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Magnetization in pristine graphene with Zeeman splitting and variable spin-orbit coupling F. Escudero^†, L. Sourrouille^†, J.S. Ardenghi^†email: [email protected], fax number: +54-291-4595142 and P. Jasen^†^†IFISUR, Departamento de Física (UNS-CONICET)Avenida Alem 1253, Bahía Blanca, Buenos Aires, Argentina========================================================================================================================================================================================================================================= The aim of this work is to describe the spin magnetization of graphene with Rashba spin-orbit coupling and Zeeman effect. It is shown that the magnetization depends critically on the spin-orbit coupling λ that is controlled with an external electric field. In turn, by manipulating the density of charge carriers, it is shown that spin up and down Landau levels mix introducing jumps in the spin magnetization. Two magnetic oscillations phases are described that can be tunable through the applied external fields. The maximum and minimum of the oscillations can be alternated by taking into account how the energy levels are filled when the Rashba-spin-orbit coupling is turned on. The results obtained are of importance to design superlattices with variable spin-orbit coupling with different configurations in which spin oscillations and spin filters can be developed. § INTRODUCTION Graphene, a one-atom-thick allotrope, has become one of the most significant topics in solid state physics due to its two-dimensional structure as well as from its unique electronic properties (<cit.>,<cit.>,<cit.>, <cit.>, <cit.>). The carbon atoms form a honey-comb lattice made of two interpenetrating triangular sublattices, A and B that create specific electronic band structure at the Fermi level: electrons move with a constant velocity about c/300. In turn, the electronic properties are dictated by the π and π ^' bands that form conical valleys touching at the two independent high symmetry points at the corner of the Brillouin zone, the so called valley pseudospin <cit.>. In the absence of defects, electrons near these symmetry points behave as massless relativistic Dirac fermions with an effective Dirac-Weyl Hamiltonian <cit.> which allows to consider graphene as a solid-state toy for relativistic quantum mechanics. When the interaction between the orbital electron motion and spin degrees of freedom is taken into account, the spin-orbit coupling (SOC) induces a gap in the spectrum. SOC is the most important interaction affecting electronic spin transport in nonmagnetic materials. The use of graphene in spintronics (<cit.> and <cit.>) would require detailed knowledge of graphene's spin-orbit coupling effects, as well as discovering ways of increasing and controlling them. The SOC in graphene consists of intrinsic and extrinsic components (<cit.> and <cit.>). The intrinsic component of SOC is weak in the graphene sheet, although considerable magnitudes can be obtained in nanoscaled graphene ( <cit.> and <cit.>). The extrinsic component, known as the Rashba spin-orbit coupling (RSOC) <cit.> can be larger than the intrinsic component <cit.>. The RSOC may be controlled by the application of external electric fields (<cit.> and <cit.>). On the other side, when a magnetic field is applied perpendicular to the graphene sheet, a discretization of the energy levels is obtained, the so called Landau levels<cit.>. These quantized energy levels still appear also for relativistic electrons, just their dependence on field and quantization parameter is different. In a conventional non-relativistic electron gas, Landau quantization produces equidistant energy levels, which is due to the parabolic dispersion law of free electrons. In graphene, the electrons have relativistic dispersion law, which strongly modifies the Landau quantization of the energy and the position of the levels. In particular, these levels are not equidistant as occurs in a conventional non-relativistic electron gas in a magnetic field. This large gap allows one to observe the quantum Hall effect in graphene, even at room temperature <cit.>. In turn, when magnetic fields are applied to solids an important effect called the de Haas-van Alphen (dHvA) <cit.> appears as oscillations of magnetization as a function of inverse magnetic field. This effect is a purely quantum mechanical phenomenon and is a powerful toof for maping the Fermi surface, i.e. the electronic states at the Fermi energy (<cit.>,<cit.> ) and because gives important information on the energy spectrum, it is one of the most important tasks in condensed matter physics. The different frequencies involved in the oscillations are related to the closed orbits that electrons perform on the Fermi surface. It has been predicted in graphene that magnetization oscillates periodically in a sawtooth pattern, in agreement with the old Peierls prediction <cit.>, although the basic aspects of the behavior of the magnetic oscillations for quasi-2D materials remains yet unclear <cit.>. In contrast to 2D conventional semiconductors, where the oscillating center of the magnetization M remains exactly at zero, in graphene the oscillating center has a positive value because the diamagnetic contribution is half reduced with that in the conventional semiconductor <cit.>. From an experimental point of view, carbon-based materials are more promising because the available samples already allows one to observe the Shubnikov-de Haas effect (<cit.> and<cit.>) and then may be easier to interpret than quantum oscillations in its transport properties. Because the dHvA signal in 2D systems are free of the k_z smearing, it should be easier to obtain much rich information about the electron processes. In addition, the SOC, which is considerably large in graphene <cit.> plays an important role in the determination of the magnetic oscillations because of the fundamental difference with conventional semiconductor 2DEG.Motivated by this phenonema, in the present paper we study the dHvA oscillations in the spin magnetization in graphene by taking into account the Rashba spin-orbit interaction modulated by a perpendicular external electric field and the Zeeman effect modulated by a constant magnetic field perpendicular to the graphene sheet. It is well known that each electron contributes with μ _B to the magnetization density if the spin is parallel to the applied magnetic field and -μ _B if it is antiparallel. Hence, if N_± is the number of electrons per area with spin parallel (+) or antiparallel (-), the magnetization density will beM=μ _B(N_+-N_-). As it was said before, the introduction of a constant magnetic field introduces the Landau levels that are splitted by the Zeeman effect. The filling of these levels is not trivial in graphene due to its square root dependence in the Landau index. The degeneracy, that depends linearly on B, defines the number of states that are completely filled, but these levels may not be sorted in ascending order with intercalated spins. In fact, whenever the n+1 Landau level with spin up is lower than the n Landau level with spin down, an enhancement of the spin magnetization will be obtained. This behavior can be alter drastically with the RSOC, because in this case, the ordering of the energy levels is not trivial. In this sense and since both the magnetic field and RSOC are externally controllable, in what follows the role of each of the parameters involved, as the electron density and the electric and magnetic field strength are discussed in relation to the maximum and minimum of the oscillations. The spin-orbit effects are important, besides the fundamental electronic and band structure and its topology, to understand spin relaxation, spin Hall effect and other effects such as weak (antilocalization). There are two main routes to implement spintronics devices: the giant magneto resistance effect (GMR) <cit.> and the spin effect field transistor (spin-FET) <cit.>. Both devices consists in a sandwich structure made of two ferromagnetic materials separated by a non-magnetic later in GMR and two dimensional electron gas in spin-FET. In this device, the spin-orbit coupling causes the electron spin to precess with a precession length determined by the strength of the spin-orbit coupling, which through a gate voltage becomes tunable. A detailed knowledge of the interplay of the the different parameters entering in the Hamiltonian is vital to fundamentally understand the spin-dependent phenomena in graphene.[ For a good review of spintronics in graphene see <cit.>.] This work will be organized as follow: In section II, pristine graphene under a constant magnetic field with spin-orbit coupling and Zeeman splitting is introduced and the magnetic oscillations are discussed. In section III, N_+ and N_- populations as a function of B, the electron density and RSOC parameter is studied and the oscillations are computed and discussed. The principal findings of this paper are highlighted in the conclusion.§ MAGNETIC OSCILLATIONS WITH ZEEMAN SPLITTING For a self-contained lecture of this paper, a brief introduction of the quantum mechanics of graphene in a constant magnetic field in the long wavelength approximation will be introduced (see <cit.>). The Hamiltonian in one of the two inequivalent corners of the Brillouin zones can be put in a compact notation as (see <cit.> and <cit.>) H=v_F(σ·π)+1/2Δ _R( σ×𝐬)-Δ _Zs_zwhere σ are the Pauli matrices acting on the pseudospin space and 𝐬 are the Pauli matrices acting on the spin space. Δ _R is the extrinsic spin-orbit coupling that arise when an external electric field is applied perpendicular to the graphene sheet or from a gate voltage or charged impurities in the substrate (<cit.>, <cit.> and <cit.>).[ Intrinsic spin-orbit coupling Δ _int that contains contributions from the p orbitals and d orbitals (see eq.(10) of <cit.>)is neglected.] Δ _Z=gμ _B/2B is the Zeeman energy that depends on the magnetic field strength. In the following we will write the extrinsic Rashba spin orbit coupling as Δ _R=ħω _y. The quasiparticle momentum must be replaced by π =𝐩-e𝐀, where e is the electron charge and 𝐀 is the vector potential which in the Landau gauge reads 𝐀=(-By,0,0). For the K valley, the third term can be written as -iΔ _R(σ _+s_+-σ _-s_-) where σ _±=σ _x± iσ _y and s_±=s_x± is_y. This term describes a coupling between pseudospin and spin states, that is, a spin-flip process can be achieved by hopping an electron in the A sublattice with spin up to the B sublattice with spin down. The eigenvalues of the Hamiltonian of eq.(<ref>) has been computed in <cit.> and <cit.> without the Zeeman term. By introducing this term the eigenvalues reads[ In <cit.> it is discussed the intrinsic and extrinsic spin-orbit couplings and the quasi-classical limit for higher Landau levels.] E_n,s,l=lħ/√(2)√(2ω _Z^2+2nω _L^2+ω _y^2-s√(16nω _Z^2ω _L^2+4nω _L^2ω _y^2+ω _y^4))where l=+1(-1) is for the conduction (valence) band, n=0,1,2,... is the Landau level index and s=± 1 is the spin. The number of conduction electrons will be given by N=n_eA where n_e is the electron density and A is the area of the graphene sheet. For simplicity, we can consider only conduction electrons, which implies that the added electrons are due to a gate voltage applied to the graphene sheet so that n_e can be varied as a function of V_G. When a magnetic field is applied, the energy is discretized and each level is degenerated with degeneracy D=BA/ϕ, where ϕ is the magnetic unit flux. There is a critical magnetic field in which the degeneracy D equals the number of electrons NB_C=n_eϕIf we would consider only valence electrons, then B_C∼ 80× 10^3T, which is an unfeasible experimentally. Nevertheless, B_C defined as in last equation will serve as an upper bound for the numerical calculations.Let us consider the first case under study: graphene under a magnetic field with Zeeman effect. In this case, the energy levels can be found and readsE_n,s,l=-sħω _z+lħω _L√(n)being s=± 1 for spin up and down respectively, n=0,1,2,... is the Landau level index and l=1 for the conduction band and l=-1 for the valence band. For the conduction band, electrons will start filling the lowest levels, so in the case in which |ħω _z| <1/2|ħω _L|, that is, when the Zeeman splitting is lower than half the separation between consecutive Landau levels, then the filling will be done considering the Landau index first and then the spin index. This implies that if we consider the energy levels in ascending order, there will be no consecutive Landau levels with same spin filled. This assumption is fulfilled for normal systems, where ω _L is cyclotron frequency ω _L=eB/m and the Landau levels are ħω _C(n+1/2).In this case, because ω _Z is proportional to B, then the condition |ħω _z| <1/2|ħω _L| implies that gμ _B/2<ħ e/m. For graphene, due to the relativistic dispersion relation, ω _L is proportional to √(B) and the Landau levels increase as √( n). Then if we write ħω _z=γ _ZB and ħω _L=γ _L√(B), then the condition for no consecutive spin up or down filling is given byB<( γ _L/2γ _Z) ^2( √(n+1)-√( n)) ^2For graphene γ _L/2γ _Z∼36.29/ 2· 0.12∼ 151.2 (see <cit.> and <cit.>).Under this regime, the ordered energy levels can be written asE_p=-(-1)^pħω _Z+ħω _c√(p/2-1/ 4(1-(-1)^p))for p even we obtain the spin up levels and for p odd the spin down levels. The filling factor v=B_C/B can be written as v=q+θ, where q=[ B_C/B], where [ x] is the floor function defined as the largest integer less than or equal to x and θ =ν -q. The spin Pauli paramagnetism is given by the remaining term for the Landau filling which is proportional to λM=μ _B(N_↑-N_↓)=Nμ _BB/B_C[1/2( 1-(-1)^[B_C/B]) +(-1)^[ B_C /B] ( B_C/B-[ B_C/B] )]where the difference (N_↑-N_↓) is proportional to θ and the factor (-1)^q selects the majority of spin up or down states in the last Landau level partially filled. In figure <ref>, the dimensionless magnetization per electron M/Nμ _B is plotted against x= B/B_C.It must be stressed that condition of eq.(<ref>) depends on n and because√(n+1)-√(n)<1 for n>1, then for higher Landau levels the upper bound for B decreases and we must take into account the lowest Landau level completely filled n=q=[ B_C/B]. This gives the condition for B B<( γ _L/2γ _Z) ^2( √([B_C/B] +1)-√([ B_C/B] )) ^2The jumps in the magnetization are well understood in terms of the filling of the Landau levels up to the Fermi level. Because the Fermi level indicates the lowest Landau level filled but the degeneration of each level is not a integer number, then when a level is completely filled, the Fermi level jumps and in consequence the magnetization. In the second regime, where B≥( γ _L/2γ _Z) ^2( √( n+1)-√(n)) ^2, the energy levels will show consecutive Landau levels with the same spin filled. In this case, it is necessary to use numerical methods to sort the energy levels in ascending order taking into account if the level belongs to a spin up or down state. In figure <ref> , a sawtooth like behavior for the spin magnetization is obtained which is given by the spin up and down population, where we have used that n_e=0.1 . When ν =2, then D=N/2 which implies that the fundamental and first excited levels are completely filled. For this particular value of B, the first excited level are filled with spin down states. This behavior is shown in figure <ref>, where both population are identical for B=B_C/2. Between this value and B=B_C (ν =1), the spin up population decreases linearly. The same behavior is expected for v=l where l=1,2, ... For those values of B, the spin up and down population are identical. The spin magnetization peaks follows a linear behavior for high magnetic fields M=B/n_eϕ _0. For B>n_eϕ _0 the spin magnetization reaches a plateau given by M=μ _BN. §.§ Spin-orbit coupling When an external electric field is applied perpendicular to the graphene sheet, the Byckhov-Rashba effect appears (see <cit.>). The parameter y that describes this interaction depends linearly with the external electric field strength. In figures <ref>, <ref>, <ref> and <ref>, a sequence of plots for different values of y are shown, where n_e=0.1 and A=10nm^-2 (N=1).[ A supplementary video has been uploaded where the spin magnetization is plotted for increasing values of y for the particular case n_e=0.1nm^-2.] As it can be seen in the supplementary material, there are set of values of y where there are no changes in the magnetization. These set of values are given when the spin up Landau energy level n is lower than the spin down energy level n-1. For simplicity, let us consider that ν =2. Then only the two first Landau levels are completely filled when y=0. In this case, because these two levels correspond to the spin up and down n=0 Landau levels, then the spin magnetization vanishes because N_+=N_-. When y>0, there is a critical value y_c^(2) where E_1,↑<E_0,↓ which implies that the two first Landau levels are spin up states. Then the magnetization jumps to a constant value M=μ _BN. The condition E_1,↑<E_0,↓ implies that√(2γ _Z^2B^2+2y^2)≥√(2γ _Z^2B^2+2γ _L^2B+y^2-√(4γ _Z^2B^2(4Bγ _L^2+y^2)+y^4))and the solutions readsy^2≥γ _L^2/2B-2γ _Z^2B^2which gives two solutions, one for positive values of y which is the case of interest. In turn, when ν =3, then D=N/3, then only the first three Landau levels are completely filled. For y=0, there is no spin mixing and therefore the magnetization is positive and maximum. When y>0 and in particular when y>y_c^(3) then E_1,↑<E_0,↓, but in this case, when we increase the magnetic field strength, the degeneration increases, which implies that only the two first Landau levels are filled completely, which correspond to spin up states. Then, the magnetization increase as B does until we reach B=B_C/2. These considerations imply that there is a critical value y_c^(3)<y<y_c^(2) defined from the equation E_1,↑=E_0,↓ when B=n_eϕ _0/2 for y^(2) and when B=n_eϕ _0/3 for y_c^(3) where the magnetization changes with respect the case y=0. The regions for y_c^(j+1)<y<y_c^(j) and n_e where the magnetization changes can be obtained from the condition E_j+1,↑=E_j,↓ when B= n_eϕ _0/j for y_c^(j) and B=n_eϕ _0/j+1 for y_c^(j+1). In figure <ref> these set of values are shown. For small values of y the regions tends to a continuum when j→∞. In turn, in figures <ref>, the spin magnetization as a function of B is shown when y>y_c^(1), where the magnetization saturates. The set of magnetic fields B_j=n_eϕ _0/2j are particularly important. For these values and y_c^(j)=√(n_eϕ _0/2j(γ _L^2-4γ _Z^2n_eϕ _0/2j)^2-16(j-1)γ _Z^2γ _L^2( n_eϕ _0/2j)^2/4(j-1)γ _L^2+2γ _L^2-8γ _Z^2n_eϕ _0/2j)which is found by computing E_j+1,↑<E_j,↓ and replacing B_j=n_eϕ _0/2j, the magnetization jumps from M=0 to M_j=μ _BN/2j (see figure <ref>). There are infinite mixes of Landau spin up and down states that are defined through the inequality E_n+r,↑≤ E_r,↓ which implies thaty^2≥16nγ _Z^2γ _L^2-(rγ _L^2-4γ _Z^2)^2/8γ _Z^2-2(2n+r)γ _L^2These set of values are located between the regions described in figure <ref>.These results are of importance to develop two-dimensional supertlattices structures by using graphene or silicene, which supports charge carriers behaving as massless Dirac fermions with graphene-like electronic band structure (<cit.> and <cit.>). Spin-orbit interaction in silicene can be 1000 times larger that of graphene, which implies that quantum spin Hall effect in silicene is experimentally accesible. Considering two semi-infinite graphene ribbons, that are employed as source and drain, and a drain voltage that introduces charge carriers with a concentration n_e it is possible to develop a superlattice of different graphene samples with specific values of y in series. By fixing the magnetic field strength in any of the critical values B_j=n_eϕ _0/2j, the possible values of y can be located over the curves in figure <ref> in such a way to increase the spin polarized density of states across the superlattice. Different configurations can be obtained by developing superlattice with alternating constant electric fields in order to obtain spin flipping, which can be used as a spin down or up filter <cit.>. By applying Landauer-Büttiker formalism, the electronic behavior of Dirac fermions in the superlattice can be studied. The conductance will shows resonant tunneling behavior depending on the number of barriers and barrier width <cit.>. In turn, the spin polarization lifetime can be controlled by the applied electric field that controls Rashba spin-orbit coupling <cit.> and <cit.>. It has been shown the angular range of the spin-inversion can be efficiently controlled by the number of barriers<cit.>. Magnons can be obtained in the superlattice structure by combining alternating applied electric fields, where the wavelength can be accomodated by the width of the middle graphene samples <cit.>. It is source of future works to develop graphene superlattice with different configurations of Rashba spin-orbit couplings and external magnetic fields.Finally, to further explore the magnetic activation by changing the RSOC, we can consider the following setup: N=2, that is, two conduction electrons only, B=B_c/2, where D=1 which implies that we have to consider the first two Landau levels only. When y=0, one electron occupy the ground state with spin up and the second electron the ground state with spin down which implies that the spin magnetization is zero. Suppose that we consider the action of a fast sudden perturbation in the system where the RSOC changes to y=y_c^(1), where as we said before, y_c^(1) is such that E_1,↑=E_0,↓ which is given in eq.(<ref> ). Then, because the first excited spin up level is below the ground state with spin down, the first two Landau levels filled are with spin up, which implies that M jumps to M=2μ _B. The transition amplitude is given by the inner product of both eigenstates. To obtain this value we can consider the anti-symmetric state for y=0|ψ _M=0⟩ =1/√(2)[ |φ _0,↑^y=0⟩⊗|φ _0,↓^y=0⟩ -|φ _0,↓^y=0⟩⊗|φ _0,↑^y=0⟩]and the anti-symmetric state for y=y_c^(1)|ψ _M=2μ _B⟩ =1/√(2)[ |φ _0,↑^y=y_c^(1)⟩⊗|φ _1,↑^y=y_c^(1)⟩ -|φ _1,↑^y=y_c^(1)⟩⊗|φ _0,↑^y=y_c^(1)⟩]where we can note that the state for the second electron is spin up. where we are considering that both electrons are not interacting. Each vector |φ _n,s^y=0⟩ can be computed by considering the diagonalization of the Hamiltonian of eq.(<ref>). This eigenfunctions are computed in <cit.> and <cit.> and we can write |φ _0,↑^y=0⟩ =1/√( 2L)([ ϕ _0 ϕ _000 ] ), |φ _0,↓^y=0⟩ =1/ √(2L)([00 ϕ _0 ϕ _0 ] ), |φ _0,↑^y=y_c^(1)⟩ = 1/√(2L)√(|β _1| ^2+1)([ β _1ϕ _000 ϕ _0 ] ) and |φ _1,↑^y=y_c^(1)⟩ =1/√(2L)√(|α _1| ^2+|α _2| ^2+|α _3| ^2+1)([ α _1ϕ _1 α _2ϕ _0 α _3ϕ _1 ϕ _0 ] ), where ϕ _n(ξ )=π ^-1/4/√(2^nn!)e^- 1/2ξ ^2H_n(ξ ) where H_n are the Hermite polynomials of order n and ξ =ξ =y/l_B-l_Bk, where l_B=√(1/eB) is the magnetic length and k is the wavevector in the x direction. 2L is the total length of the graphene sample in the x direction. The coefficients β _1, α _1, α _2 and α _3 are obtained through the diagonalization of the Hamiltonian and readsβ _1=i( √(2)γ _ZB+√(γ _L^2B-2γ _Z^2B^2)) /√(γ _L^2B-4γ _Z^2B^2) α _1=i( γ _L^2B+2√(2)γ _ZB√(γ _L^2B-2γ _Z^2B^2)) /√(γ _L^2B-2γ _Z^2B^2)( -√(2)γ _ZB+√(γ _L^2B-2γ _Z^2B^2)) α _2=2i√(2)γ _L^2B√(γ _L^2B-4γ _Z^2B^2)/2γ _L^2B-4√(2)γ _ZB√(γ _L^2B-2γ _Z^2B^2) α _3=γ _L^2B/2-γ _ZB+√(γ _L^2B+2γ _Z^2B^2)where we have to replace B=B_c/2 and y=y_c^(1). In figure<ref>, the probability amplitude |⟨ψ _M=0|ψ _M=2μ _B⟩| ^2 is plotted as a function of the electron density n_e. It is shown that the transition tends to zero as n_e→ 0 and a non-trivial zero for n_e∼ 21nm^-2. The curve in this region follows the behavior of the curve y=y_c^(1) as it can be seen in figure <ref>, although it is not exactly a semicircle. For higher values of n_e, the probability amplitude tends to zero. Then it is possible to increase the spin magnetic transition for the specific value, where the probablity amplitude is maximum for n_e. In this model we have not consider more than two electrons because in this case we should take into account the interactions between electrons in the same state, which can be done by considering the Laughlin wavefunctions <cit.> and the subtleties introduced by the fractional quantum Hall effect.§ CONCLUSIONS In this work we have examined the spin magnetization in pristine graphene with spin-orbit coupling and Zeeman splitting. The magnetization has been found as a function of the electron density, the magnetic field strength and the Rashba spin-orbit coupling parameter. We have derived and compared the maximum and minimum of the magnetization with and without spin-orbit coupling showing that for certain regions of the spin-orbit coupling parameter y and certain values of the electron density n_e, the magnetization jumps from zero to a constant value given M_j=μ _BN/2j being N the number of electrons and j=1,2,3...being an integer number that controls when a spin up state with Landau level j+1 is lower than the spin down state with Landau level j. Morever, these regions for y follows a complex relation with n_e that numerically shows certain values of y where magnetization is not altered by y. These results show that it is possible to obtain spin filters or spin oscillations by considering different superlattice configurations, where the inner graphene samples can alternate the applied electric field in such a way to fix the magnetic field strength in those values where the Landau spin up and down states mix. Finally, we have studied the probability amplitude for a sudden change of the Rashba spin-orbit coupling from the non-magnetic state M=0 to M=2μ _B when there are two electrons, B=n_eϕ _0/2 and y=y_c^(1), showing that the transition has a a peak for n_e>0 and a non-trivial zero.§ ACKNOWLEDGMENT This paper was partially supported by grants of CONICET (Argentina National Research Council) and Universidad Nacional del Sur (UNS) and by ANPCyT through PICT 1770, and PIP-CONICET Nos. 114-200901-00272 and 114-200901-00068 research grants, as well as by SGCyT-UNS., J. S. A. and L. S. are members of CONICET., F. E. is a fellow researcher at this institution.§ AUTHOR CONTRIBUTIONS All authors contributed equally to all aspects of this work.99 novo K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos and A. A. Firsov, Nature , 438, 197 (2005).intro1 A.K. Geim and K. S. Novoselov, Nature Materials,6, 183 (2007).intro2 Y. B. Zhang, Y.W. Tan, H. L. 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"authors": [
"Federico Escudero",
"Lucas Sourrouille",
"Juan Sebastián Ardenghi",
"Paula Jasen"
],
"categories": [
"cond-mat.str-el"
],
"primary_category": "cond-mat.str-el",
"published": "20170824231617",
"title": "Magnetization in pristine graphene with Zeeman splitting and variable spin-orbit coupling"
} |
empty2126Streaming Graph Challenge: Stochastic Block Partition Edward Kao, Vijay Gadepally, Michael Hurley, Michael Jones, Jeremy Kepner, Sanjeev Mohindra, Paul Monticciolo, Albert Reuther, Siddharth Samsi, William Song, Diane Staheli, Steven Smith MIT Lincoln Laboratory, Lexington, MA Received: date / Accepted: date ======================================================================================================================================================================================================================================= Pipelines combining SQL-style business intelligence (BI) queries and linear algebra (LA) are becoming increasingly common in industry. As a result, there is a growing need to unify these workloads in a single framework. Unfortunately, existing solutions either sacrifice the inherent benefits of exclusively using a relational database (e.g. logical and physical independence) or incur orders of magnitude performance gaps compared to specialized engines (or both). In this work we study applying a new type of query processing architecture to standard BI and LA benchmarks.To do this we present a new in-memory queryprocessing engine called . uses worst-case optimal joins as its core execution mechanism for both BI and LA queries. With , we show how crucial optimizations for BI and LA queries can be captured in a worst-case optimal query architecture. Using these optimizations, outperforms other relational database engines (LogicBlox, MonetDB, and HyPer) by orders of magnitude on standard LAbenchmarks, while performing on average within 31% of the best-of-breed BI (HyPer) and LA (Intel MKL) solutions on their own benchmarks. Our results show that such a single query processing architecture is capable of delivering competitive performance on both BI and LA queries. Finally, on an application which combines SQL-style querying and linear algebra, we show that can outperform popular solutions like MonetDB/SciKit-Learn, Spark, and Pandas/SciKit-Learn by up to two orders of magnitude. In this work we identify these techniques, validate their merit, and describe a novel but practical join processing architecture which uses them.§ INTRODUCTIONThe efficient processing of classic SQL-style workloads is no longer enough; machine learning algorithms are being adopted at an explosive rate. In fact, Intel projects that by 2020 the hardware cycles dedicated to machine learning tasks will grow by 12x, resulting in more servers running this than any other workload <cit.>. As a result, there is a growing need for query processing engines that are efficient on (1) the SQL-style queries at the core of most business intelligence workloads and (2) the linear algebra operations at the core of most machine learning algorithms. In this work, we explore whether a new query processing architecture is capable of delivering competitive performance in both cases.An increasingly popular workflow combines business intelligence (BI) and linear algebra (LA) queries by executing SQL queries in a relational warehouse as a means to extractfeature sets for machine learning models.Unsurprisingly, these SQL queries are similar to standard BI workloads:the data is de-normalized (via joins), filtered, and aggregated to form a single feature set<cit.>. Still, BI queriesare best processed in a relational database management system (RDBMS) and LA queries are bestprocessed in a LA package. As a result, there has been a flurry of activity around building systems capable of unifying both BI and LA querying <cit.>.At a high-level, existing approaches fall into one of three classes:* Exclusively using a relational engine. There are many inherent advantages to exclusively using a RDBMS to process both BI and LA queries. Simplifying extract-transform-load(ETL), increasing usability, and leveraging well-known optimizations are just a few <cit.>. Although it is known thatLA queries can be expressed using joins and aggregations, executing these queriesvia the pairwise join algorithms in standard RDBMSs is orders of magnitude slowerthan using a LA package (see <Ref>).Thus, others <cit.> have shown that a RDBMS must be modified to compete on LA queries. * Extending a linear algebra package. Linear algebra packages, likeBLAS <cit.> or LAPACK <cit.>, provide high-performance through low-levelprocedural interfaces and therefore lack the ability for high-level querying. Toaddress this, array databases with high-level querying, like SciDB<cit.>, have been proposed. Unfortunately, array databases arehighly specialized and are not designed for general BI querying.As a result, support for SQL-style BI querying<cit.> has recently been combined with the LA support found in popular packageslike Spark's MLlib <cit.> and Scikit-learn<cit.>.Still, these solutions lack many of the inherent benefits of a RDMBS,like a sophisticated (shared-memory) query optimizer or efficient data structures (e.g. indexes) for query execution, and therefore achieve suboptimalperformance on BI workloads (see <Ref>).* Combining a relational engine with a linear algebra package. To preserve the benefits of using a RDBMS on BI queries, while also avoiding their pairwise join algorithms on LA queries,others (e.g. Oracle's UT_NLA <cit.> and MonetDB/NumPy <cit.>) have integrated a RDBMS with a LA package.Still, these approaches just tack on an external LA package to a RDBMS—they do not fully integrate it. Therefore, users are forced to write low-level code wrapping the LA package and the LA computation is a black box to the query optimizer. Even worse, this integration, while straightforward on dense data, is complicated (and largely unsolved) on sparse data because a RDBMSand a LA package use fundamentally differentdata layouts here (e.g. column store versus compressed sparse row).Therefore, existing approaches either (1) sacrifice the inherent benefits of using only an RDBMS to process both classes of queries, (2) are unable to process both classes of queries, or (3) incur orders of magnitude performance gaps relative to the bestsingle-class approach.In this work we study an alternative approach to building a RDBMS for both BI and LA querying. In particular, we study using worst-case optimal join (WCOJ) algorithms <cit.> as the mechanism to unify these query workloads. To do this, we present a new in-memory queryprocessing engine called . uses a novel WCOJ query architecture to optimize, plan,andexecute both BI and LA queries. In contrastto previous WCOJ engines <cit.>,is designed for and evaluated on more than just graph queries. As such, is the first WCOJ engine to present an evaluation onboth BI and LA queries. In contrast to other query engines<cit.>,competes with both LA packages and RDBMSs on their own benchmarks (see <Ref>).However, designing a new query processing engine that is efficient on both BI and LA queries is a challenging task. The recently proposed WCOJs at the core of this new query processing architecture are most effective on graph querieswhere they have an asymptotic advantage over traditional pairwise join algorithms. In contrast, pairwise join algorithms are well-understood, and have the advantage of 45+ years of proven constant factor optimizations for BI workloads <cit.>. Further, LA queries, which also have the benefit of decades of optimizations<cit.>, are typically not a good match for the relational model. Therefore, it is not at all obvious whether they are good match for this new type of relational query processing architecture.Despite these challenges, we found that the unification of three new techniques in a WCOJ architecture could enable it to deliver competitive performance on bothBI and LA queries. The techniques we leverage are (1) a new mechanism to translate general SQL queries to WCOJ query plans, (2) a new cost-based query optimizer forWCOJ's, and (3) a simple storage engine optimizer for . It was not at all obvious how to identify and unify these techniques in a WCOJ framework and, as such, we are the first to do so. The three core techniques of in more detail are:* SQL to GHDs: Pushing Down Selections and Attribute Elimination. Neither the theoretical literature <cit.> nor the query compilation techniques for WCOJs <cit.> maps directly to all SQL features. In we implement a practical extension of these techniques that enables us to capture more general query workloads as well as the classic query optimizations of pushing down selections and attributeelimination. Besides providing up to 4x speedup on BIqueries, a core artifact of our attribute elimination implementation is that itenables to target BLAS packages on dense LA queries at little to no execution cost. This is because it is challenging to outperform BLASpackages, like Intel MKL <cit.> on sparse data and is usually notpossible[Intel MKL often gets peak hardware efficiency on denseLA <cit.>.] on dense data. Therefore,leverages attribute elimination to opaquely call Intel MKL on dense LAqueries while executing sparse LA queries as pure aggregate-join queries (entirely in).* Cost-Based Optimizer: Attribute Ordering. WCOJ query optimizers need to select an attribute order <cit.> ina similar manner to how traditional query optimizers select a join order<cit.>. With , we present acost-based optimizer to select a WCOJ attribute order for thefirst time. We highlight that our optimizer follows heuristics that are different from what conventional wisdom from pairwise join optimizerssuggests (i.e. highest cardinality first) and describe how to leverage theseheuristics to provide an accurate cost-estimate for a WCOJ algorithm. Using this cost-estimate, we validate that 'scost-based optimizer selects attribute orders than can be up to 8815x faster than attribute orders that previous WCOJ engines<cit.> might select.* Group By Tradeoffs: Battling Skew on Group By. Finally,we study the classic tradeoffs around executing s in the presence of skew. Although these tradeoffs are well understood for BI queries, they have yet to be applied to this new style of query processing engine and are also essential for LA queries(see <Ref>). We describe 's optimizer that exploits these tradeoffs and show that it provides up to a 875x and 185x speedup over a naive implementation on BIand LA queries respectively. We evaluate on standard BI and LA benchmarks: seven TPC-H queries[The TPC-H queries are run without theclause.] and four (two sparse, two dense)LA kernels. These benchmarks are de-facto standards for relational query processing and LA engines. Thus, each engine we compare to is designed to process one of these benchmarks efficiently using specific optimizations that enable high-performance on one typeof benchmark, but not necessarily the other. Therefore, although these engines are the state-of-the-art solutions within a benchmark, they are unable to remain competitive across benchmarks. For example, HyPeR delivers high performance on BI queries, but is notcompetitive on most LA workloads; similarly, Intel's Math Kernel Library (MKL)<cit.> delivers high performance on LA queries, but does not providesupport for BI querying. In contrast, is designed to be generic, maintaining efficiency across the queries in bothbenchmarks. *Contribution SummaryThis paper introduces the engine and demonstrates that its novel architecture can compete on standard BI and LAbenchmarks. We show that outperforms other relational engines by at least an order of magnitude on LA queries, while remaining on average within 31% ofbest-of-the-breed solutions on both BI and LA queries.Our contributions and outline are as follows.* In <Ref>, we describe the architecture.In particular, we describe how's query and data model, which is different from that of previous WCOJ engines <cit.>, preserves the theoretical benefits of a WCOJ architectureand enables it toefficiently process BI and LA queries. * In <Ref> we present the core (logical and physical) optimizations that we unify in a WCOJ query architecture for the first time. We show that these optimizationsprovide up to a three orders of magnitude performancespeedup and are necessary to compete on BI and LA queries. In more detail, we present the classic query optimizations of pushing down selections and attribute elimination in <Ref>, a cost-based optimizer for selecting an attribute order for a WCOJ algorithm in <Ref>, and, finally, a simple query execution optimizer for s in <Ref>.* In <Ref> we show that can outperform other relational engines by an order of magnitude on standard BI and LA benchmarkswhile remaining on average within 31% of a best-of-breed solution within each benchmark. For the first time, this evaluation validates that a WCOJ architecture can compete on BI and LA workloads. We argue that the inherent benefits ofsuch a unified (relational) design has the potential to outweigh its minor performance overhead. We believe represents a first step in unifying relational algebra and machine learning in a single engine. As such, we extend in <Ref> to explore some of the potential benefits ofthis approach on a full application. We show here that is up to an order of magnitudefaster than the popular (unified) solutions of MonetDB/Scikit-learn, Pandas/Scikit-learn,and Spark on a workload that combines SQL-style querying and a machine learning algorithm. We hope adds to the debate surrounding the design of unified querying systems.§.§ Related Workextends previous work in worst-case optimal join processing (and LogicBlox), relational data processing, and linear algebra processing. *EmptyHeaded is designed around many of the techniques presented in the engine<cit.>, but with many fundamental differences. First, was only evaluated in the graph andresource description framework (RDF) domains. As such, the design is unable to support BI workloads that contain many attributes. The limitations in the design stem from its storage model and query model which was specifically designed for graph and RDF workloads. In addition, the design only supported equality selections and 's on indexed attributes.Finally, the design ofwas optimized for only sparse data and achieved suboptimal performance on workloads with dense data. is designed to fix these flaws by leveraging new techniques (<Ref>) that eliminate these limitations,while preserving the theoretical benefits of such a design. *LogicBloxLogicBlox <cit.> is a full featured commercial database engine built around similar worst case optimal join <cit.> and query compilation <cit.> algorithms. Still, a systematic evaluation of the LogicBlox engine on BI and LA workloads is yet to be presented.From our conversations with LogicBlox, we learned that theyoften avoid using a WCOJ algorithm on these workloads in favor of more traditional approaches to join processing. In contrast, always uses a WCOJ algorithm (even on queries not approaching the worst-case). Further, LogicBlox uses a query optimizer that has similar benefits to 's (see<Ref>), but does so using custom algorithms <cit.>.In contrast, uses a generalization of these algorithms <cit.>that maps to well-known techniques <cit.>. Finally, LogicBlox stores its data in manner that resembles a row store, whereas resembles column store. *Relational Processing A significant amount of work has focused onbringing LA to relational data processing engines. Some have suggested treating LA objects as first class citizens in a column store <cit.>. Others, such as Oracle's UTL_NLA <cit.> and MonetDB with embedded Python <cit.> allows users to call LApackages through user defined functions. Still, the relational optimizers in these approaches do not see, and therefore are incapable of optimizing, theLA routines. Even worse, these packages are cumbersome to use and place significant burden on the user to make low-level library calls. Finally, the SimSQL project <cit.> suggests that relational engines can be modified in straightforward ways to accommodate LA workloads. Our goals are similar to the SimSQL, but explored with different mechanics. SimSQL studied the necessary modifications for a classic database architecture to support LA queries and was only evaluated on LA queries. In , we evaluate a new, theoretically advanced, query processing architecture on both BI and LAworkloads. Additionally, SimSQL was evaluated in the distributed setting and therefore compared to higher-level baselines.is evaluated in a shared memory setting against Intel MKL and HyPer<cit.>. Other high performance in-memory databases, like HyPer, focused on classic OLTP and OLAP workloads and were not designed with other workloads in mind. *Linear Algebra Processing Researchers have long studied how to implement high-performance LA kernels. IntelMKL <cit.> represents the culmination of this work on Intel CPUs.Unsurprisingly, researchers have shown<cit.> that it requires tedious low-level code optimizations to come near the performance of a package such as Intel MKL. As a result, processing these queries in a traditional RDBMS(using relational operators) is at least an order of magnitude slower than usingsuch packages (see <Ref>). In response, researchers have released array databases, like SciDB and TileDB <cit.>, which provide high-level LA querying, often bywrapping lower-level libraries.In contrast, our goal is not to design an entirely different and specialized engine for these workloads, butrather to design a single (relational) engine that processes multiple classesof queries efficiently. § LEVELHEADED ARCHITECTUREIn this section we overview the data model, storage engine, query compiler,and join algorithm at the core of the architecture. This overview presents thepreliminaries necessary to understand the optimizations presented in <Ref>. ingests structured data from delimited files on disk and has a Python front-end that accepts Pandas dataframes.The query language is a subset of the SQL 2008 syntaxwith some standard limitations that we detail in <Ref>. The input and output of each query is a table which we describe in <Ref>. The SQL queries are translated to generalized hypertree decompositions (GHD) which we describe in <Ref>. Finally, code is generated from the selected GHD using a WCOJ algorithm which we describe in <Ref>. The entire process is shown in <Ref>.§.§ Data Model The data model is relational with some minor restrictions. A core aspect of 's data model is that attributes are classified as either keys or annotations via a user-defined schema. Keys in correspond to primary or foreign keys and are the only attributes which can partake in a join. Keys cannot be aggregated. Annotations are all other attributes and can be aggregated.Both keys and annotations support filter predicates andoperations. In its current implementation, supports equality () filters on keys and range (,, ) filters on annotations. This represents the existing implementation, not fundamental restrictions. 's current implementation supports attributes with types of , , ,, and . In many regards is similar to a key-value store combined with the relational model. is not unique in this regard, and commercial databases like Google's Mesa <cit.> and Spanner <cit.> followa similar (if not identical) model. §.§ Storage Engine 's data model is tightly coupled with how it stores relations. All keyattributes from a relation are stored in a trie, which serves as the onlyphysical index in . In the trie, each level is composed of sets ofdictionary encoded (unsigned integer) values. As isstandard <cit.>, stores dense sets usinga bitset and sparse sets using unsigned integers. Annotations are stored in separate buffers attached to the trie. supports multiple annotations, and each can be reached from any level of the trie (coredifferences from ). Anexample trie is shown in <Ref> where the dimensions and are keys andis an annotation. tries can be thought of as an index on key attributes or a materialized view of the input table. §.§ Query CompilerWe briefly overview the theoretical foundation of 's query compiler by describing the essential details. *Representing Queries as Hypergraphs Like <cit.>, a query is represented using ahypergraph H = (V, E), where V is a set of vertices (attributes) and each hyperedge e ∈ E(relation) is a subset of V. A join query is represented as a subgraph of H. Example hypergraphs are shown in<Ref>. In <Ref> we describe how SQL is translated to hypergraphs in more detail. *Generalized Hypertree DecompositionsThe query compiler uses generalized hypertree decompositions (GHDs)<cit.> to represent query plans. It is useful to think ofGHDs as an analog of relational algebra for a WCOJ algorithm. Formally, given ahypergraph H = (V_H, E_H), a GHD is a tree T = (V_T, E_T) and a mappingχ : V_T → 2^V_H that associates each node of the tree with asubset of vertices in the hypergraph.The asymptotic runtime of a plan is bound by the fractional hypertree width (FHW) of the GHD<cit.> and, as is standard <cit.>, chooses a GHD with the lowest FHW to select a query plan that matches the bestworst-case guarantee. * Each edge of the hypergraph e ∈ E_H must be a subset of the vertices in one of the nodes of the tree, i.e. there exists a tree node t ∈ V_T such that e ⊆χ(t).* For each attribute v ∈ V_H, the tree nodes it is a member of, {t | v ∈χ(t)}, must form a connected subtree. This is called the running intersection property. GHDs are designed to capture the degree of cyclicity in a query. We formalize this idea by defining the fractional hypertree width (FHW) of a GHD. Each node t of a GHD defines a subgraph (V', E'). That is, V' is χ(t) and E' is the set of edges that are subsets of χ(t).To calculate the width of this node we define a linear program with one variable x_e for each hyperedge e ∈ E', with the constraints∑_e ∈ E': e ∋ v x_e ≥ 1, ∀ v ∈ V'and objective function ∑_e ∈ E' x_e. Consider a vertex v. The constraint requires that the sum of the edges the vertex is contained in is at least 1. The objective minimizes the sum of all the edges. The FHW of a GHD is then defined as the maximum width of all of the nodes V_T. The asymptotic runtime of a plan is bound by the FHW of the GHD and by choosing the GHD with the lowest FHW, chooses the theoretically optimal plan.*Capturing Aggregate-Join QueriesAggregate-join queries are common in both BI and LA workloads. To capture aggregate-join queries, uses the AJAR framework<cit.>. Despite that others <cit.> have argued that custom algorithms are necessary to capture such queries, AJAR extends the theoretical results ofGHDs to queries with aggregations. AJAR does this by associating each tuplein a one-to-one mapping with an annotation. Aggregated annotations aremembers of a commutative semiring, which is equipped with product andsum operators that satisfy a set of properties (identity and annihilation,associativity, commutativity, and distributivity). Therefore, when relations are joined,the annotations on each relation are multiplied together to form the annotationon the result. Aggregations are expressed by an aggregation orderingα = (α_1, ⊕_1), (α_2, ⊕_2), … of attributes and operators. Using AJAR, picks a query plan with the best worst-case guaranteeby going through three phases.First, it breaks the input query into characteristic hypergraphs, which are subgraphs of the input that can be decomposed to optimalGHDs. Second, for each characteristic hypergraph all possible decompositions are enumerated, and a decomposition that minimizes FHW is chosen. Finally, the chosen GHDs are combined to form an optimal GHD. To avoid unnecessary intermediate results, compressesall final GHDs with a FHW of 1 into a singlenode, as the query plans here are always equivalent to running just a WCOJ algorithm. The GHDs for TPC-Hquery 5 and matrix multiplication are shown in <Ref>. §.§ Join AlgorithmThe generic WCOJ algorithm <cit.> shown in <Ref> is the computational kernel for all queries in the engine. This algorithm is used in each node of the GHD-based query plan. The generic WCOJalgorithm can be asymptotically better than any pairwise join algorithm <cit.>.Intuitively, the generic WCOJ algorithm joinsattributes in a multiway fashion as opposed to the traditional approach of joining relations in a pairwise fashion. This means that queries in are processed one attribute equivalence class at a time, and the core operation to process an equivalence class is a set intersection.Selecting the order that these attributes are processed in is the focus of <Ref>. An unrolling of this algorithm on TPC-H query 5 and matrix multiplication is shown in <Ref>. § SQL TO GHDSThe query compilation techniques presented in <Ref> are able to capture a wide range of domains, including linear algebra, message passing, and graph queries <cit.>. However, most of this work has been theoretical and none of the current literature demonstrates how to capture general SQL-style queriesin such a framework. In this section we show how to extend this theoretical work to more complex queries.In particular we describe how translates generic SQL queries (which could not) to hypergraphs in <Ref>. Using this translation, we show how thewell-known optimization of attribute elimination is captured both logically and physically in .In <Ref> we describe the manner in which selects a GHD and optimizes it by pushing down selections. We argue that with these extensions represents the first practicalimplementation of these techniques capable of capturing general SQLworkloads. §.§ SQL to Hypergraph TranslationWe demonstrate how certain features of SQL, such as complex expressions inside aggregation functions, can be translated to operations on annotated relations using a series of simple rules to construct query hypergraphs. The rules uses to translate aSQL query to a hypergraph H = (V, E) and an aggregation ordering α are as follows: * The set V of vertices in the hypergraph contains all of the key columns in the SQL query. The set of hyperedges E is each relation in the SQL query.All attributes that appear in an equi-join condition are mapped to the same attribute in V. * All key attributes that do not appear in the output of the query must be in the aggregation ordering α.* If only the columns of a single relation appear inside of an aggregation function, the expression inside of the aggregation function is the annotation of that relation. If none of the relation's columns appear inside an aggregation function, the relation's annotation is the identity element. If the inner expression of an aggregation function touches multiple relations, those relations are constrained to be in the same node of the GHD where the expression is the output annotation. * The rules above do not capture annotations which are not aggregated, sothese annotations are added to a meta data container M that associatesthem to the hyperedge (relation) from which they originate. Consider how these rules capture TPC-H query 5 from <Ref> in a query hypergraph.By Rule 1, the equality join in this query is captured in the set ofvertices (V) and hyperedges (E) shown in the hypergraph in<Ref>. The columnsandmust be mapped to the same vertex in “custkey" ∈ V.Similarly, the columnsandare mapped to the vertex “orderkey", the columnsandare mapped to the vertex “suppkey", the columns , , andare mapped to the vertex “nationkey", the columnsandare mapped to the vertex “regionkey".By Rule 2, a valid aggregation ordering is:α = [regionkey, nationkey, suppkey, custkey, orderkey]with the aggregation operator Σ (the order is irrelevant here).To apply Rule 3, consider the expression inside the aggregation function,. Only columns on the table are involved in this expression, so the annotations on the table will be this expression for each tuple. Theand tables do not have any columns in aggregation expressions, sothey are annotated with the identity element.By Rule 4, the hypergraph does not capture theattributes , , orbut our metadatacontainer M does. M here is the following:{ ↔ ,↔ ,↔ }.Thus, the final aggregate-join query is:∑_regionkey, nationkey, suppkey, custkey, orderkey supplier(custkey, suppkey) ⋈ orders(custkey, orderkey) ⋈ nation(nationkey, regionkey) ⋈ region(regionkey) ⋈ lineitem(orderkey, suppkey) Each (orderkey, suppkey) tuple on relation lineitem is annotated with the value of, and the tuples on all other relations are annotated with the identity element. *Attribute Elimination The rules above only add the attributes that are used in the query to the hypergraph. Although this elimination of unused attributes is obvious, ensuring this physically in the trie data structure was non-trivial. To do this we ensured that any number of levels from the trie can be used during query execution. This means that annotations can be reached individually from any level of the trie. Further, the annotations are all stored in individual data buffers(like a column store) to ensure that they can be loaded in isolation. These fundamental differences with , enable to support attribute elimination both logically and physically. This is essential on dense LA kernels because it enablesto call BLAS routines by storing a single denseannotation in flat (BLAS compatible) buffer. §.§ GHD OptimizationAfter applying the rules in <Ref>, a GHD is selectedusing the process described in <Ref>. Still, needs a wayto select among multiple GHDs that the theory cannot distinguish. In thissection we explain how adapts the theoretical definition of GHDs to both select and produce practical query plans. *Choosing Among GHDs with the same FHWFor many queries, multiple GHDs have the sameFHW. Therefore, a practical implementation mustalso include principled methods to choose between query planswith the same worst-case guarantee. Fortunately, there are three intuitive characteristicsofGHD-based query plans that makes this choice relatively simple (and cheap).The first is that the smaller a GHD is (in terms of number of nodes and height), the quicker it can be executed (less generated code). The second is that fewer intermediate results (shared vertices between nodes) results in faster execution time. Finally, the lower selection constraints appear in a query plan corresponds to how early work is eliminated in the query plan. Therefore, uses the following order of heuristics to choose between GHDs with the same FHW:* Minimize |V_T| (number of nodes in the tree).* Minimize the depth (longest path from root to leaf).* Minimize the number of shared verticesbetween nodes.* Maximize the depth of selections. Although most of the queries in this paper are single-node GHDs, on the two node TPC-H query 5, using these rules to select a GHD results in a 3x performance advantage over a GHD(with the same FHW) that violates the rules above.We explain how selections are pushed down below joins in GHDs next.*Pushing Down Selections Below Joins Our mechanism of selecting a GHD ensures that selections appear as low aspossible in a GHD, but we still need a mechanism to push down selections downbelow joins in our chosen GHD. Motivated by how does this<cit.>, the querycompiler pushes down selections below joins by:* Taking as input an optimal (with respect to FHW) GHD T = (V_T, E_T)and a set of selections σ_a_i applied to attributes a_i.* For each σ_a_i: Let e_i be the edge that contains the vertexderived from a_i or that has the meta data M associated with a_i.Let t_i be the GHD node in V_T associated with e_i. If t_i contains more than one hyperedge, create a new GHD node t'_i that contains only e_i, and make t'_i a child of t_i. This means that pushes down selections by creating new GHD nodes with only the selection constraints under the original GHD nodes. On TPC-H query 5 this results in the GHD shown in <Ref> which executes 1.8x faster thanthe GHD without this optimization.§ COST-BASED OPTIMIZERAfter a GHD-based query plan is produced using the process described in<Ref>, needs to select an attribute order for the WCOJ algorithm. Similar to the classic query optimization problem of selecting a join order <cit.>, WCOJ attribute ordering can result inorders of magnitude performance differences on the same query. Unfortunately, the knowntechniques for estimating the cost of join orders are designed for Selinger-style <cit.> query optimizers using pairwise join algorithms—not a GHD-based query optimizer with a WCOJ algorithm. In this sectionwe present the first cost-based optimizer for the generic WCOJ algorithm and validate the crucial observations it uses to derive its costestimate. We show that this optimizer selects attribute ordersthat provide up to a three orders of magnitude speedup over ones that couldselect. *Optimizer Overview 's cost-based optimizerselects a key attribute order for each node in a GHD-based query plan. As is standard <cit.>,requires thatmaterialized key attributes appear before those that are projected away (with oneimportant exception described in <Ref>) and that materialized attributes always adhere to some global ordering (e.g. if attribute `a' is before attribute `b' in one GHD node order, then 'a' must be before 'b' in all GHD orders). To assign an attribute order to each GHD node, 's cost-based optimizer: traverses the GHD in a top-down fashion, considers all attribute orders[We remind the reader that the number of attributes considered here is only the numberof key (joined) attributes inside of a GHD node which is typically small.] adhering to the previously described criteria at each node, and selects the attribute order with the lowest cost estimate.For each order, a cost estimate is computedbased on two characteristics of the generic WCOJ algorithm: (1) the algorithmprocesses one attribute at a time and (2) set intersection is the bottleneckoperator. As such, assigns a set intersection cost () and a cardinality weight (<Ref>) to each key attribute (or vertex) in thehypergraph of a GHD node. Using this, the cost estimate for a given a key attribute (or hypergraph vertex) order[v_0,v_1,...,v_|V|] is:cost = ∑_i=0^|V|((v_i) ×weight(v_i))The remainder of this section discusses how the s(<Ref>) and weights (<Ref>) are derived. §.§ Intersection Cost The bottleneck of the generic WCOJ algorithm is set intersection operations.In this section, we describe how to derive a simple cost estimate, called, for the set intersections in the generic WCOJ algorithm.§.§.§ Cost Metric Recall that the sets in tries are stored using an unsigned integerlayout () if they are sparse and a bitset layout() if they are dense, a design inherited from . Thus, theintersection algorithm used is different depending on the data layout of thesets. These differentlayouts have a large impact on the set intersection performance, even with similar cardinality sets. For example,<Ref>, shows that a∩is roughly50x faster than a∩with the same cardinality sets.Therefore, uses the results from<Ref> to assign the followings: ( ∩ ) = 1,( ∩ ) = 10, ( ∩ ) = 50Unfortunately, it is too expensive for the query compiler to check (or track) the layout of each set during query compilation—set layoutsare chosen dynamically during data ingestion and a single trie can have millions of sets. To address this uses <Ref>.The sets in the first level of a trie are typically dense and therefore represented as a bitset. The sets of any other level of a trie are typicallysparse (unless the relation is completely dense) and therefore representedusing the unsigned integer layout.Empirical Validation: Consider atrie for the TPC-Hrelation where the trie levels correspond to the key attributes[,,,]in that order. At scale factor 10, each level of this trie has the following number ofandsets:* 1st level() = {0sets, 1set}* 2nd level() = {14999914sets, 86sets}* 3rd level() = {59984817sets, 0sets}* 4th level() = {59986042sets, 0sets} Thus, given a key attribute order [v_0,...,v_|V|] (where eachv_i ∈ V), the optimizer assigns an to each v_i, in order of appearance, using the following method which leverages <Ref>:* For each edge e_j with node v_i, assign l(e_j) (wherel=layout), to eitheror .As a reminder, edges are relations and vertices are attributes. Thus, for each relation this assignment guesses onedata layout for all of the relation's v_i sets.If e_j has been assigned with a previous vertex v_k where k < i,l(e_j) = (not the first trie level),otherwisel(e_j) =. * Compute the cost of intersecting the v_i attribute from each edge (relation) e_j. For a vertex with two edges, the pairwiseis used.For a vertex with N edges, where N > 2, theis the sum of pairwise s wherethesets are always processed first. For example, when N=3 and l(e_0) ≤ l(e_1) ≤ l(e_2) where< , =( l(e_0) ∩l(e_1)) +(l(l(e_0) ∩l(e_1)) ∩l(e_2)). Note,= l( ∩ ). Consider the attribute order [,,,] forone of the GHD nodes in TPC-H query 5 (see <Ref>).Thevertex is assigned anof 1 as it isclassified withintersections. Thevertex is assigned anof 10, classified with intersections. The vertex is assigned anof 11, classified with intersections. Finally,thevertex is assigned an of 50, classified with intersections.Finally, in the special case of a completely dense relation, the LevelHeadedoptimizer assigns anof 0 because no set intersection is necessary in thiscase. This is essential to estimate the cost of LA queries properly.§.§.§ Relaxing the Materialized Attributes First Rule An interesting aspect of the intersection cost metric is that the cheapest keyattribute order could have materialized key attributes come afterthose which are projected away.[We remind the reader that later (or after) in the attribute order corresponds to a lower level of the tries.] To support such key attribute orders, the execution engine must be able to combine children(in the trie) of projected away key attributes using a set union or (to materialize the result sets). Unfortunately, it is difficult to design an efficient data structure to union more than one level of a trie (materialized key attribute) in parallel (e.g. use a massive 2-dimensional buffer or incur expensive merging costs). Therefore, kept its design simpleand never considered relaxing the rule that materialized attributes must appear before those which are projected away. In we relax this rule by allowing 1-attribute unions (see <Ref>) on keys when it can lowerthe .Within a GHD node, relaxes the materialized attributes first rule underthe following conditions: * The last attribute is projected away.* The second to last attribute is materialized.* Theis improved by swapping the two attributes. These conditions ensure that 1-attribute union will only be introduced when thecan be lowered. Consider the sparse matrix multiplication query and its unrollingof the generic WCOJ algorithmfor an attribute order ofshown in <Ref>. This attribute order has acost 50assigned to theattribute.Now consider an attribute order. Here a cost 10is assigned toand theunrolling of the generic WCOJ algorithm (where () denotes accesses to keys and [] denotes accessto annotations) is the following: for $̄i ∈π_iM1dos_j ←∅for $̄k ∈π_kM1(i) ∩π_kM2 dofor $̄j ∈π_jM2(k)dos_j[j] += π_v1M1[i,k] * π_v2M2[k,j]out(i) ←s_jThis lower cost attribute orderrecovers the same loop ordering as Intel MKL on sparse matrix multiplication<cit.> and its bottleneck is the operation that unionsvalues and sums values.This is in contrast to the standard bottleneck operation of a set intersection as in the order. In <Ref> we discuss the tradeoffs around implementing this union add or operation. <Ref> shows that this order is essential to runsparse matrix multiplication as a join query without running out of memory.§.§ Weights Like classic query optimizers,also tracks the cardinality ofeach relation as this influences the s for the generic WCOJ algorithm. <Ref> shows theunsurprising fact that larger cardinality sets result in longer intersectiontimes. To take this into account when computing a cost estimateassigns weights to each vertex using <Ref>, which directly contradicts conventional wisdom from pairwise optimizers: The highest cardinality attributes should be processed first in the genericWCOJ algorithm. This enables these attributes to partake in fewer intersections (outermost loops) and ensures that they are a highertrie levels (more likely 's with lowers).Empirical Validation: We show in <Ref>that the generic WCOJ can run over 70x faster on TPC-H query 5 whenthe high cardinalityattributeis first in the attribute order instead of last. 'sgoal when assigning weights is to follow <Ref> byassigning high cardinality attributes heavier weights so that they appearearlier in the attribute order. To do this,assigns a cardinality score to each queried relation and uses this to weight to each attribute. We describe this in more detail next. *Score maintains a cardinality score for eachrelation in a query which is just the relation's cardinality relative to thehighest cardinality relation in the query. The score (out of 100) fora relationr_iis:score = ceiling(|r_i|/|r_heavy|×100)wherer_heavyis the highest cardinality relation in the query.*Weight To assign a weight to each vertexuses the highest score edge (orrelation) with the vertex when a high selectivity (equality) constraint is present, otherwisetakes the lowest score edge (or relation). The intuition forusing the highest score edge (or relation) with a highselectivity constraint is that this relation represents the amount of work that could be filtered (or eliminated) at this vertex (or attribute). The intuition for otherwise taking the lowest score edge (or relation) is that theoutput cardinality of an intersection is at most the size of the smallest set. Consider TPC-H Q5 at scale factor 10. The cardinality score for each relationhere is: score(lineitem) = 100, score(orders) = 26, score(customer)= 3,score(region) = 1, score(supplier) = 1, score(nation) = 1The weight for each vertex is (region is equality selected): weight(orderkey) = min(26,100), weight(custkey) = min(3,26)weight(suppkey) = min(1,100), weight(nationkey) = min(1,1,3) weight(regionkey) = max(1,1)These weights are then used to derive the cost estimates shown in <Ref>.§ GROUP BY TRADEOFFS Designing a skew-resistantoperator requires carefulconsideration of the tradeoffs associated with its implementation. In this section we present these tradeoffs in the context of . Although many s are automatically captured in 'stries, two classes of s (that were not supported in ) requirean additional implementation in : (1) ato unionkeys in the attribute orders from <Ref> and (2) a on annotations. In this section we present the tradeoffs around two implementationsfor each type ofand describe a simple optimizer that automatically exploits these tradeoffs toselect the implementation used during execution. In <Ref>, we show that thisoptimizerprovides up to a 875x and 185x speedup over using a singleimplementation on BI and LA queries respectively.*GROUP BY Key An important artifact of the key attribute orders from <Ref> is that a projected away attribute can appearbefore a materialized attribute in . In this case a on a key attribute is needed to union the result.selects between two implementations for this: (1) a hash map that upserts (key, value) pairs as they are encountered and (2) a bitset that unions (OR's) keys and a dense array to hold the values. <Ref> shows that the performance of these implementationsdepends greatly on the density of theoutput key attribute set. The bitset performs better when the output density is high and the hash map performs better when the output density is low. To predict the outputset's densityleverages a simple observation: the output density is correlated in a1-1 manner with the density ofprojected away attribute (set being looped over). As such,uses the density of the projected away attribute to predict theoutput set's density and select the implementation used. *GROUP BY Annotation supports SQL queries withoperations on annotations or on both annotations and keys. To support this in parallel,selects between two implementations of aparallel hash map: (1) libcuckoo <cit.> and (2) a per-thread instance of the C++ standard library unordered map. In the context of a , the crucial operation for these hash maps is an upsert and we found that each implementation worked best under different conditions. <Ref> show that when the output key size is small (many collisions), the per-threadimplementation outperforms libcuckoo by up to 4x. In contrast, libuckoo outperformsthe per-thread approach by an order of magnitude when the output size is large (few collisions).Unfortunately, predicting output cardinalities isa difficult problem <cit.>, soleverages a generalobservation to avoid this when selecting between these two approaches:libcuckoo is at worst close (<2x) to theperformance of theper-thread approach when the hash-map key tuple is wide (>3 values), and theper-threadapproach is at worst close (<2x) to the performance of libcuckoowhen the hash map key is small (<3 values). This trend is shown in<Ref>. Thus, theoptimizerselects a per-thread hash map when the hash map key is a tuple of three elements or lessandlibcuckoo otherwise.§ EXPERIMENTS We compareto state-of-the-art relational database managementengines and LA packages on standard BI and LA benchmark queries.We show thatis able to compete within 2.5x of theseengines, while sometimes outperforming them, and that the techniques fromin <Ref> can provide up to a three orders of magnitude speedup. This validates thata WCOJ architecture is a practical solution for both BI and LA queries. §.§ SetupWe describe the experimental setup for all experiments. *Environment is a shared memory engine that runs and is evaluated on a single node server.As such, we ran all experiments on a single machine with a total of 56 coreson four Intel Xeon E7-4850 v3 CPUs and 1 TB of RAM. For all engines, we chosebuffer and heap sizes that were at least an order of magnitude larger than thedataset to avoid garbage collection and swapping data out to disk. *Relational Comparisons We compare to HyPer, MonetDB, and LogicBloxon all queries to highlight the performance of other relationaldatabases. Unlike , these engines are unable to compete within anorder of magnitude of the best approaches on BI and LA queries. We compare toHyPer v0.5.0 <cit.> as HyPer is a state-of-the-art in-memoryRDBMS design. We also compare to the MonetDB Dec2016-SP5 release. MonetDB is a popular columnar store database engine and is a widely usedbaseline <cit.>. Finally, we compare to LogicBlox v4.4.5 asLogicBlox isthe first general purpose commercial engine to provide similar worst-caseoptimal join guarantees <cit.>. Our setup of LogicBlox was aided by a LogicBlox engineer. Because of this, we know that LogicBloxdoes not use a WCOJ algorithm for many of the join queries in theTPC-H benchmark. HyPer, MonetDB, and LogicBlox arefull-featured commercial strength systems (support transactions, etc.) andtherefore incur inefficiencies thatdoes not. *Linear Algebra Package Comparison We use Intel MKL v121.3.0.109 as thespecialized linear algebra baseline. This is the best baseline for LA performanceon Intel CPUs (as we use in this paper). Others <cit.> have shownthat it takes considerable effort and tedious low-level optimizations to approachthe performance of such libraries on LA queries. *Omitted Comparisons We omit a comparison to array databases like SciDB<cit.> as these engines call BLAS or LAPACK libraries (like Intel MKL) on the LA queries that we present in this paper. Therefore, Intel MKL was theproper baseline for our LA comparisons. Additionally, SciDB is notdesigned to process TPC-H queries. *MetricsFor end-to-end performance, we measure the wall-clock timefor each system to complete each query. We repeat each measurement seven times,eliminate the lowest and the highest value, and report the average. Thismeasurement excludes the time used for outputting the result, data statisticscollection, and index creation for all engines. We omit the data loading andquery compilation for all systems except for LogicBlox and MonetDB where thisis not possible. The query compilation time for these queries, especially at large scale factors, is negligible to the execution time. Still, we found that 'sunoptimized query compilation process performed within 2x of HyPer's. To minimize unavoidable differences with disk-based engines(LogicBlox and MonetDB) we place each database in the tmpfs in-memoryfile system and collect hot runs back-to-back. Between measurements for thein-memory engines (HyPer and Intel MKL), we wipe the caches andre-load the data to avoid the useof intermediate results. §.§ Experimental Results We show thatcan compete within 2x of HyPer on seven TPC-H queries andwithin 2.5x of Intel MKL on four LA queries, while outperforming MonetDB and LogicBlox by up to two orders of magnitude.§.§.§ Business Intelligence On seven queries from the TPC-H benchmark we show thatcan competewithin 2x of HyPer while outperformingMonetDB by up to an order of magnitude and LogicBlox by up to two ordersof magnitude. *Datasets We run the TPC-H queries at scale factors 1, 10,and 100. We stopped at TPC-H 100 as in-memory engines, such as HyPer,often use 2-3x more memory than the size of the input database during loading—thereforeapproaching the memory limit of our machine. For reference, on TPC-H query 1 HyPer uses 161GB of memory whereasuses 25GB (only loading the data it needs from disk).Additionally, scale factor 100 was the largest one where we could guarantee that the buffer space for disk-based engines was an order ofmagnitude large than the data.*Queries We choose TPC-H queries 1, 3, 5, 6, 8, 9 and 10 tobenchmark, as these queries exercise the core operationsof BI querying and also containing interesting joinpatterns (except 1 and 6). TPC-H queries 1 and 6 do not contain a join anddemonstrate that althoughis designed for join queries, it can alsocompete on scan queries.*Discussion In <Ref> we show thatcan outperformMonetDB by up to 80x and LogicBlox by up to 270x while remainingwithin 1.88x of the highly optimized HyPer database. Unsurprisingly, thequeries whereis the farthest off the performance of the HyPer engine isTPC-H queries 1 and 8. Here the output cardinality is small and the runtime is dominated by theoperation. As described in <Ref>,leverages existing hash mapimplementationsto perform these s,and a custom built (NUMA aware) hash map would likely close this performance gap.On queries 3 and 9, where the output cardinality is larger (and closer toworst-case), is able to compete within 11% of Hyper and sometimesoutperforms it. In , we also tested additional optimizations like indexing annotations with selection constraints and pipelining intermediate results between GHD nodes. We found thatthese optimizations couldprovide up to a 2x performance increase on queries like TPC-H query 8, but did not consistently outweigh their added complexity.§.§.§ Linear Algebra We show thatcan compete within 2.5x of Intel MKL while outperformingHyPer and LogicBlox more than 18x and 587x respectively onLA benchmarks. *Datasets We evaluate linear algebra queries on three densematrices and three sparse matrices. The first sparse matrix dataset weuse is the Harbor dataset, which is a 3D CFD model of the Charleston Harbor <cit.>. The Harbor dataset is a sparse matrix<cit.> that contains 46,835 rows and columns and2,329,092 nonzeros. The second sparse matrix dataset we use is theHV15R dataset, which is a CFD matrix of a 3D engine fan <cit.>. The HV15R matrix contains 2,017,169 rows and columns and 283,073,458 nonzerosand is a large, non-graph, sparse matrix.The final sparse matrix dataset we use is the nlpkkt240 dataset with 27,993,600rows and columns and 401,232,976 nonzeros <cit.>. Thenlpkkt240 dataset is a symmetric indefinite KKT matrix. For dense matrices, we use synthetic matrices with dimensionsof 8192x8192 (8192), 12288x12288 (12288), and 16384x16384 (16384). *Queries We run matrix dense vector multiplication and matrixmultiplication queries on both sparse (SMV,SMM) and dense (DMV,DMM) matrices.These queries were chosen because they are simple to express using joinsand aggregations in SQL and are the core operations formost machine learning algorithms. Further,Intel MKL is specifically designed to process these queries and, as a result,achieves the largest speedups over using a RDBMS here.Thus, these queries represent the mostchallenging LA baseline queries possible.For both SMM and DMM wemultiply the matrix by itself, as is standard forbenchmarking <cit.>. *Discussion<Ref> shows thatis able to compete within 2.44x of Intel MKL on both sparse and dense LA queries.On dense data,uses the attribute elimination optimization from<Ref> to store dense annotations in single buffers that are BLAS compatible and code generates to Intel MKL. Still, MKL produces only the output annotation, not the key values, soincurs a minor performance penalty (<2%) for producing the key values. On sparse data,is able to compete with MKL when executing these LA queries as pure aggregate-join queries. To do this, the attribute orderandoptimizations from <Ref> were essential.Although we tested more sophisticated optimizations, like cacheblocking (by adding additional levels to the trie), we found that the performance benefit of these optimizations did not consistently outweigh their added complexity. In contrast, other relational designs fall flat on these LA queries.Namely, HyPer usually runs out of memory on the matrix multiplication query and,on the queries which it does complete, is often an order of magnitude slowerthan Intel MKL. Similarly, LogicBlox and MonetDB are at least anorder of magnitude slower than Intel MKL on these queries. §.§ Micro-Benchmarking ResultsWe break down the performance impact of each optimization presented in <Ref>.*Attribute Elimination<Ref> show that attribute elimination can enable up to a 4.82x performance advantage on the TPC-H queries and up to a 500x performance advantage ondense LA queries. Attribute elimination is crucialon most on TPC-H queries, as these queries typically touch a small number ofattributes from schemas with many attributes. Unsurprisingly,<Ref> shows that attribute elimination provides the largestbenefit on the scan TPC-H queries (1 and 6) because it allowsto scan lessdata. On dense LA queries,calls Intel MKL with little overhead because attribute elimination enables us to store eachdense annotation in a BLAS acceptable buffer. As <Ref> shows,this yields up to a 500x speedup over processing these queries purely in .*Selections As shown in <Ref>, pushing downselections can provide up to 2.67x performance advantageon the TPC-H queries we benchmark. Although pushing down selections is generally useful, on TPC-H Q10 it actually hurts execution time.This is because this optimization adds additional GHD nodes (intermediateresults) to our query plans (something pipelining would fix). Still,the performance impact is small on this query (<12%) and on averagethis optimization provided a 90% performance gain across TPC-H queries. *Attribute OrderAs shown in <Ref>, the cost-based attribute ordering optimizer presented in <Ref> can enable up to a 8815x performance advantage on TPC-H queries and enablesto run sparse matrix multiplicationas a join query without running out of memory. <Ref> show the difference between thebest-cost and the worst-cost attribute orders.The most interesting queries here are TPC-H query 5 and TPC-H query 8. On TPC-H query 5, it is essential that the high cardinalityattribute appears first. On TPC-H query 8, it is essential that theattribute, which was connected to an equality selection, appears first. The process of assigning weights to the intersection costs in <Ref> ensures that orderssatisfying these constraints are chosen. Finally, the cost-basedattribute ordering optimizer is also crucial on sparse matrix multiplication.Here it is essential that the lower cost attribute order, with a projectedaway attribute before one that is materialized, is selected. This order not only prevents a high cost intersection, but eliminates thecomputation and materialization of annotation values do not participatein the output (due to sparsity). *GROUP BY Finally, we show in <Ref>that 'soptimizers provide up to a 875x performance advantage on TPC-H queries and up to a 185xperformance advantage on LA queries. Theannotation optimizerprovides a 875x speedup on TPC-H query 1 becauseis thebottleneck operation in this query and our optimizer properly selects to use aper-thread hash map instead of libcuckoo here. Thekey optimizerprovides a 185x speedup on SMM with the nlp240 dataset because it correctly predicts that the many of the output key attribute sets are sparse. As such 's optimizer chooses to use a standard hash map to producethese sparse sets because using a bitset is highly inefficient (due to theamount of memory it wastes). This optimizer making the opposite choice providesa 3x performance advantage on SMM with the HV15R dataset.§ EXTENSIONWe extendto show that such a unified query processing architecture could enable faster end-to-end applications.To do this, we add the ability forto process workloads that combine SQL queries andfull machine learning algorithms (as described in <cit.>). With this,we show on a full-fledged application thatcan be an order of magnitudefaster than the popular solutionsof Spark v2.0.0,MonetDB Dec2016[ Development build with embedded Python <cit.>.]/Scikit-learn v0.17.1, and Pandas v0.18.1/Scikit-learn v0.17.1 (using the same experimental setting as <Ref>). *Application We run a voter classification application<cit.> that joins and filters two tables to create a single feature set which is then used to train a logistic regression model forfive iterations. This application is a pipelineworkload that consists of three pipeline phases: (1) a SQL-processing phase, (2) a feature engineering phase where categorical variables are encoded, and (3) amachine learning phase. The dataset <cit.> consists of two tables: (1) one with information such as gender and age for 7,503,555 voters and (2) one with information about the 2,751 precincts that the voters were registered in. *Performance<Ref> shows thatoutperforms Spark, MonetDB, and Pandas on the voter classification application by up toan order of magnitude. This is largely due to 's optimized shared-memory SQL processing and ability to minimize data transformations betweenthe SQL and training phase. To expand on the cost of data transformations a bit further,in <Ref> we show the cost ofconverting from a column store to the compressed sparse row format used by most sparse library packages. This transformation is notnecessary inas it always uses a single, trie-based data structure.As a result, <Ref> shows that up to 41 SMV queries can be run inin the time that it takes for acolumn store to convert the data to a BLAS compatible format. Similarly, on the voter classification application LevelHeaded avoids expensivedata transformations (in the encoding phase) by using its trie-based datastructure for all phases. § CONCLUSIONS This paper introduced theengine and demonstrated that a query architecture built around WCOJs can compete onstandard BI and LA benchmarks. We showed thatoutperforms other relationalengines by at least an order of magnitude on LA queries, while remainingon average within 31% of best-of-the-breed solutions on BI and LAbenchmark queries. Our results are promising and suggest thatsuch a query architecture could serve as the foundation for futureunified query engines.Acknowledgments:We thank LogicBlox for their helpful conversationsand assistance in setting up our comparisons.plain | http://arxiv.org/abs/1708.07859v1 | {
"authors": [
"Christopher R. Aberger",
"Andrew Lamb",
"Kunle Olukotun",
"Christopher Ré"
],
"categories": [
"cs.DB"
],
"primary_category": "cs.DB",
"published": "20170825184930",
"title": "LevelHeaded: Making Worst-Case Optimal Joins Work in the Common Case"
} |
