitonic case. We find that the tidal deformability for boson stars with a quartic self-interaction is bounded below by Λ_min≈ 280 and for those with a solitonic interaction by Λ_min≈ 1.3. We summarize our results as ready-to-use fits for practical applications. Employing a Fisher matrix analysis, we estimate the precision with which Advanced LIGO and third-generation detectors can measure these tidal parameters using the inspiral portion of the signal. We discuss a novel strategy to improve the distinguishability between black holes/neutrons stars and boson stars by combining tidal deformability measurements of each compact object in a binary system, thereby eliminating the scaling ambiguities in each boson star model. Our analysis shows that current-generation detectors can potentially distinguish boson stars with quartic potentials from black holes, as well as from neutron-star binaries if they have either a large total mass or a large (asymmetric) mass ratio. Discriminating solitonic boson stars from black holes using only tidal effects during the inspiral will be difficult with Advanced LIGO, but third-generation detectors should be able to distinguish between binary black holes and these binary boson stars.Distinguishing Boson Stars from Black Holes and Neutron Stars from Tidal Interactions in Inspiraling Binary Systems Serguei Ossokine December 30, 2023 =================================================================================================================== § INTRODUCTIONObservations of gravitational waves (GWs) by Advanced LIGO <cit.>, soon to be joined by Advanced Virgo <cit.>, KAGRA <cit.>, and LIGO-India <cit.>, open a new window to the strong-field regime of general relativity (GR). A major target for these detectors are the GW signals produced by the coalescences of binary systems of compact bodies. Within the standard astrophysical catalog, only black holes (BHs) and neutron stars (NSs) are sufficiently compact to generate GWs detectable by current-generation ground-based instruments. To test the dynamical, non-linear regime of gravity with GWs, one compares the relative likelihood that an observed signal was produced by the coalescence of BHs or NSs as predicted by GR against the possibility that it was produced by the merger of either: (a) BHs or NSs in alternative theories of gravity or (b) exotic compact objects in GR. In this paper, we pursue tests within the second class. Several possible exotic objects have been proposed that could mimic BHs or NSs, including boson stars (BSs) <cit.>, gravastars <cit.>, quark stars <cit.>, and axion stars <cit.>.The coalescence of a binary system can be classified into three phases— the inspiral, merger, and ringdown— each of which can be modeled with different tools. The inspiral describes the early evolution of the binary and can be studied within the post-Newtonian (PN) approximation, a series expansion in powers of the relative velocity v/c (see Ref. <cit.> and references within). As the binary shrinks and eventually merges, strong, highly-dynamical gravitational fields are generated; the merger is only directly computable using numerical relativity (NR). Finally, during ringdown, the resultant object relaxes to an equilibrium state through the emission of GWs whose (complex) frequencies are given by the object's quasinormal modes (QNMs), calculable through perturbation theory (see Ref. <cit.> and references within). Complete GW signals are built by synthesizing results from these three regimes from first principles with the effective-one-body (EOB) formalism <cit.> or phenomenologically, through frequency-domain fits <cit.> of inspiral, merger and ringdown waveforms.An understanding of how exotic objects behave during each of these phases is necessary to determine whether GW detectors can distinguish them from conventional sources (i.e., BHs or NSs). Significant work in this direction has already been completed. The structure of spherically-symmetric compact objects is imprinted in the PN inspiral through tidal interactions that arise at 5PN order (i.e., as a (v/c)^10 order correction to the Newtonian dynamics). Tidal interactions are characterized by the object's tidal deformability, which has recently been computed for gravastars <cit.> and “mini” BSs <cit.>. During the completion of this work, an independent investigation on the tidal deformability of several classes of exotic compact objects, including examples of the BS models considered here, was performed in Ref. <cit.>; details of the similarities and differences to this work are discussed in Sec. <ref> below. Additional signatures of exotic objects include the magnitude of the spin-induced quadrupole moment and the absence of tidal heating. The possibility of discriminating BHs from exotic objects with these two effects was discussed in Refs. <cit.> and <cit.>, respectively—we will not consider these effects in this paper. The merger of BSs has been studied using NR in head-on collisions <cit.> and following circular orbits <cit.>. The QNMs have been computed for BSs <cit.> and gravastars <cit.>. In this paper, we compute the tidal deformability of two models of BSs: “massive” BSs <cit.> characterized by a quartic self-interaction and non-topological solitonic BSs <cit.>. The self-interactions investigated here allow for the formation of compact BSs, in contrast to the “mini” BSs considered in Ref. <cit.>. We perform an extensive analysis of the BS parameter space within these models, thereby going beyond previous work in Ref. <cit.>, which was limited to a specific choice of parameter characterizing the self-interaction for each model. Special consideration must be given to the choice of the numerical method because BSs are constructed by solving stiff differential equations—we employ relaxation methods to overcome this problem <cit.>. Our new findings show that for massive BSs, the tidal deformability Λ (defined below) is bounded below by Λ_min≈ 280 for stable configurations, while for solitonic BSs the deformability can reach Λ_min≈ 1.3. For comparison, the deformability of NSs is Λ_NS≳𝒪(10) and for BHs Λ_BH=0. We compactly summarize our results as fits for convenient use in future gravitational wave data analysis studies. In addition, we employ the Fisher matrix formalism to study the prospects for distinguishing BSs from NSs or BHs with current and future gravitational-wave detectors based on tidal effects during the inspiral. Prospective constraints on the combined tidal deformability parameters of both objects in a binary were also shown for two fiducial cases in Ref. <cit.>. Our findings are consistent with the conclusions drawn in Ref. <cit.>; we discuss a new type of analysis that can strengthen the claims made therein on the distinguishability of BSs from BHs and NSs by combining information on each body in a binary system. The paper is organized as follows. Section <ref> introduces the BS models investigated herein. We provide the necessary formalism for computing the tidal deformability in Sec. <ref>, and describe the numerical methods we employ in Sec. <ref>. In Sec. <ref>, we compute the tidal deformability, providing results that range from the weak-coupling limit to the strong-coupling limit as well as numerical fits for the tidal deformability. Finally, in Sec. <ref> we discuss the prospects of testing the existence of stellar-mass BSs using GW detectors and provide some concluding remarks in Sec. <ref>.We use the signature (-,+,+,+) for the metric and natural units ħ=G=c=1, but explicitly restore factors of the Planck massm_Planck=√(ħ c/G) in places to improve clarity. The convention for the curvature tensor is such that∇_β∇_α a_μ - ∇_β∇_β a_μ = R^ν_μαβ a_ν, where ∇_α is the covariant derivative and a_μ is a generic covector. § BOSON STAR BASICSBoson stars—self-gravitating configurations of a (classical) complex scalar field—have been studied extensively in the literature, both as potential dark matter candidates and as tractable toy models for testing generic properties of compact objects in GR. Boson stars are described by the Einstein-Klein-Gordon actionS=∫ d^4 x √(-g)[R/16 π-^αΦ∇_αΦ^*-V(|Φ|^2)] ,where ^* denotes complex conjugation. The only experimentally confirmed elementary scalar field is the Higgs boson <cit.>, which is an unlikely candidate to form a BS because it readily decays to W and Z bosons. However, other massive scalar fields have been postulated in many theories beyond the Standard Model, e.g., bosonic superpartners predicted by supersymmetric extensions <cit.>.The Einstein equations derived from the action (<ref>) are given byR_αβ-1/2 g_αβ R= 8π T^Φ_αβ,withT^Φ_αβ= _αΦ^* _βΦ+_βΦ^*_αΦ-g_αβ(∇^γΦ^* ∇_γΦ+V(|Φ|^2)).The accompanying Klein-Gordon equation is∇^α∇_αΦ=d V/d |Φ |^2Φ,along with its complex conjugate.The earliest proposals for a BS contained a single non-interacting scalar field <cit.>, that isV(|Φ |^2)= μ^2 |Φ |^2,where μ is the mass of the boson. The free Einstein-Klein-Gordon action also describes the second-quantized theory of a real scalar field; thus, this class of BS can also be interpreted as a gravitationally bound Bose-Einstein condensate <cit.>. The maximum mass for BSs with the potential given in Eq. (<ref>) is M_max≈ 0.633 m_Planck^2/μ,or in units of solar mass, M_max / M_⊙≈ 85 peV / μ. The corresponding compactness for this BS is C_max≈ 0.08 <cit.>.[Formally, BSs have no surface, so the notion of a radius (and hence compactness) is inherently ambiguous. One common convention is to define the radius as that of a shell containing a fixed fraction of the total mass of the star (e.g., R_99 where m(r=R_99)=0.99 m(r=∞)). To avoid this ambiguity, our results are given in terms of quantities that can be extracted directly from the asymptotic geometry of the BS: the total mass M and dimensionless tidal deformability Λ (defined below).] Because this maximum mass scales more slowly with μ than the Chandrasekhar limit for a degenerate fermionic star M_CH∼ m_Planck^3/m_ Fermion^2, this class of BSs is referred to as mini BSs. The tidal deformability was computed in this model in Ref. <cit.> Since the seminal work of the 1960s <cit.>, BSs with various scalar self-interactions have been studied. We consider two such models in this paper, both which reduce to mini BSs in the weak-coupling limit. The first BS model we consider is massive BSs <cit.>, with a potential given by V_ massive(|Φ|^2)=μ^2 |Φ|^2+λ/2|Φ|^4,which is repulsive for λ≥ 0. In the strong-coupling limit λ≫μ^2 / m_Planck^2, spherically symmetric BSs obtain a maximum mass of M_max≈0.044√(λ)m_Planck^3/μ^2 <cit.>. In units of the solar mass M_⊙ this reads M_max / M_⊙≈√(λ) (0.3 GeV / μ)^2. Such configurations are roughly as compact as NSs, with an effective compactness of C_max≈ 0.158 <cit.>. This BS model is a natural candidate from an effective-field-theoretical perspective because the potential in Eq. (<ref>) contains all renormalizable self-interactions for a scalar field, i.e., other interactions that scale as higher powers of |Φ| are expected to be suppressed far from the Planck scale. The “natural” values of λ∼ 1 and μ≪ m_Planck yield the strong-coupling limit of the potential (<ref>). Because it is the most theoretically plausible BS model, we investigate the strong-coupling regime of this interaction in detail in Section <ref>.The second class of BS that we consider is the solitonic BS model <cit.>, characterized by the potentialV_ solitonic(|Φ|^2)=μ^2 |Φ|^2(1-2 |Φ|^2/σ_0^2)^2 .This potential admits a false vacuum solution at |Φ |=σ_0/√(2). One can construct spherically symmetric BSs whose interior closely resembles this false vacuum state and whose exterior is nearly vacuum |Φ |≈ 0; the transition between the false vacuum and true vacuum occurs over a surface of width Δ r∼μ^-1. In the strong-coupling limit σ_0 ≪ m_Planck, the maximum mass of non-rotating BSs is M_max≈ 0.0198 m_Planck^4/(μσ_0^2), or M_max / M_⊙≈ (μ / σ_0)^2 (0.7 PeV / μ)^3 <cit.>. The corresponding compactness C_max≈ 0.349 approaches that of a BH C_BH=1/2 <cit.>.[This compactness is still lower than the theoretical Buchdahl limit of C ≤ 4/9 for isotropic perfect fluid stars that respect the strong energy condition <cit.>.] The main motivation for considering the potential (<ref>) is as a model of very compact objects that could even possess a light-ring when C>1/3. In this paper, we will only consider solitonic BSs as potential BH mimickers, as NSs could be mimicked by the more natural massive BS model.In this paper, we restrict our attention to only non-rotating BSs. Axisymmetric (rotating) BSs have been constructed for the models we consider <cit.>, but these solutions are significantly more complex than those that are spherically symmetric (non-rotating). The energy density of a rotating BS forms a toroidal topology, vanishing at the star's center. Because its angular momentum is quantized, a rotating BS cannot be constructed in the slow-rotation limit, i.e. by adding infinitesimal rotation to a spherically symmetric solution <cit.>. § TIDAL PERTURBATIONS OF SPHERICALLY-SYMMETRIC BOSON STARS We consider linear tidal perturbations of a non-rotating BSs.We work within the adiabatic limit, that is we assume that the external tidal field varies on timescales much longer than any oscillation period of the star or relaxation timescale to reach a microphysical equilibrium. These conditions are typically satisfied during the inspiral of compact binaries. Close to merger, the assumptions concerning the separation of timescales can break down and the tides can become dynamical <cit.>; we ignore these complications here. The computation of the tidal deformability of NSs in general relativity was first addressed in Refs. <cit.> and was extended in Refs. <cit.>. §.§ Background configurationHere we review the equations describing a spherically symmetric BS <cit.>, which is the background configuration that we use to compute the tidal perturbations in the following subsection. We follow the presentation in Ref. <cit.>. The metric written in polar-areal coordinates readsds_0^2=-e^v(r) dt^2+e^u(r) dr^2+r^2 (dθ^2+sin^2θ dφ^2).As an ansatz for the background scalar field, we use the decompositionΦ_0(t,r)=ϕ_0(r)e^-i ω t.Inserting Eqs. (<ref>) and (<ref>) into Eqs. (<ref>)–(<ref>) gives e^-u(-u'/r+1/r^2)-1/r^2=-8 πρ,e^-u(v'/r+1/r^2)-1/r^2=8π p_rad, ϕ”_0+(2/r+v'-u'/2)ϕ'_0=e^u(U_0-ω^2e^-v)ϕ_0, where a prime denotes differentiation with respect to r, U_0=U(ϕ_0), U(ϕ)=dV/d|Φ|^2. Because the coefficients in Eq. (<ref>) are real numbers, we can restrict ϕ_0(r) to be a real function without loss of generality. We have also defined the effective density and pressuresρ≡-T^Φ_t^t=ω^2 e^-vϕ_0^2+e^-u(ϕ_0')^2+V_0,p_rad≡ T^Φ_r^r=ω^2 e^-vϕ_0^2+e^-u(ϕ_0')^2-V_0,p_tan≡ T^Φ_θ^θ=ω^2 e^-vϕ_0^2-e^-u(ϕ_0')^2-V_0,where V_0=V(ϕ_0). Note that BSs behave as anisotropic fluid stars with pressure anisotropy given byΣ=p_rad-p_tan=2 e^-u(ϕ_0')^2.An additional relation derived from Eqs. (<ref>)–(<ref>) that will be used to simplify the perturbation equations discussed in the next subsection isp'_rad=-(p_rad+ρ)/2r[e^u(1+8π r^2 p_rad)-1]-2Σ/r.We restrict our attention to ground-state configurations of the BS, in which ϕ_0(r) has no nodes. The background fields exhibit the following asymptotic behavior lim_r→ 0 m(r) ∼ r^3, lim_r→∞ m(r)∼ M, lim_r→ 0 v(r) ∼ v^(c), lim_r→∞ v(r) ∼ 0, lim_r→ 0ϕ_0(r) ∼ϕ_0^(c), lim_r→∞ϕ_0(r)∼1/r e^-r√(μ^2-ω^2), where M is the BS mass, v^(c) and ϕ_0^(c) are constants, and m(r) is defined such thate^-u(r)=(1-2 m(r)/r).§.§ Tidal perturbationsWe now consider small perturbations to the metric and scalar field defined such thatg_αβ =g_αβ^(0)+h_αβ,Φ =Φ_0+δΦ.We restrict our attention to static perturbations in the polar sector, which describe the effect of an external electric-type tidal field. Working in the Regge-Wheeler gauge <cit.>, the perturbations take the formh_αβdx^α dx^β= ∑_l≥ |m|Y_lm(θ,φ)[e^v h_0(r)dt^2..+e^u h_2(r) dr^2 +r^2 k(r)(dθ^2+r sin^2 θ dφ^2)],andδΦ= ∑_l≥ |m|ϕ_1(r)/rY_l m(θ,φ) e^- i ω t, where Y_lm are scalar spherical harmonics.We insert the perturbed metric and scalar field from Eqs. (<ref>)–(<ref>) into the Einstein and Klein-Gordon equations, Eqs. (<ref>) and (<ref>), and expand to first order in the perturbations. For the metric functions, the (θ,ϕ)-component of the Einstein equations gives h_2=h_0, and the (r,r)- and (r, θ)-components can be used to algebraically eliminate k and k^' in favor of h_0 and its derivatives. Finally, the (t,t)-component leads to the following second-order differential equation: h_0”+e^uh_0'/r(1+e^-u-8 πr^2 V_0)-32 πe^uϕ _1/r^2[ϕ _0' (-1+e^-u-8 πr^2 p_rad)+r ϕ_0 (U_0 -2 ω ^2e^-v)]+h_0 e^u/r^2[-16 πr^2 V_0-l (l+1)-e^u(1-e^-u+8 π r^2 p_rad)^2+64 πr^2 ω ^2 ϕ _0^2 e^-v]=0,where we have also used the background equations (<ref>), (<ref>), and (<ref>). From the linear perturbations to the Klein-Gordon equation, together with the results for the metric perturbations and the background equations, we obtainϕ _1”+e^u ϕ _1'/r(1-e^-u-8 πr^2V_0)-e^uh_0 [ϕ _0' (-1+e^-u-8 πr^2 p_rad)+rϕ _0(U_0 -2 ω ^2 e^-v)] +e^uϕ _1/r^2[8 π r^2V_0-1+e^-u-l(l+1)-r^2 (U_0+2 W_0 ϕ _0^2)+r^2 e^-vω ^2-32πe^-ur^2 (ϕ _0')^2]=0, where W_0=W(ϕ_0) with W(ϕ)=dU/d|Φ|^2. These perturbation equations were also independently derived in Ref. <cit.> and are a special case of generic linear perturbations considered in the context of QNMs (see, e.g., Refs. <cit.>). As a check, we combined the three first-order and one algebraic constraint for the spacetime perturbations from Ref. <cit.> into one second-order equation for h_0, which agrees with Eq. (<ref>) in the limit of static perturbations.For the special case of mini BSs, the tidal perturbation equations were also obtained in Ref. <cit.>. The perturbations exhibit the following asymptotic behavior <cit.> lim_r→ 0 h_0(r) ∼ r^l, lim_r→∞ h_0(r) ∼ c_1 (r/M)^-(l+1)+c_2 (r/M)^l, lim_r→ 0ϕ_1(r) ∼ r^l+1, lim_r→∞ϕ_1(r) ∼ r^M μ^2/√(μ^2-ω^2)e^-r√(μ^2-ω^2).§.§ Extracting the tidal deformabilityThe BS tidal deformability can be obtained in a similar manner as with NSs <cit.>. Working in the (nearly) vacuum region far from the center of the BS, the formalism developed for NSs remains (approximately) valid. For simplicity, we consider only l=2 perturbations for the remainder of this section. The generalization of these results to arbitrary l is detailed in Ref. <cit.>.As shown in Eqs. (<ref>) and (<ref>), very far from the center of the BS, the system approaches vacuum exponentially. Neglecting the vanishingly small contributions from the scalar field, the metric perturbation reduces to the general formh_0^ vac=c_1 Q̂_22(x)+c_2 P̂_22(x)+[(ϕ_0)^1,( ϕ_1)^1],where we have defined x≡ r/M-1, P̂_22 and Q̂_22 are the associated Legendre functions of the first and second kind, respectively,normalized as in Ref. <cit.> such that P̂_22∼ x^2 and Q̂_22∼ 1/x^3 when x→∞. The coefficients c_1 and c_2 are the same as in Eq. (<ref>).In the BS's local asymptotic rest frame, the metric far from the star's center takes the form <cit.>g̅_00= -1+2 M/r+3 𝒬_ij/r^3(n^i n^j-1/3δ^ij)+(1/r^4)-ℰ_ij x^i x^j +(r^3 )+[(ϕ_0)^1,( ϕ_1)^1],where n^i=x^i/r, ℰ_ij is the external tidal field, and 𝒬_ij is the induced quadrupole moment. Working to linear order in ℰ_ij, the tidal deformability λ_Tidal is defined such that𝒬_ij=-λ_Tidalℰ_ij.For our purposes, it will be convenient to instead work with the dimensionless quantityΛ≡λ_Tidal/M^5. Comparing Eqs. (<ref>) and (<ref>), one finds that the tidal deformability can be extracted from the asymptotic behavior of h_0 usingΛ= c_1/3 c_2.From Eq. (<ref>), the logarithmic derivative y≡dlog h_0/dlog r= r h_0'/h_0,takes the formy(x)=(1+x)3 ΛQ̂_2 2'(x)+ P̂_2 2'(x)/3 ΛQ̂_2 2(x)+ P_2 2(x),or equivalentlyΛ=-1/3((1+x)P̂_2 2'(x)- y(x)P̂_2 2(x)/(1+x) Q̂_2 2'(x)-y(x)Q̂_2 2(x)).Starting from a numerical solution to the perturbation equations (<ref>) and (<ref>), one obtains the deformability Λ by first computing y from Eq. (<ref>) and then evaluating Eq. (<ref>) at a particular extraction radius x_Extract far from the center of the BS. Details concerning the numerical extraction are described in Sec. <ref> below. § SOLVING THE BACKGROUND AND PERTURBATION EQUATIONS The background equations (<ref>)–(<ref>) and perturbation equations (<ref>)–(<ref>) form systems of coupled ordinary differential equations. These equations can be simplified by rescaling the coordinates and fields by μ (the mass of the boson field). To ease the comparison with previous work, we extend the definitions given in Ref. <cit.>: for massive BSs, we user → m_Planck^2r̃/μ, m(r)→m_Planck^2m̃(r̃)/μ, λ→ 8 πμ^2 λ̃/m_Planck^2,ω→μω̃/m_Planck^2,ϕ_0(r)→ m_Planckϕ̃_0(r̃)/(8 π)^1/2, ϕ_1(r)→m_Planck^2ϕ̃_1(r̃)/μ (8 π)^1/2,while for solitonic BSs, we use r → m_Planck^2r̃/σ̃_0 μ, m(r)→m_Planck^2m̃(r̃)/σ̃_0 μ, σ_0 →m_Planckσ̃_0/(8π)^1/2,ω→σ̃_0 μω̃/m_Planck^2,ϕ_0(r)→ σ_0 ϕ̃_0(r̃)/(2)^1/2, ϕ_1(r)→m_Planck^2ϕ̃_1(r̃)/(16 π)^1/2μ,where factors of the Planck mass have been restored for clarity.Finding solutions with the proper asymptotic behavior [Eqs. (<ref>) and (<ref>)] requires one to specify boundary conditions at both r̃=0 and r̃=∞. To impose these boundary conditions precisely, we integrate over a compactified radial coordinateζ=r̃/N+r̃,as is done in Ref. <cit.>, where N is a parameter tuned so that exponential tails in the variables ϕ̃_0 and ϕ̃_1 [see Eqs. (<ref>) and (<ref>)] begin near the center of the domain ζ∈[0,1]. For massive BSs, we use N ranging from 20 to 60 depending on the body's compactness; for solitonic BSs we use N between 1 and 10.Ground-state solutions to the background equations (<ref>)–(<ref>) can be completely parameterized by the central scalar field ϕ̃_0^(c) and frequency ω̃. To determine the ground state frequency, we formally promote ω̃ to an unknown constant function of r̃ and simultaneously solve both the background equations andω̃'(r̃)=0.We impose the following boundary conditions on this combined system:u(0)= 0, ϕ̃_0(0)=ϕ̃_0^(c), ϕ̃_0'(0)=0,v(∞)= 0, ϕ̃_0(∞)=0.Here, the inner boundary conditions ensure regularity at the origin, and the outer conditions guarantee asymptotic flatness.The background and pertrubation equations are stiff, and therefore the shooting techniques usually used to solve two-point boundary value problems require signficant fine-tuning to converge to a solution <cit.>. To avoid these difficulties, we use a standard relaxation algorithm that more easily finds a solution given a reasonable initial guess <cit.>. Once a solution is found for a particular choice of the central scalar field ϕ̃_0^(c) and scalar coupling (i.e., λ for massive BSs or σ_0 for solitonic BSs), this solution can be used as an initial guess to obtain nearby solutions. By iterating this process, one can efficiently generate many BS configurations.After finding a background solution, we solve the perturbation equations (<ref>) and (<ref>). To improve numerical behavior of the perturbation equations near the boundaries, we factor out the dominant r̃ dependence and instead solve forh̅_0(r̃)≡ h_0 r̃^-2, ϕ̅_1(r̃)≡ϕ̃_1 r̃^-3.We employ the boundary conditions h̅_0(0) =h̅_0^(c), h̅_0'(0)=0, ϕ̅_1'(0) =0, ϕ̅_1(∞)=0,where the normalization h̅_0^(c) is an arbitrary non-zero constant.Finally, we compute the tidal deformability using Eq. (<ref>) in the nearly vacuum region x≫1. At very large distances, the exponential falloff of ϕ_0 and ϕ_1 is difficult to resolve numerically. This numerical error propagates through the computation of the tidal deformability inEq. (<ref>) for very large values of x. We find that extracting Λ at smaller radii provides more numerically stable results, with a typical variation of ∼ 0.1% for different choices of extraction radius x_Extract. For consistency, we extract Λ at the radius at which y attains its maximum.Figure <ref> demonstrates our procedure for computing the tidal deformability. The background and perturbation equations are solved for a massive BS with a coupling of λ̃=300 using a compactified coordinate with N=20. The profile of the effective density ρ, decomposed into its background value ρ_0 and first order correction δρ, is shown in the top panel for a star of mass 3.78m_Planck^2/μ. Note that the magnitude of the perturbation is proportional to the strength of the external tidal field; to improve readability, we have scaled δρ to match the size of ρ_0. The middle panel of Fig. <ref> shows the computed logarithmic derivative y across the entire spacetime (black). We calculate the deformability with Eq. (<ref>) using the peak value of y, located at the dot-dashed line. Comparing with the top panel, one sees that the scalar field is negligible in this region, justifying our use of formulae valid in vacuum. The bottom panel depicts the typical variation of Λ computed at different locations x_Extract—our procedure yields consistent results provided one works reasonably close to the edge of the BS. As a check, we insert the computed value of Λ back into the vacuum solution for y given in Eq. (<ref>), plotted in red in the middle panel. As expected, this curve closely matches the numerically computed solution at large radii, but deviates upon entering a region with non-negligible scalar field. § RESULTS §.§ Massive Boson StarsThe dimensionless tidal deformability of massive BSs is given as a function of the rescaled total mass M̃ [defined as in Eq. (<ref>)] in the left panel of Fig. <ref>. The deformability in the weak-coupling limit λ̃=0 is given by the dotted black curve; this limit corresponds to the mini BS model considered in Ref. <cit.>.[In Ref. <cit.>, the authors computed the quantity k_BS, related to the quantity Λ presented here by k_BS=Λ M^10. The quantity k_2^E, computed in Ref. <cit.> for mini, massive, and solitonic BSs, is related to Λ by k_2^E=(4 π/5)^1/2Λ.] One finds that the tidal deformability of the most massive stable star (colored dots) decreases from Λ∼900 in the weak-coupling limit towards Λ∼ 280 as λ̃ is increased. For large values of λ̃, the deformability exhibits a universal relation when written in terms of the rescaled mass M̃ / λ̃^1/2 in the sense that the results for large λ̃ rapidly approach a fixed curve as the coupling strength increases. This convergence towards the λ̃=∞ relation is illustrated in the right panel of Fig. <ref>, in which the x-axis is rescaled by an additional factor of λ̃^1/2 relative to the left panel; in both panels, we have added black arrows to indicate the direction of increasing λ̃. Employing this rescaling of the mass, we compute the relation Λ(M̃,λ̃) in the strong-coupling limit λ̃→∞ below. The tidal deformability in this limit is plotted in Fig. <ref> with a dashed black curve. The gap in tidal deformability between BSs, for which the lowest values are Λ≳ 280, and NSs, where for soft equations of state and large masses Λ≳ 10, can be understood by comparing the relative size or compactness C=M/R of each object. From the definitions (<ref>) and (<ref>), one expects the tidal deformability to scale as Λ∝ 1/C^5. In the strong-coupling limit, stable massive BSs can attain a compactness of C_max≈ 0.158; note that in the exact strong-coupling limit λ̃=∞, BSs develop a surface, and thus their compactness can be defined unambiguously.A NS of comparable compactness has a tidal deformability that is only ∼ 025% larger than that of BSs. However, NS models predict stable stars with approximately twice the compactness that can be attained by massive BSs, and thus, their minimum tidal deformability is correspondingly much lower. As argued in Sec. <ref>, the strong-coupling limit of massive BSs is the most plausible model investigated in this paper from an effective field theory perspective. We analyze the tidal deformability in this limit in greater detail.To study the strong-coupling limit of λ̃→∞, we employ a different set of rescalings introduced, first in Ref. <cit.>:r → m_Planck^2λ̃^1/2r̂/μ, m(r)→m_Planck^2λ̃^1/2m̂(r̂)/μ, λ→ 8 πμ^2 λ̃/m_Planck^2,ω→μω̂/m_Planck^2, ϕ_0(r)→ m_Planckϕ̂_0(r̂)/(8 πλ̃)^1/2 , ϕ_1(r)→m_Planck^2ϕ̂_1(r̂)/μ(8 π)^1/2,where we have kept the previous notation for λ̃ to emphasize that it is the same quantity as defined in Eq. (<ref>). Keeping terms only at leading order in λ̃^-1≪ 1, Eqs. (<ref>)–(<ref>) becomee^-u(-u'/r̂+1/r̂^2)-1/r̂^2=-2ϕ̂_0^2-3ϕ̂_0^4/2,e^-u(v'/r̂+1/r̂^2)-1/r̂^2=ϕ̂_0^4/2, ϕ̂_0=(ω̂^2 e^-v-1)^1/2,where a prime denotes differentiation with respect to r̂. Note that in particular, Eq. (<ref>) becomes an algebraic equation, reducing the system to a pair of first order differential equations.Turning now to the perturbation equations, we use these rescalings and find that to leading order in λ̃^-1, Eqs. (<ref>) and (<ref>) become h_0”+e^uh_0'/r̂[r̂^2/2(1-e^-2 vω̂^4)+e^-u+1 ] -e^u h_0/r̂̂̂^2[r̂^4 e^u/4(1-e^-vω̂ ^2)^4 +r̂^2 (e^u(1-e^-vω̂^2)^2+10 e^-vω̂^2(1-e^-vω̂^2)-2)+e^u(1-e^-u)^2+l(l+1)]=0 , ϕ̂_1=h_0 r̂(1+ϕ̂_0^2)/2 ϕ̂_0. As with the background fields, the equation for the scalar field ϕ̂_1 becomes algebraic in this limit. Note that the scalar perturbation diverges as one approaches the surface of the BS, defined as the shell on which ϕ̂_0 vanishes. Nevertheless, the metric perturbation h_0 remains smooth over this surface.We integrate the simplified background equations (<ref>) and (<ref>) and then the perturbation equation (<ref>) using Runge-Kutta methods. We compute the tidal deformability using Eq. (<ref>) evaluated at the surface of the BS, and plot the results in the right panel of Fig. <ref> (dashed black).§.§ Solitonic boson starsThe dimensionless tidal deformability of solitonic BSs is given as a function of the mass in Fig. <ref>. As in Fig. <ref>, the colored dots highlight the most massive stable configuration for different choices of the scalar coupling σ̃_0. To aid comparison with the massive BS model, in the left panel we rescale the mass by an additional factor of σ̃_0 relative to the definition of M̃ in Eq. (<ref>).When the coupling σ̃_0 is strong, solitonic BSs can manifest two stable phases that can be smoothly connected through a sequence of unstable configurations <cit.>. The large plot in the left panel only shows stable configurations on the more compact branch of configurations. In the weak-coupling limit σ̃_0→∞, solitonic BSs reduce to the free field model considered in Ref. <cit.>. To illustrate this limit, we show in the smaller inset the tidal deformability for both phases of BSs as well as the unstable configurations that bridge the two branches of solutions. The weak-coupling limit is depicted with a dotted black curve. We find that the tidal deformability of the less compact phase of BSs smoothly transitions from Λ→∞ in the strong-coupling limit (σ̃_0→ 0)[In the exact strong-coupling limit σ̃_0 = 0, this diffuse phase of solitonic BSs vanishes <cit.>. However, the tidal deformability of this branch of BS configurations can be made arbitrarily large by choosing σ̃_0 to be sufficiently small.] to Λ∼900 in the weak-coupling limit (σ̃_0→∞). Because their tidal deformability is so large, diffuse solitonic BSs of this kind would not serve as effective BH mimickers, and we will not discuss them for the remainder of this paper. However, it should be noted that only this phase of stable configurations exists when σ_0≳ 0.23 m_Planck.Focusing now on the more compact phase of solitonicBSs, one finds that the tidal deformability of the most massive stable star (colored dots) decreases towards Λ∼ 1.3 as σ̃_0 is decreased. As before, the relation between a rescaled mass and Λ approaches a finite limit in the strong coupling limit. We illustrate this in the right panel of Fig. <ref> by rescaling the mass by an additional factor of σ̃_0^-1 relative to the definition in Eq. (<ref>). While we do not examine the exact strong-coupling limit σ̃_0→ 0 here, we find that the minimum deformability has converged to within a few percent of Λ=1.3 for 0.03 m_Planck≤σ_0≤0.05m_Planck.§.§ Fits for the relation between M and Λ In this section we provide fits to our results for practical use in data analysis studies, focusing on the regime that is the most relevant region of the parameter space for BH and NS mimickers.For massive BS, it is convenient to express the fit in terms of the variable w = 1/1 + λ̃/ 8 ,which provides an estimate of the maximum mass in the weak-coupling limit M̃_max≈ 2 /( π√(w)) <cit.> and has a compact range 0 ≤ w ≤ 1. A fit for massive BSs that is accurate[The accuracy quoted here corresponds to the prediction for the mass at fixed Λ and coupling constant. The error in Λ at a fixed mass can be much larger, because Λ has a large gradient when varying the mass, which even diverges at the maximum mass. The applicability of our fits must be judged by the accuracy with which the masses can be measured from a GW signal.] to ∼ 1% for Λ≤ 10^5 and up to the maximum mass is given by√(w)M̃ = [ -0.529 + 22.39/logΛ - 143.5/(logΛ)^2 + 305.6/(logΛ)^3] w + [ -0.828 + 20.99/logΛ - 99.1/(logΛ)^2 + 149.7/(logΛ)^3] (1-w) .The maximum mass where the BSs become unstable can be obtained from the extremum of this fit, which also determines the lower bound for Λ.In the solitonic case, a global fit for the tidal deformability for all possible values of σ_0 is difficult to obtain due to qualitative differences between the weak- and strong-coupling regimes. However, small values of σ_0 are most interesting, since they allow for the widest range for the tidal deformability and compactness. A fit for σ_0 = 0.05 m_Planck accurate to better than 1% and valid for Λ≤ 10^4 (and again up to the maximum mass) readslog (σ_0 M̃) = -30.834 + 1079.8/logΛ + 19 - 10240/(logΛ + 19)^2 .This fit is expected to be accurate for 0 ≤σ_0 ≲ 0.05m_Planck, i.e., including the strong coupling limit σ_0 = 0, within a few percent. Notice that this fit remains valid through tidal deformabilities of the same magnitude as that of NSs. § PROSPECTIVE CONSTRAINTS §.§ Estimating the precision of tidal deformability measurementsGravitational-wave detectors will be able to probe the structure of compact objects through their tidal interactions in binary systems, in addition to effects seen in the merger and ringdown phases. In this section, we discuss the possibility of distinguishing BSs from NSs and BHs using only tidal effects. We emphasize that our results in this section are based on several approximations and should be viewed only as estimates that provide lower bounds on the errors and can be used to identify promising scenarios for future studies with Bayesian data analysis and improved waveform models.The parameter estimation method based on the Fisher information matrix is discussed in detail in Ref. <cit.>. This approximation yields only a lower bound on the errors that would be obtained from a Bayesian analysis. We assume that a detection criterion for a GW signal h(t;θ) has been met, where θ are the parameters characterizing the signal: the distance D to the source, time of merger t_c, five positional angles on the sky, plane of the orbit, orbital phase at some given time ϕ_c, as well as a set of intrinsic parameters such as orbital eccentricity, masses, spins, and tidal parameters of the bodies. Given the detector output s=h(t)+n, where n is the noise, the probability p(θ|s) that the signal is characterized by the parameters θ is p(θ|s)∝ p^(0) e^-1/2 (h(θ)-s|h(θ)-s), where p^(0) represents a priori knowledge. Here, the inner product (·|·) is determined by the statistical properties of the noise and is given by(h_1|h_2)=2∫_0^∞h̃_1^*(f)h̃_2(f)+h̃_2^*(f)h̃_1(f)/S_n(f)df, where S_n(f) is the spectral density describing the Gaussian part of the detector noise. For a measurement, one determines the set of best-fit parameters θ̂ that maximize the probability distribution function (<ref>). In the regime of large signal-to-noise ratio SNR=√((h|h)), for a given incident GW in different realizations of the noise, the probability distribution p(θ|s) is approximately given by p(θ|s)∝ p^(0) e^-1/2Γ_ijΔθ^i Δθ^j, whereΓ_ij=(∂ h/∂θ^i|∂ h/∂θ^j) , is the so-called Fisher information matrix. For a uniform prior p^(0), the distribution (<ref>) is a multivariate Gaussian with covariance matrixΣ^ij=(Γ^-1)^ij and the root-mean-square measurement errors in θ^i are given by√(⟨ (Δθ^i)^2⟩)=√((Γ^-1)^ii), where angular brackets denote an average over the probability distribution function (<ref>).We next discuss the model h̃(f, θ) for the signal. For a binary inspiral, the Fourier transform of the dominant mode of the signal has the form h̃(f, θ)= A(f,θ) e^iψ(f,θ).Using a PN expansion and the stationary-phase approximation (SPA), the phase ψ is computed from the energy balance argument by solvingd^2ψ/dΩ^2=2/dΩ/dt=2 (dE/dΩ)/Ė_ GW, where E is the energy of the binary system, Ė_ GW is the energy flux in GWs, and Ω=π f is the orbital frequency. The result is of the formψ= 3/128(π Mf)^5/2[1+α_ 1PN(ν) x+… +(α_ tidal^ Newt+α_ 5PN(ν)) x^5+ O(x^6)],with x=(π M f)^2/3, M=m_1+m_2, ν=m_1m_2/M^2, M=ν^3/5M, and the dominant tidal contribution isα_ tidal^ Newt=-39/2Λ̃.Here, Λ̃ is the weighted average of the individual tidal deformabilities, given byΛ̃(m_1,m_2,Λ_1,Λ_2)=16/13[(1+12m_2/m_1)m_1^5/M^5Λ_1+(1↔ 2)].The phasing in Eq. (<ref>) is known as the “TaylorF2 approximant.”Specifically, we use here the 3.5PN point-particle terms <cit.> and the 1PN tidal terms <cit.>. At 1PN order, a second combination of tidal deformability parameters enters into the phasing in addition to Λ̃. This additional parameter vanishes for equal-mass binaries and will be difficult to measure with Advanced LIGO <cit.>. For simplicity, we omit this term from our analysis.The tidal correction terms in Eq. (<ref>) enter with a high power of the frequency, indicating that most of the information on these effects comes from the late inspiral. This is also the regime where the PN approximation for the point-mass dynamics becomes inaccurate. To estimate the size of the systematic errors introduced by using the TaylorF2 waveform model in our analysis, we compare the model against predictions from a tidal EOB (TEOB) model. The accuracy of the TEOB waveform model has been verified for comparable-mass binaries through comparison with NR simulations; see, for example, Ref. <cit.>. For our comparison, we use the same TEOB model as in Ref. <cit.>. The point-mass part of this model—known as “SEOBNRv2”—has been calibrated with binary black hole (BBH) results from NR simulations. The added tidal effects are adiabatic quadrupolar tides including tidal terms at relative 2PN order in the EOB Hamiltonian and 1PN order in the fluxes and waveform amplitudes. The SPA phase for the TEOB model is computed by solving the EOB evolution equations to obtain Ω(t), numerically inverting this result for t(Ω), and solving Eq. (<ref>) to arrive at ψ(Ω).Figure <ref> shows the difference in predicted phase from the TEOB model and the TaylorF2 model (<ref>) for two nearly equal mass binary NS (BNS) systems. For our analysis, we consider two representative equations of state (EoS) for NSs: the relatively soft SLy model <cit.> and the stiff MS1b EoS <cit.>. Figure <ref> illustrates that the dephasing between the TaylorF2 and TEOB waveforms remains small compared to the size of tidal effects, which is on the order of ≳ 20 rad for MS1b (1.4+1.4)M_⊙. Thus, we conclude that the TaylorF2 approximant is sufficiently accurate for our purposes and leave an investigation of the measurability of tidal parameters with more sophisticated waveform models for future work.Besides the waveform model, the computation of the Fisher matrix also requires a model of the detector noise. We consider here the Advanced LIGO Zero-Detuned High Power configuration <cit.>. To assess the prospects for measurements with third-generation detectors we also use the ET-D <cit.> and Cosmic Explorer <cit.> noise curves.To compute the measurement errors we specialize to the restricted set of signal parameters θ={ϕ_c,t_c, M, ν, Λ̃}. The extrinsic parameters of the signal such as orientation on the sky enter only into the waveform's amplitude and can be treated separately; they are irrelevant for our purposes. Spin parameters are omitted because the TaylorF2 approximant inadequately captures these effects and one would instead need to use a more sophisticated model such as SEOBNR. We restrict our analysis to systems with low masses M≲ 12M_⊙ <cit.> for which the merger occurs at frequencies f_ merger>900Hz so that the information is dominated by the inspiral signal. The termination conditions for the inspiral signal employed in our analysis are the predicted merger frequencies from NR simulations: for BNSs the formula from Ref. <cit.>, and for BBH that from Ref. <cit.>. From the tidal parameter Λ̃, we can obtain bounds on the individual tidal deformabilities. We adopt the convention that m_1≥ m_2. For any realistic, stable self-gravitating body, we expect an increase in mass to also increase the body's compactness. Because the tidal deformability scales as Λ∝ 1/C^5, we assume that Λ_1≤Λ_2. At fixed values of m_1,m_2, and Λ̃, the deformability of the more massive object Λ_1 takes its maximal value when it is exactly equal to Λ_2, i.e. when Λ̃=Λ̃(m_1,m_2,Λ_1,Λ_1). Conversely, Λ_2 takes its maximal value when Λ_1 vanishes exactly so that Λ̃=Λ̃(m_1,m_2, 0, Λ_2). Substituting the expression for Λ̃ from Eq. (<ref>) and using that m_1,2=M(1±√(1-4ν))/2 leads to the following bounds on the individual deformabilities Λ_1≤ g_1(ν)Λ̃, Λ_2≤ g_2(ν)Λ̃,where the functions g_i are given by g_1(ν)≡ 13/16(1+7ν-31 ν^2),g_2(ν)≡ 13/8[1+7ν-31 ν^2-√(1-4ν)(1+9ν-11ν^2)].Thus, the expected measurement precision of ν and Λ̃ provide an estimate of the precision with which Λ_1 and Λ_2 can be measured through ΔΛ_1 ≤[(g_1(ν) ΔΛ̃)^2+(g_1'(ν) Λ̃Δν)^2]^1/2, ΔΛ_2 ≤[(g_2(ν) ΔΛ̃)^2+(g_2'(ν) Λ̃Δν)^2]^1/2, For simplicity, we have assumed in Eq. (<ref>) that the statistical uncertainty in ν and Λ̃ is uncorrelated. Note that for BBH signals, this assumption is unnecessary because Λ̃=0, and thus the second terms in Eqs. (<ref>) and (<ref>) vanish.In the following subsections, we outline two tests to distinguish conventional GW sources from BSs and discuss the prospects of successfully differentiating the two with current- and third-generation detectors. First, we investigate whether one could accurately identify each body in a binary as a BH/NS rather than a BS. This test is only applicable to objects whose tidal deformability is significantly smaller than that of a BS, e.g., BHs and very massive NSs. For bodies whose tidal deformabilities are comparable to that of BSs, we introduce a novel analysis designed to test the slightly weaker hypothesis: can the binary system of BHs or NSs be distinguished from a binary BS (BBS) system? For both tests, we will assume that the true waveforms we observe are produced by BBH or BNS systems and then assess whether the resulting measurements are also consistent with the objects being BSs. In our analyses we consider only a single detector and assume that the sources are optimally oriented; to translate our results to a sky- and inclination-averaged ensemble of signals, one should divide the expected SNR by a factor of √(2) and thus multiply the errors on ΔΛ̃ by the same factor.We consider two fiducial sets of binary systems in our analysis. First, we consider BBHs at a distance of 400 Mpc (similar to the distances at which GW150914 and GW151226 were observed <cit.>) with total masses in the range 8 M_⊙≤ M ≤ 12M_⊙. This range is determined by the assumption that the lowest BH mass is 4M_⊙ and the requirement that the merger occurs at frequencies above ∼ 900Hz so the information in the signal is dominated by the inspiral. The SNRs for these systems range from approximately 20 to 49given the sensitivity of Advanced LIGO.The second set of systems that we consider are BNSs at a distance of 200Mpc and with total masses 2M_⊙≤ M≤ M_ max, where M_ max is twice the maximum NS mass for each equation of state. The lower limit on this mass range comes from astrophysical considerations on NS formation <cit.>. The BNS distance was chosen to describe approximately one out of every ten events within the expected BNS range of ∼ 300 Mpc for Advanced LIGO and translates to SNR∼ 12-22 for the SLy equation of state. §.§ Distinguishability with a single deformability measurement A key finding from Sec. <ref> is that the tidal deformability is bounded below by Λ≳ 280 for massive BSs andΛ≳ 1.3 for solitonic BSs. By comparison, the deformability of BHs vanishes exactly, i.e. Λ=0, whereas for nearly-maximal mass NSs, the deformability can be of order Λ≈(10). Thus, a BH or high-mass NS could be distinguished from a massive BS provided that a measurement error of ΔΛ≈ 200 can be reached with GW detectors. Similarly, to distinguish a BH from a solitonic BS requires a measurement precision of ΔΛ≈ 1.The results for the measurement errors with Advanced LIGO for BBH systems at 400Mpc are shown in Fig. <ref>, for a starting frequency of 10Hz. The left panel shows the error in the combination Λ̃ that is directly computed from the Fisher matrix as a function of total mass M and mass ratio q=m_1/m_2. As discussed above, the ranges of M and q we consider stem from our assumptions on the minimum BH mass and a high merger frequency. The right panel of Fig. <ref> shows the inferred bound on the less well-measured individual deformability in the regime of unequal masses. We omit the region where the objects have nearly equal masses q→ 1 because in this regime, the 68% confidence interval ν+2Δν exceeds the physical bound ν≤1/4. Inferring the errors on the parameters of the individual objects requires a more sophisticated analysis <cit.> than that considered here. The coloring ranges from small errors in the blue shaded regions to large errors in the orange shaded regions; the labeled black lines are representative contours of constant ΔΛ. Note that the errors on the individual deformability Λ_2 are always larger than those on the combination Λ̃. We find that the tidal deformability of our fiducial BBH systems can be measured to within ΔΛ≲ 100 by Advanced LIGO, which indicates that BHs can be readily distinguished from massive BSs. However, even for ideal BBHs—high mass, low mass-ratio binaries—the tidal deformability of each BH can only be measured withinΔΛ≳ 15 by Advanced LIGO. Therefore one cannot distinguish BHs from solitonic BSs using estimates of each bodies' deformability alone. Given these findings, we also estimate the precision with which the tidal deformability could be measured with third-generation instruments. Compared to Advanced LIGO, the measurement errors in the tidal deformability decrease by factors of ∼13.5 and ∼23.5 with Einstein telescope and Cosmic Explorer, respectively. Thus, the more massive BH in the binary would be marginally distinguishable from a solitonic BS with future GW detectors, as ΔΛ_1≤ΔΛ̃≲ 1. These findings are consistent with the conclusions of Cardoso et al <cit.>, although these authors considered only equal-mass binaries at distances D=100Mpc with total masses up to 50M_⊙.However, we find that in an unequal-mass BBH case, the less massive body could not be differentiated from a solitonic BS even with third-generation detectors.Next, we consider the measurements of a BNS system, shown in Fig. <ref> assuming the SLy EoS. We restrict our analysis to systems with individual masses 1M_⊙≤ m_ NS≤ m_ max, where m_ max≈ 2.05 M_⊙ is the maximum mass for this EoS. Similar to Fig. <ref>, the left panel in Fig. <ref> shows the results for the measurement error in the combination Λ̃ directly computed from the Fisher matrix, and the right panel shows the error for the larger of the individual deformabilities. The slight warpage of the contours of constant ΔΛ̃ compared to those in Fig. <ref>, best visible for the ΔΛ̃=50 contour, is due to an additional dependence of the merger frequency on Λ̃ for BNSs that is absent for BBHs, and a small difference in the Fisher matrix elements when evaluated for Λ̃≠ 0. We see that the deformability of NSs of nearly maximal mass in BNS systems can be measured to within ΔΛ≲ 200, and thus can be distinguished from massive BSs. However, the measurement precision worsens as one decreases the NS mass, rendering lighter NSs indistinguishable from massive BSs using only each bodies deformability alone. In the next subsection, we discuss how combining the measurements of Λ for each object in a binary system can improve distinguishability from BSs even when the criteria discussed above are not met.For completeness, we also computed how well third-generation detectors could measure the tidal deformabilities in BNS systems. As in the BBH case, we find that measurement errors in Λ decrease by factors of ∼13.5 and ∼ 23.5 with the Einstein Telescope and Cosmic Explorer, respectively. However, the conclusions reached above concerning the distinguishability of BHs or NSs and BSs remain unchanged. §.§ Distinguishability with a pair of deformability measurements In the previous subsection we determined that compact objects whose tidal deformability is much smaller than that of BSs could be distinguished as such with Advanced LIGO, e.g., BHs versus massive BSs. In this subsection, we present a more refined analysis to distinguish compact objects from BSs when the deformabilities of each are of approximately the same size. In particular, we focus on the prospects of distinguishing NSs between one and two solar masses from massive BSs and distinguishing BHs from solitonic BSs. Throughout this section, we only consider the possibility that a single species of BS exists in nature; differentiation between multiple, distinct complex scalar fields goes beyond the scope of this paper. We show that combining the tidal deformability measurements of each body in a binary system can break the degeneracy in the BS model associated with choosing the boson mass μ. Utilizing the mass and deformability measurements of both bodies allows one to distinguish the binary system from a BBS system.In Figs. <ref> and <ref>, the tidal deformability of BSs was given as a function of mass rescaled by the boson mass and self-interaction strength. By simultaneously adjusting these two parameters of the BS model, one can produce stars with the same (unrescaled) mass and deformability. This degeneracy presents a significant obstacle in distinguishing BSs from other compact objects with comparable deformabilities. For example, the boson mass can be tuned for any value of the coupling λ (σ_0) so that the massive (solitonic) BS model admits stars with the exact same mass and tidal deformability as a solar mass NS. However, combining two tidal deformability measurements can break this degeneracy and improve the distinguishability between BSs and BHs or NSs. As an initial investigation into this type of analysis, we pose the following question: given a measurement (m_1,Λ_1) of a compact object in a binary, can the observation (m_2,Λ_2) of the companion exclude the possibility that both are BSs? We stress that our analysis is preliminary and that only qualitative conclusions should be drawn from it; a more thorough study goes beyond the scope of this paper. From the Fisher matrix estimates for the errors in ( M, ν, Λ̃) we obtain bounds on the uncertainty in the measurement (m_i,Λ_i) for each body in a binary, which we approximate as being characterized by a bivariate normal distribution with covariance matrix Σ=diag(Δ m_i,ΔΛ_i). Figure <ref> depicts such potential measurements by Advanced LIGO of (m_1,Λ_1) and (m_2, Λ_2), shown in black, for a (1.55 +1.35) M_⊙ BNS system at a distance of 200 Mpc with two representative equations of state for the NSs: the SLy and MS1b models discussed above. The dashed black curves in Figure <ref> show the Λ(m) relation for these fiducial NSs. Figure <ref> shows the corresponding measurements in a 6.54.5 M_⊙ BBH measured at 400 Mpc made by Advanced LIGO, Einstein Telescope, and Cosmic Explorer in blue, red, and black, respectively. The strategy to determine if the objects could be BSs is the following. Consider first the measurement (m_2, Λ_2) of the less massive body. For each point 𝐱=(m,Λ) within the 1σ ellipse, we determine the combinations of theory parameters (μ,λ)[𝐱] or (μ,σ_0)[𝐱] that could give rise to such a BS, assuming the massive or solitonic BS model, respectively. As discussed above, in general, λ or σ_0 can take any value by appropriately rescaling μ. Finally, we combine all mass-deformability curves from Figs. <ref> or <ref> that pass through the 1σ ellipise, that is we consider the model parameters (μ,λ) ∈⋃_𝐱 (μ,λ)[𝐱] or (μ,σ_0) ∈⋃_𝐱 (μ,σ_0)[𝐱] for massive and solitonic BS, respectively. These portions of BS parameter space are shown as the shaded regions in Figs. <ref> and <ref>. If the tidal deformability measurements (m_1, Λ_1) of the more massive body—indicated by the other set of crosses—lie outside of these shaded regions, one can conclude that the measurements are inconsistent with both objects being BSs. Figure <ref> demonstrates that an asymmetric BNS with masses 1.551.35M_⊙ can be distinguished from a BBS with Advanced LIGO by using this type of analysis. When considered individually, either NS measurement shown here would be consistent with a possible massive BS; by combining these measurements we improve our ability to differentiate the binary systems. This type of test can better distinguish BBSs from conventional GW sources than the analysis performed in the previous section because it utilizes measurements of both the mass and tidal deformability rather than just using the deformability alone. However the power of this type of test hinges on the asymmetric mass ratio in the system; with an equal-mass system, this procedure provides no more information than that described in Section <ref>.A similar comparison between a BBH with masses 6.54.5M_⊙ and a binary solitonic BS system is illustrated in Fig. <ref>. For simplicity, the yellow shaded region depicts all possible solitonic BSs for a particular choice of coupling σ_0=0.05 m_ Planck that are consistent with the measurement of the smaller mass by Advanced LIGO (rather than all possible values of the coupling σ_0). We see that in contrast to the massive BS case, after fixing the boson mass μ with the measurement of one body, the measurement of the companion remains within that shaded region. As with the more simplistic analysis performed inSection <ref>, we again find that Advanced LIGO will be unable to distinguish solitonic BSs from BHs. In the previous section, we showed that third-generation GW detectors will be able to distinguish marginally at least one object in a BBH system from a solitonic BS and thus determine whether a GW signal was generated by a BBS system. Using the analysis introduced in this section, we can now strengthen this conclusion. We repeat the procedure described above for a 6.54.5M_⊙ BBH at 400 Mpc but instead use the 3σ error estimates in the measurements of the bodies' mass and tidal deformability. In Fig. <ref>, all possible solitonic BSs consistent with the measurement of the smaller mass are shown in green and pink for Einstein Telescope and Cosmic Explorer, respectively. We see that while the deformability measurements of each BH considered individually are consistent with either being solitonic BSs, they cannot both be BSs. Thus, we can conclude with much greater confidence that third-generation detectors will be able to distinguish BBH systems from binary systems of solitonic BSs. To summarize, the precision expected from Advanced LIGO is potentially sufficient to differentiate between massive BSs and NSs or BHs, particularly in systems with larger mass asymmetry. Advanced LIGO is not sensitive enough to discriminate between solitonic BSs and BHs, but next-generation detectors like the Einstein Telescope or Cosmic Explorer should be able to distinguish between BBS and BBH systems. However, we emphasize again that our conclusions are based on several approximations and further studies are needed to make these precise. We also note that we have deliberately restricted our analysis to the parameter space where waveforms are inspiral-dominated in Advanced LIGO. Tighter constraints on BS parameters are expected for binaries where information can also be extracted from the merger and ringdown portion, provided that waveform models that include this regime are available.§ CONCLUSIONS Gravitational waves can be used to test whether the nature of BHs and NSs is consistent with GR and to search for exotic compact objects outside of the standard astrophysical catalog.A compact object's structure is imprinted in the GW signal produced by its coalescence with a companion in a binary system. A key target for such tests is the characteristic ringdown signal of the final remnant. However, the small SNR of that part of the GW signal complicates such efforts. Complementary information can be obtained by measuring a small but cumulative signature due to tidal effects in the inspiral that depend on the compact object's structure through its tidal deformability. This quantity may be measurable from the late inspiral and could be used to distinguish BHs or NSs from exotic compact objects. In this paper, we computed the tidal deformability Λ for two models of BSs: massive BSs, characterized by a quartic self-interaction, and solitonic BSs, whose scalar self-interaction is designed to produce very compact objects. For the quartic interaction, our results span the entire two-dimensional parameter space of such a model in terms of the mass of the boson and the coupling constant in the potential. For the solitonic case, our results span the portion of interest for BH mimickers. We presented fits to our results for both cases that can be used in future data analysis studies. We find that the deformability of massive BSs is markedly larger than that of BHs and very massive NSs; in particular, we showed that the tidal deformability Λ≳ 280 irrespective of the boson mass and the strength of the quartic self-interaction. The tidal deformability of solitonic BSs is bounded below by Λ≳ 1.3. To determine whether ground-based GW detectors can distinguish NSs and BHs from BSs, we first computed a lower bound on the expected measurement errors in Λ using the Fisher matrix formalism. We considered BBH systems located at 400 Mpc and BNS systems at 200 Mpc with generic mass ratios that merge above 900 Hz. We found that, with Advanced LIGO, BBHs could be distinguished from binary systems composed of massive BSs and that BNSs could be distinguished provided that the NSs were of nearly-maximal mass or of sufficiently different masses (i.e. a high mass ratio binary). We also demonstrated that the prospects for distinguishing solitonic BSs from BHs based only on tidal effects are bleak using current-generation detectors; however, third-generation detectors will be able to discriminate between BBH and BBS systems. We presented two different analyses to determine whether an observed GW was produced by BSs: the first relied on the minimum tidal deformability being larger than that of a NS or BH, while the second combined mass and deformability measurements of each body in a binary system to break degeneracies arising from the (unknown) mass of the fundamental boson field.Recent work by Cardoso et al. <cit.> also investigated the tidal deformabilities of BSs and the prospects of distinguishing them from BHs and NSs. Despite the topic being similar, the work in this paper is complementary: Cardoso et al. <cit.> performed a broad survey of tidal effects for different classes of exotic objects and BHs inmodified theories of gravity, while our work focuses on an in-depth analysis of BSs. Additionally, these authors computed the deformability of BSs to both axial and polar tidal perturbations with l=2,3, whereas our results are restricted to the l=2 polar case. The l=2 effects are expected to leave the dominant tidal imprint in the GW signal, with the l=3 corrections being suppressed by a relative factor of 125 Λ_3/(351Λ_2)(M Ω)^4/3∼ 4 (MΩ)^4/3 <cit.> using the values from Table I of Ref. <cit.>, where Ω is the orbital frequency of the binary. For reference, MΩ∼ 5×10^-3 for a binary with M=12M_⊙ at 900Hz. We also cover several aspects that were not considered in Ref. <cit.>, where the study of BSs was limited to a single example for a particular choice of theory parameters for each potential (quartic and solitonic). Here, we analyzed the entire parameter space of self-interaction strengths for the quartic potential and the regime of interest for BH mimickers in the solitonic case. Furthermore, we developed fitting formulae for immediate use in future data analysis studies aimed at constraining the BS parameters with GW measurements. Cardoso et al. <cit.> also discussed prospective constraints obtained from the Fisher matrix formalism for a range of future detectors, including the space-based detector LISA that we did not consider here. However, their analysis was limited to equal-mass systems, to bounds on Λ̃, and to the specific examples within each BS models. We went beyond this study by delineating a strategy for obtaining constraints on the BS parameter space from a pair of measurements and considering binaries with generic mass ratio. We also restricted our results to the regime where the signals are dominated by the inspiral. Although this choice significantly reduces the parameter space of masses surveyed compared to Cardoso et al., we imposed this restriction because full waveforms that include the late inspiral, merger and ringdown are not currently available. Another difference is that we took BBH or BNS signals to be the “true" signals around which the errors were computed and used results from NR for the merger frequency to terminate the inspiral signals, whereas the authors of Ref. <cit.> chose BS signals for this purpose and terminated them at the Schwarzschild ISCO. The purpose of this paper was to compute the tidal properties of BSs that could mimic BHs and NSs for GW detectors and to estimate the prospects of discriminating between such objects with these properties. Our analysis hinged on a number of simplifying assumptions. For example, the Fisher matrix approximation that we employed only yields lower bounds on estimates of statistical uncertainty. Additionally, we considered only a restricted set of waveform parameters, whereas including spins could also worsen the expected measurement accuracy. On the other hand, improved measurement precision is expected if one uses full inspiral-merger-ringdown waveforms or if one combines results from multiple GW events. Our conclusions should be revisited using Bayesian data analysis tools and more sophisticated waveform models, such as the EOB model. Tidal effects are a robust feature for any object, meaning that the only change needed in existing tidal waveform models is to insert the appropriate value of the tidal deformability parameter for the object under consideration. However, the merger and ringdown signals are more difficult to predict, and further developments and NR simulations are needed to model them for BSs or other exotic objects.N.S. acknowledges support from NSF Grant No. PHY-1208881. We thank Ben Lackey for useful discussions. We are grateful to Vitor Cardoso and Paolo Pani for helpful comments on this manuscript. | http://arxiv.org/abs/1704.08651v1 | {
"authors": [
"Noah Sennett",
"Tanja Hinderer",
"Jan Steinhoff",
"Alessandra Buonanno",
"Serguei Ossokine"
],
"categories": [
"gr-qc"
],
"primary_category": "gr-qc",
"published": "20170427165555",
"title": "Distinguishing Boson Stars from Black Holes and Neutron Stars from Tidal Interactions in Inspiraling Binary Systems"
} |
Department of Electrical Engineering, and Ginzton Laboratory, StanfordUniversity,Stanford,California94305,USAWe develop a class of supercell photonic crystals supporting complete photonic bandgaps based on breaking spatial symmetries of the underlying primitive photonic crystal. One member of this class based on a two-dimensional honeycomb structure supports a complete bandgap for an index-contrast ratio as low as n_high/n_low = 2.1, making this the first such 2D photonic crystal to support a complete bandgap in lossless materials at visible frequencies. The complete bandgaps found in such supercell photonic crystals do not necessarily monotonically increase as the index-contrast in the system is increased, disproving a long-held conjecture of complete bandgaps in photonic crystals.Complete photonic bandgaps in supercell photonic crystals Alexander Cerjan and Shanhui Fan December 30, 2023 =========================================================Since their discovery, photonic crystals have become an indispensable technology across the entire field of optical physics due to their ability to confine and control light of an arbitrary wavelength <cit.>. This critical feature is achieved by designing the crystal lattice to possess a complete photonic bandgap, a range of frequencies for which no light can propagate regardless of its momentum or polarization. Unlike their electronic counterparts in conventional crystals, whose band structure is limited to the crystal lattices available in atomic and molecular structures, the dielectric structure comprising a photonic crystal can be specified with nearly complete arbitrariness, yielding a vast design space for optimizing photonic crystals for specific applications that is limited only by the index of refraction of available materials at the operational wavelength. For example, photonic crystals have been developed to promote absorption in monolayer materials <cit.>, or for use in achieving high-power solid-state lasers <cit.>. Moreover, this design freedom in dielectric structures has been leveraged in numerous studies to optimize the complete bandgaps in high-index materials <cit.>.Unfortunately, similar efforts to realize new crystal structures or improve upon existing ones to achieve complete bandgaps in low-index materials have yielded only minor improvements upon traditional simple crystal structures with high symmetry <cit.>, i.e. the inverse triangular lattice in two-dimensions <cit.> and the network diamond lattice in three-dimensions <cit.>. This has led many to conclude that the known high-symmetry dielectric structures are nearly optimal for achieving low-index complete bandgaps <cit.>.However, the ability to realize complete bandgaps for low-index materials is critically important to the development of many photonics technologies operating in the visible wavelength range, such as augmented and virtual reality systems, where the highest index lossless materials have n ≈ 2.4–2.5. Currently, there are no known 2D photonic crystals which display a complete bandgap in this index contrast regime, and thus it is not possible to realize dual-polarization in-plane guiding at this index contrast using photonic crystal slabs. Although a few 3D photonic crystals do display a complete bandgap in this range, 3D photonic crystals are difficult to fabricate <cit.>.In this Letter, we demonstrate a new class of complete photonic bandgaps which are achieved by judiciously breaking symmetry, rather than promoting it. By starting with a photonic crystal possessing a large bandgap for one polarization, we show that by expanding the primitive cell of the photonic crystal to form a supercell and then slightly adjusting the dielectric structure within this supercell to break part of the translational symmetry of the original primitive cell, a bandgap in the other polarization can be opened, thus producing a complete bandgap. This method yields a two-dimensional photonic crystal based on a honeycomb lattice with a complete bandgap that persists down to an index-contrast ratio of n_high/n_low = 2.1, the lowest known index-contrast ratio in 2D photonic crystals. Such low index contrast bandgaps can also be translated into photonic crystal slabs, where they represent the first structures able to confine optical frequencies in-plane regardless of their polarization. In contrast to the complete photonic bandgaps found in traditional photonic crystals, complete bandgaps in supercell photonic crystals do not necessarily monotonically increase as a function of the index-contrast ratio, disproving a long-held conjecture in the photonic crystal literature <cit.>. To illuminate how symmetry breaking can help to realize complete photonic bandgaps, we first consider the 2D photonic crystal comprised of a network structure on a honeycomb lattice depicted in Fig. <ref>(a). The primitive cell of this system contains a pair of vertices in this network lattice, and the system can be parameterized solely in terms of the thickness, t, of the lines forming the network structure. Although in a low-index network structure, n_high/n_low = 2.4, a wide range of t yields a large transverse electric (TE) bandgap as shown in Fig. <ref>(b), no complete photonic bandgap exists for any choice of t for this choice of n_high/n_low.However, starting from the crystal structure as shown in Fig. <ref>(a), we can find a complete bandgap in a closely related supercell photonic crystal. First, we increase the size of the primitive cell to contain six vertices which form the supercell, as depicted in Fig. <ref>(c). In doing so, each of the bands in the primitive Brillouin zone fold up into three bands in the supercell Brillouin zone, shown in Fig. <ref>(d).Along the edge of the supercell Brillouin zone (M' → K'), pairs of the folded supercell bands can form lines of degeneracies, i.e. degenerate contours <cit.>, and one such degenerate contour is formed per trio of folded bands originating from the same band in the primitive Brillouin zone. From the perspective of the supercell photonic crystal, the degeneracies comprising each of the degenerate contours are accidental, and are only the result of the supercell obeying an extra set of spatial symmetries as it is a three-fold copy of the original photonic crystal. Thus by breaking these symmetries, the degeneracies forming the degenerate contours are lifted, and a gap can begin to open between the two transverse magnetic (TM) bands. The supercell is now characterized in terms of three parameters, the thickness of the center lines, t_1, the thickness of the connecting lines, t_2, and the size of the thick-lined hexagons, r, shown in Fig. <ref>(e). When the symmetry breaking becomes sufficiently strong, a complete photonic bandgap opens between the 8th and 9th bands of the system, whose maximum width at n_high/n_low = 2.4 can be found numerically to be Δω / ω̅ = 8.6%. Here, Δω is the difference between the minimum of the 9th band and the maximum of the 8th band, while ω̅ is the central frequency between the two bands. Rigorously, after the symmetry of this system is broken, the supercell containing six vertices becomes the primitive cell of the system. However, for semantic convenience, we will continue to refer to this larger primitive cell as the `supercell' and reserve `primitive cell' for the smaller system whose symmetry is intact.Previously, the lowest index-contrast ratio known to support a complete bandgap in a 2D photonic crystal was a decorated honeycomb lattice, shown as the blue curve in Fig. <ref>, which has a complete bandgap between the 3rd and 4th bands for index-contrast ratios as low as n_high/n_low = 2.6 <cit.>. This structure provides a relatively modest improvement upon the traditional triangular lattice of air holes, also shown in Fig. <ref> as the pink curve. In contrast, the supercell honeycomb lattice possesses a complete bandgap for index-contrast ratios as low as n_high/n_low = 2.1, and as such is the first 2D photonic crystal design that can realize a complete bandgap for visible wavelengths where the largest index of refraction possible in lossless materials is n ≈ 2.4–2.5, which is found in Diamond <cit.>, Titanium Dioxide <cit.>, and Strontium Titanate <cit.>. Furthermore, this supercell honeycomb structure could also be used in conjunction with high-index materials available in other frequency ranges so that the low-index material used in the structure need not be air. For example, this could enable realizing complete bandgaps in completely solid photonic crystal fibers operating in the communications band, where the high index regions are Silicon, n_high = 3.48, and the low-index regions are filled with fused silica, n_low=1.45, such that n_high/n_low = 2.4 for λ = 1.55μm.This 2D supercell honeycomb structure can also be used to design new photonic crystal slabs so as to provide confinement in three dimensions. In Fig. <ref>, we show a supercell honeycomb slab with a complete below-light-line dual-polarization bandgap of Δω / ω_0 = 4.3% for n_high/n_low = 2.4. Note that in photonic crystal slabs, to define aband gap one only considers the phase space regions below the light line, as above the light line the radiations modes form a continuum with no gaps. To the best of our knowledge, this represents the first system which could confine visible frequencies emitted from a omni-polarization source burried within the system. Likewise, this design could also be used to realize entirely solid photonic crystal slabs for communications frequencies where higher index dielectric materials are available.The procedure used above is not restricted to the honeycomb lattice. To illustrate this point, we use the same method to produce a complete bandgap in a 2D supercell square network lattice, as shown in Fig. <ref>. Unlike in the primitive honeycomb crystal, the TE bandgap in the underlying primitive square lattice is spanned by two TM bands. Thus, a complete bandgap is only realized for sufficiently strong symmetry breaking so that not only does a gap open in each degenerate contour of the folded supercell TM bands, but that a gap opens between these two folded bands. This limits the overall width of the complete bandgap, and the lowest index-contrast ratio for which this structure possesses a complete bandgap is n_high/n_low∼ 3.1, as shown as the yellow line in Fig. <ref>. However, this still represents a significant improvement upon the range of index-contrast ratios which can yield a complete bandgap when compared against other 2D photonic crystals based on a square lattice.Complete bandgaps in supercell photonic crystal possess three features which distinguish them from complete bandgaps found in traditional photonic crystals. First, as noted above, these structures have been designed by specifically breaking symmetry within the system. This is entirely distinct from what is observed for bandgaps found in traditionally designed structures, which consider the high-symmetry triangular lattice in 2D and diamond lattice in 3D to be near optimal.Second, as can be seen in Fig. <ref>, complete bandgaps in supercell structures do not necessarily monotonically increase in size as a function of the index-contrast. This disproves a long-held conjecture of complete bandgaps in photonic crystals, that the optimized bandgap (between the same two bands) always increases as the index contrast increases <cit.>. Finally, the complete bandgap in supercell crystals is found between higher order bands. This is unlike many photonic crystal structures previously considered where the complete bandgap occurs between lower-order bands. Designing two-dimensional supercell photonic crystals to possess complete bandgaps has three steps. First, a candidate primitive photonic crystal must be constructed which possesses a large bandgap for one polarization, and which is spanned by at most one or two bands in the other polarization. Second, a supercell must be generated from this primitive cell such that the degenerate contours of the folded band spanning the single-polarization bandgap lie entirely within the single-polarization bandgap. Finally, the supercell perturbation which breaks the underlying primitive cell symmetries must be designed, such that a bandgap in the degenerate contour opens before the single-polarization bandgap in the original primitive system closes.We expect these same design principles to hold for finding complete bandgaps in three-dimensional supercell photonic crystals, but in practice we have been unable to find such a structure. Although the second and third steps in the above procedure are relatively straightforward, finding good candidate primitive cell structures is much more challenging in 3D, as it is rare to find what would be a large bandgap spanned by only a single other band. For comparison, this is relatively easy in 2D, structures with isolated dielectric elements typically possess large TM bandgaps, but not TE bandgaps, while network structures typically possess large TE bandgaps, but no TM bandgaps. In conclusion, we have developed a new class of photonic crystals which support complete bandgaps which stem from breaking spatial symmetries. These structures can exhibit complete bandgaps for much lower index-contrast ratios than was previously known, enabling the confinement of visible light in two-dimensional structures. The discovery of this new class of supercell structures also provides encouragement that there may be significant improvements remaining to be discovered in designing and optimizing complete bandgaps at low index-contrasts in both two- and three-dimensional systems. 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Misdirected Registration Uncertainty ^1Graduate School of Frontier Sciences, The University of Tokyo, Japan^2Radiology Department, Brigham and Women's Hospital, Harvard Medical School, USA^3School of Engineering Science, Simon Fraser University, Canada^4Computing Science Department, University of Alberta, Canada ^5Department of Computer Science, The University of Tokyo, Japan ^6 Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, USA ^7 RIKEN Center for Advanced Intelligence Project, Japan Lecture Notes in Computer Science Authors' Instructions Misdirected Registration Uncertainty Jie Luo^1,2, Karteek Popuri^3,Dana Cobzas^4, Hongyi Ding^5 William M. Wells III^2,6 and Masashi Sugiyama^7,1====================================================================================================================Being a task of establishing spatial correspondences, medical image registration is often formalized as finding the optimal transformation that best aligns two images. Since the transformation is such an essential component of registration, most existing researches conventionally quantify the registration uncertainty, which is the confidence in the estimated spatial correspondences, by the transformation uncertainty. In this paper, we give concrete examples and reveal that using the transformation uncertainty to quantify the registration uncertainty is inappropriate and sometimes misleading. Based on this finding, we also raise attention to an important yet subtle aspect of probabilistic image registration, that is whether it is reasonable to determine the correspondence of a registered voxel solely by the mode of its transformation distribution.§ INTRODUCTION Medical image registration is a process of establishing anatomical or functional correspondences between images. It is often formalized as finding the optimal transformation that best aligns two images <cit.>. Since many important clinical decisions or analysis are based on registered images, it would be useful to quantify the intrinsic uncertainty, which is a measure of confidence in solutions, when interpreting the image registration results.Among all methods that characterize the uncertainty of non-rigid image registration, the most mainstream, or perhaps the most successful framework is probabilistic image registration (PIR) <cit.>. Unlike point-estimate registration methods that report a unique set of transformation parameters, PIR models the transformation parameters as a random variable and estimates a distribution over them. PIR methods can be broadly categorized into discrete probabilistic registration (DPR) and continuous probabilistic registration (CPR). The transformation distribution estimated by DPR and CPR have different forms. DPR discretizes the transformation space into a set of displacement vectors. Then it uses discrete optimization techniques to compute a categorical distribution as the transformation distribution <cit.>. CPR is essentially a Bayesian registration framework, with the estimated transformation given by a multivariate continuous posterior distribution <cit.>. A remarkable advantage of PIR is that its registration uncertainty can be naturally obtained from the distribution of transformation parameters, and further utilized to benefit the subsequent clinical tasks<cit.>.§.§.§ Related WorkImage registration refers to the process of finding spatial correspondences, hence the uncertainty of registration should be a measure of the confidence in spatial correspondences. However, since the transformation is such an essential component of registration, in the PIR literature, most existing works do not differentiate the transformation uncertainty from the registration uncertainty. Indeed, the conventional way to quantify the registration uncertainty is to employ summary statistics of the transformation distribution. Applications of various summary statistics have been found in previous researches: the Shannon entropy and its variants of the categorical transformation distribution were used to measure the registration uncertainty of DPR <cit.>. Meanwhile, the variance <cit.>, standard deviation <cit.>, inter-quartile range <cit.> and covariance Frobenius norm <cit.> of the transformation distribution were used to quantify the registration uncertainty of CPR. In order to visually assess the registration uncertainty, each of these summary statistics was either mapped to a color scheme, or an object overlaid on the registered image. By inspecting the color of voxels or the object's geometry, clinicians can infer the registration uncertainty, which suggests the confidence they can place in the registered image.It is acknowledged that registration uncertainty should be factored into clinical decision making. This work mainly investigates whether those summary statistics of the transformation distribution truly give insight into the registration uncertainty. If clinicians are misdirected from the registration uncertainty to the transformation uncertainty, and hence be conveyed by the false amount of uncertainty with respect to the established correspondence, it can cause detrimental effects on their performance.In the following sections, we use concrete examples and reveal that using the transformation uncertainty to quantify the registration uncertainty is inappropriate and sometimes misleading. Based on this finding, we also raise attention to an important yet subtle aspect of PIR, that is whether it is reasonable to determine the correspondence of a registered voxel solely by the mode of its transformation distribution. § MISDIRECTED REGISTRATION UNCERTAINTY Most existing works do not differentiate the transformation uncertainty from the registration uncertainty. In this section, we give concrete examples and further point out that it is inappropriate to quantify the registration uncertainty by the transformation uncertainty. For the convenience of illustration, we use Random Walker Image Registration (RWIR) method as the PIR scheme in all examples <cit.>.§.§ The RWIR Set Up In the RWIR setting, let I_f and I_m respectively be the fixed and moving image I_f,I_m: Ω_I→ℝ,Ω_I⊂ℝ^d, d=2or 3. RWIR discretizes the transformation space into a set of K displacement vectors 𝒟 = {𝐝_k}_k=1^K, 𝐝_k∈ℝ^d. These displacement vectors radiate from voxels on I_f and point to their candidate transformation locations on I_m. The corresponding label for 𝐝_k, which can be intensity values or tissue classes at those locations, are stored inℐ={I(𝐝_𝐤)}_k=1^K. For every voxel v_i, the algorithm computes a unity-sum probabilistic vector 𝒫(v_i)={P_k(v_i)}_k=1^K as the transformation distribution. P_k(v_i) is the probability of displacement vector 𝐝_k. In a standard RWIR, the algorithm takes a displacement vector that has the highest probability in 𝒫(v_i) as the most likely transformation 𝐝_m. The corresponding label of 𝐝_m in ℐ is assigned to voxel v_i as its established correspondence. Conventionally, the uncertainty of registered v_i is quantified by the Shannon entropy of the transformation distribution 𝒫(v_i). Since RWIR takes 𝐝_m as its“point-estimate", the entropy provides a measure of how disperse the rest of displacement vectors in 𝒟 are from 𝐝_m. If other displacement vectors are all equally likely to occur as 𝐝_m, then the entropy is maximal, because it is completely uncertain which displacement vector should be chosen as the most likely transformation. When the probability of 𝐝_m is much higher than the other displacement vectors, the entropy decreases, and it is more certain that 𝐝_m is the right choice. For example, assuming 𝒫(v_l) and 𝒫(v_r) are two discrete transformation distribution for voxels v_l and v_r respectively. As shown in Fig.1, 𝒫(v_l) is uniformly distributed, and its entropy is E(𝒫(v_l))=2. 𝒫(v_r) has an obvious peak, hence its entropy is E(𝒫(v_r))≈1.36, which is lower than E(𝒫(v_l).§.§ Transformation Uncertainty and Registration Uncertainty For a registered voxel, the entropy of its transformation distribution is usually mapped to a color scheme. Clinicians can infer how uncertain the registration is by the color of that voxel. However, does the conventional uncertainty measure, which is the entropy of transformation distribution, truly reflect the uncertainty of registration? In a hypothetical RWIR example, assuming v_1 on I_f is the voxel we want to register. As shown in Fig.2(a), v_1's transformation space 𝒟 = {𝐝_k}_k=1^6 is a set of 6 displacement vectors. 𝒫(v_1)={P_k(v_1)}_k=1^6 is the computed distribution of 𝒟. The corresponding labels for displacement vectors in 𝒟 are image intensities stored in ℐ={I(𝐝_𝐤)}_k=1^6. For clarity, suppose that there are only two different intensity values in ℐ, one is 50 and the other is 200. The color of squares in Fig.2(a) indicates the appearance of that intensity value. We can observe that 𝐝_3 has the highest probability in 𝒟, hence its corresponding intensity I(𝐝_3)=50 will be assign to the registered v_1. Fig.2(b) is a bar chart illustrating the transformation distribution 𝒫(v_1). Although 𝒫(v_1) has its mode at P_3(v_1) , the whole distribution is more or less uniformly distributed. The transformation distribution's entropy E(𝒫(v_1))≈2.58 is close to the maximal. Therefore, the conventional uncertainty measure will suggest that the registration uncertainty of v_1 is high. Once clinicians knew its high amount of registration uncertainty, they would place less confidence in v_1's current appearance. The conventional way to quantify the registration uncertainty seems useful. However, its correctness is questionable. In the same v_1 RWIR example, let's take into account the intensity value I(𝐝_𝐤) associated with each 𝐝_𝐤 and form an intensity distribution. As shown in Fig.3(a), even if 𝐝_1,𝐝_2,𝐝_4 and 𝐝_6 are different displacement vectors, they correspond to the same intensity value as the most likely displacement vector 𝐝_3. As we accumulate the probability for all intensity values in ℐ, it is clear that 50 is the dominate intensity. Interestingly, despite being suggested of having high registration uncertainty by the conventional uncertainty measure, the intensity distribution in Fig.3(b) indicates that the appearance of registered v_1 is quite trustworthy. In addition, the entropy of the intensity distribution is as low as 0.63, which also differs from the high entropy value computed from the transformation distribution.This counter-intuitive example implies that high transformation uncertainty does not guarantee high registration uncertainty. In fact, the amount of transformation uncertainty can hardly guarantee any useful information about the registration uncertainty at all. More precisely, in the PIR setting, the transformation R_T is modeled as a random variable. The corresponding label R_L, consisting of intensity values or tissue classes, is a function of R_T, so it is also a random variable. Even if R_T and R_L are intuitively correlated, given different hyper parameters and priors, there is no guaranteed statistical correlation between these two random variables. Therefore, it's inappropriate to measure the statistics of R_L by the summary statistics of R_T.In practice, for many PIR approaches, the likelihood term is often based on voxel intensity differences. In case there is no strong informative prior, these approaches tend to estimate “flat" transformation distribution for voxels in homogeneous intensity regions. Transformation distributions of these voxels are usually more diverse than their intensity distributions, and therefore they are typical examples of how the conventional uncertainty measure, that is using the transformation uncertianty to quantify the registration uncertainty, tends to report false results <cit.>. In the following real data example, as shown in Fig.4(a), I_f and I_m are two brain MRI images arbitrarily chosen from the CUMC12 dataset. After performing RWIR, we obtain the registered moving image I_rm. To give more insight into the misleading defect of conventional uncertainty measures, we take a closer look at two voxels, v_c at the center of a white matter area on the zoomed I_rm, and v_e near the boundary of a ventricle. As can be seen from Fig.4(b), the transformation distribution of v_c is more uniformly distributed than that of v_e. Therefore, conventional entropy-based methods will report v_c having higher registration uncertainty than v_e. However, like the hypothetical example in Fig.3, we take into account the corresponding intensities and form a new intensity distribution. Since the intensity distribution is no longer categorical, we can employ other summary statistics, such as the variance, to measure the uncertainty. It turns out that the registered v_e has larger intensity variance than v_c, which again reveals that the conventional uncertainty measure is misleading.§ IMPORTANT YET SUBTLE ISSUES IN PIR Point-estimate registration methods output a unique transformation, and establish the correspondence I_rm by assigning the corresponding label of its transformation to each voxel on I_f. PIR methods output a transformation distribution, yet they still seek to establish a “point-estimate" correspondence. Since the transformation mode is the most likely transformation, the common standard for PIR to establish the correspondence I_rm is assigning the corresponding label of its transformation mode to each voxel on I_f. However, is it reasonable to determine the correspondence solely by the transformation mode?In another hypothetical example, assuming v_2 on I_f is the voxel we want to register. As shown in Fig.5(a), the transformation 𝒟 = {𝐝_k}_k=1^4 is a set of 4 displacement vectors. 𝒫(v_2)={P_k(v_2)}_k=1^4 is the estimated distribution of 𝒟. The corresponding intensity labels of all displacement vectors in 𝒟 are stored in ℐ={I(𝐝_𝐤)}_k=1^4. In RWIR, the transformation mode 𝐝_𝐦 is the displacement vector with the highest probability. Therefore, 𝐝_3 is the transformation mode, and I(𝐝_𝐦)=I(𝐝_3) will be assigned to the registered v_2. The probability of 𝐝_3 is considerably higher than that of other displacement vectors. Based on the relatively low entropy of the transformation distribution 𝒫(v_2), the intensity of registered v_2 should be trustworthy. However, once again we take into account the intensity value I(𝐝_𝐤) associated with each 𝐝_𝐤, and form an intensity distribution. Surprisingly enough, Fig.5(c) shows that the corresponding intensity of the transformation mode I(𝐝_𝐦)=50 is no longer the most likely intensity. Displacement vectors 𝐝_1,𝐝_2 and 𝐝_4 are all less likely transformations, yet their combined corresponding intensities outweigh I(𝐝_3).The above example implies that the corresponding label of the transformation mode can differ from the most likely correspondence that is given by the full transformation distribution. This example makes sense because in the previous section we have pointed out that, in PIR, the transformation R_T and correspondence R_L are both regarded as random variables. Since there is no guaranteed statistical correlation between R_T and R_L, the mode of R_T's distribution is not guaranteed to be the mode of R_L's distribution.As illustrated in Fig.6(a), we generate another example that register a MRI image I_f, which is arbitrarily chosen from the BRATS dataset, with synthetically distorted itself using RWIR. In this example, we investigate intensity distributions of four registered voxels v_b,v_c,v_d and v_e, which are shown in Fig.6(b),(c),(d),(e) respectively. In Fig.6, the red circle indicates the Most Likely Intensity (MLI) given by the full transformation distribution, the orange circle indicates the corresponding intensity of the transformation mode I(𝐝_𝐦), and the green circle is the Ground Truth (GT) intensity. We can observe that for v_b, the MLI and I(𝐝_𝐦) are both equal to the GT. On the other hand, for v_c, v_d and v_e, their MLIs are indeed not equal to their I(𝐝_𝐦). This experiment does support our point of view that the corresponding label of the transformation mode I(𝐝_𝐦) is not guaranteed to be the most likely label given by the full transformation distribution. However, at this stage, we can not conclude which one is better with respect to the registration accuracy for PIR.As we conduct more experiments, we come across another interesting finding. As can be seen in Fig.6(c), the MLI of registered v_c is equal to the GT intensity and more accurate than I(𝐝_𝐦). Yet for v_d and v_e, unexpectedly, it is their I(𝐝_𝐦) more closer to the GT than their MLI. Voxels like v_d and v_e can be found very frequently in our experiments using other real data. This surprising result indicates that utilizing the full transformation distribution can actually give worse estimation than using the transformation mode alone. Some existing researches have reported that it was beneficial to utilize the registration uncertainty, which is information obtained from the full transformation distribution, in some PIR-based tasks <cit.>. However, the above finding make us wonder whether utilizing the full transformation distribution could always improve the performance. It is noteworthy that the above finding is based on RWIR. In PIR, the correlation between the transformation R_T and the correspondence R_L is influenced by the choice of hyper parameters and priors. Other PIR approaches that use different transformation, regularization and optimization models, hence having different hyper parameters and priors, can certainly yield different findings than RWIR. However, we still suggest that researchers should analyze and investigate the credibility of the full transformation distribution before using it. § SUMMARY Previous studies don't differentiate the transformation uncertainty from the registration uncertainty. In this paper, we point out that, in PIR the transformation R_T and the correspondence R_L are both random variables, so it is inappropriate to quantify the uncertainty of R_L by the summary statistics of R_T. We have also raised attention to an important yet subtle aspect of PIR, that is whether it is reasonable to determine the correspondence of a registered voxel solely by the mode of its transformation distribution. We reveal that the corresponding label of the transformation mode is not guaranteed to be the most likely correspondence given by the full transformation distribution. Finally, we share our concerns with respect to another intriguing finding, that is utilizing the full transformation distribution can actually give worse estimation.Findings presented in this paper are significant for the development of PIR. We feel it is necessary to share our findings to the registration community. 4Sotiras Sotiras, A., Davatzikos, C., Paragios, N.: Deformable Medical Image Registration: A Survey. IEEE. TMI. 32(7), 1153–1190 (2013)Cobzas Cobzas, D., Sen, A.: Random Walks for Deformable Registration. In: MICCAI. LNCS, vol. 6892, pp. 557–565. Springer, Toronto (2011)Simpson Simpson, I.J.A., Schnabel, J.A., Norton, I., Groves, A.R., Andersson, J.L.R., Woolrich, M.W. : Probabilistic Inference of Regularisation in Non-rigid Registration. NeuroImage. 59, 2438-2451 (2012)Risholm Risholm, P., Janoos, F., Norton, I., Golby, A.J., Wells III, W.M.: Bayesian Characterization of Uncertainty in Intra-subject Non-rigid Registration. Med. Image Anal. 17(5), 538-555 (2013)Lotfi Lotfi, P., Tang, L., Andrews, S., Hamarneh, G.: Improving Probabilistic Image Registration via Reinforcement Learning and Uncertainty Evaluation. In: MLMI. LNCS, vol. 8184, pp. 187–194. Springer, Nagoya (2013)Popuri Popuri, K., Cobzas, D., Jagersand, M.: A Variational Formulation for Discrete Registration. In: MICCAI. LNCS, vol. 8151, pp. 187–194. Springer, Nagoya (2013) Wassermann Wasserman, D., Toews, M., Niethammer, M, Wells III, W.M.: Probabilistic Diffeomorphic Registration: Representing Uncertainty. In: WBIR. LNCS, vol. 8545, pp. 72–82. Springer, London (2014)Simpson2 Simpson, I.J.A., Cardoso, M.J., Norton, I., Modat, M., Woolrich, M.W., Andersson, J.L.R, Schnabel, J.A., Ourselin, S.: Probabilistic Non-linear Registration with Spatially Adaptive Regularisation. Med. Image Anal. 26, 203-216 (2015) Heinrich Heinrich, M.P., Simpson, I.J.A., Papiez, B.W., Brady, M.: Deformable Image Registration by Combining Uncertainty Estimates From Supervoxel Belief Propagation. Med. Image Anal. 27, 57-71 (2016)Risholm2 Risholm, P., Balter, J., Wells III, W.M.: Estimation of Delivered Dose in Radiotherapy: The influence of Registration Uncertainty. In: MICCAI. LNCS, vol. 6891, pp. 548–555. Springer, Toronto (2011) Simpson3 Simpson, J.A., Woolrich, M.W., Andersson, J.R., Groves, A.R., Schnabel, J.A.: Ensemble Learning Incorporating Uncertain Registration. IEEE. TMI. 32(4), 748–756 (2013) | http://arxiv.org/abs/1704.08121v2 | {
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Institut de Ciències de l'Espai (IEEC-CSIC), C/Can Magrans,s/n, Campus UAB, 08193 Bellaterra, [email protected] of Physics, University of California, Davis, One Shields Avenue, Davis, CA95616, USADepartment of Physics & Astronomy, Louisiana State University, Baton Rouge, LA 70803, USALaboratoire AIM Paris-Saclay, CEA/Irfu Université Paris-Diderot CNRS/INSU, 91191Gif-sur-Yvette, FranceThe discovery of Proxima b, a terrestrial temperate planet, presents the opportunityof studying a potentially habitable world in optimal conditions. A key aspect to modelits habitability is to understand the radiation environment of the planet in thefull spectral domain. We characterize the X-rays to mid-IR radiative properties of Proxima withthe goal of providing the top-of-atmosphere fluxes on the planet. We also aim atconstraining the fundamental properties of the star, namely its mass, radius, effectivetemperature and luminosity. We employ observations from a large number of facilities and make use of different methodologies to piece together the full spectral energy distribution of Proxima. In thehigh-energy domain, we pay particular attention to the contribution by rotationalmodulation, activity cycle, and flares so that the data provided are representative ofthe overall radiation dose received by the atmosphere of the planet. We present the full spectrum of Proxima covering 0.7 to 30000 nm. The integration ofthe data shows that the top-of-atmosphere average XUV irradiance on Proxima b is0.293 W m^-2, i.e., nearly 60 times higher than Earth, and that the totalirradiance is 877±44 W m^-2, or64±3% of the solar constant but with a significantly redder spectrum.We also provide laws for the XUV evolution of Proxima corresponding to two scenarios, one with aconstant XUV-to-bolometric luminosity value throughout its history and another one in which Proxima left the saturation phase at an age of about 1.6 Gyr and is now in a power-law regime. Regarding the fundamental properties of Proxima, we find M=0.120±0.003 M_⊙,R=0.146±0.007 R_⊙, T_ eff=2980±80 K, and L=0.00151±0.00008 L_⊙.In addition, our analysis reveals a ∼20% excess in the 3–30 μm flux of the starthat is best interpreted as arising from warm dust in the system.The data provided here should be useful to further investigate the current atmospheric properties of Proxima b as well as its past history, with the overall aim of firmly establishing the habitability of the planet. The full spectral radiative properties of Proxima Centauri Ignasi Ribas1 Michael D. Gregg2 Tabetha S. Boyajian3 Emeline Bolmont4Received; accepted ================================================================================================================== § INTRODUCTION The discovery of a terrestrial planet candidate around the nearest star to the Sun, Proxima Centauri (hereafter Proxima), was reported by <cit.> and has opened the door to investigating the properties of a potentially habitable planetfrom nearest possible vantage point. The detailed studies of <cit.> and<cit.> show that Proxima b is likely to have undergone substantial loss ofvolatiles, including water, in particular during the first ∼100–200 Myr, whenit could have been in a runaway phase prior to entering the habitable zone. Volatileloss processes once inside the habitable zone could have also been at work. Thecalculations are highly uncertain <cit.> and reasonable doubt existsas to whether the modelling schemes currently used are adequate. There are numerousexamples in the Solar System that would contradict the hypothesis of substantialvolatile losses in the early stages of its evolution in spite of the Sun being astrong source of high-energy radiation <cit.>. The studies carried out thus farconclude that Proxima b is a viable habitable planet candidate because the presenceof surface liquid water cannot be ruled out, as the initial amount of water is uncertain and the efficiency of volatile loss processes is poorly known.A key ingredient for understanding the evolution and current state of the atmosphere of Proxima b is a proper description of the high-energy irradiation. Today, theflux that Proxima b receives in the XUV domain (X-rays to EUV) is stronger than thatreceived by the Earth by over an order of magnitude and the level of irradiation wasprobably even stronger in the past. The situation is likely to be quite different inthe UV range as Proxima has a significantly lower photospheric temperature than theSun and therefore a redder emission distribution. UV irradiation has an impact onphotolysis processes, as photoabsorption cross sections of abundant molecules peak inthe 100–300 nm range <cit.>, and is also of biological interest <cit.>.Therefore, the high-energy budget from the X-rays to the UV is important for many aspectsrelated to the study of Proxima b, including understanding its atmospheric physicalproperties, its photochemistry, and even to the first attempts to constrain a putativebiosphere on its surface. The optical and IR irradiation, on the other hand, is the main contributor to the overall energy budget, thus determining the surface temperature ofthe planet and, ultimately, its habitability.<cit.> obtained a rough XUV spectrum of Proxima and also discussed possible XUV evolution laws. Here we generalize this study by providing better estimates of theradiation environment of Proxima b and extending the analysis to the full spectral domain (X-rays to mid-IR). In Sect. <ref> we combine observations over a wide wavelengthrange to deduce the spectral energy distribution (SED) of Proxima that is representative ofthe average radiation dose. As a consequence of this analysis, we identify a conspicuousIR excess, possibly due to dust in the Proxima system, which is discussed in Sect.<ref>. Also important to understand the climate of Proxima b is a gooddetermination of the basic physical properties of its stellar host, namely its mass,radius, effective temperature and bolometric luminosity. In Sect. <ref> we useall available observational constraints to provide the best estimate of such properties.In Sect. <ref> we address the issue of the XUV evolution law and propose tworelationships that take into account the pre-main sequence evolution of Proxima. Wealso perform a new calculation of water loss during the early stages of the evolutionof Proxima b and we compare the results with our earlier estimates in <cit.>.Finally, the conclusions of our work are given in Sect. <ref>.§ SPECTRAL ENERGY DISTRIBUTIONThe aim of the study is to provide the full energy distribution at the top of theatmosphere of Proxima b by characterizing the electromagnetic spectrum of the hoststar as accurately as possible. This necessarily implies making use of a number ofdifferent facilities and also employing theoretical estimates for those wavelengthintervals that do not have observations. Some of the datasets that we consider werealready discussed in <cit.> and we just give a short description and additionalrelevant details. We have also improved the methodology in the case of the FUV range (Sect. <ref>) and this leads to a total integrated XUV flux value that differsby a few per cent from that presented by <cit.>. A summary of the wavelength intervals considered and the datasets used is provided in Table <ref>, and the full details are discussed in the sections below.One of the complications associated with the determination of the flux emitted by Proxima is the effect of stellar flares. Flare events can significantly increase the flux with a relative contribution that is stronger at shorter wavelengths. In the present study weestimate the mean XUV flux over a relatively extended timescale in an attemptto measure the overall dose on the planetary atmosphere, including the flare contribution.Our strategy is, thus, to consider long integration times to ensure proper averaging of theflare events with the quiescent flux. We apply a further correction to account for thecontribution of large (infrequent) flares, and this correction is ∼10–25%, dependingon wavelength, with larger corrections for shorter wavelengths. The basic scheme is thesame as in <cit.>, and the actual details are discussed for each wavelength intervalbelow. Of course, future detailed multiwavelength studies of Proxima flares can providemuch better constraints. §.§ X-rays: XMMIn the 0.7 to 3.8 nm range we used four XMM-Newton observations with IDs 0049350101,0551120201, 0551120301, and 0551120401. The first dataset, with a duration of 67 ks,was studied by <cit.> and contains a very strong flare with atotal energy of ≈2×10^32 erg. The other three datasets (adding toa total of 88 ks), were studied by <cit.>, and include several flares, the strongest of which has an energy of about 2×10^31 erg. As in<cit.> we adopt the total spectrum corresponding to the combined 88-ks datasetsand an additional energetic flare correction corresponding to a flux multiplicativescaling factor of 1.25. The comparison of the four individual observations with ouradopted spectrum is shown in Fig. <ref>. The observation with the strong flarehas fluxes that are 3–5 times higher than our average representative spectrum. X-ray observations of Proxima were also obtained withother facilities, namely the Chandra observatory and the Swift mission. Chandra is optimizedfor high spatial and spectral resolution, which is not relevant to the determination of theSED of Proxima, and its flux calibration has larger uncertainty than that of XMM-Newton<cit.>. As for Swift, both its sensitivity and spectral resolution are significantlybelow that of XMM-Newton. Thus, including Chandra and Swift observations in our analysiswould not contribute significantly to the quality of the derived SED but instead addcomplexity and potential for systematic errors. For these reasons, we prefer to base ourhard X-ray SED solely on XMM-Newton data.A detailed analysis of Proxima X-ray observations obtained by Swift and other facilitieswas recently published by <cit.> and they find good consistency between the differentintegrated X-ray flux measurements. Furthermore, the authors present evidence of a ∼7-yractivity cycle with an amplitude of L_ X^ max/L_ X^ min≈ 1.5and note that the XMM observations (which are the same we use) correspond to X-ray cyclemaxima (years 2001 and 2009). This implies that a correction should be made to refer themto the cycle average. We did so by adopting a multiplicative factor of 0.83 applied tothe fluxes to yield our final values. §.§ X-rays: ROSATROSAT observations were used in the wavelength range from 3.8 to 10 nm. Four suitabledatasets are available from the ROSAT archive, with IDs RP200502N00, RP200502A01,RP200502A02, and RP200502A03. Their integration times were 3.8, 7.9, 20.3, and 3.8 ks, andthe observation dates 1992.3, 1993.2, 1993.7, and 1994.2, respectively. The analysisprocedure is explained in <cit.>. We calculated the average spectrum by using theintegration time as the weight factor, and this should correspond a mean date of 1993.5,which is quite close to the midpoint of the activity cycle according to <cit.>. Comparison with the overlapping wavelength region with the XMM data indicates very goodmutual agreement. A multiplicative scaling factor of 1.25 was further applied to includethe energetic flare correction also in accordance with the procedure followed for XMM. §.§ EUV: EUVEFor the extreme-UV range, covering from 10 to 40 nm, we used the EUVE spectrumavailable from the mission archive with Data ID proxima_cen__9305211911N,corresponding to an integration time of 77 ks and observation date 1993.5. The detailsof this observation are given in <cit.> and <cit.>. We correctedthis spectrum using a multiplicative scaling factor 1.25 to account for the average fluxcontribution coming from energetic flares. No activity cycle correction was necessary because the observation is close to the actual mid point <cit.>. §.§ EUV: Lyman continuumThe interval between 40 and 92 nm (Lyman limit) cannot be observed from Earthdue to the very strong interstellar medium absorption, even for a star as nearbyas Proxima. To estimate the flux in this wavelength range we make use of thetheoretical calculations presented by <cit.>. We adopt the model corresponding to intermediate activity (1303) because it best reproduces the H Ly α flux at the stellar surface (see Sect. <ref>). We consider thewavelength intervals 40–50 nm, 50–60 nm, 60–70 nm, 70–80 nm, and 80–91 nm,and the resulting ratios of the fluxes to the integrated H Lyα flux are0.01, 0.04, 0.03, 0.05, and 0.12, respectively. In our combined spectrum we considerthese wavelength bins, yielding the appropriate integrated flux values. Note that theflux in this interval had been underestimated by about a factor of 2 in our previouscalculations in <cit.>. §.§ FUV: FUSEData from FUSE were used to obtain the flux in part of the far-UV range, from 92 to117 nm. We employed the spectrum with Data ID D1220101000 with a total integration time of 45 ks and observation date 2003.3 (another FUSE dataset exists, namelyP1860701000, but it has much shorter duration - 6 ks - and correspondingly lowersignal-to-noise ratio). All obvious spectral regions with geocoronal emission wereremoved and only the wavelength intervals with stellar features <cit.> were kept. The actual intervals are:97.4–98 nm, 99.1–101.1 nm, 103.0–103.4 nm, 103.7–103.8 nm, 110.9–113 nm. Thesewavelength ranges include most of the features from stellar origin (notably three stronglines corresponding to C iii and O vi, which account for 80% of the92–117 nm flux except for the H Lyman series) and no geocoronal contamination. Theseintervals are missing the flux from the H Lyman series from H Ly β to the HLyman limit and this needs to be considered. We calculated the ratios between the different H Lyman features using anintermediate activity model (1303) from <cit.>. The values areshown in Table <ref>. To produce a spectrum, we assumed the line profilefrom the H Ly α feature (see Sect. <ref>). For each of the H Lyman serieslines we scaled the width to match the typical width of the stellar features(C iii and O vi) and also the height of the emission to match theintegrated flux. The results that we obtain are consistent with those presented by<cit.> and <cit.> for a Sun-like star with similarscaled H Lyman α flux.<cit.> found 3 flare events in the FUSE dataset that we employed, which produce an increase of up to one order of magnitude in theinstantaneous flux. The combined effect of such flares is about 20–30% relativeto the quiescent emission, which appears to be reasonable given our X-ray estimatesbelow. Also, the observation date is close to the mid point of the activity cycle <cit.>. Thus, no further corrections were applied. §.§ FUV: HST/STIS E140MA high-quality spectrum from the StarCAT catalog <cit.> obtained withthe HST Space Telescope Imaging Spectrograph <cit.> was used tomeasure the fluxes between 117 and 170 nm(except for H Lyα). The spectrumwas produced by co-adding a number of individual observations corresponding to HSTdatasets O5EO01010, O5EO01020, O5EO01030, O5EO01040, O5EO02010, O5EO02020, andO5EO02040 and with a total integration time of 35.7 ks and observation date 2000.4. A flare analysis of these individual datasets was carried out by<cit.>, who identified a number of flare events in the strongeremission lines. These flares contribute some 25–40% of the integrated flux (Loyd,priv. comm.) and thus represent similar values to those found in X-rays. In addition,as before, the date of the observations is nearly at the mid point of the activitycycle <cit.> and no further corrections were made. The intrinsic line profileof the H Lyα feature that we adopt was calculated from the same base spectrumby <cit.>. The relative flare contribution corrected for ISM absorptionis estimated to be of ∼10% (Loyd, priv. comm.).§.§ UV to NIR: HST/STIS & HST/FOSProxima was observed with HST/STIS on 24 April, 2015 as part of the Cycle 22 incarnationof the Next Generation Spectral Library (NGSL). The specific dataset references arelisted in Table <ref>. The NGSL is an HST/STIS snapshotprogram which has compiled a spectral library of 570 representative spectraltype stars for use in spectral synthesis of galaxies and other composite stellarsystems. The spectra are obtained using the three low dispersion CCD modes ofSTIS, G230LB, G430L, and G750L, covering λλ170–1020 nm at a resolution of about 1000. For Proxima, the exposure times were 2×600 s,2×30 s, and 30 s for the three gratings.The G230LB and G430L spectra were obtained through the 02 E1 aperture,located near one edge of the CCD to reduce charge transfer losses during readout. The G750L spectrum, also observed through a 02 slit, was obtained at theregular long slit center near the middle of the CCD. This was in order to takeadvantage of the very narrow 009 slit during contemporaneous CCD fringeflat calibration exposures to improve the removal of the considerable (10–15%)fringing above 700 nm in the G750L data.To save valuable on-target time during the HST snapshots, no contemporaneouswavelength calibrations (wavecals) are carried out during NGSL observations. Instead, a generic wavelength calibration is supplied in the pipeline downloadof the data, and a linear zeropoint pixel shift is determined either from inspectionor cross-correlation of a preliminary 1D extraction of the source with a velocitytemplate spectrum.This pixel shift is then inserted into the FITS header of the 2DSTIS data; subsequent extraction of 1D spectra using the task x1d in thestsdas package of iraf takes out the first-order grating setting differencebetween the actual observation and the generic wavelength solution, typically 3–5 pixels.The 1D spectra were extracted in iraf/pyraf using the x1d task. Duringextraction, the x1d task also applies charge transfer inefficiency corrections,corrects for slit losses in the 02 slit, and applies an overall flux calibrationto units of F_λ. The G750L spectrum was defringed using the contemporaneousfringe flat obtained through the narrower slit which mimics a point source on the detectorbetter than obtaining a flat through the 02 slit.The G230LB mode of STIS suffers from contamination by scattered zero-order light fromall wavelengths to which the detector is sensitive. This is corrected for using theprocedure developed by <cit.>.Briefly, the initial combined flux calibratedspectrum is run in reverse through the G230LB sensitivity function covering allwavelengths to produce a best estimate of the G230LB counts over the entire opticalrange. From this, a pixel-by-pixel correction is calculated from a simple wavelength-dependent function dependent on the total counts in the overallcomputed spectrum.This correction is then subtracted pixel-by-pixel from the G230LBcounts spectrum to correct for the red light contamination of the UV spectrum, andthen the G230LB counts spectrum is again flux calibrated.Inspection of the modest wavelength overlap (∼200 nm) between the three lowdispersion spectra shows that the absolute calibrations of three individual gratingsagree to better than 2–3% for the observation of Proxima. The three individual grating spectra were combined into a single spectrum using the scombine and dispcortasks in iraf. The final spectrum covers 170 to 1020 nm with a constant sampling0.2 nm per pixel.§.§.§ Comparison to ground-based photometry Standard UBVRI photometry of Proxima was collected from the literature. Measurementsfrom different sources are provided in Table <ref>. In the case of the Uband, one of the measurements is very discrepant from the other two. It is possible that the value from <cit.>, which is brighter by 0.3 mag and correspondsto a single epoch, was affected by a flare. In contrast, the photometric measurements of <cit.> correspond to the average of several observations taken outside offlare activity and thus we adopt the U-band magnitude from this study. The quoteduncertainty is 0.05 mag. For the BVRI bands we adopt the photometry from <cit.>,which is the average of 24 individual measurements. The quoted uncertainty is 0.028 mag, although it is not certain to which band this value corresponds. To compare the final STIS spectrum with the ground-based photometry, we calculatedsynthetic Johnson/Cousins photometric indices by convolving the STIS spectrum withUBVRI bandpasses from <cit.>, with zero points calibrated via the STIS_008Vega spectrum from the CALSPEC Calibration database at<http://www.stsci.edu/hst/observatory/crds/calspec.html>. The calculated magnitudesare listed in Table <ref>. All bands agree with the adopted, best-reliableground-based photometry of Proxima within 3–6%. §.§.§ Comparison to HST/FOS HST observed Proxima with the Faint Object Spectrograph (FOS) on 1 July, 1996 throughthe 10 aperture for 430 s with the G570H grating (dataset Y2WY0305T) and 280 swith the G780 grating (dataset Y2WY0705T). The spectrum covers λλ450–850 nmwith a resolution of 0.09 nm and we compare the STIS and combined FOS spectra in Fig.<ref>. There is general agreement at the ∼5% level and thus the STIS spectrum agrees with the FOS data of Proxima at a similar level to thebroad-band photometry.§.§.§ Final spectrum The FOS spectrum covers a subrange of the STIS spectrum and it does so at a higherspectral resolution. One could thus consider adopting a final spectrum composed ofthree wavelength intervals: 170–460 nm (STIS), 460–840 nm (FOS), 840–1000 nm (STIS). We have calculated the comparison with broad-band photometric measurementsand this is shown in the last row of Table <ref>. As expected, thedifferences are rather minor with the all-STIS spectrum. Also, for most applicationsrequiring irradiance measurements, the increased resolution in the central part ofthe optical wavelength range is of little use. Given these considerations, theresults of the comparison between the STIS and FOS fluxes, and the interest ofpreserving homogeneity, we decided to adopt the full wavelength coverage from STIS as a fair representation of the spectral energy distribution of Proxima over thewavelength region of comparison.As occurs at high energies, Proxima is also known to experience flux variations inthe optical due to the presence of surface inhomogeneities<cit.>. Photometric monitoring of Proxima shows that thepeak-to-peak variability with respect to the mean is of the order of 5% in the Bband and 2% in the V band over timescales of months, and can be attributedto rotational modulation. This provides a viable explanation for the 3–6% difference inthe results of the comparisons between different measurements. The analysis of <cit.>using MOST satellite observations covering roughly 430 to 760 nm and taken over a timeperiod of nearly 38 days reveal frequent white-light flares. There are 5–8 measurableflares per day with a typical duration of ∼1 hour. However, the average fluxcontribution from flares in this wavelength range to the quiescent flux is only 2.6%(Davenport, priv. comm.). This relatively small effect, less than the typical uncertaintyof the absolute flux calibration, suggests that flare correction to optical (and IR)spectrophotometric and photometric observations is not necessary. The flux values forProxima that we provide should be representative of the average flux to better than 5%. §.§ IR: Model spectrumAs we have shown, spectrophotometric observations that can be calibrated to yield physical fluxes are available for most wavelength regions up to about 1 μm. Beyond this wavelength value, the measurements are in the form of broad-band magnitudes or fluxes. Weperformed a search in the literature for flux measurements of Proxima. An important source of measurements is the catalog of <cit.>, and we complemented it with subsequent references. A summary of the photometry is given in Table <ref>.In view of the uncertainties and absolute calibration of the photometric systems, we decided to adopt two independent photometric datasets, namely the 2MASS JHK photometry (although the 2MASS K band measurement has a flag indicating poor quality) and the <cit.> JHKL photometry, which has two epochs and is given in a well-calibrated standard system. No M-band photometry was used in view of the large uncertainty. The magnitudes weretransformed into physical flux units using the zero-point calibrations in <cit.>for the 2MASS system and <cit.> for the photometry from <cit.>. The fluxesare given in Table <ref>. We additionally considered the recent revision ofthe zero points for the NIR magnitudes by <cit.> but the results are very similar. Besides the photometry in the classical broad-band systems, flux measurements of Proxima coming a number of space missions also exist, namely WISE, MSX, IRAS, and Spitzer. For the WISEmission <cit.> we considered both the AllWISE and the WISE All-Sky Source catalogs.In both cases, the W1 and W2 magnitudes are saturated (17% to 25% saturated pixels),while the W3 and W4 bands are not. The agreement for the W1, W3 and W4 bands forboth catalogs is good but the W2 magnitudes are highly discrepant. The W2 magnitudefrom the WISE All-Sky Source catalog leads to an unphysical energy distribution (muchhigher flux than in all other bands). Also, the uncertainties associated to thesaturated bands of the WISE All-Sky Source catalog seem unrealistically low. We decided toadopt the AllWISE measurements and uncertainties but did not consider the saturated W1 anW2 bands in the fits. The physical fluxes for the WISE bands were calculated using zeropoints in <cit.> and are listed in Table <ref>. Proxima is included inthe MSX6C Infrared Point Source Catalog <cit.>. Measurements are only available inthe so-called A and C bands and are given in physical units. These are included inTable <ref>. Proxima was also observed by the IRAS mission in two bands andhas an entry in the IRAS catalog <cit.>, with measurements in two bands (12 μmand 25 μm). The fluxes are provided in physical units and are listed in Table<ref>. Finally, <cit.> included Proxima in their survey of the far-IR properties of M dwarfs and obtained a flux measurement in the Spitzer/MIPS 24-μm band.This is listed in Table <ref>. In addition to the bands considered above, wealso included an anchor point from the HST/STIS G750L calibrated spectrophotometry ata wavelength of 1 μm, taking advantage of the very precise flux calibration of HST/STISand to tie in with the optical measurements. For this, we considered an ad hocsquare passband of 40 nm in width and calculated the average flux in this wavelengthinterval. The fluxes in Table <ref> are quite consistent for all bands except for themeasurements of WISE W3 and IRAS_12, which correspond to nearly identical effectivewavelengths but differ by over 50%. While we initially employed the passband zero pointsfrom the literature, we explored another approach, namely the calibration of thefluxes using a standard spectrum. As before, we used the STIS_008 Vega spectrum fromthe CALSPEC Calibration database. We calculated integrated fluxes for the relevantbroad-band passbands using the definitions from the references in Table <ref>and used them to set the zero point of the magnitude scale. The fluxes for Proximacalculated in this way are also listed in Table <ref>. As expected, thecomparison between the literature zero points and those estimated using the spectrumof Vega reveals little differences in most cases (less than 3%). However, the WISEW3 band zero point is notably different (by about 20%) when comparing both methods.The value that we calculate from the Vega standard spectrum leads to closer agreement(though still far from perfect) with the IRAS_12 value. Given this circumstance, wedecided to adopt the fluxes as calculated by us from the Vega spectrum for all bandswith magnitude measurements (i.e., not for fluxes given in physical units).Our procedure to obtain the NIR SED of Proxima is to fit all flux measurements with aspectrum from a theoretical model. As already mentioned in Sect.<ref>, no specific correction for flares was made. We chose to use the BT-Settl gridfrom <cit.> in its latest version available from<https://phoenix.ens-lyon.fr/Grids/BT-Settl/CIFIST2011_2015/>. Proxima's surface gravity and metallicity are compatible within the error bars with values of log g = 5.0and [Fe/H]=0.0 <cit.>, which are part of the BT-Settl model grid, and thosewere adopted in our SED fitting procedure. The free parameters of the fit were theeffective temperature and the angular diameter. For the latter, however, we useda prior from <cit.> of θ = 1.011±0.052 mas. From the model spectra wecalculated integrated fluxes for all passbands using the definitions from the references in Table <ref>. We built a χ^2 statistic by comparing the model fluxes with the observed values and adopting the usual weight proportional to 1/σ^2, andthis was minimized via the simplex algorithm as implemented by <cit.>. Tofurther constrain the model we doubled the weight of the anchor point at 1 μm and ofthe angular diameter measurement. Using the constraints above, the SED fit yields an effective temperature of 2870 K butan angular diameter that is 2.7σ larger than the observation. Also, the residualsreveal a strong systematic difference between the bands roughly at either side of 3 μm.Such discrepancy suggests that Proxima has higher fluxes at longer wavelengths thanexpected from models. We then considered a fit only to the bands shortwards of 3 μm andthis led to an effective temperature of 3000 K and an angular diameter within 1σ ofthe measured value. In Table <ref> we list the flux residuals and resultingparameters from two fitting scenarios. We adopted the solution that fits the bands shorterthan 3 μm, the HST/STIS flux and JHK bands, our “Fit 1”. Figure<ref> illustrates this fit and the comparison between models and observations.In addition, we ran tests by considering only measurements up to 2 μm, thereby excludingthe K band; these also show the same systematic trend and yielded very similar results.§.§ Combined spectrum table1The full spectrum, covering 0.7 to 30000 nm, is provided in Table 8 and shown in Fig.<ref>. Table 8 is available at the CDS and contains the following information: Column 1 gives the wavelength in nm and Column 2 lists the top-of-atmosphere flux forProxima b in units of W m^-2 nm^-1, which are most commonly employed for planetary atmosphere work. The spectrum as obtained by adding the data from the various sources hasrather inhomogeneous wavelength steps and was resampled using different bin sizes fordifferent wavelength intervals depending on the quality of the spectrum. We calculatedthe top-of-atmosphere flux for Proxima b by using the trigonometric distance to Proximaand by adopting an orbital distance of 0.0485 AU for Proxima b. Figure <ref>also shows the top-of-atmosphere solar irradiance of the Earth for comparison, corresponding to the <cit.> solar spectrum for medium solar activity. The integrated fluxes invarious relevant intervals are listed in Table <ref>. Our results show that the XUVflux is nearly 60 times higher than Earth's value <cit.> andthe total integrated flux is 877±44 W m^-2, or 64±3% of the solar constant(i.e., top-of-atmosphere solar flux received by Earth, adopting S_⊕=1361 W m^-2,). The adopted uncertainty on the total irradiance corresponds to a relativeerror of 5% on the Proxima flux (see below).Proxima is variable over different timescales, most notably related to flare events(hours), rotational modulation (months) and activity cycle (years). In Table <ref>we provide estimates of such variability amplitude (peak), when available, with respect tothe mean flux value listed. Such estimates come from various literature sources that we haveadapted to the relevant wavelength intervals. For the bolometric flux we scale the variabilityfrom that coming from the V band. <cit.> obtain a variation of 2% with respect tothe average (4% peak to peak). This, however, is not representative of the bolometricvariability because activity-related effects are known to diminish with increasingwavelengths. In the case of Proxima, wavelengths around 1 μm would be a better proxy forflux variations of the bolometric luminosity. We have used the StarSim simulator <cit.>to estimate that variations of 2% in the V band correspond to about 0.5% around 1 μmif we assume spots with contrasts of 300–500 K <cit.>. We adopt a similar scaling forthe flare statistics obtained by <cit.>, which correspond to the MOST satellite band.It is interesting to point out that the variability of the total irradiance of Proxima isabout 25 times higher than the solar value <cit.> and this could have animpact on the climate forcing.It should be noted that no information on the rotational modulation and cycleamplitude are available for the FUV range. The only relevant data in the UV comes fromthe results of <cit.>, who find a 4% rotational modulation and similar cyclevariability for the Swift/UVOT W1 band, which has an effective wavelength of 260 nm.Regarding flares, <cit.> studied several large events in the UBV bands, and foundpeak-to-quiescence flux ratios of up to 25, 4 and 1.5, respectively. We do not include thesevalues in Table <ref> but we note that both the rotation/cycle amplitudes and theflare fluxes are strongly variable with wavelength.§ IR EXCESSThe flux residuals in Fig. <ref> show a clear systematic offset beyond≈2 μm, with the observed flux being ∼20% larger thanmodel predictions. This systematic difference can be interpreted as a mid-to-near IR excessassociated with the Proxima system, which, to our knowledge, has not been pointed out before.A possible physical explanation is the presence of dust grains, in what could be a warmring close to the star, scattering the light from Proxima. The presence of such a dustreservoir could be leftover from the formation process of the planetary system aroundProxima. Worth noting here is the K0 planet-host HD 69830 <cit.>, which was foundto have a mid-IR excess (∼50% over photosphere) and was interpreted by <cit.>as caused by small dust grains within 1 AU of the star. While no other warm disk around anold M dwarf has been reported, a cold resolved debris disk (an analog to the Kuiper Beltof our Solar System) was found by <cit.> with Herschel Space Observatoryobservations of GJ 581. Unfortunately, no far-IR measurements of Proxima are availableto investigate the presence of a cold debris disk, which could lend additional support tothe explanation of the mid-to-near IR excess that we find.The systematic trend of the residuals could alternatively be related to certain shortcomings of the theoretical models. However, this is rather unlikely, as a ∼20% flux deficitis very significant and would have been identified before in other stars <cit.>. Inaddition, one could think that the differences are related to the SED fitting procedure. Ahigher T_ eff value could yield fluxes in better agreement. Being in the Rayleigh-Jeansregime, this would mean a ∼20% increase in temperature of ∼600 K. The otherpossibility is to assume a larger angular diameter by ∼10%. Neither optioncan be valid because of the existence of strong constraints coming from the HST/STIS fluxat 1 μm and from the interferometric angular diameter determination. Finally,one could also consider a heavily spotted stellar surface (i.e., a hotter photosphere anda significant fraction of cooler spot areal coverage) that could result in a SED with anapparent IR flux excess. However, a IR flux excess that becomes significant at≈2 μm would require an unrealistically low spot temperature value(T_ spot<1500 K, T_ phot-T_ spot>1500 K; c.f., ). Thus,given the lack of an alternative explanation consistent with the data and model fits, wefind that the most likely cause of the IR excess is scattering of light from warm dustparticles close to the star.§ PHYSICAL AND RADIATIVE PROPERTIESFrom the full spectral energy distribution of Proxima we can estimate its radiative parameters. The integration of the total flux (from X-rays to 30 μm) yields aflux of 2.86×10^-8 erg s^-1 cm^-2. The uncertainty of this value should be mostly driven by the uncertainty in the HST/STIS spectrophotometricmeasurements and of the IR fit. For the former, the absolute flux scale is found to bebetter than 5%, and possibly better than 3% <cit.>. For the IR, given thediscussion on the quality of the fit, we also adopt an uncertainty of 5%. Therefore,the bolometric flux of Proxima is found to beF_ bol = (2.86±0.14)×10^-8 erg s^-1 cm^-2. This value and theangular diameter in Table <ref> lead to an effective temperature value ofT_ eff = 2980±80 K. The difference from the value in Table <ref>arises because in this calculation we consider the full wavelength range,not only the interval beyond 1 μm. In other words, the optical flux of Proxima islower than that of a 3000 K model and, hence, results in a lower T_ eff. Finally, we calculate the bolometric luminosity by using the stellar parallax of Proximafrom <cit.> and a solar luminosity value from IAU 2015 Resolution B3 onRecommended Nominal Conversion Constants for Selected Solar and Planetary Properties(<https://www.iau.org/administration/resolutions/general_assemblies/>). The bolometric luminosity of Proxima is, thus, L_ bol = (5.80±0.30)×10^30 erg s^-1 or L_ bol = 0.00151±0.0008 L_⊙.§ TIME EVOLUTION OF THE FLUX RECEIVED BY PROXIMA B §.§ Bolometric flux The total flux evolution of Proxima can be estimated from theoretical evolutionary model calculations. We employed the recent models of <cit.> that are well suited for very low mass stars and include the most up-to-date physical ingredients.We linearly interpolated the evolutionary tracks from the models corresponding to 0.110and 0.130 M_⊙ to find a good match of the model predictions with our determinedvalues for L_ bol and T_ eff at the estimated age of the star of4.8 Gyr <cit.>. A stellar mass of 0.120 M_⊙ yields the bestsimultaneous agreement of all parameters within the corresponding uncertainties, resulting in values of L_ bol = 0.00150 L_⊙ and T_ eff = 2980 K. A formal uncertainty of 0.003 M_⊙ can be estimated from the errors associated toL_ bol and T_ eff. This is obviously a model-dependent estimate andno error in metallicity and log g was assumed. The evolutionary track of Proxima is shownin Fig. <ref>, in normalized units of today's bolometric luminosity. At 10 Myr,the time when the protoplanetary disk may have dissipated <cit.> and Proxima bbecame vulnerable to XUV radiation, the stellar luminosity was a factor of 10 larger thantoday. Thus, Proxima b spent some 90–200 Myr in an orbit interior to the stellarhabitable zone and possibly in a runaway greenhouse state. A detailed discussion isprovided by <cit.>. §.§ High-energy flux The evolution of the XUV flux of Proxima with time was addressed by <cit.>. Herewe revisit the calculations by considering also the early evolution of L_ bol as thestar was contracting towards the Main Sequence. The XUV evolution of young M dwarfs is poorly constrained but some tantalizing evidence exists indicating that the saturation limit oflog (L_ X/L_ bol) ≈ -4 also applies to the pre-Main Sequence<cit.>. As we show above, the bolometric luminosity should have experienced significantchanges over the first few hundred Myr in the history of Proxima and therefore this needsto be properly taken into account in the calculations <cit.>. Different XUVevolution laws are discussed in <cit.>. One considers a saturated emission stateup to a certain age followed by a power law decrease to today's XUV flux. The other oneconsiders that Proxima has shown saturated behaviour since its birth and until today.Observational evidence is still inconclusive as to which of these two XUVevolution scenarios is correct, and we hereby further consider them both. They should be representative of the extreme cases bracketing the real evolution of Proxima over its lifetime.§.§.§ Proxima is just at the end of the saturation phaseProxima's current relative X-ray value is log (L_ X/L_ bol) = -3.83,which is very similar to the average of the distribution for stars between 0.1 and0.2 M_⊙ and ages of 0.1 to 10 Myr found by <cit.>. This circumstancesuggests that Proxima may still be today in the saturated regime andthat log (L_ X/L_ bol) = -3.83 has been satisfied during itsentire lifetime. This is in good agreement with the estimates of the saturation limitas determined from the equations in <cit.>, which extends up to arotation period of P_ rot≈80 d for a 0.146-R_⊙ star, very close toProxima's current rotation period of P_ rot=83 d <cit.>.To parameterize the bolometric flux, we consider the evolutionary model track and twodifferent regimes, from 10 to 300 Myr and from 300 Myr to today. The evolution of the stellar bolometric flux as a function of the age (τ) in Myr can be approximated as (see top panel of Fig. <ref>):L_ bol/L_ bol, current=57.38τ^-0.71 L_ bol/L_ bol, current=1.000Then, we assume that the XUV flux scales in the same way as the X-rays. This is anapproximation because the hardness ratio of the XUV spectrum may have softened as the star spun down. But in the absence of a better model, we used the evolution law from X-rays as valid for the full XUV range, and, therefore, that log (L_ XUV/L_ bol)= -3.48 has remained constant for the entire lifetime of Proxima. With this, and theexpressions in Eq. (1), we find the following relationship for the top-of-atmosphere fluxof Proxima b as a function of age (the current value is taken from Table <ref>):F_ XUV=16.81τ^-0.71 F_ XUV=0.293The proposed evolution of the top-of-atmosphere flux received by Proxima b corresponding to this scenario is illustrated in the bottom panel of Fig. <ref> with a solid line.§.§.§ Proxima has evolved off saturation and is in the power law regimeAs an alternative to the XUV evolution scenario above, one can consider the resultsof <cit.>. Although based on 4 stars (among which is Proxima), the authors suggest thatthe X-ray evolution of fully convective stars is analogous to that of more massive Sun-likestars. In this case, to model the stage after saturation, we can adopt the relationship in<cit.> by which L_ X/L_ bol∝ R_∘^-2.70∝ P_rot^-2.70, where R_∘≡ P_ rot/τ_ c is the so-called Rossbynumber <cit.> and we assume the convective turnover time (τ_ c) to beconstant during the main sequence lifetime of Proxima. From this, we can further adopt<cit.>, who find P_ rot∝τ^0.566, where τ is the stellar age,to obtain L_ X/L_ bol∝τ^-1.5. Thus, considering thatlog (L_ X/L_ bol) = -3.83 at an age of 4.8 Gyr and that saturation of Sun-likestars occurs at an average value of log (L_ X/L_ bol) = -3.13 <cit.>, wefind that the end of saturation should have happened at an age of 1.64 Gyr.As before, we further make the assumption that the total XUV flux follows the sameevolution as the X-ray flux and we can write the relationship (τ in Myr):F_ XUV=84.1τ^-0.71 F_ XUV=1.47 F_ XUV=9.74×10^4τ^-1.5This proposed evolution is illustrated in the bottom panel of Fig. <ref> witha thick dashed line. §.§ XUV dose and water loss estimates The integration of the XUV relationships presented here and the comparison withthe equivalent relationship for the Sun and the Earth <cit.> indicatesthat the total XUV dose that Proxima b has received over its lifetime is between 8 and25 times greater than Earth's. But the most critical part may be the phase atwhich the atmosphere of Proxima b was in runaway greenhouse effect, in an orbit interiorto the habitable zone. The amount of XUV irradiation during this period of time from about10 Myr until about 90–200 Myr could have caused an intense loss of water. To estimate thewater loss, we proceeded as in <cit.> and <cit.>. We use the sameunits for the water loss as in those articles: 1 EO_H corresponds to the Earth ocean'sworth of hydrogen. We also took into account the revised smaller mass for Proxima, butthis has no significant impact on the calculations. With our model, we can estimatethe current volatile losses of Proxima b: the hydrogen loss is of 0.003 EO_H/Myr, whichcorresponds to 1.5×10^7 g s^-1, the oxygen loss is 0.009 EO_H/Myr, whichcorresponds to 4.3×10^7 g s^-1.Table <ref> summarizes the results for the two prescriptions given inEqs. (2) and (3). The parameterization of the XUV flux evolution given by Eq. (2) differs fromthe one used in <cit.> as follows: it is higher during the first 100 Myr but lowerby a factor ∼2.7 during the following few Gyr. This has two consequences on the waterloss: 1) during the runaway phase, and more especially during the first 100 Myr, the loss ismore intense than in <cit.>, and 2), on the long term, the total loss is lower.The parameterization of the evolution of the XUV flux given by Eq. (3) leads to higher XUVfluxes throughout the entire lifetime of Proxima b when compared with <cit.>.If we assume synchronous rotation, our estimates indicate that Proxima b could have lostfrom 0.47 EO_H (Eq. 2) to 1.07 EO_H (Eq. 3) between 10 Myr and 90 Myr, when itreached the inner edge of the habitable zone at 1.5 S_⊕ <cit.>. Ournew calculations therefore suggest that, during that time, Proxima b may have lost morewater than previously estimated by <cit.>, by about a factor 1.25 to 3.Assuming non-synchronous rotation, the amount of water lost could range from0.9 EO_H (Eq. 2) to 1.91 EO_H (Eq. 3) between 10 Myr and 200 Myr, when it reachedthe habitable zone inner edge at 0.9 S_⊕ <cit.>. Theestimate obtained with the prescription of Eq. (2) is about the same value as previouslyprovided by <cit.> while the calculation with Eq. (3) is about a factor of 2larger.In spite of the strong volatile losses (∼0.5–2 EO_H), the planet could still have asignificant amount of water reservoir when it entered the habitable zone depending on theinitial content. What could have occurred beyond this point is quite uncertain. If we assumethat the water loss processes were still active upon entering the habitable zone, wefind that Proxima b could have lost up to 15–25 EO_H during its lifetime. However, thisneeds to be considered an extreme upper limit because the volatile loss mechanisms wouldprobably be significantly less efficient under such conditions <cit.>. § CONCLUSIONSThis paper presents a full analysis of the SED of Proxima, covering X-rays tomid-IR, with the goal of providing useful input to study the atmosphere of Proxima b.We made use of measurements covering different wavelength intervals and acquired withvarious facilities (see Tables <ref> and <ref>) to determinetop-of-atmosphere fluxes from 0.7 to 30000 nm, in steps of widths ranging from 0.05 to10 nm depending on the wavelength range. Where spectrophotometric measurements wereunavailable, we made use of theoretical models fitted using all availableconstraints. With the full spectral energy distribution and the available trigonometricdistance, we could calculate the bolometric luminosity and the effective temperature.Also, Proxima has a quite accurate interferometric angular diameter measurement and thiswas used both to constrain the SED fit in the IR and to provide an empirical determinationof the stellar radius. Interestingly, the fit of the IR SED revealed a fluxexcess ≈20% from Proxima.While the origin of this excess is uncertain, the mostnatural explanation is light scattering by dust particles in the Proxima system;additional observations can better ascertain the nature of the excess. The stellar masswas estimated by comparison with evolutionary models using the constraints provided bythe radiative properties of Proxima. All the resulting fundamental parameters aresummarized in Table <ref>. Proxima is a benchmark star, not only for us to understand the stellar lower main sequence,but also, since the discovery of Proxima b, to study its habitable planet candidate. As discussed by <cit.>, to determine the habitability of the planet it isessential to analyze the volatile loss processes that may affect its atmosphere, both currently and in the past. The detailed spectral energy distribution for Proxima presentedhere and the newly proposed XUV flux time evolution laws should help to provide thenecessary constraints to model and interpret future observations of the nearest potentiallyhabitable planet outside the Solar System. We are grateful to Rodrigo Luger for pointing out the increased XUV flux in the early evolution of Proxima, and to Parke Loyd for assistance with the flare characterization in the FUV range. We also gratefully acknowledge the insightful comments andsuggestions by an anonymous referee. I. R. acknowledges support by the Spanish Ministryof Economy and Competitiveness (MINECO) and the Fondo Europeo de Desarrollo Regional(FEDER) through grant ESP2016-80435-C2-1-R, as well as the support of the Generalitatde Catalunya/CERCA programme. M. D. G. and T. S. B. acknowledge generous support providedby NASA through grant number GO-13776 from the Space Telescope Science Institute, whichis operated by AURA, Inc., under NASA contract NAS5-26555. E. B. acknowledges funding bythe European Research Council through ERC grant SPIRE 647383.aa | http://arxiv.org/abs/1704.08449v1 | {
"authors": [
"Ignasi Ribas",
"Michael D. Gregg",
"Tabetha S. Boyajian",
"Emeline Bolmont"
],
"categories": [
"astro-ph.SR",
"astro-ph.EP"
],
"primary_category": "astro-ph.SR",
"published": "20170427064734",
"title": "The full spectral radiative properties of Proxima Centauri"
} |
0000 00 0 1 plain lemLemme[section] thm[lem]Theorem prop[lem]Proposition cor[lem]Corollary as[lem]Assumption *conjConjecture definition defin[lem]Definition exa[lem]Example xca[lem]Exercise remark re[lem]Remark equationsectionfiguresection ( [S. Shen and J. Yu]Shu Shen and Jianqing Yu This article is devoted to a study of flat orbifoldvector bundles.We construct a bijection between the isomorphic classes of properflat orbifoldvector bundles and the equivalence classes ofrepresentations of the orbifold fundamental groups ofbase orbifolds. We establish a Bismut-Zhang likeanomaly formula for the Ray-Singer metric on the determinant line of the cohomology ofa compact orbifold with coefficients in an orbifoldflat vector bundle. We show that the analytic torsion ofan acyclic unitary flat orbifold vector bundle is equal to thevalue at zero of a dynamical zeta function when the underlying orbifold isa compact locally symmetric space of reductive type, which extends one of the results obtained bythe first author for compact locally symmetric manifolds. Flat vector bundles and analytic torsion onorbifolds [ December 30, 2023 ======================================================section§ INTRODUCTION Orbifolds were introduced by Satake <cit.> under name of V-manifold as manifolds with quotient singularities.They appear naturally, for example,in the geometry of 3-manifolds,in thesymplectic reduction, in the problems on moduli spaces, andin string theory, etc. It is natural to considerthe index theoretic problem and the associated secondary invariants onorbifolds. Satake <cit.> and Kawasaki <cit.> extendedthe classical Gauss-Bonnet-Chern Theorem,the Hirzebruch signature Theorem and the Riemann-Roch-Hirzebruch Theorem. For the secondary invariants,Ma <cit.> studiedtheholomorphic torsions and Quillen metrics associated withholomorphic orbifold vector bundles, andFarsi <cit.> introduced an orbifold version eta invariant and extendedthe Atiyah-Patodi-Singer Theorem.In this article, we studyflat orbifold vector bundles and the associated secondary invariants, i.e., analytic torsions or more precisely Ray-Singer metrics. Let us recall some results onflat vector bundles on manifolds. Let Z be a connected smooth manifold, and let F be a complexflat vector bundle on Z.Equivalently, F can be obtained via a complex representation of the fundamental group π_1(Z) of Z, whichis called the holonomy representation. Denote by H^·(Z,F) the cohomology ofthe sheaf of locally constant sections of F. Assume that Z is compact. Givenmetrics g^TZ and g^F onTZ and F, the Ray-Singer metric <cit.> on the determinant line λ of H^·(Z,F) is defined by the product ofthe analytic torsionwith an L^2-metric on λ. If g^F is flat, or equivalently if the holonomy representation isunitary,then the celebrated Cheeger-Mller Theorem<cit.> tells us that in this case the Ray-Singer metriccoincides with the so-called Reidemeister metric<cit.>, which is a topological invariant of the unitarilyflat vector bundles constructed with the help of a triangulation on Z.Bismut-Zhang <cit.> and Mller <cit.> simultaneously considered generalizations of this result. In<cit.>, Mller extended it to the casewhere g^F is unimodular or equivalentlythe holonomy representationis unimodular.In <cit.>, Bismut and Zhang studied the dependence of the Ray-Singer metric on g^TZ andg^F.They gave an anomaly formula <cit.> for the variation of the logarithm of the Ray-Singer metric on g^T Z and g^F as an integral of a locally calculable Chern-Simonsform on Z. They generalized the original Cheeger-Mller Theorem to arbitrary flat vector bundles with arbitrary Hermitian metrics <cit.>.In <cit.>,Bismut and Zhangalso considered the extensions to the equivariant case. Note that both in <cit.>, the existence of a Morse function whose gradient satisfies the Smale transversality condition <cit.> plays an important role. From the dynamical side,motivated bya remarkable similarity <cit.> betweenthe analytictorsion and Weil's zeta function,Fried <cit.>showed that, when the underlying manifold is hyperbolic,the analytic torsion of an acyclic unitarily flat vector bundle is equal tothevalue at zeroof the Ruelle dynamical zeta function.In <cit.>, he conjectured similar results hold true for more general spaces.In <cit.>, following the early contribution ofFried <cit.> andMoscovici-Stanton <cit.>, theauthorshowed the Fried conjecture on closed locally symmetric manifolds of the reductive type. The proof is based onBismut's explicit semisimple orbital integral formula <cit.>. (SeeMa's talk <cit.> at Sminaire Bourbaki for an introduction.)In this article,we extend most of the above resultstoorbifolds.Now, we will describe our results in more details and explain the techniques used in the proof.§.§ Orbifold fundamental group and holonomy representation Let Z be a connectedorbifold with the associated groupoid . Following Thurston <cit.>,let X be the universal covering orbifold of Z with the deck transformation group Γ, which is calledorbifold fundamental group of Z. Then, Z=Γ\ X. In an analogous way as in the classical homotopy theory of ordinary paths on topologicalspaces, Haefliger <cit.> introduced the -paths and their homotopy theory.Hegave an explicit construction ofX and Γ following the classical methods. IfF is acomplex proper flat orbifold vector bundle of rank r, in Section <ref>, we constructed a parallel transportalong a -path. In this way, we obtain a representationρ:Γ→_r() of Γ, which is called the holonomy representation of F. Denote by^ pr_r(Z) theisomorphic classesof complex proper flat orbifoldvector bundles of rank r on Z, and denoteby Hom(Γ,GL_r ())/_∼the equivalence classesof complexrepresentations ofΓ of dimension r. We show the following theorem.The above construction descends toa well-defined bijection^pr_r(Z)≃Hom(Γ,_r())/_∼. The difficulty of the prooflies in the injectivity, which consists inshowing that F is isomorphic to the quotient of X×^r by the Γ-action induced by the deck transformation on X and by the holonomy representation on ^r. Indeed, applyingHaefliger's construction, in subsection <ref>, weshow directly that theuniversal covering orbifold of the total spaceof F isX×^r. Moreover, its deck transformation group is isomorphic toΓ with the desiredaction on X×^r.We remark thaton the universal covering orbifold there exist non trivial and non proper flat orbifold vector bundles. Thus, Theorem <ref> no longer holds true for non proper orbifold vector bundles.On the other hand,for a general orbifold vector bundle E which is not necessarily proper, there exists a propersubbundle E^ pr of E such thatC^∞(Z,E)=C^∞Z,E^ pr.Moreover, if E is flat, E^ pr is also flat.For a Γ-space V, we denote by V^Γ the set offixed points in V. By Theorem <ref> and (<ref>), we get: For any(possibly non proper) flat orbifold vector bundle F on a connected orbifold Z, there existsa representation of the orbifold fundamental group ρ:Γ→_r() such thatC^∞(Z,F)=C^∞X,^r^Γ.By abuse of notation, in this case, although ρ is not unique, we still callρ a holonomy representation of F.Waldroninformed us that in his PhD thesis <cit.> he provedTheorem<ref> ina more abstract setting using differentiable stacks. §.§ Analytic torsion on orbifoldsAssume that Z is a compact orbifold of dimension m. Let ΣZ be the strata of Z, which has a natural orbifold structure.Write Z∐Σ Z=∐_i=0^l_0Z_i as a disjoint unionof connected components. We denotem_i∈ the multiplicity ofZ_i (see (<ref>)). Let F be a complex flat orbifoldvector bundle on Z. Letλ be the determinant lineof the cohomology H^·(Z,F) (see (<ref>)).Let g^TZ and g^F be metrics on TZ and F. Denote by ^Z the associated Hodge Laplacian acting on the space Ω^·(Z,F) of smooth forms with values in F. By the orbifold Hodge Theorem <cit.>, we have the canonical isomorphismH^·(Z,F)≃^Z. As in the case of smooth manifolds, by <cit.> (or by the short time asymptotic expansions of the heat trace <cit.>),the analytic torsion T(F) is still well-defined. It is a real positive number definedby the followingweighted product of the zeta regularized determinants T(F)=∏_i=1^mdet^Z|_Ω^i(Z,F)^(-1)^ii/2.Let |·|^ RS,2_λ be the L^2-metric on λ induced by g^TZ,g^F via (<ref>).The Ray-Singer metric on λ is then given by ·^ RS_λ=T(F)|·|^ RS_λ.We remark that as in the smooth case, if Z is of even dimension andorientable, if F is unitarily flat, in Proposition <ref>, weshow thatT(F)=1.In Section <ref>, we study the dependence of ·^ RS,2_λ on g^TZ and g^F. To state our result, let us introduce some notation. Let (g^' TZ,g^' F) be another pair of metrics.Let ·^' RS,2_λ bethe Ray-Singer metric for (g^' TZ,g^' F).Let ∇^TZ and ∇^' TZ be the respective Levi-Civita connections on TZ forg^TZ and g^' TZ. Denote by o(TZ)the orientation line of Z. Consider the Euler form e(TZ,∇^TZ)∈Ω^m(Z,o(TZ))and the first odd Chern form 1/2θ(∇^F,g^F)=1/2[(g^F)^-1∇^F g^F]∈Ω^1(Z). Denote by eZ_i,∇^TZ_i∈Ω^Z_i(Z_i,o(TZ_i)), θ_i∇^F,g^F∈Ω^1(Z_i)the canonical extensions ofe(TZ,∇^TZ) andθ(∇^F,g^F) to Z_i (see subsection <ref>).Lete(TZ_i,∇^TZ_i,∇^' TZ_i)∈Ω^ Z_i-1(Z_i,o(TZ_i))/d Ω^ Z_i-2(Z_i,o(TZ_i))and θ_i(∇^F,g^F,g^' F)∈ C^∞(Z_i) be the associated Chern-Simonsforms such that d eTZ_i,∇^TZ_i,∇^'TZ_i =eZ_i,∇^' TZ_i-eZ_i,∇^TZ_i,d θ_i∇^F,g^F,g^'F =θ_i∇^F,g^' F-θ_i∇^F,g^F.In Section <ref>,we show: The following identity holds:log·^' RS,2_λ/·^RS,2_λ=∑^l_0_i=01/m_i(∫_Z_iθ_i∇^F,g^F,g^' FeZ_i,∇^TZ_i- ∫_Z_iθ_i∇^F,g^' FeTZ_i,∇^TZ_i,∇^' TZ_i).The arguments in Section <ref> are inspired byBismut-Lott <cit.>, who gave a unified proof for the family local indextheorem and the anomaly formula <cit.>. Conceptually, their proof issimpler and more natural than the original proof given by Bismut-Zhang <cit.>. Also, our proof relies onthe finite propagation speeds for the solutions of hyperbolic equations on orbifolds, which is originally due to Ma <cit.>.If Z is of odd dimensionand orientable, then all the Z_i, for 0≤ i≤ l_0, is of odd dimension. By Theorem <ref>,the Ray-Singer metric ·^ RS,2_λ does not depend on the metrics g^TZ, g^F; it becomes a topological invariant.§.§ A solution of Fried conjecture onlocally symmetric orbifoldsIn <cit.>, Friedraised the question of extending his result <cit.> to hyperbolic orbifolds on the equality betweenthe analytic torsion and the zero value of the Ruelle dynamical zeta function associated to a unitarily flatacyclicvector bundle on hyperbolic manifolds. In Section <ref>, we extendFried'sresult to more general compact odd dimensional [The even dimensional case is trivial.] locally symmetricorbifolds of the reductive type. Let G be a linear connected realreductive group with Cartan involution θ∈ Aut(G). LetK⊂ G be the set of fixed points of θ in G, so that K is a maximal compact subgroup of G. Letandbe the Lie algebras of G and K. Let =⊕ be the Cartan decomposition. Let B be an (G)-invariant and θ-invariant non degenerate bilinear form onsuch that B|_>0 and B|_<0.Recall thatan element γ∈ G is said to be semisimple if and only if γ can be conjugated to e^ak^-1 with a∈, k∈ K, (k)a=a. And γ is said to be elliptic if and only ifγ can be conjugated into K. Note that if γ is semisimple,its centralizerZ(γ) in G is still reductive with maximal compact subgroup K(γ). TakeX=G/K to be the associated symmetric space. Then, B|_ induces a G-invariant Riemannian metric g^TX on X such that (X,g^TX) is of non positive sectionalcurvature.Let d_X be the Riemannian distance on X. Let Γ⊂ Gbe a cocompact discrete subgroup of G. Set Z=Γ\ G/K. Then Z is a compact orbifold with universal covering orbifold X.To simplify the notation in Introduction, we assume that Γ acts effectively on X.Then Γ is the orbifold fundamental group of Z.Clearly,Γ contains only semisimple elements.Let Γ_+ be the subset of Γ consisting of non ellipticelements. Take [Γ] to be the set of conjugacy classes ofΓ. Denote by[Γ_+]⊂ [Γ]the set of nonelliptic conjugacy classes.Proceeding as in the proof for the manifold case <cit.>, the set of closed geodesicsofpositive lengthsconsists of a disjoint union of smooth connected compact orbifolds ∐_[γ]∈ [Γ_+] B_[γ]. Moreover,B_[γ] is diffeomorphic toΓ∩ Z(γ)\ Z(γ)/K(γ). Also, all the elements inB_[γ] have the same length l_[γ]>0. Clearly, thegeodesic flow induces a locally free ^1-action onB_[γ]. By an analogy to themultiplicity m_i ofZ_i inZ∐Σ Z, we can define the multiplicity m_[γ]of the quotient orbifold 𝕊^1\ B_[γ] (see(<ref>)). Denote byχ_orb(𝕊^1\ B_[γ])∈𝐐theorbifold Euler characteristic number <cit.> (see also (<ref>)) of 𝕊^1\ B_[γ]. In Section <ref>,we show: If Z is odd, and if F is a unitarily flatorbifoldvectorbundle on Z with holonomy ρ:Γ→U(r), then the dynamical zeta function R_ρ(σ)=exp∑_[γ]∈ [Γ_+][ρ(γ)]χ_orb𝕊^1\ B_[γ]/m_[γ]e^-σ l_[γ]iswell-defined and holomorphicon (σ)≫1, and extendsmeromorphically to . There exist explicit constants C_ρ∈ with C_ρ≠0 and r_ρ∈ (see (<ref>)) such that as σ→0, R_ρ(σ)=C_ρ T(F)^2σ^r_ρ+(σ^r_ρ+1).Moreover, if H^·(Z,F)=0, we have C_ρ=1, r_ρ=0,so that R_ρ(0)=T(F)^2. The proof of Theorem <ref> issimilar to the onegiven in <cit.>, except that in the currentcase, we also need to take account of the contribution of elliptic orbital integrals in the analytic torsion. On the other hand, let us note that apriori ellipticelements do not contribute to thedynamical zeta function. This seemingly contradictory phenomenon has already appeared in the smooth case. In fact,in the current case, the elliptic and non elliptic orbital integralsare relatedvia functional equations of certain Selberg zeta functions. We refer the readers to the papers ofGiulietti-Liverani-Pollicott <cit.> and Dyatlov-Zworski <cit.> for other points of view on the dynamical zeta function on negatively curved manifolds. Let us also mentionFedosova's recent work <cit.> on the Selberg zeta function andthe asymptotic behavior of the analytic torsion of unimodular flat orbifold vector bundles on hyperbolic orbifolds. §.§ Organisation of the articleThis articleis organized as follows. In Section <ref>, we introduce some basic notationon the determinant line and characteristic forms. Also we recall some standard terminologyongroup actions on topological spaces.In Section <ref>, we recall the definition of orbifolds, orbifold vector bundles, and the -path theory of Haefliger <cit.>. We show Theorem <ref>.In Section <ref>, we explainhow to extend the usual differential calculus and Chern-Weil theory on manifolds to orbifolds. In Section <ref>, westudythe analytic torsion andRay-Singer metricon orbifolds. Following <cit.>, we show ina unified way an orbifold version ofGauss-Bonnet-Chern Theorem and Theorem <ref>. Some estimates on heat kernels are postponed to Section <ref>.In Section <ref>, we study the analytic torsion on locally symmetric orbifoldusing the Selberg trace formula and Bismut's semisimple orbital integral formula. We show Theorem <ref>. §.§ NotationIn the whole paper, we use the superconnection formalism of Quillen<cit.> (see also <cit.>).Here we just briefly recall that if A is a _2-graded algebra, if a,b∈ A,the supercommutator [a, b] isgiven by [a,b]=ab-(-1)^ abba. If B is another _2-graded algebra, we denote by A⊗ B the super tensor algebra. If E = E^+ ⊕ E^- is a _2-graded vector space, the algebra (E) is _2-graded. If τ = ± 1 on E^±, if a ∈(E), the supertrace [a] is defined by[a]=[τ a].We make the convention that 𝐍={0,1, 2,⋯} and 𝐍^*={1,2,⋯}.If A is a finite set, we denote by |A| its cardinality. §.§ AcknowledgementThe workstarted while S.S. was visiting the University of Science and Technology of China in July, 2016. He wouldlike to thank the School of Mathematical Sciences for hospitality.S.S. wassupported by a grant from the European Research Council (E.R.C.) under European Union's Seventh Framework Program (FP7/2007-2013) /ERC grant agreement (No. 291060) and by Collaborative Research Centre “Space-Time-Matte" (SFB 647) founded by the German Research Foundation (DFG). J.Y. was partially supported by NSFC (No. 11401552, No. 11771411). subsection§ PRELIMINARYThe purpose of this section is to recall some basic definitions and constructions. This section is organized as follows. In subsection <ref>, weintroduce the basic conventions ondeterminant lines.In subsection <ref>, we recall some standard terminology of group actions on topological spaces, which will be used in the whole paper.In subsection <ref>, we recall theChern-Weil constructiononcharacteristic forms andthe associated secondary classes of Chern-Simons forms on manifolds. §.§ DeterminantsLet V be a complexvector space of finite dimension. We denote by V^* the dual space of V, and by Λ^· V the exterior algebra of V.Set V=Λ^ V V.Clearly, V is aline. We use the convention that 0=. Ifλis aline, we denote by λ^-1=λ^* the dual line.§.§ Group actionsLet L be a topologicalgroupacting continuously on a topologicalspace S. The action of L issaid to befree if for any g∈ L and g≠ 1, the set of fixed points of g in S is empty. The action of Lis said to be effective if the morphism of groups L→ Homeo(S) is injective, where Homeo(S) is the group of homeomorphisms of S. The action of L is said to be properly discontinuous if for anyx∈ S there is a neighborhood U of x such that the set {g∈ L: gU∩ U≠∅}is finite.If L acts on the right (resp. left) on the topological space S_0 (resp. S_1), denote by S_0/L (resp. L\ S_1) the quotient space, and by S_0×_L S_1 the quotient of S_0× S_1 by the left action defined by g(x_0,x_1)=(x_0g^-1,gx_1), forg∈ Land(x_0,x_1)∈ S_0× S_1.If S_2 is another left L-space, denote byS_1 [_L]× S_2 the quotient of S_1× S_2 by the evident left action of L.§.§ Characteristic forms on manifoldsLet S be amanifold. Denote by(Ω^·(S),d^S) the de Rham complex of S, and byH^·(S)its de Rham cohomology. Let E be a realvector bundle of rank r equippedwith aEuclidean metric g^E. Let ∇^E be a metric connection, andletR^E=(∇^E)^2 be the curvature of ∇^E. Then R^E is a 2-form on S with values inantisymmetric endomorphisms of E. If A is an antisymmetric matrix, denote by Pf[A] the Pfaffian <cit.> of A. Then Pf[A] is a polynomial function of A, which is a square root of [A]. Let o(E) be the orientation line of E. The Eulerformof E,∇^E is given byeE,∇^E=Pf R^E/2π ∈Ω^r(S,o(E)).The cohomology class e(E)∈ H^r(S,o(E)) of e(E,∇^E) does not depend on the choice of (g^E,∇^E). More precisely,if g^' E is another metric on E, and if ∇^' E is another connection on E which preservesg^' E, we can define a class of Chern-Simons (r-1)-form eE,∇^E, ∇^' E∈Ω^r-1(S,o(E))/dΩ^r-2(S,o(E)) such that d^SeE,∇^E, ∇^' E=eE,∇^' E-eE,∇^E.Note thatif r is odd, then e(E,∇^E)=0 ande(E,∇^E, ∇^' E)=0.Let us describe the construction of e(E,∇^E, ∇^' E).Take a smooth family (g^ E_s,∇^E_s)_s∈ of metrics and metric connectionssuch that g^ E_0,∇^E_0=g^E,∇^E,g^ E_1,∇^ E_1=g^' E,∇^' E.Setπ:× S→ S.We equipπ^*E with a Euclidean metric g^π^* E and with a metric connection ∇^π^*E definedby g^π^* E|_{s}× S=g^E_s,∇^π^*E =ds∧d/ds+1/2g^ E,-1_sd/ds g^ E_s+∇^E_s.Write eπ^*E,∇^π^*E=eE,∇^E_s+ds∧α_s∈Ω^r(× S, π^*o(E)).Sinceeπ^*E,∇^π^*E is closed, by (<ref>), for s∈, we have/ s eE,∇^E_s=d^Sα_s.Then, e(E,∇^E, ∇^' E)∈Ω^r-1(S,o(E))/dΩ^r-2(S,o(E)) is definedbythe class of∫_0^1α_s ds∈Ω^r-1(S,o(E)).Note thate(E,∇^E, ∇^' E) does not depend on the choice of smooth family(g^E_s,∇^E_s)_s∈ (c.f. <cit.>). Also, (<ref>) is a consequenceof(<ref>) and (<ref>).Let us recall the definition ofthe A-form of (E,∇^E). For x∈, setA(x)=x/2/sinh(x/2).The A-formof (E,∇^E)is given byAE,∇^E= A-R^E/2iπ ^1/2∈Ω^·(S). Let L be a compact Lie group.Assume thatL acts fiberwisely and linearly on the vector bundle E over S, which preserves (g^E,∇^E).Take g∈ L. Assume that g preserves the orientation of E. Let E(g) be the subbundle of E defined by thefixed pointsof g. Let±θ_1,⋯,±θ_s_0, 0<θ_i≤π be the district nonzero angles of the action of g on E. Let E_θ_i be the subbundle of E on which g acts by a rotation of angle θ_i. The subbundles E(g) and E_θ_i are canonically equipped with Euclidean metrics andmetric connections ∇^E(g),∇^E_θ_i.For θ∈-2π, set A_θ(x)=1/2sinhx+iθ/2.Given θ_i, let A_θ_i(E_θ_i,∇^E_θ_i) bethe corresponding multiplicative genus. The equivariantA-form of (E,∇^E) is given by A_gE,∇^E=AE(g),∇^E(g)∏_i=1^s_0A_θ_iE_θ_i,∇^E_θ_i∈Ω^·(S).Let E' bea complexvector bundlecarrying aconnection ∇^E' with curvature R^E'. Assume thatE' is equipped with a fiberwise linear action of L, which preserves ∇^E'. For g∈ L, the equivariant Chern character form of (E',∇^E') is given by ch_gE',∇^E'= gexp-R^E'/2iπ ∈Ω^ even(S). As before, A_g(E,∇^E), ch_g(E',∇^E') are closed. Their cohomology classes do not depend on the choice ofconnections. The closed forms in (<ref>) and (<ref>) on S are exactly the ones that appear in the Lefschetz fixed point formula of Atiyah-Bott <cit.>. Note that there are questions of signs to be taken care of, because of the need to distinguish between θ_i and -θ_i. We refer to the above references for more detail. Let F be aflat vector bundle on S with flat connection∇^F. Let g^F be a Hermitian metricon F. Assume that Fis equipped with a fiberwise and linear action of L which preserves∇^F and g^F. Following <cit.>, putω∇^F,g^F=g^F^-1∇^Fg^F.Then, ω(∇^F,g^F) is a 1-form on S with values in symmetric endomorphisms of F. For x∈, set h(x)=xe^x^2.Following<cit.> and <cit.>, for g∈ L,the equivariantodd Chern character form of (F,∇^F) is given by h_g∇^F,g^F=√(2iπ) ghω∇^F,g^F/2/√(2iπ) ∈Ω^ odd(S).When g=1, we denote byh∇^F,g^F=h_1∇^F,g^F.By <cit.> and <cit.>, we knowthat the cohomology classh_g(∇^F)∈ H^ odd(S) of h_g(∇^F,g^F) does not depend on g^F. If g^' F is another L-invariant Hermitian metric on F, we can define the class of Chern-Simons form h_g(∇^F,g^F,g^' F)∈Ω^ even (S)/d Ω^ odd(S)such that d^Sh_g(∇^F,g^F,g^' F)=h_g∇^ F,g^' F-h_g∇^F,g^F.More precisely, choose a smooth family of L-invariant metrics (g^F_s)_s∈ such that g_0^F=g^F,g_1^F=g^' F.Consider the projection π defined in (<ref>). Equipπ^*Fwith the following flat connection and Hermitian metric∇^π^*F=d^+∇^F,g^π^* F|_{s}× S=g^F_s. Writeh_g∇^π^* F,g^π^* F=h_g∇^F,g^ F_s+ds ∧β_s∈Ω^ odd(× S).As (<ref>),h_g∇^F,g^ F,g^' F∈Ω^ even (S)/d Ω^ odd(S) is definedby the class of ∫_0^1β_s ds ∈Ω^ even( S).By <cit.> and <cit.>, h_g∇^F,g^ F,g^' F does not depend on the choice of the smooth family of metrics (g^F_s)_s∈. Also,h_g∇^F,g^ F,g^' F satisfies (<ref>).§ TOPOLOGY OF ORBIFOLDSThe purpose of this section is to introduce some basic definitions and related constructions for orbifolds. We show Theorem <ref>, which claims a bijectionbetween the isomorphism classes of proper flatorbifoldvector bundles and the equivalent classes of representations of the orbifold fundamental group.This section is organized as follows. In subsection <ref>,we recall the definition of orbifolds and the associated groupoid .In subsection <ref>,we introducethe resolution for thesingular set of an orbifold.In subsection <ref>, we recall the definition of orbifold vector bundles. In subsection <ref>, the orbifold fundamental group and the universal covering orbifold are constructed using the -path theory of Haefliger <cit.>. Finally, in subsection <ref>, we definethe holonomy representation for aproper flatorbifold vector bundle. Werestate and show Theorem <ref>.§.§ Definition of orbifolds In this subsection, we recall the definition of orbifoldsfollowing <cit.> and <cit.>. Let Z be a topological space, and let U⊂ Z be a connected open subset of Z. Take m∈.An m-dimensional orbifold chart forU is given by a triple(U, G_U, π_U), where* U⊂𝐑^m is a connected open subset of 𝐑^m;* G_U is a finite group acting smoothly and effectively on the left onU;* π_U: U→U is a G_U-invariant continuousmapwhich induces a homeomorphism of topological spaces G_U\U≃ U. In <cit.>, it is assumed that the codimension of the fixed point set of G_U in U isbigger than or equal to 2. In this article,we do not make this assumption.Let U↪ V be an embedding of connected open subsets of Z, and let (U, G_U, π_U) and (V, G_V, π_V) be respectively orbifold charts for U and V. An embedding of orbifold charts is a smooth embedding ϕ_VU: U→V such that the diagramU[r]^ϕ_VU[d]^π_U V[d]^π_V U@^(->[r]V commutes. We recall the followingproposition. The proof was given by Satake <cit.> under the assumption thatthe codimension of the fixed point set isbigger than or equal to 2. For general cases,see <cit.> for example. Let ϕ_VU:(U, G_U, π_U)↪ (V, G_V, π_V) be an embedding oforbifold charts. The following statements hold:*if g∈G_V, then x∈U→ gϕ_VU(x)∈V is another embedding of orbifold charts. Conversely, any embedding of orbifold charts (U, G_U, π_U)↪ (V, G_V, π_V) is ofsuch form;*there exists a unique injective morphism λ_VU:G_U→ G_V of groups such thatϕ_VU is λ_VU-equivariant;*if g∈ G_V such that ϕ_VU(U)∩ gϕ_VU(U)≠∅, then g is in the image of λ_VU, and soϕ_VU(U)= g ϕ_VU(U). Let U_1, U_2⊂ Z betwo connected open subsets of Z with orbifold charts (U_1, G_U_1, π_U_1) and (U_2, G_U_2, π_U_2). The orbifold charts(U_1, G_U_1, π_U_1) and (U_2, G_U_2, π_U_2) are said to be compatible if for any z∈ U_1∩ U_2, there is an open connected neighborhood U_0⊂ U_1∩ U_2 of zwith orbifold chart (U_0, G_U_0, π_U_0) such that there exist two embeddings of orbifold chartsϕ_U_iU_0: (U_0, G_U_0, π_U_0)↪ (U_i, G_U_i, π_U_i),fori=1,2.The diffeomorphismϕ_U_2U_0ϕ^-1_U_1U_0:ϕ_U_1U_0(U_0)→ϕ_U_2U_0(U_0) iscalled a coordinate transformation. An orbifoldatlas on Z consistsof an open connectedcover ={U} of Z and compatible orbifold charts ={(U, G_U, π_U)}_U∈. An orbifold atlas (,) is called a refinement of (,), if 𝒱 is a refinement ofand if every orbifold chart in 𝒱 has an embedding into some orbifold chart in.Two orbifoldatlases are said to be equivalent if they have a common refinement.The equivalent class of an orbifold atlas is called an orbifold structure on Z. An orbifold is a second countable Hausdorff spaceequipped with anorbifold structure.It said to have dimension m, if all the orbifold charts which define the orbifold structure are of dimension m.Let U, V betwo connected open subsets of an orbifold with respectively orbifold charts (U, G_U, π_U) and (V, G_V, π_V), whichare compatible with the orbifold structure. If U⊂ V, and if U is simply connected, then there exists an embedding of orbifold charts (U, G_U, π_U)↪ (V, G_V, π_V). For any point z of an orbifold, there exists an open connectedneighborhoodU_z⊂ Z of z with a compatible orbifold chart(U_z,G_z,π_z) such that π_z^-1(z) contains onlyone point x∈U_z. Such a chart is called to becentered at x. Clearly,x is a fixed point of G_z. The isomorphism class of the group G_z does not depend on the different choicesof centered orbifold charts, and is called the local group at z. Moreover,we can choose (U_z,G_z,π_z) to be a linear chart centered at 0, which means U_z=^m,x=0∈^m,G_z⊂ O(m). In the sequel, let Z be an orbifold with orbifold atlas(,). We assume that is countable andthateach U∈ is simply connected.When we talk of an orbifold chart, we mean the one which is compatible with . Let us introduce a groupoidassociated to the orbifold Z with orbifold atlas (,). Recall that a groupoid is a category whose morphisms, which are called arrows,are isomorphisms. We define_0,the objects of , to be the countable disjoint unionof smooth manifold _0=∐_U∈U.An arrowg from x_1∈_0 to x_2∈_0, denoted by g:x_1→ x_2, is agerm of coordinate transformation g defined near x_1 such that g(x_1)=x_2. We denote by _1 the set ofarrows.This way defines a groupoid =(_0,_1). By <cit.>, _1 is equipped with a topology such thatis a proper, effective, étale Lie groupoid.For x_1,x_2∈_0, we call x_1 and x_2 in the same orbit if there is an arrowg∈_1 from x_1 to x_2.We denote by _0/_1 the orbit space equipped with the quotienttopology. The projection π_U:U→ U induces a homeomorphism of topological spaces_0/_1≃ Z. Let Y and Z be two orbifolds. Following <cit.>, we introduce:A continuous map f:Y→ Z between orbifoldsis called smooth if for any y∈ Y, there exist*an open connected neighborhood U⊂ Y of y,an open connected neighborhood V⊂ Z of f(y) such that f(U)⊂ V,*orbifold charts (U,G_U,π_U) and (V,G_V,π_V) for U and V,*a smooth map f_U:U→Vsuch that the following diagram U[r]^f_U[d]^π_U V[d]^π_V U[r]^f|_UVcommutes.We denote by C^∞(Y,Z)the space of smooth maps from Y to Z. Two orbifolds Y and Z are called isomorphic if there are smoothmaps f:Y→ Z and f':Z→ Y such that ff'=id and f'f=id. Clearly, this is the case if f:Y→ Z is a smooth homeomorphism such that each lifting f_U is a diffeomorphism. Moreover, in this case, by Proposition <ref>,there is an isomorphism of group ρ_U:G_U→ G_V such that f_U is ρ_U-equivariant. Also, any possible lifting has the form gf_U, g∈ G_U.An action of Lie groupL on Z is said to besmooth, ifthe actionL× Z→ Z is smooth. The following proposition is an extensionof <cit.>.We include a detailed proofsince someconstructions in the proof will be useful to show Theorem <ref>. Let Γ be a discrete group acting smoothly and properly discontinuouslyon the left on an orbifold X.ThenΓ\ X has a canonical orbifold structure induced from X.Let p:X→Γ\X be the natural projection. We equip Γ\ X with the quotient topology.Since X is Hausdorff and second countable, and since the Γ-action is properly discontinuous, then Γ\ X is also Hausdorff and second countable.Take z∈Γ\ X. We choose x∈ X such that p(x)=z. SetΓ_x={γ∈Γ: γ x=x}.As the Γ-action is properly discontinuous, Γ_x is a finite group, and there exists an open connected Γ_x-invariant neighborhood V_x⊂ X of x such that for γ∈Γ- Γ_x,γ V_x∩ V_x=∅.Then, p(V_x)⊂Γ\ Xis an open connected neighborhood of z. Also, we have Γ_x\ V_x≃Γ\Γ V_x=p(V_x). By taking V_x small enough, there is an orbifold chart (V_x, H_x,π_x)for V_x centered at x∈V_x (see Remark <ref>).As Γ acts smoothly on X, we can assume thathomeomorphism of V_x defined by γ_x∈Γ_x lifts to a local diffeomorphism γ_x defined near x such that π_xγ_x=γ_xπ_x holds near x. By Proposition <ref>, the lifting γ_x is not unique, and all possible liftings can be written as h_xγ_x for some h_x∈ H_x. Let G_x be the group of local diffeomorphism defined near x generated by {γ_x}_γ_x∈Γ_x and H_x. Then G_x is a finite group. By choosing V_x small enough and by (<ref>), G_x acts on V_x such thatG_x\V_x≃ p(V_x).Since the G_x-action on V_x is effective, (V_x, G_x,p ∘π_x) is an orbifold chart of Z for p(V_x). The family of open sets {p(V_x)} covers Γ\ X. It remains to show that two suchorbifold charts (V_x_1, G_x_1,p ∘π_x_1) and(V_x_2, G_x_2,p ∘π_x_2) are compatible. Its proof consists of two steps.In the fist step, we consider the case x_2=γ x_1 for some γ∈Γ. We can assume that V_x_2=γ V_x_1, and that γ|_V_x_1 lifts to γ_x_1: V_x_1→V_x_2. Then γ_x_1 defines an isomorphism betweenthe orbifold charts (V_x_1, G_x_1,p ∘π_x_1) and(V_x_2, G_x_2,p ∘π_x_2).In the second step, we consider general x_1,x_2∈ X such that p(V_x_1)∩ p(V_x_2)≠∅. Because of the first step, we can assume that V_x_1∩ V_x_2≠∅. For x_0∈ V_x_1∩ V_x_2, take an open connected neighborhood V_x_0⊂ V_x_1∩ V_x_2 of x_0 and an orbifold chart (V_x_0,H_x_0,π_x_0) of X as before. We canassume that there existtwo embeddingsϕ_V_x_iV_x_0:(V_x_0,H_x_0,π_x_0)↪ (V_x_i,H_x_i,π_x_i), for i=1,2, of orbifoldchartsof X. Then, ϕ_V_x_iV_x_0 also definetwoembeddings of orbifoldcharts of Z,(V_x_0,G_x_0,p ∘π_x_0)↪ (V_x_i,G_x_i,p ∘π_x_i). The proof our proposition is completed.By the construction, H_x is a normal subgroup of G_x, and γ_x→γ_x induces a surjective morphism of groupsΓ_x→ G_x/H_x. If the action of Γ on X is effective,then (<ref>) is anisomorphism of groups. Thus, the followingsequence of groups 1→ H_x→ G_x→Γ_x→1 is exact.§.§ Singular set of orbifoldsLet Z be an orbifold with orbifold atlas (,). PutZ_ reg={z∈ Z: G_z={1}}, Z_ sing={z∈ Z: G_z≠{1}}.Then Z=Z_ reg∪ Z_ sing. Clearly, Z_ reg is a smooth manifold. However, Z_ sing is not necessarilyan orbifold.Following <cit.>,we will introduce the orbifold resolution Σ Z forZ_ sing. Let [G_z] be the set of conjugacy classes of G_z. SetΣ Z={(z,[g]): z∈ Z, [g]∈ [G_z]- {1}}.By <cit.>, Σ Z possess a naturalorbifold structure. Indeed, take U∈ and (U,π_U,G_U)∈. For g∈ G_U, denote byU^g⊂U the set of fixed points of gin U, and by Z_G_U(g)⊂ G_U the centralizerof g in G_U. Clearly, Z_G_U(g) acts on U^g, andthe quotient Z_G_U(g)\U^g depends only onthe conjugacy class[g]∈ [G_U].The map x∈U^g→ (π_U(x),[g])∈Σ U induces an identification ∐_[g]∈ [G_U]\{1} Z_G_U(g)\U^g≃Σ U.By (<ref>), weequip Σ U withthe induced topologyandorbifold structure. The topology and the orbifold structure on Σ Z is obtained by gluing Σ U. We omit the detail. We decomposeΣ Z=∐_i=1^l_0Z_i following its connected components. If (z,[g])∈ Z_i, setm_i=|(Z_G_U(g)→ Diffeo(U^g))|∈^*.By definition, m_i is locally constant, and is called the multiplicity of Z_i.In the sequel, we writeZ_0=Z, m_0=1. §.§ Orbifold vector bundleWe recall the definition oforbifold vector bundles.A complexorbifold vector bundle E of rank r on Z consists of an orbifold , called the total space, and a smooth map π:→ Z, such that *there is an orbifold atlas (,) of Z such thatfor any U∈ and (U,G_U,π_U)∈, there exist a finite group G_U^E acting smoothly on U which induces a surjective morphism of groups G_U^E→ G_U,a representation ρ^E_U: G^E_U→_r(), and a G^E_U-invariant continuous map π^E_U:U×^r →π^-1(U)which induces a homomorphism of topological spacesU[_G^E_U]×^r ≃π^-1(U);*the triple (U×^r, G^E_U,π^E_U) is a (compatible)orbifold chart on ; *ifU_1,U_2∈ such that U_1∩ U_2≠∅, and for any z∈ U_1∩ U_2, there exist a connected open neighborhoodU_0⊂ U_1∩ U_2 of z with a simply connected orbifold chart (U_0,G_U_0,π_U_0) and the triple (G_U_0^E, ρ^E_U_0, π^E_U_0) such that(1) and (2) hold, and that theembeddings of orbifold charts ofϕ^E_U_iU_0: U_0×^r,G^E_U_0,π^E_U_0↪U_i×^r, G^E_U_i,π^E_U_i, fori=1,2,have the followingformϕ^E_U_iU_0(x,v)=(ϕ_U_iU_0(x),g^E_U_iU_0(x) v), for(x,v)∈U_0×^r,where ϕ_U_iU_0:(U_0,G_U_0,π_U_0)↪ (U_i,G_U_i,π_U_i) is an embedding of orbifold charts of Z, andg^E_U_iU_0∈ C^∞(U_0,_r()).The vector bundle E is called proper if the surjective morphismG_U^E→ G_U is an isomorphism, and is called flat ifg^E_U_iU_0 can be chosen to be constant. The embedding ϕ^E_U_iU_0 exists since U_0×^r is simply connected. By Proposition <ref>, it is uniquelydetermined by the first component ϕ_U_iU_0 when E is proper.We can define the real orbifold vector bundle in an obvious way.In the sequel, for U∈, we will denote by E_U the trivial vector bundle of rank r on U, and by E_U the restriction of E to U. Their total spaces are given respectivelyby _U=U×^r,_U=U[_G_U]×^r.Let us identifythe associated groupoid ^E=(^E_0,^E_1) for the total space of a proper orbifold vector bundle E. By (<ref>), the object of ^E is given by ^E_0=∐_U∈_U=_0×^r. If g∈_1 is represented by the germ of the transformationϕ_U_2U_0ϕ_U_1U_0^-1, denote by g^E_*the germ of transformation g^E_U_2U_0g^E,-1_U_1U_0. By Remark <ref>,g^E_* is uniquely determined by g. Thus, if g is an arrow form x, and if v∈^r, (g,v) defines an arrow from (x,v) to (gx,g^E_*v). This way givesan identification ^E_1=_1×^r. We give some examples of orbifold vector bundles.The tangent bundle TZ of an orbifold Z is a real proper orbifold vector bundle locally defined by {(TU,G_U)}_U∈.Assume Z is covered by linear charts {(U, G_U,π_U)}_U∈ (see Remark <ref>). The orientation line o(TZ) isa real proper orbifoldline bundle on Z, locally defined by (U×, G_U) where the action of g∈ G_U is given by g:(x,v)∈U×→ (gx,( sign(g))v)∈U×.Clearly, o(TZ) is flat. If o(TZ) is trivial, Z is called orientable.If E,F are orbifold vector bundles on Z, thenE^*, E,Λ^· (E), 𝒯(E)=⊕_k∈ E^⊗ k and E⊗ F aredefined in an obvious way.Let E be an orbifold vector bundle on Z. For U∈, letV_U⊂^r be subspace of ^r of the fixed points of (G_U^E→G_U). Then G_U acts on V_U, and {(U× V_U, G_U)}_U∈ defines a proper orbifold vector bundle E^ pr on Z. Clearly, if E is flat, then E^ pr is also flat.A smooth section of E is defined by a smoothmap s: Z→ E inthe sense of Definition <ref> such that π∘ s=id and thateach local lift s_Uof s|_U is G^E_U-invariant. The space of smoothsections of E is denoteby C^∞(Z,E). The space of differential forms with values in E is defined byΩ^·(Z,E)=C^∞(Z,Λ^·(T^*Z)⊗_ E). For k∈, we define C^k(Z,E) in a similar way. Also, the space of distributions '(Z,E) of E is defined by the topologicaldual ofC^∞(Z,E^*). By definition, we have C^∞(Z,E)=C^∞(Z,E^ pr).For this reason, most of results in thispaper can be extended to non proper flat vector bundles. Assume now E is proper.By (<ref>), s∈ C^∞(Z,E) can be represented by{s_U∈ C^∞(U,E_U)^G_U}_U∈ a family of G_U-invariant sections such that for any x_1∈ U_1,x_2∈ U_2 and g∈_1 from x_1 to x_2, near x_1 we haveg^*s_U_2=s_U_1.We have thesimilar description for elements of C^∞(Z,E) and D'(Z,E).We call g^Ea Hermitian metric on E, if g^E is a section inC^∞(Z,E^*⊗E^*) such that g^E is represented by a family{g^E_U}_U∈Uof G_U-invariant metrics on E_U such that (<ref>) holds. If E is the real orbifoldvector bundle TZ, g^TZ is called a Riemannianmetric on Z.Two orbifold vector bundles E and F are called isomorphic ifthere is f∈ C^∞(Z,E^*⊗ F) and g∈C^∞(Z,F^*⊗ E) such that fg=id andgf=id. Let Γ be a discrete group acting smoothly and properly discontinuously on an orbifold X. Letρ:Γ→_r() be a representation of Γ. By Proposition <ref>, Γ\ X and =X[_Γ]×^rhave canonicalorbifold structures.Theprojection X×^r→ X descends to a smoothmapof orbifoldsπ:→Γ\ X. Assume that the action of Γ on X is smooth, properly discontinuous and effective. Then (<ref>) defines canonicallya proper flat vector bundle F on Γ\ X.Recall that p:X→Γ\ X is the projection.For x∈ X, we use the same notations Γ_x, V_x, (V_x, H_x) and G_xas in the proof of Proposition <ref>. Then, Γ\ X is covered by p(V_x)≃Γ_x\ V_x≃ G_x\V_x. The stabilizer subgroupof Γ at (x,0)∈ X×^r is Γ_x. By (<ref>),if γ∈Γ-Γ_x,γ (V_x×^r)∩ (V_x×^r)= ∅. As in (<ref>), we have π^-1(p(V_x))=Γ\Γ(V_x×^r)≃ V_x_Γ_x×^r.Since the action of Γ on X is effective, by Remark <ref>, we have a morphism of groups G_x→Γ_x.The group G_x acts on ^r via the composition of G_x→Γ_x and ρ|_Γ_x:Γ_x→_r(). Thus,G_x acts on V_x×^r effectively such that V_x [_Γ_x]×^r≃V_x [_G_x]×^r.By Proposition <ref>, (V_x×^r,G_x) is an orbifold chart offor π^-1(p(V_x)). Take two (V_x_1×^r, G_x_1) and (V_x_2×^r, G_x_2) orbifold charts of . It remains to show the compatibility condition (<ref>).We proceed as in the proof of Proposition <ref>. If x_2=γ x_1, then (x,v)∈V_x_1×^r→ (γ_x_1 x,ρ(γ)v)∈V_x_2×^rdefines an isomorphism of orbifoldcharts on . For general x_1, x_2∈ X, we can assumethat V_x_1∩ V_x_2≠∅. For x_0∈ V_x_1∩ V_x_2, takeV_x_0,V_x_0 and ϕ_V_x_iV_x_0 as in the proof of Proposition <ref>.Then(x,v)∈V_x_0×^r→ (ϕ_V_x_iV_x_0(x),v)∈V_x_i×^rdefinetwo embeddings of orbifold charts of .From(<ref>) and (<ref>), we deduce that (<ref>)definesa flat orbifold vector bundle on Γ\ X. Theproperness is clear from the construction.The proof ofour propositionis completed. Take A∈_r(). Let ρ_A:γ∈Γ→ Aρ(γ) A^-1∈_r() be another representation of Γ. Then(x,v)∈X ×^r→ (x,Av) ∈ X ×^rdescends to an isomorphism betweenflat orbifold vector bundles X [_ρ]×^r andX [_ρ_A]×^r.§.§ Orbifold fundamental groups and universalcovering orbifoldIn this subsection, following <cit.>,<cit.>, we recall the constructions of the orbifold fundamental group and the universal covering orbifold. We assume that the orbifold Z is connected. Letbe the groupoid associated withsome orbifold atlas (, ). A continuous -path c=(b_1, …, b_k;g_0,…,g_k) starting at x∈_0 and ending at y∈_0 parametrized by [0,1] is given by*a partition 0=t_0<t_1<⋯<t_k=1 of [0,1];*continuouspaths b_i: [t_i-1,t_i] →_0, for 1≤ i≤ k;*arrows g_i∈_1 such that g_0:x→b_1(0), g_i:b_i(t_i)→b_i+1(t_i), for 1≤ i≤ k-1, and g_k:b_k(1)→ y.If x=y, we call that c is a -loop based at x.Two-paths c=(b_1, ⋯, b_k; g_0,⋯,g_k), c'=(b'_1, ⋯, b'_k'; g'_0,⋯,g'_k'),such thatc ending at y and c' starting at y can be composed into a-path (with a suitable reparametrization<cit.>),cc'=(b_1, ⋯, b_k,b'_1, ⋯, b'_k';g_0,⋯,g'_0g_k,⋯,g'_k').Also, we can define the inverse of a -path in an obvious way.We define an equivalence relation on -paths generated by*subdivision of the partition and adjunction by identity elements of _1 on new partition points.*for some 1≤ i_0≤ k, replacement of the triple(b_i_0, g_i_0-1, g_i_0) bythe triple (hb_i_0, hg_i_0-1, g_i_0 h^-1),where h∈_1 is well-definednear the path b_i_0([t_i_0-1,t_i_0]).The equivalent class of -paths is called the path on theorbifold Z.If Z is equipped with a Riemannian metric, then the length of a pathon Zrepresented by the -path c=(b_1,…,b_k;g_0,…,g_k) is defined by the sum of the lengths of b_i. Clearly, this definition does not depend on the choice of the representative c. The set of paths on Z with length 0 is just the orbifold Z∐Σ Z. Following <cit.>, if Z is equipped with a Riemannian metric, a -path c=(b_1,…,b_k;g_0,…,g_k) is called a -geodesic, if for all 1≤ i≤ k,b_i is a geodesic and if for all 1≤ i≤ k-1, g_i,*ḃ_i(t_i)=ḃ_i+1(t_i). The geodesic on Z is defined by the equivalence class of the -geodesics. Similarly, we can define the closed geodesic on Z by the equivalent class of the closed geodesic -paths, i.e., a -geodesic starting and ending at the same point such that g_0,*g_k,*ḃ_k(1)=ḃ_0(0).An elementary homotopy between two -paths c and c' isafamily, parametrized by s ∈ [0,1], of -pathsc^s =(b^s_1,⋯,b^s_k; g_0^s,⋯, g_k^s), over thesubdivisions 0 = t_0^s ≤t_1^s ≤⋯≤ t_k^s = 1, wheret_i^s , b^s_i and g_i^s depend continuously on the parameter s, the elements g_0^s and g_k^s are independent of s and c^0 = c, c^1 = c'. Two -paths are said to be homotopic (with fixed extremities) if one can pass from the first to the second by equivalences of -paths and elementary homotopies. The homotopy class of a -path c will be denoted by [c].As ordinary paths in topological spaces, the composition and inverse operations of -paths are well-defined for theirhomotopy classes. Take x_0∈_0.With the operations of composition and inverse of -paths, the homotopy classes of -loops based at x_0 form a group π_1^ orb(Z,x_0) called the orbifold fundamental group. As Z is connected, any two points of _0 can be connected by a -path. Thus, the isomorphic class of the group π_1^ orb(Z,x_0)does not depend on the choice of x_0. Also, it depends only on the orbifold structure of Z. In the sequel, for simplicity, we denote by Γ=π^ orb_1(Z,x_0). As a fundamental group of a manifold, Γ is countable.In the rest of this subsection, following<cit.>, we will construct the universalcovering orbifold X of Z.Let us begin with introducing a groupoid .Assume that={U_z} and ={(U_z,G_z,π_z)} where all the U_z are simply connected and are centered at x∈U_z as in Remark <ref>.Fix x_0∈_0 as before. Let _0 be the space of homotopy classes of-paths starting at x_0.The group Γ acts naturally on _0 by compositionat the starting point x_0. We denote by p:_0→_0the projection sending [c]∈_0 to its ending point. Clearly, p is Γ-invariant.Define a topology and manifold structure on _0 as follows.For x_1,x_2∈U_z, we denote by c_x_1x_2=(b_x_1x_2;id,id) a -pathstarting at x_1 and ending atx_2, where b_x_1x_2 is a pathin U_z connecting x_1 and x_2. Note that since U_z is simply connected, the homotopy class [c_x_1x_2] does not depend on the choice of b_x_1x_2. For each U_z∈, we fix a -path c_z starting at x_0 andending at x∈U_z. For a∈Γ, setV_z,a={[c]∈p^-1(U_z): c c_p([c])xc_z^-1 =a}.By (<ref>) and (<ref>), we have p^-1(U_z)=∐_a∈ΓV_z,a,_0=∐_U_z∈,a∈ΓV_z,a.Also,p:V_z,a→U_zis a bijection. We equip V_z,a with a topology and a manifold structurevia (<ref>). Clearly, the choice of c_z is irrelevant.By (<ref>), _0 is a countable disjoint union of smooth manifolds such that (<ref>) is a Galois covering with deck transformation group Γ. If y∈_0 and if g∈_1 is defined neary, we denote byc_y,g=(b_y;id, g) the -path,where b_y is the constant path at y.Set_1={([c], g)∈_0×_1: gis defined near p([c])∈_0 }.Then, ([c], g)∈_1 represents an arrow from [c] to [c][c_p[c],g]. Thisdefines a groupoid =(_0,_1). Let X=_0/_1 be the orbit space ofequipped with the quotient topology.The action of Γ on _0 descends to an effective and continuous action on X.The projection p descends toa Γ-invariant continuous mapp:X→ Z. Assume that Z is a connected orbifold. Then, the topological space X defined in (<ref>) is connected and has a canonical orbifold structure such that Γ acts smoothly, effectively andproperly discontinuously on X. Moreover, (<ref>) induces an isomorphism of orbifoldsΓ\ X→Z. Let us begin with showing that the topological space X is connected. Take any -path c=(b_1, ⋯, b_k; g_0,⋯,g_k) starting at x_0. By our construction of _1, the imagesin X of the -paths c and(b_1, ⋯, b_k; g_0,⋯,g_k-1,id) are in the same connected component of X. The same holds true forthe -paths (b_1, ⋯, b_k-1; g_0,⋯,g_k-1) and(b_1, ⋯, b_k; g_0,⋯,g_k-1, id). By induction argument, the images in X of c and the constant -path at x_0 are in the same connected component of X. So X is connected. Let us construct an orbifold atlas on X. For a∈Γ, letπ_z,a be the composition of continuous maps V_z,a↪_0→ X. SetV_z,a=π_z,a(V_z,a)⊂ X. By (<ref>),p^-1(U_z)=⋃_a∈Γ V_z,a.Recall that x∈U_z andπ_z(x)=z. By (<ref>), for g∈ G_z, c_x,g is a -loop based at x. Setr_z:g∈ G_z→ [c_z c_x,gc_z^-1]∈Γ.Then, r_z is amorphism of groups. By(<ref>) and (<ref>), p^-1(U_z)=∐_[a]∈Γ/ (r_z) V_z,a.Using the fact that p^-1(U_z) is open in X, we can deducethat V_z,a is open in X. Put H_z= r_z.By (<ref>) and (<ref>), H_z acts on V_z,a by (g,[c])∈ H_z×V_z,a→ [c][c_p([c]),g^-1]∈V_z,a.Then π_z,a induces a homeomorphismof topological spacesH_z\V_z,a≃ V_z,a.As the H_z-action on V_z,a is effective, (V_z,a,H_z,π_z,a) is an orbifold chart for V_z,a. Moreover, the compatibility of each charts is a consequence of (<ref>) and the compatibility of charts in . Hence, {(V_z,a,H_z,π_z,a)}_U_z∈,a∈Γ forms an orbifold atlas on X. As Γ is countable, X is second countable. We will show that X is Hausdorff. Indeed, take y_1,y_2∈ X and y_1≠ y_2.If p(y_1)≠ p(y_2), as Z is Hausdorff, take respectively open neighborhoods U_1 and U_2 of p(y_1) and of p(y_2) such that U_1∩ U_2=∅. Then p^-1(U_1)∩ p^-1(U_2)=∅. Assume p(y_1)=p(y_2). By adding charts in , we can assume that there is U_z∈ such thatp(y_1)=p(y_2)=z with orbifold charts (U_z,G_z,π_z) centered at x. Assume y_1, y_2 are represented by -paths c_1 and c_2 starting at x_0 and ending at x. For i=1,2, seta_i=[c_i][c^-1_z_0]∈Γ.As y_1≠y_2, then [a_1]≠ [a_2]∈Γ/(r_z). Thus,y_1∈ V_z,a_1, y_2∈ V_z,a_2, V_z,a_1∩ V_z,a_2=∅. In summary, we have shown that X is an orbifold.Note that γ V_z,a=V_z,γ a. By (<ref>), the set{γ∈Γ: γ V_z,a∩ V_z,a≠∅}=a(r_z)a^-1⊂Γis finite. Then the Γ-action on X is properly discontinuous. As Γ acts on _0,the Γ-action is smooth.We claim that (<ref>) is homeomorphism of topological space. Indeed, by the construction,(<ref>) is injective. It is surjective asZ is connected. The continuity of the inverse (<ref>)is a consequence of(<ref>).The isomorphism of orbifolds between Γ\ X and Z is a consequence of Proposition <ref> and(<ref>), (<ref>) and (<ref>). The proof our theorem is completed.By (<ref>), (<ref>), andby the covering orbifoldtheory of Thurston <cit.>, p:X→ Z is a covering orbifoldof Z. Moreover, we can show that for anyconnected covering orbifold p':Y→ Z, there exists a covering orbifold p”:X→ Y such thatthe diagramX[dd]_p [dr]^p” Y[dl]^p' Zcommutes.For this reason, X is called a universal covering orbifold of Z.As in the case of the classical covering theory of topologicalspaces, the universal covering orbifoldis unique up to covering isomorphism. Also, Γ is isomorphic tothe orbifold deck transformation group of X. If a connected covering orbifold X' of Z has a trivial orbifold fundamental group, then X' is a universal covering orbifold of Z. In particular, if Z has acovering orbifold X', which is a connected simply connected manifold, then X' is a universal covering orbifold of Z. The teardropZ_n with n2 (see Figure <ref>) is an example of an orbifold with a trivial orbifold fundamental group which is not a manifold (c.f. <cit.>). Its underlying topological space is a 2-sphere 𝕊^2, and its singular set consists of a single point, whose neighbourhood is modelled on 𝐑^2/(𝐙/n𝐙), where the cyclic group 𝐙/n𝐙 acts by rotations. §.§ Flat vector bundles and holonomy In this subsection, we still assume that Z is a connected orbifold. Let F be a proper flat orbifoldvector bundle on Z.Let (,) be an orbifold atlas as in Definition <ref>. Let bethe associated groupoid. We fix x_0∈.For a -path c=(b_1,⋯,b_k;g_0,⋯,g_k), the parallel transport τ_c of F along c is defined by τ_c= g_k,*^F⋯ g^F_0,*∈_r().Itdepends onlyon the homotopy class of c. In particular, it defines a representation, called holonomy representation of F,ρ: Γ→_r().The isomorphicclass of the representation ρ is independent of the choice of orbifold atlas on Z, of the localtrivialization of F,andof the choice of x_0. Moreover, it does not depend on the isomorphic class of F.Let (Γ,_r())/_∼ be the set of equivalent classes of complex representations of Γ of dimensionr, andlet ^ pr_r(Z) be the set of isomorphic classes of proper complexflat orbifold vector bundles of rank r on Z.By Proposition <ref> and Remark <ref>, the mapρ∈(Γ,_r())/_∼→ X_ρ×^r∈^ pr_r(Z)is well-defined. The map (<ref>) is one-one and onto, whose inverse is given by the holonomy representation (<ref>).Step 1. The holonomy representation ofX _ρ×^r is isomorphic to ρ.Assume that the orbifold Z is covered by {U_z} with simply connected orbifold charts {U_z} centered at x∈U_z and X is covered by {V_z,a} as (<ref>) such that p(V_z,a)= U_z.Take γ∈Γ. Let c=(b_1,⋯,b_k;g_0,⋯,g_k) be a -loop based at x_0 which represents γ. It is enough to show the parallel transport along c is ρ(γ). For 1≤ i≤ k, take x_i=b_i(t_i-1).Up to equivalence relation of c and up to adding charts into the orbifold atlas of Z, we can assume that there are orbifold charts U_z_i of Z centered at x_i such thatb_i:[t_i-1,t_i]→U_z_i. Also, we assume that U_z_0 is an orbifold chart of Z centered at x_0. Let c_z_1=c_x_0,g_0 as in (<ref>). For 2≤ i≤ k, set c_z_i=(b_1,⋯,b_i-1;g_0,⋯,g_i-1).By (<ref>), for 1≤ i≤ k, c_z_i is a -path starting at x_0 and ending at x_i such that [c_z_i]∈V_z_i,1.We claim that for 1≤ i≤ k, V_z_i-1,1∩ V_z_i,1≠∅,V_z_k,1∩ V_z_0,γ≠∅.Indeed,c'_z_i=(b_1,⋯,b_i-1;g_0,⋯,g_i-2,id) projects to the same element of X as c_z_i, and[c'_z_i]∈V_z_i-1,1. Also, c'_z_k+1 projects to the same element of X as c. Recall that γ V_z_0,1=V_z_0,γ. By (<ref>), (<ref>) and (<ref>), for 1≤ i≤ k-1, we haveg_i,*=1, g_k,*=ρ(γ). By (<ref>),the parallel transport along c is ρ(γ).Step 2. If F has holonomy ρ, then F is isomorphic to X _ρ×^r. We will construct the bundle isomorphism. By (<ref>) and (<ref>), the groupoid of the total spaceis given by ^F=_0×^r,_1×^r. Let usconstruct a universal covering orbifold ofby determiningits groupoid ^F.Take (x_0,0),(x_1,u)∈^F_0. Let (c,v) be a ^F-path starting at (x_0,0) and ending at (x_1,u).Then there is a partition of [0,1] given by 0=t_0<⋯<t_k=1 such thatc=(b_1,⋯,b_k;g_0,⋯,g_k) as in Definition <ref>. Also, v=(v_1,⋯,v_k), where v_i:[t_i-1,t_i]→^r is a continuous path such that v_1(0)=0,g^F_k,*v_k(1)=u and for 1≤ i≤ k-1,g^F_i,*v_i(t_i)=v_i+1(t_i). Put w:[0,1]→^r a continuous path such that for t∈ [t_i-1,t_i],w(t)=g^F,-1_0,*⋯ g^F,-1_i-1,* v_i(t).We identify (c,v) with (c,w) via (<ref>). Then, (c,v) is homotopic to (c',v') if and only if c,c' are homotopic as -path and w,w' arehomotopic as ordinary continuous paths in ^r.Since any continuous path w:[0,1]→^r such that w(0)=0 is homotopic to the path t∈ [0,1]→ tw(1), we have the identification[c,v]∈_0^F→ ([c],w(1))=([c],τ^-1_cu)∈_0×^r.In particular,we have an isomorphism of groups[c]∈Γ→ [(c,0)]∈π^ orb_1(,(x_0,0)),where 0 is the constant loop at 0∈^r.In the same way, we identify ([c,v],g)∈_1^F→(([c],g), w(1)) ∈_1×^r.We deducethat (([c],g), w(1)) represents an arrow from([c],w(1))∈^F_0 to([cc_g],w(1))∈_0^F. Therefore,the orbitspace of ^F, which is also the universal covering orbifold of , is given byX×^r.By the identification (<ref>), the projection (<ref>) is given by p_ρ:([c],w(1))∈_0×^r→(p([c]),w(1))∈_0×^r.The group Γ acts on the left on _0, and on the left on ^r by ρ. As in (<ref>),the projection (<ref>) is a Galois covering with deck transformation group Γ. And p_ρ descends to a Γ-invariant continuous mapp_ρ:X×^r→.By Theorem <ref> , p_ρ induces an isomorphism of orbifolds X [_ρ]×^r≃.Using the fact that (<ref>) is linear on the ^r,we can deduce that (<ref>) is an isomorphism oforbifold vector bundles.The proof of our theorem is completed. The properness condition is necessary. Indeed, Theorem <ref> implies that the proper flat vector bundle is trivial on the universal cover. Consider a non trivialfinite group H acting effectively on ^r. Then H\^r is a non proper orbifold vector bundle over a point. Clearly, it is not trivial. By (<ref>) and Theorem <ref>, we get Corollary <ref>.§ DIFFERENTIAL CALCULUS ON ORBIFOLDSThe purpose of this section is to explain brieflyhow to extend the usual differentialcalculus to orbifolds. To simplify our presentation, we assume thatthe underlyingorbifold is compact. We assume also thattheorbifold vector bundles are proper. By (<ref>), all theconstructions in this section extend trivially to non proper orbifold vector bundles.This section is organized as follows. In subsections <ref>-<ref>, we introducedifferential operators, integration of differential forms, integral operators andSobolev space on orbifolds.In subsection <ref>,we explain Chern-Weil theory forthe orbifold vector bundles. The Euler form, odd Chern character form, their Chern-Simons classes, and their canonical extensions to Z∐Σ Z are constructed in detail. §.§ Differential operators on orbifoldsLet Z be a compact orbifold with atlas (,). LetE be a proper orbifold vector bundle on Z such that (<ref>) holds.A differential operator D of order p is a family{D_U:C^∞(U,E_U)→C^∞(U,E_U)}_U∈ ofG_U-invariant differential operatorsof order p such thatifg∈_1 is an arrowfrom x_1 ∈U_1 to x_2∈U_2, then near x_1, we haveg^* D_U_2=D_U_1.If each D_U is elliptic, then D is called elliptic. If s∈ C^∞(Z,E) is represented by the family {s_U∈ C^∞(U,E_U)^G_U} such that (<ref>) holds. By (<ref>) and (<ref>), {D_Us_U∈ C^∞(U,E_U)^G_U}_U∈ defines a section of E, which is denoted by Ds. Clearly, D:C^∞(Z,E)→ C^∞(Z,E) is a linear operator such that (Ds)⊂(s).As in the manifold case,the differential operator acts naturally on distributions. A connection ∇^E on E is a firstorder differential operator from C^∞(Z,E) to Ω^1(Z,E) such that∇^E is represented bya family {∇^E_U}_U∈U of G_U-invariant connections on E_U such that (<ref>) holds.The curvature R^E=(∇^E)^2 is defined as usual. It is a sectionof Λ^2(T^*Z)⊗_(E). As usual, ∇^Eis called metric with respect to a Hermitian metric g^E if ∇^Eg^E=0. Let (Z,g^TZ) be a Riemannian orbifold.If g^TZ isdefined by the family{g^TU}_U∈ of Riemannian metrics, then the family ofLevi-civita connections on (U,g^TU) defines the Levi-civita connection ∇^TZ on(Z,g^TZ). Let F be aflat orbifoldvector bundle on Z. The de Rham operator d^Z: Ω^·(Z,F)→Ω^·+1(Z,F) isa first order differential operatorrepresented by the family of de Rhamoperators{d^U:Ω^·(U,^r)→Ω^·+1(U,^r)}_U∈.Clearly,(d^Z)^2=0. ThecomplexΩ^·(Z,F),d^Z iscalled the orbifold de Rham complex with values in F.Denote byH^·(Z,F) the corresponding cohomology. When F= is the trivial bundle, we denote simply by Ω^·(Z) and H^·(Z).[ By Satake <cit.>, H^·(Z) coincides with thesingular cohomology of the underlying topological space Z. Ingeneral, H^·(Z,F) coincides with the cohomology of the sheaf oflocally constant sections of F.] Clearly, ∇^F=d^Z|_C^∞(Z,F) defines aconnection on F with vanishing curvature. As in manifold case, sucha connection will be called flat. We say F is unitarily flat, ifthere exists a Hermitian metric g^F on F such that ∇^Fg^F=0. Clearly, this is equivalent to say the holonomy representation ρ is unitary. §.§ Integral operatorson orbifolds Since Z is Hausdorff and compact, there exists a (finite) partition of unity subordinate to . That meansthere is a (finite) familyof smooth functions {ϕ_i∈C^∞_c(Z,[0,1])}_i∈ Ion Zsuch thatthe supportϕ_i is contained in some U_i∈,and that∑_i∈ Iϕ_i=1.Denote by ϕ_i=π^*_U_i(ϕ_i)∈ C^∞_c(U_i)^G_U_i. Following <cit.>, for α∈Ω^·(Z,o(TZ)) which is representedby the invariantforms {α_U_i∈Ω^·(U_i,o(TU_i))^G_U_i}_i∈ I, define∫_Zα=∑_i∈ I1/|G_U_i|∫_U_iϕ_iα_U_i.By (<ref>) and (<ref>), we get:If α∈Ω^·(Z,o(TZ)), then α is integrable on Z_ reg such that∫_Zα=∫_Z_ regα. From (<ref>), we see that the definition (<ref>) doesnot depend on the choice of orbifold atlas and the partition ofunity. Also, we have the orbifold Stokes formula. The following identity holds: for α∈Ω^·(Z,o(TZ)),∫_Zd^Zα=0. Let us introduceintegraloperators. Let (E,g^E) be a Euclidean orbifold vectorbundle on Z. Fix avolumeform dv_Z∈Ω^·(Z,o(TZ)) of Z.Then, we can define thespace L^2(Z,E) of L^2-sections in anobviously way. By (<ref>), we haveL^2(Z,E)=L^2(Z_ reg,E_ reg).As in manifold case, with the help of dv_Z, we have the natural embeddingC^∞(Z,E)→𝒟'(Z,E). By our local description of smooth sections anddistributions (<ref>), theSchwartz kernel theorem still holds for orbifolds.That meansfor any continuous linear map A:C^∞(Z,E)→'(Z,E),there exists a unique p∈'(Z× Z,E⊠ E^*) such that for s_1∈ C^∞(Z,E) and s_2∈ C^∞(Z,E^*), we haveAs_1,s_2=p,s_2⊗ s_1.Assume that p is of class C^k for some k∈. Then A is called integral operator.The restriction of p to the regular part defines a bounded section p_ reg∈ C^k(Z_ reg× Z_ reg,E_ reg⊠ E^*_ reg) such thatfor s∈ C^∞(Z,E) and z∈ Z_ reg, As(z)= ∫_z'∈ Z_ reg p_ reg(z,z')s(z') dv_Z_ reg.Using (<ref>), A extends uniquely to a bounded operator onL^2(Z,E). Moreover, since p_ reg(z,z') is bounded, ∫_(z,z')∈ Z_ reg× Z_ reg |p_ reg(z,z')|^2dv_Z_ reg× Z_ reg<∞.Then A is in the Hilbert-Schmidt class.If A is in the traceclass, then A =∫_z∈ Z_ reg^E p_reg(z,z) dv_Z_ reg=∫_z∈ Z^E p(z,z) dv_Z. Now we give another description ofintegral operators. For any local chart U, thereis a G_U-invariant integral operatorA_U:C^∞(U,E_U)→C^∞(U,E_U) with integral kernelp_U∈ C^k(U×U,E_U⊠E_U^*) such that if s∈C^∞_c(U,E|_U), then As|_U is defined by the invariantsection A_Us_U(x)=∫_x'∈Up_U(x,x')s_U(x')dv_U.Then the restriction of the integral kernel p on U× U isrepresented by the invariant section (see <cit.>)∑_g∈ G_U gp_U(g^-1x,x')∈C^k(U×U,E_U⊠E_U^*)^G_U× G_U.If A is in trace class, we have [A]=∑_i∈ I1/|G_U_i|∑_g∈ G_U_i∫_x∈U_iϕ_i(x) gp_U_i(g^-1x,x) dv_U_i.§.§ Sobolev space on orbifolds Let (Z,g^TZ) be a compact Riemannianorbifold of dimension m. Let ∇^TZ be the Levi-civita connection on TZ, and letR^TZ be the corresponding curvature.Let (E,g^E) be a properHermitianorbifold vector bundle with connection ∇^E.When necessary, we identity E with E^* via g^E.Denote still by ∇^𝒯(T^*Z)⊗_ E the connection 𝒯(T^*Z)⊗_ E induced by ∇^TZ and ∇^E. For q∈, take ^q(Z,E) to be the Hilbert completion of C^∞(Z,E) under the norm defined by s^2_q=∑_j=0^q∫_Z|∇^𝒯(T^*Z)⊗_ E^js(z)|^2dv_Z. Let ^-q(Z,E) be the dual of ^q(Z,E).If q∈, we can define ^q(Z,E) by interpolation. As in the case of smooth sections, s∈^q(Z,E) can be represented by the family {s_U ∈^q(U,E_U)^G_U}_U∈of G_U-invariant sectionssuch that (<ref>) holds. Using these local descriptions, we have ⋂_q∈^q(Z,E)=C^∞(Z,E), ⋃_q∈^q(Z,E)='(Z,E).Moreover, if q>q', we have the compact embedding ^q(Z,E)↪^q'(Z,E), and if q∈ and q>m/2, we have the continuous embedding^q(Z,E)↪ C^q-[m/2](Z,E).§.§ Characteristic forms on orbifoldsAssume now (E,g^E) is areal Euclidean proper orbifold vector bundle of rank rwith a metric connection ∇^E. The Euler form e(E,∇^E)∈Ω^r(Z,o(E)) is defined by the family of closed forms {e(E_U,∇^E_U) }_U∈. Following <cit.>, theorbifold Euler characteristic number is defined by χ_ orb(Z)=∫_Z eTZ,∇^TZ.If ∇^E' is another metric connection,theclass of Chern-Simons form e(E,∇^E, ∇^' E)∈Ω^r-1(Z,o(E))/dΩ^r-2(Z,o(E))is defined by the family{e(E_U,∇^E_U,∇^'E_U)}_U∈. Clearly, (<ref>) still holds true.Let (F,∇^F) be a proper orbifoldflat vector bundle on Zwith a Hermitian metric g^F. The odd Chern character h∇^F,g^F∈Ω^ odd(Z) of (F,∇^F) is defined by the family of closed odd forms{h(∇^F_U,g^F_U) }_U∈. Ifg^' F is another Hermitian metric on F, the class of Chern-Simons form h(∇^F,g^F,g^' F)∈Ω^ even (Z)/d Ω^ odd(Z)is defined by the family{h(∇^F_U,g^F_U,g^'F_U)}_U∈. As before, (<ref>) still holds true. The degree 1-part of h(∇^F,g^ F) and the degree 0-part of h(∇^F,g^ F,g^' F)will be especially important in the formulation of Theorem <ref>. We denote by θ∇^F,g^F=2h∇^F,g^ F^[1],θ∇^F,g^F,g^' F=2h∇^F,g^ F,g^' F^[0].By (<ref>)-(<ref>) and (<ref>), we haveθ∇^F,g^F= g^F^-1∇^Fg^F ,d^Zθ∇^F,g^F,g^' F=θ∇^F,g^' F-θ∇^F,g^ F.Let ·_F and ·^'_F be the metrics on the line bundle F induced by the metricsg^F and g^' F. By <cit.>, we have θ∇^F,g^F,g^'F=log·^'_ F/·_F^2.The odd Chern characterform h(∇^F,g^ F) and the Chern-Simons class h(∇^F,g^ F,g^' F) can be extended to Z∐Σ Z.Recall that for U∈ and g∈ G_U,U^g is anorbifold chart of Z∐Σ Z. Therestriction of(F_U,∇^F_U) toU^g is a flat vector bundle. The elementg actsfiberwisely on F_U|_U^g and preserves ∇^F_U and g^F_U. The family{h_g∇^F_U,g^F_U∈Ω^ odd(U^g)}_U∈, g∈ G_Udefines a closed differential form h_Σ(∇^F,g^F)∈Ω^ oddZ∐Σ Z.Denote by h_i(∇^F,g^F)the restriction of h_Σ(∇^F,g^F) to Z_i⊂ Z∐Σ Z. Similarly, we can define h_i(∇^F,g^F,g^' F)∈Ω^ even(Z_i)/Ω^ odd(Z_i),θ_i(∇^F,g^F)∈Ω^1(Z_i), θ_i(∇^F,g^F,g^' F)∈ C^∞(Z_i).The rank of F can be extended to a locally constant functiononZ∐Σ Z in a similar way. Indeed, the family {[ρ^F_U(g)]∈ C^∞(U^g)}_U∈,g∈ G_U of constant functions defines a locally constant function ρ on Z∐Σ Z. Denote byρ_i its value at Z_i. Clearly, ρ_0=[F]. § RAY-SINGER METRIC OF ORBIFOLDSIn this section,given metrics g^TZ and g^F on TZ and F, we introducethe Ray-Singer metric on the determinant of the de Rham cohomology H^·(Z,F). We establishthe anomaly formula for the Ray-Singer metric. In particular,when Z is of odd dimension and orientable, the Ray-Singer metric is a topological invariant. In subsection <ref>, we introduce the Hodge Laplacianassociated to the metrics (g^TZ,g^F). Westate Gauss-Bonnet-Chern Theorem for compact orbifolds. In subsection <ref>, we construct the analytic torsionandthe Ray-Singer metric. We restate the anomaly formula.In subsection <ref>, following <cit.>, weinterpret the analytic torsion as a transgressionofodd Chernforms. We state Theorem<ref>, which extends the main result of Bismut-Lott, andfrom whichthe anomaly formula follows.In subsection <ref>, we prove Gauss-Bonnet-Chern Theoremand Theorem<ref> in a unified way. Using an argument due to <cit.>, which is basedon the finite propagation speeds for the solutions of hyperbolicequations <cit.>, we can turnour problem into alocal one. Sincethe orbifold locally is a quotient of a manifold by some finite group, we can then rely on the results of Bismut-Goette <cit.>, where the authors there consider somesimilar problems in the equivariant setting.§.§ Hodge LaplacianLet Z be a compact orbifold of dimension m, andletF be aproper flat orbifold vector bundle of rank r with flat connection∇^F.Putχ_ top(Z,F)=∑_i=0^m (-1)^i_H^i(Z,F), χ'_ top(Z,F)=∑_i=1^m (-1)^ii_ H^i(Z,F). Take a Riemannian metric g^TZ and a Hermitian metric g^F onF. We apply the construction of subsection <ref> to the Hermitianorbifold vector bundleE=Λ^·(T^*Z)⊗_ F with the Hermitianmetricinduced by g^TZ and g^F, and with the connection∇^Λ^·(T^*Z)⊗_ F induced by theLevi-Civita connection ∇^TZ and the flat connection ∇^F.Let d^Z,* be the formal adjointof d^Z. PutD^Z=d^Z+d^Z,*, ^Z=D^Z,2= d^Z,d^Z,* .Then d^Z,* is a first order differential operator,representedby the family of the formal adjoint d^U,*of d^U with respectto the L^2-metric defined by g^TU and g^F_U. Also, ^Z isa formally self-adjoint second order elliptic operator acting on Ω^·(Z,F), which isrepresentedby the family of HodgeLaplacian ^U acting onΩ^·(U,F_U) associated withg^TU and g^F_U.Also, the operator^Z,Ω^·(Z,F) is essentially self-adjoint. And thedomain of the self-adjoint extension is ^2(Z,Λ^·(T^*Z)⊗_ F).The following theorem is well-known (e.g., <cit.>, <cit.>).The following orthogonal decomposition holds:Ω^·(Z,F)=^Z ⊕d^Z|_Ω^·(Z,F)⊕d^Z,*|_Ω^·(Z,F). In particular, we have the canonical identification of the vector spaces ^Z≃ H^·(Z,F).In the sequel, we still denote by ^Z theself-adjointextension of the operator ^Z,Ω^·(Z,F).By (<ref>), for k≫1, the operator(1+^Z)^-k has a continuous kernel. In particular,(1+^Z)^-k is in the Hilbert-Schmidt class, and (1+^Z)^-2k is in the trace class.By the above argument,if f lies in the Schwartz space (), then f(^Z) has a smooth kernel, and is in the trace class. For t>0, the same statement holds true for the heat operator exp(-t^Z) of ^Z.In this way, most of results on compact manifolds, which have been obtained by the functional calculus of the Hodge Laplacian, still hold true for compact orbifolds. Let N^Λ^·(T^*Z) be the number operator on Λ^·(T^*Z). We write [·]= (-1)^N^Λ^·(T^*Z)· for the supertrace.By the classical argument of Mckean-Singer formula <cit.>, we get:For t>0,the following identity holds:χ_ top(Z,F)= exp-t^Z . Recall thatχ_ orb(Z) and ρ_iare defined in (<ref>) and (<ref>). When t→0, we have exp-t^Z →∑^l_0_i=0ρ_iχ_ orb(Z_i)/m_i.In particular, χ_ top(Z,F)= ∑^l_0_i=0ρ_i χ_ orb(Z_i)/m_i.Equation (<ref>) is a consequence of (<ref>) and(<ref>).The proof of (<ref>) will be given in subsection <ref>. §.§ Analytic torsion and its anomalyBy (<ref>),let P^Z be the orthogonal projection onto^Z.By theshort time asymptotic expansions of the heat trace <cit.>, proceeding asin <cit.>, the functionθ(s)=-1/Γ(s)∫_0^∞ N^Λ^·(T^*Z)exp-t ^Z1-P^Z t^s-1dtdefined on the region {s∈: (s)>m/2} is holomorphic, andhas a meromorphic extension towhich is holomorphic at s=0. The analytic torsion of F is defined byTF,g^TZ,g^F=exp(θ'(0)/2)>0.The formalism of Voros <cit.> on the regularized determinant of the resolvent of Laplacianextends to orbifolds trivially, as theproofrelies only on theshort time asymptotic expansions of the heat trace and on the functional calculus. Thus the weighted product ofzeta regularized determinantsσ→∏_i=1^m σ+^Z|_Ω^i(Z,F)^(-1)^ii is ameromorphic function onsuch thatwhen σ→ 0, we have ∏_i=1^m σ+^Z|_Ω^i(Z,F)^(-1)^ii =TZ,g^TZ,g^F^2σ^χ'_ top(Z,F)+(σ^χ'_ top(Z,F)+1). We have a generalization of <cit.>. If Z is an orientableeven dimensional compactorbifold and if F is aunitarilyflat orbifold vector bundle, then for any Riemannian metricg^TZ and any flat Hermitian metric g^F,N^Λ^·(T^*Z)-m/2exp-t^Z =0.In particular,TF,g^TZ,g^F=1.Let *^Z:Λ^·(T^*Z)→Λ^m-·(T^*Z)⊗ o(TZ) be theHodge star operator associated to g^TZ, which islocally definedby the family{*^U:Λ^·(T^*U)→Λ^m-·(T^*U)⊗_o(TU)}_U∈.Note that Z is orientable, so o(TZ) is trivial. Write⋆^Z=*^Z⊗_id_F:Λ^·(T^*Z)⊗_ F→Λ^m-·(T^*Z)⊗_ F.Clearly,we have ⋆^ZN^Λ^·(T^*Z)-m/2⋆^Z,-1=-N^Λ^·(T^*Z)-m/2.Since g^F is flat, we have ⋆^Z ^Z⋆^Z,-1=^Z.Note that when m is even, ⋆^Z is an even isomorphism ofΩ^·(Z,F). Hence, by (<ref>) and(<ref>), we get (<ref>). Equation (<ref>) is a consequence of (<ref>). Setλ=⊗_i=0^mH^i(Z,F)^(-1)^i.Then λ is a complex line. Let |·|^ RS,2_λ be the L^2-metric on λ induced via (<ref>).The Ray-Singer metric on λ is defined by·^ RS_λ=TF,g^TZ,g^F|·|^ RS_λ. Let (g^TX,g^F) and (g^' TX,g^' F) be two pairs of metrics on TX and F. Let ·^ RS, 2_λ and ·^' RS, 2_λ be the corresponding Ray-Singer metrics on λ. We restate Theorem <ref>. The following identity holds:log·^' RS,2_λ/·^ RS,2_λ = ∑_i=0^l_01/m_i∫_Z_i( θ_i∇^F,g^F,g^' FeTZ_i,∇^TZ_i-θ_i∇^F,g^FeTZ_i,∇^TZ_i,∇^' TZ_i). The proof of our theorem will be given in Remark <ref>. If allthe Z_i's areofodd dimension, then ·^RS,2_λ does not depend on g^TZ or g^F. In particular, this is the case if Z is an orientable odd dimensional orbifold. WhenZ_i isodd, eTZ_i,∇^TZ_i=0 and eTZ_i,∇^TZ_i,∇^' TZ_i=0. By Theorem <ref>, we get Corollary <ref>.If for 0≤ i≤ l_0, χ_ orb(Z_i)=0,and if F isunitarily flat, then ·^ RS,2_λ does not depend on g^TZ or onthe flat Hermitian metric g^F.Take (g^TX,g^F) and (g^' TX, g^' F) two pairsofmetrics on TX and F such that ∇^Fg^F=0 and∇^Fg^' F=0. By (<ref>), for 0≤ i≤ l_0, θ_i∇^F,g^F=θ_i∇^F,g^' F=0. By(<ref>) and (<ref>), θ_i(∇^F, g^F, g^' F) is closed. It becomes a constant c_i∈ as Z_i is connected. Using (<ref>), we get ∫_Z_iθ_iF,g^F,g^' FeTZ_i,∇^TZ_i=c_i∫_Z_ieTZ_i,∇^TZ_i=0.By (<ref>),(<ref>),and (<ref>), we get ·^ RS,2_λ=·^'RS,2_λ. If F is not proper, we can define the analytic torsion and Ray-Singer metric in the same way. Indeed, we haveH^·(X,F)=H^·(X,F^ pr),TF,g^TZ,g^F=TF^ pr,g^TZ,g^F^ pr. Also, the Ray-Singer metrics of F and F^ pr coincides. For this reason, allthe result in this section holds true for non proper flat orbifoldvector bundle.§.§ Analytic torsion as a transgressionLet (g_s^TZ,g_s^F)_s∈ be a smooth family of metrics on TZ and F such that (g^TZ_s,g^F_s)|_s=0=(g^TZ,g^F), (g^TZ_s,g^F_s)|_s=1=(g^' TZ,g^' F).Then, s∈→log T(F,g^TZ_s,g^F_s) is a smooth function.Following <cit.>, we will interpret the analytic torsionfunction logT(F,g^TZ_·,g^F_·) as a transgression for some odd Chernforms associated to certain flat superconnections on . Recall that π is defined in (<ref>), and g^π^* (TZ),g^π^*F are defined in (<ref>)with E=TZor F.Consider now a trivial infinite dimensionalvector bundle W ondefined by ×Ω^·(Z,F)→.Let g^W be a Hermitianmetric on W such thatg^W_s is the L^2-metriconΩ^·(Z,F) induced by(g^TZ_s,g^F_s).PutA'=d^+d^Z.Then A' is a flat superconnection on W.Let A” be the adjointof A' with respect to g^W.For s∈, denote by d^Z,*_s,^Z_sthe corresponding objects for(g^TZ_s,g^F_s), andby*^Z_sthe Hodge star operator with respect to g^TZ_s. Thus,A”=d^+d^Z,*_s+ds∧g^F,-1_s g^F_s/s+*^Z,-1_s *^Z_s/ s. SetA=1/2(A”+A'), B=1/2(A”-A').Then A is a superconnection on W, and B is a fibrewise first order elliptic differential operator. The curvature of A is given by A^2=-B^2.It is a fibrewise second order elliptic differential operator.Following <cit.>, weintroduce a deformation of g^W. For t>0, set g_t^W=t^N^Λ^·(T^*Z)g^W. Let A”_t be the adjoint of A' with respect to g_t^W. Clearly,A”_t=t^-N^Λ^·(T^*Z) A” t^N^Λ^·(T^*Z).We define A_t and B_t as in (<ref>), i.e.,A_t=1/2(A”_t+A'), B_t=1/2(A”_t-A').For t>0, we haveexpB_t^2 =χ_ top(Z,F). Theorem <ref> can be proved using the technique of the localfamilyindex theory as in <cit.>. Since here the parameter space is of dimension 1, we give a short proof.Byconstruction, expB_t^2 is an even form on , thus it is a function. By (<ref>), (<ref>), (<ref>) and (<ref>), we have expB_t^2 = exp-t^Z_s/4 . By(<ref>) and (<ref>), we get (<ref>). Recall that h is defined in (<ref>). Following <cit.>, for t>0, setu_t= h(B_t) ∈Ω^1(),v_t= N^Λ^·(T^*Z)/2h'(B_t) ∈ C^∞().The subspace(^Z_s)⊂Ω^·(Z,F) defines afinite dimensionalsubbundle W_0⊂ W on . By Theorem<ref>, thefiber of W_0 is H^·(Z,F). Asin <cit.>, we equip W_0 with the restricted metric g^W_0 and the induced connection ∇^W_0. Putu_0=∑_i=0^l_01/m_i∫_Z_ieπ^*(TZ_i),∇^π^*(TZ_i) h_i∇^π^*F,g^π^* F,u_∞=h∇^W_0,g^W_0,andv_0=m/4χ_ top(Z,F), v_∞=1/2χ'_ top(Z,F). For a smooth family {α_t}_t>0 of differential forms on,we say α_t=(t) if for all the compact K⊂, and for allk∈, there is C>0 such that α_t_C^k(K)≤ Ct.For t>0,u_t is a closed 1-form onsuch thatits cohomology class does notdepend on t>0 and thatthe following identity of 1-forms holds:/ t u_t=d^v_t/t.As t→ 0, we have u_t=u_0+√(t),v_t=v_0+√(t),as t→∞,u_t=u_∞+(1/√(t)),v_t=v_∞+(1/√(t)). By <cit.>, u_t isclosed and (<ref>) holds.The first equation of (<ref>) will be proved in subsection <ref>.The first equation of (<ref>) can be proved as <cit.>, whose proof is based on functional calculus.Proceeding as <cit.>, we can showthe second equations of (<ref>) and(<ref>) as consequence of the firstequations[More precisely, we need the corresponding resultson a larger parametrized space × (0,∞). We leave thedetailsto readers. ] of (<ref>) and(<ref>). The following identities in C^∞() and Ω^1() hold:logTF,g^TZ_·,g^F_·=-∫_0^∞{v_t-v_∞h'(0) -v_0-v_∞h'i√(t)/2}dt/t,and u_0-u_∞=d^log TF,g^TZ_·,g^F_·. By Theorem <ref>, proceeding as<cit.>, we get Corollary <ref>. Theorem <ref> is just (<ref>). Indeed, if ·^ RS,2_λ,sdenotes the Ray-Singer metric on λ associated to(g^TZ_s,g^F_s). By (<ref>), (<ref>), and using the fact that our base manifoldis of dimension 1, we have ds∧/ s{log·^RS,2_λ,s}=∑_i=0^l_01/m_i∫_Z_i eπ^*(TZ_i),∇^π^*(TZ_i)θ_i∇^π^*F,g^π^*F.By (<ref>) and (<ref>) with α_i,s∈Ω^ Z_i-1(Z_i,o(TZ_i)), β_i,s^[0]∈ C^∞(Z_i) defined in an obvious way, we haveeπ^*(TZ_i),∇^π^*(TZ_i) =eTZ_i,∇^TZ_i+d^Z∫_0^sα_i,s ds+ ds∧α_i,s, θ_i∇^π^*F,g^π^*F =θ_i∇^F,g^F+2d^Z∫_0^sβ^[0]_i,s ds+ 2ds ∧β^[0]_i,s.By (<ref>),we have∫_Z_i eπ^*(TZ_i),∇^π^*(TZ_i)θ_i∇^π^*F,g^π^*F=ds ∫_Z_i2 β^[0]_i,s eTZ_i,∇^π^*(TZ)- θ_i∇^F,g^F∧α_i,s.By integrating (<ref>) with respect to the variable s form 0 to 1, and by (<ref>), we get (<ref>).§.§ Estimates on heat kernels §.§.§ Proof of Theorem <ref> We follow <cit.>.Take α_0>0. Let f,g:→[0,1] besmooth even functions such that f(s)={[ 1, |s|≤α_0/2;; 0, |s|≥α_0, ]. g(s)=1-f(s).For t>0 and a∈, set F_t(a)=∫_e^2isae^-s^2f√(t)sds/√(π), G_t(a)=∫_e^2isae^-s^2g√(t)sds/√(π).By (<ref>) and (<ref>),we getexp-a^2=F_t(a)+G_t(a).Moreover, F_t and G_t are even holomorphic functions, whose restriction tolies in (). By (<ref>), we find that given m,m'∈, c>0, there existC>0, C'>0 such that if t∈ (0,1], a∈, |(a)|≤ c, |a|^m |G_t^(m')(a)|≤ Cexp(-C'/t). There exist uniquely well-defined holomorphic functions ℱ_t(a) and 𝒢_t(a) such that F_t(a)=ℱ_t(a^2), G_t(a)=𝒢_t(a^2).By (<ref>) and (<ref>), we haveexp(-a)=_t(a)+_t(a). By (<ref>), we getexp-t^Z=_tt^Z+_tt^Z.If A is a bounded operator, let A be its operator norm. IfA is in the trace class, let A_1= √(A^*A) be its trace norm.There exist c>0 and C>0 such that for t∈ (0,1], _tt^Z_1≤ Ce^-c/t.In particular, as t→ 0, we haveexp-t^Z = _tt^Z +(e^-c/t).By (<ref>) and (<ref>), for any k∈, the operator1+^Z^k _tt^Z is a bounded operator such that there existC>0 and C'>0,1+^Z^k _tt^Z≤ Cexp-C'/t.Takek∈ big enough such that (1+^Z)^-k is of trace class. Then _tt^Z_1≤(1+^Z)^-k_1(1+^Z)^k _tt^Z.By (<ref>) and (<ref>), we get (<ref>). By (<ref>) and (<ref>), we get (<ref>). Assume that Z is covered by a finite family ={U_i}_i∈ I ofconnected open setswith orbifold atlas ={(U_i,G_U_i, π_U_i)}_i∈ I. Let{ϕ_i}_i∈ I be a partition of unity subordinate to {U_i}_i∈ I.Let _tt^Z_U_i(z,z') be the smooth kernel in the sense (<ref>).By (<ref>), we have_tt^Z =∑_i∈ I1/|G_U_i|∑_g∈ G_U_i∫_U_iϕ_i(x) g_tt^Z_U_i(g^-1x,x) dv_U_i.For i∈ I, and g∈ G_U_i,let N_U_i^g/U_i be the normal bundle of U_i^g in U_i. We identityN_U_i^g/U_i with the orthogonal bundle of TU_i^g in TU_i|_U_i^g. For _0>0, set N_U_i^g/U_i,_0={(y,Y)∈ N_U_i^g/U_i:disty, (ϕ_i)<_0, |Y|<_0}.Take _0>0 small enough such that for alli∈ I,{x∈U_i: dist(x,(ϕ_i))< _0}⊂U_i,and such that alli∈ I, g∈ G_U_i, the exponential map (y,Y)∈ N_U_i^g/U_i,_0→exp_y(Y)∈U_i defines a diffeomorphism from N_U_i^g/U_i,_0 onto its image U_i,_0,g⊂U_i. Also, there exists δ_0>0 such that for all i∈ I, g∈ G_U_i if x∈(ϕ_i) and dist(g^-1 x,x)< δ_0, then x∈U_i,_0,g.Let dv_U^g be the induced Riemannian volume of U^g, and let dY be the induced Lebesgue volume on the fiber of N_U_i^g/U_i. Let k_i:N_U_i^g/U_i,_0→^*_+ be a smooth function such that on U_i,_0,g we have dv_U_i=k_i(y,Y)dv_U^g dY.Clearly,k_i(y,0)=1. For x∈ (ϕ_i) and r∈ (0,_0), let B_x^U_i(r) be the geodesic ball of center x and radius r. Using the result of the finite propagationspeeds for the solutions of hyperbolic equations on orbifolds<cit.> (see also <cit.>),by taking α_0<1/4min{δ_0, _0}, forx∈ (ϕ_i), we find the support of_tt^Z_U_i(x,·) in B^U_i_x(4α_0). Moreover,_tt^Z_U_i(x,·) depends only ontheHodge Laplacian ^U_i acting onΩ^·(U_i,F_U_i).Using (<ref>) and (<ref>), we get∫_U_iϕ_i(x) g_tt^Z_U_i(g^-1x,x) dv_Z =∫_y∈U^g_i dv_U^g_i∫_Y∈ N_U^g_i/U_i,y, |Y|<_0ϕ_i(y,Y)g_tt^Z_U_i(g^-1(y,Y),(y,Y)) k_i(y,Y)dY.Consider an isometric embedding of(U_i,g^TU_i)into a compact manifold(X_i,g^TX_i). We extendthe trivialHermitian vector bundle(F_U_i, g^F_U_i) to a trivialHermitian vector bundle (F_i,g^F_i) on X_i. Thus, when restricted on U_i, ^U_i=^X_i.Using the results of the finite propagation speeds for the solutionsof hyperbolic equations on orbifolds <cit.> and on manifolds <cit.>, for x,y∈U_i, we have ϕ_i(x)_tt^Z_U_i(x,y)=ϕ_i(x)_tt^X_i(x,y).Recall that for x∈U_i and g∈ G_U_i, g:F_g^-1x→ F_x is a linear map. In particular, g_tt^X_i(g^-1x,x) is well-defined on U_i.By (<ref>),for y∈U^g_i, we have∫_Y∈ N_U^g_i/U_i,y,|Y|<_0ϕ_i(y,Y) g_tt^Z_U_i(g^-1(y,Y),(y,Y)) k_i(y,Y)dY =∫_Y∈ N_U^g_i/U_i,y, |Y|<_0ϕ_i(y,Y) g_tt^X_i(g^-1(y,Y),(y,Y)) k_i(y,Y)dY=t^1/2N_U^g_i/U_i∫_Y∈ N_U^g_i/U_i,y, √(t)|Y|<_0ϕ_iy,√(t)Yg_tt^X_i(g^-1(y,√(t)Y),(y,√(t)Y)) k_iy,√(t)YdY. Let eTU^g,∇^TU^g ^max be the function definedon U^g such that eTU^g,∇^TU^g= eTU^g,∇^TU^g ^maxdv_U^g.There existc>0 and C>0 such that for any i∈ I, g∈ G_U_i and (y,√(t)Y)∈ N_U^g_i/U_i,_0, we havet^1/2N_U^g_i/U_i|ϕ_iy,√(t)Yk_iy,√(t)Yg_tt^X_i(g^-1(y,√(t)Y),(y,√(t)Y)) | ≤ Cexp(-c|Y|^2).As t→ 0, we havet^1/2N_U^g_i/U_i∫_Y∈N_U^g_i/U_i,y,√(t)|Y|<_0{ϕ_i(y,√(t)Y)k_iy,√(t)Yg_tt^X_i(g^-1(y,√(t)Y),(y,√(t)Y))dY} →ϕ_i(y,0)[ρ^F_U_i(g)] eTU^g,∇^TU^g ^max.Theorem <ref> is a consequence of <cit.>.By (<ref>), (<ref>), (<ref>), (<ref>)-(<ref>), and thedominated convergence Theorem, we get lim_t→ 0 exp-t^Z =∑_i∈ I1/|G_U_i|∑_g∈ G_U_i[ρ^F_U_i(g)]∫_U^g_iϕ_i(y,0)eTU_i^g,∇^TU_i^g.As ϕ_i is G_U_i-invariant,the integral on the right-hand side of (<ref>)depends only on the conjugation class of G_U_i. Thus, ∑_i∈ I1/|G_U_i|∑_g∈ G_U_i[ρ^F_U_i(g)]∫_U^g_iϕ_i(y,0)eTU_i^g,∇^TU^g_i =∑_i∈ I∑_[g]∈ [G_U_i][ρ^F_U_i(g)]/|Z_G_U_i(g)|∫_U^g_iϕ_i(y,0)eTU_i^g,∇^TU^g_i=∑_i=0^l_0ρ_iχ_ orb(Z_i)/m_i.By (<ref>) and (<ref>), we get (<ref>).§.§.§ The end of the proof of Theorem<ref> It remains to show the firstidentity of (<ref>).Following <cit.>,we introduce a new Grassmann variablez which is anticommutewith ds. Fortwo operators P,Q inthetrace class, set ^ z[P+ zQ]=[Q].By (<ref>), (<ref>), (<ref>) and (<ref>), we haveu_t=^ z_ s exp-A^2_t+ z B_t =_ s^z _tA^2_t- z B_t +_ s^ z _tA^2_t- z B_t . We follow the same strategy used in the proof of Theorem <ref>. As in (<ref>), proceeding as in <cit.>, when t→ 0, we have _ s^ z _tA^2_t- z B_t =(e^-c/t). Moreover, the principalsymbol ofthe lifting of A^2_t- zB_t on U_iis scalar, and is equal to t|ξ|^2/4for ξ∈ T^*U_i.Take α_0<min{δ_0,_0}. As in the case of _tt^Z,for x∈U_i and x∈(ϕ_i), the support of_tA^2_t- z B_t_U_i(x,·) is B^U_i_x(α_0) and its value depends only on the restriction of A^2_t- z B_t on U_i. Also,_ s^ z _tA^2_t- z B_t=∑_i∈ I1/|G_U_i|∑_g∈ G_U_i∫_y∈U^g_i{∫_Y∈ N_U^g/U_i,y, |Y|<_0ϕ_i(y,Y)_ s^ z g_tA^2_t- z B_t_U_ig^-1(y,Y),(y,Y) k_i(y,Y)dY}dv_U^g_iAs in (<ref>), we can replace A^2_t- z B_t by the corresponding operator on manifolds.Recall that h_g∇^π^* F_U_i, g^π^* F_U_i∈Ω^ odd(U_i^g) is defined in (<ref>). Proceedingas <cit.>, as t→ 0, we have∫_y∈U^g_i{∫_Y∈ N_U^g/U_i,y, |Y|<_0ϕ_i(y,Y)k_i(y,Y) _ s^ z g_tA^2_t- z B_t_U_ig^-1(y,Y),(y,Y) dY}dv_U^g_i = ∫_U^g_iϕ_i(y,0) eπ^*(TU^g_i), ∇^π^*(TU^g_i) h_g∇^π^* F_U_i, g^π^* F_U_i+(√(t)).Proceeding now as in (<ref>), by(<ref>)-(<ref>), we get the first identity of (<ref>).§ ANALYTIC TORSION ON COMPACT LOCALLY SYMMETRIC SPACELet G be a linear connected realreductive group with maximal compact subgroup K⊂ G, and let Γ⊂ G be a discretecocompact subgroupof G. The correspondinglocally symmetricspace Z=Γ\ G/K is a compact orientable orbifold. Thepurpose of this section is to show Theorem <ref> which claimsan equality between the analytic torsion of an acyclic unitarily flatorbifold vector bundle F on Z and the zero value ofthe dynamical zeta function associated to the holonomy of F.This section is organized as follows. In subsections <ref> and <ref>, werecallsome facts onreductive groups and the associatedsymmetric spaces.In subsections <ref> and <ref>, we recall thedefinition ofsemisimple elements and the semisimple orbitalintegrals. We recall the Bismut formula for semisimple orbitalintegrals <cit.>. In subsection <ref>, we introduce the discrete cocompact subgroup Γ and the associatedlocally symmetric spaces. We recall the Selberg trace formula.In subsection <ref>, we introduce a Ruelledynamicalzeta function associated to the holonomy ofa unitarily flatorbifold vector bundle on Z. We restate Theorem <ref>. Whenthe fundamental rank δ(G)∈ of G does not equal to 1or when G has noncompact center, we show Theorem <ref>.Subsections <ref>-<ref> are devoted tothe casewhere G has compact center andδ(G)=1. In subsection <ref>, we recall somenotation and results proved in<cit.>. In subsection <ref>, we introduce a class of representationsof K. In subsection <ref>, using the Bismut formula,we evaluatethe orbital integrals for the heat operators of the Casimirassociated to the K-representations constructed in subsection <ref>.In subsection <ref>,we introduce theSelbergzeta functions, which are shown to be meromorphic onand satisfy certain functional equations.In subsection <ref>,we show thatthe dynamical zeta function equals an alternating product of certain Selbergzeta functions.We show Theorem <ref>. §.§ ReductivegroupsLet G be a linear connected real reductive group <cit.>, let θ∈Aut(G) be the Cartan involution. That means G is a closed connected group of real matrices that is stable under transpose, and θ is the composition of transpose and inverse of matrices.Let K⊂ Gbe the subgroup of G fixed by θ, so that K is a maximal compact subgroup of G.Letandbethe Lie algebras of G and K.The Cartan involution θ acts naturally as Lie algebra automorphism of . Thenis the eigenspace of θ associated with the eigenvalue 1. Let ⊂ be the eigenspace with the eigenvalue -1, so that=⊕. By <cit.>, we have the diffeomorphism(Y,k)∈× K→e^Y k∈ G.Setm=, n=. Let B be a real-valuednon degenerate bilinear symmetric form onwhich is invariant under the adjoint action of G, and also under θ. Then (<ref>) is an orthogonal splittingwith respect to B. We assume B to be positive on , and negative on . The form ·,,·=-B(·,θ·) defines an (K)-invariant scalar product onsuch that the splitting (<ref>) is still orthogonal. We denote by |·| the corresponding norm. Let _=⊗_ be the complexification ofand let =√(-1)⊕ be the compact form of . Let G_ and U be the connected group of complex matrices associated to the Lie algebras _ and . By <cit.>, if G has a compact center, G_ is a linear connected complex reductive group with maximal compact subgroup U.Let 𝒰() be the enveloping algebra of . Weidentify 𝒰() with the algebra of left-invariant differential operators on G. Let C^∈𝒰() be the Casimir element. Ife_1,⋯,e_m is an orthonormalbasis of , ife_m+1,⋯,e_m+n is an orthonormalbasis of , then C^=-∑_i=1^me^2_i+∑_i=m+1^n+me_i^2.Classically, C^ is in the center of 𝒰().We define C^ similarly. Let τ be a finite dimensional representation of K on V. We denote by C^,V or C^,τ∈(V) the corresponding Casimir operator acting on V, so that C^,V = C^k,τ = ∑_i=m+1^m+nτ(e_i)^2.Let δ(G)∈ be the fundamental rank of G, that isdefined bythe difference between the complex ranks of G and K. IfT⊂ K is a maximal torus of K with Lie algebra of ⊂, set={Y∈: [Y,]=0}.Put=⊕, H=exp()T.By <cit.>, ⊂ (resp. H⊂ G) is a θ-invariant Cartan subalgebra (resp. subgroup). Therefore, δ(G)=.Moreover, up to conjugation, ⊂ (resp. H⊂ G) is the uniqueCartan subalgebra (resp. subgroup) with minimal noncompact dimension.§.§ Symmetric spaceLet ω^ be the canonical left-invariant 1-form on G with values in , and let ω^, ω^ be its components in ,, so thatω^=ω^+ω^.Set X=G/K. Then p:G→ X=G/K is a K-principle bundle equipped with the connection form ω^.Let τ be a finite dimensional orthogonal representation of K on the real Euclidean space E_τ. Let _τ be the associated Euclidean vector bundle with total spaceG×_K E_τ. It is equipped a Euclidean connection ∇^_τ induced by ω^.We identifyC^∞(X,_τ) with the K-invariant subspace C^∞(G,E_τ)^K of smooth E_τ-valued functions on G. Let C^,X,τ be the Casimir elementof G acting on C^∞(X,_τ). Observe that K acts isometrically onby adjoint action. Using the above construction, the total space of the tangent bundle TX is given by G×_K. It is equipped with a Euclidean metric g^TX and a Euclidean connection ∇^TX, which coincides with the Levi-Civita connection of the Riemannian manifold(X,g^TX).Classically, (X,g^TX) hasnon positive sectional curvature. If E_τ=Λ^·(^*) isequipped with the K-action induced by theadjoint action, then C^∞(X,_τ)=Ω^·(X). In this case, we write C^,X=C^,X,τ. By <cit.>, C^,X coincides with the Hodge Laplacian acting on Ω^·(X). Let dv_X be the Riemannian volume of (X,g^TX). Define[e(TX,∇^TX)]^max as in (<ref>). Since both dv_Xand e(TX,∇^TX) are G-invariant, we see that [e(TX,∇^TX)]^max∈ is a constant. Note that δ(G) and X have the same parity. By <cit.>, if δ(G)≠0, theneTX,∇^TX ^max=0.If δ(G)=0, G has a compact center. Then U is a compactgroup with maximal torus T. Denote by W(T,U) (resp. W(T,K)) the Weyl group of U (resp. K) with respect to T,and by (U/K) the volume of U/K induced by -B. Then, <cit.> asserts eTX,∇^TX ^max=(-1)^m/2|W(T,U)|/|W(T,K)|/(U/K).§.§ Semisimple elementsIf γ∈ G, we denote by Z(γ)⊂ G the centralizer of γ in G, and by (γ)⊂ its Lie algebra. If a∈, let Z(a)⊂ G be the stabilizer of a in G, and let (a)⊂ be its Lie algebra. Following <cit.>, γ∈ G is said to be semisimple if and only if there is g_γ∈ G, such that γ=g_γ e^ak^-1g_γ^-1 with a∈,k∈ K,(k)a=a. Set a_γ=(g_γ)a,k_γ=g_γ k g_γ^-1. Therefore, γ=e^a_γk_γ^-1. Moreover, this decomposition does not depend on the choice of g_γ. By <cit.>, we haveZ(γ)=Z(a_γ)∩ Z(k_γ), (γ)=(a_γ)∩(k_γ).By <cit.>, Z(γ) is reductive. The corresponding Cartan evolution andbilinear form are given by θ_g_γ=g_γθ g^-1_γ,B_g_γ(·,·)=B(g^-1_γ)·, (g^-1_γ)·.Let K(γ)⊂ Z(γ) be the fixed point of θ_g_γ, soK(γ) is a maximal compact subgroup Z(γ).Let (γ)⊂(γ) be the Lie algebra of K(γ). Let (γ)=(γ)⊕(γ)be the Cartan decomposition of (γ).Let X(γ)= Z(γ)/K(γ)be the associated symmetric space. Let Z^0(γ) be the connected component of the identity in Z(γ). Similarly,Z^0(γ) is reductivewith maximal compact subgroup Z^0(γ)∩ K(γ). Also, Z^0(γ)∩ K(γ) coincides with K^0(γ),the connected component of the identity in K(γ). Clearly, we have X(γ)=Z^0(γ)/K^0(γ). The semisimple element γ is called elliptic if a_γ=0.Assume now γ is semisimple and nonelliptic. Then a_γ≠0. Let ^a,(γ) (resp. ^a,(γ)) be the orthogonal spaces to a_γ in (γ) (resp. (γ)) with respect to B_g_γ. Thus,^a,(γ)=^a,(γ)⊕(γ).Moreover, ^a,(γ) is a Lie algebra. Let Z^a,,0(γ) be the connected subgroup of Z^0(γ) that is associated with the Lie algebra ^a,(γ). By <cit.>, Z^a,,0(γ) is reductive with maximal compact subgroup K^0(γ) with Cartan decomposition (<ref>), andZ^0(γ)=𝐑× Z^a,,0(γ),so that e^ta_γ∈ Z^0(γ) maps into t|a|∈.SetX^a,(γ)=Z^a,,0(γ)/K^0(γ).By (<ref>), (<ref>), and (<ref>), we haveX(γ)=𝐑× X^a,(γ),so that the action e^ta_γ on X(γ) is just the translation by t|a| on . §.§ Semisimple orbital integralRecall that τ is a finite dimensional orthogonal representation of K on the real Euclidean space E_τ.Let p_t^X,τ(x,x') be the smooth kernel of the heat operator exp(-tC^,X,τ/2) with respect to the Riemannian volume dv_X. As in <cit.>, let p_t^X,τ(g) be the equivariant representation of thesection p_t^X,τ(p1,·). Then p_t^X,τ(g) is aK× K-invariantfunction in C^∞(G,(E_τ)).Let dv_G be the left-invariant Riemannian volume on G induced bythe metric -B(·, θ·). For a semisimple element γ∈ G, denote by dv_Z^0(γ) theleft-invariant Riemannian volume on Z^0(γ) induced by -B_g_γ(·,θ_g_γ·).Clearly, the choice ofg_γ is irrelevant. Letdv_Z^0(γ)\ G be the Riemannian volume on Z^0(γ)\ G such that dv_G=dv_Z^0(γ)dv_Z^0(γ)\ G. By <cit.>,theorbital integral ^[γ] exp-tC^,X,τ/2 =1/(K^0(γ)\ K)∫_Z^0(γ)\ G^E_τ p^X,τ_t(g) dv_Z^0(γ)\ G is well-defined.In <cit.>,the volume (K_0(γ)\ K) are normalized to be 1. By<cit.>, in the definition of the orbital integral (<ref>), we canreplace K^0(γ),Z^0(γ) byK(γ),Z(γ). As the notation ^[γ] indicates, the orbital integral only depends on the conjugacy class of γ in G. However, the notation [γ] will be used later for the conjugacy class of a discrete group Γ. Here, we consider ^[γ] as an abstract symbol. We will also consider the case where E_τ=E_τ^+⊕ E_τ^- is a _2-graded representation of K. In this case, We will use thenotation ^[γ] exp(-tC^,X,τ/2) when the trace on the right-hand side of (<ref>) is replaced by the supertrace on E_τ.In <cit.>, for any semisimple element γ∈ G, Bismut gave an explicit formula for ^[γ] exp-tC^,X,τ/2.For the later use, let us recall the formula when γ is elliptic. Assume now γ∈ K. By (<ref>), we can take g_γ=1. Then (γ)⊂, (γ)⊂. Let ^(γ)⊂, ^(γ)⊂ be the orthogonal space of (γ), (γ). Take ^(γ)=^(γ)⊕^(γ). Recall that A is defined in (<ref>). Following <cit.>, for Y∈(γ), putJ_γ(Y)=A(i(Y)|_(γ))/A(i(Y)|_(γ))1/(1-(γ))|_^(γ)(1-exp(-i(Y))(γ))|_^(γ)/(1-exp(-i(Y))(γ))|_^(γ) ^1/2.Note that by <cit.>, the square root in (<ref>) is well-defined, and its sign is chosen such that J_γ(0)=((1-(γ))|_^(γ))^-1.Moreover, J_γ is an(K^0(γ))-invariant analytic function on (γ) such thatthere exist c_γ>0, C_γ>0,for Y∈(γ), |J_γ(Y)|≤ C_γexp(c_γ|Y|). Denote by dY be the Lebesgue measure on (γ) induced by -B. Recall that C^,, C^, are defined in (<ref>).By <cit.>, for t>0, we have^[γ] exp-tC^,X,τ/2 =1/(2π t)^(γ)/2expt/16^ C^, +t/48^ C^, ∫_Y∈(γ)J_γ(Y) ^E_τ τγexp(-iτ(Y)) exp-|Y|^2/2tdY.§.§ Locally symmetric spacesLet Γ⊂ G be a discrete cocompact subgroup of G. By<cit.>,the elements of Γ are semisimple.Let Γ_e⊂Γ be the subset of elliptic elements inΓ. Set Γ_+=Γ-Γ_e. Let [Γ] be the setof conjugacy classes of Γ, andlet [Γ_e]⊂ [Γ] and [Γ_+]⊂ [Γ] berespectively the subsets of [Γ] formed by the conjugacy classes of elements in Γ_e and Γ_+. Clearly, [Γ_e] is a finite set. The group Γ acts properlydiscontinuously and isometricallyon theleft on X. Take Z=Γ\ X to be the correspondinglocally symmetric space. By Proposition <ref> and Theorem <ref>, Z is a compactorbifold. Note that by (<ref>), X is a contractible manifold. By Remark <ref>, X is the universal covering orbifold of Z. The Riemannian metric g^TX on X induces a Riemannian metric g^TZ on Z. Clearly, (Z,g^TZ) has nonpositive curvature. Let Δ_Γ⊂Γ be the subgroup of the elementsin Γ that act like the identity on X. Clearly,Δ_Γ is a finite group given by Δ_Γ=Γ∩ K∩ Z(), where Z()⊂ G is the stabiliser ofin G.Thus, the orbifold fundamental group of Z is Γ/ Δ_Γ. Let F be a (possibly non proper) flat vector bundle on Z withholonomy ρ':Γ/Δ_Γ→_r() such that C^∞(Z,F)=C^∞(X,^r)^Γ/Δ_Γ.Take ρ to be the composition of the projection Γ→Γ/Δ_Γ and ρ'. ThenC^∞(Z,F)=C^∞(X,^r)^Γ.By abuse of notation, we still callρ:Γ→_r() theholonomy of F. In the rest of this section, we assume F isunitarily flat, or equivalently ρ is unitary. Let g^F be theassociate flat Hermitian metric on F.Since g^TZ and g^F are fixed in the whole section, we writeT(F)=TF,g^TZ,g^F. The group Γ acts on the Euclideanvector bundles like _τ, and preserves the corresponding connections ∇^_τ.The vector bundle_τ descends to a (possiblynon proper) orbifoldvector bundle_τon Z. The total space of _τ is given by Γ\ G ×_K E_τ, and we have the identification of vector spacesC^∞(Z,_τ)≃ C^∞(Γ\ G, E_τ)^K. By (<ref>) and (<ref>), we identify C^∞(Z, _τ⊗_ F)with the Γ-invariant subspace ofC^∞(X, _τ⊗_^r).Let C^,Z,τ,ρ be the Casimir operator of G acting on C^∞(Z, _τ⊗_ F).As we see in subsection <ref>, when E_τ=Λ^·(^*),Ω^·(Z,F)≃ C^∞(Z, _τ⊗_ F), and the Hodge Laplacian acting on Ω^·(Z,F) is given by ^Z=C^,Z,τ,ρ.For γ∈Γ, set Γ(γ)=Z(γ)∩Γ. By <cit.> (see also <cit.>), Γ(γ) is cocompact in Z(γ). Then Γ(γ)\ X(γ) is a compact locally symmetric orbifold. Clearly, it depends only on the conjugacy class of γ in Γ. Denote by (Γ(γ)\ X(γ)) the Riemannian volume of Γ(γ)\ X(γ) induced by B_g_γ.The group K(γ) acts on the right on Γ(γ)\ Z(γ). For h∈Γ(γ)\ Z(γ), let K(γ)_h be thestabilizer of h in K(γ).Since Γ(γ)\ X(γ) is connected, the cardinal of a generic stabilizer is well defined anddepends only on the conjugacy class of γ in Γ. We denote it by n_[γ]. Then, we have(Γ(γ)\ Z(γ))/(K(γ))=(Γ(γ)\ X(γ))/n_[γ].Let us note that even if K(γ) acts effectively onΓ(γ)\ Z(γ), n_[γ] is not necessarily equal to 1. For γ∈Γ, we have n_[γ]=|K∩Γ(γ)∩ Z((γ))|.In particular, if γ=e, we haven_[e]= |Δ_Γ|,and if γ∈Δ_Γ, we haven_[γ]= |Γ(γ)∩Δ_Γ|. For a generic elementg=e^fh in Z(γ) with f∈(γ) and h∈ K(γ), the stabliser of Γ(γ)g∈Γ(γ)\ Z(γ) in K(γ) is given by K(γ)_Γ(γ)g= K(γ)∩g ^-1Γ(γ)g= g ^-1(e^fK(γ)e^-f∩Γ(γ))g. Then, n_[γ]=|e^fK(γ)e^-f∩Γ(γ)|.Since e^fK(γ)e^-f is compact, sinceΓ(γ) isdiscrete, andsince f can varyin an open dense set, we can deduce that n_[γ]=|K(γ)∩ Z((γ)) ∩Γ(γ)|, from which we get (<ref>). By (<ref>) and (<ref>), we get (<ref>). If γ∈Δ_Γ, by (<ref>), we get (γ)=. Combining the result with (<ref>), we get (<ref>). There existc>0 and C>0 such that, for t>0, we have∑_[γ]∈ [Γ_+](Γ(γ)\X(γ))/n_[γ]|^[γ] exp-tC^,X,τ/2 | ≤ Cexp-c/t+Ct.For t>0, the following identity holds:exp-tC^,Z,τ,ρ/2 =∑_[γ]∈[Γ][ρ(γ)](Γ(γ)\X(γ))/n_[γ]^[γ] exp-tC^,X,τ/2 . The proofis identical to the one given in <cit.>. One difference is that we need to show an estimatelike<cit.>. This can be deduced from <cit.>, which states that Γ possesses anormal torsion free subgroup of finite index.Let us explain the reason for which the coefficients n_[γ] appear in (<ref>). Indeed, the restriction on the diagonal of the trace of the integral kernel of exp(-t C^,Z,τ,ρ/2) is given by1/|Δ_Γ |∑_γ∈Γ[ρ(γ)]^_τ γ_xp^X,τ_t(x,γ x) , where γ_x denotes the obvious element in (_τ,x,_τ,γ x) induced by γ (see <cit.>).By (<ref>), (<ref>), and (<ref>),we haveexp-t C^,Z,τ,ρ/2 =1/(K)∫_Γ\ G∑_γ∈Γ[ρ(γ)]^E_τ p_t^X,τ(g^-1γ g) dv_Γ\ G,where dv_Γ\ G is the volume form on Γ\ Ginduced by dv_G. Proceeding as in <cit.>, by Remark <ref>, (<ref>), and (<ref>), we get exp-t C^,Z,τ,ρ/2 =∑_[γ]∈ [Γ] [ρ(γ)](Γ(γ)\ Z(γ))/(K(γ))^[γ] exp(-tC^,X,τ/2) . By (<ref>) and (<ref>), we get (<ref>). By <cit.> and <cit.>, when counting with multiplicites, we have the identification of the orbifolds, Z∐Σ Z≃∐_[γ]∈ [Γ_e]Γ(γ)\ X(γ), where the multiplicity of each component Z_i of Z∐Σ Z ism_i (see <ref>) and the multiplicity of Γ(γ)\ X(γ) is n_[γ]. Note also that, by Remark <ref>, we can consider ∐_[γ]∈[Γ_e]Γ(γ)\ X(γ) as the space of paths on Z of length 0. Assume that Γ acts effectively on X. Then Z can berepresented by the action groupoid(see <cit.>)whose object is X and whose arrow isΓ× X. An arrow(γ,x)∈Γ× X maps x to γ x. By Remark<ref>, any closed geodesic on Z can be represented by theclosed -geodesic c=(b_0; id,γ^-1) where b_0:[0,1]→ X is ageodesic on X such that γ b_0=b_1 and γ_*ḃ_0(0)=ḃ_0(1). By<cit.>, the length of b_0 is given by|a_γ| (see (<ref>)). Moreover,the space of the closed-geodesics is givenby ∪_γ∈ΓX(γ). Thus, the space ofthe closedgeodesics on Z is given by ∐_[γ]∈ [Γ]Γ(γ)\ X(γ), whosecomponent has themultiplicityn_[γ].For general Γ, as in Remark <ref>, the same result stillholds ture. We extend <cit.> and <cit.> to orbifolds (see also <cit.>). Let F be a unitarily flat orbifold vector bundle on Z. If Z is odd and δ(G)≠ 1, then for any t>0, we haveN^Λ^·(T^*Z)exp-t^Z/2 =0.In particular,T(F)=1. Since Z is odd, δ(G) is odd. Since δ(G)≠ 1,δ(G)3.By <cit.>, for any γ∈ G semisimple, we have ^[γ] N^Λ^·(T^*X)exp-tC^,X/2 =0.By(<ref>), (<ref>), and (<ref>), weget(<ref>). Suppose that δ(G)=1. Let us recall some notation in<cit.>. Up to sign, we fix an elementa_1∈ suchthat B(a_1,a_1)=1. As in subsection <ref>, setM=Z^a_1,,0(e^a_1),K_M=K^0(e^a_1),and M@P@_K@_ =^a_1,(e^a_1), _=^a_1,(e^a_1), _=(e^a_1).As in subsection <ref>, M is a connected reductive group such thatδ(M)=0 with Lie algebra , with maximalcompact subgroup K_M, andwith Cartan decomposition=_⊕_.Let X_M=M/K_Mbe the corresponding symmetric space.For k∈ T, we have δ(Z^0(k))=1.Denote by M^0(k), (k),_(k), _(k), X_M(k) the analogies of M, , _, _, X_ when Gis replaced by Z^0(k).Assumethat δ(G)=1 and that G hasnoncompact center.By<cit.>, we haveG=× M,K=K_M, X=× X_M.Recall that H=exp() T. Note that if γ=e^ak^-1∈ H witha≠0, then X^a,(γ)=X_M(k). We have an extensionof<cit.>. Let γ∈ G be a semisimple element. If γ can not be conjugated into H by elements of G, then for t>0, we have ^[γ] N^Λ^·(T^*X)exp-tC^, X/2 =0. If γ=e^ak^-1∈ H with a∈ and k∈ T, then for t>0, we have ^[γ] N^Λ^·(T^*X)exp-tC^,X/2 =-1/√(2π t)e^-|a|^2/2t eTX_M(k),∇^TX_M(k) ^max. Equations (<ref>), (<ref>) with γ=1 or γ=e^ak^-1 with a≠0 are just <cit.>. Equation (<ref>) for general γ∈ H is a consequence of <cit.> and (<ref>). §.§ Ruelle dynamical zeta functions By Remark <ref> (c.f. <cit.> for manifold case), the space of theclosed geodesics on Z ofpositive lengthsconsists of a disjoint union of smooth connected compact orbifolds∐_[γ]∈ [Γ_+] B_[γ].Moreover, B_[γ] is diffeomorphic toΓ(γ)\ X(γ) with multiplicity n_[γ]. Also, all the elements inB_[γ] have the same length l_[γ]=|a_γ|>0.The group^1 acts locally freely on B_[γ] byrotation. Then^1\ B_[γ] is still an orbifold.Setm_[γ]=n_[γ]|(^1→Diffeo(B_[γ]))|∈𝐍^*.We define m_[γ] to be the multiplicity of ^1\B_[γ]. For γ∈Γ_+ such that γ=a_γ k_γ^-1 as in (<ref>), we have χ_ orb^1\B_[γ]/m_[γ]=(Γ(γ)\X(γ))/|a_γ|n_[γ] eTX^a,(γ),∇^TX^a,(γ) ^max.In particular, if δ(G)2, then for all [γ]∈ [Γ_+], we have χ_ orb^1\ B_[γ]=0.Also, if δ(G)=1 and if γ can not be conjugated into H, then (<ref>) stillholds.The proof of our propositionis identical to the one given in<cit.>. Recall thatρ:Γ→ U(r) is a unitary representation ofΓ. The Ruelle dynamical zeta function R_ρ is said to be well-defined if*for (σ)≫1, the sum Ξ_ρ(σ)=∑_[γ]∈ [Γ_+][ρ(γ)]χ_ orb^1\ B_[γ]/m_[γ]e^-σ l_[γ]converges absolutely to a holomorphic function; *the function R_ρ(σ)=exp(Ξ_ρ(σ)) has a meromorphic extension to σ∈. By (<ref>), if δ(G)2, the dynamical zeta function R_ρ is well-defined and R_ρ≡1.We restate Theorem <ref>, which is the main result of this section.If Z isodd, then the dynamical zeta function R_ρ(σ) iswell-defined.There exist explicit constants C_ρ∈ with C_ρ≠0 andr_ρ∈ (see (<ref>)) such that as σ→0, R_ρ(σ)=C_ρ T(F)^2σ^r_ρ+(σ^r_ρ+1).Moreover, if H^·(Z,F)=0, we have C_ρ=1, r_ρ=0,so that R_ρ(0)=T(F)^2. Ifδ(G)≠ 1, Theorem <ref> is a consequence of (<ref>) and (<ref>). Assume now δ(G)=1 and G hasnoncompact center. Proceeding as <cit.>, up to evident modification, we see that the dynamical zeta function R_ρ(σ) extendsmeromorphically to σ∈ such that the following identity ofmeromorphic function holds,R_ρ(σ)=∏_i=1^m σ^2+^Z|_Ω^i(Z,F)^(-1)^ii exp(σ∑_[γ]∈ [Γ_e] γ=g_γk^-1g_γ^-1 ρ(γ) (Γ(γ)\ X(γ))/n_[γ] eTX_M(k),∇^TX_M(k) ^max),from which we get (<ref>)-(<ref>). The proof for the case where δ(G)=1 and where G has compact center will be given in subsections <ref>-<ref>.By (<ref>), we have the formalidentity 2log T(F)= ∑_[γ]∈ [Γ_+][ρ(γ)]χ_ orb(^1\ B_[γ])/m_[γ].We note the similarity between (<ref>) and (<ref>).The formal identity (<ref>) can be deduced formally using the path integral argument and Bismut-Goette's V-invariant<cit.> asin <cit.>. We leave the details toreaders.§.§ Reductive group with δ(G)=1 and with compact centerFrom now on, we assume that δ(G)=1 and that G has compactcenter. Let us introduce somenotation following <cit.>.We use the notation in (<ref>)-(<ref>). LetZ()⊂ Gbe the stabilizer ofin G, andlet ()⊂ be its Lie algebra.We define (), (), ^(),^(), ^() in an obvious way as insubsection <ref>, so that ()=⊕_, ()=_,and=⊕_⊕^(), =_⊕^(). Let Z^0()be the connected component ofthe identity in Z(). By (<ref>), we haveZ^0()=× M. Set^()=^()⊕^().Recall that we have fixed a_1∈ such thatB(a_1,a_1)=1. The choice of a_1 fixes an orientation of.By<cit.>, there exists unique α∈^* such that α,a_1>0, and that for anya∈, the action of (a)on^() has only two eigenvalues ±α,a∈. Take a_0=a_1/α,a_1∈. We haveα,a_0=1.Let ⊂^() (resp. ) be the +1 (resp. -1) eigenspace of (a_0), so that ^()=⊕.Clearly, =θ, and M acts onand .As explained in <cit.>,is even. Set l=1/2.Let ()⊂ and _⊂ be respectively the compact forms of () and of . Then,()=√(-1)⊕√(-1)_⊕_, _= √(-1)_⊕_.Let ^()⊂ be the orthogonal space of (),so that=√(-1)⊕_⊕^(). Let U()⊂ U and U_M⊂ U be respectively thecorresponding connected subgroups of complexmatrices of groups associated to the Lie algebras () and_. By <cit.>, U() and U_M are compact such that U()=exp(√(-1))U_M.Clearly, U() acts on , _, ^() and preservesthe splitting (<ref>). PutY_=U/U().By <cit.>, Y_ is a Hermitian symmetricspace of the compact type. Let ω^Y_∈Ω^2(Y_) bethe canonical Kälher form on Y_ induced by B.As insubsection <ref>, U→ Y_ is a U()-principle bundleon Y_ with canonical connection. Let (TY_,∇^TY_)and (N_,∇^N_) the Hermitian vector bundle withHermitian connection induced by the representation of U() on_ and ^_. For a vector space E, we still denoteby E the corresponding trivial bundleon Y_. By (<ref>), we have an analogy of <cit.>,=√(-1)⊕ N_⊕ TY_. Takek∈ T.Denote by (k), U^0(k), Y_(k) and ω^Y_(k) the analogiesof , U, Y_ and ω^Y_ when G is replaced by Z^0(k). The embedding U^0(k)→ U induces an embedding Y_(k)→ Y_. Clearly, k acts on the left on Y_, andY_(k) is fixed by the action of k. Recall that the equivariant A-forms A_k^-1N_|_Y_(k), ∇^N_|_Y_(k) and A_k^-1TY_|_Y_(k), ∇^TY_|_Y_(k) are defined in (<ref>). Let A^__k^-1(0) and A^^()_k^-1(0) be the components of degree 0 of the form A_k^-1N_|_Y_(k), ∇^N_|_Y_(k) and A_k^-1TY_|_Y_(k), ∇^TY_|_Y_(k). Following <cit.>,setA_k^-1(0)= A^__k^-1(0)A^^()_k^-1(0). By (<ref>), as in<cit.>, the following identity of closed forms on Y_(k) holds: A_k^-1(0)=A_k^-1N_|_Y_(k), ∇^N_|_Y_(k)A_k^-1TY_|_Y_(k), ∇^TY_|_Y_(k),which generalizes <cit.>. §.§ Auxiliary virtual representations of KWe follow <cit.>. Denote by RO(K_M) and RO(K) the real representation rings of K_M and K. Since K_M andK have the same maximal torus T, the restriction RO(K)→ RO(K_M) is injective. By <cit.>,we have the identity in RO(K_M),∑_i=1^m (-1)^i-1i Λ^i(^*)|_K_M=∑_i=0^_∑_j=0^2l(-1)^i+jΛ^i(^*_)⊗Λ^j(^*).By<cit.>, each term on the right hand sideof (<ref>) has a lift to RO(K). More precisely, let us recall<cit.>. Let η be a real finite dimensional representation of M on thevector space E_η such that *the restriction η|_K_M to K_M can be lifted into RO(K); *the action ofthe Lie algebra_⊂⊗_ on E_η⊗_, induced by complexification, can be lifted to an action of Lie group U_M; *the Casimir element C^_ of _ acts on E_η⊗_ as a scalar C^_,η∈.By <cit.>, letη=η^+-η^-∈ RO(K) be a realvirtual finite dimensional representation of K on E_η=E^+_η- E^-_η such that the following identity in RO(K_M) holds:E_η|_K_M=∑_i=0^_(-1)^iΛ^i(^*_)⊗ E_η|_K_M.Note thatM acts onbyadjoint action. By<cit.>, for 0≤ j≤ 2l, the induced representation η_jof K_M on Λ^j(^*) satisfiesAssumption<ref>, such that the following identity in RO(K) holds,∑_i=1^m (-1)^i-1i Λ^i(^*)=∑_j=0^2l(-1)^jE_η_j.§.§ Evaluation of ^[γ][exp(-t C^,X,η/2)]In <cit.>, we evaluatethe orbital integral^[γ][exp(-t C^,X,η/2)] whenγ=1 or when γ is a non elliptic semisimple element.In this subsection, we evaluate^[γ][exp(-tC^,X,η/2)] when γ is elliptic. To statethe result, let us introduce some notation <cit.>.Recall that T is a maximal torus of K_M, K and U_M. Denote by W(T,U_M)and W(T,K) the corresponding Weyl groups, and denote by (K/K_M) and (U_M/K_M) the Riemannian volumes induced by -B. Setc_G=(-1)^(-1)/2|W(T,U_M)|/|W(T,K)|(K/K_M)/(U_M/K_M)∈.The constant c_Z^0(k) is defined in a similar way.As in <cit.>, by (2) ofAssumption <ref>,U_M acts on E_η⊗_.We extend this actionto U() such that exp(√(-1)) acts trivially.Denote by F_,ηthe Hermitian vector bundleon Y_ with total space U×_U() (E_η⊗_) with Hermitian connection ∇^F_,η. Note thatthe Kähler form ω^Y_(k) defines a volume form dv_Y_(k) on Y_(k). For a U^0(k)-invariant differential form β on Y_(k),as in <cit.>, define β ^max∈ such thatβ-[β]^maxdv_Y_(k)has degree smaller than Y_(k). Recall that a_0∈ is defined in (<ref>).Let γ∈ G be semisimple. If γ can not be conjugated into H by elements of G, thenfor t>0, we have ^[γ] exp-t C^,X,η/2 =0.If γ=k^-1∈ T, then for t>0, we have ^[γ][exp-tC^,X,η/2]=c_Z^0(k)/√(2π)texpt/16 C^(),^() -t/2C^_,η[exp-ω^Y_(k),2/8π^2|a_0|^2 t A_k^-1TY_|_Y_(k),∇^TY_|_Y_(k)ch_k^-1F_,η|_Y_(k),∇^F_,η|_Y_(k)]^max.If γ=e^ak^-1∈ H with a≠0, then for any t>0, we have ^[γ] exp-t C^,X,η/2=1/√(2 π t) eTX_M(k),∇^TX_M(k) ^max exp-|a|^2/2t+t/16 C^(),^() -t/2C^_,η^E_η η(k^-1) /|(1-(γ))|_^_0|^1/2.Equations (<ref>), (<ref>) with γ=1, and(<ref>) are <cit.>. It remains to show(<ref>) for a non trivialγ=k^-1∈ T.Set_^(k)=_∩^(k), _^(k)=_∩^(k),^(k)=∩^(k).By (<ref>), we have_=_(k)⊕_^(k),_=_(k)⊕_^(k).Similarly,k acts on ^() and ^(). Set^_1()=^()∩(k),^_1()=^()∩(k),_1^(k)=^()∩(k),^_2()=^()∩^(k),^_2()=^()∩^(k),_2^(k)=^()∩^(k).Then^()=^_1()⊕_2^(),^()=^_1()⊕_2^().By (<ref>) and (<ref>), we get(k)=⊕_(k)⊕_1^(),(k)= _(k)⊕_1^(),^(k)= ^_(k)⊕_2^(),^(k)= ^_(k)⊕_2^().As in the case of <cit.>, we haveisomorphisms of representations of T,^_1()≃_1^()≃(k),where the first isomorphism is given by (a_0).Moreover, (a_0) induces an isomorphism of representations of T,^_2()≃_2^().Set_(k)=√(-1)_(k)⊕_(k), ^_(k)=√(-1)^_(k)⊕^_(k),_1^()=√(-1)_1^()⊕_1^(), _2^()=√(-1)_2^()⊕_2^(). Proceeding as <cit.>, by (<ref>) and by the Weyl integral formula for Lie algebra<cit.>, we have ^[k^-1] exp-t C^,X,η/2 =1/(2π t)^(k)/2expt/16^ C^, +t/48^ C^, (K^0(k)/T)/|W(T,K^0(k))|∫_Y∈((Y))|_(k)/ J_k^-1(Y) ^E_η η(k^-1)exp(-iη(Y)) exp-|Y|^2/2tdY. Asis also the Cartan subalgebra of _(k), we will rewrite the integral on the right-hand side as an integral over _(k). By (<ref>), (<ref>)-(<ref>), for Y∈, we haveJ_k^-1(Y)=A(i(Y)|_ _(k))/A(i(Y)|__(k)) (1-(k^-1))|__2^()⊕_2^() ^-1/21/(1-(k^-1))|__^(k)⊕_^(k)(1-exp(-i(Y))(k^-1))|__^(k)/(1-exp(-i(Y))(k^-1))|__^(k) ^1/2. As in <cit.>, by (<ref>), (<ref>), and (<ref>), for Y∈, we have((Y))|_(k)//((Y))|__(k)/J_k^-1(Y)^E_η η(k^-1)exp(-iη(Y)) =(-1)^_(k)/2((Y))|_(k)A^-1i(Y)|__(k) ^E_η η(k^-1)exp(-iη(Y))(1-(k^-1))|__2^() ^-1/2(1-exp(-i(Y))(k^-1))|__^(k)/(1-(k^-1))|__^(k) ^1/2.Let U_M(k) be the centralizer of k in U_M, andlet U^0_M(k) be the connected component of the identity in U_M(k).The right-hand side of (<ref>) is (U^0_M(k))-invariant. By (<ref>), (<ref>) and (<ref>), and usingagain the Weyl integral formula <cit.>, as in <cit.>, we get^[k^-1] exp-tC^,X,η/2 =(-1)^(k)/2/(2π t)^(k)/2c_Z^0(k) expt/16^ C^, +t/48^ C^,/ (1-(k^-1))|__2^() ^1/2∫_Y∈_(k)((Y))|_(k) A^-1i(Y)|__(k)^E_η η(k^-1)exp(-iη(Y)) (1-exp(-i(Y))(k^-1))|__^(k)/(1-(k^-1))|__^(k) ^1/2exp(-|Y|^2/2t)dY.Proceeding as in <cit.>, by (<ref>), we get ^[γ] exp-tC^,X,η/2 =c_Z^0(k)/√(2π t) expt/16 C^(),^() -t/2C^_,η[exp-ω^Y_(k),2/8π^2|a_0|^2tA_k^-1(0) A^-1_k^-1N_|_Y_(k),∇^N_|_Y_(k) ch_k^-1F_,η|_Y_(k),∇^F_,η|_Y_(k)]^max.By (<ref>) and (<ref>), we get (<ref>). §.§ Selberg zeta functionsWe follow <cit.>.Recall that ρ:Γ→U(r) is a unitaryrepresentation of Γ. For σ∈, we define a formal sum Ξ_η,ρ(σ)=-∑_[γ]∈ [Γ_+] γ=g_γ e^a k^-1 g_γ^-1[ρ(γ)]χ_orb(𝕊^1\B_[γ])/m_[γ] ^E_η η(k^-1) /|(1-(e^ak^-1))|__0^|^1/2e^-σ|a| and a formal Selberg zeta function Z_η,ρ(σ)=exp(Ξ_η,ρ(σ)).The formal Selberg zeta function is said to be well defined ifthe same conditions as in Definition <ref> hold. Recall that the Casimir operator C^g,Z,η,ρ acting on C^∞(Z,_η⊗_ F) is a formally self-adjoint second order elliptic operator, which is bounded from below. Setm_η,ρ(λ)=_C^,Z,η^+,ρ-λ-_C^,Z,η^-,ρ-λ.As in<cit.>,considerthe quotient of zeta regularized determinants det_ grC^,Z,η,ρ+σ=(C^,Z,η^+,ρ+σ)/(C^,Z,η^-,ρ+σ).By Remark <ref>, it is a meromorphic function on . Its zeros and polesbelong to the set {-λ:λ∈(C^,Z,η,ρ)}. The order of zeroat σ=-λ is m_η,ρ(λ). Set σ_η=1/8 C^(),^() -C^_,η.Let P_η,ρ(σ) be the odd polynomial defined by P_η,ρ(σ)=∑_[γ]∈[Γ_e] γ=g_γk^-1 g_γ^-1c_Z^0(k)[ρ(γ)](Γ(γ)\X(γ))/n_[γ] ( ∑^(k)/2_j=0(-1)^jΓ(-j-1/2)/j!(4π)^2j+1/2|a_0|^2jσ^2j+1ω^Y_(k),2jA_k^-1TY_|_Y_(k),∇^TY_|_Y_(k)ch_k^-1_,η|_Y_(k),∇^_,η|_Y_(k) ^max). There is σ_0>0 such that ∑_[γ]∈ [Γ_+] γ=g_γ e^a k^-1 g_γ^-1|χ_ orb^1\ B_[γ]|/m_[γ]e^-σ_0|a|/|(1-(e^ak^-1))|_^_0|^1/2<∞. The Selberg zeta function Z_η,ρ(σ) has a meromorphic extension to σ∈ such thatthe following identity of meromorphic functions onholds:Z_η,ρ(σ)= det_ grC^,Z,η,ρ+σ_η+σ^2exp(P_η,ρ(σ)).The zeros and poles of Z_η,ρ(σ) belong to{± i√(λ+σ_η):λ∈(C^,Z,η,ρ)}. If λ∈(C^,Z,η,ρ) and λ≠-σ_η, the order of zeroat σ=± i√(λ+σ_η) is m_η,ρ(λ). The order of zero at σ=0 is 2m_η,ρ(-σ_η). Also,Z_η,ρ(σ)=Z_η,ρ(-σ)exp(2 P_η,ρ(σ)).Proceeding as in <cit.>, by Theorems<ref> and <ref>, our theoremfollows. §.§ Theproof ofTheorem <ref> when G has compactcenter and δ(G)=1Weapply the results of subsection <ref> to η_j.Recall that α∈^* is defined in (<ref>).Proceeding as in<cit.>, by (<ref>), we find thatR_ρ(σ) is well-defined and holomorphic on the domain σ∈ and (σ)≫1, and that R_ρ(σ)=∏_j=0^2lZ_η_j,ρ(σ+(j-l)|α|)^(-1)^j-1.By Theorem <ref> and (<ref>),R_ρ(σ) has a meromorphic extension to σ∈. For 0≤ j≤ 2l, put r_j=m_η_j,ρ(0).By the orbifold Hodge theorem <ref>, as in <cit.>, we have χ_ top'(Z,F)=2∑_j=0^l-1(-1)^j-1r_j+(-1)^l-1r_l.SetC_ρ=∏_j=0^l-1(-4(l-j)^2|α|^2)^(-1)^j-1r_j, r_ρ=2∑_j=0^l(-1)^j-1r_j. Proceeding as in <cit.>, we get (<ref>). If H^·(Z,F)=0, proceeding as in<cit.>, for all 0≤ j≤ 2l, we have r_j=0.By (<ref>), we get (<ref>), which completes the proofof Theorem <ref> in the case where G has compact center and δ(G)=1. ' DKV79Ruan_Orbifold A. Adem, J. Leida, and Y. Ruan, Orbifolds and stringy topology, Cambridge Tracts in Mathematics, vol. 171, Cambridge University Press, Cambridge, 2007. AtiyahBott67 M. F. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes. I, Ann. of Math. (2) 86 (1967), 374–407. AtiyahBott68 M. F. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes. II. Applications, Ann. of Math. (2) 88 (1968), 451–491. BGV N. Berline, E. Getzler, and M. 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Satake, The Gauss-Bonnet theorem for V-manifolds, J. Math. Soc. Japan 9 (1957), 464–492. Seeley66 R. T. Seeley, Complex powers of an elliptic operator, Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966), Amer. Math. Soc., Providence, R.I., 1967, pp. 288–307. Selberg60 A. Selberg, On discontinuous groups in higher-dimensional symmetric spaces, Contributions to function theory (internat. Colloq. Function Theory, Bombay, 1960), Tata Institute of Fundamental Research, Bombay, 1960, pp. 147–164.Shfried S. Shen, Analytic torsion, dynamical zeta functions, and the Fried conjecture, Anal. PDE 11 (2018), no. 1, 1–74. S61 S. Smale, On gradient dynamical systems, Ann. of Math. (2) 74 (1961), 199–206. S67 S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. Thurston_geo_3_maniflod W. P. Thurston, The geometry and topology of three-manifolds,available at <http://www.msri.org/publications/books/gt3m>,unpublished manuscript, 1980.Voros A. Voros, Spectral functions, special functions and the Selberg zeta function, Comm. Math. Phys. 110 (1987), no. 3, 439–465.Waldron J. Waldron, Lie algebroids over differentiable stacklie algebroids over differentiable stacks, arXiv:1511.07366.Institut de Mathmatiques de Jussieu-Paris Rive Gauche, Sorbonne Universit,4 place Jussieu, 75252 Paris Cedex 05, France [email protected] School of Mathematical Sciences, University of Science and Technology of China,96 Jinzhai Road,Hefei, Anhui 230026, P. R. China. [email protected] | http://arxiv.org/abs/1704.08369v3 | {
"authors": [
"Shu Shen",
"Jianqing Yu"
],
"categories": [
"math.DG"
],
"primary_category": "math.DG",
"published": "20170426223224",
"title": "Flat vector bundles and analytic torsion on orbifolds"
} |
Canonical RDEs and general semimartingales as rough paths Rong Du1, Lazaros Gkatzikis1, Carlo Fischione1, Ming Xiao2This work is supported by the Wireless@KTH Seed Project LTE-based Water Monitoring Networks and ICT EIT Lab project I3C.1 Automatic Control Department, 2 Communication Theory DepartmentKTH Royal Institute of Technology, Stockholm, Sweden Email: {rongd, lazarosg, carlofi, mingx}@kth.se April 25, 2017 =================================================================================================================================================================================================================================================================================================================================================================== Many physical systems, such as water/electricity distribution networks, are monitored by battery-powered Wireless Sensor Networks (WSNs). Since battery replacement of sensor nodes is generally difficult, long-term monitoring can be only achieved if the operation of the WSN nodes contributes to a long WSN lifetime. Two prominent techniques to long WSN lifetime are i) optimal sensor activation and ii) efficient data gathering and forwarding based on compressive sensing. These techniques are feasible only if the activated sensor nodes establish a connected communication network (connectivity constraint), and satisfy a compressive sensing decoding constraint (cardinality constraint). These two constraints make the problem of maximizing network lifetime via sensor node activation and compressive sensing NP-hard. To overcome this difficulty, an alternative approach that iteratively solves energy balancing problems is proposed. However, understanding whether maximizing network lifetime and energy balancing problems are aligned objectives is a fundamental open issue. The analysis reveals that the two optimization problems give different solutions, but the difference between the lifetime achieved by the energy balancing approach and the maximum lifetime is small when the initial energy at sensor nodes is significantly larger than the energy consumed for a single transmission. The lifetime achieved by the energy balancing is asymptotically optimal, and that the achievable network lifetime is at least 50% of the optimum. Analysis and numerical simulations quantify the efficiency of the proposed energy balancing approach. network lifetime, energy balancing, sensor network, cyber-physical system§ INTRODUCTIONWireless sensor networks (WSNs) are being used to monitor critical infrastructures in smart cities, such as water distribution networks, tunnels, bridges, and towers. Since sensor nodes are generally power limited, and battery replacement is difficult or even impossible, network lifetime is an important performance metric <cit.>. Several approaches have been proposedto prolong network lifetime and hence to enable long-term monitoring. For example, sensor nodes can form clusters, where participating nodestake turn to act as cluster-head to balance the energy consumption of the nodes <cit.>. The nodes can optimize routing <cit.> or use multi-hop short range communication <cit.> to save energy in the data transmission. Moreover, event-trigger mechanisms <cit.> can be used to reduce the transmitted data volume. The sensor nodes can also be put into sleep or idle mode to save energy <cit.>. The methods to be used for energy saving should depend on the characteristics of the monitoring applications.In this paper, we consider the case of using densely deployed sensors to monitor an area where node replacement is difficult. Such a dense sensor network has the following benefits: * better detection of events; * robustness to sensor failure and measurement errors because of the availability of redundant sensor nodes. Thus, the network operation is ensured even when some sensor nodes fail; * reduced energy consumption in data transmission by exploiting multi-hop short range communication. Thus, network lifetime is increased.Consequently, even though dense networks introduce a higher installation cost, they substantially reduce the maintenance cost in return, and, most importantly, may provide better monitoring performance.We consider to use data compression in the data gathering process, together with a sleep/awake mechanism for the sensing process, to prolong lifetime for such a dense sensor network. A natural question is whether the usual approach of energy balancing, i.e., preferably use the nodes with more residual energy <cit.>, would be a viable choice for maximizing network lifetime in this context.As the sensor nodes are densely deployed, their measurements exhibit spatial correlations. Such correlations enable us to use compressive sensing (CS) to accurately estimate the state of the monitored infrastructure with a minimal number of measurements<cit.>. Therefore, one may adopt a CS-based data gathering scheme, such that in every timeslot only a portion of sensor nodes is activatedto sense and transmit data hop-by-hop to the sink nodes. It follows that the expected monitoring performance of such a system can be guaranteed by CS while its energy efficiencycan be improved by turning off the rest of the sensor nodes.In our previous works <cit.>, we proposed a CSbased sensor activation scheme based on energy balancing for dense WSNs to monitor water distribution networks. The devised data gathering scheme activates only a few connected sensor nodes for sensing and data transmission, to reduce the overall energy consumption and so that the monitoring performance is guaranteed even under sensor failures. However, whether that algorithm (or any energy-balancing based one) can achieve the maximum WSN lifetime is an open issue that has not been investigated before in the CS context. In summary, we address this fundamental issue of whether, in the considered WSN scenario, the energy balancing problem is equivalent to lifetime maximization. The main results of the paper are as follows: * In order to provide insight on the complexity and the structure of the Lifetime maximization Problem, we cast it as a Multi-dimensional Knapsack problem , for which a rich literature on solution approaches exists. * We provide an easy to calculate upper bound of maximum lifetime based on a transformation to a maximum flow problem, as shown by Theorem <ref> in Section <ref> A. * We propose an algorithm that gives an approximate solution to the maximum lifetime problem. The algorithm is based on the solution to an energy balancing problem. We show that such an algorithm is asymptotically optimal (as given by Theorem <ref> in Section <ref> C) and the worst case approximation ratio (the ratio of the lifetime achieved by the algorithm to the optimal lifetime) is 50% (as shown in Theorem <ref>in Section <ref> C). The asymptotic optimality and the approximation ratio of the algorithm constitute major original contributions of this paper because for maximum lifetime problems with connectivity and cardinality constraints there are no known optimality bounds. The organization of the rest of the paper is as follows. We provide an overview of related literature in Section <ref>. The formulation of the lifetime maximization problem and the energy balancing problem are described in Section <ref>. In Section <ref>, the performance of the proposed algorithm in terms of network lifetime is analyzed.Numerical evaluations are provided in Section <ref>. The conclusions of this work are presented in Section <ref>. § RELATED WORK§.§ Lifetime maximization by flow approximationThe lifetime of a network greatly depends on the residual energy of the participating nodes. There are different models for the energy consumption of sensor nodes. In <cit.>, the energy consumption is linearly related to the receiving power, transmitting power and data transmission rate, and the expected lifetime of a sensor node is defined as the ratio of the energy capacity of the node and the average energy consumption. In <cit.>, a slice model is used where the monitored area is partitioned into slices, each of which contains all the nodes that have the same hop distance from the sink node. In this model, the energy consumption of the nodes depends also on the distance (hop count). However, the distances of a node to any node in the same slice are considered to be the same. In <cit.>, the energy consumptions of the nodes in sleep mode are assumed to be 0, whereas that of the active nodes follows an independent and identical distribution. In <cit.>, a non-linear dynamic energy consumption model, called kinetic battery model, is used. In <cit.>, given the fixed topology of a WSN, the energy consumption rate of the working sensor nodes is considered constant, whereas the consumption of the sleeping sensor nodes is 0. Similar to <cit.>, we normalize the energy consumption of the active sensor nodes to 1 and set that of the nodes in sleep mode equal to 0. One common way to prolong network lifetime is by reducing energy consumptions of each node. For example, in the event-trigger based approaches <cit.>, one could reduce the sampling rate of the sensors, to save energy of sensing and data transmission. Notice that for general networks, the data transmission constitutes the major part of a node's energy consumption. Thus, several solutions have been considered to reduce nodal energy consumption due to data transmission, such as controlling the transmission power <cit.> and compressing the measurements <cit.> to be transmitted. Besides, harvesting energy from ambient environment <cit.>, such as solar energy and vibration, or transmitting energy to the nodes wirelessly <cit.>, can also extend the lifetime of a node, but are beyond the scope of this work. Besides the nodal perspective, prolonging lifetime from the network perspective, e.g., by optimized routing, has been widely studied. In this context, the classic flowapproximation is commonly used <cit.>. In particular, the energy budget of each node is represented as the number of flows that can pass the node, which is referred to as `vertex capacity'. Then, finding the route in each wireless communication timeslot to maximize network lifetime is equivalent to finding the maximum flow from source node to sink node. In the seminal work <cit.>, the energy consumption of the network has been modelled as a function of the traffic flow routing decisions. Then the problem is cast as a linear programming problem. In a similar network setting, where every sensor node can either transmit its data to its neighbor with low energy cost, or transmit data directly to the sink node with high energy cost, maximizing network lifetime is equivalent to flow maximization and energy balancing <cit.>. In such scenario, energy balancing has been used to maximize network lifetime <cit.>. Another way to balance the energy consumption is rotating the working period of sensor nodes, i.e., allowing some sensor nodes to sleep without sacrificing in the monitoring performance. For instance, Misra et. al. <cit.> have considered finding different connected dominating sets of the WSN to prolong network lifetime. In each timeslot, only the sensor nodes in the connected dominating set are active and the other nodes are put into sleep. To rotate the working period of the nodes, it is desired to find the maximum connected domatic partition, which divides the WSN into as many as possible disjoint connected dominating sets. A similar problem has been considered in <cit.>, where the sensor nodes have the energy harvesting ability. Thus, the working schedules of the connected dominating sets are also taken into consideration.Compared to the WSN scenarios mentioned above, CS considered in this paper introduces a cardinality constraint. It is an open question how the problem of lifetime maximization of such WSN is related to the energy balancing problem.Another major difference of our study is that, in most of previous studies, the network lifetime depends on the minimal nodal lifetime, i.e., the network depletes once a node in the network depletes, since the sensor nodes in those scenarios monitor different events; here, due to the correlations of measurements, CS enables us to care only about the number of activated sensor nodes. The network is considered depleted only when either there are not enough remaining sensor nodes to satisfy the cardinality requirement, or the remaining sensor nodes, with the sink nodes, can not form a connected graph. Thus, the flow approximation and the existing solutions cannot be directly applied in our setting. In this paper we adopt the concept of energy balancing and we characterize its relation to the maximum lifetime. §.§ Compressive sensing for data gathering For the sake of completeness, we describe the operation of existing data gathering schemes based on CS. Suppose that the sink node in a network wants to collect the measurements of N sensor nodes. In work <cit.>, the proposed compressive data gathering (CDG) algorithm operates as shown in Fig. <ref> (a), in which each sensor node multiplies its local measurement d_i∈ℝ, for i=1,…,N with a random vector ϕ⃗_i of dimension M≪ N, adds the product ϕ⃗_id_i with the vector ∑_k=1^i-1ϕ⃗_id_i it receives from its neighbour, and then transmits the summation ∑_k=1^iϕ⃗_id_i to its next-hop sensor node. At the sink node, the measurements of the sensor nodes, y⃗=[ϕ⃗_1,…,ϕ⃗_N]d⃗, are recovered by solving an l_1-minimization problem as follow:x⃗̂⃗=min_x⃗x⃗_l_1s.t. y⃗-Φ⃗Ψ⃗x⃗_l_2^2≤ε ,where Ψ⃗ is the basis on which d⃗ is sparse, and where l_1 and l_2 are the Manhattan norm and the Euclidean norm respectively. Then, d⃗̂⃗=Ψ⃗x⃗̂⃗.In this case, every sensor node only transmits messages of size O(M), which balances the energy consumptions of the sensor nodes and also reduces the overall data traffic. Fig. <ref> (b) shows the Compressed Sparse Function (CSF) algorithm <cit.> for data gathering. In the figure, the grey sensor nodes sense and transmit their measurements to the next-hop sensor nodes, whereas the white ones act as relay nodes. As long as the sink node collects M out of N data measurements, the CSF algorithm can recover the remaining N-M measurements. The idea is based on the mapping of the reading of the sensor nodes to their locations (or ids). Denote the mapping function f(x) where x is the location of the id of the sensor nodes. Since the sensor nodes are densely deployed, f(x) could be represented as f(x)=Ψ⃗ (x)^Tc⃗, where Ψ⃗ (x)^T is a basis such as the type-IV DCT basis, and c⃗ is the sparse coefficient of f(x) on the basis. Then, suppose the ids of the activated sensor nodes are a_1,…,a_M, the sink node has the data of y⃗=[f(a_1),…,f(a_M)]^T=[Ψ⃗(a_1)^T,…,Ψ⃗(a_M)]^Tc⃗. The sink node estimates c⃗ by the l_1-minimization problem similar to the CDG approach, and then the mapping function f(x) is retrieved. Based on f(x), the measurements of the white nodes are also retrieved.Considering a dense network, in our previous work <cit.>, we proposed the data gathering scheme shown in Fig. <ref> (c), where only the grey nodes are active and transmit data in the CDG way to the sink node. Since each active node transmits a calculated vector based on the summation of its measurement and its received vector, the packet sizes of the nodes are the same, and the energy consumption of the active nodes is balanced. The sink node first uses l_1-minimization to estimate the measurement of the grey sensor nodes, and then we use CSF to estimate the measurement of the white sensor nodes. The overall data traffic is further reduced, and the energy consumption of the grey sensor nodes is balanced. If the activation of the sensor nodes is decided carefully in each monitoring timeslot, the energy consumption of all the sensor nodes can be approximately balanced. However, understanding whether the network lifetime can be maximized by such an energy balancing approach is a fundamental open question.Compared to our previous work <cit.>, the major difference of this paper is two-fold: 1) our previous work provides an efficient method to solve the energy balancing problem, whereas this paper characterizes the performance of energy balancing in terms of network lifetime; 2) in this paper, we achieve another upper bound of network lifetime by transforming the original problem into a maximum flow with cardinality constraint problem. This upper bound is tighter than the bound of <cit.> and provides useful insight for the structure of the problem. § PROBLEM FORMULATION AND PRELIMINARIES We consider a WSN consisting of two tiers of nodes that monitors an area of line shape. Characteristic such examples are a pipeline in a water distribution network, a tunnel, or a bridge. The first tier consists of battery-powered sensor nodes that are densely deployed in the monitored area. Their role is to sense and relay data to a set of sink nodes. The second tier consists ofsink nodes, which are grid-powered and are deployed at the two ends of the line. They collect data from the sensor nodes, and transmit the data to a remote monitoring center. Due to the length of the monitored area and the comparative small communication range of the sensor nodes, a multi-hop communication path from the sensor nodes to the sink node has to be established. Since battery replacement is not easy for the applications mentioned above, a main objective is to maximize the network lifetime. Intuitively, it is beneficial to keep alive as much of the sensor nodes as possible, which motivates the design of activation algorithms based on energy balancing, i.e., preferably activate the nodes with more residual energy. Thus, the major problem to be considered here, is whether the maximum network lifetime can be achieved by the energy balancing approach, or (if not), what is the performance of the energy balancing approach in terms of network lifetime.In the following, we use a scenario of pipeline monitoring for water distribution network to better illustrate the necessary concepts. However, all the provided results hold for any linear network. The major notations are listed in Table <ref>. A WSN for monitoring a single pipeline can be represented by a communication graph 𝒢=(𝒱,ℰ), where vertex set 𝒱 represents both the sensor nodes and the sink nodes, and edge set ℰ represents the links among nodes. Suppose there are N sensor nodes, then we denote s_l the leftmost sink node, s_r the rightmost sink node, and v_1,v_2,…,v_N the sensor nodes from left to right. For simplicity, sink nodes s_l and s_r are also represented as v_0 and v_N+1. Let r_i be the transmission range of v_i, then ⟨ v_i,v_j⟩∈ℰ if and only if the distance between v_i and v_j is smaller than or equal to r_i. Also, we denote 𝒩(v_i)={v_j|⟨ v_i,v_j⟩∈ℰ} the set of neighbours of v_i, and 𝒩_-(v_i)={v_j|v_j∈𝒩(v_i)∧ j>i} the downstream set of v_i.Our analysis relies on the following two assumptions that generally hold in water distribution networks[Those assumptions are necessary to derive the analytical results. More details on the general performance can be found in our previous work <cit.>.]: All sensor nodes and the sink nodes are characterized by the same communication range r_i=r. We suppose that the transmission power of the sensor nodes are fixed to a pre-set value. Since in a dense network, a node could have multiple neighbors for data relaying, the transmission power of a node could be set to the minimum value to save energy. Thus, the communication range of all the sensor nodes is the same. However, in the numerical simulations, we also examine the performance of Algorithm 1 when this assumption does not hold. All sensor nodes and sink nodes in the same pipeline are deployed in a line. This assumption is reasonable as the diameter of the pipeline is small compared to the length of a pipeline. Thus, a pipeline sensor network can be considered as a line. For energy saving purposes, nodes can be deactivated. Time is slotted. In a timeslot t, a sensor node can either be activated to sense and transmit data, or be in the sleeping mode. The activated sensor nodes transmit data in the CDG way <cit.>, i.e., every node transmits a vector of the same size based on the projection of its measured data and the vector it receives. Thus, the payloads of the transmitted packets at each active node are the same, and the energy consumption is approximately the same. Therefore, we may normalize the energy consumption for the active sensor nodes in a timeslot to 1 for simplicity. Then the energy budget of v_i, E_i∈𝒵^+, is characterized as the number of timeslots that a node can be activated. Let E_i(t) denote the number of timeslots that v_i can be activated from timeslot t, which can be considered as the residual energy at t, and E_i(1)=E_i.In the following, we first formulate the lifetime maximization problem. Then, we introduce an equivalent multi-dimensional knapsack problem, which allows us to solve the lifetime maximization problem for small networks. Last, we present the energy balancing problem and a solution algorithm, which are then used to study the achievableperformance of energy balancing in terms of network lifetime. §.§ Lifetime maximization problemWe define the lifetime of a WSN to be the operating time until either WSN becomes disconnected, or the monitoring performance of the WSN cannot be guaranteed. In each timeslot, the connectivity and the monitoring performance requirement of the active sensor network must be satisfied. Let binary variable x_i(t) indicate whether node v_i is active at timeslot t.Then, the energy dynamics of v_i can be written as E_i(t+1)=E_i(t)-x_i(t), and the scheduling problem considered in this paper is to determine x⃗(t)=[x_1(t),…,x_N(t)]^T,∀ t.Let 𝒢(x⃗(t)) denote the induced graph of active sensor nodes and the sink nodes. Then, the connectivity constraint is defined as follows: (Connectivity Constraint) The activation of the sensor nodes x⃗(t) satisfies the connectivity constraint if and only if the induced graph 𝒢(x⃗(t)) is connected. To check the connectivity of a subgraph, one approach could be breadth-first search. However, since the connectivity checking is not the major objective of the paper, we do not elaborate further. More details can be found in Chapter 3.2 of <cit.>.Regarding monitoring performance, our previous works <cit.> have shown that the estimation error by CS is related to the number of active nodes, m, where m is much smaller than the number of sensor nodes N, i.e., m≪ N. Thus, the requirement on monitoring performance can be specified as a cardinality constraint defined as follows: (Cardinality Constraint) The activation of the sensor nodes x⃗(t) satisfies the cardinality constraint if and only if ∑ x_i(t)≥ M_cs, where M_cs is determined by the required estimation error of the measured data.Then, the lifetime maximization problem can be formulated as an optimal control problem as follows: max_x⃗(t),t=1,…,T∑_t=1^T 1s.t.E_i(t+1)=E_i(t)-x_i(t),∀ i,1≤ t≤ T, E_i(1)=E_i,∀ i, E_i(t)≥ 0,∀ i,1≤ t≤ T+1,𝒢(x⃗(t)) is connected ,∀ t,∑_v_i∈𝒱x_i(t)≥ M_cs,∀1≤ t≤ T,x_i(t)∈{0,1},∀ i,t , where Constraint (<ref>) is the dynamic of the energy of the sensor nodes, Constraint (<ref>) is the initial state of the WSN, (<ref>) is the non-negative constraint on the energy of sensor nodes, (<ref>) is the connectivity constraint, (<ref>) is the cardinality constraint. Also, we have that Constraints (<ref>)-(<ref>) can be equivalently captured by the following energy budget constraint: ∑_t=1^Tx_i(t)≤ E_i,∀ i. ,where the proof is in Appendix <ref>.Notice that, given a feasible x⃗(t),∀ t that satisfies Constraints (<ref>), and (<ref>), we can construct E_i(1)=E_i, and E_i(t+1)=E_i(t)-x_i(t), such that E_i(t)≥ 0,∀ i,1≤ t≤ T+1 always holds. Thus, indeed we can replace Constraints (<ref>)-(<ref>) by Constraint (<ref>). This optimization problemis particularly challenging due to the binary nature of activation variables and cardinality constraint (<ref>), as we articulate below. Note that Problem (<ref>) addresses only which set of nodes should be activated, and not how the routing of measurements to the sink in a multi-hop fashion should be performed. Under Assumptions <ref> and <ref>, all vertices in a connected subgraph of 𝒢 can form a connected routing path, i.e., they are in a line topology and every node receives data from at most one node and transmits data to at most one node (see more details from the proof Theorem 1 in our previous work <cit.>). Thus, it is guaranteed that a forward to the nearest active neighbour routing is always an optimal routing strategy. In other words, the induced graph of the active sensor nodes and the sinks is a connected graph, if and only if there is a path in the induced graph from v_0 to v_N+1 that passes only through all the active nodes. Thus, it is equivalent to replace Constraint (<ref>) by the requirement of the existence of path from v_0 to v_N+1 for any t without changing the optimal solution of the original problem, as we will do in Section <ref> to simplify analysis.§.§ Knapsack approximation for small WSNsLifetime maximization Problem (1) is NP-Hard <cit.>. In order to provide insight on the complexity and the structure of the problem, we show that it can be cast as a knapsack optimization problem, for which a solution method is known. However, the method is practical only for small networks. We also use it in the numerical evaluation part as a benchmark. To begin with, we define activation profile as follow: (Activation Profile) An activation profile is a group of sensor nodes that satisfies the connectivity constraint. We say an activation profile is feasible if and only if it also satisfies the cardinality constraint. Then, we may equivalently reformulate the maximum lifetime problem as stated in the following lemma: Consider a WSN 𝒢. Let L be the total number of feasible activation profiles. Each feasible activation profile is represented by a column vector Q⃗_l=[q_l(1),…,q_l(N)]^T. Here,q_l(i)=1 if and only if sensor node v_i belongs to profile l, otherwise q_l(i)=0. Define vector z⃗=[z_1,…,z_L]^T, where z_i denotes the number of timeslots that profile i is chosen for activation. Then, the lifetime maximization Problem (<ref>) is equivalent to the following problem: T=max_z⃗ ∑_l=1^Lz_l s.t. Q⃗z⃗≤E⃗,z_l∈𝒵^+,∀ l∈{1,…,L} , where Q⃗=[Q⃗_1,…,Q⃗_L] is the set of all feasible profiles, E⃗=[E_1,…,E_N]^T is the vector of initial energies of all sensor nodes, and 𝒵^+ is the set of non-negative integers. The proof is in Appendix <ref>.Lemma <ref> shows that the maximum lifetime problem under cardinality and connectivity constraints can be turned into a multi-dimensional knapsack (MDK) problem <cit.>. In our case, the knapsack corresponds to the energy budget of each sensor nodes, and each profile corresponds to an item of value 1. There are several methods to solve MDK problems, such as branch-and-bound, dynamic programming, and heuristic algorithms. We compare them in terms of complexity and optimality in Table <ref>. Branch-and-bound, and dynamic programming can achieve optimal solution, however, the complexity of branch-and-bound is as high as exhaustive search in the worst case, whereas dynamic programming suffers from the curse of dimensionality and is not a scalable approach. Therefore, they cannot be applied in our scenario, where network size is large and nodes have limited computational and storage capability. However, we use branch-and-bound algorithm insimulation part for comparison purposes. Besides, heuristic methods such as genetic networks, even though with low complexity, cannot guarantee optimality. Thus, the transformation into an MDK problem is only valid for small scale network instances. For large scale (dense) networks, we consider approximating Problem (<ref>) by an energy balancing problem described in the next subsection. §.§ Energy balancing problemIn our previous work <cit.>, we proposed an energy balancing problem, together with a solution method. Since in this paper we investigate the fundamental properties of the energy balancing problem from the point of view of network lifetime maximization, we give the necessary details in the following:Recall that E_i(t) is the residual energy of v_i at timeslot t, we define its normalized residual energy as p_i(t)=E_i(t)/E_i. We denote 𝒱(t)={v_i∈𝒱|E_i(t)≥ 1} the set of candidate sensor nodes to be potentially activated at timeslot t. Then, we posed the energy balancing problem as a sequence of problems, and one for each timeslot t as follows: max_x⃗∑_i∈𝒱 x_ip_i s.t. ∑_v_i∈𝒱 x_i=max{M_cs,M_c}, 𝒢(x⃗)is connected, x_i∈{0,1}, ∀ i∈𝒱, where time index t is discarded for notational simplicity, and M_c is the minimum number of sensor nodes that must be activated to satisfy the connectivity constraint. The intuition behind the energy balancing approach is that, in each timeslot, the number of active sensor nodes has to be as small as possible. Out of the feasible activation profiles with the same number of active sensor nodes, the profile with maximum normalized residual energy is preferred. Recall that M_cs is the minimum number of sensor nodes that should be activated to ensure monitoring performance.When M_cs<M_c, one cannot find a route from s_l to s_r which passes exactly M_cs sensor nodes. In this case, the requirement of ∑_v_i∈𝒱x_i=M_cs contradicts to Constraint (<ref>). It gives us that ∑_v_i∈𝒱x_i should be M_c instead. Thus, even though Constraint (<ref>) is given, the M_c in Constraint (<ref>) is not redundant. In <cit.>, we developed an algorithm to solve Problem (<ref>). The details of the procedure are shown in Algorithm <ref>.First, Algorithm <ref> finds M_c by a shortest path algorithm such as Dijkstra's algorithm in Line 1, namely finds the shortest path from v_0 to v_N+1, where the weights of all the edges are 1. Then, the minimum number of sensor nodes, m, that satisfies both the connectivity and the cardinality constraints is calculated in Line 3. Knowing m, we can solve Problem (<ref>) by dynamic programming (Line 4), where g(v_i,k) represents the maximum sum of normalized residual energy of k connected sensor nodes among v_i+1 to v_N, and it is calculated asg(v_i,k)=max_v_j∈𝒩'_-(v_i){g(v_j,k-1)+p_j} ifk>0, 𝒩'_-(v_i) ≠∅ 0ifk=0, s_r∈𝒩_-(v_i) -∞ otherwise ,where p_j is the normalized residual energy of sensor node v_j. Recall that 𝒩_-(v_i)={v_j|v_j∈𝒩(v_i)∧ j>i} is the downstream set of v_i, 𝒩'_-(v_i)=𝒩_-(v_i)\{s_r} is the set of sensor nodes in the downstream set of v_i. Notice that g(v_i,k) is directly related to g(v_j,k-1), where v_j is in the neighbour of v_i. Thus, the nodes determined by this dynamic programming are connected. In our previous work <cit.>, we proposed to activate the sensor nodes as suggested by the solution of the energy balancing problem (<ref>) in each timeslot. Then update the nodal normalized residual energy to be the input of the energy balancing problem in the next timeslot, until the problem is infeasible, as described by Algorithm <ref>. However, whether this approach could lead to the maximum network lifetime has not been analyzed before. Thus, the investigation of the fundamental properties of energy balancing in terms of network lifetime with a cardinality constraint is the core contribution of this paper.§ PERFORMANCE ANALYSIS OF THE OPTIMAL ACTIVATION SCHEDULEIn this section, we analyze the performance of Algorithm <ref> in terms of network lifetime. We transform the lifetime maximization Problem (<ref>) to a maximum flow problem with a typical vertex capacity constraintsand a new cardinality constraint. Such transformationenables a better understanding ofthe maximum lifetime problem, provides us a new lifetime upper bound, and also enables us to derive the performance bound of the proposed Algorithm <ref>. §.§ Lifetime maximization as a maximum flow problemThe maximum lifetime Problem (<ref>) is a maximum flow problem with vertex capacities (see <cit.> for a description of these problems) with an additionalcardinality constraint. Let u_i,j,t denote the flow from node v_i to v_j in timeslot t.Then, the maximum flow with vertex capacity and cardinality constraints is formulated as follows: max_u⃗T s.t. ∑_v_j∈𝒩(v_i)u_i,j,t-∑_v_j:v_i∈𝒩(v_j)u_j,i,t= 1, i=0,∀ t=1,…,T -1,i=N+1,∀ t=1,…,T 0,∀ 1≤ i≤ N,∀ t=1,…,T ,∑_v_j∈𝒩(v_i)u_i,j,t≤ 1,∀ i,t∑_t=1^T∑_v_j∈𝒩(v_i)u_i,j,t≤ E_i,∀ i=1,…,N,∑_v_i∈𝒱∑_v_j∈𝒩(v_i)u_i,j,t≥ M_cs+1, ∀ t=1,…,T, u_i,j,t∈{0,1},∀ i,j,t ,Then, we have the following lemma: Under Assumptions <ref> and <ref>, Problem (<ref>) is equivalent to Problem (<ref>). The proof is in Appendix <ref>.Problem (<ref>) is a binary programming problem. If Constraint (<ref>) is relaxed to 0≤ u_i,j,k≤ 1,∀ i,j,k, the problem becomes a linear programming problem. Then, an upper bound of WSN lifetime for Problem (<ref>) can be established, as stated by the following theorem: Consider optimization Problem (<ref>) for a WSN that follows Assumptions <ref> and <ref>. The WSN lifetime is upper bounded by T̅^f, where T̅^f is the optimal value of Problem (<ref>) with Constraint (<ref>) relaxed as 0≤ u_i,j,t≤ 1,∀ i,j,t.Suppose T^* is the optimal value of Problem (<ref>), then it is also the optimal value of Problem (<ref>) according to Lemma <ref>. Further, as T̅^f is the optimal value of the relaxed Problem (<ref>), we have T^*≤T̅^f which completes the proof. This bound is quite tight if Assumptions <ref> and <ref> hold, as will be shown in Section V. Notice that the relaxed Problem (<ref>) is a linear optimization problem and solvable. Consequently, we can use a bisection approach to find T̅^f, as discussed in Appendix <ref>. Besides, one may derive a good solution by rounding the result of the relaxed Problem (<ref>)[However, the main difficulty is to determine the rules of rounding such that the result satisfies both connectivity and cardinality constraints, and leads to the maximum network lifetime. This is left as a future work.].Based on Theorem <ref>, we study the performance of Algorithm <ref> from the perspective of maximum flow problem. We first turn the maximum flow Problem (<ref>) with vertex capacities to a maximum flow problem with edge capacities according to the following remark. Problem (<ref>) can be formulated as a maximum flow problem with edge capacities <cit.> and cardinality constraints. The basic idea is to substitute each node v_i with two nodes v_i^in and v_i^out connected by an arc ⟨ v_i^in,v_i^out⟩ with capacity E_i. More details can be found in Appendix <ref>. Then, we show how the problem can be solved via a modified maximum flow algorithm. For such a purpose, we introduce some additional notations.Let f_ii be the flow on arc ⟨ v_i^in,v_i^out⟩, and f_ij the flow on arc⟨ v_i^out,v_j^in⟩. The capacity of arc ⟨ v_i^in,v_i^out⟩ is C_ii=E_i, and the capacity of arc ⟨ v_i^out,v_j^in⟩ is C_ij=+∞, as shown in Fig. <ref>. Given a route R⃗_a=⟨ s_l, v_a_1,…,v_a_k, s_r⟩, we say that arc ⟨ v_a_i,v_a_i+1⟩ belongs to the set of forward arcs of R⃗_a, which is denoted by R⃗_a^+, if and only if ⟨ v_a_i,v_a_i+1⟩∈ℰ'. Otherwise, the arc belongs to the set of backward arcs of R⃗_a, which is denoted by R⃗_a^-. Then the maximum flow increment of R⃗_a is defined as δ_a=min{{C_a_ia_j-f_a_ia_j|⟨ v_a_i,v_a_j⟩∈R⃗_a^+},{f_a_ia_j|⟨ v_a_i,v_a_j⟩∈R⃗_a^-}}. R⃗_a is said to be unblocked if and only if δ_a>0. Then we can use a modified Ford-Fulkerson Algorithm to find which nodes shouldbe activated at each timeslot, so that the corresponding route at each timeslot is feasible. §.§ A modified maximum flow algorithm based on Ford-Fulkerson AlgorithmThe derived modified Ford-Fuklerson Algorithm works as follows. We find an unblocked route R⃗_i from s_l to s_r in 𝒢' that contains no backward arcs andpasses at least M_cs arcs with capacity less than +∞. This is equivalent to passing at least M_cs sensor nodes in 𝒢, and hence it corresponds to one route in R⃗. We perform an augmentation along route R_i with increment 1, i.e., the flows f of all the arcs in the route R_i increase by 1. Then, we find an unblocked route that contains no backward arcs until there is no such unblocked route in 𝒢' again. This operation gives a sequence of routes R⃗(1),R⃗(2),…,R⃗(T). If we are unable to find an unblocked route that contains backward arcs at T+1, then activation scheme R⃗=[R⃗(1),R⃗(2),…,R⃗(T)] leads to maximum network lifetime.Notice that this algorithm does not allow choosing an unblocked route with backward arcs in each timeslot, it requires an exhaustive search, and thus is not practical. However, it suggests the following lemma: Consider a WSN satisfying Assumptions <ref> and <ref>. If an unblocked route with backward arcs exists when Algorithm <ref> terminates, we can alter one of the previous activation decisions to extend network lifetime by 1. The proof is in Appendix <ref>. This lemma shows that the existence of an unblocked route with backward arcs suggests the suboptimality of an activation. Furthermore, extending the network lifetime in such way requires an unblocked routewith backward arcs (R⃗_b in the proof) and an unblocked route with no backward arcs (R⃗_a in the proof) that we have selected before the WSN expires. This gives us the worst case approximation ratio of the lifetime achieved by Algorithm <ref> to the maximum network lifetime, as will be shown in the next subsection. §.§ Performance analysis of Algorithm <ref>According to Lemma <ref>,the more the unblocked routes we can find, the more suboptimal the activation is. Thus, we can analyze the gap of lifetime we achieve by Algorithm <ref> to the optimal lifetime value, by counting how many unblocked backward routes can be found when Algorithm <ref> terminates.Then, the performance of Algorithm <ref> can be characterized by the following lemmas. Consider a WSN 𝒢 that satisfies Assumptions <ref> and <ref>. Algorithm <ref> is applied to determine the activation of the sensor nodes in each time slot. Let the achieved WSN lifetime be t_1, i.e., on t_1+1, no unblocked routes that contain only forward arcs can be found. If an unblocked route with backward arcs, R⃗_b, can be found at t_1+1, then R⃗_b does not contain any backward arc ⟨ v_i^out,v_i^in⟩ for any i. The proof is in Appendix <ref>. According to Lemma <ref>, if one can find unblocked routes with backward arcs at the end, the backward arcs should be ⟨ v_j^in, v_i^out⟩, j>i. Thus, even though the route may contain several backward arcs, we can divide the route into several separated parts, each of which contains only one backward arc for easier analysis. Thus, we just need to focus on one of them as shown in Fig. <ref>. Consider a WSN 𝒢 that satisfies Assumptions <ref> and <ref>. Algorithm <ref> is applied to determine nodes to activate in each timeslot. If a backward arc ⟨ v_j^in,v_i^out⟩ (j>i) exists in an unblocked route R⃗_b when the WSN lifetime expires, then the maximum flow increment of ⟨ v_j^in,v_i^out⟩ is 1. The proof is in Appendix <ref>. From the proof of Lemma <ref>, we know that Algorithm <ref> could be suboptimal due to the existence of unblocked routes with backward arcs. However, the maximum flow increment of the unblocked route with backward arcs does not increase when we multiply the E_i,∀ iwith a positive integer η.Then we have the following core result: Consider a WSN that satisfies Assumptions <ref> and <ref>, with initial energy E⃗. Let T_max(η) and T_G(η) be the maximum network lifetime by Problem (<ref>) and the network lifetime achieved by Algorithm <ref>, with initial energy ηE⃗. Then lim_η→ +∞T_max(η)-T_G(η)/T_max(η)=0 . T_max(η)-T_G(η) is bounded by the number of unblocked routes with backward arcs when the network expires. According to Lemma <ref>, this number does not increase if the E_i,∀ i are multiplied by a positive integer. Thus, T_max(η)-T_G(η) is bounded. However, we know that T_max(η)≥⌊η⌋ T_max(1), and it tends to ∞ as η tends to ∞. Thus, lim_η→ +∞T_max(η)-T_G(η)/T_max(η)=0 . This concludes the proof. Furthermore, based on Lemma <ref>, we have the approximation ratio of Algorithm <ref> as shown in the following theorem. Consider a WSN that satisfies Assumptions <ref> and <ref>. Let T_max and T_G be the maximum network lifetime and the lifetime achieved by Algorithm <ref>. Then, we have T_G≥ 0.5T_max. According to Lemma <ref>, to extend the network lifetime requires us selecting an unblocked route with backward arcs and an unblocked route with no backward arcsbefore the WSN expires. We call such pair of routes an incremental pair. Furthermore, we know from Lemma <ref> that the maximum flow increment of each backward arc is at most 1. Thus, when Algorithm <ref> terminates, we have that, T_G, i.e., the summation of flows in unblocked routes without backward arcs, is larger or equal to the number of incremental pairs. Such number of incremental pairs is larger or equal to the additional network lifetime that can be extended by the backward arcs, which is T_max-T_G. This gives us T_G≥ 0.5T_max and completes the proof. The provided approximation ratio is tight. An example for T_G=0.5T_max can be shown using the topology of Fig. <ref>, where the left sink node is connected to v_h and v_i and the right sink node is connected to v_j and v_k, and the cardinality constraint requires us to pick 2 sensor nodes to activate in each time slot. The initial energy of v_i,v_j,v_h,v_k are all 1. Then, it is easy to achieve that the maximum network lifetime is 2, i.e., to activate v_i and v_k in one timeslot and to activate v_h and v_j in the other timeslot. In this case, the lifetime achieved by Algorithm <ref> could be 1, if it picks v_i and v_j at the first timeslot. Then T_G=1=0.5T_max. However, Lemma <ref> also gives us that, if the initial energy of all four sensor nodes are E≫ 1, the lifetime gap, T_max(E)-T_G(E), is always 1. In this case T_max(E)=2E and T_G(E)=2E-1. Thus, the gap is negligible when E is large enough, as suggested by Theorem <ref>. Therefore, when the nodal energy consumption in a timeslot is much smaller compared to the nodal battery capacity, which is generally true for sensor nodes for long term monitoring applications, the lifetime gap is negligible. Theorem <ref> shows that even though energy balancing is not equivalent to lifetime maximization in the considered network structure, the gap between the lifetime achieved by Algorithm <ref> to the maximum network lifetime is small when the initial energies of the sensor nodes are large enough compared to the energy consumption in an active timeslot. It follows that Algorithm <ref> can be used to derive good activation schedules for sensor nodes in terms of WSN lifetime.For an illustration of the performance of Algorithm <ref>, numerical evaluations are given in Section <ref>. § NUMERICAL EVALUATIONSIn this section, we evaluate numerically the lifetime achieved by energy balancing, and we compare it to the maximum lifetime. Suppose the length of the pipeline under study is L, we normalize it to be 1 for simplicity. Then the normalized transmission range of sensor nodes is r/L. Two sink nodes are deployed at the end point of the pipeline, one at point 0 and the other at point 1. The sensor nodes are uniformly randomly deployed in the range of (0,1). The initial energy of each node is randomly set according to a Gaussian distributionE_i∼𝒩(50,5^2). In every timeslot, the energy of the active sensor nodes is reduced by 1, whereas the energy of other sensors remains the same. Once the residual energy of a sensor node is less than 1, it is considered as expired and is excluded from the available sensor node set 𝒱. Once the sensor nodes in 𝒱 become disconnected or their numberis less than M_cs, the network has expired.We first compare the network lifetime achieved by Algorithm <ref> to the optimal solution by solving Problem (<ref>) based on Branch-and-Bound method. As the number of nodes increases, the number of possible routes increases dramatically. Consequently, the size of the variables in the MDK Problem (<ref>) also increases dramatically, and it becomes difficult to solve. Thus, we set the size of network to relatively small values, so that the MDK problem can be solved efficiently. The parameters of the network are as follows: the number of nodes, N, are randomly picked from 15 to 20, M_cs are randomly picked from 7 to 10, the normalized transmission range, r/L, is 0.25. We test 222 different random cases, among which there are 44 cases that the network lifetime by Algorithm <ref> is 1 timeslot less than the optimal, and 1 case that is 2 timeslots less than the optimal, as shown in Table <ref>. This supports our finding that balancing residual energy is effective to achieve the lifetime close to the maximum in the considered network.We further compare the performance of Algorithm <ref> to greedy based search with random (GBS+R) Algorithm and greedy based search with maximum (GBS+M)Algorithm <cit.> as shown in Fig.<ref>.We check the network lifetime when the initial energy of sensor nodes are multiplied by η. We calculate the network lifetime when η is 1, 2, 10, 20, respectively, and then divide the lifetime by the upper bound of the network lifetime.For Algorithm <ref>, it is shown that the ratio of network lifetime achieved by the algorithm to the upper bound of network lifetime, T_G(η)/T̅(η), increases slightly as η increases. However, such a trend does not exist for the GBS+R and GBS+M Algorithm. Last, we evaluate the performance of Algorithm <ref> by comparing its performance to that of the state-of-art approaches, i.e., CDG <cit.>, CSF <cit.>, CDC <cit.>, MECDA <cit.>, and also by comparing the upper bound of WSN lifetime T̅^f achieved according to Theorem <ref>. The results for equal transmission range are shown in Fig. <ref> (a) and (b). The horizontal axis represents the normalized transmission range, and the vertical axis is the average WSN lifetime. The WSN lifetime achieved by Algorithm <ref> (blue solid line with circles) is very close to the upper bound (yellow dash line marked by plus) established by Theorem <ref>. It shows that the performance of Algorithm <ref> is near optimal, and the upper bound by Theorem <ref> is tight. The result of unequal transmission range is shown in Fig. <ref> (c). In this case, the yellow dash line may not be the upper bound of the network lifetime because Assumption <ref> is not satisfied.The results also indicate that performance achieved by Algorithm <ref> is better than that of the CDG, CSF, CDC and MECDA algorithms. Also, the network lifetime achieved by CDG and CSF does not increase when the transmission range of sensor nodes increases. The reason is that, in these two algorithms, all sensor nodes must be constantly activated. Regarding the MECDA algorithm, since it is used to find the routing with the smallest energy consumptions, some sensor nodes are always activated until their energy is depleted. The network is then easier to become disconnected, and thus has smaller lifetime compared to the one achieved by Algorithm <ref>. Regarding the CDC algorithm, it is based on opportunistic routing. Therefore, the energy of the sensor nodes are more balanced than the case of MECDA. However, it does not guarantee minimum activation of sensor nodes in each timeslot. Therefore, it consumes more energy than Algorithm <ref> in each timeslot, and has less network lifetime. The average lifetime achieved by Algorithm <ref> is significantly longer than the CDG, CSF, CDC and MECDA approaches, which suggests the effectiveness of Algorithm <ref> in general scenarios.Regarding the time complexity, as the complexity ofAlgorithm 1 is O(N^2) as discussed in our previous paper <cit.>, and recall that this algorithm determines the sensor activation for a single timeslot, the overall time complexity to achieve the approximate network lifetime based on Algorithm <ref> is O(N^2E). We further test the computational time to achieve the approximate network lifetime by Algorithm <ref> and the upper bound network lifetime by testing the feasibility of Problem (<ref>) with relaxed Constraint (<ref>). In the settings, N=50 and M_cs=10, and the nodes have the same transmission range, which is the same as in Fig. <ref> (a). The average computational time to retrieve the network lifetime are 1.4031 seconds and 1.4246 seconds for the normalized transmission range of 0.3 and 0.4, respectively. To calculate the lifetime upper bound T_max, one has to test a range between the approximate lifetime T_G and ∑ E_i/M_cs, which requires 4.8974 seconds and 0.7193 seconds for he two aforementioned cases, respectively. The computational time of calculating the lifetime upper bound reduces as the transmission range increases. The reason is that the feasible testing ranges becomes smaller and the approximate lifetime is closer to the upper bound when the node's transmission range increases.Based on the results above, we conclude that the approach of energy balancing is effective for lifetime maximization. § CONCLUSIONWe considered a dense sensor network for monitoring a one dimensional strip area, such as a pipeline, a tunnel, or a bridge. Given that sensor node replacement is expensive and difficult, the problem of maximizing network lifetime by using compressive sensing was considered. The compressive sensing introduces a cardinality constraint, which makes the problem challenging. Thus, we characterized the performance of an approximation approach based on balancing the residual energy of sensor nodes. We proved that the resulting lifetime is at least 50% of the optimal and that it is near optimal when the ratio of nodal initial energy to nodal energy consumptions is large enough. Simulation results showed that the ratio of the lifetime achieved bybalancing the residual energy of the nodes to the upper bound of network lifetime is close to 1 when the WSN is dense enough.An interesting topic of future work is to study the relationship of energy balancing with lifetime maximization under cardinality constraints in a more general network structure, e.g., a WSN in 2-dimensional free spaces. Besides, deriving solution approaches that apply rounding to the results of the relaxed maximum flow problem is a promising research direction. § PROOF OF THE EQUIVALENCE OF CONSTRAINTS (<REF>)-(<REF>) TO CONSTRAINT <REF> Constraints (<ref>)-(<ref>) imply that0 ≤ E_i(T+1)=E_i(T)-x_i(T)=E_i(T-1)-x_i(T-1)-x_i(T)=…=E_i(1)-∑_t=1^T x_i(t)=E_i-∑_t=1^T x_i(t) ,where the first inequality comes from Constraint (<ref>), and the equalities come from Constraints (<ref>) and (<ref>). Thus, Constraints (<ref>)-(<ref>) can be equivalently captured by Constraint <ref> . § PROOF OF LEMMA <REF> We need to show that the solution of Problem (<ref>) can be converted to the solution of Problem (<ref>), and vice versa. Suppose the solution of Problem (<ref>) is z⃗^*=[z^*(1),…,z^*(L)]^T. Then, we have the solution for Problem (<ref>) to be, x_i(t)=1,∀ (t,i)∈{(t,i)|∑_k=1^K-1z^*(k)+1≤ t≤∑_k=1^Kz^*(k)∧ q_K(i)=1,K=1,2,…,L}, otherwise, x_i(t)=0. That is,activate all the sensor nodes in Profile Q⃗_1 for z^*(1) timeslots, and then activate all the sensor nodes in Profile Q⃗_2 for z^*(2) timeslots, and so on. On the other hand, suppose the solution for Problem (<ref>) is x⃗={x⃗(1),…,x⃗(T)}. Notice that the activated sensor nodes in each timeslot must belong to a feasible activation profile, i.e., ∀ t,∃ l,Q_l=x⃗(t). Then, the solution for Problem (<ref>) is z⃗=[z(1),…,z(L)]^T, where z(i)=∑_t=1,Q_i=x⃗(t)^T 1. This completes the proof. Since the column in matrix Q⃗ represents a feasible activation profile, the construction of Q⃗ is equivalent to enumerating all the feasible activation profiles. It consists of two steps: 1. Search the route from s_l to s_r; 2. Remove the routes that does not satisfy cardinality constraint. In detail, for the first step, denote s⃗(v_i, k) the set of routes starting from v_i to s_r with k vertices. Then, a dynamic programming based searching could proceed as s⃗(v_i,k)=⋃_v_j∈𝒩_-(v_i)s⃗(v_j,k-1). Then, the set of feasible activation profiles could be represented by ⋃_k≥ M+2s⃗(s_l,k). Notice that the number of feasible activation profiles is huge for large and dense networks, and hence this approach cannot be applied in WSNs with limited storage capacity. This approach is just for performance comparison. § PROOF OF LEMMA <REF> The sketch of the proof is based on the one-to-one mapping of the constraints. Constraint (<ref>) ensues that the flows are unit flows. Together with Constraint (<ref>), we have that there is at most one unit flow going out from each vertex in each timeslot. Therefore, a unit flow on edge v_i to v_j, u_i,j,t=1, represents the activation of both nodes v_i,v_j at timeslot t. Constraint (<ref>) represents that the output flow (the first summation) of a node should be equal to the input flow (the second summation) if the node is a sensor node. If the node is v_0 (the sink node s_l), then the difference of its output flow and its input flow should be 1 in each timeslot. If the node is v_N+1 (the sink node s_r), then the difference should be -1 in each timeslot. Furthermore, Constraints (<ref>) and (<ref>) ensure that there is at most one unit flow that goes out from each vertex, i.e, there is one outgoing edge in the trail for each active node. Together with flow conservation Constraint (<ref>), there is no cyclic in the trail from v_0 to v_N+1. This means that the trail is a path. Recall Remark <ref> that the connectivity Constraint (<ref>) is equivalent to the existence of path from v_0 to v_N+1. Thus, this flow conservation Constraint (<ref>) with (<ref>) and (<ref>) is equivalent to the connectivity Constraint (<ref>). From Constraints (<ref>), (<ref>) and (<ref>), the unit flow represents the activation profile in that timeslot, and ∑_v_j∈𝒩(v_i) u_i,j,t is either 1 (which means v_i is active) or 0 (which means v_i is inactive) for each sensor node and timeslot. Thus, the summation over time, ∑_t=1^T∑_v_j∈𝒩(v_i) u_i,j,t, is the number of timeslots that v_i is activated, and it should be smaller than E_i. Thus, Constraint (<ref>) is equivalent to the energy budget Constraint (<ref>). Besides, the summation over nodes, ∑_v_i∈𝒱∑_v_j∈𝒩(v_i) u_i,j,t represents the number of nodes that is activated in a timeslot, which corresponds to the cardinality constraint. Notice that v_l∈𝒱, and contributes to the summation. On the other hand, even though v_r∈𝒱, u_N+1,j,t=0, thus, v_r does not contribute to the summation. Therefore, the right hand side of Constraint (<ref>) should be M_cs+1. Given that the objectives of the two problems are identical, and the constraints are equivalent, it is concluded that Problem (<ref>) is equivalent to Problem (<ref>). § THE APPROACH TO SOLVE THE RELAXED PROBLEM (<REF>)We know that given a fixed T, the relaxed Problem (<ref>) is a linear optimization and thus solvable. Based on this idea, we can use a bisection approach, i.e., we can turn the problem into testing the feasibility of a linear programming problem. The idea is, given a T, we test whether the problem is feasible. If it is feasible, we increase T; otherwise we decrease it, until it converges. Notice that we have already had an upper bound of network lifetime in our previous work <cit.>, which is ∑ E_i/M_cs, the time complexity of solving the relaxed problem isthe time complexity of solving a linear programming problem, which is polynomial, multiplied by ln (E_i), and it is not high. § TRANSFORMATION TO A MAXIMUM FLOW PROBLEM WITH EDGE CAPACITIES We need to show Problem (<ref>) can be formulated as a maximum flow problem with edge capacities and cardinality constraints. The transformation follows the standard techniques <cit.>, as shown in Fig. <ref>. Given a network 𝒢={𝒱,ℰ}, we construct a new directed graph 𝒢'={𝒱',ℰ'}. The two sink nodes in 𝒱, s_l and s_r respectively, are replicated to 𝒱'. Every sensor node v_i of initial energy E_i is represented by two nodes v_i^in and v_i^out connected by a directed arc ⟨ v_i^in,v_i^out⟩ of capacity E_i in 𝒢'. For each edge ⟨ v_i,v_j⟩∈ℰ, if i<j, we construct a directed arc ⟨ v_i^out,v_j^in⟩ in 𝒢' with capacity +∞, otherwise we construct a directed arc ⟨ v_j^out,v_i^in⟩ with capacity +∞ in 𝒢'. Thus, the vertex capacity constraints in Problem (<ref>) turn to the edge capacity constraints. For the cardinality constraint, only the edges ⟨ v_i^in,v_i^out⟩ are taken into accounts. Then, the new maximum flow with edge capacity under cardinality constraint is equivalent to the Problem (<ref>). This completes the proof. § PROOF OF LEMMA <REF> Suppose a route R⃗_a from s_l to s_r contains no backward arcs, and can be divided into three parts R⃗_a^1=⟨ s_l,…,v'_1⟩, R⃗_a^2=⟨ v'_1,…,v'_2⟩, R⃗_a^3=⟨ v'_2,…,s_r⟩. The sensor nodes in R⃗_a are activated at t_1. The lifetime of an activation is t_2-1, i.e., at time t_2>t_1, we cannot find any unblocked routes from s_l to s_r that contains no backward arcs.However, if we can find a route R⃗_b that contains at least one backward arc, R⃗_b can also be divided into three parts R⃗_b^1=⟨ s_l,…,v'_2⟩, R⃗_b^2=⟨ v'_2,…,v'_1⟩ and R⃗_b^3=⟨ v'_1,…,s_r⟩, where R⃗_b^1 and R⃗_b^3 contain no backward arcs. If both ⟨R⃗_a^1,R⃗_b^3⟩ and ⟨R⃗_b^1,R⃗_a^3⟩ satisfy the cardinality constraint, we can pick route ⟨R⃗_a^1, R⃗_b^3⟩ at t_1 and ⟨R⃗_b^1,R⃗_a^3⟩ at t_2, such that the WSN lifetime increases from t_2-1 to t_2, which completes the proof. § PROOF OF LEMMA <REF> The proof is by contradiction. Suppose that a backward arc ⟨ v_i^out,v_i^in⟩ exists in R⃗_b, we divide R⃗_b into three parts, R⃗_b^1, R⃗_b^2, R⃗_b^3, where R⃗_b^1=⟨ s_l,…,v_u^out,v_j^in⟩, R⃗_b^2=⟨ v_j^in,…, v_i^out,v_i^in,…,v_k^out⟩, R⃗_b^3=⟨ v_k^out,v_w^in,…,s_r⟩, such that backward arcs only exist in R⃗_b^2. As R⃗_b is unblocked, R⃗_b^2 is unblocked. If R⃗_b leads to suboptimal network lifetime, there is a route R⃗_a=⟨ s_l,…,v_k^in,v_k^out,…,v_i^in,v_i^out,…,v_j^in,v_j^out,…,s_r⟩ was chosen to be activated at a time t_2≤ t_1, where ⟨ v_k^out,…,v_i^in,v_i^out,…,v_j^in⟩ is the inverse sequence of R⃗_b^2. Similarly, we divide R_ainto three parts, R⃗_a^1=⟨ s_l,…,v_k^out⟩, R⃗_a^2=⟨ v_k^out,…, v_j^in⟩, and R⃗_a^3=⟨ v_j^out,…,s_r⟩, where R⃗_a^2. Then we have the maximum flow increment of R⃗_a^1 and R⃗_a^3 at t_2 satisfies δ_a^1(t_2)≥ 1, δ_a^3(t_2)≥ 1, and the maximum flow increment of R⃗_b^1 and R⃗_b^3 at t_2 satisfies δ_b^1(t_2)≥δ_b^1(t_1+1)≥ 1, δ_b^3(t_2)≥δ_b^3(t_1+1)≥ 1. (Otherwise, R⃗_a is blocked at t_2 and R⃗_b is blocked at t_1+1.) Notice that route ⟨R⃗_a^1,R⃗_b^3⟩ and route ⟨R⃗_b^1,R⃗_a^3⟩ are not blocked at t_2, and they contain no backward arcs. Then, according to Line 4 to Line 5 in Algorithm <ref>, which minimize the number of nodes to be activated in each time slot, we have that |R⃗_a^1|, |R⃗_a^3|, |R⃗_b^1| and |R⃗_b^3|, the number of sensor nodes in R⃗_a^1, R⃗_a^3, R⃗_b^1 and R⃗_b^3 respectively, should satisfy |R⃗_b^3|>|R⃗_a^3|, |R⃗_b^1|>|R⃗_a^1|. Thus, the sensor node v_w lies between v_i and v_j, and the node v_ulies between v_k and v_i. From Assumption 1, we have v_u∈𝒩_-(v_k). Further, as arc ⟨ v_k^out,v_w^in⟩ is in R⃗_b^3 v_w∈𝒩_-(v_k), and hence v_u and v_w is connected. Moreover, as R⃗_b leads to the suboptimality of the network lifetime, we have that route ⟨R⃗_b^1,R⃗_a^3⟩ and route ⟨R⃗_a^1,R⃗_b^3⟩ satisfy cardinality constraint, i.e., |R⃗_b^1|+|R⃗_a^3|≥ M_cs and |R⃗_a^1|+|R⃗_b^3|≥ M_cs. Together with |R⃗_b^1|>|R⃗_a^1|, |R⃗_b^3|>|R⃗_a^3|, we have that |R⃗_b^1|+|R⃗_b^3|≥ M_cs. Then we have an unblocked route that satisfies cardinality constraint and has no backward arcs, ⟨R⃗_b^1', R⃗_b^3'⟩, at t_1+1 where R⃗_b^1' is the route R⃗_b^1 without v_j^in, and R⃗_b^3' is the route R⃗_b^3 without v_k^out. This comes in contradiction with that no unblocked route with only forward arcs can be found at t_1+1. Thus, R⃗_b does not contain any backward arc ⟨ v_i^out, v_i^in⟩ for all i, which completes the proof. § PROOF OF LEMMA <REF> We first need a lemma that is used in the proof for Lemma <ref>: Consider two positive integers E_i and E_j that satisfy E_i<E_j. For any positive integers x and y, if x and y satisfy (y-1)/E_j<x/E_i≤ y/E_j, then we have (x-1)/E_i<(y-1)/E_j. It suffices to show that (x-1)/E_i<(y-1)/E_j. Since x/E_i≤ y/E_j, we have that x-1/E_i <x/E_i-1/E_j≤y/E_j-1/E_j=y-1/E_j , where the first inequality comes from E_i<E_j. This concludes the proof. Then, Lemma <ref> is proved as follows: Proof of Lemma <ref>: Suppose that backward arc ⟨ v_j^in,v_i^out⟩ with j>i exists in an unblocked route R⃗_b when the network expires at t=t_1. Then there exists a node v_h^out and v_k^in in R⃗_b such that route ⟨ v_h^out, v_j^in, v_i^out, v_k^in⟩ is in R⃗_b and h<i<j<k. According to Assumption <ref> and <ref>, forward arcs ⟨ v_h^out, v_i^in⟩ and ⟨ v_j^out,v_k^in⟩ exist in 𝒢'. As R⃗_b leads to the suboptimality in network lifetime, we have that v_i∈𝒩_-(v_x) for any node v_x that v_h∈𝒩_-(v_x), that v_y∈𝒩_-(v_j) for any node v_y∈𝒩_-(v_k), and that there is no direct edge between v_h^out and v_k^in, as shown in Fig. <ref>. Then we focus on the part in R⃗_b that contains backward arcs. Similar to the proof for Lemma <ref>, we again divide R⃗_b into three parts, R⃗_b^1=⟨ s_l,…,v_h^in,v_h^out⟩, R⃗_b^2=⟨ v_h^out,v_j^in,v_i^out,v_k^in⟩, and R⃗_b^3=⟨ v_k^in,v_k^out,…,s_r⟩. If R⃗_b causes the suboptimality in network lifetime, we have that the maximum flow increment of R⃗_b^1, R⃗_b^2, R⃗_b^3, should satisfy δ_b^1(t_1)≥ 1, δ_b^2(t_1)≥ 1, and δ_b^3(t_1)≥ 1. Let E_i(t) be the residual energy of sensor node v_i before the activation at t-th slot, E_i(0)=E_i be the initial energy of sensor node v_i. As δ_b^2(t_1)≥ 1, we have that a route that contains v_i and v_j were chosen for activation at t_2<t_1, which means that E_i(t_2)/E_i+E_j(t_2)/E_j≥ E_h(t_2)/E_h+E_j(t_2)/E_j and E_i(t_2)/E_i+E_j(t_2)/E_j≥ E_i(t_2)/E_i+E_k(t_2)/E_k at t_2 according to Algorithm <ref>, which chooses the nodes having the maximum sum of normalized residual energy in every timeslot. Then, we divide the analysis into four cases: 1) E_h<E_i and E_k<E_j; 2) E_h< E_i and E_k≥ E_j; 3) E_h≥ E_i and E_k< E_j; 4) E_h≥ E_i and E_k≥ E_j. We will show that for case 1), there will be no flow increment; for case 2), 3) and 4), there will be at most 1. In case 1), E_h<E_i and E_k<E_j. In the initial time when none of these four nodes have been activated, then E_h(0)/E_h=E_i(0)/E_i=E_j(0)/E_j=E_k(0)/E_k=1. According to Lemma <ref>, we have 0/E_h<1/E_i<1/E_h. It means that sensor node v_h expires earlier than sensor node v_i, as the sum of residual energy of ⟨ v_h,v_j⟩, which is 1/E_h+E_j(t)/E_j, is always larger than that of ⟨ v_i,v_j⟩, which is 1/E_i+E_j(t)/E_j. Similarly, we have 0/E_k<1/E_j<1/E_k and hence sensor node v_k expires earlier than sensor node v_j. Once either v_h or v_k expires, the route R_b is blocked even when the network has not expired, which contradicts that R_b is unblocked when the network expires. Thus, there will be no flow increment in this case. In case 2), E_h<E_i and E_k≥ E_j. If E_h≤ E_j, with similar reason to case 1), we have that v_h expires first, and then R_b is blocked, which is in contradiction with thatR_b is unblocked when the network expires. It means that E_h> E_j as we can find an unblocked route R_b when the network expires. Thus, v_j expires earlier than v_h. If E_j≤ E_k≤ E_i, we have that after v_j expires, ⟨ v_h,v_j⟩ and ⟨ v_i,v_j⟩ is blocked. The algorithm will pick ⟨ v_i,v_k⟩ until v_k expires. Then R_b is blocked as ⟨ v_k^in,v_k^out⟩ is blocked. Consequently, E_k>E_i>E_h> E_j. Then, in the initial time when none of these four sensor nodes have been activated, the normalized residual energy of these four nodes is 1. Consequently, once ⟨ v_i,v_j⟩ is chosen, suppose at t_2 when all these four nodes are not activated before, we have E_i(t_2)/E_i(0)=E_j(t_2)/E_j(0)=E_k(t_2)/E_k(0)=E_h(t_2)/E_h(0)=1 , E_k(t_2+1)/E_k(0)>E_j(t_2+1)/E_j(0),E_h(t_2+1)/E_h(0)>E_i(t_2+1)/E_i(0) , It directly gives us that, in the next timeslot t_2+1, E_k(t_2+1)/E_k+E_i(t_2+1)/E_i>E_j(t_2+1)/E_j+E_i(t_2+1)/E_i Also, we have that E_k(t_2+1)/E_k(0)+E_i(t_2+1)/E_i(0) =E_k(t_2)/E_k(0)+E_i(t_2)-1/E_i(0)=2-1/E_i(0)>2-1/E_j(0) =E_h(t_2)/E_h(0)+E_j(t_2)-1/E_j(0)=E_h(t_2+1)/E_h(0)+E_j(t_2+1)/E_j(0) , where (<ref>) holds since v_k is not activated but v_i is activated at t_2, (<ref>) holds since E_i>E_j>0, and (<ref>) holds since v_h is not activated but v_j is activated at t_2. Then the algorithm will choose ⟨ v_i,v_k⟩ instead of ⟨ v_i,v_j⟩ and ⟨ v_h,v_j⟩, due to (<ref>) and (<ref>). After this activation, as E_k>E_j, we have (E_i(t_2+1)-2)/E_i+(E_k(t_2+1)-1)/E_k>(E_i(t_2+1)-2)/E_i+(E_j(t_2+1)-1)/E_j. It means that the normalized residual energy of route ⟨ v_i,v_j⟩ is still smaller than that of ⟨ v_i,v_k⟩ and ⟨ v_h,v_j⟩. The algorithm will then pick ⟨ v_i,v_k⟩ until the normalized residual energy of v_k is smaller than that of v_j. When this happens, we have that the normalized residual energy of v_i is smaller than that of v_k, and hence smaller than v_j and v_h. It means that the algorithm will then pick ⟨ v_h,v_j⟩. After that, the normalized residual energy of ⟨ v_i,v_k⟩ is again larger than that of ⟨ v_i,v_j⟩ according to Lemma <ref>. This indicates that the algorithm will always pick ⟨ v_i,v_k⟩ or ⟨ v_h,v_j⟩ instead of ⟨ v_i,v_j⟩. Consequently,the maximum flow increment of ⟨ v_j^in, v_i^out⟩ in this case is 1 as ⟨ v_i,v_j⟩ was chosen only once. As case 3) is symmetric to case 2), we have the maximum flow increment of ⟨ v_j^in,v_i^out⟩ in this case is also 1. In case 4), E_h≥ E_i and E_k≥ E_j. We have for any positive integer x, x/E_i≥ x/E_h, and x/E_j≥ x/E_k. In the initial time when none of these four sensor nodes have been activated, the normalized residual energy of these four nodes is equal to 1. Hence, once ⟨ v_i,v_j⟩ is chosen, suppose at t_2, we have E_h(t_2+1)/E_h(0)>E_i(t_2+1)/E_i(0),E_k(t_2+1)/E_k(0)>E_j(t_2+1)/E_j(0) , and thus E_i(t_2+1)/E_i(0)+E_j(t_2+1)/E_j(t_2+1)<E_i(t_2+1)/E_i(0)+E_k(t_2+1)/E_k(0) ,E_i(t_2+1)/E_i(0)+E_j(t_2+1)/E_j(t_2+1)<E_h(t_2+1)/E_h(0)+E_j(t_2+1)/E_j(0) . Thus, the algorithm will choose ⟨ v_i,v_k⟩ or ⟨ v_h,v_j⟩ instead of ⟨ v_i,v_j⟩. Due to the symmetry, the analysis when it picks ⟨ v_i,v_k⟩ is similar to that if it picks ⟨ v_h,v_j⟩. Thus, we only analyze the case if it picks ⟨ v_i,v_k⟩. Suppose that the algorithmpicks ⟨ v_i,v_k⟩, then the normalized residual energy of sensor node v_i and v_k becomes (E_i(t_2+1)-1)/E_i and (E_k(t_2+1)-1)/E_k. Suppose (E_k(t_2+1)-1)/E_k>E_j(t_2+1)/E_j, we have that the normalized residual energy of route ⟨ v_i,v_j⟩ is still less than that of ⟨ v_i,v_k⟩ and ⟨ v_h,v_j⟩. Otherwise, we have E_j(t_2+1)-1/E_j+E_h(t_2+1)/E_h > E_j(t_2+1)-1/E_j+E_i(t_2+1)/E_i > E_j(t_2+1)-1/E_j+E_i(t_2+1)-1/E_i , where the first inequality comes from (<ref>). The summation of residual energy of v_j and v_h is larger than that of v_i and v_j. Thus, the algorithm will pick route ⟨ v_h,v_j⟩. Then the normalized residual energy of these four sensor nodes are (E_h(t_2+1)-1)/E_h, (E_i(t_2+1)-1)/E_i, (E_j(t_2+1)-1)/E_j and (E_k(t_2+1)-1)/E_k. As 1/E_i>1/E_h, 1/E_j>1/E_k, we have E_i(t_2+1)-1/E_i(0)+E_j(t_2+1)-1/E_j(0)=E_i(t_2+1)/E_i(0)-1/E_i(0)+E_j(t_2+1)-1/E_j(0)<E_h(t_2+1)/E_i(0)-1/E_h(0)+E_j(t_2+1)-1/E_j(0)=E_h(t_2+1)-1/E_h(0)+E_j(t_2+1)-1/E_j , and similarly E_i(t_2+1)-1/E_i(0)+E_j(t_2+1)-1/E_j(0)<E_i(t_2+1)-1/E_i(0)+E_k(t_2+1)-1/E_k(0) . (<ref>) and (<ref>) indicate that the algorithm will always pick ⟨ v_i,v_k⟩ or ⟨ v_h,v_j⟩ instead of ⟨ v_i,v_j⟩, until v_i or v_j expires. Consequently, the maximum flow increment of ⟨ v_j^in, v_i^out⟩ in this case is 1, as ⟨ v_i,v_j⟩ was only picked once. To sum up, the maximum flow increment of ⟨ v_j^in,v_i^out⟩ is 1 for Cases 2) to 4), and it is 0 for Case 1). Thus, the maximum flow increment of⟨ v_j^in,v_i^out⟩ is 1. IEEEtran | http://arxiv.org/abs/1704.08050v1 | {
"authors": [
"Rong Du",
"Lazaros Gkatzikis",
"Carlo Fischione",
"Ming Xiao"
],
"categories": [
"cs.SY"
],
"primary_category": "cs.SY",
"published": "20170426103507",
"title": "On Maximizing Sensor Network Lifetime by Energy Balancing"
} |
fancy ℙ 𝔼 | http://arxiv.org/abs/1704.08248v1 | {
"authors": [
"Robert J. Adler",
"Sarit Agami",
"Pratyush Pranav"
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"published": "20170426105742",
"title": "Modeling and replicating statistical topology, and evidence for CMB non-homogeneity"
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http://arxiv.org/abs/1704.08007v1 | {
"authors": [
"Wenfei Liu",
"Ming Li",
"Xiaowen Tian",
"Zihuan Wang",
"Qian Liu"
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"categories": [
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"published": "20170426082252",
"title": "Transmit Filter and Artificial Noise Design for Secure MIMO-OFDM Systems"
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[4] Strong Coordination over Noisy Channels:Is Separation Sufficient?This work is supported by NSF grants CCF-1440014, CCF-1439465. Sarah A. Obead, Jörg Kliewer Department of Electrical and Computer EngineeringNew Jersey Institute of TechnologyNewark, New Jersey 07102Email: [email protected], [email protected] Badri N. Vellambi Research School of Computer Science Australian National University Acton, Australia 2601 Email: [email protected] 30, 2023 ============================================================================================================================================================================================================================================================================================================================================================================================================= We study the problem of strong coordination of actions of two agentsX and Y that communicate over a noisy communication channel such thatthe actions follow a given joint probability distribution. We proposetwo novel schemes for this noisy strong coordination problem, and deriveinner bounds for the underlying strong coordination capacity region.The first scheme is a joint coordination-channel coding scheme thatutilizes the randomness provided by the communication channel to reduce the local randomness required in generating the action sequence at agent Y. The second scheme exploits separate coordination and channel coding where localrandomness is extracted from the channel after decoding. Finally,we present an example in which the joint scheme is able to outperform the separate scheme in terms of coordination rate. § INTRODUCTION The problem of communication-based coordination of multi-agent systems arises in numerous applications including mobile robotic networks, smart traffic control, and distributed computing such as distributed games and grid computing <cit.>. Several theoretical and applied studies on multi-agent coordination have targetedquestions onhow agents exchange information and how their actions can be correlated to achieve a desired overall behavior. Two types of coordination have been addressed in the literature – empirical coordination where the histogram of induced actions is required to be close to a prescribed target distribution, and strong coordination, where the induced sequence of joint actions of all the agents is required to be statistically close (i.e., nearly indistinguishable) from a chosen target probability mass function (pmf). Recently, the capacity regions of several empirical and strong coordination network problems have been established <cit.>. Bounds for the capacity region for the point-to-point casewere obtained in <cit.> under the assumption that the nodes communicate in a bidirectional fashion in order to achieve coordination. A similar framework was adopted and improved in <cit.>. In <cit.>, the authors addressed inner and outer bounds for the capacity region of a three-terminal network in the presence of a relay. The work of <cit.> was later extended in <cit.> to derive aprecise characterization of the strong coordination region for multi-hop networks. Starkly, the majority of the recent works on coordination have considered noise-free communication channels with the exception of two works: joint empirical coordination of the channel inputs/outputs of a noisy communication channel with source and reproduction sequences is considered in <cit.>, and in <cit.>, the notion of strong coordination is used to simulate a discrete memoryless channel via another channel.In this work, we consider the point-to-point coordination setup illustrated in Fig. <ref>, where in contrast to <cit.> only source and reproduction sequences at two different nodes (X and Y) are coordinated by means of a suitable communication scheme over a discrete memoryless channel (DMC). Specifically, we propose two different novel achievable coding schemes for this noisy coordination scenario, and derive inner bounds to the underlying strong capacity region. The first scheme is a joint coordination channel coding scheme that utilizes randomness provided by the DMC to reduce the local randomness required in generating the action sequence at Node Y (see Fig. <ref>). The second scheme exploits separate coordination and channel coding where local randomness is extracted from the channel after decoding. Even though the proposed joint scheme is related to the scheme in <cit.>, the presented scheme exhibits a significantly different codebook construction adapted to our coordination framework. Our scheme requires the quantification of the amount of common randomness shared by the two nodes as well as the local randomness at each of the two nodes. This is a feature that is absent from the analysis in<cit.>. Lastly, when the noisy channel and the correlation between X to Y are both given by binary symmetric channels (BSCs), we study the effect of the capacity of the noisy channel on the sum rate of common and local randomness. We conclude this work by showing that the joint scheme outperforms the separate scheme in terms of the coordination rate in the high-capacity regime.The remainder of the paper is organized as follows: Section <ref> sets the notation. The problem of strong coordination over a noisy communication link is presented in Section <ref>. We then derive achievability results for the noisy point-to-point coordination in Section <ref> for the joint scheme and in Section <ref> for the separate scheme, respectively. In Section <ref>, we present numerical results for both schemes when the target joint distribution is described as a doubly binary symmetric source and the noisy channel is given by a BSC. § NOTATIONThroughout the paper, we denote a discrete random variable with upper-case letters (e.g., X) and its realization with lower case letters (e.g., x), respectively. The alphabet size of the random variable X is denoted as |𝒳|. We use X^n to denote the finite sequence [X_1,X_2,…,X_n].The binary entropy function is denoted as h_2(·), the indicator function by 1(w), and the counting function as N(ω|w^n)=∑_i=1^n1(w_i=ω).ℙ[A] is the probability that the event A occurs.The pmf of the discrete random variable X is denoted as P_X(x). However, we sometime use the lower case notation (e.g., p_X(x)) to distinguish target pmfs or alternative definitions. We let 𝔻(P_X(x)||Q_X(x)) denote the Kullback-Leibler divergence between two distributions P_X(x) and Q_X(x) defined over an alphabet X. T_ϵ^n(P_X) denotes the set of ϵ-strongly letter-typical sequences of length n. Finally, P^⊗n_X_1X_2… X_k denotes the joint pmf of n i.i.d. random variables X_1,X_2,…, X_k. § PROBLEM DEFINITIONThe point-to-point coordination setup we consider in this work is depicted in Fig. <ref>. Node X receives a sequence of actions X^n ∈𝒳^n specified by nature where X^n is i.i.d. according to a pmf p_X. Both nodes have access to shared randomness J at rate R_o bits/action from a common source, and each node possesses local randomness M_k at rate ρ_k, k=1,2. Thus, in designing a block scheme to coordinate n actions of the nodes, we assume J∈{1,…, 2^nR_o}, and M_k∈{1,…, 2^nρ_k}, k=1,2, and we wish to communicate a codeword A^n(I) over the rate-limited DMC P_B|A(b|a) to Node Y, where I denotes the (appropriately selected) coordination message. The codeword A^n(I) is constructed based on the input action sequence X^n, the local randomness M_1 at Node X, and the common randomness J. Node Y generates a sequence of actions Y^n∈𝒴^n based on the received codeword B^n, common randomness J, and local randomness M_2. We assume that the common randomness is independent of the action specified at Node X. A tuple (R_o, ρ_1,ρ_2) is deemed achievable if for each ϵ>0, there exist n∈ℕ and a (strong coordination) coding scheme such that the joint pmf of actions P̂_X^n,Y^n induced by this scheme and the n-fold product[This is the joint pmf of n i.i.d. copies of (X,Y)∼ p_XY.] of the desired joint pmf P^⊗ n_XY are close in total variation, i.e.,P̂_X^nY^n-P^⊗ n_XY_ TV<ϵ. We now present the two achievable coordination schemes. § JOINT COORDINATION CHANNEL CODINGThis scheme follows an approach similar to those in <cit.> where coordination codes are designed based on allied channel resolvability problems <cit.>. The structure of the allied problem pertinent to the coordination problem at hand is given in Fig. <ref>.The aim of the allied problem is to generate n symbols for two correlated sources X^n and Y^n whose joint statistics is close to P^⊗ n_XY as defined by (<ref>). To do so, we employ three independent and uniformly distributed messages I, K, and J and two codebooks 𝒜 and 𝒞 as shown in Fig. <ref>. To define the two codebooks, consider auxiliary random variables A∈𝒜 and C∈𝒞 jointly correlated with (X,Y) asP_XYABC=P_ACP_X|ACP_B|AP_Y|BC. From this factorization it can be seen that the scheme consists of tworeverse test channelsP_X|AC and P_Y|AC used to generate the sources from the codebooks. In particular,P_Y|AC=P_B|AP_Y|BC, i.e., the randomness of the DMC contributes to the randomized generation of Y^n.Generating X^n and Y^n from I, K, J represents a complex channel resolvability problem with the following ingredients: * Nested codebooks: Codebook 𝒞 of size 2^n(R_o+R_c) is generated i.i.d. according to pmf P_C, i.e., C^n_ij∼ P_C^⊗ n for all (i,j) ∈I×J. Codebook 𝒜 is generated by randomly selecting A^n_ijk∼P_A|C^⊗ n(·|C_ij^n) for all (i,j,k) ∈I×J×K. * Encoding functions: C^n: {1,2,…,2^nR_c}×{1,2,…,2^nR_o}→𝒞^n, A^n: {1,…,2^nR_c}×{1,…,2^nR_o}×{1,…,2^nR_a}→𝒜^n. * Indices: I,J,K are independent and uniformly distributed over {1,…,2^nR_c}, {1,…,2^nR_o}, and {1,…,2^nR_a}, respectively. These indices select the pair of codewords C^n_IJ and A^n_IJK from codebooks 𝒞 and 𝒜. * The selected codewords C^n_IJ and A^n_IJK are then passed through DMC P_X|AC at Node X, while at Node Y, codeword A^n_IJK is sent through DMC P_B|A whose output B^n is used to decode codeword C^n_ÎJ and both are then passed through DMC P_Y|BC to obtain Y^n. Since the codewords are randomly chosen, the induced joint pmf of the generated actions and codeword indices in the allied problem is itself a random variable and depends on the random codebook. Given a realization of the codebooks 𝖢≜ (𝒜,𝒞)={ a_ijk^n, c_ij^n: i∈{1,…, 2^nR_c} j∈{1,…, 2^nR_o}k∈{1,…, 2^nR_a}}, the code-induced joint pmf of the actions and codeword indices in the allied problem is given byP_X^nY^nIJK(x^n,y^n,i,j,k) ≜P_X|AC^⊗ n(x^n|a_ijk^n,c^n_ij)/2^n(R_c+R_o+R_a) ×(∑_b^n,îP_B|A^⊗ n(b^n|a_ijk^n) 𝖯_Î | B^nJ(î|b^n,j) P_Y|BC^⊗ n(y^n|b^n,c^n_îj) ), where 𝖯_Î|B^nJ denotes the pmf induced by the operation of decoding the index I using the common randomness and the channel output at Node Y. Note that by denoting the decoding operation as a pmf, we can even incorporate randomized decoders. Note also that the indices for the C-codeword that generate X and Y sequencesin (<ref>) can be different since the decoding of the index I at Node Y may fail. We are done if we accomplish the following tasks: (1) identify conditions on R_o, R_c,R_a under which the code-induced pmf P_X^nY^n is close to the design pmf P_XY^⊗ n in the total variation sense; and (2) devise astrong coordination scheme by inverting the operation atNode X. This will be done in following sections by subdividing the analysis of the allied problem. §.§ Resolvability constraints Assuming that the decoding of I and the codeword C^n_IJ occurs perfectly at Node Y, we see that the code-induced joint pmf induced by the scheme for the allied problem for a given realization of the codebook 𝖢 in (<ref>) isP̌_X^nY^nIJK(x^n,y^n,i,j,k)=P_X|AC^⊗ n(x^n|a_ijk^n,c^n_ij)/2^n(R_c+R_o+R_a) ×( ∑_b^n P_B|A^⊗ n(b^n|a_ijk^n)P_Y|BC^⊗ n(y^n|b^n,c^n_ij) ). The following result quantifies when the induced distribution in (<ref>) is close to the n-fold product of the design pmf P_XY. The total variation between the code-induced pmf P̌_X^nY^n in (<ref>) and the desired pmf P^⊗ n_XY asymptotically vanishes, i.e., 𝔼_𝖢[ *P̌_X^nY^n-P^⊗ n_XY_ TV] → 0 as n→∞, if R_a+R_o+R_c> I(XY;AC), R_o+R_c> I(XY;C). Note that here 𝔼_𝖢 denotes the expectation over the random realization of the codebooks.In the following, we drop the subscripts from the pmfs for simplicity.Let R≜ R_a+R_c+R_o,and choose ϵ>0. Consider the argument for 𝔼_𝖢[𝔻(P̌_X^nY^n||P^⊗ n_XY)] shown at the top of the following page. In this argument:(a) follows from the law of iterated expectation. Note that we have used (a^n_ijk,c^n_ij) to denote the codewords corresponding to the indices (i,j,k), and (a^n_i'j'k',c^n_i'j') to denote the codewords corresponding to the indices (i',j',k'), respectively.(b) follows from Jensen's inequality.(c) follows from dividing the inner summation over the indices (i',j',k') into three subsets based on the indices (i,j,k) from the outer summation.(d) follows from taking the expectation within the subsets in (c) such that when * (i',j')=(i,j),(k'≠ k): a^n_i'j'k' is conditionally independent of a^n_ijk following the nature of the codebook construction (i.e., i.i.d. at random); * (i',j')≠(i,j): both codewords (a^n_ijk,c^n_ij) are independent of (a^n_i'j'k',c^n_i'j') regardless of the value of k. As a result, the expected value of the induced distribution with respect to the input codebooks is the desired distribution P^⊗ n_XY <cit.>.(e) follows from * (i',j',k')=(i,j,k): there is only one pair of codewords (a^n_ijk,c^n_ij); * when (k'≠ k) while (i',j')=(i,j) there are (2^nR_a-1) indices in the sum; * (i',j')≠(i,j): the number of the indices is at most 2^nR. (f) results from splitting the outer summation: The first summation contains typical sequences and is bounded by using the probabilities of the typical set. The second summation contains the tuple of sequences when the pair of actions sequences x^n, y^n and codewords c^n,a^n are not ϵ-jointly typical (i.e., (x^n,y^n,a^n,c^n)∉ T_ϵ^n( P_XYAC)). This sum is upper bounded following <cit.> with μ_XY =min_x,y(P_XY(x,y)).(g) following the Chernoff bound of the probability that a sequence is not strongly typical <cit.> where μ_XYAC =min_x,y,a,c(P_XYAC(x,y,a,c)). Consequently, the contribution of typical sequences can be made asymptotically small ifR_a+R_o+R_c >I(XY;AC), R_o+R_c>I(XY;C),while the second term converges to zero exponentially fast with n<cit.>. Finally, by applying Pinsker's inequality we have𝔼_𝖢[||P̌_X^nY^n -P^⊗ n_XY||_ TV] ≤𝔼_𝖢[√(2𝔻(P̌_X^nY^n||P^⊗ n_XY)) ] ≤√(2𝔼_𝖢[𝔻(P̌_X^nY^n||P^⊗ n_XY)])⟶^n→∞ 0. Given ϵ>0, R_a, R_o, R_c satisfying (<ref>) and (<ref>), it follows from (<ref>) that there exist an n∈ℕ and a random codebook realization for which the code-induced pmf between the indices and the pair of actions satisfies||P̌_X^nY^n-P^⊗ n_XY||_ TV<ϵ.§.§ Decodability constraint Since the operation at Node Y in Fig. <ref> involves the decoding of I and thus the codeword C^n(I,J) using B^n and J, the induced distribution of the scheme for the allied problem will not match that of (<ref>) unless and until we ensure that the decoding succeeds with high probability as n→∞. The following lemma quantifies the necessary rate for this decoding to succeed asymptotically almost always.Let Î, C^n_ÎJ be the output of a typicality-based decoder that uses common randomness J to decode the index I and the sequence C^n_IJ from B^n. If the rate for the index I satisfies R_c<I(B;C) then,* 𝔼_𝖢[ℙ[Î≠ I]]→ 0 as n→∞, where ℙ[Î≠ I] is the probability that the decoding fails for a realization of the random codebook, and * lim_n→∞𝔼_𝖢[ P̌_X^nY^nIJK-P_X^nY^nIJK_ TV] =0. We start the proof of <ref>) by calculating the average probability of error, averaged over all codewords in the codebook and averaged over all random codebook realizations. 𝔼_𝖢[ℙ[Î≠ I]] = ∑_𝖢 P_𝖢(𝖼) ℙ[Î≠ I] = ∑_𝖢 P_𝖢(𝖼) ∑_i,j,k1/2^nRℙ[Î≠ I | I=iJ=j K=k] =∑_i,j,k1/2^nR∑_𝖢 P_𝖢(𝖼) ℙ[Î≠ I | I=iJ=j K=k] (a)=ℙ[Î≠ I | I=1J=1 K=1] ,wherein (a) we have used the fact that the conditional probability of error is independent of the triple of indices due to the i.i.d. nature of the codebook construction. Also, due to the random construction and the properties of jointly typical set, we have 𝔼_𝖢 [1((A_111^n,B^n,C^n_11)∈ T_ϵ^n(P_ABC))]1.We now continue the proof by constructing the sets for each j and b^n∈ℬ^n that Node Y will use to identify the transmitted index: Ŝ_j,b^n,𝖼≜{i: (b^n,c^n_ij) ∈ T_ϵ^n(P_BC)}.The set Ŝ_j,b^n,𝖼 consists of indices i∈ I such that for a given common randomness index J=j and channel realization B^n=b^n, the sequences (b^n,c_ij^n) are jointly-typical. Assuming (i,j,k)=(1,1,1) was realized, and if Ŝ_1,b^n,𝖼={1}, then the decoding will be successful. The probability of this event is divided into two steps as follows: •First, assuming (i,j,k)=(1,1,1) was realized, for successful decoding, 1 must be an element of Ŝ_J,B^n,𝖼. The probability of this event can be bounded as follows.𝔼_𝖢[ℙ [I ∈Ŝ_J,B^n,𝖢|I=1 J=1 K=1]]=∑_a^n,b^n,c^n(P_C^⊗ n(c^n) P_A|C^⊗ n(a^n|c^n) P_B|A^⊗ n(b^n|a^n) ×1((c^n,b^n)∈ T_ϵ^n(P_BC)))=∑_b^n,c^n P_BC^⊗ n(b^n,c^n)1((b^n,c^n)∈ T_ϵ^n(P_BC))(a)≥ 1-δ(ϵ)1, where (a) follows from the properties of jointly typical sets. •Next, assuming again that (i,j,k)=(1,1,1) was realized, for successful decoding no index greater than or equal to 2 must be an element of Ŝ_J,B^n,𝖼. The probability of this event can be bounded as follows: 𝔼_𝖢ℙ [Ŝ_J,B^n,𝖢∩{2,…,2^nR_c}=∅|I=1 J=1 K=1]=1- ∑_i' ≠ 1𝔼_𝖢ℙ [i' ∈Ŝ_J,B^n,𝖢|I=1 J=1 K=1] =1- ∑_i' ≠ 1ℙ[(C^n_i'1,B^n)∈ T_ϵ^n(P_BC)](a)≥1- ∑_i' ≠ 1 2^-n(I(B;C)-δ(ϵ))=1-(2^nR_c-1) 2^-n(I(B;C)-δ(ϵ))=1- 2^-n(I(B;C)-R_c-δ(ϵ)) + 2^-nI(B;C)(b)≥ 1-δ(ϵ)1,where (a) follows from the packing lemma <cit.>, and (b) results if R_c <I(B;C)-δ(ϵ). Then from (<ref>), the claim in <ref>) follows as given by 𝔼_𝖢[ℙ[Î≠ I]] = 𝔼_𝖢ℙ[Î≠ I|I=1 J=1 K=1] ≤(𝔼_𝖢ℙ[I ∉Ŝ_J,B^n,𝖢|I=1 J=1 K=1].+. 𝔼_𝖢ℙ [Ŝ_J,B^n,𝖢∩{2,…,2^nR_c}≠∅|I=1 J=1 K=1])0Finally, the proof of <ref>) follows in a straightforward manner. If the previous two conditions are met, then 𝔼_𝖢[ℙ[Î≠ I]]→ 0 and𝔼_𝖢 [P_Î|B^nJ(î|b^n,j)]→δ_IÎ,where δ_IÎ denotes the Kronecker delta. Consequently, from (<ref>) and (<ref>)lim_n→∞𝔼_𝖢[ P̌_X^nY^nIJK-P_X^nY^nIJK_ TV]=0.§.§ Independence constraintWe complete modifying the allied structure to mimic the original problem with a final step. By assumption, we have a natural independence between the action sequence X^n and the common randomness J. As a result, the joint distribution over X^n and J in the original problem is a product of the marginal distributions P^⊗ n_X and P_J. To mimic this behavior in the scheme for the allied problem, in Lemma <ref> we artificially enforce independence by ensuring that the mutual information between X^n and J vanishes. Consider the scheme for the allied problem given in Fig. <ref>. Both I(J;X^n)→ 0 and 𝔼_𝖢[||P̌_X^nJ-P^⊗ n_XP_J||_ TV] → 0 as n→∞ if the code rates satisfy R_a+R_c> I(X;AC),R_c> I(X;C).The proof of Lemma <ref> builds on the results of Section <ref> and the proof of Lemma <ref> of Section <ref>, resulting in𝔼_𝖢[||P̌_X^nJ -P^⊗ n_XP_J||_ TV] ≤𝔼_𝖢[√(2𝔻(P̌_X^nJ||P^⊗ n_XP_J)) ] ≤√(2𝔼_𝖢[𝔻(P̌_X^nJ||P^⊗ n_XP_J)])⟶^n→∞ 0. Given ϵ>0, R_a, R_c meeting (<ref>) and (<ref>), it follows from (<ref>) that there exist an n∈ℕ and a random codebook realization for which the code-induced pmf between the common randomness J and the actions of Node X satisfies ||P̌_X^nJ-P^⊗ n_XP_J||_ TV<ϵ. In the original problem of Fig. <ref>, the input action sequence X^n and the index J from the common randomness source are available and the A- and C-codewords are to be selected. Now, to devise a scheme for the strong coordination problem, we proceed as follows. We let Node X choose indices I and K (and, consequently, the A- and C-codewords) from the realized X^n and J using the conditional distribution P_IK|X^nJ. The joint pmf of the actions and the indices is then given byP̂_X^nY^nIJK≜ P_X^⊗ n P_JP_IK|X^nJ P_Y^n| IJK. Finally, we can argue thatlim_n→∞𝔼_𝖢 [P̂_X^nY^n- P_XY^⊗ n_ TV ] =0, since the total variation between the marginal pmf P̂_X^nY^n and the design pmf P_XY^⊗ n can be bounded as P̂_X^nY^n-P_XY^⊗ n_ TV(a)≤P̂_X^nY^n - P_X^nY^n_ TV+P_X^nY^n - P̌_X^nY^n_ TV+ P̌_X^nY^n -P_XY^⊗ n_ TV(b)≤P̂_X^nY^nIJK-P̌_X^nJ P_IKY^n|X^n,J_ TV + P̌_X^nY^nIJK-P_X^nY^nIJK_ TV + P̌_X^nY^n-P_XY^⊗ n_ TV(c)=P_X^⊗ n P_J -P̌_X^nJ_ TV+ P̌_X^nY^nIJK-P_X^nY^nIJK_ TV + P̌_X^nY^n -P_XY^⊗ n_ TVwhere (a) follows from the triangle inequality; (b) follows from (<ref>), (<ref>), (<ref>) and <cit.>; (c) follows from <cit.>. The terms in the RHS of (c) can be made vanishingly small provided the resolvability, decodability, and independence conditions are met. Thus, we are guaranteed that by meeting the five conditions of Lemmas <ref>-<ref>, the scheme defined by (<ref>) achieves strong coordination between Nodes X and Y by communicating over the DMC P_B|A. Note that since the operation at Nodes X and Y amount to an index selection according to P_IK|X^nJ, and a generation of Y^n using the DMC P_Y|BC, both operations are randomized. The last step is to derandomize the operations at Nodes X and Y by viewing the corresponding local randomness as the source of randomness in these operations. This is detailed next. §.§ Local randomness ratesAt Node X, local randomness is employed to randomize the selection of indices (I,K) by synthesizing the channel P_IK|X^nJ whereas Node Y utilizes its local randomnessto generate the action sequence Y^n by simulating the channel P_Y|BC. Using the arguments in <cit.>, we can argue that for any given realization of J, the minimum rate of local randomness required for the probabilistic selection of indices (I,K) can be derived by quantifying the number of A and C codewords (equivalently the pair of indices I,K) jointly typical with X^n. Quantifying the list size as in <cit.> yields ρ_1 ≥ R_a+R_c-I(X;AC). At Node Y, the necessary local randomness for the generation of the action sequence is bounded by the channel simulation rate of DMC P_Y|BC <cit.>. Thus, ρ_2 ≥ H(Y|BC).Moreover, one can always view a part of the common randomness as local randomness, which then allows us to incorporate the rate-transfer arguments given in <cit.>. Combining the rate-transfer argument with the constraints in Lemmas <ref>-<ref>, we obtain following inner bound to the strong coordination capacity region. A tuple (R_o, ρ_1,ρ_2) is achievable for the strong noisy communication setup in Fig. <ref> if for some R_a,R_c,δ_1,δ_2≥0, R_a+R_o+R_c >I(XY;AC)+δ_1 +δ_2, R_o+R_c >I(XY;C)+δ_1 +δ_2, R_a+R_c >I(X;AC), R_c >I(X;C), R_c<I(B;C), ρ_1 >R_a+R_c-I(X;AC)-δ_1, ρ_2 >H(Y|BC)-δ_2. § SEPARATE COORDINATION-CHANNEL CODING SCHEME WITH RANDOMNESS EXTRACTIONAs a basis for comparison, we will now introduce a separation-based scheme that involves randomness extraction. We first use a (2^nR_c,2^nR_o,n) noiseless coordination code with the codebook 𝒰 to generate a message I of rate R_c. Such a code exists if and only if the rates R_o, R_c satisfy <cit.>R_c+R_o ≥ I(XY;U), R_c ≥ I(X;U).This coordination message I is then communicated over the noisy channel using a rate-R_a channel code over m channel uses with codebook 𝒜. Hence, R_c= λ R_a, where λ=m/n. The probability of decoding error can be made vanishingly small if R_a <I(A;B). Then, from the decoder output Î and the common randomness message J we reconstruct the coordination sequence U^n and pass it though a test channel P_Y|U to generate the action sequence at Node Y. Note that this separation scheme is constructed as a special case of the joint coordination-channel scheme of Fig. <ref> by choosing C=U and P_AC =P_A P_U.In the following, we restrict ourselves to additive-noise DMCs, i.e., B^m=A^m(I)+Z^m,where Z is the noise random variable drawn from some finite field 𝒵, and “+” is the native addition operation in the field. To extract randomness, we exploit the additive nature of the channel to recover the realization of the channel noise from the decoded codeword. Thus, at the channel decoder output we obtainẐ^m=B^m+A^m(Î),where B^m is the channel output and A^m(Î) the corresponding decoded channel codeword. We can then utilize a randomness extractor on Ẑ^m to supplement the local randomness available at Node Y. The following lemma provides some guarantees with respect to the randomness extraction stage. Consider the separation based scheme over a finite-field additive DMC. If R_a<I(A;B) and m,n→∞ with m/n = λ, the following hold: * ℙ [ Z^m ≠Ẑ^m ] → 0, * 1/mH(Ẑ^m) → H(Z), and * I(Ẑ^m; I Î) → 0. Let P_e be the probability of decoding error (i.e., P_I_e=ℙ[I≠Î] and P_Z_e=ℙ[Z^m ≠Ẑ^m]).We first show the claim in <ref>). From the channel coding theorem we obtain that P_I_e≤ 2^-nε. Consequently, from (<ref>) and (<ref>) ℙ[Z^m ≠Ẑ^m] will follow directly as P_Z_e≤ 2^-mε.Then, the claim in <ref>) is shown as follows H(Ẑ^m) (a)≤ H(Z^m)+ H(Ẑ^m|Z^m)(b)≤ mH(Z) + h_2(P_Z_e)+ P_Z_emlog| Z| 1/mH(Ẑ^m) ≤ H(Z) + 1/m h_2(P_Z_e)+ P_Z_elog| Z| 1/mH(Ẑ^m)H(Z)where (a) follows from the chain rule of entropy; (b) follows from Fano's inequality and the fact that Z^m∼ P_Z^⊗ n; Finally, the claim in <ref>) is shown by the following chain of inequalities:I(Ẑ^m;IÎ)≤ I(Z^mẐ^m;IÎ)≤ I(Z^m Ẑ^m;I)+H(Î|I) = H(Ẑ^m|Z^m)-H(Ẑ^m|Z^mI)+ H(Î|I)≤ H(Ẑ^m|Z^m)+ H(Î|I)(a)≤ h_2(P_Z_e)+ P_Z_emlog| Z| + h_2(P_I_e)+P_I_enR_c(b)≤ϵwhere (a) follows from Fano's inequality; (b) follows fromP_I_e≤ 2^-nε, P_Z_e≤ 2^-mε and ϵ,ε→ 0 as n,m→∞ respectively. Now, similar to the joint scheme, we can quantify the local randomness at both nodes, apply the rate transfer lemma <cit.>, and set λ=1 to facilitate comparison with the joint scheme from Section <ref>. The following theorem then describes an inner bound to the strong coordination region using the separate-based scheme with randomness extraction. There exists an achievable separation based coordination-channel coding scheme for the strong setup in Fig <ref> such that (<ref>) is satisfied for δ_1≥0, δ_2≥0 if R_c+R_o≥ I(XY;U) +δ_1 +δ_2, R_c≥ I(X;U), R_c<I(A;B), ρ_1≥ R_c-I(X;U)-δ_1,ρ_2≥max(0,H(Y|U)-H(Z))-δ_2. The proof follows in a straightforward way from the proofs of both Theorem <ref> and Lemma <ref> and is therefore omitted.§ EXAMPLE In the following, we compare the performance of the joint scheme in Section <ref> and the separation-based scheme in Section <ref>using a simple example. Specifically, we let X be a Bernoulli-1/2 source, thecommunication channel P_B|A be abinary symmetric channel with crossover probability p_o (BSC(p_o)), and the conditional distribution P_Y|X be a BSC(p). §.§ Basic separation scheme with randomness extraction To derive the rate constraints for the basic separation scheme, we considerX-U-Y with U∼Bernoulli-1/2 (which is known to be optimal <cit.>),P_U|X=BSC(p_1), and P_Y|U=BSC(p_2),p_2 ∈ [0,p], p_1=p-p_21-2p_2. Using this to obtain the mutual information terms in Theorem <ref>, we get I(X;U)=1-h_2(p_1),I(A;B)=1-h_2(p_o),I(XY;U)=1+ h_2(p)-h_2(p_1)-h_2(p_2),andH(Y|U)= h_2(p_2).After a round of Fourier-Motzkin elimination by using (<ref>)-(<ref>) in Theorem <ref>, we obtain the following constraints for the achievable region using the separation-based scheme with randomness extraction: R_o+ρ_1+ρ_2 ≥ h_2(p)-min(h_2(p_2), h_2(p_o)), h_2(p_1) ≥ h_2(p_o) R_c ≥ 1-h_2(p_1).Note that (<ref>) presents the achievable sum rate constraint for the total required randomness in the system.§.§ Joint schemeThe rate constraints for the joint scheme are constructed in two stages. First, we derive the scheme for the codebook cardinalities | A|=2 and | C|=2, an extension to larger | C| is straightforward but more tedious (see Figs. <ref> and <ref>)[Note that these cardinalities are not optimal. They are, however, analytically feasible and provide a good intuition about the performance of the scheme.]. The joint scheme correlates the codebooks while ensuring that the decodability constraint (<ref>) is satisfied. To get the best tradeoff, we find the joint distributionP_AC that maximizes I(B;C). For |𝒞|=2 this is simply given by P_A|C(a|c)=δ_ac.Then, the distribution P_X(x)P_CA|X(c,a|x)P_B|A(b|a)P_Y|BC(y|b,c) that produces the boundary of the strong coordination region for the joint scheme is formed by cascading two BSCs and another symmetricchannel, yielding the Markov chain X-(C,A)-(C,B)-Y, with the channel transition matricesP_CA|X =[1-p_100 p_1 p_1 00 1-p_1],P_CB|CA =[ 1-p_o p_o0 000p_o 1-p_o], P_Y|CB =[ 1-α 1-β β α α β1-β1-α]^T for some α,β∈ [0,1]. Then, the mutual information termsin Theorem <ref> can beexpressed with p_2≜ (1-p_o)α + p_oβ asI(X;AC)=I(X;C) = 1-h_2(p_1), I(XY;AC)=I(XY;C)=1+ h_2(p)-h_2(p_1)-h_2(p_2), I(B;C)= 1-h_2(p_o), and H(Y|BC) = p_oh_2(β)+(1-p_o)h_2(α).To find the minimum achievable sum rate we first perform Fourier-Motzkin elimination on the rate constraints in Theorem <ref> and then minimize the information terms with respect to the parameters p_2, α, and β as follows:R_o+ρ_1+ρ_2=min_p_2,α,β(h_2(p)-h_2(p_2)+(1-p_o)h_2(α)+p_oh_2(β)) subject to[ h_2(p_1)>h_2(p_o),;R_c≥1-h_2(p_1),;p= p_1-2p_1p_2+p_2. ]§.§ NumericalresultsFig. <ref> presents a comparison between the minimum randomness sum rate R_o+ρ_1+ρ_2 required to achieve coordination using the joint and the separate scheme with randomness extraction when the communication channel is given by BSC(p_o). The target distribution is set as p_Y|X=BSC(0.4). The rates for the joint scheme are obtained by solving the optimization problem in (<ref>).Similar results are obtained for the joint scheme with | C|> 2. For the separate scheme we choose p_2 such that h_2(p_1)=h_2(p_0) to maximize the amount of extracted randomness. We also include the performance of the separate scheme without randomness extraction. As can be seen from Fig. <ref>, both the joint scheme and the separate scheme with randomness extraction provide the same sum rate R_o+ρ_1+ρ_2 for p_o≤ p'_o where p'_o≜1-√(1-2p)/2. We also observe that for noisy channels the joint scheme approaches the performance of the separate scheme when the cardinality of C is increased. In this regime, we let p_2=p_0 such that h_2(p_2)=h_2(p_0) in order tomaximize the amount of extracted randomness. This is done by selecting α=0 and β=1 associated with P_Y|BC. However, it can be easily shown that for p_0>p_0' this does not ensure a target distribution of P_XY^⊗ n anymore. Therefore, the optimization over the parameters α and β now results in a larger sum rate R_o+ρ_1+ρ_2 as can be seenfrom Fig. <ref>. As p_o increases further, the required totalrandomness of the joint scheme approaches the one for the basic separate scheme again.Fig. <ref> provides a comparison of the communication rate for both schemes. Note that the joint scheme provides significantly smaller rates than the separation scheme with randomness extractionfor p_o≤ p'_o, independent of the cardinality of |𝒞|.Thus, in this regime joint coordination-channel coding provides an advantage in terms of communication cost and outperforms the separation-based scheme for the same amount of randomness injected into the system. IEEEtran | http://arxiv.org/abs/1704.08771v1 | {
"authors": [
"Sarah A. Obead",
"Badri N. Vellambi",
"Jörg Kliewer"
],
"categories": [
"cs.IT",
"math.IT"
],
"primary_category": "cs.IT",
"published": "20170427230122",
"title": "Strong Coordination over Noisy Channels: Is Separation Sufficient?"
} |
firstpage–lastpage Structured Sparse Modelling with Hierarchical GP Danil Kuzin, Olga Isupova, Lyudmila Mihaylova The University of Sheffield, Sheffield, UKEmail: [email protected], [email protected], [email protected] Olga Isupova and Lyudmila Mihaylova acknowledge the support from the EC Seventh Framework Programme [FP7 2013-2017] TRAcking in compleX sensor systems (TRAX) Grant agreement no.: 607400.Received: date / Revised version: date ================================================================================================================================================================================================================================================================================================================================================================================== We present simultaneous radio through sub-mm observations of the black hole X-ray binary (BHXB) V404 Cygni during the most active phase of its June 2015 outburst.Our 4 hour long set of overlapping observations with the Very Large Array, the Sub-millimeter Array, and the James Clerk Maxwell Telescope (SCUBA-2), covers 8 different frequency bands (including the first detection of a BHXB jet at 666 GHz/450μ m), providing an unprecedented multi-frequency view of the extraordinary flaring activity seen during this period of the outburst. In particular, we detect multiple rapidly evolving flares, which reach Jy-level fluxes across all of our frequency bands. With this rich data set we performed detailed MCMC modeling of the repeated flaring events. Our custom model adapts the van der Laan synchrotron bubble model to include twin bi-polar ejections, propagating away from the black hole at bulk relativistic velocities, along a jet axis that is inclined to the line of sight. The emission predicted by our model accounts for projection effects, relativistic beaming, and the geometric time delay between the approaching and receding ejecta in each ejection event. We find that a total of 8 bi-polar, discrete jet ejection events can reproduce the emission that we observe in all of our frequency bands remarkably well.With our best fit model, we provide detailed probes of jet speed, structure, energetics, and geometry. Our analysis demonstrates the paramount importance of the mm/sub-mm bands, which offer a unique, more detailed view of the jet than can be provided by radio frequencies alone. black hole physics — ISM: jets and outflows — radio continuum: stars — stars: individual (V404 Cygni, GS 2023+338) — submillimetre: stars — X-rays: binaries§ INTRODUCTIONBlack hole X-ray binaries (BHXBs), the rapidly evolving, stellar-mass counterparts of active galactic nuclei, are ideal candidates with which to study accretion and accretion-fed outflows, such as relativistic jets. These transient binary systems, containing a black hole accreting mass from a companion star, occasionally enter into bright outburst phases lasting days to weeks, providing a real time view of the evolving relativistic jets (probed by radio through IR frequencies) and accretion flow (probed at X-ray frequencies).BHXBs display two different types of relativistic jets, dependent on the mass accretion rate in the system <cit.>. At lower mass accretion rates (< 10^-1 L_ Edd)[The Eddington luminosity is the theoretical limit where, assuming ionized hydrogen in a spherical geometry, radiation pressure balances gravity. This limit corresponds to L_ Edd = 1.26×10^38M/M_⊙ergs^-1, where M is the black hole mass.], during the hard accretion state (seeandfor a review of accretion states in BHXBs), a steady, compact synchrotron-emitting jet is believed to be present in all BHXBs. It has also been shown that the compact jet is not only present during outburst phases, but can persist down into quiescence, at < 10^-5 L_ Edd <cit.>.At higher mass accretion rates, during the transition between accretion states, discrete jet ejecta are launched (e.g., ), and the compact jet may become quenched <cit.>. A small number of BHXBs have been observed to display multiple jet ejection events within a single outburst (e.g., ).Compact jets are characterized by a flat to slightly inverted optically thick spectrum (α>0; where f_ν∝ν^α; ), extending from radio up to sub-mm or even infrared frequencies <cit.>. Around infrared frequencies the jet emission becomes optically thin (α∼-0.7; ), resulting in a spectral break. Each frequency below this break probes emission (from the optical depth, τ=1 surface) coming from a narrow range of distances downstream in the jet, where higher frequencies originate from regions along the jet axis that are closer to where the jet is launched <cit.>. The exact spectral shape (i.e., spectral index, location of the spectral break) is believed to evolve with changing jet properties such as geometry, magnetic field structure, and particle density profiles <cit.>, as well as the plasma conditions in the region where the jet is first accelerated <cit.>. In contrast to the compact jets, jet ejecta are characterized by an optically thin spectrum (α<0), give rise to bright flaring activity, and can be routinely resolved with Very Long Baseline Interferometry (VLBI; e.g., ). The accompanying flares typically have well defined rise and decay phases, where the flares are usually optically thick in the rise phase, until the self-absorption turnover in the spectrum has passed through the observing band. These jet ejection events are believed to be the result of the injection of energy and particles to create an adiabatically expanding synchrotron emitting plasma, threaded by a magnetic field (i.e., van der Laan synchrotron bubble model, hereafter referred to as the vdL model; ). In this model, as the source expands the evolving optical depth results in the distinct observational signature of the lower frequency emission being a smoothed, delayed version of the higher frequency emission. The ejection events have been linked to both X-ray spectral and timing signatures (e.g., ), although a definitive mechanism or sequence of events leading to jet ejection has not yet been identified. Additionally, an extremely rare jet phenomenon, so called jet oscillation events, has also been observed in two BHXBs, GRS 1915+105 (radio, mm, IR; ) and V4641 Sgr (optical band; ). Such rare events seem to occur only when the accretion rate is at very high fractions of the Eddington rate. These quasi-periodic oscillations (seefor a review) show lower frequency emission peaking at later times (consistent with the vdL model for expanding discrete jet ejecta), rise and decay times of the repeated flares that are similar at all frequencies, and time lags between frequencies that vary within a factor of two.Moreover, no discrete moving components were resolved with VLBI during these oscillation events (although we note this could very well be due to sensitivity limits or the difficulty of synthesis imaging of fast-moving, time-variable components).As such, the exact nature of these events remains unclear, with theories including discrete plasma ejections, internal shocks in a steady flow, or variations in the jet power in a self-absorbed, conical outflow (e.g., ). In GRS 1915+105 , these oscillations have also been clearly associated with dips in hard X-ray emission, possibly linking the launching of jet ejecta to the ejection and refilling of the inner accretion disc or coronal flow <cit.>.While several transient BHXBs may undergo an outburst period in a given year, in which the jet emission becomes bright enough for detailed multi-wavelength studies, only rare (e.g., once per decade) outbursts probe the process of accretion and the physics of accretion-fed outflows near (or above) the Eddington limit. Observing the brightest and most extreme phases of accretion during these outbursts presents us with a unique opportunity to study jet and accretion physics in unprecedented detail. On 2015 June 15, the BHXB V404 Cygni entered into one of these rare near-Eddington outbursts. In this paper we report on our simultaneous radio through sub-mm observations of V404 Cygni during the most active phase of this outburst. §.§ V404 CygniV404 Cygni (aka GS 2023+338; hereafter referred to as V404 Cyg) is a well studied BHXB that has been in a low-luminosity quiescent state since its discovery with the Ginga satellite in 1989 <cit.>. This source has been observed to undergo a total of three outbursts prior to 2015; most recently in 1989 <cit.>, and two prior to 1989 which were recorded on photographic plates <cit.>. V404 Cyg is known to display bright X-ray luminosities and high levels of multi-wavelength variability, both in outburst and quiescence<cit.>. The prolonged quiescent period of V404 Cyg, and high quiescent luminosity (L_X∼1×10^33erg s^-1; ),has allowed the complete characterization of the system. The optical extinction is low, with E(B-V) = 1.3, enabling the study of the optical counterpart, and the determination of the mass function as 6.08±0.06 M_⊙ <cit.>. Subsequent modelling determined the black hole mass to be 9.0^+0.2_-0.6 M_⊙, with an inclination angle of 67^∘^+3_-1, and an orbital period of 6.5 days <cit.>.However, we note that this inclination angle estimate is dependent on the assumed level of accretion disc contamination in the optical light curves being modelled. <cit.> assumed <3% accretion disc contamination, but given that V404 Cyg is known to be variable in quiescence in the optical, it is plausible that the accretion disc contamination may be larger <cit.>, which would imply a larger inclination angle. Further, the faint, unresolved radio emission from the quiescent jets was used to determine a model-independent parallax distance of 2.39±0.14 kpc <cit.>, making V404 Cyg one of the closest known BHXBs in the Galaxy. The close proximity, well-determined system parameters, and bright multi-wavelength activity make this system an ideal target for jet and accretion studies.On 2015 June 15[<cit.> serendipitously detected an optical precursor to this outburst on June 8/9, approximately one week prior to the first X-ray detection.], V404 Cyg entered into its fourth recorded outburst period. The source began exhibiting bright multi-wavelengthflaring activity (e.g., ) immediately following the initial detection of the outburst in X-rays <cit.>, and swiftly became the brightest BHXB outburst seen in the past decade. This flaring behaviour was strikingly similar to that seen in the previous 1989 outburst <cit.>. Towards the end of June the flaring activity began to diminish across all wavelengths (e.g., ), and the source began to decay <cit.>, reaching X-ray quiescence[V404 Cyg entered optical quiescence in mid October 2015 <cit.>.]in early to mid August <cit.>. V404 Cyg also showed a short period of renewed activity from late December 2015 to early January 2016 (e.g., ), and <cit.> present radio, optical, and X-ray monitoring during this period. We organized simultaneous observations with the Karl G. Jansky Very Large Array (VLA), the Sub-millimeter Array (SMA), and the James Clerk Maxwell Telescope (JCMT) on 2015 June 22 (approximately one week following the initial detection of the outburst), during which time some of the brightest flaring activity seen in the entire outburst was observed.This comprehensive data set gives us an unprecedented multi-frequency view of V404 Cyg, in turn allowing us to perform detailed multi-frequency light curve modelling of the flaring events. In 2 we describe the data collection and data reduction processes. 3 describes the custom procedures our team developed to extract high time resolution measurements from our data. In 4 we present our multi-frequency light curves, outline our model, and describe the modelling process.A discussion of our best fit model is presented in 5, and a summary of our work is presented in 6. § OBSERVATIONS AND DATA ANALYSIS §.§ VLA Radio ObservationsWe observed V404 Cyg with the VLA (Project Code: 15A-504) on 2015 June 22, with scans on source from 10:37:24–14:38:39 UTC (MJD=57195.442-57195.610) in both C (4-8GHz) and K (18-26GHz) band. The array was in its most extended A configuration, where we split the array into 2 sub-arrays of 14 (sub-array A) and 13 (sub-array B) antennas. Sub-array A observed the sequence C-K-C, while sub-array B observed the sequence K-C-K, with an 80 second on target and 40 second on calibrator cycle, in order to obtain truly simultaneous observations across both bands. All observations were made with an 8-bit sampler, comprised of 2 base-bands, with 8 spectral windows of 64 2 MHz channels each, giving a total bandwidth of 1.024 GHz per base-band. Flagging, calibration, and imaging of the data were carried out within the Common Astronomy Software Application package (CASA; ) using standard procedures. We used 3C48 (0137+331) as a flux calibrator, and J2025+3343 as a phase calibrator for both sub-arrays. No self-calibration was performed.Due to the rapidly changing flux density of the source, we imaged the source (with natural weighting; see the Appendix for details on our choice of weighting scheme) on timescales as short as the correlator dump time (2 seconds) using our custom CASA timing scripts (see 3.1 for details). §.§ SMA (Sub)-Millimetre ObservationsWe observed V404 Cyg with the SMA (Project Code: 2015A-S026) on 2015 June 22, with scans on source from 10:16:17–18:20:47 UTC (MJD=57195.428-57195.764), and the correlator tuned to an LO frequency of 224 GHz. The array was in the sub-compact configuration with a total of 7 antennas (out of a possible 8 antennas). These observations were made with both the ASIC and SWARM <cit.> correlators active, to yield 2 side-bands, with 48 spectral windows of 128 0.8125 MHz channels (ASIC) and an additional 2 1.664 GHz spectral windows (SWARM), giving a total bandwidth of 8.32 GHz per side-band. The SWARM correlator had a fixed resolution of 101.6 kHz per channel, and thus originally 16383 channels for each SWARM spectral window.Given the continuum nature of these observations, we performed spectral averaging, to yield 128 13 MHz channels in both SWARM spectral windows, to match the number of channels in the ASIC spectral windows, and in turn make it easier to combine ASIC and SWARM data.We used 3C454.3 (J2253+1608) as a bandpass calibrator, MWC349a and J2015+3710 as phase calibrators, and Neptune and Titan as flux calibrators[The SMA calibrator list can be found at http://sma1.sma.hawaii.edu/callist/callist.html.]. We note that only the second IF (spectral windows 25-50) was used for flux calibration in the upper side-band due to a CO line that was present in both flux calibrators at 230.55 GHz. Our observing sequence consisted of a cycle of 15 min on target and 2.5 min on each of the two phase calibrators. As CASA is unable to handle SMA data in its original format, prior to any data reduction we used the SMA scripts, sma2casa.py and smaImportFix.py, to convert the data into CASA MS format, perform the T_ sys corrections, and spectrally average the two SWARM spectral windows. Flagging, calibration, and imaging of the data were then performed in CASA using procedures outlined in the CASA Guides for SMA data reduction[Links to the SMA CASA Guides and these scripts are publicly available at https://www.cfa.harvard.edu/sma/casa.]. Due to the rapidly changing flux density of the source, we imaged the source (with natural weighting; see the Appendix for details on our choice of weighting scheme) on timescales as short as the correlator dump time (30 seconds) using our custom CASA timing scripts (see 3.1 for details). §.§ JCMT SCUBA-2 (Sub)-Millimetre ObservationsWe observed V404 Cyg with the JCMT (Project Code: M15AI54) on 2015 June 22 from 10:49:33–15:12:40 UTC (MJD=57195.451-57195.634), in the 850μ m (350 GHz) and 450μ m (666 GHz) bands. The observation consisted of eight ∼30 min scans on target with the SCUBA-2 detector <cit.>. To perform absolute flux calibration, observations of the calibrator CRL2688 were used to derive a flux conversion factor <cit.>. The daisy configuration was used to produce 3 arcmin maps of the target source region. During the observations we were in the Grade 3 weather band with a 225 GHz opacity of 0.095–0.11. Data were reduced in the StarLink package using both standard procedures outlined in the SCUBA-2 cookbook[http://starlink.eao.hawaii.edu/devdocs/sc21.htx/sc21.html] and SCUBA-2 Quickguide[https://www.eaobservatory.org/jcmt/instrumentation/continuum/scuba-2/data-reduction/reducing-scuba2-data], as well as a custom procedure to create short timescale maps (timescales shorter than the 30 minute scan timescale) to extract high time resolution flux density measurements of the rapidly evolving source (see 3.2 for details).§ HIGH TIME RESOLUTION MEASUREMENTS §.§ VLA and SMATo obtain high time resolution flux density measurements of V404 Cyg from our interferometric data sets (VLA and SMA) we developed a series of custom scripts that run within CASA.A detailed account of the development and use of these scripts will be presented in Tetarenko & Koch et al. 2017, in prep., although we provide a brief overview of the capabilities here.Our scripts split an input calibrated CASA Measurement Set into specified time intervals for analysis in the image plane or the uv plane. In the image plane analysis, each time interval is cleaned and the flux density of the target source is measured by fitting a point source in the image plane with the native CASA task . All imaging parameters (e.g., image size, pixel size, number of CLEAN iterations, CLEAN threshold) can be fully specified. In the uv plane analysis, the uvmultfit package <cit.> is used to measure flux density of the target source. In either case, an output data file and plot of the resulting light curve are produced. These scripts are publicly available on github[https://github.com/Astroua/AstroCompute_Scripts], and are being implemented as a part of an interactive service our team is developing to run on Amazon Web Services Cloud Resources.All VLA and SMA flux density measurements output from this procedure (fitting only in the image plane) are provided in a machine readable table online, which accompanies this paper. Additionally, to check that the variability we observed in V404 Cyg is dominated by intrinsic variations in the source and not due to atmospheric or instrumental effects, we also ran our calibrator sources through these scripts (see the Appendix for details).§.§ JCMT SCUBA-2To obtain high time resolution flux density measurements of V404 Cyg from our JCMT SCUBA-2 data we developed a custom procedure to produce a data cube, containing multiple maps of the target source region, at different time intervals throughout our observation.We run the StarLink Dynamic Iterative Mapmaker tool on each of the target scans, using the bright compact recipe, with the addition of the shortmap parameter. The shortmap parameter allows the Mapmaker to create a series of maps, each of which will include data from a group of adjacent time slices. The number of time slices included in each map is equivalent to the shortmap parameter value. At 850μ m we use shortmap=200 to produce 362 time slices for a 32 minute scan, resulting in 5 second time bins. At 450μ m shortmap=400 would produce the same number of time slices, where a factor of 2 is applied as the default pixel size is 2 arcsec at 450 μ m and 4 arcsec at 850μ m. However, as the noise is higher at 450 μ m, we use shortmap=4800 to produce 32 time slices for a 32 minute scan, resulting in 60 second time bins. Thetask is then used to combine all of the short maps into a cube for each scan. The sort= True and sortby= MJD-AVG parameters ensure the maps are ordered chronologically in time, with the resulting cube having the dimensions, position X (pixels), position Y (pixels), time (MJD). Using thetask we then combined the cubes from all the scans. We calibrated the combined cube into units of Jy using theandtasks. Finally, the combined cube can be viewed in , and converted to FITS format with thetask.To extract flux densities from each time slice in the combined cube, we fit a 2D gaussian[The python package gaussfitter is used in the gaussian fitting; https://github.com/keflavich/gaussfitter] with the size of the beam (FWHM of 15.35 arcsec at 850μ m and 10.21 arcsec at 450 μ m; derived using the task ) to each slice of the cube. All JCMT SCUBA-2 flux density measurements output from this procedure are provided in a machine readable table online, which accompanies this paper. As with our interferometric data sets, to check that the variability we observed in V404 Cyg is dominated by intrinsic variations in the source and not due to atmospheric or instrumental effects, we also ran this procedure on our calibrator source scans (see the Appendix for details).§ RESULTS §.§ Multi-frequency Light CurvesA composite light curve of all of our VLA, SMA and JCMT observations from June 22 is presented in Figure <ref>. We observe rapid multi-frequency variability in the form of multiple large scale flares, reaching Jy flux levels. In the SMA data, the largest flare (at ∼ 13:15 UTC) rose from ∼ 100 mJyto a peak of ∼ 5.6 Jy on a timescale of ∼ 25 min. The JCMT SCUBA-2 data appear to track the SMA data closely, with the largest flare at 350 GHz rising from ∼400 mJy to a peak of ∼ 7.2 Jy on a timescale of ∼18 min. This is the largest mm/sub-mm flare ever observed from a BHXB, far surpassing even the brightest events in GRS 1915+105 <cit.>. The VLA radio data lag the mm/sub-mm (where the lag appears to be variable among the flares; ∼20–45 min & ∼40–75 min between 350 GHz and the 18–26 GHz & 4–8 GHz bands, respectively), with flares in the 18–26 GHz band rising to a peak of ∼1.5 Jy on a timescale of ∼35 min, and flares in the 4–8 GHz band rising to a peak of ∼ 780 mJy on a timescale of ∼45 min.Upon comparing the multi-frequency emission, it is clear that the mm/sub-mm data provide a much more extreme view of the flaring activity than the radio emission. In particular, there is more structure present in the mm/sub-mm light curves when compared to the radio light curves. As such, while not immediately apparent in the radio light curves, the mm/sub-mm data suggest that each of the three main flares in the light curves is actually the result of the superposition of emission from multiple flaring components.Additionally, the lower frequency emission in the light curves appears to be a smoothed, delayed version of the high frequency emission (with the flares showing longer rise times at lower frequencies). This emission pattern is consistent with an expanding outflow structure, where the mm/sub-mm emission originates in a region (with a smaller cross-section) closer to the black hole, and has thus not been smoothed out to as high a degree as the radio emission, as the material expands and propagates outwards.Therefore, all of these observations suggest that the emission in our light curves could be dominated by emission from multiple, expanding, discrete jet ejection events <cit.>.Further, we notice that the baseline flux level at which the flaring begins at each frequency in our light curves appears to vary.This suggests that there is an additional frequency-dependent component contributing to our light curves, on top of the discrete jet ejecta. In an effort to determine the origin of this extra emission, assuming that the baseline emission is constant in time, we create a spectrum of this emission by estimating the baseline flux level at each frequency (we performed iterative sigma clipping and take the minimum of the resulting sigma clipped data). This spectrum[We note that these are only empirical initial estimates of the baseline flux at each frequency, and do not necessarily represent the flux of the compact jet in our model presented in 4.2.] is presented in Figure <ref>, where it appears as though the baseline emission could be described by a broken power-law or a single power-law (with higher frequency emission displaying a lower baseline level than lower frequency emission). This spectral shape, combined with the fact that we observe a strong compact core component (in addition to resolved ejecta components) within simultaneous high resolution radio imaging (Miller-Jones, et al. 2017, in prep), suggests that the baseline emission originates from an underlying compact jet that was not fully quenched.§.§ V404 Cyg Jet ModelGiven the morphology of our light curves outlined in the previous section, we have constructed a jet model for V404 Cyg that is capable of reproducing emission from multiple, repeated, discrete jet ejection events, on top of an underlying compact jet component. We define two coordinate frames, the observer frame and the source frame (at rest with respect to the ejecta components). We will compute our model primarily in the source frames, and then transform back to the observer frame. All variables with the subscript obs are defined in the observer frame. Schematics displaying the geometry of our model from different viewpoints are displayed in Figures <ref> & <ref>. In our model, the underlying compact jet is characterized by a broken power-law spectrum, where the flux density is independent of time and varies only with frequency according to,F_ν, cj = {[ F_ br, cj (ν / ν_ br) ^ α_1 , ν < ν_ br; F_ br, cj (ν / ν_ br) ^ α_2,ν > ν_ br; ].Here ν_ br represents the frequency of the spectral break, F_ br,cj represents the amplitude of the compact jet at the spectral break frequency, α_1 represents the spectral index at frequencies below the break, and α_2 represents the spectral index at frequencies above the break. In the case where the spectral break frequency is located below the lowest sampled frequency band, or above the highest sampled frequency band, the underlying compact jet can be characterized by a single power-law spectrum, where F_ν, cj=F_0, cj(ν/ν_0)^α. Here F_ 0,cj represents the amplitude of the compact jet at ν_ 0, and α represents the spectral index. On top of the compact jet, we define a discrete ejection event as the simultaneous launching of two identical, bi-polar plasma clouds (an approaching and receding component).Each of these clouds evolve according to the vdL model <cit.>. In this model, a population of relativistic electrons, with a power-law energy distribution (N(E)dE=KE^-pdE), is injected into a spherical cloud threaded by a magnetic field. The cloud is then allowed to expand adiabatically, while the electrons and magnetic field are assumed to be kept in equipartition.As a result of the expansion, this model predicts the flux density of each cloud will scale as,F_ν, ej=F_0 (ν/ν_0)^5/2(R/R_0)^31-exp(-τ_ν)/1-exp(-τ_0).Here R indicates the time-dependent radius of the cloud, and the synchrotron optical depth, τ_ν, at a frequency, ν, scales as,τ_ν=τ_0 (ν/ν_0)^-(p+4)/2(R/R_0)^-(2p+3).Note that the subscript 0 in all our equations indicates values at the reference frequency[We defined our reference frequency as the upper-sideband in our SMA data (230 GHz).], at the time (or radius) of the peak flux of the component. Taking the derivative of Equation 2 with respect to time[Our expression in Equation 4 differs from that of <cit.>, as he takes the derivative with respect to ν instead of time, yielding e^τ_0-([p+4]/5)τ_0-1=0.] (or radius), allows us to relate the optical depth at which the flux density of the reference frequency reaches a maximum, τ_0, to the power-law index of the electron energy distribution, p,e^τ_0-(2p/3+1)τ_0-1=0. Equation 4 has no analytic solution and thus must be solved numerically. Therefore, we choose to leave our model in terms of τ_0, and solve for p after the fitting process.To describe the time-dependence of the cloud radius, a linear expansion model is used, according to,R=R_0+β_ expc(t-t_0).Here β_ expc represents the expansion velocity of the cloud, while R_0 can be expressed in terms of the distance to the source, d, peak flux, F_0, and optical depth, τ_0, of the cloud at the reference frequency <cit.>,R_0=[F_0d^2/π1/1- exp(-τ_0)]^1/2. At the same time that the clouds are expanding, they are also propagating away from the black hole at bulk relativistic velocities, along a jet axis that is inclined to the observer's line of sight (see Figure <ref>). As such, the emission we observe will have been affected by projection effects, relativistic beaming, and a geometric time delay between the approaching and receding clouds in each ejection event. To account for these effects, we first assume that the clouds are travelling at a constant bulk velocity, β_b c, and that the jet has a conical geometry (with an observed opening angle, ϕ_ obs). In turn, the apparent observed velocity across the sky (derived via the transverse Doppler effect) is represented as <cit.>, β_ app,obs = {[ r sin i/c (t-t_ ej) - r cos i→approaching; ,; r sin i/c (t-t_ ej) + r cos i →receding; ]. where r=β_b c (t-t_ ej) is the distance travelled by the cloud away from the black hole, t_ ej represents the ejection time, c represents the speed of light, and i represents the inclination angle of the jet axis to the line of sight.Equation 7 can be simplified by substituting in our expression for r to yield,β_ app,obs=β_bΓδ_∓ sin(i),where the Doppler factor and bulk Lorentz factor are given by δ_∓=Γ^-1[1∓β_b cos i]^-1 and Γ=(1-β_b^2)^-1/2, respectively. The sign convention in the Doppler factor indicates that a δ_- should be used for the approaching cloud and a δ_+ should be used for the receding cloud.From Figures <ref> & <ref>,tan ϕ_ obs=R_ obs/r_ obs=δ_∓ β_ exp c (t-t_ej)_ obs/β_ app,obs c (t-t_ ej)_ obs=δ_∓ β_ exp/β_ app,obs. Combining Equations 8 & 9, and solving for the bulk Lorentz factor, Γ, yields,Γ=(1+β_ exp^2/ tan^2ϕ_ obs sin^2i)^1/2. Rearranging Equation 10 (and substituting in 1-Γ^2=-Γ^2β_b^2) gives the expansion velocity (to be input into Equation 5) in terms of only the bulk velocity and jet geometry (inclination and opening angle), such that,β_ exp= tan ϕ_ obs[Γ^2{1-(β_bcos i)^2}-1]^1/2.Further, we wish to write our model in terms of only the ejection time (t_ ej), rather than the time of the peak flux at the reference frequency (t_0), without introducing any additional parameters. Using our definition that R=R_0 at the instant t=t_0, the two timescales are related by,t_0=t_ ej+R_0/β_ expc. Lastly, we correct for relativistic beaming by applying a factor of δ_∓^3 <cit.> to our flux density in Equation 2, according to,F_ν, ej,obs=δ_∓^3F_ν, ej.Here F_ν, ej,obs indicates the flux density of the cloud in the observer frame, at the observing frequency ν_ obs, at the observed times since the zero point of our observations, Δ t_ obs, while F_ν, ej indicates the flux density of the clouds in the source frame, at the frequency, ν=δ_∓^-1ν_ obs, at the times, Δ t=δ_∓Δ t_ obs.All of the ejection events we model are not correlated,and thus evolve independently of each other. The total observed flux density in our model is represented as,F_ν, obs,tot=∑_iδ_-^3(F_ν, i, app)+∑_iδ_+^3(F_ν, i, rec)+F_ν, cj.§.§.§ Jet PrecessionIn addition to our VLA, SMA, and JCMT observations, we also obtained simultaneous high angular resolution radio observations with the Very Long Baseline Array (VLBA). Through imaging the VLBA data set in short 2 minute time bins, we resolve multiple discrete ejecta. Our analysis of these VLBA images has shown clear evidence of jet precession, where the position angle of the resolved ejecta change by up to 40 degrees on an hourly timescale (this result will be be reported in detail in Miller-Jones et al. 2017, in prep.). As the emission predicted by our model is highly dependent on the inclination angle of the jet axis, we account for the effect of this rapid, large scale jet precession in our model by allowing our inclination parameter, i, to vary between ejection events.§.§.§ Accelerated MotionWhile we have assumed that the jet ejecta are travelling at constant bulk velocities, it is possible that they undergo some form of accelerated motion. To test this hypothesis we generalized our model to allow the input of a custom bulk velocity profile, where we implemented simple velocity profiles to mimic a finite acceleration period where the cloud would approach a terminal velocity (e.g., a linear ramp function, a body subject to a quadratic drag force). However, in all cases, our best fit model either tended towards a constant velocity profile, or would not converge. This result, while not ruling out the possibility of accelerated motion, suggests that any potential acceleration period may have only lasted for a short enough period of time that we are not able to discern the difference between the resulting light curves for the accelerated and constant bulk motion. §.§.§ Sub-Conical Jet GeometryWhile we have assumed that the jet in our model is conical (constant opening angle), it is possible that the jet geometry could deviate from a strictly conical shape (especially on the AU size scales we are probing), where the opening angle (and in turn the expansion speed of the ejecta) could change with time. In particular, if we assume that the jet confinement mechanism is external, then the jet geometry will depend on the adiabatic indices of the two media (i.e., the jet and its surrounding medium). A relativistic plasmon confined by the internal pressure of a terminal spherical wind (made up of a Γ=5/3 gas) will expand sub-conically, according to R ∝ r^5/6. To test this scenario, we modified our model to use the above sub-conical expansion expression in place of Equation 5. In doing this we find that our best fit model still tends toward constant expansion speed/opening angle profiles for all the ejecta. This result, while not ruling out a non-conical jet geometry, could suggest that any deviations from a conical jet shape only occur on sub-AU size scales, probing timescales before the sub-mm emission peaks, and thus we are not able to discern the difference between the resulting light curves for conical/sub-conical jet geometry. §.§.§ Bi-polar vs. Single-Sided EjectionsOur jet model assumes that each ejection event takes the form of two identical, oppositely directed plasmons. However, in principle our light curves could also be fit with a collection of single-sided ejections. These unpaired components could occur as a result of Doppler boosting of highly relativistic plasmons causing us to observe only the approaching component of an ejecta pair, or intrinsically unpaired ejecta. Our simultaneous VLBA imaging may help distinguish between these two scenarios. We resolve both paired and (possibly[Given the rapid timescales of the ejections, multiple ejecta can become blended together in these images, making it difficult at times to conclusively identify and track individual components.]) unpaired ejecta components in our VLBA images, which could suggest that the emission in our light curves is produced by a combination of bi-polar and single-sided ejection events. Using these VLBA results to include stricter constraints within our model on ejecta numbers, type (single/bi-polar), and ejection times, is beyond the scope of this work, but will be considered in a future iteration of the model.§.§ Modelling Process and Best Fit ModelDue to the large number of free parameters in our model, we use a Bayesian approach for parameter estimation. In particular, we apply a Markov-Chain Monte Carlo algorithm (MCMC), implemented with the emcee package <cit.>, to fit our light curves with our jet model. This package is a pure-Python implementation of Goodman & Weare's Affine Invariant MCMC Ensemble Sampler <cit.>, running a modified version of the commonly used Metropolis-Hastings Algorithm, whereby it simultaneously evolves an ensemble of “walkers" through the parameter space. We use 500 walkers (10 × the number of dimensions in our model) for our MCMC runs. Prior distributions used for all of our parameters are listed in Table <ref>. We choose physically informative priors that reflect our knowledge of V404 Cyg (or commonly assumed values for BHXBs) where possible, and wide uninformative uniform priors when we have no pre-defined expectation for a specific parameter.For instance, the prior for the inclination angle is set as a truncated normal distribution, centered on 67 degrees (the measured inclination angle of the system), with boundaries of 0 and 90 degrees (allowed values of the inclination angle). On the other hand, the prior for the ejection time is simply a uniform distribution, sampling a wide range of possible times around our best initial guess.Before running the MCMC,the initial position of the walkers in the parameter space needs to be defined. As the performance of the emcee algorithm tends to benefit heavily from well defined initial conditions, we do an initial exploration of the parameter space using a harmony search global optimization algorithm[Implemented in the python package, pyHarmonySearch; https://github.com/gfairchild/pyHarmonySearch ]. This metaheuristic algorithm, that is similar to, but much more efficient than a brute force grid search method (which would not be computationally feasible in this case), yields a reasonable initial guess for our model, and we place our walkers in a tight ball around this initial guess in the parameter space.As our jet model can predict emission at multiple frequencies, to reduce the degeneracy in our model, we choose to simultaneously fit all of our multi-frequency data sets, except for the JCMT 666 GHz data set, due to its sparser sampling and larger uncertainty in flux calibration (see the Appendix for details). To do this, we use an iterative process whereby we start with our reference frequency data set, run the MCMC (the walkers are evolved over a series of steps, where the first 500 step “burn in" period is not retained) until convergence is reached, and use the final position of the walkers for the first run as the initial guess for the next run of the MCMC, which will include increasingly more data sets in the fit.To monitor the progress of the MCMC and ensure that correct sampling was occurring, we checked that the acceptance fraction stayed within the suggested bounds (between 0.25 and 0.75). Our criteria for convergence requires that the positions of the walkers are no longer significantly evolving. We determine whether this criteria is met by monitoring the chains of each of the walkers through the parameter space, and ensuring that, for each parameter,the intra-chain variance across samples is consistent with the inter-chain variance at a given sample.Using the multi-dimensional posterior distribution output from the converged MCMC solution, we create one dimensional histograms for each parameter. The best fit result is taken as the median of these distributions, and the uncertainties are reported as the range between the median and the 15th percentile (-), and the 85th percentile and the median (+), corresponding approximately to 1σ errors.All of the best-fit parameters and their uncertainties are reported in Table <ref>. Figure <ref> & <ref> show the best fit model overlaid on our multi-frequency light curves. Additionally, with our multi-dimensional posterior distribution we can explore the possible two-parameter correlations for our model, where a significant correlation between a pair of parameters can indicate a model degeneracy or a physical relationship between the parameters. In the Appendix section we show correlation plots (Figure <ref>), along with the one-dimensional histograms, for pairs of parameters for which we find a correlation, and discuss the significance of such a correlation.Within the Bayesian formalism, the uncertainties reported in Table <ref> are purely statistical, and only represent the credible ranges of the model parameters under the assumption that our model is correct. Given the residuals withrespect to the optimal model (Figure <ref> bottom panel), the observations contain physical or observational effects not completely accounted for in our model. To factor in how well our chosen model represents the data, we estimated an additional systematic error for our parameters (displayed in Table <ref> of the Appendix). To do this we rerun our MCMC, starting from the best fit solution, with an extra variance parameter (effectively modelling all the physical/observational effects not included in our model) in our log probability for each frequency band. This variance is equivalent to the square of the mean absolute deviation of the residuals with respect to our optimal model at each frequency (difference between the best fit model and the data). The resulting uncertainties in the parameters after this extra MCMC run will reflect the full (statistical + systematic) uncertainties. Our broad frequency coverage, in particular the high sub-mm frequencies, is crucial to the success of our modelling. Detailed substructure detected in the sub-mm bands can be used to separate out emission from different ejections, where their lower frequency counterparts are smoothed out and blended together.As such, modelling the lower frequency emission would not be possible without the critical information the high frequency sub-mm emission provides and vice versa. § DISCUSSION OF THE BEST FIT MODEL Our jet model for V404 Cyg, with a total of 8 bi-polar ejection events on top of an underlying compact jet, is able to reproduce the emission in all of our observed frequency bands, matching the flux levels, time lags between frequencies, and the overall morphology remarkably well.With such a large sample of jet ejecta, we can probe the intrinsic ejecta properties, and the distribution of these properties between the different ejection events. In particular, our model characterizes the bulk speeds, peak fluxes, the electron population injected during each event, and the jet geometry, all of which we find can vary between events, with bulk speeds of 0.08<β_b<0.86 c, peak fluxes of 986<S_0<5496 mJy, electron energy distribution indices of 1.4<p<5.6 (corresponding to 1.2<τ_0<2.6), and observed opening angles of 4.06<ϕ_ obs<9.86^∘.In the following sections we discuss these ejecta parameters and what they can tell us about jet speeds, energetics, mass loss, and geometry. Additionally, we draw comparisons between the V404 Cyg ejection events and the jet oscillation events in GRS 1915+105, as well as other multi-wavelength observations of V404 Cyg. §.§ Jet SpeedsThe bulk speeds of jet ejecta measured in BHXBs[An important caveat when considering the value of the bulk Lorentz factor (Γ), estimated using proper motions of discrete jet ejecta, is that Γ depends strongly on the assumed distance to the source <cit.>. While the distance is well known for V404 Cyg, this is not the case for the majority of BHXBs, and as a result constraints on Γ in these systems typically represent lower limits.] can vary from system to system (e.g., Γ∼ 1 in SS 433; , Γ∼2 in V4641 Sgr; ), where some systems that are known to enter high luminosity states, like V404 Cyg, have been shown to launch jet ejecta with Γ>2 (e.g., GRO J1655-40; ). However, in V404 Cyg we find that the bulk speeds of our modelled ejecta are quite low, with bulk Lorentz factors of only Γ∼ 1-1.3 (excluding ejection 5; see footnote c in Table <ref> for details).Moreover, V404 Cyg shows bulk speeds that vary substantially between ejection events, on timescales as short as minutes to hours. There is some evidence in the literature that jet speeds can vary within a BHXB[There is also evidence of jet speeds varying in neutron star XBs, most notably, Sco X-1 <cit.> and Cir X-1 <cit.>.] source. For example, <cit.> find small variations in jet speed up to 10% in SS 433, jet speeds have been reported to vary between outbursts of H1743-322 (; ), and varying proper motions have been measured in GRS 1915+105 <cit.>. However, no other source has shown variations as large, or on as rapid timescales as V404 Cyg. Performing a Monte Carlo Spearman's rank correlation test, we find no correlation between jet speed and ejection time, where, for instance, the bulk speed of the ejections (i.e., β_bc) increased or decreased throughout our observation period. However, we find a potential correlation (Spearman coefficient of 0.83±0.07 with a p-value of 0.01) between bulk speed and peak flux of our modelled ejecta, where brighter ejecta tend to have higher speeds. This correlation is consistent with what was seen in H1743-322, where higher bulk ejecta speeds corresponded to higher radio luminosity measurements (; ).The factors that govern jet speed in BHXBs are not well understood, but our measurements of surprisingly slow speeds, which can vary between sequential jet ejection events, suggest that the properties of the compact object (i.e., black hole mass) or peak luminosity of the outburst are likely not the dominant factors that affect jet speed.Additionally, given the varying bulk speeds between the ejection events, it is plausible that later, faster ejections could catch up to earlier, slower ejections. Such a collision between ejecta may result in a shock that could be as bright or even brighter than the initial ejections, and in turn produce a flaring profile that could mimic a new ejection event. While including ejecta collisions in our model is beyond the scope of this work, we briefly consider the possibility here by examining the bulk motion of all of the ejections. We find that a collision between ejection 3 and ejection 2 would occur at ∼ 11:30 (if they were ejected at the same PA), which is very close to the predicted ejection time of ejection 4. Moreover, ejection 4 has a bulk speed which is in between the bulk speeds of ejection 2 and 3, as we might expect for the bulk motion of the plasmon after such a collision. However, given that the jet appears to be rapidly precessing in V404 Cyg (Miller-Jones et al. 2017, in prep), ejection 2 and ejection 3 are launched at very different inclination angles, which would prevent such a collision from occurring.Therefore, given the precessing jet, we find this collision scenario unlikely.§.§ Jet Energetics, Mass Loss, and Particle AccelerationIn our model we assumed that the radiating electrons follow a power-law energy distribution. The power-law index of this distribution, p, informs us about the population of accelerated electrons initially injected into each discrete jet component, where the value of this energy index is governed by the electron acceleration mechanism.Fermi acceleration by a single shock can produce values of p∼2-3, which are typically found in XRBs <cit.>. However, the energy index can take on a wider range of values under certain conditions, where for example, lower values of p (which result in a more asymmetric flare profile) can be produced if the acceleration occurs in multiple shocks <cit.>, or if the electrons carry away kinetic power from the shock <cit.>, and higher values of p could be produced in the presence of oblique shocks (although this case requires highly relativistic shocks to produce large p values; ). Distributions with values of p>4 are nearly indistinguishable from a thermal (Maxwellian) distribution, which in the shock acceleration paradigm, implies very little acceleration has occurred (a shock essentially takes an input thermal distribution of electrons and builds a power-law distribution up over time). Magnetic reconnection in a relativistic plasma is another viable mechanism that can accelerate electrons into distributions with similar p values to shock acceleration. In this case, smaller p values can be produced in the case of a strongly magnetized plasma (σ>10; where σ≡ B^2/4π nmc^2 represents the magnetization parameter), and larger p values can be produced in a weakly magnetized plasma (σ∼1; ). In either theory of particle acceleration, we would expect a link between the speed (for shock acceleration) or magnetization (for magnetic reconnection), and the energy index, p.The energy indices of our modelled ejecta appear to vary between sequential ejection events, with 1.4<p<5.6(where we find no clear correlation between p values and jet speed). These p values could be produced by shock acceleration or magnetic reconnection (under the right conditions), although we would need to invoke different mechanisms to produce distributions in both the very low and very high p regimes (e.g., 1.4 in ejection 7, and 5.7 in ejection 2), which is not entirely physical for a single source. Further, this significant range seen in our energy indices suggests that our model may not be capturing all of the complexities of these ejection events, where the more extreme values of the energy index could be mimicking the effect of physics that has not been included in our model. For instance, the vdL model assumes equipartition, but as the plasmons expand they must do work, which will result in some of the magnetic field dissipating into kinetic or thermal pressure, and in turn, the assumption of equipartition may break down. Simplifications in our model such as this could also explain the lack of expected correlation between our energy indices and the speed of the ejecta. A more rigorous treatment, which, for example, calculates the full synchrotron flux (and does not rely on the equipartition assumption), is beyond the scope of this work, but will be considered in future iterations of this model.For synchrotron emitting clouds of plasma injected with our measured electron distributions, we estimate that the minimum energy[In our minimum energy calculations, we perform the full calculations outlined in <cit.>, where we integrate the electron energy distribution from ν_ min=150 MHz to ν_ max=666 GHz. The minimum frequency represents the lowest radio detection with LOFAR on June 23 & 24 <cit.>, and the maximum frequency represents our highest frequency sub-mm detection. When we consider an electron-proton plasma, we assume the ratio of the energy in the protons over that of the electrons is ϵ_p/ϵ_e=1.] needed to produce each of our modelled ejection events range from5.0×10^35<E_ min<3.5×10^38erg, with minimum energy magnetic fields[We note that while these calculation assume equipartition, the system could be far from equipartition. In this case the magnetic field would not necessarily be equivalent to the minimum energy field, but rather could be either much higher or much lower.] on the order of a few Gauss (1<B_ min<35 G). Taking into account the duration of each event, these energies correspond to a mean power into each event ranging from 4.0×10^32<P_ min<2.5×10^35ergs^-1. Due to the slow bulk speeds of the ejecta, including the kinetic energy from the bulk motion (in an electron-positron plasma E_ KE=(Γ-1)E_ min) yields only slightly higher values of 4.1×10^32<P_ tot<2.6×10^35ergs^-1. The minimum energy and power released within each of our modelled ejection events is comparatively lower than estimated for other major ejection events in BHXBs (E_ min∼1×10^43erg; e.g.,and P_ tot∼10^36-10^39ergs^-1; e.g., ). This difference is dominated by the difference in the estimated size of the emitting region, where the radii of our modelled ejecta are smaller than is normally estimated for major ejection events, and the low bulk speeds, which result in a much smaller kinetic energy contribution. Considering that the flaring activity in V404 Cyg lasted ∼ 2 weeks (and assuming our observations to be representative of this entire period), we estimate that the the total (minimum) energy (radiative + kinetic) released into jet ejections over the full flaring period is ∼3.2×10^40erg, which is more on par with typical energies estimated for major ejection events in BHXBs. This total energy is also comparable to that carried by the accretion disc wind (∼10^41erg)[A rough estimate of the energy lost in the accretion disc wind is equivalent to E_ wind∼(1/2)M_ wind v_ wind^2. Using M_ wind∼ 10^-8 M_⊙ and v_ wind∼1000km s^-1 <cit.>, we estimate E_ wind∼10^41erg.]. If we assume that the jet ejecta contain some baryonic content, in the form of one cold proton for every electron, we calculate that the mean power into each event (including the kinetic energy from bulk motion) ranges from 6.2×10^32<P_ tot<3.8×10^35ergs^-1. In this baryonic case, we estimate a total mass lost through the jet in our observation period of 9.4×10^-13 M_⊙ (corresponding to 7.2×10^-11 M_⊙ over the 2 week flaring period). To compare this jet mass loss to the mass accreted onto the black hole, we follow a procedure similar to <cit.>, using simultaneous INTEGRAL X-ray observations (only including the harder ISGRI bands, ranging from 25-200 keV) to calculate the total energy radiated (integrated X-ray luminosity) during our observations. To do this we convert the count rate into flux in the 10–1000 keV band using a power law model with photon index Γ_p∼1-2, and approximating the integral as a sum (∫ L_X dt≈∑_i L_i δ t=L̅Δ T, where L̅ is the weighted mean, δ t is size of the time bins, and Δ T is the total observation time). Assuming an accretion efficiency of 0.1, we calculate a total mass accreted during our observations of M_ acc,BH=3.4×10^-11-7.8×10^-11 M_⊙. Therefore, the mass lost in the jet is a small fraction of the total mass accreted, M_ jet=(1-3)×10^-2M_ acc,BH, and much less than the mass estimated to be lost in the accretion disc wind (∼ 1000 M_ acc,BH; ). §.§ Jet Geometry and Ejecta Size ScaleMeasurements of jet geometry in BHXBs, in particular the observed opening angle, only exist for a handful of systems, where all but one are upper limits (e.g., see Table 1 in , as well asandfor recent measurements in XTE J1752-223 and XTE J1908+094). Our simultaneous light curve modelling technique allows us to directly derive the first measurements of the jet geometry in V404 Cyg, where we model observed jet opening angles of 4.06<ϕ_ obs<9.86^∘. These measurements are consistent with the opening angle estimates for the other BHXB systems with constraints, where the majority show ϕ_ obs≲10^∘. With the opening angles, we can estimate the level of confinement of the jets in V404 Cyg by solving for the intrinsic expansion speed (using Equation 11; see last column of Table <ref>) of our modelled ejecta (β_ expc=c/√(3) indicates freely expanding components, where c/√(3) represents the speed of sound in a relativistic gas). We find intrinsic expansion speeds of0.01<β_ exp<0.1 c, indicating a highly confined jet in V404 Cyg. There are many possible mechanisms that could be responsible for confining the jets in V404 Cyg.In particular: the jet could be inertially confined <cit.>, where the ram pressure of the strong accretion disc wind detected in V404 Cyg<cit.> could inhibit the jet ejecta expansion[Although, we note that if the confinement is external, this would suggest that a very large amount of pressure surrounds the ejections. If this is supplied solely by the ram pressure from an accretion disc wind, then the mass-loss rate (proportional to the velocity ratio of the ejections to the wind) would be unrealistically large (i.e., greater then the mass accretion rate).]; the jet could be magnetically confined by a toroidal magnetic field <cit.>; the jet could contain cold protons, which may impede the jet ejecta expansion <cit.>; or a combination of these different mechanisms could be at work.Further, as we alluded to in the previous section, the initial radii of the jet ejecta (i.e., the radii of the ejecta at the time the sub-mm emission peaks)estimated by our model are noticeably smaller than those typically estimated for major ejection events in other BHXBs. This is likely a result of the much slower expansion velocities (β_ exp<<1) we find for the V404 Cyg ejecta.In particular, we infer a range of initial radii for our ejecta ranging from (0.6-1.3)× 10^12cm. These radii appear to remain similar (to within a factor of 2) between ejection events. §.§ Underlying Compact JetIn addition to the jet ejecta component, we observe an extra constant flux component in our light curves, which varies with frequency. Due to the shape of our estimated spectrum of this emission (see Figure <ref>) and the strong compact core jet present throughout the span of our simultaneous VLBA imaging (Miller-Jones et al. 2017, in prep.), we interpret this extra flux term as emission from an underlying compact jet. We believe that this compact jet was switched on during the launching of the multiple discrete ejection events. In our best fit model, this compact jet emission is characterized by a single power-law spectrum, with an optically thin spectral index of α=-0.46^+0.03_-0.03. Our suggestion of a compact jet, that has not been fully quenched, is in agreement with the findings of <cit.>, who show that V404 Cyg never fully reached a soft accretion state (where we would likely expect strong quenching of the compact jet; e.g., ), but rather remained in either a harder intermediate or very high state during our observations. Under this interpretation, based on our lowest radio frequency measurement, we place limits on the optically thick to thin jet spectral break frequency of ν_ br<5.25 GHz, and flux at the spectral break of S_ br>318 mJy. However, we note that simultaneous VLITE observations <cit.> detect V404 Cyg at a total time-averaged flux density of 186±6 mJy at 341 MHz.Given that our best fit model predicts a maximum jet ejecta flux component of ∼ 100 mJy at 341 MHz, it is clear that the 341 MHz compact jet component cannot lie along the single power-law stated above. As such, the spectral break would therefore occur within the range of 0.341<ν_ br<5.25 GHz, which is significantly lower than previous estimates for V404 Cyg made during the hard accretion state (ν_ br=1.82±0.27×10^5GHz; ). This evolution in the location of the spectral break is consistent with the pattern suggested by recent observations (e.g., ) and MHD simulations <cit.>, where, as the mass accretion rate increases during softer accretion states of BHXB outbursts (which usually occur at high luminosities; ), the jet spectral break moves toward lower radio frequencies prior to the jet switching off (or at least fading below our detection limits). Up to now we have only considered the compact jet and the ejection events as separate entities. In the presence of explosive, energetic ejection events, we might expect a compact jet to be disrupted. In particular, as the ejecta collide with the pre-existing compact jet, a shock would likely develop, due to the difference in bulk speeds between the two.In this situation, if the compact jet rapidly re-establishes itself after being destroyed by ejecta (before the ejecta propagate far enough away from the black hole to be resolved), we would observe a compact core jet which appears to never shut off. Therefore, we believe it is plausible that a compact jet is being repeatedly destroyed and re-established (on rapid timescales) following ejection events in V404 Cyg. Further, the emission from such a shock interaction could display an optically thin spectrum (similar to the interaction between the discrete ejecta and the surrounding ISM; e.g., ), like the one we observe for our baseline emission component.Thus, while we interpreted the baseline emission in our light curves as originating only from a compact jet, emission from a possible interaction of the jet ejecta with this compact jet, and/or continuous lower-level, fainter jet ejecta that never get resolved, could also be contributing to the baseline flux level we observe.Moreover, in our model we have assumed that the compact jet flux component is constant in time. However, as the accretion rate (and in turn the jet power) changes, the flux of a compact jet is expected to change as well <cit.>. If we consider the erratic X-ray behaviour observed in the source (which presumably traces a rapidly changing accretion rate), it is plausible that the compact jet component could in fact be variable as well. Exploring the possibility of a variable compact jet component in our model is left for future work.§.§ Ejecta Time LagsOur model predicts that the intrinsic time lag (in the source frame) between a certain frequency (ν) and the reference frequency (ν_0), is represented by,t_ν-ν_0, src=(R_0/β_ exp){(ν_0/ν)^p+4/4p+6-1}where the observed time lag can be obtained through the transformation, t_ν-ν_0, obs=t_ν-ν_0, src/δ_∓.Figure <ref> shows the observed time lags, predicted by our model, between each frequency band and the ejection time, for the approaching (top panel) and receding (bottom panel) components. The time lags are clearly variable between different ejection events (e.g., ∼10-30 min between the ejection and our reference frequency, 230 GHz), which is a result of the varying ejecta properties (i.e., β_ exp, p, R_0).Further, it is interesting to note that, for a different flaring event that occurred ∼ 2 days after our data set, <cit.> measured a time lag of 2.0 hours & 3.8 hours between the predicted ejection time (indicated by an r'-band polarization flare, which these authors suggest could be the signature of the launching of major jet ejection event) and the flare peaks at 16 GHz & 5 GHz, respectively. These lags are slightly higher than predicted for the approaching components of our modelled ejection events, but appear to share a similar slope across frequencies.§.§ Comparison to GRS 1915+105 GRS 1915+105 is the only other BHXB in which a similar multi-frequency variability pattern to that seen in V404 Cyg has been reported. While flaring activity has been seen in other systems, it is often only detected in one frequency band (e.g., V4641 Sgr in optical; ), or the flares in question evolved over much longer (days rather than minutes/hours) timescales (e.g., 4U 1630-47 in radio/X-ray; ). GRS 1915+105 has displayed some correlated radio, sub-mm, and IR flares (with lower frequency emission delayed from higher frequency emission), which repeated every ∼20 minutes for a ≥10 day period <cit.>. While no discrete components were resolved with VLBI during the events, the similar rise and decay times of flares at different frequencies suggest that adiabatic energy losses, likely during the expansion of discrete components, played a key role in determining the flaring profiles of these events. In fact, <cit.> found that the timing of the radio emission during these events was consistent with synchrotron emission from adiabatically expanding plasma clouds, where each event required an energy input of ∼10^39erg, and carried an estimated mass of ∼10^18g. Many studies of these jet ejection events suggest that they occur as a result of instabilities causing the repeated ejection and refilling of the inner accretion disc or coronal flow (e.g., ).In V404 Cyg our modelled ejection times appear to occur on a similar rapid timescale as seen in GRS 1915+105, where we observe groups of 2-4 ejections (separated by at most ∼20 minutes), followed by longer periods of up to ∼ 1 hour between groups (see Figure <ref>). Each group of ejections seems to correspond to a large flaring event in the light curves. Our estimates of the energetics and mass-loss of the V404 Cyg events (5.2) are also similar to those estimated for the oscillation events in GRS 1915+105, where both are consistent with being smaller-scale analogues of major ejection events seen in other BHXBs. Further, <cit.> suggested that multiple ejections in GRS 1915+105 could manifest as a single radio flare, similar to the ejection groupings we see in V404 Cyg. However, a noticeable difference in the timing of the V404 Cyg and GRS 1915+105 events is that the V404 Cyg events are not as quasi-periodic (i.e., they do not occur on as regular intervals) when compared to the GRS 1915+105 events, which occurred every ∼20 min <cit.>. The absence of quasi-periodicity in the V404 Cyg events could indicate that the jet production process is not as stable in V404 Cyg as it was during the GRS 1915+105 events.The similarity between the morphology, duration and energetics of the rapid ejection periods in V404 Cyg and GRS 1915+105 suggests that the events may have a common origin, possibly in the repeated ejection and refilling of some reservoir in the inner accretion flow. This hypothesis is consistent with the recent finding of <cit.>, who report the non-detection of the disc component in the X-ray spectra following major radio flares in V404 Cyg. Although, given the large intrinsic absorption <cit.> seen in V404 Cyg during the outburst, it is conceivable that we may not have been able to detect the soft disc component, even if it was present. Nevertheless, as both V404 Cyg and GRS 1915+105 are long period systems, with large accretion discs, a key ingredient in fuelling rapid, repeated ejection events may be a large accretion disc (as suggested by ). §.§ Alternative Emission Models Other than the vdL model, an alternative emission model that has been used to reproduce flaring light curves in XRBs is the shock-in-jet model <cit.>. This analytical model, while traditionally favoured for extragalactic sources, has been successfully applied to flaring events in Cyg X–3 <cit.>, GRO J1655-40 <cit.>, and GRS 1915+105 <cit.>. The shock-in-jet model considers a shock wave travelling downstream in a jet flow as the source of each flare in the light curve. Each shock wave will evolve through three different phases; (1) Compton losses dominate, (2) synchrotron losses dominate, and (3) adiabatic losses dominate. The main differences between the shock-in-jet model and the vdL model are that the shock-in-jet model considers an initial phase where Compton losses dominate over adiabatic losses, all shock wave events are self-similar, and the electron energy scales differently when compared to the vdL model (shock-in-jet flow expands in 2D, E∝ R^-2/3;vdL cloud expands in 3D, E∝ R^-1). These differences will result in a different flare profile between models, where for the same electron population (i.e., same p value), the shock-in-jet model flares will show a much shallower decay, and the peak fluxes at frequencies that are initially optically thin (likely IR and above) will be smaller than predicted by the vdL model (which will over predict peak fluxes at these frequencies). As our adapted vdL model is able to reproduce all our light curves (at 7 different frequencies) remarkably well, and simultaneous VLBA imagingresolves multiple, discrete components (Miller-Jones et al. 2017, in prep), we favour the expanding plasmon model over the shock-in-jet model for the V404 Cyg events (although we can not rule out the shock-in-jet model).However, for the GRS 1915+105 oscillation events, the emission has been shown to be consistent with both an expanding plasmon model (; although we note that these authors only model a single flaring event, and did not include any relativistic/projection effects in their model) and a shock-in-jet model <cit.>. If the GRS 1915+105 events are in fact a result of shock waves rather than expanding plasmons, this could explain the notable differences to the V404 Cyg events, namely the quasi-periodicity and the lack of VLBI resolved components[Although, we note that these GRS 1915+105 oscillation events were only observed with MERLIN <cit.>, which does not have the resolution to see ejection events of a few mas in size (like those of V404 Cyg).]. Additionally, as <cit.> point out, the shock-in-jet model is still consistent with the scenario that these oscillation events originated with instabilities in the inner accretion disc, as these instabilities could be the catalyst that leads to an increased injection rate of material at the base of the jet, and in turn a downstream shock wave. §.§ Connection to X-ray & OIRIf the jet ejection events in V404 Cyg are linked to processes occurring in the accretion flow, we might expect our predicted ejection times to correlate well with X-ray/OIR emission. For instance, in GRS 1915+105, IR and radio flares (which are presumably tracers of the ejection events) followed an X-ray peak and occurred during a period of spectral softening (dips in hard X-ray emission). However, the connection is not as clear in V404 Cyg. Figure <ref> displays our predicted ejection times on top of simultaneous X-ray[All X-ray data presented in this paper are taken from the INTEGRAL public data products available at http://www.isdc.unige.ch/integral/analysis#QLAsources.] <cit.> and OIR <cit.> emission. Flares in the OIR light curve appear to coincide with flares in the X-ray light curves. However, an unfortunate gap in the OIR coverage makes it difficult to confirm that such a pattern holds for the final X-ray flare.In terms of our modelled ejection times, we may be able to tentatively match groups of ejections with specific X-ray/OIR peaks, and possibly local dips in hardness (where the start/end of a steep gradient in hardness appears to correspond to ejections). But it is puzzling that the group which contains the largest number of ejections and produces the largest sub-mm flares appears to be connected to the X-ray flare with the smallest amplitude (although, if an X-ray flare is indicative of a strong dissipative process, more energy dissipated in the X-ray implies less energy would be available to the jets, and vice versa). Further, the second X-ray flare appears to have no jet ejecta counterpart.Given the extremely high intrinsic absorption during this time period <cit.>, it is entirely possible that the flaring in the X-ray light curves is not always dominated by intrinsic source variation, but rather dependent on how much of the inner accretion flow is obscured. This effect was seen in the 1989 outburst, where large changes in column density were determined to be the origin of some of the extreme X-ray variability observed <cit.>. Thus, even if the jet ejections are linked to processes in the accretion flow, we may not expect to see a clear correlation between our jet ejections and the X-ray/OIR emission. On the other hand, if the high absorption reduced the X-ray flux artificially, we would expect the high energy bands (60-200 keV) to be less affected than the lower energy bands (5-10 keV), which does not seem to be the case here. Therefore, the nature of the connection (if any) between our jet ejections and the X-ray/OIR emission is still not fully understood.§.§ The Critical Sub-mm PerspectiveTraditionally XRB jet studies have been dominated by radio frequency observations, such that there only exists a limited set of XRBs that have been observed at mm/sub-mm frequencies (e.g., ). When considering time-resolved (<1 day cadence) mm/sub-mm observations this number decreases to two (i.e., GRS 1915+105; , Cygnus X-3; ).However, our work in this paper has clearly shown the vital importance of high time resolution mm/sub-mm data in XRB jet studies. In particular, the mm/sub-mm bands can be used to isolate emission from different flaring events in the light curves, while the lower frequency counterparts of these events tend to be smoothed out and blended together.As such, we find that radio frequency observations alone can often be misleading, especially in terms of identifying and pinpointing the timing of individual rapidly variable flaring events. Including mm/sub-mm monitoring during future XRB outbursts will continue to add key insight to our understanding of jet behaviour. § SUMMARYIn this paper we present the results of our simultaneous radio through sub-mm observations of the BHXB V404 Cyg during its June 2015 outburst, with the VLA, SMA and JCMT. Our comprehensive data set, taken on 2015 June 22 (∼ 1 week following the initial detection of the outburst), extends across 8 different frequency bands (5, 7, 21, 26, 220, 230, 350, and 666 GHz).Using custom procedures developed by our team, we created high time resolution light curves of V404 Cyg in all of our sampled frequency bands. In these light curves, we detect extraordinary multi-frequency variability in the form of multiple large amplitude flaring events, reaching Jy level fluxes. Based on the overall morphology, we postulate that our light curves were dominated by emission from a relativistic jet. To understand the source of the emission we constructed a detailed jet model for V404 Cyg. Our model is capable of reproducing emission from multiple, discrete, bi-polar plasma ejection events, which travel at bulk relativistic speeds (along a jet axis inclined to the line of sight), and evolve according to the van der Laan synchrotron bubble model <cit.>, on top of an underlying compact jet.Through implementing a Bayesian MCMC technique to simultaneously fit all of our multi-frequency light curves with our jet model, we find that a total of 8 bi-polar ejection events can reproduce the emission we observe in all of our sampled frequency bands. Using our best fit model to probe the intrinsic properties of the jet ejecta, we draw the following conclusions about the ejection events in V404 Cyg: * The intrinsic properties of the jet ejecta (i.e., speeds, peak fluxes, electron energy distribution indices, opening angles) vary between different ejection events. This results in varying time lags between the flares produced by each ejection at different frequencies.* The ejecta require (minimum) energies on the order of 10^35-10^38erg. When taking into account the duration of each event, these energies correspond to a mean power into the ejection events of 10^32-10^35erg s^-1.* The ejecta carry very little mass (∼ 1% M_ acc,BH), especially when compared to that carried by the other form of outflow detected in V404 Cyg, the accretion disc wind (∼ 1000M_ acc,BH). However, despite carrying much less mass, we estimate that the ejecta carry similar energy to that of the accretion disc winds.* We place the first constraints on jet geometry in V404 Cyg, where we find that V404 Cyg contains a highly confined jet, with observed opening angles of the ejecta ranging from 4.06-9.86^∘.While we can not pin down the main jet confinement mechanism in V404 Cyg, it is possible that the ram pressure of the strong accretion disc wind detected in V404 Cyg <cit.> could contribute to inhibiting the jet ejecta expansion, and thus be a key cause of the highly confined jet in this system.* The ejecta travel at reasonably slow bulk speeds, that can vary substantially between events, on timescales as short as minutes to hours (Γ∼ 1-1.3).* Brighter ejections tend to travel at faster bulk speeds.* Our modelled ejection events appear to occur in groups of 2-4 ejections (separated by at most ∼ 20 minutes), followed by longer periods of up to ∼ 1 hour between groups. * The rapid timescale of the ejections is similar to the jet oscillation events observed in GRS 1915+105. Although the V404 Cyg events do not occur on as regular intervals as the GRS 1915+105 events, possibly suggesting the jet production process is not as stable in V404 Cyg.* We can tentatively match groups of ejections with peaks in simultaneous X-ray/OIR emission. However, the nature of the connection (if any) between our modelled ejection events and X-ray/OIR emission is still not completely clear. Based on these conclusions, it appears as though the V404 Cyg ejection events are smaller-scale analogues of major ejection events, typically seen during the hard to soft accretion state transition in BHXBs. Given the similarity between these rapid ejection events in V404 Cyg and those seen in GRS 1915+105, wepostulate that the ejection events in both systems may have a common origin, in the repeated ejection and refilling of some reservoir in the inner accretion flow. This suggests that, in agreement with the findings of <cit.> & <cit.>, the presence of a large accretion disc in both systems may be a key ingredient in producing these rare, rapid ejection events.Overall, the success of our modelling has shown that, multiple expanding plasmons, on top of a compact jet, is a good match to the emission we observe from V404 Cyg in multiple frequency bands. However, it is also apparent from our results that some simplifications within our model may not fully capture all of the physics of these ejection events (e.g., assuming equipartition, assuming a constant flux from the compact jet), and future iterations of this model will work to address these assumptions and explore their effect on the ejecta properties.In this work we have demonstrated that simultaneous multi-band photometry of outbursting BHXBs can provide a powerful probe of jet speed, structure, energetics, and geometry. Additionally, our analysis has revealed that the mm/sub-mm bands provide a critical new perspective on BHXB jets (especially in the time-domain) that can not be achieved with radio frequency observations alone. Future high time resolution, multi-band observations of more systems, including the mm/sub-mm bands, have the potential to provide invaluable insights into the underlying physics that drives jet behaviour, not only in BHXBs but across the black hole mass and power scale.§ ACKNOWLEDGEMENTSWe extend our sincere thanks to all of the NRAO, SMA, and JCMT staff involved in the scheduling and execution of these observations. Without their tireless hard work and constant support during this observing campaign we would never have obtained such an extraordinary data set.We offer a special thanks to Iain Coulson for continuing to share his JCMT expertise. We thank M. Kimura et al. for sharing their OIR data. AJT thanks Eric Koch for many helpful discussions on MCMC implementation and cloud computing, and Patrick Crumley for his helpful comments and feedback on particle acceleration mechanisms. Also, many thanks to Christian Knigge for creating the V404 mailing list and Tom Marsh for creating the V404 observations website. Both of your efforts made it much easier to organize coordinated multi-frequency observations of this outburst.AJT is supported by an NSERC Post-Graduate Doctoral Scholarship (PGSD2-490318-2016). AJT, GRS, and EWR are supported by NSERC Discovery Grants. JCAMJ is the recipient of an Australian Research Council Future Fellowship (FT140101082). SM acknowledges support from VICI grant Nr. 639.043.513/520, funded by the Netherlands Organisation for Scientific Research (NWO). TDR acknowledges support from the Netherlands Organisation for Scientific Research (NWO) Veni Fellowship, grant number 639.041.646. Cloud computing time on Amazon Web Services, used for the development and testing of our CASA timing scripts, was provided by the SKA/AWS Astro-Compute in the Cloud Program. Additionally, we acknowledge the use of Cybera Rapid Access Cloud Computing Resources, and Compute Canada WestGrid Cloud Services for this work.The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. The Sub-millimeter Array is a joint project between the Smithsonian Astrophysical Observatory and the Academia Sinica Institute of Astronomy and Astrophysics, and is funded by the Smithsonian Institution and the Academia Sinica.The James Clerk Maxwell Telescope is operated by the East Asian Observatory on behalf of The National Astronomical Observatory of Japan, Academia Sinica Institute of Astronomy and Astrophysics, the Korea Astronomy and Space Science Institute, the National Astronomical Observatories of China and the Chinese Academy of Sciences (Grant No. XDB09000000), with additional funding support from the Science and Technology Facilities Council of the United Kingdom and participating universities in the United Kingdom and Canada. The authors also wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community.We are most fortunate to have the opportunity to conduct observations from this mountain.§ IMAGE WEIGHTING SCHEMEAs we are imaging the source on very short timescales, the uv-coverage in each time-bin will be limited. While we do not need to worry about the lack of uv-coverage affecting the fidelity of the images, as the source is point-like at the VLA and SMA resolutions, the side-lobe levels may be a concern. In particular, if the amplitude is changing significantly in each time bin, this implies that we cannot deconvolve the side-lobes properly.As such, the choice of weighting scheme used while imaging could affect the quality of the images, and in turn the flux density measurements for each time bin. While the side-lobe level is not much of a concern for the VLA, which has reasonably good instantaneous uv-coverage, the SMA is only an 8-element interferometer. In this case, imaging the source with a more uniform weighting scheme minimizes the side-lobe level, and could improve the quality of the images in each time bin. On the other hand, imaging with a natural weighting scheme would maximize sensitivity, leading to lower rms noise levels. After testing different weighting schemes we find that the choice of weighting had very little effect on the output SMA light curves, where any differences in the flux measurements in each time bin were well within the rms noise. We find that the natural weighting scheme led to lower rms noise and slightly higher dynamic range in the majority of the time bin images. Therefore, we opted to use natural weighting, as the side-lobe level/rms noise trade-off appears to be optimized for natural weighting.§ CALIBRATOR LIGHT CURVESGiven the large flux variations we detected in our data of V404 Cyg, we wished to check the flux calibration accuracy of all of our observations on short time scales, and ensure that the variations we observed in V404 Cyg are dominated by intrinsic variations and not atmospheric or instrumental effects. Therefore, we ran our custom procedures to extract high time resolution measurements from our data (see 3 for details) on all of our calibrator sources. Figure <ref> displays target & calibrator light curves at all frequencies.We find that all of our interferometric calibrator sources and our JCMT 350 GHz calibrator display relatively constant fluxes throughout our observations, with any variations (<5%/<10% of the average flux density at radio/(sub)-mm frequencies) being a very small fraction of the variations we see in V404 Cyg. However, our JCMT 666 GHz calibrator scan shows noticeably larger scale variations (∼ 30% of its average flux level). While these larger variations are not unexpected at this high frequency, as the atmosphere is much more opaque, when combined with the fact that higher noise levels at this frequency prevent us from sampling timescales shorter than 60 seconds, we choose to not include the 666 GHz data set in our modelling (although see Appendix B below for a discussion of how well our best fit model agrees with the 666 GHz data).Overall, based on these results, we are confident that the high time resolution light curves of V404 Cyg used in our modelling are an accurate representation of the rapidly changing intrinsic flux of the source.§ JCMT SCUBA-2 666 GHZ MODEL COMPARISONWhile we did not include the JCMT SCUBA-2 666 GHz data in our model fitting, it is still of interest to compare our best fit model prediction for the 666GHz band to the data (see Figure <ref>). While our best fit model appears to match the timing of the flares in the 666 GHz data quite well, the model tends to over predict flux in some areas when compared to our data. It is possible that the deviations between the best fit model and the data are dominated by the higher flux calibration uncertainty in this band, especially when considering such short timescales. On the other hand, our model (and the vdL model) are only capable of predicting emission at frequencies which are initially self-absorbed (i.e. optically thick). Thus the deviations between the best fit model and the data could also suggest that the emission we observe from the jet ejecta in the 666 GHz band is initially optically thin.§ SYSTEMATIC ERRORSAs described in 4.3, we estimated additional uncertainties on our best fit parameters, to factor in how well our chosen model represents the data. Table <ref> displays these uncertainties (+ for upper confidence interval, - for lower confidence interval) for each fitted parameter. § TWO-PARAMETER CORRELATIONSWith the multi-dimensional posterior distribution output from our MCMC runs, we explored possible two-parameter correlations for our model. A significant correlation between a pair of parameters, that is common to all of the ejecta, could indicate a model degeneracy or a physical relationship between the two parameters. Out of the possible two-parameter pairs, we find interesting correlations involving the i, ϕ_ obs, F_0, and β_b parameters. Figure <ref> displays the correlation plots, along with the one-dimensional histograms of the parameters[We make use of the corner python module to make these correlation plots; https://github.com/dfm/corner.py]. The correlation between i and ϕ_ obs (first column) indicates a known degeneracy in the vdL model. The correlation between F_0 and β_b (second column) likely indicates a physical relationship between the parameters, where faster ejecta tend to have brighter fluxes. We find the same relationship when we look at the distribution of bulk speeds and fluxes across all the ejecta, and this relationship has been seen in other sources (see 5.1 for details). The final four correlations (columns 3 through 6) seem to indicate a degeneracy between all four parameters (or at least a sub-set of them), where different combinations of the four parameters could potentially produce similar flaring profiles. | http://arxiv.org/abs/1704.08726v1 | {
"authors": [
"A. J. Tetarenko",
"G. R. Sivakoff",
"J. C. A Miller-Jones",
"E. W. Rosolowsky",
"G. Petitpas",
"M. Gurwell",
"J. Wouterloot",
"R. Fender",
"S. Heinz",
"D. Maitra",
"S. B. Markoff",
"S. Migliari",
"M. P. Rupen",
"A. P. Rushton",
"D. M. Russell",
"T. D. Russell",
"C. L. Sarazin"
],
"categories": [
"astro-ph.HE"
],
"primary_category": "astro-ph.HE",
"published": "20170427193619",
"title": "Extreme Jet Ejections from the Black Hole X-ray Binary V404 Cygni"
} |
Experimental Realization of the Green-Kubo Relation in ColloidalSuspensions Enabled by Image-based Stress Measurements Itai Cohen======================================================================================================================== This paper describes the Duluth systems that participated in SemEval-2017 Task 7 : Detection and Interpretation of English Puns. The Duluth systems participated in all three subtasks,and relied on methods that included word sense disambiguation and measures of semantic relatedness. § INTRODUCTION Puns represent a broad class of humorous word play. This paper focuses on two types of puns, homographic and heterographic.A homographic pun is characterized by an oscillationbetween two senses of a single word, each of which leads toa different but valid interpretation:I'd like to tell you a chemistry joke but I'm afraid of your reaction. Here the oscillation is between two senses of reaction. The firstthat comes to mind is perhaps that of a person revealing their true feelings about something (how they react), but then the relationship tochemistry emerges and the reader realizes that reaction can also mean the chemical sense, where substances change into others. Homographic puns can also be created via compounding: He had a collection of candy that was in mint condition. The pun relies on the oscillation between the flavor mint and the compound mint condition, where candy interactswith mint and mint condition interacts with collection.A heterographic pun relies on a different kind of oscillation,that is between two words that nearly sound alike, rhyme, or are nearlyspelled the same.The best angle from which to solve a problem is the try angle.Here the oscillation is between try angle and triangle,where try suggests that the best way to solve a problem isto try harder, and triangle is (perhaps) the best kind of angle. This example illustrates one of the main challenges of heterographicpuns, and that is identifying multi word expressions that are used as akind of compound, but without being a standard or typical compound (like the very non-standard try angle).One reading treats try angle as a kind of misspelled version of triangle while the other treats them as two distinct words (try and angle). There is also a kind ofoscillation between senses here, since try angle can waver back and forth between the geometric sense and the one of making effort. During our informal study of both heterographic and homographic puns,we observed a fairly clear pattern where a punned word will occur towards the end of a sentence and has a sense that is semantically related to an earlier word, and another sense that fits the immediatecontext in which it occurs. It often seemed that the sense that fits the immediate context is a more conventional usage (as in afraid of your reaction) and the more amusing sense is that which connects to an earlier word via some type of semantic relation (chemical reaction). This is more complicated in the case of heterographic puns since the punned word can rely on pronunciation or spelling to create the effect (i.e., try angle versus triangle). In this work we focused on exploiting these long distance semantic relations, although in future work we plan to consider the use of language models to identify more conventional usages. We used two versions of the WordNet SenseRelate word sense disambiguationalgorithm[<http://senserelate.sourceforge.net>] : TargetWord <cit.> and AllWords <cit.>. Both have the goal of finding the assignment of senses in a contextthat maximizes their overall semantic relatedness <cit.> according to measures in WordNet::Similarity[<http://wn-similarity.sourceforge.net>] <cit.>. We relied on the Extended Gloss Overlaps measure (lesk)<cit.> and the Gloss vector measure(vector) <cit.>. The intuition behind a Lesk measure is that related words will be defined using some of the same words, and that recognizing these overlaps can serve as a means of identifying relationships between words <cit.>. The Extended Gloss overlap measure (hereafter simply lesk) extends this idea by considering not onlythe definitions of the words themselves, but also concatenates the definitions of words that are directly related via hypernym, hyponym, and other relations according to WordNet. The Gloss Vector measure (hereafter simply vector) extends this idea by representing each word in a concatenated definition with a vector of co-occurring words, and thencreating a representation of this definition by averaging together all of these vectors. The relatedness between two word senses can then be measured by finding the cosine between their respective vectors. § SYSTEMS The evaluation data for each subtask was individual sentences that areindependent of each other. All sentences were tokenized sothat each alphanumeric string was separated from any adjacent punctuation,and all text was converted to lowercase. Multi-word expressions(compounds) found in WordNet were identified. SemEval–2017 Task 7 <cit.> focused on pun identification, and was divided into three subtasks.§.§ Subtask 1 The problem in Subtask 1 was to identify if a sentence contains a pun (or not). We relied on the premise that a sentence will have one unambiguous assignment of senses, and that this should be true even as the parameters of a word sense disambiguation algorithm are varied. Thus, if a sentence has multiple possible assignments of senses based on the results of different runs of a word sense disambiguation algorithm, then there is a possibility that a pun exists. To investigate this hypothesis we ran theWordNet::SenseRelate::AllWords algorithm using four different configurations, and then compared the four sense tagged sentences with each other. If there were more than two differences in the sense assignments that resulted from these different runs, then the sentence is presumed to contain a pun.WordNet::SenseRelate::AllWords takes measures of semantic relatednessbetween all the pairwise combinations of words in a sentence thatoccur within a certain number of positions of each other (the window size), and assigns the sense to eachcontent word that results in the maximum relatedness among the words in that window. The assumption that underlies this method is that words in a window will be semantically related, at least toan extent, so when choices among word senses are made, those that are most related to other words in the window will be selected. The four configurations include two where the window of context is the entire sentence (a wide window) and another two where the window of context is only one word to the left and one word to the right (a narrow window). In addition these two configurations were carried out with and without compounding of words being performed prior to disambiguation. In all four configurations the Gloss Vector measureWordNet::Similarity::vector was used as the measure of semantic relatedness. If more than two sense changes result from these different configurations, then we say that a pun has occurred in the sentence.§.§ Subtask 2 In Subtask 2 the evaluation data consists of the instances fromSubtask 1 that contain puns. The task is to identify the punning word. We took two approaches to this subtask, however both were informed by our observation that punned words often occur later in sentences. The first (run 1) wasto rely on our word sense disambiguation results from Subtask 1 and identify the last word which changed senses between different runs of the WordNet::SenseRelate::AllWords disambiguation algorithm.We relied on two of the fourconfigurations used in Subtask 1. We used the narrow and wide contexts from Subtask 1 without finding compounds. We realizedthat this might cause us to miss some cases where a pun was created with a compound, but our intuition was that the more commoncases (especially for homographic puns) would be those withoutcompounds. Our second approach (run 2) was a simple baseline where the last content word in the sentence was simply assumed to be the punned word.§.§ Subtask 3 The evaluation data for Subtask 3 includes heterographic and homographic instances from Subtask 2 where the word being punned has been identified. The task is todetermine which two senses of the punned word are creating the pun.We used the word sense disambiguation algorithm WordNet::SenseRelate::TargetWord, which assigns a sense to a single word in context (whereas AllWords assigns a sense to every word in a context).However, both TargetWord and AllWords have the same underlying premise, and that is that words in a sentence should be assigned the senses that are most related to the senses of other words in that sentence. We tried various combinations of TargetWord configurations, where each would produce their own verdict on the sense of the punned word. We took the two most frequent senses assigned by these variations and used them as the sense of the punned word. Note that for the heterographic puns there was an additional step, where alternative spellings of the target word were included in the disambiguation algorithm. For example :The dentist had a bad day at the orifice. Orifice is already identified as the punned word, and one ofthe intended senses would be that of an opening, but the other is thesomewhat less obvious spelling variation office, as in a bad day at the office.For the first variation (run 1) we used both the local and global options fromTargetWord. The local option measures the semantic relatedness of the target word with all of the other members of the window of context, whereas the global option measures the relatedness among all of the words in the window of context (not just the target word). We also varied whether the lesk or vector measure was used, if a narrow or wide window was used, and if compounds were identified.We took all possible combinations of these variations, which resulted in 16 possible configurations.To thiswe added a WordNet sense one baseline with and without finding compounds, and a randomly assigned sense baseline. Thus, there were 19 variations in our run 1 ensemble. We took this approach with both the homographic and heterographicpuns, although for the heterographic puns we also replaced the target wordwith all of the words known to WordNet that differed by one edit distance.The premise of this was to detect minor misspellings that might enable a heterographic pun. For run 2 we only used the local window of context with WordNet::SenseRelate::TargetWord,but added to lesk and vector the Resnik measure (res) and the shortest path (path) measure. We carried out each of these with and without identifying compounds, which gives us a total of eight different combinations. We also tried a much more ambitious substitution method for the heterographic puns, where we queried the Datamuse API in order to find words that were rhymes, near rhymes, homonyms, spelled like, sound like, related, and means like words for the target word. This created a large set of candidate target words, and all of these were disambiguated to find out which sense of which target word was most related to the surrounding context. § RESULTS We review our results in the three subtasks in this section.Table <ref> refers to homographic results as hom and heterographic as het. Thus the first run of the Duluth systems on homographic data is denoted as Duluth-hom1, and the first run on heterographic data is Duluth-het1. The highest ranking system isindicated via High-hom and High-het. P and R as column headers stand for precision and recall, A stands for accuracy, and C is for coverage. Rank x/y indicates that this system was ranked x of y participating systems. §.§ Subtask 1 Puns were found in 71% (1,271) of the heterographic and 71% of thehomographic instances (1,607). This suggests this subtask would have a relatively high baseline performance, for example if a systemsimply predicted that every sentence contained a pun. Given this we do not want to make too strong a claim about our approach, but it does seem that focusing on sentences that have multiple possible (and valid) sense assignments is promising for pun identification. Our method tended to over-predict puns, reporting that a pun occurred in 84% (1,489 of 1,780 instances) of the heterographic data, and 80%(1,791 of 2,250 instances) of the homographic.§.§ Subtask 2 Subtask 2 consists of all the instances from Subtask 1 thatincluded a pun. This leads to 1,489 heterographic puns and 1,791 homographic. We see that our simple baseline method of choosing the last content word as the punned word (run 2) significantly outperformed ourmore elaborate method (run 1) of identifying which word experienced more changes of senses across multiple variations of the disambiguation algorithm. We can also see that run 1 did not fare very well with heterographic puns. In general we believe the difficulty that run 1 experienced was due to the overall noisiness that is characteristic of word sense disambiguation algorithms. §.§ Subtask 3 Subtask 3 consists of 1,298 homograph instances and 1,098 heterographicinstances. We see that for homographs our method fared very well, andwas the top ranked of participating systems. On the other hand ourheterographic approach was not terribly successful. We believe that the idea of generating alternative target words for heterographic puns is necessary, since without this it would be impossible to identify one of the senses of the punned word.However, our run 1 approach of simply using target word variations with an edit distance of one did not capture the variations present inheterographic puns (e.g., orifice and office have an edit distance of 2). Our run 2 approach of finding many different target words via the Datamuse API resulted in an overwhelming number of possibilities where the intended target word was verydifficult to identify. § DISCUSSION AND FUTURE WORK One limitation of our approach is the uncertain levelof accuracy of word sense disambiguation algorithms, which vary from word to word and domain to domain. Finding multiple possible senses for a single word may signal a pun or expose the limits of a particular WSD algorithm. In addition, the contexts used in this evaluation were all single sentences, and were relatively short. Whether ornot having more context available would help or hinder these approaches is an interesting question.Heterographic puns posed a host of challenges, in particular mapping clever near spellings and near pronunciations into the intended form (e.g., try angle as triangle). Simply trying to assignsenses to try angle will obviously miss the pun, and so the ability to map similar sounding phrases to the intended word is acapability that our systems were not terribly successful with. However, we were better able to identify compounds in homographic puns (e.g., mint condition) since those were written literally and could be found (if in WordNet) via a simple subsequence search. While our reliance on word sense disambiguation and semantic relatedness served us well for homographic puns, it was clearly not sufficient for heterographic. Moving forward it seems important to have a reliable mechanism to map the spelling and pronunciation variations that characterizeheterographic puns to their intended forms. While dictionaries of rhyming and sound-alike words are certainly helpful, they typically introduce too many possibilities from which to make a reliable selection. Language modeling seems like a promising way to winnow that space, so that we can get from a try angle to a triangle. natexlab#1#1[Banerjee and Pedersen(2003)]BanerjeeP03b Satanjeev Banerjee and Ted Pedersen. 2003. Extended gloss overlaps as a measure of semantic relatedness. In Proceedings of the Eighteenth International Joint Conference on Artificial Intelligence. Acapulco, pages 805–810.[Lesk(1986)]Lesk86 M.E. Lesk. 1986. Automatic sense disambiguation using machine readable dictionaries : How to tell a pine cone from an ice cream cone. In Proceedings of the 5th Annual International Conference on Systems Documentation. ACM Press, pages 24–26.[Miller et al.(2017)Miller, Hempelmann, and Gurevych]MillerHG17 Tristan Miller, Christian F. Hempelmann, and Iryna Gurevych. 2017. SemEval-2017 Task 7: Detection and interpretation of English puns. In Proceedings of the 11th International Workshop on Semantic Evaluation (SemEval-2017). Vancouver, BC.[Patwardhan et al.(2003)Patwardhan, Banerjee, and Pedersen]PatwardhanBP03 S. Patwardhan, S. Banerjee, and T. Pedersen. 2003. Using measures of semantic relatedness for word sense disambiguation. In Proceedings of the Fourth International Conference on Intelligent Text Processing and Computational Linguistics. Mexico City, pages 241–257.[Patwardhan et al.(2005)Patwardhan, Banerjee, and Pedersen]PatwardhanBP05 S. Patwardhan, S. Banerjee, and T Pedersen. 2005. SenseRelate::TargetWord - a generalized framework for word sense disambiguation. In Proceedings of the Demonstration and Interactive Poster Session of the 43rd Annual Meeting of the Association for Computational Linguistics. Ann Arbor, MI, pages 73–76.[Patwardhan and Pedersen(2006)]PatwardhanP06 S. Patwardhan and T. Pedersen. 2006. Using WordNet-based Context Vectors to Estimate the Semantic Relatedness of Concepts. In Proceedings of the EACL 2006 Workshop on Making Sense of Sense: Bringing Computational Linguistics and Psycholinguistics Together. Trento, Italy, pages 1–8.[Pedersen and Kolhatkar(2009)]PedersenK09 T. Pedersen and V. Kolhatkar. 2009. WordNet::SenseRelate::AllWords - a broad coverage word sense tagger that maximizes semantic relatedness. In Proceedings of the North American Chapter of the Association for Computational Linguistics - Human Language Technologies 2009 Conference. Boulder, CO, pages 17–20.[Pedersen et al.(2004)Pedersen, Patwardhan, and Michelizzi]PedersenPM04a T. Pedersen, S. Patwardhan, and J. Michelizzi. 2004. Wordnet::Similarity - Measuring the relatedness of concepts. In Proceedings of Fifth Annual Meeting of the North American Chapter of the Association for Computational Linguistics. Boston, MA, pages 38–41. | http://arxiv.org/abs/1704.08388v2 | {
"authors": [
"Ted Pedersen"
],
"categories": [
"cs.CL"
],
"primary_category": "cs.CL",
"published": "20170427002917",
"title": "Duluth at Semeval-2017 Task 7 : Puns upon a midnight dreary, Lexical Semantics for the weak and weary"
} |
Failsafe Mechanism Design of Multicopters Based on Supervisory Control Theory Quan Quan*, Zhiyao Zhao, Liyong Lin, Peng Wang, Walter Murray Wonham, Life Fellow, IEEE, and Kai-Yuan CaiQ. Quan, Z. Zhao P. Wang and K.-Y. Cai are with School of Automation Science and Electrical Engineering,Beihang University, Beijing 100191, China (e-mail: [email protected], [email protected], [email protected], [email protected]). L. Lin and W. M. Wonham are with the Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON M5S 3G4, Canada (e-mail: [email protected],[email protected]). The corresponding author Q. Quan is also with the Department of Electrical and Computer Engineering, University of Toronto as a visiting professor. December 30, 2023 ==================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================== In order to handle undesirable failures of a multicopter which occur in either the pre-flight process or the in-flight process, a failsafe mechanism design method based on supervisory control theory is proposed for the semi-autonomous control mode [Most multicopters have two high-level control modes: semi-autonomous control and full-autonomous control. Many open source autopilots support both modes. The semi-autonomous control mode implies that autopilots can be used to stabilize the attitude of multicopters, and also they can help multicopters to hold the altitude and position. Under such a mode, a multicopter will be still under the control of remote pilots. On the other hand, the full-autonomous control mode implies that the multicopter can follow a pre-programmed mission script stored in the autopilot which is made up of navigation commands, and also can take off and land automatically. Under such a mode, remote pilots on the ground only need to schedule the tasks <cit.>.]. Failsafe mechanism is a control logic that guides what subsequent actions the multicopter should take, by taking account of real-time information from guidance, attitude control, diagnosis, and other low-level subsystems. In order to design a failsafe mechanism for multicopters, safety issues of multicopters are introduced. Then, user requirements including functional requirements and safety requirements are textually described, where function requirements determine a general multicopter plant, and safety requirements cover the failsafe measures dealing with the presented safety issues. In order to model the user requirements by discrete-event systems, several multicopter modes and events are defined. On this basis, the multicopter plant and control specifications are modeled by automata. Then, a supervisor is synthesized by monolithic supervisory control theory. In addition, we present three examples to demonstrate the potential blocking phenomenon due to inappropriate design of control specifications. Also, we discuss the meaning of correctness and the properties of the obtained supervisor. This makes the failsafe mechanism convincingly correct and effective. Finally, based on the obtained supervisory controller generated by TCT software, an implementation method suitable for multicopters is presented, in which the supervisory controller is transformed into decision-making codes.Keywords: Multicopter, failsafe mechanism, supervisory control.§ INTRODUCTION Multicopters are well-suited to a wide range of mission scenarios, such as search and rescue <cit.>, <cit.>, package delivery <cit.>, border patrol <cit.>, military surveillance <cit.> and agricultural production <cit.>. In either pre-flight process or in-flight process, multicopter failures cannot be absolutely avoided. These failures may abort missions, crash multicopters, and moreover, injure or even kill people. In order to handle undesirable failures in industrial systems, a technique named Prognostics and Health Management (PHM) is presented <cit.>. As shown in Figure <ref>, an integrated PHM system generally contains three levels: monitoring, prediction and management <cit.>. On the one hand, the monitoring and prediction levels assess the quantitative health of the studied system, where some quantitative indices are introduced to measure system health, such as residuals <cit.>-<cit.>, data features <cit.>, <cit.> and reliability-based indices <cit.>-<cit.>. On the other hand, the management level imports the quantitative health results from the monitoring and prediction levels, and then responds to meet qualitative safety or health requirements. In our previous paper <cit.>, multicopter health is quantitatively evaluated in the face of actuator failures. This paper studies a safety decision-making logic by Supervisory Control Theory (SCT) to guarantee flight safety from a qualitative perspective.In the framework of multicopters, guidance, attitude control, PHM, and other low-level subsystems work together under the coordination of a high-level decision-making module <cit.>. In this module, a failsafe mechanism is an important part. It is a control logic that receives information from all subsystems to decide the best flight maneuver from a global perspective, and send flight instructions to low-level subsystems <cit.>. However, current academic literature covering failure-related topics of multicopters mainly focuses on fault detection techniques <cit.> -<cit.> and fault-tolerant control algorithms <cit.> -<cit.>, which belong to a study of low-level subsystems. For the study of the high-level decision-making module, most research focuses on path planning <cit.>-<cit.> and obstacle avoidance <cit.>, <cit.> of an individual multicopter, or PHM-based mission allocation of a multicopter team <cit.>, <cit.>, <cit.>. However, few studies have focused on the failsafe mechanism design of an individual multicopter subject to multiple potential failures. References <cit.>, <cit.> proposed an emergency flight planning for an energy-constrained situation. Reference <cit.> proposed a failsafe design for an uncontrollable situation. Reference <cit.> designed multiple failsafe measures dealing with different anomalies of unmanned aerial vehicles. Nevertheless, that research only considers certain ad-hoc failsafe mechanisms for certain faults or anomalies, and so far does not present a comprehensive failsafe mechanism for a multicopter. In current autopilot products (for example, DJI autopilot <cit.> and ArduPilot <cit.>), there exist comprehensive failsafe mechanisms to cope with communication, sensor and battery failures, but such mechanisms are either proprietary, or can be accessed only in part. Moreover, as far as the authors know, these failsafe mechanisms are mainly developed and synthesized according to engineering experience. As a result, such a development process lacks a theoretical foundation; this will inevitably lead to man-made mistakes, logical bugs and an incomplete treatment. Motivated by these, this paper first summarizes safety issues and user requirements for multicopters in the semi-autonomous control manner as comprehensively and systematically as possible, and then uses SCT of Discrete-Event Systems (DES) to design a failsafe mechanism of multicopters.SCT <cit.>, <cit.>, also known as Ramadge-Wonham (RW) theory, is a method for synthesizing supervisors that restrict the behavior of a plant such that as much as possible of the given control specifications are fulfilled and never violated. Currently, SCT has been developed with a solid theoretical foundation <cit.>-<cit.>, and it has been successfully applied to practical systems such as flexible manufacturing systems <cit.>-<cit.>. Thus, this paper formalizes the problem of failsafe mechanism design as a DES control problem. The solution procedure is shown in Figure <ref>. In order to obtain the expected failsafe mechanism, the following steps are performed: 1) define related modes and events by studying the user requirements (including functional and safety requirements); 2) model the multicopter plant by transforming the functional requirements to an automaton with defined modes and events; 3) analyze the safety requirements by taking the defined modes and events into account, and transform the safety requirements to automata as control specifications; 4) synthesize the supervisor by TCT software; 5) implementation the failsafe mechanism based on the obtained supervisor.The contributions of the paper mainly lie in two aspects.* First, this paper introduces SCT into a new application area. The proposed SCT based method in this paper is a scientific method with solid theoretical foundation to design the failsafe mechanism of multicopters. In the field of aircraft engineering, especially of multicopters and drones, traditional design methods are based on engineering experience. The failsafe mechanism obtained by these methods may be problematic (for example, the failsafe mechanism may contain unintended deadlocks), especially when multiple safety issues are taken into account. Compared to existing empirical design methods, the proposed method can guarantee the correctness and effectiveness of the obtained failsafe mechanism owing to the properties of supervisors. This is an urgent need for multicopter designers and manufacturers. * Second, for the application of SCT, this paper emphasizes the modeling process of the plant and control specifications with a practical application, rather than developing a new theory of SCT. We believe this work is important to both the development of SCT research and practical engineering, because SCT is presented with complex mathematical terminology and theory which many engineers may not understand. Motivated by this, this paper presents the procedure of applying SCT to an engineering problem, from requirements described textually, to specificarions in form of automata, then to a synthesized supervisor and finally to implementation on a real-time flight simulation platform of quadcopters developed by MATLAB. In addition, we present three examples to demonstrate the potential blocking phenomenon due to inappropriate design of control specifications. From the perspective of practitioners, this paper can be a guide for engineers, who are not familiar with SCT, to solve their own problems in their own projects by SCT and related software. The remainder of this paper is organized as follows. Section II presents preliminaries of SCT for the convenience of presenting the subsequent sections. Section III lists some relevant safety issues of multicopters. Also, user requirements including functional requirements and safety requirements are textually described. In order to transform the user requirements to automata, several multicopter modes and events are defined in Section IV. On this basis, a detailed modeling process of the multicopter plant and control specifications is presented in Section V, where functional requirements determine a general multicopter plant, and safety requirements are modeled as control specifications. Then, TCT software is used to perform the process of supervisor synthesis. Section VI illustrates three examples to demonstrate some possible reasons leading to a problematic supervisor, and gives a brief discussion about the scope of applications and properties of the used method. Section VII shows an implementation process of the proposed failsafe mechanism on the platform of MATLAB and FlightGear. Section VIII presents our conclusion and suggests future research.§ PRELIMINARIES OF SUPERVISORY CONTROL THEORY As SCT is well established, readers can refer to textbooks <cit.>, <cit.>, <cit.> for detailed background and knowledge. This section only reviews some basic concepts and notation.In RW theory <cit.>, <cit.>, the formal structure of DES is modeled by an automaton (generator)𝐆=(Q,Σ,δ,q_0,Q_m)where Q is the finite state set; Σ is the finite event set (also called an alphabet); δ:Q×Σ→ Q is the (partial) transition function; q_0∈ Q is the initial state; Q_m⊆ Q is the subset of marker states. Let Σ^∗ denote the set of all finite strings, including the empty string ϵ. In general, δ is extended to δ:Q×Σ^∗→ Q, and we write δ(q,s)! to mean that δ(q,s) is defined. The closed behavior of 𝐆 is the languageL(𝐆)={s∈Σ^∗|δ( q,s)!}and the marked behavior isL_m(𝐆)={s∈ L(𝐆) |δ(q_0,s)∈ Q_m}⊆ L( 𝐆).A string s_1 is a prefix of a string s, written s_1⩽ s, if there exists s_2 such that s_1s_2=s. The prefix closure of L_m(𝐆) is L_m(𝐆 ):={s_1∈Σ^∗|(∃ s∈ L_m( 𝐆))s_1⩽ s}. We say that 𝐆 is nonblocking if L_m(𝐆) =L(𝐆). The three equivalent meanings of “nonblocking” are 1) the system can always reach a marker state from every reachable state; 2) every string in the closed behavior can be extended to a string in the marked behavior; 3) every physically possible execution can be extended to completing distinguished tasks.The usual way to combine several automata into a single, more complex automaton is called synchronous product. For two automata 𝐆_i=(Q_i,Σ_i,δ_i,q_0,i,Q_m,i) ,i=1,2, the synchronous product 𝐆=(Q,Σ,δ ,q_0,Q_m) of 𝐆_1 and 𝐆_2, denoted by 𝐆_1‖𝐆_2, is constructed to have marked behavior L_m(𝐆)=L_m(𝐆_1)‖ L_m(𝐆_2) and closed behavior L( 𝐆)=L(𝐆_1)‖ L( 𝐆_2) <cit.>. The synchronous product of more than two automata can be constructed similarly.For a practical system, the plant can be modeled as an automaton 𝐆. The desired behavior of the controlled system is determined by a control specification, also modeled as an automaton 𝐄. Both the plant 𝐆 and the control specification 𝐄 may be the synchronous product of many smaller components.For supervisory control, the alphabet Σ is partitioned asΣ=Σ_c∪̇Σ_uwhere Σ_c is the subset of controllable events that can be disabled by an external supervisor, and Σ_u is the subset of uncontrollable events that cannot directly be prevented from occurring. Here Σ_c and Σ_u are disjoint subsets. A supervisory controller (supervisor) forces the plant to respect the control specification by disabling certain controllable events that are originally able to occur in the plant.To synthesize a satisfactory supervisor, SCT provides a formal method for theoretically solving the typical supervisory control problem <cit.>: Given a plant 𝐆 over alphabet Σ=Σ_c∪̇ Σ_u and control specification 𝐄, find a maximally permissive supervisor 𝐒 such that the controlled system 𝐒/𝐆 is non-blocking and meets the control specification 𝐄. That is 𝐒 satisfies[ L_m(𝐒)=sup𝒞(𝐄∩ L_m(𝐆))⊆ L_m(𝐆 ); L_m(𝐒)=L(𝐒) ]where sup𝒞(L) means the supremal controllable sublanguage of L. Equation (<ref>) means that the supervisor 𝐒 never violates the control specification 𝐄. Here, 𝐒 is a monolithic (namely fully centralized) supervisor <cit.>. If there exist several control specifications, the supervisor can be also designed in a decentralized framework. Decentralized supervisory control assigns a separate specialized supervisor to satisfy each control specification 𝐄_j. For each control specification 𝐄_j, a decentralized supervisor 𝐒_j is computed in the same way as for a monolithic supervisor. Then, all the decentralized supervisors work together to meet the control specification 𝐄=𝐄_1‖𝐄_2‖⋯. Here, if the synthesized supervisors are blocking, a coordinator is required to make the supervisors nonblocking. The main advantage of the decentralized supervisory control framework is that the synthesized supervisors are relatively small-scale, and are easier to understand, maintain and change.Related algorithms in DES and SCT can be performed on software platforms such as TCT software <cit.>, Supremica <cit.> and Discrete Event Control Kit written in MATLAB <cit.>.§ SAFETY ISSUES AND USER REQUIREMENTS This section lists some relevant safety issues of multicopters. Also, user requirements including functional requirements and safety requirements are textually described. §.§ Safety issues Major types of multicopter failures that may cause accidents will be introduced. Here, three types of failures are considered, including communication breakdown, sensor failure and propulsion system anomaly.* Communication breakdown. Communication breakdown mainly refers to a contact anomaly between the Remote Controller (RC) transmitter and the multicopter, or between the Ground Control Station (GCS) and the multicopter. In this paper, for simplicity, only RC is considered. * Sensor failure. Sensor failure mainly implies that a sensor on the multicopter cannot accurately measure related variables, or cannot work properly. This paper considers the sensor failures including barometer failure, compass failure, GPS failure, Inertial Navigation System (INS) failure. * Propulsion system anomaly. Propulsion system anomaly mainly refers to battery failure and propulsor failure caused by Electronic Speed Controllers (ESCs), motors or propellers. More information about safety issues can be found in the book <cit.>. §.§ User requirements From the commercial perspective of customers and users, a multicopter product is required to have general functions as a rotorcraft, and also be capable of coping with the relevant safety issues. Thus, functional requirements and safety requirements are listed in Tables 1-4, respectively. They are summarized by referring the material from <cit.> and the authors' knowledge and engineering experience.§.§.§ Functional requirements The following functional requirements describe what behavior the multicopter is able to perform.Table 1. Functional requirementsc]|c|c| NameDescription 1|l| FR1 1|l| The remote pilot can arm the multicopter by the RC transmitter and then allow it to take off. 1|l| FR2 1|l| After taking off, the remote pilot can manually switch the multicopter to fly normally, return to the 1|l| 1|l| base or land automatically by the RC transmitter. 1|l| FR3 1|l| The remote pilot can manually control the multicopter to land and disarm it by the RC transmitter. 1|l| FR4 1|l| When the multicopter is flying, the multicopter can realize spot hover, altitude-hold hover and attitude 1|l| 1|l| self-stabilization.1|l| FR5 1|l| When the multicopter is flying, the multicopter can automatically switch to returning to the base or land.Arm is the instruction that the propellers of the multicopter be unlocked; in this case, the multicopter can take off. Correspondingly, disarm is the instruction that the propellers of the multicopter be locked; in this case, the multicopter cannot take off.§.§.§ Safety requirements The safety requirements restrict what action the user wants the multicopter to perform under specific situations when it is on the ground, in flight, or in process of returning and landing.Table 2. Safety requirements on groundc]|c|c| NameDescription 1|l| SR1 1|l| When the remote pilot tries to arm the multicopter, if the INS and propulsors are both healthy, 1|l| 1|l| the connection to RC transmitter is normal, and the battery's capacity is adequate, then the 1|l| 1|l| multicopter can be successfully armed and take off. Otherwise, the multicopter cannot be armed.Table 3. Safety requirements in flightc]|c|c| NameDescription 1|l| SR2 1|l| If the multicopter is already on the ground, the multicopter can be manually disarmed by the1|l| 1|l| RC transmitter, or automatically disarmed if no instruction is sent to the multicopter by 1|l| 1|l| the RC transmitter.1|l| SR3 1|l| When the multicopter is flying, if the GPS or compass is unhealthy, the multicopter can only 1|l| 1|l| realize altitude-hold hover rather than spot hover. If the barometer is unhealthy, the multicopter1|l| 1|l| can only realize attitude self-stabilization. If the corresponding components are recovered, the 1|l| 1|l| multicopter should switch to an advanced hover status. 1|l| SR4 1|l| When the multicopter is flying and the connection to the RC transmitter becomes abnormal, if the 1|l| 1|l| INS, GPS, barometer, compass and propulsors are all healthy, the multicopter should switch to 1|l| 1|l| returning to the base. Otherwise, the multicopter should switch to landing. 1|l| SR5 1|l| When the multicopter is flying, if the battery's capacity becomes inadequate but the multicopter is 1|l| 1|l| able to return to the base, then the multicopter should switch to returning to the base; if the battery's 1|l| 1|l| capacity becomes inadequate and unable to return, then the multicopter should switch to landing. 1|l| SR6 1|l| When the multicopter is flying, if the INS or propulsors are unhealthy, the multicopter 1|l| 1|l| should automatically switch to landing. 1|l| SR7 1|l| When the multicopter is flying, the multicopter can be manually switched to returning to the base by 1|l| 1|l| the RC transmitter. This switch requires that the INS, GPS, barometer, compass, propulsors are 1|l| 1|l| all healthy, and the battery's capacity is able to support the multicopter to return to the base.1|l| 1|l| Otherwise, the switch cannot occur. 1|l| SR8 1|l| When the multicopter is flying, the multicopter can be manually switched to automatically 1|l| 1|l| landing by the RC transmitter.Table 4. Safety requirements on returning and landingc]|l|l| NameDescriptionSR9When the multicopter is in the process of returning to the base, the multicopter can be manually switched to normal flight or landing by the RC transmitter.SR10When the multicopter is in the process of returning to the base, if the distance to the base is less than a given threshold, the multicopter should switch to landing; if the battery's capacity becomes inadequate and unable to return to the base, the multicopter should switch to landing; if the INS, GPS, barometer, compass or propulsors are unhealthy, the multicopter should switch to landing.SR11When the multicopter is in the process of landing, the multicopter can be manually switched to normal flight by the RC transmitter. This switch requires that the INS and propulsors are both healthy, the connection to the RC transmitter is normal, and the battery's capacity is adequate. Otherwise, the switch cannot occur.SR12When the multicopter is in the process of landing, the multicopter can be manually switched to returning to the base by the RC transmitter. This switch requires that the INS, GPS, barometer, compass, propulsors are all healthy, the battery's capacity is able to support the multicopter to return to the base, and the distance to the base is not less than a given threshold. Otherwise, the switch cannot occur.SR13When the multicopter is in the process of landing, if the multicopter's altitude is lower than a given threshold, or the multicopter's throttle is less than a given thresholdover a time horizon, the multicopter can be automatically disarmed.§ MULTICOPTER MODE AND EVENT DEFINITION In order to transform the user requirements to automata, several multicopter modes and events are defined in this section. §.§ Multicopter mode Referring to <cit.>, the whole process from taking off to landing of multicopters is divided into eight multicopter modes. They form the basis of the failsafe mechanism.* POWER OFF MODE. This mode implies that a multicopter is out of power. In this mode, the remote pilot can (possibly) disassemble, maintain and replace the hardware of a multicopter. * STANDBY MODE. When a multicopter is connected to the power module, it enters a pre-flight status. In this mode, the multicopter is not armed, and the remote pilot can arm the multicopter manually. Afterwards, the multicopter will perform a safety check and then transit to the next mode according to the results of the safety check. * GROUND-ERROR MODE. This mode indicates that the multicopter has a safety problem. In this mode, the buzzer will turn on an alarm to alert the remote pilot that there exist errors in the multicopter. * LOITER MODE. Under this mode, the remote pilot can use the control sticks of the RC transmitter to control the multicopter. Horizontal location can be adjusted by the roll and pitch control sticks. When the remote pilot releases the control sticks, the multicopter will slow to a stop. Altitude can be controlled by the throttle control stick. The heading can be set with the yaw control stick. When the remote pilot releases the roll, pitch and yaw control sticks and pushes the throttle control stick to the mid-throttle deadzone, the multicopter will automatically maintain the current location, heading and altitude. * ALTITUDE-HOLD MODE. Under this mode, a multicopter maintains a consistent altitude while allowing roll, pitch and yaw to be controlled normally. When the throttle control stick is in the mid-throttle deadzone, the throttle is automatically controlled to maintain the current altitude and the attitude is also stabilized but the horizontal position drift will occur. The remote pilot will need to regularly give roll and pitch commands to keep the multicopter in place. When the throttle control stick goes outside the mid-throttle deadzone, the multicopter will descend or climb depending upon the deflection of the control stick. * STABILIZE MODE. This mode allows a remote pilot to fly the multicopter manually, but self-levels the roll and pitch axes. When the remote pilot releases the roll and pitch control sticks, the multicopter automatically stabilizes its attitude but position drift may occur. During this process, the remote pilot will need to regularly give roll, pitch and throttle commands to keep the multicopter in place as it might be pushed around by wind. * RETURN-TO-LAUNCH (RTL) MODE. Under this mode, the multicopter will return to the base location from the current position, and hover there. * AUTOMATIC-LANDING (AL) MODE. In this mode, the multicopter realizes automatic landing by adjusting the throttle according to the estimated height [Even if the barometer fails, the height estimation is acceptable within a short time. Similarly, the other estimates generated by filters could continue to be used for a short time, even if related sensors fail.].§.§ Event definition Three types of events are defined here: Manual Input Events (MIEs), Mode Control Events (MCEs) and Automatic Trigger Events (ATEs). The failsafe mechanism detects the occurrence of MIEs and ATEs, and uses MCEs to decide which mode the multicopter should stay in or switch to. Here, MIEs and MCEs are controllable, while ATEs are uncontrollable in the sense of SCT.§.§.§ MIEs MIEs are instructions from the remote pilot sent through the RC transmitter. This part defines eight MIEs as shown in Table 5.Table 5. MIE definitionc]|c|l| NameDescription MIE1Turn on the power. MIE2Turn off the power MIE3Execute arm action. This action is realized by manipulating the sticks of the RC transmitter. MIE4Execute disarm action. MIE5Other actions manipulated by the sticks of the RC transmitter. These actions correspond to normal operations by the remote pilot. Here, no manipulation on the sticks is also inclusive. MIE6Switch to normal flight. In normal flight, the multicopter can be in either LOITER MODE, ALTITUDE-HOLD MODE or STABILIZE MODE. MIE7Switch to RTL MODE. MIE8Switch to AL MODE.Here, MIE6, MIE7 and MIE8 are realized by a three-position switch (namely the flight mode switch) on the RC transmitter as shown in Figure <ref>.§.§.§ MCEs MCEs are instructions from multicopter's autopilot. As shown in Table 6, these events will control the multicopter to switch to a specified mode.Table 6. MCE definitionc]|c|c| NameDescription MCE1 1|l| Multicopter switched to POWER OFF MODE. MCE2 1|l| Multicopter switched to STANDBY MODE. MCE3 1|l| Multicopter switched to GROUND-ERROR MODE. MCE4 1|l| Multicopter switched to LOITER MODE. MCE5 1|l| Multicopter switched to ALTITUDE-HOLD MODE. MCE6 1|l| Multicopter switched to STABILIZE MODE. MCE7 1|l| Multicopter switched to RTL MODE. MCE8 1|l| Multicopter switched to AL MODE.§.§.§ ATEs ATEs are independent of the remote pilot's operations. As shown in Table 7, these events contain the health check results of onboard equipment and sensor measurements of the multicopter status.Table 7. ATE definitionc]|c|c| NameDescription ATE1 1|l| The check result of INS is healthy. ATE2 1|l| The check result of INS is unhealthy. ATE3 1|l| The check result of GPS is healthy. ATE4 1|l| The check result of GPS is unhealthy. ATE5 1|l| The check result of barometer is healthy. ATE6 1|l| The check result of barometer is unhealthy. ATE7 1|l| The check result of compass is healthy. ATE8 1|l| The check result of compass is unhealthy. ATE9 1|l| The check result of propulsors is healthy. ATE10 1|l| The check result of propulsors is unhealthy. ATE11 1|l| The check result of connection to the RC transmitter is normal. ATE12 1|l| The check result of connection to the RC transmitter is abnormal. ATE13 1|l| The measured battery's capacity is adequate. ATE14 1|l| The measured battery's capacity is inadequate, able to RTL. ATE15 1|l| The measured battery's capacity is inadequate, unable to RTL. ATE16 1|l| The measured multicopter's altitude is lower than a given threshold. ATE17 1|l| The measured multicopter's altitude is not lower than a given threshold. ATE18 1|l| The measured multicopter's distance from the base is less than a given threshold. ATE19 1|l| The measured multicopter's distance from the base is not less than a given threshold.ATE20 1|l| The measured multicopter's throttle is less than a given threshold over a time horizon.ATE21 1|l| Other throttle situation.Here, note that this paper assumes the health check of equipment above can be performed by effective fault diagnosis and health evaluation methods. For simplified presentation, the statements of “check result of” and “measured” are omitted in the subsequent sections.Remark 1. MCEs are defined to guarantee the controllability of the plant, because supervisory control restricts the behavior of a plant such that the given control specifications are fulfilled and as much as possible never violated, by enabling or disabling controllable events in the plant. According to safety requirements, the user declares which mode the multicopter should enter. This leads to the definition of controllable events related to mode transitions, namely MCEs.§ FAILSAFE MECHANISM DESIGN The failsafe mechanism uses multicopter modes and switch conditions among them to make multicopters satisfy the user's safety requirements. In this section, functional requirements are used to model a multicopter plant automaton with defined multicopter modes and events. Then, from the safety requirements, multiple control specifications are represented by automata. These control specifications should indicate the preferable failsafe measures consistent with the textually described safety requirements. After the plant and control specifications have been obtained, the supervisor is synthesized by using monolithic supervisory control. §.§ Multicopter plant modeling §.§.§ Modeling principles Modeling the multicopter plant is to mathematically describe what behavior the multicopter is able to perform with an automaton transformed from functional requirements. In this paper, the modeling principles of the multicopter plant include: i) modeling from a simple schematic diagram to a comprehensive automaton model; 2) modeling from the `on ground' component to the `in air' component; 3) events of each transition modeled mutually exclusively. Figure <ref> depicts a schematic diagram of the `on ground' component and `in air' component of the multicopter plant, respectively. The schematic diagram lists all modes which the multicopter possibly enters, after a series of MIEs and ATEs occur.§.§.§ Model details By extending the above schematic diagrams with detailed events and transitions, the plant is described by an automaton as shown in Figure <ref>. It describes the basic function of a multicopter. Specifically, Plant contains 27 states (S_0-S_26), 37 events and 63 transitions. Here, the states S_0,S_3,S_13,S_14 are marked as accepting states. The state S_0 represents POWER OFF MODE; the state S_3 represents STANDBY MODE; the state S_13 represents GROUND-ERROR MODE; the state S_14 integrates other multicopter modes. Plant can be divided into two parts: one (consists of states S_0-S_14 and transitions among them) describes the multicopter behavior on the ground (`on ground' component), and the other one (consists of states S_3,S_12-S_26 and transitions among them) describes the behavior during flight (`in air' component). These correspond to the schematic diagram shown in Figure <ref>. §.§ Control specification design §.§.§ Modeling principle In this part, control specifications are designed to restrict the behavior of Plant according to the description of the safety requirements. In order to obtain a correct and non-blocking supervisor, the control specifications must cover all possible strings (enable desirable strings and disable the others) in the plant, and the control specifications must have no conflict among themselves.§.§.§ Control specification design `on ground' Through a study of safety requirements, it can be seen that SR1 describes the intended failsafe measure when the multicopter is on the ground. In other words, SR1 restricts what action the user wants the multicopter to perform under specific situations when it is on the ground. Thus, we design a control specification to cover all possible strings in the `on ground' component of Plant. The requirements given in Tables 1-3 are different from the designed specifications. The former is textually and informally described, whereas, based on which, the latter is designed formally described in form of automaton. Several requirements may be described by one specification, or one requirement may be described by several specifications.In safety requirement SR1, the user lists the required conditions for a successful arm. In order to model it with an automaton, the key is to split the branches in the `on ground' component of Plant, and enable only one mode which the user expects the multicopter to switch to. Following this principle, a control specification named Specification 1 is designed as shown in Figure <ref>. It contains 8 states (S_0-S_7), 24 events and 68 transitions. Here, the states S_0,S_1 are marked as accepting states. The state S_1 represents STANDBY MODE, and the state S_0 integrates other multicopter modes. Here, two points need to be noted:i) The selfloops on the state S_0,S_4,S_6 are used to guarantee that the irrelevant events will not interrupt the event sequence presented in Plant, and not influence the occurrence of other control specifications.ii) SR1 itself is textually and informally described. It does not mention which mode the multicopter should enter, if it cannot be successfully armed. Furthermore, it does not take all possible strings into consideration. In this case, during the design of control specifications, it is required to appropriately infer the user's potential intention, and add the omitted part to guarantee that the control specification covers all possible strings in the `on ground' component of Plant.§.§.§ Control specification design `in ground' (Specification 7) For the `in air' component of Plant, safety requirements SR2-SR13 restrict what action the user wants the multicopter to perform under specific situations when it is in air. Thus, we design 24 control specifications to cover all possible strings in the `in air' component of Plant. The traversal relation between the designed control specifications and the structure of the `in air' component of Plant is shown in Figure <ref>. Here, because of limitation of space, we take Specification 7 as an example to demonstrate the design of control specifications for the `in air' component of Plant. This control specification is obtained by transforming “safety requirements SR7 and SR8” to an automaton model. As shown in Figure <ref>, Specification 7 contains 6 states (S_0-S_5), 31 events and 91 transitions. Here, the states S_0,S_1 are marked as accepting states. The state S_1 represents LOITER MODE, and the state S_0 integrates other multicopter modes. The details of other control specifications are presented in the support material available in http://rfly.buaa.edu.cn/resources.§.§ Supervisor synthesis on TCT software The algorithms and operations in this part are performed on TCT software. In order to synthesize a supervisor by TCT software, the modeled multicopter plant and designed control specifications are first input. The multicopter plant is named as “PLANT ”, and the 25 control specifications are named as “𝐄_j”, j=1,2,⋯,25. The input process is shown in http://rfly.buaa.edu.cn/resources.§.§.§ Control specification completion Here, note that PLANT contains 37 events, while the number of events in each 𝐄_j is less than 37 (i.e. the alphabet of each 𝐄_j is different from that of PLANT). This is because the given textual safety requirements only emphasize the events we are concerned with and ignore the remaining events. For supervisory control, the alphabet of each 𝐄_j should be equal to the alphabet of PLANT. Thus, the control specification should be completed by the following TCT instructions:𝐄𝐕𝐄𝐍𝐓𝐒=𝐚𝐥𝐥𝐞𝐯𝐞𝐧𝐭𝐬(𝐏𝐋𝐀𝐍𝐓)where 𝐄𝐕𝐄𝐍𝐓𝐒 is a selfloop automaton containing all events in the alphabet of PLANT. Then, for each 𝐄_j, we have𝐄_j=𝐬𝐲𝐧𝐜(𝐄_j,𝐄𝐕𝐄𝐍𝐓𝐒).Here, the events present in PLANT but not in 𝐄_j are added into 𝐄_j in form of selfloops.§.§.§ Supervisor synthesis In the monolithic supervisory control framework, all the control specifications should be synchronized into a monolithic one. That is𝐄=𝐬𝐲𝐧𝐜(𝐄_1,𝐄_2,⋯𝐄 _25).It turns out that 𝐄 is nonblocking, and contains 133 states and 2219 transitions. Then, a monolithic supervisor is synthesized by𝐒=𝐬𝐮𝐩𝐜𝐨𝐧(𝐏𝐋𝐀𝐍𝐓,𝐄).The obtained supervisor is the expected failsafe mechanism. It contains 784 states, 37 events and 1554 transitions. There are 8 accepting states to be marked, which correspond respectively to 8 multicopter modes. Besides the monolithic supervisory control, the supervisor can also be synthesized by decentralized supervisory control, and a supervisor reduction process can also be carried out for an easier realization in practice. The synthesis is also carried out in the software Supremica with the result same to TCT. These source files are presented in http://rfly.buaa.edu.cn/resources.§ EXAMPLES AND DISCUSSION This section illustrates three examples to demonstrate some possible reasons leading to a problematic supervisor, and gives a brief discussion about the scope of applications and properties of the method. §.§ Examples The design of control specifications is a process to understand and re-organize the safety requirements. If the designer synthesizes a blocking supervisor, he must recheck the correctness of control specifications and make modifications. Here, we illustrate three examples demonstrating the blocking phenomenon due to inappropriate design of control specifications and conflicting safety requirements.Example 1. The aim of this example is to show that missing information in control specification may lead to a blocking supervisor. In this example, we delete transitions “S_6→ATE13→S_6”, “S_6→ATE14→S_6” and “S_6→ATE15→S_6” in Specification 1. In this case, Specification 1 is changed to an automaton named Example 1 as shown in Figure <ref>. By replacing Specification 1 with Example 1, the supervisor is synthesized and turns out to be blocking. The blocking branch is depicted in Figure <ref>. The reason is that blocking occurs owing to the missing selfloops at state S_6 in Example 1. The missing selfloops make the automaton “think” that events ATE13, ATE14 and ATE15 will not occur at state S_6, while these events should occur in Plant. Thus, a blocking supervisor is synthesized. This means an uncertainty as to what should occur in the blocking point. Example 2. The aim of this example is to show that conflict in control specifications will lead to a blocking supervisor. In this example, we replace the transition “S_6→MCE2 →S_1” with a transition “S_6→MCE3→S_1” in Specification 1. In this case, Specification 1 is changed to an automaton named Example 2 as shown in Figure <ref>. By adding Example 2 to the whole control specification, the supervisor is synthesized and turns out to be blocking. The blocking branch is depicted in Figure <ref>. The reason that blocking occurs is the conflict between Specification 1 and Example 2. Specification 1 indicates a transition “S_6→MCE2→S_1”, while Example 2 has a transition “S_6→MCE3→S_1”. This conflict will “confuse” the supervisor, and make it impossible to decide which transition should occur. Thus, a blocking supervisor is synthesized. Example 3. The aim of this example is to show that conflict in user requirements will lead to a blocking supervisor. Assume we have a new safety requirement described as follows: “when the multicopter is flying, the multicopter can be manually switched to return to the base by the RC transmitter. This switch requires that the INS, GPS, barometer, compass and propulsors are all healthy. Otherwise, the switch cannot occur.” Then, this safety requirement is transformed to an automaton as shown in Figure <ref>. By adding Example 3 to the whole control specification, the supervisor is synthesized and turns out to be blocking. The blocking branch is depicted in Figure <ref>. The reason that blocking appeared is the conflict between the original SR7 and the newly presented safety requirement. In SR7, it indicates that “this switch requires that the INS, GPS, barometer, compass, propulsors are all healthy, and the battery's capacity is able to support the multicopter to return to the base”. However, the new safety requirement does not restrict the condition of battery's capacity. As in Example 2, this conflict leads to a blocking supervisor. Remark 2. From the above examples, it can be seen that an incorrect failsafe mechanism might be obtained during the design process due to conflicting safety requirements or incorrect and inappropriate design of control specifications. The mistake might be introduced inadvertently, and the designer cannot easily detect the problem by using empirical design methods. However, by relying on the SCT-based method, we can check the correctness of the obtained failsafe mechanism, and make modifications if a problematic supervisor is generated. This is a big advantage of the proposed method over empirical design methods. Once a nonblocking supervisor is obtained, the resulting failsafe mechanism is logically correct, and able to deal with all relevant safety issues appearing during flight. §.§ Discussion This paper aims to study a method to guarantee the correctness in the design of the failsafe mechanism. Actually, correctness can be interpreted in two different ways. On the one hand, correctness can be explained as absolute safety, meaning that the multicopter can cope with all possible safety problems. On the other hand, correctness is defined as consistency between the obtained failsafe mechanism and safety requirements. Given a model and a control specification for an autonomous system, synthesis approaches can automatically generate a protocol (or strategy) for controlling the system that satisfies or optimizes the property. This process is named as “correct-by-design”<cit.>. In this domain, various formal methods and techniques, such as SCT and linear temporal logic, are used to design control protocol of autonomous systems, including autonomous cars <cit.>, aircraft <cit.> -<cit.> and swarm robots <cit.>. Similar to the above literature, in this paper, we focus on the meaning of correctness that all requirements can be correctly satisfied. With a precise description of both the multicopter and its correct behavior, the proposed method allows a failsafe mechanism that guarantees the correct behavior of the system to be automatically designed.Here, the generated supervisor by SCT satisfies the following properties:i) Deterministic. This property has two aspects. First, there exists no situation that one event triggers a transition from a single source state to different target states in the obtained supervisor. This is a necessary condition for a deterministic automaton. However, this situation might occur due to man-made mistakes in an empirical failsafe mechanism design. Second, after occurrence of MIEs and ATEs, SCT can guarantee that only one MCE is enabled by disabling other MCEs due to deliberate design of control specifications. In this case, after occurrence of certain MIEs and ATEs, the mode which the multicopter should enter is deterministic.ii) Nonconflicting <cit.>. None of the control specifications conflict with any others. If there exist conflicts, the supervisor will not be successfully synthesized, because SCT cannot decide which control specification is the user's true intention. If so, the designer should check 1) the correctness of control specifications transforming from user requirements; or 2) the reasonableness of the user requirements.iii) Nonblocking. The generated supervisor is nonblocking, which can be interpreted that all possible strings in Plant are considered (either enabled or disabled) in the supervisor. If there are some strings which are not considered in the control specifications, the marker states may not be reached in some branches from the initial state. Then, the obtained supervisor might be incomplete (even empty). This is because SCT cannot compute control due to incorrect user's specifications. If so, the designer should modify the control specifications to make them consistent.iv) Logical correctness. SCT is a mature and effective tool to be used in the area of decision-making. If the plant and control specifications are correctly modeled, the logic of the generated supervisor will correctly satisfy user requirements without introducing man-made mistakes and bugs.§ IMPLEMENTATION AND SIMULATION Based on the obtained supervisory controller generated by TCT software or Supremica, an implementation method suitable for multicopter is presented, in which the supervisory controller is transformed into decision-making codes. §.§ Failsafe mechanism implementation On the one hand, we would want to avoid manual implementation of the calculated supervisors, since this may introduce errors and is also difficult for a complex case. On the other hand, we expect an easy way to generate an Application Programming Interface (API) function with events as the input and marked states as output so that it can be easily integrated into the existing program in flight boards. The information required from a synthesized supervisor is a transition matrix, which is an m×3 matrix where m is the number of transitions in the synthesized supervisor (We have developed a function to export the transition matrix based on the output file of Supremica, available in http://rfly.buaa.edu.cn/resources). As shown in Table 8, in each row, it consists of a source state, a destination state and a triggered event. For example, if the multicopter is in source state 1 and the triggered event is 1, then the destination state will be 2. By taking the synthesized supervisor of multicopters as an example, it contains 784 states, 37 events and 1554 transitions. So, the transition matrix is an 1554×3 matrix. In fact, we only need to consider 8 accepting states, namely POWER OFF MODE, STANDBY MODE, GROUND-ERROR MODE, LOITER MODE, ALTITUDE-HOLD MODE, STABILIZE MODE, RTL MODE and AL MODE. Based on them, corresponding low-level control actions exist. However, there exist many intermediate states in the transition matrix (784-8=776 intermediate states for the considered multicopter), to which no control actions correspond. Therefore, after one decision period, the system must be in an accepting state. This is a major problem we need to solve. Fortunately, this is always true.Table 8. Transition matrixc]|c|c|c| Source stateDestination stateTriggered Event 1 2 1 ⋮ ⋮ ⋮ 2 3 3In practice, the events will be detected every 0.01s for example, while the decision period may be 1s. All triggered events are collected in every decision period. By recalling Figure <ref>, since the events in every transition are mutually exclusive, one and only one event must be triggered for any transition. As a result, the system does not stop at intermediate states after feeding in all detected events. For example, by recalling the `in air' component in Figure 5, if the initial state is S14 and we collect the events MIE5, MIE6, ATE1, ATE3, ATE5, ATE7, ATE9, ATE11, ATE13, ATE16, ATE18, ATE20, then the system will go to S26 in Plant. Consequently, only one MCEi will be enabled by the autopilot according to the specifications. Therefore, the system will stop at an accepting state finally. For our case, the failsafe mechanism is implemented as shown in Table 9, where Δ>0 represents a decision-making time interval. Actually, the high-level decision-making should be a relatively slow process in practice. Thus, the failsafe mechanism implementation is not synchronized with the low-level flight control system.Table 9. Decision-making logic implementationc]|c|l| StepDescription1.Export a transition matrix from the supervisor synthesized by TCT software or Supremica; k=0;Δ>0 is a positive integer representing a decision-making time interval; the initial state s=s_0.2. k=k+13.Detect the instruction from the RC transmitter, health status of all considered equipments and flight status of the multicopter. If mod( k,Δ)=0, go to Step 4; Otherwise, go to Step 2.4.Generate an event set occurred in the decision-making time interval Δ .5.By starting at state s with the events inputed according to the occurrence order in Plant one by one, search the transition matrix when an event is inputed. After all the MIEs and ATEs are inputed completely, search the transition matrix again, and only one match will be found, where the triggered event is an MCE and the destination state is s_1 .6. s=s_1, go to Step 2. §.§ Simulation In this part, we put the failsafe mechanism into a real-time flight simulation platform of quadcopters developed by MATLAB. Although it is realized by MATLAB, this method is applicable to any programming language. The simulation diagram is shown in Figure 15. This simulation contains three main functions: i) the failsafe mechanism can determine the flight mode according to the health check result, instruction of RC transmitter and quadcopter status; ii) the remote pilot can fly the quadcopter through RC transmitter; iii) the flight status of quadcopter can be visually displayed by FlightGear. Thus, this simulation can be viewed as a semi-autonomous autopilot simulation of quadcopters. A video of this simulation is presented in https://www.youtube.com/watch?v=b1-K2xWbwF8&feature=youtu.be or http://t.cn/RXmhnu6. It contains three scenarios: i) the remote pilot manually controls the quadcopter to arm, fly, return to launch, and land; ii) anomalies of GPS, barometer, and INS are occurred during flight; iii) the connection of RC transmitter is abnormal during flight.§ CONCLUSIONS This paper proposes an SCT based method to design a failsafe mechanism of multicopters. The modeling process of system plant and control specifications is presented in detail. The failsafe mechanism is obtained by synthesizing a supervisor in a monolithic framework. It ignores the detailed dynamic behavior underlying each multicopter mode. This is reasonable because the failsafe mechanism belongs to the high-level decision-making module of a multicopter, while the dynamic behavior can be characterized and controlled in the low-level flight control system. 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"authors": [
"Quan Quan",
"Zhiyao Zhao",
"Liyong Lin",
"Peng Wang",
"Walter Murray Wonham",
"Kai-Yuan Cai"
],
"categories": [
"cs.SY"
],
"primary_category": "cs.SY",
"published": "20170427144617",
"title": "Failsafe Mechanism Design of Multicopters Based on Supervisory Control Theory"
} |
Laboratoire d'Hydrodynamique, CNRS/École polytechnique, 91128 Palaiseau, [email protected] Artificial eigenmodes in truncated flow domains Lutz Lesshafft December 30, 2023 =============================================== Whenever linear eigenmodes of open flows are computed on a numerical domain that is truncated in the streamwise direction, artificial boundary conditions may give rise to spurious pressure signals that are capable of providing unwanted perturbation feedback to upstream locations. The manifestation of such feedback in the eigenmode spectrum is analysed here for two simple configurations. First, explicitly prescribed feedback in a Ginzburg–Landau model is shown to produce a spurious eigenmode branch, named the `arc branch', that strongly resembles a characteristic family of eigenmodes typically present in open shear flow calculations. Second, corresponding mode branches in the global spectrum of an incompressible parallel jet in a truncated domain are examined. It is demonstrated that these eigenmodes of the numerical model depend on the presence of spurious forcing of a local k^+ instability wave at the inflow, caused by pressure signals that appear to be generated at the outflow. Multiple local k^+ branches result in multiple global eigenmode branches. For the particular boundary treatment chosen here, the strength of the pressure feedback from the outflow towards the inflow boundary is found to decay with the cube of the numerical domain length. It is concluded that arc-branch eigenmodes are artifacts of domain truncation, with limited value for physical analysis. It is demonstrated, for the example of a non-parallel jet, how spurious feedback may be reduced by an absorbing layer near the outflow boundary. § INTRODUCTION Linear instability analysis of open flows today is commonly carried out in a so-called `global' framework, where at least two non-homogeneous spatial directions of a steady base state are numerically resolved. In contrast to `local' analysis, where the base state is assumed to be parallel, and unbounded in the flow direction, a global discretisation of an open flow problem in a truncated numerical domain necessitates the formulation of artificial streamwise boundary conditions for flow perturbations. The question then arises in how far such boundary conditions influence the instability dynamics of the truncated flow system. This paper investigates the effect of spurious pressure feedback, due to domain truncation, on the eigenmode spectrum of incompressible open flow problems. The investigation is motivated by observations made in recent linear instability studies of jet flows <cit.>, where a prominent family of eigenmodes (black symbols in Fig. <ref>), referred to as the `arc branch' from here on,was found to present features that suggest a resonance between the inflow and outflow boundaries. Such branches are in fact ubiquitous in many, if not all, global spectra of truncated open shear flows found in the literature: boundary layers <cit.>, cylinder wakes <cit.>,jets <cit.>, plumes <cit.>, three-dimensional boundary layers with roughness elements <cit.> — all these and many others present similar characteristic branches of eigenvalues that are often described as being highly dependent on the type or position of outflow boundary conditions. Typically, no convergence with respect to the length of the numerical box can be attained for such modes.Ehrenstein & Gallaire <cit.> remark on the resemblance between arc-branch-type global eigenmode structures obtained for a flat-plate boundary layer and spatial modes as found in a local analysis. Åkervik <cit.> document the dependence of boundary layer eigenvalues on the type of outflow boundary conditions. Heaton <cit.> characterise arc branch modes in the spectrum of a Batchelor vortex as artifacts, speculating that these arise from the limited precision of their numerical scheme, not from domain truncation. Cerqueira & Sipp <cit.> demonstrate that such precision errors indeed lead to spurious `quasi-eigenmodes', but that these differ from the arc branch. In their analysis, quasi-eigenmodes appear in regions of the complex frequency plane where pseudospectrum ϵ values <cit.> are very small, below approximately 10^-12. These modes, also visible in the lower left corner of Fig. <ref>, are very sensitive to the numerical scheme, to mesh refinement and to the eigenvalue shift parameter. Arc branch modes are found to be robust with respect to those details<cit.>, but strongly dependent on the numerical domain length. Coenen <cit.> show that arc branch eigenfunctions of a jet are characterised by an integer number of wavelengths between the inflow and outflow boundaries, and they suggest an analogy with acoustic modes in a pipe of finite length. This analogy implies that arc branch modes are the result of unwanted resonance between the numerical boundaries, potentially leading to a spuriousinstability of the numerical system. The present study expands on all these observations, and it aims at a detailed characterisation of unphysical resonance due to imperfect boundary conditions in open shear flow calculations.The possibility that global instability in truncated systems may be brought about by spurious pressure feedback from boundary conditions was probably first described by Buell & Huerre<cit.>. In direct numerical simulations of perturbations in a mixing layer, unstable perturbation growth was observed at long times, in a configuration that was only convectively unstable in a local sense. Such behaviour is inconsistent with the interpretation of local convective instability, and it was demonstrated to be caused by unphysical pressure perturbations emanating from the outflow boundary, which in turn provoked the formation of vortical perturbations at the inflow boundary. Chomaz <cit.> interprets pressure feedback as a non-local operator variation, arguing that strong non-normality in the spectrum of convection-dominated flows is likely to induce a high sensitivity of global eigenmodes with respect to such feedback. While the spurious generation of acoustic pressure waves from artificial boundary conditions is an important and much-discussed problem in compressible flow simulations, especially those that aim to accurately capture the acoustic radiation from shear flows <cit.>, the question of how such artifacts may affect the global stability behaviour remains largely unexplored, both in compressible and in incompressible configurations. The problem is approached here in the following manner. Section 2 presents a global instability analysis of the Ginzburg–Landau equation with explicit feedback from a downstream sensor to an upstream actuator, in order to examine the effect of such feedback on the eigenvalue spectrum in a controlled setting. This model study fully describes the suspected mechanism behind arc branch modes. In Sect. 3 the analysis is extended to a parallel jet flow in a finite-size numerical domain. Global eigenmodes are projected onto their local counterparts, the global pressure field is examined, and spurious feedback effects are analysed. The question how spurious feedback may be reduced in a practical manner is addressed in Sect. 4, for the example of a spatially developing jet. § A GINZBURG–LANDAU MODEL PROBLEM The hypothesis put forward by Coenen <cit.>, that the arc branch may be the manifestation of a non-physical upstream scattering of perturbations from the outflow boundary, is first investigated with the help of a simplified model. The linear Ginzburg–Landau equation is written as ∂_t ψ = - U∂_x ψ + μ(x) ψ + γ∂_xxψ + f(x,ψ) . The complex scalar variable ψ(x,t) is a function of time t and of one single spatial coordinate x. Constant parameters U=6 and γ=1-ı are chosen ad hoc, whereas the coefficient μ varies linearly in x as μ(x) = U^2/8( 1- x/20). With this particular variation of μ, the system is marginally absolutely unstable at x=0, convectively unstable for 0<x<20, and stable for x>20, all in a local sense. Ginzburg–Landau systems of this form, with linearly decreasing μ(x) and without feedback, f≡ 0, have been extensively used to model the global instability behaviour of spatially developing flows in semi-infinite domains <cit.>. The instability characteristics mimic, in a very simple and qualitative manner, those of a spreading jet. However, the system (<ref>) is not intended here to predict or reproduce the dynamics of any specific flow. A forcing term f(x,ψ) is added in (<ref>) in order to provide an explicit closed-loop forcing between the upstream and the downstream end of the flow domain. Some aspects of closed-loop forcing in the Ginzburg–Landau equation are discussed by Chomaz <cit.>; in the present context, it is used to model a suspected spurious feedback in global shear flow computations. Taken to be of the form f(x,ψ) = C exp( -(x-x_a)^2/0.1^2) ψ(x_s), a forcing proportional to ψ(x_s) is applied in a close vicinity of x_a, such that a feedback loop is established between a (downstream) sensor location x_s and an (upstream) actuator location x_a. The complex coefficient C governs the amplitude and phase of the feedback, and the Gaussian spreading around x_a is introduced for reasons of numerical resolution.Equation (<ref>) is discretised on an interval 0≤ x ≤ 40, with a step size Δ x=0.1, using an upwind-biased seven-point finite difference stencil for the spatial derivatives. A homogeneous Dirichlet boundary condition for ψ is imposed at the upstream boundary, consistent with typical jet conditions. Actuator and sensor locations are chosen close to the boundaries, at x_a=1 and x_s=39, where spatial derivatives are well resolved.Temporal eigenmodes of (<ref>) are sought in the form ψ(x,t)=ψ̂(x) e^-iω t. The eigenvalues of the system without feedback, C=0, are known analyticallyto be ω_n = i{U^2/8 - U^2/4γ + γ^1/3U^4/3/160^2/3ζ_n }, where ζ_n is the n^th root of the Airy function <cit.>. These values are represented in Fig. <ref> as circles. Each frame <ref>(a–d) also displays the numerically computed eigenvalues of systems with feedback, shown asbullet symbols, for different non-zero values of the coefficient C. Already with very low-level feedback, C=10^-10, the spectrum is clearly affected: only the leading three eigenvalues of the unforced system are recovered, and the lower part of the spectrum is replaced with two new stable branches. These branches, named feedback branches in the following, move upward in the complex ω plane as the feedback coefficient is increased, masking more and more of the original unforced eigenvalues. Note that those affected original eigenvalues are not merely altered by the feedback, but they rather disappear abruptly from the spectrum as they fall below the new branches. The spectra obtained with C=10^-6 and 10^-2 resemble those of pure helium jets, as shown in figure 6 of <cit.>,and those of cylinder wakes obtained by Marquet <cit.>. The least stable original eigenvalue lies just above the feedback branches, apparently unaffected. In the strongest feedback case, C=10^2, the feedback branches have merged into one, overarching the entire spectrum of the feedback-free system, which has altogether disappeared. This picture (Fig. <ref>d) resembles the spectrum of the slowly developing jet of Coenen <cit.>, reproduced in our Fig. <ref>.The similarity between the feedback branches in the present model and the arc branch in the jet spectrum (Fig. <ref>) is also manifest in the pseudospectra.If feedback is thought of as a variation of the operator <cit.>, it should be possible to relate feedback-induced eigenvalues to the pseudospectrum.The pseudospectrum is defined here by the spectral norm of the resolvent operator, ∂_t ψ = Lψ→(ıωI+L)^-1 = ϵ^-1,at any complex frequency ω <cit.>.Pseudospectrum ϵ-contours of the Ginzburg–Landau equation are compared between the unforced setting with C=0, Fig. <ref>(a), and the forced setting with C=10^-6, Fig. <ref>(b). It is seen that the feedback branches align closely with an isocontour of the feedback-free pseudospectrum. This criterion also applies to all other cases displayed in Fig. <ref>, with different values of C, and it is fully consistent with observations made in three flow configurations <cit.>. Furthermore, the pseudospectrum of the system with feedback is identical to that of the system without feedback above the feedback branches, whereas the pseudospectrum is flattened, nearly constant, below the feedback branches. The same behaviour is found in the jet pseudospectrum shown in Fig. <ref>.Finally, the discrete distribution of feedback modes along the branch is investigated. The strong feedback case C=10^2 is considered for illustration. Eigenfunctions of successive feedback modes are presented in figure <ref>,analogous to the representation of jet results in figure 5 of <cit.>. The absolute value of ψ is traced in logarithmic scale as a function of x.The phase of ψ is always chosen such that the real part of ψ is zero in the sensor location x_s=39, with the exception of the first mode (Fig. <ref>a), which has no interior wave nodes. Only positive real frequencies are considered.Clearly, the eigenfunctions are characterised by integer numbers of wavelengths between the actuator and sensor locations, starting from zero wavelengths in the case of the lowest real frequency (Fig. <ref>a, only amplitude variations but approximately constant phase), to one wavelength in Fig. <ref>(b), and so forth with a continuously increasing count. The series could be continued further along the entire arc branch.The present model seems to reproduce very well the characteristics of arc branch eigenfunctions, as displayed in figure 5 of <cit.>.The position and spacing of eigenvalues along the branch curve, fixed approximately by an isocontour of the pseudospectrum, appears to be determined by a fitting phase relation between ψ at the actuator location and the applied feedback. To further illustrate this mechanism, the phase of the feedback coefficient C is varied. Figure <ref> shows the arc branch as it is obtained with values C=10^2 (bullets, same as before), C=10^2 ı (crosses), C=-10^2 (plus signs) and C=-10^2 ı (squares). All symbols fall onto the same curve, confirming that the position of feedback eigenmodes, which represent singularities in the pseudospectrum, is fixed by the phase of the feedback relation, such that resonance can occur between both ends of the loop.§ A PARALLEL JETThe effect that imperfect numerical boundary conditions can have on eigenmode computations in a two-dimensional flow domain is now investigated for the case of axisymmetric perturbations in a parallel round jet, governed by the incompressible axisymmetric Navier–Stokes equations,0= _r u_r + u_r/r + _x u_x,_t u_r= - u_r _r u_r - u_x _x u_r - _r p + 1/Re( _rru_r + _r u_r/r -u_r/r^2 + _xxu_r ) , _t u_x= - u_r _r u_x - u_x _x u_x- _x p + 1/Re( _rru_x + _r u_x/r+ _xxu_x ) . Boundary conditions are prescribed as u_r=u_x=0 at x=0, Re^-1_x u_x - p = 0 at x=x_max, u_r = _r u_x= 0 at r=0 and r=r_max. The stress-free outflow condition is a common and convenient choice for finite-element computations. Jet radius and centreline velocity are the dimensional length scales in this formulation, and the Reynolds number is chosen as Re=100. The advantage of using a parallel base flow profile is that the global results can be rigorously compared with local instability properties. The standard analytical model of Michalke <cit.> is adopted,U(r) = 1/2(1 + tanh[1/4θ( 1/r - r)]),and a momentum thickness θ=0.1 is chosen for this example. The axisymmetric perturbation equations that follow from linearization of (<ref>) about the base flow (<ref>) are discretized with finite elements on a domain of length x_max=20 and radial extent r_max=50, using the FreeFEM++ software. This domain is resolved with 80 equidistant elements in x, and with 360 non-equidistant elements in r. The equations are solved for eigenmodes [u_r,u_x,p]^T(r,x,t) = [û_r, û_x,p̂]^T (r,x) exp (-ıω t), where the eigenvalue ω = ω_r +ıω_i contains the angular frequency ω_r and the temporal growth rate ω_i.The global spectrum is shown in Fig. <ref>. It features a clean upper arc branch (red) with maximum growth rate at ω_1=1.111 - 0.052ı. This mode, labelled `1', is chosen for further analysis. A lower branch (blue) is also present, from which the mode labelled `2' with ω_2=0.912 - 1.211ı will be examined. Similar lower branches are visible in wake spectra <cit.> and in boundary-layer calculations <cit.>. §.§ Projection onto local instability modesFor comparison, the corresponding local instability problem, for a domain of infinite extent x∈ (-∞,∞), is solved with a standard Chebyshev collocation technique on a staggered grid <cit.>, using a coordinate transformation adapted for jet profiles <cit.>. Identical radial collocation point distributions are used in the global and local computations, such as to eliminate the need for interpolation. Consistent boundary conditions (<ref>c) are imposed in the local problem.Spatial local instability modes are computed for the ω values corresponding to the global modes labelled in Fig. <ref>. The flow is convectively unstable, with an absolute frequency ω_0=1.074-0.286ı. Both direct and adjoint modes are solved for. The adjoint local modes represent the dual basis associated with the set of direct modes, and they serve for projecting the spatial structure of the global mode onto the local direct modes. This projection is carried out in the following way, similar to the procedure used by Rodríguez <cit.>: at a given streamwise station x, the radial variations of the global mode perturbation quantities are extracted. Since the eigenvector of the spatial local problem contains auxiliary variables kû_r and kû_x <cit.>, these must also be added to the extracted slice of the global mode. This is accomplished by computing the streamwise derivative of the global û_r and û_x fields, and by augmenting the extracted vector [û_r,û_x,p̂]^T with [-ı_x û_r,-ı_x û_x]^T. This augmented global slice is finally projected onto the local modes, via scalar multiplication with the associated adjoint modes. As usual, the adjoint modes are normalised beforehand in such a way that their scalar product with the associated direct mode is unity, whereas the direct modes are by themselves scaled to have unit norm.The spatial local spectrum of the parallel jet profile, shown in Fig. <ref> for the frequency values of global modes 1 and 2, is composed of discrete k^+ and k^- modes <cit.>, which represent downstream- and upstream-propagating hydrodynamic perturbations inside and near the jet. The labels in Fig. <ref> rank all k^+ and k^- modes according to their spatial growth rate -k_i. Figure <ref> displays absolute values of the projection coefficients, obtained for global mode 1, pertaining to the three dominant local modes. These are the first two k^+ modes and the first k^- mode. Blue symbols represent the k^+_1 mode (the only one displaying unstable spatial growth). Except very near the Dirichlet inlet, the streamwise variation of this local mode amplitude is perfectly exponential, with a spatial growth rate 0.4604, as measured by a regression fit over 1≤ x ≤ 19. This value matches within 0.01% the imaginary part of the local eigenvalue. The amplitude of the global mode component û_x on the jet centerline is shown as a black line for reference. Green symbols in Fig. <ref> denote the amplitude of k^+_2, and red symbols the amplitude of k^-_1. Straight lines indicate the corresponding growth rate of the local eigenvalue for comparison. It is seen that both projections follow the amplitude variations expected from local analysis in a region close to one boundary, where their amplitude is maximal. Clearly, the k^+_2 mode originates at the upstream boundary, whereas the k^-_1 mode is forced at the downstream end. Farther away from those boundaries, both projections approximately follow the slope of the dominant k^+_1 branch. This behaviour results from imperfections in the numerical projection, which apparently only allows a clean distinction between these local modes down to amplitude ratios around 10^-3 in the present setup. Higher spatial resolution in the local and global computations does not improve this threshold. Note that these three modes are highly non-orthogonal, which makes their distinction numerically delicate.Overall, local mode contributions other than from the k^+_1 mode to the global mode 1 are tractable but rather negligible. Modulations of the centerline velocity perturbation (black line in Fig. <ref>) are instead attributable to global pressure modes, as will be shown later on.Very similar results are obtained for the global mode 2, for which the local mode amplitudes are displayed in Fig. <ref>. However, the local spectrum in this case differs from that of mode 1, as the global frequency ω_2=0.912 - 1.211ı has an imaginary part below that of the absolute frequency ω_0=1.074-0.286ı. Therefore, in the analysis of global mode 2, the local spatial modes are selected from a spectrum where pinching of the k^+_1 and the k^-_1 has already occurred (see Fig. <ref>b). In this setting, it is now the k^+_2 mode that displays the strongest downstream growth, followed by the k^+_3 mode. A mode from the mixed branch, formed from the k^+_1 and k^-_1 branches after pinching, is denoted k^±.The global mode 2, as represented by a black line in Fig. <ref>, is clearly dominated by the k^+_2 mode (blue bullet symbols), but the k^+_3 and k^± modes are again discernible down to amplitudes three orders of magnitude below k^+_2. It is not clear a priori how the k^± mode is to be interpreted, in particular with regard to its up- or downstream propagation. However, the projection results plainly show that this mode is generated at the downstream end, from where it propagates upstream; thereby, it behaves as a mode of k^- type. It is stressed that the analysis of spatial branches below the absolute growth rate is indeed meaningful in the present context. Spatial analysis below the absolute growth rate, i.e. after pinching has taken place, is usually said to be in violation of temporal causality, formally expressed by the fact that no integration path can be found in the complex k-plane that separates k^+ and k^- branches <cit.>. This argument however arises in the context of the asymptotic flow behaviour at long times, when indeed the system dynamics are determined by the absolute mode. In the same sense, a global system is asymptotically determined by only the most unstable eigenmode (here: global mode 1). Notwithstanding, global eigenmodes with lesser growth rate do exist, and they are observable in the transient system dynamics. The spatial local modes used in the analysis of global mode 2 are justified in the same way.In summary, the projections onto local spatial modes demonstrate that global mode 1 is supported by a local k^+_1 mode, whereas global mode 2 relies on a k^+_2 mode. In all likelihood, the same holds true for any global mode of the arc branch (k^+_1) and of the lower branch (k^+_2). Moreover, although not shown in Fig. <ref>, further lower branches exist at lower growth rates ω_i<-1.5, for which a match with higher local k^+ branches is anticipated. The link between the arc branch and the dominant k^+ mode has been pointed out in earlier studies <cit.>. However, the argument so far has one loose end: the presence of a local k^+ mode is contingent on it being forced upstream. The present results show that this forcing takes place immediately at the upstream boundary. The essential ingredient that can give rise to a global mode with a k^+ wave is feedback from downstream.§.§ Global pressure feedback The ellipticity of the global linear jet problem is contained in the pressure gradient and in the viscous terms. The latter will only be noticeable over distances much shorter than the numerical box length, and they are not considered in the following analysis. The pressure, as noted by Ehrenstein & Gallaire <cit.>, obeys a Poisson equation, which in the present case of parallel flow takes the formΔp̂ = -2_r U _x û_r,with homogenous Dirichlet and Neumann conditions at the upstream and lateral boundaries, respectively, and with p̂=Re^-1_x û_x at the outflow.The following analysis is restricted to the arc branch mode labelled `1' in Fig. <ref>, but results for mode 2 are not fundamentally different. The pressure amplitude of mode 1 is shown in Fig. <ref>(a) as log_10|p̂|. Its structure is somewhat irregular in the downstream near-field region of the jet, yet the characteristic wavelength 2π/k_r=4.1 of the k^+_1 mode is apparent.The stress-free outflow boundary condition is seen to result in p̂≈0, and the pressure at r≳ 5 is essentially a superposition of fundamental solutions of the homogeneous equation Δp̂ = 0, with Dirichlet conditions at the inflow and outflow. These are given byp̂_j = [A_j I_0(jπ/Lr) + B_j K_0(jπ/Lr)] sin(jπ/Lx),j ∈ℕ,where I_0 and K_0 are the modified Bessel functions of the first and second kind, respectively, and L=20 is the streamwise length of the numerical box.The K_0 functions are exponentially decaying in r, manifestly dominant in the present problem, whereas the I_0 functions grow exponentially in r. These only enter the global mode at very low amplitude (A_j/B_j ≪ 1) in order to satisfy the Neumann condition at r_max=50. Only the p̂_1 component is clearly visible in Fig. <ref>a, because it experiences the slowest radial decay, but a projection confirms that at least the first five p̂_j components enter the pressure field with comparable global norms.It is known from the analysis in Sect. <ref> that the global mode involves a strong k^+_1 wave. In the present section, the complementary part is sought that may provide the upstream-reaching part of a feedback loop. Therefore, the k^+_1 component of the global mode is subtracted from the pressure field, using the already known projection coefficients shown in Fig. <ref>. This `stripped' pressure field p̃ is presented in Fig. <ref>(b), as log_10|p̃|. A remarkably clean structure is recovered, suggestive of a solution of the Laplace equation Δp̃ = 0 which is forced at the outflow boundary near the jet axis. A small inhomogeneity is also observed at the inflow near the the jet axis.Note that the local k^+_1 mode by construction represents a particular solution of (<ref>) in a domain of infinite streamwise extent. Within the limits of the simplifying assumption that the propagating perturbations in the region of _r U≠ 0 are indeed given by the k^+_1 mode alone, in the interior of the bounded domain, the stripped pressure field p̃ only needs to satisfy the inhomogeneities that arise from the boundary conditions. These inhomogeneities on both ends of the domain are thus coupled through the Laplace equation in p̃.As a final plausibility check, the amplitudes of the supposed downstream-propagating and upstream-reaching components of the feedback loop are compared in Fig. <ref> along a path at constant r. The radial position of the critical point of the local k^+_1 mode is chosen, r_c=0.92. A thin blue line represents the pressure amplitude of the k^+_1 mode component, according to the projection carried out in Sect. <ref>, a thick red line represents the amplitude of the pressure feedback signal p̃, and black symbols mark the amplitude of the total pressure field p̂. The up- and downstream branches have approximately equal amplitude at the downstream boundary, nearly cancelling each other. Near the upstream boundary, the feedback signal is larger by about a factor 3 than the k^+_1 wave — this appears reasonable in view of the assumption that the latter is forced by the former. §.§ The influence of box lengthArc branch modes in the literature are consistently found to be sensitive to the length of the numerical box. In some instances <cit.>, a longer box yields eigenvalues with lower growth rates, in other instances <cit.> the opposite effect is observed. If the present feedback model is correct, it should allow an estimation of the influence of the numerical box length on the arc branch growth rates.Eigenmodes of the parallel jet (<ref>) have been computed in numerical domains of streamwise lengths L=10,15 and 25, for comparison with the standard configuration L=20; in all cases, the radial discretisation and the constant step size Δ x are unchanged. Resulting modes of the arc branch are displayed in Fig. <ref>(a). As L is increased, the entire branch is seen to shift to higher growth rates, and the modes are more densely spaced.The analysis so far suggests that a downstream-convecting k^+ mode and the elliptic pressure field are coupled in small regions near the inflow and outflow boundaries. In analogy with the Ginzburg–Landau model discussed in Sect. <ref>, an effective reflection coefficient C may be defined, which relates the forcing of the k^+ mode at the inflow to the k^+ amplitude at the outflow, thus lumping the narrow interaction regions into singular actuator and sensor positions in x. The effective forcing at the Dirichlet inflow boundary is modelled as-ıωq̂ = Cq̂ e^ı kL→ C = -ıωe^-ı kL,where ω is the global eigenmode frequency, and (k,q̂) denote the relevant local k^+ mode.The coefficient C includes the effects of coupling on both ends of the domain, as well as the upstream decay of the pressure signal, seen in Fig. <ref>.Numerically obtained values of C are reported in figures <ref>(b,c), in terms of their absolute value and their phase, for the arc branch in all four numerical domains. In each domain, the reflection coefficient decreases with real frequency, but only weakly so for frequencies larger than one. This effect may be caused by the slower radial decay of q̂ at low frequencies, resulting in radially more extended interaction regions on both domain ends. The phase of C varies only slightly with frequency, and it is independent of box length. If the reflection coefficients are scaled with the cube of the box length, as shown in Fig. <ref>(d), they neatly collapse onto one curve. Consequently, the effective feedback imparted by boundary reflections decreases with the domain length as L^3. This scaling factor indicates that the relevant component of the pressure signal, which couples the downstream with the upstream end of the k^+ wave, is of a quadrupole type (signals from monopole and dipole sources at x=L would decay as L and L^2, respectively). It is then straightforward to interpret the effect of an increased numerical box length on a given arc branch mode: if the exponential growth of the dominant k^+ mode over the added streamwise interval is larger than the algebraic decay of the upstream-reaching pressure signal, over the same added distance, then the spurious inlet forcing will increase in strength, resulting in a higher growth rate ω_i. If the k^+ mode in the added region is locally stable, then the inlet forcing will decrease, and the global growth rate will be lower as a result. This interpretation appears to be consistent with the two cited examples: the boundary layer investigated by Ehrenstein & Gallaire <cit.> is locally unstable at the outflow, as confirmed by the authors, and longer domain sizes are found to result in higher global growth rates. In contrast, the rapidly spreading jet considered by Garnaud <cit.> is locally stable at the outflow with respect to axisymmetric perturbations, and indeed lower global growth rates are obtained for the arc branch in longer domains. This criterion may of course be frequency-dependent: certain (complex) frequencies may be locally stable at the outflow while others are unstable, and opposite trends ought to be observed in these frequency ranges. This seems indeed to be the case in the Re=360 setting of Coenen <cit.>.§ POSSIBLE REMEDIES, TESTED FOR A NON-PARALLEL JETWhile the preceding analyses of the Ginzburg–Landau equation and of the parallel jet served to characterise the global feedback mechanism due to domain truncation, the question how such feedback may be reduced is addressed in this section for the example of a spatially developing jet. With this choice, the results from the previous section can be largely transferred to the new setting.Each type of open flow may present particular problems in view of domain truncation. In favourable configurations, the numerical boundaries can be placed in stable flow regions, as for instance in a uniform flow upstream of a solid obstacle, like a cylinder. In other cases, the need for truncation may be avoided by prescribing consistent physical boundaries, like confining solid walls, which are then part of the flow configuration.Jets belong to a more problematic category. They are created by some upstream source of momentum, and it is in general not feasible to include the entire upstream apparatus (fan, chamber, nozzle, etc.) in the calculations, which would furthermore defeat the purpose of any generic description of jet dynamics. The formulation of upstream flow and boundary conditions is therefore necessarily imperfect with respect to any flow realisation, as potentially important regions are not accounted for.For the present study, the jet is modelled as issuing from an orifice in a solid wall, very similar to the configuration of Garnaud et al.<cit.>. Upstream of the orifice, the flow develops in a straight circular pipe, from where it exits with a fairly thin boundary layer. The base flow is computed as a steady, axisymmetric solution of the incompressible Navier–Stokes equations (<ref>), with an inflow conditionU(r)=tanh [5/2(1/r-r)]imposed at x=-10. Newton–Raphson iterations are performed in order to converge to a steady flow state. The numerical domain for these base flow calculations is truncated downstream at x_max=50 and radially at r_max=50, where stress-free conditions are applied. The Reynolds number is set to 1000 in the free jet. However, in order to maintain a thin shear layer at the orifice, this value is varied exponentially inside the pipe, between Re=10^5 at x=-10 and Re=10^3 at x≥ -0.2, in the base flow calculation. The resulting flow field near the orifice at x=0 is represented in Fig. <ref>, together with the characteristic streamwise variations of the shear layer momentum thickness θ_m and of the centreline velocity. Eigenmode calculations are carried out on smaller domains, where portions of the base flow are cropped at the inflow, the outflow, or both, in order to probe the effect of domain truncation. All configurations are listed in table <ref>. The Reynolds number in these calculations is set to 1000 throughout the domain.The influence of upstream truncation on eigenmodes is considered first. Figure <ref>a shows spectra for three different domains, with homogeneous Dirichlet conditions (<ref>a) imposed at x_min=(0,-5,-10), respectively. The downstream end of the numerical domain is placed at x_max=40, where stress-free conditions (<ref>b) are applied in all three cases. Unconverged and lower-branch eigenvalues are not shown here and in the following for clarity; the criterion for convergence is that an eigenvalue could be reproduced within three-digit accuracy using two different shift values. It is seen from Fig. <ref>a that the arc branch is quite insensitive to the position of the upstream Dirichlet boundary. Even a full truncation of the upstream pipe (case 1) only results in a slight stabilisation at high frequencies. Eigenvalues obtained with pipe lengths of 5 and 10 radii (cases 2 and 3) are virtually identical. Stress-free inflow conditions (cases 4, 5 and 6, spectra shown in Fig. <ref>b) are found to perform less favourably. When applied at x=0, these conditions give rise to global instability; the inclusion of portions of the pipe has a stabilising effect, but even with x_min=-10 the growth rates along the entire arc branch are still significantly higher than those obtained with Dirichlet conditions.It is to be expected that global pressure fluctuations induce a generation of vorticity waves at the solid corner at x=0, and the independence of the eigenvalues in cases 2 and 3 (Fig. <ref>a) on the upstream boundary position suggests that this physical effect is dominant over spurious pressure-vorticity coupling at x_min. It remains to be determined to what extent the downstream boundary conditions emit spurious pressure signals, and how these may be reduced. Garnaud et al. <cit.> tested stress-free and convective outflow conditions in an almost identical jet configuration, and found no significant difference in the resulting spectra. In the present study, the potential of absorbing layers <cit.> for a stabilisation of the arc branch is examined. To this end, an artificial damping term is added to the linear perturbation equations, with the purpose of reducing perturbation amplitudes before they reach the numerical boundary at x_max. If the original global eigenvalue problem is written-iω𝐁q̂ = 𝐋q̂,then the eigenvalue problem with absorbing layer is defined as-iω𝐁q̂ = [𝐋 - λ(x)𝐁]q̂.For cases 7 and 8 (see table <ref>), the damping parameter is prescribed asλ(x) = {[0for x ≤ 19,; 0.00625x^2 - 0.2375x + 2.25625 for 19<x<21,;0.025(x-20)for x ≥ 21. ].This variation of λ is continuous in its first derivative, and it has maximum values 0.5 and 0.75 at x_max=40 and x_max=50, respectively.The effect of absorbing layers on the eigenvalue spectrum is documented in Fig. <ref>. Damping in the downstream region 19≤ x ≤ 40 (case 7) is seen to reduce the growth rates of arc branch modes significantly (compared to case 2). Eigenfunctions of selected modes, marked by circles in Fig. <ref>, are compared in Fig. <ref>. An integral amplitude measure is defined asA(x)=[∫_0^1 ( |û_r|^2 + |û_x|^2 ) r dr ]^1/2,and the amplitude curves in Fig. <ref> are normalisedwith respect to their values at the outflow boundary for comparison. While the red curve (case 2, no absorbing layer) shows a monotonic growth of perturbation amplitude throughout the domain, similar to the parallel jet results in figures <ref> and <ref>, the blue curve (case 7) reaches a maximum inside the absorbing layer and subsequently decays in x. Remarkably, the ratio A(x_max)/A(0) is identical in both cases, which is consistent with the interpretation of these modes as being the result of spurious feedback from the outflow, in the same way as detailed in Sect. <ref> for the parallel jet. The absorbing layer reduces the gain of the k^+ hydrodynamic branch of the feedback loop, and thereby the temporal modal growth rate.Based on the discussion in Sect. <ref>, stronger artificial damping of the hydrodynamic branch ought to lead to ever smaller modal growth rates; longer domains should furthermore lead to a reduced amplitude of the incident pressure signal upstream, due to its cubic decay. Both conditions are combined in case 8 (see table <ref>), of which results are included in figures <ref> and <ref>. Indeed, the temporal growth rates are further reduced, and the spatial decay of the eigenfunction amplitude in the absorbing layer is more pronounced than in the case 7.However, the shape of the arc branch in case 8 displays some differences with respect to all other cases, and similar mode patterns have been reported from jet calculations on long domains by Coenen <cit.>, where no artificial damping was applied. In the vicinity of the least stable mode (circled in Fig. <ref>), a regular spacing in ω_r is still observed, but with larger distances between consecutive modes than in cases 2 and 7. The corresponding amplitude function in Fig. <ref> shows that the ratio A(x_max)/A(0) is smaller than in the two other cases, whereas the cubic decay of the reflection coefficient with box length, as described in Sect. <ref>, should have resulted in a larger ratio. Furthermore, stronger damping through higher values of λ, tried in test calculations that are not shown here, does not decrease the maximum growth rate much further, but it quickly leads to ill-conditioned system matrices. Case 8 seems indeed to mark an efficiency limit of absorbing layers for the stabilisation of the arc branch.The above observations, in particular the increased spacing of modes in ω_r, suggest that pressure feedback in the strongly damped case 8 may originate from the interior of the domain, possibly from the location of the amplitude maximum. Such feedback may be spurious, in the sense of Heaton <cit.>, or it may be physical. This hypothesis is suggested for further examination. § CONCLUSIONS A family of linear instability eigenmodes, named the arc branch, has been analysed in view of its physical or numerical origin. Branches of this type have previously been observed in a large variety of open flows. All results presented herein lead to the conclusion that arc branch modes in incompressible flow calculations are an artifact of domain truncation, due to spurious pressure feedback between numerical inflow and outflow boundaries.A Ginzburg–Landau model was investigated first, because in this setting the effect of explicitly prescribed feedback between a downstream sensor and an upstream actuator could be examined without ambiguity. In the presence of feedback, a branch of eigenmodes was observed to arise that exhibits all typical characteristics of the arc branch described by Coenen <cit.> for the example of light jets. These modes align with a contour of the pseudospectrum, spaced at regular intervals of the real frequency, and their spatial eigenfunction is characterised by an integer number of wavelengths between actuator and sensor locations. As the strength of the feedback is increased, the branch moves steadily upward to higher growth rates in the complex frequency plane.Eigenmodes of the feedback-free system that lie below the arc branch are no longer detectable. An important observation is that this arc branch in the Ginzburg–Landau equation with feedback has no counterpart in the feedback-free spectrum: these eigenmodes are not merely affected by the presence of feedback, indeed without it they do not exist.An incompressible parallel jet in a truncated domain was examined next. No explicit feedback was prescribed in this setting, but a similar arc branch of eigenmodes was nonetheless found to dominate the spectrum. A lower branch of eigenmodes with stronger temporal decay was also described, which may correspond to subdominant branches observed in wakes and boundary layers. Both the arc-branch and the lower-branch modes in the parallel jet were shown to be composed primarily of one downstream-propagating k^+ wave, as computed from a local spatial analysis. Each of the two branches involves a different k^+ mode. This observation raises the question how a k^+ wave is generated at the upstream boundary of the numerical domain. The analogy with the Ginzburg–Landau model from Sect. <ref> suggests feedback from downstream.The most plausible mechanism for global feedback in a truncated domain is the generation of pressure perturbations due to the artificial downstream boundary condition, as described by Buell & Huerre<cit.>. The pressure field of the least stable arc branch mode was decomposed into one portion associated with the prominent k^+ wave, which cannot be involved in upstream feedback, and into a residual part that is essentially governed by a Laplace equation forced at the domain boundaries. The latter appears to be strongly dominated by a source region at the outflow, near the jet axis. A numerical evaluation of effective reflection coefficients, in analogy with the Ginzburg–Landau model ofSect. <ref>, established as a principal result that the strength of upstream feedback decays with the numerical box length to the third power. This scaling is indicative of a spurious pressure quadrupole situated at the outflow. The algebraic nature of the pressure decay allowed a prediction of the effect of box length variations on arc-branch growth rates, depending on the local stability or instability of the flow, apparently consistent with common observations. While this study has been limited to incompressible flow settings, similar mechanisms will also be present in compressible calculations, with the difference that spurious pressure signals are thenpropagated by a wave equation. This may in fact result in stronger feedback, as the far-field acoustic pressure only decays with the first power of the distance from the source.Finally, possible strategies for a reduction of spurious pressure feedback have been examined for the example of a spreading jet. On the one hand, upstream boundary conditions must be chosen that minimise the unphysical conversion of pressure feedback from downstream to vortical perturbations. It has been found, in this particular flow example, that Dirichlet conditions perform much better than stress-free conditions in this regard. On the other hand, the generation of spurious pressure signals from the outflow boundary must be reduced. It has been demonstrated that artificial damping in a downstream absorbing layer provides an efficient means to achieve this. An advantage of this technique is that it is straightforward to implement in any usual open shear flow problem. It has been noted, however, that increasingly strong damping does not reduce the growth rate of the arc branch to arbitrarily low levels. Discussions with Xavier Garnaud and Wilfried Coenen greatly helped shape the ideas presented in this paper. The study was supported by the Agence Nationale de la Recherche under the Cool Jazz project, grant number ANR-12-BS09-0024. plain 10Akervik:2008p1176 E. Åkervik, U. Ehrenstein, F. Gallaire, and D. Henningson. Global two-dimensional stability measures of the flat plate boundary-layer flow. Eur. J. Mech. B/Fluids, 27:1–13, Jul 2008.BuellHuerre J.C. Buell and P. Huerre. Inflow/outflow boundary conditions and global dynamics of spatial mixing layers. Proc. 2nd Summer Prog., Stanford Univ. Cent. Turbul. Res., pages 19–27, 1988.cerqueira S. Cerqueira and D. Sipp. Eigenvalue sensitivity, singular values and discrete frequency selection mechanism in noise amplifiers: the case of flow induced by radial wall injection. J. 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Aspectral collocation method using a staggered grid for the stability of cylindrical flows. Int. J. Numer. Meth. Fluids, 12:825–833, Apr 1991.Kurz2016 H. Kurz and M. Kloker. Mechanisms of flow tripping by discrete roughness elements in a swept-wing boundary layer. Journal of Fluid Mechanics, 796:158–194, 2016.Lesshafft:2007p54 L. Lesshafft and P. Huerre. Linear impulse response in hot round jets. Phys. Fluids, 19(2):024102, Jan 2007.Loiseau2014investigation J.-C. Loiseau, J.-C. Robinet, S. Cherubini, and E. Leriche. Investigation of the roughness-induced transition: global stability analyses and direct numerical simulations. J. Fluid Mech., 760:175–211, 2014.Marquet:2008p1127 O. Marquet, D. Sipp, and L. Jacquin. Sensitivity analysis and passive control of cylinder flow. J. Fluid Mech., 615:221–252, Jan 2008.M84 A. Michalke. Survey on jet instability theory. Prog. Aerospace Sci., 21:159–199, 1984.Nichols:2011p1075 J.W. Nichols and S.K. Lele. Global modes and transient response of a cold supersonic jet. J. Fluid Mech., 669:225–241, 2011.dani D. Rodríguez, A. V. G. Cavalieri, T. Colonius, and P. Jordan. A study of linear wavepacket models for subsonic turbulent jets using local eigenmode decomposition ofdata. Eur. J. Mech. B/Fluids, 49:308–321, 2015.Sipp2007 D. Sipp and A. Lebedev. Global stability of base and mean flows: a general approach and its applications to cylinder and open cavity flows. J. Fluid Mech., 593:333–358, 2007.TrefethenBook L. N. Trefethen and M. Embree. Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. Princeton University Press, 2005. see also . | http://arxiv.org/abs/1704.08450v2 | {
"authors": [
"Lutz Lesshafft"
],
"categories": [
"physics.flu-dyn"
],
"primary_category": "physics.flu-dyn",
"published": "20170427070552",
"title": "Artificial eigenmodes in truncated flow domains"
} |
Entanglement generation between a charge qubit and its bosonic environment during pure dephasing - dependence on environment size Katarzyna Roszak Accepted 2017 XXX. Received 2017 Apr; in original form 2016 Feb =================================================================================================================================We use the database leak of Mt. Gox exchange to analyze the dynamics of the price of bitcoin from June 2011 to November 2013. This gives us a rare opportunity to study an emerging retail-focused, highly speculative and unregulated market with trader identifiers at a tick transaction level. Jumps are frequent events and they cluster in time. The order flow imbalance and the preponderance of aggressive traders, as well as a widening of the bid-ask spread predict them. Jumps have short-term positive impact on market activity and illiquidity and induce a persistent change in the price.JEL classification: C58, G12, G14.Keywords: Jumps, Liquidity, High-frequency data, Bitcoin.2 § INTRODUCTION Bitcoin, a distributed digital currency, was created in 2009 and is the most popular cryptocurrency with a multi-billion dollar capitalization since 2013.It is the first such currency to gain relatively widespread adoption. The technology provides an infrastructure for maintaining a public accounting ledger and for processing transactions with no central authority. Unlike traditional currencies, which rely on central banks, bitcoin relies on a decentralized computer network to validate transactions and grow money supply (see <cit.> and <cit.> for further background on the bitcoin and its technology). Each bitcoin is effectively a (divisible) unit which is transferred between pseudonymous addresses through this network.Its promising potential and scarcity have driven the market price of bitcoin to parity with the U.S. dollar in February 2011 and above $1,000in November 2013. It is estimated that by the end of our period of study in 2013, bitcoin had approximately one million users worldwide with a three-digit annual growth. Mt. Gox was the largest exchange platform to provide bitcoin trading for U.S. dollar until it went bankrupt early 2014 as a result of the theft of client funds by hackers.[The Japanese courts are holding pre-trial hearings, and the claim process enters its fourth year. Japanese police have found part of the missing bitcoins, and the 24,000 or so claimants are waiting for a final settlement.] An important part of Mt. Gox internal database leaked, revealing a full history of trades on the period April 2011–November 2013. This data set gives us a rare opportunity to observe the emergence of a retail-focused, highly-speculative and unregulated market at a tick frequency with trader identifiers at the transaction level.Bitcoin has experienced numerous episodes of extreme volatility and apparent discontinuities in the price process. On one hand, the absence of solid history and exhaustive legal framework make bitcoin a very speculative investment. Because it does not rely on the stabilizing policy of a central bank, the reaction to new information, whether fundamental or speculative, results in high volatility relative to established currencies. On the other hand, the relative illiquidity of the market with no official market makers makes it fundamentally fragile to large trading volumes and to market imperfections, and thus more prone to large swings than other traded assets. The focus of this paper is to contribute to the growing literature on the analysis of jumps and their potential explanation. Jumps are sporadic events of a larger amplitude than what a continuous diffusion process can explain. Analyzing their distributional properties is important because of the consequences in applications including derivatives pricing and risk management. <cit.> consider U.S. large-cap stocks, equity indexes, and currency pairs. They conclude that jumps in financial asset prices are often erroneously identified and are, in fact, rare events accounting for a very small proportion of the total price variation. They show that measures of jump variation based on low-frequency data tend to spuriously assign a burst of volatility to the jump component. <cit.> test for the presence of jumps in Dow Jones stocks at high frequency. They explain that the repetition of the jump test over a large number of days leads to a number of spurious detections because of multiple testing issues. They correct for this bias, reducing even further the number of remaining detections in comparison to the findings of <cit.>. They find an average of 3 to 4 remaining jumps a year and relate them to macroeconomic news, prescheduled company-specific announcements, and stories from news agencies which include a variety of unscheduled and uncategorized events. They conclude that the vast majority of news do not cause jumps but may generate a market reaction in the form of bursts of volatility. They conjecture that jumps might be related to liquidity issues and order flow imbalances but the limited number of detected jumps in their study poses a challenge for getting statistically significant empirical evidence.Our main contributions are to assess the presence of jumps in a highly-speculative emerging market with low liquidity, and to determine whether liquidity is a main driver of jump occurence. The information of the trader identifier and the direction of trade, i.e., whether the transaction are initiated by a buyer or a seller, provided by our data records is key for our empirical analysis. Such information is rarely available for other markets and is related to the unique way that the Mt. Gox database stores the knowledge about successive trades. Our first contribution is to detect the presence of jumps in the bitcoin market, and to study their dynamics. We apply the jump detection test of <cit.> to the tick data and control for multiple testing across days using the False Discovery Rate (henceforth, FDR) technique <cit.>. We identify 124 days including at least one jump during the period, or approximately one detection day per week. The number of detections is significantly larger than what previous research observes for large-cap assets and indices, suggesting that the intensity of jump occurrence largely varies depending on the market characteristics, such as its liquidity or the specificities of the participants. We investigate the dynamics of durations between jumps. <cit.> cannot reject the hypothesis that jump arrivals follow a Poisson process. We apply a runs test on jump detections date and strongly reject the independence of inter-jump durations. Hence, jump dynamics do not support the jump process used by <cit.> and subsequent models based on compound Poisson processes with constant intensity.Our second contribution is to perform a systematic event study for the identified jumps to characterize the market conditions preceding and following a discontinuity. We seek to determine, if not the cause, the main factors driving the occurrence of jumps as well as their impact on market conditions. Such an empirical analysis is made possible because of a sufficientlylarge number of detected jumps, which is not the case for large-cap markets. We use a probit regression model and find that discontinuities are anticipated by abnormal trading activity and liquidity conditions: the order flow imbalance, the proportion of aggressive traders and the bid ask spread have significant predicting power over jumps. Those findings support the hypothesis that jumps occur when trading activity clashes with a liquidity shock, and there is no stabilizing mechanism either induced by a central bank or by market makers whose mandate is to provide liquidity. We perform a post-jump analysis of the market conditions and find that most indicators are exacerbated, including the trading volume, the number of traders, the order flow imbalance, the bid-ask spread, the realized variance, the microstructure noise variance and the proportion of aggressive traders. These factors however revert to their anterior level in less than half an hour. Comparing the price levels before and after jumps reveals a significant, persistent impact: positive (negative) jumps occur during locally bearish (bullish) trends.The rest of the paper is organized as follows. Section 2 reviews the data and our cleaning procedure. Section 3 defines our methodology for detecting jumps. Section 4 presents our empirical results. Section 5 concludes. § DATA ON THE BITCOIN MARKET Let us first briefly introduce the bitcoin. Bitcoin is a novel form of electronic money that is based on a decentralised network of participating computers. It has no physical counterpart; it is merely arbitrary (divisible) units that exist on this network. There is no central bank and there are no interest rates. The system has a pre-programmed money supply that grows at a decreasing rate until reaching a fixed limit. This semi-fixed supply exacerbates volatility and deflationary pressure.Each user of bitcoin can generate an address (like an email address or account number) through which to make and receive transactions, making bitcoin pseudonymous. The crucial aspect that makes bitcoin work is that it solves the double-spending problem without relying on a central authority. In other words, it is possible to send a bitcoin securely, without then being able to spend that bitcoin again, without someone else being able to forge a transaction, and also without your being able to claim that bitcoin back (i.e., a chargeback). These transactions get recorded in a decentralised ledger (known as the blockchain), which is maintained by a network of computers (called 'miners').Miners maintain consensus in the blockchain through solving difficult mathematical problems, and are rewarded with bitcoins and optional (voluntary) transaction fees. The additionalrewarded bitcoins are the mechanism that increases the bitcoin money supply. For our empirical study, we use transaction-level data with trader identifiers for the Mt. Gox bitcoin exchange. We conduct our analysis over the uninterrupted period from June 26, 2011 to November 29, 2013. Mt. Gox was the leading bitcoin trading platform during that period and processed the majority of trading orders.We extract the data from the Mt. Gox database leak of March 2014, following Mt. Gox suspension of its operation and bankruptcy filing. This data set is available on the BitTorrent network and includes a history of all executed trades. The data is organized as a series of comma-separated files with each row listing a time stamp, a trade ID, a user ID, a transaction type (buy or sell), the currency of the fiat leg, the fiat and bitcoin amounts, and the fiat and bitcoin transaction fees. A subset of the trades additionally reveals the country and state of residence of the user. We ignore these last pieces of information as they are only available for a limited number of trades. A heuristic analysis of trade IDs reveals that they correspond to the concatenation of a POSIX timestamp and a microsecond timestamp. We parse the timestamps accordingly to define the execution time of each trade with a microsecond precision.The respective legs of the trades are split across multiple lines. We initiate the cleaning procedure by aggregating trade entries according to their trade IDs. We filter out trades whose bid leg or ask leg are missing, and remove all duplicates. We also remove from the sample trades for which the same user identifier appears on both legs. Those trades are either due to a bug in the order book matching algorithm, or are simple data errors. Finally, we only consider U.S. dollar-denominated trades and filter out trades whose fiat amount is smaller than $0.10 to avoid numerical errors in the computation of the price. We define the tick-time price series as the ratio of the bitcoin amount over the fiat amount for the chronological trades series, rounded to the third decimal.We confirm the authenticity of the remaining data by comparing them to the data set published by Mt. Gox in 2013 and its subsequent updates. However, the comparison also reveals two problems related to multi-currency trades.[On August 27, 2011, Mt. Gox implemented a form of order book aggregation across currencies, with the exchange acting as intermediary. For exemple, a market buy order in USD could match a limit sell order in EUR, triggering a pair of trades between the users and Mt. Gox. The two legs share the same trade ID, which allows us to identify them easily. The published data set further distinguishes the primary and non-primary legs of a multi-currency trade. The primary leg is the one where Mt. Gox is selling bitcoins in exchange for fiat. All missing trades are non-primary legs.] First, 92,174 trades have a systematic data error whereby the fiat amount is the same in the primary and the secondary currency, and thus incorrect by a factor corresponding to the exchange rate between the two currencies. We correct this error by copying the fiat amount from the published data set and updating the price. Second, 129,081 trades corresponding to secondary legs of multi-currency trades are missing from the data set, representing less than 2% of all trades. We find in unreported robustness checks that the impact of the missing trades have a negligible effect on our results.A visual analysis of the remaining tick data reveals frequent outliers on the whole time period. We eliminate obvious data errors such as trade prices reported at zero or above $10,000. We fetch daily high and low prices from the external data source Bitcoin Charts[See http://www.bitcoincharts.com.] and remove trades whose exchange price lies outside of the high-low interval with a 20% margin. We also discard `bounceback' outliers as defined in <cit.>. The resulting set of trades is used for our analysis of the bitcoin market.The data set only includes information on executed trades. It lacks limit orders, and consequently provides no explicit information on the bid-ask spread across time or the depth of the order book. The published data set provides an additional field specifying whether orders are initiated by the buyer or the seller, that is, if they are aggressive bids or aggressive asks. This recording is important for our analysis of the potential determinants of jump occurence. We define the best bid series as the price series of aggressive ask orders, and the best ask series as the price series of aggressive bid orders. In the rare occurrences where the best bid price gets higher than the best ask price, we update the best ask to the value of the bid price; reciprocally, we update the best bid price if the best ask price crosses it.We construct calendar-time price series by computing the median of the tick-time prices within each interval of 5 minutes. In the case where no trade occurs, we propagate the price from the previous period. We build the calendar-time volume series by summing the respective volumes within each interval, and the trades number series by taking the number of trades on each period.The final data set contains 6.4 million transactions involving 90,382 unique traders. The transactions amount to a total volume of $2.1 billion, or on average $2.4 million per day. Figure <ref> shows the time series of the price and volume on a logarithmic scale during the period. The price of bitcoin increases from $16 on June 26, 2011 to an all-time high of $1,207 on November 29, 2013. Volume increases significantly during the period as well, and the linear correlation between price and volume exceeds 70%. The price of bitcoin has experienced several booms and busts. The clearest example is the crash of April 10, 2013 which saw the bitcoin value drop by 61% in only hours for no obvious reason, after doubling over the previous week. No stabilizing mechanisms mitigate those large swings. There are no central banks, no market makers, and no circuit breakers in the bitcoin market.§ METHODOLOGY Many pricing models rely on the assumption that the dynamics of the underlying asset follow a continuous trajectory. For instance, <cit.> propose a diffusion model with constant volatility and <cit.> augments it with a second factor to allow for heteroskedasticity. The empirical literature challenges continuous models <cit.>. The probability of large moves disappears asymptotically as the horizon shrinks, which does not provide consistent short-term skewness and kurtosis.There are mainly two approaches to overcome this limitation.[Another alternative would be to consider Lévy jumps of infinite activity <cit.>.] First, we can introduce a jump component in the price process <cit.>. Jumps are discontinuous price changes occurring instantaneously, no matter the frequency of observations. Alternatively, we can consider models with highly dynamic volatility, such as the two-factor stochastic volatility model of <cit.> and <cit.>. The probability of sudden moves asymptotically still vanishes, yet those models allow for bursts of volatility leading to significant changes on short-term horizons.Identifying whether a price process is continuous or has jumps is important because of the implications for financial management such as pricing, hedging and risk assessment. For deep out-of-the-money call options, there may be relatively low probability that the stock price exceeds the strike price prior to expiration if we exclude the possibility of jumps. However, the presence of jumps in the price dynamics significantly increases this probability, and hence, makes the option more valuable. The converse holds for deep in-the-money call options. This phenomenon is exacerbated with short-maturity options. <cit.>, <cit.>, <cit.>, <cit.> develop statistical tools to test for the presence of jumps. Their modeling approach assumes that the data is not contaminated by microstructure noise, preventing a high-frequency analysis. <cit.> show that it is crucial to test for jumps at a high frequency to avoid misclassification of bursts of volatility as jumps. <cit.> emphasize the multiple testing issue in jump analysis. After correcting for this bias, they find that jumps are extremely rare events in large-cap stocks.We follow <cit.> to test for the presence of jumps in the bitcoin market at a tick frequency. We define a complete probability space (Ω,ℱ_t,), where Ω is the set of events of the bitcoin market, {ℱ_t : t∈ [0,T] } is the right-continuous information filtration for market participants, andis the physical measure. We denote the log-price P and model its dynamics on a given day asP_t = σW_t + a Y_t J_t,where W_t is a Brownian motion, J_t is a jump counting process, Y_t is the size of the jump, σ is the volatility assumed to be constant on a one-day period, and a is 0 under the null hypothesis of no jump and 1 otherwise.[We omit the drift term in our log-price model as it has no impact in the jump detection test asymptotically, as explained in Mykland and Zhang (2009).]The log-price P stands for the unobservable, fundamental price in an ideal market. The bitcoin market is relatively illiquid and is subject to multiple frictions such as trading fees. Consequently, the observed price is contaminated by noise. We define the observed price P̃ asP̃_t_i = P_t_i + U_t_i,where t_i is the time of observation[We assume that Assumption A of <cit.> about the density of the sampling grid holds.], i=1,...,n, with n being the number of observations per day. Here U denotes the market microstructure noise with mean 0 and variance q^2. Figure <ref> shows the autocorrelation function at a tick frequency of the observed log-returns on June 10, 2013.[We observe a similar pattern of significantly negative 1–3 lag coefficients throughout the sample.] The significant dependence in the first lags suggests that the microstructure noise has serial dependence. We therefore allow U to have a (k-1)-serial dependence, with k = 4.We define the block size as M=C (n/k)^1/2, where x denotes the integer part of the number x, and follow the recommendations of <cit.>, Section 5.4, for specifying the parameter C. We compute the averaged log-price over the block size M asP̂_t_j = 1/M∑_i=j/k^j/k+M-1P̃_t_ik,and test for the presence of jumps between t_j and t_j+kM using the asymptotically normal statistic ℒ defined asℒ(t_j) = P̂_t_j+km-P̂_t_j,for j = 0, kM, 2kM, …The asymptotic variance of the test statistic is given by V = lim_n →∞ V_n = 2/3 0.2^2 σ^2 T + 2 q^2 where the limit holds in probability. We estimate the volatility σ̂ using the consistent estimator of <cit.>, which is robust to the presence of noise and jumps. We use Proposition 1 of <cit.> to estimate the noise variance q̂^2, that is,q̂^2 = 1/2(n-k)∑_m=1^n-k(P̃_t_m - P̃_t_m+k)^2.Our estimate of the asymptotic variance is therefore V̂_n =2/3 0.2^2 σ̂^2 T + 2 q̂^2.<cit.> show the convergence in distribution of the test statisticsB_n^-1( √(M)/√(V_n)max_j|ℒ(t_j)| - A_n ) ⟶ξ,for j = 0, kM, 2kM, …, where ξ follows a standard Gumbel distribution with cumulative distribution function (ξ≤ x) = exp( -e^-x), and the constants are as followsA_n=( 2 log[]n/kM)^1/2 - log(π) + log( log( []n/kM) )/2 ( 2log([]n/kM) )^1/2, B_n=1/( 2log( []n/kM) )^1/2. We test the presence of jumps on a given day by identifying a divergence of the test statistic from the Gumbel distribution. As emphasized in <cit.>, it is crucial to account for multiple testing when applying a statistical test more than once. Indeed, if the rejection threshold is fixed, the proportion of rejections converges to the size of the test under the null hypothesis because of type I errors, preventing any statistical inference. The FDR ensures that at most a certain expected fraction of the rejected null hypotheses correspond to spurious detections. The FDR approach results in a threshold for the p-value that is inherently adaptive to the data. It is higher when there are few true jumps, i.e., the signal is sparse, and lower when there are many jumps, i.e., the signal is dense. Setting the FDR target parameter to 0 is equivalent to a strict control of the family-wise error rate. It is very conservative as it asymptotically admits no spurious detection due to multiple testing. We prefer a FDR target level of 10%, which results in a more liberal threshold than with family-wise error rate control. The power of the test is therefore improved, at the cost of accepting that up to 10% of detected jump days may be spurious. We refer to <cit.> and <cit.> for further discussion, background, and applications of the FDR methodology in finance (see also <cit.> for multiple testing issues in factor modeling). § EMPIRICAL RESULTS In this section, we study the dynamics of jump arrivals on the bitcoin market. We aim to assess the presence of jumps and their distributional properties. We qualify market conditions favoring the apparition of discontinuities and show that jumps have a positive impact on market activity and illiquidity. §.§ Jump distributionWe apply the high-frequency jump detection test of <cit.> with FDR control at a 10% level and find 124 jump days in the period June 2011 to November 2013, or approximately one jump date per week. Table <ref> reports the summary statistics for the jumps detected from 5-min intervals and Figure <ref> shows the histogram of jump sizes. In 70 cases, the jump has a positive size, and in 54 cases, a negative size. This contrasts with the common idea that jumps depict mainly price crashes. The average size of a positive jump is 4.7%, and that of a negative jump is -4.1%. We observe discontinuities of up to a 32% move within a 5-min interval, emphasizing the importance of modeling jumps on this market. Figure <ref> shows the p-values of the jump test statistics, as well as the 1% confidence threshold and the FDR threshold. We see that a fixed level of 1% is too permissive and leads to many spurious detections. Interestingly, the thresholding only discards 35% of rejections, where <cit.> marked up to 95% as spurious detections on Dow Jones stocks. This is due to the adaptiveness of the FDR control, which is less strict where there are many true jumps in the data.A widely-used assumption is that jump arrival times follow a simple Poisson process, or equivalently that durations between successive jumps are independent and exponentially distributed. We study the dynamics of jump arrivals to assess whether this assumption is consistent with empirical data. Figure <ref> shows the number of jump detections per quarter on the whole data set. It suggests that the frequency of days with jumps varies across time. Because our test only indicates whether at least one jump occurred on a given date but does not give the exact number of jumps within that day, we cannot test the null hypothesis of exponential inter-jump durations, however. We follow the approach of <cit.> and use the runs test of <cit.>. The runs test measures the randomness of detections by comparing the number of sequences of consecutive days with jumps and without jump against its sampling distribution under the hypothesis of random arrival. Table <ref> reports the results of the runs test on the full sample and on three sub-periods of 296 days. We strongly reject the hypothesis of independent jump durations on the full sample, indicating significant clustering in jump times. Applying the runs test over three sub-periods reveals that clustering is not equally present on the whole sample. On the period June 26, 2011 to April 16, 2012, which corresponds to the early bitcoin trading days, we observe a strong rejection of the hypothesis of independent runs. On the second period, we only reject at a 10% level, and we cannot reject on the last period. The dynamics of jumps on the bitcoin market contrast with previous literature on high-frequency jump analysis. <cit.> and <cit.> identify a small number of jumps on large markets such as Dow Jones constituents, market-wide U.S. equity indices and foreign currencies. <cit.> do not identify clustering in the few remaining jumps. We investigate the hypothesis that the relative illiquidity of the bitcoin market coupled with abnormal market activity is key to understanding sudden moves. §.§ Jump predictabilityFigure <ref> shows an example of a 5% positive jump that occurred on June 10, 2013. The highlighted region emphasizes the time interval with the maximum absolute value of ℒ(t_j) during that day. As illustrated in Panel (c), the jump occurs after an apparent increase in the trading volume and the order flow imbalance. Panel (d) also reveals multiple spikes in the bid-ask spread as well as a general widening of the spread shortly before the discontinuity. In this section, we investigate the conjecture that the relative illiquidity of the bitcoin market coupled with abnormal market activity is key to understanding sudden moves. Specifically, we hypothesize that jumps are the result of liquidity drying up in certain market conditions, in conjunction with a regime change in the order flow.[For a study on the importance of the order flow on price discovery, see, e.g., <cit.>, <cit.>, <cit.> and <cit.>.]We consider a regular time series at a 5-minute frequency on the whole sample. For each 5-minute period i, we set Y_t_i = 1 if a jump was identified during the period i and 0 otherwise, and compute the following statistics using the tick data:[Our results are robust to the choice of frequency. We get similar estimates at a 10-minute and 20-minute frequency, but the significance of estimates decreases strongly at 20-minute. We also try alternative measures of the spread such as the maximum spread on the period or the average spread with no qualitative change. We note that because the jump test of <cit.> only reveals the largest jump of the day, we might have several time indices i for which Y_t_i is incorrectly set to 0 in the regression.] * MS_i is the bid-ask spread, calculated as the median of the ratio of the bid-ask difference to the mid-price. We use the bid-ask spread factor as a proxy for market illiquidity. * OF_i is the absolute order flow imbalance, defined as the absolute value of the difference between the aggressive buy volume and the aggressive sell volume. A large OF_i thus indicates excessive buying pressure in the market. * WR_i is the `whale'[The term `whale' is frequently used to describe the big money bitcoin players that show their hand in the bitcoin market. The large players being referred to are institutions such as hedge funds and bitcoin investment funds.] index calculated as the ratio of the number of unique passive traders to the total number of unique traders during the period. The ratio is large when few aggressive traders are responsible for most of the transactions. * P_i is the median observed price. * RV_i is the realized variance of the latent price during the period, given by the noise-robust estimator of <cit.>. * NV_i is the variance of the microstructure noise, estimated as in <cit.>.The order flow imbalance OF_i and the whale ratio WR_i quantify two different aspects of the trading pressure that were not directly observable by market participants. The former measures excess directional volume, irrespective of the number of traders responsible for the divergence. For the latter, we take advantage of the richness of our data set that allows us to track the activity of each individually identified trader. The whale index thus gives us a measure of the imbalance between liquidity providers and liquidity takers: a large estimate indicates that few traders are responsible for most of the liquidity taking.[ph!] [Jump predictability]Jump predictability The table displays estimates of the probit regression model in Equation <ref>. On Panel A, we compute statistics for periods of 5 minutes. On Panel B, we compute statistics for periods of 10 minutes. First four columns show estimates for the model including fixed effects; last four columns do not include fixed effects. The `Marg. prob.' columns shows the marginal probability change induced by a one-standard deviation change in the values of the covariates from their respective sample averages. 4cWith fixed effects 4cWithout fixed effects (lr)2-5 (lr)6-9 Coefficient Est. Std error p-value Marg. prob. Est. Std error p-value Marg. prob. 9cPanel A: 5-minute periods Intercept -3.61 0.13 <0.01 -3.76 0.12 <0.01 Realized variance 2.28 6.54 0.73 4.12% 2.10 6.37 0.74 4.50% Noise variance -2876.16 876.80 <0.01 -56.65% -2793.26 851.57 <0.01 -57.90% Abs. order flow 0.00 0.00 <0.01 12.16% 0.00 0.00 <0.01 12.48% Med. spread 23.54 2.84 <0.01 52.13% 23.56 2.83 <0.01 52.33% Med. price -0.00 0.00 0.06 -34.70% -0.00 0.00 0.02 -34.51% Whales 0.60 0.17 <0.01 50.91% 0.68 0.17 <0.01 43.56% Adj. R^2 0.07 0.07 9cPanel B: 10-minute periods Intercept -3.60 0.14 <0.01 -3.78 0.13 <0.01 Realized variance -2.54 8.09 0.75 -7.76% -2.50 7.55 0.74 -7.91% Noise variance -2015.06 1655.32 0.22 -37.39% -1871.53 1419.32 0.19 -39.76% Abs. order flow 0.00 0.00 0.01 11.58% 0.00 0.00 0.01 12.20% Med. spread 23.10 3.48 <0.01 45.04% 22.77 3.45 <0.01 46.05% Med. price -0.00 0.00 0.12 -24.74% -0.00 0.00 0.06 -25.22% Whales 0.83 0.19 <0.01 69.24% 0.96 0.19 <0.01 58.08% Adj. R^2 0.05 0.06We apply a binary probit model to assess the predictive power of these statistics on the probability a jump in the next period and verify our hypothesis. Formally,ℙ[ J_i+1 | MS_i, OF_i, WR_i, P_i, RV_i, NV_i ]=Φ( β_0 + β_11_297:592,i + β_21_593:888,i+ β_MSMS_i + β_OFOF_i+ β_WRWR_i + β_PP_i + β_RVRV_i + β_NVNV_i ),where Φ is the Gaussian cumulative distribution function and 1_t_1:t_2,i=1 if t_1≤ i≤ t_2, zero otherwise. We add fixed effects for the same sub-periods as in Section <ref> to control for the changing market conditions associated with the rapid development of the market for bitcoin. Table <ref> exhibits the parameter estimates and their respective significance levels. The adjusted pseudo-R^2=0.07 confirms the predictive power of the regression, and the unreported likelihood ratio test rejects the constant model at the 0.1% level. The estimates for β_MS and β_OF are both positive and significant, showing the strong impact of market illiquidity and order flow on jump risk. This confirms the results of <cit.>, who find that illiquidity factors and order flow imbalance play a positive role in the occurrence of jumps in the U.S. Treasury market. The estimate of β_WR is significantly positive as well, indicating that it is not only an imbalance in volume that increases jump risk, but also an asymmetry in the number of aggressive traders relative to their passive counterparts. For β_P, it is significantly negative, supporting the intuition that jumps have less probability of occurring as the bitcoin market develops and its size increases. Microstructure noise variance plays a negative role in the occurrence of jumps. We can explain the negative sign by the probit model capturing the dominant effect that very large values (or at least above the time series average)of microstructure noise variance are not being followed by a jump most of the time. When the microstructure noise variance is large, the market participants do not get a clear signal of the fundamental value of the asset and do not seem to adjust their expectations in an abrupt way. Yet, in contrast to <cit.>, realized variance has no significant impact on jump risk. Setting aside the obvious differences between the markets for U.S. Treasuries and bitcoin, we believe that the divergence is explained by our use of robust-to-noise estimators and multiple testing adjustments for jump detection on 5-min intervals.The positive impact of the realized variance in their empirical results from jump detection on 5-min intervals for many consecutive days could be a consequence of spurious detections.Panel B of Table <ref> reports the estimation of the same model for periods of 10 minutes. The results are consistent with the estimation with 5-minute periods, albeit less categorical, with a slightly lower adjusted pseudo-R^2 and the coefficient for microstructure noise variance losing significance, which again highlights the importance of considering high-frequency data for such an analysis.Our findings thus indicate that jumps are systematically associated with market conditions characterized by a low level of liquidity and the presence of few large and active directional traders. §.§ Jump impact We perform a post-jump analysis of the market dynamics. On Figure <ref>, we plot the average dynamics of the whale index, the bid-ask spread, the noise variance and the absolute order flow around jumps. The graphs show that these measures are affected before and after a jump. The whale ratio surges right before a jump, as shown already in Section <ref>, but quickly reverts to its previous level. The bid-ask spread and the microstructure noise variance gradually increase and peak right around the jump, followed by a slow reversion. The order flow imbalance massively increases before the occurrence but falls to below-average levels right after that. This figure illustrates the intuition of the previous section about the influence of market forces on price discontinuities: aggressive traders placing massive orders, in conjunction with market illiquidity are a significant signal for the occurrence of jumps.The figure emphasizes the market reaction and dynamics after the jumps. We aim to determine if market conditions are affected and how persistent the possible subsequent changes are. We consider the same set of statistics as in the model of Equation (<ref>), and include additionally the trading volume and the number of traders.For each jump, we compute the statistics on four consecutive spans of 15 minutes following the detection period. We compare the statistics to a reference period preceding respective jumps by one hour. We define the test statistics as the log-ratio of the post-jump measure over the reference measure for each period. We run a Student t-test to assess changes in the means. Table <ref> gathers the results of t-tests, grouped by their respective spans. We find that all measures are exacerbated in the 15 minutes immediately following a jump. The trading volume and the absolute order flow imbalance are abnormally high. At the same time, the number of active traders, and the proportion of aggressive traders are significantly larger. Liquidity proxies including the bid-ask spread and the microstructure noise variance see an increase too, as well as the realized variance. However, the impact of jumps dampens after 30 minutes already. After 45 minutes, all measures revert to anterior levels except the market price: a positive jump generally induces a persistent lower price—and reciprocally, a negative jump induces a higher price. Figure <ref> illustrates this feature by showing the (rescaled) average price around positive and negative jumps, respectively. Jumps tend to occur in episodes of massive price trends and act in an opposite direction to allow for an abrupt and quick price correction. This type of correction is not observed on other markets with stability and liquidity providing mechanisms. § CONCLUSION The presence of jumps in the dynamics of asset prices remains a debated question in the empirical literature. While many jumps may be detected in low-frequency data, recent studies based instead on high-frequency data have shown that most are in fact misidentified bursts of volatility in continuous price paths. True jumps in large-cap stock prices appear to be rare which prevents systematic studies of their properties.In this paper, we have been able to conduct such a study for the bitcoin-to-U.S. dollar (BTC/USD) exchange rate using transaction-level data obtained from Mt. Gox exchange , the leading platform during the sample period of June 2011 to November 2013. We contribute to the literature in several ways. First, in contrast tolarge-cap stock markets, we find that jumps are frequent: out of the 888 sample days, we identify 124 jump days, or on average one jump day per week. In contrast to the intuition that relates jumps to crash events, most jumps are in fact positive. They are economically significant, with a mean size of 4.65% for positive jumps and -4.14% for negative ones. Second, we show that jumps cluster in time: we find runs of jump days that are incompatible with the classical assumption of independent Poisson arrival times. Third, we estimate a binary probit model of jump occurrence using covariates that proxy for illiquidity and market activity, including the `whale' index, a novel measure of the concentration of order flow across traders that exploits a unique feature of our data set which allows us to identify individual traders. We find that illiquidity, order flow imbalance, and the preponderance of aggressive traders are significant factors driving the occurrence of jumps. Finally, we test for the effect of jumps on several market measures and find that jumps have a positive impact on market activity as proxied by volume and number of traders and a negative impact on liquidity. The measured impacts disappear gradually and are no longer significant after an hour, except for the effect on the price level which is persistent.We have thus shown that jumps are an essential component of the price dynamics of the BTC/USD exchange rate. They are associated with several identified factors, some of which are directly observable from available market data. These conclusions have immediate implications for the modeling of the exchange rate. Further research could seek to verify whether we can extend our conclusions to other financial markets that share characteristics with the studied market, but whose detailed transaction level records are still unavailable. jf | http://arxiv.org/abs/1704.08175v2 | {
"authors": [
"Olivier Scaillet",
"Adrien Treccani",
"Christopher Trevisan"
],
"categories": [
"q-fin.ST"
],
"primary_category": "q-fin.ST",
"published": "20170426160419",
"title": "High-Frequency Jump Analysis of the Bitcoin Market"
} |