Condensed Matter Theory Center and Joint Quantum Institute, Department of Physics, University of Maryland, College Park, Maryland 20742-4111, USA We study a pair of capacitively coupled singlet-triplet spin qubits. We characterize the two-qubit decoherence through two complementary measures,the decay time of coupled-qubit oscillations and the fidelity of entangledstate preparation. We provide a quantitative map of their dependence on charge noise and fieldnoise, and we highlight the magnetic field gradient across eachsinglet-triplet qubit as an effective tool to suppressdecoherence due to charge noise.Decoherence of two coupled singlet-triplet spin qubits S. Das Sarma December 30, 2023 ======================================================§ INTRODUCTION Localized electron spins in semiconductor quantum dots provide a promisingarchitecture for quantum computation <cit.>. Compared with alternative platforms (e.g., trapped ions, trapped atoms, orsuperconducting circuits), semiconductor spin qubits have the threefold advantage of (1) fast single-qubit gateoperations, (2) long coherence time <cit.>, and (3) potential for scalability <cit.> because of advancedfabrication technology developed for semiconductor integrated circuits <cit.>. There has been significant progress in the experimental development ofhigh-fidelity single spin qubit gate operations in different platforms usingdifferent materials. Unfortunately, despite considerable experimentalefforts <cit.>, it remainschallenging to engineer high-fidelity two-qubit entangling gates for semiconductor spin qubits,as decoherence due to environmental noises is exacerbated bycomparatively weak coupling between localized electron spins. In fact, the experimental progress in developing two-qubit gates insemiconductor spin quantum computing platforms has been disappointing so farwhen compared with the corresponding situation in superconducting and ion trapplatforms. The relatively low entangling-gate fidelity of spin qubits (0.9 in GaAs as reported in Ref. Nichol17) is currently the main obstacle to unlocking their aforementioned advantage asa platform for large-scale quantum computing, and it calls for a better understanding of the effect of environmental noises onthe dynamics of two coupled semiconductor spin qubits in order to enhance thecoupled-qubit fidelity.In this paper, we undertake this challenge and study the decoherence of two coupledsinglet-triplet spin qubits <cit.>, which are among the actively studied spin qubits with the added advantage thattwo-qubit gate operations have been demonstrated in GaAs-based singlet-tripletqubits <cit.>. Each singlet-triplet qubit consists of a pair of exchange-coupled electronspins localized in a double quantum dot, and the two qubits interact via an Ising-type capacitive coupling <cit.>. Compared with the exchange-coupled spin qubits <cit.> studied in aprevious paper <cit.>, where two localized electron spins are coupled through the Heisenbergcoupling <cit.>, the singlet-triplet system we consider here enjoys full two-axis control through purely electrical gating <cit.> and is protected from homogeneous magnetic field fluctuations ineach double quantum dot <cit.>. It also operates in a larger active Hilbert space due to the lack of spinconservation, and has more complicated dynamics and richer physics. This makes it harder to extract useful insights from analytical solutions <cit.>. Instead, we study the coupled singlet-triplet qubits through numerical calculations in order to provide quantitative insight into the detrimental role of(electric) charge and (magnetic) field noise on the two-qubit Ising gateoperations. We mention that sophisticated dynamical decoupling schemes have already beendeveloped for semiconductor singlet-triplet qubits enabling efficient andfault-tolerant gate operations <cit.>, and substantial progress is likely in the near future once two-qubitentangling gates achieve higher fidelity, making our current theoreticalanalysis timely.We consider coupled-qubit decoherence from two different types ofenvironmental noises, charge noise from charge fluctuations in each qubit device <cit.>, and field (Overhauser) noise due to nuclear spin dynamics in thesemiconductor background <cit.>. We assume that the noises are slow relative to the qubit dynamics, andwe model them in the quasistatic bath approximation by averaging observables overtime-independent but randomly distributed disorder configurations. The quasistatic bath approximation has been used extensively in thesemiconductor spin qubit studies and is generally considered to be quite validin most situations <cit.>.The decoherence of the coupled qubits is examined quantitatively through apair of complementary probes. First, we extract a characteristic time scale, the two-qubit coherence time,from the envelope decay of the coupled-qubit oscillations. This measures the persistence of the initial state information in the presenceof environmental noises. It also provides a direct physical measure of the time duration of coherentgate oscillations. Second, we compute the fidelity of preparing an entangled state through timeevolution from an unentangled product state. This quantifies the ability of the coupled qubits to carry out a precise unitarytransformation despite the fluctuations in coupling parameters, and serves asa simple proxy for gate fidelity. We also note in this context the interesting possibility of singlet-tripletsemiconductor qubits being effective quantum sensors because of their delicatedependence on charge and field noises.In particular, the type of theoreticalanalysis presented in the current work can be inverted and used for aquantitative determination of background charge and/or field fluctuations fromthe singlet-triplet entanglement information as described in our study.We study the dependence of these two fidelity measures on charge noise,field noise, as well as the magnetic field gradient across eachsinglet-triplet qubit. When the average magnetic field gradient is zero, we find that the coupledsinglet-triplet qubits are significantly more susceptible to charge noise thanfield noise. This difference is manifested in both the two-qubit coherence time andthe entanglement fidelity, although less pronounced in thelatter. The situation changes dramatically when a strong magnetic field gradient isapplied in each qubit. As the magnetic field gradient increases, the coupled-qubit system becomes moresensitive to field noise and less to charge noise. In the regime dominated by the magnetic field gradient, charge noise becomesrelatively inconsequential and the decoherence of the coupled qubits is mainlydriven by field noise, in sharp contrast to the situation without a strongmagnetic field gradient. The change of noise sensitivity driven by the magnetic field gradient is a uniquefeature of the singlet-triplet system and has no direct counterpart in asystem of two exchange-only qubits. We mention that the magnetic field gradient induced strong suppression of thecharge noise effect on the two-qubit Ising gate operations for singlet-tripletqubits provides encouraging prospects for Si-based quantum computing platformssince isotopic purification enables the elimination of the nuclear field noisein Si systems <cit.>.The paper is organized as follows. In Sec. <ref>, we describe the Ising model of two single-triplet spinqubits and discuss the two-qubit coherence time T_2^*. We provide an operational definition of T_2^* and examine its dependence onboth charge noise and field noise, as well as the magnetic field gradient. In Sec. <ref>, we introduce the fidelity of entangled statepreparation and examine its parametric dependence. In Sec. <ref>, we summarize our numerical results and highlightthe implications for future experiments. § TWO-QUBIT COHERENCE TIMEThe system of two capacitively coupled singlet-triplet spin qubits is described by the following Ising-type Hamiltonian <cit.>H=ε J_1J_2 σ_1^zσ^z_2 +J_1σ_1^z+J_2σ_2^z +h_1σ^x_1+h_2σ^x_2.Here we work in the singlet-triplet basis, with the σ^z_i=+1 (-1) eigenstate denoting the singlet (triplet) state of the two electrons in thedouble quantum dot constituting the ith spin qubit (i=1,2), respectively. For each spin qubit, the Zeeman h_iσ_i^x term is controlled by themagnetic field gradient h_i across the corresponding double quantum dot, and the J_iσ_i^z term is controlled by the intraqubit exchangecoupling J_i between the two electrons in the qubit. The coupling ε J_1J_2 σ_1^zσ_2^z between the two qubits comes from the capacitive dipole-dipole interaction, with a strength approximately proportional to the product of intraqubit exchange couplings J_1J_2 asargued empirically in Ref. Shulman12.We employ the quasistatic bath approximation and model the environmentalnoises by averaging observables over time-independent but randomlydistributed model parameters. Specifically, the couplings J_1 and J_2 are drawn from a Gaussiandistribution with mean J_0 and variance σ_J^2 but restricted tonon-negative values, and the transverse fields h_1 and h_2 are drawn from a Gaussiandistribution with mean h_0 and variance σ_h^2. Physically, the parameter J_0 is the (average) exchange coupling between thetwo electrons in each double quantum dot,and the parameter h_0 is the (average) magnetic field gradient appliedbetween the two electrons. In a typical experiment <cit.>, the intraqubit exchangeJ_0 is on the order of 10^2MHz, and the (quasistatic equivalent of) chargenoise σ_J is on the order of 10^-2∼ 10^-1J_0. The field noise σ_h may range from up to J_0 in GaAs to essentially negligible in isotopically purified ^28Si.For our numerical calculations, we fix the interqubit coupling parameter ε to 0.1J_0^-1, andfocus on the effect of the remaining dimensionless parameters σ_J/J_0,σ_h/J_0, and h_0/J_0. For the disorder average, we typically use a sample size between 10^4 and10^5 for each parameter set. We note that the dependence on h_0 is an important newelement in the physics of Ising-coupled singlet-triplet qubits with no analogin the corresponding exchange coupled spin qubits studied inRef. Throckmorton17. §.§ Decay of coupled-qubit oscillations In the following we introduce an operational definition for thetwo-qubit coherence time T_2^* from the envelope decay of the coupled-qubitoscillations. Without loss of generality, we consider the product initial state|ψ(0)⟩=|↑⟩_x⊗|↑⟩_x of the coupled qubits, and we compute the dynamics of the disorder-averaged return probability to theinitial stateR(t)=|⟨ψ(0)|ψ(t)|⟩|^2 .Here, the double bracket denotes the average over disorder realizations. As the initial state |ψ(0)⟩ is not an eigenstate of the coupledqubits, the return probability is oscillatory in time. The oscillations have a typical frequency on the order of J_0, driven by the intraqubit exchange coupling J_iσ_i^z terms in theHamiltonian [Eq. (<ref>)], and they are further modulated by beats with frequency around ε J_0^2due to the interqubit coupling ε J_1J_2σ_1^zσ_2^z term. The oscillations in R(t) are damped by environmental noises through disorderaveraging, in a fashion mathematically similar to (although physically distinct from) thedecaying Rabi oscillations of a single qubit in the presence of environmental noises. Very loosely one can think of these oscillations as the two-qubit Rabioscillations decaying due to charge and field noises. A few representative examples of the decaying R(t) curves are shown inFig. <ref>, using noise parameters approximatelyconsistent with experimental situations.We extract from R(t) a characteristic time scale T_2^* associated with thedecay of the oscillation envelope of the coupled qubits, and use it as aquantified measure of the coupled-qubit decoherence. Compared with the exchange-only case <cit.>, the R(t)oscillations here have morecomplicated wave forms, with extra beats in the decay envelope. Since these additional features are irrelevant to our maingoal of characterizing the damping effect of environmental noises, we disregard them and adopt a simple fitting procedure that focuses only on thedecay envelope. Operationally, we take the upper envelope of the R(t) oscillations andperform on it a least-squares fit to an exponential decay of the formR(∞)+A e^-t/T_2^*,where R(∞) is estimated from the asymptotic value of R(t) and A isa nuisance parameter of no interest in the current work. The fitted upper envelope of R(t) and the coherence time T_2^* are shownin Fig. <ref> for a few representative examples.Compared with the exchange-only case studied inRef. Throckmorton17, here we are using a slightly differentoperational definition for the two-qubit coherence time. This is necessitated by the irregular wave forms of the coupled-qubitoscillations allowed by a larger Hilbert space. We emphasize that this alternative choice only introduces moderate variationsin the numerical value of T_2^* and does not affect our conclusionsqualitatively. In addition, it is worth emphasizing that the coherence time in this papermeasures the decay rate of the oscillation envelope of thedisorder-averaged return probability R(t), rather than the decay rate ofR(t) itself. As we noted in a previous paper <cit.>, the latterdefinition is more appropriate for a large number of coupled qubits, whereasthe definition adopted here provides a more precise measure of the decoherenceprocess within a low-dimensional Hilbert space appropriate for just two coupled qubits. §.§ Quality factor The two-qubit coherence time T_2^* as defined above measures the time ittakes for the envelope of the damped oscillations in R(t) to decay to 1/e of its initial value. Hence, the dimensionless combination J_0T_2^* can be thought of as the number ofappreciable oscillations in R(t) before it saturates to the asymptotic value R(∞). In the results presented in Fig. <ref>, the dimensionless parameter J_0T_2^* varies from 69 [Fig. <ref>(a)] to 4 [Fig. <ref>(f)], with the results of Fig. <ref>(f)being the most representative of the current experimental state ofthe arts in GaAs singlet-triplet qubits <cit.> where only a few (≲ 5) two-qubit gate oscillations have so far been achievedexperimentally. (We mention, however, that the experiments <cit.> aremostly in the h_0>0 regime more appropriate for the discussion in the nextsubsection of this paper.) To convert this into a number with a normalization comparable with otherfidelity measures, we further define the quality factor <cit.>Q=exp(-1/J_0T_2^*).This quantity is essentially the exponential decay factor for thereturn probability oscillation envelope over Δ t=1/J_0, the intraqubit exchange coupling time scale.In the rest of this section we present numerical results on the decoherence of twosinglet-triplet qubits using the quality factor Q as a quantitative measureof coherence. We will make comparisons with the exchange-only case studied inRef. Throckmorton17 where appropriate, and explain how theadditional tunability of the singlet-triplet system through the magnetic fieldgradient may be exploited to suppress decoherence due to charge noise. §.§ Noise dependenceWe first consider the case where the magnetic field gradient is zero onaverage, h_0 = 0, and examine the variation of the coherence time with respect to both thecharge noise σ_J and the field noise σ_h. Within the h_0=0 parameter subspace, we find that the singlet-triplet qubitsbehave similarly to the exchange-only qubits as reported inRef. Throckmorton17.Figures <ref> and <ref> show the dependence of thequality factor Q on the charge noise σ_J as well asthe field noise σ_h. We find that the quality factor for the coupled qubits is suppressed wheneither type of noise increases, and the system is significantly moresusceptible to charge noise than field noise. As marked by a contour line in Fig. <ref>, to achieve a quality factor Q higher than 0.9 (corresponding to acoherence time T_2^*∼ 10J_0^-1), the maximum allowed charge noiseσ_J is around 0.045J_0, while the maximum allowed field noiseσ_h is around 1.0J_0.The order of magnitude difference between the sensitivity to charge noise andthe sensitivity to field noise as measured by two-qubit coherence time is in agreement with the results previous reported on the exchange-onlyqubits <cit.>. Quantitatively, we find that the singlet-triplet system is about 3 (1.5) timesmore sensitive to the charge (field) noise compared with the exchange-onlysystem in the regime with a quality factor Q≥ 0.9. This is consistent with the intuitive observation that the exchange-onlysystem enjoys an additional protection due to the spin S_z conservation. The fact that charge noise is the dominant decohering mechanism forsinglet-triplet qubits (and is even more detrimental here than for exchange-only qubits) is, however, only true for h_0=0 as we discuss next. §.§ Effect of the magnetic field gradient Experimentally, charge noise in GaAs-based spin-qubit devices is typicallymuch weaker in absolute strength than field noise, due to the strong Overhauser nuclear spin fluctuations. In Si-based spin-qubit devices, however, the nuclear spin fluctuations can besignificantly suppressed thanks to isotope purification of ^28Si <cit.>. In this case, the strong sensitivity to charge noise may pose a seriousobstacle to the fidelity of coupled qubits, since there is no known way to systematically reduce the charge noise insemiconductor structures. From our numerical results, we find that this problem may be alleviatedthrough the additional tunable parameter in the singlet-triplet system, namely, the average magnetic field gradient h_0 across each singlet-triplet qubit.Figure <ref> shows the effect of h_0 on the noisedependence of the quality factor Q. As the magnetic field gradient h_0 goes up, the coherence of the coupled-qubitdynamics becomes less sensitive to the charge noise σ_J, but morevulnerable to the field noise σ_h. When h_0 is higher than J_0, the coupled qubits become more susceptible tofield noise than charge noise, in sharp contrast to the situation forh_0=0. The sensitivity to σ_h saturates when h_0 is more than a few timesstronger than J_0. For reference, we note that the entangling gate experiments reported inRef. Nichol17 were carried out at an effective h_0∼ 5J_0,albeit under a different setup with individual qubits driven by an oscillatoryJ_i(t). Comparing the numerical results in Figs. <ref>and <ref>(e), we find that the maximum allowed charge noiseto achieve a high quality factor Q≥ 0.99 increases by more than 10 times as the magnetic field gradient h_0 is cranked upfrom zero to 5J_0. This enhanced stability against charge noise is consistent with theexperimental observation in Ref. Nichol17 that a magnetic field gradient h_0∼ 5J_0 increases the two-qubit coherence time by an orderof magnitude in a device dominated by charge noise. We mention here that the GaAs system used in Ref. Nichol17 obviouslyalso has considerable field noise, arising from nuclear spin fluctuations inGa and As, contributing to decoherence.§ FIDELITY OF ENTANGLED STATE PREPARATIONThe two-qubit coherence time T_2^* measures the persistence of two-qubitoscillations in the presence of environmental noises. This is acharacterization of how well the system retains the initial non-eigenstateinformation. In this section, we study a different aspect of two-qubit fidelity, namely,the fidelity F_E of preparing an entangled state. We investigate how well the system produces an entangled state starting froman initial product state under the influence of environmental noises. This analysis is less sophisticated than a full-blown gate fidelity calculation using randomized benchmarking.Nevertheless, it provides useful insights through a perspective complementaryto the two-qubit coherence time T_2^*, and provides a single fidelity number (the entanglement fidelity, F_E) similar to the full-blown numerically intensive Clifford gate randomizedbenchmarking calculation which is beyond the scope of the current work. §.§ Producing an entangled state We choose the product initial state|ϕ(0)⟩=|↑⟩_x⊗|↓⟩_x and let the systemevolve under the Hamiltonian H in Eq. (<ref>) for a fixed amount oftime t (to be specified). In the clean limit, the resulting state is|E(t)⟩= e^-i[ ε J_0^2σ_1^zσ_2^z +J_0(σ_1^z+σ_2^z) +h_0(σ_1^x+σ_2^x) ]t |ϕ(0)⟩.This is in general an entangled state, and as we discuss below, with a properchoice of the evolution time t, |E(t)⟩ is in fact maximally entangledfor both h_0=0 and h_0≫ J_0. It should be noted that the Hamiltonian H is not always effective atgenerating entanglement starting from an arbitrary initial state. Theparticular initial state|ϕ(0)⟩=|↑⟩_x⊗|↓⟩_x chosen here provides a simple setup to discuss the effect of noise on entangled statepreparation.In the presence of environmental noises, the time-evolved state e^-iHt|ϕ(0)⟩ depends on the disorderrealization and deviates from its clean limit |E(t)⟩. Using the latter as a reference, we define the fidelity of entangled statepreparation <cit.> F_E as the disorder-averaged overlapF_E=|⟨E(t)|e^-iHt|ϕ(0)|⟩|^2.Similarly to the two-qubit coherence time T_2^*, this is a function of thefield noise σ_h, the charge noise σ_J, and the magnetic field gradienth_0.We want to make sure that F_E indeed measures the fidelity associated withgenerating an entangled state. To this end, we now discuss the choice of the evolution time t thatmaximizes the entanglement between the two qubits in the referencestate |E(t)⟩. In the absence of a magnetic field gradient h_0, the reduced density matrixafter tracing out one qubit in |E(t)⟩ takes the simple form1/2([ 1 -e^-2i J_0tcos (2ε J_0^2 t);-e^2i J_0tcos (2ε J_0^2 t) 1; ]).This suggests setting the evolution time t in Eq. (<ref>) tot_0=π/4ε J_0^2,where ε J_0 measures the strength of the interqubit Isingcoupling (set to 0.1 in this paper). This choice ensures that the reference state |E(t_0)⟩ is maximallyentangled between the two qubits for h_0=0, with entanglement entropyS_E=log 2.The situation for h_0≠ 0 is less obvious. Figure <ref>(a) shows the dependence of the entanglemententropy S_E between the two qubits on the evolution time t, for variousvalues of h_0. We find that for both h_0=0 and h_0≫ J_0, the entanglement entropy of |E(t)⟩ peaks at t=t_0, whereas forintermediate h_0∼ J_0, the entanglement entropy has irregular dynamicsbut still reaches a moderate level at t=t_0. Figure <ref>(b) shows the entanglement entropy at t=t_0 asa function of the magnetic field gradient h_0. We find that the reference state at t=t_0 is nearly maximally entangled fora wide range of h_0 except for a small window near J_0. This justifies our operational definition of the entanglement fidelity usingthe evolution time t_0 defined in Eq. (<ref>).§.§ Noise dependence of F_EWe now examine how the fidelity of entangled state preparation is affected byenvironmental noises. First we consider the case of zero magnetic gradient h_0=0. Figure <ref> shows the dependence of the entanglement fidelity F_E asa function of the charge noise σ_J and the field noise σ_h. We find that F_E decays monotonically when either type of noiseincreases, and the system is more susceptible to the chargenoise σ_J than the field noise σ_h. To reach F_E higher than 0.9, the maximum allowed charge noise σ_J is around 0.03J_0, while the maximum allowed field noise σ_h isaround 0.18J_0. We observe that the fidelity of entangled state generation has a charge noisedependence comparable to that of the quality factor Q associated withthe two-qubit coherence time T_2^*, but it has a field noise dependenceabout 5 times stronger than that of the quality factor Q. This suggests that field noise is more effective at disrupting theprecise preparing of an entangled state than damping the coupled-qubitoscillations. This is germane for future progress in the subject since field noise canessentially be eliminated is Si qubits through isotopic purification.Compared with the exchange-only qubits studied inRef. Throckmorton17, the entanglement fidelity F_E for the singlet-tripletqubits computed here is significantly more susceptible to charge noise. Intuitively, this is consistent with the fact that the Ising Hamiltonian ofthe singlet-triplet system has a weaker (by a factor of ε J_0)interqubit coupling and thus is less effective at entangling the two qubitsthan the Heisenberg Hamiltonian for the exchange-only system. The longerevolution time strengthens the effect of charge noise as it modifies thequbit precession frequency. This particular damaging aspect of charge noise can be partially rectified byhaving stronger interqubit coupling through careful qubit geometry engineering.The noise dependence of F_E changes qualitatively when we turn on themagnetic field gradient h_0. The progression is shown in Fig. <ref>. As the magnetic field gradient h_0 increases, the entanglement fidelity F_Equickly developsmore sensitivity to the field noise σ_h while becoming less susceptible tothe charge noise σ_J. At the turning point h_0=J_0, the noise dependence of F_E is approximatelysymmetric with respect to σ_J and σ_h. As the magnetic field gradient h_0 increases further, the sensitivity to charge noise is quickly suppressed, while the sensitivity to field noise reaches a plateau. For h_0≫ J_0, the entanglement fidelity F_E is limited mainly by the field noiseσ_h (again implying a considerable advantage for isotopically purifiedSi qubits). Overall, we find that the fidelity of entangled state preparation has anoise dependence qualitatively similar to that of the coherence time qualityfactor.§ CONCLUSIONIn this paper we have studied the decoherence of two singlet-tripletspin qubits with capacitive coupling under the influence ofquasistatic environmental noises. We consider two complementary decoherence measures for coupled qubits, namely,the two-qubit coherence time characterizing the persistence of coupled-qubitoscillations, and the fidelity of entangled state preparation. Through numerical calculations, we provide a quantitative map of thedependence of each decoherence measure on charge noise, field noise, andthe intraqubit magnetic field gradient.We find that the noise dependence of the coupled-qubit coherence changesqualitatively as the magnetic field gradient increases. When the (average) magnetic field gradient vanishes, the coupled-qubit systemis more susceptible to charge noise than field noise. For the two-qubit coherence time to be longerthan 10J_0^-1, the maximum allowed charge noise is an order of magnitudelower than the maximum allowed field noise. The fidelity of entangled statepreparation has a similar (although less pronounced) bias in its noisesensitivity. In contrast, when the coupled-qubit system is dominated by a strong magneticfield gradient, the sensitivity to charge noise is strongly suppressed and becomesmuch weaker than the sensitivity to field noise, as visible in both the two-qubitcoherence time and the entanglement fidelity.Our results highlight the impact of the magnetic field gradient on the noisedependence of the coupled-qubit system. Increasing the magnetic field gradient h_0 proves to be an effective measure toprotect against charge noise the coherence of coupled singlet-triplet qubitsin terms of both the persistence of coupled-qubit oscillations and the precisepreparation of entangled states. In addition, our work points to clear advantages for Si-based qubits over GaAsqubits since isotopic purification could eliminate field noise in Si (but notin GaAs). Elimination of field noise would enhance fidelity, and working in a largefield gradient would suppress the charge noise, eventually leading tohigh-fidelity singlet-triplet semiconductor spin qubits suitable for quantumerror correction protocols. 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Das Sarma, 10.1103/PhysRevB.67.033301 journal journal Phys. Rev. B volume 67, pages 033301 (year 2003)NoStop [Witzel et al.(2010)Witzel, Carroll, Morello, Cywi ńński, and Das Sarma]Witzel10 author author W. M. Witzel, author M. S. Carroll, author A. Morello, author L. Cywi ńński,and author S. Das Sarma, 10.1103/PhysRevLett.105.187602 journal journal Phys. Rev. Lett. volume 105, pages 187602 (year 2010)NoStop [Martins et al.(2016)Martins, Malinowski, Nissen, Barnes, Fallahi, Gardner, Manfra, Marcus, and Kuemmeth]Martins16 author author F. Martins, author F. K. Malinowski, author P. D. Nissen, author E. Barnes, author S. Fallahi, author G. C. Gardner, author M. J. Manfra, author C. M. Marcus,and author F. Kuemmeth, 10.1103/PhysRevLett.116.116801 journal journal Phys. Rev. Lett. volume 116, pages 116801 (year 2016)NoStop [Wu et al.(2017)Wu, Deng, Li, and Das Sarma]Wu17 author author Y.-L. Wu, author D.-L. Deng, author X. Li,and author S. Das Sarma, 10.1103/PhysRevB.95.014202 journal journal Phys. Rev. B volume 95, pages 014202 (year 2017)NoStop [Nielsen and Chuang(2000)]Nelsen00 author author M. A. Nielsen and author I. L. Chuang, @nooptitle Quantum Computation and Quantum Information (publisher Cambridge University Press,address Cambridge, year 2000)NoStop | http://arxiv.org/abs/1708.07539v2 | {
"authors": [
"Yang-Le Wu",
"S. Das Sarma"
],
"categories": [
"cond-mat.mes-hall",
"quant-ph"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170824200404",
"title": "Decoherence of two coupled singlet-triplet spin qubits"
} |
A magnetic version of the Smilansky-Solomyakmodel] A magnetic version of the Smilansky-Solomyakmodel Department of Mathematics, University of Ostrava, 30. dubna 22, 70103 Ostrava, Czech Republic Nuclear Physics Institute, Academy of Sciences of the Czech Republic, Hlavní 130, 25068 Řež near Prague, Czech Republic [email protected], [email protected] Nuclear Physics Institute, Academy of Sciences of the Czech Republic, Hlavní 130, 25068 Řež near Prague, Czech Republic Doppler Institute, Czech Technical University, Břehová 7, 11519 Prague, Czech Republic [email protected] We analyze spectral properties of two mutually related families of magnetic Schrödinger operators, H_Sm(A)=(i ∇ +A)^2+ω^2 y^2+λ y δ(x) and H(A)=(i ∇ +A)^2+ω^2 y^2+ λ y^2 V(x y) in L^2(^2), with the parameters ω>0 and λ<0, where A is a vector potential corresponding to a homogeneous magnetic field perpendicular to the plane and V is a regular nonnegative and compactly supported potential. We show that the spectral properties of the operators depend crucially on the one-dimensional Schrödinger operators L= -d^2/dx^2 +ω^2 +λδ (x) and L (V)= - d^2/dx^2 +ω^2 +λ V(x), respectively. Depending on whether the operators L and L(V) are positive or not, the spectrum of H_Sm(A) and H(V) exhibits a sharp transition.Keywords: Discrete spectrum, essential spectrum, Smilansky-Solomyak model, spectral transition, homogeneous magnetic field J. Phys. A.: Math. Theor. [ Pavel Exner December 30, 2023 ===================== § INTRODUCTION Irreversible dynamics of quantum systems is usually described through a coupling of the object, regarded as an open or unstable system, to another one that plays the role of a `heat bath'. The latter is usually supposed to be a `large' system having an infinite numbers of degrees of freedom and the Hamiltonian with a continuous spectrum, moreover, the presence (or absence) of irreversible modes is determined by the energies involved rather than the coupling strength between the object and the bath. While this is all true in many cases, it need not be true in general. This was demonstrated by Uzy Smilansky using a simple model <cit.> which was subsequently analyzed in detail and generalized by Mikhail Solomyak and coauthors <cit.>, see also <cit.> and <cit.>.In the simplest case the model is described by the HamiltonianH_Sm=-∂^2/∂ x^2 +1/2( -∂^2/∂ y^2+y^2 ) +λ yδ(x)in L^2(ℝ^2) with the natural domain and exhibits a transition between two types of spectral behavior: for |λ|≤√(2) the operator (<ref>) is bounded from below, while for |λ|>√(2) its spectrum fills the real line <cit.>. The factor 1/2 is not important and can be changed by a scaling of one of the variables. If we replace it by one, for instance, the critical value of the coupling constant will be λ=2. The transition between the two regimes can be interpreted also dynamically <cit.>: in the supercritical regime the y-dependent binding energy of δ interaction wins over the oscillator potential and the wave packet can escape to infinity along the singular channel.While mathematically we deal with the same object, from the physical point of view one can interpret it in two different ways. In the original Smilansky paper <cit.> it was meant as a system of two one dimensional components, a particle motion on a line to which a heat bath consisting of a single harmonic oscillator is coupled in a coordinate dependent way. In the generalizations mentioned above the line was replaced by other simple configuration spaces, a loop (in other words, segment with periodic boundary conditions) or a graph, and the bath could be anharmonic or multidimensional (but still with a finite number of degrees of freedom).Another point of view, which can be associated with the work of Solomyak and coauthors, is to associate the Hamiltonian (<ref>) with a two-dimensional system in which the particle moves in the potential which is the sum of the oscillator `channel' and the singular component with the position-dependent coupling strength. Viewed from this angle, the system brings to mind motion in a potential with channels which are below unbounded and narrowing towards infinity. In this situation one may also observe a jump transition from a below bounded to below unbounded spectrum as was first noted in <cit.>, a class of models of this type was analyzed recently in <cit.>. The analogy becomes even more convincing when we recall that the model (<ref>) has a `regular' analogue <cit.> in which the δ interaction is replaced by a non-singular potential properly scaled. In this case, of course, the `first', two subsystem, interpretation is lost unless we try to interpret the potential as a sort of nonlocal position-dependent subsystem coupling.The main question we want to address in the present paper is what will happen with the model in its two-dimensional version when the particle it describes is charged and exposed to a homogeneous magnetic field perpendicular to the plane, in other words, what are the properties of the operatorH_Sm(A)=(i ∇ +A)^2+ω^2 y^2+λ y δ(x)with λ∈ℝ and ω>0, where the vector potential A corresponds to the indicated magnetic field of intensity B>0. We may consider non-positive λ only; as in the nonmagnetic case this is due to mirror symmetry but the argument is a bit trickier. The magnetic field changes direction when observed in a mirror, however, switching the sign of both the variables we return to the original A and at the same time the last term on the right-hand side of (<ref>) changes sign. It is clear that now the two-subsystem interpretation is ultimately lost, therefore it is appropriate to speak of (<ref>) as of the Hamiltonian of the magnetic Smilansky-Solomyak model.The dynamics of the model combines the influence of several forces and its properties are not a priori obvious. For λ=0 the spectrum is absolutely continuous and the particle moves along the parabolic channel provided its energy is larger than √(ω^2+B^2), and moreover, the transport is stable against localized perturbations <cit.>. If both ω and λ vanish, operator (<ref>) is the Landau Hamiltonian the spectrum of which is known to be pure point, consisting of the Landau levels (2n+1)B,n=0,1,2,…. Was the singular term position independent, just λδ(x), it would make the spectrum absolutely continuous corresponding to transport in the y direction as one could check in a way similar to the Iwatsuka model <cit.>, <cit.>, or to magnetic transport along a barrier <cit.>. The operator (<ref>) with ω=0 has not been analyzed to the best of our knowledge, but one can expect that it will exhibit some transport properties again; at least we will show, as a byproduct of our results here, that its spectrum covers the whole real axis whenever λ 0.The oscillator potential, however, acts against a transport in the y direction. We are going to show that the resulting behavior is determined by the balance of the two forces, in a way to a large degree similar to the nonmagnetic case, A=0. To be specific, we introduce thecomparison operator,L=-d^2/dx^2+ω^2+λδ(x)on L^2() with the usual domain <cit.>; our goal is to establish a correspondence between the spectral regime of H_Sm(A) and the positivity of the operator (<ref>). We are going to show that the spectrum is bounded from below provided infσ(L)>0 — we speak here about the subcritical case — and it has a purely discrete character below √(ω^2+B^2) being nonempty whenever λ 0. In the critical case, infσ(L)=0, the operator H_Sm(A) remains positive but its spectrum is purely essential and equal to [0,∞). Finally, if one passes to the supercritical regime, infσ(L)<0, an abrupt transition occurs and the spectrum fills now the whole real line.In a similar way, the `regular' version of the model <cit.> mentioned above has also its magnetic counterpart. The dynamics is this case described by the HamiltonianH(A)=(i ∇ +A)^2+ω^2 y^2+ λ y^2 V(x y) ,where V is a nonnegative, sufficiently smooth function with supp(V)⊂[-s_0, s_0] for some s_0>0, furthermore, λ≤ 0 andω>0, and the magnetic potential A corresponds as before to a homogeneous magnetic field of the intensity B>0. Note that the analogy is not complete because both parts of the scalar potential are mirror symmetric with respect to the x axis, however, the effect which we are interested in depends on the presence of an attractive interaction in the y direction, irrespective whether is one- or two-sided. Spectral properties of the operator (<ref>) will be the topic of the second part of the paper. The abrupt spectral transition occurs here again. The comparison operatorwill beL(V)=-d^2/dx^2+ω^2+λ V(x)on L^2(ℝ) with the domain ℋ^2(), and its spectral threshold will be shown to be decisive: H(A) will be bounded from below provided if L(V) is nonnegative, and its spectrum will fill the whole real line in the opposite case.§ SPECTRUM OF H_SM(A) Before coming to our proper subject we note that in order to interpret H_Sm(A) as a quantum mechanical Hamiltonian, one has check its self-adjointness. In the subcritical case, when L is strictly positive, we can consider first the domain 𝒟_0 consisting of the family of functions v twice differentiable except at the y axis, x=0, continuous there and satisfying the matching conditions ∂ v/∂ x(0+, y)-∂ v/∂ x(0-, y)=λ y v(0, y) and to identify H_Sm(A) with the Friedrichs extension of such an operator. In other words, we will consider the quadratic formQ (H_Sm (A))[u]=∫_ℝ^2[ |i ∂ u/∂ x-B y u|^2+|∂ u/∂ y|^2+ω^2 y^2 |u|^2 ] dx dy+λ∫_ℝy |u(0, y)|^2 dyand demonstrate that it is closed on 𝒟:=ℋ^1(ℝ^2)∩{∫_ℝ^2y^2 |u(x,y)|^2 dx dy<∞} and bounded from below provided infσ(L)>0, and therefore associated with a unique self-adjoint operator.The approach based on quadratic forms fails, of course, if we cannot ensure that the operator is bounded from below. To make things simple, we can bypass this trouble by noting that H_Sm(A) is essentially self-adjoint on 𝒟_0. Indeed, it is easy to check that such a operator is densely defined and symmetric. To ensure that its deficiency indices coincide, it is by <cit.> sufficient to check that its commutes with a conjugation, i.e. an antilinear map L^2(^2) → L^2(^2) which is norm preserving and idempotent; choosing 𝒞:(𝒞v)(x,y) = u(-x,y) we get the claim. Then we know that H_Sm(A)↾ D_0 has self-adjoint extension and will show that the indicated spectral properties hold for any such extension.§.§ The subcritical caseAs we have indicated, to establish the self-adjointness of H_Sm(A) one has to check that the form (<ref>) is bounded from below and closed if infσ(L)= ω^2-1/4λ^2>0. First we will show that the operator is in fact positive even if the last inequality is not sharp. Let λ≥-2ω, then H_Sm(A)↾𝒟_0 ≥ 0. For every u∈𝒟_0 the form (<ref>) can be estimated form below by neglecting the `transverse' contribution to the kinetic energyQ( H_Sm (A))[u] ≥∫_ℝ^2[|i ∂ u/∂ x-B y u|^2+ω^2 y^2 |u|^2] dx dy +λ∫_ℝy |u(0, y)|^2 dy .For any fixed y the form u(·,y)↦∫_ℝ|i ∂ u/∂ x-B y u|^2+ω^2 y^2 |u|^2 dx+λ y |u(0, y)|^2 corresponds to the essentially self-adjoint operator(i d/dx-B y)^2+ω^2 y^2+λ y δ(x)whose closure has ℋ^1(ℝ) as its form domain. This operator is unitarily equivalent to y^2 L which is positive by assumption if y>0, and to y^2L≥0 if y<0, whereL:=-d^2/d x^2+ω^2-λδ(x) ;this establishes the sought claim.The form (<ref>) is closed if λ>-2ω. Let {u_n}_n=1^∞⊂𝒟 be a sequence converging to some u∈ L^2(^2) and satisfyingQ(H_Sm)(A)[u_n-u_m]→ 0as m,n→∞. By the assumption one can choose α∈(|λ|/2ω,1) and rewrite the form value in question asQ( H_Sm(A))[u_n-u_m] = (1-α)(∫_ℝ^2|(i ∇+A)(u_n-u_m)|^2 dx dy+ω^2∫_ℝ^2 y^2 |u_n-u_m|^2 dx dy)+α(∫_ℝ^2|(i ∇+A)(u_n-u_m)|^2 dx dy+ω^2 ∫_ℝ^2y^2 |u_n-u_m|^2 dx dy+λ/α∫_ℝy |u_n(0, y)-u_m(0, y)|^2 dx) .In the same way as in the proof of the previous proposition one can check that the second summand on the right-hand side of (<ref>) is nonnegative, hence neglecting it we estimate Q( H_Sm(A))[u_n-u_m] from below by the first summand which in view of (<ref>) tends to zero as m,n→∞. Since α 1 by construction, this means that the sequence {Q_0(A)[u_n]} is Cauchy, where Q_0(A) is the `unperturbed' formu↦ Q_0(A)[u]:=∫_ℝ^2[|(i ∇+A)u|^2+ω^2 y^2 |u|^2] dx dydefined on 𝒟. It is not difficult to verify that Q_0(A) is closed and this in turn implies that the limit function u belongs to 𝒟 and∫_ℝ^2[|(i ∇+A)(u_n-u)|^2+ω^2 y^2 |u_n-u|^2] dx dy→0as n→∞ ,cf. <cit.>. It remains to check that ∫_ℝy |u_n(0, y)-u(0, y)|^2 dy → 0 holds as well. Using a couple of simple estimates,∫_ℝ|y||v(0, y)|^2 dy ≤2/ω∫_ℝ^2(|∂ v/∂ x|^2+ω^2y^2 |v(x, y)|^2) dx dy≤1/2ω∫_ℝ^2(|∂ v/∂ x|^2+|∂ v/∂ y+i B x v|^2(x, y)+ ω^2y^2 |v(x, y)|^2) dx dy = 1/2ωQ_0(a)[v] ,and inserting v=u_n-u, we conclude the proof. This guarantees that in the subcritical case there is a unique self-adjoint operator H_Sm(A) associated with the form (<ref>).§.§ The essential spectrumIn fact will first show that the essential spectrum is nonempty independently of λ.σ_ess (H_Sm(A))⊃ [√(ω^2+B^2), ∞). It is sufficient to construct a Weyl sequence for any number μ>√(ω^2+B^2). To this aim, we fix first a positive number ε and construct a function ϕsuch thatH_Sm (A) ϕ-μϕ_L^2 (ℝ^2)<εϕ .We employ the functionsφ_k, α, m(x, y):= 1/√(2π vol(E))(∫_E g(y-ξ B/ω^2+B^2)e^i ξ (x-α k) dξ) η(x/k) χ(y/k) ,where g is the normalized eigenfunction associated with the principal eigenvalue of the harmonic oscillator modified by the presence of the magnetic field, h_osc= -d^2/dy^2+(ω^2+B^2) y^2 on L^2(ℝ), the functions η∈ C_0^∞ (1, m) , χ∈ C_0^∞(-1, 1) are supposed to satisfy the following requirements,η(z)≥1/2if z∈(3/2, m/2) ,χ(z)≥1/2if z∈(-1/2, 1/2) ,and the set E is defined byE=(δ_1(ε), δ_2(ε)):= {ξ: √((μ̃-ε) (ω^2+B^2))/ω<ξ<√((μ̃+ε) (ω^2+B^2))/ω} ,where μ̃:=μ-√(ω^2+B^2) and k, m, α ∈ℕ are positive integers to be chosen later. Note that supp (ψ_k,α,m)⊂ [k,mk]×[-k,k], and therefore∂ψ_k,α,m/∂ x(0+, y)=∂ψ_k,α,m/∂ x(0-, y)=λ y ψ_k,α,m(0,y)=0 ,which means that the functions ψ_k,α,m belong to the domain of H_Sm(A) as needed.First we observe that φ_k, α, m_L^2(ℝ^2)≥1/8 because∫_ℝ^2 |φ_k, α, m(x,y)|^2 dx dy=1/2π vol(E)∫_ℝ^2|∫_E g(y-ξ B/ω^2+B^2) e^i ξ (x-α k) dξ|^2 η^2(x/k) χ^2(y/k) dx dy =1/2π vol(E)∫_k^m k ∫_-k^k|∫_E g(y-ξ B/ω^2+B^2) e^i ξ (x-α k) dξ|^2 η^2(x/k) χ^2(y/k) dx dy ≥1/32π vol(E)∫_-k/2^k/2 ∫_3k/2^m k/2|∫_E g(y-ξ B/ω^2+B^2) e^i ξ (x-α k) dξ|^2 dy dx =1/32π vol (E)∫_-k/2^k/2 ∫_(3-2α) k/2^(m-2α) k/2|∫_E g(y-ξ B/ω^2+B^2) e^i ξ x dξ|^2 dy dx .By choosingα=α(k) and m=m(α, k) large enough one is able to guarantee that for everyy∈(-k/2, k/2) we have1/2π∫_(3-2α)k/2^(m-2α) k/2|∫_E g(y-ξ B/ω^2+B^2) e^i ξ x dξ|^2 dx ≥1/4π∫_ℝ|∫_E g(y-ξ B/ω^2+B^2) e^i ξ x dξ|^2 dx =1/2∫_Eg^2(y-x B/ω^2+B^2) dx ,where in the last step we have employed Plancherel formula. This estimate together with (<ref>) gives for large k the inequalities∫_ℝ^2 |φ_k, α, m(x,y)|^2 dx dy≥1/32 vol(E)∫_-k/2^k/2∫_E |g(y-x B/ω^2+B^2)|^2 dx dy ≥1/32 vol(E)∫_E ∫_-k/2-x B/(ω^2+B^2)^k/2-x B/(ω^2+B^2)g^2(z) dz dx ≥1/64 ∫_ℝ|g(z)|^2 dz=1/64. Our next aim is to show the validity of (<ref>) with an appropriate choice of k, α(k) and m(α, k). By a straightforward calculation one gets√(2π) ∂^2φ_k, α, m/∂ y^2 =1/√(vol(E))(∫_E g”(y-ξ B/ω^2+B^2)e^i ξ (x-α k) dξ) η(x/k) χ(y/k) +2/k √(vol(E))(∫_E g^'(y-ξ B/ω^2+B^2)e^i ξ (x-α k) dξ) η(x/k) χ^'(y/k) +1/k^2 √(vol(E))(∫_E g(y-ξ B/ω^2+B^2)e^i ξ (x-α k) dξ) η(x/k) χ”(y/k) ,and√(2π) ∂^2φ_k, α, m/∂ x^2 =-1/√(vol(E))(∫_E ξ^2 g(y-ξ B/ω^2+B^2) e^i ξ (x-α k) dξ) η(x/k) χ(y/k) +2i/k √(vol(E))(∫_E ξ g(y-ξ B/ω^2+B^2) e^i ξ (x-α k) dξ) η^'(x/k) χ(y/k) +1/k^2 √(vol(E))(∫_E g(y-ξ B/ω^2+B^2) e^i ξ (x-α k) dξ) η”(x/k) χ(y/k) , √(2π)y ∂φ_k, α, m/∂ x =i y/√(vol(E))(∫_E ξ g(y-ξ B/ω^2+B^2)e^i ξ (x-α k) dξ) η(x/k) χ(y/k) +y/k √(vol(E))(∫_Eg(y-ξ B/ω^2+B^2)e^i ξ (x-α k dξ) η^'(x/k) χ(y/k).We want to show that choosing k sufficiently large one can make the last two terms on the right-hand side of the first equation (<ref>) as small as one wishes in the L^2 norm, and the same for the last two terms of the second equation (<ref>) and the last term of (<ref>). This follows from the following estimates,1/k^2 vol (E)∫_ℝ^2|∫_E g^'(y-ξ B/ω^2+B^2)e^i ξ (x-α k) dξ|^2 η^2(x/k) (χ^')^2(y/k) dx dy≤η_∞^2 χ^'_∞^2/k^2 vol (E)∫_ℝ^2|∫_E g^'(y-ξ B/ω^2+B^2)e^i ξ x dξ|^2 dx dy =η_∞^2 χ^'_∞^2/k^2 vol (E)∫_ℝ∫_E(g^'(y-x B/ω^2+B^2))^2 dx dy = η_∞^2 χ^'^2_∞/k^2∫_ℝ(g^')^2(z) dz ,where in the last step we employed again Plancherel formula, and similarly,1/k^4 vol (E)∫_ℝ^2|∫_E g(y-ξ B/ω^2+B^2)e^i ξ (x-α k) dξ|^2 η^2(x/k) (χ”)^2(y/k) dx dy ≤η_∞^2 χ”^2_∞/k^4∫_ℝ g^2(z) dz1/k^2 vol (E)∫_ℝ^2|∫_E ξ g(y-ξ B/ω^2+B^2)e^i ξ (x-α k) dξ|^2 (η^')^2(x/k) χ^2(y/k) dx dy ≤η^'_∞^2 χ^2_∞δ^2_2(ε)/k^2∫_ℝ g^2(z) dz , 1/k^4 vol (E)∫_ℝ^2|∫_E g(y-ξ B/ω^2+B^2)e^i ξ (x-α k) dξ|^2 (η”)^2(x/k) χ^2(y/k) dx dy ≤η”_∞^2 χ^2_∞/k^4∫_ℝ g^2(z) dz , 1/k^2 vol (E)∫_ℝ^2y^2|∫_E g(y-ξ B/ω^2+B^2)e^i ξ (x-α k) dξ|^2 (η^')^2(x/k) χ^2(y/k) dx dy ≤η^'_∞^2 χ^2_∞/k^2∫_ℝ(|z|+δ_2 (ε)B/ω^2+B^2)^2 g^2(z) dz ,and consequently, all these integrals are at least 𝒪(k^-2) as k→∞. Then we can estimate the norm on left-hand side of (<ref>) as∫_ℝ^2|H_Sm(A) φ_k, α, m-μφ_k, α, m|^2(x,y) dx dy =∫_ℝ^2|-∂^2φ_k, α, m/∂ x^2- ∂^2φ_k, α, m/∂ y^2+2 i By∂φ_k, α, m/∂ x +(ω^2+B^2)y^2φ_k, α, m-μφ_k, α, m|^2 dx dy =1/2π vol (E)∫_ℝ^2|∫_E (-g”(y-ξ B/ω^2+B^2) +ξ^2 g(y-ξ B/ω^2+B^2)-2B y ξ g(y-ξ B/ω^2+B^2)+(ω^2+B^2)y^2 g(y-ξ B/ω^2+B^2)-μ g(y-ξ B/ω^2+B^2)) e^i ξ (x-α k) dξ|^2 η^2(x/k) χ^2(y/k) dx dy+𝒪(k^-2) =1/2πvol (E)∫_ℝ^2|∫_E (-g”(y-ξ B/ω^2+B^2)+((B^2+ω^2) (y-ξ B/ω^2+B^2)^2+ω^2 ξ^2/ω^2+B^2-μ)g(y-ξ B/ω^2+B^2) ) e^i ξ (x-α k) dξ|^2 η^2(x/k) χ^2(y/k) dx dy+𝒪(k^-2)=1/2πvol (E)∫_ℝ^2|∫_E (ω^2 ξ^2/ω^2+B^2-μ̃) g(y-ξ B/ω^2+B^2) ) e^i ξ (x-α k) dξ|^2×η^2(x/k) χ^2(y/k) dx dy+𝒪(k^-2) ≤η_∞^2 χ_∞^2/2πvol (E)∫_ℝ^2|∫_E (ω^2 ξ^2/ω^2+B^2-μ̃) g(y-ξ B/ω^2+B^2) ) e^i ξ x dξ|^2 dx dy +𝒪(k^-2) ≤η_∞^2 χ_∞^2/vol (E)∫_E×ℝ(ω^2 x^2/ω^2+B^2-μ̃)^2 g^2(y-x B/ω^2+B^2)dx dy +𝒪(k^-2) ≤ε^2 η_∞^2 χ_∞^2/vol (E)∫_E×ℝ g^2(y-xB/ω^2+B^2) dx dy +𝒪(k^-2) ≤ε^2 η_∞^2 χ_∞^2 ∫_ℝ g^2(z) dz+𝒪(k^-2) .Hence choosing a large enough k we can achieve that∫_ℝ^2|H_Sm (A)φ_k, α, m-μφ_k, α, m|^2(x,y) dx dy <64ε^2 η_∞^2 χ_∞^2 ∫_ℝ^2|φ_k, α, m|^2 dx dy+𝒪(k^-2)≤65ε^2 η_∞^2 χ_∞^2 ∫_ℝ^2|φ_k, α, m|^2 dx dy .To complete the proof we choose a sequence {ε_j}_j=1^∞ such that ε_j↘0 holds as j→∞ and to any given j we construct a function {φ_ε_j}_j=1^∞={φ_k(ε_j), α(k(ε_j)), m(α(k(ε_j)), k(ε_j))}_j=1^∞ with the parameters chosen in such a way that k(ε_j) >m(α(k(ε_j-1)), k(ε_j-1)) k(ε_j-1). The norms of H_Sm (A)φ_ε_j satisfy the inequality (<ref>) with 65ε_j^2 η_∞^2 χ_∞^2 φ_ε_j_L^2(ℝ^2)^2 on the right-hand side, and at the same time the sequence converges by construction weakly to zero. §.§ Subcritical case: the essential spectrum thresholdNext we are going to show that in the subcritical case the inclusion in Theorem <ref> is in fact an equality.Let λ>-2 ω, then the spectrum of H_Sm (A) below √(ω^2+B^2) is purely discrete. We employ a Neumann bracketing in a way similar to <cit.>. Let h^(±)_n (A) and h_0 (A) be the Neumann restrictions of operator H_Sm(A) to the regionsG^(±)_n=ℝ×{y: ± y ≥ n}and G_0=ℝ× [-n, n], where n∈ℕ will be chosen later. In view of the minimax principle <cit.> we have the inequalityH_Sm(A)≥(h^(+)_n (A)⊕ h^(-)_n (A))⊕ h^(0) (A) .To prove the claim we are going first to demonstrate that for a sufficiently large n the spectra of h_n^(±)(A) below √(ω^2+B^2) are empty, and secondly, that for any Λ<√(ω^2+B^2) the spectrum of h_0(A) below Λ is purely discrete.The quadratic form Q(h_n^(±) (A)) corresponding to h_n^(±) (A) coincides withQ(h_n^(±) (A))=∫_ℝ∫_± y≥ n|∂ u/∂ x-i B y u|^2 dx dy+∫_ℝ∫_± y≥ n|∂ u/∂ y|^2 dx dy+ω^2∫_ℝ∫_± y≥ ny^2 |u|^2 dx dy +λ∫_ℝ∫_± y≥ ny |u(0, y)|^2 dy ≥∫_ℝ∫_± y≥ n|∂ u/∂ x-i B y u|^2 dx dy +ω^2∫_ℝ∫_± y≥ ny^2 |u|^2 dx dy+λ∫_ℝ∫_± y≥ ny |u(0, y)|^2 dydefined on ℋ^1(ℝ×{y: ± y≥ n}). As before the quadratic formsu(·,y)↦∫_ℝ|i ∂ u/∂ x-B y u|^2+ω^2 y^2 |u|^2 dx+λ y |u(0, y)|^2are for any fixed y unitarily equivalent to y^2 L if y>0 and y^2L if y<0, where L is given in (<ref>). Thus we haveinfσ (h_n^±)≥ n^2 infσ (L)=n^2 (ω^2-λ^2/4)if y>0 ,infσ (h_n^±)≥ n^2 infσ (L)=n^2 ; ω^2 if y<0recall that we suppose λ<0. This concludes the proof of our first claim provided one chooses a sufficiently large n. It remains to inspect the essential spectrum of h_0 (A). To this aim we employ the following auxiliary result. Under our assumptionsinfσ_ess (h_0 (A)) =infσ_ess(h̃_0 (A)) ,where h̃_0 (A) is the Neumann operator (i ∂/∂ x- B y)^2-∂^2/∂ y^2+ω^2 y^2 definedon ℋ^1(G_0). We are going to show that for any real number μ from the resolvent sets of both h_0 (A)and h̃_0 (A)the operatorW:=(h_0 (A)-μ𝕀)^-1-(h̃_0 (A)-μ𝕀)^-1is compact in L^2(G_0). We proceed as in <cit.>; for any fixed f, g∈ L^2(G_0) we put u:=(h_0 (A)-μ𝕀)^-1 f and v:=(h̃_0 (A)-μ𝕀)^-1 g. Then(Wf, g)=((h_0 (A)-μ𝕀)^-1 f, g)-((h̃_0 (A)-μ𝕀)^-1 f, g)=(u, g)-(f, v)=(u,(h_0(A)-μ𝕀) v) -((h_0 (A)-μ𝕀) u,v)=(h̃_0 (A) u, v)-(h_0 (A) u,v) =-α∫_-n^n y u(0, y) v(0,y) dy .Let {f_n}_n=1^∞ and {g_n}_n=1^∞ bebounded sequences in L^2(G_0). Then it follows from the Sobolev trace theorem <cit.> that the ℋ^1(G_0) bounded functions u_n:=(h_0 (A)-μ𝕀)^-1 f_n and v_n=(h̃_0 (A)-μ𝕀)^-1 g_n are also bounded in ℋ^1/2(x=0, |y|≤ n). In view of the compact embedding ℋ^1/2(x=0, |y|≤ n) ↪ L^2(x=0, |y|≤ n) the sequences {u_n} and {v_n}_n=1^∞ are compact in L^2(x=0, |y|≤ n). In combination with the inequality(W(f_n-f_m), W(f_n-f_m)) ≤|α|n √(∫_-n^n|(u_n-u_m)(0, y)|^2 dy)√(∫_- n^n |(ṽ_n-ṽ_m)(0, y)|^2 dy)with ṽ_n=(h̃_0 (A)-μ𝕀)^-1 (W(f_n-f_m)), this establishes our claim. Let us now return to the proof of the theorem. In view of the lemma it is sufficient to inspect the threshold of σ_ess(h̃_0 (A)). Using the partial Fourier transformation F_ξ (u(x, y)) given byF_ξ (u(x, y))=u(x, y)=1/√(2 π)∫_ℝ u(ξ, y) e^-i ξ x dξand the Landau gauge for the vector potential, A=(-B y, 0), one is able to rewrite the quadratic form Q_0 of h̃_0 (A) asQ_0 (u)=∫_ℝ× [-n, n](|i ∇ u+A u)|^2(x, y)+ω^2 y^2|u|^2(x, y)) dx dy=∫_ℝ× [-n, n][|∂u/∂ y|^2(ξ, y)+((ω^2+B^2) (y-ξ B/ω^2+B^2)^2 +ω^2 ξ^2/ω^2+B^2)|u|^2(ξ,y) ] dξ dy .Thus h̃_0 (A) is unitarily equivalent to the direct integral∫_ℝ^⊕ h(ξ) dξ ,where the fibers h(ξ) =-d^2/dy^2 +(ω^2+B^2) (y-ξ B/ω^2+B^2)^2+ω^2 ξ^2/ω^2+B^2 are one-dimensional Neumann operators defined on L^2(-n, n). By a simple change of variables we arrive at the Neumann harmonic oscillators l (ξ)=-d^2/dy^2+(ω^2+B^2) y^2+ω^2 ξ^2/ω^2+B^2on the interval [-n-ξ B/ω^2+B^2, n-ξ B/ω^2+B^2]. Similarly as in <cit.> one can check thatinfσ (l (ξ))≥√(ω^2+B^2)+𝒪(n^-1)holds for large n uniformly in ξ∈ℝ. The spectrum of σ (h̃_0 (A)) is determined by those of the fiber operators <cit.>, in particular we getσ (h̃_0 (A))=σ_ess(h̃_0 (A))⊂[√(ω^2+B^2)+𝒪(n^-1), ∞) .By virtue of Lemma <ref> this yieldsinfσ_ess (h_0 (A))≥√(ω^2+B^2)+𝒪(n^-1) ,and therefore for any Λ<√(ω^2+B^2) one can choose n large enough to ensure that the spectrum of h_0(A) below Λ is purely discrete which concludes the proof. §.§ Subcritical case: existence of the discrete spectrumThe above results localize exactly the essential spectrum, however, they tell us nothing about the existence of the discrete spectrum. This is the question we are going to address now.Let λ>-2ω, then the discrete spectrum of H_Sm (A) is nonempty and contained in the interval (0, √(ω^2+B^2)). To demonstrate the non-emptiness of the discrete spectrum one needs to construct a normalized function u∈Dom(Q(H_Sm (A))) such that Q(H_Sm (A))(u)<√(ω^2+B^2). On the other hand, since λ>-2ω>-2√(ω^2+B^2) the non-magnetic operator H=-Δ+(ω^2+B^2) y^2+λ yδ(x) has a nonempty finite set of eigenvaluesbelow √(ω^2+B^2), and moreover, the corresponding eigenfunctions can be chosen real-valued <cit.>. It is easy to see that for any such eigenfunction u we haveQ(H(A))(u)=Q(H)(u)<√(ω^2+B^2) ,and since by Proposition <ref> operator H_Sm (A) is positive, the claimed is proved. §.§ The supercritical caseLet us now turn to the situation when the coupling constant surpasses the critical value. As discussed in the opening of Sec. <ref>, we know that operator H_Sm(A)↾𝒟_0 has self-adjoint extensions and in the following we will use the symbol H_Sm (A) for any of them.σ(H_Sm (A))=ℝ holds provided λ<-2ω. To check that any real number μ belongs to the spectrum of H_Sm (A) we employ Weyl's criterion finding a sequence {ψ_k}_k=1^∞⊂ D(H_Sm (A)) such that ψ_k=1 satisfyingH_Sm (A)ψ_k-μψ_k→0 as k→∞ ;note that one need not require that {ψ_k} contains no convergent subsequence because the spectrum covering the whole real axis cannot be anything else than essential. To this aim we modify the method of <cit.> without repetition of the parts that do not change. With scaling transformations in mind we may suppose that infσ(L)=ω^2-1/4λ^2=-1 corresponding to the single eigenvalue of the operator which is simple and associated with the normalized eigenfunction h=√(|λ|/2) e^-|λ| |t|/2.As in <cit.> we will first show that 0∈σ(H_Sm (A)). We fix an ε>0 and choose a positive integer k=k(ε) to which we associate a function χ_k⊂ C_0^2(1,k) such that∫_1^k1/zχ_k^2(z) dz=1 and∫_1^k z(χ'_k(z))^2 dz<ε ;we know from <cit.> that such functions can be constructed. Then we defineψ_k(x,y):=h(x y) ^iy^2/2χ_k(y/n_k) +f(x y)/y^2 ^iy^2/2χ_k(y/n_k) ,where the smooth function f∈Dom (L) and the positive integer n_k∈ℕ will be chosen later. The functions (<ref>) belong to Dom(H_Sm(A)) by construction and have the following property <cit.>.ψ_k_L^2(ℝ^2)≥1/2 holds provided n_k is large enough.Next we have to show that the functions ψ_k approximate the generalized eigenfunction corresponding to zero energy.H_Sm (A)ψ_k_L^2(ℝ^2)^2<cε holds with a c independent of k provided n_k is large enough. We have to estimate the following integral,∫_ℝ^2|H_Sm (A)ψ_k|^2(x,y) dx dy =∫_ℝ^2|-∂^2ψ_k/∂ x^2- ∂^2ψ_k/∂ y^2+2i B x ∂ψ_k/∂ y+B^2 x^2 ψ_k +ω^2 y^2ψ_k|^2 dx dy .We know from <cit.> that the claim is valid if B=0, hence it remains to deal with the additional terms associated with the magnetic field. We have∂ψ_k/∂ y= (x h^'(x y) χ_k(y/n_k)+1/n_kh(x y) χ_k^'(y/n_k)+x/y^2 f^'(x y) χ_k(y/n_k) +i/yf(x y) χ_k(y/n_k) +1/n_k y^2f(x y) χ_k^'(y/n_k)-2/y^3f(x y) χ_k(y/n_k)+i y h(x y) χ_k(y/n_k)) ^iy^2/2which allows us to check that choosing n_k large enough one can make norms of all the terms in the expression of x ∂ψ_k/∂ y except of the last one small enough, because∫_ℝ^2|x^2 h^'(x y) ^iy^2/2χ_k (y/n_k)|^2 dx dy≤1/n_k^4∫_ℝ|h^'(t)|^2 dt∫_1^k|χ_k(z)|^2 dz , 1/n_k^2∫_ℝ^2|h(x y) ^iy^2/2χ^'_k (y/n_k) |^2 dx dy ≤1/n_k^2∫_ℝ|h(t)|^2 dt ∫_1^k|χ^'_k(z)|^2 dz ,∫_ℝ^2|x/y^2f^'(x y) ^iy^2/2χ_k(y/n_k)|^2 dx dy≤1/n_k^4∫_ℝ|f^'(t)|^2 dt∫_1^k|χ_k(z)|^2 dz , ∫_ℝ^2|i/y f(x y) ^iy^2/2χ_k(y/n_k)|^2 dx dy≤1/n_k^2∫_ℝ|f(t)|^2 dt∫_1^k|χ_k(z)|^2 dz , ∫_ℝ^2|1/n_k y^2 f(x y) ^iy^2/2χ_k^'(y/n_k)|^2 dx dy≤1/n_k^6∫_ℝ|f(t)|^2 dt ∫_1^k|χ^'_k(z)|^2 dz , ∫_ℝ^2|2/y^3 f(x y) ^iy^2/2χ_k(y/n_k)|^2 dx dy ≤4/n_k^6∫_ℝ|f(t)|^2 dt ∫_1^k|χ_k(z)|^2 dz .In a similar way one can check that for large n_k the integral ∫_ℝ^2|x^2 ψ_k|^2 dx dy is less than ε. This yields the estimate∫_ℝ^2 |H_Sm (A)ψ_k|^2(x,y) dx dy≤ 21∫_n_k^kn_k∫_ℝ|y^2(h”(xy)-ω^2h(xy)- h(xy))χ_k(y/n_k)+ih(xy)χ_k(y/n_k) +f”(xy)χ_k(y/n_k)+2ixyh'(xy)χ_k(y/n_k) +2iy/n_kh(xy)χ'_k(y/n_k)-f(xy)χ_k(y/n_k) -ω^2 f(xy)χ_k(y/n_k) +2B x y h(x y)χ_k(y/n_k)|^2 dx dy +21ε ,where the coefficient in front of the integrals comes from the number of the summands. Using the fact that Lh=-h and applying the Cauchy inequality, the above inequality implies1/21∫_ℝ^2|H_Sm (A)ψ_k|^2(x,y) dx dy<∫_n_k^kn_k∫_ℝ|(f”(xy)+2ixyh'(xy)+ih(xy)-f(xy)-ω^2 f(xy)+2B x y h(x y))χ_k(y/n_k)+2iy/n_kh(xy)χ_k' (y/n_k)|^2 dx dy +ε ≤2∫_1^k1/z|χ_k(z)|^2 dz ∫_ℝ|-f”(t)+f(t)(1+ω^2))-2ith'(t) -ih(t)-2B t h(t)|^2 dt +8∫_1^kz|χ_k'(z)|^2 dz+ε ≤2∫_ℝ|-f”(t)+f(t)(1+ω^2)-2ith'(t)-ih(t)-2B t h(t)|^2dt+9ε .It is easy to check that∫_ℝ(2ith'(t)+ih(t)+2B t h(t)) h(t) dt=0 ,which together with the simplicity of the eigenvalue -1 establishes the existence of the solution for the differential equation-f”(t)+f(t)(1+ω^2)-2ith'(t)-ih(t)-2B t h(t)=0belonging to Dom(L) which will use in the Ansatz (<ref>). With this choice the last integral in the above estimate vanishes, which gives∫_ℝ^2|H_Sm (A)ψ_k|^2(x,y) dx dy≤ 189εconcluding this the proof of the lemma. Now we can complete the proof of the theorem. We fix a sequence {ε_j}_j=1^∞ such that ε_j↘0 holds as j→∞ and to any j we construct a function ψ_k(ε_j). As we have mentioned it is not necessary that such a sequence converges weakly to zero, but we can achieve that at no extra expense by choosing the corresponding numbers in such a way that n_k(ε_j)>k(ε_j-1) n_k(ε_j-1). The norms of H_Sm (A)ψ_k(ε_j) satisfy inequality which (<ref>) with 189ε_j on the right-hand side, which yields the desired result.For any nonzero real number μ we use the same procedure replacing (<ref>) withψ_k(x,y)=h(xy) ^iϵ_μ(y)χ_k(y/n_k)+ f(xy)/y^2 ^iϵ_μ(y)χ_k(y/n_k) ,where ϵ_μ(y):= ∫_√(|μ|)^y √(t^2+μ) dt and the functions f, χ_k are the same as above. Repeating the estimates with the modified exponential function one can check that to any positive ε one can choose the integer n_k large enough so that∂^2ψ_k/∂ y^2^-iε_μ(y)-2i B x ∂ψ_k/∂ y ^-iε_μ(y) +μψ_k^-iε_μ(y) - ^-iy^2/2(∂^2/∂ y^2(ψ_k ^-iε_μ(y)+iy^2/2) -2i B x ∂/∂ y(ψ_k ^-iε_μ(y)+iy^2/2))_L^2(ℝ^2) <εholds. Using further the identity ∂^2ψ_k/∂ x^2 ^-iε_μ(y)= ^-iy^2/2∂^2/∂ x^2(ψ_k ^-iε_μ(y)+iy^2/2) we getH_Sm (A)ψ_k-μψ_k_L^2(ℝ^2)=(H_Sm (A)ψ_k) ^-iε_μ(y) -μψ_k^-iε_μ(y)_L^2(ℝ^2)<^-iy^2/2H_Sm (A) (ψ_k^-iε_μ(y)+iy^2/2)_L^2(ℝ^2)+ε ;now we can use the result of the first part of proof to conclude the proof.In view of Theorem <ref> we could have restricted our attention to the numbers μ<√(ω^2+B^2) only. Avoiding this restriction makes sense, however, showing that in the supercritical case one can construct for any μ a Weyl sequence with the support in the vicinity of the y axis. Looking at the problem from the dynamical point of view as in <cit.>, this fact is connected with the existence of states escaping to infinity along the singular channel. §.§ The critical caseIf the two competing forces are in exact balance, λ=-2ω, the quadratic form is still positive by Proposition <ref>, hence H_Sm(A) can be defined as the Friedrich's extension of the operator initially defined on set 𝒟_0.Let λ=-2ω, then under the stated assumptions we haveσ (H_Sm (A))=σ_ess (H_Sm)=[0, ∞) . In view of Proposition <ref> it is enough to show that σ_ess (H_Sm) (A)⊃ [0, ∞). We proceed as in the previous section: to any μ≥ 0 we are again going to construct a sequence {ψ_n}_n=1^∞⊂ D(H_Sm (A)) of unit vectors, ψ_n=1, such thatH_Sm (A)ψ_n-μψ_n→ 0 as n→∞ .holds. The operator L_0=-d^2/dx^2+λδ(x) on L^2(ℝ) has a single eigenvalue equal to -1/4λ^2, hence its spectral threshold of is an isolated eigenvalue corresponding to the normalized eigenfunction h, the same as in the previous section. Given a smooth function χ with supp χ⊂[1, 2] and satisfying ∫_1^2 χ^2(z) dz=1, we putψ_n(x,y):=h(x y) ^i √(μ) yχ(y/n) ,where n∈ℕ is to be chosen later. For the moment we just note that choosing n large enough one can achieve that ψ_n_L^2(ℝ^2)≥1/√(2) as the following estimates show,∫_ℝ^2|h(xy) ^i√(μ) y χ(y/n)|^2 dx dy= ∫_n^2n∫_ℝ|h(xy) χ(y/n) |^2 dx dy =∫_n^2n∫_ℝ1/y|h(t) χ(y/n) |^2 dt dy =∫_n^2n1/y|χ(y/n) |^2 dy=∫_1^21/z|χ(z) |^2 dz≥1/2 .Nextwe are going to show that H_Sm (A)ψ_n -μψ_n _L^2(ℝ^2)^2<ε holds for a suitably chosen n=n(ε). By a straightforward computation we express ∂^2ψ_n/∂ x^2, ∂^2ψ_n/∂ y^2, and x ∂ψ_n/∂ y; in the same way as in the previous section one can check that the norms of the last two can be made as small as we wish by choosing n sufficiently large. Moreover, we have∫_ℝ^2|x^2 h(x y) ^i √(μ) yχ(y/n) |^2 dx dy≤1/n^4∫_1^2|χ(z)|^2/zdz ∫_ℝ|h(t)|^2 dt ,hence this term too can be made small. This allows us to estimate the expression in question as∫_ℝ^2|H_Sm (A)ψ_n-μψ_n|^2(x,y) dx dy=∫_ℝ^2|-∂^2ψ_n/∂ x^2- ∂^2ψ_n/∂ y^2+2i B x∂ψ_n/∂ y+B^2 x^2ψ_n+ω^2y^2ψ_n-μψ_n|^2 dx dy=∫_n^2n∫_ℝ|y^2(-h”(xy) +ω^2h(xy))χ(y/n)|^2dx dy+εfor all sufficiently large n, and using the fact that Lh=0 holds by assumption, we get from here∫_ℝ^2|H_Sm (A)ψ_n-μψ_n|^2(x,y) dx dy<ε ,which is what we have set out to demonstrate.§ SPECTRUM OF H(A) Now we pass to the `regular' version of the magnetic Smilansky-Solomyak model described by the Hamiltonian (<ref>). As before, the first question to address concerns its self-adjointness. In this case we can check that H(A) is essentially self-adjoint on C_0^∞(ℝ^2) with a reference to <cit.>: it is sufficient to find a sequence of non-overlapping annular regions A_m={z∈ℝ^2:a_m<|z|<b_m} and a sequence of positive numbers ν_m such that(b_m-a_m)^2 ν_m>K , V(z)≥ -k ν^2_m (b_m-a_m)^2for z∈ A_m and∑_m=1^∞ν_m^-1=∞ ,where K and k are positive constants independent of m. It can be seen easily that for a_m=m, b_m=m+1, and ν_m=m+1, m=0, 1, 2,…, the requirement (<ref>) is satisfied if we choose K=1/2 and k=|λ|V_∞.§.§ Subcritical case: positivity and essential spectrumAs before we show first that the operator is positive in the subcritical situation.H (A)≥0 holds provided infσ (L(V))≥0. The argument is mimicking the reasoning used in Proposition <ref>. For any u∈Dom (Q (H (A)))⊂ℋ^1(ℝ^2) one hasQ( H (A))(u)=∫_ℝ^2|i ∂ u/∂ x-B y u|^2+|∂ u/∂ y|^2+(ω^2 y^2+λ y^2 V(x y)) |u|^2 dx dy ≥∫_ℝ^2|i ∂ u/∂ x-B y u|^2+(ω^2 y^2+λ y^2 V(x y)) |u|^2 dx dy .Furthermore, the quadratic formsu(·,y) ↦∫_ℝ|i ∂ u/∂ x-B y u|^2+(ω^2 y^2+λ y^2 V(x y)) |u|^2 dxwith a fixed y correspond to the operators(id/d x-B y)^2+ω^2 y^2+λ y^2 V(x y)onℋ^1(ℝ) ,which are unitarily equivalent to y^2 L(V)≥0. As in the case of H_Sm(A) the `unperturbed' essential spectrum is preserved independently of the value the coupling constant λ may take. σ_ess (H (A))⊃[√(ω^2+B^2), ∞). As before we have to construct a Weyl sequence for any μ≥√(ω^2+B^2), in other words, to find to any ε>0 a function ϕsuch thatH (A) ϕ-μϕ_L^2 (ℝ^2)<εϕ .We employ the functions defined by (<ref>) which obviously belong to the domain of H(A). The only change on the right-hand side of (<ref>) comes now from the addition of the term λ y^2 V(x y)φ_k, α, m. Using the fact that V is by assumption compactly supported, we infer that1/2π vol (E)∫_ℝ^2y^2 |∫_E g(y-ξ B/ω^2+B^2)e^i ξ (x-α k) dξ|^2 V^2(x y) η^2(x/k) χ^2(y/k) dx dy ≤η_∞^2 χ^2_∞V_∞^2/vol (E)∫_E×{|y|≤s_0/k}y^2 g^2(y-x B/ω^2+B^2) dx dy≤s_0^2 η_∞^2 χ^2_∞V_∞^2/k^2 vol (E) ∫_E×ℝ g^2(y-x B/ω^2+B^2) dx dy=s_0^2 η_∞^2 χ^2_∞V_∞^2/k^2 ∫_ℝ g^2(z) dz .Consequently, choosing k large enough one can achieve that the above integral will be sufficiently small, which together with the inequality (<ref>) implies the validity of <ref>. The rest of the argument is the same as in Theorem <ref>. §.§ Subcritical case: essential spectrum thresholdWhile the value of λ was irrelevant in the previous theorem, it becomes important if we ask about the essential spectrum threshold. Let inf σ(L(V))>0, then the spectrum of H (A) below √(ω^2+B^2) is purely discrete. We employ Neumann bracketing combined with the minimax principle in a way similar to that used in <cit.>. By h^(±)_n (A, V) and h_0 (A, V) we denote the Neumann restrictions of operator H (A) to the stripsG^(±)_n=ℝ×{y:1+ln n<± y≤ 1+ln(n+1)},n≥ n_0 ,and G_0=ℝ× [-1-ln n_0,1+ln n_0], where n_0∈ℕ will be chosen later. It allows us to estimated the operator from below,H(A)≥(⊕_n=n_0^∞h^(+)_n (A, V)⊕ h^(-)_n (A, V))⊕ h_0 (A, V) .To prove the result we have to show first that the spectral thresholds of h_n^(±) (A, V) tend to infinity as n→∞, and secondly, that for any Λ<√(ω^2+B^2) one can choose n_0 in such a way that the spectrum of h_0 (A, V) below Λ is purely discrete. The function V is by assumption compactly supported with a bounded derivative, hence we haveV(x y)-V(x (1+ln n))=𝒪(1/nln n) , y^2-(1+ln n)^2=𝒪(ln n/n)for any (x,y) ∈ G_n^(+) and an analogous relation for G^(-)_n, and consequently,y^2V(xy)-(1+ln n)^2 V(± x(1+ln n))=𝒪(ln n/n)holds for any (x,y) ∈ G_n^(±). Moreover, for any function u∈ℋ^1(G_n) and any fixed positive ε we have∫_G_n|i ∂ u/∂ x-B y u|^2 dx dy =∫_G_n|i ∂ u/∂ x-B ln n u+B(ln n-y)u|^2 dx dy ≥∫_G_n|i ∂ u/∂ x-B ln n u|^2 dx dy -2∫_G_n√(ε)|i ∂ u/∂ x-B ln n u| B/√(ε)|ln n-y| |u| dx dy ≥(1-ε) ∫_G_n|i ∂ u/∂ x-B ln n u|^2 dx dy -B^2/ε∫_G_n(ln (n+1)-ln n)^2 |u|^2 dx dy ,and therefore(i ∂/∂ x-B y)^2≥(1-ε) (i ∂/∂ x-B ln n)^2-B^2/ε(ln (n+1)-ln n)^2 ,which together with (<ref>) implies the asymptotic inequalitiesinfσ(h_n^(±) (A, V))≥(1-ε) infσ(l_n^(±) (A, V))+𝒪(ln n/n) ,in which the operatorsl_n^(±) (A, V):=(i∂/∂ x-B ln n)^2-∂^2/∂ y^2+ω^2(1+ln n)^2+λ/1-ε (1+ln n)^2 V(± x(1+ln n))with Neumann conditions are defined on G^(±)_n.Since l_n^± (A, V)is easily seen to be unitarily equivalent to the non-magneticoperator l̃_n^±(V)=-∂^2/∂ x^2-∂^2/∂ y^2+ω^2(1+ln n)^2+λ/1-ε (1+ln n)^2 V(± x(1+ln n)) defined on the same domain, their spectra coincide. The operator l̃_n^±(V) allows the separation of variables. Since the principal eigenvalue of -d^2/dy^2 on an any interval with Neumann boundary conditions is zero, we haveinfσ (l̃_n^±(V))=infσ (l_n(V)) ,where l_n(V)=-d^2/dx^2+ω^2(1+ln n)^2+λ/1-ε (1+ln n)^2 V(± x(1+ln n)). By the change of variable, x=t/1+ln n, the last named operator is in turn unitarily equivalent to (1+ln n)^2 L_ε(V) with L_ε(V) =d^2/dt^2+ω^2+λ/1-ε V, and therefore in view of inequality (<ref>) the relation infσ (l̃_n^±(V))=(1+ln n)^2 infσ (L_ε(V)) holds, which together with the first order of perturbation-theory argument and (<ref>) concludes the proof of the discreteness of ⊕_n=1^∞ h^(+)_n (A, V)⊕ h^(-)_n (A, V).It remains to inspect the spectrum of h_0 (A, V). To proceed with the proof we need the following auxiliary result.Under our assumptionsinfσ_ess (h_0 (A, V))=infσ_ess(h̃_0 (A)) ,where h̃_0 (A) is the operator (i ∂/∂ x-By)^2-∂^2/∂ y^2+ω^2 y^2 on L^2(G_0) with Neumann boundary conditions. From the minimax principle <cit.> it follows thatinfσ_ess (h_0 (A, V))≤infσ_ess(h̃_0 (A)) .To establish the opposite inequality it is enough to check that the spectrum of h_0(A, V) is purely discrete below infσ_ess (h̃_0 (A)). Given a k∈ℕ, we introduce the operator h_1 (A,V)=(i ∂/∂ x-By)^2 -∂^2/∂ x^2+ω^2 y^2+λ y^2 V(x y)χ_{|x|≤ k} (x) for some large k∈ℕ. It differs from h_0 (A, V) by the potential term in the region {|x|>k}×ℝ, however, the potential V is compactly supported by assumption, and therefore only y∈(-s_0/k, s_0/k) must be considered and we geth_0(A, V)≥ h_1(A, V)-s_0^2 |λ| V_∞/k^2 ,hence infσ_ess (h_0 (A, V))≥infσ_ess (h_1 (A, V))+𝒪(k^-2). Since k can be chosen arbitrarily large, the identity (<ref>) would follow if we check thatσ_ess (h_1 (A,V))=σ_ess (h̃_0 (A)) .To this aim we use the stability of the essential spectrum against compact perturbations <cit.>, specifically, we check the compactness of the resolvent difference (h_1 (A, V)-z 𝕀)^-1-(h̃_0 (A)-z 𝕀)^-1 as an operator on L^2(G_0) for z belonging to both the resolvent sets of h_1 (A, V) and h̃_0 (A). Using the resolvent identity we write the difference in question as(h_1 (A, V)-z 𝕀)^-1 (h_1 (A, V)-h̃_0 (A)) (h̃_0 (A)-z 𝕀)^-1 .It is easy to realize that for any bounded 𝒰⊂ L^2(G_0) the set (h̃_0 (A)-z 𝕀)^-1𝒰 is uniformly ℋ^1 bounded. Furthermore, h_1 (A, V)-h̃_0 (A) is by construction a compactly supported potential in {|x|≤ k}×{|y|≤ n_0}, which implies its boundedness in H^1(Ω), where Ω={|x|≤ k}×{|y|≤ n_0})∩{(x, y): x y∈suppV}. Finally, using the embedding theorems for Sobolev spaces on bounded domains we conclude that (h_1 (A, V)-h̃_0 (A)) 𝒱 is compact in L^2(Ω) which implies the same also for (h_1 (A, V)-z 𝕀)^-1 ((h_1 (A, V)-h̃_0 (A)) 𝒱), and thus the claim we have set out to prove. The rest is simple: combining the preceding lemma with the inclusion (<ref>) we verify the claim of Theorem <ref>. §.§ Subcritical case: existence of the discrete spectrumAs in the previous section, proving that the spectrum below √(ω^2+B^2)) says nothing about its existence, it has to be checked separately.Let infσ (L(V))>0, then the discrete spectrum of H (A) is non-empty and contained in the interval (0, √(ω^2+B^2)). We have to construct a normalized trial function ϕ such that the corresponding value of the quadratic form Q (H(A)) will be less than √(ω^2+B^2). This time we employ the letter h to denote the normalized ground-state eigenfunction of the one-dimensional harmonic oscillator, h_osc=-d^2/dy^2+(ω^2+B^2)y^2 on L^2(ℝ), and setϕ(x, y):=1/√(k) h(y)χ(x/k) ,where χ(z) is a real-valued smooth function with supp (χ)=[-1, 1] such that∫_-1^1χ^2(z) dz=1 ,min_|z|≤1/2 χ(z)=:α>0 ,and k is a natural number to be chosen later. A straightforward computation yieldsQ (H(A))[ϕ]=∫_ℝ^2|∂ϕ/∂x|^2 dx dy +∫_ℝ^2|∂ϕ/∂y|^2 dx dy +∫_ℝ^2(ω^2+B^2) y^2|ϕ|^2 dx dy+ λ∫_ℝ^2y^2 V(x y) |ϕ^2| dx dy ,because the contribution from the terms containing the first derivatives is easily seen to vanish, and thereforeQ (H(A))[ϕ] =1/k^3∫_ℝ^2h^2(y) (χ^')^2 (x/k) dx dy +1/k∫_ℝ^2(h^')^2(y) χ^2(x/k) dx dy +1/k∫_ℝ^2(ω^2+B^2) y^2h^2(y) χ^2(x/k) dx dy +λ/k∫_ℝ^2y^2 V(x y)h^2(y) χ^2 (x/k) dx dy =𝒪(k^-2)+1/k∫_ℝ^2((h^')^2(y) +(ω^2+B^2) y^2 h^2(y)) χ^2(x/k) dx dy +λ/k∫_ℝ^2y^2 V(x y)h^2(y) χ^2(x/k) dx dy =𝒪(k^-2)+√(ω^2+B^2)/k∫_ℝ^2 h^2(y) χ^2(x/k) dx dy +λ/k∫_ℝ^2y^2 V(xy)h^2(y) χ^2(x/k ) dx dy =𝒪(k^-2)+√(ω^2+B^2)+λ/k∫_ℝ^2y^2 V(x y)h^2(y) χ^2(x/k) dx dy .We need to estimate the last term on the right-hand side of (<ref>). One has|λ|/k∫_ℝ^2y^2 V(x y)h^2(y) χ^2(x/k) dx dy=|λ|/k∫_-k^k ∫_ℝ y^2 V(x y)h^2(y) χ^2(x/k) dx dy ≥|λ|/k∫_-k/2^k/2∫_0^∞ y^2 V(x y)h^2(y) χ^2(x/k) dx dy≥α^2 |λ|/k∫_-k/2^k/2∫_0^∞ y^2 V(x y)h^2(y) dx dy=α^2 λ/k∫_0^∞∫_-k y/2^k y/2yV(t)h^2(y) dt dy≥α^2 |λ|/k∫_1^∞∫_-k/2^k/2y V(t)h^2(y) dt dy ≥α^2 |λ|/k∫_1^∞ y h^2(y) dy ∫_-k/2^k/2V(t) dt .If k is chosen large enough the above estimate implies|λ|/k∫_ℝ^2y^2 V(x y)h^2(y) χ^2(x/k) dx dy≥α^2 |λ|/k∫_1^∞ y h^2(y) dy ∫_-s_0^s_0V(t) dt ,hence in combination with (<ref>) we infer thatQ (H (A))[ϕ] ≤𝒪(k^-2)+√(ω^2+B^2)-α^2 |λ|/k∫_1^∞ y h^2(y) dy ∫_-a^aV(t) dt ,where the right-hand side is obviously less than √(ω^2+B^2) for all k large enough. §.§ The supercritical and critical casesLet us turn next to the case where the `escape to infinity' is possible.Under our hypotheses, assuming in addition that the potential V is symmetric with respect to the origin, σ(H(A))=ℝ holds if infσ(L(V))<0.The proof is completely the same as in Theorem <ref>, the only difference concerns the substitution of the function h into the normalized eigenfunction of L(V). The symmetry of the potential V is needed to guarantee the existence of the solutions of the differential equation (<ref>).Finally, in the critical case we haveσ (H(A))=σ_ess (H(A))=[0, ∞) holds provided infσ(L(V))=0.The proof repeats the almost exactly the argument leading to Theorem <ref>.§.§ AcknowledgementsThe authors are grateful to V. Lotoreichik for a useful discussion. The research has been supported by the Czech Science Foundation (GAČR) within the project 17-01706S. D.B. also acknowledges a support from the projects 01211/2016/RRC and the Czech-Austrian Grant CZ 02/2017.§.§ References 10[AGHH05]AGHH05 Albeverio S, Gesztesy F, Høegh-Krohn R and Holden H 2005 Solvable Models in Quantum Mechanics, 2nd edition (Providence, R.I.:AMS Chelsea Publishing)[BE14]BE14 Barseghyan D and Exner P 2014 A regular version of Smilansky model, J. Math. Phys. 55 042104[BE17]BE17 Barseghyan D and Exner P 2017 A regular analogue of the Smilansky model: spectral properties, Rep. Math. 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"authors": [
"Diana Barseghyan",
"Pavel Exner"
],
"categories": [
"math.SP",
"math-ph",
"math.MP"
],
"primary_category": "math.SP",
"published": "20170824123222",
"title": "A magnetic version of the Smilansky-Solomyak model"
} |
An Algorithm of Parking Planning for Smart Parking System Xuejian Zhao Wuhan UniversityHubei, ChinaEmail: [email protected] Kui Zhao Zhejiang UniversityZhejiang, ChinaEmail: [email protected] Feng Hai Wuhan UniversityHubei, ChinaTelephone: (+86) 027-68753031December 30, 2023 ============================================================================================================================================================================================================================== There are so many vehicles in the world and the number of vehicles is increasingrapidly. To alleviate the parking problems caused by that,the smart parking system has been developed. The parkingplanning is one of the most important parts of it. An effectiveparking planning strategy makes the better use of parking resources possible.In this paper, we present a feasible method to do parking planning.We transform the parking planning problem into a kind of linear assignment problem.We take vehicles as jobs and parking spaces as agents.We take distances between vehicles and parking spaces ascosts for agents doing jobs. Then we design an algorithm for this particular assignment problem and solve the parking planning problem. The methodproposed can give timely and efficient guide information to vehicles fora real time smart parking system. Finally, we show theeffectiveness of the method with experiments over some data,which can simulate the situation of doing parking planning in the real world. § INTRODUCTIONThere are so many vehicles in the world, especially in China.According to the annual report presented by J. Wang <cit.>, there are almost onebillion private cars in China till 2013. In other words, there are 20 private cars per 100 families. By contrast,the parking resources are too limited. Taking Beijing as anexample, in 2013, there are 4.075 million private cars, but the number ofparking spaces is only about 1.622 million [http://www.gov.cn/gzdt/2013-04/10/content_2374428.htm]. That is to say, on average, each private car has only 0.398 parking spaces.If we take other types of vehicles into consideration, the situation will getworse with no doubt. The problem can be alleviated via many methods. But inthis paper we focuse on the smart parking system, primarily onthe algorithm of parking planning.§.§ Related WorkT. Litman<cit.> presents a comprehensive implementation guideon parking management, including parking planning practices. The study of M. Idris<cit.> reviews the evolution of parking spaceoccupancy detection. The detection technology makes the status of the parking spaceavailable for the parking management system.S. Zeitman<cit.> invents a communication system for theparking management system. They provide a method to establish communicationbetween municipality, driver and parking spaces. S. Shaheen<cit.>documents the research and feasibility analysis for thedesign and implementation of parking management field test. Their report givesa description of the parking field test and its technology in detail.R. Lu<cit.> provides a solution for smart parking scheme through taking advantage of communicationbetween vehicules. Our method is different since we study the problem of processingparking planning on a real time smart parking system, then sending the guide information to vehicles.§.§ ContributionsIn this paper we present an algorithm to process parking planning for areal time smart parking system. First, basing on some limiting conditions inthe real world, we transform the parking planning problem which is an on-lineproblem into an off-line problem.Second, we establish the mathematical model by regarding this off-line problemas a kind of linear assignment problem. Third, we design an algorithm to solve this particular linear assignment problem.Last, we evaluate our algorithm by some simulation experiments.The experimental results show that our method is both timely and efficient.Outline The remainder of this paper is organized as follows.In Section 2 we describe the parking planningproblem with mathematical model. In Section 3 wepresent the details about the method of parking planning.In Section 4 we present the experimentalevaluation of the method. Finally, in Section 5we conclude and outline some prospects for future work.§ ESTABLISHING MATHEMATICAL MODELThe smart parking system receives parking queries one by one.If the smart parking system processes each parkingquery immediately when it arrives, the problem to be solved is an on-line problem.We can solve this on-line problem easily through greedy method. For instance,we guide the vehicle, which is querying parking space, to the nearest availableparking space. The experimental results presented inSection 5 reveal greedy method is veryinefficient in some situations and that makes it less feasible. In order to make parking planning strategy be efficient in most situations, instead of processing it immediatelywe hold parking queries in a queue for a while and the number of queries we hold is a controllable parameter. By transformingthe on-line problem into an off-line problem, we get more information andthen we can get efficient solution in most situations.Let P denote the set of all vehicles having parking query in the queue.Let S denote the set of all available parking spaces included in the smart parking system.Let D denote the set of d_ij, and d_ij is the distance between thevehicle p_i(p_i∈P) and the parking space s_j(s_j∈S).We can achieve D through many methods, such as GPS<cit.>.Let M and N be the size of P and S, respectively.So the size of D is M×N. We take vehicles as jobs and parking spaces as agents.We take distances between vehicles and parking spaces as costs for agentsdoing jobs. We save the solution in X where x_ij∈X. That is, x_ij={[0, if p_i will not be guided to s_j;;1, if p_i will be guided to s_j. ]. Let C be the total cost for all vehicles in P going to the parking spacesassigned to them by the smart parking system. That is, C=∑_i=1^M∑_j=1^Nd_ij×x_ij. We aim to make C minimum on condition that each vehicle gets exactly one parkingspace and each parking space can be assigned to only one vehicle at most. That is, {[∑_i=1^Mx_ij=1;; ∑_j=1^Nx_ij≤ 1. ]. This optimization problem is a linear assignment problem. § PROCESSING PARKING PLANNINGWhen M is equal to N, the well-known Hungarian method<cit.>, proposed byH. W. Kuhn in 1955, can solve this optimization problem.The time complexity of that is O(N^4).In 1971, N. Tomizawa<cit.> improved it to achievean O(N^3) running time. We describe the latter one asAlgorithm <ref> without detailed steps. In Algorithm <ref>, D_N×N is the distance square matrix andwe save the solution in X_N×N. The space complexity of Algorithm <ref> is O(N^2).§.§ Available Parking Spaces are EnoughIn most situations, the available parking spaces are enough to satisfy all parking queriesin the queue. That is to say, M and N satisfy M≤ N. In fact,M and N satisfy M≪N in most situations. We can simply extend P to P' by adding N-M virtual vehicles.At the same time, we extend the old distance matrix D_M× Nto a new distance square matrix D'_N×N bysetting all distances between virtual vehicles andparking spaces zero. If we have d'_ij∈ D'_N× N, that is,d'_ij={[0, if p_i'∈ P' and p_i'∉P;; d_ij,otherwise. ]. We can solve this new problem by running Algorithm <ref> on D'_N× N.H. W. Kuhn proved that the solution of this new problem includes the solutionof the original one<cit.>. However, the value of N is alwaysvery large. In practice, the value of N is almost certainly in the millions.Because its time complexity is O(N^3), the time consumed by Algorithm <ref>is too long for a real time smart parking system. If we notice that the value of M is always small and its upper bound isunder our control, we can design an approximation algorithm to solve the originalproblem. First, we select a subset of S, denoted S'. Second,we extend P to P' which has the same size with S' by adding virtual vehicles. We constructa new distance matrix D' between P' and S'. Finally, we runAlgorithm <ref> on D' and convert the result into final solution. §.§.§ Construct the Set S'Let SUB_i be the top M nearest available parking spaces for p_i. We construct the set S' from the union ofSUB_i (1≤i≤M). That is, S'=⋃_i=1^MSUB_i.We describe it as Algorithm <ref> in detail. In Algorithm <ref>, P represents the set of vehicles thathave query in the queue and D_M×N is the original distance matrix.We save the subset in S' picked from S, denoting the set of all available parking spaces.For each vehicle in P, we save the top nearest M available parking spacesin S-Set temporarily. To the best of our knowledge,we can get the top M nearest availableparking spaces by using Heap, which is a common and useful data structure.The time complexity of it is O(MN+Mlog_2N). We check whether s_j has been in S' or not by using Hash method.The time consumed by that is constant, thus it can be omitted.So the the total time cost by Algorithm <ref>is O(MN+M^2log_2N). The space complexity is O(N+M^2). §.§.§ Extend P to P' and Construct D'Let N' be the size of S' and we extend P to P' by adding N'-M virtualvehicles. The distances between virtual vehicles and parking spaces in S'are all zero. We describe the method of constructing D' inAlgorithm <ref>. In Algorithm <ref>, S' is the subset picked from the set of all availableparking spaces by algorithm <ref>, D_M×N is the original distance matrix.We save the new distance square matrix in D'_N'× N'. We use MAP_P tomap the row index in D'_N'× N' to the vehicle in P'.We use MAP_P to map the column index in D'_N'× N' to the parking space in S'.They help us to construct the new distance square matrix D'_N'× N'.The time complexity of Algorithm <ref>is O(MN') and the space complexity is O(N'^2).If we notice O(N')=O(M^2) and substitute it into O(MN') and O(N'^2),the time and space complexity are O(M^3) and O(M^4), respectively. §.§.§ Get the Final SolutionFirst, we run Algorithm <ref> on D'_N'× N' and save its solutionin X_N'×N'. It costs O(N'^3) running timeand O(N'^2) running space. If we notice O(N')=O(M^2) and substitute itinto O(N'^3) and O(N'^2), the running time and space are O(M^6) and O(M^4), respectively.Because N'≥M is obviously always right, the solution X always exists.Let X' be the final solution for the original problem. We can construct X' from Xby the method shown in Algorithm <ref>. In Algorithm <ref>, M is the number of vehiclesin set P, MAP_P and MAP_S are used to map the index in D'_N'× N',X_N'×N' is the temporary solution from Algorithm <ref>. We save the final solution in X'. The time complexity ofAlgorithm <ref> is O(MN') and the space complexity isO(M). If we notice O(N')=O(M^2) and substitute it into O(MN'),the time complexity is O(M^3). The total running time and space to process parking planning in most solutions are O(MN+M^2log_2N+M^6) and O(M^4), respectively.Because M and N satisfy M≪N in most situations,the method proposed by us is much better than the naive Hungarian Method,whose running time and space are O(N^3) and O(N^2), respectively.In practice, the size of S' denoted by N' is very close to M rather than M^2.This implied fact makes the method we proposed more efficient than what it looks like on the level of theory.We will show more details about that in Section 4. §.§ Available Parking Spaces are Not EnoughIn a few situations, the relation between M and N is M>N. In other words,the available parking spaces are not enough to satisfy all queries in the queue.In order to distribute parking spacesfairly, we should follow the principle of first come first served.So we select the top N parking queries in the queue and assign the N available parking spacesto them using Algorithm <ref>. Because thevalue of M is always small and M and N satisfy M>N, the value of N is also small in this situation.So we can get the result from Algorithm <ref> quickly enough in this special situation. Moreover, we will respond the other M-N parking queries with the information that there areno more available parking spaces.§.§ Relation with Greedy Method We notice that our method covers the greedy method. When we set M=1,our method is exactly the greedy method. However,the greedy method is very inefficient in some situations. When we set M a right value, our method is efficient in most situations.We will see that our method is controllable andthat point makes it more feasible than the simple greedy method. § EXPERIMENTAL EVALUATION OF THE ALGORITHMWe simulate different situations of parking planning in the real world byconstructing different distance matrices D_M× N. Because the running space isnot a critical problem for any method, we put our attentions on the efficiency andrunning time of the algorithm.§.§ Efficiency of Our MethodThe new method proposed by us is an approximation algorithm. So the total cost Cwill be a little higher than the global optimal solution. We run our method on many differentdistance matrices and calculate an average. Fig.1. shows the result we have got.It can be seen from Fig.1. thatthe more parking queries we hold the less waste our method has.When M and N satisfy M≪N, the waste of our method is less than 11%.It is proved that our method is efficient enough for a real time smart parkingsystem.§.§ Running Time of Our Method We run our method on an old personal laptop. The CPU of it isIntel(R) Core(TM)2 Duo CPU P8800 @ 2.66GHz and the memory of itis Kingston(R) 8.00 GB @ 1066MHz. We set the value of N as 1.6 millionto simulate the situation of parking planning in Beijing. We run our method on manydifferent distance matrices and record the average running time in TABLE 1 in seconds.As we can see, our method is quick enough for a real time smart parking system. TABLE 1 and Fig.2. also tellus that if we want to get results more quickly we shouldhold as few parking queries as possible.In the real world, one parking place often has many available parking spaces.For instance, the number of parking spaces in each parking place in Beijing is about 3 hundred.As we can see from Fig.3., such clustering phenomenon makes N', the size of S',very close to M and very far away from M^2. So the time complexityof our method is close to O(MN+Mlog_2N+M^3) rather than O(MN+M^2log_2N+M^6).This gives us the reason why the running time taken by our methodin practice is much less than what it looks like in theory.Above all, our method can give timely and efficient solutionsfor a real time smart parking system on condition that we choose a right value for M.§.§ How to Choose the Best Value for MAlthough a small value for M makes it possible for our method to get solutions quickly,it may also lead our method to the annoying instability. In other words,our method with an extremely small M is very inefficient in some special situations.To be specific, we construct a special distance matrix D as an example.When we set M=1, our method is exactly the greedy method,the total cost of it to do parking planning is 50069. When we set M=2, the totalcostis 23549. When we set M=3, the total cost is 8525. When we set M=N,that is 6, the total cost is 209, which is the minimum cost in theory.This example helps us to illustrate that the value of M should not be too small if wewant the algorithm to be efficient in most situations. D= ( 12345614916253618276412521611681256625129613224310243125777616472940961562546656) From the above, when we try to choose the best value for M, we should take the waste, running time andstability into consideration based on the real situation.§ CONCLUSIONWe have presented a novel method of parking planning for smart parking system. We transform parking planning into a kind of linear assignment problem by holding parking queries in a queue for a while.We develop a new approximation algorithm to solve this particular linear assignmentproblem. The experimental results on simulation clearly show our method is a feasiblemethod which can give timely and efficient solutions for a real time smart parkingsystem. As future work, we consider to develop methods to get an adaptive number of parking queries holden in the queue, which is denoted by M. We also consider to find ways to construct the subset of available parking spaces denoted by S' more effective. For example, we can decidethe value of M according to the operation of the smart parking system and thehistorical parking data. We can try to find new ways to construct better S'to reduce the waste and running time. All of them may also help to improve the stability.IEEEtran | http://arxiv.org/abs/1708.07932v1 | {
"authors": [
"Xuejian Zhao",
"Kui Zhao",
"Feng Ha"
],
"categories": [
"cs.DS"
],
"primary_category": "cs.DS",
"published": "20170826052639",
"title": "An Algorithm of Parking Planning for Smart Parking System"
} |