Properties of Strong and Weak Propellers from MHD Simulations [ December 30, 2023 =============================================================siconxxxxxxxx–x [2]Division of Optimization and Systems Theory, Department of Mathematics, KTHRoyal Institute of Technology, 100 44 Stockholm, Sweden. ([email protected], [email protected]) [3]Department of Automation and School of Mathematics, Shanghai Jiao Tong University, 200240 Shanghai, China. ([email protected]) In our companion paper <cit.> we discussed the multidimensional rational covariance extension problem (RCEP), which has important applications in image processing, and spectral estimation in radar, sonar, and medical imaging. This is an inverse problem where a power spectrum with a rational absolutely continuous partis reconstructed from a finite set of moments. However, in most applications these moments are determined from observed data and are therefore only approximate, and RCEP may not have a solution. In this paper we extend the results <cit.> to handle approximate covariance matching. We consider two problems, one with a soft constraint and the other one with a hard constraint, and show that they are connected via a homeomorphism. We also demonstrate that the problems are well-posed and illustrate the theory by examples in spectral estimation and texture generation.Approximate covariance extension, trigonometric moment problem, convex optimization, multidimensional spectral estimation, texture generation. § INTRODUCTION Trigonometric moment problems are ubiquitous in systems and control, such as spectral estimation, signal processing, system identification, image processing and remote sensing <cit.>. In the(truncated) multidimensional trigonometric moment problem we seek a nonnegative measureon ^d satisfyingthe moment equation c_ = ∫_^d() for all ∈Λ,where :=(-π,π], :=(θ_1,…, θ_d)∈^d, and (,):=∑_j=1^d k_jθ_j is the scalar product in ^d. Here Λ⊂ℤ^d is a finite index set satisfying 0 ∈Λ and - Λ = Λ. A necessary condition for (<ref>) to have a solution is that the sequence c:= [ c_|:=(k_1,…, k_d) ∈Λ ]satisfy the symmetry condition c_-=c̅_. The space of sequences (<ref>) with this symmertywill be denotedand will be represented by vectorsc, formed by ordering the coefficient in some prescribed manner, e.g., lexiographical. Note thatis isomorphic to ^|Λ|, where |Λ| is the cardinality of Λ.However, as we shall see below, not all c∈ are bona fide moments for nonnegative measures . In manyof the applications mentioned above there is a natural complexity constraint prescribed by design specifications. In the context of finite-dimensional systems these constraints often arise in the requirement that transfer functions be rational. This leads to the rational covariance extension problem, whichhas beenstudied in various degrees of generality in <cit.> and can be posed as follows. Define e^i:=(e^iθ_1,…, e^iθ_d) and let () = Φ(e^i)() + (),be the (unique) Lebesgue decomposition of dμ(see, e.g., <cit.>),wheredm ():=(1/2π)^d∏_j=1^d dθ_jis the (normalized) Lebesgue measure andis a singular measure. Then given a c∈, we are interested in parameterizing solutions to (<ref>) such that the absolutely continuous part of the measure (<ref>)takes the formΦ(e^i)=P(e^i)/Q(e^i), p, q ∈\{0}, whereis the closure of the convex coneof the coefficients p∈ corresponding to trigonometric polynomialsP(e^i)= ∑_∈Λp_ e^-i(,),p_-=p̅_that are positive for all ∈^d. The reason for referring to this problem as a rational covariance extension problem is that the numbers (<ref>) correspond to covariances c_:= {y( t+)y( t)} of adiscrete-time, zero-mean, and homogeneous[Homogeneitygeneralizes stationarity in the case d=1. ]stochastic process{y( t);t∈^d}. The corresponding power spectrum, representing the energy distribution across frequencies, is defined as the nonnegative measure dμ on ^d whose Fourier coefficients are the covariances (<ref>). A scalar version of this problem (d=1) was first posed by Kalman <cit.> and has been extensively studied and solved in the literature <cit.>. It has been generalized to more general scalar moment problems <cit.> and to the multidimensional setting <cit.>. Also worth mentioning here is work by Lang and McClellan <cit.> considering the multidimensional maximum entropy problem, which hence has certain overlap with the above literature.The multidimensional rational covariance extension problem posed above has a solution if and only if c∈, whereis the open convex cone := { c |⟨ c, p ⟩ > 0, for all p ∈∖{0}},where ⟨ c, p ⟩ := ∑_∈Λ c_p̅_ is the inner product in(Theorem <ref>).However, the covariances [ c_|:=(k_1,…, k_d) ∈Λ] are generally determined from statisticaldata. Therefore the condition c∈ may not be satisfied, and testing this condition is difficult in the multidimensional case.Therefore we may want to find a positive measureanda corresponding r∈, namelyr_ = ∫_^d(), ∈Λ,so that r is closeto c in some norm, e.g.,the Euclidian norm _2. This is an ill-posed inverse problem which in general has an infinite number ofsolutions . As we already mentioned, we are interested in rational solutions(<ref>), and to obtain such solutions we useregularization as in <cit.>. Hence, we seek a dμthat minimizesλ(P, )+1/2r-c_2^2subject to (<ref>), where λ∈ is a regularization parameter and(P, ) :=∫_^d(P logP/Φ +dμ -Pdm )is the nomalized Kullback-Leibler divergence <cit.> <cit.>. As will be explained in Section <ref>, (P, ) is always nonnegative and has the property (P,P)=0. In this paper we shall consider a more general problem in the spirit of <cit.>. To this end, for any Hermitian, positive definite matrix M,we define the weighted vector norm x_M:=(x^*Mx)^1/2 and consider the problemmin_≥ 0, r(P, ) + 1/2r - c_W^-1^2 r_ = ∫_^d(), ∈Λ,which is the same as the problem above with W=λ I. We shall refer to W as the weight matrix.Using the same principle as in <cit.>, we shall also consider the problem to minimize (P, ) subject to (<ref>) and the hard constraint r - c ^2 ≤λ .Since (<ref>) are bona fide moments and hence r∈, while c∉ in general, this problem will not have a solution if the distance from c tois greater than √(λ). Hence the choice of λ must be made with some care. Analogously with the rational covariance extension with softconstraints in (<ref>), we shall consider the more general problem min_≥ 0, r(P, )r_ = ∫_^d(), ∈Λ, r - c _W^-1^2 ≤ 1,to which we shall refer as the rational covariance extension problem with hard constraints. Again this problem reduces to the simpler problem by setting W=λ I.As we shall see, the soft-constrained problem (<ref>) always has a solution, while the hard-constrained problem(<ref>) may fail to have a solution for some weight matrices W. However, in Section <ref> we show that the two problems are in fact equivalent in the sense that whenever (<ref>) has a solution there is a corresponding W in (<ref>) that gives the same solution, and any solution of (<ref>) can also be obtained from (<ref>) by a suitable choice of W. The reason for considering both formulations is thatone formulation might be more suitable than the other for the particular application at hand. For example, an absolute error estimate for the covariances is more naturally incorporated in the formulation with hard constraints. A possible choice of the weight matrix W in either formulation would be the covariance matrix of the estimated moments, as suggested in <cit.>. This corresponds to the Mahalanobis distanceand could be a natural way to incorporate uncertainty of the covariance estimates in the spectral estimation procedure.Previous work in this direction can be found in <cit.>, where <cit.> consider the problem of selecting an appropriate covariances sequence to match in a given confidence region. The two approximation problems considered here are similar to the ones considered in <cit.> and <cit.>. (For more details, also see <cit.>.) We begin in Section <ref>by reviewing the regular multidimensional rational covariance extension problem for exact covariance matchingin a broader perspective. In Section <ref> we present our main results on approximate rational covariance extension with softconstraints, and in Section <ref> we show that the dual solution is well-posed. In Section <ref> we investigate conditions under which there are solutions without a singular part. The approximate rational covariance extension with hardconstraints is considered inSection <ref>, and in Section <ref> we establish a homeomorhism between the weight matrices in the two problems, showing that the problems are actually equivalent when solutions exist. We also show that under certain conditions the homeomorphism can be extended to hold between all sets of parameters, allowing us to carry over results from the soft-constrained setting to the hard-constrained one.In Section <ref> we discuss the properties of various covariance estimators, in Section <ref> we give a 2D example from spectral estimation, and in Section <ref>we apply our theory to system identification and texture reconstruction.Some of the results of this paper were announced in <cit.> without proofs.§ RATIONAL COVARIANCE EXTENSION WITH EXACT MATCHING The trigonometric moment problem to determine a positive measuresatisfying (<ref>) is an inverse problem that has a solution if and only if c∈ <cit.>, whereis the closure of , and then in general it has infinitely many solutions. However, the nature of possible rational solutions (<ref>) will depend on the location of c in . To clarify this point we need the following lemma. ∖{0}⊂.Obviously the inner product ⟨ q, p ⟩ := ∑_∈Λ q_p̅_ can be expressed in the integral form⟨ q, p ⟩ = ∫_^d Q(e^i)P(e^i)(),and therefore ⟨ q,p⟩ >0 for all q,p∈∖{0}, as P and Q can have zeros only on sets of measure zero. Hence the statement of the lemma follows. Therefore, under certain particular conditions, the multidimensional rational covariance extension problem has a very simple solution with a polynomial spectral density, namely= P(e^i)dm(),p ∈\{0}.The multidimensional rational covariance extension problem has a unique polynomial solution (<ref>) if and only if c∈\{0}, namely P=C, where C(e^i):=∑_∈Λc_e^-i(,).The proof of Proposition <ref> is immediate by noting that any such C is a bona fide spectral density and noting thatc_=∫_^de^i(,)C(e^i)dm(). As seen from the following result presented in <cit.>, the other extreme occurs for c∈∂:=∖, when only singular solutions exist. For any c∈∂ there is a solutionof (<ref>) with support in at most |Λ|-1 points. There is no solution with a absolutely continuous part Φ.However, for any c∈, there is a rational solution (<ref>) parametrized by p∈\{0}, as demonstrated in <cit.> by considering a primal-dual pair of convex optimization problems.In that paper the primal problem is a weighted maximum entropy problem, but as also noted in <cit.>, it is equivalent tomin_≥ 0 ∫_^d P logP/Φ()c_ = ∫_^d(), ∈Λ,where Φ dm is the absolutely continuous part of dμ. This amounts to minimizing the (regular) Kullback-Leibler divergence between P and , subject tomatching the given data <cit.>. In the present case of exact covariance matching, this problem is equivalent to minimizing (<ref>) subject to (<ref>), since P is fixed and the total mass ofis determined by the 0:th momentc_ 0 = ∫_^d.Hence both ∫_^d and ∫_^d P are constants in this case. Hence problem (<ref>) is the natural extension of (<ref>) for the case where the covariance sequence is not known exactly. The primal problem (<ref>) is a problem in infinite dimensions, but with a finite number of constraints. The dual to this problem will then have a finite number of variables but an infinite number of constraints and is given bymin_q∈ ⟨ c, q ⟩ - ∫_^d P log Q().In particular, Theorem2.1 in <cit.>, based on corresponding analysis in <cit.>, reads as follows. Problem (<ref>) has a solution if and only if c ∈. For every c ∈ and p ∈∖{0} the functional in (<ref>) is strictly convex and has a unique minimizer q̂∈∖{0}. Moreover, there exists a unique ĉ∈∂ and a (not necessarily unique) nonnegative singular measure ν̂ with support (ν̂) ⊆{∈^d|Q̂(e^i) = 0 }such that c_ = ∫_^d( P/Q̂+ ν̂),∈Λ, ĉ_ = ∫_^dν̂, ∈Λ. For any such ν̂, the measure μ̂() = P(e^i)/Q̂(e^i)() + ν̂()is an optimal solution to the problem(<ref>). Moreover, ν̂ can be chosen with support in at most |Λ|-1 points, where |Λ| is the cardinality of the index set Λ. If c∈∂, only a singular measure with finite support would match the moment condition (Proposition <ref>). In this case, the problem (<ref>) makes no sense, since any feasible solution has infinite objective value.In <cit.> we also derived the KKT conditions q̂∈, ĉ∈∂, ⟨ĉ, q̂⟩ = 0c_ = ∫_^dP/Q̂ + ĉ_,∈Λ, which are necessary and sufficient for optimality of the primal and dual problems. Since(<ref>)is an inverse problem, we are interested in how the solution depends on the parameters of the problem. From Propositions 7.3 and 7.4 in <cit.> we have the following result. Let c,p and q̂be as in Theorem <ref>. Then the map (c,p)↦q̂ is continuous. To get a full description of well-posedness of the solution we would like to extend this continuity result to the map (c,p)↦ (q̂,ĉ). However, such a generalization is only possible under certain conditions. The following result is a consequence of Proposition <ref> and <cit.>. Let c, p, q̂ and ĉ be as in Theorem <ref>. Then,ford≤ 2 and all (c,p)∈×, the mapping (c,p)→(q̂, ĉ)is continuous. Corollary 2.3 in <cit.> actually ensures that ĉ=0 for d≤ 2 and p∈. However, in Section <ref> we present a generalization of Proposition <ref> to cases with d≥3, where then ĉ may be nonzero. (The proof of this generalization can be found in <cit.>.) Here we shall also consider an example where continuity fails when p belongs to the boundary ∂:= ∖, i.e.,the corresponding nonnegative trigonometric polynomial P(e^i) is zero in at least one point.§ APPROXIMATE COVARIANCE EXTENSION WITH SOFT CONSTRAINTS To handle the case with noisy covariance data, when c may not even belong to , werelax the exact covariance matching constraint (<ref>) in the primal problem (<ref>) to obtain the problem (<ref>). In this caseit is natural to reformulate the objective function in (<ref>) to include a term that also accounts for changes in the total mass of . Consequently, we have exchanged the objective function in (<ref>) by the normalized Kullback-Leibler divergence (<ref>) plus a term that ensures approximate data matching.Using the normalized Kullback-Leibler divergence, as proposed in<cit.> <cit.>, is an advantage in the approximate covariance matching problem since this divergence is always nonnegative, precisely as is the case for probability densities. To see this, observe that, in view of the basic inequalityx - 1 ≥log x,(P, )= ∫_^d(P ( - logΦ/P)+dμ -Pdm ) ≥∫_^d(P (1 - Φ/P) + Φ -Pdm ) + ∫_^d≥ 0, sinceis a nonnegative measure.Moreover, (P, P)=0, as can be seen by taking =Pdm in (<ref>).The problem under consideration is to find a nonnegative measure =Φ dm +ν minimizing(P, )+1/2r - c_W^-1^2subject to (<ref>). To derive the dual of this problem we consider the corresponding maximization problem and form the Lagrangian ℒ(Φ,ν, r, q) =-(P, ) - 1/2r - c_W^-1^2 + ∑_∈Λ q_^* ( r_ - ∫_^d() )=-(P, ) - 1/2r - c_W^-1^2+⟨ r,q⟩ - ∫_^dQ,where q := [ q_|:=(k_1,…, k_d) ∈Λ] are Lagrange multitipliers and Q is the corresponding trigonometric polynomial (<ref>). However, (P, )=∫_^dP(log P -1)dm - ∫_^dPlogΦ dm + r_,and therefore ℒ(Φ,ν, r, q) = ∫_^dPlogΦ dm -∫_^dQΦ dm -∫_^d Qν-∫_^dP(log P -1)dm+⟨ r,q-e⟩ - 1/2r - c_W^-1^2 ,wheree := [e_]_∈Λ, e_ 0 = 1 and e_ = 0 for ∈Λ∖{ 0 }, and hence r_=⟨ r,e⟩.In deriving the dual functionalφ(q)=sup_Φ≥ 0,ν≥ 0,rℒ(Φ,ν, r, q),to be minimized, we only need to consider q∈∖{0}, as φ will take infinite values for q∉. In fact, following along the lines of <cit.>, we note that, if Q(e^i_0)<0, (<ref>) will tend to infinity when ν(_0)→∞. Moreover, since p∈∖{0}, there is a neighborhood where P(e^i)>0, letting Φ tend to infinity in this neighborhood, (<ref>) will tend to infinity if Q≡ 0. We also note that the nonnegative function Φ can only be zero on a set of measure zero; otherwise the first term in (<ref>) will be -∞.The directional derivative[Formally, the Gateaux differential <cit.>.] of the Lagrangian (<ref>) in any feasible direction δΦ, i.e., any direction δΦ such that Φ +εδΦ≥ 0 for sufficiencly small ε >0,is easily seen to beδℒ(Φ,ν, r, q;δΦ)= ∫_^d(P/Φ-Q)δΦ. In particular, the direction δΦ:=Φ sign(P-QΦ) is feasible since (1±ε)Φ≥ 0 for 0<ε < 1. Therefore, any maximizing Φ must satisfy ∫_^d|P-QΦ|≤ 0 and hence(<ref>). Moreover, a maximizing choice of ν will require that ∫_^d Q ν=0,as this nonnegative term can be made zero by the simple choice ν≡ 0, and consequently (<ref>) must hold.Finally, the directional derivative δℒ(Φ,ν, r, q;δr)= ⟨δr, q-e + W^-1(r-c)⟩ is zero for all δ r∈ ifr=c+W(q-e). Inserting this together with (<ref>) and (<ref>) into(<ref>) then yields the dual functional φ(q)=⟨c, q⟩- ∫_^d PlogQ +1/2q-e_W^2 +c_0.Consequently the dual of the (primal) optimization problem (<ref>) is equivalent to min_q∈ ⟨ c, q ⟩ - ∫_^d P log Q + 1/2q - e_W^2 . For every p ∈∖{0} the functional in (<ref>) is strictly convex and has a unique minimizer q̂∈∖{0}. Moreover, there exists a unique r̂∈, a unique ĉ∈∂ and a (not necessarily unique) nonnegative singular measure ν̂ with support (ν̂) ⊆{∈^d|Q̂(e^i) = 0 }such that r̂_ = ∫_𝕋^d( P/Q̂+ ν̂)for all ∈Λ, ĉ_ = ∫_𝕋^dν̂,for all ∈Λ, and the measure μ̂() = P(e^i)/Q̂(e^i)() + ν̂()is an optimal solution to the primal problem (<ref>). Moreover, ν̂ can be chosen with support in at most |Λ|-1 points.The objective functionalof the dual problem (<ref>) can be written as the sum of two terms, namely _1(q) =⟨c̃, q ⟩- ∫_^d P log(Q) and_2(q)=⟨c-c̃, q ⟩+1/2q - e_W^2,where c̃∈. The functional _1 is strictly convex (Theorem <ref>), and trivially the same holds for _2 since it is a positive definite quadratic form. Consequently, =_1+_2 is strictly convex, as claimed. Moreover, _1 is lower semicontinuous<cit.> with compact sublevel sets _1^-1(-∞,ρ] <cit.>. Likewise,_2 is continuous with compact sublevel sets. Thereforeis lower semicontinuous with compact sublevel sets and therefore has a minimum q̂, which must be unique by strict convexity. In view of (<ref>), the optimal value of r is given byr̂=c +W(q̂-e)and is hence unique. Since therefore the linear term c+W(q-e)in the gradient oftakes the value r̂ at the optimal point, the analysis in <cit.> applies with obvious modifications, showing that there is a ĉ∈, which then must be unique, such thatr̂_ = ∫_𝕋^d P/Q̂+ĉ_.Moreover, there is a discrete measure ν̂ with support in at most |Λ|-1 points such that (<ref>) holds; see, e.g., <cit.>. Then (<ref>) holds as well. In view of (<ref>), ⟨ĉ,q̂⟩ =∫_𝕋^dQ̂ν̂=0,and consequently ĉ∈∂, and the support of ν̂ must satisfy (<ref>). Finally, let r be given in terms ofby (<ref>), and let () be the corresponding primal functional in (<ref>).Then, for any such , ()= ℒ(Φ,ν, r, q̂)≤ℒ(Φ̂,ν̂, r̂, q̂) =(μ̂),and henceis an optimal solution to the primal problem (<ref>), as claimed.We collect the KKT conditions in the following corollary.The conditions q̂∈, ĉ∈∂, ⟨ĉ, q̂⟩ = 0 r̂_ = ∫_^dP/Q̂ + ĉ_,∈Λr̂- c = W(q̂ - e). are necessary and sufficient conditions for optimality of the dual pair(<ref>) and(<ref>) of optimization problems. § ON THE WELL-POSEDNESS OF THE SOFT-CONSTRAINED PROBLEM In the previous sections we have shown that the primal and dual optimization problems are well-defined. Next we investigate the well-posedness of the primal problemas an inverse problem. Thus, we firstestablish continuity of the solutions q̂ in terms of the parameters W, c and p.§.§ Continuity of q̂ with respect to c, p and WWe start considering the continuity of the optimal solution with respect to the parameters. The parameter set of interest is𝒫={()| c∈, p∈∖{0}, W>0}. Let _(q)=⟨ c, q ⟩ - ∫_^d P log Q + 1/2q - e_W^2.Then the map ()↦q̂:=_q∈_(q) is continuous on𝒫. Following the procedure in<cit.> we use the continuity of the optimal value (Lemma <ref>) to show continuity of the optimal solution. To this end, let () be a sequence of parameters in 𝒫 converging to ()∈𝒫 as k→∞. Moreover, defining _k(q):=_ (q) and (q):=_ (q)for simplicity of notation, let q̂_k = _q ∈𝔓̅_+𝕁_k(q) and q̂ = _q ∈𝔓̅_+𝕁(q). By Lemma <ref>, (q̂_k) is bounded, and hence there is a subsequence, which for simplicity we also call (q̂_k), converging to a limitq_∞. If we can show that q_∞ = q̂, then the theorem follows.To this end,choosing a q_0 ∈𝔓_+, we have𝕁_k(q̂_k)= 𝕁_k (q̂_k + ε q_0) - ⟨, ε q_0 ⟩ + ∫_^dlog( Q̂_k + ε Q_0/Q̂_k)+ 1/2q̂_k-e_^2 -1/2q̂_k+ε q_0-e_^2 ≥𝕁_k (q̂_k + ε q_0) - ⟨, ε q_0 ⟩ + 1/2q̂_k-e_^2-1/2q̂_k+ε q_0-e_^2.Consequently, by Lemma <ref>, 𝕁(q̂) = lim_k →∞𝕁_k(q̂_k) ≥lim_k →∞𝕁_k(q̂_k + ε q_0) - ε⟨, q_0 ⟩ + 1/2q̂_k-e_^2 -1/2q̂_k+ε q_0-e_^2.However q̂_k + ε q_0 ∈𝔓_+, and, since (,q)↦𝕁_(q) is continuous in 𝒫×, we obtain𝕁(q̂)≥lim_k →∞(𝕁_k(q̂_k + ε q_0) - ε⟨, q_0 ⟩ + 1/2q̂_k-e_^2-1/2q̂_k+ε q_0-e_^2)= 𝕁(q_∞ + ε q_0) - ε⟨ c, q_0 ⟩+ 1/2q_∞-e_W^2 -1/2q_∞+ε q_0-e_W^2 .Letting ε→ 0 in (<ref>), we obtain the inequality 𝕁(q̂)≥𝕁(q_∞). By strict convexity ofthe optimal solution is unique, and hence q̂=q_∞. §.§ Continuity of ĉ with respect to q̂We have now established continuity from () to q̂. In the same way as in Proposition <ref> we are also interested in continuity of the map ()↦ (q̂,ĉ). This would follow if we could show that the map from q̂ to ĉ is continuous. From the KKT condition (<ref>), it is seen that r̂ is continuous in c, W and q̂. In view of (<ref>), i.e.,r̂_= ∫_^d P/Q̂+ ĉ_,∈Λ continuity of ĉ wouldfollow if ∫_^d PQ̂^-1 is continuous in (p,q̂) whenever it is finite. If p∈, this follows from the continuity the map q̂↦Q̂^-1 in L_1(^d). For the case d≤ 2, this is trivial since if∫_^dQ̂^-1 is finite, then q̂∈ and Q̂ is bounded away from zero (cf., Proposition <ref>).However, for the cased>2 the optimal q̂ may belong to the boundary ∂, i.e., Q̂ is zero in some point.The following proposition shows L_1 continuity of q̂↦Q̂^-1for certain cases.For d≥3, let q̂∈ and suppose that the Hessian ∇_ Q̂ is positive definite in each pointwhere Q̂ is zero. Then Q̂^-1∈ L_1(^d) and the mapping from the coefficient vector q∈ to Q^-1 is L_1 continuous in the point q̂.The proof of this proposition is given in <cit.>.From Propositions <ref> and <ref> the following continuity result follows directly. For all c∈, p∈, W>0, the mapping (c,p,W)→(q̂, ĉ) is continuous in any point () for which the Hessian ∇_ Q̂ is positive definite in each point where Q̂ is zero. The conditionp ∈ is needed, since we may have pole-zero cancelations in P/Q̂ when p∈∂, and then ∫_^d P/Q̂ may be finite even if Q̂^-1∉L_1. The following example shows that this may lead to discontinuities in the map p↦ĉ (cf. Example 3.8 in <cit.>). Letc = [ 1; 3; 1 ] =[ 0; 2; 0 ] +[ 1; 1; 1 ]=∫_-π^π[ e^-iθ; 1;e^iθ ] ( 2 +ν_0),where =dθ/2π and ν_0 is the singular measure δ_0(θ)dθ with support in θ=0. Since :=2+ν_0 is positive, c ∈. Moreover, sinceT_c = [ 3 1; 1 3 ] > 0we have that c ∈ (see, e.g., <cit.>). Thus we know <cit.> that for each p ∈ we have a unique q̂∈ such that P/ Q̂ matches c, and hence ĉ = 0.However, for p = 2 (-1, 2, -1)' we have that q̂ = (-1, 2, -1)' and ĉ = (1, 1, 1)' (Theorem <ref>). Then, for the sequence (p_k), where p_k=2 (-1, 2 + 1/k, -1) ∈, we have ĉ_k = 0, so lim_k →∞ĉ_k = lim_k →∞[ 0; 0; 0 ]≠[ 1; 1; 1 ],which shows that the mapping p →ĉ is not continuous. § TUNING TO AVOID A SINGULAR PART In many situations we prefer solutions where there is no singular measure dν in the optimal solution.An interesting question is therefore for what prior P andweight W we obtain ν̂=0. The following result provides a sufficient condition.Let c∈and let p be the Fourier coefficients of the prior P. If the weight satisfies[Here A_2,1=max_c≠ 0Ac_1/c_2 denotes the subordinate (induced) matrix norm.] W^-1/2_2,1<c-p_W^-1^-1, then the optimal solution of (<ref>) is on the form μ̂=(P/Q̂) dm,i.e., the singular part ν̂ vanishes.Note that for a scalar weight, W=λ I the bound (<ref>) simplifies to λ> |Λ|^1/2c-p_2, where |Λ| is the cardinality of index set Λ. For the proof of Proposition <ref> we need the following lemma. Condition (<ref>) implies W^-1(r̂-c)_1<1,where r̂ is the optimal value of r in problem (<ref>).Let 𝕀(,r):= (P, ) + 1/2r - c_W^-1^2be the cost function of problem (<ref>), and let (μ̂,r̂) be the optimal solution. Clearly, 𝕀(Pdm,p)≥𝕀(μ̂,r̂), and consequently r̂-c_W^-1 ≤p-c_W^-1,since (P, μ̂)≥ 0 and (P,P)=0. Therefore,W^-1(r̂ - c)_1≤W^-1/2_2,1W^-1/2(r̂ - c)_2= W^-1/2_2,1r̂ - c_W^-1≤W^-1/2_2,1p - c_W^-1, which is less than one by (<ref>). Hence(<ref>) implies (<ref>). Suppose the optimal solution has a nonzero singular part ν̂, and form thedirectional derivative of (<ref>) at (μ̂,r̂) in the direction -ν̂. Then Φ in (<ref>) does not vary, and δ𝕀(μ̂,r̂;-ν̂,δr)=-∫_^dν̂ +δr^*W^-1(r̂ - c),whereδr_= -∫_^d ν̂.Then |δ r_|≤∫ν̂ for all ∈Λ, and hence|δr^*W^-1(r̂ - c)|≤W^-1(r̂-c)_1 ∫_^d ν̂ <∫_^d ν̂,by (<ref>) (Lemma <ref>). Consequently,δ𝕀(μ̂,r̂;-ν̂,δr)<0whenever ν̂ 0, which contradicts optimality. Hence ν̂ must be zero.The condition of Proposition <ref> is just sufficient and is in general conservative. To illustrate this, we consider a simple one-dimensional example (d=1).Consider a covariance sequence (1,c_1), where c_1 0, and a prior P(e^iθ)=1-cosθ, and setW =λ I. Then, since c=[ c_1; 1; c_1 ] and p=[ -1/2;1; -1/2 ],the sufficient condition (<ref>) for an absolutely continuous solution is λ > √(32)|1+2c_1|.We want to investigate how restrictive this condition is. Clearly we will have a singular partif and only if Q̂ = q_0 P, in which case we have q̂=q_0[ -1/2;1; -1/2 ]and ĉ=β[ 1; 1; 1 ]for some β >0. In fact, it follows from ⟨ĉ, q̂⟩ = 0 in (<ref>) that ĉ_1=ĉ_0. Moreover, (<ref>) and(<ref>) yieldr̂ = ∫P/Q̂[e^iθ; 1; e^-iθ ] dm +ĉ= [ β; β+1/q_0; β ]c = r̂-λ(q-e)=[ β+λ q_0/2,; β+1/q_0-λq_0+λ; β+λq_0/2 ].By eliminating β, we get c_1=1-1/q_0-3/2q_0 λ+λ,and solving for q_0 yieldsq_0=λ+c_1-1+ (6λ+ (λ+c_1-1)^2 )^1/2/3λ(note that λ >0 and q_0>0). Again, using (<ref>) we haveβ =c_1-λq_0/2=c_1-1/6( λ+c_1-1+ (6λ+ (λ+c_1-1)^2)^1/2).We are interested in λ for which β >0, i.e.,6c_1-(λ+c_1-1)>(6λ+ (λ+c_1-1)^2)^1/2,which is equivalent to the two conditions 1+5c_1>λ 2c_1(1+2c_1)>λ(1+2c_1), which could be seen by noting that the left member of (<ref>) must be positive and then squaring both sides. To find out whether this has a solution we consider three cases, namely c_1<-1/2, -1/2<c_1<0, and c_1>0. For c_1<-1/2, condition (<ref>) becomes2c_1<λ <1+5c_1, which is impossible since 1+5c_1<2c_1. Condition (<ref>) cannot be satisfied when -1/2<c_1<0, because then λ would be negative which contradicts λ >0. When c_1>0,Condition (<ref>) is satisfied if and only if λ <2c_1.Consequently, there is no singular part if either c_1 is negative orλ≥ 2c_1. This shows that the condition (<ref>) is not tight. § COVARIANCE EXTENSION WITH HARD CONSTRAINTSThe alternative optimization problem(<ref>) amounts to minimizing (P, ) subject to the hard constraintr - c _W^-1^2 ≤ 1, where r_ = ∫_^d. Hard constraints of this type were used in <cit.> in the context of entropy maximization. In general the data c∉, whereas, by definition, r∈. Consequently, a necessary condition for the existence of a solution is thatand the strictly convex set={ r|r - c _W^-1^2 ≤ 1}have a nonempty intersection. In the case that ∩⊂∂, this intersection only contains one point <cit.>. In this case, any solution to the moment problem contains only a singular part (Proposition <ref>), and then the primal problem (<ref>) has a unique feasible point r, but the objective function is infinite. Moreover, (P, )≥ 0 is strictly convex with (P, P)= 0, so if p∈ then (<ref>) has the trivial unique optimal solution =P, and r̂=p. The remaining case, p∉∩∅ needs further analysis. To this end, setting =Φ+ν, we consider the Lagrangian ℒ(Φ,ν, r, q, γ) =- (P, ) + ∑_∈Λ q_^* ( r_ - ∫_^d() ) +γ(1- r - c _W^-1^2 ) =- (P, ) + ⟨ r,q⟩ - ∫_^d Q+γ(1- r - c _W^-1^2 ),where γ≥ 0. Therefore, in view of (<ref>), ℒ(Φ,ν, r, q,γ) = ∫_^dPlogΦ dm -∫_^dQΦ dm -∫_^d Qν -∫_^dP(log P -1)dm +⟨ r,q-e⟩ +γ(1-r - c _W^-1^2 ) ,where, as before,e := [e_]_∈Λ, e_ 0 = 1 and e_ = 0 for ∈Λ∖{ 0 }, and hence r_=⟨ r,e⟩. This Lagrangian differs from that in (<ref>) only in the last term that does not depend on Φ. Therefore, in deriving the dual functionalφ(q,γ)=sup_Φ≥ 0,ν≥ 0,rℒ(Φ,ν, r, q,γ), we only need to consider q∈∖{0}, and a first variation in Φ yields (<ref>) and(<ref>). The directional derivative δℒ(Φ,ν, r, q,γ;δr)= q-e + 2γW^-1(r-c)is zero for r=c+1/2γW(q-e). Thus inserting (<ref>) and (<ref>) and (<ref>) into(<ref>) yields the dual functionalφ(q,γ)=⟨ c, q⟩ - ∫_^d Plog Q+1/4γq-e_W^2 +γ -c_to be minimized over all q∈∖{0} and γ≥ 0. Since dφ/dγ= -1/4γ^2q-e_W^2 +1, there is a stationary pointγ =1/2q-e_Wthat is nonnegative as required. For γ=0 we must have q=e, and consequently φ(q,γ) tends to zero as γ→ 0. By weak duality zero is therefore a lower bound for the minimization problem (<ref>), and (P, )=0, which corresponds to the trivial unique solution =P and r̂=p mentioned above. This solution is only feasible if p∈.Therefore we can restrict our attention to the case γ>0. Inserting (<ref>) into (<ref>) and removing the constant term c_, we obtain the modified dual functional(q)=⟨ c, q⟩ - ∫_^d Plog Q+ q-e_W.Moreover, combining (<ref>) and (<ref>), we obtainr-c_W^-1=1,which also follows from complementary slackness since γ>0 and restricts r to the boundary of .Suppose that p∈∖{0}, p∉ and ∩∅. Then the modified dual problem min_q∈(q) has a unique solution q̂∈∖{0}. Moreover, there exists a unique r̂∈, a unique ĉ∈∂ and a (not necessarily unique) nonnegative singular measure ν̂ with support (ν̂) ⊆{∈𝕋^d|Q̂(e^i) = 0 }such that r̂_ = ∫_𝕋^d( P/Q̂+ ν̂)for all ∈Λ, ĉ_ = ∫_𝕋^dν̂,for all ∈Λ, and the measure μ̂() = P(e^i)/Q̂(e^i)() + ν̂()is an optimal solution to the primal problem (<ref>). Moreover, r̂-c_W^-1=1,and ν̂ can be chosen with support in at most |Λ|-1 points. If p∈, the unique optimal solution is =P, and then r̂=p. If ∩⊂∂, any solution to the moment problem will have only a singular part.Finally, if ∩=∅, then the problem (<ref>) will have no solution. We begin by showing that the functionalhas a minimum under the stated conditions. To this end, we first establish that the functionalhas compact sublevel sets ^-1(-∞,ρ], i.e., q_∞ is bounded for all q suchthat(q)≤ρ, where ρ is sufficiently large for the sublevel set to be nonempty. The functional (<ref>) can be decomposed in a linear and a logarithmic term as(q)=h(q) - ∫_^d PlogQ +c_,where h(q):= ⟨ c, q-e⟩ +q-e_W. The integral term will tend to -∞ as q_∞→∞.Therefore we need to have the linear term to tend to +∞ as q_∞→∞, in which case we can appeal to the fact that linear growth is faster than logarithmic growth.However, if c∉ as is generally assumed, there is a q∈ such that⟨ c, q⟩< 0, so we need to ensure that the positive term q-e_W dominates. Let r̃∈∩∅. Then,by Theorem <ref>,there is a positive measure μ̃=Φ̃dm+ ν̃ with a nonzero Φ̃ such that r̃=∫_^d μ̃, and r̃ satisfies the constraints in the primal problem(<ref>). Consequently, φ(q,γ)≥ℒ(Φ̃,ν̃, r̃, q,γ)≥-(P,μ̃) for all q∈ and γ≥ 0, which in particular implies that (q)≥ -(P,μ̃)for all q∈.Now, if there is a q∈ such that h(q)≤ 0, then (λ q)→ -∞ as λ→∞, which contradicts (<ref>). Therefore, h(q)> 0 for all q∈. Then, since h is continuous, it hasa minimum ε on the compact set K:={q∈∖{0}|q-e_∞ =1}. As e∉K,ϵ >0. Therefore,h(q) ≥εq-e_∞≥εq_∞- εe_∞≥ε/|Λ| Q_∞- εe_∞,since Q_∞≤ |Λ|q_∞ <cit.>.Likewise, ∫_𝕋^d P log Q=∫_𝕋^dP log[ Q/Q_∞] +∫_𝕋^dPlogQ_∞≤∫_𝕋^dP logQ_∞,since Q/Q_∞≤ 1. Henceρ≥(q) ≥ε/|Λ |Q_∞ -∫_𝕋^dP logQ_∞- εe_∞.Comparing linear and logarithmic growthwe see that the sublevel set is bounded from above and below.Moreover, a trivial modification of <cit.> shows thatis lower semi-continuous, and hence^-1(-∞,ρ] is compact. Consequently, the problem (<ref>) has an optimal solution q̂.Next we show that q̂ is unique. For this we return to the original dual problem to find a minimum of (<ref>). The solution q̂ is a minimizer of φ(q,γ̂), where γ̂=1/2q̂-e_W,and (q̂)=φ(q̂,γ̂)+c_. To show that φ is strictly convex, we form the HessianH=[ ∫_𝕋^dP/Q^20;00 ] +1/2γ^3[ γ^2 W-γ(q-e)^*W;-γW(q-e) (q-e)^*W(q-e) ] and the quadratic form[ x; ξ ]^*H[ x; ξ ]= x^* (∫_𝕋^dP/Q^2) x + 1/2γ^3 [γx - ξ(q-e)]^*W[γx - ξ(q-e)],which is positive for all nonzero (x,ξ), since (q-e) 0 and γ>0. Consequently, φ has a unique minimizer (q̂,γ̂), where q̂ is the unique minimizer of .It follows from (<ref>) and (<ref>) that r̂=c + W(q̂-e)/q̂-e_W,which consequently is unique. Moreover, h(q̂)=⟨r̂,q̂⟩ -r̂_, and hence we can follow the same line of proof as in Theorem <ref> to show that there is a unique ĉ∈∂ such that ⟨ĉ,q̂⟩=0 and a positive discrete measure ν̂ with support in |Λ| -1 points so that (<ref>) and (<ref>) hold. Next, let ()=-(Pdm,) be the primal functional in (<ref>), whereis restricted to the set of positive measures :=Φ + dνsuch that r, given by (<ref>), satisfies the constraint r-c_W≤ 1.In view of (<ref>), ()= ℒ(Φ,ν, r, q̂,γ̂)≤ℒ(Φ̂,ν̂, r̂, q̂,γ̂) =(μ̂)for any such , and henceis an optimal solution to the primal problem (<ref>).Finally, the cases p∈, ∩⊂∂, and ∩=∅ have already been discussed above.Suppose that p∈∖{0} and ∩∅. The KKT conditions q̂∈, ĉ∈∂, ⟨ĉ, q̂⟩ = 0 r̂_ = ∫_^dP/Q̂ + ĉ_,∈Λ(r̂-c)q̂-e_W = W(q̂-e), r̂∈ are necessary and sufficient conditions for optimality of the dual pair(<ref>) and(<ref>) of optimization problems.The corollary follows by noting that, if p∈, then we obtain the trivial solution q̂=e, which corresponds to the primal optimal solution =P.The conditionW > cc^*is sufficient for the pair (<ref>) and (<ref>) of dual problems to have optimal solutions. If W>cc^*, then (q-e)^*W(q-e)≥⟨ c, q-e⟩ ^2 with equality only for q=e. Hence, if q e, q-e_W > |⟨ c, q-e⟩|, i.e., h(q) > 0 for all q∈∖{0} except q=e. Then we proceed as in the proof of Theorem <ref>.Condition (<ref>) guarantees that 0 ∈ int() and hence in particular that ∩∅ as required in Theorem <ref>. To see this, note that 0∈ and that r=0 satisfies the hard constraint in (<ref>) if c^*W^-1c≤ 0. However, since W>cc^*, there is a W_0 > 0 such that W = W_0 + cc^*. Then the well-known Matrix Inversion Lemma (see, e.g., <cit.>) yields(W_0 + cc^*)^-1 = W_0^-1 - W_0^-1 c (1 + c^*W_0^-1c)^-1 c^* W_0^-1,and thereforec^*W^-1c = c^*W_0^-1c - c^*W_0^-1 c (1 + c^*W_0^-1c)^-1 c^* W_0^-1c = c^*W_0^-1c/1 + c^*W_0^-1c < 1,which establishes that 0 ∈ int(). However, for ∩ to be nonempty, r=0 need not be contained in this set. Hence, condition (<ref>) is not necessary, although it is easily testable. In fact, this provides an alternative proof of Proposition <ref>. § ON THE EQUIVALENCE BETWEEN THE TWO PROBLEMSClearly ∩ is always nonempty if c∈. Then both the problem (<ref>) with soft constraints and the problem (<ref>) with hard constraintshave a solution for any choice of W. On the other hand, if c ∉, the problem with soft constraints will always have a solution, while the problem with hard constraints may fail to have one for certain choices of W. However, if the weight matrix in the hard-constrained problem – let us denote it W_ hard – is chosen in the set 𝒲:={W > 0 |∩∅, p∉}, then it can be seenfrom Corollaries <ref> and <ref> that we obtain exactly the same solution q̂ in the soft-constrained problem by choosing W_ soft=W_ hard/q̂-e_W_ hard.We note that(<ref>) can be written W_ hard=α W_ soft, where α :=q̂-e_W_ hard. Therefore, substituting W_ hard in (<ref>), we obtainW_soft=αW_soft/q̂-e_αW_soft=α^1/2W_soft/q̂-e_W_soft,which yieldsα=q̂-e_W_ soft^2. Hence the inverse of (<ref>) is given by W_ hard=W_ softq̂-e_W_ soft^2.By Theorem <ref> q̂ is continuous in W_ soft, and hence, by (<ref>), the corresponding W_ hard varies continuously with W_ soft. In fact, this can be strengthened to a homeomorphism between the two weight matrices.The map (<ref>) is a homeomorphism between 𝒲 and the space of all (Hermitian positive definite) weight matrices, and the inverse is given by (<ref>). By <cit.>, a continuous mapbetween two spaces of the same dimension is a homeomorphism if and only if it is injective and proper, i.e., the preimageof any compact set is compact. To see that 𝒲 is open, we observe thatis continuous in W and thatis an open set. As noted above, the map (<ref>) – let us call it f – is continuousandalso injective, as it can be inverted. Hence it only remains to show that f is proper. To this end, we take a compact set K⊂𝒲 andshow that f^-1(K) is also compact. There are two ways this could fail. First, the preimage could contain a singular semidefinite matrix. However this is impossible by (<ref>), since q̂_∞ is bounded for W_hard∈ K (Lemma <ref>) anda nonzero scaling of a singular matrix cannot be nonsingular. Secondly, W_ soft_F could tend to infinity. However, this is also impossible. To see this, we first show that there is a κ >0 such that p-r_W_ hard^-1≥κ for all r∈𝔖_W_ hard and all W_ hard∈ K. To this end, we observe that the minimum of p-r_W^-1 over all W∈ K and r satisfying the constraint r-c_W^-1≤ 1 is bounded byκ:= min_W∈K p-c_W^-1 -1 by the triangle inequality p-r_W^-1≥p-c_W^-1-c-r_W^-1≥p-c_W^-1-1.The minimum is attained, since K is compact, and positive, sincep∉⋃_W∈ K𝔖_W. Now, from Corollary <ref> we see that q̂=e if and only if r̂=p. The map from q̂↦r̂ is continuous in q=e. In fact, Q̂ is uniformly positive in a neighborhood of e and hence the corresponding ĉ=0.Due to this continuity, if q̂→ e, then r̂→ p, which cannot happen since p-r_W^-1≥κ for all W∈ K. Thus, since q̂-e_W is bounded away from zero, the preimage f^-1(K) of K is bounded. Finally, consider a convergent sequence (W_k)in f^-1(K) converging to a limit W_∞. Since the sequence is bounded and cannot converge to a singular matrix, we must have W_∞ >0, i.e.,W_∞∈ f^-1(𝒲). By continuity, f(W_k) tends to the limit f(W_∞), which must belong to K since it is compact.Hence the preimage W_∞ must belong to f^-1(K).Therefore, f^-1(K) is compact as claimed.It is illustrative to consider the simple case when W=λ I. Then the two maps (<ref>) and (<ref>) becomeλ_ soft =√(λ_ hard)/q̂-e_2 λ_ hard =λ_ soft^2q̂-e_2^2Whereas the range of λ_ soft is the semi-infinite interval (0,∞), for the homeomorphism to holdλ_ hard is confined to λ_min <λ< λ_max,where λ_ min is the distance from c to the coneand λ_ max=c-p. When λ_ soft→∞, λ_ hard→λ_ max and q̂→ e. If λ_ hard≥λ_ max, then the coresponding problem has the trivial unique solution q̂= e, corresponding to the primal solution =P.Note that Theorem <ref> implies that some continuity results in one of the problems can be automatically transferred to the other problem. In particular, we have the following result.Let _W(q)=⟨ c, q ⟩ - ∫_^d P log Q + q - e_W.Then the map W↦q̂:=_q∈_W(q) is continuous.The theorem follows by noting that W↦q̂:=_q∈_W(q) can be seen as a composition of two continuous maps, namely the one in Theorem <ref>and the one in Theorem <ref>. Next we shall vary also c and p, and to this end we introduce a more explicit notation forand 𝒲, namely = in (<ref>) and 𝒲_c,p:={W>0|∩∅, p∉}.Then the corresponding set of parameters (<ref>) for the problem with hard constraints is given by𝒫_ hard={()| c∈, p∈∖{0}, W∈𝒲_c,p},the interior of which isint(𝒫_ hard)={()| c∈, p∈, W∈𝒲_c,p}.Theorem <ref> can now be modified accordingly to yield the following theorem, the proof of which is deferred to the appendix.Let the map (c,p,W_ hard)↦ W_ soft be given by (<ref>) and the map (c,p,W_ soft)↦ W_ hard by (<ref>). Then the map that sends (c,p,W_ hard)∈ int(𝒫_ hard) to (c,p,W_ soft)∈ int(𝒫) is a homeomorphism. Note that this theorem is not a strict amplification of Theorem <ref> as we have given up the possibility for p to be on the boundary ∂. The same is true for the following modification of Theorem <ref>. Let _(q) be as in (<ref>). Then the map ()↦q̂:=_q∈_(q) is continuous onint(𝒫_ hard). The theorem follows immediately by noting that (_hard)↦q̂ can be seen as a composition of two continuous maps, namely (_hard)↦(_soft) of Theorem <ref> and (_soft)↦q̂ of Theorem <ref>. Theorem <ref>is a counterpart of Theorem <ref> for the problem with hard constraints, except that p is restricted to the interior . It should be possible to extend the result to hold for all p∈∖{0} via a direct proof along the lines of the proof of Theorem <ref>.§ ESTIMATING COVARIANCES FROM DATA For a scalar stationary stochastic process {y(t); t∈}, it is well-known that the biased covariance estimate c_k = 1/N∑_t = 0^N - k -1y_ty̅_t+k ,based on an observation record { y_t }_t = 0^N-1, yields a positive definite Toeplitz matrix, which is equivalent to c ∈ <cit.> In fact, these estimates correspond to the ones obtained from the periodogram estimate of the spectrum (see, e.g., <cit.>). On the other hand, the Toeplitz matrix of the unbiased estimate c_k = 1/N-k∑_t = 0^N -k-1y_t y̅_t+kis in general not positive definite.The same holds in higher dimensions (d>1) where the observation record is {y_}_∈^d_N with^d_N={(ℓ_1,…, ℓ_d) |0≤ℓ_j≤ N_j-1, j=1,…, d}. The unbiased estimate is then given byc_ = 1/∏_j=1^d (N_j-|k_j|)∑_∈^d_N y_y̅_+,and the biased estimate byc_ = 1/∏_j=1^d N_j∑_∈^d_N y_y̅_+,where we define y_=0 for ∉^d_N. The sequence of unbiased covariance estimates does not in general belong to , butthe biased covariance estimates yields c∈ also in the multidimensional setting. In fact, this can be seen by noting that the biased estimate corresponds to the Fourier coefficients of the periodogram <cit.>, i.e.,if the estimates c_ are given by (<ref>), then Φ_ periodogram() := 1/∏_j=1^d N_j| ∑_∈^d_N y_|^2=∑_∈^d_N-^d_N c_ ,where ^d_N-^d_N denotes the Minkowski set difference. This leads to the following lemma. Given the observed data { y_}_∈^d_N, let { c_}_∈Λ be given by (<ref>).Then c ∈.Given { y_}_∈^d_N, let c = { c_}_∈^d_N, where c_ be given by (<ref>).In view of(<ref>) and (<ref>) we have⟨ c, p ⟩ = ∫_^d1/∏_j=1^d N_j| ∑_∈^d_N y_|^2 P(e^i) (),which is positive for all p ∈∖{0}. Consequently c ∈.An advantage of the approximate procedures to the rational covariance extension problem is that they can also be used for cases where the biased estimate is not available, e.g., where the covariance is estimated from snapshots. § APPLICATION TO SPECTRAL ESTIMATION As long as we use the biased estimate (<ref>), we may apply exact covariance matching as outlined in Section <ref>, whereas in general approximate covariance matching will be required for biased covariance estimates. However, as will be seen in the following example, approximate covariance matching may sometimes be better even if c∈.In this application it is easy to determine a bound on the acceptable error in the covariance matching, so we use the procedure with hard constraints. Given data generated from a two-dimensional stochastic system, we test three different procedures, namely (i) using the biased estimate and exact matching, (ii) using the biased estimate and the approximate matching (<ref>), and (iii)using the unbiased estimate and the approximate matching (<ref>). The procedures are then evaluated by checking the size of the errorbetween the matched covariances and the true ones from the dynamical system. §.§ An exampleLet y_(t_1, t_2) be the steady-state output of a two-dimensional recursive filter driven by a white noise input u_(t_1, t_2). Let the transfer function of the recursive filter beb(e^iθ_1, e^iθ_2)/a(e^iθ_1, e^iθ_2) = ∑_∈Λ_+ b_ e^-i(,)/∑_∈Λ_+ a_ e^-i(,),where Λ_+={(k_1,k_2)∈^2| 0≤ k_1≤ 2, 0≤ k_2≤ 2 } and the coefficients are given by b_(k_1,k_2)=B_k_1+1, k_2+1 and a_(k_1,k_2)=A_k_1+1, k_2+1, whereB = [0.9-0.20 0.05;0.2 - 0.30 0.05;-0.05-0.05 0.10 ], A = [ -1.0 -0.1 -0.1; -0.2 -0.2 -0.1; -0.4 -0.1 -0.2 ].The spectral density Φ of y_(t_1, t_2), which is shown in Fig. <ref> and is similar to the one considered in <cit.>, is given byΦ(e^iθ_1, e^iθ_2) = P(e^iθ_1, e^iθ_2)/Q(e^iθ_1, e^iθ_2) = |b(e^iθ_1, e^iθ_2)/a(e^iθ_1, e^iθ_2)|^2,and hence the index set Λ of the coefficients of the trigonometric polynomials P and Q is given by Λ=Λ_+-Λ_+={(k_1,k_2)∈^2 ||k_1|≤ 2, |k_2|≤ 2 }. Using this example, we perform two different simulation studies. §.§ First simulation studyThe system was simulated for 500 time steps along each dimension, starting from y_(t_1, t_2) = u_(t_1, t_2) = 0 whenever either t_1 < 0 or t_2 < 0. Then covariances were estimated from the 9 × 9 last samples, using both the biased and the unbiased estimator. With this covariance data we investigate thethree procedures (i), (ii) and (iii)described above. In each case, both the maximum entropy (ME) solutions and solutions with the true numerator are computed.[Maximum entropy: P≡ 1. True numerator: P=P_true.] The weighting matrix is taken to be W = λ I, where λ is λ_biased :=c_true - c_biased_2^2 in procedure (ii) and λ_unbiased :=c_true - c_unbiased_2^2 in procedure (iii). [Note that this is the smallest λ for which the true covariance sequence belongs to the uncertainty set {r | r-c_2^2≤λ}.] The norm of the error [Here we use the norm of the covariance estimation error as measure of fit. However, note that this is not the only way to compare accuracy of the different methods. The reason for this choice is that comparing theaccuracy of the spectral estimates is not straightforward since it depends on the selected metric or distortion measure.]between the matched covariances and the true ones, r̂ - c_true_2, is shown in Table. <ref>. The means and standard deviations are computed over the 100 runs.The biased covariance estimates belong to the cone(Lemma <ref>), and therefore procedure (i) can be used. The corresponding error in Table <ref> is the statistical error in estimating the covariance. This error is quite large because of ashort data record. Using approximate covariance matching in this case seems to give a worse match. However, approximate matching of the unbiased covariances gives as good a fit as exact matching of the biased ones.§.§ Second simulation studyIn this simulation the setup is the same as the previous one, except that the simulation data has been discarded if the unbiased estimate belongs to . To obtain 100 such data sets, 414 simulations of the system were needed. (As a comparison, in the previous experiment 23 out of the 100 runs resulted in an unbiased estimate outside .) Again, the norm of the error between matched covariances and the true ones is shown in Table <ref>, and the means and standard deviations are computed over the 100 runs. As before, the biased covariance estimates belong to the cone , and therefore procedure (i) can be used. Comparing this with the results from procedure (ii)suggests that there may be an advantage not to enforce exact matching, although we know that the data belongs to the cone.Regarding procedure (iii), we know that the unbiased covariance estimates do not belong to the cone , hence we need to use approximate covariance matching. In this example, this procedure turns out to give the smallest estimation error.§ APPLICATION TO SYSTEM IDENTIFICATION AND TEXTURE RECONSTRUCTION Next we apply the theory of this paper to texture generation via Wiener system identification. Wiener systems form a class of nonlinear dynamical systems consisting of a linear dynamic part composed with a static nonlinearity as illustrated in Figure <ref>. This is a subclass of so called block-oriented systems <cit.>, and Wiener system identification is a well-researched area (see, e.g., <cit.> and references therein) that is still very active <cit.>. Here, we useWiener systems to model and generate textures. Using dynamical systems for modeling of images and textures is not new and has been considered in, e.g., <cit.>. The setup presented here is motivated by <cit.>, where thresholded Gaussian random fields are used to model porous materials for design of surface structures in pharmaceutical film coatings. Hence we let the static nonlinearity, call it f, be a thresholding with unknown thresholding parameter τ. In our previous work <cit.> we applied exact covariance matching to such a problem. However, in general there is no guarantee that the estimated covariance sequence c belongs to the cone . Consequently, here we shall use approximate covariance matching instead.The Wiener system identification can be separated into two parts. We start by identifyingthe nonlinear part. Using the notations of Figure <ref>, let {u_;∈^d} be a zero-mean Gaussian white noise input, and let {x_;∈^d} be the stationary output of the linear system, which we assume to be normalized so that c_0:=[x_^2]=1. Moreover, let y_=f(x_) where f is the static nonlinearityf(x)=1 x>τ 0with unknown thresholding parameter τ. Since [y_] = 1-ϕ(τ), where ϕ(τ) is the Gaussian cumulative distribution function, an estimate of τ is given by τ_ est= ϕ^-1(1-[y_]).Now, let c_^x := [x_+ x_] be the covariances of x_, and let c_^y:= [y_+ y_] - [y_ + ][y_] be the covariances of y_. As was explained in <cit.>, by using results from <cit.> one can obtain a relation between c_^y and c_^x, given by c_^y = ∫_0^c_^x1/2π√(1-s^2)exp(-τ^2/1+s)ds .This is an invertible map, which we compute numerically, and given τ_ est we can thus get estimates of the covariances c_^x from estimates of the covariances c_^y. However, even if c^y is is a biased estimate so that c^y ∈, c^x may not be a bona fide covariance sequence.§.§ Identifying the linear system Solving (<ref>) or (<ref>) for a given sequence of covariance estimates c, we obtain an estimate of the absolutely continuous part of the power spectrum Φ of that process. In the case d=1, Φ =P/Q can be factorized as Φ(e^iθ) = P(e^iθ)/Q(e^iθ) = |b(e^iθ)|^2/|a(e^iθ)|^2,which provides a transfer function of a corresponding linear system, which fed by awhite noise inputwill produce an autoregressive-moving-average (ARMA) process with an output signal with precisely thepower distributionΦ in steady state. For d≥ 2, a spectral factorization of this kind is not possible in general<cit.>, but instead there is always a factorization as a sum-of-several-squares <cit.>,Φ(e^i) = P(e^i)/Q(e^i) = ∑_k = 1^ℓ |b_k(e^i)|^2/∑_k = 1^m |a_k(e^i)|^2,the interpretation of which in terms of a dynamical system is unclear when m > 1. Therefore we resort to a heuristic and apply the factorization procedure in <cit.> although some of the conditionsrequired to ensure the existence of a spectral factor may not be met. (See <cit.> for a more detailed discussion.) §.§ Simulation resultsThe method, which is summarized in Algorithm <ref>, is tested on some textures from the Outex database<cit.> (available online at <http://www.outex.oulu.fi/>). These textures are color images and have thus been converted to binary textures by first converting them to black-and-white and then thresholding them. [The algorithm has been implemented and tested in Matlab, version R2015b. The textures have been normalized to account for light inhomogenities using a reference image available in the database. The conversion from color images to black-and-white images was done with the built-in function , and the threshold level was set to the mean value of the maximum and minimum pixel value in the black-and-white image.] Three such textures are shown in Figure <ref> through <ref>.indent=12pt In this example there is no natural bound on the error, so we use the problem with soft constraints, for which we choose the weight W = λ I with λ=0.01 for all data sets. Moreover, we do maximum-entropy reconstructions, i.e., we set the prior to P ≡ 1. The optimization problems are then solved by first discretizing the grid ^2, in this case in 50 × 50 points (cf. <cit.>), and solving the corresponding problems using the CVX toolbox <cit.>. The reconstructions are shown in Figures <ref> - <ref>. Each reconstruction seems to provide a reasonable visual representation of the structure of the corresponding original. This is especially the case for the second texture. § CONCLUSIONSIn this work we extend the results of our previous paper <cit.>on the multidimensional rational covariance extension problem to allow for approximate covariance matching. We have provided two formulations to this problem, and we have shown that they are connected via a homeomorphism. In both formulations we have usedweighted 2-norms to quantify the missmatch of the estimated covariances. However,we expect that by suitable modifications of the proofs similar results can be derived for other norms, since all norms have directional derivatives in each point <cit.>. These resultsprovide a procedure for multidimensional spectral estimation, but in order to obtain a complete theory for multidimensional system identification and realization theorythere are still some open problems, such as spectral factorization and interpretations in terms of multidimensional stochastic systems, as briefly discussed in Section <ref>.§ DEFERRED PROOFS Let B_ρ(x^(0)) denote the closed ball { x∈ X|x-x^(0)_X≤ρ}, where X is either a set of vectors or a set of matrices depending on then context. Thenorm ·_X is the Euclidean norm for vectors and Frobenius norm for matrices. Let 𝒫 be givenby (<ref>) and _ by (<ref>). Furthermore, let q̂:=min_q∈_(q). Thenthe map ()↦_(q̂) is continuous for()∈𝒫. Moreover, for any compact K⊂𝒫, the corresponding set of optimal solutions q̂ is bounded.The proof follows along the lines of Lemma 7.2 andProposition 7.4 in <cit.>. Let ()∈𝒫 be arbitrary and let B̃_ρ():=B_ρ() ×(B_ρ()∩) × B_ρ(),where ρ>0 is chosen so that B̃_ρ()⊂𝒫, i.e., ρ<_2 and W>0 for all W-_F≤ρ.First we will show that the minimizer q̂_ of 𝕁_ is bounded for all()∈B̃_ρ(). To this end, note that by optimality𝕁_(q̂_)≤𝕁_(e)=⟨ c, e ⟩ - ∫_^d P log 1 + 1/2e - e_W^2= c_ ,and hence 𝕁_c, p, W(p) is bounded from above on the compact set B̃_ρ(). Consequently, by using the same inequality as in the proof of <cit.>, we see that c_≥𝕁_(q̂_)≥⟨ c, q̂_⟩ -P_1 logQ̂__∞ + 1/2q̂_ - e_W^2.Due to norm equivalence between Q_∞ and q_W, and since the quadratic term is dominating, the norm of q̂_ is bounded in the set B̃_ρ().Now, let K⊂𝒫 be compact. We want to show that q̂ is bounded on K. Assume it is not. Then let ()∈ K be a sequence with q̂_k→∞. Since K is compact there is a converging subsequence ()→ ()∈ K with q̂_k→∞. Since ()∈ K there is a ρ>0 such that B̃_ρ()⊂𝒫. However, all but finitely many points () belong to B̃_ρ(), andsince q̂_k is bounded for all ()∈B̃_ρ(), we cannot have q̂_k→∞.Next, let (), ()∈B̃_ρ() and let q̂_1, q̂_2∈𝔓̅_+ be the unique minimizers of 𝕁_ and 𝕁_, respectively.Choose a q_0 ∈𝔓_+ and note that Q_0 is strictly positive and bounded. Byoptimality,𝕁_(q̂_1)≤𝕁_(q̂_2+ε q_0)𝕁_(q̂_2)≤𝕁_(q̂_1+ε q_0)for all ε >0. Hence, ifwe can show that, for any δ >0, there is an ε>0 and a ρ̃>0 such that|𝕁_(q̂_1+ε q_0) - 𝕁_(q̂_1)|≤δ |𝕁_(q̂_2+ε q_0)-𝕁_(q̂_2)|≤δ hold whenever -_2≤ρ̃, -_2≤ρ̃ and -_F≤ρ̃, then this would imply that𝕁_(q̂_2)-δ≤𝕁_(q̂_1)≤𝕁_(q̂_2)+δ,showing that the optimal value is continuous in .The lower bound is obtained by using (<ref>) and (<ref>), and the upper bound is obtained from (<ref>) and (<ref>). To prove (<ref>),we note that|𝕁_(q̂_1+ε q_0) - 𝕁_(q̂_1)|xx=|⟨-, q̂_1⟩+⟨, ε q_0 ⟩ - ∫_^dlog(1 + ε Q_0/Q̂_1) xxxx-∫_^d (-) log(Q̂_1+ε Q_0)+12q̂_1+ε q_0 -e_^2-12q̂_1 -e_^2| xx≤-_2q̂_1_2+ ε( ⟨, q_0 ⟩ + ∫_^dQ_0/Q̂_1)xxxx+-_2 log(Q̂_1+ε Q_0)_∞ +12|q̂_1+ε q_0 -e_^2- q̂_1 -e_^2|xxxx+12|q̂_1 -e_^2 -q̂_1 -e_^2|.Next we observe that0≤∫_^dQ_0/Q̂_1 =⟨r̂_1-ĉ_1,q_0⟩≤⟨ c_1+(q̂_1-e),q_0⟩by the KKT conditions (<ref>) and the fact that q_0∈, ĉ_1∈. Hence ε can be selected small enough forthe second and fourth term in (<ref>) each to be bounded by δ/5 for any (), ()∈ B_ρ(). Each of the remaining terms can now be bounded by δ/5 by selecting ρ̃ sufficiently small.Hence (<ref>) follows. This also proves (<ref>).Let 𝒫_ hard be givenby (<ref>) and _ by (<ref>). Furthermore, let q̂:=min_q∈_(q). Then for any compact K⊂𝒫_ hard, the corresponding set of optimal solutions q̂ is bounded.The proof follows closely the proof of the corresponding part of Lemma <ref>. Let ()∈𝒫_ hard be arbitrary and let B̃_ρ():=B_ρ() ×(B_ρ()∩) × B_ρ(),where ρ>0 is chosen so that B̃_ρ()⊂𝒫_ hard.To see that the minimizer q̂_ of 𝕁_ is bounded for all()∈B̃_ρ(), first note that by optimality𝕁_(q̂_)≤𝕁_(e)=⟨ c, e ⟩ - ∫_^d P log 1+ e - e_W= c_ ,and hence 𝕁_c, p, W(p) is bounded from above on the compact set B̃_ρ().Now let h(q):= ⟨ c, q-e⟩ +q-e_W, as in the proof of Theorem <ref>. Following the same line of argument as in that proof, we see that h(q)>0for all q∈ and ()∈B̃_ρ(). Since h is continuous in the arguments (q, ), it has a minimum ε>0 on the compact set of tuples (q, ) such that q∈∖{0}, q-e_∞ =1, and ()∈B̃_ρ() hold. Thus the second half of inequality (<ref>) still holds, i.e., 𝕁_(q) ≥ε/|Λ |Q_∞ -∫_𝕋^dP logQ_∞- εe_∞for all q. This is true in particular for q̂_, thus c_≥𝕁_(q̂_)≥ε/|Λ |Q__∞ -∫_𝕋^dP logQ__∞- εe_∞.Since the linear growth dominates the logarithmic growth, the norm of q̂_ is bounded on the set B̃_ρ(). The proof now follows verbatim from the argument in the second paragraph in the proof of Lemma <ref>. This is a modification of the proof of Theorem <ref>, again utilizing <cit.>,where we replace the map f defined by𝒲∋ W_ hard↦ W_ soft∈{W| W>0} and redefine it with the map int(𝒫_ hard)∋ (c,p,W_ hard)↦ (c,p,W_ soft)∈ int(𝒫). To show that f is a homeomorphism we need to show that the map is proper.To this end, we take a compact set K⊂ int(𝒫_ hard) and show that f^-1(K) is also compact.Again, there are two ways this could fail. First, the preimage could contain a singular semidefinite matrix. However this is impossible by (<ref>), since q̂_∞ is bounded for (c,p,W_hard)∈ K (Lemma <ref>) and a nonzero scaling of a singular matrix cannot be nonsingular.Secondly, W_ soft_F could tend to infinity. However, this is also impossible. To see this, we first show that there is a κ >0 such that p-r_W_ hard^-1≥κ for all r∈𝔖_c,W_ hard and all (c,p,W_ hard)∈ K.Again, using the triangle inequality p-r_W_ hard^-1≥p-c_W_ hard^-1-c-r_W_ hard^-1, we observe that the minimum of p-r_W_ hard^-1 over all (c,p,W_ hard)∈ K and r satisfying the constraint r-c_W_ hard^-1≤ 1 is bounded by κ:= min_(c,p,W_ hard)∈ Kp-c_W_ hard^-1 -1. The minimum is attained,as K is compact, and positive, sincep∉⋃_(c,W_ hard)∈ K. The remaining part of the proof now follows with minor modifications from the proof of Theorem <ref> by noting that q̂ is bounded away from e, and hence the preimage f^-1(K) is bounded. Therefore the limit of a sequence in the preimage must belong to f^-1(K), and hence f^-1(K) is compact as claimed.siam | http://arxiv.org/abs/1704.08326v2 | {
"authors": [
"Axel Ringh",
"Johan Karlsson",
"Anders Lindquist"
],
"categories": [
"math.OC",
"math.FA",
"42A70, 30E05, 49N45, 90C25, 62M15, 74E25"
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"primary_category": "math.OC",
"published": "20170426194846",
"title": "Multidimensional Rational Covariance Extension with Approximate Covariance Matching"
} |
0.9same theoremTheoremalgorithm[theorem]Algorithm axiom[theorem]Axiom claim[theorem]Claim conclusion[theorem]Conclusion condition[theorem]Condition conjecture[theorem]Conjecture corollary[theorem]Corollary criterion[theorem]Criterion definition[theorem]Definition example[theorem]Example exercise[theorem]Exercise lemma[theorem]Lemma notation[theorem]Notation problem[theorem]Problem proposition[theorem]Proposition remark[theorem]Remark solution[theorem]Solution summary[theorem]Summary [email protected]^1Research Laboratory of Electronics,Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA ^2Department of Physics,Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA Noisy entanglement-assisted classical capacity as a security framework for two-way quantum key distribution protocolsJeffrey H. Shapiro^1 December 30, 2023 ======================================================================================================================= Quantum key distribution (QKD) offers unconditional security against eavesdropping <cit.>, but state-of-the-art secret key rates (SKRs) for QKD are only ∼1 Mbps on a 50-km-long fiber link <cit.>, i.e., orders of magnitude lower than classical fiber-communication rates. Floodlight QKD (FL-QKD) is a Gaussian two-way quantum key distribution protocol (TW-QKD) that is theorized to be capable of Gbps SKRs over metropolitan-area distances <cit.>.In FL-QKD, Alice sends quantum states to Bob. While they estimate the correlation between eavesdropper Eve and Bob, Bob encodes a raw key on the light he receives and sends that modulated light back to Alice.Unfortunately, until now FL-QKD's security proof is limited to frequency-domain collective attacks <cit.>.More generally, the security of TW-QKD against coherent attacks is still an open problem. In this paper, we use the noisy entanglement-assisted capacity <cit.> to create a coherent-attack security framework for Gaussian TW-QKD protocols in the asymptotic region.We use Eve's disturbance of Alice and Bob's Gaussian-state covariance matrix—which can be bounded from homodyne measurements—to quantify her intrusion on a Gaussian TW-QKD protocol, such as those in Refs. <cit.>, and obtain therefrom unconditional security against a coherent attack. Our results pave the way towards high-rate QKD with unconditional security. [11]l0.35 < g r a p h i c s > [caption]Single-use schematic of a Gaussian TW-QKD protocol.Gaussian TW-QKD protocols.— Figure. <ref> shows a single use of a general Gaussian TW-QKD protocol.Alice prepares a signal-reference pair (Y,W) in a two-mode squeezed vacuum (TMSV) state with mean photon number N_S.She measures a portion of W for security checking, and sends the signal Y to Bob through a forward channel that is controlled by Eve.In general, Eve performs a unitary operation on Y and her pure-state input V, retaining its E output and delivering its signal output S to Bob.(Note that V and E can have multiple modes per channel use.)In Eve's coherent attack, her unitary operation can act jointly on all channel uses <cit.>. Figure <ref> contains a schematic plot of the protocol after Bob receives S. He measures a portion of S for security checking, and encodes a random symbol x on the remainder.Alice and Bob's security checking uses homodyne measurements <cit.> to estimate constraints on the covariance matrix of the joint state ρ̂_SW; in the asymptotic regime these estimates will be perfect.Bob encodes x with a unitary Û_x composed of a phase shift θ_x and a displacement d_x that are easily realized with linear optics. Conditioned on the message x, the encoded mode has annihilation operator â_S^'(x)=e^iθ_xâ_S+d_x. The d_x's are assumed to be zero-mean Gaussian random variables, implying that encoding on a vacuum state will average to produce a thermal state. The encoding scheme is symmetric, i.e.,∑_x P_X(x) e^iθ_xd_x=0, and energy constrained, viz., E_X=∑_x P_X(x) |d_x|^2. Thus it includes the random-displacement encoding scheme used in Refs. <cit.> and the phase encoding employed in FL-QKD <cit.>. The unconditional state of (S^',W) is non-Gaussian in general.Bob's encoded signal passes through channel Ψ that models the part of the return channel that is not under Eve's control, e.g., loss in Bob's terminal, but we will allow Ψ to be any Gaussian channel without excess noise:a pure-loss channel (transmissivity η), a quantum-limited amplifier (gain G_B), or a quantum-limited phase conjugator (gain G_B).After Ψ, Bob sends its output B to Alice through a channel controlled by Eve. Alice jointly measures the light she receives with part of W to obtain a raw key from which the secret key will be distilled after Alice and Bob use their covariance-matrix constraints to bound the information gained by Eve. Bounding eavesdropper's information gain.— In the asymptotic regime, a QKD protocol's secret-key efficiency (SKE), in bits per channel use, against a coherent attack is given by the Devetak-Winter formula <cit.>SKE= max[ξ I_AB-I_E,0], where I_AB is Alice and Bob's[10]l0.35 < g r a p h i c s > [caption]Gaussian TW-QKD protocol from Eve's perspective. Shannon information in bits per channel use, ξ is their reconciliation efficiency, and I_E is Eve's Holevo-information gain in bits per channel use.(Note that Alice and Bob's SKR equals R SKE, where R is Bob's symbol rate.)The maximization in the Devetak-Winter formula needs to be performed over all possible attacks that pass the security checking measurements, i.e., that are consistent with Alice and Bob's measured covariance-matrix constraints.We will perform that maximization on χ_E≡ I_E/M_E, Eve's Holevo information in bits per mode, where M_E is the number of modes used per encoded symbol. Thus, because ξ I_AB can be inferred from Alice and Bob's reconciliation step, the asymptotic security proof of the TW-QKD protocols rests on putting an upper bound on χ_E.Bounding χ_E for a TW-QKD protocol is complicated by Eve's simultaneously attacking the forward and backward channels <cit.>. Consequently, the usual techniques, such as the entropic uncertainty principle <cit.>, are not applicable here because of loss.Recognizing that the TW-QKD protocol shown in Fig. <ref> can be regarded as noisy entanglement-assisted classical communication from Bob to Eve, we use the noisy entanglement-assisted classical capacity formula <cit.> to place on upper bound on χ_E.Thus we establish a new security framework for TW-QKD protocols. Consider a multiple channel uses QKD session over M mode pairs. We use the same notation as Fig. <ref> with subscripts indicating the different mode pairs, i.e., S=S_1S_2⋯ S_M, W=W_1W_2⋯ W_M, and B=B_1B_2⋯ B_M. For Gaussian protocols, the Û_x's are [13]l0.53[ TMSV protocol with random displacement.] < g r a p h i c s > [ FL-QKD protocol. G_B=10^6, N_S is optimized.] < g r a p h i c s >[caption]Secret-key rates versus path length L.covariant with Ψ, thus Eve's information gain is upper bounded by a maximization, given the covariance-matrix constraints, over multiple mode pairs, i.e., the multi-letter formula <cit.>,χ_E^(M) = max_ρ̂_ SW F[ρ̂_ SW], F[ρ̂_ SW] ≡ S(ρ̂_B)-E_(Ψ^⊗ M)^c⊗ℐ[ρ̂_ SW],where each ρ̂_B_m=∑_x P_X(x) Ψ[Û_x^†ρ̂_S_mÛ_x^†], i.e., we have assumed independent encoding on each mode pair. With dependent encoding, Eq. (<ref>) is still an upper bound. A trace-preserving completely-positive map ϕ has complementary channel we denote as ϕ^c, and the entropy gain of ϕ on state ρ̂ is E_ϕ[ρ̂]≡ S(ϕ[ρ̂])-S(ρ̂). To reduce Eq. (<ref>) to a single mode-pair (single-letter) formula, we use the subadditivity of F[ρ̂_ S W] <cit.>,and, because Ψ is a Gaussian channel, this also ensures that the maximum of Eq. (<ref>) is achieved by a Gaussian-state ρ̂_ SW <cit.> under the given covariance-matrix constraints.At this point we introduce the covariance-matrix constraints that Alice and Bob will obtain from their security checking.The first will be the total mean photon number of the signal received by Bob, ∑_n=1^M ⟨â^†_S_nâ_S_n|=⟩Mκ_S N_S. The second will be the total cross correlation between Alice's retained and Bob's received modes,∑_m,n=1^M( |⟨â_S_mâ_W_n||⟩^2+|⟨â_S_mâ_W_n^†||⟩^2)=(1-f_E)κ_S MN_S(N_S+1). Here, f_E, κ_S quantify Eve's intrusion on the quantum channels, and 0≤ f_E≤ 1, required by physics.By constraining the covariance matrix we can bound χ_E, because χ_E decreases with increasing κ_S and it increases with increasing f_E. The total mean photon number constrains the covariance matrix's diagonal elements, while ∑_m,n=1^M (|⟨â_S_mâ_W_n||⟩^2+|⟨â_S_mâ_W_n^†||⟩^2)≥ |∑_n=1^M ⟨â_S_nâ_W_n||⟩^2+|∑_n=1^M ⟨â_S_nâ_W_n^†||⟩^2 implies that the covariance matrix's off-diagonal elements give a lower bound on the total correlation.By using optimization techniques similar to those in Ref. <cit.>, we can show that our constraints permit Eq. (<ref>) to be reduced to a single-letter formula that can be evaluated as a function of the intrusion parameters κ_S, f_E. With Eve's information gain in hand, the SKE can then be obtained from the Devetak-Winter formula. In the examples that follow, we will use κ_S equal to the one-way fiber loss κ_S=10^-0.02 L that Alice and Bob will see when they are connected by L km of fiber. TMSV protocol with random displacement <cit.>.— In this protocol, Alice has access to the full TMSV, Bob encodes each mode using random displacements with power E_X, and Ψ is the noiseless identity channel.Figure <ref> compares our SKE lower bound with the SKE result from Refs. <cit.> when f_E=0, ξ=1 andE_X≫1, N_S≫1.Our lower bound, which applies for a coherent attack in the asymptotic regime, is much lower than the one from Refs. <cit.>, which only applies for a special class of collective attacks.We believe that much of this gap is due to our giving Eve all the light on the backward channel, which is an overly conservative assumption given the short distances involved, e.g., κ_S=0.63 for L=10 km. For TW-QKD protocols like FL-QKD, which are capable of long-distance operation, we expect that our SKE lower bound will be tighter at those long distances, e.g., when κ_S = 0.1 for L=50 km. FL-QKD protocol <cit.>.— FL-QKD offers Gbps SKRs at long distances by virtue of three features.First, Alice uses low-brightness amplified spontaneous emission light (ASE), together with TMSV light, in her transmission to Bob, while retaining a high-brightness ASE reference as a homodyne-detection local oscillator for measuring Bob's encoded message.Nevertheless, even with only partial access to the purification W, Alice can still establish asymptotic security.Second, Bob uses a high-gain (G_B ≫ 1) amplifier as his Ψ, which overcomes the backward-channel loss issue that plagues previous TW-QKD protocols <cit.>. Finally, Bob uses multi-mode encoding, M_E ≫ 1, that allows Alice to decode Bob's message despite the low-brightness of the signal light she transmitted.Previous work <cit.> has only proven Fl-QKD's security against afrequency-domain collective attack. Here we apply our framework to obtain its asymptotic SKE against a coherent attack.FL-QKD, uses phase encoding, so its E_X=0.Although alphabets larger than binary are known to be beneficial <cit.>, here we will consider binary encoding with phases θ_0=0, θ_1=π representing the bit values 0 and 1. Figure <ref> plots FL-QKD's SKE against a coherent attack in the asymptotic regime assuming f_E=0, ξ=1 for a variety of M_E values where we have optimized over the source brightness at each distance.The red line corresponds to the operating point of M_E=200 as used in Refs. <cit.> for the frequency-domain collective attack. We see that with M_E≫1 and R=10 Gbps, FL-QKD provides Gbps SKRs at long distances. Note that FL-QKD's SKE against the coherent attack—as determined here—coincides with the SKE obtained in Ref. <cit.> against the frequency-domain collective attack, and hence the SKE incurred with f_E>0 for a coherent attack can be found from that reference. 17fxundefined [1]ifx#1 fnum [1] #1firstoftwosecondoftwo fx [1] #1firstoftwosecondoftwonoop [0]secondoftworef[1]@startlink#1@href href[1]#1@endlink anitize@url [0] ` 12 `$12 `&12 `#12 `1̂2 `_12 `%12 startlink[1] endlink[0]rl [1]href #1 @bib@innerbibempty[Bennett and Brassard(2014)]Bennett20147 authorauthorC. H. Bennett and authorG. Brassard,http://dx.doi.org/10.1016/j.tcs.2014.05.025journaljournalTheor. Comput. Sci. volume560(1), pages7(year2014)NoStop lucamarini2013efficientM. Lucamarini et al.,Opt. Express21, 24550 (2013).[Zhuang et al.(2016a)Zhuang, Zhang, Dove, Wong, and Shapiro]Quntao_2015 authorauthorQ. Zhuang, authorZ. Zhang, authorJ. Dove, authorF. N. C. Wong,and authorJ. H. Shapiro, @noop journaljournalPhys. Rev. 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A volume92,pages052317 (year2015)NoStop[Berta et al.(2010)Berta, Christandl, Colbeck, Renes,and Renner]berta2010uncertainty authorauthorM. Berta, authorM. Christandl, authorR. Colbeck, authorJ. M. Renes,and authorR. Renner, @noop journaljournalNat. Phys. volume6, pages659 (year2010)NoStop[Zhang et al.(2017)Zhang, Zhuang, Wong, and Shapiro]zhang2016floodlight authorauthorZ. Zhang, authorQ. Zhuang, authorF. N. C. Wong,andauthorJ. H. Shapiro,10.1103/PhysRevA.95.012332journaljournalPhys. Rev. A volume95, pages012332 (year2017)NoStop | http://arxiv.org/abs/1704.08169v2 | {
"authors": [
"Quntao Zhuang",
"Zheshen Zhang",
"Jeffrey H. Shapiro"
],
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"primary_category": "quant-ph",
"published": "20170426154933",
"title": "Noisy entanglement-assisted classical capacity as a security framework for two-way quantum key distribution protocols"
} |
Local discontinuous Galerkin methods for the time tempered fractional diffusion equation Fengqun Zhao label1 December 30, 2023 ======================================================================================== Deep Reinforcement Learning (RL) recently emerged as one of the most competitive approaches for learning in sequential decision making problems with fully observable environments, e.g., computer Go. However, very little work has been done in deep RL to handle partially observable environments. We propose a new architecture called Action-specific Deep Recurrent Q-Network (ADRQN[ADRQN is implemented in Caffe. The source is available at https://github.com/bit1029public/ADRQN.]) to enhance learning performance in partially observable domains. Actions are encoded by a fully connected layer and coupled with a convolutional observation to form an action-observation pair. The time series of action-observation pairs are then integrated by an LSTM layer that learns latent states based on which a fully connected layer computes Q-values as in conventional Deep Q-Networks (DQNs). We demonstrate the effectiveness of our new architecture in several partially observable domains, including flickering Atari games. § INTRODUCTION Deep reinforcement <cit.> learning combines reinforcement learning and deep learning, and has shown great success in solving a number of very challenging tasks that can be easily modeled as conventional reinforcement learning problems, but cannot be solved by conventional reinforcement learning approaches due to the high dimensionality of the state space, e.g., continuous control <cit.>, high-dimensional robot control <cit.>, and Atari Learning Environment benchmarks (ALE) <cit.>. Specifically, Deep Q-Networks (DQNs) can be used to effectively and efficiently play many Atari 2600 games <cit.>. The idea is to extract convolved features of the frames as states and to approximate the Q-function over the states with a deep neural network. For instance, a DQN is trained to estimate the expected value of a policy based on the the past four frames as input in Atari 2600 games. For most Atari 2600 games, four frames are sufficient to approximate the current state of the game. Generally speaking, the Atari games are treated as MDP problems, and DQNs are used to approximate the value functions of these MDPs. However, for some other real-world tasks including some Atari 2600 games, the problem is really a partially observable Markov decision process (POMDP) where the state of the environment may be partially observable or even unobservable, to the point where arbitrarily long histories of observations are needed to extract sufficient features for optimal action selection. Unfortunately, DQN is not suitable for those problems due to the assumption of complete observability of the state.While POMDPs can naturally model planning tasks with uncertain action effects and partial state observability, finding an optimal policy is notoriously difficult. Some previous POMDP techniques focus on identifying a finite subset of beliefs that are sufficient to approximate all reachable beliefs in order to decrease computational complexity <cit.>.Recent advances in deep learning suggest a new way of thinking for solving POMDP problems. However, very little work leverages deep reinforcement learning in partially observable environments. Among this work, <cit.> adopted DQNs to solve conventional POMDP problems.A policy is obtained with a DQN that mapsconcatenated observation-belief vector pairs to an optimal action. Their work (we call it DBQN) is designed for model-based representations of the environment where the transition, observation and reward functions are already known. Thus, the belief can be estimated precisely with Bayes' theorem and can serve as input to the neural network. However, in most real-world POMDP problems, the environment dynamics are unknown. To address this, <cit.>adapted the fully connected structure of DQN with a recurrent network <cit.>, and called the new architecture Deep Recurrent Q-Network (DRQN). The proposed model recurrently integrates arbitrarily long histories of observations to find an optimal policy that is robust to partial observability. However, DRQNs consider only observation histories without explicitly including actions as part of the histories. This impacts negatively the performance of the approach as demonstrated in Sec. 4. <cit.> combined DRQN with handcrafted features to jointly supervise the learning process of 3D games in partially observable environments, however the approach suffers from the same problem as DRQN since it overlooks action histories. <cit.> extended DRQN to handle partially observable multi-agent reinforcement learning problems by proposing a deep distributed recurrent Q-networks (DDRQN). The action history is explicitly processed by an LSTM layer and fed as input to a Q-network. In DDRQN, each action is forcibly decoupled from its associated observation despite the fact that action-observation pairs are the key to belief updating.As a result, the decoupling of actions and observations in DDRQN impacts negatively belief inference.In this paper, we propose a new architecture called Action-based Deep Recurrent Q-Network (ADRQN) to improve learning performance in partially observable domains. Actions are encoded via a fully connected layer and coupled with their associated observations to form action-observation pairs. The time series of action-observation pairs is processed by an LSTM layer that learns latent states based on which a fully connected layer computes Q-values as in conventional DQNs. We demonstrate the effectiveness of our new architecture in several Atari 2600 games. Table <ref> summarizes the main differences between ADRQN and other state-of-the-art deep Q-learning techniques.§ BACKGROUNDIn this section, we give a brief review of Deep Q-Networks (DQNs), Partially Observable Markov Decision Processes (POMDPs) and Deep Recurrent Q-networks (DRQNs). §.§ Deep Q-Networks A sequential decision problem with known environment dynamics is usually formalized as a Markov Decision Process (MDP), which is characterized by a 4-tuple ⟨ S,A,P,R ⟩. At each step, an agent selects an action a_t ∈ A to execute with respect to its fully observable current state s_t ∈ S and based on its policy π.It receives an immediate reward r_t ∼ R(s_t,a_t) and transitions to a new state s_t+1. The objective of reinforcement learning is to find the policy that maximizes the expected discounted rewards R_tR_t = r_t + γr_t + 1 + γ ^2r_t + 2 +⋯where γ∈ [0,1] is the discount factor. In MDPs, an optimal policy can be computed by value iteration <cit.>.Q-Learning <cit.> was proposed as a model-free technique for reinforcement learning problems with unknown dynamics.It estimates the value of executing an action in a given state followed by an optimal policy π. This value is called the state-action value, or simply Q-value as defined below:Q^π(s,a) = E^π(R_t|s_t = s,a_t = a)The Q-values can be learned iteratively according to the following rule while the agent is interacting with the environment:Q(s,a) = Q(s,a) + α (r + γmax_a'Q(s',a') - Q(s,a))In tasks with a large number of states, a common trick is to use a function approximator to estimate the Q-function. For instance, DQN <cit.> uses a neural network parameterized by θ to represent Q(s,a;θ).Neural networks with at least one (non-linear) hidden layer and sufficiently many nodes can approximate any function arbitrarily closely. DQN is optimized by minimizing the following loss function:L(θ _i) = E_s,a,r,s'[ ( (y_i^target - Q(s,a;θ _i))^2]where y_i^target = r + γmax_a' Q(s',a';θ^-_i) denotes the target value of the action a_t given state s_t. Here θ _i^ - is cloned from θ_i every fixed numbers of iterations. DQN uses experience replay <cit.> to store previous samples e_t=⟨ s_t,a_t,r_t,s_t+1⟩ up to a fixed size memory D_t. The Q-network is then trained by uniformly sampling mini-batches of past experiences from the replay memory. An important factor for the efficiency of DQN in AlphaGo and the Atari games is the assumption of full state observability that allows the neural network to use only one (or a few) observation(s) as input. Thus, DQN suffers inaccuracy in tasks with partially observable states. §.§ Partially Observable Markov Decision Processes (POMDPs) POMDPs generalize MDPs for planning under partial observability.A POMDP is mathematically defined as a tuple ⟨𝒮,𝒜,𝒵,T,O,R⟩, consisting of a finite set of states 𝒮, a finite set of actions 𝒜, a transition function T : 𝒮×𝒜→Π(𝒮), where Π(𝒮) represent the set of probability distributions on 𝒮, a reward function depending on the state and the action just performed R :𝒮×𝒜→ℛ, a finite set of observations 𝒵 and an observation function O :𝒮×𝒜→Π(𝒵), where Π(𝒵) represent the set of probability distribution on 𝒵. As the true states are not fully observable any more, a belief is used to estimate the current state, defined as the probability mass function over the states and denoted as b=(b(s_1),b(s_2),...b(s_|S|)), where s_i∈𝒮, b(s_i)≥ 0, and ∑_s_i∈𝒮b(s_i)=1. Given the current belief b_t, performing action a_t and get the next observation z_t+1, then the next belief b_t+1 = SE(b_t,a_t,z_t+1) is estimated as follows: b_t+1(s_j) = O(s_j,a_t,z_t+1)∑_s_i∈𝒮T(s_i,a_t,s_j)b_t(s_i)/P(z_t+1|a_t,b_t) P(z_t+1|a_t,b_t)=∑_s_j∈𝒮O(s_j,a_t,z_t+1)∑_s_i∈𝒮T(s_i,a_t,s_j)b_t(s_i) The expected immediate reward for an agent performing action a at the belief state b is computed as ρ(b,a)=∑_s_i∈𝒮b(s_i)R(s_i,a). The transition function among beliefs becomes:τ(b,a,b')=∑_z ∈𝒵 p(b'| b,a,z) P(z|b, a)where p(b'| b,a,z)=1 if b'=SE(b,a,z), and 0 otherwise. An optimal policy π:ℛ^|𝒮|→𝒜 can be computed by value iteration:V(b) = max_a[ρ(b,a)+γ∑_b'τ(b,a,b')V(b')]where γ is a discount factor for the past history. In practice, it is common to have the optimal policy represented by a set of linear functions (so called α-vectors) over the belief space, with the maximum “envelop” of the α-vectors forming the value function <cit.>.§.§ DRQNDQN assumes that the agent has full knowledge of the state of the environment, i.e., the agent's observation is equivalent to the state of environment. However, in practice, this is rarely true, even in some Atari games. For example, one frame of Pong does not reveal the ball's velocity and its moving direction. In the game of Asteroids, parts of the image are concealed for several consecutive frames to challenge the player. Hence, these games are naturally POMDP problems. While the strategy of DQN is to utilize several consecutive frames as the input in the hope of deducing the full state, this works well only when a short limited history is sufficient to characterize the state of the game. For more complex games, the performance of DQN decreases sharply.To tackle problems with partially observable environments by deep reinforcement learning, <cit.> proposed a framework called Deep Recurrent Q-Learning (DRQN) in which an LSTM layer was used to replace the first post-convolutional fully-connected layer of the conventional DQN. The recurrent structure is able to integrate an arbitrarily long history to better estimate the current state instead of utilizing a fixed-length history as in DQNs. Thus, DRQNs estimate the function Q(o_t,h_t-1|Θ) instead of Q(s_t,a_t|Θ), where Θ denotes the parameters of entire network, h_t-1 denotes the output of the LSTM layer at the previous step, i.e., h_t = LSTM(h_t-1,o_t). DRQN matches DQN's performance on standard MDP problems and outperforms DQN in partially observable domains. Regarding the training process, DRQN only considers the convolutional features of the history of observations instead of explicitly incorporating the actions. However as we argued in Sec. 1 and Eq. <ref> in Sec. 2.2, the action performed is crucial for belief estimation. To that effect, we propose a new architecture for deep reinforcement learning with actions incorporated in histories to further improve the performance of deep RL in POMDPs. In the sequel, we will refer to our proposed architecture as ADRQN (action-specific deep recurrent Q-learning Network).§ ADRQN - ACTION-SPECIFIC DEEP RECURRENT Q-NETWORK Inspired by the aforementioned works, our goal is to propose a model-free deep RL approach that incorporates the influence of the performed action through time. More specifically, we couple the performed action and the obtained observation as the input to the Q-network. The architecture[`IP' means inner product, i.e., fully connect layer]. of our model is shown in Fig. <ref>.Our first attempt to couple the action and observation was to concatenate a fixed representation of the action with the observation vector directly, i.e., we utilized one-hot vectors to represent each action a ∈𝒜. However, such concatenation did not yield good performance because the lengths of the action and observation vectors differ largely which leads to numerical instability. In Atari games as well as many real-world POMDPs, the number of actions is far less than the dimensionality of the state representation. In conventional DQNs and DRQNs, the suggested dimensionality of the convolved features encoding the current state or observation is 3136 (after passing several convolutional and reshaping layers), while the number of actions is only 18 for the Atari games. To address this imbalance, we embed the one-hot action vectors by a fully connected layer into a higher dimensional vector (512-D is our experiments).Thus the representations of the actions and the observations are now more balanced.Compared with DQN and DRQN, our model is able to remember the past actions, particularly the last performed action. Thus, we modified (s_t, a_t, r_t,s_t+1) the transition in the experience replay mechanism of DQN to ⟨{a_t-1,o_t},a_t,r_t,o_t+1⟩ in order to allow the framework to fetch the action-observation pair more conveniently. During the decision process for a given frame within training or the updating process of the neural network, the LSTM layer requires a sequence of action-observation pairs as its input. Thus, we store the transitions sequentially ⟨{a_t-1,o_t},a_t,r_t,o_t+1⟩ within each episode in the replay memory.Ideally, the best strategy to estimate precisely the current state for a model-free POMDP problem is to integrate the entire transition history of each episode, which usually includes thousands of transitions for an Atari game. This also means that the LSTM layer needs to be unrolled for a large number of time steps which will increase significantly the training cost. In our experiments, the LSTM layer is unrolled for 10 time steps during training. We take this setting for three points. From the perspective of learning from whole history, this setting can be more satisfy to our original idea compared to a smaller setting of the length. Second is that this setting can also guarantee the efficiency of experiments. It allows that our experiments can be finished within several days. And last point is to make a better comparison to DRQN as it take a same setting of 10. Such empirical setting has proved its efficacy to our framework.The entire process of our proposed approach is presented in Algorithm 1. First, we initialize the parameters of the Q-network and the Target network with θ and θ^- respectively. For each episode, the first selected action is initialized to no-operation, the first hidden layer's input is initialized with a zero vector and the first observation of each episode is initialized with the preprocessed first frame. At each time step, if the observation does not indicate “game over” (the end of the episode), we select an action based on the ϵ-greedy strategy and execute the action.Accordingly, the immediate reward and the next observation of the screen will be obtained. The transition, once obtained, will be sequentially stored in the history of the current episode. To update the Q-network, we randomly sample a sequence of transitions ⟨ a_j-1,o_j,a_j,r_j,o_j+1⟩ to fit the unrolled LSTM layer. Then, the hidden layer h_j-1 of the Q-network and the hidden layer h_j of the target network are obtained. The difference between these two network Q-values (i.e., Q-value y_j and Q-network value Q(h_j-1,a_j-1,o_j,a_j;θ)) is used as the loss function to update the network parameters θ via back propagation.§ EXPERIMENTS We evaluate the training and testing performance of ADRQN with several Atari games and their flickering version. §.§ Experiments setup* Flickering Atari game: <cit.> introduced a flickering version of the Atari games, which modified the Atari games by obscuring the entire screen with a certain probability at each time step, which introduces partial observability and therefore yields a POMDP. In their framework, before a frame is sent to the neural network as input, each raw screen is either fully observable or fully obscured with black pixels. * Frame skip Scheme. We adopted the frame skip technique <cit.>. This mechanism is commonly used in most previous works of deep reinforcement learning to efficiently simulate the environment. In this mechanism, an agent performs the selected action a_i for k+1 consecutive frames and treats the transition from the first frame f_0 to frame f_k+1 as the effect of action a_i, i.e. ⟨ f_0,a_i, f_1,...,a_i,f_k+1⟩=⟨ f_0, a_i, f_k+1⟩. Thus, a large number of frames can be skipped to accelerate the training process, but with a tolerable performance loss. In our experiments, k is set to 4. * Hyperparameters. In our experiments, we also adopt experience replay and set the replay memory size to D=400,000 (i.e., storing 400,000 transitions). When selecting an action, we follow the ϵ-greedy policy with ϵ=1- 0.9*iter/explore, where iter is the current number of iterations performed and explore is the number of iterations that epsilon reaches to a given value. In our setting,ϵ will reach 0.1 and explore was set to 1,000,000. The discount factor γ was set to 0.99. The target network is updated by cloning the weights of the Q-network every 10,000 iterations. And we unrolled the LSTM to 10 time step when training as we said in Section <ref>. * Random Updates. As suggested in <cit.>, random updates can achieve the similar performance as conventional sequential updates of the entire episode, but with much lower training cost. Random updates consist of selecting randomly a series of transitions from one episode as the input. In our framework, this corresponds to utilizing a sequence of action-observation pairs to perform the updates. The initial hidden input h_0 of the RNN is set to the zero vector at the beginning of the update. §.§ TrainingCompared to the other approaches, the key idea of our framework is to construct action-observation pairs as input to the LSTM layer to retain more representative features for the Q-network to learn recurrently. The actions are first encoded with one-hot vectors,then processed by a fully connected layer to construct a higher-dimensional vector that is concatenated with the output of the third convolutional layer for better numerical stability. As the LSTM layer will be unrolled for 10 time steps, we need to ensure that there are enough transitions to be stored in the replay memory D so that we can sample a minibatch of the transition sequences of length 10 each time we update the entire network. In our experiments, we update the entire network until the replay memory is full. Moreover, the scores obtained by playing the games are not always stable since small changes of the weights may have a significant impact on the outcome <cit.>. And, it may bring the instability to the policy which the network have learned. Thus, we adopt an adaptive setting as done in most previous works by replacing negative rewards by -1 and positive rewards by +1.When training the flickering versions of Atari games, we set the probability of obscuring a frame to 0.5 as a compromise. A lower probability may prevent learning due to a large loss of information, and a higher probability may be less convincing that the transformed version is a POMDP. Besides, it is a fair setting to evaluate generalization performance since it divides the test interval into two subintervals evenly. The games Pong and Frostbite are both trained under the setting of full observation and a 0.5 probability of obscuring a frame. The training performances of Pong and Frostbite are shown in Fig. <ref> and <ref> respectively. The reported scores are based on averages of 10 and 100 episodes respectively.We compared our model ADRQN with DRQN and DDRQN on several Atari games. As DDRQN is proposed for addressing multi-agent POMDP problems and it disabled experience replay mechanism, we adapted it to a single-agent version which also adopts the replay memory mechanism. Fig. <ref> and Fig. <ref> show that ADRQN matches the performance obtained by DRQN and DDRQN, and even performs slightly better than them in the conventional setting (full observation), while the improvements of ADRQN over DRQN and DDRQN in partially observable settings are very obvious.Additionally, when trained under the setting of the observation probability as 0.5, ADRQN can obtain significant improvements in Frostbite(Fig. <ref>). We believe the gain comes from the use of ⟨action, observation⟩ pair which speeds up the training process in partially observable environments. In general, our proposed model can lead to higher scores in partially observable settings during the training process, which supports the argument that action-observation pairs are important in POMDP problems. §.§ Testing EvaluationWe also evaluate the well-learned DRQN, DDRQN and ADRQN models on the five games and their flickering version. We replay each game based on the learned model 50 times to obtain the average scores as our final results. The epsilon value used in testing evaluation for ϵ-greedy is set to 0, as we consider the model has been well trained and there is no need for the exploration. Table <ref> summarizes the results obtained with full observations. The results shows that ADRQN has a similar performance with DRQN and DDRQN in general in full observable environments. While Table <ref> demonstrates that ADRQN significantly outperforms DRQN and DDRQN when trained with half of the frames obscured in all five games. And particularly, that ADRQN can get a similar score as it get in full observable setting in the game Chopper Command and game Frostbite which further demonstrates the advantage of our model. An interesting observation is that the testing results are generally better than the results obtained during the training process for all three models. This may be explained by the fact that different distribution of the concealed frames may render the problem more or less observable, to the extent that better results may be obtained in some cases. Another explanation is that no exploration is needed for selecting action or making a decision after a policy is well trained, thus leads to better results. §.§ Generalization Performance To further demonstrate the advantages of ADRQN in dealing with environmental changes, we evaluate the generalization performance of DRQN, DDRQN and ADRQN on the flicking versions of the games.POMDP to MDP Generalization: After being trained with observation probability of 0.5, we test the learned policy in settings with observation probabilities (0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0). The results are shown in Fig. <ref>. We observe that all of three models consistently perform better with the increase of observation probability in game Pong, and also perform better with the observation probability changed in game Frostbite. ADRQN consistently outperforms DRQN and DDRQN in these two games, and has a large advantage in game Frostbite which further demonstrates that our model can better respond to environmental changes.MDP to POMDP Generalization: After being trained as an MDP by setting the observation probability to 1, we test the learned policy with different settings of the obscured probability. As shown in Fig.<ref>, all the performances achieved by the three models drop sharply as we increase the amount of missing information.Fig.<ref> clearly demonstrate that ADRQN is relatively robust compared to DRQN and DDRQN. § BRIEF DISCUSSIONSomeone may argue that action information could be viewed as redundant in the context of a fixed policy. That is if you know the entire sequence of observations and there is a fixed deterministic policy, then one can fill in the missing actions directly. We agree with this argument and we guess that's why the existing state-of-the-art work i.e. DRQN overlooked the action observation pair. However, with the adoption of a recurrent model, the observation history is inevitably truncated to ensure training efficiency. Thus, a proper action sequence cannot be inferred from the truncated history. For a complex partially observable task, the set of observations is infinite. It would be even harder to infer the actions from the finite observation history. Intuitively, an explicit incorporation of the coupled action-observation pair would help a lot especially during the training process.§ CONCLUSION In this paper, we propose an action-specific deep recurrent Q-Network (ADRQN) to enhance the learning performance in partially observable domains. We first encode actions via MLP and couple them with the features of observations extracted from CNN to form action-observation pairs. Then an LSTM layer is utilized to integrate a time series of action-observation pairs to infer the latent states. Finally, a fully connected layer will address Q-values computing to guide the overall learning process which is similar to conventional DQNs. We have demonstrated the effectiveness of our proposed approach in several POMDP problems in comparison to the state-of-the-art approaches.named | http://arxiv.org/abs/1704.07978v6 | {
"authors": [
"Pengfei Zhu",
"Xin Li",
"Pascal Poupart",
"Guanghui Miao"
],
"categories": [
"cs.LG"
],
"primary_category": "cs.LG",
"published": "20170426055507",
"title": "On Improving Deep Reinforcement Learning for POMDPs"
} |
http://arxiv.org/abs/1704.08162v5 | {
"authors": [
"C. Jebaratnam",
"Debarshi Das",
"Arup Roy",
"Amit Mukherjee",
"Some Sankar Bhattacharya",
"Bihalan Bhattacharya",
"Alberto Riccardi",
"Debasis Sarkar"
],
"categories": [
"quant-ph"
],
"primary_category": "quant-ph",
"published": "20170426152900",
"title": "Tripartite entanglement detection through tripartite quantum steering in one-sided and two-sided device-independent scenarios"
} |
|
PIs at LIGO]The Influence of Dual-Recycling on Parametric Instabilities at Advanced [email protected] University of Birmingham, UKLaser interferometers with high circulating power and suspended optics, such as the LIGO gravitational wave detectors, experience an optomechanical coupling effect known as a parametric instability: the runaway excitation of a mechanical resonance in a mirror driven by the optical field. This can saturate the interferometer sensing and control systems and limit the observation time of the detector. Current mitigation techniques at the LIGO sites are successfully suppressing all observed parametric instabilities, and focus on the behaviour of the instabilities in the Fabry-Perot arm cavities of the interferometer, where the instabilities are first generated. In this paper we model the full dual-recycled Advanced LIGO design with inherent imperfections. We find that the addition of the power- and signal-recycling cavities shapes the interferometer response to mechanical modes, resulting in up to four times as many peaks. Changes to the accumulated phase or Gouy phase in the signal-recycling cavity have a significant impact on the parametric gain, and therefore which modes require suppression.[ A. Freise December 30, 2023 =====================§ PARAMETRIC INSTABILITY IN COMPLEX INTERFEROMETERSPrior to the first detection of a gravitational wave signal in September 2015 <cit.>, the two LIGO detectors went through the first stage of a major upgrade, which aimed to boost their sensitivity by a factor of 10 <cit.>. One major aspect of this upgrade was a significant increase in optical power circulating in the arm cavities, which is expected to reduce shot noise and therefore improve detector sensitivity.During the first observational run (O1) of Advanced LIGO, circulating power approaching 100 kW was consistently achieved; the target power is 750 kW resulting in a design sensitivity of 200 Mpc binary neutron star range <cit.>.In this paper we discuss a phenomenon known as parametric instabilities <cit.>, a consequence of using high optical power in the interferometer.Parametric instabilities result from an interaction between the radiation pressure of the optical field and the natural vibrational modes of the mirrors. In the presence of positive optical feedback this can couple energy from the field into the mirror mode, resulting in exponential growth of the mechanical oscillation.Parametric instabilities (PIs) have now been observed in prototype optical cavities <cit.>, and at LIGO <cit.>, where the mechanical modes have been observed to ring up until the interferometer control systems failed. Current mitigation strategies are focused on technologies already built into the interferometers: using ring heaters to change the optical gain of problematic higher order optical modes <cit.>, and using electrostatic drivers to actively damp the mechanical mode that is unstable <cit.>. So far these techniques are proving successful; however, as the circulating power is increased towards the design level the severity and number of PIs will increase, resulting in more unstable modes.Over many years we have developed a detailed simulation model using <cit.> to describe behaviours in the Advanced LIGO detectors <cit.>. In this paper, we present numerical analyses of PIs in the full design configuration of Advanced LIGO <cit.>. This complements existing extensive analytical and numerical modelling of PIs <cit.>.First, we present an overview of how is used to model PIs throughout this work. In section <ref> we study the parametric gain of specific mechanical modes and how this changes when recycling cavities are introduced, including inherent defects such as astigmatism. Section <ref> then focuses on parameters of the signal-recycling cavity and consequences for the parametric gain in a realistic interferometer configuration.We find that changes to the tuning or accumulated Gouy phase of the signal-recycling cavity have a significant impact on parametric gain, and therefore which modes will require suppression. However whether this has consequences for the current mitigation scheme is not yet known.§.§ Numerically Modelling PIs <cit.> is a fast, frequency-domain interferometer simulation tool. It is particularly suited to modelling parametric instabilities as it easily provides the required mechanical-to-optical transfer functions in imperfect and arbitrary interferometer configurations using Hermite-Gaussian beams.Previously this has been used to apply limits to the number and type of higher order modes used in simulation <cit.>, and investigate the potential use of higher order Laguerre-Gauss modes to reduce thermal noise in future gravitational wave detector designs <cit.>. It is also actively used in LIGO commissioning and design modelling <cit.>. and its Python wrapper PyKat <cit.> are open source and freely available for others to use in future studies.Parametric instabilities can be considered as a feedback system <cit.> resulting from the linear interaction between the optical field and a vibrational mode of a suspended mirror within the interferometer. The figure of merit for determining the stability of a vibrational mode in an interferometer is called the parametric gain, ℝ, where ℝ≥ 1 corresponds to an instability. Typically this is evaluated by calculating individual optical transfer functions of higher order optical modes through the interferometer and then summing the effect of each of these optical modes, as described in Appendix <ref>. The open loop transfer function for a motion back onto itself is T(Ω) = p(Ω)/Δ p(Ω), where p(Ω) describes the amplitude motion spectrum of a mechanical mode at a frequency Ω. This exists in as a diagonal element in the inverted interferometer matrix. A single sparse matrix solution can then be used to evaluate T(Ω) at the frequency of the mth mechanical mode, ω_m. Using the frequency-domain equation of state, we find that the parametric gain of the mth mechanical mode, ℝ_m, can then be directly extracted from the real part of this transfer function, as described in <cit.>: ℝ_m= 1- (1/T(ω_m)) Since computes the light field amplitudes of all Hermite-Gauss (HG) modes to a specified order, T(ω_m) contains components from all these optical fields directly. In order to calculate the spatial overlap between the optical fields and the mechanical modes, must be supplied with a surface motion map. Typically these maps are produced using a finite element modelling package, which computes the mechanical resonant modes of the bulk optic. From the bulk modes, the normalised motion of the front surface and the corresponding resonant frequency can be extracted and used as inputs into models. Further details about using finite element modelling tools with are provided in chapter 3 of <cit.>.§.§ General MethodA complete model of the core optics in Advanced LIGO forms the basis of the simulation, as depicted in figure <ref>. This model uses design parameters given in <cit.>. Key frequencies derived from this design are listed in Table <ref>. The detector is based on a Michelson interferometer. Fabry-Perot cavities in the arms, formed by the Input- and End Test Masses (I- and ETMs) are used to amplify gravitational wave signals, and the Power Recycling Cavity formed between the Power Recycling Mirror (PRM) and arm cavities increases the circulating power to decrease shot noise. The Signal Recycling Cavity, formed by the Signal Recycling Mirror (SRM) and arm cavities, can be tuned to amplify or resonantly extract signal sidebands; currently the LIGO detectors operate in this Dual Recycled configuration using Resonant Sideband Extraction (RSE). As in the design, the X- and Y-arm cavities are identical. Parameters within this core model, such as mirror positions, angles and curvatures, may then be varied to change the response of the interferometer, and consequently the parametric gain. We have used feedback loops, mimicking those used at the detector sites, to check that these parameter sweeps do not move the model away from an operating point for the interferometer linear degrees of freedom.This means that both arm cavities and the power-recycling cavity are resonant for the carrier field, and the inner Michelson (formed by the beamsplitter and Input Test Masses) is tuned to a dark fringe on transmission. We refer to in-phase signals that are common to both arms and therefore reflected back towards the laser as `common' signals, while `differential' signals, with a 180^∘ phase difference between arms, are transmitted from the beamsplitter to the detection port.Surface motion maps for the Advanced LIGO mirrors were produced using . Examples are shown in figure <ref>, where listed mode numbers are those generated by . In particular, mode 37 has strong spatial overlaps with HG03 and HG21 optical modes and is associated with the first observation of a PI in a LIGO detector <cit.>. This observation was made at the Livingston detector (which is dual recycled and tuned for resonant sideband extraction), operated with 50 kW circulating arm power resulting in a parametric gain of ℝ=2. Minor (0.15%) adjustments have therefore been made to the radius of curvature of the four test masses in our model to reflect the observed resonant frequency and parametric gain of this mode. Each simulation of a parametric instability applies one surface motion map to the End Test Mass of the X-arm (ETMX), as shown in figure <ref>. The simulation also takes the resonant mechanical mode frequency and Q-factor as inputs; by default we use a Q-factor of 10^7 and the resonant frequency computed by . This means that we can explore the combined parameter space of mechanical mode frequency and interferometer parameters. Note that since we only study effects due to ETMX, modes from different test masses and any cross-coupling between these are not considered in this study.§ PARAMETRIC INSTABILITY IN INCREASINGLY COMPLEX INTERFEROMETERS Figure <ref> depicts the parametric gain of mode 37 (see figure <ref>) as a function of the resonant frequency of the mechanical mode, using the method described in section <ref>. We compare a single X-arm cavity to Michelson Interferometers with just Fabry-Perot arms (FPMi),Power-Recycling (PRFPMi),and Dual-Recycling (DRFPMi). We find that the presence of the power- and signal-recycling cavities significantly shapes the optical response and resulting parametric gain, in agreement with <cit.>. To allow direct comparison, the input power was adjusted to maintain a constant power circulating in the arm(s) in all cases. In both the single arm cavity and FPMi cases we see a typical single broad peak. This corresponds to an overlap between the mechanical mode frequency and the 5.16kHz mode separation frequency of the arm cavity, which allows a 3rd order optical mode to resonate.Introducing the power-recycling mirror results in a cavity coupling between the X-arm and both the power-recycling cavity (PRC) and Y-arm. The condition for resonance is therefore complicated. We see the introduction of two new peaks, since the PRC includes spherically curved mirrors at non-normal incidence, producing an astigmatic beam. This results in the HG03 and HG21 modes picking up different amounts of Gouy phase in the cavity. We describe these peaks as common peaks due to their association with the reflection port of the Michelson. The frequency separation between these common peaks and the original single cavity peak is 182 Hz, while the separation between the two common peaks is 36 Hz. Similarly, adding the signal-recycling mirror produces an additional set of couplings via the signal-recycling cavity (SRC). This time the new resonance condition results in two differential peaks, offset from the single cavity resonance by 30 Hz. These two peaks are unresolved due to the low finesse of the SRC, appearing as a broadening of the peak when compared to the non-astigmatic case. We also see that the original broad peak is supressed.Our model allows us to treat the resonant frequency of each mechanical mode as a tuneable parameter, as discussed in section <ref>. Changes of frequency on the scales explored here are not something we expect to see in reality, however plots of this kind are useful diagnostic tools. They allow us to explore the response of the interferometer to a mechanical mode, whose resonant frequency may shift and is unknown prior to measurement, without changing the interferometer state. Experimental work at both the Hanford and Livingston detectors has attributed a mirror motion with the shape of mode 37 to observed parametric instabilities at 15.53kHz. We see that in our model this falls within a differential peak in parametric gain, indicating that properties of the signal-recycling cavity could also be used to influence the gain of this mode in the interferometer, as shown in section <ref>. However, other modes will match different resonant conditions in the interferometer, for example resonating via the PRC. Improving the behaviour for one mechanical mode may worsen the situation for another. The internal properties of the arm cavities can be used to suppress parametric instabilities. Unlike the SRC properties discussed below, changes to the radii of curvature (RoCs) of the test masses are known to influence parametric gain by altering the optical response. They are therefore one focus of efforts to suppress PIs at the detector sites. Figure <ref> plots the parametric gain of modes 37, 41 and 257 on ETMX when the RoCs of all four test masses are increased simultaneously by the same amount. Once again we find four peaks in the trace for mode 37. We can also see that a significant change in RoC is expected to stabilise mode 37, but overcompensation could result in instability through mode 41.§ THE SIGNAL-RECYCLING CAVITY §.§ TuningIn addition to Advanced LIGO's current broadband operation using resonant sideband extraction (RSE), the tuning of the signal-recycling cavity can be adjusted to produce an operational mode that is optimised for a particular gravitational wave source <cit.>. In particular, an SRC detuning of ϕ =16^∘ is proposed for optimal binary neutron star detection <cit.>, where a tuning of 360^∘ corresponds to a mirror displacement of one wavelength. We find that the parametric gain of some mechanical modes has a strong dependence on the tuning of the signal-recycling cavity length.Figure <ref> depicts the parametric gain on ETMX for modes 37, 41 and 257 as a function of position of the signal-recycling mirror (SRM), expressed as tuning relative to RSE. Each mechanical mode is modelled at its determined frequency (see figure <ref>). Detuning the SRC causes a minor alteration to the operating point of the interferometer (see Section <ref>); however, actively tuning the interferometer linear degrees of freedom to maintain operating point did not significantly change our results. For mode 37, we find a broad peak in parametric gain, resulting in instability for the nominal tuning and an increase in parametric gain for negative detunings. The SRM and Input Test Mass in the X-arm (ITMX) can be viewed as a compound mirror with a frequency-dependent reflectivity determined by the phase accumulated in the SRC. Changes to the position of the SRM therefore alter the effective reflectivity of ITMX as seen by the higher order optical modes. Figure <ref> directly tracks the interferometer response to the tuning of the SRC.The animation in <ref> plots parametric gain as a function of mechanical mode frequency for different choices of SRC tuning, including depicting the power-recycled and single cavity responses for reference. Figure <ref> extracts the mechanical frequency corresponding to each of the three found peaks. We see that the common mode peaks are left unaltered, while the frequency of the differential peak doublet strongly depends on the position of the signal-recycling mirror. The differential mode peak frequency is not depicted for tunings of -70^∘ and -60^∘ since at these values the peak coincides with the common mode peaks and cannot be resolved. In this sense, the differential mode peak appears suppressed. We note that although mode 37 falls within the differential peak, and is influenced by the SRC, other mechanical modes will fall within the common mode peaks, and thus be independent of parameters in the SRC, as shown in figure <ref>. Figure <ref> also shows the behaviour of the parametric gain as a function of SRC tuning for modes 41 and 257, both at full design power. For mode 41, we find that the mode is stable for all planned SRC tunings. However, we find that mode 257 could become unstable if the operational mode were to be switched away from RSE to positive detunings. The set of mechanical modes that could result in parametric instability can change depending on the SRC tuning.§.§ Gouy PhaseProposed upgrade plans for Advanced LIGO include replacing the SRM with another mirror of a different curvature. This would alter the Gouy phase accumulated in the SRC and therefore the optical gain of higher order optical modes in this cavity.We find that changes in the Gouy phase have the same effect as SRC tuning for mode 37. In figure <ref>, we directly set the value of the Gouy phase accumulated in a single pass through the space between the SRM and telescope mirror SR2, Ψ (see figure <ref>)[This allows us to mimic the effect of changing the radius of curvature of one of these mirrors without additional design time to re-mode-match the model.]. As in figure <ref>, we see that only the differential mode doublet is affected by the change, and that the frequency range for which ℝ>1 changes with Gouy phase, to the point where we can suppress the differential peak of mode 37. Note that the periodicity of this behaviour is double that of the SRC tuning case; this is due to setting the one-way rather than round-trip phase.§.§ Consequences for Advanced LIGO InterferometersWe have calculated the parametric gain of 800 mechanical eigenmodes for discrete tunings in -100^∘≥ϕ≥ +100^∘ and Gouy phases in 0^∘≥Ψ≥ +100^∘. This is summarised in figure <ref>. In all cases the mechanical modes are modelled at their calculated frequency and the power circulating in the X-arm cavity is 750kW. In the case of SRC tuning, we find a minimum of 1 and maximum of 6 unstable modes in our model. The summed ℝ plots are dominated by the gain of just one or two modes, as can be seen by comparing the shape of the lower plot in figure <ref> to the trace for mode 37 in figure <ref>. We also find that the current tuning of the SRC sits at a local minimum in terms of number of modes, however the gain of this mode is relatively high when compared to other minima in the upper trace: at ϕ≃ -40^∘ we are closer to suppressing all modes. As expected from comparing figures <ref> and <ref>, this behaviour is also shown for the case of Gouy phase changes. Relative to the design operating point as show in figure <ref>, the number of unstable modes increases for small positive SRC detuning but the gain of a single PI increases for the equivalent negative detuning. The position and radii of curvature of the mirrors in the signal-recycling cavity alter the parametric gain of mechanical modes and have a strong influence on which PIs will appear. This may have consequences for the current mitigation scheme, such as targeting new mode shapes or requiring stronger actuation to damp higher gain modes. However as seen in our results, the number of additional PIs that could arise in the current configuration are limited. § CONCLUSION We have used to study parametric instabilities in the context of the full Advanced LIGO design. The dual-recycled configuration of the interferometer greatly expands the parameter space which determines the resulting optical gain of the system. In particular, we have shown that parameters outside the Fabry-Perot arm cavities can also affect the parametric gain of a mechanical mode, to the extent that the number and gain of unstable modes may change.By contrasting figures <ref> and <ref>, the complexity of the picture is clear: while the instability of mode 41 is minimally affected by SRC tuning and strongly affected by changes in RoC, the reverse appears true of mode 257, and mode 37 is affected by both. The list of important optical parameters is therefore extensive, and all will influence the likely number of PIs that will affect gravitational wave detectors as the operating power increases. For the parameters in the Advanced LIGO design, we find that the tuning and Gouy phase accumulated in the signal-recycling cavity will influence the total number of parametric instabilities, and the gain of these modes. For differential modes parametric instability depends on properties in the SRC, while for common modes instability depends on the PRC. Therefore if parameters in the SRC are to be changed, a PI mitigation scheme based on per-mode damping is expected to remain effective for common mode PIs, but may require changes for differential modes.§ ACKNOWLEDGEMENTSThis work was supported by the Science and Technology Facilities Council Consolidated Grant (number ST/N000633/1) and H.M. is supported by UK Science and Technology Facilities Council Ernest Rutherford Fellowship (Grant number ST/M005844/11). § PARAMETRIC INSTABILITY Throughout this work we consider the linear interaction between the optical field and a vibrational mode of a suspended mirror within the interferometer. As explained by <cit.> these parametric instabilities can be described as a feedback system, depicted schematically in figure <ref>. An excitation acts on a particular vibrational mode of the mirror, causing the reflecting surface of the optic to move. The incoming optical field (`pump') is phase modulated on reflection, resulting in scattering of the optical field into higher order optical modes (HOMs) at sideband frequencies determined by the frequency of the mirror surface motion. We describe the resulting optical modes in the Hermite-Gauss (HG) basis. The lower sideband corresponds to the Stokes mode - energy coupling from the optical field into the mechanical oscillation -while the upper sideband corresponds to the reverse process, the anti-Stokes mode. The strength of the interaction depends on the spatial overlap of the mechanical mode, m, with the incoming and scattered optical field modes, denoted B_m,n for the nth optical mode <cit.>. The resulting optical fields then propagate through the interferometer, where they may be amplified or suppressed depending on their frequency and the particular configuration of the interferometer. On returning to the mirror, radiation pressure will act on the mirror surface. If the upper sideband dominates, the motion will be damped; if instead the lower sideband dominates, energy is coupled out of the optical field into the mirror and it `rings up', resulting in a parametric instability.The figure of merit for determining the stability of a vibrational mode, m, in an interferometer is called the parametric gain, ℝ, where ℝ≥ 1 corresponds to an instability. In the case of a single dominant incident field of wavelength λ and power P, this is given by: ℝ_m = 8 π Q_m P/M ω_m^2 c λ∑_n=0^∞[G_n]B^2_m,n.where 0≥ B_n,m≥ 1 is as above, c is the speed of light, M is the mass of the mirror, and vibrational mode m has angular resonant frequency ω_m and quality factor Q_m. For the fused silica LIGO test masses M= 40 kg, ω_m ∼ 10× 2πkHz and Q_m∼10^7. The incident optical field has λ = 1064nm and P=750kW.G_n is the optical transfer function of the nth HOM through the interferometer and back to the mirror. Changes to the interferometer configuration will therefore affect which vibrational modes are likely to become unstable in an interferometer. The linear dependence of ℝ on incident power indicates that vibrational modes that are stable for low powers may become unstable once the detectors are upgraded to full design power and sensitivity.§ A `FOREST OF MODES'Figure <ref> illustrates the importance of including the full DRFPMi interferometer in PI studies for LIGO. Building the interferometer in stages as described in section <ref>, we plot all mechanical modes up to 60kHz that are found to be unstable within 2 kHz of their frequency. Each point then marks the peak value of ℝ found for each eigenmode. This allows for inaccuracies in our simple mechanical model and a range of interferometer parameters,creating a 'worst case scenario' for PIs at LIGO. Critically, we find that the dual-recycled interferometer could suffer from twice the number of PIs when compared to the single cavity case.Note also that this plot refers to PIs exclusively due to mechanical modes in ETMX. Modes from different test masses, and any cross-coupling between these, are not considered in this study. Therefore the total number of PIs could at worst be quadruple that depicted. This approach has also been used to study the influence of SRC tuning, as shown in figure <ref>. We find that in a `worst case scenario', whereby all interferometer parameters combine to maximise the number of PIs, the influence of SRC tuning on the total number of PIs is diluted. By allowing the mechanical frequency to sweep over a 4kHz range, peaks in parametric gain due to both power- and signal-recycling are included, and a value of ℝ>1 anywhere in this range is treated as a count of 1 unstable mode. Since the majority of modes are able to resonate in the PRC (given an appropriate choice of mechanical frequency), changing the tuning of the SRC just influences the minority of modes that are only resonant via the SRC.myunsrt | http://arxiv.org/abs/1704.08595v2 | {
"authors": [
"A. C. Green",
"D. D. Brown",
"M. Dovale-Álvarez",
"C. Collins",
"H. Miao",
"C. Mow-Lowry",
"A. Freise"
],
"categories": [
"gr-qc",
"astro-ph.IM",
"physics.optics"
],
"primary_category": "gr-qc",
"published": "20170427142426",
"title": "The Influence of Dual-Recycling on Parametric Instabilities at Advanced LIGO"
} |
CNR,CNR]Daniela Cabiddu, Marco Attene [CNR]CNR-IMATI Genova, ItalyWe focus on the analysis of planar shapes and solid objects having thin features and propose a new mathematical model to characterize them. Based on our model, that we call an epsilon-shape, we show how thin parts can be effectively and efficiently detected by an algorithm, and propose a novel approach to thicken these features while leaving all the other parts of the shape unchanged. When compared with state-of-the-art solutions, our proposal proves to be particularly flexible, efficient and stable, and does not require any unintuitive parameter to fine-tune the process. Furthermore, our method is able to detect thin features both in the object and in its complement, thus providing a useful tool to detect thin cavities and narrow channels. We discuss the importance of this kind of analysis in the design of robust structures and in the creation of geometry to be fabricated with modern additive manufacturing technology.§ INTRODUCTIONThickness information is important in a number of shape-related applications, including surface segmentation, shape retrieval and simulation. By aggregating points with a similar local thickness, Shapira and colleagues <cit.> developed a pose-invariant shape partitioning algorithm, whereas thickness was used in histogram form as a shape descriptor for retrieval purposes in <cit.>. While considering shape thickness in general, a particular relevance is assumed by thin parts such as thin walls or wire-like features. These parts represent weak locations in physical structures, can be an obstacle for thermal dissipation, and can be subject to unwanted melting when traversed by intense electric currents. Furthermore, thin features are a serious issue in many manufacturing technologies. In plastic injection molding, for example, thin regions can restrict the flow of molten plastic <cit.>, causing the mold to be only partially filled. In modern additive manufacturing technologies, too thin features may lead to small layers of the surface being peeled, thin shape parts being fully removed, and shape break-up in several parts due to narrow connections <cit.>.The intuitive concept of “shape thickness” has been formalized in several ways in the literature, and diverse algorithms exist for its actual evaluation <cit.> <cit.> <cit.>. However, existing algorithms require a trade-off between accuracy and efficiency that can be hard to establish for demanding applications such as industrial 3D printing.In this paper we introduce the concept of “ϵ-shape” which is a mathematical model for objects having no features thinner than a threshold value ϵ. Based on this concept, we describe an algorithm to exactly and efficiently detect thin features. Differently from existing methods, our approach focuses on thin features only, meaning that parts of the geometry which are thick enough are not subject to any complex analysis: this is the key to achieve efficiency without sacrificing the precision. Furthermore, our formulation allows to detect thin portions in both the model and its complementary part. Note that this characteristic is particularly important when analyzing 3D mechanisms <cit.>: if parts which are supposed to move independently are separated by a too small space, they risk to be glued when printed.To demonstrate the usefulness of our analysis in practice, we describe a novel algorithm to thicken those parts of the shape which are thinner than a given threshold value (e.g. the layer thickness for 3D printing applications) while leaving all the other parts unaltered.§ RELATED WORKSThe medial axis of a shape is the locus of points having more than one closest point on the shape’s boundary. The medial axis may be enriched by considering each of its points endowed with its minimum distance from the shape's boundary <cit.>, leading to the so-called Medial Axis Transform (MAT). From a purely theoretical point of view, the MAT should be the reference tool to compute thickness information.Unfortunately, as observed in <cit.>, computing the medial axis and the MAT of a surface mesh can be an expensive process, and the medial axis itself is hard to manipulate <cit.>. Furthermore, algorithms based on the MAT are too sensitive and tend to confuse surface noise with small features unless approximations are employed <cit.> <cit.>. This motivated the emergence of the numerous alternative methods described in the remainder of this section. §.§ Voxel-based methodsThe analysis of medical 3D images has been an early calling application for thickness analysis. For Hildebrand and Rüegsegger <cit.>, the thickness related to a surface point (i.e. a skin voxel) is defined as the radius of the largest sphere centered inside the object and touching that point. This definition was turned into an algorithm in <cit.> where, after having computed the discrete medial axis, a distance transform from it is employed to associate a thickness value to all the skin voxels. In a more efficient approach based on PDEs <cit.>, Yezzy and colleagues compute thickness as the minimum-length surface-to-surface path between pairs of surface points. Telea and Jalba <cit.> observe that the extension of Yezzi's method to higher-genus models is not evident, and propose an alternative algorithm based on the top-hat transform from multi-scale morphology <cit.>.All these methods are suitable when the input comes in voxel form, but for objects represented in vector form (e.g. triangle meshes) an initial “voxelization” step is required which necessarily introduces a distortion. In principle, for 3D printing applications it is sufficient to keep this distortion below the printer’s resolution <cit.>. Even if this solution may work for low-resolution devices such as low-cost FDM-based printer's, it is too demanding for industrial printing where the layer thickness can be as small as 14 microns <cit.>, which means that a 10^3 centimeters cube would require more than 364 billion voxels. §.§ Mesh-based methodsIn <cit.> the concept of thickness at a point is defined based on spheres centered within the object and touching that point tangentially. Differently from the medial axis, these spheres may be not completely contained in the object. Though this method can be applied both on NURBS models and on piecewise-linear meshes, it has been shown to work well on relatively simple shapes only.The Shape Diameter Function (SDF) was introduced in <cit.> as a tool to perform mesh segmentation, and today is one of the most diffused methods to evaluate the thickness at a mesh point. The idea is to take a point on the surface and (1) compute the surface normal at that point, (2) cast a ray in the opposite direction, and (3) measure the distance of the first intersection of this ray with the mesh. Since this basic approach is too sensitive to small shape features and noise, Shapira and colleagues propose to use a number of casting directions within a cone around the estimated surface normal, and to keep a statistically relevant average of all the distances. Unfortunately this workaround may lead to completely unexpected results exactly in those cases that we consider to be critical. Imagine a tubular shape with a very narrow bottleneck on one side (see Fig. <ref>). In this case only a minority of the rays hits the bottleneck tip, and thus the resulting average would be far from the expectation. This is even amplified in practice because, in an attempt to make the evaluation more accurate, the SDF method filters those diameters which are too far from the median value, which are considered to be outliers. Note that this is a serious issue in all those applications where potential structural weaknesses must be detected (e.g. industrial 3D printing). Though some improvements are possible to make the SDF calculation faster <cit.> and less sensitive to noise <cit.>, this intrinsic limitation remains. §.§ ThickeningAutomatic detection of thin parts is an important tool for a designer who is producing a shape model. If too thin parts are actually detected, the designer may take countermeasures such as thickening those parts. However, in contexts where the user is not expert enough (i.e. home users of consumer 3D printers), automatic thickening algorithms would help a lot. This assumption motivated the work by Wang and Chen <cit.> in the context of solid fabrication: models created by unexperienced designers may represent thin features by zero-thickness triangle strips, and <cit.> presents an algorithm to thicken these features so that they become solid and printable while keeping the outer surface geometry unaltered.A generalization of morphological operators is used in <cit.>, where thickening can be obtained by applying an overall dilation to the input.The aforementioned methods do not adjust the amount of thickening depending on the local needs (i.e. the entire model is uniformly thickened). Conversely, based on an approximate medial axis, <cit.> cleverly analyses the geometry to predict where the printed prototype undergoes excessive stress due to manipulation and, besides adaptively thickening the weak parts, the method also adds little supporting structures to improve the overall object robustness. Note that while we focus on geometry only, in <cit.> the objective is to meet physical requirements, and some of the thin features may remain if they do not represent a weak part according to mechanical criteria. A comprehensive worst-case simulation is performed in <cit.>, but in this case the algorithm proposed is limited to the analysis and does not include a thickening phase. §.§ Summary of contributionsThe method presented in this article strives to overcome the limitations of the existing approaches discussed so far. In particular, we provide an original well-defined mathematical model to represent thickness information in any dimension and, based on this model, define an efficient algorithm to detect thin features in 2D and 3D objects represented in piecewise-linear form. We show how the method can be used to detect thin features in both the object and its complementary part, and demonstrate how to speed up the calculation by limiting the complex analysis within thin areas, which are the subject of our investigation. Finally, we show how thickness information can be exploited to adaptively thickening the object only where necessary, while leaving all the other parts of the shape unchanged.§ EPSILON SHAPESLet S be a compact n-manifold with boundary, embedded in ℝ^n. Let x be a point lying on the boundary of S and let ϵ be a real non-negative number (ϵ∈ℝ^+∪{0}). Let Ω_ϵ^x be a closed n-ball centered in x and having radius equal to ϵ.We define two operators that we call ϵ-sum and ϵ-difference. An ϵ-sum σ_ϵ(S, x) is the result of the union of S and Ω_ϵ^x σ_ϵ(S, x) = S ∪Ω_ϵ^x while an ϵ-difference δ_ϵ (S, x) is the result of the difference between S and the interior of Ω_ϵ^x δ_ϵ(S, x) = S - int(Ω_ϵ^x) ϵ-sums and ϵ-differences are grouped under a common definition of ϵ-modifications. An ϵ-modification of S is topologically neutral if the interior of its result is homeomorphic with the interior of S. It is worth noticing that the boundary of a topologically neutral ϵ-modification may not be an (n-1)-manifold.A shape S is an ϵ-shape if all its ϵ'-modifications are neutral, for each ϵ' ≤ϵ. If S is a ϵ-shape, then S is also an ϵ'-shape for each ϵ' < ϵ. Let S be an ϵ-shape. S is maximal if no ϵ' > ϵ exists so that S is also an ϵ'-shape. Although any compact n-manifold is an ϵ-shape for some ϵ≥ 0, it is worth observing that there are some such manifolds for which this is true only for ϵ = 0. An example in 2D is represented by polygons with acute angles, where any arbitrarily small disk induces a non-neutral ϵ-difference if its center is sufficiently close to the acute angle (Figure <ref>). However, even in these cases, we can analyze the boundary in a point-wise manner and construct what we call an ϵ-map.We say that an ϵ-sum σ_ϵ(S,x) is strongly neutral if all the σ_ϵ'(S, x) with ϵ' ≤ϵ are topologically neutral. An analogous concept is defined for ϵ-differences. A positive ϵ-map is a function E^+: ∂ S →ℝ^+∪{0} that maps each point x of the boundary of S to the maximum value of ϵ for which the ϵ-sum at x is strongly neutral. A negative ϵ-map E^- is defined as E^+ while replacing ϵ-sums with ϵ-differences. An ϵ-map E: ∂ S →ℝ^+∪{0} is defined as E(x) = min(E^+(x), E^-(x)). The minimum of E(S) is the value ϵ for which S is a maximal ϵ-shape. We extend this concept by saying that, if ϵ is the minimum of E^+(S), then S is a positive ϵ-shape. If ϵ is the minimum of E^-(S), then S is a negative ϵ-shape.A positive ϵ-map represents our formalization of the intuitive concept of shape thickness, whereas a negative ϵ-map describes the thickness of the shape's complement. The following Sections <ref> and <ref> describe an algorithm to compute these maps in 2D and 3D respectively.§ PLANAR SHAPE ANALYSISIn our scenario the input is a single polygon P, possibly non-simply connected. Hence the boundary of P, ∂ P, is a 1-manifold made of a number of edges connected to each other at vertices. In a straightforward approach, the value of E(v_i) may be associated to each vertex v_i of P. However, we observe that a so-defined ϵ-map is not sufficient to represent all the thin features, since local minima of E(P) may correspond to points lying in the middle of some edges (see Figure <ref>). To enable a comprehensive and conservative representation of all the thin features, the aforementioned ϵ-map must be encoded while considering internal edge points as well. If a local minimum happens to be on a point x in the middle of an edge, we split that edge at x so that E(x) is represented.Our algorithm to compute E(P) is based on a region growing approach. Intuitively, we imagine to have an infinitely small disk centered at a point x on ∂ P, and imagine to grow its radius in a continuous manner. Initially, the portion of ∂ P contained in the disk is made of a single open line, and we keep growing the radius as long as the topology of this restricted ∂ P does not change. It might happen, for example, that the line splits into separate components, or that it becomes closed (e.g. when the disk becomes large enough to contain the whole P). The maximum value reached by the radius is the value of E(x) that we eventually associate to x. To turn this intuitive idea into a working algorithm, we first create a constrained triangulation of P's convex hull, where all the edges of P are also edges of the triangulation. Then, we discretize the growing process by calculating relevant “events” that may occur either when the growing disk hits an edge of the triangulation, or when it hits one of its vertices. This approach is close in spirit to Chen and Han's algorithm <cit.> to calculate geodesic distances on polyhedra, where the topology of the evolving front is guaranteed not to change between two subsequent events.To simplify the exposition, we first describe how to compute E at a single vertex (Section <ref>), and then show how to extend the method to find possible local minima of E along an edge (Section <ref>). Finally, we describe an optimized procedure to compute the value of E on the whole polygon while possibly splitting its edges at local minima (Section <ref>).§.§ Thickness for a Single VertexIn the remainder, M denotes the constrained triangulaton of P, a mesh edge is an edge of M, whereas a polygon edge is an edge of P. So, a polygon edge is also a mesh edge but the opposite may not be true. Analogously, we refer to mesh vertices and polygon vertices, which is useful because the triangulation may (or may not) have additional vertices that do not belong to the original P. A generic polygon element indicates either a polygon edge or a polygon vertex. An example of this algorithm is shown in Figure <ref>. Let x be a vertex of P. Let R(x) be our growing region which is initially empty, and ∂ R(x) be the boundary of this region. Also, let A(x) = e_1 ∪ e_2 bet the portion of P made of the two polygon edges incident at x. In the first iteration, R(x) includes all the triangles incident at x (Figure <ref>). Iteratively, the algorithm computes the point y on ∂ R(x) - A(x) which is closest to x. At any iteration there might be multiple points at the same distance from x. In this case the algorithm randomly selects one of them, and all the others are detected in the subsequent iterations. Depending on the nature of the selected point y, the algorithm may either grow the region or terminate by associating E(x)=d(x,y) to x, where d(x,y) is the Euclidian distance between x and y.Specifically, the closest point y may be a vertex of ∂ R(x) (Figures <ref> and Figure <ref>), or it may be in the middle of one of its edges (Figure <ref>). Notice that y is not necessarilya polygon element. Imagine to build a disk centered in x and whose radius equals the distance from x to y. The algorithm terminates if y is what we call the antipodean of x, that is, y is on ∂ P and the topology of ∂ P restricted to the disk is no longer a single open line due to the inclusion of y. For this to happen, one of the following conditions must hold: * y is a point in the middle of a polygon edge; * y is a polygon vertex and its two incident polygon edges are either both internal or both external wrt the disk. Herewith, one such edge is external if y is its closest point to x, whereas it is internal in the other cases.If none of these conditions holds, the algorithm grows R(x) by adding all the triangles whose boundary contains y. Thus, if y is a vertex, R(x) grows on all the triangles which are incident at y but are not in R(x) yet. If y is in the middle of an edge e, R(x) includes the triangle incident at e which is not in R(x) yet.§.§ Minimum Thickness for a Single EdgeThe region growing approach described in Section <ref> can be extended to compute the minimum E along a single edge e = ⟨ v_1, v_2 ⟩. In this case, the minimum thickness may be either at one of the two endpoints or inbetween. Intuitively, an arbitrarily small disk is centered on e and is grown iteratively by exploiting M to discretize the process. The center of the disk is not fixed, but is moved along e at each iteration. At the last iteration, the disk radius represents the minimum value of E along e, while the position of the disk center shows where the local minimum lies. Before starting the region growing, we perform an acute angle test: let e_1 be the polygon edge that shares v_1 with e. If the angle formed by e and e_1 at v_1 is acute, we assign a zero value of E to v_1. Similarly, we measure the angle at the other endpoint v_2 and set E(v_2) to zero if such an angle is acute. If either of the endpoints have been set to zero, the algorithm terminates. Indeed, in this case the minimum for the edge has already been found.Otherwise, the algorithm initializes a region R(e) with all the triangles that share at least a vertex with e. Let A(e) be the union of the two polygon edges different than e that are incident at one of the endpoints of e. At each iteration, the point y on ∂ R(e) - A(e) which is closest to e is computed, and the corresponding point x on e which is closest to y is determined. If y is the “antipodean” of x the algorithm terminates, otherwise it proceeds with the region growing as described in Section <ref>. Notice that in this description we have implicitly extended the definition of antipodean of a vertex to any point on the boundary of P: now, x is not necessarily a vertex, and this fact has a subtle though major impact on our process. Indeed, the algorithm should terminate when the disk becomes tangent to a polygon edge in the middle, or in any other case when ∂ P within the disk is no longer a single open line. But unfortunately, this is not sufficient. Consider the disk in Figure <ref>: such a disk is tangent to the polygon edge yz in one of its endpoints and no termination condition holds. However, any arbitrarily small displacement of the disk along e while possibly growing the disk itself would cause an event which is a stop condition of the algorithm (i.e. the disk would become tangent to yz in the middle). To turn this observation into practice and guarantee to detect these events, we say that a point y on a polygon edge e' of ∂ P is the antipodean of a point x in the middle of an edge e if the segment x-y is orthogonal to e'. Hence, our region growing algorithm terminates either if y is a polygon edge endpoint that satisfies this orthogonality condition, or if it is a polygon vertex satisfying the conditions given in Section <ref>. §.§ Global thin feature detection A comprehensive ϵ-map can be built by splitting all the edges at local minima and by associating the value of E(x) to each vertex x, based on the procedures described in Sections <ref> and <ref>.Specifically, if our procedure detects a local minimum in the middle of an edge, that edge is split, the triangulation is updated accordingly, and the algorithm is run again to compute the local minima along the two resulting sub-edges. This procedure is guaranteed to converge because P is piecewise-linear and E(P) must necessarily have a finite number of local minima. Particular/degenerate cases such as, e.g. parallel edges, are avoided through Simulation of Simplicity <cit.>. One might argue that computing E at edges only is sufficient because any vertex is the end-point of some edges, and therefore its value of E would be computed by the procedure in Section <ref>. Unfortunately this is not true for all the vertices, and a counter-example is shown in Figure <ref>. Thus, the algorithm described in Section <ref> must be necessarily run to compute E at any polygon vertex which is not a local minimum for any edge. Only after this step the ϵ-map is guaranteed to be correct at all the vertices.We observe that many applications need to detect and possibly process thin features only, while thickness information on other (thick enough) parts of the shape is not used (Section <ref>).In these cases our algorithm can be significantly accelerated by specifying a threshold thickness beyond which a feature is no longer considered to be thin, and by stopping the region growing as soon as the radius exceeds such a threshold value. Hence, in this case the resulting ϵ-map exactly represents the thickness at any relevant point bounding a thin feature, whereas a generic thick attribute may replace the value of E for all the other boundary points of the input shape. § SOLID OBJECT ANALYSISIn 3D we analyze a polyhedron P bounded by a 2-manifold triangle mesh M, and use a constrained tetrahedrization T of P's convex hull to discretize the ball-growing process.We observe that the minimum of the ϵ-map within a triangle can be either on its border or in the middle of its internal area. Thus, before computing E at vertices and possibly splitting edges as we do in 2D, in 3D we may need to split triangles as well to represent some of the local minima.Vertices, edges and triangles are qualified as polyhedral if they belong to M, whereas they are mesh elements if they belong to T. Thus, similarly to the 2D case, a polyhedral element is also a mesh element, but not necessarily vice-versa. §.§ Thickness at a vertex We proceed with the region growing as we do for the 2D case, with the difference that in this case R(x) is a tetrahedral mesh and its boundary is a triangle mesh. Thus, A(x) is the union of all the polyhedral triangles incident at x, R(x) is initialized with theset of all the tetrahedra incident at x, and the algorithm terminates if the closest point y on ∂ R(x) - A(x) is the antipodean of x, that is, when the topology on ∂ P restricted to the ball is no longer a single disk. To define the conditions that characterize an antipodean point, we qualify a point y on ∂ R(x) based on its incident elements as follows.If t is a polyhedral triangle having y on its boundary (either alongan edge or on one of its three vertices), t may be either completely out of the ball (i.e. its minimum distance from x is exactly the ball's radius), or it may be partly or completely contained in it (i.e. its minimum distance from x is smaller than the ball's radius). In the former case we say that t is external, whereas in the latter case we say that t is an internal triangle wrt the ball. We use an analogous terminology to characterize an edge incident at y wrt to the ball.Having said that, y is the antipodean of x if one of the following conditions holds: * y is in the middle of a polyhedral triangle; * y is in the middle of a polyhedral edge whose two incident polyhedral triangles are either both external or both internal; * y is a polyhedral vertex whose incident polyhedral edges are e_1, ..., e_n in radial order around y and either: all the e_i's are internal or all the e_i's are external or there are more than two switches in the ordered chain of the e_i's (internal/external or vice-versa, including the possible switch between e_n and e_1). Thus, at the end of each iteration, we check whether y satisfies one of the aforementioned conditions. If so, the algorithm terminates. Otherwise, we grow the region on all the tetrahedra having y on the boundary and not being already part of R(x). §.§ Minimum thickness along an edge During the analysis of a polyhedral edge e = ⟨ v_1, v_2 ⟩ we consider the possibility to move the ball's center along the edge. The position of the ball's center at the last iteration represents the minimum of E along the edge, while the radius indicates its actual value. Note that the minimum may correspond to an edge endpoint or may lie in the middle.Before starting the iterative region growing, we check for acute configurations as follows. Let t_1 and t_2 be the two polyhedral triangles incident at e. We consider all the edges which are incident at v_1 but are not edges of either t_1 or t_2. If any such edge forms an acute angle with e, E(v_1) is set to zero, otherwise we consider all the triangles incident at v_1 but not incident at e. If the projection of e on the plane of any such triangle intersects the interior of the triangle itself, E(v_1) is set to zero. Analogous checks are performed on v_2. If either E(v_1) or E(v_2) is set to zero due to these checks, the algorithm terminates.Otherwise, a region R(e) is initialized with all the tetrahedra that share at least a vertex with e, and the set A(e) is made of all the polyhedral triangles that share at least a vertex with e. At each iteration, we calculate the point y on ∂ R(e) - A(e) which is closest to e, and determine the corresponding point x on e which is closest to y. If y is the "antipodean" of x the algorithm terminates, otherwise it proceeds with the region growing as described in Section <ref>.Similarly to the 2D case, the definition of antipodean was implicitly extended and a clarification is necessary to avoid missing borderline cases. Imagine a ball centered on e and tangent to a polyhedral triangle. The tangency point may be either in the interior of the triangle or on its boundary. In the former case the algorithm terminates, whereas in the latter the region growing would take place if we consider only the conditions given in Section <ref>. Nevertheless, if we apply an arbitrarily small translation of the ball center along e while possibly growing its radius, the ball would become tangent to the same triangle in the middle (i.e. the algorithm should terminate). Once again, to cope with these cases we extend the definition of antipodean point, and we say that a point y on a polyhedral element y' of ∂ P is the antipodean of a point x in the middle of an edge e if the segment x-y is orthogonal to y'. Note that y' may be either an edge or a triangle.Hence, our region growing algorithm terminates if y is on a polyhedral element that satisfies this orthogonality condition, or if it is a polyhedral vertex satisfying the conditions given in Section <ref>.§.§ Minimum thickness on a triangle A similar approach can be exploited to compute the minimum E on a single polyhedral triangle t = ⟨ v_1, v_2, v_3 ⟩. The minimum may correspond to a triangle vertex, or lie in the middle of a triangle edge, or be a point in the middle of the triangle itself. Thus, the ball's growth process considers the possibility to move the center on the whole t. As we do for edges, before starting the iterative region growing we check for acute configurations as follows. Let t_1, t_2 and t_3 be the three polyhedral triangles adjacent to t. We consider all the polyhedral edges which are incident at v_1 but are not edges of either t_1, t_2 or t_3. If the projection of any such edge on the plane of t intersects the interior of t, E(v_1) is set to zero. Analogous checks are performed on v_2 and v_3.Furthermore, we consider each polyhedral triangle t_i that shares at least a vertex with t, and if the angle formed by the normal at t and the normal at t_i is obtuse, the value of E for all their shared vertices is set to zero. If the value of E for at least one of the three vertices is set to zero due to these checks, the algorithm terminates. Otherwise, the initial region R(t) includes all the tetrahedra which are incident to at least one vertex of t, and the set A(t) is made of all the polyhedral triangles that share at least a vertex with t.At each iteration, we calculate the point y on ∂ R(t) - A(t) which is closest to t, and determine the corresponding point x on t which is closest to y. The same arguments and procedure described in Section <ref> apply here. §.§ Global thin feature detectionThe algorithm to compute the overall ϵ-map is similar to the 2D version described in Section <ref>. We first analyze each polyhedral triangle (Section <ref>), possibly split it at its local minimum, and re-iterate the analysis on the resulting sub-triangles. We observe that these sub-triangles can be either three or just two (the latter case occurs if the minimum is on one of the three edges). When all the triangles are processed, we analyze each polyhedral edge (Section <ref>), possibly split it at its local minimum, and re-iterate on the two resulting sub-edges. The tetrahedization is updated upon each split operation performed on P. Finally, we process all the polyhedral vertices (Section <ref>).Note that in the 2D case we might have assumed that all the vertices are polygonal vertices. Indeed, a planar polygon can always be triangulated. A corresponding statement cannot be done for the 3D case due to the existence of polyhedra that do not admit a tetrahedrization. In these cases, a constrained tetrahedrization can be calculated only if a number of so-called Steiner points are added to the set of input vertices.§ THICKENINGWhen thickness information is available, the overall geometry can be adaptively changed in an attempt to replace thin features with thicker structures. For some applications a minimum thickness may be well defined (e.g. the layer thickness for 3D printing), whereas for some others this value can be empirically set by the user based on his/her experience. In any case, in this section we assume that such a threshold value is available and exploit it, in combination with thickness information, to modify the shape locally. While doing this operation, we strive to modify the geometry only where necessary, and no more than necessary.Our thickening problem can be formulated as follows: given a shape S and a threshold thickness ϵ, we wish to find the shape S' which is most similar to S while being an ϵ-shape. Clearly, if S has no feature thinner than ϵ, then S' must coincide with S. Though an exact solution to this problem appears to be extremely complicated, herewith we present a heuristic approach that proved to be both efficient and effective in all of our experiments. The basic idea is the following: if the value E(x) of the ϵ-map at a point x on S is less than the threshold ϵ, we consider both x and its antipodean y, and move both the points away from each other as long as their distance becomes equal to ϵ. Stated differently, the new position for x will be x + (ϵ-E(x))/2x-y, whereas the new position for y will be y + (ϵ-E(x))/2y-x, where a denotes the normalized vector a/||a||.Since the objective in this section is to thicken too thin features, herewith we consider the positive ϵ-map E^+ only. On piecewise-linear shapes, this map can be computed as described in Sections <ref> and <ref> by disregarding the outer part of the constrained triangulation/tetrahedrization, and by considering internal angles only when checking for acuteness. Furthermore, a partial ϵ-map can be computed as described in Section <ref>. Indeed, by using ϵ as a threshold to stop the process, we can achieve a much faster computation.After such a computation, each vertex can be categorized in three ways, depending on the value of its partial ϵ-map: in the remainder, an acute vertex is a vertex mapped to a zero value, while a thin vertex is a vertex mapped to a positive value which is lower than the thickness threshold ϵ. Any other vertex is just thick and is not considered by the algorithm. While computing the ϵ-map we keep track of the antipodean point for each thin vertex. If such a point does not coincide with a vertex, before proceeding with the thickening we split the simplex that contains it, so as to have a vertex to displace.For the sake of simplicity, we now assume that there are no acute vertices. Their treatment is described later in Sections <ref> and <ref>. Thus, each thin vertex x with antipodean y is associated to a displacement vector δ x = (ϵ-E(x))/2x-y. Similarly, each antipodean vertex y of x is associated to a displacement vector δ y = (ϵ-E(x))/2y-x. If the same vertex has several “roles” (e.g. it is the antipodean of two different vertices), its displacement vectors are summed.When this displacement vector field is complete, the actual displacement may take place. However, even this operation must be undertaken with a particular care. Indeed, if two thin features are separated by an insufficient space, their uncontrolled growth might bring one to intersect the other. To avoid this, we keep the original triangulation/tetrahedrization (outer parts included) and, for each single displacement, we check that no flip occurs among the simplexes incident at the displaced vertex. We observe that the need for such a check reveals that our problem may have no solution if we do not allow topological changes. However, if topological changes are acceptable, we just let the surfaces intersect with each other and track the so-called outer hull in a second phase as described in <cit.>.The following two subsections describe how to pre-process the model so as to remove possible acute vertices and make the model ready to be thickened as proposed. §.§ Pre-processing 2D shapes Clearly, our thickening approach is not suitable for acute vertices which are mapped to a zero value and for which no antipodean point exists. Thus, preprocessing is performed to remove acute vertices before thickening. To remove acute angles, we imagine to cut off a small portion of the shape around each acute vertex, and to fill the generated hole. Specifically, let v be an acute vertex. A cutting line is defined, which intersects the internal angle bisector and is perpendicular to it. The distance between v and the cutting line may be arbitrarily small. Then, for each edge e_i = ⟨ v , v_i ⟩ incident at v,v is replaced by the intersection point between the cutting line and e_i. Finally, an additional edge is added to close the polygon (Figure <ref>). Then, the two algorithms described in Section <ref> and <ref> are exploited to update the ϵ-map with thickness information related to the modified parts of the shape. §.§ Pre-processing 3D models Getting rid of acute vertices in 3D is not as easy as in 2D due to the possible presence of saddles. To solve this problem, we just consider the set of all the triangles which are incident at acute vertices and, on such a set only, we adaptively run a subdivision step using the modified interpolating Butterfly scheme by Zorin and colleagues <cit.>. Then, the three methods proposed in Section <ref> are exploited to update the ϵ-map restrictedly to the modified part of the model. The process is repeated as long as acute angles are found, and convergence is guaranteed because the subdivision generates a C^1 continuous surface at the limit (Fig. <ref>). To reduce the area of the modification, before running the aforementioned procedure we subdivide the considered triangles without changing the geometry (i.e. each edge is split at its midpoint) and keep working only on the subtriangles which are incident at acute vertices. This can be done as many times as necessary to keep the changes within a tolerable limit. § RESULTS AND DISCUSSIONWe implemented both our analysis and thickening methods in C++ with the support of Tetgen <cit.> for the computation of the constrained tetrahedrizations.This section reports the results of some of our experiments on a set of 3D input meshes coming from <cit.>. Our prototype software provides the possibility to set an analysis direction, that is, the user can choose whether to compute a positive, a negative, or a bi-directional ϵ-map. Also, if the complete ϵ-map is not necessary, the user can set a threshold value to compute only a partial ϵ-map. In the latter case the region growing is interrupted as soon as the ball radius reaches the threshold.Figure <ref> depicts two complete and bi-directional ϵ-maps, whereas Figures <ref> and <ref> show a positive and a negative complete map respectively. Figures <ref> and <ref> depict examples of positive and negative partial ϵ-maps respectively. Note that the two partial maps highlight thin features detected in the model and in its complementary part respectively. As expected, the thin bridge connecting the two fingers on the right is detected as the thinnest feature in the object, while the cavity generated by the bridge and the gap between fingers is detected as thinnest features in its complementary part. An additional partial negative ϵ-map is shown in Figure <ref>.During the thickening, the result of the analysis is exploited to modify the input shape only where thickness is lower than the input threshold, while any other part is kept unchanged. Figure <ref> shows both the complete and the partial positive ϵ-maps for Model 372. Note that the latter is sufficient to run the thickening. In the example in Figure <ref>, only the handle and the spout of the teapot are edited as expected. In Figures <ref>, two positive ϵ-maps are shown, which represent thickness information before and after the application of our thickening algorithm. Our heuristic approach actually increases the minimum thickness of the shape, but in some cases it is not sufficient to achieve the desired thickness threshold in one single iteration. In these cases, the whole algorithm can be run again, and we could verify that this process converged to an actual (positive) ϵ-shape is all of our experiments. Figure <ref> show some additional results. Execution time Experiments were run on a Linux machine (1.9GHz Intel Core i7, 16GB RAM). Table <ref> shows the execution times referred to the computation of complete and partial ϵ-maps. As expected, our algorithm is sensibly accelerated when a thickness threshold is specified.At a theoretical level, the time needed to characterize a single vertex grows as the number of tetrahedra to be visited increases. The worst case is represented by a convex polyhedron whose n vertices are all on the surface of a sphere. In this case, for each vertex O(n) tetrahedra must be visited, which means that the overall complexity is O(n^2). In contrast, if a shape is very thin everywhere and the tetrahedral mesh has no unnecessary internal Steiner points , then the antipodean is normally found in the first few iterations. The use of a threshold to compute a partial map simulates this behaviour. These theoretical considerations are confirmed by our experiments. As an example, the computational times referred to Model 192 (i.e. the hand) and Model 372 (i.e. the teapot) are sensibly different, even if both models have about 13.5K triangles.Table <ref> reports the time spent by our algorithm to compute the complete bi-directional ϵ-map for a set of test models. These values were measured while running a sequential version of the algorithm, where simplexes are analyzed one after the other. However, it should be considered that our method is embarrassing parallel, that is, several simplexes of the same degree can be simultaneously analyzed. This is true for both the computation of complete and partial ϵ-maps. Our implementation considers this aspect and exploits OpenMP <cit.> to enable such a parallelism. Thanks to this parallel implementation, we succeeded in reducing the running time by a factor of 2.5 on our 4-core machine.§.§ ComparisonThe Shape Diameter Function (SDF) can be considered the state-of-the-art tool for thickness evaluation on meshes. Its implementation available in CGAL <cit.> has been exploited to analyze our dataset and compare the results. Figure <ref> shows a comparison between complete ϵ-maps and SDF. As expected, in some cases the two thickness-based descriptors provide a similar representation of the input shape. Nevertheless, some relevant differences are visible. Consider thickness evaluation at the extremities of the shape (e.g. the fingertips of the hand in Figure <ref>) and at some points closest to very narrow features (Figure <ref>). In both cases, the statistic average of the distances applied by the SDF provides an unexpected result. In the former case, most of the rays casted from an extremal point touch the surface closest to the point itself. The opposite happens when a narrow bottleneck is present. By increasing the number of casted rays a more precise result may be achieved but, though this would have a significant impact on the performances, no guarantee can be given in any case.Our method solves this limitation and computes the exact thickness value even at these points according to our definition. Note that this exact value is guaranteed to be calculated and does not depend on any user-defined input parameter. Also, through our method narrow features can be efficiently detected by means of a partial ϵ-map. § CONCLUSIONS AND FUTURE WORKWe have shown that both 2D polygons and 3D polyhedra can be automatically characterized based on their local thickness. Our novel mathematical formulation allows to implement effective algorithms that associate an exact thickness value to any relevant point of the object, without the need to set any unintuitive parameter, and without the need to rely on estimates or approximations. Furthermore, we have shown how our analysis can be used to thicken thin features, so that models which are inappropriate for certain applications become suitable thanks to an automatic local editing.While the analysis is rigorously defined, our thickening algorithm is still based on heuristics and cannot guarantee a successful result in all the cases. Stated differently, if ϵ is the threshold thickness used to perform the thickening, we would like to guarantee that the thickened model is an ϵ-shape. Therefore, an interesting direction for future research is represented by the study of thickening methods that provide such a guarantee, and it is easy to see that such methods must necessarily be free to change the local topology.§ ACKNOWLEDGMENTThis project has received funding from the European Unions Horizon 2020 research and innovation programme under grant agreement No 680448 (CAxMan). Thanks are due to all the members of the Shapes and Semantics Modeling Group at IMATI-CNR and in particular to Silvia Biasotti for helpful discussions.§ REFERENCES model3-num-names | http://arxiv.org/abs/1704.08049v2 | {
"authors": [
"Daniela Cabiddu",
"Marco Attene"
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"cs.CG",
"cs.GR"
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"published": "20170426103315",
"title": "Epsilon-shapes: characterizing, detecting and thickening thin features in geometric models"
} |
elsarticle-num | http://arxiv.org/abs/1704.08206v2 | {
"authors": [
"Sebastian M. Dawid",
"Rafał Gonsior",
"Jan Kwapisz",
"Kamil Serafin",
"Mariusz Tobolski",
"Stanisław D. Głazek"
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"quant-ph"
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"primary_category": "quant-ph",
"published": "20170426164327",
"title": "Renormalization group procedure for potential $-g/r^2$"
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[email protected]@ut.edu.co Departamento de Física, Facultad de Ciencias, Universidad del Tolima, 730006299, Ibagué, [email protected] Instituto de Física, Universidad de Antioquia, Calle 70 No. 52-21, Medellín, [email protected] Laboratório de Física Teórica e Computacional, Universidade Cruzeiro do Sul, Rua Galvão Bueno, 868, 01506-000 São Paulo, SP, Brazil We calculate the masses and weak decay constants of flavorlessground and radially excited J^P=1^- mesons and the corresponding quantities for the K^*,within a Poincaré covariant continuumframework based on the Bethe-Salpeter equation. We use in both, the quark's gap equation and the meson bound-state equation, an infrared massive and finite interaction in the leading symmetry-preserving truncation. While our numerical results are in rather good agreement with experimental values where they are available, no singleparametrization of the QCD inspired interaction reproduces simultaneously the ground and excited mass spectrum, which confirms earlier work on pseudoscalar mesons.This feature being a consequence of the lowest truncation, we pin down the range and strength of the interaction in both cases to identify common qualitative featuresthat may help to tune future interaction models beyond the rainbow-ladder approximation. 12.38.-t 11.10.St 11.15.Tk,14.40.Pq 13.20.Gd14.40.Df Mass spectrum and decay constants of radially excited vector mesons Bruno El-Bennich December 30, 2023 =================================================================== § INTRODUCTION Vector mesons play an important role in the physics of strong interactions and hadron phenomenology. Since these mesons and the photon share the same quantum numbers, J^PC=1^– , flavorless neutral vector mesons can directly couple to the photon via an electromagnetic current. Historically,this led to the vector meson dominance model. Compared to other mesons their production can be measured with very high precision, for instance in electron-positron collisions via the process e^+e^- →γ^* →q̅q which provides a much cleaner signal than hadronic reactions. Notwithstandingcomplications with hadronic final states, vector mesons are abundant decay products in electroproduction of excited nucleons <cit.>,N^*, and exclusive vector meson production reactions are responsible for a large fraction of the total hadronic cross section and precede di-lepton decaysin relativistic heavy-ion collisions <cit.>. In flavor physics, exclusive B decays with final-state vector mesons, e.g. B→ Vπ, B → Vℓν_ℓ,B→ Vμ^+μ^- orB → Vγ, are a central component of the LHCb experimental program and theB → K^*μ^+μ^- andB_s → K^*μ^+μ^- decays are of particular interest, as their angular distributions are very sensitive probes of new physics <cit.>. In the latter cases, a precise knowledge of the pseudoscalar and vector meson light-cone distributionamplitudes is essential <cit.>. A characteristic feature of the 1^– ground-state vector mesons is their predominant occurrence as pure q̅q states: in the case of theω(782) and ϕ(1020) mesons the vector flavor-nonet mixing angle is close to ideal mixing, i.e. ϕ(1020)is nearly a pure |s̅s⟩state and ω =(u̅u+d̅d)/√2. This ideal mixing is not evident in pseudoscalar and scalar meson multiplets. Thus, vector mesons as decayproducts of heavier non-vector mesons are a very good probe of their flavor content measured in their respective decay rates into different types ofmesons. The study of vector mesons is complementary to that of light pseudoscalar mesons, the Goldstone bosons, as their masses are more in agreement with the sum oftheir typical constituent quark masses. This stands in contrast to the light pseudoscalar's properties best described by the dichotomy of dynamical chiral symmetrybreaking (DCSB), which generates a light-quark mass consistent with typical empirical constituent masses <cit.> even in the chiral limit,yet also produces a very light Goldstone boson due to explicit breaking of chiral symmetry of non-zerocurrent-quark masses. These characteristic featuresof the light pseudoscalar octet are dictated by an axialvector Ward-Green-Takahashi identity which relatesdynamical quantities in the chiral limit. Most chiefly, it implies that the leading Lorentz covariant in the pseudoscalar quark-antiquark γ_5 channel is equalto B(p^2)/f_π, where B(p^2) is the scalarcomponent of the chiral quark self energy.As a corollary, the the two-body problem is solved almost completelyonce a nontrivial solution of the gap equation is found.This remarkable fact facilitates the phenomenology of light pseudoscalar and is taken advantage of within the joint approach of the Dyson-Schwinger equation (DSE)and Bethe-Salpeter equation (BSE) in continuum Quantum Chromodynamics (QCD) <cit.>. That is because in both, the two-point and four-point Green functions, the axialvector Ward-Green-Takahashi identity ispreserved by their simplest approximation, namely the rainbow-ladder (RL) truncation. There is no reason to expect a priori this truncation to be as successfulin describing vector mesons whose solutions contain double the amount of covariants and whose higher masses sample the quark propagator in larger domain of thecomplex p^2 plane. Moreover, the axialvector Ward-Green- Takahashiidentity does not constrain the transverse components of the vector meson Bethe-Salpeteramplitude (BSA). Nonetheless, the simplest truncation carried out with the Maris-Tandy model <cit.>for the quark-gluon interaction functionworks remarkably well for the lowest ground-state vector mesons, such as the ρ, ω, K^* and ϕ mesons. We here reassess the seminal work on vector mesons by Maris and Tandy <cit.> within a modern understanding of the QCDinteractions <cit.>. This ansatz produces an infrared behavior of the interaction, commonly described by a “dressing function" 𝒢(k^2),that mirrors the decoupling solution found in DSE and lattice studies of the gluon propagator <cit.>. This solution is a boundedand regular function of spacelike momenta with a maximum value at k^2 = 0. Our aim is to compute the mass spectrum of the ground and radially excited statesof the light, strange and charm vector mesons as well as of vector charmonia and bottomia <cit.>which were also the object of lattice-regularized QCDstudies <cit.>;in addition we compute their weak decay constants whose precise knowledge is important in hadronicobservables measured by LHCb and FAIR-GSI, for example.§ BOUND STATES IN THE VECTOR CHANNEL In analogy with previous work on pseudoscalar ground and excited states <cit.>, we employ the RL truncation in both, the quark's DSE and the vector meson's BSE,which is the leading term in a symmetry-preserving truncation scheme. The following two sections detail their respective kernels and lay out the setup for the numerical implementation to compute vector meson properties. §.§ Quark Gap EquationThe quark's gap equation is generally described by the DSE,[We employ throughout a Euclidean metric in our notation:{γ_μ,γ_ν} = 2δ_μν; γ_μ^† = γ_μ; γ_5=γ_4γ_1γ_2γ_3, tr[γ_4γ_μγ_νγ_ργ_σ]=-4 ϵ_μνρσ; σ_μν=(i/2)[γ_μ,γ_ν];a· b = ∑_i=1^4 a_i b_i; and P_μ timelike ⇒ P^2<0.]S^-1(p)= Z_2 (i γ·p + m^bm)+ Z_1g^2∫^Λ_k D^μν (q) λ^a/2γ_μS(k)Γ^a_ν (k,p),where q=k-p, Z_1,2(μ,Λ ) are the vertex and quark wave-function renormalization constants, respectively, and ∫_k^Λ≡∫^Λ d^4k/(2π)^4represents throughout a Poincaré-invariant regularization of the integral with the regularization mass scale,Λ. Radiative gluon corrections in the second termof Eq. (<ref>) add to the current-quark bare mass, m^bm(Λ), where the integral is over the dressed gluon propagator, D_μν(q), and the dressed quark-gluon vertex, Γ^a_ν (k,p); the SU(3) matrices, λ^a, are in the fundamental representation. The gluon propagator is purelytransversal in Landau gauge,D^ab_μν (q) =δ^ab( g_μν - k_μ k_ν/q^2) Δ ( q^2) /q^2 , where Δ ( k^2) is the gluon-dressing function. In RL approximation, the quark-gluon vertex is simply given by its perturbative limit,Γ^a_μ (k, p) =λ^a/2Z_1 γ_μ,and since we neglect the three-gluon vertex and work in the “Abelian" version of QCD which enforces a Ward-Green-Takahashi identity <cit.>,Z_1=Z_2, we re-express the kernel of Eq. (<ref>),Z_1g^2D_μν (q)Γ_μ (k, p)=Z_2^2 𝒢 (q^2) D_μν^free (q)γ_μ ,where we suppress color factors and D_μν^free(q) :=( g_μν -q_μ q_ν/q^2)/q^2 is the free gluon propagator. An effective modelcoupling,whose momentum-dependenceis congruent with DSE- and lattice-QCD results and yields successful explanations of numerous hadronobservables <cit.>, is given by the sum of two scale-distinct contributions:𝒢 (q^2)/q^2 = 8π^2/ω^4De^- q^2/ω^2+8π^2 γ_m ℱ(q^2) /ln [τ + (1 + q^2/Λ_QCD^2)^ 2 ] ,The first term is an infrared-massive and finite ansatz for the interaction, whereγ_m = 12/(33 - 2N_f ), N_f = 4, Λ_QCD = 0.234 GeV; τ = e^2 - 1;and ℱ(q^2) = [1 - exp(-q^2/4m^2_t) ]/q^2, m_t = 0.5 GeV.The parameters ω and D control the width and strength of the interaction, respectively. At first sight they seem to be independent, yet a large collection of observables of ground-state vector and isospin-nonzero pseudoscalar mesons are practically insensitiveto variationsof ω∈ [0.4, 0.6] GeV, as long as Dω = constant. The second term in Eq. (<ref>) is a bounded, monotonically decreasingcontinuation of the perturbative-QCDrunning coupling for allspacelike values of q^2. The most important feature of thisansatz is that it provides sufficient strength to realize DCSB and implements a confined-gluoninteraction <cit.>. At k^2 ≳ 2 GeV^2, the perturbative component dominates the interaction. In Figure <ref> we plot the interaction, 𝒢 (q^2) in Eq. (<ref>)for a typical value, w D = (0.8 GeV)^3 and ω =0.4 <cit.>, employed in RL approximation as well as for other values of ω to illustrate the decrease of and shift towardslarger k^2 of its strength. For ω≃ 0.6, the functional form of the interaction is more akin to that employed in combination with a ghost-dressed Ball-Chiuvertex <cit.>.The solutions for spacelike momenta, p^2>0, of the gap equation (<ref>) include a vector and a scalar piece,S_f^-1(p) = i γ·p A_f(p^2) +1_D B_f(p^2),for a given flavor, f, which requires a renormalization condition for the quark's wave function,. Z_f (p^2) = 1/A_f (p^2) |_p^2 = 4 GeV^2 = 1. This imposed condition is supported by lattice-QCD simulations of the dressed-quark propagator. The mass function,M_f(p^2)=B_f(p^2, μ^2)/A_f(p^2, μ^2), isrenormalization-point independent. In order to reproduce the quark-mass value in perturbative QCD, another renormalization condition is imposed,.S^-1_f(p)|_p^2=μ^2= i γ·p+ 1_D m_f(μ ) ,at a large spacelike renormalization point, μ^2≫Λ_QCD^2, where m_f(μ ) is the renormalized running quark mass:Z_m^f(μ,Λ )m_f(μ)=m_f^ bm (Λ) .Here,Z_m^f(μ,Λ ) =Z_4^f(μ,Λ )/Z_2^f (μ,Λ ) is the flavor dependent mass-renormalization constant andZ_4^f(μ,Λ ) is associated with the mass term in Lagrangian. In particular, m_f(μ )is nothing else but the dressed-quark massfunction evaluated at one particular deep spacelike point, p^2=μ^2, namely: m_f(μ)= M_f(μ ).§.§ Vector Bound-State Equation The wave function of a bound state of a quark of flavor, f, and an antiquark of flavor, g̅, in the 1^- channel is related to their BSA, Γ_Vμ^fg̅(p,P), which for a relative momentum, p, and total momentum, P, is obtained from thehomogeneous BSE,Γ_μ^V(p,P)= ∫^Λ_k 𝒦 (p,k,P) S_f (k_+) Γ_μ^V (k,P)S_g̅ (k_-),where k_+ = k+η_+ P, k_- =k- η_- P; η_+ +η_- =1.We employ a ladder truncation of the BSE kernel consistent with that of thequark's DSE (<ref>),𝒦 (p,k,P) =- Z_2^2 𝒢 (q^2 )λ^a/2γ_μ D^free_μν (q)λ^a/2γ_ν,which satisfies an axial-vector Ward-Green-Takahashi identity <cit.> and consequently the pseudoscalar mesons aremassless in the chiral limit. The BSE defines an eigenvalue problem with physical on-shell solutions for P^2 = -M_V_0^2 for the ground state and for the radially excited states, P^2 = - M^2_V_n, M_V_n+1^2 > M_V_n^2, n=1,2,3 ....As we are interested in radially excited J^P = 1^- states, the question arises whether they can be described by the interaction in Eq. (<ref>)withexactly the same parameter set as for ground states, whether the parameters have to be adjusted or whether the truncation fails to achieve at least a reasonable description of their mass spectrum. The masses and weak decay constants of excited states are very sensitive to the strength and width of the long-range term in Eq. (<ref>), which provides more support at large inter-quark separation than, e.g., the Maris-Tandymodel <cit.>. In here, weextend the studies of Refs. <cit.> to the excited states of vector mesons with an interaction that differs from those employed in Ref. <cit.>. We also refer to the discussion in Ref. <cit.>, where it is pointed out that beyond-RL contributions are important in heavy-light mesons due to the strikingly different impact of the quark-gluonvertex dressing for a light and a heavy quark. The effects of DSCB and the importance of other quark-gluontensor structures are increasingly moreimportant for lighter quarks <cit.>. Thus, one does not expect the RL truncation to accurately describe either the groundnor the excited states of charm and beauty mesons. On the other hand,the RL approximation describes very well equal-mass bound states, such asquarkonia <cit.>.The normalization condition for the Bethe-Salpeter amplitude is,2P_μ = ∂/∂ P_μN_c/3∫^Λ_k Tr_D [Γ̅_ν^V(k, - K)S_f (k_+) ×Γ_ν^V (k, K) S_g̅(k_-) ]_K=P^P^2=-M^2_V,The charge-conjugated BSA is defined as Γ̅(k,-P) := C Γ^T (-k,-P) C^T, where C is the charge conjugation operator. Finally, the weak decay constant for 1^- meson is defined as,f_VM_Vϵ^λ_μ (P)= ⟨ 0 | q̅^g̅γ_μ q^f |V(P,λ) ⟩ ,where ϵ^λ_μ (P) is the meson's polarization vector satisfyingϵ^λ_μ· P = 0 and normalized such that ϵ^λ_μ^* ·ϵ^λ_μ = 3; Eq. (<ref>) can be expressed as,f_VM_V= Z_2N_c/3∫^Λ_kTr_D[γ_μS_f (k_+) Γ_Vμ^fg̅ (k, K) S_g̅(k_-)]. § NUMERICAL IMPLEMENTATION §.§ Quark Propagators on the Complex Plane In solving the BSE (<ref>), the quark propagators with momentum (k± P)^2 = k^2 + 2i η_± |k| M_V - η_±^2 M_V^2,where k is collinear with P = (0⃗,i M_V ) in the meson'srest frame, must necessarily be treated in the complex plane <cit.>. Complex-conjugate pole positions of the propagators depend on the analytical form of theinteraction andcan be represented by analytical expressions based on a complex-conjugate pole model <cit.>:S(p) = ∑_i^n[ z_i/iγ·p + m_i + z_i^*/iγ·p + m_i^* ] ;m_i, z_i ∈ℂ .The propagator in Eq. (<ref>) is pole-less on the real timelike axis and therefore has no Källén-Lehmann representation, which is a sufficient condition to implementconfinement <cit.>. The numerical DSE solutions we obtain on the complex plane <cit.> can be fitted with n=3complex-conjugate poles. We solve the BSE (<ref>) both ways, employing full numerical DSE solutions for the quark in the complex plane and the pole model in Eq. (<ref>) and find agreement at the one-percent level for the vector meson masses and decay constants. With respect to the current-quark masses given by,Z_4^f (μ, Λ ) m_f (μ) = Z_2^f (μ, Λ ) m_f^mb (Λ),where Z_4^f (μ, Λ ) isassociated with the mass term in the QCD Lagrangian, m_u=m_d(μ) , m_s(μ) and m_c(μ)are fixed in Eq. (<ref>)by requiringthat the pion and kaon BSEs produce m_π = 0.138 GeV and m_K = 0.493 GeV, respectively. This, in turn, yields m_u,d(μ) = 3.4 MeV,m_s(μ) = 82 MeV,m_c(μ) =0.828 GeVand m_b(μ) = 3.86 GeV for μ =19 GeV. §.§ Solving the Bethe-Salpeter Equation The general Poincaré-invariant form of the solutions of Eq. (<ref>) in the vector meson channel and for the eigenvalue trajectory, P^2= -M_V_n^2, in a orthogonal base withrespect to the Dirac trace is given by:Γ^V_n_μ (q;P)=∑^8_α=1 T^α_μ(q, P) ℱ^n_α(q^2,q· P;P^2) ,with the dimensionless orthogonal Dirac basis <cit.>,T^1_μ(q,P) = γ^T_μ,T^2_μ(q,P) = 6/q^2√(5) [q^T_μ(γ^T· q)-1/3γ^T_μ(q^T)^2 ] , T^3_μ(q,P) = 2/qP[q^T_μ(γ· P)] ,T^4_μ(q,P) = i√(2)/qP [γ^T_μ(γ· P)(γ^T· q)+q^T_μ(γ· P) ] , T^5_μ(q,P) = 2/qq^T_μ,T^6_μ(q,P)= i/q√(2) [ γ^T_μ(γ^T· q)-(γ^T · q)γ^T_μ ] ,T^7_μ(q,P) = i√(3)/P√(5) ( 1-cos^2θ ) [γ^T_μ(γ· P)- (γ· P)γ^T_μ ]-1/√(2)T^8_μ(q,P) , T^8_μ(q,P) = i 2√(6)/q^2P√(5) q^T_μ γ^T · q γ· P.The C-parity properties of this basis are elucidated elsewhere <cit.> and the transverse projection, V^T, is defined by,V^T_μ= V_μ-P_μ(P· V)/P^2,with q· P=qPcosθ. These covariants satisfy the orthonormality condition, 112 Tr_D [ T^α_μ(q,P)T^β_μ(q, P)] = f_α(cosθ)δ^αβ, where the functions f_α(z) are given byf_1(z)=1, f_α(z)= 4/3(1-z^2) withα=3,4,5,6andf_α(z)= 8/5(1-z^2)^2 forα=2,7,8. The normalization constants f_α satisfy: ∫_0^π dθ sin^2θf_α(cosθ)=π/2 . Making use of the covariant decomposition in Eq. (<ref>) and the orthogonality relations (<ref>), the homogeneous BSE (<ref>) with the kernel (<ref>)can be recast in a set of eightcoupled-integral equations,ℱ^n_α(p^2,p· P, P^2)f_α(z)= -43∫^Λ_k𝒢( q^2)D^free_μν(q) ℱ^n_β (k^2,k· P;P^2)× 112 Tr_D [ T^α_ρ(p;P) γ_μS_f (k_+)T^β_ρ(k;P)S_g̅(k_-)γ_ν ].This equation can be posed as an eigenvalue problem for a set of eigenvectors ℱ^n := {ℱ_α^n; α =1,...,8 }:λ_n (P^2) ℱ^n = 𝒦 (p,k,P) ℱ^n .For every solution eigenvector, ℱ^n, there exists a mass, M_V_n, such that λ_n(-M^2_V_n)=1 <cit.>. The set of masses, M_V_n, represents the radially excited meson spectrum of quark-antiquark bound states with J^P= 1^-.In order to improve a faster convergence in solving the coupled equations (<ref>),we expand the eigenfunction into Chebyshev polynomials,ℱ_α^n(k^2,k· P;P^2)=∑^∞_m=0ℱ_α m^n(k^2;P^2)U_m(z_k),where theU_m(z) areChebyshev polynomials of second kind and the angles, z_k = P · k/(√P^2√k^2) and z_p = P · p/(√P^2√p^2), and momenta, k and p, are discretized <cit.>. We employ three Chebyshev polynomials for the ground and five for excitedstates.We solve the eigenvalue problem posed in Eq. (<ref>) by means of the implicitly restarted Arnoldi method, as implemented in thelibrary <cit.> which computes the eigenvalue spectrum for a given N× N matrix. A practical implementation requires a mapping of the BSE kernel 𝒦_αβ (p,k,P)onto such a square matrix and is described in detail in Ref. <cit.>.We obtain the eigenvalue spectrum,λ_n (P^2), of the kernel in Eq. (<ref>) and the associated eigenvectors, ℱ^n of the vector meson's BSAs where the root, M_V_n, of the equation λ_n (P^2=-M_V_n^2)-1=0 is found by employing theNumerical Recipe <cit.> subroutinesand . We verify thesolutions with the commonly used iterative procedure and find excellent agreement of theorder 10^-16.§ DISCUSSION OF RESULTSWe summarize our results for the mass spectrum and weak decay constants of the flavor-singlet andlight-flavored vector mesons inTables <ref>and <ref>, where the DSE and BSE are solved for two interaction, 𝒢(q^2), parameter sets inEq. (<ref>), namely ω = 0.4 GeV and ω = 0.6 GeV and the fixed value ω D =(0.8 GeV)^3; see also discussion in Ref. <cit.>. In Table <ref>, we list the 1^- masses for the ground state and first radial excitation following the Particle Data Group (PDG)conventions <cit.>, whereas in Table <ref> this is done for the weak decay constants. A direct comparison of the mass and decay constant entries in both model-interaction columns reveals that the values obtained with ω = 0.4 GeVare in much better agreement with experimental values of the 1^- ground states, namely in case of the ρ, K^*(892), ϕ(1020) andJ/ψ,whoseω dependence in the range ω∈ [0.4,0.6] GeV is rather weak. On the other hand, for ω = 0.6 GeVthe masses obtained for the radially excited states, ϕ(1680) andψ(2S), are only marginally better. It turns out that in the RL approximation, a realistic description of the radially excited vector meson masses is not possible with ω D =(0.8 GeV)^3. We thus choose ω D = (1.1 GeV)^3 and ω = 0.6 GeV for which the numerical mass and decay constant values ofthe radially excited states are presented in Table <ref> and compare well with experimental values, yet the ground states are no longer insensitive to ω variations for ω D = (1.1 GeV)^3 <cit.>. In order to maintain m_π = 0.138 GeV,ω must increase beyond our reference value, ω = 0.6 GeV, for the excited spectrum. These resultsconfirm an analogous trend observed for pseudoscalar mesons <cit.>. Nonetheless, the ground states are noticeably lessdependent on the ω D values than the radial excitations where largemass differences are observed between both parameter sets.This agrees with the observations made in Refs. <cit.> and extends them to the strange and charm vector mesons: the quantity r_ω := 1/ωis a length scale that measures the range of the interaction's infrared component in Eq. (<ref>).The radially excited states were shown to be more sensitive to long-range characteristics of 𝒢(q^2) than the ground states and weconfirmthat the mass of the radially excited states is lowered when r_ω decreases except in case of the ϕ(1680) andψ(2S) mesons. The masses of the ground and the radial excitation states of the vector mesons we find correspond to the first and third eigenvalues (from highest to lowest), respectively. This is because the second eigenvalue does not correspond to 1^– states since the even Chebyshev moments are strongly suppressed. The exceptions are the ρ(1450) using Dω = (1.1 GeV)^3and the K^*(1410) with Dω = (0.8 GeV)^3and ω =0.6where the radialexcitation does correspond to the second eigenvalue. (NB: the radial excitations have identical quantum numbers as the groundstate; therefore, the odd Chebyshev polynomials must be suppressed as it occurs for the ground states).In summary, we do not find a parameter set that describes equally well the entiremass spectrum of ground and excited states, which demonstrates the insufficiency of this truncation and confirms our finding in the pseudoscalar channel <cit.>.§ CONCLUSION We computed the BSAs for the ground and first excited states of the flavor-singlet and light-flavored vectormesons with an interaction ansatz that is massive and finite in the infrared and massless in the ultraviolet domain. This interaction is qualitatively in accordancewith the so-called decoupling solutions of the gluon's dressing function and thus supersedes the Maris-Tandy model <cit.> that vanishes at small momentum squared. In conjunction with the RL truncation, the latter proved to be a successful interaction model for the flavorlesslight pseudoscalar and vector mesons as well as quarkonia.Motivated by the successful application of this interaction to the mass spectrum of light vector mesons as well assome of their excited states in Ref. <cit.>,we extend this study to the strange and charm sectors and obtain the masses of ground and radially excited states as presented inTable <ref> and also compute their weak decay constants. The numerical values obtained are in good agreement with experimental data in case of ground states,but the same parametrization yields values that compare poorly with experiment for the excited states.We thus confirm our earlier observation that nosingle parametrizationof Eq (<ref>) is suitable to reproduce the mass spectrum of both, the ground and excited states in RL truncation.Although not explicitly detailed here, this approximation also fails to produce the correct masses for the D_(s) and B_(s) vector mesons and the discrepancy is even more pronounced than in the case of charmed pseudoscalar mesons <cit.>. Reasons for this wereput forward, e.g., in Ref. <cit.>. It has thereforebecome strikingly clear that a unified description of flavored pseudoscalar and vector mesons, quarkonia and their radial excitations can only be achieved within a treatment of the BSE beyond the leading truncation.The work of F. Mojica and C. E. Vera was supported by Comité Central de Investigaciones, Universidad del Tolima, under project no. 60220516. E. Rojas acknowledges financial support from Patrimonio Autónomo Fondo Nacional de Financiamiento para la Ciencia, la Tecnología y laInnovación, Francisco José de Caldas and Sostenibilidad-UDEA. B. El-Bennich is supported by the São Paulo Research Foundation(FAPESP), grant no. 2016/03154-7, and by Conselho Nacional de Desenvolvimento Científico e Tecnolǵico (CNPq), grant no. 458371/2014-9. § EXTRACTION OF THE DECAY CONSTANTS Following Ref. <cit.>,we can extract thedecay constant of thevector mesons fromthe experimental value <cit.> forthe partial width of the ρ, V ⟶e^+e^- decay, where V denotes the ϕ and heavy-flavoredmesons. f_ρ^2= 3 m_ρ/2πα^2Γ_ρ→ e^+e^-,f_V^2= 3 m_V/4πα^2 Q^2Γ_V→ e^+e^-.In this expression, Q is the charge of the quarksin the meson andα is the fine structure constant.99 Aznauryan:2012baI. G. Aznauryan et al.,Int. J. Mod. Phys. 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C 92, no. 5, 055203 (2015). | http://arxiv.org/abs/1704.08593v1 | {
"authors": [
"Fredy F. Mojica",
"Carlos E. Vera",
"Eduardo Rojas",
"Bruno El-Bennich"
],
"categories": [
"hep-ph"
],
"primary_category": "hep-ph",
"published": "20170427142144",
"title": "Mass spectrum and decay constants of radially excited vector mesons"
} |
[pages=1,]erratum_DisorderFRG.pdf [pages=2,]erratum_DisorderFRG.pdfDahlem Center for Complex Quantum Systems and Institut fr Theoretische Physik, Freie Universitt Berlin, D-14195, Berlin, GermanyDisorder effects are especially pronounced around nodal points in linearly dispersing bandstructures as present in graphene or Weyl semimetals. Despite the enormous experimental and numerical progress, even a simple quantity like the average density of states cannot be assessed quantitatively by analytical means. We demonstrate how this important problem can be solved employing the functional renormalization group method and, for the two dimensional case, demonstrate excellent agreement with reference data from numerical simulations based on tight-binding models. In three dimensions our analytic results also improve drastically on existing approaches. Quantitative analytical theory for disordered nodal points Bjrn Sbierski, Kevin A. Madsen, Piet W. Brouwer, Christoph Karrasch December 30, 2023 =======================================================================§ INTRODUCTION Two dimensional graphene <cit.> and three dimensional Weyl materials <cit.> are important examples of Dirac type semimetals. Their electronic structure features a nodal degeneracy point where two linearly dispersing Bloch bands meet. Due to the vanishing density of states (DOS), disorder effects can be expected to be particularly pronounced in these materials and have been actively studied, for reviews see Refs. <cit.>. Despite all this effort on the disorder problem for nodal points, analytical results, even for a quantity as simple as the DOS, are at best qualitatively correct but fail widely in their quantitative predictions, even for weak disorder. This is surprising insofar as exact answers can be obtained with ease from numerical simulations of non-interacting lattice Hamiltonians. The scope of this work is to show how tremendous progress on this long-standing problem can be achieved by employing a variant of the functional renormalization group (fRG).We consider the minimal continuum model of a single disordered node in d=2,3 dimensions, H_d=H_0,d+U_d,where H_0,2=ħ v(σ_xk_x+σ_yk_y) is a d=2 Dirac Hamiltonian and H_0,3=ħ v(σ_xk_x+σ_yk_y+σ_zk_z) a d=3 Weyl Hamiltonian written with the standard Pauli matrices σ_i=x,y,z. The disorder potential U_d(𝐫), taken to be proportional to the unit matrix, is commonly assumed to have Gaussian correlations and zero mean. Explicitly, we assume a smooth form of the correlator𝒦_d(𝐫-𝐫^')=⟨ U_d(𝐫)U_d(𝐫^')⟩ =K(ħ v)^2/(2π)^d/2ξ^2e^-|𝐫-𝐫^'|^2/2ξ^2,where ⟨ ...⟩ denotes the disorder average. As H_0,d is lacking any scale, the disorder correlation length ξ serves as the fundamental scale in the problem. The dimensionless parameter K measures the disorder strength. In the Brillouin zone of real materials, nodal points usually come in pairs. This is enforced by symmetry (graphene) or topology (Weyl). However, these pairs can have a sizable k-space separation Δ k. If ξΔ k≫1 the intra-node scattering dominates over inter-node scattering and the model (<ref>) is a reasonable low-energy approximation for realistic materials.While Eq. (<ref>) with the correlator (<ref>) has the advantage that it can be easily approximated in tight-binding models if ξ≫ a (a being the lattice scale) another common choice for 𝒦_d more convenient for analytical calculations is the white noise limit ξ→0, 𝒦_d^GWN(𝐫)=K(ħ v)^2ξ^d-2δ(𝐫),along with the prescription that 1/ξ serves as an ultraviolet cutoff for the clean dispersion H_0,d. We will use the white noise approximation to make contact with known results.The bulk DOS can be calculated as ν(E)=-1/πIm Tr∫_𝐤G_𝐤^R(E),where ∫_𝐤=(2π)^-d∫ d𝐤and G_𝐤^R(E) is the retarded (matrix-valued) Green function. For the clean Hamiltonian H_0,d, one has ν_0,d(E)=|E|^d-1/(2π)^d-1(ħ v)^d, vanishing at the degeneracy point. If disorder is thought of as a local chemical potential creating carriers from conduction or valence bands, a finite ν_d(E=0) can be expected (since disorder is a self-averaging quantity, we omit ⟨ ...⟩). In the following, we distinguish between 'numerical' approaches based on explicit generation of a large number of random disorder realizations U_d(𝐫) in Eq. (<ref>) and 'analytical' methods starting from Eq. (<ref>). While the former are well established, up to now there is no known analytical method that could reproduce numerical results with reasonable accuracy, not even for small K.The scope of this work is to show how this long-standing problem can be solved by a variant of the functional renormalization group (fRG) which allows to rewrite the disorder problem as an — in principle infinite — hierarchy of coupled self-consistency equations for vertex functions. We apply this technique to calculate the DOS ν_d(E=0) and find that even a simple truncation of the above hierarchy yields results in very good quantitative agreement with numerically exact data obtained from the kernel polynomial method at much higher computational costs. We acknowledge an earlier study by Katanin <cit.> with similar objectives but a different variant of the fRG. However, our results go significantly beyond those of Ref. <cit.>, where only d=2 was investigated without comparison to numerically exact results. § EXACT NUMERICAL DOSTo gauge the quality of analytical approaches discussed in the remaining sections, let us start by obtaining numerically exact DOS data for the Dirac and Weyl systems with smooth disorder, described by Eqns. (<ref>) and (<ref>). We apply the kernel polynomial method (KPM) <cit.>, a numerically efficient tool to approximate the DOS of large lattice Hamiltonians H represented as sparse matrices. The DOS ν(E) as a function of energy E is expanded in Chebyshev polynomials and the expansion coefficients μ^(n) are expressed as a trace over a polynomial in H. Using recursion properties of Chebyshev polynomials, the μ^(n) can be efficiently computed (up to order N) involving only sparse matrix-vector products and a statistical evaluation of the trace.The clean nodal Hamiltonian H_0,d is approximated as the low energy theory of the following tight-binding models on a square/cubic lattice (with constant a, size L^d)H_0,d^L=ħ v/aσ_xcos ak_x+σ_ycos ak_y (d=2) σ_xsin ak_x+σ_ysin ak_y-σ_zcos ak_z (d=3),which feature four/eight nodal points for d=2 and d=3, respectively, with minimal mutual distance Δ k=π/a. We apply periodic boundary conditions and add a correlated disorder potential as in Eq. (<ref>). If our disordered lattice model would faithfully emulate the continuum Hamiltonian (<ref>), the DOS at zero energy must be of the scaling form ν_d(E=0)=(ħ v)^-1ξ^1-df(K) with f(K) a dimensionless function. We have checked that the KPM data based on the lattice Hamiltonian Eq. (<ref>) fulfills this scaling condition once ξ≫ a so that (i) the smooth disorder correlations are well represented on the discrete lattice, (ii) the disorder induced energy scale is well below the scale of order ħ v/a where H_0,d^L deviates from H_0,d and (iii) the inter-node scattering rate is sufficiently suppressed compared to the intra-node rate (the factor is exp[-(Δ k)^2ξ^2/2]). Moreover, we require L≫ξ to suppress finite-size effects. Thus, the KPM data (normalized to a single node) shown as dots in Fig. <ref> (d=2) and Fig. <ref> (d=3) can be regarded as the exact zero energy DOS of the continuum model Eq. (<ref>). Simulation parameters are given in the figure captions. In spite of the abundant literature on similar numerical studies for the DOS of disordered 2d Dirac (see Refs. <cit.>) and 3d Weyl systems (see Refs. <cit.>), we are not aware of existing high-precision data obtained for a smooth disorder correlator and with the required scaling properties fulfilled.§ DISORDERED D=2 DIRAC NODEWe proceed by discussing existing analytical approaches to the disorder problem in the d=2 Dirac case. The self-consistent Born approximation (SCBA) determines the disorder induced self-energy Σ≡ G^-1-G_0^-1 (where G_0 is the Green function of the clean system) according to the diagram in Fig. <ref>(i) <cit.>. The corresponding self-consistent equation can be solved in closed form for the white noise correlator Eq. (<ref>) and yields a disorder induced scale Γ=ħ v/ξe^-2π/K (for K≲1) exponentially small in K appearing in the imaginary self-energy Σ=± iΓ and a DOS ν_2(E=0)ħ vξ∝ e^-2π/K/K <cit.>. In Fig. <ref> (bottom panel), this result (dashed line) compares well to the DOS obtained from the SCBA with smooth disorder correlator (<ref>) (blue line). However, comparing to the exact KPM-DOS in Fig. <ref> (dots), we find that albeit the exponential form is correctly predicted by the SCBA, the slope (prefactor in the exponent) is roughly a factor 2 off. The failure of the SCBA can be attributed to interference corrections from multiple disorder scattering events <cit.>, see diagrams (ii.a) and (ii.b) for the lowest order corrections. While unimportant in ordinary metals (where 1/k_Fl≪1 with k_F Fermi wavevector and l the mean free path serves as a small parameter), for Dirac materials these diagrams provide corrections of order ln[ħ v/ξΓ]. Accordingly, their contribution vanishes for strong disorder where the SCBA becomes reliable, c.f. Fig. <ref>.To go beyond the SCBA, Refs. <cit.> used the super-symmetry method. Alternatively, the replica trick <cit.> can be employed: It takes a disorder average over R copies (replicas) of the original problem seeing the same disorder potential. The resulting action S=S_d,0+S_d,dis is translational invariant but contains, besides the free part S_d,0=∑_α=1^R∫_ω∫_𝐤∑_σ,σ^'ψ̅_ω𝐤σ^'^α(iω-H_0,d)_σ^'σψ_ω𝐤σ^α an attractive inter-replica interaction which is elastic (i.e. without frequency transfer)S_d,dis=∑_α,β=1^R∫_ω_1,ω_2∫_𝐤_1^',𝐤_1,𝐤_2^',𝐤_22πδ_𝐤_1^'-𝐤_1+𝐤_2^'-𝐤_2 × -𝒦_d(𝐤_1^'-𝐤_1)/2∑_σ,σ^'ψ̅_ω_1𝐤_1^'σ^αψ_ω_1𝐤_1σ^αψ̅_ω_2𝐤_2^'σ^'^βψ_ω_2𝐤_2σ^'^β.Assuming the white noise correlator (<ref>) that comes with the UV cutoff 1/ξ in k-space, this action is susceptible to a Wilsonian momentum-shell RG analysis <cit.>. Successively integrating out high energy modes down to λ^-1/ξ (λ≥1) perturbatively, the action can be approximately mapped to itself with rescaled momenta, fields and coupling constants. If the velocity is kept constant, the two-loop RG equation for the flowing disorder strength K̃(λ) reads <cit.>dK̃/d lnλ=K̃^2/π+K̃^3/(2π^2).Starting with the initial condition K̃(1)=K the flow is to strong coupling where the perturbation theory leading to Eq. (<ref>) breaks down. To find the energy scale Γ where this happens (and below which the DOS is presumably constant), let us assert K̃(ħ v/Γξ)∼1 which, in the limit of K≪1, leads to Γ∝ħ v/ξ√(1/K)e^-π/K <cit.> correcting for the factor 2 in the exponent as found from the SCBA. The DOS at the nodal point is expected to be governed by this emergent energy scale ν_2(E=0)ħ vξ∝Γ, in agreement with the KPM results in Fig. <ref>.The Wilsonian RG calculation gave the correct exponential scale governing the disorder problem. However, it is not quantitative in the sense that numerical estimates for, say, the DOS could be obtained in the strong coupling limit. We will now show how the fRG method overcomes the difficulties mentioned above and use it to obtain quantitative results for the disorder induced DOS at the nodal point without any fitting parameters. § FRG APPROACHThe fRG <cit.> introduces a flow parameter Λ in the bare propagator and rewrites the many-body problem in a hierarchy of coupled flow equations for vertex functions with respect to Λ. The flow parameter is chosen such that for Λ=∞, the vertex functions are known exactly and for Λ=0 the original problem is retained. We relegate a detailed discussion of technicalities to the appendix and only highlight the most important points and modifications related to use of the fRG with the replicated action. To actually calculate expectation values and vertex functions from the replicated action, the replica limit ⟨ O⟩ =R→0lim1/R∑_α=1^R⟨𝒪(ψ̅^α,ψ^α)⟩ _ψ is required, where ⟨𝒪(ψ̅,ψ)⟩ _ψ=∫ D(ψ̅,ψ)𝒪(ψ̅,ψ)e^-S[ψ̅,ψ] stands for the standard functional average over a polynomial of fields 𝒪(ψ̅,ψ) <cit.>. In a peturbative expansion (which is also at the heart of the fRG flow equations), thus only diagrams without closed fermion loops have a finite contribution in the replica limit. This also means that mixing of replica indices in the relevant diagrams is avoided. One can also show that the elastic nature of the interaction vertex derived from (<ref>) is maintained along the flow. As a consequence, on the right hand side (rhs) of the flow equations the frequency integral as required for inelastic (true) interactions, is absent. Thus introducing Λ via a Matsubara frequency cutoff scheme results in a Dirac delta function on the rhs which allows for a direct integration of the corresponding flow equations and results in a self-consistent hierarchy of equations for the vertices. So far no approximations have been made. To proceed, we truncate the hierarchy to order K^2. This is a pragmatic choice, that still goes beyond all diagrammatic schemes previously applied to disordered Dirac materials explicitly. Subsequently, we eliminate the interaction vertex in favor of the self-energy. The remaining self-consistency equation readsΣ(𝐤) =K(ħ v)^2∫_𝐪G(𝐪)e^-1/2ξ^2|𝐪-𝐤|^2 +K^2(ħ v)^4∫_𝐪,𝐩e^-1/2ξ^2(|𝐤-𝐩|^2+|𝐪-𝐩|^2)× G(𝐩)· G(𝐪)·[G(𝐤+𝐪-𝐩)+G(𝐩)],and is displayed in Fig. <ref> diagrammatically: The term of order K represents the SCBA approximation, c.f. diagram (i), the two second order terms are shown in diagrams (ii.a) and (ii.b) respectively. Although these diagrams would also appear in perturbation theory, the fRG approach (i) rigorously justifies the use of the self-energy dressed propagators and (ii) indicates how we could consistently go beyond order K^2 by allowing feedback for the vertex self-consistency equation. To solve Eq. (<ref>), we parameterize the self-energy using polar (d=2) or spherical (d=3) coordinates and proceed by iteration. We compute the DOS from Eq. (<ref>). Further details are given in the appendix. In the d=2 Dirac case, the resulting DOS (red line) shows excellent agreement with the numerically exact KPM data and justifies the used order K^2 truncation a posteriori, well capable of capturing the exponential scale derived from Eq. (<ref>).On the pragmatic side, let us note that our fRG method also has advantages over the KPM method besides being analytic. For example, in Fig. <ref>, the KPM data for ν_2(E) shows a dip around E=0 that can only be resolved for small K if the system size L and expansion order N is taken large. In comparison, the solution of Eq. (<ref>) requires only a small fraction of computational effort.§ DISORDERED D=3 WEYL NODEWe now turn to the disorder induced DOS for a d=3 Weyl node. Here, weak disorder is irrelevant so that the DOS is maintained at zero. Only for K>K_c, disorder induces a finite DOS, see Fig. <ref> for the KPM data (dots). These qualitative features were correctly predicted by the SCBA (blue line, see Refs. <cit.>) and by the momentum shell RG treatment, see Refs. <cit.>. From the KPM, we find K_c^KPM=4±0.5 (the precision is limited by finite size effects) while K_c^SCBA≃11 (blue line) is off by more than a factor two. The one-loop RG result K_c^RG_1=π^2≃10 can be improved with respect to the KPM value by adding two-loop corrections K_c^RG_2=π^2/2≃5. However, quantitative predictions for the DOS in the strong-disorder phase cannot be obtained with the RG approach.When compared to the d=2 case, the additional challenge for the fRG approach in the Weyl case is that the interesting disorder strengths K K_c are not numerically small. Thus we assume that our 𝒪(K^2) truncation of the fRG equations might cause a sizable error. Surprisingly, the fRG results (red line) yield K_c^fRG≃6 and predict the available exact DOS for K>7 within an error of a few percent. On the one hand, we expect that the remaining numerical error of the fRG method could be systematically reduced by considering the fRG flow of the interaction vertex, which we leave for future research. On the other hand, this might not improve the accuracy for K≃ K_c where rare region effects which lie beyond any order of perturbation theory, are expected to dominate the DOS <cit.>. However, it is known that their influence can be suppressed by choosing a different disorder model <cit.>. § CONCLUSIONWe applied the fRG to treat the disorder problem at nodal points in two and three dimensions. From the resulting hierarchy of self-consistency equations, we calculate the bulk DOS and show that it is superior in accuracy to any other existing analytical approach. Suprisingly, for two dimensions, a truncation of the self-consistency equation at second order of K is sufficient, while in three dimensions the accuracy could probably be increased with increasing order. We leave this suggestion for future work, along with the calculation of other experimentally relevant transport properties from fRG. More complicated disorder models, in particular vector disorder in two dimensions and its characteristic ν(E) behavior or scattering between multiple nodal points, as present in realistic materials, could also be studied in the future.§ ACKNOWLEDGMENTS We thank Pavel Ostrovsky and Voker Meden for useful discussions. Numerical computations were done on the HPC cluster of Fachbereich Physik at FU Berlin. Financial support was granted by the Deutsche Forschungsgemeinschaft through the Emmy Noether program (KA 3360/2-1) and the CRC/Transregio 183 (Project A02). apsrev4-1 § APPENDIX: FRG WITH REPLICATED ACTION AND SOLUTION OF SELF-CONSISTENCY EQUATIONfRG flow equations and vertex structure for the replica interaction.—The fRG flow equations for the self energy Σ and the interaction vertex Γ have the form <cit.>∂_ΛΣ_Λ(1^';1)=-∫_2,2^'[Ġ_Λ]_2,2^'Γ_Λ(1^',2^';1,2)and, in three-particle vertex truncation,∂_ΛΓ_Λ(1^',2^';1,2) =∫_3,3^',4,4^' Γ_Λ(1^',2^';3,4)([G_Λ]_3,3^'[Ġ_Λ]_4,4^')Γ_Λ(3^',4^';1,2) +Γ_Λ(1^',4^';3,2)([G_Λ]_3,3^'[Ġ_Λ]_4,4^'+[G_Λ]_4,4^'[Ġ_Λ]_3,3^')Γ_Λ(2^',3^';4,1) -Γ_Λ(1^',4^';3,1)([G_Λ]_3,3^'[Ġ_Λ]_4,4^'+[G_Λ]_4,4^'[Ġ_Λ]_3,3^')Γ_Λ(2^',3^';4,2),where Ġ_Λ=G_Λ(∂_Λ[G_0,Λ^-1])G_Λ is the single-scale propagator and the multi-index {α_1iω_1𝐤_1σ_1}≡1 includes the relevant single-particle indices: replica index, Matsubara frequency, momentum and spin, respectively. We also use the notation 1_α_j≡{α_jiω_1𝐤_1σ_1} and 1_α_jiω_k≡{α_jiω_k𝐤_1σ_1} at our convenience and also abbreviate integrals and sums on the rhs as ∫_1≡∑_α_1 1/2π∫ dω_1 1/(2π)^d∫d𝐤_1 ∑_σ_1.Starting from the inter-replica interaction S_dis, Eq. (<ref>) in the main text, we find the bare vertex by anti-symmetrizationΓ_∞(1^',2^';1,2) = 2πδ_iω_1^'-iω_1δ_α_1^',α_1 2πδ_iω_2^'-iω_2δ_α_2^',α_2 A_∞(𝐤_1^'σ_1^',𝐤_2^'σ_2^';𝐤_1σ_1,𝐤_2σ_2) - 2πδ_iω_2^'-iω_1δ_α_2^',α_1 2πδ_iω_1^'-iω_2δ_α_1^',α_2 A_∞(𝐤_1^'σ_1^',𝐤_2^'σ_2^';𝐤_2σ_2,𝐤_1σ_1),where we defined A_∞(𝐤_1^'σ_1^',𝐤_2^'σ_2^';𝐤_1σ_1,𝐤_2σ_2)≡-2πδ_𝐤_1^'+𝐤_2^'-𝐤_1-𝐤_2 K(ħ v)^2e^-1/2ξ^2|𝐤_1^'-𝐤_1|^2δ_σ_1^'σ_1δ_σ_2^'σ_2,symmetric under the simultaneous exchange 𝐤_1^'σ_1^'↔𝐤_2^'σ_2^' and 𝐤_1σ_1↔𝐤_2σ_2.It is easy to see from the vertex flow equations (<ref>) that the locking of the replica and frequency indices, as present in the bare vertex (<ref>), is preserved in the flow (since the Green functions are frequency- and replica-diagonal). This means the flowing vertex is always of the form Γ_Λ(1_α_1iω_1^',2_α_2iω_2^';1_α_1iω_1,2_α_2iω_2) or Γ_Λ(1_α_1iω_1^',2_α_2iω_2^';1_α_2iω_2,2_α_1iω_1). Hence, in analogy to the bare vertex, we can write Γ_Λ(1_α_1^'iω_1^'^',2_α_2^'iω_2^'^';1_α_1iω_1,2_α_2iω_2) = 2πδ_iω_1^'-iω_12πδ_iω_2^'-iω_2δ_α_1^',α_1δ_α_2^',α_2A_Λ^α_1^'iω_1^',α_2^'iω_2^'(𝐤_1^'σ_1^',𝐤_2^'σ_2^';𝐤_1σ_1,𝐤_2σ_2) - 2πδ_iω_2^'-iω_12πδ_iω_1^'-iω_2δ_α_2^',α_1δ_α_1^',α_2A_Λ^α_1^'iω_1^',α_2^'iω_2^'(𝐤_1^'σ_1^',𝐤_2^'σ_2^';𝐤_2σ_2,𝐤_1σ_1),with A_Λ^α_1^'iω_1^',α_2^'iω_2^'(𝐤_1^'σ_1^',𝐤_2^'σ_2^';𝐤_1σ_1,𝐤_2σ_2) symmetric under the simultaneous exchange α_1^'iω_1^'↔α_2^'iω_2^' as well as 𝐤_1^'σ_1^'↔𝐤_2^'σ_2^' and 𝐤_1σ_1↔𝐤_2σ_2 like A_∞. Next, we need to leave out all terms on the rhs of Eqs. (<ref>) and (<ref>) where the sums ∑_α_3,α_4 on the rhs provide an extra factor of R as these vanish in the replica limit, R→0lim1/R∑_α=1^R⟨𝒪(ψ̅^α,ψ^α)⟩ _ψ∝R→0lim1/R∑_α=1^RR∝R→0limR=0. Note that fixing 1_α_2iω_2,2_α_1iω_1 in the first line of Eq. (<ref>) we can associate α_1 with the replica index from multi-index 3 or 4, in the second line we have no such choice and the third line is always ∝ R and vanishes. If we would draw diagrams to represent Eq. (<ref>), the replica limit condition is equivalent of leaving out diagrams with internal fermion loops. We find∂_ΛΓ_Λ(1_α_1iω_1^',2_α_2iω_2^';1_α_1iω_1,2_α_2iω_2)=∫_3,3^',4,4^' Γ_Λ(1_α_1iω_1^',2_α_2iω_2^';3_α_1iω_1,4_α_2iω_2)([G_Λ(α_1iω_1)]_3,3^'[Ġ_Λ(α_2iω_2)]_4,4^'+Ġ↔ G)Γ_Λ(3_α_1iω_1^',4_α_2iω_2^';1_α_1iω_1,2_α_2iω_2) +Γ_Λ(1_α_1iω_1^',4_α_2iω_2^';3_α_1iω_1,2_α_2iω_2)([G_Λ(α_1iω_1)]_3,3^'[Ġ_Λ(α_2iω_2)]_4,4^'+Ġ↔ G)Γ_Λ(3_α_1iω_1^',2_α_2iω_2^';1_α_1iω_1,4_α_2iω_2), Self-energy and vertex flow.—Eventually, for the DOS we are interested in the Green function which involves the self-energy. Employing the replica-frequency locking of the vertex for the self-energy flow Eq. (<ref>), we find∂_ΛΣ_Λ(α_1iω_1𝐤_1)_σ_1^',σ_1=-∫_22^'α_2,iω_2[Ġ_Λ(α_2iω_2)]_2,2^'Γ_Λ(1_α_1iω_1^',2_α_2iω_2^';1_α_1iω_1,2_α_2iω_2).Applying Eq. (<ref>), we find that only the second part avoids the replica sum leading to ∝ R. The Green function locks all frequencies and replica indices appearing on the rhs of the self-energy flow equation,∂_ΛΣ_Λ(α iω𝐤_1)_σ_1^',σ_1=∑_σ_2,σ_2^'∫_𝐤_2[Ġ_Λ(α iω𝐤_2)]_σ_2,σ_2^'A_Λ^α iω,α iω(𝐤_1σ_1^',𝐤_2σ_2^';𝐤_2σ_2,𝐤_1σ_1). In Eq. (<ref>), the function A only appears with equal replica and frequency indices. We insert this structure in Eq. (<ref>) and obtain ∂_ΛA_Λ^α iω,α iω(𝐤_1^'σ_1^',𝐤_2^'σ_2^';𝐤_1σ_1,𝐤_2σ_2)=∫_𝐤_3,𝐤_4∑_σ_3^'σ_3,σ_4^'σ_4A_Λ^α iω,α iω(𝐤_1^'σ_1^',𝐤_2^'σ_2^';𝐤_3σ_3,𝐤_4σ_4)([G_Λ(α iω𝐤_3)]_σ_3,σ_3^'[Ġ_Λ(α iω𝐤_4)]_σ_4,σ_4^'+Ġ↔ G)A_Λ^α iω,α iω(𝐤_3σ_3^',𝐤_4σ_4^';𝐤_1σ_1,𝐤_2σ_2) + A_Λ^α iω,α iω(𝐤_1^'σ_1^',𝐤_4σ_4^';𝐤_3σ_3,𝐤_2σ_2)([G_Λ(α iω𝐤_3)]_σ_3,σ_3^'[Ġ_Λ(α iω𝐤_4)]_σ_4,σ_4^'+Ġ↔ G)A_Λ^α iω,α iω(𝐤_3σ_3^',𝐤_2^'σ_2^';𝐤_1σ_1,𝐤_4σ_4).We can now drop the replica index α from our intermediate flow equations (<ref>) and (<ref>) and proceed to specify the flow parameter Λ which was general so far. Matsubara frequency cutoff.—In its standard application to systems with inelastic (true) interactions, the fRG flow equations contain frequency integrals on the rhs <cit.>. This integral is absent in Eq. (<ref>) due to the elastic structure of the disorder induced interaction vertex. We can take this to our advantage and choose a Matsubara cutoff scheme which will allow exact integration of the flow equations. In the Matsubara cutoff scheme a multiplicative cutoff to the bare Green function is employed G_0,Λ(1_iω_1)=θ(|iω_1|-Λ)G_0(1_iω_1), the corresponding single scale propagator reads Ġ_Λ(1_iω_1)=δ(|iω_1|-Λ)G̃_Λ(1_iω_1) and Ġ_Λ(1_iω_1)G_Λ(1_iω_2)=δ(|iω_1|-Λ)Θ(|iω_2|-Λ)G̃_Λ(1_iω_1)G̃_Λ(1_iω_2) where G̃_Λ(1_iω_1)=[G_0^-1(1_iω_1)-Σ_Λ(1_iω_1)]^-1 <cit.> and θ(0)=1/2 is understood by Morris Lemma <cit.>.We find∂_ΛΣ_Λ(iω𝐤_1)_σ_1^',σ_1 = δ(|iω|-Λ)∑_σ_2,σ_2^'∫_𝐤_2G̃_Λ(iω,𝐤_2)_σ_2,σ_2^'A_Λ^iω,iω(𝐤_1σ_1^',𝐤_2σ_2^';𝐤_2σ_2,𝐤_1σ_1), ∂_ΛA_Λ^iω,iω(𝐤_1^'σ_1^',𝐤_2^'σ_2^';𝐤_1σ_1,𝐤_2σ_2) = 2δ(|iω|-Λ)Θ(|iω|-Λ)∫_𝐤_3,𝐤_4∑_σ_3^'σ_3,σ_4^'σ_4A_Λ^iω,iω(𝐤_1^'σ_1^',𝐤_2^'σ_2^';𝐤_3σ_3,𝐤_4σ_4)([G̃_Λ(iω𝐤_3)]_σ_3,σ_3^'[G̃_Λ(iω𝐤_4)]_σ_4,σ_4^')A_Λ^iω,iω(𝐤_3σ_3^',𝐤_4σ_4^';𝐤_1σ_1,𝐤_2σ_2) + A_Λ^iω,iω(𝐤_1^'σ_1^',𝐤_4σ_4^';𝐤_3σ_3,𝐤_2σ_2)([G̃_Λ(iω𝐤_3)]_σ_3,σ_3^'[G̃_Λ(iω𝐤_4)]_σ_4,σ_4^')A_Λ^iω,iω(𝐤_3σ_3^',𝐤_2^'σ_2^';𝐤_1σ_1,𝐤_4σ_4),Assuming |ω|>0, we can now integrate both flow equations exactly over Λ from Λ=∞ to Λ=0 to find the physical self-energy Σ=Σ_Λ=0 and vertex function A=A_Λ=0. The initial condition for the interaction vertex is the bare interaction. Writing simply G instead of G̃_Λ=0, we find Σ(iω𝐤_1)_σ_1^',σ_1 = -∑_σ_2,σ_2^'∫_𝐤_2G(iω𝐤_2)_σ_2,σ_2^'A^iω,iω(𝐤_1σ_1^',𝐤_2σ_2^';𝐤_2σ_2,𝐤_1σ_1),A^iω,iω(𝐤_1^'σ_1^',𝐤_2^'σ_2^';𝐤_1σ_1,𝐤_2σ_2) = A_∞(𝐤_1^'σ_1^',𝐤_2^'σ_2^';𝐤_1σ_1,𝐤_2σ_2)-∫_𝐤_3,𝐤_4∑_σ_3^'σ_3,σ_4^'σ_4 A^iω,iω(𝐤_1^'σ_1^',𝐤_2^'σ_2^';𝐤_3σ_3,𝐤_4σ_4)([G(iω𝐤_3)]_σ_3,σ_3^'[G(iω𝐤_4)]_σ_4,σ_4^')A^iω,iω(𝐤_3σ_3^',𝐤_4σ_4^';𝐤_1σ_1,𝐤_2σ_2)+A^iω,iω(𝐤_1^'σ_1^',𝐤_4σ_4^';𝐤_3σ_3,𝐤_2σ_2)([G(iω𝐤_3)]_σ_3,σ_3^'[G(iω𝐤_4)]_σ_4,σ_4^')A^iω,iω(𝐤_3σ_3^',𝐤_2^'σ_2^';𝐤_1σ_1,𝐤_4σ_4).Instead of the usual coupled fRG flow equations that have to be integrated, we thus have rephrased the disorder problem in terms of the coupled self-consistent Eqns. (<ref>) and (<ref>). Note that the above derivation did not depend on the three (or N-) particle vertex truncation in Eq. (<ref>) and thus, an extended set of coupled self-consistency equations would still be exact.We turn back to our initial goal to find the DOS at the nodal point E=0. For this, we need the retarded real frequency self-energy, see Eq. (<ref>) in the main text, that is connected to Σ(iω) by an analytical continuation iω=0+i0^+ where 0^+ is a positive real infinitesimal. After this step, we drop the frequency variable from now on. Let us emphasize that the appearance of a single frequency in the hierarchy of self-consistent equations is a remnant of the elastic nature of disorder scattering. Solution correct to order K^2.—Even the set of self-consistency equations (<ref>) and (<ref>) (with the three-particle vertex dropped) is difficult to solve without further approximations. To obtain the self-energy correct to at least 𝒪(K^2), on the rhs of Eq. (<ref>), is is sufficient to use the bare vertex Eq. (<ref>). This is a pragmatic approach, which, however still goes beyond existing studies in the literature. We obtain from Eqns. (<ref>) and (<ref>)A(𝐤_1^'σ_1^',𝐤_2^'σ_2^';𝐤_1σ_1,𝐤_2σ_2)=A_∞(𝐤_1^'σ_1^',𝐤_2^'σ_2^';𝐤_1σ_1,𝐤_2σ_2)-K^2(ħ v)^4∫_𝐤_3×e^-1/2ξ^2|𝐤_1^'-𝐤_3|^2[G(𝐤_3)]_σ_1^',σ_1([G(𝐤_1^'+𝐤_2^'-𝐤_3)]_σ_2^',σ_2+[G(𝐤_3+𝐤_2-𝐤_1^')]_σ_2^'σ_2)e^-1/2ξ^2|𝐤_1-𝐤_3|^2,and further specialize to the spin-momentum structure needed for the self-energy flow Eq. (<ref>)A(𝐤_1σ_1^',𝐤_2σ_2^';𝐤_2σ_2,𝐤_1σ_1)=-K(ħ v)^2e^-1/2ξ^2|𝐤_2-𝐤_1|^2δ_σ_1^'σ_1δ_σ_2^'σ_2-K^2(ħ v)^4∫_𝐤_3×e^-1/2ξ^2|𝐤_1-𝐤_3|^2[G(𝐤_3)]_σ_1^',σ_1([G(𝐤_1^'+𝐤_2^'-𝐤_3)]_σ_2^',σ_2+[G(𝐤_3+𝐤_2-𝐤_1^')]_σ_2^'σ_2)e^-1/2ξ^2|𝐤_2-𝐤_3|^2. We combine Eq. (<ref>) with (<ref>) and find the final self-consistency equation. Relabeling 𝐤_1→𝐤, 𝐤_2→𝐪 and 𝐤_3→𝐩 and using “·” to indicate matrix products for the 2x2 matrix-valued Green functions, we arrive at Eq. (<ref>) from the main text: Σ(𝐤) =K(ħ v)^2∫_𝐪e^-1/2ξ^2|𝐪-𝐤|^2 G(𝐪) +K^2(ħ v)^4∫_𝐪,𝐩e^-1/2ξ^2(|𝐤-𝐩|^2+|𝐪-𝐩|^2) G(𝐩)· G(𝐪)·[G(𝐤+𝐪-𝐩)+G(𝐩)].If the feedback of the flowing vertex A to the rhs of its own flow equation would be considered, this would yield two equations for Σ and A to be solved self-consistently.Numerical solution of self-consistency equations.—The self-consistency equation (<ref>) can be solved numerically by iteration. We use dimensionless units (measuring momenta in 1/ξ and energies in ħ v/ξ) and the dimensionless self-energy in d=2 (at the nodal point) can be parametrized asΣ_d=2(𝐱=𝐤ξ)/ħ v/ξ=m_2(x){σ_xcos[ϕ]+σ_ysin[ϕ]} +iM_2(x),with x,ϕ polar coordinates. The term M_2(x) has to be purely real (to avoid a spontaneous creation of chemical potential) and >0 for the retarded self energy. As a result, on the rhs of Eq. (<ref>), we can chose 𝐤 in say, the x-direction and also take only the σ_x component of the product of Green functions (it can be checked that all other components vanish). The final self-consistency loop is then only for the functions m_2(x) and M_2(x), which turn out to be rather smooth. They can be discretized on a geometric grid for the variable x, the angular integrations can be done using a linearly spaced integration grid for the angles. We made sure that our results are converged with respect to the resolution of the discretization grids. Once m_2, M_2 do not change any more under insertion on the rhs of Eq. (<ref>), the DOS is computed from Eq. (<ref>) using interpolation of the integrand and quadrature integration. Likewise, in d=3, the same strategy is applied using a parametrization in spherical coordinates x,ϕ,θ:Σ_d=3(𝐱=𝐤ξ)/ħ v/ξ=m_3(x)(sin[θ]{σ_xcos[ϕ]+σ_ysin[ϕ]} +σ_zcos[θ])+iM_3(x). | http://arxiv.org/abs/1704.08457v3 | {
"authors": [
"Björn Sbierski",
"Kevin A. Madsen",
"Piet W. Brouwer",
"Christoph Karrasch"
],
"categories": [
"cond-mat.mes-hall",
"cond-mat.dis-nn"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170427072421",
"title": "Quantitative analytical theory for disordered nodal points [Article and Erratum]"
} |
Moscow Institute of Physics and Technology, 141700 Dolgoprudny, RussiaI.E. Tamm Department of Theoretical Physics, P.N. Lebedev Physical Institute, 119991 Moscow, Russia National Research University Higher School of Economics, 101000 Moscow, RussiaInstitute of Nanotechnology, Karlsruhe Institute of Nanotechnology (KIT), 76021, Karlsruhe, Germany I.E. Tamm Department of Theoretical Physics, P.N. Lebedev Physical Institute, 119991 Moscow, Russia We argue that quantum fluctuations of the phase of the order parameter may strongly affect the electron density of states (DOS) in ultrathin superconducting wires. We demonstrate that the effect of such fluctuations is equivalent to that of a quantum dissipative environment formed by sound-like plasma modes propagating along the wire. We derive a non-perturbative expression for the local electron DOS in superconducting nanowires which fully accounts for quantum phase fluctuations. At any non-zero temperature these fluctuations smear out the square-root singularity in DOS near the superconducting gap and generate quasiparticle states at subgap energies. Furthermore, at sufficiently large values of the wire impedance this singularity is suppressed down to T=0 in which case DOS tends to zero at subgap energies and exhibits the power-law behavior above the gap. Our predictions can be directly tested in tunneling experiments with superconducting nanowires. Quantum phase fluctuations and density of states in superconducting nanowires Andrei D. Zaikin December 30, 2023 =============================================================================§ INTRODUCTION Fluctuations play an important role in a reduced dimension. Of particular interest are fluctuation effects in low dimensional superconducting structures <cit.> in which case the system behavior can be markedly different from that in the bulk limit. For instance, it is well known that properties of quasi-one-dimensional superconducting wires cannot be adequately described within the standard Bardeen-Cooper-Schriffer (BCS) mean field approach even if the temperature T becomes arbitrarily low. Perhaps one of the most striking low temperature features of ultrathin superconducting wires is the presence of nontrivial fluctuations of the order parameter – the so-called quantum phase slips (QPS) <cit.>. Such quantum fluctuations correspond to temporal local suppression of the superconducting order parameter accompanied by the phase slippage process which, in turn, generates voltage fluctuations in the system. As a result, ultrathin superconducting wires acquire QPS-induced non-vanishing resistance down to lowest T <cit.>. Subsequently this theoretical predictionreceived its convincing experimental confirmation <cit.>. More recently it was predicted <cit.> that QPS can also cause non-equilibrium (shot) voltage noise in superconducting nanowires.The magnitude of quantum phase slip effects in such nanowires is controlled by the QPS amplitude γ_QPS∼ (g_ξΔ/ξ)exp (-ag_ξ), where Δ is the superconducting order parameter and a ∼ 1 is an unimportant numerical prefactor. The key parameter here is dimensionless conductance g_ξ =R_q/R_ξ, where R_q =2π/e^2 ≃ 25.8 KΩ is the quantum resistance unit and R_ξ is the normal state resistance of the wire segment of length equal to the superconducting coherence length ξ. The same parameter g_ξ (which is directly related to the so-called Ginzburg number in one dimension <cit.> asg_ξ∼ Gi_1D^-3/2) controls the magnitude of small (Gaussian) fluctuations of the order parameter in superconducting nanowires. E.g., it is straightforward to demonstrate <cit.> that in the presence of such fluctuations the mean field value of order parameter Δ acquires a negative correction Δ→Δ -δΔ with δΔ∼Δ /g_ξ. Thus, by choosing the dimensionless conductance g_ξ sufficiently large one can essentially suppress both QPS effects and Gaussian fluctuations of the absolute value |Δ| in superconducting wires.Is the condition g_ξ≫ 1 sufficient to disregard fluctuation effects in such wires? The answer to this question is clearly negative. The point here is that even at very large values of g_ξ there remain non-vanishing fluctuations of the phase φ of the order parameter. In the limit g_ξ≫ 1 such phase fluctuations are essentially decoupled from those of |Δ | being controlled by the dimensionless parameter g=R_q/Z_ w, where Z_ w=√(ℒ_ kin/C) is the wire impedance, ℒ_ kin=1/(πσ_NΔ s) and C are respectively the kinetic wire inductance (times length) and the geometric wire capacitance (per length), σ_N is the normal state Drude conductance of the wire and s is the wire cross section. The parameter g is different from (although not unrelated to) g_ξ (e.g., g ∝√(s) while g_ξ∝ s) and, hence, by properly choosing the system parameters one can select the wires where only phase fluctuations can play a significant role. Such kind of wires will be addressed below in this paper.To be specific, we will analyze the effect of phase fluctuations on the electron density of states (DOS) of superconducting nanowires. In order to understand the basic physics behind this effect let us recall that such wires host sound-like plasma modes <cit.> which can be described in terms of phase fluctuations of the superconducting order parameter. These so-called Mooij-Schön modes can propagate along the wire with the velocity v=1/√(ℒ_ kinC) and interact with electrons inside the wire, thereby forming an effective environment for such electrons and affecting the superconducting DOS.The structure of the paper is as follows. In Sec. II we define our model and specify the basic formalism employed in our analysis. In Sec. III we derive the general expression for the quasiclassical electron Green function in superconducting nanowires in the presence of phase fluctuations. This expression is then employed to evaluate the electron DOS in such nanowires in Sec. IV. Sec. V is devoted to a brief discussion of our key observations.§ THE MODEL AND BASIC FORMALISMBelow we will analyze the structure displayed in Fig. 1. A long superconducting wire with sufficiently small cross section s is attached to two big superconducting reservoirs. As usually, superconducting properties of the system are described by the order parameter field Δ (x, t)=|Δ (x, t)|exp (iφ (x, t)) which in general depends both on the coordinate along the wire x and on time t. The wire parameters are chosen in a way to enable one to disregard all fluctuations of the absolute value of the order parameter which is set to be equal to a constant value |Δ (x, t)|=Δ independent of both x and t. As we already discussed above, for this purpose we need to set the dimensionless conductance g_ξ to be large g_ξ≫ 1. On the other hand, we will allow for fluctuations of the phase variable φ (x, t) along the wire.In what follows we will assume our superconducting wire to remain in thermodynamic equilibrium at temperature T well below the superconducting gap, i.e. T ≪Δ. We will perform our analysis in the most relevant diffusive limit implying that the elastic electron mean free path ℓ is much smaller than the superconducting coherence length ξ.We will operate with the quasiclassical electron Green functionǦ(t,t^',x)=[ G^R(t,t^',x) G^A(t,t^',x);0 G^K(t,t^',x) ],which has the matrix structure in both Keldysh and Nambu spaces and satisfies the Usadel equation <cit.>[ ∂_t σ_3 -iΔ̌+i eV̌σ_0,Ǧ] -D/2∂̂[ Ǧ,∂̂Ǧ]=0together with the standard normalization condition Ǧ^2=1̌.In Eq. (<ref>) we introduce the covariant spatial derivative ∂̂(…)=∂_x(…)+ie[Ǎ_xσ_3,(…)], [a,b]=ab-ba denotes the commutator, V and A are the scalar and vector potentials of the electromagnetic field, D=v_Fℓ /3 is the diffusion coefficient, _a and _a (together with the unity matrices _0=_0=1̂) stand for the Pauli matrices respectively in Keldysh and Nambu spaces and Δ̌ is the order parameter matrix to be defined below. We also note that all matrix products are understood as convolutions(A B)(t_1,t_2,x)=∫ dtA(t_1,t,x)B(t,t_2,x),while taking the trace implies integration over both time and space coordinatesA=∫ dtdx A(t,t,x). The electron DOS ν(E,x) is related to the quasiclassical Green function (<ref>) by means of the equationν(E,x)=ν_0 _3/4( G^R(E,x)-G^A(E,x) ),where ν_0 stands for DOS in a normal metal at the Fermi level and(E,x)=∫ d(t-t^')e^iE(t-t^')(t,t^',x). It will be convenient for us to perform the rotation in the Keldysh space expressing initial field variables, e.g., the phase of the order parameter _F,B on the forward and backward branches of the Keldysh time contour in terms of their classical and quantum components _±= (_F±_B)/2. We also define the matrices=[ _+ _-; _- _+ ]andΔ̌=_0⊗[0Δ_+; -Δ^*_+0 ]+_1⊗[0Δ_-; -Δ^*_-0 ],where Δ_± are defined analogously to φ_±.§ GREEN FUNCTIONS IN THE PRESENCE OF PHASE FLUCTUATIONSThe task at hand is to average the Green function (<ref>) over both the fluctuating phase variable φ and the electromagnetic field. The latter step is easily accomplished within the saddle point approximation which allows to directly link the potentials V and Ato the phase variable<cit.>. Employing the gauge transformationeV̌→Φ̌≡ eV̌+/2, eǍ_x→≡ eǍ_x-∂_x /2, Δ_±→ |Δ|_±,we expel the phase of the order parameter from Δ (x,t) and get(t,t^',x)= e^i/2(t,x)_3(t,t^',x) e^-i/2(t^',x)_3,whereobeys Eq. (<ref>) combined with Eqs. (<ref>)-(<ref>). It is also necessary to keep in mind that under the condition g_ξ≫ 1 adopted here one has |Δ|_+= Δ and|Δ|_-=0.As usually, magnetic effects associated withremain weak and can be neglected by setting =0 <cit.>. Likewise, one can disregard the effects related to weak (∝Φ) penetration of the fluctuating electric field inside the wire as compared to those caused by the gauge factors in (<ref>). This conclusion can be drawn from the equation <cit.>Φ∼ /(4E_Cν_Fs),E_C=e^2/(2C)combined with the observation that the condition E_Cν_F s ≫ 1 is usually well satisfied in generic metallic wires.Thus, we may setequal to the Green function Ł of a uniform superconductor in thermodynamic equilibrium, i.e.=Ł=[ Λ^R Λ^K; 0 Λ^A ],whereΛ^R_ϵ =1/√((+i0)^2-Δ^2)[ Δ; -Δ- ],Λ^A=-_3 (Λ^R)^†_3 andΛ^K_ϵ=Λ^R_ϵ F_ϵ- F_ϵΛ^A_ϵ,F_ϵ =tanh/2T.Then we obtain(t,t^',x)≃ e^i/2(t,x)_3Ł(t-t^' ) e^-i/2(t^',x)_3.Here Ł(t-t^') is the inverse Fourier transform of Ł_ϵ.What remains is to average the Green function (<ref>) over all possible phase configurations. This averaging is accomplished with the aid of the path integral⟨Ǧ⟩_ (t-t^')=∫ D exp(iS_ eff[])(t,t^',x).Here S_ eff[] is the effective action which accounts for phase fluctuations in a superconducting wire. At low energies this action reads <cit.>S_ eff[]=C/4e^2[ [ _+ _- ]^-1[ _+; _- ]],where=[ ^K ^R; ^A0 ]is the equilibrium Keldysh matrix propagator for plasma modes and^R,A(ω,k)=1/(ω± i0)^2-(kv)^2, ^K(ω,k)=(^R(ω,k)-^A(ω,k))ω/2T. § DENSITY OF STATESLet us now implement the above program and evaluate the electron DOS in superconducting nanowires. Making use of the structure of Ł in the Nambu space and performing Gaussian integration, we getν(E)=ν_0∫ d(t-t^') e^iE(t-t^') ×⟨_3_3/4 e^i/2(t,x)_3Ł(t-t^') e^ -i/2(t^',x)_3⟩_=ν_0∫ dte^iEt(_3_3/4 _a Ł(t)_b ^ab(t)),where a,b={0,1},(t)=[ ^K(t) ^R(t); ^A(t) 0 ]= e^ iE_C(^K(t)-^K(0)) ×[ cos(E_C(^R(t)-^A(t)) )isin(E_C^R(t) );isin(E_C^A(t) )0 ]and(t)=(t,0)=∫dω dk/(2π)^2e^-iω t(ω,k).Note that Eq. (<ref>) accounts for all emission and absorption processes of multiple plasmons in our system via an auxiliary propagator . This propagator obeys the standard causality requirements and satisfies bosonic fluctuation-dissipation theorem (FDT) because plasmons remain in thermodynamic equilibrium, cf. Eq. (<ref>).Taking the trace in the Keldysh space, employing the FDT relation for the bare Green function and the fluctuation propagator and, finally, evaluating the trace in the Nambu space, from Eq. (<ref>) we obtain⟨ν⟩_(E)= ν_0/4∫ dt e^-iEt(_3 ( Λ^R(t)-Λ^A(t) )^K(t). . +_3Λ^K(t) ( ^R(t)-^A(t) ) ) =∫d/2πν_BCS()^K(E-)( 1+F_ϵ F_E-ϵ),where ν_BCS() is the BCS density of states in a bulk superconductor.It is easy to observe that for ϵ≳ E+2T the combination 1+F_ϵ F_E-ϵ decays as ∝exp ((E-ϵ)/T). Hence, at subgap energies the electron DOS is suppressed by the factor ∼exp ((E-Δ)/T) and at T → 0 the superconducting gap Δ is not affected by the Mooij-Schön plasmons.Evaluating ^K in Eq. (<ref>), one finds^K(t)=exp(-1/g∫_0^ω_c dω 1-cos(ω t)/ω(ω/2T)) ×cos(1/g∫_0^ω_c dω sin(ω t)/ω).Here and below we define∫_-ω_c,0^ω_c dω (...)=∫_-∞,0^∞ dωe^-̣|ω|/ω_c(...),where ω_c ∼Δ sets the high frequency cutoff which follows naturally from the fact that the effective action defined in Eqs. (<ref>)-(<ref>) remains applicable only at energies well below the superconducting gap.It is straightforward to observe that ^K(t=0)=1 and, hence,∫ dE (ν(E)-ν_BCS(E))=0.This identity implies that phase fluctuations can only redistribute the electron states among different energies but do not affect the total (energy integrated) DOS.At low temperatures Eq. (<ref>) can be evaluated explicitly. We obtain^K(t)=(sinh(π T t)/π T t√(1+(ω_c t)^2))^-1/g ×cos(arctan(ω_c t)/g).In order to recover ^K(ω) it is convenient to express it in terms of the Matsubara propagator for the phase fluctuations continued analytically to the complex plane. For this purpose let us define^K(t)=1/2∑_± e^-((0)-(t± i0))/g,where we introduced the propagator(t± i0)=∫_-ω_c^ω_cdω/2ω e^-iω t( (ω/2T)∓ 1 ).This propagator is periodic in imaginary time and has cuts at Im t =β n with β=1/T and n∈ℤ. Shifting the integration contour, one obtains^K(ω)=cosh( βω/2)∫ dt e^-iω t ^K(t+iβ/2),where^K(t+iβ/2)=exp(-1/g∫_0^ω_cdω/ωcosh(ω/2T)-cos(ω t)/sinh(ω/2T))These integrals can easily be evaluated with the result^K(ω)≃cosh( βω/2)(2π T/ω_c)^1/g|Γ(%̣ṣ/̣%̣ṣ12g+iω/2π T)|^2/2π T Γ(1/g),where ω is supposed to be well below the superconducting gap Δ. For ω≪ T Eq. (<ref>) reduces to^K(ω)≃1/gω_c(2π T/ω_c)^1/g2π T/ω^2+(π T/g)^2,whereas at higher frequencies T ≪ω≪Δ we find^K(ω)≃π/ω_cΓ(1/g)(ω/ω_c)^1/g-1. Making use of the above expressions, at energies in the vicinity of the superconducting gap Δ we recover the following result for the electron DOS:ν(Δ+ω)=ν_0√(Δ)/√(2)(2π T/Δ)^1/g∑_k=0^∞Γ(k+1/g)/k!Γ(1/g) × Re(e^-iπ/2g/√(ω+2iπ T(1/2g+k))).The energy dependent density of states ν (E) for superconducting nanowires in the presence of phase fluctuations is also displayed in Fig. 2 at different temperatures and two different values of the dimensionless conductance g. One observes that at any nonzero T the BCS singularity at E →Δ is smeared due to interactions between electrons and Mooij-Schön plasmons. For the same reason, as we already indicated above, the electron DOS at subgap energies 0<E<Δ remains non-zero at any non-zero T, i.e.ν (E) ∝exp ((E-Δ)/T). We also point out a qualitative difference in the energy dependence of DOS displayed in top and bottom panels of Fig. 2 at energies slightly above the gap. While at bigger values of gthe function ν (E) demonstrates a non-monotonous behavior at such energies (top panel), at smaller g DOS decreases monotonously with decreasing energy at all E not far from the gap (bottom panel). In the zero temperature limit T → 0 and for E-Δ≪Δ we obtainν(E)≃ν_0 √(π)θ(E-Δ)/√(2)Γ(1/2+1/g)( E-Δ/Δ)^1/g- 1/2.We observe that while at E<Δ the electron DOS (<ref>) vanishes at all values of g, the behavior of ν (E) at overgap energies is markedly different depending on the dimensionless conductance g. For g > 2 (i.e. for relatively thicker wires) the DOS singularity at E →Δ survives though becoming progressively weaker with decreasing g. On the other hand,at g≤ 2 (corresponding to relatively thinner wires) the DOS singularity vanishes completely due to intensive phase fluctuations and ν (E) tends to zero at E →Δ as a power law (<ref>). This behavior is also illustrated in Fig. 3.§ DISCUSSIONIn this paper we argued that fluctuations of the phase of the order parameter may significantly affect low temperature properties of superconducting nanowires. While the dramatic effect of spin-wave-like fluctuations on long-range phase coherence in quasi-one-dimensional systems is well known for a long time <cit.>, here we demonstrated that local properties of superconducting nanowires, such as the electron density of states, can also be sensitive to phase fluctuations. We deliberately chose the wire parameters in a way to minimize fluctuations of the absolute value of the order parameter and specifically addressed the effect of small phase fluctuations associated with low energy sound-like plasma modes propagating along the wire. These Mooij-Schön plasmons form an effective quantum dissipative environment for electrons inside the wire. Previously various ground state properties of superconducing nanorings affected by such an environment were explored by several authors <cit.>. Here we adopted a physically similar standpoint in order to investigate the behavior of the electron DOS in long superconducting nanowires.The coupling strength between electrons and the effective plasmon environment is controlled by the dimensionless parameter g representing the ratio between the quantum resistance unit R_q and the wire impedance Z_ w. Provided g ≫ 1, i.e. the impedance Z_ w remains much smaller than R_q, phase fluctuations weakly affect the electron DOS except in the immediate vicinity of the superconducting gap Δ. For larger values Z_ w∼ R_q the effect of phase fluctuations becomes strong and should be treated non-perturbatively in 1/g at all energies. Another important parameter is temperature which is restricted here to be sufficiently low T ≪Δ.Our analysis demonstrates that at any nonzero T the electron DOS depends on temperature and substantially deviates from that derived from the standard BCS theory. In particular, at T >0 the BCS square-root singularity in DOS at E=Δ gets totally smeared and ν (E) differs from zero also at subgap energies, cf. Eq. (<ref>).This behavior can be interpreted in terms of a depairing effect due to the interaction between electrons and Mooij-Schön plasmons. We also note that our results are consistent with the phenomenological Dynes formula <cit.>ν (E)≃ν_0 Re(E+iΓ/√((E+iΓ )^2-Δ^2))describing smearing of the BCS singularity in DOS in the immediate vicinity of the superconducting gap.At T=0 and subgap energies the electron DOS vanishes as in the BCS theory, while the BCS singularity in DOS at E →Δ becomes weaker for any finite g >2 and eventually disappears for g ≤ 2. Thus, we conclude that even in the absence of fluctuations of the absolute value of the order parameter |Δ | quantum fluctuations of its phase φ may result in qualitative modifications of the ground state properties of quasi-one-dimensional superconducting wires.The local electron DOS in superconducting nanowires can be probed in a standard manner by performing a tunneling experiment, as it is also illustrated in Fig. 1. Attaching a normal or superconducting electrode to our wire and measuring the differential conductance of the corresponding tunnel junction one gets a direct access to the energy dependent electron DOS of a superconducting nanowire. E.g., in the case of a normal electrode at T → 0 and eV > Δ one findsdI/dV ∝ν (eV) ∝ (V -Δ /e)^1/g- 1/2.This power law dependence of the differential conductance resembles one encountered in small normal tunnel junctions at low voltages dI/dV ∝ V^2/g_N <cit.>, where g_N is the dimensionless conductance of normal leads. In fact, both the dependence (<ref>) and the zero bias anomaly in normal metallic junctions <cit.> are caused by Coulomb interaction and are controlled by the impedance of the corresponding effective electromagnetic environment.Finally, we remark that in superconducting nanowires with not too large values of g_ξ it is also necessary to account for quantum fluctuations of the absolute value of the order parameter |Δ |. Such fluctuations combined with those of the phase φ result in a reduction of the superconducting gap <cit.> and cause Berezinskii-Kosterlitz-Thouless-like (superconductor-insulator) quantum phase transition for QPS <cit.> at λ≡ g/8 =2. A complete analysis of quantum fluctuations and their impact on the electron DOS in superconducting nanowires should include all the effects controlled by both parameters g and g_ξ. This analysis will be worked out elsewhere. Acknowledgements We would like to thank K.Yu. Arutyunov for encouragement and useful discussions. This work was supported by the Russian Science Foundation under grant No. 16-12-10521.AGZ K.Yu. Arutyunov, D.S. Golubev, and A.D. Zaikin, Phys. Rep. 464, 1 (2008). LV A.I. Larkin and A.A. Varlamov, Theory of fluctuations in superconductors (Clarendon, Oxford, 2005). ZGOZ A.D. Zaikin, D.S. Golubev, A. van Otterlo, and G.T. Zimanyi, Phys. Rev. Lett. 78, 1552 (1997). GZQPS D.S. Golubev and A.D. Zaikin, Phys. Rev. B 64, 014504 (2001). BT A. Bezryadin, C.N. Lau, and M. Tinkham, Nature 404, 971 (2000). Lau C.N. Lau, N. Markovic, M. Bockrath, A. Bezryadin, and M. Tinkham, Phys. Rev. Lett. 87,217003 (2001). Zgi08 M. Zgirski, K.P. Riikonen, V. Touboltsev, and K.Y. Arutyunov, Phys. Rev. B 77,054508 (2008). SZ16 A.G. Semenov and A.D. Zaikin, Phys. Rev. B 94, 014512 (2016). GZTAPS D.S. Golubev and A.D. Zaikin, Phys. Rev. B 78, 144502 (2008). Mooij J.E. Mooij and G. Schön, Phys. Rev. Lett. 55, 114 (1985). Buisson B. Camarota, F. Parage, F. Balestro, P. Delsing, and O. Buisson, Phys. Rev. Lett. 86, 480 (2001). Usadel K.D. Usadel, Phys. Rev. Lett. 25, 507, (1970). bel W. Belzig, F. Wilhelm, C. Bruder, G. Schön, and A.D. Zaikin, Superlatt. Microstruct. 25, 1251 (1999). OGZB A. van Otterlo, D.S.Golubev, A.D.Zaikin, and G.Blatter, Eur. Phys. J. B 10, 131 (1999). HMW P.C. Hohenberg, Phys. Rev. 158, 383 (1967); N.D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133 (1966). HG F.W.J. Hekking and L.I. Glazman,Phys. Rev. B 55, 6551 (1997). SZ13 A.G. Semenov and A.D. Zaikin, Phys. Rev. B 88, 054505 (2013). dynes78 R.C. Dynes, V. Narayanamurti and J.P. Garno, Phys. Rev. Lett. 41, 1509 (1978). PZ88 S.V. Panyukov and A.D. Zaikin, J. Low Temp. Phys. 73, 1 (1988). | http://arxiv.org/abs/1704.08004v1 | {
"authors": [
"Alexey Radkevich",
"Andrew G. Semenov",
"Andrei D. Zaikin"
],
"categories": [
"cond-mat.mes-hall",
"cond-mat.supr-con"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170426080754",
"title": "Quantum phase fluctuations and density of states in superconducting nanowires"
} |
[ [===== This paper describes the Duluth system that participated inSemEval-2017 Task 6 #HashtagWars:Learning a Sense of Humor. The system participated inSubtasks A and B using N-gramlanguage models, ranking highly in the task evaluation.This paper discusses the results ofour system in the development and evaluation stagesand from two post-evaluation runs.§ INTRODUCTIONHumor is an expression ofhuman uniqueness and intelligence and has drawnattention in diverse areas such as linguistics, psychology, philosophy andcomputer science. Computational humor draws from all of thesefields and is a relatively new area of study.There is some history of systems that are able to generatehumor (e.g., <cit.>, <cit.>).However, humor detection remains a less exploredand challenging problem (e.g., <cit.>,<cit.>, <cit.>, <cit.>). SemEval-2017 Task 6 <cit.>also focuses on humor detection by asking participants todevelop systems thatlearn a sense of humor from the Comedy Central TV show,@midnight with Chris Hardwick. Our systemranks tweets according to how funny they are by training N-gram language modelson two different corpora. One consisting of funny tweetsprovided by the task organizers, andthe other on a freely available research corpus of news data.The funny tweet data is made up of tweets that are intended to be humorous responses to a hashtag given by host Chris Hardwick during the program. § BACKGROUND Training Language Models (LMs) is a straightforwardway to collect a set of rules byutilizing the fact that words do not appear in an arbitrary order;we in fact can gain useful information about a word by knowingthe company it keeps <cit.>. A statistical languagemodel estimates the probability of a sequence of words or an upcomingword.An N-gram is a contiguous sequence of N words: a unigram is asingle word, a bigram is a two-word sequence,and a trigram is a three-word sequence. For example, in the tweet tears in Ramen #SingleLifeIn3Words“tears”, “in”, “Ramen” and “#SingleLifeIn3Words” are unigrams; “tears in”,“in Ramen” and “Ramen #SingleLifeIn3Words”are bigrams and “tears in Ramen” and “in Ramen#SingleLifeIn3Words” are trigrams.An N-gram model can predict the next word from a sequence of N-1 previous words. A trigram Language Model (LM) predicts the conditional probability of the next word using the following approximation: P(w_n|w_1^n-1)≈ P(w_n|w_n-2, w_n-1) The assumption that the probability of a word depends only on a small number of previous wordsis called a Markov assumption <cit.>. Given this assumptionthe probability of a sentence can be estimated as follows:P(w_1^n)≈∏_k=1^n P(w_k|w_k-2, w_k-1)In a study on how phrasing affects memorability, <cit.> take a language model approach to measure the distinctiveness of memorable movie quotes. They do this by evaluating a quote with respect to a “common language” model built from the newswire sectionsof the Brown corpus <cit.>. They find that movie quotes which are less like“common language” are more distinctive and therefore more memorable. The intuition behind our approach is that humor should in some way be memorable or distinct, and so tweets thatdiverge from a “common language” model would be expected to be funnier. In order to evaluate how funny a tweet is, we train language models on two datasets:the tweet data and the news data.Tweets that are more probable according to the tweet data language modelare ranked as being funnier. However, tweets that have a lower probability according to the news languagemodel are considered the funnier since they are the least like the(unfunny) news corpus. We relied on both bigrams and trigrams when training our models. We use KenLM <cit.> as our language modeling tool. Language models are estimated using modified Kneser-Ney smoothingwithout pruning. KenLM also implements a back-off technique so if an N-gram is not found, KenLM applies the lower order N-gram's probabilityalong with its back-off weights. § METHOD Our system[https://xinru1414.github.io/HumorDetection-SemEval2017-Task6/] estimated tweet probability using N-gram LMs.Specifically, it solved the comparison (Subtask A) and semi-ranking (Subtask B) subtasks in four steps:* Corpus preparation and pre-processing: Collected all trainingdata into a single file. Pre-processing included filtering and tokenization.* Language model training: Built N-gram language models using KenLM.* Tweet scoring: Computed log probability for each tweet basedon trained N-gram language model. * Tweet prediction: Based on the log probability scores. * Subtask A – Given two tweets, compare and predict which one is funnier. * Subtask B – Given a set of tweets associated with one hashtag, ranktweets from the funniest to the least funny.§.§ Corpus Preparation and Pre-processing The tweet data was provided by the task organizers. It consists of 106 hashtag files made up of about 21,000 tokens. The hashtag files were further divided into a development set trial_dir of 6 hashtags and a training set of 100 hashtags train_dir.We also obtained 6.2 GB of English news data with about two million tokens from the NewsCommentary Corpus and the News Crawl Corpus from 2008, 2010 and 2011[http://www.statmt.org/wmt11/featured-translation-task.html].Each tweet and each sentence from the news data is found on a single line in their respective files.§.§.§ Preparation During the development of our system we trained our language models solely on the 100 hashtag files from train_dir and then evaluated our performance on the 6 hashtag files found in trial_dir. That data was formatted such that each tweet was found on a single line.§.§.§ Pre-processing Pre-processing consists of two steps: filtering and tokenization. The filtering step was only for the tweet training corpus.We experimented with various filtering and tokenziation combinations during the development stage to determine the best setting. * Filtering removes the following elements from the tweets: URLs, tokens starting with the “@” symbol (Twitter user names), and tokens starting with the “#” symbol (Hashtags). * Tokenization: Text in all training data was split on white space and punctuation§.§ Language Model Training Once we had the corpora ready, we used the KenLM Toolkit to train the N-gram language models on each corpus.We trained using both bigrams and trigrams on the tweet and news data. Our language models accounted for unknown words and were built both with andwithout considering sentence or tweet boundaries.§.§ Tweet Scoring After training the N-gram language models, the next step was scoring.For each hashtag file that neededto be evaluated, the logarithm of the probabilitywas assigned to each tweet in thehashtag file based on the trained language model. The larger the probability, the more likely that tweet was accordingto the language model. Table 1 shows an example of twoscored tweets from hashtag file Bad_Job_In_5_Words.tsvbased on the tweet data trigram language model. Note that KenLM reports the log of the probability of the N-gramsrather than the actual probabilities so the value closer to 0 (-19) has the higher probability and isassociated with the tweet judged to be funnier. §.§ Tweet Prediction The system sorts all the tweets for each hashtag and ordersthem based on their log probability score, where the funniest tweet should be listed first. If the scores are based on the tweet language model then they are sorted in ascending order since the log probability value closest to 0 indicates the tweet that is mostlike the (funny) tweets model.However, if the log probability scores are based on the news data then they are sorted in descending order since the largest value will have thesmallest probability associated with it and is therefore least like the (unfunny) news model.For Subtask A, the system goes through the sorted list of tweets in a hashtag file and compares each pair of tweets. For each pair, if the first tweet was funnierthan the second, the system would output the tweet_ids for the pairfollowed by a “1”. If the second tweet is funnier it outputs the tweet_idsfollowed by a “0”. For Subtask B, the system outputs all the tweet_ids for a hashtag file starting from the funniest. § EXPERIMENTS AND RESULTS In this section we present the results from our development stage (Table 2),the evaluation stage (Table 3), and two post-evaluation results(Table 3). Since we implemented both bigram and trigam language models during thedevelopment stage but only results from trigram language models were submitted to the task,we evaluated bigram language models in the post-evaluation stage. Note that the accuracy anddistance measurements listed in Table 2 and Table 3 are defined by the task organizers<cit.>. Table 2 shows results from the development stage. These results showthat for the tweet data the best setting is to keepthe # and @, omit sentence boundaries, be case sensitive, and ignoretokenization. While using these settings the trigram language model performed better on Subtask B (.887) and the bigramlanguage model performed better on Subtask A (.548). We decided to rely on trigram language models for the task evaluation since the advantage of bigrams on Subtask A was very slight (.548 versus .543).For the news data, we found that the best setting was toperform tokenization, omit sentence boundaries, and to be case sensitive. Given that trigrams performed most effectively in thedevelopment stage, we decided to use those during the evaluation. Table 3 shows the results of our system during the task evaluation. We submitted two runs, one with a trigram language model trained on the tweet data, and another with a trigram language model trained on the news data. In addition, after the evaluation was concluded we also decided to run the bigram language models as well. Contrary to what we observed in the development data, the bigram languagemodel actually performed somewhat better than the trigram language model. In addition, and also contrary to what we observed with the development data, the news data proved generally more effective in the post–evaluation runs than the tweet data. § DISCUSSION AND FUTURE WORK We relied on bigram and trigram language models becausetweets are short and concise, and often only consist of just a few words. The performance of our system was not consistent when comparing the development to the evaluation results. During development language models trained on the tweet data performed better. However during the evaluation and post-evaluation stage,language models trained on the news data weresignificantly more effective. We also observed that bigram language models performed slightly better than trigram models on the evaluation data. This suggests that going forward we should also consider both the use of unigram and character–level language models. These results suggest that there are onlyslight differences between bigram and trigram models, and that the type and quantity of corpora used to train themodels is what really determines the results. The task description paper <cit.>reported system by system results for each hashtag.We were surprised to find that ourperformance on the hashtag file#BreakUpIn5Words in the evaluationstage was significantlybetter than any othersystem on both Subtask A(with accuracy of 0.913) and Subtask B(with distance score of 0.636). While we still do notfully understand the cause of these results, there is clearly something about the language used in this hashtag that is distinct from the other hashtags, and is somehow better represented or captured by a language model. Reaching a better understanding of thisresult is a high priority for future work. The tweet data was significantly smaller than the news data, and so certainly we believe that this was a factor in the performance during the evaluation stage, where the models built from the news data were significantly more effective. Going forward we plan to collect more tweet data, particularly those that participate in#HashtagWars. We also intend to do some experiments where wecut the amount of news data and then build models to see howthose compare. While our language models performed well, there is some evidence that neural network models can outperformstandard back-off N-gram models <cit.>.We would like to experiment with deep learning methods suchas recurrent neural networks, sincethese networks are capable offorming short term memory and may be better suited for dealingwith sequence data. natexlab#1#1[Danescu-Niculescu-Mizil et al.(2012)Danescu-Niculescu-Mizil, Cheng, Kleinberg, and Lee]hello Cristian Danescu-Niculescu-Mizil, Justin Cheng, Jon Kleinberg, and Lillian Lee. 2012. You had me at hello: How phrasing affects memorability. In Proceedings of the 50th Annual Meeting of the Association for Computational Linguistics: Long Papers - Volume 1. Association for Computational Linguistics, Stroudsburg, PA, USA, ACL '12, pages 892–901.[Firth(1968)]Firth57 J. Firth. 1968. A synopsis of linguistic theory 1930-1955. In F. Palmer, editor, Selected Papers of J. R. Firth, Longman.[Heafield et al.(2013)Heafield, Pouzyrevsky, Clark, and Koehn]Heafield-estimate Kenneth Heafield, Ivan Pouzyrevsky, Jonathan H. Clark, and Philipp Koehn. 2013. Scalable modified Kneser-Ney language model estimation. In Proceedings of the 51st Annual Meeting of the Association for Computational Linguistics. Sofia, Bulgaria, pages 690–696.[Kucera and Francis(1967)]BC Henry Kucera and W. Nelson Francis. 1967. Computational Analysis of Present-day American English. Brown University Press, Providence, RI, USA.[Markov(2006)]markov1954theory A. A. Markov. 2006. An example of statistical investigation of the text Eugene Onegin concerning the connection of samples in chains. Science in Context 19(4):591–600.[Mihalcea and Strapparava(2006)]Learning:To:Laugh Rada Mihalcea and Carlo Strapparava. 2006. Learning to laugh (automatically): Computational models for humor recognition. Computational Intelligence 22(2):126–142.[Mikolov et al.(2011)Mikolov, Kombrink, Burget, Černockỳ, and Khudanpur]mikolov2011extensions Tomáš Mikolov, Stefan Kombrink, Lukáš Burget, Jan Černockỳ, and Sanjeev Khudanpur. 2011. Extensions of recurrent neural network language model. In Acoustics, Speech and Signal Processing (ICASSP), 2011 IEEE International Conference on. IEEE, pages 5528–5531.[Miller and Gurevych(2015)]MillerG15 Tristan Miller and Iryna Gurevych. 2015. Automatic disambiguation of English puns. In Proceedings of the 53rd Annual Meeting of the Association for Computational Linguistics and the 7th International Joint Conference on Natural Language Processing (Volume 1: Long Papers). Association for Computational Linguistics, Beijing, China, pages 719–729.[Özbal and Strapparava(2012)]ozbal2012computational Gözde Özbal and Carlo Strapparava. 2012. A computational approach to the automation of creative naming. In Proceedings of the 50th Annual Meeting of the Association for Computational Linguistics: Long Papers-Volume 1. Association for Computational Linguistics, pages 703–711.[Potash et al.(2017)Potash, Romanov, and Rumshisky]PotashRR17 Peter Potash, Alexey Romanov, and Anna Rumshisky. 2017. SemEval-2017 Task 6: #HashtagWars: learning a sense of humor. In Proceedings of the 11th International Workshop on Semantic Evaluation (SemEval-2017). Vancouver, BC.[Shahaf et al.(2015)Shahaf, Horvitz, and Mankoff]ShahafHM15 Dafna Shahaf, Eric Horvitz, and Robert Mankoff. 2015. Inside jokes: Identifying humorous cartoon captions. In Proceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. ACM, New York, NY, USA, KDD '15, pages 1065–1074.[Stock and Strapparava(2003)]StockS03 Oliviero Stock and Carlo Strapparava. 2003. Getting serious about the development of computational humor. In Proceedings of the Eighteenth International Joint Conference on Artificial Intelligence. Acapulco, pages 59–64.[Zhang and Liu(2014)]Recognizing:Humor:On:Twitter Renxian Zhang and Naishi Liu. 2014. Recognizing humor on Twitter. In Proceedings of the 23rd ACM International Conference on Conference on Information and Knowledge Management. ACM, New York, NY, USA, CIKM '14, pages 889–898. | http://arxiv.org/abs/1704.08390v1 | {
"authors": [
"Xinru Yan",
"Ted Pedersen"
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"cs.CL"
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"published": "20170427004033",
"title": "Duluth at SemEval-2017 Task 6: Language Models in Humor Detection"
} |
#1 #1 #1#1 #1(#1) #1(#1) | http://arxiv.org/abs/1704.08650v1 | {
"authors": [
"Mehedi Masud",
"Mary Bishai",
"Poonam Mehta"
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"categories": [
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"primary_category": "hep-ph",
"published": "20170427164358",
"title": "Extricating New Physics Scenarios at DUNE with High Energy Beams"
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=6.0in=9.25in =-0.3in =-0.20in Institute for Particle Physics Phenomenology, University of Durham,South Rd, DURHAM,DH13LE, UNITED KINGDOM#1#1#1#1 #1#1and #1 Submitted to #1Abstract PresentedPRESENTED ATACKNOWLEDGEMENTSCreating the Baryon Asymmetry from Lepto-Bubbles Silvia Pascoli, Jessica Turner, Ye-Ling Zhou We propose a new mechanism of baryogenesis which proceeds via a CP-violating phase transition. During this phase transition, the coupling of the Weinberg operator is dynamically realised and subsequently a lepton asymmetry is generated via the non-zero interference of this operator at different times. This new scenario of leptogenesis provides adirect connection between the baryon asymmetry, low energy neutrino parameters and leptonic flavour models.NuPhys2016, Prospects in Neutrino PhysicsBarbican Centre, London, UK,December 12–14, 2016 footnote§ INTRODUCTIONIn spite of the abundance of baryogenesis models, thematter anti-matter asymmetryremains an unresolved issue.Common to all dynamical mechanisms of baryogenesis is the necessity to satisfy Sakharov'sconditions <cit.>: processes must violate baryon number, C and CP-symmetry and finally provide a departure from thermal equilibrium. Popular scenarios of high-scale thermal leptogenesis use the out-of-equilibrium decays of heavy particles, such as sterile neutrinos, to produce a lepton asymmetry which is subsequently converted to a baryon asymmetry via electroweak sphaleron processes <cit.>. In addition to providing a possible solution to the baryon asymmetry, thermal leptogenesis also explains small neutrino masses.There are also low-scale leptogenesis mechanismsin which the mass of the sterile neutrinos is below the electroweak scale. The lepton asymmetry is generated via the oscillations of these sterile states <cit.>. In bothhigh and low-scale leptogenesis, the details of the neutrino mass model must be specified.In this paper we propose a completely new scenario of leptogenesis which proceeds via a CP-violating phase transition and henceforth will be referred to as the CPPT mechanism. This phase transition produces bubbles which are leptonically CP-violating, namelylepto-bubbles. The generation of the lepton asymmetry occurs below the neutrino mass generation scale and therefore the particular neutrino mass model need not be specified. In addition to providing a novel solution to the matter anti-matter asymmetry, our mechanism establishes a connection between this asymmetry and the flavour structure of the lepton sector <cit.>. This short paper is structured as follows: in sec:mech and sec:CTP we discuss our basic assumptions, outline the CPPT mechanism and the computational method namely the Closed-Time Path Formalism. In sec:calc we give details of the lepton asymmetry calculation and estimate the temperature at which the phase transition occurs and finally we discuss and summarise in sec:discussion and sec:conclusion respectively.§ THE CPPT MECHANISMWe assume neutrinos are Majorana in nature and therefore, to leading order, their mass model reduces to the lepton number violating Weinberg operatorsmallℒ_W=λ_αβ/Λℓ_α LH Cℓ_β L H + h.c. ,small where λ_αβ=λ_βα is a model-dependent coupling,Λthe scale of new physicsandC is the charge conjugation matrix. The Weinberg operator can be UV-completed in a number of ways ranging fromloop effects to introducing heavy new degrees of freedom such as sterile neutrinos. However, unlike typical scenarios of leptogenesis, the details of the UV-completion of the dimension five operator need not be specified in this mechanism. We postulate the coupling of the Weinberg operator is functionally dependent upon a SM-singlet scalar, ϕ, such that λ_αβ=λ^0_αβ+λ^1_αβ⟨ϕ⟩/v_ϕ.Associated to ϕ is a finite temperature scalar potential, which is symmetric under a leptonic flavour symmetry at sufficiently high temperatures.As the temperature of the Universe lowers, the minima at the origin of this potential becomes metastable and a phase transition occurs. As a result, the minima changes from the vacua at the origin to a deeper, true vacua which is stable and non-zero, ⟨ϕ⟩.The ensemble expectation value (EEV) of ϕ spontaneously breaks the high-scale flavour symmetry and results in theobserved pattern ofleptonic masses and mixing. Assuming a first order phase transition, (lepto) bubbles of the leptonically CP-violating broken phase spontaneously nucleate. At a fixed space point within the bubble wall, the coupling λ is time-dependent e.g.λ(t_1) ≠λ(t_2) for t_1≠ t_2. As a consequence,the lepton asymmetry arises from thenon-zero interference of the Weinberg operator at different times. § THE CLOSED-TIME PATH FORMALISMWeapply the closed-time path (CTP) formalism to calculate the lepton asymmetry. Unlike zero temperature methods, the CTP formalism properly accounts for finite density effects andresolves unitarity issues. The basic building blocks of the CTP formalism are the Green functions and the corresponding self-energy corrections.As we will focus on the self-energy corrections from the Weinberg operator to the lepton propagators, it is sufficient to present theHiggs (Δ) and lepton (S)propagators smallΔ^T,T(x_1,x_2)= ⟨ T[H(x_1) H^*(x_2)] ⟩, ⟨T[H(x_1) H^*(x_2)] ⟩ , Δ^<,>(x_1,x_2)= ⟨ H^*(x_2) H(x_1) ⟩, ⟨ H(x_1) H^*(x_2) ⟩ ,S^T,T_αβ(x_1,x_2)= ⟨ T[ℓ_α(x_1) ℓ_β(x_2)] ⟩, ⟨T[ℓ_α(x_1) ℓ_β(x_2)] ⟩ , S^<,>_αβ(x_1,x_2)=-⟨ℓ_β(x_2) ℓ_α(x_1) ⟩ , ⟨ℓ_α(x_1) ℓ_β(x_2) ⟩ ,small where T (T) denotes time (anti-time) ordering, Greek indices are flavour indices and spinor and electroweak indices have been suppressed. The lepton asymmetry is written in terms of the leptonic Wightman propagators, S^<,>, such that smalln_L(x)=- 1/2∑_αtr{γ^0 [S^<_αα(x,x) + S^>_αα(x,x) ] } .small The Kadanoff-Baym(KB) equations are used to calculate the time evolution of the lepton asymmetry and we follow the conventions of <cit.> and write this equation as smalli ∂S^<,> - Σ^H⊙ S^<,> - Σ^<,>⊙ S^H = 1/2[Σ^>⊙ S^< - Σ^<⊙ S^>],small where the symbol ⊙ presents a convolution, Σ is the self-energy of the lepton and S^H (Σ^H) is Hermitian parts of propagator (self-energy) given by S^H=S^T-1/2(S^>+S^<) (Σ^H=Σ^T-1/2(Σ^>+Σ^<)). The self-energy contribution (Σ^HS^<,>) and broadening of the on-shell dispersion relation (Σ^<,> S^H) are given on the LHS of eq:KB. Whilst, the collision term that includesCP-violating source (1/2 (Σ^> S^< - Σ^< S^>)) is shown on the RHS <cit.>.As wefocus on the generation of an initial asymmetry, we consider only the collision term. smallsmall § CALCULATING THE LEPTON ASYMMETRYWe calculate the lepton asymmetry using similar techniques applied in <cit.>. The calculation involves application of the KB equation (<ref>) to find the time evolution of the lepton propagator.The self-energy correction is calculated to leading order, in the time-independent flavour basis,as is shown in fig:feyn. The first step is to Fourier transform the Green functions of the Higgs and leptonic propagator respectivelysmallΔ_q⃗(t_1,t_2) = ∫ d^3 r e^i q⃗·r⃗Δ(x_1,x_2) and S_k⃗(t_1,t_2) = ∫ d^3 r e^i k⃗·r⃗ S(x_1,x_2) ,small where t_1=x_1^0, t_2=x_2^0 and r=x_1-x_2.The lepton asymmetry, at a fixed space point in the bubble wall,is given by n_L(x) = ∫d^3 k/(2π)^2 L_k⃗ with smallL_k⃗≡ f_ℓk⃗ - f_ℓk⃗ = - ∫_t_i^t_f dt_1 ∂_t_1tr[γ_0 S^<_k⃗(t_1,t_1) + γ_0 S^>_k⃗(t_1,t_1)] =-∫_t_i^t_f dt_1∫_t_i^t_f dt_2tr[Σ^>_k⃗(t_1,t_2)S_k⃗^<(t_2,t_1) -Σ^<_k⃗(t_1,t_2)S_k⃗^>(t_2,t_1)],small where t_i (t_f) is the initial (final) time and Σ^<_k⃗(t_1,t_2) is the CP-violating two-loop self-energy correction as shown in fig:feyn. Subsequently, (<ref>) may be re-expressed as smallL_k⃗αβ = ∑_γδ12/Λ^2∫_t_i^t_f dt_1∫_t_i^t_f dt_2Im{λ^*_αγ(t_1)λ_βδ(t_2)}∫_q, q^' M_αβγδ(t_1,t_2,k,k^',q,q^'),small where ∫_q, q^'=∫d^3q/(2π)^3d^3q^'/(2π)^3 . The lepton asymmetry has factorised into two parts: one part is a function of the time-dependent couplings, λ(t), which allows for a connection between the lepton asymmetry, the leptonic flavour model and low energy neutrino parameters. The other, M_αβγδ(t_1,t_2,k,k^',q,q^'),is the finite temperature matrix element which is calculated using CTP Feynman rules (for further details see <cit.>). For the present calculation, we will ignore the differing thermal widths of the charged lepton propagators and using the limitt_i (t_f)→ -∞ (+∞), the total lepton asymmetry L_k⃗≡∑_α L_k⃗αα is given by smallL_k⃗ = 12/Λ^2∫_-∞^+∞dt_1∫_-∞^+∞dt_2Im{tr[ λ^*(t_1)λ(t_2)]}∫_q, q^' M,small where thefinite temperature matrix element, decomposed in terms of the lepton and Higgs propagators, is expressed as smallM = Im{Δ^<_q⃗(t_1,t_2)Δ^<_q⃗^⃗'⃗(t_1,t_2) tr[ S^<_k⃗(t_1,t_2)S^<_k⃗^⃗'⃗(t_1,t_2) P_L] } .small Throughout we assume theHiggs and leptonic propagators are almost in thermal equilibrium as the scale of the CPPT mechanism is significantly higher than the electroweak scale. The time-varying coupling is functionally dependent upon the EEV of the scalar, ϕ.We make an ansatz for the coupling smallλ(t) =λ^0+ λ^1 f(t),small whereλ^0 (λ^0+λ^1) is the value of the coupling at t=-∞ (t=+∞) and f(t)varies continuously from 0 to 1. As expected, the lepton asymmetry is not sensitive to theprecise functional form of f(t); it has been shown a tanh and step function produce the same result <cit.>.Performing a change of integration variables from t_1, t_2 tot̃=(t_1+t_2)/2,y=t_1-t_2 and using ∫_-∞^+∞ dt̃ [f(t̃+y/2)-f(t̃-y/2)] = y, the lepton asymmetry may be written assmallL_k⃗=- 12/v^4_HIm{tr[ m^0_νm_ν^* ]}∫_-∞^+∞dy y ∫_q,q^' M,small where v_H is the Higgs vacuum expectation value and m_ν^0 ≡λ^0 v_H^2/Λ (m_ν≡ (λ^0 + λ^1) v_H^2/Λ) is the effective neutrino mass matrix before (after) the phase transition. As ϕ only interacts with the leptons and Higgs in the thermal bath, it is reasonable to assume a fast-moving bubble wall. Consequently, the lepton asymmetry (<ref>) is not dependent upon the bubbleproperties. Using the calculation ofM (full details given in <cit.>) the lepton asymmetry can be rewritten assmallL_k⃗ = 3 Im{tr[ m^0_νm^*_ν]}T^2/( 2π)^4v^4_HF( x_1,x_γ) .smallF( x_1,x_γ) is a loop factor given bysmallF( x_1,x_γ) = 1/x_1∫_0^+∞dx∫_0^+∞x_2dx_2∫_|x_1-x |^|x_1+x |dx_3∫_|x_2-x |^|x_2+x |dx_4∑_η_2,η_3,η_4=±1[1-(x^2_1+x^2-x^2_3)(x^2_2+x^2-x^2_4)/4η_2x_1x_2x^2] ×X_η_2η_3η_4x_γsinh X_η_2η_3η_4/(X^2_η_2η_3η_4+x^2_γ)^2cosh x_1cosh x_2sinh x_3sinh x_4 ,small where thefour momentum of the leptons and Higgs shown in fig:feyn are defined as k = |k⃗|, k = |k⃗^⃗'⃗|,q = |q⃗| andq = |q⃗^⃗'⃗| and correspondinglyx_1=k/2T,x_2=k^'/2T,x_3=q/2T,x_4=q^'/2T,x=p/2T andX_η_2η_3η_4=x_1+η_2x_2+η_3x_3+η_4x_4. The loop factor is dependent upon the lepton energy and the thermal width normalised by the temperature, i.e., x_1 and x_γ as shown in fig:loopfactor. § DISCUSSION The lepton asymmetry produced during the CPPT mechanism is partially converted into a final baryon asymmetry via sphaleron processes which are unsuppressed above the electroweak scale. The final baryon asymmetry is roughly given by η_B≈1/3η_B-L and may be written as smalln_B/n_γ≈ - Im{tr[ m^0_νm^*_ν]}T^2/8π^2ζ(3)v_H^4F( x_γ),small whereF( x_γ)=∫_0^∞x_1dx_1F( x_1,x_γ), n_γ = 2ζ( 3)T^3/π^2 and ζ( 3) = 1.202. In order to produce a positive baryon to photon ratio, Im{tr[ m^0_νm^*_ν]} should be negative.It is worth noting the baryon asymmetry is dependent upon three quantities: the self-energy correction to the lepton propagatorrepresented by the loop factor F( x_1,x_γ), the effective neutrino mass matrices (m_ν and m^0_ν) and finally the temperature, T, of the phase transition.First, the loop factor is shown as a function of thetemperature normalised lepton energy (x_1) as shown in fig:loopfactor. For Standard Model values of thetemperature normalised lepton thermal width, x_γ∼ 0.1, the loop factor provides an 𝒪(10 )enhancement to the lepton asymmetry. Second, the lepton asymmetry is crucially reliant on the effective neutrino masses. The structure of m^0_ν is determined by the particular high-scale flavour symmetryand m_ν is the neutrino mass matrix which is diagonalised by the PMNS matrix U_PMNS^Tm_νU_PMNS=diag(m_1, m_2, m_3). This establishes a connection between the lepton asymmetry, low-energy neutrino parameters and the flavour symmetry. Finally,in order to estimate the phase transition temperature, we assume Im{tr[ m^0_νm_ν^* ]} is of the same order as m_ν^2∼(0.1eV)^2 and we have calculated that F( x_γ)∼𝒪(100 ), hence the temperature for successful leptogenesis issmallT ∼ 3 √(η_B) v_H^2/m_ν .small In order to produce the observed baryon to photon ratio (η_B=(6.19±0.15) ×10^-10 <cit.>) the temperature of the phase transition is approximately T∼ 10^11 GeV. The energy scale of this mechanism is similar to that of high-scale thermal leptogenesis. However, there are several improvements to this calculation which may lower the temperature.These improvements include accounting for the differing charged lepton thermal widths andcalculating the fully time evolved asymmetry.§ CONCLUSIONCPPT is a completely new and novel mechanism that could simultaneously explainthe observed baryon asymmetry andthepattern of mixing in the lepton sector. Unlike conventional scenarios of leptogenesis, which specify a particular neutrino mass generation mechanism, CPPT allows for relative model independence as the new physics responsible for neutrino masses has already been integrated out before the CP-violating phase transition occurs. There are several interesting aspects of theour mechanism that could be further explored. These include studying this mechanism in the context of a particular flavour model andpredicting the associatedgravitational wave spectra.99 Sakharov:1967djA. D. Sakharov,Pisma Zh. Eksp. Teor. Fiz.5, 32 (1967) [JETP Lett.5, 24 (1967)]t [Sov. Phys. Usp.34, 392 (1991)] [Usp. Fiz. Nauk 161, 61 (1991)].Fukugita:1986hrM. Fukugita and T. Yanagida,Phys. Lett. B 174, 45 (1986).Akhmedov:1998qxE. K. Akhmedov, V. A. Rubakov and A. Y. Smirnov,Phys. Rev. Lett.81, 1359 (1998)[hep-ph/9803255].Pascoli:2016tivS. Pascoli and Y. L. Zhou,arXiv:1611.04817 [hep-ph]. Pascoli:2016gkfS. Pascoli, J. Turner and Y. L. Zhou,arXiv:1609.07969 [hep-ph].Prokopec:2003pjT. Prokopec, M. G. Schmidt and S. Weinstock,Annals Phys.314, 208 (2004)[hep-ph/0312110]. Garbrecht:2008cbB. Garbrecht and T. Konstandin,Phys. Rev. D 79, 085003 (2009)[arXiv:0810.4016 [hep-ph]].Anisimov:2010dkA. Anisimov, W. Buchmuller, M. Drewes and S. Mendizabal,Annals Phys.326, 1998 (2011) Erratum: [Annals Phys.338, 376 (2011)][arXiv:1012.5821 [hep-ph]].Agashe:2014kdaK. A. Olive et al. [Particle Data Group],Chin. Phys. C 38, 090001 (2014). | http://arxiv.org/abs/1704.08322v1 | {
"authors": [
"Silvia Pascoli",
"Jessica Turner",
"Ye-Ling Zhou"
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"primary_category": "hep-ph",
"published": "20170426194446",
"title": "Creating the Baryon Asymmetry from Lepto-Bubbles"
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Dark Energy Density in SUGRA models and degenerate vacua R. Nevzorov^* and A. W. Thomas December 30, 2023 ======================================================== We present Monte Carlo (MC) simulation studies of phase separation in binary (AB) mixtures with bond-disorder that is introduced in two different ways: (i) at randomly selected lattice sites and (ii) at regularly selected sites. The Ising model with spin exchange (Kawasaki) dynamics represents the segregation kinetics in conserved binary mixtures. We find that the dynamical scaling changes significantly by varying the number of disordered sites in the case where bond-disorder is introduced at the randomly selected sites. On the other hand, when we introduce the bond-disorder in a regular fashion, the system follows the dynamical scaling for the modest number of disordered sites. For higher number of disordered sites, the evolution morphology illustrates a lamellar pattern formation. Our MC results are consistent with the Lifshitz-Slyozov (LS) power-law growth in all the cases.§ INTRODUCTION A binary (AB) mixture, which is homogeneous (or disordered) at high temperatures becomes thermodynamically unstable when rapidly quenched inside the coexistence curve. Then, the binary (AB) mixture undergoes phase separation (or ordering) via the formation and growth of domains enriched in either component. Much research interest has focused on this far-from-equilibrium evolution <cit.>. The domain morphologies are usually quantified by two important properties: a) the domain growth law (characteristic domain size L(t) grows with time t), which depends on general system properties, e.g., the nature of conservation laws governing the domain evolution, the presence of hydrodynamic velocity fields, the presence of quenched or annealed disorder, etc. b) the correlation function or its Fourier transform, the structure factor, which is a measure of the domain morphology <cit.>. There now exists a good understanding of phase separation dynamics for binary mixtures <cit.>. Normally, for a pure and isotropic system, domain growth follows a power-law behavior, L(t) ∼ t^ϕ where ϕ is referred to as the growth exponent. For the case with nonconserved order parameter (ordering of a magnet into up and down phases), the system obeys the Lifshitz-Cahn-Allen (LCA) growth law with ϕ=1/2 <cit.>. For the case with conserved order parameter (diffusion driven phase separation of an AB mixture into A-rich and B-rich phases). The system obeys the Lifshitz-Slyozov (LS) growth law with ϕ=1/3 <cit.>. However, including the hydrodynamic effects in a system with conserved order parameter (e.g., segregation of a binary fluid), there appear to be various domain growth regimes, depending on the dimensionality and system parameters <cit.>. In reality, the experimental systems are neither pure nor isotropic. Usually, they always endure impurities (annealed or quenched) within the system. An important set of results has been well documented from both analytical and numerical studies on phase ordering in systems with quenched disorder <cit.>. The quench disorder (considered as an immobile impurity) is introduced into the pure Ising model by either random spin-spin exchange interaction, i.e., random-bond Ising model (RBIM) <cit.> or by introducing a site-dependent random-field Ising model (RFIM) <cit.>. In general, sites of quenched disorder act as traps for domain boundaries with the energy barrier being dependent on the domain size. In this regard, a significant contribution is made by Huse and Henley (HH)<cit.> to understand the growth law for the bond disorder case. They argued that the energy barrier follows power-law dependence on domain size: E_b(L) ≃ϵ L^ψ. Here, ϵ is the disorder strength and ψ is the barrier exponent that depends on the roughening exponent ζ and the pinning exponent χ as ψ=χ/(2-ζ); the roughening and pinning exponents are related as χ=2ζ+d-3, where d is the system dimensionality. Consequently, the normal power law growth (L(t)∼ t^ϕ) of the characteristic domain size changes over to a logarithmic growth L(t)∼ (ln t)^ϕ. A few numerical simulations <cit.> and experiments <cit.> were performed to test the HH proposal. Nevertheless, to date, no definite confirmation of logarithmic growth in the asymptotic regime is observed. Later, Paul, Puri, and Rieger (PPR) <cit.> reconsidered this problem via extensive Monte Carlo (MC) simulations of the RBIM with nonconserved (Glauber) spin-flip kinetics, and conserved (Kawasaki) spin exchange kinetics. In contrast to HH scenario, PPR observed the normal power-law domain growth with temperature and disorder dependent growth exponent, similar to the one seen in the experiments <cit.> on domain growth in the disordered system. PPR proposed that the growth exponents can be understood in the framework of a logarithmic domain size dependence of trapping barrier (E_b(L) ≃ϵln(1+L)) rather than power-law <cit.>. At early times, domains coarsening is not affected by disorder due to small energy barriers, and therefore, the system evolves like a pure system. At late times, the disorder traps become effective at a crossover length scale, and it can only move by thermal activation over the corresponding energy barrier. Thus, thermal fluctuations drive the asymptotic domain growth in disordered systems <cit.>. This should be contrasted with the pure case, where thermal fluctuations are irrelevant. In these cases, quench disorder was introduced by uniformly varying the strength of the spin-spin exchange interaction between zero and one at all the lattice sites.In this paper, we present MC simulation of domain coarsening in binary mixtures with quenched disorder using conserved (Kawasaki) spin-exchange kinetics. Here, we introduce the disorder in two different ways: a) at randomly selected lattice sites and b) at regularly selected lattice sites. We consider the strength of the spin-spin exchange interaction equal to zero at these selected sites (equivalent to have sites at T≫ T_c called disordered sites) and equal to one at the rest of the sites. By varying the number of selected sites, we discuss the effect of disorder on the domain growth law and the dynamical scaling. Our simulations are aimed to gain a conceptual understanding of these disordered systems where theoretical calculations are challenging at present. This paper is organized as follows. In Sec. <ref>, we describe the methodology we used to simulate the system. In Sec. <ref>, we present the results and discussion for both the cases of introducing disorder. Finally, Sec. <ref> concludes this paper with the summary of our results. § METHODOLOGY Let us start with a description of Monte Carlo (MC) simulations for the study of phase separation in binary (AB) mixtures. The Hamiltonian for the Ising system is described by H=-∑_<ij>J_ijS_iS_j,S_i=± 1.Here, S_i denotes the spin variable at site i. We consider two state spins: S_i=+1 when a lattice site i is occupied by an A atom and S_i=-1 when occupied by a B atom. The subscript <ij> in Eq. <ref> denotes a sum over nearest-neighbor pairs only. The term J_ij denotes the strength of the spin-spin exchange interaction between nearest-neighbor spins. We consider the case where J_ij≥ 0 so that the system is locally ferromagnetic. The case where a system has both J_ij≥ 0 (ferromagnetic) and J_ij≤ 0 (antiferromagnetic) is relevant to spin glasses. Normally, in MC simulation for a pure phase-separating binary (AB) mixture, we consider J_ij=1 with a critical temperature T_c≃2.269/k_B for a d=2 square lattice. Further, J_ij=0 corresponds to the maximally disordered system, equivalent to the system at T≫ T_c where all proposed spin exchanges will be accepted. In our MC simulation, spins are placed on a square lattice (L_x × L_y) with periodic boundary conditions in both the directions. We assign random initial orientations: up (S_i=+1) or down (S_i=-1) to each spin and rapidly quench the system to T<T_c. The quench disorder is introduced via exchange coupling as J_ij=1-ϵ, where ϵ quantifies the degree of disorder. In this paper, we considered only two values of the degree of disorder, ϵ =0 (pure system) and ϵ =1 (disordered sites corresponding to impurities in the system). Notably, inPPR's study <cit.> J_ij is uniformly distributed in the interval [1-ϵ,1], where the limit ϵ=0 corresponds to the pure case and ϵ=1 corresponds to the maximally disordered case with J_ij∈ [0,1]. We perform our MC simulations for two different cases corresponding to the way we introduce disorder into the system. In Case 1 we randomly selected a fraction of sites with ϵ =1 and in Case 2, we picked the same fraction of sites in a regular fashion. The remaining lattice sites are set to ϵ =0. Shortly, we present the results for three different percentages of disorder sites (ϵ =1) namely at 2%, 5% and 10% of total sites, N, for both the cases and compare them with the pure case (ϵ =0). The initial condition of the system corresponds to a critical quench with 50% A (up) and 50% B (down) spins.We place the Ising system in contact with a heat bath to associate stochastic dynamics. The resultant dynamical model is referred to as a Kinetic Ising Model. We consider spin-exchange (Kawasaki) kinetics, an appropriate model to study the phase separation in AB mixtures <cit.>. It is straight forward to implement MC simulation of the Ising model with spin-exchange Kinetics. In a single step of MC dynamics, a randomly selected spin S_i is exchanged with a randomly chosen nearest-neighbor S_j (S_i↔ S_j). The change in energy Δ H that would occur if the spins were exchanged is computed. The step is then accepted or rejected with then Metropolis acceptance probability<cit.>: P=exp(-βΔ H)forΔ H ≥ 0,1 forΔ H ≤ 0.Here, β=(k_BT)^-1 denotes the inverse temperature; k_B is the Boltzmann constant. One Monte Carlo step (MCS) is completed when this algorithm is performed N times (where N is the total number of spins), regardless of whether the move is accepted or rejected. Noticeably, if at least one of the spin in the randomly chosen spin pair belongs to the disordered site, the proposed spin exchange will be accepted. The morphology of the evolving system is usually characterized by studying the two-point (r⃗=r⃗_⃗1⃗-r⃗_⃗2⃗) equal-time correlation function:C(r⃗,t) = 1/N∑_i=1[⟨ S_i(t) S_i+r⃗(t)⟩ - ⟨ S_i(t)⟩⟨ S_i+r⃗(t)⟩],which measures the overlap of the spin configuration at distance (r⃗). Here, the angular brackets denote an average over different initial configurations and different noise realizations. However, most experiments study the structure factor, which is the Fourier transform of the correlation function,S(k⃗,t)=∑_r⃗exp(ik⃗·r⃗) C(r⃗,t),where k⃗ is the scattering wave-vector. Since the system under consideration is isotropic, we can improve statistics by spherically averaging the correlation function and the structure factor. The corresponding quantities are denoted as C(r,t) and S(k,t), respectively, where r is the separation between two spatial points and k is the magnitude of the wave-vector.It is now a well-established fact that the domain coarsening during phase separation is a scaling phenomenon. The correlation function and the structure factor exhibit the dynamical scaling form <cit.>C(r,t)= g[r/L(t)],S(k,t)= L(t)^d f[kL(t)].Here, g(x) and f(p) are the scaling functions. The characteristic length scale L(t) (in the units of lattice spacing) is defined from the correlation function as the distance over which it decays to(say) zero or any fraction of its maximum value [C(r=0,t)=1]; we find that the decay of C(r,t) to 0.1 gives a good measure of average domain size L(t). There are few different definitions of the length scale, but all these are equivalent in the scaling regime i.e., they differ only by constant multiplicative factors <cit.>.§ NUMERICAL RESULTS Using our MC simulations, we present results for the structure and dynamics of phase separating symmetric binary mixture (50%A and 50%B) with the bond disorder. We discuss both the cases of introducing the disorder (Case1: at randomly selected sites, and Case2: at regularly selected sites). The simulations are performed on a system of N=L_x× L_y particles of type A and B confined to a square lattice (d=2, L_x=L_y=512) such that the number density ρ=1.0. We quench the system from high-temperature homogeneous phase to a temperature T=1.0 (T<T_c) and then monitor the evolution of the system at various Monte Carlo steps. In presenting these results, our purpose is two-fold: first, we analyze the effects of bond-disorder on the domain coarsening and how the number of disordered sites (N_1) influences the characteristic features of the domains morphology and scaling behavior. Secondly, we intend to study how the different ways of introducing the same disorder affect phase separating kinetics in the system. §.§ Disorder at randomly selected sites We present evolution morphologies of AB mixture obtained from our MC simulations for Case1 in Fig. <ref> at t=4 × 10^5 and t=1.6 × 10^6 MCS. Fig. <ref> display the evolution pictures for four different percentages of disordered sites: (a) 0% (N_1 = 0; pure case), (b) 2% (N_1 = N/50), (c) 5% (N_1 = N/20), and (d) 10% (N_1 = N/10), respectively. Immediately after the quench, the system starts evolving via the emergence and growth of domains, namely A-rich (marked in blue) and B-rich (unmarked) regions. As expected, for a symmetric (critical) composition, a bicontinuous domain structure is seen for the pure case (Fig. <ref>a). However, with the increase of disordered sites (N_1), the roughening of domain walls increases <cit.>; this is because of the disordered sites at which all the proposed spin exchanges are accepted and hence, domains look more fuzzier with increasing N_1. To study the domain morphology, we plot the scaled correlation function [C(r,t) vs. r/L(t)] in Fig. <ref>a at three different times during the evolution. Here, we considered Case1 with 5% of disordered sites (see Fig. <ref>c); L(t) is defined as the distance over which C(r,t) decays to 0.1 of its maximum value (C(0,t)=1). A neat data collapse demonstrates the dynamical scaling of the domains morphology and confirms that the system for a given N_1 belongs to the same dynamical universality class. An excellent data collapse of the structure factor (log-log plot of S(k,t)L^-2 vs. kL in Fig. <ref>b), obtained from the Fourier transform of the correlation function data sets presented in Fig. <ref>a, also demonstrate the dynamical scaling. However, for large k values, S(k,t) deviates from the well-known Porod's law, S(k,t)∼ k^-(d+1), which results from scattering off sharp interfaces <cit.>. For other values of N_1, the correlation function and the structure factor exhibit the similar scaling behavior (not shown here).We now discuss how the evolution morphology depends on the number of disordered sites, N_1. Fig. <ref>a shows the scaled correlation function for three different values of N_1 at t=1.6 × 10^6 MCS when the system is already in the scaling regime (see the evolution snapshots in Fig. <ref>). The scaled correlation function for a pure binary mixture (denoted in the black symbols) is also included as a reference. Our results suggest that the data sets do not collapse onto a master function and therefore, does not belong to the same dynamical universality class. Thus, the scaling functions clearly depend upon the number of disordered sites, N_1. In Fig. <ref>b, we present the scaling plot of the structure factor, S(k,t)L^-2 vs. kL on a log-log scale, corresponding to the data sets in Fig. <ref>a. For the pure system, the structure factor tail obeys the Porod's law, S(k,t)∼ k^-(d+1) (indicated by the black symbols) as there are large regions of pure phases separated by sharp interfaces <cit.>. A black solid line shows the slope (-3) of the structure factor tail. The structure factor data at three different values of N_1 =2%, 5%, and 10% are demonstrated by the red, green, and blue curves, respectively. Corresponding slopes of the structure factor tail are -2.2 (red dashed line), -0.92 (green dashed line), and -0.48 (blue dashed line), respectively. A deviation of the structure factor tail from the Porod's law to a lower noninteger exponent suggest a fractal architecture in the domains or interfaces as a consequence of interfacial roughening caused by quenched disorder <cit.>. Notice that the structure factor peak shifted to a lower k values with increasing N_1 that correspond to a large-scale structure in the system which is evident in Fig. <ref>d.This further confirms the N_1 dependent scaling functions.The results of the time dependence of average domain size L(t) vs. t are displayed in Fig. <ref> for the morphologies shown in Fig. <ref>. For the pure case (ϵ=0), coarsening morphology follows the standard Lifshitz-Slyozov (LS) growth law: L(t)∼ t^1/3 (black symbols); the black solid line represent the expected growth exponent (ϕ=1/3) in Fig. <ref>a. For all values of N_1 ≠ 0, our data clearly follows the LS growth law for an extended period, although, the prefactors of the power-law growth varies with N_1. However, on the time scale of our simulation, the domain growth law for N_1=10% crosses over to the saturation beyond t>10^6, which is a sign of the presence of frozen morphologies. Another concurrent way of extracting the growth law exponent is to define an effective growth exponent <cit.>,ϕ_eff(t)= log_αL(α t)/log_αL(t).Here, we chose α=10 <cit.>. The corresponding plots of time variation of ϕ_eff are shown in Fig. <ref>b. Notice that in the pure case (ϵ=0), numerical growth exponent data consistent with LS growth exponent. However, for other cases, the asymptotic growth exponents slightly deviate from the expected values with N_1.Overall, we find that the system with the disorder at randomly selected sites follows the expected LS power-law growth: L(t)∼ t^ϕ with ϕ = 1/3. For a fixed number of disorder sites (N_1), the system displayed the dynamical scaling at various time steps. However, the system deviates significantly from the dynamical scaling for different N_1 values at a fixed time step.§.§ Disorder at regularly selected sites We now examine Case2, where the disorder is introduced at the regularly selected sites by keeping the other numerical details same as in Case1. The entire system consists of N=L_x × L_y sites. The set of indexes i = 1 ⋯ L_x and j = 1 ⋯ L_y, defines the respective positions of the sites in x, and y directions. We sweep the entire lattice sites (N) by tracing all the indexes in y-direction (1 ⋯ Ly) at each fixed i. In the process, every m^th site is selected to introduce the quenched disorder. The total number of disordered sites in the system are N_1=N/m. We investigate the domain morphologies and the corresponding scaling properties by varying the number of disordered sites (N_1) and compare them with the pure case (ϵ=0) as described for the Case1. Fig. <ref> shows the evolution morphologies at t=4 × 10^5 and t=1.6 × 10^6 MCS for the number of disordered sites (a) N_1= 0 (0%), (b) N_1= N/50 (2%), (c) N_1= N/20 (5%), and (d) N_1= N/10 (10%), respectively. After the temperature quench, A-rich (marked in blue) and B-rich (unmarked) domains started growing with the passage of time. In this process, where we select disordered sites in a regular manner,stripped pattern morphology is observed (see Fig. <ref>b-d). In Fig. <ref>d, we find that with N_1=10% the evolution of stripped pattern resulting in a lamellar pattern at late times. Furthermore, we believe that even with a lower number of disorder sites (N_1=2% and 5%) lamellar pattern could be observed at late times t≫1.6× 10^6 MCS (see Fig. <ref>b-c), whereas such lamellar patterns occurred earlier for N_1=10%. The reason for the stripped pattern could be due to the melting of domains near the regularly chosen disordered sites for which J=0. Hence the evolution of such systems leads to the stripe/lamellar pattern formations. Fig. <ref>b-d also reveals the dependence of stripe orientation on the number of disordered sites, N_1. Thus, by the combination of phase separation phenomenon of a binary mixture and the introduction of disorder at the regularly selected sites, one can guide the typical morphology of the coexisting A and B phases into an ordered stripped/lamellar pattern.Next, we present the scaling plots of the correlation function (C(r,t) vs. r/L(t) in Fig. <ref>a) and the structure factor (S(k,t)L^-2 vs. kL in Fig. <ref>b), defined in Eq. (<ref>). Fig. <ref> corresponds to the morphologies shown in Fig. <ref>c with 5% disordered sites. We plot the scaling functions at three time instants as indicated by the symbols. The dynamics regarding the correlation function and the structure factor at different times has shown a perfect congruence with each other witnessing the universality in their behavior as well as confirming the validity of dynamical scaling. We also observed that unlike the previous case, here the structure factor data obeys the Porod's law (S(k,t)∼ k^-3 as k →∞) which results from scattering off sharp interfaces.We now discuss whether the evolution morphology depends on the number of disordered sites present in the system. Fig. <ref> shows a comparison of the scaled correlation function and the corresponding structure factor at four different percentages of disordered sites (N_1=0%, 2%, 5% and 10%) for t=1.6× 10^6 MCS. At lower values of N_1, particularly at (2% and 5%), excellent data collapse with the pure case (N_1=0%) suggest that they belong to the same dynamical universality class i.e. the morphologies are equivalent and their statistical properties are independent of N_1. However, for N_1=10% the interconnected morphology of A and B phases transformed into an ordered lamellar pattern, hence the deviation from the dynamical scaling. Notice that the scaled correlation function for N_1=10% (shown by the blue symbols) in Fig. <ref> exhibits a crossover due to the formation of lamellar morphology. In Fig. <ref>b, the structure factor data sets also manifest the excellent data collapse on the master curve for N_1=0%, 2%, 5%. However, notice that the structure factor for N_1=10% shows a distinct shoulder, which characterizes the lamellar structure in Fig. <ref>d. The scaled structure factor shows a Porod tail S(k,t)∼ k^-(d+1) as k →∞ for all the values N_1.Finally, we turn our attention to the time dependence of domain size for the evolution shown in Fig. <ref>. We plot L(t) vs. t on a log-log scale in Fig. <ref>a for various N_1 values. The corresponding plots of ϕ_eff vs. t are shown in Fig. <ref>b. We find that, after an initial transient, our data is consistent with the power-law growth for all the percentages of disorder introduced at regularly selected sites. The slight upward trend of the curves for N_1≠ 0 in the log-log plot suggest that the growth cannot be slower than a power-law growth. This is verified in Fig. <ref>b where we show that the variation of effective growth exponent with the number of disordered sites.§ CONCLUSIONS We have undertaken extensive Monte Carlo simulations to study the segregation kinetics in binary mixtures with bond-disorder. Our studies are based on kinetic Ising model with the conserved (Kawasaki) spin-exchange dynamics. We presented results for two different cases of introducing bond-disorder in the system: (i) at randomly selected sites, and (ii) at regularly selected sites, where the exchange interaction J=1-ϵ with ϵ =1 and remaining sites have ϵ = 0. We discussed the characteristic features of domains morphologies of phase separating (AB) mixtures with critical composition (50% A and 50% B) for a broad range of percentages of the disorder sites N_1= 0%, 2%, 5%, and 10%. When the disorder is incorporated at randomly selected sites (Case1), the scaling functions C(r,t) and S(k,t) appear to be dependent on the number of disordered sites. We observe that the domain growth law is always consistent with the Lifshitz-Slyozov (LS) growth law. However, on the time scale of our simulation, the data for a higher number of disorder sites (10%) have crossed over to a saturation regime. We have not accessed this crossover regime for lower percentages of disorder sites; nevertheless, we cannot rule out the possibility of saturation of growth law at even later times than those investigated here. Next, in the Case2 where we introduced disorder at sites selected in a regular manner, evolution morphologies lead to a stripped/lamellae pattern formation. In this case, for the lower percentages (2% and 5%) of disorder sites, domains morphologies, which are mostly connected stripes, showing a good scaling behavior. Whereas for 10% disordered sites, these system does not fall into the same universality class as the morphology is now a lamellar pattern. Hence, we observed a corresponding crossover in the scaling functions. The domain growth law, in this case, is also consistent with LS growth law on the time scale of our simulation as in the Case1. Overall, we believe that the results presented here will provoke a fresh interest in this significant problem, particularly, the experimental studies on the kinetics of phase separation in disordered binary mixtures.§ ACKNOWLEDGEMENTSA.S. is thankful to Prof. Sanjay Puri, SPS, JNU, India for the fruitful discussion and providing the computational facilities. A.S. is grateful to CSIR, New Delhi for the financial support. A.C. acknowledges the financial support from grant number BT/BI/03/004/2003(C) ofGovernment of India Ministry of Science and Technology, Department of Biotechnology, Bioinformatics division, and DST-PURSE grant given to JNU by the Department of Science and Technology, Government of India.120 ab94A. J. Bray, Adv. Phys., 1994, 43, 357-459.pwphi9S. Puri and V. Wadhawan (eds.), Kinetics of Phase Transitions, CRC Press, Boca Raton, FL, 2009.bf2013K. Binder and P. Fratzl, in Materials Science and Technology, 2013, Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, Germany.aophi2A. 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"authors": [
"Awaneesh Singh",
"Amrita Singh",
"Anirban Chakraborti"
],
"categories": [
"cond-mat.stat-mech"
],
"primary_category": "cond-mat.stat-mech",
"published": "20170427063555",
"title": "Effect of bond-disorder on the phase-separation kinetics of binary mixtures: a Monte Carlo simulation study"
} |
http://arxiv.org/abs/1704.07988v1 | {
"authors": [
"Zihuan Wang",
"Ming Li",
"Xiaowen Tian",
"Qian Liu"
],
"categories": [
"cs.IT",
"math.IT"
],
"primary_category": "cs.IT",
"published": "20170426070903",
"title": "Joint Hybrid Precoder and Combiner Design for mmWave Spatial Multiplexing Transmission"
} |
|
http://arxiv.org/abs/1704.08454v1 | {
"authors": [
"Denis Lacroix",
"Antoine Boulet",
"Marcella Grasso",
"C. -J. Yang"
],
"categories": [
"nucl-th",
"cond-mat.quant-gas",
"cond-mat.str-el",
"nucl-ex"
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"primary_category": "nucl-th",
"published": "20170427071707",
"title": "From bare interactions, low--energy constants and unitary gas to nuclear density functionals without free parameters: application to neutron matter"
} |
|
[email protected] I. I. Mechnikov Odesa National University, Dvoryanska str, 2, Odesa, Ukraine, 65082 In this paper we develop conditions for various types of stability in social networks governed by Imitation of Success principle. Considering so-called Prisoner's Dilemma as the base of node-to-node game in the network we obtain well-known Hopfield neural network model. Asymptotic behavior of the originalmodel and dynamic Hopfield model have a certain correspondence. To obtain more general results, we consider Hopfield model dynamic system on time scales.Developed stability conditions combine main parameters of network structure such as network size and maximum relative nodes' degree with the main characteristics of time scale, nodes' inertia and resistance, rate of input-output response.time scale stability asymptotical stability Hopfield neural network social network Prisoner's Dilemma 34N05 37B25 91D30§ INTRODUCTIONA social network is the set of people or groups of people with some pattern of links or interconnection between them. Processes taking place on social networks often may be interpreted as information transition.The aim of this paper is to consider asymptotic properties of collective opinion formation in social networks with general topology. Transition of opinion between linked nodes will be modelled by game-theoretical mechanism. Total payoff may be a key factor to choose one of the two alternative strategies, cooperation or defection in opinion propagation.Such type of dynamics is called Imitation of Success.An opposite (in some sense) kind of model is for example the Voter Model.Last named model assume imitation of a behavior of uniformly random chosen neighbor node and games payoff has no affects on state updating of the particular node. In paper <cit.> was considered model called WeakImitation of Success. This updating rule is mixture of IS and VM rules: dependently of some parameter ε behavior of updating node may be close to one of the two types of dynamics.The analysis of the total payoff function for so-called Prisoner's Dilemma leads us to well-known Hopfield neural network model <cit.>. Asymptotic behavior of the direct node-to-node model and dynamic Hopfield model have a certain correspondence. To obtain more general results, we consider Hopfield model on time scale. This problem is discussed in detail in <cit.>, but we develop more direct and precise conditions for stability of the social network behavior.§ PRELIMINARY RESULTS. We now present some basic information about time scales according to <cit.>. A time scale is defined as a nonempty closed subset of the set of real numbers and denoted by 𝕋.The properties of the time scale are determined by the following three functions: (i) the forward-jump operator: σ(t) = inf{s ∈𝕋: s > t}; (ii) the backward-jump operator: ρ(t) = sup{s ∈𝕋: s < t } (in this case, we set inf∅= sup𝕋 and sup∅ = inf𝕋); (iii) the granularity function μ(t) = σ(t) - t. The behavior of the forward- and backward-jump operators at a given point of the time scale specifies the type of this point. The corresponding classification of points of the time scale is presented in Table 1. We define a set 𝕋^κ in the following way:𝕋^κ = 𝕋∖{M}, if ∃ right scattered pointM ∈𝕋: M = sup𝕋, sup𝕋<∞ 𝕋 , otherwise.In what follows, we set [a, b] = {t ∈𝕋: a ⩽ t ⩽ b }. Let f:𝕋→ℝ and t ∈𝕋^κ. The number f^Δ(t) is called Δ-derivative of function f at the point t, if ∀ε > 0 there exists a neighborhood U of the point t(i. e., U = (t - δ, t + δ) ∩𝕋, δ < 0) such that f(σ(t)) - f(s) - f^Δ(t)(σ(t)-s)⩽εσ(t) - s∀ s ∈ U. If f^Δ(t) exists ∀ t ∈𝕋^κ, then f:𝕋→ℝ is called Δ-differentiable on𝕋^κ. The function f^Δ(t): 𝕋^κ→ℝ is called the delta-derivative of a function f on𝕋^κ. If f is differentiable with respect to t then f(σ(t)) = f(t) + μ(t)f^Δ(t). The function f:𝕋→ℝ is called regular if it has finite right limits at all right-dense points of the time scale 𝕋 and finite left limits at all points left-dense points of 𝕋. The function f:𝕋→ℝ is called rd-continuous if it is continuous at the right-dense points and has finite left limits at the left-dense points.The set of these functions is denoted by C_rd = C_rd(𝕋) = C_rd(𝕋; ℝ). The indefinite integral on the time scale takes the form∫f(t) Δ t = F(t) + C,where C is integration constant and F(t) is the preprimitive for f(t). If the relation F^Δ(t) = f(t) where f:𝕋→ℝis an rd-continuous function, is true for all t ∈𝕋^κ thenF(t) is called the primitive of the function f(t). If t_0 ∈𝕋 then F(t) = ∫_t_0^tf(s) Δ s for all t. For all r,s ∈𝕋 the definite Δ-integral is defined as follows:∫_r^sf(t)Δ t = F(s) - F(r). For any regular function f(t) there exists a function F differentiable in the domain D and such that the equality F^Δ(t) = f(t) holds for all t ∈ D. This function is defined ambiguously. It is called the preprimitive of f(t).A function p:𝕋→ℝ is called regressive (positive regressive) if1 + μ(t)p(t) ≠ 0,(1 + μ(t)p(t) > 0),t ∈𝕋^κ.The set of regressive (positive regressive) and rd-continuous functions is denoted byℛ = ℛ(𝕋) (ℛ^+ = ℛ^+(𝕋)).For any p, q ∈ℛ by definition put(p ⊕ q)(t) = p(t) + q(t) + p(t)q(t)μ(t),t ∈𝕋^κ. It is easy to see that the pair (ℛ, ⊕) is an Abelian group.As shown in [Bohner Peterson], a function p from the class ℛ can be associated with a function e_p(t, t_0) which is the unique solution of Cauchy problemy^Δ = p(t)y,y(t_0) = 1.The function e_p(t,t_0) is an analog, by its properties, of the exponential function defined on ℝ.Let us consider dynamic system on time scale 𝕋:x^Δ = f(t, x) x(t_0) = x_0.In following formulations we denote solution of (<ref>) by x(t; t_0, x_0). The equilibrium state x = x^* of the system (<ref>) is uniformly stable if ∀ε > 0 there exists δ = δ(ε), such that x_0 - x^* < δx(t; t_0, x_0) - x^* < ε, ∀ t ∈ [t_0, +∞]_𝕋, t_0 ∈𝕋. The equilibrium state x = x^* of the system (<ref>) is uniformly asymptotically stable if it is uniformly stable and there exists Δ > 0 such that x_0 - x^* < Δlim_t → +∞x(t; t_0, x_0) - x^* = 0, ∀ t_0 ∈𝕋. The equilibrium state x = x^* of the system (<ref>) is uniformly exponentially stable if there exist constants α, β > 0 (β∈ℛ^+) such that x(t, t_0, x_0)⩽x(t_0)α e_-β(t, t_0), t ⩾ t_0 ∈𝕋, for all t_0 ∈𝕋 and x(t_0) ∈ℝ^n. In further definitions and theorems <ref> – <ref>we assume f(t, 0) = 0 for all t ∈𝕋, t ⩾ t_0 and x_0 = 0 so that x = 0 is a solution to equation (<ref>). For more details see <cit.>. A function ψ : [0, r] →[0, ∞) is of class 𝒦 if it is well-defined, continuous, and strictly increasing on [0, r] with ψ(0) = 0. A continuous function P : ℝ^n →ℝ with P(0) = 0 is called positive definite (negative definite) on D if there exists a function ψ∈𝒦, such that ψ(x) ⩽ P(x) (ψ(x) ⩽ -P(x)) for x ∈ D. A continuous function P : ℝ^n →ℝ with P(0) = 0 is called positive semidefinite (negative semidefinite) on D if P(x) ⩾ 0 (P(x) ⩽ 0) for x ∈ D. A continuous function Q : [t_0, ∞)×ℝ^n →ℝ with Q(t, 0) = 0 is called positive definite (negative definite) on [t_0, ∞) × D if there exists a function ψ∈𝒦, such that ψ(x) ⩽ Q(t, x) (ψ(x) ⩽ -Q(t, x)) for t ∈𝕋, t ⩾ t_0, x ∈ D. A continuous function Q : [t_0, ∞)×ℝ^n →ℝ with Q(t, 0) = 0 is called positive semidefinite (negative semidefinite) on [t_0, ∞) × D if Q(t, x) ⩾ 0 (Q(t, x) ⩽ 0) for t ∈𝕋, t ⩾ t_0, x ∈ D. In what follows by V^Δ(t,x) we denote the full Δ-derivative forfunction V(x(t)) along solution of (<ref>). If there exists a continuously differentiable positive-definite function V in a neighborhood of zero with V^Δ(t, x) negative semidefinite, then the equilibrium solution x = 0 of equation (<ref>) is stable. If there exists a continuously differentiable, positive definite function V in a neighborhood of zero and there exists a ξ∈ C_rd([t_0, ∞); [0, ∞)) and a ψ∈𝒦, such that V^Δ(t, x) ⩽ -ξ(t)ψ(x), where lim_t →∞∫_t_0^tξ(s) Δ s = ∞, then the equilibrium solution x = 0 to equation (<ref>) is asymptotically stable. If there exists a continuously differentiable, positive definite function V in a neighborhood of zero and there exists a ξ∈ C_rd([t_0, ∞); [0, ∞)) and a ψ∈𝒦, such that V^Δ(t, x) ⩽ξ(t)ψ(x), where (<ref>) holds, then the equilibrium solution x = 0 to equation (<ref>) is unstable.Here and elsewhere we shall use spectral matrix norm as a norm by default:A_2 = √(λ_max(A^^*A)). Now we formulate a base model for Hopfield network dynamics and few important results about stability of its solutions. Indeed, let us consider dynamic equation of the typex^Δ(t) = -Bx(t) + Ag(x(t)) + J,where t ∈𝕋, sup𝕋 = +∞, x(t) ∈ℝ^n,A = (a_ij), i,j = 1,n, B = diag(b_i),b_i > 0, i = 1,n, J = (J_1, …, J_n)^T,g(x) = (g_1(x_1), …, g_n(x_n))^T. Also, b̅ = max_i{b_i}, b = min_i{b_i}. Conceptual meaning of model's components will be clarified below. We assume on system (<ref>) as follows. S_1. The vector-function f(x) = -Bx + Ag(x) + J is regressive. S_2. There exist positive constants M_i > 0, i = 1,n, such that |g_i(x)| ⩽ M_i for all x ∈ℝ. S_3. There exist positive constants λ_i > 0, i = 1,n such that |g_i(x') - g_i(x”)| ⩽λ_i |x' - x”| for all x', x”∈ℝ. In what follows we denote Λ = diag(λ_i), L = max_iλ_i. An n × n matrix A that can be expressed in the form A = sE - B, where E is an identity matrix, B = (b_ij) with b_ij⩾ 0, 1 ⩽ i, j ⩽ n, and s ⩾ρ(B), the maximum of the moduli of the eigenvalues of B, is called an M-matrix. It should be noted that M-matrix can be characterized in many other ways. Detailed description of forty such ways one can find in <cit.>. For our purpose we find useful the following definition. An n × n matrix A with non-negative diagonal elements and non-positive off-diagonal ones is called M-matrix when real part of each eigenvalue of A is positive. <cit.> Let assumption S_3 be fulfilled. If for every fixed t ∈𝕋 the matrix (I - μ(t)B)Λ^-1 - μ(t)|A| is an M-matrix, the function f(x) = -Bx + Ag(x) + J is regressive. <cit.> If for system (<ref>) conditions S_1 - S_3 are satisfied then there exists an equilibrium state x = x^* of system (1) and moreover, x^*⩽ r_0, where r_0 = ( ∑_i=1^n 1b_i^2( ∑_j=1^nM_j|a_ij| + |J_i| )^2 )^1/2. Besides, if the matrix BΛ^-1 - |A| is an M-matrix, this equilibrium state is unique. And last result we need is so-called Gershgorin circle theorem.Let A be a complex n× n matrix, with entries a_ij.For i∈{1,… ,n} let ρ_i be the sum of the absolute values of the non-diagonal entries in the i-th row.Let D(a_ii, ρ_i) be the closed disc centered at a_ii with radius ρ_i. Such a disc is called a Gershgorin disc. Every eigenvalue ν of A lies within at least one of the Gershgorin discs D(a_ii, ρ_i), i. e. there exists i ∈{1, …, n} such that |ν - a_ii| ⩽ρ_i = ∑_j ≠ i|a_ij|.§ MAIN RESULTS §.§ Node-to-node game setup.Let us define a set of nodes V = {1, 2, …, n}. Each member of V is interpreted as player in some matrix game with its neighbors.This game repeats at time steps, discrete or continuous. Set of i-th node neighbors we denote by Ω_i, k_i = |Ω_i|. Here we consider only one type of matrix game is known as Prisoner's Dilemma. Each node has two strategies: cooperate (C) and defect (D).Payoff matrix is illustrated below:P = [ b-c b;-c 0 ].Here b is a benefit provided by node to its co-player, c is a cost of cooperation and hereafter we assume b > c. In this case the strategy of mutual defection is the only Nash equilibrium, while mutual cooperation is more acceptable social outcome. Current state of i-th node at moment t we denote by S_i(t) ∈{0,1}, where zero state represent the defection strategy. Easy to show that at theinstant of time t node i gets total payoff equal to-k_icS_i(t) + ∑_j ∈Ω_ibS_j(t). This equation remains correct regardless of the nature of time.Hence in what follows we assume t ∈𝕋, where 𝕋 is time scale.§.§ Hopfield network setup. Assume reaction of each node in network is governed by simple threshold rule:S_i(t) = 0,if -k_icS_i(t) + ∑_j ∈Ω_ibS_j(t) < U_i, 1,if -k_icS_i(t) + ∑_j ∈Ω_ibS_j(t)⩾ U_i,where U_i is individually payoff threshold for cooperation. With the aim of using Hopfield neurons model we transform last threshold rule to the rule with continuous responses. In the end this transformation will lead us todynamical system on time scale modelling asymptotic behavior of network.Let the state variable S_i for i-th “neuron” have the range [0, 1] and be a continuous and strictly increasing function of the total payoff u_i.In biological terms S_i and u_i are output and input signal of i-th “neuron” respectively.Input–output relation we denote by g_i(u_i), so S_i(t) = g_i(u_i(t)) and u_i(t) = g^-1(S_i(t)). If some node having non-zero payoff abruptly loses all connections in the network, its behavior may be described by simple dynamic equation:C_i u_i^Δ(t) = -u_i(t)R_i, u_i(0) = u_i0.By analogy with electrical circuit theory in last equation C_i is calledcapacitance of i-th node and R_i is called its resistance.Obviously, without communication node's payoff will decay to zero as individual intention of cooperation do. With communication the node gets additional payoff playing with its neighbours, so dynamic equation becomes as follows:C_i u_i^Δ(t) = -c k_i S_i(t) + ∑_j ∈Ω_ib S_j(t) - u_i(t)R_i, i = 1,n,or, using input–output relation and adjacency matrix of network D = (d_ij), i,j = 1,n,C_i u_i^Δ(t) = -c k_i g_i(u_i) + ∑_jbd_ij g_j(u_j) - u_iR_i, i = 1,n. Let us introduce two matrices:A = [ -k_1cC_1 bd_12C_1… bd_1nC_1; bd_21C_2 -k_2cC_2… bd_2nC_2;…………; bd_n1C_n bd_n2C_n… -k_ncC_n ], B = [ 1R_1C_1 0 … 0; 0 1R_2C_2 0 …; …; 0 0 … 1R_nC_n ].Then we can rewrite equation in vector form:u^Δ(t) = -Bu(t) + Ag(u(t)). It is easy to see that without significant changes in arguments we can consider more general model with constant input for every node in network.By denoting this input as vector J = (J_i), we finally get our main equation as follows:u^Δ(t) = -Bu(t) + Ag(u(t)) + J.§.§ Stability condition for network game For system (<ref>) assume that conditions S_1-S_3 are valid and k_i < λ_i/R_i(c + b),i = 1,n. Then there exists unique equilibrium state u = u^* of system (<ref>) and u^*⩽ r_0. Let us show Q = BΛ^-1 - |A| be an M-matrix. Using inequality for number of neighbors it is easy to derive estimation for real part of eigenvalues ν of matrix Q. Indeed, by Gershgorin circle theorem Reν > 0 if and only if Q_ii - ρ_i > 0 for all i=1,n , where Q_ii = λ_i/R_iC_i - k_i c/C_i, ρ_i = ∑_j ≠ i|Q_ij| = b/C_i∑_j ≠ i d_ij = k_i b/C_i. We can write Q_ii - ρ_i = λ_i/R_iC_i - k_i c/C_i - k_i b/C_i = λ_i/R_iC_i - k_ic+b/C_i > 0 and it is obvious that the upper bound for k_i in theorem's statement does guarantee last inequality. Hence Q is an M-matrix and now to end the proof it remains to apply lemma <ref>. It is interesting to notice that existence of unique stable state in network does not depend on nodes' “capacitance”. For every particular node in network ratio λ_i/R_i describes its potential activity. If λ_i > k_iR_i then node's output reaction can conquer withits overall “resistance” to accept neighbor's behaviour. Notice that in theorem key role plays overall payoff scale of the game expressed as sum b + c. It is particularly remarkable that existence of the unique equilibrium state in network is robust against vanishing of any particular node. Indeed, nodes fulfil conditions of theorem <ref> independently. So if there exists the unique equilibrium state, vanishing any particular node does not affect on the conditions for all other nodes. On the other hand, a new node may easily violate conditions of theorem and break the existence of equilibrium. In two theorems below we use Lyapunov method to formulate sufficient conditions for asymptotic stability of stable state in network dynamics. Let u = u^* bethe unique stable state of (<ref>), i. e. -Bu^* + Ag(u^*) + J = 0.By introducing new variable z = u - u^* we obtain dynamical system on time scale 𝕋z^Δ(t) = -Bz(t) + Ah(z(t)),where h(z) = g(z + u^*) - g(u^*). If conditions S_1-S_3 are valid for system (<ref>), it is easy to see that Σ_1. The vector-function f(z) = -Bz + Ah(z) is regressive. Σ_2. |h_i(z)| ⩽ 2M_i, i = 1,n for all z ∈ℝ. Σ_3. |h_i(z') - h_i(z”)| ⩽λ_i |z' - z”|, i = 1,n for all z', z”∈ℝ.Conditions Σ_1–Σ_3 guarantee existence and uniqueness for solution of (<ref>) on t ∈[t_0, +∞) for any initial values z(t_0) = z_0. Under the conditions of theorem <ref> assume that sup𝕋 = +∞ and μ(t) ⩽μ^* for all t ∈𝕋. If inequality √(n)(b + c)max_1 ⩽ i ⩽ nk_iC_i⩽- 1 - μ^*b̅ + √(1 + 2μ^*(b̅ + b))/μ^*L holds, then unique equilibrium state u = u^* of system (<ref>) is uniformly asymptotically stable. Clearly, stability of the trivial solution z = 0 of (<ref>) is equivalent to stability of the stable state u^* of (<ref>). Let us choose the V(z) = z^Tz as a Lyapunov function. It can easily be checked that V(z) is positive definite. If z(t) is Δ-differentiable at the moment t ∈𝕋^κ, the full Δ-derivative of V(z(t)) along solution of (<ref>) be as follows V^Δ(z(t)) = (z^T(t) z(t))^Δ = z^T(t)z^Δ(t) + [z^T(t)]^Δ z(σ(t)) = = z^T(t)z^Δ(t) + [z^T(t)]^Δ[z(t) + μ(t)z^Δ(t)] = = 2z^T(t)[-Bz(t) + Ah(z(t))] + μ(t)-Bz(t) + Ah(z(t))_2^2. Since B is diagonal matrix with all positive diagonal elements, it follows that maximal eigenvalue of B is b̅ = max_i{b_i} and the same one of -B is b = min_i{b_i}. Using properties of matrix and vector norms and the fact that h(z(t))⩽ Lz(t) it's easy to obtain following estimation: V^Δ(z(t)) ⩽ -2bz(t)^2 + 2z(t)A_2h(z(t)) + μ(t)(b̅z(t) + A_2h(z(t)))^2 ⩽ -2bz(t)^2 + 2LA_2z(t)^2 + μ(t)(b̅z(t) + LA_2z(t))^2 ⩽ -(2b - 2LA_2 - μ(t)(b̅ + LA_2)^2) z(t)^2. It is obvious that ψ(z) = z^2 belongs to class 𝒦. Let us prove that the function ξ(t) = 2b - 2LA_2 - μ(t)(b̅ + LA_2)^2 under theorems' assumptions belongs to C_rd([t_0, ∞); [0, ∞)) and fulfills condition (<ref>). Indeed, we have ξ(t) ⩾ 2b - 2LA_2 - μ^*(b̅ + LA_2)^2. Hence, lim_t →∞∫_t_0^t ξ(s) Δ s= lim_t →∞∫_t_0^t (2b - 2LA_2 - μ(t)(b̅ + LA_2)^2) Δ s ⩾⩾(2b - 2LA_2 - μ^*(b̅ + LA_2)^2) lim_t →∞∫_t_0^t Δ s = ∞. Solving quadratic inequality 2b - 2LA_2 - μ^*(b̅ + LA_2)^2 ⩾ 0 with respect to A_2, we get -1 - μ^*b̅ - √(1 + 2μ^*(b̅ + b))/μ^*L⩽A_2 ⩽-1 - μ^*b̅ + √(1 + 2μ^*(b̅ + b))/μ^*L. Clearly, by definition of matrix norm left inequality always holds. We have A_2 ⩽√(n)A_∞ = √(n)max_1 ⩽ i ⩽ n∑_j=1^n|a_ij| = = √(n)max_1 ⩽ i ⩽ n{bd_i1C_i + … + bd_i,j-1C_i + k_icC_i + bd_i,j+1C_i + … + bd_inC_i} = = √(n)max_1 ⩽ i ⩽ n{k_icC_i + bC_i∑_j≠ id_ij} = = √(n)max_1 ⩽ i ⩽ n{k_icC_i + k_ibC_i} = = √(n)(b + c)max_1 ⩽ i ⩽ nk_iC_i. Now it is easy to see that inequality (<ref>) guarantees non-negativity of ξ(t) and condition (<ref>). To conclude the proof, it remains to use theorem <ref>. We stress that the left side of inequality (<ref>) gathers main parameters of network structure (size and maximum relative nodes' degree). On the other hand, the right side combines main characteristics of time scale, nodes' inertia and resistance, rate of input-output response. Obviously, for any given matrix A inequality (<ref>) can be checked directly. Under the conditions of theorem <ref> assume that sup𝕋 = +∞ and μ(t) ⩽μ^* for all t ∈𝕋. If inequality 2b - 2LA_2 - μ(t)(b̅ + LA_2)^2 ⩾ 0 holds, then unique equilibrium state u = u^* of system (<ref>) is uniformly asymptotically stable. Under the conditions of theorem <ref> assume that sup𝕋 = +∞ and μ(t) ⩽μ^* for all t ∈𝕋. Let C_* denote the minimal “capacitance” in the network and K^* denote the largest node's degree: C_* = min_1 ⩽ i ⩽ n C_i , K^* = max_1 ⩽ j ⩽ nk_j. If inequality (b + c)K^*/C_*⩽- 1 - μ^*b̅ + √(1 + 2μ^*(b̅ + b))/μ^*L holds, then unique equilibrium state u = u^* of system (<ref>) is uniformly asymptotically stable. By repeating the same steps as in previous theorem, we obtain A_2 ⩽-1 - μ^*b̅ + √(1 + 2μ^*(b̅ + b))/μ^*L. Now if we recall matrix norm inequality A_2^2 ⩽A_1 ·A_∞, we get A_2^2 ⩽A_1 ·A_∞ = = max_1 ⩽ j ⩽ n∑_i=1^n|a_ij| ·(b + c)max_1 ⩽ i ⩽ nk_iC_i⩽= max_1 ⩽ j ⩽ n{bd_1jC_1 + … + bd_j-1,jC_j-1 + k_jcC_j + bd_j+1,jC_j+1 + … + bd_njC_n}·(b + c)max_1 ⩽ i ⩽ nk_iC_i⩽= max_1 ⩽ i ⩽ n{k_jcC_j + bC_*∑_i ≠ jd_ij}·(b + c)K^*C_*⩽= max_1 ⩽ i ⩽ n{k_jcC_* + bk_jC_*}·(b + c)K^*C_*⩽= (b + c)K^*C_*·(b + c)K^*C_* = (b + c)^2(K^*C_*)^2. It is obvious that inequality (<ref>) guarantees non-negativity of ξ(t) and condition (<ref>). To conclude the proof, it remains to use theorem <ref>. For large network, i. e. n ≫ 1, size-dependent condition (<ref>) is unlikely to be fulfilled. In the same time condition (<ref>) can be valid regardless of network's size. Under the conditions of theorem <ref> assume that -b∈ℛ^+ and b - LA_2 > 0, where b = min_i{b_i}. Then 1) solution z = 0 of the following system is exponentially stable: z^Δ(t) = -Bz(t),z(t_0) = z_0,t ⩾ t_0 ∈𝕋; 2) unique equilibrium state z = 0 of the system (<ref>) is exponentially stable on t ⩾ t_0 ∈𝕋 and the following estimation holds: z(t)⩽z_0·e_-(b - LA_2)(t, t_0). Since B = diag(b_i) it is easy to obtain fundamental matrix Φ_-B(t, t_0) = diag(e_-b_i(t, t_0)). Φ_-B(t, t_0) = √(λ_max(Φ_-B^TΦ_-B)) = √(λ_max(diag(e^2_-b_i(t, t_0)))) = = max_1 ⩽ i ⩽ n|e_-b_i(t, t_0)| = e_-b(t, t_0). It proofs 1) (see <cit.>). The solution of (<ref>) satisfies the variation of constants formula <cit.> z(t) = Φ_-B(t, t_0)z_0 + ∫_t_0^t Φ_-B(t, σ(s))Ah(z(s)) Δ s. Hence we have z(t) ⩽Φ_-B(t, t_0) z_0 + ∫_t_0^t Φ_-B(t, σ(s)) · A h(z(s)) Δ s ⩽⩽ e_-b(t, t_0)z_0 + ∫_t_0^t e_-b(t, σ(s)) ·A_2 h(z(s)) Δ s ⩽⩽ e_-b(t, t_0)z_0 + ∫_t_0^t e_-b(t, s)/1 - bμ(s)·A_2 Lz(s) Δ s. Multiplying both sides of inequality by 1e_-b(t, t_0) > 0 (due to -b∈ℛ^+) we obtain z(t)/e_-b(t, t_0) ⩽z_0 + ∫_t_0^t LA_2/1 - bμ(s)·e_-b(t, s)/e_-b(t, t_0)·z(s) Δ s = = z_0 + ∫_t_0^t LA_2/1 - bμ(s)·z(s)/e_-b(s, t_0) Δ s. Further, by using Grownall's inequality z(t)/e_-b(t, t_0)⩽z_0· e_LA_2/1 - bμ(s)(t, t_0), or z(t)⩽z_0· e_-b⊕LA_2/1 - bμ(s)(t, t_0) = z_0· e_-(b - LA_2)(t, t_0). By conditions of theorem we have -(b - LA_2) ∈ℛ^+.Therefore, the last estimate means that the solution z = 0 of (<ref>) is exponentially stable. This completes the proof of theorem. Proving Theorems <ref>, <ref> we obtain two variants of majorization for A_2. Both of them can be easily used to finddirect conditions of exponentially stability expressed in the terms of network's structure and nodes' internal properties. Under the conditions of theorem <ref> assume that -b∈ℛ^+ and b/L > min{√(n)(b + c)max_1 ⩽ i ⩽ nk_i/C_i, (b + c)K^*/C_*}. Then unique equilibrium state z = 0 of the system (<ref>) is exponentially stable on t ⩾ t_0 ∈𝕋 and the estimation (<ref>) holds. Obviously, the exponential convergence of the solution for (<ref>) to zero and the solution u(t) for (<ref>) to the unique equilibrium u^* are the same. § DISCUSSION.In this paper we developed conditions for various types of stability in social networks governed by Imitation of Success principle.There isn't direct, one-to-one correspondence between considered Hopfield neural network model and original game-based model.Hence all obtained results can be considered only as the base of understanding of opinion propagation in social network.We limited ourselves to the one type of node-to-node game, Prisoner's Dilemma. Moreover, arguing we deliberately choose few key elements such as M-matrixcharacterization, spectral norm estimation etc. Choosing this elements we were guided by the aim to obtain simple, fast-checkable and meaningful conditions.It is an open problem to study network dynamics based on the another interestingmatrix game types. Perhaps, taking into account network's topology or considering particular type of time scale one can develop more specific, precise, and useful results.elsarticle-harv | http://arxiv.org/abs/1704.08171v1 | {
"authors": [
"Aleksey Ogulenko"
],
"categories": [
"math.DS",
"34N05, 37B25, 91D30"
],
"primary_category": "math.DS",
"published": "20170426155203",
"title": "Asymptotical properties of social network dynamics on time scales"
} |
[email protected] LIGO Laboratory, California Institute of Technology, Pasadena, CA 91125, USA The Pennsylvania State University, University Park, PA 16802, USA Carleton College, Northfield, MN 55057, USA Artemis, Université Côte d'Azur, Observatoire Côte d'Azur, CNRS, CS 34229, F-06304 Nice Cedex 4, France LIGO Laboratory, California Institute of Technology, Pasadena, CA 91125, USA University of Minnesota, Minneapolis, MN 55455, USA Artemis, Université Côte d'Azur, Observatoire Côte d'Azur, CNRS, CS 34229, F-06304 Nice Cedex 4, France Centre Scientifique de Monaco, 8 Quai Antoine 1er, MC 98000, Monaco Artemis, Université Côte d'Azur, Observatoire Côte d'Azur, CNRS, CS 34229, F-06304 Nice Cedex 4, France King's College London, University of London, Strand, London WC2R 2LS, UK Carleton College, Northfield, MN 55057, USA School of Physics and Astronomy, Monash University, Clayton, Victoria 3800, Australia OzGrav: The ARC Centre of Excellence for Gravitational-Wave Discovery, Hawthorn, Victoria 3122, AustraliaThe direct observation of gravitational waves with Advanced LIGO and Advanced Virgo offers novel opportunities to test general relativity in strong-field, highly dynamical regimes. One such opportunity is the measurement of gravitational-wave polarizations. While general relativity predicts only two tensor gravitational-wave polarizations, general metric theories of gravity allow for up to four additional vector and scalar modes. The detection of these alternative polarizations would represent a clear violation of general relativity. The LIGO-Virgo detection of the binary black hole merger GW170814 has recently offered the first direct constraints on the polarization of gravitational waves. The current generation of ground-based detectors, however, is limited in its ability to sensitively determine the polarization content of transient gravitational-wave signals. Observation of the stochastic gravitational-wave background, in contrast, offers a means of directly measuring generic gravitational-wave polarizations. The stochastic background, arising from the superposition of many individually unresolvable gravitational-wave signals, may be detectable by Advanced LIGO at design-sensitivity. In this paper, we present a Bayesian method with which to detect and characterize the polarization of the stochastic background. We explore prospects for estimating parameters of the background, and quantify the limits that Advanced LIGO can place on vector and scalar polarizations in the absence of a detection. Finally, we investigate how the introduction of new terrestrial detectors like Advanced Virgo aid in our ability to detect or constrain alternative polarizations in the stochastic background. We find that, although the addition of Advanced Virgo does not notably improve detection prospects, it may dramatically improve our ability to estimate the parameters of backgrounds of mixed polarization.Polarization-based Tests of Gravity with the Stochastic Gravitational-Wave Background Eric Thrane December 30, 2023 =====================================================================================§ INTRODUCTIONThe recent Advanced LIGO-Virgo observations of coalescing binary black holes have initiated the era of gravitational-wave astronomy <cit.>. Beyond their role as astrophysical messengers, gravitational waves offer unique opportunities to test gravity in previously unexplored regimes <cit.>. The direct detection of gravitational waves has already enabled novel experimental checks on general relativity, placing the best model-independent dynamical bound to date on the graviton mass and limiting deviations of post-Newtonian coefficients from their predicted values <cit.>.The measurement of gravitational-wave polarizations represents another avenue by which to test general relativity. While general relativity allows for the existence of only two gravitational-wave polarizations (the tensor plus and cross modes), general metric theories of gravity may allow for up to four additional polarizations: the x and y vector modes, and the breathing and longitudinal scalar modes <cit.>. The effects of all six polarizations on a ring of freely-falling test particles are shown in Fig. <ref>. The detection of these alternative polarization modes would represent a clear violation of general relativity, while their non-detection may serve to experimentally constrain extended theories of gravity. Few experimental constraints exist on the polarization of gravitational waves <cit.>. Very recently, though, the simultaneous detection of GW170814 with the Advanced LIGO and Virgo detectors has allowed for the first direct study of a gravitational wave's polarization <cit.>. When analyzed with models assuming pure tensor, pure vector, and pure scalar polarization, GW170814 significantly favored the purely-tensor model over either alternative <cit.>. This result represents a significant first step in polarization-based tests of gravity. Further tests with additional detectors, though, will be needed to sensitively test general relativity and its alternatives.In particular, many alternate theories of gravity yield signals of mixed polarizations, yielding vector and/or scalar modes in addition to standard tensor polarizations. When allowing generically for all six polarization modes, the three-detector Advanced LIGO-Virgo network is generally unable to distinguish the polarization of transient gravitational-wave signals, like those from binary black holes <cit.>. First, two LIGO detectors are nearly co-oriented, leaving Advanced LIGO largely sensitive to only a single polarization mode <cit.>. Second, even if the LIGO detectors were more favorably-oriented, a network of at least six detectors is generically required to uniquely determine the polarization content of a gravitational-wave transient <cit.>. Some progress can be made via the construction of “null-streams” <cit.>, but this method is infeasible at present without an independent measure of a gravitational wave's source position (such as an electromagnetic counterpart). Future detectors like KAGRA <cit.> or LIGO-India <cit.> will therefore be necessary to break existing degeneracies and confidently distinguish vector or scalar polarizations in gravitational-wave transients. It should be noted that the scalar longitudinal and breathing modes induce perfectly-degenerate responses in quadrupolar detectors like Advanced LIGO and Virgo. Thus a network quadrupolar detectors can at most measure five independent polarization degrees of freedom <cit.>. Beyond the direct detection of binary coalescences, another target for current and future detectors is the observation of the astrophysical stochastic gravitational-wave background, formed via the superposition of all gravitational-wave sources that are too weak or too distant to individually resolve <cit.>. Although the strength of the background remains highly uncertain, it may be detected by Advanced LIGO in as few as two years of coincident observation at design-sensitivity <cit.>.Unlike direct searches for binary black holes, Advanced LIGO searches for long-lived sources like the stochastic background and rotating neutron stars <cit.> are currently capable of directly measuring generic gravitational-wave polarizations without the introduction of additional detectors or identification of an electromagnetic counterpart.The observation of the stochastic background would therefore enable novel checks on general relativity not possible with transient searches using the current generation of gravitational-wave detectors.In this paper, we explore the means by which Advanced LIGO can detect and identify alternative polarizations in the stochastic background. First, in Sect. <ref>, we consider possible theorized sources which might produce a background of alternative polarizations. We note, though, that stochastic searches are largely unmodeled, requiring few assumptions about potential sources or theories giving rise to alternative polarization modes (see, however, Sect. <ref>).In Sect. <ref> we discuss the tools used for detecting the stochastic background and compare the efficacy of standard methods with those optimized for alternative polarizations. In Sect. <ref> we then propose a Bayesian method with which to both detect generically-polarized backgrounds and determine if alternative polarization modes are present. Next, in Sect. <ref> we explore prospects for estimating the polarization content of the stochastic background. We quantify the limits that Advanced LIGO can place on the presence of alternative polarizations in the stochastic background, limits which may be translated into constraints on specific alternative theories of gravity.As new detectors are brought online in the coming years, searches for alternative polarizations in the stochastic background will become ever more sensitive. In both Sects. <ref> and <ref>, we therefore investigate how the addition of Advanced Virgo improves our ability to detect or constrain backgrounds of alternative polarizations. Finally, in Sect. <ref> we ask if our proposed search is robust against unexpectedly complex backgrounds of standard tensor polarizations.§ EXTENDED THEORIES OF GRAVITY AND ALTERNATIVE POLARIZATION MODES Searches for the stochastic background are largely unmodeled, making minimal assumptions about the source of a measured background. Nevertheless, it is interesting to consider which sources might give rise to a detectable background of alternative polarization modes. In this section we briefly consider several possibilities that have been proposed in the literature. We will focus mainly on scalar-tensor theories, which predict both tensor and scalar-polarized gravitational waves <cit.>. Our discussion below is not meant to be exhaustive; there may well exist additional sources that can give rise to backgrounds of extra polarization modes. In particular, we do not discuss possible sources of vector modes, predicted by various alternative theories of gravity (see Ref. <cit.> and references therein). Note that, while advanced detectors may not be sensitive to the sources described below, these sources may become increasingly relevant for third generation detectors (or beyond).Core-collapse supernovae (CCSNe) represent one potential source of scalar gravitational waves. Although spherically-symmetric stellar collapses do not radiate gravitational waves in general relativity, they do emit scalar breathing modes in canonical scalar-tensor theories. While the direct observation of gravitational waves from CCSNe is expected to place strong constraints on scalar-tensor theories <cit.>, only supernovae within the Milky Way are likely to be directly detectable using current instruments <cit.>. Such events are rare, occurring at a rate between (0.6-10.5)×10^-2^-1 <cit.>. The stochastic gravitational-wave background, on the other hand, is dominated by distant undetected sources, and so in principle it is possible that a CCSNe background of breathing modes could be detected before the observation of a single Galactic supernova <cit.>. However, realistic simulations of monopole emission from CCSNe predict only weak scalar emission <cit.>. Nevertheless, certain extreme phenomenological supernovae models predict gravitational radiation many orders of magnitude stronger than in more conventional models <cit.>. According to such models, CCSNe may contribute non-negligibly to the stochastic background.Compact binary coalescences may also contribute to a stochastic background of scalar gravitational waves. In many scalar-tensor theories, bodies may carry a “scalar charge" that sources the emission of scalar gravitational waves <cit.>. Monopole scalar radiation is suppressed due to conservation of scalar charge, but in a general scalar-tensor theory there is generally no conservation law suppressing dipole radiation. Scalar dipole radiation from compact binaries is enhanced by a factor of (v/c)^-2 relative to ordinary quadrupole tensor radiation (where v is the orbital velocity of the binary and c the speed of light), and thus represents a potentially promising source of scalar gravitational waves. Electromagnetic observations of binary neutron stars place stringent constraints on anomalous energy loss beyond that predicted by general relativity; these constraints may be translated into a strong limit on the presence of additional scalar-dipole radiation <cit.>. Such limits, though, are strongly model-dependent, assuming a priori only small deviations from general relativity. Additionally, pure vacuum solutions like binary black holes are not necessarily subject to these constraints. If, for example, the scalar field interacts with curvature only through a linear coupling to the Gauss-Bonnet term, scalar radiation is produced by binary black holes but not by binary neutron stars <cit.>. Alternatively, binary black holes can avoid the no-hair theorem and obtain a scalar charge if moving through a time-dependent or spatially-varying background scalar field <cit.>.A variety of exotic sources may generically contribute to stochastic backgrounds of alternative polarizations as well. Cosmic strings, for instance, generically radiate alternative polarizations in extended theories of gravity and may therefore contribute extra polarization modes to the stochastic gravitational-wave background <cit.>. Another potential source of stochastic backgrounds of alternative polarizations are the so-called “bubble walls” generated by first order phase transitions in the early Universe <cit.>. In scalar-tensor theories, bubbles are expected to produce strong monopolar emission <cit.>. Gravitational waves from bubbles are heavily redshifted, though, and today may have frequencies too low for Advanced LIGO to detect <cit.>. Bubble walls may therefore be a more promising target for future space-based detectors like LISA than for current ground-based instruments.Finally, we note that it is also possible for alternative polarizations to be generated more effectively from sources at very large distances.There are several ways in which this might occur. First, modifications to the gravitational-wave dispersion relation can lead to mixing between different polarizations in vacuum (an effect analogous to neutrino oscillations). This can cause mixing between the usual tensor modes <cit.>, and also between tensor modes and other polarizations <cit.>. Thus alternative polarizations can be generated during propagation, even if only tensor modes are produced at the source. This effect would build with the distance to a given gravitational-wave source. Such behavior is among the effects arising from generic Lorentz-violating theories of gravity <cit.>. While birefringence and dispersion of the standard plus and cross modes have been explored observationally in this context <cit.>, the phenomenological implications of additional polarization modes remain an open issue at present. Secondly, in many alternative theories fundamental constants (such as Newton's constant G) are elevated to dynamical fields; these fields may have behaved differently at earlier stages in the Universe's evolution <cit.>. As a consequence, local constraints on scalar emission may not apply to emission from remote sources. Additionally, it is in principle possible for local sources to be affected by screening mechanisms that do not affect some remote sources <cit.>.§ STOCHASTIC BACKGROUNDS OF ALTERNATIVE POLARIZATIONS The stochastic background introduces a weak, correlated signal into networks of gravitational-wave detectors. Searches for the stochastic background therefore measure the cross-correlation Ĉ(f) ∝s̃_1^*(f) s̃_2(f)between the strain s̃_1(f) and s̃_2(f) measured by pairs of detectors (see Ref. <cit.> for a comprehensive review of stochastic background detection methods).We will make several assumptions about the background. First, we will assume that the stochastic background is isotropic, stationary, and Gaussian. Second, we assume that there are no correlations between different tensor, vector, and scalar polarization modes. We can therefore express the total measured cross-power ⟨Ĉ(f)⟩ as a sum of three terms due to each polarization sector. Finally, we assume that the tensor and vector sectors are individually unpolarized, with equal power in the tensor plus and cross modes and equal power in the vector-x and vector-y modes. This follows from the fact that we expect gravitational-wave sources to be isotropically distributed and randomly oriented with respect to the Earth. In contrast, we cannot assume that the scalar sector is unpolarized. Scalar breathing and longitudinal modes cannot be rotated into one another via a coordinate transformation (as can the tensor plus and cross modes, for instance), and so source isotropy does not imply equal power in each scalar polarization. However, the responses of the LIGO detectors to breathing and longitudinal modes are completely degenerate, and so Advanced LIGO is sensitive only to the total power in scalar modes rather than the individual energies in the breathing and longitudinal polarizations <cit.>.The above assumptions are not all equally justifiable, and may be broken by various alternative theories of gravity. For instance, one should not expect an unpolarized background in any theory that includes parity-odd gravitational couplings, like Chern-Simons gravity <cit.>, even in the absence of non-tensorial modes <cit.>. Furthermore, different polarizations may not be statistically independent, as is the case for the breathing and longitudinal modes in linearized massive gravity <cit.>. Finally, we should expect a departure from isotropy in any theory violating Lorentz invariance, like those within the standard model extension framework <cit.>. These exceptions notwithstanding, for simplicity we will proceed under the assumptions listed above, leaving more generic cases for future work.Under our assumptions, the measured cross-power due to the background is given by <cit.> ⟨s̃_1^*(f) s̃_2(f')⟩ = δ(f-f') γ_a(f) H^a(f),where repeated indices denote summation over tensor, vector, and scalar modes (a∈{T,V,S}). The overlap reduction functions γ_a(f) quantify the sensitivity of detector pairs to isotropic backgrounds of each polarization <cit.> (see Appendix <ref> for details). The functions H^a(f), meanwhile, encode the spectral shape of the stochastic background within each polarization sector.In the left side of Fig. <ref>, we show the overlap reduction functions for the Hanford-Livingston (H1-L1) Advanced LIGO network. The overlap reduction functions are normalized such that γ_T(f) = 1 for coincident and coaligned detectors. For the Advanced LIGO network, the tensor overlap reduction function has magnitude |γ_T(0)| = 0.89 at f=0, representing reduced sensitivity due to the separation and relative rotation of the H1 and L1 detectors. Additionally, the H1-L1 tensor overlap reduction function decays rapidly to zero above f≈64. Standard Advanced LIGO searches for the stochastic background therefore have negligible sensitivity at frequencies above ∼64.Relative to γ_T(f), the H1-L1 vector overlap reduction function γ_V(f) is of comparable magnitude at low frequencies, but remains non-negligible at frequencies above 64 Hz. As a result, we will see that Advanced LIGO is in many cases more sensitive to vector-polarized backgrounds than standard tensor backgrounds. The scalar overlap reduction function, meanwhile, is smallest in magnitude, with |γ_S(0)| a factor of three small than |γ_T(0)| and |γ_V(0)|. Advanced LIGO is therefore least sensitive to scalar-polarized backgrounds. This reflects a generic feature of quadrupole gravitational-wave detectors, which geometrically have a smaller response to scalar modes than to vector and tensor polarizations <cit.>. For an extreme example of the opposite case, see pulsar timing arrays, which are orders of magnitude more sensitive to longitudinal polarizations than standard tensor-polarized signals <cit.>.For comparison, the right side of Fig. <ref> shows the overlap reduction functions for the Hanford-Virgo (H1-V1) baseline. As the separation between Hanford and Virgo is much greater than that between Hanford and Livingston, the Hanford-Virgo overlap reduction functions are generally much smaller in amplitude and more rapidly oscillatory, translating into weaker sensitivity to the stochastic background. Note, however, that the H1-V1 tensor overlap reduction function remains larger in amplitude than H1-L1's at frequencies f≳200, implying heightened relative sensitivity to tensor backgrounds at high frequencies <cit.>.The functions H^a(f) appearing in Eq. (<ref>) are theory-independent; they are observable quantities that can be directly measured in the detector frame. Stochastic backgrounds are not conventionally described by H(f), though, but by their gravitational-wave energy-density<cit.> Ω(f) = 1/ρ_cdρ(f)/dln f,defined as the fraction of the critical energy density ρ_c = 3H_0^2 c^2/(8π G) contained in gravitational waves per logarithmic frequency interval dln f. Here, H_0 is the Hubble constant and G is Newton's constant. Within general relativity, the background's energy-density is related to H(f) via <cit.> Ω(f) = 20π^2/3 H_0^2 f^3 H(f).Eq. (<ref>) is a consequence of Isaacson's formula for the effective stress-energy of gravitational waves <cit.>. Alternate theories of gravity, though, can predict different expressions for the stress-energy of gravitational-waves and hence different relationships between H^a(f) and Ω^a(f) <cit.>. For ease of comparison to previous studies, we will use Eq. (<ref>) to define the canonical energy-density Ω^a(f) in polarization a. If we allow Isaacson's formula to hold, then Ω^a(f) may be directly interpreted as a physical energy density. If not, though, then Ω^a(f) can instead be understood as a function of the observable H^a(f).We will choose to normalize the cross-correlation statistic Ĉ(f) such that ⟨Ĉ(f) ⟩ = γ_a(f) Ω^a(f).Its variance is then <cit.> σ^2(f) = 1/2 T df(10π^2/3H_0^2)^2 f^6 P_1(f) P_2(f).Here, T is the total coincident observation time between detectors, df is the frequency bin-width considered, and P_i(f) is the noise power spectral density of detector i. Note that the normalization of our cross-correlation measurement, with the overlap reduction functions appearing in ⟨Ĉ(f)⟩ rather than σ^2(f), differs from the convention normally adopted in the literature. Standard stochastic searches typically define a statistic Ŷ(f) ∝s̃^*_1(f) s̃_2(f)/γ_T(f), such that ⟨Ŷ(f) ⟩ = Ω^T(f) in the presence of a pure tensor background<cit.>. Our choice of normalization, though, will prove more convenient when studying stochastic backgrounds of mixed gravitational-wave polarizations. To emphasize this distinction, though, we denote our cross-power estimators by Ĉ(f), rather than the more common Ŷ(f).A spectrum of cross-correlation measurements Ĉ(f) may be combined to obtain a single broadband signal-to-noise ratio (SNR), given by SNR^2 = ( Ĉ | γ_a Ω^a_M )^2 /( γ_b Ω^b_M| γ_c Ω^c_M ) ,where we have defined the inner product ( A | B) = ( 3 H_0^2/10π^2)^2 2 T ∫_0^∞Ã^*(f) B̃(f)/f^6 P_1(f) P_2(f) df.In Eq. (<ref>), Ω^a_M(f) is our adopted model for the energy-density spectrum of the stochastic background. The expected SNR is maximized when this model is equal to the background's true energy-density spectrum. The resulting optimal SNR is given by SNR^2_opt = ( γ_aΩ^a| γ_bΩ^b )(see Appendix <ref> for details).Conventionally, stochastic energy-density spectra are modeled as power laws, such that Ω^a_M(f) = Ω^a_0 (f/f_0)^α_a,where Ω^a_0 is the background's amplitude at a reference frequency f_0 and α_a is its spectral index (or slope) <cit.>. The predicted tensor stochastic background from compact binary coalescences, for instance, is well-modeled by a power law of slope α_T=2/3 in the sensitivity band of Advanced LIGO <cit.>. For reference, slopes of α=0 and α=3 correspond to scale-invariant energy and strain spectra, respectively. While we will largely assume power-law models in our analysis, in Sect. <ref> we will explore the potential consequences if this assumption is in fact incorrect (as would be the case, for instance, for a background of unexpectedly massive binary black holes <cit.>). Throughout this paper we will use the reference frequency f_0=25.With the above formalism in hand, we can quantify Advanced LIGO's sensitivity to stochastic backgrounds of alternative polarizations. Plotted on the left side of Fig. <ref> are power-law integrated (PI) curves representing Advanced LIGO's optimal sensitivity to power-law backgrounds of pure tensor (solid blue), vector (solid red), and scalar (solid green) modes <cit.>. The PI curves are defined such that a power-law spectrum drawn tangent to the PI curve will be marginally detectable with ⟨⟩ = 3 after three years of observation with design-sensitivity Advanced LIGO. In general, energy-density spectra lying above and below the PI curves are expected to have optimal SNRs greater and less than 3, respectively. In the right side of Fig. <ref>, meanwhile, the solid curves trace the power-law amplitudes required for marginal detection (⟨⟩ = 3 after three years of observation) as a function of spectral index. Incidentally, the left and right-hand subplots of Fig. <ref> are Legendre transforms of one another.For spectral indices α_a≲0, Advanced LIGO is approximately equally sensitive to tensor and vector-polarized backgrounds, with reduced sensitivity to scalar signals. When α_a=0, for instance, the minimum optimally-detectable tensor and vector amplitudes are Ω^T_0 = 1.1×10^-9 and Ω^V_0=1.5×10^-9, while the minimum detectable scalar amplitude is Ω^S_0 = 4.4×10^-9, a factor of three larger. This relative sensitivity is due to the fact that the tensor and vector overlap reduction functions are of comparable magnitude at low frequencies, while the scalar overlap reduction function is reduced in size (see Fig. <ref>).At high frequencies, on the other hand, Advanced LIGO's tensor overlap reduction function decays more rapidly than the vector and scalar overlap reduction functions. As a result, Advanced LIGO is more sensitive to vector and scalar backgrounds of large, positive slope than to tensor backgrounds of similar spectral shape. In Fig. <ref>.a, for instance, the vector and scalar PI curves are seen to lie an order of magnitude below the tensor PI curve at frequencies above f∼300. The constraints that Advanced LIGO can place on positively-sloped vector and scalar backgrounds are therefore as much as an order of magnitude more stringent than those that can be placed on tensor backgrounds of similar slope.We emphasize that the Advanced LIGO network's relative sensitivities to tensor, vector, and scalar-polarized backgrounds are due purely to its geometry, rather than properties of the backgrounds themselves. If we were instead to consider the Hanford-Virgo baseline, for instance, the right-hand side of Fig. <ref> shows that at high frequencies the H1-V1 pair is least sensitive to scalar polarizations, whereas the H1-L1 baseline is least sensitive to tensor modes.So far we have discussed only Advanced LIGO's optimal sensitivity to stochastic backgrounds of alternative polarizations. Existing stochastic searches, though, are not optimized for such backgrounds, instead using models Ω^a_M(f) that allow only for tensor gravitational-wave polarizations. The dashed curves in Fig. <ref> illustrate Advanced LIGO's “naive” sensitivity to backgrounds of alternative polarizations when incorrectly assuming a purely-tensor model. Note that the “naive” curves on the right side of Fig. <ref> are not smooth, with sharp kinks at α_a∼2; more on this below. The loss in sensitivity between the optimal and naive searches varies greatly with different spectral indices. Sensitivity loss is relatively minimal for slopes α_a≲0. When α_S=0, for example, the minimum detectable scalar amplitude rises from Ω^S_0=4.4×10^-9 in the optimal case to 5.3×10^-9 in the naive case, an increase of 20%. Thus, a flat scalar background that is optimally detectable by Advanced LIGO may still be detected using existing techniques tailored to tensor polarizations. The SNR penalty is more severe for stochastic backgrounds of moderate positive slope. For α_S = 2, Advanced LIGO can optimally detect a scalar background of amplitude Ω^S_0 = 1.3×10^-9, while existing methods would detect only a background of amplitude Ω^S_0 = 4.4×10^-9, a factor of 3.4 larger.Since the SNR of the stochastic search accumulates only as SNR∝√(T), even a small decrease in sensitivity can result in a somewhat severe increase in the time required to make a detection. To illustrate this, Fig. <ref> shows the ratio T_Naive/T_Optimal between the observing times required for Advanced LIGO to detect vector (red) and scalar (green) backgrounds using existing “naive” methods and optimal methods. Although we noted above that existing methods incur little sensitivity loss to flat scalar backgrounds, the detection of such backgrounds would nevertheless require at least 50% more observing time with existing searches. Since the stochastic background is expected to be optimally detected only after several years, even a 50% increase potentially translates into years of additional observation time, a requirement which may well stress standard experimental lifetimes and operational funding cycles. Naive detection of a scalar background with α_S=2, for comparison, would require nearly twelve times the observing time.Figs. <ref> and <ref> both show conspicuous kinks occurring at α_S≈1.75 and α_V≈2.5. These features are due to severe systematic parameter biases incurred when recovering vector and scalar backgrounds with a purely tensorial model. For vector and scalar backgrounds of with α_a ≳ 3, the best-fit slope α_T (which maximizes the recovered SNR) is biased towards large values. Meanwhile, vector and scalar backgrounds with α_a ≲ 1 bias α_T in the opposite direction, towards smaller values. The sharp kinks in Fig. <ref> and <ref> occur at the transition between these two regimes. Such biases indicate another pitfall of existing search methods designed only for tensor-polarizations. Even if a vector or scalar-polarized background is recovered with minimal SNR loss, without some independent confirmation we may remain entirely unaware that the detected background indeed violates general relativity (see Sect. <ref> below). Furthermore, we would suffer from severe “stealth bias,” unknowingly recovering heavily-biased estimates of the amplitude and spectral index of the stochastic background <cit.>. § IDENTIFYING ALTERNATIVE POLARIZATIONS We have seen in Sect. <ref> that, even when using existing methods assuming only standard tensor polarizations, Advanced LIGO may still be capable of detecting a stochastic background of vector or scalar modes (albeit after potentially much longer observation times). Detection is only the first of two hurdles, though. Once the stochastic background has been detected, we will still need to establish whether it is entirely tensor-polarized, or if it contains vector or scalar-polarized gravitational waves.Since tensor, vector, and scalar gravitational-wave polarizations each enter into cross-correlation measurements [Eq. (<ref>)] with unique overlap reduction functions, the polarization content of a detected stochastic background is in principle discernible from the spectral shape of Ĉ(f). As an example, Fig. <ref> shows simulated cross-correlation measurements Ĉ(f) for both purely tensor (blue) and purely scalar-polarized (green) backgrounds after three years of observation with design-sensitivity Advanced LIGO. The left-hand side shows simulated measurements of extremely strong backgrounds, with spectra Ω^T(f) = 5×10^-8 (f/f_0)^2/3 and Ω^S(f) = 1.8×10^-7 (f/f_0)^2/3; amplitudes are chosen such that each background has expected ⟨⟩ = 150 after three years of observation. The dashed curves trace the expectation values ⟨Ĉ(f) ⟩ of the cross-correlation spectra for each case, while the solid curves show a particular instantiation of measured values. The alternating signs (positive or negative) of each spectrum are determined by the tensor and scalar overlap reduction functions, which have zero-crossings at different characteristic frequencies (see Fig. <ref>). As a result, tensor and scalar-polarized signals each impart a unique shape to the cross-correlation spectra, offering a means of discriminating between the two cases.As mentioned above, though, the backgrounds shown on the left side of Fig <ref> are unphysically loud, with =152 and 148 for the simulated tensor and scalar backgrounds, respectively. A tensor background of this amplitude would have been detectable with the standard isotropic search over Advanced LIGO's O1 observing run <cit.>.Since stochastic searches accumulate SNR over time, the first detection of the stochastic background will necessarily be marginal; in this case the presence of alternative gravitational-wave polarizations would not be clear. To demonstrate this, the right side of Fig. <ref> shows the simulated recovery of weaker tensor and scalar backgrounds of spectral shape Ω^T(f) = 1.7×10^-9 (f/f_0)^2/3 and Ω^S(f) = 6.1×10^-9(f/f_0)^2/3, again after three years of observation with Advanced LIGO. These amplitudes correspond to expected ⟨⟩ = 5 after three years. While Advanced LIGO would still make a very confident detection of each background, with =6.7 and 7.8 for the simulated tensor and scalar cases, the backgrounds' polarization content is no longer obvious.Interestingly, even when naively searching for purely-tensor polarized backgrounds, design-sensitivity Advanced LIGO still detect the “quiet” scalar example (on the right side of Fig. <ref>) with =5.0. When assuming a priori that the stochastic background is purely tensor-polarized, any vector or scalar components detected with existing techniques may therefore be mistaken for ordinary tensor modes. Not only would vector or scalar components fail to be identified, but, as discussed in Sect. <ref>, they would heavily bias parameter estimation of the tensor energy-density spectrum. If we wish to test general relativity with the stochastic background, we will therefore need to develop new tools in order to formally quantify the presence (or absence) of vector or scalar polarizations. Additionally, while we have so far investigated only backgrounds of pure tensor, vector, or scalar polarization, most plausible alternative theories of gravity will predict backgrounds of mixed polarization, with vector or scalar components in addition to a tensor component. Any realistic approach must therefore be able to handle a stochastic background of completely generic polarization content.Our approach will be to detect and classify the stochastic background using Bayesian model selection, adapting the method used in Ref. <cit.> to study the polarization content of continuous gravitational-wave sources. First, we will define an odds ratiobetween signal (SIG) and noise (N) hypotheses to determine if a stochastic background (of any polarization) has been observed. Once a background is detected, we then construct a second odds ratioto determine if the background contains only tensor polarization (GR hypothesis) or if there is evidence of alternative polarizations (the NGR hypothesis). We describe the definition and construction ofandin Appendix <ref>. Unlike existing detection methods that assume a pure tensor background, our scheme allows for the detection of generically-polarized stochastic backgrounds. It encapsulates the optimal detection of tensor, vector, and scalar polarizations as described in Sect. <ref>, and moreover enables the detection of more complex backgrounds of mixed polarization.To compute the odds ratiosand , we use thepackage <cit.>, which implements a Python wrapper for the nested sampling software<cit.>. , an implementation of the nested sampling algorithm <cit.>, is designed to efficiently evaluate Bayesian evidences [see Eq. (<ref>)] in high-dimensional parameter spaces, even in the case of large and possibly-curving parameter degeneracies. At little additional computational cost,also returns posterior probabilities for each model parameter, allowing for parameter estimation in addition to model selection. Details associated with runningare given in Appendix <ref>.Our approach fundamentally differs from the strategy proposed by Nishizawa et al. in Refs. <cit.>. Nishizawa et al. endeavor to separate and measure the background's tensor, vector, and scalar content within each frequency bin. To solve for these three unknowns, three pairs of gravitational-wave detectors are required to break the degeneracy between polarizations. A nice feature of this method is that it allows for the separation of polarization modes without the need for a parametrized model of the background's energy-density spectrum. However, it has several drawbacks. First, the Nishizawa et al. component separation scheme requires at least three detectors. Even then, this method is not very sensitive; covariances between polarization modes mean that only very loud backgrounds can be separated and independently detected with reasonable confidence. Finally, Nishizawa et al. are largely concerned with the detection of a background, not the characterization of its spectral shape. Ref. <cit.> does discuss parameter estimation on the stochastic background using a Fisher matrix formalism, but there are very well-known problems with this approach <cit.>.Our method is more aggressive. Rather than attempting to resolve the relative polarization content within each frequency bin, we assume a power-law model for the energy-density in each polarization mode (see Appendix <ref>). This allows us to confidently detect far weaker signals than the Nishizawa et al. approach. While this approach is potentially susceptible to bias if our model poorly fits the true background, it is a reasonable model for astrophysically plausible scenarios. Even if the true background differs significantly from this model, we find in Sect. <ref> that potential bias is negligible. Another advantage of our method is that it can be used with only two detectors and hence can be applied today, rather than waiting for the construction of future gravitational-wave detectors. Finally, in Sect. <ref>, we show that our Bayesian approach allows for full parameter estimation on the stochastic background, which properly takes into account the full degeneracies between background parameters (something a Fisher matrix analysis cannot do).§.§ Backgrounds of Single PolarizationsAs a first demonstration of this machinery, we explore the simple cases of purely tensor, vector, or scalar-polarized stochastic backgrounds. Shown in Fig. <ref> are distributions of odds ratiosandobtained for simulated observations of both tensor and scalar backgrounds, each of slope α=2/3 (the characteristic slope of a tensor binary black hole background). For each polarization, we consider two choices of amplitude, corresponding to ⟨⟩ = 5 and 10 after three years of observation with design-sensitivity Advanced LIGO. For comparison, the hatched grey distributions show odds ratios obtained in the presence of pure Gaussian noise.As seen in the left-hand side of Fig. <ref>, Gaussian noise yields a narrow odds ratio distribution centered at ln≈-1.0 . In contrast, the simulated observations of tensor and scalar backgrounds yield large, positive odds ratios, well-separated from Gaussian noise. Note that the tensor and scalar distributions lie nearly on top of one another, asdepends primarily on the optimal SNR of a background and not its polarization content.The right-hand side of Fig. <ref>, in turn, shows the odds ratiosquantifying the evidence for alternative polarization modes. In the case of pure Gaussian noise, we again see a narrow distribution of odds ratios, centered at ln≈-0.4. In the absence of informative data, our analysis thus slightly favors the GR hypothesis. This can be understood as a consequence of the implicit Bayesian “Occam's factor," which penalizes the more complex NGR hypothesis over the simpler GR hypothesis. Simulated observations of scalar backgrounds, in turn, yield large positive values for ln, correctly preferencing the NGR hypothesis. In contrast, pure tensor backgrounds yield negative ln. Interestingly, the recovered odds ratios do not grow increasingly negative with larger tensor amplitudes, but instead saturate at ln≈-1.4. This reflects the fact that a non-detection of vector or scalar polarizations can never strictly rule out their presence, but only place an upper limit on their amplitudes. In other words, a strong detection of a pure tensor stochastic background cannot provide evidence for the GR hypothesis, but at best only offers no evidence against it. This behavior is in part due to our choice of amplitude priors, which allow for finite but immeasurably small vector and scalar energy densities (see Appendix <ref>).Figure <ref> illustrates more generally how(left column) and(right column) scale with the amplitudes of purely tensor (blue), vector (red), and scalar (green) stochastic backgrounds. Black points mark odds ratios computed from individual realizations of simulated data, while the solid curves and shaded regions trace their smoothed mean and standard deviation. We again see ln increasing monotonically with injected amplitude for all three polarizations. Specifically,depends inversely on the noise-hypothesis likelihood [defined by Eq. (<ref>)] and therefore scales as ln∝^2.As seen earlier in Fig. <ref>, ln saturates at -1.4 for loud tensor backgrounds. In the case of vector and scalar backgrounds, on the other hand, ln grows quadratically with increasing amplitude. In particular, ln is proportional to the squared SNR of the residuals between the observed Ĉ(f) and the best-fit tensor model. We begin to see a strong preference for the NGR hypothesis when these residuals become statistically significant.§.§ Backgrounds of Mixed PolarizationSo far we have considered only cases of pure tensor, vector, or scalar polarization. Plausible alternative theories of gravity, however, would typically predict a mixed background of multiple polarization modes. How does our Bayesian machinery handle a background of mixed polarization? To answer this question, we will investigate backgrounds of mixed tensor and scalar polarization. Figure <ref> shows values ofand(left and right-hand sides, respectively) as a function of the amplitude of each polarization. While we allow the amplitudes to vary, we fix the tensor and scalar slopes to α_T = 2/3 (as predicted for binary black hole backgrounds) and α_S = 0.In the left side Fig. <ref>, the recovered values of ln simply trace contours of total energy. Thus the detectability of a mixed background depends only on its total measured energy, rather than its polarization content. Meanwhile, three distinct regions are observed in the right-hand subplot. First, for small tensor and scalar amplitudes (logΩ_0^T≲-9.0 and logΩ_0^S≲-8.5), we obtain ln≈-0.4. In this region, the mixed background simply cannot be detected and so we recover the slight Occam's bias towards the GR hypothesis as noted above. Secondly, for small scalar and large tensor amplitudes (logΩ_0^T ≳-9.0), the recovered odds ratios decrease to ln≈-1.4. This corresponds to the detection of the tensor component alone; the decrease in odds ratios is the same behavior previously seen in Figs. <ref> and <ref>. Finally, when Ω_0^S is large, the scalar component is detectable and the recovered ln increases rapidly to large, positive values. The threshold value of Ω_0^S at which ln becomes positive shows only little dependence on the amplitude of any tensor background which might also be present. When Ω_0^T is small, for instance, scalar amplitudes of size logΩ_0^S≳-7.9 are required to preference the NGR model. When Ω_0^T is large, this requirement increases only slightly to logΩ_0^S≳-7.8. Thus, we should expect Advanced LIGO to be able to both detect and identify as non-tensorial a flat scalar background of amplitude logΩ_0^S ≳-8, regardless of the presence of an additional tensor component.It should be pointed out that positive log indicates only that there exists evidence for alternative polarizations. From the odds ratio alone we cannot infer which specific polarizations – vector and/or scalar – are present in the background. While we found above that Advanced LIGO can identify mixed tensor-scalar backgrounds as non-tensorial when logΩ^S_0≳-8, this does not imply that we can successfully identify the scalar component as such, only that our measurements are not consistent with tensor polarization alone (see Sect. <ref>). The future addition of new gravitational wave detectors will extend the reach of stochastic searches and help to break degeneracies between backgrounds of different polarizations. This expansion recently began with the completion of Advanced Virgo, which joined Advanced LIGO during its O2 observing run in August 2017 <cit.>. It is therefore interesting to investigate how the introduction of Advanced Virgo will improve the above results.Given detectors indexed by i∈{1,2,...}, the total SNR of a stochastic background is the quadrature sum of SNRs from each detector pair <cit.>: ^2 = ∑_i∑_j>i^2_ij,where each _ij is computed following Eq. (<ref>). Naively, the SNR with which a background is observed is expected to increase as ∝√(N), where N is the total number of available detector pairs (three in the case of the Advanced LIGO-Virgo network). However, both the Hanford-Virgo and Livingston-Virgo pairs exhibit reduced sensitivity to the stochastic background due to their large physical separations. This fact is reflected in their respective overlap reduction functions, which are a factor of several smaller in magnitude than the Hanford-Livingston overlap reduction functions (see Fig. <ref>).Given three independent detector pairs (and hence three independent measurements at each frequency), one can in principle directly solve for the unknown tensor, vector, and scalar contributions to the background in each frequency bin <cit.>. This component separation scheme can be performed without resorting to a model for the stochastic energy-density spectrum. However, frequency-by-frequency component separation is unlikely to be successful using the LIGO-Virgo network, due to the large uncertainties in the measured background at each frequency. Instead, when considering joint Advanced LIGO-Virgo observations we will again apply the Bayesian framework introduced above, leveraging measurements made at many frequencies in order to constrain the power-law amplitude and slope of each polarization mode.To quantify the extent to which Advanced Virgo aids in the detection of the stochastic background, we again consider simulated observations of a mixed tensor (slope α_T=2/3) and scalar (slope α_S=0) background, this time with a three-detector Advanced LIGO-Virgo network. Our Bayesian formalism is easily extended to accommodate the case of multiple detector pairs; details are given in Appendix <ref>. The odds ratios obtained from our simulated Advanced LIGO-Virgo observations are shown in Fig. <ref> for various tensor and scalar amplitudes. The inclusion of Advanced Virgo yields no clear improvement over the Advanced LIGO results in Fig. <ref>. Due to its large distance from LIGO, Advanced Virgo does not contribute more than a small fraction of the total observed SNR. As a result, the combined Hanford-Livingston-Virgo network both detects (as indicated with ) and identifies (via ) the scalar background component with virtually the same sensitivity as the Hanford-Livingston network alone.§ PARAMETER ESTIMATION ON MIXED BACKGROUNDS Parameter estimation will be the final step in a search for a stochastic background of generic polarization. If a gravitational-wave background is detected (as inferred from ), how well can Advanced LIGO constrain the properties of the background? Alternatively, if no detection is made, what upper limits can Advanced LIGO place on the background amplitudes of each polarization mode? We investigate these questions through three case studies: an observation of pure Gaussian noise, a standard tensor stochastic background, and a background of mixed tensor and scalar polarizations. The simulated background parameters used for each case are listed in Table <ref>.When performing model selection above, the odds ratiosandwere constructed by independently allowing for each combination of tensor, vector, and scalar modes (see Appendix <ref>). Parameter estimation, meanwhile, must be performed in the context of a specific background model. For the case studies below, we will adopt the broadest possible hypothesis, allowing for all three polarization modes (the TVS hypothesis in Appendix <ref>). This choice will allow us to place simultaneous constraints on the presence of tensor, vector, and scalar polarizations in the stochastic background. Parameter estimation is achieved using , which returns samples drawn from the measured posterior distributions.There are several key subtleties that must be understood when interpreting the parameter estimation results presented below. First, whereas standard tensor upper limits are conventionally defined with respect to a single, fixed slope <cit.>, we will quote amplitude limits obtained after marginalization over spectral index. This approach concisely combines information from the entire posterior parameter space to offer a single limit on each polarization considered. As a result, however, our simulated upper limits presented here should not be directly compared to those from standard searches for tensor backgrounds. Secondly, parameter estimation results are contingent upon the choice of a specific model. While we will demonstrate parameter estimation results under our TVS hypothesis (see Appendix <ref>), other hypotheses may be better suited to answering other experimental questions. For example, if we were specifically interested in constraining scalar-tensor theories (which a priori do not allow vector polarizations), we would instead perform parameter estimation under the TS hypothesis. And if our goal was to perform a standard stochastic search for a purely tensor-polarized background, we would restrict to the T hypothesis. Although these various hypotheses all contain an analogous parameter Ω^T_0, the resulting upper limits on Ω^T_0 will generically be different in each case. In short, different experimental questions will yield different answers.§.§ Case 1: Gaussian Noise First, we consider the case of pure noise, producing a simulated three-year observation of Gaussian noise at Advanced LIGO's design sensitivity. The resulting TVS posteriors are shown in Fig. <ref>. The colored histograms along the diagonal show the marginalized 1D posteriors for the amplitudes and slopes of the tensor, vector, and scalar components (blue, green, and red, respectively). The priors placed on each parameter are indicated with a dashed grey curve. Above each posterior we quote the median posterior value as well as ±34% credible limits. The remaining subplots illustrate the joint 2D posteriors between each pair of parameters.For this simulated Advanced LIGO observation, we obtain log=-1.1, consistent with a null detection. Accordingly, the posteriors on Ω^T_0, Ω^V_0, and Ω^S_0 are each consistent with the lower bound of our amplitude prior (at logΩ_Min = -13). Meanwhile, the posteriors on spectral indices α_T, α_V, and α_S simply recover our chosen prior. The 95% credible upper limits on each amplitude are logΩ^T_0 < -9.8, logΩ^V_0 < -9.7, and logΩ^S_0 < -9.3.In Fig. <ref> we show the posteriors obtained if we additionally include design-sensitivity Advanced Virgo (incorporating simulated measurements for the HV and LV detector pairs). For reference, the grey histograms show the posteriors from Fig. <ref> obtained by Advanced LIGO alone. The Advanced LIGO-Virgo posteriors are virtually identical to those obtained from Advanced LIGO alone, with 95% credible upper limits of logΩ^T_0 < -9.9, logΩ^V_0 < -9.6, and logΩ^S_0 < -9.4. In the case of a null-detection, then, the inclusion of Advanced Virgo does not notably improve the upper limits placed on the amplitudes of tensor, vector, and scalar backgrounds. §.§ Case 2: Tensor Background Next, we produce a simulated observation of a pure tensor background with amplitude logΩ^T_0 = -8.78 and spectral index α_T = 2/3. The amplitude is chosen such that the background would be detected by Advanced LIGO with expected ⟨⟩ = 5 after three years of observation at design-sensitivity. The odds ratios obtained for this simulated observation are log=8.4 and log=-1.4, indicating a strong detection consistent with general relativity.The corresponding parameter posteriors are shown in Fig. <ref>. In this case, the injected parameter values are shown via dot-dashed black lines. The logΩ^T_0 posterior is strongly peaked near the true value, with a central 68% credible interval of -9.0≤logΩ^T_0≤-8.7 and a median value of logΩ^T_0 = -8.8. The vector and scalar amplitudes, in turn, are consistent with the lower bound on our prior, with 95% credible upper limits of logΩ^V_0 < -9.2 and logΩ^S_0 < -9.0.The parameter estimation results when additionally including Advanced Virgo are given in Fig. <ref>. Once again, the grey histograms show parameter estimation results from Advanced LIGO alone. Although Virgo does not improve our confidence in the detection, it can serve to break degeneracies present between different polarization modes. We begin to see this behavior in Fig. <ref>, in which the vector and scalar log-amplitude posteriors are pushed to smaller values in the joint LIGO-Virgo analysis. When including Advanced Virgo, we obtain a marginally tighter 68% credible interval of -8.9≤logΩ^T_0≤-8.7 on the tensor amplitude, and slightly improved upper limits of logΩ^V_0 < -9.3 and logΩ^S_0 < -9.2 on vector and scalar amplitudes.§.§ Case 3: Tensor and Scalar BackgroundsAs discussed above, most alternative theories of gravity would predict a stochastic background of mixed polarization. For our final case study, we therefore consider a mixed background with both tensor (logΩ^T_0 = -8.48 and α_T=2/3) and scalar (logΩ^S_0 = -7.83 and α_S = 0) components. The amplitudes are chosen such that each component is individually observable with ⟨⟩ = 10 after three years of observation. Analysis with yields odds ratios log = 193.5 and log=16.1, representing an extremely loud detection with very strong evidence for the presence alternative polarizations.The posteriors obtained for this data are shown in Fig. <ref>. Despite the strength of the simulated stochastic signal, we see that parameter estimation results are dominated by degeneracies between the different polarization modes. Although the tensor and scalar amplitude posteriors are locally peaked about their true values, much of the background's energy is misattributed to vector modes, illustrating that potential severe degeneracies persist even at high SNRs. These degeneracies are exacerbated for backgrounds with small or negative spectral indices, as in the present case. Such backgrounds preferentially weight low frequencies where the Advanced LIGO overlap reduction functions are all similar (see Fig. <ref>). This example serves to illustrate that, while Advanced LIGO can likely identify the presence of alternative polarizations through the odds ratio , Advanced LIGO alone is unable to determine which modes (vector or scalar) have been detected.In contrast, the degeneracies in Fig. <ref> are completely broken with the inclusion of Advanced Virgo. Whereas the Ω^V_0 posterior is strongly peaked in Fig. <ref>, we see in Fig. <ref> that the posterior is instead entirely consistent with our lower prior bound when including Advanced Virgo. The tensor and scalar amplitude posteriors, meanwhile, are each more strongly-peaked about their correct values and are now inconsistent with the lower amplitude bound. Thus, while Advanced Virgo generally does not improve our ability to detect a stochastic background, we see that it can significantly improve prospects for simultaneous parameter estimation of multiple polarizations.§ BROKEN TENSOR SPECTRA The stochastic search presented here offers a means to search for alternative gravitational-wave polarizations in a nearly model-independent way. Unlike direct searches for compact binary coalescences, our search makes minimal assumptions about the source and nature of the stochastic background. We do, however, make one notable assumption: that the energy density spectra Ω^a(f) are well-described by power laws in the Advanced LIGO frequency band. This is expected to be a reasonable approximation for most predicted astrophysical sources of gravitational waves. The backgrounds expected from stellar-mass binary black holes <cit.>, core-collapse supernovae <cit.>, and rotating neutron stars <cit.>, for instance, are all well-modelled by power laws in the Advanced LIGO band. It may be, however, that the stochastic background is in fact not well-described by a single power law. This may be the case if, for instance, the background is dominated by high-mass binary black holes, an excess of systems at high redshift, or previously-unexpected sources of gravitational waves <cit.>.Given that our search allows only for power-law background models, how would we interpret a non-power-law background? In particular, if the stochastic background is purely tensorial (obeying general relativity) but is not well-described by a power-law, would our search mistakenly claim evidence for alternative polarizations?To investigate this question, we consider simulated Advanced LIGO observations of pure tensor backgrounds described by broken power laws: Ω^T(f) = Ω_0 (f/f_k)^α_1(f<f_k)Ω_0 (f/f_k)^α_2(f≥ f_k).Here, Ω_0 is the background's amplitude at the “knee frequency" f_k, while α_1 and α_2 are the slopes below and above the knee frequency, respectively. We will set the knee frequency to f_k=30, placing the backgrounds' knees in the most sensitive band of the stochastic search. The odds ratioswe obtain for these broken power laws are shown in Fig. <ref> as a function of the two slopes α_1 and α_2. Each simulation assumes three years of observation at design-sensitivity, and the amplitudes Ω_0 are scaled such that each background has expected ⟨⟩ = 5 after this time. Any trends in Fig. <ref> are therefore due to the backgrounds' spectral shapes rather than their amplitudes.If tensor broken power laws are indeed misclassified by our search, we should expect large, positive ln values in Fig. <ref>. Instead, we see that broken power laws are not systematically misclassified. When α_1 and α_2 are each positive, we recover ln≈-1.5, correctly classifying backgrounds as tensorial despite the fact that they are not described by power laws. When α_1<0, meanwhile, we recover odds ratios scattered about ln≈ 0. This simply reflects the fact that when α_1 is negative the majority of a background's SNR is collected at low frequencies where Advanced LIGO's tensor, vector, and scalar overlap reduction functions are degenerate. In such a case we do not show preference for either model over the other. Note that we find ln≈0 even along the line α_2=α_1 (for α_1<0), where the background is described by a single power law.We expect broken power laws to be most problematic when α_1>0 and α_2<0; in this case a background's SNR is dominated by a small frequency band around the knee itself. This would be the case if, for instance, the stochastic background were dominated by unexpectedly massive binary black hole mergers <cit.>. Figure <ref> does suggest a larger scatter in log for such backgrounds. Even in this region, however, there is not a systematic bias towards larger values of , and the largest recovered odds ratios have log≲2.5, well below the level required to confidently claim evidence for the presence of alternative polarizations.Despite the fact that we assume purely power-law models for the stochastic energy-density spectra, our search appears reasonably robust against broken power law spectra that are otherwise purely tensor-polarized. In particular, in order to be mistakenly classified by our search, a tensor stochastic background would have to emulate the pattern of positive and negative cross-power associated with the vector and/or scalar overlap reduction functions (see, for instance, Fig. <ref>). This is simply not easy to do without a pathological background. While we have demonstrated this only for Advanced LIGO, we find similarly robust results for three-detector Advanced LIGO-Virgo observations.Nevertheless, when interpreting odds ratiosit should be kept in mind that the true stochastic background may deviate from a power law. Even if a broken tensor background is not misclassified in our analysis, the parameter estimation results we obtain would likely be incorrect (another example of so-called “stealth bias”). It should be pointed out, though, that our analysis is not fundamentally restricted to power-law models. While we adopt power-law models here for computational simplicity, our analysis can be straightforwardly expanded in the future to include more complex models for the stochastic energy-density spectrum.§ DISCUSSION The direct detection of gravitational waves by Advanced LIGO and Virgo has opened up new and unique prospects for testing general relativity. One such avenue is the search for vector and scalar gravitational-wave polarizations, predicted by some alternative theories of gravity but prohibited by general relativity. Observation of vector or scalar polarizations in the stochastic background would therefore represent a clear violation of general relativity. While the first preliminary measurements have recently been made of the polarization of GW170814, our ability to study the polarization of transient gravitational-wave signals is currently limited by the number and orientation of current-generation detectors. In contrast, searches for long-duration sources like the stochastic background offer a promising means of directly measuring gravitational-wave polarizations with existing detectors. In this paper, we explored a procedure by which Advanced LIGO can detect or constrain the presence of vector and scalar polarizations in the stochastic background. In Sect. <ref>, we found that a stochastic background dominated by alternative polarization modes may be missed by current searches optimized only for tensor polarizations. In particular, backgrounds of vector and scalar polarizations with large, positive slopes may take up to ten times as long to detect with current methods, relative to a search optimized for alternative polarizations. In Sect. <ref>, we therefore proposed a Bayesian method with which to detect a generically-polarized stochastic background. This method relies on the construction of two odds ratios (see Appendix <ref>). The first serves to determine if a stochastic background has been detected, while the second quantifies evidence for the presence of alternative polarizations in the background. This search has the advantage of being entirely generic; it is capable of detecting and identifying stochastic backgrounds containing any combination of gravitational-wave polarizations. With this method, we demonstrated flat scalar-polarized backgrounds of amplitude Ω^S_0≈2×10^-8 can be confidently identified as non-tensorial with Advanced LIGO.In Sect. <ref>, we then considered the ability of Advanced LIGO to perform simultaneous parameter estimation on tensor, vector, and scalar components of the stochastic background. After three years of observation at design sensitivity, Advanced LIGO will be able to limit the amplitudes of tensor, vector, and scalar polarizations to Ω^T_0 < 1.6×10^-10, Ω^V_0<2.0×10^-10, and Ω^S_0<5.0×10^-10, respectively, at 95% credibility. If, however, a stochastic background of mixed polarization is detected, Advanced LIGO alone cannot precisely determine the parameters of the tensor, vector, and/or scalar components simultaneously due to large degeneracies between modes.We also considered how the addition of Advanced Virgo to the Hanford-Livingston network affects the search for alternative polarizations. In Sect. <ref>, we found that addition of Advanced Virgo does not particularly increase our ability to detect or identify backgrounds of alternative polarizations. However, we found in Sect.<ref> that Advanced Virgo does significantly improve our ability to perform parameter estimation on power-law backgrounds, breaking the degeneracies that plagued the Hanford-Livingston analysis.Relative to other modeled searches for gravitational waves, the stochastic search described here has the advantage of being nearly model-independent. We have, however, made one large assumption: that the tensor, vector, and scalar energy-density spectra are well-described by power laws in the Advanced LIGO band. Finally, in Sect. <ref> we explored the implications of this assumption, asking the question: would tensor backgrounds not described by power laws be mistaken for alternative polarizations in our search? We found that our proposed Bayesian method is reasonably robust against this possibility. In particular, even pure tensor backgrounds with sharply-broken power law spectra are not systematically misidentified by our search.The non-detection of alternative polarizations in the stochastic background may yield interesting experimental constraints on extended theories of gravity. Meanwhile, any experimental evidence for alternative polarizations in the stochastic background would be a remarkable step forward for experimental tests of gravity. Of course, if future stochastic searches do yield evidence for alternative polarizations, careful study would be required to verify that this result is not due to unmodeled effects like non-Gaussianity or anisotropy in the stochastic background <cit.>. Comparison to polarization measurements of other long-lived sources like rotating neutron stars <cit.> will additionally aid in the interpretation of stochastic search results.Several future developments may further improve the ability of ground-based detectors to detect alternative polarization modes in the stochastic background. First, the continued expansion of the ground-based detector network will improve our ability to both resolve the stochastic background and accurately determine its polarization content. Secondly, while we presently assume that the stochastic background is Gaussian, the background contribution from binary black holes is expected to be highly non-Gaussian <cit.>. Future stochastic searches may therefore be aided by the development of novel data analysis techniques optimized for non-Gaussian backgrounds <cit.>. We would like to thank Thomas Dent, Gregg Harry, Joe Romano, and Alan Weinstein for their careful reading of this manuscript, as well as many members of the LIGO-Virgo Collaboration Stochastic Backgrounds working group for helpful comments and conversation. T. C. and M. I. are members of the LIGO Laboratory, supported by funding from the U. S. National Science Foundation. LIGO was constructed by the California Institute of Technology and Massachusetts Institute of Technology with funding from the National Science Foundation and operates under cooperative agreement PHY-0757058. N.C. is supported by NSF grant PHY-1505373. The work of A.M. was supported in part by the NSF grant PHY-1204944 at the University of Minnesota. M.S. is partially supported by STFC (UK) under the research grant ST/L000326/1. E.T. is supported through ARC FT150100281 and CE170100004. This paper carries the LIGO Document Number LIGO-P1700059 and King's College London report number KCL-PH-TH/2017-25.§ OVERLAP REDUCTION FUNCTIONS The sensitivity of a two-detector network to a stochastic gravitational-wave background is quantified by the overlap reduction function <cit.> γ(f) ∝∑_A ∫ e^2π i f Ω·Δ𝐱/c F^A_1(Ω) F^A_2(Ω) dΩ,where Δ𝐱 is the displacement vector between detectors, c is the speed of light, and F^A_1/2(Ω) are the antenna patterns describing the response of each detector to gravitational-waves of polarization A propagating from the direction Ω. The overlap reduction function is effectively the sky-averaged product of the two detectors' antenna patterns, weighted by the additional phase accumulated as a gravitational wave propagates from one site to the other.In the standard stochastic search, the summation in Eq. (<ref>) is taken over the tensor plus and cross polarizations. When extending the stochastic search to generic gravitational-wave polarizations, we now must consider three separate overlap reduction functions for the tensor, vector, and scalar modes <cit.>: γ_T(f)= 5/8π∑_A={+,×}∫ e^2π i f Ω·Δ𝐱/c F^A_1(Ω) F^A_2(Ω) dΩ,γ_V(f)= 5/8π∑_A={x,y}∫ e^2π i f Ω·Δ𝐱/c F^A_1(Ω) F^A_2(Ω) dΩ,γ_S(f)= 5/12π∑_A={b,l}∫ e^2π i f Ω·Δ𝐱/c F^A_1(Ω) F^A_2(Ω) dΩ.We normalize these functions such that γ_T(f) = 1 for coincident, co-aligned detectors; detectors that are rotated or separated relative to one another have γ_T(f) < 1. The amplitudes of γ_V(f) and γ_S(f), meanwhile, express relative sensitivities to vector and scalar backgrounds.Note that the normalization of γ_S(f) differs from that of Nishizawa et al. in Ref. <cit.>. This difference is due to Nishizawa et al.'s definition of the longitudinal polarization tensor as ∼e^l = √(2)Ω⊗Ω,rather than the more common e^l = Ω⊗Ω(to distinguish between these two conventions, the quantities adopted by Nishizawa et al. will be underscored with tildes). As a consequence, Nishizawa et al. obtain a longitudinal antenna pattern ∼F^l(Ω) = 1/√(2)sin^2θcos2ϕ,which differs by a factor of √(2) from the conventional form F^l(Ω) = 1/2sin^2θcos2ϕ.Correspondingly, the quantity ∼Ω^l(f) defined by Nishizawa et al. is actually half of the canonical energy density in longitudinal gravitational waves: ∼Ω^l(f)= 1/2Ω^l(f). While each overlap reduction function may be calculated numerically via Eq. (<ref>), they may also be analytically expanded in terms of spherical Bessel functions <cit.>. See Ref. <cit.> for definitions of the tensor, vector, and scalar overlap reduction functions in this analytic form. Note, however, that these definitions follow Nishizawa et al.'s normalization convention as discussed above; the analytic expression given for γ_S(f) must be divided by 3 to match our Eq. (<ref>). § OPTIMAL SIGNAL-TO-NOISE RATIO Searches for the stochastic background rely on measurements Ĉ(f) of the cross-power between two detectors. As discussed in Sect. <ref>, the expectation value and variance of Ĉ(f) are given by Eqs. (<ref>) and (<ref>), respectively. Here, we derive the optimal broadband signal-to-noise ratio [Eq. (<ref>)], which combines a spectrum of cross-correlation measurements into a single detection statistic.Given a measured spectrum Ĉ(f) and associated uncertainties σ^2(f), a single broadband statistic may be formed via the weighted sum Ĉ = ∑_f Ĉ(f) w(f)/σ^2(f)/∑_f w(f)/σ^2(f),where w(f) is a set of yet-undefined weights. The mean and variance of Ĉ are ⟨Ĉ⟩ = ∑_f γ_a(f) Ω^a(f) w(f)/σ^2(f)/∑_f w(f)/σ^2(f),and σ^2 = ∑_f w^2(f)/σ^2(f)/(∑_f w(f)/σ^2(f))^2,where γ_a(f) Ω^a(f) denotes summation ∑_a γ_a(f) Ω^a(f) over polarization modes a∈{T,V,S}.We define a broadband signal-to-noise ratio by SNR = Ĉ/σ. In the limit df→0, this quantity may be written SNR = (Ĉ |w)/√((w | w)),where we have substituted Eq. (<ref>) for σ^2(f) and made use of the inner product defined in Eq. (<ref>). The expected SNR is maximized when the chosen weights are equal to the true background, such that w(f) = γ_a(f)Ω^a_gw(f). In this case, the optimal expected SNR of the stochastic background becomes ⟨SNR_opt⟩ = √(( γ_aΩ^a_gw | γ_bΩ^b_gw)).§ ODDS RATIO CONSTRUCTION Here, we describe the construction of the odds ratiosandintroduced in Sect. <ref>. Given data Ĉ(f), the Bayesian evidence for some hypothesis 𝒜 with parameters θ_A is defined P(Ĉ | 𝒜) = ∫ℒ(Ĉ | θ_A,𝒜) π(θ_A|𝒜) dθ_A.Here, the likelihood ℒ(Ĉ |θ_A,𝒜) gives the conditional probability of the measured data under hypothesis 𝒜 for fixed parameter values, while π(θ_A | 𝒜) is the prior probability set on these parameters. When selecting between two such hypotheses 𝒜 and ℬ, we may define an odds ratio 𝒪^𝒜_ℬ = P(Ĉ | 𝒜) / P(Ĉ | ℬ) π(𝒜)/π(ℬ).The first factor in Eq. (<ref>), called the Bayes factor, is the ratio between the Bayesian evidences for hypotheses 𝒜 and ℬ. The second term, meanwhile, is the ratio between the prior probabilities π(𝒜) and π(ℬ) assigned to each hypothesis.To construct odds ratios for our stochastic background analysis, we will first need the likelihood ℒ({Ĉ}|θ,𝒜) of a measured cross-power spectrum under model 𝒜 with some parameters θ. In the presence of Gaussian noise, the likelihood of measuring a specific Ĉ(f) within a single frequency bin is <cit.> ℒ(Ĉ(f) | θ,𝒜) ∝exp( - [Ĉ(f) - γ_a(f)Ω^a_𝒜 (θ; f) ]^2 / 2σ^2(f)),with variance σ^2(f) given by Eq. (<ref>). Here, Ω^a_𝒜(θ;f) is our model for the energy-density spectrum under hypothesis 𝒜 and with parameters θ, evaluated at the given frequency f. The full likelihood ℒ({Ĉ}|θ,𝒜) for a spectrum of cross-correlation measurements is the product of the individual likelihoods in each frequency bin: ℒ ({Ĉ} | θ,𝒜) ∝∏_f ℒ(Ĉ(f) | θ,𝒜) = 𝒩exp[ -1/2( Ĉ-γ_aΩ^a_𝒜 | Ĉ-γ_bΩ^b_𝒜) ],where 𝒩 is a normalization coefficient and we have used the inner product defined by Eq. (<ref>).As discussed in Sect. <ref>, we will seek to detect and characterize a generic stochastic background via the construction of two odds ratios: , which indicates whether a background of any polarization is present, and , which quantifies evidence for the presence of alternative polarization modes. First consider . Under the noise hypothesis (N), we assume that no signal is present [such that Ω_n^a(f) = 0]. From Eq. (<ref>), the corresponding likelihood is simply ℒ({Ĉ} | N ) = 𝒩exp[ - 1/2( Ĉ | Ĉ)]. The signal hypothesis (SIG) is somewhat more complex. The signal hypothesis is ultimately the union of seven distinct sub-hypotheses that together describe all possible combinations of tensor, vector, and scalar polarizations <cit.>. To understand this, first define a “TVS” hypothesis that allows for the simultaneous presence of tensor, vector, and scalar polarization. In this case, we will model the stochastic energy-density spectrum as a sum of three power laws Ω_tvs(f) = Ω^T_0 (f/f_0)^α_T + Ω^V_0 (f/f_0)^α_V + Ω^S_0 (f/f_0)^α_S,with free parameters Ω^a_0 and α_a setting the amplitude and spectral index of each polarization sector. The priors on these parameters are given by Eqs. (<ref>) and (<ref>) below.In defining the TVS hypothesis, we have made the explicit assumption that tensor, vector, and scalar radiation are each present. This is not the only possibility, of course. A second distinct hypothesis, for instance, is that only tensor and vector polarizations exist. This is our “TV" hypothesis. We model the corresponding energy spectrum as Ω_tv(f) = Ω^T_0 (f/f_0)^α_T + Ω^V_0 (f/f_0)^α_V.In a similar fashion, we must ultimately define seven such hypotheses, denoted TVS, TV, TS, VS, T, V, and S, to encompass all combinations of tensor, vector, and scalar gravitational-wave backgrounds. Our complete signal hypothesis is given by the union of these seven sub-hypotheses <cit.>. For each signal sub-hypothesis, we adopt the log-amplitude and slope priors given below in Eqs. (<ref>) and (<ref>).Each of the signal sub-hypotheses are logically independent <cit.>, and so the odds ratiobetween signal and noise hypotheses is given by the sum of odds ratios between the noise hypothesis and each of the seven signal sub-hypotheses: = ∑_𝒜∈{,,,...}𝒪^𝒜_n.As illustrated in Fig. <ref>, we assign equal prior probability to the signal and noise hypotheses. Within the signal hypothesis, we weight each of the signal sub-hypotheses equally, such that the prior odds between e.g. the T and N hypothesis is π(T)/π(N)=1/7. We note that our choice of prior probabilities is not unique; there may exist other valid choices as well. Our analysis can easily accommodate different choices of prior weight.The odds ratiois constructed similarly. In this case, we are selecting between the hypothesis that the stochastic background is purely tensor-polarized (GR), or the hypothesis that additional polarization modes are present (NGR). The GR hypothesis is identical to our tensor-only hypothesis T from above: Ω_gr(f) = Ω^T_0 (f/f_0)^α_T.The NGR hypothesis, on the other hand, will be the union of the six signal sub-hypotheses that are inconsistent with general relativity: V, S, TV, TS, VS, and TVS. The complete odds ratio between NGR and GR hypothesis is then = ∑_𝒜∈{,,,...}𝒪^𝒜_.As shown in Fig. <ref>, we have assigned equal priors to the GR and NGR hypotheses as well as identical priors to the six NGR sub-hypotheses.In computing the odds ratiosand , we also need priors for the various parameters governing each model for the stochastic background. In the various energy-density models presented above, we have defined two classes of parameters: amplitudes Ω^a_0 and spectral indices α_a of the background's various polarization components. For each amplitude parameter, we will use the prior π(Ω_0) ∝ 1/Ω_0( Ω_Min≤Ω_0 ≤Ω_Max) 0(Otherwise) .This corresponds to a uniform prior in the log-amplitudes between logΩ_Min and logΩ_Max. In order for this prior to be normalizable, we cannot let it extend all the way to Ω_Min = 0 (logΩ_Min→ -∞). Instead, we must choose a finite lower bound. While this lower bound is somewhat arbitrary, our results depend only weakly on the specific choice of bound <cit.>. In this paper, we take Ω_Min = 10^-13, an amplitude that is indistinguishable from noise with Advanced LIGO. Our upper bound, meanwhile, is Ω_Max = 10^-6, consistent with upper limits placed by Initial LIGO and Virgo <cit.>.We adopt a triangular prior on α, centered at zero: π(α) = 1/α_Max(1-|α|/α_Max) ( |α| ≤α_Max) 0(Otherwise) .This prior has several desirable properties. First, it captures a natural tendency for spectral index posteriors to peak symmetrically about α = 0. As a result, our α posteriors reliably recover this prior in the absence of informative data (see Fig. <ref>, for example). Second, this prior preferentially weights shallower energy-density spectra. This quantifies our expectation that the stochastic background's energy density be distributed somewhat uniformly across logarithmic frequency intervals (at least in the LIGO band), rather than entirely at very high or very low frequencies.Alternatively, Eq. (<ref>) can be viewed as corresponding to equal priors on the background strength at two different frequencies. To understand this, first note that α may be written as a function of background amplitudes Ω_0 and Ω_1 at two frequencies f_0 and f_1: α(Ω_1,Ω_2) = log(Ω_1/Ω_0)/log(f_1/f_0).The prior probability of a particular slope α is equal to the probability of drawing any two amplitudes Ω_1 and Ω_2 satisfying log(Ω_1/Ω_2) = αlog(f_1/f_2). This is given by the convolution π(α) = ∫π(logΩ_1) π(logΩ_0 = logΩ_1 - αlog(f_1/f_0)) dlogΩ_1.For simplicity, we will set f_1 = 10 f_0 (such that log(f_1/f_0) = 1) and place identical log-uniform priors [Eq. (<ref>)] on each amplitude. Under these assumptions, Eq. (<ref>) yields Eq. (<ref>).In Sects. <ref> and <ref>, we additionally considered the performance of the three-detector Advanced LIGO-Virgo network. The Bayesian framework considered here is easily extended to accommodate multiple detector pairs. The three LIGO and Virgo detectors allow for the measurement of three cross-correlation spectra: Ĉ^hl(f), Ĉ^hv(f), and Ĉ^lv(f). In the small signal limit (Ω^a(f) ≪ 1), the correlations between these measurements vanish at leading order, and so the three baselines can be treated as statistically independent <cit.>. We can therefore factorize the joint likelihood for the three sets: ℒ({ Ĉ^hl, Ĉ^hv, Ĉ^lv} | θ,𝒜) = ℒ ({Ĉ^hl}|θ,𝒜) ℒ ({Ĉ^hv}|θ,𝒜) ℒ ({Ĉ^lv}|θ,𝒜) = 𝒩exp{ -1/2[ [t] ( Ĉ^hl-γ^hl_aΩ^a_𝒜 | Ĉ^hl-γ^hl_bΩ^b_𝒜) + ( Ĉ^hv-γ^hv_aΩ^a_𝒜 | Ĉ^hv-γ^hv_bΩ^b_𝒜) + ( Ĉ^lv-γ^lv_aΩ^a_𝒜 | Ĉ^lv-γ^lv_bΩ^b_𝒜) ]},substituting likelihoods of the form (<ref>) for each pair of detectors. Note that we have explicitly distinguished between the overlap reduction functions for each baseline, and 𝒩 is again a normalization constant. Other than the above change to the likelihood, all other details of the odds ratio construction is unchanged when including three detectors. § EVALUATING BAYESIAN EVIDENCES WITHHere we summarize details associated with usingto evaluate Bayesian evidences for various models of the stochastic background. Thealgorithm allows for several user-defined parameters, including the number n of live points used to sample the prior volume and the sampling efficiency ϵ, which governs acceptance rate of new proposed live points (see e.g. Ref. <cit.> for details).also provides the option to run in Default or Importance Nested Sampling (INS) modes, each of which use different methods to evaluate evidences <cit.>.To set the number of live points, we investigated the convergence of 's evidence estimates with increasing values of n. For a single simulated observation of a tensorial background (with amplitude Ω_0^T = 2×10^-8 and slope α_T=2/3), for instance, Fig. <ref> shows the recovered evidence for the T hypothesis (see Appendix <ref> above) as a function of n, using both the Default (blue) and INS modes (green). The results are reasonably stable for n≳1000; we choose n=2000 live points. Meanwhile, our recovered evidence estimates do not exhibit noticeable dependence on the sampling efficiency; we choose the recommended values ϵ=0.3 for evidence evaluation and ϵ=0.8 for parameter estimation <cit.>.In addition to computing Bayesian evidences,also returns an estimate of the numerical error associated with each evidence calculation. See, for instance, the error bars in Fig. <ref>. To gauge the accuracy of these error estimates, we construct a single simulated Advanced LIGO observation of a purely-tensorial stochastic background (again with Ω_0^T = 2×10^-8 and α_T=2/3). We then useto compute the corresponding TVS evidence 500 times, in both Default and INS modes. The resulting distributions of evidences are shown in Fig. <ref>. The dashed error bars show the averaged ±1σ intervals reported by , while the solid bars show the ±1σ intervals obtained manually from the distributions. We see that the errors reported by 's Default mode appear to accurately reflect the numerical error in the evidence calculation, while the errors reported by the INS mode are underestimated by a factor of ∼2.Additionally, Fig. <ref> illustrates several systematic differences between the Default and INS results. First, Default mode appears significantly more precise than INS mode, giving rise to a much narrower distribution of evidences. Not only is the INS evidence distribution wider, but it exhibits a large tail extending several units in evidence above the mean. We find that similarly long tails also appear for other pairs of injected signals and recovered models. For this reason, we choose to use 's Default mode in all evidence calculations. Typical numerical errors in Default mode are of order δ(evidence) ∼ 0.1, and so the uncertainty associated with a log-odds ratio is δ(ln𝒪)∼√(2)δ(evidence), again of order 0.1. Additionally, we see that the peaks of the Default and INS distributions do not coincide. In general, the peaks of evidence distributions from the Default and INS modes lie ∼0.3 units apart. Thus there may be additional systematic uncertainties in a given evidence calculation. However, as long as we consistently use one mode or the other (in our case, Default mode), any uniform systematic offset in the evidences will simply cancel when we ultimately compute a log-odds ratio. | http://arxiv.org/abs/1704.08373v2 | {
"authors": [
"Thomas Callister",
"A. Sylvia Biscoveanu",
"Nelson Christensen",
"Maximiliano Isi",
"Andrew Matas",
"Olivier Minazzoli",
"Tania Regimbau",
"Mairi Sakellariadou",
"Jay Tasson",
"Eric Thrane"
],
"categories": [
"gr-qc",
"astro-ph.HE"
],
"primary_category": "gr-qc",
"published": "20170426225516",
"title": "Polarization-based Tests of Gravity with the Stochastic Gravitational-Wave Background"
} |
Sunspots rotation and magnetic transients associated with flares in NOAA AR 11429 ^* * Supported by the National Natural Science Foundation of China.Jianchuan Zheng1 Zhiliang Yang1 Jianpeng Guo1, 2 Kaiming Guo1 Hui Huang1 Xuan Song1 Weixing Wan2, 3 December 30, 2023 ====================================================================================================================================================== We say that an algorithm is stable if small changes in the input result in small changes in the output. This kind of algorithm stability is particularly relevant when analyzing and visualizing time-varying data. Stability in general plays an important role in a wide variety of areas, such as numerical analysis, machine learning, and topology, but is poorly understood in the context of (combinatorial) algorithms.In this paper we present a framework for analyzing the stability of algorithms. We focus in particular on the tradeoff between the stability of an algorithm and the quality of the solution it computes.Our framework allows for three types of stability analysis with increasing degrees of complexity: event stability, topological stability, and Lipschitz stability.We demonstrate the use of our stability framework by applying it to kinetic Euclidean minimum spanning trees.§ INTRODUCTION With recent advances in technology, vast amounts of time-varying data are being processed and analyzed on a daily basis. Time-varying data play an important role in a many different application areas, such as computer graphics, robotics, simulation, and visualization. There is a great need for algorithms that can operate efficiently on time-varying data and that can offer guarantees on the quality of the results. A specific relevant subset of time-varying data are motion data: geometric time-varying data. To deal with the challenges of motion data, Basch <cit.> introduced the kinetic data structures (KDS) framework in 1999. Kinetic data structures are data structures that efficiently maintain a structure on a set of moving objects.The performance of a particular algorithm is usually judged with respect to a variety of criteria, with the two most common being solution quality and running time. In the context of algorithms for time-varying data, a third important criterion is stability. Whenever analysis results on time-varying data need to be communicated to humans, for example via visual representations, it is important that these results are stable: small changes in the data result in small changes in the output. These changes in the output are continuous or discrete depending on the algorithm: graphs usually undergo discrete changes while a convex hull of moving points changes continuously. Sudden changes in the visual representation of data disrupt the mental map <cit.> of the user and prevent the recognition of temporal patterns. Stability also plays a role if changing the result is costly in practice, for example in physical network design, where frequent total overhauls of the network are simply infeasible.The stability of algorithms or methods has been well-studied in a variety of research areas, such as numerical analysis <cit.>, machine learning <cit.>, control systems <cit.>, and topology <cit.>. In contrast, the stability of combinatorial algorithms for time-varying data has received little attention in the theoretical computer science community so far. Here it is of particular interest to understand the tradeoffs between solution quality, running time, and stability. As an example, consider maintaining a minimum spanning tree of a set of moving points. If the points move, it might have to frequently change significantly. On the other hand, if we start with an MST for the input point set and then never change it combinatorially as the points move, the spanning tree we maintain is very stable – but over time it can devolve to a low quality and very long spanning tree.Our goal, and the focus of this paper, is to understand the possible tradeoffs between solution quality and stability. This is in contrast to earlier work on stability in other research areas, such as the ones mentioned above, where stability is usually considered in isolation. Since there are currently no suitable tools available to formally analyze tradeoffs involving stability, we introduce a new analysis framework. We believe that there are many interesting and relevant questions to be solved in the general area of algorithmic stability analysis and we hope that our framework is a first meaningful step towards tackling them.Results and organization We present a framework to measure and analyze the stability of algorithms.As a first step, we limit ourselves to analyzing the tradeoff between stability and solution quality, omitting running time from consideration.Our framework allows for three types of stability analysis of increasing degrees of complexity: event stability, topological stability, and Lipschitz stability.It can be applied both to motion data and to more general time-varying data. We demonstrate the use of our stability framework by applying it to the problem of kinetic Euclidean minimum spanning trees (EMSTs). Some of our results for kinetic EMSTs are directly more widely applicable.In Section <ref> we give an overview of our framework for stability. In Sections <ref>, <ref>, and <ref> we describe event stability, topological stability, and Lipschitz stability, respectively. In each of these sections we first describe the respective type of stability analysis in a generic setting, followed by specific results using that type of stability analysis on the kinetic EMST problem. In Section <ref> we make some concluding remarks on our stability framework. Omitted proofs can be found in Appendix <ref>. Related work Stability is a natural point of concern in more visual and applied research areas such as graph drawing, (geo-)visualization, and automated cartography. For example, in dynamic map labelling <cit.>, the consistent dynamic labelling model allows a label to appear and disappear only once, making it very stable. There are very few theoretical results, with the noteworthy exception of so-called simultaneous embeddings <cit.> in graph drawing, which can be seen as a very restricted model of stability. However, none of these results offer any real structural insight into the tradeoff between solution quality and stability.In computational geometry there are a few results on the tradeoff between solution quality and stability. pecifically, Durocher and Kirkpatrick <cit.> study the stability of centers of kinetic point sets, and define the notion of κ-stable center functions, which is closely related to our concept of Lipschitz stability. In later work <cit.> they consider the tradeoff between the solution quality of Euclidean 2-centers and a bound on the velocity with which they can move. De Berg <cit.> show similar results in the black-box KDS model. One can argue that the KDS framework <cit.> already indirectly considers stability in a limited form, namely as the number of external events. However, the goal of a KDS is typically to reduce the running time of the algorithm, and rarely to sacrifice the running time or quality of the results to reduce the number of external events.Kinetic Euclidean minimum spanning trees and related structures have been studied extensively. Katoh <cit.> proved an upper bound of O(n^3 2^α(n)) for the number of external events of EMSTs of n linearly moving points, where α(n) is the inverse Ackermann function. Rahmati <cit.> present a kinetic data structure for EMSTs in the plane that processes O(n^3 β_2s + 2^2(n) log n) events in the worst case, where s is a measure for the complexity of the point trajectories and β_s(n) is an extremely slow-growing function. The best known lower bound for external events of EMSTs in d dimensions is Ω(n^d) <cit.>. Since the EMST is a subset of the Delaunay triangulation, we can also consider to kinetically maintain the Delaunay triangulation instead. Fu and Lee <cit.>, and Guibas <cit.> show that the Delaunay triangulation undergoes O(n^2 λ_s+2(n)) external events (near-cubic), where λ_s(n) is the maximum length of an (n, s)-Davenport-Schinzel sequence <cit.>. On the other hand, the best lower bound for external events of the Delaunay triangulation is only Ω(n^2) <cit.>. Rubin improves the upper bound to O(n^2+ε), for any ε > 0, if the number of degenerate events is limited <cit.>, or if the points move along a straight line with unit speed <cit.>). Agarwal <cit.> also consider a more stable version of the Delaunay triangulation, which undergoes at most a nearly quadratic number of external events. However, external events for EMSTs do not necessarily coincide with external events of the Delaunay triangulation <cit.>. To further reduce the number of external events, we can consider approximations of the EMST, for example via spanners or well-separated pair decompositions <cit.>. However, kinetic t-spanners already undergo Ω(n^2/t^2) external events <cit.>. Our stability framework allows us to reduce the number of external events even further and to still state something meaningful about the quality of the resulting EMSTs.§ STABILITY FRAMEWORK Intuitively, we can say that an algorithm is stable if small changes in the input lead to small changes in the output. More formally, we can formulate this concept generically as follows. Let Π be an optimization problem that, given an input instance I from a set ℐ, asks for a feasible solution S from a set 𝒮 that minimizes (or maximizes) some optimization function fℐ×𝒮→. An algorithm 𝒜 for Π can be seen as a function 𝒜ℐ→𝒮. Similarly, the optimal solutions for Π can be described by a function ℐ→𝒮. To define the stability of an algorithm, we need to quantify changes in the input instances and in the solutions. We can do so by imposing a metric[A metric would typically be the most suitable solution, but any dissimilarity function is sufficient.] on ℐ and 𝒮. Let d_ℐℐ×ℐ→_≥ 0 be a metric for ℐ and let d_𝒮𝒮×𝒮→_≥ 0 be a metric for 𝒮. We can then define the stability of an algorithm 𝒜ℐ→𝒮 as follows. (𝒜) = max_I, I' ∈ℐd_𝒮(𝒜(I),𝒜(I'))/d_ℐ(I,I') This definition for stability is closely related to that of the multiplicative distortion of metric embeddings, where 𝒜 induces a metric embedding from the metric space (ℐ, d_ℐ) into (𝒮, d_𝒮). The lower the value for (𝒜), the more stable we consider the algorithm 𝒜 to be. As with the distortion of metric embeddings, there are many other ways to define the stability of an algorithm given the metrics, but the above definition is sufficient for our purpose.For many optimization problems, the functionmay be very unstable. This suggests an interesting tradeoff between the stability of an algorithm and the solution quality. Unfortunately, the generic formulation of stability provided above is very unwieldy. It is not always clear how to define metrics d_ℐ and d_𝒮 such that meaningful results can be derived. Additionally, it is not obvious how to deal with optimization problems with continuous input and discrete solutions, where the algorithm is inherently discontinuous, and thus the stability is unbounded by definition. Finally, analyses of this form are often very complex, and it is not straightforward to formulate a simplified version of the problem. In our framework we hence distinguish three types of stability analysis: event stability, topological stability, and Lipschitz stability.Event stability follows the setting of kinetic data structures (KDS). That is, the input (a set of moving objects) changes continuously as a function over time. However, contrary to typical KDSs where a constraint is imposed on the solution quality, we aim to enforce the stability of the algorithm. For event stability we simply disallow the algorithm to change the solution too rapidly. Doing so directly is problematic, but we formalize this approach using the concept of k-optimal solutions. As a result, we can obtain a tradeoff between stability and quality that can be tuned by the parameter k. Note that event stability captures only how often the solution changes, but not how much the solution changes at each event.Topological stability takes a first step towards the generic setup described above. However, instead of measuring the amount of change in the solution using a metric, we merely require the solution to behave continuously. To do so we only need to define a topology on the solution space 𝒮 that captures stable behavior. Surprisingly, even though we completely ignore the amount of change in a single time step, this type of analysis still provides meaningful information on the tradeoff between solution quality and stability. In fact, the resulting tradeoff can be seen as a lower bound for any analysis involving metrics that follow the used topology.Lipschitz stability finally captures the generic setup described above. As the name suggests, we require the algorithm to be Lipschitz continuous and we provide an upper bound on the Lipschitz constant, which is equivalent to (𝒜). We are then again interested in the quality of the solutions that can be obtained with any Lipschitz stable algorithm. Given the complexity of this type of analysis, a complete tradeoff for any value of the Lipschitz constant is typically out of reach, but some results may be obtained for values that are sufficiently small or large.Remark Our framework makes the assumption that an algorithm is a function 𝒜ℐ→𝒮. However, in a kinetic setting this is not necessarily true, since the algorithm has history. More precisely, for some input instance I, a kinetic algorithm may produce different solutions for I based on the instances processed earlier. We generally allow this behavior, and for event stability this behavior is even crucial. However, for the sake of simplicity, we will treat an algorithm as a function. We also generally assume in our analysis that the input is time-varying, that is, the input is a function over time, or follows a trajectory through the input space ℐ. Again, for the sake of simplicity, this is not always directly reflected in our definitions. Beyond that, we operate in the black-box model, in the sense that the algorithm does not know anything about future instances.While these are the conditions under which we use our framework, it can be applied in a variety of algorithmic settings, such as streaming algorithms and algorithms with dynamic input.§ EVENT STABILITY The simplest and most intuitive form of stability is event stability. Similarly to the number of external events in KDSs, event stability captures only how often the solution changes. §.§ Event stability analysisLet Π be an optimization problem with a set of input instances ℐ, a set of solutions 𝒮, and optimization function fℐ×𝒮→. Following the framework of kinetic data structures, we assume that the input instances include certain parameters that can change as a function of time. To apply the event stability analysis, we require that all solutions have a combinatorial description, that is, the solution description does not use the time-varying parameters of the input instance. We further require that every solution S ∈𝒮 is feasible for every input instance I ∈ℐ. This automatically disallows any insertions or deletions of elements. Note that an insertion or a deletion would typically force an event, and thus including this aspect in our stability analysis does not seem useful.For example, in the setting of kinetic EMSTs, the input instances would consist of a fixed set of points. The coordinates of these points can then change as a function over time. A solution of the kinetic EMST problem consists of the combinatorial description of a tree graph on the set of input points. Note that every tree graph describes a feasible solution for any input instance, if we do not insist on any additional restrictions like, e.g., planarity. The minimization function f then simply measures the total length of the tree, for which we do need to use the time-varying parameters of the problem instance.Rather than directly restricting the quality of the solutions, we aim to restrict the stability of any algorithm. To that end, we introduce the concept of k-optimal solutions. Let d_ℐ be a metric on the input instances, and let ℐ→𝒮 describe the optimal solutions. We say that a solution S ∈𝒮 is k-optimal for an instance I ∈ℐ if there exists an input instance I' ∈ℐ such that f(I', S) = f(I', OPT(I')) and d_ℐ(I, I') ≤ k. With this definition any optimal solution is always 0-optimal. Note that this definition requires a form of normalization on the metric d_ℐ, similar to that of e.g. smoothed analysis <cit.>. We therefore require that there exists a constant c such that every solution S ∈𝒮 is c-optimal for every instance I ∈ℐ. For technical reasons we require the latter condition to hold only for some time interval [0, T] of interest.Following the framework of kinetic data structures, we typically require the functions of the time-varying parameters to be well-behaved (e.g., polynomial functions), for otherwise we cannot derive meaningful bounds. The event stability analysis then considers two aspects. First, we analyze how often the solution needs to change to maintain a k-optimal solution for every point in time. Second, we analyze how well a k-optimal solution approximates an optimal solution. Typically we are not able to directly obtain good bounds on the approximation ratio, but given certain reasonable assumptions, good approximation bounds as a function of k can be provided. §.§ Event stability for EMSTsOur input consists of a set of points P = {p_1, …, p_n} where each point p_i has a trajectory described by the function x_i [0, T] →^d. The goal is to maintain a combinatorial description of a short spanning tree on P that does not change often. We generally assume that the functions x_i are polynomials with bounded degree s.To properly use the concept of k-optimal solutions, we first need to normalize the coordinates. We simply assume that x_i(t) ∈ [0, 1]^d for t ∈ [0, T]. This assumption may seem overly restrictive for kinetic point sets, but note that we are only interested in relative positions, and thus the frame of reference may move with the points. Next, we define the metric d_ℐ along the trajectory as follows. d_ℐ(t, t') = max_i x_i(t) - x_i(t') Note that this metric, and the resulting definition of k-optimal solutions, is not specific to EMSTs and can be used in general for problems with kinetic point sets as input. In our case a-b denotes the distance between a and b in the (Euclidean) ℓ_2 norm. Now let OPT(t) be the EMST at time t. Then, by definition, OPT(t) is k-optimal at time t' if d_ℐ(t, t') ≤ k. Our approach is now very simple: we compute the EMST and keep that solution as long as it is k-optimal, after which we compute the new EMST, and so forth. Below we analyze how often we need to recompute the EMST, and how well a k-optimal solution approximates the EMST.Number of events To derive an upper bound on the number of events, we first need to bound the speed of any point with a polynomial trajectory and bounded coordinates. For this we can use a classic result known as the Markov Brothers' inequality. Let h(t) be a polynomial with degree at most s such that h(t) ∈ [0, 1] for t ∈ [0, T], then |h'(t)|≤ s^2 / T for all t ∈ [0, T].Let P be a kinetic point set with polynomial trajectories x_i(t) ∈ [0, 1]^d (t ∈ [0, T]) of degree at most s, then we need only O(s^2/k) changes to maintain a k-optimal solution. By Lemma <ref> the velocity of any point is at most s^2 / T. Now assume that we have computed an optimal solution S for some time t. The solution S remains k-optimal until one of the points has moved at least k units. Since the velocity of the points is bounded, this takes at least Δ t = k T / s^2 time, at which point we can recompute the optimal solution. Since the total time interval is of length T, this can happen at most T / Δ t = s^2 / k times. Next we show that this upper bound is tight up to a factor of s, using Chebyshev polynomials of the first kind <cit.>. A Chebyshev polynomial of degree s with range [0, 1] and domain [0, T] will pass through the entire range exactly s times. Let P be a kinetic point set with n points with polynomial trajectories x_i(t) ∈ [0, 1]^d (t ∈ [0, T]) of degree at most s, then we need Ω(min(s/k, s n)) changes in the worst case to maintain a k-optimal solution. We can restrict ourselves to d=1. Let p_1 move along a Chebyshev polynomial of degree s, and let the remaining points be stationary and placed equidistantly along the interval [0, 1]. As soon as p_1 meets one of the other points, then p_1 can travel at most k units before the solution is no longer k-optimal (see Fig.<ref>). Therefore, p_1 moving through the entire interval requires Ω(min(1/k, n)) changes to the solution. Doing so s times gives the desired bound. It is important to notice that this behavior is fairly special for polynomial trajectories. If we allow more general trajectories, then this bound on the number of changes breaks down. Let P be a kinetic point set with n points with pseudo-algebraic trajectories x_i(t) ∈ [0, 1]^d (t ∈ [0, T]) of degree at most s, then we need Ω(min(sn/k, sn^2)) changes in the worst case to maintain a k-optimal solution. We can restrict ourselves to d=1. Any two pseudo-algebraic trajectories of degree at most s can cross each other at most s times. We make n/2 points stationary and place them equidistantly along the interval [0, 1]. The other n/2 points follow trajectories that take them through the entire interval s times, in such a way that every point moves through the entire interval completely before another point does so. The resulting trajectories are clearly pseudo-algebraic, and each time a point moves through the entire interval it requires Ω(min(1/k, n)) changes to the solution. As a result, the total number of changes is Ω(min(sn/k, sn^2)). We can show the same lower bound for algebraic trajectories of degree at most s, but this is slightly more involved. The result can be found in Appendix <ref>.Approximation factor We cannot expect k-optimal solutions to be a good approximation of optimal EMSTs in general: if all points are within distance k from each other, then all solutions are k-optimal. We therefore need to make the assumption that the points are spread out reasonably throughout the motion. To quantify this, we use a measure inspired by the order-l spread, as defined in <cit.>. Let mindist_l(P) be the smallest distance in P between a point and its l-th nearest neighbor. We assume that mindist_l(P) ≥ 1/Δ_l throughout the motion, for some value of Δ_l. We can use this assumption to give a lower bound on the length of the EMST. Pick an arbitrary point and remove all points from P that are within distance 1/Δ_l, and repeat this process until the smallest distance is at least 1/Δ_l. By our assumption, we remove at most l-1 points for each chosen point, so we are left with at least n/l points. The length of the EMST on P is at least the length of the EMST on the remaining n/l points, which has length Ω(n/lΔ_l). A k-optimal solution of the EMST problem on a set of n points P is an O(1 + klΔ_l)-approximation of the EMST, under the assumption that mindist_l(P) ≥ 1/Δ_l. Let S be a k-optimal solution of P and letbe an optimal solution of P. By definition there is a point set P' for which the length of solution S is at most that of . Since d_ℐ(P, P') ≤ k, the length of each edge can grow or shrink by at most 2k when moving from P' to P. Therefore we can state that f(P, S) ≤ f(P, ) + 4 k n. Now, using the lower bound on the length of an EMST, we obtain the following. f(P, ) + 4kn≤ f(P, ) + 4k O(f(P, ) l Δ_l) = O(1 + k l Δ_l)· f(P, )Note that there is a clear tradeoff between the approximation ratio and how restrictive the assumption on the spread is. Regardless, we can obtain a decent approximation while only processing a small number of events. If we choose reasonable values k = O(1/n), l = O(1), and Δ_l = O(n), then our results show that, under the assumptions, a constant-factor approximation of the EMST can be maintained while processing only O(n) events. § TOPOLOGICAL STABILITYThe event stability analysis has two major drawbacks: (1) it is only applicable to problems for which the solutions are always feasible and described combinatorially, and (2) it does not distinguish between small and large structural changes. Topological stability analysis is applicable to a wide variety of problems and enforces continuous changes to the solution. §.§ Topological stability analysisLet Π be an optimization problem with input instances ℐ, solutions 𝒮, and optimization function f. An algorithm 𝒜ℐ→𝒮 is topologically stable if, for any (continuous) path π [0,1] →ℐ in ℐ, 𝒜π is a (continuous) path in 𝒮. To properly define a (continuous) path in ℐ and 𝒮 we need to specify a topology 𝒯_ℐ on ℐ and a topology 𝒯_𝒮 on 𝒮. Alternatively we could specify metrics d_ℐ and d_𝒮, but this is typically more involved. We then want to analyze the approximation ratio of any topologically stable algorithm with respect to . That is, we are interested in the ratio (Π, 𝒯_ℐ, 𝒯_𝒮) = inf_𝒜sup_I ∈ℐf(I, 𝒜(I))/f(I, (I)) where the infimum is taken over all topologically stable algorithms. Naturally, ifis already topologically stable, then this type of analysis does not provide any insight and the ratio is simply 1. However, in many cases,is not topologically stable. The above analysis can also be applied if the solution space (or the input space) is discrete. In such cases, continuity can often be defined using the graph topology of so-called flip graphs, for example, based on edge flips for triangulations or rotations in rooted binary trees. We can represent a graph as a topological space by representing vertices by points, and representing every edge of the graph by a copy of the unit interval [0,1]. These intervals are glued together at the vertices. In other words, we consider the corresponding simplicial 1-complex. Although the points in the interior of the edges of this topological space do not represent proper spanning trees, we can still use this topological space in Equation <ref> by extending f over the edges via linear interpolation. It is not hard to see that we need to consider only the vertices of the flip graph (which represent proper spanning trees) to compute the topological stability ratio. §.§ Topological stability of EMSTsWe use the same setting of the kinetic EMST problem as in Section <ref>, except that we do not restrict the trajectories of the points and we do not normalize the coordinates. We merely require that the trajectories are continuous. To define this properly, we need to define a topology on the input space, but for a kinetic point set with n points in d dimensions we can simply use the standard topology on ^dn as 𝒯_ℐ. To apply topological stability analysis, we also need to specify a topology on the (discrete) solution space. As the points move, the minimum spanning tree may have to change at some point in time by removing one edge and inserting another edge. Since these two edges may be very far apart, we do not consider this operation to be stable or continuous. Instead we specify the topology of 𝒮 using a flip graph, where the operations are either edge slides or edge rotations <cit.>. The optimization function f, measuring the quality of the EMST, is naturally defined for the vertices of the flip graph as the length of the spanning tree, and we use linear interpolation to define f on the edges of the flip graph. For edge slides and rotations we provide upper and lower bounds on (EMST, 𝒯_ℐ, 𝒯_𝒮). Edge slides An edge slide is defined as the operation of moving one endpoint of an edge to one of its neighboring vertices along the edge to that neighbor. More formally, an edge (u,v) in the tree can be replaced by (u,w) if w is a neighbor of v and w ≠ u. Since this operation is very local, we consider it to be stable. Note that after every edge slide the tree must still be connected.If 𝒯_𝒮 is the topology on 𝒮 defined by edge slides, then (EMST, 𝒯_ℐ, 𝒯_𝒮) ≤3/2. [10]r0pt< g r a p h i c s > x is the longest edge during an edge slide from e to e'. Proof. Consider a point in time where the EMST has to be updated by removing an edge e and inserting an edge e', where |e| = |e'|. Note that e and e' form a cycle C with other edges of the EMST. We now slide edge e to edge e' by sliding it along the vertices of C. Let x be the longest intermediate edge when sliding from e to e' (see Fig. <ref>). To allow x to be as long as possible with respect to the length of the EMST, the EMST should be fully contained in C. By the triangle inequality we get that 2 |x| ≤ |C|. Since the length of the EMST is = |C| - |e|, we get that |x| ≤/2 + |e|/2. Thus, the length of the intermediate tree is |C| - 2|e| + |x| =- |e| + |x| ≤3/2. If 𝒯_𝒮 is the topology on 𝒮 defined by edge slides, then, for any ε > 0, (EMST, 𝒯_ℐ, 𝒯_𝒮) ≥π +1/π - ε. Consider a point in time where the EMST has to be updated by removing an edge e and inserting an edge e', where |e| is very small. Let the remaining points be arranged in a circle, as shown in Figure <ref>(a), such that the farthest distance between any two points is /π - ε, whereis the length of the EMST. We can make this construction for any ε > 0 by using enough points and making e and e' arbitrarily short. Simply sliding e to e' will always grow e to be the diameter of the circle at some point. Alternatively, e can take a shortcut by sliding over another edge f as a chord (see Figure <ref>(b)). This is only beneficial if |e|+|f|</π - ε. However, if f helps e to avoid becoming a diameter of the circle, then e and f, as chords, must span an angle larger than π together. As a result, |e|+|f|≥/π - ε by triangle inequality. A motion of the points that forces e to slide to e' in this particular configuration looks as follows. The points start at e and move at constant speed along the circle, half of the points clockwise and the other half counter clockwise. The speeds are assigned in such a way that at some point all points are evenly spread along the circle. Once all points are evenly spread, they start moving towards e', again along the circle. It is easy to see that, using the arguments above, any additional edges inside the circle must have total length of at least the diameter of the circle at some point throughout the motion. On the other hand,is largest when the points are evenly spread along the circle. Thus, for any ε > 0, (EMST, 𝒯_ℐ, 𝒯_𝒮) ≥π +1/π - ε≈ 1.318 - ε. Edge rotations Edge rotations are a generalization of edge slides, that allow one endpoint of an edge to move to any other vertex. These operations are clearly not as stable as edge slides, but they are still more stable than the deletion and insertion of arbitrary edges. If 𝒯_𝒮 is the topology on 𝒮 defined by edge rotations, then (EMST, 𝒯_ℐ, 𝒯_𝒮) ≤4/3. Consider a point in time where the EMST has to be updated by removing an edge e = (u, v) and inserting an edge e' = (u', v'), where |e| = |e'|. Note that e and e' form a cycle C with other edges of the EMST. We now rotate edge e to edge e' along some of the vertices of C. Let x be the longest intermediate edge when rotating from e to e'. To allow x to be as long as possible with respect to the length of the EMST, the EMST should be fully contained in C. We argue that |x| ≤/3 + |e|, whereis the length of the EMST. Removing e and e' from C will split C into two parts, where we assume that u and u' (v and v') are in the left (right) part. First assume that one of the two parts has length at most /3. Then we can rotate e to (u, v'), and then to e', which implies that |x| = |(u, v')| ≤/3 + |e| by the triangle inequality (see Fig. <ref>). Now assume that both parts have length at least /3. Let e_L = (u_L, v_L) be the edge in the left part that contains the midpoint of that part, and let e_R = (u_R, v_R) be the edge in the right part that contains the midpoint of that part, where u_L and u_R are closest to e (see Fig. <ref>). Furthermore, let Z be the length of C∖{e, e', e_L, e_R}. Now consider the potential edges (u, v_R), (v, v_L), (u', u_R), and (v', u_L). By the triangle inequality, the sum of the lengths of these edges is at most 4|e| + 2|e_L| + 2|e_R| + Z. Thus, one of these potential edges has length at most |e| + |e_L|/2 + |e_R|/2 + Z/4. Without loss of generality let (u, v_R) be that edge (the construction is fully symmetric). We can now rotate e to (u, v_R), then to (u', v_R), and finally to e'. As each part of C has length at most 2/3, we get that |(u', v_R)| ≤/3 + |e| by construction. Furthermore we have that = |e| + |e_L| + |e_R| + Z. Thus, |(u, v_R)| ≤ |e| + |e_L|/2 + |e_R|/2 + Z/4 = /3 + 2|e|/3 + |e_L|/6 + |e_R|/6 - Z/12. Since e needs to be removed to update the EMST, it must be the longest edge in C. Therefore |(u, v_R)| ≤/3 + |e|, which shows that |x| ≤/3 + |e|. Since the length of the intermediate tree is - |e| + |x| ≤4/3, we obtain that (EMST, 𝒯_ℐ, 𝒯_𝒮) ≤4/3. If 𝒯_𝒮 is defined by edge rotations, then, (EMST, 𝒯_ℐ, 𝒯_𝒮) ≥10-2√(2)/9-2√(2).[11]r0pt< g r a p h i c s > Lower bound construction for edge rotations. Proof. Consider a point in time where the EMST has to be updated by removing an edge e and inserting an edge e'. Let the remaining points be arranged in a diamond shape as shown in Figure <ref>, where the side length of the diamond is 2, and |e| = |e'| = 1. As a result, the distance between an endpoint of e and the left or right corner of the diamond is 2 - 1/2√(2). Now we define a top-connector as an edge that intersects the vertical diagonal of the diamond, but is completely above the horizontal diagonal of the diamond. A bottom-connector is defined analogously, but must be completely below the horizontal diagonal. Finally, a cross-connector is an edge that crosses or touches both diagonals of the diamond. Note that a cross-connector has length at least 2, and a top- or bottom-connector has length at least |e| = 1. In the considered update, we start with a top-connector and end with a bottom-connector. Since we cannot rotate from a top-connector to a bottom-connector in one step, we must reach a state that either has both a top-connector and a bottom-connector, or a single cross-connector. In both options the length of the spanning tree is 10 - 2 √(2), while the minimum spanning tree has length 9 - 2 √(2).To force the update from e to e' in this configuration, we can use the following motion. The points start at the endpoints of e and move with constant speeds to a position where the points are evenly spread around the left and right corner of the diamond. Then the points move with constant speeds to the endpoints of e'. The argument above still implies that we need edges of total length at least 2 intersecting the vertical diagonal of the diamond at some point during the motion. On the other hand, OPT ≤ 9 - 2 √(2) throughout the motion. Thus (EMST, 𝒯_ℐ, 𝒯_𝒮) ≥10-2√(2)/9-2√(2)≈ 1.162. § LIPSCHITZ STABILITYThe major drawback of topological stability analysis is that it still does not fully capture stable behavior; the algorithm must be continuous, but we can still make many changes to the solution in an arbitrarily small time step. In Lipschitz stability analysis we additionally limit how fast the solution can change. §.§ Lipschitz stability analysisLet Π be an optimization problem with input instances ℐ, solutions 𝒮, and optimization function f. To quantify how fast a solution changes as the input changes, we need to specify metrics d_ℐ and d_𝒮 on ℐ and 𝒮, respectively. An algorithm 𝒜ℐ→𝒮 is K-Lipschitz stable if for any I, I' ∈ℐ we have that d_𝒮(𝒜(I), 𝒜(I')) ≤ K d_ℐ(I, I'). We are then again interested in the approximation ratio of any K-Lipschitz stable algorithm with respect to . That is, we are interested in the ratio (Π, K, d_ℐ, d_𝒮) = inf_𝒜sup_I ∈ℐf(I, 𝒜(I))/f(I, (I)) where the infimum is taken over all K-Lipschitz stable algorithms. It is easy to see that (Π, K, d_ℐ, d_𝒮) is lower bounded by (Π, 𝒯_ℐ, 𝒯_𝒮) for the corresponding topologies 𝒯_ℐ and 𝒯_𝒮 of d_ℐ and d_𝒮, respectively. As already mentioned in Section <ref>, analyses of this type are often quite hard. First, we often need to be very careful when choosing the metrics d_ℐ and d_𝒮, as they should behave similarly with respect to scale. For example, let the input consist of a set of points in the plane and let c I for I ∈ℐ be the instance obtained by scaling all coordinates of the points in I by the factor c. Now assume that d_ℐ depends linearly on scale, that is d_ℐ(c I, c I') ∼ c d_ℐ(I, I'), and that d_𝒮 is independent of scale. Then, for some fixed K, we can reduce the effective speed of any K-Lipschitz stable algorithm arbitrarily by scaling down the instances sufficiently, rendering the analysis meaningless. We further need to be careful with discrete solution spaces. However, using the flip graphs as mentioned in Section <ref> we can extend a discrete solution space to a continuous space by including the edges.Typically it will be hard to fully describe (Π, K, d_ℐ, d_𝒮) as a function of K. However, it may be possible to obtain interesting results for certain values of K. One value of interest is the value of K from which the approximation ratio equals or approaches the approximation ratio of the corresponding topological stability analysis. Another potential value of interest is the value of K below which any K-Lipschitz stable algorithm performs asymptotically as bad as a constant algorithm always computing the same solution regardless of instance.§.§ Lipschitz stability of EMSTsWe use the same setting of the kinetic EMST problem as in Section <ref>, except that, instead of topologies, we specify metrics for ℐ and 𝒮. For d_ℐ we can simply use the metric in Equation <ref>, which implies that points move with a bounded speed. For d_𝒮 we use a metric inspired by the edge slides of Section <ref>. To that end, we need to define how long a particular edge slide takes, or equivalently, how “far” an edge slide is. To make sure that d_ℐ and d_𝒮 behave similarly with respect to scale, we let d_𝒮 measure the distance the sliding endpoint has traveled during an edge slide. However, this creates an interesting problem: the edge on which the endpoint is sliding may be moving and stretching/shrinking during the operation (see Fig. <ref>). This influences how long it takes to perform the edge slide. We need to be more specific: (1) As the points are moving, the relative position (between 0 and 1 from starting endpoint to finishing endpoint) of a sliding endpoint is maintained without cost in d_𝒮, and (2) d_𝒮 measures the difference in relative position multiplied by the length L(t) of the edge on which the endpoint is sliding. More tangibly, an edge slide performed by a K-Lipschitz stable algorithm can be performed in t^* time such that ∫_0^t^*K/L(t)d t = 1, where L(t) describes the length of the edge on which the endpoint slides as a function of time. Finally, the optimization function f simply computes a linear interpolation of the cost on the edges of the flip graph defined by edge slides.We now give an upper bound on K below which any K-Lipschitz stable algorithm for kinetic EMST performs asymptotically as bad as any fixed tree. Given the complexity of the problem, our bound is fairly crude. We provide it anyway to demonstrate the use of our framework, but we believe that a stronger bound exists. First we show the asymptotic approximation ratio of any spanning tree. Any spanning tree on a set of n points P is an O(n)-approximation of the EMST.Let d_𝒮 be the metric for edge slides, then (EMST, c/log n, d_ℐ, d_𝒮) = Ω(n) for a small enough constant c > 0, where n is the number of points.Consider the instance where n points are placed equidistantly vertically above each other with distance 1/n between two consecutive points. Now let 𝒜 be any (c/log n)-Lipschitz stable algorithm for the kinetic EMST problem and let T be the tree computed by 𝒜 on this point set. We now color the points red and blue based on a 2-coloring of T. We then move the red points to the left by 1/2 and the blue points to the right by 1/2 in the time interval [0, 1] (see Fig. <ref> left). This way every edge of T will be stretched to a length of Ω(1) and thus the length of T will be Ω(n). On the other hand, the length of the EMST in the final configuration is = O(1). Therefore, we must perform an edge slide (see Fig. <ref> middle). However, we show that 𝒜 cannot complete any edge slide. Consider any edge of T and let x be the initial (vertical) distance between its endpoints. Then the length of this edge can be described as L(t) = √(x^2 + t^2) (see Fig. <ref> right). Now assume that we want to slide an endpoint over this edge. To finish this edge slide before t = 1, we require that ∫_0^1 c/log n √(x^2 + t^2)d t ≥ 1. This solves to c log(1/x + √(1 + 1/x^2)) ≥log n. However, since x ≥ 1/n, we get that c log(1/x + √(1 + 1/x^2)) ≤ c log(n + √(1 + n^2)) < log n for c small enough. Finally, since one edge slide can reduce the length of only one edge to o(1), the cost of the solution at t = 1 computed by 𝒜 is Ω(n). Thus, (EMST, c/log n, d_ℐ, d_𝒮) = Ω(n) for a small enough constant c > 0. § CONCLUSIONWe presented a framework for algorithm stability, which includes three types of stability analysis, namely event stability, topological stability, and Lipschitz stability. We also demonstrated the use of this framework by applying the different types of analysis to the kinetic EMST problem, deriving new interesting results. We believe that, by providing different types of stability analysis with increasing degrees of complexity, we make stability analysis for algorithms more accessible, and make it easier to formulate interesting open problems with regard to algorithm stability.However, the framework that we have presented is not designed to offer a complete picture on algorithm stability. In particular, we do not consider the algorithmic aspect of stability. For example, if we already know how the input will change over time, can we efficiently compute a stable function of solutions over time that is optimal with regard to solution quality? Or, in a more restricted sense, can we efficiently compute the one solution that is best for all inputs over time? Even in the black-box model we can consider designing efficient algorithms that are K-Lipschitz stable and perform well with regard to solution quality. We leave such problems for future work.Acknowledgements W. Meulemans and J. Wulms are (partially) supported by the Netherlands eScience Center (NLeSC) under grant number 027.015.G02. B. Speckmann and K. Verbeek are supported by the Netherlands Organisation for Scientific Research (NWO) under project no. 639.023.208 and no. 639.021.541, respectively.§ OMITTED PROOFS Let P be a kinetic point set with n points with algebraic trajectories x_i(t) ∈ [0, 1]^d (t ∈ [0, T]) of degree at most s, then we need Ω(min(sn/k, sn^2)) changes in the worst case to maintain a k-optimal solution. We can restrict ourselves to d=1. We make n/2 points stationary and place them equidistantly along the interval [0, 1]. The other n/2 points follow trajectories that take them through the entire interval s/4 times, in such a way that every point moves through the entire interval completely before another point does so. The trajectory of a non-stationary point p_i is x_i(t) = ∑_j=0^s/41/(t-10· j-10· i· s/4)^4+1. The trajectory consists of s/4 moves through the stationary points, one such move every 10 time units (see Fig. <ref>). The i-th point will be finished 10* s/4 time units after it starts its first move through the stationary points, while the (i+1)-st point starts 10 units after the i-th point finishes. The resulting trajectories are clearly algebraic, and each time a point moves through the entire interval it requires Ω(min(1/k, n)) changes to the solution. As a result, the total number of changes is Ω(min(sn/k, sn^2)).lemmalem:napproxtree Any spanning tree on a set of n points P is an O(n)-approximation of the EMST. Let T be an EMST on point set P with total edge length . Additionally let u,v∈ P, and observe that the path along T from u to v is at least the Euclidean distance between u and v, d(u,v)≤path_T(u,v). Furthermore, any path along an EMST is at most as long as the total length of an EMST, path_T(u,v)≤. If we now take an arbitrary spanning tree T' on the same point set P, then we know that each edge (u',v') in this spanning tree has at most length d(u',v')≤path_T(u',v')≤. Since T' has n-1 edges, its total length is O(n)·. | http://arxiv.org/abs/1704.08000v2 | {
"authors": [
"Wouter Meulemans",
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"Kevin Verbeek",
"Jules Wulms"
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firstpage–lastpage 2017Non-equilibrium quantum thermodynamics in Coulomb crystals G. De Chiara December 30, 2023 ========================================================== The nature of ultraluminous X-ray sources (ULXs) – off-nuclear extra-galactic sources with luminosity, assumed isotropic, ≳ 10^39 erg s^-1 – is still debated. One possibility is that ULXs are stellar black holes accreting beyond the Eddington limit. This view has been recently reinforced by the discovery of ultrafast outflows at ∼0.1-0.2c in the high resolution spectra of a handful of ULXs, as predicted by models of supercritical accretion discs. Under the assumption that ULXs are powered by super-Eddington accretion onto black holes, we use the properties of the observed outflows to self-consistently constrain their masses and accretion rates. We find masses ≲ 100 M_ and typical accretion rates ∼ 10^-5 M_ yr^-1, i.e. ≈ 10 times larger than the Eddington limit calculated with a radiative efficiency of 0.1. However, the emitted luminosity is only ≈ 10% beyond the Eddington luminosity, because most of the energy released in the inner part of the accretion disc is used to accelerate the wind, which implies radiative efficiency ∼ 0.01. Our results are consistent with a formation model where ULXs are black hole remnants of massive stars evolved in low-metallicity environments.accretion, accretion discs – black hole physics – X-rays: binaries – binaries: close§ INTRODUCTION Ultraluminous X-ray sources (ULXs) are non-nuclear, point-like, extragalactic sources with X-ray luminosity, assumed isotropic, L_ X≳ 10^39 erg s^-1; they are preferentially hosted by star-forming galaxies (seefor a review). There is no consensus yet on the nature of ULXs. Indeed, their luminosities are larger than the Eddington luminosity of ∼ 10 M_ black holes (BHs) in Galactic binaries, suggesting that they could represent a different class of objects, unless various combinations of significant beaming (e.g. ) and super-Eddington accretion (e.g. ) are advocated. On the other hand, ULXs could also be powered by sub-Eddington accretion onto intermediate-mass black holes (IMBHs) with masses 100 ≲ M_∙ /M_≲ 10^5; however, IMBHs might be required only to explain the most luminous ULXs (e.g. ) and perhaps some of those showing quasi-periodic oscillations (e.g. ; but see also ), but they are unlikely to account for the majority of the ULX population (e.g. ). An intermediate possibility is represented by massive stellar BHs (20 ≲ M_∙ /M_≲ 80) produced by low metallicity (Z ≲ 0.4 Z_), massive (≳ 40 M_) stars <cit.>, which are expected in ULX host galaxies owing to their low metallicity and high star formation rate <cit.>. Moreover, some ULXs are powered by accretion onto neutron stars <cit.>;therefore, ULXs likely represent an heterogeneous class of objects.Unfortunately, dynamical masses are not available for most ULXs. Recently, <cit.> claimed the detection of the orbital modulation in M101 ULX-1, inferring a mass >5 M_ and likely ∼ 30 M_. However, the putative companion is likely a Wolf-Rayet star, which makes this dynamical measurement uncertain <cit.>. In addition to that, hard X-ray observations with NuSTAR of a few ULXs show a downturn in the spectra at energies ≳ 10 keV, which excludes sub-Eddington accretion onto IMBHs, whereas it favours super-Eddington accretion on lighter accretors <cit.>. If super-Eddington accretion is common among ULXs with luminosities ∼ 10^40 erg s^-1, an expected feature is the presence of optically-thick outflows from the inner, geometrically-thick, radiation-dominated regions of the accretion disc <cit.>. Remarkably, recent work on high-resolution soft (∼0.4-1.8 keV) X-ray spectra revealed for the first time blueshifted, high-excitation absorption lines compatible with velocity offsets as high as ∼ 0.25 c in 3 ULXs <cit.>. These results were strengthened by the high-energy counterpart of the NGC 1313 X-1 outflow observed in moderate-resolution broadband (∼3-20 keV)X-ray spectra <cit.>. The discovery of ultrafast outflows from ULXs strongly hints that at least a fraction of them may be powered by super-Eddington accretion. However, as the nature of ULXs remains mostly unknown, it is unclear whether this would make them a peculiar class of accretors or an evolutionary phase of more common sources (e.g. high-mass X-ray binaries).In this Letter, we assume that ULXs are BHs accreting beyond the Eddington limit and therefore that they are able to launch radiation-driven outflows from the inner part of the accretion disc. We neglect here the case of ULXs powered by accreting neutron stars. We use the observed properties of the outflows to self-consistently constrain the expected mass and accretion rate of the powering BH within such a framework. Our results suggest that the few ULXs with observed outflows could be associated with BHs with masses between ∼ 10 and ∼ 100 M_⊙, and typical accretion rates ∼ 10^-5 M_ yr^-1. If some ULXs are powered by BHs and ultrafast outflows caused by super-Eddington accretion are common, then our results suggest that their origin could be consistent with being the remnant of massive metal-poor stars.§ SUPERCRITICAL ACCRETION DISCS WITH OUTFLOWS When the mass accretion rate within an accretion disc becomes supercritical, the excess of heat released by viscosity may both inflate the inner part of the disc, producing outflows, and be advected within the flow, increasing its entropy (e.g. ). By supercritical we mean an accretion disc sustaining an accretion rate Ṁ > Ṁ_ Edd, where we defineṀ_ Edd≡L_ Edd/η̃ c^2 = 4 π G M_∙/η̃κ_ es c≈ 1.86 × 10^18 m_∙ g s^-1,where M_∙ = m_∙ M_ is the central BH mass, κ_ es is the electron scattering opacity, and η̃ = 1/12 is the (newtonian) radiative efficiency of a thin disc extending inward toR_ in = 3 R_ S = 6 G M_∙/c^2. We stress that we adopt η̃ as a definition, and it can be different from the actual radiative efficiency η≡ L / (Ṁ c^2) of a disc emitting L.We base our analysis on the model of supercritical accretion disc with outflows presented by <cit.> and <cit.>. The disc is composed of two regions separated by the spherisation radius R_ sp. Outside R_ sp, the disc is geometrically thin and it can sustain a constant accretion rate ṁ = Ṁ / Ṁ_ Edd > 1 because it is locally sub-Eddington. In this region, the local energy balance is Q^+ = Q_ rad + Q_ adv≈ Q_ rad, where Q^+, Q_ adv, and Q_ rad are the heat flux released by viscosity, the energy flux advected with the flow, and the energy flux that can eventually be radiated away, respectively. The emitted luminosity in photons is L_γ(R > R_ sp) = ∫_R > R_ sp 2 Q_ rad dA ≈ L_ Edd. Inside R_ sp, the disc is locally super-critical, despite a significant energy fraction is advected with the flow. In response, the disc becomes geometrically thick under the dominant effect of radiation pressure and a radiation-driven outflow mainly coupled with radiation through electron scattering is unavoidably launched. Specifically, a fraction ϵ_ w of the available energy Q_ rad is transferred to the kinetic luminosity of the wind, while the remaining can be radiated away. The bolometric luminosity emitted by the disc is thereforeL_γ≈ (1- ϵ_ w) ∫_R_ in^R_ sp 2 Q_ rad dA + L_ Edd = 1 - ϵ_ w/ϵ_ w P_ w + L_ Edd,where we used the definition of the wind kinetic luminosity P_ w = ϵ_ w∫_R_ in^R_ sp 2 Q_ rad dA, and the additional L_ Edd comes from the outer part of the disc. The wind is radiation dominated and energy driven, and it is expected to be mainly accelerated at expense of the advected internal energy <cit.>. The conservation of energy expressed by equation (<ref>) already suggests an upper limit on the BH mass as L_γ≥ L_ Edd, which implies M_∙≲ M_∙,Edd≈ 72 l_γ, 40 M_, where L_γ = l_γ,40× 10^40 erg s^-1 and M_∙,Edd is the mass corresponding to L_γ = L_ Edd. However, this limit is not general and applies only to sources where radiation-driven outflows are detected.The spherisation radius R_ sp determines the structure of the disc by setting where the transition between the outer geometrically thin and the inner geometrically thick disc occurs. The value of R_ sp depends on the mass, angular momentum, and energy conservation within the disc under the effect of viscosity, advection, and radiation, and it can be self-consistently calculated by solving the accretion disc equations. <cit.> provide a fitting formula that depends on ṁ and ϵ_ w,r_ sp≈ṁ[1.34 - 0.4 ϵ_ w + 0.1 ϵ_ w^2 - (1.1 - 0.7 ϵ_ w) ṁ^-2/3],where we have defined r_ sp = R_ sp / R_ in.We can explicitly write the expression for P_ w by considering the structure of the wind. Within R_ sp the scaling of the accretion rate is approximately linear,dṀ /dR ≈Ṁ_ Edd (ṁ - ṁ_ in) / r_ sp, where ṁ_ in = ṁ_ in(ϵ_w, ṁ) is the mass flow effectively reaching R_ in in units of Ṁ_ Edd [see equation (23) of ]. The rest of the mass accretion rate is accelerated in the outflow, which reaches an asymptotic velocity v_ w(R) ≈√(2 G M_∙ / R) within R_ sp. Therefore, we finally obtainP_ w = 1/2∫_R_ in^R_ sp dṀ/ dR v^2_ w(R)dR ≈1/2Ṁ_ w v_∞^2 - log(3 β_∞^2)/1 - 3 β_∞^2,where we defined the mass outflow rateṀ_ w = ∫_R_ in^R_ sp dṀ/ dR dR ≈Ṁ_ Edd (ṁ - ṁ_ in) r_ sp - 1/r_ sp,and β^2_∞ = v_∞^2/c^2 = 1/(3 r_ sp), i.e. v_∞ = v_ w(R_ sp). The latter relation assumes that the radiative acceleration beyond R_ sp is negligible; we checked that such simplification does not significantly affect our mass estimates below, while it tends to decrease the inferred accretion rate by a factor < 1.5.§ CONSTRAINING THE BLACK HOLE MASS Recent observations have discovered fast outflows in the deepest X-ray spectra of ULXs (e.g. ). They are identified through high ionisation Fe, O, and Ne absorption lines produced by gas outflowing at ≈ 0.1-0.2 c with photoionisation parameters ξ = L_ ion / (n r^2) ranging from ∼ 10^2 to ≳ 10^4 erg cm s^-1, where n is the density of the absorbing gas at radial distance r from the source emitting the ionising luminosity L_ ion in the energy band 1-1000 Ry. We can use these results to constrain the parameters of a supercritical accretion disc potentially able to produce such winds and eventually the mass of the central BH as follows.The conservation of mass in the outflow may simply read asṀ_ w≈ 4 πΩ_ w r^2 C_ wρ_ w v_∞,where we introduced a clumping factor C_ w to phenomenologically capture the effect of a multi-phase wind and the fraction Ω_ w of the full sphere occupied by the outflow. Under the simplifying assumption of a steady outflow, the density in the wind scales as ρ_ w∝ r^-2 at radii larger than R_ sp. We can rewrite the density as a function of the photoionisation parameter ξ as ρ_ w = μ m_ p L_ ion / (ξ r^2) ≈μ m_ p f (L_γ - L_ Edd) / (ξ r^2), where μ≈ 0.59 is the mean molecular weight of a fully ionised gas, m_ p is the proton mass, and we express L_ ion≈ f (L_γ - L_ Edd), which is the luminosity produced within R_ sp that, after streaming through the wind beyond the wind photosphere, is able to photoionise the outflow, where f ∼ 1 is an adjustable parameter to account for e.g. a small additional contribution of ionising photons from the outer disc or the conversion between bolometric and ionising luminosity. If we introduce this definition in equation (<ref>) and we then use it both in the equation of energy conservation, equation (<ref>), and in the definition of Ṁ_ w, equation (<ref>), we can solve for ϵ_ w:ϵ_ w(ξ, β_∞) = 2 πΩ_ w C_ w f ξ̃^-1 B(β_∞)/1 + 2 πΩ_ w C_ w f ξ̃^-1 B(β_∞),and we obtainΓ = 1 + ṁ - ṁ_ in(ϵ_ w, ṁ)/4 π C_ wΩ_ w f η̃ ξ̃ 1 - 3 β_∞^2/β_∞,where B(x) = -x^3 log(3 x^2)/(1 - 3 x^2), and we normalise ξ = ξ̃μ m_ p c^3 and L_γ = Γ L_ Edd. The set of equation (<ref>), (<ref>), and (<ref>), with the relation r_ sp = 1/(3 β_∞^2), fully characterises our problem. After solving equation (<ref>) numerically, we get ϵ_ w, ṁ, and Γ as a function of the observable quantities β_∞ and ξ. Then, we rescale ṁ and Γ to physical values through the definition of equation (<ref>) by choosing the value of the emitted bolometric luminosity L_γ, and we finally obtain M_∙ and Ṁ that are consistently required to have a supercritical accretion disc launching outflows at v_∞.Figure <ref> summarises the results of our calculations, showing the values of ϵ_ w, ṁ, Γ, and η in the ξ-β_∞ plane. We adopt fiducial values for the parameters Ω_ w = 0.5, C_ w = 0.3, and f = 1. For values of β_∞ above ≈0.05-0.1, the lines of constant ϵ_ w roughly scale as β_∞∝ [ϵ_ wξ / (1- ϵ_ w)] ^1/3. Mildly relativistic winds (β_∞≳ 0.05) typically require ≳ 90% of the energy produced within R_ sp to accelerate the outflows, unless logξ≫ 4 and ϵ_ w can reduce to ∼ 50%. The remaining energy is released in photons and it contributes to bolometric luminosities ∼ 10% in excess of L_ Edd, as shown by the isocontours of Γ. ϵ_ w can even exceed 99% for fast outflows ∼ 0.2 c at logξ < 3, locking the bolometric luminosity at about L_ Edd. On the other hand, the accretion rate ṁ mainly depends on the outflow velocity because ξ enters only weakly in equation (<ref>) through ϵ_ w and ṁ∼ r_ sp∼β_∞^-2. Fast winds with β_∞≳ 0.1 are typically associated with ṁ∼ 10, where ∼ 50 % reaches the central BH while the remaining is accelerated in the outflows. The differences between the isocontours of ṁ and Γ determines the actual radiative efficiency of the disc, η = η̃ Γ / ṁ, which is typically ∼ 0.001 and increases up to ∼ 0.01 for winds as fast as ∼ 0.2 c. This trend is in qualitative agreement with numerical simulations of supercritical accretion disc with radiation-driven outflows that show η∼ 0.04, larger than the equivalent slim disc <cit.>; however, we also infer η's slightly lower than what numerical simulations predict. Figure <ref> shows the corresponding BH mass in the ξ-β_∞ plane for a source scaled to L_γ = l_γ, 40× 10^40 erg s^-1. ULXs associated to outflows faster than ∼ 0.1 c are expected to be powered by BHs with masses M_∙∼ 40-70 l_γ, 40 M_. In fact, the isocontours follow those of the luminosity Eddington ratio Γ and the mass asymptotically tends to the limiting mass M_∙,Edd when β_∞ increases at constant ξ (i.e. faster wind with higher P_ w), while it decreases when ξ increases at constant β_∞ (i.e. less dense wind with lower P_ w). These trends arise because, for a given L_γ, a wind that has a larger P_ w requires a higher fraction ϵ_ w of the energy produced within R_ sp, which implies that η reduces and Γ→ 1^+, i.e. the emitted luminosity mostly comes from the outer, thinner disc. We note that, for high β_∞ and low ξ, our mass estimates tend to saturate at about M_∙,Edd.Both Figures <ref> and <ref> show the positions in the ξ-β_∞ of different observations of ULX outflows detected in three nearby star-forming low-metallicity spiral/dwarf galaxies. The observations are summarised in Table <ref>; the table also shows the inferred radiative efficiency η, BH mass M_∙, and accretion rate Ṁ. All these quantities have been derived by assuming L_γ = L_ X, which could introduce a systematic error in the mass estimates. However, if we take the results with caution as order of magnitude estimates, we see that our calculations constrain the BH masses for those ULXs grossly to range between ∼ 10 and ∼ 100 M_, with accretion rates ∼ 10^-5 M_ yr^-1 (corresponding to ṁ∼ 10) and η∼ 0.01.Finally, we note that the derived masses depend on three free parameters, namely Ω_ w, C_ w, and f. The fiducial values that we adopt for C_ w and Ω_ w, the latter corresponding to 60from the rotation axes of the disc, are grossly expected from theoretical and numerical models <cit.>. The fudge factor f is more uncertain, as it may account for different effects. We tested the sensitivity of our results to these parameters by varying C_ w and Ω_ w between 0.2 and 0.8, and f between 0.2 and 3. As they always appear together as C_ wΩ_ w f in equation (<ref>) and (<ref>), changing each of them independently has the same effect. We find that M_∙ and Ṁ change by up to ≈15-20%, i.e. they do not significantly affect our estimates.§ DISCUSSION AND CONCLUSIONSIn this Letter, we attempted to constrain the mass and the accretion rate of 3 ULXs through the properties of their observed ultrafast outflows, under the assumption that the latter are caused by super-Eddington accretion onto BHs. We find masses between ∼ 10 and ∼ 100 M_ and accretion rates ∼ 10^-5 M_ yr^-1, about 10 times larger than Ṁ_ Edd in equation (<ref>). However, the bolometric luminosity results to be only up to ∼ 10% in excess of L_ Edd, implying a typical radiative efficiency η∼ 0.01, because ≳ 90% of the luminosity produced within R_ sp is required to accelerate the outflow.Bearing in mind that our small sample may not be representative of the whole population of ULXs, it is nonetheless interesting to note that the inferred masses lie between typical galactic binaries and the presumed low-mass tail of IMBHs (e.g. ). However, they should not be regarded as exotic; they are well consistent with the expected remnants of low-metallicity massive stars. Indeed, <cit.> have shown that such massive stellar BHs may statistically account for a large fraction of ULXs as well as for the correlation between the number of ULXs and the star formation rate of their host. Therefore, this might support the speculation that many ULXs could be powered by super-Eddington accretion on ∼ 10-100 M_ BHs (e.g. ). However, a future larger sample of ultrafast outflows in ULX spectra as well as more robust dynamical mass estimates are necessary to eventually confirm this scenario.Our investigation represents a tentative new approach to exploit recent observations of ultrafast outflows to constrain the mass of a few ULXs under the plausible assumption of super-Eddington accreting BHs. However, we emphasise again that the derived masses should be taken as indicative. Indeed, a factor of ∼ 2-3 to reduce the derived masses may still be accommodated, because we have effectively neglected geometric beaming effects when we associate the bolometric luminosity to the observed one. They might be particularly relevant for the radiation produced within R_ sp that is mainly released in a rather narrow funnel along the disc rotation axis <cit.>. As a consequence, our masses should be considered as upper limits because the true L_γ, and therefore M_∙, could be lower by the beaming factor b ≲ 0.5-0.7 <cit.>. Moreover, we assumed L_γ = L_ X for the sake of simplicity, but in fact the bolometric correction of the inferred X-ray luminosity might vary from system to system. This might depend e.g. on the line of sight through the outflow, as hinted by the change in spectral hardness among the ULXs considered in Table <ref>, as well as by the conjectured connection with ultraluminous supersoft sources <cit.>. According to this scenario, NGC 1313 X-1 is likely to be seen more face-on because of the harder spectrum, and therefore L_γ≈ L_ X is a reasonable assumption (but it is more likely to suffer from beaming though), while NGC 55 ULX and NGC 5408 X-1 have softer spectra that may come from the reprocessing of the harder radiation from the inner edge of the disc (and perhaps by a close hot corona) by the optically thick wind, for which L_ X≲ L_γ. This latter correction might partially compensate the effect of beaming, but note that L_ X as well is assumed isotropic and neglects beaming effects. Nonetheless, even when reduced by a factor ∼2-3, our inferred masses are consistent with previous estimates of similar systems as well as with the aforementioned scenario of ULX formation.Our calculations also neglect general relativistic effects, such as spinning BHs, and magnetic fields. We explored the impact of a spinning BH by changing the inner boundary condition and the normalising radiative efficiency η̃. We find values of M_∙ up to ≈ 20 % lower than in the no spin cases, owing to larger values of Γ and η at similar accretion rates. Magnetic fields might also contribute to accelerate the outflow <cit.>, effectively lowering ϵ_ w and increasing L_γ for the same accretion rate.This might also lower the value of the inferred central mass.Regardless of their masses, BHs are not the only possible accretors powering ULXs. Indeed, the light curves of 3 ULXs show sinusoidal pulses with a period ≈ 1 s that are a distinctive signature of neutron stars <cit.>. Recently, <cit.> and <cit.> argued that neutron stars may actually power a fraction of ULXs larger than previously expected. According to their analysis, pulsations may only be observed during a rather short phase of the ULX evolution, preventing an easy detection. However, the variability of the optical spectrum has revealed the existence of a massive stellar BH, likely ∼ 20-30 M_, in M101 ULX-1 <cit.> and perhaps also in X-ray binaries associated to Wolf-Rayet stars (; but see also ). Therefore, it appears evident that ULXs comprise a diverse variety of accreting objects, and whether the majority is represented by BHs or neutron stars must be scrutinised further. Clearly, our analysis can be applied only to the subset of ULXs with outflows that probably host an accreting BH; nonetheless, it would be possible to extend this treatment to neutron stars by modifying the inner boundary conditions in modelling the accretion disc owing to the magnetic fields.Finally, we step on to more speculative grounds by noting that the BH masses we infer, when considered as upper limits as discussed above, are not too dissimilar from those of the binary BHs detected by Advanced LIGO <cit.>. While this might be just a coincidence, it is nonetheless worth to stress that such observations at least unambiguously demonstrate the existence in nature of rather heavy stellar BHs. According to stellar evolution models, the most natural pathway to form ≳ 30 M_ stellar BH is the evolution of massive stars in Z < 0.1-0.5 Z_ environments <cit.>. Despite the evolution of binary stars is complicated by several physical processes that make difficult to predict the final outcome <cit.>, it is still conceivable to imagine an evolutionary connection between binary massive stars, whose fraction can be as high as 70%<cit.>, an intermediate phase when the binary turns into an ULX, and finally the merger of heavy stellar BH binaries (see also ). While this may sound attractive, further observations are required to better assess the puzzle of the nature of ULXs and consequently this potential connection.§ ACKNOWLEDGEMENTS We acknowledge useful discussions with Massimo Dotti, Michela Mapelli, and Elena M. Rossi. D.F. acknowledges support by ERC Starting Grant 638707 “Black holes and their host galaxies: coevolution across cosmic time”. C.P. and A.C.F. acknowledge support by ERC Advanced Grant 340442 “Accreting black holes and cosmic feedback”.mnras | http://arxiv.org/abs/1704.08255v1 | {
"authors": [
"Davide Fiacconi",
"Ciro Pinto",
"Dominic J. Walton",
"Andrew C. Fabian"
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"published": "20170426180001",
"title": "Constraining the mass of accreting black holes in ultraluminous X-ray sources with ultrafast outflows"
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Structured Sparse Modelling with Hierarchical GP Danil Kuzin, Olga Isupova, Lyudmila Mihaylova The University of Sheffield, Sheffield, UKEmail: [email protected], [email protected], [email protected] Olga Isupova and Lyudmila Mihaylova acknowledge the support from the EC Seventh Framework Programme [FP7 2013-2017] TRAcking in compleX sensor systems (TRAX) Grant agreement no.: 607400.Received: date / Revised version: date ================================================================================================================================================================================================================================================================================================================================================================================== In this paper a new Bayesian model for sparse linear regression with a spatio-temporal structure is proposed. It incorporates the structural assumptions based on a hierarchical Gaussian process prior for spike and slab coefficients. We design an inference algorithm based on Expectation Propagation and evaluate the model over the real data. § INTRODUCTION Sparse regression problems arise often in various applications, e.g., model selection, compressive sensing, EEG source localisation and gene modelling <cit.>. One of the Bayesian approaches to force the coefficients being zeros is the spike and slab prior <cit.>: each component is modelled as a mixture of spike, that is the delta-function in zero, and slab, that is some vague distribution. Following the Bayesian approach, latent variables that are indicators of spikes are added to the model <cit.> and the relevant distribution is placed over them <cit.>.In this model each component is modelled to be spike or slab independently. However, in many applications non-zero elements tend to appear in groups forming an unknown structure: wavelet coefficients of images are usually organised in trees <cit.>, chromosomes have a spatial structure along the genome <cit.>. We propose an extension of the spike and slab model by imposing a hierarchical Gaussian process (GP) prior on the latent variables. Such hierarchical prior allows to model spatial structural dependencies for coefficients that can evolve in time. The new model is flexible as spatial and temporal dependencies are decoupled by different levels of the hierarchical GP prior. § PROPOSED MODEL The observations 𝐲_t ∈ℝ^K are collected with the design matrix 𝐗∈ℝ^K× N from the unknown coefficients β_t ∈ℝ^N at every time moment t∈ [1, …, T], with independent noise:𝐲_t∼𝒩(𝐲_t | 𝐗β_t, σ^2 𝐈).We consider the case when K < N, therefore the problem of recovery of β_t from 𝐲_t is ill-posed and regularisation is required. Sparsity The vectors β_t are assumed to be sparse, that is implemented in the model using the spike and slab approach:β_it∼ω_itδ_0(β_it)+(1-ω_it)𝒩(β_it | 0, σ_β^2),where ω_it are the latent indicators of spike and slab.Spatial clustering Non-zero elements in β_t are assumed to be clustered in groups at every timestamp. Therefore spatial dependencies for the positions of spikes in β_it are modelled with the GP:ω_it ∼Ber(ω_it | Φ(γ_it)),Φ(·)is the standard Gaussian cdf γ_t∼𝒩(γ_t | μ_t, Σ_0),Σ_0(i,j) = α_Σexp(-(i-j)^2/2 l^2_Σ). GPs specify prior over an unknown structure. This is particularly useful as it allows to avoid a specification of any structural patterns — structural modelling is governed only by the GP covariance function. Temporal evolution Clusters of spikes in β_t are assumed to evolve in time. This evolution is addressed with the hierarchical GP dynamic system model <cit.>. The mean for the spatial GP changes over time according to the top-level temporal GP:μ_t ∼𝒩 (μ_t | μ_t-1, 𝐖), 𝐖(i,j) = α_Wexp(-(i-j)^22 l^2_W).This allows to implicitly specify the prior over the transition function of the structure. The rate of the evolution is controlled with the top-level GP covariance function. The exact posterior of the parameters is intractable, therefore approximate inference methods are required. Inference is based on Expectation Propagation <cit.> in this paper.The structural assumptions in sparse models are studied in the literature. The group lasso <cit.> provides sparse solutions for predefined groups of coefficients. Group constraints for sparse models include smooth relevance vector machines <cit.>, Boltzmann machine prior <cit.>; spatio-temporal coupling of the parameters <cit.>. In <cit.> a spatio-temporal structure is modelled with a one-level GP prior. In contrast to that model the new one introduces an additional level of a GP prior for temporal dependencies, therefore the temporal and spatial structures are decoupled, adding flexibility to the model. The high-level GP controls the change of spike groups in time while the low-level GP allows the local changes within each group.§ NUMERICAL EXPERIMENTS The performance of the proposed hierarchical GP algorithm is compared with the one-level GP prior introduced in <cit.>. We apply both algorithms to the problem of object detection in video sequences. The Convoy dataset <cit.> is used where the frame difference is applied for moving object detection. The sparse observations are obtained as 𝐲_t = 𝐗β_t, where 𝐗 is the matrix with i.i.d. Gaussian elements, the number of observations is ≈ 40% of the dimension of the hidden signal β_t. This procedure corresponds to compressive sensing observations <cit.>. The reconstruction results on the sample frames are presented in Figure <ref>. The obtained performance measures values can be found in Table <ref>. The F-measure compares binary masks computed by thresholding the true vectors β and the posterior estimates β. NMSE represents normalised mean squared error between β and β. The proposed algorithm shows better results in terms of both measures. § CONCLUSION In this work we propose a new approach for spatial structure modelling in sparse models that allows to capture complex temporal evolution of data patterns. We also develop an efficient inference method based on EP. The numerical experiments show superiority of the proposed model over the current state-of-the-art. | http://arxiv.org/abs/1704.08727v1 | {
"authors": [
"Danil Kuzin",
"Olga Isupova",
"Lyudmila Mihaylova"
],
"categories": [
"stat.ML"
],
"primary_category": "stat.ML",
"published": "20170427193641",
"title": "Structured Sparse Modelling with Hierarchical GP"
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Research School of Chemistry, Australian National University, ACT 2601, [email protected] Laboratoire de Chimie et Physique Quantiques, Université de Toulouse, CNRS, UPS, France Research School of Chemistry, Australian National University, ACT 2601, AustraliaWe report the three main ingredients to calculate three- and four-electron integrals over Gaussian basis functions involving Gaussian geminal operators: fundamental integrals, upper bounds, and recurrence relations. In particular, we consider the three- and four-electron integrals that may arise in explicitly-correlated F12 methods. A straightforward method to obtain the fundamental integrals is given. We derive vertical, transfer and horizontal recurrence relations to build up angular momentum over the centers. Strong, simple and scaling-consistent upper bounds are also reported. This latest ingredient allows to compute only the N^2 significant three- and four-electron integrals, avoiding the computation of the very large number of negligible integrals.Three- and four-electron integrals involving Gaussian geminals:fundamental integrals, upper bounds and recurrence relations Pierre-François Loos ============================================================================================================================= § INTRODUCTION It is well known that highly-accurate wave functions require the fulfilment (or near-fulfilment) of the electron-electron cusp conditions. <cit.> For correlated wave functions expanded in terms of products of one-electron Gaussian basis functions, the energy converges as L^-3, where L is the maximum angular momentum of the basis set. <cit.> This slow convergence can be tracked down to the inadequacy of these products to properly model the Coulomb correlation hole. <cit.>In the late 20's, Hylleraas solved this issue for the helium atom by introducing explicitly the interelectronic distance r_12 = _1 - _2 as an additional two-electron basis function. <cit.> As Kutzelnigg later showed, this leads to a prominent improvement of the energy convergence from L^-3 to L^-7. <cit.>Around the same time, Slater, while studying the Rydberg series of helium, <cit.> suggested a new correlation factor known nowadays as a Slater geminal:S_12 = exp(-λ_12r_12 ).Unfortunately, the increase in mathematical complexity brought by r_12 or S_12 has been found to be rapidly computationally overwhelming.In 1960, Boys <cit.> and Singer <cit.> independently proposed to use the Gaussian geminal (GG) correlation factorG_12 = exp(-λ_12r_12^2 ),as “there are explicit formulas for all of the necessary many-dimensional integrals". <cit.> Interestingly, in the same article, a visionary Boys argued that, even if GGs do not fulfil the electron-electron cusp conditions, they could be used to fit S_12.During the following years, variational calculations involving GGs flourished, giving birth to various methods, such as the exponentially-correlated Gaussian method. <cit.>However, this method was restricted to fairly small systems as it requires the optimization of a large number of non-linear parameters. In the 70's, the first MP2 calculations including GGs appeared thanks to the work by Pan and King, <cit.> Adamowicz and Sadlej, <cit.> and later Szalewicz et al. <cit.>Even if these methods represented a substantial step forward in terms of computational burden, they still require the optimization of a large number of non-linear parameters.In 1985, Kutzelnigg derived a first form of the MP2-R12 equations using r_12 as a correlation factor. <cit.>Kutzelnigg's idea, which was more formally described together with Klopper in 1987,<cit.> dredged up an old problem: in addition to two-electron integrals (traditional ones and new ones), three-electron and four-electron integrals were required. At that time, the only way to evaluate them would have been via an expensive one- or two-dimensional Gauss-Legendre quadrature. <cit.> Nevertheless, the success of the R12 method was triggered by the decision to avoid the computation of these three- and four-electron integrals through the use of the resolution of the identity (RI) approximation. <cit.> In this way, three- and four-electron integrals are approximated as linear combinations of products of two-electron integrals. Several key developments and improvements of the original MP2-R12 approach have been proposed in the last decade. <cit.> Of course, the accuracy of the RI approximation relies entirely on the assumption that the auxiliary basis set is sufficiently large, i.e. N_RI≫ N, where N and N_RI are the number of basis functions in the primary and auxiliary basis sets, respectively.In 1996, Persson and Taylor killed two birds with one stone. Using a pre-optimized GG expansion fitting r_12 r_12≈∑_k a_k[1-exp(-λ_k r_12^2) ],they avoided the non-linear optimization, and eschewed the RI approximation thanks to the analytical integrability of three- and four-electron integrals over GGs. <cit.> They were able to show that a six- or ten-term fit introduces a 0.5 mE_h or 20 μE_h error, respectively. <cit.> Unfortunately, further improvements were unsuccessful due to the failure of r_12 in modelling the correct behaviour of the wave function for intermediate and large r_12. <cit.>In fact, Ten-no showed that S_12 is near-optimal at describing the correlation hole, and that a 10-term GG fit of S_12 yields very similar results. This suggests that, albeit not catching the cusp per se, the Coulomb correlation hole can be accurately represented by GGs. <cit.> Methods for evaluating many-electron integrals involving GGs have already been developed.As mentioned previously, Persson and Taylor <cit.> derived recurrence relations based on Hermite Gaussians, analogously to the work of McMurchie and Davidson for two-electron integrals. <cit.> These recurrence relations were implemented by Dahle. <cit.> Saito and Suzuki <cit.> also proposed an approach based on the work by Obara and Saika. <cit.> More recently, a general formulation using Rys polynomials <cit.> was published by Komornicki and King. <cit.> Even if limited to the three-center case, it is worth mentioning that May has also developed recurrence relations for two types of three-electron integrals. <cit.>These recurrence relations were implemented by Womack using automatically-generated code. <cit.> Recently, we have developed recurrence relations for three- and four-electron integrals for generic correlation factors. <cit.>A major limitation of all these approaches is that they do not include any integral screening.[Komornicki and King mentioned the crucial importance of an effective integral screening in Ref. Komornicki11.] Indeed, a remarkable consequence of the short-range nature of the Slater and Gaussian correlation factors is that, even if formally scaling as N^6 and N^8, there are only N^2 significant (i.e. greater than a given threshold) three- and four-electron integrals in a large system. <cit.> Therefore, it is paramount to devise rigorous upper bounds to avoid computing the large number of negligible integrals.The present manuscript is organized as follows. In Sec. <ref>, we discuss Gaussian basis functions, many-electron integrals and the structure of the three- and four-electron operators considered here. The next three sections contain the main ingredients for the efficient computation of three- and four-electron integrals involving GGs: i) fundamental integrals (FIs) in Sec. <ref>, ii) upper bounds (UBs) in Sec. <ref>, and iii) recurrence relations (RRs) in Sec. <ref>.In Sec. <ref>, we give an overall view of our algorithm which is an extension of the late-contraction path of PRISM (see Refs. HGP, Gill94b and references therein).Note that the RRs developed in this study differ from the ones reported in our previous studies (Refs. 3ERI1, 4ERI1) as they are specifically tailored for the unique factorization properties brought by the association of Gaussian basis functions and GGs. Atomic units are used throughout. § GENERALITIES §.§ Gaussian functions A primitive Gaussian function (PGF) is specified by an orbital exponent α, a center =(A_x,A_y,A_z), and angular momentum =(a_x,a_y,a_z):φ_^() = (x-A_x)^a_x (y-A_y)^a_y (z-A_z)^a_z e^-α-^2.A contracted Gaussian function (CGF) is defined as a sum of PGFsψ_^() =∑_k=1^K_a D_ak (x-A_x)^a_x (y-A_y)^a_y (z-A_z)^a_z e^-α_k -^2,where K_a is the degree of contraction and the D_ak are contraction coefficients. A CGF-pair |⟩≡ψ_^()ψ_^() = ∑_i=1^K_a∑_j=1^K_b_ijis a two-fold sum of PGF-pairs =φ_^() φ_^().A primitive shell a is a set of PGFs sharing the same total angular momentum a, exponent α and center .Similarly, a contracted shell |a⟩ is a set of CGFs sharing the same PGFs and total angular momentum.A contracted shell-pair is the set of CGF-pairs obtained by the tensor product |a b⟩=|a⟩⊗|b⟩. Similarly, a primitive shell-pair a b=a⊗b is the set of PGF-pairs. Finally, primitive and contracted shell-quartets, -sextets and -octets are obtained in an analogous way. For example, a_1 b_1 a_2 b_2 = a_1 b_1⊗a_2 b_2 and |a_1 a_2 b_1 b_2⟩ = |a_1 b_1⟩⊗|a_2 b_2⟩. Note that 1 is a set of three p-type PGFs, a 11≡pp shell-pair is a set of nine PGF-pairs, and a 2222≡dddd shell-quartet is a set of 1,296 PGF-quartets.§.§ Many-electron integrals Throughout this paper, we use physicists notations, and we write the integral over a n-electron operator f_1 ⋯ n of CGFs as ⟨_1 ⋯_n|_1 ⋯_n⟩ ≡_1 ⋯_nf_1⋯ n_1 ⋯_n = ψ__1^_1(_1) ⋯ψ__n^_n(_n) f_1⋯ n ψ__1^_1(_1) ⋯ψ__n^_n(_n) d _1⋯ d _n.Additionally, square-bracketed integrals denote integrals over PGFs:_1 ⋯_n_1 ⋯_n = φ__1^_1( _1) ⋯φ__n^_n(_n) f_1 ⋯ n φ__1^_1( _1) ⋯φ__n^_n(_n) d _1⋯ d _n.The FIs (i.e. the integral in which all 2n basis functions are s-type PGFs) is defined as ≡⋯⋯ with =(0,0,0). The Gaussian product rule reduces it from 2n to n centers:= ( ∏_i=1^n S_i ) φ_^_1(_1) ⋯φ_^_n(_n) f_1 ⋯ n d _1⋯ d _n, whereζ_i = α_i + β_i, _i = α_i _i + β_i _i/ζ_i, S_i= exp(-α_i β_i/ζ_i_i_i^2), and _i_i = _i - _i. We also define the quantity _ij = _i - _j which will be used later on.For conciseness, we will adopt a notation in which missing indices represent s-type Gaussians. For example, _2_3 is a shorthand for _2_3.We will also use unbold indices, e.g. a_1a_2a_3a_4b_1b_2b_3b_4 to indicate a complete class of integrals from a shell-octet.§.§ Three- and four-electron operators In this study, we are particularly interested in the “master” four-electron operator C_12G_13G_14G_23G_34 (where C_12 = r_12^-1 is the Coulomb operator) because the three types of three-electron integrals and the three types of four-electron integrals that can be required in F12 calculations can be easily generated from it (see Fig. <ref>). These three types of three-electron integrals are composed by a single type of integrals over the cyclic operator C_12G_13G_23, and two types of integrals over the three-electron chain (or 3-chain) operators C_12G_23 and G_13G_23. F12 calculations may also require three types of four-electron integrals: two types of integrals over the 4-chain operators C_12G_14G_23 and C_12G_13G_34, as well as one type over the trident operator C_12G_13G_14. Explicitly-correlated methods also requires two-electron integrals.However, their computation has been thoroughly studied in the literature. <cit.> Similarly, the nuclear attraction integrals can be easily obtained by taking the large-exponent limit of a s-type shell-pair.Starting with the “master” operator C_12G_13G_14G_23G_34, we will show that one can easily obtain all the FIs as well as the RRs required to compute three- and four-electron integrals within F12 calculations. This is illustrated in Fig. <ref> where we have used a diagrammatic representation of the operators. The number N_sig of significant integrals in a large system with N CGFs is also reported. § FUNDAMENTAL INTEGRALS Following Persson and Taylor, <cit.> the ^m are derived starting from the momentumless integral (<ref>) using the following Gaussian integral representation for the Coulomb operatorC_12 = 2/√(π)∫_0^∞exp(-u^2 r_12^2) du.After a lengthy derivation, one can show that the closed-form expression of the FIs is^m =2/√(π)_G√(δ_0/δ_1-δ_0)(δ_1/δ_1-δ_0 )^m F_m [ δ_1 ( Y_1-Y_0 )/δ_1-δ_0 ],where m is an auxiliary index, F_m(t) is the generalized Boys function, and_G = ( ∏_i=1^4 S_i ) ( π^4/δ_0 )^3/2exp(-Y_0)is the FI of the “pure” GG operator G_13G_14G_23G_34 from which one can easily get the FI of the 3-chain operator G_13G_23 by setting _14 = _34 = 0. While the FIs involving a Coulomb operator contain an auxiliary index m, the FIsover “pure” GG operators (like G_13G_23) do not, thanks to the factorization properties of GGs. <cit.>The various quantities required to compute (<ref>) areδ_u = ζ + _u = ζ ++ u^2 ,where ζ= [ ζ_1 0 0 0; 0 ζ_2 0 0; 0 0 ζ_3 0; 0 0 0 ζ_4; ], = [1 -100; -1100;0000;0000;], = [ _13+_14 0-_13-_14; 0 _23-_23 0;-_13-_23 _13+_23+_34-_34;-_14 0-_34 _14+_34; ], and Δ_u = ζ·δ_u^-1·ζ, ^k = [ 0 _12^k _13^k _14^k; 0 0 0 _24^k; 0 0 0 _34^k; 0 0 0 0; ], δ_u = (δ_u), Y_u =( Δ_u ·^2). The generalized Boys function F_m(t) in Eq. (<ref>) can be computed efficiently using well-established algorithms. <cit.> § UPPER BOUNDS In this section, we report UBs for primitive and contracted three- and four-electron integrals. Our UBs are required to be simple (i.e. significantly computationally cheaper that the true integral), strong (i.e. as close as possible to the true integral in the threshold region 10^-14-10^-8), and scaling-consistent (i.e. the number of significant integrals N_sig=𝒪(N_UB), where N_UB is the number of integrals estimated by the UB). We refer the interested reader to Refs. Gill94a, GG16 for additional information about UBs. A detailed study of these UBs (as well as their overall performances) will be reported in a forthcoming paper. <cit.>Our screening algorithms are based on primitive, [B_m], and contracted, B_m, shell-mtuplet bounds.These are based on shell-mtuplet information only: shell-pair (m=2), shell-quartet (m=4), shell-sextet (m=6) and shell-octet (m=8).Thus, for each category of three- and four-electron integrals, we will report from shell-pair to shell-sextet (or shell-octet) bounds. Figure <ref> is a schematic representation of the overall screening scheme for contracted four-electron integrals. First, we use a primitive shell-pair bound B_2 to create a list of significant primitive shell-pairs. For a given contracted shell-pair, if at least one of its primitive shell-pairs has survived, a contracted shell-pair bound B_2 is used to decide whether or not this contracted shell-pair is worth keeping. The second step consists in using a shell-quartet bound B_4 to create a list of significant contracted shell-quartets by pairing the contracted shell-pairs with themselves. Then, we combine the significant shell-quartets and shell-pairs, and a shell-sextet bound B_6 identifies the significant contracted shell-sextets. Finally, the shell-sextets are paired with the shell-pairs.If the resulting shell-octet quantity is found to be significant, the contracted integral class ⟨a_1a_2a_3a_4|b_1b_2b_3b_4⟩ must be computed via RRs, as discussed in the next section. The number of significant shell-mtuplets generated at each step is given in Table <ref>. As one can see, the size of any shell-mtuplet list is, at worst, quadratic in a large system.During the shell-pair screening, either a contracted or a primitive path is followed depending on the degree of contraction of the integral class K_tot=∏_i=1^n K_a_iK_b_i. If K_tot>1, the contracted path is enforced, otherwise the primitive path is followed. This enables to adopt the more effective primitive bounds for primitive integral classes which are usually associated with medium and high angular momentum PGFs and, therefore, are more expensive to evaluate via RRs. The scheme for primitive four-electron integrals differs only by the use of primitive bounds instead of contracted ones. The three-electron integrals screening scheme can be easily deduced from Fig. <ref>.Note that we bound an entire class of integrals with a single UB. This is a particularly desirable feature, especially when dealing with three- or four-electron integrals where the size of a class can be extremely large.For example, the simple pppppp and pppppppp classes are made of 729 and 4,096 integrals! §.§ Primitive bounds In this section we present UBs for primitive three- and four-electron integrals. Without loss of generality, we assume that the geminal operators are ordered by decreasing exponent, i.e. _13≤_14≤_23≤_34.All the required primitive bounds have the formB_m= max I_m, m=2, min I_m, m>2,where m is the shell multiplicity.The bound sets I_m are reported in Table <ref>.They also require the bound factors in Table <ref>, which are easily computed with the following quantities: h = D_a D_bγ_aγ_b√(a^a b^b/e^a+b)(4 αβ)^3/4/^a+b/2 e^- (1-)^2/α^-1+β^-1, _u^13,14,23,34 = (_u^13,14,23,34), ^13,14,23,34 =(_0^13,14,23,34·^2), where γ_a= ∏^2_i=0[ Γ(⌊a+i/3⌋+1/2 )]^-1/2, _u^13,14,23,34 =+ _u = [ _1000;0 _200;00 _30;000 _4;] + _u, _u^13,14,23,34 = ·(_u^13,14,23,34)^-1·, Γ(x) and ⌊ x ⌋ are the Gamma and floor functions respectively, <cit.> and _i= (1-_i)ζ_i. We point out that bound factors 1^12 or 1^12,13 reported in Table <ref> can be obtained from 1^12,13,14by setting _13=_14=0 or _14=0, respectively. The parameter σ_1 is obtained by solving the quadratic equation1^13,14,23,34_1 = 0. Factors of the kind 1^13,14,23,34 are also required for the bounds in Table <ref>. They are defined as the largest factor 1^13,14,23,34 within a given system and basis set, and can be pre-computed and stored with the remaining basis set information. §.§ Contracted bounds Contracted integral bounds are straightforward variations of primitive ones. While contracting at the shell-pair level (m=2) only requires K^2 computational work, contracting at the shell-quartet, -sextet or -octet level would require K^4, K^6 or K^8 work, respectively.Therefore, as sketched in Fig. <ref>, we use a primitive bound for a first screening of the shell-pairs, then contracted bounds are used to screen shell-pairs, -quartets, -sextets and -octets. Considering K-fold CGFs, the contraction step never exceeds 𝒪(K^2) computational cost. Bound factors such as1^13,14 = ∑_ij^K1^13,14_ij,and its maximum within the basis set 1^13,14, are computed within the shell-pair loop. In Eq. (<ref>), i and j refer to the PGFs a_1_i and b_1_j in the contracted shells |a_1⟩ and |b_1⟩, respectively (see Eq. (<ref>)).The expressions of the contracted bounds are identical to the primitive bounds, with the only exception that the contracted factors 13^12,13,23, 123^12,13,23 and 134^12,13,14,34 are bound by13^12,13,23 ≤min1^12,133^23, 1^123^13,23exp(-Y̌^13) , 123^12,13,23 ≤2min1^12,133^23, 1^123^13,23exp(-Y̌^13,23), 134^12,13,14,34 ≤min1^12,133^344^14, 1^12,143^134^34exp(-Y̌^13,14,34),whereY̌^13,14,23,34=( _0^13,14,23,34·Y̌^2),can be evaluated with the following expressions δ̌_u^13,14,23,34 = ζ̌ + _u = [ ζ̌_1000;0 ζ̌_200;00 ζ̌_30;000 ζ̌_4;] + _u, _u^13,14,23,34 = ζ̌·(δ̌_u^13,14,23,34)^-1·ζ̌, Y̌^k = [ 0 Y̌_12^k Y̌_13^k Y̌_14^k; 0 0 0 Y̌_24^k; 0 0 0 Y̌_34^k; 0 0 0 0; ], where ζ̌_i is the smallest effective exponent _i in the contracted shell-pair |a_ib_i⟩, andY̌_ij= max 0, _i _i^+/2-_j _j^+/2 - _i _i/2 - _j _j/2 is the distance between two spheres of diameters _i _i and _j _j (where _i _i^+ = _i + _i). § RECURRENCE RELATIONS §.§ Vertical recurrence relations Following Obara and Saika, <cit.> vertical RRs (VRRs) are obtained by differentiation of Eq. (<ref>) with respect to the centers coordinates. <cit.> For the integrals considered in this study, one can show that⋯_i^+ ⋯^m= ( _i_i - _i Y_0 ) ⋯_i ⋯^m - ( _i Y_1 - _i Y_0 ) ⋯_i ⋯^m+1 + ∑_j=1^n _j {( δ_ij/2ζ_i - _ij Y_0 )⋯_j^- ⋯^m - ( _ij Y_1 - _ij Y_0 ) ⋯_j^- ⋯^m+1},where δ_ij is the Kronecker delta, <cit.>_i = ∇_A_i/2α_i, _ij= _i _j,and _i Y_u = (Δ_u ·_i ^2), (_i ^2)_kl= κ_ikl ()_kl, _ij Y_u = (Δ_u ·_ij^2), (_ij^2)_kl= κ_iklκ_jkl/2, with_ij= 1, if i ≤ j, 0, otherwise, κ_ijk= _ijδ_ki - δ_ij_ki/ζ_i.One can easily derive VRRs for other three- and four-electron operators following the simple rules given in Fig. <ref>. The number of terms for each of these VRRs is reported in Table <ref> for various two-, three- and four-electron operators.Note that for a pure GG operator, we have m = 0 and Y_1 = Y_0. Therefore, Eq. (<ref>) reduces to a simpler expression:⋯_i^+ ⋯ = ( _i_i - _i Y_0 ) ⋯_i ⋯+ ∑_j=1^n _j ( δ_ij/2ζ_i - _ij Y_0 )⋯_j^- ⋯. §.§ Transfer recurrence relations Transfer RRs (TRRs) redistribute angular momentum between centers referring to different electrons. <cit.> Using the translational invariance, one can derive ⋯_i^+ ⋯ = ∑_j=1^n _j/2ζ_i ⋯_j^- ⋯ - ∑_j ≠ i^n ζ_j/ζ_i ⋯_j^+ ⋯ - ∑_j=1^n β_j _j _j/ζ_i ⋯_j ⋯.Note that Eq. (<ref>) can only be used to build up angular momentum on the last center. Moreover, to increase the momentum by one unit on this last center, one must increase the momentum by the same amount on all the other centers (as evidenced by the second term in the right-hand side of (<ref>)). Therefore, the TRR is computationally expensive for three- and four-electron integrals due to the large number of centers (see below). As mentioned by Ahlrichs, <cit.> the TRR can be beneficial for very high angular momentum two-electron integral classes.§.§ Horizontal recurrence relations The so-called horizontal RRs (HRRs) enable to shift momentum between centers over the same electronic coordinate: <cit.>⟨⋯_i ⋯|⋯_i^+ ⋯⟩ = ⟨⋯_i^+ ⋯|⋯_i ⋯⟩ + _i _i ⟨⋯_i ⋯|⋯_i ⋯⟩.Note that HRRs can be applied to contracted integrals because they are independent of the contraction coefficients and exponents. § ALGORITHM In this Section, we propose a recursive algorithm for the computation of a class of three- or four-electron integrals of arbitrary angular momentum. The present recursive algorithm is based on a late-contraction scheme inspired by the Head-Gordon-Pople algorithm <cit.> following a BOVVVVCCCCHHHH path. The general skeleton of the algorithm is shown in Fig. <ref> for the trident operator C_12G_13G_14. We will use this example to illustrate each step.Based on the shell data, the first step of the algorithm (step B) is to decide whether or not a given class of integrals is significant or negligible. If the integral class is found to be significant by the screening algorithm presented in Sec. <ref> and depicted in Fig. <ref>, an initial set of FIs is computed (step O) via the formulae gathered in Sec. <ref>.Starting with these FIs, angular momentum is then built up over the different bra centers _1, _2, _3 and _4 using the VRRs derived in Sec. <ref>. To minimize the computational cost, one has to think carefully how to perform this step. Indeed, the cost depends on the order in which this increase in angular momentum is performed. This is illustrated in Fig. <ref>, where we have represented the various possible pathways for the 3-chain operator C_12G_23 (left) and the trident operator C_12G_13G_14 (right). The red path corresponds to the path generating the least intermediates (i.e. requiring the smallest number of classes in order to compute a given class). Different paths are compared in Table <ref> for various two-, three- and four-electron operators, where we have reported the number of intermediates generated by each path for various integral classes.Taking the 3-chain operator C_12G_23 as an example, one can see that, to compute a ppp class, it is more advantageous to build momentum over center _3, then over centers _2, and finally over center _1 using VRRs with 4, 6 and 6 terms, respectively. The alternative path corresponding to building momentum over _3, _1, and then _2 with 4-, 5- and 7-term VRRs is slightly more expensive for a ppp class but becomes affordable for high angular momentum classes. For both paths, using the TRR instead of the last VRR implies a large increase in the number of intermediates.For the trident operator, we successively build angular momentum over _4, _3, _1 and _2 using VRRs with 4, 6, 8 and 7 terms. The pathway using VRRs with 4, 6, 6, and 9 terms is more expensive due to the large number of terms of the VRR building up momentum over the last center. Again, using the TRR instead of the last VRR significantly increases the number of intermediates.The path involving the minimal number of intermediates is given in Table <ref> for various two-, three- and four-electron operators.It is interesting to point out that it is never beneficial to use the TRR derived in Eq. (<ref>) (see Sec. <ref>).One can easily show that, for operators involving the Coulomb operator, the number of intermediates required to compute a n-electron integral class a … a increases as a^n+1 for the VRR-only paths (see Table <ref>). This number is reduced to a^n if one uses the TRR to build up angular momentum on the last center.However, the prefactor is much larger and the crossover happens for extremely high angular momentum for three- and four-electron integrals. For “pure”GG operators, such as G_12 or G_13G_23, the number of intermediates required to compute a class a … a increases as a^n for any type of paths.Finally, we note that the optimal path for the trident C_12G_13G_14 and the 4-chain C_12G_13G_34 is similar, thanks to their similar structure. Indeed, these two operators can be seen as two “linked” GGs (G_13G_14 or G_13G_34) interacting with the Coulomb operator C_12 (see Fig. <ref>), while the other 4-chain operator C_12G_14G_23 can be seen as two “unlinked” GGs (G_14 and G_23) interacting with the Coulomb operator.When angular momentum has been built over all the bra centers, following the HGP algorithm, <cit.> we contract _1 _2 _3_4 to form ⟨_1 _2 _3_4|⟩ (step CCCC). We can perform the contraction at this point because all of the subsequent RRs are independent of the contraction coefficients and exponents. More details about this contraction step can be found in Ref. Gill94b.The last step of the algorithm (step HHHH) shifts momentum from the bra center _1, _2, _3 and _4 to the ket centers_1, _2, _3 and _4 using the two-term HRRs given by Eq. (<ref>) in Sec. <ref>. § CONCLUDING REMARKS We have presented the three main ingredients to compute three- and four-electron integrals involving GGs. Firstly, a straightforward method to compute the FIs is given. Secondly, scaling-consistent UBs are reported, as they allow to evaluate only the N^2 significant integrals in a large system.Finally, the significant integrals are computed via a recursive scheme based on vertical and horizontal RRs, which can be viewed as an extension of the PRISM late-contraction path to three- and four-electron integrals. 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"authors": [
"Giuseppe M. J. Barca",
"Pierre-François Loos"
],
"categories": [
"physics.chem-ph"
],
"primary_category": "physics.chem-ph",
"published": "20170426104418",
"title": "Three- and four-electron integrals involving Gaussian geminals: fundamental integrals, upper bounds and recurrence relations"
} |
Nanoscience Center, Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen Ø, Denmark Department of Physics, Oxford University, Oxford, OX1 3PU, United KingdomNanoscience Center, Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen Ø, Denmark Interdisciplinary Nanoscience Center - INANO-Kemi, 8000 Aarhus C, DenmarkNanoscience Center, Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen Ø, Denmark Institute of Energy Conversion, Technical University of Denmark, 4000 Roskilde, DenmarkNanoscience Center, Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen Ø, DenmarkNanoscience Center, Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen Ø, DenmarkInstitut Max Von Laue Paul Langevin, 38042 Grenoble, FranceHelmholtz-Zentrum Berlin, 14109 Berlin, Germany Department of Physics, Technical University of Denmark, 2800 Kgs. Lyngby, DenmarkInstitute of Energy Conversion, Technical University of Denmark, 4000 Roskilde, DenmarkDepartment of Physics and Institute of Materials Science, University of Connecticut, Connecticut 06269, USA Present address: Naval Surface Warfare Center Indian Head EOD Technology Division, Indian Head, MD 20640 USA Nanoscience Center, Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen Ø, DenmarkDepartment of Physics and Institute of Materials Science, University of Connecticut, Connecticut 06269, USANanoscience Center, Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen Ø, DenmarkWe present detailed neutron scattering studies of the static and dynamic stripes in an optimally doped high-temperature superconductor, La_2CuO_4+y. We observe that the dynamic stripes do not disperse towards the static stripes in the limit of vanishing energy transfer. Therefore, the dynamic stripes observed in neutron scattering experiments are not the Goldstone modes associated with the broken symmetry of the simultaneously observed static stripes, and the signals originate from different domains in the sample.These observations support real-space electronic phase separation in the crystal, where the static stripes in one phase are pinned versions of the dynamic stripes in the other, having slightly different periods.Our results explainearlier observations of unusual dispersions in underdopedLa_2-xSr_xCuO_4 (x=0.07)and La_2-xBa_xCuO_4 (x=0.095).Distinct nature of static and dynamic magnetic stripes in cuprate superconductors K. Lefmann December 30, 2023 ================================================================================= An imperative open question in materials physics is the nature of high-temperature superconductivity. Unlike conventional superconductors, where the Cooper pairing mechanism is well-established <cit.>, the pairing mechanism in high-temperature superconductors (HTS) still sparks controversy <cit.>. A comprehensive description of the electronic behavior inside HTS is indispensable to push this field of research onward. Hence, the magnetic structures which appear close to as well as inside the superconducting phase are still being studied intensively <cit.>. In many HTS compounds, experiments indicate a modulated magnetic structure,consistent with superconducting "stripes" of charge separated by magnetic regions as sketched in Fig. <ref>a <cit.>. Magnetic excitations, referred to as "dynamic stripes", are found with similar periodicity, and are therefore thought to be related tothe Goldstone modes of the static stripes <cit.>.Here we present evidence that this model is incomplete for a family of HTS. We find that the dynamic stripes do not disperse towards the static stripes in the limit of vanishing energy transfer and interpret this in terms of electronic phase separation, where static and dynamic stripes populate different spatial regions of the HTS. Compounds based on the La_2CuO_4 family were the first high-temperature superconductors (HTS) to be discovered <cit.>. They become superconducting upon doping with electrons or holes, with a maximum critical temperature, T_c≈ 40 K, whether the dopant is Sr (La_2-xSr_xCuO_4, LSCO), Ba (La_2-xBa_xCuO_4, LBCO), or O (La_2CuO_4+y, LCO+O).The generic crystal structure of these compounds is illustrated in Fig. <ref>b. They consist of planes of CuO separated by layers of La/Sr/Ba. Each Cu atom is at the center of an octahedron of oxygen atoms.At elevated temperatures these materials are in the high-temperature tetragonal (HTT) phase. Upon lowering the temperature, the crystals enter the low-temperature orthorhombic phase (LTO) where the oxygen octahedra tilt around the tetragonal a axes, leading to a change in lattice parameters, a<b and to possible twinning <cit.>, see Supplementary Material for details <cit.>.Since the first discovery, a multitude of HTS have been found in the cuprate family. The amplitude and period of the stripe order modulations vary strongly with the choice and amount of dopant, with static stripes being particularly pronounced in LCO+O <cit.>.The spin stripes can be measured using magnetic neutron scattering, where they are observed as pairs of intensity peaks at incommensurate (IC) wave vector transfers, e.g. at Q=(1+δ_h,δ_k,0) and Q=(1-δ_h,-δ_k,0) for stripes along the (110) direction, see Fig. <ref>c. Here, the components of the scattering vector are given in terms of(2π/a, 2π/b,2π/c), where a,b and c are the orthorhombic lattice constants. The real-space modulation period is L ∼ (2π /δ) a, and we refer to δ as the incommensurability of the stripes. Typically δ_h ≈δ_k, indicating that the modulation is approximately along the Cu-O-Cu bonds (the (110) and (11̅0) directions), although variations have been reported, indicating a kink in the stripes after a number of unit cells <cit.>.Typically, stripes areobserved not only at the above mentioned positions, but also at Q=(1-δ_h,δ_k,0) and Q=(1+δ_h,-δ_k,0), giving rise to a quartet of peaks around the (100) position, as illustrated in Fig. <ref>d. This indicates that the compound exhibits stripes (approximately) along both the (110) and (11̅0) directions, most likely by the stripes in adjacent layers alternating between the (110) and (11̅0) directions <cit.>.Inelastic neutron scattering has shown the presence of dynamic stripes, which at low energies have similar modulation period as the static stripes <cit.>.The modulation period of the stripes is found to be almost constant up to energy transfers of Δ E ∼ 10-15 meV <cit.>. In the cuprates an hourglass shaped dispersion develops at higher energies <cit.>.The incommensurability of the stripes varies with doping. In the LSCO-type cuprates, δ increases linearly with doping and saturates at a maximal value of δ =1/8 <cit.>. In some cuprates, similar stripes of charge with half the modulation period have been observed using X-ray diffraction, validating the picture of magnetic and charge stripes in Fig. <ref>a <cit.>. However, the energy resolution of X-rays does not allow to distinguish between static and dynamic stripes.We have used elastic and inelastic scattering of low energy neutrons to accurately measure the reciprocal space position of the static and dynamic stripes in highly oxygenated LCO+O in the LTO phase.The experiments were performed at the cold-neutron triple axis spectrometers FLEXX at HZB, Berlin <cit.>, and ThALES at ILL, Grenoble <cit.>. The elastic energy resolution in the ThALES experiment was 0.24 meV (Full Width at Half Max, FWHM), while the Q resolution was 0.05 r.l.u. (FWHM). For further details on the experiments, see the Supplementary Material <cit.>. Panel (d) of Fig. <ref> shows how we probe two of the four IC peaks in our neutron scattering experiments. The actual data for a series of scans are shown in panels (e) and (f) as 2D colorplots. Fig. <ref> shows examples of the scans through the center of the peaks at 0 and 1.5 meV energy transfer, probing the static and dynamic stripes, respectively. The inset illustrates the direction of the scans in reciprocal space. To eliminate errors from minor misalignments, we determine the incommensurability along k, δ_k, as half the distance between the peak centers.In Fig. <ref> we display δ_kfor all energy transfers probed in the experiment at two temperatures. As expected, the dynamic stripes appear at the same reciprocal space position in the normal phase (45 K) as in the SC phase (2 K) (within the instrument resolution), whereas the static stripes are only present at low temperature. The elastic stripes are found to be rotated by 7^∘ from the Cu-O-Cu bond directions, while the observed inelastic stripes are rotated by 3^∘. The inelastic dispersion appears continuous and steep, consistent with earlier cuprate results <cit.>. However, the elastic signal shows a large and significant difference in δ_k, appearing as a discontinuity in the dispersion relation at vanishing energy transfer.Similar observations have been briefly remarked upon in underdopedLa_2-xSr_xCuO_4 (x=0.07) <cit.> andLa_2-xBa_xCuO_4 (x=0.095) <cit.>. In both cases the observation was left unexplained.To rule out that these surprising differences in δ_k and the stripe rotation are artifacts caused by experimental non-idealities, we have performed a virtual ray-tracing experiment using a close model of our experiment, further detailed in the Supplementary Material <cit.>. This method is known to accurately reproduce experimental effects like peak broadening and displacement<cit.>. The virtual experiments exclude misalignment of the instrument as a cause of the effect and show that the experimental resolution can cause a tiny shift in the observed incommensurability, see Fig. <ref>. The experimentally observed shift in peak position is, however, more than an order of magnitude larger than what can be explained by instrument effects, and is therefore a genuine property of the sample. Hence, in order not to violate the Goldstone's theorem, the static and dynamic stripes must originate from different regions in the sample. There are two probable ways this can occur:First, the dynamic stripes could be transverse fluctuations from the static stripe order, resembling ordinary spin waves.Due to the neutron scattering selection rules, the scattering observed in the elastic and inelastic channels stem from different twin domains as explained in detail in the Supplementary Material <cit.>. This results in a shift in the observed peak position between the elastic and inelastic channels, comparable to the observed shift. The magnitude and direction of the shift due to twinning depends heavily onδ_h and δ_k and requires δ_h<δ_k.Secondly, the static and dynamic spin response may originate from different microscopic regions which are not related by twinning. This suggests a real-space electronic phase separation of the crystal into regions with two different spin structures; one domain type which has static stripe order and associated dynamic stripes, and another type of domain where only dynamic stripes are present.At first glance the twinning model seems to provide an explanation of our data. However, it fails to explain the similar observations in LBCO (where δ_h=δ_k) mentioned above <cit.>, as the model requires δ_h<δ_k. Furthermore, the model relies on the assumption that the four twin domains display only one type of stripe order with associated transverse excitations.Most likely these assumptions are too simplified and relaxing any of themreduces the effect of twinning on the observed signal. We therefore turn to the second model: electronic phase separation.Muon spin rotation experiments on highly oxygenatedLCO+O show that the material electronically phase separates into a magnetic (A) and a superconducting (B) phase of roughly equal volume<cit.> and transition temperature T_c≈ T_N∼ 40 K with the present slow cooling conditions. Based on these experiments we propose the following properties of the two phases:Phase A is underdoped (resembling LSCO with n_h=0.125) and has static magnetism (and weak fluctuations), responsible for the observed static signal and a small fraction of the dynamic signal. Phase B is optimally doped (resembling LSCO with n_h=0.16) and superconducting with strong fluctuations, responsible for (the majority of) the observed dynamic signal. We note that no spin gap was observed below T_c in our experiments. Absence of a spin gap was also observed in the experiments on strongly underdoped LSCO <cit.> and LBCO <cit.>, mentioned above. Both materials were suggested not to be d-wave superconductors but instead display Pair Density Wave (PDW) superconductivity <cit.>. Our results are consistent with this interpretation. A PDW state would require some degree of magnetic order in the SC phase, but this may be extremely weak and thus effectively invisible in our experiments.The simultaneous observation of gapless excitations and a shift in incommensurability in all three compounds suggests a connection between the two effects. The gapless excitations are likely a result of a PDW state in the sample, while the shift is caused by electronic phase separation. At present it is unclear whether these two behaviours are related. The critical temperature of the superconducting phase, T_c, coincides with the Néel temperature of the magnetic phase, T_N, such that above this temperature superconductivity and the static magnetism disappear, but strong stripe fluctuations remain. The fact that T_c≈ T_N is likely not coincidental, but it is unclear whether the electronic phase separation is caused by, or is the cause of, the close proximity of T_c and T_N. We suggest a scenario where the ground state energy for phase A and phase B are very close and lowering the temperature below T_c will cause an electronic phase separation with concurrent static magnetism and superconductivity.The spatial distribution of impurity potentials as well as inhomogeneous hole doping becomes important parameters that can tip a region towards becoming type A or B, e.g. by pinning.The relative population of each phase is primarily controlled by the total number of holes, but can also be influenced by an applied magnetic field or by crash cooling <cit.>. In the case of LCO+O, crash cooling can further inflict a lowering of T_c which has been explained by disconnection of the optimally superconducting pathways <cit.>.The crucial point is that although the two phases are closely related, there is no a priori reason why the stripe order in phase A and the stripe dynamics in phase B should have the same incommensurability. Indeed, our results show that this is not the case. This indicates that other properties of the stripes may not be identical either, and one should thus be extremely careful when interpreting neutron scattering data on stripes.Phase separation has been suggested to occur in a number of cuprates or related compounds, a few of which we will mention here. LSCO with x=0.12has been suggested to phase separate into microscopic superconducting regions with gapped dynamic stripes and non-superconducting regions with static stripes <cit.>. Spontaneous, microscopicphase separation has also been observed in purely oxygen doped LCO+O crystals <cit.> and in crystals doped with both oxygen and strontium <cit.>. Furthermore, recent studies of La_5/3Sr_1/3CoO_4 show evidence of microscopic phase separation into components with different local hole concentration <cit.>. In the latter material the upper and lower parts of the hourglass dispersionare even proposed to originate from different nano-scale structures in the sample <cit.>. No discrepancy in the incommensurability between static and dynamic stripes was reported in these studies.The idea of dynamic and static stripes having different origin is supported by a number of other observations.For example, the static and dynamic stripes exhibit different behaviors as function of temperature. In underdoped LSCO and in LCO+O as evidenced in this experiment, the static stripes vanish above T_c, but the dynamic stripes remain to far higher temperatures <cit.>. In contrast, in the optimally doped region, the static stripes are altogether absent, while the dynamic stribe exist above a certain energy gap <cit.>. In the heavily overdoped region it has been shown that substituting small amounts of Fe for Cu induces static magnetism <cit.>. The incommensurability of the induced magnetic order is governed by nesting of the underlying Fermi surface and differs from the 1/8 periodicity of the low-energy dynamic stripes.When applying a magnetic field, the static stripes are in general strengthened <cit.>, with a few exceptions <cit.>. In many cases this happens with an accompanying change in the dynamic stripe spectrum <cit.>, but in other cases, the dynamic stripe spectrum is unchanged <cit.>.Hence, the coupling between static and dynamic stripes is not simple and unique. In conclusion we have found that the dynamic stripes do not disperse towards the static stripes in the limit of vanishing energy transfer in a HTSC. The effect is subtle and requires high flux and good resolution such as provided by the ThALES spectrometer in order to be observed. Our findings are, however, of prime importance, since they suggest that the observed static and dynamic stripes originate from different electronic phases in the sample, where one of these phases is likely to be a competitor for superconductivity with the development of static stripe order.Our observations are relevant for all compounds displaying stripe order. As an example, the structurally similar, but non-superconducting compound (La,Sr)_2NiO_4 (LSNO) displays magnetic and charge stripes with the dynamic stripes persisting at higher temperatures than the ordering temperature <cit.>. In some of these compounds it has also beenobserved that the ordering vector of the static and dynamic stripes do not coincide at vanishing energy transfer <cit.>. We speculate that a similar electronic phase separation could be in play here, as we suggest for LCO+O. 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Sc. thesis, University of Copenhagen (2015). § SAMPLE DETAILSThe samples were prepared by growing stoichiometric LCO crystals at the Technical University of Denmark in an optical image furnace using the travelling solvent float zone technique. After annealing and characterization, chosen crystals were super-oxygenated in an aqueous bath at the University of Connecticut. The resulting LCO+O crystals were cut in pieces suitable for neutron scattering experiments and smaller pieces of the sample were characterized by resistivity and susceptibility measurements to find a superconducting transition temperature of 40 ± 1 K, typical for LCO+O .In equilibrium LCO+O spontaneously separates into a series of superconducting and magnetic line phases. If out of equilibrium due to gross oxygen inhomogeneity, crystals often contain both T_c=30 K and T_c = 40 K superconducting regions <cit.>.The superconducting transition temperature for our sample was measured by a vibrating sample magnetometer. The data is shown in Fig. <ref> and show that our sample clearly has only one superconducting transition near T=40 K, and therefore just one superconducting phase that is similar to what was obtained by Lee et al. <cit.>.The orthorhombic lattice parameters are a=5.33 Å, b=5.40 Å, c=13.2 Å. The spins align along the b axis in LCO+O, just as for the parent compound <cit.>.§ DETAILS ON NEUTRON SCATTERING EXPERIMENTS ON LCO+ONeutron scattering experiments were performed at the cold-neutron triple axis spectrometers FLEXX at HZB, Berlin <cit.>, and ThALES at ILL, Grenoble <cit.>. The spectrometers were configured to run at a constant final energy of 5.0 meV. At FLEXX the sample was mounted inside a magnet. The results of applying a magnetic field will be reported in a following publication.In both experiments, a velocity selector in the incident beam before the monochromator removed second-order contamination, while a cooled Be-filter between sample and analyzer further reduced background. The sample was aligned in the (a,b) plane. At FLEXX, several cylindrical crystals were co-aligned, resulting in a total sample mass of ∼ 9 g. At ThALES, only the largest crystal of mass 3.44 g was used in order to improve Q resolution.In both experiments, we used vertically focusing monochromators, leading to relatively loose vertical collimation along the c-direction, where the stripe signal from cuprates is nearly constant <cit.>. At the ThALES experiment there was a small offset in the A4 angle which was corrected for in the subsequent data analysis. The effect reported in the main paper was also seen in the experiment at FLEXX. An example of the data is shown in Fig. <ref>, showing the same difference between elastic and inelastic stripe positions as found at the ThALES experiment. In most of the experiment, however, only a single peak was measured due to time constraints. We here show additional data from the ThALES experiment. The measurements at Δ E= 0 meV and 1.5 meV were taken as grid scans in the (h,k) plane around the (100) position. The data were fitted to a pair of two-dimensional Gaussians, as seen in Fig. <ref>. The fitted peak positions are given in Tab. <ref>From the fits, the shift in the incommensurability between Δ E=0 meV and 1.5 meV is (-0.0070(15),0.0090(9),0). The elastic data around the (010) peak, similar to the data around (100) except for a 90 degree rotation are shown in Fig. <ref>. The incommensurability is the same within error bars for the two data sets. Thestatic stripes are rotated by approximately 7^∘ away from the Cu-O-Cu directions; a value that is approximately twice what was found earlier <cit.>. The peak positions of the fitted two-dimensional Gaussians are shown in Fig. <ref>. The peak positions from the (010) peak were rotated 90 degrees for comparison. In this figure, it is observed that the inelastic peaks move together in k and to larger h, compared to the elastic peaks. The data show that the distance between the incommensurate peaks is dependent on where exactly on the peak the measurement is performed.The shape of the resolution function and the peaks implies that the observed value of δ_k increases as function of h. This could lead to systematic errors, if different parts of the peak were probed at different energies. By making the grid scans shown in Figs. <ref> and <ref>, we measured exactly this effect. Each scan in the grid was fitted individually. The resulting value of δ_k, determined as a function of h, is shown in Fig. <ref>. It is seen that δ_k increases with increasing h, as we move through the peak. However, it is also evident that at any given position in h, a significant shift in δ_k happens between the elastic and inelastic signals.§ VIRTUAL EXPERIMENTS The full neutron scattering experiment at ThALES was simulated using the Monte Carlo (MC) ray tracing program McStas <cit.>, which has previously been shown to produce very accurate results regarding, in particular, instrument resolution <cit.>. Here follows a more detailed description of the simulation method.The guide system at ILL has been simulated by E. Farhi <cit.>, and we adopted his McStas model. The remainder of the instrument was simulated using standard components from McStas: slits, graphite crystals, and a detector. For the purpose of these simulations, asample model was written, simulating scattering from static and dynamic stripes. This sample scatters elastically at a user-defined position in (h,k)-space (with the scattering being independent of l, as is true to a good approximation near l=0 where the experiment was performed <cit.>). Furthermore, the sample scatters inelastically at a (possibly different) user-defined position in (h,k)-space. The absolute scattering cross section of the elastic and inelastic scattering can be set individually. For simplicity, and since we are not interested in absolute intensities, the cross section was kept independent of energy transfer, and no incoherent background was simulated. A combination of two such samples, rotated with respect to each other, was used to simulate the two measured peaks at q=(1+δ_h,±δ_k,0).A small offset in the A4 angle in the experiment caused the lattice parameters to appear slightly larger than their actual values. This was accounted for in the simulations. Two sets of simulations were made: one with both the elastic and inelastic peaks at δ_h=δ_k = 1/8, and one with δ_h= 0.0973, δ_k=0.1222 for the elastic peaks and δ_h=0.1036, δ_k=0.1133 for the inelastic peaks, as found in the experiments. For each set of simulations, similar scans to the ones used in the experiments were simulated.Grid scans, similar to the ones shown in Fig <ref> were simulated at Δ E=0 and 1.5 meV, see Fig. <ref>, where the simulations are shown on top of fits to two-dimensional Gaussians. The tilt of the peaks in the h,k-plane slightly deviates from the data, although the width of the peaks is reproduced correctly. This deviation is likely caused by small inaccuracies in the description of the ThALES instrument, and does not influence our conclusions. Single scans are shown in Fig. <ref> for the two sets of simulations. It is evident that the resolution of the instrument does not cause a shift in the distance between the incommensurate peaks. In particular, if we assume the traditional model of steeplydispersing stripes, the simulated data do not match the actual data (Fig. <ref> a), whereas a model with a significant difference in the peak distance between the elastic and inelastic signals agrees with the data (Fig. <ref> b).In Fig. <ref>, we show that the experimental observation of δ_k depending on h is reproduced in the simulations. Here it is also clear that a shift in peak position between the elastic and inelastic data is needed to explain the data. There is a small effect of the resolution, causing the peaks to shift slightly towards smaller h and δ_k at higher energies. There is also a small difference between the simulated and measured values of δ_k as function of h for the inelastic data. Both of these effects are too small by at least an order of magnitude to explain the experimental results.We further checked the effect of the instrument not being perfectly calibrated, so that the actual value of k_f differed slightly from the set value. This was done by increasing/decreasing E_f by 0.05 meV, while keeping all other parameters constant. Here, E_f refers to the energy that the analyzer was aligned to select. This did indeed cause a small shift of the peak position in h, but there was no change in δ_k, as expected.In total, our simulations show that the experimental observations are not caused by instrumental effects such asresolution or misalignment. § TWINNINGThe stripe modulations in LCO+O are roughly along the Cu-O-Cu bonds <cit.>, 45^∘ to the orthorhombic a (and b) axis, but parallel to the tetragonal a axis as shown in Fig. 1 in the main paper. Twinning occurs when cooling through the transition from a tetragonal to an orthorhombic unit cell, where a≠ b. In LCO+O, a=5.33 Å, b=5.40 Å at low temperature <cit.>. The twinning is caused by the oxygen octahedra tilting around different axes, which slightly rotates the crystallographic axes of the different domains, see Fig. <ref>a. The results in reciprocal space is that each peak is split into two as shown in Fig. <ref>b. Similarly, a domain wall can run along the (11̅0) direction, creating a second set of twins, giving rise to a total of four close-lying peaks illustrated in Fig. <ref>c <cit.>. Two of the peaks originate from domains with the a axis along the experimental "(100)" direction, and two with the b axis along this direction, see Fig. <ref> where the direction of the spins in each domain is also shown. Each set of two peaks with similar orientation is split by an angle Δ = 90^∘ - 2 tan^-1(a/b),which in our case is around 0.7^∘, or ∼ 0.01 r.l.u. at the position of the IC peaks - too small to separate the peaks with our resolution, see Fig. <ref>. The twinning can, however, easily be observed at e.g. the nuclear (200) Bragg peak. To estimate the effect on twinning on the observed peak positions we assume that the twin domains are equal in size and have the same stripe order with spins aligned along the local b axis of the twin domain <cit.>. The stripe order in each domain will give rise to a quartet of peaks centered around the (100), (010), (1̅00) and (01̅0) positions, leading to a total of 16 peaks around the experimental (100) position, as illustrated in Fig. <ref>a. Each group of four peaks around (1±δ_h, ±δ_k,0) overlap and cannot be distinguished within the instrument resolution, and will be observed as a single peak. The position of the signal observed with neutron scattering is the average of the position of the four peaks that contribute, weighted by their relative intensity. In magnetic neutron scattering, the intensity is proportional to the component ofperpendicular to . In this experiment, due to twinning, we measure both ≈ (100) and ≈ (010). Since the spins lie along the local (010) direction, there are two domains which have ⊥ (red circle and blue triangle in Fig. <ref>), while two have ∥ (green diamond and yellow square in Fig. <ref>).In the measurement of the elastic stripe signal, only the two former peakswill have measurable intensity, while the two latterwill be suppressed. In this discussion we assume for simplicity that ∥ (100); the error in the calculated peak position from this assumption is around 1% and thus negligible in this context.Considering the dynamic signal and assuming the fluctuations to be transverse, each of the four peaks will have a non-zero contribution to the observed neutron scattering signal. The position of this signal will thus be different than that of the elastic signal where only two of the domains contribute.The resulting observed shift in peak position is illustrated by the arrow in Fig. <ref>a for δ_h=δ_k and Fig. <ref>a for δ_h<δ_k.In Fig. <ref>c we zoom in on one of the peak positions for δ_h<δ_k, showing where the expected peaks from the four domains will be in the model described here, along with the data. We also show where the peak would be observed in a neutron scattering experiment assuming the model explained above.We note that in this model it is a requirement that δ_h<δ_k as is true in our case. If δ_h=δ_k, as is usually the case in cuprates, the shift would be in the opposite direction, as evident in Fig. <ref>a. | http://arxiv.org/abs/1704.08528v6 | {
"authors": [
"H. Jacobsen",
"S. L. Holm",
"M. -E. Lacatusu",
"A. T. Rømer",
"M. Bertelsen",
"M. Boehm",
"R. Toft-Petersen",
"J. -C. Grivel",
"S. B. Emery",
"L. Udby",
"B. O. Wells",
"K. Lefmann"
],
"categories": [
"cond-mat.supr-con"
],
"primary_category": "cond-mat.supr-con",
"published": "20170427120913",
"title": "Distinct nature of static and dynamic magnetic stripes in cuprate superconductors"
} |
Turku Centre for Quantum Physics, Department of Physics and Astronomy, University of Turku, FI-20014 Turun yliopisto, FinlandTurku Centre for Quantum Physics, Department of Physics and Astronomy, University of Turku, FI-20014 Turun yliopisto, FinlandInstitute for Complex Quantum Systems and Center for Integrated Quantum Science and Technologies, Universitat Ulm, D-89069 Ulm, Germany Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, AustriaTurku Centre for Quantum Physics, Department of Physics and Astronomy, University of Turku, FI-20014 Turun yliopisto, Finland Center for Quantum Engineering, Department of Applied Physics, Aalto University School of Science, P.O. Box 11000, FIN-00076 Aalto, Finland CentreforTheoreticalAtomic,MolecularandOpticalPhysics, Queen'sUniversityBelfast,BelfastBT7 1NN,UnitedKingdom We present an in-depth study of the non-equilibrium statistics of the irreversible work produced during sudden quenches in proximity to the structural linear-zigzag transition of ion Coulomb crystals in 1+1 dimensions. By employing both an analytical approach based on a harmonic expansionand numerical simulations, we show the divergence of the average irreversible work in proximity to the transition. We show that the non-analytic behaviour of the work fluctuations can be characterized in terms of the critical exponents of the quantum Ising chain.Due to the technological advancements in trapped ion experiments, our results can be readily verified. Non-equilibrium quantum thermodynamics in Coulomb crystals G. De Chiara December 30, 2023 ==========================================================§ INTRODUCTION The recently renovated interest in non-equilibrium thermodynamics of quantum systems, spurred from tremendous advances in experimental techniques, has found a plethora of interesting developments and applications <cit.>. From a theoretical perspective,several important achievements are already available in the literature. Prominent examples are the quantum generalisation of fluctuation relations such as the celebrated Jarzynski equality <cit.> and the design of a single-atom thermal machine <cit.>, recently realised with a trapped ion <cit.>.In recent years, much interest has been devoted to the analysis of the quantum work extracted from, or absorbed by, a quantum system.While many definitions of work in a quantum setting have been proposed, the most popular one in the literature, based on two-time measurements <cit.>, fulfils the Jarzynski equality but not the first law of thermodynamics when the system exhibits energy coherences <cit.>. Furthermore, several proposals have been put forward to estimate work in quantum systems without the need of realising energy projections. These include schemes based on Ramsey interferometry <cit.> experimentally realised in an NMR setting <cit.>. Other methods employ the aid of auxiliary continuous quantum systems <cit.>. Some others are based on population imbalance and coherence in bosonic Josephson junctions <cit.>.In the many-body scenario, fluctuations of the work and of its irreversible contribution have been calculated mainly for spin chains in proximity to a quantum phase transition <cit.>. In this context it has been shown that all the moments of the work probability distribution are singular when the system is dynamically driven close to the phase transition <cit.>. In this context, most studies so far havebeen limited to archetypal examples of strongly correlated systems in condensed matter <cit.>. However, testing such predictions on experimental platforms poses additional challenges. In fact, the full probability distribution seems to be unaccessible to observation because of the complexity of energy projection and because Ramsey schemes would involve, unrealistically, an ancilla coupled to the whole many-body system.In this work, we analyse the out-of-equilibrium quantum thermodynamics of a model specifically tailored to an experimental setup. Precisely, we consider ion Coulomb crystals (ICC): Many-body quantum systems of cold atomic ions confined to highly anisotropic traps and mutually interacting via Coulomb repulsion <cit.>. Here we estimate the statistics of the irreversible work production during sudden quenches in proximity to the phase transitions in such ICC. Our results can be experimentally tested in current experiments by measuring, as we show in this work, the transverse displacement distribution of the ions positions. At equilibrium, ICC exhibit different structural arrangements, generally depending on the spatial properties of the trapping potential <cit.>. The structural transitions occurring between these configurations are typically phase transitions of the first order <cit.>. Here, we focus on quasi one-dimensional arrangements obtained for strongly anisotropic traps, which exhibit a transition from a linear to a zigzag configuration <cit.>. In the limit of ultracold ions, quantum fluctuations become relevant and the linear-zigzag transition becomes a quantum phase transition of the Ising universality classin 1+1 dimensions, at the thermodynamical limit <cit.>. The production of defects during a quick change of the trap anisotropy has been studied theoretically <cit.> and experimentally <cit.>.With this setup in mind we analyse the fluctuations of the work performed upon the ICC system by changing suddenly the transverse confinement frequency near the linear-zigzag transition point. We compare analytical results from calculations based on a harmonic expansion with numerical calculations based on the density matrix renormalization group algorithm (DMRG) <cit.> in the matrix product state formalism <cit.>. We show that when approaching the critical point such fluctuations display a singularity, and they exhibit a universal scaling compatible with the quantum Ising model. § LINEAR-ZIGZAG MODEL FOR ICCIons confined to a fully anisotropic 3D trap and interacting via repulsive Coulomb interaction undergo a structural phase transition in which their spatial geometry changes from a one-dimensional linear chain to a planar zigzag configuration <cit.>. The control parameter of such a transition is the frequency ω of the transverse harmonic trapping. This transition is driven by a mechanical instability of the chain that is associated with a soft mode at the boundary of the Brillouin zone whose frequency vanishes at the critical transverse trapping frequency. In the following paragraph we briefly review the analytical approximations needed to recast the linear-zigzag transition into a simple short-range model. For convenience, we restrict the motion of the ions to the XY-plane in which the zigzag structure develops. Here X is the direction parallel to the trap axis and Y is the direction perpendicular to the trap axis that emerges as a result of spontaneous symmetry breaking or because of a small anisotropy in the transverse confinement.The strong repulsion between the ions makes them practically distinguishable particles, which in turn allows us to write a Hamiltonian in a first quantizationH_0 = ∑_j=1^L[P_x,j^2+P_y,j^2/2M+Mω_0^2/2Y_j^2+V_L(X_j)]+Q^2/8πϵ_0∑_i j[(X_i-X_j)^2+(Y_i-Y_j)^2]^-1/2in which Q is the ion charge, M is the mass, (X_j,Y_j) and (P_x,j,P_y,j) are the position and momentum of the j-th ion, respectively, and V_L(x) is the longitudinal component of the confining potential. The quantum nature of this model stems from the commutator [X_i,P_x,j] = [Y_i,P_y,j] = i ħδ_i,j. Asshown in <cit.>, when the chain is sufficiently close to criticality, the longitudinal and transversal components of H effectively decouple. One can therefore fix the average equilibrium positions of the ions along the longitudinal direction to x_j = j a, with a the effective Wigner lattice spacing. Afterwards, the transverse dynamics Hamiltonian H_y can be Taylor-expanded at fourth order in the displacements y_j. The resulting theory, still long-range, can then be recast into a short-range model through an expansion of the scattering matrix of the harmonic modes, at second order in δ k around the soft mode (δ k = k - π/a) <cit.>. This mapping effectively simplifies the Hamiltonian of Eq. (<ref>) intoH(ω) =1/2∑_j=1^L[- g^2∂^2/∂y_j^2+(ω^2-h_1)y_j^2+ h_2(y_j+y_j+1)^2+h_3 y_j^4], in which g=√(ħ^2/Ma^2E_0) plays the role of an effective Planck constant measuring the impact of quantumfluctuations <cit.>. Here all quantities have been expressed in dimensionless scales, according to: H=H_0/E_0 with E_0=Q^2/(4πϵ_0a), y_j = Y_j/a, and ω=ω_0/√(E_0/Ma^2). Finally, h_1=7ζ(3)/2, h_2=ln 2, and h_3=93ζ(5)/8, with ζ being the Riemann function, are universal constants <cit.>. The Hamiltonian of Eq. (<ref>) captures accurately the dynamics of the linear-zigzag quantum phase transition, and the critical point, identified by transverse frequency ω = ω_C(g), can becomputed as a function of g, and it was estimated to scale as ω_C(g) ≈ h_1 - 3 h_3 g | ln g | / 2π+ 𝒪(g) for small g <cit.>. In what follows we are going to extensively study the non-equilibrium statistics of the irreversible work generated after sudden changes of the transverse frequency from ω_i=ω to ω_f such that |ω_i^2-ω_f^2|=Δω. First, we are going to present analytical results obtained using an approximated harmonic version of Eq. (<ref>) and then compare them with numerical simulations, based on the DMRG algorithm.§ NON-EQUILIBRIUM QUANTUM THERMODYNAMICS In quantum mechanics work is not a quantum observable <cit.> but a generalised measurement <cit.>. As such, it is strongly affected by quantum fluctuations arising in the measurement process.The key figure in this respect is the probability distribution of the work generated when the system is subject to a time-dependent Hamiltonian, but is otherwise isolated by sources of heat or dissipation.In this paradigm the time-dependent Hamiltonian H_i=H( ω_i) is controlled via the frequency ω. In turn, ω is quenched in time according to a certain time-dependent protocol ω(t), within the time window [t_i, t_f] [and accordingly, ω_i = ω(t_i) and ω_f = ω(t_f)].At the beginning of the protocol the system is assumed at equilibrium in the Gibbs state ρ_i=e^-β H ( ω_i)/𝒵_i, where 𝒵_i =Tr[e^-β H ( ω_i) ] is the partition function, and the inverse temperature β is also expressed in dimensionless units. For later convenience we also define the final equilibrium partition function 𝒵_f =Tre^-β H ( ω_f).Two sets of energy measurements are then performed, the first prior to the protocol and corresponding to the eigenstates of H_i, and the second one right after the protocol and corresponding to the eigenstates of H_f. One can define the work distribution performed during the H_i→ H_f transformation asP_F(W)≡∑_n,m̅p^0_n p^t_f_m̅|nδ [W-( ϵ_m̅-ϵ_n)],in which ϵ_n and ϵ_m̅are the eigenvalues of the initial and final Hamiltonian respectively, p^0_n=e^-βϵ_n/𝒵 is the initial probability distribution in the energy levels, and p^t_f_m̅|n=|⟨ϵ_m̅| U |ϵ_n⟩|^2is the transition probability for the system to evolve from the state |ϵ_n⟩ to |ϵ_m̅⟩, after the time evolution U = U(t_i → t_f) = 𝒯exp∫_t_i^t_f - i H(t) dt.For a sudden quench (U=1), the average work is simply ⟨ W ⟩= Tr[ρ_i( H(ω_f)- H(ω_i) )] whilethe free energy difference is Δ F=-β^-1ln(Z_f/Z_i). Because of the relation ⟨ W ⟩≥Δ F, we define the irreversible work as the extra work needed to perform the transformation: ≡⟨ W ⟩- Δ F. § HARMONIC APPROXIMATION For small quantum fluctuations, corresponding to small values of g, the dynamics of the Wigner crystal described by the Hamiltonian (<ref>) can be expressed in terms of small quantum displacements, coupled harmonically, around the classical equilibrium ion positions. In the linear phase the classical equilibrium positions are y_j=0 while in the zigzag phase these are y_j=(-1)^j b/2 where the zigzag width b is determined by ω <cit.>. In this regime one can find the normal frequencies associated with a normal mode at momentum k∈[-π,π] of the harmonic chain of oscillators. In the linear phase these readω_k^2=g^2[ω^2-h_1+ 4 h_2 cos^2 k/2 ] At zero temperature, this semiclassical model predicts a critical transverse frequency at ω_C=√(h_1), for which the frequency of the soft-mode at k=π and the quadratic term in Eq. (<ref>) vanish. For ω> ω_C the chain spatial configuration is linear, while it is zig-zag in the opposite case.Instead of calculating directly the work probability distribution of Eq. (<ref>), we analytically compute its Fourier transform, namely the characteristic or moment-generating function:χ_F(t)≡∫ dW e^iWt P_F (W) = Tr [ e^i H _f t U(t_f,0)^† e^-i H_i tU(t_f,0)ρ_i]similarly to the methods reported in <cit.>. We extractthe average irreversible workand its statistical variance σ^2_W=⟨ W^2⟩ - ⟨ W⟩^2 by computing the first two moments of χ_F(t). For an instantaneous quench within the same phase one finds=∑_k[1/2(Ω_kω_k^f-ω_k^i) βω_k^i/2 -1/βlnsinh(βω_k^f/2)/sinh ( βω_k^i/2)], σ^2_W=∑_k ω_k^f^2 cosh (βω_k^i) (Ω_k^2-1)+(ω_k^fΩ_k^2-ω_k^i)^2/4 sinh ^2( βω_k^i/2),in which ω_k^i(f) is the initial (final) frequency of the k mode and Ω_k=(ω_k^i2+ω_k^f2)/2ω_k^iω_k^f. The irreversible work and the work variance, at zero temperature, are shown in Fig. <ref> for small quenches within the same phase and for chain lengths ranging from 60 to 144 ions. It is interesting to note the extensiveness of both the irreversible work and its variance. In agreement with previous results <cit.> bothand σ^2_W diverge at the critical frequency as a consequence of the vanishing of the lowest eigenfrequency.While the harmonic approximation works well far from the critical point, the vanishing soft mode frequency causes an unphysical divergence, even for a finite number of ions, as the chain approaches criticality, as evidenced in Fig. <ref>.On both the linear and the zigzag sides of the transition, the irreversible work and the statistical variance are monotonically increasing functions for ω→ω_C respectively.Due to the vanishing excitation gap at k=π (soft mode), any quenchclose to the critical point, no matter how small, will always require an amount of work much larger than the mere energy difference between the two equilibrium configurations: ⟨ W⟩≫Δ F. In order to understand better this divergence, we isolate the contribution to the irreversible work and to the variance of the soft mode. Limiting the sums in Eqs. (<ref>) and (<ref>) to k=π and expanding up to second order in Δω^2 we obtain ^ soft = g Δω ^2/8γ_W |ω ^2-h_1| ^3/2+𝒪(Δω^3)σ_ soft^2 = g^2Δω ^2 /γ_σ |ω ^2-h_1|+𝒪(Δω^3)Both expressions are valid in the linear phase by taking γ_W=2 and γ_σ=8 and in the zigzag phase by taking γ_W=√(2) and γ_σ=4. The expression for σ_ soft^2 is exact in the linear phase. These results are shown in the inset of Fig. <ref>. The scaling found for the soft mode is however modified when adding the other modes. In the thermodynamic limit and close to criticality we can expand Eq. (<ref>) in the linear phase, at T=0, as(T=0)= ∑_k (ω_k^f-ω_k^i )^2/4 ω_k^i≃ L ∫ dk(ω_k^f-ω_k^i )^2/4 ω_k^i∼ Lln(ω^2-h_1)+A,in which the constant A depends on a small momentum cut-off introduced when turning the summation in Eq. (<ref>) into an integral. A similar expression holds in the zigzag phase <cit.>.We remark that the results of this section have been obtained assuming the short range effective model of Eq. (<ref>). We obtain similar expressions, within the harmonic approximation, for the long-range model of Eq. (<ref>) finding the same scaling with renormalised parameters, e.g. the critical frequency is modified and has a weak finite-size correction ∼ 1/L^2. So far, however, we have neglected the non-linear couplings between the normal modes. To overcome this,in the next section we solve numerically the full anharmonic problem.§ FULL ANHARMONIC MODELIn this section we present numerical results from the treatment of the full short-range Hamiltonian (<ref>). These results rely on the assumption that the initial state is prepared at zero temperature β→∞, and thus can be found via variational methods. In fact, the quantum many-body ground states |Ψ_G(ω)⟩ of the Hamiltonian from Eq. (<ref>) are simulated with the DMRG algorithm, using a numerical technique for continuous-variables quantum systems analogous to Refs. <cit.>. For any given ω, we evaluate the corresponding ground state energy E_G(ω) = ⟨Ψ_G(ω) | H(ω) |Ψ_G(ω)⟩ and the total fluctuation of the transverse displacement operators 𝒴^2(ω) = ∑_j ⟨Ψ_G(ω) | y^2_j |Ψ_G(ω)⟩. Such data are sufficient to evaluate the average work ⟨ W ⟩arising from a sudden quench where ω is instantaneously driven from ω_i to ω_f. In fact, we can simplify⟨ W ⟩ = ⟨Ψ_G(ω_i) | [H(ω_f) - H(ω_i)]|Ψ_G(ω_i)⟩= 1/2Δω𝒴^2(ω)which we can easily compute from the equilibrium data we acquired. Moreover, this expression for the average work shows how to measure it in an experiment by estimating the average quadratic transverse displacement of the ions. Fig. <ref> displays the irreversible work generated by a small quench of the Hamiltonian (<ref>), at T=0 and for several values of L. The first feature we notice is the disappearance of the divergence at the critical point. This is a clear signature of the finite-size effects in the quantum many-body system: at finite size L the energy gap remains finite for all ω, thus actually smearing out the non-analyticity. On the other hand, the harmonic approach exhibits a critical behaviour even at finite size. Moreover,by increasing the number of ions two features appear: the peak inbecomes increasingly sharper and its position slowly shifts towards larger frequencies, in contrast to the harmonic theory.As for the value of the peak itself, we expect it to grow linearly in L, being an extensive quantity and as already derived from the harmonic theory, see Eq. (<ref>). Fig. <ref> fully confirms this prediction. In this case wecan also draw a directconnection between our findings and recently results in the study of infinitesimal quenches and ground state fidelity susceptibility for the quantum Ising model <cit.>. In this respect, one can investigate the finite-size effect on the maxima positions by plotting ω_max as a function of 1/L, see Fig. <ref>, left inset. To gain further insight into the behaviour of the irreversible work, we adopt the finite-size scaling ansatzwith a model-dependent scaling function f1-e^-^(max)/L=f[L^1/ν(ω^2-ω^2_⋆)]which is typical of quantities that diverges logarithmically with the control parameter as in the case of the specific heat in the 2D classical Ising model <cit.>.In Fig. <ref> we show the rescaled data and we obtain a collapse of the irreversible work as in the Ising model with the same critical exponent ν=1. We remark that given the magnitude of the data shown in Fig. <ref>, a similar collapse plot (not shown) is obtained by expanding the exponential function: 1-exp[(-^(max))/L]≃ (^(max)-)/L.§ CONCLUSIONSIn this work we have investigated the irreversible work production associated with infinitesimal quenches of the transverse frequency around a structural phase transition of Coulomb crystals. We have employed different approaches starting from a harmonic approximation that allows us to obtain analytical results but in turn generates an unphysical vanishing gap at k=π at the critical point, which is known to drive the structural phase transition. We isolated the contribution of the soft mode to the irreversible work finding a power law scaling of the irreversible work. Finally, we have studied the full anharmonic model through DMRG T=0 simulations. With these different approaches we have observed the extensiveness of the irreversible work, and how they generate different scaling properties. Interestingly, the scaling laws recently found for the irreversible work of the Ising model, in terms critical exponents and collapse ansatz, are recovered in the full anaharmonic model when the g parameter is big enough to appreciate shifts from the classical critical point due to pure quantum effects. Beyond the fundamental interest, it has been found that such critical behaviours could be used to design a quantum Otto engine where it has been showed that, for a working substance around criticality, the Carnot point can be reached <cit.>. In this sense our setup is of particular experimental interest, since we have also shown how the calculated quantities can be related to the fluctuations in the transverse displacement of the ions.We acknowledge support from the Horizon 2020 EU collaborative projects QuProCS (Grant Agreement 641277) and TherMiQ (Grant Agreement 618074). PS gratefully acknowledges support from the EU via UQUAM and RYSQ, the DFG via SFB/TRR 21, and the Baden-Wrttemberg Stiftung via Eliteprogramm for Postdocs. apsrev4-1 | http://arxiv.org/abs/1704.08253v2 | {
"authors": [
"F. Cosco",
"M. Borrelli",
"P. Silvi",
"S. Maniscalco",
"G. De Chiara"
],
"categories": [
"quant-ph",
"cond-mat.quant-gas",
"cond-mat.stat-mech"
],
"primary_category": "quant-ph",
"published": "20170426180001",
"title": "Non-equilibrium quantum thermodynamics in Coulomb crystals"
} |
Department ofPhysics and Institute of Theoretical Physics, Nanjing Normal University, Nanjing 210097, ChinaSchool of Physics, Southeast University, Nanjing 210094,ChinaInvoked by the recent observation ofY(4390) at BESIII, which is about 40 MeV below the D^*(2010)D̅_1(2420) threshold,we investigatepossible bound and resonance states from theD^*(2010)D̅_1(2420)interaction with the one-boson-exchange model in a quasipotential Bethe-Salpeter equation approach. A bound state withquantum number 0^-(1^–) is produced at 4384 MeV from theD^*(2010)D̅_1(2420) interaction, which can be related toexperimentally observed Y(4390). Another state with quantum number 1^+(1^+) is also produced at 4461+i39 MeV from this interaction. Different from the 0^-(1^–) state,the 1^+(1^+)state is a resonance stateabove the D^*(2010)D̅_1(2420) threshold. This resonance state can be related to the first observed charged charmonium-like state Z(4430), whichhas a mass about 4475 MeV measuredabove the thresholdas observed at Belle and LHCb.Our result suggests thatY(4390) is an isoscalar partner of theZ(4430)as a hadronic-molecular state fromthe D^*(2010)D̅_1(2420) interaction.Interpretation ofY(4390) as an isoscalar partner ofZ(4430) fromD^*(2010)D̅_1(2420) interaction Jun He1 [email protected] (corresponding author) Dian-Yong Chen2 [email protected]: date / Revised version: date ====================================================================================================§ INTRODUCTION A recent measurement of the cross section of e^+e^-→π^+π^- h_c at center-of-mass energies from 3.896 up to 4.600 GeV suggested a new resonance structure near the D^*(2010)D̅_1(2420) [thereafter we denote it as D^*D̅_1] threshold, Y(4390), which has a mass of 4391.5^+6.3_-6.8± 1.0 MeVand a width of 139.5^+16.2_-20.6±0.6 MeV <cit.>. Afterobservation ofY(4390), a few interpretations of its internal structure were proposed, such as a 3^3D_1 charmonium statein the conventional quark model <cit.>. A QCD sum rule calculation favors an assignment of the Y(4390) as a DD̅_1 molecular state <cit.>.However, the DD̅_1threshold is much lower thanY(4390). In fact, in the literature,Y(4260), which is about 130 MeV lower thanY(4390), has been interpreted as a DD̅_1 molecular state <cit.>. Considering the D^* meson is about 140 MeV heavier than the D meson, it is reasonable to discuss an assignment ofY(4390) as a D^*D̅_1 molecular state. In the history ofstudy ofexotic states, the D^*D̅_1 molecular state has been applied to interpret the first observed charged charmonium-like state near 4.43 GeVwith a mass of 4433±4(stat)±2(syst) MeV and a width of 45^+18_-13 (stat)^+30_-13(syst) MeV reported byBelle Collaboration <cit.>. The mass measured by the Belle Collaboration, about 4430 MeV, is close to the D^*D̅_1 threshold, so it had ever been popular to explainZ(4430) as an S-wave D^*D̅_1 molecular state with spin parity J^P=0^- <cit.>. However,a higher mass of 4485^+22+28_-22-11 MeV and a larger width of 200^+41+26_-46-35 MeV were reported by a new measurementat Belle Collaboration through a full amplitude analysis of B^0→ψ' K^+π^- decay and a spin parity of J^P=1^+ was favored over other hypotheses <cit.>. A new LHCb experiment in the B^0→ψ'π^- K^+ decay confirmed the existence of the 1^+ resonant structure Z(4430)with a mass of 4475±7^+15_-25 MeV and a width of 172±13^+37_-34 MeV <cit.>.The new Belle and LHCb results support that the spin parity of the Z(4430) is 1^+ instead of 0^- which was suggested by previous hadronic-molecular-state studies. If insisting on the interpretation ofZ(4430) as aD^*D̅_1 molecular state, oneshould go beyondS wave, at least to P wave,to reproduce experimentally observedpositive parity. Besides, the new measured mass of the Z(4430) is higher than the D^*D̅_1 threshold, which suggests that the Z(4430) can not be a bound state. To explain the new observation ofZ(4430), Barnes et al.suggested that the Z(4430) is either a D^*D̅_1 state dominated by long-range π exchange, or a DD̅^*(1S, 2S) state with short-range components <cit.>.It has also been suggested that the Z(4430) may be from the S-wave DD̅'^*_1(2600) interaction, which has a threshold about 4470 MeV, to avoid the difficulties mentioned above <cit.>. In Ref. <cit.>, the D^*D̅_1 and DD̅'^*(2600) interactions werestudied by solving the quasipotential Bethe-Salpeter equation for vertex which is only valid for the bound state problem. It is found thatthe DD̅'^*(2600) interaction is too weak to produce a bound state.An isovectorbound state with quantum number J^P=1^+ can be produced from the D^*D̅_1 interaction,which corresponds toZ(4430).Such a picture was confirmed by a lattice calculation where a state with 1^+(1^+-) is also produced from the D^*D̅_1 interaction <cit.>. IfZ(4430) is from the D^*D̅_1 instead ofDD̅'^*(2600) interaction, the new observed Z(4430) massat Belle and LHCb above the D^*D̅_1 threshold suggests thatZ(4430) should be a resonance state above the threshold instead of a bound state below the threshold.In Refs. <cit.>, we develop a quasipotential Bethe-Salpeter equation for amplitude to study the resonance state above the threshold. With such formalism,it is found that a state corresponding toP wave should be taken as serious as these corresponding toS wave <cit.>. Such an idea was applied to interpret the puzzling parities oftwo LHCb hidden-charmed pentaquarks andY(4274) <cit.>. It was found thata P-wave state is usually higher than an S-wave state because ofweaker interaction but is still hopefully to be observed. If we turn to the case of Y(4390) and Z(4430), it is very natural to assign this two states as an S-wave D^*D̅_1 bound state and a P-waveD^*D̅_1 resonance state, respectively.Hence, in this work we will investigate the D^*D̅_1 interaction with the Bethe-Salpeter equation for the amplitude to study the possibility of interpreting the Y(4390) and Z(4430) ashadronic molecular-states form the D^*D̅_1interaction.In the next section, the formalism adopted in the current work is presented. The interaction potential is constructed with an effective Lagrangian and the quasipotential Bethe-salpeter equation will be introduced briefly. The numerical results are given in Section <ref>. A brief summaryis given in the last section. § FORMALISM In the energy region ofY(4390) and Z(4430), besides theD^*D̅_1threshold, there areother three thresholds of channels, D^* D̅'_1(2430), DD̅'^*(2600), and D^*D̅'^*(2550).The large width ofD'_1(2430), Γ=384^+130_-110 MeV <cit.>, which means a very short lifetime, makes it difficult to bindthe D^* meson and itself together to form a state with a width of about 170 MeV. TheD^*D̅'^*(2550) interaction has also been related to the Z(4430) in the literature. However, its threshold is about 100 MeV higher than the Z(4430) mass.The calculation in Ref. <cit.> suggested that the DD̅'^*_1(2600) interaction and its coupling to the D^*D̅_1 interaction are very weak. Hence, in this work, we only consider the D^*D̅_1 interaction.For a loosely bound system, long-range interaction by the π exchange should be more important than short-range interaction by exchanges of heavier mesons. The Z(4430) locates higher than the D^*D̅_1 threshold, so in the current work it will be seen as a resonance state where the interaction is even weaker than a loosely bound state. Hence, the dominance of the π exchange is well satisfied in the case ofZ_c(4430). However, in the case ofY(4390), the exchanges by heavier mesons may be involved because a binding energy about 40 MeV is found.For the heavier pseudoscalar mesons which caneasily be introduced as π meson in the frame of this work, their contribution is obviously much smaller than the π meson because their mass is much heavier than the π meson and the coupling constants are the same as those of the π meson. An explicit calculation suggested the medium-range σ exchange is very small partly due to its larger mass <cit.>. For the vector-meson exchange, it is difficult to determine the coupling constants involved with the existent information in the literature. And it is beyond the scope of this work to calculate such coupling constants. Furthermore, all vector mesons have much larger mass than π meson, which also leads to a suppression effect on their contributions as in the case of the σ meson exchange. Hence, in the current work, we do not include heavier-meson exchanges to avoid more not-well-determined coupling constants being introduced in the calculation with an assumption that the contributions form the heavier-meson exchanges are suppressed by the heavier mass as the σ meson exchange. The direct diagram of the π exchangewas also found negligiblecompared with cross diagram by the π exchange in an explicit calculation <cit.>. Hence, in this work, we will only considerthe cross diagramof the D^*D̅_1 interaction by π exchange as shown in Fig. <ref>. The explicit flavor structures for isovectors (T) or isoscalars (S) |D^*D̅_1⟩ are <cit.>|D^*D̅_1⟩^+_T = 1/√(2)(|D^*+D̅^0_1⟩+c|D^+_1D̅^*0⟩),|D^*D̅_1⟩^-_T = 1/√(2)(|D^*-D^0_1⟩+c|D^-_1D^*0⟩),|D^*D̅_1⟩^0_T = 1/2[(|D^*+D^-_1⟩-|D^*0D̅^0_1⟩)+ c(|D^+_1D^*-⟩-|D^0_1D̅^*0⟩)],|D^*D̅_1⟩^0_S = 1/2[(|D^*+D^-_1⟩+|D^*0D̅^0_1⟩)+ c(|D^+_1D^*-⟩+|D^0_1D̅^*0⟩)],where c=± corresponds to C-parity C=∓. For the isovector state,c is related to the G-parity. The involved effective Lagrangians describing the interaction betweena light pseudoscalar meson ℙ and heavy flavor mesons can beconstructed with the help of the chiraland heavy quark symmetries <cit.>,ℒ_D_1D^*ℙ = i√(2/3)h'/ f_π√(m_D_1m_D^*)·{[-1/4m_D_1m_D^*D_1b^α∂^ρ∂^λ D^*†_α a∂_ρ∂_λℙ_ba-D_1b^αD^*†_α a∂^ρ∂_ρℙ_ba+3D_1b^α D^*†β_a∂_α∂_βℙ_ba]-[-1/4m_D_1m_D*D^*†_α a∂^ρ∂^λ D_1b^α·∂_ρ∂_λℙ_ab- D^*†_α aD_1b^α∂^ρ∂_ρℙ_ab +3D^*†β_aD_1b^α∂_α∂_βℙ_ab]}.With the above Lagrangians, we can obtain the potential for the cross diagram bythe π exchange,i V_λ'_1λ'_2,λ_1λ_2 =f_I2/3h'^2m_D_1m_D^*/ f^2_π(q^2-m_π^2) ϵ_D_1,λ'_2^†μϵ^ν_D^*,λ_1ϵ_D_1,λ_2^ρϵ^†σ_D^*,λ'_1·{[q^2-(p'^2_2-p_1^2)^2/4m_D_1m_D^*]g_μν-3q_μ q_ν}·{[q^2-(p^2_2-p'^2_1)^2/4m_D_1m_D^*]g_ρσ-3q_ρ q_σ},where p^(')_1,2 and λ^(')_1,2 arethe initial (final) momentum and the helicity for constituent 1 or 2.And the flavor factor f_I=-c/2 and 3c/2 for I=1 and 0, respectively. Withavailable experimental information, Casalbuoni et al. extracted h'=0.55 GeV^-1 from the olddata of decay width Γ_tot(D_1(2420))≈ 6 MeV <cit.>.Compared with the new suggested value of the decay width inPDG, 25±6 MeV <cit.>, avalue of1.1 GeV^-1 can be obtained for the coupling constant h'. In this work, we will adopt this new value of h' in the calculation.The adoption of such value of h' does not affect theanalysis above as regards the relative magnitude ofthe contributions from different interaction channels and different exchanges.The scattering amplitude of the D^*D̅_1 interaction can be obtained by solving Bethe-Salpeter equation with the above potential. The Bethe-Salpeter equation is usually reduced to three-dimensional equation with a quasipotential approximation. To avoid the unphysical singularity fromthe OBE interaction below the threshold, the off-shellnessof two constituent hadrons should be kept.Here we adopt the most economic treatment, that is, the covariant spectator theory <cit.>,which was explained explicitly in the appendices of Ref. <cit.> and applied to a study ofthe X(3250), the Z(3900) and the LHCb pentaquarks and its strange partners <cit.>. In such a treatment,we put the heavier constituent, D_1 meson here,on shell <cit.>. Then the partial-wave Bethe-Salpeter equation with fixed spin parity J^P reads <cit.>i M^J^P_λ'_1λ'_2,λ_1λ_2( p', p)=i V^J^P_λ'_1λ'_2,λ_1λ_2( p', p)+∑_λ”_1λ”_2≥0∫ p”^2d p”/(2π)^3· i V^J^P_λ'_1λ'_2,λ”_1λ”_2( p', p”) G_0( p”)i M^J^P_λ”_1λ”_2,λ_1λ_2( p”, p),Written down in the center-of-mass frame where P=(W, 0), the reduced propagatorisG_0 =δ^+(k”^ 2_2-m_2^2)/k”^ 2_1-m_1^2=δ^+(k”^0_2-E_2( p”))/2E_2( p”)[(W-E_2( p”))^2-E_1^2( p”)],where the momentum of D^* mesonk”_1=(k”^0_1,- p”)=(W-E_2( p”),- p”) and the momentum of the D_1 meson k”_2=(k_2^0, p”)=(E_2( p”), p”) with E_1,2( p”)=√( M_1,2^ 2+ p”^2). Here and hereafter we will adopta definition p=| p|. The potential kernel V_λ'_1λ'_2λ_1λ_2 obtained inprevious section,the partial-wave potential with fixed spin parity J^P can be calculated asi V_λ'_1λ'_2λ_1λ_2^J^P( p', p)= 2π∫ dcosθ [d^J_λ_1-λ_2λ'_1-λ'_2(θ) i V_λ'_1λ'_2λ_1λ_2( p', p)+ η d^J_λ_2-λ_1λ'_1-λ'_2(θ) i V_λ'_1λ'_2-λ_1-λ_2( p', p)],where η=PP_1P_2(-1)^J-J_1-J_2 with P_(1,2) and J_(1,2) being the parity and spin ofconstituent 1 or 2. Here without loss ofgeneralitythe initial and final relative momenta can be chosen as p=(0,0, p)and p'=( p'sinθ,0, p'cosθ), and the d^J_λλ'(θ) is the Wigner d-matrix.To guarantee the convergence of the integral in Eq. (<ref>), a regularization should be introduced. In this work we will introduce an exponential regularization by a replacement of the propagator as G_0( p)→ G_0(p)[e^-(k”^2_1-m_1^2)^2/Λ^4]^2,where k”_1 and m_1 are the momentum and mass of the lighter one of two constituent mesons. We would like to recall that the exponentialfactor e^-(k”^2_2-m_2^2)^2/Λ^4 for particle 2vanishes, which is only because the particle 2 is put on shell in the quasipotential approximation adopted in the current work. With such treatment, the contributions at large momentum p” will be suppressed heavily at the energies higher than 2 GeV as shown in Fig. 1 of Ref. <cit.>, and convergence of the integral is guaranteed. By multiplying the exponential factor on both sides of the Eq. (<ref>), it is easy to found thatthe exponential factor canalso be seen as a form factor to reflect the off-shell effect of particle 1 in a form of e^-(k^2-m^2)^2/Λ^4. It is also the reason why a square of the exponential factor is introduced in Eq. (<ref>). The interested reader is referred to Ref. <cit.> for further information as regardsthe regularization. A sharp cutoff of the momentum ofp” at certain value p”^max, namely cutoff regularization,is also often adopted in the literature <cit.>. The exponential regularization can be seen as a soft version of the cutoff regularization. A comparison of the exponential regularizationand thecutoff regularization as adopted in the chiral unitary approach <cit.> was made in Ref. <cit.> and it was found that the different treatments do not affect the conclusion. Because the current treatmentguarantees the convergence of the integration, we do not introduce the form factor for the exchanged meson, which is redundant and its effect can be absorbed intovariation of the cutoffs Λ as discussed in Ref. <cit.>. The integral equation (<ref>) can be solved bydiscretizing the momenta p, p', and p” by the Gauss quadrature with a weight w( p_i). After such treatment, the integral equation can be transformed to a matrix equation <cit.>M_ik = V_ik+∑_j=0^N V_ijG_jM_jk.The propagator G is a diagonal matrix withG_j>0 = w( p”_j) p”^2_j/(2π)^3G_0( p”_j), G_j=0 = -i p”_o/32π^2 W+∑_j [w( p_j)/(2π)^3 p”^2_o/2W( p”^2_j- p”^2_o)],with on-shell momentump”_o=1/2W√([W^2-(M_1+M_2)^2][W^2-(M_1-M_2)^2]). The scattering amplitude M can be solved asM=(1- V G)^-1V. Obviously,thepole ofscattering amplitude we wanted can be found at |1-VG|=0 afteranalytic continuation total energy W into the complex plane as z. In the current work, the pole is searched by scanningthe value of |1-V(z)G(z)| by variation of real and imaginary parts of z, Re(z) and Im(z), in complex plane to findposition of z with|1-V(z)G(z)|=0. § THE NUMERICAL RESULTS With potential in Eq. (<ref>), the pole from the scattering amplitude can be found at |1-V(z)G(z)|=0 at complex plane by a continuation of the real center-of-mass energy W to a complex z.In this work,only free parameter is the regularization cutoff Λ. Byvaryingthe cutoff, we try to found abound state with 0^-(1^–) and a resonance state with 1^+(1^+) which correspond toY(4390) andZ(4430), respectively,with the same cutoff.In Fig. <ref>,thelog |1-V(z)G(z)| is plottedwith variations ofRe(z) and Im(z). It is found that with cutoff Λ=1.4 GeV two states expected can be produced from the D^*D̅_1 interaction. Under the D^*D̅_1 threshold,a bound state with quantum number 0^-(1^–)can be found at z=4384 MeV, which can beobviously related toY(4390) with a mass of 4391 MeV observed at BESIII. Since only theD^*D̅_1 interaction is considered in this work, no width is produced and the pole is atreal axis.This state has a negative parity, so can be produced from the D^*D̅_1 interaction in S wave. For the state with1^+(1^+), the P wave should be introduced to produce its positive parity.As discussed in Ref. <cit.>, the P-wave interaction isusually weakerthan the S-wave interaction. Furthermore,for the D^*D̅_1 interaction considered in this work, the flavor factor for the isoscalar sector is three times larger than that for the isovector sector, which makes the isovector interaction weaker.Hence, one can expect that the 1^+(1^+) state is considerably higher than the0^-(1^–) state. The resultin Fig. <ref> confirmssuch surmise. The expected 1^+(1^+) state is found at z=4461+i39 which is much higher thanthe 0^-(1^–) state, even above the D^*D̅_1 threshold.Obviously, this pole can be related to the charged charmonium-like state Z(4430) whosemassis about 4475 MeV as suggested by the new LHCb experiment. Though only one-channel is included in this work, the resonance state carries a width as suggested by the scattering theory.The above results show that the experimentally observed Y(4390) and Z(4430) can be reproduced from theD^*D̅_1 interaction with the same cutoff Λ=1.4 GeV.In the rest part of this section, we will study whether there exist other possible states produced from this interaction.Here, we only consider theD^*D̅_1 interaction with spin parties 0^±, 1^±, 2^±, and 3^-. Other partial waves are not considered because their spin parities cannotbe constructed with S and P waves. Because a coupled-channel effect is not included in this work, we allow the regularization cutoff to deviate from the value above, 1.4 GeV,by 0.5 GeV, i.e. from 0.9 to 1.9 GeV.Only poles in an energy range4.35<Re(z)<4.50 GeV are searched for in the calculation. The isovector states from theD^*D̅_1 interaction with typical cutoffs are listed in Table <ref>. In the isovector sector, besides the 1^+(1^+) state corresponding toZ(4430), there exist othertwo possible states with 1^+(0^-) and 1^+(2^-) produced from theD^*D̅_1 interaction. With the decrease of the regularization cutoff, the interaction gradually weaken. As a result, the poles ofthese states will run to and cross the threshold at certain cutoff, and then the bound state becomes a resonance state. If we fix the cutoff at 1.4 GeV as in the case of reproducingY(4390) and Y(4430),1^+(0^-) is a bound state around 4.4 GeV.1^+(2^-) is a resonance state much higherZ(4430). A dependence of the results on the cutoff can be found inTable 1, which isfromneglecting of the coupled-channel effect and other approximations adopted in our approaches.It is also the reason why we will vary the cutoff in the calculation, that is, the effectsof the approximationscan be absorbed into the variation of the cutoff. The results of the isoscalar sector is listed in Table <ref>. Ninestates with 0^+(0^±+), 0^± (1^+±), 0^± (1^-±), 0^±(2^+±) and 0^+(2^-+) are produced inthis sector, whichare much more than three states in the isoscalar sector. It is reasonable because the flavor factor for the isoscalar sector is three times larger than that for the isovector sector, which means stronger interaction in this sector. Generally speaking, the spin-negative states are more binding than the positive states which reflects the P-wave interaction is usually weaker than the S-wave state. In our calculation, more than one state is found in some cases, which can be seen as excited state. As in the study of the hydrogen energy leveland the hadron spectrum in the constituent quark model, it is natural to find radial excited states besides the ground state. For theY(4390) andZ(4430) which we focused on in this work, only one state was found in a considerable large range of the Re(z),and it is not so meaningful to present the results of excited states for other states even which ground state has not yet been observed in the experiment. So, in Tables <ref> and <ref>, only the results of the ground state are presented.§ SUMMARY In this work, theD^*D̅_1 interaction is investigated in a quasipotential Bethe-Salpeter equation approach, andbound and resonance states are searched for to interpretY(4390) observed recently at BESIII and the first observed charged charmonium-like state Z(4430).A bound state with 0^-(1^–) at 4384 GeV and a resonance state with 1^+(1^+) at 4461+i39 MeV are produced fromthe D^*D̅_1 interaction which can be related toY(4390) andZ(4430), respectively.Hence,Y(4390) is an isoscalar partner ofZ(4430) and a partner ofY(4260) by replacing the D meson by the D^* meson in the hadronic-molecular state picture. 10ptAcknowledgement This project is supported by the National Natural Science Foundation of China (Grants No. 11675228and No. 11375240), the Major State Basic Research Development Program in China under grant 2014CB845405, and the Fundamental Research Funds for the Central Universities.23BESIII:2016adj M. 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He, “D̅Σ^*_c and D̅^*Σ_c interactions and the LHCb hidden-charmed pentaquarks,” Phys. Lett. B 753, 547 (2016)He:2017aps J. He, “Nucleon resonances N(1875) and N(2100) as strange partners of LHCb pentaquarks,” Phys.Rev. D95 (2017), 074031Gross:1999pd F. Gross,“Charge conjugation invariance of the spectator equations,” Few Body Syst.30, 21 (2001) He:2015yva J. He, “Internal structures of the nucleon resonances N(1875) and N(2120),” Phys. Rev. C 91, 018201 (2015) Oller:1998hw J. A. Oller, E. Oset and J. R. Pelaez, “Meson meson interaction in a nonperturbative chiral approach,” Phys. Rev. D 59, 074001 (1999) Erratum: [Phys. Rev. D 60, 099906 (1999)] Erratum: [Phys. Rev. D 75, 099903 (2007)] Lu:2016nlp P. L. Lü and J. He, “Hadronic molecular states from the KK̅^∗ interaction,” Eur. Phys. J. A 52, 359 (2016) | http://arxiv.org/abs/1704.08776v2 | {
"authors": [
"Jun He",
"Dian-Yong Chen"
],
"categories": [
"hep-ph",
"hep-ex",
"hep-lat",
"nucl-th"
],
"primary_category": "hep-ph",
"published": "20170427235843",
"title": "Interpretation of $Y(4390)$ as an isoscalar partner of $Z(4430)$ from $D^*(2010)\\bar{D}_1(2420)$ interaction"
} |
http://arxiv.org/abs/1704.08601v1 | {
"authors": [
"Matt Holzer",
"Ratna Khatri"
],
"categories": [
"nlin.PS"
],
"primary_category": "nlin.PS",
"published": "20170427143348",
"title": "Pattern formation, traveling fronts and consensus versus fragmentation in a model of opinion dynamics"
} |
|
update pict/./ Fig | http://arxiv.org/abs/1704.07994v2 | {
"authors": [
"Wei Liu"
],
"categories": [
"physics.optics"
],
"primary_category": "physics.optics",
"published": "20170426072855",
"title": "Superscattering pattern shaping for radially anisotropic nanowires"
} |
mymainaddress]Dilini Subasinghemycorrespondingauthor [mycorrespondingauthor]Corresponding author [email protected],mysecondaryaddress]Margaret Campbell-Brownmymainaddress,mysecondaryaddress]Edward Stokan[mymainaddress]Department of Physics and Astronomy, University of Western Ontario, London, Canada, N6A 3K7 [mysecondaryaddress]Centre for Planetary Science and Exploration, University of Western Ontario, London, Canada, N6A 5B7The luminous efficiency of meteors is poorly known, but critical for determining the meteoroid mass. We present an uncertainty analysis of the luminous efficiency as determined by the classical ablation equations, and suggest a possible method for determining the luminous efficiency of real meteor events. We find that a two-term exponential fit to simulated lag data is able to reproduce simulated luminous efficiencies reasonably well. eteors meteoroids optical masses[2010] 00-0199-00 § INTRODUCTIONDetermining the mass of a meteoroid, a basic property, is currently very difficult to do. Because most meteoroids are too small to reach the ground, meteoroid mass needs to be determined through observations. The simplest method is to use the total luminous energy emitted during ablation. The large uncertainty associated with mass is due to many unknown variables, such as the bulk density, shape, and luminous efficiency, and their (possible) changes during ablation. Spacecraft hazard estimates rely on accurate meteoroid masses: while rare, collisions and damage to satellites by meteoroids have occurred <cit.>. There are two coupled differential equations in classical meteor physics that describe the state of the meteoroid and allow mass to be determined: the luminous intensity equation and the drag equation. The luminous intensity, given in Equation <ref>, assumes the brightness (or luminous intensity, I) of a meteor is proportional to the change in kinetic energy. I =-τ dE_k/ dt = -τ(v^2/2 dm/ dt + mv dv/ dt)The proportionality constant, τ, is the luminous efficiency, the fraction of kinetic energy dissipated as meteor light. The m refers to the total meteoroid mass, including any fragments. Despite their small masses (< 10^-4 kg), the majority of small meteoroids do fragment <cit.>, and that light is taken into account when calculating the photometric mass. Equation <ref> may be rearranged to solve for the photometric mass, but there is typically a large associated uncertainty, due to the vast range in luminous efficiency values. The second term in Equation <ref> is often neglected, as the deceleration for fast, faint meteors is negligible, relative to the first term. Using typical values, it can be shown that for slow meteors the deceleration term is almost equal in importance to the mass loss term, but becomes significantly less important at higher speeds (i.e. the deceleration term is about 40% of the mass loss term for a meteor moving at 11 km/s, but only 1% for a meteor travelling at 70 km/s). The drag equation given in Equation <ref>, can also be used to determine the mass of a meteoroid, and is derived through conservation of momentum.dv/dt = -Γρ_atm v^2 A/m^1/3ρ_m^2/3 The mass in this equation is called the dynamic mass, as it is based on the deceleration of the largest, brightest fragment (or group of similarly sized fragments). The other variables in Equation <ref> are the drag coefficient Γ, the atmospheric density ρ_atm, the velocity v, the shape factor A, and the meteoroid density ρ_m. Previous studies have found that the dynamic mass of faint meteors is consistently smaller than the photometric mass, and is thus not an accurate measure of the true meteoroid mass for fragmenting meteors <cit.>. Again, this is because the photometric mass considers the mass of all light producing fragments, and the dynamic mass only considers the largest, brightest fragment.Since most meteoroids do fragment, it is therefore useful to better understand the luminous efficiency to determine the meteoroid mass through the luminous intensity equation. The goal of this study is ultimately to examine faint meteoroids that do not appear to fragment, to determine their luminous efficiencies. In those cases, the dynamic mass, found by the deceleration, is equivalent to the photometric mass, and we can solve for the luminous efficiency. This luminous efficiency can then be used to find the masses of other meteoroids, even those that fragment. It has been suggested that the luminous efficiency depends on meteoroid speed and height, camera spectral response (an iron-rich meteoroid may emit strongly in the blue portion of the visible spectrum, but may not be detected if the camera system is not sensitive to that range), meteoroid and atmospheric composition, and possibly meteoroid mass, among other factors, but the extent to which it depends on each variable is unknown <cit.>.§.§ Previous luminous efficiency studiesAs a meteoroid enters the atmosphere, it heats up through collisions with atmospheric atoms and molecules. This results in meteoroid ablation and the release of meteoric atoms and molecules into the atmosphere. Evaporated meteoritic material interacts with atmospheric molecules or other ablated atoms, leading to the excitation of the meteoric and atmospheric species. The luminosity observed is due to the decay of these excited states and is emitted in spectral lines. Many of the early luminous efficiency studies were done by Opik, who used a theoretical approach to determine luminous efficiencies for various atoms. Uncertainty in the approach used led to questions of the validity of his work: he is mentioned here for completeness. <cit.> combined the drag equation with the luminous intensity equation to solve for the luminous efficiency. This method explicitly equates the photometric mass with the dynamic mass, which is problematic since these masses are not equivalent for meteoroids that fragment, and studies have shown that the majority of observed meteoroids show fragmentation <cit.>. <cit.> attempted to correct for fragmentation, and assumed that luminous efficiency can be described as shown in Equation<ref>, with luminous efficiency proportional to speed raised to some power. He found for the 413 Super Schmidt meteors he studied, that n=1.01 ± 0.15 and 1.24 ± 0.22 for fragmenting and non-fragmenting meteors respectively. He further investigated whether luminous efficiency depends on mass (he found that it does not), and confirmed that luminous efficiency does not depend on the atmospheric density. He used a single non-fragmenting meteor, suggested to be asteroidal in origin (based on orbital characteristics), to conclude that in the photographic bandpass, the constant τ_0 in Equation <ref>, is log_10τ_0 = -4.37 ± 0.08 for n = 1. These results, along with the following studies, are illustrated in Figure <ref>.τ = τ_0v^n Many lab experiments were performed in the sixties and seventies, with the obvious advantage of being able to control many aspects of the ablation process such as the mass and composition of the ablating particles, and the gas density in which the particles ablate. One of the limitations of lab experiments for luminous efficiency estimates is the difficulty in reaching all valid meteor speeds – <cit.> reached speeds between 15 - 40 km/s, while <cit.> explored speeds between 11 - 47 km/s. The experimental lab set up involved charging and accelerating particles in a Van deGraaf generator (detectors measured the charge and velocity), and then observing as the particles ablated in a gas region meant to simulate free molecular flow (13.3 Pa). <cit.> used 167 iron and 120 copper spherical simulated meteors, with diameters between 0.05 and 1 micron, and their results are shown in Figure <ref>. <cit.> used essentially the same methods as <cit.>, but studied silicon and aluminium particles with similar diameters, as they ablated in a gas region of air, nitrogen, or oxygen, at a pressure around 27 Pa. These results are not applicable to optical meteors directly, as these lab studies used particles much smaller than the millimetre sized objects that most optical cameras observe, and the pressures at which the micron sized particles ablated correspond to heights much lower (between 55 - 65 km) than those at which optical cameras typically observe (around 90 - 110 km). Artificial meteors are another method of determining the luminous efficiency. In this method, objects of known mass and composition are subjected to atmospheric re-entry, and observed as they ablate. <cit.> used iron and nickel objects, launched between 1962 and 1967, observing a total of ten artificial meteors. These artificial meteoroids had either a disk or cone shape, and their masses ranged between 0.64 - 5.66 grams. The average begin and end heights were 76 and 66 km, respectively. These artificial meteoroids were observed optically, and the luminous intensity and velocity were collected. Combined with the measured initial meteoroid mass, the luminous efficiency was calculated using a simplified version of Equation <ref>, in which the second term (related to the deceleration) is ignored. <cit.> found that n=1.9 ± 0.4 in Equation <ref> for four artificial meteors, including one from <cit.>. <cit.> also formulated a luminous efficiency relationship for meteors of stony composition, assuming that 15% of the mass is iron, which is the main emitter in their blue sensitive cameras: that between 20 and 30 km/s, the luminous efficiency increases monotonically; and that above 30 km/s, n=1. This may not be applicable to other more red-sensitive optical systems. They noted that this work was a first approximation. A slight reworking of the <cit.> results was done by <cit.>, who increased the proportion of iron by weight from 15% to 28%. The luminous efficiency suggested by <cit.> is a piece-wise function (shown in Figure <ref>), and was used for fireball analysis. <cit.> defined an excitation coefficient, which is the average number of times a meteoric atom is excited during ablation. In combining theory and lab measurements, they found that their primary excitation probability is unphysical beyond 42 km/s (they assumed ionised atoms are unavailable for excitation). They referred to scattering and diffusion cross-sections to describe the excitation coefficient, but found that the values were higher than experimental values suggested. Simultaneous optical and radar observations of meteors were used by <cit.> to determine luminous efficiency for the bandpass of their GEN-III image intensifiers. The ratio of the ionisation coefficient β (the number of electrons produced per ablating atom) to the luminous efficiency τ can be determined through radar and video measurements, and assuming a value for either β or τ allows the other to be determined. <cit.> determined an expression for β using both theory and observations, which <cit.> used to determine a peak bolometric value of τ = 5.9% at 41 km/s, for their Gen-III bandpass (470 - 850 nm). § METHODThe purpose of this work is to develop a method, using simulated data, to calculate luminous efficiency from non-fragmenting meteors observed with a high-resolution optical system, and to investigate the sensitivity of the method to the various assumed parameters. Equating the dynamic and photometric masses is appropriate, provided the meteoroid does not fragment, and allows for the determination of the luminous efficiency. The classical meteor ablation equations apply to a solid, single, non-fragmenting body. The Canadian Automated Meteor Observatory (CAMO; discussed below) has at best, a resolution of 3 m/pixel in its narrow-field optical camera, which means it can confirm that the meteor events collected do not significantly fragment on that scale. The dynamic mass can then be equated to the photometric mass to solve for the luminous efficiency: rearranging Equations <ref> and <ref> gives us: m= - Γ^3 ρ_atm^3 v^6 A^3/ρ_m^2 (dv/dt)^3τ = - I/v^2/2 dm/ dt + mv dv/ dt Assumptions must be made for certain parameters: the drag coefficient Γ, which can range from 0 - 2; the shape factor A, given by surfacearea/volume^2/3; and the meteoroid density ρ_m, which can range from 1000 - 8000 kg/m^3. For the drag coefficient and the shape factor, typical values were used (Γ = 1; A = 1.21 (sphere)). An atmospheric density profile was taken from the NRLMSIS E-00 model <cit.>.§.§ Future application to real dataThe Canadian Automated Meteor Observatory is a two station, image intensified video system, located in Ontario, Canada <cit.>. The two stations are approximately 45 km apart, with one station in Tavistock, Ontario, Canada (43.265^∘N, 80.772^∘W), and the other in Elginfield, Ontario, Canada (43.193^∘N, 81.316^∘W). Sky conditions permitting, the camera systems run each night. The guided system, used for data collection, consists of two cameras: a wide-field camera, with a field of view of 28^∘, and a narrow-field camera, with a field of view of 1^∘.5. The wide-field cameras, which run at 80 frames per second, allow for orbit determination, as well as light curve measurements; and the narrow-field cameras, which run at 110 frames per second, provide high-resolution observations of the meteoroid. To reduce the possibility of image saturation, the cameras each have 12 bit image depth. Meteors are detected in the wide-field camera in real time with the All Sky and Guided Automatic Realtime Detection (ASGARD) software<cit.>, and ASGARD directs a pair of mirrors to track the meteor and direct the image into the narrow-field camera. With the high-resolution narrow-field cameras, meteors that appear to show single body ablation can be selected and studied to determine their luminous efficiencies. In a future work, we will analyze a number of events and apply this luminous efficiency determination method to them. The meteor events will be reduced using mirfit: software designed to process meteor events recorded with the CAMO tracking system, and provide high-precision position measurements (sub-metre scale). § SENSITIVITY ANALYSISOne of the main difficulties in solving for luminous efficiency is determining the measured deceleration of the meteor, needed for both the dynamic mass (Equation <ref>) and the luminous efficiency (Equation <ref>). Small uncertainties in the measured position result in large point-to-point errors in the speed, and very large scatter in the deceleration.To test the sensitivity of our technique to the assumptions made and the fitting techniques used, we simulated meteors using the model of <cit.>. We used the classical ablation model to investigate different smoothing and fitting algorithms. The lag is the distance that the meteoroid falls behind an object with a constant velocity (equal to the initial meteoroid velocity, which is determined by fitting the first half of the distance-time data), andrequires a monotonically increasing form. As a first attempt, we expect an exponential relationship between the meteoroid lag and time, based on the atmospheric density encountered by the meteoroid increasingly roughly exponentially with time. A two term exponential will provide a better fit than a single term exponential (more terms and/or higher order terms will fit the data better, but it is important to note that adding more terms will eventually overfit the data and does not have any physical justification). A classically modelled meteor with the following parameters was investigated for fitting: mass of 10^-5 kg, density of 2000kg/m^3, and initial speed of 30km/s. Because meteors show very little deceleration at the beginning of their ablation, a comparison of fitting the lag from the full curve versus the second half of the lag was done and the results are shown in Figure <ref>. Fitting only the second half of the lag curve gives smaller residuals (relative to the model lag), and is more accurate at later times, when the meteoroid deceleration is more apparent and easier to fit. The RMSE value for fitting the entire lag profile was 3.49, and for fitting only the second half of the lag profile was 0.018. Because meteor ablation can last from less than a second to a few seconds, the decision was made to fit the second half of the ablation profile, rather than the last second, or half second. A comparison of the derived deceleration (based on the second derivative of the two-term exponential fit to the lag) to the model deceleration was also done, and is shown in Figure <ref>. When fitting the entire lag profile, the derived deceleration matches the simulated deceleration well towards the beginning of the ablation profile, but the absolute relative error is large towards the end where deceleration is greatest, and which is of greatest interest for finding luminous efficiency. In Figure <ref>, only the second half of the lag data was fit, but the fit was extended backwards for comparison purposes. The relative percentage error is smaller when the deceleration is greatest, compared to when the entire lag is fit, as shown in Figure <ref>.Based on Figures <ref> and <ref>, a two-term exponential fit y = ae^bx + ce^dx to the second half of the lag data is able to visually reproduce a classically modelled meteoroid reasonably well. To investigate this method for other parameters, a set of simulated meteors were created, each with different parameters (speed, mass, shape factor, meteoroid density, drag coefficient) and tested to see if the luminous efficiency used to simulate the meteor could be extracted from simulated observations with this method. The simulated meteors were generated with the ablation model of <cit.>. Calculation of the luminous efficiency was done blind, with no knowledge of the value used in the simulation until the analysis was complete. There are three variables in Equations <ref> and <ref> that are assumed to be constant with time: the drag coefficient, the shape factor, and the meteoroid density. These variables cannot be measured and values must be assumed. A representative event was simulated with the following parameters: initial speed 30 km/s; shape factor 1.21 (sphere); drag coefficient 1; meteoroid density 2000 kg/m^3, and a mass of 10^-5 kg. Any difference between an assumed constant term and the simulated value will change the luminous efficiency by a scaling factor, and the variation and uncertainty in the calculated luminous efficiency (for a range of physically possible values) is shown in Figure <ref>. A more complicated parameter to deal with is the atmospheric density. Each of the simulated meteor events used the same atmospheric density profile, not specific to any date or location. However, with real meteor events, the atmospheric density on that day, at that time and location needs to be used. To investigate the variation in luminous efficiency due to variations in atmospheric density, four days of data (each from a different season) from the NRLMSIS E - 00 Atmosphere Model <cit.> were compared. The four days of modelled data and the simulated atmospheric density are shown in Figure <ref>, and the resulting luminous efficiency estimates (keeping everything the same except for the atmospheric density profile) are shown in Figure <ref>. The resulting luminous efficiency profiles show values that range from 0.2% up to 1%. However, not all luminous efficiency profiles have valid solutions at all heights: when the atmospheric density used in calculating the luminous efficiency is lower than the modelled atmospheric density, we end up with an unphysical situation where the meteoroid is gaining rather than losing energy as it ablates, and a singularity appears in our luminous efficiency profile. After investigating our simulated representative meteor to determine how meteoroid density, drag coefficient, shape factor, and atmospheric density model affect our calculated luminous efficiency values, the full parameter space of mass, speed, meteoroid density, zenith angle, and shape factor, was explored. Fifty meteors were simulated for each mass - speed group. The simulated meteors had a mass of 10^-4, 10^-5, or 10^-6 kg. The speeds used were 11, 20, 30, 40, 50, 60, and 70 km/s. This meant there were 21 possible groups; however some of the low mass - low speed groups did not produce enough light that they would be detected by the CAMO optical system. This reduced the number of mass - speed groups to 18. The luminous efficiency for each meteor in this set of simulated meteors was 0.7%, constant over time. To investigate how sensitive our results are to the chosen two-term exponential fit of the meteoroid lag, we analyzed each meteor in our mass - speed groups according to our method: the simulated position was used to determine the lag, which was fit by a two-term exponential function. This function was then numerically differentiated (i.e. finite differenced) to determine the deceleration. By using all values (drag coefficient; atmospheric density; velocity; shape factor; meteoroid density; intensity) directly from the simulation, with the exception of the determined lag, we were able to see how sensitive our derived luminous efficiency values were to our fit to the meteoroid lag. Our derived luminous efficiencies did not come out as constant values over the ablation due to the sensitivity of this method to small variations in deceleration. The mean and standard deviation for the luminous efficiency of each meteor was determined, and the average of those values in each mass - speed group are given in Table <ref>. Fitting a two-term exponential to the lag, to find the deceleration and the luminous efficiency seems to work for most mass - speed groups. Table <ref> shows that almost all of the mass - speed groups investigated show luminous efficiency ranges that include the true value of 0.7%. This is not the case for high-mass, low-velocity meteors (11 km/s). In fact, for each mass group, the lowest speed that produces a luminous efficiency profile does not produce a mean luminous efficiency range that includes the value that was used in the simulation.A comparison of the fitted lag, the corresponding deceleration, and the resulting luminous efficiency profile of a typical event are shown in Figure <ref>. While the simulated lag appears to be fit well by the two term exponential, the resulting deceleration from the fit deviates from the simulated deceleration. This may be due to temperature fluctuations in the ablation model. The luminous efficiency derived using only the ablation model output is unable to produce the exact luminous efficiency (constant 0.7% over the ablation period) used in the simulation, as shown in Figure <ref>.§ DISCUSSIONOur method for determining the luminous efficiency uses only the luminous intensity and drag equations, while the ablation model by <cit.> is more sophisticated. <cit.> use the classical form of the drag equation, but their mass loss equation is not classical: they instead use the Knudsen-Langmuir formula with the Clausius-Clapeyron equation to simulate the meteoroid ablating as soon as it begins heating up. When the meteoroid becomes very hot, a spallation term is included. The third differential equation used in the <cit.> model is the temperature equation, which describes the energy gained (through collisions with the atmospheric atoms) and lost (through radiation and evaporation of material). As seen in the previous figures, uncertainties in each of the variables of Equations <ref> and <ref> yield corresponding variances in the computed luminous efficiency. Figures <ref> and <ref> were created assuming the drag coefficient and shape factor are constant over the ablation. This is not necessarily true for real meteor events, but for simplicity, was assumed for this work, both in the modelling and analysis. If a real meteor event has a constant drag coefficient or shape factor, but an incorrect value is assumed in the analysis, the difference will be a simple scaling factor; if the value changes over the course of the flight the luminous efficiency will be off by an amount proportional to the difference in the assumed value and the average of the true value.It is obvious that variations in the atmospheric density over the course of a year (even as much as a factor of two) can change the derived luminous efficiency profile. The solid red line in Figure <ref> indicates the calculated luminous efficiency using the same atmospheric density model that was used in the simulation. A constant luminous efficiency of 0.7% was used in the simulation, but this method is unable to exactly reproduce that: the calculated luminous efficiency is quite close to, but not exactly, a constant 0.7%. We find that small rounding errors in the ablation model cause the small variations we see in the luminous efficiency.One of the most challenging aspects of this work is determining which functional form to fit to the lag; while more complex functional forms are able to fit the lag better (a combination of an exponential with a polynomial, for example), they do not necessarily provide a better fit to the deceleration, to which the luminous efficiency is very sensitive. Various combinations of exponential fits with polynomials (lag = ae^bx^2 + cx + d; lag = ae^bx + cx + d; lag = ae^bx + e^cx + dx + f; etc.) were tried. Much better results were obtained when the modelled deceleration was fit directly, but this approach will not work for real data. Even very precise observations from CAMO have enough noise in the measured lag that finite differencing produces wildly oscillating decelerations. A smooth fit to the lag is crucial in order to obtain a useful second derivative.We found that the luminous efficiency calculated by fitting the lag with a two-term exponential did not reproduce the model's constant 0.7%(see Table <ref>).On average, this fitting method does return the correct luminous efficiency, except in the lowest speed groups. In particular, high mass (10^-4 kg) meteors with initial speeds of 11 km/s had a much lower mean luminous efficiency, because there was poor agreement between the simulated lag and the two-term exponential functional fits by visual inspection. Visual inspection also determined that visually good fits to the lag data may or may not produce a good match to the simulated deceleration.§ CONCLUSIONAn attempt at quantifying the uncertainty in using the classical meteor ablation equations to determine the luminous efficiency of meteors has been made. Certain parameters (drag coefficient; meteoroid density; shape factor) were assumed to be constant. The wrong drag coefficient could produce errors of roughly a factor of 2; the meteoroid density can vary by a factor of 8, but is much more likely to be within a range of a factor of 2; and the shape factor may be different from a sphere, but is not very likely to be as extreme as the end values modelled here, which correspond to an oriented needle and a disk with its largest dimension oriented to the airstream. It's much more likely that the shape factor will be within a factor of 2 of a sphere, and therefore these three parameters together are each a small random effect on the luminous efficiency. The atmospheric density over the course of a year changes by a factor of 2 in the height range that meteors are detected with our optical system, and these variations cause similar factor-of-2 discrepancies in the luminous efficiency computed for simulated events. The possibility of using radar echo decay measurements to verify atmospheric density profiles at the location of the optical cameras is being investigated. Simulated meteor events were studied by examining how different functional fits to the simulated meteoroid lag and derived deceleration affected the luminous efficiency computed for each simulated meteor. A simple two-term exponential fit to the lag provides reasonable decelerations, which in turn provide an average luminous efficiency value close to what was used in the simulation. This method however, was only tested on simulated events that were free of noise. In a future work, we will test the method with noise that approximates the noise observed with the CAMO optical system, and then on actual meteor events recorded by CAMO that show single-body ablation. Measuring luminous efficiencies requires precise measurements and a thorough knowledge of the sources of uncertainty. The high-resolution CAMO tracking system will allow luminous efficiencies to be calculated much more accurately than previous observational attempts, and should be able to reveal the order of magnitude of the luminous efficiency and any trend in luminous efficiency with speed.§ ACKNOWLEDGEMENTSThis work was supported by the NASA Meteoroid Environment Office [grant NNX11AB76A]. DS thanks the province of Ontario for scholarship funding. | http://arxiv.org/abs/1704.08656v1 | {