Topological Tutte Polynomial] Topological Tutte PolynomialOhio State University - Mansfield,1760 University Drive,Mansfield, OH 44906, USA [email protected] This is a survey of recent works on topological extensions of the Tutte polynomial.[ SERGEI CHMUTOV==================Mathematics Subject Classifications: 05C31, 05C22, 05C10, 57M15, 57M25, 57M30§ INTRODUCTIONWe survey recent works on topological extensions of the Tutte polynomial. The extensions deal with graphs on surfaces. Graphs embedded into surfaces are the main objects of the topological graph theory. There are several books devoted to this subject <cit.>. Cellular embeddings of graphs can be described as ribbon graphs. Oriented ribbon graphs appear under different names such as rotation systems <cit.>, maps, fat graphs, cyclic graphs, dessins d'enfants <cit.>.Since the pioneering paper of L. Heffter in 1891 <cit.> they occur in various parts of mathematics ranging from graph theory, combinatorics, and topology to representation theory, Galois theory, algebraic geometry, and quantum field theory <cit.>.For example, ribbon graphs are used to enumerate cells in the cell decomposition of the moduli spaces of complex algebraic curves <cit.>. The absolute Galois groupAut(ℚ/ℚ) faithfully acts on the set of ribbon graphs (see <cit.>and references therein). Ribbon graphs are the main combinatorial objects of the Vassiliev knot invariant theory<cit.>.They are very useful for Hamiltonicity of the Cayley graphs <cit.>.M. Las Vergnas <cit.> found a generalization of the Tutte polynomial to cellularly embedded graphs as an application of his matroid perspectives. Recently his polynomial was extended to not necessarily cellularly embedded graphs <cit.>.In 2001 B. Bollobás and O. Riordan <cit.>, motivated by some problems of knot theory, introduced a different generalization of the Tutte polynomial for ribbon graphs. For non-planar graphs, there is no duality relation for the Tutte polynomial but there is one for the Bollobás-Riordan polynomial. In <cit.>, this was proved for a one-variable specialization.J. Ellis-Monaghan and I. Sarmiento <cit.> extended it to a two-variable relation.Independently the duality formula was discovered in <cit.>.In knot theory the classical Thistlethwaite's theorem relates the Jones polynomial and the Tuttepolynomial of a corresponding planar graph <cit.>.Igor Pak suggested using the Bollobás-Riordan polynomial for Thistlethwaite type theorems. This idea was first realized in <cit.> for a special class of (checkerboard colorable) virtual links. Then it was realized for classical links in <cit.>, and for arbitrary virtual links in <cit.>.Formally all three theorems from <cit.> were different. They used different constructions of a ribbon graph from a link diagram and different substitutions in the Bollobás-Riordan polynomials of these graphs. An attempt to understand and unify these theorems led to the discovery of partial duality in <cit.> (called there generalized duality) and to a proof of an invariance of a certain specializations of theBollobás-Riordan polynomial under it.A different generalization of Thistlethwaite theorem for virtual links was found in <cit.> based on the relative Tutte polynomial of plane graphs. The equivalence of this approach to that of <cit.> was clarified in <cit.>.In 2011 V. Krushkal <cit.> found a four variable generalization of the Tutte polynomial. It can be reduced to the Bollobás-Riordan polynomial under a certain substitution. It turns out <cit.> that it also can be reduced to the Las Vergnas polynomial under a different substitution, while the Las Vergnas and the Bollobás-Riordan polynomials are essentially independent of each other. Generalizing the classical spanning tree expansion of the Tutte polynomial, Clark Butler found <cit.> an elegant quasi-tree expansion of the Krushkal polynomial.The next diagram represents various relations between these polynomials.[ ; K_G,Σ(X,Y,A,B) ][d]^[ A:=YZ^2; B:=Y^-1 ][rd]^(.7)[ X:=x-1, Y:=y-1,; A:=z^-1, B:=z ][ ; T_G,H(X,Y,ψ) ]@/_1.5pc/[rd]^[ ψ (H_F):≡ 1,; X:=x-1, Y:=y-1 ] [; BR_G(X,Y,Z) ]@/_2.5pc/[l]_-18ptZ:=(XY)^-1/2[d]_[ X:=x-1, Y:=y-1,;Z:=1 ] [; LV_G(x,y,z) ][ld]^ z:=1/y-1[; T(;x,y) ] Both the relative Tutte polynomial of <cit.> and the Las Vergnas polynomial of <cit.> may be formulated for matroids.Since the Bollobás-Riordan polynomial specializes to the relative Tutte polynomial one may expect that the relative Tutte and the Las Vergnas polynomials are also independent. This may signify the existence of a more general matroid polynomial which would be a matroidal counterpart of the Krushkal polynomial. Very recently I. Moffatt and B. Smith found such a polynomial.The Tutte polynomial has also been extended to higher dimensional cell complexes. We do not pursue this connection here, but the interested reader may see<cit.>.In <cit.> this polynomial was related to the cellular spanning trees of<cit.>, the arithmeric matroids of <cit.> and the simplicial chromatic polynomial<cit.>.§ RIBBON GRAPHS. The topological objects we are dealing with are graphs embedded into a surface. The embedding is assumed to be cellular, that means each connected component of the complement of the graph, a face, is homeomorphic to a disc. cellular embeddingembedding!cellular Considering a small regular neighborhood of the graph on a surface we come to the equivalent notion of ribbon graphs.A ribbon graphribbon graphis an abstract (not necessarily orientable) compact surface withboundary, decomposed into a number of closed topological discs of two types,vertex-discs and edge-ribbons, satisfying the following natural conditions:the discs of the same type are pairwise disjoint; the vertex-discs and the edge-ribbons intersect in disjoint line segments, each such line segment lies on the boundary of precisely one vertex and precisely one edge, and every edge contains exactly two such line segments.We consider ribbon graphs up to a homeomorphism of the corresponding surfaces preserving the decomposition into vertex-discs and edge-ribbons. It is important to note that a ribbon graph is an abstract two-dimensional surface with boundary; its embedding into the 3-space shown in pictures is irrelevant.Here are three examples.-35rg-ex14000-30rg-ex24000-30rg-ex3180045 Gluing a disc to each boundary component of a ribbon graph G we get a closed surface without boundary. Placing a vertex at the center of each vertex-disc and connecting the vertices by edges along the edge-ribbons we get a graph, the core graph of G, cellularly embedded into the surface. Thus ribbon graphs and cellularly embedded graphs are the same objects.In this paper we will use the following notation for the major parameters of a ribbon graph.∙ v(G) := |V(G)| denotes the number of vertices of a ribbon graph G;∙ e(G) := |E(G)| denotes the number of edges of G;∙ k(G) denotes the number of connected components of G;∙ r(G):=v(G)-k(G) denotes the rank of G;∙ n(G):=e(G)-r(G) denotes the nullity of G;∙ bc(G) denotes the number of connected components of the boundary of the surface of G.Only the last parameter bc(G) is a topological one. All the others parameters are standard in graph theory.A spanning subgraph F of a ribbon graph G isa ribbon graph consisting of all the vertices of G and a subset of the edges of G.§ THE BOLLOBÁS–RIORDAN POLYNOMIAL.The Bollobás–Riordan polynomial, originally defined in <cit.>, was generalized to a multivariable polynomial of weighted ribbon graphs in <cit.>. We will use a slightly more general doubly weightedBollobás–Riordan polynomial of a ribbon graph G with weights (x_e,y_e) of an edge e∈ G. The doubly weighted Tutte polynomial is useful in connection with the knot theory<cit.>. Double weights also allow to specialize the Bollobás–Riordan polynomial to different signed versions of the Tutte polynomial, see the Section <ref>.Bollobás–Riordan polynomialBR_G(X,Y,Z):=∑_F⊆ G (∏_e∈ Fx_e)(∏_e∉Fy_e) X^r(G)-r(F)Y^n(F)Z^k(F)-bc(F)+n(F),where the sum runs over all spanning subgraphs F.The original Bollobás–Riordan polynomial of <cit.> used an additional variable w responsible for orientability of the ribbon subgraph F. We set w=1 and replace their x-1 by our X, but we introduce weights of edges.Note that the exponent k(F)-bc(F)+n(F) of the variable Z is equal to 2k(F)-χ(F), where χ(F) is the Euler characteristic of the surface F obtained by gluing a disc to each boundary component of F. For orientable F, it is twice the genus of F. For non-orientable F, according to theclassical classification theorem for surfaces, F can be represented by a connected sum of several real projective planes; the number of them is equal to the exponent of Z. In particular, for a planar ribbon graph G (i.e. when the surface G has genus zero) theBollobás-Riordan polynomial BR_G does not contain Z.In this case, and if all weights are equal to 1, it is equal to the classical Tutte polynomialT(;x,y) of the core graphof G with shifted variables X=x-1, Y=y-1:BR_G(x-1,y-1,Z) = T(;x,y) .Similarly, a specialization Z=1 (and x_e=1, y_e=1 for all edges e) of the Bollobás-Riordan polynomial of an arbitrary ribbon graph G gives the Tutte polynomial of the core graph:BR_G(x-1,y-1,1) = T(;x,y) .So one may think of the Bollobás-Riordan polynomial as a generalization of the Tutte polynomial to graphs cellularly embedded into a surface and capturing topological information of the embedding. Let the following ribbon graph G have all weights equal to 1.[ 8rg-ex32(7,-5)G50305 8rgBBB4500 8rgBBA450013rgBAB450013rgBAA4500; (k,r,n,bc) term ofBR_G(1,1,1,2) Y(1,1,0,1) 1(1,1,0,1) 1 (2,0,0,2) X-10pt(0,25); 8rg-ex3248405 8rgABA480013rgAAB480013rgAAA4800;2-5 (1,1,2,1) Y^2Z^2 (1,1,1,1) YZ (1,1,1,1) YZ (2,0,1,2) XYZ-10pt(0,25) ] BR_G(X,Y,Z) = Y+2+X+Y^2Z^2+2YZ+XYZ §.§ Properties.One can easily prove BR_G={[x_e· BR_G/e + y_e· BR_G-e ;; x_e· BR_G/e + y_eX· BR_G-e ; x_eY· BR_G/e+y_e· BR_G-e ;x_eYZ· BR_G/e+y_e· BR_G-e]. BR_G_1⊔ G_2 = BR_G_1· BR_G_2, Here a trivial loop means a loop such that the deleting it and cutting its vertex-disc along a chord connecting the endpoints of the loop separates the the ribbon graph. §.§ A signed version of the Bollobás-Riordan polynomial.Signed graphs are weighted graphs with single weights of ±1 on their edges. They are very important in particular in knot theory. There is a vast literature on signed graphs<cit.>. In <cit.>L. Kauffman introduced a generalization of the Tutte polynomial to signed graphs. The signed version of the Bollobás-Riordan polynomial was introduced in <cit.>and used in <cit.>(a version of it was also used in <cit.>).This signed version can be obtained by changing the sign weights to double weights and using the Bollobás-Riordan polynomial above. We set the weights of positive edges to 1, x_+:=y_+:=1, and the weights of negative edges to x_-:=(X/Y)^1/2, y_-:=(Y/X)^1/2. Thus the signed version becomesBR_G(x,y,z) = ∑_F ∈ G X^r(G)-r(F)+s(F) Y^n(F)-s(F)Z^k(F)-bc(F)+n(F) , Bollobás–Riordan polynomial!signed versionwhere s(F) := e_-(F)-e_-( F)/2, e_-(F) denotes the number of negative edges in F, andF = G-F is the complement to F in G, i.e. the spanning subgraph of G with exactly those (signed) edges of G that do not belong to F.As above, specializing to Z=1 gives Kauffman's Tutte polynomial of signed graphs <cit.>. The book of C. Godsil and G. Royle <cit.>introduced a slightly different signed version of the Tutte polynomial, R(M;α,β,x,y), following a result of K. Murasugi <cit.>. It is formulated in terms of signed matroids. Its application to the cycle matroid of a ribbon graph also can be obtained as a specialization Z:=1, X:=x, Y:=y of our double weighted Bollobás-Riordan polynomial with weights of positive edgesx_+:=β, y_+:=α, and weights of negative edgesx_-:=α, y_-:=β. §.§ The dichromatic version of the Bollobás-Riordan polynomial. 𝐛 The dichromatic version of the Tutte polynomial Z_G(a,b) (see for example<cit.>),also known as the partition function of the Potts model in statistical mechanics, can be defined as a generating function of vertex colorings of a graph G into a colors:Z_G(a,b) := ∑_c∈ Col(G)(1+b)^ ,where Col(G):={c:V(G)→{1,…,a}} is the set of colorings, and an edge e is colored properly by c if its endpoints are colored in different colors. Also, it can be expressed as a generating function of all spanning subgraphs:Z_G(a,b) = ∑_F⊆ E()a^k(F) b^e(F).The Tutte polynomial is equivalent to the dichromatic polynomial because of the identities[ Z_G(a,b)=a^k(G)b^r(G) T(G;1+ab^-1,1+b) ;; T(G;x,y)= (x-1)^-k(G)(y-1)^-v(G) Z_G((x-1)(y-1),y-1) . ] A multivariable version of dichromatic polynomial was introduced in <cit.>and used in <cit.>.The multivariable Tutte polynomial also appears as a very special case of Zaslavsky's colored Tutte polynomial of a matriod<cit.>.It can be defined as Z_G(a,):=∑_F⊆ E(G) a^k(F)∏_e∈ Fb_e ,The sum runs over all spanning subgraphs of G, which we identify with subsets F of E(G); :={b_e} is the set of variables (weights).The multivariable dichromatic polynomial was generalized to ribbon graphs in<cit.>asZ_G(a,,c):=∑_F⊆ E(G) a^k(F)(∏_e∈ Fb_e) c^bc(F). Bollobás–Riordan polynomial!dichromatic versionIt is equivalent to the Bollobás-Riordan polynomial BR_G(X,Y,Z) due to the relationsZ_G(a,,c)= (ac)^k(G)BR_G(ac,c,c^-1) , BR_G(X,Y,Z)= (∏_e∈ E(G)y_e) (YZ)^-v(G)X^-k(G) Z_G(XYZ^2,{x_eYZ/y_e},Z^-1) ,where on the right-hand side of the first equation we use the Bollobás-Riordan polynomial with weights x_e:=b_e, y_e:=1. §.§ The arrow version of the Bollobás-Riordan polynomial. The arrow version of the Jones polynomial was introduced in <cit.> as far reaching generalization of the classical Jones polynomial for virtual links. The Thistlethwaite-type theorem for this polynomial was obtained in <cit.>.An arrow ribbon graph is a ribbon graph together with an arrow structure on it which is a (possibly empty) set of arrows tangent to the boundaries of the (vertex- and edge-) discs of the decomposition. Two arrow graphs are equivalent if there is a homeomorphism between the corresponding surfaces respecting the decompositions and the orientations of the arrows. The endpoints of segments along which the edges are attached to the vertices divide the boundaries of (vertex- and edge-) discs into arcs. The arrows may slide along these arcs but may not slide over the end-points of the arcs or each other. Each arc may contain several arrows. So, if there are several arrows on an arc, then only their order on the arc is relevant, not their actual position on the arc. Here are some examples.-42ag-ex14000-34ag-ex24000-42ag-ex375040 #1BR_#1 In the presence of an arrow structure on a ribbon graph G we can extend the Bollobás-Riordan polynomial to the arrow Bollobás-Riordan polynomial<cit.>adding new variables K_c reflecting the arrow structure:G:=∑_F⊆ E(G)(∏_e∈ Fx_e) (∏_e∉Fy_e)X^r(G)-r(F)Y^n(F)Z^k(F)-bc(F)+n(F)∏_f∈∂(F) K_c(f). Bollobás–Riordan polynomial!arrow versionHere the independent variables K_c(f) are assigned to each boundary component withc(f) being equal to half of the number of arrows along the boundary component f remaining after cancellations of all pairs of arrows which are next to each other and pointing in the same direction.-25ar-can1502030 -2toto2500K_1-25ar-can2502030 -2toto2500K_1/2-25ar-can3502030 -2toto2500K_2We set K_0=1. One may note that whenever the number of arrows is odd on a boundary component, the associated variable is always K_1/2. Thus the arrow Bollobás-Riordan polynomial G becomes a polynomial in infinitely many variablesx_e, y_e, X, Y, Z, K_1/2, K_1, K_2, …. Note that, for a concrete graph G only finitely many K's appear in G. For the arrow graph G shown on the leftmost column in the table, there are eight spanning subgraphs. Their parameters and the corresponding monomial in K's are shown. [ 8ag-ex3(23,30)e_1(40,24)e_2(40,-5)e_356305 8agBBB5200 8agBBA520015agBAB520015agBAA5200; (k,r,n,bc)(1,1,1,2)(1,1,0,1)(1,1,0,1)(2,0,0,2)(0,12); ∏ K_c(f)K_1^2K_1K_1K_1/2^2; 8agABB56400 8agABA560010agAAB560010agAAA5600;2-5(1,1,2,1)(1,1,1,1)(1,1,1,1)(2,0,1,2)(0,12); K_1K_11K_1/2^2 ][G= y_1x_2x_3YK_1^2+y_1y_2x_3K_1+y_1x_2y_3K_1+ y_1y_2y_3XK_1/2^2;+x_1x_2x_3Y^2Z^2K_1+x_1y_2x_3YZK_1+x_1x_2y_3YZ+x_1y_2y_3XYZK_1/2^2 . ] § THE RELATIVE TUTTE POLYNOMIAL OF PLANAR GRAPHS WITH DISTINGUISHED EDGES.A relativegraph is agraph G with a distinguished subsetH ⊆ E(G) of edges. The edges H are called the 0-edges of G. Edges in E(G)\ H will be referred to as regular edges.relative graphWe will use only plane relative graphs. The medial graph M(G), see for example <cit.> or<cit.>, of a ribbon graph G is a 4-valent graph embedded in the same surface as G and constructed in a standard way. Consider the core graphof G as naturally embedded in G. Place a vertex at the center of each edge of . These will be the vertices of M(G). The centers of edges ofadjacent in the cyclic ordering around a vertex ofare connected by arcs following the boundary of G and not intersecting . These will be the edges of M(G).medial graphHere is an example of the construction for a plane graph. -60med22502070The medial graph can be considered as an image of an immersion of several circles into the plane having only double points at the vertices of M(G) as intersections (and self-intersections). The number of these circles is denoted by δ(G).The last ingredient in definition of the relative Tutte polynomial is a function ψ(G) of two variables d and w defined for plane ribbon graph G as followsψ(G):=d^δ(G)-k(G)w^v(G)-k(G) .This function is block-invariant, which means that its value on a one-point join of two graphs G_1 and G_2 does depend on the choice of identifying vertices of G_1 and G_2.Let G be a relative plane graph with the distinguished set of 0-edges H. We define the relative Tutte polynomial relative Tutte polynomial by the equationT_G,H(X,Y,ψ):=∑_F ⊆ G∖ H (∏_e ∈ Fx_e) (∏_e ∈Fy_e) X^k(F∪ H)-k(G)Y^n(F)ψ (H_F) ,here, abusing notation, we use F, H, F ∪ H etc. both for denoting the subsets of edges and the spanning subgraph of G with those edges;F:=G ∖ (F∪ H); andH_F is the plane graph obtained from F ∪ H by contracting all edges of F.Remarks. 1. The relative Tutte polynomial was introduced by Y. Diao and G. Hetyei in <cit.>,who used the notion of activities to produce the most general form of it. The all subset formula we use was discovered by a group of undergraduate students (M. Carnovale, Y. Dong, J. Jeffries) at the Ohio State University summer program “Knots and Graphs" in 2009. However, similar expressions may be traced back to L. Traldi<cit.>for the non-relative case, and to S. Chaiken <cit.>for the relative case of matroids. 2. The function ψ in <cit.> can be obtained from ours by the substitution w=1. 3. Another difference with <cit.> is that we are using a doubly weighted version of the relative Tutte polynomial with weights (x_e,y_e) of an edge e∈ G∖ H.4. In the process of constructing the graph H_F by contracting the edges of F in F ∪ H, we may come to a situation when we have to contract a loop. Then the contraction of a loop actually means its deletion. Since G and F ∪ H are plane graphs, then the graphH_F is also embedded in the plane.5. While the medial graph of the planar graph H_F depends on the embedding of H_F in the plane, the number δ(H_F) does not (see <cit.>). It depends only on the abstract graph H_F. This was proved in the paper <cit.>,which showed that δ(H_F)= log_2|T(H_F;-1,-1)| + 1.§.§ From ribbon graphs to relative plane graphs. Here we describe a construction from <cit.>relating ribbon graphs and planar relative graphs.The manner in which we draw ribbon graphs suggests that we consider a projection π: G→^2 with singularites oftwo types. The first occurs when two edge-ribbons cross over each other. The second occurs when an edge ribbon twists over itself.Away from these singularities the projection is one-to-one. In fact we use this idea on the medial graph M(G).-30singular2(45,60)R(25,40)π(-5,-10) Possible singularities of the projection4000-20singular(60,50)R(30,30)π605045The image B:=π(M(G)) may then be considered as a regular 4-valent planar graph whose vertices are divided into two types. The vertices which are images of vertices of M(G) will be called regular vertices, and the vertices that arise from the singularities of the projection will be called 0-vertices.The faces, the connected components of the complement of B in the plane, can be colored in white and green in checkerboard manner. Place a vertex on each green face and connect the vertices corresponding to the faces sharing a vertex of B by an edge through the corresponding vertex of B. We obtain get a plane graphC(B) whose medial graph is B, M(C(B))=B. This procedure is similar to taking the plane dual, only now we place a vertex only on green faces, not on every face, and th edges of C(B) go not across the edges of B but across the vertices of B. We consider C(B) as a relative plane graph whose 0-edges are those which pass through the0-vertices of B.The constructed relative plane graph C(B) clearly depends on the projection π and on the position of the vertices of the medial graph on the edge-ribbons. However the invariants we will work with will not be affected by this ambiguity. The figure below shows two possibilities for C(B) depending on the position of a vertex of the medial graph M(G).-100rg1a(225,65)B(285,65)C(B)(40,22)π(G)(137,75)M(G)(53,135)G(33,100)π32035100In this figure the 0-edges of C(B) are drawn as dashed lines.§.§ From relative plane graphs to ribbon graphs.Conversely, from a relative plane graph C we may construct a ribbon graph G. Consider the spanning subgraph H of C whose edges are the 0-edges of C. Construct its medial graph M(H). Consider it as an immersion of a collection of (̣H) circles with only transverse double points as singularities.We construct a ribbon graph G with (̣H) vertices corresponding to the circles of the medial graph M(G). The edges of G correspond to regular edges of C. They glued to thevertex-discs according to the intersection of a small neighborhood of a regular edge with the arcs of the medial graph. Namely, we can label small arcs on M(G) at which the regular edgesintersect M(G) by arrows following counterclockwise orientation of the plane. Then we glue ribbons along the corresponding arcs of the vertex-discs.-80pg-rg1(38,10)C(173,4)H320070 -80pg-rg2(314,16)G320080§.§ Relation to the Bollobás–Riordan polynomial. Suppose G is a ribbon graph, and C is a relative plane graph associated to a projection of G with H as a spanning subgraph corresponding to the zero-edges. Or, equivalently, assume C is a relative plane graph and G is the ribbon graph arising from C. Then under the substitution w=√(X/Y), d=√(XY),X^ Y^Ṯ_C,H(X,Y,ψ)=BR_G( X,Y,1/√(XY) ) ,where :=k(C)-k(G)-$̱ and:̱=-1/2(v(G)-v(C)). Remarks.1. It is an interesting consequence of this theorem that the specialization(w=√(X/Y), d=√(XY)) of the relative Tutte polynomial does not depend on the variouschoices made in the construction of the relative plane graph. It is not difficult to describe a sequence of moves on relative plane graphs relating the graphs with different choices of the regular edges. It would be interesting to find such moves for different choices of the projectionπand, more generally, the moves preserving the relative Tutte polynomial. 2. The construction ofCfromGand back can be generalized to a wider class of projectionsπ. We can require that only the restriction ofπto the boundary ofGbe an immersion with onlyordinary double points as singularities. The theorem holds in this topologically more general situation. However, from the point of view of graph theory it is more natural to restrict ourselves to the class of projections above.§.§ Dual relative plane graphs LetCbe a relative plane graph with 0-edgesH. Thedual ofC, denotedC^*is formed by taking the dual ofCas a plane graph, and labeling the edges ofC^*which intersect 0-edges ofCas the 0-edges ofG^*. Note that for relative plane graphs(C^*)^*=C, as with usual planar duality.relative graph!dualThe following theorem generalizes the classical relation for the Tutte polynomials of dual plane graphs,T(C;x,y)=T(C^*;y,x), to relative plane graphs. Under the substitution w=√(X/Y), d=√(XY), we have X^a(C,H)Y^b(C)T_C,H(X,Y,ψ)= Y^a(C^*,H^*)X^b(C^*)T_C^*,H^*(Y,X,ψ)with the correspondence on the edge weights being x_e=y_e^*, y_e=x_e^*, where e^* is the edge of C^* that intersects e, anda(C,H) = (|E(C ∖ H)|-v(C))/2+k(C) , b(C) = v(C)/2 . § THE LAS VERGNAS POLYNOMIAL. 𝒞 M. Las Vergnas <cit.> came to his polynomials of graphs on surfaces through his general approach to matroid perspectives. It turns out thatfor a (ribbon) graph on a surface with the edge setE, there is a geometric (Poincaré) dual graphG^*. For this graph there is a dual matroid to its cycle matroid,((G^*))^*, which is the bond matroid ofG^*. Then the natural bijection of the ground sets((G^*))^*→(G)forms a matroid perspective according to <cit.>. The Tutte polynomial of a perspectiveM→M'is a polynomialT_M→M'(x,y,z)in three variables defined byT_M→ M' := ∑_F⊆ M (x-1)^r_M'(E)-r_M'(F) (y-1)^n_M(F) z^(r_M(E)-r_M(F))-(r_M'(E)-r_M'(F)),where in the rank and nullity functions, we use the subscript to indicate at which matroid the function fuction is considered in. Properties (<cit.>). The usual Tutte polynomial of matroidsMandM'can be recovered from the Tutte polynomial of matroid perspective in the following ways:[T(M;x,y) = T_M→M(x,y,z) ;; T(M;x,y) = T_M→M'(x,y,x-1) ;; T(M';x,y) = (y-1)^r(M)-r(M') T_M→M'(x,y,1/y-1) . ]TheLas Vergnas polynomial of G, LV_G(x,y,z),is the Las Vergnas polynomial of the matroid perspective ((G^*))^*→(G).Las Vergnas polynomialIfGis a planar graph, then the Whitney planarity criteria claims that the matroid((G^*))^*is isomorphic to(G). So the matroid perspective is the identity, and then the Las Vergnas polynomial is equal to the classical Tutte polynomial ofG. Let G be a graph with one vertex and two loops embedded into a torus as shown. Then the dual graph G^* is isomorphic to G. -20rg-torus(0,45)G10000-20rg-torus-d(80,50)G^*1003525 In this case the bond matroid M=((G^*))^* has rank 2, and the cycle matroid M'=(G) has rank 0.For any subset F: r_M(F)=|F|, n_M(F)=0, and r_M'(F)=0. We have LV_G(x,y,z)=z^2+2z+1 .§ THE KRUSHKAL POLYNOMIAL. Σϰ̨ im The most general topological polynomial for graphs on surfaces was discovered by V. Krushkal in his research in topological quantum field theories and the algebraic and combinatorial properties of models of statistical mechanics <cit.>. Originally V. Krushkal<cit.> defined his polynomial for orientable surfaces only. Here we follow the exposition of Clark Butler <cit.> who extended the definition to non-orientable surfaces. Let G be a graph embedded in a surface . The embedding does not have to be cellular and the surfacedoes not have to be orientable. Then the Krushkal polynomial is defined byK_G,(X, Y, A, B) := ∑_F⊆ G X^k(F)-k(G) Y^k(∖ F)-k() A^s(F)/2 B^s^⊥(F)/2, Krushkal polynomial where the sum runs over all spanning subgraphs considered as ribbon graphs, and the parameterss(F) and s^⊥(F) are defined by s(F):= 2k(F)-χ(F), s^⊥(F):= 2k(∖ F)-χ(∖ F)Here χ(F) (resp. χ(∖ F)) stands for the Euler characteristic of the surface F(resp. ∖ F) obtained by gluing a disc to each boundary component of the ribbon graph F (resp.∖ F).As was outlined in Section <ref>, for orientable surfaces the parameters s ands^⊥ are equal to twice the genus of the corresponding surfaces, and for non-orientable surfaces they are equal to the number of Möbius bands one has to glue into k(F) and∖ F spheres to obtain a homeomerphic surfacesRemarks.1.In the orientable case V. Krushkal<cit.> used a different expression for the exponent$̨ of Y, as the dimension of the kernel of the map of the first homology groups(̨F)=((H_1(F;) → H_1(;)))induced by inclusion F↪. It was noted in <cit.> that this dimension is equal to(̨F)=k(∖ F)-k(). C. Butler observed in <cit.> that considering homology groups with coefficients in _2:=/2 leads to a topological interpretation of the exponents in the Krushkal polynomial for a general (not necessarily orientable) surface :[k(∖ F)-k() = ((H_1(F;_2) → H_1(;_2))),;s(F) =H_1(F;_2),;s^⊥(F) =H_1(∖ F;_2). ] 2. In the orientable case V. Krushkal <cit.> indicated that the parameters s(F) and s^⊥(F) have another interpretation in terms of the symplectic bilinear form on the vector space H_1(;) given by the intersection number. For a given spanning subgraph F, let V be its image in the homology group:H_1(;)⊃ V := V(F) := (H_1(F;) → H_1(;)) .For the subspace V we can define its orthogonal complement V^⊥ in H_1(;) with respect to the symplectic intersection form. Thens(F)=(V/(V∩ V^⊥)) ,s^⊥(F)=(V^⊥/(V∩ V^⊥)) .§.§ Relation to the previous polynomials.Let G be a ribbon graph, or equivalently a cellular embedding of the graph in a surface:=G. The Tutte polynomial is a specialization of the Krushkal polynomial:T(G;x,y)=(y-1)^s()/2 K_G,(x-1, y-1, y-1, (y-1)^-1). The unweighted (with all edge weights equal to 1, x_e=y_e=1) Bollobás-Riordan polynomial is a specialization of the Krushkal polynomial:BR_G(X,Y,Z)=Y^s(G)/2 K_G,(X,Y,YZ^2,Y^-1). The Las Vergnas polynomial is a specialization of the Krushkal polynomial:LV_G(x,y,z) = z^s(G)/2 K_G,(x,y,z^-1,z). §.§ Properties.K_G,={[ K_G/e, + K_G-e,;; (1+ X)· K_G/e; (1+ Y)· K_G-e; ]. K_G_1⊔ G_2,_1⊔_2 = K_G_1,_1· K_G_2,_2, §.§ The contraction/deletion properties of these polynomials. The contraction/deletion properties for BR_G, for K_G,, and for LV_G, are not quite the same. The problem arises when deletion of an edge of a ribbon graph changes its genus. The genus might decrease by 1 with the removal of an edge. For example, if we delete a loop from the ribbon graph G of Example <ref>, then the resulting graph with a single loop will have genus zero. So, while in the Bollobás-Riordan approach it is considered as a ribbon graph embedded into a sphere, in the Krushkal approach it is still embedded into the torus. We cannot apply the substitution (<ref>) to that graph since its embedding on the torus is no longer cellular. Thus the Krushkal polynomial does not satisfy the contraction/deletion property in the sense of the Bollobás and Riordan polynomial. The Las Vergnas polynomial LV_G,(x,y,z) does not satisfy either the contraction/deletion property in the sense of Bollobás and Riordan. The contraction/deletion property for it discussed in <cit.>. This is an example of a calculation of the three polynomials. Here G is a graph on the torus with two vertices and three edges a, b, and c considered as a ribbon graph. Its dual G^* has one vertex and three loops. We use the same symbols a, b, c to denote the corresponding edges of G^* as well.G=-22G-torus(52,33)a(15,15)a(46,8)b(70,24)b(28,13)c(85,35)c903035=-13cG-rg(37,32)a(15,-7)b(33,-6)c5000 G^*=-22Gd-torus(28,50)a(72,16)b(7,26)c903535The matroid M'=(G) is of rank 1, and for any nonempty subset F, r_M'(F)=1. The cycle matroid (G^*) of the dual graph is of rank zero because G^* has only loops. So its dual M=((G^*))^* has rank 3, all subsets F are independent and r_M(F)=|F|. The next table shows the values of various parameters and the contributions of all eight subsets F⊆{a,b,c} to the three polynomials.[ F∅{a}{b}{a,b}{c}{a,c}{b,c}{a,b,c};k(F)21111111;——— (̨F)00000000;——— s(F)00000002;——— s^⊥(F)22202000;———(20,0)60pt90Krushkal K_G, XBBB1B11A;r_M(F)01121223;———r_M'(F)01111111;——— n_M(F)00000000; ———(20,0)50pt90Las Vergnas LV_G (x-1)z^2z^2z^2zz^2zz1;k(F)21111111;——— n(F)00010112;———bc(F)21121221;———(20,0)50pt900pt0 BR_GX11Y1YY Y^2Z^2;]ThusK_G,=3+3B+XB +A, LV_G=3z+3z^2+(x-1)z^2+1, BR_G=3Y+3+X+Y^2Z^2.One can readily confirm the relations (<ref>) and (<ref>) from here.Now if we contract the edge c, the graph G/c will still be cellularlyembedded in the same torus . The right four columns of the table above give the following polynomials:K_G/c,=B+2+A, LV_G/c=z^2+2z+1, BR_G/c=1+2Y+Y^2Z^2.Meanwhile if we delete the edge c, then K_G-c,=XB+2B+1 .But the graph G-c is not cellularly embedded into the torusany more. Thus the Las Vergnas and the Bollobás-Riordan polynomials are not defined for it. The ribbon graph G-c, after gluing discs to its two boundary components, results in the sphere G-c=S^2. Thus the graph G-c embeds cellularly into the sphere S^2. For this embedding we haveK_G-c,S^2=X+2+Y, LV_G-c=(x-1)+2+(y-1), BR_G-c=X+2+Y.Therefore K_G,=K_G-c,+K_G/c, BR_G=BR_G-c+BR_G/c ,butK_G,≠K_G-c,S^2+K_G/c, LV_G≠LV_G-c+LV_G/c . §.§ Duality for the Krushkal polynomial.The classical relation, T(G;x,y)=T(G^*;y,x), for the Tutte polynomials of dual plane graphs can be generalized to a relation for the Krushkal polynomial of arbitrary ribbon graphs.Let G be a ribbon graph naturally embedded into the surface =G, and G^*⊂ be its Poincaré (geometric) dual ribbon graph.ThenK_G,(X, Y, A, B)=K_G^*,(Y, X, B, A) . Krushkal polynomial!duality § QUASI-TREES. A quasi-tree is a ribbon graph with one boundary component. quasi-treeQuasi-trees, as well as a notion of activities relative to a spanning quasi-tree, wereintroduced and used for a quasi-tree expansion of the Bollobas-Riordan polynomial in<cit.>Their results were generalized to non-orientable ribbon graphs in <cit.>(see also an unpublished manuscript <cit.>). We denote the set of all spanning quasi-trees of a given ribbon graph G by 𝒬_G. If G is a plane graph of genus zero, then a spanning quasi-tree is a spanning tree. But for non-plane graphs the set of spanning quasi-trees is bigger than the set of spanning trees. In Example <ref>, the whole graph consisting of a vertex and two loops on a torus is a quasi-tree but, of course, not a tree. The table in Example <ref> shows that the ribbon graph G has four spanning quasi-trees {a}, {b}, {c}, and {a,b,c}. §.§ Quasi-tree activities. Let ≺ be a total order on edges E(G) of a ribbon graph G, and Q be a spanning quasi-tree of G. Tracing the boundary component of Q we will get a round trip passing the boundary arcs of each edge-ribbon twice, on opposite sides of a ribbon separating it from faces for edges in Q and on opposite sides of a ribbon separating it from vertices for edges not in Q. This can be encoded in a chord diagram C_G(Q) consisting of a circle corresponding to the boundary of Q with pairs of arcs corresponding to the same edge-ribbon connected bychords.Thus the set of chords inherits the total order ≺. An edge is called live if the corresponding chord is smaller than any chord intersecting it relative to the order ≺.Otherwise it is called dead. edge!live/dead activities! w.r.t. quasi-tree quasi-tree!activitiesFor a plane graph G a spanning quasi-tree is a spanning tree and the notion of live/dead consides with the classical Tutte's notion of active/inactive. But for the higher genus the notions are different, so we follow the terminology live/dead of <cit.>,where a reader can find further discussion and examples. Also, this notion is different from the embedding-activities introduced by O. Bernardi in <cit.>. In Example <ref> of a single vertex and two loops there are two spanning quasi-trees consisting of a vertex without edges and the whole graph. Both of them have a chord diagram depicted on the left in the picture below. In Example <ref> there are four spanning quasi-trees. All of them have the same chord diagram, depicted on the right in the picture below.-10cd-240300-10cd-34000In either case there is only one live edge, the smallest in the order ≺. It could be either internal or external depending on whether it belongs or does not belong to the spanning quasi-tree.Here we consider a more complicated example of a non-orientable ribbon graph G following<cit.>.-20rg-ex4(10,20)1(60,37)2(60,-5)3(116,50)41203020The edges are labeled by natural numbers, so their order is the obvious one:1≺ 2≺ 3≺ 4.There are four spanning quasi-treesQ_{2}, Q_{3}, Q_{2,3}, and Q_{2,3,4}, where the subscript indicates the set of edges included in the quasi-tree. Their chord diagrams look like this:C_G(Q_{2})= -17cd-4-q2(18,28)1(28,22)2(11,29)3(26,12)4402030,C_G(Q_{3})=-17cd-4-q3(18,28)1(32,21)2(11,30)3(20,4)44000, C_G(Q_{2,3})=-17cd-4-q23(17,28)1(28,20)2(11,27)3(4,12)440020,C_G(Q_{2,3,4})=-17cd-4-q234(17,28)1(28,19)2(11,27)3(14,8)44000.Thus 1 is always an externally live edge, 2 is also live, and 3 is internally live for Q_{2,3,4}. We depict the chords corresponding to dead edges by dashed lines. Also some chords on the pictures are marked by a cross. These correspond to non-orientable edges according to the next definition.An edge of a non-orientable ribbon graph G can be classified as orientable ornon-orientable relative to a spanning quasi-tree Q for G. Choose an orientation on the boundary circle ∂ Q of Q and on the boundary of each edge-ribbon considered as a topological disc. Then, tracing ∂ Q in the direction of the orientation we meet exactly two arcs of each-ribbon edge. If the orientations of these two arcs are coherent, either both along the tracing, or both against the tracing, then we call the edge orientable relative toQ. Otherwise we call it non-orientable. edge!orientable/non-orientableIt is easy to see that this property does not depend on the choice of the orientations. Essentially the chord diagram C_G(Q) depicts a partial dual ribbon graph G^E(Q) in the sense of <cit.>. All edges of G^E(Q) are loops. For loops we can define orientability in the usual topological sense which coincides with our definition above. For the graph G in Example <ref> consider a spanning quasi-treeand choose the orientations as follows.-20rg-ex4(10,20)1(60,37)2(60,-5)3(116,50)41204020 -2toto2500 -20rg-ex4or(14,20)1(60,40)2(60,-4)3(128,50)413000We observe that the two relevant arcs of edge 1 are oriented coherently to the direction of ∂ Q_{2,3}. For the other edges this is not true. So 1 is an orientable edge relative to Q_{2,3}, and the other three edges are not.Thus each edge has three bits of additional information relative to a spanning tree Q: internal/external, live/dead, and orientable/non-orientable. This information is used in the quasi-tree expansion of the Krushkal polynomial in next subsection. §.§ Quasi-tree expansion of the Krushkal polynomial. With each spanning quasi-tree Q of a ribbon graph G we associate∙ a spanning ribbon subgraph F(Q) by deleting the (internally) live orientable edges of Q;∙ an abstract (not embedded) graph Γ(Q) whose vertices are the connected components of F(Q) and whose edges are the internally live orientable edges of Q.We will also need a dual construction. For a ribbon graph G, regarded as a graph cellularly embedded into the surface =G, we consider the Poincaré dual graph ribbon graph G^*. A spanning subgraph F for G determines a spanning subgraph F^⋆ containing all edges of G^* which do not intersect edges of F. Note that F^⋆ is not a dual ribbon graph for F, its edge set is rather the complement of the set of edges of G^* corresponding to the edges of F. The asterisk applied to an entire graph means the dual graph, but the star applied to a subgraph means the spanning subgraph given by this construction.The spanning subgraphs F and F^⋆ have common boundary and their gluing along this common boundary gives the whole surface . In particular, for a spanning quasi-tree Q for G, the subgraph Q^⋆ is a quasi-tree for G^*. Moreover, these quasi-trees have the same chord diagrams, C_G(Q)=C_G^*(Q^⋆). Also the natural bijection of edges of G and G^* leads to the total order ≺^* on edges ofG^* induced by ≺. Consequently the property of an edge of being live/dead relative toQ is mappeded by the bijection to the same property relative to Q^⋆. Also one may check that the property of an edge of being orientable/non-orientable relative to Q is preserved by the bijection to the same property relative to Q^⋆. But its property of being internal/external is changed to the opposite. Now we can apply definition <ref> to the quasi-tree of G^* just constructed. ∙ We obtain a spanning ribbon subgraph F(Q^⋆) by deleting the internally live orientable edges of Q^⋆. They correspond to externally live orientable edges of Q.∙ We obtain an abstract graph Γ(Q^⋆) consists of vertices that are the connected components of F(Q^⋆) and edges that are the internally live orientable edges ofQ^⋆. For a ribbon graph G, the Krushkal polynomial has the following expansion over the set of quasi-trees. K_G(X,Y,A,B)=∑_Q∈𝒬_G A^s(F(Q))/2T_Q · B^s(F(Q^⋆))/2T_Q^⋆ ,where T_Q=T(Γ(Q);X+1,A+1) and T_Q^⋆=T(Γ(Q^⋆);Y+1,B+1) stand for the classical Tutte polynomial of the abstract graphs Γ(Q) and Γ(Q^⋆), and the parameter s was defined in <ref>. Krushkal polynomial!quasi-tree expansionFor the graph G on the torus with a single vertex and two loops, parallel and meridian, in Example <ref> we have two spanning quasi-trees consisting of a vertex only and the whole graph, as in Example <ref>. For Q being a vertex, both F(Q) and Γ(Q) consist of one vertex. So s(F(Q))=0 andT_Q=1. The dual quasi-tree Q^⋆=D^* is the whole graph in this case, and one of its loops is internally active.So F(Q^⋆) consists of one vertex with a loop attached to it.The abstract graph Γ(Q^⋆) is the same. Thus s(F(Q^⋆))=0 andT_Q^⋆=T(Γ(Q^⋆);Y+1,B+1)=B+1. Therefore the contribution of the single vertex spanning quasi-tree is B+1. For the second quasi-tree Q, the whole graph, the situation is completely symmetrical. Its dual quasi-tree Q^⋆ consists of a single vertex, and its contribution is A+1. SoK_G(X,Y,A,B)=A+2+B,as was mentioned in Example <ref>.For the graph G in Examples <ref> and <ref> with edges labeled a, b, and c there are four spanning quasi-trees Q_{a}, Q_{b}, Q_{c}, and Q_{a,b,c}. For each of them a is a live edge and b and c are dead relative to the total order a≺ b≺ c. The next tables show the ribbon graphs F(Q), F(Q^⋆) and abstract graphs Γ(Q), Γ(Q^⋆) and their contributions to the Krushkal polynomial for each of these spanning quasi-trees.[QQ_{a}Q_{b}Q_{c}Q_{a,b,c}; F(Q)-5F_Q_a501525 -13F_Q_b(15,-7)b5000 -13F_Q_c(33,-6)c5000 -13F_Q_abc(15,-7)b(33,-6)c5000; (0,15) A^s(F(Q))/21111; Γ(Q)0Ga_Q_a302015 0Ga_Q_b600 0Ga_Q_b600-10Ga_Q_abc2000;T_QX+111A+1; ;Q^⋆Q^⋆_{a}Q^⋆_{b}Q^⋆_{c}Q^⋆_{a,b,c}; F(Q^⋆) -12F_Qst_a(15,-7)b(33,-6)c502525 -12F_Qst_b(15,-5)c3000 -12F_Qst_c(20,-6)b3000-2F_Qst_abc1400; (0,15) B^s(F(Q^⋆))/2B111; Γ(Q^⋆) 0Ga_Q_b62015-10Ga_Q_abc2000-10Ga_Q_abc2000 0Ga_Q_b600;T_Q^⋆1B+1B+11;]So the Krushkal polynomial is equal toK_G=(X+1)B+(B+1)+(B+1)+(A+1)=XB+A+3B+3,which coincides with the direct calculation in Example <ref>Now we consider the example <ref> of a non-orientable ribbon graph G and its dual G^* from <cit.>.G=-20rg-ex4(10,20)1(60,37)2(60,-5)3(116,50)41203035 G^*=-40rgd-ex4(25,31)1(38,60)2(38,10)3(100,50)410000It has four spanning quasi-trees Q_{2}, Q_{3}, Q_{2,3}, and Q_{2,3,4}. Again we indicate the ribbon graphs F(Q), F(Q^⋆) and abstract graphs Γ(Q), Γ(Q^⋆) and their contributions to the Krushkal polynomial.[ Q Q_{2} Q_{3} Q_{2,3} Q_{2,3,4};F(Q) -5F_Q_a501525-10F_Q_3(15,-7)35000 -16F_Q_23(30,22)2(15,-6)35000-20F_Q_234(55,20)45500;(0,15) A^s(F(Q))/2 1 1 A^1/2 1;Γ(Q) 0Ga_Q_a3020150Ga_Q_b6000Ga_Q_b600-5Ga_Q_2343500; T_Q X+1 1 1 X+A+2;; Q^⋆ Q^⋆_{2} Q^⋆_{3} Q^⋆_{2,3} Q^⋆_{2,3,4};F(Q^⋆) -18F_Qst_2(18,6)3(57,26)4572525-18F_Qst_3(22,32)2(53,26)45700 -12F_Qst_23(53,26)45700 -5F_Q_a5000;(0,15) B^s(F(Q^⋆))/2 B B B^1/2 1;Γ(Q^⋆) 0Ga_Q_a302015 0Ga_Q_a3300 0Ga_Q_a302015 0Ga_Q_a302015; T_Q^⋆ Y+1 Y+1 Y+1 Y+1; ]So the Krushkal polynomial is equal toK_G=(X+1)(Y+1)B+(Y+1)B+(Y+1)A^1/2B^1/2+(X+A+2)(Y+1).One may check this answer by a direct calculation according to the definition <ref>. Clark Butler's Theorem <ref> specializes to the quasi-tree expansion of the Bollobás-Riordan polynomial from <cit.>in the orientable case and from <cit.>in the non-orientable case. 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"authors": [
"Sergei Chmutov"
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"title": "Topological Tutte Polynomial"
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shapes.gates.logic.US,trees,positioning,arrowsthmTheorem[section] prop[thm]Proposition lem[thm]Lemma cor[thm]Corollary conj[thm]Conjecture *mainMain Theorem claimClaim[thm]definition defn[thm]Definition rem[thm]Remark conv[thm]Convention exmp[thm]Example ques[thm]Question prob[thm]Problem cons[thm]Construction assu[thm]Assumptions *sassuStanding Assumptions commentyes [t]1.25in[footnote]yesThe relative hyperbolicity and manifold structure of certain RACGs]On the relative hyperbolicity and manifold structure of certain right-angled Coxeter groups Department of Mathematics1326 Stevenson CenterVanderbilt UniversityNashville, TN 37240 USA [email protected] Department of Mathematical SciencesUniversity of Wisconsin-MilwaukeeP.O. Box 413Milwaukee, WI 53201USA [email protected] of MathematicsThe University of Georgia1023 D. W. Brooks DriveAthens, GA 30605USA [email protected] this article, we study the manifold structure and the relatively hyperbolic structure of right-angled Coxeter groups with planar nerves. We then apply our results to the quasi-isometry problem for this class of right-angled Coxeter groups. [2000] 20F67,20F65[ Hung Cong Tran December 30, 2023 =====================§ INTRODUCTIONFor each finite simplicial graph Γ the associated right-angled Coxeter group G_Γ has generating set S equal to the vertices of Γ, relations s^2=1 for each s in S and relations st = ts whenever s and t are adjacent vertices. The graph Γ is the defining graph of right-angled Coxeter group G_Γ and its flag complex Δ=Δ(Γ) is the defining nerve of the group. Therefore, sometimes we also denote the right-angled Coxeter group G_Γ by G_Δ. In geometric group theory, groups acting on (0) cube complexes are fundamental objects and right-angled Coxeter groups provide a rich source of such groups. The coarse geometry of right-angled Coxeter groups has been studied by Caprace <cit.>, Dani-Thomas <cit.>, Dani-Stark-Thomas <cit.>, Behrstock-Hagen-Sisto <cit.>, Levcovitz <cit.> and others. In this paper, we will study the boundary of relatively hyperbolic right-angled Coxeter groups and the geometry of right-angled Coxeter groups with planar nerves. In this paper, we study the geometry of right-angled Coxeter groups of planar defining nerves. We will focus our attention on one-ended non-hyperbolic groups. We will also exclude the virtually ^2 groups. Thus, we assume the nerves of our groups satisfy the following conditions.The planar flag complex Δ⊂S^2: * is connected with no separating vertices and no separating edges (G_Δ is one-ended);* contains at least one induced 4-cycle (G_Δ is not hyperbolic);* is not a 4-cycle and not a cone of a 4-cycle (G_Δ is not virtually ^2). §.§ Manifold group structure and relatively hyperbolic structureDavis-Okun <cit.> and Droms <cit.> proved that the one-ended right-angled Coxeter group G_Δ is virtually the fundamental group of a 3–manifold M if the defining nerve Δ is planar. So, we can learn a lot about the geometry of G_Δ if we know the manifold type of M. This is the main motivation for proving the following:Let Δ⊂S^2 be a flag complex satisfying Standing Assumptions. Then the right-angled Coxeter group G_Δ is virtually the fundamental group of a 3-manifold M with empty or toroidal boundary if and only if Δ=S^2 or the boundary of each region in S^2-Δ is a 4-cycle. Moreover, in the case G_Δ is virtually the fundamental group of a 3-manifold M with empty or toroidal boundary, there are four mutually exclusive cases:* If Δ is a suspension of some n-cycle (n≥ 4) or some broken line (i.e a finite disjoint union of vertices and finite trees with vertex degrees 1 or 2), then M is a Seifert manifold;*If the 1-skeleton of Δ is 𝒞ℱ𝒮 (see Definition <ref>) and does not satisfy (1), then M is a graph manifold; *If the 1-skeleton of Δ has no separating induced 4-cycle and is not 𝒞ℱ𝒮, then M is a hyperbolic manifold with boundary; * If the 1-skeleton of Δ contains a separating induced 4-cycle and is not 𝒞ℱ𝒮, then M is a mixed manifold.In Theorem <ref> we characterize the associated manifold M of G_Δ with empty or toroidal boundary using the work in <cit.> on the Euler characteristic of torsion free finite index subgroups of right-angled Coxeter groups. For further classification of such a manifold M we investigate the relatively hyperbolic structure of G_Δ. As a consequence, we obtain a key result on all possible divergence of G_Δ.Let Γ be a graph whose flag complex Δ is planar. There is a collectionof 𝒞ℱ𝒮 subgraphs of Γ such that the right-angled Coxeter group G_Γ is relatively hyperbolic with respect to the collection =G_JJ∈. In particular, if G_Γ is one-ended, then the divergence of G_Γ is linear, quadratic, or exponential. For the proof of Theorem <ref> we carefully investigate the tree structure of the defining nerve Δ and then follow an almost identical strategy to the proof of Theorem 1.6 in <cit.>. The result concerning the divergence of the group follows directly from <cit.> and <cit.>.We also characterize right-angled Coxeter groups with planar nerves such that they have non-trivial minimal peripheral structures with the Sierpinski carpet as their Bowditch boundaries. This result will contribute to the study of the coarse geometry of our groups and we will discuss it in the next section. Let Δ⊂S^2 be a planar flag complex satisfying Standing Assumptions. Then G_Δ has a non-trivial minimal peripheral structure with Bowditch boundary the Sierpinski carpet if and only if all following conditions hold: * The boundary of some region of S^2-Δ is an n-cycle with n≥ 5;* The 1-skeleton of Δ has no separating induced 4-cycle, no cut pair, and no separating induced path of length 2.§.§ Coarse geometry Theorem <ref> divides right-angled Coxeter groups with nerves satisfying Standing Assumptions into two main types: the ones which are virtually the fundamental group of a 3–manifold with empty or toroidal boundary which we call type A, and the ones which are not virtually the fundamental group of such a 3–manifold which we will call type B. Behrstock and Neumann summarize the rigidity results of 3–manifolds with empty or toroidal boundary from many authors (see <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, and <cit.>) in Theorem A of <cit.>. Thus, a right-angled Coxeter group of type A and a right-angled Coxeter group of type B are never quasi-isometric by Theorem A of <cit.>.Theorem <ref> further divides right-angled Coxeter groups of type A into 4 subtypes (see Figure <ref>). It is well-known that the fundamental groups of the associated manifolds in the four different subtypes are not quasi-isometric (see Theorem 5.4 in <cit.>), so two right-angled Coxeter groups of type A are not quasi-isometric if they are of different subtypes. Thus, we must study each subtype to understand the quasi-isometry classification of groups of type A. The collection of groups of subtype A.1 contains exactly three quasi-isometry equivalence classes. The first quasi-isometry equivalence class of groups of type A.1 contains only one right-angled Coxeter group with nerve a suspension of a 4–cycle. This right-angled Coxeter group is virtually ^3. The second quasi-isometry equivalence class consists of right-angled Coxeter groups whose nerve is a suspension of some n–cycle (n≥ 5). Each right-angled Coxeter groups in this second class is virtually the fundamental group of a closed Seifert manifold. The last quasi-isometry equivalence class consists of right-angled Coxeter groups with nerve a suspension of some broken line. Each right-angled Coxeter group in this class is virtually the fundamental group of a Seifert manifold with non-empty boundary.The quasi-isometry classification of groups of subtype A.2 is much more complicated. We give a complete quasi-isometry classification of groups of subtype A.2. Let Δ⊂S^2 and Δ'⊂S^2 be two planar flag complexes satisfying Standing Assumptions such that their 1–skeletons are 𝒞ℱ𝒮 and not join graphs. Let T_r and T_r' be the visual decomposition trees of Δ and Δ' respectively. Then two groups G_Δ and G_Δ' are quasi-isometric if and only if T_r and T_r' are bisimilar. The above theorem strengthens Theorem 1.1 of <cit.> by removing the condition “triangle free” from the hypothesis of the theorem. We remark that the visual decomposition trees of the defining nerves in Theorems <ref> are colored trees whose vertices are colored black and white, and they are constructed in Sections <ref> and <ref>. We also refer the reader to Section <ref> for the notion of bisimilarity among two-colored graphs.We also classify groups of subtype A.2 which are quasi-isometric to right-angled Artin groups. The following theorem is an extension of Theorem 1.2 of <cit.>.Let Δ⊂S^2 be a planar flag complex satisfying Standing Assumptions with the 1–skeleton a non-join 𝒞ℱ𝒮. Let T_r be a visual decomposition tree of Δ. Then the following are equivalent: * The right-angled Coxeter group G_Δ is quasi-isometric to a right-angled Artin group.* The right-angled Coxeter group G_Δ is quasi-isometric to the right-angled Artin group of a tree of diameter at least 3.* The right-angled Coxeter group G_Δ is quasi-isometric to the right-angled Artin group of a tree of diameter exactly 3.* All vertices of the tree T_r are black. All right-angled Coxeter groups of subtype A.3 are virtually fundamental groups of hyperbolic 3–manifolds. So far, the authors do not know a complete quasi-isometric classification for the groups in this subtype. By Schwartz Rigidity Theorem <cit.> (see also, for example, Theorem 24.1 <cit.>) the fundamental groups of two hyperbolic 3–manifolds are quasi-isometric if and only if they are commensurable. Therefore, we turn the quasi-isometry classification problem for right-angled Coxeter groups of type A.3 into a commensurability problem. Classify all right-angled Coxeter groups which are virtually the fundamental group of a hyperbolic 3–manifold up to commensurability.All right-angled Coxeter groups of subtype A.4 are virtually the fundamental groups of mixed 3–manifolds. Therefore, the last conclusion of Theorem <ref> potentially allows us to use the work of Kapovich-Leeb <cit.> to understand the geometry of right-angled Coxeter groups of subtype A.4. The complete quasi-isometry classification of fundamental groups of mixed manifolds remains an open question. As a consequence, the authors do not know the complete quasi-isometry classification of groups of subtype A.4.We note that all right-angled Coxeter groups of type B are non-trivially relatively hyperbolic (see Proposition <ref>). So, we divide these groups into three different subtypes based on their Bowditch boundary with respect to the minimal peripheral structure and whether they split over 2–ended subgroups (see Figure <ref>). Right-angled Coxeter groups of subtype B.1 all have minimal peripheral structure whose Bowditch boundary is a Sierpinski carpet (see Theorem <ref>). Meanwhile, the Bowditch boundary of right-angled Coxeter groups of subtypes B.2 and B.3 with respect to a minimal peripheral structure must have a cut point or a non-parabolic cut pair by Theorems <ref> and <ref>. Therefore, a right-angled Coxeter group of subtype B.1 is not quasi-isometric to one of subtype B.2 and subtype B.3. A right-angled Coxeter group of subtype B.2 always splits over a 2–ended subgroup while a right-angled Coxeter group of subtype B.3 does not. Therefore, they are never quasi-isometric by the work of Papasoglu <cit.>. (Papasoglu showed that among 1-ended finitely presented groups that are not commensurable to surface groups, having a splitting over a 2-ended subgroup is a quasi-isometry invariant.) We hope to use the work of Cashen-Martin <cit.> to understand the coarse geometry of right-angled Coxeter groups of subtype B.2. It seems hard to obtain the complete quasi-isometry classification of groups of type B. However, the quasi-isometry classification of the ones of subtype B.1 seems reasonable. Classify up to quasi-isometry all relatively hyperbolic right-angled Coxeter groups whose Bowditch boundary is a Sierpinski carpet with respect to some minimal peripheral structure.§.§ Outline of the paper In Section <ref>, we review some concepts in geometric group theory: (0) spaces, δ–hyperbolic spaces, (0) spaces with isolated flats, 3–manifold groups, relatively hyperbolic groups, (0) boundaries, Gromov boundaries, Bowditch boundaries, and peripheral splitting of relatively hyperbolic groups. We also recall some necessary results related to these concepts. In Section <ref>, we give descriptions of cut points and non-parabolic cut pairs of Bowditch boundaries of relatively hyperbolic right-angled Coxeter groups (see Theorems <ref> and <ref>). In Section <ref>, we study the coarse geometry of right-angled Coxeter groups with planar defining nerves. We first analyze the tree structure of planar flag complexes (see Subsection <ref>). Then we use this structure to study the relatively hyperbolic structure, group divergence, and manifold structure of right-angled Coxeter groups with planar nerves. We give the proofs of Theorem <ref>, Theorem <ref>, and Theorem <ref> in Subsection <ref>. We improve the tree structure of planar flag complexes with 1–skeletons 𝒞𝒮ℱ graphs and give a complete quasi-isometry classification of right-angled Coxeter groups which are virtually graph manifold groups in Subsection <ref>. The proof of Theorem <ref> is also given in this subsection. §.§ AcknowledgmentsAll three authors would like to thank Chris Hruska for suggestions and insights. The authors are also grateful for the insightful comments of Jason Behrstock, Ivan Levcovitz, and Mike Mihalik that have helped improve the exposition of this paper. The first author would like to thank Genevieve Walsh for helpful conversations concerning to this paper. Lastly, all three authors are particularly grateful for the referee's helpful comments and suggestions.§ PRELIMINARIES In this section, we review some concepts in geometric group theory: (0) spaces, δ–hyperbolic spaces, (0) spaces with isolated flats, relatively hyperbolic groups, (0) boundaries, Gromov boundaries, Bowditch boundaries, and peripheral splitting of relatively hyperbolic groups. We also use the work of Behrstock-Druţu-Mosher <cit.> and Groff <cit.> to prove that the Bowditch boundary is a quasi-isometry invariant among relatively hyperbolic groups with minimal peripheral structures. We discuss the work of Caprace <cit.>, Behrstock-Hagen-Sisto <cit.>, Dani-Thomas <cit.>, <cit.>, and <cit.> on peripheral structures of relatively hyperbolic right-angled Coxeter groups and divergence of right-angled Coxeter groups. We also discuss the work of Gersten <cit.> and Kapovich–Leeb <cit.> on divergence of 3–manifold groups. We also mention the concept of colored graphs and the bisimilarity equivalence relation on such graphs. §.§ (0) spaces, δ–hyperbolic spaces, and relatively hyperbolic groupsWe first discuss (0) spaces, δ–hyperbolic spaces, Gromov boundaries, and (0) boundaries. We refer the reader to the book <cit.> for more details. We say that a geodesic triangle Δ in a geodesic space X satisfies the CAT(0) inequality if d(x,y) ≤ d(x̅,y̅) for all points x, y on the edges of Δ and the corresponding points x̅, y̅ on the edges of the comparison triangle Δ̅ in Euclidean space ^2.A geodesic space X is said to be a CAT(0) space if every triangle in X satisfies the (0) inequality.If X is a (0) space, then the CAT(0) boundary of X, denoted ∂ X, is defined to be the set of all equivalence classes of geodesic rays in X, where two rays c and c' are equivalent if the Hausdorff distance between them is finite.We note that for any x ∈ X and ξ∈∂ X there is a unique geodesic ray α_x,ξ: [0,∞) → X with α_x,ξ(0)=x and α_x,ξ(∞)= ξ. The CAT(0) boundary has a natural topology with a basis of local neighborhoods (not necessarily open) of any point ξ∈∂ X given by the sets U(ξ;x, R, ϵ) = ξ'∈∂ Xd(α_x,ξ(R),α_x,ξ'(R)≤ϵ), where x ∈ X, ξ∈∂ X, R >0 and ϵ >0.A geodesic metric space (X, d) is δ–hyperbolic if every geodesic triangle with vertices in X is δ–thin in the sense that each side lies in the δ–neighborhood of the union of other sides. If X is a δ–hyperbolic space, then we could build the Gromov boundary of X, denoted ∂ X, in the same way as for a (0) space. That is, the Gromov boundary of X is defined to be the set of all equivalence classes of geodesic rays in X, where two rays c and c' are equivalent if the Hausdorff distance between them is finite. However, the topology on it is slightly different from the topology on the boundary of a (0) space (see for example <cit.> for details).We now review relatively hyperbolic groups and related concepts. Let T be any graph with the vertex set V. We define the combinatorial horoball based at T, ℋ(=ℋ(T)) to be the following graph: * ℋ^(0)= V×{{0}∪}.* ℋ^(1) = {((t, n), (t, n + 1))}∪((t_1, n), (t_2, n))d_T(t_1,t_2)≤ 2^n. We call edges of the first set vertical and of the second horizontal. In <cit.>, the combinatorial horoball is described as a 2-complex, but we only require the 1-skeleton for the horoball in this paper.Let ℋ be the horoball based at some graph T. Let D: ℋ→ [0,∞) be defined by extending the map on vertices (t,n)→ n linearly across edges. We call D the depth function for ℋ and refer to vertices v with D(v)=n as vertices of depth n or depth n vertices.Because T ×{0} is homeomorphic to T, we identify T with D^-1(0). Let G be a finitely generated group anda finite collection of finitely generated subgroups of G. Let S be a finite generating set of G such that S∩ P generates P for each P∈. For each left coset gP of subgroup P∈ let ℋ(gP) be the horoball based at a copy of the subgraph T_gP with vertex set gP of the Cayley graph Γ(G,S). The cusped space X(G,,S) is the union of Γ(G,S) with ℋ(gP) for every left coset of P∈, identifying the subgraph T_gP with the depth 0 subset of ℋ(gP). We suppress mention of S andwhen they are clear from the context. Let G be a finitely generated group anda finite collection of finitely generated subgroups of G. Let S be a finite generating set of G such that S∩ P generates P for each P∈. If the cusped space X(G,,S) is δ–hyperbolic then we say that G is hyperbolic relative toor that (G,) is relatively hyperbolic. Collectionis a peripheral structure, each group P ∈ is a peripheral subgroup and its left cosets are peripheral left cosets. The peripheral structureis minimal if for any other peripheral structure Q on G, each P∈ is conjugate into some Q∈Q.Replacing S for some other finite generating set S' may change the value of δ, but does not affect the hyperbolicity of the cusped space for some δ' (see <cit.>). As a consequence, the concept of relatively hyperbolic group does not depend on the choice of finite generating set.We say that a finitely generated group is non-trivially relatively hyperbolic if it is relatively hyperbolic with respect to some collection of proper subgroups.§.§ The Bowditch boundary We now discuss the Bowditch boundary of a relatively hyperbolic group and prove that it is a quasi-isometry invariant when the peripheral structure is minimal. We also recall peripheral splitting of relatively hyperbolic groups.Let (G,) be a finitely generated relatively hyperbolic group. Let S be a finite generating set of G such that S∩ P generates P for each P∈. The Bowditch boundary, denoted ∂(G,), is the Gromov boundary of the associated cusped space, X(G,,S).There is a natural topological action of G on the Bowditch boundary ∂(G,) that satisfies certain properties (see <cit.>).Bowditch has shown that the Bowditch boundary does not depend on the choice of finite generating set (see <cit.>). More precisely, if S and T are finite generating sets for G as in the above definition, then the Gromov boundaries of the cusped spaces X(G,,S) and X(G,,T) are G–equivariantly homeomorphic (see <cit.>).For each peripheral left coset gP the limit set of the associated horoball ℋ(gP) consists of a single point in ∂(G,), called the parabolic point labelled by gP, denoted v_gP. The stabilizer of the point v_gP is the subgroup gPg^-1. We call each infinite subgroup of gPg^-1 a parabolic subgroup and subgroup gPg^-1 a maximal parabolic subgroup.The homeomorphism type of the Bowditch boundary was already known to be a quasi-isometry invariant of a relatively hyperbolic group with minimal peripheral structure (see Groff <cit.>). However, we combine the work of Groff <cit.> and Behrstock-Druţu-Mosher <cit.> to elaborate on the homeomorphism between Bowditch boundaries induced by a quasi-isometry between two relatively hyperbolic groups with minimal peripheral structures.Let (G_1,_1) and (G_2,_2) be finitely generated non-trivially relatively hyperbolic groups with minimal peripheral structures. If G_1 and G_2 are quasi-isometric, then there is a homeomorphism f from ∂(G_1,_1) to ∂(G_2,_2) that maps the set of parabolic points of ∂(G_1,_1) bijectively onto the set of parabolic points of ∂(G_2,_2). Moreover, if parabolic point v of ∂(G_1,_1) is labelled by some peripheral left coset g_1P_1 in G_1 and the parabolic point f(v) of ∂(G_2,_2) is labelled by some peripheral left coset g_2P_2 in G_2, then P_1 and P_2 are quasi-isometric.Fix generating sets S_1 and S_2 as in Definition <ref> for G_1 and G_2 respectively, then there is a quasi-isometry q:Γ(G_1,S_1)→Γ(G_2,S_2). Since the peripheral structures are minimal, the map q takes a peripheral left coset g_1P_1 of G_1 to within a uniform bounded distance of the corresponding peripheral left coset g_2P_2 of G_2 by Theorem 4.1 in <cit.>. In particular, P_1 and P_2 are quasi-isometric. Using the proof of Theorems 6.3 in <cit.>, we can extend q to the quasi-isometry q̂:X(G_1,_1,S_1)→ X(G_2,_2,S_2) between cusped spaces such that q̂ restricts to a quasi-isometric embedding on each individual horoball of X(G_1,_1,S_1) and the image of the horoball lies in some neighborhood of a horoball of X(G_2,_2,S_2). Therefore, there is a homeomorphism f induced by q̂ from ∂(G_1,_1) to ∂(G_2,_2) that maps the set of parabolic points of ∂(G_1,_1) bijectively onto the set of parabolic points of ∂(G_2,_2). Moreover, if the parabolic point v of ∂(G_1,_1) is labelled by some peripheral left coset g_1P_1 in G_1 and the parabolic point f(v) of ∂(G_2,_2) is labelled by some peripheral left coset g_2P_2 in G_2, then by the above observation P_1 and P_2 are quasi-isometric. Let G be a group. By a splitting of G, over a given class of subgroups, we mean a presentation of G as a finite graph of groups, where each edge group belongs to this class. Such a splitting is said to be relative to another classof subgroups if each element ofis conjugate into one of the vertex groups. A splitting is said to be trivial if there exists a vertex group equal to G.Assume G is hyperbolic relative to a collection . A peripheral splitting of (G,) is a representation of G as a finite bipartite graph of groups, whereconsists of precisely of the (conjugacy classes of) vertex groups of one color. Obviously, any peripheral splitting of (G,) is relative toand over subgroups of elements of . Peripheral splittings of (G,) are closely related to cut points in the Bowditch boundary ∂(G,) (<cit.>). Given a compact connected metric space X, a point x ∈ X is a global cut point (or just simply cut point) if X-{x} is not connected. If {a, b}⊂ X contains no cut points and X-{a, b} is not connected, then {a, b} is a cut pair. A point x ∈ X is a local cut point if X-{x} is not connected, or X-{x} is connected and has more than one end. §.§ (0) spaces with isolated flats In this section, we discuss the work of Hruska-Kleiner <cit.> on (0) spaces with isolated flats. A k–flat in a (0) space X is an isometrically embedded copy of Euclidean space ^k for some k ≥ 2. In particular, note that a geodesic line is not considered to be a flat.Let X be a (0) space, G a group acting geometrically on X, and ℱ a G–invariant set of flats in X. We say that X has isolated flats with respect to ℱ if the following two conditions hold. * There is a constant D such that every flat F ⊂ X lies in a D–neighborhood of some F' ∈ℱ.* For each positive r < ∞ there is a constant ρ = ρ(r) < ∞ so that for any two distinct flats F, F' ∈ℱ we have (N_r(F) ∩ N_r(F′))<ρ.We say X has isolated flats if it has isolated flats with respect to some G–invariant set of flats. Suppose X has isolated flats with respect to ℱ. For each F ∈ℱ the stabilizer _G(F) is virtually abelian and acts cocompactly on F. The set of stabilizers of flats F ∈ℱ is precisely the set of maximal virtually abelian subgroups of G of rank at least two. These stabilizers lie in only finitely many conjugacy classes.Let X have isolated flats with respect to ℱ. Then G is relatively hyperbolic with respect to the collection of all maximal virtually abelian subgroups of rank at least two. The previous theorem also has the following converse.Let G be a group acting geometrically on a (0) space X. Suppose G is relatively hyperbolic with respect to a family of virtually abelian subgroups. Then X has isolated flats A group G that admits an action on a (0) space with isolated flats has a “well-defined” (0) boundary, often denoted by ∂ G, by the following theorem. Let G act properly, cocompactly, and isometrically on (0) spaces X and Y. If X has isolated flats, then so does Y, and there is a G–equivariant homeomorphism ∂ X →∂ Y.§.§ Right-angled Coxeter groups and their relatively hyperbolic structures In this section, we review the concepts of right-angled Coxeter groups and Davis complexes. We also review the work of Caprace <cit.> and Behrstock-Hagen-Sisto <cit.> on peripheral structures of relatively hyperbolic right-angled Coxeter groups. Given a finite simplicial graph Γ, the associated right-angled Coxeter group G_Γ is generated by the set S of vertices of Γ and has relations s^2 = 1 for all s in S and st = ts whenever s and t are adjacent vertices. Graph Γ is the defining graph of a right-angled Coxeter group G_Γ and its flag complex Δ=Δ(Γ) is the defining nerve of the group. Therefore, sometimes we also denote the right-angled Coxeter group G_Γ by G_Δ where Δ is the flag complex of Γ. Let S_1 be a subset of S. The subgroup of G_Γ generated by S_1 is a right-angled Coxeter group G_Γ_1, where Γ_1 is the induced subgraph of Γ with vertex set S_1 (i.e. Γ_1 is the union of all edges of Γ with both endpoints in S_1). The subgroup G_Γ_1 is called a special subgroup of G_Γ.The right-angled Coxeter group G_Γ is one-ended if and only if Γ is not equal to a complete graph, is connected and has no separating complete subgraphs (see Theorem 8.7.2 in <cit.>).Given a finite simplicial graph Γ, the associated Davis complex Σ_Γ is a cube complex constructed as follows. For every k–clique, T ⊂Γ, the special subgroup G_T is isomorphic to the direct product of k copies of Z_2. Hence, the Cayley graph of G_T is isomorphic to the 1–skeleton of a k–cube. The Davis complex Σ_Γ has 1–skeleton the Cayley graph of G_Γ, where edges are given unit length. Additionally, for each k–clique, T ⊂Γ, and coset gG_T, we glue a unit k–cube to gG_T ⊂Σ_Γ. The Davis complex Σ_Γ is a (0) space and the group G_Γ acts properly and cocompactly on the Davis complex Σ_Γ (see <cit.>). Let Γ be a simplicial graph andbe a collection of induced subgraphs of Γ. Then the right-angled Coxeter group G_Γ is hyperbolic relative to the collection =G_JJ∈ if and only if the following three conditions hold: * If σ is an induced 4-cycle of Γ, then σ is an induced 4-cycle of some J∈.* For all J_1, J_2 inwith J_1≠ J_2, the intersection J_1∩ J_2 is empty or J_1∩ J_2 is a complete subgraph of Γ.* If a vertex s commutes with two non-adjacent vertices of some J in , then s lies in J. Let 𝒯 be the class consisting of the finite simplicial graphs Λ such that G_Λ is strongly algebraically thick. Then for any finite simplicial graph Γ either: Γ∈𝒯, or there exists a collectionof induced subgraphs of Γ such that ⊂𝒯 and G_Γ is hyperbolic relative to the collection =G_JJ ∈ and this peripheral structure is minimal.In Theorem <ref> we use the notion of strong algebraic thickness which is introduced in <cit.> and is a sufficient condition for a group to be non-hyperbolic relative to any collection of proper subgroups. We refer the reader to <cit.> for more details. §.§ Divergence of right-angled Coxeter groups and 3–manifold groupsRoughly speaking, divergence is a quasi-isometry invariant that measures the circumference of a ball of radius n as a function of n. We refer the reader to <cit.> for a precise definition. In this section, we state some theorems about divergence of certain right-angled Coxeter groups and 3-manifold groups which will be used later in this paper. §.§.§ Divergence of right-angled Coxeter groupsThe divergence of a right-angled Coxeter group is either exponential (if the group is relatively hyperbolic) or bounded above by a polynomial (if the group is strongly algebraically thick).Before continuing, we will take a brief detour to define a property of graphs that will be relevant to our study of right-angled Coxeter groups. Given a graph Γ, define Γ^4 as the graph whose vertices are induced 4–cycles of Γ. Two vertices in Γ^4 are adjacent if and only if the corresponding induced 4-cycles in Γ have two nonadjacent vertices in common. A graph Γ is 𝒞ℱ𝒮 if Γ is the join Ω*K where K is a (possibly empty) clique and Ω is a non-empty subgraph such that Ω^4 has a connected component T such that every vertex of Ω is contained in a 4–cycle that is a vertex of T. If Γ is 𝒞ℱ𝒮, then we will say that the right-angled Coxeter group G_Γ is 𝒞ℱ𝒮.Let Γ be a finite, simplicial, connected graph which has no separating complete subgraph. Let G_Γ be the associated right-angled Coxeter group. * The group G_Γ has linear divergence if and only if Γ is a join of two graphs of diameters at least 2.* The group G_Γ has quadratic divergence if and only if Γ is 𝒞ℱ𝒮 and is not a join of two graphs of diameters at least 2.§.§.§ Divergence of 3–manifold groups Let M be a compact, connected, orientable 3–manifold with empty or toroidal boundary. The 3–manifold M is geometric if its interior admits a geometric structure in the sense of Thurston. These structures are the 3–sphere, Euclidean 3–space, hyperbolic 3-space, S^2×, ℍ^2×, SL(2,), Nil and Sol. We note that a geometric 3–manifold M is Seifert fibered if its geometry is neither Sol nor hyperbolic. A non-geometric 3–manifold can be cut along tori called JSJ tori so that each component is a hyperbolic 3-manifold or a Seifert fibered space (<cit.>, <cit.>).Each component of this JSJ decomposition is a piece. The manifold M is called a graph manifold if all the pieces are Seifert fibered, otherwise it is a mixed manifold.Let M be a non-geometric manifold. Then M is a graph manifold if and only if the divergence of π_1(M) is quadratic, and M is a mixed manifold if and only if the divergence of π_1(M) is exponential. Let M be a compact, orientable 3–manifold whose fundamental group has linear divergence.Then M has the geometry of Sol, or M is a Seifert manifold. However, if the universal cover M̃ of M is a fattened tree crossed with , then M must be a Seifert manifold.§.§ Two-colored graphs and their bisimilarity classes In this section, we briefly discuss two-colored graphs and the bisimilar equivalence relation on them. These materials are introduced by Behrstock-Neumann <cit.> to study the quasi-isometry classification of graph manifolds. In this paper, we also use two-colored graphs and the bisimilar equivalence relation to study the quasi-isometry classification of right-angled Coxeter groups G_Γ whose defining graphs Γ are 𝒞ℱ𝒮 and satisfy Standing Assumptions.A two-colored graph is a graph Γ with a “coloring” c:V(Γ) →{b,w}. A weak covering of a two-colored graph Γ' is a graph homomorphism f:Γ→Γ' which respects colors and has the property that for each v∈ V(Γ) and for each edge e' ∈ E(Γ') at f(v), there exists an e∈ E(Γ) at v with f(e)=e'. Two-colored graphs Γ_1 and Γ_2 are bisimilar, written Γ_1∼Γ_2 if Γ_1 and Γ_2 weakly cover some common two-colored graph.The bisimilarity relation ∼ is an equivalence relation. Moreover, each equivalence class has a unique minimal element up to isomorphism.§ BOWDITCH BOUNDARIES OF RELATIVELY HYPERBOLIC RIGHT-ANGLED COXETER GROUPSIn this section, we give descriptions of cut points and non-parabolic cut pairs of Bowditch boundaries of relatively hyperbolic right-angled Coxeter groups (see Theorems <ref> and <ref>). The Bowditch boundary of relatively hyperbolic groups with minimal peripheral structures is a quasi-isometry invariant (see Theorem <ref>). So, these results can be applied, in certain cases, to differentiate two relatively hyperbolic RACGs in terms of quasi-isometry equivalence.In <cit.>, the third author investigates the connection between the Bowditch boundary of a relatively hyperbolic group (G,) and the boundary of a (0) space X on which G acts geometrically. For relatively hyperbolic right-angled Coxeter groups, the relevant result from <cit.> can be stated as follows:Let Γ be a finite simplicial graph. Assume that the right-angled Coxeter group G_Γ is relatively hyperbolic with respect to a collectionof its subgroups. Then the Bowditch boundary ∂ (G_Γ,) is obtained from the (0) boundary ∂Σ_Γ by identifying the limit set of each peripheral left coset to a point. Moreover, this quotient map is G_Γ-equivariant. We now introduce some definitions concerning defining graphs of right-angled Coxeter groups that we will use to “visualize” cut points and non-parabolic cut points in the Bowditch boundary. Let Γ_1 and Γ_2 be two graphs, the join of Γ_1 and Γ_2, denoted Γ_1*Γ_2, is the graph obtained by connecting every vertex of Γ_1 to every vertex of Γ_2 by an edge. If Γ_2 consists of distinct vertices u and v, then the join Γ_1*{u,v} is the suspension of Γ_1. Let Γ be a simplicial graph. A pair of non-adjacent vertices {a, b} in Γ is called a cut pair if {a, b} separates Γ. An induced subgraph Γ_1 of Γ is a complete subgraph suspension if Γ_1 is a suspension of a complete subgraph σ of Γ. If σ is a single vertex, then Γ_1 is a vertex suspension.An induced subgraph Γ_1 of Γ is separating if Γ_1 separates Γ. Thus, we may also consider a cut pair as a separating complete subgraph suspension which is a suspension of the empty graph. We will need the following lemma to visualize cut points in the Bowditch boundary of a relatively hyperbolic right-angled Coxeter group. Let Γ be a simplicial graph anda collection of induced proper subgraphs of Γ. Assume that the right-angled Coxeter group G_Γ is one-ended, hyperbolic relative to the collection =G_JJ∈, and suppose each subgroup inis one-ended. Let J_0 be an element insuch that some induced subgraph of J_0 separates the graph Γ. Then we can write Γ=Γ_1∪Γ_2 such that the following conditions hold: * Γ_1, Γ_2 are both proper induced subgraphs of Γ;* Γ_1∩Γ_2 is an induced subgraph of J_0.* For each J in , J lies completely inside either Γ_1 or Γ_2 Since each subgroup inis one-ended, each graph J inis connected and J has no separating complete subgraph by Theorem 8.7.2 in <cit.>. Let L be an induced subgraph of J_0 that separates the graph Γ. Let Γ'_1 be L together with some of the components of Γ-L, and let Γ'_2 be L together with the remaining components of Γ-L. Then, Γ'_1, Γ'_2 are both proper induced subgraphs of Γ, L=Γ'_1∩Γ'_2, and Γ=Γ'_1∪Γ'_2. Since J_0 is a proper subgraph of Γ, Γ'_1-J_0 ≠∅ or Γ'_2-J_0≠∅ (say Γ'_1-J_0≠∅).Let Γ_1=Γ'_1, Γ_2=Γ'_2∪ J_0. Then, Γ_1 is an induced proper subgraph of Γ. We now prove that Γ_2 is also an induced proper subgraph. Choose a vertex w∈Γ_1-J_0=Γ'_1-J_0. Since Γ_1∩Γ_2=Γ'_1∩(Γ'_2∪ J_0)⊂ J_0, the vertex w does not belong to Γ_2. Therefore, Γ_2 is a proper subgraph of Γ. We now prove that Γ_2 is induced. Let e be an arbitrary edge with endpoints u and v in Γ_2. If e is an edge of Γ'_2, then e is the edge of Γ_2. Otherwise, e is an edge of Γ'_1=Γ_1, because Γ=Γ'_1∪Γ'_2. In particular, u and v are also the vertices of Γ_1. Again, Γ_1∩Γ_2 is a subgraph of J_0. Then u and v are vertices of J_0. Therefore, e is an edge of J_0, because J_0 is an induced subgraph. Thus, e is also an edge of Γ_2. Thus, Γ_2 is an induced subgraph. This implies that Γ_1∩Γ_2 is an induced subgraph. We already checked that Γ_1∩Γ_2 is a subgraph of J_0, and by construction Γ=Γ_1∪Γ_2.Now, we prove that for each J in , J lies completely inside either Γ_1 or Γ_2. By the construction, J_0 is a subgraph of Γ_2. Therefore, we only need to check the case where J≠ J_0. It suffices to show that J lies completely inside either Γ'_1 or Γ'_2. By Theorem <ref>, for each J≠ J_0 inthe intersection J∩ J_0 is empty or it is a complete subgraph of Γ. Also the intersection J∩ L is an induced subgraph of J∩ J_0 if J∩ L≠∅. Therefore, J∩ L is empty or it is a complete subgraph of Γ. Recall that each graph inis connected and has no separating complete subgraph by our assumption that the peripheral subgroups are 1–ended. Therefore, J-L=J-(J∩ L) is connected for each J≠ J_0 in . By the construction J-L lies completely inside either Γ'_1 or Γ'_2. Thus, J also lies completely inside either Γ'_1 or Γ'_2. Therefore, J lies completely inside either Γ_1 or Γ_2. The following theorem describes cut points in the Bowditch boundary of a relatively hyperbolic right-angled Coxeter group using the defining graph.Let Γ be a simplicial graph andbe a collection of induced proper subgraphs of Γ. Assume that the right-angled Coxeter group G_Γ is one-ended, G_Γ is hyperbolic relative to the collection =G_JJ∈, and suppose each subgroup inis also one-ended. Then each parabolic point v_gG_J_0 is a global cut point if and only if some induced subgraph of J_0 separates the graph Γ.We first assume that some induced subgraph of J_0 separates the graph Γ. Let Γ_1 and Γ_2 be graphs satisfying all conditions of Lemma <ref>. This implies that G_Γ=G_Γ_1*_G_KG_Γ_2, G_Γ_1≠ G_Γ, and G_Γ_2≠ G_Γ. Since J lies completely inside either Γ_1 or Γ_2 for each J∈, each peripheral subgroup inmust be a subgroup of G_Γ_1 or G_Γ_2. Therefore, G_Γ splits non-trivially relative toover the parabolic subgroup G_K≤ G_J_0. By the claim following Theorem 1.2 of <cit.> the parabolic point v_G_J_0 labelled by G_J_0 is a global cut point of ∂(G,). Also, the group G_Γ acts as a group of homeomorphisms on ∂(G,) and gv_G_J_0=v_gG_J_0. Thus, each parabolic point v_gG_J_0 is also a global cut point.Now, we assume that some parabolic point v_gG_J_0 is a global cut point. Again, the group G_Γ acts as a group of homeomorphisms on ∂(G,) and gv_G_J_0=v_gG_J_0. Therefore, v_G_J_0 is also a global cut point. Thus the maximal peripheral splitting 𝒢 of (G,) is non-trivial, and G_J_0 is a vertex stabilizer which is adjacent to a component vertex group in 𝒢 (see <cit.> for details concerning maximal peripheral splittings). So, by Theorem 3.3 of <cit.> G_Γ splits non-trivially over a subgroup H of G_J_0. Theorem 1 of <cit.> implies that there is some induced subgraph K of Γ which separates Γ such that G_K is contained in some conjugate of H. Therefore, G_K is also contained in some conjugate of the peripheral subgroup G_J_0. Moreover, G_K and G_J_0 are both special subgroups of G_Γ. Thus, K is an induced subgraph of J_0. (This is a standard fact, and we leave the details to the reader.) Now, we discuss a few examples related to cut points in Bowditch boundaries of relatively hyperbolic right-angled Coxeter groups. These examples illustrate an application of Theorem <ref> to the problem of quasi-isometry classification of right-angled Coxeter groups.Let Γ_1, Γ_2, and Γ_3 be the graphs in Figure <ref>. Observe that all groups G_Γ_i are one-ended. We will prove that groups G_Γ_1, G_Γ_2, and G_Γ_3 are not pairwise quasi-isometric by investigating their minimal peripheral structures.In Γ_1, let K_1^(1) and K_1^(2) be induced subgraphs generated by {a_1,a_2,a_3,a_4,a_5} and {a_6,a_7,a_8,a_9,a_10}, respectively. Observe that Γ_1 has only six induced 4-cycles which are not subgraphs of K_1^(1) and K_1^(2). Denote these cycles by L_1^(i) (i=1,2,⋯, 6). Let _1 be the set of all graphs L_1^(i) and K_1^(j). By Theorems <ref> and <ref>, _1 is the minimal peripheral structure of Γ_1. Moreover, no induced subgraph of a graph in _1 separates Γ_1. Therefore by Theorem <ref>, G_Γ_1 is hyperbolic relative to the collection _1=G_JJ∈_1 and the Bowditch boundary ∂ (G_Γ_1,_1) has no global cut point.Similarly, let K_2^(1) and K_2^(2) be the induced subgraphs of Γ_2 generated by {b_1,b_2,b_3,b_4,b_5} and {b_6,b_7,b_8,b_9,b_10}, respectively. Observe that Γ_2 has only four induced 4-cycles, denoted L_2^(i) (i=1,2,⋯, 4), such that each of them is not a subgraph of K_2^(1) and K_2^(2). Let _2 be the set of all graphs L_2^(i) and K_2^(j). Then by Theorems <ref> and <ref>, _2 is the minimal peripheral structure of Γ_2. Moreover, K_2^(1) and K_2^(2) are the only graphs in _2 which contain induced subgraphs that separate Γ_2. Therefore, G_Γ_2 is hyperbolic relative to the collection _2=G_JJ∈_2, the Bowditch boundary ∂ (G_Γ_2,_2) has global cut points and each of them is labelled by some left coset of G_K_2^(1) or G_K_2^(2) by Theorem <ref>.Finally, let K_3^(1) be an induced subgraph of Γ_3 generated by {c_6,c_7,c_8,c_9,c_10}. The graph Γ_3 has only three induced 4-cycles, denoted L_3^(i) (i=1,2,3), such that each of them is not a subgraph of K_3^(1). Assume that L_3^(1) is the induced 4-cycle generated by {c_1,c_2,c_5,c_4}. Let _3 be the set of all graphs L_3^(i) and K_3^(1). Again, by Theorem <ref> and <ref> we have that _3 is the minimal peripheral structure of Γ_3. Moreover, K_3^(1) and L_3^(1) are the only graphs in _3 which contain induced subgraphs that separate Γ_3. Therefore, G_Γ_3 is hyperbolic relative to the collection _3=G_JJ∈_3, the Bowditch boundary ∂ (G_Γ_3,_3) has global cut points and each of them is labelled by some left coset of G_K_3^(1) or G_L_3^(1) by Theorem <ref>.Note that all the groups in _i are one-ended. The Bowditch boundary ∂ (G_Γ_1,_1) has no global cut point, but the Bowditch boundaries ∂ (G_Γ_2,_2) and ∂ (G_Γ_3,_3) do. So, G_Γ_1 cannot be quasi-isometric to G_Γ_2 or G_Γ_3. Additionally, the Bowditch boundary ∂ (G_Γ_3,_3) has global cut points labelled by a left coset of G_L_3^(1). Meanwhile, no global cut point of the Bowditch boundary ∂ (G_Γ_2,_2) is labelled by the left coset of a peripheral subgroup which is quasi-isometric to G_L_3^(1). Therefore, G_Γ_2 and G_Γ_3 are not quasi-isometric. In the remainder of this section, we describe non-parabolic cut pairs in Bowditch boundaries of relatively hyperbolic right-angled Coxeter groups in terms of their defining graphs.Let Γ be a simplicial graph andbe a collection of induced proper subgraphs of Γ. Assume that the right-angled Coxeter group G_Γ is one-ended, hyperbolic relative to the collection =G_JJ∈, and suppose each subgroup inis also one-ended. If Γ has a separating complete subgraph suspension whose non-adjacent vertices do not lie in the same subgraph J∈, then the (0) boundary ∂Σ_Γ has a cut pair and the Bowditch boundary ∂(G_Γ,) has a non-parabolic cut pair.Let K be a separating complete subgraph suspension of Γ whose non-adjacent vertices u and v do not both lie in the same subgraph J∈. Let T be the set of all vertices of Γ which are both adjacent to u and v. Then T is a vertex set of a complete subgraph σ of Γ. Otherwise, the two vertices u and v both lie in the same 4-cycle. Thus, u and v lie in the same subgraph J∈, a contradiction.Let K̅=σ*{u,v}. We can easily verify the following properties of K̅. * For each J∈ the intersection K̅∩ J is empty or a complete subgraph.* No vertex outside K̅ is adjacent to the unique pair of nonadjacent vertices {u,v} of K̅.* K is an induced subgraph of K̅.Therefore, the collection =∪{K̅} satisfies all the conditions of Theorem <ref>, which implies that G_Γ is hyperbolic relative to the collection =G_JJ∈.Using an argument similar to that of Lemma <ref>, we can write Γ=Γ_1∪Γ_2 such that the following conditions hold: * Γ_1, Γ_2 are both proper induced subgraphs of Γ;* Γ_1∩Γ_2 is an induced subgraph L of K̅.* For each J in , J lies completely inside either Γ_1 or Γ_2Therefore, we can prove that the Bowditch boundary ∂ (G,) has a global cut point v_G_K̅ stabilized by the subgroup G_K̅ by using an argument similar to the one in Theorem <ref>.By Theorem <ref>, the Bowditch boundary ∂ (G_Γ,) is obtained from the (0) boundary ∂Σ_Γ by identifying the limit set of each peripheral left coset of a subgroup into a point. Let f be this quotient map. Since G_K̅ is two-ended, its limit set consists of two points w_1 and w_2 in ∂Σ_Γ. Therefore, f(w_1)=f(w_2)=v_G_K̅ and f(∂Σ_Γ-{w_1,w_2})=∂(G_Γ,)-{v_G_K̅}. Since ∂(G_Γ,)-{v_G_K̅} is not connected, the space ∂Σ_Γ-{w_1,w_2} is also not connected. This implies that {w_1,w_2} is a cut pair of the (0) boundary ∂Σ_Γ.Again, by Theorem <ref> the Bowditch boundary ∂ (G_Γ,) is obtained from the (0) boundary ∂Σ_Γ by identifying the limit set of each peripheral left coset of a subgroup into a point. Let h be this quotient map. The two points w_1 and w_2 do not lie in limit sets of peripheral left cosets of subgroups in . Therefore, h(w_1)≠ h(w_2) and they are non-parabolic points in the Bowditch boundary ∂ (G_Γ,). Moreover, h(∂Σ_Γ-{w_1,w_2})=∂(G_Γ,)-{h(w_1),h(w_2)} and the limit set of each peripheral left coset of a subgroup inlies completely inside ∂Σ_Γ-{w_1,w_2}.We observe that for any two points s_1, s_2 ∈∂Σ_Γ -{w_1,w_2} satisfying h(s_1)=h(s_2) the two points s_1 and s_2 both lie in some limit set C of a peripheral left coset of a subgroup in . Also, each subgroup inis one-ended, so C is connected. Therefore, s_1 and s_2 lie in the same connected component of ∂Σ_Γ -{w_1,w_2}. This implies that if U and V are different components of ∂Σ_Γ -{w_1,w_2}, then h(U)∩ h(V)=∅. Therefore, ∂(G_Γ,)-{h(w_1),h(w_2)} is not connected. This implies that {h(w_1),h(w_2)} is a non-parabolic cut pair of the Bowditch boundary ∂(G_Γ,). The following theorem describes non-parabolic cut pairs in Bowditch boundaries of relatively hyperbolic right-angled Coxeter groups in terms of their defining graphs.Let Γ be a simplicial graph andbe a collection of induced proper subgraphs of Γ. Assume that the right-angled Coxeter group G_Γ is one-ended, G_Γ is hyperbolic relative to the collection =G_JJ∈, and suppose each subgroup inis one-ended. If the Bowditch boundary ∂ (G_Γ,) has a non-parabolic cut pair, then Γ has a separating complete subgraph suspension. Moreover, if Γ has a separating complete subgraph suspension whose non-adjacent vertices do not lie in the same subgraph J ∈, then the Bowditch boundary ∂ (G_Γ,) has a non-parabolic cut pair.Since G_Γ is one-ended, G_Γ does not split over a finite group by the characterization of one-endedness in Stallings's theorem. Therefore by Proposition 10.1 in <cit.> the Bowditch boundary ∂ (G_Γ,) is connected. If the Bowditch boundary ∂ (G_Γ,) is a circle, then G_Γ is virtually a surface group, and the peripheral subgroups are the boundary subgroups of that surface by Theorem 6B in <cit.>. This is a contradiction because each peripheral subgroup is one-ended. Therefore, the Bowditch boundary ∂ (G_Γ,) is not a circle. We now assume that the Bowditch boundary ∂ (G_Γ,) has a non-parabolic cut pair {u,v}. Then u is obviously a non-parabolic local cut point. Therefore by Theorem 1.1 in <cit.>, G_Γ splits over a two-ended subgroup H.Since G_Γ splits over a two-ended subgroup H, there is an induced subgraph K of Γ which separates Γ such that G_K is contained in some conjugate of H by Theorem 1 in <cit.>. Because the group G_Γ is one-ended, the group G_K is two-ended. This implies that K is a complete subgraph suspension. The remaining conclusion is obtained from Proposition <ref>. Now, we discuss a few examples related to cut points in Bowditch boundaries of relatively hyperbolic right-angled Coxeter groups. These examples illustrate an application of Theorem <ref> to the problem of quasi-isometry classification of right-angled Coxeter groups.Let Γ_1 and Γ_2 be the graphs in Figure <ref>. Then G_Γ_1 and G_Γ_2 are both one-ended. Let _1 and _2 be the sets of all induced 4-cycles of Γ_1 and Γ_2, respectively. By Theorems <ref> and <ref>, the collection _i is the minimal peripheral structure of Γ_i for each i. Also, subgroups in each _i=G_JJ∈_i are virtually ^2 and thus one-ended. Moreover, for each i no induced subgraph of a graph in _i separates Γ_i. Thus by Theorem <ref>, both Bowditch boundaries ∂ (G_Γ_1,_1) and ∂ (G_Γ_2,_2) have no cut points. So in this case, we cannot use cut points to differentiate G_Γ_1 and G_Γ_2 up to quasi-isometry. However, the graph Γ_1 has no separating complete subgraph suspension, so by Theorem <ref> the Bowditch boundary ∂ (G_Γ_1,_1) has no non-parabolic cut pair. Meanwhile, Γ_2 has a cut pair (u,v) such that u and v do not lie in the same subgraph in _2. Again, by Theorem <ref> the Bowditch boundary ∂ (G_Γ_2,_2) has a non-parabolic cut pair. By Theorem <ref>, G_Γ_1 and G_Γ_2 are not quasi-isometric.§ GEOMETRIC STRUCTURE OF RIGHT-ANGLED COXETER GROUPS WITH PLANAR DEFINING NERVESIn this section, we study the coarse geometry of right-angled Coxeter groups with planar defining nerves. We first analyze the tree structure of planar flag complexes. Then we use this structure to study the relatively hyperbolic structure, group divergence, and manifold structure of right-angled Coxeter groups with planar nerves. Finally, we give a complete quasi-isometry classification of right-angled Coxeter groups which are virtually graph manifold groups. A simplicial complex Δ is called flag if any complete subgraph of the 1-skeleton of Δ is the 1-skeleton of a simplex of Δ. Let Γ be a finite simplicial graph. The flag complex of Γ is the flag complex with 1-skeleton Γ. A simplicial subcomplex B of a simplicial complex Δ is called full if every simplex in Δ whose vertices all belong to B is itself in B.The flag complex of Δ is planar if it can be embedded into the 2-dimensional sphere S^2. From now on every time we consider a flag complex it will be as a subspace of the 2-dimensional sphere S^2.The following lemma provides necessary and sufficient conditions for right-angled Coxeter groups of planar nerves to be one-ended.Let Δ⊂S^2 be a non-simplex planar flag complex.Then G_Δ is one-ended if and only if Δ is connected has no separating vertex and no separating edge.Assume that our flag complex Δ is not a simplex. It is known that the right-angled Coxeter group G_Δ is one-ended if and only if Δ is connected with no separating simplex. When Δ⊂S^2 is a connected flag complex and Δ has a separating simplex, we observe that Δ has a separating vertex or a separating edge.The following lemma provides necessary and sufficient conditions for when a one-ended right-angled Coxeter group with planar nerve splits over a two-ended subgroup.Let Δ⊂S^2 be a non-simplex connected flag complex with no separating vertex and no separating edge. Then G_Δ splits over a two-ended subgroup if and only if Δ has a cut pair or a separating induced path of length 2. By Lemma <ref>, the right-angled Coxeter group G_Δ is one-ended. Hence, G_Δ splits over a two-ended subgroup if and only if Δ contains a separating full subcomplex which is a cut pair or a suspension of a simplex. Since Δ⊂S^2 is a connected flag complex with no separating vertex and no separating edge, it is clear that if Δ has a separating full subcomplex which is a suspension of a simplex then Δ has a cut pair or a separating induced path of length 2. We now restrict our attention to right-angled Coxeter groups which are one-ended, non-hyperbolic, and not virtually ^2. Therefore, we need the following assumptions on defining nerves of right-angled Coxeter groups.The planar flag complex Δ⊂S^2: * is connected with no separating vertices and no separating edges (G_Δ is one-ended);* contains at least one induced 4-cycle (G_Δ is not hyperbolic);* is not a 4-cycle and not a cone of a 4-cycle (G_Δ is not virtually ^2).§.§ Decomposition of planar flag complexes In <cit.>, the second and thirds authors describe a tree-like decomposition for triangle-free planar graphs. This decomposition has “nice” vertex graphs and is one of the key ideas of <cit.>. The techniques of <cit.> also apply to planar flag complexes. Starting with some terminology, we review the basics of this construction in the setting of planar flag complexes.An induced 4–cycle σ of a flag complex Δ separates Δ if Δ-σ has at least two components. Let Δ⊂S^2 be a planar flag complex. An induced 4–cycle σ strongly separates Δ if Δ has non-empty intersection with both components of S^2-σ. The complex Δ is called prime if Δ satisfies the following conditions: * Δ is connected with no separating vertex and no separating edge;* Δ is not a 4–cycle but contains at least one induced 4-cycle;* Δ has no strongly separating induced 4-cycle (i.e. each induced 4-cycle bounds a region of S^2-Δ). The suspension of a non-triangle graph (not necessarily connected) with 3 vertices is prime.The notion of strongly separating in Definition <ref> depends on the choice of embedding map of the ambient flag complex into the sphere S^2. This notion is also based on the Jordan Curve Theorem that S^2-σ has two components.A prime flag complex Δ is called special if it the suspension of a non-triangle graph (not necessarily connected) with 3 vertices. We remark that there are only three possible special prime flag complexes (see Figure <ref>). The following lemma will help us understand the structure of prime flag complexes and it can be compared to Lemma 3.7 in <cit.>.Let Δ⊂S^2 be a prime flag complex. If Δ contains a full subcomplex B which is a special prime flag complex, then Δ is exactly the complex B. In particular, the 1-skeleton of Δ is 𝒞ℱ𝒮 if and only if Δ is special.Assume by way of contradiction that Δ is not equal B. Then there is a vertex of v of Δ which is not a vertex of B. Therefore, v is a point in S^2-B. Since B is a special prime flag complex, B is one of the complexes in Figure <ref>. By working on each case of B in Figure <ref>, we always can find an induced 4-cycle σ in B such that v and the unique vertex u of B-σ lies in different components of S^2-σ. This implies that σ strongly separates Δ. Therefore, Δ is not prime which is a contradiction.We now prove that the 1-skeleton of Δ is 𝒞ℱ𝒮 if and only if Δ is special. If Δ is special, we can see from Figure <ref> that the 1-skeleton of Δ is 𝒞ℱ𝒮. We now assume that the 1-skeleton of Δ is 𝒞ℱ𝒮. Then Δ contains a full subcomplex B which is a special prime flag complex. Therefore, Δ must be exactly the complex B as we proved above. Thus, Δ is special. The following lemma will play an important role in understanding the relatively hyperbolic structure of right-angled Coxeter groups of planar defining nerves (see Theorem <ref>).Let Δ⊂S^2 be a non-special prime flag complex. Let α_1 and α_2 be two distinct induced 4-cycles of Δ which have non-empty intersection. Then α_1∩α_2 is a vertex or an edge of Δ. Assume by way of contradiction that α_1∩α_2 contains two non-adjacent vertices, then the subcomplex of Δ induced by α_1∪α_2 contains a special full subcomplex B. In particular, Δ contains the special full subcomplex B. Also Δ is a prime flag complex. Therefore, by Lemma <ref>, Δ=B is special which is a contradiction. Therefore, α_1∩α_2 is a vertex or an edge of Δ.The following definition is an extension of the concept of strong visual decomposition of triangle free planar graphs (see Definition 3.4 in <cit.>) to planar flag complexes. Let Δ⊂S^2 be a planar flag complex and σ a strongly separating 4-cycle of Δ. Then S^2-σ has two components U_1 and U_2 which both intersect Δ. For each i=1,2, let Δ_i be σ together with components of Δ-σ in U_i. Then, Δ=Δ_1∪Δ_2 and Δ_1∩Δ_2=σ. We call the pair (Δ_1,Δ_2) a strong visual decomposition of Δ with respect to the given embedding of Δ into S^2. If the embedding is clear from the context, we just say the pair (Δ_1,Δ_2) is a strong visual decomposition of Δ along σ. The proof of the following proposition is analogous to the proof of Proposition 3.11 in <cit.>. We leave the details to the reader. Let Δ⊂S^2 be a planar flag complex satisfying Standing Assumptions. Then there is a finite tree T that encodes the structure of Δ as follows: * Each vertex v of T is associated to a full subcomplex Δ_v of Δ which is prime. Moreover, Δ_v≠Δ_v' if v≠ v' and ⋃_v∈ V(T)^Δ_v=Δ.* Each edge e of T is associated to an induced 4–cycle Δ_e of Δ. Moreover, Δ_e≠Δ_e' if e≠ e'. * Two vertices v_1 and v_2 of T are endpoints of the same edge e if and only if Δ_v_1∩Δ_v_2=Δ_e. Moreover, if V_1 and V_2 are vertex sets of two components of T removed the midpoint of e, then (⋃_v∈ V_1Δ_v,⋃_v∈ V_2Δ_v) is a strong visual decomposition of Δ along Δ_e.Moreover, the 1-skeleton of Δ is 𝒞ℱ𝒮 if and only if the 1-skeleton of Δ_v is also 𝒞ℱ𝒮 for each vertex v of T (i.e. Δ_v is a special prime complex by Lemma <ref>).§.§ Relatively hyperbolic structure and manifold structure of RACGs with planar defining nervesIn this subsection, we are going to discuss the proof of Theorem <ref>, Theorem <ref>, and Theorem <ref>. Before proving Theorem <ref>, we discuss some key concepts in the following remark.Let Δ⊂S^2 be a planar flag complex satisfying Standing Assumptions. Then the divergence of the right-angled Coxeter group G_Δ is linear if and only if the 1–skeleton of Δ is a join of two graphs of diameters at least 2 (see Theorem <ref>). Since Δ⊂S^2 is a planar flag complex satisfying Standing Assumptions, the 1–skeleton of Δ is a join of two graphs of diameters at least 2 if and only if Δ is a suspension of a graph K where K is an n–cycle for n≥ 4 or K has at least 3 vertices and it is a finite disjoint union of vertices and finite trees with vertex degrees 1 or 2. If K has at least 3 vertices and it is a finite disjoint union of vertices and finite trees with vertex degrees 1 or 2, then we call K a broken line.We begin with a few words about the proof of Theorem <ref>. The proof of Theorem <ref> is completely analogous to that of Theorem 1.6 of <cit.>, and we leave the details to the reader. The key ideas behind the proof of Theorem 1.6 of <cit.> are the tree structure of graphs and the fact that the intersection of two induced 4–cycles in a prime graph which is not a suspension of three points is empty, a vertex, or an edge. In the case of planar flag complexes, the analogous facts are addressed by Proposition <ref> and Lemma <ref>. We will use Theorem <ref> to prove Theorem <ref>. We first show that G_Δ is virtually the fundamental group of a 3-manifold M with empty or toroidal boundary if and only if the boundary of each region, if it exists, in S^2-Δ is a 4-cycle.We are now going to prove the necessity. Suppose that the boundary of each region, if it exists, in S^2-Δ is a 4-cycle. Let Γ be the 1-skeleton of Δ. Then the boundary of each region in S^2-Γ is a 3-cycle or a 4-cycle. The group G_Δ acts by reflections on a simply-connected 3-manifold N with fundamental domain a ball whose boundary is the cell structure on S^2 that is dual to Γ. Since the boundary of each region in S^2-Γ is a 3-cycle or a 4-cycle, the stabilizer of each vertex is _2^3 or D_∞× D_∞. Let H be a torsion-free finite index subgroup of G_Δ. The quotient space N/H has the property that the link of each vertex associated to a region with 4-cycle boundary (if it exists) is a torus, and we can thus remove a finite neighborhood from such a vertex (if it exists) to obtain a desired manifold M whose boundary is empty or a union of tori.We now prove sufficiency. Assume that G_Δ has a finite index subgroup H such that H is the fundamental group of a 3-manifold M with empty or toroidal boundary. Then the boundary of each region in S^2-Γ must be a 3-cycle or a 4-cycle, otherwise the Euler characteristic χ(H) is negative which is a contradiction (see Section 3.b in <cit.>). Therefore, the boundary of each region, if it exists, in S^2-Δ is a 4-cycle. Next, we consider all possible types of 3–manifold M via graph theoretic properties on Δ.The fact (<ref>) is clearly true since a right-angled Coxeter group induced by an n-cycle (n≥ 4) or a broken line is virtually a surface group. For the fact (<ref>) we first observe that the 1-skeleton of Δ is not a join of two graphs of diameter at least 2.Therefore, by Theorem <ref> the divergence of G_Δ (also the divergence of π_1(M)) is quadratic. By Theorem <ref> M must be a graph manifold. For the facts (<ref>) and (<ref>) we note that if the 1-skeleton of Δ is not 𝒞ℱ𝒮, then then the divergence of G_Δ (also the divergence of π_1(M)) is exponential by Theorem <ref>. Therefore, M must be a hyperbolic manifold with boundary or a mixed manifold. If, in addition, Δ contains at least a separating induced 4-cycle, then M contains at least a JSJ torus and it must be a mixed manifold which proves (<ref>).For (<ref>) we see that Δ is non-special prime flag complex. Therefore, in this case, G_Δ is hyperbolic relative to the collection of all right-angled Coxeter subgroups G_σ where σ is an induced 4-cycles of Δ. Due to the peripheral structure of G_Δ, the JSJ decomposition of M cannot have a Seifert piece. Moreover, the JSJ decomposition of M must consist of a single piece, otherwise π_1(M) would split over ^2. This implies that M is a hyperbolic manifold with boundary. We now study the relatively hyperbolic structure of right-angled Coxeter groups which are not virtually the fundamental group of a 3-manifold with empty or toroidal boundary. We prove that each such group is hyperbolic relative to a collection of its proper subgroups and we then study its Bowditch boundary with respect to the minimal peripheral structure. The following two lemmas will help us study the relatively hyperbolic structure of right-angled Coxeter groups which are not virtually the fundamental group of a 3-manifold with empty or toroidal boundary.Let Δ⊂S^2 be a planar flag complex. Let (Δ_1,Δ_2) be a strong visual decomposition of Δ along some induced 4–cycle σ. If R is a region of S^2-Δ, then R is a region of either S^2-Δ_1 or S^2-Δ_2. Let U and V be two regions of S^2-σ such that Δ_1-σ⊂ U and Δ_2-σ⊂ V. Since R is a connected subset in S^2-σ, R lies inside either U or V (say U). We now prove that R is also a region of S^2-Δ_1. We first observe that S^2-Δ⊂S^2-Δ_1⊂ (S^2-Δ)∪ V. Therefore R lies in some region R_1 of S^2-Δ_1. Since R_1 is a connected subset of S^2-σ,R_1 lies either in U or V, say U. Moreover, R_1∩ U contains the non-empty set R. Therefore R_1 must lie in U which implies that R_1∩ V=∅. Thus R_1 is a connected subset of S^2-Δ. This implies that R_1 must lie in some region of S^2-Δ and the region must be R. Therefore, R=R_1 is also a region of S^2-Δ_1. Let Δ⊂S^2 be a planar flag complex satisfying Standing Assumptions. Let T be a finite tree that encodes the structure of Δ as in Proposition <ref>. Then for each region R of S^2-Δ there is a vertex v of T such that R is also a region of S^2-Δ_v. The lemma can be proved by induction on the number of vertices of tree T by using Lemma <ref>. In the rest of this subsection, we prove Theorem <ref>. The following proposition shows the existence of a non-trivial relatively hyperbolic structure of right-angled Coxeter groups which are not virtually the fundamental group of a 3-manifold with empty or toroidal boundary.Let Δ⊂S^2 be a connected flag complex with no separating vertex and no separating edge. Assume that the boundary of some region in S^2-Δ is an n-cycle for some n≥ 5. Then the 1-skeleton of Δ is not 𝒞ℱ𝒮. In particular, the divergence of G_Δ is exponential (i.e. G_Δ is non-trivially relatively hyperbolic). Since the boundary of some region in S^2-Δ is an n-cycle for some n≥ 5, it follows that Δ is not a simplex, a 4-cycle, or cone on a 4-cycle. If Δ does not contain an induced 4-cycle, then it is clear that the 1-skeleton of Δ is not 𝒞ℱ𝒮. Therefore, we assume that Δ contains at least one induced 4-cycle. In this case, we note that Δ satisfies Standing Assumption.Let T be a tree that encodes the structure of Δ as in Proposition <ref>. By Lemma <ref> there is a vertex v of T such that the boundary of some region in S^2-Δ_v is an n-cycle for some n≥ 5. This implies that Δ_v is not a special prime complex. Therefore, by Proposition <ref> the 1-skeleton of Δ is not 𝒞ℱ𝒮. By Theorem <ref> the divergence of right-angled Coxeter group G_Δ is exponential. We now study the Bowditch boundary of the relatively hyperbolic right-angled Coxeter groups in Proposition <ref>. We begin with the definition of Sierpinski carpet and recall a result due to Świa̧tkowski <cit.> which gives sufficient conditions for when the (0) boundary of a right-angled Coxeter group satisfying Standing Assumptions is a Sierpinski carpet. Let D_1, D_2,⋯ be a sequence of open disks in a 2-dimensional sphere S^2 such that * D̅_i ∩D̅_j =∅ for i≠ j,* diam(D_i) → 0 with respect to the round metric on S^2, and* ⋃ D_i is dense. Then X= S^2-⋃ D_i is a Sierpinski carpet. The circles C=∂D̅_i ⊂ X are called peripheral circles. Let Δ⊂S^2 be a planar flag complex satisfying Standing Assumptions. Suppose also that the following two conditions hold: * For any two distinct regions of S^2-Δ, the intersection of their boundaries is empty, or a vertex, or an edge;* The boundary σ of each region of S^2-Δ satisfies the condition that for any two vertices of σ staying at distance 2 in Δ any induced path of length 2 in the 1-skeleton of Δ connecting these vertices is entirely contained in σ.Then the (0) boundary ∂Σ_Δ is the Sierpinski carpet. We now give a proof of Theorem <ref>.We assume that Δ satisfies both Conditions (1) and (2) of Theorem <ref> and we will prove that G_Δ has a non-trivial minimal peripheral structure with Bowditch boundary the Sierpinski carpet. We first show that the (0) boundary ∂Σ_Δ is the Sierpinski carpet by using Theorem <ref>. Then we will use Theorem <ref> to prove that G_Δ has a non-trivial minimal peripheral structure with Bowditch boundary the Sierpinski carpet. Since the 1-skeleton of Δ has no cut pair, Δ satisfies Condition (1) of Theorem <ref>. Also, the 1-skeleton of Δ has no separating induced path of length 2. Now, we show that Δ also satisfies Condition (2) of Theorem <ref>. By the way of contradiction that the boundary σ of some region of S^2-Δ does not satisfy Condition (2). Then there is an induced path α of length 2 in the 1–skeleton of Δ such that σ∩α consists of two distinct vertices u and v. We now claim that α separates the 1–skeleton of Δ which leads to a contradiction. We first assume that Δ=α∪σ. Then Δ-α=σ-{u,v} is disconnected. Therefore, α separates Δ which is also its 1–skeleton. We now assume that there is a point w in Δ-(α∪σ). We observe that σ is the union of two paths σ_1 and σ_2 such that σ_1∩σ_2={u,v}. Since σ bounds a region R in S^2-Δ, the path α lies completely outside the region R. Therefore, S^2-(α∪σ) contains exactly three regions and R is one of them. Since R is also a region in S^2-Δ, the point w can not lie in R. Therefore, w must lie in the region of S^2-(α∪σ) which is bounded by α∪σ_1 or α∪σ_2 (say α∪σ_1). Therefore, α separates w from a vertex of σ_2 in the 1–skeleton of Δ. This implies that α separates the 1–skeleton of Δ which leads to a contradiction. Thus, Δ satisfies Condition (2) of Theorem <ref>. Therefore, the (0) boundary ∂Σ_Δ is the Sierpinski carpet. Since the 1-skeleton of Δ has no separating induced 4-cycle, each induced 4-cycle of Δ must bound a region of S^2-Δ. Therefore, the intersection of two induced 4-cycles of Δ is empty, or a vertex, or an edge. Thus by Theorem <ref> the group G_Δ is relatively hyperbolic with respect to the collectionof all right-angled Coxeter groups induced by some induced 4-cycle of Δ. We now prove the Bowditch boundary ∂ (G_Γ,) is a Sierpinski carpet. We first claim that if σ is the boundary of a region in S^2-Δ, then the limit set of subgroup G_σ is a peripheral circle of the Sierpinski carpet ∂Σ_Γ. We see that the intersection of σ with an induced 4-cycle of Δ is empty, or a vertex, or an edge. Therefore by Theorem <ref> again G_Δ is relatively hyperbolic with respect to the collection =∪{G_σ}. By Theorem <ref>, the Bowditch boundary ∂ (G_Γ,) is obtained from the (0) boundary ∂Σ_Γ by identifying the limit set of each peripheral left coset of a subgroup into a point. Let f be this quotient map. Let v_G_K be the point in ∂ (G_Γ,) that is the image of the limit set of subgroup G_σ under the map f. Suppose by way of contradiction that the limit set of the subgroup G_σ is a separating circle of the Sierpinski carpet ∂Σ_Γ. Then the point v_G_K is a global cut point of ∂ (G_Γ,). We know that σ bounds a region of S^2-Δ while the intersection of the boundaries of two distinct regions of S^2-Δ is empty, or a vertex, or an edge. This implies that no induced subgraphs of σ separates the 1-skeleton of Δ. Therefore by Theorem <ref> point v_G_K is not a global cut point of ∂ (G_Γ,) which is a contradiction. Thus the limit set of subgroup G_σ is a peripheral circle of the Sierpinski carpet ∂Σ_Γ.By Theorem <ref> again, the Bowditch boundary ∂ (G_Γ,) is obtained from the Sierpinski carpet ∂Σ_Γ by identifying each peripheral circle of ∂Σ_Γ which is the limit set of a peripheral left coset of a subgroup into a point. Let h be this quotient map. Then h is G_Γ–equivariant. By Condition (1) some boundary γ of a region of S^2-Δ is an n-cycle with n≥ 5. Therefore, the limit set of subgroup G_γ is a peripheral circle C of the Sierpinski carpet ∂Σ_Γ by the above argument. Since G_γ is not a group in , the peripheral circle C is not collapsed to a point via the map h. We now prove that the Bowditch boundary ∂ (G_Γ,) is a Sierpinski carpet. Let ℒ be the collection of all translates of the peripheral circle C by group elements in G_Δ. Then all peripheral circles in ℒ survive via the map h. Therefore, we can consider ℒ as a collection of pairwise disjoint circles in ∂ (G_Γ,). Moreover, ℒ is a G_Γ-invariant collection in ∂ (G_Γ,) since the quotient map h is G_Γ-equivariant. Also the action of G_Γ on ∂ (G_Γ,) is minimal (i.e. the orbit of each single point is dense in ∂ (G_Γ,)) by <cit.>. Therefore, the union of all circles in ℒ is dense in ∂ (G_Γ,). Fix metrics on (0) boundary ∂Σ_Γ and Bowditch boundary ∂ (G_Γ,). We consider ℒ as a sequence (C_n) of circles and we need to prove that (C_n)→ 0 with respect to the metric on ∂ (G_Γ,). Let ϵ be an arbitrary positive number and let {U_α}_α∈Λ be a cover of ∂ (G_Γ,) consisting of the open ball with diameter ϵ. Then {h^-1(U_α)}_α∈Λ is an open cover of ∂Σ_Γ. Since (C_n)→ 0 with respect to the metric on ∂Σ_Γ and ∂Σ_Γ is a compact space, then each set C_n lies in some member of the cover {h^-1(U_α)}_α∈Λ for each n sufficiently large. Therefore, each set C_n=h(C_n) also lies in some member of the cover {U_α}_α∈Λ for each n sufficiently large. This implies that the diameter of each such circle C_n is less than ϵ with respect to the metric on ∂ (G_Γ,). This implies that (C_n)→ 0 with respect to the metric on ∂ (G_Γ,). Therefore, the Bowditch boundary ∂ (G_Γ,) is a Sierpinski carpet by <cit.>.We now assume that G_Δ has a non-trivial minimal peripheral structure with Bowditch boundary the Sierpinski carpet. We will prove that Δ satisfies both Conditions (1) and (2). In fact, if Δ does not satisfy Condition (1), then G_Γ is virtually a 3-manifold group with empty boundary or tori boundary by Theorem <ref>.Therefore, the Bowditch boundary of G_Δ with respect to a non-trivial minimal peripheral structure is not the Sierpinski carpet which is a contradiction. Therefore, Δ must satisfy Condition (1). Also Δ must satisfy Condition (2). Otherwise by Theorem <ref> and Theorem <ref> the Bowditch boundary of G_Δ with respect to a non-trivial minimal peripheral structure contains a global cut point or a cut pair which is a contradiction.§.§ Quasi-isometry classification of virtually graph manifold group RACGsIn this subsection, we prove Theorem <ref>. This theorem generalizes Theorem 1.1 in <cit.> by removing the condition “triangle-free” from the hypotheses.In the rest of this section, we will assume that the planar flag complex Δ⊂S^2 satisfies Standing Assumptions and the 1-skeleton of Δ is a non-join 𝒞ℱ𝒮 graph. In particular, Δ is not the suspension of an n-cycle. Moreover, Δ also does not contain a proper full subcomplex which is the suspension of an n-cycle since Δ⊂S^2 is a planar flag complex. Let T be a tree that encodes the structure of Δ as in Proposition <ref>. Since the 1–skeleton of Δ is 𝒞ℱ𝒮, it is shown in Proposition <ref> that each vertex subcomplex Δ_v is the suspension of a non-triangle graph with 3 vertices. For the purpose of obtaining the quasi-isometry classification of our 𝒞ℱ𝒮 right-angled Coxeter groups, the tree structure T in Proposition <ref> is not the correct tree structure to use. So, we now modify the tree T to obtain a new two-colored new tree that encodes the structure of Δ. Step 1: We color an edge of T by one of two colors: red and blue as follows. Let e be an edge of T with vertices v_1 and v_2. If either Δ_v_1 or Δ_v_2 is a suspension of a path of length 2 we color the edge e red. Next, we consider two cases. If Δ_v_1 and Δ_v_2 have the same suspension points, then we color the edge e red. Otherwise, we color e blue. Step 2: Let ℛ be the union of all red edges of T. We remark that ℛ is not necessarily connected. We form a new tree T_r from the tree T by collapsing each component C of ℛ to a vertex labelled by v_C and we associate each such new vertex v_C to the complex Δ_v_C=⋃_v∈ V(C)Δ_v. For each vertex v of T_r which is also a vertex of T we still assign v the complex Δ_v as in the previous tree T structure. We observe that for each vertex v of the new tree T_r the associated vertex complex Δ_v is either one of the first two complexes in Figure <ref> or a union of at least two special complexes which all share a pair of suspension vertices. Therefore, Δ_v is the suspension of a broken line ℓ_v which is distinct from a path of length 2. We call the number of induced 4-cycles of Δ_v⊂S^2 that bound a region in S^2-Δ_v the weight of v denoted by w(v).By construction, each edge e of the new tree T_r is also an edge of the old tree T. Therefore, we still assign the edge e in the new tree T_r the complex Δ_v as in the previous tree T structure. We observe that for each edge e of T_r with vertices v_1 and v_2 the induced 4-cycle Δ_e bounds a region in both S^2-Δ_v_1 and S^2-Δ_v_2. Therefore, the weight of each vertex v is always greater than or equal to the degree of v in T_r. Moreover, if v_1 and v_2 are two adjacent vertices in T_r, then suspension vertices of Δ_v_1 are vertices of ℓ_v_2 and similarly suspension vertices of Δ_v_2 are vertices of ℓ_v_1. Step 3: We now color vertices of T_r. For each vertex v of T_r, the complex Δ_v is a suspension of a broken line set ℓ_v. Observe that in Step 2 that the weight of each vertex v is always greater than or equal to the degree of v in T. Therefore, we now color v black if its weight is strictly greater than its degree. Otherwise, we color v white. We now summarize some key properties of the tree T_r. * Each vertex v of T_r is associated to a full subcomplex Δ_v of Δ that is a suspension of a broken line ℓ_v which is distinct from a path of length 2.The weight w(v) of each vertex v is required to be greater than or equal its degree in T_r. We color v black if its weight is strictly greater than its degree. Otherwise, we color v white.* Δ_v≠Δ_v' if v≠ v' and ⋃_v∈ V(T_r)^Δ_v=Δ.* Each edge e of T_r is associated to an induced 4–cycle Δ_e of Δ. If e≠ e', then Δ_e≠Δ_e'.* Two vertices v_1 and v_2 of T_r are endpoints of the same edge e if and only if Δ_v_1∩Δ_v_2=Δ_e. Moreover, the induced 4-cycle Δ_e bounds a region in both S^2-Δ_v_1 and S^2-Δ_v_2. Also, suspension vertices of Δ_v_1 are vertices of ℓ_v_2. Similarly, suspension vertices of Δ_v_2 are vertices of ℓ_v_1. Lastly, if V_1 and V_2 are vertex sets of two components of T_r removed the midpoint of e, then (⋃_v∈ V_1Δ_v)∩(⋃_v∈ V_2Δ_v)=Γ_e. Let Δ⊂S^2 be a planar flag complex satisfying Standing Assumptions with the 1-skeleton a non-join 𝒞ℱ𝒮 graph. A tree that encodes the structure of Δ carrying Properties (1), (2), (3), and (4) as above is called a visual decomposition tree of Δ.The existence of a visual decomposition tree for a planar flag complex satisfying Standing Assumptions with the 1-skeleton a non-join 𝒞ℱ𝒮 graph is guaranteed by Construction <ref>.Let Δ⊂S^2 be a planar flag complex satisfying Standing Assumptions with the 1-skeleton a non-join 𝒞ℱ𝒮 graph. Let T_r be a visual decomposition tree of Δ. Since Δ is planar, G_Δ is virtually a 3–manifold group by Theorem <ref>. However, for the purpose of obtaining a quasi-isometry classification we will construct explicitly a 3–manifold Y on which G_Δ acts properly and cocompactly. We note that the construction of the manifold Y is associated to the graph T_r. Therefore, we can import the work of Behrstock-Neumann <cit.> to prove Theorem <ref>.We now construct a 3-manifold Y on which the right-angled Coxeter group G_Δ acts properly and cocompactly. For each vertex v of T_r, the complex Δ_v is a suspension of a broken line ℓ_v of Δ. We now let b and c be suspension vertices and let Σ_v be the Davis complex associated to the broken line ℓ_v. We fatten the 1-skeleton of Σ_v to obtain a universal cover of a hyperbolic surface with boundary as follows:Let n be the number vertices ofℓ_v. We replace each vertex of the 1-skeleton Σ^(1)_v by a regular n–gon with sides labelled by vertices of ℓ_v. We assume that two edges of such an n–gon labeled by a_1 and a_2 in V(ℓ_v) are adjacent if and only if either a_1 and a_2 are vertices of an edge of ℓ_v or the set {a_1, a_2,b,c} forms an induced 4–cycle of Δ_v that bounds a region in S^2-Δ_v. We also assume the length side of the n–gon is 1/2. We replace each edge E labelled by a_i by a strip E×[-1/4,1/4]. We label each side of length 1 of the strip E×[-1/4,1/4] by a_i and we identify the edge E to E×{0} of the strip. If u is an endpoint of the edge E of Σ^(1)_v, then the edge {u}×[-1/4,1/4] is identified to the side labelled by a_i of the n–gon that replaces u. We observe that for each pair of adjacent vertices a_1 and a_2 of ℓ_v we have an induced 4–cycle in Σ^(1)_v with one pair of opposite sides labeled by a_1 and the other pair of opposite sides labeled by a_2. We note that this 4–cycle bounds a 2–cell in Σ_v. In the “fattening” of Σ^(1)_v constructed above we also have such 4–cycles and we also fill each with a 2–cell as in Σ_v. We denote F(Σ_v) to be the resulting space (See Figure <ref>). Clearly, G_ℓ_v acts properly and cocompactly on the space F(Σ_v). Additionally, F(Σ_v) is a simply connected surface with boundary and each boundary component of F(Σ_v) is an infinite concatenation of edges labelled by a_1 and a_2 where a_1 and a_2 are vertices of ℓ_v such that the set {a_1, a_2,b,c} forms an induced 4–cycle that bounds a region in S^2-Δ_v. We denote such a boundary by α_a_1,a_2.The group G_{b,c} acts on the line α that is a concatenation of edges labelled by b and c by edge reflections. Let P_v=F(Σ_v)×α and equip P_v with the product metric. Then, the group G_Δ_v acts properly and cocompactly on P_v in the obvious way. The space P_v is the universal cover of the trivial circle bundle of a hyperbolic surface with nonempty boundary.Moreover, for each pair of vertices a_1 and a_2 of ℓ_v such that the set {a_1, a_2,b,c} forms an induced 4–cycle that bounds a region in S^2-Δ_v the right-angled Coxeter groups generated by {a_1,a_2,b,c} acts on the boundary α_a_1,a_2×α in a way analogous to its action on the Davis complex. We label this plane by {a_1,a_2,b,c}. If v_1 and v_2 are two adjacent vertices in T_r, then the pair of suspension vertices (a_1,a_2) of Δ_v_1 is a pair of vertices of ℓ_v_2 and the pair of suspension vertices (b_1,b_2) of Δ_v_2 is a pair of of vertices of ℓ_v_1. Moreover, the set {a_1, a_2,b_1,b_2} forms an induced 4–cycle that bounds a region in both S^2-Δ_v_1 and S^2-Δ_v_2. Therefore, the spaces P_v_1 and P_v_2 have two Euclidean planes which are labeled by {a_1,a_2,b_1,b_2} as constructed above. Thus, using the Bass-Serre tree T̃_r of the decomposition of G_Δ as tree T_r of subgroups we can form a 3-manifold Y by gluing copies of P_v appropriately and we obtain a proper, cocompact action of G_Δ on Y. We are ready to prove the quasi-isometry classification theorem. The proof is identical with the proof of (3) in Theorem 1.1 <cit.>. We can color vertices of the Bass-Serre tree T̃_r so that the quotient map q: T̃_r→ T_r preserves the vertex coloring. Moreover, the map q: T̃_r→ T_r is a week covering. Therefore, the trees T̃_r and T_r are bisimilar. (See Definitions <ref> and <ref> for the definitions of weak covering and bisimilarity.) We observe that a vertex of T̃_r is colored black if and only if the corresponding Seifert manifold contains a component of the boundary of Y. Using the proof of Theorem 3.2 in <cit.>, we obtain the proof of Theorem <ref>.Let Δ and Δ' be flag complexes in Figure <ref>. A visual decomposition tree T_r of Δ is shown in the same figure with the following information. The complex Δ_u_1 is the suspension of the broken line with three vertices a_1, a_3, and a_5 with two suspension vertices a_6 and a_7. The complex Δ_u_2 is the suspension of the broken line with three vertices a_2, a_6, and a_7 with two suspension vertices a_1 and a_3. The complex Δ_u_3 is the suspension of the broken line with three vertices a_4, a_6, and a_7 with two suspension vertices a_3 and a_5. The complex Δ_u_4 is the suspension of the broken line with three vertices a_6, a_7, and a_8 with two suspension vertices a_1 and a_5. We note, that the vertices u_2, u_3, and u_4 all have weight 2 and the vertex u_1 has weight 3. By comparing their degrees, the vertices u_2, u_3, and u_4 are colored black while u_1 is colored white.Similarly, a visual decomposition tree T_r' of Δ' is also shown in Figure <ref> with the following information. The complex Δ_v_1 is the suspension of the broken line with four vertices b_1, b_3, b_5, and b_9 with two suspension vertices b_6 and b_7. The complex Δ_v_2 is the suspension of the broken line with three vertices b_2, b_6, and b_7 with two suspension vertices b_1 and b_3. The complex Δ_v_3 is the suspension of the broken line with three vertices b_4, b_6, and b_7 with two suspension vertices b_3 and b_5. The complex Δ_v_4 is the suspension of the broken line with three vertices b_6, b_7, and b_8 with two suspension vertices b_1 and b_9. We observe that the vertices v_2, v_3, and v_4 all have weight 2 and the vertex v_1 has weight 4. By comparing their degrees we see that all vertices of T_r' are colored black. Thus, the visual decomposition trees T_r and T_r' are not bisimilar even though they are isomorphic if we ignore the vertex colors. Therefore, the groups G_Γ and G_Γ' are not quasi-isometric. alpha | http://arxiv.org/abs/1708.07818v4 | {
"authors": [
"Matthew Haulmark",
"Hoang Thanh Nguyen",
"Hung Cong Tran"
],
"categories": [
"math.GR",
"20F65, 20F67, 20F69"
],
"primary_category": "math.GR",
"published": "20170825173532",
"title": "On the relative hyperbolicity and manifold structure of certain right-angled Coxeter groups"
} |
maketitle 0.5 -1 Limited Feedback Scheme forCommunications in 5G Cellular Networks with Reliability and Cellular Secrecy Outage ConstraintsFaezeh Alavi, Nader Mokari, Mohammad R. Javan, and Kanapathippillai Cumanan December 30, 2023 ============================================================================================================================ In this paper, we propose a device to device (D2D) communication scenario underlaying a cellular network where both D2D and cellular users (CUs) are discrete power-rate systems with limited feedback from the receivers.It is assumed that there exists an adversary which wants to eavesdrop on the information transmission from the base station (BS) to CUs. Since D2D communication shares the same spectrum with cellular network, cross interference must be considered. However, when secrecy capacity is considered, the interference caused by D2D communication can help to improve the secrecy communications by confusing the eavesdroppers. Since both systems share the same spectrum, cross interference must be considered. We formulate the proposed resource allocation into an optimization problem whose objective is to maximize the average transmission rate of D2D pair in the presence of the cellular communications under average transmission power constraint. For the cellular network, we require a minimum average achievable secrecy rate in the absence of D2D communication as well as a maximum secrecy outage probability in the presence of D2D communication which should be satisfied. Due to high complexity convex optimization methods, to solve the proposed optimization problem, we apply Particle Swarm Optimization (PSO) which is an evolutionary approach. Moreover, we model and study the error in the feedback channel and the imperfectness of channel distribution information (CDI) using parametric and nonparametric methods. Finally, the impact of different system parameters on the performance of the proposed scheme is investigated through simulations. The performance of the proposed scheme is evaluated using numerical results for different scenarios.Index Terms– Device to Device (D2D) communications, Limited Rate Feedback, Physical (PHY) layer security, Particle swarm optimization. § INTRODUCTION§.§ Background and Motivation The growth of the cellular networks and the number of users as well as the emergence of the new multimedia based services result in growing demands for high data rate and capacities which is beyond the capability of forth generation (4G) wireless networks. Recently, the fifth generation (5G) cellular network has triggered a great attention to provide high data rate and low latency services in a power and spectrally efficient manner. Introducing new applications like context-aware applications requires the direct communications of neighboring devices. In this context, device to device (D2D) communication has been considered as a promising technique for 5G wireless networks <cit.>. D2D communication operates as an underlay network to a cellular network <cit.> and enablesreusing the cellular resources which increases the spectral efficiency and the system capacity. In D2D communications, two neighboring devices use the cellular bandwidth to communicate directly without the help of cellular base station (BS).Although D2D communications can improve the spectral efficiency,it should provide access to licensed spectrum with a controlled interference to avoid the uncertainties of the cellular network performance. Therefore, interference management is a critical issue for D2D underlaying cellular networks without considering it, the effectiveness of D2D communication links will be deteriorated. In this sense, several papers have proposed mechanisms for interference mitigation and avoidance. To perform interference management in D2D underlaying cellular network, one approach is to consider cooperative communications. In this way, a D2D user equipped with multiple antennas acts as an in-band relay to a cellular link where the multi-antenna relay is able to help decoding messages, cancelling interference, and providing multiplexing gain in the network <cit.>. In <cit.>, power control problem for the D2D users is investigated in order to optimize the energy efficiency of the user equipments (UEs) as well as to ensure that the quality of service (QoS) of D2D devices and UEs does not fall below the acceptable target. The problem of interference management through multi rate power control for D2D communications is studied in <cit.>. The transmission power levels of D2D users are optimized to maximize the cell throughput while preserving the signal-to-interference-plus-noise ratio (SINR) performance for the cellular user.In <cit.>, authors guarantee the reliability of D2D links and mitigate the interference from the cellular link to the D2D receivers. A pricing framework has been suggested in <cit.> where BS protects itself by utilizing game theory approach. To increase the security of wireless transmission, physical layer securityhas been developed based on information theoretic concepts <cit.>. From the physical layer point of view, the security is quantified by the secrecy rate which is defined as the difference of achievable rate between the legitimate receiver and the rate overheard by eavesdroppers <cit.>. In this sense, unlike the previous work on D2D underlaying cellular networks in which the focus is on the interference mitigation and avoidance, the interference works well when secrecy capacity of the cellular communication is taken into consideration <cit.>. In other words, it can be assumed that the D2D communication works as a friendly jammer and its interference is helpful for the secure cellular network to improve secrecy capacity. In practice, since the eavesdropper is a passive attacker, obtaining its channel state information (CSI) is impossible in many situations. In this case, the secrecy outage probability can be used as a security performance criterion.The performance of previous works is based on the fact that the perfect CSI of all links is available. However, due to the estimation errors and feedback delay, perfect CSI may not be available. In addition, the feedback channel has a limited capacity since transmitting unlimited feedback information between transmitters and receivers means passing a huge amount of bits for signaling. To tackle this issue, the limited feedback channel model can be employed. In the limited feedback channel, the space of channel gains is divided into a finite number of regions, and instead of channel gain values, the index of the fading region in which the actual channel gain lies is feedbacked <cit.>.In <cit.>, the authors study the effect of the feedback information on the performance of the D2D underlaying cellular networks and develop user selection strategies based on limited feedback. §.§ Contributions and Organization In this paper, we study D2D communications in the presence of the cellular communications while there exists a malicious user which wants to eavesdrop the information transmitted from the BS to CU. We assume that the legitimate transmitters do not have the perfect values of the channel power gains and the knowledge about their respective direct channel power gains is obtained via their dedicated limited rate feedback channel. In other words, we assume that the space of the channel gains is divided into a finite number of regions. Then given the actual value of the channel gains, the receiver determines the index of the region in which the channel gain lies and feedbacks the index of that region to the corresponding receiver. Note that, the cellular system is superior to D2D communication and D2D pair uses the spectrum of cellular networks in an opportunistic manner. The concurrent transmission of cellular network and D2D pair, if exists, degrades the performance of both systems due to the cross interference between these two systems. Therefore, in this paper, we consider the performance of the cellular system in both the presence and the absence of D2D communication. Precisely, we require that the average transmission rate of cellular user in the absence of the D2D communication should be above a predefined threshold while its performance in the presence of the D2D communication, in terms of outage probability, satisfies a predefined threshold. Our objective is to maximize the average achievable data rate of the D2D pair in the presence of the cellular communication while individual constraints on the average transmission power of the cellular BS and the D2D pair should be satisfied. Due to non-convexity and nonlinearity of the proposed problem, to find the optimal solution of the problem, we use particle swarm optimization (PSO) method which is an evolutionary algorithm <cit.>.In reality, the feedback channels can be affected by the noise which makes the transmitter select an incorrect code word from the designed code book. Therefore, in this paper, we consider the effect of error in the feedback channel on the performance of the proposed scheme by incorporating such error into the problem formulation. We further study the effect of channel distribution information (CDI) imperfectness. Parametric and nonparametric methods are investigated in estimating the CDI of the channels. The contributions of this paper are as follows: * We develop a mathematical model for the secure communication in D2D communication underlaying the cellular network in which the knowledge of transmitters about the CSIs is obtained via a limited rate feedback channel. In our model, we consider the cross interference between the cellular network and the D2D pair explicitly and formulate the resource allocation problem as an optimization framework. * To solve this optimization problem and obtain its solutions which are the fading regions' boundaries and transmission power levels, we use PSO algorithm which is an evolutionary algorithm. * We further consider the effect of the noise in the feedback channel and incorporate it into our optimization problem. In this case, the error in the feedback channel would lead the transmitters to choose the incorrect code-words. We formulate the corresponding optimization problem and solve it using the PSO approach. * We also consider the effect of the CDI imperfectness in our proposed scheme. In this case, the CDI's parameters are not perfectly known and parametric and non-parametric approaches are used to estimate the CDI parameters.Finally, the performance of the proposed scheme in different scenarios is investigated via simulations.The paper is organized as follows. System model is described in Section <ref>. Limited rate feedback schemes are proposed in Section <ref>. The limited rate feedback resource allocation problem is formulated and solved in Section <ref>. In Section <ref>, practical considerations, i.e., noisy feedback channel and CDI estimation error, are investigated. Simulation and numerical results are provided in Section <ref> and finally conclusions are drawn in Section <ref>.§ SYSTEM MODEL We consider a D2D communication scenario underlaying an existing cellular network. It is assumed the downlink transmission in the cellular network where the BS transmits information to a cellular user while at the same time; the existing D2D pair performs its own transmission on the same channel. Such a scenario can be interpreted as there are many cellular users in the network each of which is assigned to a channel over which the BS sends information to them. Assuming this assignment is performed based on some network parameters and is fixed, two cellular users exploit one of the available cellular channels to perform their information transmission directly. In this paper, we assume that this assignment is predefined.In addition to cellular user and D2D pair, we assume that there exists a malicious user which wants to eavesdrop on the information transmission of cellular network, i.e., from the BS to the CU. However, the malicious user does not eavesdrop on the D2D pair. Such assumption can be justified when the malicious user is not aware of the existence of D2D pair as such sharing can be performed opportunistically (i.e., D2D pair may or may not exist at any time) when the malicious user is not interested in D2D pair information, or when the D2D pair applies upper layer security measures, e.g., cryptography. In such case, the malicious user treats the signals from D2D pair as noise. Let h^BC, h^BD, h^DD, h^DC, h^BE, and h^DE denote, respectively, the noise normalized channel power gain of the channel from BS to CU, from BS to the receiver of D2D pair (RD2D), from the transmitter of D2D pair (TD2D) to RD2D, from TD2D to CU, from BS to the eavesdropper, and from the TD2D to the eavesdropper. We assume that all channels undergo independent block fading with Rayleigh distribution meaning that the channel power gains, i.e., h^BC, h^BD, h^DD, h^DC, h^BE, and h^DE, are exponentially distributed with the mean of h̅^BC, h̅^BD, h̅^DD, h̅^DC, h̅^BE, and h̅^DE, respectively.In this paper, we assume that the eavesdropper has complete knowledge about the instantaneous channel power gains and the CDI of the channels from BS to CU and from BS to itself. We further assume that the legitimate receivers, i.e., the cellular user and the RD2D, know the CDI of all channels and only the instantaneous channel power gains of their respective channels. We assume that, the legitimate transmitters do not have perfect values of channel power gains and the knowledge of legitimate transmitters about their respective direct channel power gains is obtained via their respective limited rate feedback channels. In this case, the space of h^BC is divided into a finite number of M regions, i.e., [0,h^BC(1)), [h^BC(1),h^BC(2)), ⋯, [h^BC(M-1), h^BC(M)) where h^BC(M)=∞. Similarly, for D2D pair, the space of h^DD is divided into a finite number of N regions, i.e., [0,h^DD(1)), [h^DD(1),h^DD(2)), ⋯, [h^DD(N-1), h^DD(N)) where h^DD(N)=∞. The receiver, i.e., CU, measures the channel power gain h^BC and feedbacks the index m if h^BC lies in the region [h^BC(m),h^BC(m+1)). Similarly, RD2D measures the channel power gain h^DD and feedbacks the index n if h^DD lies in the region [h^DD(n),h^DD(n+1)). In this paper, we assume that the feedback links are confidential. This means that, the feedbacked index of the cellular network could not be overheard by D2D pair and that of D2D pair could not be overheard by the cellular network. In this case, the power-rate tuples will depend only on the corresponding feedbacked index, i.e., we have (p^BC(m), r^BC(m),r_S^BC(m)) for cellular communication which are the transmit power, the transmission rate and the secrecy rate at which BS transmits information to CU, respectively. Moreover, we consider (p^DD(n), r^DD(n)) for D2D communications which are the transmit power and the transmission rate at which TD2D transmits information to RD2D, respectively.In fact, the proposed schemes operate in two phases.In the first phase (off-line phase), several parameters that are later used for resource allocation are computed. It is done before the communication established and based on the CDIs of the network's links the optimum boundary regions and code-books are designed by the base station. At the end of this phase, all code-words are informed to users by BS. However, the channel partitioning structure is kept at the CU. In the second phase (on-line phase) that is employed during communication, the transmitters use the parameters obtained in off-line phase. In fact, in the on-line phase, the CU and D2D receiver measure the related CSIs and based on them find the related channel partition and boundary region.Then, they transmit back the index to the base station and D2D transmitter in order to select the corresponding code-word from the obtained code-book. Note that the code-book, which contains a set of code-words, is designed off-line and known by each node. The computational burden takes place during the initialization (off-line) phase and requires a negligible burden during the transmission (online) phase and it is certainly desirable from an implementation perspective <cit.>. In this paper, we consider that there is a central processing unit and some assignments are performed in this step. Then, the network uses the information prepared in the central processing<cit.>.For the resource allocation, two approaches could be adopted. One is to consider the problem of pairing the D2D and cellular links as well as the designing limited feedback scheme jointly. In this way, the outcome of the resource allocation problem is which D2D link is paired with which cellular link as well as code-books (power allocation and boundary regions). However, this approach is much complex and would be computationally prohibitive. Another approach which could lower the complexity of the scheme is to consider the pairing problem and limited rate feedback design separately. In this case, one first solves the problem of pairing D2D and cellular link. Then, given this pairing result, the problem of designing a limited rate feedback scheme could be formulated and solved. In this paper, we assumed the second approach and assumed that the D2D and cellular links are paired and the pairing result is available based on which we design the limited rate feedback scheme. The pairing process could be performed based on network parameters as well some degrees of the required QoS level. For example, one could formulate a problem in which the aim is to pair the D2D and cellular links based on the average channel gains instead of instantaneous or long term channel considerations.Several authors have studied the problem of pairing D2D links with CUs for spectrum sharing and focus on the selection of the D2D link and CUs as a pair for better performance <cit.>.However, in our paper, we present the resource allocation in D2D underlaying cellular network and focus on devising a limited rate feedback model as well as power allocation problem encompassing different performance metrics. In this way, first, the D2D link and cellular communication link are scheduled based on the mean of channel power gain. In the next step, the resource allocation can be obtained based on the proposed scheme in this paper. To obtain the best optimum solution, they should be solved at the same time; however, it causes a high computational complexity. To reduce the complexity, they can be considered separately at the cost of a slight performance loss. However, we can extend this approach for solving the problem at the same time as future works. § LIMITED RATE FEEDBACK SCHEMES§.§ Capacity of Links When the feedback links are confidential, the feedbacked indices cannot be heard by any party other than the eavesdropper. In this case, we assume that, the cellular network quantizes the main channel power gain, i.e., h^BC, independent of the index feedbacked by the RD2D. Knowing that the index m is feedbacked by CU, from the designed code book 𝒞^BC, BS chooses transmit power level p^BC(m) to send its information. In other word, it chooses the tuple (p^BC(m), r^BC(m),r_S^BC(m)) where r^BC(m)=log(1+h^BC(m) p^BC(m)) is the transmission rate over BS to CU.In this case, in the absence of D2D transmission, the capacity of the link between BS and CU and its corresponding secrecy capacity are, respectively, given by C^C(m)=log(1+h^BC p^BC(m)),C_S^C(m)=[log(1+h^BC p^BC(m))-log(1+h^BE p^BC(m))]^+, while knowing indices n, the D2D pair chooses transmit power level p^DD(n) to send its information. In other word, the D2D pair has chosen (p^DD(n), r^DD(n)) for concurrent transmission with cellular network and the capacity of the link between BS and CU and its corresponding secrecy capacity are, respectively, given by Ĉ^C(m,n)=log(1+ĥ^BC p^BC(m)),Ĉ_S^C(m,n)=[log(1+ĥ^BC p^BC(m))-log(1+ĥ^BE p^BC(m))]^+, where ĥ^BC=h^BC/1+h^DCp^DD(n) and ĥ^BE=h^BE/1+h^DEp^DD(n) are the effective channel gains between BS and CU and between BS and eavesdropper, respectively. Note that, the transmission capacity in (<ref>) and the secrecy capacity in (<ref>) can be achieved only if we have full knowledge of CSIs, i.e., the perfect values of ĥ^BC and ĥ^BE.On the other hand, given that the index n is feedbacked by RD2D, from the designed code book 𝒞^DD, TD2D chooses the tuple (p^DD(n), r^DD(n)) where r^DD(n)=log(1+h^DD(n) p^DD(n)). In this case, in the absence of cellular transmission, the capacity of D2D link is given byC^D(n)=log(1+h^DD p^DD(n)),while given that BS has chosen the tuple (p^BC(m), r^BC(m),r_S^BC(m)) for concurrent transmission with D2D pair, the capacity of the D2D link is given byĈ^D(m,n)=log(1+ĥ^DD p^DD(n)),where ĥ^DD=h^DD/1+h^BDp^BC(m) is the effective channel gain between TD2D and RD2D. The perfect value of ĥ^DD is needed to achieve the transmission capacity in (<ref>). §.§ Outage Events In our model, there are two types of outage, namely reliability outage which corresponds to the case where the transmission rate exceeds the channel capacity and secrecy outage whose definition depends on the availability of CSI at the transmitter. More precisely, consider the case where CU feedbacks the index m, i.e., BS chooses the tuple (p^BC(m), r^BC(m),r_S^BC(m)). In the absence of D2D transmission, reliability outage for cellular communication occurs if r^BC(m) > C^C(m) where C^C(m) is given by (<ref>). This event corresponds to the case where h^BC(m) > h^BC which never occurs. In addition, the secrecy outage occurs if r_S^BC(m) > C_S^C(m) where C_S^C(m) is given by (<ref>). In this paper, however, we are interested in the outage event in the presence of D2D pair communication. In this case, reliability outage for cellular communication occurs if r^BC(m) > Ĉ^C(m,n) where Ĉ^C(m,n) is given by (<ref>). This event corresponds to the case where h^BC(m) > ĥ^BC which is possible. However, as we assume that only the knowledge of direct channels is available, the secrecy outage does not correspond to the event r_S^BC(m) > Ĉ_S^C(m,n) where Ĉ_S^C(m,n) is given by (<ref>). Note that, given that the indices m and n are feedbacked, the transmission rate is fixed to r^BC(m)=log(1+h^BC(m) p^BC(m)). In this case, any secrecy rate given by r_S^BC(m)≤(r^BC(m) -r^e(m)) is achievable where r^e(m) is the maximum allowable equivocation rate of the eavesdropper. Now, assume for the feedbacked index m, the secrecy rate is fixed to r_S^BC(m) and hence we have r^e(m)=(r^BC(m) -r_S^BC(m)). Therefore, the secrecy outage occurs if the instantaneous capacity of the eavesdropper exceeds the value of r^e(m), i.e., we have Ĉ^BE(m,n)=log(1+ĥ^BE p^BC(m))>r^e(m) where the dependence of the value of Ĉ^BE(m,n) on the feedbacked index n is through ĥ^BE=h^BE/1+h^DEp^DD(n). Therefore, given that D2D pair chooses (p^DD(n), r^DD(n)), the outage probability for cellular communication using tuple (p^BC(m), r^BC(m),r_S^BC(m)) is given byP^outage_p^BC(m), r^BC(m),r_S^BC(m),p^DD(n), r^DD(n)= 1-P^success_p^BC(m), r^BC(m),r_S^BC(m),p^DD(n), r^DD(n), whereP^success_p^BC(m), r^BC(m),r_S^BC(m),p^DD(n), r^DD(n)=(r^BC(m)≤Ĉ^C(m,n),Ĉ^BE(m,n)≤ r^BC(m)-r_S^BC(m)), and we assumed that h^BC∈ [h^BC(m),h^BC(m+1)) and h^DD∈ [h^DD(n),h^DD(n+1)).Using the above explanations and defining ℛ^BC_m= [h^BC_m, h^BC_m+1) and ℛ^DD_n= [h^DD_n, h^DD_n+1), the outage probability for cellular communication when it uses the tuple (p^BC(m), r^BC(m),r_S^BC(m)) is given by P^outage_p^BC(m), r^BC(m),r_S^BC(m)=∑_n=1^N-1(h^DD∈ℛ^DD_n)P^outage_p^BC(m), r^BC(m),r_S^BC(m),p^DD(n), r^DD(n), and the outage probability of cellular link code book, i.e., 𝒞^BC, is given by P^outage_𝒞^BC=∑_m=1^M-1(h^BC∈ℛ^BC_m)P^outage_p^BC(m), r^BC(m),r_S^BC(m). Details on obtaining the above probabilities are deferred to Appendix <ref>.Similarly, for the D2D pair, assume that the transmitter chooses the pair (p^DD(n), r^DD(n)). In the absence of cellular transmission, reliability outage occurs if r^DD(n) > C^D(n) where C^D(m) is given by (<ref>). This event corresponds to the case where h^DD(n) > h^DD which never occurs. On the other hand, in the presence of cellular communication, reliability outage for D2D communication occurs if r^DD(n) > Ĉ^D(m,n) where Ĉ^D(m,n) is given by (<ref>). This event corresponds to the case where h^DD(n) > ĥ^DD which is possible. Note that, as we assumedthe malicious user is not interested in D2D communication, only reliable transmission is considered for D2D pair and no secrecy rate is defined. §.§ Transmit Powers and Achievable Rates As we assumed, the transmitters only know the region number in which the channel power gains of direct channels lay. This means that it is impossible to know the value of effective channel gains when a concurrent transmission is running. Therefore, for cellular network the value of direct channel gain, i.e., h^BC, and for D2D pair, the value of h^DD are quantized. Given that h^BC lies in the region ℛ^BC_m, BS chooses tuple (p^BC(m), r^BC(m),r_S^BC(m)). Note that, regardless of the channel power gains of other links, i.e., h^BD, h^DD, h^DC, h^BE, and h^DE, BS transmits with power level p^BC(m). Therefore, in this case, the average transmission power of BS only depends on h^BC and is given by P̅^C=∑_m=1^M-1(h^BC∈ℛ^BC_m )p^BC(m), which is the same for both cases where D2D pair is not transmitting or concurrently transmits information. Similarly, given that h^DD lies in the region ℛ^DD_n, TD2D chooses the pair (p^DD(n), r^DD(n)). As we assumed that the feedback link of cellular network is confidential, the transmission power of D2D pair, i.e., p^DD(n), only depends on h^DD. Therefore, in this case, the average transmission power of D2D pair is given by P̅^D=∑_n=1^N-1(h^DD∈ℛ^DD_n )p^DD(n), which does not depend on whether BS is transmitting concurrently or not.In addition, for cellular communication, the transmission is assumed successful if no outage occurs, i.e., we have both the reliable and secure communications. We define the average achievable secrecy rate for cellular communication as the adopted secrecy rate, i.e., r_S^BC(m), times the probability of success, i.e., no outage occurs, summed over all regions. When the D2D pair is absent, the average achievable secrecy rate is given by R̅_S^C=∑_m=1^M-1(h^BC∈ℛ^BC_m ,r_S^BC(m)≤ C_S^C(m))r_S^BC(m), where C_S^C(m) is given by (<ref>). Note that, it is not required to include the term r^BC(m)≤ C^C(m) in (<ref>) because it is always satisfied. Please refer to Appendix <ref> for more details on obtaining the probability terms in (<ref>).Since, we need only reliable transmission for D2D communication, the average achievable rate of D2D pair is defined as the adopted data rate. i.e., r^DD(n), times the probability of succeed, i.e., no outage occurs, summed over all region. The average transmission rate which is achievable by D2D pair in the presence of cellular communication is given by R̅^D= ∑_m=1^M-1∑_n=1^N-1(h^BC∈ℛ^BC_m ,h^DD∈ℛ^DD_n ,r^DD(n)≤Ĉ^D(m,n))r^DD(n), where Ĉ^D(m,n) is given by (<ref>). Details on obtaining probability terms in (<ref>) can be found in Appendix <ref>.§ LIMITED RATE FEEDBACK RESOURCE ALLOCATION PROBLEM §.§ Problem Formulation In this paper, we assume that D2D pair opportunistically uses the cellular network resources to maximize its average transmission rate. Note that, generally, D2D communication is underlay to cellular communication which means the later one is superior and should be protected against the side effects of concurrent transmission of D2D pair.There are several approaches to achieve this. One approach is to limit the amount of interference that D2D pair produces on the cellular receiver. Such approach can be seen exactly the same as the notion of interference temperature in cognitive radio networks <cit.>. However, note that this approach is effective when it is used in its instantaneous form (i.e., the exact amount of interference D2D pair produces) and not the averaged one (i.e., the average amount of interference D2D pair produces). However, since we only know the direct channel power gains using limited rate feedback, applying instantaneous interference constraint is not possible. Another approach is to maintain the average achievable rate of the cellular link above a predefined threshold <cit.>. Moreover, the reliability of the cellular network is much of our concern, particularly, in the case that the resource is shared with D2D links. To this end, outage based approach is the next approach in which the outage probability for cellular communication is kept below a predefined threshold <cit.>.In this paper, we combine the last two approaches. More precisely, our objective is to maximize the average achievable data rate for D2D pair in the presence of cellular communication, i.e., (<ref>), while it is required to maintain a minimum amount of the average achievable data rate of cellular link in the absence ofD2D communication, i.e., (<ref>), and the outage probability for cellular communication in the presence of D2D communication, i.e., (<ref>), is kept below a predefined threshold. In this way, we take into account the performance of cellular communication both in the absence and presence of D2D communication. Indeed, by doing so, we require that the average transmission rate of cellular link in the absence of D2D pair to stay above a predefined threshold while its performance in the presence of D2D communication, which is given by the outage probability, remains as satisfactory as is required.In addition, the average transmit power of the cellular link and D2D pair, which are, respectively, given by (<ref>) and (<ref>), should not exceed a predefined value. Mathematically, defining 𝒜={h^BC,h^DD,p^BC, p^DD,r_S^BC}, we aim to solve the optimization problem which is given bymax_𝒜∑_m=1^M-1∑_n=1^N-1(h^BC∈ℛ^BC_m, h^DD∈ℛ^DD_n,r^DD(n)≤Ĉ^D(m,n))r^DD(n),s.t.:∑_m=1^M-1(h^BC∈ℛ^ BC_m ,r_S^BC(m)≤ C_S^C(m))r_S^BC(m)≥R̅_S^Cmin,∑_m=1^M-1( h^BC∈ℛ^BC_m )P^outage_p^BC(m), r^BC(m),r_S^BC(m)≤P^outage,max_𝒞^BC,∑_m=1^M-1(h^BC∈ℛ^BC_m )p^BC(m)≤P̅^C,max,∑_n=1^N-1(h^DD∈ℛ^DD_n )p^DD(n)≤P̅^D,max.This optimization problem is nonlinear and non-convex and it is hard to solve it, hence, we utilize the PSO method which has been used to solve highly non-linear mixed integer optimization problems in various research <cit.>. Since PSO is a computational intelligence-based technique and has global search ability, it can converge to the optimal solution and not largely affected by the size and non-linearity of the problem <cit.>. §.§ Particle Swarm Optimization MethodIn this paper, to solve the optimization problem, we apply relatively new technique, PSO algorithm which is a computational intelligence-based technique.PSO is based on a moment of the swarm which searches to find the best optimal solution by updating generations <cit.>. This method is not largely affected by the size and nonlinearity of the problem, and can converge to the optimal solution. In PSO algorithm, all particles which are the potential solutions, move towards its optimum value. For each iteration all the particles in this swarm are updated by its position and velocity for optimization ability and based on them the aim function for the system is evaluated. PSO starts with the random initialization of swarm of particles in the search space. Then, by adjusting the path of each particle to its own best location and the best particle of the swarm at each step, the global best solution is found. The path of each particle in the search space is adjusted by its velocity, according to moving experience of that particle and other particles in the search space.In this paper, we consider different particles for each variable, i.e, 𝒜={h^BC,h^DD,p^BC, p^DD,r_S^BC}, which denote a solution of the problem. The PSO algorithm consists of 𝒜_i as the vector of i^th particle in d dimension, i.e, for {h^BC, p^BC, r_S^BC}, d is equal to M-1 and for{h^DD, p^DD}, d is equal to N-1 <cit.>.The position and the velocity of the i^th particle in the d dimensional search space can be shown as X_i=[x_i,1, x_i,2,… , x_i,d]^T and V_i=[v_i,1, v_i,2,… , v_i,d]^T, respectively. A best position of each particle is denoted by pbest (P_i=[p_i,1, p_i,2,… , p_i,d]^T), corresponding to the personal best objective value obtained at time t. The global best particle, i.e., gbest (p_g), shows the best particle at time t in the entire swarm. The new velocity of each particle can be obtained as follows <cit.>: v_i,j(t+1)=wv_i,j(t)+c_1r_1(p_i,j-x_i,j(t))+c_2r_2(p_g-x_i,j(t)), j=1,…,d, where c_1 and c_2 are constants called acceleration coefficients, w is the inertia factor, r_1 and r_2 are two independent random numbers uniformly distributed in [0, 1]. Thus, the position of each particle is updated in each step as follows:x_i,j(t+1)=x_i,j(t)+v_i,j(t+1). The standard form of PSO uses (<ref>) to calculate the new velocity of each particle based on its previous velocity and the distance of its current position from both its best position and global best position. To control search of particles outside the search space [X_i^min,X_i^max], we can limit the value of V_i to the range [V_i^min,V_i^max] and according to (<ref>), each particle moves to a new position. The process is repeated until a stopping criterion is satisfied. This algorithm is summarized in Table.<ref> <cit.>.§ PRACTICAL CONSIDERATION§.§ CDI Estimation ErrorThe most practical assumption made in this paper is that the instantaneous channel power gains of the eavesdropper's links, i.e., h^BE and h^DE, are not available which is mostly due to the fact that the eavesdropper is passive and hence acquiring its channel power gains are not possible. Generally, the CDI of a channel depends on the environmental property of the communication channel. If the propagation environment is known, one can assume that the channel CDIs, including those of the eavesdropper, are available. The statistical property of the signal propagation in the coverage area of the network can be easily obtained as the legitimate users are present and can be involved in finding the required statistical properties. Since for small geographical areas, a unified distribution can be applied to all channels[This assumption is reasonable when the size of the area under investigation is small which is the case for nowadays cellular networks specially for small cells.], we can have the CDIs of eavesdropper's links at hand.Due to the availability of limited statistical data, the distribution function is hard to drive and cannot be fit into the known ones, e.g., Rayleigh distribution. In such cases, schemes developed based on the availability of the perfect CDI may exhibit performance worse than that expected. Therefore, the imperfectness of CDIs should be taken into account. Generally, such consideration can be performed by assuming that the true distribution differs from the nominal distribution by the value known as Kullback–Leibler distance <cit.> and incorporate such inaccuracy into problem formulation <cit.>. We investigate imperfect CDI through two parametric and nonparametric methods.§.§.§ Parametric MethodIn parametric methods, the effect of the imperfect CDI is studied through the performance loss by simulations as in Section <ref>. This means that, we solve the optimization problem (<ref>) with the available channel CDIs and obtain the channel quantization and code books for cellular link and D2D pair. Then, we evaluate the performance loss due to imperfect CDI in terms of changes in the average achievable rates. In other words, we consider the imperfect channel power gain of each channel i which is exponentially distributed with the mean of ḧ̅̈^i : ḧ̅̈^i=(1-Δ) h̅^i, where Δ is percent error of imperfect CDI.§.§.§ Non-Parametric MethodAnother way to estimate CDI is nonparametric method which estimates the density based on the received samples from the channel. In this paper, we adopt two nonparametric methods: kernel density estimation (KDE) and robust KDE (RKDE).2.1. Kernel Density Estimation (KDE): One of the most well-known non-parametric density estimation methods is kernel density estimation <cit.>. When the samples, referred to as the nominal data, are noise free, KDE can provide a good estimate of the density. A set of observations {x_1,...,x_L}∈ℝ^j is used to estimate a random vector xwith a density f(x) where L is the number of observation vectors. Moreover, each x_i = x_i1, . . . , x_ij, i = 1, . . . ,L is a sequence of j data in the vector x_i. The kernel density estimate of f(x) given by f̂_KDE(x)=1/L∑_i=1^L k_δ(x, x_i), where k_δ(x, x_i) is the kernel function which commonly is a Gaussian kernel: k_δ(x, x_i)=(1/√(2Π)δ)^j exp(-∥x-x_i∥^2/2δ^2), where δ is the smoothing parameter and referred to as the bandwidth. It is set to the median distance of a training point x_i to its nearest neighbor.2.2. Robust Kernel Density Estimation: In practice, the channel gain samples might include contaminated data, referred to as outlier data, which makes it necessary to use robust density estimation methods such as robust KDE (RKDE). In the presence of the contaminated samples, RKDE can give robustness to contamination of the training sequence and estimate the density. Contaminated data consists of realizations from both a nominal or clean distribution in addition to outlying or anomalous measurements. In an increasing number of applications, data arises from high dimensional or high-throughput systems where the nominal distribution itself may be quite complex and not amenable to parametric modelling.The RKDE has the following form: f̂_RKDE(x)=∑_i=1^L ω_i k_δ(x, x_i), where k_δ(x, x_i) is a kernel function and ω_i are nonnegative weights that sum to one. The RKDE can be implemented based on the iteratively reweighed least square (IRWLS) <cit.> algorithm in which the main goal is to find the optimal value of ω_i. §.§ Noisy Feedback Channel So far, we assumed that the feedback channels are error free meaning that the received index is the same as the feedbacked one. However, in reality, the feedback channel could be affected by the noise which makes transmitter to select an incorrect code word from the designed code book. Note that, designing limited rate feedback systems with incorporating feedback error is complicated, especially for our scheme with two interfering links. In this paper, to consider the feedback error, we utilize the scheme which is commonly used in the literature <cit.>. We consider the memoryless feedback channel which characterized by index transition probabilities ρ^C_m,m' (m,m'=0,⋯,M-1) for cellular link which is the probability of receiving index m in BS given the index m' was sent by CU, and ρ^C_n,n' (n,n'=0,⋯,N-1) for D2D pair which is the probability of receiving index n in TD2D given the index n' was sent by RD2D. It is assumed b_M=log_2(M) bits feedback for cellular link and b_N=log_2(N) bits feedback for D2D pair. Let m_1 m_2 ⋯ m_b_M, m'_1 m'_2 ⋯ m'_b_M, n_1 n_2 ⋯ n_b_M, and n'_1 n'_2 ⋯ n'_b_M indicate the binary display of indices m, m', n, and n', respectively. We assume that the cellular and D2D pair's feedback channel can be considered as, respectively, b_M and b_N independent use of binary symmetric channel (BSC) to sent each of the feedback bits presented in binary representations of cellular link and D2D pair's feedbacked indices. Let q^C and q^D represent the cross over probabilities of the feedback channels of cellular link and D2D pair, respectively. The index transition probabilities of the feedback channels of cellular link and D2D pair can be obtained, respectively, by ρ^C_m,m'=(q^C)^d_m,m'(1-q^C)^b_M-d_m,m', ρ^D_n,n'=(q^D)^d_n,n'(1-q^D)^b_N-d_m,m', where d_m,m' and d_n,n' denote the Hamming distances between, indices m and m' and indices n and n', respectively <cit.>.With the above definitions and assumptions, the average transmission powers in (<ref>) and (<ref>), average transmission data rates in (<ref>) and (<ref>), and the outage probabilities in (<ref>), (<ref>), (<ref>), and (<ref>) should be manipulated to incorporate the effect of noisy feedback channel. Note that, choosing the transmit power level from a code book only depends on the corresponding channel region index which is feedbacked by the respective transmitter. This mean that, in (<ref>), we should only consider the noise effect of the feedback channel of cellular link, and in (<ref>), we should only consider the noise effect of the feedback channel of D2D pair. Therefore, the average transmit powers of cellular link and D2D pair, when noisy channel feedback is assumed, are given, respectively, byP̅^C=∑_m=1^M-1∑_m'=1^M-1ρ^C_m,m'(h^BC∈ℛ^BC_m')p^BC(m),P̅^D=∑_n=1^N-1∑_n'=1^N-1ρ^D_n,n'(h^DD∈ℛ^DD_n')p^DD(n). For the average transmission data rate in (<ref>), weassumed the D2D pair is absent, hence, it is not affected by the noise in feedback channel of D2D pair. Therefore, the average transmission data rate can be written as R̅_S^C=∑_m=1^M-1∑_m'=1^M-1(h^BC∈ℛ^BC_m', m'→m,r^BC(m)≤ C^C(m/m') ,C^BE(m)≤ r^e(m))r_S^BC(m), where r^e(m)=r^BC(m)-r_S^BC(m) and C^BE(m)=log(1+h^BE p^BC(m)) where we note that actually, the value of C^BE(m) does not depend on m'. In (<ref>), m'→ m is the event that the feedbacked index m' is received as m. Here, we highlight that, in (<ref>), C^C(m/m') means that its value is given by (<ref>) with h^BC∈ℛ^BC_m' and p^BC(m). Note that, here, in contrast to (<ref>), we must include r^BC(m)≤ C^C(m/m') in (<ref>) because reliability outage can occur when the feedback is noisy.From (<ref>), we know that the event r^BC(m)≤ C^C(m/m') occurs when m≤ m'. Therefore, (<ref>) can be rewritten as follows: R̅_S^C=∑_m=1^M-1∑_m'=m^M-1ρ^C_m,m'(h^BC∈ℛ^BC_m' ,Ĉ^BE(m)≤ r^e(m))r_S^BC(m). The remaining steps are similar to those in obtaining probability terms in (<ref>) in Appendix <ref>, and hence omitted. However, as we assumed in (<ref>) that both the cellular link and D2D pair transmit simultaneously, the cross effect of noisy feedback channel should be considered. In other words, given that the transmitted index m' was received as m by BS and the transmitted index n' was received as n by transmitter of D2D pair, the average data rate of D2D pair in the presence of cellular communication with noisy feedback channels is given by R̅^D=∑_m=1^M-1∑_m'=1^M-1∑_n=1^N-1∑_n'=1^N-1(h^BC∈ℛ^BC_m',h^DD∈ℛ^DD_n', (m',n')→(m,n),r^DD(n)≤Ĉ^D(m,n/n'))r^DD(n). Note that, in (<ref>), the value of Ĉ^D(m,n/n') does not depend on m'. In addition, the effect of noise in feedback channel of cellular link on the value of Ĉ^D(m,n/n') appears through the choice of transmit power level p^BC(m) which affects the value of the effective channel gain ĥ^DD=h^DD/1+h^BDp^BC(m) with h^DD∈ℛ^DD_n'. Obtaining probability terms in (<ref>) is similar to obtaining probability terms in (<ref>) in Appendix <ref>, and hence omitted.Like (<ref>), for the outage probabilities in (<ref>), (<ref>), (<ref>), and (<ref>), we should consider the cross effect of noisy feedback channels. If we consider the noisy feedback channel effect in the outage probability of cellular communication in the presence of D2D pair, we observe that given the feedback indices m' and n' were received by the corresponding receiver as m and n, respectively, BS uses the code word (p^BC(m), r^BC(m),r_S^BC(m)) while we have h^BC∈ℛ^BC_m' and the transmitter of D2D pair uses the code word p^DD(n), r^DD(n) while we have h^DD∈ℛ^DD_n'. In this case, the outage probability is given by P^outage(m/m',n)_p^BC(m), r^BC(m),r_S^BC(m),p^DD(n), r^DD(n)=1-P^success(m/m',n)_p^BC(m), r^BC(m),r_S^BC(m),p^DD(n), r^DD(n), where P^success(m/m',n)_p^BC(m), r^BC(m),r_S^BC(m),p^DD(n), r^DD(n)=(r^BC(m)≤Ĉ^C(m/m',n),Ĉ^BE(m,n)≤ r^BC(m)-r_S^BC(m)), where Ĉ^C(m/m',n) is given by (<ref>) with ĥ^BC=h^BC/1+h^DCp^DD(n) and h^BC∈ℛ^BC_m', and Ĉ^BE(m,n)=log(1+ĥ^BE p^BC(m)) with ĥ^BE=h^BE/1+h^DEp^DD(n). To obtain (<ref>) one can follow the similar steps as those for (<ref>) in Appendix <ref>.Using the above explanations, the outage probability for cellular communication when it uses the tuple (p^BC(m),r^BC(m),r_S^BC(m)) and under noisy feedback channel model, is given by P^outage(m/m')_p^BC(m), r^BC(m),r_S^BC(m)= ∑_n=1^N-1∑_n'=1^N-1(ρ^D_n,n'(h^DD∈ℛ^DD_n')× P^outage(m/m',n)_p^BC(m), r^BC(m),r_S^BC(m),p^DD(n), r^DD(n)), and the outage probability of cellular link code book, i.e., 𝒞^BC, is given by P^outage_𝒞^BC=∑_m=1^M-1∑_m'=1^M-1ρ^C_m,m'(h^BC∈ℛ^BC_m') P^outage_p^BC(m), r^BC(m),r_S^BC(m). To take the noisy feedback channel model into consideration, in the optimization problem (<ref>), we must use (<ref>), (<ref>), (<ref>), (<ref>), and (<ref>).§ SIMULATION RESULTSIn this section, numerical results are presented to evaluate the performance of the proposed limited feedback scheme in a D2D communication through the simulations under various system parameters. The channel gain is an exponential random variable with the probability density function (PDF) given by f(h)=1/σexp(-h/σ), where σ can be used to model the average channel gain as σ=s(d/d_0)^-γ where d is the distance between the transmitter and the receiver, d_0 is the reference distance, γ is the amplitude path-loss exponent, and s characterizes the shadowing effect. The users are assumed to be uniformly distributed in a cell of radius 100 m. The small-scale channel fading is assumed to be Rayleigh distributed. The path-loss exponent is equal to 4, and the shadowing effect follows a log-normal distribution, i.e., 10log_10(s)∼ N(0,8 dB). System parameters are equal to P_D^max=10 dB, P_C^max=5 dB, P_outage^max=0.1, R^C_S_min=0.1 bps/Hz, q^C=q^D=0.25. We set the coefficients c_1=c_2=1.496 and w=0.729 for PSO algorithm and simulated for 1000 iterations. §.§ ConvergenceFig. <ref> shows the convergence of the algorithm. For the limited feedback scheme, we consider 1, 2, and 3 bits to display the results clearly. To demonstrate the performance of the proposed system, the results are obtained for non-noisy and noisy limited-feedback schemes for both the perfect and imperfect CDI. The PSO method, generally, does not guarantee to achieve global optimum for n-dimensional functions. It is difficult to prove and show mathematically that PSO can guarantee global optima in our problem. However, we have used different searches to show the reliability of the PSO in Fig. <ref>. As it is shown, with different random initialization of swarm of particles in the different part of problem space, all of the solutions converge to the same point.§.§ The Effect of the System ParametersIn Fig. <ref> and Fig. <ref>, the D2D average rate is plotted versus the maximum transmit power of D2D user (P_D^max) and the different number of BC and D2D feedback bits (M, N). In Fig. <ref>, the D2D average rate is studied for perfect CDI and parametric CDI estimation method as well as noisy feedback. Obviously with increasing P_D^max, the average rate of D2D increases due to increasing the feasibility set of the resource allocation problem with the relaxation of constraint on the transmit power of D2D user. As we can see, some curves are flattened when the D2D power constraint is increased. This is because the cellular rate constraint becomes the dominant factor in the optimization problem and D2D rate can not increase with increasing the transmit power. To study the effect of percent error of imperfect CDI, in Fig. <ref> the D2D average rate is obtained for different errors. As it is shown, by increasing the error, the D2D achievable rate decreases.Fig.<ref> describes the performance of D2D communication in terms of the maximum outage probability for cellular communication (P_outage^max) and different number of feedback bits. As the maximum outage probability limit increases, the D2D average rate increases. Specifically, for smaller P_outage^max, the overall D2D rate increases fairly rapidly. Hence, if the cellular communication can withstand slight secrecy outage probability, simultaneous D2D communication can be a great advantage. Similar to that of the previous case, the curves become flat since it is limited by D2D power constraint.In Fig. <ref>, the effect of the minimum required secrecy rate of the cellular network R_S^C^min on the D2D rate is illustrated. Obviously, when the minimum secrecy rate of the cellular network increases, the operation of D2D communication is limited. Therefore, the D2D average rate is reduced.In Fig. <ref> the effect of q^C and q^D is studied. As it is shown, by growing the error probability, i.e, the quality of the feedback link degrades, we see the decline in the rate of D2D.As it is seen in all figures, the increasing number of feedback bits results in the improvement of the D2D performance, and the average rate increases. Also, the resultsdemonstrate that the performances of the limited-feedback scheme without noise have the better performance in comparison with the noisy case. §.§ CDI Estimation ErrorTo check out the effect of CDI estimation on the performance of the system, |Δ r|/r is define as the percent of the difference between the average rate of D2D obtained based on the perfect and estimated CDI.Fig.<ref> demonstrates |Δ r|/r as a function of the total number of users for different numbers of the nominal data where the number of outlier data is set to κ=10. As the figure shows, for small L both KDE and RKDE methods perform very poor. As L grows, the performance of both methods improves, and for L = 200, the average rate obtained based on RKDE is very close to that of the perfect CDI case.In Fig. <ref>, |Δ r|/r is plotted versus the number of feedback bits for the different number of outlier data κ where the number of nominal data is set to L=200. As it is seen, the value of |Δ r|/r is close to zero for RKDE method with κ = 10. As κ grows, |Δ r|/r increases implying the divergence from the actual pdf. It is also observed that the value of |Δ r|/r for KDE is far away from zero and the performance degrades faster compared to that of RKDE as κ grows.§ CONCLUSIONSIn this paper, we studied a limited-feedback radio resource allocation problem for the D2D communication scenario underlaying an existing cellular network with the objective of maximizing the D2D average rate subject to average users transmit power limitations, the average secrecy rate and outage probability threshold for the cellular network. Through the PSO algorithm, the appropriate code book for the channel partitioning was designed. In addition, we solved the problem when the feedback channel is noisy. To investigate the effect of the CDI imperfectness on the performance, we applied both the parametric and non-parametric methods. Using simulations, we studied the impact of the system parameters, such as themaximum allowable transmit power of D2D user, the number of feedback bits, and the minimum secrecy rate of cellular network, on the achievable rate of D2D. As it was shown, by more feedback bits, better D2D performance can be achieved.equationsection § FINDING OUTAGE PROBABILITY IN (<REF>)To compute the outage probability in (<ref>), we should compute the success probability in (<ref>). Note that, we have h^BC∈ℛ^BC_m= [h^BC(m),h^BC(m+1)] and h^DD∈ℛ^DD_n= [h^DD(n),h^DD(n+1)]. The success probability can be written as follows P^success_p^BC(m), r^BC(m),r_S^BC(m),p^DD(n), r^DD(n)=(r^BC(m)≤Ĉ^C(m,n),Ĉ^BE(m,n)≤ r^e(m)) =(h^BC(m)≤ĥ^BC,ĥ^BE≤ 2^r^e(m)-1)=(h^BC(m)≤ĥ^BC) (ĥ^BE≤ 2^r^e(m)-1)=(1-F^m,n_ĥ^BC(h^BC(m)))F_ĥ^BE^n(2^r^e(m)-1),where ĥ^BC =h^BC/1+h^DCp^DD(n) depends on m and n, ĥ^BE=h^BE/1+h^DEp^DD(n) depends on n, and the variables ĥ^BC and ĥ^BC are independent and hence the product term in (<ref>) follows. Note that in the above equations, we have implicitly assumed that h^BC∈ℛ^BC_m and h^DD∈ℛ^DD_n meaning that the above probability is a conditional probability.To compute (<ref>), we need to find PDFs of ĥ^BC and ĥ^BE. The CDF of ĥ^BC can be computed as follows:F^m,n_ĥ^BC(x)=(ĥ^BC≤ x| h^BC∈ℛ^BC_m,h^DD∈ℛ^DD_n)=(h^BC/1+h^DCp^DD(n)≤ x | h^BC∈ℛ^BC_m,h^DD∈ℛ^DD_n)=(h^BC≤(1+h^DCp^DD(n))x | h^BC∈ℛ^BC_m,h^DD∈ℛ^DD_n)={[(h^BC≤ x | h^BC∈ℛ^BC_m,h^DD∈ℛ^DD_n),; (h^BC/x-1/p^DD(n)≤ h^DC | h^BC∈ℛ^BC_m,h^DD∈ℛ^DD_n). ].In (<ref>), for the cases n=0 and n≠ 0, respectively, we have the followings:F^m,0_ĥ^BC(x)=(h^BC≤ x,h^BC∈ℛ^BC_m )/(h^BC∈ℛ^BC_m )=1/(h^BC∈ℛ^BC_m)×_h^BC(m)^xf_h^BC(h^BC) dh^BC, ,F^m,n_ĥ^BC(x)=(h^BC/x-1/p^DD(n)≤ h^DC, h^BC∈ℛ^BC_m)/(h^BC∈ℛ^BC_m)=1/(h^BC∈ℛ^BC_m)×_h^BC(m)^h^BC(m+1)_h^BC/x-1/p^DD(n)^∞f_h^DC(h^DC) f_h^BC(h^BC) dh^DC dh^BC,. The CDF of ĥ^BE is given byF_ĥ^BE^n(x) = 1-h̅^BE/h̅^BE+h̅^DEp^DD(n)xexp( -x/h̅^BE), .Finally, we will have the following: P^outage_𝒞^BC = ∑_n=1^N-1∑_m=1^M-1(h^DD∈ [h^DD(n),h^DD(n+1)))×(h^BC∈ [h^BC(m),h^BC(m+1)))× P^success_p^BC(m), r^BC(m),r_S^BC(m),p^DD(n), r^DD(n). § FINDING SUCCESS PROBABILITY IN (<REF>) To obtain the success probability in (<ref>), i.e., (h^BC∈ℛ^BC_m,r_S^BC(m)≤ C_S^C(m)), we first define new random variables x_m=1+h^BC p^BC(m) and y_m=1+h^BE p^BC(m) with respective distributions f_x_m(x_m)=1/p^BC(m)f_h^BC(x_m-1/p^BC(m)) and f_y_m(y_m)=1/p^BC(m)f_h^BE(y_m-1/p^BC(m)). Therefore, we have(r_S^BC(m)≤ C_S^C(m),h^BC∈ℛ^BC_m))=Pr(y_m≤ 2^-r_S^BC(m) x_m,h^BC∈ℛ^BC_m) =∫_1+h^BC(m) p^BC(m)^1+h^BC(m+1) p^BC(m)∫_1^2^-r_S^BC(m) x_mf_y_m(y_m) f_x_m(x_m) dy_m dx_m. § FINDING SUCCESS PROBABILITY IN (<REF>)To obtain the success probability in (<ref>), i.e., (h^BC∈ℛ^BC_m,h^DD∈ℛ^DD_n,r^DD(n)≤Ĉ^D(n)), first, we introduce the cumulative distribution function ofh, which is equal to F^h_m(x)=1-e^-x/h̅ and G^h_m= F^h_m(h(m+1))-F^h_m(h(m))=e^-h(m)/h̅-e^-h(m+1)/h̅,which is the probability that h falls into the region of [h(m),h(m+1)). Therefre, for h^BC we have the following:G^h^BC_m=F^h^BC_m(h^BC(m+1))-F^h^BC_m(h^BC(m) )=e^-h^BC(m)/h̅^BC-e^-h^BC(m+1)/h̅^BC.Then, we have the following:(h^BC∈ℛ^BC_m,h^DD∈ℛ^DD_n,r^DD(n)≤Ĉ^D(n))=G^h^BC_m(h^DD∈ℛ^DD_n,h^DD(n)≤ĥ^DD),and given that ĥ^DD=h^DD/1+h^BDp^BC(m), we will have the following:(h^DD(n)≤ĥ^DD, h^DD∈ℛ^DD_n) =(0≤ h^BD≤(h^DD/h^DD(n)-1)/p^BC(m), h^DD∈ℛ^DD_n)=∫^h^DD(n+1)_h^DD(n)[1-e^(h^DD/h^DD(n)-1)/h̅^BDp^BC(m)] 1/h̅^DD e^-h^DD/h̅^DD d h^DD.Finally, the success probability in (<ref>) is equal to:(h^BC∈ℛ^BC_m,h^DD∈ℛ^DD_n,r^DD(n)≤Ĉ^D(n))=G^h^BC_m∫^h^DD(n+1)_h^DD(n)[1-e^(h^DD/h^DD(n)-1)/h̅^BDp^BC(m)] 1/h̅^DD e^-h^DD/h̅^DDdh^DD.IEEEtran | http://arxiv.org/abs/1708.07980v1 | {
"authors": [
"Faezeh Alavi",
"Nader Mokari",
"Mohammad R. Javan",
"Kanapathippillai Cumanan"
],
"categories": [
"cs.IT",
"math.IT"
],
"primary_category": "cs.IT",
"published": "20170826144335",
"title": "Limited Feedback Scheme for Device to Device Communications in 5G Cellular Networks with Reliability and Cellular Secrecy Outage Constraints"
} |