"authors": [
"Dilini Subasinghe",
"Margaret Campbell-Brown",
"Edward Stokan"
],
"categories": [
"astro-ph.EP"
],
"primary_category": "astro-ph.EP",
"published": "20170427165939",
"title": "Luminous Efficiency Estimates of Meteors -I. Uncertainty analysis"
} |
[email protected] and Beverly Sackler School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel We use Newtonian and overdamped Langevin dynamics to study long flexible polymers dragged by an external force at a constant velocity v. The work W performed by that force depends on the initial state of the polymer and the details of the process. The Jarzynski equality can be used to relate the non-equilibrium work distribution P(W) obtained from repeated experiments to the equilibrium free energy difference Δ F between the initial and final states. We use the power law dependence of the geometrical and dynamical characteristics of the polymer on the number of monomers Nto suggest the existence of a critical velocity v_c(N), such that for v<v_c the reconstruction of Δ F is an easy task, while for v significantly exceeding v_c it becomes practically impossible. We demonstrate the existence of such v_c analytically for an ideal polymer in free space and numerically for a polymer which is being dragged away from a repulsive wall. Our results suggest that the distribution of the dissipated work W_ d=W-Δ F in properly scaled variables approaches a limiting shape for large N. 05.70.Ln 05.40.-a 82.37.-j 82.37.Gk 36.20.Ey Nonequilibrium interactions between ideal polymers and a repulsive surface Yacov Kantor December 30, 2023 ==========================================================================§ INTRODUCTIONEquilibrium interactions between a single polymer and a repulsive surface have been a subject of intensive study for several decades <cit.> and benefited from the relation between the statistical mechanics of polymers and the general theory of phase transitions <cit.>. Current experimental methods allow a detailed study of biological macromolecules <cit.>. In particular, atomic force microscopy <cit.> is an important tool in measuring force-displacement curves of biomolecules, and reconstruction of their free energy and spatial structure <cit.>.A long polymer held by its end at a distance h from a repulsive flat surface (wall), experiences an equilibrium repulsive force f_eq(h), i.e., to keep the polymer in place an external force f=-f_eq towards the wall must be applied at the end of the polymer. If h is significantly larger than the microscopic length scale a, such as monomer size or persistence length, but is much smaller than the root-mean-square (rms) end-to-end distance R of the polymer, then the expression for the force, at temperature T, has a particularly simple formf_ eq(h)= Ak_ BT/h,where the dimensionless prefactor A can be related to the critical exponents of the polymer <cit.>. (For non-flat scale-free repulsive surfaces, such as cones, the prefactor A depends on the surface geometry <cit.>). In many cases the polymer size R is related to the number of monomers N by R ≈ aN^ν. In particular, for an ideal polymer, in which the interactions between non-adjacent monomers are neglected, ν=1/2, while for polymers in good solvents ν≈ 0.59 <cit.>. Thus, the work W performed by an external agent while moving a polymer slowly away from a surface at fixed T, as well as the free energy difference Δ F=F_ f-F_ i, between the final free energy of the polymer in free space F_ f and the initial free energy when the polymer is attached to the surface F_ i, isW=Δ F=_0^∞f(h)dh≈-_a^Rf_ eq(h) dh=- Aν k_ BTln N.For an ideal polymer near a flat surface Aν = 1/2 <cit.>. The negative sign reflects the need to push the polymer towards the wall as we slowly move it away.Equation (<ref>) for the force and the resulting Eq. (<ref>) correspond to quasistatic motions. However, if the change is performed at a finite rate, then the work W of the external agent will depend on the details of the experimental protocol, as well as on the microscopic initial state of the system and the specific realization of thermal noise, if such is present. Consider a situation where the initial state, such as a polymer attached to the wall, corresponds to an equilibrium situation at temperature T, i.e., is selected from a canonical ensemble. When an external agent follows an experimental protocol and performs work W, the system reaches a new non-equilibrium state, such as having a polymer far away from a wall. If we proceed to equilibrate the system at temperature T, it settles into a statecharacterized by the free energy F_ f. Repeated non-equilibrium experiments result in the work distribution P(W). A remarkable relation derived by Jarzynski <cit.> relates the exact distribution P(W) to the change of the free energy Δ F between the final equilibrated state and the initial equilibrium state by⟨ e^-β W⟩= _-∞^∞e^-β WP(W)dW = e^-βΔ F,where β=1/k_BT and ⟨·⟩ denotes averaging over the initial states and over realizations of thermal noise, if such is present.At first sight, the Jarzynski equality (JE) provides a tool for easy calculation of free energies from nonequilibrium measurements, and it has been used to reconstruct free energies in certain situations <cit.>. However, it has been observed that for a system significantly out of equilibrium, the successful use of Eq. (<ref>) requires an accurate knowledge of the probabilities of nontypical (rare) events <cit.>. (This can be explicitly observed in the rare cases of exactly solvable systems, such as a one-dimensional Jepsen gas<cit.>.) From the mathematical point of view, this happens when the integrand of Eq. (<ref>)G(W)≡ e^-β W P(W) has a peak centered well below the position of the peak of P(W), i.e., the distance between the peaks exceeds the width of P(W), as illustrated in Fig. <ref>. In the latter situation, the function G(W) is reconstructed from the tail of P(W), which cannot be accurately estimated with a moderate number of repeated experiments. The separation of G and P increases with departure from equilibrium in the experiment. A convenient measure of this departure is the mean of the dissipated work W_ d≡ W -Δ F. This ⟨ W_ d⟩ vanishes in quasistatic isothermal processes and increases with increasing rate of the processes, and when it exceeds several k_BT the free energy reconstruction becomes unreliable. (It has been shown that the number of repeated experiments required for a reliable reconstruction of Δ F increases exponentially with ⟨ W_ d⟩ of a reverse process <cit.>.) Thus, the borderline between easy measurements and practically impossible ones is rather abrupt. In this paper we consider the problem of a polymer, which is initially in equilibrium near a flat repulsive wall, and is being dragged away with a constant velocity v. The final state is when the polymer is in equilibrium far away from the wall, such that we can treat it as being in free space. The dynamics of the system will be either overdamped Langevin dynamics, in which the inertia term is neglected, or Newtonian dynamics in which friction and thermal noise are absent. We argue that there is a critical pulling velocity v_c, such that for v<v_c reconstruction of Δ F is possible by using JE, while for v>v_c such reconstruction is practically impossible. In our discussion we will focus only on these two extreme cases, and we will not consider the range of velocities around v_c for which the ability of reconstruction strongly depends on the number of experiments. In Sec. <ref> we present a heuristic argument for calculating v_c in rather general circumstances. In the remainder of the paper, we provide supporting evidence for the analytically solvable case of ideal polymers in free space (Sec. <ref>) and for the numerically solved case of an ideal polymer near a flat repulsive surface (Sec. <ref>). In Sec. <ref> we summarize our results, and discuss their possible generalizations. Since our work relies on the known results of a dragged harmonic oscillator, we provide a short summary of these properties in Appendix <ref>. § CRITICAL DRAGGING VELOCITY: A HEURISTIC ARGUMENT Consider a situation in which a large polymer is being dragged away from a repulsive wall by moving its end monomer at a constant velocity v. In the initial (equilibrium) state the end monomer is attached to the wall, and in the final state the end-monomer is at a distance significantly exceeding the polymer size R, so that interactions with the wall can be ignored.Many equilibrium properties of polymers have a simple power-law dependence on the number of monomers N. In many cases dynamic features also have that property <cit.>. We would like to take advantage of these scaling properties of polymers to estimate the critical velocity v_c, which defines the borderline between “fast" and “slow" dragging.Let us first consider the simple case of a polymer prepared in thermal equilibrium, and subsequently disconnected from the thermal bath, i.e., its subsequent motion will be determined by Newtonian dynamics (ND). For a polymer at equilibrium, the velocity of its center of mass is 𝐯_cm=1/N∑_i=1^N𝐯_i, where 𝐯_i is the velocity of the ith monomer. For a polymer at equilibrium the average⟨𝐯_cm ⟩ =0. However, the typical or the rms velocity isv_cm=√(1/N^2∑_i,j=1^N⟨𝐯_i·𝐯_j⟩)=v_th/√(N),where v_th=√(d/ β m) is the rms thermal velocity of a single monomer, while m is the mass of the monomer, and d is the spatial dimension. [In further (approximate) calculations we will omit d.] The time t it would take the polymer to move a distance equal to its own size R=aN^ν would bet=R/v_cm≈ a√(β m)N^ν+1/2.We expect that this will also be the time scale of the slowest internal motion of the polymer. It is natural to define a “slow motion" velocity v, such that during the time t the polymer is not dragged more than R, i.e., we must require v<v_cm. In other words, for the ND the borderline critical velocity v_c coincides with the typical velocity of the center of mass v_cm, orv_c≈1√(β m)N^-1/2.The ND approach neglects the interactions of a polymer with the surrounding fluid and therefore its practical usefulness is limited. However, it presents a theoretically important case that formed an essential part of the original proof of JE <cit.>, and provides important insights into the relations betweenthe “regular" mechanics and the thermal physics. It also can be viewed as a limiting case of a general Langevin equation.Alternatively, we can consider motion of the polymer in a very viscous fluid, where the inertia can be neglected on sufficiently long time scales. In this example we neglect hydrodynamic interactions, i.e., we consider the “free-draining" <cit.> regime. Such motion can be described by the overdamped Langevin dynamics (OLD). In the overdamped regime, the center of mass of an N-monomer polymer in free space, performs diffusion characterized by a diffusion constant D, which is N times smaller than the diffusion constant D_0 of a single monomer. Therefore, the time t it takes it to diffuse a distance R=aN^ν, ist ≈R^2D =a^2 N^2νD_0/N= a^2/D_0 N^2ν+1.This is also the slowest relaxation time of an internal mode of the polymer <cit.>.If the polymer is being dragged with a velocity v, we would consider such motion “slow" if during the same time t, the distance vt that the polymer is dragged does not exceed R. This means that we need to have v<D_0/aN^-1-ν, orv_c ≈D_0/aN^-1-ν. When hydrodynamic interactions are important (the Zimm regime <cit.>), a long polymer is not “transparent" to the surrounding liquid, and can be treated as a sphere of radius R diffusing in a liquid of viscosity η <cit.> with a diffusion constant D≈ 1/βη R, where we omitted a dimensionless prefactor. The time it takes for such a polymer to diffuse a distance R is t≈ R^2/D≈βη R^3, which leads tov_c ≈k_BT/η R^2≈k_BT/η a^2N^-2ν. The arguments presented in this section are equally valid for a polymer in free space or near a repulsive wall, because in both cases the polymer will have similar relaxation times. § GAUSSIAN POLYMER IN FREE SPACE The arguments presented in the previous section were valid for a broad class of polymer types and interactions between monomers. In this section we consider a simple model of an ideal polymer, in which we neglect the interactions between non-adjacent monomers of the chain, that are usually important in good solvents. The model only retains the linear connectivity of the monomers and is analytically solvable. Ideal polymers rarely represent experimental systems, but the scaling properties of their static and dynamic characteristics provide guidance to the treatment of more realistic and complicated models <cit.>.Consider a linear chain of N identical monomers of mass m connected bysprings with constants k, such that the potential energy is given by 1/2k∑_i=1^N(x_i-x_i-1)^2, where x_i (i=1,...,N) are the positions of the monomers, while x_0 is the position of the end point to which thefirst monomer is connected, as depicted in Fig. <ref>. Such an energy describes the Gaussian model of an ideal polymer, which in the polymer literature is well known as Rouse model <cit.>, although the latter term is also used to describe the type of dynamics, rather than the polymer structure. (The term “Gaussian" refers to the functional shape of the Boltzmann weight of this energy.) The microscopic length a is given by the rms separation between two consecutive monomers, i.e., a^2≡⟨(x_i-x_i-1)^2⟩ =1/β k. We can consider motion in three-dimensional space, with monomers positioned at r_i. However, the particular form of the potential 1/2k( r_i- r_i-1)^2 splits into three independent parts and the motions in different space directions are independent. Thus, the only non-trivial part of the problem is in the direction parallel to the velocity with which the point at r_0 is being dragged. This reduces the problem to a single space dimension.The problem of stretching a Gaussian chain, when Δ F increases as a result of the external work, has already been solved for OLD, and the distribution of work has been calculated analytically <cit.>. We apply the same technique to a problem with slightly different boundary conditions, for both ND and OLD cases, and have a different goal: We consider the particular case of dragging the polymer in free space, where the free energy difference Δ F between the equilibrated final and initial states vanishes. In this section we find analytical expressions for P(W), as well as v_c. The ND and OLD cases can be viewed as two extremes of the general Langevin equation for the system. §.§ Analytical calculation of P(W) In the absence of friction and thermal noise, the equation of motion of the nth monomer (1≤ n ≤ N-1) is governed by the ND equationsẍ_n=-ω^2 (2x_n-x_n+1-x_n-1),and for n=N ẍ_N=-ω^2(x_N-x_N-1),where ω≡√(k/m). It is more convenient to work in a reference frame which moves with x_0, i.e., with a constant velocity v such that the position of the nth monomer (in the moving system) is x̃_n=x_n-vt. In this reference frame the equations of motion can be separated into N independent (Rouse <cit.>) eigenmodes by decomposing the position of the nth monomer into its discrete Fourier components,x̃_n=A∑_q=1^Nx̃_qsin(α_qn),where x̃_q is the amplitude of the qth mode, and A=√(2/N+1/2), while α_q=π(q-1/2)/N+1/2 was chosen to satisfy the boundary conditions, where one end of the polymer is fixed (x̃_0=0) and the other end (x̃_N) is free. The equations of motion remain the same in the moving system, with x replaced by x̃. Substituting Eq. (<ref>) into Eq. (<ref>) produces the equation of motion for the qth eigenmode in the moving reference frame,ẍ̃̈_q=-4ω^2sin^2(α_q/2)x̃_q,which is a simple harmonic oscillator with frequency ω_q defined byω_q^2≡ 4ω^2sin^2(α_q /2). The constant pulling velocity can be represented (for any n=1,…,N) asv=A∑_q=1^Nv_qsin(α_qn).In this particular case of constant pulling velocity, v_q can be simply expressed via the inverse transform as v_q=A∑_n=1^Nvsin(α_qn)=1/2Av(α_q/2) and used to transform the solution back to the laboratory frame,x_n=x̃_n+vt = A∑_q=1^Nx̃_qsin(α_qn)+A∑_q=1^Nv_qsin(α_qn)t = A∑_q=1^Nx_q(x̃_q+v_qt)sin(α_qn).Here, we defined x_q=x̃_q+v_qt, and the equation for the qth eigenmode in the laboratory frame is given byẍ_q=-ω_q^2(x_q-v_qt).This can be viewed as N independent simple harmonic oscillators with frequencies ω_q, being pulled by effective velocities v_q. Note that, for large N, the frequency of the lowest mode ω_q=1∼ω N^-1 corresponds to the time it would take the polymer to move a distance R, as in Eq. (<ref>) with ν=1/2.We now examine the other extreme of this problem, in which the polymer is moving in a very viscous fluid, so that the inertia term can be neglected, and its motion is described by OLD. The equation of motion of the nth monomer, for 1≤ n ≤ N-1, is given byγẋ_n=-k(2x_n-x_n+1-x_n-1)+η_n(t),and for n=Nγẋ_N=-k(x_N-x_N-1)+η_N(t),where γ is the friction coefficient and η_n(t) is the thermal noise associated with the nth monomer. The thermal noise is chosen to be white Gaussian noise which satisfies ⟨η_n(t)⟩ =0 and ⟨η_n(t)η_n'(t')⟩ =2γ k_ BTδ(t-t')δ_nn'. The same decomposition that was applied to the position x_n and the pulling velocity v in the ND case, can be applied in this case too, while the decomposition of the noise isη_n(t)=A∑_q=1^Nη_q(t)sin(α_qn),where η_q(t) is the effective thermal noise acting on the qth eigenmode, which satisfies ⟨η_q(t)⟩ =0 and ⟨η_q(t)η_q'(t')⟩ =2γ k_BTδ(t-t')δ_qq'.Similarly to the ND case, the system is decomposed into N independent (Rouse) eigenmodes, where each one represents an independent overdamped harmonic oscillator that is being dragged with an effective velocity v_q. Each eigenmode satisfies ẋ_q=-1/τ_q(x_q-v_qt)+1/γη_q,whereτ_q≡τ/4sin^2α_q/2is the relaxation time of the qth eigenmode, and τ≡γ /k. As we can see, the largest relaxation time τ_q=1∼τ N^2 (for large N) coincides with the time it takes the center of mass of the polymer to diffuse a distance R [Eq. (<ref>)]. During the time τ a monomer moves an approximate distance a≈√(D_0 τ), where D_0=k_BT/γ.Both in the ND and OLD cases we can treat the system as N independent harmonic oscillators, and write the total work W done on the system (by an external agent) during the pulling, as a sum of works W_q done on each single effective oscillator, i.e.,W=∑_q=1^NW_q.Each W_q has a Gaussian distribution with mean μ_q and variance σ_q^2, such that μ_q=β/2σ_q^2 = 2mv_q^2sin^2(ω_q t/2) for the ND case, and μ_q=β/2σ_q^2=γτ_q v_q^2(e^-t / τ_q+t / τ_q-1) for the OLD case (as shown in the Appendix). Therefore, W also has a Gaussian distribution characterized by mean μ=∑_q=1^Nμ_q and variance σ^2=∑_q=1^Nσ_q^2.In the ND case the mean work isμ_ND(t)=β/2σ^2_ND(t)= ∑_q=1^N 2m v_q^2sin^2(ω_qt/2).For small q the frequencies ω_q have almost integer ratios with ω_q=1 and, not surprisingly, μ_ND(t) is “almost" a periodic function. For N≫ 1, it looks as a triangular wave depicted in Fig. <ref> of amplitude 2Nmv^2 with period T=2 π/ω_q=1≈ 4N/ω. However, higher frequencies have more complicated dependence on q [see Eq. (<ref>)], and after many oscillations the appearance of the periodicity vanishes. It is interesting to note that in terms of dimensionless variable y≡β W, the probability distribution of the work W has a very simple form,P̃(y)=1/√(4πμ̃)exp[-(y-μ̃)^2/4μ̃],where we used the relation (<ref>) between the mean and the variance and the reduced mean μ̃≡βμ_ ND. For large N and moderate times it is convenient to express μ̃ via triangular wave function S (as in Fig. <ref>) of unit amplitude and period leading to μ̃=2β Nmv^2S(t/T). If we measure the pulling velocity in units of v_c [as defined in Eq. (<ref>)], i.e., u≡ v/v_c, and the total pulling length L in units of polymer size R, ℓ≡ L/R, then the expression for the reduced mean further simplifies toμ̃=2u^2S(ℓ/4u),where we used the fact that R=aN^1/2 and the mean separation between the monomers is a=1/ω√(β m).In the OLD case the mean work isμ_ OLD(t)=β/2σ^2_ OLD(t)=∑_q=1^Nγτ_qv_q^2 (e^-t/τ_q+t/τ_q-1),which is depicted in Fig. <ref>. For short times (t≪τ_q=N≈τ), it increases parabolically with time: μ_ OLD(t) ≈1/2γ/τv^2t^2=1/2k(vt)^2. This corresponds to an external force stretching a single spring of the first monomer connected to x_0, since during time t<τ only x_0 moves, and the rest of the system “does not know” yet that it is being pulled. In the long time regime (t≫τ_q=1≈τ N^2) the mean work grows linearly with time as: μ_ OLD(t) ≈γ Nv^2t. This represents the work against friction performed by dragging N monomers together. (Other eigenmodes are already relaxed in the system.)§.§ Analysis of the results The JE in Eq. (<ref>) can be cast in the form of a cumulant expansion <cit.>:Δ F=-1/βln⟨ e^-β W⟩ = μ-β/2σ^2+...If P(W) is a Gaussian, as in the case of our model in free space, this expansion terminates at the second term. In addition, in free space displacement of the polymer does not modify its free energy, i.e., Δ F = 0. From Eq. (<ref>) we conclude that in free space μ=β/2σ^2, which coincides with the analytical results obtained by a direct calculation in the previous subsection. In the particular case of Gaussian P(W), G(W)≡ e^-β W P(W) is another Gaussian shifted by 2μ towards lower values of W. For slow motion we must have μ < σ, i.e., the mean work (and the shift between P and G) does not exceed the width of the distribution. At the critical velocity this relation becomes an equality. Due to the relation between μ and σ, this condition becomesμ≈ 2k_BT. In the case of ND, the mean value of work is bounded by 2Nmv^2, and therefore, from Eq. (<ref>) we find that v_c≈ N^-1/2/√(β m), which is exactly v_c that we found in Eq. (<ref>).In the OLD case, μ_OLD increases monotonically with time, making the free energy reconstruction more difficult as t grows. We would like to drag the polymer a distance at least equal to its size, i.e., L=v t ∼ a√(N). (In free space it is a somewhat arbitrary choice, but in the next section we will consider a polymer being dragged away from a wall, and then such a choice of distance becomes crucial.) For such L the condition inEq. (<ref>) translates into a γ N^3/2v_c≈ k_BT, which defines the critical velocity,v_c≈1/a γβN^-3/2.Substituting a≈√(D_0 τ) brings us back to the same critical velocitythat was found in Eq. (<ref>), with ν=1/2. In terms of dimensionless variable y=β W the probability distribution of work is given by Eq. (<ref>), with reduced mean μ̃≡βμ_ OLD. For times larger than the relaxation time of the polymer and for N≫1, we get μ̃=βγ Nv^2t, which can be conveniently expressed via relative distance ℓ=L/R and relative velocity u=v/v_c, where v_c was derived in Eq. (<ref>), leading toμ̃=uℓ.We note that both in ND and OLD cases for large N the work distribution is described by Eqs. (<ref>), (<ref>) and (<ref>), which for fixed dimensionless u and ℓ are independent of N. This is a direct consequence of the scaling of internal relaxation times and internal length scales related to scale-invariant internal structures of the polymer <cit.>, when only the polymer size R and the largest relaxation time determine the time and length scales of the internal modes.Hydrodynamic interactions cannot be accurately treated even for ideal polymers, since the equations of motion do not split into a set of independent equations for each q mode, as in Eq. (<ref>). However, it has been shown <cit.> that close to equilibrium such interactions can be mimicked by replacing fixed γ by a power law of q in the Fourier space, i.e., modifying τ_q in Eq. (<ref>) by an extra power of q. [This change also requires a proper change in the noise correlation ⟨η_q(t)η_q'(t')⟩.] While we expect these changes to correctly reproduce the near-equilibrium behavior of the system, as well as the value of v_c in Eq. (<ref>), we do not expect them to produce a good approximation for P(W) for v≫ v_c.§ POLYMER NEAR A WALL In this section we consider the Gaussian (Rouse) chain dragged away from a repulsive wall.At time t=0 the polymer is in equilibrium near the wall at x_0=0, and is being dragged away from the wall at a constant velocity v, i.e., x_0=vt, as illustrated in Fig. <ref>. If the final distance L of the polymer from the wall is significantly larger than R, then in the final equilibrated state the free energy will be equal to its value in free space, and in accordance with Eq. (<ref>) Δ F=-1/2k_BTln N. Unlike the simple case considered in the previous section, we can no longer expect a simple relation between μ and σ, and P(W) will not have a Gaussian form.Our results rely on a numerical solution of Newton's equations in the ND case, and on a solution of an overdamped Langevin equation in the OLD case <cit.>. In both cases the calculation begins by choosing a properly weighted initial configuration. The coordinates of the monomers are then advanced in time until x_0 reaches the value L=5R. Integration of the external force that needs to be applied to x_0 to keep it moving at a constant velocity v produces the work W of the external agent. Each calculation was repeated N=10^3 times. Every time a new initial equilibrium state was selected, and, in the OLD case a new thermal noise function was generated. Such calculations produce a numerical estimate of P(W) and can be used to produce a numerical estimate of Δ F. We repeated the calculations for chain sizes N ranging from 10 to 100, and for each chain size repeated the calculation for dragging velocities ranging from 0.1v_c to 5v_c. In this section we present a partial set of our results. Since the exactly known Δ F is proportional to -ln N, the graphs of P(W) shift towards more negative values of W with increasing N. For an easy comparison of the results with different chain sizes, it is convenient to present all the functions in terms of the dissipated work W_ d=W-Δ F, rather than the entire work W. Furthermore, we will use the dimensionless variable y≡β W_d. The probability density P̃(y) is simply related to P(W) by P̃(y)=P(W_ d+Δ F)/β. Similarly, a new shifted function G̃(y)≡ e^-yP̃(y) can be used. In this notation the probability density is normalized, i.e., ∫_-∞^∞P̃(y)dy=1 and the relation (<ref>) has the simple form ∫_-∞^∞G̃(y)dy=1. While these two integrals impose some restrictions on the shape of P̃, there is still plenty of room for dependence of this function on N or v and on the type of dynamics (ND or OLD). Normally the term “dissipation" implies positive W_d or y>0. In macroscopic systems for the average W_ d this is referred to as the Clausius inequality. However, a particular experiment may violate this inequality <cit.>. This is very unlikely, and it can be shown that the probability of y<-ζ (for ζ>0) is bounded by e^-ζ <cit.>. This means that W_ d can be only a few times -k_BT, independently of the system size. This inequality further restricts the possible shapes ofP̃. The solid lines in Figs. <ref> and <ref> depict the the probability distributions of the dissipated work W_ d for short polymers (N=10), for the ND and OLD cases, respectively. These histograms are results of N=10^3 repeated numerical experiments. The size of the bin was selected for convenient presentation of the results. (Calculation of Δ F from the data is performed directly from the set of measured W_ ds rather than from these histograms.) All distributions have a tail in the negative y region but most of such “Clausius-inequality-violating" events are within one unit away from 0. For small velocities v=0.1v_c (graphs (a)) the distributions represent a process that is rather close to quasistatic, i.e., they are narrow and close to 0, and the mean dissipation satisfies ⟨ y⟩=β⟨ W_ d⟩≪ 1. When the polymer is dragged away at a large velocity v=2v_c [graphs (b)] the distributions are shifted towards larger y values, corresponding to an external agent pulling the polymer away from the wall, in contrast with the low-velocity case when the force is mostly towards the wall to maintain a constant velocity.The areaunder the solid lines in all the graphs is 1 due to normalization. The dotted histograms inFigs. <ref> and <ref> represent the shifted functions G̃, and they were constructed from the values of the solid histograms by multiplying them by e^-y. To correctly reproduce the known value of Δ F,the area under G̃ must be 1. This indeed happens in the low-velocity graphs (a), where the area does not deviate from 1 by more that 0.1 even for a moderate number N of experiments. At high velocities [graphs (b)] the reconstructed G̃ is significantly shifted compared to P̃, and is reconstructed from the poor quality tail of P̃. For y<-1 most bins correspond either to 0 or to 1 event found in that range, which explains the noisy behavior of G̃. The area under the G̃ curve at high velocities significantly deviates from 1. This means that the reconstructed free energy difference will have significant errors. We used our data sets to directly evaluate the free energy difference from the expressionΔ F_ num=-1/βln(1/ N∑_i=1^ Ne^-β W_i),where W_i is the work associated with the ith repetition of the calculation. When these estimates of the free energy difference were compared with the exactly known Δ F, we saw (for N ranging from 10 to 100) a fast deterioration of accuracy when v exceeded v_c. This result confirms our expectation that v_c serves as a borderline velocity between “slow" and “fast" processes for all values of N in the problem of a polymer near a wall.In the previous section we found analytically that the work distribution for a polymer of large N in free space is described by Eqs. (<ref>), (<ref>), and (<ref>), which for fixed dimensionless u and ℓ are independent of N. We argued that in free space this is a direct consequence of the scaling of internal relaxation times and internal length scales <cit.>. Such a lack of N dependence is rather natural even in the presence of a wall, since the equations of motion can be reformulated in properly scaled variables in the N→∞ limit, indicating the existence of such a limit. In free space, Δ F=0, or W_ d=W. In the presence of the wall, a simple free space result is no longer possible, due to the N dependence of Δ F. Nevertheless, by considering W_ d we eliminate the leading N dependence, and may hope to get an N-independent limit. Figure <ref> depicts P̃(y) in the ND case for several values of N, when each case has been calculated with the same relative velocity u. Since v_c decreases with increasing N, the velocity v was also decreased. We note that three different Ns produce rather similar graphs. A change of u produces different graphs, but again, they seem to be almost independent of N. In the third paragraph of this section we mentioned that there were several constraints on the shape of P̃(y). Therefore, the similarity of the graphs is not surprising. Nevertheless it is possible that in these scaled variables there is an N→∞ limit of this graph which our numerical graphs are approaching. We could see this property explicitly in the solution of a polymer in free space. Due to scaling of the dynamical properties of a polymer,it is possible that similar features exist in the more complicated case of a polymer near a wall. Similar behavior is also observed in the OLD case, although the graphs for the same u values differ slightly from the ND shapes discussed above. § DISCUSSIONWe studied the problem of a flexible polymer being dragged with a constant velocity both in free space and in the vicinity of the wall, and argued that there exists a critical N-dependent pulling velocity v_c that separates the region of “easy" reconstruction of Δ F by means of the Jarzynski equality from the region of “impossible" reconstruction. The existence of a maximal deviation from equilibrium for which the reconstruction of Δ F is possible is well known from previous studies <cit.>. In the context of unfolding of a large molecule this was typically viewed as an event of a “single particle" escaping from one well into another <cit.>, or a sequence of such events <cit.>. We attempted to integrate the well-known static and dynamic scaling properties of polymers into the description of their non-equilibrium motion. Our heuristic argument in Sec. <ref> produced in the ND case the result v_c∼ N^-1/2 which was independent of the polymer type, while for OLD the result depended on the exponent ν. The numerical support of our claim is limited to the two simple cases of free space and repulsive wall for ideal polymers.We observed the lack of N-dependence of the dissipated work distribution, but the range of Ns was rather limited, and much longer polymers need to be studied.While our calculations were limited to the ideal polymers, some of the concepts can be generalized to more realistic models, such as the polymers in good solvents, when the interactions between non-adjacent monomers are important. In the latter case, the Rouse modes, such as x̃_q in Eq. (<ref>), are no longer the exact eigenmodes of the system. Nevertheless, the properly modified concept of Rouse modes is used to describe dynamics of the polymers <cit.>. We expect that this and other generalized concepts can be used to produce results described inSec. <ref>. Confirmation of this expectation will require detailed numerical simulations.While the “free draining" regime provides an adequate description of the motion of polymers of moderate length, in experiments with longer polymers the hydrodynamic interactions play an important role. We briefly mentioned this regime in Sec. <ref>. In accordance with Eq. (<ref>) the typical critical velocity of a 1μm size polymer will be slightly above 1 μm/s, which is one order of magnitude larger than the typical speeds in many experiments (see, e.g., <cit.>). The use of a constant dragging velocity significantly simplified our derivations. By contrast, in real experiments the force, rather than the speed, is controlled. However, in homogeneous polymers these two should exhibit a similar behavior. We thank Y. Hammer and M. Kardar for numerous discussions. We also thank them and D. J. Bergman for valuable comments about the manuscript. This work was supported by Israel Science Foundation Grant No. 453/17. § DRAGGED HARMONIC OSCILLATOROur analytical treatment of the Gaussian polymer in free space relies on a decomposition into Rouse modes that are treated as simple dragged harmonic oscillators. The harmonic oscillator (HO) was one of the first systems used to demonstrate the workings of the JE <cit.>. The theoretical treatment of a dragged HO <cit.> followed an experimental study of the translation of a particle in a harmonic optical trap <cit.>, which was designed to test violations of the second law of thermodynamics with findings consistent with a fluctuation theorem of Evans et al. <cit.>. Consider the motion of a particle of mass m at position x attached by a spring with force constant k to a point x_0, which moves with velocity v, i.e., at time t its position is x_0=vt. In the ND case the equation of motion ism ẍ=-k(x-vt),and its solution is given byx(t)=x^0 cos(ω t)+(p^0/mω -v/ω)sin(ω t)+vt,where ω=√(k/m), and x^0, p^0 are the initial position and momentum of the particle, respectively, which are selected from a Gaussian distribution corresponding to the temperature of the system. For the OLD case the equation of motion isγẋ=-k(x-vt)+η (t),where γ is the friction constant and random function η(t) represents white Gaussian noise which satisfies ⟨η(t)⟩=0 and ⟨η(t)η(t')⟩ =2γ k_ BTδ(t-t'). The solution for this equation is given byx(t)=x^0 e^-t/τ+vt+_0^t(1/γη(t') -v)e^-(t-t')/τdt',where τ=γ/k is the relaxation time of the oscillator. Note, that both in ND and in OLD cases the position x(t) has a Gaussian distribution since it is a linear combination of Gaussian variables. The work done on the oscillator during the dragging of x_0 at a constant velocity v isW =-v_0^t k [x(t')-vt' ] d t'.Since x(t) is known as function of the initial conditions and the realization of noise η(t) (in the OLD case), the distribution P(W) can be easily determined. Since x(t) has a Gaussian distribution, the distribution of work P(W) is also a Gaussian characterized by its mean μ = ⟨ W⟩ and variance σ^2 = ⟨ W^2⟩ -⟨ W⟩ ^2.Direct calculation of μ and σ, both in the ND and in the OLD cases, finds that these quantities are simply related: μ(t)=β/2σ^2(t). [This can also be viewed as a consequence of the JE for a Gaussian work distribution in a situation where the equilibrium free energy of an oscillator is independent of its position, as explained in the paragraph following Eq. (<ref>).] In the ND caseμ(t)= 2mv^2sin^2(ω t/2).This μ(t) is periodic and vanishes after each complete period of the oscillator. 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"title": "Nonequilibrium interactions between ideal polymers and a repulsive surface"
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http://arxiv.org/abs/1704.08188v2 | {
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[][email protected] de Física Atómica, Molecular y Nuclear, Universidad de Granada, Granada 18071, SpainInstituto Carlos I de Física Teórica y Computacional, Universidad de Granada, Granada 18071, SpainWe introduce a biparametric Fisher-Rényi complexity measure for general probability distributions and we discuss its properties. This notion, which is composed of two entropy-like components (the Rényi entropy and the biparametric Fisher information), generalizes the basic Fisher-Shannon measure and the previous complexity quantifiers of Fisher-Rényi type. Then, we illustrate the usefulness of this notion by carrying out a information-theoretical analysis of the spectral energy density of a d-dimensional blackbody at temperature T. It is shown that the biparametric Fisher-Rényi measure of this quantum system has a universal character in the sense that it does not depend on temperature nor on any physical constant (e.g., Planck constant, speed of light, Boltzmann constant), but only on the space dimensionality d. Moreover, it decreases when d is increasing, but exhibits a non trivial behavior for a fixed d and a varying parameter, which somehow brings up a non standard structure of the blackbody d-dimensional density distribution.The biparametric Fisher-Rényi complexity measure and its application to the multidimensional blackbody radiation D. Puertas-Centeno, I. V. Toranzo, J. S. Dehesa December 30, 2023 ================================================================================================================§ INTRODUCTION General quantification of complexity attributed to many-body systems, the task which is closely connected with evolution from order to disorder, is among the most important scientific challenges in the theory of complex systems <cit.>. The fundamental issue is to find one quantifier which is able to capture the intuitive idea that complexity lies between perfect order and perfect disorder. Most probably this idea cannot be formalized by a single complexity quantifier because of the so many facets of the term complexity. Based on Information Theory and Density Functional Theory, various computable and operationally meaningful density-dependent measures have been proposed: the entropy and complexity measures of the one-body probability density of the system. The former ones (mainly the Fisher information and the Shannon entropy, and their generalizations like Rényi and Tsallis entropies) capture a single macroscopic facet of the internal disorder of the system. The latter ones capture two or more macroscopic facets of the quantum probability density which characterize the system, the most relevant ones being, up until now, the complexity measures of Crámer-Rao <cit.>, Fisher-Shannon <cit.> and LMC (López-Ruiz-Mancini-Calvet) <cit.>, which are composed by two entropic factors. These three basic measures, which are dimensionless, have been shown to satisfy a number of interesting properties: bounded from below by unity, invariant under translation and scaling transformation, and monotonicity (see e.g. <cit.>). Recently, they have been generalized in various directions such as the measures of Fisher-Rényi <cit.> and LMC-Rényi<cit.> types. The aim of this article is twofold. First, we introduce a novel class of biparametric measures of complexities (namely, the generalized Fisher-Rényi measures) for continuous probability densities, which generalizes the basic Fisher-Shannon and its extensions of Fisher-Rényi type. Second, we illustrate the utility of these novel complexity quantifiers by applying them to the generalized Planck radiation law (see below). Beyond the temperature, we will focus on the dependence of these quantifiers on the space dimensionality d and the complexity parameters.The cosmic microwave, neutrino and gravitational backgrounds (cmb, cnb and cgb, respectively) give information about the universe at different times after the big bang. The cnb and cgb have been claimed to give information at one minute after the big bang and during the big bang, respectively, and to have been seen in recent experiments with controversial results, still under a careful examination <cit.> (see <cit.> for a brief summary). The cmb, originated at around 380,000 years after the big bang at a temperature of around 3000 Kelvin, is the only cosmic radiation background which is well established. It was first detected in 1964 by the antennae manipulation works of Penzias and Wilson <cit.>, and later confirmed by satellite observations in a very detailed way <cit.>. It is known that the frequency distribution of the cmb which presently bathes our (three-dimensional) universe follows the Planck’s blackbody radiation law given by the (unnormalized) spectral density ρ_T^(3)(ν) = 8 π h/c^3ν^3 (e^h ν/k_B T-1)^-1 at the temperature T_0 = 2.7255(6) Kelvin, where h and k_B denote the Planck and Boltzmann constants, respectively. In the last few years there is an increasingly strong interest in the analysis of the quantum effects of the space dimensionality in the blackbody radiation <cit.> and, in general, for natural systems and phenomena of different types in various fields from high energy physics and condensed matter to quantum information and computation (see e.g. <cit.> and the monographs <cit.>). This is not surprising because of the fundamental role that the spatial dimensionality plays in the solutions of the associated wave equations <cit.>. In the present work we adopt an information-theoretical approach to investigate the complexity effects of the spatial dimensionality in the spectral energy density per unit of frequency of a blackbody at temperature T which has been found <cit.> to be given in a d-dimensional space by the generalized Planck radiation lawρ_T^(d)(ν) = 1/Γ(d+1) ζ(d+1) (h/k_BT)^d+1ν^d/e^h ν/k_B T-1,(normalized to unity), where Γ(x) and ζ (x) denote the Euler gamma function and the Riemann zeta function <cit.>, respectively. This investigation will be done by means of a novel class of biparametric measures of complexity of Fisher-Rényi type which allows us to go further beyond the 2014-dated work <cit.> based on the entropy-like measures of ρ_T^(d)(ν) and the three basic two-component complexity measures of Crámer-Rao, Fisher-Shannon and LMC types.The structure of the work is the following. First, we define and explore the main properties of the biparametric complexities and their entropy-like components (Rényi entropy, biparametric Fisher information) of a general continuous one-dimensional probability distribution. Second, we determine and discuss the values of the previous entropy-like and complexity measures for the spectral density ρ_T^(d)(ν) which characterizes the multidimensional blackbody distribution. Then, we numerically discuss the dependence of these blackbody spectral quantifiers on the universe dimensionality and the complexity parameters. Let us advance that the resulting blackbody biparametric complexities are mathematical constants (i.e. they are dimensionless), independent of the temperature T and of the physical constants (Planck's constant, speed of light and Boltzmann's constant), so that they only depend on the spatial dimensionality. Finally, some concluding remarks are given. § THE BIPARAMETRIC FISHER-RÉNYI COMPLEXITY MEASURE OF A GENERAL DENSITY In this Section we define and discuss the meaning of a class of biparametric complexity measures of Fisher-Rényi type for a one-dimensional continuous probability distribution ρ(x), x ∈Δ⊆ℝ. Heretoforth we assume that the density is normalized to unity, so that ∫_Δρ(x) dx=1.First, we define the biparametric Fisher-Rényi complexity measureof the density ρ(x) as C^(p,λ)_FR[ρ] =𝒦_FR(p,λ) ϕ_p,λ[ρ] × N_λ[ρ],with p^-1 + q^-1 = 1, λ > (p+1)^-1 and wherethe symbols 𝒦_FR(p,λ), ϕ_p,λ[ρ] and N_λ[ρ] denote a normalization factor, the biparametric Fisher information <cit.> and the Rényi entropy power, N_λ[ρ] = exp(R_λ[ρ] ), respectively. Moreover, the symbol R_λ[ρ] denotes the Rényi entropy of order λ defined <cit.> asR_λ[ρ]=1/1-λln(∫_Δ[ρ(x)]^λdx),with λ>0 and λ≠1.This quantity is known to quantify various λ-dependent aspects of the spreading of the density ρ(x) all over its support Δ. In particular, when λ→1 the Rényi entropy tends to the Shannon entropy S[ρ], which measures the total spreading ofρ(x). On the other hand, the biparametric Fisher information F_p,λ[ρ] <cit.> is given byϕ_p,λ[ρ] = (∫_Δ|[ρ(x)]^λ-2ρ'(x)|^qρ(x)dx)^1/qλ , with 1/p+1/q =1, p∈[1,∞) and λ∈ℝ.Note that for the values (p,λ)=(2,1) the square of this generalized measure reduces to the standard Fisher information, i.e., ϕ_2,1[ρ]^2 = F[ρ] = ∫_Δ|ρ'(x)|^2/ρ(x) dx. So, while F[ρ] quantifies the gradient content of ρ(x), the generalized Fisher information ϕ_p,λ[ρ] with p≠ 2 and λ≠ 1 measures the (p,λ)-dependent aspects of the density fluctuations other than the gradient content.The normalization factor, 𝒦_FR(p,λ), in Eq. (<ref>) is given by𝒦_FR(p,λ)=(ϕ_p,λ[G]N_λ[G])^-1= [λ^1/q/p^1/p a_p,λ e_λ(-1/pλ)^λ-1/p+1]^1/λ=a_p,λ^1/λ((pλ+λ-1)^qλ-λ+1/q/pλ^λ)^1/λ-λ^2,where G denotes the generalized Gaussian distribution G(x) <cit.> given by G(x)=a_p,λe_λ(|x|^p)^-1where e_λ(x) denotes the modified q-exponential function <cit.>:e_λ(x)=(1+(1-λ)x)_ +^1/1-λ,which for λ→ 1 reduces to the standard exponential one, e_1(x)≡ e^x. For p ∈ (0,∞), λ >1-p and with the notation t_+ = max{t,0} for any real t, the constant a_p,λ is given by a_p,λ = {[ p(1-λ)^1/p/2B(1/p,1/1-λ-1/p) ifλ < 1,;p/2Γ(1/p) if λ =1,; p(λ-1)^1/p/2B(1/p,λ/λ-1) if λ >1. ].Moreover, the symbol N_λ[G] denotes the Rényi entropic power (<ref>) of the generalized Gaussian distribution G(x) given byN_λ[G] =[a_p,λ e_λ(-1/pλ)]^-1,and ϕ_p,λ[G] represents the biparametric Fisher information of the generalized Gaussian distribution <cit.> which, for 1≤ p ≤∞ and λ> 1/1+p, is given by ϕ_p,λ[G]={[ p^1/pλ/λ^1/qλ[a_p,λ e_λ(-1/pλ)^1/q]^λ-1/λ,p<∞; 2^(1-λ)/λλ^-1/λ, p=∞, ]. Note that, from (<ref>), (<ref>) and (<ref>) we can state that the biparametric complexity measure C^(p,λ)_FR[ρ] quantifies the combined balance of the λ-dependent spreading facet of the probability distribution ρ(x) and the (p,λ)-dependent oscillatory facet of ρ(x). It is then clear that this quantity is much richer than e.g. the Fisher-Shannon measure which quantifies a single spreading aspect of the distribution (namely, its total spreading given by the Shannon entropy power) together with a single oscillatory facet (which corresponds to the gradient content as given by the standard Fisher information).Indeed, the generalized complexity measure C^(p,λ)_FR[ρ] includes the basic Fisher-Shannon complexity measure <cit.>, C_FS[ρ]=1/2 π e F[ρ]exp(2 S[ρ]), and the various one-dimensional complexity measures of Fisher-Rényi types recently published in the literature <cit.>. Most important is to point out that the novel complexity quantifier C^(p,λ)_FR[ρ] includes theone-parameter Fisher-Rényi complexity measure C_FR^(λ) <cit.>, sinceC_FR^(λ)=(C_FR^(2,λ))^2λ. These two measures of complexity present a number of similarities and differences, which are worth mentioning. First, following the lines of<cit.> it is straightforward to show that the biparametric measure, like the monoparametric one, has the following important properties: a universal unity lower bound (C_FR^(p,λ)≥1), invariance under scaling and translation transformations and monotonicity The latter property, defined by Rudnicki et al [12], can be proved by use of rearrangements (a powerful tool of functional analysis) and operating in the same manner as done in Ref. [16] for the one-parameter Fisher-Rényi complexity. Moreover, the biparametric measure has the following behavior under replication transformationC_FR^ (p,λ)[ρ̃] =n^1/λC_FR^ (p,λ)[ρ],where the density ρ̃ representing n replications of ρ is given byρ̃(x) = ∑_m=1^nρ_m(x);ρ_m(x)= n^-1/2ρ(n^1/2(x-b_m)),where the points b_m are chosen such that the supports Δ_m of each density ρ_m are disjoints. This property shows that this biparametric complexity quantifier, as opposed to the monoparametric one, becomes replication invariant in the limit λ→∞; this limiting property is an effect of the power 1/λ which has the biparametric measure but not the monoparametric one. In this limit, the minimizer distribution of this complexity measure has the form of a Dirac-like delta. Moreover, this power effect makes that the biparametric measure is well defined in the limit λ→∞, which does not happen in the monoparametric case. Another important difference between the bi- and uni-parametric complexity quantifiers is that the former one has two degrees of freedom; this means that it does not only depend on λ but also on the parameter p. So, in particular when λ=1, we can readily show that this quantifier is minimized for Freud-like probability distributions of the form e^-|x|^p <cit.>, which has a great physical relevance in the theory of sub- and super-diffusive systems. There exist other instances of the biparametric complexity measure C^(p,λ)_FR[ρ] which are relevant for different reasons. Let us just mention three of them. First, when λ=1+1/p>1/1+p, ∀ p>1, we have that the resulting measure C_FR^[p][ρ] ≡ C_FR^(p,1+1/p)[ρ]is composed by two particularly relevant entropic factors: the generalized Fisher information ϕ_p,1+1/p[ρ]=(∫_Δ |ρ'(x)|^qdx)^1/2q-1,which is a pure functional of the derivative of the density ρ, and the Rényi entropic powerN_1+1/p[ρ]=(∫_Δ[ρ(x)]^1+1/p dx)^-1/p=⟨[ρ(x)]^1/p⟩^-pMoreover, this complexity measure is minimized by the distribution e_p,1+1/p(x)=a_p,1+1/p(1-|x|^p/p)_+^pNote furthermore that the support of this distribution is [-p^1/p,p^1/p], which boils down to [-1,1] for both values p=1 and p=∞, and it becomes longest for p=e. Second, when λ = 2 the corresponding complexity measure C_FR^(p,2)[ρ] is proportional to the ratio ⟨|ρ'(x)|^q⟩^1/2q/D[ρ], since the Fisher-information-factor is ϕ_p,2[ρ]=(∫_Δρ(x) |ρ'(x)|^qdx)^1/2q=⟨|ρ'(x)|^q⟩^1/2qand the Rényi-entropic-power-factor is the inverse of the disequilibrium D(ρ) asN_2[ρ]=(∫_Δ[ρ(x)]^2 dx)^-1=D[ρ]^-1Moreover, the resulting measure C_FR^(p,2)[ρ] is minimized by the distributione_p,2(x)=a_p,2(1-|x|^p)_+,whose support [-1,1] remains invariant. Third, most remarkable, is that the measure in the limit p→∞ is also well defined, corresponding to a step function ∀λ>0. In the latter case the complexity measure is given by C_FR^(∞,λ)=(λ/2)^1/λ ϕ_∞,λ[ρ] N_λ[ρ].where the generalized Fisher-like information ϕ_∞,λ[ρ] is given byϕ_∞,λ[ρ]=(∫_Δ [ρ(x)]^λ-1 |ρ(x)'| dx)^1/λ(so that (ϕ_∞,λ[ρ])^λ corresponds to the total variation of ρ^λ/λ <cit.>) and N_λ[ρ] is the previously defined Rényi entropy power. The measure C_FR^(∞,λ) has all the properties previously pointed out for the general biparametric Fisher-Rényi complexity. Moreover, it is minimized by the uniform distribution (as the basic LMC complexity). As well, within the set of all possible step-permutations of a generic distribution ρ composed of N step functions, the measure C_FR^(∞,λ) gets minimized by all the monotonically increasing or decreasing distributions.Finally, let us point out that when 1/1+p < λ < 1 the resulting complexity measureshave heavy-tailed distributions as minimizers.§ APPLICATION TO THE D-DIMENSIONAL BLACKBODYIn this section, the biparametric Fisher-Rényi complexity measure C_FR^(p,λ) is investigated for the d-dimensional blackbody frequency distribution at temperature T, ρ(ν) ≡ρ_T^(d)(ν), given by Eq. (<ref>). That is, C_FR^(p,λ)[ρ_T^(d)] =𝒦_FR(p,λ)ϕ_p,λ[ρ_T^(d)] × N_λ[ρ_T^(d)],with λ p >d/d-1, where 𝒦_FR(p,λ) is the normalization constant given by Eq. (<ref>), andϕ_p,λ[ρ_T^(d)] and N_λ[ρ_T^(d)] are the generalized Fisher information and power Rényi entropy of the d-dimensional blackbody density, respectively, defined in the previous section whose values will be first expressed in the following.The Rényi entropy power is given by N_λ[ρ_T^(d)] = exp(R_λ[ρ_T^(d)] ), where the Rényi entropy for the d-dimensional blackbody density, defined in (<ref>), has been shown <cit.> to be given by R_λ[ρ_T^(d)]=1/1-λln A_R(λ,d) + ln(k _BT/h),with λ >0 and λ≠ 1, where the constant A_R(λ,d) has the valueA_R(λ,d)=Γ (λ d+1)ζ_λ(λ d+1,λ)/Γ^λ (d+1) ζ^λ(d+1),with λ∈ℕ∖{1}, and the symbol ζ_λ(s,a) denotes the modified Riemann zeta function or Barnes zeta function <cit.>.The biparametric Fisher information ϕ_p,λ[ρ], defined in (<ref>), for the d-dimensional blackbody density at temperature T has been recently obtained <cit.> asϕ_p,λ[ρ_T^(d)] = [A_F(p,λ,d)]^1/qλh/k_BT,with q ∈ (1,∞),λ >0, 1/q + 1/p =1 and where A_F(p,λ,d) denotes the proportionality constant,A_F(p,λ,d)=I(q,λ,d)/(Γ(d+1)ζ(d+1))^qλ -q+1 , with I(q,λ,d)=∫_ℝ^+x^q(dλ -d-1)+d/(e^x-1)^qλ +1|d(e^x-1)-xe^x|^qdx.For d>λ p/λ p-1 (so that ϕ_p,λ[ρ_T^(d)] is well-defined), q even and qλ∈ℕ, the integral I(p,λ,d) in (<ref>) is analytically solvable giving rise to the following value for the proportionality constant A_F(p,λ,d)=(Γ(d+1) ζ(d+1))^-α ×∑_i=0^q(-1)^q-iq id ^i(α d -i)! ζ_α+q-i(1+α d -i ,α), with α≡ qλ -q+1. For the standard case (p=2,λ=1) one obtains thatA_F(2,1,d)=1/2ζ(d+1)(ζ(d)-d-3/d-1ζ(d-1) ),with d>2.The insertion of (<ref>) and (<ref>) into (<ref>) allows us to obtain finally theexpressionC_FR^(p,λ)[ρ_T^(d)] =𝒦_FR(p,λ) A_F(p,λ,d)^1/qλ A_R(λ,d)^1/1-λ,for the biparametric Fisher-Rényi complexity measure of the d-dimensional blackbody at temperature T, where the constants A_R(p,λ,d) and A_F(λ,d) are given by Eqs. (<ref>) and (<ref>), respectively. Note that this two-parameter complexity quantifier does not depend on T nor on any physical constants (e.g., Boltzmann and Planck constants), but it does depend on the universe dimensionality only; thus, having a universal character.For a better understanding of how the biparametric Fisher-Rényi complexity measure C_FR^(p,λ)[ρ_T^(d)] is able to characterize the multidimensional blackbody distribution, we study its dependence on the spatial dimensionality d and the parameters p and λ in Figures 1 and 2. In Fig. <ref> we represent the (p,λ)-chart of the Fisher-Rényi complexity for the three-dimensional blackbody distribution, C_FR^(p,λ)[ρ_T^(3)], in terms of p and λ. This quantity has no physical units and it does not depend on the blackbody temperature, what highlights the universal character of the chart. We observe that the (p,λ)-Fisher-Rényi complexity (i) presents a relative minimum at around (p=2.20,λ=1.74), (ii) for all fixed λ it hasan absolute minimum whose location depends on λ, appearing a complexity horizontal asymptote at values bigger than unity, and opposite (iii) for fixed p it has the three following regimes. For 1≤ p ≲ 1.55 the λ-behavior of the complexity is monotonically decreasing and strictly concave; for 1.55≲ p ≲ 4.3 this behavior breaks down, appearing two critical points (see also Figure <ref> left), and finally for 4.3 ≲ p the critical points dissapear and the concave behaviour go back slowly when p is increasing. Nevertheless when λ is large enough the complexity goes to unity independently of the p-value.These phenomena can be also observed in the functions C_FR^(2,λ)[ρ_T^(3)] and C_FR^(p,1)[ρ_T^(3)] of Fig. <ref>, which corresponds to the cuts at p=2 y λ=1 of the (p,λ)-chart, respectively.Besides we study in Fig. 2 the dimensionality dependence of the Fisher-Rényi complexity measures C_FR^(2,λ)[ρ_T^(d)] and C_FR^(p,1)[ρ_T^(d)] when d= 3(+), 4(×),5 (*), 6(□) in terms of the corresponding parameters λ and p, respectively. Note that the dimensionality behavior is qualitatively similar in each case. The complexity C_FR^(2,λ)[ρ_T^(d)] as a function of λ has, for all dimensionalities, a minimum and a maximum within the interval (0,4) and then it monotonically decreases to unity. On the opposite, the complexity C_FR^(p,1)[ρ_T^(d)] as a function of p has only a minimum at p<4 for all dimensionalities and then it monotonically grows when p is increasing. In both cases the minimum location decreases when the dimensionality is augmenting. Quantitatively, we observe that for λ≥ 5 in the left graph and for p≥ 4 in the right graph the corresponding complexities do not practically depend on the dimensionality. § CONCLUSIONS AND OPEN PROBLEMS First we have shown in this paper that the Rényi entropy power, N_λ[ρ], (that generalizes the Shannon entropy power) and the biparametric Fisher information, ϕ_p,λ[ρ], (which generalizes the standard Fisher information) allow us to construct a novel class of generalized complexity measures for a general probability density ρ(x), the biparametric Fisher-Rényi complexities C^(p,λ)_FR[ρ]. They quantify jointly the λ-dependent spreading aspects and the (p,λ)-dependent oscillatory facets of ρ(x), so being much richer than the basic Fisher-Shannon measure and all its extensions of Fisher-Rényi type. Second, we have pointed out a number of properties of these quantifiers, such as universal lower bound, scaling and translation invariance and monotonicity, among others. Third, we have applied these complexity quantifiers to the d-dimensional blackbody radiation distribution at temperature T. We have found that they do not depend on the temperature nor on any physical constant (Planck constant, speed of light and Boltzmann constant) but only on the spatial dimensionality, which gives them a universal character. We are aware that the full power of the novel complexity quantifiers here proposed will be shown in multimodal/anisotropic probability distributions so abundant in natural phenomena, such as e.g. the ones which are being observed in the present observational missions of the various cosmic frequency backgrounds of the universe from the known microwave and infrared ones to the emergent cosmic neutrino and gravitational backgrounds; this is because in such cases this novel class of quantifiers may be used to detect and quantify the inherent anisotropies because of the sensitivity of its generalized-Fisher-information factor which captures the density fluctuations in a multi-faceted manner. Moreover, the application of this class of complexity quantifiers to fractal phenomena as well as to the non-linear blackbody radiation laws <cit.>, which presumably takes into account the small deviations from the Planck radiation formula that have been recently detected in the cosmic microwave radiation, are two relevant problems which deserve to be explored. § ACKNOWLEDGMENTSThis work was partially supported by the grants FQM-7276 and FQM-207 of the Junta de Andalucía and the MINECO-FEDER (Ministerio de Economía y Competitividad, and the European Regional Development Fund) grants FIS2014-54497P and FIS2014-59311P. The work of I. V. Toranzo was financed by the program FPU of the Spanish Ministerio de Educación. § REFERENCES badiiBadii R, Politi A 1997Complexity: Hierarchical Structure and Scaling in Physics (Henry Holt, New York) sen2012 Sen K D(ed.) 2012Statistical Complexity. 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Publ., Dordrecht) dong DongS H 2011 Wave Equations in Higher Dimensions (Springer, Berlin) weinberg Weinberg S. 1986 Physics in Higher Dimensions (World Scientific, Singapore) dehesa1 DehesaJ S, López-Rosa S and Manzano D 2012 Entropy and complexity analyses of D-dimensional quantum systems (In K.D. Sen (ed.) Statistical Complexity, Springer, Berlin) olverOlver F W J, Lozier D W, Boisvert R Fand Clark C W 2010 NIST Handbook of Mathematical Functions (Cambridge University Press, Cambridge) toranzo14 Toranzo I V and Dehesa J S 2014 Eur. Phys. J. D 68 316 lutwak Lutwak E, Yang D and Zhang G 2005 IEEE Trans. Inf. Theor. 51 473 renyi_70 Rényi A 1970 Probability Theory (North Holland, Amsterdam) borges_physABorges E P 2004 Physica A 340 95 david_bbd Puertas-Centeno D, Toranzo I V and Dehesa J S 2017 Preprint barnes Barnes E W 1899 Quart. J. Math. 31 264 plastinoMartínez S, Pennini F, Plastino A and Tessone C J 2002 Physica A 309 85 tsallis Tsallis C, Sa Barreto F C and Loh E D 1995 Phys. Rev. E 52 1447 valluri Valluri S R, Gil M, Jeffrey D J and Basu S 2009 J. Math. Phys. 50 102 | http://arxiv.org/abs/1704.08452v1 | {
"authors": [
"D. Puertas-Centeno",
"I. V. Toranzo",
"J. S. Dehesa"
],
"categories": [
"quant-ph"
],
"primary_category": "quant-ph",
"published": "20170427071359",
"title": "The biparametric Fisher-Rényi complexity measure and its application to the multidimensional blackbody radiation"
} |
Smoothed nonparametric two-sample testsTaku MORIYAMA Yoshihiko MAESONO ======================================= We propose new smoothed median and the Wilcoxon's rank sum test. As is pointed out by Maesono et al.<cit.>, some nonparametric discrete tests have a problem with their significance probability. Because of this problem, the selection of the median and the Wilcoxon's test can be biased too, however, we show new smoothed tests are free from the problem. Significance probabilities and local asymptotic powers of the new tests are studied, and we show that they inherit good properties of the discrete tests. Keywords: median test Wilcoxon's rank sum test kernel estimator significance probability Let X_1,X_2,⋯,X_m be independently and identically distributed random variables (i.i.d.) from distribution function F(x) and Y_1,Y_2,⋯,Y_n be i.i.d. from F(x-θ) where θ is unknown location parameter. We assume that m+n =N and λ_N := m/N →λ s.t. 0< λ <1. We consider `2-sample problem' whose null hypothesis H_0 : θ=0 and alternative H_1 : θ >0. There are many nonparametric tests based on linear order statistic (see Hájek et al.<cit.>). The median and the Wilcoxon's rank sum test are widely used and investigated well among them. Moreover, the median test is known for its low cost especially in survival analysis because it only needs the number of the lower values than the combined sample median. However, Freidlin & Gastwirth<cit.> have reported that the median test is numerically inferior even in double exponential case in spite of its theoretical most powerfulness. We first investigate the median's power again and find theoretical and numerical superiority when the distribution has heavy tail.As is shown by Maesono et al.<cit.>, the sign test and the Wilcoxon's signed rank test have a problem of their significance probabilities, and it comes from their discreteness of p-values. Because of this, the discrete tests can let an distorted statistical decision, and we show that the median and the Wilcoxon's rank sum test are also the same. Using a smooth function in a similar way as Maesono et al.<cit.>, we propose the smoothed test statistics of them, and the smoothed median test also needs only the lower values than the combined median. It is proved that their Pitman's A.R.E. are respectively same, and they are asymptotically nonparametric. § THE MEDIAN AND THE WILCOXON'S RANK SUM TEST§.§ The median and the Wilcoxon's rank sum testIn this section, we introduce the median and the Wilcoxon's rank sum test statistic and their properties. Some numerical experiments on their power are studied and the problem about their p-values is brought up here.Let us define that ψ(x)=1 (x≥0), =0 (x<0) and Z_1=X_1, ⋯, Z_m=X_m, ⋯, Z_m+1=Y_1, ⋯, Z_N=Y_n. There are various forms of the median test statistic, and here we defineM=M( X, Y)=∑_j=1^nψ(Y_j -Z)where X=(X_1,X_2,⋯,X_m)^T and Z denotes the sample median of {Z_1,Z_2,⋯,Z_N}. We put Z=Z_((N+1)/2) if N is odd and Z=(Z_(N/2) +Z_((N/2)+1))/2 else where Z_(1) < ⋯ < Z_(N) are the order statistics. The Wilcoxon's rank sum test statistic is given byW_2=W_2( X, Y)=∑_1≤ i≤ m∑_1≤ j≤ nψ(Y_j-X_i).For observed values x=(x_1,x_2,⋯,x_m)^T and y=(y_1,y_2,⋯,y_n)^T, we put m=m( x, y) and w_2=w_2( x, y) which are the realized values of M and W_2.If the p-valueP_0(M≥ m) orP_0(W_2≥ w_2)is small enough, we reject the null hypothesis H_0.Here P_0(·) denotes a probability under the null hypothesis H_0.As we can see easily, the median test only needs the number of the lower values than the combined median, and we can finish the observation when the median is obtained. If the data size is large or the tail of the distribution is heavy, it is possible to save much cost and time. However, not many statisticians take the great merit into account. We show the smoothed median test inherits the superiority in Section 3.1.§.§ Local power The median and the Wilcoxon's rank sum test are one of linear rank tests which are defined as followsS=S( X, Y) =∑_j=1^n a_m,n(R_j)where R_j (j=1,⋯,n) denotes rank of the observation Y_j in the combined sample Z_1, ⋯, Z_N. The Wilcoxon's test is given by a_m,n(u) = m+ 1/2(n+1) -u,so the test statistic isW_2=W_2( X, Y)= mn+ 1/2n(n+1) -∑_j=1^n R_j.The last termU=∑_j=1^n R_jis led as locally most powerful rank test in logistic case, and W_2 is equivalent to the test statistic U (Hájek et al. <cit.>). The median test statistic is obtained bya_m,n(u) = ψ(u- 1/2(m+n+1)),and soM=M( X, Y)=∑_j=1^n ψ(R_j - 1/2(m+n+1)).M is asymptotically equivalent to the locally most powerful rank test statistic when the underlying distribution is the double exponential.Table 1 shows Pitman's asymptotic relative efficiencies (A.R.E.) of the combinations of the two-sample t-test T_2, the median and the Wilcoxon's test. Pitman's A.R.E.s are given by a ratio of the values related to asymptotic local power of the two tests. We confirm that T_2 is the most powerful in normal case, M is in double exponential case and so on. Note that ARE(M|W_2) = 1.33 in the double exponential case.Nevertheless, the numerical weakness of the median test's power was reported. Freidlin & Gastwirth <cit.> shows the empirical local power of some two-sample linear rank tests, and in their tables, the median test is inferior to the Wilcoxon's test even in the double exponential case.We investigate their powers both theoretically and numerically in heavy tailed case. Table 2 shows Pitman's ARE(M|W_2) of T(2) (T distribution with 2 degrees of freedom), T(1) (Cauchy distribution) and T(1/2). Distribution's tail gets heavy in accordance with decrease in the degree, and we can prove the monotonic increase of ARE(M|W_2). Table 3 and 4 are the numerical results of the ratio of their empirical local power in the t distributions. We find that the power of the median test is rather stronger especially in T(1/2). If to obtain complete data needs much cost and time in such case, the median test has great superiority.§.§ Significance probability We will show that the median test and the Wilcoxon's rank sum test have a problem with their significance probabilities. Maesono et al.<cit.> reports that the sign and the Wilcoxon's signed rank test can make `distorted' statistical results, and this problem comes from their p-values' discreteness. The median and the Wilcoxon's test are also discrete, and we study their significance probabilities. Table 5 shows the ratio of frequency of exact p-value of W_2 smaller than that of M in the following tale area Ω_αΩ_α={ x∈ R^n | m( x)-E_0(M)/√(V_0(M)) > v_1-α,. orw_2( x)-E_0(W_2)/√(V_0(W_2)) > v_1-α} where v_1-α is a (1-α)th quantile of the standard normal distribution N(0,1), and E_0(·) and V_0(·) stand for an expectation and a variance under H_0, respectively.We count samples that an exact p-value of the test is smaller than the other in Ω_α, and calculate the ratio of the frequency. [1]U_m^* and U_n^* are random numbers from the discrete uniform distribution U^*(5,40)Because the values in Table 5 are larger than 1, we find that W_2 tends to have smaller p-value than M, and they can let us to use W_2 if we wants the small p-value. This comes from that the possible p-values of M are more sparse than W_2 like the sign and Wilcoxon's signed rank test as is discussed by Maesono et al.<cit.>.In order to conquer the problem, we propose the smoothed median test M and theWilcoxon's test W_2. The discrete tests are distribution-free, but the smoothed tests are not. However, we can confirm that they are asymptotically distribution-free and their Pitman's efficiencies of them are the same.§ SMOOTHED MEDIAN TEST§.§ Smoothed median test Hereafter, we assume that N is odd for the brevity, and we consider to make M smooth appropriately. A possible way is to define them as kernel type statistics, and we introduce a kernel distribution estimator first. The empirical distribution function of a population distribution function F is given byF_n(x) = 1/n∑_i=1^nψ(x -X_i).Let k(u) be a kernel function which satisfies∫_-∞^∞k(t)dt=1,and K(t) is an integral of k(t) s.t.K(t)=∫_-∞^tk(u)du.In this paper, we assume that the kernel k is a symmetric function around the origin. The kernel distribution estimator of F is given byF(x)=1/n∑_i=1^n K(x-X_i/h)where h is a bandwidth which satisfies h→ 0 (n →∞).The median test statistic is given byM = n-m+1/2 -∑_i=1^mψ^*(Z -X_i)where ψ^*(x)=1 (x>0), =0 (x≤0), and we put M^† as the second term. Applying the kernel smoothing to M^†, we define the following smoothed test statisticM=∑_i=1^m K^*(Z -X_i/h)where K^* is an integral of the kernel k^*(t) which satisfies∫_0^∞k^*(t)dt=1 and k^*(t)=0 for t ≤ 0and h is a bandwidth. In addition, we assume that A_1,1^*=0 whereA_i,j^* = ∫_-∞^∞ t^i k^*^j(t) dt.As we see easily from the definition, the smoothed median test does not need the values of the data larger than the combined median. The above condition of k^* is not tough, and we can easilyconstruct the following simple polynomial type onek^*(t)=(-6t+4) I(0<u<1).We can easily construct a kernel function which satisfies A_1,1^*=A_2,1^*=0 too. It is not a problem that k^* may take negative value, because M is test statistic. §.§ Asymptotic properties The median test statistic M^† exactly follows the hypergeometric distribution HG(N,m,(N-1)/2) under H_0, and so we easily find thatE_0[M^†] = m/2( 1-1/m+n), V_0[M^†] =mn/4(m+n).Using the following joint distribution of Z and U (∈{1,⋯,r}) which is the number of {X_1, ⋯, X_m} less than Zh(u,z) =m m-1unr-uF^u(z)[1-F(z)]^m-u-1F^r-u(z- θ)[1-F(z-θ)]^n-r+u f(z)+ n mun-1r-uF^u(z)[1-F(z)]^m-uF^r-u(z- θ)[1-F(z-θ)]^n-r+u-1 f(z-θ)under H_1, Mood <cit.> obtains the following asymptotic expectationE_θ[M^†]= mF(z_θ,N) + o(N)where z_θ,N stands for the median of the distribution G_θ,N defined asG_θ,N(x)= λ_N F(x) +(1-λ_N) F(x-θ).Then we find the following Pitman efficiency is given bye_P[M^†] = lim_N[(N V_0[M^†])^-1/2∂/∂θ E_θ[M^†] |_θ=0]= 2√(λ(1-λ)) f(z_0)where z_0 is the median of the population distribution of F.Now, we clarify the difference of the asymptotics between the odd and even. If N is odd, the exact distribution of Z is given byF_Z^O(z) = 1/β((N+1)/2,(N+1)/2)∫_-∞^z [F(x)]^(N-1)/2 [1-F(x)]^(N-1)/2 f(x) dxwhere β(·,·) is the beta function. For an even number, the exact distribution is (Desu & Rodine <cit.>)F_Z^E(z) = 2/β(N/2,N/2)∫_-∞^z [F(x)]^(N/2)-1{ [1-F(x)]^(N/2) - [1-F(2z -x)]^(N/2)}f(x) dx.From the above, we have the following lemma.The difference of the distribution of Z between N=N and N=N+1 is| F_Z^O(z) -F_Z^E(z) | = o(N^-2) (negligible). Proof. The direct computation gives the result. See Appendices.Hereafter, we denote the l-th derivative of f by f^(l). About the Pitman's efficiency, we obtain the following result.Let us assume that f^(1) exists and is continuous at a neighborhood of both z_0 and z_θ,N, and both f(z_0) and f(z_θ,N) are positive. In addition, we assume that h=o(N^-1/2) or that A_1,1^*=0 and h=o(N^-1/4). Then, we havee_P[M] = e_P[M^†]. Proof. See Appendices.Since the main term of M is a two-sample U statistic, it is easy to prove the following asymptotic normality. Let us assume that f^(1) exists and is continuous at a neighborhood of z_0, and f(z_0) >0. If h=o(N^-1/2), or A_1,1^*=0 and h=o(N^-1/4) holds, we havesup_-∞ < x < ∞| P_0 [ V_1^-1/2 (M-E_1) < x] -Φ(x) | = o(1)whereE_1=m/2(1- 1/m+n), V_1=mn/4(m+n),Φ is the standard normal distribution function and ϵ is any positive number.Proof. The evaluation of the difference between M and M^† is the main, and the detail is inAppendices.Note that the main terms of the asymptotic expectation and variance of M do not depend on F under H_0.Next, we study the local power of M, and obtain the following result. Under the same assumptions of Theorem 2.3 and that √(N) h →∞, if A_1,1^*=0 and A_1,1,1^* is positive, we havelim_N →∞1/h (LP_ξ/√(N), α[M] - LP_ξ/√(N), α[M^†]) > 0.whereLP_ξ/√(N), α[M] = P_ξ/√(N)[ V_0[M]^-1/2(M -E_0[M]) > v_1-α]and LP_ξ/√(N), α[M^†] is also the same.Proof. See Appendices. We can construct the following polynomial type kernelk^*(t)=[( ± 3√(4353) -3/17) u^2+ ( ∓ 3√(4353) -99/17) u + (±√(4353) +135/34)] I(0<u<1).which satisfies A_1,1^*=0 and A_1,1,1^*=1 (>0). Similarly, we have the following exponential typek^*(t) = [e^-t + (613 -2√(207586)/58)*(2e^-2t)+(3√(207586) -1137/58)*(3e^-3t) +(524 -√(207586)/58)*(4e^-4t)]which satisfies A_1,1^*=0 and A_1,1,1^*=1 (>0). We want to use a kernel whose value of A_1,1,1^* is larger, however, in practice, the estimated value of V_1^⋆ which we need can be negative. § SMOOTHED WILCOXON'S RANK SUM TEST§.§ Smoothed Wilcoxon's rank sum test Here, we give the smoothed test W_2 in the same manner. In the same way, we can define the smoothed test statistics of W_2W_2=∑_i=1^m∑_j=1^n K(Y_j-X_i/h). The following moments of the Wilcoxon's test statistic W_2 are easy to obtainE_θ[W_2] = mn ∫_-∞^∞ f(y) F(y+θ) dy, V_0[W_2] =mn(m+n)/12,and then we find the following Pitman efficiencye_P[W_2] = √( 12λ(1-λ))∫ f^2(x) dx. Using variable changes and the Taylor expansion, we obtain the following asymptotic expectation of W_2E_θ[W_2]=mn ∫_-∞^∞∫_-∞^∞ K(x -y/h) f(x-θ) f(y)dx dy=mn ∫_-∞^∞∫_-∞^∞ k(v) f(y +hv -θ) F(y)dv dy=mn [∫_-∞^∞ f(y) F(y+θ) dy +O(h^2) ]. Hereafter, we assume that h = o(N^-1/2) or that A_1,1=⋯=A_3,1=0 and h = o(N^-1/4) in order to ignore the residual term. It is easy to obtain such kernels using Jones & Signorini <cit.>.Further, it is easy to see that V_0[W_2]= mn(m+n)/12 + o(N^3). Combining the above results, we can obtain the following result. Let us assume that f^(1) exists and is continuous at a neighborhood of both 0 and θ. Then, we havee_P[W_2] = e_P[W_2]. Proof. See Appendices.Using the asymptotic results of two-sample U-statistics, we can obtain the following error bound of the normal approximation. Let us assume that f^(1) exists and is continuous around a neighborhood of 0 and h = o(N^-1/2), or that such f^(3) exists, A_2,1=0 and h = o(N^-1/4). Then, we havesup_-∞ < x < ∞|P_0 [ V_2^-1/2 (W_2 -E_2) < x] -Φ(x) | = o(N^-1/2)whereE_2=mn/2, V_2=mn(m+n)/12. Proof. We need the theory of edgeworth expansion of thetwo-sample U-statistics. See Appendices.In the same manner as the local power of M, we can obtain the power of W_2, and find the following result.Under the assumptions of Theorem 3.1, we havelim_N →∞1/h (LP_ξ/√(N), α[W_2] - LP_ξ/√(N), α[W_2]) = 0whereLP_ξ/√(N), α[W_2] = P_ξ/√(N)[ V_0[W_2]^-1/2(W_2 -E_0[W_2]) > v_1-α]and LP_ξ/√(N), α[W_2] is also the same.Proof. See Appendices.§ SIMULATION STUDYIn this section, we compare the significance probabilities of M and W_2 by simulation because the distributions of them depend on F. For 100,000 times random samples from the standard normal distribution, we estimate the significance probabilities in the tale areaΩ_α={ x∈ R^n | m( x)-E_1/√(V_1) > v_1-α,. orw_2( x)-E_2/√(V_2) > v_1-α}.Similarly as Table 5, Table 6 shows the ratio of the samples that the significance probability of W_2 is smaller than M. Here we use the exponential type kernel as introduced in Remark 2.6 in M, and the Epanechnikov kernel in W_2. The both bandwidths are same, and h=N^-1/4/ log N as we will explain later. Comparing Table 5 and 6, we can see that the differences of the p-values of M and W_2 is smaller than those of M and W_2. [2]U_m^* and U_n^* are random numbers from the discrete uniform distribution U^*(5,40)Next we will compare the powers of the smoothed sign, the smoothed Wilcoxon's and the two-sample T-test. The Pitman's efficiency of the T-test is given by e_P[T_2] = √(λ(1-λ)/V_0[X_1]). In Table 7 and Table 8, we also use the exponential type kernel, the Epanechnikov kernel and h=N^-1/4/ log N which attains almost maximum order under the assumptions, to focus on the first orders of their local power. We simulate their power when θ= 0.1, 0.5 and the significance level α=0.01, 0.05, based on 100,000 repetitions.In order to check the size condition, we also simulate the case θ=0 in Table 9. When the underlying distribution F is the heavy-tailed T-distributions, the simulation results show that the smoothed median test is superior than the other tests.When the underlying distribution F is the logistic, the simulation results show that the smoothed Wilcoxon's test is best. The student t-test is superior than others, when F is normal. These simulation studies coincide with the Pitman's A.R.E.s, and justify the continuation of the ordinal tests. Also all of the empirical sizes of the proposed smoothed tests is close to those of the significance levels, and we can conclude they are asymptotically nonparametric. One possible way of choosing their bandwidths is to reduce the errors of the approximations. It is especially important to `fix' the type I error in statistical testing, however there are no tradeoffs about the bandwidths in their normal approximations. Although the best h is as small as possible from the asymptotic result, `smoothed' bootstrap method can give a numerical solution. Under the assumptions, there are no same values, and the appropriately smoothed resampling never returns `ties' similarly. If we choose the squared loss at the significance level α, using L repetition, the best bandwidth of M is given by h = min_h>0( ♯{ l : V_1^-1/2(M_⋆^(l) -E_1) > v_1-α} - Lα)^2where M_⋆^(l) is the lth test statistic given by {X_⋆,1^(l), ⋯, X_⋆,m^(l), X_⋆,m+1^(l), ⋯, X_⋆,m+n^(l)} and X_⋆,1^(l), ⋯, X_⋆,m+n^(l) is i.i.d. sample from smooth F estimated by the original {X_1, ⋯, X_m}. The minimizer is not decided uniquely because the function is discrete, so we choose the middle. If L is enough large, the effect of choosing the middle is negligible. As we confirmed in Table 9, the normal approximation of p-values of W_2 is good enough, but that of M is not. To see improvement of the size condition of M, we do the numerical study, lastly. Using the standard normal density as the kernel function and cross-validated bandwidth, we obtain the resampled data, numerically optimal bandwidth and the results in Table 10 based on 100,000 repetitions. The bandwidth is calculated every time, using 1,000 resampled data sets. As we can see, the size conditions are improved and the numerical powers are also the same in these cases.The kernel function does not affect much both theoretical and numerical result as same as the kernel density estimation under the assumption. The better choice of the bandwidths should be discussed more, but we postpone it for future work. gSTA § APPENDICES Proof of Lemma 2.2By the direct computation, we can findF_Z^O(z) -F_Z^E(z) = 1/β(N/2,N/2)∫_-∞^z [F(x)]^(N/2)-1 f(x) {β(N/2,N/2)/β((N+1)/2,(N+1)/2) (√(F(x)) [1-F(x)]^(N-1)/2) -2( [1-F(x)]^(N/2)[1-F(2z -x)]^(N/2)) } dx.Sincelim_N →∞Γ(N + η)/Γ(N) N^η =1 (η∈𝐑),we can obtainlim_N →∞β(N/2,N/2)/β((N+1)/2,(N+1)/2) =1.When [1-F(x)]^N↛0, it is easily to see F^N(x) → 0 and vice versa. If both fails, both [F(x)]^N and [1-F(x)]^N shrinks to 0. From the above, we can see the result. Proof of Theorem 2.3 Hereafter, we assume that h = o(N^-1/2) or that A_1,1^*=A_1,2^*=0 and h = o(N^-1/3), to ignore the residual term.We utilize the following Bahadur representation (Bahadur <cit.>) of the combined median Z under H_0Z = z_0 + 1/f(z_0)[1/2 -F_(N)(z_0) ] + R_Nwhere z_0 is the median of F, F_(N) stands for the empirical distribution function of {Z_1, ⋯ ,Z_N} and R_N is the residual which satisfies R_N=o_P(N^-3/4log N). By the following moment evaluation of the residual term of Z (Reiss <cit.>)N^l/2E[(Z-z_0)^l] = (p(1-p))^l/2κ_l/f^l(z_0) + O(N^-1/2)andE[Z-z_0] = -f^(1)(z_0)/f^3(z_0) E[(B-z_0)^2] + o(N^-1)where κ_l is the l-th moment of the standard normal distribution and B stands for the beta distribution Beta(N/2,N/2), we findE[R_N] = -f^(1)(z_0)/4N f^3(z_0) + O(N^-2)andE[R_N^2]= o(N^-1). Hereafter we put N is odd. One of {X_i}_i=1, ⋯ m may be Z but we don't need the differentiability of K^*. This is because the probability of Z-X_i takes any finite points (≠ 0) is 0, and to smooth K^* near 0 and 1 appropriately is possible. Using the representation, we can see the following asymptotic expansion under H_0M = ∑_i=1^m [ K^*(z_0 -X_i/h) +1/hk^*(z_0 -X_i/h) (Z -z_0) + 1/h^2k^*^(1)(z_0 -X_i/h) (Z -z_0)^2 ] + o_P(1) = ∑_i=1^m [ K^*(z_0 -X_i/h) +1/hk^*(z_0 -X_i/h) [ 1/f(z_0)( 1/2 -F_(N)(z_0) ) + R_N ] + 1/f^2(z_0) h^2k^*^(1)(z_0 -X_i/h) [1/2 -F_(N)(z_0) ]^2 ] + o_P(1)where k^*^(1) is the derivative of k^*, F_(N) is the empirical distribution function of {Z_1, ⋯, Z_N} and R_N is a residual which satisfies R_N = o_P(N^-3/4log N).Using conditional expectation given X_j and the Taylor expansion, we obtainE_0[M]=m ∫_-∞^z_0[K^*(z_0 -x/h) +1/hk^*(z_0 -x/h) {1/f(z_0) N(1/2 -ψ(z_0-x) ) + E[R_N]}]f(x)dx + m/f^2(z_0) N h^2 E[ k^*^(1)(z_0 -X_i/h) (1/2 -ψ(X_j-z_0) )^2 ]_i ≠ j + o(1 +log N (N^1/4h^2 +N^-1/4 h^-1/2))=m ∫_0^∞ k^*(v) [F(z_0 -hv)+ f(z_0 -hv) {-1/2f(z_0) N -f^(1)(z_0)/4N f^3(z_0)} + m/4N f^2(z_0) f^(1)(z_0 -hv)] dv+ o(1 +(log N) (N^1/4h^2 +N^-1/4 h^-1/2))= m/2[1 + h^2 A_2,1^* f^(1)(z_0) - 1/m+n]+ o(1 + (log N) N^-1/4 h^-1/2). Using the stochastic expansion and the above results, we haveM^2= [ ∑_i=1^m { K^*(z_0 -X_i/h) +1/hk^*(z_0 -X_i/h) [ 1/f(z_0)( 1/2 -F_(N)(z_0) ) + R_N ] + 1/f^2(z_0) h^2k^*^(1)(z_0 -X_i/h)[1/2 -F_(N)(z_0) ]^2 } + O_P(√(N) R_N + N R_N^2 +N^-1/2) ]^2. From the Bahadur representation, we can obtainE [ ∑_i=1^m K^*(z_0 -X_i/h) O_P(√(N) R_N + N R_N^2 +N^-1/2) ] =O(N^1/2log N),so the expectation of the squared value is given byE_0[M^2] = ∑_i=1^m ∑_j=1^m ( E[ K^*(z_0 -X_i/h) K^*(z_0 -X_j/h) ]+2/h E[K^*(z_0 -X_i/h) k^*(z_0 -X_j/h) ( 1/f(z_0) N (1 -ψ(z_0-X_i) -ψ(z_0-X_j)) +R_N ) ]+ 1/f^2(z_0) (Nh)^2∑_k=1^N ∑_l=1^N { E[K^*(z_0 -X_i/h) k^*^(1)(z_0 -X_j/h) {1/2 -ψ(z_0-Z_k)}{1/2 -ψ(z_0-Z_l)}] + 2 E[k^*(z_0 -X_i/h) k^*(z_0 -X_j/h) {1/2 -ψ(z_0-Z_k)}{1/2 -ψ(z_0-Z_l)}] }) + O(N^1/2log N). Calculating the expectations, we have the variance as followsV_0[M]=E_0[M^2] - E_0[M]^2 =(m^2 -m) E[ K^*(z_0 -X_i/h) ]^2 +m E[ K^*^2(z_0 -X_i/h) ] +2 m^2/f(z_0) Nh E[K^*(z_0 -X_i/h) k^*(z_0 -X_j/h) { 1 -ψ(z_0-X_i) -ψ(z_0-X_j) }]_i ≠ j +2 m^2/h E[K^*(z_0 -X_i/h) k^*(z_0 -X_j/h) R_N ]_i ≠ j + 2 m^2 N/f^2(z_0) (Nh)^2 E[K^*(z_0 -X_i/h) k^*^(1)(z_0 -X_j/h) {1/2 -ψ(z_0-Z_k)}^2 ]_i ≠ j + m^2 N/f^2(z_0) (Nh)^2 E[k^*(z_0 -X_i/h) k^*(z_0 -X_j/h) {1/2 -ψ(z_0-Z_k)}^2 ]_i ≠ j -m^2/4[1 +h^2 A_2,1^* f^(1)(z_0) - 1/m+n]^2+ O(Nh^2 +N^1/2log N) = m/4 +m^2/2N -m^2/N -m^2 f^(1)(z_0)/4 Nf^2(z_0) +m^2 f^(1)(z_0)/4 Nf^2(z_0) +m^2/4N + O(Nh +N^1/2log N) = m/4- m^2/4N + O(Nh +N^1/2log N) = mn/4(m+n) + O(Nh +N^1/2log N). Using the Bahadur representation for quantiles of two samples(Liu & Yin <cit.>) under H_1, and both f(z_θ,N) and f(z_θ,N -θ) are positive, we haveM = ∑_i=1^m [ K^*(z_θ,N -X_i/h) +1/g_θ,N(z_θ,N) hk^*(z_θ,N -X_i/h) {1/2 -G_θ,(N)(z_θ,N)}]+ o_P(N^1/2)whereG_θ,(N)(x) = λ_N F_X,(m)(x) +(1-λ_N) F_Y,(n)(x)and F_X,(m), F_Y,(n) are the empirical distribution function of {X_1, ⋯ ,X_m} and {Y_1, ⋯ ,Y_n} respectively. Therefore, we see the following asymptotic expectation under H_1E_θ[M]=m ∫_-∞^∞ k^*(v) F(z_θ,N -hv)dy+ o(N^1/2)=m F(z_θ,N) +o(N^1/2).Combining the above results, we can prove that the Pitman efficiency is same as the discrete one.Proof of Theorem 2.4Using the result of Lahiri & Chatterjee <cit.>, we have the following Berry-Esseen boundsup_-∞ < x < ∞|P_0 [ V_3^-1/2 (M^† -E_3) < x] -Φ(x) | = O (1/√(N))where E_3 and V_3 are the expectation and the variance of M^†. From the calculation of the proof of Theorem 2.3, we haveE_0[ D^2 ] = O( ∑_i,j=1 (A_i,1^* h^i + A_j,1,1^* h^j) )where D=V_1^-1/2 (M -E_0[M]) -V_3^-1/2 (M^† -E_3). Therefore, we can obtain D = O_P(√(h)) and P[|D|>N^-ϵ h^1/4] = N^ϵ h^1/4. Combiningsup_|t|<N^-ϵ h^1/2, -∞ < x < ∞| Φ(x+t)-Φ(x) | = O (N^-ϵ h^1/4)and the above result, we get sup_-∞ < x < ∞|P_0 [ V_1^-1/2 (M -E_1) < x] -Φ(x) | = sup_-∞ < x < ∞|P_0 [ {V_3^-1/2 (M^† -E_3) + D + O(√(N)∑_i=1 A_i,1^* h^i)} < x] -Φ(x) | =O (∑_i=1(√(N) A_i,1^* h^i) + N^ϵ h^1/4 +1/√(N)).This completes the proof.Proof of Theorem 2.5Put θ = N^-1/2ξ and z_θ,N = z_0 + N^-1/2η + o(N^-1/2 h) where η is a constant number and z_0 is defined before. Since the following expansion holdsN/2 =m F(z_θ,N) + n F(z_θ,N -θ)=N F(z_θ,N) - n/√(N) f(z_θ,N) ξ + O(N^-1),we have√(N) f(z_0) η = n/√(N) f(z_0) ξ + o(N^1/2 h).Then, we find z_θ,N = z_0 + N^-1/2 (1- λ) ξ + o(N^-1/2 h).CombiningV_0[M] = mn/4(m+n) - 2(mh) A_1,1,1^* f(z_0) + O(Nh^2 +log N/√(N))where A_i,j,l^* = ∫ t^i k^*^j(t) K^*^l(t) dt, andE_θ[M] -E_0[M]=m F(z_θ,N) - m/2 + O(N h^2)= λ (1 -λ) √(N) f(z_0) ξ + o(N^1/2 h),we can find that the local power of M is given byLP_ξ/√(N), α[M] = P_ξ/√(N)[ V_1^⋆^-1/2(M -E_1) > v_1-α] = 1- Φ( v_1-α -c) + O(N^-1/2 + h^2)whereV_1^⋆ = V_1 - 2(mh) A_1,1,1^* f(z_0), c = ξ[ 2√(λ (1- λ)) f(z_0) + 8h √(λ/1 -λ) A_1,1,1^* f^2(z_0) ]and v_1-α is the (1-α)th quantile of the standard normal distribution. M^† doesn't have the second term of c of order h, so we can find the result. Proof of Theorem 3.1, 3.2 and 3.3We can easily see that the main term of the variance vanishes, andV_θ[W_2]= ∑_i=1^m ∑_k=1^m ∑_j=1^n ∑_l=1^n E[ K(X_i-Y_j/h) K(X_k-Y_l/h) ] - [mn ∫_-∞^∞ f(y) F(y-θ) dy ]^2 +o(N^3)=m^2 n Cov_θ( K(X_i-Y_j/h), K(X_k-Y_j/h) ) + m n^2 Cov_θ( K(X_i-Y_j/h), K(X_i-Y_l/h) ) +o(N^3). By the direct calculation, we haveE_θ[ K(X_i-Y_j/h) K(X_i-Y_l/h) ] = ∫⋯∫ K(x -y/h) K(x -w/h) f(x) f(y-θ) f(w-θ)dx dy dw = ∫_-∞^∞f(x) F(x-θ) F(x-θ)dx + O(h^2)andE_θ[ K(X_i-Y_j/h) K(X_k-Y_j/h) ] = ∫⋯∫ K(x -y/h) K(u -y/h) f(x) f(u) f(y-θ) dx du dy = ∫_-∞^∞{1-F(y)}^2 f(y-θ) dy + O(h^2). Now, we can easily obtain the following equationV_θ[W_2]=m^2 n ( ∫_-∞^∞ f(x) F(x-θ) F(x-θ)dx - [ ∫_-∞^∞ f(y) F(y-θ) dy ]^2 )+ m n^2 ( ∫_-∞^∞{1-F(y)}^2 f(y-θ) dy- [∫_-∞^∞ f(y) F(y-θ) dy ]^2 ) +o(N^3). From the above, Theorem 3.1 is proved. Next, using the result of Maesono <cit.> and García-Soidán et al <cit.>, we can obtainsup_-∞ < x < ∞|P_0 [ V_2^-1/2 (W_2 -E_2) < x] -Q_m,n(x) | = o(N^-1/2)whereQ_m,n(x)= Φ(x) -ϕ(x) x^2 -1/6 τ_m,n^3[1/m^2E_0[g_1,0^3 (X)] + 1/n^2 E_0[g_0,1^3 (Y)] + 6/mn E_0[g_1,0 (X) g_0,1 (Y) g_1,1 (X,Y)] ], g_1,0(X) = ∫ k(v) F(X +hv) dv -1/2 + O(C(h)), g_0,1(Y) = ∫ k(v) F(Y -hv) dv -1/2 + O(C(h)), g_1,1(X,Y) = W(Y-X/h) -g_1,0(X) -g_0,1(Y) -1/2 +O(C(h))and C(h) = ∑_i=1 A_i,1 h^i. By the direct computation, we can get the followingsτ_m,n = m+n/12mn +O(C(h)), E_0[g_1,0^3 (X)] = O(C(h)), E_0[g_0,1^3 (Y)] = O(C(h)), E_0[g_1,0 (X) g_0,1 (Y) g_1,1 (X,Y)] = O(C(h)). Then, we complete the proof of Theorem 3.2. In the similar manner of the proof of M, we can easily obtain the Theorem 3.3. | http://arxiv.org/abs/1704.07977v1 | {
"authors": [
"Taku Moriyama",
"Yoshihiko Maesono"
],
"categories": [
"math.ST",
"stat.TH"
],
"primary_category": "math.ST",
"published": "20170426053618",
"title": "Smoothed nonparametric two-sample tests"
} |
=1 | http://arxiv.org/abs/1704.08721v1 | {
"authors": [
"A. Caputo",
"P. Hernandez",
"J. Lopez-Pavon",
"J. Salvado"
],
"categories": [
"hep-ph"
],
"primary_category": "hep-ph",
"published": "20170427191614",
"title": "The seesaw portal in testable models of neutrino masses"
} |
Brow-serID in-fra-struc-ture brow-ser doc-u-ment Chro-mi-um meth-od sec-ond-ary Java-Script Mo-zil-la post-Mes-sageThe Web SSO Standard OpenID Connect: In-Depth Formal Security Analysis and Security Guidelines Daniel Fett, Ralf Küsters, and Guido SchmitzUniversity of Stuttgart,Germany Email:December 30, 2023 ===============================================================================================================================Web-based single sign-on (SSO) services such as Google Sign-In and Log In with Paypal are based on the OpenID Connect protocol. This protocol enables so-called relying parties to delegate user authentication to so-called identity providers. OpenID Connect is one of the newest and most widely deployed single sign-on protocols on the web. Despite its importance, it has not received much attention from security researchers so far, and in particular, has not undergone any rigorous security analysis.In this paper, we carry out the first in-depth security analysis of OpenID Connect. To this end, we use a comprehensive generic model of the web to develop a detailed formal model of OpenID Connect. Based on this model, we then precisely formalize and prove central security properties for OpenID Connect, including authentication, authorization, and session integrity properties.In our modeling of OpenID Connect, we employ security measures in order to avoid attacks on OpenID Connect that have been discovered previously and new attack variants that we document for the first time in this paper. Based on these security measures, we propose security guidelines for implementors of OpenID Connect. Our formal analysis demonstrates that these guidelines are in fact effective and sufficient. § INTRODUCTION OpenID Connect is a protocol for delegated authentication in the web: A user can log into a relying party (RP) by authenticating herself at a so-called identity provider (IdP). For example, a user may sign into the website tripadvisor.com using her Google account.Although the names might suggest otherwise, OpenID Connect (or OIDC for short) is not based on the older OpenID protocol. Instead, it builds upon the OAuth 2.0 framework, which defines a protocol for delegated authorization (e.g., a user may grant a third party website access to her resources at Facebook). While OAuth 2.0 was not designed to provide authentication, it has often been used for this purpose as well, leading to several severe security flaws in the past <cit.>.OIDC was created not only to retrofit authentication into OAuth 2.0 by using cryptographically secured tokens and a precisely defined method for user authentication, but also to enable additional important features. For example, using the Discovery extension, RPs can automatically identify the IdP that is responsible for a given identity. With the Dynamic Client Registration extension, RPs do not need a manual set-up process to work with a specific IdP, but can instead register themselves at the IdP on the fly.Created by the OpenID Foundation and standardized only in November 2014, OIDC is already very widely used. Among others, it is used and supported by Google, Amazon, Paypal, Salesforce, Oracle, Microsoft, Symantec, Verizon, Deutsche Telekom, PingIdentity, RSA Security, VMWare, and IBM. Many corporate and end-user single sign-on solutions are based on OIDC, for example, well-known services such as Google Sign-In and Log In with Paypal. Despite its wide use, OpenID Connect has not received much attention from security researchers so far (in contrast to OpenID and OAuth 2.0). In particular, there have been no formal analysis efforts for OpenID Connect until now. In fact, the only previous works on the security of OpenID Connect are a large-scale study of deployments of Google's implementation of OIDC performed by Li and Mitchell <cit.> and aninformal evaluation by Mainka et al. <cit.>.In this work, we aim to fill the gap and formally verify the security of OpenID Connect.Contributions of this Paper. We provide the first in-depth formal security analysis of OpenID Connect. Based on a comprehensive formal web model and strong attacker models, we analyze the security of all flows available in the OIDC standard, including many of the optional features of OIDC and the important Discovery and Dynamic Client Registration extensions. More specifically, our contributions are as follows.Attacks on OIDC and Security Guidelines We first compile an overview of attacks on OIDC, common pitfalls, and their respective mitigations. Most of these attacks were documented before, but we point out new attack variants and aspects.Starting from these attacks and pitfalls, we then derive security guidelines for implementors of OIDC. Our guidelines are backed-up by our formal security analysis, showing that the mitigations that we propose are in fact effective and sufficient. Formal model of OIDC Our formal analysis of OIDC is based on the expressive Dolev-Yao style model of the web infrastructure (FKS model) proposed by Fett, Küsters, and Schmitz <cit.>. This web model is designed independently of a specific web application and closely mimics published (de-facto) standards and specifications for the web, for instance, the HTTP/1.1 and HTML5 standards and associated (proposed) standards. It is the most comprehensive web model to date. Among others, HTTP(S) requests and responses, including several headers, such as cookie, location, referer, authorization, strict transport security (STS), and origin headers, are modeled. The model of web browsers captures the concepts of windows, documents, and iframes, including the complex navigation rules, as well as modern technologies, such as web storage, web messaging (via postMessage), and referrer policies. JavaScript is modeled in an abstract way by so-called scripts which can be sent around and, among others, can create iframes, access other windows, and initiate XMLHttpRequests. Browsers may be corrupted dynamically by the adversary.The FKS model has already been used to analyze the security of the BrowserID single sign-on system <cit.>, the security and privacy of the SPRESSO SSO system <cit.>, and the security of OAuth 2.0 <cit.>, each time uncovering new and severe attacks that have been missed by previous analysis attempts.Using the generic FKS model, we build a formal model of OIDC, closely following the standard. We employ the defenses and mitigations discussed earlier in order to create a model with state-of-the-art security features in place. Our model includes RPs and IdPs that (simultaneously) support all modes of OIDC and can be dynamically corrupted by the adversary. Formalization of security properties Based on this model of OIDC, we formalize four main security properties of OIDC: authentication, authorization, session integrity for authentication, and session integrity for authorization. We also formalize further OIDC specific properties.Proof of Security for OpenID Connect Using the model and the formalized security properties, we then show, by a manual yet detailed proof, that OIDC in fact satisfies the security properties. This is the first proof of security of OIDC. Being based on an expressive and comprehensive formal model of the web, including a strong attacker model, as well as on a modeling of OpenID Connect which closely follows the standard, our security analysis covers a wide range of attacks. Structure of this Paper. We provide an informal description of OIDC in Section <ref>. Attacks and security guidelines are discussed in Section <ref>. In Section <ref>, we briefly recall the FKS model. The model and analysis of OIDC are then presented in Section <ref>. Related work is discussed in Section <ref>. We conclude in Section <ref>. All details of our work, including the proofs, are provided in the appendix.§ OPENID CONNECTThe OpenID Connect protocol allows users to authenticate to RPs using their existing account at an IdP.[Note that the OIDC standard also uses the terms client for RP and OpenID provider (OP) for the IdP. We here use the more common terms RP and IdP.] (Typically, this is an email account at the IdP.) OIDC was defined by the OpenID Foundation in a Core document <cit.> and in extension documents (e.g., <cit.>). Supporting technologies were standardized at the IETF, e.g., <cit.>. (Recall that OpenID Connect is not to be confused with the older OpenID standards, which are very different to OpenID Connect.)Central to OIDC is a cryptographically signed document, the id token. It is created by the user's IdP and serves as a one-time proof of the user's identity to the RP.A high-level overview of OIDC is given in Figure <ref>. First, the user requests to be logged in at some RP and provides her email address oichl-start. RP now retrieves operational information (e.g., some URLs) for the remaining protocol flow (discovery, oichl-discovery) and registers itself at the IdP oichl-registration. The user is then redirected to the IdP, where she authenticates herself oichl-auth (e.g., using a password). The IdP issues an id token to RP oichl-token, which RP can then verify to ensure itself of the user's identity. (The way of how the IdP sends the id token to the RP is subject to the different modes of OIDC, which are described in detail later in this section. In short, the id token is either relayed via the user's browser or it is fetched by the RP from the IdP directly.) The id token includes an identifier for the IdP (the issuer),[The issuer identifier of an IdP is an HTTPS URL without any query or fragment components.] a user identifier (unique at the respective IdP), and is signed by the IdP. The RP uses the issuer identifier and the user identifier to determine the user's identity. Finally, the RP may set a session cookie in the user's browser which allows the user to access the services of RP oichl-set-service-cookie.Before we explain the modes of operation of OIDC, we first present some basic concepts used in OIDC. At the end of this section, we discuss the relationship of OIDC to OAuth 2.0. §.§ Basic Concepts We have seen above that id tokens are essential to OIDC. Also, to allow users to use any IdP to authenticate to any RP, the RP needs to discover some information about the IdP. Additionally, the IdP and the RP need to establish some sort of relationship between each other. The process to establish such a relationship is called registration. Both, discovery and registration, can be either a manual task or a fully automatic process. Further, OIDC allows users to authorize an RP to access user's data at IdP on the user's behalf. All of these concepts are described in the following.§.§.§ Authentication and ID TokensThe goal of OIDC is to authenticate a user to an RP, i.e., the RP gets assured of the identity of the user interacting with the RP. This assurance is based on id tokens. As briefly mentioned before, an id token is a document signed by the IdP. It contains several claims, i.e., information about the user and further meta data. More precisely, an id token contains a user identifier (unique at the respective IdP) and the issuer identifier of the IdP. Both identifiers in combination serve as a global user identifier for authentication. Also, every id token contains an identifier for the RP at the IdP, which is assigned during registration (see below). The id token may also contain a nonce chosen by the RP during the authentication flow as well as an expiration timestamp and a timestamp of the user's authentication at the IdP to prevent replay attacks. Further, an id token may contain information about the particular method of authentication and other claims, such as data about the user and a hash of some data sent outside of the id token.When an RP validates an id token, it checks in particular whether the signature of the token is correct (we will explain below how RP obtains the public key of the IdP), the issuer identifier is the one of the currently used IdP, the id token is issued for this RP, the nonce is the one RP has chosen during this login flow, and the token has not expired yet. If the id token is valid, the RP trusts the claims contained in the id token and is confident in the user's identity.§.§.§ Discovery and RegistrationThe OIDC protocol is heavily based on redirection of the user's browser: An RP redirects the user's browser to some IdP and vice-versa. Hence, both parties, the RP and the IdP, need some information about the respective URLs (so-called endpoints) pointing to each other. Also, the RP needs a public key of the IdP to verify the signature of id tokens. Further, an RP can contact the IdP directly to exchange protocol information. This exchange mayinclude authentication of the RP at the IdP.More specifically, an RP and an IdP need to exchange the following information: (1) a URL where the user can authenticate to the IdP (authorization endpoint), (2) one or more URLs at RP where the user's browser can be redirected to by the IdP after authentication (redirection endpoint), (3) a URL where the RP can contact the IdP in order to retrieve an id token (token endpoint), (4) the issuer identifier of the IdP, (5) the public key of the IdP to verify the id token's signature, (6) an identifier of the RP at IdP (client id), and optionally (7) a secret used by RP to authenticate itself to the token endpoint (client secret). (Recall that client is another term for RP, and in particular does not refer to the browser.)This information can be exchanged manually by the administrator of the RP and the administrator of the IdP, but OIDC also allows one to completely automate the discovery of IdPs <cit.> and dynamically register RPs at an IdP <cit.>.During the automated discovery, the RP first determines which IdP is responsible for the email address provided by the user who wants to log in using the WebFinger protocol <cit.>. As a result, the RP learns the issuer identifier of the IdP and can retrieve the URLs of the authorization endpoint and the token endpoint from the IdP. Furthermore, the RP receives a URL where it can retrieve the public key to verify the signature of the id token (JWKS URI), and a URL where the RP can register itself at the IdP (client registration endpoint).If the RP has not registered itself at this IdP before, it starts the registration ad-hoc at the client registration endpoint: The RP sends its redirection endpoint URLs to the IdP and receives a new client id and (optionally) a client secret in return.§.§.§ Authorization and Access TokensOIDC allows users to authorize RPs to access the user's data stored at IdPs or act on the user's behalf at IdPs. For example, a photo printing service (the RP) might access or manage the user's photos on Google Drive (the IdP). For authorization, the RP receives a so-called access token (besides the id token). Access tokens follow the concept of so-called bearer tokens, i.e., they are used as the only authentication component in requests from an RP to an IdP. In our example, the photo printing service would have to add the access token to each HTTP request to Google Drive.§.§ Modes OIDC defines threemodes: the authorization code mode, the implicit mode, and the hybrid mode. While in the authorization code mode, the id token is retrieved by an RP from an IdP in direct server-to-server communication (back channel), in the implicit mode, the id token is relayed from an IdP to an RP via the user's browser (front channel). The hybrid mode is a combination of both modes and allows id tokens to be exchanged via the front and the back channel at the same time.We now provide a detailed description of all three modes.§.§.§ Authorization Code Mode In this mode, an RP redirects the user's browser to an IdP. At the IdP, the user authenticates and then the IdP issues a so-called authorization code to the RP. The RP now uses this code to obtain an id token from the IdP.Step-by-Step Protocol Flow The protocol flow is depicted in Figure <ref>. First, the user starts the login process by entering her email address[Note that OIDC also allows other types of user ids, such as personal URLs.] in her browser (at some web page of an RP), which sends the email address to the RP in oicacf-start-req.Now, the RP uses the OIDC discovery extension <cit.> to gather information about the IdP: As the first step (in this extension), the RP uses the WebFinger mechanism <cit.> to discover information about which IdP is responsible for this email address. For this discovery, the RP contacts the server of the email domain in oicacf-wf-req (in the figure, the server of the user's email domain is depicted as the same party as the IdP). The result of the WebFinger request in oicacf-wf-resp contains the issuer identifier of the IdP (which is also a URL). With this information, the RP can continue the discovery by requesting the OIDC configuration from the IdP in oicacf-conf-req and oicacf-conf-resp. This configuration contains meta data about the IdP, including all endpoints at the IdP and a URL where the RP can retrieve the public key of the IdP (used to later verify the id token's signature). If the RP does not know this public key yet, the RP retrieves the key (Steps oicacf-jwks-req and oicacf-jwks-resp). This concludes the OIDC discovery in this login flow.Next, if the RP is not registered at the IdP yet, the RP starts the OIDC dynamic client registration extension <cit.>: In Step oicacf-reg-req the RP contacts the IdP and provides its redirect URIs. In return, the IdP issues a client id and (optionally) a client secret to the RP in Step oicacf-reg-resp. This concludes the registration.Now, the core part of the OIDC protocol starts: the RP redirects the user's browser to the IdP in oicacf-start-resp. This redirect contains the information that the authorization code mode is used. Also, this redirect contains the client id of the RP, a redirect URI, and a state value, which serves as a Cross-Site Request Forgery (CSRF) token when the browser is later redirected back to the RP. The redirect may also optionally include a nonce, which will be included in the id token issued later in this flow. This data is sent to the IdP by the browser oicacf-idp-auth-req-1. The user authenticates to the IdP oicacf-idp-auth-resp-1, oicacf-idp-auth-req-2, and the IdP redirects the user's browser back to the RP in oicacf-idp-auth-resp-2 and oicacf-redir-ep-req (using the redirect URI from the request in oicacf-idp-auth-req-1). This redirect contains an authorization code, the state value as received in oicacf-start-resp, and the issuer identifier.[The issuer identifier will be included in this message in an upcoming revision of OIDC to mitigate the IdP Mix-Up attack, see Section <ref>.] If the state value and the issuer identifier are correct, the RP contacts the IdP in oicacf-token-req at the token endpoint with the received authorization code, its client id, its client secret (if any), and the redirect URI used to obtain the authorization code. If these values are correct, the IdP responds with a fresh access token and an id token to the RP in oicacf-token-resp. If the id token is valid, then the RP considers the user to be logged in (under the identifier composed from the user id in the id token and the issuer identifier). Hence, the RP may set a session cookie at the user's browser in oicacf-redir-ep-resp.§.§.§ Implicit Mode The implicit mode (depicted in Figure <ref> in Appendix <ref>) is similar to the authorization code mode, but instead of providing an authorization code, the IdP issues an id token right away to the RP (via the user's browser) when the user authenticates to the IdP. Hence, the Steps oicacf-start-req–oicacf-idp-auth-req-2 of the authorization code mode (Figure <ref>) are the same. After these steps, the IdP redirects the user's browser to the redirection endpoint at the RP, providing an id token, (optionally) an access token, the state value, and the issuer identifier. These values are not provided as a URL parameter but in the URL fragment instead. Hence, the browser does not send them to the RP at first. Instead, the RP has to provide a JavaScript that retrieves these values from the fragment and sends them to the RP. If the id token is valid, the issuer is correct, and the state matches the one previously chosen by the RP, the RP considers the user to be logged in and issues a session cookie. §.§.§ Hybrid ModeThe hybrid mode (depicted in Figure <ref> in Appendix <ref>) is a combination of the authorization code mode and the implicit mode: First, this mode works like the implicit mode, but when IdP redirects the browser back to RP, the IdP issues an authorization code, and either an id token or an access token or both.[The choice of the IdP to issue either an id token or an access token or both depends on the IdP's configuration and the request in Step oichf-idp-auth-req-1 in Figure <ref>.] The RP then retrieves these values as in the implicit mode (as they are sent in the fragment like in the implicit mode) and uses the authorization code to obtain a (potentially second) id token and a (potentially second) access token from IdP.§.§ Relationship to OAuth 2.0Technically, OIDC is derived from OAuth 2.0. It goes, however, far beyond what was specified in OAuth 2.0 and introduces many new concepts: OIDC defines a method for authentication (while retaining the option for authorization) using a new type of tokens, the id token. Some messages and tokens in OIDC can be cryptographically signed or encrypted while OAuth 2.0 does neither use signing nor encryption. The new hybrid flow combines features of the implicit mode and the authorization code mode. Importantly, with ad-hoc discovery and dynamic registration, OIDC standardizes and automates a process that is completely out of the scope of OAuth 2.0.These new features and their interplay potentially introduce new security flaws. It is therefore not sufficient to analyze the security of OAuth 2.0 to derive any guarantees for OIDC. OIDC rather requires a new security analysis. (See Section <ref> for a more detailed discussion. In Section <ref> we describe attacks that cannot be applied to OAuth 2.0.)§ ATTACKS AND SECURITY GUIDELINESIn this section, we present a concise overview of known attacks on OIDC and present additions that have not been documented so far. We also summarize mitigations and implementation guidelines that have to be implemented to avoid these attacks.The main focus of this work is to prove central security properties of OIDC, by which these mitigations and implementation guidelines are backed up. Moreover, further (potentially unknown types of) attacks on OIDC that can be captured by our security analysis are ruled out as well.The rest of the section is structured as follows: we first present the attacks, mitigations and guidelines, then point out differences to OAuth 2.0, and finally conclude with a brief discussion. §.§ Attacks, Mitigations, and Guidelines(Mitigations and guidelines are presented along with every class of attack.)§.§.§ IdP Mix-Up AttacksIn two previously reported attacks <cit.>, the aim was to confuse the RP about the identity of the IdP. In both attacks, the user was tricked into using an honest IdP to authenticate to an honest RP, while the RP is made to believe that the user authenticated to the attacker. The RP therefore, after successful user authentication, tries to use the authorization code or access token at the attacker, which then can impersonate the user or access the user's data at the IdP. We present a detailed description of an application of the IdP Mix-Up attack to OpenID Connect in Appendix <ref>.The IETF OAuth Working Group drafted a proposal for a mitigation technique <cit.> that is based on a proposal in <cit.> and that also applies to OpenID Connect. The proposal is that the IdP puts its identity into the response from the authorization endpoint. (This is already included in our description of OIDC above, see the issuer in Step oicacf-idp-auth-resp-2 in Figure <ref>.) The RP can then check that the user authenticated to the expected IdP. §.§.§ Attacks on the State ParameterThe state parameter is used in OIDC to protect against attacks on session integrity, i.e., attacks in which an attacker forces a user to be logged in at some RP (under the attacker's account). Such attacks can arise from session swapping or CSRF vulnerabilities. OIDC recommends the use of the state parameter. It should contain a nonce that is bound to the user's session. Attacks that can result from omitting or incorrectly using state were described in the context of OAuth 2.0 in <cit.>.The nonce for the state value should be chosen freshly for each login attempt to prevent an attack described in <cit.> (Section 5.1) where the same state value is used first in a user-initiated login flow with a malicious IdP and then in a login flow with an honest IdP (forcefully initiated by the attacker with the attacker's account and the user's browser).§.§.§ Code/Token/State Leakage Care should be taken that a value of state or an authorization code is not inadvertently sent to an untrusted third party through the Referer header. The state and the authorization code parameters are part of the redirection endpoint URI (at the RP), the state parameter is also part of the authorization endpoint URI (at the IdP). If, on either of these two pages, a user clicks on a link to an external page, or if one of these pages embeds external resources (images, scripts, etc.), then the third party will receive the full URI of the endpoint, including these parameters, in the Referer header that is automatically sent by the browser.Documents delivered at the respective endpoints should therefore be vetted carefully for links to external pages and resources. In modern browsers, referrer policies <cit.> can be used to suppress the Referer header. As a second line of defense, both parameters should be made single-use, i.e., state should expire after it has been used at the redirection endpoint and authorization code after it has been redeemed at IdP.In a related attack, an attacker that has access to logfiles or browsing histories (e.g., through malicious browser extensions) can steal authentication codes, access tokens, id tokens, or state values and re-use these to impersonate a user or to break session integrity. A subset of these attacks was dubbed Cut-and-Paste Attacks by the IETF OAuth working group <cit.>.There are drafts for RFCs that tackle specific aspects of these leakage attacks, e.g., <cit.> which discusses binding the state parameter to the browser instance, and <cit.> which discusses binding the access token to a TLS session. Since these mitigations are still very early IETF drafts, subject to change, and not easy to implement in the majority of the existing OIDC implementations, we did not model them.In our analysis, we assume that implementations keep logfiles and browsing histories (of honest browsers) secret and employ referrer policies as described above.§.§.§ Naïve RP Session Integrity AttackSo far, we have assumed that after Step oicacf-start-resp (Figure <ref>), the RP remembers the user's choice (which IdP is used) in a session; more precisely, the user's choice is stored in RP's session data. This way, in Step oicacf-redir-ep-req, the RP looks up the user's selected IdP in the session data. In <cit.>, this is called explicit user intention tracking.There is, however, an alternative to storing the IdP in the session. As pointed out by <cit.>, some implementations put the identity of the IdP into the redirect_uri (cf. Step oicacf-start-resp), e.g., by appending it as the last part of the path or in a parameter. Then, in Step oicacf-redir-ep-req, the RP can retrieve this information from the URI. This is called naïve user intention tracking.RPs that use naïve user intention tracking are susceptible to the naïve RP session integrity attack described in <cit.>: An attacker obtains an authorization code, id token, or access token for his own account at an honest IdP (HIdP). He then waits for a user that wants to log in at some RP using the attacker's IdP (AIdP) such that AIdP obtains a valid state for this RP. AIdP then redirects the user to the redirection endpoint URI of RP using the identity of HIdP plus the obtained state value and code or (id) token. Since the RP cannot see that the user originally wanted to log in using AIdP instead of HIdP, the user will now be logged in under the attacker's identity.Therefore, an RP should always use sessions to store the user's chosen IdP (explicit user intention tracking), which, as mentioned, is also what we do in our formal OIDC model.§.§.§ 307 Redirect AttackAlthough OIDC explicitly allows for any redirection method to be used for the redirection in Step oicacf-idp-auth-resp-2 of Figure <ref>, IdPs should not use an HTTP 307 status code for redirection. Otherwise, credentials entered by the user at an IdP will be repeated by the browser in the request to RP (Step oicacf-redir-ep-req of Figure <ref>), and hence, malicious RPs would learn these credentials and could impersonate the user at the IdP. This attack was presented in <cit.>. In our model, we exclusively use the 303 status code, which prevents re-sending of form data.§.§.§ Injection AttacksIt is well known that Cross-Site Scripting (XSS) and SQL Injection attacks on RPs or IdPs can lead to theft of access tokens, id tokens, and authorization codes (see, for example, <cit.>). XSS attacks can, for example, give an attacker access to session ids. Besides using proper escaping (and Content Security Policies <cit.> as a second line of defense), OIDC endpoints should therefore be put on domains separate from other, potentially more vulnerable, web pages on IdPs and RPs.[Since scripts on one origin can often access documents on the same origin, origins of the OIDC endpoints should be free from untrusted scripts.] (See Third-Party Resources below for another motivation for this separation.)In OIDC implementations, data that can come from untrusted sources (e.g., client ids, user attributes, state and nonce values, redirection URIs) must be treated as such: For example, a malicious IdP might try to inject user attributes containing malicious JavaScript to the RP. If the RP displays this data without applying proper escaping, the JavaScript is executed.We emphasize that in a similar manner, attackers can try to inject additional parameters into URIs by appending them to existing parameter values, e.g., the state. Since data is often passed around in OIDC, proper escaping of such parameters can be overlooked easily.As a result of such parameter injection attacks or independently, parameter pollution attacks can be a threat for OIDC implementations. In these attacks, an attacker introduces duplicate parameters into URLs (see, e.g., <cit.>). For example, a simple parameter pollution attack could be launched as follows: A malicious RP could redirect a user to an honest IdP, using a client id of some honest RP but appending two redirection URI parameters, one pointing to the honest RP and one pointing to the attacker's RP. Now, if the IdP checks the first redirection URI parameter, but afterwards redirects to the URI in the second parameter, the attacker learns authentication data that belongs to the honest RP and can impersonate the user.Mitigations against all these kinds of injection attacks are well known: implementations have to vet incoming data carefully, and properly escape any output data. In our model, we assume that these mitigations are implemented.§.§.§ CSRF Attacks and Third-Party Login InitiationSome endpoints need protection against CSRF in addition to the protection that the state parameter provides, e.g., by checking the origin header. Our analysis shows that the RP only needs to protect the URI on which the login flow is started (otherwise, an attacker could start a login flow using his own identity in a user's browser) and for the IdP to protect the URI where the user submits her credentials (otherwise, an attacker could submit his credentials instead). In the OIDC Core standard <cit.>, a so-called login initiation endpoint is described which allows a third party to start a login flow by redirecting a user to this endpoint, passing the identity of an IdP in the request. The RP will then start a login flow at the given IdP. Members of the OIDC foundation confirmed to us that this endpoint is essentially an intentional CSRF protection bypass. We therefore recommend login initiation endpoints not to be implemented (they are not a mandatory feature), or to require explicit confirmation by the user.§.§.§ Server-Side Request Forgery (SSRF) SSRF attacks can arise when an attacker can instruct a server to send HTTP(S) requests to other hosts, causing unwanted side-effects or revealing information <cit.>. For example, if an attacker can instruct a server behind a firewall to send requests to other hosts behind this firewall, the attacker might be able to call services or to scan the internal network (using timing attacks). He might also instruct the server to retrieve very large documents from other sources, thereby creating Denial of Service attacks.SSRF attacks on OIDC were described for the first time in <cit.>, in the context of the OIDC Discovery extension: An attacker could set up a malicious discovery service that, when queried by an RP, answers with links to arbitrary, network-internal or external servers (in Step oicacf-conf-resp of Figure <ref>).We here, for the first time, point out that not only RPs can be vulnerable to SSRF, but also IdPs. OIDC defines a way to indirectly pass parameters for the authorization request (cf. Step oicacf-idp-auth-req-1 in Figure <ref>). To this end, the IdP accepts a new parameter, request_uri in the authorization request. This parameter contains a URI from which the IdP retrieves the additional parameters (e.g., redirect_uri). The attacker can use this feature to easily mount an SSRF attack against the IdP even without any OIDC extensions: He can put an arbitrary URI in an authorization request causing the IdP to contact this URI.This new attack vector shows that not only RPs but also IdPs have to protect themselves against SSRF by using appropriate filtering and limiting mechanisms to restrict unwanted requests that originate from a web server (cf. <cit.>).SSRF attacks typically depend on an application specific context, such as the structure of and (vulnerable) services in an internal network.In our model, attackers can trigger SSRF requests, but the model does not contain vulnerable applications/services aside from OIDC. (Our analysis focuses on the security of the OIDC standard itself, rather than on specific applications and services.) Timing and performance properties, while sometimes relevant for SSRF attacks, are also outside of our analysis.§.§.§ Third-Party ResourcesRPs and IdPs that include third-party resources, e.g., tracking or advertisement scripts, subject their users to token theft and other attacks by these third parties. If possible, RPs and IdPs should therefore avoid including third-party resources on any web resources delivered from the same origins<ref> as the OIDC endpoints (see also Section <ref>). For newer browsers, subresource integrity <cit.> can help to reduce the risks associated with embedding third-party resources. With subresource integrity, websites can instruct supporting web browsers to reject third-party content if this content does not match a specific hash. In our model, we assume that websites do not include untrusted third-party resources.§.§.§ Transport Layer SecurityThe security of OIDC depends on the confidentiality and integrity of the transport layer. In other words, RPs and IdPs should use HTTPS. Endpoint URIs that are provided for the end user and that are communicated, e.g., in the discovery phase of the protocol, should only use thescheme. HTTPS Strict Transport Security and Public Key Pinning can be used to further strengthen the security of the OIDC endpoints. (In our model, we assume that users enter their passwords only over HTTPS web sites because otherwise, any authentication could be broken trivially.)§.§.§ Session HandlingSessions are typically identified by a nonce that is stored in the user's browser as a cookie. It is a well known best practice that cookies should make use of the secure attribute (i.e., the cookie is only ever used over HTTPS connections) and the HttpOnly flag (i.e., the cookie is not accessible by JavaScript). Additionally, after the login, the RP should replace the session id of the user by a freshly chosen nonce in order to prevent session fixation attacks: Otherwise, a network attacker could set a login session cookie that is bound to a known state value into the user's browser (see <cit.>), lure the user into logging in at the corresponding RP, and then use the session cookie to access the user's data at the RP (session fixation, see <cit.>). In our model, RPs use two kinds of sessions: Login sessions (which are valid until just before a user is authenticated at the RP) and service sessions (which signify that a user is already signed in to the RP). For both sessions, the secure and HttpOnly flags are used. §.§ Relationship to OAuth 2.0Many, but not all of the attacks described above can also be applied to OAuth 2.0. The following attacks in particular are only applicable to OIDC: (1) Server-side request forgery attacks are facilitated by the ad-hoc discovery and dynamic registration features. (2) The same features enable new ways to carry out injection attacks. (3) The new OIDC feature third-party login initiation enables new CSRF attacks. (4) Attacks on the id token only apply to OIDC, since there is no such token in OAuth 2.0.It is interesting to note that on the other hand, some attacks on OAuth 2.0 cannot be applied to OIDC (see <cit.> for further discussions on these attacks): (1) OIDC setups are less prone to open redirector attacks since placeholders are not allowed in redirection URIs. (2) TLS is mandatory for some messages in OIDC, while it is optional in OAuth 2.0. (3) The nonce value can prevent some replay attacks when the state value is not used or leaks to an attacker. §.§ DiscussionIn this section, our focus was to provide a concise overview of known attacks on OIDC and present some additions, namely SSRF at IdPs and third-party login initiation, along with mitigations and implementation guidelines. Our formal analysis of OIDC, which is the main focus of our work and is presented in the next sections, shows that the mitigations and implementation guidelines presented above are effective and that we can exclude other, potentially unknown types of attacks.§ THE FKS WEB MODELOur formal security analysis of OIDC is based on the FKS model, a generic Dolev-Yao style web model proposed by Fett et al. in <cit.>. Here, we only briefly recall this model following the description in <cit.> (see Appendices <ref> ff. for a full description, and<cit.> for comparison with other models and discussion of its scope and limitations).The FKS model is designed independently of a specific web application and closely mimics published (de-facto) standards and specifications for the web, for example, the HTTP/1.1 and HTML5 standards and associated (proposed) standards. The FKS model defines a general communication model, and, based on it, web systems consisting of web browsers, DNS servers, and web servers as well as web and network attackers.Communication ModelThe main entities in the model are (atomic) processes, which are used to model browsers, servers, and attackers. Each process listens to one or more (IP) addresses.Processes communicate via events, which consist of a message as well as a receiver and a sender address. In every step of a run, one event is chosen non-deterministically from a “pool” of waiting events and is delivered to one of the processes that listens to the event's receiver address. The process can then handle the event and output new events, which are added to the pool of events, and so on.As usual in Dolev-Yao models (see, e.g., <cit.>), messages are expressed as formal terms over a signature Σ. The signature contains constants (for (IP) addresses, strings, nonces) as well as sequence, projection, and function symbols (e.g., for encryption/decryption and signatures). For example, in the web model, an HTTP request is represented as a term r containing a nonce, an HTTP method, a domain name, a path, URI parameters, request headers, and a message body. For instance, an HTTP request for the URI <http://ex.com/show?p=1> is represented as r := ⟨, n_1, , ex.com, /show, p,1, , ⟩ where the body and the list of request headers is empty. An HTTPS request for r is of the form rk'(k_ex.com), where k' is a fresh symmetric key (a nonce) generated by the sender of the request (typically a browser); the responder is supposed to use this key to encrypt the response.The equational theory associated with Σ is defined as usual in Dolev-Yao models. The theory induces a congruence relation ≡ on terms, capturing the meaning of the function symbols in Σ. For instance, the equation in the equational theory which captures asymmetric decryption is x(y)y=x. With this, we have that, for example, rk'(k_ex.com)k_ex.com≡r,k' , i.e., these two terms are equivalent w.r.t. the equational theory.A (Dolev-Yao) process consists of a set of addresses the process listens to, a set of states (terms), an initial state, and a relation that takes an event and a state as input and (non-deterministically) returns a new state and a sequence of events. The relation models a computation step of the process.It is required that the output can be computed (formally, derived in the usual Dolev-Yao style) from the input event and the state.The so-called attacker process is a Dolev-Yao process which records all messages it receives and outputs all events it can possibly derive from its recorded messages. Hence, an attacker processcarries out all attacks any Dolev-Yao process could possibly perform. Attackers can corrupt other parties. A script models JavaScript running in a browser. Scripts are defined similarly to Dolev-Yao processes. When triggered by a browser, a script is provided with state information. The script then outputs a term representing a new internal state and a command to be interpreted by the browser (see also the specification of browsers below). We give an annotated example for a script in Algorithm <ref> in the appendix. Similarly to an attacker process, the so-called attacker script outputs everything that is derivable from the input.A system is a set of processes. A configuration of this system consists of the states of all processes in the system, the pool of waiting events, and a sequence of unused nonces. Systems induce runs, i.e., sequences of configurations, where each configuration is obtained by delivering one of the waiting events of the preceding configuration to a process, which then performs a computation step. The transition from one configuration to the next configuration in a run is called a processing step. We write, for example, Q = (S, E, N)(S', E', N') to denote the transition from the configuration (S, E, N) to the configuration (S', E', N'), where S and S' are the states of the processes in the system, E and E' are pools of waiting events, and N and N' are sequences of unused nonces.A web system formalizes the web infrastructure and web applications. It contains a system consisting of honest and attacker processes. Honest processes can be web browsers, web servers, or DNS servers. Attackers can be either web attackers (who can listen to and send messages from their own addresses only) or network attackers (who may listen to and spoof all addresses and therefore are the most powerful attackers). A web system further contains a set of scripts (comprising honest scripts and the attacker script).In our analysis of OIDC, we consider either one network attacker or a set of web attackers (see Section <ref>). In our OIDC model, we need to specify only the behavior of servers and scripts. These are not defined by the FKS model since they depend on the specific application, unless they become corrupted, in which case they behave like attacker processes and attacker scripts; browsers are specified by the FKS model (see below). The modeling of OIDC servers and scripts is outlined in Section <ref> and with full details provided in Appendices <ref> and <ref>.Web BrowsersAn honest browser is thought to be used by one honest user, who is modeled as part of the browser. User actions, such as following a link, are modeled as non-deterministic actions of the web browser. User credentials are stored in the initial state of the browser and are given to selected web pages when needed. Besides user credentials, the state of a web browser contains (among others) a tree of windows and documents, cookies, and web storage data (localStorage and sessionStorage).A window inside a browser contains a set of documents (one being active at any time), modeling the history of documents presented in this window. Each represents one loaded web page and contains (among others) a script and a list of subwindows (modeling iframes). The script, when triggered by the browser, is provided with all data it has access to, such as a (limited) view on other documents and windows, certain cookies, and web storage data. Scripts then output a command and a new state. This way, scripts can navigate or create windows, send XMLHttpRequests and postMessages, submit forms, set/change cookies and web storage data, and create iframes. Navigation and security rules ensure that scripts can manipulate only specific aspects of the browser's state, according to the relevant web standards.A browser can output messages on the network of different types, namely DNS and HTTP(S) (including XMLHttpRequests), and it processes the responses. Several HTTP(S) headers are modeled, including, for example, cookie, location, strict transport security (STS), and origin headers. A browser, at any time, can also receive a so-called trigger message upon which the browser non-deterministically choses an action, for instance, to trigger a script in some document. The script now outputs a command, as described above, which is then further processed by the browser. Browsers can also become corrupted, i.e., be taken over by web and network attackers. Once corrupted, a browser behaves like an attacker process.§ ANALYSISWe now present our security analysis of OIDC, including a formal model of OIDC, the specifications of central security properties, and our theorem which establishes the security of OIDC in our model. More precisely, our formal model of OIDC uses the FKS model as a foundation and is derived by closely following the OIDC standards Core, Discovery, and Dynamic Client Registration <cit.>. (As mentioned above, the goal in this work is to analyze OIDC itself instead of concrete implementations.) We then formalize the main security properties for OIDC, namely authentication, authorization, session integrity for authentication, and session integrity for authorization. We also formalize secondary security properties that capture important aspects of the security of OIDC, for example, regarding the outcome of the dynamic client registration. We then state and prove our main theorem. Finally, we discuss the relationship of our work to the analysis of OAuth 2.0 presented in <cit.> and conclude with a discussion of the results.We refer the reader to Appendices <ref>–<ref> for full details, including definitions, specifications, and proofs. To provide an intuition of the abstraction level, syntax, and concepts that we use for the modeling without reading all details, we extensively annotated Algorithms <ref>, <ref>, and <ref> in Appendix <ref>.§.§ Model Our model of OIDC includes all features that are commonly found in real-world implementations, for example, all three modes, a detailed model of the Discovery mechanism <cit.> (including the WebFinger protocol <cit.>), and Dynamic Client Registration <cit.> (including dynamic exchange of signing keys). RPs, IdPs, and, as usual in the FKS model, browsers can be corrupted by the adversary dynamically. We do not model less used features, in particular OIDC logout, self-issued OIDC providers (“personal, self-hosted OPs that issue self-signed ID Tokens”, <cit.>), and ACR/AMR (Authentication Class/Methods Reference) values that can be used to indicate the level of trust in the authentication of the user to the IdP.Since the FKS model has no notion of time, we overapproximate by never letting tokens, e.g., id tokens, expire. Moreover, we subsume user claims (information about the user that can be retrieved from IdPs) by user identifiers, and hence, in our model users have identities, but no other properties.We have two versions of our OIDC model, one with a network attacker and one with an unbounded number of web attackers, as explained next. The reason for having two versions is that while the authentication and authorization properties can be proven assuming a network attacker, such an attacker could easily break session integrity. Hence, for session integrity we need to assume web attackers (see the explanations for session integrity in Section <ref>). §.§.§ OIDC Web System with a Network AttackerWe model OIDC as a class of web systems (in the sense of Section <ref>) which can contain an unbounded finite number of RPs, IdPs, browsers, and one network attacker. More formally, an OIDC web system with a network attacker (^n) consists of a network attacker, a finite set of web browsers, a finite set of web servers for the RPs, and a finite set of web servers for the IdPs. Recall that in ^n, since we have a network attacker, we do not need to consider web attackers (as the network attacker subsumes all web attackers). All non-attacker parties are initially honest, but can become corrupted dynamically upon receiving a special message and then behave just like a web attacker process. As already mentioned in Section <ref>,to model OIDC based on the FKS model, we have to specify the protocol specific behavior only, i.e., the servers for RPs and IdPs as well as the scripts that they use. We start with a description of the servers. Web Servers Since RPs and IdPs both are web servers, we developed a generic model for HTTPS server processes for the FKS model. We call these processes HTTPS server base processes. Their definition covers decrypting received HTTPS messages and handling HTTP(S) requests to external webservers, including DNS resolution.RPs and IdPs are derived from this HTTPS server base process. Their models follow the OIDC standard closely and include the mitigations discussed in Section <ref>.An RP waits for users to start a login flow and then non-deterministically decides which mode to use. If needed, it starts the discovery and dynamic registration phase of the protocol, and finally redirects the user to the IdP for user authentication. Afterwards, it processes the received tokens and uses them according to their type (e.g., with an access token, the RP would retrieve an id token from the IdP). If an id token is received that passes all checks, the user will be logged in. As mentioned briefly in Section <ref>, RPs manage two kinds of sessions: The login sessions, which are used only during the user login phase, and service sessions.The IdP provides several endpoints according to its role in the login process, including its OIDC configuration endpoint and endpoints for receiving authentication and token requests.Scripts Three scripts (altogether 30 lines of code) can be sent from honest IdPs and RPs to web browsers. The script script_rp_index is sent by an RP when the user visits the RP's web site. It starts the login process. The script script_rp_get_fragment is sent by an RP during an implicit or hybrid mode flow to retrieve the data from the URI fragment. It extracts the access token, authorization code, and state from the fragment part of its own URI and sends this information in the body of a POST request back to the RP. IdP sends the script script_idp_form for user authentication at the IdP. §.§.§ OIDC Web System with Web AttackersWe also consider a class of web systems where the network attacker is replaced by an unbounded finite set of web attackers and a DNS server is introduced.We denote such a system by ^w and call it an OIDC web system with web attackers. Such web systems are used to analyze session integrity, see below. §.§ Main Security PropertiesOur primary security properties capture authentication, authorization and session integrity for authentication and authorization. We will present these security properties in the following, with full details in Appendix <ref>. Authentication PropertyThe most important property for OIDC is the authentication property. In short, it captures that a network attacker (and therefore also web attackers) should be unable to log in as an honest user at an honest RP using an honest IdP. Before we define this property in more detail, recall that in our modeling, an RP uses two kinds of sessions: login sessions, which are only used for the login flow, and service sessions, which are used after a user/browser was logged in (see Section <ref> for details). When a login session has finished successfully (i.e., the RP received a valid id token), the RP uses a fresh nonce as the service session id, stores this id in the session data of the login session, and sends the service session id as a cookie to the browser. In the same step, the RP also stores the issuer, say d, that was used in the login flow and the identity (email address) of the user, say id, as a pair d, id, referred to as a global user identifier in Section <ref>. Now, our authentication property defines that a network attacker should be unable to get hold of a service session id by which the attacker would be considered to be logged in at an honest RP under an identity governed by an honest IdP for an honest user/browser.In order to define the authentication property formally, we first need to define the precise notion of a service session. In the following, as introduced in Section <ref>, (S, E, N) denotes a configuration in the run ρ with its components S, a mapping from processes to states of these processes, E, a set of events in the network that are waiting to be delivered to some party, and N, a set of nonces that have not been used yet. By 𝗀𝗈𝗏𝖾𝗋𝗇𝗈𝗋(id) we denote the IdP that is responsible for a given user identity (email address) id, and by 𝖽𝗈𝗆(𝗀𝗈𝗏𝖾𝗋𝗇𝗈𝗋(id)), we denote the set of domains that are owned by this IdP. By S(r).sessions[lsid] we denote a data structure in the state of r that contains information about the login session identified by lsid. This data structure contains, for example, the identity for which the login session with the id lsid was started and the service session id that was issued after the login session. We can now define that there is a service session identified by a nonce n for an identity id at some RP r iff there exists a login session (identified by some nonce lsid) such that n is the service session associated with this login session, and r has stored that the service session is logged in for the id id using an issuer d (which is some domain of the governor of id). We say that there is a service session identified by a nonce n for an identity id at some RP r in a configuration (S, E, N) of a run ρ of an OIDC web system iff there exists some login session id lsid and a domain d ∈𝖽𝗈𝗆(𝗀𝗈𝗏𝖾𝗋𝗇𝗈𝗋(id)) such that S(r).sessions[lsid][loggedInAs] ≡d, id and S(r).sessions[lsid][serviceSessionId] ≡ n. By d_∅(S(attacker)) we denote all terms that can be computed (derived in the usual Dolev-Yao style, see Section <ref>) from the attacker's knowledge in the state S. We can now define that an OIDC web system with a network attacker is secure w.r.t. authentication iff the attacker can never get hold of a service session id (n) that was issued by an honest RP r for an identity id of an honest user (browser) at some honest IdP (governor of id).Let ^n be an OIDC web system with a network attacker. We say that ^n is secure w.r.t. authentication iff for every run ρ of ^n, every configuration (S, E, N) in ρ, every r∈RP that is honest in S, every browser b that is honest in S, every identity id∈𝖨𝖣 with 𝗀𝗈𝗏𝖾𝗋𝗇𝗈𝗋(id) being an honest IdP, every service session identified by some nonce n for id at r, we have thatn is not derivable from the attackers knowledge in S (i.e., n ∉d_∅(S(attacker))). Authorization Property Intuitively, authorization for OIDC means that a network attacker should not be able to obtain or use a protected resource available to some honest RP at an IdP for some user unless certain parties involved in the authorization process are corrupted. As the access control for such protected resources relies only on access tokens, we require that an attacker does not learn access tokens that would allow him to gain unauthorized access to these resources.To define the authorization property formally, we need to reason about the state of an honest IdP, say i. In this state, i creates recordscontaining data about successful authentications of users at i. Such records are stored in S(i).records. One such record, say x, contains the authenticated user's identity in x[subject], two[In the hybrid mode, IdPs can issue two access tokens, cf. Section <ref>.] access tokens in x[access_tokens], and the client id of the RP in x[client_id].We can now define the authorization property. It defines that an OIDC web system with a network attacker is secure w.r.t. authorization iff the attacker cannot get hold of an access token that is stored in one of i's records for an identity of an honest user/browser b and an honest RP r. Let ^n be an OIDC web system with a network attacker. We say that ^n is secure w.r.t. authorization iff for every run ρ of ^n, every configuration (S, E, N) in ρ, every r∈RP that is honest in S, every i∈IdP that is honest in S, every browser b that is honest in S, every identity id∈𝖨𝖣 owned by b and 𝗀𝗈𝗏𝖾𝗋𝗇𝗈𝗋(id) = i, every nonce n, every term x ∈ S(i).records with x[subject] ≡id, n ∈ x[access_tokens], and the client id x[client_id] having been issued by i to r,[See Definition <ref> in Appendix <ref>.] we have that n is not derivable from the attackers knowledge in S (i.e., n ∉d_∅(S(attacker))). Session Integrity for Authentication The two session integrity properties capture that an attacker should be unable to forcefully log a user/browser in at some RP. This includes attacks such as CSRF and session swapping. Note that we define these properties over ^w, i.e., we consider web attackers instead of a network attacker. The reason is that OIDC deployments typically use cookies to track the login sessions of users. Since a network attacker can put cookies into browsers over unencrypted connections and these cookies are then also used for encrypted connections, cookies have no integrity in the presence of a network attacker (see also <cit.>). In particular, a network attacker could easily break the session integrity of typical OIDC deployments.For session integrity for authentication we say that a user/browser that is logged in at some RP must have expressed her wish to be logged in to that RP in the beginning of the login flow. Note that not even a malicious IdP should be able to forcefully log in its users (more precisely, its user's browsers) at an honest RP. If the IdP is honest, then the user must additionally have authenticated herself at the IdP with the same user account that RP uses for her identification. This excludes, for example, cases where (1) the user is forcefully logged in to an RP by an attacker that plays the role of an IdP, and (2) where an attacker can force an honest user to be logged in at some RP under a false identity issued by an honest IdP.In our formal definition of session integrity for authentication (below), 𝗅𝗈𝗀𝗀𝖾𝖽𝖨𝗇_ρ^Q(b, r, u, i, lsid) denotes that in the processing step Q (see below), the browser b was authenticated (logged in) to an RP r using the IdP i and the identity u in an RP login session with the session id lsid. (Here, the processing step Q corresponds to Step oicacf-redir-ep-resp in Figure <ref>.) The user authentication in the processing step Q is characterized by the browser b receiving the service session id cookie that results from the login session lsid.By 𝗌𝗍𝖺𝗋𝗍𝖾𝖽_ρ^Q'(b, r, lsid) we denote that the browser b, in the processing step Q' triggered the script script_rp_index to start a login session which has the session id lsid at the RP r. (Compare Section <ref> on how browsers handle scripts.) Here, Q' corresponds to Step oicacf-start-req in Figure <ref>.By 𝖺𝗎𝗍𝗁𝖾𝗇𝗍𝗂𝖼𝖺𝗍𝖾𝖽_ρ^Q”(b, r, u, i, lsid) we denote that in the processing step Q”, the user/browser b authenticated to the IdP i. In this case, authentication means that the user filled out the login form (in script_idp_form) at the IdP i and, by this, consented to be logged in at r (as in Step oicacf-idp-auth-req-2 in Figure <ref>).Using these notations, we can now define security w.r.t. session integrity for authentication of an OIDC web system with web attackers in a straightforward way: Let ^w be an OIDC web system with web attackers. We say that ^w is secure w.r.t. session integrity for authentication iff for every run ρ of ^w, every processing step Q in ρ with Q = (S, E, N)(S', E', N') (for some S, S', E, E', N, N'), every browser b that is honest in S, every i∈IdP, every identity u that is owned by b, every r∈RP that is honest in S, every nonce lsid, with 𝗅𝗈𝗀𝗀𝖾𝖽𝖨𝗇_ρ^Q(b, r, u, i, lsid), we have that (1) there exists a processing step Q' in ρ (before Q) such that 𝗌𝗍𝖺𝗋𝗍𝖾𝖽_ρ^Q'(b, r, lsid), and (2) if i is honest in S, then there exists a processing step Q” in ρ (before Q) such that 𝖺𝗎𝗍𝗁𝖾𝗇𝗍𝗂𝖼𝖺𝗍𝖾𝖽_ρ^Q”(b, r, u, i, lsid).Session Integrity for Authorization For session integrity for authorization we say that if an RP uses some access token at some IdP in a session with a user, then that user expressed her wish to authorize the RP to interact with some IdP. Note that one cannot guarantee that the IdP with which RP interacts is the one the user authorized the RP to interact with. This is because the IdP might be malicious. In this case, for example in the discovery phase, the malicious IdP might just claim (in Step oicacf-wf-resp in Figure <ref>) that some other IdP is responsible for the authentication of the user. If, however, the IdP the user is logged in with is honest, then it should be guaranteed that the user authenticated to that IdP and that the IdP the RP interacts with on behalf of the user is the one intended by the user.For the formal definition, we use two additional predicates: 𝗎𝗌𝖾𝖽𝖠𝗎𝗍𝗁𝗈𝗋𝗂𝗓𝖺𝗍𝗂𝗈𝗇_ρ^Q(b, r, i, lsid) means that the RP r, in a login session (session id lsid) with the browser b used some access token to access services at the IdP i.By 𝖺𝖼𝗍𝗌𝖮𝗇𝖴𝗌𝖾𝗋𝗌𝖡𝖾𝗁𝖺𝗅𝖿_ρ^Q(b, r, u, i, lsid) we denote that the RP r not only used some access token, but used one that is bound to the user's identity at the IdP i.Again, starting from our informal definition above, we define security w.r.t. session integrity for authorization of an OIDC web system with web attackers in a straightforward way (and similarly to session integrity for authentication): Let ^w be an OIDC web system with web attackers. We say that ^w is secure w.r.t. session integrity for authentication iff for every run ρ of ^w, every processing step Q in ρ with Q = (S, E, N)(S', E', N')(for some S, S', E, E', N, N'), every browser b that is honest in S, every i∈IdP, every identity u that is owned by b, every r∈RP that is honest in S, every nonce lsid, we have that (1) if 𝗎𝗌𝖾𝖽𝖠𝗎𝗍𝗁𝗈𝗋𝗂𝗓𝖺𝗍𝗂𝗈𝗇_ρ^Q(b, r, i, lsid), then there exists a processing step Q' in ρ (before Q) such that 𝗌𝗍𝖺𝗋𝗍𝖾𝖽_ρ^Q'(b, r, lsid), and (2) if i is honest in S and 𝖺𝖼𝗍𝗌𝖮𝗇𝖴𝗌𝖾𝗋𝗌𝖡𝖾𝗁𝖺𝗅𝖿_ρ^Q(b, r, u, i, lsid), then there exists a processing step Q” in ρ (before Q) such that 𝖺𝗎𝗍𝗁𝖾𝗇𝗍𝗂𝖼𝖺𝗍𝖾𝖽_ρ^Q”(b, r, u, i, lsid).§.§ Secondary Security PropertiesWe define the following secondary security properties that capture specific aspects of OIDC. We use these secondary security properties during our proof of the above main security properties. Nonetheless, these secondary security properties are important and interesting in their own right.We define and prove the following properties (see the corresponding lemmas in Appendices <ref> and <ref> for details):Integrity of Issuer Cache:If a relying party requests the issuer identifier from an identity provider (cf. Steps oicacf-wf-req–oicacf-wf-resp in Figure <ref>), then the RP will only receive an origin that belongs to this IdP in the response. In other words, honest IdPs do not use attacker-controlled domains as issuer identifiers, and the attacker is unable to alter this information on the way to the RP or in the issuer cache at the RP.Integrity of OIDC Configuration Cache:(1) Honest IdPs only use endpoints under their control in their OIDC configuration document (cf. Steps oicacf-conf-req–oicacf-conf-resp in Figure <ref>) and (2) this information (which is stored at the RP in the so-called OIDC configuration cache) cannot be altered by an attacker.Integrity of JWKS Cache:RPs receive only “correct” signing keys from honest IdPs, i.e., keys that belong to the respective IdP (cf. Steps oicacf-jwks-req–oicacf-jwks-resp in Figure <ref>). Integrity of Client Registration: Honest RPs register only redirection URIs that point to themselves and that these URIs always use HTTPS. Recall that when an RP registers at an IdP, the IdP issues a freshly chosen client id to the RP and then stores RP's redirection URIs. Third Parties Do Not Learn Passwords: Attackers cannot learn user passwords. More precisely, we define that 𝗌𝖾𝖼𝗋𝖾𝗍𝖮𝖿𝖨𝖣(id), which denotes the password for a given identity id, is not known to any party except for the browser b owning the id and the identity provider i governing the id (as long as b and i are honest).Attacker Does Not Learn ID Tokens: Attackers cannot learn id tokens that were issued by honest IdPs for honest RPs and identities of honest browsers.Third Parties Do Not Learn State: If an honest browser logs in at an honest RP using an honest IdP, then the attacker cannot learn the state value used in this login flow.§.§ TheoremThe following theorem states that OIDC is secure w.r.t. authentication and authorization in presence of the network attacker, and that OIDC is secure w.r.t. session integrity for authentication and authorization in presence of web attackers. For the proof we refer the reader to Appendix <ref>. Let ^n be an OIDC web system with a network attacker. Then, ^n is secure w.r.t. authentication and authorization. Let ^w be an OIDC web system with web attackers. Then, ^w is secure w.r.t. session integrity for authentication and authorization.§.§ Comparison to OAuth 2.0As described in Section <ref>, OIDC is based on OAuth 2.0. Since a formal proof for the security of OAuth 2.0 was conducted in <cit.>, one might be tempted to think that a proof for the security of OIDC requires little more than an extension of the proof in <cit.>. The specific set of features of OIDC introduces, however, important differences that call for new formulations of security properties and require new proofs:Dynamic Discovery and Registration: Due to the dynamic discovery and registration, RPs can directly influence and manipulate the configuration data that is stored in IdPs. In OAuth, this configuration data is fixed and assumed to be “correct”, greatly limiting the options of the attacker. See, for example, the variant <cit.> of the IdP Mix-up attack that only works in OIDC (mentioned in Section <ref>).Different set of modes: Compared to OAuth, OIDC introduces the hybrid mode, but does not use the resource owner password credentials mode and the client credentials mode.New endpoints, messages, and parameters: With additional endpoints (and associated HTTPS messages), the attack surface of OIDC is, also for this reason, larger than that of OAuth. The registration endpoints, for example, could be used in ways that break the security of the protocol, which is not possible in OAuth where these endpoints do not exist. In a similar vein, new parameters like nonce, request_uri, and the id token, are contained in several messages (some of which are also present in the original OAuth flow) and potentially change the security requirements for these messages.Authentication mechanism: The authentication mechanisms employed by OIDC and OAuth are quite different. This shows, in particular, in the fact that OIDC uses the id token mechanism for authentication, while OAuth uses a different, non-standardized mechanism. Additionally, unlike in OAuth, authentication can happen multiple times during one OIDC flow (see the description of the hybrid mode in Section <ref>). This greatly influences (the formulation of) security properties, and hence, also the security proofs.In summary, taking all these differences into account, our security proofs had to be carried out from scratch. At the same time, our proof is more modular than the one in <cit.> due to the secondary security properties we identified. Moreover, our security properties are similar to the ones by Fett et al. in <cit.> only on a high level. The underlying definitions in many aspects differ from the ones used for OAuth.[As an example, in <cit.>, the definitions rely on a notion of OAuth sessions which are defined by connected HTTP(S) messages, i.e., messages that are created by a browser or server in response to another message. In our model, the attacker is involved in each flow of the protocol (for providing the client id, without receiving any prior message), making it hard to apply the notion of OAuth sessions. We instead define the properties using the existing session identifiers. (See Definitions <ref>, <ref>, <ref>–<ref> in Appendix <ref> for details.)] §.§ DiscussionUsing our detailed formal model, we have shown that OIDC enjoys a high level of security regarding authentication, authorization, and session integrity. To achieve this security, it is essential that implementors follow the security guidelines that we stated in Section <ref>. Clearly, in practice, this is not always feasible—for example, many RPs want to include third-party resources for advertisement or user tracking on their origins. As pointed out, however, not following the security guidelines we outline can lead to severe attacks.We have shown the security of OIDC in the most comprehensive model of the web infrastructure to date. Being a model, however, some features of the web are not included in the FKS model, for example browser plugins. Such technologies can under certain circumstances also undermine the security of OIDC in a manner that is not reflected in our model. Also, user-centric attacks such as phishing or clickjacking attacks are also not covered in the model. Nonetheless, our formal analysis and the guidelines (along with the attacks when these guidelines are not followed) provide a clear picture of the security provided by OIDC for a large class of adversaries.§ RELATED WORKAs already mentioned in the introduction, the only previous works on the security of OIDC are <cit.>. None of these works establish security guarantees for the OIDC standard: In <cit.>, the authors find implementation errors in deployments of Google Sign-In (which, as mentioned before, is based on OIDC). In <cit.>, the authors describe a variant of the IdP Mix-Up attack (see Section <ref>), highlight the possibility of SSRF attacks at RPs, and show some implementation-specific flaws. In our work, however, we aim at establishing and proving security properties for OIDC.In general, there have been only few formal analysis efforts for web applications, standards, and browsers so far. Most of the existing efforts are based on formal representations of (parts of) web browsers or very limited models of web mechanisms and applications <cit.>.Only <cit.> and <cit.> were based on a generic formal model of the web infrastructure. In <cit.>, Bansal, Bhargavan, Delignat-Lavaud, and Maffeis analyze the security of OAuth 2.0 with the tool ProVerif in the applied pi-calculus and the WebSpi library. They identify previously unknown attacks on the OAuth 2.0 implementations of Facebook, Yahoo, Twitter, and many other websites. They do not, however, establish security guarantees for OAuth 2.0 and their model is much less expressive than the FKS model.The relationship of our work to <cit.> has been discussed in detail throughout the paper. § CONCLUSIONDespite being the foundation for many popular and critical login services, OpenID Connect had not been subjected to a detailed security analysis, let alone a formal analysis, before. In this work, we filled this gap.We developed a detailed and comprehensive formal model of OIDC based on the FKS model, a generic and expressive formal model of the web infrastructure. Using this model, we stated central security properties of OIDC regarding authentication, authorization, and session integrity, and were able to show that OIDC fulfills these properties in our model. By this, we could, for the first time, provide solid security guarantees for one of the most widely deployed single sign-on systems.To avoid previously known and newly described attacks, we analyzed OIDC with a set of practical and reasonable security measures and best practices in place. We documented these security measures so that they can now serve as guidelines for secure implementations of OIDC.§ ACKNOWLEDGEMENTS This work was partially supported by Deutsche Forschungsgemeinschaft (DFG) throughGrant KU 1434/10-1.abbrv10AbadiFournet-POPL-2001 M. Abadi and C. Fournet. 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The honest RP assumes that the login uses the attacker's IdP and interacts with this IdP, while the user's browser interacts with an honest IdP and relays the data acquired at this IdP to the RP. As a result, the attacker learns information such as authorization codes and access tokens he is not supposed to know and that allow him to break the authentication and authorization properties. There exist several variants of this attack <cit.>. Here, we describe two variants of this attack using the hybrid mode of OIDC. The normal flow of the hybrid mode is depicted in Figure <ref>, the attack is depicted in Figures <ref> and <ref> (without the mitigation against the Mix-Up attack presented in Section <ref>). To start the login flow, the user selects an IdP at RP (by entering her email address) in Step oichf-att-start-req. This step is the only difference between the two variants that we describe: In Variant 1, the user selects a malicious IdP, say AIdP. In Variant 2, the user selects an honest IdP, but the request is intercepted by the attacker and altered such that the attacker replaces the honest IdP by AIdP (email is replaced by email' in Steps oichf-att-start-req and oichf-att-start-req-manipulated in Figure <ref>).[This initial request is often unencrypted in practice, see <cit.>.]Now, RP starts with the discovery phase of the protocol. As RP thinks that the user wants to login with AIdP, it retrieves the OIDC configuration from AIdP (Steps oichf-att-conf-req and oichf-att-conf-resp). In this configuration, the attacker does not let all endpoint URLs point to himself, as would be usual for OIDC, but instead sets the authorization endpoint to be the one of HIdP. Next, the RP registers itself at AIdP (Steps oichf-att-reg-req and oichf-att-reg-resp). In this step, AIdP issues the same client id to RP which RP is registered with at HIdP (client id are always public). This is important as HIdP will later redirect the user's browser back to RP and checks the redirect URI based on the client id.Next, RP redirects the user's browser to HIdP (Variant 1) or AIdP (Variant 2) in Step oichf-att-start-resp. In Variant 1 of the attack, a vigilant user might now be able to detect that she tried to log in using AIdP but instead is redirected to HIdP. This does not happen in Variant 2, but here the attacker needs to replace the redirection to AIdP by a redirection to HIdP (which should not be any problem if he succeeded in altering the first step of the protocol).The user then authenticates at HIdP and is redirected back to RP along with an authorization code and an access token (depending on the sub-mode of the hybrid flow, IdPs do not send id tokens in this step). Now, RP retrieves the authorization code and the access token from the user's browser and continues the login flow. As RP still assumes that AIdP is used in this case, it tries to redeem the authorization code for an id token (and a second access token) at AIdP in Step oichf-att-token-req. As the authorization code has not been redeemed at HIdP yet, the code is still valid and the attacker may start a second login flow (pretending to be the user) at RP (Steps oichf-att-start-req-attff.). The attacker skips the authentication at HIdP and returns to RP with the authorization code he has learned before. RP now redeems this code at HIdP and receives an id token issued for the honest user and consequently assumes that the attacker has the identity of the user and logs the attacker in.In another variation of the attack, if HIdP does not issue client secrets to RPs, the attacker can also redeem the authorization code by himself (Steps oichf-att-token-req-att2f.). In this case, the attacker receives an access token valid for the user's account. With this access token, he can retrieve data of the user or act on the user's behalf at HIdP. (As he redeems the authorization code, he cannot use it to log himself into the RP in this case.)In any case, the attacker can also respond to the authorization code sent to his token endpoint in Step oichf-att-token-resp-att with a mock access token and a mock id token (which will not be used in the following). In the next step, the RP might then use the access token learned from the honest IdP in Step oichf-att-redir-ep-token-req to retrieve data of the user from AIdP (Steps oichf-att-token-resp-attff.).[Depending on the RP implementation, the RP might choose to use the mock access token or the one learned from the honest IdP in this step. In the real-world implementation mod_auth_openidc, the access token from the honest IdP was used.] Then the attacker learns also this access token, which (as described in the paragraph above) grants him unauthorized access to the user's account at HIdP.This shows that, using the IdP Mix-Up attack, an attacker can successfully impersonate users at RPs and access their data at honest IdPs. The mitigation presented in Section <ref> would have prevented the attack in Step oichf-att-redir-ep-reqff. § THE FKS WEB MODEL In this and the following two sections, we present the FKS model for the web infrastructure as proposed in <cit.>, <cit.>, and <cit.>, along with the addition of a generic model for HTTPS web servers that harmonizes the behavior of such servers and facilitates easier proofs. §.§ Communication Model We here present details and definitions on the basic concepts of the communication model.§.§.§ Terms, Messages and Events The signature Σ for the terms and messages considered in this work is the union of the following pairwise disjoint sets of function symbols: * constants C =∪ 𝕊∪{,,} where the three sets are pairwise disjoint, 𝕊 is interpreted to be the set of ASCII strings (including the empty string ε), andis interpreted to be a set of (IP) addresses,* function symbols for public keys, (a)symmetric encryption/decryption, and signatures: 𝗉𝗎𝖻(·), ··, ··, ··, ··, ··, ···, and ·,* n-ary sequences , ·, ·,·, ·,·,·, etc., and* projection symbols π_i(·) for all i ∈ℕ. For strings (elements in 𝕊), we use a specific font. For example,andare strings. We denote by ⊆𝕊 the set of domains, e.g., example.com∈.We denote by ⊆𝕊 the set of methods used in HTTP requests, e.g., , ∈.The equational theory associated with the signature Σ is given in Figure <ref>.By X={x_0,x_1,…} we denote a set of variables and bywe denote an infinite set of constants (nonces) such that Σ, X, andare pairwise disjoint. For N⊆, we define the set _N(X) of terms over Σ∪ N∪ X inductively as usual: (1) If t∈ N∪ X, then t is a term. (2) If f∈Σ is an n-ary function symbol in Σ for some n≥ 0 and t_1,…,t_n are terms, then f(t_1,…,t_n) is a term. By ≡ we denote the congruence relation on (X) induced by the theory associated with Σ. For example, we have that π_1(a,b(k)k)≡a. By _N=_N(∅), we denote the set of all terms over Σ∪ N without variables, called ground terms. The setof messages (over ) is defined to be the set of ground terms _. We define the set V_process = {ν_1, ν_2, …} of variables (called placeholders). The set ^ν := _(V_process) is called the set of protomessages, i.e., messages that can contain placeholders.For example, k∈ and (k) are messages, where k typically models a private key and (k) the corresponding public key. For constants a, b, c and the nonce k∈, the message a,b,c(k) is interpreted to be the message a,b,c (the sequence of constants a, b, c) encrypted by the public key (k).Let t be a term. The normal form of t is acquired by reducing the function symbols from left to right as far as possible using the equational theory shown in Figure <ref>. For a term t, we denote its normal form as t.Let pattern∈({*}) be a term containing the wildcard (variable *). We say that a term t matches pattern iff t can be acquired from pattern by replacing each occurrence of the wildcard with an arbitrary term (which may be different for each instance of the wildcard). We write t ∼pattern. For a sequence of patterns patterns we write t ∼̇patterns to denote that t matches at least one pattern in patterns.For a term t' we write t'| pattern to denote the term that is acquired from t' by removing all immediate subterms of t' that do not match pattern. For example, for a pattern p = ⊤,* we have that ⊤,42∼ p, ,42≁p, and,⊤,⊤,23,a,b,⊤, |p = ⊤,23,⊤, . Let N⊆, τ∈_N({x_1,…,x_n}), and t_1,…,t_n∈_N. By τ[t_1/x_1,…,t_n/x_n] we denote the (ground) term obtained from τ by replacing all occurrences of x_i in τ by t_i, for all i∈{1,…,n}.An event (overand ) is a term of the form a, f, m, for a, f∈ and m ∈, where a is interpreted to be the receiver address and f is the sender address. We denote bythe set of all events. Events overand ^ν are called protoevents and are denoted ^ν. By 2^ (or 2^^ν, respectively) we denote the set of all sequences of (proto)events, including the empty sequence (e.g., , a, f, m, a', f', m', …, etc.). §.§.§ Notations For a sequence t = t_1,…,t_n and a set s we use ts to say that t_1,…,t_n ∈ s.We define . xt. ∃ i: . t_i = x.. For a term y we write ty to denote the sequence t_1,…,t_n,y. For a sequence r = r_1, …, r_m we write t ∪ r to denote the sequence t_1, …, t_n, r_1, …, r_m.For a finite set M with M = {m_1, …,m_n} we use M to denote the term of the form m_1,…,m_n. (The order of the elements does not matter; one is chosen arbitrarily.)A dictionary over X and Y is a term of the formk_1, v_1, …, k_n,v_nwhere k_1, …,k_n ∈ X, v_1,…,v_n ∈ Y. We call every term k_i,v_i, i∈{1,…,n}, an element of the dictionary with key k_i and value v_i.We often write [k_1: v_1, …, k_i:v_i,…,k_n:v_n] instead of k_1, v_1, …, k_n,v_n. We denote the set of all dictionaries over X and Y by [X × Y].We note that the empty dictionary is equivalent to the empty sequence, i.e.,[] =.Figure <ref> shows the short notation for dictionary operations. For a dictionary z = [k_1: v_1, k_2: v_2,…, k_n:v_n] we write k ∈ z to say that there exists i such that k=k_i. We write z[k_j] to refer to the value v_j. (Note that if a dictionary contains two elements k, v and k, v', then the notations and operations for dictionaries apply non-deterministically to one of both elements.) If k ∉z, we set z[k] :=.Given a term t = t_1,…,t_n, we can refer to any subterm using a sequence of integers. The subterm is determined by repeated application of the projection π_i for the integers i in the sequence. We call such a sequence a pointer: A pointer is a sequence of non-negative integers. We write τ.p for the application of the pointer p to the term τ. This operator is applied from left to right. For pointers consisting of a single integer, we may omit the sequence braces for brevity.For the term τ = a,b,c,d,e,f and the pointer p = 3,1, the subterm of τ at the position p is c = 13τ. Also, τ.3.3,1 = τ.3.p = τ.3.3.1 = e. To improve readability, we try to avoid writing, e.g., o2 or 2o in this document. Instead, we will use the names of the components of a sequence that is of a defined form as pointers that point to the corresponding subterms. E.g., if an Origin term is defined as host, protocol and o is an Origin term, then we can write oprotocol instead of 2o or o2. See also Example <ref>. §.§.§ Atomic Processes, Systems and RunsAn atomic process takes its current state and an event as input, and then (non-deterministically) outputs a new state and a set of events. A (generic)is a tuplep = (I^p, Z^p, R^p, s^p_0)where I^p ⊆, Z^p ∈ is a set of states, R^p⊆ (× Z^p) × (2^^ν×(V_process)) (input event and old state map to sequence of output events and new state), and s^p_0∈ Z^p is the initial state of p. For any new state s and any sequence of nonces (η_1, η_2, …) we demand that s[η_1/ν_1, η_2/ν_2, …] ∈ Z^p. A systemis a (possibly infinite) set of .A configuration of a systemis a tuple (S, E, N) where the state of the system S maps every atomic process p∈ to its current state S(p)∈ Z^p, the sequence of waiting events E is an infinite sequence[Here: Not in the sense of terms as defined earlier.] (e_1, e_2, …) of events waiting to be delivered, and N is an infinite sequence of nonces (n_1, n_2, …).For a term a = a_1, …, a_i and a sequence b = (b_1, b_2, …), we define the concatenation as a · b := (a_1, …, a_i, b_1, b_2, …). For a sequence X and a set or sequence Y we define X ∖ Y to be the sequence X where for each element in Y, a non-deterministically chosen occurence of that element in X is removed.A processing step of the systemis of the form(S,E,N)(S', E', N')where * (S,E,N) and (S',E',N') are configurations of ,* e_in = a, f, m∈ E is an event,* p ∈ is a process,* E_out is a sequence (term) of eventssuch that there exists* a sequence (term) E^ν_out⊆ 2^^ν of protoevents,* a term s^ν∈_(V_process), * a sequence (v_1, v_2, …, v_i) of all placeholders appearing in E^ν_out (ordered lexicographically),* a sequence N^ν = (η_1, η_2, …, η_i) of the first i elements in N with * ((e_in, S(p)), (E^ν_out, s^ν)) ∈ R^p and a ∈ I^p,* E_out = E^ν_out[m_1/v_1, …, m_i/v_i]* S'(p) = s^ν[m_1/v_1, …, m_i/v_i] and S'(p') = S(p') for all p' ≠ p* E' = E_out· (E ∖{e_in}) * N' = N ∖ N^ν We may omit the superscript and/or subscript of the arrow. Intuitively, for a processing step, we select one of the processes in , and call it with one of the events in the list of waiting events E. In its output (new state and output events), we replace any occurences of placeholders ν_x by “fresh” nonces from N (which we then remove from N). The output events are then prepended to the list of waiting events, and the state of the process is reflected in the new configuration. Letbe a system, E^0 be sequence of events, and N^0 be a sequence of nonces. A run ρ of a systeminitiated by E^0 with nonces N^0 is a finite sequence of configurations ((S^0, E^0, N^0),…, (S^n, E^n, N^n)) or an infinite sequence of configurations ((S^0, E^0, N^0),…) such that S^0(p) = s_0^p for all p ∈ and (S^i, E^i, N^i)(S^i+1, E^i+1, N^i+1) for all 0 ≤ i < n (finite run) or for all i ≥ 0 (infinite run).We denote the state S^n(p) of a process p at the end of a run ρ by ρ(p). Usually, we will initiate runs with a set E^0 containing infinite trigger events of the form a, a, TRIGGER for each a ∈, interleaved by address.§.§.§ Atomic Dolev-Yao Processes We next define atomic Dolev-Yao processes, for which we require that the messages and states that they output can be computed (more formally, derived) from the current input event and state. For this purpose, we first define what it means to derive a message from given messages.Let M be a set of ground terms. We say that a term m can be derived from M with placeholders V if there exist n≥ 0, m_1,…,m_n∈ M, and τ∈_∅({x_1,…,x_n}∪ V) such that m≡τ[m_1/x_1,…,m_n/x_n]. We denote by d_V(M) the set of all messages that can be derived from M with variables V.For example, a∈ d_{}({a,b,c(k), k}).An atomic Dolev-Yao process (or simply, a DY process) is a tuple p = (I^p, Z^p, R^p, s^p_0) such that (I^p, Z^p, R^p, s^p_0) is an atomic process and (1) Z^p ⊆_ (and hence, s^p_0∈_), and (2) for all events e ∈, sequences of protoevents E, s∈_, s'∈_(V_process), with ((e, s), (E, s')) ∈ R^p it holds true that E, s' ∈ d_V_process({e,s}).An (atomic) attacker process for a set of sender addresses A⊆ is an atomic DY process p = (I, Z, R, s_0) such that for all events e, and s∈_ we have that ((e, s), (E,s')) ∈ R iff s'=e, E, s and E=a_1, f_1, m_1, …, a_n, f_n, m_n with n ∈ℕ, a_1,…,a_n∈, f_0,…,f_n∈ A, m_1,…,m_n∈ d_V_process({e,s}).§.§ ScriptsWe define scripts, which model client-side scripting technologies, such as JavaScript. Scripts are defined similarly to DY processes.By V_script = {λ_1, …} we denote an infinite set of variables used in scripts.A script is a relation R⊆×(V_script) such that for all s ∈, s' ∈(V_script) with (s, s') ∈ R it follows that s'∈ d_V_script(s). A script is called by the browser which provides it with state information (such as the script's last state and limited information about the browser's state) s. The script then outputs a term s', which represents the new internal state and some command which is interpreted by the browser. The term s' may contain variables λ_1, … which the browser will replace by (otherwise unused) placeholders ν_1,… which will be replaced by nonces once the browser DY process finishes (effectively providing the script with a way to get “fresh” nonces).Similarly to an attacker process, we define the attacker script : The attacker scriptoutputs everything that is derivable from the input, i.e., ={(s, s')| s∈, s'∈ d_V_script(s)}. §.§ Web System The web infrastructure and web applications are formalized by what is called a web system. A web system contains, among others, a (possibly infinite) set of DY processes, modeling web browsers, web servers, DNS servers, and attackers (which may corrupt other entities, such as browsers).A web system =(, ,𝗌𝖼𝗋𝗂𝗉𝗍, E^0) is a tuple with its components defined as follows:The first component, , denotes a system (a set of DY processes) and is partitioned into the sets 𝖧𝗈𝗇, 𝖶𝖾𝖻, and 𝖭𝖾𝗍 of honest, web attacker, and network attacker processes, respectively.Every p∈𝖶𝖾𝖻∪𝖭𝖾𝗍 is an attacker process for some set of sender addresses A⊆. For a web attacker p∈𝖶𝖾𝖻, we require its set of addresses I^p to be disjoint from the set of addresses of all other web attackers and honest processes, i.e., I^p∩ I^p' = ∅ for all p' ∈𝖧𝗈𝗇∪𝖶𝖾𝖻. Hence, a web attacker cannot listen to traffic intended for other processes. Also, we require that A=I^p, i.e., a web attacker can only use sender addresses it owns. Conversely, a network attacker may listen to all addresses (i.e., no restrictions on I^p) and may spoof all addresses (i.e., the set A may be ). Every p ∈𝖧𝗈𝗇 is a DY process which models either a web server, a web browser, or a DNS server, as further described in the following subsections. Just as for web attackers, we require that p does not spoof sender addresses and that its set of addresses I^p is disjoint from those of other honest processes and the web attackers. The second component, , is a finite set of scripts such that ∈. The third component, 𝗌𝖼𝗋𝗂𝗉𝗍, is an injective mapping fromto 𝕊, i.e., by 𝗌𝖼𝗋𝗂𝗉𝗍 every s∈ is assigned its string representation 𝗌𝖼𝗋𝗂𝗉𝗍(s). Finally, E^0 is an(infinite) sequence of events, containing an infinite number of events of the form a,a, for every a ∈⋃_p∈ I^p.A run ofis a run ofinitiated by E^0. § MESSAGE AND DATA FORMATS We now provide some more details about data and message formats that are needed for the formal treatment of the web model and the analysispresented in the following.§.§ URLs A URL is a term of the form, protocol, host, path, parameters, fragmentwith protocol ∈{, } (for plain (HTTP) and secure (HTTPS)), host∈, path∈𝕊, parameters∈𝕊, and fragment∈. The set of all valid URLs is . The fragment part of a URL can be omitted when writing the URL. Its value is then defined to be .For the URL u = , a, b, c, d, uprotocol = a. If, in the algorithm described later, we say upath := e then u = , a, b, c, e afterwards. §.§ Origins An origin is a term of the form host, protocol with host∈ and protocol∈{, }. We writefor the set of all origins.For example, FOO, is the HTTPS origin for the domain FOO, while BAR, is the HTTP origin for the domain BAR. §.§ Cookies A cookie is a term of the form name, content where name∈, and content is a term of the form value, secure, session, httpOnly where value∈, secure, session, httpOnly∈{, }. We writefor the set of all cookies and ^ν for the set of all cookies where names and values are defined over (V). If the secure attribute of a cookie is set, the browser will not transfer this cookie over unencrypted HTTP connections. If the session flag is set, this cookie will be deleted as soon as the browser is closed. The httpOnly attribute controls whether JavaScript has access to this cookie.Note that cookies of the form described here are only contained in HTTP(S) requests. In responses, only the components name and value are transferred as a pairing of the form name, value. §.§ HTTP Messages An HTTP request is a term of the form shown in (<ref>). An HTTP response is a term of the form shown in (<ref>).nonce=nonce, method=method, xhost=host, xpath=path, parameters=parameters, headers=headers, xbody=body nonce=nonce, status=status, headers=headers, xbody=bodyThe components are defined as follows: * nonce∈ serves to map each response to the corresponding request * method∈ is one of the HTTP methods.* host∈ is the host name in the HOST header of HTTP/1.1.* path∈𝕊 is a string indicating the requested resource at the server side* status∈𝕊 is the HTTP status code (i.e., a number between 100 and 505, as defined by the HTTP standard)* parameters∈𝕊 contains URL parameters* headers∈𝕊, containing request/response headers. The dictionary elements are terms of one of the following forms:* Origin, o where o is an origin,* SetCookie, c where c is a sequence of cookies,* Cookie, c where c ∈, 𝕊 (note that in this header, only names and values of cookies are transferred),* Location, l where l ∈,* Referer, r where r ∈,* StrictTransportSecurity,,* Authorization, username,password where username, password∈𝕊,* ReferrerPolicy, p where p ∈{noreferrer, origin} * body∈ in requests and responses. We write / for the set of all HTTP requests or responses, respectively. [HTTP Request and Response]r :=⟨,n_1,,example.com,/show,index,1,[Origin: example.com, ],foo, bar⟩ s := nonce=n_1, status=200, headers=SetCookie,SID,n_2,,,, xbody=somescript,xAn HTTPrequest for the URL <http://example.com/show?index=1> is shown in (<ref>), with an Origin header and a body that contains foo,bar. A possible response is shown in (<ref>), which contains an httpOnly cookie with name SID and value n_2 as well as the string representation somescript of the script 𝗌𝖼𝗋𝗂𝗉𝗍^-1(somescript) (which should be an element of ) and its initial state x. §.§.§ Encrypted HTTP Messages For HTTPS, requests are encrypted using the public key of the server.Such a request contains an (ephemeral) symmetric key chosen by the client that issued the request. The server is supported to encrypt the response using the symmetric key. An encrypted HTTP request is of the form m, k'k, where k ∈ terms, k' ∈, and m ∈. The corresponding encrypted HTTP response would be of the form m'k', where m' ∈. We call the sets of all encrypted HTTP requests and responsesor , respectively. We say that an HTTP(S) response matches or corresponds to an HTTP(S) request if both terms contain the same nonce.rk'(k_example.com) sk'The term (<ref>) shows an encrypted request (with r as in (<ref>)). It is encrypted using the public key (k_example.com).The term (<ref>) is a response (with s as in (<ref>)). It is encrypted symmetrically using the (symmetric) key k' that was sent in the request (<ref>).§.§ DNS Messages A DNS request is a term of the form , domain, n where domain ∈, n∈. We call the set of all DNS requests .A DNS response is a term of the form , domain, result, n with domain ∈, result∈, n∈. We call the set of all DNS responses . DNS servers are supposed to include the nonce they received in a DNS request in the DNS response that they send back so that the party which issued the request can match it with the request. §.§ DNS Servers Here, we consider a flat DNS model in which DNS queries are answered directly by one DNS server and always with the same address for a domain. A full (hierarchical) DNS system with recursive DNS resolution, DNS caches, etc. could also be modeled to cover certain attacks on the DNS system itself. A DNS server d (in a flat DNS model) is modeled in a straightforward way as an atomic DY process (I^d, {s^d_0}, R^d, s^d_0). It has a finite set of addresses I^d and its initial (and only) state s^d_0 encodes a mapping from domain names to addresses of the forms^d_0=⟨domain_1,a_1,domain_2, a_2, …⟩ .DNS queries are answered according to this table (otherwise ignored).§ DETAILED DESCRIPTION OF THE BROWSER MODEL Following the informal description of the browser model in Section <ref>, we now present a formal model. We start by introducing some notation and terminology.§.§ Notation and Terminology (Web Browser State) Before we can define the state of a web browser, we first have to define windows and documents. A window is a term of the form w = nonce, documents, opener with nonce∈, documents (defined below), opener∈∪{} where dactive = for exactly one d documents if documents is not empty (we then call d the active document of w). We writefor the set of all windows. We write wactivedocument to denote the active document inside window w if it exists andelse.We will refer to the window nonce as (window) reference. The documents contained in a window term to the left of the active document are the previously viewed documents (available to the user via the “back” button) and the documents in the window term to the right of the currently active document are documents available via the “forward” button.A window a may have opened a top-level window b (i.e., a window term which is not a subterm of a document term). In this case, the opener part of the term b is the nonce of a, i.e., bopener = anonce.A document d is a term of the formnonce, location, headers, referrer, script, scriptstate,scriptinputs, subwindows, activewhere nonce∈, location∈, headers∈𝕊, referrer∈∪{}, script∈, scriptstate∈, scriptinputs∈, subwindows, active∈{, }. A limited document is a term of the form nonce, subwindows with nonce, subwindows as above. A window w subwindows is called a subwindow (of d). We writefor the set of all documents. For a document term d we write d.origin to denote the origin of the document, i.e., the term d.location.host, d.location.protocol∈.We will refer to the document nonce as (document) reference. For two window terms w and w' we write ww' if w w'activedocumentsubwindows. We writefor the transitive closure.§.§ Web Browser State We can now define the set of states of web browsers. Note that we use the dictionary notation that we introduced in Definition <ref>.The set of states Z_webbrowser of a web browser atomic Dolev-Yao process consists of the terms of the form⟨windows, ids, secrets, cookies, localStorage, sessionStorage, keyMapping,sts, DNSaddress, pendingDNS, pendingRequests, isCorrupted ⟩where * windows,* ids,* secrets∈,* cookies is a dictionary overand sequences of , * localStorage∈,* sessionStorage∈OR for OR := {o,r|o ∈,r ∈},* keyMapping∈,* sts,* DNSaddress∈,* pendingDNS∈,* pendingRequests∈ ,* and isCorrupted∈{, , }. §.§ Description of the Web Browser Relation We will now define the relation R_webbrowser of a standard HTTP browser. We first introduce some notations and then describe the functions that are used for defining the browser main algorithm. We then define the browser relation.§.§.§ Helper FunctionsIn the following description of the web browser relation R_webbrowser we use the helper functions 𝖲𝗎𝖻𝗐𝗂𝗇𝖽𝗈𝗐𝗌, 𝖣𝗈𝖼𝗌, 𝖢𝗅𝖾𝖺𝗇, 𝖢𝗈𝗈𝗄𝗂𝖾𝖬𝖾𝗋𝗀𝖾 and 𝖠𝖽𝖽𝖢𝗈𝗈𝗄𝗂𝖾. Subwindows Given a browser state s, 𝖲𝗎𝖻𝗐𝗂𝗇𝖽𝗈𝗐𝗌(s) denotes the set of all pointers[Recall the definition of a pointer in Definition <ref>.] to windows in the window list swindows, their active documents, and (recursively) the subwindows of these documents. We exclude subwindows of inactive documents and their subwindows. With 𝖣𝗈𝖼𝗌(s) we denote the set of pointers to all active documents in the set of windows referenced by 𝖲𝗎𝖻𝗐𝗂𝗇𝖽𝗈𝗐𝗌(s). For a browser state s we denote by 𝖲𝗎𝖻𝗐𝗂𝗇𝖽𝗈𝗐𝗌(s) the minimal set of pointers that satisfies the following conditions: (1) For all windows w swindows there is a p∈𝖲𝗎𝖻𝗐𝗂𝗇𝖽𝗈𝗐𝗌(s) such that sp = w. (2) For all p∈𝖲𝗎𝖻𝗐𝗂𝗇𝖽𝗈𝗐𝗌(s), the active document d of the window sp and every subwindow w of d there is a pointer p'∈𝖲𝗎𝖻𝗐𝗂𝗇𝖽𝗈𝗐𝗌(s) such that sp' = w.Given a browser state s, the set 𝖣𝗈𝖼𝗌(s) of pointers to active documents is the minimal set such that for every p∈𝖲𝗎𝖻𝗐𝗂𝗇𝖽𝗈𝗐𝗌(s), there is a pointer p'∈𝖣𝗈𝖼𝗌(s) with sp' = spactivedocument. By 𝖲𝗎𝖻𝗐𝗂𝗇𝖽𝗈𝗐𝗌^+(s) and 𝖣𝗈𝖼𝗌^+(s) we denote the respective sets that also include the inactive documents and their subwindows. Clean The function 𝖢𝗅𝖾𝖺𝗇 will be used to determine which information about windows and documents the script running in the document d has access to.Let s be a browser state and d a document. By 𝖢𝗅𝖾𝖺𝗇(s, d) we denote the term that equals swindows but with (1) all inactive documents removed (including their subwindows etc.), (2) all subterms that represent non-same-origin documents w.r.t. d replaced by a limited document d' with the same nonce and the same subwindow list, and (3) the values of the subterms headers for all documents set to . (Note that non-same-origin documents on all levels are replaced by their corresponding limited document.)CookieMerge The function 𝖢𝗈𝗈𝗄𝗂𝖾𝖬𝖾𝗋𝗀𝖾 merges two sequences of cookies together: When used in the browser, oldcookies is the sequence of existing cookies for some origin, newcookies is a sequence of new cookies that was output by some script. The sequences are merged into a set of cookies using an algorithm that is based on the Storage Mechanism algorithm described in RFC6265. For a sequence of cookies (with pairwise different names) oldcookies and a sequence of cookies newcookies, the set 𝖢𝗈𝗈𝗄𝗂𝖾𝖬𝖾𝗋𝗀𝖾(oldcookies, newcookies) is defined by the following algorithm: From newcookies remove all cookies c that have c.content.httpOnly≡. For any c, c' newcookies, cname≡c'name, remove the cookie that appears left of the other in newcookies. Let m be the set of cookies that have a name that either appears in oldcookies or in newcookies, but not in both. For all pairs of cookies (c_old, c_new) with c_oldoldcookies, c_newnewcookies, c_oldname≡c_newname, add c_new to m if c_oldcontenthttpOnly≡ and add c_old to m otherwise. The result of 𝖢𝗈𝗈𝗄𝗂𝖾𝖬𝖾𝗋𝗀𝖾(oldcookies, newcookies) is m.AddCookieThe function 𝖠𝖽𝖽𝖢𝗈𝗈𝗄𝗂𝖾 adds a cookie c received in an HTTP response to the sequence of cookies contained in the sequence oldcookies. It is again based on the algorithm described in RFC6265 but simplified for the use in the browser model. For a sequence of cookies (with pairwise different names) oldcookies and a cookie c, the sequence 𝖠𝖽𝖽𝖢𝗈𝗈𝗄𝗂𝖾(oldcookies, c) is defined by the following algorithm: Let m := oldcookies. Remove any c' from m that has cname≡c'name. Append c to m and return m.NavigableWindowsThe function 𝖭𝖺𝗏𝗂𝗀𝖺𝖻𝗅𝖾𝖶𝗂𝗇𝖽𝗈𝗐𝗌 returns a set of windows that a document is allowed to navigate. We closely follow <cit.>, Section 5.1.4 for this definition.The set 𝖭𝖺𝗏𝗂𝗀𝖺𝖻𝗅𝖾𝖶𝗂𝗇𝖽𝗈𝗐𝗌(w, s') is the set W⊆𝖲𝗎𝖻𝗐𝗂𝗇𝖽𝗈𝗐𝗌(s') of pointers to windows that the active document in w is allowed to navigate. The set W is defined to be the minimal set such that for every w'∈𝖲𝗎𝖻𝗐𝗂𝗇𝖽𝗈𝗐𝗌(s') the following is true:* If s'w'activedocumentorigin≡s'wactivedocumentorigin (i.e., the active documents in w and w' are same-origin), then w'∈W, and* If s'ws'w' ∧ ∄ w”∈𝖲𝗎𝖻𝗐𝗂𝗇𝖽𝗈𝗐𝗌(s') with s'w's'w” (w' is a top-level window and w is an ancestor window of w'), then w'∈W, and* If ∃ p∈𝖲𝗎𝖻𝗐𝗂𝗇𝖽𝗈𝗐𝗌(s') such that s'w's'p∧ s'pactivedocumentorigin = s'wactivedocumentorigin (w' is not a top-level window but there is an ancestor window p of w' with an active document that has the same origin as the active document in w), then w'∈W, and* If ∃ p∈𝖲𝗎𝖻𝗐𝗂𝗇𝖽𝗈𝗐𝗌(s') such that s'w'opener = s'pnonce ∧ p∈W (w' is a top-level window—it has an opener—and w is allowed to navigate the opener window of w', p), then w'∈W. §.§.§ Notations for Functions and AlgorithmsWe use the following notations to describe the browser algorithms: Non-deterministic chosing and iteration The notation let n ← N is used to describe that n is chosen non-deterministically from the set N. We write for each s ∈ M do to denote that the following commands (until end for) are repeated for every element in M, where the variable s is the current element. The order in which the elements are processed is chosen non-deterministically.We write, for example,for some variables x,y, a string Constant, and some term t to express that x := 2t, and y := 3t if Constant≡1t and if |Constant,x,y| = |t|, and that otherwise x and y are not set and doSomethingElse is executed. Stop without output We write stop (without further parameters) to denote that there is no output and no change in the state. Placeholders In several places throughout the algorithms presented next we use placeholders to generate “fresh” nonces as described in our communication model (see Definition <ref>). Figure <ref> shows a list of all placeholders used. §.§.§ Functions In the description of the following functions, we use a, f, m, and s as read-only global input variables. All other variables are local variables or arguments.* The function 𝖦𝖤𝖳𝖭𝖠𝖵𝖨𝖦𝖠𝖡𝖫𝖤𝖶𝖨𝖭𝖣𝖮𝖶 (Algorithm <ref>) is called by the browser to determine the window that is actually navigated when a script in the window s'.w provides a window reference for navigation (e.g., for opening a link). When it is given a window reference (nonce) window, this function returns a pointer to a selected window term in s': * If window is the string , a new window is created and a pointer to that window is returned.* If window is a nonce (reference) and there is a window term with a reference of that value in the windows in s', a pointer w' to that window term is returned, as long as the window is navigable by the current window's document (as defined by 𝖭𝖺𝗏𝗂𝗀𝖺𝖻𝗅𝖾𝖶𝗂𝗇𝖽𝗈𝗐𝗌 above).In all other cases, w is returned instead (the script navigates its own window).* The function 𝖦𝖤𝖳𝖶𝖨𝖭𝖣𝖮𝖶 (Algorithm <ref>) takes a window reference as input and returns a pointer to a window as above, but it checks only that the active documents in both windows are same-origin. It creates no new windows.* The function 𝖢𝖠𝖭𝖢𝖤𝖫𝖭𝖠𝖵 (Algorithm <ref>) is used to stop any pending requests for a specific window. From the pending requests and pending DNS requests it removes any requests with the given window reference n. * The function 𝖧𝖳𝖳𝖯_𝖲𝖤𝖭𝖣 (Algorithm <ref>) takes an HTTP request message as input, adds cookie and origin headers to the message, creates a DNS request for the hostname given in the request and stores the request in s'pendingDNS until the DNS resolution finishes. For normal HTTP requests, reference is a window reference. For , reference is a value of the form document, nonce where document is a document reference and nonce is some nonce that was chosen by the script that initiated the request. url contains the full URL of the request (this is mainly used to retrieve the protocol that should be used for this message, and to store the fragment identifier for use after the document was loaded). origin is the origin header value that is to be added to the HTTP request.* The functions 𝖭𝖠𝖵𝖡𝖠𝖢𝖪 (Algorithm <ref>) and 𝖭𝖠𝖵𝖥𝖮𝖱𝖶𝖠𝖱𝖣 (Algorithm <ref>), navigate a window forward or backward. More precisely, they deactivate one document and activate that document's succeeding document or preceding document, respectively. If no such successor/predecessor exists, the functions do not change the state.* The function 𝖱𝖴𝖭𝖲𝖢𝖱𝖨𝖯𝖳 (Algorithm <ref>) performs a script execution step of the script in the document s'd (which is part of the window s'w). A new script and document state is chosen according to the relation defined by the script and the new script and document state is saved. Afterwards, the command that the script issued is interpreted.* The function 𝖯𝖱𝖮𝖢𝖤𝖲𝖲𝖱𝖤𝖲𝖯𝖮𝖭𝖲𝖤 (Algorithm <ref>) is responsible for processing an HTTP response (response) that was received as the response to a request (request) that was sent earlier. In reference, either a window or a document reference is given (see explanation for Algorithm <ref> above). requestUrl contains the URL used when retrieving the document.The function first saves any cookies that were contained in the response to the browser state, then checks whether a redirection is requested (Location header). If that is not the case, the function creates a new document (for normal requests) or delivers the contents of the response to the respective receiver (for responses). §.§.§ DefinitionWe can now define the relation R_webbrowser of a web browser atomic process as follows:The pair ((a,f,m, s), (M, s')) belongs to R_webbrowser iff the non-deterministic Algorithm <ref> (or any of the functions called therein), when given (a,f,m, s) as input, terminates with stop M, s', i.e., with output M and s'. Recall that a,f,m is an (input) event and s is a (browser) state, M is a sequence of (output) protoevents, and s' is a new (browser) state (potentially with placeholders for nonces). §.§ Definition of Web BrowsersFinally, we define web browser atomic Dolev-Yao processes as follows:A web browser atomic Dolev-Yao process is an atomic Dolev-Yao process of the form p = (I^p, Z_webbrowser, R_webbrowser, s_o^p) for a set I^p of addresses, Z_webbrowser and R_webbrowser as defined above, and an initial state s_0^p ∈ Z_webbrowser. § GENERIC HTTPS SERVER MODELThis model will be used as the base for all servers in the following. It makes use of placeholder algorithms that are later superseded by more detailed algorithms to describe a concrete relation for an HTTPS server. The state of each HTTPS server that is an instantiation of this relation must contain at least the following subterms: pendingDNS∈, pendingRequests∈ (both containing arbitrary terms), DNSaddress∈ (containing the IP address of a DNS server), keyMapping∈ (containing a mapping from domains to public keys), tlskeys∈ (containing a mapping from domains to private keys), and corrupt∈ (eitherif the server is not corrupted, or an arbitrary term otherwise). We note that in concrete instantiations of the generic HTTPS server model, there is no need to extract information from these subterms or alter these subterms. Let ν_n0 and ν_n1 denote placeholders for nonces that are not used in the concrete instantiation of the server. We now define the default functions of the generic web server in Algorithms <ref>–<ref>, and the main relation in Algorithm <ref>.§ FORMAL MODEL OF OPENID CONNECT WITH A NETWORK ATTACKERWe here present the full details of our formal model of OIDC which we use to analyze the authentication and authorization properties. This model contains a network attacker. We will later derive from this model a model where the network attacker is replaced by a web attacker. We use this modified model for the session integrity properties.We model OIDC as a web system (in the sense of Appendix <ref>). We call a web system ^n=(, , 𝗌𝖼𝗋𝗂𝗉𝗍, E^0) an OIDC web system with a network attacker if it is of the form described in what follows. §.§ OutlineThe system =𝖧𝗈𝗇∪𝖭𝖾𝗍 consists of a network attacker process (in 𝖭𝖾𝗍), a finite set B of web browsers, a finite set RP of web servers for the relying parties, a finite set IDP of web servers for the identity providers, with 𝖧𝗈𝗇 := B∪RP∪IDP. More details on the processes inare provided below. We do not model DNS servers, as they are subsumed by the network attacker.Figure <ref> shows the set of scriptsand their respective string representations that are defined by the mapping 𝗌𝖼𝗋𝗂𝗉𝗍.The set E^0 contains only the trigger events as specified in Appendix <ref>.This outlines ^n. We will now define the DY processes in ^n and their addresses, domain names, and secrets in more detail.§.§ Addresses and Domain NamesThe setcontains for the network attacker in Net, every relying party in RP, every identity provider in IDP, and every browser in B a finite set of addresses each. Bywe denote the corresponding assignment from a process to its address. The setcontains a finite set of domains for every relying party in RP, every identity provider in IDP, and the network attacker in Net. Browsers (in B) do not have a domain.Byandwe denote the assignments from atomic processes to sets ofand , respectively. §.§ Keys and SecretsThe setof nonces is partitioned into five sets, an infinite sequence N, an finite set K_TLS, an finite set K_sign, and a finite set 𝖯𝖺𝗌𝗌𝗐𝗈𝗋𝖽𝗌. We therefore have =N_infinite sequence∪̇K_TLS_finite∪̇K_sign_finite∪̇𝖯𝖺𝗌𝗌𝗐𝗈𝗋𝖽𝗌_finite.These sets are used as follows: * The set N contains the nonces that are available for each DY process in(it can be used to create a run of ). * The set K_TLS contains the keys that will be used for TLS encryption. Let → K_TLS be an injective mapping that assigns a (different) private key to every domain. For an atomic DY process p we define tlskeys^p = {d, (d)| d ∈(p)}. * The set K_sign contains the keys that will be used by IdPs for signing id tokens. Let 𝗌𝗂𝗀𝗇𝗄𝖾𝗒IDP→ K_sign be an injective mapping that assigns a (different) signing key to every IdP.* The set 𝖯𝖺𝗌𝗌𝗐𝗈𝗋𝖽𝗌 is the set of passwords (secrets) the browsers share with the identity providers. These are the passwords the users use to log in at the IdPs. §.§ Identities and PasswordsIdentites consist, similar to email addresses, of a user name and a domain part. For our model, this is defined as follows:An identity (email address) i is a term of the form name,domain with name∈𝕊 and domain∈.Letbe the finite set of identities. We say that an ID is governed by the DY process to which the domain of the ID belongs. Formally, we define the mapping : →, name, domain↦^-1(domain). By ^y we denote the set 𝗀𝗈𝗏𝖾𝗋𝗇𝗈𝗋^-1(y). The governor of an ID will usually be an IdP, but could also be the attacker. Besides , we define the following mappings:* By :→𝖯𝖺𝗌𝗌𝗐𝗈𝗋𝖽𝗌 we denote the bijective mapping that assigns secrets to all identities. * Let : 𝖯𝖺𝗌𝗌𝗐𝗈𝗋𝖽𝗌→B denote the mapping that assigns to each secret a browser that owns this secret. Now, we define the mapping : →B, i ↦((i)), which assigns to each identity the browser that owns this identity (we say that the identity belongs to the browser).§.§ CorruptionRPs and IdPs can become corrupted: If they receive the message , they start collecting all incoming messages in their state and (upon triggering) send out all messages that are derivable from their state and collected input messages, just like the attacker process. We say that an RP or an IdP is honest if the according part of their state (s.corrupt) is , and that they are corrupted otherwise.We are now ready to define the processes inas well as the scripts inin more detail.§.§ Network AttackersAs mentioned, the network attacker na is modeled to be a network attacker as specified in Appendix <ref>. We allow it to listen to/spoof all available IP addresses, and hence, define I^na =. The initial state is s_0^na = attdoms, tlskeys, signkeys, where attdoms is a sequence of all domains along with the corresponding private keys owned by the attacker na, tlskeys is a sequence of all domains and the corresponding public keys, and signkeys is a sequence containing all public signing keys for all IdPs. §.§ BrowsersEach b ∈B is a web browser atomic Dolev-Yao process as defined in Definition <ref>, with I^b := (b) being its addresses.To define the inital state, first let _b := ^-1(b) bethe set of all IDs of b. We then define the set of passwords that a browser b gives to an origin o: If the origin belongs to an IdP, then the user's passwords of this IdP are contained in the set. To define this mapping in the initial state, we first define for some process p𝖲𝖾𝖼𝗋𝖾𝗍𝗌^b,p = { s | b = 𝗈𝗐𝗇𝖾𝗋𝖮𝖿𝖲𝖾𝖼𝗋𝖾𝗍(s)∧(∃i : s = 𝗌𝖾𝖼𝗋𝖾𝗍𝖮𝖿𝖨𝖣(i) ∧ i ∈𝖨𝖣^p) } .Then, the initial state s_0^b is defined as follows: the key mapping maps every domain to its public (TLS) key, according to the mapping ; the DNS address is an address of the network attacker; the list of secrets contains an entry d,, 𝖲𝖾𝖼𝗋𝖾𝗍𝗌^b,p for each p ∈RP∪IDP and d ∈𝖽𝗈𝗆(p); ids is _b; sts is empty. §.§ Relying Parties A relying party r ∈RP is a web server modeled as an atomic DY process (I^r, Z^r, R^r, s^r_0) with the addresses I^r := (r).Next, we define the set Z^r of states of r and the initial state s^r_0 of r. A state s∈ Z^r of an RP r is a term of the form ⟨DNSAddress, pendingDNS, pendingRequests, corrupt, keyMapping, tlskeys,sessions, issuerCache, oidcConfigCache, jwksCache, clientCredentialsCache⟩ with DNSaddress∈, pendingDNS∈, pendingRequests∈, corrupt∈, keyMapping∈ , tlskeys∈K_TLS (all former components as in Definition <ref>), sessions∈, issuerCache∈, oidcConfigCache∈, and jwksCache∈.An initial state s^r_0 of r is a state of r with s^r_0.pendingDNS≡, s^r_0.pendingRequests≡, s^r_0.corrupt≡,s^r_0.keyMapping being the same as the keymapping for browsers above, s^r_0.tlskeys≡tlskeys^r, s^r_0.sessions≡, s^r_0.issuerCache≡, s^r_0.oidcConfigCache≡, s^r_0.jwksCache≡, and s^r_0.clientCredentialsCache≡.We now specify the relation R^r: This relation is based on our model of generic HTTPS servers (see Appendix <ref>). Hence we only need to specify algorithms that differ from or do not exist in the generic server model. These algorithms are defined in Algorithms <ref>–<ref>. (Note that in several places throughout these algorithms we use placeholders to generate “fresh” nonces as described in our communication model (see Definition <ref>). Figure <ref> shows a list of all placeholders used.)The scripts that are used by the RP are described in Algorithms <ref> and <ref>. In these scripts, to extract the current URL of a document, the function 𝖦𝖤𝖳𝖴𝖱𝖫(tree,docnonce) is used. We define this function as follows: It searches for the document with the identifier docnonce in the (cleaned) tree tree of the browser's windows and documents. It then returns the URL u of that document. If no document with nonce docnonce is found in the tree tree,is returned.§.§ Identity Providers An identity provider i ∈IDP is a web server modeled as an atomic process (I^i, Z^i, R^i, s_0^i) with the addresses I^i := (i). Next, we define the set Z^i of states of i and the initial state s^i_0 of i. A state s∈ Z^i of an IdP i is a term of the form ⟨DNSAddress, pendingDNS, pendingRequests, corrupt, keyMapping, tlskeys,registrationRequests, (sequence of terms) clients, (dict from nonces to terms) records, (sequence of terms) jwk⟩ (signing key (only one)) with DNSaddress∈, pendingDNS∈, pendingRequests∈, corrupt∈, keyMapping∈ , tlskeys∈K_TLS (all former components as in Definition <ref>), registrationRequests∈, clients∈, records∈, and jwk∈ K_sign.An initial state s^i_0 of i is a state of i with s^i_0.pendingDNS≡, s^i_0.pendingRequests≡, s^i_0.corrupt≡, s^i_0.keyMapping being the same as the keymapping for browsers above, s^i_0.tlskeys≡tlskeys^i, s^i_0.registrationRequests≡, s^i_0.clients≡an, s^i_0.records≡, and s^i_0.jwk≡𝗌𝗂𝗀𝗇𝗄𝖾𝗒(i). We now specify the relation R^i: As for the RPs above, this relation is based on our model of generic HTTPS servers (see Appendix <ref>). We specify algorithms that differ from or do not exist in the generic server model in Algorithms <ref> and <ref>. As above, Figure <ref> shows a list of all placeholders used. Algorithm <ref> shows the script 𝑠𝑐𝑟𝑖𝑝𝑡_𝑖𝑑𝑝_𝑓𝑜𝑟𝑚 that is used by IdPs.§ FORMAL MODEL OF OPENID CONNECT WITH WEB ATTACKERSWe now derive ^w (an OIDC web system with web attackers) from ^n by replacing the network attacker with a finite set of web attackers. (Note that we more generally speak of an OIDC web system if it is not important what kind of attacker the web system contains.) An OIDC web system with web attackers, ^w, is an OIDC web system ^n=(, , 𝗌𝖼𝗋𝗂𝗉𝗍, E^0) with the following changes: * We have = 𝖧𝗈𝗇∪𝖶𝖾𝖻, in particular, there is no network attacker. The set 𝖶𝖾𝖻 contains a finite number of web attacker processes. The set 𝖧𝗈𝗇 is as described above, and additionally contains a DNS server d as defined below.* The set of IP addresses 𝖨𝖯𝗌 contains no IP addresses for the network attacker, but instead a finite set of IP addresses for each web attacker.* The set of Domains 𝖣𝗈𝗆𝗌 contains no domains for the network attacker, but instead a finite set of domains for each web attacker. * All honest parties use the DNS server d as their DNS server. §.§ DNS ServerThe DNS server d is a DNS server as defined in Definition <ref>. Its initial state s_0^d contains only pairings D, i such that i ∈𝖺𝖽𝖽𝗋(𝖽𝗈𝗆^-1(D)), i.e., any domain is resolved to an IP address belonging to the owner of that domain (as defined in Appendix <ref>). §.§ Web AttackersWeb attackers, as opposed to network attackers, can only use their own IP addresses for listening to and sending messages. Therefore, for any web attacker process w we have that I^w = 𝖺𝖽𝖽𝗋(w). The inital states of web attackers are defined parallel to those of network attackers, i.e., the initial state for a web attacker process w is s_0^w = attdoms^w, tlskeys, signkeys, where attdoms^w is a sequence of all domains along with the corresponding private keys owned by the attacker w, tlskeys is a sequence of all domains and the corresponding public keys, and signkeys is a sequence containing all public signing keys for all IdPs.§ FORMAL SECURITY PROPERTIES The security properties for OIDC are formally defined as follows.§.§ AuthenticationIntuitively, authentication for ^n means that an attacker should not be able to login at an (honest) RP under the identity of a user unless certain parties involved in the login process are corrupted. As explained above, being logged in at an RP under some user identity means to have obtained a service token for this identity from the RP.We say that there is a service session identified by a nonce n for an identity id at some RP r in a configuration (S, E, N) of a run ρ of an OIDC web system iff there exists some session id x and a domain d ∈𝖽𝗈𝗆(𝗀𝗈𝗏𝖾𝗋𝗇𝗈𝗋(id)) such that S(r).sessions[x][loggedInAs] ≡d, id and S(r).sessions[x][serviceSessionId] ≡ n.Let ^n be an OIDC web system with a network attacker. We say that ^n is secure w.r.t. authentication iff for every run ρ of ^n, every configuration (S, E, N) in ρ, every r∈RP that is honest in S, every browser b that is honest in S, every identity id∈𝖨𝖣 with 𝗀𝗈𝗏𝖾𝗋𝗇𝗈𝗋(id) being an honest IdP, every service session identified by some nonce n for id at r, n is not derivable from the attackers knowledge in S (i.e., n ∉d_∅(S(attacker))).§.§ AuthorizationIntuitively, authorization for ^n means that an attacker should not be able to obtain or use a protected resource available to some honest RP at an IdP for some user unless certain parties involved in the authorization process are corrupted.We say that a client id c has been issued to r by i iff i has sent a response to a registration request from r in Line <ref> of Algorithm <ref> and this response contains c in itsbody under the dictionary key client_id.Let ^n be an OIDC web system with a network attacker. We say that ^n is secure w.r.t. authorization iff for every run ρ of ^n, every configuration (S, E, N) in ρ, every r∈RP that is honest in S, every i∈IdP that is honest in S, every browser b that is honest in S, every identity id∈𝖨𝖣^i owned by b, every nonce n, every term xS(i).records with x[subject] ≡id, nx[access_tokens], and the client id x[client_id] has been issued by i to r, we have that n is not derivable from the attackers knowledge in S (i.e., n ∉d_∅(S(attacker))).§.§ Session Integrity for Authentication and AuthorizationThe two session integrity properties capture that an attacker should be unable to forcefully log a user in to some RP. This includes attacks such as CSRF and session swapping. §.§.§ Session Integrity Property for AuthenticationThis security property captures that (a) a user should only be logged in when the user actually expressed the wish to start an OIDC flow before, and (b) if a user expressed the wish to start an OIDC flow using some honest identity provider and a specific identity, then user is not logged in under a different identity.We first need to define notations for the processing steps that represent important events during a flow of an OIDC web system. For a run ρ of an OIDC web system with web attacker ^w we say that a browser b was authenticated to an RP r using an IdP i and an identity u in a login session identified by a nonce lsid in processing step Q in ρ withQ = (S, E, N)(S', E', N') (for some S, S', E, E', N, N') and some event y,y',m∈ E_out such that m is an HTTPS response matching an HTTPS request sent by b to r and we have that in the headers of m there is a header of the form SetCookie, [serviceSessionId:ssid,⊤,⊤,⊤] for some nonce ssid and we have that there is a term g such that S(r).sessions[lsid] ≡ g, g[serviceSessionId] ≡ssid, and g[loggedInAs] ≡d, u with d ∈𝖽𝗈𝗆(i). We then write 𝗅𝗈𝗀𝗀𝖾𝖽𝖨𝗇_ρ^Q(b, r, u, i, lsid). For a run ρ of an OIDC web system with web attacker ^w we say that the user of the browser b started a login session identified by a nonce lsid at the RP r in a processing step Q in ρ if (1) in that processing step, the browser b was triggered, selected a document loaded from an origin of r, executed the script script_rp_index in that document, and in that script, executed the Line <ref> of Algorithm <ref>, and (2) r sends an HTTPS response corresponding to the HTTPS request sent by b in Q and in that response, there is a header of the form SetCookie, [sessionId:lsid,⊤,⊤,⊤]. We then write 𝗌𝗍𝖺𝗋𝗍𝖾𝖽_ρ^Q(b, r, lsid). For a run ρ of an OIDC web system with web attacker ^w we say that the user of the browser b authenticated to an IdP i using an identity u for a login session identified by a nonce lsid at the RP r if there is a processing step Q in ρ withQ = (S, E, N)(S', E', N')(for some S, S', E, E', N, N') in which the browser b was triggered, selected a document loaded from an origin of i, executed the script script_idp_form in that document, and in that script, (1) in Line <ref> of Algorithm <ref>, selected the identity u, and (2) we have that the scriptstate of that document, when triggered, contains a nonce s such that scriptstate[state] ≡ s and S(r).sessions[lsid][state] ≡ s. We then write 𝖺𝗎𝗍𝗁𝖾𝗇𝗍𝗂𝖼𝖺𝗍𝖾𝖽_ρ^Q(b, r, u, i, lsid).For a run ρ of an OIDC web system with web attacker ^w we say that the RP r uses some access token t in a login session identified by the nonce lsid established with the browser b at an IdP i if there is a processing step Q in ρ withQ = (S, E, N)(S', E', N') (for some S, S', E, E', N, N') in which (1) r calls the function 𝖴𝖲𝖤_𝖠𝖢𝖢𝖤𝖲𝖲_𝖳𝖮𝖪𝖤𝖭 with the first two parameters being lsid and t, (2) S(r).issuerCache[S(r).sessions[lsid][identity]] ∈𝖽𝗈𝗆(i), and (3) sessionid, lsid, y, z, z' S(b).cookies[d] for d ∈𝖽𝗈𝗆(r), y, z, z' ∈. We then write 𝗎𝗌𝖾𝖽𝖠𝗎𝗍𝗁𝗈𝗋𝗂𝗓𝖺𝗍𝗂𝗈𝗇_ρ^Q(b, r, i, lsid). For a run ρ of an OIDC web system with web attacker ^w we say that the RP r acts on behalf of the user with the identity u at an honest IdP i in a login session identified by the nonce lsid established with the browser b if there is a processing step Q in ρ withQ = (S, E, N)(S', E', N') (for some S, S', E, E', N, N') in which (1) r calls the function 𝖴𝖲𝖤_𝖠𝖢𝖢𝖤𝖲𝖲_𝖳𝖮𝖪𝖤𝖭 with the first two parameters being lsid and t, (2) we have that there is a term g such that gS(i).records with tg[access_tokens] and g[subject] ≡ u, and (3) sessionid, lsid, y, z, z' S(b).cookies[d] for d ∈𝖽𝗈𝗆(r), y, z, z' ∈. We then write 𝖺𝖼𝗍𝗌𝖮𝗇𝖴𝗌𝖾𝗋𝗌𝖡𝖾𝗁𝖺𝗅𝖿_ρ^Q(b, r, u, i, lsid). For session integrity for authentication we say that a user that is logged in at some RP must have expressed her wish to be logged in to that RP in the beginning of the login flow. If the IdP is honest, then the user must also have authenticated herself at the IdP with the same user account that RP uses for her identification. This excludes, for example, cases where (1) the user is forcefully logged in to an RP by an attacker that plays the role of an IdP, and (2) where an attacker can force a user to be logged in at some RP under a false identity issued by an honest IdP. Let ^w be an OIDC web system with web attackers. We say that ^w is secure w.r.t. session integrity for authentication iff for every run ρ of ^w, every processing step Q in ρ withQ = (S, E, N)(S', E', N')(for some S, S', E, E', N, N'), every browser b that is honest in S, every i∈IdP, every identity u that is owned by b, every r∈RP that is honest in S, every nonce lsid, and 𝗅𝗈𝗀𝗀𝖾𝖽𝖨𝗇_ρ^Q(b, r, u, i, lsid) we have that (1) there exists a processing step Q' in ρ (before Q) such that 𝗌𝗍𝖺𝗋𝗍𝖾𝖽_ρ^Q'(b, r, lsid), and (2) if i is honest in S, then there exists a processing step Q” in ρ (before Q) such that 𝖺𝗎𝗍𝗁𝖾𝗇𝗍𝗂𝖼𝖺𝗍𝖾𝖽_ρ^Q”(b, r, u, i, lsid). For session integrity for authorization we say that if an RP uses some access token at some IdP in a session with a user, then that user expressed her wish to authorize the RP to interact with some IdP. If the IdP is honest, and the RP acts on the user's behalf at the IdP (i.e., the access token is bound to the user's identity), then the user authenticated to the IdP using that identity. Let ^w be an OIDC web system with web attackers. We say that ^w is secure w.r.t. session integrity for authentication iff for every run ρ of ^w, every processing step Q in ρ withQ = (S, E, N)(S', E', N') (for some S, S', E, E', N, N'), every browser b that is honest in S, every i∈IdP, every identity u that is owned by b, every r∈RP that is honest in S, every nonce lsid, we have that (1) if 𝗎𝗌𝖾𝖽𝖠𝗎𝗍𝗁𝗈𝗋𝗂𝗓𝖺𝗍𝗂𝗈𝗇_ρ^Q(b, r, i, lsid) then there exists a processing step Q' in ρ (before Q) such that 𝗌𝗍𝖺𝗋𝗍𝖾𝖽_ρ^Q'(b, r, lsid), and (2) if i is honest in S and 𝖺𝖼𝗍𝗌𝖮𝗇𝖴𝗌𝖾𝗋𝗌𝖡𝖾𝗁𝖺𝗅𝖿_ρ^Q(b, r, u, i, lsid) then there exists a processing step Q” in ρ (before Q) such that 𝖺𝗎𝗍𝗁𝖾𝗇𝗍𝗂𝖼𝖺𝗍𝖾𝖽_ρ^Q”(b, r, u, i, lsid).§ PROOF OF THEOREM <REF>Before we prove Theorem <ref>, in order to provide a quick overview, we first provide a proof sketch. We then show some general properties of OIDC web systems with a network attacker, and then proceed to prove the authentication, authorization, and session integrity properties separately. §.§ Proof SketchFor authentication and authorization, we first show that the secondary security properties from Section <ref> hold true (see Lemmas <ref>–<ref> below). We then assume that the authentication/authorization properties do not hold, i.e., that there is a run ρ of ^n that does not satisfy authentication or authorization, respectively. Using Lemmas <ref>–<ref>, it then only requires a few steps to lead the respective assumption to a contradication and thereby show that ^n enjoys authentication/authorization.For the session integrity properties, we follow a similar scheme. We first show Lemma <ref>, which essentially says that a web attacker is unable to get hold of the state value that is used in a session between an honest browser b, an honest RP r, and an honest IdP i. (Recall that the state value is essential for session integrity.) We then show session integrity for authentication/authorization by starting from the latest “known” processing steps in the respective flows (e.g., for authentication, 𝗅𝗈𝗀𝗀𝖾𝖽𝖨𝗇_ρ^Q(b, r, u, i, lsid)) and tracking through the OIDC flows to show the existence of the earlier processing steps (e.g., 𝗌𝗍𝖺𝗋𝗍𝖾𝖽_ρ^Q'(b, r, lsid)) and their respective properties.§.§ Properties of ^n Let ^n = (, , 𝗌𝖼𝗋𝗂𝗉𝗍, E^0) be an OIDC web system with a network attacker. Let ρ be a run of ^n. We write s_x = (S^x,E^x,N^x) for the states in ρ.We say that a term t is derivably contained in (a term) t' for (a set of DY processes) P (in a processing step s_i → s_i+1 of a run ρ=(s_0,s_1,…)) if t is derivable from t' with the knowledge available to P, i.e.,t ∈ d_∅({t'}∪⋃_p∈ PS^i+1(p))We say that a set of processes P leaks a term t (in a processing step s_i → s_i+1) to a set of processes P' if there exists a message m that is emitted (in s_i → s_i+1) by some p ∈ P and t is derivably contained in m for P' in the processing step s_i → s_i+1. If we omit P', we define P' := ∖ P. If P is a set with a single element, we omit the set notation.We say that an DY process p created a message m (at some point) in a run if m is derivably contained in a message emitted by p in some processing step and if there is no earlier processing step where m is derivably contained in a message emitted by some DY process p'. We say that a browser b accepted a message (as a response to some request) if the browser decrypted the message (if it was an HTTPS message) and called the function 𝖯𝖱𝖮𝖢𝖤𝖲𝖲𝖱𝖤𝖲𝖯𝖮𝖭𝖲𝖤, passing the message and the request (see Algorithm <ref>).We say that an atomic DY process p knows a term t in some state s=(S,E,N) of a run if it can derive the term from its knowledge, i.e., t ∈ d_∅(S(p)). We say that a script initiated a request r if a browser triggered the script (in Line <ref> of Algorithm <ref>) and the first component of the command output of the script relation is either ,, , orsuch that the browser issues the request r in the same step as a result. The following lemma captures properties of RP when it uses HTTPS. For example, the lemma says that other parties cannot decrypt messages encrypted by RP. §.§ Proof of AuthenticationWe here want to show that every OIDC web system is secure w.r.t. authentication, and therefore assume that there exists an OIDC web system that is not secure w.r.t. authentication. We then lead this to a contradiction, thereby showing that all OIDC web systems are secure w.r.t. authentication. In detail, we assume:For any run ρ of an OIDC web system ^n with a network attacker or an OIDC web system ^w with web attackers, every configuration (S, E, N) in ρ, every IdP i that is honest in S, every identity id∈𝖨𝖣^i, every relying party r that is honest in S, we have that S(r).issuerCache[id] ≡ (not set) or S(r).issuerCache[id] ∈𝖽𝗈𝗆(i).Initially, the issuer cache of an honest relying party is empty (according to Definition <ref>). This issuer cache can only be modified in Line <ref> of Algorithm <ref>. There, the value of S^l(r).issuerCache[id'] (for some l<j) is taken from an HTTPS response. The value of id' is taken from session data (Line <ref>) which is identified by a session id that is taken from the internal reference data of the incoming message. This internal reference data must have been created previously in Algorithm <ref> (𝖧𝖳𝖳𝖯𝖲_𝖲𝖨𝖬𝖯𝖫𝖤_𝖲𝖤𝖭𝖣) which must have been called in Line <ref> of Algorithm <ref> (since this is the only place where the reference data for a webfinger request is created). In this algorithm, it is easy to see that the request to which the request is sent (see Line <ref>) is the domain part of the identity. We therefore have that a webfinger request must have been sent (using HTTPS) to the IdP i. (Note that an attacker can neither decrypt any information from this request, nor spoof a response to this request. The request must therefore have been responded to by the honest IdP. add reference to lemma attacker cannot spoof or decrypt https messages)Since the path of this request is /.wk/webfinger, the IdP can respond to this request only in Lines <ref>ff. of Algorithm <ref>. Since the IdP there chooses an issuer value that is one of its own domains (see Line <ref>), we finally have that S(r).issuerCache[id] ≡ (if the response is blocked or the webfinger request was never sent) or we have that S(r).issuerCache[id] ∈𝖽𝗈𝗆(i), which proves the lemma.For any run ρ of an OIDC web system ^n with a network attacker or an OIDC web system ^w with web attackers, every configuration (S, E, N) in ρ, every IdP i that is honest in S, every domain d ∈𝖽𝗈𝗆(i), every relying party r that is honest in S, l ∈{1,2,3,4} we have that S(r).oidcConfigCache[d] ≡ (not set) or S(r).oidcConfigCache[d] ≡ [ issuer: d, auth_ep: u_1, token_ep: u_2, jwks_ep: u_3, reg_ep: u_4] with u_l being URLs, u_l.host∈𝖽𝗈𝗆(i), and u_l.protocol≡.This proof proceeds analog to the one for Lemma <ref> with the following changes: First, the OIDC configuration cache is filled only in Line <ref> of Algorithm <ref>. It requires a request that was created in Line <ref> of Algorithm <ref>. This request was not sent to the domain contained in an ID (as above) but instead to the issuer (in this case, d). The issuer responds to this request in Lines <ref>ff. of Algorithm <ref>. There, the issuer only choses the redirection endpoint URIs such that the host is the domain of the incoming request and the protocol is HTTPS (). This proves the lemma. For any run ρ of an OIDC web system ^n with a network attacker or an OIDC web system ^w with web attackers, every configuration (S, E, N) in ρ, every IdP i that is honest in S, every domain d ∈𝖽𝗈𝗆(i), every relying party r that is honest in S, we have that S(r).jwksCache[d] ≡ (not set) or S(r).jwksCache[d] ≡𝗉𝗎𝖻(S(i).jwks).This proof proceeds analog to the one for Lemma <ref>. The relevant HTTPS request by r is created in Line <ref> of Algorithm <ref>, and responded to by the IdP i in Lines <ref>ff. of Algorithm <ref>. There, the IdP chooses its own signature verification key to send in the response. This proves the lemma. For any run ρ of an OIDC web system ^n with a network attacker or an OIDC web system ^w with web attackers, every configuration (S, E, N) in ρ, every IdP i that is honest in S, every domain d ∈𝖽𝗈𝗆(i), every relying party r that is honest in S, every client id c that has been issued to r by i, every URL uS(i).clients[c][redirect_uris] we have that u.host∈𝖽𝗈𝗆(r) and u.protocol≡. From Definition <ref> it follows that an HTTPS request must have been sent from r to i in Lines <ref>ff. of Algorithm <ref>. This request must have been processed by i in Lines <ref>ff. of Algorithm <ref>, and, after receiving the client id from some other party (usually the attacker), in Algorithm <ref>. From the latter algorithm it is easy to see that the redirection endpoint data must have been taken from r's initial registration request to create the dictionary stored in S(i).clients[c]. This data, however, was chosen by r in Line <ref> of Algorithm <ref> such that u.host∈𝖽𝗈𝗆(r) and u.protocol≡ for every uS(i).clients[c][redirect_uris]. For any run ρ of an OIDC web system ^n with a network attacker or an OIDC web system ^w with web attackers, every configuration (S, E, N) in ρ, every IdP i that is honest in S, every identity id∈𝖨𝖣^i, every browser b with b = 𝗈𝗐𝗇𝖾𝗋𝖮𝖿𝖨𝖣(id) that is honest in S, every p ∈∖{b, i} we have that 𝗌𝖾𝖼𝗋𝖾𝗍𝖮𝖿𝖨𝖣(id) ∉d_∅(S^l(p)).Let s := 𝗌𝖾𝖼𝗋𝖾𝗍𝖮𝖿𝖨𝖣(id). Initially, in S^0, s is only contained in S^0(b).secrets[d,] with d ∈𝖽𝗈𝗆(i) and in no other states of any atomic processes (or in any waiting events). By the definition of the browser, we can see that only scripts loaded from the origins d, can access s. We know that i is an honest IdP. Now, the only script that an honest IdP sends to the browser is script_idp_form. This scripts sends the form data only to its own origin, which means, that the form data is sent over HTTPS and to the honest IdP. In this request, the script uses the path /auth2. There, identity and password are checked, but not used otherwise. Therefore, the form data cannot leak from the honest IdP. It could, however, leak from the browser itself. The form data is sent via POST, and therefore, not used in any referer headers. A redirection response from the server contains the status code 303, which implies that the browser does not send the form data again when following the redirection. Since there are also no other scripts from the same origin running in the browser which could access the form data, the password s cannot leak from the browser either. This proves Lemma <ref>. For any run ρ of an OIDC web system ^n with a network attacker or an OIDC web system ^w with web attackers, every configuration (S, E, N) in ρ, every IdP i that is honest in S, every domain d ∈𝖽𝗈𝗆(i), every identity id∈𝖨𝖣^i with b = 𝗈𝗐𝗇𝖾𝗋𝖮𝖿𝖨𝖣(id) being an honest browser (in S), every relying party r that is honest in S, every client id c that has been issued to r by i, every term y, every id token t = 𝗌𝗂𝗀([iss: d, sub: id, aud: c, nonce: y ] , k) with k= S(i).jwks, every attacker process a we have that t ∉d_∅(S(a)).The signing key k is only known to i initially and at least up to S (since i is honest). Therefore, only i can create t. There are two places where an honest IdP can create such a token in Algorithm <ref>: In Line <ref> (immediately after receiving the user credentials) and in Lines <ref>ff. (after receiving an access token).We now distinguish between these two cases to show that in either case, the attacker cannot get hold of an id token. We start with the first case.ID token was created in Line <ref>.To create t, the IdP i must have received a request to the path /auth2 in Lines <ref>ff. of Algorithm <ref>. It is clear that i sends the response to this request to the sender of the request, and, if that sender is honest, the response cannot be read by an attacker. The request must contain 𝗌𝖾𝖼𝗋𝖾𝗍𝖮𝖿𝖨𝖽(id). Only b and i know this secret (per Lemma <ref>). Since i does not send requests to itself, the request must have been sent from b. Since the origin header in the request must be a domain of i, we know that the request was not initiated by a script other than i's own scripts, in particular, it must have been initiated by script_idp_form.Now it is easy to see that this script does not use the token t in any way after the token was returned from i, since the script uses a form post to transmit the credentials to i, and the window is subsequently navigated away. Instead, i provides an empty script in its response to b. This response contains a location redirect header. It is now crucial to check that this location redirect does not cause the id token to be leaked to the attacker: With Lemma <ref> we have that the redirection URIs that are registered at i for the client id c only point to domains of r (and use HTTPS).We therefore know that b will send an HTTPS request (say m) containing t to r. We have to check whether r or a script delivered by r to b will leak t. Algorithm <ref> processes all HTTPS requests delivered to r. As i redirected b using the 303 status code, the request to r must be a GET request. Hence, r does not process this request in Lines <ref>ff. of Algorithm <ref>.Lines <ref>ff. do only respond with a script and do not use t in any way. We are left with Lines <ref>ff. to be analyzed.As in m the id token t is always contained in a dictionary under the key id_token and this dictionary is either in the parameters, the fragment, or the body of m, it is now easy to see that r does not store or send out t in any way.We now have to check if a script delivered by r to b leads to t being leaked. First note that r always sets the header ReferrerPolicy to origin in every HTTP(S) response r sends out. Hence, t can never leak using the Referer header.There are only two scripts that r may deliver: (1) The script script_rp_index either issues a FORM command to the browser, which does not contain t, or this script issues a HREF command to the browser for some URL, which also does not contain t. (2) The script script_rp_get_fragment takes the fragment of the current URL (which may be a dictionary that contains t under the key id_token) and the iss parameter and issues an HTTPS request to r for the path /redirect_ep, which will be processed by r in Lines <ref>ff. of Algorithm <ref>. Now, the same reasoning as above applies. ID token was created in Lines <ref>ff. In this case, the id token is created by i only when an HTTPS request was received by i that matches the following criteria: (a) it must be for the path /token, (b) it must contain the client id c in the body (under the key client_id), and (c) it must contain a authorization code in the body (under the key code) that occurs in one of i's internal records with a matching subject, issuer, and nonce. To be more precise, the request must contain a code code such that there is a record rec with rec S(i).records and rec[issuer] ≡ d, rec[subject] ≡id, rec[client_id] ≡ c, and rec[code] ≡ c. Such a record can only be created and the authorization code code issued under exactly the same circumstances that allow an id token (of the above form) to be created in Line <ref>. With exactly the same reasoning as above, this time for the code instead of the id token, we can follow that code does not leak to the attacker. We have therefore shown that no attacker process can get hold of the id token t. This proves the lemma.There exists an OIDC web system ^n with a network attacker such that there exists a run ρ of ^n, a configuration (S, E, N) in ρ, some r∈RP that is honest in S, some identity id∈𝖨𝖣 with 𝗀𝗈𝗏𝖾𝗋𝗇𝗈𝗋(id) being an honest IdP (in S) and 𝗈𝗐𝗇𝖾𝗋𝖮𝖿𝖨𝖣(id) being an honest browser (in S), some service session identified by some nonce n for id at r, and n is derivable from the attackers knowledge in S (i.e., n ∉d_∅(S(attacker))).Assumption <ref> is a contradiction.We first recall how the service session identified by some nonce n for id at r is defined. It means that there is some session id x and a domain d ∈𝖽𝗈𝗆(𝗀𝗈𝗏𝖾𝗋𝗇𝗈𝗋(id)) with S(r).sessions[x][loggedInAs] ≡d, id and S(r).sessions[x][serviceSessionId] ≡ n. Now the assumption is that n is derivable from the attacker's knowledge. Since we have that S(r).sessions[x][serviceSessionId] ≡ n, we can check where and how, in general, service session ids can be created. It is easy to see that this can only happen in Algorithm <ref>, where, in Line <ref>, the RP chooses a fresh nonce as the value for the service session id, in this case x. In the line before, it sets the value for S(r).sessions[x][loggedInAs], in this case d, id. In the Lines <ref>ff., r performs several checks to ensure the integrity and authenticity of the id token.The function function 𝖢𝖧𝖤𝖢𝖪_𝖨𝖣_𝖳𝖮𝖪𝖤𝖭 can be called in either (a) Line <ref> of Algorithm <ref> or (b) in Line <ref> of Algorithm <ref>.We can now distinguish between these two cases. Case (a). In this case, we can easily see that the same party that finally receives the service session id x, must have provided, in an HTTPS request, an id token (say, t') with the following properties (for some l < j):𝖾𝗑𝗍𝗋𝖺𝖼𝗍𝗆𝗌𝗀(t')[iss] ≡ d 𝖾𝗑𝗍𝗋𝖺𝖼𝗍𝗆𝗌𝗀(t')[sub] ≡id 𝖾𝗑𝗍𝗋𝖺𝖼𝗍𝗆𝗌𝗀(t')[aud] ≡ S^l(r).clientCredentialsCache[d][client_id] 𝖼𝗁𝖾𝖼𝗄𝗌𝗂𝗀(t', 𝗉𝗎𝖻(S^l(i).jwks)) ≡ .The attacker (and, by extension, any other party except for i, b, and r), however, cannot know such an id token (see Lemma <ref>). Since r and i do not send requests to r, the id token must have been sent by b to r. As the service session id x is only contained in a set-cookie header with the httpOnly and secure flags set, b will only ever send the service session id x to r (contained in a cookie header). As b does not leak x in any other way and as r does not leak information sent in cookie headers, the service session id x does not leak.Case (b). Otherwise, the party that finally receives the service session id x needs to provide a code csuch that, when this code is sent to the token endpoint of i (Algorithm <ref>), i responds with an id token matching the criteria listed in Case (a). This, however, would mean that an attacker, knowing this code, could do the same, violating Lemma <ref>. (Note that for every run where a client secret is associated with the client id there is also a run where the client secret is not used; the client secret does not prevent the attacker from requesting an id token at the token endpoint for a valid code.) <- kann man noch ausfuehrenWe therefore have shown that the attacker cannot know x, proving the lemma and showing that Assumption <ref> is, in fact, a contradiction. §.§ Proof of AuthorizationAs above, we assume that there exists an OIDC web system that is not secure w.r.t. authorization and lead this to a contradiction. There exists an OIDC web system with a network attacker ^n, a run ρ of ^n, a state (S^j, E^j, N^j) in ρ, a relying party r∈RP that is honest in S^j, an identity provider i∈IdP that is honest in S^j, a browser b that is honest in S^j, an identity id∈𝖨𝖣^i owned by b, a nonce n, a term xS^j(i).records with x[subject] ≡id, nx[access_tokens], and the client id x[client_id] has been issued by i to r, and n is derivable from the attackers knowledge in S^j (i.e., n ∈ d_∅(S^j(attacker))).Assumption <ref> is a contridiction.We have that n ∈ d_∅(S^j(attacker)) and therefore, there must have been a message from a third party to attacker (or any other corrupted party, which could have forwarded n to the attacker) that contained n. We can now distinguish between the parties that could have sent n to the attacker (or to the corrupted party):The access token n was sentby the browser b:We now track different cases in which the access token n can get into b's knowledge. We will omit the cases in which b learns n from any dishonest party as in such a case there is a different run ρ' of ^n in which this dishonest party immediately sends n to the attacker.(I) First, we analyze the case in which b has learned n from an honest (in S^j) identity provider, say i'. In this case, b must have received an HTTPS response from i' (honest identity providers do not send out unencrypted HTTP responses). Honest identity providers send out HTTPS responses in Lines <ref>, <ref>, <ref>, <ref>, <ref>, and <ref> of Algorithm <ref> and Line <ref> of Algorithm <ref>. It is easy to see that i' does not send out n in Lines <ref>, <ref>, <ref>, and <ref> of Algorithm <ref> and Line <ref> of Algorithm <ref> (given that the attacker does not know n), leaving Lines <ref>, and <ref> of Algorithm <ref> to analyze.(a) If i' sends out n in Line <ref> of Algorithm <ref>, b must have sent an HTTPS POST request bearing an Origin header for one of the domains of i' to i'. As i' only delivers the script script_idp_form, only this script could have caused this request (using a FORM command). Hence, b will navigate the corresponding window to the location indicated in the Location header of the HTTPS response assembled in Lines <ref>ff. of Algorithm <ref>. The body of this response can consist of an authorization code (a fresh nonce), an access token (a fresh nonce), and an id token consisting of one domain of i, a valid user name for i', a client id, and a nonce (say n') from the request.We now reason why i' must be i, and the access token in the response must be n. In the id token, only the client id and the nonce n' could be n. As the client id is always set by the attacker during registration, the client id cannot be n. The nonce n' originates from the request sent by b on the command of script_idp_form. In this request, the nonce n' must be contained in the URL, which is the URL from which the script was loaded before. Hence, the browser must have been navigated to this URL. As the attacker does not know n at this point, only honest scripts or honest web servers could have navigated the browser to such an URL (containing n). Honest relying parties only populate the parameter nonce (bearing n') in such a redirect with a fresh nonce, honest identity providers do not populate such an URL parameter by themselves, but could have used this parameter in a redirect based on a registered redirect URL. As honest parties never register such a redirect URL, n' cannot be n. Hence, only the access token in the response above can be n. As the access token is a fresh nonce, we must have that i' is i and that i creates the term xS^j(i).records with x[subject] ≡id, nx[access_tokens] (i will never create such a term at any other time), and the client id x[client_id] has been issued by i to r. Hence, the location redirect issued by i must point to an URL of r with the path /redirect_ep (see Lemma <ref>) and this URL contains the parameter isswith a domain of i. The access token n is only contained in the fragment of this URL under the key access_token.Now, b sends an HTTPS request to r. This request does not contain n (as it is placed in the fragment part of the URL). The relying party r can (regardless of the path) send out only the scripts script_rp_index and script_rp_get_fragment as a response to such a request. The script script_rp_index ignores the fragment of its URL. The script script_rp_get_fragement takes the fragment of the URL and uses it as the body of a POST request to its own origin (which is r) with path /redirect_uri. When r processes this POST request, r only ever uses n in Line <ref> of Algorithm <ref>. There, the access token n and the value of the parameter iss (a domain of i) is processed by Algorithm <ref>. From Lemma <ref>, we know that r will only send n to the token endpoint of i in an HTTPS request. This request is then processed by i in Lines <ref>ff. of Algorithm <ref>. There, i only checks n, but does not send out n.If b sends out a response, the same reasoning as above applies. Hence, we have that n does not leak to the attacker in this case. (b) If i' sends out n in Line <ref> of Algorithm <ref>, we have that the response does not contain a script or a redirect. The browser would only interpret such a response if the request was caused by an XMLHTTPREQUEST command of a script. Honest scripts do not issue such a command, leaving only the attacker script as the only possible source for such a request. If i' is not i, it is easy to see that this response cannot contain n. The identity provider i only sends out n (taken from the subterm records from its state) if the request contains a valid authorization code for this access token. With the same reasoning as for the authentication property above, the attacker cannot know a valid id token for any user id owned by b. If the attacker would know a valid authorization code, he could retrieve a valid id token (for such a user id) from i. Hence, the attacker cannot know a valid authorization code. As this reasoning also applies for the attacker script, the attacker script could not have caused a request to i revealing n.(II) Now, we analyze the case in which b received n from some honest (in S^j) relying party, say r'.In this case, b must have received an HTTPS response from r' (honest relying parties do not send out unencrypted HTTP responses). Honest relying parties only send out such HTTPS responses in Lines <ref> and <ref> of Algorithm <ref>, Line <ref> of Algorithm <ref>, and Line <ref> of Algorithm <ref>. In the former three cases, r' only sends out fixed information and fresh nonces (either chosen by r' directly before sending out the message or the HTTPS nonce and key chosen by b when creating the request). In the latter case, r' (besides the pieces of information as before) also adds information from its OpenID Connect configuration cache (i.e., client id and authorization URL). From Lemma <ref> and <ref> we know that if r' gathered this information from an honest party, this information cannot contain n. As the attacker does not know n at this point, this registration information cannot contain n if r' gathered this information from a dishonest party. Hence, b cannot have learned n from any r'.(III) b cannot have learned n from a different honest (in S^j) browser as honest browsers do not create messages that can be interpreted by honest browsers.(IV) b cannot have learned n from the attacker, as the attacker does not know n at this point. The access token n was sent by the IdP i: We can see that access tokens are sent by the IdP only after a request to the path (endpoints) /auth2 or to the path and /token.In case of a request to the path /auth2, a pair of access tokens is created and the first access token in the pair is returned from the endpoint. If the attacker would be able to learn n from this endpoint such that there exists a record xS^j(i).records with x[subject] ≡id, then the attacker would need to provide the user's credentials to the IdP i. The attacker cannot know these credentials (Lemma <ref>), therefore the attacker cannot request n from this endpoint.In case of a request to the path /token, the attacker would need to provide an authorization code that is contained in the same record (in this case x) as n. Now, recall that we have that x[subject] ≡id and c := x[client_id] has been issued to r by i. We can now see that if the attacker would be able to send a request to the endpoint /token which would cause a response that contains n, the attacker would also be able to learn an id token of the form shown in Lemma <ref> (the issuer is a domain of i, the subject is id, and the audience is c). This would be a contradiction to Lemma <ref>.We can conclude that the access token n was not sent by the IdP i. The access token n was sent by the RP r:The only place where the (honest) RP uses an access token is in Algorithm <ref>. There, the access token is sent to the domain of the token endpoint (compare Algorithm <ref>, where the authorization code is sent to that endpoint). We can now see that the access token is always sent to i: If the access token would be sent to the attacker, so would the authorization code in Algorithm <ref>, and Lemma <ref> would not hold true. <- etwas kurz §.§ Proof of Session IntegrityBefore we prove this property, we highlight that in the absence of a network attacker and with the DNS server as defined for ^w, HTTP(S) requests by (honest) parties can only be answered by the owner of the domain the request was sent to, and neither the requests nor the responses can be read or altered by any attacker unless he is the intended receiver.We further show the following lemma, which says that an attacker (under the assumption above) cannot learn a state value that is used in a login session between an honest browser, an honest IdP, and an honest RP. There exists no run ρ of an OIDC web system with web attackers ^w, no configuration (S, E, N) of ρ, no r∈RP that is honest in S, no i ∈IDP that is honest in S, no browser b that is honest in S, no nonce lsid∈, no domain h ∈𝖽𝗈𝗆(r) of r, no terms g, x, y, z ∈, no cookie c := sessionId, lsid, x, y, z, no atomic DY process p ∈∖{b,i,r} such that (1) S(r).sessions[lsid] ≡ g, (2) g[state] ∈ d_∅(S(p)), (3) S(r).issuerCache[g[identity]] ∈𝖽𝗈𝗆(i), and (4) cS(b).cookies[h]. To prove Lemma <ref>, we track where the login session identified by lsid is created and used.Login session ids are only chosen in Line <ref> of Algorithm <ref>. After the session id was chosen, its value is sent over the network to the party that requested the login (in Line <ref> of Algorithm <ref>). We have that for lsid, this party must be b because only r can set the cookie c for the domain h in the state of b[Note that we have only web attackers.] and Line <ref> of Algorithm <ref> is actually the only place where r does so.Since b is honest, b follows the location redirect contained in the response sent by r. This location redirect contains state (as a URL parameter). The redirect points to some domain of i. (This follows from Lemma <ref>.) The browser therefore sends (among others) state in a GET request to i. Of all the endpoints at i where the request can be received, the authorization endpoint is the only endpoint where state could potentially leak to another party. (For all other endpoints, the value is dropped.) If the request is received at the authorization endpoint, state is only sent back to b in the initial scriptstate of script_idp_form. In this case, the script sends state back to i in a POST request to the authorization endpoint. Now, i redirects the browser b back to the redirection URI that was passed alongside state from r via the browser to i. This redirection URI was chosen in Line <ref> of Algorithm <ref> and therefore points to one of r's domains.The value state is appended to this URI (either as a parameter or in the fragment). The redirection to the redirection URI is then sent to the browser b. Therefore, b now sends a GET request to r.If state is contained in the parameter, then state is immediately sent to r where it is compared to the stored login session records but neither stored nor sent out again. In each case, a script is sent back to b. The scripts that r can send out are script_rp_index and script_rp_get_fragment, none of which cause requests that contain state (recall that we are in the case where state is contained in the URI parameter, not in the fragment). Also, since both scripts are always delivered with a restrictive Referrer Policy header, any requests that are caused by these scripts (e.g., the start of a new login flow) do not contain state in the referer header.[We note that, as discussed earlier, without the Referrer Policy, state could leak to a malicious IdP or other parties.]If state is contained in the fragment, then state is not immediately sent to r, but instead, a request without state is sent to r. Since this is a GET request, r either answers with a response that only contains the string ok but no script (Lines <ref>ff. of Algorithm <ref>),a response containing script_rp_index (Lines <ref>ff. of Algorithm <ref>), ora response containing script_rp_get_fragment (Line <ref> of Algorithm <ref>).In case of the ok response, state is not used anymore by the browser. In case of script_rp_index, the fragment is not used. (As above, there is no other way in which state can be sent out, also because the fragment part of an URL is stripped in the referer header.) In the case of script_rp_get_fragment being loaded into the browser, the script sends state in the body of an HTTPS request to r (using the path /redirect_ep). When r receives this request, it does not send out state to any party (see Lines <ref>ff. of Algorithm <ref>).This shows that state cannot be known to any party except for b, i, and r. §.§.§ Proof of Session Integrity for AuthenticationTo prove that every OIDC web system with web attackers is secure w.r.t. session integrity for authentication, we assume that there exists an OIDC web system with web attackers which is not secure w.r.t. session integrity for authentication and lead this to a contradiction. There exists an ^w be an OIDC web system with web attackers, a run ρ of ^w, a processing step Q in ρ withQ = (S, E, N)(S', E', N') (for some S, S', E, E', N, N'), a browser b that is honest in S, an IdP i∈IdP, an identity u that is owned by b, an RP r∈RP that is honest in S, a nonce lsid, with 𝗅𝗈𝗀𝗀𝖾𝖽𝖨𝗇_ρ^Q(b, r, u, i, lsid) and (1) there exists no processing step Q' in ρ (before Q) such that 𝗌𝗍𝖺𝗋𝗍𝖾𝖽_ρ^Q'(b, r, lsid), or (2) i is honest in S, and there exists no processing step Q” in ρ (before Q) such that 𝖺𝗎𝗍𝗁𝖾𝗇𝗍𝗂𝖼𝖺𝗍𝖾𝖽_ρ^Q”(b, r, u, i, lsid).Assumption <ref> is a contradiction.(1) We have that 𝗅𝗈𝗀𝗀𝖾𝖽𝖨𝗇_ρ^Q(b, r, u, i, lsid). With Definition <ref> we have that r sent out the service session id belonging to lsid to b. (This can only happen when the function 𝖢𝖧𝖤𝖢𝖪_𝖨𝖣_𝖳𝖮𝖪𝖤𝖭 (Algorithm <ref>) was called with lsid as the first parameter.) This means that r must have received a request from b containing a cookie with the name sessionId and the value lsid: The response by r (which we know was sent to b) was sent in Line <ref> in Algorithm <ref>. There, r looks up the address of b using the login session record under the key redirectEpRequest. This key is only ever created in Line <ref> of Algorithm <ref>. This line is only ever called when r receives an HTTPS request from b with the cookie as described.We can now track how the cookie was stored in b: Since the cookie is stored under a domain of r and we have no network attacker, the cookie must have been set by r. This can only happen in Line <ref> in Algorithm <ref>. Similar to the redirectEpRequest session entry above, r sends this cookie as a response to a stored request, in this case, using the key startRequest in the session data. This key is only ever created in Lines <ref>ff. of Algorithm <ref>. Hence, there must have been a request from b to r containing a POST request for the path /startLogin with an origin header for an origin of r. There are only two scripts which could potentially send such a request, script_rp_index and script_rp_get_fragment. It is easy to see that only the former send requests of the kind described. We therefore have a processing step Q' that happens before Q in ρ with 𝗌𝗍𝖺𝗋𝗍𝖾𝖽_ρ^Q'(b, r, lsid).(2) Again, we have that 𝗅𝗈𝗀𝗀𝖾𝖽𝖨𝗇_ρ^Q(b, r, u, i, lsid). Now, however, i is honest.We first highlight that if r receives an HTTPS request, say m, which contains state such that S(r).sessions[lsid][state] ≡state and contains a cookie with the name sessionId and the value lsid then this request must have come from the browser b and be caused by a redirection from i or a script from r. From 𝗅𝗈𝗀𝗀𝖾𝖽𝖨𝗇_ρ^Q(b, r, u, i, lsid) it follows that there is a term g such that S(r).sessions[lsid] ≡ g, and g[loggedInAs] ≡d, u with d ∈𝖽𝗈𝗆(i). From the Algorithm <ref> we have that S(r).issuerCache[g[identity]] ≡ d. With Lemma <ref> we have that only b, r, and i know state.We can now show that m must have been caused by i by means of a Location redirect that was sent to b or by the script script_rp_get_fragment. First, neither r nor i send requests that contain cookies. The request must therefore have originated from b. Since no attacker knows state, the request cannot have been caused by any attacker scripts or by redirects from parties other than r or i (otherwise, there would be runs where the attacker learns state).Redirects from r can be excluded, since r only sends a redirection in Line <ref> in Algorithm <ref> but there, a freshly chosen state value is used, hence, there is only one processing step in which r uses state for this redirect. This is the processing step where r adds state to the session data stored under the key lsid. Since this is a session in which the honest IdP i is used, and with Lemma <ref>, we have that r does not redirect to itself (but to i instead).The scripts script_rp_index and script_idp_form do not send requests with the state parameter. Therefore, the remaining causes for the request m are either the script script_rp_get_fragment or a location redirect from i.If the request m was caused by script_rp_get_fragment, then it is easy to see from the definition of script_rp_get_fragment (Algorithm <ref>) that this script only sends data from the fragment part of its own URI (except for the iss parameter) and it sends this data only to its own origin. This script therefore must have been sent to b by r, which only sends this script after receiving HTTPS request to the redirection endpoint (/redirect_ep). With the same reasoning as above this must have been caused by a location redirect from i. For clarity, by m_redir we denote the response by i to the browser b containing this redirection. We now show that for m_redir to take place, there must have been a processing step Q” (before Q) with 𝖺𝗎𝗍𝗁𝖾𝗇𝗍𝗂𝖼𝖺𝗍𝖾𝖽_ρ^Q”(b, r, u', i, lsid) for some identity u'.In the honest IdP i, there is only one place where a redirection happens, namely in Line <ref> in Algorithm <ref>. To reach this point, i must have received the login data for u' in the HTTPS request corresponding to m_redir. This must be a POST request with an origin header containing an origin of i. As i only uses script_idp_form, the request must have been caused by this script. Hence, we have 𝖺𝗎𝗍𝗁𝖾𝗇𝗍𝗂𝖼𝖺𝗍𝖾𝖽_ρ^Q”(b, r, u', i, lsid).We now only need to show that u' = u. With 𝗅𝗈𝗀𝗀𝖾𝖽𝖨𝗇_ρ^Q(b, r, u, i, lsid), we know that r must have called the function 𝖢𝖧𝖤𝖢𝖪_𝖨𝖣_𝖳𝖮𝖪𝖤𝖭 (Algorithm <ref>). We further have that S(r).sessions[lsid][loggedInAs] ≡d,u. We therefore have that i must have created an id token with the issuer d and the identity u. 𝖢𝖧𝖤𝖢𝖪_𝖨𝖣_𝖳𝖮𝖪𝖤𝖭 can be called in Line <ref> in Algorithm <ref> and in Line <ref> in Algorithm <ref>. We now distinguish between these two cases.𝖢𝖧𝖤𝖢𝖪_𝖨𝖣_𝖳𝖮𝖪𝖤𝖭 was called in Line <ref> in Algorithm <ref>: When the function is called in this line, there must have been an HTTPS request reference with the string TOKEN (cf. generic web server model, Algorithm <ref>). Such a reference is only created in Line <ref> of Algorithm <ref>. With Lemma <ref> we know that this HTTPS request was sent to the token endpoint (path /token) of i (because the issuer, stored in the login session record, is i). Since the token endpoint returned an id token of the form described above, and i is honest, there must have been a record in i, say v, with v[subject] ≡ u. In the request to the token endpoint, r must have sent a nonce c such that v[code] ≡ c. This request, as already mentioned, must have been sent in Line <ref> of Algorithm <ref>. This means, that there must have been an HTTPS request to i containing the session id lsid as a cookie, c, and state. Such a request can only be the request m as shown above, hence there must have been the HTTPS response m_redir containing the values c and state. Recall that we have the record v as shown above in the state of i. Such a record is only created in i if 𝖺𝗎𝗍𝗁𝖾𝗇𝗍𝗂𝖼𝖺𝗍𝖾𝖽_ρ^Q”(b, r, u, i, lsid). Therefore, u = u' in this case.𝖢𝖧𝖤𝖢𝖪_𝖨𝖣_𝖳𝖮𝖪𝖤𝖭 was called in Line <ref> in Algorithm <ref>: In this case, the id token must have been contained in m and m_redir as above. Such an id token is only sent out in m_redir by i if 𝖺𝗎𝗍𝗁𝖾𝗇𝗍𝗂𝖼𝖺𝗍𝖾𝖽_ρ^Q”(b, r, u, i, lsid). Therefore, u = u' in every case.§.§.§ Proof of Session Integrity for AuthorizationTo prove that every OIDC web system with web attackers is secure w.r.t. session integrity for authorization, we assume that there exists an OIDC web system with web attackers which is not secure w.r.t. session integrity for authorization and lead this to a contradiction.There is a OIDC web system ^w with web attackers, a run ρ of ^w, a processing step Q in ρ withQ = (S, E, N)(S', E', N') (for some S, S', E, E', N, N') a browser b that is honest in S, an IdP i∈IdP, an identity u that is owned by b, an RP r∈RP that is honest in S, a nonce lsid, with (1) 𝗎𝗌𝖾𝖽𝖠𝗎𝗍𝗁𝗈𝗋𝗂𝗓𝖺𝗍𝗂𝗈𝗇_ρ^Q(b, r, i, lsid) and there exists no processing step Q' in ρ (before Q) such that 𝗌𝗍𝖺𝗋𝗍𝖾𝖽_ρ^Q'(b, r, lsid), or (2) i is honest in S and 𝖺𝖼𝗍𝗌𝖮𝗇𝖴𝗌𝖾𝗋𝗌𝖡𝖾𝗁𝖺𝗅𝖿_ρ^Q(b, r, u, i, lsid) and there exists no processing step Q” in ρ (before Q) such that 𝖺𝗎𝗍𝗁𝖾𝗇𝗍𝗂𝖼𝖺𝗍𝖾𝖽_ρ^Q”(b, r, u, i, lsid).Assumption <ref> is a contradiction.(1) We have that 𝗎𝗌𝖾𝖽𝖠𝗎𝗍𝗁𝗈𝗋𝗂𝗓𝖺𝗍𝗂𝗈𝗇_ρ^Q(b, r, i, lsid). With Definition <ref> we have that r sent out the access token belonging to lsid to i. This can only happen when the function 𝖴𝖲𝖤_𝖠𝖢𝖢𝖤𝖲𝖲_𝖳𝖮𝖪𝖤𝖭 (Algorithm <ref>) was called with lsid. This function is called in Line <ref> of Algorithm <ref> and in Line <ref> of Algorithm <ref>. In both cases, there must have been a request, say m, to r containing a cookie with the session id lsid. In the former case, this is the request that is processed in the same processing step as calling the function 𝖴𝖲𝖤_𝖠𝖢𝖢𝖤𝖲𝖲_𝖳𝖮𝖪𝖤𝖭. In the latter case, there must have been an HTTPS request reference with the string TOKEN (cf. generic web server model, Algorithm <ref>). Such a reference is only created in Line <ref> of Algorithm <ref>. To get to this point in the algorithm, a request as described above must have been received. Since we have web attackers (and no network attacker), it is easy to see that this request must have been sent by b. With the same reasoning as in the proof for session integrity for authentication, we now have that there exists a processing step Q' in ρ (before Q) such that 𝗌𝗍𝖺𝗋𝗍𝖾𝖽_ρ^Q'(b, r, lsid).(2) We have that i is honest and 𝖺𝖼𝗍𝗌𝖮𝗇𝖴𝗌𝖾𝗋𝗌𝖡𝖾𝗁𝖺𝗅𝖿_ρ^Q(b, r, u, i, lsid). From (1) we know that r must have received a request m from b containing a cookie with the session id lsid. Therefore, we know that m_redir exists just as in the proof for Lemma <ref> (2). As in that proof, we have that 𝖺𝗎𝗍𝗁𝖾𝗇𝗍𝗂𝖼𝖺𝗍𝖾𝖽_ρ^Q”(b, r, u', i, lsid) for some identity u'. We therefore need to show that u = u'.With 𝖺𝖼𝗍𝗌𝖮𝗇𝖴𝗌𝖾𝗋𝗌𝖡𝖾𝗁𝖺𝗅𝖿_ρ^Q(b, r, u, i, lsid), we know that r must have called the function 𝖴𝖲𝖤_𝖠𝖢𝖢𝖤𝖲𝖲_𝖳𝖮𝖪𝖤𝖭 with some access token t (Algorithm <ref>). We further have that there is a term g such that gS(i).records with tg[access_tokens] and g[subject] ≡ u.𝖴𝖲𝖤_𝖠𝖢𝖢𝖤𝖲𝖲_𝖳𝖮𝖪𝖤𝖭 can be called in Line <ref> in Algorithm <ref> and in Line <ref> in Algorithm <ref>. We now distinguish between these two cases.𝖴𝖲𝖤_𝖠𝖢𝖢𝖤𝖲𝖲_𝖳𝖮𝖪𝖤𝖭 was called in Line <ref> in Algorithm <ref>: When the function is called in this line, there must have been an HTTPS request reference with the string TOKEN (cf. generic web server model, Algorithm <ref>). Such a reference is only created in Line <ref> of Algorithm <ref>. With Lemma <ref> we know that this HTTPS request was sent to the token endpoint (path /token) of i (because the issuer, stored in the login session record, is i). Since the token endpoint returned the access token t, and i is honest, there must have been a record in i, say v, with v[subject] ≡ u. In the request to the token endpoint, r must have sent a nonce c such that v[code] ≡ c. This request, as already mentioned, must have been sent in Line <ref> of Algorithm <ref>. This means, that there must have been an HTTPS request to i containing the session id lsid as a cookie, c, and state. Such a request can only be the request m as shown above, hence there must have been the HTTPS response m_redir containing the values c and state. Recall that we have the record v as shown above in the state of i. Such a record is only created in i if 𝖺𝗎𝗍𝗁𝖾𝗇𝗍𝗂𝖼𝖺𝗍𝖾𝖽_ρ^Q”(b, r, u, i, lsid). Therefore, u = u' in this case.𝖴𝖲𝖤_𝖠𝖢𝖢𝖤𝖲𝖲_𝖳𝖮𝖪𝖤𝖭 was called in Line <ref> in Algorithm <ref>: In this case, the access token t must have been contained in m and m_redir as above. This access token is only sent out in m_redir by i if 𝖺𝗎𝗍𝗁𝖾𝗇𝗍𝗂𝖼𝖺𝗍𝖾𝖽_ρ^Q”(b, r, u, i, lsid). Therefore, u = u' in every case. §.§ Proof of Theorem <ref>With Lemmas <ref>, <ref>, <ref>, and <ref>, Theorem <ref> follows immediately. | http://arxiv.org/abs/1704.08539v1 | {
"authors": [
"Daniel Fett",
"Ralf Kuesters",
"Guido Schmitz"
],
"categories": [
"cs.CR"
],
"primary_category": "cs.CR",
"published": "20170427124333",
"title": "The Web SSO Standard OpenID Connect: In-Depth Formal Security Analysis and Security Guidelines"
} |
=11cm .04.03 .12e_α e_β e_α^β/α e_β^α/β loglog μ_S→S μ_S→Jθ_c θ_cs θ_sj z_s z_cs z_sjτ_1 τ_2 z_cutnn̅nn̅n̅[1]Sec. <ref>[2]Secs. <ref> and <ref>[1]App. <ref>[1]Table <ref>[2]Tables <ref> and <ref>[1]Fig. <ref>[2]Figs. <ref> and <ref>[1]Eq. (<ref>)[2]Eqs. (<ref>) and (<ref>)[1]Ref. <cit.>[1]Refs. <cit.>[1] O(#1)[1] O(#1)JHEPPhysics Department, Reed College, Portland, OR 97202, [email protected]@reed.edu Machine learning techniques are increasingly being applied toward data analyses at the Large Hadron Collider, especially with applications for discrimination of jets with different originating particles.Previous studies of the power of machine learning to jet physics have typically employed image recognition, natural language processing, or other algorithms that have been extensively developed in computer science.While these studies have demonstrated impressive discrimination power, often exceeding that of widely-used observables, they have been formulated in a non-constructive manner and it is not clear what additional information the machines are learning.In this paper, we study machine learning for jet physics constructively, expressing all of the information in a jet onto sets of observables that completely and minimally span N-body phase space.For concreteness, we study the application of machine learning for discrimination of boosted, hadronic decays of Z bosons from jets initiated by QCD processes.Our results demonstrate that the information in a jet that is useful for discrimination power of QCD jets from Z bosons is saturated by only considering observables that are sensitive to 4-body (8 dimensional) phase space. How Much Information is in a Jet? Kaustuv Datta and Andrew Larkoski December 30, 2023 ===================================== § INTRODUCTION The problem of discrimination and identification of high energy jet-like objects observed at the Large Hadron Collider (LHC) is fundamental for both Standard Model physics and searches as the lower bound on new physics mass scales increase.Heavy particles of the Standard Model, like the W, Z, and H bosons or the top quark, can be produced with large Lorentz boosts and dominantly decay through hadrons.They will therefore appear collimated in the detector and similar to that of jets initiated by light QCD partons.The past several years have seen a huge number of observables and techniques devoted to jet identification <cit.>, and many have become standard tools in the ATLAS and CMS experiments.The list of observables for jet discrimination is a bit dizzying, and in many cases there is no organizing principle for which observables work well in what situations.[There has been some effort in the past to identify and quantify (over)complete bases of jet observables <cit.>.]Motivated by the large number of variables that define the structure of a jet, several groups have recently applied machine learning methods to the problem of jet identification <cit.>.Rather than developing clever observables that identify certain physics aspects of the jets, the idea of the machine learning approach is to have a computer construct an approximation to the optimal classifier that discriminates signal from background.For example, deOliveira:2015xxd interpreted the jet detected by the calorimetry as an image, with the pixels corresponding to the calorimeter cells and the “color” of the pixel corresponding to the deposited transverse momentum in the cell.These techniques have outperformed standard jet discrimination observables and show that there is additional information in jets to exploit.However, this comes with a significant cost.Machine learning methods applied to jet physics typically have hundreds of input variables with thousands of correlations between them.Thus, in one sense this problem seems ideally suited for machine learning, but it also lacks the immediate physical interpretation and intuition that individual observables have.Previous studies have shown that the computer is learning information about what discriminates jets of different origins, but it has not been clearly demonstrated what information standard observables are missing.Along these same lines, the improvement of discrimination performance of machine learning over standard observables is relatively small, suggesting that standard observables capture the vast majority of useful information in jets.In this paper, we approach machine learning for jet discrimination from a different perspective.We construct an observable basis that completely and minimally spans the phase space for the substructure of a jet.[By “span” we do not mean in the vector space sense.Rather, the measurement of the basis of observables defines a system of equations that can be inverted to uniquely determine the phase space variables.]For a jet with M particles, the phase space is 3M-4 dimensional, and so we identify 3M-4 infrared and collinear (IRC) safe jet substructure observables that span the phase space.[Note that this will completely define the phase space of the jet substructure; that is the relative configuration of emissions in the jet.It will not identify the total jet (p_T,η,ϕ).This may be useful information, but is explicitly sensitive to global event properties which is beyond the scope of this paper.We thank Ben Nachman for emphasizing this point.]These basis observables are then passed to a machine learning algorithm for identification of relevant discrimination information.[Because we input a finite number of IRC safe observables to the machine, its output classifier will in general be Sudakov safe <cit.>.]A general jet will have an arbitrary number of particles in it, and so we will observe how the discrimination power depends on the dimension of phase space that we assume.That is, we will assume that the jet has 2 particles, 3 particles, 4 particles, etc., as defined by the set of basis observables and observe how the discrimination power improves.This method is constructive in the following sense.With some number of assumed particles in the jet, the discrimination power will saturate, which then immediately tells us what reduced set of observables are necessary to effectively extract all information that is useful for discrimination.This approach has the additional advantage that the identified observables can be calculated theoretically from first principles, without relying on parton shower modeling.As it is a widely-studied problem in jet substructure, we will apply this approach to the discrimination of boosted, hadronically decaying Z bosons from jets initiated by light quarks or gluons.The results of our study are shown in fig:introplot.Here, we plot the simulated signal (Z boson) efficiency versus the background (QCD jet) rejection rate as determined by a deep neural network, for observables that are sensitive to 2-, 3-, 4-, 5- and 6-body phase space.To identify the phase space variables, we choose to measure the jet mass and the N-subjettiness observables <cit.>, but this choice is not special.This plot demonstrates that observables sensitive to 4-body phase space saturate the discrimination power.4-body phase space is only 8 dimensional, suggesting that very few observables are necessary to identify all interesting structure of these jets.We anticipate that this approach can be applied to other discrimination problems in jet substructure, as well, and greatly reduce the dimensionality of the variable space that is being studied.The outline of this paper is as follows.In sec:basis, we define the observable basis that is used to identify all variables of M-body phase space.As mentioned above, we choose to use the N-subjettiness observables.In this section, we also prove that the set of observables is complete and minimal.In sec:deeplearn, we discuss our event simulation and machine learning implementation.We present the results of our study, and compare discrimination power from the M-body phase space observables to standard observables as a benchmark.We conclude in sec:conc.Additional details are in the appendices. § OBSERVABLE BASIS In this section, we specify the basis of IRC safe observables that we use to identify structure in the jet.For simplicity, we will exclusively use the N-subjettiness observables <cit.>, however this choice is not special.One could equivalently use the originally-defined N-point energy correlation functions <cit.>, or their generalization to different angular dependence <cit.>.Our choice of using the N-subjettiness observbles in this analysis is mostly practical: the evaluation time for the N-subjettiness observables is significantly less than for the energy correlation functions.We also emphasize that the particular choice of observables below is to just ensure that they actually span the phase space for emissions in a jet.There may be a more optimal choice of a basis of observables, but optimization of the basis is beyond this paper.The N-subjettiness observable τ_N^(β) is a measure of the radiation about N axes in the jet, specified by an angular exponent β>0:τ_N^(β) = 1/p_T J∑_i∈Jet p_Timin{ R_1i^β,R_2i^β,…,R_Ni^β} .In this expression, p_TJ is the transverse momentum of the jet of interest, p_Ti is the transverse momentum of particle i in the jet, and R_Ki, for K=1,2,…,N, is the angle in pseudorapidity and azimuth between particle i and axis K in the jet.There are numerous possible choices for the N axes in the jet; in our numerical implementation, we choose to define them according to the exclusive k_T algorithm <cit.> with standard E-scheme recombination <cit.>.Note that τ_N^(β) = 0 for a jet with N or fewer particles in it.To identify structure in the jet, we need to measure an appropriate number of different N-subjettiness observables.This requires an organizing principle to ensure that the basis of observables is complete and minimal.Our approach to ensuring this is to identify the set of N-subjettiness observables that can completely specify the coordinates of M-body phase space.Ensuring that the set is minimal is then straightforward: as M-body phase space is 3M-4 dimensional, we only measure 3M-4N-subjettiness observables.A jet also has an overall energy scale.To ensure sensitivity to this energy scale, we will also measure the jet mass, m_J.We will describe how to do this for low dimensional phase space, and then generalize to arbitrary M-body phase space.We will work in the limit where the jet is narrow and so all particles in the jet can be considered as relatively collinear.This simplifies the expressions for the values of the N-subjettiness observables to illustrate their content, but does not affect their ability to span the phase space variables.*2-Body Phase Space: 2-body phase space is 3·2-4=2 dimensional.For a jet with two particles, the phase space can be completely specified by the transverse momentum fraction z of one of the particles:z=p_T1/p_TJ ,1-z=p_T2/p_TJ ,and the splitting angle θ between the particles.This configuration is shown in fig:2body.To uniquely identify the z and θ of this jet, we can measure two 1-subjettiness observables, defined by different angular exponents α≠β.For concreteness, we will measure τ_1^(1) and τ_1^(2).To determine the measured values of the 1-subjettiness observables, we need to determine the angle between the individual particles of the jet and the axis.Because E-scheme recombination conserves momentum, the angles between the particles 1 and 2 and the axis are:θ_1 = (1-z)θ ,θ_2 = zθ .It then follows that the values of the 1-subjettiness observables are:τ_1^(1) = 2z(1-z)θ ,τ_1^(2) = z(1-z)θ^2 .These expressions can be inverted to find z and θ individually:z(1-z)=(τ_1^(1))^2/4τ_1^(2) ,θ=2τ_1^(2)/τ_1^(1) .Note the symmetry for z↔ 1-z: this is to be expected because we have not assumed an ordering of the transverse momenta of particles 1 and 2. *3-Body Phase Space:3-body phase space is 3· 3-4=5 dimensional, and so to completely determine the configuration of a jet with three particles, we need to measure 5 N-subjettiness observables.The 5 phase space variables can be defined to be the 3 pairwise angles between the particles i and j in the jet: θ_12, θ_13, and θ_23, and two of the transverse momentum fractions, say, z_1 and z_2.We define the momentum fractions as: z_1=p_T1/p_TJ ,z_2=p_T2/p_TJ , 1-z_1-z_2 = p_T3/p_TJ .This configuration is shown in fig:3body.To determine the phase space variables, we will measure a collection of 1- and 2-subjettiness observables.Our choice for which collection of 1- and 2-subjettiness observables is the following.We will measure three 1-subjettiness observables τ_1^(0.5), τ_1^(1), and τ_1^(2) and two 2-subjettiness observables τ_2^(1) and τ_2^(2).To motivate this collection of observables, note that one of the axes for measuring 2-subjettiness necessarily lies along the direction of a particle.Therefore, measuring 2-subjettiness is only sensitive to one relative energy fraction and one angle between pairs of particles, as illustrated explicitly in the 2-body case in eq:2bodynsub.Because 2-subjettiness is only sensitive to two phase space variables, we only measure two 2-subjettiness observables.The axis for the 1-subjettiness observables, however, is necessarily displaced from the direction of any particle in the jet.[This is an important point, and the reason why we use E-scheme recombination as opposed to winner-take-all (WTA) recombination <cit.>, for example, to define the N-subjettiness axes.Because the axes defined by the WTA scheme necessarily lie along the direction of particles, there are non-degenerate configurations of particles for which measuring 5 N-subjettiness observables do not span the full 3-body phase space.]This is because the E-scheme recombination conserves momentum, and so this axis can only degenerate to the direction of a particle in the jet if another particle has 0 energy or is exactly collinear to another particle.Therefore, this collection of 5 N-subjettiness observables will generically span the full 3-body phase space.In app:3body, we present the explicit expressions for the 1- and 2-subjettiness observables in terms of the phase space coordinates. *M-Body Phase Space:For M-body phase space, we can define the coordinates of that phase space by M-1 transverse momentum fractions z_i, for i=1,…,M-1, and 2M-3 pairwise angles θ_ij between particles i and j.The remaining M2 - (2M-3) = 1/2(M-2)(M-3) ,pairwise angles angles are then uniquely determined by the geometry of points in a plane.[The proof of this is an application of the Euler Characteristic formula:V-E+F=2 .The number of vertices V is just the number of particles in the jet, M.The number of faces F is equal to the number of triangles that tesselate the plane, with vertices located at the particles.This is F=M-1, as we include the face outside the region where the points are located.It then follows that the number of edges E, that is, the number of pairwise angles necessary to uniquely specify their distribution, is E=2M-3.]To determine all of these phase space variables, we extend the set of N-subjettinesses that were measured in the 2- and 3-body case.In this case, the 3M-4 observables we measure are:{τ_1^(0.5),τ_1^(1),τ_1^(2),τ_2^(0.5),τ_2^(1),τ_2^(2),…,τ_M-2^(0.5),τ_M-2^(1),τ_M-2^(2),τ_M-1^(1),τ_M-1^(2)} .Note that there are 3(M-2)+2=3M-4 observables, and these will span the space of phase space variables for generic momenta configurations, when all particles have non-zero energy and are a finite angle from one another.As we observed in the 3-body phase space case, for a collection of M particles, all but one of the axes for the measurement of (M-1)-subjettiness lies along the direction of a particle.Therefore, we only measure two (M-1)-subjettiness observables.Stepping back another clustering as relevant for (M-2)-subjettiness, there are two possibilities:*Either M-3 axes lie along the direction of M-3 particles in the jet, and the three remaining particles are all clustered around the last axis.Then, the measurement of (M-2)-subjettiness is sensitive to the phase space configuration of 3 particles in the jet.By measuring three (M-2)-subjettinesses and two (M-1)-subjettinesses, this then completely specifies the phase space configuration of those three particles. *The other possibility is that M-4 axes lie along particles in the jet, while there are two particles clustered around each of the two remaining axes.About each axis, you are sensitive to the phase space configuration of two particles, which corresponds to a total of 4 phase space variables.Additionally, you are sensitive to the relative contribution of the two pairs of particles to the total (M-2)-subjettiness value.This configuration therefore is described by 5 phase space variables, and can be completely specified by the measurement of three (M-2)-subjettinesses and two (M-1)-subjettinesses. This argument can be continued at further stages in the declustering.Each time an axis is removed, three new phase space variables are introduced.These can be completely specified by the measurement of three additional N-subjettiness observables.This then proves that the collection of N-subjettiness observables given above uniquely determines M-body phase space. In the next section, we will study the information contained in this basis and use it to identify the features that are exploited in the discrimination of hadronically decaying Z boson jets from QCD jets. § DEEP LEARNING IMPLEMENTATION In this section, we describe our event simulation and implementation of machine learning to the N-subjettiness basis of observables introduced in the previous section.We generate pp→ Z+ jet and pp→ ZZ events at the 13 TeV LHC with MadGraph5 v2.5.4 <cit.>.The Z boson in pp→ Z+ jet events is decayed to neutrinos, while one Z boson in pp→ ZZ events is decayed to neutrinos, while the other is decayed to quarks.These tree-level events are then showered in Pythia v8.223 <cit.> with default settings.In app:herwig, we will show results showered with Herwig v7.0.4 <cit.>, however with one-tenth the number of events as the Pythia samples.Ignoring the neutrinos in the showered and hadronized events, we use FastJet v3.2.1 <cit.> to cluster the jets.On the clustered anti-k_T<cit.> jets with radius R=0.8 and minimum p_T of 500 GeV, we then measure the basis of N-subjettiness observables using the code provided in FastJet contrib v1.026.We emphasize that observables are measured on the particles as a proof of concept; we do not apply any detector simulation.The precise set of observables we measure on the jet that we use for discrimination are the following.We measure the jet mass and the collection of N-subjettiness observables sufficient to completely determine up through 6-body phase space.That is, we measure the collection of N-subjettiness observables defined with k_T axes:{τ_1^(0.5),τ_1^(1),τ_1^(2),τ_2^(0.5),τ_2^(1),τ_2^(2),τ_3^(0.5),τ_3^(1),τ_3^(2),τ_4^(0.5),τ_4^(1),τ_4^(2),τ_5^(1),τ_5^(2)} .We will see that this collection of N-subjettiness observables is more than sufficient to describe all of the information useful for discrimination in the jet.Additionally, for comparison, we will measure a collection of standard observables that have been defined for discrimination of boosted, hadronic decays of Z bosons from jets initiated by QCD.We measurethe N-subjettiness ratios τ_2,1^(1) and τ_2,1^(2) with one-pass winner-take-all (WTA) axes <cit.>, and (generalized) energy correlation function ratios D_2^(1) and D_2^(2)<cit.> and N_2^(1) and N_2^(2)<cit.>.The discrimination power of these observables will provide a benchmark for the information extracted in the machine learning of the collection of N-subjettiness observables.All deep learning analysis was carried out on the NVIDIA DIGITS DevBox, with four GeForce GTX TitanX GPUs, built on the 28 nm Maxwell architecture. The specifications of the GPU are listed in Table <ref>. Only one GPU was used during training and testing. ccccccCUDA cores Base/Boost. clock (MHz) Memory size (GB) Memory clock (Gbps) Interface width Memory Bandwidth (GB/s) 3072 1000/1075 12 7.0 384-bit 336.5 Manufacturer specifications of the GTX TitanX.The dataset consisted of 7,868,000 events, split evenly between Z and QCD jets, stored in the compressed HDF5 format<cit.>.The data was shuffled to ensure each data file had approximately a 1:1 ratio of both classes of events.No mass cuts were imposed on the events fed to networks with the expectation that they would automatically learn the optimal cuts on mass and the observable phase space. The training and validation data consisted of 6,144,000 events and 1,536,000 events respectively, while 188,000 events were set aside for predictions.All networks were trained using the highly modular Keras <cit.> deep learning libraries and tested using the relevant scikit-learn <cit.> packages. At the time of training, data from the relevant columns of N-subjettiness variables was fed to the neural networks with the aid of a custom-designed data generator, which creates an archive of pre-processed data files. A single neural network architecture, consisted exclusively of five fully connected layers, was utilized for all analyses.The first two Dense layers consisted of 10000 and 1000 nodes, respectively, and were assigned a Dropout <cit.> regularization of 0.2, while next two Dense layers consisted of 100 nodes each, and were assigned a Dropout regularization of 0.1 to prevent over-fitting on training data by making each node more `independent'. The input layer and all hidden layers utilized the ReLU activation function <cit.>, while the output layer, consisting of a single node, used a sigmoid activation. The network was compiled with the binary cross-entropy loss minimization function, using the Adam optimization <cit.>. Models were trained with Keras' default EarlyStopping, with a patience threshold of 5, to negate possible over-fitting.For each set of observables, the typical number of training epochs was about 60.To further eliminate errors due to under-training or over-training of networks, the same architecture was trained 25 different times for each round of analysis. The model that trained best for a given variable basis was picked based on a metric of maximizing the area under the signal vs. background efficiency curve. Before showing the results from the deep neural network, we first show plots of the collection of observables sensitive to two-prong structure measured on the jets.In fig:massplot, we plot the mass of the signal and background jets as defined by the simulation and jet finding from earlier.Applying a mass cut around the Z boson peak, we then measure the two-prong jet observables.In fig:obsdistros, we show the distributions of the N-subjettiness and energy correlation function ratios τ_2,1^(β), D_2^(β), and N_2^(β).As was extensively studied in the original works, these plots make clear the separation power that these observables enable.When we compare these observables to the discrimination power of the M-body phase space observables, we relax the hard mass cut, and let the machine learn the optimal mass and observable cuts dynamically.In fig:introplot, we plot the signal jet (Z boson) efficiency versus the background jet (QCD)rejection rate for the collection of observables that minimally span M-body phase space, along with the jet mass.The observables that are passed to the neural network to specify M-body phase space are, explicitly:2-body: τ_1^(1) ,τ_1^(2)3-body: τ_1^(0.5) ,τ_1^(1) ,τ_1^(2) ,τ_2^(1) ,τ_2^(2)4-body: τ_1^(0.5) ,τ_1^(1) ,τ_1^(2) ,τ_2^(0.5) ,τ_2^(1) ,τ_2^(2) ,τ_3^(1) ,τ_3^(2)5-body: τ_1^(0.5) ,τ_1^(1) ,τ_1^(2) ,τ_2^(0.5) ,τ_2^(1) ,τ_2^(2) ,τ_3^(0.5) ,τ_3^(1) ,τ_3^(2) ,τ_4^(1) ,τ_4^(2)6-body: τ_1^(0.5) ,τ_1^(1) ,τ_1^(2) ,τ_2^(0.5) ,τ_2^(1) ,τ_2^(2) ,τ_3^(0.5) ,τ_3^(1) ,τ_3^(2) ,τ_4^(0.5) ,τ_4^(1) ,τ_4^(2) ,τ_5^(1) ,τ_5^(2)Significant gains in discrimination power are observed by including observables sensitive to higher-body phase space, until enough observables to specify at least 4-body phase space are included.Including observables sensitive to 5- and 6-body phase space does not improve discrimination power, and therefore suggests that there is only an extremely limited amount of information in a jet useful for discrimination.To see what information is necessary to accomplish the maximal discrimination power, in fig:obsroc we plot the signal efficiency versus background rejection rate for the collection of N-subjettiness and energy correlation function ratios plotted earlier.For comparison, we also include the corresponding curves for the jet mass, jet mass plus 3-body phase space observables, and jet mass plus 4-body phase space observables.The discrimination power of all of these observables are comparable, and this illustrates that they appear to capture most of the information contained in the 3-body phase space observables.Then, to match the maximum discrimination power (as represented by the jet mass plus 4-body phase space curve), one just needs to augment the measurement of jet mass and an N-subjettiness or energy correlation function ratio with observables that are sensitive to some 3- and 4-body phase space information.We leave the construction of these optimal 3- and 4-body phase space observables for this purpose to future work.As a cross check that our minimal basis of N-subjettiness observables listed above does capture the maximal amount of information useful for discrimination, in fig:overplot, we compare our minimal basis to an overcomplete basis of observables.Here, we measure the mass and the following collection of N-subjettiness observables on the jet:{τ_1^(0.25),τ_1^(0.5),τ_1^(1),τ_1^(2),τ_1^(4),τ_2^(0.25),τ_2^(0.5),τ_2^(1),τ_2^(2),τ_2^(4),τ_3^(0.25),τ_3^(0.5),τ_3^(1),τ_3^(2),τ_3^(4),..τ_4^(0.25),τ_4^(0.5),τ_4^(1),τ_4^(2),τ_4^(4)} .From our arguments in sec:basis, this is an overcomplete basis for 5-body phase space and therefore should not contain any additional information useful for discrimination.This is illustrated in fig:overplot where we plot the discrimination power of this overcomplete basis as determined by the neural network described earlier.For comparison, we also show the discrimination power of the jet mass, the jet mass plus the 3-body observable basis, and the jet mass plus the 4-body observable basis as determined by the neural network described earlier.As expected, no improvement of discrimination power is accomplished when more observables beyond the minimal set are included.The apparent slight decrease in discrimination power using the overcomplete basis is likely due to suboptimal training because of the large number of input observables.In app:morearchs, we present results for the signal vs. background efficiency as determined by a neural network with an additional hidden layer and the result of a boosted decision tree.These different classification networks demonstrate the same conclusion, that discrimination power saturates once enough observables are measured to resolve 4-body phase space.Additionally, these results show that the discrimination power of the overcomplete basis is just marginally better than that accomplished by the 4-body observable basis.This is consistent with our observation that 4-body phase space is essentially saturating all useful discrimination information. § CONCLUSIONS Motivated by both the enormous data sets produced by the ATLAS and CMS experiments as well as their exceptional resolution, deep learning approaches to physics at the LHC are seeing an increased interest.This is especially true for jet physics, where the identification of the initiating particle of a jet is of fundamental importance.Previous applications of deep learning to jet physics applied techniques from computer science (like image recognition or natural language processing) and demonstrated impressive discrimination power.While the effectiveness of these methods is exceptional, they often lack a physical interpretation and are not presented in a constructive manner.The deep neural network is definitely identifying relevant structure in the jets, but what this is or if it is just a feature of the simulated data is not identified.Other recent efforts to reduce dependence on modeling have been studied in the context of weak supervision in Ref. <cit.>.In this paper, we have approached the problem of machine learning for jet physics in a physically clear, constructive manner.Instead of providing the machine with the energy deposits in calorimeter cells of the jet, we measure a basis of observables on the jet that completely and minimally spans M-body phase space.The effective resolution to the emissions in the jet is increased by increasing the number of observables measured on the jet.We demonstrated that the information useful for discrimination of a jet initiated by a boosted, hadronically-decaying Z boson from a jet initiated by a light QCD parton is saturated when enough observables are measured to span 4-body phase space.As 4-body phase space is only 8 dimensional, the amount of useful information in the jet is quite small.Additionally, this procedure is constructive in the sense that one can then form observables that are non-zero for a jet with four constituents to optimally discriminate signal from background.Similar constructions of observable bases for identifying particular phase space regions has been studied recently to resum non-global logarithms <cit.> and calculate multi-differential cross sections on jets <cit.>.Important for our analysis is that we use an IRC safe basis of observables that span the M-body phase space, namely, the N-subjettiness observables.This is vital for constructibility, as in principle the cross section for the measurement of multiple N-subjettiness observables on a jet can be calculated in the perturbation theory of QCD.[Actually calculating distributions of N-subjettiness observables in practice may be a significant challenge, however <cit.>.]It would be possible to additionally include information that is not IRC safe, for example, jet charge.Nevertheless, some non-IRC safe information is already included in this approach, like the jet constituent multiplicity.Additionally, included in the basis of M-body phase space observables are techniques like jet grooming that systematically remove radiation from the jet.This could enable a systematic study of how jet grooming methods affect the optimal discrimination observables, which has been addressed recently <cit.>.An advantage of our approach is that the jet data is preprocessed in a useful way at the same time that the basis observables are being measured.In applications of image processing to jets, one typically has to perform a series of transformations to ensure that different jets can be compared (see the discussion in, e.g., Ref. <cit.>).Jets must be rotated and rescaled appropriately so that (approximate) symmetries do not wash out the ability to discriminate.By instead measuring a collection of IRC safe observables like N-subjettiness on which we train, this preprocessing step is unnecessary, as the value of the observable is only sensitive to relative angles between particles and energy fractions.From our results, it would also be interesting to study in detail the information for discrimination that is missed when using standard jet observables like N-subjettiness ratios τ_2,1^(β) or energy correlation function ratios D_2^(β) or N_2^(β).The construction and justification of these particular observables exploited properties of QCD in the soft and/or collinear limits.These observables appear to be sensitive to most of the 3-body phase space information available for discrimination of boosted, hadronically decaying Z bosons from QCD jets.Observables that are sensitive to the remaining information for discrimination could be constructed by studying in detail the differences between how the decays of Z bosons and QCD fill 4-body phase space.We anticipate that these methods can also be used for discrimination of many different types of jets, including quark versus gluon and QCD versus top quark discrimination, as well as for multi-label classification of jets.The ultimate goal of such a program would be to design an anti-QCD tagger which could identify, using only a few observables that are sensitive to a small phase space, if a jet was likely initiated by a light QCD parton.This could open the door to new classes of observables that are sensitive to exotic configurations within jets. We thank Kyle Cranmer, Michael Kagan, Ian Moult, Ben Nachman, Duff Neill, Justin Pilot, Francesco Rubbo, Ariel Schwartzman, Jesse Thaler, and Daniel Whiteson for comments on the draft.We also thank our anonymous referee for suggesting the studies presented in app:morearchs.§ EXPLICIT EXPRESSIONS FOR 3-BODY PHASE SPACE In this appendix, we present the explicit expressions for the 1- and 2-subjettiness observables measured on a jet with three particles.The configuration of particles in the jet is shown in fig:3body.We will start with the evaluation of the 2-subjettiness observables, and then the 1-subjettiness observables. §.§ 2-subjettiness For measuring 2-subjettiness, we identify two axes defined by the exclusive k_T algorithm with E-scheme recombination.For three particles, one of the axes must necessarily lie along the direction of one particle in the jet, which we can take to be particle 3 without loss of generality.Then, only particles 1 and 2 can contribute to 2-jettiness.Call the axis about which particles 1 and 2 are clustered Â.Then, from fig:3body, the angle that particles 1 and 2 make with  are:θ_1 = z_2/z_1+z_2θ_12 ,θ_2 = z_1/z_1+z_2θ_12 .The 2-subjettiness observables that we measure are then:τ_2^(1) = z_1·z_2/z_1+z_2θ_12 + z_2·z_1/z_1+z_2θ_12 = 2z_1 z_2/z_1+z_2θ_12 , τ_2^(2) = z_1(z_2/z_1+z_2θ_12)^2 + z_2·(z_1/z_1+z_2θ_12)^2=z_1z_2/z_1+z_2θ_12^2 .Therefore the values of the 2-subjettiness observables can be inverted to determine the relative momentum fractionz_1z_2/z_1+z_2 ,and the pairwise angle θ_12. §.§ 1-subjettiness Now, we would like to calculate the value of 1-subjettiness on this configuration of particles.This requires determining the angle between each of the three particles and their direction of net momentum.To determine these angles, we consider the distribution of particles in the jet in a plane, as displayed in fig:3bodyplane.We set particle 1 at the origin (0,0) of the plane, particle 2 along the horizontal axis at (θ_12,0), and particle 3 at a generic point in the plane.The horizontal and vertical coordinates of particle 3 can be calculated to be:particle 3: ( θ_12^2+θ_13^2-θ_23^3/2θ_12,√(2θ_12^2θ_23^2+2θ_12^2θ_13^2+2θ_13^2θ_23^2-θ_12^4-θ_23^4-θ_13^4)/2θ_12) . With this expression, we can determine the location of the jet axis.With E-scheme recombination, the jet axis is located at the momentum-weighted centroid of the three particles:jet center: ( z_2θ_12+z_3 θ_12^2+θ_13^2-θ_23^3/2θ_12, z_3 √(2θ_12^2θ_23^2+2θ_12^2θ_13^2+2θ_13^2θ_23^2-θ_12^4-θ_23^4-θ_13^4)/2θ_12) .Here, for conciseness, we express z_3 = 1-z_1-z_2.It then follows that the angle from each particle to this jet axis  is:θ_1Â^2= z_2^2θ_12^2 + z_3^2θ_13^2 +z_2 z_3 (θ_12^2+θ_13^2-θ_23^2) , θ_2Â^2= z_1^2θ_12^2 + z_3^2θ_23^2 +z_1 z_3 (θ_12^2+θ_23^2-θ_13^2) , θ_3Â^2= z_1^2θ_13^2 + z_2^2θ_23^2 +z_1z_2 (θ_13^2+θ_23^2-θ_12^2) .The values of the three 1-subjettiness observables are then:τ_1^(0.5) = z_1θ_1Â^0.5+z_2θ_2Â^0.5+z_3θ_3Â^0.5 , τ_1^(1) = z_1θ_1Â+z_2θ_2Â+z_3θ_3 , τ_1^(2) = z_1θ_1Â^2+z_2θ_2Â^2+z_3θ_3Â^2 =z_1 z_2 θ_12^2+z_1 z_3 θ_13^2+z_2 z_3 θ_23^2 .For τ_1^(2), the expression simplifies significantly in terms of the momentum fractions and pairwise angles. § HERWIG RESULTS In this appendix, we present discrimination results for jets showered in Herwig 7.0.4 <cit.> from events generated in MadGraph.The number of events showered in Herwig is about a factor of 10 fewer than that shown in the main body of the paper with Pythia, and so the neural network training is not as efficient.Nevertheless, the conclusions drawn from this reduced Herwig sample are the same as from Pythia; namely, that observables sensitive to 4-body phase space saturate discrimination power.On the sample of jets from pp→ ZZ, with one Z decaying hadronically, and pp→ Z+ jet, we identify the same jets and measure the same collection of N-subjettiness observables as described in the main text.These observables are then passed through the deep neural network as described in sec:deeplearn, with 390,000 events each for pp→ ZZ and pp→ Z+ jet processes. These events were divided into 684,000 used for training, 76,000 for validation, and 20,000 for testing.In fig:massplot_hfig:obsdistros_h, we show validation plots on the jets showered with Herwig, to be compared with fig:massplotfig:obsdistros from Pythia.The jet mass distribution in fig:massplot_h agrees qualitatively well with the corresponding plot from Pythia; though the Herwig sample seems to lack the small shoulder of the Z boson mass distribution present in Pythia.With a cut on the jet mass around the location of the Z boson peak, we then measure the same selection of one- versus two-prong discriminant variables in the Herwig sample.Again, good qualitative agreement is seem with Pythia, though the effects of finite statistics are much more evident.fig:herwig shows the signal efficiency vs. background rejection rate for the collections of observables that resolve M-body phase space as determined by the neural network.Just like in the Pythia samples, the discrimination power is observed to increase as more N-subjettiness observables are included.The discrimination power is observed to saturate with observables that are sensitive to 3- or 4-body phase space.This difference from when the Pythia events saturated could be due to the smaller jet sample size, though it could also be due to differences between the Pythia and Herwig parton showers.It has been observed in numerous other studies <cit.> that the discrimination performance differs significantly between jets showered in Pythia versus Herwig.The exact reason for the discrepancy is beyond this paper, but the existence of a saturation point also in Herwig demonstrates that there is only a very limited amount of information in the jet for discrimination.§ RESULTS WITH OTHER ARCHITECTURES In this appendix, we show discrimination results for a neural network with one more hidden layer than the network studied in the body of the paper, as well as the output of a boosted decision tree. §.§ A Deeper Neural Network The neural network used in this appendix is identical to the network studied in the body of the paper, except with the addition of another layer.Immediately after the input layer, we have included an additional Dense layer of 1000 nodes, with a Dropout regularization of 0.2.The typical number of training epochs of this new neural network was about 50 for each collection of observables.We show the discrimination performance as identified by this network in fig:deeperplotfig:deeperoverplot.In fig:deeperplot, we show the discrimination power as more observables are added to resolve higher-body phase space.As with the other studies in this paper, we see that the discrimination power is saturated when 4-body phase space is resolved.Additionally in fig:deeperoverplot, we compare the discrimination power of 3- and 4-body phase space observables to the overcomplete 5-body phase space observables described in sec:deeplearn.The overcomplete basis of observables is observed to be only very slightly better than 4-body phase space basis, suggesting that essentially all useful discrimination information has been extracted. §.§ Boosted Decision Tree Because our observable basis is quite small, we can input them to a boosted decision tree to evaluate the discrimination power.We used ROOT's TMVA package <cit.> to train and test the boosted decision trees.Each collection of phase space observables studied elsewhere in this paper were input to the boosted decision trees, and forests of 2500 trees were used.We also trained on forests of 850 trees, and observed no significant improvement in discrimination power in extending to forests of 2500 trees, suggesting that the boosted decision trees are extracting all the information that they can.The results of the boosted decision trees are shown in fig:bdtplot.These results are again consistent with what we found earlier; namely, that discrimination power is observed to saturate once 4-body phase space is resolved. | http://arxiv.org/abs/1704.08249v2 | {
"authors": [
"Kaustuv Datta",
"Andrew Larkoski"
],
"categories": [
"hep-ph",
"hep-ex"
],
"primary_category": "hep-ph",
"published": "20170426180000",
"title": "How Much Information is in a Jet?"
} |
APS/123-QED Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana 47907, USACorresponding author: [email protected] Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana 47907, USA School of Electrical and Computer Engineering, Purdue University, West Lafayette, Indiana 47907, USA Purdue Quantum Center, Purdue University, West Lafayette, Indiana 47907, USA Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana 47907, USA The linear momentum and angular momentum of virtual photons of quantum vacuum fluctuations can induce the Casimir force and the Casimir torque, respectively. While the Casimir force has been measured extensively, the Casimir torque has not been observed experimentally though it was predicted over forty years ago. Here we propose to detect the Casimir torque with an optically levitated nanorod near a birefringent plate in vacuum. The axis of the nanorod tends to align with the polarization direction of the linearly polarized optical tweezer. When its axis is not parallel or perpendicular to the optical axis of the birefringent crystal, it will experience a Casimir torque that shifts its orientation slightly. We calculate the Casimir torque and Casimir force acting on a levitated nanorod near a birefringent crystal.We also investigate the effects of thermal noise and photon recoils on the torque and force detection. We prove that a levitated nanorod in vacuum will be capable of detecting the Casimir torque under realistic conditions, and will be an important tool in precision measurements.Valid PACS appear here Detecting Casimir torque with an optically levitated nanorod Tongcang Li December 30, 2023 ============================================================ § INTRODUCTION A remarkable prediction of quantum electrodynamics (QED) is that there are an infinite number of virtual photons in vacuum due to the zero-point energy that never vanishes, even in the absence of electromagnetic sources and at a temperature of absolute zero. In 1948, Casimir predicted an atrractive force between two ideal metal plates due to the linear momentum of virtual photons <cit.>. The number of electromagnetic modes between two metal plates is less than the number of modes outside the plates, thus the plates experience an attractive force, which is Casimir force. Casimir force has already been measured many times throughout the years <cit.>. Besides the linear momentum, the angular momentum carried by virtual photons can generate the Casimir torque (or van der Waals torque) for anisotropic materials<cit.>. Despite significant interests about the van der Waals and Casimir torque <cit.>, the torque has not been measured experimentally though it was predicted over 40 years ago, mainly due to the lack of a suitable tool <cit.>.Here we propose a method to measure the Casimir torque with a nanorod levitated by a linearly polarized optical tweezer in vacuum near a birefringent plate. The relative orientation between the nanorod and the birefringent crystal could be manipulated by the polarization of the trapping laser beam. When the long axis of the nanorod is not aligned with a principle axis of the birefringent plate, there will be a Casimir torque acting on the nanorod, which tries to minimize the energy, as shown in FIG. <ref>.(a),(b). Here d is the separation between the nanorod and the plate, and θ is the angle between the long axis of the nanorod and the optical axis of the birefringent plate. Casimir torque is related to separation d and relative orientation θ, which will be shown in the next section.An optically levitated nanoparticle in vacuum can have an ultrahigh mechanical quality factor (Q>10^9) as it is well-isolated from the thermal environment, which is excellent for precision measurements <cit.>. Optical levitation of a silica (SiO_2) nanosphere in vacuum at 10^-8 torr <cit.>, andforce sensing at10^-21N level with a levitated nanosphere <cit.> have been demonstrated in two separate experiments.The libration of an optically levitated nonspherical nanoparticle in vacuum has also been observed <cit.>, which provides a solid foundation for this proposal.We are going to detect the torsional vibration of the nanorod and measure its orientation with the laser polarization in a scheme similar to those reported in Ref. <cit.>. The nanorod will be levitated using an optical tweezer formed by a linearly polarized 1064 nm laser beam near a birefringent plate (FIG. <ref>).The torsional vibration of the nanorod will dynamically change the polarization of the laser beam, which can be detected with a polarizing beam splitter (PBS) and a balanced detector. The birefringent plate will cause a static change of the polarization of the laser, which can be canceled by a tunable waveplate as shown in FIG. <ref>.(c).In this paper, we will show that a silica nanorod with a length of 200 nm and a diameter of 40 nm levitated by a 100 mW optical tweezer in vacuum at 10^-7 torr will have torque detection sensitivity about 10^-28 Nm/√( Hz) at room temperature. The Casimir torque between a nanorod with the same size and a birefringent plate separated by 266 nm is calculated to be on the order of 10^-25 Nm. The Casimir torque is 3 orders of magnitude larger than the minimum torque we can detect in 1 second, and thus will be detectable with our system. A levitated nanorod in vacuum will be several orders more sensitive than the state-of-the-art torque sensor. The best reported torque sensitivity is 2.9× 10^-24 Nm/√( Hz), which was achieved by cooling a cavity-optomechanical torque sensor to 25mKin a dilution refrigerator <cit.>. The force detection sensitivity will be limited by the thermal noise when the pressure is above 10^-7 torr. When the pressure is below 10^-7 torr, the force sensitivity is mainly limited by photon recoil, which is about 10^-21 N/√( Hz). Our calculated turning point of the force sensitivity around 10^-7 torr is consistent with the experimental observation of photon recoils around 10^-7 torr <cit.>. The exact turning point depends on the size and shape of the nanoparticle, as well as the intensity of the trapping laser.Compared to the recent proposal of detecting the effects of Casimir torque with a liquid crystal <cit.>, our method with a levitated nanorod in vacuum will be able to measure the Casimir torque at a much larger separation (d>200 nm), where retardation is significant. We will also be able to investigate the Casimir torque as a function of relative orientation in detail.As an ultrasensitive nanoscale torsion balance <cit.>,our systemwill also enable many other precision measurements.§ TRAPPING POTENTIALWe consider a silicananorod with a length of l=200 nm in the long axis and a diameter of2a=40 nm trapped with a linearly polarized Gaussian beam in vacuum. The electric field of the beam can be described under the paraxial approximation asE_x(x,y,z)=E_0ω_0/ω(z) exp{-(x^2+y^2)/[ω(z)]^2} × exp(ikz+ikx^2+y^2/2R(z)-iζ(z)),E_y(x,y,z)=E_z(x,y,z)=0 ,where E_0 is the electric field amplitude at the origin, ω(z) is the radius at which the field amplitudes fall to 1/e of their axial values at the plane z along the beam, R(z) is the radius of curvature of the beam's wavefronts at z and ζ(z) is the Gouy phase at z.In the case of rods with a large apsect ratio, the components of the polarizability tensor <cit.> parallel and perpendicular to the symmetry axis are α_∥=Vϵ_0(ϵ_r-1) and α_⊥=2Vϵ_0(ϵ_r-1)/(ϵ_r+1). Here V is the volume of the object and ϵ_r is the relative permittivity of the object. The absorption of electromagnetic field of the silica is negligible, so we assume ϵ_r is real.When the size of the nanorod is much smaller than the wavelength of the laser (here we choose wavelength as 1064 nm), we can apply the Rayleigh approximation. The induced dipole will be 𝐩=α_xE_x𝐱̂_N+α_yE_yŷ_N+α_zE_zẑ_N, where the instantaneous electric field of the laser beam 𝐄 is decomposed into components along the principle axes of the nanorod. The long axis of the nanorod will tend to align with the polarization of the laser. When the vibration amplitude is small, the optical potential is harmonic around the laser focus and the vibrations of the trapped nanorod along different directions are uncoupled. Here we focus on its center-of-mass motion along z axis and its rotation around z axis. Thus the potential energy of the nanorod in the optical tweezer isU(z,ϕ)=-1/2cϵ_0[α_∥-(α_∥-α_⊥)sin^2ϕ)]I_laser(z),where c is the speed of light, ϵ_0 is the vacuum permittivity, ϕ is the angle between the long axis of the nanorod and the polarization of the Gaussian beam, and I_laser(z) is the laser intensity at the location of the nanorod. In free space, the peak laser intensity at the focus is given by I_laser=Pk_0^2 NA^2/(2π), where P is the laser power, NA is the numerical aperture of the objective lens, and k_0 is the magnitude of the wave vector of the laser beam. We assume NA=0.85 in this paper.Here we also need to consider the reflection from the substrate, which will form a standing wave with the incident wave and strengthen the trapping potential at the 1/4 wavelength point (FIG. <ref>). We assume that the center of Gaussian beam is 1/4 wavelength away from the surface of the substrate. In our case, the wavelength is 1064 nm, so the center of the beam will be 266 nm from the substrate surface. The refractive index is n_0=1 for vacuum and are n_o=2.269 and n_e=2.305 for ordinary and extraordinary axis of the birefringent crystal BaTiO_3 at 1064 nm<cit.>, respectively. If the laser is perpendicular to the surface, the reflectance along ordinary and extraordinary axis are R_o=0.16 and R_e=0.15. For a laser beam focused by a NA=0.85 objective lens, the angular aperture is 58^∘. The angle of incidence varies a lot at the surface of birefringent crystaland then the reflectance will become location and orientation dependent. However, only reflected wave from light with a smallincident angle will interfere with the incident wave and contributes to the trapping potential near z axis, where the nanorod is trapped. Furthermore, when alinearly polarized laser beam is focused before hitting a surface, 50% of the laser will be parallel to the incident plane (p-polarized), while the other 50% of the laser will be perpendicular to the incident plane (s-polarized). The average reflectance of a 50%p-polarized and 50%s-polarized laser at the maximum incident angle 58^∘ is 0.19, which is still close toR_o=0.16 and R_e=0.15.Therefore, we use R_o and R_e in the calculation for simplicity. We assume that the linearly polarized laser is polarized at π/4 relative to both optical axes of the birefringent crystal, and the axis of the nanorod is aligned with the polarization of the Gaussian beam (ϕ≈ 0). Then the potential near z=0 (or d=d_0) is U(z)=U(d-d_0)≈ -1/4α_∥E_0^2 [ω_0^2/[ω(d-d_0)]^2+R_o+R_e/2ω_0^2/[ω(d+d_0)]^2 +(√(R_o)+√(R_e)) cos(2kd) ω_0^2/ω(d-d_0)ω(d+d_0)], where d_0=266 nm is the distance from the center of the Gaussian beam to the birefringent crystal. We use Eq. <ref> to calculate the trapping potential and the result is shown in FIG. <ref> (a). Here the laser power is 100 mW and the waist radius is approximately 400 nm.The potential energy at the center of the beam is around -2.2× 10^4K, which allow us to avoid losing the nanorod from thermal motion at room temperature. § CASIMIR INTERACTION The Casimir force and the van der Waals force have the same physical origin, as they both arise from quantum fluctuations. Casimir forces between macroscopic surfaces involve separations typically larger than 100 nm where retardation effect plays an important role, while van der Waals forces often refer to separations smaller than a few nm where retardation is negligible<cit.>. To calcualte the Casimir interaction between a nanorod and a birefringent plate, we follow the method in Ref. <cit.> by assuming that a half space is a dilute assembly of anisotropic cylinders. With that we could extract the interaction between a cylinder and one semi-infinite half space from the interaction free energy between two half spaces.We notice that Ref. <cit.> has two typos. In Eq. 5 about the function N in Ref. <cit.>, the first term in the third line, should be ρ_3^2(ϵ_3-ϵ_1,⊥)(Q^2+ρ_1,⊥ρ_3) instead of ρ_3^3(ϵ_3-ϵ_1,⊥)(Q^2+ρ_1,⊥ρ_3). In Eq.7for the function f̃(ϕ) in Ref. <cit.>, the term inside the square root should be Q^2((ϵ_1,∥/ϵ_1,⊥)-1)cos^2ϕ+ρ_1,⊥^2 instead of Q^2((ϵ_1,∥/ϵ_1,⊥)-1)cos^2ϕ+ρ_1,∥^2. The corrected interaction free energy per unit length of the cylinder, g(d,θ), between a single cylinder and a half-space substrate is g(d,θ)=k_BTa^2/4π∑_n=0^∞'∫_0^∞QdQ∫_0^2πdϕ[e^-2dρ_3N/D],where N=(Δ_∥/2-Δ_⊥){Q^2 sin^2(ϕ+θ)×[f̃(ϕ)ϵ_1,⊥(Q^2 sin^2ϕ(ρ_1,⊥+ρ_3)+ρ_1,⊥ρ_3(ρ_3-ρ_1,⊥)) +(ϵ_1,⊥-ϵ_3)(ρ_3(ρ_1,⊥+2ρ_3)-Q^2)]-2f̃(ϕ)ϵ_1,⊥ρ_1,⊥ρ_3^2[2Q^2 sinϕcosθsin(ϕ+θ)+ρ_3^2 sin^2θ]+ f̃(ϕ)ϵ_1,⊥ρ_3^2[Q^2sin^2ϕ(ρ_1,⊥-ρ_3) +ρ_1,⊥ρ_3(ρ_1,⊥+ρ_3)] +ρ_3^2(ϵ_3-ϵ_1,⊥)(Q^2+ρ_1,⊥ρ_3)}+ 2f̃(ϕ)Δ_⊥ϵ_1,⊥[Q^2sin^2ϕ(Q^2ρ_1,⊥-ρ_3^3)+ρ_1,⊥ρ_3^2(Q^2cos(2ϕ)+ρ_1,⊥ρ_3)] -Δ_⊥(ϵ_1,⊥-ϵ_3)×[(Q^2+ρ_3^2)(Q^2+ρ_1,⊥ρ_3)+(Q^2-ρ_3^2)(Q^2-ρ_1,⊥ρ_3)] andD=ρ_3(ρ_1,⊥+ρ_3){ϵ_1,⊥f̃(ϕ)[Q^2sin^2ϕ-ρ_1,⊥ρ_3]+ϵ_1,⊥ρ_3+ϵ_3ρ_1,⊥}.In the equations above,ρ_1,⊥=√(Q^2+ϵ_1,⊥ω_n^2/c^2) , ρ_3=√(Q^2+ϵ_3ω_n^2/c^2), f̃(ϕ)=√(Q^2((ϵ_1,∥/ϵ_1,⊥)-1)cos^2ϕ+ρ_1,⊥^2)-ρ_1,⊥/Q^2 sin^2ϕ-ρ_1,⊥^2 . Here Δ_⊥=(ϵ_2,⊥-ϵ_3)/(ϵ_2,⊥+ϵ_3), Δ_∥=(ϵ_2,∥-ϵ_3)/ϵ_3 are the relative anisotropy measures of the cylinder, d is the separation between the cylinder and the half-space, a is the radius of the cylinder, k_B is Boltzmann constant, T is temperature, ϵ_3 is the dielectric response of the isotropic medium between the cylinder and the half space, ϵ_1,⊥ and ϵ_1,∥ are the dielectric responses of the birefringent material,ϵ_2,⊥ and ϵ_2,∥ are thedielectric responses of the cylinder material. Subscript n is the index for the Matsubara frequencies, which are ω_n=2nπ k_BT/ħ, and the prime on the summation in Eq. <ref> means that the weight of the n=0 term is 1/2. All the dielectric responses should be considered as functions of discrete imaginary Matsubara frequencies, i.e., as ϵ_3≡ϵ_3^(n)=ϵ_3(iω_n), ϵ_1,⊥(iω_n),ϵ_1,∥(iω_n), ϵ_2,⊥(iω_n) and ϵ_2,∥(iω_n). The dielectric properties of many materials are well described by a multiple oscillator model (the so-called Ninham-Parsegian representation)<cit.>. For most inorganic materials, only two undamped oscillators are commonly used to describe the dielectric function<cit.>,ϵ(iξ)=1+C_IR/1+(ξ/ω_IR)^2+C_UV/1+(ξ/ω_UV)^2,where ω_IR and ω_UV are the characteristic absorption angular frequencies in the infrared and ultraviolet range, respectively, and C_IR and C_UV are the corresponding absorption strengths. For the birefringent materials, there are separate functions describing dielectric functions for the ordinary and extraordinary axis. The model parameters used for our calculations are summarized in Table <ref>.We could use Eq. <ref>- <ref> and parameter data in Table <ref> to calculate the Casimir free energy G(d,θ)=g(d,θ)× l, where l is thelength of the cylinder <cit.>. FIG. <ref>.(a) inset shows that the Casimir free energy is very small for separation d>100 nm, compared to the optical trapping potential. So the nanorod will be trapped near the center of the laser beamwithout being attracted to the substrate by the Casimir force. When d<100 nm the size of the nanorod is comparable to the separation between the nanorod and the birefringent crystal, thus the dilute cylinder approximation will fail. Therefore, we only consider the situation when d>100 nm.Then the retarded Casimir (or Casimir-Lifshitz) force is given byF=-∂ G(d,θ)/∂ d,and the torque induced by the birefringent plates is given by <cit.>M=-∂ G(d,θ)/∂θ.UsingEq. <ref>, <ref>, we have calculated the Casimir force and torque expected for different separations at relative orientation θ=π/4, both for the barium titanate and calcite as the birefringent crystal. The results obtained for T=300 K are reported in FIG. <ref>.(b) and FIG. <ref>.(a).The force and torque both decrease as the separation increases. The force follows the same power dependence of the separation for different birefringent materials. However, there is no single power law dependence that describes the torque at all separations regardless of the choice of materials. That is because the Casimir torque is directly determined by the dielectric response difference between ordinary and extraordinary axes, which is discrepant between barium titanate and calcite (in Table.1).We have also calculated the Casimir torque at d=266 nm as a function of the relative orientation. From the results reported in FIG. <ref>.(b), one can clearly see that the torque oscillates sinusoidallywith periodicity of π: M=M_0 sin(2θ). The maximum magnitude of the torque occurs at θ=π/4 and θ=3π/4. For different birefringent crystals, the maximum magnitudes of the torque are different, but have the same periodicity. As expected, materials with less birefringence give rise to a smaller torque. Thus Casimir torque between silica nanorod and calcite is smaller than that with barium titanate. § EFFECTS OF THERMAL PHOTONSSeveral papers have reported measurements of the thermal Casimir force <cit.>, which is due to thermal photons (blackbody radiation) at finite temperature rather than quantum vacuum fluctuations of the electromagnetic field. At room temperature, the thermal Casimir force is typically much smaller than the conventional Casimir force due to quantum vacuum fluctuations. The measurements of thermal Casimir force could test different models of materials. In fact, it is still under debate about how to calculate the thermal Casimir force between real materials <cit.>.It is thus interesting to see how thermal photons affect the Casimir torque and whether thermal Casimir torque will be detectable with our proposed method.When the separation between the nanorod and the birefringent crystal is relatively small and the temperature is relatively low, the blackbody radiation could be neglected. But when the temperature and the separation increase, a small fraction of the torque will come from thermal photons. To single out the effects of thermal photons, we assume the dielectric functions of materials are independent of temperature. The effect of temperature is only included in the Bose-Einstein distribution of thermal photons. In other words, we let the explicit temperature T in Eq. <ref> and the Matsubara frequencies ω_n = 2 n π k_B T/ ħbe a variable, while assuming all parameters listed in Table 1to be constants. While this is a crude approximation, it can help us to understand the effects of thermal photons on the Casimir torque. In real experiments, the properties of materials will depend on temperature. So the situation will be more complex.The calculated results of the Casimir force and torque for d=266 nm as a function of temperature are shown in FIG. <ref>.(a)-(d). We can see that the thermal Casimir effect is very small (less than a few percent). So the measured torque will mainly come from quantum fluctuations. The purpose of this calculation is to estimate the magnitude of the effect of thermal photons. Since we did not consider the change of the dielectric constants of the real materials as a function of temperature, the temperature dependence of the experimental results is expected to bedifferent from FIG. <ref>. To avoid complications, it will be better to do the experiment at a fixed temperature. Because the effect of thermal photons is very small for separations considered here, the measured temperature dependence of the Casimir torque will be most likely due to the temperature dependence of the dielectric functions, instead ofthermal photons. § TORQUE MEASUREMENT METHODFor d=266 nm, the maximum magnitude of the Casimir torque on a silica nanorod (l=200 nm, a=20 nm)is around 3.2× 10^-25 Nm for barium titanate and around 4.6× 10^-26 Nm for calcite (Fig. <ref>). In order to prove that our optically levitated nanorod system is able todetect the Casimir force and torque, we calculated the sensitivity of the force and the torque and the results will be shown in subsection A. and B.In a real system, there are some other effects, such as stray fields, surface roughness and patch potential on the surface, which may introduce errors to the measurement. We will analyze these effects in subsection C. and D. §.§ Torque sensitivityTo understand the limit of torque sensitivity in the quantum regime, one must consider the noise limit which comes from thermal fluctuations, as well as from photon recoil. In air, the interaction between the nanorod and the thermal environment dominates the noise, thus the photon recoil from the laser can be neglected. However, in high vacuum, the dominant source of noise can come from the unavoidable photon recoil in the optical trap and sets an ultimate bound for the sensitivity.For small oscillation amplitudes, the equation of motion of a harmonic torsional oscillator isθ̈+γθ̇+Ω_r^2θ =M(t)/I ,where θ is the angular deflection of the oscillator, Ω_r is the frequency of rotational motion, M is a fluctuating torque, I is the moment of inertia around the torsion axis, and γ is the damping rate of the torsional motion which can be written as γ=γ_th+γ_rad. Here γ_th accounts for the interaction with the background gas, γ_rad refers to the interaction with the radiation field.When the torque fluctuation is from Brownian noise, the angular fluctuations of an oscillator excited by such a stochastic torque could be calculated. The thermal noise limited minimum torque that can be measured with a torsion balance is<cit.>M_th=√(4k_BTIγ_th/Δ t),where k_B is the Boltzmann constant, T is the environment temperature,and Δ t is the measurement time. The damping coefficient from thermal noise is γ_th=f_r/I. I=ρπ a^2l^3/12 is the moment of inertia of the nanorod around its center and perpendicular to its axis, ρ is the density of the nanorod, a is the nanorod radius and l is the nanorod length. f_r=k_BT/D_r is the rotational friction drag coefficient. D_r is the rotational diffusion coefficient for a rod in the free molecular regime and can be represented as <cit.>D_r=k_BTK_n/{πμ l^3[(1/6+1/8β^3)+f(π-2/48+1/8β+1/8β^2+π-4/81/8β^3)] } ,where β=l/a is the rod aspect ratio, K_n=λ/a is the Knudsen number, λ=μ/p√(π k_BT/2m_gas) is the mean free path, m_gas is the molecular mass, μ is the gas viscosity and f is the momentum accommodation, where we choose f=0.9. Thus the minimum detectable torque due to thermal fluctuations M_th decreases with the square root of the measurement duration Δ t, while increases with the square root of the pressure p.Apart from fluctuations due to contact with the background gas, the unavoidable photon recoil from the optical trap also contributes to the noise limit. The shot noise due to photon recoil can be understood as momentum or angular momentum kicks from the scattered photons. The photon recoil limited minimum torque isM_rad=√(4I/Δ td/dtK_R),where rotational shot noise heating rate of a nanorod from a linearly polarized trapping beam is <cit.>d/dtK_R = 8π J_p/3(k_0^2/4πϵ_0)^2 (α_⊥-α_∥)^2ħ^2/2I.Here the photon flux J_p is equal to the laser intensity over the energy of a photon, which means J_p=I_laser/ħω_0. ω_0 is the frequency of incident beam, k_0 is the incoming wave vector. Therefore, the total torque limit is given byM_min=√(M_th^2+M_rad^2), We calculate the torque limit by using Eq. <ref> - <ref> and the result of the calculation is shown in FIG. <ref>.(a). The torque detection sensitivity of a levitated nanorod will be limited by the thermal noise when the pressure is above 10^-7 torr. When the pressure is below 10^-7 torr, the torque sensitivity is mainly limited by photon recoil from the 100 mW trapping laser, andis around 10^-28 Nm/√( Hz). Thus the Casimir torque will be 3 orders larger than the minimum torque our system can detect in 1 second, and is expected to be measurable. §.§ Force sensitivity Similar to the torque sensitivity,both thermal noise and photon recoil needs to be considered to determine force sensitivity. For small oscillation amplitudes, the nanorod's motion is described by three independent harmonic oscillators (for three directions), each with its own oscillation frequency Ω_0i and damping rate γ_i, which is a result of the asymmetric shape of the optical potential. For example, the motion along y is described by,ÿ+γ_yẏ+Ω_0y ^2 y=1/mF_y(t),where y is the motion of the center of mass, m is the nanorod mass, F_y is a fluctuating force along y axis acting on the nanorod. The thermal noise limited minimum force in one direction i that can be measured with a force balance isF_th(i)=√(4k_BTmγ_i/Δ t),Here m is the mass, γ_i is the damping coefficient of the translational motion due to the background gas. For a nanorod, damping coefficients are directly related to the drag coefficients at each directions, which means that γ_⊥=K_⊥/m (component perpendicular to the axial direction) and γ_∥=K_∥/m (component parallel to the axial direction). In the free molecular regime, the drag force along different directions for a cylindrical particle are expressed by F_⊥=K_⊥V_⊥ and F_∥=K_∥V_∥ and drag coefficients are<cit.>K_⊥=2πμ a^2/λ[(π-2/4β+1/2)f+2β], K_∥= 2πμ a^2/λ[(β+π/4-1)f+2], In our system, we only consider the motion perpendicular to the axis, which will affect the measurement of Casimir force. The force inducted by thermal fluctuations isF_th=√(4k_BT/Δ tK_⊥),while the photon recoil limited minimum force isF_rad=√(4m/Δ td/dtK_T).Here the translational shot noise heating rate of a nanorod from a linearly polarized trapping beam is<cit.>dK_T/dt=8π J_p/3(k_0 ^2/4πϵ_0)^2α_⊥^2ħ ^2 k_0^2/2m,Therefore, the total force limit will beF_min=√(F_th^2+F_rad^2),Then we use Eq. <ref>- <ref> to calculate the force sensitivity limit and the result is shown in FIG. <ref>.(b). The force detection sensitivity will be limited by the thermal noise when the pressure is above 10^-7 torr. When the pressure is below 10^-7 torr, the force sensitivity is mainly limited by photon recoil, which is about 10^-21 N/√( Hz). The Casimir force is approximately 10^-16 N at d=266 nm, therefore, it is expected to be measurable. §.§ Pulsed measurement schemeSince the reflectancesalong the ordinary and extraordinary axes of the birefringent plate are different, the reflected lightwill not have the same polarization as the incident light (Fig. <ref>). Thus there will be an optical torque from the laser reflected by the birefringent plate. To eliminate this effect, we will apply a pulsed measurement scheme, which means to switch the optical tweezer on and off repeatedly to detect the Casimir torque by observing the rotation of the nanorod. We could extract the contribution whichcomes from the Casimir effect to the rotation during the period when the laser is off.When the laser is off, the naonorod will experience the torqueattributed to the Casimir effect. We can use this method to extract the torque from the Casimir part. However, when the laser is off, the nanorod will fall to the substrate by gravity as well as the Casimir force between it and the substrate. Therefore, we may lose the nanorod when the off-period is too long, while increasing the length of the off-period can amplify the signal observed from the Casimir torque.FIG. <ref> shows the simulation for this method. Initially the polarization of the laser is set to be 45^∘ relative to the optical axis of the birefringent plate, and the center of the laser beam is set at a distance of 266 nm from the substrate. FIG. <ref>.(a) is the separation evolution during the pulse measurement. During the period 0 to t_1 we will keep the laser on. During this time, the nanorod is trapped stably around the center of the beam (d=266nm), which is an equilibrium position. At t=t_1=10μ s, we will turn off the laser. So at this moment the nanorod will fall to the substrate with a acceleration of about 200 m/s^2 due to gravity and the Casimir force. We notice that, the nanorod will only fall 10 nm for a 10 μ s period. At t=t_2=20μ s, we turn on the laser again. The trapping force from the laser will pull back the nanorod. Then the nanorod will do harmonic oscillations around the equilibrium position (d=266 nm). When time reaches t_3=40 μ s, we will intentionally apply feedback coolingto the nanorod <cit.>. So the amplitude of the nanorod will decay to almost zero. At the end of a measurement cycle, the nanorod will come back to the initial situation. FIG. <ref>.(b) is the angle evolution during a pulsed measurement cycle. When the laser is on, the nanorod experiences an optical torque from the incident laser beam and from the reflected light, as well as a relatively small Casimir torque. Since the optical torque provided by the trapping laser can be far larger than theCasimir torque, the initial relative orientation is approximately 45^∘. When the laser is off during t_1=10 μ s and t_2=20 μ s, the torquecomes only from the Casimir effect. We will repeat this sequence many times (could be millions of times <cit.>) and average the results to extract the signal from the noise. The effective measurement time will be t_2-t_1 times the number of measurement cycles. §.§ Other effects In real experiments,there could be external stray fields, surface roughness and surface patch potentials that could affect measurements. If the nanorod has a permanent electric or magnetic dipole, there may be a torque on the nanorod due to stray electricor magnetic fields. Different from the Casimir torque, such torque due to a permanent dipole has a period of 2π. So if we rotate the nanorod by 180^∘, the dipole torque will change its sign, while the Casimir torque will be the same (FIG. <ref>(b)). Thus we can cancel out this dipole torque on the nanorod by a careful design. Roughness can change the effective separation between the nanorod and the plate, andinduce an additional torque on the nanorod. However, afterpolishing, the roughness could be controlled to be less than 3 nm for the regime near the nanorod <cit.>. FIG. <ref>(b) and FIG. <ref>(a) show the dependence of Casimir force and torque on the separation. When the separation changes 3 nm at an average separation about 266 nm, the force and the torque don't changemuch. Therefore, the torque generated by surface roughness can be neglected. Besides, there could be an inhomogeneous surface patch potential along the surface of real materials. Such patch potential can introduce a force and a torque, which can affect Casimir force and torque measurements. Luckily, it has been experimentally demonstrated that mostlevitated nanoparticles have zero electric charge, which can be verified by driving the particle with an AC electric field<cit.>. When there is no charge on the nanorod, the electric field of a patch potential can still cause a force and a torque due to the induced dipole. Here we analyze this situation and consider that there is a roundpatch with a diameter of 5 μm on a large birefringent plate. The patch is assumed to have a potential of 10 mV,while the plate is kept at zero potential. Then the potential at any point above the plane is given by <cit.><cit.>Φ(ρ,z)=V_0r_0∫_0^∞e^-λ zJ_1(λ r_0)J_0(λρ)dλ,where r_0=2.5 μm is the radius of the patch, V_0=10 mV is the fixed potential of the patch, J_0 and J_1 are the zero-order and first-order Bessel function of the first kind,ρ and z are the position of the potential in polar coordinates.The long axis of the rod aligns with the laser polarization, which is assumed to be y direction in FIG. <ref>.(a). Then the induced torque along z axis from the patch potential on the nanorod becomesT_z=(α_⊥-α_∥)E_xE_y, where α_∥ and α_⊥ are the components of the DC polarizability tensor parallel and perpendicular to theaxis of the nanorod, E_x and E_y are the electric field along x axis and y axis, respectively. Here the positive torque is in the direction along the positive z axis.The calculated results of the electric potential, electric field and induced torque on the nanorod due to the patch potential are shown in FIG. <ref>. FIG. <ref>(a) is the calculated patch potential in the plane 266 nm above the birefringent plate. FIG. <ref>.(b) is the calculated electric field in the plane 266 nm above the birefringent plate. We can see that the electric field reaches the maximum value at the edge of the patch and is always pointing away from the center of the patch. FIG. <ref>.(c) shows the induced torque from the patch potential and it also reaches the maximum at the edge. The torque at the edge could reach 2× 10^-25 Nm, which is at the same order as the Casimir torque.FIG. <ref>.(d) is the calculated potential in a plane perpendicular to the birefringent plate (y=0, shown as the white dashed line in (a)). Here the height above the plate ranges from 100 nm to 500 nm. FIG. <ref>.(e) is the calculated patch potential in the X-O-Z plane (y=0, shown as the white dashed line in (b)).Here we only consider the electric field along x axis.We can see that the electric fieldis anti-symmetric with respect to the z axis and is large at the edge of the patch.FIG. <ref>.(f) shows the induced torque from the patch potential in a plane perpendicular to the plate but has a distance of 1.77 μm from the center of the patch (y=1.77 μ m, shown as the black dashed line in (d)). In this plane, the maximum electric field is 45^∘ relative to the plane.The torque also reaches the maximum at the edge. FIG. <ref> provides a more detailed profile of the induced torque at a height of 266 nm and 1.77 μm from the center of the patch (corresponding to a horizontal line in FIG. <ref>(f)). Comparing the induced torque by the patch potential with the Casimir torque between silica nanaorod and two birefringent plates, we can see that the maximum value of the induced torque is at the same order as the Casimir torque. However, when the nanorod is not close to the edge of the patch, the induced torque is far smaller than the Casimir torque. Besides, the induced torque has different signs at different positions, while the Casimir torque is independent of the location for a single crystal birefringent plate. Therefore, we can cancel out the torque from the patch potential by measuring the torque at multiple locations along a line (1D) or along a 2D array.One possible way is to measure the torque at points that are equally spaced on the birefringent plate.For 1D scan, we will measure the points along the direction which is perpendicular to the axis of the nanorod. We assume that the axis of the nanorod is trapped along y direction, then we will measure the torque along positions that are equally spaced along x axis. In this way, the average of measured torque will be M_1D(x)=1/N∑_i=0^N-1M(x+L/Ni), where x∈(0,L/N) is the position of the first measurement point, N is the total number ofmeasurements, L is the measurement range, and M(x+L/Ni) is the torque measured at the i-th point, which includes both Casimir torque and the torque from patch potential. We let x be a variable to simulate the situation when we do not know the exact location of the patch. The measurement range L should be larger than the size of a patch. In the situation when there are many patches, L will be the larger the better for a single crystal birefringent plate.Here we simulate the results for the situation we discussed before in FIG. <ref> and set N to be 1, 2, 3, 4, 8 and 16. The results are shown in FIG. <ref>. We compare the effect from the patch at z=266 nm, y=1.77 μm with the Casimir torque between the silica nanorod and the birefringent plates (barium titanate and calcite). Here we set the measurement range L=10 μm. FIG. <ref>.(a) shows the expected measured torque (blue solid line) when the patch-induced torque is included, which means M=M_Casimir+M_patch. The red dashed line shows the Casimir torque between the nanorod and barium titanate plate. (b)-(f) show the average of the torque measured at N equally spaced positions on the plate when N=2, 3, 4, 8, 16, respectively. We use Eq. <ref> to the get the average torque. We can see that the average torque from the patch decays very fast when we increase N. FIG. <ref>(g)-(l) showsimilar results when the birefringent plate is a calcite plate. In real experiments, we may not know the position of the patch. However, as shown in Fig. <ref>(f),(l),when N is large, the torque from the patch potential will be negligible compared to the Casimir torque and will only have a weak dependence of the measurement position.Therefore, we can use this method to minimize the effect from patch potential.We also consider a two-dimensional scan, which means we will measure the torque following a 2D array in the X-O-Y plane. In this way, the averaged torque will be M_2D(x,y) =1/K^2∑_i=0^K-1∑_j=0^K-1M(x+L/Ki,y+L/Kj), where (x,y) is the position of the first measurement point on the plate and x∈(0,L/K),y∈(0,L/K). K is the number of measurements along one axis, N=K^2 is the total number ofmeasurements, L is the measurement range in one dimension and M(x+L/Ki,y+L/Kj) is the torque measured at the i-th point along x axis and the j-th point along y axis. Here we also choose measurement range L to be 10 μm. FIG. <ref> shows the relation between the number of measurements and the averaged torque from patch potential, both for one-dimensional (blue dotted line) and two-dimensional scans (black dashed line). Vertical axis shows the maximum of the averaged torque from the patch potential, horizontal axis corresponds to the number of measurements. The red solid line showsM=M_0/N, where M_0=2.65× 10^-25Nm is the maximum value of the patch-induced torque at 266 nm above the birefringent plate (no averaging). For the two-dimensional averaging method, the maximum averaged torque from the patch potential approximately follows the 1/N law. The 1D averaged torque from the patch along x axis decays much faster than the 2D averaged torque from the patch potential. When N=30, the maximum value of the 1D averaged torque from the patch potential is about 3× 10^-28 Nm, which is three orders smaller than the Casimir torque between a silica nanorod and a barium titanate plate. This further proves that we can decrease the effect from patch potential by measuring the torque at multiple locations. Meanwhile, we can reduce the surface patch potential by careful preparation of the sample, as done in an experiment that measured the Casimir force which improved the flatness to be less than 3 nm over mm^2 area<cit.>.We can determine the topography and observe patch potential on the surface by using Kelvin probe force microscopy and choose the area with relatively small roughness and patch potential to do the measurement<cit.>. We can also measure the toque due to the surface patch potential directly by utilizing the angular dependence of the Casimir torque. As shown in FIG. <ref>.(b), the Casimir torque will be maximum when the relative angle between the nanorod and the optical axis of a birefringent crystal is 45^∘, and it will be 0 when the angle is 0^∘ or 90^∘. So we candirectly measure the the torque due to the patch potential by setting the angle to be 0^∘and 90^∘when the Casimir torque is zero. We can then subtract the torque due to the surface patch potential from the total measured torque to obtain the Casimir torque at 45^∘. § CONCLUSIONIn this paper, we show that the calculated Casimir forceis on the order of 10^-16 N and the torque is on the order of 10^-25 Nm between an optically levitated silica nanorod (l=200nm, a=20nm) and a birefringent crystal separated by 266 nm. Considering noise from thermal interaction and photon recoil, we get the sensitivity of our system, which is on the order of 10^-28 Nm/√(Hz) at 10^-7 torr. Therefore, the system will allow us to measure the Casimir torque and test the fundamental prediction of quantum electrodynamics <cit.>. Besides its fascinating origin, the QED torque between anisotropic surfaces is expected to be important for the anisotropic growth of some crystals <cit.> and biological membranes. Our system will enable many other precision measurements, such as a measurement of the torque on a single nuclear spin <cit.>. It can also study electrostatics of surfaces. § ACKNOWLEDGMENTS We thank C. Zhong, F. Robicheaux, E. Fischbach, Z.-Q. Yin, J. Ahn, and J. Bangfor helpful discussions. This work is partly supported by the National Science Foundation under grant No. PHY-1555035.100 Casimir H. B. G. Casimir, “On the attraction between two perfectly conducting plates”, Proc. K. Ned. Akad. Wet. 51, 793 (1948). 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Robicheaux, “Decoherence of rotational degrees of freedom”, Phys. Rev. A 94, 052109 (2016). decoherenceB. A. Stickler, B. Papendell, and K. Hornberger, “Spatio-orientational decoherence of nanoparticles”, Phys. Rev. A 94, 033828 (2016). MingdongLi2M. Li, G. W. Mulholland, and M. R. Zachariah, “The effect of orientation on the mobility and dynamic shape factor of charged axially symmetric particles in an electric field”, Aerosol Sci. Technol. 46 (9), 1035 (2012).Hoang2017 T. M. Hoang, R. Pan, J. Ahn, J. Bang, H. T. Quan, T. Li. "Experimental test of the universal differential fluctuation theorem with a levitated nanosphere", arXiv:1706.09587 (2017)surfaceeffect J. L. Garrett, D. Somers, and J. N. Munday, “The effect of patch potentials in Casimir force measurements determined by heterodyne Kelvin probe force microscopy”, J. Phys.: Condens. Matter 27, 214012 (2015).netchargeM. Frimmer, K. Luszcz, S. Ferreiro, V. Jain, E. Hebestreit, and L. Novotny, “Controlling the net charge on a nanoparticle optically levitated in vacuum”, Phys. Rev. A 95, 061801 (2017).KPFMR. O. Behunin, D. A. R. Dalvit, R. S. Decca, C. Genet, I. W. Jung, A. Lambrecht, A. Liscio, D. Lopez, S. Reynaud, G. Schnoering, G. Voisin, and Y. Zeng, “Kelvin probe force microscopy of metallic surfaces used in Casimir force measurements”, Phys. Rev. A 90, 062115 (2014). Jackson J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1999). Surfacetrap P.-J. Wang, T. Li, C. Noel, A. Chuang, X. Zhang, and H. Häffner, “Surface traps for freely rotating ion ring crystals”, J. Phys. B 48, 205002 (2015). CasimirTang D. Garcia-Sanchez, K. Y. Fong, H. Bhaskaran, S. Lamoreaux, and H. X. Tang, “Casimir Force and In Situ Surface Potential Measurements on Nanomembranes”,Phys. Rev. Lett. 109, 027202 (2012). | http://arxiv.org/abs/1704.08770v2 | {
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"Tongcang Li"
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"title": "Detecting Casimir torque with an optically levitated nanorod"
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Technical ReportMillimeter Wave Communication in Vehicular Networks: Coverage and Connectivity Analysis Marco Giordani Andrea Zanella Michele Zorzi E-mail: {giordani, zanella, zorzi}@dei.unipd.it December 30, 2023 ===========================================================================================================style1In this technical report (TR), we will report the mathematical model we developed to carry out the preliminary coverage and connectivity analysis in mmWave-based vehicular networks, proposed in our work <cit.>. The purpose is to exemplify some of the complex and interesting tradeoffs that have to be considered when designing solutions for mmWave automotive scenarios. The rest of TR is organized as follows.In Section <ref>, we describe the scenario used to carry out our simulation results, presenting the main setting parameters and the implemented mmWave channel model. In Section <ref> we report the mathematical model used todevelop our connectivity and coverage analysis. Finally, in Section <ref>, we show our main findings in terms of throughput.§ SIMULATION SETTINGS Weconsider a simple but representative V2I scenario, where a single Automotive Node (AN, i.e., a car) moves along a road at constant speed V and Infrastructure Nodes (INs, i.e., static mmWave Base Stations) are randomly distributed according to a Poisson Point Process (PPP) of parameter ρ nodes/km, so that the distance d between consecutive nodes is an exponential random variable of mean 𝔼[d]= 1/ρ km (see, e.g., <cit.>). §.§ Millimeter Wave Channel ModelAs assessed in <cit.>, in order to overcome the increased isotropic path loss experienced at higher frequencies, next-generation mmWave automotive communication must provide mechanisms by which the vehicles and the infrastructuresdetermine suitabledirections of transmission for spreading around their sensors information, thus exploiting beamforming (BF) gain at both the transmitter and the receiver side. To provide a realistic assessment of mmWave micro and picocellular networks in a dense urban deployment, we considered the channel model obtained from recent real-world measurements at 28 GHz in New York City. Further details on the channel model and its parameters can be found in <cit.>.[As pointed out in <cit.>, available measurements at mmWaves in the V2X context are still very limited, and realistic scenarios are indeed hard to simulate. Moreover, current models for mmWave cellular systems (e.g., <cit.>) present many limitations for their applicability to a V2X context, due to the more challenging propagation characteristics of highly mobile vehicular nodes. Simulating more realistic scenarios for further validating the presented results, like considering channel models specifically tailored to a V2X context, is of great interest an will be part of our future analysis.]The link budget for the mmWave propagation channel is defined as:P_RX = P_TX + G_BF - PL - ξwhere P_RX is the total received power expressed in dBm, P_TX is the transmit power, G_BF is the gain obtained using BF techniques, PL represents the pathloss in dB and ξ∼ N(0,σ^2) is the shadowing in dB, whose parameter σ^2 comes from the measurements in <cit.>. Based on the real-environment measurements of <cit.>, the pathloss can be modeled through three different states: Line-of-Sight (LoS), Non-Line-Of-Sight (NLoS) and outage. Based on the distance d between the transmitter and the receiver, the probability to be in one of the states (P_LoS, P_NLoS, P_out) is computed by:P_out(d)= max(0,1-e^-a_outd+b_out)P_LoS(d)= (1-P_out(d) )e^-a_LoSdP_NLoS(d)= 1- P_out(d) - P_LoS(d)where parameters a_out = 0.0334m^-1, b_out = 5.2and a_LoS = 0.0149m^-1 have been obtained in <cit.> for a carrier frequency of 28 GHz. The pathloss is finally obtained by:PL(d)[dB] = α + β 10 log_10(d) where d is the distance between receiver and transmitter, and the value of the parameters α and β are given in <cit.>.fancyTo generate random realizations ofthe large-scale parameters, the mmWave channel is defined as a combination of a random number K∼max{Poisson(λ),1 }of path clusters, for which the parameter λ can be found in <cit.>, eachcorresponding to a macro-level scattering path. Each cluster is further composed of several subpaths L_k ∼ U[1,10]. The time-varyingchannel matrix is described as follows:H(t,f) = 1/√(L)∑_k=1^K∑_l=1^L_k g_kl(t,f) u_rx(θ_kl^rx,ϕ_kl^rx)u^*_tx(θ_kl^tx,ϕ_kl^tx)where g_kl(t,f) refers to the small-scale fading over time and frequency on the l^th subpath of the k^th cluster and u_rx(·), u_tx(·) are the spatial signatures for the receiver and transmitter antenna arrays and are functions of the central azimuth (horizontal) and elevation (vertical) Angle of Arrival (AoA) and Angle of Departure (AoD),respectively θ_kl^rx, ϕ_kl^rx, θ_kl^tx, ϕ_kl^tx [Such angles can be generated as wrapped Gaussian around the cluster central angles with standard deviation given by the rms angular spread for the cluster given in <cit.>.].The small-scale fading in Equation (<ref>) describes the rapid fluctuations of the amplitude of a radio signal over a short period of time or travel distance. It is generated based on the number of clusters, the number of subpaths in each cluster, the Doppler shift, the power spread, and the delay spread, as: g_kl(t,f)=√(P_lk)e^2π i f_d cos(ω_kl)t-2π i τ_klf,where: * P_lk is the power spread ofsubpath l incluster k, as defined in <cit.>;* f_d is the maximum Doppler shift and is related to theuser speed (v) and to the carrier frequency f as f_d = f v / c, where c is the speed of light;* ω_klis the angle of arrival of subpath l in cluster k with respectto the direction of motion;* τ_kl gives the delay spread of subpath l incluster k;* f is the carrier frequency. Due to the high pathloss experienced at mmWaves, multiple antenna elements with beamforming are essential to provide an acceptablecommunication range. The BF gain parameter in (<ref>) from transmitter i to receiver j is thus given by:G_BF(t,f)_ij = |w^*_rx_ijH(t,f)_ijw_tx_ij|^2where H(t,f)_ij is the channel matrix of the ij^th link, w_tx_ij∈ℂ^n_𝕋x is the BF vector of transmitter i when transmitting to receiver j, and w_rx_ij∈ℂ^n_ℝx is the BF vector of receiver j when receiving from transmitter i.Both vectors are complex, with length equal to the number of antenna elements in the array, and are chosen according to the specific direction that links BS and UE.The channel quality is measured in terms of Signal-to-Interference-plus-Noise-Ratio (SINR). By referring to the mmWave statistical channel described above, the SINR between a transmitted j and a test RX can computed in the following way:Γ = SINR_ j,RX = P_ TX/PL_ j,UEG_ j,RX/∑_ k≠ jP_ TX/PL_ k,RXG_ k,RX+W_ tot× N_0where G_ i,RX and PL_ i,RX are the BF gain and the pathloss obtained between transmitter i and the test RX, respectively, and W_ tot× N_0 is the thermal noise.§.§ System Model We say that an AN is within coverage () of a certain IN if, assuming perfect beam alignment, the SINR is the best possible for the AN, and it exceeds a minimum threshold, which we set to Γ_0=-5 dB. Due to the stochastic nature of the signal propagation and of the interference, the coverage range of an IN is a random variable, whose exact characterization is still unknown but clearly depends on a number of factors, such as beamwidth,propagation environment, and level of interference that, in turn, depends on the spatial density of the nodes.To gain some insights on these complex relationships, weperformed a number of simulations and evaluated the mean coverage rangewhen varying the node density and the antenna configuration of the nodes. Tab. <ref> collects the main simulation parameters, which are based on realistic system design considerations.A set of two dimensional antenna arrays is used at both the INs and the ANs. INs are equipped with a Uniform Planar Array (UPA) of 2 × 2 or 8 × 8 elements, while the ANs exploit an array of 2 × 2 or 4 × 4 antennas. The spacing of the elements is set to λ/2, where λ is the wavelength. In general,as depicted in Fig. <ref>,increases with the number of antennas, thanks to the narrower beams that can be realized, which increase the beamforming gain.On the other hand,decreases as the density of nodes increases, because of the larger amount of interference received at the AN from the INs due to their reduced distance.As we pointed out in <cit.>,a directional beam pair needs to be determined toenable the transmission between two ANs, thus beam tracking heavily affects the connectivity performance of a V2X mmWave scenario. In thisanalysis, according to the procedure described in <cit.>, we assume thatmeasurement reports are periodically exchanged among thenodes so that, at the beginning of everyslot of duration , ANs and INs identify the best directions for their respectivebeams.Such configuration is kept fixed for the whole slot duration, during which nodes may lose the alignment due to the AN mobility. Incase the connectivity is lost during a slot, it can only be recovered at the beginning of the subsequent slot, when the beam tracking procedure is performed again. It is hence of interest to evaluate the AN connectivity, i.e., the fraction of slots in which the AN and the IN remain connected (since the AN is in theIN's coverage range), as a function of the following parameters: (i) the MIMO configuration; (ii) the vehicle speed V; (iii) the slot duration T_ RTO; (iv) the node density ρ.At the beginning of a time slot, the AN can be eitherin a connected (C) state, if it is within the coverage range of an IN, or in an idle (I) state, if there are no INs within a distance .Starting from state C, the AN can either maintain connectivity to the serving IN for the whole slot duration, or lose the beam alignment and get disconnected. Starting from state I, instead, the AN can either remain out-of-range for the whole slot of duration , or enter the coverage range of a new IN within(catch-up). Even in this second case, however, the connection to the IN will be established only at the beginning of the following slot, when the beam alignment procedure will be performed.Therefore,preservation of the connectivity during a slot requires that the AN is within the coverage range of the IN at the beginning of the slot and does not lose beam alignment in the slot period . In this case, the vehicle can potentially send data with a rate R(d) that depends on the distance d to the serving IN. § COVERAGE AND CONNECTIVITY ANALYSISIn order to determine the average throughputa vehicular node will experience in the considered simple automotive scenario, as a function of several V2X parameters, we first need tocompute the average portion of slot in which the VN is both within the coverage of an infrastructure node and properly aligned. To do so, in this section we will: (i) evaluate the probability for the VN to be within , at the beginning of the slot; (ii) evaluate the probability for the VN not to misalign within the slot; (iii) finally evaluate the mean communication duration.§.§ Probability of Starting the Communication The communication can start, within the slot of duration T_ RTO, only if the vehicle is already within the coverage range R_ comm, with probability ℙ_D (R_ comm):P_ start = ℙ[d ≤ 2 R_ comm] = 1-e^-2ρ R_ commFrom the results in Figure <ref>, we deduce that:* P_ start increases with the IN density ρ, since the mean distance 𝔼[d] = 1/ρ from the IN decreases, so it's more likely for the AN to fall within the coverage range of the IN.On the other hand, although the communication range R_ comm is reduced when considering denser networks, due to the increased interference perceived by the vehicular node, P_ start stillincreases, making the reduction of the distance dominant to the increased interference[We observe that the increasing behavior of P_ start saturates when INs are particularly dense.]. * P_ start increases when increasing the MIMO order, that is when packing more antenna elements in the MIMO array. In fact, keeping ρ fixed, beams are narrower, interference is reduced, the achieved BF gain is higher, and the increased R_ comm makes the discoverable range of the IN increased as well.§.§ Probability of NOT Leaving the CommunicationConstraining on the probability of having started the communication within the slot of duration, the vehicle loses its ability to communicate with the IN with probability 1-P_ NL, whereis defined as:P_ NL = P_ start·ℙ(T_ L > T_ RTO). If thevehicle was already in the communication range at the beginning of the slot, it does not leave the communication with probability ℙ(T_ L > T_ RTO), that is if it covers a distance smaller than d|d<, within , moving at relative speed V>0:ℙ(T_ L > T_ RTO) ==1-ℙ(Overtaking the IN within| vehicle is connected) = =ℙ[ dV >| d<] = e^-ρ V-e^-ρ/1-e^-ρ On the other hand, if the AN overtakes the IN (as in Figure <ref>) during the slot, although theoretically being under the coverage of the infrastructure, itneeds to adapt its beam orientation to be perfectly aligned; however, this operation can be triggered only at the beginning of the subsequent slot, making the AN misaligned and thus disconnected for the whole remaining slot period. In Figure <ref>, we plot the probability of not leaving the communication (P_ NL) and we state that: * P_ NL increases with ρ, for sparse networks (ρ small). On one hand, P_ start increases but, on the other hand, the AN is closer and closer to the IN (the mean distance 𝔼[d] = 1/ρ from the IN decreases) and, within the same time slot T_ RTO, it is more and more likely for the AN to overtake the IN and being misaligned.However, when the infrastructure nodes are quite scattered,the mean distance 𝔼[d] = 1/ρ is relatively large and so the increasing behavior of P_ start in Eq. (<ref>) is dominant. * P_ NL decreases with increasing values of ρ, for dense networks (ρ large). In fact, P_ start has reached a quasi-steady state (see Figure <ref>) and almost does not increase as ρ increases, whereas the mean distance 𝔼[d] = 1/ρ keeps reducing, thus increasing the chances for the AN to leave its serving IN's connectivity range and be misaligned. * P_ NL increases when V decreases since, when the VN is slower, it has less chances to leave the communication range of the INwithin the same time slot of duration T_ RTO.* P_ NL increases when T_ RTO decreases, since the ANcovers a shorter distance V · T_ RTO moving at the same speed V, thus reducing the chances toleave its serving IN's communication range.§.§ Mean Communication Duration The communication is "active" onlywhen the vehicle is both within the coverage range of the IN and correctly aligned, during the slot of duration .Constraining on the probability of having started the communication, the mean communication duration is: 𝔼[T_ comm] =P_ start·[ℙ(T_ L > T_ RTO)T_ RTO +(1-ℙ(T_ L > T_ RTO))𝔼[T_ L] ]. In particular, if thevehicle was already in the communication range at the beginning of the slot: * if the vehicle never overtakes its serving IN and never becomes misaligned within the slot (with probability ℙ(T_ L > T_ RTO)), it communicates for the whole slot, so for a duration ;* if, at some point, the vehiclebecomes misaligned with its IN, although being insideduring the slot (with probability 1-ℙ(T_ L > T_ RTO)), it communicates only for the time 𝔼[T_ L]< in which it was correctly able to be served (during this time window, the VN has covered a distance d|d< V, moving at speed V), where:𝔼[T_ L] = =𝔼[Time in which the VN is aligned within| VN is connected but leaves] == 𝔼[d|d< V | d < ]V== 1/V1/ℙ[d< V | d<]∫_0^ Vηp_d(η)dη(a) == 1/V1-e^-ρ/1-e^-ρ V1-e^-ρ V(ρ V +1)/ρ, where step (a) is based on the fact that, in realistic vehicular environments, V ≪.In Figure <ref>, we plot the communication duration ratio, that is the ratio between the portion of time slot in which the VN is both within the communication rangeof its serving IN and properly aligned (𝔼[T_ comm]) and the whole slot duration () (i.e., 𝔼[T_ comm]/=1 if the VN is connected for the whole time slot). We observe that results agree with the considerations we made in the previous subsection and, in particular: * 𝔼[T_ comm]/ increases with ρ, for sparse networks (ρ small), since the mean distance 𝔼[d] = 1/ρ is relatively large and it is very unlikely for the AN to overtake its serving IN and become misaligned.The increasing behavior of the communication duration ratio is therefore caused by the increased probability of being withinat the beginning of the slot. * 𝔼[T_ comm]/ decreases with increasing values ofρ, for dense networks (ρ large). In fact P_ start has reached a quasi-steady state (see Figure <ref>) and almost does not increase as ρ increases, while the mean distance 𝔼[d] = 1/ρ keeps reducing, thus increasing the chances for the AN to overtake the IN and disconnect. * 𝔼[T_ comm]/ increasesfor lower values of V and . Therefore, we assess that considering slower vehicular nodes or updating more frequently the AN-IN beam pair, respectively, reflects an increased average communication duration[Of course, more frequent beam-alignment updates has some overhead issues which must be considered.]. § THROUGHPUT ANALYSIS A non-zero throughput can be perceived (within the slot of duration ) only when the vehicle is within coverage and properly aligned with the infrastructure, that is for a time 𝔼[T_ comm].In this period, the vehicle perceives a rate 𝔼[R(d)] that depends on the mean distance 𝔼[d] = 1/ρ, proportional to the nodes spatial density ρ. The throughput is therefore defined as: B = 𝔼[R(d)] ·𝔼[T_ comm]/T_ RTO In Figure <ref> and <ref>, we report the average throughput, when varying some system parameters. It is rather interesting to observe that, in all considered configurations, the throughput exhibits a similar pattern when varying the node density ρ. In particular: * We note that, as the node density is increased, the average distance between the AN and the IN decreases and, hence, the average bit rate experienced by the AN in case of connectivity becomes larger. On the other hand, the smaller coverage range will increase the probability of losing connectivity during a slot and will determine an increment of the frequency of handovers. * For small values of ρ, B initially increases with ρ. In this region, the SINR increases with ρ because the reduction of the mean distance to the serving IN is more significant than the increase of the interference coming from the neighboring INs.Moreover, the distance between adjacent INs is still sufficiently large to allow for a loose beam alignment (thanks to the widening of the beam with the distance), so that the connectivity between the AN and the IN is maintained for a relatively large number of slots.* After a certain value of ρ (approximately 40 nodes/km in our scenario), B starts decreasing. In this region, the interference from close-by INs becomes dominant and the perceived SINR degrades.Moreover, the closer the distance between the IN and the AN, the smaller the beam widening and, hence, the higher the risk of losing connectivity during a slot.* the throughput grows as V decreases since a slower AN is less likely to lose connectivity to the serving IN during a slot.* Similarly, the throughput grows asdecreases, because the beam alignment is repeated more frequently, reducing the disconnection time. However, the overhead (which is not accounted for in this simple analysis) would also increase, thus limiting or even nullifying the gain.* Finally, as shown in Figure <ref>, the throughput grows with the MIMO array size due to the increased achievable communication rangeof the nodes.In conclusion, we have presented a preliminary connectivity and coverage study in a simple automotive scenario using mmWave communication link, and show how the performance of common directional beam tracking protocols can be improved by accounting for the specificities of the automotive scenario.IEEEtran | http://arxiv.org/abs/1705.06960v1 | {
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"Michele Zorzi"
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"published": "20170426152314",
"title": "Technical Report - MillimeterWave Communication in Vehicular Networks: Coverage and Connectivity Analysis"
} |
lemmaLemma proofProof | http://arxiv.org/abs/1704.08572v1 | {
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"Nuria González-Prelcic",
"Kiran Venugopal",
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"primary_category": "cs.IT",
"published": "20170427135119",
"title": "Frequency-domain Compressive Channel Estimation for Frequency-Selective Hybrid mmWave MIMO Systems"
} |
Local h-vectors of Quasi-Geometric and Barycentric Subdivisions]Local h-vectors of Quasi-Geometric and Barycentric Subdivisions Martina Juhnke-Kubitzke & Richard Sieg, Universität Osnabrück, FB Mathematik/ Informatik, 49069 Osnabrück, Germany [email protected], [email protected] Satoshi Murai, Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Suita, Osaka, 565-0871, Japan [email protected] In this paper, we answer two questions on local h-vectors, which were asked by Athanasiadis. First, we characterize all possible local h-vectors of quasi-geometric subdivisions of a simplex. Second, we prove that the local γ-vector of the barycentric subdivision of any CW-regular subdivision of a simplex is nonnegative. Along the way, we derive a new recurrence formula for the derangement polynomials. [2000]05E45, 05A05The first and the third author were partially supported by the German Research Council DFG-GRK 1916. The second author was partially supported by JSPS KAKENHI JP16K05102. [ Richard Sieg================§ INTRODUCTIONThe classification of face numbers of (triangulated) spaces is an important and central topic not only in algebraic and geometric combinatorics but also in other fields, as e.g., commutative and homological algebraand discrete, algebraic and toric geometry. The studied classes of spaces comprise abstract simplicial complexes, triangulated spheres and (pseudo)manifolds but also not necessarily simplicial objects such as (boundaries of) polytopes and Boolean cell complexes. In 1992,Stanley <cit.> introduced the so-called local h-vector of a topological subdivision of a (d-1)-dimensional simplex as a tool to study face numbers of subdivisions of simplicial complexes. His original motivation was the question, posed by Kalai and himself, if the (classical) h-vector increases under subdivision of a Cohen-Macaulay complex. Using local h-vectors Stanley could provide an affirmative answer to this question for so-called quasi-geometric subdivisions. The crucial property of those, that he used, is that their local h-vectors are nonnegative; a property, which is no longer true if one considers arbitrary topological subdivisions. Complementing results by Chan <cit.>, one of our main results shows that – except for the conditions that are true for any local h-vector – nonnegativity already characterizes local h-vectors of quasi-geometric subdivisions entirely. More precisely, we show the following: Let ℓ=(ℓ_0,…,ℓ_d)∈^d+1. The following conditions are equivalent: (1) There exists a quasi-geometric subdivision Γ of the (d-1)-simplex such that the local h-vector of Γ is equal to ℓ. (2) ℓ is symmetric (i.e., ℓ_i=ℓ_d-i for 0≤ i≤ d), ℓ_0=0 and ℓ_i≥ 0 for 1≤ i≤ d-1.We want to remark, that it already follows from <cit.> that the local h-vector of any quasi-geometric subdivision satisfies (2). To prove Theorem <ref>, it therefore suffices to construct a quasi-geometric subdivision Γ having a prescribed vector ℓ=(ℓ_0,…,ℓ_d)∈^d+1, satisfying (2), as its local h-vector. For this, we will extend constructions by Chan <cit.> who provided characterizations forlocal h-vectors of regular and topological subdivisions.As the local h-vector is symmetric <cit.> it makes sense to define a local γ-vector, which was introduced by Athanasiadis in <cit.> andis defined in the same way as is the usual γ-vector for homology spheres. The central conjecture for local γ-vectors is the following, due to Athanasiadis <cit.>.The local γ-vector of a flag vertex-induced homology subdivision of a simplex is nonnegative.It can be shown that this conjecture is indeed a strengthening of Gal's conjecture for flag homology spheres <cit.> and, in particular, implies the Charney-Davis conjecture <cit.>.Conjecture <ref> is known to be true in small dimensions <cit.> and for various special classes of subdivisions, including barycentric, edgewise and cluster subdivisions of the simplex <cit.> but besides it is still widely open.We add more evidence to it by showing the following:Let Γ be a CW-regular subdivision of a simplex. The local γ-vector of the barycentric subdivision (Γ) of Γ is nonnegative. The proof is based on an expression of the local h-vector which involves differences of h-vectors of restrictions of the subdivision and their boundary as well as derangement polynomials (Theorem <ref>). The nonnegativity is then concluded from a result on these differences by Ehrenborg and Karu <cit.>. As a byproduct of our proof we obtain a new recurrence formula for the derangement polynomials (Corollary 4.2).We point out that the local h-vector, and therefore the local γ-vector, not only depends on the combinatorial type of (Γ) but also on the subdivision map. In Theorem <ref>, we are considering the natural subdivision map of the barycentric subdivision, which will be explained in Section <ref>.Theorems <ref> and <ref> are motivated by questions asked by Athanasiadis <cit.>. Indeed, Theorem <ref> partially solves <cit.> and Theorem <ref> answers <cit.>. The paper is structured as follows. In Section <ref>, we provide the necessary background, including basic facts on simplicial complexes, topological subdivisions and more specifically barycentric subdivisions. Section <ref> is devoted to the characterization of local h-polynomials in the quasi-geometric case (Theorem <ref>). Finally, in Section <ref> we prove Theorem <ref>. § PRELIMINARIES First we provide some background material on simplicial complexes, their subdivisions and (local) h-vectors.§.§ Simplicial complexes and their face numbers Given a finite set V, a simplicial complex Δ on V is a family of subsets of V which is closed under inclusion, i.e., G∈Δ and F⊆ G implies F∈Δ. The elements of Δ are called faces and the inclusion-maximal faces are called facets of Δ. The dimension of a face F∈Δ is given by |F|-1 and the dimension of Δ is the maximal dimension of its facets. If all facets of Δ have the same dimension, Δ is called pure. We define the link of a face F∈Δ to be_Δ(F)={ G∈Δ| G∩ F=∅, G∪ F∈Δ}. The f-vector f(Δ)=(f_-1(Δ),f_0(Δ),…,f_d-1(Δ)) of a (d-1)-dimensional simplicial complex Δ encodes the number of i-dimensional faces (-1≤ i≤ d-1), i.e., f_i(Δ)=| {F∈Δ|(F)=i }|-1≤ i≤ d-1.Often it is more convenient to work with the h-vector h(Δ)=(h_0(Δ),…,h_d(Δ)) of Δ, which is defined byh_i(Δ)=∑_j=0^i(-1)^i-jd-ji-jf_j-1(Δ) 0≤ i≤ d.Regarding this vector as a sequence of coefficients yields the h-polynomialh(Δ,x)=∑_i=0^dh_i(Δ)x^i.We refer the reader to <cit.> for further background material. §.§ Subdivisions and local h-vectorsThe notion of local h-vectors goes back to Stanley <cit.> and we recommend this article as a detailed reference (also see <cit.>). Throughout this section, V will always denote a nonempty finite set of cardinality d. A topological subdivision of a simplicial complex Δ is a pair (Γ, σ), where Γ is asimplicial complex and σ is a map σ:Γ→Δ such that, for any face F ∈Δ, (1) Γ_F:=σ^-1(2^F) is a subcomplex of Γ which is homeomorphic to a ball of dimension (F). Γ_F is called the restriction of Γ to F.(2) σ^-1(F) consists of the interior faces of Γ_F. Following Stanley <cit.>, we call the face σ(G)∈Δ the carrier of G∈Γ.We also want to warn the reader not to confuse the notation Γ_F with the induced subcomplex of Γ on vertex set F, which might even consist just of the vertices in F. Also, note that it directly follows from condition (1) that σ is inclusion-preserving, i.e., σ(G)⊆σ(F) if G⊆ F. In what follows, we often just write subdivision instead of topological subdivision if we are referring to a subdivision without additional properties, and we say that Γ is a subdivision of Δ without referring to the map σ if this one is clear from the context.Let Γ be a subdivision (Γ,σ) of a simplicial complex Δ. We say that (Γ,σ) is quasi-geometric if there do not exist E∈Γ and F∈Δ with (F)<(E) such that σ(v)⊆ F for any vertex v of E. The subdivision (Γ,σ) is vertex-induced if for all faces E∈Γ and F∈Δ such that every vertex of E is a vertex of Γ_F, we have E∈Γ_F. Moreover, (Γ,σ) is called geometric if the subdivision Γ admits a geometric realization that geometrically subdivides a geometric realization of Δ. Finally, we say that (Γ,σ) is regular if the subdivision is induced by a weight function, i.e., it can be obtained via a projection of the lower hull of a polytope (see <cit.>). We have the following relations between those properties: {}⊋{}⊋{}⊋{}⊋{}, where all containments are strict. Figure <ref> shows examples of subdivisions of the 2-simplex that are (a) regular, (b) geometric but not regular, (c) quasi-geometric but not vertex-induced and (d) not even quasi-geometric. A subdivision that is vertex-induced but not geometric is harder to depict but can be found in <cit.>.In 1992, Stanley introduced the local h-vector of a subdivision of a simplex as a tool to study the classical h-vector of the subdivision of a simplicial complex. Let Γ be a subdivision of 2^V. Then ℓ_V(Γ,x)=∑_F⊆ V(-1)^d-|F|h(Γ_F,x)=∑_i=0^dℓ_i(Γ)x^iis called the local h-polynomial of Γ (with respect to V) and the vector ℓ_V(Γ)=(ℓ_0(Γ),ℓ_1(Γ),…,ℓ_d(Γ)) is referred to as the local h-vector of Γ (with respect to V). To make this article self-contained, we now summarize some of the most important properties of local h-vectors that will be used later on (see also <cit.>). For this, we recall that a sequence (a_0,a_1,…,a_m)∈^m+1 is called unimodal if there exists 0≤ s≤ m such that a_0≤ a_1≤⋯≤ a_s≥ a_s+1≥⋯≥ a_m.(1) Let Δ be a pure simplicial complex and let Γ be a subdivision of Δ. Then:h(Γ,x)=∑_F∈Δℓ_F(Γ_F,x)h(_Δ(F),x). (2)Let V≠∅ and let Γ be a subdivision of 2^V. Then: (a) The local h-vector is symmetric, i.e., ℓ_i(Γ)=ℓ_d-i(Γ) for 0≤ i≤ d. Furthermore, ℓ_0(Γ)=0 and ℓ_1(Γ)≥0.(b) If Γ is a quasi-geometric, then ℓ_i(Γ)≥0 for 0≤ i≤ d.(c) If Γ is a regular, then ℓ_V(Γ) is unimodal.It was shown by Chan <cit.> that the conditions in (2(a)) already characterize local h-vectors of topological subdivisions. Adding unimodality, one obtains the characterization of local h-vectors of regular subdivisions. We will complete this picture by showing in the next section that indeed every vector satisfying the conditions in (2(a)) and (2(b)) occurs as the local h-vector of a quasi-geometric subdivision.In the following, let Γ be a subdivision of 2^V.As by Theorem <ref> (2(a)) ℓ_V(Γ) is symmetric, we can express the local h-polynomial ℓ_V(Γ,x) uniquely in the polynomial basis {x^k(1+x)^d-2k| 0≤ k≤⌊ d/2⌋}, i.e., ℓ_V(Γ,x)=∑_k=0^⌊ d/2⌋ξ_k(Γ) x^k(1+x)^d-2k,where ξ_k(Γ)∈ are uniquely determined. The sequence ξ_V(Γ)=(ξ_0(Γ),…,ξ_⌊ d/2⌋(Γ)) is called the local γ-vector of Γ (with respect to V).The local γ-vector is known to be nonnegative for flag vertex-induced subdivisions in dimension ≤ 3 <cit.> and for special classes of subdivisions including barycentric, edgewise and cluster subdivisions of the simplex<cit.>. More generally, any symmetric polynomial p(x) with center of symmetry n/2 is called γ-nonnegative if the coefficients γ_k given byp(x)=∑_k=0^n/2γ_k x^k(1+x)^n-2k.are nonnegative. §.§ CW-regular subdivisionsThe definition of topological subdivisions can be naturally extended to regular CW-complexes <cit.>. For a regular CW-complex Γ, we write P(Γ) for the face poset of Γ. A CW-regular subdivision of a simplicial complex Δ is a pair (Γ,σ), where Γ is a regular CW-complex and σ : P(Γ) →Δ is a map satisfying the conditions (1) and (2) of topological subdivisions.Given a regular CW-complex Γ, its barycentric subdivision (Γ) is the simplicial complex, whose i-dimensional faces are given by chainsτ_0 ⪇τ_1 ⪇⋯⪇τ_i,where τ_j ∈ P(Γ) is a non-empty face of Γ (0≤ j≤ i). It is well-known that Γ and (Γ) are homeomorphic. Then, for a CW-regular subdivision (Γ,σ) of a simplicial complex Δ, (Γ) can be naturally considered as a subdivision of Δ by the map σ'({τ_0,τ_1,…,τ_i}) :=σ(max{τ_0,τ_1,…,τ_i}),since(σ')^-1(2^F)={{τ_0,τ_1,…,τ_i}∈(Γ): σ(max{τ_0,τ_1,…,τ_i}) ⊆ F}= (Γ_F),where Γ_F={τ∈ P(Γ): σ(τ) ⊆ F}. When we consider the local h-polynomials of (Γ), we always consider the local h-polynomial using the above map σ'.§.§ Derangement polynomialLet V be a set of cardinality d. It is easy to see that the barycentric subdivision (2^V) of a (d-1)-simplex 2^V is a special instance of a regular subdivision. Its h-vector h((2^V))=(h_0((2^V)),…,h_d((2^V))) is given by h_i((2^V))=A(d,i),where A(d,i) are the Eulerian numbers, counting the number of permutations in the symmetric group S_d on d elements with exactly i descents. Recall that a descent of a permutation π∈ S_d is an index 1≤ k≤ d-1 such that π(k)>π(k+1).Similarly, there is an expression of the local h-vector ℓ_V((2^V)) of (2^V) involving permutation statistics. An excedance of a permutation π is an index 1≤ i≤ d such that π(i)>i. We write (π) for the number of excedances of π, i.e.,(π)=|{ i∈[d] |π(i)>i }|. A permutation π∈ S_d is called a derangement if it does not have any fixed point. We denote by _d the set of derangements in S_d. The derangement polynomial of order d is defined by_d(x)=∑_π∈_dx^(π).It is convenient to set _0(x)=1.These polynomials were first studied by Brenti in <cit.>.It is not hard to see (see also <cit.>) that the local h-polynomial of (2^V) is given by ℓ_V((2^V),x)=_d(x).It was shown in <cit.> that derangement polynomials, and thus the local h-polynomial of (2^V) is γ-nonnegative. § CHARACTERIZATION OF LOCAL H-POLYNOMIALSIn this section, we provide a characterization of local h-vectors of quasi-geometric subdivisions. This complements work by Chan <cit.>, who showed that local h-vectors of topological and regular subdivisions are completely characterized by their properties inTheorem <ref> (2).In particular, we prove Theorem <ref> by extending her main idea. As local h-vectors of quasi-geometric subdivisions are known to be symmetric and nonnegative with first entry equal to 0 (see Theorem <ref> (2)), we only need to show that the conditions in (2b) are also sufficient. Let ℓ=(ℓ_0,…,ℓ_d)∈^d+1 be fixed and assume that ℓ satisfies the conditions in Theorem <ref> (2). We will explicitly construct a quasi-geometric subdivision Γ of a (d-1)-simplex 2^V with ℓ_V(Γ)=ℓ. The basic idea is to find operations on a subdivision Δ of a simplex that preserve quasi-geometricity, change the local h-vector of Δ in a prescribed way and such that ℓ can be realized as local h-vector of a subdivision obtained by successively applying these operations. In <cit.> Chan already provided three operations that suffice to construct all local h-vectors of arbitrary topological subdivisions. Though one of these does not necessarily preserve quasi-geometricity,we recall her constructions and their effects on the local h-vector, since we will use them in what follows.Let V be a set with |V|=d and let (Γ,σ) be a subdivision of the (d-1)-simplex 2^V. (O1) Let (_Γ(F),σ') be obtained from Γ by stellar subdivision of a facet F of Γ, where σ'(z)=σ(F) for the new vertex z. Then:ℓ_V(_Γ(F))=ℓ_V(Γ)+(0,1,…,1,0). (O2) Let d≥ 4 and G be a (d-2)-dimensional face of Γ with (d-2)-dimensional carrier σ(G). Let (P_Γ(G),σ') be the subdivision of 2^V obtained from Γ by adding a new vertex w with carrier σ'(w)=σ(G) and one new facet G∪{w} (with carrier V). (Note that σ'(G)=V.) Then:ℓ_V(_Γ(G))=ℓ_V(Γ)+(0,0,-1,…,-1,0,0).We will say that _Γ(G) is obtained from Γ by pushing G into the interior. (O3) Let Ω be the subdivision of a 1-simplex 2^{d+1,d+2} into two edges. Then Γ^*1:=Γ∗Ω is a subdivision of the (d+1)-simplex 2^V∪{d+1,d+2} with ℓ_V'(Γ^*1)=(0,ℓ_V(Γ),0). The operations are depicted in Figure <ref>.Chan showed that – starting from a 2- or 3-simplex – these constructions suffice to generate any symmetric vector ℓ∈^d+1 with ℓ_0=0 and ℓ_1≥0. Moreover, both, the stellar subdivision (O1) and the join operation (O3), maintain regularity of a subdivision and any symmetric and unimodal vector ℓ∈^d+1 with ℓ_0=0 can be constructed by their successive application <cit.>. It is straight forward to show that that stellar subdivision (O1) and the join operation (O3) behave well with respect to quasi-geometricity. Let Γ be a quasi-geometric subdivision of 2^V. Then: (1) If F is a facet of Γ, then _Γ(F) is a quasi-geometric subdivision of 2^V.(2) Γ^*1 is a quasi-geometric subdivision of 2^V∪{d+1,d+2}.Even though, _Γ(G) (if defined) might not be quasi-geometric (even if Γ is), the next lemma shows that this obstruction can be “repaired” with justone additional stellar subdivision.Let Γ be a quasi-geometric subdivision of 2^V and let d=|V|≥4. LetG be a (d-2)-dimensional face of Γ with (d-2)-dimensional carrier. Let w be the new vertex of _Γ(G). Then the subdivision __Γ(G)(G∪{w}) obtained by first pushing G into the interior of Γ and then stellarly subdividing the new facet G∪{w} is a quasi-geometric subdivision of 2^V. Moreover,ℓ_V(__Γ(G)(G∪{w}))=ℓ_V(Γ)+(0,1,0,…,0,1,0).Figure <ref> shows __Γ(G)(G∪{w}) in the case that Γ is just a 3-simplex. To simplify notation, we set Γ':=__Γ(G)(G∪{w}).The claim about the local h-vector follows immediately from the definition of the used operations (see also <cit.>). It remains to verify that Γ' is a quasi-geometric subdivision of 2^V. We denote by σ_1:Γ→ 2^V the map corresponding to the subdivision Γ of 2^V and by σ_2:Γ'→Γ the subdivision map of Γ' (as a subdivision of Γ). The subdivision map of Γ' as a subdivision of 2^V is then given by σ:=σ_2∘σ_1. Let z be the newly added vertex when applying stellar subdivision to G∪{w}. We first note that by definition of Γ' and σ we haveσ(E)=σ_1(E),∀ E∈Γ∩Γ'∖{G},σ(w)=σ_1(G) σ(z)=V.Given a face E∈Γ', we need to show that the following condition, referred to as condition (QG) in the sequel, is satisfied: (QG) For all F⊆ V such that σ'(v)⊆ Ffor all v∈ E,it holds that F≥ E.Let E∈Γ∩Γ'∖{G}. In this case, we have σ(E)=σ_1(E) and σ(u)=σ_1(u) for all u∈ E. As Γ is quasi-geometric, it follows, that E satisfies condition (QG).Similarly, we have σ(u)=σ_1(u) for all u∈ G and as Γ is quasi-geometric,condition (QG) holds for G. It remains to consider faces E∈Γ'∖Γ. First assume z∈ E. As σ(z)=V by construction and |V|=d, those faces satisfy condition (QG). Suppose that z∉ E. As E∉Γ, we must have that w∈ E. Since z∉ E, we can further conclude that E≤ d-2. (Indeed, any facet containing w also contains z.) Combining this with the fact that σ(w)=σ_1(G) is of dimension d-2 (by assumption) we get that E meets condition (QG). The claim follows. We can finally provide the proof of Theorem <ref>, i.e., the desired characterization of local h-vectors of quasi-geometric subdivisions. The “only if”-part follows directly from Theorem <ref> (b).We now show the “if”-part. If d is even we start with a 2-simplex and take ℓ_d/2 times the stellar subdivision (O1) of a facet. Similarly, if d is odd, we start with the 3-simplex and take the stellar subdivision (O1) of a facet ℓ_d-1/2 times. In the next step, we apply once (O3) and then ℓ_⌊d/2⌋ -1 times the operation defined in Lemma <ref>. We continue in this way and, by Lemmas <ref> and <ref>, this yields a quasi-geometric subdivision Γ of the (d-1)-simplex whose local h-vector is equal to ℓ.Chan and Stanley originally conjectured that all local h-vectors of quasi-geometric subdivisions are unimodal. However,Athanasiadis disproved this conjectured by providing a counterexample to it (<cit.> and <cit.>). This example is obtained by applying the operation defined in Lemma <ref> to the 3-simplex (see Figure <ref>) and has local h-vector (0,1,0,1,0). Nevertheless, no geometric or even just vertex-induced subdivisions of the (d-1)-simplex are known whose local h-vector is not unimodal. Already, Athanasiadis in <cit.> asked if such examples exist or if all vertex-induced subdivision of the (d-1)-simplex have unimodal local h-vector. Based on a lot of experiments and a great vain effort to construct counterexamples we are inclined to believe that the latter is indeed the case. § LOCAL Γ-VECTORS OF BARYCENTRIC SUBDIVISIONS In this section, weprovide the proof of Theorem <ref>, i.e., we show that the local γ-vector of the barycentric subdivision of any CW-regular subdivision of a simplex is nonnegative. This answers Question 6.2 in <cit.> in the affirmative. As an example of the type of subdivision we are interested in, Figure <ref> depicts the barycentric subdivision of the stellar subdivision of the 2-simplex. Let V be a finite set with |V|=d and let Δ be a subdivision of the simplex 2^V. The local h-polynomial of Δ can be written asℓ_V(Δ,x)=h(Δ,x)-h(∂Δ,x)+∑_F⊊ Vℓ_F(Δ_F,x)(x+⋯+x^d-|F|-1).Here, ∂Δ denotes the boundary of Δ.First, note that for any F⊊ V the link of F in 2^V respectively in ∂(2^V) is a (|V|-|F|-1)-simplex respectively its boundary. Hence, h(_2^V(F),x)=1h(_∂(2^V)(F),x)=1+x+⋯+x^d-|F|-1.Applying (<ref>) to Δ, we obtainh(Δ,x) =∑_F⊆ Vℓ_F(Δ_F,x)h(_2^V(F),x)=ℓ_V(Δ,x)+∑_F⊊ Vℓ_F(Δ_F,x).Similarly, viewing ∂Δ as a subdivision of ∂(2^V) and using that (∂Δ)_F=Δ_F for F ⊊ V,the h-polynomial of ∂Δ can be written in the following way:h(∂Δ,x) =∑_F⊊ Vℓ_F(Δ_F,x)h(_∂ (2^V)(F),x)=∑_F⊊ Vℓ_F(Δ_F,x)(1+x+⋯+x^d-|F|-1).Subtracting (4.2) from (4.1) yieldsℓ_V(Δ,x)=h(Δ,x)-h(∂Δ,x)+∑_F⊊ Vℓ_F(Δ_F,x)(x+⋯+x^d-|F|-1),as desired.Using (<ref>) and the fact that (2^V) has the same h-polynomial as its boundary(since it is just the cone over it), we get the followingrecurrence formula for the derangement polynomials as a special case of Lemma <ref> when Δ=(2^V). We recall that _0(x)=1 by definition. For every d∈,_d(x)=∑_k=0^d-2dk_k(x)(x+⋯+x^d-1-k). To the best of our knowledge this formula seems to be new, as we could not find it in the literature. Note that it directly implies the unimodality of _n(x). We also found a purely combinatorial proof of Corollary <ref> using a similar recurrence formula for the Eulerian polynomials. This proof will appear in the PhD thesis of the third author.The next theorem is crucial in the proof of Theorem <ref>.Let V≠∅ be a finite set and let Δ be a subdivision of the simplex 2^V. The local h-polynomial of Δ can be written asℓ_V(Δ,x)=∑_F⊆ V[h(Δ_F,x)-h(∂(Δ_F),x)]·_|V∖ F|(x).To simplify notation, we seth_F(x)=h(Δ_F,x)-h(∂(Δ_F),x).We show the claim by induction on |V|. If |V|=1, both sides in (<ref>) are equal to 0 and the claim is trivially true. Assume |V|≥ 2. By Lemma <ref> and the induction hypothesis, we haveℓ_V(Δ,x)= h_V(x)+∑_F⊊ V[∑_G⊆ Fh_G(x)·_|F∖ G|(x) ](x+⋯+x^|V|-|F|-1)= h_V(x)+∑_G⊊ Vh_G(x)[∑_G⊆ F⊊ V_|F∖ G|(x) (x+⋯+x^|V|-|F|-1)]= h_V(x)+∑_G⊊ Vh_G(x)[∑_F⊊ V∖ G_|F|(x)(x+⋯+x^|V|-|G|-1-|F|)]= h_V(x)+∑_G⊊ Vh_G(x)·_|V∖ G|(x),where the last equality follows from Corollary <ref>.We illustrate Theorem <ref> using two examples.(1) Let Δ=2^V be the trivial subdivision of the simplex 2^V with |V|=d≥ 1. Then ℓ_V(Δ,x)=0 andh(Δ_F,x)-h(∂(Δ_F),x)= -x-x^2-…-x^|F|-1∅ F⊆ V. However, h(Δ_∅,x)-h(∂(Δ_∅),x)=1 and all negative terms on the right-hand side of (<ref>) cancel out. We retrieve the recurrence formula from Corollary <ref>:0=ℓ_V(Δ,x) =_d(x)+∑_∅ F⊆ V(-x-…-x^|F|-1)_|V∖ F|(x)=_d(x)+∑_k=2^ddk(-x-…-x^k-1)_d-k(x) =_d(x)-∑_k=0^d-2dk(x+…+x^d-k-1)_k(x). (2) Let Δ be the barycentric subdivision of the stellar subdivision of the 2-simplex, as depicted in Figure <ref>. Then _0(x)=1, _1(x)=0 andh(Δ,x)-h(∂Δ,x)=(1+10x+7x^2)-(1+4x+x^2)=6x+6x^2. The right-hand side of (<ref>) is ∑_F⊆ V[h(Δ_F,x)-h(∂(Δ_F),x)]·_|V∖ F|(x)= 6x+6x^2+∑_F⊆ V, |F|≤ 1[h(Δ_F,x)-h(∂(Δ_F),x)]·_|V∖ F|(x)= 6x+6x^2+_3(x)=7x+7x^2,which is indeed equal to the local h-polynomial of Δ (see Figure <ref>). We now prove Theorem <ref>. Let Γ be a CW-regular subdivision of a simplex 2^V. Then, by applying the special case of Theorem <ref> when Δ=(Γ), we obtainℓ_V((Γ),x)=∑_F ⊆ V[ h((Γ_F),x) - h(∂((Γ_F)),x) ] ·_|V∖ F|(x).Since we already know that _k(x) is γ-nonnegative and since the product of two γ-nonnegative polynomials is γ-nonnegative, the next result due to Ehrenborg and Karu <cit.> completes the proof of Theorem <ref>. Let Γ be a regular CW-complex which is homeomorphic to a ball. Then h((Γ),x)-h(∂((Γ)),x) is γ-nonnegative. Theorem <ref> is an immediate consequence of <cit.>, but since this result is written in the language of 𝐜𝐝-indices, we explain how Theorem <ref> can be deduced from it. Before the proof, we recall flag h-numbers and 𝐚𝐛-indices. Let Γ be a regular CW-complex of dimension d-1. An S-chain of Γ, where S ⊆ [d]={1,2,…,d}, is a chain of Γτ_0 ⪇τ_1 ⪇⋯⪇τ_iwith S={τ_0+1,…,τ_i+1}. Let f_S(Γ) be the number of S-chains of Γ. Then, for S ⊆ [d], we define h_S(Γ) by h_S(Γ)= ∑_T ⊆ S (-1)^|S|-|T| f_T(Γ). Note that one has f_i((Γ))=∑_S⊆ [d] |S|=i+1 f_S(Γ)and h_i((Γ))=∑_S⊆ [d] |S|=i h_S(Γ).Let ℤ⟨𝐚, 𝐛⟩ and ℤ⟨𝐜, 𝐝⟩ be noncommutative polynomial rings, where 𝐚,𝐛,𝐜,𝐝 are variables with 𝐚=𝐛 = 𝐜=1 and 𝐝=2. We say that a polynomial f ∈ℤ⟨𝐜, 𝐝⟩ is nonnegative if all coefficients of monomials in f are nonnegative. For S ⊆ [d], we define the noncommutative monomial u_S=u_1u_2⋯ u_d ∈ℤ⟨𝐚, 𝐛⟩ by u_i=𝐚 if i ∉S and u_i=𝐛 if i ∈ S. The homogeneous polynomial Ψ_Γ(𝐚, 𝐛) = ∑_S ⊆ [d] h_S(Γ) u_Sis called the 𝐚𝐛-index of Γ. Note that by substituting 𝐚=1 to Ψ_Γ(𝐚 ,𝐛) we obtain the h-polynomial of (Γ), that is, Ψ_Γ (1,𝐛)= ∑_i=0^d h_i((Γ)) 𝐛^i.Let Γ be a regular CW-complex which is homeomorphic to a (d-1)-dimensional ball. Then the face poset of Γ is near-Gorenstein* in the sense of <cit.> (indeed, the near-Gorenstein* property is an abstraction of being a ball), and <cit.> says that there is a nonnegative homogeneous polynomial Φ(𝐜, 𝐝) ∈ℤ⟨𝐜, 𝐝⟩ of degree d such thatΦ(𝐚 +𝐛, 𝐚𝐛 + 𝐛𝐚)= Ψ_Γ(𝐚, 𝐛) - Ψ_∂Γ(𝐚, 𝐛) ·𝐚.By substituting 𝐚=1 in the above equation, we see thatΦ(1+ 𝐛, 2 𝐛)= Ψ_Γ (1, 𝐛)- Ψ_∂Γ (1,𝐛) = ∑_i=0^d ( h_i((Γ))-h_i (∂ ((Γ)) ) ) 𝐛^icoincides with the polynomial h((Γ),𝐛)-h(∂ ((Γ)),𝐛). On the other hand, since Φ(𝐜,𝐝) is homogeneous and nonnegative, there exist nonnegative integers γ_0,γ_1,γ_2,… such thatΦ(1+ 𝐛, 2 𝐛)= ∑_k=0^⌊ d/2 ⌋γ_k (1+ 𝐛)^d-2k (2 𝐛)^k.The two equations (<ref>) and (<ref>) guarantee the γ-nonnegativity of h((Γ),𝐛)-h(∂ ((Γ)),𝐛). As γ-nonnegativity implies unimodality, we obtain the following immediate corollary. Let Γ be a CW-regular subdivision of a simplex 2^V. Then ℓ_V((Γ)) is unimodal. The polynomials h(Δ_F,x)-h(∂ (Δ_F),x) in Theorem <ref> are always symmetric. Indeed, if Δ is a simplicial complex which is homeomorphic to a (d-1)-ball with h(Δ)=(h_0,h_1,…,h_d), then h_d=0 and h_i(∂Δ)= ∑_j=0^i (h_j-h_d-j) for all i (see <cit.>). Henceh_i(Δ)-h_i(∂Δ)= ∑_j=0^i-1 (h_j-h_d-j-1),and therefore h_i(Δ)-h_i(∂Δ)=h_d-i(Δ)-h_d-i(∂Δ) for all i=0,1,…,d. Moreover, (<ref>) says that if h_j ≥ h_d-j-1 holds for j < d/2, then h(Δ,x)-h(∂Δ,x) is unimodal.For example, it follows from <cit.> that, if Δ is the rth edgewise subdivision of any topological subdivision of a simplex and if r is sufficiently large, then we have h_j(Δ_F)≥ h_|F|-j-1(Δ_F) for j<|F|/2 (see <cit.> for a connection between edgewise subdivisions and Veronese subrings). As explained above, this implies the unimodality of h(Δ_F,x)-h(∂ (Δ_F),x), and hence the unimodality of the local h-polynomial of Δ by Theorem <ref>. Although the polynomials h(Δ_F,x)-h(∂ (Δ_F),x) may have negative coefficients in general, the previous remark suggests to study the following problem: Find classes of subdivisions Δ such that h(Δ_F,x)-h(∂ (Δ_F),x) is nonnegative, unimodal or γ-nonnegative. Moreover, for those classes try to find a combinatorial interpretation of the coefficients of h(Δ_F,x)-h(∂ (Δ_F),x) respectively the coefficients of its γ-polynomial. Possible subdivisions one might consider include chromatic subdivisions (see <cit.>), interval subdivisions (see <cit.>) or partial barycentric subdivisions (see <cit.>). With respect to the second part, one could e.g., study the barycentric subdivision of the cubical barycentric subdivision of the simplex (see <cit.>) which belongs to the class of subdivisions considered in Theorem<ref> (this question was raised by Christos Athanasiadis).In personal communication with Christos Athanasiadis, he asked us if Theorem <ref> can be extended to relative local γ-vectors (see <cit.> for the precise definition). We include this question as a reference for future research. Given a CW-regular subdivision Γ of a simplex and E∈Γ, is the relative local γ-vector of the barycentric subdivision (Γ) of Γ at E nonnegative?§ ACKNOWLEDGMENT We wish to thank Christos Athanasiadis for his useful comments. The third author wants to express his gratitude to Isabella Novik for her hospitality at the University of Washington and interesting discussions about the subject.plain | http://arxiv.org/abs/1704.08057v1 | {
"authors": [
"Martina Juhnke-Kubitzke",
"Satoshi Murai",
"Richard Sieg"
],
"categories": [
"math.CO",
"05E45, 05A05"
],
"primary_category": "math.CO",
"published": "20170426111552",
"title": "Local $h$-vectors of Quasi-Geometric and Barycentric Subdivisions"
} |
Multifractal Analysis of Pulsar Timing Residuals:Assessmentof Gravitational Wave Detection I. Eghdami1, H. Panahi1 and S. M. S. Movahed2,3 December 30, 2023 ============================================================================================== Complex networks in different areas exhibit degree distributions with heavy upper tail. A preferential attachment mechanism in a growth process produces a graph with this feature. We herein investigate a variant of the simple preferential attachment model, whose modifications are interesting for two main reasons: to analyze more realistic models and to study the robustness of the scale free behavior of the degree distribution.We introduce and study a model which takes into account two different attachment rules: a preferential attachment mechanism (with probability 1-p) that stresses the rich get richer system, and a uniform choice (with probability p) for the most recent nodes. The latter highlights a trend to select one of the last added nodes when no information is available.The recent nodes can be either a given fixed number or a proportion (α n) of the total number of existing nodes. In the first case, we prove that this model exhibits an asymptotically power-law degree distribution. The same result is then illustrated through simulations in the second case.When the window of recent nodes has constant size, we herein prove that the presence of the uniform rule delays the starting time from which the asymptotic regime starts to hold.The mean number of nodes of degree k and the asymptotic degree distribution are also determined analytically. Finally, a sensitivity analysis on the parameters of the model is performed.Keywords: Barabási–Albert random graph; Preferential and Uniform attachments; degree distribution. MSC2010: 05C80, 90B15, 60J80. § INTRODUCTION Networks grow according to different paradigms and the preferential attachment mechanism is one of the simplest rules that explains some of the observed features exhibited by real networks. Barabási-Albert proposed it to model the World-Wide Web (WWW) <cit.>. However, it is also used for a variety of other phenomena such as citation networks or some genetic networks <cit.>. The preferential attachment model connects new and existing nodeswith probabilities proportional to the number of links already present. This rule is also known as rich get richer because it rewards with new links nodes with an higher number of incoming links. Networks growing according with the preferential attachment rule exhibit the scale free property and their degree distribution are characterized by power law tails <cit.>.Other models for different real worldnetworks request the use of different growth paradigms and do not present the scale free property. For example, some networks exhibitthe small-world phenomenon, in which sub-networks have connections between almost any two nodes within them. Furthermore, most pairs of nodes are connected by at least one short path.On the other hand, one of the most studied models, the Erdös-Rényi random graph, does not exhibit either the power-law behavior for the degree distribution of its nodes, nor the small-world phenomenon <cit.>. Mathematically, the growth of networks can be modeled through random graph processes, i.e. a family (G^t)_t∈ℕ of random graphs (defined on a common probability space), where t is interpreted as time. Different features of the model are then described as properties of the corresponding random graph process. In particular, the interest often focuses on the degree, (v,t), of a vertex v at time t, that is on the total number of incoming and (or) outgoing edges to and (or) from v, respectively.In this framework, new nodes of the Barabási-Albert modellink with higher probabilities with nodes of higher degree.An important feature of preferential attachment models is an asymptotic power-law degree distribution:the fraction P(k) of vertices in the networkwith degree k, goesas P(k)∼ k^-γ, with γ>0, for large values of k. Real world modeling instances motivated the proposal of generalizations of the Barabási-Albert model, see e.g. <cit.>. A common characteristic of many of these models is the presence of the same attachment rule for all the nodes of the network. However, this hypothesis is not always realistic. Consider, for example, a website where registered members can submit contents, such as text and posts.Furthermore, registered users can vote previous posts to determine their position on the site's pages. Then, the submissions with the most positive votes appear on the front page together with a fixed number of the most recent posts (see www.reddit.com). This is a network in which we identify nodes with posts and links with votes. Let's consider the case in which a user submits a new post, but also votes on some previous submissions. Moreover only positive votes are allowed. It is reasonable to assume that the user tends to select and vote either the most recent posts or the most popular posts. Hence, the user votes the posts according to two different rules: with uniform probability if the user decides to select a post recently published, andwith probability proportional to the number of votes, otherwise.This structure of network growth arises also when we consider some social networks. A new subject may connect choosing uniformly from people who recently joined the group (for instance new schoolmates) or preferring famous people present in the network since a long period of time (for example schoolmates belonging to a rock band). A third instance arises in the case of citation networks. Typically, authors of a new paper cite recent work on the same subject as well as the most important papers on the considered topic.Having this type of networks in our mind, in Section 2 we propose a generalization of the Barabási-Albert model which takes into account the two different attachment rules for new nodes of the network. We called itUniform-Preferential-Attachment model (UPA model) to pinpoint the double nature of the attachment rule.In <cit.>, Magner et al. introduced a model in which new vertices choose the nodes for their links within windows. Inspired by their idea of introducingwindows, weformulate our model. We apply the preferential attachment rule to any node but we re-inforce this rule with a uniform choice for the most recent nodes. To define recent nodes we introduce windows either of fixed or linear in time amplitude. Our first result in Section 3 is the proof of recursive formulas for the expectation of the number of nodes with a given degree at a fixed time, in the case of windows of fixed amplitude. Our studytakes advantage of the existing methodsused for example in <cit.>, together with the Azuma-Hoeffding Inequality (see Lemma 4.2.3 in <cit.>), but some of the relationships requested by these techniques are not trivial in our case. Since the mixing of uniform and preferential attachment rules suggests the possible disappearance of the scale free property we investigated this possibility. In Section 3, we also determine the degree distribution by employing a a rigorous mathematical approach in the case of fixed size windows. The use of this distribution allowed us to prove the asymptotic scale free property. These results are illustrated in Section 4 through a sensitivity analysis for the different parameters. Furthermore, in Section 4 we use simulations to study the case in which the size of the windows grows linearly in time. We show that an asymptoticallypower-law degree distribution is preserved.Finally, Section 5 contains some concluding remarks, while the Appendix reports some auxiliary lemmas and their proofs.§ UNIFORM-PREFERENTIAL-ATTACHMENT MODEL (UPA MODEL) In order to model instances in which recent nodes play also an important role, we propose that every new node v_t+1 selects its neighbour within a limited window { v_t-w+1, ... v_t} ofnodes of size w∈ℕ (i.e. the w youngest nodes of the network) or among all nodes { v_0, ... v_t}. The former happens with probability p ∈[ 0, 1 ] and the latter with probability (1-p). Furthermore, the attachment rules are different in these two cases. When there is the window, the neighbour is chosen with a uniform distribution (that is, every node within the window has probability 1/w to be selected), but in absence of window, the neighbour is chosen according to a preferential attachment mechanism (that is v_j, j=0,…,t has a probability proportional to (v_j,t) to be chosen). We first fix a probability 0 ≤ p ≤ 1 and a natural number l≥ 1. The process starts at time t=l, with l+1adjacent vertices, growing monotonically by adding at each discrete time step a new vertex v_t+1 together with a directed edge connecting this with some of the vertices already present.As far as the window size is concerned we consider two cases: * For all t≥ l, w:=w(t)=l, l∈ℕ, that is the window size is fixed; and * for each t≥ l, w:=w(t)=α t, 0<α<1, that is the size of the window is a linear function of the size of the network. We study the first case analytically. Instead, for the second case we limit ourselves to numerical results. The algorithmic description of the UPA model is: (a) At the starting period t = l, l∈ℕ, the initial graph G^l has l+1 nodes (v_0, v_1, ... v_l), where every node v_j, 1 ≤ j ≤ l, is connected to v_0. (b) Given G^t, at time t+1 add a new node v_t+1 together with an outgoing edge. Such edge links v_t+1 with an existing node chosen either within a window, or among all nodes present in the network at time t, as follows: * with probability p, v_t+1chooses its neighbour in the set { v_t-w+1, ...,v_t}, and each node within this window has probability1/w of being chosen. * with probability 1-p, the neighbour of v_t+1 is chosen from the set { v_0, ..., v_t}, and each node v_j, j=0,…,t, has probability (v_j)/2t of being chosen. Here, (v_j) indicates the total number of incoming links to v_j.Note that: * When p = 0 the UPA model reduces to the usual preferential attachment model of Barabási-Albert <cit.>. * The initial degrees of the nodes of the UPA model, are (v_j) = 1 for 1 ≤ j ≤ l and (v_0) = l. § MAIN RESULTSIn this section we analyze the empirical degree distribution of a vertex in the UPA model with fixed window size, that is, for all t≥ l, w(t)=l, l∈ℕ.Let us denote with N(k,t), the number of vertices with degree k at time t in G^t. We prove recurrence equations for the expected degree of each node at time t in Sub-Section <ref> and we determine the asymptotic degree distribution in Sub-Section <ref>. For simplicitywe will write l to refer to the window size. §.§ Recursive formulae for [N(k,t)] Herein, we distinguish the case of windows of size l=1 from that with l>1. The first case is considered in Lemma <ref> while the second is the subject of Lemma <ref>. In the UPA model with fixed windows size l=1, it holds:N(k,t+1)= ( 1-1-p/2t)[N(1,t)] + (1-p),ifk=1( 1-1-p/t)[N(2,t)] + 1-p/2t[N(1,t)] + p, ifk=2 ( 1-(1-p)k/2t)[N(k,t)] + (1-p)(k-1)/2t[N(k-1,t)], ifk>2, with initial conditions N(k,1)=2 for k=1 and N(k,1)=0 otherwise.We start calculating the probability that a node with degree k at time t receives a new link from v_t+1. To achieve this we distinguishtwo cases, k > 1 and k = 1. Let us consider a node with degre k>1. By construction its degree can increase only when it isselected through a preferential attachment mechanism (which happens with probability 1-p). On the contrary when k=1we should also consider the effect of the window (which happens with probability p). Thus, the probability that the new link exiting from v_t+1 attains a node with degree k>1 at time t is given by(1-p) N(k,t)k/∑_k'=1^+∞N(k',t)k'=(1-p) N(k,t)k/2t,whilethe probability that the new link attains a node with degree k=1 at time t is(1-p) N(1,t)/2t + p . Let 𝒢_s be the σ-field generated by the appearance of edges up to time s. Observe that conditioning on 𝒢_t, N(k,t+1), k≥ 1, is a random variable depending on N(k,t). Thus, if k=1 at time t+1 the number of nodes of degree 1, N(1,t+1), remains unchanged or increases by 1. The first possibility happens either in presence of a window, or when v_t+1 is attached to an existent node of degree 1 at time t in absence of window. The second possibility arises if the selected node has degree larger than 1. That is,N(1,t+1) = N(1,t) w.p. p+(1-p)N(1,t)/2tN(1,t)+1 w.p. (1-p)(2t-N(1,t))/2t where the abbreviation w.p. means “with probability”. Similarly, conditioned on 𝒢_tthe probability distribution of N(k,t+1), k≥2, is N(2,t+1) = N(2,t)+1 w.p. p+(1-p)N(1,t)/2tN(2,t)-1 w.p. (1-p) · 2N(2,t)/2tN(2,t) w.p. (1-p)[ 2t-N(1,t)-2N(2,t) ]/2t,and for k > 2N(k,t+1) = N(k,t)+1 w.p. (1-p)N(k-1,t)(k-1)/2tN(k,t)-1 w.p. (1-p) N(k,t) k/2tN(k,t) w.p. p+(1-p)[ 2t-N(k,t)k-N(k-1,t)(k-1) ]/2t.Taking the conditional expectations given 𝒢_tof (<ref>), (<ref>) and (<ref>), respectively, we get[ N(k,t+1) |𝒢_t]= ( 1-1-p/2t)N(1,t) + (1-p), ifk=1 ( 1-1-p/t)N(2,t) + 1-p/2tN(1,t) + p, ifk=2 ( 1-(1-p)k/2t)N(k,t) + (1-p)(k-1)/2tN(k-1,t),if k>2.Finally, taking expectation of both sides of (<ref>) we obtain the desired result. To switch to the case of window size l>1, let us define M_m(t) as the degree of the m-th node from the left, inside the window in G^t. That is, M_1(t)=d(v_t-l+1),M_2(t)=d(v_t-l+2),…,M_m(t)=d(v_t-l+m),…,M_l(t)=d(v_t) (see equations (<ref>) and (<ref>) for recursive formulae for M_m(t)). Then, it holds:In the UPA model with fixed windows size l>1: [N(k,t+1)] = [N(1,t)] + 1 - p∑_m=1^lℙ(M_m(t)=1) /l -(1-p)[N(1,t)]/2t,ifk=1[N(k,t)] + p/l[ ∑_m=1^lℙ(M_m(t)=k-1) - ∑_m=1^lℙ(M_m(t)=k) ] +1-p/2t{ (k-1)[N(k-1,t)] - k[N(k,t)] }, ifk>1 with initial conditions N(k,l)=l for k=1, N(k,l)=1 for k=l, and N(k,l)=0 otherwise. We proceed in the same way as in the proof of Lemma <ref>. Recall that at time t+1, v_t+1 appears with degree 1, and the number of nodes with degree 1 remains unchanged if and only if the neighbour chosen by v_t+1 is a vertex with degree 1 too. This happenswith probability N(1,t)/2t if there is no window, and with probability ∑_m=1^l1(M_m(t)=1)/l if the selected node lies inside the window. Thus, the random variable N(1,t+1) given 𝒢_t is given byN(1,t+1) =N(1,t) w.p. p∑_m=1^l1(M_m(t)=1)/l + (1-p)1 · N(1,t)/2tN(1,t)+1 w.p. 1-[p∑_m=1^l1(M_m(t)=1)/l + (1-p)1 · N(1,t)/2t] . With a similar reasoning, for k ≥ 2, we obtainN(k,t+1) given 𝒢_t, N(k,t+1) =N(k,t)-1 w.p. p∑_m=1^l1(M_m(t)=k)/l+(1-p)kN(k,t)/2tN(k,t)+1 w.p. p∑_m=1^l1(M_m(t)=k-1)/l+(1-p)(k-1)N(k-1,t)/2tN(k,t) w.p. 1 - p∑_m=1^l1(M_m(t)=k)/l-(1-p)kN(k,t)/2t -p∑_m=1^l1(M_m(t)=k-1)/l-(1-p)(k-1)N(k-1,t)/2t. Taking the conditional expectation of (<ref>) and (<ref>) we get(N(1,t+1) |𝒢_t) = N(1,t) + 1 - p∑_m=1^l1(M_m(t)=1) /l -(1-p)N(1,t)/2t,and[N(k,t+1) |𝒢_t] = N(k,t) +p/l[ ∑_m=1^l1(M_m(t)=k-1) - ∑_m=1^l1(M_m(t)=k) ] +1-p/2t[ (k-1)N(k-1,t) - kN(k,t) ] .Finally, taking theexpectation of both sides of (<ref>) and (<ref>) we obtain (<ref>).§.§ Asymptotic degree distribution Consider the UPA model with fixed windows size l ∈ℕ. Then, N(k,t)/t→P(k) in probability as t→∞. Furthermore, for l=1 it holds:P(k)=2(1-p)/3-pif k=1 (1-p)^2/(2-p)(3-p) +p/2-pif k=2 (2/1-p+2)(2/1-p+1)B(k,1+2/1-p)P(2) if k>2,while for l>1 we have:P(k)=2/(3-p)(1-p/l)^lik k=1 2/2+k(1-p)(p/l(H_k-1-H_k)+(1-p)(k-1)/2P(k-1))if k=2,…,l+1 B(k,l+2+2/1-p)/B(l+1,k+1+2/1-p)P(l+1)if k>l+1, where B(x,y) is the Beta function and H_k=(p/l)^k-1∑_m=1^l-(k-1)l-ml-m-(k-1)(1-p/l)^l-m-(k-1)ifk=1,…,l. 0 ifk>l.The proof includes the following steps: (1) we determine recursivelly [N(k,t)], k=1,2,…; (2) we prove the existence of P(k) := lim_t →∞[N(t,k)]/t, (3) we determine an explicit expression for P(k), (4) we use the Azuma-Hoeffding Inequality (see Lemma 4.2.3 in <cit.>) to prove convergence in probability of N(k,t)/t to P(k). The first step is solved by the results of Sub-Section 3.1. For the proof of steps (2) and (3) it is necessary to distinguish the cases of windows size l=1 and l > 1, respectively. The proof of the second and third steps request some technical results that are reported and proved in the Appendix. In particular, Lemma <ref> and Lemma <ref> in Appendix prove the existence of P(k), in the cases of windows size l=1 and l > 1, respectively. Furthermore, Lemma <ref> and Lemma <ref> in Appendix give the explicit expression of such limits in the cases of windows size l=1 and l > 1, respectively. It remains to prove step (4).Define X_s = E( N(k,t) |𝒢_s ), s ≤ t, and observe that X_s is a martingale. Furthermore, | X_s-X_s-1|≤ 2. This happens because the degree of any vertex v_k, k≠ i,j is not affected by the rising of v_s-1 and v_s, whether these last vertices link to v_i and v_j, i<s-1 andj<s, respectively. Furthermore, they do not affect the probabilities of the vertices v_k that will be chosen later.Therefore, applying the Azuma-Hoeffding inequality with X_t=N(k,t), X_0=[N(k,t)] and taking x = √(tlog(t)),we obtainP(| N(k,t)/t - [N(k,t)]/t| > √(log(t)/t)) ≤ t^-1/8.Hence, as t→∞ we have N(k,t)/t→P(k) in probability. Recalling that B(x,y)=Γ(x)Γ(y)/Γ(x+y) and using that Γ(x+a)/Γ(x+b)∼ x^a-b[1+(a-b)(a+b-1)2x], for x large enough (see <cit.>),we get the following. As k→∞, for l=1 N(k,t)/t∼ C_p [k^-(1+2/1-p)-3-p/(1-p)^2 k^-(2+2/1-p)],where C_p=Γ(1+2(1-p))(2(1-p)+2)(2(1-p)+1) P(2),and for l>1, N(k,t)/t∼ C_p,l[k^-(1+2/1-p)-3-p/(1-p)^2 k^-(2+2/1-p)],where C_p,l=Γ(l+2+2(1-p))(Γ(l+1))^-1 P(l+1). Note that the value of p is part of the exponent of the power law in (<ref>) and (<ref>). Moreover, when the UPA model corresponds to the Barabási-Albert model, that is when l=1 and p=0, the power law exponent is equal to -3, as expected. § EXAMPLES OF THE UPA MODEL AND ITS GENERALIZATION In this section we show through some examples, our analytical and asymptotic results. We then study the UPA model when the windows size grows linearly in time. In this case, we use simulations to study the empirical degree distribution. We divided this section into three Sub-sections. In Sub-Section <ref> we study the mean number of nodes having a certain degree in different instances, while in Sub-Section <ref> we illustrate our results on the empirical degree distribution. Finally, Sub-Section <ref> illustrates the case with time dependent windows size. §.§ Mean number of nodesFrom Corollary <ref> we know that the presence of uniform attachment is not sufficient to asymptotically destroy the scale free feature. Herein we investigate the role of the presence of the windows for fixed times and using different weights for the two types of attachment rule. We use Lemmas <ref> and <ref> with formulas (<ref>) and (<ref>) to compute[N(k,t)] as a function of k and t for different values of l and of p. In Fig. 1a, we first fix t=101 and p=0.5, to study the mean number of nodes having degree k, with k=1,2,⋯, for different sizes of the window. Since the grade k=1 is forced by the model hypothesis (N(1,l)=l), the figure is of interest for k ≥ 2. As expected, the effect of the size of the window is stronger for lower degrees, and it tends to disappear as the degree of the node increases. The same result is observed increasing the considered time (see inset of Fig. 1a). Fig. 1b illustrates the increase of [N(10,t)] as t increases, for different sizes of the window.First of all, we can notice that the size of the window penalizes the growth of [N(10,t)]. Each of the curves of [N(10,t)], corresponding to l=10 and l=100, crosses the analogous curve for l=1 after a certain time. After such times, the curves corresponding to l=10 and l=100 grow faster than in the case of l=1. This fact can be explained by noting that for l=1, the uniform rule does not privilege the choice of a vertex with degree 10. In this case, the uniform rule determines the choice of the last added vertex, which has degree 1. This also determines the faster increase of [N(10,t)] for l=10 than for l=100, as t increases enough. This phenomenon does not disappear if we increase the value of k (see inset of Fig. 1b). The shapes of the curves do not change but there is a remarkable change of scale. As far as the role of p is concerned, we note that it only affects the nodes with low degree (see Fig. 2a). When p is large, the uniform attachment mechanism helps links to nodes of lower degree but its role tends to disappear for nodes of higher degree. Again, the shape of the curves remains the same if we change t but the scale changes (see Fig. 2a and its inset). Fig. 2b illustrates [N(10,t)] as a function of t for different values of p. We observe that for small values of t, the curves of [N(10,t)] decrease, and after some time they start to increase. That is just an effect of the initial condition, [N(10,10)]=1. At the beginning of the process there are few vertices, and only one with degree 10, v_0. Then, v_0 can easily increase its degree by receiving a new link.A stronger presence of uniform attachment rule (greater p) decreases the speed of the growth of the degree of the nodes. Note that for p=1, pure uniform attachment, [N(10,t)] remains constant when t grows. For the case k=20 (see inset of Fig. 2b), the initial condition is [N(20,10)]=0. Here we observe that for small values of t, the curves of [N(20,t)] increase reaching a local maximum, then they decrease and after some time they start to increase again. This behavior of the curves for small values of t is a consequence of the zero value of the initial condition, and of the fact that l=10. In this range of t, [N(20,t)] is only affected by the preferential attachment rule.Fig. 1a, 1b, 2a and 2b suggest a role of the two attachment rules for the growth rate of low degree nodes. During the initial development of the network, lower degree nodes receive more links thanks to the presence of the window. However, the considered examples suggest a marginal asymptotic role of the uniform attachment in making significant changes to long term dynamics. §.§ Asymptotic degree distributionBy using Theorem <ref>, we herein illustrate the role of the UPA model parameters on the shape of the asymptotic degree distribution. In the case of the classical Barabási-Albert model, the preferential attachment rule determines a scale free behavior of the degree distribution while the uniform attachment implies a uniform asymptotic distribution. To illustrate the tail behavior of the UPA model weuse (<ref>) and (<ref>), given in Corollary <ref>. In Fig. 3a and 3b we compare the analytical results obtained by (<ref>), and the asymptotic approximations obtained by (<ref>), only taking the first term. In the inset the comparison is with the asymptotic approximations (<ref>), but taking account both the first and the second term.More precisely, inFig. 3a we highlight the effect of the presence of windows by keeping its size fixed to l=100 while we vary the probability of the presence of windows, p. In Fig. 3b we highlight the effect of the size of the windows by keeping fixed p=0.5 and varying the size of the windows, l. In Fig. 3a we observe that starting from log k=3 (or log k=3.5), for p=0.8 (or p=0.5), the analytical curves fit the corresponding linear straight lines of the asymptotic approximations(in log-log scale). However, different weights for the uniform attachment change in a significant way the slope of the straight lines in the log-log plot of Fig. 3a. Accounting both the first and second term in (<ref>), we get a good approximation of (<ref>) from log k=2 (or log k=2.5), for p=0.8 (or p=0.5),(see inset of Fig. 3a). We draw the solid lines in the inset (asymptotic results taking the first and the second term) only when the asymptotic approximation starts to hold. Results of Corollary <ref>, formulae (<ref>) and (<ref>) show a limited role of the windows size on the asymptotic degree distribution. Only the coefficient C_p,l in (<ref>) depends on l and this dependency is weak. Fig. 3b illustrates this property. Note that it becomes impossible to distinguish different solid lines corresponding to different values of l. §.§ Case of linear windows sizeOur results show the impossibility to destroy the scale free behavior by introducing a second uniform attachment rule. However, it might be hypothesided that this result is determined by the fixed size of the windows. When the network increases its size, a decrement of the contribution of the uniform attachment to the global dynamics seems intuitive. For this reason we decided to investigate the dynamics of a network governed by the same rules as the one studied in the previous sections but withwindows size that grows linearly with time (and hence with the size of the network).Unfortunately, this model cannot be studied analitically with the methodology used for the case of fixed size of the window. Therefore, we used simulations to analyze its asymptotic degree distribution. In order to choose the dimension of the network for this study, we compared analytical results with simulations for the UPA model with fixed size window. This study combined with some memory capacity restrictions, suggested to taken≥100000 as a value for which analytical results fit simulations well.Hence, we used n=300000 to analyze the asymptotic degree distribution of the network with linear size of the windows. Herein we focus on the effect of the size of the windows by keeping fixed p=0.8 and letting l=α n, for different values of α. In addition, we compare these results with the analytical results obtained by (<ref>) for the case l=100, see Fig. 4. We do not observe a remarkable difference between the case of fixed and linear windows size.Hence, we conclude that the scale free behavior is mantained also in the case of linear windows size.§ DISCUSSION AND CONCLUDING REMARKSIn this paper we propose a new model to describe the growth of networks in which the preferential attachment co-exists with the preferential attachment rule. Hypothesizing that the uniform attachment rule works for fixed windows sizewe were able to perform an analytical study of the mean number of nodes characterized by a fixed degree. Furthermore, analytical results were proved for the degree distribution.From our results we can conclude that the scale free behavior does not disappear by merging preferential and uniform attachment rules. Furthermore, we show that different weights between preferential and uniform attachment rules change the exponent of the power law while the value of l affects only the intercept of the log-log lines and the rate of convergence to the scale free behavior. That is, depending on the value of l, it changes the starting value of t to observe the power law behavior. This is due to the fact that when there is a window, some connections are “wasted” and do not contribute to the rich-get-richer mechanism. The obtained analytical results are strongly related with the finiteness of the windows size, requested by Theorem <ref>. This finiteness is indeed required. In fact, if the windows size is finite, then at each step an old node is always kept out from the window, so the degree distribution inside the window is bounded and controllable. Hence, Theorem <ref> still holds when the windows shape is unconventional but finite. However, the simulation study of the model in presence of linearly growing windows size shows that the scale free behavior is preserved also in presence of unbounded windows size. This result suggest a robustness of the scale free feature that explains the incredible number of natural instances in which the phenomenon is observed. We conclude this paper conjecturing the necessity to investigate the introduction of alternative attachment rules beside the preferential attachment. It seems interesting to understand which type of attachment could destroy the asymptotic power law behavior of the degree distribution.§ APPENDIXHere we complete the proof of Theorem <ref>, that is, the proof of steps (2) and (3). We use the results of subsection <ref>. We divide this section in two parts,l = 1 and l > 1.PART 1. l = 1 In the UPA Model with fixed window size l=1,P(k), k≥1, exists. LetS_T := sup_t ≥ T[N(k,t)]/t and I_T := inf_t ≥ T[N(k,t)]/t.We divide the proof in two cases, k=1 and k>1. Case k=1: We start by observing that the sequence of frequencies ( (N(1,t))/t )_t≥ l is bounded, so we only need to prove that it is monotone. By Lemma <ref> we obtain that[N(1,t+1)]/t+1⪋[N(1,t)]/t⇔2(1-p)/3-p⪋[N(1,t)]/t. Let us now consider the following events:A := {∃ Ts.t.S_T := sup_t ≥ T[N(1,t)]/t≤2(1-p)/3-p} ,B := {∃ T's.t.I_T' := inf_t ≥ T'[N(1,t)]/t≥2(1-p)/3-p}andC := {∀ T S_T > 2(1-p)/3-p > I_T } .Note that the events A and B are mutually exclusive since their bound is determined by the same constant, while C=(A∪ B)^c. Therefore, A, B and C form a partition. Next we are going to prove monotonicity when A or B holds. Moreover, the union of these eventsalways happen.In fact, note that if A holdsthen [N(1,t)]/t ≤ S_T ≤ 2(1-p)/(3-p) for t ≥ T. Thus, the sequence ([N(1,t)]/t )_t is monotone increasing. In the same way, if B holds, [N(1,t)]/t ≥ I_T'≥ 2(1-p)/(3-p) for t ≥ T'. Thus, the sequence ( [N(1,t)]/t )_t is monotone decreasing.Now observe that if C holds, for every T≥0, S_T is always larger than 2(1-p)/(3-p) and I_T smaller than 2(1-p)/(3-p). This means that if we are in situation C,the sequence ([N(1,t)]/t)_t should cross the value 2(1-p)/(3-p) infinitely many times. Assume that C holds, then by the definition of inf we can state that there exists t'≥ l such that [N(1,t')] < t'2(1-p)/3-p. Multiplying both sides by (1-(1-p/2t')) and adding (1-p), we obtain by Lemma <ref> [N(1,t'+1)] < t' ( 1-1-p/2t') 2(1-p)/3-p + (1-p)=2(1-p)/3-p(t'+1). Therefore, if [N(1,t')]/t' < 2(1-p)/(3-p) then we also have that [N(1,t'+1)]/(t'+1) < 2(1-p)/(3-p). In other words, if at a certain point the sequence ( [N(1,t)]/t )_t is on the left of 2(1-p)/(3-p) then it will remain on the left of 2(1-p)/(3-p) forever, so, the sequence ( [N(1,t)]/t )_t will cross the value 2(1-p)/(3-p) a finite number of times, which is a contradiction. Case k>1:By Lemma <ref> we obtain that [N(k,t+1)]/t+1⪌[N(2,t)]/t⇔1/2-p( 1-p/2[N(1,t)]/t +p ) ⪌[N(2,t)]/t,ifk=2[N(k,t)]/t⇔(1-p)(k-1)/2+(1-p)k([N(k-1,t)]/t)⪌[N(k,t)]/t,ifk>2. Since we have already proved that P(1) exists, then for each ϵ∈ℝ, let us now defineg(ϵ) := 1/2-p(1-p/2 (P(1) + ϵ) + p ).Next we are going to write the proof when k=2. For k>2 we perform exactly the same reasoning, but replacing g(ϵ) byh(ϵ) := (1-p)(k-1)/2+(1-p)k(P(k-1) + ϵ). Thus, in order to prove that the sequence ([N(2,t)]/t)_t converges, we are going to prove thatlim_T →∞(S_T-I_T) = 0 .We begin with an arbitrary time T_1. Then, by definition, I_T_1≤ S_T_1, so, there exists the midpointM_T_1:=(S_T_1 + I_T_1)/2. We consider two cases M_T_1≥ g(0) and M_T_1<g(0), and what we are going to show is that in both cases there exists a time T_2>T_1 such that S_T_2 - I_T_2≤ (S_T_1 - I_T_1)/2. Note that using this repeatedly we obtain (<ref>), and thus we have the convergence. It is not difficult to verify that g(ϵ) is the equation of a line with positive slope, so it is a continuous and growing function. Then if M_T_1≥ g(0), there exists a δ≥ 0 such thatM_T_1 = g(δ). Moreover, by definition of I_T_1, there is a t'>T_1 such that [N(2,t')] ≤ t'g(δ),and this could happen either for a limited number or infinitely many t'. If this happens only a limited number of times, then there exists a T_2 > T_1 such that I_T_2≥ g(δ), so, S_T_2 - I_T_2≤1/2(S_T_1 - I_T_1). Otherwise, if this happens for infinitely many t', then by Lemma <ref> and multiplying both sides of (<ref>) by [1-(1-p)/t')] and adding (1-p)[N(1,t)]/2t'+p, we get that [N(2,t'+1)]≤(1-1-p/t')t'g(δ)+1-p/2t'[N(1,t')] + p. Since P(1) exists, then by definition,∀β > 0 ∃ Ts.t. ∀ t ≥ T P(1)-β < [N(1,t)]/t < P(1)+β .Using this we have that there exists a T', such that for any t' ≥ T', the right side of (<ref>) is less than or equal(1-1-p/t')t'g(δ) + 1-p/2(P(1)+δ) + p=t'g(δ) - (1-p)g(δ) + 1-p/2(P(1)+δ) + p=t'g(δ) - (1-p)g(δ) + (2-p)g(δ)=(t'+1)g(δ),where the second equality follows by (<ref>). Thus, we have provedthat for large values of t',[N(2,t')]/t' ≤ g(δ) ⇔ [N(2,t'+1)]/(t'+1) ≤ g(δ). Hence, there exists a T_2>t'>T_1 such that S_T_2≤ g(δ), thenS_T_2 - I_T_2≤ (S_T_1 - I_T_1)/2 . In the same way we analyze the case M_T_1 < g(0). If this condition is verified, then there exists a δ≥ 0 such thatM_T_1 = g(-δ) . Furthermore, by definition of S_T_1, there is a t' such that [N(2,t')] ≥ t'g(-δ) ,what could happen for either a limited number or infinitely many t'. If this happens only a limited number of times, then there exists a T_2 > T_1 such that S_T_2≤ g(-δ), so S_T_2 - I_T_2≤1/2(S_T_1 - I_T_1). Otherwise, if this happens for infinitely many t', then by Proposition <ref> and multiplying both sides of (<ref>) by [1-(1-p)/t')] and adding (1-p)[N(1,t)]/2t'+p, we get that [N(2,t'+1)]≥(1-1-p/t')t'g(-δ)+1-p/2t'[N(1,t')] + p.Moreover, by (<ref>), there exists a T' such that for t' ≥ T', the right side of (<ref>) is greater than or equal(1-1-p/t')t'g(-δ) + 1-p/2(P(1)-δ) + p=t'g(-δ) - (1-p)g(-δ) + 1-p/2(P(1)-δ) + p=t'g(-δ) - (1-p)g(-δ) + (2-p)g(-δ)=(t'+1)g(-δ),where the second equality follows by (<ref>). Then, we have proved that for large values of t',[N(2,t')]/t' ≥ g(-δ) ⇔ [N(2,t'+1)]/(t'+1) ≥ g(-δ). Hence, there exists a T_2 such that I_T_2≥ g(-δ), thenS_T_2 - I_T_2≤ (S_T_1 - I_T_1)/2 . In the UPA model with fixed window size l=1, P(k), k≥1, is given by (<ref>). We have proved that the limit P(k) exists for every k ∈ℕ. Thus, by Lemma <ref> we can state thatQ(k) := lim_t →∞( [N(k,t+1)] - [N(k,t)] ) exists for every k ∈ℕ. That is∀ϵ > 0 ∃ Ts.t. ∀ t ≥ T : Q(k)-ϵ≤[N(k,t+1)] - [N(k,t)] ≤ Q(k) + ϵ.Using this we obtain that(t-T)(Q(k)-ϵ) ≤ ∑_i=0^t-T-1[[N(k,T+i+1)]-[N(k,T+i)]] = [N(k,t)]-[N(k,T)] ≤ (t-T)(Q(k) + ϵ). Furthermore, dividing by t+1 and taking the limit as t →∞ we get Q(k)-ϵ≤P(k) ≤ Q(k)+ϵ ,or, equivalentlyP(k) - ϵ≤ Q(k) ≤P(k) + ϵ . Since ϵ is chosen arbitrarily, then Q(k) = P(k).Thus, by (<ref>) we obtain that P(1)=2(1-p)/(3-p) and P(2)=(1-p)P(1)/2(2-p)+p/(2-p)=(1-p)^2/(2-p)(3-p)+p/(2-p). Now, for k > 2, by Lemma <ref> we get[N(k,t+1)]-[N(k,t)] = -(1-p)k/2t[(k,t)]+(1-p)(k-1)/2t[N(k-1,t).Then, taking the limit as t → +∞ and using again (<ref>), we obtainP(k) = -(1-p)k/2P(k)+(1-p)(k-1)/2P(k-1)=(1-p)(k-1)/2+(1-p)kP(k-1).Now, iterating (<ref>) we haveP(k) =(1-p)^k-2(k-1)(k-2)… (k-(k-2))/[2+(1-p)k]… [2+(1-p)(k-(k-3))]P(2)= (k-1)!/[ 2/1-p+k ] ⋯[ 2/1-p+(k-(k-3)) ]P(2)= Γ(k)Γ( 3+2/1-p)/Γ( k+1+2/1-p)P(2)= P(2) (2/1-p+2)(2/1-p+1)B(k,1+2/1-p),where Γ(x) and B(x,y) are the Gamma and the Beta functions, respectively. PART 2. l > 1Now we are going to prove the existence of P(k). To achieve this, we use the Lemma <ref> and the following result.For k∈ℕ, the following limit exits, H_k = lim_t →∞∑_m=1^lℙ(M_m(t) = k) .We start by recalling that at the starting period t=l the initial graph has l+1 nodes, (v_0,v_1,…,v_l), so that deg(v_0)=l and deg(v_j)=1, 1≤ j≤ l. That is, ℙ(M_m(l)=1)=1, for 1≤ m≤ l. Furthermore, given the graph process at time t, we add at time t+1 a new vertex, v_t+1, together with an edge going out from this, so v_t+1 has degree one for all t≥ l, which means that for m=l, ℙ(M_l(t+1)=1)=1 for all t≥ l. Now note that since ℙ(M_m(t+1) = 1 | M_m+1(t) ≠ 1) = 0, t≥ l,then for 1≤ m < l, ℙ(M_m(t+1) = 1)= ℙ(M_m(t+1) = 1 | M_m+1(t) = 1) ℙ(M_m+1(t) = 1) = I_1(t)ℙ(M_m+1(t) = 1), and for k>1, ℙ(M_m(t+1) = k)= ℙ(M_m(t+1) = k | M_m+1(t) = k) ℙ(M_m+1(t) = k) + ℙ(M_m(t+1) = k | M_m+1(t) =k-1) ℙ(M_m+1(t) =k-1= I_k(t)ℙ(M_m+1(t) = k)+L_k-1(t)ℙ(M_m+1(t) =k-1),where,I_k(t):=ℙ(M_m(t+1) = k | M_m+1(t) = k)= pl-1/l + (1-p)2t-k/2t,andL_k-1(t):=ℙ(M_m(t+1) = k | M_m+1(t) =k-1)=p/l + (1-p)k-1/2t,k∈ℕ. Thus by (<ref>) and (<ref>) we can construct for 1≤ m≤ l, the evolution of ℙ(M_m(t) = k) over time. In fact if k=1, thenat timet=l, ℙ(M_m(l) = 1)=1, while for t>l, ℙ(M_l(t) = 1)=1 and by (<ref>) we get thatℙ(M_m(t) = 1)=∏_h=1^l-mI_1(t-h),1≤ m< l,with the convention that I_1(x)=1 when x<l. In this case note that t-h≥ l if t≥ 2l-1, and lim_t →∞I_1(t) = 1-p/l. Then, we are able to write H_1 =∑_m=1^l(1-p/l)^l-m = l/p - l/p(1-p/l)^l.For the case k>1, by (<ref>) we also note that for every 1≤ m≤ l, and for every t≥ l, ℙ(M_m(t) = k) is a finite product of functions I_k(·) and L_k-1(·) calculated at different instants. Furthermore, sincelim_t →∞ I_k(t)= 1 - p/l and lim_t →∞ L_k-1(t) = p/l,so, both of these limits exist. Then, by a procedure similar to that we made to findH_1, weconclude that the limit H_k also exists for k > 1, and H_k =(p/l)^k-1∑_m=1^l-(k-1)l-ml-m-(k-1)(1-p/l)^l-m-(k-1). Now we are ready to prove the existence of P(k). In the UPA model with fixed window size l>1, the limit P(k) exists for every k ∈ℕ.We divide the proof in 2 cases, k=1 and k>1. Let H_k(t)=∑_m=1^lℙ(M_m(t) = k) and a_t=∏_J=l^t( 1-1-p/2J).Applying (<ref>) recursively, we get that for k=1[N(1,t+1)] =[N(1,l)]a_t +∑_J=0^t-l-1( 1-pH_1(t-J-1)/l) ∏_k=0^J( 1-1-p/2(t-k))+(1-pH_1(t)/l).Now observe the sequence {a_t}_t ∈ℕ is decreasing(except when p=1, in this case a_t=1, t ∈ℕ) and bounded from below by 0. Then lim_t → +∞a_t exists and is finite.Hence,lim_t → +∞a_t/t+1= 0.Furthermore, by Lemma <ref>lim_t →∞1/t+1(1-pH_1(t)/l) = 0,andfor every ϵ > 0 there exists a t such that H_1-ϵ≤ H_1(t) ≤ H_1+ϵ, ∀ t ≥t.Note that once we chose ϵ, t is fixed and we can split the summation∑_J=0^t-l-1( 1-pH_1(t-J-1)/l) ∏_k=0^J( 1-1-p/2(t-k))=∑_J=0^t-t-1( 1-pH_1(t-J-1)/l) ∏_k=0^J( 1-1-p/2(t-k)) +∑_J=t-t^t-l-1( 1-pH_1(t-J-1)/l) ∏_k=0^J( 1-1-p/2(t-k)),where the second term in the split has a fixed number of elements, given by (t-l-1)-(t-t) = t-l-1. Hence,lim_t →∞1/t+1∑_J=t-t^t-l-1( 1-pH_1(t-J-1)/l) ∏_k=0^J( 1-1-p/2(t-k)) = 0 .Let now b_t = ∑_J=0^t-2( ∏_k=0^J( 1-1-p/2(t-k)) ) . Iterating the last equation of (<ref>), when l=1 we get[N(1,t+1)] =[N(1,1)]a_t+ (1-p)b_t+(1-p).By (<ref>) andsince P(1) exists when l=1, then we have that b_t/(t+1) converges as t→∞. Therefore,∃ Ls.t. lim_t →∞1/t+1∑_J=0^t-t-1∏_k=0^J( 1-1-p/2(t-k)) = L.Thus, by (<ref>) and (<ref>), the first term of the split in (<ref>) satisfies( 1-p(H_1+ϵ)/l) L ≤ lim_t →∞1/t+1∑_J=0^t-t-1( 1-pH_1(t-J-1)/l) ∏_k=0^J( 1-1-p/2(t-k))≤ ( 1-p(H_1-ϵ)/l) L.Since ϵ is chosen arbitrary, then the limit above exists. Hence by (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>)lim_t→∞[N(1,t+1)]/t+1=(1-pH_1/l).For the case k > 1, by Lemma <ref> we know that H_kexists for k > 1. Then, we perform the same reasoning of Lemma <ref> when k>1, but replacing g(ϵ) byh(ϵ) := ( p/l( H_k-1-H_k + 2ϵ) + (1-p)(k-1)/2(P(k-1)+ϵ) ) 2/2+(1-p)k. In the UPA model with fixed window size l>1, P(k), k≥1 is given by (<ref>). Following the same ideas of the first lines of the proof of Lemma <ref>, we obtain that for every k ∈ℕ P(k)=lim_t→ +∞( [N(k,t+1)]-[N(k,t)] ).Now observethat within a window of size l, the maximum degree of a node is l, then ℙ(M_m(t)=k)=0 for k>l, 1≤ m≤ l. Therefore for k>l+1, by Lemma (<ref>) [N(k,t+1)]-[N(k,t)]= -(1-p)k/2t[N(k,t)] + (1-p)(k-1)/2t[N(k-1,t)].Then, taking the limit as t→∞ in (<ref>) and using (<ref>), when k>l+1 we getP(k) = (1-p)(k-1)/2+(1-p)kP(k-1).Note that (<ref>) is equal to (<ref>). Then, as we did in the proof of Lemma <ref>, iterating (<ref>) we obtain P(k) = Γ( k ) Γ( l+2+ 2/1-p)/Γ( l+1 ) Γ( k+1+2/1-p)P(l+1)=B(k,l+2+2/1-p)/B(l+1,k+1+2/1-p)P(l+1), For 1≤ k ≤ l+1, we use Lemma <ref> and (<ref>) and obtain P(k)=2/(3-p)(1-p/l)^l,if k=1 2/2+k(1-p)(p/l(H_k-1-H_k)+(1-p)(k-1)/2)P(k-1),if k=2,…,l+1,where H_k is given by (<ref>) and (<ref>), withH_k=0 for any k>l.plain | http://arxiv.org/abs/1704.08597v1 | {
"authors": [
"Angelica Pachon",
"Laura Sacerdote",
"Shuyi Yang"
],
"categories": [
"math.PR",
"05C80, 90B15, 60J80"
],
"primary_category": "math.PR",
"published": "20170427142703",
"title": "Scale-free behavior of networks with the copresence of preferential and uniform attachment rules"
} |
Secure Precise Wireless Transmission with Random-Subcarrier-Selection-based Directional Modulation Transmit Antenna Array Feng Shu, Xiaomin Wu, Jinsong Hu, Riqing Chen, and Jiangzhou Wang This work was supported in part by the National Natural Science Foundation of China (Nos. 61472190, and 61501238), the Open Research Fund of National Key Laboratory of Electromagnetic Environment, China Research Institute of Radiowave Propagation (No. 201500013). Feng Shu, Xiaomin Wu, Jinsong Hu are with School of Electronic and Optical Engineering, Nanjing University of Science and Technology, 210094, CHINA. E-mail:{shufeng, xiaoming.wu}@njust.edu.cn Feng Shu and Riqing Chen are with the College of Computer and Information Sciences, Fujian Agriculture and Forestry University, Fuzhou 350002, China. E-mail: [email protected]. Feng Shu is also with the National Key Laboratory of Electromagnetic Environment, China Research Institute of Radiowave Propagation, Qingdao 266107, China Jiangzhou Wang iswiththe School of Engineering and Digital Arts, University of Kent, Canterbury Kent CT2 7NT, United KingdomDecember 30, 2023 ================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================== In this paper, a practical wireless transmission scheme is proposed to transmit confidential messages to the desired user securely and precisely by the joint use of multiple techniques including artificial noise (AN) projection, phase alignment (PA)/beamforming, and random subcarrier selection (RSCS) based on OFDM, and directional modulation (DM), namely RSCS-OFDM-DM. This RSCS-OFDM-DM scheme provides an extremely low-complexity structures for the transmitter anddesired receiver and makes the secure and precise wireless transmission realizable in practice. For illegal eavesdroppers, the receive power of confidential messages is so weak that their receivers cannot intercept these confidential messages successfully once it is corrupted by AN. In such a scheme, the design of phase alignment/beamforming vector and AN projection matrix depend intimately on the desired direction angle and distance. It is particularly noted that the use of RSCS leads to a significant outcome that the receive power of confidential messages mainly concentrates on the small neighboring region around the desired receiver and only small fraction of its powerleaks out to the remaining large broad regions. This concept is called secure precise transmission. The probability density function of real-time receive signal-to-interference-and-noise ratio (SINR) is derived.Also, the average SINR and its tight upper bound are attained.The approximate closed-form expression for average secrecy rate is derived by analyzing the first-null positions of SINR and clarifying the wiretap region. From simulation and analysis, it follows that the proposed scheme actually can achieve a secure and precise wireless transmission of confidential messages in line-of-propagation channel, and the derived theoretical formula of average secrecy rateis verified to coincide with the exact one well for medium and large scale transmit antenna array or in the low and medium SNR regions.Secure precise transmission, phase alignment, random-subcarrier-selection, directional modulation, SINR§ INTRODUCTIONIn the last decade, secure physical-layer wireless transmission has increasingly become an extremely important research field in both academia and industry <cit.>. As a key-less physical layer secure transmit way, directional modulation (DM) is attracting an ever-growing interest in <cit.> andhas made substantial progression in many aspects by using antenna array with the help of aided artificial noise (AN) <cit.>. In <cit.>, the authors propose an actively driven DM array of utilizing analog radio frequency (RF) phase shifters or switches, which exists some fundamental weaknesses such as requirement of high-speed RF switches and high complexity of design process. To overcome those disadvantages, instead of the RF frontend,the DM is synthesized on the baseband bythe AN-aided orthogonal vector method in <cit.>.To enhance security,the symbol-level precoder in <cit.> is proposed by using the concept of constructive interference in directional modulation with the goal of reducing the energy consumption at transmitter. Taking direction measurement error into account,the authors in <cit.> propose two new robust DM synthesis methods for two different application scenarios: single-desired user and multi-user broadcasting by exploiting the statistical properties of direction measurement error. Compared to existing non-robust methods, the proposes robust methods can achieve an order-of-magnitude bit error rate (BER) performance improvement along desired drections. In general, the DM may achieve a high performance gain along the desired directions via beamforming and provide a secure transmission by seriously degrading the undesired directions by artificial noise(AN) projection.However, the major disadvantage of DM is that its beamforming method only depends on direction angle and is independent of distance. Therefore, as an eavesdroppermoves within the main beam of the desired direction, it can readily intercept the confidential messages towards the desired direction. This will result in a serious secure problem.To address such a problem, the authors in <cit.> propose a novel frequency diversity array model. The frequency increments are randomly (instead of linearly) assigned to all antenna-array elements, which is called a random frequency diverse array (RFDA) <cit.>. In <cit.>, the authors propose a RFDA based directional modulation with aided AN. This scheme can achieve a secure precise wireless transmission. That is, the confidential beamforming vectors for desired users and AN projectionmatrix for eavesdroppers rely heavily on both direction and distance. In other words, by making use of RFDA plus DM, the confidential messages can be securely and precisely transmitted towards a given position (direction and distance) with only a small fraction of confidential power radiating outside the small neighborhood around the desired position. Actually, the frequency diverse array has been widely investigated in RADAR field <cit.>. In <cit.>, the RFDA concept has also been employed to construct OFDM transmitter and achieved secure wireless communication in free space. However, as the number N_T of transmit antennas tends to medium-scale and large-scale, the desired transmitter and receiver require a high-complexity transmit and receive structure. For example, at desired receiver, N_T RF chains, or one RF chain with N_T matched filter are required to coherently combineN_T parallel random independent subchannels of all N_T frequency bands. This N_T-RF-chain structure will dramatically increase the circuit cost at both transmitter and receiver for medium-scale and large-scale directional modulation system. This motivates us propose a low-complexity structure of using random-subcarrier-selection (RSCS) array based on OFDM technique instead of RFDA with the help of AN projection and phase-alignment at transmitter. The proposed scheme can easily implement the DM synthesis at transmitter and coherent combining at receiver using IFFT/FFT operations, respectively. Thus, this new structure can significantly reduce the circuit cost and make the secure precise communication feasible and applicable. Our main contributions are summarized as follows:∙ To address the problem of circuit complexity at receiver, we propose a new scheme of replacing random frequency diverse by random subcarrier selection based on OFDM and DM (RSCS-OFDM-DM). At desired receiver, the FFT/IFFT operation will reduce N_T RF chains to only one. This simple structure will significantly save the circuit budget in the medium-scale and large-scale systems.∙At transmitter,phase alignment in frequency-domain is combined with RSCS toconstructively synthesize the N_T frequency-shifted versions of transmit signal at desired receiver and destructively at eavesdroppers. The AN projection is used to further degrade the performance of eavesdroppers.∙In the scenario that AN and channel noise have approximately the same receive power, the probability density function (PDF) of receive signal-to-interference-and-noise ratio (SINR) is first derived. Subsequently, the average SINR expression is presented by using the derived PDF.∙The approximate closed-form expression for average secrecy rate is derived by analyzing the first-null position and clarifying the wiretap region. This theoretical formula is verified to be valid in medium and large scale transmit antenna array and almost independent of SNR. Additionally, in the low and medium SNR regions, the derived expression is shown to coincide with the exact one well.∙By simulation and analysis,we find that the receive confidential message power forms a high SINR peak around the desired position and shows a very small leakage to other wide regions. In other words, this gathers almost all confidential message power at a small neighbourhood around the desired position and at the same time leads to an extremely low receive SINR at eavesdropper positions by AN projection, RSCS and phase alignment at transmitter, which may successfully destroy the interception of eavesdroppers.The remainder are organized as follows: Section II describes the system model of the proposed RSCS-OFDM-DM and proves its feasibility. The AN-aided and range-angle-dependent beamforming scheme for RSCS-OFDM-DM is presented in Section III. Here, we analyze the PDF of the receive SINR and its means. Also, we derive the theoretical formula for average secrecy rate by using the desired SINR and upper bound of eavesdropper SINR.Section IV presents the simulation results and analysis. Finally, we make a conclusion in Section V.Notations: throughout the paper, matrices, vectors, and scalars are denoted by letters of bold upper case, bold lower case, and lower case, respectively. Signs (·)^T, and (·)^H denote transpose and conjugate transpose respectively. The operation · and |·| denote the norm of a vector and modulus of a complex number. The notation 𝔼{·} refers to the expectation operation. Matrices I_N denotes the N× N identity matrix and 0_M× N denotes M× N matrix of all zeros.§ SYSTEM MODELIn this section, we propose a secure precise system structure as shown in Fig. <ref>. This system consists of OFDM, random-subcarrier-selection (RSCS) and DM. In Fig. <ref>, the RSCS-OFDM-DM communication system includes a legitimate transmitter equipped with one N_T-element linear antenna array,a desired receiver, and several eavesdroppers, which lie at different positions from the desired position and is not shown in Fig. <ref>, with each receiver havingsingle receive antenna. The distinct frequency shifts of the same signal/symbol over different transmit antennas areradiated as indicated in Fig. <ref>. For the convenience of implementation of both transmitter and receiver, all frequency shifts are designed to be orthogonal with each otherin this paper. Thus, the subcarriersin OFDM systems are a natural choice as shown in Fig. 2. Suppose there are N subcarriers in our OFDM systems, the set of subcarriers is the followingS_sub={f_m| f_m=f_c+mΔ f, (m=0,1,…,N-1)},where f_c is the reference frequency, and Δ f is the subchannel bandwidth. We assume NΔ f≪ f_c in this paper. In our system, the total bandwidth is B=NΔ f. The corresponding total subcarrier index set is defined as followsS_N={0,1,2,…,N-1}.N_T random subcarriers is chosen from S_N,allocated to N_T transmit antennas individually, and represented as S_N_TS_N_T⊆ S_N, |S_N_T|=N_T.Now, we define a chosen subcarrier index function η(∙) as a mapping fromthe set of transmit antenna indices {1,2,⋯, N_T} tothe chosen subcarrier setS_N_T={η(n)| n∈{1,2,⋯, N_T}},where η(n)∈ S_N. Here, T_s denote the period of OFDM symbol. Fig. 2 sketches two different kinds of RSCS patterns: block-level and symbol-level. The first pattern shown in Fig. 2 (a) is called block-level pattern. Every block is made up of several even one thousand OFDM symbols. Within one block, the same RSCS pattern is used but the RSCS pattern will vary from one block to another. The second pattern shown in Fig. 2 (b) is a symbol-level type. In this type, the RSCS changes from one OFDM symbol to another. For desired receivers, the latter provides a more secure protection but also place a large computational complexity and uncertainty on the receive active subcarrier test. The former can strike a good balance among receiver complexity, security and performance by choosing a proper block size.A large block size means a high successful detection probability of active subcarriers and less security because of less randomness. In other words, a better performance can be achieved. Determining the block size is a challenging problem beyond the scope of our paper due to the length limit on paper .Let Δ T be the sampling interval, we have the next definitionT_s=NΔ T=N/B, Δ T=1/B.With the above definitions, the transmit RF signal vector from N_T transmit antennas is expressed in vector form as follows𝐬(t)=[s_1(t),s_2(t),…,s_n(t),…,s_N_T(t)]^T,withs_n(t)=x_ke^j(2π f_nt+ϕ_n), (n=1,2,…,N_T)where f_n is randomly chosen from subcarrier set S_sub, x_k is the kth transmitted complex digital modulation symbol with 𝔼{x_k^*x_k}=1, ϕ_n is the initial phase, and the time variable t∈((k-1)T_s,kT_s). In far-field scenario, after the transmit signal s_n(t) from element n experiences the line-of -propagation (LoP) channel, thecorresponding receive signal at an arbitrary position(θ,R), where R and θ are the distance andangle with respect to the firstelement of transmit antenna array, respectively, and the first element is chosen as a reference antenna,can be expressed ass'_n(θ,R;t) =x_ke^j(2π f_n(t-R^(n)/c)+ϕ_n)=x_ke^j(2π (f_c+Δ f_η(n))(t-R^(n)/c)+ϕ_n),withR^(n)=R-(n-1)dcosθ,where c is the light speed,and d=c/(2f_c) is the spacing between two elements of transmit antenna array, and Δ f_η(n) is the subcarrier frequency increment of the nth element. Hence, the overall received superimposed receive signal from all array elementscan be written asr_k'(θ,R;t)=ρ∑_n=1^N_Ts'_n(θ,R;t)+n_k'(t)=ρ x_ke^j2π f_c(t-R/c) ·∑_n=1^N_Te^j[2π(Δ f_η(n) t-Δ f_η(n)R^(n)/c+f_c(n-1)dcosθ/c)+ϕ_n] +n_k'(t),where ρ is the loss path factor in free space or LoP channel, and proportional to 1/R^2, n'_k(t) is the received channel noise. Therefore, the received useful signals at desired position (θ_D,R_D) and undesired position (θ_E,R_E) are expressed asr_k'(θ_D,R_D;t)=ρ x_ke^j2π f_c(t-R_D/c) ·∑_n=1^N_Te^j[2π(Δ f_η(n)t-Δ f_η(n)R_D^(n)/c+f_c(n-1)dcosθ_D/c)+ϕ_n] +n'_k(t),andr_k'(θ_E,R_E;t)=ρ x_ke^j2π f_c(t-R_E/c) ·∑_n=1^N_Te^j[2π(Δ f_η(n)t-Δ f_η(n)R_E^(n)/c+f_c(n-1)dcosθ_E/c)+ϕ_n] +n'_k(t). In what follows,the received RF signal in (<ref>) is first down-converted to the following analog baseband signalr_k(θ,R,t)=ρ x_ke^j2π f_c(t-R/c)e^-j2π f_c(t-R/c) ·∑_n=1^N_Te^j[2π(Δ f_η(n)t-Δ f_η(n)R^(n)/c+f_c(n-1)dcosθ/c)+ϕ_n] +n_k(t)=ρ x_k∑_n=1^N_Te^j[2π(Δ f_η(n)t-Δ f_η(n)R^(n)/c+f_c(n-1)dcosθ/c)+ϕ_n] +n_k(t),by using a signal down-conversion operation. Sampling the above analog complex baseband signal by signal bandwidthB Hz in complex field, then we get the received sampling signal sequence for kth OFDM symbol as follows𝐫_k[N]=[r_k[0],r_k[1],…,r_k[m],…,r_k[N-1]],where r_k[m]=r_k(t)|_t=mΔ T denotes the mth sample of the signal with Δ T=1/B as followsr_k[m]=ρ x_k∑_n=1,η(n)∈ S_N^N_T(e^j[2π(Δ f_η(n)mΔ T-Δ f_η(n)R^(n)/c+f_c(n-1)dcosθ/c)+ϕ_n]) +n_k(mΔ T)=ρ x_k∑_n=1,η(n)∈ S_N^N_T(e^j[2π(η(n)(Δ fmΔ T-Δ fR-(n-1)dcosθ/c)+(n-1)cosθ/2)+ϕ_n]) +n_k(mΔ T),where Δ f_η(n)=η(n)Δ f. Hence, the finite sequence 𝐫_k[N] has N samples, and we divide the digital frequency interval [0, 2π] into N equidistant points. Thus, the form of N-points discrete Fourier transform (DFT) for sample signal isy(q)=y(e^jω)|_ω=2π q/N=∑_m=0^N-1r_k[m]e^-j2π/Nmq=ρ x_k∑_m=0^N-1∑_n=1,η(n)∈ S_N^N_Te^-j2π/Nmq· e^j2π(η(n)Δ fmΔ T-η(n)Δ fR-(n-1)dcosθ/c+f_c(n-1)dcosθ/c)+jϕ_n +∑^N-1_m=0n_k(mΔ T)e^-j2π/Nmq,where 0≤ q≤ N-1, and ω∈[0, 2π]. Exchanging the order of double summations yields equation (<ref>).Considering 0≤|(η(n)-q)|≤ N, we havesin[π(η(n)-q)]/sin[π/N(η(n)-q)]e^jπ/N(η(n)-q)(N-1) ={[N,q=η(n);0, q≠η(n). ].The received frequency-domain symbol over subchannel q is rewritten asy(q)=ρ x_k∑_n=1^N_T·(e^-j2π(Δ f_η(n)R^(n)/c-(n-1)cosθ/2)+jϕ_nNδ[2πΔ f(η(n)-q)])+∑^N_T_n=1n_k(n),withδ(q=η(n))={[1,q=η(n);0, q≠η(n). ].Based on (<ref>), all the received signals from N_T subchannels are as followsr_k(θ,R,q)=ρ x_k∑_n=1^N_T·(e^-j2π(η(n)Δ fR^(n)/c-(n-1)cosθ/2)+jϕ_nNδ[2πΔ f(η(n)-q)])+∑^N_T_n=1n_k(n)=𝐚^H(θ,R)diag{𝐱(q)}𝐯_k+∑^N_T_n=1n_k(n),where the vector 𝐚^H(θ,R) is as follows for a target at the angle θ and range R𝐚^H(θ,R) =[e^-j2πψ_1,e^-j2πψ_2,...,e^-j2πψ_n,...,e^-j2πψ_N_T],where the function ψ_n is defined byψ_n=η(n)Δ fR^(n)/c-f_c(n-1)dcosθ/c. In (<ref>), the vector 𝐯_k is designed according to the knowledge of desired receiver position (θ_D,R_D) and defined as𝐯_k=[e^jϕ_1,e^jϕ_2,…,e^jϕ_N_T]^T,and𝐱(q)=ρ x_kN {δ(2πΔ f(η(1)-q)),…,δ(2πΔ f(η(N_T)-q))}.Observing (<ref>), to realize the constructive combining ofthe N_T terms at the desired receiver (θ_D, R_D), it is very apparent that all initial phases for N_T transmit antennas satisfy the following identityϕ_1-2πψ_D1=ϕ_2-2πψ_D2=⋯ϕ_N_T-2πψ_DN_T=θ_0,where θ_0 is a constant phase, andψ_Dn=Δ f_η(n)R^(n)_D/c-f_c(n-1)dcosθ/cwithR^(n)_D=R_D-(n-1)dcosθ.This condition will guarantee that all N_T signals from N_T antennas with random orthogonal frequencyshifts of the same signal is coherently combined at the desired receiver. The final receive combined signal at desired receiver is given byr_k(θ_D,R_D)=ρζ N_T x_k+∑^N_T_n=1n_k(n),where ζ=e^jθ_0N. Additionally, the corresponding signal at eavesdropper isr_k(θ_E,R_E)=𝐚^H(θ_E,R_E)diag{𝐱(q)}𝐯_k+∑^N_T_n=1n_k(n)=ρ x_k∑_n=1^N_Te^-j2π(Δ f_η(n)R_E-(n-1)dcosθ/c-(n-1)cosθ_E/2)+jϕ_n · Nδ[2πΔ f(η(n)-q)]+∑^N_T_n=1n_k(n).Since all phases of signals received from all transmit antennas, ϕ_n-2π(Δ f_η(n)R_E-(n-1)dcosθ_E/c-(n-1)cosθ_E/2),are viewed as independently identical distributed (iid) random variables due to random-subcarrier-selection,the received signal sum r_k(θ_E,R_E)→ 0 as N_T tends to medium-scale or large-scale.§ PROPOSED AN-AIDED SECURE PRECISE TRANSMISSION METHODIn the previous section, without the help of AN, we establish the principle of secure precise transmission by using RSCS, multiple transmit antennas, and phase alignment, which may transmit confidential messages to the given desired direction and distance with a weak signal leakage to other large region outside the small neighborhood around the given position. In this section, this scheme, in combination with AN, can further improve the security of transmitting confidential message. Below, we will show how to design the precoding vector for the confidential message and construct the AN projection matrix. Here, the AN projection matrix enforces the AN into the eavesdropper steering space or the null-space of the desired steering vector with a small fraction or no leakage of AN powerto the desired steering space. Conversely, the precoding vector for the confidential message is to drive the major power of confidential message to the desired steering space with a slight or no leakage to its null-space.§.§ General beamforming and AN projection schemeIn terms of the derivation and analysis in Section II, we first construct the k baseband frequency-domain vector corresponding to transmit antenna m with other N_T-1 antennas being idle or non-active as follows𝐬_k,m=√(P_S)[β_1v_k(m)x_k+β_2w̃(m)]𝐞_η(m),where P_S is the total transmit power, 𝐞_η(n) is an N× 1 vector with only element η(n) being one and others being zeros, v_k,m denotes the mth element of the phase alignment/beamforming vector 𝐯_k which is to implement the coherent combining at the desired receiver, and w̃_m denotes the mth element of the AN vector 𝐰̃, which equals to 𝐓𝐰_k. As we can see, 𝐓 is an N_T× N_T orthogonal projection matrix and 𝐰_k∼𝒞𝒩(0,𝐈_N) is an N_T× 1random vector. In (<ref>), β_1 andβ_2 are two power allocation (PA)factors and satisfies the constraint β_1^2+β_1^2=1, which ensure that the total transmit power is equal to P_S. In accordance with (<ref>), the kth total transmit spatial-frequency codeword from is rewritten as𝐒_k=√(P_S)β_1𝐄_kdiag(𝐯_k) x_k +√(P_S)β_2𝐄_kdiag(𝐓𝐰_k),where 𝐄_k is an N× N_T matrix of random-subcarrier-selection for transmit antenna array defined as𝐄_k=[𝐞_η(1), ⋯, 𝐞_η(m), ⋯, 𝐞_η(N_T)],which is to force more AN into the null-space as possible as it can and at the same time less even no AN into the useful signal space consisting of the desired steering vectors. There are several various rules to optimize 𝐯_k and 𝐓 like null-space projection, leakage and so on. Experiencing the LoP channel, the N× 1 received vector of frequency-domain data symbols at position(θ,R)is given by𝐲_k(θ,R)=ρ√(P_S)∑_m=1^N_T a^*(θ,R)(m)[β_1v_k(m)x_k+β_2w̃(m)]𝐞_η(m)+𝐧_k=ρ√(P_S)β_1{∑_m=1^N_Ta^*(θ,R)(m)[v_k(m)x_k]𝐞_η(m)}x_k_Useful confidential message+ρ√(P_S)β_2∑_m=1^N_Ta^*(θ,R)(m)w̃(m)𝐞_η(m)_Artifical noise+𝐧_k_Channel noise.In the case of symbol-level precoding, making a summation of all N subchannels yieldsz_k(θ,R)=∑_n=1^N𝐲_k(θ,R)(n)=ρ√(P_S)β_1{∑_m=1^N_Ta^*(θ,R)(m)v_k(m)}x_k_Useful confidential message u_k(θ,R)+ρ√(P_S)β_2∑_m=1^N_Ta^*(θ,R)(m)𝐓_m𝐰_k_Artifical noise w̅_k(θ,R)+∑_n=1^N𝐧_k(n)_Channel noise n̅_k,where the final projection vector 𝐓_m is the mth row of the AN projection matrix 𝐓. The useful confidential signal u_k(θ,R), AN w̅_k(θ,R), and channel noise n̅_k are assumed to be random variables with zero mean. Now, we define the average receive SINR asSINR(θ,R) =𝔼{u_k(θ,R)u^*_k(θ,R)}/𝔼{w̅_k(θ,R)w̅^*_k(θ,R)}+𝔼{n̅_kn̅_k^*}=ρ^2P_Sβ_1^2𝐚^H(θ,R)𝐯_k^2/ρ^2P_Sβ_2^2𝐚^H(θ,R)𝐓𝐰_k^2+Nσ^2_n,which is also proved to be valid for 𝔼{w̅_k(θ,R)w̅^*_k(θ,R)}≈ 𝔼{n̅_kn̅_k^*} in Appendix A.In the block-level precoding case, assume the receiver at position (θ,R)is so smart that it can identify the set of the N_T active subcarrier indices by energy detection algorithm similar to cognitive radio <cit.>, the correspondingN_T active subchannels are summed togetherz_k(θ,R) =ρ√(P_S)β_1{∑_m=1^N_Ta^*(θ,R)(m)v_k(m)}x_k_Useful confidential message +ρ√(P_S)β_2∑_m=1^N_Ta^*(θ,R)(m)𝐓_m𝐰_k_Artifical noise+∑_n=1^N_T𝐧_k(n)_Channel noise.According to the above model and similar to Appendix A, the average SINR is given bySINR(θ,R)=ρ^2P_Sβ_1^2𝐚^H(θ,R)𝐯_k^2/ρ^2P_Sβ_2^2𝐚^H(θ,R)𝐓𝐰_k^2+N_Tσ^2_n.§.§ Analysis of receive SINR performance in the case of phase alignment and null-space projectionIn particular, when the ideal desired direction angle and distance are available, we use a simple beamforming form𝐯_k=1/√(N_T)𝐚(θ_D,R_D),which provides a maximum coherent combining at desired receiver, and𝐚^H(θ_D,R_D)𝐓=0_1× N_T,which projects the AN onto the null-space of the desired steering vector 𝐚(θ_D,R_D). For simplification, 𝐓_m is taken to be the mth row of the following projection matrix𝐓=𝐈_N_T-1/N_T𝐚(θ_D,R_D)𝐚^H(θ_D,R_D).By using the above beamforming scheme, the desired receive coherent combining signal isz_k(θ_D,R_D)=ρ√(P_SN_T)β_1x_k_Useful confidential message+∑_n=1^N_T𝐧_k(n)_Channel noise,Due to phase alignment at transmitter, that is, constant-envelop beamformer with all elements having the same magnitude, we have the following average SINRSINR(θ_D,R_D)=ρ^2P_Sβ_1^2N_T/N_Tσ^2_n=ρ^2P_Sβ_1^2/σ^2_n,since the AN projection matrix at desired receive position is perpendicular to the desired steering vector. At the eavesdroppers, the receive combining signal isz_k(θ_E,R_E)=ρ√(P_S/N_T)β_1𝐚^H(θ_E,R_E)𝐚(θ_D,R_D)x_k_Useful confidential message+ρ√(P_S)β_2𝐚^H(θ_E,R_E)𝐓𝐰_k_Artifical noise+∑_n=1^N_T𝐧_k(n)_Channel noise.The SINR expression at undesired position (θ_E,R_E) reduces toSINR(θ_E,R_E) =ρ^2P_Sβ_1^2𝐚^H(θ_E,R_E)𝐯_k^2/ρ^2P_Sβ_2^2𝐚^H(θ_E,R_E)𝐓𝐰_k^2+N_Tσ^2_n.In a practical system, it is preferred that the SINR at desired position is maximized given that the SINR at eavesdropper position SINR(θ_E,R_E) is lower than the predefined value like 0dB. §.§ Analysis of average secrecy rateWith the help of the definition of average SINR in the previous subsection, the average secrecy rate of the system is written asSR=log_2(1+SINR(θ_D,R_D)) -max_(θ_E,R_E)∈wiretap area{log_2(1+SINR(θ_E,R_E))}subject to: wiretap area= {(θ_E,R_E)||R_E-R_D|≥Δ R, |θ_E-θ_D|≥Δθ},where Δ R and Δθ can refer to the first null position around the main beam of the desired receiver, which is defined in Appendix B. The constraint in the above definition denotes the wiretap region where its complementary set is the desired region. The above definition of average secrecy rate means the achievable rate at the desired receiver minus the maximum rate achievable by eavesdroppers within the wiretap region. In Appendix B, we derive a tight upper bound on (<ref>)SINR(θ_E,R_E) =ρ^2P_Sβ_1^2𝐚^H(θ_E,R_E)𝐯_k^2/ρ^2P_Sβ_2^2𝐚^H(θ_E,R_E)𝐓𝐰_k^2+N_Tσ_n^2 =ρ^2P_Sβ_1^2𝐚^H(θ_E,R_E)𝐚(θ_D,R_D)^2/N_Tρ^2P_Sβ_2^2𝐚^H(θ_E,R_E)𝐓𝐰_k^2+N_T^2σ_n^2.as followsSINR(θ_E,R_E) ≤max{SINR(λ_max1), SINR(λ_max2)},which is achievable and a tight upper bound, where parameters λ_max1 andλ_max2 are given byλ_max1= 1/N_T∑_n=1^N_Te^-j2π(Δ f_η(n)R_E_sidelobe-R_D/c)·1/N_T∑_n=1^N_Te^j2π(Δ f_η(n)R_E_sidelobe-R_D/c), λ_max2= 1/N_T∑_n=1^N_Te^j2π(1/2(cosθ_E_sidelobe-cosθ_D)(n-1))·1/N_T∑_n=1^N_Te^-j2π(1/2(cosθ_E_sidelobe-cosθ_D)(n-1)),where θ_E_sidelobe and R_E_sidelobeare defined in Appendix B. Its detailed proving process is presented in Appendix B. Evidently, by adjusting the values of parameters β_1 and β_2, we may make the SINR value low enough to be unable to be correctly detected by eavesdropper.Theorem 1: In a secure precise communication shown in Fig. 1, if the phase-alignment precoder for confidential messages in (<ref>)is adopted and the AN null-space projection method in (<ref>) is used, the average secrecy rateis approximated by the following formulaSR=log_2(1+SINR(θ_D,R_D)) -log_2(1+max{SINR(λ_max1), SINR(λ_max2)}),where the first term in the right hand side of the above equation denotes the rate achieved by desired user and the second one is theupper bound of rate achievable by eavesdropper. ▪Proof: The proof of Theorem 1 is straightforward by combining (<ref>), and (<ref>).▪§ SIMULATION RESULTS AND ANALYSISIn what follows, the proposed scheme RSCS-OFDM-DM is evaluated in terms of the averageSINR and secrecy rate performance. In our simulation, system parameters are chosen as follows: the carrier frequency f_c=3GHz, the total signal bandwidth B changes from 5MHz to 100MHz, the number of total subcarriers N=1024, P_S/σ_n^2=10dB,the antenna element spacing is half of the wavelength (i.e., d=c/2f_c), the position of the desired receiver is chosen to be (60^∘,500m).Fig. <ref> illustrates the 3-D performance surface of SINR versus direction angle θ and distance R of the proposedmethod of phase alignment/beamforming plus null-space projection in Section III for three different bandwidths: 5MHz, 20MHz, and 100MHz. Observing three parts in Fig. 3,the receive power forms a high useful signal energy peak only around the desired receiver position (60^∘,500m) due to the joint use of AN projection and PA. Otherwise, in other undesired region of excluding the main useful power peak, the receive SINR is so low that it is smaller than one, due to a weak receive confidential signal corrupted by AN.Additionally, we also find an very interesting fact that the width of useful main peak along distance dimension decreases accordingly as the signal bandwidth grows from 5MHz to 100MHz.In summary, a larger signal bandwidthgenerates a narrower confidential power peak along distance dimension but has no influence on the width of peak along direction dimension. This means that a larger security along distance dimension can be offered by using a larger signal bandwidth. Fig. <ref> draws the 3-D performance surfaceof SINR versus direction angle and distance of the proposed method for three different numbers of transmit antennas: 8, 32 and 128. Via three parts (a), (b), and (c) inFig. 4, we find the following fact that only single energy peakis synthesized around the desired receiver position (60^∘,500m) similar to Fig. <ref> by beamforming and AN projection. Furthermore, with the increasing in the number of transmit antennas, the width of the peak along direction dimension becomes narrower and narrower. This can be readily explained. Given a fixed antenna spacing, increasing the number of transmit antenna array will increase antenna array size and provide a high-spatial-angle-resolution, i.e., a narrower peak along direction dimension.As the number of transmit antennas tends to large-scale, the peak converges to an extremely narrow peak along direction dimension and perpendicular to the plane along direction dimension. If bothlarge signal bandwidth B and large-scale transmit antenna array are used, thepeak is formed to narrowalong both direction and distance dimensions. Then, the ultimate aim of the true secure precise transmission is achieved in terms of the outcomes in Fig. 3 and Fig. 4. That is, the peak converges to a line perpendicular to the 2-D plane of direction and distance dimensions and at the desired position. In Fig. 5, we will evaluate the impact of power allocation parameters β_1 and β_2 on the SINR performance. Three different typicalscenarios are considered as follows: (1) β_1<β_2 with β_1^2=0.1 and β_2^2=0.9, (2) β_1=β_2, with β_1^2=β_2^2=0.5, and (3) β_1>β_2 with β_1^2=0.9 and β_2^2=0.1, which corresponds to three different parts in Fig. 5. From the three parts, the maximum value of SINR peak decreases as the value of β_1 decreases. The result is obvious intuitively. Also, the SINR value outside the main peak approaches zero. Thus, increasing the value of parameter β_2 will further degrade the SINR performance of eavesdropper regions. Conversely, increasing the value of parameter β_1 will improve the SINR performance of desired receiver but also lower the security. Additionally, the values of β_1 and β_1 are also intimately related to energy efficiency. Thus, how to choose the values of β_1 and β_1 depends on application scenarios and should consider a balance among energy efficiency, security and performance.Fig. 6 demonstrates the curves of derived theoretical and numerical average secrecy rates versus SNR for three different numbers N_T of transmit antennas.Parts (a), (b), and (c) correspond to N_T=8, 32, 128, respectively, where each part includes two different power allocation strategies. In this figure, the derived theoretical curve is given by (<ref>) while the simulated numerical curve is the exact SR values by sufficient Monte Carlo simulation runs. Observing the three parts, the gap of the simulated curve and the theoretical one disappears as the number of transmit antennas increases from 8 to 128. Thus, the derived theoretical SR expression achieves a good approximation to the exact SR. This also verifies the validness of our expression in (<ref>).Additionally,in the low and medium SNR regions, the theoretical curve is also very close tothe numerical one such that they overlap together for three parts.Below, we make an investigation concerning the impact of power allocation parameters β_1 and β_2on the secrecy rate performance in Fig. 7. Parts (a), (b), and (c) illustrate the secrecy rate versus β_1 in three different SNR scenarios: (1) SNR=0dB, (2) SNR=10dB, and (3) SNR=20dB, respectively. Observing the three parts, we first find that the gap between AN-aided method and non-AN-aided method decreases as the number of transmit antennas increases, where the non-AN-aided implies β_2=0 and no AN. In Part (a), that is, in low SNR region and with large number of antennas, the influence of AN is weak due to a large channel noise. However, as SNR increases , the AN becomes more and more important.In Parts (b) and (c),as the value of β_1 varies from 0 to 1, the average SR curveis a concave function of β_1. There exists an optimal power allocation strategy of maximizing the SR.As SNR increases, the optimal power allocation factor β_1 decreases. A high SNR means a good desired channel quality. In other words, for a high SNR scenario,more power may be used to transmit artificial noise(AN) to corrupt the eavesdroppers while less power is adopted to transmit confidential message.§ CONCLUSIONIn this paper, the RSCS-OFDM-DM scheme is proposed to achieve an ultimate goal of secure precise transmission of confidential messages. To accomplish this goal, several important tools are utilized like: random subcarrier selection, phase alignment/beamforming for confidential messages, and null-space projection of AN. Using this scheme, we obtain the following interesting results: 1) the proposed scheme can generate a high SINR peak only at the desired position (θ_D,R_D) and a low flat SINR plane for other eavesdropper regions with SINR being far less than the former, 2) the widths of main peak along angle and distance dimensions are intimately related tothe number of antennas N_T andbandwidth B, respectively. Increasing B and N_T will achieve a narrower and narrower main peak along both direction and distance dimensions at desired position, 3) increasing the fraction of AN power in the total transmit power will gradually degrade the SINR performance of eavesdropper at the expense of that of desired user. The final result is as follows: at eavesdroppers, the received power of confidential messages is very weak and corrupted by AN, thus their receive SINR is so poor that the confidential messages can not be successfully and reliably intercepted outside main peak at the desired position, and the transmit confidential power mainly gathers in a small neighbourhood around the desired position, where only a small fraction of the total power leaks out to eavesdropper regions. The proposed RSCS-OFDM-DM scheme is easy to be implemented compared to the conventional RFDA proposed in <cit.>. Finally, we also derive the tight upper bound for SINR in wiretap region. By using this bound, we attain the approximate expression foraverage secrecy rate. Simulation results and analysis confirm that the derived theoretical expression of average secrecy rate is a good approximation to the exact average SR. Due to the low-complexity and high security of the proposed scheme, it will be potentially applied in the future scenarios including unmanned aerial vehicle communications, satellite communications, mm-Wave communications and so on.§ APPROXIMATE DERIVATION OF MEAN OF SINR(Θ,R) FOR Σ^2_CN/2≈Σ^2_AN/2In the position (θ,R), consideru_k(θ,R)=ρ√(P_S)β_1{∑_m=1^N_Ta^*(θ,R)(m)v_k(m)}x_k,andg_k(θ,R)=w̅_k(θ,R)+n̅_k=ρ√(P_S)β_2∑_m=1^N_Ta^*(θ,R)(m)𝐓_m𝐰_k_g_k,AN(θ,R)+∑_n=1^N_T𝐧_k(n)_g_k,CN(θ,R).where channel noise is abbreviated as CN. Firstly, let us address the problem of PDF of random variable u_k(θ,R). Three random variables u_k(θ,R), g_k,AN(θ,R), and g_k,CN(θ,R) are the sum of N_T random variables. In order to simplify the following deriving, they are approximately modelled as complex Gaussian distributions in terms of the central limit theorem in probability theory. From(<ref>) and (<ref>), their means and variances are𝔼{u_k(θ,R)}=0, 𝔼{g_k,AN(θ,R)}=0, 𝔼{g_k,CN(θ,R)}=0,andσ_u^2 =𝔼{u_k(θ,R)u_k^*(θ,R)}=ρ^2P_Sβ_1^2𝐚^H(θ,R)𝐯_k^2, σ^2_AN=N_TP_Sρ^2β_2^2σ^2_w∑_m=1^N_Ta^*(θ,R)(m)^2𝐓_m𝐓^H_m,σ_CN^2=N_Tσ^2_n.Considering the real-time SINR γ is approximately given byγ=u_k(θ,R)u^*_k(θ,R)/g_k,AN(θ,R)g_k,AN^*(θ,R)+g_k,CNg_k,CN^*.In accordance with the above definition, let define the three new random variables as followsα=u_k(θ,R)u_k^*(θ,R)/σ_u^2/2,β=g_k,AN(θ,R)g_k,AN^*(θ,R)/σ_AN^2/2,λ=g_k,CNg_k,CN^*/σ_CN^2/2,which are three Chi-squared distribution with two degrees of freedom. Without losing generality, putting the above variables in (<ref>) yieldsγ=σ_u^2/2α/σ_AN^2/2β+σ_CN^2/2λ=e∙α/aβ+bλ,where e=σ^2_u/2, a=σ^2_AN/2, b=σ^2_CN/2, three random variables α, β and λ have the same PDF as follows <cit.>f_α(x)=f_β(x)=f_λ(x)=1/2exp(-1/2x).Below, we discuss one typical situation: σ^2_CN/2≈σ^2_AN/2. In this scenario, we will give an approximation of the PDF of random variable γ. The distribution of aβ+bλ/(a+b) is approximately regarded as one Chi-squared distribution with four degrees of freedom. Then, the SINR in (<ref>) is rewritten byγ=σ^2_u/2α/σ^2_AN/2β+σ^2_CN/2λ=eα/aβ+bλ=e/2(a+b)∙α/2/q/4_γ̃,From the definition of F-distribution, we have the PDF of γ̃ as followsf_γ̃(x)=1/(1+1/2x)^3,which leads to the PDF of random variable γf_γ(x)=2(a+b)/e1/(1+(a+b)/ex)^3, x>0which yields the average value of SINR γγ̅=𝔼{γ}=2(a+b)/e∫^+∞_0x/(1+(a+b)/ex)^3dx.Let us define a new integral variable x'=(1+(a+b)/ex), then the above integral can be converted intoγ̅=2e/a+b∫^+∞_1x'-1/(x')^3dx'=e/a+b.by checking the integral table in <cit.>. This completes the proof of the PDF of SINR and its mean. ▪§ TIGHT UPPER BOUND OF SINR(Θ_E,R_E) In this Appendix, a tight upper bound for the receive SINR in eavesdropper region is proved. By means of the SINR definition in (<ref>),the SINR SINR(θ_E,R_E) is rewritten asSINR(θ_E,R_E)=ρ^2P_Sβ_1^2𝐚^H(θ_E,R_E)1/√(N_T)𝐚(θ_D,R_D)^2/ρ^2P_Sβ_2^2𝐚^H(θ_E,R_E)𝐓𝐰_k^2+N_Tσ_n^2 =ρ^2P_Sβ_1^2/N_T𝐚^H(θ_E,R_E)𝐚(θ_D,R_D)^2/ρ^2P_Sβ_2^2𝐚^H(θ_E,R_E)𝐓𝐰_k^2+N_Tσ_n^2,where𝐚^H(θ_E,R_E)𝐚(θ_D,R_D)=∑_n=1^N_Te^-j2πΔθ_n, 𝐚^H(θ_D,R_D)𝐚(θ_E,R_E)=∑_n=1^N_Te^j2πΔθ_n,withΔθ_n=Δ f_η(n)R_E-R_D/c-1/2(cosθ_E-cosθ_D)(n-1),and𝐚^H(θ_E,R_E)𝐓𝐰_k^2=𝐚^H(θ_E,R_E)𝐓𝐚(θ_E,R_E)=N_T-1/N_T𝐚^H(θ_E,R_E)𝐚(θ_D.R_D)𝐚^H(θ_D,R_D)𝐚(θ_E,R_E).Let us defineλ=1/N_T∑_n=1^N_Te^-j2πΔθ_n·1/N_T∑_n=1^N_Te^j2πΔθ_n.which ensures that 0≤λ<1. Then,SINR(θ_E,R_E)=μ_1λ/μ_2(1-λ)+μ_3,where μ_1=ρ^2P_Sβ_1^2, μ_2=ρ^2P_Sβ_2^2 and μ_3=σ_n^2. Obviously, the above function SINR(θ_E,R_E) is an increasing function of independent variable λ for given fixed μ_1=ρ^2P_Sβ_1^2, μ_2=ρ^2P_Sβ_2^2 and μ_3=σ_n^2. As λ varies from 0 to 1, therange ofthis function is[μ_1λ/μ_2+μ_3, μ_1λ/μ_3].where λ=1 implies that the receiver locates at the desired position.In what follows, we will determine the maximum value of λwhen the eavesdropper lies in wiretap region. It is certain that the maximum value of λ is smaller than one since the eavesdropper is assumed to be outside the main-beam around the desired position (θ_D,R_D). Here, we define the wiretap region as all region outside main-peak of SINR around the desired position (θ_D,R_D). Now, to make the wiretap region approximately clear, we should compute the first nulls of SINR along phase and distance dimensions. In such positions, λ=0. In other words, the lowest value of SINR is reached.The null points of array pattern along the direction dimension satisfy the following condition <cit.>∑_n=1^N_Te^-j2πΔθ_n=∑_n=1^N_Te^j2π1/2(cosθ_E-cosθ_D)(n-1) =1-e^jπ(cosθ_E-cosθ_D)N_T/1-e^jπ(cosθ_E-cosθ_D)=0,which yieldsN_Tπ(cosθ_E-cosθ_D)=± 2Kπ, K≠ mN_T (m=1,2,3,...).In the same manner, the null points along the distance dimension are2π N_TB/N_TR_E-R_D/c=±2Kπ, K≠ mN_T (m=1,2,3,...).When K=1, we will get the first-null positionθ_E_zero=arccos(cosθ_D±2/N_T),R_E_zero=R_D±c/B.Then, Δθ=|θ_E_zero-θ_D| and Δ R=|R_E_zero-R_D|. While λ=λ_max, we can get the maximum value of SINR(θ_E,R_E) in the eavesdropper region. Hence, once the eavesdropper located at the peak position of the first side lobe, λ will achieve the maximum value. The maximum value of the sidelobe falls between the first andsecond nulls. As for the direction dimension, the maximum values of sidelobes are approximately equivalent tosin(π(cosθ_E-cosθ_D)N_T/2)=1.which yieldsπ(cosθ_E-cosθ_D)N_T/2=±(2K+1)π/2, (K=1,2,3...).Similar to the above analysis,thecondition associated with the distance dimension satisfies2πB/N_TR_E-R_D/cN_T/2=±(2K+1)π/2, (K=1,2,3...).When K=1, we will get two maximum values corresponding to two first sidelobes along phase and distance dimensions as followsθ_E_sidelobe=arccos(cosθ_D±3/N_T),R_E_sidelobe=R_D±3c/2B,which yieldsλ_max1= 1/N_T∑_n=1^N_Te^-j2π(Δ f_η(n)R_E_sidelobe-R_D/c)·1/N_T∑_n=1^N_Te^j2π(Δ f_η(n)R_E_sidelobe-R_D/c), λ_max2= 1/N_T∑_n=1^N_Te^j2π(1/2(cosθ_E_sidelobe-cosθ_D)(n-1))·1/N_T∑_n=1^N_Te^-j2π(1/2(cosθ_E_sidelobe-cosθ_D)(n-1)).Hence,SINR(θ_E,R_E) ≤max{SINR(λ_max1), SINR(λ_max2)}.This completes the proof of the tight upper bound. ▪ IEEEtran | http://arxiv.org/abs/1704.07996v2 | {
"authors": [
"Feng Shu",
"Xiaomin Wu",
"Jinsong Hu",
"Riqing Chen",
"Jiangzhou Wang"
],
"categories": [
"cs.IT",
"math.IT"
],
"primary_category": "cs.IT",
"published": "20170426074343",
"title": "Secure Precise Wireless Transmission with Random-Subcarrier-Selection-based Directional Modulation Transmit Antenna Array"
} |
[ 1/2rCenter for Cosmology and Particle Physics, Department of Physics, New York University, New York, NY, 10003, [email protected] the metric formalism, we study the derivative mixings of spin-2 fields in massive bi-Gravity. Necessary (but not sufficient) criteria are given for such mixings to be ghost free. Examples satisfying those criteria are studied and it is shown that in the decoupling limit they host a ghost.Possible Derivative Interactions in Massive bi-Gravity Giorgi Tukhashvili Received 1 March 2017 /Accepted 21 April 2017 ======================================================§ INTRODUCTION In 1940, N. Rosen published a pair of papers <cit.> in where he proposed a view on gravitation different from Einstein's General Theory of Relativity. Per Rosen's interpretation, the tensor, g_μν, which describes the gravitational field, has no connection with geometry and it propagates on a flat background described by the tensor g̃^0_μν. By flat we mean that the corresponding Riemann tensor vanishes, R^λ_μνρ (g̃^0)=0[The most general parametrization of such a metric can be done as g̃^0_μν = ∂_μΦ^a ∂_νΦ^b η_a b. Where η_a b is the Minkowski metric.]. One should note that both objects, g_μν and g̃^0_μν, are tensors with respect to the same General Coordinate Transformations (diff). It turns out that such a formalism has some advantages and the predictions, in general, coincide with Einstein's General Theory of Relativity (GR). This model is defined by the following action:S = ∫ d^4 x√(g)g^μν( Δ^ρ_νλΔ^λ_μρ - Δ^ρ_λρΔ^λ_μν). Δ^λ_μν≡Γ^λ_μν (g) - Γ^λ_μν (g̃^0).This was the very first attempt to create a bi-metric theory of gravity. Flatness of g̃^0_μν is a crucial part of this theory, since promoting g̃^0_μν to a dynamical field would generate a ghost even at the linear level. The study of the interacting spin-2 fields dates back to 1958, when H. A. Buchdahl <cit.> studied the interaction between the gravity and higher spin (>3/2) fields. He argued that these interactions were strongly constrained and precluded any interesting solutions. C. Aragone and S. Deser <cit.> showed that the inconsistency was caused by the broken diff invariance. The result was generalized by N. Boulanger et. al.<cit.> who showed that, “in the massless case, there is no ghost free coupling, with at most two derivatives of the fields, that can mix different spin-2 fields”. In other words, the most general action for massless spin-2 fields is a sum of Einstein-Hilbert terms:S = ∑_n1/2 M^2_n∫ d^4 x √(-g^[n]) R^[n].Interactions through the mass term were introduced by C. J. Isham, A. Salam, and J. Strathdee (ISS) <cit.>, who developed the first theory of massive bi-Gravity by analogy with the Vector Meson Dominance model <cit.>. By then the correct form of the mass term was not known, so the ISS model hosts the Boulware-Deser (BD) ghost <cit.>. After the discovery of the ghost free massive gravity <cit.>, the ISS model was reformulated in <cit.> and it was shown to be ghost free. As we already mentioned, in the massless case, the presence of the ghost in the mixing terms is due to the broken diffeomorphism invariance. For the massive case, in order to make this invariance explicit, one needs to introduce the Stückelberg fields in the de Rham-Gabadadze-Tolley (dRGT) <cit.> potential. We think that these new degrees of freedom might help to avoid the ghost coming from the derivative interactions (≡ kinetic mixings) as well. There were no successful attempts to construct such terms, both in massive gravity and massive bi-Gravity <cit.>.[See <cit.> for a potential loophole.] Within this paper, we will work out the necessary (but not sufficient) criteria for the kinetic mixings to be ghost free. Then we will build terms that satisfy those criteria and we'll try to study them using the following examples: M_h M_f/2√(-g) g^μνR̃_μν+βM_h M_f/2√(-g̃)g̃^μν R_μν.We will show that in the Λ = ( m^2 M_p )^1/3 decoupling limit, (<ref>) generates both strongly coupled terms and a ghost. In general, conventions through the paper coincide with those of <cit.>. To denote the strong coupling scale, we will use Λ = ( m^2 M_p )^1/3 instead of usual convention Λ_3. The objects (Christoffel symbol, Ricci tensor, etc.) with tilde are defined with respect to (wrt) the metric g̃_μν, while those without tilde wrt g_μν. Minkowski metric is mostly minus.§ EXTENDED BI-GRAVITY The first theory of massive Bi-Gravity, the f Dominance or Tensor Meson Dominance was introduced by C. J. Isham, A. Salam, and J. Strathdee <cit.>. The idea is based on the Vector Meson Dominance (VMD) model <cit.>. Within the VMD model the electromagnetic field couples directly to leptons and through a vector meson to hadrons (Fig. <ref>). The VMD model successfully managed to qualitatively explain the hadronic form factors.In a complete analogy with VMD, C. J. Isham et. al.<cit.> postulated the existence of a new spin-2 massive particle, g̃_μν, that was coupled to the hadrons, while the graviton, g_μν, was coupled to leptons (Fig. <ref>). On the Lagrangian level this theory can be expressed as:ℒ_ISS = - 1/2 M^2_h√(-g) g^μν R_μν + ℒ( g_μν , leptons) -- 1/2 M^2_f√(-g̃)g̃^μνR̃_μν + ℒ( g̃_μν , hadrons) + ℒ_mass( g_μν , g̃_μν).The mixing of the fields happens through the mass term, ℒ_mass, without which the two worlds do not communicate with each other. Choosing this term properly is a crucial step in the model. The original choice for ℒ_mass by C. J. Isham et. al. exhibits the BD ghost. Recently a ghost free form of ℒ_mass was found by C. de Rham, G. Gabadadze and A. J. Tolley <cit.> and this opened new frontiers for massive bi-Gravity. Using the new mass term, to which we will refer as ℒ_dRGT, the ISS model was reformulated in <cit.> and it was shown to be ghost free. The spectrum of this model consists of one massive and one massless spin-2 fields i.e. 7 healthy degrees of freedom (DoF). After including the dRGT mass term we need new DoF to recover the full diffeomorphism invariance, diff ⊗diff, one might expect that these new DoF could help to avoid the ghost coming from the kinetic mixings as well. The following criteria need to be satisfied for the kinetic mixings to be ghost free (h and f are the fluctuations of g and g̃ around Minkowski background respectively): *On the linear level generate (∂ h)^2, (∂ f)^2 and ∂ h ∂ f.*Have at least one copy of diff invariance nonlinearly.*On the linear level have two copies of diff invariance. These criteria are necessary but not sufficient for the ghost to be absent. Let's discuss them in more detail. Criterion <ref> is obvious since we are building the kinetic mixings. <ref> assures that we get a scalar under the diagonal subgroup, diag(diff ⊗diff), and we can use the same Stückelberg fields to recover the two copies of diffeomorphism invariance, both, in the mass term and in the kinetic mixings. In other words: <ref> helps us to avoid new DoF. Criterion <ref> guarantees that, up to a field redefinition, on the linear level we get Einstein-Hilbert action, which is unique to avoid Ostrogradski instability.It is easy to find the building blocks of this model. The object that transforms covariantly under diff and generates terms like (∂ h)^2, (∂ f)^2 on the linear level is the Riemann tensor. In addition to the Riemann tensor we have Δ^ρ_μν = Γ^ρ_μν - Γ̃^ρ_μν which transforms covariantly under diag(diff ⊗diff), so terms like ∂Δ and ΔΔ are also allowed.[Following relation holds: √(-g) g^μν R_μν= ∂_μ( )^μ + √(-g) g^μνR̃_μν + √(-g) g^μν( Δ^ρ_νλΔ^λ_μρ - Δ^ρ_λρΔ^λ_μν). This is a general version of the formula given in <cit.>.] All we need is to build scalars out of these tensors. There exist infinitely many terms that satisfy these criteria. Here are some of them: √(-g) g^μνR̃_μν,√(-g̃)g̃^μν R_μν, √(-g̃) g^μν R_μν + √(-g)g̃^μν R_μν, √(-g̃)g̃^αμg̃^βν R_αβμν - √(-g)g̃^αμg̃^βν R_αβμν+ √(-g)g̃^μν R_μν, √(-g)g̃^μα_1 g_α_1 β_1g̃^β_1 α_2⋯ g_α_n β_ng̃^β_n ν R_μν - (n+1) √(-g)g̃^μν R_μν, ⋯ The particular combinations in (<ref>),(<ref>),(<ref>) are chosen in order to satisfy criterion <ref>.[On the linear level √(-g) g^μνR̃_μν= - h^μνG̃^(1)_μν + η^μνR̃^(2)_μν, which obviously has two copies of diff invariance due to the Bianchi identity (G̃^(1)_μν and R̃^(2)_μν are the linearized Einstein tensor and quadratic Ricci tensor respectively). Same is true for the other terms.] Because these terms are healthy at the linear level and it's possible to recover the nonlinear diff invariance without introducing new DoF, we think that it is worthy to study their nonlinear effects as well. We will try to study the new terms by considering the example of (<ref>). From now on, our starting point will be the following action:S_EbG = ∫ d^4 x [ M_h^2/2√(-g) g^μν R_μν + (α - β ) M_h M_f/2√(-g) g^μνR̃_μν + . + βM_h M_f/2√(-g̃)g̃^μν R_μν + γM_f^2/2√(-g̃)g̃^μνR̃_μν + . + m^2 M_h^2/4√(-g)( ℒ_2 [𝒦] + α_3 ℒ_3 [𝒦] + α_4 ℒ_4 [𝒦] ) + ℒ( g_μν, matter fields) ],where the term in the parenthesis in the third line is the usual dRGT mass term with 𝒦^α_β = δ^α_β - √(g^αλg̃_λβ) and for simplicity only one metric is coupled to matter. We will refer to the model defined by (<ref>) as Extended Bi-Gravity (EbG) and assume that α≠β≠ 0. To the best of our knowledge this model has not been studied before.[In terms of Tetrads ∫ d^4 x √(-g̃)g̃^μν R_μν = 1/2∫ε_a b c d f_α^a f^βb e_a^α e_β b R^a b∧ f^c∧ f^d and ∫ d^4 x √(-g) g^μνR̃_μν = 1/2∫ε_a b c d e_α^a e^β b f_a^α f_βbR̃^a b∧ e^c ∧ e^d (e and f are tetrads corresponding to metrics g and g̃ respectively) which are different from those discussed in <cit.>: ∫ε_a b c d R^a b∧( e^c ∧ f^d + β f^c ∧ f^d ). Although they break the diff ⊗diff to the diagonal, the new terms preserve the two copies of Local Lorentz Transformations. ] §.§ The Linear Theory Now we'll investigate the linear limit of (<ref>) and we'll show that in this limit there are one massive and one massless spin-2 modes, i.e. seven healthy degrees of freedom. Let's expand the fields around the Minkowski background, with h_μν and f_μν being the fluctuations of g_μν and g̃_μν respectively. Up to quadratic order in fields (<ref>) takes the form:𝒮_EbG = ∫ d^4 x[ - 1/4( [ h^μν; f^μν ])^T ( [1 - β/ κ α; α γ - ακ + βκ ]) ℰ^αβ_μν( [ h_αβ; f_αβ ]) - . . - m^2/4( [ h^μν; f^μν ])^T ( [ 1-κ;-κ κ^2 ]) ℱ^αβ_μν( [ h_αβ; f_αβ ]) + 1/2 M_h h_μν T^μν]. ℰ^αβ_μν≡ - 1/2( δ^α_μδ^β_ν∂^2 + η^αβ∂_μ∂_ν + η_μν∂^α∂^β - η_μνη^αβ∂^2 - δ^α_ν∂^β∂_μ - δ^β_μ∂^α∂_ν), ℱ^αβ_μν≡1/2( δ_μ^αδ_ν^β - η_μνη^αβ).In order for both kinetic terms to have the correct sign, we need to constrain the parameters (κ≡M_h/M_f>0): κ > β, γ > κ( α - β + α^2/κ - β).After diagonalizing kinetic and mass terms and canonically normalizing the fields, only one mode turns out to be massive. For the final action, we get:S = ∫ d^4 x [ - 1/4𝔞^μνℰ^αβ_μν𝔞_αβ - m_^2/4𝔞^μνℱ^αβ_μν𝔞_αβ - . . - 1/4𝔟^μνℰ^αβ_μν𝔟_αβ + t_a/ 2 M_h 𝔞_μν T^μν + t_b/ 2 M_h 𝔟_μν T^μν]. μ^2 ≡κ/κ - β·γ + κ(α +κ ) /γ -κ(α -β + α ^2/κ -β).Here we defined the effective mass as m_^2 =μ^2 m^2. The parameters t_a/b are functions of α,β,γ and κ. Considering the constraints in (<ref>), it's trivial to show that m_^2 > 0. The EoM's corresponding to (<ref>) are: ℰ^αβ_μν𝔞_αβ + m_eff^2 ℱ^αβ_μν𝔞_αβ = t_a/M_h T_μν,ℰ^αβ_μν𝔟_αβ = t_b/ M_h T_μν.Assuming that the source T^μν is conserved and imposing the linear de Donder gauge for the 𝔟_μν, (<ref>) reduce to:(- m_eff^2 ) 𝔞_μν =- 2 t_a/ M_h( T_μν - 1/3 T η_μν + 1/3 m_eff^2∂_μ∂_ν T ),𝔟_μν = - 2 t_b/ M_h( T_μν - 1/2 T η_μν).For the exchange amplitude between two sources (Fig. <ref>) we get the sum:A^tot =- 2 | t_a |/ M_h∫ d^4 x T̃^μν1/ - m_eff^2 ( T_μν - 1/3 T η_μν + 1/3 m_eff^2∂_μ∂_ν T ) - 2 | t_b |/ M_h∫ d^4 x T̃^μν1/( T_μν - 1/2 T η_μν).§.§ Decoupling Limit The full Λ = (m^2 M_h)^1/3 decoupling limit (dl) of (<ref>), with parameters α=β=0 and γ=1, was studied in <cit.>. Before I move to general case I'll try to briefly review their results. Neglecting the vector modes the dl bi-Gravity action is:S_bG =∫ d^4 x [ - 1/4 h^μνℰ_μν^αβ h_αβ- 1/4 f^μνℰ_μν^αβ f_αβ + .+ 1/8 h^μν( 2 X^(1)_μν - 1/Λ^3 (2 + 3 α_3) X^(2)_μν + 1/Λ^6 ( α_3 + 4 α_4 ) X^(3)_μν) + . κ/8 f^μν( 2 Y^(1)_μν + 1/Λ^3 (4 + 3 α_3) Y^(2)_μν + 1/Λ^6 ( 2 + 3 α_3 + 4 α_4 ) Y^(3)_μν) ].X^μν_(n)≡ -1/(3-n)!ε^μμ_1 …μ_n σ_1 …σ_3-nε^νν_1 …ν_n_σ_1 …σ_3-n∂_μ_1∂_ν_1π⋯∂_μ_n∂_ν_nπ.Here Y^(i)_μν are defined in a fashion similar to X^(i)_μν with π→ρ, where ρ(x) is the Galileon field from the point of view of the g̃ metric and is related non-locally to π(x): ρ (x) = - π (x) + 1/2 Λ^3( ∂_μπ (x) )^2 - 1/2 Λ^6∂^μπ (x) ∂^νπ (x) ∂_μ∂_νπ (x) + ⋯ Although this definition involves higher derivatives, because of its nonlocal structure, the Ostrogradski ghost is absent. Making the field redefinitions,h_μν =ĥ_μν + η_μνπ + 1/2 Λ^3( 2 + 3 α_3 ) ∂_μπ∂_νπ, f_μν =f̂_μν + κη_μνρ - κ/2 Λ^3( 4 + 3 α_3 ) ∂_μρ∂_νρ,we partially unmix the tensor modes from the scalars. After unmixing, the bi-Gravity Action becomes:S_bG=∫ d^4 x [ - 1/4ĥ^μνℰ_μν^αβĥ_αβ - 1/4f̂^μνℰ_μν^αβf̂_αβ -. - 1/8 Λ^6( α_3 + 4 α_4 )ĥ^μν X_μν^(3) + κ/8 Λ^6( 2 + 3 α_3 + 4 α_4 )f̂^μν Y_μν^(3) ++ ∑_n=2^5c_n/Λ^3(n-2)ℒ^(n)_gal[ π] +∑_n=2^5. c̃_n/Λ^3(n-2)ℒ^(n)_gal[ ρ] ].We are now going to study the Λ = (m^2 M_h)^1/3 decoupling limit of the theory defined by (<ref>). We start by introducing the Stückelberg fields which recover the full diffeomorphism invariance diff ⊗diff. According to the Stückelberg trick we make the following substitutions (choose to Stückelberguise g̃_μν): g̃_αβ (x) ⟶g̃_a b (Φ) ∂Φ^a/∂ x^α∂Φ^b/∂ x^β.Now the theory is invariant under the two copies of the diffeomorphisms. Note that we attributed the Greek index to diff and the Latin index to diff. The vectors from the diff and diff sectors transform respectively as:A_μ⟶ A'_μ = ∂ x^α/∂ y^μ A_α, Ψ_a ⟶Ψ'_a = ∂Φ^i/∂ Y^aΨ_i.After the Stückelberg substitution, the Lagrangian density for the new kinetic terms takes the form: Δℒ= (α - β ) M_h M_f/2√(-g) g^μν∂Φ^a/∂ x^μ∂Φ^b/∂ x^νR̃_a b + βM_h M_f/2√(-g̃)| ∂Φ/∂ x| g̃^a b∂ x^μ/∂Φ^a∂ x^ν/∂Φ^bR_μν.[This is the only way to recover two copies of diffeomorphism invariance without introducing new fields and without explicitly breaking the Poincaré symmetry.] Next we define the canonically normalized fields <cit.>:g_μν =η_μν + 1/M_h h_μν,g̃_a b =η_a b + 1/M_f f_a b,∂Φ^a/∂ x^μ =∂_μ( x^a + m/Λ^3 B^a +1/Λ^3∂^a π)and take the following limit:M_h → +∞,M_f → +∞,m → 0, Λ→const, κ→const.The parameters α,β,γ are kept constant in the decoupling limit. As we will see these two terms contain parts that become strongly coupled in the (<ref>) limit and host the BD ghost as well. The complete decoupling limit for these terms:M_g M_f√(-g)g^μν∂Φ^a/∂ x^μ∂Φ^b/∂ x^νR̃_a b = M_gη^μνR̃_a b^(10)ϕ_0_μ^a ϕ_0_ν^b + Λ^3/mη^μν( 2 R̃_a b^(10)ϕ_0_μ^a ϕ_1_ν^b . ++ . R̃_a b^(11)ϕ_0_μ^a ϕ_0_ν^b )+ 1/2 h η^μνR̃_a b^(10)ϕ_0_μ^a ϕ_0_ν^b - h^μνR̃_a b^(10)ϕ_0_μ^a ϕ_0_ν^b +κη^μνR̃_a b^(20)ϕ_0_μ^a ϕ_0_ν^b ++ Λ^3 η^μν( R̃_a b^(10)ϕ_1_μ^a ϕ_1_ν^b + 2 R̃_a b^(11)ϕ_1_μ^a ϕ_0_ν^b + R̃_a b^(12)ϕ_0_μ^a ϕ_0_ν^b ),M_g M_f√(-g̃)| ∂Φ/∂ x| g̃^a b∂ x^μ/∂Φ^a∂ x^ν/∂Φ^bR_μν = M_fη^a b R_μν^(1)ψ_0_a b^μν +Λ^3/κmη^a b R_μν^(1)ψ_1_a b^μν + + 1/2 f η^a b R_μν^(1)ψ_0_a b^μν - f^a b R_μν^(1)ψ_0_a b^μν +1/κη^a b R_μν^(2)ψ_0_a b^μν + Λ^3/κη^a b R_μν^(1)ψ_2_a b^μν. In addition to the f_a b, the Ricci tensor R̃_a b = R̃_a b( Φ) also depends on the vector (B^a) and the scalar (π) modes. In fact wrt the scalar mode it's an infinite series. One should also keep in mind that f_a b( Φ) = f_a b( x ) + 1/Λ^3∂^c π∂_cf_a b + 1/2Λ^6∂^c π∂^d π∂_c ∂_df_a b + ⋯ + vector modes. The first upper index on the rhs of (<ref>) and (<ref>) (in parenthesis) corresponds to the order of spin 2 field, while the second upper index indicates the order of the spin 1 field (B^a). Lower indices indicate the order of spin 1 field. Other notations:∂Φ^a/∂ x^μ≡ ϕ_0_μ^a + m ·ϕ_1_μ^a,∂ x^μ/∂Φ^a≡ ϕ̃_0_a^μ + m ·ϕ̃_1_a^μ + m^2 ·ϕ̃_2_a^μ + 𝒪 (m^3),| ∂Φ/∂ x| ≡ ϕ_0 + m ·ϕ_1 + m^2 ·ϕ_2,ψ_0_a b^μν≡ ϕ_0 ϕ̃_0_a^μϕ̃_0_b^ν,ψ_1_a b^μν≡ ϕ_1 ϕ̃_0_a^μϕ̃_0_b^ν +ϕ_0 ϕ̃_0_a^μϕ̃_1_b^ν +ϕ_0 ϕ̃_1_a^μϕ̃_0_b^ν,ψ_2_a b^μν≡ ϕ_0ϕ̃_1_a^μϕ̃_1_b^ν + ϕ_0 ϕ̃_0_a^μϕ̃_2_b^ν+ ϕ_0 ϕ̃_2_a^μϕ̃_0_b^ν+ ϕ_1 ϕ̃_0_a^μϕ̃_1_b^ν + ϕ_1 ϕ̃_1_a^μϕ̃_0_b^ν + ϕ_2 ϕ̃_0_a^μϕ̃_0_b^ν.The highlighted terms in (<ref>) and (<ref>) are not total derivatives and become strongly coupled in the limit (<ref>). This is the first problem encountered within this model. From now on we will forget the existence of strongly coupled terms, since they are irrelevant for the rest of our analysis. §.§ Analysis in the Decoupling LimitLet's try to prove the existence of the ghost. We first unmix the scalar from the spin-2 modes at the quadratic level. Assuming that at the beginning only h_μν is coupled to the source, the linear Lagrangian after the unmixing becomes:ℒ = - 1/4( [ h^μν; f^μν ])^T ( [1 - β/ κ α; α γ - ακ + βκ ]) ℰ^αβ_μν( [ h_αβ; f_αβ ]) -- 3/4μ^2 ( ∂π)^2 + 1/2 M_h h^μν T_μν + a_1/2 M_hπ T.Let's take a static and spherically symmetric source with the following ansatz:T_μν = s^4 θ( r_* - r ) η_μ 0η_ν 0.Classical solutions of this theory:h_0 0 ^in (r) = b_2 s^4/2 M_h( r^2/3 - r_*^2 ), h_0 0 ^out (r) = - b_2 s^4 r_*^3/3 M_h r,f_0 0 ^in (r) = b_1 s^4/2 M_h( r^2/3 - r_*^2 ), f_0 0 ^out (r) = - b_1 s^4 r_*^4/3 M_h r, π^in (r) = a_1 s^4/6 μ^2 M_h( r^2/3 - r_*^2 ),π^out (r) = -a_1 s^4 r_*^3/9 μ^2 M_h r.for which we introduce the following notations:a_1 =γ +κβ/γ +κ( α +κ)μ^2, a_2 =β - α - κ/γ +κ( α +κ)μ^2, b_1 = - α/γ +κ( α +κ)μ^2, b_2 =γ +κ( β - α)/γ +κ( α +κ)μ^2.Let's assume that we are inside the source near the wall (r<r_* and r≈ r_*) and study the fluctuation of the scalar field, while freezing the tensors to their classical values: π = π_c + π̅.Corrections to the kinetic term (those that might excite the ghost) coming from the cubic level (Π_μν = ∂_μ∂_νπ): ℒ^(3) = 1/2 Λ^3[ ( α - β) a_2 + β a_1/κ] ×[- 1/2 h_c [ Π̅]^2 + 2 h^μν_c Π̅_μν[ Π̅]- κ(h_c → f_c)].Note that the 3π vertex is a total derivative. After the derivation of the EoM's we find four time derivatives acting on π̅ and therefore giving rise to the instability with the mass:M_ghost^2 = - 3 μ^2/a_1[ ( α - β) a_2 + β a_1/κ]^-11/r^2_*M_h/s(Λ/s)^3.In order for this solution to be valid we need to set three conditions:182 Condition for the next order corrections to be small: s^4/M_h Λ^3≪ 1. 183 Condition for the mass of the ghost to be small compared to the cutoff scale: 1/r_*^2·M_h Λ^3/s^4≪Λ^2. 184 Condition for the absence of the black hole: s^4 r_*^2/ M_h ≪ M_h.It is possible to satisfy all these conditions. For instance:M_h = 10^19 GeV,s = 10^-2 GeV,r_* = 10^15 GeV^-1,Λ = 10^-7 GeV,or in more familiar units, the mass of the source =s^4 r_*^3 = 10^10 Kg; radius of the source r_* = 10 cm and Schwarzschild radius for the source r_g = 10^-15 cm. We can avoid this ghost by setting ( α - β) a_2 + β a_1/κ = 0,but in this case it will re-emerge at the quartic level. To make this clear let's forget about the source and consider the following exact solution to the classical equations of motion:h^μν_c =0,f^μν_c =0, π_c = 1/2( a · x )^2,where a_μ = ( a_0, -𝐚_0 ) is some light-like vector. Since there are no 3 π interactions at the cubic level, the scalar kinetic term will start receiving corrections from the quartic order:ℒ^(4)_π =1/2 Λ^6 a_2( α - β) (π_c Π_c^μν[ Π] Π_μν - 6 π_c Π_c^μνΠ^2_μν - 9/2∂^μπ_c ∂^νπ_c Π^2_μν) μ^2 -- 1/2 Λ^6 a_1 β/κ( π_c Π_c^μν[ Π] Π_μν - 6 π_c Π_c^μνΠ^2_μν - 3/2∂^μπ_c ∂^νπ_c Π^2_μν) μ^2.The mass of the ghost in this case is:M_ghost^2 = - 3/2 [ 14 ( α - β) a_2 - 8 β a_1/κ]^-1Λ^6/a_0^2 ( a · x )^2.To justify our approximation we set:172 Condition for the possible next order corrections to be small: a_0^2/Λ^3≪ 1. 173 Condition for the mass to be small compared to the cutoff scale: Λ^6/a_0^2 ( a · x )^2≪Λ^2.By properly choosing a_0 and x it is possible to fulfill both of these conditions.We can get rid of this ghost by setting:14 ( α - β) a_2 - 8 β a_1/κ = 0,but this condition is not consistent with (<ref>), since in order for both, (<ref>) and (<ref>) to hold we need α = β and this condition goes against our assumptions. At this point one might still think that α = β≠ 0 might be a solution. Note that the Lagrangian (<ref>) without the matter part and dRGT potential is invariant under g ↔g̃ (up to a redefinition of constants). As a consequence of this fact if we go back and Stückelberguise the field g instead of g̃, replace g →g̃ in the matter part and run the same analysis the consistency condition (<ref>) will be replaced with β=0.§ CONCLUSIONS AND DISCUSSION We proposed necessary criteria for derivative mixings of spin-2 fields to be ghost free and studied few examples (<ref>). We proved that for any choice of parameters (α, β and γ) the nonlinear model hosts a ghost. The Action (<ref>) is not the most general, in fact as we mentioned there exist infinitely many terms that satisfy criteria <ref>,<ref> and <ref>. These infinitely many terms bring infinitely many parameters into the action and this makes it hard to prove the existence of ghost in the general case.I'm indebted to Prof. Gregory Gabadadze for proposing this project and guiding me through it. I would like to thank Anna-Maria Taki and Siqing Yu for the feedback on a draft version of this paper. The project was supported by the NYU MacCracken scholarship and by the NSF grant PHY-1316452. § APPENDIX: EQUATIONS OF MOTIONThe equations of motion corresponding to (<ref>)1/√(-g)δ S_EbG/δ g^μν =R_μν - 1/2 g_μν R + α - β/κ[ R̃_μν - 1/2 g_μν g^αβR̃_αβ] ++ β/ 2κ[∇_α∇_β( a g̃^αβ g_μν) +g^αβ∇_α∇_β( a g̃^λρ g_μρ g_νλ) - ∇_α∇_μ( a g̃^αβg_νβ) -. . - ∇_α∇_ν( a g̃^αβg_μβ) ]+ m^2 /2 U_μν + 1/M_h^2 T_μν = 0,1/√(-g̃)δ S_EbG/δg̃^μν= R̃_μν - 1/2g̃_μνR̃ +β κ/γ[ R_μν - 1/2g̃_μνg̃^αβ R_αβ] ++ κ (α - β)/ 2 γ[∇̃_α∇̃_β( 1/a g^αβg̃_μν) +g̃^αβ∇̃_α∇̃_β( 1/a g^λρg̃_μρg̃_νλ) - . . - ∇̃_α∇̃_μ( 1/a g^αβg̃_νβ) - ∇̃_α∇̃_ν( 1/a g^αβg̃_μβ) ] +m^2 κ^2/2 γŨ_μν = 0.Here a^2 ≡g̃/g, ∇ (∇̃) stands for the covariant derivative wrt g (g̃), T_μν is the Energy momentum tensor for the matter fields and U (Ũ) are defined as: 1/√(-g)δ S_dRGT/δ g^μν≡m^2 M_h^2/4 U_μν, 1/√(-g̃)δ S_dRGT/δg̃^μν≡m^2 M_h^2/4Ũ_μν. 99Rosen:1940zza N. Rosen,Phys. Rev.57, 147 (1940). doi:10.1103/PhysRev.57.147Rosen:1940zz N. Rosen,Phys. 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Rev. Lett.54 (1985) 1215.Fasiello:2013woa M. Fasiello and A. J. Tolley,JCAP1312, 002 (2013) doi:10.1088/1475-7516/2013/12/002 [arXiv:1308.1647 [hep-th]].Gabadadze:2013ria G. Gabadadze, K. Hinterbichler, D. Pirtskhalava and Y. Shang,Phys. Rev. D88, no. 8, 084003 (2013) doi:10.1103/PhysRevD.88.084003 [arXiv:1307.2245]. ] | http://arxiv.org/abs/1704.08603v1 | {
"authors": [
"Giorgi Tukhashvili"
],
"categories": [
"hep-th",
"gr-qc"
],
"primary_category": "hep-th",
"published": "20170427143815",
"title": "Possible Derivative Interactions in Massive bi-Gravity"
} |
ARTICLE TEMPLATEAn efficient method to compute solitary wave solutions of fractional Korteweg-de Vries equationsA. DuránCONTACT A. Durán. Email: [email protected] Department of Applied Mathematics, University of Valladolid, Paseo Belén, 15, 47011, Valladolid, Spain.December 30, 2023 =============================================================================================================================================================== Considered here is an efficient technique to compute approximate profiles of solitary wave solutions of fractional Korteweg-de Vries equations. The numerical method is based on a fixed-point iterative algorithm along with extrapolation techniques of acceleration. This combination improves the performance in both the velocity of convergence and the computation of profiles for limiting values of the fractional parameter. The algorithm is described and numerical experiments of validation are presented. The accuracy attained by the procedure can be used to investigate additional properties of the waves. This approach is illustrated here by analyzing the speed-amplitude relation.Fractional KdV equations; solitary waves; Petviashvili method; acceleration techniques§ INTRODUCTION The paper is concerned with the computation of solitary wave solutions of the fractional Korteweg-de Vries (fKdV) equationu_t+u^pu_x-(D^αu)_x=0.In (<ref>), u=u(x,t) is a real-valued function of x∈ℝ, t≥ 0, p∈ℕ, α∈ℝ and D^α stands for the linear operator represented by the symbol(D^αg)(ξ)=β(ξ)g(ξ),β(ξ)=|ξ|^α,whereg(ξ)=∫_-∞^∞g(x)e^-iξ xdx,ξ∈ℝis the Fourier transform, defined on the space of squared integrable functions g∈ L^2(ℝ). Equation (<ref>) is relevant as a dispersive and nonlinbear perturbation of Burgers' inviscid equation. The parameters α and p govern, respectively, the dispersion and the nonlinear effects and in this sense (<ref>) is suitable to investigate the relations between nonlinearity and dispersion that lead to different dynamics, such as existence and stability of solitary waves, blow-up phenomena, etc, <cit.>. The case α=2 corresponds to the classical Korteweg-de Vries (KdV, when p=1) and generalized Korteweg-de Vries (gKdV, when p≥2) equations, while α=1 leads to the Benjamin-Ono (BO) equation and generalized (gBO) versions.The parameters α and p also determine several mathematical properties of the initial value problem for (<ref>), (<ref>). The main results on well-posedness in the literature concern the case p=1, for which the Cauchy problem, when α≥ 1, is proved to have global solutions in suitable functional spaces, <cit.>. When -1<α<0, suitable smooth initial data lead the corresponding solution to blow up at finite time, <cit.>. As far as the case 0<α<1 is concerned, see <cit.> for local well-posedness results. Global weak solutions, without uniqueness, inL^∞(ℝ,H^α/2(ℝ))={v:ℝ→ H^α/2(ℝ)/max_t∈ℝ||v(t)||_H^α/2(ℝ)<∞},for initial data in the Sobolev space H^α/2(ℝ) (with the usual norm ||·||_H^α/2(ℝ), are proved to exists in <cit.> (see also <cit.>) for α>1/2. Also, Klein and Saut, <cit.> conjecture several cases of blow-up with different structure: first, no hyperbolic blow-up (blow-up of the spatial gradient with bounded sup-norm) exists; when 1/2<α<1, the solution is global; when 1/3<α<1/2, there is a sort of nonlinear dispersive blow-up, <cit.>, and, finally, blow-up of different type occurs when 0<α<1/3.On the other hand, at least formally, the following quantities are preserved by smooth enough, decaying solutions of (<ref>) C(u)=∫_-∞^∞ u(x,t)dx, M(u)=∫_-∞^∞ u^2(x,t)dx, M(u)=∫_-∞^∞(1/2|D^α/2u(x,t)|^2- 1/(p+1)(p+2)u^p+2(x,t))dx. The quantity (<ref>) is well defined when α≥ 1/3 and provides a Hamiltonian structure to (<ref>), see <cit.> for the case p=1 and <cit.> for p>1.An additional, relevant point on the dynamics of (<ref>) concerns the existence and stability of solitary wave solutions. They are solutions of the form u(x,t)=ϕ(x-ct) for some speed c>0 and profile ϕ_c=ϕ_c(X) with ϕ_c(X)→ 0 as |X|→∞. Substituting into (<ref>) and integrating once, the profile ϕ_c must satisfyD^αϕ_c+cϕ_c-ϕ_c^p+1/p+1=0.Some known results on existence and stability of solutions of (<ref>) are now summarized. For the case p=1, solitary waves are proved to exist when α>1/3, <cit.>, with an asymptotic decay as 1/x^1+α, |x|→∞; orbital stability, <cit.>,for α>1/2 is proved in <cit.>, while some spectral instability analysis can be obtained from <cit.>. In the case p>1, as mentioned in <cit.> (see also references therein), existence of solutions of (<ref>) holds for α>1 and any p, with orbital stability when p<2α. For 0<α<1, existence can be derived for 1<p<2α/(1-α), <cit.>. Results on orbital stability when 1/2<α<1, p<2α, as well as a linear instability criterium, can be seen in <cit.>.Since no explicit formulas for solitary wave solutions of (<ref>) are, except in the classical cases α=1,2, unknown, then some numerical method for the generation of approximate profiles is required. In this sense, Klein and Saut, <cit.>, solve numerically (<ref>) for p=1 to construct approximate solitary wave profiles. The numerical method to this end is based on transforming (<ref>) into the corresponding algebraic equations for the Fourier transform of the profile, which are iteratively solved by Newton iteration. The procedure is implemented by approximating each profile with a trigonometric interpolant polynomial on a long enough interval, in such a way that the Newton method is applied to the system for the corresponding Fourier components. As mentioned in <cit.>, the algebraic decrease of the modulus of the Fourier coefficients, due to the loss of smoothness of the periodic approximations of the profiles at the boundary of the computational domain (the solitary wave profile decreases slowly at infinity) leads to a slow convergence of the iteration. This is addressed by taking a large computational domain and a high number of Fourier modes. The resolution is also performed by using GMRES, <cit.>, to compute iteratively the inverse of the Jacobian matrix.In this paper, an alternative to construct numerically solitary wave profiles of (<ref>) is proposed. The technique was successfully applied to the numerical generation of periodic traveling wave solutions of the fKdV equation in <cit.>. The main points of this approach here are the following: * The method is also based on the implementation of (<ref>) in Fourier space for the periodic approximation on a long enough interval.* The algebraic system for the discrete Fourier coefficients of the trigonometric interpolant is however iteratively solved by the Petviashvili method, <cit.>, a fixed point type algorithm which may overcome some of the limitations of the Newton iteration.* In order to improve the slow convergence due to the periodic approximation, the Petviashvili method is complemented with the use of acceleration techniques based on extrapolation, <cit.>, which have shown a relevant performance in the numerical generation of solitary waves, <cit.>.The main contributions are the following: * The combination of the Petviashvili method with extrapolation improves the computation of the solitary wave profiles. The improvement is observed in mainly two points: the first one is that the method is able to generate numerical profiles for limiting values of α. A second point of improvement is found in the efficiency, since with a high number of Fourier modes and on a long interval, the extrapolation technique accelerates the convergence in a relevant way. * These new advantages can be used to study computationally additional properties of the waves, such as the speed-amplitude relation. * The method can also be applied to compute approximate solitary wave solutions of other generalizations of (<ref>) of the form, <cit.> u_t+(f(u))_x-( Mu)_x=0, where M is a linear, pseudo-differential operator associated to a continuous, even, real-valued Fourier symbol β(ξ) and f is a smooth, nonlinear, real-valued function. A relevant example of (<ref>) is the extended Whitham equation, <cit.>, for which β(ξ)=(1+γ|ξ|^2)^1/2(tanhξ/ξ)^1/2, f(u)=u^2/2. In (<ref>), the parameter γ≥ 0 controls the surface tension effects in the model. The case γ=0 leads to the classical Whitham equation, <cit.>. The computation of traveling-wave solutions of equations of the form (<ref>), kncluding the extended Whitham equation (<ref>), has been made in the literature with different techniques, see e. g. the references in <cit.> and, more recently, the method introduced and performed in <cit.>, based on continuation with spectral projection. The technique can also be applied to study the solitary wave solutions of fractional BBM type equations u_t+u_x+(f(u))_x+( Mu)_t=0, with M and f defined as in (<ref>), see <cit.>. The structure of the paper is as follows. In Section <ref> the numerical method to compute approximate solitary profiles, based on the Petviashvili method and accelerating techniques, is described, along with some implementations details. The purpose of Section <ref> is two-fold: a first group of experiments validates the efficiency of the method and studies its performance. This is used, in a second part, to analyze computationally additional properties of the solitary waves. The illustration is focused on the speed-amplitude relation and its dependence on the parameters α and p. § NUMERICAL GENERATION OF SOLITARY WAVES§.§ The Petviashvili methodIn order to describe the numerical method to compute solitary waves of (<ref>), observe that (<ref>) can be written in the formℒϕ=𝒩(ϕ),ℒ=D^α+c,𝒩(ϕ)=ϕ^p+1/p+1, c>0.Note now that the nonlinear term 𝒩 is homogeneous of degree p+1, in the sense that𝒩(λ u)=λ^p+1𝒩(u),λ, u∈ℝ.Thus, differentiating (<ref>) with respect to λ and evaluating at λ=1, we have𝒩^'(u)u=(p+1)𝒩(u), u∈ℝ.On the other hand, the operator ℒ is invertible for c>0 and if ϕ=ϕ_c satisfies (<ref>) then, using (<ref>), we haveℒ^-1𝒩^'(ϕ_c)ϕ_c=(p+1) ℒ^-1𝒩(ϕ_c)=(p+1)ϕ_c.This means that ϕ_c is an eigenfunction of the iteration operator ℒ^-1𝒩^'(ϕ_c) with eigenvalue p+1>1. Therefore, for a given initial ϕ_0, the classical fixed point iterationℒϕ_n+1=𝒩(ϕ_n), n=0,1,…will not be, in general, convergent. An alternative iterative method of fixed-point type is the Petviashvili method, <cit.>, which is formulated as m(ϕ_n)=(ℒϕ_n,ϕ_n)/(𝒩(ϕ_n),ϕ_n), ℒϕ_n+1=m(ϕ_n)^ϵ𝒩(ϕ_n), n=0,1,…, for some parameter ϵ. The term (<ref>) is called the stabilizing factor. The convergence of (<ref>) for equations of the form (<ref>) was studied in <cit.>. Pelinovsky and Stepanyants prove the convergence for 1<ϵ<(p+2)/p, under some hypotheses on the spectrum of the linearized operator ℒ-𝒩^'(ϕ_c) at the profile ϕ_c, with the fastest rate of convergence given by ϵ^*=(p+1)/p. The inclusion of the stabilizing factor modifies the spectrum of ℒ^-1𝒩^'(ϕ_c) in such a way that the eigenvalue λ=p+1 becomes, for the values of ϵ considered, an eigenvalue of the iteration operator of (<ref>) at ϕ_c with magnitude below one (and which is equals zero in the case of choosing ϵ=ϵ^*). The rest of the spectrum is preserved, see <cit.> and references therein.The implementation of (<ref>) is typically carried out ´by Fourier pseudospectral approximation, <cit.>. Let us consider the periodic problem of (<ref>) on a sufficiently long interval (-l,l). This is discretized by a uniform grid x_j=-l+jh, j=0,…,N-1, h=2l/N, in such a way that (<ref>) can be approximated by the discrete system ℒ_h= 𝒩_h(ϕ_h), ℒ_h=γ c I_N+D_N^α,𝒩_h(ϕ_h)=ϕ_h.^p+1/p+1, where ϕ_h is a N-vector approximation to the profile ϕ at the collocation points x_j, I_N is the N× N identity matrix and D_N^α defined as the N× N matrixD_N^α=F_N^-1Λ_N^αF_N,with F_N the discrete Fourier transform matrix on ℂ^N and Λ_N^α the N× N diagonal matrix with diagonal entries of the form |kπ/l|^α, k=0,…,N-1, <cit.>. The dot in (<ref>) stands for the Hadamard product. Finally, system (<ref>) is implemented for the discrete Fourier coefficients, ϕ_h=(1/N)F_Nϕ_h, of ϕ_h and the resulting algebraic system is iteratively solved with the discrete version of (<ref>) m(ϕ_n)=∑_k=0^N-1(c+|kπ/l|^α)|ϕ_n(k)|^2/∑_k=0^N-1𝒩(ϕ_n)(k)ϕ_n(k), ϕ_n+1(k)=m(ϕ_n)^ϵ𝒩(ϕ_n)(k)/c+|kπ/l|^α, for k=0,…,N-1, n=0,1,… and where ϕ_n=(ϕ_n(0),…,ϕ_n(N-1))^T.A final point concerns the way how the iteration is controlled. This is done by using three strategies: * Since in the case of convergence, the sequence of the stabilizing factors m_n:=m(ϕ_n), must go to one, see (<ref>), then a first control is given by the differences |1-m_n|,n=0,1,… * A second group of control parameters is given by the sequence of the Euclidean errors between two consecutive iterationsERROR_c(n)=||ϕ_n-ϕ_n-1||, n=0,1,… * Finally, the sequence of the residual errors (also in Euclidean norm) RES(n)=||ℒ_hϕ_n-𝒩_h(ϕ_n)||, n=0,1,…,is also considered.Thus, the iteration is run up to one of (<ref>)-(<ref>) is below a fixed tolerance parameter tol which, for the experiments below, has been taken as 10^-10. §.§ Acceleration techniquesAs mentioned in the Introduction, the loss of smoothness at the boundary due to the periodic approximation to the equations for the profiles implies an algebraic decrease of the modulus of the Fourier coefficients and, consequently, a slow iteration. In order to improve the velocity of convergence, our proposal here is including some acceleration method in the iterative process, <cit.>. In this sense, the so-called Vector Extrapolation Methods (VEM), <cit.>, introduce a final stage of extrapolation at the end of each iteration of (<ref>). This is usually carried out in a cycling way: from the last iterate ψ_0=ϕ_n at stage n, a number mw (called width of extrapolation) of iterations ψ_1,…,ψ_mw of (<ref>) is computed, and the next iteration ϕ_n+1 is derived as a suitable extrapolation formula from ψ_0,…,ψ_mw, see <cit.> for details. The coefficients of the extrapolation are functions of previous steps of the iteration and their derivation makes use of different criteria, leading to different methods. The one considered for the experimentsin this paper is the minimal polynomial extrapolation (MPE), a polynomial method which computes the coefficients by setting orthogonality conditions on the generalized residual, see <cit.>. The procedure (<ref>), accelerated with MPE is now illustrated by the following numerical results.§ NUMERICAL EXPERIMENTS§.§ Some experiments of validationSome experiments on the performance of the iterative scheme are first presented. Figure <ref> shows the form of approximate solitary wave profiles of (<ref>) with p=1, c=1, mw=6 and several values of α. The initial iteration is a squared hyperbolic secant and for the implementation, an interval with l=2048 and N=2^18 Fourier modes were taken. As it is known (at least for p=1, <cit.>), the more peaked the profile the smaller α is. (This behaviour is independent of the nonlinearity parameter, although as p increases, the amplitude of the corresponding profile has been observed to decrease, see figures <ref> and <ref> below.) In all the computations, the minimum value of the profile is below 10^-5. From Figure <ref>, note that the limiting case α_l=1/3, for p=1, of α looks to be computable. This also holds for any p>1, for which α_l=p/(p+2), see Figure <ref>.The algebraic decay at infinity of some of the profiles can be observed in Figure <ref>, which displays the corresponding phase portraits and where the derivative has been computed by using pseudospetral differentiation, see <cit.>.In order to check the accuracy of the computed profiles, several experiments are made. Figure <ref> displays the behaviour of the residual error and the stabilizing factor as function of the number of iterations and for each of the waves computed in Figure <ref>. The results confirm the convergence of the sequences (<ref>), (<ref>) and consequently of the iteration. (The second control sequence (<ref>) behaves even better and the corresponding results are not shown.) As far as the performance is concerned, note that in all the cases, the tolerance tol=10^-10 is attained in less than 50 iterations.A second experiment to check the accuracy is as follows. The computed profiles were taken as initial condition of a numerical method to approximate the periodic initial value problem associated to (<ref>). The numerical scheme is based on a pseudospectral Fourier discretization in space and a fourth order, diagonally implicit, Runge-Kutta composition method, described in <cit.> (see also references therein) as time integrator. The scheme has relevant geometric properties, <cit.>, and has been shown to be efficient in nonlinear wave problems, <cit.>.The evolution of the corresponding numerical solution was monitored and, in the case of α=0.7, is represented at several times in Figures <ref>(a),(b). They show that the initial approximate profile evolves in a solitary way without relevant disturbances, suggesting that the computed profile represents a solitary wave of (<ref>) with a high degree of accuracy. This is also confirmed by the evolution of the amplitude and speed of the numerical approximation during the integration, displayed in Figures <ref>(c) and (d). (The computation of amplitude and speed was made in the standard way, see e. g. <cit.>.)The following experiments complement the illustration of the performance of the method. Figure <ref> shows the number of iterations required to get a residual error below the tolerance as function of the fractional parameter α and for several values of the width of extrapolation mw, in the approximation to a solitary wave profile of (<ref>) with p=1 and c=1. Note that, for moderate values of mw, as mw increases, the number of iterations decreases and in this sense the performance improves. However, observe that the parameter mw cannot be fixed a priori in this sort of nonlinear problems, <cit.>. Also, from some value of mw, the number of iterations and the computational time will not improve anymore. For one of the values of mw considered, the number of iterations does not increase as α decreases; it is maximum when α is close to 0.6. §.§ Application to study additional properties of the solitary wavesThe final group of experiments is concerned with the model (<ref>). Once the accuracy and performance of the iterative method have been checked, the procedure can be used to obtain some additional information about the solitary waves. This idea is focused here on the search for the speed-amplitude relation and its dependence on the parameters α and p. Figure <ref>(a) displays this relation for α=0.8 fixed and several values of p. The amplitude is an increasing function of the speed but the increment depends on p. When α=2 (gKdV case) the solitary waves are known explicitly and the amplitude has the exact formula, <cit.>Amp=(c/2(p+1)(p+2))^1/p,see Figure <ref>(b).A comparison of both figures suggests to consider the experiment of studying the speed-amplitude relation for a fixed value of p and as function of α. This is illustrated by Figure <ref>. Note that for a fixed value of c and α, increasing p typically leads to a solitary wave of smaller amplitude. When c and p are fixed, the taller the wave the smaller value of α is (cf. Figures <ref> and <ref>). Due to the peaked form of the profiles, the maximum is, for small values of α, not easy to compute with a minimum of accuracy. However, the most reliable numerical results suggest that a similar relation to (<ref>) holds for any α. This is observed in Table <ref> where, for several values of p and α, the numerical data speed-amplitude were fitted to a power function f(x)=ax^b. The accuracy of the results was guaranteed by a goodness of fit where some statistical parameters are around some fixed tolerance. Explicitly, the following statistics were used: the sum of squares due to error SSE (with a tolerance threshold of 10^-6), the R-squared (which in all the cases is equals 1) and the root mean squared error RSME (around 10^-4). Note that while the coefficients b suggest an amplitude as a power 1/p of the speed as in (<ref>), the dispersion parameter α looks to affect only the coefficient a of the fit.§ ACKNOWLEDGEMENT(S)This work was supported bySpanish Ministerio de Economía y Competitividad under the Research Grant MTM2014-54710-P.14 AlvarezD2014 J. Alvarez and A. Durán, Petviashvili type methods for traveling wave computations: I. Analysis of convergence, J. Comput. Appl. Math. 266 (2014), pp 39-51. AlvarezD2016 J. Alvarez and A. Durán, Petviashvili type methods for traveling wave computations: II. Acceleration with extrapolation methods, Math. Comput. 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Sidi, Convergence and stability of minimal polynomial and reduced rank extrapolation algorithms, SIAM J. Numer. Anal. 23 (1986), pp 197-209. Sidi2003 A. Sidi, Practical Extrapolation Methods, Theory and Applications, Cambridge University Press, New York, 2003. sidifs A. Sidi, W. F. Ford, D. A. Smith, Acceleration of convergence of vector sequences, SIAM J. Numer. Anal. 23 (1986), pp 178-196. smithfs D. A. Smith, W. F. Ford, A. Sidi, Extrapolation methods for vector sequences, SIAM Rev. 29 (1987), pp 199-233. Weinstein1987 M. Weinstein, Existence and dynamic stability of solitary-wave solutions of equations arising in long wave propagation, Commun. Partial Differ. Eq. 12 (1987), pp 1133-1173. Weinstein1986 M. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Commun. Pure Appl. Math. 39 (1986), pp 51-68. Whitham1967 G. B. Whitham, Variational methods and applications to water waves, Proc. R. Soc. A 299 (1967), pp 6-25. | http://arxiv.org/abs/1704.08654v1 | {
"authors": [
"A. Duran"
],
"categories": [
"math.NA"
],
"primary_category": "math.NA",
"published": "20170427165839",
"title": "An efficient method to compute solitary wave solutions of fractional Korteweg-de Vries equations"
} |
School of Physics and Information Technology, Shaanxi Normal University, Xi'an 710119, People's Republic of China [email protected] a complementary study to that performed on the transverse momentum (p_ T) spectra of charged pions, kaons and protons in proton-proton (pp) collisions at LHC energies 0.9, 2.76 and 7 TeV, we present a scaling behaviour in the p_ T spectra of strange particles (K_S^0, Λ, Ξ and ϕ)at these three energies. This scaling behaviour is exhibited when the spectra are expressed in a suitable scaling variable z=p_ T/K, where the scaling parameter K is determined by the quality factor method and increases with the center of mass energy (√(s)). The rates at which K increases with ln√(s) for these strange particles are found to be identical within errors. In the framework of the colour string percolation model, we argue that these strange particles are produced through the decay of clusters that are formed by the colour strings overlapping. We observe that the strange mesons and baryons are produced from clusters with different size distributions, while the strange mesons (baryons) K_S^0 and ϕ (Λ and Ξ) originate from clusters with the same size distributions. The cluster's size distributions for strange mesons are more dispersed than those for strange baryons.The scaling behaviour of the p_ T spectra for these strange particles can be explained by the colour string percolation model in a quantitative way.13.85.Ni, 13.87.Fh Liwen Yang, et al. Universal scaling of strange particle p_ T spectra in pp collisions Universal scaling of strange particle p_ T spectra in pp collisions Liwen Yang, Yanyun Wang, Wenhui Hao, Na Liu, Xiaoling Du, Wenchao ZhangReceived: date / Revised version: date ============================================================================== § INTRODUCTION The transverse momentum (p_ T) spectra of final state particles are important observables in high energy collisions. They play an essential role in understanding the mechanism of particle productions. In many studies, searching for a scaling behaviour of the p_ T spectra is useful to reveal the mechanism. In ref. <cit.>, a scaling behaviour was presented in the pion p_ T spectra in Au-Au collisions at the Relativistic Heavy Ion Collider (RHIC). It was independent of the centrality of the collision. This scaling behaviour was later extended to the proton and anti-proton p_ T spectra with different centralities in Au-Au collisions at RHIC <cit.>.Recently, a similar scaling behaviour was found in the p_ T spectra of inclusive charged hadrons as well as identified charged hadrons (charged pions, kaons and protons) in proton-proton (pp) collisions at the Large Hadron Collider (LHC) <cit.>. This scaling behaviour was independent of the center of mass energy (√(s)). It was exhibited when the spectra were expressed in a suitable scaling variable z=p_ T/K, where K is the scaling parameter relying on √(s). In pp collisions, the hadrons produced are predominantly pions, kaons and protons. As the strange quark is heavier than the up and down quarks, the strange particles such as K_S^0, Λ, Ξ and ϕ only constitute a small fraction of final state particles. However, the investigation of their spectra is an important ingredient in understanding the mechanism of particle production in high energy collisions. Thus, in this paper, we will focus on the p_ T spectra of K_S^0, Λ, Ξ and ϕ produced in pp collisions at 0.9, 2.76 and 7 TeV <cit.>. The p_ T spectra of Ω are not considered in this work, as their spectra at 0.9 TeV are not available so far. A scaling behaviour independent of the collision energy will be searched for among these strange particle spectra. If the scaling behaviour exists, then one may ask two questions: (1) Is the dependence of the scaling parameter K on √(s) for K_S^0, Λ, Ξ and ϕ the same as that for charged pions, kaons and protons? (2) Can the string percolation model utilized in ref. <cit.> be adopted to explain the scaling behaviour of strange particles?The organization of the paper is as follows. In sect. <ref>, the method to search for the scaling behaviour will be described briefly. In sect. <ref>, the scaling behaviour of the K_S^0, Λ, Ξ and ϕ spectra will be presented.In sect. <ref>,we will discuss the scaling behaviour of the strange particle spectra in the framework of the colour string percolation model. Finally, the conclusion is given in sect. <ref>.§ METHOD TO SEARCH FOR THE SCALING BEHAVIOUR As done in ref. <cit.>, we will search for the scaling behaviour of the K_S^0 p_ T spectra with the following steps. A scaling variable, z=p_ T/K, and a scaled p_ T spectrum, Φ(z)=A(2π p_ T)^-1d^2N/dp_ Tdy|_p_ T=Kz will be defined first. Here y is the rapidity of K_S^0, (2π p_ T)^-1d^2N/dp_ Tdy is the invariant yield of K_S^0. With suitable scaling parameters K and A that depend on √(s), the data points of the K_S^0 p_ T spectra at 0.9, 2.76 and 7 TeV can be coalesced into one curve. In ref. <cit.>, K and A for the charged pion, kaon and proton spectra at 2.76 TeV were set to be 1. This choice was made due to reason that the p_ T coverage of the spectra at 2.76 TeV is much larger than the coverage at 0.9 and 2.76 TeV. In this work, the K_S^0 spectrum at 2.76 TeV covers a p_ T range from 0.225 to 19 GeV/c, which is larger the ranges of the spectra at 0.9 and 7 TeV, 0.1 to 9 GeV/c and0.1 to 9 GeV/c. Therefore, to keep the similarity and consistency with ref. <cit.>, we prefer to set the K and A for the K_S^0 spectrum at 2.76 TeV to be 1. K and A values at 0.9 and 7 TeV will be determined by the quality factor method <cit.>. Obviously, the scaling function Φ(z) depends on the choice of K and A at 2.76 TeV. This arbitrariness could be eliminated if the spectra are presented in u=z/⟨ z ⟩=p_ T/⟨ p_ T⟩. Here ⟨ z ⟩=∫^∞_0zΦ(z)zdz/∫^∞_0Φ(z)zdz. Thenormalized scaling function then is Ψ(u)=⟨ z ⟩^2Φ(⟨ z ⟩ u)/∫^∞_0Φ(z)zdz. With Ψ(u), the spectra at 0.9 and 7 TeV can be parameterized as f(p_ T)=∫^∞_0Φ(z)zdz/(A⟨ z ⟩^2)Ψ(p_ T/(K⟨ z ⟩)), where K and A are the scaling parameters at these energies. The methods to search for the scaling behaviour of the Λ, Ξ and ϕ spectra are similar to that for the K_S^0 spectra.§ SCALING BEHAVIOUR OF THE K_S^0, Λ, Ξ AND Φ P_ T SPECTRA The K_S^0, Λ and Ξ p_ T spectra in pp collisions at 0.9 and 7 TeV were published by the CMScollaboration <cit.>. Here Λ and Ξ refer to (Λ+ Λ̅)/2 and (Ξ^++Ξ^-)/2 respectively. For the K_S^0, Λ and Ξ p_ T spectra at 2.76 TeV, so far there are no official data. As the K_S^0 spectrum is theoretically the same as the charged kaon spectrum, and the charged kaon spectrum at 2.76 TeV were officially published in ref. <cit.>, we utilize the charged kaon spectrum instead of the K_S^0 spectrum at this energy. For the Λ and Ξ spectra at 2.76 TeV, we use the preliminary results of the ALICE collaboration at this energy instead. They are publicly available in refs. <cit.>.The ϕ spectra at 0.9, 2.76 and 7 TeV were published by the ALICE collaboration <cit.>. Since the scaling parameters K and A at 2.76 TeV are chosen to be 1, the scaling function Φ(z) is exactly the K_S^0, Λ, Ξ or ϕ p_ T spectrum at this energy. As described in ref. <cit.>, due to the reason that the temperature of the hadronizing system fluctuates from event to event, the p_ T spectrum of final state hadrons produced in high energy collisions follows a non-extensive statistical distribution, the Tsallis distribution <cit.>. Thus, the scaling function Φ(z) for strange particles can be parameterized as follows <cit.>Φ(z)=C_q[1-(1-q)√(m^2+z^2)-m/z_0]^1/1-q,where C_q, q and z_0 are free parameters, 1-q is a measure of the non-extensivity, m is the strange particle mass. In eq. (<ref>), 1/(q-1) determines the power law behaviour of Φ(z) in the high p_ T region, while z_0 controls the exponential behaviour in the lowp_ T region.C_q, q and z_0 are determined by the least squares fitting of Φ(z) to the K_S^0, Λ, Ξ and ϕ p_ T spectra at 2.76 TeV. The statistical and systematic errors of the data points have been added in quadrature in the fits. Table <ref> tabulates C_q, q, z_0 and their uncertainties returned by the fits. The χ^2s per degrees of freedom (dof), named reduced χ^2s, for these fits are also given in the table.As described in sect. <ref>, the scaling parameters K and A at 0.9 and 7 TeV will be evaluated with the quality factor (QF) method. Compared with the method utilized in ref. <cit.>, this method is more robust since it does not rely on the shape of the scaling function. To define the quality factor, a set of data points (ρ^i, τ^i) is considered first. Here ρ^i=p_ T^i/K, τ^i=log(A(2π p^i_ T)^-1d^2N^i/dp^i_ Tdy^i), ρ^i are ordered, τ^i are rescaled so that they are in the range between 0 and 1. Then, the QF is introduced as follows <cit.>QF(K,A)=[∑_i=2^n(τ^i-τ^i-1)^2/(ρ^i-ρ^i-1)^2+1/n^2]^-1,where n is the number of data points and 1/n^2 keeps the sum finite in the case oftwo points taking the same ρ value. It is obvious that a large contribution to the sum in the QF is given if two successive data points are close in ρ and far in τ. Therefore, a set of data points are expected to lie close to a single curve if they have a small sum (a large QF) in eq. (<ref>). The best set of (K,A) at 0.9 (7) TeV is chosen to be the one which globally maximizes the QF of the data points at 0.9 (7) and 2.76 TeV. Table <ref> tabulates K and A for the K_S^0, Λ, Ξ and ϕ spectra at 0.9, 2.76 and 7 TeV. Also shown in the table is the maximum QF (QF_max). In order to determine the uncertainties of K and A at 0.9 and 7 TeV, we utilize the method mentioned in ref. <cit.>. Let's take the determination of the uncertainty of K (A) for K_S^0 at 0.9 TeV as an example. In fig. <ref> we first plot the QF as a function of K (A) with A (K) fixed to the value 0.24 (0.92) returned by the QF method. The peak value with QF >(QF_max-0.01) shows a good scaling and we make a Gaussian fit to this bump. The standard deviation of the Gaussian fit, σ_K(A), is taken as the uncertainty of K (A) for K_S^0 at 0.9 TeV. The mean value of the Gaussian fit, μ_K(A), is consistent with the value of K (A) returned by the QF method, thus this method to determine the uncertainties of scaling parameters is robust. The errors of K and A for K_S^0 at 7 TeV, Λ, Ξ and ϕ at 0.9 and 7 TeV are determined by makingGaussian fits to the peaks with QF>(QF_max-0.01).Using the scaling parameters K and A in table <ref>, now we can shift the K_S^0 p_ T spectra at 0.9 and 7 TeV to the spectrum at 2.76 TeV. They are shown in the upper panel of fig. <ref>. On a log scale, most of the data points at different energies appear consistent with the universal curve which is described by Φ(z) in eq. (<ref>) with parameters in the second row of table <ref>. In order to see how well the data points agree with the fitted curve, a ratio, R= (data-fitted)/data, is evaluated at 0.9, 2.76 and 7 TeV. The uncertainty of R is determined to be (fitted/data)×(Δ data/data), where Δ data is the total uncertainty of the data point. The R distribution is shown in the lower panel of the figure. Except for the last three points in the high p_ T region at 0.9 TeV, all the other points have R values in the range between -0.3 and 0.3, which implies that the agreement between the data points and the fitted curve is within 30%. This agreement roughly corresponds to the systematic errors on R and the accuracy of the fits. If we take into account the systematic errors on R, then this agreement is within 22%. In the upper panels of figs. <ref>, <ref> and <ref>, we present the scaling behaviour of the Λ, Ξ and ϕ p_ T spectra at 0.9, 2.76 and 7 TeV. In the lower panels of these figures are the R distributions for these spectra. For the Λ spectra, except for the second-to-last pointat 0.9 TeV and the last point at 7 TeV, all the other points agree with the fitted curve within 30%. Taking into account the systematic uncertainties of R, this agreement is within 11%. For the Ξ spectra, except for the points with z= 4.1, 4.3 and 6.4 GeV/c at 0.9 TeV, all the other points are consistent with the fitted curve within 20%. Taking into account the systematic errors of R, this consistency is within 3%. For the ϕ spectra, all the points are in agreement with the fitted curve within 30%. With the consideration of the systematic errors of R, this agreement is within 18%.From the above statement, we have shown that the p_ T spectra of K_S^0, Λ, Ξ and ϕ at 0.9, 2.76 and 7 TeV exhibit a scaling behaviour independent of √(s). As described in sect. <ref>, the scaling function Φ(z) relies on K and A chosen at 2.76 TeV. In order to get rid of this reliance, we utilize the scaling variable u=z/⟨ z ⟩ instead. The ⟨ z ⟩ values for the K_S^0, Λ, Ξ and ϕ p_ T spectra are determinedas 0.701±0.008, 0.97±0.03, 1.12±0.01 and 1.04±0.02 GeV/c, where the errors are due to the uncertainties of C_q, q and z_0 in table <ref>. The corresponding normalized scaling function Ψ(u) isΨ(u)=C^'_q[1-(1-q^')√((m^')^2+u^2)-m^'/u_0]^1/1-q^'.Here C^'_q=⟨ z ⟩^2C_q/∫^∞_0Φ(z)zdz, q^'=q, u_0=z_0/⟨ z ⟩ and m^'=m/⟨ z ⟩. Their valuesare presented in table <ref>. As described in sect. <ref>, with Ψ(u), the spectra of K_S^0, Λ, Ξ and ϕ at 0.9 (7) TeV can be parameterized as f(p_ T)=∫^∞_0Φ(z)zdz/(A⟨ z ⟩^2)Ψ(p_ T/(K⟨ z ⟩)), where K and A are the scaling parameters of these strange particles at 0.9 (7) TeV in table <ref>. In ref. <cit.>, the CMS collaboration have presented the relative production versus p_ T between different strange particle species, N(Λ)/N(K_S^0) and N(Ξ)/N(Λ), at 0.9 and 7 TeV. In the upper (lower) panel of fig. <ref>, we show that the N(Λ)/N(K_S^0) (N(Ξ)/N(Λ)) distributions in data at 0.9 and 7 TeV are well described by f_Λ(p_ T)/f_K_S^0(p_ T)(f_Ξ(p_ T)/f_Λ(p_ T)). This agreement is a definite indication that the scaling behaviour exists in the p_ T spectra of strange particles at 0.9, 2.76 and 7 TeV.The p_ T dependence of the relative production can be explained as follows. Atlow p_ T, f(p_ T) inclines to be an exponential distribution which is controlled by the parameter z_0=u_0⟨ z ⟩. For N(Λ)/N(K_S^0) (N(Ξ)/N(Λ)), the z_0 value for Λ (Ξ) is larger than that for K_S^0 (Λ), therefore both N(Λ)/N(K_S^0) and N(Ξ)/N(Λ) grow withp_ T. At high p_ T, f(p_ T) prefers to be a power law distribution which is dominated by 1/(q^'-1). q^' value for Λ (Ξ) is smaller than (almost equal to) that for K_S^0 (Λ), thus N(Λ)/N(K_S^0) decreases with p_ T while N(Ξ)/N(Λ) appears to be flat. § DISCUSSIONSIn sect. <ref>, we have shown that there is indeed a scaling behaviour in the K_S^0, Λ, Ξ and ϕ p_ T spectra in pp collisions at 0.9, 2.76 and 7 TeV. This scaling behaviour appears when the spectra are presented in terms of the scaling variable z. Now we would like to discuss this scaling behaviour in terms of the colour string percolation (CSP) model <cit.>.In this model, colour strings are stretched between the partons of the projectile and target protons in pp collisions. These strings then will split into new ones by the production of sea qq̅ pairs from the vacuum. Strange particles such as K_S^0, Λ, Ξ and ϕ are produced through the hadronization of these new strings. In the transverse plane, the colour strings look like discs, each of which has an area, S_1=π r_0^2, r_0≈ 0.2 fm. When the collision energy increases, the number of strings grows and they interact with each other and start to overlap to form clusters. A cluster with n strings is assumed to behave as a single string. The colour field of the cluster Q_n is the vectorial sum of the colour charge of each individual Q_1 string, Q_n=∑_1^nQ_1. Since the individual string colour fields are oriented arbitrarily, the average value of Q_1i·Q_1j is zero and Q_n^2=nQ_1^2. Q_n also depends on the transverse area of each individual string S_1 and the transverse area of the cluster S_n. Thus, Q_n=√(nS_n/S_1) Q_1. As the multiplicity of strange particles produced from the cluster is proportional to its colour charge, μ_n=√(nS_n/S_1)μ_1, where μ_1 is the multiplicity of strange particles produced by a single string. Since the transverse momentum is conserved before and after the overlapping, μ_n⟨ p_ T^2⟩_n=nμ_1⟨ p_ T^2⟩_1, where ⟨ p_ T^2⟩_n is the mean p_ T^2 of strange particles produced by the cluster, ⟨ p_ T^2⟩_1 is the mean p_ T^2 of strange particles produced by a single string. Therefore, ⟨ p_ T^2⟩_n=√(nS_1/S_n)⟨ p_ T^2⟩_1, where nS_1/S_n is the degree of string overlap. For the case where strings just get in touch with each other, S_n=nS_1, nS_1/S_n=1 and ⟨ p_ T^2⟩_n=⟨ p_ T^2⟩_1, which means that the n strings fragment into strange hadrons independently. For the case in which strings maximally overlap with each other, S_n=S_1, nS_1/S_n=n and ⟨ p_ T^2⟩_n=√(n)⟨ p_ T^2⟩_1, which means that the mean p_ T^2 is maximally enhanced due to the percolation. The p_ T spectra of strange particles produced in pp collisions can be written as a superposition of the p_ T distribution produced by each cluster, g(x, p_ T), weighted with the cluster's size distribution W(x),d^2N/2π p_ Tdp_ Tdy=C∫_0^∞W(x)g(x,p_ T)dx,where C is a normalization parameter which characterizes the total number of clusters formed for strange particles before hadronization. W(x) is supposed to be a gamma distribution,W(x)=γ/Γ(κ)(γ x)^κ-1exp(-γ x),where x is proportional to 1/⟨ p_ T^2⟩_n, κ and γ are free parameters. κ is related to the dispersion of the size distribution, 1/κ=(⟨ x^2⟩-⟨ x⟩^2)/⟨ x⟩^2. It depends on the density of the strings, η=(r_0/R)^2N_s, where R is the effective radius of the interaction region, N_s is the average number of strings of the cluster. γ is related to the mean x, ⟨ x⟩=κ/γ.In order to see whether the CSP model can describe the scaling behaviour of strange particle p_ T spectra, we attempt to fit eq. (<ref>) to the combination of the scaled data points at 0.9, 2.76 and 7 TeV with the least squares method. Here the cluster's fragmentation function in the CSP fit is chosen as the Schwinger formula <cit.>g(x, p_ T)=exp(-p_ T^2x).C, γ and κ returned by the fits are listed in table <ref>. From the table, we see that the dispersion of the cluster's size distribution (1/κ) for strange mesons (K_S^0 and ϕ) is larger than that of strange baryons (Λ and Ξ), while the dispersion of the cluster's size distribution for K_S^0 (Λ) is almost equal to that of ϕ (Ξ) when considering the errors. This implies that the strange mesons and baryons are produced from clusters with different size distributions, while the strange mesons (baryons) K_S^0 and ϕ (Λ and Ξ) originate from clusters with the same size distributions. The cluster's size distributions for strange mesons are more dispersed than those for strange baryons. The difference between the cluster's size distributions of strange mesons and baryons could be explained as follows. As described in ref. <cit.>, since additional quarks required to form a baryon are provided by the quarks of the overlapping strings that form the cluster, the baryons probe a higher string density than mesons for the same energy of collisions. When η is above the critical string density at which the string percolation appears, κ increases with η <cit.>.Therefore the κ values for strange baryons are larger than those for strange mesons. The fit results for K_S^0, Λ, Ξ and ϕ are presented in the upper panels of figs. <ref>, <ref>, <ref> and <ref> respectively. The R distributions are shown in the lower panels of these figures. For the K_S^0 spectra, except for the last two points at 0.9 TeV and the last three points at 2.76 TeV, all the other points agree with the CSP fit within 30%. For the Λ spectra, except for the points with z= 6.4 and 8.2 GeV/c at 0.9 TeV, all the other data points are consistent with the CSP fit within 30%. For the Ξ spectra, except for the points at z= 4.1, 4.3 and 6.4 GeV/c at 0.9 TeV, all the other data points agree with the CSP fit within 20%. For the ϕ spectra, except for the last point at 2.76 TeV, all the other points are consistent with the CSP fit within 30%.From the above statement, we see that the CSP model can successfully describe the scaling behaviour of the strange particle p_ T spectra at 0.9, 2.76 and 7 TeV. The reason is as follows. W(x) in eq. (<ref>) and g(x, p_ T) in eq. (<ref>) are invariant under the transformation x → x^' = λ x, γ→γ^' = γ/λ and p_ T→ p_ T^' = p_ T/√(λ). Here λ=⟨ S_n/nS_1⟩^1/2, where the average is taken over all the clusters decaying into strange particles <cit.>. As a result, the strange particle p_ T spectra in eq. (<ref>) are also invariant. This invariance is exactly the scaling behaviour we are looking for. Comparing the p_ T^' transformation in the CSP model p_ T^'→ p_ T^'√(λ) with the one utilized to search for the scaling behaviour p_ T→ p_ T/K, we deduce that the scaling parameter K is proportional to ⟨ nS_1/S_n⟩^1/4. As the degree of string overlap nS_1/S_n nonlinearly grows with √(s) <cit.>, the scaling parameter K should also increase with √(s) in a nonlinear trend. That's indeed what we observed in table <ref>. Therefore the CSP model can qualitatively explain the scaling behaviour for the K_S^0, Λ, Ξand ϕ p_ T spectra separately.In order to determine the nonlinear trend with which Kincreases with √(s), we fit the K values at 0.9, 2.76 and 7 TeV for K_S^0, Λ, Ξ and ϕ in table <ref> with a function K=αln(√(s))+β, where √(s) is in TeV, α and β are free parameters and α characterizes the rate at which K changes with ln√(s). In sect. <ref>, the scaling parameter K at 2.76 TeV is set to be 1 and it is not assigned to an uncertainty. Here, in order to do the fit, we take its uncertainty as the relative error of ⟨ p_ T⟩ at this energy. The α values returnedby the fits for K_S^0, Λ, Ξand ϕ are 0.109±0.024, 0.120±0.028, 0.131±0.031 and 0.085±0.026. They are consistent within uncertainties. This can be explained by the CSP model as follows. The values of ⟨ z ⟩ are the same for K_S^0 (Λ, Ξ or ϕ) at 0.9, 2.76 and 7 TeV. As K= ⟨ p_ T⟩/⟨ z ⟩, the ratio between the values of K should be equal to the ratio between the values of ⟨ p_ T⟩. ⟨ p_ T⟩ is evaluated in terms of the CSP model as <cit.>⟨ p_ T⟩=∫_0^∞∫_0^∞W(x)g(x, p_ T)p_ T^2dxdp_ T/∫_0^∞∫_0^∞W(x)g(x, p_ T)p_ Tdxdp_ T.Plugging W(x) in eq. (<ref>) and g(x, p_ T) in eq. (<ref>) into eq. (<ref>), we get⟨ p_ T⟩=√(γπ)(κ-1)Γ(κ-3/2)/2Γ(κ),which depends on γ and κ. In order to determine the values of γ and κ at 0.9, 2.76 and 7 TeV, we fit the strange particle spectra at these three energies to eq. (<ref>) with the least squares method. They are tabulated in table <ref>.With these γ and κ values, we can calculate the ratios between the values of ⟨ p_ T⟩ at 0.9 (7) and 2.76 TeV for K_S^0, Λ, Ξ and ϕ. They are 0.93±0.03, 0.87±0.04, 0.86±0.04 and 0.95±0.38 (1.14±0.03, 1.08±0.04, 1.08±0.04 and 1.03±0.03), where uncertainties are due to the errors of γ and κ at 0.9 (7) and 2.76 TeV. Comparing these ratios with the scaling parameters K at 0.9 and 7 TeV in table <ref>, we find they are indeed consistent within uncertainties. Therefore, the CSP model can also explain the scaling behaviour of the K_S^0, Λ, Ξ and ϕ p_ T spectra in a quantitative way.Finally, we would like to see whether the energy dependence of the scaling parameter K for the strange particles K_S^0, Λ, Ξ and ϕ is the same as that for charged pions, kaons and protons. We fit K=αln(√(s))+β to the K values at 0.9, 2.76 and 7 TeV for charged pions, kaons and protons in ref. <cit.>.The values of α for charged pions, kaons and protons are 0.0638±0.0008, 0.085±0.015 and 0.088±0.003. The α value for charged pions is smaller than those for the strange particles while the α values for charged kaons and protons are comparable to those for the strange particles.§ CONCLUSIONS In this paper, we have presented the scaling behaviour of the K_S^0, Λ, Ξp_ T and ϕ p_ T spectra at 0.9, 2.76 and 7 TeV. This scaling behaviour appears when the spectra are shown in terms of the scaling variable z=p_ T/K. The scaling parameter K is determined by the quality factor method and it increases with energy. The rates at which K increases with ln√(s) for these strange particles are found to be identical within errors. In the framework of the CSP model, the strange particles are produced through the decay of clusters that are formed by the strings overlapping. We find that the strange mesons and baryons are produced from clusters with different size distributions, while the strange mesons (baryons) K_S^0 and ϕ (Λ and Ξ) originate from clusters with the same size distributions. The cluster's size distributions for strange mesons are more dispersed than those for strange baryons. The scaling behaviour of the p_ T spectra for these strange particles can be explained by the colour string percolation model quantitatively. § ACKNOWLEDGEMENTSLiwen Yang, Yanyun Wang, Na Liu, Xiaoling Du and Wenchao Zhang were supported by the Fundamental Research Funds for the Central Universities of China under Grant No. GK201502006, by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, by Natural Science Basic Research Plan in Shaanxi Province of China under Grant No. 2017JM1040,and by the National Natural Science Foundation of China under Grant Nos. 11447024 and 11505108. Wenhui Hao was supported by the National Student's Platform for Innovation and Entrepreneurship Training Program under Grant No. 201710718043.pion_spectrum R. C. Hwa and C. B. Yang, Phys. Rev. Lett.90, 212301 (2003).proton_antiproton_spectra W. C. Zhang, Y. Zeng, W. X. Nie, L. L. Zhu and C. B. Yang, Phys. Rev. C 76, 044910 (2007). inclusive_scaling W. C. Zhang and C. B. Yang, J. Phys. G: Nucl. Part. Phys. 41, 105006 (2014). pi_k_p_scaling W. C. Zhang, J. Phys. G: Nucl. Part. Phys. 43, 015003 (2016). strange_production_1 V. Khachatryan et al. (CMS Collaboration),J. High Energy Phys. 05, 064 (2011). strange_production_2 L. D. Hanratty, CERN-THESIS-2014-103 (2014). strange_production_3 D. Colella (for the ALICE Collaboration),J. Phys.: Conf. Ser. 509, 012090 (2014). strange_production_4 K. Aamodt et al. (ALICE Collaboration), Eur. Phys. J. C 71, 1594 (2011). strange_production_5 J. Adam et al. (ALICE Collaboration), Phys. Rev. C 95, 064606 (2017). strange_production_6 B. Abelev et al. (ALICE Collaboration), Eur. Phys. J. C 72, 2183 (2012). QF_1F. Gelis et al., Phys. Lett. B 647, 376-379 (2007). QF_2G. Beuf et al., Phys. Rev. D 78, 074004 (2008). charged_kaon_spectra_2_76_TeV B. Abelev et al. (ALICE Collaboration), Phys. Lett. B 736, 196-207 (2014). Tsallis_distribution_1M. Rybczynski,Z. Wlodarczyk and G. Wilk, J. Phys. G: Nucl. Part. Phys.39, 095004 (2012). Tsallis_distribution_2C. Tsallis, J. Stat. Phys. 52, 479 (1988). string_perco_model_1L. Cunqueiro et al., Eur. Phys. J. C 53, 585-589 (2008). string_perco_model_2J. Dias de Deuset al., Eur. Phys. J. C 41, 229-241 (2005). schwinger_formulaJ. Schwinger, Phys. Rev. 82, 664 (1951). string_perco_model_3 J. Dias de Deus et al., Phys. Lett. B 601, 125-131 (2004). | http://arxiv.org/abs/1704.08138v2 | {