On the asymptotic variance of the debiased Lasso.1in Sara van de GeerSeminar for Statistics, ETH Zürich.1in August 21, 2018 MSC 2010 subject classifications: Primary 62J07; secondary 62E20.Keywords and phrases: asymptotic efficiency, asymptotic variance, Cramér Rao lower bound,debiasing, Lasso, sparsity. Abstract. We consider the high-dimensional linear regression model Y = X β^0 + ϵ with Gaussian noise ϵ and Gaussian random design X. We assume that Σ:=X^T X / n is non-singular and write its inverse as Θ := Σ^-1. The parameter of interest is the first component β_1^0 of β^0.We show that in the high-dimensional case the asymptotic variance of a debiased Lasso estimator can be smaller than Θ_1,1.For some special such cases we establish asymptotic efficiency. The conditions include β^0 beingsparseand the first column Θ_1 of Θ being not sparse. These conditions depend on whether Σ is known or not.§ INTRODUCTIONLet Y be an n-vector of observations and X ∈^n × p an input matrix. The linear model isY = X β^0 + ϵ ,where β^0 ∈^p is a vector of unknown coefficients and ϵ∈^n is unobservable noise. We examine the high-dimensional case with p ≫ n. The parameter of interest in this paper is a component of β^0, say the firstcomponent β_1^0. We consider the asymptotic properties of debiased estimators of the one-dimensional parameter β_1^0 under scenarios where certain sparsity assumptions fail to hold. The paper shows that the asymptotic variance of the debiased estimator can be smaller than the “usual" value for the low-dimensional case. For simplicity we examine Gaussian data: the rows of (X,Y) ∈^n× (p+1) are i.i.d. copies of a zero-mean Gaussian row vector ( x,y ) ∈^p+1, where x= ( x_1 , … ,x_p) has covariance matrix Σ :=x^Tx. We assume the inverse of Σ exists and write it as Θ:= Σ^-1. The vector β^0 of regression coefficients isβ^0 = Θ x^Ty. We denote the first column of Θ byΘ_1 ∈^p and the first element of this vector by Θ_1,1. Our main aim is to present examples where lack of sparsity in Θ_1 can result in a small asymptotic variance of a suitably debiased estimator. In particular, the asymptotic variance can be smaller than Θ_1,1. For the case Σ known, this means applying for instance a (noiseless) node-wise Lasso, instead of an exact orthogonalization of the first variable with respect to the others, may reduce the asymptotic variance (as follows from combining Theorem <ref> with Lemma <ref>). If Σ is unknown, the high dimensionality of the problem already excludes exact empirical projections for orthogonalization. The (noisy) Lasso is designedto deal with approximate orthogonalization in the high-dimensional case. Using the node-wise Lasso, we find that one may again profit from non-sparsity of the (now unknown) vector Θ_1 (see Theorem <ref>).We look at specific examples or constructions of covariance matrices Σ.The results illustrate thatasymptotic efficiency claims require some caution. The high-dimensional situation exhibits new phenomena that do not occur in low dimensions. Throughout, the minimal eigenvalue of Σ, denoted by Λ_ min^2, is required to stay away from zero, i.e. 1/Λ_ min^2 = 𝒪 (1). We further consider only Σ's with all 1's on the diagonal and assume for simplicity that σ_ϵ^2 :=(y -xβ^0 )^2 is known and that its value is σ_ϵ^2=1.Let a given subset B of ^p be the model class for β^0. An interesting research goal is to construct for the model B a regularestimator of β_1^0 with asymptotic variance that achieves the asymptotic Cramér Rao lower bound (given here in Proposition <ref>). One then needs to decide which model class Bone considers as relevant. In high-dimensional statistics it is commonly assumed that β^0 is sparse in some sense.Let 0< r ≤ 1, definefor a vector b ∈^p its ℓ_r-“norm" b _r by b _r^r := ∑_j=1^p | b_j |^r and let b _0^0 be its number of non-zero entries. A sparse modelis for exampleB := { b ∈^p : b _0^0 ≤ s } for some (“small") s ∈.Alternatively one may believe only in ℓ_1-sparsity. Then B = { b ∈^p : b _1 ≤√(s)} for some s >0. These are the two extremes of weakly sparse models of the formB = { b ∈^p :b _r^r ≤ s^2-r2} , for some s>0 and 0≤ r ≤1. Throughout, the value of s is allowed to depend on n, butr is fixed for all n.Constructing estimators that achieve the asymptotic Cramér Rao lower bound for model (<ref>), (<ref>), (<ref>) or some other sparse modelis to our understandingquite ambitious, especially if one wants to do this for all possible covariance matrices Σ.See e.g. Example <ref> for some details concerning model (<ref>).However, for special cases of Σ's the problem can be solved. One such special case is where the first row of Θ_1 of Θ is sparse in an appropriate sense. This is the situation considered in previous work such as<cit.> and <cit.>where Σ is unknown. In this paper we consider knownand unknown Σ and in both cases do not require sparsity of Θ_1. The paper <cit.> also does not require sparsity of Θ_1when Σ is known andit turns out that for certain non-sparse vectors Θ_1 their estimator is not asymptotically efficient (see Theorem <ref> or Remark <ref> following this theorem). The debiased Lasso defined in this paper in equation (<ref>) below is based on a direction Θ̃_1 ∈^p where typically Θ̃_1 is thought of as some estimate of Θ_1. As we do notassume sparsity of Θ_1 a reliable estimator of Θ_1 may not be available.Nevertheless, we show that this does not rule out good theoretical performance. We present a class of covariance matrices Σ for which a debiasedLasso has asymptotic variance smaller than Θ_1,1. This phenomenon is tied to the high-dimensional situation, see Remark <ref>.For special cases, we establish thatan asymptotic Cramér Rao bound smaller than Θ_1,1 canbe achieved.In other words,there exist cases where adebiased Lasso profits from sparsity of β^0 combined with non-sparsity of Θ_1. This is good news: the asymptotic variance can be small for two reasons: either Θ_1 is sparse in which case the asymptotic variance Θ_1,1 is typically small (as it is the inverse of the residuals of regressing the first variableon only a few of the other variables), or Θ_1 is not sparsebut then the asymptotic variance can be smaller than Θ_1,1. This paper presentscases where the latter situation indeed occurs. §.§ The Lasso, debiased Lasso and sparsity assumptions The Lasso (<cit.>) is defined as β̂:= min_b ∈^p{ Y - X b _2^2/n + 2 λ b _1 }with λ>0 a tuning parameter. (We will throughout take of order √(log p / n ) but not too small.)A debiased Lasso is given byb̂_1 = β̂_1 + Θ̃_1^T X^T (Y- X β̂)/n .The p-dimensional vector Θ̃_1 is typically some estimate of the first columnΘ_1 ofthe precision matrix Θ, but in our case it will rather be estimating a sparse approximation.We refer to Θ̃_1 as a direction. The estimator β̂ is commonly taken to be the Lasso given in (<ref>) although this is not a must.The debiased Lasso (<ref>) was introduced in <cit.> and further developed in <cit.> and<cit.> for example.Related work is <cit.>and <cit.>.The theory for the Lasso (<ref>) and debiased Lasso (<ref>)requires some form of sparsity of β^0. Consider for some s ∈ one of the sparsity models (<ref>), (<ref>) or, more generally, model (<ref>). Two prevalent sparsity assumptions are:(i) s = o ( n / log p ),(ii) s = o ( √(n) / log p ). Sparsity variant (i) is for example invoked to establish ℓ_2-consistency of the Lasso β̂ (see <cit.> or the monographs <cit.>, <cit.> and <cit.>, and their references). Which sparsity variant is needed to establish asymptotic normality of the debiased Lasso(<ref>) depends to a large extent on whether Σ is known or not. In <cit.>, <cit.>, <cit.>.<cit.> one can find refined results on this issue.This caseΣ known is treated in Section <ref>. We introduce and apply there the concept of an eligible pair, see Definition <ref>. An eligible pair is a sparse approximation of Θ_1 together with a parameter describing the order of approximation and sparsity.We allow for sparsity variant (i) in model (<ref>) as in <cit.>, see Example <ref>. Sparsity variant (i) will also be allowed for the models (<ref>) and (<ref>), see Examples <ref> and <ref>. Eligible pairs will also play a crucial role in Section <ref>where Σ is unknown.Let us discuss some of the literature for this caseand for the sparsity model (<ref>).From the papers <cit.> and <cit.>we know that for the minimax bias of an estimator of β_1^0 to be of order 1/ √(n), the assumption s=O( √(n / log p) ) is necessary.Thus, up to log-termsthis needs the second sparsityvariant.When considering asymptotic Cramér Rao lower bounds,one also needs to restrict oneself to a certain class of estimators,for instance estimators with bias of small order 1/√(n) or asymptotically linear estimators. In <cit.> such restrictions are studied. One can show asymptotic linearity of the debiased Lasso under model (<ref>) with sparsity variant (ii)and in addition Θ_1 _0 = o (√(n) / log p ). If Θ_1 is not sparse nor can be approximated by a sparse vector, then it is unclear whether an asymptotically linear estimator exists. We refer to Remark <ref> for more details.In summary, modulo log-terms, sparsity variant (ii) cannot be relaxed as far as minimax ratesfor the bias are concerned, and sparsity variant (ii) with in addition sparsityof order o (√(n) / log p )for Θ_1 or its sparse approximationappears to be needed for establishing asymptotic linearity. We note thatthe paper <cit.> establishes asymptotic normality under (among others) the assumptionmin ( s, Θ_1 _0^0) = o( √(n) / log p ). Bias and asymptotic linearityare not considered (these issues are not within the scope of that paper).In our setup however, Θ_1 is not sparse at all, sovariant (ii) is in line with (<ref>).Tables <ref> and <ref> presented in Subsection <ref> summarizes the sparsity conditions applied in this paper. One sees that models (<ref>) and (<ref>) are special cases of model (<ref>), with r=0 and r=1 respectively. However, when r=0 the asymptotic efficiency depends on β^0 and also quite severely on the value of s. For the case Σ unknown, model (<ref>) is too large. §.§ The asymptotic Cramér Rao lower boundWe briefly review the Cramér Rao lower bound and refer to <cit.> for details. Let the model be β^0 ∈ B, where B is a given class of regression coefficients. LetH_β^0 := { h ∈^p:β^0 + h/√(n)∈ B}. We call H_β^0 the set of model directions.An estimator T (or actually: sequence of estimators) is called regular at β^0 if for all fixed ρ>0 and R >0 and all sequences h∈ H_β^0 with | h_1 | ≥ρ and h^T Σ h ≤ R^2, it holds that√(n) ( T- (β_1^0+ h_1/√(n))V_β^0 ) D_β^0 + h/√(n)⟶ N (0,1)where V_β^0^2 = 𝒪 (1) is some constant (depending on n and possibly on β^0, but not depending on ρ, R or h), called the asymptotic variance (it is defined up to smaller order terms). Regularity is important in practice. It means that the asymptotics is not just pointwise but remains valid in neighbourhoods. Typically, the class B is not a cone. This is the reason why we do not restrict ourselves to model directions h ∈ H_β^0 with h_1=1 (say).One may also opt for defining the set of possible directionsH_β^0 differently, sayH_β^0 :=B. Then regularity concerns parameter values that fall outside the parameter space.For example under model (<ref>) one then has to deal with sparsity 2s instead of s. With our choice of H_β^0 we stay inside the parameter space but the set of possible directions then depends on β^0. In model (<ref>) or (<ref>) one can move away from this dependence when β^0 is a proper “interior point" of B (see Examples <ref> and <ref>). Suppose T is asymptotically linear at β^0 with influence function i_β^0: ^p+1→:T- β_1^0 = 1n∑_i=1^ni_β^0 (X_i , Y_i) + o__β^0(1/ √(n) )where _β^0 i_β^0( x ,y ) = 0 and V_β^0^2 := _β^0i_β^0^2 ( x ,y )= 𝒪 (1). Assume the Lindeberg conditionlim_n →∞_β^0 i_β^0^2 ( x ,y )l{ i_β^0^2 ( x ,y )> ηn V_β^0^2} =0 ∀ η >0 .Assume further that T is regular at β^0. Then for all fixed ρ>0 and R>0V_β^0^2 + o(1) ≥max_h ∈ H_β^0:|h_1| ≥ρ,h^T Σ h ≤ R^2 h_1^2h^T Σ h.This proposition is as Theorem 9 in<cit.> but tailored for the particular situation. A proof is given in Section <ref>. We remark that such results are not a direct consequence of the Le Cam theory, as we are dealing with triangular arrays.Assume the conditions of Proposition <ref> and that for some fixed ρ>0 and R >0 and some sequence h ∈ H_β^0, with |h_1| ≥ρ and h^T Σ h ≤ R^2,it is true thatV_β^0^2 =h_1^2h^T Σ h+ o(1) .Then T is asymptotically efficient. This is our main tool to arrive at asymptotic efficiency for some special Σ's.The restriction to directions in H_β^0 means that the Cramér Rao lower bound for the asymptotic variance V_β^0^2 can be orders of magnitude smaller than Θ_1,1. Under the sparse model (<ref>)[A more natural model in this context might be B := { b ∈^p : b_-1_0^0 ≤ s-1 } where, for b ∈^p, b_-1 := ( b_2 , … , b_p)^T.]we haveH_β^0⊆ H_β^0 ,whereH_β^0:= { h _0^0 ≤ s- s_0 }and s_0 = |S_0| withS_0 := {β_j^0 ≠ 0 } being the set of active coefficients of β^0. Some special casesa) If Θ_1∈ H_β^0, then V_β^0^2 + o(1)= Θ_1,1. Note that the condition on Θ_1 depends on β^0 (via s_0).b)Suppose { 1 }∈ S_0, s=s_0 and that the following “betamin" condition holds: | β_j^0 | > m_n/√(n) for all j ∈ S_0, where m_n is some sequence satisfyingm_n →∞. Then we see that H_β^0∩{ h^T Σ h ≤ R^2} ={ h_-S_0 _0 = 0 }∩{ h^T Σ h ≤ R^2}.The lower bound is thenV_β^0^2+ o(1) ≥ ( Σ_S_0, S_0 ^-1)_1,1 ,where Σ_S_0 , S_0 is the matrix of covariances of the variables in S_0. This lower bound corresponds to the case S_0 is known. The bound could be achieved if one has a consistent estimate Ŝ of S_0. For this one needs betamin conditions in order to have no false negatives. Estimation of β_1^0 using some estimator Ŝ of S_0 typically means that the estimatorof β_1^0 is not regular. Imposing further conditions beyond model (<ref>) to make it regular may diminish the lower bound.c) More generally, if{ 1 }∈ S_0 and | β_j^0 | > m_n /√(n) for all j ∈ S_0 and some sequence m_n →∞ then the lower bound corresponds to knowing the set S_0 up to s-s_0 additional variables. d) Suppose that β^0 is an interior point in the sense that it stays away from the boundary: for some fixed 0 < η< 1 it holds that s_0 ≤ (1- η) s (so that 1- s_0 / s stays away from zero). Then { h : h _0^0 ≤η s }⊂ H_β^0 .The lower bound can then still depend on η. A rescaling argument as applied in the next examples, Example <ref> and more generally Example <ref>,does not work here. This signifies that model (<ref>) less suitable in our context: the exact value of s plays a too prominent role. Consider model (<ref>). The situation is then more like a classical one. Suppose β^0 stays away from the boundary of the parameter space, i.e., for a fixed 0<η < 1 it holds that β^0 _1 ≤ (1-η )√(s). Then{ h _1 ≤η√(ns)}⊂ H_β^0 .By a rescaling argument the dependence on η in the left hand side plays no role in the lower bound: for all M>0 fixedV_β^0^2 +o(1) ≥ ( min_c ∈^p-1: c _1 ≤ M√(ns) (x_1 -x_-1 c )^2)^-1 ,where x_1 is the first entry of x and x_-1 := ( x_2 , … ,x_p ).In other words, unlike in model (<ref>), the exact value of s does not play a prominent role. We thus see that in order to be able to improve over Θ_1,1 we must have that Θ_1 is rather non-sparse: Θ_1 _1 should be of larger order than √(ns).For example if s = o(n /log p), say s= n/(m_n log p ) for some sequence m_n →∞ slowly, the model directions have ℓ_1-norm of order n / √(m_n log p). To improve over Θ_1,1 one thus must have Θ_1_1 of larger order, say Θ_1_1= n / √(log p ). We get back to this in Example <ref>.We now turn to model (<ref>). Using the same arguments as in Example <ref> one sees that if β^0 stays away from the boundary, one may use model directionswith ·_r^r of order √(n^r s^2-r). The lower bound is thenV_β^0^2 +o(1) ≥ ( min_c ∈^p-1: c _r^r ≤ M√(n^r s^2-r) (x_1 -x_-1 c )^2)^-1where M >0 is any fixed constant. It is clear, and illustrated by Examples <ref> , <ref> and <ref>, that the lower bound of Proposition <ref> depends on the model B.The sparse model (<ref>) is perhaps too stringent. One may want to take the model B as the largest set for which a regular estimators exist. This points in the directionofmodel (<ref>). Wewill see that when Σ is known this model is indeed useful but when Σ is unknown it is too large.§.§ Notations and definitionsWe consider an asymptotic framework with triangular arrays of observations. Thus, unless explicitly stated otherwise, all quantities depend on n although we do not (always) express this in our notation. Let x_1 be the first entry of x and x_-1:= ( x_2 , …,x_p) be this vector with the first entry excluded, so that x = (x_1 ,x_-1 ). Write Σ_-1,-1:=x_-1^Tx_-1∈^(p-1)× (p-1).For vectors b ∈^p be use a similar notation: b_1∈ is the first coefficient and b_-1∈^p-1 forms the rest of the coefficients. Apart from the regression (projection) xβ^0 of y on x, we consider the regression x_-1γ^0 of x_1 on x_-1, and an approximation γ^♯ of γ^0 which is accompanied by a parameter λ^♯ to form an “eligible pair" (γ^♯, λ^♯) as given in Definition <ref>. When Σ is known we can invoke the noiseless Lasso γ_ Lasso with tuning parameter λ_ Lasso(an approximate projection of x_1 on x_-1) to approximate γ^0. See (<ref>) for its definition. For the case Σ is unknown we apply the notation Σ̂:= X^T X / n. We let X_1∈^n be the first column of X, and the matrix X_-1∈^n × (p-1) be the remaining columns and we write Σ̂_-1,-1:= X_-1^T X_-1 / n.We do a approximate regression of X_1 on X_-1invoking the noisy Lasso γ̂ with tuning parameter λ^ Lasso as given in (<ref>). The various vectors of coefficients and their “lambda parameter" are summarized in Table <ref>. Here we also addthe Lasso β̂for the estimation of β^0, as defined in (<ref>) with tuning parameter λ. We write for S ∈{1 , … , p }, x_S :={ x_j}_j ∈ S and x_-S :={ x_j}_j ∉ S,j ≠ 1. We further let for S ⊂{ 1 , … , p} with cardinality s, the matrix Σ_S,S:=x_S^Tx_S ∈^ s× s be thecovariance sub-matrix formed by the variables in S and Σ_-S,-S:=x_-S^Tx_-S∈^(p-1- s) ×( p-1- s) andΣ_S,-S :=x_S^T x_-S=: Σ_-S, S^T ∈^ s× (p-1- s). Note that the vector of coefficients γ^0 of the regression of x_1 on x_-1 is a vector in ^p-1. We will index its entries by { 2 , … , p }: γ^0 = (γ_2^0 , … , γ_p^0 )^T. Also the various other “gamma" parameters" γ will be indexed by { 2, … , p }. It should be clear from the context when this indexing applies.For c=(c_2, … , c_p) and S ⊂{ 2 , … , p } we write c_S := {γ_j :j ∈ S } and γ_-S := { c_j : j ∉ S, j ≠ 1}. We use the same notation for the (p-1)-dimensional vector c_S which has the entries not in S set to zero, and we then let c_-S= c- c_S.For a positive semi-definite matrix A we let Λ_ min^2 (A) be its smallest eigenvalue and Λ_ max^2 (A) be its largest eigenvalue.The smallest eigenvalue of Σ is written shorthand as Λ_ min^2:= Λ_ min^2 (Σ). Recall we assume throughout thatΛ_ min^2 stays away from zero: 1/ Λ_ min^2 = 𝒪 (1). We use the shorthand notation ≫ 0 for “strictly positive and staying away from zero". Thus throughout we assume Λ_ min^2 ≫ 0. In order to be able to construct confidence intervals one needs some uniformity inunknown parameters. We introduce the following definition.Let B be the model for β^0.Let { Z_n } be a sequence of real-valued random variables depending on (X,Y) and { r_n } a sequence of positive numbers. We say that { Z_n } is 𝒪__β^0 ( r_n) uniformly in β^0 ∈ B iflim_M →∞lim sup_n →∞sup_β^0 ∈ B_β^0 ( |Z_n|> M r_n) = 0 .We say that Z_n= o__β^0(r_n) uniformly in β^0 ∈ B if lim_n →∞sup_β_0 ∈ B_β^0 (| Z_n| > η r_n) = 0 ,∀ η>0 .§.§ Organization of the rest of the paper Section <ref> contains the results for the case Σ known and applying a debiased Lasso using sample splitting. Here we also introduce the concept of an eligible pair (γ^♯ , λ^♯) in Definition <ref>.Section <ref> contains results and constructions for eligible pairs. Section <ref> considers the case Σ unknown and a debiased Lasso (without sample splitting). Section <ref> concludes and Section <ref> collects the proofs. Section <ref> (included for completeness) contains some elementary probability inequalities for products of Gaussians, which are applied in Section <ref>.In Tables <ref> and <ref> we summarize the (sparsity) conditionswe use, see Examples <ref>, <ref> and <ref> for the case Σ known andExamples <ref> and <ref> for the case Σ unknown. The particular cases r=0 and r=1 follow from the general case 0 ≤ r ≤ 1 when Σ is known. When Σ is unknown the case r=0 also follows from the general case 0≤ r < 1. With r=1 the model is then too large. We have displayed the extreme cases separately so that the conditions for these can be read off directly . In particular for r=0 one sees the standard sparsity conditions known from the literature. For r=1 (Σ known) one sees that unlike in the other cases there is no logarithmic gap between conditions for asymptotic normality and asymptotic efficiency.§ THE CASE Σ KNOWN Before presenting “eligible pairs" in Definition <ref>, we provide the motivation that led us to this concept.Recall thedebiased Lasso given in (<ref>). If Σ is known we choose the direction Θ̃_1=: Θ_1^♯ non-random,depending on Σ.We invoke the decompositionb̂_1 - β_1^0 =Θ_1^♯ T X^T ϵ / n +( e_1 -Σ̂Θ^♯ )^T ( β̂- β^0 )_ remainder .The remainderis( e_1- Σ̂Θ^♯)^T ( β̂- β^0 )=Θ_1^♯ T ( Σ - Σ̂)(β̂- β^0 )_:=(i) +(e_1-ΣΘ^♯)^T (β̂- β^0 )_:=(ii) . The first term (i) can be handled assuming Θ_1^♯ TΣΘ_1^♯ = 𝒪 (1) andΣ^1/2 (β̂- β^0)_2 = o_ (1). This goes along the lines oftechniques as in <cit.>, applying the conditionsused there. One then arrives at (i)= o_ (1/√(n) ).(We will however alternatively use a sample splitting technique later on in Theorem <ref> to simplify the derivations.) The second term (ii) is additional bias and will be our major concern. If Θ_1^♯= Θ_1 thisterm vanishes. However, as we will see it is useful to applyinstead of Θ_1some sparse approximation of Θ_1. In fact, weaim at a sparse approximation Θ_1^♯ withΘ_1,1^♯ being smaller than Θ_1,1 and theirdifference not vanishing.We will assume conditionsthat ensure the additional bias is negligible and invoke that |(e_1-ΣΘ_1^♯)^T (β̂- β^0 ) | ≤Σ ( Θ_1^♯ - Θ_1) _∞β̂- β^0 _1 (recall that by the definition of Θ_1 it is true that e_1 = ΣΘ_1). One may think of applying insteadthe Cauchy-Schwarz inequality |(e_1-ΣΘ_1^♯)^T (β̂- β^0 ) |≤Σ^1/2 ( Θ_1^♯- Θ_1) _2Σ^1/2 (β̂- β^0) _2. Thisleads to requiring thatΣ^1/2 ( Θ_1^♯- Θ_1) _2→ 0 fast enough. But we actually wantΣ^1/2 ( Θ_1^♯- Θ_1) _2↛0 in order to be able to arriveat an improvement over the asymptotic variance. Consider now for some p≥ 1 the generaldual norm inequality |(e_1-ΣΘ^♯)^T (β̂- β^0 ) |≤Σ ( Θ_1^♯ - Θ_1) _ pβ̂- β^0 _ qwhere 1/p + 1/q =1. Choosing p≤ 2 here again works against our aim to improve the asymptotic variance. Thus we need to choose p>2 (and therefore q < 2). This certifies the choice p=∞ as being quite natural. We note thatΣ^1/2 ( Θ_1^♯ - Θ_1) _2 ≤Σ (Θ_1^♯ - Θ_1) _2 / Λ_ min. This means we want Σ (Θ_1^♯ - Θ_1) _2 ↛0 as weassume that Λ_ min stays away from zero. Now it is clear that if for some vectorv ∈^p, it holds that v _∞≤λ_0^♯, thenv _2 ≤√(p)λ_0^♯. So v _2 ↛0 impliespλ^♯ 2↛0. In other words, we can only improve the asymptotic variancein the high-dimensional case. Taking the dual norm inequality (<ref>) as starting point we now need Σ (Θ_1^♯- Θ_1)_∞≤λ_0^♯ for some constant λ_0^♯ small enough, such that uniformly in β^0 ∈ B λ_0^♯β̂- β^0 _1 = o__β^0 (1/√(n)) .With the above considerations as motivation, wenow concentrate on an ℓ_∞-condition as given ininequality (<ref>). We fix some λ^♯ and construct vectors Θ_1^♯for which inequality (<ref>) holds. It is based on replacing the vector of coefficients γ^0 of the regressionof x_1 on x_-1 by a sparse approximation γ^♯.Let γ^♯∈^p-1 and λ^♯ >0.We say that the pair (γ^♯ , λ^♯ ) is eligible ifΣ_-1,-1 (γ^♯- γ^0) _∞≤λ^♯ and λ^♯γ^♯_1 → 0 . The two conditions in Definition <ref> will allow us to arrive at(<ref>) as is shown in the next lemma. Suppose (γ^♯, λ^♯) is an eligible pair. Then1- γ^0TΣ_-1,-1γ^♯≥Λ_ min^2 -o(1)i.e., 1- γ^0TΣ_-1,-1γ^♯≫ 0 eventually.Let (for n sufficiently large)Θ_1^♯ := [ 1- γ^♯ ]/ (1- γ^0TΣ_-1, -1γ^♯ ).Then we have Σ( Θ_1^♯ - Θ_1)_∞≤λ_0^♯ whereλ_0^♯:= λ^♯ /(1- γ^0TΣ_-1, -1γ^♯) =𝒪 (λ^♯ ). Moreover,Θ_1^♯ T ΣΘ_1^♯ = Θ_1,1^♯ +o(1)≤ Θ_1,1 + o(1).Finally, in order to have a non-vanishingimprovementof Θ_1,1^♯ over Θ_1,1 it must be true thatγ^0 is not sparse, in the sense that λ^♯γ^0 _1≫ 0 .The first condition (<ref>) of Definition<ref> can be rewritten as x_-1γ^0 =x_-1γ^♯ + ξ^0,|x_j ξ^0 | ≤λ^♯ ∀ j ∈{2 , … , p } . The second condition (<ref>) in this definition can be thought of as a sparsity condition on γ^♯. The two conditions together require that the regression of x_1 on x_-1 is sparse when one relaxes the orthogonality condition of residuals to approximate orthogonality.[Condition (<ref>) is of the same nature as the condition λβ^0 _1 = o(1) (which follows from the classical condition λ^2 β^0 _0^0 =o(1) if β^0 _2 = 𝒪 (1)) when applying the Lasso (<ref>) with tuning parameter λ.]One may think of γ^0 as a “least squares estimate" of γ^♯ in a noisy regression model. We refer to Subsection <ref> which gives a very natural interpretation of eligible pairs. Further, for Θ_1^♯ defined in (<ref>)one has the equivalence Θ_1,1 - Θ_1,1^♯≫ 0⇔( ξ^0)^2 ≫ 0 .To have Θ_1,1^♯ improving over Θ_1,1 we see from the above lemma that we aim at a situation where γ^0, and hence Θ_1,is not sparse, but where γ^0 can be replaced by a sparse vector γ^♯. For some special Σ's, we give examples of eligible pairs in Section <ref>. That section also discusses for a given λ^♯ uniqueness of the vectors γ^♯ for which the pair (γ^♯, λ^♯) is eligible. Moreover, we show cases where x_-1γ^♯ is an approximation of the projection of x_1 on a subset of x_S of the other variables for some S ⊂{ 2 , … , p }, see Lemma <ref>. This may help to understand why the Cramér Rao lower bound can be achieved in those cases. Lemma <ref> essentially has all the ingredients to prove asymptotic normality of the debiased Lasso (<ref>) with direction Θ̃_1= Θ_1^♯ and Θ_1^♯ given in (<ref>) in this lemma.It can be done along the lines of Theorem 3.8 in <cit.>, assuming the conditions stated there. However, as the authors point out,when using instead the sample splitting approach their Assumption (iii) is not needed.Sampling splitting it is also mathematically less involved. We present it here.Assume the sample size n is even. Define the matrices(X_I, Y_I):={X_i,1, … , X_i,p, Y_i}_1 ≤ i ≤ n/2∈^n/2 × (p+1),(X_II, Y_I):= {X_i,1, … , X_i,p, Y_i}_n/2 < i ≤ n∈^n/2 × (p+1).Let β̂_I be an estimator of β^0 based on the first half (X_I, Y_I) of the sample, for instance the Lasso estimator min{ Y_I - X_I b_2^2 /n+λ b _1 }.Similarly, let β̂_II be an estimator of β^0 based on the second half (X_II, Y_II) of the sample.Let (γ^♯, λ^♯) be an eligible pair. We then define the two debiased estimatorsb̂_I,1^♯ := β̂_II,1 + 2Θ_1^♯ T X_I^T (Y_I - X_I β̂_II )/n b̂_II,1^♯ := β̂_I,1 + 2Θ_1^♯ T X_II^T (Y_II - X_IIβ̂_I)/nwhere Θ_1^♯ is given in (<ref>) in Lemma <ref>. The final estimator b̂_1^♯ is obtainedby averaging these two:b̂_1^♯ := b̂_I,1^♯+b̂_II,1^♯ 2 . Let now B be a given model class for the unknown vector of regression coefficients β^0. Let (γ^♯, λ^♯) be an eligible pair, Θ_1^♯ be given in (<ref>) andb̂_1^♯ be given in(<ref>) with b̂_I,1^♯ and b̂_II,1^♯ the debiased estimators based on Θ_1^♯ using the splitted sample. Suppose that uniformly in β^0 ∈ BΣ^1/2 (β̂_I - β^0) _2 = o__β^0 (1) , Σ^1/2 (β̂_II - β^0 )_2 = o__β^0 (1)and√(n)λ^♯β̂_I - β^0 _1 = o__β^0 (1) , √(n)λ^♯β̂_II - β^0 _1 = o__β^0 (1) .Then, uniformly in β^0 ∈ B,b̂_1^♯ - β_1^0 = Θ_1^♯ TX^T ϵ /n + o__β^0 (1/ √(n) ) ,and lim_n →∞sup_β_0 ∈ B_β^0 ( √(n) ( b̂_1^♯ - β_1^0 ) Θ_1,1^♯≤ z)= Φ(z) ,∀ z ∈.Theorem <ref> shows that under its conditions the estimator b̂_1^♯ is uniformly asymptotically linear and regular. It means that for this estimator the Cramér Rao lower bound as given in Subsection <ref> is relevant. Depending among other things on B and Σ it does or does not reach the Cramér Rao lower bound, see Remark <ref>. Lemmas <ref>, <ref> and <ref> in Subsection <ref> present examples of eligible pairs (γ^♯, λ^♯) where Θ_1,1 - Θ_1,1^♯(with Θ_1,1^♯ given in (<ref>)) is non-vanishing:Θ_1,1 - Θ_1,1^♯≫ 0.Thus for those cases Theorem <ref> shows an asymptotic variance remaining strictly smaller than Θ_1,1. Moreover, the constructionsof Lemmas <ref>, <ref> and <ref> allow for directions Θ_1^♯(depending on β^0 _0^0∈ H_β^0 in model (<ref>). In models (<ref>) and (<ref>) the direction Θ_1^♯ lies in H_β^0 after scaling where only the scaling depends on β^0. For these cases (special constructions of Σ) the Cramér Rao lower bound is therefore achieved.We now discuss the requirements (<ref>) and (<ref>) for the models (<ref>), (<ref>) and (<ref>). The overall picture is summarized in Table <ref>.Consider modelmodel (<ref>) with s= o( n / log p ) (sparsity variant (i)). For theLasso estimatorβ̂ given in (<ref>), with appropriate choice of the tuning parameter λ≍√(log p / n ), one has uniformly in β^0 ∈ BΣ^1/2 (β̂- β^0 )_2^2 = 𝒪__β^0 ( s log p / n) , β̂- β^0 _1 = O__β^0 ( s √(log p / n)).This follows from e.g. Theorem 6.1 in <cit.>,together with results for Gaussian quadratic forms as given in (<ref>).So for β̂_I and β̂_II Lasso's based on the splitted sample with suitable tuning parameter λ, the requirement on λ^♯ becomesλ^♯ s √(log p)= o( 1).This is guaranteedwhen λ^♯= 𝒪 ( √(log p) /n ). The smaller the sparsity s, the more room there is for improvement (i.e., the larger the collection of covariance matrices Σ that allow for improvement over Θ_1,1). For example, if in fact s= o( √(n ) / log p ) we can take λ^♯= 𝒪 ( √(log p /n) ). Finally, if γ^♯_0^0 = s_♯ with s_♯≤ s- s_0 -1 the estimator b̂_1^♯ is asymptotically efficient. In view of Theorem <ref> and the statements of Example <ref> and Remark <ref>, we see that for model (<ref>) the debiased estimator of Theorem 3.8 in <cit.>, which uses Θ̃_1= Θ_1, is in certain cases asymptoticallyinefficient as a choice Θ̃_1 = Θ_1^♯≠Θ_1 can give an improvement in the asymptotic variance (and is then efficient for certain cases). In this example we take the model (<ref>) with 0< s = o( n / log p ).Let β̂ be again the Lasso estimatorgiven in (<ref>) with appropriate choice of the tuning parameter λ≍√(log p / n ). One may use a “slow rates" result: uniformly in β^0 ∈ B it is true thatΣ^1/2 ( β̂- β^0 ) _2 = o__β^0 (1) ,β̂- β^0 _1 = 𝒪__β^0(√(s)) ,see for example <cit.>, Theorem 6.3, and combine this with results for quadratic forms as given in (<ref>).(The arguments for establishing these “slow rates" are in fact as in Lemma <ref> for the node-wise Lasso.) Taking β̂_I and β̂_II again as appropriate Lasso's, condition (<ref>)of Theorem <ref> holds ifλ^♯√(ns)=o(1) .Then for γ^♯_1 = 𝒪 (√(ns)) we get an eligible pair ( γ^♯, λ^♯). Since when β^0 stays away from theboundary, such γ^♯ form a model direction after scaling, we see that the Cramér Rao lower bound is achieved. Recall now that in Example <ref> we concluded that, in view of Corollary <ref>, in order to be able to improve over Θ_1,1 we must have γ^0 _1 of order larger than √(ns).As pointed out in Remark <ref>one may apply one of theLemmas <ref>, <ref> or <ref> to create vectors γ^0 and eligible pairs (γ^♯, λ^♯) where λ^♯√(ns)→ 0, γ^♯_1 can take any value of order √(ns) andΘ_1,1 - Θ_1,1^♯≫ 0.The results for model (<ref>) are a special case of those for model (<ref>). One needs again s= o( n / log p) and one may for example apply Corollary 2.4 in <cit.>, again together with (<ref>). For the Lasso β̂ in (<ref>) with λ≍√(log p / n) appropriately chosen one gets uniformly in β^0 ∈ BΣ^1/2 ( β̂- β^0 ) _2^2 = 𝒪__β^0( s log p / n )^2-r2 ,β̂- β^0 _1 = 𝒪__β^0( ( log p / n)^1-r2s^2-r2 ).The requirement (<ref>) on λ^♯ is thusλ^♯ ( log p)^1- r2√(n^rs^2-r )= o(1) .If γ^♯_r^r = 𝒪 ( √( n^r s^2-r) ) then (γ^♯, λ^♯) is an eligible pair and the Cramér Rao lower bound is achieved whenever β^0 stays away from the boundary.In order to be able to improve over Θ_1,1 we now need γ^0 _r^r of larger order √( n^r s^2-r) by Corollary <ref>.Remark <ref> can be taken into the considerations here too. § FINDING ELIGIBLE PAIRS The main results of this section can be found in Subsection <ref> where for any λ^♯ we construct eligible pairs (γ^♯, λ^♯) by choosing γ^0 appropriately. The results can be seen as existence proofs. Before doing these constructions we discuss uniqueness in Subsection <ref>, in Subsection <ref> the noiseless Lasso as a practical method for improving over Θ_1,1 and in Subsection <ref> we examine whether or not projections on a subset of the variables can lead to eligible pairs. For the latter we impose rather string conditions. We show in Subsection <ref> that eligible pairs are more flexible than projections. Nevertheless in the final part of this section we return to projections as they come up naturally when imposing non-sparsity constraints on γ^0 §.§ UniquenessFix λ^♯ and let (γ^♯, λ^♯) and (γ^♭, λ^♯) be two eligible pairs. ThenΣ_-1,-1^1/2(γ^♯ - γ^♭ )_2^2 → 0 .This lemma tells us that asymptotically it makes no difference to debias using (γ^♯, λ^♯) or (γ^♭, λ^♯).§.§ Using the LassoFix some tuning parameter λ_ Lasso and consider the noiseless Lassoγ_ Lasso:= min_c ∈^p-1{ (x_1 -x_-1 c )^2 + 2λ_ Lasso c _1 } .One may verify that(γ_ Lasso, λ_ Lasso) is an eligible pair if λ_ Lassoγ^0 _1 → 0. But the latter is exactly what we want to avoid.If ( γ^♯ , λ^♯) is an eligible pair then the noiseless Lasso can find it if one chooses λ_ Lasso= 𝒪 ( λ^♯) larger than λ^♯, as follows from the next lemma. It says that given λ^♯ one may use the noiseless Lasso for constructing a direction, Θ_1,Lasso say, with which one has same improvement over Θ_1,1 as with Θ_1^♯. Suppose ( γ^♯, λ^♯) is an eligible pair. Let λ_ Lasso > λ^♯and λ_ Lassoγ^♯_1 → 0. Let γ_ Lasso be the noiseless Lasso defined in (<ref>). ThenΣ_-1,-1^1/2(γ_ Lasso - γ^♯ )_2^2 → 0 .In addition, if λ_ Lasso≥ 2 λ^♯ (say) then ( γ_ Lasso , λ_ Lasso) is an eligible pair.§.§ Using projections In this subsection we investigate (rather straightforwardly) conditions such that the coefficients of a projection of x_-1γ^0can be joined with a λ^♯ to form an eligible pair.Fix some set S⊂{ 2, … , p } with cardinality s.The value of s need not be s, where s is the sparsity used in the model class B. Let x_S γ_S^S be the projection of x_-1γ^0 on x_S.Let γ^S ∈^p-1 be the vector γ_S^S ∈^ s completed with zeroes.Then γ^S _1 ≤√( s)γ^S _2 = 𝒪 (√( s) ) so that when λ^♯√( s)→ 0we have for γ^S the sparsity condition (<ref>) of Definition <ref>: λ^♯γ^S _1 → 0.Note that γ^0TΣ_-1,-1γ^S = γ^STΣ_-1,-1γ^S(=(x_S γ_S^S )^2) exactly in this case. Furthermorev^S := Σ_-1,-1 ( γ^0 - γ^S) = [ 0 v_-S^S ]wherev_-S^S =x_-S x_-1γ^-S ,withx_-1γ^-S = ( x_-Sγ_-S)A x_S =:(x_-S -x_S^T Σ_S,S^-1Σ_S,-S ) γ_-S^0being the anti-projection of x_-Sγ_-S^0 on x_S.We check whether the pair (γ^S , λ^♯) is an eligible pair, which is the case ifλ^♯√( s) = o(1) and v_-S^S _∞≤λ^♯. We briefly discuss some conditions that may help ensuring the latter.Let A _1 := max_j ∑_k |a_j,k | be the ℓ_1-operator norm of the matrixA. It holds thatv_-S^S _∞≤Σ_-S, -S - Σ_-S, SΣ_S,S^-1Σ_S,-S_1 γ_-S^0 _∞.To arrive at a sparse approximation of γ^0 one may consider putting its p- s-1 smallest-in-absolute-value coefficients to zero. To this end, consider the ordered sequence | γ^0 |_(1)≥⋯≥ |γ^0|_(p-1),write |γ^0|_(j) =: |γ_r_j ^0 |, j= 1, … , p-1, and let S= { r_1 , … r_ s}. Then clearly γ_-S^0 _∞≤γ^0 _2 / √( s) = 𝒪 (1/ √(s)). The condition (<ref>) excludes 1/ √( s) = 𝒪 ( λ^♯). We therefore examine situations where the coefficients in γ^0 decrease at a rate quicker than 1/ √( s)Let N be some integer and 0 ≠ v ∈^N be a vector.We callκ (v) :=v _1v _∞ v _2the sparsity index of v. We say that v is asymptotically sparse if κ (v)→ 0. A vector v with v _2=𝒪 (1) can have somerelatively large coefficients, but it cannot have too many of these. Ifin addition v _1 is large it cannot have many zeroes either. Asymptotic non-sparseness of the vector v with the large coefficients removed means that there are many very small non-zero coefficients.Let v_j = 1/ √(N) for all j. Thenv _1 = 1/v_∞ = √(N). Thus κ (v)=1and v is not (asymptotically) sparse.Let s≤ N and v_j=0 for j ≤ s and v_j = 1/ √(j log N ) for j> s. Thenκ(v)≍√( N /( slog^2 N) ). The vector v is not asymptotically sparse if s= 𝒪(N/ (log^2 N) ).Suppose that γ_-S^0 is not asymptotically sparse. Letλ^S = κ (γ_-S^0)Cγ_-S^0 _1,whereC:= Σ_-S, -S - Σ_-S, SΣ_S,S^-1Σ_S,-S_1.Assume that λ^S √( s)→ 0. Then ( γ^S , λ^S) is an eligible pair and λ^S γ^0 _1 ↛0.§.§ Approximate projectionsRecall that the first condition (<ref>) of Definition <ref> can be written asx_-1γ^0 =x_-1γ^♯ + ξ^0where ξ^0 is “noise" satisfying x_-1^T ξ^0 _∞≤λ^♯.Denote the active set of γ^♯by S = { j∈{ 2 , … , p }: γ_j^♯≠ 0 } and its cardinality by s := |S|.Then λ^♯√( s)→ 0 implies that 1/ Θ_1,1^♯ is asymptotically the squared residual of the projection of x_1 on x_S as is shown in the next lemma. In that sense, eligible pairs are more flexible than projections.If s is small enough, then for model (<ref>),(<ref>) or (<ref>)one has that Θ_1^♯ is a model direction (after rescaling). The estimator b̂_1^♯ in Theorem <ref> then reaches the Cramér Rao lower bound. Suppose for some set Swith cardinality s that γ^♯ = γ_S^♯ and assumethe first condition (<ref>) of Definition <ref>. If λ^♯√( s)→ 0then the second condition (<ref>) holds too so that (γ^♯, λ^♯) is an eligible pair. Moreover, 1/ Θ_1,1^♯ = ( x_1 -x_S γ_S^S )^2 + o(1) whereγ_S^S = Σ_S,S^-1 x_S^Tx_1 .§.§ Reverse engineeringIn this subsection we fix λ^♯ and then construct vectors γ^0 ∈^p-1 such that there is an eligible pair (γ^♯, λ^♯ ). These constructions are in a sense equivalent but approach the problem from different angles. In these constructions the vector γ^♯ is having active set S= {γ_j^♯≠ 0 } with cardinality s:= |S|. The sparsity of γ^♯ is then measured in terms ofthe value of s. More general constructions are possible, but in this way we can apply the results to any of the models (<ref>), (<ref>) or (<ref>). In view of Lemma <ref> is means that the constructions correspond to an approximate projection on x_S. Throughout this subsection, the matrix Σ_-1, -1 is is assumed to have 1's one the diagonal and smallest eigenvalue Λ_ min^2 (Σ_-1, -1)≫ 0.§.§.§ Which γ^0's are allowed?We let for γ^0 ∈^pΣ (γ^0):= [1 γ^0TΣ_-1,-1Σ_-1, -1γ^0 Σ_-1, -1 ] . We say that the vector γ^0 is allowed if Σ (γ^0) is positive definite, with Λ_ min^2 (Σ (γ^0)) ≫ 0 andΣ_-1, -1γ^0 _∞≤ 1.Suppose that 1- Σ_-1,-1^1/2γ^0 _2^2≫ 0 .Then γ^0 is allowed.§.§.§ Regression: γ^0 as least squares estimate of γ^♯In this subsection we create γ^0 using random noise. We then arrive at an eligible pair “with high probability". Let N∈ be a given sequence with N > p. Take a matrix Z_-1∈^N × (p-1) which has Z_-1^T Z_-1 / N = Σ_-1,-1.Let ξ∈^N have i.i.d. standard Gaussian entries, let γ^♯∈^p-1 be a vector satisfying γ^♯_1= o ( √(log p / N ) ) and defineZ_1 := Z_-1γ^♯ + ξ . Let γ^0 := (Z_-1^T Z_-1 )^-1 Z_-1^T Z_1 be the leastsquares estimator of γ^♯. Finally let λ^♯≍√(log p / N).be appropriately chosen. Then ( γ^♯ , λ^♯) is with high probability an eligible pair. Indeedthe first condition (<ref>) of Definition <ref> follows from Σ_-1,-1 (γ^0- γ^♯ ) _∞=Z_-1^T ξ/N _∞ = 𝒪_ ( √(log p / N) ) so that for appropriate λ^♯≍√(log p / N), with high probabilityΣ_-1,-1 (γ^0- γ^♯ ) _∞≤λ^♯ . The second condition (<ref>) of Definition <ref> follows from the conditionγ^♯_1= o ( √(log p / N ) ) so λ^♯γ^♯_1 = o(1). We also see that if p/N ≫ 0 then with high probability Θ_1,1^♯ as givenin (<ref>) is an improvement over Θ_1,1, since Σ_-1,-1 ^1/2 ( γ^0 - γ^♯ )_2^2 = χ_p-1^2 / N where χ_p-1^2 has the chi-squared distribution with p-1 degrees of freedom. It follows thatΣ_-1,-1 ^1/2 ( γ^0 - γ^♯ )_2^2= p + 𝒪 (√(p) )N .With high probability this stays away from zero so that alsoΘ_1,1 - Θ_1,1^♯≫ 0 .For appropriate N with 1- p/N ≫ 0 the vector γ^0 is with high probability also allowed by Lemma <ref>.With this choice of N we have λ_♯≍√(log p / p ). Recall that according to Remark <ref> it must be true that p ≥ 1/ λ^♯ 2. In the present context we in fact have p ≍log p / λ^♯ 2.§.§.§ Creating γ^0 directly We first recall that Θ_1,1 - Θ_1,1^♯≫ 0 ⇔Σ_-1,-1^1/2 (γ^0 - γ^♯) _2^2 ≫ 0with Θ_1^♯ given in (<ref>). It will be the case in the constructions of this subsection. Fix some set S⊂{ 2 , … , p } with cardinality s and fix some λ^♯. We will assumeλ^♯√( s)→ 0.Suppose there exists a vector z ∈^p-1 with z _∞≤ 1 ,and 1 - λ^♯2Σ_-1,-1^-1/2 z _2^2 ≫ 0, λ^♯ 2Σ_-1,-1^-1/2 z _2^2 ≫ 0.Let γ^♯ be a vector in ^p-1 withγ_-S^♯=0 (i.e. γ^♯ = γ_S^♯) and1-λ^♯2Σ_-1,-1^-1/2 z _2^2 -Σ_-1,-1^1/2γ_S^♯_2^2≫ 0.Defineγ^0 :=γ^♯+ λ^♯Σ_-1,-1^-1 z .Then, if λ^♯√( s)→ 0, the pair (γ^♯, λ^♯) is an eligible pair. Moreover, γ^0 is eventually allowed, λ^♯γ^0 _1 ↛0 and in factΣ_-1,-1^1/2 (γ^0 - γ^♯) _2^2 ≫ 0. Recall that Λ_ min^2 (Σ_-1,-1 ) ≫ 0. To check condition (<ref>) for some z _∞≤ 1 one may want to impose in addition that Λ_ max (Σ_-1,-1 ) = 𝒪 (1).Then the condition is true if z _2≍ 1/λ^♯ which is true for instance if1/ λ^♯2 of the coefficients of z stay away from zero.This is only possibleif p > 1/ λ^♯ 2 i.e., in high-dimensional situations (see alsoRemark <ref>). One may also consider takingz= Z_-1^T ξ/(N λ^♯) where Z_-1, ξ and λ^♯ areis in Subsection <ref>. Then λ^♯ 2Σ_-1,-1^1/2 z _2^2 = χ_p-1^2 / N as there and one arrives at eligible pairs “with high probability".We now examine the following question: can one choose γ_S^♯ in Lemma <ref> equal to γ_S^0? As we will see this will only be possible if a form of the irrepresentable conditionholds. The “usual" irrepresentable condition (that implies the absence of false positives of the Lasso, see <cit.>) involves the coefficients of theprojection of the “large" collection x_-S on the “small" collection x_S. In our case, we reverse the roles of S and -S. Fix a vector z_-S∈^p- s-1 with z_-S_∞≤ 1. We say that the reversed irrepresentable condition holds for (S, z_-S) ifΣ_S, -SΣ_-S, -S^-1 z_-S_∞≤ 1 .a) Assume the reversed irrepresentable condition holds for (S , z_-S),and that in addition1- λ^♯ 2Σ_-1,-1^-1/2z_-S_2^2 ≫ 0,λ^♯2Σ_-1,-1^-1/2z_-S_2^2 ≫ 0.Let γ_S^0 satisfy1- λ^♯2Σ_-1,-1^-1/2z_-S_2^2- Σ_-1,-1^1/2γ_S^0 _2^2≫ 0.Define γ_-S^0 :=λ^♯Σ_-S,-S^-1 z_-S andγ^♯:= γ_S^0 Then, if λ^♯√( s)→ 0, the pair (γ^♯, λ^♯) is an eligible pair, γ^0 is eventually allowed andλ^♯γ^0 _1 ↛0. In factΣ_-1,-1^1/2 (γ^0 - γ^♯) _2^2 ≫ 0.b) Conversely, if for some γ^0 and for γ^♯ := γ_S^0the pair (γ^♯, λ^♯) is an eligible pair, then the reversed irrepresentable condition holds for (S , z_-S) with appropriate z_-S satisfying z_-S_∞≤ 1. §.§.§ Creating γ^0 using a non-sparsity restrictionFix some set S⊂{2 , … , p } with cardinality s and some λ^♯>0. Let w ∈^p-1 be a vector of strictly positive weights with w _∞≤ 1 and define the matrix W as the diagonal matrix with w on its diagonal.Letc^0 ∈min{Σ_-1,-1^1/2c_2^2 : λ^♯ ( W c )_-S_1 = 1 }andζ_S := 0,ζ_j:=sign (c_j^0 ),j ∉ S .The random variable x_-1 c^0 is orthogonal to (i.e. independent of) x_S. MoreoverΣ_-1,-1 c^0 _∞ =w_-S_∞λ^♯Σ_-1,-1^-1/2 W ζ_2^2andΣ_-1,-1^1/2 c^0 _2^2 = 1 λ^♯2Σ_-1,-1 ^-1/2 W ζ_2^2Suppose that1-(λ^♯2Σ_-1,-1^-1/2 W ζ_2^2)^-1≫ 0 ,λ^♯2Σ_-1,-1^-1/2 W ζ_2^2 = 𝒪 (1) .Let γ^♯ be a vector satisfying γ_-S^♯=0 and0<1- (λ^♯2Σ_-1,-1^-1/2Wζ_2^2)^-1- Σ_-1,-1^1/2γ_S^♯_2^2 ≫ 0.Define γ_-S^0 = c_-S^0 and x_-1γ^0 :=x_S γ_S^♯+ ( x_-S A x_S ) γ_-S^0.Then x_S γ_S^♯ is the projection of x_-1γ^0 on x_S. Moreover, if λ^♯√( s)→ 0, the pair (γ^♯, λ^♯) is eligible, γ^0 is allowed and λ^♯γ^0 _1 ↛0. In factΣ_-1,-1^1/2 (γ^0 - γ^♯) _2^2 ≫ 0.As inRemark <ref>,to deal with the requirement(<ref>) one may want to impose the conditionΛ_ max^2 ( Σ_-1,-1 )=𝒪 (1).§ THE CASE Σ UNKNOWN As we will see the concept of an eligible pair will also play a crucial role when Σ is unknown.We use in this section the noisy Lassoγ̂∈min_c ∈^p-1{ X_-1 - X_-1 c _2^2 / n + 2λ^ Lasso c _1 } .We require the the tuning parameter λ^ Lasso to beof order √(log p / n). Then in the debiased Lasso given in (<ref>) we applyΘ̃_1 := Θ̂_1whereΘ̂_1 := [ 1- γ̂ ]/ ( X_1 - X_-1γ̂_2^2 / n +λ^ Lassoγ̂_1).Let (γ^♯, λ^♯) be an eligible pair, with λ^♯ = 𝒪 ( √(log p / n) ).In Lemma <ref>, we in fact need λ^ Lasso to be at least as large as λ^♯. The latter is allowed to be of small order √(log p / n), yet we will assume λ^ Lassoγ^♯_1 → 0.Let η_n → 0 be a sequencesuch that (inf_c:λ^ Lasso c _1 ≤ 4η_n^2,Σ_-1,-1^1/2 c_2= 1X_-1 c _2^2/n≥12 ) → 1 .Such a sequence exists, see for example Chapter 16 in <cit.> and its references.Defineε := X_1- X_-1γ^0 and ε^♯ := X_1- X_-1γ^♯= ε + X_-1 (γ^0 - γ^♯ ).Choose λ_ε^♯≍√(log p / n) in such a way that(X^T ϵ^♯_∞/n≥λ_ε^♯ ) → 0(see Lemma <ref>). This implies λ_ϵ^♯≥λ^♯. The following lemma establishes the so-called “slow rate". The proof is standard (see for example <cit.>, Theorem 6.3), up to the replacement of ε by ε^♯. The result is the noisy counterpart of Lemma <ref>Let λ^ Lasso≍√(log p / n ) satisfy λ^ Lasso≥ 2 λ_ε^♯ and suppose λ^ Lasso γ^♯_1 ≤η_n^2. Then we haveλ^ Lasso γ̂_1 = o_(1) , X_-1( γ̂- γ^♯) _2^2/n= o_ (1) ,Σ_-1,-1^1/2( γ̂- γ^♯) _2^2 = o_ (1) . Let b̂_1 be the debaised Lasso given in (<ref>)with Θ̃_1 equal to Θ̂_1 given in (<ref>):b̂_1 := β̂_1 + Θ̂_1 X^T (Y- X β̂)/n .Assume that λ^♯ = 𝒪 (√(log p / n)), that √(log p / n)γ^♯_1 = o(1) andλ^ Lasso≍√(log p / n) is sufficiently large(depending only on λ^♯). Suppose that uniformly in β^0 ∈ B√(log p )β̂- β^0 _1= o__β^0 (1). Then uniformly in β^0 ∈ B b̂_1 - β_1^0 = Θ̂_1^T X^T ϵ / n + o__β^0 (1/√(n)) . Moreover,lim_n →∞sup_β^0 ∈ B ( √(n)(b̂_1 - β_1^0 )√(Θ̂_1^T Σ̂Θ̂_1 )≤ z)=Φ (z)∀ z ∈ ,and for Θ_1^♯ given in (<ref>)Θ̂_1^T Σ̂Θ̂_1= Θ_1,1^♯ + o_ (1) .Recall that Lemmas <ref>, <ref> and <ref> present examples of eligible pairs (γ^♯, λ^♯) where Θ_1,1^♯remains strictly smaller than Θ_1,1. By Slutsky's Theorem we conclude that the asymptotic variance of b̂_1 is (up to smaller order terms) equal to Θ_1,1^♯.Note that Theorem <ref> only requires sparsity of γ^♯ in ℓ_1-sense: it requires γ^♯_1= o( √(n / log p)). Indeed, we in fact base the result on “slow rates" for the Lasso γ̂ given in Lemma <ref>Assume the conditions of Theorem <ref>. and that in fact√(log p)γ̂- γ^♯_1 = o_ (1) .Then also √(log p)Θ̂_1- Θ_1^♯_1 = o_ (1), which impliesΘ̂_1 X^T ϵ /n = Θ_1^♯ X^T ϵ / n + o_ (1/ √(n)) .Thus, then the estimator b̂_1 is asymptotically linear, uniformly in β^0 ∈ B.The uniform asymptotic linearity of b̂_1 implies in turn that the Cramér Rao lower bound of Subsection <ref> applies.The results of the following two examples are summarized in Table <ref>.Assume the sparse model (<ref>) with s = o ( √(n) / log p ). As stated in Example <ref>, for theLasso estimatorβ̂ given in (<ref>), with appropriate choice of the tuning parameter λ≍√(log p / n ), one has uniformly in β^0 ∈ Bβ̂- β^0 _1 = O__β^0 ( s √(log p / n) )= o__β^0 (1/√(log p) ) .Suppose now as in Theorem <ref> that λ^♯ = 𝒪 (√(log p / n )) and √(log p / n )γ^♯_1 = o(1).In fact, assume that γ^♯_0^0 is small enough so that Θ_1^♯ is a model direction. Then one obtains by the same argumentsif λ^ Lasso≍√(log p /n ) is suitably chosenγ̂- γ^♯_1 = O_ ( s √(log p / n))= o__β^0 (1/ √(log p) ).Thus then (<ref>) is met so that we have asymptotic linearity. It means that the Cramér Rao lower bound applies and is achieved. The model (<ref>) is too large for us to be able to apply when Σ is unknown. We now turn to the model (<ref>). For the model (<ref>) the rates for the Lasso β̂ are given in Example <ref>. One sees that for 0 ≤ r < 1 the requirement on s becomess = o(n^1-r2-r / log p ) .By the same arguments, if for a fixed 0 ≤ r≤ 1γ^♯_ r^ r= o(n^1- r 2 / log^2-r 2 p )one findsγ̂- γ^♯_1 =o_ (1/ √(log p) )which yields asymptotic linearity so that the Cramér Rao lower bound applies. If such γ^♯ is in addition a model direction after scaling, i.e. if γ^♯_r^r = 𝒪 ( n^r2 s^2-r2 ), then the Cramer Rao lower bound is achieved whenever β^0 stays away from the boundary.§ CONCLUSIONThis paper illustrates that Θ_1,1 can be larger than the asymptotic Cramér Rao lower bound, that for certain Σ the asymptotic variance of a debiased Lasso is smaller than Θ_1,1 and that in special such cases the asymptotic Cramér Rao lower bound is achieved.In Examples <ref> and <ref> we showed that if β^0 stays away from the boundary, then the asymptotic Cramér Rao lower bound is(min_ c _r^r ≤ M n^r2 s^2-r2 (x_1 -x_-1 c )^2)^-1where M is any fixed value. When Σ is known, Theorem <ref> shows that up to log-terms this lower bound is achieved as soon as there exists an eligible pair (γ^♯, λ^♯). with γ^♯_r^r = 𝒪 (n^r2 s^2-r2 ). When Σ is unknown the situation is more involved and in particular for model (<ref>) sparsity variant (i) is replaced by the stronger variant (ii). Model (<ref>) is too large for the case Σ unknown and model (<ref>) requires more sparsity then model (<ref>): the larger r the smaller s is required to be.Model (<ref>) however appears for both known and unknown Σ too stringent as results depend on the exact value of s, not only on its order. Model (<ref>) (with Σ known) or more generally model (<ref>) (with 0<r<1 if Σ is unknown) do not suffer from such a dependence as long as β^0 stays away from the boundary.§ PROOFS §.§ Proof for Section <ref>The proof of Proposition<ref> relies on the results in <cit.>, which allow the arguments to follow those of the low-dimensional case. These arguments are then rather standard.Proof of Proposition <ref>.Let h ∈ H_β^0, |h_1 | ≥ρ and h^T Σ h ≤ R^2. The log-likelihood ratio L (h) forβ^0+ h / √(n) with respect to β^0 isL (h)=h^T X^T ϵ / √(n) + h^T Σ̂h / 2 = h^T X^T ϵ / √(n) + h^T Σ h / 2+ o_ (1)since h^T Σ h = 𝒪 (1).Let for (x , y) ∈^p+1l_β_0 (x,y):= [ i_β^0 (x,y)x^T (y- xβ^0) ]and defineΩ_β^0 := _β^0 l_β^0 ( x,y ) l_β^0 ( x,y )^T. By the Lindeberg condition, we can apply Lindeberg's central limit theorem to conclude that for any sequence a:= (1, c^T)^T ∈^p+1 with c^T Σ c= 𝒪 (1) it holds thata^T∑_i=1^nl_β^0 (X_i, Y_i) √( n a^TΩ_β^0a) ) D_β^0⟶ N (0,1).Therefore, by Wold's device([1 00h^T ]Ω_β^0[1 00h ] )^-1/2[1 00h^T ]∑_i=1^n l_β^0(X_i, Y_i) / √(n) D_β^0⟶N (0,[1 001 ] ) . We now applya slight modification of Lemmas 16 and 23 in <cit.>, where wedrop the assumption of bounded eigenvalues of Σ (which is possible because we have h^T Σ h ≤ R^2 = 𝒪 (1)). The asymptotic linearity of T and the 2-dimensional central limit theorem just obtained imply thatat the alternative β^0 + h / √(n) it holds thatT - (β_1^0 + h_1 / √(n)) +h_1- h^T_β^0 i_β^0( x,y )x^T ( y-xβ^0 ) √(n V_β^0^2 ) D_β^0+ h / √(n)⟶ N (0 ,1) . As T is assumed to be regular at β^0 we conclude that h^T _β^0 i_β^0( x,y )x^T ( y-xβ^0 ) = h_1 + o(1) .But by the Cauchy-Schwarz inequality (h^T _β^0 i_β^0( x,y )x^T ( y-xβ^0 ))^2 ≤ V_β^0^2 h^T Σ h .Moreover, (h_1 +o(1) )^2= h_1^2 + o(1) .so that we obtainV_β^0^2≥ h_1^2 +o(1)h^T Σ h=h_1^2h^T Σ h + o(1)where in the last step we used h^T Σ h ≥ h_2^2 / Λ_ min^2 ≥ρ^2 so that1/h^T Σ h= 𝒪 (1). Since the result is true for all h ∈ H_β^0, |h_1 | ≥ρ and h^T Σ h ≤ R^2, we may maximize the right hand side of (<ref>) over all such h.⊔ -12mu ⊓ §.§ Proofs for Section <ref> Proof of Lemma <ref>.Because x_-1γ^0 is the projection of x_1 on x_-1 we know that(x_1 -x_-1γ^♯ )^2 ≥ (x_1 -x_-1γ^0 )^2 = 1/ Θ_1,1 .MoreoverΘ_1,1 =e_1^T Θ e_1 ≤Λ_ max^2 (Θ ) = 1/ Λ_ min^2 .Thus(x_1 -x_-1γ^♯ )^2≥Λ_ min^2 .We now rewrite(x_1 -x_-1γ^♯ )^2=1+ γ^♯ TΣ_-1,-1γ^♯ - 2x_1^Tx_-1γ^♯=1+γ^♯ TΣ_-1,-1γ^♯ - 2 γ^0TΣ_-1,-1γ^♯where in the second equality we used that x_1 -x_-1γ^0 is the anti-projection of x_1 on x_-1and hence orthogonal to x_-1γ^♯.For the cross-product we have by the two conditions on the pair (γ^♯, λ^♯)γ^0TΣ_-1,-1γ^♯ =γ^♯ TΣ_-1, -1γ^♯ + ( γ^0 - γ^♯ )^T Σ_-1, -1γ^♯_ | · | ≤Σ_-1, -1( γ^0 - γ^♯ ) _∞γ^♯_1 ≤λ^♯γ^♯_1= o(1) = γ^♯ TΣ_-1, -1γ^♯ + o(1) .Thus(x_1 -x_-1γ^♯ )^2 = 1- γ^0 TΣ_-1, -1γ^♯+o(1) .Combining this with inequality (<ref>) proves the first result of the lemma.The second result:Σ( Θ_1^♯- Θ_1 )_∞≤λ_0^♯follows trivially from this.For the third result, we compute and re-use the already obtained results:Θ_1^♯ΣΘ_1^♯= 1 - γ^0 TΣ_-1,-1γ^♯ - γ^♯ TΣ_-1,-1 (γ^0 - γ^♯ ) (1 - γ^0 TΣ_-1,-1γ^♯)^2 = 1 1 - γ^0 TΣ_-1,-1γ^♯_= Θ_1,1^♯+o(1) =1/( x_1-x_-1γ^♯ )^2 + o(1) ≤ 1/( x_1-x_-1γ^0 )^2 _= Θ_1,1 +o(1) .To show the final statement of the lemma, assume on the contrarythat γ^0 is sparse: λ^♯γ^0 _1 → 0 . Then Θ_1,1^♯ =11- γ^0TΣ_-1,-1γ^♯ =11- γ^0TΣ_-1,-1γ^0 + γ^0TΣ_-1,-1(γ^0 -γ^♯_|· | ≤λ^♯γ^0 _1 = o(1) )= Θ_1,1 + o(1) . ⊔ -12mu ⊓Proof of Theorem <ref>. We use the decomposition of the beginning of this section applied to b̂_I,1^♯b̂_I,1^♯ - β_1^0 =2 Θ_1^♯ T X_I^T ϵ_I / n +( e_1^T - Θ_1^♯ TΣ̂_I) ( β̂_II - β_1^0 )_ remainder_Iwhere remainder_Iis( e_1^T-Θ_1^♯ TΣ̂_I ) ( β̂_II - β^0 )=Θ_1^♯ T ( Σ - Σ̂_I )(β̂_II - β^0 )_:=(i) +(e_1^T-Θ_1^♯ TΣ) (β̂_II - β^0 )_:=(ii) . Here, ϵ_I := Y_I - X_I β^0 and Σ̂_I:= 2X_I^T X_I /n.But, given (X_II , Y_II), Θ̂_1^♯ TΣ̂_I (β̂_II - β^0) =2 Θ_1^♯ T X_I^T X_I(β̂_II - β^0) /n is the average of n/2 i.i.d. random variables which are the product of a random variable with theN (0, Θ_1^♯ TΣΘ_1^♯)-distribution and a N (0, Σ_1/2 ( β̂_II - β^0) _2^2)-distributed random variable. Since the variances satisfy Θ_1^♯ TΣΘ_1^♯= Θ_1,1^♯ +o(1) = 𝒪 (1) and Σ_1/2 ( β̂_II - β^0) _2^2=o__β^0 (1) uniformly in β^0 ∈ B we see that (i) = o__β^0 (1/ √(n) ) uniformly in β^0 ∈ B.For the term (ii) we use that ΣΘ_1^♯ -e_1_∞ = 𝒪 (λ^♯ ) and the assumptionλ^♯β̂_II - β^0 _1 =o__β^0(1/ √(n) ) uniformlyin β^0 ∈ B. This gives that uniformlyin β^0 ∈ Bb̂_I,1^♯ - β_1^0 = 2 Θ_1^♯ TX_I^T ϵ_I /n +o__β^0 (1/ √(n)). In the same way one derives thatuniformlyin β^0 ∈ Bb̂_II,1^♯ - β_1^0 = 2 Θ_1^♯ T X_II^T ϵ_II/n +o__β^0 (1/ √(n)) with ϵ_II := Y_II - X_IIβ^0. Sinceb̂_1^♯ = (b_I,1^♯ + b_II,1^♯ )/2 is the average of thetwo, this proves the asymptotic linearity. Furthervar ( Θ_1^♯ X^T ϵ / √(n)) = Θ_1^♯ TΣΘ_1^♯=Θ_1,1^♯+o(1) by Lemma <ref>. The central limit theorem completes the proof. ⊔ -12mu ⊓ §.§ Proofs for Section <ref> Proof of Lemma <ref>. We have(γ^♯ - γ^♭ )^TΣ_-1,-1 (γ^♯ - γ^♭ )≤ γ^♯ - γ^♭_1 Σ_-1,-1 (γ^♯ - γ^♭ )_∞≤ λ^♯γ^♯ - γ^♭_1 ≤ λ^♯γ^♯_1 +λ^♯γ^♭_1 → 0⊔ -12mu ⊓ Proof of Lemma <ref>.By the KKT conditions(γ_ Lasso - γ^♯)^T Σ_-1, -1 ( γ_ Lasso - γ^0) ≤λ_ Lassoγ^♯_1-λ_ Lassoγ_ Lasso_1 .Therefore,(γ_ Lasso - γ^♯ )^T Σ_-1,-1 (γ_ Lasso - γ^♯ )=(γ_ Lasso - γ^♯)^T Σ_-1, -1 (γ_ Lasso - γ^0)+(γ_ Lasso - γ^♯)^T Σ_-1, -1 ( γ^0 - γ^♯)≤ λ_ Lassoγ^♯_1-λ_ Lassoγ_ Lasso_1 + γ_ Lasso - γ^♯_1 Σ_-1, -1 ( γ^0 - γ^♯)_∞≤ λ_ Lassoγ^♯_1-λ_ Lassoγ_ Lasso_1 + λ^♯γ_ Lasso - γ^♯_1≤( λ_ Lasso + λ^♯ )γ^♯_1 - ( λ_ Lasso - λ^♯ ) γ_ Lasso_1 .Thus(γ_ Lasso - γ^♯ )^TΣ_-1,-1(γ_ Lasso - γ^♯ ) +( λ_ Lasso - λ^♯ ) γ_ Lasso_1≤( λ_ Lasso + λ^♯ )γ^♯_1 → 0 where we used that λ_ Lasso > λ^♯ and λ_ Lassoγ^♯_1 → 0. We also know that by the KKT conditions Σ_-1, -1 ( γ_ Lasso- γ^0) _∞≤λ_ Lasso . Ifλ_ Lasso≥ 2 λ^♯, we obtain from the above λ_ Lassoγ_ Lasso_1 ≤ 3 λ_ Lassoγ^♯_1/2→ 0 . So ( γ_ Lasso, λ_ Lasso) is an eligible pair.⊔ -12mu ⊓.1inProof of Lemma <ref>.Note thatx_-S^Tx_-1γ^-S =x_-S^T[ (x_-Sγ_-S^0 )A x_S]=x_-S^T[x_-S -x_S Σ_S,S^-1Σ_S,-S ] γ_-S^0= [ Σ_-S, -S - Σ_-S, SΣ_S,S^-1Σ_S,-S ] γ_-S^0 . We therefore havev_-S^S _∞ =[ Σ_-S, -S - Σ_-S, SΣ_S,S^-1Σ_S,-S ]γ_-S^0 _∞≤ Σ_-S, -S - Σ_-S, SΣ_S,S^-1Σ_S,-S_1γ_-S^0 _∞ .⊔ -12mu ⊓Proof of Lemma <ref>.We have γ_-S^0 _∞ = λ^S/C ,so that by Lemma <ref>Σ_-1,-1γ^S _∞≤λ^S .Moreoverλ^S γ^S _1 ≤λ^S √( s)→ 0 .Thus ( γ^S , λ^S) is an eligible pair. Finallyλ^S γ^0 _1 ≥λ^S γ_-S^0 _1 = κ (γ_-S^0)C ↛0 .⊔ -12mu ⊓Proof of Lemma <ref>. Writex_-1γ^0 =x_-1γ^♯ + ξ^0 , x_-1^T ξ^0 _∞≤λ^♯.It holds thatγ_S^S = Σ_S,S^-1 x_S^Tx_-1γ^0 .SoΣ_-1,-1^1/2 ( γ_S^S - γ^♯ ) _2^2=( x_S^T ξ^0)^TΣ_S,S^-1 (x_S^T ξ^0)≤x_S ^Tξ^0 _2 / Λ_ min^2 ≤s x_S^Tξ^0_∞^2 / Λ_ min^2 → 0 .⊔ -12mu ⊓ Proof of Lemma <ref>.For a ∈ and c ∈^p-1 satisfying a^2 + Σ_-1,-1^1/2 c _2^2=1,[ a c ]^T Σ (γ^0) [ a c ]= a^2+ 2 a γ^0TΣ_-1,-1 c + c^T Σ_-1,-1 c=1+ 2 a γ^0TΣ_-1,-1 c ≥1- 2|a|Σ_-1, -1^1/2γ^0 _2 Σ_-1,-1^1/2 c _2 =1- 2 |a| √(1- a^2)Σ_-1,-1^1/2γ^0 _2 ≥1- Σ_-1,-1^1/2γ^0 _2.But then1a^2 +c _2^2 [ a c ]^T Σ (γ^0) [ a c ]= a^2 +Σ_-1,-1^1/2 c _2^2a^2 +c _2^2 [ a c ]^T Σ (γ^0) [ a c ]≥a^2 + Λ_ min^2 (Σ_-1,-1 )c _2a^2 +c _2(1- Σ_-1,-1^1/2γ^0 _2) ≥(1- Σ_-1,-1^1/2γ^0 _2) Λ_ min^2 (Σ_-1,-1 ) . Hence Λ_ min^2 (Σ (γ^0)) is positive definite and Λ_ min^2 (Σ (γ^0)) ≥ (1- Σ_-1,-1^1/2γ^0 _2 ) Λ_ min^2 (Σ_-1,-1 ) .It further holds for all j∈{ 2 , … , p } that|x_j^Tx_-1γ^0 |≤√( ( x_-1γ^0 )^2) < 1 . ⊔ -12mu ⊓Proof of Lemma <ref>. By definitionΣ_-1,-1 ( γ^0 - γ^♯ ) = λ^♯ z ,so thatΣ_-1,-1 ( γ^0 - γ^♯ ) _∞≤λ^♯ .Moreover, λ^♯γ^♯_1 ≤λ√( s)γ^♯_2 → 0, since γ^♯_2 ≤Σ_-1,-1^1/2γ^♯_2 / Λ_ min (Σ_-1,-1 ) = 𝒪 (1) .So ( γ^♯, λ^♯) is an eligible pair. We further have(γ^0 - γ^♯)^T Σ_-1,-1 (γ^0 - γ^♯) = λ^♯2 z^T Σ_-1, -1^-1 z .Thusγ^0TΣ_-1,-1γ^0= γ^♯ TΣ_-1-1γ^♯+ 2 ( γ^0 - γ^♯)^T Σ_-1,-1γ^♯ + λ^♯2 z^T Σ_-1,-1^-1 z <1+ o(1)and 1- Σ_-1, -1^1/2γ^0 _2^2 ≫ 0where the positivity is true for large enough n.Therefore, by Lemma <ref>,γ^0 is eventually allowed. Finally, λ^♯γ^0 - γ^♯_1 ≥λ^♯ z^T ( γ^0 - γ^♯) = λ^♯2 z^T Σ_-1,-1^-1 z ≫ 0 .So, since λ^♯γ^♯_1 → 0, it must be true that λ^♯γ^0 _1 ≫ 0. We in fact haveγ^0TΣ_-1,-1γ^0 - γ^♯ TΣ_-1,-1γ^♯ = λ^♯2 z^T Σ_-1, -1^-1 z + o(1) so thatγ^0TΣ_-1,-1γ^0 - γ^♯ TΣ_-1,-1γ^♯≫ 0 .⊔ -12mu ⊓Proof of Lemma <ref>. ((a) ⇒)For z_S :=Σ_S,-SΣ_-S,-S^-1 z_-S the equality Σ_-1,-1 (γ^0 - γ^♯) = λ^♯ zholds. By assumption z_-S_∞≤ 1 and by thereversed irrepresentable condition also z_S _∞≤ 1. ThusΣ_-1,-1(γ^0 - γ^♯) _∞≤λ^♯ . Moreover, λ^♯γ^♯_1 =λ^♯γ_S^0 _1 ≤λ^♯√( s)γ_S^0 _2 → 0 . So ( γ^♯ , λ^♯) is eligible.To see make sure that γ^0 is allowed we bound γ^0TΣ_-1, -1γ^0:γ^0TΣ_-1,-1γ^0 ≤γ_S^0TΣ_S,Sγ_S^0 + 2 λ^♯γ_S^0 _1 + λ^♯ 2 z_-S^TΣ_-1,-1^-1 z_-S.Therefore, since λ^♯γ_S^0 _1→ 0, and in view ofLemma <ref>, the vector γ^0 is for large enough n allowed.Finally, we haveλ^♯γ_0 _1 ≥ λ^♯γ_-S^0 _1≥ λ^♯z_-S^T γ_-S^0 =λ^♯ 2 z_-S^T Σ_-S, -S^-1 z_-S≫ 0.In factγ^0TΣ_-1,-1γ^0 - γ^♯ TΣ_-1,-1γ^♯ = λ^♯ 2 z_-S^T Σ_-S, -S^-1 z_-S+ o(1) ≫ 0 .((b)⇐) If (γ^♯, λ^♯) is an eligible pair we haveΣ_-1,-1 ( γ^♯ - γ^0 ) _∞≤λ^♯.Define now c =γ^0 - γ^♯ and z=Σ_-1,-1^-1 c / λ^♯.Then Σ_-1,-1 c = λ^♯ z and z _∞≤ 1, c_S=0. It follows thatΣ_S,-SΣ_-S,-S^-1 z_-S = z_Sso thatΣ_S,-SΣ_-S,-S^-1 z_-S_∞≤ 1.⊔ -12mu ⊓Proof of Lemma <ref>.One readily verifies that all c_j^0 with j ∉ S are non-zero.One thus has the LagrangrianΣ_-1, -1c^0 = λ̃W ζwhere λ̃ is the Lagrangian parameter. Since ζ_S =0 this says that(Σ_-1,-1 c^0)_S = 0so we know that x_-1 c^0 is orthogonal to x_S. The restriction givesΣ_-1,-1^1/2 c^0 _2^2 = λ̃ (W c^0)_-S_∞ = λ̃/ λ^♯ ,soλ̃= λ^♯Σ_-1,-1^1/2 c^0 _2^2,and inserting this back yieldsΣ_-1,-1 c^0 =λ^♯Σ_-1,-1^1/2 c^0 _2^2 W ζ.It follows thatc^0 =λ^♯Σ_-1,-1^1/2 c^0_2^2 Σ_-1,-1^-1 Wζ .But thenΣ_-1,-1^1/2 c^0 _2 =λ^♯Σ_-1,-1^1/2 c^0 _2^2 Σ_-1,-1^-1/2 W ζ_2orΣ_-1,-1^1/2 c^0_2 = 1 λ^♯Σ_-1,-1^-1/2 W ζ_2.So now we haveΣ_-1,-1 c^0=1 λ^♯Σ_-1,-1^-1/2ζ_2^2 W ζand henceΣ_-1,-1 c^0_∞ =w_-S_∞λ^♯Σ_-1,-1^-1/2ζ_2^2 .⊔ -12mu ⊓Proof of Lemma <ref>. It holds thatΣ_-1,-1 (γ^0 - γ^♯ ) =x_-1^Tx_-1 (γ^0 - γ^♯) =x_-1^T ( x_-S A x_S ) γ_-S^0 = Σ_-1,-1 c^0 .So Σ_-1,-1 (γ^0 - γ^♯ ) _∞ =w_-S_∞λ^♯Σ_-1,-1^1/2 Wζ_2^2=λ^♯ w_-S_∞λ^♯2Σ_-1,-1^1/2Wζ_2^2 ≤λ^♯.Moreoverλ^♯γ^♯_1 ≤√( s)γ^♯_2 → 0 .Thus (γ^♯, λ^♯) is an eligible pair.Furthermoreγ^0TΣ_-1, -1γ^0 = γ^♯ TΣ_-1,-1γ^♯ + c^0TΣ_-1,-1 c^0 = γ^♯ TΣ_-1,-1γ^♯ +1 λ^♯2Σ_-1,-1 ^-1/2 W ζ_2^2.So by Lemma <ref> γ^0 is allowed.Finally, λ^♯γ^0 _1 ≥λ^♯ (W γ^0 )_-S_1 = 1 and in factγ^0TΣ_-1, -1γ^0 - γ^♯ TΣ_-1,-1γ^♯ =1 λ^♯2Σ_-1,-1 ^-1/2 W ζ_2^2 ≫ 0. ⊔ -12mu ⊓ §.§ Proofs for Section <ref>Proof of Lemma <ref>.We recall the notation Σ̂_-1,-1 := X_-1^T X_-1 / n. The event{ X_-1^T ϵ^♯_∞ /n ≤λ_ε^♯}∩{inf_c:λ^ Lasso c _1 ≤ 4 η_n^2,c^T Σ_-1,-1 c = 1c^T Σ̂_-1,-1 c≥12}has probability converging to one so in the rest of the proof we may assume that we are on this event.By the KKT conditionsΣ̂_-1,-1 ( γ̂- γ^0) = X_-1^T ε / n - λ^ Lassoζ̂where ζ̂∈∂γ̂_1, with ∂ c _1 the sub-differential of the map c ↦ c _1.ThusΣ̂_-1,-1 ( γ̂- γ^♯) = X_-1^T ε^♯ / n -λ^ Lassoζ̂.Therefore( γ̂- γ^♯)^T Σ̂_-1,-1 ( γ̂- γ^♯)= ( γ̂- γ^♯)^T X_-1^T ε^♯ / n - λ^ Lasso( γ̂- γ^♯)^Tζ̂≤ λ_ε^♯γ̂- γ^♯_1+ λ^ Lassoγ^♯_1 - λ^ Lassoγ̂_1 ≤( λ^ Lasso + λ_ε^♯ ) γ^♯_1- (λ^ Lasso - λ_ε^♯ )γ̂_1 ≤( λ^ Lasso + λ_ε^♯ ) γ^♯_1≤3 λ^ Lassoγ^♯_1 /2 ≤ 3 η_n^2 /2 .It moreover follows from the above thatγ̂_1 ≤ ( λ^ Lasso+ λ_ε^♯λ^ Lasso - λ_ε^♯ )γ^♯_1 and soλ^ Lassoγ̂_1≤λ^ Lasso ( λ^ Lasso + λ_ελ^ Lasso -λ_ε^♯ ) ≤ 3 η_n^2 ,and also λ^ Lassoγ̂- γ^♯_1 ≤λ^ Lasso ( 2λ^ Lassoλ^ Lasso -λ_ε^♯ )γ^♯_1 ≤ 4 η_n^2 . If Σ^1/2 ( γ̂- γ^♯ )_2 ≤ 2 η_n we are done.Otherwise, if Σ^1/2 ( γ̂- γ^♯ )_2 ≥ 2 η_n it holds that 4 η_n^2 ≤ 2 η_n Σ^1/2 ( γ̂- γ^♯ )_2. But then12( γ̂- γ^♯)^TΣ_-1,-1 ( γ̂- γ^♯)≤( γ̂- γ^♯)^TΣ̂_-1,-1 ( γ̂- γ^♯) ≤(λ^ Lasso + λ_ε^♯ ) γ^♯_1 ≤2 λ^ Lassoγ^♯_1≤ 2 η_n^2 .⊔ -12mu ⊓Proof of Theorem <ref>. We rewriteβ̂_1 - β_1^0 = Θ̂_1 X^T ϵ / n + ( e_1^T - Θ̂_1^T Σ̂) ( β̂- β^0) _ remainder .By the KKT conditionse_1^T - Θ̂_1^T Σ̂_∞≤λ^ Lasso/( X_1 - X_-1γ̂_2^2 / n +λ^ Lassoγ̂_1). But by Lemma <ref> X_1 - X_-1γ̂_2^2 / n +λ^ Lassoγ̂_1 =(x_1 -x_-1γ^♯ )^2 + o(1) which stays away from zero.Moreover, by assumption, √(n)λ^ Lassoβ̂- β_0 _1 = o__β^0 (1)uniformly in β^0 ∈ B. Thus, for the remainder we find|( e_1^T - Θ̂_1^T Σ̂) ( β̂- β^0) |≤ e_1^T - Θ̂_1^T Σ̂_∞β̂- β^0 _1 = o__β^0 ( 1/ √(n) )uniformly in β^0 ∈ B. For the main term, we have after standardizationΘ̂_1^T X^T ϵ /√(n)√(Θ̂_1^T Σ̂Θ̂_1)∼ N (0,1) .It further holds thatΘ̂_1^T Σ̂Θ̂_1 = Θ_1,1^♯ + o_ (1) by Lemma <ref>. which stays away from zero. Therefore, forthe standardized remainder term√(n)|( e_1^T - Θ̂_1^T Σ̂) ( β̂- β^0) | √(Θ̂_1^T Σ̂Θ̂_1) = o__β^0 (1 )uniformly in β^0 ∈ B.The final result thus follows from Slutsky's Theorem. ⊔ -12mu ⊓§ PROBABILITY INEQUALITIES In this section we present some probability inequalities for products of Gaussians. Such results are known (for example as Hanson-Wright inequalities for sub-Gaussians, see<cit.>) and only presented here for completeness. Let U and W be two independent N (0,1)-distributed random variables.Then for all L > 1 exp [ UWL ] ≤exp [ 12 L^2 - 2L] . Proof. We have for L>1 exp [ UWL]≤ exp [ (U + W)^2 -(U-W)^2 4L] =exp [ (U+W)^2-24L ] exp [2- (U-W)^2 4L ] =exp [ (U+W)^2-24L ] exp [ 2-(U-W)^2 4L ] ≤ exp [ 14L^2 - 4L ]exp [ 2-(U-W)^2 4L ] . (see Lemma 1 and its proof in <cit.>) orSection8.4 in <cit.>). Butexp [ 2-(U-W)^2 4L ]=1 √(2 π)∫exp [ 1-v^22L ] exp [ -v^22]dv=exp [12L ] 1 √(2 π)∫exp [ - 12 ( 1+ 1L ) v^2]dv=exp [12L ]( 1+ 1L )^-1/2=exp [ 12L - 12log (1+ 1/L ) ] . Since log (1+ 1/L) ≥1/L - 12 (1/L^2 ) we obtain exp [ 2-(U+W)^2 4L ]≤ exp [ 14 L^2] ≤ exp [14L^2 - 4L] . It follows that exp [ UW L]≤exp [ 24L^2 - 4L ]=exp [ 12L^2 - 2L ] . ⊔ -12mu ⊓Let U= (U_1 , … , U_n)^T and W= (W_1 , … , W_n)^T be two independent standard Gaussian n-dimensional random vectors. Then for all t >0( U^T W /n ≥√(2t/n) +t/n) ≤exp [-t] . Proof. By Lemma <ref> and using the independence exp [U^T WL ] ≤exp [ n2L^2 - 2L ]. This gives for all t>0 ( U^T W ≥√(2nt) +t) ≤exp [-t] (see e.g.Lemma 8.3 in <cit.>).⊔ -12mu ⊓Let { (U_i, V_i) }_i=1^n be i.i.d. two-dimensional Gaussians with mean zero. Suppose var (U_1) =1. Define λ^♯ :=U_1V_1 andσ^♯ 2=V_1^2.Then for all t >0 ( U^T V / n ≥λ^♯ + ( √(2)σ^♯ + 2λ^♯) √(t/n) +( σ^♯ + 2 λ^♯) t/n) ≤ 2 exp[-t] .Proof. For all i the projection of V_i on U_i is [ U_iV_i /var(U_i) ] U_i = λ^♯ U_i. Hencewe may write for all i V_i = λ^♯ U_i + W_i , where W_i is a zero-mean Gaussian random variable independent of U_i. It followsthat U^T V/n= λ^♯ U _2^2 /n + U^T W/n . Since var ( W_i ) ≤σ^♯2 for all i we see from Lemma <ref>that ( U^T W / n ≥√(2)σ^♯√( t/n) + σ^♯ t/n) ≤exp [-t] . Moreover (see Lemma in <cit.>, also given in <cit.> as Lemma 8.6) (U _2^2/n-1 ≥ 2 √(t/n) + 2 t/n) ≤exp[-t] . Thus ( U^T V / n ≥λ^♯ + (√(2)σ^♯ + 2 λ^♯) √(t/n) +( σ^♯ + 2 λ^♯) t/n) ≤ 2 exp[-t] . ⊔ -12mu ⊓plainnat | http://arxiv.org/abs/1708.07986v2 | {
"authors": [
"Sara van de Geer"
],
"categories": [
"math.ST",
"stat.TH",
"62J07"
],
"primary_category": "math.ST",
"published": "20170826153057",
"title": "On the efficiency of the de-biased Lasso"
} |
Critical role of water in defect aggregation and chemical degradation of perovskite solar cells [ Received: date / Accepted: date ===============================================================================================[1]College of Information Engineering, Shenzhen University, Shenzhen, China. (mailto:[email protected]@sina.com, mailto:[email protected]@ee.udel.edu, mailto:[email protected]@szu.edu.cn mailto:[email protected]@cqu.edu.cn)[2]Department of Electrical and Computer Engineering, University of Delaware, Newark, DE 19716, USA. [3]College of Mathematics and Statistics,Chongqing University,Chongqing 401331, China. This research is partially supported by the research grants from the Chinese NSF project (61372078), National Natural Science Foundation of China (No. 61701320, No. 61561019), Guangdong NSF project (2014A030313549), Key project of Guangdong (2016KZDXM006) and the Shenzhen NSF project (JCYJ20160226192223251). definition LemmaLemma RemarkRemarkTheoremTheorem DefinitionDefinition *keywordskeywordsIn this paper, we propose a vector total variation (VTV) of feature image model for image restoration.TheVTV imposes different smoothing powers on different features (e.g. edges and cartoons) based on choosing various regularization parameters. Thus, the model can simultaneously preserve edges and remove noises. Next, the existence ofsolution for the model is proved and the split Bregman algorithm is used to solve the model. At last, we use the wavelet filter banks to explicitly define the feature operator and present some experimental results to show itsadvantage over the related methodsin both quality and efficiency. Image restoration, wavelet, vectore total variation, variational method， § INTRODUCTIONImage restoration including image denoising, deblurring, inpainting etc. is a procedure of improving the quality of a given image that is degraded in various ways during the process of acquisition and commutation. Since many advanced applications in computer vision depend heavily on the input of high quality images, image restoration becomes an indispensable and preprocessing step of these applications. Therefore, image restoration is a basic but very important area in image processing and analysis.Image restoration can be modeled as a linear inverse problem:f = Au + ηwhere fis the observed images, Ais alinear operator and ηrepresents the additive noise whose type depends on its probability density function (PDF). The goal of image restoration is to find the unknown true image ufrom the observed imagef. The problem is usually an ill-posed inverse problem and often solved by imposing a prior regularizationassumption. Among all regularization-based methods for image restoration, variational based <cit.><cit.><cit.><cit.><cit.><cit.><cit.> and wavelet frame based <cit.><cit.><cit.><cit.><cit.><cit.><cit.><cit.><cit.><cit.><cit.><cit.><cit.><cit.><cit.> methods have attracted a lot of attention in the past.The common assumption of the regularization based methods is that images can be sparsely approximated in some transformed domains. Such transforms can be gradient operator, wavelet frame transform, Fourier transform, Gabor transformetc. In order to utilize the sparsity, one solves (1) by finding a sparse solution in the corresponding transformed domain. Typically, the l_1norm is used as a penalty of the sparsity.One of the well-known variational approaches is the Rudin-Osher-Fatemi (ROF) <cit.> model, which imposes the total variation (TV) regularization on the imageu:inf_u ∈ BV(Ω ){λ∫_Ω|∇ u|+ ∫_Ω(u - f)^2 dx}.Here, Ω is the image domian, BV(Ω) is the bounded variational space,∫_Ω|∇ u|and ∫_Ω(u - f)^2 dx are the TV (regularization) term and fidelity (fitting) term, respectively. The ROF model performs well for removing noise while preserving edges. However, it tends to produce piecewise constant results (called staircase effect in the literature). After the ROF model was proposed, the TV regularization has been extended to many other image restoration applications (such as deblurring, inpainting, superresolution, etc.) and has been modified in a variety of ways to improve its performance <cit.><cit.><cit.><cit.>.To solve the ROF model, many algorithms have been proposed <cit.><cit.><cit.><cit.><cit.> <cit.><cit.>and the split Bregman method <cit.> is the one of the most widely used ones . Wavelet frames represent images as a summarization of smooth components (i.e. cartoon), and local features (i.e. singularities). In wavelet frame domain, smooth image components are the coefficient images obtained from low-pass filters, while local features are those obtained from high-pass filters. Thus, we can impose different strengths of regularization on smooth components and local features separately when restoring images in the wavelet frame transform domain. There exist plenty of wavelet frame based image restoration models in the literature, such as, the synthesis based approach <cit.><cit.><cit.><cit.>, the analysis based approach <cit.><cit.><cit.><cit.>, and the balanced approach <cit.><cit.><cit.>. A typical analysis based model of l_1norm regularization has a similar form with the ROF model:min_u λWu_1 + 1/2Au - f_2^2,where Wrepresents the wavelet frame transform. Since the l_1norm regularizations usually exist in the wavelet frame based models, the split Bregman algorithm is also a widely used method to solve them.In this paper, we propose a vector total variation (VTV) of feature image modelfor image restoration where the VTV is used as the regularization. The vector images are generated by mapping the original image into the feature spaces which represent different features of the image such as edges and cartoons. By choosing different regularization parameters, we can impose different smoothing powers on different features. Thus, the model can simultaneously preserve edges and remove noises. Next, the existence ofsolution for the model is proved and the split Bregman algorithm is used to solve the model. At last, we use the wavelet filter banks to explicitly define the feature operator andpresent some experimental results to show itsadvantage over the related methodsin both quality and efficiency.The rest of the paper is organized as follows. In Section 2.1, we describe the proposed model, prove the existence of its solution and use the convolution to define the feature operators. In Section 2.2, we use the split Bregamn algorithm to solve the model. In Section 3, we present some experimental results to show the advantage of our model. In Section 4, we conclude this paper.§ THE WAVELETCOEFFICIENT TOTAL VARIATIONAL MODEL§.§ Proposed modelBefore describing the model, we briefly introduce some mathematical definitions related to our paper.Let Ω be a bounded open subset of R^2, f∈ L^2(Ω,R),F=(F_1,F_2,...,F_m) be an operator such that Ff=(F_1f,F_2f,...,F_mf)∈ L^2(Ω,R^m) and F_if∈ L^2(Ω,R). In this paper, we call F_if a feature image of f and F a feature operator. The inner product inL^2(Ω,R^m) is defined as <u,v>_L^2(Ω,R^m)=∫_Ω∑_i=1^mu_iv_i, where u=(u_1,u_2,...,u_m),v=(v_1,v_2,...,v_m)∈ L^1(Ω,R^m). BV(Ω) is the subspace of L^1(Ω,R) such that the following quantity is finite: |f|_TV=sup_η∈ K<f,divη>_L^2(Ω,R), where divη=∂η_1/∂ x+∂η_2/∂ y, K={η=(η_1,η_2)∈ C_0^1(Ω,R^2):η_L^∞(Ω,R^2)≤ 1}, C_0^1(Ω,R^2) is the space of continuously differentiable functions with compact support in Ω, η_L^∞(Ω,R^2)=sup_x∈Ω√(η_1^2+η_2^2) and the quantity |f|_TV is called the total variation of f. The vector total variation of function g=(g_1,g_2,...,g_m)∈ L^1(Ω,R^m) is defined as |g|_VTV=sup_ξ∈ P<g,divξ>_L^2(Ω,R^m),where ξ=(ξ_1,ξ_2,...,ξ_m), divξ=(divξ_1,divξ_2,...,divξ_m), P={ξ∈ C_0^1(Ω,R^2× m):ξ(x)_L^∞(Ω,R^2× m)≤ 1} and ξ(x)_L^∞(Ω,R^2× m)=max_i=1,2...,mξ_i(x)_L^∞(Ω,R^2). For g smooth enough, we have <cit.><cit.>|g|_VTV= ∑_i=1^m|g_i|_TV= ∑_i=1^m∫_Ω|∇ g_i|.For more details about the vector total variation, refer to <cit.><cit.><cit.><cit.><cit.>.In this paper, we propose the following model for image recovery:inf_u ∈ BV(Ω ){ E(u) = | λ⃗Fu|_VTV +1/2Au - f_L^2(Ω,R)^2},where A is a linear operator, f is an observed image, F is a feature operator, λ⃗=(λ_1,λ_2,...,λ_m)∈ (R^+)^m is the regularization parameter vector and λ⃗Fu=(λ_1F_1u,λ_2F_2u,...,λ_mF_mu) represents the feature images. The ideal case is that different F_mu can represent different features of the image u, such as edges, cartoons, etc. and thus we can impose different smooth strengths on them by choosing different λ_i. Clearly, if m=1 and F=I, then the proposed model is reduced to the ROF model. In the following, we show the existencce of the solution for problem (<ref>). If F_i is linear, then the adjoint operator F^* of F is F^*g=∑_i=1^mF_i^*g_i, whereg=(g_1,g_2,...,g_m)∈ L^2(Ω,R^m) and F_i^* is the adjoint operator of F_i. Let f∈L^2(Ω,R) and g ∈ L^2(Ω,R^m), then <Ff,g>= ∑_i=1^m<F_if,g_i> = ∑_i=1^m<f,F_i^*g_i> = <f,∑_i=1^mF_i^*g_i>Thus, F^*g=∑_i=1^mF_i^*g_i. If F is linear and bounded and for any ξ∈ C_0^1(Ω,R^2× m), the operator div and F^* are commutative, i.e., F^*divξ=div F^*ξ, then for any f∈ BV(Ω), we have |Ff|_VTV≤F|f|_TV. |Ff|_VTV= sup_ξ∈ P<Ff,divξ>= sup_ξ∈ P<f,F^*divξ>= sup_ξ∈ PF^*ξ_L^∞(Ω,R^2)<f,div(F^*ξ/F^*ξ_L^∞(Ω,R^2))> ≤ sup_ξ∈ PF^*ξ_L^∞(Ω,R^2× m)<f,div(F^*ξ/F^*ξ_L^∞(Ω,R^2))> ≤ F^*|f|_TV= F|f|_TV If F is linear and bounded,invertible, and for any η∈ C_0^1(Ω,R^2), the operator div and (F^-1)^* are commutative, i.e., (F^-1)^*divη=div (F^-1)^*η, then for any f∈ BV(Ω), we have |f|_TV≤F^-1|Ff|_VTV. |f|_TV= sup_η∈ K<F^-1Ff,divη> = sup_η∈ K<Ff,(F^-1)^*divη> = sup_η∈ K(F^-1)^*η_L^∞(Ω,R^2× m)<Ff,div((F^-1)^*η/(F^-1)^*η_L^∞(Ω,R^2× m))> ≤ sup_η∈ K(F^-1)^*η_L^∞(Ω,R^2)<Ff,div((F^-1)^*η/(F^-1)^*η_L^∞(Ω,R^2× m))> ≤ (F^-1)^*sup_η∈ K<Ff,div((F^-1)^*η/(F^-1)^*η_L^∞(Ω,R^2× m))> ≤ (F^-1)|Ff|_VTV From the proof, we can see the condition we need is that there exists F_L^-1: L^2(Ω,R^m)→ L^2(Ω,R) such that F_L^-1F=I and (F_L^-1)^*divη=div(F_L^-1)^*η for any η∈ C_0^1(Ω,R^2). If F islinear and bounded, and invertible, thenfor any ξ∈ C_0^1(Ω,R^2× m) and η∈ C_0^1(Ω,R^2), the following two conditions are equivalent: (a) F^*divξ=div F^*ξ, (b) (F^-1)^*divη=div (F^-1)^*η. By (a) ⇒ (b): Let ξ=(F^-1)^*η in (a), then F^*div(F^-1)^*η=divF^*(F^-1)^*η=div(F^-1F)^*η=divη. Multiplying (F^-1)^* on both sides of the above equation, we obtain div(F^-1)^*η=(F^-1)^*divη, (b) ⇒ (a): Let η=F^*ξ in (b), then we have (F^-1)^*divF^*ξ=div (F^-1)^*F^*ξ=divξ. Therefore,divF^*ξ=F^*divξ. By 1=F^-1F≤F^-1F, we only have F^-1≥F^-1. IfF^-1=F^-1, then combining Lemma 2 and Lemma 3, we can obtain |Ff|_VTV=F|f|_TV. If the conditions of Remark 1 are satisfied, λ_min=min_i=1,2,...mλ_i>0 andA⊗ 1=∫_Ω A(x)dx 0, then Problem (<ref>) admits a solution. Let u_n be a minimizing sequence of Problem (<ref>), then there exists a constant M>0 such that E(u_n)≤ M, i.e.| λ⃗Fu|_VTV + 1/2Au - f_L^2(Ω,R)^2≤ M.Therefore, | λ⃗Fu|_VTV≤ M Au - f_L^2(Ω,R)^2≤ 2MFrom equation (<ref>), by Lemma 3, we have |u|_TV ≤F^-1|Fu|_VTV≤1/λ_minF^-1|λ⃗Fu|_VTV≤M/λ_minF^-1As proved in in <cit.><cit.>, from equation (<ref>), we can obtainu_L^2(Ω,R)≤ M Combining equations (<ref>) and (<ref>), we have that the sequence {u_n} is bounded in BV(Ω).Thus, there exists u^*∈ BV(Ω) such that u_nconverges to u^*weakly in L^2(Ω )and strongly inL^1(Ω ). Since the operator Fis linear and bounded, we have that Fu_n also converges to Fu^*weakly in L^2(Ω )and strongly inL^1(Ω ). Then a standard process can show that u^*is a minimizer ofE(u). If there exists a F_k such that F_k is invertible and F_kdivη=div F_kη for any η∈ C_0^1(Ω,R^2), λ_min=min_i=1,2,...mλ_i> 0 andA⊗ 1=∫_Ω A(x)dx 0, then Problem (<ref>) admits a solution. Let u_n be a minimizing sequence of Problem (<ref>), then there exists a constant M>0 such that E(u_n)≤ M, i.e. | λ⃗Fu|_VTV + 1/2Au - f_L^2(Ω,R)^2≤ M. Therefore, | λ⃗Fu|_VTV=∑_i^m|λ_iF_iu|_TV≤ M and so |F_ku|_TV≤M/λ_k Since F_k is invertible and F_kdivη=div F_kη for any η∈ C_0^1(Ω,R^2), like the proof in Lemma 2, we can obtain |u|_TV≤F_k^-1|F_ku|_TV≤M/λ_kF_k^-1 The left part of the proof is the same as that in Theorem 1 and so is omitted. To assure the existence of the solution, we can intentionally design a F_k such that F_k satisfies the conditions in Theorem 2. The simplest F_k satisfying the conditions is F_k=I. In the following, we give the definition of the feature operator F. Insignal processing, a filter of convolution is usually used to extract some feature of a signal. In this paper, we also use the convolution to define F_i.Let {K_i(x)} be a family of functions with a compact support E=[-r,r]× [-r,r] such that K_i(x)=0 for x∈ R^2-E and K_i(x)∈ L^∞(R^2,R). For any f∈L^2(Ω,R), we define F_if asF_if(x)=f⊗ K_i=∫_Ω f(y)K_i(x-y)dy.Since |∫_Ω f(y)K_i(x-y)dy|^2 ≤∫_Ω f^2(y)dy∫_Ω K_i^2(x-y)dy≤K_i^2_L^∞(R^2,R)f^2_L^2(Ω,R)we have F_if∈ L^2(Ω,R). By the Fubini's theorem,we haveF_i^*f(x)=f⊗K̂_i=∫_Ω f(y)K_i(y-x)dy,where K̂_i(x)=K_i(-x).For any η=(η_1,η_2)∈ C_0^1(Ω,R^2), we haveF_i^*divη =∫_Ωdivη(y)K_i(y-x)dy=-∫_Ωη(y)∇_y (K_i(y-x))dy=∫_Ωη(y)∇_x (K_i(y-x))dy=∫_Ωη_1(y)∂K_i(y - x)/∂x_1+ η_2(y)∂K_i(y - x)/∂x_2dy=div(∫_Ωη_1(y)K_i(y-x)dx,∫_Ωη_2(y)K_i(y-x)dx)=div(F_i^*η_1,F_i^*η_2)=divF_i^*ηwhere we assume x=(x_1,x_2). Combining the above equation with Lemma 1, we thus have F^*divη=divF^*η. §.§ AlgorithmIn this section, we utilize the split Bregman algorithm <cit.> to solve problem (<ref>). Before describing the algorithm, we need to give the discrete version of Problem (<ref>). Let f be amatrix, such as an image of size s× t, K_i a matrix convolution filter of size (2r+1)× (2r+1). Then the discrete convolution of f and K_i is defined byF_if(k,l)=(f⊗ K_i)(k,l)=∑_p=k-r,q=l-r^p=k+r,q=l+rf(k,l)K_i(k-p,k-q)=∑_p,q=-r^rf(k-p,l-q)K_i(p,q)where we assume the center of K_i is theorigin of coordinates and extend f circularly.By equation (<ref>), we have F_i^*f(k,l)=(f⊗K̂_i)(k,l)=∑_p=k-r,q=l-r^p=k+r,q=l+rf(k,l)K_i(k-p,l-q)=∑_p,q=-r^rf(k-p,l-q)K_i(-p,-q) Let g=(g_1,g_2,...,g_r), g_i be a matrix of size s× t. Define the p-norm of g asg_1 =∑_i=1^rg_i_1 g_2 =√(∑_i=1^rg_i_2^2)Then by equation (<ref>),the discrete version of Problem (<ref>) ismin_u{E(u)=∑_i=1^mλ_i∇F_iu_1+1/2Au - f_2^2}Letd = ∇ (Fu), i.e. d_i=∇ (F_iu), d^0 = b^0 = 0, then Problem (<ref>) is equivalent to the following iterations:for j=0,1,2,…u^j + 1 = min1/2Au - f_2^2 + ∑_i=1^mγ_i/2∇ (F_iu) - d^j_i + b^j_i_2^2d^j + 1 =min∑_i=1^m(λ_i d_i _1 + γ_i /2d_i - ∇ ( F_iu^j + 1) - b_i^j_2^2)b_i^j + 1 = b_i^j + (∇ ( F_iu^j + 1) - d_i^j + 1) The KKT condition for problem (<ref>) is [ A^*(Au - f) + ∑_i=1^mγ_iF_i^*∇ ^* (∇ F_iu - d_i^j + b_i^j) = 0 ]Therefore, the solution for Problem (<ref>) is u^j + 1 = FFT^ - 1(FFT(∑_i=1^mγ_iF_i^*∇ ^* (d_i^j - b_i^j) + A^*f)/FFT(A^*A +∑_i=1^mγ_iF_i^*∇ ^*∇ F_i)) The solution for problem (<ref>) can be explicitly solved:d_i^j + 1 = TH(∇ ( F_iu^j + 1) + b_i^j,λ_i/γ_i )whereTH(x,T) =sgn (x)max (| x | - T,0).At last, the algorithm is summarized in Algorithm <ref>. To implement Algorithm 1, if we don't consider the existence of solution, we only need F_i^* but don't need F^-1 or F_i^-1.The proof of the convergence of the split Bregman algorithm was given in <cit.>. Theconvergence of our algorithm can be proved accordingly and so is not listed here. § EXPERIMENTAL RESULTSIn this section, we conduct some numerical experiments on image denoising and image deblurring using Algorithm <ref>. To implement Algorithm 1, we need to constructK_i explicitly. In our experiments, the piecewise linear B-spline wavelet frame is used for constructing K_i. The filter banksof the B-spline wavelet frame are[h_1( - 1),h_1(0),h_1(1)] = 1/4[1,2,1][h_2( - 1),h_2(0),h_2(1)] = √(2)/4[ 1,0,-1] [h_3( - 1),h_3(0),h_3(1)] = 1/4[ - 1,2, - 1].Then the K_i used in our experiment are K_1=h_1^† h_1 and K_3(i-1)+j=h_i^† h_j,i,j=1,2,3.Clearly, the feature image generated by K_1 have more even region than those by the other K_i. So in the experiments, we set a larger λ_1 and γ_1 to impose a more smooth effect on it.By the unitary extension principle (UEP) in <cit.>,we have that F^*F=I. Thus, F_L^-1=F^*. By Lemma 1 and (<ref>), we also have∑_i=1^mF_i^*F_i=1 If we set the parameter vector γas γ_1>γ_2=γ_3=...=γ_m, then by equation (<ref>), we have∑_i=1^mγ_iF_i^*∇ ^*∇ F_i =(γ_1-γ_2)F_1^*F_1∇^*∇+γ_2∇ ^*∇∑_i=1^mF_i^* F_i=(γ_1-γ_2)F_1^*F_1∇^*∇+γ_2∇ ^*∇,where we can interchange these operators since they are all convolutions. Thus equation (<ref>) in Algorithm 1 can be reduced tou^j + 1 = FFT^ - 1(FFT(∑_i=1^mλ_iF_i^*∇ ^* (d_i^j - b_i^j) + A^*f)/FFT(A^*A +(γ_1-γ_2)F_1^*F_1∇^*∇+γ_2∇ ^*∇)) In Figure <ref>, we show the Fast Fourier transform (FFT) of ∇ ^*∇ and F_1^*F_1, from which we can see thatthe effect of the FFT of F_1^*F_1 is opposite to that of ∇ ^*∇. Since the effect of∇ ^*∇ is smoothing the image, the term F_1^*F_1 may slow down the convergence speed and cause artifacts in the denoised images. Thus in the numerical simulations, we neglect the term (γ_1-γ_2)F_1^*F_1∇^*∇for computational efficiency. Equation (<ref>) is thus changed intou^j + 1 = FFT^ - 1(FFT(γ∑_i=1^mλ_iF_i^*∇ ^* (d_i^j - b_i^j) + A^*f)/FFT(A^*A + γ _1∇ ^*∇ ))Equation (<ref>) can be seen as an analogy of the original split Bregman algorithm who solves u^j+1 byu^j + 1 = FFT^ - 1(FFT(λ∇ ^* (d^j - b^j) + A^*f)/FFT(A^*A + γ _1∇ ^*∇ ))By experiments, we find that using equation (<ref>) can indeed get better results than using equation (<ref>). To demonstrate it, we giveboth results of using equations (<ref>) and(<ref>)in the compared experiments. In Section <ref>, we compare the experimental results of our model with the related methods (such as split Bregman algorithm (SB) <cit.>, dual tree complex wavelet transform (DTCWT) <cit.>, local contextual hidden markov model (LHMM) <cit.> and thel_ominimization in wavelet frame based model ( l_o-WF) <cit.>. In Section <ref>, we perform the comparison on the simulation of image debluring with our model, thel_o-WF model and the EDWF<cit.> model. For measuring the image quality quantitatively, we use the index peak signal to noise ratio (PSNR) defined byPSNR: = 10log _10(255*255*N/ref - ũ_2^2)where refand ũare the true image and recovered image, respectively.§.§ Image denoising In this subsection, we compare the performances of our model withSB, DTCWT, LHMM andl_o-WF on image denoising. Six images (all of size 256*256) shown in Figure <ref> aretested. The noisy images are generated by the MATLAB command `imnoise' with `type=Guassian', m = 0and variance v = 0.01. In the implementation of image denoising, the parameters of our algorithm using equation (<ref>) are set as:λ _1 =λ _2==...=λ _9 = 0.2,γ _1 = 8,γ _2=γ _3=...=γ _9 = 4,tol = 1e - 4; the parameters of our algorithm using equation (<ref>) are set as:λ _1 = 2,λ _2=λ _3=...=λ _9 = 1.5,γ _1 = 12,γ _2=γ _3=...=γ _9 = 4.5,tol = 5e - 4. ForSB algorithm, the parameters are set as:λ= 19,gamma= 1,tol = 5e - 4. ForDTCWT, the decomposition levelL = 4, thresholdT = 20. For l_o-WF, the piecewise linear B-spline wavelet frame is also used and the decomposition levelL = 1,λ= 350,μ= 1,γ= 0.2,tol = 5e - 4. The LHMM model has no free parameters. For fair comparison, all the paramters are uniformly set forthe six tested images and we try our best to tune them to get the highest average PSNR.In Figure <ref>, we show the feature (coefficient) images of a noisy image and its denoised version by our method using equation (<ref>). From Figure <ref>, we can see that the noises in every feature images are all removed.In Figure <ref> - Figure <ref>, we show the visual comparisons of the results from the four compared methods. From the results of SB, we can see that the recovered imageshave a lot of artifacts in the homogeneous regions. The reason is that the regularization parameter λ is chosen optimal for PSNR value. In order to preserve thesharp edges in images, a relatively small λ should be chosen. However, itleads to lack of smoothness inhomogeneous regions. This visual effect is especially clear in the `cameraman' image (see the sky and grass). From the results of DTCWT and LCHMM, we can observe that the restored images introduce too much artifacts. one of the reasons is that they both use the decimated wavelet transform. From the last three results, we can see that the denoised images byl_o-WF and ours have similar visual effects, but our method has a slightly higher PSNR and less runtime averagely which can be seen in Table <ref>. Table <ref> lists the PSNR andtotal run-time of the compared five methods for image denoising. Since the LCHMM is coded by C++ and the others by matlab, we don't list the runtime of LCHMM.From Table <ref>, we can see that our model outperforms the other image restoration methods in terms of PSNR, averagely. In addition, from Table <ref>, we can see that the DTCWT is the fastest (It spends less than 1 second for the six test images). However, the quality of the DTCWT is the worst. To show the convergence of the our algorithm using equation (<ref>), we plot the relative error ||u^n+1-u^n||/||u^n|| versus the iterations of the six test images in Figure <ref>. From Figure <ref>, we can observethe relative error is monotonically decreasing with the iterations, which numerically proves that thealgorithm is convergent. §.§ Image debluring In this subsection, we apply the proposed model to image deblurring and compare the results with those ofl_o-WF and EDWF. We also test the same six images for image deblurring. The blurred images are generated by convolution with the blur kernel Aand added by a Gaussian noiseη, where Aand ηare generated by the MATLAB command `A = fspecial('motion', 9, 0) ' and `η= 5*randn(size(u)) ', respectively. In the implementation of image deblurring, the parameters of our model using equation (<ref>) are set as:λ _1 =0.006, λ _2==...=λ _9 = 0.004,γ _1 = 0.4,γ _2=γ _3=...=γ _9 = 0.1,tol = 5e - 4; the parameters of our algorithm using equation (<ref>) are set as:λ _1 = 0.004,λ _2=λ _3=...=λ _9 = 0.002,γ _1 = 0.1,γ _2=γ _3=...=γ _9 = 0.4,tol = 5e - 4. Forl_o-WF, the piecewise linear B-spline wavelet frame is also used and the decomposition level L = 1,λ= 30,μ= 0.01,γ= 0.003,tol = 5e - 4. ForEDWF, λ=1, γ=1.5, ρ=0.2, v=0.15, μ_1=1, μ_2=1.In Figure <ref>, the visual results of the three models are presented and Table <ref>, summarizes the PSNR andtotal run-time of the three methods. From Table <ref>, we can see that the PSNR of our model using equation (<ref>) is averagely the largest.§ CONCLUSIONSIn this paper, we proposed a vector total variation (VTV) of feature image model for image restoration.TheVTV imposes different smoothing powers on different features and thus can simultaneously preserve edges and remove noises. Next, the existence ofsolution for the model was proved and the split Bregman algorithm was used to solve the model. At last, we used the wavelet filter banks to explicitly define the feature operator andpresented some experimental results to show itsadvantage over the related methodsin both quality and efficiency.unsrt | http://arxiv.org/abs/1708.07601v2 | {
"authors": [
"Wei Wang",
"Xiang-Gen Xia",
"Shengli Zhang",
"Chuanjiang He"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170825022223",
"title": "A wavelet frame coefficient total variational model for image restoration"
} |
Hamiltonian Maker-Breaker games on small graphs Miloš Stojaković [Department of Mathematics and Informatics, Faculty of Sciences, University of Novi Sad, Serbia. Partly supported by Ministry of Education and Science, Republic of Serbia, and Provincial Secretariat for Science, Province of Vojvodina. {milos.stojakovic, nikola.trkulja}@dmi.uns.ac.rs] Nikola Trkulja [1] ================================================================================================================================================================================================================================================================================================================================================One of the most recent members of the Paxos family of protocols is Generalized Paxos.This variant of Paxos has the characteristic that it departs from the original specification of consensus, allowing for a weaker safety condition where different processes can have a different views on a sequence being agreed upon.However, much like the original Paxos counterpart, Generalized Paxos does not have a simple implementation. Furthermore, with the recent practical adoption of Byzantine fault tolerant protocols, it is timely and important to understand how Generalized Paxos can be implemented in the Byzantine model. In this paper, we make two main contributions. First, we provide a description of Generalized Paxos that is easier to understand, based on a simpler specification and the pseudocode for a solution that can be readily implemented. Second, we extend the protocol to the Byzantine fault model. § INTRODUCTION The evolution of the Paxos <cit.> protocol is an unique chapter in the history of Computer Science. It was first described in 1989 through a technical report <cit.>, and was only published a decade later <cit.>. Another long wait took place until the protocol started to be studied in depth and used by researchers in various fields, namely the distributed algorithms <cit.> and the distributed systems <cit.> research communities. And finally, another decade later, the protocol made its way to the core of the implementation of the services that are used by millions of people over the Internet, in particular since Paxos-based state machine replication is the key component of Google's Chubby lock service <cit.>, or the open source ZooKeeper project <cit.>, used by Yahoo! among others. Arguably, the complexity of the presentation may have stood in the way of a faster adoption of the protocol, and several attempts have been made at writing more concise explanations of it <cit.>. More recently, several variants of Paxos have been proposed and studied. Two important lines of research can be highlighted in this regard. First, a series of papers hardened the protocol against malicious adversaries by solving consensus in a Byzantine fault model <cit.>. The importance of this line of research is now being confirmed as these protocols are now in widespread use in the context of cryptocurrencies and distributed ledger schemes such as blockchain <cit.>. Second, many proposals target improving the Paxos protocol by eliminating communication costs <cit.>, including an important evolution of the protocol called Generalized Paxos <cit.>, which has the noteworthy aspect of having lower communication costs by leveraging a more general specification than traditional consensus that can lead to a weaker requirement in terms of ordering of commands across replicas. In particular, instead of forcing all processes to agree on the same value, it allows processes to pick an increasing sequence of commands that differs from process to process in that commutative commands may appear in a different order. The practical importance of such weaker specifications is underlined by a significant research activity on the corresponding weaker consistency models for replicated systems <cit.>.In this paper, we draw a parallel between the evolution of the Paxos protocol and the current status of Generalized Paxos. In particular, we argue that, much in the same way that the clarification of the Paxos protocol contributed to its practical adoption, it is also important to simplify the description of Generalized Paxos. Furthermore, we believe that evolving this protocol to the Byzantine model is an important task, since it will contribute to the understanding and also open the possibility of adopting Generalized Paxos in scenarios such as a Blockchain deployment.As such, the paper makes several contributions, which are listed next.* We present a simplified version of the specification of Generalized Consensus, which is focused on the most commonly used case of the solutions to this problem, which is to agree on a sequence of commands; * we extend the Generalized Paxos protocol to the Byzantine model;* we present a description of the Byzantine Generalized Paxos protocol that is more accessible than the original description, namely including the respective pseudocode, in order to to make iteasier to implement; * we prove the correctness of the Byzantine Generalize Paxos protocol; * and we discuss several extensions to the protocol in the context of relaxed consistency models and fault tolerance. The remainder of the paper is organized as follows: Section <ref> gives an overview of Paxos and its family of related protocols. Section <ref> introduces the model and simplified specification of Generalized Paxos. Section <ref> presents the Generalized Paxos protocol that is resilient against Byzantine failures. Section <ref> presents a correctness proof for Byzantine Generalized Paxos. Section <ref> concludes the paper with a discussion of several optimizations and practical considerations. § BACKGROUND AND RELATED WORK §.§ Paxos and its variants The Paxos protocol family solves consensus by finding an equilibrium in face of the well-known FLP impossibility result <cit.>. It does this by always guaranteeing safety despite asynchrony, butforegoing progress during the temporary periods of asynchrony, or if more than f faults occur for a system of N > 2f replicas <cit.>. The classic form of Paxos uses a set of proposers, acceptors and learners, runs in a sequence of ballots, and employs two phases (numbered 1 and 2), with a similar message pattern: proposer to acceptors (phase 1a or 2a), acceptors to proposer (phase 1b or 2b), and, in phase 2b, also acceptors to learners. To ensure progress during synchronous periods, proposals are serialized by a distinguished proposer, which is called the leader.Paxos is most commonly deployed as Multi (Decree)-Paxos, which provides an optimization of the basic message pattern by omitting the first phase of messages from all but the first ballot for each leader <cit.>. This means that a leader only needs to send a phase 1a message once and subsequent proposals may be sent directly in phase 2a messages. This reduces the message pattern in the common case from five message delays to just three (from proposing to learning). Fast Paxos observes that it is possible to improve on the previous latency (in the common case) by allowing proposers to propose values directly to acceptors <cit.>. To this end, the protocol distinguishes between fast and classic ballots, where fast ballots bypass the leader by sending proposals directly to acceptors and classic ballots work as in the original Paxos protocol. The reduced latency of fast ballots comes at the added cost of using a quorum size of N-e instead of a classic majority quorum, where e is the number of faults that can be tolerated while using fast ballots. In addition, instead of the usual requirement that N> 2f, to ensure that fast and classic quorums intersect, a new requirement must be met: N > 2e+f. This means that if we wish to tolerate the same number of faults for classic and fast ballots (i.e., e=f), then the minimum number of replicas is 3f+1 instead of the usual 2f+1. Since fast ballots only take two message steps (phase 2a messages between a proposer and the acceptors, and phase 2b messages between acceptors and learners), there is the possibility of two proposers concurrently proposing values and generating a conflict, which must be resolved by falling back to a recovery protocol.=-1 Generalized Paxos improves the performance of Fast Paxos by addressing the issue of collisions. In particular, it allows acceptors to accept different sequences of commands as long as non-commutative operations are totally ordered <cit.>. In the original description, non-commutativity between operations is generically represented as an interference relation. In this context, Generalized Paxos abstracts the traditional consensus problem of agreeing on a single value to the problem of agreeing on an increasing set of values. C-structs provide this increasing sequence abstraction and allow the definition of different consensus problems. If we define the sequence of learned commands of a learner l_i as a c-struct learned_l_i, then the consistency requirement for generalized consensus can be defined as: learned_l_1 and learned_l_2 must have a common upper bound, for all learners l_1 and l_2. This means that, for any learned_l_1 and learned_l_2, there must exist some c-struct of which they are both prefixes. This prohibits interfering commands from being concurrently accepted because no subsequent c-struct would extend them both. More recently, other Paxos variants have been proposed to address specific issues. For example, Mencius <cit.> avoids the latency penalty in wide-area deployments of having a single leader, through which every proposal must go through. In Mencius, the leader of each round rotates between every process: the leader of round i is process p_k, such that k = n mod i.Another variant is Egalitarian Paxos (EPaxos), which achieves a better throughput than Paxos by removing the bottleneck caused by having a leader <cit.>. To avoid choosing a leader, the proposal of commands for a command slot is done in a decentralized manner, taking advantage of the commutativity observations made by Generalized Paxos <cit.>. Conflicts between commands are handled by having replicas reply with a command dependency, which then leads to falling back to using another protocol phase with f+⌊f+1/2⌋ replicas.§.§ Byzantine fault tolerant replicationConsensus in the Byzantine model was originally defined by Lamport et al. <cit.>. Almost two decades later, a surge of research in the area started with the PBFT protocol, which solves consensus for state machine replication with 3f+1 replicas while tolerating up to f Byzantine faults <cit.>. In PBFT, the system moves through configurations called views, in which one replica is the primary and the remaining replicas are the backups. The protocol proceeds in a sequence of steps, where messages are sent from the client to the primary, from the primary to the backups, followed by two all-to-all steps between the replicas, with the last step proceeding in parallel with sending a reply to the clients. Zeno is a Byzantine fault tolerance state machine replication protocol that trades availability for consistency <cit.>. In particular, it offers eventual consistency by allowing state machine commands to execute in a weak quorum off+1 replicas. This ensures that at least one correct replica will execute the request and commit it to the linear history, but does not guarantee the intersection property that is required for linearizability. The closest related work is Fast Byzantine Paxos (FaB), which solves consensus in the Byzantine setting within two message communication steps in the common case, while requiring 5f+1 acceptors to ensure safety and liveness <cit.>. A variant that is proposed in the same paper is the Parameterized FaB Paxos protocol, which generalizes FaB by decoupling replication for fault tolerance from replication for performance. As such, the Parameterized FaB Paxos requires 3f+2t+1 replicas to solve consensus, preserving safety while tolerating up to f faults and completing in two steps despite up to t faults. Therefore, FaB Paxos is a special case of Parameterized FaB Paxos where t=f. It has also been shown that N>5f is a lower bound on the number of acceptors required to guarantee 2-step execution in the Byzantine model. In this sense, the FaB protocol is tight since it requires 5f+1 acceptors to provide the same guarantees.In comparison to the FaB Paxos protocol, our BGP protocol requires a lower number of acceptors than what is stipulated by FaB Paxos' lower bound <cit.>. However, this does not constitute a violation of the result since BGP does not guarantee a two step execution in the Byzantine scenario. Instead, BGP only provides a two communication step latency when proposed sequences are universally commutative with any other sequence. In the common case, BGP requires three messages steps for a sequence to be learned. In other words, Byzantine Generalized Paxos introduces an additional broadcast phase to decrease the requirements regarding the minimum number of acceptor processes. This may be a sensible trade-off in systems that target datacenter environments where communication between machines is fast and a high percentage of costs is directly related to equipment. The fast communication links would mitigate the latency cost of having an additional phase between the acceptors and the high cost of equipment and power consumption makes the reduced number of acceptor processes attractive.=-1 § MODELWe consider an asynchronous system in which a set of n ∈ℕ processes communicate bysending and receiving messages. Each process executes an algorithm assigned to it, but may fail in two different ways. First, it may stop executing it by crashing. Second, it may stop following the algorithm assigned to it, in which case it is considered Byzantine. We say that a non-Byzantine process is correct. This paper considers the authenticated Byzantine model: every process can produce cryptographic digital signatures <cit.>.Furthermore, for clarity of exposition, we assume authenticated perfect links <cit.>,where a message that is sent by a non-faulty sender is eventually received and messages cannot be forged(such links can be implemented trivially using retransmission, elimination of duplicates, and point-to-point message authentication codes <cit.>.) A process may be a learner, proposer or acceptor. Informally, proposers provide input values that must be agreed upon by learners, the acceptors help the learners agree on a value, and learners learn commands by appending them to a local sequence of commands to be executed, learned_l . Our protocols require a minimum number of acceptor processes (N), which is a function of the maximum number of tolerated Byzantine faults (f), namely N ≥ 3f+1. We assume that acceptor processes have identifiers in the set {0,...,N-1}. In contrast, the number of proposer and learner processes can be set arbitrarily.=-1Problem Statement. In our simplified specification of Generalized Paxos, each learner l maintains a monotonically increasing sequence of commands learned_l.We define two learned sequences of commands to be equivalent ()if one can be transformed into the other by permuting the elements in a way such that the order of non-commutative pairs is preserved. A sequence x is defined to be an eq-prefix of another sequence y (x ⊑ y), if the subsequence of y that contains all the elements in x is equivalent () to x.We present the requirements for this consensus problem, stated in terms of learned sequences of commands for a correct learner l, learned_l.To simplify the original specification, instead of using c-structs (as explained in Section <ref>), we specialize to agreeing on equivalent sequences of commands:* Nontriviality. If all proposers are correct, learned_l can only contain proposed commands.* Stability. If learned_l = s then, at all later times, s ⊑ learned_l, for any sequence s and correct learner l.* Consistency. At any time and for any two correct learners l_i and l_j, learned_l_i and learned_l_j can subsequently be extended to equivalent sequences.* Liveness. For any proposal s from a correct proposer, and correct learner l, eventually learned_l contains s.§ PROTOCOLThis section presents our Byzantine fault tolerant Generalized Paxos Protocol (or BGP, for short).§.§ OverviewWe modularize our protocol explanation according to the following main components, which are also present in other protocols of the Paxos family: * View Change – The goal of this subprotocol is to ensure that, at any given moment, one of the proposers is chosen as a distinguished leader, who runs a specific version of the agreement subprotocol. To achieve this, the view change subprotocol continuously replaces leaders, until one is found that can ensure progress (i.e., commands are eventually appended to the current sequence). * Agreement – Given a fixed leader, this subprotocol extends the current sequence with a new command or set of commands. Analogously to Fast Paxos <cit.> and Generalized Paxos <cit.>, choosing this extension can be done through two variants of the protocol: using either classic ballots or fast ballots, with the characteristic that fast ballots complete in fewer communication steps, but may have to fall back to using a classic ballot when there is contention among concurrent requests. §.§ View ChangeThe goal of the view change subprotocol is to elect a distinguished proposer process, called the leader, that carries through the agreement protocol (i.e., enables proposed commands to eventually be learned by all the learners). The overall design of this subprotocol is similar to the corresponding part of existing BFT state machine replication protocols <cit.>.In this subprotocol, the system moves through sequentially numbered views, and the leader for each view is chosen in a rotating fashion using the simple equation leader(view)=view mod N. The protocol works continuously by having acceptor processes monitor whether progress is being made on adding commands to the current sequence, and, if not, by multicasting a signed suspicion message for the current view to all acceptors suspecting the current leader. Then, if enough suspicions are collected, processes can move to the subsequent view. However, the required number of suspicions must be chosen in a way that prevents Byzantine processes from triggering view changes spuriously. To this end, acceptor processes will multicast a view change message indicating their commitment to starting a new view only after hearing that f+1 processes suspect the leader to be faulty. This message contains the new view number, the f+1 signed suspicions, and is signed by the acceptor that sends it. This way, if a process receives a view-change message without previously receiving f+1 suspicions, it can also multicast a view-change message, after verifying that the suspicions are correctly signed by f+1 distinct processes.This guarantees that if one correct process receives the f+1 suspicions and multicasts the view-change message, then all correct processes, upon receiving this message, will be able to validate the proof of f+1 suspicions and also multicast the view-change message.Finally, an acceptor process must wait for N-f view-change messages to start participating in the new view (i.e., update its view number and the corresponding leader process). At this point, the acceptor also assembles the N-f view-change messages proving that others are committing to the new view, and sends them to the new leader. This allows the new leader to start its leadership role in the new view once it validates the N-f signatures contained in a single message. §.§ Agreement ProtocolThe consensus protocol allows learner processes to agree on equivalent sequences of commands (according to the definition of equivalence present in Section <ref>). An important conceptual distinction between Fast Paxos <cit.> and our protocol is that ballots correspond to an extension to the sequence of learned commands of a single ongoing consensus instance, instead of being a separate instance of consensus,. Proposers can try to extend the current sequence by either single commands or sequences of commands. We use the term proposal to denote either the command or sequence of commands that was proposed.Ballots can either be classic or fast. In classic ballots, a leader proposes a single proposal to be appended to the commands learned by the learners. The protocol is then similar to the one used by classic Paxos <cit.>, with a first phase where each acceptor conveys to the leader the sequences that the acceptor has already voted for (so that the leader can resend commands that may not have gathered enough votes), followed by a second phase where the leader instructs and gathers support for appending the new proposal to the current sequence of learned commands. Fast ballots, in turn, allow any proposer to contact all acceptors directly in order to extend the current sequence (in case there are no conflicts between concurrent proposals). However, both types of ballots contain an additional round, called the verification phase, in which acceptors broadcast proofs among each other indicating their committal to a sequence. This additional round comes after the acceptors receive a proposal and before they send their votes to the learners.Next, we present the protocol for each type of ballot in detail. We start by describing fast ballots since their structure has consequences that implicate classic ballots. Figures <ref> and <ref> illustrate the message pattern for fast and classic ballots, respectively. In these illustrations, arrows that are composed of solid lines represent messages that can be sent multiple times per ballot (once per proposal) while arrows composed of dotted lines represent messages that are sent only once per ballot.§.§.§ Fast BallotsFast ballots leverage the weaker specification of generalized consensus (compared to classic consensus) in terms of command ordering at different replicas, to allow for the faster execution of commands in some cases. The basic idea of fast ballots is that proposers contact the acceptors directly, bypassing the leader, and then the acceptors send their vote for the current sequence to the learners. If a conflict exists and progress isn't being made, the protocol reverts to using a classic ballot. This is where generalized consensus allows us to avoid falling back to this slow path, namely in the case where commands that ordered differently at different acceptors commute. However, this concurrency introduces safety problems even when a quorum is reached for some sequence. If we keep the original Fast Paxos message pattern <cit.>, it's possible for one sequence s to be learned at one learner l_1 while another non-commutative sequence s' is learned before s at another learner l_2. Suppose s obtains a quorum of votes and is learned by l_1 but the same votes are delayed indefinitely before reaching l_2. In the next classic ballot, when the leader gathers a quorum of phase 1b messages it must arbitrate an order for the commands that it received from the acceptors and it doesn't know the order in which they were learned. This is because, of the N-f messages it received, f may not have participated in the quorum and another f may be Byzantine and lie about their vote, which only leaves one correct acceptor that participated in the quorum and a single vote isn't enough to determine if the sequence was learned or not. If the leader picks the wrong sequence, it would be proposing a sequence s' that is non-commutative to a learned sequence s. Since the learning of s was delayed before reaching l_2, l_2 could learn s' and be in a conflicting state with respect to l_1, violating consistency. In order to prevent this, sequences accepted by a quorum of acceptors must be monotonic extensions of previous accepted sequences. Regardless of the order in which a learner learns a set of monotonically increasing sequences, the resulting state will be the same. The additional verification phase is what allows acceptors to prove to the leader that some sequence was accepted by a quorum. By gathering N-f proofs for some sequence, an acceptor can prove that at least f+1 correct acceptors voted for that sequence. Since there are only another 2f acceptors in the system, no other non-commutative value may have been voted for by a quorum. An interesting alternative to requiring N-f proofs from each acceptor, would be for the leader to wait for 2f+1 matching phase 1b messages. Since at least f+1 of those would be correct, only that sequence could've been learned since any other non-commutative sequence would obtain at most 2f votes. Zyzzyva uses a similar approach of waiting for 3f+1 to commit requests in single round-trip in executions where no faults occur <cit.>. However, this approach is unsuitable for BGP since it's possible for a sequence to be chosen by a quorum without the leader being aware of more than f+1 votes in its quorum. Since f+1 votes aren't enough to ensure the leader that the sequence was chosen by a quorum, the leader wouldn't be able to pick a learned sequence.Next, we explain each of the protocol's steps for fast ballots in greater detail.Step 1: Proposer to acceptors. To initiate a fast ballot, the leader informs both proposers and acceptors that the proposals may be sent directly to the acceptors. Unlike classic ballots, where the sequence proposed by the leader consists of the commands received from the proposers appended to previously proposed commands, in a fast ballot, proposals can be sent to the acceptors in the form of either a single command or a sequence to be appended to the command history. These proposals are sent directly from the proposers to the acceptors.Step 2: Acceptors to acceptors. Acceptors append the proposals they receive to the proposals they have previously accepted in the current ballot and broadcast the resulting sequence and the current ballot to the other acceptors, along with a signed tuple of these two values. Intuitively, this broadcast corresponds to a verification phase where acceptors gather proofs that a sequence gathered enough support to be committed. This proofs will be sent to the leader in the subsequent classic ballot in order for it to pick a sequence that preserves consistency. To ensure safety, correct learners must learn non-commutative commands in a total order. When an acceptor gathers N-f proofs for equivalent values, it proceeds to the next phase. That is, sequences do not necessarily have to be equal in order to be learned since commutative commands may be reordered. Recall that a sequence is equivalent to another if it can be transformed into the second one by reordering its elements without changing the order of any pair of non-commutative commands (in the pseudocode, proofs for equivalent sequences are being treated as belonging to the same index of the proofs variable, to simplify the presentation). By requiring N-f votes for a sequence of commands, we ensure that, given two sequences where non-commutative commands are differently ordered, only one sequence will receive enough votes even if f Byzantine acceptors vote for both sequences. Outside the set of (up to) f Byzantine acceptors, the remaining 2f+1 correct acceptors will only vote for a single sequence, which means there are only enough correct processes to commit one of them. Note that the fact that proposals are sent as extensions to previous sequences is critical to the safety of the protocol. In particular, since the votes from acceptors can be reordered by the network before being delivered to the learners, if these values were single commands, it would be impossible to guarantee that non-commutative commands would be learned in a total order. Step 3: Acceptors to learners. Similarly to what happens in classic ballots, the fast ballot equivalent of the phase 2b message, which is sent from acceptors to learners, contains the current ballot number, the command sequence and the N-f proofs gathered in the verification round. One could think that, since acceptors are already gathering proofs that a value will eventually be committed, learners are not required to gather N-f votes and they can wait for a single phase 2b message and validate the N-f proofs contained in it. However, this is not the case due to the possibility of learners learning sequences without the leader being aware of it. If we allowed the learners to learn after witnessing N-f proofs for just one acceptor then that would raise the possibility of that acceptor not being present in the quorum of phase 1b messages. Therefore, the leader wouldn't be aware that some value was proven and learned. The only way to guarantee that at least one correct acceptor will relay the latest proven sequence to the leader is by forcing the learner to require N-f phase 2b messages since only then will one correct acceptor be in the intersection of the two quorums. Arbitrating an order after a conflict. When, in a fast ballot, non-commutative commands are concurrently proposed, these commands may be incorporated into the sequences of various acceptors in different orders and, therefore, the sequences sent by the acceptors in phase 2b messages will not be equivalent and will not be learned. In this case, the leader subsequently runs a classic ballot and gathers these unlearned sequences in phase 1b. Then, the leader will arbitrate a single serialization for every previously proposed command, which it will then send to the acceptors. Therefore, if non-commutative commands are concurrently proposed in a fast ballot, they will be included in the subsequent classic ballot and the learners will learn them in a total order, thus preserving consistency.§.§.§ Classic Ballots Classic ballots work in a way that is very close to the original Paxos protocol <cit.>. Therefore, throughout our description, we will highlight the points where BGP departs from that original protocol, either due to the Byzantine fault model, or due to behaviors that are particular to our specification of the consensus problem.In this part of the protocol, the leader continuously collects proposals by assembling all commands that are received from the proposers since the previous ballot in a sequence (this differs from classic Paxos, where it suffices to keep a single proposed value that the leader attempts to reach agreement on). When the next ballot is triggered, the leader starts the first phase by sending phase 1a messages to all acceptors containing just the ballot number. Similarly to classic Paxos, acceptors reply with a phase 1b message to the leader, which reports all sequences of commands they voted for. In classic Paxos, acceptors also promise not to participate in lower-numbered ballots, in order to prevent safety violations <cit.>. However, in BGP this promise is already implicit, given (1) there is only one leader per view and it is the only process allowed to propose in a classic ballot and (2) acceptors replying to that message must be in the same view as that leader.As previously mentioned, phase 1b messages contain N-f proofs for each learned sequence. By waiting for N-f such messages, the leader is guaranteed that, for any learned sequence s, at least one of the messages will be from a correct acceptor that, due to the quorum intersection property, participated in the verification phase of s. Note that waiting for N-f phase 1b messages is not what makes the leader be sure that a certain sequence was learned in a previous ballot. The leader can be sure that some sequence was learned because each phase 1b message contains cryptographic proofs from 2f+1 acceptors stating that they would vote for that sequence. Since there are only 3f+1 acceptors in the system, no other non-commutative sequence could've been learned. Even though each phase 1b message relays enough proofs to ensure the leader that some sequence was learned, the leader still needs to wait for N-f such messages to be sure that he is aware of any sequence that was previously learned. Note that, since each command is signed by the proposer (this signature and its check are not explicit in the pseudocode), a Byzantine acceptor can't relay made-up commands. However, it can omit commands from its phase 1b message, which is why it's necessary for the leader to be sure that at least one correct acceptor in its quorum took part in the verification quorum of any learned sequence. After gathering a quorum of N-f phase 1b messages, the leader initiates phase 2a where it assembles a proposal and sends it to the acceptors. This proposal sequence must be carefully constructed in order to ensure all of the intended properties. In particular, the proposal cannot contain already learned non-commutative commands in different relative orders than the one in which they were learned, in order to preserve consistency, and it must contain unlearned proposals from both the current and the previous ballots, in order to preserve liveness (this differs from sending a single value with the highest ballot number as in the classic specification). Due to the importance and required detail of the leader's value picking rule, it will be described next in its own subsection. The acceptors reply to phase 2a messages by broadcasting their verification messages containing the current ballot, the proposed sequence and proof of their committal to that sequence. After receiving N-f verification messages, an acceptor sends its phase 2b messages to the learners, containing the ballot, the proposal from the leader and the N-f proofs gathered in the verification phase. As is the case in the fast ballot, when a learner receives a phase 2b vote, it validates the N-f proofs contained in it. Waiting for a quorum of N-f messages for a sequence ensures the learners that at least one of those messages was sent by a correct acceptor that will relay the sequence to the leader in the next classic ballot (the learning of sequences also differs from the original protocol in the quorum size, due to the fault model, and in that the learners would wait for a quorum of matching values instead of equivalent sequences, due to the consensus specification.)§.§.§ Leader Value Picking RulePhase 2a is crucial for the correct functioning of the protocol because it requires the leader to pick a value that allows new commands to be learned, ensuring progress, while at the same time preserving a total order of non-commutative commands at different learners, ensuring consistency. The value picked by the leader is composed of three pieces: (1) the subsequence that has proven to be accepted by a majority of acceptors in the previous fast ballot, (2) the subsequence that has been proposed in the previous fast ballot but for which a quorum hasn't been gathered and (3) new proposals sent to the leader in the current classic ballot. The first part of the sequence will be the largest of the N-f proven sequences sent in the phase 1b messages. The leader can pick such a value deterministically because, for any two proven sequences, they are either equivalent or one can be extended to the other. The leader is sure of this because for the quorums of any two proven sequences there is at least one correct acceptor that voted in both and votes from correct acceptors are always extensions of previous votes from the same ballot. If there are multiple sequences with the maximum size then they are equivalent (by same reasoning applied previously) and any can be picked. The second part of the sequence is simply the concatenation of unproven sequences of commands in an arbitrary order. Since these commands are guaranteed to not have been learned at any learner, they can be appended to the leader's sequence in any order. Since N-f phase 2b messages are required for a learner to learn a sequence and the intersection between the leader's quorum and the quorum gathered by a learner for any sequence contains at least one correct acceptor, the leader can be sure that if a sequence of commands is unproven in all of the gathered phase 1b messages, then that sequence wasn't learned and can be safely appended to the leader's sequence in any order. The third part consists simply of commands sent by proposers to the leader with the intent of being learned at the current ballot. These values can be appended in any order and without any restriction since they're being proposed for the first time.§.§.§ Byzantine LeaderThe correctness of the protocol is heavily dependent on the guarantee that the sequence accepted by a quorum of acceptors is an extension of previousproven sequences. Otherwise, if the network rearranges phase 2b messages such that they're seen by different learners in different orders, they will result in a state divergence. If, however, every vote is a prefix of all subsequent votes then, regardless of the order in which the sequences are learned, the final state will be the same. This state equivalence between learners is ensured by the correct execution of the protocol since every vote in a fast ballot is equal to the previous vote with a sequence appended at the end (Algorithm <ref> lines {43-46}) and every vote in a classic ballot is equal to all the learned votes concatenated with unlearned votes and new proposals (Algorithm <ref> lines {42-45}) which means that new votes will be extensions of previous proven sequences. However, this begs the question of how the protocol fares when Byzantine faults occur. In particular, the worst case scenario occurs when both f acceptors and the leader are Byzantine (remember that a process can have multiple roles, such as leader and acceptor). In this scenario, the leader can purposely send phase 2a messages for a sequence that is not prefixed by the previously accepted values. Coupled with an asynchronous network, this malicious message can be delivered before the correct votes of the previous ballot, resulting in different learners learning sequences that may not be extensible to equivalent sequences. To prevent this scenario, the acceptors must ensure that the proposals they receive from the leader are prefixed by the values they have previously voted for. Since an acceptor votes for its val_a sequence after receiving N-f verification votes for an equivalent sequence and stores it in its proven variable, the acceptor can verify that it is a prefix of the leader's proposed value (i.e., proven ⊑ value). A practical implementation of this condition is simply to verify that thesubsequence of value starting at the index 0 up to index length(proven)-1 is equivalent to the acceptor's proven sequence. §.§ Checkpointing BGP includes an additional feature that deals with the indefinite accumulation of state at the acceptors and learners. This is of great practical importance since it can be used to prevent the storage of commands sequences from depleting the system's resources. This feature is implemented by a special command C^*, proposed by the leader, which causes both acceptors and learners to safely discard previously stored commands. However, the reason why acceptors accumulate state continuously is because each new proven sequence must contain any previous proven sequence. This ensures that an asynchronous network can't reorder messages and cause learners to learn in different orders. In order to safely discard state, we must implement a mechanism that allows us to deal with reordered messages that don't contain the entire history of learned commands.To this end, when a learner learns a sequence that contains a checkpointing command C^* at the end, it discards every command in its learned sequence except C^* and sends a message to the acceptors notifying them that it executed the checkpoint for some command C^*. Acceptors stop participating in the protocol after sending phase 2b messages with checkpointing commands and wait for N-f notifications from learners. After gathering a quorum of notifications, the acceptors discard their state, except for the command C^*, and resume their participation in the protocol. Note that, since the acceptors also leave the checkpointing command in their sequence of proven commands, every valid subsequent sequence will begin with C^*. The purpose of this command is to allow a learner to detect when an incoming message was reordered. The learner can check the first position of an incoming sequence against the first position of its learned and, if a mismatch is detected, it knows that either a pre and post-checkpoint message has been reordered. When performing this check, two possible anomalies that can occur: either (1) the first position of the incoming sequence contains a C^* command and the learner's learned sequence doesn't, in which case the incoming sequence was sent post-checkpoint and the learner is missing a sequence containing the respective checkpoint command; or (2) the first position of the learned sequence contains a checkpoint command and the incoming sequence doesn't, in which case the incoming sequence was assembled pre-checkpoint and the learner has already executed the checkpoint. In the first case, the learner can simply store the post-checkpoint sequences until it receives the sequence containing the appropriate C^* command at which point it can learn the stored sequences. Note that the order in which the post-checkpoint sequences are executed is irrelevant since they're extensions of each other. In the second case, the learner receives sequences sent before the checkpoint sequence that it has already executed. In this scenario, the learner can simply discard these sequences since it knows that it executed a subsequent sequence (i.e., the one containing the checkpoint command) and proven sequences are guaranteed to be extensions of previous proven sequences. For brevity, this extension to the protocol isn't included in the pseudocode description. § CORRECTNESS PROOFSThis section argues for the correctness of the Byzantine Generalized Paxos protocol in terms of the specified consensus properties.§.§.§ Consistency At any time and for any two correct learners l_i and l_j, learned_l_i and learned_l_j can subsequently be extended to equivalent sequences Proof: -1. At any given instant, ∀ s,s' ∈𝒫, ∀ l_i,l_j ∈ℒ, learned(l_j,s)learned(l_i,s') ∃σ_1,σ_2 ∈𝒫∪{}, s ∙σ_1s' ∙σ_2Proof: -*31.1. At any given instant, ∀ s,s' ∈𝒫, ∀ l_i,l_j ∈ℒ, learned(l_i,s)learned(l_j,s')(maj_accepted(s,b)(min_accepted(s,b)s ∙σ_1x ∙σ_2))(maj_accepted(s',b')(min_accepted(s',b')s' ∙σ_1x ∙σ_2)), ∃σ_1, σ_2 ∈𝒫∪{}, ∀ x ∈𝒫,∀ b,b' ∈ℬ -*4Proof: A sequence can only be learned in some ballot b if the learner gathers N-f votes (i.e., maj_accepted(s,b)), each containing N-f valid proofs, or if it is universally commutative (i.e., s ∙σ_1x ∙σ_2, ∃σ_1, σ_2 ∈𝒫∪{}, ∀ x ∈𝒫) and the learner gathers f+1 votes (i.e., min_accepted(s,b)). The first case requires gathering N-f votes from each acceptor and validating that each proof corresponds to the correct ballot and value (Algorithm <ref>, lines {1-12}). The second case requires that the sequence must be commutative with any other and at least f+1 matching values are gathered (Algorithm <ref>, {14-18}). This is encoded in the logical expression s ∙σ_1x ∙σ_2 which is true if the accepted sequence s and any other sequence x can be extended to an equivalent sequence, therefore making it impossible to result in a conflict.-*31.2. At any given instant, ∀ s,s' ∈𝒫,∀ b,b' ∈ℬ, maj_accepted(s,b)maj_accepted(s',b') ∃σ_1,σ_2 ∈𝒫∪{}, s ∙σ_1s' ∙σ_2-*4Proof: We divide the following proof in two main cases: (1.2.1.) sequences s and s' are accepted in the same ballot b and (1.2.2.) sequences s and s' are accepted in different ballots b and b'.-*51.2.1. At any given instant, ∀ s,s' ∈𝒫,∀ b ∈ℬ, maj_accepted(s,b)maj_accepted(s',b) ∃σ_1,σ_2 ∈𝒫∪{}, s ∙σ_1s' ∙σ_2 Proof: Proved by contradiction.-*71.2.1.1. At any given instant, ∀ s,s' ∈𝒫, ∀σ_1,σ_2 ∈𝒫∪{},∀ b ∈ℬ, maj_accepted(s,b)maj_accepted(s',b) ∧ s ∙σ_1 s' ∙σ_2 Proof: Contradiction assumption.-*71.2.1.2. Take a pair proposals s and s' that meet the conditions of 1.2.1 (and are certain to exist by the previous point), then s and s' contain non-commutative commands.-*8Proof: The statement ∀ s,s' ∈𝒫, ∀σ_1,σ_2 ∈𝒫∪{}, s ∙σ_1 s' ∙σ_2 is trivially false because it implies that, for any combination of sequences and suffixes, the extended sequences would never be equivalent. Since there must be some s,s',σ_1 and σ_2 for which the extensions are equivalent (e.g., s=s' and σ_1=σ_2), then the statement is false.1.2.1.3. A contradiction is found, Q.E.D. -*51.2.2. At any given instant, ∀ s,s' ∈𝒫,∀ b,b' ∈ℬ, maj_accepted(s,b)maj_accepted(s',b')b ≠ b' ∃σ_1,σ_2 ∈𝒫∪{}, s ∙σ_1s' ∙σ_2-*6Proof: To prove that values accepted in different ballots are extensible to equivalent sequences, it suffices to prove that for any sequences s and s' accepted at ballots b and b', respectively, such that b < b' then s ⊑ s'. By Algorithm <ref> lines {11-16,35,46}, any correct acceptor only votes for a value in variable val_a when it receives 2f+1 proofs for a matching value. Therefore, we prove that a value val_a that receives 2f+1 verification messages is always an extension of a previous val_a that received 2f+1 verification messages. By Algorithm <ref> lines {32,43}, val_a only changes when a leader sends a proposal in a classic ballot or when a proposer sends a sequence in a fast ballot. -*6In the first case, val_a is substituted by the leader's proposal which means we must prove that this proposal is an extension of any val_a that previously obtained 2f+1 verification votes. By Algorithm <ref> lines {24-39,41-47}, the leader's proposal is prefixed by the largest of the proven sequences (i.e., val_a sequences that received 2f+1 votes in the verification phase) relayed by a quorum of acceptors in phase 1b messages. Note that, the verification in Algorithm <ref> line {27} prevents a Byzantine leader from sending a sequence that isn't an extension of previous proved sequences. Since the verification phase prevents non-commutative sequences from being accepted by a quorum, every proven sequence in a ballot is extensible to equivalent sequences which means that the largest proven sequence is simply the most up-to-date sequence of the previous ballot.-*6To prove that the leader can only propose extensions to previous values by picking the largest proven sequence as its proposal's prefix, we need to assert that a proven sequence is an extension any previous sequence. However, since that is the same result that we are trying to prove, we must use induction to do so: -*71. Base Case: In the first ballot, any proven sequence will be an extension of the empty commandand, therefore, an extension of the previous sequence. -*72. Induction Hypothesis: Assume that, for some ballot b, any sequence that gathers 2f+1 verification votes from acceptors is an extension of previous proven sequences. -*73. Inductive Step: By the quorum intersection property, in a classic ballot b+1, the phase 1b quorum will contain ballot b's proven sequences. Given the largest proven sequence s in the phase 1b quorum (which, by our hypothesis, is an extension of any previous proven sequences), by picking s as the prefix of its phase 2a proposal (Algorithm <ref>, lines {41-47}), the leader will assemble a proposal that is an extension of any previous proven sequence. -*6In the second case, a proposer's proposal c is appended to an acceptor's val_a variable. By definition of the append operation, val_a ⊑ val_a ∙ c which means that the acceptor's new value val_a ∙ c is an extension of previous ones. -*31.3. For any pair of proposals s and s', at any given instant, ∀ x ∈𝒫, ∃σ_1,σ_2,σ_3,σ_4 ∈𝒫∪{}, ∀ b,b' ∈ℬ, (maj_accepted(s,b)(min_accepted(s,b)s ∙σ_1x ∙σ_2))(maj_accepted(s',b')(min_accepted(s',b')s ∙σ_1x ∙σ_2))s ∙σ_3s' ∙σ_4Proof: By 1.2 and by definition of s ∙σ_1x ∙σ_2.-*31.4. At any given instant, ∀ s,s' ∈𝒫, ∀ l_i,l_j ∈ℒ, learned(l_i,s)learned(l_j,s') ∃σ_1,σ_2 ∈𝒫∪{}, s ∙σ_1s' ∙σ_2 Proof: By 1.1 and 1.3.1.5. Q.E.D. -*32. At any given instant, ∀ l_i,l_j ∈ℒ, learned(l_j,learned_j)learned(l_i,learned_i) ∃σ_1,σ_2 ∈𝒫∪{}, learned_i ∙σ_1learned_j ∙σ_2Proof: By 1.3. Q.E.D. §.§.§ Nontriviality If all proposers are correct, learned_l can only contain proposed commands.Proof: -1. At any given instant, ∀ l_i ∈ℒ, ∀ s ∈𝒫, learned(l_i,s) ∀ x ∈𝒫, ∃σ∈𝒫, ∀ b ∈ℬ,maj_accepted(s,b)(min_accepted(s,b) (sx ∙σ xs ∙σ)) -*2Proof: By Algorithm <ref> lines {16,30,41} and Algorithm <ref> lines {1-18}, if a correct learner learned a sequence s at any given instant then either N-f or f+1 (if s is universally commutative) acceptors must have executed phase 2b for s.2. At any given instant, ∀ s ∈𝒫, ∀ b ∈ℬ, maj_accepted(s,b)min_accepted(s,b)proposed(s) -*2Proof: By Algorithm <ref> lines {18-23}, for either N-f or f+1 acceptors to accept a proposal it must have been proposed by a proposer (note that the leader is considered a distinguished proposer).3. At any given instant, ∀ s ∈𝒫, ∀ l_i ∈ℒ, learned(l_i,s)proposed(s)Proof: By 1 and 2.4. Q.E.D.§.§.§ Stability If learned_l = s then, at all later times, s ⊑ learned_l, for any sequence s and correct learner l=-1 Proof: By Algorithm <ref> lines {12,18,20-26}, a correct learner can only append new commands to its learned command sequence.§.§.§ Liveness For any proposal s from a correct proposer, and correct learner l, eventually learned_l contains sProof: -1. ∀ l_i ∈ℒ,∀ s,x ∈𝒫, ∃σ∈𝒫, ∀ b ∈ℬ, maj_accepted(s,b)(min_accepted(s,b) (sx ∙σ xs ∙σ))e learned(l_i,s)-*2Proof: By Algorithm <ref> lines {10-15,28-29,41-42} and Algorithm <ref> lines {1-18}, when either N-f or f+1 (if s is universally commutative) acceptors accept a sequence s (or some equivalent sequence), eventually s will be learned by any correct learner.-2. ∀ s ∈𝒫, proposed(s) e∀ x ∈𝒫, ∃σ∈𝒫, ∀ b ∈ℬ, maj_accepted(s,b)(min_accepted(s,b) (sx ∙σ xs ∙σ)) -*2Proof: A proposed sequence is either conflict-free when its incorporated into every acceptor's current sequence or it creates conflicting sequences at different acceptors. In the first case, it's accepted by a quorum (Algorithm <ref>, lines {10-15,28-29,41-42}) and, in the second case, it's sent in phase 1b messages to the in leader in the next ballot (Algorithm <ref>, lines {1-4}) and incorporated in the next proposal (Algorithm <ref>, lines {24-47}). 3. ∀ l_i ∈ℒ, ∀ s ∈𝒫, proposed(s) e learned(l_i,s) Proof: By 1 and 2. 4. Q.E.D. § CONCLUSION AND DISCUSSION We presented a simplified description of the Generalized Paxos specification and protocol,and an implementation of Generalized Paxos that is resilient against Byzantine faults. We now draw some lessons and outline some extensions to our protocol that present interesting directions for future work and hopefully a better understanding of its practical applicability. Handling faults in the fast case. A result that was stated in the original Generalized Paxos paper <cit.> is that to tolerate f crash faults and allow for fast ballots whenever there are up to e crash faults, the total system size N must uphold two conditions: N > 2f and N > 2e+f. Additionally, the fast and classic quorums must be of size N-e and N-f, respectively. This implies that there is a price to pay in terms of number of replicas and quorum size for being able to run fast operations during faulty periods. An interesting observation from our work is that, since Byzantine fault tolerance already requires a total system size of 3f+1 and a quorum size of 2f+1, we are able to amortize the cost of both features, i.e., we are able to tolerate the maximum number of faults for fast execution without paying a price in terms of the replication factor and quorum size. Extending the protocol to universally commutative commands.A downside of the use of commutative commands in the context of Generalized Paxos is that the commutativity check is done at runtime, to determine if non-commutative commands have been proposed concurrently. This raises the possibility of extending the protocol to handle commands that are universally commutative, i.e., commute with every other command. For these commands, it is known before executing them that they will not generate any conflicts, and therefore it is not necessary to check them against concurrently executing commands.This allows us to optimize the protocol by decreasing the number of phase 2b messages required to learn to a smaller f+1 quorum. Since, by definition, these sequences are guaranteed to never produce conflicts, the N-f quorum is not required to prevent learners from learning conflicting sequences. Instead, a quorum of f+1 is sufficient to ensure that a correct acceptor saw the command and will eventually propagate it to a quorum of N-f acceptors.This optimization is particularly useful in the context ofgeo-replicated systems, since it can be significantly faster to wait for the f+1st message instead of the N-fth one.The usefulness of this optimization is severely reduced if these sequences are processed like any other, by being appended to previous sequences at the leader and acceptors. New proposals are appended to previous proven sequences to maintain the invariant that subsequent proven sequences are extensions of previous ones. Since the previous proven sequences to which a proposal will be appended to are probably not universally commutative, the resulting sequence will not be as well. We can increase this optimization's applicability by sending these sequences immediately to the learners, without appending them to previously accepted ones. This special handling has the added benefit of bypassing the verification phase, resulting in reduced latency for the requests and less traffic generated per sequence. This extension can also be easily implemented by adding a single check in Algorithm <ref> lines {19-20}, Algorithm <ref> lines {29-30,40-41} and Algorithm <ref> lines {14-18}.Generalized Paxos and weak consistency.The key distinguishing feature of the specification of Generalized Paxos <cit.> is allowing learners to learn concurrent proposals in a different order, when the proposals commute. This idea is closely related to the work on weaker consistency models like eventual or causal consistency <cit.>, or consistency models that mix strong and weak consistency levels like RedBlue <cit.>, which attempt to decrease the cost of executing operations by reducing coordination requirements between replicas.The link between the two models becomes clearer with the introduction ofuniversally commutative commands in the previous paragraph. In the case of weakly consistent replication, weakly consistent requests can be executed as if they were universally commutative, even if in practice that may not be the case. E.g., checkingthe balance of a bank account and making a deposit do not commute since the output of the former depends on their relative order. However, some systems prefer to run both as weakly consistent operations, even though it may cause executions that are not explained by a sequential execution, since the semantics are still acceptable given that the final state that is reached is the same and no invariantsof the application are violated <cit.>. Acknowledgements. This work was supported by the European Research Council (ERC-2012-StG-307732) and FCT (UID/CEC/50021/2013). plain | http://arxiv.org/abs/1708.07575v3 | {
"authors": [
"Miguel Pires",
"Srivatsan Ravi",
"Rodrigo Rodrigues"
],
"categories": [
"cs.DC"
],
"primary_category": "cs.DC",
"published": "20170824232205",
"title": "Generalized Paxos Made Byzantine (and Less Complex)"
} |
CloudScan - A configuration-free invoice analysis system using recurrent neural networksRasmus Berg Palm DTU Compute Technical University of Denmark [email protected] Winther DTU Compute Technical University of Denmark [email protected] Florian Laws Tradeshift Copenhagen, Denmark [email protected] 30, 2023 =============================================================================================================================================================================================================================== We present CloudScan; an invoice analysis system that requires zero configuration or upfront annotation.In contrast to previous work, CloudScan does not rely on templates of invoice layout, instead it learns a single global model of invoices that naturally generalizes to unseen invoice layouts.The model is trained using data automatically extracted from end-user provided feedback. This automatic training data extraction removes the requirement for users to annotate the data precisely.We describe a recurrent neural network model that can capture long range context and compare it to a baseline logistic regression model corresponding to the current CloudScan production system.We train and evaluate the system on 8 important fields using a dataset of 326,471 invoices. The recurrent neural network and baseline model achieve 0.891 and 0.887 average F1 scores respectively on seen invoice layouts. For the harder task of unseen invoice layouts, the recurrent neural network model outperforms the baseline with 0.840 average F1 compared to 0.788. § INTRODUCTION Invoices, orders, credit notes and similar business documents carry the information needed for trade to occur between companies and much of it is on paper or in semi-structured formats such as PDFs <cit.>. In order to manage this information effectively, companies use IT systems to extract and digitize the relevant information contained in these documents. Traditionally this has been achieved using humans that manually extract the relevant information and input it into an IT system. This is a labor intensive and expensive process <cit.>.The field of information extraction addresses the challenge of automatically extracting such information and several commercial solutions exists that assist in this. Here we present CloudScan, a commercial solution by Tradeshift, free for small businesses, for extracting structured information from unstructured invoices.Powerful information extraction techniques exists given that we can observe invoices from the same template beforehand, e.g. rule, keyword or layout based techniques. A template is a distinct invoice layout, typically unique to each sender. A number of systems have been proposed that rely on first classifying the template, e.g. Intellix <cit.>, ITESOFT <cit.>, smartFIX <cit.> and others <cit.>. As these systems rely on having seen the template beforehand, they cannot accurately handle documents from unseen templates. Instead they focus on requiring as few examples from a template as possible.What is harder, and more useful, is a system that can accurately handle invoices from completely unseen templates, with no prior annotation, configuration or setup. This is the goal of CloudScan: to be a simple, configuration and maintenance free invoice analysis system that can convert documents from both previously seen and unseen templates with high levels of accuracy.CloudScan was built from the ground up with this goal in mind. There is no notion of template in the system. Instead every invoice is processed by the same system built around a single machine learning model. CloudScan does not rely on any system integration or prior knowledge, e.g. databases of orders or customer names, meaning there is no setup required in order to use it.CloudScan automatically extracts the training data from end-user provided feedback. The end-user provided feedback required is the correct value for each field, rather than the map from words on the page to fields. It is a subtle difference, but this separates the concerns of reviewing and correcting values using a graphical user interface from concerns related to acquiring training data. Automatically extracting the training data this way also results in a very large dataset which allows us to use methods that require such large datasets.In this paper we describe how CloudScan works, and investigate how well it accomplishes the goal it aims to achieve. We evaluate CloudScan using a large dataset of 326,471 invoices and report competitive results on both seen and unseen templates. We establish two classification baselines using logistic regression and recurrent neural networks, respectively. § RELATED WORK The most directly related works are Intellix <cit.> by DocuWare and the work by ITESOFT <cit.>. Both systems require that relevant fields are annotated for a template manually beforehand, which creates a database of templates, fields and automatically extracted keywords and positions for each field. When new documents are received, both systems classify the template automatically using address lookups or machine learning classifiers. Once the template is classified the keywords and positions for each field are used to propose field candidates which are then scored using heuristics such as proximity and uniqueness of the keywords. Having scored the candidates the best one for each field is chosen.smartFIX <cit.> uses manually configured rules for each template. Cesarini et al. <cit.> learns a database of keywords for each template and fall back to a global database of keywords. Esser et al. <cit.> uses a database of absolute positions of fields for each template. Medvet et al. <cit.> uses a database of manually created (field, pattern, parser) triplets for each template, designs a probabilistic model for finding the most similar pattern in a template, and extracts the value with the associated parser.Unfortunately we cannot compare ourselves directly to the works described as the datasets used are not publicly available and the evaluation methods are substantially different. However, the described systems all rely on having an annotated example from the same template in order to accurately extract information.To the best of our knowledge CloudScan is the first invoice analysis system that is built for and capable of accurately converting invoices from unseen templates.The previous works described can be configured to handle arbitrary document classes, not just invoices, as is the case for CloudScan. Additionally, they allow the user to define which set of fields are to be extracted per class or template, whereas CloudScan assumes a single fixed set of fields to be extracted from all invoices.Our automatic training data extraction is closely related to the idea of distant supervision <cit.> where relations are extracted from unstructured text automatically using heuristics.The field of Natural Language Processing (NLP) offers a wealth of related work. Named Entity Recognition (NER) is the task of extracting named entities, usually persons or locations, from unstructured text. See Nadeau and Sekine <cit.> for a survey of NER approaches. Our system can be seen as a NER system in which we have 8 different entities. In recent years, neural architectures have been demonstrated to achieve state-of-the-art performance on NER tasks, e.g. Lample et al. <cit.>, who combine word and character level RNNs, and Conditional Random Fields (CRFs).Slot Filling is another related NLP task in which pre-defined slots must be filled from natural text. Our system can be seen as a slot filling task with 8 slots, and the text of a single invoice as input. Neural architectures are also used here, e.g. <cit.> uses bi-directional RNNs and word embedding to achieve competitive results on the ATIS (Airline Travel Information Systems) benchmark dataset. In both NER and Slot Filling tasks, a commonly used approach is to classify individual tokens with the entities or slots of interest, an approach that we adopt in our proposed RNN model.§ CLOUDSCAN§.§ Overview CloudScan is a cloud based software as a service invoice analysis system offered by Tradeshift. Users can upload their unstructured PDF invoices and the CloudScan engine converts them into structured XML invoices. The CloudScan engine contains 6 steps. See Figure <ref>.* Text Extractor. Input is a PDF invoice. Extracts words and their positions from the PDF. If the PDF has embedded text, the text is extracted, otherwise a commercial OCR engine is used. The output of this step is a structured representation of words and lines in hOCR format <cit.>. * N-grammer. Creates N-grams of words on the same line. Output is a list of N-grams up to length 4. * Feature Calculator. Calculates features for every N-gram. Features fall in three categories: text, numeric and boolean. Examples of text features are the raw text of the N-gram, and the text after replacing all letters with "x", all numbers with "0" and all other characters with ".". Examples of numeric features are the normalized position on the page, the width and height and number of words to the left. Boolean features include whether the N-gram parses as a date or an amount or whether it matches a known country, city or zip code. These parsers and small databases of countries, cities and zip codes are built into the system, and does not require any configuration on the part of the user. The output is a feature vector for every N-gram. For a complete list of features see table <ref>. * Classifier. Classifies each N-gram feature vector into 32 fields of interest, e.g. invoice number, total, date, etc. and one additional field 'undefined'. The undefined field is used for all N-grams that does not have a corresponding field in the output document, e.g. terms and conditions. The output is a vector of 33 probabilities for each N-gram. * Post Processor. Decides which N-grams are to be used for the fields in the output document. For all fields, we first filter out N-gram candidates that does not fit the syntax of the field after parsing with the associated parser. E.g. the N-gram "Foo Bar" would not fit the Total field after parsing with the associated parser since no amount could be extracted. The parsers can handle simple OCR errorsand various formats, e.g. "100,0o" would be parsed to "100.00". The parsers are based on regular expressions.For fields with no semantic connection to other fields, e.g. the invoice number, date, etc. we use the Hungarian algorithm <cit.>. The Hungarian algorithm solves the assignment problem, in which N agents are to be assigned to M tasks, such that each task has exactly one agent assigned and no agent is assigned to more than one task. Given that each assignment has a cost, the Hungarian algorithm finds the assignments that minimizes the total cost. We use 1 minus the probability of an N-gram being a field as the cost.For the assignment of the Total, Line Total, Tax Total and Tax Percentage we define and minimize a cost function based on the field probabilities and whether the candidate totals adds up.The output is a mapping from the fields of interest to the chosen N-grams. * Document Builder. Builds a Universal Business Language (UBL) <cit.> invoice with the fields having the values of the found N-grams. UBL is a XML based invoice file format. Output is a UBL invoice.§.§ Extracting training data from end-user provided feedbackThe UBL invoice produced by the engine is presented to the user along with the original PDF invoice in a graphical user interface (GUI). The user can correct any field in the UBL invoice, either by copy and pasting from the PDF, or by directly typing in the correction. See figure <ref>.Once the user has corrected any mistakes and accepted the invoice we add the resulting UBL to our data collection. We will extract training data from these validated UBL documents, even though they might deviate from the PDF content due to OCR error, user error or the user intentionally deviating from the PDF content. We discuss these issues later.The classifier is trained on N-grams and their labels, which are automatically extracted from the validated UBL invoices and the corresponding PDFs. For each field in the validated UBL document we consider all N-grams in the PDF and check whether the text content, after parsing, matches the field value. If it does, we extract it as a single training example of N-gram and label equal to the field. If an N-gram does not match any fields we assign the 'undefined' label. For N-grams that match multiple fields, we assign all matched fields as labels. This ambiguity turns the multi-class problem into a multi-label problem. See Algorithm <ref> for details. 0.5em Using automatically extracted pairs like this results in a noisy, but big data set of millions of pairs. Most importantly, however, it introduces no limitations on how users correct potential errors, and requires no training. For instance, we could have required users to select the word matching a field, which would result in much higher quality training data. However in a high volume enterprise setup, this could reduce throughput significantly. Our automatic training data generation decouples the concerns of reviewing and correcting fields from creating training data, allowing the GUI to focus solely on reviewing and correcting fields. The user would need to review the field values and correct potential errors regardless, so as long as we do not limit how the user does it, we are not imposing any additional burdens. In short, the machine learning demands have lower priority than the user experience in this regard.As long as we get a PDF and a corresponding UBL invoice we can extract training data, and the system should learn and improve for the next invoice.§ EXPERIMENTSWe perform two experiments meant to test 1) the expected performance on the next invoice, and 2) the harder task of expected performance on the next invoice from an unseen template. These are two different measures of generalization performance.The data set consists of 326,471 pairs of validated UBL invoices and corresponding PDFs from 8911 senders to 1013 receivers obtained from use of CloudScan. We assume each sender corresponds to a distinct template.For the first experiment we split the invoices into a training, validation and test set randomly, using 70%, 10% and 20% respectively. For the second experiment we split the senders into a training, validation and test set randomly, using 70%, 10% and 20% respectively. All the invoices from the senders in a set then comprise the documents of that set. This split ensures that there are no invoices sharing templates between the three sets for the second experiment.While the system captures 32 fields we only report on eight of them: Invoice number, Issue Date, Currency, Order ID, Total, Line Total, Tax Total and Tax Percent. We only report on these eight fields as they are the ones we have primarily designed the system for. A large part of the remaining fields are related to the sender and receiver of the invoice and used for identifying these. We plan to remove these fields entirely and approach the problem of sender and receiver identification as a document classification problem instead. Preliminary experiments based on a simple bag-of-words model show promising results. The last remaining fields are related to the line items and used for extracting these. Table extraction is a challenging research question in itself, and we are not yet ready to discuss our solution. Also, while not directly comparable, related work <cit.> also restricts evaluation to header fields.Performance is measured by comparing the fields of the generated and validated UBL. Note we are not only measuring the classifier performance, but rather the performance of the entire system. The end-to-end performance is what is interesting to the user after all. Furthermore, this is the strictest possible way to measure performance, as it will penalize errors from any source, e.g. OCR errors and inconsistencies between the validated UBL and the PDF. For instance, the date in the validated UBL might not correspond to the date on the PDF. In this case, even if the date on the PDF is found, it will be counted as an error, as it does not match the date in the validated UBL.In order to show the upper limit of the system under this measure we include a ceiling analysis where we replace the classifier output with the correct labels directly. This corresponds to using an oracle classifier. We use the MUC-5 definitions of recall, precision and F1, without partial matches <cit.>.We perform experiments with two classifiers 1) The production baseline system using a logistic regression classifier, and 2) a Recurrent Neural Network (RNN) model. We hypothesize the RNN model can capture context better. §.§ BaselineThe baseline is the current production system, which uses a logistic regression classifier to classify each N-gram individually.In order to capture some context, we concatenate the feature vectors for the closest N-grams in the top, bottom, left and right directions to the normal feature vectors. So if the feature vector for an N-gram had M entries, after this it would have 5M entries.All 5M features are then mapped to a binary vector of size 2^22 using the hashing trick <cit.>. To be specific, for each feature we concatenate the feature name and value, hash it, take the remainder with respect to the binary vector size and set that index in the binary vector to 1. The logistic regression classifier is trained using stochastic gradient descent for 10 epochs after which we see little improvement. This baseline system is derived from the heavily optimized winning solution of a competition Tradeshift held[https://www.kaggle.com/c/tradeshift-text-classification]. §.§ LSTM modelIn order to accurately classify N-grams the context is critical, however when classifying each N-gram in isolation, as in the baseline model, we have to engineer features to capture this context, and deciding how much and which context to capture is not trivial.A Recurrent Neural Network (RNN) can model the entire invoice and we hypothesize that this ability to take the entire invoice into account in a principled manner will improve the performance significantly. Further, it frees us from having to explicitly engineer features that capture context. As such we only use the original M features, not the 5M features of the baseline model. In general terms, a RNN can be described as follows. h_t= f(h_t-1, x_t) y_t= g(h_t) Where h_t is the hidden state at step t, f is a neural network that maps the previous hidden state h_t-1, and the input x_t to h_t and g is a neural network that maps the hidden state h_t to the output of the model y_t. Several variants have been proposed, most notably the Long Short Term Memory (LSTM) <cit.> which is good at modeling long term dependencies.A RNN models a sequence, i.e. x and y are ordered and as such we need to impose an ordering on the invoice. We chose to model the words instead of N-grams, as they fit the RNN sequence model more naturally and we use the standard left-to-right reading order as the ordering. Since the labels can span multiple words we re-label the words using the IOB labeling scheme <cit.>. The sequence of words "Total Amount: 12 200 USD" would be labeled "O O B-Total I-Total B-Currency".We hash the text of the word into a binary vector of size 2^18 which is embedded in a trainable 500 dimensional distributed representation using an embedding layer <cit.>. Using hashing instead of a fixed size dictionary is somewhat unorthodox but we did not observe any difference from using a dictionary, and hashing was easier to implement. It is possible we could have gotten better results using more advanced techniques like byte pair encoding <cit.>.We normalize the numerical and boolean features to have zero mean and unit variance and form the final feature vector for each word by concatenating the word embedding and the normalized numerical features.From input to output, the model has: two dense layers with 600 rectified linear units each, a single bidirectional LSTM layer with 400 units, and two more dense layers with 600 rectified linear units each, and a final dense output layer with 65 logistic units (32 classes that can each be 'beginning' or 'inside' plus the 'outside' class).Following Gal <cit.>, we apply dropout on the recurrent units and on the word embedding using a dropout fraction of 0.5 for both. Without this dropout the model severely overfits.The model is trained with the Adam optimizer <cit.> using minibatches of size 96 until the validation performance has not improved on the validation set for 5 epochs. Model architecture and hyper-parameters were chosen based on the performance on the validation set. For computational reasons we do not train on invoices with more than 1000 words, which constitutes approximately 5% of the training set, although we do test on them. The LSTM model was implemented in Theano <cit.> and Lasagne <cit.>.After classification we assign each word the IOB label with highest classification probability, and chunk the IOB labeled words back into labeled N-grams. During chunking, words with I labels without matching B labels are ignored. For example, the sequence of IOB labels [B-Currency, O, B-Total, I-Total, O, I-Total, O] would be chunked into [Currency, O, Total, O, O]. The labeled N-grams are used as input for the Post Processor and further processing is identical to the baseline system.§ RESULTS The results of the ceiling analysis seen in Table <ref> show that we can achieve very competitive results with CloudScan using an oracle classifier. This validates the overall system design, including the use of automatically generated training data, and leaves us with the challenge of constructing a good classifier.The attentive reader might wonder why the precision is not 1 exactly for all fields, when using the oracle classifier. For the 'Number' and 'Order ID' fields this is due to the automatic training data generation algorithm disregarding spaces when finding matching N-grams, whereas the comparison during evaluation is strict. For instance the automatic training data generator might generate the N-gram ("16 2054": Invoice Number) from (Invoice Number: "162054") in the validated UBL. When the oracle classifier classifies the N-gram "16 2054" as Invoice Number the produced UBL will be (Invoice Number: "16 2054"). When this is compared to the expected UBL of (Invoice Number: "162054") it is counted as incorrect. This is an annoying artifact of the evaluation method and training data generation. We could disregard spaces when comparing strings during evaluation, but we would risk regarding some actual errors as correct then. For the total fields and the tax percent, the post processor will attempt to calculate missing numbers from found numbers, which might result in errors.As it stands the recall rate is the limiting factor of the system. The low recall rate can have two explanations: 1) The information is present in the PDF but we cannot read or parse it, e.g. it might be an OCR error or a strange date format, in which case the OCR engine or parsing should be improved, or 2) the information is legitimately not present in the PDF, in which case there is nothing to do, except change the validated UBL to match the PDF.Table <ref> shows the results of experiment 1 measuring the expected performance on the next received invoice for the baseline and LSTM model. The LSTM model is slightly better than the baseline system with an average F1 of 0.891 compared to 0.887. In general the performance of the models is very similar, and close to the theoretical maximum performance given by the ceiling analysis. This means the classifiers both perform close to optimally for this experiment. The gains that can be had from improving upon the LSTM model further are just 0.034 average F1.More interesting are the results in Table <ref> which measures the expected performance on the next invoice from an unseen template. This measures the generalization performance of the system across templates which is a much harder task due to the plurality of invoice layouts and reflects the experience a new user will have the first time they use the system. On this harder task the LSTM model clearly outperform the baseline system with an average F1 of 0.840 compared to 0.788. Notably the 0.840 average F1 of the LSTM model is getting close to the 0.891 average F1 of experiment 1, indicating that the LSTM model is largely learning a template invariant model of invoices, i.e. it is picking up on general patterns rather than just memorizing specific templates.We hypothesized that it is the ability of LSTMs to model context directly that leads to increased performance, although there are several other possibilities given the differences between the two models. For instance, it could simply be that the LSTM model has more parameters, the non-linear feature combinations, or the word embedding.To test our hypothesis we trained a third model that is identical to the LSTM model, except that the bidirectional LSTM layer was replaced with a feedforward layer with an equivalent number of parameters. We trained the network with and without dropout, with all other hyper parameters kept equal. The best model got an average F1 of 0.702 on the experiment 2 split, which is markedly worse than both the LSTM and baseline model. Given that the only difference between this model and the LSTM model is the lack of recurrent connections we feel fairly confident that our hypothesis is true. The feedforward model is likely worse than the baseline model because it does not have the additional context features of the baseline model.Table <ref> shows examples of words and the two closest words in the learned word embedding. It shows that the learned embeddings are language agnostic, e.g. the closest word to "Total" is "Betrag" which is German for "Sum" or "Amount". The embedding also captures common abbreviations, capitalization, currency symbols and even semantic similarities such as cities. Learning these similarities versus encoding them by hand is a major advantage as it happens automatically as it is needed. If a new abbreviation, language, currency, etc. is encountered it will automatically be learned.§ DISCUSSION We have presented our goals for CloudScan and described how it works. We hypothesized that the ability of a LSTM to model context directly would improve performance. We carried out experiments to test our hypothesis and evaluated CloudScan's performance on a large realistic dataset. We validated our hypothesis and showed competitive results of 0.891 average F1 on documents from seen templates, and 0.840 on documents from unseen templates using a single LSTM model. These numbers should be compared to a ceiling of F1=0.925 for an ideal system baseline where an oracle classifier is used.Unfortunately it is hard to compare to other vendors directly as no large publicly available datasets exists due to the sensitive nature of invoices. We sincerely wish such a dataset existed and believe it would drive the field forward significantly, as seen in other fields, e.g. the large effect ImageNet <cit.> had on the computer vision field. Unfortunately we are not able to release our own dataset due to privacy restrictions.A drawback of the LSTM model is that we have to decide upon an ordering of the words, when there is none naturally. We chose the left to right reading order which worked well, but in line with the general theme of CloudScan we would prefer a model which could learn this ordering or did not require one.CloudScan works only on the word level, meaning it does not take any image features into account, e.g. the lines, logos, background, etc. We could likely improve the performance if we included these image features in the model.With the improved results from the LSTM model we are getting close to the theoretical maximum given by the ceiling analysis. For unseen templates we can at maximum improve the average F1 by 0.085 by improving the classifier. This corresponds roughly to the 0.075 average F1 that can at maximum be gained from fixing the errors made under the ceiling analysis. An informal review of the errors made by the system under the ceiling analysis indicates the greatest source of errors are OCR errors and discrepancies between the validated UBL and the PDF.As such, in order to substantially improve CloudScan we believe a two pronged strategy is required: 1) improve the classifier and 2) correct discrepancies between the validated UBL and PDF. Importantly, the second does not delay the turnaround time for the users, can be done at our own pace and only needs to be done for the cases where the automatic training data generation fails. As for the OCR errors we will rely on further advances in OCR technology.§ ACKNOWLEDGMENTWe would like to thank Ángel Diego Cuñado Alonso and Johannes Ulén for our fruitful discussions, and their great work on CloudScan. This research was supported by the NVIDIA Corporation with the donation of TITAN X GPUs. This work is partly funded by the Innovation Fund Denmark (IFD) under File No. 5016-00101B. IEEEtran | http://arxiv.org/abs/1708.07403v1 | {
"authors": [
"Rasmus Berg Palm",
"Ole Winther",
"Florian Laws"
],
"categories": [
"cs.CL"
],
"primary_category": "cs.CL",
"published": "20170824134006",
"title": "CloudScan - A configuration-free invoice analysis system using recurrent neural networks"
} |
In this paper we generalize the Perron Identity for Markov minima. We express the values of binary quadratic forms with positive discriminant in terms of continued fractions associated to broken lines passing through the points where the values are computed.Topological invariants for Floquet-Bloch systems with chiral, time-reversal, or particle-hole symmetry Holger Fehske December 30, 2023 ========================================================================================================psf § INTRODUCTIONConsider a binary quadratic form f with positive discriminant Δ(f). In this paper we give a geometric interpretation and generalization of the Perron Identity relating the minimal value of |f| at integer points except the origin and their corresponding continued fractions:min_^2∖{(0,0)}|f|=inf_i∈(√(Δ(f))/a_i+[0;a_i+1:a_i+2:…]+[0;a_i-1:a_i-2:…]).Here [a_0;a_1:…] and [0;a_-1:a_-2:…]are regular continued fractions of the slopes of linear factors of corresponding reduced linear forms. Recall that a continued fraction is regular if all its elements are non negative. We discuss this in more detail further in Section <ref>. The Perron Identity was shown by A. Markov in his paper on minima of binary quadratic forms and the Markov spectrum below 3 in <cit.>. The statement holds for the entire Markov spectrum (see, e.g., the books by O. Perron <cit.>, and T. Cusick and M. Flahive <cit.>). Recently Markov numbers were used in relation to Federer-Gromovs stable norm,(<cit.>). There is not much known about higher dimensional analogue of Markov spectrum. It is believed to be discrete (which is equivalent to Oppenheim conjecture on best approximations, see in Chapter 18 of <cit.>). Various values of three-dimensional Markov spectrum were constructedby H. Davenport in <cit.>. In this paper we show the geometric interpretation of the Perron Identity in terms of sails of the form (Remark <ref>) and generalize this expression in the spirit of integer geometry. This establishes a relationship between non-regular continued fractions and the values of the corresponding binary quadratic form at any point on the plane (Theorem <ref> and Corollary <ref>). The result of this paper is based on recent results of the first author in geometric theory of continued fractions for arbitrary broken lines, see <cit.>. Organization of the paper. We start in Section <ref> with necessary definitions and background. We discuss reduced forms, LLS sequences, and formulate the classical Perron Identity. In Section <ref> we formulate and prove the Generalized Perron Identity for finite broken lines. Finally in Section <ref> we prove the Generalized Perron Identity for infinite broken lines, and discuss the relation with the classical the Perron Identity. Acknowledgement. The first author is partially supported by EPSRC grant EP/N014499/1 (LCMH). § BASIC NOTIONS AND DEFINITIONSIn this section we give necessary notions and definitions. We start in Subsection <ref> with classical definitions of Markov minima and Markov spectrum. Further in Subsection <ref> we discuss reduced forms of quadratic binary forms with positive discriminant. In Subsection <ref> we discuss the classical Perron Identity. Finally in Subsection <ref> we introduce LLS sequences for broken lines, which is the central notion in the formulation of the main results. §.§ Markov minima and Markov spectrum Let f be a binary quadratic form with positive discriminant. Recall that in this case f is decomposable into two real factors, namelyf(x,y)=(ax-b y)(cx-d y),for some real numbers a, b, c, and d. The discriminant of this form isΔ(f)=(ad-bc)^2. The Markov minimum of the form f is the following number:m(f)=min_^2∖{(0,0)}|f|.The set of all possible values of Δ(f)/m(f) is called Markov Spectrum. (Note that Δ(f)/m(f) is invariant under multiplication of the form f by a non-zero scalar.) The spectrum below 3 correspond to special forms with integer coefficients, we refer an interested reader to an excellent book <cit.> by T. Cusick and M. Flahive on Markov spectrum and related subjects. §.§ Reduced forms, and LLS-sequencesIt is clear that m(f) is invariant under the action of the group of (2,). Therefore in order to study the Markov spectrum one can restrict to so called reduced forms which are simple to describe. There are several ways to pick reduced forms,although the algorithmic part is rather similar to all of them, it is a subject of a Gauss reduction theory (see, e.g., <cit.>, <cit.>, <cit.>, and <cit.>). We consider the following family of reduced forms. For every α≥ 1 and 1> β≥ 0 setf_α,β=(y-α x)(y+β x).Every form is multiple to some reduced form in appropriate basis of integer lattice ^2. However such representation is not unique. The following notion provides a complete invariant distinguishing different classes of reduced forms.Let α≥ 1, 1> β≥ 0 and letα=[a_0;a_1:…] andβ=[0;a_-1:a_-2:…]be the regular continued fractions for α and β. Then the sequence(… a_-2,a_-1,a_0, a_1,a_2,…)is called the LLS sequence of the form f_α,β.This sequence can be either finite or infinite from one or both sides. The name for the LLS sequence (Lattice Length-Sine sequence) is due its lattice trigonometric properties, e.g., see in <cit.> and <cit.>. Two reduced forms are equivalent (i.e., multiple to each other after (2,)-change of coordinates) if and only if they have the same LLS sequence up to shifts of sequence by k-elements for some integer k and a reversing of the order of a sequence.This statement follows directly from geometric properties of continued fractions. As we do not use this statement in the proof of the results of this paper we skip the proof here. We refer an interested reader to <cit.>. Due to Proposition <ref> we can extend the notion of LLS-sequence toany binary quadratic form with positive discriminant.Let f be a binary quadratic form with positive discriminant. The LLS sequence for f is the LLS sequencefor any reduced form f_α,β equivalent to f. We denote it by (f).§.§ Classical Perron Identity We are coming to one of the most mysterious statements in theory of Markov minima. It is known as the Perron Identity.Let f be a binary quadratic form with positive discriminant Δ (f). Let also(f)=(… a_-2,a_-1,a_0, a_1,a_2,…)Then we have the following result by A. Markov in <cit.>:m(f)/√(Δ(f))=inf_i∈(1/a_i+[0;a_i+1:a_i+2:…]+[0;a_i-1:a_i-2:…]). This result is based on the following observation. Let α≥ 1, 1> β≥ 0 and letα=[a_0;a_1:…] andβ=[0;a_-1:a_-2:…]be the regular continued fractions for α and β. Thenf_α,β(0,1)=1/a_0+[0;a_1:a_2:…]+[0;a_-1:a_-2:…]. Our goal is to investigate the lattice geometry behind this expression. It will lead us to a more general rule relating continued fractions whose elements are arbitrary non zero real numbers, andthe values of the corresponding binary form at any point on the plane(see Theorem <ref>, Corollary <ref> and Remark <ref>). §.§ LLS sequences for broken lines We start with the following general definition.Consider a quadratic binary form f with positive discriminant. A broken line A_0… A_n is an f-broken line if the following conditions hold: * A_0,A_n O belong to the two distinct loci of linear factors of f; * all edges of the broken line are of positive length; * for every k=1,…, n the line A_k-1A_k does not pass through the origin. Recall the definition of oriented Euclidean area for parallelograms. Consider three points A, B, C in the plane. Then the determinant for the matrix of vectors AB and AC is called the the oriented Euclidean area for the parallelogram spanned by AB and AC and denoted by(AB,AC). Let 𝒜=A_0A_1… A_n be a broken line with A_0,A_n O. Then the sign function of the determinant (OA_1,OA_n) is called the signature of 𝒜 with respect to the origin and denoted by (𝒜). We conclude this section with the following important definition. Given an f-broken line 𝒜 =A_0… A_n define a_2k =(OA_k,OA_k+1),k=0,…, n; a_2k-1 =(A_kA_k-1,A_kA_k+1)/ a_2k-2a_2k , k=1,…, n. The sequence (a_0,…, a_2n) is called the LLS sequence for the broken line and denoted by (𝒜). The expression [a_0;…: a_2n] is said to be the continued fraction for the broken line A_0… A_n. Note that the values a_i 0 may be negative. The LLS sequence encodes the integer angles and integer lengths of the broken line (see <cit.> for further details). § GENERALIZED PERRON IDENTITY FOR FINITE BROKEN LINES Now we are in position to formulate and to prove the main result of this paper. (Generalized Perron Identity: case of finite broken lines.) Consider a binary quadratic form with positive discriminant f. Let 𝒜=A_0… A_n+m be an f-broken line (here n and m are arbitrary positive integers), and let(𝒜)=(a_0,a_1,…,a_2n+2m).Thenf(A_n)=(𝒜) ·√(Δ(f))/a_2n-1+[0;a_2n-2:…:a_0]+[0;a_2n:…:a_2n+2m]. Let us first consider the following example. Consider the following binary quadratic formf(x,y)=(x+y)(x-2y). Let 𝒜=A_0… A_7 be the broken line with vertices[ A_0=(2,-2), A_1=(4,-1), A_2=(3,-2),A_3=(2,0),;A_4=(3,1),A_5=(4,0), A_6=(3,-1),A_7=(4,2), ]see Figure <ref>. Let us check Theorem <ref> for the broken line 𝒜 at point A_4=(3,1). We leave the computations of LLS-sequences to a reader as an exercise, the result is as follows:(𝒜)=(6, -1/30,-5,-3/20,4, 3/8,2, -1/4,-4: 1/8: -4: -1/20: 10)(here we denote the elements of (𝒜) by a_0,…, a_12). Finally we have Δ(f)=9 and (𝒜)=1.According to Theorem <ref> we expect the following.[ f(A_4)= (𝒜) ·√(Δ(f))/a_7+[0;a_6:…:a_0]+[0;a_8:…:a_12]; = 1 · 3/-1/4+[0; 2: 3/8: 4: -3/20: -5: -1/30: 6]+ [0; -4: 1/8: -4: -1/20: 10]; =4. ]Indeed, direct computation shows thatf(A_4)=(3+1)(3-2· 1)=4. We start the proof with three lemmas. Consider a binary quadratic form with positive discriminant f. Let P O and Q O annulate distinct linear factors of f. Then for every point A it holdsf(A)=(POQ)·(OP,OA)·(OA,OQ)/(OP,OQ)·√(Δ (f)). Consider the following binary quadratic formf(x,y)=(x+y)(x-2y). Let PAQ be an f-broken line, with P=(2,1), A=(3,0), and Q=(2,-2), see Figure <ref>. Direct calculations show that[ (OP,OA)=6, (OA,OQ)=3, (OP,OQ)=6,; (POQ)=1,f(A)=9, Δ (f)=9. ]Therefore, we have(POQ)·(OP,OA)·(OA,OQ)/(OP,OQ)·√(Δ (f)) = 1·6· 3/6·√(9)= 9=f(A). The statement is straightforward for the formf_α(x,y)=α xy.Assume that P=(p,0), Q=(0,q), and A=(x,y). Then we havef_α(A)=α xy= py· qx/pq·α =(OP,OA)·(OA,OQ)/(OP,OQ)·√(Δ(f)).For P=(0,p) and Q=(q,0) we havef_α(A) =α xy= (-px)· (-qx)/-pq·α=-(OP,OA)·(OA,OQ)/(OP,OQ)·√(Δ(f)).This conclude the proof for the case of f_α. The general case follows from the invariance of the expressions of the equality of the lemma under the group of linear area preservingtransformations (i.e., whose determinants equal 1) of the plane. Now we prove a particular case of Theorem <ref>. Let f be a binary quadratic form with positive discriminant. Consider an oriented f-broken line ℬ=B_0B_1B_2 with (ℬ)=(b_0,b_1,b_2). Thenf(B_1)=(ℬ)·√(Δ(f))/b_1+[0;b_0]+[0;b_2].Set B_i=(x_i,y_i) for i=0,1,2. Then Definition <ref> impliesb_0 =(OB_0, OB_1)=x_0y_1-x_1y_0,b_2 =(OB_1,OB_2)=x_1y_2-y_1x_2,b_1 =(B_1B_0,B_1B_2)/b_0b_2=x_0y_2-x_2y_0-x_0y_1+x_1y_0-x_1y_2+y_1x_2/b_0b_2.After a substitution and simplification we get1/b_1+[0;b_0]+[0;b_2] = (x_0y_1-x_1y_0)(x_1y_2-y_1x_2)/x_0y_2-x_2y_0= (OB_0,OB_1)·(OB_1,OB_2)/(OB_1,OB_2).Finally recall that(ℬ)=(B_0B_1B_2).Now Lemma <ref> follows directly from Lemma <ref>.For the proof of general case we need the following important result. (<cit.>.) Consider a broken line A_0… A_n that has the LLS sequence (a_0,…,a_2n), with A_0=(1,0), A_1=(1,a_0), and A_n=(x,y). Let α=[a_0;a_1:…:a_2n] be the corresponding continued fraction for this broken line. Theny/x=α. For the case of an infinite value for α=[a_0;a_1:…: a_2n], x/y=0.For a proof of Lemma <ref> we refer to <cit.>. As a consequence of Lemma <ref> we have the following statement. Consider two broken lines A_0… A_n and B_0… B_m that have the LLS sequences (a_0,…,a_2n), and (b_0,…,b_2m) respectively. Suppose that the following hold: * A_0=B_0;* the points A_n, B_m, and O are in a line;* the points A_0=B_0, A_1, and B_1 are in a line.Then[a_0;a_1:…:a_2n]=[b_0;b_1:…:b_2n]. In coordinates of the basise_1=OA_0,e_2=A_0A_1/|A_0A_1||OA_0|the coincidence of continued fractions follows from Lemma <ref>. Let f be a binary quadratic form with positive discriminant. Denote the linear factors of f by f_1 and f_2. Consider an f-broken line 𝒜= A_0… A_n+m. Without loss of generality we assume that A_0 and A_n+m annulate f_1 and f_2 respectively. Denote by B the intersection of the line A_nA_n-1 with the line f_1=0. Denote by C the intersection of the line A_nA_n+1 with the line f_2=0. (See Figure <ref>.) Then the continued fraction for the broken line BA_nC is [b_0:a_2n-1:b_2] for some real numbers b_0 and b_2. By Corollary <ref> we have[ [b_0]=[a_2n-2;…:a_0];; [b_2]=[a_2n;…:a_2n+2m].; ]By construction(BA_nC)=(𝒜).Therefore by Lemma <ref> we havef(A_n) =(BA_nC)·√(Δ(f))/a_2n-1+[0;b_0]+[0;b_2]= (𝒜)·√(Δ(f))/a_2n-1+[0;a_2n-2:…:a_0]+[0;a_2n:…:a_2n+2m].This concludes the proof of Theorem <ref>. § GENERALIZED PERRON IDENTITY FOR ASYMPTOTIC INFINITE BROKEN LINESIn this section we extend the Generalized Perron Identity (of Theorem <ref>) to the case of certain infinite broken lines and discuss the relation to the classical Perron Identity. We start with the following definition. Consider a binary quadratic form f with positive discriminant. An infinite in both sides broken line … A_-2A_-1A_0A_1A_2 … is an asymptotic f-broken line if the following conditions hold (here we assume that A_k=(x_k,y_k) for every integer k): * the two side infinite sequence (y_n/x_n) converges to different slopes of the linear factors in the kernel of f as n increases and decreases respectively;* all edges of the broken line are of positive length;* for every admissible k the line A_k-1A_k does not pass through the origin. Here and below one can consider one side infinite broken lines. All the proofs are similar, so we leavethis case as an exercise. The signature of an asymptotic f-broken line is defined as a determinant for two vectors in the kernel of f, the first with the starting limit direction and the second with the end limit direction. Finally we have a definition of LLS-sequences similar to Definition <ref>.Given an asymptotic f-broken line 𝒜 =… A_-2A_-1A_0A_1A_2 … define a_2k =(OA_k,OA_k+1),k∈; a_2k-1 =(A_kA_k-1,A_kA_k+1)/ a_2k-2a_2k , k∈. The sequence (…, a_-2,a_-1,a_0, a_1,a_2…) is called the LLS sequence for the broken line and denoted by (𝒜). Let us extend the Generalized Perron Identity (of Theorem <ref>) to the case of asymptotic f-broken line.(Generalized Perron Identity: case of infinite broken lines.) Consider a binary quadratic form with positive discriminant f. Let 𝒜=… A_-2A_-1A_0A_1A_2… be an asymptotic f-broken line, and let(𝒜)=(… a_-2,a_-1,a_0,a_1,a_2,…).Assume also that both continued fractions[0;a_-1:a_-2:…]and [0;a_1:a_2:…]converge. Thenf(A_0)=(𝒜) ·√(Δ(f))/a_0+[0;a_-1:a_-2:…]+[0;a_1:a_2:…]. Without loss of generality we consider the formf=λ f_α,β=λ(y-α x)(y+β x),for some nonzero λ and arbitrary αβ. Let 𝒜=… A_-2A_-1A_0A_1A_2… be an asymptotic f-broken line, where A_k=(x_k,y_k) for all integer k. Also we assume that x_k 0 for all k (otherwise, switch to another coordinate system, where the last condition holds). Set[ 𝒜_n=A_-n… A_-2A_-1A_0A_1A_2… A_n;; α_n=y_-n/x_-n; β=y_n/x_n. ] First of all, by definition (𝒜_n) coincides with (𝒜) for all admissible entries. Secondly, we immediately have thatlim_n→∞λ f_α_n,β_n(A_0)=λ f_α,β(A_0). Thirdly, the sequence of signatures stabilizes as n tends to infinity. In other wordslim_n→∞(𝒜_n)=(𝒜). Fourthly,lim_n→∞Δ(λ f_α_n,β_n)=Δ(λ f_α,β). Finally since both continued fractions[0;a_-1:a_-2:…],and [0;a_1:a_2:…]converge and by the above four observations we havef(A_0) = lim_n→∞λ f_α_n,β_n(A_0) = lim_n→∞(𝒜_n) ·√(Δ(λ f_α_n,β_n))/a_0+[0;a_-1:a_-2:…:a_2-2n]+[0;a_1:a_2:…:a_2n-2]. = (𝒜) ·√(Δ(f))/a_0+[0;a_-1:a_-2:…]+[0;a_1:a_2:…].The second equality holds as it holds for the elements in the limits for every positive integer n by Theorem <ref>.This concludes the proof of the corollary. We conclude this paper with the following important remark.(Lattice geometry of the Perron Identity.) Let f be a binary quadratic form with positive discriminant. Consider an angle in the complement to the kernel of f. The sail of this angle is the boundary of the convex hull of all integer points inside the angle except the origin. Note that each form f has four angles in the complement, and, therefore, it has four sails. It is important that the sail of any angle in the complement to the set f=0 is an asymptotic f-broken line, so the Corollary <ref> holds for each of four sails of f. From general theory of geometric continued fractions, the Markov minimum is an accumulation point of the values at vertices of all sails. For every vertex V_i of a sail there exists a reduced form f_α_i,β_i. with α_i≥ 1 and 1≥β_i > 1 such that V_i corresponds to (0,1). In particular we have f(V_i)=f_α_i,β_i(0,1).The point (0,1) is a vertex of the sail for f_α_i,β_i. Then from general theory of continued fractions (see Part 1 of <cit.>) the sequence (f_α,β) coincides with thesequence for the sail containing (0,1). Hence the expressions in the Perron Identity (<ref>) for which the minimum is computed, i.e.,√(Δ(f))/a_i+[0;a_i+1:a_i+2:…]+[0;a_i-1:a_i-2:…]for i=…, -2,-1,0,1,2,… correspond to the formula of Corollary <ref>for vertices V_i of all four sails.We consider the vertex V_i, with the sail containing it, as an f-broken line. plain | http://arxiv.org/abs/1708.07396v2 | {
"authors": [
"Oleg Karpenkov",
"Matty van-Son"
],
"categories": [
"math.NT",
"11H99"
],
"primary_category": "math.NT",
"published": "20170824132956",
"title": "Perron identity for arbitrary broken lines"
} |
[ Pavel Exner December 30, 2023 ===================== A finite set A ⊂ℝ^d is called diameter-Ramsey if for every r ∈, there exists some n ∈ and a finite set B ⊂ℝ^n with A=B such that whenever B is coloured with r colours, there is a monochromatic set A' ⊂ B which is congruent to A. We prove that sets of diameter 1 with circumradius larger than 1/√(2) are not diameter-Ramsey. In particular, we obtain that triangles with an angle larger than 135^∘ are not diameter-Ramsey, improving a result of Frankl, Pach, Reiher and Rödl. Furthermore, we deduce that there are simplices which are almost regular but not diameter-Ramsey. § INTRODUCTIONIn this note, we discuss questions related to Euclidean Ramsey theory, a field introduced in <cit.> by Erdős, Graham, Montgomery, Rothschild, Spencer and Straus. A finite set A ⊂ℝ^d is called Ramsey if for every r ∈, there exists some n ∈ such that in every colouring of ℝ^n with r colours, there is a monochromatic set A' ⊂^n which is congruent to A. The problem of classifying which sets are Ramsey has been widely studied and is still open (see <cit.> for more details). The diameter of a set P ⊂^d is defined by P := sup{x-y : x,y ∈ P }, where · denotes the Euclidean norm. Recently,Frankl, Pach, Reiher and Rödl <cit.> introduced the following stronger property.A finite set A ⊂ℝ^d is called diameter-Ramsey if for every r ∈, there exists some n ∈ and a finite set B ⊂ℝ^n with A=B such that whenever B is coloured with r colours, there is a monochromatic set A' ⊂ B which is congruent to A. It follows from the definition that every diameter-Ramsey set is Ramsey. A set A ⊂^d is called spherical, if it lies on some d-dimensional sphere and the circumradius of A, denoted by A, is the radius of the smallest sphere containing A. (Note that if A is spherical and is not contained in a proper subspace of ℝ^d, then there is a unique sphere that contains it.) In <cit.> it was proved that every Ramsey set must be spherical. Our main result states that every diameter-Ramsey set must also have a small circumradius. If A ⊂ℝ^d is a finite, spherical set with circumradius strictly larger than A/√(2), then A is not diameter-Ramsey.Frankl, Pach, Reiher and Rödl <cit.> proved that acute and right-angled triangles are diameter-Ramsey, while triangles having an angle larger than 150^∘ are not. <ref> implies the following improvement. Triangles with an angle larger than 135^∘ are not diameter-Ramsey.Let us call a d-simplex A = {p_1, …, p_d+1} -almost regular if1/d+12∑_1 ≤ i < j ≤ d+1 A^2 - p_i -p_j^2 ≤· A^2.In <cit.> it was further proved that -almost regular simplices are diameter-Ramsey for every ≤ 1/d+12. This is a rather small class of simplices since 1/d+12 tends to zero, but another corollary of <ref> shows that one cannot hope for much more. For every d ∈ and every > √(d)/d+12, there is an -almost regular d-simplex which is not diameter-Ramsey.For d ∈ and r ≥ 0, we denote the closed d-dimensional ball of radius r centred at the origin by B_d(r). We will deduce <ref> from the following result. For every finite, spherical set A ⊂^d and every positive number r <A, there is some k = k(A,r) ∈ such that the following holds. For every D ∈, there is a colouring of B_D(r) with k colours and with no monochromatic, congruent copy of A. A result of Matoušek and Rödl <cit.> shows that the conclusion of <ref> does not hold whenever r >A. We do not know what happens when r =A. After completing this work, we have learnt that Theorem <ref> has independently been proved byFrankl, Pach, Reiher and Rödl, with a similar proof (János Pach, private communication). § PROOFS §.§ Proof of the main theoremFix some finite, spherical A ⊂^d and some positive number r <A. The following claim is the key step of the proof. There exists a constant c = c(A,r) > 0 such that for every D ∈ and for every congruent copy A' of A in B_D(r) we have max_x,y ∈ A'(x -y ) ≥ c. First observe that it is sufficient to prove the claim for D = d+1. For D < d+1, this follows immediately from B_D(r) ⊂ B_d+1(r), and for D>d+1 we can consider the at most (d+1)-dimensional subspace spanned by the vertices of A' and the origin.Let E={e:A→ B_D(r)}⊂ B_D(r)^|A| be the set of all embeddings of A to B_D. It is easy to see that, if e_1, e_2, …∈ E and the pointwise limit e := lim_n e_n exists, then e ∈ E. Therefore, E is a closed subset of a compact metric space and hence E is compact as well. Define f:E→ℝ byf(e) := max_x,y ∈ e(A)( x -y ). Clearly, f(e)≥ 0 for every e ∈ E, and f(e) = 0, if and only if e(A) lies on a sphere around the origin. But since e(A) > r for every embedding e ∈ E, this is not the case, and hence f(e)>0 for all e ∈ E. Finally, since f is continuous, there is a constant c >0 such that f(e) ≥ c for all e ∈ E. Let k = ⌊ r/c ⌋ + 1 now, and fix some D ∈. We will colour points in B_D(r) by their distance to the origin: Define χ : B_D(r) →{0, …, k-1 } by χ(x) = ⌊ 1/c ·x⌋, and let A' ⊂ B_D(r) be a congruent copy of A. It follows immediately from <ref> that there are x,y ∈ A' with x-y≥ c, and hence χ (x) ≠χ (y). This finishes the proof of the main theorem.§.§ Implications of the main theoremIn this section we will deduce <ref> and then <ref>. In order to do so we will use the following classical result. Every bounded set A ⊂^d can be covered by a closed ball of radius √(d/(2d+2))· A. In particular, every finite set B ⊂^n can be covered by a ball of radius B /√(2), and hence <ref> follows immediately from <ref>.Furthermore, if T is a triangle with an angle α>135^∘ and diameter a, it is folklore that the circumradius of T is a/(2sinα)>a/√(2). Thus, we obtain <ref> as a corollary of <ref>.To prove <ref>, we show that we can move one vertex of the regular d-simplex by just a little bit to obtain a simplex of circumradius strictly larger than 1/√(2). We will use the elementary geometric fact that the circumradius of a d-dimensional unit simplex is √(d/(2d+2)). Let δ >0, r^2 = 1/2 + δ, a^2 = 1/(2d) +δ, and define H_a := { x ∈^d : x_d = a }. Then B:=B_d(r) ∩ H_a is a (d-1)-dimensional ball of radius √(r^2 - a^2) = √((d-1)/(2d)), and hence there is a (d-1)-dimensional unit simplex A'={p_1, …, p_d} contained in the boundary of B. Finally, let A = A' ∪{p_d+1}, where p_d+1 = (0, …, 0, r).By construction, we have A > 1/√(2), p_i-p_j^2 = 1 for all 1 ≤ i < j ≤ d, and p_i - p_d+1^2 = 1 - 1/√(d) + O(δ) for all 1 ≤ i ≤ d. Hence the theorem follows from <ref> after choosing δ > 0 small enough. § REMARKS In <cit.> it was asked whether there exists an obtuse triangle which is diameter-Ramsey. Although we could not answer this question, we think the answer is no. More generally, we think the following statement is true. A simplex is diameter-Ramsey if and only if its circumcentre is contained in its convex hull. Furthermore, it would be interesting to close the gap between <ref> and the related result from <cit.>. For every d ∈, determine the largest = (d) >0, such that every -almost-regular simplex is diameter-Ramsey. Note that, provided <ref> is true, a similar construction as in the proof of <ref> shows that the result in <cit.> is best possible, i.e. (d) = 1/d2.§ ACKNOWLEDGEMENTThe authors would like to thank their supervisors Peter Allen, Julia Böttcher, Jozef Skokan and Konrad Swanepoel for their helpful comments on this note.amsplain | http://arxiv.org/abs/1708.07373v2 | {
"authors": [
"Jan Corsten",
"Nóra Frankl"
],
"categories": [
"math.CO",
"05D10"
],
"primary_category": "math.CO",
"published": "20170824122911",
"title": "A note on diameter-Ramsey sets"
} |