"authors": [
"Liwen Yang",
"Yanyun Wang",
"Wenhui Hao",
"Na Liu",
"Xiaoling Du",
"Wenchao Zhang"
],
"categories": [
"hep-ph"
],
"primary_category": "hep-ph",
"published": "20170426142517",
"title": "Universal scaling of strange particle $p_{\\rm T}$ spectra in pp collisions"
} |
= 9999 Three-dimensional structure of the magnetic field in the disk of the Milky Way A. Ordog1 J.C. Brown1R. Kothes2T.L. Landecker2 ===============================================================================================SHO EJIRIIn this paper, we study the Albanese morphisms in positive characteristic. We prove that the Albanese morphism of a variety with nef anti-canonical divisor is an algebraic fiber space,under the assumption that the general fiber is F-pure.Furthermore, we consider a notion of F-splitting for morphisms, and investigate it in the case of Albanese morphisms. We show that an F-split variety has F-split Albanese morphism, and that the F-split Albanese morphism is an algebraic fiber space.As an application, we provide a new characterization of abelian varieties. § INTRODUCTIONThe Albanese morphism is an important toolin the study of a variety with non-positive Kodaira dimension. In characteristic zero, Kawamata proved thatthe Albanese morphism of a smooth projective variety with Kodaira dimension zerois an algebraic fiber space <cit.>. Zhang showed that the same holds in the case when the anti-canonical divisor is nef <cit.>. In the same case, Lu, Tu, Zhang and Zheng proved thatthe Albanese morphism is a flat morphism with reduced fibers <cit.>,and Cao showed recently that it is actually locally isotrivial <cit.>. In positive characteristic, Hacon and Patakfalvi proved that the Albanese morphism of a smooth projective variety Xis surjective if the S-Kodaira dimension κ_S(X) of X is zero <cit.> (cf. <cit.>). Here, the S-Kodaira dimension is a positive characteristic analogue of the usual Kodaira dimension.Wang showed that the Albanese morphism of a threefold with semi-ample anti-canonical divisoris surjective if the general fiber is F-pure <cit.>. In this paper, we generalize his result to varieties of arbitrary dimension,which can be viewed as a positive characteristic counterpart of the above result of Zhang.Let X be a normal projective variety over an algebraically closed field of characteristic p>0,and Δ an effective ℚ-Weil divisor on X such that-m(K_X+Δ) is a nef Cartier divisor for an integer m>0 not divisible by p.Let a:X→ A denote the Albanese morphism of X, and X_η the geometric generic fiber over the image of a. If (X_η,Δ|_X_η) is F-pure, then a is an algebraic fiber space. We also study the relationship between the Albanese morphism of X and Frobenius splittings on X. The notion of a Frobenius split variety was introduced by Mehta and Ramanathan <cit.>,which is considered to be closely related to a variety of Calabi–Yau type <cit.>.We consider a generalization of this notion to pairs (f,Γ)consisting of a morphism f:V→ W between varietiesand an effective ℚ-Weil divisor Γ on V (Definition <ref>). In this paper, we focus on a Frobenius splitting of the Albanese morphism.Let X be a normal projective variety over an algebraically closed field k of characteristic p>0,let Δ be an effective ℚ-Weil divisor on X,and let a:X→ A be the Albanese morphism of X.Then there is the following relationship between Frobenius splittings of (X,Δ) and that of (a,Δ): The pair (X,Δ) is F-split if and only if (a,Δ) is F-split and A is ordinary. We study the Albanese morphism a under the assumption that (a,Δ) is locally F-split (Definition <ref>),which is weaker than the assumption that the pair is F-split. For instance, a flat morphism with normal F-split fibers is locally F-split, but not necessarily F-split.The next theorem shows that, if (a,Δ) is locally F-split,then a is an algebraic fiber space whose fibers satisfy certain geometric properties. Assume that (a,Δ) is locally F-split and that mΔ is Cartier for an integer m>0 not divisible by p.Then the following hold: (1) The morphism a is an algebraic fiber space. (2) The support of Δ does not contain any irreducible component of any fiber. (3) For every scheme-theoretic point z∈ A, the pair (X_ z,Δ_ z) is F-split,where X_ z is the geometric fiber over z. In particular, X_ z is reduced.(4) The morphism a is smooth in codimension one.In other words, there exists an open subset U of X such that codim(X∖ U)≥ 2 and a|_U:U→ A is a smooth morphism.In particular, the general fiber of a is normal.One can recover the result of Hacon and Patakfalvi when K_X is numerically trivial,because in this case, the condition κ_S(X)=0 is equivalent to saying that X is F-split. As a corollary of Theorem <ref>, we provide a new characterization of abelian varieties.Assume that (a,Δ) is locally F-split (resp. (X,Δ) is F-split). Then A≤ X.Furthermore, the equality holds if and only if X is an abelian variety (resp. ordinary abelian variety) and Δ=0. Applying this theorem, we also give a necessary and sufficient conditionfor a normal projective variety to have F-split Albanese morphism (Theorem <ref>). We conclude this paper with a classification of minimal surfaceswhose Albanese morphisms are F-split or locally F-split (Theorem <ref>). The author wishes to express his gratitude to his supervisor Professor Shunsuke Takagi for suggesting problems,valuable comments and helpful advice. He is deeply grateful to Professors Zsolt Patakfalvi and Yoshinori Gongyo for fruitful discussions and valuable comments.He would like to thank Professors Osamu Fujino, Nobuo Hara and Doctor Yuan Wang for stimulating discussions, questions and comments.He also would like to thank the reviewer for a careful reading and helpful suggestions. Part of this work was carried out during his visit to Princeton University with support fromThe University of Tokyo/Princeton University Strategic Partnership Teaching and Research Collaboration Grant,and from the Program for Leading Graduate Schools, MEXT, Japan. He was also supported by JSPS KAKENHI Grant Number 15J09117.He would like to thank Professor Adrian Langer for pointing out an error in Proposition <ref>. § NOTATION AND CONVENTIONSThroughout this paper, we fix an algebraically closed field k of characteristic p>0. By k-scheme, we mean a separated scheme of finite type over Spec k.An integral k-scheme is called a variety over k. Let X be a normal variety.A Weil divisor D is said to be ℚ-Cartier (resp. ℤ_(p)-Cartier) ifmD is Cartier for some 0<m∈ℤ (resp. 0<m∈ℤ∖ pℤ). Here, ℤ_(p) is the localization of ℤ at (p)=pℤ. Let φ:S→ T be a morphism of schemes and T' a T-scheme.Then, S_T' and φ_T':S_T'→ T' denote respectivelythe fiber product S×_TT' and its second projection. For a Cartier, ℤ_(p)-Cartier or ℚ-Cartier divisor D on S (resp. an Ø_S-module 𝒢),the pullback of D (resp. 𝒢) to S_T' is written as D_T' (resp. 𝒢_T'), if it is well-defined. Similarly, for a homomorphism α:ℱ→𝒢 of Ø_S-modules,α_T':ℱ_T'→𝒢_T' is the pullback of α to S_T'. Let f:X→ Z be a morphism between k-schemes.Then, F_X denotes the absolute Frobenius morphism of X.The source of F_X^e is often written as X^e. The morphism f:X→ Z is denoted by f^(e):X^e→ Z^ewhen we regard X and Z as X^e as Z^e, respectively.We write the induced morphism (F_X^e, f^(e)):X^e→ X×_Z Z^e=:X_Z^e as F^(e)_X/Z.§ TRACE MAPS OF RELATIVE FROBENIUS MORPHISMS In this section,given a morphism between varieties over an algebraically closed field k,we consider the relative Frobenius morphism and its trace map.§.§ Base change by Frobenius morphismsLet f:X→ Z be a morphism (not necessarily surjective) between k-schemes.For each integer e>0, we have the following diagram:@R=25pt@C=25ptX^e [dr]^F_X^e[d]_F_X/Z^(e)X_Z^e[r]_(F_Z^e)_X[d]_f_Z^eX [d]^f Z^e [r]^F_Z^eZ We first consider the properties of X_Z^e when Z is a smooth variety.With the notation above, assume that Z is a smooth variety.(1) If X is a Gorenstein k-scheme of pure dimension, then so is X_Z^e.Furthermore, the dualizing sheaf ø_X_Z^e of X_Z^e is isomorphic tof_Z^e^*ø_Z^e^1-p^e⊗(ø_X)_Z^e,where ø_X is the dualizing sheaf of X. (2) Suppose that f is dominant and separable.If X is a variety, then so is X_Z^e. We first note that since (F_Z^e)_X is homeomorphic,if X is of pure dimension (resp. irreducible), then so is X_Z^e.Now, since F_Z^e is a Gorenstein morphism <cit.>,the base change (F_Z^e)_X has the same property,so X_Z^e is a Gorenstein k-scheme.Furthermore, by <cit.>, we have ø_X_Z^e≅ø_(F_Z^e)_X⊗(ø_X)_Z^e≅f_Z^e^*ø_F_Z^e⊗(ø_X)_Z^e≅f_Z^e^*ø_Z^e^1-p^e⊗(ø_X)_Z^e, which proves (1).Next, we show (2). We may assume that X=Spec A and Z=Spec B.Let K be the function field of X.The separability of f implies that F^e_*B⊗_BK is reduced.Since F_Z^e is flat, F^e_*B⊗_BA→ F^e_*B⊗_BK is injective, so F^e_*B⊗_BA is reduced. §.§ Trace maps(<ref>.1)Let π:X→ Z be a finite surjective morphism between Gorenstein k-schemes of pure dimension,and let ø_X and ø_Z be dualizing sheaves of X and Z, respectively.We denote by Tr_π:π_*ø_X→ø_Z the morphism obtained by applying the functor ℋom_Z(,ø_Z)to the natural morphism π^#:Ø_Z→π_*Ø_X.This is called the trace map of π.(<ref>.2)Let f:X→ Z be a morphism (not necessarily surjective) from a Gorenstein variety X to a smooth variety Z.Then it follows from Lemma <ref> that, ø_X^e⊗F_X/Z^(e)^*ø_X_Z^e^-1≅ø_X^e⊗f^(e)^*ø_Z^e^p^e-1⊗ø_X^e^-p^e≅ø_X^e/Z^e^1-p^e. 1Hence, by the projection formula, we get(F_X/Z^(e)_*ø_X^e)⊗ø_X_Z^e^-1≅F_X/Z^(e)_*ø_X^e/Z^e^1-p^e. Defineϕ^(e)_X/Z:=Tr_F_X/Z^(e)⊗ø_X_Z^e^-1:F_X/Z^(e)_*ø_X^e/Z^e^1-p^e→Ø_X_Z^e.Note that we now have the following isomorphisms:F_X/Z^(e)_*ø_X^e/Z^e^1-p^e ≅F_X/Z^(e)_*ℋom(F_X/Z^(e)^*ø_X_Z^e,ø_X^e) ≅ℋom((F_X/Z^(e)_*Ø_X^e)⊗ø_X_Z^e,ø_X_Z^e) ≅ℋom(F_X/Z^(e)_*Ø_X^e,Ø_X_Z^e). (<ref>.3)Let f:X→ Z be a morphism from a normal variety X to a smooth variety Z.Let ι:U→ X be the open immersion from the regular locus U of X.We then have ι_Z^e_*F_U/Z^(e)_*ø_U^e/Z^e^1-p^e≅F_X/Z^(e)_*ø_X^e/Z^e^1-p^e, ι_Z^e_*Ø_U_Z^e≅Ø_X_Z^e.Hence, we can define ϕ^(e)_X/Z:=ι_*ϕ^(e)_U/Z:F_X/Z^(e)_*ø_X^e/Z^e^1-p^e→Ø_X_Z^e.Let K_X/Z be a Weil divisor on X such that Ø_X(K_X/Z)≅ø_X/Z.Let Δ be an effective ℚ-Weil divisor on X.For every e>0, we define ℒ^(e)_(X/Z,Δ):=Ø_X^e( ⌊ (1-p^e)(K_X^e/Z^e+Δ) ⌋ ) ⊆Ø_X^e( (1-p^e)K_X^e/Z^e ),ϕ^(e)_(X/Z,Δ):F_X/Z^(e)_*ℒ_(X/Z,Δ)^(e)↪F_X/Z^(e)_* Ø_X^e((1-p^e)K_X^e/Z^e) Ø_X_Z^e. It is easily seen that the above morphism is also obtained by applying the functorℋom_X_Z^e(,Ø_X_Z^e)to the natural morphism Ø_X_Z^e→F_X/Z^(e)_*Ø_X^e↪F_X/Z^(e)_*Ø_X^e(⌈(p^e-1)Δ⌉).When Z=Spec k, we may identify Z^e, X_Z^e and F_X/Z^(e) with Z, X and F_X^e, respectively.In this case, we denote ϕ_(X/Z,Δ)^(e) by ϕ_(X,Δ)^(e).§ VARIETIES WITH NEF ANTI-CANONICAL DIVISORIn this section, we prove Theorem <ref> which states thatthe Albanese morphism of a normal projective varietywith nef anti-canonical divisor is an algebraic fiber spaceif the geometric generic fiber is F-pure.Throughout this section, we work over an algebraically closed field k of characteristic p>0. To begin with, we recall the Albanese morphism of a normal projective variety X.This is a morphism a:X→ A to an abelian variety A (called the Albanese variety) such thatevery morphism b:X→ B to another abelian variety B factors uniquely through a. The existence of the Albanese morphism for a normal projective variety is proved for instance in <cit.>.We must remark that the above morphism is different in generalfrom the Albanese map that is defined by using the Albanese morphismof a resolution of singularities.We next recall the definition of weak positivity.Note that our definition is slightly different from the original one introduced by Viehweg <cit.>. Let 𝒢 be a coherent sheaf on a normal quasi-projective variety Y,and let η denote the generic point of Y.We say that 𝒢 is weakly positive iffor every ample line bundle ℋ on Y and each positive number α,there exists a positive number β such that the natural map H^0( Y, ( S^αβ𝒢)^**⊗ℋ^β)⊗Ø_Y,η→( ( S^αβ𝒢)^**⊗ℋ^β)_η is surjective. Here, S^αβ() and ()^** denotethe αβ-th symmetric product and the double dual, respectively.It follows from the definition that a line bundle on a projective variety is weakly positive if and only if it is pseudo-effective.Theorem <ref> is an application of Theorems <ref> and <ref> below.Let f:X→ Y be a surjective morphism between normal varieties. Let Δ be an effective divisor on X such that m(K_X+Δ) isan integral Cartier divisor for an integer m>0 not divisible by p. Assume that (X_η,Δ|_X_η) is F-pure, where X_η is the geometric generic fiber. If -(K_X+Δ+f^*D) is a nef ℚ-Cartier divisor on X for some ℚ-Cartier divisor D on Y,then Ø_Y(-n(K_Y+D)) is weakly positive for an integer n>0 such that nD is integral.Let X be a normal projective variety with κ(X,K_X)=0.Let a:X→ A be the Albanese morphism of X.If a:X→Im(a) is generically finite and separable, then a is surjective. The next lemma is also used in the proof of Theorem <ref>. Let D be an effective Weil divisor on a normal projective variety Y. If Ø_Y(-D) is weakly positive, then D=0.Thanks to <cit.>, we can find a regular alteration π:Y' → Y of Y,that is, a generically finite morphism π from a smooth projective variety Y' to Y.Let V denote the regular locus of Y and set V':=π^-1(V).Let D'≥0 be a divisor on Y' such that D'|_V'=(π|_V')^*(D|_V). Take an ample Cartier divisor H on Y and α∈ℤ_>0.By the weak positivity of Ø_Y(-D), we see that h^0(Y, Ø_Y(-αβ D +β H))0 for some β∈ℤ_>0,so there is a rational function φ∈ K(Y)with β( -α D +H ) +(φ) ≥ 0. We then have a divisor E ≥0 on Y'such that Supp E ⊆ Y' ∖ V' and β( -α D' +π^*H) +(π^*φ) +E ≥0. Since ( π(Supp E) ) ≤ d-2,where d:= Y,we get E · (π^* H)^d-1=0,and so (-α D' +π^* H)· (π^*H)^d-1≥ 0. This means (-D') · (π^*H)^d-1≥ 0,which can occur only when D'=0,and hence D=0. Let Z be the normalization of Im(a) and f:X→ Z be the induced morphism.We now have the natural morphism Ω_A^1|_Z→Ω_Z^1 that is generically surjective,so H^0(Z,ø_Z)0.We see from Theorem <ref> that ø_Z^-1 is weakly positive,so we get ø_Z≅Ø_Z by Lemma <ref>,and hence a is surjective (i.e., Z=A) by Theorem <ref>. Let a:XYA be the Stein factorization of a.Since the geometric generic fiber of a is F-pure, it is reduced, so a is separable,which implies h is a separable finite morphism.Therefore, we get ø_Y≅ø_Y/A≅Ø_Y(R), where R is the ramification divisor of h. We then find that R=0 by the same argument as before,and so h is an étale morphism by the Zariski–Nagata purity theorem.Hence, we see that Y is an abelian variety by <cit.> and that h is an isomorphism,which is our assertion.§ FROBENIUS SPLIT MORPHISMSIn this section, we introduce and study the notion of an F-split morphism. We fix an algebraically closed field k of characteristic p>0.Let X be a normal variety and Δ an effective ℚ-Weil divisor on X.Let f:X→ Z be a projective morphism to a smooth variety Z.We say that f is sharply F-split (F-split for short)with respect to Δ if there exists an e>0 such that the composite Ø_X_Z^eF_X/Z^(e)_*Ø_X^e↪F_X/Z^(e)_*Ø_X^e(⌈(p^e-1)Δ⌉) *(<ref>.1)_eof the natural homomorphism F_X/Z^(e)^♯ andthe natural inclusion F_X/Z^(e)_*Ø_X^e↪F_X/Z^(e)_*Ø_X^e(⌈(p^e-1)Δ⌉)is injective and splits as an Ø_X_Z^e-module homomorphism. We say that f is locally sharply F-split (locally F-split for short)with respect to Δ if there exists an open covering {V_i} of Z such thatf|_f^-1(V_i):f^-1(V_i)→ V_i is F-split with respect to Δ|_f^-1(V_i) for every i.We often say that the pair (f,Δ) is F-split (resp. locally F-split)if f is F-split (resp. locally F-split) with respect to Δ.We simply say f is F-split (resp. locally F-split) if so is (f,0). (1) If the morphism (<ref>.1)_e splits, then (<ref>.1)_en also splits for every integer n>0. (2) When Z=Spec k, where we recall that k is assumed to be algebraically closed,it is easily seen that (f,Δ) is F-split if and only if (X,Δ) is F-split. (3) Let Δ' be an effective ℚ-divisor on X with Δ'≤Δ.If (f,Δ) is F-split (resp. locally F-split), then so is (f,Δ'). (4) In <cit.>, Hashimoto has dealt with morphisms with local splittings of (<ref>.1)_eAs an example, we consider the case of projective bundles.Let X be the projective bundle associated with a vector bundle ℰ on a normal projective variety Z,and let f:X→ Z be the projection.Then f is locally F-split.Furthermore, if ℰ≅⊕_i=1^nℒ_i, where ℒ_1,…,ℒ_n are line bundles on Z, then f is F-split. The first statement follows from the second.For every m≥0, there exists the natural injective morphism ψ_m:⊕_m_1+⋯+m_n=mℒ_i^m_ip≅F_Z^*S^mℰ→ S^mpℰ.Then, the image of ψ_m is obviously⊕_m_1+⋯+m_n=mℒ_i^m_ip⊆ S^mpℰ, and hence ψ_m splits. The morphism Ø_X_Z^1→F_X/Z^(1)_*Ø_X^1 corresponds to the morphismψ:=⊕_m≥0ψ_m :⊕_m≥0 S^mF_Z^1^*ℰ→⊕_m≥0 S^mpℰ⊆⊕_m≥0 S^mℰ.Since ψ_m splits for every m≥0, we see that ψ also splits,and hence so does Ø_X_Z^1→F_X/Z^(1)_*Ø_X^1.Note that, as we see in Theorem <ref>,there exists a projective bundle over an elliptic curvewhose projection is not an F-split morphism.Next, we prove that if f:X→ Z is F-split with respect to Δ,then there exists a ℤ_(p)-Weil divisor Δ'≥Δ on X such that K_X/Z+Δ'∼_ℤ_(p)0.Let X, Δ, Z and f be as in Definition <ref>.Assume that (f,Δ) is F-split. Then there exists e ∈ℤ_>0 such thatϕ^(e)_(X/Z,Δ):F_X/Z^(e)_* ℒ^(e)_(X/Z,Δ)→Ø_X_Z^e splits as a homomorphism of Ø_X_Z^e-module. Here, we recall that ℒ^(e)_(X/Z,Δ):=Ø_X^e(⌊(1-p^e)(K_X^e/Z^e+Δ)⌋). Then there exists an element s∈ H^0(X^e,⌊(1-p^e)(K_X^e/Z^e+Δ)⌋)such that ϕ^(e)_(X/Z,Δ) sends s to 1.Let E be an effective Weil divisor on X^e defined by s.Set Δ':=(p^e-1)^-1⌈(p^e-1)Δ+E⌉≥Δ.Then by the choice of E we have ℒ^(e)_(X/Z,Δ') := Ø_X^e((1-p^e)(K_X^e/Z^e+Δ')) = Ø_X^e(⌊(1-p^e)(K_X^e/Z^e+Δ)-E⌋)≅Ø_X^e,and ϕ^(e)_(X/Z,Δ):F_X/Z^(e)_*ℒ^(e)_(X/Z,Δ')→Ø_X_Z^e splits. The following lemma shows that an F-split morphism is surjective: Let f:X→ Z be a projective morphism between normal varieties that is locally F-split.Then for each i, the sheaf 𝒢^i:=R^if_*Ø_X is a vector bundle satisfying F_Z^e^*𝒢^i≅𝒢^i for some e>0.In particular, f is surjective. Applying the functor R^if_Z^e_* to Ø_X_Z^e→F_X/Z^(e)_*Ø_X^e,we obtain the morphism R^if_Z^e_*Ø_X_Z^e→ R^if^(e)_*Ø_X^e=𝒢^iwhich is injective and splits locally.Since F_Z is flat, the sheaf R^if_Z^e_*Ø_X_Z^e is isomorphic to F_Z^e^*R^if_*Ø_X=F_Z^e^*𝒢^i,so we get the morphism Φ^i:F_Z^e^*𝒢^i→𝒢^i.We can easily check that Φ^i is an isomorphism,and so the lemma below shows 𝒢^i is locally free. Let M be a finitely generated module over a regular local ring R of positive characteristic.If F_R^e^*M≅ M for some e>0, then M is free. The next proposition shows thatthe local F-splitting of (f,Δ) requires certain conditions on Δ and the fibers of f. Let X, Δ, Z and f be as in Definition <ref>.Assume that (f,Δ) is locally F-split and Δ is ℤ_(p)-Cartier.Then the following hold:(1) Let f:XYZ be the Stein factorization. Then h is étale. (2) The support of Δ does not contain any irreducible component of any fiber. (3) For every scheme-theoretic point z∈ Z,the pair (X_ z,Δ_ z) is F-split,where X_ z is the geometric fiber over z. In particular, X_ z is reduced.(4) There exists an open subset U⊆ X such thatcodim(X∖ U)≥2 and f|_U:U→ Z is a smooth morphism. In particular, the general fiber of f is normal. We first prove (2) and (3).Take a point z∈ Z.Restricting the homomorphism (<ref>.1)_e to X_ z^e,we obtain the homomorphism of Ø_X_ z^e-modulesØ_X_ z^eF_X_ z/ z^(e)_*Ø_(X_ z)^e→F_X_ z/ z^(e)_* ( Ø_(X_ z)^e( (p^e-1)Δ|_(X_ z)^e) ) which is injective and splits for some e>0.This implies that the homomorphismØ_X_ z→(Ø_X(p^e-1)Δ)|_X_ zis not zero over each irreducible component.Hence, Supp Δ does not contain any component of X_ zand (X_ z,Δ_ z) is F-split, so (2) and (3) hold. We show (4). Let π:W→ X_Z^e be the normalization of X_Z^e.Then, F_X/Z^(e):X^e→ X_Z^e factors through W, and we have the morphisms Ø_X_Z^eπ_*Ø_W→F_X/Z^(e)_*Ø_X^e of Ø_X_Z^e-modules, which implies π^# is injective and splits.Since π_*Ø_W/Ø_X_Z^e is a torsion Ø_X_Z^e-module and π_*Ø_W is torsion-free,we find that π_*Ø_W/Ø_X_Z^e=0, so π is an isomorphism, which means X_Z^e is normal. Now, F_X/Z^(e)_*Ø_X^e is a torsion-free sheaf on a normal variety,so it is locally free over an open subset U whose complement X_Z^e∖ U has codimension at least two. Therefore, for every z∈ Z we see thatF_U_ z/ z^(e)_*Ø_(U_ z)^e≅( F_U/Z^(e)_*Ø_U^e) |_U_ z^eis locally free, and so U_ z is regular by Kunz's theorem.Hence f|_U:U→ Z is smooth. This means that the general fiber satisfies S_2 and R_1, i.e. it is normal.We prove (1). Proposition <ref> (1) shows that h is F-split,so we see from Proposition <ref> that h is étale.Note that (1) is used in Section <ref> but not in this section,so the use of Propositions <ref> and <ref> does not cause a logical problem.On the contrary to Proposition <ref> (3), the morphism f is not necessarily F-spliteven if every fiber is F-split (see Theorem <ref> for example).However, if K_X is ℤ_(p)-linearly trivial relative to f,then the converse holds as seen in Theorem <ref> below. This is used in the proofs of Proposition <ref> and Theorem <ref>.Let f:X→ Z be a surjective projective morphism from a normal variety Xto a smooth variety Z such that f_*Ø_X≅Ø_Z.Let Δ be an effective ℤ_(p)-Weil divisor on X.Suppose that K_X+Δ∼_ℤ_(p) f^*C for some Cartier divisor C on Z. Let X_η denote the geometric generic fiber of f.( i) If (X_η,Δ_η) is not F-split,then so is (X_ z,Δ_ z) for general z∈ Z. ( ii) If (X_η,Δ_η) is F-split,then there exists an effective ℤ_(p)-Weil divisor Δ_Z on Z such that the following hold: (1) The divisor (K_Z+Δ_Z) is ℤ_(p)-linearly equivalent to C.(2) The pair (X,Δ) is F-split if and only if so is (Z,Δ_Z).(3) The following are equivalent: (3-1) (f,Δ) is F-split;(3-2) (f,Δ) is locally F-split;(3-3) (X_ z,Δ|_X_ z) is F-split for every codimension one point z∈ Z; (3-4) Δ_Z=0.Here, X_ z is the geometric fiber over z. Obviously, (i) (resp. (1)) follows from <cit.> (resp. <cit.>). By <cit.>, we see that S^0(X,Δ,Ø_X)≅ S^0(Z,Δ_Z,Ø_Z)(see <cit.> or <cit.> for the definition of S^0),so (2) is a consequence of the fact that(X,Δ) is F-split if and only if S^0(X,Δ,Ø_X)=H^0(X,Ø_X). To prove (3), we recall the construction on Δ_Z.Replacing X and Z by their smooth loci, we may assume that X and Z are smooth.For an e>0 with a|(p^e-1), we have f^(e)_*ℒ^(e)_(X/Z,Δ)=f^(e)_*Ø_X^e((1-p^e)(K_X^e/Z^e+Δ))≅Ø_Z^e((1-p^e)(C-K_Z^e))by the projection formula. We then set θ^(e):Ø_Z^e((1-p^e)(C-K_Z^e))≅f^(e)_*ℒ^(e)_(X/Z,Δ)f_Z^e_*Ø_X_Z^e≅Ø_Z^e.Since( f_Z^e_*ϕ^(e)_(X/Z,Δ)) ⊗ k ( η^e ) ≅ H^0 ( X_η^e,ϕ^(e)_(X_η/η,Δ_η))is surjective because of the assumption, θ^(e) is non-zero.Hence there exists an effective divisor E on Z such that Ø_Z^e(-E) is equal to the image of θ^(e).Define Δ_Z:=(p^e-1)^-1E.By definition, Δ_Z=0 if and only if θ^(e) is surjective. Furthermore, by the argument similar to the above, we see that for a codimension one point z∈ Z,the pair (X_ z,Δ|_X_ z) is F-splitif and only if θ^(e)⊗ k( z) is non-zero,i.e. Δ is zero around z. We now prove (3). Obviously, (3-1)⇒(3-2).Proposition <ref> (3) shows (3-2)⇒(3-3),and (3-3)⇒(3-4) follows from the above argument.If Δ_Z=0, i.e., if θ^(e) is an isomorphism,then H^0(X_Z^e,ϕ^(e)_(X/Z,Δ))≅ H^0(Z^e,θ^(e)) is also an isomorphism, so ϕ^(e)_(X/Z,Δ) splits. This proves (3-4)⇒(3-1). Proposition <ref> below deals with the case of finite morphisms.Note that, in the case when Δ=0, Proposition <ref> has already been proved in <cit.>. Let X, Δ, Z and f be as in Definition <ref>.Assume that X= Z. Then the following are equivalent: (1) (f,Δ) is F-split;(2) (f,Δ) is locally F-split;(3) f is étale and Δ=0.Obviously, (1)⇒(2). Let f be étale and Δ=0.Then F_X/Z^(e):X^e→ X_Z^e is a finite morphism of degree one between normal varieties,so it is an isomorphism, which implies (3)⇒(1). We show (2)⇒(3). By Lemma <ref> and Proposition <ref> (4),the morphism f is surjective and separable, so it follows from the assumption that f is generically finite.Let e>0 be an integer such that the morphism Ø_X_Z^eF_X/Z^(e)_*Ø_X^e(⌈(p^e-1)Δ⌉)splits. Since F_X/Z^(e) is a finite morphism of degree one,F_X/Z^(e)_*Ø_X^e(⌈(p^e-1)Δ⌉) is a torsion-free sheaf of rank one.Note that as f is separable, X_Z^e is a variety. Therefore, Coker α=0, so it is an isomorphism,which means that Δ=0 and F_X/Z^(e) is an isomorphism. Then, F_X_ z/ z^(e) is also an isomorphism for every z∈ Z,where z is the algebraic closure of z. Hence, X_ z is isomorphic to a disjoint union of copies of the spectrum of k( z).In particular, f is finite. The local freeness of f_*Ø_X, which follows from Lemma <ref>, implies f is flat,and so we conclude that f is étale. The following lemma is used in the proofs of Proposition <ref> and Theorem <ref>. Let X, Δ, Z and f be as in Definition <ref>.Assume that (f,Δ) is locally F-split and that Δ is a ℤ_(p)-Weil divisor.Then the Iitaka–Kodaira dimension κ(X,K_X/Z+Δ) of K_X/Z+Δ is non-positive.Furthermore, if (f,Δ) is F-split, then κ(X,-(K_X/Z+Δ))≥0.The second statement follows from Observation <ref>. By Lemma <ref>, the morphism f is surjective. Assume that κ(X,K_X/Z+Δ)≥0.Then κ(X_η,K_X_η/η+Δ_η)≥0,where η is the geometric generic point of Z. Since (X_η, Δ_η) is F-split,we have H^0(X_η,(1-p^e)(K_X_η+Δ_η))0 for some e>0,so (1-p^e)(K_X_η+Δ_η)∼ 0. The morphism f_Z^e_*ϕ^(e)_(X/Z,Δ):f^(e)_*Ø_X^e((1-p^e)(K_X^e/Z^e+Δ))→f_Z^e_*Ø_X_Z^eis then a surjective morphism between torsion-free coherent sheaves of the same rank,and so it is an isomorphism. Hence, H^0(X,(1-p^e)(K_X/Z+Δ))0, which implies κ(X,K_X/Z)=0.This is our assertion. Finally, we consider the composition of (locally) F-split morphisms, which is used frequently in Section <ref>.Let X, Δ, Z and f be as in Definition <ref>,and Y be a normal variety.Assume that f:X→ Z can be factored into projective morphisms g:X→ Y with g_*Ø_X≅Ø_Y and h:Y→ Z.Suppose that Z is projective.(1) If (f,Δ) is F-split, then so is h.(2) Assume that Y is smooth. If (g,Δ) and h are F-split, then so is (f,Δ). (3) The converse of (2) holds if K_Y∼_ℤ_(p)h^*K_Z.Let e>0 be an integer.Now we have the following commutative diagram: @R=25pt@C=50ptX^e [r]|F_X/Y^(e)@/^20pt/[rr]^F_X/Z^(e)[dr]_g^(e) X_Y^e[d]^g_Y^e[r]|π^(e)X_Z^e[r]^(F_Z^e)_X[d]^g_Z^eX [d]_g@/^30pt/[dd]^f Y^e [r]^F_Y/Z^(e)[dr]_h^(e)Y_Z^e[d]^h_Z^e[r]^(F_Z^e)_YY [d]_hZ^e [r]^F_Z^eZ. Here, π^(e):=(F_Y/Z^(e))_X. We first show (1). The above diagram induces the commutative diagram of Ø_Y_Z^e-modules @R=15pt@C=20ptØ_Y_Z^e[r] [d]_≅ F_Y/Z^(e)_*Ø_Y^e[d]^≅ g_Z^e_*Ø_X_Z^e[r]g_Z^e_*F_X/Z^(e)_*Ø_X^e, where the left vertical morphism is an isomorphism because of the flatness of (F_Z^e)_Y.Since the lower horizontal morphism splits, so does the upper one. Next, we show (2) and (3).As explained in Observation <ref>,if (g,Δ) (resp. (f,Δ)) is F-split,then there exists an effective ℤ_(p)-Weil divisor Δ'≥Δ on Xsuch that K_X/Y+Δ' (resp. K_X/Z+Δ') is ℤ_(p)-linearly trivialand that (g,Δ') (resp. (f,Δ')) is also F-split. Therefore, we may assume that Δ is a ℤ_(p)-Weil divisorand that (p^e-1)(K_X/Y+Δ)∼0 (resp. (p^e-1)(K_X/Y+Δ)∼(p^e-1)(f^*K_Z-g^*K_Y)) for every e>0 divisible enough.In particular, ℒ^(e)_(X/Y,Δ) (resp. ℒ^(e)_(X/Z,Δ)) is isomorphic tothe pullback by g^(e) of a line bundle on Y^(e). Let V⊆ Y be an open subset such thatX_V:=g^-1(V) is flat over V and codim(Y∖ V)≥2. Let u:U→ X_V be the open immersion of the regular locus of X_V.Set g':=g∘ u:U→ Y. We then have g'_*Ø_U≅ g_*Ø_X≅Ø_Y because of the assumptions. In addition, by the flatness of F_Z^e, we see that g'_Z^e_*Ø_U_Z^e≅Ø_Y_Z^e. Hence, by the projection formula, we see that H^0(U_Z^e,(g_Z^e^*ℒ)|_U_Z^e)≅ H^0(Y_Z^e,g'_Z^e_*(g'_Z^e^*ℒ))≅ H^0(Y_Z^e,ℒ) ≅ H^0(X_Z^e,g_Z^e^*ℒ) for every line bundle ℒ on Y_Z^e,and so we get the following commutative diagram: @R=25pt@C=85pt H^0(U^e,ℒ^(e)_(X/Z,Δ)|_U^e) [r]^H^0(U_Z^e,ϕ_(U/Z,Δ|_U)^(e))[d]_≅H^0(U_Z^e,Ø_U_Z^e) [d]^≅ H^0(X^e,ℒ^(e)_(X/Z,Δ)) [r]^H^0(X_Z^e, ϕ_(X/Z,Δ)^(e))H^0(X_Z^e,Ø_X_Z^e). In particular, H^0(U_Y^e,Ø_U_Y^e)≅ H^0(Y^e,Ø_Y^e)≅ k. Since the splitting of ϕ^(e)_(X/Z,Δ) is clearly equivalent tothe surjectivity of H^0(X_Z^e,ϕ_(X/Z,Δ)^(e)),we see that the F-splitting of (f,Δ) and that of (f|_U:U→ Z,Δ|_U) are equivalent. By an argument similar to the above,we find that the F-splitting of (g,Δ) and that of (g|_U,Δ|_U) are also equivalent. Assume that we can choose V=Y and U=X, i.e. X and Y are regular and g is flat. Let e>0 be an integer.By the flatness of g, we have the following commutative diagram: @R=25pt@C=85ptg_Z^e^*Ø_Y_Z^e[r]^g_Z^e^*(F_Y/Z^(e)^♯)[d]_≅ g_Z^e^*F_Y/Z^(e)_*Ø_Y^e[d]^≅ Ø_X_Z^e[r]^π^(e)^♯ π^(e)_*Ø_X_Y^e. This implies that ℋom (π^(e)^♯,Ø_X_Z^e)≅g_Z^e^*ℋom (F_Y/Z^(e)^♯,Ø_V_Z^e) =g_Z^e^*ϕ_Y/Z^(e). Applying the functor ℋom(,Ø_X_Z^e)and the Grothendieck duality to the natural morphism Ø_X_Z^eπ^(e)_*Ø_X_Y^e→F_X/Z^(e)_*Ø_X^e(⌈(p^e-1)Δ⌉), we obtain the morphism ϕ^(e)_(X/Z,Δ):F_X/Z^(e)_*ℒ^(e)_(X/Z,Δ)g_Z^e^*F_Y/Z^(e)_*ℒ^(e)_Y/ZØ_X_Z^e. Note that ø_π^(e)≅ø_X_Y^e⊗π^(e)^*ø_X_Z^e≅g_Z^e^*ø_Y^e/Z^e^1-p^eg_Z^e^*F_Y/Z^(e)_*ℒ^(e)_Y/Z≅π^(e)_*g_Y^e^*ℒ^(e)_Y/Z. We now prove the assertion.If (g,Δ) is F-split and h is F-split,then both of ϕ^(e)_(X/Y,Δ) and ϕ^(e)_Y/Z split for every e>0 divisible enough.Therefore, ϕ^(e)_(X/Z,Δ) also splits, i.e. (f,Δ) is F-split. Conversely, suppose that (f,Δ) is F-split and that (p^e-1)K_Y/Z∼ 0 for an e>0. Then, ℒ^(e)_Y/Z≅Ø_Y_Z^e and ø_π^(e)≅Ø_X_Y^e.Fix an e>0 divisible enough.Since H^0(X_Z^e,ϕ_(X/Z,Δ)^(e)) is surjective,H^0(X_Z^e,π^(e)_*ϕ_(X/Y,Δ)^(e)) is a non-zero morphism,and hence so is H^0(X_Y^e,ϕ^(e)_(X/Y,Δ)).This morphism is surjective because of H^0(X_Y^e,Ø_X_Y^e)≅ H^0(Y^e,Ø_Y^e)≅ k.Thus, ϕ^(e)_(X/Y,Δ) splits, and so (g,Δ) is F-split. Note that the F-splitting of h follows directly from (1). § VARIETIES WITH F-SPLIT ALBANESE MORPHISMIn this section, we proveTheorems <ref>, <ref>, <ref> and <ref>. Throughout this section, we fix an algebraically closed field k of characteristic p>0,and we denote by X and Δ respectivelya normal projective variety over k and an effective ℚ-Weil divisor on X.Suppose that (a,Δ) is locally F-split,and let XZA be the Stein factorization of a.Thanks to Proposition <ref>,we only need to show that the étale morphism g is an isomorphism.Applying <cit.>, we see that Z is an abelian variety,so the universal property of a tells us that g is an isomorphism, which completes the proof.The next lemma is used to prove Theorems <ref> and <ref>.Let ℱ be a coherent sheaf of rank r on a normal variety Y.Let ℱ' be an indecomposable direct summand of ℱ whose rank is r'.Set I:={ℒ∈Pic(Y)|ℱ⊗ℒ≅ℱ} and I':={ℒ∈ I|ℱ'⊗ℒ≅ℱ'}.Then ⊕_[ℒ]∈ I/I'ℱ'⊗ℒ is a direct summand of ℱ.In particular, #(I/I')≤ r/r'.For every ℒ∈ I, the sheaf ℱ'⊗ℒ is also a direct summand of ℱ.Furthermore, ℱ⊗ℒ≅ℱ⊗ℒ' if and only if ℒ'⊗ℒ^-1∈ I.Hence, the Krull–Schmidt theorem <cit.> tells us that⊕_[ℒ]∈ I/I'ℱ'⊗ℒ is a direct summand of ℱ,and so r'#(I/I')≤ r, which is our claim. To prove Theorem <ref>,we recall a characterization of ordinary abelian varieties which was established by Sannai and Tanaka.A smooth projective variety Y is an ordinary abelian varietyif and only if K_Y is pseudo-effective and F_Y^e_*Ø_Y is isomorphic to a direct sum of line bundles for infinitely many e>0.It was shown in <cit.> thatwe actually need to check the decomposition of F_Y^e_*Ø_X only for e=2 in the above theorem.For convenience, we use the following notation:Let φ:S→ T be a morphism of schemes.We denote by Pic(S)[φ] (resp. Pic^0(S)[φ])the kernel of the induced homomorphism φ^*:Pic(T)→Pic(S) (resp. φ^*:Pic^0(T)→Pic^0(S)). Then, Pic(X)[F_X^e] is the set of p^e-torsion line bundles for every e>0.We denote it by Pic(X)[p^e].We first prove that if (X,Δ) is F-split,then (a,Δ) is F-split and A is ordinary. We have the following commutative diagram @R=20pt @C=20ptH^1(X,Ø_X) [r]^F_X^*H^1(X,Ø_X)H^1(A,Ø_A) [u]^a^*[r]^F_A^*H^1(A,Ø_A) [u]_a^*. Since X is F-split, the upper horizontal arrow is bijective.By <cit.>, we see that the vertical arrows are injective,so the lower horizontal arrow is also injective, which implies A is ordinary.(Note that, although X is assumed to be smooth in <cit.>,the proof does not use smoothness of X.)Let XZA be the Stein factorization of a.Proposition <ref> (1) then shows that Z is F-split,so Ø_Z is a direct summand of ℱ^(e):=F_Z^e_*Ø_Z^e for each e>0.Since a^*:Pic^0(A)→Pic^0(X) is bijective,g^*:Pic^0(A)→Pic^0(Z) is injective, so p^e· A=#Pic^0(A)[F_A^e]≤#Pic^0(Z)[F_Z^e]. By the projection formula and Lemma <ref> (set ℱ:=ℱ^(e) and ℱ':=Ø_Z),we obtainp^e· A≤#{ℒ∈Pic(Z)|ℱ^(e)⊗ℒ≅ℱ^(e)}≤ rank ℱ^(e) =p^e· Z.This implies that Z= A and that⊕_ℒ∈Pic(Z)[p^e]ℒ⊆ℱ^(e) is a direct summand of maximum rank.This inclusion is an isomorphism because of the torsion-freeness of ℱ^(e).In particular, F_Z^e is flat, i.e. Z is smooth. We now only need to prove that K_Z is pseudo-effective.If this holds, then Theorem <ref> shows that Z is an ordinary abelian variety,so we see from Proposition <ref> (3) that (a,Δ) is F-split, which is our assertion. Fix e>0. We now have (ℱ^(e))^*≅ℱ^(e) and F_Z^e^*ℱ^(e)≅⊕Ø_Z^e.By (<ref>.2) of Subsection <ref>, we haveF_Z^e_*ø_Z^e^1-p^e≅ℋom(F_Z^e_*Ø_Z^e,Ø_Z)=(ℱ^(e))^*≅ℱ^(e),so we have surjective morphisms ⊕Ø_Z^e≅F_Z^e^*ℱ^(e)≅F_Z^e^*F_Z^e_*ø_Z^e^1-p^e→ø_Z^e^1-p^e,which implies that ø_Z^1-p^e is globally generated.Since H^0(Z^e,ø_Z^e^1-p^e)≅ H^0(Z,ℱ^(e))≅ k,we get ø_Z^1-p^e≅Ø_Z, i.e. ø_Z^p^e-1≅Ø_Z, and so K_Z is pseudo-effective. The converse follows directly from Proposition <ref>.Assume that (a,Δ) is locally F-split.Lemma <ref> shows the first statement.We show the second. Suppose A= X.Then Proposition <ref> tells us that a is an isomorphism and Δ=0.The remainder of this section is devoted to prove Theorem <ref> below.We continue to use the same notation as that introduced at the begging of this section. Let γ_A denote the p-rank of A.Let f:X→ B be a morphism to an abelian variety B of p-rank γ_B.Suppose that (f,Δ) is F-split.Then (a,Δ) is F-split and γ_A=γ_B+ A- B.In particular, if B is ordinary, then (X,Δ) is F-split. This theorem is a consequence of Proposition <ref>.To prove the proposition, we need the lemma below.Let f:X→ Z be an F-split morphism to a smooth projective variety Z.Let X_z be the general fiber of f.Thenh^1(X,Ø_X)≤ h^1(X_z,Ø_X_z)+h^1(Z,Ø_Z). Set 𝒢^i:=R^if_*Ø_X.Lemma <ref> then shows thatrank 𝒢^i=h^i(X_z,Ø_X_z) and F_Z^e^*𝒢^i≅𝒢^i for some e>0.Applying <cit.>,we find an étale cover π:Z'→ Z such that π^*𝒢^i≅⊕Ø_Z' for each i,so h^0(Z,𝒢^i) ≤ h^0(Z',π^*𝒢^i)=rank 𝒢^i=h^i(X_z,Ø_X_z). By the Leray spectral sequence, we conclude thath^1(X,Ø_X)≤ h^0(Z,𝒢^1)+h^1(Z,Ø_Z) ≤ h^1(X_z,Ø_X_z)+h^1(Z,Ø_Z). Let f:X→ Z be a separable surjective morphism to a smooth projective variety Z such that f_*Ø_X≅Ø_Z. (1) We consider the following commutative diagram: @R=25pt@C=25ptPic(X^e) Pic(X_Z^e) [u]^F_X/Z^(e)^* Pic(X) [l]^(F_Z^e)_X^*[ul]_F_X^e^* Pic(Z^e) [u]^f_Z^e^* Pic(Z) [l]_F_Z^e^*[u]_f^*Clearly, f^* induces an injective homomorphism Pic(Z)[p^e]Pic(X)[(F_Z^e)_X].This is actually an isomorphism.Indeed, for every ℒ∈Pic(X)[(F_Z^e)_X], we see from the flatness of F_Z^e that F_Z^e^*f_*ℒ≅f_Z^e_*ℒ_Z^e≅f_Z^e_*Ø_X_Z^e≅F_Z^e^*f_*Ø_X ≅Ø_Z^e,so f_*ℒ is a p^e-torsion line bundlesuch that the natural morphism f^*f_*ℒ→ℒ is an isomorphism.Hence, we now have the exact sequence0→Pic(Z)[p^e] Pic(X)[p^e] Pic(X_Z^e)[F_X/Z^(e)]. (2)Set ℱ:=F_X/Z^(e)_*Ø_X^e and I:={ℒ∈Pic(X_Z^e)|ℱ⊗ℒ≅ℱ}.Then, we have Pic(X_Z^e)[F_X/Z^(e)]⊆ I by the projection formula.Let ℱ' be an indecomposable direct summand of ℱ and let I':={ℒ∈Pic(X_Z^e)|ℱ'⊗ℒ≅ℱ'}. Then by Lemma <ref>, we obtain that ⊕_[ℒ]∈ I/I'ℱ'⊗ℒ is a direct summand of ℱ.In particular,rank ℱ'·#(I/I')≤ rank ℱ=p^e( X- Z). Let f:X→ Z be an F-split morphism to an abelian variety Z.Suppose that the Albanese morphism a:X→ A of X is a finite morphism.Then, a is an isomorphism, or equivalently, X is an abelian variety.Let f:XZ'Z be the Stein factorization.Then, Proposition <ref> (1) shows that π is étale,so Z' is an abelian variety by <cit.>.Combining this with Proposition <ref> (3),we see that (f',Δ) is F-split.We may assume that f_*Ø_X≅Ø_Z by replacing Z by Z'. By the universal property of a, we can write f:XAZ.Let z∈ Z be a general point.Proposition <ref> then tells us that X_z is integral, normal and F-split. Recall that a is assumed to be finite.The induced morphism X_z→ (A_z)_red is then a finite morphism to an abelian variety,so Theorem <ref> implies that X_z is an ordinary abelian variety. Therefore, by Lemma <ref>, we have A≤ h^1(X,Ø_X)≤ h^1(X_z,Ø_X_z)+h^1(Z,Ø_Z)= X_z+ Z= X, and so a is a surjective finite morphism.Since f is F-split, it is separable, and hence so is g, which implies that A_z is reduced. We may assume X_z→ A_z is an isogeny of abelian varieties.Considering p-torsion points, we find that A_z is also ordinary,so Theorem <ref> (ii) says that g is F-split, and γ_A=γ_A_z+γ_Z = A_z+γ_Z = A- Z+γ_Z,<ref>.1where γ denotes the p-rank.The morphism a:X→ A is separable. If the claim holds, then X is an abelian variety.Indeed, since f is F-split, Lemma <ref> says that κ(X,K_X/A)=κ(X,K_X)=κ(X,K_X/Z)≤0, so the Zariski–Nagata purity theorem implies a is étale,and hence it follows from <cit.> that X is an abelian variety. Let s:Y→ A be the normalization of A in the separable closure of K(X)/K(A).Then, there is the purely inseparable finite cover i:X→ Y with a=s∘ i. We prove i is an isomorphism. There exist an e>0 and a morphism b:Y^e→ X such that the following diagram commutes: @R=25pt@C=25pt X^e [r]^F_X^e[d]_i^(e)X [d]^i Y^e [r]_F_Y^e[ur]^bY.This induces the following commutative diagram:@R=25pt@C=25pt X^e [r]^F_X/Z^(e)[d]_i^(e)X_Z^e[d]^i_Z^e[r]^(F_Z^e)_XX[d]^i @/^25pt/[dd]_a @/^50pt/[ddd]^f Y^e [r]_F_Y/Z^(e)[ur]^b_Z^eY_Z^e[r]_(F_Z^e)_Y[d]|s_Z^eY [d]^sA_Z^e[r]_(F_Z^e)_A[d]_g_Z^eA [d]^gZ^e [r]_F_Z^eZ.Note that since f and g∘ s are separable,Lemma <ref> says that X_Z^e and Y_Z^e are varieties. The splitting of F_X/Z^(e)^#:Ø_X_Z^e→F_X/Z^(e)_*Ø_X^e induces that of b_Z^e^#:Ø_X_Z^e→b_Z^e_*Ø_Y^e.Pushing forward via i_Z^e, we find thati_Z^e_*Ø_X_Z^e is isomorphic to a direct summand of ℱ:=F_Y/Z^(e)_*Ø_Y^e. Let ℱ' be the indecomposable direct summand of i_Z^e_*Ø_X_Z^e with H^0(Y_Z^e,ℱ')0. Set I:={ℒ∈Pic(Y_Z^e)|ℱ⊗ℒ≅ℱ}I':={ℒ∈ I|ℱ'⊗ℒ≅ℱ'}.For any p^e-torsion line bundle ℒ on Y, we have ℒ_Z^e=(F_Z^e)_Y^*ℒ∈ I.Indeed,ℱ⊗ℒ_Z^e = (F_Y/Z^(e)_*Ø_Y^e)⊗_Ø_Y_Z^eℒ_Z^e≅F_Y/Z^(e)_*(F_Y/Z^(e)^*ℒ_Z^e)≅F_Y/Z^(e)_*(F_Y^e^*ℒ)≅F_Y/Z^(e)_*Ø_Y^e=ℱ Consider the morphism Φ:Pic(A)[p^e]→ I/I' obtained by Pic(A)[p^e] Pic(Y)[p^e]I I/I'. We show that the following sequence is exact: 0→Pic(Z)[p^e]Pic(A)[p^e] I/I'. <ref>.2Take ℳ∈Pic(Z)[p^e].Then, (F_Z^e)_Y^*s^*g^*ℳ≅s_Z^e^*g_Z^e^*F_Z^e^*ℳ≅s_Z^e^*g_Z^e^*Ø_Z^e≅Ø_Y_Z^e∈ I'. Take ℳ∈Pic(A)[p^e] such that 𝒩:=(F_Z^e)_Y^*s^*ℳ∈ I'.We then have 0 H^0(Y_Z^e,ℱ') ≅ H^0(Y_Z^e,ℱ'⊗𝒩)⊆ H^0(Y_Z^e,(i_Z^e_*Ø_X_Z^e)⊗𝒩) ≅ H^0(X_Z^e,i_Z^e^*𝒩)so we get i_Z^e^*𝒩≅Ø_X_Z^e, which implies that(F_Z^e)_X^*a^*ℳ≅i_Z^e^*(F_Z^e)_Y^*s^*ℳ =i_Z^e^*𝒩≅Ø_X_Z^e.Observation <ref> (1) tells us thata^*ℳ≅ a^*g^*𝒫 for some 𝒫∈Pic(Z)[p^e],and the injectivity of a^*:Pic^0(A)→Pic^0(X) shows that ℳ≅ g^*𝒫.Hence, we see that the sequence <ref> is exact.Now, we have the following inequalities: p^e( A- Z)·rank ℱ'= p^e(γ_A-γ_Z)·rank ℱ' ≤#(I/I') ·rank ℱ' ≤rank ℱ=p^e( A- Z) This implies that rank ℱ'=1 and#(I/I')=p^e( A- Z), so Φ is surjective.Furthermore, it follows from the torsion-freeness of ℱ thatℱ≅⊕_[ℒ]∈ I/I'ℱ'⊗ℒ. Since i_Z^e_*Ø_X_Z^e is a direct summand of ℱ,it is isomorphic to ⊕_[ℒ]∈ Jℱ'⊗ℒfor some J⊆ I/I'.However, we can prove that this J must be {[Ø_Y_Z^e]} as follows. Take ℳ∈Pic(A)[p^e] with Φ(ℳ)∈ J.Set 𝒩:=(F_Z^e)_Y^*s^*ℳ.We then have 0 H^0(Y_Z^e,ℱ')=H^0(Y_Z^e,ℱ'⊗𝒩⊗𝒩^-1) ⊆ H^0(Y_Z^e,(i_Z^e_*Ø_X_Z^e)⊗𝒩^-1) =H^0(X_Z^e,i_Z^e^*𝒩^-1),so (F_Z^e)_X^*a^*ℳ≅i_Z^e^*𝒩≅Ø_X_Z^e.By an argument similar to the above,we find some 𝒫∈Pic(Z)[p^e] with ℳ≅ g^*𝒫,so the exactness of (<ref>) shows that Φ(ℳ)=[Ø_Y_Z^e].We then see that J={[Ø_Y_Z^e]} by the surjectivity of Φ,so i_Z^e is an isomorphism, and hence so is i, which proves our claim. Let XX'A be the Stein factorization of a.Then, f can be written as f:XX'A B.Proposition <ref> (1) says that h∘ g':X'→ B is F-split.Since the morphism g':X'→ A is finite and is the Albanese morphism of X',we see from Proposition <ref> that g' is an isomorphism.Therefore, Proposition <ref> (3) implies that (a,Δ) is F-split.Since h:A→ B is an F-split morphism whose closed fibers A_z are ordinary abelian varieties, we obtainγ_A=γ_A_z+γ_B= A_z+γ_B= A- B+γ_B.§ MINIMAL SURFACES WITH F-SPLIT ALBANESE MORPHISM Fix an algebraically closed field k of characteristic p>0.In this section, we study the Albanese morphismof a minimal surface over k in terms of F-splitting.Here, by a minimal surface, we mean a smooth projective surface with no (-1)-curves. Note that it follows from Proposition <ref> (1) thatif a smooth projective surface has F-split (resp. locally F-split) Albanese morphism,then so does its minimal surface.Throughout this section, X denotes a smooth projective minimal surfaceand a:X→ A denotes the Albanese morphism of X. The morphism a is locally F-split if and only ifone of the following conditions holds: (0) b_1(X)=0 and X is F-split;(1-1) b_1(X)=2, κ(X)=-∞ andX is the projective bundle ℙ(ℰ) associated with a rank two vector bundle ℰ on A; (1-2) b_1(X)=2, κ(X)=0 andX is a hyperelliptic surface such that every closed fiber of a is an ordinary elliptic curve; (2) X is an abelian surface. In the case of (1-1), the morphism a is F-split if and only if either ( a) ℰ is decomposable, ( b) ℰ is indecomposable, p>2 and ℰ is odd, or( c) ℰ is indecomposable, p=2 and A is ordinary.In the case of (0), (1-2) and (2), the morphism a is F-split. Note that the first Betti number b_1(X) is equal to 2 A.By Theorem <ref>, we see that b_1(X)=0, 2 or 4.If b_1(X)=0, then the F-splitting of a is equivalent to the F-splitting of X.If b_1(X)=4, then X is an abelian surface as shown in Theorem <ref>.The case when b_1(X)=2 is dealt with in the remainder of this section.As shown by Lemma <ref>, we have κ(X)≤0.§.§ The case b_1(X)=2 and κ(X)=0 In this case, according to Bombieri and Mumford's classification of minimal surfaces with Kodaira dimension zero <cit.>,we find that X is a hyperelliptic or quasi-hyperelliptic surface.If a is locally F-split, then Proposition <ref> shows that a has normal geometric generic fiber, so X is hyperelliptic. Hence, X can be obtained as the quotient ofthe product E_0× E_1 of two elliptic curves E_0 and E_1by an action of a finite group B <cit.>. Then, A≅ E_1/B and every closed fiber of a is isomorphic to E_0.The followings are equivalent: (1) a is F-split;(2) a is locally F-split;(3) E_0 is ordinary. The implication (1)⇒(2) is obvious.If a is locally F-split, Proposition <ref> says that the general fiber is F-split,so E_0 is F-split, which shows (2)⇒(3).We prove (3)⇒(1). Assume that E_0 is F-split. Since a is flat and every fiber has trivial canonical bundle,K_X∼ a^*C for a Cartier divisor C on A. We see from the assumption that C∼_ℚ0.Theorem <ref> (ii)-(3) then tells us that a is F-split, which is our claim. §.§ The case b_1(X)=2 and κ(X)=-∞ In this case, X is a ruled surface over an elliptic curve.We start with recalling some facts about vector bundles on elliptic curves.In the following theorem and lemmas, we let C be an elliptic curve. Let ℰ_C(r,d) be the set of isomorphism classes of indecomposable vector bundleson C of rank r and of degree d.(1)<cit.>For every ℰ, ℰ'∈ℰ_C(r,d), there exists an ℒ∈Pic^0(C) such that ℰ⊗ℒ≅ℰ'.Take ℒ_1, ℒ_2∈Pic^0(C).Then, ℰ⊗ℒ_1≅ℰ⊗ℒ_2 if and only if ℒ_1^r'≅ℒ_2^r', where r':=r/(r,d).Furthermore, when d=0, there exists a unique element ℰ_r,0 in ℰ_C(r,0) such that H^0(C,ℰ_r,0)0. (2)<cit.>Let π:C'→ C be an isogeny of degree r and ℒ a line bundle of degree d on C'. If r and d are coprime, then π_*ℒ∈ℰ_C(r,d). (3)<cit.>Let r>0 and d be coprime integers and suppose that ℰ∈ℰ_C(rh,dh) for some h>0.When C is ordinary, F_C^*ℰ is indecomposable.When C is supersingular, F_C^*ℰ is indecomposable if and only ifeither h 1 and p∤ r, or h=1. Let ℱ be a vector bundle on C of rank rand ℒ a line bundle such that ℱ⊗ℒ≅ℱ. Then ℒ^r≅Ø_C.This follows from (ℱ)⊗ℒ^r≅(ℱ⊗ℒ)≅ℱ.Let π:C'→ C be a finite morphism of degree d from an elliptic curve C'.Let ℒ be a line bundle on C such that π^*ℒ≅Ø_C'.Then ℒ^d≅Ø_C.The projection formula shows (π_*Ø_C')⊗ℒ≅π_*Ø_C',so the claim follows from Lemma <ref>. In characteristic zero, the pullback of ℰ_r,0to another elliptic curve via a finite morphism is again indecomposable. However, the lemma below shows that, in positive characteristic,the pullback of ℰ_r,0 can be trivial. There exists a finite morphism π:C'→ C from an elliptic curve C' such that the following conditions hold: (1) π^*ℰ_2,0≅Ø_C'⊕Ø_C';(2) π_*Ø_C'≅ℰ_p,0; (3) π^*:Pic^0(C)→Pic^0(C') is injective.Since ℰ_2,0 is obtained as a nontrivial extension ξ of Ø_C by Ø_C,it is enough to find a finite morphism π:C'→ C froman elliptic curve such that π^* kills H^1(C,Ø_C). If C is ordinary, i.e., F_C^*:H^1(C,Ø_C)→ H^1(C,Ø_C) is an isomorphism, then we may assume that F_C^*ξ_1=ξ_1. In this case, ξ_1 defines an étale cover π:C'→ C of degree p such that π^*ξ_1=0.If C is supersingular, or equivalently F_C^*ξ_1=0, then we set π:=F_C.Next, we prove (2).Let ℱ be an indecomposable direct summand of π_*Ø_C'.Theorem <ref> says that ℱ≅ℰ_r,0⊗ℒ for an ℒ∈Pic^0(C),where r:=rank ℱ.Then, by the projection formula, we have ℰ_r,0≅ℱ⊗ℒ^-1⊆ (π_*Ø_C')⊗ℒ^-1≅π_*π^*ℒ^-1.This means that π^* ℒ^-1 is a numerically trivial line bundle having non-zero global sections,so it is trivial, and hence Lemma <ref> implies that ℒ is p-torsion. If C is ordinary, then π is étale, so the commutative diagram C' [r]^F_C'[d]_πC' [d]^πC [r]^F_CCis cartesian, which implies F_C^*π_*Ø_C'≅π_*Ø_C'.From this, we find π_*Ø_C'≅ℰ_p,0. If C is supersingular, then C has no nontrivial p-torsion line bundle, so ℒ≅Ø_C. This means that π_*Ø_C'≅ℰ_p,0. Finally, we show (3). Take ℒ∈Pic(C) with π^*ℒ≅Ø_C'.Combining (2) with the projection formula,we get ℰ_p,0⊗ℒ≅ℰ_p,0,so Theorem <ref> (1) says that ℒ≅Ø_C,which is our assertion.The m-th symmetric product S^mℰ_2,0 of ℰ_2,0 is isomorphic to a direct sum of vector bundles of the form ℰ_r,0.Let ℱ be an indecomposable direct summand of S^mℰ_2,0.Theorem <ref> says that ℱ≅ℰ_r,0⊗ℒ for an ℒ∈Pic^0(C),where r:=rank ℱ.Let π:C'→ C be as in Lemma <ref>.Since π^*S^mℰ_2,0 is trivial, we have π^*ℒ≅Ø_C',so Lemma <ref> (3) shows that ℒ≅Ø_C,which completes the proof.We now return to the study of the Albanese morphism a:X→ A of X.We may regard a:X→ A as the natural projection ℙ(ℰ)→ Aof the projective bundle ℙ(ℰ) for a vector bundle ℰ on A of rank two.If ℰ is decomposable, then a is F-split as shown in Example <ref>.Hence, we assume that ℰ is indecomposable in the remaining of this section.We only need to consider the case when ℰ=0 or 1. §.§.§ The case when ℰ is an indecomposable vector bundle of degree 0. In this case, we may assume that ℰ=ℰ_2,0 by Theorem <ref> (1). Lemma <ref> then says that we have a finite morphism π:A'→ Afrom an elliptic curve A' such that π^*ℰ_2,0≅Ø_A'^⊕ 2.In particular, X_A'≅ℙ(π^*ℰ_2,0)≅ℙ^1× A^1.We show the following: The morphism a:X→ A is F-split if and only if A is ordinary and p=2. To prove Proposition <ref>, we prepare the claims below. There exists an algebraic fiber space g:X→ Y≅ℙ^1 such that g^*Ø_Y(1)≅Ø_X(p).The p-th symmetric power S^pℰ_2,0 of ℰ_2,0 is isomorphic to ℰ_p,0⊕Ø_A. Since π^* preserves the Iitaka-Kodaira dimension (cf. <cit.>),we have 1≤κ(X,Ø_X(1)) =κ(X_A',Ø_X_A'(1)) =κ(ℙ^1,Ø_ℙ^1(1)) =1, so Ø_X(1) is semi-ample.Let g:X→ Y be the Iitaka fibration associated to Ø_X(1).Then, Y≅ℙ^1 and a general fiber B of g is an elliptic curve. Now we have the following commutative diagram:B_A'[r] [d] B [d] X_A'[r]^π_X[d]_a_A'X [d]^a[r]^g YA' [r]_πA. By the construction, we have Ø_X(B)≅Ø_X(m)⊗ a^*ℒfor an m≥ 2 and a torsion line bundle ℒ on A. Take l∈ℤ.We consider the exact sequence 0→Ø_X(l)⊗Ø_X(-B)→Ø_X(l)→Ø_B→ 0<ref>.1of Ø_X-modules. Pushing forward via a, when l=m-1, we get S^m-1ℰ_2,0≅ a_*Ø_B.This shows H^0(A,S^m-1ℰ_2,0)≅ k,so Lemma <ref> says S^m-1ℰ_2,0≅ℰ_m,0,which means π^*a_*Ø_B≅Ø_A'^⊕ m.Since π:A'→ A and a|_B:B→ A are flat, we obtain p≥ H^0(B,(a|_B)^*ℰ_p,0)= H^0(B,(a|_B)^*π_*Ø_A')= H^0(B_A',Ø_B_A') = H^0(A',π^*a_*Ø_B)=m≥2.If p>m, then (a|_B):Ø_A→ (a|_B)_*Ø_B splits,so (a|_B)^*ℰ_p,0≅ℰ_p,0, which contradicts the above.Hence, p=m.Pushing forward (<ref>) via a again, when l=p, we obtain the exact sequence 0→ℒ^-1→ S^pℰ_2,0→ a_*Ø_B→0.<ref>.2Combining this with Lemma <ref>,we see that ℒ≅Ø_A, i.e., g^*Ø_Y(1) ≅Ø_X(B) ≅Ø_X(p),which is Claim <ref>. Also, since H^0(A,S^pℰ_2,0)≅ H^0(X,Ø_X(p))≅ H^0(Y,Ø_Y(1))=k^⊕ 2,we see that (<ref>) splits, which is Claim <ref>.We now start the proof of Proposition <ref>. We use the same notation as the proof of Claims <ref> and <ref>. We first prove that (a,B) is F-split, assuming that A is ordinary and p=2. We now have ø_X⊗Ø_X(B)≅Ø_X(-2)⊗Ø_X(p)≅Ø_X.Hence, thanks to Theorem <ref> (ii)-(3),it is enough to show that (X_z,B|_X_z) is F-split for a fiber X_z≅ℙ^1 of a.Since π^*a_*Ø_B≅Ø_A'⊗Ø_A',we see that B_A' is a disjoint union of two sections of a_A':X_A'→ A'.This implies that the divisor B|_X_z is a sum of two distinct (reduced) points. Using the assumption that p=2, we find that (X_z,B|_X_z) is F-split. Next, we show that A is ordinary and p=2, assuming a is F-split. Fix an e>0 such that ϕ^(e)_X/A splits. By Claim <ref>, we have H^0(X,ø_X^1-p^e)=H^0(X,Ø_X(2p^e-2)) =H^0(X,Ø_X(p-2)⊗ g^*Ø_Y(2p^e-1-1))=V· g^*W,where V:=H^0(X,Ø_X(p-2)) and W:=H^0(Y,Ø_Y(2p^e-1-1)). Therefore, we can choose v∈ V and w∈ W such that v· g^*w∈ H^0(X,ø_X^1-p^e)gives a splitting of ϕ^(e)_X/A. The pullback (π_X^*v)·(π_X^*g^*w) then gives a splitting of ϕ^(e)_X_A'/A'. Suppose that A is supersingular. Then, by definition, π=F_A.One can easily check that g∘π_X factors through F_Y:Y→ Y,so there is some u∈ H^0(Ø_X_A'(2p^e-1-1)) with π_X^*g^*w=u^p. This implies that the pullback of v· u^p to a fiber of a_A' inducesa splitting of ℙ^1, which is a contradiction.Hence, A must be ordinary. We show p=2.It follows from the F-splitting of a thatØ_X_A^1→F_X/A^(1)_*Ø_X^1 splits.Applying the functor a_A^1_*(⊗Ø_X_A^1(1)) to this, we get ℰ_2,0≅F_A^*ℰ_2,0≅a_A^1_*Ø_X_A^1(1) a^(1)_*Ø_X^1(p) ≅ S^pℰ_2,0≅ℰ_p,0⊕Ø_A, so ℰ_2,0≅ℰ_p,0, and thus p=2, which completes the proof.§.§.§ The case when ℰ is an indecomposable vector bundle of degree 1. Suppose that ℰ=1. Then, a is F-split if and only if A is ordinary or p>2.We first prove that a is F-split, assuming that A is ordinary or p>2.When p>2, we take the étale cover ρ:A'→ A of degree twocorresponding to a torsion line bundle ℒ of order two. Then ρ_A'_*Ø_A'×_AA'≅ρ^*ρ_*Ø_A'≅ρ^*(Ø_A⊕ℒ) ≅Ø_A'⊕Ø_A'. Hence A'×_A A' is a disjoint union of two copies of A'.By Theorem <ref> (1) and (2), we see thatρ_*ℳ≅ℰ for a line bundle ℳ on A' of degree 1. Then, ρ^*ℰ≅ρ^*ρ_*ℳ≅ρ_A'_*ℳ_A'≅ℳ⊕ℳ.Therefore, X':=X_A'≅ℙ(ℳ⊕ℳ) is F-split over A'. We now have the following commutative diagram: @R=25pt@C=25ptX'^e[r] [d]_F_X'/A'^(e)X^e [d]^F_X/A^(e) X'_A'^e[r]_(ρ^(e))_XX_A^e.This induces the commutative diagram of Ø_X_A^e-modules @R=25pt@C=25ptØ_X_A^e[r] [d] [dr](ρ^(e))_X_*Ø_X'_A'^e[d]F_X/A^(e)_*Ø_X^e[r](ρ^(e))_X_*F_X'/A'^(e)_*Ø_X'^e. By the above argument, the right vertical morphism splits. Since p∤ρ, the upper horizontal morphism splits,so the diagonal one also splits,and hence so does the left vertical one, i.e., a:X→ A is F-split. When p=2 and A is ordinary, Theorem <ref> (3) tells us that F_A^*ℰ∈ℰ_A(2,2).Proposition <ref> then shows that a_A^1:X_A^1→ A^1 is F-split. Replacing ρ by F_A, we can prove the assertion by the same argument as the above. Next, we assume that p=2 and A is supersingular.Theorem <ref> (3) then shows F_A^*ℰ∈ℰ_A(2,2).Hence, Proposition <ref> tells us that a_A^1:X_A^1→ A^1 is not F-split.This requires that a:X→ A is not F-split, which completes the proof. abbrv | http://arxiv.org/abs/1704.08652v2 | {
"authors": [
"Sho Ejiri"
],
"categories": [
"math.AG",
"14D06, 14J40"
],
"primary_category": "math.AG",
"published": "20170427165617",
"title": "When is the Albanese morphism an algebraic fiber space in positive characteristic?"
} |
Circumbinary discs: Numerical and physical behaviour Daniel Thun 1 Wilhelm Kley 1 Giovanni Picogna 2==================================================== We study the problem of finding the cycle of minimum cost-to-time ratio in a directed graph with n nodes and m edges. This problem has a long history in combinatorial optimization and has recently seen interesting applications in the context of quantitative verification. We focus on strongly polynomial algorithms to cover the use-case where the weights are relatively large compared to the size of the graph. Our main result is an algorithm with running time Õ (m^3/4 n^3/2), which gives the first improvement over Megiddo's Õ (n^3) algorithm [JACM'83]Megiddo83 for sparse graphs.[We use the notation Õ(·) to hide factors that are polylogarithmic in n.] We further demonstrate how to obtain both an algorithm with running time n^3 / 2^Ω(√(log n)) on general graphs and an algorithm with running time Õ (n) on constant treewidth graphs. To obtain our main result, we develop a parallel algorithm for negative cycle detection and single-source shortest paths that might be of independent interest.§ INTRODUCTION We revisit the problem of computing the cycle of minimum cost-to-time ratio (short: minimum ratio cycle) of a directed graph in which every edge has a cost and a transit time. The problem has a long history in combinatorial optimization and has recently become relevant to the computer-aided verification community in the context of quantitative verification and synthesis of reactive systems <cit.>. The shift from quantitative to qualitative properties is motivated by the necessity of taking into account the resource consumption of systems (such as embedded systems) and not just their correctness. For algorithmic purposes, these systems are usually modeled as directed graphs where vertices correspond to states of the system and edges correspond to transitions between states. Weights on the edges model the resource consumption of transitions. In our case, we allow two types of resources (called cost and transit time) and are interested in optimizing the ratio between the two quantities. By giving improved algorithms for finding the minimum ratio cycle we contribute to the algorithmic progress that is needed to make the ideas of quantitative verification and synthesis applicable.From a purely theoretic point of view, the minimum ratio problem is an interesting generalization of the minimum mean cycle problem.[In the minimum cycle mean problem we assume the transit time of each edge is 1.] A natural question is whether the running time for the more general problem can match the running time of computing the minimum cycle mean (modulo lower order terms). In terms of weakly polynomial algorithms, the answer to this question is yes, since a binary search over all possible values reduces the problem to negative cycle detection. In terms of strongly polynomial algorithms, with running time independent of the encoding size of the edge weights, the fastest algorithm for the minimum ratio cycle problem is due to Megiddo <cit.> and runs in time Õ (n^3), whereas the minimum mean cycle can be computed in O (m n) time with Karp's algorithm <cit.>. This has left an undesirable gap in the case of sparse graphs for more than three decades. Our results.We improve upon this situation by giving a strongly polynomial time algorithm for computing the minimum ratio cycle in time O (m^3/4 n^3/2log^2 n) (Theorem <ref> in Section <ref>). We obtain this result by designing a suitable parallel negative cycle detection algorithm and combining it with Megiddo's parametric search technique <cit.>. We first present a slightly simpler randomized version of our algorithm with one-sided error and the same running time (Theorem <ref> in Section <ref>).As a side result, we develop a new parallel algorithm for negative cycle detection and single-source shortest paths (SSSP) that we use as a subroutine in the minimum ratio cycle algorithm. This new algorithm has work Õ (mn + n^3 h^-3) and depth O(h) for any logn≤ h ≤ n. Our algorithm uses techniques from the parallel transitive closure algorithm of Ullman and Yannakakis <cit.> (in particular as reviewed in <cit.>) and our contribution lies in extending these techniques to directed graphs with positive and negative edge weights. In particular, we partially answer an open question by Shi and Spencer <cit.> who previously gave similar trade-offs for single-source shortest paths in undirected graphs with positive edge weights. We further demonstrate how the parametric search technique can be applied to obtain minimum ratio cycle algorithms with running time Õ (n) on constant treewidth graphs (Corollary <ref> in Section <ref>). Our algorithms do not use fast matrix multiplication. We finally show that if fast matrix multiplication is allowed then slight further improvements are possible, specifically we present ann^3 / 2^Ω(√(log n)) time algorithm on general graphs (Corollary <ref> in Section <ref>). Prior Work. The minimum ratio problem was introduced to combinatorial optimization in the 1960s by Dantzig, Blattner, and Rao <cit.> and Lawler <cit.>. The existing algorithms can be classified according to their running time bounds as follows: strongly polynomial algorithms, weakly polynomial algorithms, and pseudopolynomial algorithms. In terms of strongly polynomial algorithms for finding the minimum ratio cycle we are aware of the following two results: * O (n^3 logn + m n log^2n) time using Megiddo's second algorithm <cit.> together with Cole's technique to reduce a factor of loglogn <cit.>,* O (m n^2) time using Burn's primal-dual algorithm <cit.>.For the class of weakly polynomial algorithms, the best algorithm is to follow Lawler's binary search approach <cit.>, which solves the problem by performing O (log(n W)) calls to a negative cycle detection algorithm. Here W = O (C T) if the costs are given as integers from 1 to C and the transit times are given as integers from 1 to T. Using an idea for efficient search of rationals <cit.>, a somewhat more refined analysis by Chatterjee et al. <cit.> reveals that it suffices to call the negative cycle detection algorithm O (log(|a · b |)) times when the value of the minimum ratio cycle is ab. Since the initial publication of Lawler's idea, the state of the art in negative cycle detection algorithms has become more diverse. Each of the following five algorithms gives the best known running time for some range of parameters (and the running times have to be multiplied by the factor log(n W) or log(|a · b |) to obtain an algorithm for the minimum ratio problem): * O (mn) time using a variant of the Bellman-Ford algorithm <cit.>,* n^3 / 2^Ω(√(log n)) time using a recent all-pairs shortest paths (APSP) algorithm by Williams <cit.>,* Õ (n^ω W) time using fast matrix multiplication <cit.>, where 2 ≤ω < 2.3728639 is the matrix multiplication coefficient <cit.>,* O (m √(n)log W) time using Goldberg's scaling algorithm <cit.>,* Õ (m^10/7log W) time using the interior point method based algorithm of Cohen et al. <cit.>The third group of minimum ratio cycle algorithms has a pseudopolynomial running time bound. After some initial progress <cit.>, the state of the art is an algorithm by Hartmann and Orlin <cit.> that has a running time of O (m n T).[Note that the more fine-grained analysis of Hartmann and Orlin actually gives a running time of O (m (∑_u ∈ Vmax_e = (u, v) t (e))).] Other algorithmic approaches, without claiming any running time bounds superior to those reported above, were given by Fox <cit.>, v. Golitschek <cit.>, and Dasdan, Irani, and Gupta <cit.>.Recently, the minimum ratio problem has been studied specifically for the special case of constant treewidth graphs by Chatterjee, Ibsen-Jensen, and Pavlogiannis <cit.>. The state of the art for negative cycle detection on constant treewidth graphs is an algorithm by Chaudhuri and Zaroliagis with running time O (n) <cit.>, which by Lawler's binary search approach implies an algorithm for the minimum ratio problem with running time O (n log(n W)). Chatterjee et al. <cit.> report a running time of O (n log(|a · b |)) based on the more refined binary search mentioned above and additionally give an algorithm that uses O (logn) space (and hence polynomial time).As a subroutine in our minimum ratio cycle algorithm, we use a new parallel algorithm for negative cycle detection and single-source shortest paths. The parallel SSSP problem has received considerable attention in the literature <cit.>, but we are not aware of any parallel SSSP algorithm that works in the presence of negative edge weights (and thus solves the negative cycle detection problem). To the best of our knowledge, the only strongly polynomial bounds reported in the literature are as follows: For weighted, directed graphs with non-negative edge weights, Broda, Träff, and Zaroliagis <cit.> give an implementation of Dijkstra's algorithm with O (m logn) work and O (n) depth. For weighted, undirected graphs with positive edge weights, Shi and Spencer <cit.> gave (1) an algorithm with O (n^3 t^-2lognlog(n t^-1) + m logn) work and O (t logn) depth and (2) an algorithm with O ((n^3 t^-3 + m n t^-1) logn) work and O (t logn) depth, for any logn≤ t ≤ n.§ PRELIMINARIES In the following, we review some of the tools that we use in designing our algorithm. §.§ Parametric Search We first explain the parametric search technique as outlined in <cit.>. Assume we are given a property 𝒫 of real numbers that is monotone in the following way: if 𝒫 (λ) holds, then also 𝒫 (λ') holds for all λ' < λ. Our goal is to find λ^*, the maximum λ such that 𝒫 (λ) holds. In this paper for example, we will associate with each λ a weighted graph G_λ and 𝒫 is the property that G_λ has no negative cycle.Assume further that we are given an algorithm 𝒜 for deciding, for a given λ, whether 𝒫 (λ) holds. If λ were known to only assume integer or rational values, we could solve this problem by performing binary search with O (logW) calls to the decision algorithm, where W is the number of possible values for λ. However, this solution has the drawback of not yielding a strongly polynomial algorithm.In parametric search we run the decision algorithm `generically' at the maximum λ^*. As the algorithm does not know λ^*, we need to take care of its control flow ourselves and any time the algorithm performs a comparison we have to `manually' evaluate the comparison on behalf of the algorithm. If each comparison takes the form of testing the sign of an associated low-degree polynomial p (λ), this can be done as follows. We first determine all roots of p (λ) and check if 𝒫 (λ') holds for each such root λ' using another instance of the decision algorithm 𝒜. This gives us an interval between successive roots containing λ^* and we can thus resolve the comparison. With every comparison we make, the interval containing λ^* shrinks and at the end of this process we can output a single candidate. If the decision algorithm 𝒜 has a running time of T, then the overall algorithm for computing λ^* has a running time of O (T^2).A more sophisticated use of the technique is possible, if in addition to a sequential decision algorithm 𝒜_s we have an efficient parallel decision algorithm 𝒜_p. The parallel algorithm performs its computations simultaneously on P_p processors. The number of parallel computation steps until the last processor is finished is called the depth D_p of the algorithm, and the number of operations performed by all processors in total is called the work W_p of the algorithm.[To be precise, we use an abstract model of parallel computation as formalized in <cit.> to avoid distraction by details such as read or write collisions typical to PRAM models.] For parametric search, we actually only need parallelism w.r.t. comparisons involving the input values.We denote by the comparison depth of 𝒜_p the number of parallel comparisons (involving input values) until the last processor is finished.We proceed similar to before: We run 𝒜_p `generically' at the maximum λ^* and (conceptually) distribute the work among P_p processors. Now in each parallel step, we might have to resolve up to P_p comparisons. We first determine all roots of the polynomials associated to these comparisons. We then perform a binary search among these roots to determine the interval of successive roots containing λ^* and repeat this process of resolving comparisons at every parallel step to eventually find out the value of λ^*. If the sequential decision algorithm 𝒜_s has a running time of T_s and the parallel decision algorithm runs on P_p processors in D_p parallel steps, then the overall algorithm for computing λ^* has a running time of O (P_p D_p + D_p T_slogP_p). Formally, the guarantees of the technique we just described can be summarized as follows. Let 𝒫 be a property of real numbers such that if 𝒫 (λ) holds, then also 𝒫 (λ') holds for all λ' < λ and let 𝒜_p and 𝒜_s be algorithms deciding for a given λ whether 𝒫 (λ) holds such that * the control flow of 𝒜_p is only governed by comparisons that test the sign of an associated polynomial in λ of constant degree,* 𝒜_p is a parallel algorithm with work W_p and comparison depth D_p, and* 𝒜_s is a sequential algorithm with running time T_s.Then there is a (sequential) algorithm for finding the maximum value λ such that 𝒫 (λ) holds with running time O (W_p + D_p T_slogW_p). Note that 𝒜_p and 𝒜_s need not necessarily be different algorithms. In most cases however, the fastest sequential algorithm might be the better choice for minimizing running time.§.§ Characterization of Minimum Ratio Cycle We consider a directed graph G = (V, E, c, t), in which every edge e = (u, v) has a cost c (e) and a transit time t (e). We want to find the cycle C that minimizes the cost-to-time ratio ∑_e ∈ C c (e) / ∑_e ∈ C t (e).For any real λ define the graph G_λ = (V, E, w_λ) as the modification of G with weight w_λ (e) = c (e) - λ t (e) for every edge e ∈ E. The following structural lemma is the foundation of many algorithmic approaches towards the problem. Let λ^* be the value of the minimum ratio cycle of G. * For λ > λ^*, the value of the minimum weight cycle in G_λ is < 0.* The value of the the minimum weight cycle in G_λ^* is 0. Each minimum weight cycle in G_λ^* is a minimum ratio cycle in G and vice versa.* For λ < λ^*, the value of the minimum weight cycle in G_λ is > 0.The obvious algorithmic idea now is to find the right value of λ with a suitable search strategy and reduce the problem to a series of negative cycle detection instances.§.§ Characterization of Negative CycleA potential function p V →ℝ assigns a value to each vertex of a weighted directed graph G = (V, E, w). We call a potential function p valid if for every edge e = (u, v) ∈ E, the condition p(u) + w (e) ≥ p (v) holds. The following two lemmas outline an approach for negative cycle detection. A weighted directed graph contains a negative cycle if and only if it has no valid potential function.Let G = (V, E, w) be a weighted directed graph and let G' = (V', E', w') be the supergraph of G consisting of the vertices V' = V ∪{ s' } (i.e. with an additional super-source s'), the edges E' = E ∪{ s' }× V and the weight function w' given by w' (s', v) = 0 for every vertex v ∈ V and w' (u, v) = w (u, v) for all pairs of vertices u, v ∈ V. If G does not contain a negative cycle, then the potential function p defined by p (v) = _G' (s', v) for every vertex v ∈ V is valid for G.Thus, an obvious strategy for negative cycle detection is to design a single-source shortest paths algorithm that is correct whenever the graph contains no negative cycle. If the graph contains no negative cycle, then the distances computed by the algorithm can be verified to be a valid potential. If the graph does contain a negative cycle, then the distances computed by the algorithm will not be a valid potential (because a valid potential does not exist) and we can verify that the potential is not valid.§.§ Computing Shortest Paths in Parallel In our algorithm we use two building blocks for computing shortest paths in the presence of negative edge weights in parallel. The first such building block was also used by Megiddo <cit.>. By repeated squaring of the min-plus matrix product, all-pairs shortest paths in a directed graph with real edge weights can be computed using work O (n^3 logn) and depth O (logn). The second building block is a subroutine for computing the following restricted version of shortest paths. The shortest h-hop path from a vertex s to a vertex t is the path of minimum weight among all paths from s to t with at most h edges. Note that a shortest h-hop path from s to t does not exist, if all paths from s to t use more than h edges.Furthermore, if there is a shortest path from s to t with at most h edges, then the h-hop shortest path from s to t is a shortest path as well. Shortest h-hop paths can be computed by running h iterations of the Bellman-Ford algorithm <cit.>.[The first explicit use of the Bellman-Ford algorithm to compute shortest h-hop paths that we are aware of is in Thorup's dynamic APSP algorithm <cit.>.] Similar to shortest paths, shortest h-hop paths need not be unique. We can enforce uniqueness by putting some arbitrary but fixed order on the vertices of the graph and sorting paths according to the induced lexicographic order on the sequence of vertices of the paths. Note that the Bellman-Ford algorithm can easily be adapted to optimizing lexicographically as its second objective. By performing h iterations of the Bellman-Ford algorithm, the lexicographically smallest shortest h-hop path from a designated source vertex s to each other vertex in a directed graph with real edge weights can be computed using work O (m h) and depth O (h). We denote by π (s, t) the lexicographically smallest shortest path from s to t and by π^h (s, t) the lexicographically smallest shortest h-hop path from s to t. We denote by V (π^h (s, t)) and E (π^h (s, t)) the set of nodes and edges of π^h (s, t), respectively. §.§ Approximate Hitting SetsGiven a collection of sets 𝒮⊆ 2^U over a universe U, a hitting set is a set T ⊆ H that has non-empty intersection with every set of 𝒮 (i.e., S ∩ T ≠∅ for every S ∈𝒮). Computing a hitting set of minimum size is an -hard problem. For our purpose however, rough approximations are good enough. The first method to get a sufficiently small hitting set uses a simple randomized sampling idea and was introduced to the design of graph algorithms by Ullman and Yannakakis <cit.>. We use the following formulation. Let c ≥ 1, let U be a set of size s and let 𝒮 = { S_1, S_2, …, S_k } be a collection of sets over the universe U of size at least q. Let T be a subset of U that was obtained by choosing each element of U independently with probability p = min (x / q, 1) where x = c ln(k s) + 1. Then, with high probability (whp), i.e., probability at least 1 - 1/s^c, the following two properties hold: * For every 1 ≤ i ≤ k, the set S_i contains an element of T, i.e., S_i ∩ T ≠∅.* |T| ≤ 3 x s / q = O (c s log(k s) / q).The second method is to use a heuristic to compute an approximately minimum hitting set. In the sequential model, a simple greedy algorithm computes an O(logn)-approximation <cit.>. We use the following formulation. Let U be a set of size s and let 𝒮 = { S_1, S_2, …, S_k } be a collection of sets over the universe U of size at least q. Consider the simple greedy algorithm that picks an element u in U that is contained in the largest number of sets in 𝒮 and then removes all sets containing u from 𝒮, repeating this step until 𝒮 = ∅. Then the set T of elements picked by this algorithm satisfies: * For every 1 ≤ i ≤ k, the set S_i contains an element of T, i.e., S_i ∩ T ≠∅.* |T| ≤ O (s log(k) / q).We follow the standard proof of the approximation ratio O(log n) for the greedy set cover heuristic.The first statement is immediate, since we only remove sets when they are hit by the picked element. Since each of the k sets contains at least q elements, on average each element in U is contained in at least k q / s sets. Thus, the element u picked by the greedy algorithm is contained in at least k q / s sets. The remaining number of sets is thus at most k - k q / s = k(1-q/s). Note that the remaining sets still have size at least q, since they do not contain the picked element u. Inductively, we thus obtain that after i iterations the number of remaining sets is at most k(1-q/s)^i, so after O(log(k) · s/q) iterations the number of remaining sets is less than 1 and the process stops. The above greedy algorithm is however inherently sequential and thus researchers have studied more sophisticated algorithms for the parallel model. The state of the art in terms of deterministic algorithms is an algorithm by Berger et al. <cit.>[Berger et al. actually give an approximation algorithm for the following slightly more general problem: Given a hypergraph H = (V, E) and a cost function c V →ℝ on the vertices, find a minimum cost subset R ⊆ V that covers H, i.e., an R that minimizes c (R) = ∑_v ∈ R c (v) subject to the constraint e ∩ R ≠∅ for all e ∈ E.]. Let 𝒮 = { S_1, S_2, …, S_k } be a collection of sets over the universe U, let n = |U| and m = ∑_1 ≤ i ≤ k |S_i|. For 0 < ϵ < 1, there is an algorithm with work O ((m + n) ϵ^-6log^4nlogmlog^6(nm)) and depth O (ϵ^-6log^4nlogmlog^6(nm)) that produces a hitting set of 𝒮 of size at most (1 + ϵ) (1 + lnΔ) ·𝑂𝑃𝑇, where Δ is the maximum number of occurrences of any element of U in 𝒮 and 𝑂𝑃𝑇 is the size of a minimum hitting set. § RANDOMIZED ALGORITHM FOR GENERAL GRAPHS§.§ A Parallel SSSP Algorithm In the following we design a parallel SSSP algorithm that can be used to check for negative cycles. Formally, we will in this subsection prove the following statement. There is an algorithm that, given a weighted directed graph G = (V, E, w) containing no negative cycles, computes the shortest paths from a designated source vertex s to all other vertices spending O (m n logn + n^3 h^-3log^4n) work with O (h + logn) depth for any 1 ≤ h ≤ n. The algorithm is correct with high probability and all its comparisons are performed on sums of edge weights on both sides. The algorithm proceeds in the following steps: * Let C ⊆ V be a set containing each vertex v independently with probability p = min (3 c h^-1lnn, 1) for a sufficiently large constant c. * If | C | > 9 c n h^-1lnn, then terminate. * For every vertex x ∈ C ∪{ s } and every vertex v ∈ V, compute the shortest h-hop path from x to v in G and its weight _G^h (x, v). * Construct the graph H = (C ∪{ s }, (C ∪{ s })^2, w_H) whose set of vertices is C ∪{ s }, whose set of edges is (C ∪{ s })^2 and for every pair of vertices x, y ∈ C∪{ s } the weight of the edge (x, y) is w_H (x, y) = _G^h (x, y). * For every vertex x ∈ C, compute the shortest path from s to x in H and its weight _H (s, x). * For every vertex t ∈ V, set δ (t) = min_x ∈ C ∪{ s } (_H (s, x) + _G^h (x, t)). §.§.§ Correctness In order to prove the correctness of the algorithm, we first observe that as a direct consequence of Lemma <ref> the randomly selected vertices in C with high probability hit all lexicographically smallest shortest ⌊ h/2 ⌋-hop paths of the graph. Consider the collection of sets𝒮 = { V(π^⌊ h/2 ⌋ (u, v)) | u, v ∈ Vwith _G^⌊ h/2 ⌋ (u, v) < ∞ and |E(π^⌊ h/2 ⌋ (u, v))| = ⌊ h/2 ⌋}containing the vertices of the lexicographically smallest shortest ⌊ h/2 ⌋-hop paths with exactly ⌊ h/2 ⌋ edges between all pairs of vertices. Then, with high probability, C is a hitting set of 𝒮 of size at most 9 c n h^-1lnn.If G contains no negative cycle, then δ (t) = _G (s, t) for every vertex t ∈ V with high probability.First note that the algorithm incorrectly terminates in Step <ref> only with small probability. We now need to show that, for every vertex t ∈ V, δ (t) := min_x ∈ C ∪{s} (_H (s, x) + _G^h (x, t)) = _G (s, t). First observe that every edge in H corresponds to a path in G (of the same weight). Thus, the value δ (t) corresponds to some path in G from s to t (of the same weight) which implies that _G (s, t) ≤δ (t) (as no path can have weight less than the distance).Now let π (s, t) be the lexicographically smallest shortest path from s to t in G. Subdivide π into consecutive subpaths π_1, …, π_k such that π_i for 1 ≤ i ≤ k-1 has exactly ⌊ h/2 ⌋ edges, and π_k has at most ⌊ h/2 ⌋ edges. Note that if π itself has at most ⌊ h/2 ⌋ edges, then k = 1. Since every subpath of a lexicographically smallest shortest path is also a lexicographically smallest shortest path, the paths π_1, …, π_k are lexicographically smallest shortest paths as well. As the subpaths π_1, …, π_k-1 consist of exactly ⌊ h/2 ⌋ edges, each of them is contained in the collection of sets 𝒮 of Observation <ref>. Therefore, each subpath π_i, for 1 ≤ i ≤ k-1, contains a vertex x_i ∈ C with high probability.Set x_0 = s and x_k = t, and observe that for every 0 ≤ i ≤ k-1, the subpath of π (s, t) from x_i to x_i+1 is a shortest path from x_i to x_i+1 with at most h edges and thus _G^h (x_i, x_i+1) = _G (x_i, x_i+1). We now get the following chain of inequalities:_G (s, t) = ∑_0 ≤ i ≤ k-1_G (x_i, x_i+1)= ∑_0 ≤ i ≤ k-1_G^h (x_i, x_i+1) = ( ∑_0 ≤ i ≤ k-2 w_H (x_i, x_i+1) ) + _G^h (x_k-1, t) ≥_H (x_0, x_k-1) + _G^h (x_k-1, t) = _H (s, x_k-1) + _G^h (x_k-1, t) ≥min_x ∈ C ∪{s} (_H (s, x) + _G^h (x, t)) = δ(t) .Note that we have formally argued only that the algorithm correctly computes the distances from s. It can easily be checked that the shortest paths can be obtained by replacing the edges of H with their corresponding paths in G. §.§.§ Running TimeThe algorithm above can be implemented with O (m n logn + n^3 h^-3log^4n) and O (h + logn) depth such that all its comparisons are performed on sums of edge weights on both sides.Clearly, in Steps <ref>–<ref>, the algorithm spends O (m + n) work with O (1) depth. Step <ref> can be carried out by running h iterations of Bellman-Ford for every vertex x ∈ C in parallel (see Lemma <ref>), thus spending O (|C| · m h) work with O (h) depth. Step <ref> can be carried out by spending O (|C|^2) work with O(1) depth. Step <ref> can be carried out by running the min-plus matrix multiplication based APSP algorithm (see Lemma <ref>), thus spending O (|C|^3 logn) work with O (logn) depth. The naive implementation of Step <ref> spends O (n |C|) work with O (|C|) depth. Using a bottom-up `tournament' approach where in each round we pair up all values and let the maximum value of each pair proceed to the next round, this can be improved to work O (n |C|) and depth O (logn).It follows that by carrying out the steps of the algorithm sequentially as explained above, the overall work is O (|C| · m h + |C|^3 logn) and the depth is O (h + logn). As the algorithm ensures that | C | ≤ 9 c n h^-1lnn for some constant c, the work is O (m n logn + n^3 h^-3log^4n) and the depth is O (h + logn).§.§.§ Extension to Negative Cycle Detection To check whether a weighted graph G = (V, E, w) contains a negative cycle, we first construct the graph G' (with an additional super-source s') as defined in Lemma <ref>. We then run the SSSP algorithm of Theorem <ref> from s' in G' and set p (v) = _G' (s', t) for every vertex t ∈ V. We then check whether the function p defined in this way is a valid potential function for G testing for every edge e = (u, v) (in parallel) whether p (u) + w (u, v) ≥ p (v). If this is the case, then we output that G contains no negative cycle, otherwise we output that G contains a negative cycle. There is a randomized algorithm that checks whether a given weighted directed graph contains a negative cycle with O (m n logn + n^3 h^-3log^4n) work and O (h + logn) depth for any 1 ≤ h ≤ n. The algorithm is correct with high probability and all its comparisons are performed on sums of edge weights on both sides.Constructing the graph G' and checking whether p is a valid potential can both be carried out with O (m + n) work and O(1) depth. Thus, the overall work and depth bounds are asymptotically equal to the SSSP algorithm of Theorem <ref>.If G contains no negative cycle, then the SSSP algorithm correctly computes the distances from s' in G'. Thus, the potential p is valid by Lemma <ref> and our algorithm correctly outputs that there is no negative cycle. If G contains a negative cycle, then it does not have any valid potential by Lemma <ref>. Thus, the potential p defined by the algorithm cannot be valid and the algorithm outputs correctly that G contains a negative cycle.§.§ Finding the Minimum Ratio Cycle Using the negative cycle detection algorithm as a subroutine, we obtain an algorithm for computing a minimum ratio cycle in time Õ (n^3/2 m^3/4). There is a randomized one-sided-error Monte Carlo algorithm for computing a minimum ratio cycle with running time O (n^3/2 m^3/4log^2n).By Lemma <ref> we can compute the value of the minimum ratio cycle by finding the largest value of λ such that G_λ contains no negative-weight cycle. We want to apply Theorem <ref> to find this maximum λ^* by parametric search. As the sequential negative cycle detection algorithm A_s we use Orlin's minimum weight cycle algorithm <cit.> with running time T(n, m) = O (m n). The parallel negative cycle detection algorithm A_p of Corollary <ref> has work W (n, m) = O (m n logn + n^3 h^-3log^4n) and depth D (n, m) = O (h + logn), for any choice of 1 ≤ h ≤ n. Any comparison the latter algorithm performs is comparing sums of edge weights of the graph. Since in G_λ edge weights are linear functions in λ, the control flow only depends on testing the sign of degree-1 polynomials in λ. Thus, Theorem <ref> is applicable[Formally, Theorem <ref> only applies to deterministic algorithms. However, only step <ref> of our parallel algorithm is randomized, but this step does not depend on λ. All remaining steps are deterministic. We can thus first perform steps <ref> and <ref>, and invoke Theorem <ref> only on the remaining algorithm. The output guarantee then holds with high probability.] and we arrive at a sequential algorithm for finding the value of the minimum ratio cycle with running time O (m n logn (h + logn) + n^3 h^-3log^4n). Finally, to output the minimum ratio cycle and not just its value, we run Orlin's algorithm for finding the minimum weight cycle in G_λ^*, which takes time O (m n). By setting h = n^1/2 m^-1/4logn the overall running time becomes O (n^3/2 m^3/4log^2n). § DETERMINISTIC ALGORITHM FOR GENERAL GRAPHS We now present a deterministic variant of our minimum ratio cycle algorithm, with the same running time as the randomized algorithm up to logarithmic factors. §.§ Deterministic SSSP and Negative Cycle Detection We can derandomize our SSSP algorithm by combining a preprocessing step with the parallel hitting set approximation algorithm of <cit.>. Formally, we will prove the following statement. There is a deterministic algorithm that, given a weighted directed graph containing no negative cycles, computes the shortest paths from a designated source vertex s to all other vertices spending O (m n log^2n + n^3 h^-3log^7n + n^2 h log^11n) work with O (h + log^11n) depth for any 1 ≤ h ≤ n. From this, using Lemmas <ref> and <ref> analogously to the proof of Corollary <ref>, we get the following corollary for negative cycle detection.There is a deterministic algorithm that checks whether a given weighted directed graph contains a negative cycle with O (m n log^2n + n^3 h^-3log^7n + n^2 h log^11n) work and O (h + log^11n) depth for any 1 ≤ h ≤ n. Our deterministic SSSP algorithm does the following: * For all pairs of vertices u,v ∈ V, compute the shortest ⌊ h/2 ⌋-hop path π^⌊ h/2 ⌋ (u, v) from u to v in G.[Note that in case there are multiple shortest ⌊ h/2 ⌋-hop paths from u to v, any tie-breaking is fine for the algorithm and its analysis.] * Compute an O (logn)-approximate set cover C of the system of sets 𝒮 = { V(π^⌊ h/2 ⌋ (u, v)) | u, v ∈ Vwith _G^⌊ h/2 ⌋ (u, v) < ∞ and |E(π^⌊ h/2 ⌋ (u, v))| = ⌊ h/2 ⌋}. * Proceed with steps <ref> to <ref> of the algorithm in Section <ref>.§.§.§ Correctness Correctness is immediate: In the previous proof of Lemma <ref> we relied on the fact that C is a hitting set of 𝒮. In the above algorithm, this property is guaranteed directly. §.§.§ Running Time Step <ref> can be carried out by running h iterations of the Bellman-Ford algorithm for every vertex v ∈ V. By Lemma <ref> this uses O (m n h) work and O (h) depth. We carry out Step <ref> by running the algorithm of Theorem <ref> to compute an O (logn)-approximate hitting set of 𝒮 with work O (n^2 h log^11n) and depth O (log^11n).Lemma <ref> gives a randomized process that computes a hitting set of 𝒮 of expected size O (n h^-1logn). By the probabilistic method, this implies that there exists a hitting set of size O (n h^-1logn). We can therefore use the algorithm of Theorem <ref> to compute a hitting set 𝒮 of size O (n h^-1log^2n). The work is O (n^2 h log^11n) and the depth is O (log^11n). Carrying out the remaining steps with a hitting set C of size O (n h^-1log^2n) uses work O (m h |C| + |C|^3 logn) = O (m n log^2n + n^3 h^-3log^7n) and depth O (h + logn). Thus, our overall SSSP algorithm has work O (m n log^2n + n^3 h^-3log^7n + n^2 h log^11n) and depth O (h + log^11n). §.§ Minimum Ratio Cycle We again obtain a minimum ratio cycle algorithm by applying parametric search (Theorem <ref>). We obtain the same running time bound as for the randomized algorithm.There is a deterministic algorithm for computing a minimum ratio cycle with running time O (n^3/2 m^3/4log^2 n).The proof is analogous to the proof of Theorem <ref>, with the only exception that we use the deterministic parallel negative cycle detection algorithm of Corollary <ref>. However, we do not necessarily need to run the algorithm of Theorem <ref> to compute an approximate hitting set. Instead we can also run the greedy set cover heuristic (Lemma <ref>) for this purpose. The reason is that at this stage, the greedy heuristic does not need to perform any comparisons involving the edge weights of the input graph, which are the only operations that are costly in the parametric search technique. This means that finding an approximate hitting set C of size O(n h^-1log n) can be implemented with O (∑_S ∈𝒮 |S|) = O (n^2 h) work and O (1) comparison depth. Thus, we use a parallel negative cycle detection algorithm A_p which has work W (n, m) = O (m h |C| + |C|^3 logn + n^2 h) = O (m n log n + n^3 h^-3log^4 n + n^2 h) and depth D (n, m) = O (h + log n), for any choice of 1 ≤ h ≤ n. We thus obtain a sequential minimum ratio cycle algorithm with running time O (m n log n + n^3 h^-3log^4 n + n^2 h + m n logn (h + log n)), for any choice of 1 ≤ h ≤ n. Note that the summands m n log n and n^2 h are both dominated by the last summand m n logn (h + log n). Setting h = n^1/2 m^-1/4log n to optimize the remaining summands, the running time becomes O (n^3/2 m^3/4log^2n). § NEAR-LINEAR TIME ALGORITHM FOR CONSTANT TREEWIDTH GRAPHS In the following we demonstrate how to obtain a nearly-linear time algorithm(in the strongly polynomial sense) for graphs of constant treewidth. We can use the following results of Chaudhuri and Zaroliagis <cit.> who studied the shortest paths problem in graphs of constant treewidth.[The first result of Chaudhuri and Zaroliagis <cit.> has recently been complemented with a space-time trade-off by Chatterjee, Ibsen-Jensen, and Pavlogiannis <cit.> at the cost of polynomial preprocessing time that is too large for our purposes.] There is a deterministic algorithm that, given a weighted directed graph containing no negative cycles, computes a data structure that after O (n) preprocessing time can answer, for any pair of vertices, distance queries in time O (α(n)), where α (·) is the inverse Ackermann function. It can also report a corresponding shortest path in time O (ℓα(n)), where ℓ is the number of edges of the reported path.There is a deterministic negative cycle algorithm for weighted directed graphs of constant treewidth with O (n) work and O (log^2n) depth.We now apply the reduction of Theorem <ref> to the algorithm of Theorem <ref> to find λ^*, the value of the minimum ratio cycle, in time O (n log^3n) (using T_s (n) = W_p (n) = O (n), and D_p (n) = O (log^2n)). We then use the algorithm of Theorem <ref> to find a minimum weight cycle in G_λ^* in time O (n α(n)): Each edge e = (u, v) together with the shortest path from v to u (if it exists) defines a cycle and we need to find the one of minimum weight by asking the corresponding distance queries. For the edge e = (u, v) defining the minimum weight cycle we query for the corresponding shortest path from v to u. This takes time O (n) as a graph of constant treewidth has O (n) edges. We thus arrive at the following guarantees of the overall algorithm. There is a deterministic algorithm that computes the minimum ratio cycle in a directed graph of constant treewidth in time O (n log^3n). § SLIGHTLY FASTER ALGORITHM FOR DENSE GRAPHS All our previous algorithm do not make use of fast matrix multiplication. We now show that if we allow fast matrix multiplication, despite the hidden constant factors being galactic, then slight further improvements are possible. Specifically, we sketch how the running time of n^3 / 2^Ω(√(log n)) of Williams's recent APSP algorithm <cit.> (with a deterministic version by Chan and Williams <cit.>) can be salvaged for the minimum ratio problem. In particular, we explain why Williams' algorithm for min-plus matrix multiplication parallelizes well enough. There is a deterministic algorithm that checks whether a given weighted directed graph contains a negative cycle with n^3 / 2^Ω(√(log n)) work and O (logn) comparison depth.First, note that the value of the minimum weight cycle in a directed graph can be found by computing min_e = (u, v) ∈ E w (u, v) + _G (v, u), i.e., the cycle of minimum weight among all cycles consisting of first an edge e = (u, v) and then the shortest path from u to v is the global minimum weight cycle. If all pairwise distances are already given, then computing the value of the minimum weight cycle (and thus also checking for a negative cycle) can therefore be done with O (n^2) work and O (logn) depth (again by a `tournament' approach).The APSP problem can in turn be reduced to min-plus matrix multiplication <cit.>. Let M be the adjacency matrix of the graph where additionally all diagonal entries are set to 0. Recall that the the all-pairs distance matrix is given M^n-1, where matrix multiplication is performed in the min-plus semiring. By repeated squaring, this matrix can be computed with O (logn) min-plus matrix multiplications. Williams's principal approach for computing the min-plus product C of two matrices A and B is as follows. (A1) Split the matrices A and B into rectangular submatrices of dimensions n × d and d × n, respectively, where d = 2^Θ(√(logn)), as follows: For every 1 ≤ k ≤⌈ n/d ⌉ - 1, A_k contains the k-th group of d consecutive columns of A and B_k contains the k-th group of d consecutive rows of B; for k = ⌈ n/d ⌉, A_k contains the remaining columns of A and B_k contains the remaining rows of B.(A2) For each 1 ≤ k ≤⌈ n/d ⌉, compute C_k, the min-plus product of A_k and B_k (using the algorithm described below).(A3) Determine the min-plus product of A and B by taking the entrywise minimum C := min_1 ≤ k ≤⌈ n/d ⌉ C_k. To carry out Step (A2), Williams first uses a preprocessing stage applied to each pair of matrices A_k and B_k (for 1 ≤ k ≤⌈ n/d ⌉) individually. It consists of the following three steps: (B1) Compute matrices A_k^* and B_k^* of dimensions n × d and d × n, respectively, as follows: Set A_k^* [i, p] := A_k [i, p] · (n + 1) + p, for every 1 ≤ i ≤ n and 1 ≤ p ≤ d, and set B_k^* [q, j] := B_k [q, j] · (n + 1) + q, for every 1 ≤ q ≤ d and 1 ≤ j ≤ n.(B2) Compute matrices A_k' and B_k' of dimensions n × d^2 and d^2 × n, respectively, as follows: Set A'_k [i, (p, q)] := A_k^* [i, p] - A_k^* [i, q], for every 1 ≤ i ≤ n and 1 ≤ p, q ≤ d, and set B_k' [(p, q), j] := B_k^* [q, j] - B_k^* [p, j], for every 1 ≤ j ≤ n, 1 ≤ p, q ≤ d.(B3) For every pair p, q (1 ≤ p, q ≤ d), compute and sort the set S_k^p, q := { A_k' [i, (p, q)] | 1 ≤ i ≤ n }∪{ B_k' [(p, q), j] | 1 ≤ j ≤ n }, where ties are broken such that entries of A_k' have precedence over entries of B_k'. Then compute matrices A_k” and B_k” of dimensions n × d^2 and d^2 × n, respectively, as follows: Set A_k” [i, (p, q)] to the rank of the value A_k' [i, (p, q)] in the sorted order of S_k^p, q, for every 1 ≤ i ≤ n and 1 ≤ p, q ≤ d, and set B_k” [(p, q), j] to the rank of the value B'_k [(p, q), j] in the sorted order of S_k^p, q, for every 1 ≤ j ≤ n and 1 ≤ p, q ≤ d.This type of preprocessing is also known as Fredman's trick <cit.>. As Williams shows, the problem of computing C_k now amounts to finding, for every 1 ≤ i ≤ n and 1 ≤ j ≤ n, the unique p^* such that A”_i, (p^*, q)≤ B”_(p^*, q), j for all 1 ≤ q ≤ d. Using tools from circuit complexity and fast rectangular matrix multiplication, this can be done in time Õ (n^2), either with a randomized algorithm <cit.>, or, with slightly worse constants in the choice of d (and thus the exponent of the overall algorithm), with a deterministic algorithm <cit.>. The crucial observation for our application is that after the preprocessing stage no comparisons involving the input values are performed anymore since all computations are performed with regard to the matrices A_k” and B_k”, which only contain the ranks (i.e., integer values from 1 to 2 n).The claimed work bound follows from Williams's running time analysis. We can bound the comparison depth as follows. First note that apart from Steps (A3) and (B3) we only incur O (log n) overhead in the depth. Step (A3) can be implemented with O (logn) depth by using a tournament approach for finding the respective minima. For Step (B3) we can use a parallel version of merge sort on n items that has work O (n logn) and depth O (logn) <cit.>.We now apply the reduction of Theorem <ref> to the algorithm of Theorem <ref> to find λ^*, the value of the minimum ratio cycle, in time n^3 / 2^Ω(√(log n)) (using T_s (n) = W_p (n) = n^3 / 2^Ω(√(log n)), and D_p (n) = O (logn)). We then use Williams' APSP algorithm to find a minimum weight cycle in G_λ^* in time n^3 / 2^Ω(√(log n)). We thus arrive at the following guarantees of the overall algorithm. There is deterministic algorithm for computing a minimum ratio cycle with running time n^3 / 2^Ω(√(log n)). § CONCLUSIONWe have presented a faster strongly polynomial algorithm for finding a cycle of minimum cost-to-time ratio, a problem which has a long history in combinatorial optimization and recently became relevant in the context of quantitative verification. Our approach combines parametric search with new parallelizable single-source shortest path algorithms and also yields small improvements for graphs of constant treewidth and in the dense regime. The main open problem is to push the running time down to Õ (m n), nearly matching the strongly polynomial upper bound for the less general problem of finding a minimum mean cycle. [heading=bibintoc] | http://arxiv.org/abs/1704.08122v1 | {
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#1#2#3#4#1 #2, #4 (19#3)2#1#2#3#4#1 #2, #4 (20#3)NaturePhys. Rev. Lett.Eur. Phys. J.AEurophys. Lett. Phys. Rev.Phys. Rev.CPhys. Rev.DJ. Appl. Phys.Am. J. Phys.Nucl. Instr. and Meth. Phys. ANucl. Phys. ANucl. Phys.BNucl. Phys.B (Proc. Suppl.)New J.Phys.Eur. Phys. J.CPhys. Lett. BPhysicsMod. Phys. Lett. APhys. Rep.Z. Phys. CZ. Phys. AProg. Part. Nucl. Phys.J. Phys. GComput. Phys. Commun.Acta Physica Polon. BAIP Conf. Proc.J. High Energy Phys.Prog. Sci. CultureNuovo Cim.Suppl. Nuovo Cim.Sov. J. Nucl. Phys.Sov. Phys. JETPJETP Lett. Prog. Theor. Phys.Prog. Theor. Phys. Suppl.Izv. Akad. Nauk: Ser. Fiz.J. Phys. Conf. Ser.Adv. High Energy Phys.Int.J. Mod. Phys. EAnn. Rev. Nucl. Part. Sci.J. Astrophys. Astron. Γ(1+1/ν) Γ(1+2/ν) Γ(1+3/ν) Γ(1+4/ν) Γ(1+5/ν)<cit.>[] | http://arxiv.org/abs/1704.08377v2 | {
"authors": [
"Ranjit K. Nayak",
"Sadhana Dash",
"Edward K. Sarkisyan-Grinbaum",
"Marek Tasevsky"
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[email protected] www.quantum-photonics.dk Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen, Denmark 42.50.-p, 78.67.Hc, 78.47.-p, 42.50.Ct, 78.67.PtPlanar photonic nanostructures have recently attracted a great deal of attention for quantum optics applications. In this article, we carry out full 3D numerical simulations to fully account for all radiation channels and thereby quantify the coupling efficiency of a quantum emitter embedded in a photonic-crystal waveguide. We utilize mixed boundary conditions by combining active Dirichlet boundary conditions for the guided mode and perfectly-matched layers for the radiation modes. In this way, the leakage from the quantum emitter to the surrounding environment can be determined and the spectral and spatial dependence of the coupling to the radiation modes can be quantified. The spatial maps of the coupling efficiency, the β-factor, reveal that even for moderately slow light, near-unity β is achievable that is remarkably robust to the position of the emitter in the waveguide. Our results show that photonic-crystal waveguides constitute a suitable platform to achieve deterministic interfacing of a single photon and a single quantum emitter, which has a range of applications for photonic quantum technology. Numerical modelling of the coupling efficiency of single quantum emitters in photonic-crystal waveguides Peter Lodahl December 30, 2023 ========================================================================================================§ INTRODUCTION Enhancing the spontaneous emission rate of a quantum emitter by placing it in an optical cavity was first suggested by Purcell <cit.>. In the following decades it was realized that the spontaneous emission rate of a quantum emitter can also be suppressed by placing it in a photonic bandgap <cit.>. This has led to a significant research effort in manipulating the photonic environment surrounding quantum emitters to suppress coupling to unwanted radiation modes and to boost coupling to specific localized modes. The spontaneous emission rate of a quantum emitter scales with the projected local density of optical states (LDOS). Significant enhancement of spontaneous emission rates have been demonstrated in optical cavities <cit.>, nanophotonic waveguides <cit.> and with surface plasmon modes <cit.>, while suppression of spontaneous emission has been measured in the bandgap region of a photonic crystal <cit.>. Recently there has been a growing interest in quantum emitters coupled to planar nanostructures. Indeed different quantum emitters such as quantum dots <cit.>, diamond color centers <cit.>, and atoms <cit.> have been efficiently coupled to planar nanostructures. Planar photonic crystals typically only possess a bandgap for a single polarization and for in-plane guided propagation. Nevertheless, this partial bandgap can greatly reduce the LDOS for emitters oriented in plane by suppressing the coupling rate to the radiation modes and therefore decrease the spontaneous emission rate of embedded quantum emitters <cit.>. By implementingwaveguides or cavities in the band gap frequency region, the spontaneous emission can preferentially be directed with very high efficiency into a single mode. Combination of suppression of the coupling to the radiation modes and the enhancement of coupling to a photonic-crystal waveguide (PCW) mode has been predicted to enable a deterministic single-photon source <cit.>. The fraction of emitted light that is coupled into the waveguideis defined as the β-factor. The coupling of single quantum dots to PCWs has been studied by several groups <cit.> and a record value of β > 98.4% was recently achieved <cit.>. An important feature of PCWs is their wide bandwidth contrary to cavities. However, the β-factor depends significantly on the spatial position of the emitter in the PCW due to the coupling to waveguide mode as well as the coupling to radiation modes. The spatial and spectral dependencies of the coupling to the PCW guided modeare well understood <cit.> since they can be obtained from eigenfunctions computed using standard techniques, e.g.,the plane-wave expansion method <cit.>. In contrast, the spatial and spectral dependencies of the radiation continuum in PCWs have thus far only been quantified at certain spatial positions <cit.>. A full mapping of the radiation modes is essential in order to find the β-factor and thereby determine how large coupling efficiencies may be obtained under experimentally realistic conditions. In this article, we develop the necessary tools to carry out a detailed analysis of the LDOS in a PCW. The main challenge in modeling an infinite PCW in a finite computation domain is that, along the propagation direction, open boundary conditions are required. Although perfectly matched layers are typically good approximations for open boundaries, they fail in inhomogeneous dielectric structures <cit.>, particularly at low group velocities. To overcome this problem, we derive Dirichlet boundary conditions, whose phase and amplitude match a propagating PCW mode excited by a dipole at an arbitrary point inside the waveguide. Armed with these boundary conditions, we compute the LDOS contribution from the radiation continuum for a range of frequencies across the waveguide band. We show that the coupling to the radiation continuum is highly suppressed in a PCW. We map out the dependence of the coupling to the radiation continuum on the position, frequency, and orientation of the dipole. The resulting β-factor is remarkably robust to spatial position and spectral tuning of the emitter, which has been confirmed experimentally <cit.>.This paper is arranged in the following sections: Section <ref> discusses the different decay channels for an emitter embedded in a PCW and introduces the parameters that govern the emitter dynamics. Section <ref> includes the details of the simulations. We present and discuss the results of the numerical simulations in Section <ref>. Section <ref> sums up our results and gives an overview of various applications that could benefit from an efficient light matter-interface. The two appendices include the convergence tests, and a short overview of the decay dynamics of an emitter in a photonic crystal and the comparison to a PCW.§ ELECTRODYNAMICS OF A QUANTUM EMITTER IN A PCW Figure <ref>a, shows the band diagram of the TE modes of a PCW membrane<cit.>. Inside the band gap, light is mainly guided by three highly confined waveguide modes and by matching the PCW to the targeted emitter, the emitter is typically coupled to a single propagating mode (cf. Fig. <ref>a, solid black lines). The waveguide modes are highly dispersive and the group velocity of the wave is reduced as its frequency approaches the band edge, where the slow-down factor n_g (also known as the group index) ideally diverges. Due to the partial band gap of the 2D PCW membrane, there exist a continuum of modes that are not guided by the waveguide and leak to the surrounding environment (blue area in Fig. <ref>a). In real PCWs unavoidable fabrication imperfections influence light transport leading to multiple scattering effects. As a consequence, the guided mode is coupled to leaky modes or back-scattered to the oppositely propagating mode in the waveguide <cit.>, which has been quantitatively studied in Ref. <cit.>. Effects of disorder become dominating for long waveguides and large group indices, and may be eliminated by reducing both. In the present work we only consider PCWs where effects of disorder are negligible, which in practice means that we consider quantum emitters coupled only to moderately large values of n_g (in experiments typicallyn_g ∼ 100 can be achieved).An emitter embedded in a PCW can emit photons either to the guided modes of the PCW or to the radiation continuum, as schematically illustrated in Fig. <ref>b. The spontaneous emission rate of an excited emitter can be related to the transition dipole moment d, and the projected LDOS as <cit.>:γ=πω/ħϵ_0|d|^2ρ(ω_0,𝐫_0,𝐧_𝐝),whereρ(ω_0,𝐫_0,𝐧_𝐝)=∑_𝐤|𝐧_𝐝·𝐮^*_𝐤(𝐫_0)|^2δ(ω_0-ω_𝐤),where 𝐧_𝐝 is the orientation of the the dipole moment, ρ(ω_0,𝐫_0,𝐧_𝐝) is the projected LDOS, and 𝐮_𝐤 denotes the electric field eigenmode functions.The modes in Fig. <ref>a are classified into three categories: the guided modes of the PCW, the slab guided modes present outside the bandgap region (the gray region in Fig. <ref>b), and the continuum of radiation modes. At each frequency, the total decay rate of the emitter can be written as a sum of the contribution from these three sets of modes in addition to any residual contributions from coupling to transverse magnetic (TM) modes, i.e. γ_total(ω)=γ_wg(ω)+ γ_rad(ω)+γ_slab(ω)+γ_TM(ω), where the explicit dependence on spatial position and dipole orientation has been omitted for brevity. The β-factor quantifying the fraction of radiation coupled to the primary waveguide mode is defined as β=γ_wg/γ_total. For the purpose of this paper, we limit the discussion to the experimentally relevant situation of dipole emitters located in the center of the membrane, whereby no coupling to TM modes is present, i.e. γ_TM = 0. The frequency range of interest for a PCW single-photon source is mainly the primary guided mode in the bandgap region, where also γ_slab=0. The contribution to the β-factor describing the coupling to the waveguide, γ_wg, is straightforwardly determined by computing the eigenvalue and the corresponding eigenvectors of the electric and magnetic fields 𝐄_pg(ω,r), 𝐇_pg(ω,r) <cit.>. The corresponding Purcell factor is defined asF_p^wg= γ_wg/γ_0 = 6π^2c^3ϵ_0|𝐄_pg·𝐧_𝐝^*|^2/ω^2∫_unitcell d^3r nRe[ 𝐄_𝐩𝐠×𝐇_𝐩𝐠^*]/a,where n is the refractive index of the membrane material, γ_0=nω^3 d^2/3πϵ_0ħ c^3 is the decay rate of an emitter in a homogenous material of refractive index n.§ COMPUTING THE COUPLING OF A DIPOLE TO THE RADIATION CONTINUUM In this section we detail how to extract the contribution to β of coupling to the radiation continuum, i.e. γ_rad, which numerically is the most challenging part of the problem.The coupling is quantified by the Purcell factor of coupling to the radiation modes, which is denoted F_p^rad. It is given by F_p^mode=γ^mode/γ_0=P^mode/P_0, where γ^mode is the rate of coupling to radiation modes and P_0 and P^mode are the power emitted from the dipole in the reference medium and the nanophotonic structure, respectively. The total power emitted from the dipole can be extracted by integrating Poynting's vector over a closed surface around the current source, i.e. P_total=1/2Re[∮ d𝐬𝐄×𝐇^* ]. Thereby the Purcell factor can be determined.A main consideration in numerical simulations of optical problems is to ensure proper convergence, i.e., the computed quantities must not depend on the physical size of the computational domain. At the same time, it is desirable to limit the geometrical size of the simulation domain to the minimum possible in order to make the simulation efficient. A general approach to tackle these problems has been to introduce an absorber in the boundaries of a finite simulation domain and adiabatically absorb the incoming wave<cit.>. This can be applied when the geometry of the computational domain is invariant in the direction perpendicular to the boundary and the solutions are propagating waves rather than evanescent fields. In the case of a PCW the simulation domain is invariant along y and z at the boundaries, hence we can apply such perfectly-matched-layer (PML) boundary conditions. However, this is not applicable along the propagation direction (x) in the PCW. The generalization of PMLs to photonic- crystal waveguides is challenging<cit.>, particularly for slowly-propagating Bloch modes.Instead of using PMLs along the direction of the waveguide (x), a better choice is to introduce Dirichlet boundary conditions for the purpose of mimicking an open system. This corresponds to setting E|_x±=C_± at the two ends of the waveguide (x_±). In general, C_± havecontributions both from the primary mode of the waveguide and the radiation modes, however the main contribution stems from the guided mode of the waveguides that are extended by many optical wavelengths, i.e. the contributions from radiation modes are negligible. This is checked explicitlyby running a convergence test while varying the length of the simulation domain. The electric fields at the right and left boundaries (x_±) can be written asE|_x±=-|A^r(l)_0|e^-iϕ(𝐫_0)𝐄^*_pge^± ikx_±,where 𝐫_0 is the position of the dipole in the unitcell and ϕ(𝐫_0) is the phase of the projection of 𝐄_𝐩𝐠 on the dipole at the position 𝐫_0. These boundary conditions are referred to as active boundary conditions. The amplitudes A^r(l)_0 can be calculated from the knowledge of the eigenvectors of the PCW using the Green function formalism. For linear dipoles, these amplitudes simplify to:|A^r_0|=|A^l_0|=√(F_p^wgP_0/1/a ∫_unitcelld^3rRe[𝐄_𝐩𝐠^*×𝐇_𝐩𝐠]), ϕ=arg(-i 𝐄_𝐩𝐠(𝐫_0) ·𝐝).γ_rad can subsequently be calculated as the difference between γ_wg and γ_total. However, for a PCW typically γ_rad≪γ_wg and even small reflections, and numerical inaccuracies in γ_wg or γ_total limit the obtainable precision of γ_rad. This can be circumvented by calculating γ_rad directly by integrating Poynting's vector over a box surrounding the current source and leaving out the integration over the boundaries normal to waveguide direction. This is indicated by the green box in Fig. <ref>, which illustrates the geometry of the computation domain. Due to the symmetry of the structure and the position of the dipole being in the center of the slab, the solutions of Maxwell's equations are eigenvectors of the mirror symmetry operator about the z=0 plane. As a result, the simulation domain can be cut in half along this symmetry plane with the following boundary conditions: E_z(z=0)=0 and ∂/∂ z{E_x,E_y}|_z=0=0.The PCW membrane has a length of l=(2n+1)a and a width of w=√(3)(2m+1)a and is surrounded by an air box of hight D_z. The refractive index of the PCW slab is chosen to be 3.5 corresponding to the refractive index ofGaAs. The simulation domain is encapsulated by PMLs on all sides (blue box in Fig. <ref>). The width of the PML layer is W_PML. Active boundary conditions override the PMLs on the two ends of simulation domain normal to the waveguide direction (red plane in Fig. <ref>). The height of these planes are h_bnd and they cover the full waveguide in the y direction. The green box in Fig. <ref> resembles the box that captures the radiation modes. l_b, w_b and h_b are the length, width and height of the radiation box. Note that Fig. <ref> is not to scale. Table <ref> presents the parameter values that were used in the computations presented in the article. To establish these numbers we have carried out rigorous convergence tests. Appendix <ref> contains the results of some of the convergence tests for the most sensitive parameters, l, l_bnd and h_bnd. From the convergence test results, we estimate that the values of γ_rad are accurate to within 5%.The simulation procedure can be summarized as follows: We first carry out an eigenvalue calculation to determine the eigenfrequency, the group index n_g, eigenvector of the primary guided mode, and F_p^wg for a given dipole position. Using Eq. (<ref>) we determine the correct amplitudes for the respective boundaries of the waveguide. Subsequently a finite element frequency domain simulation of a dipole in a PCW is performed with the correct boundary conditions. The total power emitted from the waveguide is calculated by integrating Poynting's vector over a small box around the dipole. The coupling rate, γ_rad is extracted by integrating Poynting's vector over the radiation box. We repeat all the simulations for n_g=5, 20, 58, 120. These correspond to realistic values of the slow-down factor of light, which have been obtained experimentally in GaAs PCWs<cit.>.§ RESULTS The Purcell factor of a quantum emitter coupled to a waveguide is an important figure-of-merit determining the rate of photon generation and ability to overcome decoherence processes. Figure <ref> shows the position and frequency dependence of F_p^wg for x- and y-oriented dipoles.The four columns correspond to dipoles at different frequencies, n_g=5, 20, 58, and 120, respectively. At n_g=5 the Purcell factor is less than one, but scales linearly with the group index and reaches 23 at n_g=120. At the bandedge of the waveguide, the group index and consequently the Purcell-factor diverge. However, in practice this van-Hove singularity in the LDOS is damped by Anderson localization of light induced by unavoidablefabrication disorder. Experimentally F_p^wg=24 has been reported for quantum emitters coupled to PCWs<cit.>. The actual excitation of the waveguide mode by a dipole emitter is shown in Fig. <ref>, which plots |E| for a y-oriented dipole in the antinode of the E_y field for n_g=5 and n_g=58, corresponding to fast and slow light propagation in the PCW. The plots are zoom-ins around the position of the dipole and it should be mentioned that the color bars have been saturated since |𝐄|diverges at the position of the point source. Furthermore, a 'chevron feature' in the field profile is observed close to the dipole, which is a manifestation of dipole-induced light localization coming from the coupling to evanescent modes of the PCW<cit.>. We observe that the active boundary conditions suppress the reflections from the boundaries of the simulation domain very effectively, i.e. the field intensities on the right hand side and left hand side of the simulation domain are uniform as expected from an infinite system.The field profiles plotted in Fig. <ref>are subsequently integrated, as detailed in the previous section, in order to extract coupling to radiation modes. Subsequently we discuss the results of the computations of the coupling to radiation modes. Figure <ref> shows the position dependence of the Purcell factor associated with the coupling to radiation continuum, F_p^rad=γ_rad/γ_0, inside one unit cell. In this case it is desirable to reduce F_p^rad as much as possible below unity, so that the parasitic coupling to non-guided modes is reduced. We find that the suppression is better than a factor of 10 for most spatial positions in Fig. <ref>, and importantly F_p^rad has a complex spatial structure. On the contrary, the frequency dependence of F_p^rad is rather weak and, e.g., changes only about 10% for a y-oriented dipole between n_g=5 and n_g=120. The smallest achievable Purcell factor isF_p^rad = 0.005, i.e. suppression of radiation modes by a factor of 200 relative to the emission rate of a homogeneous medium. The strong suppression of radiation modes in 2D photonic-crystal membranes was first predicted in Ref. <cit.> for photonic crystals without defects. Interestingly the suppression achieved in a PCW reaches the value obtainable in a photonic crystal without defects demonstrating that the missing row of holes in the PCW does not induce additional leakage of the light from the membrane, see Appendix <ref> for further details. Finally, the spatial map of the β-factor and its frequency dependence is investigated, see Fig. <ref>. Here the green and the blue contours correspond to β=0.80 and β=0.96. Note that the implemented color bar showing the magnitude of β is highly nonlinear. Even at low n_g, cf. Fig. <ref>(a and b), a large β-factor can be achieved (higher than 96%) although limited to relatively small spatial regions in the PCW. Increasing n_g by moving into the slow-light region, cf. Fig. <ref>(c-h), increases β significantly, and we find β≥ 0.96 for a very wide range ofdipole positions. More quantitatively, for any dipole located within ± a from the center of the waveguide β≥ 0.96 at the experimentally achievable value of n_g=58. This is a remarkably robustness towards spatial and spectral detuning, which was already confirmed experimentally where the statistics of the β-factor of more than 70 different quantum dots in a PCW has been reported <cit.>. § CONCLUSIONSWe have presented detailed numerical calculations of the β-factor in a PCW. A key step has been to adopt mixed boundary conditions, i.e., activeDirichlet boundary conditions at the terminations of the waveguide and PMLs at the other boundaries to treat the radiation modes. Based on this approach we calculated the coupling rate from a quantum emitter to different optical channels in a PCW. Our results show that the coupling from the emitter to the radiation continuum is highly suppressed compared to an emitter in homogenous medium. The spatial dependence of γ_rad quantifies that a suppression factor larger than 10 is achieved for most regions in the PCW and for all frequencies of the waveguide band. As a direct consequence, the β-factor is close to unity for essentially all emitter locations in the PCW even for moderately-slow light propagation. The detailed simulations confirm the remarkable robustness of the PCW platform against spatial and spectral inhomogeneities and consequently also fabrication imperfections. Such a high coupling efficiency is of importance for a wide range of photonic quantum technology applications including on-demand single-photon sources, multi-qubit gates<cit.>, and single-photon transistors<cit.>. § ACKNOWLEDGMENTS We would like to thank Yuntian Chen for fruitful discussion regarding the finite-element simulations. We gratefully acknowledge financial support from the European Research Council (ERC Advanced Grant "SCALE"), Innovation Fund Denmark (Quantum Innovation Center "Qubiz"), and the Danish Council for Independent Research.§ APPENDIX §.§ Influence of the simulation parameters on γ_radFigure <ref> presents some of the convergence tests carried out to ensure the validity of the simulations and to justify the choice of the radiation box size. We choose γ_box/γ_rad as the target parameter for the convergence tests, where γ_box is the amount of radiation captured by the radiation box of size h_b and l_b, and γ_rad is the value reported in Fig. <ref>.From Fig. <ref>a, we conclude that for l_b/a>25 and h_b/2d_z>4 the value of γ_box is independent of the size of the box to within 5%. Furthermore, the convergence of γ_rad with the size of the actual simulation domain is plotted in Fig. <ref>(b) displaying a similar precision. These convergence tests were carried out for a y-oriented dipole at the E_y-antinode and with n_g=58. We repeated the same tests for dipoles at a few more positions, orientations, and frequencies with very similar results. §.§ Position and frequency dependence of coupling to radiation modes in a photonic crystal. As a comparison, we present the position and frequency dependence of F_p^rad for dipoles located in a photonic-crystal membrane without any waveguide defect region. These simulations were carried out in a similar fashion as for the PCW case, however they did not require active boundary conditions as the photonic crystal already suppresses the light propagation and hence PML boundary conditions are adequate.Figure <ref> maps out the position dependence of F_p^rad inside the bandgap of a photonic crystal for two orthogonal dipole orientations. Furthermore, the frequency dependence of F_p^rad for an emitter in the photonic crystal is displayed. The bandgap of the photonic crystal extends from a/λ=0.256 to a/λ=0.360. The main feature is the inhibition of spontaneous emission inside the bandgap of the photonic crystal, which reaches values as high as 168. These values are very similar to what is found inPCWs (see Fig. <ref>), and hence we conclude that the missing row of holes in the PCW does not significantly alter the coupling to the radiation modes. We mention that these results compare very well to the values reported in <cit.>. | http://arxiv.org/abs/1704.08576v2 | {
"authors": [
"Alisa Javadi",
"Sahand Mahmoodian",
"Immo Söllner",
"Peter Lodahl"
],
"categories": [
"quant-ph",
"physics.optics"
],
"primary_category": "quant-ph",
"published": "20170427140220",
"title": "Numerical modelling of the coupling efficiency of single quantum emitters in photonic-crystal waveguides"
} |
Journal ofClass Files, Vol. 13, No. 9, September 2014 Shell et al.: Bare Demo of IEEEtran.cls for Computer Society JournalsDespite the rapid progress, existing works on action understanding focus strictly on one type of action agent, which we call actor—a human adult, ignoring the diversity of actions performed by other actors. To overcome this narrow viewpoint, our paper marks the first effort in the computer vision community to jointly consider algorithmic understanding of various types of actors undergoing various actions. To begin with, we collect a large annotated Actor-Action Dataset (A2D) that consists of 3782 short videos and 31 temporally untrimmed long videos. We formulate the general actor-action understanding problem and instantiate it at various granularities: video-level single- and multiple-label actor-action recognition, and pixel-level actor-action segmentation. We propose and examine a comprehensive set of graphical models that consider the various types of interplay among actors and actions. Our findings have led us to conclusive evidence that the joint modeling of actor and action improves performance over modeling each of them independently, and further improvement can be obtained by considering the multi-scale natural in video understanding. Hence, our paper concludes the argument of the value of explicit consideration of various actors in comprehensive action understanding and provides a dataset and a benchmark for later works exploring this new problem. video analysis, action recognition, fine-grained activity, actor-action segmentation. Action Understandingwith Multiple Classes of Actors Chenliang Xu, Member, IEEE, Caiming Xiong, and Jason J. Corso, Senior Member, IEEEC. Xu is with the Department of Computer Science, University of Rochester, Rochester, NY 14627. Email: [email protected]. C. Xiong is a senior researcher at Salesforce MetaMind, San Francisco, CA 94105. E-mail: [email protected]. J. J. Corso is with the Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI 48109. E-mail: [email protected]: date / Revised version: date ========================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================== § INTRODUCTIONAction is the heart of video understanding. As such, it has received a significant amount of attention in the last decade. The emphasis has moved from small datasets of a handful of actions <cit.> to large datasets with many dozens of actions <cit.>; from constrained domains like sporting <cit.> to videos in-the-wild <cit.>. Notable methods have demonstrated that low-level features <cit.>, mid-level atoms <cit.>, high-level exemplars <cit.>, structured models <cit.>, attributes <cit.>, and even deeply-learned features <cit.> can be used for action recognition. Impressive methods have even pushed toward action recognition for multiple views <cit.>, event recognition <cit.>, group-based activities <cit.>, and even human-object interactions <cit.>. However, these many works emphasize a small subset of the broader action understanding problem. First, aside from Iwashita et al. <cit.> who study egocentric animal activities, these existing methods all assume the agent of the action, which we call the actor, is a human adult. Although looking at people is certainly a relevant application domain for computer vision, it is not the only one; consider recent advances in video-to-text <cit.> that can be used for semantic indexing large video databases <cit.>, or advances in autonomous vehicles <cit.>. In these applications, understanding both the actor and the action are critical for success: e.g., the autonomous vehicle needs to distinguish between a child, a deer and a squirrel running into the road so it can accurately make an avoidance plan. Applications like these are abundant and growing. Furthermore, if we look to the philosophy of action <cit.>, we find an axiomatic definition of action: “first, action is what an agent can do; second, action requires an intention; third, action requires a bodily movement guided by an agent or agents; and fourth, action leads to side-effects”. For comprehensive action understanding, we need to consider not only the action, but also the agent and the movement of the agent. To that end, our recent work has begun to move in this direction. In the visual-psychophysical study <cit.>, we jointly consider different types of actors performing actions to assess the degree to which a supervoxel segmentation <cit.> retains sufficient semantic information for humans to recognize it. Subsequently, in <cit.>, we consider a larger problem with seven classes of actors and eight classes of actions and assess the state of the art for joint actor-action understanding; this article is an extended version of this latter work. Second, in addition to the limited prior work on the different agents of action, the prior literature largely focuses on action recognition, which is posed as the classification of a temporally pre-trimmed clip into one of k action classes from a closed-world setting. The direct utility of results based on this problem formulation is limited. The community has indeed begun to move beyond this simplified problem into action detection <cit.>, action localization <cit.>, action segmentation <cit.>, and even actionness ranking <cit.>. Again, all of these works do so strictly in the context of human actors and furthermore, many assume that videos are temporally pre-trimmed to contain only single type of action.In this paper, we overcome both of these narrow viewpoints and introduce a new level of generality to the action understanding problem by considering multiple different classes of actors undergoing multiple different classes of actions. To be exact, we consider seven actor classes (adult, baby, ball, bird, car, cat, and dog) and eight action classes (climb, crawl, eat, fly, jump, roll, run, and walk) not including the no-action class, which we also consider. We formulate a general actor-action understanding framework and implement it for three specific problems: actor-action recognition with single- and multiple-label; and actor-action segmentation. These three problems cover different levels of modeling and hence allow us to analyze the new problem thoroughly.To support these new actor-action understanding problems, we have created a new dataset, which we call the Actor-Action Dataset or A2D. The A2D has 3782 short videos with at least 99 instances per valid actor-action tuple and 31 long videos for a selected subset of actor-action tuples. The short videos are temporally trimmed to a few seconds such that the individual actors perform only one type of action in each video, and the videos are labeled at the pixel-level for actors and actions (densely in space over actors, sparsely in time, see Fig. <ref>). The long videos range from a half-minute to two minutes and the individual actors may perform multiple actions over time, where action-tagged bounding boxes are annotated on the actors (densely in time, see Fig. <ref>).We thoroughly analyze empirical performance of both baseline and state-of-the-art graphical models in the context of modeling the interplay of actors and actions. The baseline models include a naïve Bayes model (independent over actors and actions) and a joint product-space model (each actor-action pair is considered as one class). We adapt the bilayer model from <cit.> that considers the compatibilities of actors performing actions; we propose a trilayer model, where two sets of conditional classifier-based edges are added to connectthe bilayer and the joint product-space. These edges explore even finer detail, i.e., the diversity and similarity of various actions performed by various actors. The above models are evaluated at both video-level recognition tasks and pixel-level segmentation task. Observing the performance gap between video-level and pixel-level tasks (the latter is harder), we further explore a multi-scale approach to incorporate video-level recognition responses in optimizing pixel-level actor-action segmentation.Our experiments demonstrate that inference jointly over actors and actions outperforms inference independently over them, and hence, supports the explicit consideration of various actors in comprehensive action understanding. In other words, although a bird and an adult can both eat, the space-time appearance of a bird eating and an adult eating are different in significant ways. Furthermore, the various mannerisms of the way birds eat and adults eat mutually reinforces inference over the constituent parts. This result is analogous to Sadeghi and Farhadi's visual phrases work <cit.> in which it is demonstrated that joint detection over small groups of objects in images is more robust than separate detection over each object followed a merging process, and to Gupta et al.'s <cit.> work on human object-interactions in which considering specific objects while modeling human actions leads to better inferences for both parts. A secondary founding is that the performances of all models considered in actor-action segmentation get improved when both pixel-level and video-level evidences are considered, which further suggests the necessity of multi-scale modeling in understanding videos.Our paper presents the first effort in the computer vision community to jointly consider various types of actors undergoing various actions.As such, we pose three goals: * we seek to formulate the general actor-action understanding problem and instantiate it at various granularities; * we assess whether it is beneficial to explicitly jointly consider actors and actions in this new problem-space using both short and long videos; * we explore a multi-scale approach that bridges the gap between recognition andsegmentation tasks.Notice that an early version of our work appears in <cit.>. In this journal article, we make the following additions: 1) we extend the A2D dataset with 31 long videos, a total of 40 minutes, that contain lots of actor-action changes in time; 2) we conduct experiments on long videos using newly devised metric to evaluate temporal performance of segmentation models; 3) we apply a new inference scheme that achieves 10 times faster inference than the conference paper;4) we explore a multi-scale approach to incorporate video-level recognition responses via label costs in inferring pixel-level segmentation; and 5) we provide more detail of our methods and a new related work section.The rest of the paper is organized as follows: Sec. <ref> discusses related work; Sec. <ref> introduces the extended A2D dataset; Sec. <ref> formulates the general actor-action understanding problem and describes various models to consider actor-action interactions; Sec. <ref> describes a multi-scale approach to actor-action segmentation; Sec. <ref> and Sec. <ref> detail the experiments conducted on short videos and long videos, respectively; and Sec. <ref> concludes our paper.§ RELATED WORKOur work explores a new dimension in fine-grained action understanding. A related work is Bojanowski et al. <cit.>, where they focus on finding different human actors in movies, but these are the actor-names and not different types of actors, like dog and cat as we consider in this paper. In <cit.>, Iwashita et al. study egocentric animal activities, such as turn head, walk and body shake, from a view-point of an animal—a dog. Our work differs from them by explicitly considering the types of actors in modeling actions. Similarly, our work also differs from the existing works on actions and objects, such as <cit.>, which are strictly focused on interactions between human actors manipulating various physical objects.In addition to the increased diversity in acitivities, the literature in action understanding is moving away from simple video classification to fine-grained output. For example, methods like <cit.> detect human actions with bounding box tubes or 3D volumes in videos; and methods like <cit.> even provide pixel-level segmentation of human actions. The shift of research interest is also observed in the relaxed assumptions of videos. For instance, temporally untrimmed videos are considered in <cit.> and <cit.>, and online streaming videos are explored in <cit.> and <cit.>. Our work analyzes actors and actions at various granularities, e.g., both video-level and pixel-level, and we experiment with both short and long videos. §.§ From Segmentation Perspective We now discuss related work in segmentation, which is a major emphasis of our broader view ofaction understanding, e.g., the actor-action segmentation task. Semantic segmentation methods can now densely label more than a dozen classes in images <cit.> and videos <cit.> undergoing rapid motion; semantic segmentation methods have even been unified with object detectors and scene classifiers <cit.>, extended to 3D <cit.> and posed jointly with attributes <cit.>, stereo <cit.> and SFM <cit.>. Although the underlying optimization problems in these methods tend to be expensive, average per-class accuracy scores have significantly increased, e.g., from 67% in <cit.> to nearly 80% in <cit.> on the popular MSRC semantic segmentation benchmark. Further works have moved beyond full supervision to weakly supervised object discovery and learning <cit.>. However, these existing works in semantic segmentation focus on labeling pixels/voxels as various objects or background-stuff classes, and they do not consider the joint label-space of what actions these “objects” may be doing. Our work differs from them by directly considering this actor-action problem, while also building on the various advances made in these papers. Other related works include video object segmentation <cit.> and joint temporal segmentation with action recognition <cit.>. The video object segmentation methods are class-independent and assume a single dominant object (actor) in the video; they are hence not directly comparable to our work although one can foresee a potential method using video object segmentation as a precursor to the actor-action understanding problem. §.§ From Model Perspective Multi-task learning has been effective in many applications, such as object detection <cit.>, and classification <cit.>. The goal is to learn models or shared representations jointly that outperforms learning them separately for each task. Recently, multi-task learning has also been adapted to action classification. For example, in <cit.>, Zhou et al. classify human actions in videos with shared latent tasks. Our paper differs from them by explicitly modeling the relationship and interactions amongactors and actions under a unified graphical model. Notice that it is possible to use multi-task learning to train a shared deep representation for actors and actions extending frameworks such as the multi-task network cascades <cit.>, which has a different focus than this work; hence, we leave it to future work. § A2D V2—THE EXTENDED ACTOR-ACTION DATASETThe A2D v2 has 3782 short videos from the initial A2D <cit.> and 31 new long videos. All videos are collected from YouTube; and they are hence unconstrained “in-the-wild” videos with varying characteristics. We select seven classes of actors performing eight different actions. Our choice of actors covers articulated ones, such as adult, baby, bird, cat and dog, as well as rigid ones, such as ball and car. The eight actions are climbing, crawling, eating, flying, jumping, rolling, running, and walking. A single action-class can be performed by mutiple actors, but none of the actors can perform all eight actions.For example, we do not consider adult-flying or ball-running in the dataset. In some cases, we have pushed the semantics of the given action term to maintain a small set of actions: e.g., car-running means the car is moving and ball-jumping means the ball is bouncing. One additional action label none is added to account for actions other than the eight listed ones and actors that are not performing an action. Therefore, we have in total 43 valid actor-action tuples (see Fig. <ref> colored entries). This allows us to explore the valid combinations of actor-action tuples, rather than brute-force enumeration.§.§ Short VideosWe use various text-searches generated from actor-action tuples to query the YouTube database.Resulting videos are then manually trimmed to contain at least one instance of the primary actor-action tuple, and they are also encouraged to contain optionally many actor-action tuples. The trimmed short videos have an average length of 136 frames, with a minimum of 24 frames and a maximum of 332 frames. We split them into 3036 training videos and 746 testing videos divided evenly over all actor-action tuples. Figure <ref> (numbers) shows the counts of short videos for each actor-action tuple. Notice that one-third of the short videos have more than one actor performing different actions (see Fig. <ref> for exact counts), which further distinguishes our dataset from existing action classification datasets, e.g., <cit.>. To support the broader set of action understanding problems in consideration, we label three to five frames for each short video with dense pixel-level actor-action labels. Figure <ref> shows sampled frame examples. The selected frames are evenly distributed over a video. We start by collecting crowd-sourced annotations from MTurk using the LabelMe toolbox <cit.>, then we manually filter each video to ensure the labeling quality as well as the temporal coherence of labels. In total, we have collected 11936 labeled frames. Video-level labels are computed directly from these pixel-level labels for the recognition tasks. To the best of our knowledge, our dataset is the first video dataset that contains both pixel-level actor and action annotations.§.§ Long VideosWe add 31 temporally untrimmed long videos to A2D v2, where a single actor has many changes of actions over time in a video. These videos are more realistic comparing to manually trimmed short videos. The long videos also contain multiple actors, thus they are very different than the untrimmed videos used in existing action localization challenges such as the ActivityNet <cit.>, which allows us to analyze the actor-action problems in both space and time.The long videos have an average length of 1888 frames, with a minimum of 593 frames and a maximum of 3605 frames. The actors in long videos may perform a series of actions. For example, a dog may perform walking, then running, then walking in a video. The statistics are shown in Fig. <ref>. For long videos, we reserve 8 videos for validation and use the remaining 23 videos for testing. We use VATIC annotation tool <cit.> to track actors with bounding boxes through a video, and label actions densely in time. Figure <ref> shows examples on two long videos. § ACTOR-ACTION UNDERSTANDING PROBLEMSWithout loss of generality, let 𝒱 = {v_1, …, v_n} denote a video with n voxels in space-time lattice Λ^3 or n supervoxels in a video segmentation <cit.> represented as a graph 𝒢 = (𝒱,ℰ), where the neighborhood structure is given by the supervoxel segmentation method; when necessary we write ℰ(v), where v ∈𝒱, to denote the subset of 𝒱 that are neighbors with v. We use 𝒳 to denote the set of actor labels: {adult, baby, ball, bird, car, cat, dog}, and we use 𝒴 to denote the set of action labels: {climbing, crawling, eating, flying, jumping, rolling, running, walking, none[The none action means either there is no action present or the action is not one of those we have considered.]}. Consider a set of random variables 𝐱 for actor and another 𝐲 for action; the specific dimensionality of 𝐱 and 𝐲 will be defined later.Then, the general actor-action understanding problem is specified as a posterior maximization: (𝐱^*,𝐲^*) = _𝐱,𝐲P(𝐱,𝐲|𝒱) . Specific instantiations of this optimization problem give rise to various actor-action understanding problems, which we specify next, and specific models for a given instantiation will vary the underlying relationship between 𝐱 and 𝐲 allowing us to deeply understand their interplay.§.§ Single-Label Actor-Action RecognitionThis is the coarsest level of granularity we consider and it instantiates the standard action recognition problem <cit.>. Here, 𝐱 and 𝐲 are simply scalars x and y, respectively, depicting the single actor and action label to be specified for a given video 𝒱. We consider three models for this case: Naïve Bayes-Based: Assume independence across actions and actors, and then train a set of classifiers over actor space 𝒳 and a separate set of classifiers over action space 𝒴. This is the simplest approach and is not able to enforce actor-action tupleexistence: e.g., it may infer invalid adult-flying. Joint Product Space: Create a new label space 𝒵 that is the joint product space of actors and actions: 𝒵 = 𝒳×𝒴.Directly learn a classifier for each actor-action tuple in this joint product space. Clearly, this approach enforces actor-action tuple existence, and we expect it to be able to exploit cross-actor-action features to learn more discriminative classifiers. However, it may not be able to exploit the commonality across different actors or actions, such as the similar manner in which a dog and a cat walk. Trilayer: The trilayer model unifies the naïve Bayes and the joint product space models. It learns classifiers over the actor space 𝒳, the action space 𝒴 and the joint actor-action space 𝒵. During inference, it separately infers the naïve Bayes terms and the joint product space terms and then takes a linear combination of them to yield the final score. It models not only the cross-actor-action but also the common characteristics among the same actor performing different actions as well as the different actors performing the same action. In all cases, we extract local features (see Sec. <ref> for details) and train a set of one-versus-all classifiers, as is standard in contemporary action recognition methods, and although not strictly probabilistic, can be interpreted as such to implement Eq. <ref>.§.§ Multiple-Label Actor-Action RecognitionAs noted in Fig. <ref>, about one-third of the short videos in A2D v2 have more than one actor and/or action present in a given video. In many realistic video understanding applications, we find such multiple-label cases. We address this explicitly by instantiating Eq. <ref> for the multiple-label case. Here, 𝐱 and 𝐲 are binary vectors of dimension |𝒳| and |𝒴| respectively. The x_i takes value one if the ith actor-type is present in the video and zero otherwise. We define 𝐲 similarly. This general definition does not tie specific elements of 𝐱 to those in 𝐲. Therefore, it allows us to compare independent multiple-label performance over actors and actions within that of the actor-action tuples. We again consider a naïve Bayes pair of multiple-label actor and action classifiers, multiple-label actor-action classifiers over the joint product space, as well as the trilayer model that unifies the above classifiers.§.§ Actor-Action SegmentationActor-action segmentation is the most fine-grained instantiation that we consider, and it subsumes other coarser problems like detection and localization, which we do not consider in this paper. Here, we seek a label for actor and action per-voxel over the entire video. We define two sets of random variables 𝐱 = {x_1, …, x_n} and 𝐲 = {y_1, …, y_n} to have dimensionality in the number of voxels or supervoxels, and assign each x_i ∈𝒳 and each y_i ∈𝒴.The objective function in Eq. <ref> remains the same, but the way we define the graphical model implementing P(𝐱,𝐲|𝒱) leads to acutely different assumptions on the relationship between actor and action variables.We explore this relationship in the remainder of this section. We start by again introducing a naïve Bayes-based model in Sec. <ref> that treats the two classes of labels separately, and a joint product space model in Sec. <ref> that considers actors and actions together in a tuple [𝐱, 𝐲]. We then explore a bilayer model in Sec. <ref>, inspired byLadický et al. <cit.>, that considers the inter-set relationship between actor and action variables. Following that, we introduce a trilayer model in Sec. <ref> that considers both intra- and inter-set relationships. Figure <ref> illustrates these various graphical models. Finally, we show that these models can be efficiently solved by graph cuts inference in Sec. <ref>.§.§.§ Naïve Bayes-based ModelFirst, let us consider a naïve Bayes-based model, similar to the one used for actor-action recognition earlier:P(𝐱,𝐲|𝒱) = P(𝐱|𝒱)P(𝐲|𝒱) = ∏_i∈𝒱 P(x_i)P(y_i) ∏_i∈𝒱∏_j∈ℰ(i) P(x_i,x_j)P(y_i,y_j) ∝∏_i∈𝒱ϕ_i (x_i) ψ_i (y_i) ∏_i∈𝒱∏_j∈ℰ(i)ϕ_ij (x_i,x_j)ψ_ij(y_i,y_j) where ϕ_i and ψ_i encode the separate potential functions defined on actor and action nodes alone, respectively, and ϕ_ij and ψ_ij are the pairwise potential functions within sets of actor nodes and sets of action nodes, respectively. We train classifiers {f_c|c ∈𝒳} over actors and {g_c|c ∈𝒴} on sets of actions using features described in Sec. <ref>, and ϕ_i and ψ_i are the classification scores for supervoxel i. The pairwise edge potentials have the form of a contrast-sensitive Potts model <cit.>: ϕ_ij = {[1 ; exp(-θ/(1+χ_ij^2)) otherwise, ]. where χ_ij^2 is the χ^2 distance between feature histograms of nodes i and j, θ is a parameter to be learned from the training data. ψ_ij is defined analogously.Actor-action semantic segmentation is obtained by solving these two flat CRFs independently. §.§.§ Joint Product SpaceWe consider a new set of random variables 𝐳 = {z_1, …, z_n} defined again on all supervoxels in a video and take labels from the actor-action product space 𝒵 = 𝒳×𝒴. This formulation jointly captures the actor-action tuples as unique entities but cannot model the common actor and action behaviors among different tuples as later models below do; we hence have a single-layer graphical model: P(𝐱,𝐲|𝒱) ≐ P(𝐳|𝒱) = ∏_i∈𝒱 P(z_i)∏_i∈𝒱∏_j∈ℰ(i) P(z_i,z_j) ∝∏_i∈𝒱φ_i(z_i) ∏_i∈𝒱∏_j∈ℰ(i)φ_ij (z_i, z_j) = ∏_i∈𝒱φ_i([x_i,y_i]) ∏_i∈𝒱∏_j∈ℰ(i)φ_ij ([x_i,y_i], [x_j,y_j]) , where φ_i is the potential function for joint actor-action product space label, and φ_ij is the inter-node potential function between nodes with the tuple [𝐱, 𝐲]. To be specific, φ_i contains the classification scores on the node i from running trained actor-action classifiers {h_c|c∈𝒵}, and φ_ij has the same form as Eq. <ref>. Figure <ref> (b) illustrates this model as a one layer CRF defined on the actor-action product space. §.§.§ Bilayer ModelGiven the actor nodes 𝐱 and action nodes 𝐲, the bilayer model connects each pair of random variables {(x_i, y_i)}_i=1^n with an edge that encodes the potential function for the tuple [x_i, y_i], directly capturing the covariance across the actor and action labels. Therefore, we have: P(𝐱,𝐲|𝒱) = P(𝐱|𝒱) P(𝐲|𝒱)∏_i∈𝒱 P(x_i,y_i)∝ ∏_i∈𝒱ϕ_i(x_i) ψ_i(y_i) ξ_i(x_i,y_i) ·∏_i∈𝒱∏_j∈ℰ(i)ϕ_ij(x_i,x_j)ψ_ij(y_i,y_j) , where ϕ_· and ψ_· are defined as earlier, ξ_i(x_i,y_i) is a learned potential function over the product space of labels, which can be exactly the same as φ_i in Eq. <ref> above or a compatibility term like the contrast sensitive Potts model, Eq. <ref> above. We choose the former in this paper. Fig. <ref> (c) illustrates this model. We note that additional links can be constructed by connecting corresponding edges between neighboring nodes across layers and encoding the occurrence among the bilayer edges, such as the joint object class segmentation and dense stereo reconstruction model in Ladický et al. <cit.>. However, their model is not directly suitable here. §.§.§ Trilayer ModelSo far we have introduced three baseline formulations of Eq. <ref> for actor-action segmentation that relate the actor and action terms in different ways. The naïve Bayes model (Eq. <ref>) does not consider any relationship between actor 𝐱 and action 𝐲 variables. The joint product space model (Eq. <ref>) combines features across actors and actions as well as inter-node interactions in the neighborhood of an actor-action node. The bilayer model (Eq. <ref>) adds actor-action interactions among separate actor and action nodes, but it does not consider how these interactions vary spatiotemporally. Therefore, we introduce a new trilayer model that explicitly models such variations (see Fig. <ref> (d)) by combining nodes 𝐱 and 𝐲 with the joint product space nodes 𝐳:P(𝐱,𝐲, 𝐳|𝒱) = P(𝐱|𝒱)P(𝐲|𝒱) P(𝐳|𝒱) ∏_i∈𝒱P(x_i,z_i) P(y_i,z_i) ∝ ∏_i∈𝒱ϕ_i(x_i) ψ_i(y_i) φ_i(z_i) μ_i(x_i,z_i) ν_i(y_i,z_i)·∏_i∈𝒱∏_j∈ℰ(i)ϕ_ij(x_i,x_j)ψ_ij(y_i,y_j) φ_ij(z_i, z_j) , where we further define: μ_i(x_i,z_i)={[ w(y_i'|x_i); 0 otherwise ].ν_i(y_i,z_i)={[ w( x_i'|y_i) ;0otherwise ]. . Terms w(y_i'|x_i) and w(x_i'|y_i) are classification scores of conditional classifiers, which are explicitly trained for this trilayer model. These conditional classifiers are the main reason for the increased performance found in this method: separate classifiers for the same action conditioned on the type of actor are able to exploit the characteristics unique to that actor-action tuple. For example, when we train a conditional classifier for action eating given actor adult, we use all other actions performed by adult as negative training samples. Therefore our trilayer model considers all relationships in the individual actor and action spaces as well as the joint product space. In other words, the previous three baseline models are all special cases of the trilayer model.It can be shown that the solution (𝐱^*,𝐲^*,𝐳^*) maximizing Eq. <ref> also maximizes Eq. <ref> (see Appendix). §.§.§ InferenceUntil now we have described all models solving Eq. <ref>, and they can be optimized using loopy belief propagation in our previous paper <cit.>. In this section, we show that they can also be optimized using the graph cuts <cit.>, which greatly improves the inference efficiency. We use the trilayer model here as an illustration example and we note that the bilayer model can be solved in a similar way. We define a new set of random variables 𝐋 = {(L^o,L^a)_1, …, (L^o,L^a)_n | L^o ∈𝒳, L^a ∈𝒴} on voxels in a video or supervoxels of a video segmentation. Therefore, we can rewrite Eq. <ref> in the following form: P(𝐱,𝐲,𝐳|𝒱)≐ P(𝐋|𝒱) = ∏_i∈𝒱 P(L_i)∏_i∈𝒱∏_j∈ℰ(i) P(L_i,L_j) ∝∏_i∈𝒱Φ_i(L_i) ∏_i∈𝒱∏_j∈ℰ(i)Φ_ij (L_i, L_j) , where we define the unary term as: Φ_i(L_i) ∝φ_i(z_i)·μ_i(x_i,z_i) ϕ_i(x_i)·ν_i(y_i,z_i) ψ_i(y_i), and the pairwise term as: Φ_ij(L_i,L_j) ∝{[ ϕ_ij(x_i,x_j) φ_ij(z_i, z_j) ; ψ_ij(y_i,y_j) φ_ij(z_i, z_j) ; ϕ_ij(x_i,x_j) ψ_ij(y_i,y_j) φ_ij(z_i, z_j) ;1. ]. Figure <ref> shows a visualization example of the trilayer model after the rewritten. The edges linking (x_i,z_i) and (y_i,z_i) become a part of the unary potential in Eq. <ref>. The pairwise term as defined in Eq. <ref> satisfies the submodular property. Therefore, Eq. <ref> can be efficiently optimized using α-expansion or αβ-swap based multi-label graph cuts.§ BRIDGING THE GAP BETWEEN RECOGNITION AND SEGMENTATIONWe have considered the video-level actor-action recognition in Sec. <ref> and Sec. <ref>, and the pixel-level actor-action segmentation in Sec. <ref>. Apparently, the latter is a much harder task; this is also backed up in our experiments in Sec. <ref>. Therefore, it makes sense to use the results obtained from the easier task, i.e., the video-level actor-action recognition, to guide the harder tasks, i.e., pixel-level actor-action segmentation. In this section, we describe a simple but effective approach, inspired by <cit.> to make use of the recognition responses in optimizing segmentation. The idea is to model the responses from video-level recognition as label costs for pixel-level segmentation. Let us define Φ_𝒱 (l) as the classification score, in the range of (0,1], for a given actor-action label l from video-level recognition, and Φ_𝒱 (·) can be any model considered at the video-level (see Sec. <ref>). We extend the pixel-level trilayer segmentation formulation in Eq. <ref>, such that:P(𝐱,𝐲,𝐳|𝒱)≐ P(𝐋|𝒱)= ∏_i∈𝒱 P(L_i)∏_i∈𝒱∏_j∈ℰ(i) P(L_i,L_j) · P(𝐋) ∝∏_i∈𝒱Φ_i(L_i) ∏_i∈𝒱∏_j∈ℰ(i)Φ_ij (L_i, L_j) ·∏_l∈ℒδ_l (𝐋) Φ_𝒱 (l) , where δ_l (𝐋) has the form: δ_l (𝐋) ={[ 1; 1 / Φ_𝒱 (l)otherwise. ]. In other words, δ_l (𝐋) Φ_𝒱 (l) represent a penalty term: when a certain actor-action label l is presented in the labeling field, it has to pay a penalty that is proportional to its video-level classification score Φ_𝒱 (l); when label l is not presented, the penalty is canceled. The formulation in Eq. <ref> has two utilities. First, it enforces a compact video labeling that is parsimony to the number of different labels used to describe the video. In other words, we prefer the output video labeling to be as clean as possible so that it can be unambiguously used in other applications, e.g., we do not want too many fragmented segment noises to confuse a real-time robotic system. Second, rather than uniformly penalizing over all labels, the amount of penalty is proportional to its video-level recognition response, i.e., less penalty if the actor-action label is supported at the video-level and more penalty if it is not. Therefore, the modeling achieves our goal of guiding harder task using relatively more reliable responses from easier task. Equation <ref> is described in the context of the trilayer segmentation model. It is trivial to extend to other segmentation models. For the naïve Bayes segmentation model, we use separate scores for actors and actions. For all other models in actor-action segmentation, we use the trilayer model from the video-level actor-action recognition for implementing Φ_𝒱 (·). Notice that Eq. <ref> can also be efficiently solved by graph cuts inference following <cit.>. § EXPERIMENTS ON SHORT VIDEOSWe thoroughly study each of the instantiations of the actor-action understanding problem with the overarching goal of assessing if the joint modeling of actor and action improves performance over modeling each of them independently, despite the large space. We discuss the experimental results obtained on A2D short videos in this section and leave the discussion of experiments on long videos in Sec. <ref>. We follow the training and testing splits discussed in Sec. <ref>; for assigning a single-label to a short video to evaluate the single-label actor-action recognition, we choose the label associated with the query for which we searched and selected that video from YouTube. §.§ Single-Label Actor-Action RecognitionFollowing the typical action recognition setup, e.g., <cit.>, we use the dense trajectory features (trajectories, HoG, HoF, MBHx and MBHy) <cit.> and train a set of one-versus-all SVM models (with RBF-χ^2 kernels from LIBSVM <cit.>) for the label sets of actors, actions and joint actor-action labels. Specifically, when training the eating classifier, the other seven actions are negative examples; when we train the bird-eating classifier, we use the 35 other actor-action labels as negative examples. Notice that for video-level actor-action recognition, we do not consider actors with none action since they are not considered as a dominant action for videos. To evaluate the actor-action tuple for the naïve Bayes model, we first train and test actor and action classifiers independently, and then score them together (i.e., a video is correct if and only if both actor and action are correct).Table <ref>-left shows the classification accuracy of the naïve Bayes, joint product space and trilayer models in terms of classifying actor labels alone, action labels alone and joint actor-action labels. We note that the scores are not directly comparable along the columns (e.g., the space of independent actors and actions is significantly smaller than that of actor-action tuples); the point of comparison is along the rows. We observe that the independent model outperforms the joint product space model in evaluating actions alone; for this, we suspect that the regularity across different actors for the same action is underexploited in the naïve Bayes model and exploited in the joint product space model, but that results in more inter-class overlap in the latter case. For example, a cat-running and a dog-running have both similar and different signatures in space-time: the naïve Bayes model does not need to distinguish between them; the joint product space model does, but its effort is not appreciated in evaluating actions alone ignoring the actors. Furthermore, we find that when we consider both actor and action in evaluation, it is clearly beneficial to jointly model them. This phenomenon occurs in all of our experiments. Finally, the trilayer model outperforms the other two models in terms of both individual actor or action tasksas well as the joint actor-action task. The reason is that the trilayer model incorporates both types of relationships that are separately modeled in the naïve Bayes and joint product space models.§.§ Multiple-Label Actor-Action RecognitionFor the multiple-label case, we use the same dense trajectory features as in Sec. <ref>, and we train one-versus-all SVM models again for the label sets of actor, action and actor-action tuples, but with different training regimen to capture the multiple-label setting. For example, when training the adult classifier, we use all videos containing actor adult as positive samples no matter the other actors that coexist in the video, and we use the rest of videos as negative samples. For evaluation, we adapt the approach from HOHA2 <cit.>. We treat multiple-label actor-action recognition as a retrieval problem and compute mean average precision (mAP) given the classifier scores.Table <ref>-right shows the performance of the three methods on this task. Again, we observe that the joint product space model has higher mAP than the naïve Bayes model in the joint actor-action task, and lower mAP in individual tasks. We also observe that the trilayer model further improves the scores following the same trend as in the single-label recognition case. Furthermore, we observe large improvement in both individual tasks when comparing the trilayer model to the other two. This implies that the “side” information of the actor when doing action recognition (and vice versa) provides useful information to improve the inference task in a careful modeling, thereby answering the core question in the paper.§.§ Actor-Action Segmentation§.§.§ Experiment Setup We use TSP <cit.> to obtain supervoxel segmentations due to its strong performance on the supervoxel benchmark <cit.>. In our experiments, we set k=400 yielding about 400 supervoxels touching each frame. We compute histograms of textons and dense SIFT descriptors over each supervoxel volume, dilated by 10 pixels. We also compute color histograms in both RGB and HSV color spaces and dense optical flow histograms. We extract feature histograms from the entire supervoxel 3D volume, rather than a single representative superpixel <cit.>. Furthermore, we inject the dense trajectory features <cit.> to supervoxels by assigning each trajectory to the supervoxels it intersects in the video.Frames in A2D short videos are sparsely labeled; to obtain a supervoxel's ground-truth label, we look at all labeled frames in a video and take a majority vote over labeled pixels. We train one-versus-all SVM classifiers (linear kernels) for actors, actions, and actor-actions as well as conditional classifiers separately. The parameters of the graphical model are tuned by empirical search, and graph cut is used for inference as in Sec. <ref>. The inference output is a dense labeling of video voxels in space-time, but, as our dataset is sparsely labeled in time, we compute the average per-class segmentation accuracy only against those frames for which we have ground-truth labels. We choose average per-class accuracy over global pixel accuracy because our goal is to compare the labelings of actors and actions other than background classes, which still dominates the majority of pixels in a video and all algorithms are performing quite well (all over 82%, see the second column in Tab. <ref>). §.§.§ Benchmark State-of-The-Art CRFs Before we discuss the performance of segmentation models proposed in this paper, we benchmark two strong CRF models that are originally proposed for image segmentation. The first is the robust P^N model <cit.> that defines a multi-label random field on a superpixel hierarchy and performs inference exhaustively from finer levels to coarser levels. We apply their supplied code off-the-shelf as a baseline. The average per-class accuracy is 13.7% for the joint actor-action tuple, 47.2% for actor alone and 34.49% for action alone. We suspect that the modeling at pixel and superpixel levels can not well capture the motion changes of actions, which explains why the actor score is high but the other scores are comparatively lower. The second method is the fully-connected CRF <cit.> that builds a pairwise dense random filed and imposes Gaussian mixture kernels to regularize the pairwise interactions. We extend their algorithm to use the same supervoxels and features as models considered in this paper and implement it based on the joint product space unaries. Its average per-class accuracy achieves 25.4% for the joint actor-action tuple, 44.8% for actor alone and 45.5% for action alone. With the video-oriented features, the fully-connected CRF has surpassed many baseline segmentation models considered in this paper, e.g., the naïve Bayes and joint product space models, but still has a clear gap to the trilayer model. This is expected as the dense edges have already modeled significant pairwise interactions but itlacks the explicit consideration of various actor and action dependencies as in the trilayer model.§.§.§ Results & Discussion Overall Performance. We first analyze the overall performance of various segmentation models in Table <ref>. From left to right are results obtained with unary classifiers alone, unary and pairwise interactions, and full model with video-level recognition as label costs, respectively. We evaluate the actor-action tuples as well as individual actor and action tasks. Notice that the conditional model is a variation of bilayer model with different aggregation—we infer the actor label first then the action label conditioned on the actor. We also note that the bilayer model has the same unary scores as the naïve Bayes model (using actor ϕ_i and action ψ_i outputs independently) and the actor unary of the conditional model is the same as that of the naïve Bayes model (followed by the conditional classifier for action). Although for the individual tasks, the naïve Bayes model is not the worst one, it performs worst when we consider the actor-action tuples, which is expected as it dose not encode any interactions between the two sets of labels. The joint product space model outperforms the naïve Bayes model for the actor-action tuples, but it has worse performance for the individual actor and action tasks, which is also observed in the actor-action recognition task. The conditional model has better actor-action scores, albeit it uses the inferred actors from the naïve Bayes model, which indicates that knowing actors can help with action inference. We also observe that the bilayer model has a poor unary performance of 16.35% (for actor-action tuple) that is the same as the naïve Bayes model but after the edges between the actor and action nodes are added, it improves dramatically to 23.43%, which suggests that the performance boost comes from the interactions of the actor and action nodes in the bilayer model. We also observe that the trilayer model has not only much better performance in the joint actor-action task, but also better scores for the individual actor and action tasks, as it is the only model that considers all three types of tasks together—the individual actor and action tasks, the joint space actor-action task and the conditional tasks. Furthermore, all models have received significant performance boost when the extra information from video-level recognition is used. Individual Actor-Action Tuples. We compare the performance for individual actor-action tuples in Table <ref>. The models considered here are full models with video-level recognition as label costs. We observe that the trilayer model has leading scores for more actor-action tuples than the other models, and the margin is significant for labels such as bird-flying and adult-running. We also observe a systematic increase in performance as more complex actor-action variable interactions are included. We note that the tuples with none action are sampled with greater variation than the action classes (see Fig. <ref> for examples), which contributes to the poor performance of none over all actors. Interestingly, the naïve Bayes model has relatively better performance on the none action classes. We suspect that the label-variation for none leads to high-entropy over its classifier density and hence when joint modeling, the actor inference pushes the action variable away from the none action class.Qualitative Results. We first show intermediate results leading to the full model of trilayer segmentation in Fig. <ref>, which include unaryresponses alone, and unary responses and pairwise interactions. The full model contains unary responses, pairwise interactions and label costs from video-level recognition. The first row shows examples where both pairwise and full model are mostly correct. The next two rows contain examples where the video-level recognition via label costs help recover partially correct and even wrong segmentations produced by the pairwise model. Notice that our model differs from a uniform distribution of label costs, which can only enforce the compactness of segmentation and is hard to correct segmentations, where most of local predictions are wrong, e.g., the bird-walking and dog-rolling examples. One limitation of our model is that when video-level recognition generates poor predictions, it may also result in incorrect pixel-level segmentation, as shown in the last row of the figure. Finally, we show the comparison of all actor-action interaction models in Fig. <ref>, where they use the same video-level recognition via label costs. The point of comparison is at the various ways they model the interplay of actors and actions. Recall that the naïve Bayes model considers the actor and action tasks independent of each other; it gets many partially correct segmentations, e.g., the cat-eating v.s. ground-truth dog-eating and the dog-rolling v.s. ground-truth cat-rolling, but none of them agrees completely with ground-truth labels. The bilayer model adds a compatibility term between actors and actions, and correctly recovers the bird-flying in the first video, while the conditional model fails since it infers action after inferring actor. The joint product space model correctly recovers the bird-flying and the baby-rolling in the first two videos. The trilayer model considers all situations in other models, and recovers the correct segmentations in most videos. The baby-crawling example in the last video is particularly hard, where the trilayer model mislabels part of it as adult-crawling suggesting the need of object-level awareness in the future work.§ EXPERIMENTS ON LONG VIDEOSIn this section, we evaluate our segmentation models on the 31 untrimmed long videos, which contain over 58K frames with dense annotations in time as introduced in Sec. <ref>. We focus the evaluation on the temporal performance, e.g., the continuity of performing one action and the sensitivity of action changes, rather than the exact spatial location of segmentation as we evaluate on the short videos. This provides us a complementary viewpoint to assess our overarching goal, i.e., whether the joint modeling of actor and action improves performance over modeling each of them independently. §.§ Evaluation Metric Despite the large actor-action label space, actors in A2D short videos only perform one type of actions throughout an entire video; this assumption certainly does not hold in reality: most of our queried YouTube videos contain actors performing a series of actions. Take the video shown in the top row of Fig. <ref> as an example: the blue-clothe baby performs walking, rolling, walking, then none; and the blue ball performs rolling, jumping, then none. For videos like this, we have manually labeled bounding boxes on actors throughout the video and recorded the start and finish timestamps of actions we consider in the paper. We define an actor-action track as a tube composed of bounding boxes on actor that performs one type of actions for a continues period of time defined by the start and finish timestamps of the action. Therefore, in the previous example, the blue-clothe baby have four actor-action tracks and the blue ball has three actor-action tracks. There are a total of 727 actor-action tracks for A2D long videos (128 for validation and 599 for testing). Our evaluation metric is, indeed, designed to measure the recall of such actor-action tracks. Formally, let us define an actor-action track as T = {𝐁^t_1_l, …, 𝐁^t_m_l}, where 𝐁^t_i_l denotes the bounding box on frame t_i with actor-action label l, and t_1 and t_m are the start and finish timestamps of the track. The actor-action video segmentation to be evaluated is denoted as a sequence of image segmentations 𝐋 = {𝐋^1, …, 𝐋^n}, where n is the total number of frames in video. The temporal recall of the actor-action track is defined as: R(𝐋, T) = 1/m∑_i=1^m 1[ | 𝐋^t_i(𝐁^t_i_l) | >0] , where | 𝐋^t_i(𝐁^t_i_l) | denotes the sum of pixels that match actor-action label l within the segmentation region at 𝐋^t_i cropped by the bounding box 𝐁^t_i_l. Intuitively, Eq. <ref> measures how well in temporal domain the track is covered by segmentation, and it counts a match as long as the segmentation overlaps with bounding box region spatially in a frame.We evaluate over all actor-action tracks {T_1, …, T_K} in the testing set and we have: Recall(σ) = 1/K∑_k=1^K 1[R(𝐋, T_k) ≥σ] , where σ is a recall threshold meaning that we only count a positive recall if larger or equal to σ of the track is covered by the correct segmentation. To generate a plot, we vary σ from 0 to 1 by a step size of 0.1. Notice that we apply 1[R(𝐋, T_k) > σ] for σ=0. At the time of writing, there is not a well-established evaluation metric for measuring the quality of spatiotemporal segmentations in untrimmed long videos. The metric proposed in this section aims to measure the temporal performance of actor-action segmentation models. It is complementary to the spatial-oriented evaluation in Sec. <ref> as the metric here does not consider how well the segmentation overlaps with ground-truth spatially. For a comprehensive understanding, both evaluations defined here and in Sec. <ref> should be considered. §.§ Results & Discussion We evaluate our actor-action segmentation models on long videos, where we only model unary and pairwise interactions as described in Sec. <ref>. We do not consider video-level actor-action recognition or its utility as label costs in segmentation, since the task is ill-posed in dealing with temporally untrimmed videos where a single actor may perform multiple different actions. Again, we use TSP <cit.> to obtain supervoxel segmentations and compute the same set of features as we do for short videos. Since we have limited number of long videos, the unary classifiers are trained on the training set of short videos where ground-truth segmentations are available; the parameters are empirically tuned using validation set of long videos.The recall plot of the actor-action tracks for all segmentation models is shown in Fig. <ref>. In general, the recall decreases when σ increases; this is expected as the measure of positive recall is getting more strict as σ increases. The trilayer segmentation model outperforms other models by a significant margin. It achieves a 45% recall of all tracks when σ=0. Even in the extreme case, when σ=1 meaning that the track region at every frame is covered by at least one correct segment, the trilayer model still get roughly 10% of all actor-action tracks correct. For other models, the bilayer model outperforms the joint product space and conditional models. Interestingly, the naïve Bayes model performs as good as the bilayer model for σ>0.2. We suspect that the noisy segmentations produced by the independent modeling of actors and actions have tricked our evaluation metric when the spatial overlapping of segments is not strictly examined. Furthermore, when combined with the spatial-oriented evaluation in Table <ref>, it is clear that the bilayer model outperforms the naïve Bayes model. In summary, the results we obtained on long videos have confirmed our findings from short videos that the explicitly joint modeling of actors and actions outperforms modeling them independently. We also note that the current methods have difficulties in dealing with untrimmed long videos as seen from relatively poor performance in general. We have released the newly collected long videos along with their annotations and evaluation metric in A2D v2 and look forward to seeing future methods toward solving the challenges of segmenting untrimmed videos.§ CONCLUSIONOur thorough assessment of all instantiations of the actor-action understanding problem on short videos and actor-action segmentation on long videos provides strong evidence that the joint modeling of actor and action improves performance over modeling each of them independently. We find that for both individual actor and action understanding and joint actor-action understanding, it is beneficial to jointly consider actor and action. A proper modeling of the interactions between actor and action results in dramatic improvement over the baseline models of the naïve Bayes and joint product space models, as we observe from the bilayer and trilayer models. Our paper set out with three goals: first, we sought to motivate and develop a new, more challenging, and more relevant actor-action understanding problem; second, we sought to assess whether joint modeling of actors and actions improves performance for this new problem using both short and long videos; third, we sought to explore a multi-scale modeling to bridge the gap between recognition and segmentation tasks. We achieved these goals through the following four contributions:* New actor-action understanding problem and fully labeled dataset of 3782 short videos and 31 long videos.* Thorough evaluation of actor-action recognition and segmentation problems using state-of-the-art features and models. The experiments unilaterally demonstrate a benefit for jointly modeling actors and actions.* A new trilayer model that combines independent actor and action variations and product-space interactions.* Improved the performance of pixel-level segmentation via label costs from video-level recognition responses.Our full dataset, computed features, codebase, and evaluation regimen are released at <http://www.cs.rochester.edu/ cxu22/a2d/> to support further inquiries into this new and important problem in video understanding.Limitations. There are two directions for our future work. First, our models lack the concept of modeling object as a whole entity; this results in fragmented segments in our actor-action segmentation. It would be beneficial to incorporate such concept in modeling actors and actions to achieve a holistic video understanding. Second, deep learning and deep representations are emerging topics in video understanding. It is our future work to explore the actor-action problem in these contexts.§ ACKNOWLEDGMENTS § ACKNOWLEDGMENTThis work has been supported in part by Google, Samsung, DARPA W32P4Q-15-C-0070 and ARO W911NF-15-1-0354.IEEEtran[ < g r a p h i c s > ]Chenliang Xuis an assistant professor of Computer Science at the University of Rochester. He received his Ph.D. degree at the University of Michigan, Ann Arbor in 2016, and M.S. degree from SUNY Buffalo in 2012, both in Computer Science. He received his B.S. degree in Information and Computing Science from Nanjing University of Aeronautics and Astronautics in 2010. His research interests include computer vision, robot perception, and machine learning. [ < g r a p h i c s > ]Caiming Xiong is a senior researcher at Salesforce MetaMind. Before that, he was Postdoctoral Researcher in the Department of Statistics, University of California, Los Angeles. He received a Ph.D. in the Department of Computer Science and Engineering, SUNY Buffalo in 2014, and got the B.S. and M.S. of Computer Science degree from Huazhong University of Science and Technology in the year 2005 and 2007 in China. His research interests include video understanding, action recognition, metric learning and active clustering, and human-robot interaction. [ < g r a p h i c s > ]Jason J. Corso is an associate professor of Electrical Engineering and Computer Science at the University of Michigan.He received his PhD and MSE degrees at The Johns Hopkins University in 2005 and 2002, respectively, and the BS Degree with honors from Loyola College In Maryland in 2000, all in Computer Science.He spent two years as a post-doctoral fellow at the University of California, Los Angeles. From 2007-14 he was a member of the Computer Science and Engineering faculty at SUNY Buffalo.He is the recipient of a Google Faculty Research Award 2015, the Army Research Office Young Investigator Award 2010, NSF CAREER award 2009, SUNY Buffalo Young Investigator Award 2011, a member of the 2009 DARPA Computer Science Study Group, and a recipient of the Link Foundation Fellowship in Advanced Simulation and Training 2003.Corso has authored more than ninety peer-reviewed papers on topics of his research interest including computer vision, robot perception, data science, and medical imaging.He is a member of the AAAI, IEEE and the ACM. | http://arxiv.org/abs/1704.08723v1 | {
"authors": [
"Chenliang Xu",
"Caiming Xiong",
"Jason J. Corso"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170427192050",
"title": "Action Understanding with Multiple Classes of Actors"
} |
#1 1Limit theorems in the context of multivariate long-range dependenceMarie-Christine Dükermyth3pt[1] December 30, 2023 =================================================================== [1]mythRuhr-Universität Bochum, Faculty of Mathematics, Bochum, Germany,<[email protected]> Limit theorems in the context of multivariate long-range dependenceMarie-Christine Dükermyth3pt[1] December 30, 2023 =================================================================== This article considers multivariate linear processes whose components are either short- or long-range dependent. The functional central limit theorems for the sample mean and the sample autocovariances for these processes are investigated, paying special attention to the mixed cases of short- and long-range dependent series. The resulting limit processes can involve multivariate Brownian motion marginals, operator fractional Brownian motions and matrix-valued versions of the so-called Rosenblatt process.Keywords: Long-range dependence; multivariate time series; linear processes; sample autocovariances; functional central limit theorem; operator self-similar processes.§ INTRODUCTIONIn this work, we are interested in the asymptotic behavior of the sample mean and the sample autocovariances for multivariate linear processes under several assumptions on their dependence structure, with the focus on long-range dependence.For long-range dependent time series, the autocovariance decays power-like as the time lag increases. Limit theorems for univariate long-range dependent time series were studied by a number of authors. See, for example, <cit.> and for an overview of long-range dependence, <cit.>. Limit theorems for multivariate processes under long-range dependence were studied in <cit.>. Vectors of univariate long-range dependent time series were considered in <cit.>.The present work develops limit theorems for two-sided multivariate linear processes, whose components are allowed to be either short- or long-range dependent. Under short-range dependence, the dependence parameters take values in a range different from long-range dependence, so that the autocovariances decay faster for the corresponding components.We investigate functional limit theorems for these processes for the sample mean, as well as for the sample autocovariances, i.e. we prove weak convergence in a multivariate product space of D[0,1], the space of càdlàg functions on [0,1] equipped with the uniform metric. Depending on the dependence structure, the limit involves Brownian motion marginals, operator fractional Brownian motions and matrix-valued versions of the so-called Rosenblatt process. Our setting is as follows. We consider an p-dimensional second-order stationary time series {X_n}_n∈ with X_n=(X_1,n,…,X_p,n)', where the prime indicates transposition. We suppose throughout the paper that X_n can be represented as a multivariate linear processX_n = ∑_j∈_jε_n-j,where {_j}_j ∈ is a sequence of matrices with _j=(ψ_kl,j)_k,l=1,…,p∈ ^p × p and{ε_j}_j∈ is a sequence of mean zero independent and identically distributed (i.i.d.) random vectors with covariance matrix (ε_0ε_0')=I_p.Write the entries of the matrices {_j}_j ∈ asψ_kl,j=C_kl(j)|j|^d_k-1,j ∈^*,for k,l ∈{1,…, p}, where ^*=∖{0}.We further assume that p_1 components of (<ref>) are multivariate long-range dependent and p_2:=p-p_1 components are multivariate short-range dependent in the following sense. The first p_1 and the last p_2 components of the linear process (<ref>) are called, respectively, (L) multivariate long-range dependent, if d_k∈ (0,1/2) for k ∈{1,…,p_1}, and C_kl(j) in (<ref>) satisfiesC_kl(j) ∼α^+_kl asj →∞ andC_kl(j) ∼α^-_kl asj → -∞,where the matrices A^+=(α^+_kl)_k=1,…,p_1;l=1,…,p,A^-=(α^-_kl)_k=1,…,p_1;l=1,…,p∈^p_1× pare assumed to have full rank; (S) multivariate short-range dependent, if d_k<0 fork ∈{p_1+1,…, p}, and C_kl(j) in (<ref>) satisfiessup_ j ∈^* | C_kl(j) | ≤βfor all k ∈{p_1+1,…, p}, l ∈{1,…,p} and some constant β >0 and the matrix∑_j ∈ (ψ_kl,j)_k=p_1+1,…,p;l=1,…,p∈^p_2× pis assumed to have full rank. The definition (L) allows for quite general long-range dependent, linear time series, for example, multivariate FARIMA series; see <cit.>. For univariate time series, the short-range dependence definition (S) was introduced in <cit.>. It allows for exponentially decaying coefficients ψ_kl,j. The multivariate setting includes, for example, vector ARMA models. Under the introduced setting, we are interested in the asymptotic behavior of the vector-valued sample mean processS_⌊ Nt ⌋ = ∑_n=1^⌊ Nt ⌋ X_n,t ∈ [0,1],and the asymptotic behavior of the sample autocovariance process((Γ_N,ℓ-Γ_ℓ)(t), ℓ=0,…,L),t ∈ [0,1],with(Γ_N,ℓ-Γ_ℓ)(t) = 1/N∑_n=1^⌊ Nt ⌋( X_n X_n+ℓ'-(X_0 X_ℓ') ), Γ_N,ℓ(t) = 1/N∑_n=1^⌊ Nt ⌋ X_n X'_n+ℓ.In order to prove convergence of the sample mean process under different assumptions on the dependence structure, we first consider a more general result, which is of independent interest. It states that the linear process in (<ref>) satisfies the central limit theorem when ∑_j∈_j_F^2 < ∞, where ·_F denotes the Frobenius norm.In the univariate case, this result was proven in <cit.>. Multivariate processes under less flexible assumptions on the dependence structure were studied by <cit.>, who considered one-sided multivariate long-range dependent linear processes and derived limit theorems for the sample mean and sample autocovariances in terms of the convergence of the finite-dimensional distributions (f.d.d.). The works <cit.> and <cit.> characterized a class of processes which converge to an operator fractional Brownian motion, and the latter also considered limit theorems for functionals of Gaussian vectors. In <cit.>, limits of a vector of normalized sums of functions of long-range dependent stationary Gaussian series were studied and <cit.> investigated the limit of normalized partial sums of a vector of multilinear polynomial-form processes.The two latter works also allowed for mixture cases, where vectors of both univariate short- and long-range dependent time series were studied. The works <cit.> are perhaps the closest to this study.However, our work is based on a linear process generated by a multivariate i.i.d. sequence and allows for multivariate short- and long-range dependence. Considering the sample autocovariance matrix of such a process leads to a matrix-valued process, whose entries depend on different combinations of short- and long-range dependent parameters. In contrast, <cit.> considered vectors of univariate processes.The dependence structure in <cit.> is determined by the Hermite rank of the respective function applied to a univariate long-range dependent process.The work <cit.> supposed that each component can be represented as a univariate multilinear polynomial form process obtained by applying an off-diagonal multi-linear polynomial-form filter to an i.i.d. sequence. They allowed the components to be either short- or long-range dependent.See Remarks <ref> and <ref> for more details.The rest of the paper is organized as follows. In Section <ref>, some properties of multivariate short- and long-range dependent time series are reviewed and the processes resulting as limits of (<ref>) and (<ref>) are given. In Section <ref>, we present the limit theorems concerning the sample mean process (<ref>). In Section <ref>, the functional limit theorems for the sample autocovariances (<ref>) are presented. In the last section, we provide the detailed proofs. § PRELIMINARIES In this section, we introduce some further notation, give more details about short- and long-range dependence and define the resulting limit processes. The autocovariances of a second-order stationary zero mean time series {X_n}_n ∈ at lag ℓ are denoted and defined as Γ_ℓ = (γ_kl(ℓ))_k,l=1,…,p = (X_0 X_ℓ') = ∑_j∈_j_j+ℓ'.The autocovariance Γ_ℓ takes values in the space of real-valued matrices ^p × p equipped with the Frobenius inner product⟨ A, B ⟩=∑_k,l=1^p a_klb_kl forA=(a_kl)_k,l=1,…,p, B= (b_kl)_k,l=1,…,p, which induces the Frobenius norm A _F^2=⟨ A, A ⟩=∑_k,l=1^p |a_kl|^2. The convergence of the finite-dimensional distributions and that in law are denoted by f.d.d.⟶ and ℒ⟶, respectively. We use the notation D[0,1]^p for the p-dimensional product space of D[0,1]. Furthermore, we write a^G=diag(a^g_1,…,a^g_p) for a > 0 and a diagonal matrix G=diag(g_1,…,g_p). Also, we setD=(d_1,…,d_p), D_p_1=(d_1,…,d_p_1) andD^c_p_2=(d_p_1+1,…,d_p) with 0 ≤ p_1≤ p, p_2 = p-p_1 and D_0=D^c_p+1=0 for the dependence parameters introduced in (<ref>). In proving our convergence results, we will use the following two propositions on autocovariances of linear processes satisfying the long- or short-range dependence definition (L) and (S) in Section 1. For this purpose, let {X^L_n}_n ∈ denote the p_1-dimensional process satisfying (L) and {X^S_n}_n ∈ the p_2-dimensional process satisfying (S), which combined together make the process {X_n}_n ∈. The respective autocovariances are denoted byΓ_p_1,ℓ=(X^L_0X^L'_ℓ)and Γ_p_2,ℓ=(X^S_0X^S'_ℓ). The autocovariances of the process {X^L_n}_n ∈ satisfyΓ_p_1,ℓ= ℓ^D_p_1-1/2I_p_1R(ℓ)ℓ^D_p_1-1/2I_p_1= (R_kl(ℓ)ℓ^d_k+d_l-1)_k,l=1,…, p_1,where R(ℓ)= (R_kl(ℓ))_k,l=1,…,p_1 is a function satisfyingR(ℓ) ∼ R=(R_kl)_k,l=1,…, p_1 as ℓ→∞withR_kl= B(d_k,d_l) ( c_1,klsin(π d_k)/sin(π (d_k+d_l))+ c_2,kl+c_3,klsin(π d_l)/sin(π (d_k+d_l))) ,where B denotes the beta function andc_1,kl =(A^-(A^-)')_kl,c_2,kl =(A^-(A^+)')_kl,c_3,kl =(A^+(A^+)')_kl.The proof is given in <cit.>. The autocovariances of the process {X^S_n}_n ∈ are absolutelysummable in the sense that ∑_ℓ∈Γ_p_2,ℓ_F < ∞. Note that Γ_p_2,-ℓ=Γ_p_2,ℓ'. As in the proof of <cit.>, one has|Γ_p_2,ℓ| ≤ℓ^D^c_p_2-1/2I_p_2 T(ℓ) ℓ^D^c_p_2-1/2I_p_2,ℓ>0,for some ^p_2× p_2-valued function T(ℓ), whose components are slowly varying functions. Indeed, note that|γ_kl(ℓ)| ≤ℓ^d_k+d_l-1∑_i=1^3 T_i,kl(ℓ), ℓ>0,where for example,T_1,kl(ℓ)= p β^2 ∑_j=ℓ+1^∞(j/ℓ)^d_k-1(j/ℓ-1)^d_l-11/ℓ.Then, there is a constant C such that∑_ℓ∈Γ_p_2,ℓ_F≤ 2C ∑_ℓ=0^∞( ∑_k,l=p_1+1^p|ℓ^d_k+d_l-1| ^2) ^1/2.The absolute summability follows, since 1-d_k-d_l>1.We now turn to the processes resulting as limits of the sample mean and autocovariance processes.In connection to the sample mean process, we follow <cit.> to introduce operator fractional Brownian motions (OFBMs). OFBMs are multivariate extensions of the univariate fractional Brownian motion and denoted here as ℬ^(p)_H(t)=(ℬ_1,H(t),…,ℬ_p,H(t))' ∈^p with t ∈ and some symmetric matrix H ∈^p × p. They are Gaussian, operator self-similar with exponent H and have stationary increments. Additionally, it shall be assumed that they are proper, that is, for each t ∈, the distribution of ℬ^(p)_H(t) is not contained in a proper subspace of ^p. The process {ℬ^(p)_H(t)}_t ∈ is operator self-similar(<cit.>) if {ℬ^(p)_H(ct)}_t ∈f.d.d.={c^Hℬ^(p)_H(t)}_t ∈ for every c>0. To introduce an integral representation for OFBMs, let W(dx)=(W_1(dx),…,W_p(dx)) be a multivariate, real-valued Gaussian random measure satisfyingW(dx)=0,W(dx)W'(dx)=I_p dx, W_k(dx)W_l(dy)=0,x ≠ y.Then, if the eigenvalues of the symmetric matrix H denoted by h_k satisfy 0<h_k<1 and h_k≠1/2 for k=1,…, p, in the time domain, the OFBM ℬ^(p)_H(t) admits the representationℬ^(p)_H(t)f.d.d.=ℬ^(p)_H,M^+,M^-(t) withℬ^(p)_H,M^+,M^-(t) = ∫_(((t-x)^H-1/2I_+-(-x)_+^H-1/2I) M^+ +((t-x)^H-1/2I_--(-x)_-^H-1/2I) M^-) W(dx),where x_+=max(0,x), x_-=max(-x,0), M^+,M^-∈^p × p; see <cit.>. According to <cit.>, the corresponding cross-covariance function is given byℬ^(p)_H(t) ℬ^(p)'_H(u) = |t|^HR|t|^H+|u|^HR'|u|^H-|t-u|^HR(t-u)|t-u|^H,if 0<h_k<1 and h_k + h_l≠ 1 for k,l ∈{1,…,p}, where R(t)= R, if t>0, R', if t<0and R=(R_kl)_k,l=1,…, p is defined asR_kl= B(h_k+1/2, h_l+1/2) ( c_1,klcos(π h_k )/sin(π (h_k+h_l))+ c_2,kl+c_3,klcos(π h_l)/sin(π (h_k+h_l)))withc_1,kl=(M^-(M^-)')_kl, c_2,kl=(M^-(M^+)')_kl, c_3,kl=(M^+(M^+)')_kl.In connection to the sample autocovariance process, the limit process will possibly be non-Gaussian. We will represent it by means of double Wiener-Itô integrals, as a matrix-valued generalization of the univariate Rosenblatt process. For the sake of simplicity, we define it in a vectorized form using theoperator. Theoperator transforms a matrix into a vector by stacking the columns of the matrix one underneath the other. Let L^2(^2, ^p^2 × p^2) denote the space of all functions f:^2 →^p^2 × p^2 equipped with the norm ‖ f ‖^2= ∫_^2 f(x_1,x_2) ^2_F dx_1dx_2<∞. Then, for f ∈ L^2(^2, ^p^2 × p^2), we define a double Wiener-Itô integral with respect to a multivariate real-valued Gaussian random measure W(dx) asI_2(f)= ∫_^2' f(x_1,x_2) (W(dx_1)W'(dx_2) ),where ∫_^2' means that integration excludes the diagonals. See, for example <cit.>, for more information about multiple Wiener-Itô integrals.Then, the ^p^2-valued process {Z(t)}_t ∈ is defined asZ(t)=I_2(f_H,t),with f_H,t: ^2 →^p^2 × p^2 given by f_H,t(x_1,x_2):=f_H,t,M^+,M^-(x_1,x_2) withf_H,t,M^+,M^-(x_1,x_2) = ∑_s_1,s_2∈{+,-}∫_0^t ((v-x_2)^H-I_p_s_2⊗ (v-x_1)^H-I_p_s_1) (M^s_2⊗M^s_1)dv,where ⊗ denotes the Kronecker product and M^+, M^-∈^p × p. The eigenvalues of the symmetric matrix H ∈^p × p are assumed to satisfy h_k + h_l∈ (0,1/2) for k,l=1,…, p. Like the OFBM, the process {Z(t)}_t ∈ is supposed to be proper. It is also operator self-similar and has stationary increments; see Lemma <ref>. More precisely, the process {Z(t)}_t ∈ is operator self-similar with scaling family {Δ_c:^p^2→^p^2 | c>0}, where Δ_c = c^H⊗ c^H that means Z(ct) f.d.d.=Δ_c Z(t). § CONVERGENCE OF THE SAMPLE MEAN PROCESS In this section, we state the convergence results for the vector-valued sample mean process. The following theorem gives the asymptotic normality for a large class of multivariate linear processes and will serve as a helpful tool to investigate the functional limit theorems under different assumptions on the dependence structure. Let {X_n}_n ∈ be a stationary linear process (<ref>) with ∑_j∈_j_F^2 < ∞ and set Σ_N^2:=(S_NS_N'). Suppose there is a nonsingular matrix Σ_N^2 such that Σ_N^2∼Σ_N^2 componentwise, as N →∞, andlim_N→∞(Σ_N^-1 S_N) =I_p.If each diagonal entry of the matrix Σ_N^2=(σ^2_kl(N))_k,l=1,…, p goes to infinity as N →∞, thenΣ_N^-1∑_n=1^N X_nℒ⟶𝒩(0,I_p).The conditions in Theorem <ref> are satisfied under multivariate long- as well as under multivariate short-range dependence. The assumptions on the matrices A^+, A^- in (L) and ∑_j ∈ (ψ_kl,j)_k=p_1+1,…,p;l=1,…,p in (S) to have full rank ensure that there is a nonsingular matrix Σ_N^2 with Σ_N^2∼Σ_N^2 and lim_N→∞(Σ_N^-1 S_N) =I_p. The matrix Σ_N^2 satisfies σ^2_kk(N) →∞ as N →∞. See Lemma <ref> for t=u=1.The following result is the functional central limit theorem for the sample mean process, allowing the multivariate linear process to admit either short- or long-range dependence. The limit process in the result is Gaussian and given by𝒢(t)= [ C_H^-1ℬ^(p_1)_H(t); 𝒲^(p_2)(t) ]with C_H=H-1/2I_p_1 and H=D_p_1+1/2I_p_1, where{ℬ^(p_1)_H(t)}_ t ∈ [0,1] is an ^p_1-valued OFBM restricted to the unit interval and {𝒲^(p_2)(t)}_t ∈ [0,1] is an^p_2-valued multivariate Brownian motionwith 𝒲^(p_2)(t)=(𝒲_p_1+1(t),…, 𝒲_p(t))'.The cross-covariances of ℬ^(p_1)_H(t) are given in (<ref>). The corresponding matrix R is defined in (<ref>) and depends on the parameters c_i,kl, i=1,2,3, which are defined in terms of the matrices A^+,A^- arising in (L). The cross-covariances of {𝒲^(p_2)(t)}_t ∈ [0,1] can be written as𝒲^(p_2)(t)𝒲^(p_2)'(u) = min(t,u) ∑_ℓ∈Γ_p_2,ℓ,t,u ∈ [0,1].The cross-covariance structure between ℬ^(p_1)_k,H(t) and 𝒲^(p_2)_l(t) for k = 1,…,p_1 and l = p_1+1,…,p is given byℬ^(p_1)_k,H(t)𝒲^(p_2)_l(u) = ∑_l=1^ph^-1_k((t^h_kα^-_kl + u^h_kα^+_kl- |t-u|^h_kα_kl(t-u)) ∑_j ∈ψ_kl,j),for d_k+d_l≥ 0, where α_kl(t)=α_kl^+1_{t>0}+ α_kl^-1_{t<0} andh_k=d_k+1. Otherwise, when d_k+d_l < 0, the components are uncorrelated. Whenever we refer to the process {𝒢(t)}_t ∈ [0,1] in (<ref>), we mean the process with the previously described cross-covariance structure. Let {X_n}_n ∈ be a stationary linear process(<ref>) whose components satisfy (L) and (S), with ε_0^2+δ<∞ for some δ>0. Then,A_N^-1(H)S_⌊ Nt ⌋ℒ⟶𝒢(t),t ∈ [0,1],in D[0,1]^p, where {𝒢(t)}_t ∈ [0,1] is a Gaussian process given in (<ref>). The normalizationA_N(H)=diag(N^H,N^1/2I_p_2) is such that there is a non-singular matrix C(H) ∈^p × p withlim_N →∞(A_N^-1(H) ∑_n=1^N X_n )=C(H).The dependence parameters d_l for l ∈{p_1+1,…, p} determine the short-range dependent components,while the dependence parameters d_k with k ∈{1,…, p_1} determine the long-range dependent components. Choosing d_l with l ∈{p_1+1,…, p} small enough to get d_k+d_l<0 for d_k with k ∈{1,…, p_1}, yields an asymptotic independence between the short- and long-range dependent components.As noted in the previous remark, the asymptotic independence in Theorem <ref> depends on the interplay between the dependence parameters of the long- and short-range dependent components. In contrast, <cit.> proved that the short- and long-range dependent components of a vector of functions of a univariate long-range dependent process are always asymptotically independent; see Theorem 5 in <cit.>.By expressing the sample mean process in (<ref>) componentwise, each component can be viewed as the sample mean process of the sum of different linear processes. An implication of Theorem 3.5 in <cit.> is that a vector of univariate linear processes which are either short- and long-range dependent converges to a vector whose components are either a univariate Brownian motion or fractional Brownian motion, which leads to non-proper limiting process. In contrast, our assumptions (L) and (S) on the linear process (<ref>) ensure a proper limiting process. § CONVERGENCE OF THE SAMPLE AUTOCOVARIANCE PROCESS We first introduce some notation. For simplicity, we write Y_N,ℓ(t)=(Γ_N,ℓ-Γ_ℓ)(t) and denote the kl-th component of Y_N,ℓ(t) byY_kl,N,ℓ(t)= 1/N∑_n=1^⌊ Nt ⌋( X_k,n X_l,n+ℓ-(X_k,0 X_l,ℓ) ).When k=l, it is well-known (see <cit.>) that the normalized limit ofY_kk,N,ℓ(t) is the Rosenblatt process when 2d_k∈ (1/2,1) and the usual Brownian motion when 2d_k∈ (0,1/2). This suggests to consider the index setsI_L ={(k,l) ∈{1,…,p_1}^2 | d_k+d_l∈ (1/2,1) },I_S ={(k,l) ∈{1,…,p}^2 | d_k+d_l∈ (-∞,1/2) },where the subscripts L and S refer to the long- and short-range dependence of the expected limits, that is the Rosenblatt process and Brownian motion, respectively. For a matrix M=(M_kl)_k,l=1,…, p and an index set I⊂{1,…, p}^2, we set_I(M)=E_I(M),where E_I∈{0,1}^|I| × p^2 denotes an elimination matrix, which transforms (M) into a vector including only the matrix elements with indices in I. Then, Y_N,ℓ(t) is partitioned intoY_N,ℓ^L(t)=_I_L(Y_N,ℓ(t)),Y_N,ℓ^S(t)=_I_S(Y_N,ℓ(t)) and Γ_N,ℓ(t) given in (<ref>) intoΓ^L_N,ℓ(t)= _I_L(Γ_N,ℓ(t)), Γ^S_N,ℓ(t)= _I_S(Γ_N,ℓ(t)). The limit process {Z^L(t)}_t ∈ [0,1] of Γ^L_N,ℓ(t) in the result given below is defined asZ^L(t)=E_I_L Z(t),where Z(t) is given in (<ref>). The corresponding function is defined in (<ref>) with H=D, M^+=((A^+)' 0_p × p_2)' and M^-=((A^-)' 0_p × p_2)', where A^+,A^- are given in (L) and 0_p × p_2 denotes an p × p_2 matrix with all entries equal to zero. The limit process {G^S_ℓ(t)}_t ∈ [0,1] of Γ^S_N,ℓ(t) is a multivariate Brownian motion with cross-covariances(G^S_ℓ_1(t),G^S_ℓ_2(u))= min(t,u)E_I_S(∑_r ∈(Γ_r+ℓ_2⊗Γ_r-ℓ_1+ K_p ( Γ_r⊗Γ_r+ℓ_2-ℓ_1 ))+∑_r∈∑_i∈(_i+ℓ_1⊗_i) Σ (_i+r⊗_i+r+ℓ_2 )' ) E_I_S',where Σ:=σ^* -(I_p) ((I_p) )' -I_p^2-K_pwith σ^* := ((ε_0ε_0') ((ε_0ε_0'))').Furthermore, K_p denotes the commutation matrix, which transforms (M) into (M') for a matrix M=(M_kl)_k,l=1,…,p; see <cit.> for more details on these kind of operations. In order to characterize the joint distribution of the processes {Z^L(t)}_t ∈ [0,1] in (<ref>) and {G^S_ℓ(t)}_t ∈ [0,1] in (<ref>) we give the cross-covariance structure between the two Gaussian processes W(t) and G^S_ℓ(t), where W(t) induces the random measure in the integral representation (<ref>). The cross-covariance structure is given by(G^S_ℓ(t)W'(u)) = min{t,u}∑_ r ∈∑_i ∈ E_I_S(_r+ℓ-i⊗_r-i)Σ,where Σ=((ε_0ε_0 ')ε_0' ). See also Remark 4.3. for more information about the dependence structure. The following theorem gives the joint convergence of Γ^L_N,ℓ(t) and Γ^S_N,ℓ(t). Let {X_n}_n ∈ be a stationary linear process(<ref>) whose components satisfy (L) and (S), with Eε_0^5 < ∞.Then,( B_N^-1(D) [ Y_N,ℓ^L(t); Y_N,ℓ^S(t) ] , ℓ=0,…,L ) ℒ⟶( [ Z^L(t); G^S_ℓ(t) ] ,ℓ=0,…,L),t ∈ [0,1],in D[0,1]^p^2, where {Z^L(t)}_t ∈ [0,1] is defined in (<ref>),{G^S_ℓ(t)}_t ∈ [0,1] in (<ref>). Furthermore, Z^L(t) and G^S_ℓ(t) are uncorrelated but not independent. The normalization B_N(D)=(Δ_N,N^1/2I_|I_S|) with Δ_N = E_I_L( N^D-1/2I_p⊗ N^D-1/2I_p )E_I_L' is such that there are non-singular matricesC_1(D_p), C_2 withlim_N →∞(Δ_N^-1Γ^L_N,ℓ(1), Δ_N^-1Γ^L_N,ℓ(1))=C_1(D_p)and lim_N →∞ N( Γ^S_N,ℓ(1), Γ^S_N,ℓ(1)) = C_2. The sum of two dependence parameters d_k+d_l determines if the corresponding component of the sample autcovariance process is long- or short-range dependent. The case when a component behaves long-range dependent is characterized by d_k+d_l∈ (1/2,1) and can only occur when the sample autocovariances between two long-range dependent components are considered.As stated in Theorem <ref>, the processes {Z^L(t)}_t ∈ [0,1] and {G^S_ℓ(t)}_t ∈ [0,1] are uncorrelated but not independent.To understand why these processes are not independent, note that the sample autocovariances in (<ref>) can be separated into diagonal and off-diagonal parts; see (<ref>). While the diagonal terms are asymptotically negligible for the the long-range dependent components (see Lemma <ref>), the diagonals are crucial for the asymptotic behavior of the short-range dependent components. According to (<ref>), the diagonals in the short-range dependent components influence the resulting dependence structure and lead to dependent limiting processes {Z^L(t)}_t ∈ [0,1] and {G^S_ℓ(t)}_t ∈ [0,1].In contrast to Remark <ref>, <cit.> studied vectors of univariate multilinear polynomial form processes whose filters depend on either short- or long-range dependent components and exclude the diagonals. The resulting limit theorem gives asymptotic independence between the short- and long-range dependent components when the linear forms are at least of order two; see Theorem 3.5 in <cit.>.Our setting can also be compared to <cit.> when the Hermite rank in each component is supposed to be two, which is the same as considering a vector of univariate sample autocovariances. In this case, the short- and long-range dependent components are asymptotically independent; see Theorem 5 in <cit.>.Furthermore, our setting ensures a proper limiting process, while the limits in <cit.> are not necessarily proper. § PROOFS§.§ Proof of Theorem <ref>We first state a lemma from <cit.>, which gives sufficient conditions for two linear processes with values in an arbitrary Hilbert space to have the same convergence behavior.Letand 𝔼 be two Hilbert spaces and {ε_j}_ j ∈ a sequence of i.i.d. random variables with values in 𝔼. Define {X_n}_ n ∈ with X_n=∑_j∈ D_njε_j and D_nj∈ L(,𝔼), the space of bounded linear operators fromto 𝔼. Similarly, define {Y_n}_ n ∈ with Y_n=∑_j∈ D_njε_j, where D_nj is the same operator as in X_n and ε_j is a sequence of Gaussian random elements with values in 𝔼, zero mean and the same covariance operator as ε_j. The notation ·_op stands for the operator norm. Iflim_n →∞sup_j ∈‖ D_nj‖_op=0 and lim sup_n →∞∑_j ∈‖ D_nj‖_op^2<∞,thenlim_n →∞ϱ_3 (X_n,Y_n)=0,where the metric ϱ_k is defined by ϱ_k (X,Y)=sup_f ∈ F_k|f(X) -f(Y) |,for the set F_k of all k times Frèchet differentiable functions f:→ such thatsup_x ∈| f^(i)(x) |≤ 1 for i∈{0,…,k}.The proof of Lemma <ref> is given in <cit.>. The processes X_n and Y_n have the same convergence behavior if lim_n →∞ϱ_3 (X_n,Y_n)=0, since the metric induces the weak topology on the set of probability measures on ; see <cit.>. We next rewrite the normalized sample mean Σ^-1_N∑_n=1^N X_n as a linear process and prove for it the conditions (<ref>). Thus, let Σ_N^-1=: (Σ_kl,N)_k,l=1,…,p, which exists since (a'S_N) ≠ 0 fora ∈\{0} by assumption. For λ= (λ_1,…,λ_p) ∈^p, writeλ'Σ^-1_N∑_n=1^N X_n = ∑_j∈∑_n=1^N∑_k=1^p∑_i=1^pλ_iΣ_ik,N∑_l=1^pψ_kl,n-jε_l,j=: ∑_l=1^p∑_j∈ B_l,Njε_l,j = :∑_j∈( B_1,Nj,…,B_p,Nj)[ ε_1,j; ⋮; ε_p,j ] =:∑_j ∈ B_Nj' ε_j,which is the form needed to apply Lemma <ref>. Since B_Nj∈^p, the operator norm is ‖ B_Nj‖_op=max_1 ≤ l ≤ p| B_l,Nj|. To show that B_Nj satisfies the conditions (<ref>), we need the following auxiliary lemma concerning the variances of one entry of the sample mean,ω^2_kl(N):= (∑_n=1^N∑_j∈ψ_kl, n-jε_l,j)^2= ∑_j∈ (∑_n=1^Nψ_kl, n-j)^2. The sequence of matrix entries(Σ_ik,Nω^1/2_kl(N))_N ≥ 1converges to zero for each k,l,i ∈{1,…,p}. Define the matrix Ω^2_N=(Ω_kl,N)_k,l=1,…,p byΩ_kl,N=1, ifk=l, σ_kl^2(N)/σ_kk(N)σ_ll(N),ifk ≠ l,so that Σ_N^2=diag(σ_11(N),…,σ_pp(N)) Ω^2_Ndiag(σ_11(N),…,σ_pp(N)). Then, there is a matrixC ∈^p × p, whose diagonal entries are equal to one and the off-diagonal elements are constants c_ij such that lim _N →∞Ω_kl,N=C componentwise. This implieslim_N →∞Σ_N^-2= lim_N →∞(Σ_N^2)^-1 = lim_N →∞ (diag(σ_11(N),…,σ_pp(N)) Ω^2_Ndiag(σ_11(N),…,σ_pp(N)))^-1 = lim_N →∞diag( 1/σ_11(N),…,1/σ_pp(N)) C^-1diag( 1/σ_11(N),…,1/σ_pp(N)),since matrix inversion is a continuous transformation. This leads tolim_N →∞ (Σ_ik,Nω^1/2_kl(N))_i,l=1,…,p = (lim_N →∞(ω_1l(N),…,ω_pl(N))Σ_N^-2)^1/2 = (lim_N →∞diag(ω_1l(N)/σ_11(N),…,ω_pl(N)/σ_pp(N)) C^-1diag( 1/σ_11(N),…,1/σ_pp(N)) ) ^1/2 = lim_N →∞( ω_kl(N)c_ik/σ_kk(N)σ_ii(N))_i,l=1,…, pwith C^-1=(c_ij)_i,j=1,…, p and finallylim_N →∞ω_kl(N) /σ_kk(N)σ_ii(N) = lim_N →∞(∑_j∈ (∑_n=1^Nψ_kl,n-j)^2)^1/2/σ_kk(N)σ_ii(N) = 0with σ^2_kk(N) ∼∑_m=1^p∑_j∈ ( ∑_n=1^Nψ_km,n-j)^2. The sequence of matrices (B_Nj')_N ≥ 1,j ∈ in (<ref>) satisfies the conditions (<ref>). To prove the conditions (<ref>), we consider| B_l,Nj| for each l ∈{1,…, p} instead ofmax_1 ≤ l ≤ p| B_l,Nj|. Then,sup_j ∈| B_l,Nj| = sup_j ∈|∑_n=1^N∑_i=1^p∑_k=1^pλ_iΣ_ik,Nψ_kl,n-j|≤sup_j ∈∑_i=1^p∑_k=1^p |λ_i||Σ_ik,Nω^1/2_kl(N) | | ∑_n=1^Nψ_kl,n-j |/ω^1/2_kl(N). By using the inequality | ∑_n=1^Nψ_kl,n-j |^2/ω^2_kl(N)≤4/ω_kl(N)( ∑_j ∈ψ^2_kl,j/ω_kl(N) + ( ∑_j ∈ψ_kl,j^2)^1/2)from <cit.>, we further get thatsup_j ∈| B_l,Nj|≤ 2 ∑_i=1^p∑_k=1^p |λ_i||Σ_ik,Nω^1/2_kl(N) | ( ∑_j ∈ψ_kl,j^2/ω_kl(N) + ( ∑_j ∈ψ_kl,j^2)^1/2) ^1/2→ 0,as N →∞ since Σ_ik,Nω^1/2_kl(N) converges to zero for eachi,k,l ∈{1,…,p} by Lemma <ref>. This proves the first condition in (<ref>). The second condition holds, sincelim sup_N →∞∑_j ∈ |B_l,Nj |^2 = lim sup_N →∞∑_j ∈ | ∑_n=1^N∑_i=1^pλ_i∑_k=1^pΣ_ik,Nψ_kl,n-j |^2 ≤lim sup_N →∞∑_j ∈∑_l=1^p|∑_n=1^N∑_i=1^pλ_i∑_k=1^pΣ_ik,Nψ_kl,n-j |^2 = ∑_i=1^pλ_i^2 < ∞,where we used the fact that the variances of the normalized sample mean satisfyλ' λ = lim_N →∞(λ' Σ_N^-1∑_n=1^N X_n)^2 = lim_N →∞∑_j ∈∑_l=1^p|∑_n=1^N∑_i=1^pλ_i∑_k=1^pΣ_ik,Nψ_kl,n-j |^2. By Lemma <ref>, the variables λ' Σ_N^-1∑_n=1^N X_n behave like Gaussian. The variances are given by (<ref>), so λ' Σ_N^-1∑_n=1^N X_n converges in distribution to λ'Z where Z follows the 𝒩(0,I_p) distribution. §.§ Proof of Theorem <ref> In order to prove Theorem <ref>, we first present an auxiliary result regarding the limit processes covariance structure (Section <ref>). We then investigate the convergence of the finite-dimensional distributions and tightness in D[0,1]^p (Sections <ref> and <ref>), which establish Theorem <ref>.§.§.§ Auxiliary result We examine the asymptotic covariance structure in the following auxiliary result. Under the assumptions in Theorem <ref>,lim_N →∞(A^-1 _N(H)S_⌊ Nt ⌋, A^-1_N(H)S_⌊ Nu ⌋) = (𝒢(t),𝒢(u)),where {𝒢(t)}_t ∈ [0,1] is defined in(<ref>). Furthermore, A_N(H)=diag(N^H,N^1/2I_p_2).The proof is divided into three parts: (i) we examine the covariances between the long-range dependent components, (ii) the covariances between the short-range dependent components and (iii), we consider the mixture terms. For each part, note that interchanging the order of summation and assuming t<u leads to(∑_n=1^⌊ Nt ⌋ X_k,n∑_n=1^⌊ Nu ⌋ X_l,n) = ⌊ Nt ⌋γ_kl(0) + ∑_n=1^⌊ Nt ⌋( ⌊ Nt ⌋ - n )(γ_kl(n)+ γ_lk(n)) + ∑_n=1^m_N-1nγ_kl(n) + m_N∑_n=m_N^⌊ Nu ⌋ -m_Nγ_kl(n)+∑_n=⌊ Nu ⌋-m_N+1^⌊ Nu ⌋-1( ⌊ Nu ⌋ - n )γ_kl(n), where m_N=min(⌊ Nt ⌋,⌊ Nu ⌋-⌊ Nt ⌋). Part (i): By Proposition <ref>, the underlying process {X^L_n}_n ∈ satisfies (<ref>) with (<ref>). The proof follows by applying(<ref>) and similar arguments as in the univariate case (see for example the proof of Proposition 2.8.8 in <cit.>) to each component so thatlim_N →∞(N^-H∑_n=1^⌊ Nt ⌋ X^L_n (N^-H∑_n=1^⌊ Nu ⌋ X^L'_n) ) = lim_N →∞( N^-(1+d_k+d_l)( ∑_n=1^⌊ Nt ⌋ X_k,n∑_n=1^⌊ Nu ⌋ X_l,n) ) _k,l=1,…,p_1 = ( 1/(d_k+d_l)(1+d_k+d_l)=( (R_klt^1+d_k+d_l+R_lku^1+d_k+d_l- R_kl(t-u)|t-u|^1+d_k+d_l) ) _k,l=1,…,p_1 = C_H^-1(t^HRt^H+u^HR'u^H-|t-u|^HR(t-u)|t-u|^H) C_H^-1,where C_H=H-1/2I_p_1 with H=D_p_1+1/2I_p_1, R_kl is defined by (<ref>) andR_kl(t)=R_kl, if t>0,R_lk, if t<0.The matrix R is given in (<ref>) and depends onthe parameters c_i,kl, i=1,2,3, which are defined in terms ofthe matrices A^+,A^- arising in (L). Using the basic properties of the beta and gamma function gives1/(d_k+d_l)(1+d_k+d_l) R_kl= 1/(h_k-1/2)(h_l-1/2)R_kl,since h_k=d_k+1/2. Part (ii): By Proposition <ref>, the autocovariances of the process {X^S_n}_n ∈ are absolutely summable. Then, the relation (<ref>) and standard arguments under univariate short-range dependence (see e.g. <cit.>) yieldlim_N →∞(N^-1/2∑_n=1^⌊ Nu ⌋ X^S_n (N^-1/2∑_n=1^⌊ Nt ⌋ X^S_n)' ) = lim_N →∞N^-1( ⌊ Nt ⌋γ_kl(0) + ∑_n=1^⌊ Nt ⌋( ⌊ Nt ⌋ - n ) (γ_kl(n)+ γ_lk(n)) )_k=p_1+1,…,p; l=1,…,p_1 = t ∑_n ∈Γ_p_2,nfor t<u, since Γ_p_2,-n=Γ_p_2,n'. Part (iii): For the covariances between the sample means of {X^L_n}_n ∈ and {X^S_n}_n ∈, we distinguish two cases: d_k+d_l<0 andd_k+d_l≥ 0, where d_k are associated with the components satisfying (L) and d_l with the components satisfying (S). When d_k+d_l<0, the autocovariances are absolutelysummable following the proof of Proposition <ref>, so thatN^-(1+d_k)∑_n=1^⌊ Nt ⌋( ⌊ Nt ⌋ - n ) γ_kl(n)= N^-(1+d_k)∑_n=1^⌊ Nt ⌋( ⌊ Nt ⌋ - n ) ∑_j ∈∑_m=1^pψ_km,jψ_lm,j+n∼ t ∑_m=1^p N^-d_k∑_n=1^∞∑_j ∈ψ_km,jψ_lm,j+n→ 0.When d_k+d_l≥ 0, we consider only the summand ∑_n=1^⌊ Nt ⌋( ⌊ Nt ⌋ - n ) γ_kl(n) in (<ref>) in detail since the others can be dealt with analogously. Write the kl-th component of the autocovariance function asγ_kl(n)=:∑_m=1^p(∑_j = -∞^-n-1ψ_km,jψ_lm,j+n + ∑_j = -n^0ψ_km,jψ_lm,j+n + ∑_j =0^∞ψ_km,jψ_lm,j+n) =: γ_1,kl(n)+γ_2,kl(n)+γ_3,kl(n).Recall that k ∈{1,…, p_1} for the long-range dependent components and l ∈{p_1+1,…, p} for the short-range dependent components. For γ_1,kl(n), we haveN^-(1+d_k)∑_n=1^⌊ Nt ⌋( ⌊ Nt ⌋ - n ) γ_1,kl(n) = N^-(1+d_k)∑_n=1^⌊ Nt ⌋( ⌊ Nt ⌋ - n ) ∑_j = -∞^-n-1∑_m=1^pψ_km,jψ_lm,j+n = ∑_m=1^p N^-(1+d_k)∑_n=1^⌊ Nt ⌋( ⌊ Nt ⌋ - n ) ∑_j = -∞^-1ψ_km,j-nψ_lm,j∼∑_m=1^pα_km^-1/d_k(d_k+1)t^d_k+1∑_j = -∞^-1ψ_lm,j.For γ_2,kl(n), we getγ_2,kl(n)= ∑_j = -n^0∑_m=1^pψ_km,jψ_lm,j+n = ∑_m=1^p∑_j = 0^nψ_km,j-nψ_lm,j= ∑_m=1^p∑_j = 0^n(ψ_kl,j-n -ψ_km,-n) ψ_lm,j+∑_m=1^p∑_j = 0^nψ_kl,-nψ_lm,j.The last term in this expression determines the limit as∑_m=1^p N^-(1+d_k)∑_n=1^⌊ Nt ⌋( ⌊ Nt ⌋ - n ) ψ_km,-n∑_j = 0^nψ_lm,j∼∑_m=1^pα^-_km1/d_k(d_k+1) t^d_k+1∑_j = 0^∞ψ_lm,j,whereas the other term is asymptotically negligible since|N^-(1+d_k)∑_n=1^⌊ Nt ⌋( ⌊ Nt ⌋ - n ) ∑_j = 0^n (ψ_km,j-n -ψ_lm,-n) ψ_lm,j| ∼ |N^-(1+d_k)∑_n=1^⌊ Nt ⌋( ⌊ Nt ⌋ - n ) ∑_j = 0^nα_km^-((n-j)^d_k-1 -n^d_k-1) j^d_l-1C_lm(j)| ≤ N^-(1+d_k)∑_n=1^⌊ Nt ⌋( ⌊ Nt ⌋ - n ) ∑_j = 0^n |α_km^-β |((n-j)^d_k-1 -n^d_k-1) j^d_l-1∼ N^-(1+d_k)∑_n=1^⌊ Nt ⌋( ⌊ Nt ⌋ - n ) n^d_k+d_l-1 |α_km^-β | ∫_0^1 ((1-x)^d_k-1 -1) x^d_l-1 dx → 0,where the integral is finite since d_k+d_l≥ 0 (see <cit.>). For γ_3,kl(n) note thatγ_3,kl(n) = ∑_m=1^p∑_j = 0^∞ψ_km,jψ_lm,j+n = ∑_m=1^p∑_j = 0^∞ C_km(j) j^d_k-1 C_lm(j+n) (j+n)^d_l-1and|∑_m=1^p N^-(1+d_k)∑_n=1^⌊ Nt ⌋ (⌊ Nt ⌋-n) ∑_j = 0^∞ C_km(j) j^d_k-1 C_lm(j+n) (j+n)^d_l-1| ∼ |∑_m=1^p N^-(1+d_k)∑_n=1^⌊ Nt ⌋ (⌊ Nt ⌋-n) ∑_j = 0^∞α_km^+ j^d_k-1 C_lm(j+n) (j+n)^d_l-1| ≤∑_m=1^p N^-(1+d_k)∑_n=1^⌊ Nt ⌋ (⌊ Nt ⌋-n) |α_km^+β | ∑_j = 0^∞ j^d_k-1(j+n)^d_l-1∼∑_m=1^p N^-(1+d_k)∑_n=1^⌊ Nt ⌋ (⌊ Nt ⌋-n) |α_km^+β | n^d_k+d_l-1∫_0^∞ x^d_k-1(x+1)^d_l-1 dx → 0 ,where the integral is finite since d_k+d_l<1. Combining the results for γ_1,kl(n),γ_2,kl(n) and γ_3,kl(n) yieldsN^-(1+d_k)∑_n=1^⌊ Nt ⌋( ⌊ Nt ⌋ - n ) γ_kl(n)∼∑_m=1^p1/d_k(d_k+1) t^d_k+1α^-_km∑_j ∈ψ_lm,j.Dealing similarly with the other summands in (<ref>) giveslim_N →∞(N^-H∑_n=1^⌊ Nt ⌋ X^L_n N^-1/2∑_n=1^⌊ Nu ⌋ X^S'_n) = C_H^-1H^-1 ( t^H A^- + u^H A^+- |t-u|^H A(t-u)) ( ∑_j ∈ψ_lm,j)_m = 1,…,p l = p_2,…,p ,where C_H=H-1/2I_p_1 with H=D_p_1+1/2I_p_1, H=D_p_1+I_p_1 and A(t)=A^+, ift>0,A^- , ift<0. We conclude the section with a comment on the properness of the process {𝒢(t)}_t ∈ [0,1], since it is a consequence of the previous Lemma <ref>. The matrices A^+, A^- and Λ:=∑_j ∈ (ψ_kl,j)_k=p_1+1,…,p;l=1,…,p are assumed to have full rank by condition (L) and (S), respectively. For this reason, A^+(A^+)', A^-(A^-)' andΛΛ' are positive definite. This and the positive-semi definiteness of 𝒢(t)𝒢'(t) imply, that𝒢(t) 𝒢'(t) is positive definite. So, one can infer that {𝒢(t)}_t ∈ [0,1] is proper. §.§.§ Convergence of the finite-dimensional distributions We prove the convergence of the finite-dimensional distributions by adapting the proof of Theorem <ref>. It is enough to verify thatλ'A^-1_N(H) S_⌊ Nt ⌋f.d.d.⟶λ' 𝒢(t),where 𝒢(t) is defined in (<ref>) and λ∈^p. The left-hand side of (<ref>) can be written asλ' A^-1_N(H) S_⌊ Nt ⌋ = ∑_j∈∑_k=1^p∑_n=1^⌊ Nt ⌋λ_k a^-1_k(N) ∑_l=1^pψ_kl,n-jε_l,j =: ∑_l=1^p∑_j∈ B_l,Nj(t) ε_l,j,wherea_k(N)=N^1/2+d_k, if k ∈{1,…,p_1} , N^1/2, if k ∈{p_1+1,…,p }.For the convergence of the finite-dimensional distributions, consider( ∑_l=1^p∑_j ∈ B_l,Nj(t_1) ε_l,j,…, ∑_l=1^p∑_j ∈ B_l,Nj(t_z) ε_l,j) = ∑_j ∈[ B_1,Nj(t_1) … B_p,Nj(t_1); ⋮ ⋱ ⋮; B_1,Nj(t_z) … B_p,Nj(t_z) ][ ε_1,j; ⋮; ε_p,j ] =: ∑_j ∈ B_Njε_j,where t_1,…, t_z∈ [0,1] and z ∈. This representation allows us to proceed as in the proof of Theorem <ref>. The sequence of matrices (B_Nj)_N ≥ 1,j ∈ satisfies the conditions (<ref>). The operator norm of the matrix B_Nj is given byB_Nj_op= max_1 ≤ l ≤ p∑_i=1^z | B_l,Nj(t_i) |.It is enough to prove the statement for | B_l,Nj(t) | for all l ∈{1,…,p} and t ∈ [0,1]. By using the inequality (<ref>),| ∑_n=1^⌊ Nt ⌋ψ_kl,n-j |^2/ω^2_kl(⌊ Nt ⌋)≤4/ω_kl(⌊ Nt ⌋)( ∑_j ∈ψ_kl,j^2/ω_kl(⌊ Nt ⌋) + (∑_j ∈ψ_kl,j^2)^2).Then,sup_j ∈ |B_l,Nj(t)| = sup_j ∈ | ∑_n=1^⌊ Nt ⌋∑_k=1^pλ_k a^-1_k(N) ψ_kl,n-j | ≤ 4∑_k=1^p |λ_k| |a^-1_k(N) ω^1/2_kl(N) | ( ω_kl(⌊ Nt ⌋)/ω_kl(N)) ^1/2( ∑_j ∈ψ_kl,j^2/ω_kl(⌊ Nt ⌋) + ( ∑_j ∈ψ_kl,j^2 )^1/2) ^1/2→ 0,since |a^-1_k(N) ω^1/2_kl(N) | converges to zero for eachi,k,l ∈{1,…,p} by Lemma <ref>. Moreover, for ω^2_kl(N) given in (<ref>), ω^2_kl(⌊ Nt ⌋)/ω^2_kl(N)is bounded for the long- as well as for the short-range dependent components by Lemma <ref>. The second condition of Lemma <ref> follows also by Lemma <ref>, sincelim sup_N →∞∑_j ∈ |B_l,Nj(t)|^2 = lim sup_N →∞∑_j ∈ | ∑_n=1^⌊ Nt ⌋∑_k=1^pλ_ka^-1_k(N) ψ_kl,n-j |^2 ≤lim sup_N →∞∑_j ∈∑_l=1^p| ∑_n=1^⌊ Nt ⌋∑_k=1^pλ_k a^-1_k(N)ψ_kl,n-j |^2 = lim sup_N →∞λ' E| A^-1_N(H) S_⌊ Nt ⌋ |^2 λ < ∞. By Lemma <ref> the process A_N^-1(H)S_⌊ Nt ⌋ can be treated as linear with Gaussian innovations. By <cit.> it suffices to establish the componentwise convergence behavior of the cross-covariances as we did in Lemma <ref>. So, the sample mean process converges to the multivariate Gaussian process {𝒢(t)}_t ∈ [0,1]defined in (<ref>). §.§.§ Tightness By <cit.>, it suffices to prove tightness of each component. Each component is a sum of a sample mean process of univariate linear processes,S_⌊ Nt ⌋= ∑_l=1^p∑_n=1^⌊ Nt ⌋∑_j ∈ψ_kl,jε_l,n-j.Since sums of tight processes are tight (see <cit.>), what remains is to prove tightness for the sample mean process of a univariate linear process∑_n=1^⌊ Nt ⌋∑_j ∈ψ_kl,jε_l,n-j.For the long-range dependent components, this follows by Proposition 4.4.2 in <cit.> and for the short-range dependent ones by Proposition 4.4.4 in <cit.> and the assumption ε_0^2+δ<∞ for some δ>0. §.§ Proof of Theorem <ref>As in the previous section, we first give some auxiliary results regarding the limit processes covariance structure (Section <ref>). We then investigate the convergence of the finite-dimensional distributions and tightness (Sections <ref> and <ref>), which establish Theorem <ref>. In Section <ref> we investigate the properties of the process {Z(t)}_t∈ defined in (<ref>). §.§.§ Auxiliary results Lemma <ref> provides the limiting covariance structure for the components which are short-range dependent. Let Γ_N,ℓ^S(t) be defined by (<ref>). Then, for ℓ_1,ℓ_2≥ 0 and t,u ∈ [0,1]lim_N →∞ N(Γ^S_N,ℓ_1(t), Γ^S_N,ℓ_2(u))= (G^S_ℓ_1(t),G^S_ℓ_2(u)) , where the right-hand side is given in (<ref>). First, note that((ε_i_1ε_i_2') ((ε_i_3ε_i_4'))') =σ^* , i_1=i_2=i_3=i_4, (I_p) ((I_p) )', i_1=i_2≠ i_3=i_4, I_p^2, i_1=i_3≠ i_2=i_4, K_p, i_1=i_4≠ i_2=i_3, 0, i_1≠ i_2≠ i_3≠ i_4with σ^* := ((ε_iε_i') ((ε_iε_i'))'). We investigate the covariances(Γ^S_N,ℓ_1(t), Γ^S_N,ℓ_2(u)) = ( 1/N^2∑_n=1^⌊ Nt ⌋∑_l=1^⌊ Nu ⌋_I_S(X_n X_n+ℓ_1') (_I_S(X_l X_l+ℓ_2'))' ) - ⌊ Nt ⌋⌊ Nu ⌋/N^2Γ^S_ℓ_1 (Γ^S_ℓ_2)'.Setting r=l-n and by (<ref>), we have(_I_S(X_n X_n+ℓ_1') (_I_S(X_l X_l+ℓ_2'))') = E_I_S∑_i_1,i_2,i_3,i_4∈ (_i_2+ℓ_1⊗_i_1) ( (ε_n-i_1ε_n-i_2 ') ((ε_n-i_3ε_n-i_4'))' ) (_i_4+r⊗_i_3+r+ℓ_2)' E_I_S' = ∑_i ≠ j (_I_S(_i_i+ℓ_1') (_I_S(_j+r+ℓ_2_j+r'))'+ E_I_S (_i+ℓ_1_i+r' ⊗_j_j+r+ℓ_2')E_I_S'+ E_I_S K_p (_i_i+r' ⊗_j+ℓ_1_j+r+ℓ_2') E_I_S' + E_I_S∑_i∈(_i+ℓ_1⊗_i) σ^*(_i+r⊗_i+r+ℓ_2 )'E_I_S'= Γ^S_ℓ_1 (Γ^S_ℓ_2)'+ E_I_S(Γ_r+ℓ_2⊗Γ_r-ℓ_1)E_I_S'+ E_I_S K_p (Γ_r⊗Γ_r+ℓ_2-ℓ_1)E_I_S' + E_I_S∑_i∈(_i+ℓ_1⊗_i) (σ^* - (I_p) ((I_p) )' - I_p^2 - K_p) (_i+r⊗_i+r+ℓ_2 )' E_I_S'.Interchanging the order of summation, assuming t<u and defining m_N=min(⌊ Nt ⌋,⌊ Nu ⌋-⌊ Nt ⌋), we getlim_N →∞N(Γ^S_N,ℓ_1(t), Γ^S_N,ℓ_2(u)) = ⌊ Nt ⌋/N∑_|r|<⌊ Nt ⌋( 1-|r|/⌊ Nt ⌋)T_r + ∑_r=1^m_N-1r/NT_r + m_N/N∑_r=m_N^⌊ Nu ⌋ -m_NT_r + ⌊ Nu ⌋/N∑_r=⌊ Nu ⌋-m_N+1^⌊ Nu ⌋-1( 1-r/⌊ Nu ⌋)T_r,where T_r=E_I_ST_r E_I_S' withT_r=(Γ_r+ℓ_2⊗Γ_r-ℓ_1)+K_p (Γ_r⊗Γ_r+ℓ_2-ℓ_1) + ∑_i∈(_i+ℓ_1⊗_i) Σ (_i+r⊗_i+r+ℓ_2 )'and Σ defined in (<ref>). In particular, T_r is absolutely summable, since∑_r∈∑_i∈ |E_I_S (_i+ℓ_1⊗_i) Σ (_i+r⊗_i+r+ℓ_2 )' E_I_S'|<∞and∑_r∈ E_I_S(Γ_r⊗Γ_r)E_I_S'<∞componentwise. The latter inequalities hold, since d_k_1+d_k_2+d_k_3+d_k_4<1 by construction. Applying the dominated convergence theorem yieldslim_N →∞ N(Γ^S_N,ℓ_1(t), Γ^S_N,ℓ_2(u)) = lim_N →∞⌊ Nt ⌋/N∑_|r|<⌊ Nt ⌋T_r = t E_I_S(∑_r ∈(Γ_r+ℓ_2⊗Γ_r-ℓ_1+ K_p ( Γ_r⊗Γ_r+ℓ_2-ℓ_1) ) t E_I_S( ) +∑_r∈∑_i∈(_i+ℓ_1⊗_i) Σ (_i+r⊗_i+r+ℓ_2 )' ) E_I_S' .The following lemma deals with the normalization for the long-range dependent components and gives the covariance structure of Z^L(t). Let Γ_N,ℓ^L(t) be defined by (<ref>) and Δ_N = E_I_L( N^D-1/2I_p⊗ N^D-1/2I_p )E_I_L'. Then, for ℓ_1, ℓ_2≥ 0,lim_N →∞(Δ_N^-1Γ^L_N,ℓ_1(t), Δ_N^-1Γ^L_N,ℓ_2(u)) = E_I_L (t^D I(R)t^D + u^D I(R)'u^D -|t-u|^D I(R,t-u)|t-u|^D)E_I_L' with D= (D_p_1⊕ D_p_1)-1/2I_p_1^2and I(R)=∫_0^1 (1-x)x^D(R ⊗ R) x^Ddx, where ⊕ denotes the Kronecker sum defined as D_p_1⊕ D_p_1=(I_p_1⊗ D_p_1) + (D_p_1⊗ I_p_1) and I(R,t)= I(R) 1_{t>0}+ I(R)' 1_{t<0}.Note that one gets for (Δ_N^-1Γ^L_N,ℓ_1(t),Δ_N^-1Γ^L_N,ℓ_2(u)) with t<u the same expression as in (<ref>) by replacing the subscript “S” by “L” and adjusting the normalization sequence such that T_r= 1/NΔ_N^-1E_I_LT_r E_I_L' Δ_N^-1 with T_r as in (<ref>). We consider the summands separately. By Proposition <ref>, the autocovariances of the underlying process {X^L_n}_n ∈ satisfy (<ref>) with (<ref>) and we get for the first summand of T_r lim_N →∞1/N^2∑_r=0^⌊ Nt ⌋(⌊ Nt ⌋- r ) Δ_N^-1E_I_L (Γ_r+ℓ_2⊗Γ_r-ℓ_1)E_I_L'Δ_N^-1 = lim_N →∞1/N^2∑_r=0^⌊ Nt ⌋(⌊ Nt ⌋- r ) Δ_N^-1 ( (r+ℓ_2)^D_p_1-1/2I_p_1 R(r+ℓ_2) (r+ℓ_2)^D_p_1-1/2I_p_1⊗(r-ℓ_1)^D_p_1-1/2I_p_1 R(r-ℓ_1) (r-ℓ_1)^D_p_1-1/2I_p_1 )Δ_N^-1 = ∫_0^t (t-x) (x^D_p_1-1/2I_p_1⊗ x^D_p_1-1/2I_p_1)(R ⊗ R) (x^D_p_1-1/2I_p_1⊗ x^D_p_1-1/2I_p_1)dx= t^(D_p_1⊕ D_p_1)-1/2I_p_1∫_0^1 (1-x)x^(D_p_1⊕ D_p_1)-I_p_1(R ⊗ R) x^(D_p_1⊕ D_p_1)-I_p_1dx t^(D_p_1⊕ D_p_1)-1/2I_p_1.The second term of T_r can be dealt with analogously. We consider the last summand componentwise for indices taking values in I_L. Define^i(k;l) := ^i(k_1,…,k_4;l_1,…,l_4) := ψ_k_1l_1,iψ_k_2l_2,i+ℓ_1ψ_k_3l_3,i+r+ℓ_2ψ_k_4l_4,i+ras part of the component, and consider∑_i∈^i(k;l) = ∑_i=-∞^-r-1^i(k;l) +∑_i=0^∞^i(k;l) +∑_i=-r^0^i(k;l). For example, for the last term, note that∑_i=-r^0^i(p;q) = ∑_i=-r^0 C_p_1q_1(i)C_p_2q_2(i+k_1)C_p_3q_3(i+r+k_2)C_p_4q_4(i+r) ∑_i=-r^0 C_ll_1(i)C_ml_2(i+h)× |i|^d_p_1-1 |i+k_1|^d_p_2-1 |i+r+k_2|^d_p_3-1 |i+r|^d_p_4-1 ∼∑_i=0^r C_p_1q_1(-i)C_p_2q_2(k_1-i)C_p_3q_3(r+k_2-i)C_p_4q_4(r-i) i^d_p_1+d_p_2-2(r-i)^d_p_3+d_p_4-2= r^d_p_1+d_p_2+d_p_3+d_p_4-3∑_i=0^r C_p_1q_1(-i)C_p_2q_2(k_1-i)C_p_3q_3(r+k_2-i)C_p_4q_4(r-i) r^1-d_l-2d_m-d_l∑_i=0^r C_ll_1(-i)L_ml_2(h-i)×( i/r) ^d_p_1+d_p_2-2(r-i/r) ^d_p_3+d_p_4-21/r ∼ r^d_p_1+d_p_2+d_p_3+d_p_4-3α_p_1q_1^-α_p_2q_2^-α_p_3q_3^+α_p_4q_4^+∫_0^1 x^d_p_1+d_p_2-2 (1-x)^d_p_3+d_p_4-2 dx, as r →∞. The first and second terms yield similarly∑_i=-∞^-r-1^i(k;l)∼ r^d_k_1+d_k_2+d_k_3+d_k_4-3α_k_1l_1^-α_k_3l_2^-α_k_3l_3^-α_k_4l_4^-∫_1^∞ x^d_k_1+d_k_2-2 (x-1)^d_k_3+d_k_4-2 dx ∑_i=0^∞^i(k;l)∼ r^d_k_1+d_k_2+d_k_3+d_k_4-3α_k_1l_1^+α_k_3l_2^+α_k_3l_3^+α_k_4l_4^+∫_0^∞ x^d_k_1+d_k_2-2 (x+1)^d_k_3+d_k_4-2dx. Then,∑_i∈^i(k;l)∼ r^d_k_1+d_k_2+d_k_3+d_k_4-3 C(d_k_1,d_k_2,d_k_3,d_k_4)andlim_N →∞1/N^2∑_|r|<N(N-|r| ) ∑_i∈ N^2-d_k_1-d_k_2-d_k_3-d_k_4×ψ_k_1l_1,iψ_k_2l_2,i+ℓ_1ψ_k_3l_3,i+r+ℓ_2ψ_k_4l_4,i+rΣ_k_1k_2k_3k_4 =0,where Σ_k_1k_2k_3k_4 denotes a component of Σ in (<ref>). The next lemma gives the covariance between the long- and the short-range dependent components of the sample autocovariances.Let Γ_N,ℓ^L(t) and Γ_N,ℓ^S(t) be defined by (<ref>) and Δ_N = E_I_L( N^D-1/2I_p⊗ N^D-1/2I_p )E_I_L'. Then, for ℓ_1,ℓ_2≥ 0, lim_N →∞(Δ_N^-1Γ^L_N,ℓ_1(t), N^1/2Γ^S_N,ℓ_2(u)) =0 We follow the proof of Lemma <ref>. Note that one gets for (Δ_N^-1Γ^L_N,ℓ_1(t), N^1/2Γ^S_N,ℓ_2(u)) with t<u the same expression as in (<ref>) by settingT_r= N^-1/2Δ_N^-1E_I_LT_r E_I_S' with T_r as in (<ref>). Then, since ∑_i=1^4 d_k_i<1 for k_1,k_2∈ I_L and k_3,k_4∈ I_S,∑_r∈∑_i∈ |E_I_L (_i+ℓ_1⊗_i) Σ (_i+r⊗_i+r+ℓ_2 )' E_I_S'|<∞and∑_r∈ E_I_L(Γ_r⊗Γ_r)E_I_S'<∞componentwise. This implies that T_r is absolutely summable over r andlim_N →∞(Δ_N^-1Γ^L_N,ℓ_1(t), N^1/2Γ^S_N,ℓ_2(u)) = lim_N →∞Δ_N^-1/N^1/2⌊ Nt ⌋/N∑_|r|<⌊ Nt ⌋( 1-|r|/⌊ Nt ⌋) E_I_LT_r E_I_S' = 0. We conclude the section with a comment on the properness of the limiting process ((Z^L(t))' , (G_ℓ^S(t))')' defined in (<ref>) and (<ref>), since it is in parts a consequence of the previous Lemmas <ref> and <ref>. Since Z^L(t) and G_ℓ^S(t) are uncorrelated by Lemma <ref>, it is enough to prove properness for each of those processes separately. The matrix Λ:=∑_j ∈ (ψ_kl,j)_k=p_1+1,…,p;l=1,…,p is assumed to have full rank by condition (S), soΛΛ' is positive definite. The matrix ΛΛ' is also equal to ∑_n ∈Γ_p_2,n. For this reason, ( G^S(t) (G^S(t))' ) is positive definite, which follows by Lemma <ref>.We refrain from using Lemma <ref> to infer that Z^L(t) is proper. Instead, we calculate the covariances of Z^L(t) directlyZ^L(t) (Z^L(t))'= E_I_L ( I_2(f_t,D)I'_2(f_t,D) ) E_I_L'=E_I_L∑_s_1,s_2∈{+,-}∑_r_1,r_2∈{+,-}∫_^2'∫_0^t∫_0^t ((v_1-x_2)^D-I_p_s_2⊗ (v_1-x_1)^D-I_p_s_1) ×((M^s_2(M^r_2)' ⊗M^s_1(M^r_1)') + (M^s_2(M^r_1)' ⊗M^s_1(M^r_2)') ) × ((v_2-x_2)^D-I_p_r_2⊗ (v_2-x_1)^D-I_p_r_1) dv_1dv_2 dx_1 dx_2 E_I_L',where we used Theorem 2.2 and 7.9 in <cit.>. Note that M^+=((A^+)' 0_p × p_2)' and M^-=((A^-)' 0_p × p_2)'. Since the elimination matrix E_I_L is applied from both sides, the zero rows and columns are eliminated (see Remark <ref>). For this reason, the function is positive definite, since A^+(A^+)' and A^-(A^-)' are positive definite as a consequence of condition (L). §.§.§ Convergence of the finite-dimensional distributions The proof of the convergence of the finite-dimensional distributions is structured as follows. First, we focus on the pure long-range dependence part with k,l ∈ I_L. Note that the sample autocovariances can be separated into the diagonal and off-diagonal parts (Γ_N,ℓ-Γ_ℓ)(t) = 1/N∑_n=1^⌊ Nt ⌋∑_j+ℓ≠ i_jε_n-jε_n+k-i' _i ' + 1/N∑_n=1^⌊ Nt ⌋∑_j∈_j (ε_n-jε_n-j'-I) _j+ℓ' =: O_N,ℓ(t)+D_N,ℓ(t).The following Lemma <ref> will imply that the sought convergence can be proved only for the off-diagonal part of the long-range dependent components. We will then provide a convergence result, Lemma <ref>, for multivariate second-order linear forms with respect to the components of a multivariate i.i.d. process. Finally, a series of lemmas, Lemma <ref> - <ref>, follow the idea of <cit.> to prove the joint convergence of Y_N,ℓ^L(t) and Y_N,ℓ^S(t). Let {X_n}_n ∈ be as in Theorem <ref>. Then N^1/2(_I_L(D_N,ℓ(t)), ℓ=0,…,L) ℒ⟶ (C_k(G(t)), ℓ=0,…,L),t ∈ [0,1],in D[0,1]^|I_L|, where {G(t)}_t ∈ [0,1] is an ^p × p-valued Brownian motion and C_ℓ:^p × p→^|I_L| withC_ℓ(X)=_I_L( ∑_j ∈_j X _j+ℓ' ),X ∈^p × p. Consider the linear combination ∑_ℓ=0^Lμ_ℓ(_I_L(D_N,ℓ(t)) of the diagonal term for μ_ℓ∈, ℓ∈{0,…,L }. It is short-range dependent in the sense that∑_j ∈ a_j_F < ∞,where a_j:^p × p→^p × p witha_j(·)= ∑_ℓ=0^Lμ_ℓ_I_L(_j (·) _j+ℓ'). Using the Cramér-Wold theorem and the results in <cit.>, we get∑_ℓ=0^Lμ_ℓ(_I_L(D_N,ℓ(t)) ℒ⟶AG(t),t ∈ [0,1],for all μ_ℓ∈, ℓ∈{0,…,L }, where A=∑_j∈ a_j. To investigate the asymptotic behavior of the off-diagonal terms for the long-range dependent components denoted by O_N,ℓ^L(t)=_I_L(O_N,ℓ(t)), we proveΔ_N^-1 O_N,ℓ^L(t) f.d.d.⟶ Z^L(t).The process Z^L(t) is defined in (<ref>). Rewriting the left-hand side of (<ref>) yieldsΔ_N^-1 O_N,ℓ^L(t) = Δ_N^-1_I_L(1/N∑_n=1^⌊ Nt ⌋∑_j+ℓ≠ i_jε_n-jε_n+ℓ-i' _i ' ) = Δ_N^-1 E_I_L(1/N∑_n=1^⌊ Nt ⌋∑_i_1≠ i_2_n-i_1ε_i_1ε_i_2' _n+ℓ-i_2 ' ) = ∑_i_1≠ i_2Δ_N^-1 E_I_L1/N∑_n=1^⌊ Nt ⌋ (_n+ℓ-i_2⊗_n-i_1) (ε_i_1ε_i_2') = ∑_i_1≠ i_2 C_N(i_1,i_2)(ε_i_1ε_i_2') ,whereC_N(i_1,i_2) = Δ_N^-1 E_I_L1/N∑_n=1^⌊ Nt ⌋ (_n+ℓ-i_2⊗_n-i_1). The following lemma provides a generalization of Proposition 14.3.2 in <cit.>. It uses the space of simple functions S_M(^2, ^p^2 × p^2) defined as follows. Partition the space ^2 into cubes of size 1/M with M∈. Let (Δ):=Δ_1×Δ_2⊂^2 be such thatΔ_1, Δ_2∈ I_M, whereI_M:= { (j/M, j+1/M ], j ∈}, M∈. We write(Δ) ∈{Δ_M} and(Δ) ∈{Δ_M^diag}, if Δ_1=Δ_2.This space S_M(^2, ^p^2 × p^2) then consists of ^p^2 × p^2-valued functions f = f(x_1,x_2) on ^2 satisfying f(x_1,x_2)=f^Δ_1Δ_2, x ∈ (Δ), (Δ) ∈{Δ_M}, 0, x ∈ (Δ), (Δ) ∈{Δ_M^diag} ,where f^Δ_1Δ_2∈^p × p. Set ‖ f ‖^2= ∫_^2 f(x_1,x_2) ^2_F dx_1dx_2 for f:^2 →^p^2 × p^2. Consider the off-diagonal tupleQ_2(C_N)=∑_i_1≠ i_2C_N(i_1,i_2) (ε_i_1ε_i_2') .Assume that the weights C_N are such that the functionsC_N(x_1,x_2)= N C_N([x_1N],[x_2N]),x_1, x_2∈satisfy||C_N-f|| → 0for a function f ∈ L^2(^2, ^p^2 × p^2). Then Q_2(C_N) f.d.d.⟶ I_2(f).Let f_ε be in S_M(^2, ^p × p) and defineC_N,ε(i_1,i_2):= N^-1 f_ε( i_1/N,i_2/N) , i_1,i_2∈.It is enough to prove that for all ε>0, there exists f_ε∈ S_M(^2, ^p^2 × p^2), M ≥ 1,such thatQ_2(C_N)-Q_2(C_N,ε) _F≤ε, I_2(f_ε)-I_2(f) _F≤ε, Q_2(C_N,ε) f.d.d.⟶ I_2(f_ε) ,as N →∞. Note that((ε_i_1ε_i_2') ((ε_j_1ε_j_2'))')=I_p^2, ifi_1=j_1, i_2=j_2, K_p, ifi_1=j_2, i_2=j_1, 0, otherwise.Then, for (<ref>), Q_2(C_N)_F = ∑_i_1≠ i_2C_N(i_1,i_2)(ε_i_1ε_i_2') _F^2 ≤ 2 ∑_i_1≠ i_2 C_N(i_1,i_2) _F^2 = 2 ∫_^2'N^2C_N([x_1N],[x_2N]) _F^2 dx_1 dx_2 = 2 C_N^2 .This impliesQ_2(C_N)-Q_2(C_N,ε)_F≤ 2 C_N - C_N,ε^2.The latter bound could be approximated by finding simple functions f_ε such thatC_N - C_N,ε^2 < εas N →∞. By assumption, there is N_0≥ 1 such thatC_N -C_N_0^2 ≤ 2C_N -f ^2+ 2 f-C_N_0^2 ≤ε/6for all N ≥ N_0. Given N_0≥ 1 and ε>0, there exist simple functionsf_ε such thatC_N_0 -f_ε^2 ≤ε/6.The function C_N,ε derived from C_N,ε satisfiesf_ε-C_N,ε^2 = ∫_^2 f_ε(x_1,x_2)- f_ε(⌊ x_1N ⌋/N ,⌊ x_2N ⌋/N)_F^2 dx_1dx_2→0as N →∞. Hence, there is N_0≥ 1 such thatC_N -C_N,ε^2 ≤ 3C_N -C_N_0,ε^2+ 3C_N_0 -C_ε^2+ 3C_ε -C_N,ε^2 ≤ε /2.This proves (<ref>) sinceI(f_ε)-I(f) ≤ 2 f_ε-f ^2 ≤ 2 (2f_ε-C_N_0^2 + 2C_N_0-f ^2).Finally, for (<ref>), note thatQ_2(C_ε,N)= ∑_i_1≠i_2C_N,ε(i_1,i_2)(ε_i_1ε_i_2') = ∑_i_1≠ i_2 N^-1f_ε(i_1/N,i_2/N)(ε_i_1ε_i_2') = ∑_(Δ) ∈{Δ_M}f_ε^Δ_1Δ_2 N^-1∑_i_1≠ i_2(ε_i_1ε_i_2') 1 _{i_1/N∈Δ_1,i_2/N∈Δ_2}= ∑_(Δ) ∈{Δ_M}f_ε^Δ_1Δ_2(W_N (Δ_1) W'_N (Δ_2)), whereW_N(Δ_i) = N^-1/2∑_j : j/N∈Δ_iε_j.Now, define the vectorW(Δ_i):=(W_1,N(Δ_i),…,W_p,N(Δ_i)). Since the intervals Δ_i are disjoint, (W(Δ_i))_i ∈ areindependent random vectors. Since {ε_j}_ j ∈ are i.i.d.the central limit theorem applies and hence(W_N(Δ_-J),…,W_N(Δ_J)) f.d.d.⟶ (W(Δ_-J),…,W(Δ_J)). Using thecontinuous mapping theorem yieldsQ_2(C_N,ε)f.d.d.⟶∑_(Δ) ∈{Δ_M}f_ε^Δ_1Δ_2(W (Δ_1) W' (Δ_2)) = I_2(f_ε). As noted above, we prove Theorem <ref> through a number of lemmas following the idea of <cit.>. We introduce the κ-truncated versions of the quantities of interestX_n^(κ) =∑_j=-κ^κ_jε_n-j,Γ_N,ℓ^S,(κ) = _I_S(1/N∑_n=1^N X_n^(κ) X_n+ℓ^(κ)')andΓ_ℓ^S,(κ) = _I_S((X_0^(κ)X_ℓ^(κ)')) = _I_S(∑_j=-κ-ℓ^κ-ℓ_j_j+ℓ').The κ-truncated version ofY_N,ℓ(t) is written asY_N,ℓ^(κ)(t) = 1/N∑_n=1^⌊ Nt ⌋( X_n^(κ) X_n+ℓ^(κ)'-(X_0^(κ) X_ℓ^(κ)') ).The truncated version of the limit process {G^S_ℓ(t)}_t ∈ [0,1] is defined by its covariance structure(G^S,(κ)_ℓ_1(t),G^S,(κ)_ℓ_2(u)) = min(t,u)E_I_S(∑_r ∈( Γ_r+ℓ_2^(κ)⊗Γ_r-ℓ_1^(κ)+ K_p (Γ_r^(κ)⊗Γ_r+ℓ_2-ℓ_1^(κ) )) + ∑_r ∈∑_i=-κ^κ(_i+ℓ_1⊗_i) Σ (_i+r⊗_i+r+ℓ_2 )' ) E_I_S'. Suppose the assumptions of Theorem <ref> and setY_N,ℓ^S,(κ)(t)=_I_S(Y_N,ℓ^(κ)(t))andW_N(t)=N^-1/2∑_n=1^⌊ Nt ⌋ε_nwith ε_0∈^p. Then,(N^1/2Y_N,ℓ^S,(κ)(t), W_N(t)) f.d.d.⟶(G^S,(κ)_ℓ(t),W(t)),where G^S,(κ)_ℓ(t) is defined by (<ref>) and W(t) is a standard Brownian motion. By the Cramér-Wold theorem, we can prove thatμ' N^1/2 Y_N,ℓ^S,(κ)(t)+ ν' W_N(t) f.d.d.⟶μ' G^S,(κ)_ℓ(t)+ν'W(t)with μ∈^|I_S| and ν∈^p. The left-hand side of (<ref>) can be written asμ' N^1/2 Y^S,(κ)_N,ℓ(t)+ ν' W_N(t) = N^-1/2∑_n=1^⌊ Nt ⌋ Q_n,ℓ^(κ)withQ_n,ℓ^(κ) = μ' _I_S( X_n^(κ) X_n+ℓ^(κ)'-(X_0^(κ) X_ℓ^(κ)') ) + ν' ∑_i=(2κ+ℓ-1)n+1^(2κ+ℓ)nε_i .Following <cit.> and <cit.>, we shall use the notion ofm-dependence. Recall that a stationary sequence {Y_j}_j ∈ is m-dependent, where m is a non-negative integer if the random sequences {Y_j}_j ≤ 0 and {Y_j}_j ≥ m+1 are independent. Define the sequence {Y_n}_n ∈ of ^p^2(ℓ+1)-valued random variables by Y_n= (Z_n,Z_n+1,… ,Z_n+ℓ),where Z_n+ℓ:=_I_S(X_n^(κ) X_n+ℓ^(κ)'). Since the process {Y_n}_n ∈ is (2κ+ℓ)-dependent, so is {Q_n,ℓ^(κ)}_n ∈. For any λ∈^ℓ+1, the sequence (λ'Q_n,ℓ^(κ)) is (2κ+ℓ)-dependent as well. Then, the convergence (<ref>) follows by the functional central limit theoremfor m-dependent processes in <cit.>. Since thejoint asymptotic normality is proven, it is left to verify that the asymptotic covariance structure of (N^1/2 Y_N,ℓ^S,(κ)(t), W_N(t)) coincides with that of (G^S,(κ)_ℓ(t),W(t)).For (N^1/2Y_N,ℓ^S,(κ)(t),N^1/2Y_N,ℓ^S,(κ)(u)), the relation follows by Lemma <ref> and (<ref>). By similar arguments as in Lemma<ref> for t<u,lim_N →∞(N^1/2 Y_N,ℓ^S,(κ)(t)W_N'(u)) = lim_N →∞⌊ Nt ⌋/N∑_|r|<⌊ Nt ⌋(1-|r|/⌊ Nt ⌋) ∑_i=-κ^κ E_I_S (_r+ℓ-i⊗_r-i)Σ= t ∑_ r ∈∑_i=-κ^κ E_I_S(_r+ℓ-i⊗_r-i)Σ,where Σ=((ε_0ε_0 ')ε_0' ). Since N^1/2 Y_N,ℓ^S,(κ)(t) and W_N'(t) are not asymptotically uncorrelated as shown in (<ref>), one can infer that the resulting limits Z^L(t) of the long-range dependent components and G_ℓ^S(t) of the short-range dependent components in Theorem <ref> are not independent. However, Lemma <ref> gives uncorrelatedness between Z^L(t) and G_ℓ^S(t). Using the same notation as in the proof of Lemma <ref>, the joint convergence in the previous lemma still holds by replacing W_N(t) by(W_N(Δ_-J),…,W_N(Δ_J)) withW_N(Δ_i):=(W_1,N(Δ_i),…,W_p,N(Δ_i)) andW_N(Δ_i) as in (<ref>), since the intervals Δ_i are disjoint. Replace (N^1/2 Y_N,ℓ^S,(κ)(t), W_N(t)) by(N^1/2 Y_N,ℓ^S,(κ)(t), Δ_N^-1 O^L_N,ℓ(t)) in Lemma <ref>. Assume that the weights C_N defined in (<ref>) are such that for a function f ∈ L^2(^2, ^p^2 × p^2) the functionsC_N(x_1,x_2)= N C_N([x_1N],[x_2N]),x_1, x_2∈satisfy||C_N-f|| → 0.Then,(N^1/2Y_N,ℓ^S,(κ)(t), Δ_N^-1 O^L_N,ℓ(t)) f.d.d.⟶(G^S,(κ)_ℓ(t),Z^L(t)). We prove the lemma by combining the previous Lemmas <ref> and <ref>. As in (<ref>), the sum O^L_N,ℓ(t) can be represented as Q_2(C_N) with Q_2 defined in (<ref>). By Lemma <ref>, for all ε>0,there exists f_ε∈ S_M(^2,^p^2 × p^2), M ≥ 1, such that (<ref>), (<ref>) and (<ref>) are satisfied.As in the proof of Lemma <ref> by applying the continuous mapping theorem to the result in Lemma <ref>, we get(N^1/2 Y^S,(κ)_N,ℓ(t), Q_2(C_N,ε)) f.d.d.⟶(G^S,(κ)_ℓ(t), I_2(f_ε)).Now, defineR^ε,(κ)_N,ℓ(t)= N^1/2 Y_N,ℓ^S,(κ)(t)+ Q_2(C_N,ε), R^(κ)_N,ℓ(t)= N^1/2 Y_N,ℓ^S,(κ)(t)+ Q_2(C_N), R^ε,(κ)_ℓ(t)= G_ℓ^S,(κ)(t)+I_2(f_ε), R_ℓ^(κ)(t)= G_ℓ^S,(κ)(t)+I_2(f).Then, by (<ref>), (<ref>) and (<ref>),R^ε,(κ)_N,ℓ(t) f.d.d.⟶ R^ε,(κ)_ℓ(t), asN →∞, R^ε,(κ)_ℓ(t) f.d.d.⟶ R_ℓ^(κ)(t), as ε→ 0, lim_ε→ 0lim sup_N →∞R^ε,(κ)_N,ℓ(t)-R^(κ)_N,ℓ(t)_F=0for allt ∈ [0,1],which finally impliesR^(κ)_N,ℓ(t)f.d.d.⟶R^(κ)_ℓ(t)by <cit.>. In the following lemma the truncated qunatities get replaced by their non-truncated originals. Replace (N^1/2 Y_N,ℓ^S,(κ)(t), Δ_N^-1 O^L_N,ℓ(t)) by (N^1/2Y_N,ℓ^S(t), Δ_N^-1 O^L_N,ℓ(t)) in Lemma <ref>. Then,(N^1/2 Y_N,ℓ^S(t), Δ_N^-1 O^L_N,ℓ(t)) f.d.d.⟶(G^S_ℓ(t),Z^L(t)). DefineR_N,ℓ(t) = μ' N^1/2 Y_N,ℓ^S(t)+λ' Q_2(C_ N), R_ℓ(t) = μ' G_ℓ^S(t)+λ' I_2(f)for μ∈^|I_S|, λ∈^|I_L|. We prove thatR_N,ℓ^(κ)(t) f.d.d.⟶ R_ℓ^(κ)(t), asN →∞, R_ℓ^(κ)(t) f.d.d.⟶ R_ℓ(t), asl →∞ ,lim_κ→∞lim sup_N →∞(R_N,ℓ^(κ)(t) -R_N,ℓ(t))=0for allt ∈ [0,1]. The convergence (<ref>) follows by Lemma<ref>, (<ref>) is a consequence oflim_κ→∞(G_ℓ_1^S,(κ)(t),G^S,(κ)_ℓ_2(u)) = (G_ℓ_1^S(t),G^S_ℓ_2(u)),while (<ref>) follows fromlim_κ→∞lim sup_N →∞(μ' N^1/2 Y_N,ℓ^S,(κ)(t))^2= (μ'G_ℓ^S(t))^2, lim sup_N →∞(μ' N^1/2 Y_N,ℓ^S(t))^2= (μ' G_ℓ^S(t))^2, lim_κ→∞lim sup_N →∞(μ' N^1/2 Y_N,ℓ^S,(κ)(t) μ' N^1/2 Y_N,ℓ^S(t))= (μ'G_ℓ^S(t))^2 .To conclude the proof of Theorem <ref>, it remains to verify that C_N defined in (<ref>) satisfies the assumptions of Lemma <ref>.WriteNC_N([x_1N],[x_2N])= ∑_n=1^⌊ Nt ⌋Δ_N^-1 E_I_L (_n+ℓ-[x_2N]⊗_n-[x_1N]) = N ∫_0^tΔ_N^-1 E_I_L (_[vN]+ℓ-[x_2N]⊗_[vN]-[x_1N]) dv.Then, considering the expression componentwiseN^2-d_l_1-d_l_2ψ_l_1q_1, [vN]-[x_1N]ψ_l_2q_2, [vN]+ℓ-[x_2N]= N^2-d_l_1-d_l_2 C_l_1q_1([vN]-[x_1N]) |[vN]-[x_1N]|^d_l_1-1= l N^2-d_l_1-d_l_2× C_l_2q_2([vN]+ℓ-[x_2N]) |[vN]+ℓ-[x_2N]|^d_l_2-1 = p_N^(l_1,q_1)(v,x_1)p_N^(l_2,q_2)(v,x_2) [ν_l_1l_2q_1q_2^(+, +)(v,x_1,x_2)+ ν_l_1l_2q_1q_2^(-, -)(v,x_1,x_2) =l p_n^(l_1,q_1)(v,x_1)p_n^(l_2,q_2)(v,x_2) [ + ν_l_1l_2q_1q_2^(+, -)(v,x_1,x_2)+ ν_l_1l_2q_1q_2^(-, +)(v,x_1,x_2) ] , whereν_l_1l_2q_1q_2^(s_1,s_2)(v,x_1,x_2) := (v-x_1)_s_1^d_l_1-1 (v-x_2)_s_2^d_l_2-1α_l_1q_1^s_1α_l_2q_2^s_2, fors_1,s_2∈{+,-}andp_N^(l_i,q_i)(v,x_i) := N^1-d_l_i C_l_iq_i([vN]-[x_iN]) |[vN]-[x_iN]|^d_l_i-1/ (v-x_i)_+^d_l_i-1α_l_iq_i^++ (v-x_i)_-^d_l_i-1α_l_iq_i^-→ 1.Furthermore, there are constants C_1, C_2, such thatsup_N ≥ 1sup_v,x_i p_N^(l_i,q_i)(v,x_1) ≤ C_i, i=1,2,which impliessup_x_1,x_2| ∫_0^t N^2-d_l_1-d_l_2ψ_l_1q_1, [vN]-[x_1N]ψ_l_2q_2,[vN]+ℓ-[x_2N] dv|≤ Csup_x_1,x_2 |∑_s_1,s_2∈{+,-}∫_0^tν_l_1l_2q_1q_2^(s_1,s_2 )(v,x_1,x_2) dv|and by the dominated convergence theorem∫_0^t N^2-d_l_1-d_l_2ψ_l_1q_1,[vN]-[x_1N]ψ_l_2q_2,[vN]+ℓ-[x_2N] dv →∑_s_1,s_2∈{+,-}∫_0^tν_l_1l_2q_1q_2^(s_1,s_2 )(v,x_1,x_2)dv.Then, applying again the dominated convergence theorem leads toC_N -f_H,t^2 = ∫_^2∫_0^t N Δ_N^-1 E_I_L (_[vN]+ℓ-[x_2N]⊗_[vN]-[x_1N]) dv- E_I_L f_H,t(x_1,x_2) ^2_F dx_1dx_2→ 0.This shows that the conditions in Lemma <ref> are satisfied. §.§.§ Tightness Under the assumptions in Theorem <ref> the sample autocovariance process is tight in D[0,1]^p^2.By Lemma 1 in <cit.>, it is enough to prove tightness in each componenta^-1_kl(N)Y_kl,N,ℓ(t) = a^-1_kl(N) ∑_n=1^⌊ Nt ⌋ (X_k,nX_l,n+ℓ - (X_k,0X_l,ℓ))= ∑_r,s=1^p a^-1_kl(N) ∑_n=1^⌊ Nt ⌋( ∑_j+ℓ≠ i ψ_kr,iψ_ls,jε_r,n-jε_s,n+ℓ-i + ∑_j ∈ψ_kr,jψ_ls,j+ℓ (ε_r,n-jε_s,n-j-1) )wherea_kl(N)=N^d_k+d_l-1, ifk,l ∈ I_L, N^1/2, if k,l ∈ I_S. By <cit.> it is enough to prove tightness of one summanda^-1_kl(N) ∑_n=1^⌊ Nt ⌋∑_j+ℓ≠ i ψ_kr,iψ_ls,jε_r,n-jε_s,n+ℓ-i + a^-1_kl(N) ∑_n=1^⌊ Nt ⌋∑_j ∈ψ_kr,jψ_ls,j+ℓ (ε_r,n-jε_s,n-j-1). Note that for fixed r,s ∈{1,…,p} the first summand is the sample mean process of a univariate bilinear polynomial-form process. The second summand is the sample mean process of a univariate linear process generated by an i.i.d. sequence {ε_r,jε_s,j}_j ∈.In the case k,l ∈ I_S and under the assumption ε_0^5<∞, the first summand is tight by Theorem 3.8 (2.d.) in <cit.>,The second summand is tight by Proposition 4.4.4 in <cit.>.For k,l ∈ I_L, the first summand is tight by Theorem 4.8.2 in <cit.> and the second by Proposition 4.4.4 in <cit.>. §.§.§ Properties of the limit process The next lemma provides some properties of the process {Z(t)}_t ∈ defined in (<ref>).The process {Z(t)}_t ∈ is operator self-similar with scaling family {Δ_c:^p × p →^p × p | c>0}, where Δ_c = c^H⊗ c^H, and has stationary increments. We getI_2(f_H,ct)f.d.d.= (c^H-I_p⊗ c^H-I_p ) ∑_s_1,s_2∈{+,-}∫_^2' ∫_0^ct((v-x_2/c)_s_1^H-I_p M^s_1⊗(v-x_1/c)_s_2^H-I_p M^s_2)(c^H-I_p⊗ c^H-I_p ) ∑_s_1,s_2∈{+,-}∫_^2'× dv (W(dx_1)W'(dx_2) ) f.d.d.= (c^H-1/2I_p⊗ c^H-1/2I_p) ∫_^2 ' f_H,t(x_1,x_2)(W(dcx_1)W'(dcx_2) ) f.d.d.=(c^H⊗ c^H) I_2(f_H,t) ,since W(d(cx))f.d.d.= c^1/2I_pW(dx) and by Theorem 2.2 in <cit.>. Thus, Z(ct)f.d.d.= (c^H⊗ c^H) Z(t). Similarly, we can prove that the process has stationary increments, since for any ℓ∈I_2(f_H,t+ℓ)f.d.d.=∑_s_1,s_2∈{+,-}∫_^2' ∫_-ℓ^t ((v-(x_2-ℓ))_s_1^H-I_pM^s_1⊗ (v-(x_1-ℓ))_s_2^H-I_pM^s_2) ∑_s_1,s_2∈{+,-}∫_^2'× dv (W(dx_1)W'(dx_2) ) f.d.d.=∫_^2' (f_H,t(x_1,x_2)-f_H,-ℓ(x_1,x_2)) (W(d(x_1+ℓ))W'(d(x_2+ℓ))) f.d.d.=I_2(f_H,t)-I_2(f_H,-ℓ),since W(d(x+ℓ))f.d.d.= W(dx). ThusI_2(f_H,t+ℓ)-I_2(f_H,ℓ) f.d.d.= I_2(f_H,t)-I_2(f_H,0) and Z(t+ℓ)-Z(ℓ)f.d.d.= Z(t)-Z(0). Acknowledgements: The author would like to thank the three anonymous referees and the two editors for their comments and their advice that led to a substantial revision and improvement of the original version of this paper. Parts of this work were finalized during a stay in the Department of Statistics and Operation Research at the University of North Carolina, Chapel Hill.The author thanks the department for its hospitality and, in particular, Vladas Pipiras for his support. The author would also like to thank the Research Training Group 2131 - High-dimensional Phenomena in Probability - Fluctuations and Discontinuity for financial support.plainnat | http://arxiv.org/abs/1704.08609v4 | {
"authors": [
"Marie-Christine Düker"
],
"categories": [
"math.PR"
],
"primary_category": "math.PR",
"published": "20170427145549",
"title": "Limit theorems for multivariate long-range dependent processes"
} |
Canonical RDEs and semimartingales t1Affiliated to TU Berlinand supported by DFG research unit FOR2402 when this project was commenced. Currently supported by a Junior Research Fellowship of St John's College, Oxford. t2Partially supported by the European Research Council through CoG-683166 and DFG research unit FOR2402 I. Chevyrev and P. K. FrizUniversity of Oxfordm1, TU Berlinm2 and WIASm3I. ChevyrevMathematical InstituteUniversity of OxfordAndrew Wiles BuildingRadcliffe Observatory QuarterWoodstock RoadOxford OX2 6GGUnited Kingdome1P. K. FrizInstitut für MathematikTechnische Universität BerlinStrasse des 17. Juni 13610623 BerlinGermanyandWeierstraß–Institut für AngewandteAnalysis und StochastikMohrenstrasse 3910117 BerlinGermanye2 In the spirit of Marcus canonical stochastic differential equations, we study a similar notion of rough differential equations (RDEs), notably dropping the assumption of continuity prevalent in the rough path literature. A new metric is exhibited in which the solution map is a continuous function of the driving rough path and a so-called path function, which directly models the effect of the jump on the system.In a second part, we show that general multidimensional semimartingales admit canonically defined rough path lifts. An extension of Lépingle's BDG inequality to this setting is given, and in turn leads to a number of novel limit theorems for semimartingale driven differential equations, both in law and in probability, conveniently phrased via Kurtz–Protter's uniformly-controlled-variations (UCV) condition. A number of examples illustrate the scope of our results. [class=MSC] [Primary ]60H99 [; secondary ]60H10Càdlàg rough paths stochastic and rough differential equations with jumps Marcus canonical equations general semimartingales limit theorems § INTRODUCTION Itô stochastic integrals are well-known to violate a first order chain rule of Newton–Leibniz type, as is manifest from Itô's formula. In a number of applications, is is important to have a chain rule which, in the context of continuous semimartingales, was achieved in a satisfactory way by Stratonovich stochastic integration, which - loosely speaking - replaces left-point evaluation (in Itô–Riemann sums) by a symmetric mid-point evaluation. In the case of stochastic integration against general semimartingales (Lévy processes as an important special case), one can check that the Stratonovich integral no longer gives a chain rule – a more sophisticated approach is necessary to take care of jumps and the mechanism for doing this was developed by Marcus <cit.>. The resulting “Marcus canonical integration” and “Marcus canonical (stochastic differential) equations” (in the terminology of <cit.>) was then investigated in a number of works, including <cit.>, see also <cit.> and the references therein.On the other hand, continuousstochastic integration has been understood for some time in the context of rough path theory, see <cit.>, <cit.>. Loosely speaking, given a multidimensional continuous semimartingale X, the Stratonovich integral ∫ f(X) ∘ dX can be given a robust (pathwise) meaning in terms of = (X, ∫ X ⊗∘ dX), a.e. realization of which constitutes a geometric rough path of finite p-variation for any p>2. In contrast to the popular class of Hölder rough paths (usually sufficient to deal with Brownian motion, see e.g. <cit.>), p-variation has the advantage that it immediately allows for jumps. This also prompts the remark that Young theory, somewhat the origin of Lyons' rough paths, by no means requires continuity. Extensions of rough path theory to a general p-variation setting (for possibly discontinuous paths) were then explored in <cit.>, <cit.> and <cit.>. However, none of these works provided a proper extension of Lyons' main result in rough path analysis: continuity of the solution map as a function of the driving rough path.The first contribution of this paper is exactly that. We introduce a new metric on the space of càdlàg rough paths, and a type of (Marcus) canonical rough differential equation, for which one has the desired stability result. (Experts in la théorie générale des processus will recognize our topology as a p-variation rough paths variant of Skorokhod's strong M_1 topology.) In fact, we reserve the prefix “Marcus” to situations in which jumps only arise in the d-dimensional driving signal, and are handled (in the spirit of Marcus) by connecting X_t- and X_t by a straight line. (As a straight line has no area, this creates no jump in the area.) A “general” rough path (level N, over ^d), however, can have jumps of arbitrary value _t-^-1⊗_t ∈ G^N (^d), and there are many (different) ways to implement Marcus's idea of continuously (parametrized over a fictitious time interval) connecting _t- and _t. This is really a modelling choice, no different than choosing the driving signal and/or the driving vector fields. The notion of a “path function” ϕ helps us to formalize this, and indeed one may view (,ϕ) as the correct/extended rough driver. In the second part of the paper, we show how general (càdlàg) semimartingales fit into the theory. In particular, we show that the canoncial lift of a semimartingale indeed is a.s. a (geometric) rough path of finite p-variation for any p>2 (several special cases, including Lévy processes, were discussed in <cit.> but the general case remained open). Our result is further made quantitative by establishing a BDG inequality for general local martingale rough paths. (We thus extend simultaneously the classical p-variation BDG inequality <cit.>, and its version for continuous local martingale rough paths <cit.>). This BDG inequality turns out to be a powerful tool, especially in conjunction with uniform tightness (UT) and uniformly controlled variation (UCV) of semimartingale sequences. (Introduced by <cit.> and <cit.> respectively, these conditions are at the heart of basic convergence theorems for stochastic integrals in Skorokhod topology; we work only with UCV in this article, but note that UCV and UT are equivalent under extra assumptions, e.g., convergence in law <cit.>. See Section <ref> for the definition of UCV, and also <cit.> for some links to continuous semimartingale rough paths.)As an example of an application to general semimartingale theory, we are able to state a criterion for convergence in law (resp. in probability) of Marcus SDEs, which is an analogue of the celebrated criterion for Itô SDEs due to Kurtz–Protter <cit.> (we emphasize however that neither criterion is a simple consequence of the other). Loosely speaking, the result asserts that if X, (X^n)_n ≥ 1 are ^d-valued semimartingales such that (X^n)_n ≥ 1 satisfies UCV and X^n → X in law (resp. in probability) for the Skorokhod topology, then the solutions to Marcus SDEs driven by X^n (along fixed vector fields) converge in law (resp. in probability) to the Marcus SDE driven by X (see Theorem <ref> for a precise formulation). Our theorem (which crucially involves rough paths in the proof, but not in the statement) entails a pleasantly elegant approach to the Wong–Zakai theorem for SDEs with jumps (Kurtz–Protter–Pardoux <cit.>, with novel interest from physics <cit.>) and in fact gives a number of novel limit theorems for Marcus canonical SDEs (see Theorem <ref>). We remark further that homeomorphism and diffeomorphism properties of solution flows are straightforward, in contrast to rather lengthy and technical considerations required in a classical setting (see, e.g., <cit.> and the references therein). At last, we discuss the impact of more general path functions, noting that the “Marcus choice” really corresponds to the special case of the linear path function.The paper is organized as follows. In Section <ref> we collect some necessary preparatory material, including basic properties of path functions. In Section <ref> we give meaning to canonical RDEs, for which drivers are (rough path, path function) pairs (,ϕ), and introduce the metric α_ for which the direct analogue of Lyons' universal limit theorem holds. Section <ref> is then devoted to applications to càdlàg semimartingale theory, particularly in connection with the UCV condition and Wong–Zakai type approximations. We briefly comment in Section <ref> on the further scope of the theory.§ PREPARATORY MATERIAL §.§ Wiener and Skorokhod spaceThroughout the paper, we denote by C([s,t],E) and D([s,t],E) the space of continuous and càdlàg functions (paths) respectively from an interval [s,t] into a metric space (E,d). Unless otherwise stated, we equip C([s,t],E) and D([s,t],E) respectively with the uniform metricd_∞;[s,t](,) = sup_u∈ [s,t]d(_u,_u)and the Skorokhod metricσ_∞;[s,t] (,) = inf_λ∈Λ_[s,t] |λ| ∨ d_∞;[s,t](∘λ,),where Λ_[s,t] denotes the set of all strictly increasing bijections of [s,t] to itself, and |λ| := sup_u ∈ [s,t]|λ(u) - u|.When we omit the interval [s,t] from our notation, we will always assume it is [0,T].We recall that if E is Polish, then so is the Skorokhod space D([0,T],E) with topology induced by σ_∞, also known as the J_1-topology.We always let = (t_0 = s < t_1 < … < t_k-1 < t_k = t) denote a partition of [s,t], and notation such as ∑_t_i ∈ denotes summation over all points in(possibly without the initial/final point depending on the indexing). We let || = max_t_i ∈ |t_i+1-t_i| denote the mesh-size of a partition.For p > 0, we define the p-variation of a path ∈ D([s,t],E) by_;[s,t] := sup_⊂ [s,t]( ∑_t_i ∈ d(_t_i,_t_i+1)^p )^1/p.We use superscript notation such as D^([s,t],E) to denote subspaces of paths of finite p-variation. For continuousonly p ≥ 1 is interesting, for otherwiseis constant. At least when E = ^d (or G^N(^d), see below) there is an immediate p-variation metric and topology. Due to the fact that convergence in J_1 topology to a continuous limit is equivalent to uniform convergence, a discontinuous path cannot be approximated by a sequence of continuous paths in the metric σ_∞. The same will be true for a J_1/p-variation (rough path) metric σ_ below. That said, we will propose below a useful SM_1/p-variation (rough path) metric α_ under which the space of continuous rough paths is not closed.§.§ Rough paths For N ≥ 1, we let G^N(^d) ⊂ T^N(^d) ≡∑_k=0^N (^d)^⊗ k denote the step-N free nilpotent Lie group over ^d, embedded into the truncated tensor algebra (T^N(^d),+,⊗), which we equip with the Carnot-Carathéodory normand the induced (left-invariant) metric d. Recall that the step-N free nilpotent Lie algebra 𝔤^N(^d) = log G^N (^d)⊂ T^N(^d) is the Lie algebra of G^N(^d). The space C^([0,T],G^N(^d)), with N= p, is the classical space of (continuous, weakly) geometric p-rough paths as introduced by Lyons. Unless otherwise stated, we always suppose a path : [s,t] → G^N(^d) starts from the identity _s = 1_G^N(^d). We denote the increments of a path by _s,t = ^-1_s_t. We consider on D^([s,t], G^N(^d)) the inhomogeneous p-variation metricρ_;[s,t](,) = max_1 ≤ k ≤ Nsup_⊂ [s,t](∑_t_i ∈_t_i,t_i+1^k - _t_i,t_i+1^k^p/k)^k/p.Unless otherwise stated, we shall always assume that p and N satisfy p ≤ N.We let C^0,([s,t], G^N(^d)) denote the closure in C^([s,t], G^N(^d)) under the metric ρ_ of the lifts of smooth paths C^∞([s,t],^d). Recall in particular that C^0,([s,t], G^N(^d)) is precisely the space of absolutely continuous paths.We let V = (V_1,…, V_d) denote a collection of vector fields in^γ+m-1(^e) with γ > p and m ≥ 1. For a geometric p-rough path ∈ C^([s,t], G^N(^d)), we let π_(V)(s,y_s;) ∈ C^([s,t],^e) denote the solution to the RDEdy_t = V(y_t) d_t,y_s ∈^e.We let U^_t← s : ^e →^e denote the associated flow map y ↦π_(V)(s,y;)_t, which we recall is an element of ^m(^e). For further details on the theory of (continuous) rough paths theory, we refer to <cit.>.For the purpose of his paper we have (cf. <cit.>) Let 1 ≤ p < N+1. Any ∈ D^([0,T],G^N(^d)) is called a general (càdlàg, weakly) geometric p-rough path overℝ^d. Define Δ_t := _t-^-1⊗_t and sayis Marcus-like iffor all t ∈ [0,T]logΔ_t ∈ℝ^d⊕{ 0}⊕ ... ⊕{ 0}⊂𝔤^N(ℝ^d),where log is taken in T^N(^d).As we will see later, any canonical lift of a general d-dimensional semimartingale X (with area given by 12∫ [ X^-, dX]) gives rise to a Marcus-like general geometric p-rough path for p>2. The model case of Lévy processes was studied in <cit.>.§.§ Path functionsWe briefly review and elaborate on the concept of a path function introduced in <cit.>. Let (E,d) be a metric space. A path function on E is a map ϕ : J → C([0,1],E) defined on a subset J ⊆ E× E for which ϕ(x,y)_0 = x and ϕ(x,y)_1 = y for all (x,y) ∈ J.For a path ∈ D([0,T], E), we say that t ∈ [0,T] is a jump time ofif _t-≠_t. We call the pair (,ϕ) admissible if (_t-,_t) ∈ J for all jumps times t of . We say that two admissible pairs (,ϕ) and (,ϕ̅) are equivalent, and write (,ϕ) ∼ (,ϕ̅), if = and ϕ(_t-,_t) is a reparametrization of ϕ̅(_t-,_t).We denote by ([0,T],E) the set of all admissible pairs (,ϕ), and by ([0,T],E) = ([0,T],E)/∼ the set of all equivalence classes of admissible pairs. For a fixed path function ϕ, let _ϕ([0,T],E) denote the set of all ∈ D([0,T], E) such that (,ϕ) is admissible. We will often simply say that ϕ is a path function on E and keep implicit the fact there is an underlying domain of definition J. We point out that situations where J ≠ E× E arise naturally when studying solution maps of canonical càdlàg RDEs, see Theorem <ref> and the discussion before it.In the case that E is a Lie group with identity element 1_E (taken in this article to always be G^N(^d)), we shall often assume that ϕ is left-invariant, which is to say that there exists a subset B ⊆ E such that J = {(x,y) ∈ E× Ex^-1y ∈ B} andϕ(x,y)_t = xϕ(1_E,x^-1y)_t,∀ (x,y) ∈ J,∀ t ∈ [0,1].In this case, it is equivalent to view ϕ as a map ϕ : B → C([0,1],E) such that ϕ(x)_0 = 1_E and ϕ(x)_1 = x for all x ∈ B, for which ϕ(x,y)_t = xϕ(x^-1y)_t. Whenever we write ϕ with only one argument as ϕ(x), we shall always mean that it is left-invariant.[log-linear and Marcus path function]The prototypical example of a (left-invariant) path function ϕ on G^N(^d), which we shall often refer to in the paper, is the log-linear path functionϕ(x)_t = e^tlog x, ∀ x ∈ G^N(^d),∀ t ∈ [0,1],where log is taken in T^N(^d). Since J = G^N(^d) × G^N(^d), we have (,ϕ) is admissible for all ∈ D([0,T],G^N(^d)). When N=1 we see a familiar special case: since G^1(^d) ≅^d, one has ϕ(x)_t = t x, and then ϕ (x,y) = x + t(y - x). This is precisely the “Marcus interpolation” of a càdlàg path before and after its jump. See parametric plots in Figures <ref> and <ref>. For (,ϕ) ∈([0,T],E) we now construct a continuous path ^ϕ∈ C([0,T], E) as follows. Fix a convergent series of strictly positive numbers ∑_k=1^∞ r_k. Let t_1, t_2, … be the jump times ofordered so that d(_t_1-,_t_1) ≥ d(_t_2-,_t_2) ≥…, and t_j < t_j+1 if d(_t_j-,_t_j) = d(_t_j+1-,_t_j+1). Let 0 ≤ m ≤∞ be the number of jumps of .Let r = ∑_k=1^m r_k and define the strictly increasing (càdlàg) functionτ : [0,T] → [0, T+r], τ(t) = t + ∑_k=1^m r_kt_k ≤ t.Note that τ(t-) < τ(t) if and only if t = t_k for some 1 ≤ k < m+1. Moreover, note that the interval [τ(t_k-), τ(t_k)) is of length r_k.Define ∈ C([0,T+r],E) by_s = _t ϕ(_t_k-, _t_k)_(s-τ(t_k-))/r_k .Denote by τ_r(t) = t(T+r)/T the increasing linear bijection from [0,T] to [0,T+r]. We finally define^ϕ = ∘τ_r ∈ C([0,T], E).We note that one can recover = ^ϕ∘τ_ via the time changeτ_ := τ_r^-1∘τ,for which it holds thatsup_t ∈ [0,T]|τ_(t) - t| ≤∑_k=1^∞ r_k. The construction of ^ϕ involves an ad-hoc choice, namely the sequence (r_n) and the increasing bijection τ_r. If ^ϕ is constructed similarly, but via a sequence (r̅_n), followed by another reparametrisation given by τ̅_r̅, then ^ϕ and ^ϕ are reparametrizations of one another.The construction above is similar to ones appearing in <cit.>, and is a simplification of the construction in <cit.>. The primary difference is that in <cit.> the added fictitious time r_k for the jump t_k depended further on the size of the jump d(_t_k-,_t_k). This extra dependence was used to show continuity of the map ↦^ϕ from D([0,T],E) → C([0,T],E), which we will not require here.§.§ A generalisation of Skorokhod's SM_1 topologyFor (,ϕ) ∈([0,T],E) and δ > 0, let ^ϕ,δ∈ C([0,T],E) be constructed in the same procedure as ^ϕ but using the series ∑_k=1^∞δ r_k instead of ∑_k=1^∞ r_k. For all (,ϕ), (, ϕ̅) ∈([0,T],E), the limit lim_δ→ 0σ_∞;[0,T] (^ϕ,δ,^ϕ̅,δ)exists, is independent of the choice of series ∑_k=1^∞ r_k, and induces a pseudometric on the set of equivalence classes ([0,T],E).To show that the limit exists, note that for every δ, δ̅> 0, there exists λ∈Λ such that |λ| < 2(δ + δ̅)∑ r_k and ^ϕ,δ̅ = ^ϕ,δ∘λ. Since |λ∘λ̅| ≤ |λ| + |λ̅|, it follows that for every δ, δ̅> 0|σ_∞(^ϕ,δ,^ϕ̅,δ) - σ_∞(^ϕ,δ̅, ^ϕ̅,δ̅)| < 4(δ̅+ δ)∑ r_k,from which the existence of the limit follows. The fact that (<ref>) is independent of the series ∑ r_k and is zero if (,ϕ)∼ (,ϕ̅) is straightforward. Define the pseudometric α_∞ on ([0,T],E) byα_∞(,) := α_∞;[0,T]((, ϕ), (, ϕ̅)) := lim_δ→ 0σ_∞;[0,T] (^ϕ,δ,^ϕ̅,δ).(Usually no confusion will arise by using the abusive notation on the left-hand side.) Note that for any fixed ϕ, α_∞ induces a genuine metric on the space _ϕ([0,T],E) ⊆ D([0,T],E). We note also that the strong M_1 (i.e., SM_1) topology on the space D([0,T],^d) is a special case of the topology induced by the metric α_∞, as demonstrated by the following result. For E=^d and ϕ the linear path function, it holds that α_∞ induces the SM_1 topology on the space D([0,T],^d). It is straightforward to verify that α_∞ in this case is equivalent to the metric d_s (see <cit.>) which induces the SM_1 topology.The reader may wonder if convergence in the (Skorokhod J_1) metric σ_∞ implies, as in the classical setting, convergence in the (Skorokhod SM_1-type) pseudometric α_∞. In essence, the answer is yes, however this requires “reasonable” path functions, see Lemma <ref>. It is trivial to see that uniform convergence of paths on [0,T] implies convergence on any subinterval of [0,T], while this fails for both Skorokhod J_1 and (S)M_1 metrics. The observation generalizes to our setting and, in particular, the α_∞ metric does not behave well under restriction. Indeed, while for any (,ϕ) ∈([0,T],E) the jumps of |_[s,t] still belong to J, so that (|_[s,t])^ϕ is well-defined, it does not hold that α_∞(^n,) → 0 implies that α_∞(^n|_[s,t],|_[s,t]) → 0. We now collect several useful definitions and lemmas concerning path functions. Let (,ϕ) ∈([0,T],E) for whichlim_n →∞sup_s ∈ [0,1] d(_t_n-, ϕ(_t_n-,_t_n)_s) = 0,where the limit is taken over some enumeration of jump times of . Let (^k,ϕ^k)_k ≥ 1 be a sequence in ([0,T],E) such that α_∞(^k,) → 0. Then ^k_t →_t for every continuity point of .Note that condition (<ref>) is satisfied whenever ϕ is either endpoint continuous or (,ϕ) has finite p-variation (see Definitions <ref> and <ref> below). In particular, since càdlàg paths are uniquely determined by their continuity points, it follows from Lemma <ref> that α_∞ is a genuine metric on the space ^([0,T],E) introduced in Definition <ref>.Suppose t is a continuity point of . Then using (<ref>) and the definition of α_∞, it holds that for every ε>0 there exists δ> 0 such that for all k sufficiently large and δ_k sufficiently small we havesup_s ∈ [t-δ,t+δ]d(_t, (^k)^ϕ_k,δ_k_s) < ε,from which the conclusion follows.For p ≥ 1, we define the p-variation of (,ϕ) ∈([0,T],E) as(,ϕ)_;[0,T] := ^ϕ_;[0,T]and let ^([0,T],E) denote all (,ϕ) ∈([0,T],E) of finite p-variation.Moreover, a path function ϕ : J → C([0,1], E) is called p-approximating if there exists a function η_ : [0,∞) → [1,∞) such that for all r ∈ [0,∞)sup_(x,y) ∈ J; d(x,y) ≤ rϕ(x,y)_; [0,1]≤η_(r) d(x,y).We say that η_ is a p-variation modulus of ϕ. Due to the invariance of p-variation norms under reparametrizations, and our previous Remark <ref>, we see that there is no ambiguity in the definition of (,ϕ)_;[0,T] and that ^([0,T],E) is well-defined. The following lemma gives a simple criterion for a pair (,ϕ) ∈([0,T],E) to have finite p-variation. Let p ≥ 1 and set R = 1+2^p + 3^p-1. Then for every (,ϕ) ∈([0,T],E), it holds that_;[0,T]^p ∨(∑_tϕ(_t-,_t)_;[0,1]^p ) ≤^ϕ_;[0,T]^p≤ R_;[0,T]^p + (R+3^p-1)∑_tϕ(_t-,_t)_;[0,1]^p,where the summations are over the jump times of .In particular, if ϕ has a p-variation modulus η_, then for all ∈_ϕ([0,T],E),^ϕ_;[0,T]^p ≤[R + η_(r)^p(R + 3^p-1) ] _;[0,T]^p,where r = sup_t ∈ [0,T] d(_t-,_t). A path function ϕ : J → C([0,1], E) is called endpoint continuous if * (x,x) ∈ J whenever (x,y) ∈ J, * ϕ(x,x) ≡ x for all (x,x) ∈ J, and * ϕ is continuous with C([0,1],E) equipped with the uniform topology.Moreover, we say that a function η_∞ : [0,∞) → [0,∞) is a uniform modulus of ϕ if η_∞(r) ≥ r for all r ≥ 0, lim_r → 0η_∞(r) = η_∞(0)= 0, and for all (x,y), (x̅,y̅) ∈ Jd_∞;[0,1](ϕ(x,y), ϕ(x̅, y̅)) ≤η_∞(max{d(x,x̅), d(y,y̅)}). In general, it is hard to find an explicit uniform modulus of a path function (or even show that one exists). But evidently if ϕ is restricted to J ∩ (K × K) for a compact K ⊆ E, then a uniform modulus exists whenever ϕ is endpoint continuous.Let ϕ be the log-linear path function on G^N(^d). Then clearly ϕ is endpoint continuous and there exists a constant C ≥ 1 such that for all p ≥ N and x,y ∈ G^N(^d)ϕ(x,y)_;[0,1]≤ C d(x,y),so that the constant C is a p-variation modulus of ϕ.Suppose ϕ has a uniform modulus η_∞. Then for all ,∈_ϕ([0,T],E), it holds that α_∞(,) ≤η_∞(σ_∞(,)).Suppose there exists λ∈Λ such that |λ| < r and d_∞(,∘λ) < r. Then for all δ > 0 sufficiently small there exists λ_δ∈Λ such that |λ_δ| < r and d_∞(^ϕ,δ,^ϕ,δ∘λ_δ) < η_∞(r), and the conclusion follows. § CANONICAL RDES DRIVEN BY GENERAL ROUGH PATHSTo ease notation, we assume throughout this section that all path spaces, unless otherwise stated, are defined on the interval [0,T] and take values in G^N(^d). For example ^ will be shorthand for ^([0,T],G^N(^d)). §.§ Notion of solutionFollowing the notation of Section <ref>, let 1 ≤ p < N+1 and fix a family of vector fields V = (V_1,…, V_d) in ^γ+m-1(^e) for some γ > p and m ≥ 1. For ∈ D^, we would like to solve the RDE“ dy_t = V(y_t)d_t ” .Our notion of solution to this equation will depend on a path function ϕ defined on a subset J ⊆ G^N(^d) × G^N(^d), and therefore the fundamental input to an RDE will be a pair (,ϕ) ∈^.Consider (,ϕ) ∈^ and let ỹ∈ C^([0,T],^e) be the solution to the continuous RDEdỹ_t = V(ỹ_t)d_t^ϕ, ỹ_0 = y_0 ∈^e.We define the solution y ∈ D^([0,T],^e) to the canonical RDEd y_t = V(y_t)♢ d(_t,ϕ),y_0 ∈^e,by y = ỹ∘τ_ (where τ_ is given by (<ref>)).In the particular case that ϕ is the log-linear path function from Example <ref>, we denote the RDE simply bydy_t = V(y_t) ♢ d_t,y_0 ∈^e. While the continuous RDE solution ỹ clearly depends (up to reparametrization) on the choice of representative (,ϕ) ∈^ as well as the choice of (r_k), it is easy to see that y is independent of these choices, and is therefore well-defined on ^.Observe that every continuity point t ofis also a continuity point τ_, and is therefore also a continuity point of y.For the log-linear path function ϕ, the solution y agrees precisely with the solution to the rough canonical equation considered in <cit.>. Furthermore, we shall see in Section <ref> that all semimartingales admit a canonical lift to a càdlàg geometric p-rough path, and that, for the log-linear path function, the solution y agrees with the Marcus solution of the associated SDE (see Proposition <ref> below).§.§ Skorokhod-type p-variation metric We now introduce a metric α_ on ^ for which the RDE solution map is locally Lipschitz continuous. We first define an auxiliary metric σ_ on D^ which is independent of any path function. Recall the inhomogeneous p-variation metric ρ_ from Section <ref>. For p ≥ 1 and ,∈ D^, defineσ_(,) = inf_λ∈Λmax{|λ|, ρ_(∘λ, )}. Note that σ_ and ρ_ induce the same topology on C^0,. Indeed, it is sufficient to show that ρ_(,∘λ^n) → 0 for all ∈ C^0, and |λ^n| → 0, which follows from writing _t = ∫_0^t _s ds and applying dominated convergence.Furthermore, for p' > p ≥ 1, σ_ and ρ_ induce the same topology on C^. Indeed, it again suffices to show that ρ_(,∘λ^n) → 0 for all ∈ C^ and |λ^n| → 0, which follows from d_∞;[0,T](, ∘λ^n) → 0 and interpolation <cit.>.However, note that σ_ and ρ_ do not induce the same topology on C^. This can be seen from the fact that C^0, is dense in C^ under σ_ (see Proposition <ref> part <ref>), or from the following direct example: consider the -valued Cantor function _t = μ([0,t]), where μ is the Cantor distribution, and shifts ^n_t = _t-α_n (with ^n_t = 0 for t ∈ [0,α_n]), where α_n ↓ 0. Clearly σ_(,^n) ≤α_n ∨_;[0,α_n]→ 0. However, choosing α_n irrational, one can show that μ and μ(· - α_n) are mutually singular measures (see, e.g., <cit.>), so that ρ_(,^n) = ^n_;[0,1] + _;[0,1]→ 2.We note that for the case 1 = N ≤ p < 2, the metric σ_ already appears in the works of Simon <cit.> and Williams <cit.> where in particular a continuity statement for RDE solutions in the Young regime appears in terms of σ_ (cf. Remark <ref>).A drawback of the metric σ_ is that the space of continuous rough paths C^ is closed under σ_. In particular, this implies that σ_ is unable to describe situations in which continuous drivers approximate a discontinuous one (e.g. the Wong–Zakai theorem in <cit.>). We are thus motivated to introduce the following metric whose relation with σ_ is analogous to that of α_∞ with σ_∞. For (,ϕ), (,ϕ̅) ∈^, define the metricα_(,) = lim_δ→ 0σ_(^ϕ,δ,^ϕ̅,δ),where ^ϕ,δ is defined as at the start of Section <ref>.Note that the limit (<ref>) exists, is independent of the choice of series ∑_k=1^∞ r_k, and induces a well-defined metric on ^, all of which follows from the same argument as in Lemma <ref> and Remark <ref>.In light of Remark <ref>, it may seem possible to define an equivalent topology as that induced by α_ (at least on C^0,) by replacing σ_ by ρ_ in (<ref>) for the definition of α_ (and thus avoid introducing σ_ altogether). However one can readily check that doing so will induce a completely different topology even on C^0, (in fact the same remark applies to replacing σ_∞ by d_∞ when defining α_∞ in Definition <ref>). We record several basic properties of the metric space (^, α_). For a path function ϕ, let _ϕ^0, = _ϕ^0,([0,T],G^N(^d)) denote the closure of C^0, in the metric space (_ϕ^, α_). Let p ≥ 1 and ϕ a path function defined on a subset J ⊆ G^N(^d)× G^N(^d).*The space (^0,_ϕ, α_) is a separable metric space. *It holds that C^ is dense in (_ϕ^, α_). *It holds that _ϕ^0, = _ϕ^. *If p > 1, the closure of C^0, in (C^, σ_) is precisely C^0,. In particular, _ϕ^0,⊊_ϕ^. *For every p' > p, _ϕ^⊊_ϕ^0,. <ref> Recall that (C^0,,ρ_) is a separable space, and therefore so is (C^0,,σ_) (see Remark <ref>). Since the metrics σ_ and α_ coincide on C^0,, it follows that _ϕ^0, is separable.<ref> For every ∈_ϕ^ and δ > 0, it holds that ^ϕ, δ∈ C^([0,T],G^N(^d)). One can readily see that lim_δ→ 0α_(, ^ϕ, δ) = 0, from which the claim follows.<ref> By point <ref>, it suffices to show that C^0, is dense in C^ under σ_. This in turn follows from the fact that any ∈ C^([s,t],G^N(^d)) can be reparametrized to be in C^([s,t],G^N(^d)) ≅ L^∞([s,t],^d) (where the isometry is via the weak derivative) and thus lies in C^0,([s,t],G^N(^d)) ≅ L^1([s,t],^d).<ref> Recall by Wiener's characterization <cit.> (which relies on p>1) that ∈ C^0, if and only if lim_δ→ 0sup_||<δ∑_t_i ∈_;[t_i,t_i+1]^p = 0.As a consequence, for any λ∈Λ, it holds that ∘λ∈ C^0, if and only if ∈ C^0,. Therefore, for any sequences (^n)_n ≥ 1 in C^0, and (λ^n)_n ≥ 1 in Λ for which ρ_(,^n∘λ_n) → 0, it holds that ∈ C^0,, from which the conclusion follows.<ref> Since C^⊊ C^0,, the conclusion follows from <ref>. We now record an interpolation estimate which will be helpful later. It turns out to be simpler to state in terms of a homogeneous version of the distance α_. See <cit.> for the definition and basic properties of the metrics d_0 and d_. For (,ϕ),(,ϕ̅) ∈^ we defineβ_(,) = lim_δ→ 0inf_λ∈Λ |λ| ∨ d_(^ϕ,δ∘λ, ^ϕ̅,δ)as well asα_0(,) = lim_δ→ 0inf_λ∈Λ |λ| ∨ d_0(^ϕ,δ∘λ, ^ϕ̅,δ)(which are well-defined metrics on ^ by the same argument as in Lemma <ref> and Remark <ref>). Observe that the d_0/d_∞ estimate <cit.> impliesα_∞(,) ≤α_0(,) ≤ Cmax{α_∞(,), α_∞(,)^1/N( ^ϕ_∞ + ^ϕ̅_∞)^1-1/N}.Moreover, to move from β_ to α_, it holds that the identity map𝕀 : (^, β_) ↔ (^, α_)is Lipschitz on bounded sets in the → direction, and 1/N-Hölder on bounded sets in the ← direction <cit.>.Finally, the following result now follows directly from the usual interpolation estimate for the homogeneous metric d_ <cit.>. Let 1 ≤ p < p'. For all (,ϕ),(,ϕ̅) ∈^ it holds thatβ_(,) ≤(^ϕ_ + ^ϕ̅_)^p'/pα_0(,)^1-p'/p.As a consequence, we obtain the following useful embedding result. Let 1 ≤ p < p' and ϕ a p-approximating, endpoint continuous path function defined on a subset J ⊂ G^N(^d) × G^N(^d). Then the identity map𝕀 : (_ϕ^, σ_∞) → (_ϕ^, α_)is uniformly continuous on sets of bounded p-variation.This is a combination of Remark <ref> and Lemmas <ref>, <ref>, and <ref>.§.§ Continuity of the solution map An advantage of the metric α_ is it allows us to directly carry over continuity statements about the classical (continuous) RDE solution map to the discontinuous setting. Recall the RDE (<ref>)dy_t = V(y_t)♢ d(_t,ϕ),y_0 ∈^e,which is well-defined for any admissible pair (,ϕ) ∈^.Consider the driver-solution space E := G^N(^d) ×^e. Given that ϕ is defined on J ⊂ G^N(^d) × G^N(^d), we obtain a natural path function ϕ_(V) defined on the following (necessarily strict)subset of E × EW := { ((x,y),(x̅,y̅)) | (x,x̅) ∈ J, π_(V)(0,y;ϕ(x,x̅))_1 = y̅}and which is given byϕ_(V)((x,y),(x̅, y̅))_t = (ϕ(x,x̅)_t, π_(V)(0,y;ϕ(x,x̅))_t),where we recall π_(V) from Section <ref> is the solution map for the (continuous) RDE driven along V. Therefore, an admissible pair (,ϕ) ∈^ yields an admissible pair ((,y), ϕ_(V)) ∈^([0,T],E) as the solution to the RDE. The following is now a consequence of Lyons' classical rough path universal limit theorem. For vector fields V=(V_1,…, V_d) in ^γ+m-1(^e) with γ > p and m ≥ 1, the solution map of the RDE (<ref>)^e ×(^, α_)→( ^([0,T],E), α_) (y_0, (,ϕ))↦ ((,y),ϕ_(V))is locally Lipschitz. In particular,lim_n →∞ |y_0^n - y_0| + α_(^n,) = 0 implies thatsup_n y^n_ < ∞, andlim_n →∞ y^n_t = y_t for all continuity points t of .Furthermore the flow map(_ϕ^, α_) →^m(^e) ↦ U^_T← 0≡ U^^ϕ_T ← 0is uniformly continuous on sets of bounded p-variation (see Section <ref> for the definition of U^^ϕ_T ← 0).Note that one cannot replace U^^n_T ← 0 by U^^n_t ← 0 for any (fixed) t ∈ [0,T] in the final statement of Theorem <ref>. This is a manifestation of the fact that α_ does not behave well under restrictions to subintervals [0,t] ⊂ [0,T] (cf. Remark <ref>).Though we don't address this here, the Lipschitz constant appearing in Theorem <ref> can be made to depend explicitly on V and the p-variation of ^ϕ.Note that in the second statement of Theorem <ref>, one cannot replaceby y in “for all continuity points t of ”. Note also that this type of convergence is the one considered in the Wong–Zakai theorem of <cit.>.The claim that the solution map is locally Lipschitz and that the associated flows converge follows from the corresponding result for continuous rough paths (see, e.g., <cit.>).To make this explicit, consider 𝐱 with path function ϕ and then z=( 𝐱,y) with path function ϕ _( V). Write also z^δ=z^ϕ_( V) ,δ and 𝐱 ^δ= 𝐱^ϕ ,δ. By definitionα _p-var( z_1,z_2) = lim_δ→ 0inf_λ∈Λmax{|λ| ,ρ _p-var( z_1^δ∘λ ,z_2^δ) }= lim_δ→ 0inf_λ∈Λmax{|λ| ,ρ _p-var( 𝐱 _1^δ∘λ ,𝐱_2^δ) +| y_1^δ∘λ -y_2^δ|}.On the other hand, for i=1,2, we have 𝐱_i^δ=𝐱 _i^ϕ _i,δ∈ C^p-var. Write ỹ_i=π _( V) (0,y_i,0,𝐱_i^δ) for the unique RDE solution to dy=V( y) d𝐱_i^δ and note that ỹ _i≡ y_i^δ, by the very definition of the path function ϕ _(V). Note also that y_i^δ∘λ≡ỹ^i∘λ =π _( V) (0,y_0^i,𝐱_i^δ∘λ )for every time change λ∈Λ. It is then a direct consequence of the (local Lipschitz) continuity of the Itô-Lyons map, in the setting of continuous rough paths, that | y_1^δ∘λ -y_2^δ| _p-var;[ 0,T] ≲ρ _p-var( 𝐱 _1^δ∘λ ,𝐱_2^δ) +| y_1,0-y_2,0|.As a consequence,ρ _p-var( z_1^δ∘λ ,z_2^δ) ≲ρ _p-var( 𝐱_1^δ∘λ ,𝐱_2^δ) +| y_1,0-y_2,0|.Finally, take lim_δ→ 0inf_λ∈Λmax( |λ| ,·) on both sides to see thatα _p-var( z_1,z_2) ≲α _p-var( 𝐱_1,𝐱_2) +| y_1,0-y_2,0| .The claim of a.e. pointwise convergence follows from Lemma <ref>, while uniform continuity of (,ϕ) ↦ U^_T ← 0 follows as above (cf. <cit.>).An important feature of solutions to continuous RDEs (<ref>) is that they can be canonically treated as geometric p-rough paths ∈ C^([0,T], G^N(^e)), and the solution map (_0,) ↦ remains locally Lipschitz for the metric ρ_ (see <cit.>). This allows one to use the solutionas the driving signal in a secondary RDEdz_t = W(z_t) d_t,z_0 ∈^n.At least with the notion of canonical RDEs considered in this article, we cannot expect this functorial nature to be completely preserved due to the fact that in general no path function ψ can be defined on G^N(^e) to capture the information of howtraversed a jump (_t-,_t) (which, in our context, clearly impacts the solution of (<ref>)).However, at the cost of retaining the original driving signal , we can readily solve the secondary RDE (<ref>). Indeed, for ^γ vector fields V=(V_1,…, V_d) on ^e and W=(W_1,…, W_e) on ^n, we consider the ^γ vector fields U=(U_1,…,U_d) on ^e⊕^n defined by U_i(y,z) = V_i(y) + ∑_k=1^e W_k(z) V_i(y)^k, where V_i(y)^k is the k-th coordinate of V_i(y) ∈^e. Then the natural solution to (<ref>) is given by the larger RDEd(y_t,z_t) = U(y_t,z_t) ♢ d(_t,ϕ),(y_0,z_0) ∈^e ⊕^n.As a consistency check, one can readily see that, in the continuous setting, the second component of the solution to (<ref>) coincides with the solution z_t of (<ref>). If we further assume that drivers converge in Skorokhod topology, then more can be said about convergence of RDE solutions. Let notation be as in Theorem <ref>.*Suppose that (<ref>) holds and that lim_n →∞^n = in the Skorokhod topology. Then lim_n →∞ y^n = y in the Skorokhod topology. *Suppose that ϕ is an endpoint continuous, p-approximating path function defined on (a subset of) G^N(^d) × G^N(^d). Then on sets of bounded p-variation, the solution map^e ×(_ϕ^, σ_∞) →( D^([0,T],^e), σ_∞) (y_0, )↦ yis continuous. <ref> By Theorem <ref>, it suffices to show that (y^n)_n ≥ 1 is compact in the Skorokhod space D([0,T],^e). Recall that for a Polish space E, a subset A⊂ D([0,T],E) is compact if and only if {_t |∈ A, t ∈ [0,T]} is compact andlim_ε→ 0sup_∈ Aω'_(ε) = 0,where ω'_(ε) := inf_||_min > εmax_t_i ∈sup_s,t ∈ [t_i,t_i+1) d(_s,_t),||_min := min_t_i ∈|t_i+1 - t_i|,see, e.g., <cit.>. In particular, using that ^ϕ_< ∞ and applying the first inequality in Lemma <ref> to , we see that lim_ε→ 0lim_δ→ 0ω^_^ϕ,δ(ε)=0, whereω^_(ε) := inf_||_min > εmax_t_i ∈_;[t_i,t_i+1).It now follows from (<ref>) thatlim_ε→ 0sup_n lim_δ→ 0ω^_(^n)^ϕ^n,δ(ε) = 0,from which we see that lim_ε→ 0sup_n ω^_y^n(ε) = 0. Since ω'_y(ε) ≤ω^_y(ε), it follows that (y^n)_n ≥ 1 is compact in D([0,T],^e) as required.<ref> This follows directly from taking p < p' < γ and applying <ref> and Proposition <ref>.We suspect that under suitable conditions on a fixed path function ϕ, the solution map _ϕ^→ D^([0,T],^e) remains locally uniformly (or even Lipschitz) continuous in the classical p-variation metric ρ_ defined by (<ref>) (and thus trivially for the metric σ_). However there seems to be no easy way to derive this as a consequence of our main Theorem <ref>. We suspect that this could be done by carefully relating canonical with “Itô-type”non-canonical RDEs (see <cit.>), followed by a proper stability analysis of the latter. The study of such non-canonical equations was recently carried out in <cit.>.§.§ Young pairing and translation operator In this section we extend the Young pairing S_N(,h) and rough path translation operator T_h() to the càdlàg setting.Given a path (x,h) in ^d+d' (smooth), its canonical level-2 rough path lift is given by ((x,h), (∫ x ⊗ dx, ∫ x ⊗ dh, ∫ h ⊗ dx, ∫ h⊗ dh)). This extends immediately to a p-rough path , for p ∈ (2,3), with ∫ x ⊗ dx replaced by the a priori level-2 information ^2. For h of finite q-variation with 1/p+1/q > 1, all other cross-integrals remain defined. This is the Young pairing of a p-rough pathwith a q-variation path h called S_2 (, h). The mapping (,h) ↦ S_2(,h) is continuous, and the general construction, for continuous (rough) paths, is found in <cit.>. The important operation of translating rough paths in some h-direction, formally ∫ (+h) ⊗ d(+h), can be algebraically formulated in terms of the Young pairing and has found many applications <cit.>.We first illustrate the difficulty of continuously extending S_N(,h) and T_h() to càdlàg paths by showing that the addition map (,h) ↦+h is not continuous as a map ^([0,T],^d) ×^([0,T],^d) →([0,T],^d) where we equip ^([0,T],^d) and ([0,T],^d) with α_ and α_∞ respectively (which is reminiscent of the fact that D([0,T],^d) is not a topological vector space). Consider the sequences (^n)_n ≥ 1, (h^n)_n ≥ 1, and (h̅^n)_n ≥ 1, of continuous piecewise linear paths from [0,2] to ^2, where ^n is constant on [0,1-1/n], moves linearly from 0 to e_1 over [1-1/n,1] and remains constant on [1,2], while h^n and h̅^n do the same, except from 0 to e_2 and over the intervals [1-2/n,1-1/n] and [1,1+1/n] respectively. One can see that ^n converges to (t ≥ 1 e_1, ϕ) in α_, and h^n and h̅^n both converge to (t ≥ 1 e_2, ϕ) in α_, where ϕ is the linear path function on ^2. However ^n + h^n converges to (t ≥ 1(e_1+e_2), ϕ_2,1) while ^n + h̅^n converges to (t ≥ 1(e_1+e_2), ϕ_1,2) in α_∞, where ϕ_i,j is the “Hoff” path function which moves first in the i-th coordinate, and then in the j-th coordinate. Note that the limiting path in the above example is unambiguously defined (namely t ≥ 1(e_1+e_2)) whereas the corresponding path function is not. In the following definition we circumvent thisproblem by choosing a priori the (log-)linear path function. Let p ≥ 1 and 1 ≤ q < 2 such that 1/p + 1/q > 1. For integers d,d' ≥ 1, let ϕ and ϕ̅ be the log-linear path functions on G^N(^d+d') and G^N(^d)×^d' respectively.For ∈ D^([0,T],G^N(^d)) and h ∈ D^([0,T],^d'), consider the continuous G^N(^d)×^d'-valued path (,ĥ) = (,h)^ϕ̅. We define (S_N(,h),ϕ) ∈^([0,T],G^N(^d+d')) byS_N(,h) = S_N(,ĥ) ∘τ_(,h),where we recall that τ_(,h) is defined by (<ref>) for the càdlàg path (,h). In the case that d=d', we let ϕ̂ denote the log-linear path function on G^N(^d) and define (T_h(),ϕ̂) ∈^([0,T],G^N(^d)) by T_h() = T_ĥ() ∘τ_(,h).Due to the choice of log-linear path function, note thatand ĥ have finite p- and q-variation respectively, so that S_N(,ĥ) and T_ĥ() are well-defined as continuous rough paths. Moreover, we have the relation T_h() = ( S_N(,h)) (see <cit.>). We now record a simple result in the case N=2 which will be helpful in the remainder of the paper. Recall from Definition <ref> that ∈ D^([0,T],G^2(^d)) is called Marcus-like if log(Δ_t) ∈^d⊕{0}⊂^2(^d), i.e., = exp(x, A) where exp is taken in T^2(^d) and A_t-,t = 0 for all t ∈ [0,T].Let notation be as in Definition <ref>. Suppose that ∈ D^([0,T],G^2(^d)) is Marcus-like. Then for every h ∈ D^([0,T],^d') it holds that S_2(,h) is a weakly geometric Marcus-like p-rough path with the anti-symmetric part of its second level determined for i∈{1,…, d} and i' ∈{1…, d'} byS_2(,h)^i,d+i'_s,t - S_2(,h)^d+i',i_s,t = ∫_(s,t] x^i_s,u-⊗ dh^i'_u - h_s,u- ^i'⊗ dx^i_u,where the integrals are well-defined Young integrals (all other components of the anti-symmetric part are given canonically byand h). FurthermoreS_2(,h)^ϕ_∨(T_h()^ϕ̂_≲(,h)^ϕ̅_≲_ + h_. By definition of := (Z,) := S_2(,h) and the choice of log-linear path functions, we see thatis indeed Marcus-like. Observe that := S_2(,ĥ) satisfies the linear RDE in T^N(^d+d') driven by itself d_t = _t⊗ d. It follows by <cit.> and the definition of τ_(,h) thatsatisfies the canonical rough equation d_t = _t⊗♢ d_t (in the sense of <cit.>), so that_t= _0 + ∫_0^t _s-⊗ d_s + ∑_0<s≤ t_s-⊗exp(Δ Z_s) - _s- - _s-⊗Δ Z_s = 1_G^N(^d+d') + ∫_0^t _s-⊗ d_s + ∑_0<s≤ t1/2(Δ Z_s)^⊗ 2,where we have used thatis Marcus-like. In particular, the last term on the RHS only contributes to the symmetric part of , so that (<ref>) follows by identifying ∫_0^t_s-⊗ d_s in the appropriate components with well-defined Young integrals. The inequality (<ref>) follows from standard (continuous) rough path estimates <cit.> along with the fact thatis Marcus-like (so that upon appropriately restricting domains, we may assume ϕ, ϕ̅, and ϕ̂ are 1-approximating). § GENERAL MULTIDIMENSIONAL SEMIMARTINGALES AS ROUGH PATHS §.§ Enhanced p-variation BDG inequality The main result of this subsection is the BDG inequality Theorem <ref> for enhanced càdlàg local martingales. The proof largely follows a classical argument found in <cit.> with the exception of Lemma <ref> which constitutes our main novel input.Let X be an ^d-valued semimartingale with X_0 = 0 and (^d)^⊗ 2-valued bracket [X]. Consider the ^(2)(^d) = 𝔰𝔬(^d)-valued (“area”) processA_t^i,j = 1/2∫_0^t X_r-^i dX^j_r - X^j_r- dX^i_ras an Itô integral, and the “Marcus lift” (terminology from <cit.>)of X to a G^2(^d)-valued process given by_t := exp(X_t + A_t),where exp is taken in the truncated tensor algebra T^2(^d).Note that, if X = Y + K where Y is (another) semimartingale (w.r.t. the same filtration) and K is adapted, of bounded variation, then the respective Marcus lifts of X and Y are precisely related by the translation operator; that is, = T_K. This can be seen by combining (<ref>) with the fact that the second level of is given precisely by the Marcus canonical integral ∫ X⊗♢ dX; see <cit.> for details.Recall that a function F : [0,∞) → [0,∞) is called moderate if it is continuous and increasing, F(x)=0 if and only if x=0, and there exists c>0 such that F(2x) ≤ cF(x) for all x > 0. For any convex moderate function F there exist c,C > 0 such that for any ^d-valued local martingale Xc F ([X]_∞)≤[sup_s,t ≥ 0 F (_s,t^2)] ≤ C F ([X]_∞). For a process M, denote M_t^* := sup_0 ≤ s ≤ t|M_s|. Following the proof of <cit.>, it suffices to show thatF (|A_∞^*|)≤ c_2 F ([X]_∞).However we have[A]_∞^1/2≤ c_1|X_∞^*|[X]_∞^1/2≤ c_1(|X_∞^*|^2 + [X]_∞),so one can apply the classical BDG inequality (e.g., <cit.>) to A and F, and to X and F(|·|^2), which is also a convex moderate function, to obtain F(|A^*_∞|)≤ c_2F([A]_∞^1/2)≤ c_3F(|X^*_∞|^2) + F([X]_∞)≤ c_4F([X]_∞).The following lemma is the crucial step in establishing finite p-variation of the lift of a local martingale. For every 2 < q < p < r there exists C = C(p,q,r) such that for every ^d-valued local martingale X[_^p] ≤ C[[X]_∞^q/2 + [X]_∞^r/2]. For δ > 0 define the increasing sequence of stopping times (τ^δ_j)_j=0^∞ by τ^δ_0 = 0 and for j ≥ 1τ^δ_j = inf{t ≥τ^δ_j-1|sup_u,v ∈ [τ^b_j-1,t] d(_u,_v) > δ}(where inf of the empty set is ∞). Define further ν(δ) := inf{j ≥ 0 |τ^δ_j = ∞}-1. Observe that (cf. <cit.>)^p_≤∑_k = -∞^∞ 2^p(k+1)ν(2^k). Fix δ > 0 and denote τ_j := τ^δ_j. For j=0,1,… consider the sequence of local martingales Y^j_t := X_(τ_j+t)∧τ_j+1 - X_τ_j. Denote by ^j_t the lift of Y^j_t, which coincides with _τ_j, (τ_j + t)∧τ_j+1.It holds that∑_j=0^∞ [Y^j]_∞ = ∑_j=0^ν(δ) [Y^j]_∞ = [X]_∞,and moreover sup_s,t ≥ 0^j_s,t≥δ for all j = 0, …, ν(δ)-1. Thus by the uniform enhanced BDG inequality (Lemma <ref>) with F = |·|^α, α≥ 1,[X]_∞^α≥[∑_j=0^∞[Y^j]_∞^α] ≥ c_α[∑_j=0^∞sup_s,t ≥ 0^j_s,t^2α] ≥ c_αδ^2αν(δ).It follows that for 2 ≤ 2α < p[∑_k ≤ 0 2^p(k+1)ν(2^k)] ≤ c_α^-1[X]_∞^α∑_k ≤ 0 2^p(k+1)-2α k≤ C(α, p) [X]_∞^α,and likewise for 2β > p[∑_k > 0 2^p(k+1)ν(2^k)] ≤ C(β, p) [[X]_∞^β].Taking 2α = q and 2β = r, the conclusion follows from the estimate (<ref>). For every ^d-valued semimartingale X, p > 2, and T > 0, it holds that a.s._;[0,T] < ∞. Note that X can be decomposed into a process K of finite variation and a local martingale L with jump sizes bounded by a positive constant (see, e.g., <cit.>). Denoting bythe lift of L, it follows from a localization argument and Lemma <ref>, that _;[0,T] < ∞ a.s.. The conclusion follows by observing that = T_K().For all p > 2, there exists a constant A > 0 such that for every ^d-valued local martingale X and λ > 0,[_ > λ] ≤A/λ^2[X]_∞. This crucially uses thathas finite p-variation for every local martingale X, and follows in exactly the same manner as <cit.> or <cit.>. Suppose X and Y are non-negative random variables, F is a moderate function, and β > 1 and δ, ε, γ, η > 0 such that γε < 1,F(βλ) ≤γ F(λ),F(δ^-1λ) ≤η F(λ), ∀λ > 0,andX > βλ, Y < δλ≤εX > λ, ∀λ > 0.ThenF(X)≤γη/1-γεF(Y). Let X be an ^d-valued local martingale and D an adapted, non-decreasing process such that a.s., |Δ X_t| ≤ D_t- for all t ≥ 0. Then for every moderate function F (not necessarily convex), there exists C = C(F) > 0 such that[F(_)] ≤ C[F([X]_∞^1/2 + D_∞)]. We follow closely the proof of <cit.> and <cit.>. Sinceis Marcus-like, i.e., log(Δ_t) ∈^d, there exists a constant c>0 such that for all t ≥ 0Δ_t = c|Δ X_t| ≤ c[X]_t^1/2.Let δ > 0, β > cδ + 1, λ > 0, and define the stopping timesT= inf{ t ≥ 0 |_;[0,t] > βλ}, S= inf{ t ≥ 0 |_;[0,t] > λ}, R= inf{ t ≥ 0 | D_t ∨[X]_t^1/2 > δλ}.Define the local martingale N_t = X_(t + S)∧ R - X_S∧ R with liftand note that_≥_;[0,R] - _;[0,R∧ S].On the event {T < ∞, R = ∞}, we have Δ X_S ≤ D_S-≤δλ, and so from (<ref>)_≥βλ - λ - Δ_S≥ (β - 1 - cδ)λ.By Lemma <ref>, it follows thatT < ∞, R = ∞≤[_ > (β - cδ - 1)λ] ≤A/(β - cδ - 1)^2λ^2[N]_∞.On the event {S = ∞}, it holds that N ≡ 0, whilst on {S < ∞}, we have D_R-≤δλ and thus[N]_∞ = [X]_R - [X]_R∧ S≤[X]_R- + |Δ X_R|^2 ≤ 2δ^2λ^2.It follows that[N]_∞≤ 2δ^2λ^2S < ∞,and thus we have for all λ > 0[_ > βλ, D_∞∨ [X]_∞^1/2≤δλ] ≤2Aδ^2/(β - cδ - 1)^2[_ > λ].The conclusion now follows by applying Lemma <ref>.For every convex moderate function F and p > 2 there exists c, C > 0 such that for every ^d-valued local martingale Xc[F([X]_∞^1/2)] ≤[F (_)] ≤ C[F ([X]_∞^1/2)]. This again follows very closely the proof of <cit.>. We may suppose [[X]_∞^1/2] < ∞ (otherwise all concerned quantities are infinite).Let D_t := sup_0 ≤ s ≤ t |Δ X_t| and K^1_t := ∑_s ≤ tΔ X_s |Δ X_s| ≥ 2D_s-. Note that K^1 is of integrable variation since|K^1|_≤ 4D_∞≤ 4[X]_∞^1/2,so there exists a unique previsible process K^2 such that K^1-K^2 is a martingale (K^2 is the dual previsible projection of K^1 <cit.> and is a special case of the Doob–Meyer decomposition). Finally, define the martingale L := X - (K^1-K^2); recall that X = (K^1 - K^2) + L is called the Davis decomposition of X.Observe that |Δ L_t| ≤ 4D_t-. Indeed, |Δ(L-K^2)_t| ≤ 2D_t- by construction, and, if T is a stopping time, then either T is totally inaccessible, in which case Δ K^2_T = 0 since K^2 is previsible, or T is previsible, in which case |Δ K^2_T| = |Δ (K^2-L)_T|_T-| ≤2D_T-|_T-=2D_T-, whereis the underlying filtration (cf. <cit.>). Hence, by Lemma <ref>, we have[F(_)] ≤ C_1 [F([L]_∞^1/2 + D_∞)].Since D_∞≤[X]_∞^1/2 and[L]_∞^1/2≤[X]_∞^1/2 + [K]_∞^1/2≤[X]_∞^1/2 + |K|_,we have [F(_)] ≤ C_2 [F([X]_∞^1/2 + |K|_)].Furthermore, since F is convex, it follows from the Garsia-Neveu lemma (using the argument provided by <cit.>) thatF(|K^2|_)≤ C_3F(|K^1|_),and thusF(|K|_)≤ C_4[F([X]_∞^1/2)].Finally, as = T_K(), we obtain[F(_)] ≤ C_5[F(|K|_ + _)] ≤ C_6[F([X]_∞^1/2)].It was seen in <cit.> that every (level M) càdlàg p-rough path 𝐗 admits a unique minimal jump extension 𝐗̅ as a level-N rough path, N≥ M. (This generalizes Lyons' fundamental extension theorem to the jump setting.) Moreover, it is clear from the proof of <cit.>𝐗̅_p-var≲𝐗_p-var(with a multiplicative constant that depends on N,M). Applied with M=2 and Marcus lift 𝐗=𝐗( ω) of an ^d-valued local martingale, we obtain the following form of the BDG inequality.For every N ≥ 1, convex moderate function F, and p > 2, there exists c, C > 0 such that for every ^d-valued local martingale Xc[ F( [ X] _∞^1/2) ] ≤[ F( 𝐗̅_) ] ≤ C[ F( [ X]_∞^1/2) ].This is a useful result in the study of expected signatures, which is, loosely speaking, the study of 𝔼[ 𝐗̅_0,T], with 𝐗̅_0,T∈ G^N(^d) ⊂ T^N(^d) and component-wise expectation in the linear space T^N(^d). Since the norm of π_m( 𝐗̅_0,T), the projection to (^d) ^⊗ m, is bounded (up to a constant) by 𝐗̅_p-var;[ 0,T] ^m, we see that the very existence of the expected signature is guaranteed by the existence of all moments of [ X] _0,T. In a Lévy setting with triplet ( a,b,K), this clearly holds whenever K( dy) 1_[ | y| >1] has moments of all orders. One can also apply this with a stopping time T=T(ω), e.g., the exit time of Brownian motion from a bounded domain. In either case, the expected signature is seen to exist (see also <cit.> and <cit.> for more on this).A motivation for the study of expected signatures comes from one of the main results of <cit.> which provides a solution to the moment problem for (random) signatures, i.e., determines conditions under which the sequence of expectations ([π_m𝐗̅_0,T])_m ≥ 0 uniquely determines the law of the full signature of 𝐗̅ (see <cit.> where the moment problem was discussed for the Lévy case, and <cit.> for other families of random geometric rough paths).§.§ Convergence of semimartingales and the UCV condition As an application of the BDG inequality, we obtain a convergence criterion for lifted local martingales in the rough path space (^_ϕ([0,T],G^2(^d)), α_) with a fixed path function ϕ, which is the main result of this subsection.We first recall the uniformly controlled variation (UCV) condition for a sequence of semimartingales (X^n)_n ≥ 1. For X ∈ D([0,T], ^d) and δ > 0, we defineX^δ_t = X_t - ∑_s ≤ t (1-δ/|Δ X_s|)^+Δ X_s.Note that X ↦ X^δ is a continuous function on the Skorokhod space and sup_t ∈ [0,T]|Δ X_t^δ| ≤ 1 with Δ X^δ_t = Δ X_t whenever |Δ X_t| ≤δ. We say that a sequence of semimartingales (X^n)_n ≥ 1 satisfies UCV if there exists δ > 0 such that for all α > 0 there exist decompositions X^n,δ = M^n,δ + K^n,δ and stopping times τ^n,α such that for all t ≥ 0sup_n ≥ 1τ^n,α≤α≤1/αandsup_n ≥ 1[M^n,δ]_t ∧τ^n,α + K^n,δ_;[0,t∧τ^n,α]] < ∞.Recall the following result of Kurtz–Protter <cit.> (see also <cit.> Theorem 7.10 and p. 30). Let X, (X^n)_n ≥ 1, H, (H^n)_n ≥ 1 be càdlàg adapted processes (with respect to some filtrations ^n). Suppose (H^n, X^n)_n ≥ 1 converges in law (resp. in probability) to (H,X) in the Skorokhod topology as n →∞, and that (X^n)_n ≥ 1 is a sequence of càdlàg semimartingales satisfying UCV. Then X is a semimartingale (with respect to some filtration ) and (H^n, X^n, ∫_0^· H^n_t- dX^n_t) converge in law (resp. in probability) to (H,X,∫_0^· H_t-dX_t) in the Skorokhod topology as n →∞. We can now state the main result which allows us to pass from convergence in the Skorokhod topology to convergence in rough path topology (see also Corollary <ref>).Let (X^n)_n ≥ 1 be a sequence of semimartingales such that X^n converges in law (resp. in probability) to a semimartingale X in the Skorokhod topology. Suppose moreover that (X^n)_n ≥ 1 satisfies the UCV condition. Then the lifted processes (^n)_n ≥ 1 converge in law (resp. in probability) to the lifted processin the Skorokhod space D([0,T], G^2(^d)). Moreover, for every p > 2, (^n_)_n ≥ 1 is a tight collection of real random variables.Since the stochastic area is given by the Itô integral (<ref>), the convergence in law (resp. in probability) of (^n)_n ≥ 1 tois an immediate consequence of Theorem <ref>.Let δ > 0 for which we can apply the UCV condition to (X^n)_n ≥ 1. We next claim that (^n,δ_;[0,T])_n ≥ 1 is tight. Indeed, for ε > 0 choose α > T so that 1/α < ε/2. Let X^n,δ = M^n,δ + K^n,δ be the decomposition from the UCV condition along with the stopping times τ^n,α. Then there exists C > 0 such that for all n ≥ 1[K^n,δ_;[0,T] > C] ≤τ^n,α≤α + C^-1[K^n,δ_;[0,T∧τ^n,δ]] < ε,andsup_n ≥ 1[^n,δ_;[0,T] > C]≤sup_n ≥ 1τ^n,α≤α + C^-2[^n,δ^2_;[0,T∧τ^n,δ]] ≤sup_n ≥ 1τ^n,α≤α + C^-2|[M]_T∧τ^n,δ|< ε,where in the final line we used the enhanced BDG inequality Theorem <ref>. Using the fact that ^n,δ = T_K^n,δ(^n,δ) proves that (^n,δ_;[0,T])_n ≥ 1 is tight as claimed.To conclude, observe that ^n = T_L^n(^n,δ) where L^n := X^n - X^n,δ is a process of bounded variation for whichL^n_;[0,T]≤∑_|Δ X^n_t| > δ |Δ X^n_t|.Since (X^n)_n ≥ 1 is tight and ∑_|Δ X^n_t| > δ |Δ X^n_t| is a continuous function of X^n (for the Skorokhod topology), it follows that (^n_;[0,T])_n ≥ 1 is tight as required.Follow the notation of Theorem <ref>. Let p > 2 and ϕ an endpoint continuous, p-approximating path function defined on J ⊂ G^2(^d) × G^2(^d) such that , ^n ∈_ϕ([0,T],G^2(^d)) a.s.. Then for every p' > p, (^n,ϕ) → (,ϕ) in law (resp. in probability) in the metric space (^_ϕ([0,T],G^2(^d)), α_).As we shall see in Proposition <ref>, a simple way to apply Corollary <ref> is to assume that ϕ comes from the lift of a (left-invariant) path function ϕ : ^d → C^([0,T],^d) (denoted by the same symbol) which is endpoint continuous and q-approximating for some 1 ≤ q < 2 (so that a canonical lift indeed exists), and does not create area, i.e.,log S_2(ϕ(x))_0,1 = x ∈^d ⊕{0}⊂^2(^d).Sinceis Marcus-like, i.e., log(Δ_t) ∈^d ⊕{0}, it indeed follows that ∈_ϕ([0,T],G^2(^d)) so that ^ϕ is well defined (which corresponds to interpolating the jumps ofusing ϕ); the same of courses applies to ^n. Consider first the case of convergence in probability. By Theorem <ref>, (^n_)_n ≥ 1 is tight, so for every ε > 0 we can find R > 0 such thatsup_n ≥ 1[max{^n, , ^n_, _} > R] ≤ε.The conclusion now follows from Proposition <ref>. For the case of convergence in law, the proof follows in a similar way upon applying the Skorokhod representation theorem <cit.> to the space D([0,T],G^N(^d)) and the sequence (^n)_n ≥ 1. As an application of Corollary <ref>, along with the fact that piecewise constant approximations satisfy UCV <cit.>, we obtain the following Wong–Zakai-type result (which shall be substantially generalized in Section <ref> using different methods). The following result resembles the Wong–Zakai theorem of <cit.>, where it was shown that ODEs driven by approximations of the form X^h_t = h^-1∫_t-h^t X_s ds converge to an SDE of the Marcus type. Here we are able to complement <cit.> by showing this for the case of genuine piecewise linear (and a variety of other) approximations. Moreover the deterministic nature of our rough path approach is able to handle anticipating initial data (see Remark <ref>).Let X be a semimartingale with Marcus lift , and let _n ⊂ [0,T] be a sequence of deterministic partitions such that |_n| → 0. Let X^[D_n] be the piecewise constant approximations of X along the partition _n, and let ^[_n] be their (Marcus) lifts.Let ϕ : ^d → C^([0,1], ^d) be an endpoint continuous, q-approximating path function for some 1 ≤ q < 2 such that ϕ does not create area, i.e., (<ref>) holds. By an abuse of notation, let ϕ : exp(^d ⊕{0}) → C^([0,1],G^2(^d)) denote also the canonical lift of ϕ, treated as a path function defined on exp(^d ⊕{0}) ⊂ G^2(^d).*Consider the admissible pairs (^[_n],ϕ) and (,ϕ) in ([0,T],G^2(^d)). Then for every p > 2, α_(^[_n], ) → 0 in probability as n →∞. *Let X^_n,ϕ be the piecewise-ϕ interpolation of X along the partition _n. Let U^_T ← 0∈^m(^e) denote the flow associated to the RDE (<ref>). Then U^X^_n,ϕ_T ← 0 converges in probability to U^^ϕ_T ← 0 as n →∞ (as ^m(^e)-valued random variables). <ref> follows immediately from Corollary <ref> and the fact that (X^[_n])_n ≥ 1 satisfies UCV <cit.>.For <ref>, note that, since ϕ does not create area, (^[_n])^ϕ coincides (up to reparametrization) with the canonical lift of X^_n,ϕ. The conclusion now follows from <ref> and Theorem <ref>. We now record a relation between canonical RDEs and Marcus SDEs. Let X : [0,T] →^d be a semimartingale andits Marcus lift. Then for vector fields V = (V_1,…, V_d) in ^γ(^e) for some γ > 2, it holds that the canonical RDEdY_t = V(Y_t) ♢ d_t,Y_0 ∈^e(i.e., the path function ϕ is the taken to be log-linear) coincides a.s. with the Marcus SDEdY_t = V(Y_t)♢ dX_t,Y_0 ∈^e. Recall that, by definition, the Marcus SDE satisfies <cit.>Y_t = Y_0 + ∫_0^t V(Y_s-)dX_s + 1/2∫_0^t V' V(Y_s)d[X]^c_s + ∑_0 < s ≤ t{ e^VΔ X_s (Y_s-) - Y_s- - V(Y_s-)Δ X_s },where e^W(y) denotes the flow at time 1 along the vector field W from y, i.e., the solution at t=1 to z_0 = y, ż_t = W(z_t).Recall likewise that, by definition, the rough canonical equation (in the sense of <cit.>) satisfiesY_t = Y_0 + ∫_0^t V(Y_s-)d_s + ∑_0 < s ≤ t{ e^VΔ X_s (Y_s-) - Y_s- - V(Y_s-)Δ X_s - V' V(Y_s-)1/2(Δ X_s)^⊗ 2},which agrees with the solution to the canonical RDE (<ref>) from Definition <ref> (see <cit.>; we point out that <cit.> and <cit.> are not restricted to the case of finite activity). It remains to verify that a.s.∫_0^t V(Y_s-)d_s =∫_0^t V(Y_s-)dX_s + 1/2V'V(Y_s)d[X]^c_s+ ∑_0<s≤ t V'V(Y_s-)1/2(Δ X_s)^⊗ 2.To this end, observe that∫_0^t V(Y_s-) d_s = lim_|| → 0∑_t_i ∈ V(Y_t_i-)X_t_i,t_i+1 + V'V(Y_t_i-)^(2)_t_i,t_i+1 ,where ^i,j_s,t = ∫_s^t X^i_s,u-dX^j_u +1/2[X^i,X^j]^c_s,t+1/2∑_r ∈ (s,t]Δ X^i_rΔ X^j_r. It follows from Lemma <ref> that for a (deterministic) sequence of partitions with |_n| → 0, we have a.s.∫_0^t V(Y_s-)d_s = lim_n→∞∑_t_i ∈_nV(Y_t_i-)dX_t_i,t_i+1 +1/2V'V(Y_t_i-)[X]_t_i,t_i+1 + ∑_r∈ (t_i,t_i+1]V'V(Y_t_i-)1/2(Δ X_r)^⊗ 2from which the conclusion readily follows. Observe that a simple special case of Proposition <ref> (which does not require any probabilistic considerations) is a piecewise constant path X^[], which is constant between the points of a partition ⊂ [0,T]. In this case, the solution to dY =V(Y) ♢ dX^[],Y_0 ∈^e,agrees, for all t ∈, with the ODE solution dỸ =V(Ỹ) dX^,Y_0 ∈^e,where X^ is the piecewise linear path obtained from X^[] by connecting with a straight line consecutive points X^[]_t_n⇝ X^[]_t_n+1 for all t_n ∈. We are now ready to state the precise criterion for convergence in law (resp. in probability) of Marcus SDEs which was advertised in the introduction and which is analogous to the same criterion for Itô SDEs <cit.>. Let V = (V_1,…, V_d) be a collection of ^γ vector fields on ^e for some γ > 2. Let Y_0, (Y_0^n)_n ≥ 1 be a collection of (random) initial conditions in ^e and X, (X^n)_n ≥ 1 be a collection of semimartingales such that (X^n)_n ≥ 1 satisfies UCV and (Y^n_0, X^n) → (Y_0, X) in law (resp. in probability) as n →∞ (as ^e × D([0,T], ^d)-valued random variables). Then the solutions to the Marcus SDEsdY^n_t = V(Y^n_t)♢ dX^n_t,Y^n_0 ∈^e,converge in law (resp. in probability) as n →∞ (in the Skorokhod topology) to the solution of the Marcus SDEdY_t = V(Y_t)♢ dX_t,Y_0 ∈^e. This is an immediate consequence of Corollary <ref>, Proposition <ref>, and the deterministic continuity of the solution map (part <ref> of Proposition <ref>).We have not been explicit about filtrations, but of course, every semimartingale X^n above is adapted to some filtration {^n_t }_t ≥ 0. In the same vain, as is standard in the context of SDEs, the initial datum Y_0^n is assumed to be ^n_0-measurable, so that (<ref>) makes sense as a bona fide integral equation (as recalled in the proof of Proposition <ref>).Situations where Y^0_n is independent of the driving noise X^n are then immediately handled. If, on the other hand, Y^0_n depends in some anticipating fashion on the driving noise, then classical SDE theory (Marcus or Itô) breaks down and ideas from anticipating stochastic calculus are necessary (such as composing the stochastic flow with anticipating initial data; in the Marcus context this would be possible thanks to <cit.>). Our(essentially deterministic) rough path approach bypasses such problems entirely. We shall not pursue further application of rough paths to “anticipating Marcus SDEs” here, but note that this could be done analogously to <cit.>. §.§ Examples We now give a list of examples to which Corollary <ref>, Proposition <ref>, and Theorem <ref> apply. The main criterion of application is of course the UCV condition. For further examples of sequences of semimartingales satisfying UCV, see <cit.>. We note that in the framework of Theorem <ref>, the UCV condition cannot in general be ommited (but see Theorem <ref> below) as seen, e.g., in homogenization theory <cit.> and non-standard approximations to Brownian motion <cit.>.[Piecewise constant approximations] Let X be a càdlàg semimartingale and X^[_k] be its piecewise constant approximation (see Figure <ref>) along a sequence of deterministic partitions _k ⊂ [0,T] such that |_k| → 0. Then by Theorem <ref>, the solutions todY^[_k]_t = V(Y^[_k]_t) ♢ dX^[_k]_t,Y^[_k]_0 = y_0 ∈^econverge in probability (for the Skorokhod topology) to the solution ofdY_t = V(Y_t) ♢ dX_t,Y_0 = y_0 ∈^e.Moreover, if X is continuous, then Y^[_k] converges in probability for the uniform topology on [0,T] to Y (which is now also the solution to the Stratonovich SDE).[Piecewise linear approximations] Let X be a càdlàg semimartingale and now let X^_k be its piecewise linear (i.e., Wong–Zakai) approximation (see Figure <ref>) along a sequence of deterministic partitions _k ⊂ [0,T] such that |_k| → 0. Consider the solutions to random ODEsdY^_k_t = V(Y^_k_t) dX^_k_t,Y^_k_0 = y_0 ∈^e,and the Marcus SDEdY_t = V(Y_t) ♢ dX_t,Y_0 = y_0 ∈^e.Then, by Proposition <ref>, it holds that Y^_k_T → Y_T in probability.Moreover, if X is continuous, then in light of the last part of Example <ref> and Remark <ref>, Y^_k converges in probability for the uniform topology on [0,T] to Y (which, we emphasize again, is now the solution to the Stratonovich SDE), which agrees with the classical Wong–Zakai theorem for continuous semimartingales.[Donsker approximations to Brownian motion] Consider an ^d-valued random walk X^n with iid increments and finite second moments, rescaled so that X^n→ B in law. Here we treat X^n as either piecewise constant or interpolated using any sufficiently nice path function ϕ which does not create area, i.e., satisfies (<ref>) (e.g., piecewise linear). Then (X^n)_n ≥ 1 satisfies UCV, so by Corollary <ref> we again have convergence of the Marcus SDEs (or random ODEs in case of continuous interpolations)dY^n_t = V(Y^n_t) ♢ dX^n_tY_0 = y_0 ∈^ein law for the uniform topology on [0,T] to the Stratonovich limitdY_t = V(Y_t)∘ dB_tY_0 = y_0 ∈^e.This is a special case of <cit.> (see also Example <ref>) which improves the main result of Breuillard et al. <cit.> in the sense that no additional moment assumptions are required (highlighting a benefit of p-variation vs. Hölder topology). [Null array approximations to Lévy processes] Generalizing Example <ref>, consider a null array of ^d-valued random variables X_n1,…, X_nn, i.e., lim_n →∞sup_k |X_nk|∧ 1 = 0, and, for every n ≥ 1, X_n1,…, X_nn are independent. Consider the associated random walkX^n : [0,1] →^d,X^n_t = ∑_k=1^tn X_nk.Suppose X^n → X in law for a Lévy process X (see <cit.> for necessary and sufficient conditions for this to occur), which in particular implies that for some h > 0 * ∑_k=1^n [X_nk|X_nk| < h] → b^h, * ∑_k=1^n [X^i_nkX^j_nk|X_nj| < h] → a^h_i,j, and * ∑_k=1^n [f(X_nk)] →ν(f) for every f ∈ C_b(^d) which is identically zero on a neighbourhood of zero,where b^h, a^h, and ν are determined by the Lévy triplet of X (in particular ν is the Lévy measure of X). As a consequence, it is immediate to verify that (X^n)_n ≥ 1 satisfies UCV. By Theorem <ref>, the solutions todY^n_t = V(Y^n_t) ♢ dX^n_t,Y^n_0 = y_0 ∈^e,converge in law (for the Skorokhod topology) to the solution of the Marcus SDEdY_t = V(Y_t) ♢ dX_t,Y_0 = y_0 ∈^e.If, once more, X^n are interpolated using any sufficiently nice path function ϕ (which in particular does not create area (<ref>)), an application of Corollary <ref> implies that Y^n_T (now solutions to random ODEs) converge in law to Y_T (now the solution to the random RDE driven by ^ϕ). Note that if ϕ is allowed to create area and X_n1,…, X_nn are further assumed iid for every n ≥ 1, then this is precisely the case addressed in <cit.> (though in this case one must consider a non-Marcus lift of X similar to the upcoming Theorem <ref>). [Martingale CLT] Let (X^n)_n ≥ 1 be a sequence of ^d-valued càdlàg local martingales. Suppose that, as n→∞, [sup_t∈ (0,T] |Δ X_t^n| ] → 0,[ X^n,X^n] _t→ C( t)∀ t∈ (0,T],where t↦ C( t) ∈ℝ^d× d is continuous and deterministic. Then X^n → X, where X is a continuous Gaussian process with independent increments and X(t)X(t)^T = C(t) <cit.>, and moreover the UCV condition is satisfied <cit.>. Therefore solutions todY^n_t = V(Y^n_t) ♢ dX^n_t,Y^n_0 = y_0 ∈^e,converge in law for the uniform topology on [0,T] to the solution of the Stratonovich SDEdY_t = V(Y_t) ∘ dX_t,Y_0 = y_0 ∈^e. §.§ Wong–Zakai revisitedIn this subsection we significantly expand Proposition <ref> by showing convergence in probability of very general (area-creating) interpolations of càdlàg semimartingales.If the interpolation creates area, we in general no longer expect to converge to the Marcus lift of X (which is the reason one cannot apply Proposition <ref>), and therefore we first modify the liftappropriately.Throughout the section, we fix a (left-invariant) q-approximating path function ϕ : ^d → C^([0,1], ^d) for some 1 ≤ q < 2 (which we take to be defined on the entire space ^d only for simplicity). Consider the two mapsψ : ^d → G^2(^d) a : ^d →^(2)(^d) ≅𝔰𝔬(^d)defined uniquely byS_2(ϕ(x))_0,1 = ψ(x) = exp(x + a(x))(so that a(x) is the area generated by the path ϕ(x) : [0,1] →^d).Let us also fix a càdlàg semimartingale X : [0,T] →^d and a sequence of deterministic partitions _k ⊂ [0,T] such that lim_k →∞ |_k| = 0. Consider the following assumption. There exists a càdlàg bounded variation process B : [0,T] →^(2)(^d) such thatsup_t ∈ [0,T]|B_t - ∑__k ∋ t_j ≤ t a(X_t_j - X_t_j-1)| → 0 in probability as k →∞.Before stating the Wong–Zakai theorem, we give several examples of ϕ for which Assumption <ref> is satisfied.[No area] If ϕ(x) does not create area for all x ∈^d, so that a ≡ 0 (e.g., when ϕ is the linear interpolation on ^d), then evidently Assumption <ref> is satisfied with B_t ≡ 0. [Hoff-type process]Suppose that ^d = ^2, so that ^(2)(^2) ≅. Let ϕ travel to (x,y) first linearly along the x-coordinate and then linearly along the y-coordinate:ϕ(x,y)_t = t ∈ [0,1/2] 2tx + t ∈ (1/2,1] (x + (2t-1)y).Then a(x,y) = 1/2xy, so that Assumption <ref> is satisfied with B_t = 1/2[X^1,X^2]_t. [Regular a] Suppose more generally that a is twice differentiable at 0, so that by Lemma <ref>a(X_s,t) = 1/2D^2a(0) (X_s,t^⊗ 2) + o(|X_s,t|^2).We see in this case that Assumption <ref> is satisfied withB_t = 1/2∑_i,j = 1^d D^2a(0)^i,j [X^i,X^j]^c_t + ∑_s ≤ t a(Δ X_s).For a partition ⊂ [0,T], let X^,ϕ be the piecewise-ϕ interpolation of X along , and ^,ϕ its canonical lift.The following is the main result of this subsection. Suppose that Assumption <ref> is satisfied. Let : [0,T] → G^2(^d) be the modified level-2 lift of X defined by:= exp(X + A̅), A̅_t := A_t + B_t,where = exp(X + A) is the Marcus lift of X. Suppose further that ϕ is endpoint continuous.*Consider the admissible pair (,ϕ) ∈([0,T],G^2(^d)). Then for every p > 2, it holds thatα_(^_k,ϕ,) → 0 in probability as k →∞.*Let U^_T ← 0∈^m(^e) denote the flow associated to the RDE (<ref>). Then U^X^_k,ϕ_T ← 0 converges in probability to U^^ϕ_T ← 0 (as a ^m(^e)-valued random variable). Note that the jumps of B_t must necessarily be of the form a(Δ X_t). Hence Δ_t ∈ψ(^d), so that indeed ∈_ϕ([0,T],G^2(^d)) and ^ϕ is well-defined (this is an abuse of notation since ϕ is a path function on ^d, but because ϕ(x) of finite q-variation, it can be canonically lifted to a path function ϕ : ψ(^d) → G^2(^d)). For the proof of the theorem, we require several lemmas. Let η_ be a q-variation modulus of ϕ (see Definition <ref>). Then|a(x)| ≤η_(r)|x|^2, ∀ r > 0, ∀ x ∈^ds.t.|x| ≤ r. This is immediate from the property|x| + |a (x)|^1/2≍ψ(x)≤ϕ(x)_;[0,1]≤η_(r)|x|.The following lemma essentially involves no probability. Suppose ϕ is endpoint continuous. Then for a.e. sample path ∈ D([0,T], G^2(^d)), it holds that for all ε > 0 there exists r > 0, such that for all partitions ⊂ [0,T] with || < r, there exists δ_0 > 0, such that for all δ < δ_0, there exists λ∈Λ such that|λ| < 2||, _t_n = ^ϕ,δ_λ(t_n), ∀ t_n ∈,andmax_t_n ∈sup_t ∈ [t_n,t_n+1] d(^,ϕ_t_n,t, ^ϕ,δ_λ(t_n),λ(t)) < ε. Sinceis càdlàg, for every ε > 0 we can find r > 0 sufficiently small and a partition = (s_0,…,s_m) so that |s_i+1 - s_i| > 2r andsup_u,v ∈ [s_i,s_i+1)_u,v < ε.Then whenever || < r, for every t_n ∈, there exists at most one s_i ∈ such that s_i ∈ [t_n,t_n+1]. Now using the fact that ϕ is q-approximating and endpoint continuous, the claim readily follows.The family of real random variables (^,ϕ_;[0,T])_⊂ [0,T] is tight.As the proof of Lemma <ref> will reveal, the biggest difficulty is overcome by the enhanced BDG inequality for càdlàg local martingales (Theorem <ref>). We wish to emphasize that the lemma is even helpful in the context of a rough paths proof of the Wong–Zakai theorem for continuous semimartingales with piecewise linear interpolations (so that a ≡ 0), since an analogous tightness result is still needed in this case and is non-trivial (cf. <cit.>).We also mention that part <ref> of Proposition <ref> in particular shows that (^[_k]_)_k ≥ 1 is tight, which can significantly simply the proof of Lemma <ref> (at least if one restricts attention to the family (^_k,ϕ_)_k ≥ 1). However we give a direct proof of the general result here.We can decompose X = L + K (non-uniquely) where K is of bounded variation and L is a local martingale with jumps bounded by some M > 0 (e.g., <cit.>). Using the localizing sequence τ_m = inf{ t ∈ [0,T] | |L_t| + |K_t| > m }, we may further suppose that L is bounded.For a partition ⊂ [0,T], consider the piecewise constant path ^[] : [0,T] → G^2(^d) which is constant on [t_n,t_n+1) and ^[]_t_n = ^,ϕ_t_n for every t_n ∈. Consider also the piecewise constant semimartingale X^[] which is constant on [t_n,t_n+1) and X^[]_t_n = X_t_n for every t_n ∈. Let ^[] be the (Marcus) lift of X^[]. By definition of a : ^d →^(2)(^d), we have for all t_n ∈^[]_t_n = ^[]_t_n⊗exp(∑_∋ t_k ≤ t_n a(X_t_k - X_t_k-1)).It follows that^[]^p_;[0,T]≤ C(^[]^p_;[0,T] + Y^[]_p/2;[0,T]^p/2),where Y^[]_t = ∑_∋ t_n ≤ t a(X_t_n - X_t_n-1) ∈^(2)(^d).Observe that ^,ϕ is a reparametrization of (^[])^ϕ (where we use the same abuse of notation as in Remark <ref>), so in particular^,ϕ_;[0,T] = (^[])^ϕ_;[0,T].Moreover, defining the piecewise constant local martingale L^[], its lift ^[], and the piecewise constant path of bounded variation K^[] in the same way as X^[], we note that ^[] = T_K^[](^[]). Following Lemma <ref>, it suffices to show that the families(Y^[]_p/2)_⊂ [0,T],(^[]_)_⊂ [0,T],(K^[]_)_⊂ [0,T]are tight. This in turn follows respectively from Lemma <ref>, the enhanced BDG inequality Theorem <ref>, and the fact that K^[]_≤K_.Note that <ref> follows directly from <ref> and Theorem <ref>. To show <ref>, note that _ < ∞ a.s. (by Corollary <ref>), so following Lemma <ref> and interpolation (Lemma <ref>), it suffices to show that to show thatα_∞(^_k,ϕ,) → 0 in probability as k →∞. Let A̅, A̅^ϕ,δ and A^_k,ϕ denote the stochastic area of , ^ϕ,δ and ^_k,ϕ respectively. We have for t ∈ [t_n,t_n+1] ⊂_k^_k,ϕ_t = ψ(X_0,t_1) …ψ(X_t_n-1,t_n) S_2(ϕ(X_t_n,t_n+1))_t-t_n/t_n+1-t_n,and so by the Campbell–Baker–Hausdorff formulaA^_k,ϕ_t = A^_k,ϕ_t_n,t + ∑_j=0^n-1 a(X_t_j,t_j+1) + 1/2[X_0,t_n, X^_k,ϕ_t_n,t] + 1/2∑_j=0^n-1 [X_0,t_j,X_t_j,t_j+1].LikewiseA̅_t = A̅_t_n,t + ∑_j=0^n-1A̅_t_j,t_j+1 + 1/2[X_0,t_n, X_t_n,t] + 1/2∑_j=0^n-1 [X_0,t_j,X_t_j,t_j+1].Recalling further that A̅_t_j,t_j+1 = A_t_j,t_j+1 + B_t_j,t_j+1 and X^_k,ϕ_t_n = X_t_n, it follows that for all t_n ∈_kd(^_k,ϕ_t_n, _t_n) ≍|∑_j=0^n-1 A_t_j,t_j+1 + B_t_j,t_j+1 - a(X_t_j+1 - X_t_j)|^1/2.Combining Lemma <ref> with Assumption <ref>, we see that the proof is complete once we showmax_t_n ∈_k|∑_j=0^n-1 A_t_j,t_j+1| → 0 in probability as k →∞,which in turn follows from Lemma <ref>.§.§ Appendix: Vanishing areas Let X : [0,T] →^d be a càdlàg semimartingale, Y : [0,T] →((^d)^⊗ 2,) a locally bounded previsible process, and (_k)_k ≥ 1 a sequence of deterministic partitions of [0,T] such that lim_k →∞ |_k| = 0. Define 𝕏_s,t := ∫_s^t (X_r--X_s) ⊗ dX_r as Itô integrals. Thenmax_t_n ∈_k|∑_j=0^n-1 Y_t_j𝕏_t_j,t_j+1| → 0 in probability as k →∞. In the case Y ≡ 1, observe that Lemma <ref> is an immediate consequence of the convergence in part <ref> of Proposition <ref> (where ϕ is taken as the piecewise linear interpolation).As in the proof of Lemma <ref>, we can decompose X = L + K, where K is of bounded variation and L is a local martingale with bounded jumps (e.g., <cit.>).Using the localizing sequenceτ_m = inf{ t ∈ [0,T] | |Y_t| + |L_t| + |K_t| > m },we may further suppose that Y and L are uniformly bounded and that X is bounded on [0,τ_m) by m > 0 and is constant on [τ_m, T]. We now write𝕏_s,t = ∫_s^t (X_r- - X_s)(dL_r + dK_r). For a fixed càdlàg sample path X, for every ε > 0 we can find r > 0 sufficiently small and a partition = (s_0,…,s_m) so that |s_i+1 - s_i| > r andsup_u,v ∈ [s_i,s_i+1)X_u,v < ε.Then whenever || < r/2, for every t_n ∈ D, there exists at most one s_i ∈ such that s_i ∈ [t_n,t_n+1].Since K is a process of finite variation, if s_i ∈ [t_n,t_n+1] then|∫_t_n^t_n+1(X_s- - X_t_n)dK_s| ≤ε|K|_;[t_n,s_i] + 2m |K|_;[s_i,t_n+1],and if no s_i ∈ is in [t_n,t_n+1], then the upper bound is ε|K|_;[t_n,t_n+1]. Denoting by [t_n,t_n+1] the interval in _k containing s_i ∈, we then have∑_t_n ∈_k|Y_t_n∫_t_n^t_n+1 (X_s- - X_t_n) dK_s| ≤ C|Y|_∞(ε|K|_ + 2m ∑_s_i ∈ |K|_;[s_i,t_n+1]),from which it follows that the LHS converges to zero a.s. as k →∞. It remains to show thatmax_t_n ∈_k|∑_j=0^n-1 Y_t_j∫_t_j^t_j+1 (X_s- - X_t_n)dL_s | → 0in probability as k →∞. By Itô isometry[(Y_t_n∫_t_n^t(X_s- - X_t_n)dL_s)^2] ≤ C [|Y_t_n|^2∫_t_n^t |X_s- - X_t_n|^2 d|[L]_s|].As before, if s_i ∈ is in [t_n,t_n+1], then∫_t_n^t|X_s- - X_t_n|^2d|[L]_s| ≤ε^2 |[L]_t_n,s_i| + (2m)^2|[L]_s_i,t_n+1|,and if no s_i ∈ is in [t_n,t_n+1], then the upper bound is ε^2[L]_t_n,t_n+1. Hence∑_t_n ∈_k |Y_t_n|^2 ∫_t_n^t_n+1 |X_s- - X_t_n|^2 d|[L]_s| ≤ C|Y|^2_∞(ε^2 |[L]_∞| + (2m)^2∑_s_i ∈ |[L]_s_i,t_n+1|),from which it follows that the LHS converges to zero a.s. as k →∞. As L is bounded (so in particular bounded in L^2), we obtain by dominated convergence[∑_t_n∈_k( Y_t_n∫_t_n^t_n+1 (X_s--X_t_n)dL_s )^2] ≤ C [ ∑_t_n∈_k |Y_t_n|^2 ∫_t_n^t_n+1 |X_s- - X_t_n|^2 d|[L]_s|]→ 0.Finally, applying the classical BDG inequality to the discrete-time martingale∑_j=0^n Y_t_j∫_t_j^t_j+1 (X_s--X_t_n)dL_s,we see that (<ref>) holds in L^2, and thus in probability as desired.§ BEYOND SEMIMARTINGALES We have seen in the previous section that (general) multi-dimensional semimartingales give rise to (càdlàg) geometric p-rough paths. Marcus lifts of general semimartingales provide us with concrete and important examples of driving signals for canonical RDEs, providing a decisive and long-awaited <cit.> rough path view on classical stochastic differential equations with jumps. We now discuss several examples to which the theory of Section <ref> can be applied which fall outside the scope of classical semimartingale theory.§.§ Semimartingales perturbed by paths of finite q-variation Keeping focus on ^d-valued processes and their (canonical) lifts, we remark that any process with a decomposition X = Y+B, where Y is a semimartingale and B is a process with finite q-variation for some q < 2, admits a canonical Marcus lift given by = T_B(). Note that due to the deterministic nature of the (Young) integrals used to construct T_B(), we require no adaptedness assumptions on B. We summarize the existence of the lift in the following proposition, which is an immediate consequence of Proposition <ref>. Let 1 ≤ q < 2 and consider an D^([0,T],^d)-valued random variable B, and an ^d-valued semimartingale Y. Write A_Y for the area of Y and define its Marcus lift 𝐘=exp( Y+A_Y). Then the process X := Y+B admits a canonical lift, given by =T_B, which is a Marcus-like geometric p-rough path for any p > 2. We mention that the class of paths with such a decomposition contains some well-studied processes.[PII] The important class of processes with independent increments (PII) goes beyond semimartingale theory. In fact, every such process X can be decomposed (non-uniquely) as X=Y+B, where Y is a PII and a semimartingale and B is a deterministic càdlàg path. Moreover, X is a semimartingale if and only if B has finite variation on compacts. Provided that the process B has finite q-variation for some q < 2, we immediately see that X admits a lift to a (càdlàg) geometric p-rough path for any p>2.We note that there is a natural interest in differential equations driven by PIIs (under the assumptions of Proposition <ref> this is meaningful!) since the resulting (pathwise) solutions Z to canonical RDEsdZ = V(Z) ♢ dwill be (time-inhomogeneous) Markov processes. A further study and characterization of such processes seems desirable. (For instance, they may not be nicely characterized by their generator. Consider the case V≡ 1, X = B ∈ C^q-var([0,T],) ∖ C^1([0,T],) with q ∈ (1,2).)§.§ Markovian and Gaussian càdlàg rough paths One can use Dirichlet forms to construct Markovian rough paths which are not lifts of semimartingales. In the continuous setting this has been developed in detail in <cit.>. Including a non-local term in the Dirichlet form will allow to extend this construction to the jump case, but we will not investigate this here. We also note that Gaussian càdlàg rough paths can be constructed; as in the continuous theory, the key condition is finite ρ-variation of the covariance (cf. <cit.>) but without assuming its continuity. §.§ Group-valued processesThis point was already made in the context of Lévy rough paths <cit.>, which substantially generalizes the notion of the Marcus lift of an ^d-valued Lévy process (such processes, for example, arise naturally as limits of stochastic flows, see <cit.>). In the same spirit, one can defined “genuine semimartingale rough paths” as G^N(^d)-valued process with local characteristics modelled after Lévy (rough path) triplets.(Remark that the Lie group G^N(^d) is, in particular, a differentiable manifold so that the theory of manifold-valued semimartingales applies. The issue is to identify those which constitute (geometric) rough paths, which can be done analyzing the local characteristics, as was done in the Lévy case in the afore-mentioned papers. In the same spirit, the afore-mentioned Dirichlet-form construction also extends to the group setting.) imsart-number | http://arxiv.org/abs/1704.08053v2 | {
"authors": [
"Ilya Chevyrev",
"Peter K. Friz"
],
"categories": [
"math.PR",
"60H99 (Primary) 60H10 (Secondary)"
],
"primary_category": "math.PR",
"published": "20170426104610",
"title": "Canonical RDEs and general semimartingales as rough paths"
} |
LMU [email protected] [email protected] evolutionary edit distance between two individuals in a population, i.e., the amount of applications of any genetic operator it would take the evolutionary process to generate one individual starting from the other, seems like a promising estimate for the diversity between said individuals. We introduce genealogical diversity, i.e., estimating two individuals' degree of relatedness by analyzing large, unused parts of their genome, as a computationally efficient method to approximate that measure for diversity.<ccs2012> <concept> <concept_id>10010147.10010178.10010205.10010206</concept_id> <concept_desc>Computing methodologies Heuristic function construction</concept_desc> <concept_significance>500</concept_significance> </concept> <concept> <concept_id>10010147.10010257.10010293.10011809.10011812</concept_id> <concept_desc>Computing methodologies Genetic algorithms</concept_desc> <concept_significance>500</concept_significance> </concept> </ccs2012> [500]Computing methodologies Heuristic function construction [500]Computing methodologies Genetic algorithmsGenealogical Distance as a Diversity Estimate in Evolutionary Algorithms Lenz Belzner December 30, 2023 ========================================================================§ INTRODUCTION Diversity has been a central point of research in the area of evolutionary algorithms. It is a well-known fact that maintaining a certain level of diversity aids the evolutionary process in preventing premature convergence, i.e., the phenomenon that the population focuses too quickly on a local optimum at hand and never reaches more fruitful areas of the fitness landscape <cit.>. Diversity thus plays a key role in adjusting the exploration-exploitation trade-off found in any kind of metaheuristic search algorithm.We encountered this problem from an industry point of view when designing learning components for a system that needs to guarantee certain levels of quality despite being subjected to the probabilistic nature of its physical environment and probabilistic behavior of its machine learning parts <cit.>. Of course, any general solution for this kind of challenge is yet to be found. However, we believe that the engineering of (hopefully) reliable learning components can be supported by exposing all handles that search algorithms offer to the engineer at site. For scenarios like this one, we consider it most helpful to employ approaches that allow the engineer to actively control properties like diversity of the evolutionary search process instead of just observing diversity and adjust it indirectly via other parameters (like, e.g., the mutation rate).Among the copious amount of different techniques to introduce diversity-awareness to evolutionary algorithms, many do not immediately make the job of adjusting a given evolutionary algorithm easier but instead require additional engineering effort: For example, one may need to define a distance metric specifically for the search domain at hand or adjust lots of hyperparameters in island or niching models. We thus attempt to define a more domain-independent and almost parameter-free measurement for diversity by utilizing the genetic operators already defined within any given evolutionary process.We discuss related work in the following Section <ref>. We then explain the target metric called “evolutionary edit distance” in Section <ref>. Section <ref> continues by introducing the notion of “genealogical diversity” as means to approximate that concept. We improve this approach in Section <ref> by using a much simpler and computationally more efficient data structure. To support our ideas, we describe a basic evaluation scenario in which we have applied both approaches in Section <ref>. We end with a short conclusion in Section <ref>.§ RELATED WORKThe importance of diversity for evolutionary algorithms is discussed throughout the body of literature on evolutionary computing ranging from entry level <cit.> to specialized papers <cit.>. In many cases, authors refer to diversity as a measure of the evolutionary algorithm's performance and try to configure the hyperparameters of the evolutionary algorithm as to achieve an optimal trade-off between exploration and exploration for the scenario at hand <cit.>. This measure can then be used to interact with the evolutionary process by adjusting its parameters <cit.> and/or actively altering the current population, for example via episodes of “hypermutation” <cit.> or migration of individuals from other (sub-)populations <cit.>. On top of that, there exist a few approaches that include diversity into the evolutionary algorithm's objective function allowing us to use evolution's optimization abilities to explicitly achieve higher diversity of solutions <cit.>.An extensive overview of current approaches to increase diversity in evolutionary algorithms is provided in <cit.>, which also defines a helpful taxonomy of said approaches. Whenever a diversity objective can be quantified, it can be used to build a classic multi-objective optimization problem and to apply the vast amount of techniques developed to solve these kinds of problems using evolutionary algorithms <cit.>.The authors of <cit.> address the very important issue of how to efficiently compute diversity estimates requiring to compare every individual of a population to every other. They develop an approach to reduce the complexity of said computation to linear time. However, it might still be interesting to analyze how well certain metrics scale even beyond that, as for example in the present paper we chose to sample the test set for diversity from the population to further save computation time. § EVOLUTIONARY EDIT DISTANCEAs described in the previous Section, there exists a vast amount of approaches to compute a population's diversity (and an individual's diversity with respect to that population). We found, among other things, that from an engineering point of view, many (if not most) of these approaches require the designer of the evolutionary algorithm to adjust certain functions or parameters based on the problem domain <cit.>. This gave rise to the idea of using the genetic operators already programmed for the problem domain to define a domain-independent notion of diversity.The concept this line of thought is based on could be called evolutionary edit distance: Given two individuals x_1 and x_2 we want to estimate how many applications of a genetic operator it would take to turn one of these individuals into the other.[Because of the probabilistic nature of evolutionary algorithms, the evolutionary edit distance would actually be a distribution over integers. If a scalar value is needed, we could then compute the expected evolutionary edit distance.] First, we start off by defining a lower bound on the number of operator applications, i.e., the minimal evolutionary edit distance.We can assume that a given evolutionary process provides the genetic operator o: 𝒟^* →𝒟 where 𝒟 is the problem domain in which our individuals live and 𝒟^* is a list of arbitrary many elements of 𝒟. Most evolutionary algorithms define exactly two instances of genetic operators called mutation m: 𝒟→𝒟 and recombination c: 𝒟×𝒟→𝒟, but we describe the more general case for now. However, in the general case genetic operators perform in a probabilistic manner, meaning that their exact results depend on chance. We describe this behavior mathematically by adding an index s ∈𝒮 to o which represents the seed of a pseudo-random number generator (using seeds of type 𝒮). Then, we can define the minimal evolutionary edit distance mdist: 𝒟×𝒟→ℕ as follows: mdist(x_1, x_2) =0ifx_1 = x_2 min_s ∈𝒮 1 + mdist(o_s(x_1), x_2)otherwise Note that as long as we assume the genetic operator o to be symmetric (which they usually are), mdist is symmetric as well.The minimal edit distance is not an accurate estimate of the actual effort it would take the evolutionary process to turn x_1 into x_2 since the required indexed instances o_s of the genetic operator o may be arbitrarily unlikely to occur in the process. Instead, we want to estimate the expected amount of applications of o given a realistic occurrence of instances of the genetic operator. Sadly, the complexity of this problem is equal to running an evolutionary algorithm optimizing its individuals to look like x_2 and thus potentially equally expensive regarding computational effort as the evolutionary process we are trying to augment with diversity.However, if we want to use mdist to compute the diversity of individuals for a given evolutionary process, we never want to compare arbitrary solution candidates x_1, x_2 ∈𝒟 but will only ever compare individuals within the current population P ⊆𝒟 or at most individuals from the set X with P ⊆ X ⊆𝒟, which is the set of all individuals ever generated by the evolutionary process. Each of those individuals has been generated through the repetitive application of the genetic operators already and so we have a set of concrete instances of o instead of having to reason about all o_s that could be used by the evolutionary process. We write the set of all actually generated instances of o as 𝒪 = {(x_0, o_s_0, x'_0), (x_1, o_s_1, x'_1)..., (x_k, o_s_k, x'_k)} where k+1 is the total amount of evolutionary operations performed and for all (x_i, o_s_i, x'_i) ∈𝒪 the evolutionary process actually constructed x'_i ∈ X by computing o_s_i(x_i).We can thus define the factual evolutionary edit distance edist: X × X →ℕ as the total amount of operations it actually took to turn x into x': edist(x_1, x_2) =0ifx_1 = x_21if ∃ s ∈𝒮: (x_1, o_s, x_2) ∈𝒪1if ∃ s ∈𝒮: (x_2, o_s, x_1) ∈𝒪edist'(x_1, x_2) otherwiseedist'(x_1, x_2) =min_x ∈ X1 +min_s ∈𝒮, (x, o_s, x_1) ∈𝒪edist(o_s(x), x_1) + min_s ∈𝒮, (x, o_s, x_2) ∈𝒪edist(o_s(x), x_2)) Note that edist can only be defined this way when we assume that its parameters x_1 and x_2 have actually been generated through the application of genetic operators from a single base individual only. This is an unrealistic assumption: Completely unrelated individuals can be generated during evolution. Furthermore, defining the evolutionary edit distance this way requires multiple iterations through the whole set of X since we neglect many restrictions present in most genetic operators o.§ PATHS IN THE GENEALOGICAL TREEIn the context of evolutionary processes it seems natural to think of individuals as forming genealogical relationships between each other. These relations correspond to the genetic operators applied to an individual x to create the individual x'. Connecting all individuals (of all generations of the evolutionary process) to their respective children yields a directed, acyclic and usually non-connected graph. Starting from a single individual x, recursively traversing all incoming edges in reverse direction yields a connected subgraph containing all of x's ancestors. We call this graph the genealogical tree of x.Formally, we write 𝔊(x) = (𝔙_x, 𝔈_x) for the genealogical tree of x consisting of vertices 𝔙_x and edges 𝔈_x. For an evolutionary process producing (over all generations) the set of individuals X, it holds for all x_1, x_2 ∈ X that (x_1, x_2) ∈𝔈_x_2 iff x_2 is the result of a variation of x_1. If we consider an evolutionary process with two-parent recombination as its only variation operator, our notion of a genealogical tree is exactly the same as in human (or animal) genealogy.However, most evolutionary algorithms also feature a mutation operator that works independently from recombination. For the genealogical tree, we treat it like a one-parent recombination in that we consider a mutated individual an ancestor of the original one. This approach does not reflect the fact that a single mutation usually has a much smaller impact on the genome of an individual than recombination has. We tackle this issue in Section <ref>.Given these graphs, we can then trivially define the ancestral distance from an individual x_1 ∈ X to another individual x_2 ∈ X as follows: adist(x_1, x_2) = ∞ ifx_1 ∉𝔙_x_20ifx_1 = x_2 min_x ∈ X, (x, x_2) ∈𝔈_x_2 1 + adist(x_1, x)otherwise Note that adist as defined here is still not symmetric, i.e., it returns the amount of variation steps it took to get from x_1 to x_2, which is a finite number iff x_1 is an ancestor of x_2. This also usually means that if adist(x_1, x_2) is finite, adist(x_2, x_1) = ∞.Given two individuals x_1 and x_2, we can use these definitions to compute their latest common ancestor L(x_1, x_2), i.e., the individual with the closest relationship to either x_1 or x_2 that appears in the respective other individual's genealogical tree. Formally, if a (latest) common ancestor exists it is given via: L(x_1, x_2) = _x ∈𝔙_x_1∩𝔙_x_2min(adist(x, x_1), adist(x, x_2)) Note that L is symmetric, so L(x_1, x_2) = L(x_2, x_1). For our definition of genealogical distance we consider the ancestral distance from the latest common ancestor to the given individuals. However, we also want to normalize the distance values with respect to the maximally achievable distance for a certain individual's age. The main benefit here is that when normalizing genealogical distance on a scale of [0; 1], e.g., we can assign a finite distance to two completely unrelated individuals. For this reason we define a function E which computes the earliest ancestor of a given individual: E(x) = _x' ∈𝔙_xadist(x', x) Note that for all x' ∈𝔙_x it holds that adist(x', x) is finite. We can now use the ancestral distance to an individual's earliest ancestor to normalize the distance to the latest common ancestor with respect to the age of the evolutionary process. Note that if x_1 and x_2 share no common ancestor, we set gdist(x_1, x_2) = 1 and otherwise: gdist(x_1, x_2)=min(adist(L(x_1, x_2), x_1), adist(L(x_1, x_2), x_2))/max(adist(E(x_1, x_2), x_1), adist(E(x_1, x_2), x_2)) This genealogical distance function gdist then describes for two individuals x_1, x_2 how close their latest common ancestor is in comparison to their combined “evolutionary age”, i.e., the total amount of variation operations they went through.Following up from the previous Section, we claim that this genealogical distance correlates to the factual evolutionary edit distance between two individuals. It is not an exact depiction, though, because for cousins, e.g., we choose the minimum distance to their common ancestor instead of adding both paths through which they originated from their ancestor. Our reason for doing so is that we want to treat the comparison of cousins to cousins and of parents to children the same way, but the ancestral distance from child to parent is ∞. In the end, we are not interested in the exact values but only in the comparison of various degrees of relatedness, which is why lowering the overall numbers using min instead of summation seems reasonable.In effect, the metric of gdist still appears to be needlessly exact for the application purpose inside the highly stochastic nature of an evolutionary algorithm. And while a lot of algorithmic optimizations and caching of ancestry values can help to cut down the computation time of the employed metric, comparing two individuals still takes linear time with respect to the node count of their ancestral trees, which in turn is likely to grow over time of the evolutionary process. We tackle these issues by introducing a faster and more heuristic approach in the following Section. § ESTIMATING GENEALOGICAL DISTANCE ON THE GENOMEAt first, it seems impossible or at east overly difficult to estimate the genealogical distance (or for that matter, the evolutionary edit distance) of two individuals without knowing about their ancestry inside the evolutionary process. However, life sciences are facing that problem per usual and have found a way to estimate the relationship between two different genomes rather accurately. They do so by computing the similarity in genetic material between two given genomes.To most artificial evolutionary processes, this approach is not directly applicable for a few reasons:* Most evolutionary algorithms use genomes that are much smaller than that of living beings. Thus, it is much harder to derive statistical similarity estimates and the analysis is much more prone to be influenced by sampling error.* In many cases, the genomes used are not homogeneous but include various fields of different data types. Comparing similarity between different types of data requires a rather complex combined similarity metric.* The way genomes are usually structured in evolutionary algorithms means that most to all parts of the genome are subject to selection pressure reducing the variety found between different genomes. The last point may seem odd because, obviously, genomes found in nature are subject to selection pressure as well. However, biology has found that, in fact, most parts of the human genome are not expressed at all when building the phenotype (i.e., a human body) <cit.> and are thus not directly subjected to selection pressure.We can, however, mitigate these problems making a rather simple addition to an arbitrary evolutionary algorithm: Add more genes. As these additional genes do not carry any meaning for the solution candidate encoded by the genome, they are not subjected to selection pressure (iii). We can choose any data type we want for them, so we can adhere to a type that allows for an easy comparison between individuals (ii). And finally, we can choose a comparatively large size for these genes so that they allow for a subtle comparison (i). For the lack of a better name, we call these additional genes by the name trash genes to emphasize that they do not directly contribute to the individual's fitness.For our experiments thus far, we have chosen a simple bit vector of a fixed length τ to encode the added trash genes. Choosing τ too small (2^τ < n where n is the population size) can obviously be detrimental to the distance estimate, but choosing very large τ (2^τ≫ n) has not shown any negative effects in our preliminary experiments. Using bit vectors comes with the advantagethat genetic operators like mutation and recombination are trivially defined on this kind of data structure.Formally, to any individual x ∈ X we assign a bit vector T(x) = ⟨ t_0, ..., t_τ-1⟩ with t_i ∈{0, 1} for all i, which is initialized at random when the individual x is created. Every time a mutation operation is performed on x, we perform a random single bit flip on T(x).[Note that this works for typical mutation operators on the non-trash genes, which tend to change very little about the genome as well. If more invasive mutation operators are employed, a likewise operation on the bit vector could be provided.] For each recombination of x_1 and x_2, we generate the child's trash bit vector via uniform crossover of T(x_1) and T(x_2).We can then compute a trash bit distance tdist between two individuals x_1 and x_2 simply by returning the Hamming distance between their respective trash genes: tdist(x_1, x_2) = 1/τ* H(T(x_1), T(x_2))= 1/τ*∑_i=0^τ - 1 |T(x_1)_i - T(x_2)_i| This metric clearly is symmetric. Again, we normalize the output by dividing it by τ. Furthermore, trash bit vectors allow for a more detailed distinction between the impact of various genetic operators: The expected distance between two randomly generated individuals x_1 and x_2 is 𝔼(tdist(x_1, x_2)) = 0.5. However, the distance between parents and children is reasonably lower: If x is the result of mutating x', we expect their trash bit distance to be 𝔼(tdist(x, x')) = 1/τ. The trash bit distance between a parent x' of a recombination operator and its child x is 𝔼(tdist(x, x')) = 0.25 since the child shares about half of its trash bits with this one parent x' by the nature of crossover, resulting in a Hamming distance of 0 on this subset, and the other half with the other parent, say x”, with which the first parent x' naturally shares about half of its trash bits when no other assumptions about the parents' ancestry apply. This means that for the subset of trash bits inherited from x”, x and x' have a trash bit distance of 0.5, resulting in a 0.25 average for the whole bit vector of x.[These numbers correspond closely to the degrees of genetic relationship mentioned in <cit.>.] If the parents are related (or share improbable amounts of trash bits by chance), lower numbers for tdist can be achieved.These examples should illustrate that the computed trash bit diversity is able to express genealogical relations between individuals. It stresses recombination over mutation but in doing so reflects the impact the respective operators have on the individual's actual genome. We thus propose trash bit vectors as a much simpler and more efficient implementation of genealogical diversity.As is clear from the usage of the “expected value” 𝔼 in these computations, the actual distance between parents and offspring is now always subject to random effects. However, so is their similarity on the non-trash genes as well.[For example, a child generated via uniform crossover has very slim chance of not inheriting any gene material from one parent at all. The same effect can now happen not only on the fitness-relevant genes but also on the genes used for diversity marking.] This kind of probabilistic behavior is an intrinsic part of evolutionary algorithms. However, it may make sense to base the recombination on the trash bit vector on the recombination of the non-trash genes so that probabilistic tendencies are kept in sync. This is up to future research.Finally, the computational effort to compute the trash bit distance is at most times negligible. Computing the distance between two individuals is an operation that can be performed in O(τ) and while we expect there to be a lose connection between population size and the optimal τ, for a given evolution process with a fixed population size, this means that trash bit diversity can be computed in constant time. Trivially, this also means the concept scales with population size and age.§ EXPERIMENTTo verify the practical applicability of the concept of genealogical diversity and its realizations presented in the previous Sections <ref> and <ref>, respectively, we constructed a simple experimental setup: We define a simple routing task in which a robot has to choose a sequence of 10 continuous actions a ∈ℝ×ℝ to reach a marked area. Each action takes exactly one time step and can move the robot across a Manhattan square of 0.5 at most. For each time step the robot remains inside the designated target area, it is rewarded a bonus of +1. In order to reach that area, the robot has to find a path around an obstacle blocking the direct way. Figure <ref> shows a simple visualization of the setup described here.We solved this scenario with four different evolutionary algorithms. All of these use a population size of 20 individuals and have been executed for 1000 generations. For this kind of continuous optimization problem, that is not enough time for them to fully converge. We constructed a standard setup of an evolutionary algorithm with a continuous mutation operator working on a single action at a time and activated with a probability of 0.2. We employ uniform crossover with a probability of 0.3 per individual. A recombination partner is selected from a two-player tournament and offspring is added to the population before the selection step. Furthermore, 2 new individuals per generation are generated randomly.Within this setup, we define a standard genetic algorithm using a fitness function that simply returns the aforementioned bonus for each individual. It performs well but seems to suffer from premature convergence in this setup (see Figure <ref> for all plots). This is the baseline approach all diversity-enabled versions of the genetic algorithm can be tested against.To introduce the diversity of the solutions to the genetic algorithm, we choose the approach to explicitly include the distance of the individual x to other individuals of P in x's fitness. But we do not construct a multi-objective optimization problem (as in <cit.>, e.g.) but simply define a weighting function to flatten these objectives. Formally, given the original fitness function f and an average diversity measure d of a single individual with respect to the population P ⊆ X, we define an adapted fitness function f' as follows: f'(x, P) = f(x) + λ * d(x, P) It is important to note that while we use f' for the purpose of selection inside the evolutionary algorithm, all external analysis (plotting, e.g.) is performed on the value of f only in order to keep the results comparable. Also note that we reduce the computational effort to calculate any distance metric d used in this paper by not evaluating a given individual's diversity against the whole population P but only against a randomly chosen subset of 5 individuals. In our experiments, this approach has been sufficiently stable.Furthermore, we determined the optimal λ for each algorithm using grid search on this hyperparameter. In a scenario like this, where higher diversity yields better results overall, it appears reasonable to think that λ could be determined adaptively during the evolutionary process. This is still up to further research.For evaluation purposes, we provided a domain-specific distance function. In this simple scenario, this can be defined quickly as well and we chose to use the sum of all differences between actions at the same position in the sequence. Figure <ref> shows that this approach takes a bit longer to learn but can then evade local optima better, showing a curve that we would expect from a more diverse genetic algorithm.Finally, we implemented both genealogical distance metrics presented in this paper. We can see in Figure <ref> that both approaches in fact perform comparably, even though trash bit vectors require much less computational effort. For this experiment, we used τ = 32.§ CONCLUSIONIn this paper, we have introduced the expected evolutionary edit distance as a promising target for diversity-aware optimization within evolutionary algorithms. Having found that it cannot be reasonably computed within another evolutionary process, we developed approaches to estimate that distance more efficiently. To do so, we introduced the notion of genealogical diversity and presented a method to estimate it accurately using very little computational resources.The experimental results show the initial viability of the approach used here and allow for many future applications. Some of these have been realized in <cit.>. Other promising directions for future work have been mentioned throughout and include plans to omit the hyperparameter λ by using genealogical diversity within a true multi-objective evolutionary algorithm or by opening λ for self-adaptation by the evolutionary process. Furthermore, from a biological point of view, a genealogical selection process is less common in survivor selection than it is in parent selection. Testing if the metaphor to biology holds in that case would be an immediate next step of research. ACM-Reference-Format | http://arxiv.org/abs/1704.08774v1 | {
"authors": [
"Thomas Gabor",
"Lenz Belzner"
],
"categories": [
"cs.NE"
],
"primary_category": "cs.NE",
"published": "20170427233836",
"title": "Genealogical Distance as a Diversity Estimate in Evolutionary Algorithms"
} |
Experimental Physics V, Center for Electronic Correlations and Magnetism, Institute of Physics, University of Augsburg,86135 Augsburg, Germany Experimental Physics V, Center for Electronic Correlations and Magnetism, Institute of Physics, University of Augsburg,86135 Augsburg, Germany [Present address: ]Helmholtz-Zentrum Dresden-Rossendorf,Bautzner Landstraße 400, 01328 Dresden, GermanyExperimental Physics V, Center for Electronic Correlations and Magnetism, Institute of Physics, University of Augsburg,86135 Augsburg, GermanyExperimental Physics V, Center for Electronic Correlations and Magnetism, Institute of Physics, University of Augsburg,86135 Augsburg, Germany1. Physikalisches Institut, Universität Stuttgart, 70550 Stuttgart, Germany Department of Physics, Budapest University of Technology and Economics and MTA-BME Lendület Magneto-optical Spectroscopy Research Group, 1111 Budapest, HungaryDepartment of Physics, Budapest University of Technology and Economics and MTA-BME Lendület Magneto-optical Spectroscopy Research Group, 1111 Budapest, Hungary Experimental Physics V, Center for Electronic Correlations and Magnetism, Institute of Physics, University of Augsburg,86135 Augsburg, GermanyDepartment of Physics, Budapest University of Technology and Economics and MTA-BME Lendület Magneto-optical Spectroscopy Research Group, 1111 Budapest, HungaryExperimental Physics V, Center for Electronic Correlations and Magnetism, Institute of Physics, University of Augsburg,86135 Augsburg, GermanyInstitute of Applied Physics, Academy of Sciences of Moldova, MD-2028 Chisinau, Republic of Moldova Experimental Physics V, Center for ElectronicCorrelations and Magnetism, Institute of Physics, University of Augsburg, 86135 Augsburg, Germany We report on optical spectroscopy on the lacunar spinels GaV_4S_8 and GeV_4S_8 in the spectral range from 100 to 23 000 cm^-1 and for temperatures from 5 to 300 K. These multiferroic spinel systems reveal Jahn-Teller driven ferroelectricity and complex magnetic order at low temperatures. We study the infrared-active phonon modes and the low-lying electronic excitations in the cubic high-temperature phase, as well as in the orbitally and in the magnetically ordered low-temperature phases. We compare the phonon modes in these two compounds, which undergo different symmetry-lowering Jahn-Teller transitions into ferroelectric and orbitally ordered phases, and exhibit different magnetic ground states. We follow the splitting of the phonon modes at the structural phase transition and detect additional splittings at the onset of antiferromagnetic order in GeV_4S_8. We observe electronic transitions within the d-derived bands of the V_4 clusters and document a significant influence of the structural and magnetic phase transitions on the narrow electronic band gaps.Optical conductivity in multiferroic GaV_4S_8 andGeV_4S_8: Phonons and electronic transitions A. Loidl December 30, 2023 ===============================================================================================§ INTRODUCTION The cluster compounds studied belong to the family of lacunar spinels AV_4S_8, with A = Ga or Ge, consisting of weakly linked cubane (V_4S_4)^n+ and tetrahedral (AS_4)^n- clusters, arranged in a fcc structure with F43m cubic non-centrosymmetric symmetry at high-temperatures. This structure is derived from a cubic normal spinel AV_2S_4 by removing every second A-site ion.The V_4 units form stable, strongly bonded molecules with a unique and collective electron distribution and well-defined spin. In the gallium compound with n = 5, vanadium exhibits an average valence of 3.25 and the V_4 molecule is characterized by 7 electrons. In the germanium lacunar spinel with n = 4, the four V^3+ ions constitute a molecule with 8 electrons.<cit.>In a molecular orbital scheme of V_4 clusters, the highest occupied cluster orbitals are triply degenerate with one or with two unpaired electrons in case of the Ga and Ge compounds, respectively. Hence, each V_4 cluster carries a spin S = 1/2 for the Ga and S = 1 for the Ge compound. Due to partial occupation of the triply degenerate electronic levels, both compounds are Jahn-Teller (JT) active. However, the different electronic configurations of the two compounds result in specific JT distortions and consequently in different crystal symmetries of the orbitally ordered phases: In GaV_4S_8 the V_4 tetrahedra are elongated along one of the cubic body diagonals, resulting in a rhombohedral structure with R3m symmetry,<cit.> while in GeV_4S_8 the vanadium tetrahedra distort with one long and one short V–V bond on adjacent sites, into an orthorhombic Imm2 phase.<cit.>In both compounds, the orbitally ordered phases are ferroelectric.<cit.>In the cubic paramagnetic high-temperature phase, both lacunar spinels exhibit moderate antiferromagnetic (AFM) exchange interactions, indicating stable magnetic moments below room temperature. However, in the orbitally ordered phase GaV_4S_8 exhibits ferromagnetic exchange and subsequently at T = 12.7 K, undergoes a phase transition into a cycloidal phase with ferroelectric excess polarization.<cit.> In addition, at low temperatures and moderate magnetic fields the Ga compound hosts a Néel-type skyrmion lattice, before a fully spin-polarized state is reached at slightly higher magnetic fields.<cit.>In clear distinction, in the orbitally ordered phase of GeV_4S_8, the magnetic exchange remains unaltered and the Ge compound becomes antiferromagnetic below 14.6 K.<cit.> In literature, there exist no reports on further symmetry lowering at the magnetic transition in GaV_4S_8. AFM order in GeV_4S_8 seems to be commensurate and has been associated with the space group Pmn2_1, which is also compatible with ferroelectricity.<cit.> However, there also exist contradictory reports in literature.<cit.> Despite a plethora of interesting phenomena emerging in these compounds, not much is known about phonon properties and about low-lying electronic and orbital transitions in lacunar spinels.The phonon modes of GaV_4S_8 have been studied by Hlinka et al.<cit.> by Raman and infrared spectroscopy. These authors interpreted the phonon spectra at selected temperatures with the aid of ab-initio calculations. By combining group-theory analysis and first-principle calculations of electron-phonon coupling constants, Xu and Xiang<cit.>presented a case study on orbital-order driven ferroelectricity using GaV_4S_8 as a prototypical example. The polar relaxational dynamics at this orbital-order driven ferroelectric transition was studied very recently by THz and broadband dielectric spectroscopy.<cit.>The ferroelectric transition was characterized as order-disorder type, but having first order character. Band structure calculations and resulting magnetic exchange interactions for GaV_4S_8 have also been published by Zhang et al.<cit.> Combined first-principle calculations and an experimental study of the phonon modes in GeV_4S_8 were performed by Cannuccia et al. <cit.> In this work, based on the size of the electronic band gap and on phonon eigenfrequencies, the authors speculated that the room temperature space group in the germanium compound may not be F43m, but rather I4m2 and that this space group probably results from dynamic JT distortions, which are present already far above the structural phase transition. Recent THz studies were reported by Warren et al.<cit.> on polycrystalline and by Reschke et al. <cit.> on single crystalline GeV_4S_8. In the present manuscript, we provide a detailed investigation of the infrared (IR) active phonon modes of GaV_4S_8 and GeV_4S_8 for temperatures ranging from 5 to 300 K. Specifically, we followed the temperature dependencies of eigenfrequencies and damping, focusing on the behavior close to the structural and magnetic phase transitions. In addition, we traced the low-energy electronic transitions below 4 eV to gain insight into the nature of the electronic band gaps and to check for possible temperature-dependent band shifts and for the influence of phase transitions on the band-gap energies. The influence of magnetic order on optical phonon frequencies<cit.> and anomalous absorption edge shifts in magnetic semiconductors<cit.> has been studied in detail, mainly on spinel compounds already some time ago. In many of these compounds, anomalous absorption-edge shifts were observed when entering the low-temperature magnetic phases due to spin-lattice coupling. § EXPERIMENTAL DETAILS Optical spectroscopy has been performed on naturally grown (111) surfaces of well-characterized single crystals. Experimental details concerning sample preparation and characterization of GaV_4S_8 and GeV_4S_8 are given in Refs. widmann:2016a and widmann:2016b, respectively. The optical reflectivity measurements were carried out using the Bruker Fourier-transform spectrometers IFS 113v and IFS 66v/S, both of them equipped with He-flow cryostats (CryoVac). With the set of mirrors and detectors used for these experiments, we were able to cover a frequency range from 100 to 23 000 cm^-1. In the low frequency region (up to 6 000 cm^-1), a gold mirror was used as reference, while at higher frequencies a silver mirror was utilized. For the evaluation of phonon eigenfrequencies and damping, we directly analyzed the measured reflectivity with a standard Lorentz model for the complex dielectric function with the program RefFIT developed by A. Kuzmenko,<cit.>to obtain values of eigenfrequency ω_j, oscillator strength Δϵ_j, and damping parameter γ_j. The complex dielectric constant has been derived from the reflectivity spectra by means of Kramers-Kronig transformation with a constant extrapolation towards low frequencies and a smooth ω^-1.5 high-frequency extrapolation, followed by a ω^-4 extrapolation beyond 8 × 10^5 cm^-1. In analyzing the optical conductivity, we carefully checked the influence of these extrapolation procedures on the results below 2 × 10^4 cm^-1, relevant for this work.§ RESULTS AND DISCUSSION Figure <ref> shows the frequency dependence of the reflectivities measured in GaV_4S_8 and GeV_4S_8 for frequencies up to 23 000 cm^-1 for a series of selected temperatures between 10 and 300 K. Phonons are visible for wavenumbers below 600 cm^-1, while the fingerprints of electronic transitions appear between 3 000 cm^-1 and 6 000 cm^-1.These weak peak-like structures, being the manifestations of the gap edge in the reflectivity, likely correspond to transitions within the d bands of the vanadium V_4 molecules. From detailed band-structure calculations it is clear that the electronic density of states near the chemical potential is dominated by bonding states of the V_4 molecular clusters and that the insulating band gap in both compounds is determined by electronic transitions within vanadium-derived d bands.<cit.>LDA + U calculations yield narrow band gaps of 140 meV and 200 meV for the Ga and Ge compound, respectively.<cit.> Already a first inspection of the reflectivity data makes clear that –- in agreement with the theoretical predictions – the electronic transitions with vanadium d band character are at higher frequencies in the Ge compound when compared to GaV_4S_8. In both materials, these transitions become significantly more pronounced on decreasing temperatures. From resistivity measurements, a band gap of 240 meV has been deduced for the Ga compound,<cit.> while in GeV_4S_8 the gap was found to be about 300 meV.<cit.> From these measurements it was concluded that, in both compounds, the energy gaps are strongly temperature dependent and that at low temperatures hopping transport dominates the dc resistivity. Concerning phonon excitations, in the cubic high-temperature phase, in Fig.<ref> only four phonon modes out of the six IR allowed modes are observed. §.§ Phonons The phonon spectra of GaV_4S_8 and GeV_4S_8 were measured in the far-infrared (FIR) frequency regime between 5 and 300 K. Figure <ref> shows a comparative study of the phonon response of GaV_4S_8 [Figs. <ref>(a), (c), and (e)] and GeV_4S_8 [Figs. <ref>(b), (d), and (f)], in the cubic high-temperature phases [Figs. <ref>(a) and (b)], in the paramagnetic and orbitally ordered [Figs. <ref>(c) and (d)], as well as in the magnetically and orbitally ordered [Figs. <ref>(e) and (f)] phases. In GaV_4S_8, four phonon modes can be observed at room temperature. On lowering the temperature, two additional modes appear just below the structural phase transition at T_JT = 44 K, due to the lowering of the crystal symmetry from cubic to rhombohedral. At the magnetic transition (T_C = 12.7 K) no further phonon splitting is observed in GaV_4S_8. In GeV_4S_8 also four phonon modes appear in the high-temperature cubic phase. Below the symmetry-lowering structural phase transition at T_JT = 30.5 K, where the crystal symmetry changes from cubic to orthorhombic, three additional modes appear in the phonon spectra of GeV_4S_8. In contrast to GaV_4S_8, in GeV_4S_8 a further small splitting of three phonon modes is detected below the antiferromagnetic phase transition at T_N = 14.6 K. This splitting is best seen in the lowest frequency mode, just above 300 cm^-1, where a shoulder evolves at the high-frequency flank, just at the onset of AFM order. To further document this additional phonon splitting in the Ge compound, Fig. <ref> shows enlarged views of the reflectivity from 300 to 340 cm^-1 [Fig. <ref>(a)] as well as from 430 to 500 cm^-1 [Fig. <ref>(b)], in the high temperature cubic phase (300 K), in the paramagnetic and orbitally ordered phase (25 K) and in the AFM ground state (8 K). On decreasing temperatures, the phonon mode which appears close to 305 cm^-1 in the high-temperature cubic phase [Fig. <ref>(a)], hardens with no indications of splitting at the JT transition and softens again in the magnetically ordered phase. The evolution of a shoulder indicates a splitting of this mode with the onset of AFM order. The mode close to 320 cm^-1 [Fig. <ref>(a)] splits at the JT transition into three modes, with one mode undergoing a further splitting at the AFM phase transition. The phonon mode close to 450 cm^-1 [Fig. <ref>(b)] undergoes a splitting in two modes at the JT transition, with the lower frequency mode undergoing a further splitting at the onset of AFM order. We offer three possibilities to explain the emergence of new modes in the antiferromagnetic phase: i) although the symmetry is orthorhombic already below T_JT, an extra term in the distortion upon magnetic ordering is needed to clearly observe the mode splitting. This interpretation seems to be consistent with x-ray scattering results below the Néel temperature: In the magnetically ordered phase only a change of the diffraction profiles was reported,<cit.> indicating further distortions without change of symmetry. ii) This mode splitting could be reminiscent of exchange-driven phonon splitting that was found in a number of spinel compounds.<cit.> Finallyiii), antiferromagnetic spin order induces in principle a doubling of the unit cell, and any weak spin-lattice coupling could be responsible for additional modes. Indeed, neutron diffraction revealed a doubling of either the orthorhombic a or b axis below T_N,<cit.> strongly favoring this scenario.Comparing the IR spectra of the two compounds, at 300 K the eigenfrequencies of the four experimentally observed modes are very similar, but the intensities exhibit marked differences and document significant alterations of the oscillator strengths and hence, of the ionic plasma frequencies of the modes.The primitive cell of lacunar spinels contains one formula unit, which gives a total of 39 modes. Group theory predicts 3 A_1 + 3 E + 3 F_1 + 6 F_2 zone center optical phonon modes in the cubic state, out of which only the F_2 modes are IR active.As clear from Figs. <ref>(a) and (b) only four modes are detected. Reflectivity spectra in the low-frequency range down to 100 cm^-1 (not shown) do not reveal additional phonon excitations. Therefore we conclude that these two modes have too low optical weight. This is in accord with results from Hlinka et al.<cit.>, where also only four IR active modes were detected. Two further F_2 modes, which were theoretically predicted<cit.> close to 133 and 203 cm^-1 are unobservable. Despite significantly different ionic plasma frequencies, this is also true for the Ge compound. In THz transmission experiments, at low temperatures anexcitation close to 90 cm^-1 was observed forGeV_4S_8 (Ref. reschke:2017b), which might be assigned to a calculated B_2 phonon mode at 67.4 cm^-1,obtained by density functional theory for the Imm2 structure.<cit.>For a detailed analysis of the phonon spectra, we fitted all reflectivity data using the RefFIT fit routine.<cit.> To describe the experimental data, the following sum of Lorentz oscillators was used for the complex dielectric function ϵ(ω): ϵ(ω) = ϵ_∞ + ∑_jΔϵ_jω^2_j/ω^2_j - ω^2 - i γ_j ω Here, ω_j, γ_j and Δϵ_j denote eigenfrequency, damping and the dielectric strength of mode j, respectively. ϵ_∞ corresponds to high-frequency electronic contributions to the dielectric constant. It is useful to define the ionic plasma frequency Ω_j of mode j, which is related to the oscillator strength via Ω^2_j = Δϵ_j ω^2_j From the dielectric function ϵ(ω), the reflectivity R(ω) can directly be calculated according to R(ω) = | 1 - √(ϵ(ω))/1 +√(ϵ(ω))|^2 Using this approach, we fitted the reflectivity data and the fit results are shown as solid lines in Fig. <ref>. Overall, the fitted spectraare in satisfactory agreement with the experimental data. Stronger deviations between the fits and experimental data appear in the orbitally and magnetically ordered phases at the lowest wave numbers shown in Fig. <ref>.These deviations could result from strong relaxational modes<cit.> or from low-lying optical phonons<cit.> beyond the experimentally accessible spectral range. One could also speculate that low-lying molecular excitations of the vanadium clusters may become relevant at low temperatures. In systems with strong spin-phonon coupling the line shapes of the phonon modes can become asymmetric.<cit.> In these cases Fano-type functions have to be used to obtain better fits. As strong spin-phonon coupling can be expected in the compounds under consideration, we also triedto improve the fit quality by using Fano line shapes. However, we found no significant improvements of the fits. Figures <ref> and <ref> show the temperature dependences of eigenfrequency ω_j and damping γ_j of the phonon modes of GaV_4S_8 and GeV_4S_8, respectively. For T > T_JT significant temperature dependences of the eigenfrequencies and damping constants can be observed in both compounds. Neglecting effects, which seem to be strongly correlated with the occurrence of phase transitions, for all phonon modes the eigenfrequencies decrease with increasing temperature, while damping effects rather increase. This continuous decrease of eigenfrequencies and continuous inrease of damping effects on increasing temperatures, is the expected behavior of canonical anharmonic dielectric solids, which mainly result from phonon-phonon interactions.<cit.>In the present work, we do not want to present detailed calculations of phonon frequency shifts and damping constants.Our main aim is to derive a phenomenological description of the anharmonic behavior and to determine in this way possible shifts in eigenfrequencies or changes in damping, due to the occurrence of orbital order at the JT transition or due to the onset of magnetic order. Following the work of Balkanski et al.<cit.>, the temperature dependence of the eigenfrequencies of an anharmonic solid can be described by the following expression: ω_j(T) = ω_0j·( 1 - c_j/exp (ħΩ/k_B T) - 1)In this simplified anharmonic model the damping increases with increasing temperature as: γ_j(T) = γ_0j·( 1 + d_j/exp (ħΩ/k_B T) -1)Here ω_0j and γ_0j denote eigenfrequency and the damping of mode j at T = 0 K. ħΩ corresponds to the energy of the characteristic frequency of the decaying phonon modes. The parameters c_j and d_j are additional parameters, which describe the strengths of the anharmonic contributions for eigenfrequencies and damping for different modes. This ansatz for the temperature dependence of frequency shift and damping is only based on three-phonon processes and has to be viewed as a mere parametrization of anharmonic contributions. It has been shown experimentally that four phonon processes play a minor role only and usually can be neglected.<cit.> For GaV_4S_8, the temperature dependence of the eigenfrequencies above the structural phase transition can be described by Eq. (<ref>), with an average characteristic frequency Ω of 243 cm^-1 for all four phonon modes, as indicated by the solid lines in Fig. <ref>. Deviations from anharmonic behavior mainly occur near the structural phase transition. This especially is true for the high-frequency phonon mode [Fig. <ref>(a)], where the eigenfrequency softens when approaching the phase transition from high temperatures and strongly decreases below T_JT. For the phonon modes shown in Figs. <ref>(e) and (g), the eigenfrequencies decrease for T < T_JT. Only the phonon mode close to 370 cm^-1 is almost unaffected by the structural phase transitions [Fig. <ref>(c)]. Due to the symmetry-lowering phase transition, for T < T_JT two additional phonons appear at about 329 cm^-1 and 336 cm^-1 in the orbitally ordered phase [Fig. <ref>(e)]. Below T_JT, GaV_4S_8 has a rhombohedral structure with R3m symmetry. According to group theory and using the Wyckoff positions from Ref. pocha:2000, we expect a total of 21 IR active optical modes at the zone center. Only six modes have been identified, documenting that most of the modes have very low optical weight. It is interesting that the eigenfrequencies remain almost unchanged when crossing the magnetic phase transition. In addition, no further phonon modes were detected in the low-temperature magnetic phase with cycloidal spin structure. Concerning the temperature dependence of the phonon damping, the phonon modes close to 304 [Fig. <ref>(h)] and 370 cm^-1 [Fig. <ref>(d)] can be approximately described by purely anharmonic effects for all temperatures using Eq. (<ref>). Here we used the same characteristic frequency Ω, which was deduced already for the temperature dependence of the frequency shifts.Significant damping effects appear for the phonon modes at 330 cm^-1 [Fig. <ref>(f)] and 440 cm^-1 [Fig. <ref>(b)]. For these modes, the damping is extremely strong in the high-temperature cubic phase and becomes suppressed in the orbitally ordered phase. These phonon modes probably are strongly susceptible to orbital reorientations when approaching the Jahn-Teller transition.As revealed by Fig. <ref>, the temperature dependence of the phonon modes in GeV_4S_8 behave significantly different compared to the Ga compound. The eigenfrequencies almost follow an anharmonic behavior in the cubic phase, undergo slight softening when approaching the structural phase transitions and show significant jumps or splittings in the orbitally ordered phase (T < T_JT = 30.5 K). Additional modes also emerge at the magnetic phase transition. Clear indications of symmetry-lowering splittings are found in GeV_4S_8, upon passing the antiferromagnetic phase transition at 14.6 K. We would like to recall that no splitting of phonon modes was observed in GaV_4S_8 when passing the magnetic phase transition, where cycloidal spin order is established at low temperatures. In the Ge compound, we observe a total of 7 modes in the orbitally ordered phase, whereas 10 modes appear in the antiferromagnetic phase. The orbitally ordered phase in GeV_4S_8 is orthorhombic with Imm2 symmetry. Using the Wyckoff positions of Ref. bichler:2008, we expect 30 IR active modes at the zone center. From those 30 modes, only a small fraction is observed. The rest seems to have too low optical weight. While the splitting of phonon modes or the appearance of new modes at a symmetry lowering structural phase transition follows quite naturally on the basis of symmetry considerations, the splitting of modes at the onset of AFM order is only rarely observed. The influence of magnetic super-exchange interactions on optical phonon frequencies has been studied in detail almost 50 years ago.<cit.> Later-on, the splitting of phonon modes in antiferromagnets due to spin-phonon coupling has been explained in terms of a spin Jahn-Teller effect<cit.> and a linear dependence of the phonon splitting on the exchange coupling has been documented for a number of transition-metal oxides.<cit.> It demands further modeling and a detailed knowledge of the low-temperature structure in GeV_4S_8 to arrive at definite conclusions about the origin of the appearance of a number of new phonon modes in the AFM phase. The damping for the modes close to 450 and 330 cm^-1 [Figs. <ref>(b) and (f)] seems to be dominated by strong orbital fluctuations above the phase transition and it becomes suppressed below T_JT. Only the mode close to 370 cm^-1 [Fig. <ref>(d)] can roughly be described by anharmonic effects in the cubic high-temperature phase. Here the damping becomes strongly enhanced in the orbitally ordered and paramagnetic phase, but is suppressed again when antiferromagnetic order is established.Without detailed microscopic ab-initio modeling it seems almost impossible to explain jumps in eigenfrequencies and damping. One should have in mind that lacunar spinels belong to the rare class of materials, in which the JT transition induces ferroelectricity, which is rather unusual and has been explained in a case study on GaV_4S_8 using group theory analysis in Ref. xu:2015. In this study JT distortions were predicted in a model including electron-phonon coupling. However, even in this case no definite predictions about changes of phonon eigenfrequencies at the Jahn-Teller transition can be made. To do so, the symmetry of low and high temperature phases have to be considered as well as the vibrational pattern and eigenvectors of these specific modes. Things become even more difficult as the JT phase transitions in both compounds under consideration are strongly of first order.<cit.> Overall, the experimentally observed behavior indicates that in both compounds the high-frequency phonon mode and the mode close to 330 cm^-1 are strongly influenced by the JT transition, despite the fact that in the two compounds orbital order induces very different symmetries in the orbitally ordered phases. This is true for both, eigenfrequencies and damping. Changes of eigenfrequencies and damping on passing the onset of magnetic order are minor in the Ga compound. In the Ge compound we identified additional splitting of the phonon modes when passing the AFM ordering temperature. Based on this fact, we conclude that spin-phonon coupling will be strong. In principle, it should be possible to analyze phonon splittings or phonon shifts induced by magnetic exchange interactions. It has been shown by Baltensperger and Helman<cit.> and by Brüesch and D'Ambrogio<cit.> that magnetic exchange interactions influence the force constants and hence the frequencies of lattice vibrations. The frequency shifts of optical phonon modes can be positive or negative, depending on the dominant exchange interactions.<cit.> However, such an analysis is far beyond the scope of this work. Table <ref> summarizes the main experimental results of this phonon investigation in the high-temperature cubic phase at 100 K. The frequencies of the longitudinal optical (LO) modes can be calculated via the ionic plasma frequency, Ω_j, and oscillator strength Δϵ_j. In this case the transverse optical (TO) mode corresponds to the eigenfrequency ω_j. The results in GaV_4S_8 are in relatively good agreement with those of Ref. hlinka:2016 deduced at 80 K.Optical phonon frequencies for GeV_4S_8 at room temperature, as well as in the ferroelectric phase, were also reported by Cannuccia et al.<cit.> At room temperature these authors found transverse optical phonon modes at 305, 323, 366 and 450 cm^-1, not too far from the values reported in this work (see Tab. <ref> and Fig. <ref>). In the orbitally ordered and ferroelectric phase, they observed 6 modes compared to 7 modes, as documented in Fig. <ref> of the present work. Obviously, they missed the splitting of the mode close to 450 cm^-1, as shown in Fig. <ref>(b). We would like to recall that from the observed eigenfrequencies in the Ge compound, Cannuccia et al. argued about a possible I4m2 structure which is stabilized by strong dynamic JT distortions already far above the structural phase transition. Reference cannuccia:2017 did not report on further splittings when passing the magnetic phase transition. As became already clear from a first inspection of Fig. <ref>, the eigenfrequencies of the Ga and the Ge compound are very similar, which certainly stems from the fact that the two elements are neighbors in the periodic table of elements with very similar masses. However, the oscillator strengths are significantly different. The difference is enormous for the mode close to 320 cm^-1, being almost by a factor of 20 stronger for the Ga compound, signaling that very different effective charges are involved in this lattice vibration. Hence, the overall dipolar strength in the Ge compound is by more than a factor of two lower. This strongly reduced phononic dipolar strength signals very different ionicity of the two compounds, with GaV_4S_8 being the one with higher ionicity, while GeV_4S_8 obviously is dominated by significantly stronger covalent bonds. It is also interesting that the high-frequency dielectric constant (ϵ_∞) of the Ga compounds is significantly higher in comparison with the Ge compound. Generally speaking, smaller high-frequency dielectric constants usually correspond to larger band gaps and this indeed seems to be the case for these two compounds. The values of the high-frequency dielectric constants from this optical study, plus the oscillator strengths of the IR active modes (20.7 and 13.2 for the Ga and Ge compound, respectively) can be compared with results from THz spectroscopy: At THz frequencies, values of 15 and 9.2 were determined for the dielectric constants of GaV_4S_8 (Ref. wang:2015) and GeV_4S_8 (Ref. warren:2017), respectively.While the absolute values differ by about 30 %, the ratios of the dielectric constants nicely agree.§.§ Electronic transitions To get a more detailed understanding of the low-lying electronic transitions and specifically, to understand the temperature dependence of the electrical resistivity, which exhibits no simple Arrhenius-like behavior over an extended temperature range,<cit.> we studied the optical conductivity of both compounds in the wave-number regime where the conductivity is dominated by electronic transitions within the d bands of the vanadium V_4 molecules. At this point, it is rather unclear if the vanadium d electrons are delocalized (metallic) within the vanadium clusters and only can hop from cluster to cluster, resulting in a semiconducting behavior of the electrical resistance. Note that in the Ga compound the average valence of the vanadium is 3.25, while it is 4 in GeV_4S_8, resulting in V_4 molecules with 7, respectively 8 electrons. It is interesting to mention that some type of charge ordering has been reported for the Ge compound at low temperatures.<cit.> Figure <ref> shows the frequency dependence of the optical conductivity of GaV_4S_8 up to 5 000 cm^-1 for a series of temperatures between 8 K and room temperature. Already at first sight, we can identify a strong smearing out of the band edge close to 3 000 cm^-1 with increasing temperature. While at 8 K the electronic transition seems to be well defined, resulting in a step-like band edge close to 3 000 cm^-1, at 300 K the band edge shows an unusual soft-edge behavior. Similar soft-edge behavior was identified in the optical conductivity of the prototypical correlated electron system, the antiferromagnetic insulator V_2O_3, where the conductivity just above the band gap raises with a conductivity exponent of 3/2.<cit.>However, we think that in the case of lacunar spinels, with a Jahn-Teller transition inducing orbital order at low temperatures, this soft-edge behavior probably emerges from strong orbital fluctuations dominating the cubic high-temperature phase. This we will outline and discuss later in more detail. Interestingly, an isosbestic point<cit.> with temperature-independent conductivity appears close to 3 000 cm^-1. Isosbestic points often occur in electronically correlated materials and isosbestic features have been observed and analyzed in THz spectra of iron-based superconductors<cit.> and of GeV_4S_8.<cit.> As outlined earlier, ferromagnetic and semiconducting spinel compounds very often show an unusual shift of the absorption edge as function of temperature.<cit.> To document this anomalous behavior of the band edge, the temperature dependence of the band gap has been deduced at a constant conductivity value.<cit.> We were unable to fit unambiguously our σ(ω) curves with a specific frequency dependence for a direct or indirect gap. In order to avoid possible errors by a wrong extrapolation of the wave number dependence of the conductivity, here we also follow the procedure of Ref. harbeke:1966 to get a rough estimate of the band-edge. Therefore, the apparent energy gap Ẽ_g was extractedby taking the intersection of σ(ω) with a horizontal line at a constant conductivity value.In the inset of Fig. <ref>, we show the temperature dependent shift of this absorption edge at constant conductivities of 200 and 400 Ω^-1cm^-1, i.e. just below and above the isosbestic point.Due to significant smearing effects of the band edge, Ẽ_g below the isosbestic point decreases, while above it increases with increasing temperature. No significant anomalies can be detected, neither at the structural nor at the magnetic phase transition. The apparent energy gap as determined from these measurements at room temperature and at lower conductivities is close to 350 meV (≈ 2 800 cm^-1), significantly higher than the value of 240 meV as determined from the temperature dependence of the electrical resistance in the temperature range 70 to 150 K.<cit.> The frequency and temperature dependence of the optical conductivity for GeV_4S_8 is shown in Fig. <ref>. We find a similar evolution of a soft-edge behavior of the band gap as function of temperature: A well-defined step-like band edge in the magnetically and orbitally ordered phase at 10 K evolves into a soft-edge behavior in the high-temperature cubic phase at room temperature. The latter, however, is not as drastic as observed in the Ga compound and the average band gap now has shifted to approximately 4 000 cm^-1. Again, an isosbestic point appears in the frequency- and temperature-dependent conductivity, but now at somewhat higher wave numbers close to 5 000 cm^-1. In the Ge compound this isosbestic point also signals a strong shift of optical weight from lower to higher wave numbers for decreasing temperatures. We followed the same procedure as outlined above to determine an apparent temperature-dependent band gap. Here both values of constant conductivity are chosen below the isosbestic point. The results are shown in the inset of Fig. <ref>. In the case of GeV_4S_8, we find a blue shift of the apparent band edge with decreasing temperature, which amounts almost 10 % and certainly cannot be explained by thermal expansion. This blue shift saturates at low temperatures below the Jahn-Teller transition. Quite astonishingly, in the temperature dependence of the band edge we find clear signatures of the structural and the magnetic phase transitions: The apparent band gap slightly decreases at the Jahn-Teller transition and again increases roughly by the same amount when entering the antiferromagnetic phase. The apparent band gap as determined at the conductivity of 150 Ω^-1cm^-1 and at room temperature approximately amounts 450 meV and is significantly larger than the gap values of 300 meV determined from electrical resistance measurements.<cit.>To further elucidate the energy gaps of GaV_4S_8 and GeV_4S_8, we analyzed the energy dependence of the optical conductivities in more detail. In canonical semiconductors, strict power-law dependences are expected, with exponents depending on the nature of the band gap, with electronic transitions being direct or indirect and IR allowed or forbidden. However, we think that the tremendous smearing of the band energies results from orbital fluctuations in the cubic high-temperature phases. Hence, it seems more appropriate just to perform linear extrapolations of the optical conductivities to zero to get an estimate of the true band gaps that dominate these materials. In Fig. <ref>, this procedure is documented for the Ga compound for a series of temperatures in the magnetic, the paramagnetic and orbitally ordered, and in the high-temperature cubic phases. For the highest and the lowest temperature, dashed lines indicate the linear extrapolation, which was used to analyze the temperature dependence of the band gap. The obtained values are indicated in the inset of Fig. <ref>. First of all, because of this extrapolation towards zero conductivity, now the band gaps are much closer to the values as determined from the electrical resistance. At room temperature, the value close to 250 meV corresponds nicely to the values of 240 meV, determined from the temperature dependence of the resistivity determined at lower temperatures.<cit.> As in the Ge compound, we find a blue shift of the band gap on decreasing temperatures and it reaches values of approximately 350 meV at the lowest temperatures. Astonishingly, this linear extrapolation yields clear anomalies in the temperature dependence of the band gap at the structural as well as at the magnetic phase transition. In contrast, no anomalies were detected in the apparent band gaps (inset of Fig. <ref>). A very similar evaluation is shown in Fig. <ref> for the optical conductivities in GeV_4S_8. Again, the extrapolation procedure to determine the temperature dependence of the band gap is documented by dashed lines for the lowest and highest temperatures.The results are indicated in the inset of Fig. <ref>. On decreasing temperature, the band gap increases from 350 meV up to 475 meV at the lowest temperatures. From electrical resistivity measurements the insulating gap in the Ge compound was of the order of 300 meV,<cit.> not too far from the values determined from these optical experiments. In the case of the Ge compound, the strong blue shift saturates below 100 K. Again, both anomalies, the structural as well as the magnetic phase transition can easily be identified in the temperature dependence of the gap energies. The energy gap seems to be slightly suppressed in the orbitally ordered but paramagnetic phase.Figures <ref> and <ref> document an extreme smearing out of the band edges. At low temperatures both lacunar spinels under consideration exhibit well-defined band edges at low temperatures and a drastic soft-edge behavior at room temperature. These smearing-out effects are closely linked with transfer of optical weight and with the occurrence of isosbestic points. Hence, to unravel the nature of this band-edge softening it seems important to estimate the temperature dependence of the optical weight for a given spectral range for both compounds. The optical or spectral weight is rigorously defined as the area under the conductivity spectrum and only depends on the electronic density or, naively, on the number of electrons. Integrating up to the highest frequencies it must be constant for a given material. Figure <ref> shows the spectral weight for both compounds, accounting for the optical conductivities up to wave numbers of 5 000 cm^-1.This upper limit is chosen rather arbitrarily, but in both compounds the estimated wave number regime covers the main smearing effects of the band edges and also covers both isosbestic points. We have to admit that the experimental uncertainties are drastically different in the two compounds. The scatter of the data is large in the Ga compound. In contrast, the temperature dependence of the optical weight is well defined in GeV_4S_8. Nevertheless, a number of important conclusions can be drawn from this figure. As can be already detected in the raw data presented in Figs. <ref> and <ref>, the transfer of optical weight behaves rather different in the two compounds. For the Ga compound, optical weight is steadily decreasing with increasing temperature. There are indications of step-like drops at the magnetic and at the structural phase transition, but these steps are almost within experimental uncertainty. Overall and beyond experimental uncertainty, with increasing temperature optical weight is shifted from low to high energies. We would like to add, that the optical weight in GaV_4S_8 calculated up to 10 000 and 15 000 cm^-1 reveals a very similar temperature dependence. We conclude that in this case transfer of optical weight must be to much higher energies beyond 15000 cm^-1 (≈ 2 eV).The temperature dependence of the optical weight is completely different for the Ge compound. At low temperatures, the spectral weight is a factor of four lower compared to the GaV_4S_8. On increasing temperatures, it increases at T_N and subsequently decreases at T_JT and then steadily increases up to room temperature gaining almost a factor of two. In contrast to the Ga compound, with increasing temperatures spectral weight is transferred from high to low energies. It is interesting to note that in the Ge compound, when the optical weight is integrated up to 10 000 cm^-1, the spectral weight is constant and temperature independent. This fact documents that in the latter compound optical weight is shifted only at lower wave numbers. To summarize, in GaV_4S_8 a transfer of optical weight from low to high energies takes place on increasing temperatures. In this compound, the transfer of optical weight obviously involves wave numbers significantly larger than 15 000 cm^-1. In GeV_4S_8 this transfer is observed at low temperatures, involving wave numbers below 10 000 cm^-1 only. At the moment we have no reasonable explanation for the transfer of optical weight in these two compounds and we have no explanations for the drastically different behavior. Significant band-edge shifts have been observed for a number of spinel compounds<cit.> and in europium chalcogenides<cit.> and have been explained by magneto-elastic couplings.<cit.>§ SUMMARY AND CONCLUSIONS We have presented a detailed study of the optical reflectivity in the lacunar spinels GaV_4S_8 and GeV_4S_8. Phonons and low-lying electronic transitions are studied as function of temperature. The lacunar spinels of this study are cluster compounds with V_4 molecules with a unique electronic distribution and well-defined spin. Both compounds show orbital order induced ferroelectricity and complex magnetic phases at low temperatures. Different Jahn-Teller distortions result in different crystal symmetries of the orbitally ordered phases and in cycloidal spin order in the Ga and in antiferromagnetic order in the Ge compound.In both compounds, we found four phonon modes in the high-temperature cubic phase, with similar eigenfrequencies but significantly different dielectric strengths. In the temperature dependence of the phonon frequencies and damping coefficients, we detected significant deviations from the canonical behavior of anharmonic solids. At the Jahn-Teller transitions, the splitting of phonon modes is different in the two compounds due to differences in the symmetry of the orbitally ordered phases. No further splitting of modes is observed for the Ga compound in passing to the cycloidal spin order at low temperatures. In marked contrast, the Ge compound, which undergoes antiferromagnetic spin ordering, reveals significant splitting at the magnetic phase boundary. These findings seem to indicate that, in contrast to GaV_4S_8, the magnetic transition in GeV_4S_8 is accompanied by lattice distortions and that spin and lattice degrees of freedom are strongly coupled in this compound. This notion is well consistent with measurements of the dielectric constant ϵ' in both materials: While in GaV_4S_8, ϵ'(T) does not exhibit any significant anomalies at the magnetic transition,<cit.> in the Ge compound both, the absolute values and the dispersion effects of ϵ', become strongly suppressed below the antiferromagnetic phase transition atT_N (Refs. widmann:2016b, singh:2014). This indicates that polar lattice distortions are strongly involved in this transition, which is also in accord with the findings of thermal-expansion measurements performed for GeV_4S_8 (Ref. widmann:2016b).The lowest electronic transition in the Ga and in the Ge compound appears close to 3 000 and 4 000 cm^-1, respectively. Both compounds reveal a smearing out of the band edge on increasing temperatures. We think that this behavior mainly arises from orbital fluctuations when approaching the structural Jahn-Teller transition. At the present stage, it seems unclear if a smearing out of electronic transitions can appear while the phonon modes seem to be much less affected. Of course, as documented in Figs. <ref> and <ref>, some modes reveal a rather high damping in the high-temperature cubic phase, but overall all phonon excitations are rather well defined at all temperatures. It is clear that the relevant electronic transitions as documented in Figs. <ref> and <ref> represent transitions between the d-derived bands of the vanadium ions, which directly participate in the orbital dynamics. However, it also could play a role that orbital fluctuations appear on a time scale faster than vibrational but slower than electronic frequencies. From a linear extrapolation of the optical conductivities, we arrive at an energy gap of 260 meV for the Ga and of 360 meV for the Ge compound at room temperature. These values are in reasonable agreement with band gaps as determined from the temperature dependence of the electrical resistance.<cit.> In both compounds we found considerable shift of spectral weight. The shift of spectral weight on decreasing temperature is completely different and exactly opposite for the two compounds under consideration.We acknowledge partial support by the Deutsche Forschungsgemeinschaft (DFG) via the Transregional Collaborative Research Center TRR 80 “From Electronic Correlations to Functionality” (Augsburg, Munich, Stuttgart). 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"authors": [
"S. Reschke",
"F. Mayr",
"Zhe Wang",
"P. Lunkenheimer",
"W. Li",
"D. Szaller",
"S. Bordács",
"I. Kézsmárki",
"V. Tsurkan",
"A. Loidl"
],
"categories": [
"cond-mat.str-el"
],
"primary_category": "cond-mat.str-el",
"published": "20170427143351",
"title": "Optical conductivity in multiferroic GaV$_4$S$_8$ and GeV$_4$S$_8$: Phonons and electronic transitions"
} |
The Geometry of F_4-Models Mboyo Esole^, Patrick Jefferson^, Monica Jinwoo Kang^ ^ Department of Mathematics, Northeastern University360 Huttington Avenue, Boston, MA 02115, USA ^ Department of Physics, Jefferson Physical Laboratory, Harvard University 17 Oxford Street, Cambridge, MA 02138, [email protected],[email protected], [email protected]: We study the geometry of elliptic fibrations satisfying the conditions of Step 8 of Tate's algorithm. We call such geometriesF_4-models, as the dual graph of their special fiber is the twisted affine Dynkin diagram F_4^t.These geometries are used in string theory to model gauge theories with the exceptional Lie group F_4 on a smooth divisor S of the base.Starting with a singular Weierstrass model of an F_4-model,we present a crepant resolution of its singularities.We study the fiber structure of this smooth elliptic fibration and identify the fibral divisors up to isomorphism as schemes over S. These are ℙ^1-bundles over S or double covers of ℙ^1-bundles over S.We compute basic topological invariants such as the double and triple intersection numbers of the fibral divisors and the Euler characteristic of the F_4-model. In the case of Calabi-Yau threefolds, we computethe linear form induced by the second Chern class andthe Hodge numbers. We alsoexplore the meaning of these geometries for the physics of gauge theories in five and six-dimensional minimal supergravity theories with eight supercharges. We also introduce the notion of“frozen representations” and explore the role of the Stein factorization in the study of fibral divisors of elliptic fibrations.Keywords: Elliptic fibrations, Crepant morphism, Resolution of singularities, Weierstrass models§ INTRODUCTIONAn elliptic fibration is a proper projective morphism φ: Y⟶ B between normal varieties such that the generic fiber is a nonsingular projective curve of genus one and the fibration is endowed with a rational section.Under mild assumptions, an elliptic fibration is birational to a potentially singular Weierstrass model <cit.>.The locus of points of B over which the elliptic fiber is singular is called the discriminant locus.The discriminant locus of a Weierstrass model is a Cartier divisor that we denote by Δ.The type of the fiber over the generic point of an irreducible component of the discriminant locusof an elliptic fibration is well understood following the work of Kodaira <cit.>, Néron<cit.>, and Tate<cit.>.Fibers over higher dimensional loci are not classified and are the subject of much interdisciplinary research by both mathematicians and physicists <cit.>.Crepant resolutions of a Weierstrass model are relative minimal models in the sense of Mori's program <cit.>. Different crepant resolutions of the same Weierstrass model areconnected to each other by a sequence of flops. §.§ G-models and F-theoryA classical problem in the study of elliptic fibrations is understanding the geometry of the crepant resolutions of Weierstrass models and their flop transitions. A natural set of singular Weierstrass models to start with are the G-models.The constructions of G-models are deeply connected to the classification of singular fibers of Weierstrass models and provide an interesting scene to explore higher dimensional elliptic fibrations witha view inspired by their applications to physics. We follow the definitions and notation of Appendix Cof <cit.>.The framework of G-models naturally includes a geometric formulation of basic notions of representation theory such as the theory of root systems and weights of representations. The data characterizing a G-model can be understood in the framework of gauge theories, which provides a natural language to talk about the geometry of these elliptic fibrations. F-theory enables a description of gauge theories in string theory and M-theory via geometric engineering based on elliptic fibrations <cit.>.The data of a gauge theory that can be extracted from an elliptic fibration are itsLie algebra, itsLie group, and the set of irreducible representations defining how charged particles transform under the action of the gauge group. In F-theory, the Lie algebrais determined by the dual graphs of the fibers over the generic points of the irreducible components of the discriminant locus of the elliptic fibration. The Mordell-Weil group of the elliptic fibration isconjectured to be isomorphic to the first homotopy group of the gauge group <cit.>. Hence, the Lie group depends on both the singular fibers and the Mordell-Weil group of the elliptic fibration.In F-theory, the singular fibers responsible for non-simply laced Lie algebras are not affine Dynkin diagrams, buttwisted affine Dynkin diagrams, as presented in Table <ref> and Figure <ref>, respectively, on pages figure.F4 and Table:DualGraph. These twisted affine Dynkin diagrams are the Dynkin duals of the corresponding affine Dynkin diagrams.The Dynkin dual of a Dynkin diagram is obtained by inverting all the arrows. In the language of Cartan matrices, two Dynkin diagramsare dual to each other if their Cartan matrices are transposes of each other. The Langlands duality interchangesB_n and C_n,but preserves all the other simple Lie algebras. For affine Dynkin diagrams, only the ADE series are preserved under the Langlands duality. In particular, the Langlands duals of B_n, C_n, G_2 andF_4 are respectively denoted in the notation of Carter asB^t_n, C^t_n, G^t_2 andF^t_4.§.§ F_4-models: definition and first properties One of the major achievements ofF-theoryis the geometric engineering of exceptional Lie groups.These elliptic fibrations play an essential role in the study of superconformal field theories even in absence of a Lagrangian description. The study of non-simply laced Lie algebras in F-theory started inMay of 1996 during the second string revolutionwitha paperofAspinwall and Gross <cit.>, followed shortly afterwards by the classic F-theory paper of Bershadsky, Intriligator, Kachru, Morrison, Sadov, and Vafa<cit.>.M-theory compactifications giving rise to non-simply laced gauge groups are studiedin <cit.>.In this paper,we study the geometry of F_4-models, namely,G-models with G=F_4, the exceptional simple Lie group of rank 4 and dimension 52.F_4 is asimply connected and non-simply lacedLie group. An F_4-model is mathematically constructed as follows.Let B be a smooth projective variety of dimension two or higher. Let S be an effective Cartier divisor in B defined as the zero scheme of a section s of a line bundle 𝒮.Since S is smooth, the residue field of its generic point is a discrete valuation ring. We denote the valuation with respect to S asv_S.An F_4-model is defined by the crepant resolution of the followingWeierstrass model y^2z=x^3+ s^3+αa_4,3+α x z^2 + s^4 a_6,4 z^3, α∈ℤ_≥ 0,where v_S(a_6,4)=0, and either v_S(a_4,3+α)=0 or a_4,3+α=0. We assume thata_6,4 is generic; in particular, a_6,4 is not a perfect square modulo s. This ensures that the generic fiber is of typeIV^*ns rather thanIV^*s.The defining equation (<ref>) is of type (c6) in Néron's classification of minimal Weierstrass models, definedover a perfect residue field of characteristic different from 2 and 3 <cit.>.This corresponds to Step 8 of Tate's algorithm <cit.>. Hence, the geometric fiber over the generic point of S is of Kodaira type IV^*. As discussed earlier, the generic fiber over Shas a dual graph that isthe twisted Dynkin diagram F^t_4, which isthe Langlands dual of the affine Dynkin diagramF_4.The structure of the generic fiber of the F_4-model is due to the arithmetic restriction that a_6,4 is not aperfect square modulo s.The innocent arithmetic condition which characterizes a F_4-model has surprisingly deep topological implications for the elliptic fibration.Whena_6,4is aperfect square modulo s, the elliptic fibration is called an E_6-model; the generic fiberIV^*ns is replaced by the fiberIV^*s whose dual graphis theaffine Dynkin diagram E_6. Both E_6-models and F_4-models share the same geometric fiber over the generic point of S; however, the generic fiber type is different.Both are characterized byStep 8 of Tate's algorithm, but whilethegeneric fiber of an E_6-model is already made of geometrically irreducible components, the generic fiber of an F_4-model requires a quadratic field extension to make all its componentsgeometrically irreducible. The Weierstrass model of an E_6-model has more complicated singularities, and thus requires amore involved crepant resolution.Ina sense, the F_4-model is a more rigidversion of the E_6-model as some fibral divisors are glued together by the arithmetic condition on a_6,4. It follows from the Shioda-Tate-Wazir theorem <cit.> that the Picard number of an E_6-model is bigger than the one for an F_4-model. Furthermore, their Poincaré-Euler characteristics are also different, as was recently analyzed in <cit.>. A crepant resolution of the Weierstrass model of an E_6-model has fourteen distinct minimal models <cit.>, while that of an F_4-modelhas only one <cit.>. An F_4-model is a flat elliptic fibration, while this is notthe case for an E_6-model of dimension fouror higher, as certain fibers over codimension-three points contain rational surfaces <cit.>. F_4-models are also studied from different points of view in <cit.>.The discriminant locus of the elliptic fibration (<ref>) is Δ= s^8 (4s^1+3α a_4,3+α^3+27 a_6,4^2). The discriminant is composed of two irreducible components not intersecting transversally. The first one is the divisor S, and the fiber above its generic point is of type IV^*ns. The second component is a singular divisor, and the fiber over its generic point is a nodal curve, i.e. a Kodaira fiber of type I_1. The fiber I_1 degenerates to a cuspidal curve over a_4,3+α=a_6,4=0. The two components of the discriminant locus collide at s=a_6,4=0, where we expect the singular fiber IV^*ns to degenerate, further producinga non-Kodaira fiber. §.§ Representations associated to an F_4-model and flops. Todetermine the irreducible representations associated with a given G-model, we adopt the approach of Aspinwall and Gross<cit.>, which is deeply rooted in geometry.The representation associated to a G-model ischaracterized byits weights, which are geometrically given bythe intersection numbers, with a negative sign, of the fibral divisors withcurves appearing over codimension-two loci over which the generic fiber IV^*ns degenerates. This approach,also used in <cit.> and closely related to the approach of<cit.>, transcends its application to physics and allows an intrinsic determination of a representation for any G-model. The representation induced by the weights of vertical curves over codimension-two points is not alwaysphysical as it is possible that no hypermultiplet is charged under thatrepresentation. In such a case, the representation is said to be “frozen" as discussed in <ref>.For F_4-models, we find that over V(s,a_6,4), the fiber IV^*ns degenerates to a non-Kodaira fiber of type 1-2-3-4-2. This fiber consists ofa chain of rational curves intersecting transversally; each number gives the multiplicity of the corresponding rational curve. We refer to thisnon-Kodaira fiber by the symbolIV^*_(2).The last two nodes, of multiplicities 4 and 2,have weights in therepresentation 26 of F_4, namely 0 1-2 1 and 0 01 -2 as illustrated inFigure<ref> on page Figure:Degeneration. The representation 26 is a quasi-minuscule fundamental representation comprised oftwo zero weights and 24 non-zero weights that form a unique Weyl orbit.The possibility of flops between different crepant resolutions of the same singular Weierstrass modelcan be explained by the relative minimal model program <cit.>. For each non-simply laced Lie algebra 𝔤_0, there is a specificquasi-minuscule representation 𝐑_0 defined by the branching rule 𝔤=𝔤_0⊕𝐑_0, where 𝔤_0 is defined by afolding of 𝔤 of degree d. The degree d is the ratio of the squared lengthsoflong roots and short roots of 𝐑_0. In other words, except for G_2, for which d=3, all other non-simply laced Lie algebras have d=2.This branching rule is related to the arithmetic degeneration K^ns→ K^s, where K is a Kodaira fiber. For F_4, we have E_6=F_4⊕26, and the representation 𝐑_0 coincides with the representation we find by computing weights of the curves overV(s,a_6,4). §.§ Counting hypermultiplets:Witten's genus formula M-theory compactified on an elliptically fibered Calabi-Yau threefoldY gives risetoa five-dimensional supergravity theory with eight supercharges coupled to h^2,1(Y)+1 neutral hypermultiplets and h^1,1(Y)-1 vector multiplets <cit.>. Taking into account the graviphoton, there are a total of h^1,1(Y) gauge fields. Thekinetic terms, the Chern-Simons coefficients of the vector multiplets, and the graviphoton are all completely determined by the intersection ring of the Calabi-Yau variety.Witten determined the number of states appearing when a curve collapses to a point using a quantization argument <cit.>. In particular, he showed that the number of hypermultiplets transforming in the adjoint representationisthe genus of the curve S over which the gauge group is localized. Aspinwall, Katz, and Morrison subsequently applied Witten's quantization argument tothe case of non-simply lacedgroups in <cit.>. If S^' is a d cover of S with ramification divisor R, then number n_𝐑_0 of hypermultiplets transforming inthe representation 𝐑_0 is given by <cit.> n_𝐑_0=g'-g,where g' is the genus of S' and g is the genus of S. This method is consistent with the six-dimensional anomaly cancellation conditions <cit.>. Computing g'-g is a classical exercise whose answer is given by the following theorem. Let f:S'→ S be a finite, separable morphism of curves of degree d branched with ramification divisor R. If g' is thegenus of S', g is the genus of S,and R is the ramification divisor, then g'-g=(d-1)( g-1) +1/2deg R. Physically, this is the number of charged hypermultiplets in the representation 𝐑_0, as expected from Witten's quantization argument[ This is a direct application ofthe Riemann-Hurwitz's theorem (Theorem <ref>) in the context of Witten's genus for non-simply laced groups<cit.>. Seesection 3 of <cit.>. ]:n_𝐑_0=(d-1)( g-1)+1/2degR.For an F_4-model, 𝐑_0=26,the ramification locus R isV(s,a_6,4), and its degree isdeg R= 12 (1-g)+2 S^2.Hence, by Witten's argument, we expect for a generic F_4-model the following multiplicities: n_52=g, n_26= 5(1-g) +S^2.We later derivein Theorem <ref> the number of matter representations from a direct comparison of the triple intersection numbers and the one-loopIntrilligator-Morrison-Seiberg prepotential of a five-dimensional gauge theory <cit.>.This matches exactly the number n_26 derived by Witten's genus formula. This provides aconfirmation of the number of charged hypermultiplets from a purely five-dimensional point of view, thereby avoiding a six-dimensional argumentbased on cancellations of anomalies <cit.> and thesubtleties of the Kaluza-Klein circle compactification <cit.>.§.§ Arithmetic versus geometric degenerations Given an elliptic fibration φ:Y→ B, ifS is an irreducible component of the discriminant locus, the generic fiber over S can degenerate further over subvarieties of S. We distinguish between two types of degenerations <cit.>.A degeneration is said to be arithmetic if it modifies the type of the fiber without changing the type of the geometric fiber. A degeneration is said to be geometric if it modifies the geometric type of the fiber.[ Arithmetic degeneration] Let K be a Kodaira fiber. Then, K^ns→ K^s or K^ns→ K^ss are arithmetic degenerations.[Geometric degeneration] Let K be a Kodaira fiber. Consider the non-split fiber K^ns. Denote by K_(d) the non-Kodaira fibers defined in the limit where d non-split curves of K^ns coincide.Then the fiber K^ns→ K_(d) is a geometric degeneration.Arithmetic degenerations and geometric degenerations are respectively responsible for non-localized and localized matter in physics.For example, in the case of an F_4-model over a base of dimension three or higher, we have the degeneration IV^*ns→IV^*s over the intersection of S with any double cover of a_6,4 as it is clear from the explicit resolution of singularities. Such a double cover has equation ζ^2=a_6,4 where ζis a section of ℒ^⊗ 3⊗𝒮^⊗ 2.We can have such an arithmetic degeneration over any point of S away from V(s,a_6,4).In the case of an F_4-model, we get a fiber of type 1-2-3-4-2, which is of the type IV^*_(2) discussed in Example <ref>. The geometry of the fiber shows explicitly thatlocalized matter fields at V(s,a_6,4) are in the representation 26and do not come from an enhancementF_4→E_6. That is because over the locusV(s,a_6,4), the generic fiber is thenon-Kodaira fiber 1-2-3-4-2. Sucha fibercan only be seen as the result of an enhancement of type IV^*ns (F_4) to either an incomplete III^* (E_7) or an incomplete II^* (E_8), depending on the valuation ofthe Weierstrass coefficient a_4. We have either the enhancement F_4⟶E_7 if v(a_4)=3, or F_4⟶E_8 if v(a_4)≥ 4. [The fiber 1-2-3-4-2 also appears in Miranda's models atthe transverse collision II+IV^* where it is presented as a contraction of a fiber of type II^* (E_8), see Table 14.1 on page 130 of <cit.>. ]§.§ Frozen representations As we have explained before, we identify representations by their weights, and we compute the weights geometrically by the intersection of fibral divisors with vertical curves located over codimension-two pointsonthe base.It is important to keep in mind that the presence of a given weight is a necessary conditionbutnota sufficient conditionfor the existence of hypermultiplets transforming underthe corresponding representation.We always have to keep in mindthat the geometric representations deducedby the weights of vertical curves over codimension-two points are not necessarilycarried by physical states. Arepresentation 𝐑 deduced geometrically on an elliptic fibration issaid to be frozenwhen the elliptic fibration has vertical curves (over a codimension-two locus of the base) carrying the weights of therepresentation 𝐑, butno hypermultiplet is charged under the representation 𝐑.It is known that the adjoint representation is frozen when the gauge group is on a curve of genus zero <cit.>.However, it is less appreciated that other representations can also be frozen. A natural candidate is the representation 𝐑_0 discussed in <ref> for the case of a non-simply laced gauge group. We will discuss the existence of a frozen representation foran F_4-model in <ref>.In particular, Theorem <ref> asserts that the representation 26 is frozen if and only if the curve S has genus zero and self-intersection -5. It is a folklore theorem of D-brane model building that the number of representations appearing at the transverse collision of two branes is the number of collision points.In other words, one would expect one hypermultiplet for each intersection points[ There are some subtleties: when overa collision point the same weight is induced by n distinctvertical curves, we can have up n hypermultiplets localized at that point. See for example <cit.>.]. While this is true for localized matter fields, it is not usually true whenthe same representationappears both as localizedandnon-localized.As a rule of thumb, whenlocalized and non-localized matter fields transforming inthe same representation coexist, the number of representations is given by assuming that all the matter is non-localized <cit.>. The notion of frozen representationdiscussed in this paper should not be confused with the frozen singularities of ref. <cit.>. §.§ Summary of resultsThe purpose ofthis paper is to study the geometry of F_4-models.We define an F_4-model as an elliptic fibration over a smooth variety of dimension two or higher such that the singular fiber over thegeneric point of a chosenirreducible Cartier divisor S is of typeIV^*ns, and the fibers are irreducible (smooth elliptic curves, type II or type I_1) away from S. Such an F_4-model is realized by a crepant resolution ofa Weierstrass model, whose coefficients have valuations with respect to S, matching the generic case of Step 8 of Tate's algorithm. AnF_4-model canis always birationaltoa singular short Weierstrass equation whose singularities are due to the valuations v_S(c_4)≥ 3 and v_S(c_6)=4 of its coefficients.Such a singular Weierstrass equation can betraced back to Néron's seminal paper <cit.>where it corresponds to type c6. The road map to the rest of thepaper is the following.In section <ref>, we summarize our most common conventions and basic definitions.We also discuss Step 8 of Tate's algorithm, which characterizes the Weierstrass model of F_4-models.In section <ref> (see Theorem <ref> on page Thm:blowups), wepresent a crepant resolution of the singular Weierstrass model, giving a flat fibration.The resolution is given by a sequence of four blowups with centers that are regular monomial ideals.In section<ref>, we analyze in details the degeneration of the singular fiber and determine the geometry of the fibral divisors (see Theorem <ref> onpage Thm:fibralGeom andFigure <ref> on pageFig:FibralDiv).The generic fiber over S degenerates along V(a_6,4)∩ S to produce a non-Kodaira fiber of type 1-2-3-4-2.Thisnon-Kodaira fiber appears asan incomplete Kodaira fiber of type III^*or II^* resulting from the (non-transverse) collision of the divisor S with the remaining factor of the discriminant locus. Such a collision is not of Miranda-type since it involves twofibers of different j-invariants<cit.>. We show that the fibral divisors D_3 and D_4 corresponding to the root α_3 and α_4 of the F_4 Dynkin diagram are not ℙ^1-bundles over the divisor S, but rather double covers of ℙ^1-bundles over S, with ramification locusV(a_6,4). The geometry of these fibral divisors is illustrated in Figure <ref>.The difference is important since it affectsthe computation of triple intersection numbers and the degeneration of the fibers in codimension two, which is responsible for the appearance of weights of therepresentation 26.We use the Stein factorization to have more control on the geometry of D_3 and D_4. Consider the morphism f: D_3→ S.Thegeometric generic fiber is not connected and consists of two rational curves.Since the morphism is proper, we consider itsStein factorizationD_3f'⟶S'π⟶ S.By definition, the morphism π: S'→ S is a finite map of degree two; each geometric point of the fiber represents a connected component of the fiber of D_3→ S. The morphism f': D_3→ S' has connected fibers that are all smooth rational curves. Hence, f': D_3→ S' givesD_3 the structure of a ℙ^1-bundle over S' rather than over S.Wedetermine in<ref> the geometric weights that identify the representation naturally associated to the degeneration of the generic fiber over the codimension-two loci. The last two nodes of the fiber 1-2-3-4-2 are responsible for generating the representation 26. In section <ref>, we compute the following topological invariants: the Euler characteristic of the elliptic fibration, the Hodge numbers in the Calabi-Yau threefold case,the double and triple intersection numbers of the fibral divisors(Theorem <ref> on page Thm:degeneration), and the linear form induced in the Chow ring by the second Chern class in the case of a Calabi threefold.In section<ref>, we leverage our understanding of the geometry to make a few statements on the physics of F_4 gauge theories in different dimensions.We specialize to the case of a Calabi-Yau threefold and consider an M-theory compactified on an F_4-model that is also a Calabi-Yau threefold.We compute the number ofhypermultiplets in the adjoint and fundamental representations using the triple intersection numbers.We then match them to the coefficients of thefive-dimensional cubic prepotential computed at the one-loop level in <cit.>. We check that the resulting spectrum is consistent with an anomaly-free parent six-dimensional gauge theory.Finally,in <ref> we discuss in detail the existence of frozen representations for an F_4-model.§ BASIC CONVENTIONS ANDDEFINITIONS We work over the complex numbers and assume that B is a nonsingular variety, ℒ is a line bundle over B, and S=V(s) is a smooth irreducible subvariety of B given by the zero scheme of a section s of a line bundle 𝒮.We use the conventions of Carterand denote an affine Dynkin diagram by 𝔤̃, where 𝔤 is the Dynkin diagram of a simple Lie algebra <cit.>. We write 𝔤̃^t for the twisted Dynkin diagram whose Cartan matrix is the transpose of the Cartan matrix of 𝔤̃. This notation is only relevant when 𝔤is not simply laced, that is, for 𝔤= G_2, F_4, B_3+k, or C_2+k.Given a vector bundle 𝒱, we denote by ℙ[𝒱] the projective bundle of lines of 𝒱. In intersection theory, we follow the conventions of Fulton <cit.>.We denote the geometric fibers of an elliptic surface by Kodaira symbols. To denote a generic fiber, we decorate the Kodaira fiber by an index “ns” , “ss” , or “s” that characterizes the degree of the field extension necessary to move from the generic fiber to the geometric generic fiber. §.§ Geometric weights Let φ: Y→ B be a smoothflat elliptic fibration whose discriminant has a unique component S over which the generic fiber is reducible with dual graph the affine Dynkin diagram 𝔤̃^t.We denote the irreducible components of the generic fiber over S asC_a.If 𝔤 is not simply laced, the curves C_a are not all geometrically irreducible. Let D_a be the fibral divisors over S. By definition, φ^* (S)=∑_a m_a D_a. The curve C_a can also be thought of as the generic fiber of D_a over S. Let C be a vertical curve of the elliptic fibration. We define the weight of a vertical C with respect to a fibral divisor D_a as the intersection number ϖ_a(C):=- ∫_Y D_a · C. Using intersection of curves with fibral divisors to determine a representation from an elliptic fibration is a particularly robust algorithmsince the intersections of divisors and curves are well-defined even in the presence of singularities <cit.>.We can ignore the intersection number of the divisor touching the section of the elliptic fibration as it is fixed in terms of the others, thus allowing us to write ϖ(C)=ϖ_1(C),ϖ_2(C),⋯ , ϖ_n(C).We interpret ϖ(C) as the weight of the vertical curve C in the basis of fundamental weights. This interpretation implies that the fibral divisors play the role of co-roots of 𝔤, while vertical curves are identified with elements of the weight latticeof 𝔤.The notion of a saturated set of weights is introduced inBourbaki (Groups and Lie Algebras,Chap.VIII.7. Sect. 2.) and providesthe algorithm to determine a representation from a subset of its weights. See <cit.> for more details. It follows from the general theory of elliptic fibration that the intersection of the generic fibers with the fibral divisors gives the invariant form of the affine Lie algebra 𝔤̃^t, where 𝔤 is the Lie algebra of G.The matrixϖ_a(C_b) is the invariant form of the Lie algebra 𝔤̃ in the normalizationwhere short roots have diagonal entries 2. §.§ Step 8 of Tate's algorithmWe follow the notation of Fulton <cit.>. The terminology is borrowed from <cit.>.Let Y_0⟶ B be aWeierstrass model over a smooth base B, in which we choose a smooth Cartier divisor S⊂ B.The local ring𝒪_B,η in B of the generic point η of S is a discrete valuation ring with valuation v_S given by the multiplicity along S.Using Tate's algorithm, the valuation of the coefficients of the Weierstrass model with respect to v_S determines the type of the singular fiber over the generic point of S.Kodaira fibers refer to the type of the geometric fiber over the generic point of irreducible components of the discriminant locus of the Weierstrass model. AnF_4-model describesthe generic case ofStep 8 ofTate's algorithm, which characterizes theKodaira fiber of type IV^*ns inF-theory notation or IV^*_2 in the notation of Liu. By definition, Kodaira fibers classify geometric fibers over the generic point of a component of the discriminant locus of an elliptic fibration.When the elliptic fibration is given by a Weierstrass model, this can be expressed in the language of a discrete valuation ring.Let S be the relevant component. We assume that S is smooth with generic point η.The local ringat η defines a discrete valuation ring with valuation v that is essentially the multiplicity along S.We then have the following characterization:Ifv(a_1)≥ 1, v(a_2)≥ 2, v(a_3)≥ 2, v(a_4)≥ 3, v(a_6)≥ 4 and the quadric polynomial Q(T)=T^2 + a_3,2T-a_6,4 has two distinct solutions, then thegeometric special fiber is of Kodaira type IV^*. If the roots of Q(T) are rational in the residue field, the generic fiber is of typeIV^*s,otherwise (if the solutions are not rational in the residue field) the generic fiber is oftypeIV^*ns.The discriminant of the quadric Q(T)=T^2 + a_3,2T-a_6,4, is exactly b_6,4. It follows that the fiber is of type IV^* if and only if v(b_6,4)=0. Moreover, the fiber is of either type IV^*s orIV^*ns, depending respectively on whether or not b_6,4 is a perfect square.In view of the multiplicities, we can safely complete the square in y and the cube in x and write the Tate equation of aIV^*ns model as y^2 z=x^3 + a_4,3+α s^3+αx z^2 + s^4 a_6,4 z^3, α∈ℤ_≥ 0,where a_6,4 isnot a perfect square modulo s. The simplest way to identify a fiber of type IV^*ns is to use the short Weierstrass equation since it does not require performing any translation.v(c_4)≥ 3,v(c_6)=4c_6 not a square modulo sIV^*ns. The condition on c_4 and c_6 can be traced back to Néron and forces the discriminant to have valuation 8. Néron also points out that a fiber of typeIV^* is uniquely identified by the valuation of its j-invariant and its discriminant locus:v(j)>0 and v(Δ)=8IV^*. This implies in particular that a fiber of type IV^* has a vanishing j-invariant. § CREPANT RESOLUTIONLetX_0=ℙ[𝒪_B⊕ℒ^⊗ 2⊕ℒ^⊗ 3] be the projective bundle in which the singular Weierstrass model is defined asa hypersurface.The tautological line bundle of the projective bundle X_0 is denoted 𝒪(-1) and its dual 𝒪(1) has first Chern class H=c_1 ( 𝒪(1) ). Let X be a nonsingular variety. Let Z⊂ X be a complete intersection defined by the transverse intersection of r hypersurfaces Z_i=V(g_i), where g_i is a section of the line bundle ℐ_i and (g_1, ⋯, g_r) is a regular sequence. We denote the blowup of a nonsingular variety X along the complete intersection Z by(X0) at (0,-.3)X; (X1) at (3,-.3)X.; [big arrow] (X1) – node[above,midway](g_1,⋯ ,g_r|e_1) (X0);The blowup of X with center Z is the morphism f:X=𝐏𝐫𝐨𝐣_X(⊕_d ℐ^d)→ X. The exceptional divisor of f is the pre-image of the center Z, that is, Z=𝐏𝐫𝐨𝐣_X(⊕_d ℐ^d/ℐ^d+1). The exceptional divisor is f-relativelyample. IfZ is a complete intersection, then ℐ/ℐ^2 is locally free. Hence,𝐒𝐲𝐦^d ( ℐ/ℐ^2)=ℐ^d/ℐ^d+1 and Z=ℙ_X(ℐ/ℐ^2). The normal sheaf N_Z|X is 𝒪_Z̃(-1), E_1=c_1 (𝒪_Z(1))is the first Chern class of the exceptional divisor Z=V(e_1), and [Z]=E_1∩ [X]. We abuse notation and use the same symbols for x, y, s, e_i and their successive proper transforms. We also do not write the obvious pullbacks. Let Z⊂ X be asmooth complete intersection of n+1 hypersurfaces meeting transversally. Let Y be a hypersurface in X singular along Z. If Y has multiplicity n along Z, then the blowup of X along Z restricts to a crepant morphism Y⟶ Y for the proper transform of Y.Let X=Bl_Z Xbe the blowup of X along Z and E be the class of the exceptional divisor. Thenc_1(TX)=f^* c_1(TX)-n E. Since Y has a multiplicity n along Z, we havef^* Y=Y+n E, where Y is the proper transform of Y.By adjunction, c_1(Y)= c_1(X)-Y=f^* c_1(X)-f^* Y=f^* c_1(Y).Consider thefollowingWeierstrass equation where S=V(s) is a Cartier divisor of the base B :ℰ_0: y^2z=x^3 +s^3+α fxz^2 + s^4 gz^3, α∈ℤ_≥ 0,where f, g, and s are respectively assumed to be generic sections ofℒ^⊗ 4⊗𝒮^-⊗ (3+α),ℒ^⊗ 6⊗𝒮^-⊗ 4, and 𝒮.LetX_0=ℙ_B[𝒪_B⊕ℒ^⊗ 2⊕ℒ^⊗ 3] be the ambient space in which ℰ_0 is defined.The following sequence of blowups provides a crepant resolution of the singular Weierstrass model ℰ_0: (X0) at (0,0)X_0; (X1) at (2.5,0)X_1; (X2) at (5,0)X_2; (X3) at (8,0)X_3; (X4) at (11,0)X_4.; [big arrow] (X1) – node[above,midway](x,y,s|e_1) (X0); [big arrow] (X2) – node[above,midway](y,e_1|e_2) (X1); [big arrow] (X3) – node[above,midway](x,e_2|e_3) (X2); [big arrow] (X4) – node[above,midway](e_2,e_3|e_4) (X3);We describe this sequence of blowups starting with the projective bundle X_0, which serves as the ambient space of the Weierstrass equation.The first blowupX_1⟶ X_0 is centered at the regular monomial ideal (x,y,s), where s is a section of 𝒮=𝒪_B(S).The exceptional divisor E_1 of the first blowup is a ℙ^2 bundle. The second blowupX_2⟶ X_1, parametrized by [x:s],is centered along the fiber of E_1 defined by the proper transform of V(y) andits exceptional divisor is E_2. The third blowup X_3⟶ X_2 is centered in E_2 along the fiber over V(x) and has exceptional divisor E_3. The last blowup X_4⟶ X_3 is centered in E_3 along the fiber given by V(e_2) and has exceptional divisor E_4. We recall that blowup up of a divisor is an isomorphism away from the singular locus. The Weierstrass model has a singular scheme supported on the ideal (x,y,s).* First blowup. Since the generic point of this ideal is a double point singularity of the Weierstrass model and the ideal has length 3, blowing up (x,y,s) is a crepant morphism. * Second blowup. We are in X_1 and the singular locus is supported on (y, x,e_1). At this point, we could choose to blowup again (x,y,e_1) since it is a locus of double points and the ideal has length 3.However, we could also blowup (y,e_1), which is anon-Cartier Weil divisor. This is clearly crepant since (y,e_1) has length 2 and multiplicity one. Blowing up this divisor is not an isomorphism since it contains (x,y,e_1), the support of the singular locus.* Third blowup. We blowup the ideal(x,e_2), which corresponds to a non-Cartier Weil divisorof multiplicity one. * Fourth blowup. We finally blowup (e_2,e_3), which is also a non-Cartier Weil divisor of multiplicity one. This is crepant because the ideal has length 2 and the defining equation has multiplicity one along (e_2,e_3).After the fourth blowup, we check using the Jacobian criterion that there are no singularities left.We can also simplifycomputations by noticing that the defining equation is a double cover and therefore, the singularities should be on the branch locus.Moreover, certain variables cannot vanish at the same time due to the centers of the blowups.In particular, each of (x,y,s), (y,e_1), (s,e_3), (s,e_4), (x,e_2), (x,e_4), and (e_2, e_3) corresponds to the empty set in X_4. The divisor classes of the different variables in X_i are given in the following table: .95x yzs e_1 e_2e_3e_4 X_02L+H3L+H H S - ---X_12L+H-E_13L+H-E_1 H S-E_1 E_1--- X_22L+H-E_13L+H-E_1-E_2 H S-E_1 E_1-E_2 E_2-- X_32L+H-E_1-E_33L+H-E_1-E_2 H S-E_1 E_1-E_2 E_2-E_3 E_3- X_42L+H-E_1-E_33L+H-E_1-E_2 H S-E_1 E_1-E_2 E_2-E_3-E_4 E_3-E_4 E_4The proper transform of the Weierstrass model is a smooth elliptic fibration φ:Y⟶ B Y: e_2 y^2 z =e_1( e_3^2 e_4 x^3+ e_1 e_2 e_3 (e_1 e_2 e_3 e_4^2)^α s^3+α a_4,3+αxz^2+ e_1 e_2 s^4a_6,4z^3).The successive relative “projective coordinates” for the fibers ofX_i over X_i-1 are (i=1,2,3,4) [e_1 e_2 e_3^2 e_4^3 x : e_1 e_2^2 e_3^2 e_4^4 y : z], [ e_3 e_4 x : e_2 e_3 e_4^2 y : s],[y:e_1], [ x:e_2 e_4], [e_2 :e_3].These projective coordinates are not independent of each other, as we have a tower ofprojective bundles defined over subvarieties of projective bundles.The interdependence between the different projective bundles are capturedby the following scalings: X_0/B [e_1 e_2 e_3^2 e_4^3 x : e_1 e_2^2 e_3^2 e_4^4 y :z] X_1/ X_0[ℓ_1 ( e_3 e_4 x) : ℓ_1 (e_3 e_4^2 e_2 y) : ℓ_1 s] X_2/ X_1 [ℓ_1 ℓ_2 y:ℓ_1^-1ℓ_2 e_1] X_3/X_2 [ℓ_1ℓ_3 x:ℓ_2^-1ℓ_3 (e_2 e_4)] X_4/X_3 [ℓ_4 ℓ_3 ℓ_2^-1 e_2 :ℓ_4 ℓ_3^-1 e_3] where ℓ_1, ℓ_2, ℓ_3, and ℓ_4 are used to denote the scalings of each blowup. § FIBER STRUCTURE In this section, we explore the geometry of the crepant resolutionY→ℰ_0 obtained in the previous section. Composing with the projection of ℰ_0 to the base B, we have a surjective morphism φ:Y→ B, which is an elliptic fibration over B.We denote by η a generic point of S.We study in details the generic fiber Y_η of the elliptic fibration and its specialization.Its dual graph isthe twisted Dynkin diagram F_4^t, namely, the dual of the affine Dynkin diagram F̃_4. We call C_a the irreducible components of the generic fiber, and D_a the irreducible fibral divisors.We can think of C_a as the generic fiber of D_a over S. Given a section u of a line bundle, we denote by V(u) the vanishing scheme of u. As a set of point, V(u) is defined by the equationu=0. If I is an ideal sheaf, we also denote by V(I) its zero scheme. §.§ Structure of the generic fiberAfter the blowup, the generic fiber over S is composed of five curves since the total transform of s is se_1 e_2 e_3 e_4^2. The irreducible components of the generic curve Y_η are the following five curves:C_0 :s= e_2 y^2-e_1e_3^2 e_4 x^3=0C_1 :e_1=e_2=0C_2 :e_2=e_4=0 C_3 :e_4=y^2-e_1^2s^4a_6,4z^2=0C_4 :e_3=y^2-e_1^2 s^4a_6,4z^2=0Their respective multiplicities are 1, 2, 3, 2, and 1.The curve C_a is the generic fiber of the fibral divisor D_a (a=0,1,2,3,4).The fibral divisors can also be defined as the irreducible components of φ^* S:φ^* S= D_0 + 2 D_1+ 3 D_2 + 2 D_3 + D_4.Furthermore, we have the following relations: V(s)=D_0,V(e_1) = D_1,V(e_2) = D_1+D_2, V(e_4)= D_2+D_3,V(e_3) =D_4.Denoting by E_i the exceptional divisor of the ith blowup and by S the class of S,we identify the classesof thefive fibral divisors to be D_0=S-E_1, D_1 = E_1-E_2, D_2 =2 E_2 - E_1-E_3-E_4,D_3 = 2E_4-2E_2+E_1+E_3, D_4 = E_3-E_4. The curve C_0 is the normalization of a cuspidal curve.The curves C_1 and C_2 are smooth rational curves.The curves C_3 and C_4 are not geometrically irreducible.After a field extension that includes the square root of a_6,4, they split into two smooth rational curves.Hence, D_0, D_1, and D_2 are ℙ^1-bundles whileD_3 and D_4 are double coverings of ℙ^1-bundles. Geometrically, when D_3 and D_4are seen asfamilies of curves over S, D_3 and D_4 arefamilies of pair of lines .In the next subsection, we determine what these ℙ^1-bundles areup to an isomorphism. §.§ Fibral divisors In this section, we study the geometry of the fibral divisors. We recall that for aℙ^1-bundle, all fibers aresmooth projective curves with no multiplicities. A conic bundle has a discriminant locus, over which the fiber is reduciblewhen it is composed of two rational curves meeting transversally or is a double line. In the case of an F_4-model, the fibral divisors D_0, D_1, and D_2 are ℙ^1-bundles while D_3 and D_4 are double covers of ℙ^1-bundles.The generic fiber of D_3 and D_4 is geometrically composed of two non-intersecting rational curves. D_3 and D_4 have V(a_6,4) as adiscriminant locus.Over the discriminant locus of these double covers, the fiber is composed of a double rational curve.We can also simply describe D_3 and D_4 as flat double coverings of ℙ^1-bundles over Sor as geometrically reducible conic bundles over S. The fibral divisors D_0, D_1, and D_2 are ℙ^1-bundles.D_3 and D_4 are double covers of ℙ^1-bundles branched at V(a_6,4). The corresponding projective bundles are[We do not write explicitly the obvious pullback of line bundles. For example, if σ: S↪ B is the embedding of S in B and ℒ is a line bundle on B, we abuse notation by writing ℙ_S[ℒ⊕𝒪_S] forℙ_S[σ^* ℒ⊕𝒪_S].](see Figure <ref>) ∙D_0is isomorphic to ℙ_S[ℒ⊕𝒪_S] ∙D_1 is isomorphic to ℙ_S[ℒ^⊗ 2⊕𝒮]∙D_2 is isomorphic to ℙ_S[ℒ^⊗ 3⊕𝒮^⊗ 2]∙ D_3 is isomorphic toa double covering of ℙ_S[ℒ^⊗ 4⊕𝒮^⊗ 3]ramified in V(a_6,4)∩ S ∙ D_4is isomorphic toa double covering ofℙ_S[ℒ^⊗ 2⊕𝒮^⊗ 2]ramified in V(a_6,4)∩ Swhere ℒ is the fundamental line bundle of the Weierstrass model and S is the zero scheme of a regular section of the line bundle 𝒮=𝒪_B(S).The strategy for this proof is as follows. We use the knowledge of the explicit sequence of blowups to parametrize each curve.Since each blowup has a center that is a complete intersection with normal crossing, each successive blowup gives aprojective bundles. The successive blowups give a tower of projective bundles over projective bundles.We keep track of the projective coordinates of each projective bundle relative to its base. An important part of the proof is to properly normalize the relative projective coordinates when working in a given patch, as they are twisted with respect to previous blowups. We show that D_0, D_1, and D_2 are ℙ^1-bundles over S while D_3 and D_4 areconic bundles defined by a double cover of a ℙ^1-bundle over S.The fiberC_0 can be studied after the first blowup since the remainingblowups are away from C_0. We can work in the patch x≠ 0. We use the defining equation of C_0 to solve for e_1 since x is a unit.We then observe that C_0 has theparametrization C_0 [t^2:t^3:1] [1:t:0],t=y/x.This is the usual normalization of a cuspidal cubic curve. It follows that C_0 is a rational curve parametrized by t. Since t=y/x is a section of ℒ, it follows that the fibral divisorD_0 is isomorphic to theℙ^1-bundle ℙ_S[ℒ⊕𝒪_S] over S. D_1 is the Cartier divisor V(e_1) in Y, which corresponds to the complete intersection V(e_1,e_2) in X_4. The generic fiber of D_1 over S is the rational curve C_1, which is parametrized asC_1[0:0 :z] [ℓ_1 ( e_3 e_4 x) :0 : ℓ_1 s] [ℓ_1 ℓ_2 y:0] [ℓ_1ℓ_3 x:0] [0:ℓ_4ℓ_3^-1 e_3].We can use ℓ_4, ℓ_3, and ℓ_2 to fix the scalings. ButC_1 is parametrized by [x:s] and D_1 is isomorphic to the ℙ^1-bundleℙ_S[ℒ^⊗ 2⊕𝒮] over S.C_2 is defined as the generic fiber with e_2=e_4=0. This gives C_2 [0:0 :z][0:0 : ℓ_1 s] [ℓ_1 ℓ_2 y:ℓ_1^-1ℓ_2 e_1] [ℓ_1ℓ_3 x:0] [0 :ℓ_4 ℓ_3^-1e_3]. Fixing the scaling as ℓ_1=s^-1,ℓ_3=s x^-1, we see that C_2 isa rational curve parametrized by [y:s^2] and D_2 is isomorphic to ℙ_S[ℒ^⊗ 3⊕𝒮^⊗ 2]. For C_3 take ℓ_1=s^-1, ℓ_3=sx^-1,ℓ_2=s^-1, ℓ_3=s x^-1, C_3 [0 : 0:1] [0 : 0: 1] [y s^-2:1] [1:0] [ ℓ_4 s^2 x^-1:ℓ_4 x s^-1].The double cover is (y/s^2)^2=a_6,4. This is clearly a double cover of D_3^+, where D_3^+ is ℙ^1-bundle over S, whose fiber isparametrized by [ s^2 x^-1:x s^-1]. Such a ℙ^1-bundle is isomorphic to ℙ_S[ℒ^⊗ 4⊕𝒮^⊗ 3]. * For C_4, take ℓ_1=s^-1, ℓ_2=s^-1, ℓ_4=sℓ_3^-1, C_4[0 : 0:1] [0 : 0: 1] [y s^-2:1] [ s^-1 x: s][1:0] The double cover is again (y/s^2)^2=a_6,4. The crepant resolution defined in Theorem<ref> has the following properties:2pt* The resolved variety is a flat elliptic fibration over the base B. * The fiber over the generic point of Shas dual graph F_4^t and the geometric generic fiber is of Kodaira type IV^*. * The fiber degenerates over V(s,a_6,4) as V(s,a_6,4) C_3 ⟶ 2 C_3', C_4⟶ 2 C_4'. where C_3 and C_4 are generic curves defined over S, and C_3' and C_4' are generic curves over V(s, a_6,4). The generic fiber over V(s, a_6,4) is a non-Kodaira fiber composed of five geometrically irreduciblerational curves. The reduced curvesmeettransversallywith multiplicities 1-2-3-4-2. The special fiber is the fiber over the generic point of S. Note that C_2 and C_3 intersect at a divisor of degree two, composed of two points that are non-split. Hence, the dual graph of this fiber is the twisted affine Dynkin diagram of type F_4^t. All the curves are geometrically irreducible with the exception of C_3 and C_4,which are the double covers of a geometrically irreducible rational curve and the branching locus is a_6,4=0.Each of these two curves splits into two geometricallyirreducible curves in a field extension that includes a square root ofa_6,4. Theydegenerate into a doublerational curve over a_6,4=0. Over the branching locus, the singular fiber is a chain 1-2-3-4-2.The geometric generic fiber has a dual graph that is a E_6 affine Dynkin diagram.The fibers C_a are fibers of fibral divisors D_a. The matrix of intersection numbers deg(D_a· C_a) is the opposite of the invariant form of the twisted affine Dynkin diagram of type F̃_4^t, normalized in such a way that the shortroots have length square 2:deg (D_a · C_b)= ([ -21000;1 -2100;01 -220;002 -42;0002 -4 ]) This matrix has a kernel generated by the vector (1,2,3,2,1).The entries of this vector give the multiplicities of the curve C_a, or equivalently, of the fibral divisors D_a.IfB is the total space of the line bundle 𝒪_ℙ^1(-n) with n∈ℤ_≥ 0, the Picard group of B is generated by one element, which we call 𝒪(1).In particular, the compact curve ℙ^1 is a section of 𝒪(-n).A local Calabi-Yau threefold can be defined by a Weierstrass model with ℒ=𝒪(2-n). Consider the case of the F_4-model, defined with S a regular section of 𝒮=𝒪(-n).This requires that 1≤ n≤ 5.Denoting the Hirzebruch surface of degree d by 𝔽_d, we haveD_0=𝔽_n-2, D_1=𝔽_n-4, D_2=𝔽_n-6, D_3 a double cover of 𝔽_n-8, and D_4a double cover of 𝔽_4. In particular for n=5, the divisors areD_0=𝔽_3, D_1=𝔽_1, D_2=𝔽_1, D_3a double cover of 𝔽_3, and D_4a double cover of 𝔽_4. §.§Representation associated to the elliptic fibration In this subsection, we compute the weights of the vertical curves appearing over codimension-two points. There is only one case to consider.The generic fiber overV(a_6,4)∩ S is a fiber of type 1-2-3-4-2 resulting from the following specialization: C_3 ⟶ 2 C_3', C_4⟶ 2 C_4'. * The intersection numbers of the generic curves C_3' and C_4' with the fibral divisors D_a for a=0,1,…, 4 are ϖ(C_3')=(0,0,1,-2,1), ϖ ( C_4')= (0,0,0,1,-2). * The representation associated to an F_4-modelisthe quasi-minuscule representation 26 of F_4. By the linearity of the intersection product,the geometric weightsφ(C)=(D_a· C) of C=C_3' and C=C_4' are half of the geometric weights of C_3 and C_4': ϖ(C_3')=(0,0,1,-2,1), ϖ ( C_4')= (0,0,0,1,-2). Ignoring the weight of D_0, we get the following two weights of F_4: 01-210 01 -2 . These two weights are quasi-minuscule and are in the same Weyl orbit, which consists of the non-zero weights of the representation 26 of F_4. See Theorem <ref> for the proof. This is afundamentalrepresentation corresponding to the fundamental weight α_4. This representation is also quasi-minuscule. Since the weight system is invariantunder a change of signs,the representation is quaternionicand we can consider half of the representation. Both weights coming from the degeneration of the main fiber are in the same half quaternionic set of weights. An important consequence of Theorem <ref> is that the elliptic fibration does not have flop transitions to another smooth elliptic fibrationsince all the curves move in families. The reduced discriminant has two components, namely S=V(s) and Δ'=V( a^3_ 4,3+α s^1+3α+27 a_6,4^2)intersecting non-transversally at V(s,a_6,4).Their intersection isexactly the locus over which the fiber IV^*ns degenerates.The generic fiber overΔ' is of Kodaira fiber I_1. Hence, what we are witnessing is a collision of type IV^*ns+I_1, leading to an incomplete III^* or an incompleteII^*IV^*ns+I_1⟶ 1-2-3-4-2 (incompleteIII^* orincompleteII^*).This is clearly not a collision of Miranda models since the fibers have different j-invariants and do not intersect transversally. The j-invariants offibers of type IV^*and I_1 are, respectively, zero andinfinity. By using an elliptic surface whose bases pass through thecollision point, the singular fiber at the collision point isof Kodaira type III^* for α=0 and Kodaira type II^* for α>0. Interestingly, we can think of the singular fiber 1-2-3-4-2 as a contraction of a fiber of type III^* or a fiber of type II^*,as expected from the analysis of Cattaneo<cit.>. § TOPOLOGICAL INVARIANTS In this section we compute several topological invariants of the crepant resolution.Using the pushforward theorem of <cit.>, we can compute the Euler characteristic of an F_4-model.We need to know the classes of the centers of the sequence of blowupsthat define the crepant resolution.The center of the nth blowup is a smooth complete intersection of d_n divisors of classes Z_i^(n), where i=1,2,⋯, d_n.We recall that H=c_1(𝒪(1)), L=c_1(ℒ), and S=[S]. The classes associated to thecenters of each blowup are <cit.> [Z^(1)_1 = H+2 L Z_2^(1) =H + 3 LZ_3^(1) = S; Z^(2)_1 =Z^(1)_2 - E_1Z^(2)_2 = E_1 ;Z^(3)_1 =Z^(1)_1 -E_1 Z^(3)_2 =E_2 ;Z^(4)_1 = E_2 - E_3 Z^(4)_2 =E_3] TheEuler characteristic of an F_4-model obtained by a crepant resolution of the Weierstrass modely^2z=x^3+ s^3+αa_4,3+α x z^2 + s^4 a_6,4 z^3 (α∈ℤ_≥ 0) over the base B is χ(Y)=∫ 12 (L + 3 S L-2 S^2)/(1+ S)(1+ 6 L-4 S )c(B), where L=c_1(ℒ) and S is the class of V(s). In particular, denoting c_i(TB) simply as c_i: χ(Y) Y=312(c_1 L-6 L^2+6 L S-2 S^2) Y=3 and c_1(TY)=012( 5 c_1^2-6 c_1 S+2 S^2)Y=412( -6 c_1 L^2+c_2 L+36 L^3+6 c_1 L S-60 L^2 S-2 c_1 S^2+34 L S^2-6 S^3) Y=4 and c_1(TY) 12 t^3 (30 c_1^3+ c_1 c_2- 54 c_1^2 S+32 c_1 S^2 - 6 S^3) Let Ybe a Calabi-Yau threefold that is an F_4-model obtained from a crepant resolution. Thenthe Hodge numbers of Y are h^1,1(Y)= 15 - K^2, h^2,1(Y)= 15+ 29 K^2 + 36 S K + 12 S^2. With an explicit resolution of singularities, it is straightforward to compute intersection numbers of divisors.In particular, we evaluate the triple intersection numbers of the fibral divisors D_a, where a=0,1,2,3,4.The result isφ_* ((∑ D_a ϕ_a )^3·φ^* M)=6ℱ(L,S,ϕ) M,where M is an arbitrary element of A_d-2(B). In particular, if the base B is a surface, M is just a point. Let D_a (a=0,1,2,3,4) be the fibral divisor of an F_4-model obtained by a crepant resolution of singularities. The triple intersection numbers φ_* ((∑ D_a ϕ_a )^3·φ^* M)=6ℱ(L,S,ϕ) M, where M is an element of A_d-2(B) (d= B), are given by6 ℱ(L,S,ϕ)=4 (L-S) S ϕ _0^3 +3 (S-2 L) S ϕ _0^2 ϕ _1+3 L S ϕ _0 ϕ _1^2 + 4(L-S) S ϕ _1^3+4 (L-S) S ϕ _2^3 +8 (S-2 L)S ϕ _3^3+8(S-2 L) S ϕ _4^3 +3 (2 S-3 L)S ϕ _1^2 ϕ _2 +3(2 L-S) S ϕ _1 ϕ _2^2+6 (3 S-4 L)S ϕ _2^2 ϕ _3 +12 (3 L-2 S)S ϕ _2 ϕ _3^2+12 (S-L)S ϕ _3^2 ϕ _4 +6 (4 L-3 S)S ϕ _3 ϕ _4^2. Use equation (<ref>) andsuccessively apply thepushforward formula of <cit.>.In the case of a Calabi-Yau threefold, we have L=-K_B. It follows that we can express the coefficient in terms of the genus of S and its self-intersection using the relation 2-2g=-K_B· S -S^2:6ℱ=- 8 (g-1) ϕ _0^3-3 (-4 g+4+S^2)ϕ _0^2ϕ _1+3(-2 g+2+S^2)ϕ _1^2 ϕ _0 -8 (g-1) ϕ _1^3 -8 (g-1) ϕ _2^3+8(4 g-4-S^2)ϕ _3^3+ 8(4 g-4-S^2) ϕ _4^3 3(6 g-6-S^2) ϕ _1^2 ϕ _2 +3 (-4 g+4+S^2) ϕ _1ϕ _2^2+6 (8 g-8-S^2) ϕ _2^2 ϕ _3 +12 (-6 g+6+S^2)ϕ _2ϕ _3^2 +24 (g-1) ϕ _3^2 ϕ _4 +6(-8 g+8+S^2)ϕ _3 ϕ _4^2. In the case of a threefold, the second Chern class defines a linear form on H^2(Y, ℤ). In particular, for the fibral divisors we have∫_Y c_2(TY)· (∑_a D_a ϕ_a)= 2 S (S-L) (ϕ _0+ϕ _1+ϕ_2) +4S(2L-S)( ϕ _3+ ϕ _4).Imposing the Calabi-Yau condition, we can rewrite this as∫_Y c_2(TY)· (∑_a D_a ϕ_a)= 4(g-1) (ϕ _0+ϕ _1+ϕ_2) +4(4-4g+S^2)( ϕ _3+ ϕ _4).§.§ Stein factorization and the geometry of non-simply laced G-modelsTo understand the geometry of a fibral divisor D, it is important to see the divisor D asa relative scheme with respect to the appropriate base.The choice of the base is crucial to having the correct physical interpretation. In particular, to discuss the matter content of the theory, the base has to be a component of the discriminant locus.However, to study the possible contractions of D, the base can be an arbitrary subvariety of the elliptic fibration. Let S be the irreduciblecomponent of the discriminant locus supporting the gauge group.In the case of G-models with G anon-simply laced groups, Stein factorization illuminates the discussion of the geometry of the fibral divisors D, whose generic fibers over S are not geometrically irreducible.The elliptic fibration φ:Y→ B pulls back to a fibration D→ S. If the generic fiber of this fibration is not geometrically irreducible, the generic fiber is not geometrically connected. We recall the following two classical theorems on morphisms that are consequences of the theorem of formal functions. Let f: X→ Y be a proper morphism of Noetherian schemes such that f_* 𝒪_X≅𝒪_Y. Then all fibers are geometrically connected and non-empty. Let f: X→ S be a proper morphism with S a Noetherian scheme. Then there exists a factorization(X) at (-2,0)X; (Y1) at (2,0)S'; (Y0) at (0,-2)S; [big arrow] (X) – node[below=.1cm,left]f (Y0); [big arrow] (X) – node[above=.1]f' (Y1); [big arrow] (Y1) – node[below=.2cm,right]π (Y0);such that* π:S'→ S is a finite morphism and f':X→ S' is a proper morphism with geometrically connected fibers.* f'_* 𝒪_X≅𝒪_S'. * S' is the normalization of S in X.* S'=Spec_S (f_* 𝒪_X). The Stein factorization on f: D→ S is the decomposition f=π'∘ f', whereπ':S→ S is a finite map of degree d and f':D→S is a morphism with connected fibers.We expectf':D→S to be a ℙ^1-bundle and π': S→ Sa smooth d-cover of S. In the case of F_4-models, D is a ℙ^1-bundle for the fibral divisor D_0, D_1, and D_2(corresponding to the affine root, and the small roots of F_4^t.The remaining two fibral divisors (namely D_3 and D_4) are not ℙ^1-bundles over S but rather double covers of ℙ^1 bundles over S with a ramification locus a_6,4=0. The crepant resolution naturally defines a double cover π: D→D, whereD is a ℙ^1-bundle p:D→ S. Let f=π∘ p: D→ S be the composition.The key to understanding the different perspective on the geometry of D is to consider the Stein factorization of f. Let D be the reduced fibral divisor D_3 or D_4 of an F_4-model.By definition, f:D→ S has a generic fiber that is not geometrically connected. Consider the Stein factorization of the morphism f: D→ S. It gives a factorizationf=π'∘ f' with π' a finite map and f' a proper morphism with geometrically connected fibers. (X) at (-2,0)D; (Y) at (0,0)D; (S0) at (0,-2)S; (S1) at (-2,-2)S; [big arrow] (X) – node[above,midway]π (Y); [big arrow] (Y) – node[above,right]p (S0); [big arrow] (X) – node[below=.2cm,left]f (S0); [big arrow] (X) – node[above,left]f' (S1); [big arrow] (S1) – node[below=.1cm]π' (S0); In particular, the morphismf':D→S endows D the structure of a ℙ^1-bundle over the double coverS of S.This structure illustrates that D can contract to S. It is important to not confuse the role of the morphisms f':D→Sand f:D→ S in F-theory. One might naively assume that the existence of a ℙ^1-bundle f':D→S means that the divisor D does not produce new curves leading to localized matter representations.However, it is important to keep in mind that it isthe morphism f:D→ S over the curve S that is relevant for studyingthe singular fibers of the elliptic fibration as S. The morphism f:D→ S contains singular fibers that are double lines. The intersection numbers of these lines with the fibral divisors give two weightsof the representation 26,namely 01-21 for D_3 and 0 01 -2 for D_4. The same weights are obtained over any closed points away from a_6,4 and are attributed to non-localized matter.§ APPLICATION TO M-THEORY AND F-THEORY IN 5 AND 6 DIMENSIONSIn this section, we study the aspects of five-dimensional gauge theories with gauge group F_4 using the geometry of the F_4-model.We consider an M-theory compactified on an F_4-model φ: Y→ B. We assume then that the variety Y is a Calabi-Yau threefold and the base B is a rational surface. Then, the resulting theory is a five-dimensional 𝒩=1 supersymmetric theory with eight supersymmetric generators, whose matter content contains n_H hypermultiplets and n_V vector multiplets. We haven_H^0 neutral hypermultiplets, n_52 hypermultiplets transforming in the adjoint representation, and n_26 hypermultiplets transforming in the fundamental representation 26.We have n_V vector multiplets whose kinetic terms and Chern-Simons terms are controlled by a cubic prepotential. For the F_4 gauge theory withboth adjoint and fundamental matters, there is a unique Coulomb phase. Since F_4does not have a non-trivial third order Casimir, the classical part of the prepotential vanishes, and the quantum corrections fully determine the prepotential. The number of vector multiplets is the dimension of F_4. Then, we can determine the quantum contribution to the prepotential and hence determine n_52 andn_26. Since we know the Hodge numbers of F_4-models on a Calabi-Yau threefold, we can compute n_H^0=h^2,1(Y)-1 as well.We also check that the data we collected geometrically for the five-dimensional gauge theory will satisfy the anomaly cancellation conditions in the uplifted six-dimensional theory with the same gauge group F_4, the same matter contents, and an addition ofn_T=h^1,1(B)-1 tensor multiplets. §.§ Intriligator-Morrison-Seiberg potentialIn this paper, the Intrilligator-Morrison-Seiberg (IMS) prepotential is the quantum contribution to the prepotential of a five-dimensional gauge theory with the matter fields in the representations 𝐑_i of the gauge group. Let ϕ be in the Cartan subalgebra of a Lie algebra 𝔤.Theweights are in the dual space of the Cartan subalgebra.We denote the evaluation of aweight on ϕ as a scalar product ⟨μ,ϕ⟩.We recall that the roots are the weights of the adjoint representation of𝔤. Denoting the fundamental roots by α and the weights of 𝐑_i by ϖ we have 6ℱ_IMS = 1/2( ∑_α |⟨α, ϕ⟩|^3-∑_𝐑_i∑_ϖ∈ W_i n_𝐑_i |⟨ϖ, ϕ⟩|^3).For all simple groups with the exception of SU(N) with N≥ 3, this is the full cubic prepotential as there are non-trivial third Casimir invariants. One complication to the formula is dealing with the absolute values. For a given choice of a group G and representations 𝐑_i, we have to determine a Weyl chamber to remove the absolute values in the sum over the roots. We then consider the arrangement of hyperplanes ⟨ϖ, ϕ⟩=0, where φ runs through all the weights of all the representations 𝐑_i. If none of these hyperplanes intersect the interior of the Weyl chamber, we can safely remove the absolute values in the sum over the weights. Otherwise, we have hyperplanes partitioning the fundamental Weyl chamber into subchambers. Each of these subchambers is defined by the signs of the linear forms ⟨ϖ, ϕ⟩. Two such subchambers are adjacent when they differ by the sign of a unique linear form. Within each of these subchambers, the prepotential is a cubic polynomial; in particular, it has smooth second derivatives. But as we go from one subchamber to an adjacent one, we have to go through one of the walls defined by the weights and the second derivative will not be well-defined. Physically, we think of the Weyl chamber as the ambient space and each of the subchambers is called a Coulomb phase of the gauge theory. The transition from one chamber to an adjacent chamber is a phase transition that geometrically corresponds to a flop between different crepant resolutions of the same singular Weierstrass model. The number of chambers of such a hyperplane arrangement is physically the number of phases of the Coulomb branch of the gauge theory.In the case of F_4 with the matter fields in the representation 52⊕26, there is a unique chamber as the hyperplanes ⟨ϖ, ϕ⟩=0have no intersections with the interior of the fundamental Weyl chamber.The explicit computation of 6ℱ_IMS is presented in the theorem below. Theprepotential for a gauge theorywith the gauge group F_4 coupled to n_52hypermultipletsin the adjoint representation andn_26hypermultipletsin the fundamental representation is 6ℱ_IMS =-8 (n_52-1)ϕ _1^3-8 (n_52-1)ϕ _2^3-8(n_52+n_26-1) ϕ _3^3-8(n_52+n_26-1)ϕ _4^3-3(-n_52+n_26+1)ϕ _1^2ϕ _2 +3 (n_52+n_26-1)ϕ _1ϕ _2^2+12 (-n_52+n_26+1) ϕ _2 ϕ _3^2-6(-3 n_52+n_26+3)ϕ _2^2 ϕ _3+6(-3 n_52+n_26+3)ϕ _3 ϕ _4^2 +24(n_52-1) ϕ _3^2 ϕ _4. The triple intersection polynomialof the elliptic fibration defined by thecrepant resolution of the F_4-model Weierstrassmatches the IMS potential if and only if n_52=g,n_26=5 (1-g) + S^2. These numbers are obtained by comparing the coefficients of ϕ_1^3 and ϕ_4^3 in ℱ_IMS and ℱ. A direct check shows that all the other coefficients match. §.§ Six-dimensional uplift and anomaly cancellation conditionsIn this section, we review the basics of anomaly cancellation in six-dimensional supergravity with the idea of applying it to the matter content we have identified in the previous section. Consider an N=1six-dimensional theory coupled to n_T tensor multiplets, n_V vectors, and n_H hypermultiplets.The pure gravitational anomaly (proportional to trR^4) is canceled by the vanishing of its coefficient:n_H-n_V+29n_T-273=0.If the gauge group is a simple group G, n_H=n_H^ch+n_H^0,n^ch_H =∑_𝐑 n_ R( dim𝐑 -dim𝐑_0 ),n_H^0=h^2,1(Y)+1,n_V=G, where dim𝐑_0 is the number of zero weights in 𝐑 and (dim𝐑 -dim𝐑_0) is the charged dimension of 𝐑. Assuming the gravitational anomaly is canceled, theanomaly polynomial is a function of the matter content of the theory. It depends on the representations and their multiplicities I_8= 9-n_T/8 (tr R^2)^2+1/6X^(2)tr R^2-2/3 X^(4),where X^(n) contains the information on the representations and their multiplicities and is given byX^(n)=tr_adjF^n -∑_𝐑n_𝐑tr_𝐑F^n. To simplify the expression of I_8, we choose a reference representation 𝐅,and introduce thecoefficients A_ R, B_ R, and C_ R defined by thetrace identitiestr_ RF^2 = A_ RtrF^2 , tr_ RF^4 =B_ Rtr F^4+C_ R (trF^2)^2.In a theory with only one quartic Casimir, we simply have B_R=0.It is very useful toreformulate X^(n) in terms of a unique representation F:X^(2) =(A_adj -∑_𝐑n_𝐑 A_𝐑) tr_𝐅F^2,X^(4)=(B_adj -∑_𝐑n_𝐑 B_𝐑) tr_𝐅F^4 + (C_adj -∑_𝐑n_𝐑 C_𝐑)( tr_𝐅 F^2)^2 . To have a chance to cancel the anomaly in a theory with at least two quartic Casimirs, the coefficient of tr_𝐅F^4 must vanish. This condition is irrelevant in a theory with only one quartic Casimir since tr_𝐅F^4 is proportional to (tr_𝐅F^2)^2. If the following three conditions are satisfied, we can deduce that the anomalies are canceled by the Green-Schwartz mechanism: * n_H-n_V+29n_T-273=0.* B_adj -∑_𝐑n_𝐑 B_𝐑=0 (in a theory with at least two quartic Casimirs).* I_8 factorizes. §.§ Cancellations of six-dimensional anomalies for an F_4-model In this section, we prove that the data we computed on the F_4-model as seen in a M-theory compactification on a Calabi-Yau threefold Y will satisfy the anomaly cancellation conditions of a six-dimensional theory with the same gauge group and same number of vector and hypermultiplets.Moreover, in a (1,0) six-dimensional gauge theory, we can also have tensor multiplets.We assume here that the tensor multiplets are massless and have numbers n_T=h^1,1(B), which is n_T=9-K^2 since we assume that the base is a rational surface. We recall the data we will need, which were computed in previous sections: n_T=9-K^2, h^2,1(Y)= 15+ 29 K^2 + 72(g-1) -24 S^2,n_V=52, n_52=g,n_26=5 - 5 g + S^2.We recall that n_V isgiven by the dimension of the Lie algebra of F_4.The numbers n_52 and n_26 were computed using the Intrilligator-Morrison-Seiberg prepotential. The Hodge numbers of Y were computed in <cit.>. We compute first thepure gravitational anomaly.We need to satify equation (<ref>). Using the data of (<ref>), wecomputen_H=29 K^2+64 and check that thegravitational anomaly cancels: n_H -n_V +29 n_T -273=(29 K^2+64)-52+29(9-K^2)-273=0.We will now show that the anomaly polynomialI_8 is a perfect square.Since F_4 does not have a fourth Casimir, B_𝐑=0. Taking 26 as our reference representation <cit.> ,tr_52F^2=3 tr_26F^2, tr_52F^4=5/12 ( tr_26 F^2)^2tr_26F^4=1/12 ( tr_26 F^2)^2.From these trace identities, we can immediately read off the coefficients A_𝐑, B_𝐑, and C_𝐑:A_52=3, B_52=0,C_52=5/12, A_26=1,B_26=0, C_26=1/12.Hence, the forms X^(2) and X^(4) are X^(2)=(3-3 n_52-n_26) trF^2,X^(4)=1/12( 5-5 n_52- n_26)( tr F^2)^2.After plugging in the values of n_52 and n_26, we have X^(2)=K· S trF^2,X^(4)=-1/12 S^2( tr F^2)^2. We can now prove that the anomaly polynomial is a perfect square:I_8= K^2/8 (tr R^2)^2+1/6 K S (trF^2)(tr R^2)+1/18S^2( tr F^2)^2=1/72(3Ktr R^2+2 Str F^2)^2.This shows that the anomalies can be canceled by the traditional Green-Schwarz mechanism. §.§ Frozen representations Motivated by the counting of charged hypermultiplets in M-theory compactifications, we introduce the notion of frozen representations.When a vertical curve of an elliptic fibration carries the weight of a representation 𝐑, the representation 𝐑 is said to be the geometric representation induced by the vertical curve.The existence of a vertical curve carrying a weight of the representation 𝐑is a necessary but not sufficient condition for hypermultiplets to be charged under the representation 𝐑.It is possible that a geometric representation is not physical in the sense that no hypermultiplet is charged under𝐑, so that n_𝐑=0.A geometric representation 𝐑 is said to be frozen if it is induced by the weights of vertical curves of the elliptic fibration but no hypermultiplet is charged under 𝐑. Witten has proven that a curve of genus g supporting a Lie groupproduces ghypermultiplets in the adjoint representation.It follows that the adjoint representation is frozen when the genus is zero.This is because the adjoint hypermultipletsare counted by holomorphic forms on the curve S.When the genus is zero, there are no such forms. Hence, even though we clearly witness vertical rational curves carrying the weights of the adjoint representation, no adjoint hypermultiplet are to be seen. For non-simply laced groups, frozen representations can occur when the curve defined by the Stein factorization has the same genus asthe base curve S.The representation is frozen if and only ifg=0 anddeg R =2 (d-1). The number of points over which thefiber IV^*ns degenerates to the non-Kodaira fiber IV^*_2 is the degree of the ramification divisor R. Thisisan interesting geometric invariant to keep in mind.If B is a surface and ℒ=𝒪_B(-K_B) so that the F_4-model is a Calabi-Yau threefold, then the number of points over which the generic fiber IV^*ns over S degeneratesfurther to the non-Kodaira fiber 1-2-3-4-2 is degR=12(1-g)+2 S^2.The number of intersection points is the intersection product of V(a_6,4) and the curve S. Note thatthis is a transverse intersection and the class of V(a_6,4) is 6L-4S. Then usingL=-K_B and 2g-2=(K+S)· S, we havedeg R=(6L-4S)· S =12 (1-g)+2S^2. The degree of the ramification locus Ris the number of points in the reduced intersection of the two components of the discriminant locus,namely S and Δ'.The degree of R has to be positive as otherwise D_3 and D_4 are not irreducible and we havean E_6-model rather than an F_4-model. This gives a constraint on the self-intersection of S:S^2>6(g-1).For example, if B=ℙ^2,the bound is respected when S is a smooth curve of degree 1, 2, 3, or 4. for an F_4-model, the number of charged hypermultiplets are n_26=1/2deg R+g-1 n_52=g. In particular, the adjoint representation is frozen if and only if g=0. The representation 26 is frozen if and only ifS^2=-5 and g=0, which also forces the adjoint representation to be frozen. The number of representation n_26is computed in Theorem <ref>.The representation 26 is frozen if and only if deg R=2(1-g). Since the degree of R has to be positive, we also see that g=0, hence by using Lemma <ref>, we conclude that S^2=-5 and deg R=2. We consider twoimportant examples. [Frozen adjoint representation] n_52=0 if and only ifg=0. In this case, deg R=12+2S^2 and n_26=5+S^2.This matches what is found in Table 3 of <cit.>using n=S^2 as the instanton number.[Frozen representation26] n_26=0 if and only ifS^2=-5 and g=0.For example, take Bto be the quasi-projective surface given bythe total space of the line bundle 𝒪_ℙ^1(-5).To construct a local Calabi-Yau threefold, use ℒ=𝒪_B(-3). This is the non-Higgsable model of <cit.>. The defining equation of such a Weierstrass model is y^2 z= x^3 + f_3 s^3 x z^2 + g_2 s^4 z^3,where g_2 and f_3 are respectively sections of 𝒪_ℙ^1(2) and 𝒪_ℙ^1(3). The IV^*ns fiber degenerates further at the two pointsg_2=0. Over these points we have the non-Kodaira fiber of type 1-2-3-4-2, each carryingthe weights ofthe representation 26.§.§ Geometry of fibral divisors in the case of frozen representations The case of an F_4-model with both representationsfrozen has been recently studied in<cit.>. Such a model does not have any charged hypermultiplets as explained firstin <cit.>. It follows fromthe geometry of the crepant resolution, the generic fiber over the base curve S does degenerate at two points V(g_2).Our description is consistent with the analyses of<cit.> and the Hirzebruch surfaces identified in <cit.>.We discussin detail the geometry of the fibral divisors of an F_4-model in the case where all representations are frozen.This means that S is a curve of genus zero and self-intersection -5. For example, the base could be 𝔽_5 or the total space of 𝒪_ℙ^1(-5).The key is Theorem<ref>,which we specialize to this situation in the following lemma. Let S⊂ B be a smooth rational curve of self-intersection -5.An F_4-model over B with gauge group supported on S has fibral divisors D_0, D_1, D_2, D_3 and D_4 such thatD_0→ S, D_1→ S, D_2→ S are respectively, Hirzebruch surfaces 𝔽_3,𝔽_1, and𝔽_1.D_3→ S and D_4→ S are not Hirzebruch surfaces over S but double covers of 𝔽_3 and 𝔽_4 Hirzebruch surfaces.Considering theStein factorization D_3 S' S and D_4 S' S whereS' S is a double cover of S branched at two points. The morphismD_3 S' presents D_3 as an Hirzebruch surface 𝔽_6 over S' and D_4 S' presents D_4 as a Hirzebruch surface 𝔽_8 over S'. We useTheorem <ref>.The Calabi-Yau condition implies that ℒ=𝒪(-K_B).D_0, D_1, and D_2 are Hirzebruch surfaces. We compute their degrees by intersection theory as follows. We recall that for a Hirzebruch surface 𝔽_n=𝔽_-n. We follow the following strategy.Given two line bundles ℒ_1 and ℒ_2 and a Hirzebruch surface ℙ_S[ℒ_1⊕ℒ_2]→ S over a smooth rational curve S, then the degree of the Hirzebruch surface isdegℒ_1-degℒ_2]=∫_S c_1(ℒ_1)-∫_Sc_1(ℒ_2). We know that D_0 is ℙ_S[ℒ⊕𝒪_S]. To compute the degree of this Hirzebruch surface, we compute ∫_S c_1(ℒ)=-S· K_B=-3. that is, D_0 is an 𝔽_3.We know that D_1 is ℙ_S[ℒ^⊗ 2⊕𝒮], which isisomorphic to ℙ_S[(ℒ^⊗ 2⊗𝒮^-1)⊕𝒪_S] .To compute the degree of this Hirzebruch surface, we compute ∫_S [2c_1(ℒ)-S]=S· (-2K_B-S)=-6+5=-1. Hence D_1→ S is an 𝔽_1.D_2 is ℙ_S[ℒ^⊗ 3⊕𝒮^⊗ 2], which is a Hirzebruch surface of degree S·(-3K_B-2S)=-9+10=1.Hence, D_2→ S is also an 𝔽_1.Asis clear from the crepant resolution, the fibral divisors D_3→ S and D_4→ S are not Hirzebruch surfaces over S.Their fibers consist of two generically disconnected rational curves ramified over R=V(s,g_2).They can be respectively described as double covers ofaℙ_S[ℒ^⊗ 4⊕𝒮^⊗ 3] and aℙ_S[ℒ^⊗ 2⊕𝒮^⊗ 2]; These are isomorphic to,respectively,𝔽_3 and 𝔽_4.In each case, the branch locus of the double cover consists of two fibers of the Hirzebruch surface, and these are the fibers over V(g_2). The absence of charged multipletsis justified by the phenomena of frozen representations rather than the absence of degenerations supporting the weights of the representation 26.The Stein factorization of D_3→ Sis D_3 S' S, where S' S is a double cover branched at two points and f:D_3→ S' is a proper morphism with connected fibers.In particular, f:D_3→ S' is a ℙ^1-bundle. Since S' has genus zero, this is a Hirzebruch surface.The degree of this Hirzebruch surface is ∫_D_3 S'^2, which is 2∫_𝔽_3 S^2=2· (-3)=-6. Hence f:D_3→ S' is an 𝔽_6-surface. We show in the same way that f:D_3→ S' is an 𝔽_8-surface. In other words, whileD_3→ S and D_4→ S are not ℙ^1-bundles over S, D_3→ S' and D_3→ S' are ℙ^1-bundles over S'.As discussed in the appendix, with respect to S', D_3 and D_4 have the structure of an 𝔽_6 and an 𝔽_8 since the base curve S' has self-intersection § CONCLUSION In this paper, we studied the geometry of F_4-models. Our starting point is asingular Weierstrass model characterized by the valuations with respect to a smooth divisor S=V(s) given byStep 8 of Tate's algorithm:v_S(a_1)≥ 1,v_S(a_2)≥ 2,v_S(a_3)≥ 2,v_S(a_4)≥ 3,v_S(a_6)≥ 3,v_S(b_6)=4. The last condition ensures that the polynomial Q(T)=T^2 + a_3,2T-a_6,4 has two distinct solutions modulo s. We focus on the case where Q(T) has no rational solutions modulo s. The generic fiber over S is called a fiber of type IV^*ns.Without loss of generality, the Weierstrass model can be written in the following canonical form: y^2z= x^3 + a_4,3+αs^3+α x z^2+ a_6,4 s^4 z^3,α∈ℤ_≥ 0.A crepant resolution of this Weierstrass model is called an F_4-model. Such elliptic fibrations are used to engineer F_4 gauge theories in F-theory and M-theory.While it is common to take α=0in the F-theory literature, here,we keep α unfixed to keep the geometry as general as possible. This allows us to cover local enhancements of the type F_4 → E_7 and F_4 → E_8 over s=a_6,4=0, depending on the valuation of a_4.While the generic fiber is a twisted affine Dynkin diagram F̃_4^t, the geometric fiber is the affine Ẽ_6 diagram. Thus we have the natural enhancement F_4 → E_6,which is non-local and appears over any closed point of S away from a_6,4=0.The crepant resolution that we have considered consists of a sequence offour blowups centered at regular monomial ideals. We answered several questions regarding the geometry and topology of the resulting smoothelliptic fibration. In particular, we identified the geometry of the fibral divisors of the F_4-modelas ℙ^1-bundles for the three divisors D_0, D_1, and D_2, while the remainders, namely D_3 and D_4, aredouble-coverings of ℙ^1-bundles with discriminant locus a_6,4=0. This is illustrated in Figure <ref>. The singular fibers of these conic bundles consist of double lines and play an important role in determining the geometry of the singular fiber over s=a_6,4=0, which is a non-Kodaira fiber of type 1-2-3-4-2.This fiber can bethought of as an incomplete E_7 or E_8 ifv(a_4)=3 or v(a_4)≥ 4, respectively.To identify the representation associated with this singular fiber over s=a_6,4=0, we computed the intersection numbers of the new rational curves with the fibral divisors.These intersection numbers are interpreted as weights of F_4. We identified the corresponding representation asthe 26 of F_4. We also compute the triple intersection number of the fibral divisors. We finally specialize to the case ofCalabi-Yau threefolds. The Euler characteristic of an F_4-model and the Hodge numbers in the Calabi-Yau threefold case have been presentedin<cit.>. We also computedthe linear form induced by the second Chern class. In the final section, we studied details of M-theory compactified on a Calabi-Yau threefold that is an F_4-model, for whichthe resulting theory is a five-dimensional gauge theory with eight supercharges.Such a theory has vector multiplets characterized by a cubic prepotential. The classical part of the cubic prepotential vanishes but there is a quantum correction coming from an exact one-loop contribution.This one-loop term depends explicitly on the number of charged hypermultiplets. It is known that this correction term matches exactly with the triple intersection numbers of the fibral divisors of the elliptic Calabi-Yau threefold. We computed the number ofhypermultiplets in the adjoint representation and in the fundamental representation via a direct comparison:n_52=g,n_26=5 (1-g) + S^2. We checked that they satisfy the genus formula of Aspinwall-Katz-Morrison– here they are derived from the triple intersection numbers. The same number were computed by Grassi and Morrison using Witten's genus formula.With theknowledge of the Hodge numbers, matter representation, and their multiplicities, we checked explicitly that these data are compatible with a six-dimensional (1,0)supergravity theory free of gravitational, gauge, and mixed anomalies. We also computed the weights of vertical curves of an F_4 models and proved that over the locus V(s,a_6,4) the generic fiber over the divisor S has weights of the quasi-minuscule representation 26.We introduced the notion of a frozen representation, which explains that the existence of geometric weights carried by vertical curves does not imply the existence of hypermultiplets charged under the corresponding representation.If the base is a surface,the divisor S is a curve. Therepresentation 26 is frozen if and only ifS has genus zero and self-intersection -5. § ACKNOWLEDGEMENTSThe authors are grateful toPaolo Aluffi, Michele del Zotto, Antonella Grassi, Jim Halverson, Jonathan Heckman, Ravi Jagadeesan, Sheldon Katz, Craig Lawrie, David Morrison,Kenji Matsuki, Shu-Heng Shao, and Shing-Tung Yaufor helpful discussions.The authors are thankful to all the participants of Series of Lectures by Kenji Matsuki in March 2017 on “Beginners Introduction to the Minimal Model Program” hold at Northeastern University supported by Northeastern University andthe National Science Foundation (NSF) grant DMS-1603247.M.E. is supported in part by the National Science Foundation (NSF) grant DMS-1701635“Elliptic Fibrations and String Theory”. P.J.issupported by NSF grant PHY-1067976. P.J.is thankful to Cumrun Vafafor his guidance and constant support.M.J.K. would like to acknowledge partial support from NSF grant PHY-1352084. M.J.K. would like to extend her gratitude toDaniel Jafferis for his tutelage and continued support.§ WEIERSTRASS MODELS AND DELIGNE'S FORMULAIRE In this section, we follow the notation of <cit.>.Letℒ be a line bundle overa normal quasi-projective varietyB.We define the projective bundle (of lines)π: X_0=ℙ_B[𝒪_B⊕ℒ^⊗ 2⊕ℒ^⊗ 3]⟶ B. The relative projective coordinates of X_0 over B are denoted [z:x:y],where z, x, and y are definedby the natural injection of 𝒪_B, ℒ^⊗ 2, and ℒ^⊗ 3 into 𝒪_B⊕ℒ^⊗ 2⊕ℒ^⊗ 3, respectively. Hence,z is a section of 𝒪_X_0(1), x is a section of 𝒪_X_0(1)⊗π^∗ℒ^⊗ 2, and y is a section of𝒪_X_0(1)⊗π^∗ℒ^⊗ 3. AWeierstrass model is an elliptic fibration φ: Y→ Bcut out by the zero locus ofa section of the line bundle 𝒪(3)⊗π^∗ℒ^⊗ 6 in X_0. The most general Weierstrass equation is written in the notation of Tate asy^2z+ a_1 xy z + a_3yz^2 -(x^3+ a_2 x^2 z + a_4 x z^2 + a_6 z^3) =0, where a_i is a section of π^∗ℒ^⊗ i.The line bundle ℒ is called the fundamental line bundle of the Weierstrass model φ:Y→ B. It can be defined directly from Y asℒ=R^1 φ_∗ Y.Following Tate and Deligne, we introduce the following quantities b_2= a_1^2 + 4 a_2b_4= a_1 a_3 + 2 a_4b_6= a_3^2 + 4 a_6b_8= a_1^2 a_6 - a_1 a_3 a_4 + 4 a_2 a_6 + a_2 a_3^2 - a_4^2c_4= b_2^2 - 24 b_4c_6= -b_2^3 + 36 b_2 b_4 - 216 b_6 Δ = -b_2^2 b_8 - 8 b_4^3 - 27 b_6^2 + 9 b_2 b_4 b_6j= c_4^3/ΔTheb_i (i=2,3,4,6) and c_i (i=4,6) aresections of π^∗ℒ^⊗ i.The discriminant Δ is a section of π^∗ℒ^⊗ 12.They satisfy the two relations1728 Δ=c_4^3-c_6^2,4b_8 = b_2 b_6 - b_4^2.Completing the square in y gives zy^2 =x^3 +14b_2 x^2 + 12 b_4 x + 14 b_6.Completing the cube in x gives the short form of the Weierstrass equationzy^2 =x^3 -148 c_4 x z^2 -1864 c_6 z^3. § REPRESENTATION THEORY OF F_4 F_4 is studied in Planche VIII of <cit.>. The exceptional group F_4 has rank 4, Coxeter number 12, dimension 52, and a trivial center.Its root system consists of 48 roots, half of which are short roots. The Weyl group W(F_4) of F_4is the semi-direct product of the symmetric group 𝔖_3 with the semi-direct product of the symmetric group 𝔖_4 and (ℤ/2ℤ)^3. Hence, W(F_4) has dimension 3! 4! 2^3=2^7× 3^2=1152.W(F_4) is also a solvable group isomorphic to the symmetry group of the 24-cell.The long roots of F_4 form a sublattice of index 4. The outer automorphism group of F_4 is trivial. Hence, F_4 has neither complex nor quaternionic representations, and all its representations are real.Its smallest representation has dimension 26 and is usually called the fundamental representation of F_4. F_4 is not simply laced and can be described from E_6 by a ℤ/2ℤ folding. Highest weight Dimension (1,0,0,0) 52 (0,1,0,0) 1274(0,0,1,0) 273(0,0,0,1) 26 F_4 contains both B_3 and C_3, asis clear by removing the first or last node. It is less trivial to see that F_4 also containsB_4. One way to see it is to remember that F_4=su(3,𝕆) while 𝔰𝔬_9=𝔰𝔲(2,𝕆). The coset manifold F_4/Spin(9) is the octonionic projective plane 𝕆ℙ^2. It follows that we have the isomorphism <cit.>π_i(F_4)=π_i(Spin(9)),i≤ 6. The compact real form of F_4 can be described as the automorphism group of the Jordan Lie algebra J_3 of dimension 27 <cit.>.Maybe a more geometrically familiar picture is to describe thecompact real form of F_4 as the Killing superalgebra of the 8-sphere S^8 <cit.>.In this form, the Lie algebra F_4 decomposes (as a vector space) as the direct sum of the Lie algebra B_4≅𝔰𝔬_9 and itsspin representation.§ A DOUBLE COVER OF A RULED SURFACE BRANCH ALONG 2B FIBERS A ruled surface is by definition a ℙ^1-bundle over a smooth curve of genus g.Let p: Y→ S be the projection of a ruledsurface Y to its base curve S. There exists a non-negativenumber n such that S^2=-n. We will now construct the double cover X of Y branched along 2b fibers. 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"authors": [
"Mboyo Esole",
"Patrick Jefferson",
"Monica Jinwoo Kang"
],
"categories": [
"hep-th",
"math.AG"
],
"primary_category": "hep-th",
"published": "20170426180000",
"title": "The Geometry of F$_4$-Models"
} |
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