1Department of Astrophysics, Princeton University, Princeton, NJ 08540, USA2Department of Astronomy and Astrophysics, University of California, Santa Cruz, 1156 High Street, Santa Cruz, CA 95064, USA3European Southern Observatory, Karl-Schwarzschild-Str. 2, D-85741 Garching, Germany4Institute of Cosmology and Gravitation, University of Portsmouth, Dennis Sciama Building, Burnaby Road, Portsmouth PO1 3FX, UK; South East Physics Network5National Astronomical Observatories, Chinese Academy of Sciences, 20A Datun Road, Chaoyang District, Beijing 100012, China6PITT PACC, Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 15260, USA7Apache Point Observatory, P.O. Box 59, Sunspot, NM 883498Department of Physics and Astronomy, University of Utah, 115 S. 1400 E., Salt Lake City, UT 84112, USA9Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA10Institute of Astronomy and Astrophysics, Academia Sinica, Taipei 10617, Taiwan 11School of Physics and Astronomy, The University of Nottingham, University Park, Nottingham, NG7 2RD, UK12Kavli Institute for the Physics and Mathematics of the Universe (WPI), Tokyo Institutes for Advanced Study, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa-shi, Chiba, 277-8583, Japan13Department of Physical Science, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8526, Japan 14Hiroshima Astrophysical Science Center, Hiroshima University, Higashi-Hiroshima, Kagamiyama 1-3-1, 739-8526, Japan 15Core Research for Energetic Universe, Hiroshima University, 1-3-1, Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8526, Japan16Department of Astronomy and Astrophysics, The Pennsylvania State University,University Park, PA 16802 17Institute for Gravitation and the Cosmos, The Pennsylvania State University, University Park, PA 16802 18School of Physical Sciences, The Open University, Milton Keynes,MK7 6AA, UK 19Department of Physics, University of North Carolina, Asheville, NC 28804, USA20Department of Physics and Astronomy, University of Kentucky, 505 Rose Street, Lexington, KY 40506-0057, USA21McDonald Observatory, The University of Texas at Austin, 1 University Station, Austin, TX 78712, USA Probing the kinematic morphology-density relation of early-type galaxies with MaNGA J. E. Greene1, A. Leauthaud2, E. Emsellem3, D. Goddard4, J. Ge5, B. H. Andrews6, J. Brinkman7, J. R. Brownstein8, J. Greco1, D. Law9,Y.-T. Lin10, K. L. Masters4, M. Merrifield11, S. More12, N. Okabe13,14,15, D. P. Schneider16,17, D. Thomas4, D. A. Wake18,19, R. Yan20, N. Drory20 Received: date / Accepted: date =========================================================================================================================================================================================================================================================================================== The “kinematic” morphology-density relation for early-type galaxies posits that those galaxies with low angular momentum are preferentially found in the highest-density regions of the universe. We use a large sample of galaxy groups with halo masses 10^12.5 < M_ halo < 10^14.5 h^-1M_⊙ observed with the Mapping Nearby Galaxies at APO (MaNGA) survey to examine whether there is a correlation between local environment and rotational support that is independent of stellar mass. We find no compelling evidence for a relationship between the angular momentum content of early-type galaxies and either local overdensity or radial position within the group at fixed stellar mass. § INTRODUCTION The angular momentum content of galaxies can serve as a probe of their assembly histories. Although early-type galaxies are dynamically hot systems, many of them show some rotation <cit.>. The evolution in angular momentum of galaxies is influenced in complex ways by mergers (both major or minor), gas accretion, and internal processes such as star formation that either turn gas into stars or expel gas <cit.>. Mergers, for instance, might increase or decrease the angular momentum depending on the configuration <cit.>. Tracking the evolution in angular momentum content with galaxy properties is a tracer of the factors that dominate galaxy evolution. It is standard to trace the galaxy angular momentum using the ratio of velocity to dispersion support. In the context of integral-field spectroscopy (IFS), a luminosity-weighted two-dimensional measurement (λ_R) is used as a proxy for the angular momentum content.We adopt the definition of λ_R from <cit.>, with R the flux-weighted radial coordinate, V the radial velocity, and σ the stellar velocity dispersion: λ_R = ⟨ R | V | ⟩/⟨ R √(V^2 + σ^2)⟩. Slowly rotating galaxies are those that fall below the expectations for a mildly anisotropic oblate rotator <cit.>. Stellar mass is the primary determinant of whether a galaxy is a slow rotator <cit.>. Secondary correlations with environment may reveal the physical processes that determine the distribution of angular momentum in galaxies.Early studies examined the relationship between angular momentum as traced by λ_R and environment by measuring the fraction of galaxies with lowas a function of local overdensity. <cit.> showed that a tiny fraction (<5%) of early-type galaxies in low-density environments are slow rotators, and tied this low fraction to a kinematic version of the morphology-density relation <cit.>. The fraction of early-type slowly rotating galaxies indeed rises dramatically in the densest environments <cit.>. As examples, we present measurements from <cit.> and <cit.> in Figure <ref> (upper left).Because massive galaxies preferentially reside in overdense regions, it is difficult to determine from these small early samples whether or not environment plays a role independent of mass. Two larger IFS surveys have recently published relevant studies. <cit.> examined a mass-selected sample of galaxies from MASSIVE <cit.>. The majority of these galaxies are central galaxies living in a wide range of halo masses. <cit.> do not find a compelling environmental dependence ofat fixed M_*. <cit.> examine eight clusters observed by SAMI <cit.>. By number, their sample is dominated by satellite galaxies, containing only eight brightest-cluster galaxies. They also argue that trends with local overdensity can be explained by the strong dependence on stellar mass. We present a bridge between the MASSIVE and SAMI samples. We exploit the large number of IFS cubes afforded by the MaNGA survey <cit.>, part of the Sloan Digital Sky Survey IV <cit.>, to build a sample of galaxies withmeasurements <cit.>. Roughly 2/3 of the galaxies in our MaNGA sample are central galaxies, spanning a wide range in inferred host halo mass (and thus environment). We consider the kinematic morphology-density relation of early-type galaxies using a number of different complementary probes of global and local environment, including halo mass, designation as central or satellite, local overdensity, and radial location in the cluster. We assume a flat ΛCDM cosmology with Ω_ m=0.238, Ω_Λ=0.762, H_0=100 h^-1 km s^-1 Mpc^-1, in order to follow the convention of our group catalog <cit.>. Halo masses are defined as M_200b≡ M(<R_200b)=200ρ̅4/3π R_200b^3 where R_200b is the radius at which the mean interior density is equal to 200 times the mean matter density of the universe (ρ̅). Stellar mass is denoted M_* and has been derived using a Chabrier Initial Mass Function (IMF). We use units in which h=1.§ SAMPLE AND DATA§.§ The MaNGA Survey MaNGA will ultimately obtain integral-field spectroscopy of 10,000 nearby galaxies with the 2.5m Sloan Foundation Telescope <cit.> and the BOSS spectrographs <cit.>. Fibers are joined into 17 hexagonal fiber bundles to perform a multi-object IFS survey <cit.>. Each fiber has a diameter of 2, while the bundles range in diameter from 12 to 32 with a 56% filling factor. The BOSS spectrographs have a wavelength coverage of 3600-10,300Å and a spectral resolution of σ_r ≈ 70 . The relative spectrophotometry is accurate to a few percent <cit.>. The survey design is described in <cit.>, the observing strategy in <cit.>, and the data reduction pipeline in <cit.>. The MaNGA sample is selected from the NASA-Sloan Atlas (NSA; Blanton M. http://www.nsatlas.org) with redshifts mostly from the SDSS Data Release 7 MAIN galaxy sample <cit.>. The sample is built in i-band absolute magnitude (M_i)-complete shells, with more luminous galaxies observed in more distant M_i shells such that the spatial coverage (in terms of R_e) is roughly constant across the sample <cit.>. We focus on the combination of the Primary sample (∼ 50% of the total sample) and Secondary sample (∼ 40% of the total sample), selected such that 80% of the galaxies in each M_i shell can be covered to 1.5 R_e (2.5 R_e) by the largest MaNGA IFU for the Primary (Secondary) sample, respectively. To compare different stellar mass bins, it is necessary to reweight the galaxy distributions in the sample to account for the different volumes probed by each mass shell. We apply these volume weights whenever population means are presented. §.§ Galaxy Sample We work with the MaNGA data derived from the data-release pipeline (DRP) v2.0.1 (MaNGA Product Launch [MPL] 5) sample that are also in the <cit.> group catalog (Y07; updated to DR7).Y07 use an iterative, adaptive group finder to assign galaxies to halos. Briefly, they first use a friends-of-friends algorithm to identify potential groups, followed by abundance matching to assign likely halo masses to each group based on the total galaxy luminosity. They iterate their group selection based on the initial estimate for halo mass. We select the central galaxy as the most luminous one, while all other galaxies are designated as satellite galaxies. We select central and satellite galaxies that reside in halos more massive than 10^12.5 h^-1 M_⊙ <cit.> and additionally require that the satellite galaxies have stellar masses M_*>10^10 h^-2 M_⊙. We also visually remove all galaxies with spiral structure, focusing only on early-type (E/S0) galaxies (for details see Paper I). The final sample comprises 379 early-type centrals with a minimum stellar mass of 3 × 10^10 and a median stellar mass of 10^11 . There are 159 early-type satellite galaxies with a minimum stellar mass of 10^10 and a median stellar mass of 3 × 10^10 .§.§ Identifying Slow Rotators The kinematic measurements we use to derivecome from the MaNGA Data Analysis Pipeline (DAP), and are measured on Voronoi-binned data <cit.> with a signal-to-noise ratio of at least 10 per 70 km s^-1 spectral pixel. The kinematics are measured using the penalized pixel-fitting code pPXF <cit.>, with emission lines masked. The stellar templates are drawn from the MILES library<cit.> and are convolved with a Gaussian line-of-sight velocity distribution to derive the velocity and velocity dispersion of the stars. The velocity dispersions are reliable above σ>40 <cit.>, while the small number of unresolved spaxels are removed from analysis. The central σ values range from 50 to 400 , with only 43 galaxies having dispersions <100 for the sample, so we are not working in this low dispersion regime. An eighth-order additive polynomial is included to account for flux calibration and stellar population mismatch. We adopt the “DONOTUSE” flags from the DAP (meaning that there are known catastrophic problems with these data), and flag all bins with σ >500 or V>400 . We only keep galaxies for which at least 50% of their spaxels are unflagged.We calculatewithin elliptical isophotes. We adopt the position angle and galaxy flattening ϵ = 1-b/a from the NASA Sloan Atlas <cit.>[http://www.sdss.org/dr13/manga/manga-target-selection/nsa/]. While it is standard in the literature to compare galaxies at , we show in Paper I that at the spatial resolution of MaNGA these measurements can be biased by 10-50%, depending on the input , with lower values ofsuffering more severely. As described in detail in Paper I, we adopt the outermost measurement of().matches λ(1.5 R_e) with ∼ 20% scatter and no bias, but allows us to include galaxies with limited radial coverage. Finally, the inner 2 of data are excluded, since the low spatial resolution of MaNGA tends to lower λ_R. In Paper I, we use simulations to show that excluding the central region brings the measured λ_R value closer to the true value, and we estimate that the residual impact of low spatial resolution leads to at most a systematic increase in slow-rotator fraction of 10% from our measured values.To determine whether a galaxy is a slow or fast rotator further requires comparison with the intrinsic shape of the galaxy, since the amount of rotation needed to support an oblate galaxy rises with ellipticity. <cit.> report an empirically motivated division between slow and fast rotators of <0.31 √(ϵ). For measurements at λ(1.5 R_e), we simply scale by the typical ratio of /λ(1.5R_e) from our data, to define slow rotators as those with <0.35 √(ϵ).§ THE RELATIONSHIP BETWEEN Λ AND ENVIRONMENT We now demonstrate that at fixed M_* there is no residual dependence ofon either local or global environment. We discuss the different measures of environment used in the analysis, describe how we normalize for M_*, and examineas a function of local overdensity at fixed M_*. §.§ Measures of Environment In Paper I we examinedin bins of M_200b, finding no evidence for any residual dependence ofon halo mass M_200b at fixed M_*. Nor was there any significant evidence for differences between central and satellite galaxies at fixed mass. Therefore, in this paper we do not reconsider M_200b or the satellite/central distinction. We do note, however, that there is considerable scatter in the halo masses and the central designations in group catalogs <cit.>. It is useful, therefore, to consider local overdensity measures as a complement to the group-based measures of environment.The first papers investigating links betweenand environment focused on a local overdensity measurement <cit.>. The local overdensity measurement employed here (Σ_n) is simply the number of galaxies (in our case n=5) with M_r>-20.3 mag, divided by the volume required to enclose that number of neighbors above the magnitude limit. We adopt the Σ_5 measurements from <cit.>. Σ_n is a complex measure of environment. For large overdensities, with more than three to five members, the region of measurement falls within the parent halo. However, when the number of members of a group approaches the overdensity measure, the distance to the n^ th nearest neighbor may fall in a different halo. In this case, the measure becomes more sensitive to very large scale clustering than to local overdensity <cit.>. Furthermore, the physical processes impacting galaxies at group centers (e.g., merging) may be different than those dominating the satellite galaxies living further from the group center (e.g., stripping). We therefore prefer using the radial group position, R/R_200b, which ensures only intra-group comparisons are made. For comparison with previous literature, we examine both measures.§.§ Controlling for stellar mass We employ a simple mass-weighting scheme to ensure flat mass distributions in all environmental bins. In each environment bin, we inversely weight each galaxy based on the number of galaxies at that mass, such that the final mass distribution is flat. We do this in two mass bins, divided at the median stellar mass. We do not employ the MaNGA weights here, since we are looking at renormalized mass distributions.Throughout the rest of the paper, we will display the weighted slow-rotator fractions as a function of Σ_n and R/R_200b. In these figures, the bin sizes are chosen to ensure a constant number of objects per bin, and are plotted at the weighted bin center. The shaded regions indicate the error in the mean, derived via bootstrapping. §.§ Trends with Σ_5 We first examine whether there is a trend betweenand local overdensity at fixed M_*. Figure <ref> (top left) presents two clusters that illustrate the range of results from individual studies of clusters, and the slow-rotator fraction is shown for our full sample of satellite and central galaxies. There is no strong trend with local overdensity in our data, which are roughly consistent with the Virgo results in a similar Σ range (although the Virgo results employ Σ_3 rather than Σ_5). We see a rising slow rotator fraction at low Σ_5. This rising fraction reflects the increasing dominance of higher-mass central galaxies at low overdensity in our sample. Figure <ref> (top right) shows that there is a trend (albeit weak) between local overdensity and stellar mass as satellites grow more dominant at higher overdensity.To garner reasonable results, we must examine the trends with overdensity at fixed mass. Note the full relationship between stellar mass and local overdensity is complicated for satellites <cit.>, and is not fully probed by our early-type, high-mass sample.In Figure <ref> (bottom left), we impose mass-weighting. While higher-mass galaxies show a higher slow-rotator fraction, no residual trend is seen with Σ_5 for either bin. Since slow-rotator fraction is assigned as a binary (noisy) division, it is useful to examine the full distribution of /√(ϵ) (Figure <ref>; bottom right). Again, we see a split in samples based on stellar mass, but no residual trend in /√(ϵ) as a function of Σ_5. We thus turn to examineas a function of R/R_200b. §.§ Trends with R/R_200b R/R_200b is complementary to Σ_5, since the radial distance is always measured within the group halo, while the local overdensity can extend to neighboring halos at low density. Figure <ref> (top left) presents the slow-rotator fraction as a function of R/R_200b for all satellites in the sample. The central galaxies by construction are found at the group center, and have a 40% slow-rotator fraction averaged over all masses. There is no compelling trend in slow-rotator fraction as a function of radius. In terms of mass, the central galaxies are more massive than the satellites, but there is not a strong mass segregation within the satellite galaxies (Figure <ref>, top right). The true radial mass trend is washed out by our narrow range in stellar mass and morphology <cit.>.We then consider the mass-weighted slow-rotator fraction as a function of radius (Figure <ref>; bottom left). Again, the high- and low-mass–weighted samples have higher and lower slow-rotator fractions, respectively, but there is no compelling additional evidence for a radial trend in slow-rotator fraction when we control the distribution in stellar mass. We similarly see a very flat mean /√(ϵ) as a function of radius (Figure <ref>; bottom right).In short, there is no evidence that large-scale or local environment plays a driving role in the distribution of , for an array of environmental measures. Of course, it is possible that a stronger radial trend might be apparent if we could consider only the most massive halos; such analysis required larger samples and should be possible with MaNGA soon.§ DISCUSSION & SUMMARY Combining the results from both local environmental indicators, we conclude that angular momentum content at fixed stellar mass is not influenced at a level that we can detect here by the local density or by the radial position within the halo. In Paper I we argued that angular momentum at fixed mass was not influenced by the halo mass or central/satellite distinction. Thus, we conclude that local processes (accretion, star-formation, and merging) determine the angular momentum content of early-type galaxies. This result is in accord with recent results from Illustris <cit.> suggesting that only major mergers (not minor ones) have the capacity to significantly alter λ_R in the most massive galaxies, while accretion and consumption of gas can alter λ_R in lower-mass galaxies. Thus, lower-mass galaxies are spun up by gaseous processes, while massive galaxies are not. In this scenario, stellar mass is the dominant factor setting λ_R, since even if the total level of accretion and merging is higher in denser environments, stellar mass determines whether λ_R is impacted at all by external factors at late times.Our findings are in accord with two more recent studies <cit.>. We suggest that stellar mass dependence also drives earlier results from individual clusters that found evidence for a rising slow-rotator fraction towards the most overdense regions <cit.>. Our work fits into a broader conversation about the role of environment in establishing internal galaxy properties. Galaxy mass functions are a function of environment <cit.>. Once stellar mass is controlled, however, there are only subtle remaining differences as a function of environment for many internal galaxy properties, including morphology and color <cit.>, star formation rates <cit.>, stellar populations <cit.> and gradients therein <cit.>. As the MaNGA survey progresses, the larger sample size will enable yet more sensitive searches for subtle trends between environment, stellar kinematics, gas content, and stellar populations.J.E.G. is partially supported by NSF AST-1411642. We thank M. Cappellari foruseful discussions. We thank the referee for constructive comments that improved this manuscript and we thank Ada Elizabeth Greene Silverman for waiting an extra week to come into this world while her mother submitted the initial manuscript. Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS-IV acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. The SDSS web site is www.sdss.org.SDSS-IV is managed by the Astrophysical Research Consortium for theParticipating Institutions of the SDSS Collaboration including theBrazilian Participation Group, the Carnegie Institution for Science,Carnegie Mellon University, the Chilean Participation Group, the French Participation Group, Harvard-Smithsonian Center for Astrophysics,Instituto de Astrofísica de Canarias, The Johns Hopkins University,Kavli Institute for the Physics and Mathematics of the Universe (IPMU) /University of Tokyo, Lawrence Berkeley National Laboratory,Leibniz Institut für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA Heidelberg),Max-Planck-Institut für Astrophysik (MPA Garching),Max-Planck-Institut für Extraterrestrische Physik (MPE),National Astronomical Observatories of China, New Mexico State University,New York University, University of Notre Dame,Observatário Nacional / MCTI, The Ohio State University,Pennsylvania State University, Shanghai Astronomical Observatory,United Kingdom Participation Group, Universidad Nacional Autónoma de México, University of Arizona,University of Colorado Boulder, University of Oxford, University of Portsmouth,University of Utah, University of Virginia, University of Washington, University of Wisconsin,Vanderbilt University, and Yale University.This research made use of Marvin, a core Python package and web framework for MaNGA data, developed by Brian Cherinka, Jose Sanchez-Gallego, and Brett Andrews (MaNGA Collaboration, 2017).52 natexlab#1#1[Abazajian et al.(2009)]abazajianetal2009 Abazajian, K. 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A., Weijmans, A.-M., Xiao, T., & Zhang, K. 2016b, , 151, 8[Yang et al.(2009)Yang, Mo, & van den Bosch]yangetal2009 Yang, X., Mo, H. J., & van den Bosch, F. C. 2009, , 695, 900[Yang et al.(2007)Yang, Mo, van den Bosch, Pasquali, Li, & Barden]yangetal2007 Yang, X., Mo, H. J., van den Bosch, F. C., Pasquali, A., Li, C., & Barden, M. 2007, , 671, 153 | http://arxiv.org/abs/1708.07843v1 | {
"authors": [
"J. E. Greene",
"A. Leauthaud",
"E. Emsellem",
"D. Goddard",
"J. Ge",
"B. H. Andrews",
"J. Brinkman",
"J. R. Brownstein",
"J. P. Greco",
"D. Law",
"Y. -T. Lin",
"K. L. Masters",
"M. Merrifield",
"S. More",
"N. Okabe",
"D. P. Schneider",
"D. Thomas",
"D. A. Wake",
"R. Yan",
"N. Drory"
],
"categories": [
"astro-ph.GA"
],
"primary_category": "astro-ph.GA",
"published": "20170825180014",
"title": "Probing the kinematic morphology-density relation of early-type galaxies with MaNGA"
} |
Chordality of Clutters with Vertex Decomposable Dual and Ascent of Clutters Ashkan Nikseresht[While this paper was under review, the author's affiliation has beenchanged to: Department of Mathematics, Shiraz University, 71457-13565, Shiraz, Iran] Department of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS),P.O. Box 45195-1159, Zanjan, Iran.E-mail: [email protected]==================================================================================================================================================================================================================================================================================================================================================In this paper, we consider the generalization of chordal graphs to clutters proposed by Bigdeli, et al inJ. Combin. Theory, Series A (2017). Assume thatis a d-dimensional uniform clutter. It is known that ifis chordal, then I() has a linear resolution over all fields. The converse has recently been rejected, but the following question which poses a weaker version of the converse is still open: “if I() has linear quotients, isnecessarily chordal?”. Here, by introducing the concept of the ascent of a clutter, we split this question into two simpler questions and present some clues in support of an affirmative answer. In particular, we show that if I() is the Stanley-Reisner ideal of a simplicial complex with a vertex decomposable Alexander dual, thenis chordal. Keywords and Phrases:chordal clutter, linear resolution, vertex decomposable simplicial complex, ascent of a clutter, squarefree monomial ideal.2010 Mathematical Subject Classification: 13F55, 05E40, 05C65. § INTRODUCTION In this paper all rings are commutative with identity, K is a field and S=K[x_1,…, x_n] is the polynomial ring in n indeterminates over K. Given an arbitrary ideal I of S, we can get a monomial ideal by taking its initial ideal or its generic initial ideal with respect to some monomial order. Many of the properties of I is similar or at least related to its (generic) initial ideal, especially if I is graded (see, for example, <cit.>). Also if I is a monomial ideal of S then its polarization is a squarefree monomial ideal, again with algebraic properties similar to I (see <cit.>). Therefore, if we know the algebraic properties of squarefree monomial ideals well, then we can understand many of the algebraic properties of much larger classes of ideals.On the other hand, squarefree monomial ideals have a combinatorial nature and correspond to combinatorial objects such as simplicial complexes, graphs, clutters and posets. Many researchers have tried to characterize algebraic properties of squarefree monomial ideals, using their combinatorial counterparts, see for example <cit.> and Part III of <cit.> and their references. In this regard, two important objectives are to classify combinatorially squarefree monomial ideals which are Cohen-Macaulay or those which have a linear resolution (either over every field or over a fixed field). Indeed these two tasks are equivalent under taking Alexander dual.A well-known theorem of Fröberg states that if I is a squarefree monomial ideal generated in degree two, then I has a linear resolution I is the edge ideal of the complement of a choral graph. Motivated by this, many have tried to generalize the concept of chordality to clutters or simplicial complexes in such a way that at least one side of Fröberg's theorem stays true in degrees >2 (see, for instance, <cit.>). One of the most promising such generalizations seems to be the concept of chordal clutters presented in <cit.>. In <cit.> it is shown that many other classes of “chordal clutters” defined by other researchers, including the class defined in <cit.> which we call W-chordal clutters, is strictly contained in the class defined in <cit.>. They also show that if a clutter is chordal, then the circuit ideal of its complement has a linear resolution over every field. Several clues were presented in <cit.> supporting the correctness of the converse (see <cit.>). But recently a counterexample to the converse was presented by Eric Babson (see Example <ref>). Despite this it is still unknown whether the following statement which is a weaker version of the converse is true or not: “if the circuit ideal of the complement of a clutter has linear quotients, then that clutter is chordal.” Moreover, in <cit.>, it is shown that many numerical invariants of the ideal corresponding to a chordal clutter can be combinatorially read off the clutter.Here, after presenting a brief review of the main concepts and setting the notations, in Section 3 we prove that if the Alexander dual of the clique complex of a clutteris vertex decomposable, thenis chordal. As clique complexes of W-chordal clutters are vertex decomposable, this generalizes the results of Subsection 3.1 of <cit.>. Then in Section 4, we define the notion of the ascent of a clutter and show how we can use this concept to divide the question “ischordal, given that the circuit ideal of the complement ofhaslinear quotients?” into two simpler questions.§ PRELIMINARIES, NOTATIONS AND A COUNTEREXAMPLE Algebraic background. Suppose that I is a graded ideal of S considered with the standard grading. This grading induces a natural grading on _i^S(K,I). We denote the degree j part of _i^S(K,I) by _i^S(K,I)_j and its dimension over K is denoted by β_ij^S(I)= β_ij(I). These β_ij's are called the graded Betti numbers of I. If there is a d≥ 0 such that β_ij(I)=0 for j ≠ i+d, it is said that I has a linear (or d-linear) resolution. Also we say that I has linear quotients with respect to an ordered system of homogenous generators f_1, …, f_t, if for each i the ideal f_1, …, f_i-1: f_i is generated by linear forms. It is known that if I is generated in degree d and has linear quotients with respect to some system of homogenous generators, then I has a d-linear resolution (see <cit.>).Now assume that I is a squarefree monomial ideal, that is, I is generated by some squarefree monomials. Then I has a unique smallest generating set consisting of squarefree monomials, say x_F_1, …, x_F_t where x_F=∏_i∈ F x_i for F [n]={1,…,n}. In this case, if I has linear quotients with respect to a permutation of this minimal system of generators, we simply say that I has linear quotients. Also a specific order of minimal generators of I, for which the aforementioned colon ideals are linear, is called an admissible order. According to <cit.>, x_F_1, …, x_F_t is an admissible order for the ideal they generate, for each i and all j<i, there is a l∈ F_j F_i and a k<i such that F_k F_i={l}. For more details on these algebraic concepts see <cit.>. Clutters. A clutteron the vertex set V= () is a family of subsets of V which are pairwise incomparable under inclusion. We call the elements ofcircuits. For any subset F of V we set F=|F|-1. If all circuits ofhave the same dimension d, we say thatis a d-dimensional uniform clutter or a d-clutter for short. If v∈(), then by -v we mean the clutter on (){v} with circuits {F∈|v∉ F}. For simplicity, we write for example ab, Ex or Eab instead of {a,b}, E∪{x} or E∪{a,b} for a,b,x∈ V and E V. If we assume that V=[n], then I() is defined as the ideal of S generated by {x_F|F∈}.From now on, we always assume thatis a d-clutter with |()|=n.We callcomplete, when all d-dimensional subsets of V=() are in . A clique ofis a subset A of V such that _A is complete, where _A={F∈|F A} is the induced d-clutter on the vertex set A. The complementofis the d-clutter with the same vertex set asand circuits {F V|F=d, F ∉}.The set {e V| e=d-1, ∃ F∈ e F} is called the set of maximal subcircuits ofand is denoted by (). For a (d-1)-dimensional subset e of V, the closed neighborhood N_[e] is defined as e∪{v∈ V| ev∈}. If e∈(), then -e means the clutter on the vertex set V with circuits {F∈|eF}.A maximal subcircuit e ofis called a simplicial maximal subcircuit, when N_[e] is a clique. We denote the set of all simplicial maximal subcircuits ofby (). If for each 1≤ i< t there is an e_i∈(_i-1), where _0= and _i= _i-1-e_i, such that _t has no circuits, thenis called chordal (see <cit.>). In the case that d=1 (that is,is graph) this notion coincides with the usual notion of chordal graphs. Theorem 3.3 of <cit.> states that ifis chordal and not complete, then I() has a linear resolution over every field. There is an example showing that the converse is not true, that is, there is a non-chordal clutterwith I() having a linear resolution over every field (see Example <ref>). But it is still unknown whether there is a non-chordal clutterwith I() having linear quotients. For further reference, we label this statement:enumi*If I() has linear quotients, thenis chordal.enumiSimplicial complexes. A simplicial complex Δ on the vertex set V=(Δ) is a family of subsets of V (called faces of Δ) such that if A B∈Δ, then A∈Δ. The dimension of Δ is defined as Δ= max_F∈Δ F. The set of maximal faces of Δ which are called facets is denoted by (Δ). If |(Δ)|=1, then Δ is called a simplex.If all facets of Δ have the same dimension, we say that Δ is pure. In this case (Δ) is a d-dimensional uniform clutter. Also ifis a clutter, thendenotes the simplicial complex Δ with (Δ)=. It should be clear that (Δ) = Δ and () =, for any simplicial complex Δ and any (not necessarily uniform) clutter . Another simplicial complex associated to a d-clutteris the clique complex Δ() ofdefined as the family of all subsets L of () with the property that L is a clique in . Note that all subsets of () with size ≤ d are cliques by assumption.For a face F of Δ, we define _Δ F= {G F|F G∈Δ}, which is a simplicial complex on the vertex set V F. Also if v∈ V, Δ-v is the simplicial complex on the vertex set V{v} with faces {F∈Δ|v∉ F}.Assuming that (Δ)=[n], the ideal of S generated by {x_F|F is a minimal non-face of Δ} is called the Stanley-Reisner ideal of Δ and is denoted by I_Δ. When SI_Δ is a Cohen-Macaulay ring, Δ is said to be Cohen-Macaulay over K.Let A be a commutative ring with identity and -1≤ d≤Δ and denote by C_d(Δ)= C_d(Δ, A) the free A-module whose basis is the set of all d-dimensional faces of Δ. Consider the A-homomorphism _d: C_d(Δ) →C_d-1(Δ) defined by _d({v_0, …, v_d}) =∑_i=0^d (-1)^i {v_0, …, v_i-1,v_i+1,…, v_d},where v_0<⋯ <v_d for a fixed total order < on (Δ). Then (C_, _) is a complex of free A-modules and A-homomorphisms called the augmented oriented chain complex of Δ over A. We denote the i-th homology of this complex by H_i(Δ; A).By the Alexander dual of a simplicial complex Δ we mean Δ^∨= {(Δ)F| F(Δ) , F∉Δ} and also we set ^∨={() F| F∈}. Then it follows from the Eagon-Reiner theorem (<cit.>) and the lemma below that I() has a linear resolution over K, ^∨ is Cohen-Macaulay over K. For more details on simplicial complexes and related algebraic concepts the reader is referred to <cit.>. We frequently use the following lemma in the sequel without any further mention. Letbe a d-clutter. Then * I()=I_Δ();* ^∨= (Δ())^∨.When working with both clutters and simplicial complexes, one should notice the differences between the similar concepts and notations defined for these objects. For example, we say that a d-dimensional simplicial complex Δ is i-complete to mean that Δ has all possible faces of dimension i, where i≤ d. But when we are talking about a d-clutter , there is no need to mention the dimension and say thatis i-complete, since we must have i=d which is implicit inand it is completely meaningless to say thatis i-complete for i<d. Therefore we just say thatis complete. Another difference arises from the concept of Alexander dual which is illustrated in the following example.Let ={125,235,345} be a 2-clutter on [5], Γ= and Δ=Δ(). Note that 14 ∉Γ because it is not contained in any facet of Γ. But by definition, 14 is a clique ofand hence 14∈Δ={125,235,345,13,14,24}. Now for example 245∈ and hence [5] 245=13∈^∨. Indeed, ^∨={13,15,23,24,25,35,45}, which by <ref> is equal to the set of facets of Δ^∨. But as 14∉Γ, we see that [5] 14= 235∈Γ^∨. In fact, ^∨= Γ^∨= {235,245,135}≠^∨.Recall that Δ_W={F∈Δ|F W} where W V.The ideal I_Δ has a t-linear resolution over K, H_i(Δ_W; K)=0 for all W(Δ) and i≠ t-2.Note that in the case that Δ= Δ(), then since all possible faces of dimension less than d are in Δ, it follows that H_i(Δ_W; K)=0 for all W and i<d-1. Hence I() has a (d+1)-linear resolution over K, H_i(Δ_W; K)=0 for all W(Δ) and i≥ d. Using this we can state an example of a non-chordal clutterwith I() having a linear resolution over every field. This example is due to Eric Babson who visited IASBS as a lecturer in the first Research School on Commutative Algebra and Algebraic Geometry in 2017.[Babson] Letbe the 2-clutter shown in Fig-Dunce which is a triangulation of the dunce hat. Then Δ= Δ() is obtained by adding the missing edges (1-dimensional subsets of vertices) to Δ'=, sincehas no cliques on more that 3 vertices. Therefore, H_t(Δ; K)=H_t(Δ'; K) for every t≥ 2. It is known that the dunce hat is a contractible space and hence 0=H_t(Δ'; K) for all t≥ 2. Also it is easy to verify that for any W [8], _W is chordal (indeed, any proper subclutter ofhas an edge which is contained in exactly one triangle and hence is simplicial) and thus has a linear resolution over every field. Therefore, H_t(Δ_W; K)=H_t(Δ(_W); K)=0 for all t≥ 2. Consequently, by Föberg's theorem I_Δ= I() has a linear resolution over every field.§ CLUTTERS WITH VERTEX DECOMPOSABLE DUAL ARE CHORDALA vertex v of a nonempty simplicial complex Δ is called a shedding vertex, when no face (or equivalently facet) of _Δ(v) is a facet of Δ-v.Recall that a nonempty simplicial complex Δ is called vertex decomposable, when either it is a simplex or there is a shedding vertex v∈ V(Δ) such that both _Δ v and Δ- v are vertex decomposable. This concept was first introduced in <cit.> in connection with the Hirsch conjecture which has applications in the analysis of the simplex method in linear programming. Suppose that Δ is a pure d-dimensional simplicial complex. Then v is a shedding vertex Δ-v is pure with (Δ-v)=d. (): Assume that Δ-v is not pure or (Δ-v)≠ d. Since every facet of Δ-v has dimension d-1 or d, there is a facet F of Δ-v such that F= d-1.As F∈Δ and Δ is pure, we see that F F' for some F'∈(Δ). Hence F'=Fv∈Δ and F∈_Δ(v), which contradicts the shedding property for v.(): Every face of _Δ(v) has dimension ≤ d-1 and by assumption Δ-v is pure and of dimension d. Thus _Δ(v) ∩(Δ-v)= and v is a shedding vertex.Suppose that Γ is as in Example dual-exam. By choosing 1 and then 4 as the shedding vertex, we see that Γ is vertex decomposable. Here Γ-5= 12, 23, 34 is pure with dimension 1= (Γ)-1 and Γ-2= 15, 345 is not pure. So by the previous lemma, neither 5 is a shedding vertex of Γ nor 2. Also Γ^∨ which is indeed isomorphic to Γ is vertex decomposable. Now assume that Δ= 12,34 which is a 1-dimensional simplicial complex. Then Δ-1= 2, 34 is not pure, so 1 is not a shedding vertex of Δ. Similarly we see that Δ has no shedding vertex and is not vertex decomposable. It is well-known that if Δ is pure and vertex decomposable, then it is shellable and Cohen-Macaulay (see <cit.>), hence I_Δ^∨ has a linear resolution and in fact linear quotients. So ifis the clutter with I()=I_Δ^∨ and if statement <ref> is true,should be chordal. In this section, we prove that this is indeed the case. By <ref>, in the above situation we have Δ= (Δ()) ^∨=^∨. First we need some lemmas. In the sequel, we assume thatis a d-clutter and |()|=n. Also recall that the pure i-skeleton of a simplicial complex Δ, is the simplicial complex whose facets are i-dimensional faces of Δ. Letbe a d-clutter. Suppose that Δ= and Γ=^∨. Also assume that v∈(Γ) (=(Δ)) and Γ-v is pure of dimension =(Γ) (that is, v is a shedding vertex of Γ) and set = (_Δ(v)). Then *_Δ(v) is the pure (d-1)-skeleton of (Γ-v)^∨;*^∨= Γ-v;*e∈() ev∈().<ref> Suppose that F∈ (Γ-v)^∨ and F=d-1. Then A=(Γ-v) F∉Γ-v and hence A∉Γ and A∉^∨. But (A)=(n-1)- (F)-2= n-d-2=(Γ)= (^∨). Thus Fv= (Δ) A∈, that is, F is a facet of _Δ(v). So the pure (d-1)-skeleton of (Γ-v)^∨ is contained in _Δ(v). The proof of the reverse inclusion is similar.<ref> Noting that ((Γ-v))^∨ is exactly the set of (d-1)-dimensional faces of (Γ-v)^∨, we see that part <ref> indeed states = ((Γ-v))^∨, which is equivalent to <ref>.<ref> Assume that e∈(). Then for x∈() e we have ex∈ exv∈(Δ)=. Therefore, _[e]=_[ev]{v}. If ev∈() and A is a (d-1)-dimensional subset of _[e], then Av_[ev] which is a clique and hence Av∈, that is, A∈. So _[e] is a clique and e∈().Conversely, assume that e∈() and A is a d-dimensional subset of _[ev]. We must show A∈. If v∈ A, then A{v}_[e] and hence A{v}∈, which means A∈. Thus suppose v∉ A. If A∉, then v∈ B=() A∈^∨=(Γ). Consequently, B{v}∈Γ-v and there is a F∈(Γ-v)=^∨ containing B{v}. So A'= () F∉. Since (Γ-v)=Γ= n-d-2= |F|-1 and ()=(){v}, we see that |A'|=d. Also A' A_[ev]{v}=_[e], and as e is simplicial, we get A'∈, a contradiction from which the result follows.Suppose thatis a 2-clutter on [6] with circuits 123, 124, 134, 234, 345, 346, 126 and Δ, Γ andare defined as in <ref>. ThenΓ= ^∨= 136, 146, 236, 246, 346, 135, 235, 145, 245, 123, 124,134,234andΓ-6=135, 235, 145, 245, 123, 124,134,234 .So 6 is a shedding vertex of Γ. Note that (Γ-6) is a 3-clutter on [5], thus (Γ-6)^∨ =12, 34 which is exactly _Δ(6). Also = {12,34} is indeed a graph and ()={1,2,3,4}. As claimed in the previous lemma, simplicial maximal subcircuits ofwhich contain the vertex 6 are 16, 26, 36,46.Letbe a d-clutter. Assume that Δ= and Γ=^∨. Let v be a shedding vertex of Γ and v∈ e∈(). Then v is a shedding vertex of (-e)^∨= (Δ(-e))^∨. Furthermore, if Δ'=-e, then (_Δ' v)= (_Δ v)-e', where e'=e{v}. Let Γ'= (-e)^∨ and suppose that F∈(_Γ' v). We have to show that F is not a facet of Γ'-v. Note that ^∨ (-e)^∨ and hence Γ-vΓ'-v. Therefore, it suffices to show that there is a G∈(Γ)=^∨ with v∉ G such that F G. We know that Fv∈(Γ')=(-e)^∨, so Fv∉-e where=(-e)=(). As v∈ e, e Fv and thus Fv∉. It follows that Fv∈Γ, that is, F∈_Γ v. No face of _Γ v is a facet of Γ-v, because v is a shedding vertex of Γ. Consequently, F is strictly contained in the facet G of Γ-v, as claimed. The proof of the “furthermore” statement, which follows from definitions, is left to the reader.Let , Δ, Γ and v=6 be as in Example vdec-ex and set e=26, Γ'= (-e)^∨ and Δ'=-e. Then -e={126} and (-e)^∨= ^∨∪{345} and the facets of Γ' (resp. Γ'-6) are obtained by adding the face 345 to the set of facets of Γ (resp. Γ-6). Thus Γ'-6 is pure and of dimension 2 and hence 6 is still a shedding vertex in Γ'. Also the only facet of _Δ(6) which does not contain the vertex 2= e{v} is 34 and _Δ'(6)= 34. Suppose that v∈() has not appeared in any circuit of . Then chordality (and many other properties) ofand -v are equivalent, although ()≠(-v). In this case, we misuse the notation and write =-v. Letbe a d-clutter. Assume that Δ=, v is a shedding vertex of Γ=^∨ and =(_Δ v). Ifis chordal, then there is a sequence e_1, …, e_t with e_i∈(_i-1), where _0= and _i=_i-1-e_i, such that _t=-v. We prove the statement by induction on ||. If =, then =-v and the claim holds trivially. So assume ≠ and e'∈() be such that -e is chordal. By <ref>, e=e'v∈(). Let Δ'=-e and '=(_Δ'v), then according to <ref>, '=-e' is chordal and v is a shedding vertex of (-e)^∨. Thus by applying the induction hypothesis on -e, the assertion follows.Let's use the notations of Examples vdec-ex and vdec-ex2. We saw that ={12, 34}, which is a chordal graph with 2∈() and 4∈(-2). Now we see that 26∈() and 46∈(-26) and in -26-46 the vertex 6 has not appeared in any circuit. In our notations, this means that -26-46=-6, as asserted in the previous lemma.Suppose thatis obtained from a complete clutter by deleting exactly one circuit. Thenis chordal. Let =_0-F, where _0 is a complete clutter and F∈_0. By <cit.>, _0 is chordal and there is a sequence of simplicial maximal subcircuits e_1,…, e_t which sends _0 to the empty clutter. By symmetry of _0, e_1 can be any maximal subcircuit of _0 and we can assume e_1 F. Hence -e= _0-e is chordal and it is easy to see that e∈(), that is,is also chordal. Now we are ready to state the main result of this section.Assume thatis a d-clutter and ^∨ is vertex decomposable. Thenis chordal. We use induction on the number of vertices of . Let Δ= and Γ= ^∨. If Γ is a simplex, thenis obtained from a complete clutter by deleting one circuit and by <ref> is chordal. Thus assume that Γ is not a simplex. So there is a shedding vertex v of Γ such that both Γ-v and _Γ v are vertex decomposable. Setting =(_Δ v), it follows from <ref> that ^∨= Γ-v and is vertex decomposable. Therefore,is chordal by induction hypothesis and by <ref>, there is a sequence e_1, …, e_t with e_i∈(_i-1), where _0= and _i=_i-1-e_i, such that _t=-v. Consequently, we just need to show that -v is chordal. For this we show that (-v)^∨=_Γ v, which is vertex decomposable and the result follows by the induction hypothesis:F∈(_Γ v) Fv∈(Γ)=^∨ Fv∉({v}) F∉-vF∈ (-v)^∨.With notations as in Example vdec-ex3, we saw that -26-46=-6 and 26∈() and 46∈(-26). Now in (-6)^∨=13,14,23,24,34 the vertices 1,2,3,4 are shedding vertices. Applying <ref> with -6 instead ofand with v=1, we find for example 12∈(-6) and 13∈(-6-12) with =(-6)-12-13= -6-1= {234,345}. Applying <ref> again we can find say 23∈() and finally 35∈(-23) to get thatis chordal. In <cit.>, a vertex of a not necessarily uniform clutter,is called a simplicial vertex if for every e_1,e_2∈ with v∈ e_1,e_2, there is an e_3∈ such that e_3 (e_1∪ e_2){v}. Also the contraction /v is defined as the clutter of minimal sets of {e{v}| e∈}. Now if every clutter obtained fromby a sequence of deletions or contractions of vertices has a simplicial vertex, then Woodroofe in <cit.> callschordal and we call it W-chordal. In <cit.> it is proved that every W-chordal clutter which is uniformis chordal. As a corollary of the above theorem, we can get the following slightly stronger version of Corollary 3.7 of <cit.>. Suppose thatis a (not necessarily uniform) W-chordal clutter, d=minł{|F|| F∈}̊ and =ł{ F∈| |F|=d }̊. Thenis chordal. Theorem 6.9 of <cit.> states that the Alexander dual of the independence complex of(for the definition of independence complex, see <cit.>) is vertex decomposable. But independence complex ofis exactly Δ() and hence the result follows from Theorem vdec main. § CHORDALITY AND ASCENT OF CLUTTERSIn <cit.>, the concept of chorded simplicial complexes is defined and it is proved that I_Δ has a (d+1)-linear resolution over a field of characteristic 2, Δ is chorded and it is the clique complex of a d-clutter. We briefly recall this concept. Let Δ be a simplicial complex. We say that Δ is d-path connected, when it is pure and for each pair F and G of d-faces of Δ there is a sequence F_0,…, F_k of d-faces of Δ, with F_0=F, F_k=G and |F_i∩ F_i-1|=d. Also a pure d-dimensional simplicial complex Δ is called a d-cycle, when Δ is d-path connected and every (d-1)-face of Δ is contained in an even number of d-faces of Δ. A d-cycle Δ is called face-minimal, if there is no d-cycle on a strict subset of the d-faces of Δ. Finally, we say that a pure d-dimensional simplicial complex Δ is d-chorded, when for every face-minimal d-cycle ΩΔ which is not d-complete, there exists a family {Ω_1, …,Ω_k} of 2≤ k d-cycles with each Ω_iΔ such that enumi *Ω∪_i=1^k Ω_i,*each d-face of Ω is contained in an odd number of Ω_i's,*each d-face in ∪_i=1^k Ω_iΩ is contained in an even number of Ω_i's,*(Ω_i)(Ω).enumi We refer the reader to <cit.>, for several examples of these concepts.Suppose that Δ is the clique complex of a d-clutter. Connon and Faridi, first in <cit.> prove that if I_Δ has a (d+1)-linear resolution over a field of characteristic 2, then Δ is d-chorded. Then in <cit.>, the same authors show that I_Δ has a (d+1)-linear resolution over a field of characteristic 2, Δ is d-chorded, and moreover Δ^[m] is m-chorded for all m>d (Δ^[m] is the pure m-skeleton of Δ). Now if I_Δ=I() for a d-clutter , then Δ=Δ() and hence (Δ^[m]) is just the set of cliques ofwith dimension m. Therefore, the aforementioned results show that to study when I() has a linear resolution, it may be useful to consider the higher dimensional clutters (Δ()^[m]). Motivated by this observation, we define the ascent of a clutter as follows. Letbe a d-clutter. We call the family of all(d+1)-dimensional cliques of(that is, cliques on d+2 vertices), the ascent ofand denote it by ^+. In other words, ^+=Δ()^[d+1].Letbe as in Example counter Ex. Then =() is a 1-clutter (that is, a graph) on [8]. The circuits ofare the edges in Fig-Dunce. Now ^+ is the set of cliques of size 3 of , in particular, all of the triangles in Fig-Dunce are in ^+. Thus ^+. Note that all of the edges 12, 13, 23, 24, 34 are in , so 123 and 234 are in ^+ and ^+≠. This shows thatis not of the form (')^+ for some 1-clutter ' (else ' should contain ()= and ^+'^+, a contradiction). Now ^+ means the set of all 4-subsets of [8] such as A, with the property that all 3-subsets of A are in . But there is no such 4-subset of [8] and hence ^+=. Moreover, one could check that for example^+⊋ ∪{123, 23a,12a,13a|a=4,5,7,8}^++= (^+)^+ ⊋{123a, 136a|a=4,5,7,8}^+++={123ab, 136ab|ab=45,78,48},^++++= Our first result, considers how the property of having a linear resolution behaves under ascension. Recall that throughout the paper, we assume thatis a d-clutter on vertex set V with |V|=n and K is an arbitrary field. Let Δ=Δ(). Then I() has a linear resolution over K, I(^̱+̱) has a linear resolution over K and H_d(Δ_W;K)=0 for all W V. Note that if Δ'= Δ(^+), then Δ'_W and Δ_W differ only in d-dimensional faces and hence H_t(Δ'_W; K)=H_t(Δ_W; K) for each t> d and W V. Thus the result follows directly form Fröberg's theorem <ref>. Note that Δ^[d]=, thus Δ is d-chordedis so. The previous simple result shows why the concept of ascent of a clutter can be useful. For example, we show that a main theorem of <cit.> can simply follow <ref> and the results of <cit.>. First we state the needed results of <cit.> as a lemma. Letbe a d-clutter. Suppose that K is a field of characteristic 2 and Δ=Δ(). Then H_d(Δ_W;K)=0 for all W Vis d-chorded. () is <cit.> and the proof for () is the proof of part 2 of <cit.>. We inductively define a CF-chordal clutter. We consider , CF-chordal and we say that a non empty d-clutteris CF-chordal, whenis d-chorded and ^+ is CF-chordal. It is easy to check thatis CF-chordal Δ()^[m] is m-chorded for all m≥ d. In <cit.> simplicial complexes with this property are called chorded. Thus the next result is indeed Theorem 18 of <cit.>, which is one of the two main theorems of that paper. Before stating this result, it should be mentioned that an example of a clutterwithd-chorded but ^+ not (d+1)-chorded, is presented in <cit.>. This example shows thatcan be d-chorded whileis not CF-chordal. Letbe a d-clutter. Suppose that K is a field of characteristic 2. Then I() has a linear resolution over K,is CF-chordal. This follows from <ref> and <ref> and a simple induction on n-d. As the following result shows, similar to having a linear resolution, chordality is also preserved under ascension.Letbe a d-clutter. Ifis chordal, then so is ^+. We use induction on ||. The result is clear for ||=0,1, where ^+=. Choose e∈() such that -e is chordal. If e is contained in only one circuit, then it is not contained in any clique on more than d+1 vertices. Thus ^+=(-e)^+ and the claim follows the induction hypothesis. So we assume that _[e] is a clique on at least d+2 vertices. Let F be any d+1 subset of _[e] with e F. If v∈_^+[F] F, then Fv is a clique inand hence ev∈. This shows that _^+[F]_[e]. Let {v_1, …, v_t}= _[e] e, F_i=ev_i and _i= - F_1- ⋯- F_i. Note that _i^+= ^+ -F_1- ⋯- F_i. We show that F_i∈(^+_i-1), if F_i∈(_i-1^+). Then as _t=-e is chordal it follows by the induction hypothesis that _t^+ and hence ^+ are chordal.Suppose that G__i-1^+[F_i] and |G|=d+2. Then by the above argument G_[e] and hence G∈^+. If v_j∈ G for some j<i, then F_iv_j∈_i-1^+. But F_j= ev_j F_iv_j and F_j∉_i-1, a contradiction. Therefore, no v_j∈ G for j<i and hence no F_j G for such a j. Thus G is a clique in _i-1, that is, G∈_i-1^+ and hence F_i∈(_i-1^+).Suppose thatis the 2-clutter shown in Fig octa. The circuits ofare the faces of the hollow octahedron and the four triangles shown in a darker color. It is not hard to check thatis chordal, for example, the sequence 12, 14, 64, 62, 13, 23, 63, 43, is a sequence of consecutive simplicial maximal subcircuits, by deletion of which, we reach to the empty clutter. Hence ^+={1235,1345, 2356, 3456} is also chordal by <ref>. Indeed, all maximal subcircuits of ^+ are simplicial except for the four darker ones. Next we state a theorem that shows some connections between deleting elements of (^+) and having a linear resolution.Letbe a d-clutter. Suppose that ^+≠. Then, considering the following statements, we have <ref><ref><ref>. Moreover, if (^+)≠, then (<ref>, <ref> and <ref> are equivalent. *There is a F∈(^+) such that I(-̱F̱) has a linear resolution over K and also I(^̱+̱) and I(-̱v̱) have linear resolutions over K for each v∈ V.*I() has a linear resolution over K.*For all F∈(^+), I(-̱F̱) has a linear resolution over K and also I(^̱+̱) and I(-̱v̱) have linear resolutions over K for each v∈ V.<ref><ref>: Set Δ=Δ(). According to <ref>, we have to show that H_d(Δ_W; K)=0 for all W V. If W V, say v∈ V W, then Δ_W= Δ(_W)= Δ((-v)_W). Thus by Fröberg's theorem, as I(-̱v̱) has a linear resolution over K, it follows that H_d(Δ_W; K)=0. Consequently it remains to show that H_d(Δ; K)=0.Assume that x= ∑_i=1^l a_iF_i∈ C_d(Δ; K) with _d(x)=0 where 0≠ a_i∈ K and F_i∈. We have to show that x∈_d+1. Let F∈(^+) be such that I(-̱F̱) has a linear resolution over K. Assume that for some i, we have F_i=F. As F∈(^+), there is a G∈^+ such that F G. Now y= _d+1(a_iG) has a term ± a_iF_i. Hence in one of the two elements x∓ y, the coefficient of F is zero. Call this element z. Then _d(z)=_d(x)∓_d(_d+1(a_iG))= 0. Thus we can assume that for no i, F_i=F. Then x∈ C_d(Δ(-F); K) and as I(-F) has a linear resolution, H_d(Δ(-F); K)=0. So there are G_i's in (-F)^+^+ and b_j's in K such that x=_d+1(∑ b_jG_j)∈_d+1.<ref><ref>: As Δ(-v)_W = Δ()_W{v} and by Fröberg's theorem, I(-̱v̱) has a linear resolution over K for each v∈ V. According to <ref>, I(^̱+̱) also has a linear resolution over K. Now let F∈(^+) and also assume that V=[n]. Let L= x_F and I=I(). We have to show I(-̱F̱)=I+L has a linear resolution over K.First we compute the minimal generating set of I∩ L. Suppose that u is one of the minimal generators of the squarefree monomial ideal I∩ L. Then as x_F|u, we should have u=x_F∪ A for some A [n] F. Since u∈ I, there is a G_0∈ such that G_0 F∪ A. Assume |A|>1 and v∈ A. Then for no G∈ we have G Fv, else x_Fv∈ I∩ L which contradicts u being a minimal generator. This means that Fv is a clique ofand hence v∈_^+[F]. Since v∈ A was arbitrary, we have F∪ A N_^+[F] which is a clique, because F is simplicial. But this contradicts G_0 F∪ A and it follows that |A|=1, say A={a}. As G_0 Fa, we get a∉_^+[F]. On the other hand, for an arbitrary a∈ B=[n]_^+[F], it is easy to see that x_Fa∈ I∩ L. Consequently, I∩ L= x_Fa| a∈ B =x_F x_a| a∈ B.Since multiplying in x_F is an S-isomorphism of degree d+1 from x_a| a∈ B to I∩ L and by <cit.>, I∩ L has a (d+2)-linear resolution over K. Also I and L have (d+1)-linear resolutions over K. Now consider the exact sequence0→ I∩ L→ I L → I+L → 0.Writing the (i+j)-th degree part of the long exact sequence of ^S(K,-) applied on the above sequence, we get the exact sequence⋯→_i^S(K, I L)_i+j→_i^S(K, I+L)_i+j→_i-1^S(K, I∩ L)_i-1+(j+1)→⋯.If j≠ d+1 then both flanking terms are zero and hence the middle term is also zero. This means that I+L has a (d+1)-linear resolution, as required. In the following example, we show how the above theorem can be used.Suppose thatis as in Example counter Ex. Let =∪{278}. Then ^+={1278}, 278∈(^+) and ^+ is chordal. Hence I(^̱+̱) has a linear resolution over every field. Also it is easy to verify that for any W [8], _W is chordal (indeed, if W≠{1,2,7,8}, then either _W is empty or it has an edge which is contained in exactly one triangle and hence is simplicial and if W= {1,2,7,8}, _W is complete) and thus has a linear resolution over every field. Thus, as I(-̱2̱7̱8̱)= I() has a linear resolution over any field (see Example counter Ex) and according to <ref>, we get that I() has a linear resolution over every field. Note thatin <ref><ref>, against part <ref>, we cannot replace F∈(^+) with F∈(^+). For example, ifis as in Example chorded octa, then -135 is not 2-chorded, and hence I(-̱1̱3̱5̱) has not a linear resolution over _2, althoughis chordal and I() has a linear resolution over every field.Also note that <ref><ref> does not necessarily imply <ref><ref>, since we may have a cluttersatisfying <ref><ref>, but with (^+)=. Although, it should be mentioned that the author could not find such an example. Thus we raise the following question.Are the three statements of <ref> equivalent for a d-clutterwith a non-empty ascent?In the proof of <ref><ref> of <ref>, we proved that if there exists F∈(^+) such that I(-̱F̱) has a linear resolution over K, then H_d(Δ(); K)=0.The following is also a corollary to the proof of <ref>. Recall that β_i(I)=∑_jβ_ij(I).Letbe a d-clutter. Suppose that I= I() has a linear resolution over K, F∈(^+) and J=I(-̱F̱). Then J has a linear resolution over K with Betti numbers β_i(J)= β_i(I)+ti, where t=n-|_^+[F]|. Suppose that L= x_F and B=[n]_^+[F] as in the proof of <ref><ref>. So t=|B| and if L'=x_a|a∈ B, then I∩ L=x_F L' and β_i(I∩ L)= β_i(L') which by <cit.> is equal to ∑_k=1^t k-1i= ∑_k=i^t-1ki= ti . Also β_0(L)=1 and β_i(L)=0 for all i>0. Thus by taking _K of the long exact sequence of _i^S(K,-) in the proof of <ref>, we get β_i(J)= β_i(I+L)= β_i(I)+β_i(L)+ β_i-1(I∩ L), where the last term is assumed to be zero if i=0. If in <ref> we restrict to the case that char K=2 and by using <ref>, we get the following.Letbe a d-clutter. Suppose that ^+≠. *If ^+ is CF-chordal and there is a F∈(^+) such that -F is CF-chordal and -v is CF-chordal for all v∈(), thenis CF-chordal.*Ifis CF-chordal, then for all F∈(^+) and all v∈(), -F and -v are CF-chordal.Moreover, if (^+)≠, then the converses of both <ref> and <ref> hold. In the rest of the paper, we study if we can get some results similar to <ref> for having linear quotients or being chordal instead of having a linear resolution. For having linear quotients we have:Letbe a d-clutter. Suppose that I() has linear quotients. Then I(^̱+̱), I(-̱F̱)and I(-̱v̱) have linear quotients for each F∈(^+) and v∈(). If we denote the ideal generated by all squarefree monomials in I with degree t by I_[t], then it is easy to see that I(^̱+̱)= I()_[d+2]. Hence the fact that I(^̱+̱) has linear quotients follows from <cit.>. But for the convenience of the readers, we present a direct shorter proof for this. Assume that x_F_1, …, x_F_t is an admissible order of I().Then ^̱+̱={F_iv|v∈ V F_i, 1≤ i≤ t}. For each 1≤ i≤ t, let _i= {F_iv|v∈ V F_i} (∪_j<i_j) and suppose that _i={F_i1, …, F_ir_i}. We prove that F_11, …, F_1r_1, …, F_t1, …, F_tr_t corresponds to an admissible order for I(^̱+̱).Consider two circuits F_ij, F_i'j' of ^̱+̱ with i≤ i'. If i=i', then |F_ij F_i'j'|=1 and hence the admissibility condition holds trivially for them. Thus we assume i<i'. Therefore by assumption there is a l∈ F_i F_i' and a k< i' such that F_k F_i'= {l}. Let F_i'j' F_i'={v}. Note that as F_i'j'∈_i', there is no j<i' with F_i'j'∈_j by the definition of _i's. So v≠ l, else F_i'j'= F_k∪ (F_i' F_k) and as |F_i' F_k|=1, we have F_i'j'∈_j for some j≤ k, a contradiction. Consequently, v∉ F_k and F_kv which is a circuit of ^̱+̱, should appear in some _j with j≤ k. Noting that F_kv F_i'j'= {l}, the proof is concluded.Now assume that F∈(^+). We show that if we add x_F to an admissible order of I(), we get an admissible order of I(-̱F̱)= I()+x_F. We just need to show if G∈, then there is a G'∈ and an l∈ G such that G F={l}. Since _^+[F] is a clique of , G_^+[F], say l∈ G_^+[F]. Then Fl∉^+ and hence there is a G' Fl with G'∈. So G' F={l}, as required.Finally, note that if x_F_1, …, x_F_t is an admissible order for I(), then by deleting x_F_i's with v∈ F_i, we get an admissible order for I(-̱v̱) and I(-̱v̱) has linear quotients.Letbe the clutter of Example chorded octa. It is not hard to check that the following is an admissible order for I(): 162, 163, 164, 165, 124, 624, 245, 234. Thus by the previous result, I(^̱+̱) has linear quotients. Indeed, by the proof of <ref>, we get the following admissible order for I(^̱+̱): 1623, 1624, 1625, 1634, 1635, 1645, 1243, 1245, 6243, 6245, 2453. Also note that the assumptionthat F is simplicial, is crucial in <ref>. For example, I(-̱1̱3̱5̱) has not a linear resolution and hence has not linear quotients. We do not know whether the converse of the above theorem is correct or if a statement similar to <ref><ref> of <ref> holds for having linear quotients.Next consider the “chordal version” of <ref>, that is, consider the statements obtained by replacing “I() has a linear resolution over K” with “ is chordal” in the three parts of <ref>, where the replacement occurs for all cluttersappearing in these assertions. Then clearly <ref> <ref> holds for the chordal version. Also <ref> shows that part of <ref> <ref> is true for chordality. We will also show that ifis chordal and F∈(^+), then -F is d-chorded, which is weaker than being chordal. But before proving this, we utilize <ref> to show that if the chordal version of <ref><ref> (or <ref> <ref>) holds, then we can reduce proving statement <ref> to verifying it only for clutters with empty ascent.In what follows, by a free maximal subcircuit of , we mean a maximal subcircuit which is contained in exactly one circuit of . In the particular case that =1, that isis a graph, free maximal subcircuits are exactly leaves (or free vertices) of the graph. Also a simplicial complex Δ is called extendably shellable, if any shelling of a subcomplex of Δ could be continued to a shelling of Δ. To see a brief literature review of this concept and some related results consult <cit.>.Consider the following statements and also statement <ref> of Section 2 on a uniform clutter . enumi*If ≠^+ is chordal and -F and -v are chordal for each F∈(^+) and v∈(), then ()≠.*If ^+= and I() has linear quotients, thenhas a free maximal subcircuit.*Simon's Conjecture (<cit.>): Every d-skeleton of a simplex is extendably shellable.enumi Then: <ref> + <ref><ref><ref> + <ref>. Assume that <ref> and <ref> hold. We claim that if I() has linear quotients, then ()≠. Then it follows from <cit.> that <ref> is correct. To prove the claim, we use induction on (n-d, ||) considered with lexicographical order. If ^+=, then as every free maximal subcircuit is simplicial, the claim holds by <ref>. If ^+≠, then applying <ref> and using the induction hypothesis, we see that ^+, -F and -v are chordal for every F∈(^+) and v∈(). Consequently, the claim follows from <ref>.Now suppose that statement <ref> is correct. If ^+=, then |_[e]|=d+1for every e∈() and hence every simplicial maximal subcircuit ofis a free maximal subcircuit. Therefore, <ref> holds as a special case of <ref>.Finally <ref><ref> follows from Theorem 2.5 and Corollary 3.7 of <cit.>.The “chordal version” of <ref><ref> in <ref>, partly states that ifis chordal, then for every F∈(^+), -F should be chordal. The author could neither prove nor reject this in the general case, but we prove a weaker result. In particular, we will show in the case that =1, this is true. We need the following lemmas which present an equivalent condition for being d-chorded. The set of facets of a d-cycle is a disjoint union of the set of facets of a family of face-minimal d-cycles. Suppose that Ω is a d-cycle. We use induction on |(Ω)|. If Ω is face-minimal, then the result is trivial. Thus we can assume that there is a d-cycle Ω' whose facets form a strict subset of (Ω). Let Γ= (Ω)(Ω'). Note that each (d-1)-face of Γ is contained in an even number of facets of Ω and an even number of facets of Ω'. Hence each (d-1)-face of Γ is contained in an even number of facets of Γ. Suppose that Γ_1, …, Γ_t are d-path connected components of Γ, that is, the maximal pure d-dimensional subcomplexes of Γ which are d-path connected. Note that if i≠ j, then Γ_i and Γ_j do not share a (d-1)-face, else Γ_i ∪Γ_j is d-path connected, contradicting maximality of both Γ_i and Γ_j. Consequently, for every i, each (d-1)-face of Γ_i is contained in an even number of d-faces of Γ_i and Γ_i is a d-cycle. Noting that (Γ_i) are mutually disjoint, and by applying induction hypothesis on Γ_i's, it follows that (Γ) is a disjoint union of facets of a familyof face-minimal d-cycles. So (Ω) is the disjoint union of facets of the family ∪{Ω'} of face-minimal d-cycles, as required. Letbe a d-clutter. The simplicial complexis d-chorded the facets of every d-cycle Ω ofis the symmetric difference of a family of complete subclutters of , each on a (d+2)-subset of (Ω). (): By induction on |(Ω)|. Because of <ref>, we can assume that Ω is face-minimal. If Ω is d-complete, then by face-minimality |(Ω)|=d+2 and we are done. Thus we assume that Ω is not d-complete. Suppose that Ω_i's are d-cycles insatisfying <ref>–<ref> in the definition of a d-chorded complex. Let _i=(Ω_i). Since each d-face in ∪_i=1^k _i(Ω) appears in an even number of _i's by <ref>, such d-faces are not in the symmetric difference of _i's. Also by <ref> and <ref>, we see that each element of (Ω_i) is in an odd number of _i's and hence is in their symmetric difference. Therefore, (Ω)=_1 ⊕⋯⊕_k, where ⊕ denotes symmetric difference. Also by <ref> of the definition of a d-chorded complex, each _i is a cycle on an smaller number of vertices in (Ω). Thus by applying the induction hypothesis on _i's we get a decomposition of Ω as the symmetric difference of a set of complete subclutters ofon (d+2)-subsets of (Ω). (): Suppose that Ω is a face-minimal cycle ofwhich is not d-complete. Then (Ω)= _1 ⊕⋯⊕_t where _i's are complete subclutters ofwith size d+2 and (_i)(Ω). If we set Ω_i=_i, then Ω_i's clearly satisfy <ref>–<ref> of the definition of a d-chorded clutter. This means thatis d-chorded.Letbe a d-clutter. Suppose thatis d-chorded (for example, ifis chordal) and F∈(^+). Then -F is d-chorded. Suppose that Ω is a d-cycle of -F and = (Ω). Sinceis d-chorded and according to <ref>, there are complete subclutters _1, …, _t ofon (d+2)-subsets of () such that = ⊕_i=1^t _i. We choose _i's in a way that m= |{i|F∈_i}| is minimum possible. Assume that m≥ 1. Because F∉, m is at least 2. Thus there are two _i, say _1,_2, containing F. Therefore there exist v_1 ≠ v_2∈() such that (_i)= Fv_i for i=1,2. Let W=Fv_1v_2 and '= _W. Then since F∈(^+) and W_^+(F), ' is complete. Now if '= _1 ⊕_2, then ' is a d-cycle in '-F. By <ref>, '-F is chordal and hence has a linear resolution over every field. Thus by <cit.> '-F is d-chorded. Consequently according to <ref>, '='_1 ⋯'_t', where '_i's are complete subclutters of '-F (and hence ) on (d+2)-subsets of (')(). By replacing ⊕_i=1^t''_i instead of _1 ⊕_2 in the decomposition of , we get a decomposition with less terms containing F. This contradicts the choice of the decomposition ofand hence m=0.Therefore, the decomposition = ⊕_i=1^t _i is in -F and the result follows by <ref>. It should be mentioned that the previous result is not correct for arbitrary F∈(^+). This can be seen by noting that I(-̱1̱3̱5̱) is not 2-chorded, whereis as in Example chorded octa.Ifis a graph, we call a F∈(^+) a simplicial edge of .Suppose thatis a chordal graph and F is a simplicial edge of . Then -F is chordal. Hence there is an ordering F_1, …, F_t, F_t+1, …, F_m of edges ofsuch that for 1≤ i≤ t, F_i is a simplicial edge of the chordal graph _i= -F_1-⋯-F_i-1 and for t<i≤ m, _i is a tree and F_i is a leaf edge of _i. The first statement is just <ref> in the case that =1. Now ifis chordal and ^+≠, then (^+)≠ by <ref>. So starting with , we can delete simplicial edges until we reach a chordal graph ' with (')^+=. But a chordal graph without any cliques on more than two vertices is a tree and hence we can delete leaf edges from ' until there is no more edges and the statement is established. Letbe the graph in Fig. Then ^+ and -F are chordal and also I(^̱+̱) and I(-̱F̱) have linear quotients for each F∈(^+). Despite thisis not chordal, since -e is a cycle of length 5. Thus the converses of <ref> and <ref>do not hold. Also edges ofcan be ordered as in <ref>, so the converse of <ref> is not true, either.Clearly Statement <ref> holds when =1. Using the concept of d-cycles, we show that <ref> holds when ^∨≤ 1 or equivalently, if n-d≤ 3. As in <cit.>, by a CF-tree we mean a d-clutterwith the property thathas no d-cycles.Assume thatis a d-clutter on n vertices with n≤ d+3. If ^+= and I() has a linear resolution over every field, thenis chordal. In particular, statement <ref> of <ref> holds for . Suppose that ^+=and I() has a linear resolution over every field. According to <cit.>,is d-chorded. But sincehas no cliques of size d+2, it follows from <ref> thathas no d-cycles, that is,is a CF-tree. Now the result follows from <cit.>.The proof of Corollary 3.7 of <cit.> uses Alexander dual. We end this paper mentioning that more generally, one can get a statement equivalent to <ref> by passing to the Alexander dual of . Indeed, by arguments quite similar to <cit.> one can see that <ref> holds for all d-clutters on n vertices, statement (ii) of <cit.> holds, when we replace “Cohen-Macaulay over _2” in that statement with “shellable”. Acknowledgements The author would like to thank Rashid Zaare-Nahandi and Ali Akbar Yazdan Pour for their helpful discussions and comments. Also the author is thankful to the reviewer whose comments led to great improvements in the presentation of the paper. 99simp ord M. Bigdeli, J. Herzog,A. A. Yazdan Pour and R. Zaare-Nahandi, Simplicial orders and chordality, J. Algebraic Combin., 45(4), 1021–1039, 2017.chordal M. Bigdeli, A. A. Yazdan Pour and R. Zaare-Nahandi, Stability of Betti numbers under reduction processes: towards chordality of clutters, J. Combin. Theory Ser. A, 145, 129–149, 2017.strong chordal —, Decomposable clutters and a generalization of Simon's conjecture, preprint, arXiv:1807.11012. bjorner A. Björner and K. Eriksson, Extendable shellability for rank 3 matroid complexes, Discrete Math. 132, 373–376, (1994). CF2 E. Connon and S. Faridi, A criterion for a monomial ideal to have a linear resolution in characteristic 2, Elec. J. Combin. 22(1), #P1.63, 2015. CF1 —, Chorded complexes and a necessary codition for a monomial ideal to have a linear resolution, J. Combin. Theory Ser. A 120, 1714–1731, 2013. froberg R. Fröberg, Rings with monomial relations having linear resolutions, J. Pure Appl. Algebra 38, 235–241, 1985.hibi J. Herzogand T. Hibi, Monomial Ideals, Springer-Verlag, London , 2011. our chordal A. Nikseresht and R. Zaare-Nahandi, On generalization of cycles and chordality to clutters from an algebraic viewpoint, Algebra Colloq. 24(4), 611–624, 2017.provan J. Provan and A. Billera, Decompositions of simplicail complexes related to the diameters ofconvex polyhedra, Math. Oper. Res. 5(4), 576–594, 1980.simon R. S. Simon, Combinatorial properties of cleanness, J. Algebra 167,361–388, 1994.solyman A. Soleyman Jahan and X. Zheng, Ideals with linear quotients, J. Combin. Theory Ser. A 117, 104–110, 2010. w-chordal R. Woodroofe, Chordal and sequentially Cohen-Macaulay clutters, Elec. J. Combin. 18(1), #P208, 2011. | http://arxiv.org/abs/1708.07372v3 | {
"authors": [
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"published": "20170824122829",
"title": "Chordality of Clutters with Vertex Decomposable Dual and Ascent of Clutters"
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Department of Physics, University of Windsor, Windsor ON, Canada * Present affiliation: Department of Physics and Astronomy, Queen’s University, Kingston ON, CanadaThe ability to prepare desired states without allowing time for the atomic system to spontaneously decay to the ground state is limited by the fact that Hamiltonian controls cannot affect the purity of the state in question. In this paper, we discuss how a continuous off-resonant electromagnetic wave can be used to prepare a desired final quantum state. With the aid of plasmonic surface-enhancement, by placing a two-level atomic system near a gold nanoparticle, preparation is accomplished in a reduced period of time and with lower electric field intensity while the system purity is improved when compared to an isolated system.PACS-keydiscribing text of that key PACS-keydiscribing text of that key Quantum Dynamics of a Two-level System Near a Gold Nanoparticle Somayeh M.A. Mirzaee*,and Chitra RanganReceived: date / Revised version: date =============================================================== § INTRODUCTION The issue of quantum computation due to its ability to perform certain types of calculations much faster than classical computers has attracted much attention recently. The basic concepts of quantum computation are quantum operations (gates) on quantum bits (qubits) and registers (arrays of qubits). A qubit can be a two-level system which can be prepared in arbitrary superposition of its two eigenstates, usually denoted as |0⟩ and |1⟩ . In that sense quantum state engineering is needed to control preparation and manipulation of these quantum states.In quantum information systems, decoherence, due to the interaction of the quantum state with the environment, represents a fundamental issue that must always be addressed as it results in a leakage of system information to the environment <cit.>. Therefore it is desirable to have quantum states that can be prepared and purified in a short period of time to avoid this loss of information.A great deal of recent research has been devoted to proposing solutions, such as entanglement purification<cit.>, that allow for the preservation of this purity of states <cit.>. These methods include quantum error-correcting codes<cit.>, decoherence-free subspaces<cit.>, strategies based on feedback or stochastic control <cit.> as well as dynamical decoupling <cit.>. These methods all share a common strategy: they fight against de-coherence from the environment by decoupling the quantum system from the environment. We show here that dissipation, if engineered, can have exactly the opposite effect: it can be a well-developed resource for universal quantum computation. To overcome noise, purification techniques, which generate qubits with higher purities from qubits with lower purities, have been proposed<cit.>. As the purification dynamics of quantum systems are quantfied by the average purification rate and the mean time needed to reach the required level of purity<cit.>, designing systems that optimize these quantities are highly desirable.This problem of finding effective purification schemes for quantum states is difficult due to fact that the purity of a quantum system cannot be changed by Hamiltonian controls alone<cit.> and require some other effects in order to modify the purity of the states in question. One such effect that can be used to modify the purity of states is through the inclusion of spontaneous decay. However, using spontaneous decay to purify quantum states requires weak driving fields and long time scales in order to reach a desired state of purity.In this work, we show that the presence of a gold nanoparticle (GNP) near individual qubits can be used to enhance the purification rates of these qubits when using an incident electric field as in laser cooling. This enhancement in the purification rate arises due to the modification of the total decay rate of a two-level quantum system coupled to a nonlocal plasmonic particle <cit.>, or to a localized surface plasmon mode<cit.> of a GNP plus local electric field enhancement. Moreover, we will prove that the proximate GNP can use to preserve the purity of the quantum information (QI) carriers in presence of decoherence processes, which can be used in transmission process.§ SYSTEM MODELA schematic illustration of our model is given in figure 1. We model the qubit as a molecular two-level system with states |g⟩ and |e⟩. This qubit is placed a specific distance ’d’ from the surface of a GNP. This quantum system is driven by an electromagnetic wave that is incident on the entire system.The Hamiltonian of the two-level quantum system that is driven by an electromagnetic wave, with the field-matter interaction of the system treated in the electric-dipole approximation described by Ĥ = Ĥ_a + μ⃗·E⃗(t) , where H_a is the Hamiltonian of a two-level system, μ⃗ is the transition dipole moment of the system and E(t) is electric field of the wave interacting with the system. This electric field magnitude varies in time as E(t) = E_L cos(ω_L t) where ω_L is the frequency of the incident wave. Making the rotating wave approximation (RWA), the matrix form of the Hamiltonian can be written as: H_RWA = (0ħΩ_ge/2 ħΩ_ge^*/2-ħ△ ) ;where△ represents the detuning between the frequency of the incident field and frequency of the state transition. The Rabi frequency, Ω_ge = μ_ge M_L E_L/ħ, depends on the amplitude of the local electric field (M_LE_L), where the incident electric field is modified by the interaction of the electromagnetic wave with the GNP. This interaction is reflected in the inclusion of the frequency-dependent field enhancement term, M_L, which affectively increases or decreases the electric field intensity experienced by the two-level system.In order to study the time-dependent response of the qubit to both the environment and the incident electromagnetic wave, we use a density matrix representation of the qubit’s state. For our two-level system this density matrix is of the form:ρ =( ρ_gg ρ_ge ρ_eg ρ_ee ) ; This density matrix evolves in time under the LVN equation which takes the form:ρ̇ = - i/ħ [H_RWA,ρ] - L ; In this evolution equation, the Lindblad term, L, models the decoherence in the system. This term is linear in the state density operator and is of the form:L̂ =∑_d M_dγ_eg/2 (σ_d^†σ_d ρ + ρσ_d^†σ_d - 2 σ_d ρσ_d^†);In this equation, σ_d are the Lindblad operators and we assume that the only decoherence mechanism present is spontaneous emission.Therefore γ_eg represents the decay rate from the excited states to the ground state and we take σ_d^† = |g⟩⟨e|.The factor M_d represents a modification of the decay rates that arises from an interaction between qubit and the environment <cit.>.In this work we quantify the purity of a state as Purity = Tr (ρ^2) <cit.>.Under this description, states with a higher purity are closer to being pure states. It is important to note that if only Hamiltonian controls are applied to the system this state purity cannot change and therefore decay must be included to change the purity of measured states <cit.>. Due to the increased decay rate, brought on by proximity to the GNP, this change in purity occurs on a shorter timescale. The enhancement quantities M_L and M_d are reflective of the effect of the GNP and next we show how these quantities are computed.§ EFFECT OF A GNP ON A PROXIMATE TWO-LEVEL SYSTEMLight incident onto a GNP can excite localized surface plasmon resonances that enhance the electric field around the nanoparticle <cit.>. In addition to these resonances, metal particles amplify the decay rates of excited states in nearby molecules by offering non-radiative decay channels and by coupling radiative emissions to a localized surface plasmon mode <cit.>. These interactions can greatly affect the response of a qubit system to an incident electric field.In order to calculate the field enhancement and polarization around a GNP, we solve Maxwell’s equations for the interaction of the incident electromagnetic field with the GNP. For a point GNP, this calculation can be done analytically <cit.>. In order to extend our formalism to GNPs of irregular size and configuration we solve Maxwell’s equations numerically <cit.>. We do this by performing a three-dimensional finite-Difference Time Domain (FDTD) calculation using the FDTD Solutions software package from Lumerical Solutions Inc. This also allows us to determine the vector components of the electromagnetic field at different locations as opposed to an average field enhancement. The dielectric function of gold was chosen to be “Au (gold) - Johnson and Christy” from the material database. Figure 2 presents the calculated electric field components relative the incident electric field amplitude for the case of a GNP of radius 20 nm that is illuminated by a z-polarized plane wave of λ = 542 nm propagating along the z direction; this wavelength corresponds to the maximum electromagnetic enhancement around the GNP. We have seen that, as the particle diameter is no longer small compared to excitation wavelength, plasmonic modes of higher order than the dipolar mode becomes important and contribute to the electric field distribution. We assumed that the parallel orientation of dipole moment relative to the nanoparticle surface is excited only by the tangential component of electric field, while only the radial component contributes to the dipole with the perpendicular orientation. Therefore, unlike the literatures that focus on total electric field enhancement, we have calculated the electric field components around the GNP. These electric field components can be used to determine the electric field enhancement M_L at any given point relative to the GNP. We will place the qubit away from the GNP along its z axis as this area represents the strongest area of electric field enhancement. We will also assume that the qubit polarization is along the z direction as the field in the y direction in ten times weaker than the field in the z direction.This numerical simulation of Maxwell’s equations can also be used to calculate the enhancement of the decay of the excited state (M_d). To calculate the decay rate enhancement factor in the presence of a gold nanoparticle, the same model as Gresten-Nitzan model<cit.> was employed. The system of radiating dipole interacting with an isolated GNP is solved using the finite-difference time-domain (FDTD) method. The quantum system is simulated be a point radiating dipole source in the near fiield of the nanoparticle. The parallel and perpendicular orientations of the radiating dipole moment relative to the nanoparticle surface are examined and by placing these model dipoles at varying distances from the GNP and in various orientations with respect to the GNP surface, we can examine the effect of GNP-qubit separation on the decay rates of the qubit system. The analysis of the FDTD results relies on the fact that, for an atomic dipole transition, the normalized quantum mechanical decay rate can be related to the normalized classical power radiated by the dipole. In the presence of absorption, this classical approach allows one to calculate separately the radiative decay rate (proportional to the far field radiated power) and the nonradiative decay rate (proportional to the power absorbed by the environment) <cit.>. In the literature of surface-enhanced fluorescence, it is normally assumed that the dipole placed a distance away from a GNP experiences enhanced spontaneous emission <cit.>. Figure 3 shows the radiative and nonradiative decay rate modification experienced by the qubit system at various distances from the 20 nm radius GNP for the qubit orientations both parallel and perpendicular to the GNP surface. From figures 2 and 3 we can see that the interaction between a qubit and the GNP is not only distance dependent; it is effected by both the orientation of the qubit as well as the location of the qubit around the GNP. By placing our qubit system at different locations, we are given a degree of control over the decay rates and local fields experienced by the GNP.§ ATOMIC SYSTEM PREPARATION IN DESIRED STATEIn order to prepare a qubit in a desired state, we investigate the dynamics of a two-level system when it is excited by a classical electric field in a dissipative environment. Our idea is to engineer those couplings, so that the environments drive the system to a desired final state. Equation 3 describes the evolution of the state densities over time. By rearranging Equation 3 into differential equations for both the ground state populations and coherences, we find that at steady state (ρ̇=0 ) conditions, the final state populations of the system are :ρ_gg =1 - Ω_ge^2/4/△^2+γ^2/4+Ω_ge^2/2 ; ρ_ge = 1/2△Ω_ge+i1/4Ω_geγ/△^2+γ^2/4+Ω_ge^2/2 ; Where γ represents the total decay rate. These equations indicate which final state the system will end up in, regardless of initial conditions, as long as the system evolves to reach a steady state. In order to illustrate this, we have evolved a qubit at a distance of 5 nm from a GNP for a variety of initial conditions. For this system, the energy separation is 2.38eV and the dipole moment is chosen to be μ_ge=1eÅ, where e is the charge of an electron. We assume that the frequency of our incident electric field is chosen to yield a photon energy of 2.29eV which leaves us with a detuning△= 0.09eV. Figure 4 shows this system evolving, under a chosen laser intensity, to a desired final ground state population ρ_gg = 0.75, for a variety of initial mixed states. Each state is initialized with no coherence, ρ_ge = 0 and evolves to the same final state in the same amount of time without any active control. This independence of the final state on initial conditions indicates that a continuous electromagnetic wave can be used to purify a mixed quantum state if the purity of the initial state is lower than the final state that will be reached. It also places a fundamental limit on the amount of purification that can be done on the system.This differs from pulsed excitation schemes in that the purity can be changed and the initial conditions do not need to be known; avoiding the need to allow the system to decay completely to the ground state.In addition the GNP also helps to greatly reduce the required electromagnetic intensity required to reach the desired final state. As both field enhancement and decay enhancement are dependent on the distance, we can examine this effect by placing the qubit closer to the GNP. Figure 5 shows the state dynamics and purity by placing the qubit at a distance 2ẑnm from the GNP. Figure 6 shows, this placement reduces the required electromagnetic field due to the electromagnetic field enhancement effect versus the required electric field for the bare qubit, which is around 5×10^9 V/m.Moreover, the main benefit of a GNP is to help reduce the time required to reach the steady-state condition by enhancing the decay rate. The time the system takes to reach a final steady state depends on the spontaneous rate of decay experienced by the upper state. When placed near a gold nanoparticle, the time taken to reach the steady state is inversely proportional to the amount of decay enhancement. For the evolution of the state populationsρ̇_̇ėė=i*Ω_ge/2( ρ_eg- ρ_ge)-γρ_gg. Therefore the excited state populations will experience an overall exponential convergence as there exists a term in the evolution equation that is linear with respect to γ. Therefore an increase in the decay rate (γ) should linearly correspond to a reduced convergence time due to this timescale. Consider a system placed at a distance of 2nm versus 5nm. At these distances the decay rate is roughly 34 times larger for the 2nm case (figure 5) than the 5nm case (figure 4) and therefore the corresponding convergence time is roughly 34 times larger in figure 5 than in figure 4 as well. Figure 3 shows that, the decay rate enhancement of a qubit placed at a distance of 2 nm from a GNP is approximately 2.4×10^4 times of a decay rate of a bare qubit, causing the same order of differences in the needed time to reach the steady state in case of bare qubit. § LIMITATION ON PURIFICATION IN A TWO-LEVEL SYSTEMFor a system with a specific decay rate, enhancement and detuning, choosing the required strength of the electromagnetic field to reach a particular desired ground state population will also set the steady state coherence. As the steady-state purity (P) of the two-level system is dependent on only the coherence and ground state population, it will be fixed as well. In fact the steady-state purity of the qubit can be shown, using Equations 5 and 6, to be entirely dependant on the choice of a desired ground state populationρ_gg with:Purity = 4ρ_gg-2ρ_gg^2-1;Under this relationshipit is not possible to reach a completely pure state that is not in the ground state configuration using only spontaneous decay and a two-level system.This limitation restricts this method of state preparation to those in which high levels of purity are not required.Define the purification factor as F(P)=P_output/P_input is beneficial to show how effective is this method to prepare a state while improving its purity.As an example, consider the initial mixed state of ρ_gg=0.75and ρ_ee=0.25 with purity of about 0.625 before going through the process proximate to a GNP. The purity of the output state after the preparation process in presense of a GNP is about 0.875, which is giving F(P)=1.4. Result of purification factors for different initial state purities are presented in figure 7. It can be seen that the F(P) corresponding to this method is always greater than one and just in case that F_input≈1 or the intial system is in a totally mixed state, F(P) approaches one.§ CONCLUSION In this work we show that an incident electromagnetic wave can be used to prepare a qubit in a specific state, regardless of the initial conditions of the qubit. The presence of a proximate GNP can be used to decrease both the required intensity of the incident field as well as the time needed to produce a desired population distribution in a qubit. This process can also be used to cool qubits and increase their state purity in a reduced timescale. The presence of a GNP accomplished this by enhancing both the local electric field around it as well as enhancing the decay rates of the desired states. This reduction in required time, electric field intensity and the ability to modify system purity may represent an improvement in the practical control and use of quantum information systems. § AUTHORS CONTRIBUTIONSAll the authors were involved in the preparation of the manuscript. All the authors have read and approved the final manuscript. § ACKNOWLEDGEMENTSThe authors gratefully acknowledge support from the Discovery Grant program of the Natural Sciences and Engineering Research Council of Canada (Grant No. 311880). Calculations were performed on the SharcNet supercomputing platform. We thank Chris DiLoreto for helpful comments and suggestions. 1a D. Giulini, Decoherence and the Apppearance of a Classical World in Quantum Theory (Springer, Berlin, 1996) 2a C.H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J.A. Smolin, W.K. Wootters, Phys. Rev. Lett. 76, 722 (1996).3a E.J. Griffith, C.D. Hill, J.F. Ralph, H.M. Wiseman, K. Jacobs, Phys. Rev. B 75(014511) (2007)4a M. Kleinmann, H. 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"Chitra Rangan"
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"title": "Quantum Dynamics of a Two-level System Near a Gold Nanoparticle"
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Institute of Materials Physics and Technology, Hamburg University of Technology, Hamburg, Germany Institute of Materials Research, Materials Mechanics, Helmholtz-Zentrum Geesthacht, Geesthacht, Germany Institute of Materials Physics and Technology, Hamburg University of Technology, Hamburg, Germany Institute of Materials Research, Materials Mechanics, Helmholtz-Zentrum Geesthacht, Geesthacht, Germany Corresponding author, email: [email protected] Institute of Materials Research, Materials Mechanics, Helmholtz-Zentrum Geesthacht, Geesthacht, Germany Institute of Materials Physics and Technology, Hamburg University of Technology, Hamburg, Germany X-ray diffraction studies of nanoporous gold face the poorly understood diffraction scenario where large coherent crystals are riddled with nanoscale holes. Theoretical considerations derived in this study show that the ligament size of the porous network influences the scattering despite being quasi single crystalline. Virtual diffraction of artificially generated samples confirms the results but also shows a loss of long-range coherency and the appearance of microstrain due to thermal relaxation. Subsequently, a large set of laboratory X-ray investigations of nanoporous gold fabricated by different approaches and synthesis parameters reveal a clear correlation between ligament size and size of the coherent scattering domains as well as extremely high microstrains in samples with ligament sizes below [10]nm. X-Ray Studies of Nanoporous Gold: Powder Diffraction by Large Crystals with Small Holes Jürgen Markmann Received: date / Revised version: date ======================================================================================= § INTRODUCTION Nanoscale metal networks made by dealloying are of current interest as model systems for fundamental studies of size and interface effects in the nanosciences <cit.>. They are also under consideration as materials basis for technologically relevant functional materials, for instance in actuation <cit.>, sensing <cit.>, energy storage <cit.>, microfluidics <cit.>, catalysis<cit.>, or lightweight structural materials <cit.>. Almost invariably, such studies require precise data for the characteristic size of the struts or ligaments of the metal network. This characteristic structure size is an indispensable basis for discussing size effects and it implies key microstructural characteristics such as the volume-specific surface area or the abundance of low-coordinated atomic surface configurations. Size also controls mechanical strength <cit.> as well as fluid transport <cit.>, and its link to the stiffness is under discussion <cit.>. Actuation amplitudes <cit.> and signal strength or sensitivity in sensing <cit.> depend on the specific surface area, and the number of low coordinated sites is a key parameter for catalysis <cit.>. Catalysis is also known to depend sensitively on the interatomic spacing at the surface and, thereby, on the lattice strain <cit.>. These remarks motivate a substantial interest in experimental techniques for quantifying the ligament size, the mean lattice parameter and the microstrain, that is, the variance of local lattice spacings.X-ray powder diffraction is a classic approach to the above-mentioned task. The evolution of lattice parameters and of the coherent scattering length of nanoporous gold (NPG) during dealloying have been studied by that technique <cit.>. Yet, so far there has been no dedicated analysis with respect to standard wide-angle X-ray patterns and its use as a microstructural characterisation method.The classic approach to size effect on powder diffraction, as embodied in the Scherrer equation and in many later refinements, such as the Warren-Averbach and Williamson-Hall approaches and their variants <cit.>, analyses scattering by a statistically relevant ensemble of very small crystalline particles in random orientation, for which the interparticle interference can be ignored. Dealloying-made nanoporous metals distinguish themselves from that scenario by one fundamental aspect: neighbouring ligaments are part of the same (porous) crystal and therefore scatter coherently. Studies of millimetre-size nanoporous samples using ion challenging contrast <cit.>, electron backscatter diffraction imaging <cit.>, X-ray Laue scattering <cit.>, and Bragg coherent X-ray diffractive imaging <cit.> show that dealloying conserves the grain size of the initial polycrystalline aggregates which typically range between 10 to [100] m. The individual ligaments, typically 10 to [100]nm in size, then share the same coherent crystal lattice. Even though microstructural defects that may degrade the lattice coherence – specifically nonuniform strain fields <cit.>, stacking faults <cit.>, and twins <cit.> – have been reported for nanoporous gold, the crystal size is clearly much larger than the ligament size. This raises the question whether and how the apparent coherent scattering lengths – as indicated by the Bragg reflection broadening – relate to the ligament size of dealloying-made nanoporous metal.Here, we present an X-ray powder diffraction study of dealloying-made nanoporous gold along with a discussion of the most elementary concepts of scattering by porous crystals. We performed virtual X-ray diffraction on atomistically modelled NPG in order to confirm that the contributions from nanoporosity and grain size are separately discernible if no relaxation of the structure was performed. Surprisingly, the contribution of the (large) grain size vanishes after thermal relaxation of the atomistically modelled structure. As in the simulation, the diffraction patterns of experimental NPG structures show no contributions from the grain size. We therefore employ samples that result from different dealloying and processing procedures to analyse their impacts on the diffraction patterns. The assessment of the results of our X-ray analysis is based on systematic scanning electron microscopy (SEM) data for ligament size and composition. We report and discuss the two following surprising findings: First, even though our theory suggests otherwise, experimental coherent scattering domain sizes as derived from Williamson-Hall analysis are strongly correlated to the "true" ligament sizes as derived by SEM. Second, the correlation between microstrain and Williamson-Hall ligament size in nanoporous gold is numerically for all practical purposes identical to what is found in a screening of other and differently derived nanocrystalline FCC metals. This latter finding is astounding since the microstrain should have quite different origin in nanoporous as compared to nanocrystalline metals.§ THEORY: SCATTERING BY POROUS CRYSTALSIn wide-angle powder diffraction, the scattering vectors and the relative intensities of the Bragg reflections are determined by the atomic structure, i.e. by the symmetry and the lattice parameter of the crystal lattice. The microstructure can be taken, in the simplest approach, to affect the reflection breadth through size and microstrain effects. Unfortunately, the state-of-the-art on microstrain in nanoporous metal does not a priori provide a basis for discussion, so that we will inspect that issue later, along with the experimental results of our work. In contrast, the information on the geometry of dealloying-made metal network structures, as obtained by electron microscopy and small angle scattering, provide an excellent basis for discussing size effects, and this will be the sole focus of the present analysis. Our approach to the impact of porosity on the wide-angle powder diffraction pattern will be based on analogous treatments for isolated particles and nanocrystalline solids, see for instance the detailed description in <cit.>.We analyse the diffraction by a polycrystalline aggregateconsisting of many randomly oriented crystallites (or “grains”). Furthermore, each crystal contains pores, much smaller than the grain size. Because of the random orientation, (the structure is statistically isotropic and the interference function, P, depends on the scattering vector only through its magnitude, q = 4 sin /, wheredenotes half the scattering angle andthe wavelength.Experimental diffraction data also depend on the atomic form factor, absorption, polarization, and incoherent scattering, all of which will be ignored for conciseness. We thus focus on P which, for the isotropic problem, relates to the real-space structure through the radial distribution function (RDF) (r) – the orientation-averaged mean density of neighbouring atoms at distance r from a mean central atom – via P(q) = 4∫_0^∞( (r) - ) sin q r /q r r^2 d r , wheredenotes the mean atomic density, as derived e.g. in <cit.>. Figure <ref> shows schematic graphics of the functions which are discussed here.The RDF of the massive (no pores) crystal lattice, ^ V(r), may be assumed known. This hypothetical massive lattice may contain defects such as dislocations, stacking faults and twin boundaries, which will be reflected in its RDF. A porous crystal may be constructed by cutting away atoms from the massive one. Since this process does not displace any of the remaining atoms, the peaks which represent the successive atomic neighbour shells in the RDF retain their positions but loose height. This loss is embodied by the microstructural envelope function H(r) via(r) = ^ V(r) H(r) The envelope function is the autocorrelation function of a microstructural phase field p(𝐱) which takes the values p=1 and p=0 for positions 𝐱 in the solid phase and in the pore, respectively. It is useful to normalise H to the volume, V^ S, of the solid phase <cit.>, so that H(𝐫) = 1/V^ S∫ p(𝐱) p(𝐱 + 𝐫 ) d^3 𝐱 and specifically H(0) = 1.As we are concerned with isotropic microstructures, we take H to depend on the interatomic spacing 𝐫 exclusively through its magnitude, r. The convolution theorem of Fourier transform implies that the interference function of the porous crystal is P(q) = P^ V(q) ⊗ W(q) with P^ V(q) the interference function of the massive crystal and with <cit.> W(q) = 1/∫_0^∞ H(r) cos q rd r. Equations <ref> and <ref> imply that the effect of porosity is to broaden the very narrow diffraction peaks of the massive crystal by convolution with the cosine-transform of the envelope function. This, in turn, implies that discussing the scattering of porous crystals requires an analysis of the envelope function. Autocorrelation functions such as H(r) of equation <ref> are best known from small-angle scattering studies, since the interference function near the origin of reciprocal space, P^ O(q), is governed by the autocorrelation function of the scattering length density (see, e.g., Section 3.2 in reference <cit.>), which agrees with H(r) except for a constant factor. Designating the scattering length density by N and a characteristic scattering volume by V^ S, the small-angle scattering signal depends on H as P^ O(q) = 4 N^2/V^ S∫_0^∞H(r) sin q r /q r r^2 d r .By definition, H(0) =1. For isolated particles, it is required that H=0 for distances larger than the longest intercept in the particle (the diameter, for the special case of a spherical particle). For the porous crystal, the value of H will converge to that of the solid volume fraction, , at spacings much larger than the characteristic pore- or ligament size. The graph of H(r) at intermediate distances r depends on the degree of order in the pore or network structure. Dealloying-made network structures such as nanoporous gold exhibit significant order, as is evidenced by a pronounced peak in their small-angle scattering at q in the order of the reciprocal ligament size <cit.>.As the microstructure of NPG resembles that of spinodally decomposed solutions <cit.>, the Berk model for small-angle scattering by bicontinuous microstructures <cit.> might provide an instructive approach <cit.>. The model works with the description of spinodal decomposition in terms of level sets in random superpositions of concentration waves <cit.> and achieves an excellent representation of the small-angle scattering interference peak in spinodal microstructures <cit.>. The algebra of the Berk model is unwieldy and the expressions for P(q) are poorly suited for the concise and qualitative discussion of the present case. Therefore, we introduce a much simpler expression as a toy envelope function that has no stringent relation to the geometry of spinodal or other bicontinuous microstructures while it reproduces important characteristics of their scattering.Consider the following envelope function: H̃(r) = exp(- r/L) 2 L/3 π rsin( 3 π r/2 L) .and its transforms W̃(δ q) = L/3 π(arctan(3 π/2 - L δ q )+arctan(3 π/2 + L δ q )) . and P̃^ O(q) = 128 π/16 L^4 q^4+8 ( 4 - 9 π ^2 ) L^2 q^2 + ( 4 + 9 π ^2 )^2. where we have taken N = 1 and, somewhat arbitrarily, V^ S= L^3. We now inspect the properties of the three above toy functions depicted in figure <ref>, bearing in mind that an approximation of the network structure is an array of cylinders with diameter (“ligament size”) L, characteristic spacing d = 2 L between neighbouring ligaments and volume-specific surface area α given by α = 4 / L . For use below we write the Scherrer formula <cit.> in its formδ q_size= 2 π K / L with K the Scherrer constant and L the relevant measure for size. The following may be noted: * As would be expected for structures with a characteristic size L, the function H̃(r) decays to near zero on the scale r → L (see figure <ref>a)).* The initial slope of H̃(r) is L^-1. This is consistent with the requirement that dH̃ / dr |_r = 0 = -α / 4 for microstructures with discrete surfaces <cit.> and with the volume-specific surface area of the idealised cylindrical ligaments, equation <ref>.* A minimum in H̃(r) near r = L (more precisely, at r = 0.911 L, see <ref>a)) agrees with the expectation for bicontinuous structures such as that in the central column of figure <ref>.* The function W̃(δ q) has integral breadth of 1.73 × 2 π / L and full width at half maximum of 1.53 × 2 π / L. Scherrer constants for equiaxed crystallites take values around 1, see for instance <cit.>. The present numerical values, 1.53 and 1.73, are larger but of similar order of magnitude. Within the scenario of our toy function, these values represent the Scherrer constants that relate the reflection breath to the ligament diameter.* Asymptotically at large momentum transfer, W̃(δ q) varies as (L/δ q)^-2, consistent with the Lorentzian behaviour expected by scattering from microstructures with discrete surfaces.* The small-angle scattering function P̃ ^ O(q) has a pronounced interference peak at q_ I = 1.47 π / L.The presence of the maximum agrees with observations for dealloying-made nanoporous gold and for spinodally decomposed microstructures, and its q-value agrees well with the estimate q_ I = 1.23 × 2 π / d for objects with a characteristic distance d = 2L.* Asymptotically at large q, the small-angle scattering intensity varies as P̃ ^ O(q) = 2 πα q^-4. These are the expected power law and prefactor for a structure with discrete surfaces and specific surface area α. As an essential conclusion, the envelope function of the porous crystal can be separated into the sum of two contributions: H(r) = (1- ) Ĥ(r) +where the information on the characteristic size and correlation of the ligament network is contained in the r-dependent part, Ĥ(r), whereas the constant represents the mean solid fraction. The former is expected to behave qualitatively as our toy function H̃(r). As a consequence of equations <ref> and <ref>, the Bragg reflection shapes will then appear as schematically illustrated in figure <ref>. The constant part of H, which transforms into a δ-function, contributes a sharp central part of the reflection, whereas the r-dependent part contributes an additive peak, of width in the order of the reciprocal ligament size, which appears as a broad foot in each Bragg peak.§ VIRTUAL DIFFRACTION§.§ Sample preparation and thermal relaxation As a verification of the theory in the previous Section, we performed a “virtual”<cit.> diffraction experiment, in which the powder diffraction pattern was computed based on an atomistic model of nanoporous gold obtained via molecular dynamics simulation. The diffraction pattern was then evaluated by the identical procedures as experimental data, see Section V below.We created a nanoporous gold microstructure via imitating spinodal decomposition of a binary mixture on an FCC lattice, using the Metropolis Monte Carlo algorithm <cit.>. Details of the simulation can be found in reference <cit.>. Figure <ref> shows a snapshot of the as-created structure. The corresponding simulation box has the size of 150 × 150 × 150 lattice spacings (61.2 × 61.2 × 61.2 nm) and ⟨100|$⟩ edges. The solid fraction of this structure is0.2992. The alpha-shape surface reconstruction algorithm (with a probe radius of3Å ) <cit.> gives the ratio of surface area per solid volumeα=0.997/nm. According to equation <ref>, assuming cylindrical ligaments, this corresponds to a ligament size of [4]nm.Molecular dynamics (MD) simulations were carried out with the simulation code Lammps<cit.>, using an embedded-atom method potential for gold <cit.>. First, the atom positions were athermally relaxed via an energy minimization using the conjugate gradient algorithm. The relative change in energy and force tolerance at convergence were less than10^-12and10^-4eV/Å, respectively. Then, the virtual sample was thermally relaxed for 1 ns at 300 K under zero external load. Periodic boundary conditions were applied in all simulations.Two states of the model were investigated in the virtual diffraction experiment: Firstly, the nanoporous structure in its unrelaxed state, that is, the truncated ideal lattice. This represents an idealised porous crystal as it is created by the Monte Carlo algorithm. All atoms occupy sites of the periodic crystal lattice and there are no strain and no thermal displacement. This structure emulates the one investigated in the theory of Section II. Secondly, the nanoporous structure in its relaxed state. The structure is represented by a snapshot of the configuration that emerges from the relaxation by MD as described above. It incorporates static lattice strain as well as the instantaneous atomic displacement due to thermal vibrations at the instant of the snapshot.Figure <ref> shows the amount of local von Mises strain in a colour coded map cut along a crystallographic⟨100 ⟩plane. In order to eliminate contributions of the thermal vibrations to the local strain of the atoms, the structure was, after its relaxation at [300]K, quenched at [0.01]K followed by an additional relaxation for [1]ns at [0.01]K. Strain gradients inside the ligaments are clearly discernible even far from the surface. Regions of high and low strain divide the volume into regions of a size which correlates to the ligament size. §.§ Diffraction pattern simulation The calculation of the X-ray diffraction used the procedures presented in reference <cit.> using wavelengthsλ_1=[1.54056]Åandλ_2=[1.54439]Åwith an intensity ratio of 2:1 for Cu K_αradiation and the atomic form factor of gold <cit.>.This kind of virtual diffraction on atom positions as calculated by molecular dynamics simulation or density functional theory is an established procedure and was done in similar ways in <cit.>. Basically, we calculate a powder pattern of cuboid particles, which are identical to the MD simulated nanoporous gold. This represents well the situation in the lab experiment described in the following section where coarse grained polycrystalline gold is penetrated by a nanoporous network. The simulation box represents a single grain and the use of the Debye formula to calculate the diffraction pattern (compare equation (<ref>) and ref <cit.>) perfectly mimics the polycrystalline character of the samples.The first step is the determination of the orientation-averaged radial distribution function (RDF). This function is discontinuous for the as-constructed sample, with non-zero values only in the discrete neighbouring atom shells. This is shown in the inset of figure <ref>. The reorganization of the RDF into a coarse histogram with a grid size of [5]Å results in a smooth function looking strikingly similar to the one shown in the centre of figure <ref> of the nanoporous microstructure without the crystalline contribution.In contrast to the situation described in the preceding paragraph, we do not use periodic boundary conditions when we calculate the RDF. This means the RDF does not reach a constant value corresponding to the mean solid fraction but decays to zero for values larger than the box diagonal. This second envelope function is necessary because it provides a smooth transition for the Fourier transform necessary to calculate the scattering function. Otherwise, heavy oscillations are produced by the calculation. But size of the MD box is one order of magnitude larger than the ligament size. Therefore, this additional envelope function does not influence the broad part of the peaks but only the sharp peaks representing the scattering signal of the crystalline host material. For the thermally relaxed structure, this additional broadening can be readily neglected.The RDF for a snapshot of the relaxed sample is shown in figure <ref>. As a consequence of the relaxation,, the discrete shells have been replaced by broadened peaks.The next step was the calculation of scattered intensity versus diffraction angle from the RDFs according to the procedure in <cit.>. Figure <ref> compares the diffraction patterns for the unrelaxed and relaxed samples.In view of the considerations in Section II it is remarkable that the unrelaxed sample indeed shows Bragg reflection superimposing a narrow central component and a broad basis. This is particularly evident for the (200) and (311) peaks. The (111) and (220) peaks exhibit satellite maxima, displaced by∼0.6 - 0.7^∘on the2θscale from the peak centre. The angle shift corresponds to a distance of roughly [12 - 14]nm in real space, much larger than the ligament size of [4]nm. This means, these satellites most probably are higher order reflections of the simulation box itself.The red line in figure <ref> represents the virtual diffraction pattern of the relaxed sample. It is seen that the sharp central peak has been lost and that uniformly broadened reflections have emerged. We emphasize that the broadening cannot be the signature of thermal disorder, since that signature would not be broadening but a reduction in the Debye-Waller factor or, in other words, a reduction in peak height <cit.>.Therefore, the broadening can be attributed to a loss of long-range lattice coherency. This effect may be attributed to either a reduction in the coherently scattering domain size or to microstrain. For the relaxed sample, both values can be determined from the dependency of the peak width on the peak position. The analysis procedure is described in Section IV below. Here, it suggests a domain size of[7.8±0.4]nmand a microstrain of0.61±0.07%.§ EXPERIMENTAL PROCEDURES§.§ Preparation of nanoporous samplesAg-Au alloys, at ratio, Ag_75Au_25(subscripts in atomic%), were produced by arc-melting the pure metals. A thermal annealing with the ingot sealed in fused silica under Ar was performed for[5]dat[925]^∘ Cin order to homogenise the Ag-Au alloy by bulk diffusion. The following step-wise cold-rolling procedure (each step to[60]%reduction of the initial thickness) of the alloy ingot (interrupted by a[5]minthermal relaxation period at[650]^∘ C) achieved a foil thickness of[150]μm. For samples denoted as “Alloy, cold-worked”, the recovery after the last rolling step was omitted. Afterwards, all foils were laser-cut to give circular discs of[5]mmdiameter.Selective Ag dissolution (chemically and electrochemically) created NPG samples. We used three different dealloying protocols as exposed in details in reference <cit.>. In brief: *Route A: dealloying in [1]MHClO4 against a coiled Ag wire as counter electrode (CE) under a stepped dealloying potential regime (potentials were [1050]mV for [24]h, [1100]mV for [8]h, [1150]mV for [8]h and [1200]mV for [10]h vs. a Ag / AgCl reference electrode (RE)). * “Fast dealloying” refers to samples made by Route A but at constant and unusually positive dealloying potential in order to achieve a brute force, very fast corrosion. Dealloying potentials were [1200]mV, “[1400]mV” or “[1600]mV”. The dealloying was here more strongly in the oxygen species adsorption regime, so that surfaces of the samples are likely covered by adsorbed oxygen species or oxide. *Route B involved potentiostatic dealloying according to Wittstock et al. <cit.> in [5]MHNO3 against a Pt CE under a constant dealloying potential of [60]mV for [24]h vs. a Pt pseudo-RE. *Route C samples were produced under open-circuit conditions according to Rouya at al. <cit.>, i.e. by immersion into thermostated ([65]^∘C) semi-concentrated HNO3 for [24]h. Further NPG modifications were achieved using electrochemical and thermal treatments, i.e. samples AD(A) (as produced by route A) that are noted with “[-500...+800]mV” or “[-500...+1300]mV” underwent an additional electrochemical treatment, i.e. potential cycling (CV) within the respective potential window for[15]cyclesat[5]mV/s. Samples AD-A noted with “[10]min” or “[6]h” underwent an additional thermal treatment at[300]^∘Cfor the respective duration under air.All electrochemical processes were controlled by a potentiostat (Metrohm Autolab PGStat10).Scanning electron microscopy (SEM) was performed to estimate the bulk ligament size,L_SEM, after sample fracture and cross-view recording at a Leo Gemini 1530 equipped with an energy-dispersive X-ray spectroscopy (EDS) sensor by Oxford instruments.L_SEMwas estimated by averaging 20 diameters readings and taking the standard deviation as the absolute error. §.§ Diffraction X-ray diffraction used a powder diffractometer (Bruker D8 Advance) in-focusing Bragg-Brentano geometry, with Ni-filtered CuK_radiation and a linear position-sensitive detector (LynxEye). Bragg reflection parameters were obtained from fits byK__1/2doublets of split Pearson VII functions.Average lattice parameters were refined using the appropriate variant of the Nelson-Riley approach along with a correction for stacking fault displacements. The algorithm also supplied an estimate of the stacking-fault probability, see details in reference <cit.>. The Williamson-Hall analysis assumed Gaussian strain and Cauchy-type size broadening <cit.>, following the procedures of reference <cit.> which were validated in virtual experiments based on MD-generated samples of nanocrystalline metal <cit.>. The instrumental line broadening, as obtained using the NIST Standard Reference Material SRM 660b, was corrected for.§ RESULTS AND ANALYSIS§.§ Sample characterization Figure <ref> illustrates the sample geometry and the microstructure. Part A shows the master alloy while part B shows a nanoporous sample after dealloying by Route A. The nanoporosity is seen in the scanning electron micrograph, part C. The ligament size,L_SEM, for this example was determined as[21 ±4]nm. Nanoporous gold samples made by dealloying contain residual silver. The residual silver atom fraction,x_Ag, monotonously decreases with increasing ligament size [53]. Table <ref> indicatesx_Agvalues for the samples of the present study. §.§ Experimental diffraction patterns and data analysis Diffraction patterns were recorded in the angle range30^∘to130^∘, covering the Bragg reflections up to (420). Figure <ref> exemplifies the results. All peaks correspond to the FCC crystal lattice of gold. As compared to the master alloy (blue), the peaks of NPG made by slow dealloying (green) are broadened, and fast dealloying (red) leads to even stronger broadening.Similar results were obtained for all samples.The relative intensities of (111) and (200) reflections varied between samples; we attribute this to statistical variation arising from the small number of irradiated crystallites. As the most notable observation in the context of our discussion, the reflection profiles of the dealloyed samples appear uniformly broadened; in no case did we observe the bimodal peak profile which is predicted for the strain-free porous crystal with a truncated lattice (as in the right column of figure <ref>).As a basis for the evaluation of lattice parameters and for the Williamson-Hall analysis, all Bragg reflections in each data set where fitted withK__1/2doublets of split Pearson VII functions. Figure <ref> exemplifies the typical quality of the fit, here for the (200) reflection of a slowly dealloyed sample withL_SEM=[38]nm. The fit function is seen to provide a good representation of the experimental data, except for a small deviation near the peak maximum. Similar results could be obtained in each case.The evaluation of lattice parameters was based on the maximum intensity positions from the fit. Figure <ref>a) exemplifies the single-peak lattice parameters after correction for height misalignment and stacking-fault induced peak shifts <cit.>; the refined lattice parameter and its confidence intervals were determined as the mean and variance of such data sets. Figure <ref>b) exemplifies the modified (Cauchy/Gauss) Williamson-Hall analysis; combinations of peak width and position inq-space are plotted so that the slope of the straight line of best fit scales with the apparent grain size while the ordinate intercept measures the microstrain. We analysed both, the integral reflection breadth (red data points) and the full width at half maximum (FWHM, blue data points), applying separate Scherrer constantsKas established in reference <cit.>. It is seen that the two separate measures for the peak broadening provide consistent results, and that the data agrees well with the straight-line behaviour expected for a combination of Cauchy-type size broadening and Gaussian strain broadening. The apparent grain size,D, and microstrain,⟨ε^2 ⟩^1/2, are obtained from the straight line of best fit to the data. In the example, they areD = [37.2±1.2]nmand⟨ε^2 ⟩^1/2 = (0.086 ±0.016)%. §.§ Lattice parameter Figure <ref>a) shows the mean lattice parameters,a, of the initial alloy and resulting porous samples with their standard errors as determined from their peak positions in dependence of the Williamson-Hall coherently scattering domain sizeD_W-Has evaluated out of the FWHM and their dependency onq. Theoreticalavalues for bulk Au ([407.82]pm), bulk Ag ([408.53]pm) and the master alloy ([408.35]pm) are indicated for comparison. Regarding the data of the alloy (note that data points for cold-worked and recovered master alloy samples superimpose on the scale of the figure), we find a negligible deviation to the theoreticala.Values forafound for the nanoporous samples were compared with the bulk Au value. In principle, the residual silver increases the lattice parameter value, yet the silver fraction, between[1.3]at.%and[6.4]at.%, is too small to affect the experiment. Focusing on the slowly dealloyed samples – that is, on samples with clean surfaces – we find the lattice parameter to decrease with decreasingD_W-H. At the smallest ligament size, the crystal lattice is compressed by[0.03]%. Lattice parameters of the “fast dealloyed” samples– that is, samples with a trend for oxygen-covered surfaces – show a large scatter and an upward deviation from this trend.Figure <ref>b) displays the results for the slowly dealloyed (clean surface) samples in isolation. The lattice parameter is here plotted versus the inverse of the “true” ligament sizeL_SEM. The straight line of best fit (dashed line) confirms the trend of compression at small size.§.§ Ligament size, Apparent Grain Size, and Microstrain Next, we inspected whether the Williamson-Hall analysis achieves a valid separation of size and strain induced broadening. More specifically, we are interested where the “Williamson-Hall size”,D_W-H, agrees with the “true” ligament size as determined by SEM. By inspection of figure <ref> it is obvious that the two measures for the size are indeed highly correlated. Except for the largest ligament size (100 nm), which approaches the resolution limit of the diffractometer, the Williamson-Hall size appears to agree with the true ligament size except for a constant upward shift by in the order of 15 nm. The correlation between the true and Williamson-Hall ligament sizes is remarkable in view of the failure of our theory – in Section <ref>– to predict a link between a simple peak-width parameter andL.The Williamson-Hall analysis separates the contributions of size and microstrain to the peak broadening. Figure <ref>a) displays the results of the analysis for the microstrain, plotted versusD_W-H. It is seen that the results for the master alloy vary substantially, depending on whether the alloy is in its cold-worked state or recovered. The amount of variation affords an estimate of the contribution of lattice dislocations (from the cold working) to the microstrain. The microstrain values of the nanoporous samples cover a wider range, including states of very low microstrain, less than the recovered master alloy, and –especially for the fast dealloying samples– states of very high microstrain, higher than the cold-worked master alloys. It is also observed that the microstrain values of the nanoporous samples systematically increase with decreasing ligament size. This is further emphasised by comparing the present data to the trend line from earlier work<cit.> on nanocrystalline FCC metals (dashed line in the figure). The data for the size-dependence of the microstrain of the nanostructured samples is in excellent agreement with that completely independent data. The figure also shows the results of the Williamson-Hall analysis of our virtual diffraction experiment. This data is also precisely consistent with the trend.On top of the microstrain data from the Williamson Hall analysis, we have also obtained estimates for the stacking fault densityα_sf. This parameter, which represents the probability of a stacking fault in (111) direction, manifests itself in relative shifts in the Bragg reflection positions according to <cit.> and is obtained along with our lattice parameter refinement, see reference <cit.> for details. The light blue columns in figure <ref>B) show the results. The most obvious finding is thatα_sfin the nanoporous material is systematically much lower than in the master alloy. This applies even when the nanoporous material is compared to the recovered master alloy.The observation on the stacking faults confirms and emphasises the observation based on the microstrain data: Even though dealloying involves a massive rearrangement of the constituents of the alloy at low homologous temperature –a situation that might typically be expected to lead to defect accumulation– the crystal lattice in the nanoporous product phase is typically more perfect than that of the initial alloy.In order to emphasise this point we have deliberately omitted the recovery step and compared the microstrain in the cold worked master alloy to that of the nanoporous material obtained by dealloying that master alloy. Dealloying was found to decrease the microstrain from initially(0.20±0.03)%(cold-worked master alloy) to(0.08±0.05)%. This is essentially the same as after a five-minute,650^∘Canneal of the master alloy, which gave the microstrain value(0.09±0.02)%.§ DISCUSSION Our theory for the powder diffraction of nanoporous crystals suggests that each Bragg reflection should have two distinct components, a narrow central component that represents the long-range coherency of the crystal lattice plus a broad foot that arises from the partial loss of atomic neighbours due to the nanopores. Our virtual diffraction experiment verified the prediction by inspecting a 61-nm size single crystalline box from which atoms were removed to create the pore space. Indeed, the virtual diffraction pattern of the unrelaxed nanoporous body confirmed the theory. Yet, thermal relaxation by MD-simulation lead to a complete loss of the narrow component. Williamson-Hall analysis of the peak broadening revealed large microstrain, suggesting elastic distortion of the crystal lattice is the origin of the loss of the metal component. The elastic strain fields of lattice dislocations might be considered as an explanation. Yet the microstrain value of 0.61% is unusually high. Furthermore, it is known that the dislocation cores tend to move into the pore space of nanoporous gold, reducing the amount of microstrain in the crystal lattice <cit.>. One may therefore speculate that the observed broadening relates to strain gradients at the scale of characteristic ligament dimensions. In the experimental part of our study we have analysed powder diffraction from disc-shaped samples of NPG, exploring various synthesis conditions as investigated in the earlier study reference <cit.>. We found that the apparent domain size,D_W-H, from Williamson-Hall analysis of our diffraction data is systematically correlated with the ligament size,L_SEM, as determined by SEM images. The correlation is, roughly,D_W-H = L_SEM + 15 nm.The observation of a systematically too high apparent domain size is somewhat unexpected in view of diffraction studies on nanoparticles, where the apparent domain size is invariably reduced whenever there are stacking faults or twin boundaries. On the other hand, x-ray diffraction and electron microscopy analysis typically work with different averages over the size distribution. The volume-weighted averaging in diffraction studies supplies larger values of the mean domain size when compared to the area-weighted analysis off electron micrographs <cit.>. Furthermore, the ligament size deduced from the SEM images represents a mean ligamentdiameter, while the diffraction is also sensitive to the longer coherence length along the ligamentaxis.The next point of the discussion treats the lattice parameters and specifically the observation that samples which were dealloyed at moderate dealloying potential and for which we expect clean surfaces showed a systematic trend for decreasing lattice parameter with decreasing ligament size.The amount of residual Ag in these samples ranges from 1.3 to 6.4 at%, with more silver <cit.> and hence larger lattice parameter at smaller ligament size, see table <ref>. Magnitude and sign of this trend are not consistent with the observations. Yet, the trend for lesser lattice parameter at lower ligament size is naturally consistent with the action of the relevant capillary force, the surface stress,f. While reversible variations in macroscopic dimension <cit.> and in lattice parameter <cit.> of NPG due to changes infhave been well documented, the absolute value offin NPG appears not to have been measured. We now present an analysis that affords an estimate of that quantity.The surface stressfis known to result in a pressure,p, in the underlying solid according to <cit.> p=2/3α⟨ f ⟩ .The surface area per solid volume,α, is connected to the mean ligament sizeLvia roughlyα= 4/L, as stated in equation <ref> for cylindrical ligaments. The surface-induced pressure changes the volume of ligaments with an elastic bulk modulus ofBaccording top= - B Δ V/V_0≈ -3 B Δ a/a_0 .Combining these two equations (as in reference <cit.> for nanocrystalline materials) leads to a relation for the lattice parameter,a = a_0 - a_0 8/9B ⟨ f ⟩1/L .This equation can be compared to the regression line in figure <ref>. Taking the bulk modulus for Au as [171]GPa<cit.> this suggests a mean surface stress value of⟨f ⟩= [3.1]N/m. Typical values for unreconstructed Au surfaces determined via ab-initio studies as presented in <cit.> range between2.0 - [3.3]N/m. This remarkable agreement strongly hints to surface stresses as the origin of the reduction in lattice parameter in NPG at small ligament size.Contrary to the slowly dealloyed samples, the ones produced by fast dealloying show largeraand a possible trend for increasedaat faster dealloying. Dealloyed NPG is known to develop silver-rich clusters in the centre of its ligaments <cit.>. Taking into account the short time for structure development, it is conceivable that ligaments in fast-dealloyed NPG have a very high silver concentration in their centre, which mainly contributes to the X-ray scattering signal while a gold rich outer shell, if thin enough, does not produce noteworthy coherent scattering signal. This would result in an increased lattice parameter.Alternatively, the trend for larger lattice parameter in the fast dealloying samples might be linked to the higher oxygen coverage in the samples along with the trend for oxygen adsorption to make the surface stress of nanoporous gold less tensile. This is supported by in-situ dilatometry data for NPG <cit.> and by in-situ x-ray studies, which show lattice parameter expansion during oxysorption <cit.>.The large microstrain at small ligament size are also of interest. In their diffraction study of NPG, Schofield et al. <cit.> observe an asymmetry of the Bragg reflections, with a broad component shifted to lesser diffraction angles. They discuss this component as representing regions of positive (expansion) strain. In fact, when the ligaments of NPG are modelled as idealised long cylinders, then the action of surface stress is to compress the cylinder axially and expand it in the radial direction <cit.>. The Bragg reflection asymmetry in reference <cit.> is therefore naturally consistent with surface-induced strain in nanoscale structures – the same region tends to be strained differently in different orientations. Yet, our study failed to observe a significant peak asymmetry in either, the experimental or the virtual diffraction patterns. The origin of this discrepancy is unclear. One observation is that reference <cit.> reported the asymmetry only for samples investigated very shortly after dealloying; this might point towards a transient state of very freshly dealloyed NPG. Such states were not explored in the present study.The high microstrain values in the experimental part of our study are remarkable. Perfectly consistent behaviour was obtained by virtual diffraction. Furthermore, an independent determination of the microstrain based on evaluating atomic positions in relaxed nanocrystalline materials confirmed the same observation in that case. It is therefore significant that the microstrain in nanocrystalline materials –that is, massive (not porous) polycrystals with nanoscale grain size– follow precisely the same trend as the present data. This is seen by comparing the present data to the dashed line in figure <ref>; the comparison is on an absolute scale with no fitting parameters. Local lattice strains within the given grain arrangement (which we referred to as microstructural constraints) were caused by the discrete nature of grain sizes in different crystallographic directions and the corresponding lattice plane spacing. Since the pore space around the ligaments is empty, that explanation cannot apply here. As a source of the high microstrain values we may, once again, inspect the influence of surface stress. The surface-induced strains increase reciprocal to the ligament size <cit.>. The anisotropy of the strain in the bulk which is necessary to cause X-ray peak broadening does not necessarily result from an anisotropy of that surface stress state. An aspect in the ligament shape, as e.g. a cylindrical shape, causes expansive strain along the axis of the cylinder and compressive strain perpendicular to it, simply because of the geometrical conditions even for an isotropic surface stress. Such an anisotropy of the strain state in the ligaments results in microstrain when the signal is spatially averaged (what is done during calculation of the autocorrelation function in virtual diffraction and what is the case in the experimental samples because of their polycrystallinity), quite analogous to the nanocrystalline case. Another probably contributing factor comes up when following the argumentation line presented in <cit.>. The anisotropy of the gold crystal lattice itself already totally suffices to cause a considerable increase of microstrain with increasing stress which, in this case, is not externally applied but internally induced by the ligament surface.§ CONCLUSIONS Imaging techniques, such as electron microscopy provide only limited access to structural parameters and are in many cases sources for misinterpretations of synthesis-property relationships and fundamental corrosion processes. In contrast, diffraction experiments provide a simple access to numerical values of ligament size, lattice parameter and microstrain which are representative for the whole sample. The good correlation between the ligament size determined by analysis of SEM pictures and coherently scattering domain sizes as determined by the analysis of X-ray peak broadening as shown in figure <ref> proofs the applicability of X-ray line profile analysis as a suitable tool to investigate microstructural properties of nanoporous metals.With respect to the named synthesis-property relationships we outline the following points as key results from our investigations:*Both grain size and ligament size should contribute to the peak broadening in X-ray diffraction. For NPG fabricated by dealloying thermal relaxation annihilates the lattice coherency over the (large) grain size leaving the ligament size as the only source of domain size broadening convoluted with microstrain in the X-ray reflection peaks.*Ligament sizes as derived by Williamson-Hall analysis of diffraction data are consistent with these observed by SEM over a range of 5 - [90]nm with a systematicunderestimation of about [15]nm. Nevertheless, X-ray diffraction as a non-destructive technique can be utilised to determine ligament sizes in a faster way that is less prone to subjective errors.*NPG with very small ligament size (below [10]nm) shows extraordinary high values of microstrain in simulated as well as in real samples. Surface stresses acting on the ligaments are the most probable reason for that.*The lattice parameters in NPG can be smaller than the theoretical value of Au. A systematic dependence of this deviation of the lattice parameter on the ligament size can be observed in case of slow dealloying techniques. A mean surface stress of ⟨ f ⟩ = [3.1]N/m in agreement with ab-initio calculations could be evaluated.*Comparing values of microstrain and stacking fault densities between the cold-worked initial alloy and the final NPG leads to the conclusion that dealloying represents a strategy to relax the Au lattice even more than the thermal treatment of the parent alloy. This work was supported by German Research Foundation (DFG) through FOR2213 “NAGOCAT”, subproject 3. The authors gratefully acknowledge the computing time granted by the John von Neumann Institute for Computing (NIC) and provided on the supercomputer JURECA<cit.> at Jülich Supercomputing Centre (JSC), project HHH34.apsrev | http://arxiv.org/abs/1708.07789v2 | {
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"title": "X-Ray Studies of Nanoporous Gold: Powder Diffraction by Large Crystals with Small Holes"
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[pages=1-last]ms.pdf | http://arxiv.org/abs/1708.07808v1 | {
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Shell et al.: Bare Demo of IEEEtran.cls for IEEE Transactions on Magnetics JournalsMultilevel converters have found many applications within renewable energy systems thanks to their unique capability of generating multiple voltage levels. However, these converters need multiple DC sources and the voltage balancing over capacitors for these systems is cumbersome. In this work, a new grid-tie multicell inverter with high level of safety has been designed, engineered and optimized for integrating energy storage devices to the electric grid. The multilevel converter proposed in this work is capable of maintaining the flying capacitors voltage in the desired value. The solar cells are the primary energy sources for proposed inverter where the maximum power density is obtained. Finally, the performance of the inverter and its control method simulated using PSCAD/EMTDC software package and good agreement achieved with experimental data. Multicell inverter; Grid-tie multilevel inverter; Maximum Power Point Tracking; Flying capacitor; Energy storage devices. Design, Engineering and Optimization of a Grid-Tie Multicell Inverter for Energy Storage Applications A. Ashraf Gandomi 1, S. Saeidabadi 1, M. Sabahi 1,M. Babazadeh 1, and Y. Ashraf Gandomi 2 1Department of Electrical and Computer Engineering,University of Tabriz, Tabriz 2Electrochemical Energy Storage and Conversion Laboratory,Department of Mechanical, Aerospace and Biomedical Engineering,University of Tennessee, Knoxville, Tennessee, USAManuscript submitted August 26, 2017. Corresponding author: A. Ashraf Gandomi (email: [email protected]). December 30, 2023 =============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================§ INTRODUCTION The application of renewable energy sources (i.e. solar wind energy) into an electric grid requires high performance energy storage devices along with various types of power electronics (i.e. rectifiers, converters and inverters).Figure 1 includes the schematic of a hybrid energy storage system in which a renewable energy source (here photovoltaic modules) along with an energy storage device has been implemented to the electric grid via the utilization the multilevel inverters <cit.>.As shown in Fig. 1, battery-based devices and hydrogen-based energy storage technologies are promising. A good review on the battery- and hydrogen-based energy storage has been provided recently <cit.>. For hydrogen based energy storage, an electrolyzer along with a polymer electrolyte membrane (PEM) fuel cells are utilized. Although being very promising, the water management for electrolyzers and PEM fuel cells remains a challenging issue to be addressed <cit.>. Also, the hydrogen gas storage technologies and infrastructure for regenerative hydrogen fuel cell systems is required to be addressed appropriately <cit.>. Several battery technologies are developed for energy storage technologies that redox flow batteries are among the most promising devices . However engineering the redox flow batteries for optimum operation requires detailed mathematical models <cit.> and experimental diagnostics <cit.>.The application of multilevel converters for the renewable energy systems has been increased recently <cit.>. This is mostly because of high energy output (in the range of MW), high efficiency and low electromagnetic interference being offered via their application <cit.>.Three-level converter first proposed by Nabae <cit.> and the converters with increased number of levels developed later to obtain higher levels of voltage and reduce the harmonic content of the output voltage. However, increasing the number of voltage levels results in increased system complexity and severe voltage balancing problems. Different topologies for these types of converters have been utilized including Neutral Point Clamped (NPC) and variant types of multicell inverters <cit.>. The NPC topology suffers from diode clamped and capacitor voltage balancing issues where the multicell inverters do not suffer from the same issue. Therefore, multicell inverters are promising candidates to generate increased voltage levels for high power generation applications. The multicell inverters have different types including the Cascaded Multicell (CM), Flying Capacitor Multicell (FCM) and Stacked Multicell (SM) inverters <cit.>. The FCM inverters along with the SM inverters have many unique advantages for medium voltage applications such as operation without transformer and naturally capacitor voltage balancing ability. A comparison of the different components being used with these different inverters for generating levels had been provided in Table. I.Table. I. comparison of different inverter’s componentsThe advantages of DFCM compared to conventional structures are eliminating the common point of DC sources, reducing the number of input DC sources, switches and capacitors. However, this structure is an island and in the grid-tie form of this inverter with presented control method, some difficulties arise including the power transfer from grid to input source and the possibility of discharging flying capacitors in case of connecting to the Photovoltaic (PV) cells. In this paper a new topology for grid-tie multicell inverter has been developed. The proposed inverter has been designed based upon the DFCM inverter and accordingly shares the common advantages associated with the DFCMs including the elimination of common point of DC source, reduced number of DC sources and increased number of voltage levels. Also, the new design assures higher level of safety and via its utilization, the common problems related to the DFCMs has been eliminated (while extracting the maximum power from PV cells).§ PROPOSED MULTILEVEL INVERTER The proposed inverter is shown in Fig. 2. In this structure independent of the number of inverter cells and generated voltage levels, only the switches S_n and P_n in cell_n are bidirectional. The oscillations of the PV cells voltage is mostly caused by climatic oscillations. In case of low PV cells and because of the cascaded diode of PV cells would not be functional and therefore, the possibility of discharging flying capacitors or grid voltage on the PV cells increases. Discharging the grid in the positive half cycle, the negative half cycle and discharging the flying capacitors are shown in Fig. 3. According to Fig. 3, in the proposed inverter, by using only two bidirectional switches, there would be no chance of discharging the grid and the flying capacitors on the PV cells. Table II, includes the switching states within the proposed inverter for generating 5-level output voltage. In this table, when the switch is on, the corresponding state number is one and for the off position it is zero. Also X in this table due to modes of charging or discharging of flying capacitor C_1, is either one or zero. Based on this table, in the states 2 and 4, charging or discharging of C_1happens naturally, while in the other states, due to value of X, C_1 can be charged. The voltage balancing algorithm of C_1 in positive half cycle is shown in Fig. 3. Based on this figure, if the measured value of voltage of capacitor C_1 is lower than its set value (v_C1), P_2 (or S_2) will be on in the state 1 (or 3). This can continues until the voltage of C_1 reaches its specified value then P_2 will be off.§ PROPOSED CONTROL METHOD In the control method, it is assumed that the inverter output current tracks the favorite sinusoidal reference current. Also as it is shown in Fig. 5, it is assumed that the proposed inverter (base on IEEE 1547 standard) does not exchange any reactive power with the grid and only active power exchanges, therefore: v_s(t)=V_msin(wt)i_r(t)=I_m.sin(wt)Figure 6 shows five output voltage levels that have been considered for the proposed inverter. The proposed control method should be capable of injecting power from inverter to grid at different operating conditions and therefore, the maximum voltage generated through the inverter (E_1) should be greater than the grid’s maximum voltage (V_m).Due to generating output voltage with low Total Harmonic Distortion (THD), the values of output voltages are considered as follows:E_2=E_1/2,E_3=0,E_4=-E_1/2,E_5=-E_1The grid voltage and reference current is calculated according to (4) and (5) assuming the circuit current in the constant t_k is i_s(t_k):v_s(t_k)=V_msin(wt_k)i_r(t_k)=I_msin(wt_k)The equivalent circuits of the proposed inverter are shown in Fig. 7.In Fig. 7, R_l and L_l are the line resistance and inductance between inverter and grid and V_0. According to Fig. 7(a):L_1di_s(t)/dt+R_li_s(t)=V_0+v_s(t)=V_0+V_msin(wt)Then:i_s(t_k)=I_ki_s(t)=K_1e^-R_lt/L_l+K_2sin(wt)+K_3cos(wt)+V_0/R_l K_1=I_k-V_m/R_l^3+L_l^2w^2R_l+V_mL_lw/R_l^2+L_l^2w^2/e^-R_lt/L_l K_2=V_m/R_l^3+L_l^2w^2R_l K_3=-V_mL_lw/R_l^2+L_l^2w^2In constant t_k, i_s(t_k) is measured. Control system using (8), calculates five current for a five-level output voltage of inverter during the next switching period. Therefore, for five output voltage levels, five cost functions can be written as follow:ΔI=|i_r)t_k+T_s)-i_s(t_k+T_s)|From (12), proper switching state and consequently proper output voltage can be applied. Initially, when the inverter connects to the network, the initial voltage of the capacitor is zero. At this point, no impulse current is drawn until the capacitor voltage reaches the proper value where the grid current is bypassed by S_1, S_2 and P_3 and simultaneously input source charged the capacitor by S_2 and P_2. When the capacitor voltage reaches its proper value, control method will follow their normal operation. In Fig. 8(a) and (b), upper waveforms are inverter output voltage and the lower waveforms are the inverter output current. Based on Fig. 8(a), if the initial control method does not apply, 60% of the original period (0.012 seconds) would be required for inverter to reach its own stable performance. This situation also results in over-drawn current as high as four times of the maximum nominal current through the circuit, which potentially damages the whole circuit. However, this situation has improved in Fig. 8(b) where the response time has decreased to about 0.0005 seconds. Therefore, the inverter only requires approximately 2.5% of the original period to reach its own stable performance. Also in this case, the drawn current is within the safe range.§ FLYING CAPACITOR VOLTAGE BALANCING METHODIn this section, the method of capacitor voltage balancing is investigated. Based on Table II, the capacitor is discharged in the duration that output voltage equals to E/2 with positive current of circuit and output voltage equal to -E/2 with negative current of circuit. Based on the equivalent circuit of capacitor discharging states (Fig. 7(b) and (c)):v_s(t)=V_msin(wt) i_s(t)=I_msin(wt) v_s(t)=V_c(t)+L_ldi_L(t)/dt+R_li_s(t) v_c(t)=√((V_m-R_lI_m)^2+(L_lI_m)^2)sin[wt+arcsin(-L_lI_m/√((V_m-R_lI_m)^2+(L_lI_m)^2))]In the equations formulated earlier ((13) to (16)), is the grid voltage and v_c(t) is the capacitor voltage. Also i_s(t) is the circuit current that should be the sine wave because of tracking the reference sine wave. According to Table II, unlike the case of the capacitor discharging state that was possible only in two operating modes, charging of capacitor happens in all five operating modes. When the inverter output voltage is equal to E/2 with negative current of circuit and equal to -E/2with positive current of circuit, capacitor is charged naturally. But in the duration that output voltage is equal to E or zero, charging of capacitor is controlled by switch P_2 and within the range that output voltage is equal to -E or zero, charging of capacitor is controlled by switch S_2. The capacitor charging circuit in all five modes is shown in Fig. 7(d). In this figure, R_s is the resistance of switches per current loop, therefore the following equations for a capacitor charging state can is obtained: E=R_si_c(t)+v_c(t) i_c(t)=cdv_c(t)/dt v_c(t_1)=V_1 v_c(t)=E+V_1-E/e^-t_1/R_sCe^-t/R_sCIn the (17) and (18), v_c(t) and i_c(t) are the voltage and current of capacitor respectively. The value of R_s is very small, therefore the time of capacitor charging is short and capacitor is charged rapidly. However, capacitor discharging state is a swinging relation. Thus, capacitor voltage can be adjusted to the desired value just by controlling the charging state of capacitor voltage.§ DETERMINATION OF CAPACITANCE OF FLYING CAPACITORThe capacitor charging and discharging states is shown in Fig. 7(b)-(d). The value of R_s is very small, therefore the time of capacitor charging is short and capacitor’s capacitance determination is applied only for discharging state of capacitor.i_c=cΔV_c/ΔtHere, Δt is the time of single switching duty cycle. ΔV_c is the capacitor rated voltage drop in one switching duty cycle which is assumed five percent of capacitor rated voltage. Consequently:T_s=Δt Δv_c=%5E/2Circuit current in Fig. 7, (i_s), is the sine wave that can be formulated as follow:i_s=i_msin(wt)i_c in (21) is equal to maximum value of current in (24), I_m, and therefore (25) is utilized to calculate the capacitor’s capacitance.c=i_cΔt/Δv_c§ MPPT IN THE PROPOSED INVERTERMaximum power of PV cells in any constant is shown by P_PV. The efficiency of the system is η, the output is P_ac. Therefore:P_ac=ηP_pvAnd:P_ac=V_sI_sHere, V_s and I_s are the RMS value of grid voltage and current respectively. Therefore, if one assigns any values for P_PV and V_s, the appropriate value for I_ref capable of tracking I_s determined and this guarantees the maximum output power from solar cells. § VOLTAGE LOSS CALCULATIONIn this section, the loss calculation has been done for the proposed topology. The loss calculation contains switches losses and capacitor losses.§.§ Switches lossesMainly conduction and switching losses are calculated for switches. Every power switch contains a transistor (MOSFET) and a diode, and therefore, the instantaneous conduction loss of these elements is calculated using (28) and (29), respectively [21]:P_c,T(t)=[R_Ti(t)]i(t) P_c,D(t)=[V_D+R_Di(t)]i(t)Where V_D is considered as forward voltage drop of the diode. R_T and R_D are equivalent resistance of the transistor and diode. At any time N_T(t) transistor and N_D(t) diodes are in current path, so the average conduction power losses can be written as:P_C=1/2π∫_0^2π[N_T(t)P_c,T(t)+N_D(t)P_c,D(t)]dtThe switching losses occur during turn on and turn off period of switches. For simplicity, the linear approximation of voltage and current of switches during the switching period is considered. Based on this assumption, the switching loss is calculated as follows:E_off,J=∫_0^t_off(v(t)i(t))dt=∫_0^t_off[(V_sw,Jt/t_off)(-I”(t-t_off/t_off)]dt=1/6V_sw,JI”t_off E_on,J=∫_0^t_on(v(t)i(t))dt=∫_0^t_on[(V_sw,Jt/t_on)(-I'(t-t_on/t_on)]dt=1/6V_sw,JI't_onE_off,J and E_on,J are turn off and turn on loss of the switch J, I’’ is the current through the switch before turning off, I’ is the current through the switches after turning on and V_sw,J is the off-state voltage on the switch. The switching power loss is equal to the sum of all turns off and turns on energy losses in the fundamental frequency of the output voltage; as a result, average switching power loss can be calculated as follow:P_sw=f∑_J=1^N_switch(∑_k=1^N_on,JE_on,JK+∑_k=1^N_off,JE_off,JK)f is the fundamental frequency, N_on,J and N_off,J are the number of turning on and off the J^th switch during fundamental frequency. E_on,JK is the energy loss of the J^th switch during the k^th turning on and E_off,JK is the energy loss of the J^th switch during the k^th turning off.§.§ Capacitor lossesThe capacitor losses consist of capacitor voltage ripple losses P_rip and conduction losses P_cond. The voltage ripple of capacitor is calculated as follow:ΔV_rip=1/C∫_t_-^t_+i_CdtWhere i_C, the transient current of the capacitor, equals to the line current. t_- to t_+ is the interval that the capacitor is connected in series with DC voltage sources. So, voltage ripple loss is written as follow: P_rip=CΔV_rip^2ff is the frequency of the output voltage. The conduction losses can be written as follow:P_cond=2f∫_t_-^t_+r_Ci_C^2dtWhere r_c is the internal resistance of the capacitor. Total loss of the proposed topology is equal:P_loss=P_C+P_sw+P_rip+P_condFinally, the efficiency of converter is written as follow:η=P_out/P_in=P_in-P_loss/P_inWhere P_out and P_in are output power and input power of converter. § SIMULATION RESULTS AND EXPERIMENTAL VERIFICATIONThe simulation has been done by PSCAD / EMTDC. The circuit used in the simulation and the experimental set up and the value of elements used in both of them are shown in Fig. 9 and Table III. Due to the limitation of laboratory components, DC voltage source is used instead of PV cells.For further analysis of inverter’s performance, input DC voltage source is changing from 60 V to 65 V in 0.04 s and from 65V to 55 V in 0.06 s, the results are shown in Fig. 10. According to Fig. 10, the proposed inverter with proposed control method produces 5-level voltage and tracks the reference current properly in the variation of DC source. Also, Flying capacitor’s voltage is balanced in its specified value (half of the input DC source).In the experimental circuit, inverter output voltage and current for 5-level output voltage is shown in Fig. 11. This figure indicates that the proposed inverter produces an output voltage waveform with 5 levels with maximum output level of 60 V. Also in this inverter, output current of approximately 2 A tracks the reference current properly that is in phase with grid voltage, therefore the proposed inverter can exchange active power with the grid. In the laboratory condition the measured output and input power are about 48 (W) and 52.8 (W), respectively. Therefore, the efficiency is equal 90.91%. Based on the efficiency calculation, it is about 91.13%. Accordingly, the computed efficiency has good accordance with the measured efficiency. Voltage and voltage ripples of flying capacitor are shown in Fig. 12. Base on Fig. 12, the voltage of flying capacitor is balanced on the desired value with 0.2 ripples.§ CONCLUSIONIn the proposed inverter compared to the structure of DFCM, independent of the inverter cell number and the number of generated voltage levels, by replacing only two unidirectional switches with two bidirectional switches, network connectivity with high safety is provided. Also, this inverter with proposed control method has unique abilities such as capacitor voltage balancing in any desired value and obtaining maximum power from PV cells. Theoretically there is no limitation for the number of flying capacitors for balancing the voltage. The proposed control method tracks maximum power point of PV cells through assuring the equal magnitude for the input and output power. Through this unique controlling methodology, the MPPT is performed just by the inverter itself. Finally, the mathematical analysis and simulation results are used to verify the operation of proposed inverter and control method. The design, engineering and optimization procedure explained in this work provides further insight for designing multilevel inverters to be used in hybrid systems equipped with various energy storage devices (i.e. redox flow batteries and regenerative hydrogen fuel cell systems). ieeetran 10url@samestyle gandomi2015dc A. A. Gandomi, K. Varesi, and S. H. Hosseini, “Dc-ac buck and buck-boost inverters for renewable energy applications,” in Power Electronics, Drives Systems & Technologies Conference (PEDSTC), 2015 6th.1em plus 0.5em minus 0.4emIEEE, 2015, pp. 77–82.saeidabadi2017novel S. Saeidabadi, A. A. Gandomi, and S. H. Hosseini, “A novel transformerless photovoltaic grid-connected current source inverter with ground leakage current elimination,” in Power Electronics, Drive Systems & Technologies Conference (PEDSTC), 2017 8th.1em plus 0.5em minus 0.4emIEEE, 2017, pp. 61–66.gandomi2017maximum A. A. Gandomi, S. Saeidabadi, and M. Sabahi, “Maximum power point tracking control method in high gain transformer-based inverters for photovoltaic application,” in Power Electronics, Drive Systems & Technologies Conference (PEDSTC), 2017 8th.1em plus 0.5em minus 0.4emIEEE, 2017, pp. 137–142.pellow2015hydrogen M. A. Pellow, C. J. Emmott, C. J. Barnhart, and S. M. Benson, “Hydrogen or batteries for grid storage? a net energy analysis,” Energy & Environmental Science, vol. 8, no. 7, pp. 1938–1952, 2015.gandomi2013assessing Y. A. Gandomi and M. M. 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Aouda, “Modeling of multilevel converters,” IEEE transactions on industrial electronics, vol. 44, no. 3, pp. 356–364, 1997. | http://arxiv.org/abs/1708.08008v1 | {
"authors": [
"A. Ashraf Gandomi",
"S. Saeidabadi",
"M. Sabahi",
"M. Babazadeh",
"Y. Ashraf Gandomi"
],
"categories": [
"physics.app-ph"
],
"primary_category": "physics.app-ph",
"published": "20170826183810",
"title": "Design, Engineering and Optimization of a Grid-Tie Multicell Inverter for Energy Storage Applications"
} |
We show a version of the DT/PT correspondence relating local curve counting invariants, encoding the contribution of a fixed smooth curve in a Calabi–Yau threefold. We exploit a local study of the Hilbert–Chow morphism about the cycle of a smooth curve. We determine, via Quot schemes, the global Donaldson–Thomas theory of a general Abel–Jacobi curve of genus 3.Self-forces in arbitrary dimensions Abraham I. Harte^1, Peter Taylor^1 and Éanna É. Flanagan^2,3 December 30, 2023 ================================================================§ INTRODUCTION Let Y be a smooth, projective Calabi–Yau threefold. The Donaldson–Thomas (DT) invariants of Y are enumerative invariants attached to the Hilbert scheme I_m(Y,β)=Z⊂ Y|χ(𝒪_Z)=m, [Z]=β,viewed as a moduli space of ideal sheaves <cit.>. These numbers are insensitive to small deformations of the complex structure of Y, but they do change when we perturb the stability condition that defines them: the rules that govern these changes are the so called wall-crossing formulas. It is in this spirit that one can interpret the “DT/PT correspondence”, an equality of generating functions_β(q)=_0(q)·_β(q)first conjectured in <cit.> and later proved by Bridgeland <cit.> and Toda <cit.>.The left hand side of (<ref>) is the Laurent series encodingDonaldson–Thomas invariants of Y in the class β∈ H_2(Y,ℤ), whereas _β encodes the Pandharipande–Thomas (PT)invariants of Y, defined through the moduli space of stable pairs <cit.>,P_m(Y,β)=(F,s)|χ(F)=m, [ F]=β.Recall that a pair (F,s), consiting of a one-dimensional coherent sheaf F and a section s∈ H^0(Y,F), is said to be stable whenF is pure and the cokernel of s is zero-dimensional. Finally, the formula <cit.>_0(q)=M(-q)^χ(Y)determines the zero-dimensional DT theory of Y. Here M(q)=∏_k>0 (1-q^k)^-k is the MacMahon function. §.§ Main resultWe prove a variant of (<ref>) in this note.Let C⊂ Y be a smooth curve of genus g embedded in class β. For integers n≥ 0, we define “local” DT invariants_n,C=∫_I_n(Y,C)ν_I χwhere ν_I:I_1-g+n(Y,β)→ is the Behrend function <cit.> on the Hilbert scheme and I_n(Y,C)⊂ I_1-g+n(Y,β) is the closed subset parametrizing subschemes Z⊂ Y containing C. For instance, a generic point of I_n(Y,C) represents a subscheme consisting of C along with n distinct points in Y∖ C. Similarly, we consider P_n(Y,C)⊂ P_1-g+n(Y,C), the closed subset parametrizing stable pairs (F,s) such that F=C. The local invariants on the stable pair side_n,C=∫_P_n(Y,C)ν_P χhave been studied in <cit.>. By smoothness of C, the local moduli space P_n(Y,C) can be identified with the symmetric product ^nC. Let us form the generating functions_C(q)=∑_n≥ 0_n,Cq^1-g+n, _C(q)=∑_n≥ 0_n,Cq^1-g+n.We say that the DT/PT correspondence holds for C if one has_C(q)=_0(q)·_C(q).We can view (<ref>) as a wall-crossing formula relating the local curve counting invariants attached to C. Let n_g,C be the BPS number of C⊂ Y <cit.>. Using the known value of the PT side <cit.>,_C(q)=n_g,C· q^1-g(1+q)^2g-2,we proved that the DT/PT correspondence (<ref>) holds for C smooth and rigid in <cit.>. The goal of this note is to extend the result to all smooth curves.Let Y be a smooth, projective Calabi–Yau threefold, C⊂ Y a smooth curve. Then the DT/PT correspondence (<ref>) holds for C. In fact, the conclusion of the theorem holds for all Cohen–Macaulay curves, by recent work of Oberdieck <cit.>. While he works with motivic Hall algebras, our method involves a local study of the Hilbert–Chow morphism, and builds upon previous calculations <cit.>, especially the weighted Euler characteristic of the Quot scheme _n(ℐ_C), as we explain in Section <ref>.The local invariants do not depend on the scheme structure one may put on I_n(Y,C) and P_n(Y,C). However, on the DT side, we will exploit the Hilbert–Chow morphism to endow I_n(Y,C) with a natural scheme structure, and we will prove that it agrees with the Quot scheme studied in <cit.>. So, in the local theory, we can think of the moduli spaces_n(ℐ_C)and^nCas living on opposite sides of the wall separating DT and PT theory from one another. Conventions. All schemes are defined over . The Calabi–Yau condition for us is simply the existence of a trivialization of the canonical line bundle. The Hilbert–Chow morphism _r(X/S)→_r(X/S) is the one constructed by D. Rydh in <cit.>. § THE DT/PT CORRESPONDENCEIn this section we outline our strategy to deduce Theorem <ref>.Let Y be a smooth projective variety, not necessarily Calabi–Yau. We consider the Hilbert–Chow morphism_1(Y)→_1(Y)constructed in <cit.>, sending a 1-dimensional subscheme of Y to its fundamental cycle.We recall its definition in Section <ref>. Let I_m(Y,β)⊂_1(Y) be the component parametrizing subschemes Z⊂ Y such that χ(𝒪_Z)=m∈, [Z]=β∈ H_2(Y,).Similarly, we let _1(Y,β)⊂_1(Y) be the component parametrizing 1-cycles of degree β. Then (<ref>) restricts to a morphism𝗁_m:I_m(Y,β)→_1(Y,β).Fix an integer n≥ 0. For a Cohen–Macaulay curve C⊂ Y of arithmetic genus g embedded in class β, we letI_n(Y,C)⊂ I_1-g+n(Y,β)denote the scheme-theoretic fibre of 𝗁_1-g+n, over the cycle of C.We will use that (<ref>) is an isomorphism around normal schemes, at least in characteristic zero <cit.>. Thus, for a smooth curve C⊂ Y, we will identify Chow with Hilb locally around the cycle [C]∈_1(Y) and the ideal sheaf ℐ_C∈_1(Y). For this reason, we do not need the representability of the global Chow functor, as around the point [C]∈_1(Y,β) we can work with the Hilbert scheme I_1-g(Y,β) instead. Consider the Quot scheme _n(ℐ_C)parametrizing quotients of length n of the ideal sheaf ℐ_C⊂𝒪_Y.We proved in <cit.> that the association [θ:ℐ_C↠ℰ]↦θ defines a closed immersion_n(ℐ_C)↪ I_1-g+n(Y,β).More precisely, for a scheme S, an S-valued point of the Quot scheme is a flat quotient ℰ=ℐ_C× S/ℐ_Z, and in the short exact sequence0→ℰ→𝒪_Z→𝒪_C× S→ 0over Y× S, the middle term is S-flat, so Z defines an S-point of I_1-g+n(Y,β). The S-valued points of the image of (<ref>) consist precisely of those flat families Z⊂ Y× S→ S such that Z contains C× S as a closed subscheme. This will be used implicitly in the proof of Theorem <ref>.The schemes I_n(Y,C) and _n(ℐ_C) have the same -valued points: they both parametrize subschemes Z⊂ Y consisting of C together with “n points”, possibly embedded. The first step towards Theorem <ref> is the following result, whose proof is postponed to the next section. Let Y be a smooth projective variety, C⊂ Y a smooth curve of genus g. Then I_n(Y,C)=_n(ℐ_C) as subschemes of I_1-g+n(Y,β). As an application of Theorem <ref>, in Section <ref> we compute the reduced Donaldson–Thomas theory of a general Abel–Jacobi curve of genus 3.To proceed towards Theorem <ref>, we need to examine the local structure of the Hilbert scheme around subschemes Z⊂ Y whose maximal Cohen–Macaulay subscheme C⊂ Z is smooth. The result, given below, will be proven in the next section.Let Y be a smooth projective variety, C⊂ Y a smooth curve of genus g.Then, locally analytically around I_n(Y,C), the Hilbert scheme I_1-g+n(Y,β) is isomorphic to I_n(Y,C)×_1(Y,β). Roughly speaking, this means that the Hilbert–Chow morphism, locally about the cycle[C]∈_1(Y,β), behaves like a fibration with typical fibre I_n(Y,C).To obtain this, we first identifywithlocally around C, cf. Remark <ref>. We then need to trivialize the universal curve 𝒞→, which can be done since smooth maps are analytically locally trivial (on the source). However, even if we had 𝒞 = C×, we would not be done: the fibre of Hilbert–Chow (which is the Quot scheme by Theorem <ref>) depends on the embedding of the curve into Y, not just on the abstract curve. So to prove Theorem <ref> we need to trivialize (locally) the embedding of the universal curve into Y×. This is taken care of by a local-analytic version of the tubular neighborhood theorem. After this step, Theorem <ref> follows easily. Granting Theorems <ref> and <ref>, we can prove the DT/PT correspondence for smooth curves. So now we assume C is a smooth curve embedded in class β in a smooth, projective Calabi–Yau threefold Y. By <cit.>, the Hilbert–Chow morphism 𝗁_1-g:I_1-g(Y,β)→_1(Y,β)is (in characteristic zero) an isomorphism over the locus of normal schemes. Under this local identification, the cycle [C] corresponds to the ideal sheaf ℐ_C. We let ν(ℐ_C) be the value of the Behrend function on I_1-g(Y,β) at the point corresponding to ℐ_C. Since the Behrend function can be computed locally analytically <cit.>, Theorem <ref> implies the identityν_I|_I_n(Y,C)=ν(ℐ_C)·ν_I_n(Y,C),where ν_I is the Behrend function of I=I_1-g+n(Y,β). After integration, we find_n,C=ν(ℐ_C)·χ̃(I_n(Y,C)),where χ̃(I_n(Y,C)), by Theorem <ref>, agrees with the weighted Euler characteristic of the Quot scheme _n(ℐ_C). But we proved in <cit.> that the relation_n,C=ν(ℐ_C)·χ̃(_n(ℐ_C))is equivalent to the local DT/PT correspondence (<ref>), so the theorem follows.As observed in <cit.>, the local DT/PT correspondence says that the local invariants are determined by the topological Euler characteristic of the corresponding moduli space, along with the BPS number of the fixed smooth curve C⊂ Y. The latter can be computed asn_g,C=ν(ℐ_C).For any integer n≥ 0, the formulas are_n,C =n_g,C· (-1)^nχ(I_n(Y,C)), _n,C =n_g,C· (-1)^nχ(P_n(Y,C)).In particular, the local invariants differ by the Euler characteristic of the corresponding moduli space by the same constant.§ PROOFSIt remains to prove Theorems <ref> and <ref>. For Theorem <ref>, we need to review some definitions and results from <cit.>.§.§ The fibre of Hilbert–ChowRydh has developed a powerful theory of relative cycles and has defined a Hilbert–Chow morphism_r(X/S)→_r(X/S)for every algebraic space X locally of finite type over an arbitrary scheme S. For us X is always a scheme, projective over S. We quickly recall the definition of (<ref>). First of all, the Hilbert scheme _r(X/S) parametrizes S-subschemes of X that are proper and of dimension r over S,but not necessarily equidimensional, while _r(X/S) parametrizes equidimensional, proper relative cycles of dimension r. We refer to <cit.> for the definition of relative cycles on X/S. Cycles have a (not necessarily equidimensional) support, whichis a locally closed subset Z⊂ X. Rydh shows <cit.> that if α is a relative cycle on f:X→ S with support Z,then, for every r≥ 0, on the same family there is a unique equidimensional relative cycle α_r with support Z_r=x∈ Z|_xZ_f(x)=r⊂ Z.Cycles are called equidimensional when their support is equidimensional over the base. The essential tool for the definition of (<ref>) is the norm family, defined by the following result. Let X→ S be a locally finitely presented morphism, ℱ a finitely presented 𝒪_X-module which is flat over S.Then there is a canonical relative cycle 𝒩_ℱ on X/S, with support equal to ℱ.This construction commutes with arbitraty base change. When Z⊂ X is a subscheme which is flat and of finite presentationover S, we write 𝒩_Z=𝒩_𝒪_Z. The Hilbert–Chow functor (<ref>) is defined by Z↦ (𝒩_Z)_r.Even though we do not recall here the full definition of relative cycle, the main idea is the following. For a locally closed subset Z⊂ X, Rydh defines a projection of X/S adapted to Z to be a commutative diagramUrpdXddBd TrgSwhere U→ X×_ST is étale, B→ T is smooth and p^-1(Z)→ B is finite. A relative cycle α on X/S with support Z⊂ X is the datum, for every projection adapted to Z, of a proper family of zero-cycles on U/B, which Rydh defines as a morphismα_U/B/T:B→Γ^⋆(U/B)to the scheme of divided powers. We refer to <cit.> for the additional compatibility conditions that these data should satisfy.Let now ℱ be a flat family of coherent sheaves on X/S. If 𝗉=(U,B,T,p,g) denotes a projection of X/S adapted to ℱ⊂ X as in (<ref>), then the zero-cycle defining the norm family 𝒩_ℱ at 𝗉 is(𝒩_ℱ)_U/B/T=𝒩_p^∗ℱ/B,constructed in <cit.>. For us ℱ will always be a structure sheaf, so it will be easy to compare these zero-cycles.If Z⊂ X is a subscheme that is smooth over S, then the norm family 𝒩_Z is an example of a smooth relative cycle, cf. <cit.>. The next result states an equivalence, in characteristic zero, between smooth relative cycles and subschemes smooth over the base.If S is of characteristic zero, then for every smooth relative cycle α on X/S there is a unique subscheme Z⊂ X, smooth over S, such that α=𝒩_Z. We can now prove Theorem <ref>. We fix Y to be a smooth projective variety, C⊂ Y a smooth curve of genus g in class β,and we denote by I_n(Y,C) the fibre over [C] of the Hilbert–Chow morphismI_1-g+n(Y,β)→_1(Y,β),as in Definition <ref>. We need to show the equalityI_n(Y,C)=_n(ℐ_C)as subschemes of I_1-g+n(Y,β). Let S be a scheme over , and set X=Y×_ S. Then a family Z⊂ X→ Sin the Hilbert scheme is an S-valued point of I_n(Y,C) when (𝒩_Z)_1=𝒩_C× S. The closed immersion (<ref>) from the Quot scheme to the Hilbert scheme factors through I_n(Y,C). Indeed, any S-point ℐ_C× S↠ℐ_C× S/ℐ_Z of the Quot scheme gives a closed immersion C× S↪ Z whose relative ideal is zero-dimensional over S, thus we have (𝒩_Z)_1=(𝒩_C× S)_1=𝒩_C× S, where in the second equality we used that 𝒩_C× S is equidimensional of dimension one over S. So we obtain a closed immersion ι:_n(ℐ_C)↪ I_n(Y,C).For every scheme S, we have an injective map of setsι(S):_n(ℐ_C)(S)↪ I_n(Y,C)(S),and since ι() is a bijection, so far ι is just a bijective closed immersion. We need to show ι(S) is onto, and for the moment we deal with the case where S is a fat point. In other words, assume S is the spectrum of a local artinian -algebra with residue field . Let Z⊂ X→ S be an S-valued point of I_n(Y,C). Consider the finite subscheme F⊂ Y⊂ X given by the support of ℐ_C/ℐ_Z_0, where Z_0 is the closed fibre of Z→ S. Form the open set V=X∖ F⊂ X. Then we have, as relative cycles on V/S, (𝒩_Z)_1|_V=𝒩_C× S|_V=𝒩_(C× S)∩ V.We claim the left hand side equals the relative cycle 𝒩_Z∩ V. For sure, these two cycles have the same support, as Z∩ V=Z_1∩ V, and they are determined by the same set of projections; indeed, being equidimensional of dimension one, they are determined by (compatible data of) relative zero-cycles for every projection 𝗉_V/S=(U,B,T,p,g) such that B/T is smooth of relative dimension one. Let us focus on (𝒩_Z)_1 first. Here r=1 is the maximal relative dimension of a point in Z, so the zero-cycle corresponding to a projection 𝗉_X/S as in (<ref>), and adapted to Z_1, is the same as the one defined by the norm family of Z (cf. the proof of <cit.>), namely 𝒩_p^∗𝒪_Z/B. Now we restrict to the open subset i:V→ X. By definition of pullback, the zero-cycle attached to a projection 𝗉_V/S (adapted to Z_1∩ V) is the cycle corresponding to the projection (U,B,T,i∘ p,g) for the full family Z/S, namely𝒩_(i∘ p)^∗𝒪_Z/B=𝒩_p^∗𝒪_Z∩ V/B.The latter is precisely the zero-cycle defining the norm family of Z∩ V/S at the same projection 𝗉_V/S, so the claim is proved, 𝒩_Z∩ V=(𝒩_Z)_1|_V.By the equivalence between smooth cycles and smooth subschemes stated in Theorem <ref>, we conclude that Z∩ V and (C× S)∩ V are the same (smooth) family over S. Moreover, the closure(C× S)∩ V⊂ Zequals C× S, because the open subscheme (C× S)∩ V⊂ C× S is fibrewise dense (intersecting with V is only deleting a finite number of points in the special fibre). We have thus reconstructed a closed immersion C× S↪ Z, giving a well-defined S-valued point of _n(ℐ_C). So ι(S) is onto, and thus a bijection, whenever S is a fat point. This implies ι is étale, by a simple application of the formal criterion for étale maps. The theorem follows because we already know ι is a bijective closed immersion.§.§ Local triviality of Hilbert–Chow In this section we prove Theorem <ref>. The main tool used in the proof is the following local analytic version of the tubular neighborhood theorem. Let S be a scheme, j:X→ Y a closed immersion over S. Assume X and Y are both smooth over S, of relative dimension d and n respectively. Then j is locally analytically isomorphic to the standard linear embedding ^d× S→^n× S. Let x∈ X and y=j(x)∈ Y. Let ℐ⊂𝒪_Y be the ideal sheaf of X in Y. The relative smoothness of X, given that of Y, is characterized by the Jacobian criterion <cit.>, asserting that the short exact sequence0→ℐ/ℐ^2→ j^∗Ω_Y/S→Ω_X/S→ 0is split locally around x∈ X. According to loc. cit. this is also equivalent to the following: whenever we choose local sections t_1,…,t_n and g_1,…,g_N of 𝒪_Y,y such that t_1,…, t_n constitute a free generating system for Ω_Y/S,y and g_1,…,g_N generate ℐ_y, after a suitable relabeling we may assume g_d+1,…,g_n generate ℐ about y and t_1,…, t_d, g_d+1,…, g_ngenerate Ω_Y/S locally around y. In particular, f_i=t_i∘ j, for i=1,…,d, define a local system of parameters at x. By this choice of local basis for Ω_Y/S around y, we can find open neighborhoods x∈ U⊂ X and y∈ V⊂ Y fitting in a commutative diagramU[hook]rj[swap]dét Vdét ^d_S[hook]r ^n_Swhere the vertical maps are defined by the local systems of parameters (f_1,…,f_d) and (t_1,…,t_d,g_d+1,…,g_n) respectively, and the lower immersion is defined by sending t_i↦ f_i for i=1,…,d and g_k↦ 0. Using the analytic topology, the inverse function theorem allows us to translate the étale maps into local analytic isomorphisms, and the statement follows. Note that Lemma <ref> does not hold globally. For a closed immersion X⊂ Y of smooth complex projective varieties, it is not true in general that one can find a global tubular neighborhood. The obstruction lies in ^1(N_X/Y,T_X).Before the proof of Theorem <ref>, we introduce the following notation. If Z⊂ Y is a 1-dimensional subscheme corresponding to a point in the fibre I_n(Y,C)of Hilbert–Chow, we can attach to Z its “finite part”, the finite subset F_Z⊂ Z which is the support of the maximal zero-dimensional subsheaf of 𝒪_Z, namely the quotient ℐ_C/ℐ_Z.By <cit.> the Hilbert–Chow map is a local isomorphism around normal schemes, so we may identify an open neighborhood of the cycle of C in the Chow scheme with an open neighborhood U of [C] in the Hilbert scheme I_1-g(Y,β). We then consider the Hilbert–Chow map 𝗁=𝗁_1-g+n:I_1-g+n(Y,β)→_1(Y,β)and we fix a point in the fibre [Z_0]∈ I_n(Y,C). It is easy to reduce to the case where the finite part F_0=F_Z_0⊂ Z_0 is confined on C, that is, Z_0 has only embedded points.We need to show that the Hilbert scheme is locally analytically isomorphic to U× I_n(Y,C) about [Z_0]. By Lemma <ref>, the universal embedding 𝒞⊂ Y× U, locally around the finite set of points F_0⊂ C⊂𝒞, is locally analytically isomorphic to the embedding of the zero section C× U⊂ C× U×^2 of the trivial rank 2 bundle.In particular we can find, in C× U×^2 and in Y× U, analytic open neighborhoods V and V' of F_0, fitting in a commutative diagram(C× U)∩ Vd[hook]rV[hook]ropendC× U×^2 𝒞∩ V'[hook]rV'[hook]ropenY× Uwhere the vertical maps are analytic isomorphisms. Now consider the open subset A={(Z,u)∈ I_n(Y,C)× U | F_Z⊂ V_u}⊂ I_n(Y,C)× U.Letting φ denote the isomorphism V → V', given a pair (Z,u)∈ A we can look at Z'=𝒞_u∪φ(F_Z), which is a new subscheme of Y, mapping to u under Hilbert–Chow. The association (Z,u)↦ Z' defines an isomorphism between A and the open subset B⊂𝗁^-1(U) parametrizing subschemes Z'⊂ Y such that F_Z' is contained in V'_u, where u is the image of [Z'] under Hilbert–Chow. Note that [Z_0]∈ B corresponds to (Z_0,C)∈ A under this isomorphism. The theorem is proved.§ THE DT THEORY OF AN ABEL–JACOBI CURVEIn this section we fix a non-hyperelliptic curve C of genus 3, embedded in its Jacobian Y=( C,Θ)via an Abel–Jacobi map. We let β=[C]∈ H_2(Y,) be the corresponding curve class. For n≥ 0, we let ℋ^n_C⊂ I_n-2(Y,β)be the component of the Hilbert scheme parametrizing subschemes Z⊂ Y whose fundamental cycle is algebraically equivalent to [C]. Let -1:Y→ Y be the automorphism y↦ -y, and let -C denote the image of C. As C is non-hyperelliptic, the cycle of C is not algebraically equivalent to the cycle of -C <cit.>.The Hilbert schemeI_n-2(Y,β) consists of two connected components, which are interchanged by -1. Moreover, the Abel–Jacobi embedding C⊂ Y hasunobstructed deformations, and there is an isomorphism Y → ℋ^0_C given by translations <cit.>. As remarked in <cit.>, the morphismℋ^1_C→ℋ^0_C× Ysending T_x(C)∪ y↦ (T_x(C),y), where T_x denotes translation by x, is the Albanese map. It can be easily checked that ℋ^1_C is isomorphic to the blow-up_𝒰(ℋ^0_C× Y),where 𝒰 is the universal family. In particular, ℋ^1_C is smooth of dimension 6.The quotient of the Hilbert scheme by the translation action of Y gives a Deligne–Mumford stack I_m(Y,β)/Y.In fact, since the Y-action is free, this is an algebraic space. The reduced Donaldson–Thomas invariants^Y_m,β=∫_I_m(Y,β)/Yν χ∈were introduced in <cit.> for arbitrary abelian threefolds. We consider their generating function_β(p)=∑_m∈^Y_m,βp^m.We state the following result as a corollary of Theorem <ref>. Let C⊂ Y be non-hyperelliptic, embedded in class β. Then_β(p)=2p^-2(1+p)^4. As the Hilbert–Chow morphism is an isomorphism around normal schemes, we have an isomorphismI_-2(Y,β) → _1(Y,β).On the other hand, the Hilbert scheme is the disjoint union of two copies of ℋ^0_C, where ℋ^0_C≅ Y because C is not hyperelliptic. Focusing on the component parametrizing translates of C, the Hilbert–Chow morphism ℋ^n_C→ℋ^0_C induces an isomorphism Y×_n(ℐ_C) → ℋ^n_Cby Theorem <ref>. This shows that the quotient space ℋ^n_C/Y is isomorphic to the Quotscheme _n(ℐ_C). Keeping into account the second componentof I_n-2(Y,β), still isomorphic to ℋ^n_C, we find ^Y_n-2,β=2·χ̃(_n(ℐ_C)),where χ̃ denotes the Behrend weighted Euler characteristic.Then _β(p)=∑_n≥ 0^Y_n-2,βp^n-2=2p^-2∑_n≥ 0χ̃(_n(ℐ_C))p^n=2p^-2(1+p)^4,where the last equality follows from <cit.>.If one considers homology classes of type (1,1,d) for all d≥ 0, on an arbitrary abelian threefold Y, one has the formula∑_d≥ 0∑_m∈ℤ^Y_m,(1,1,d)(-p)^mq^d=-K(p,q)^2,where K is the Jacobi theta functionK(p,q)=(p^1/2-p^-1/2)∏_m≥ 1(1-pq^m)(1-p^-1q^m)/(1-q^m)^2.Relation (<ref>) was conjectured in <cit.> and proved in <cit.>. Corollary <ref> confirms the coefficient of q via Quot schemes, when Y is the Jacobian of a general curve. Indeed, in this case the Abel–Jacobi class is of type (1,1,1). The local DT theory of a general Abel–Jacobi curve C of genus 3 is determined as follows.Using again the isomorphism Y≅ℋ^0_C, we can compute the BPS numbern_3,C=ν(ℐ_C)=-1,thus the DT/PT correspondence at C (Theorem <ref>) yields_C(q)=_C(q)=-q^-2(1+q)^4.In other words, the global theory is related to the local one by_β(q)=-2·_C(q).Acknowledgements. I would like to thank my advisors Martin G. Gulbrandsen and Lars H. Halle, and Letterio Gatto, for many valuable discussions. Thanks to David Rydh for sharing many useful comments and a key example that was useful to understand the norm family. Finally, I wish to thank Richard Thomas for patiently sharing his insights and commenting on a previous draft of this work.spmpsci | http://arxiv.org/abs/1708.07812v2 | {
"authors": [
"Andrea T. Ricolfi"
],
"categories": [
"math.AG"
],
"primary_category": "math.AG",
"published": "20170825170924",
"title": "The DT/PT correspondence for smooth curves"
} |
Department of Physics, Tokyo Institute of Technology, Ookayama 2-12-1, Meguro-ku, 152-8550 TokyoDepartment of Physics, Tokyo Institute of Technology, Ookayama 2-12-1, Meguro-ku, 152-8550 TokyoWe show that at low pressures the spectral widths of the power spectra of laser-trapped particles are nearly independent from pressures and, due to the nonlinearities of the trap, reflect the thermal distribution of particles. In the experiments with nanoparticles trapped in an optical lattice, we identify two distinct features of the widths. First, the widths along an optical lattice are much broader than those in the other directions. Second, the spectral widths are narrower for larger nanoparticles. We develop a theory of thermal broadening and show that the spectral widths normalized by the frequencies of the center-of-mass motion directly reveal the ratio of the thermal energy to the trap depth. The presented model provides a good understanding of the observed features. Our model holds also for smaller particles such as atoms and molecules and can be readily extended to the general case with a single-beam optical trap. Thermal broadening of the power spectra of laser-trapped particles in vacuum K. Aikawa December 30, 2023 ============================================================================ § INTRODUCTIONRecent studies on optomechanical systems have shown remarkable progresses in controlling the motional state of nano- and micro-mechanical oscillators via light <cit.>. These systems are promising platforms for diverse applications ranging from the investigation of macroscopic quantum phenomena <cit.> and the sensitive detection of weak forces <cit.> and small masses <cit.> to the studies on non-equilibrium physics at the nanoscale <cit.>. Among various optomechanical systems, optically levitated nanoparticles have a salient advantage that they are not in touch with any other objects and can potentially have an extremely high quality factor of 10^12 <cit.>. Great efforts have been made to cool the center-of-mass motion of nanoparticles and have demonstrated temperatures of the order of [1]mK, close to its quantum ground state <cit.>.In studies with optomechanical systems, a power spectral density (PSD) of the motion of the oscillator plays a central role in understanding and controlling its behavior. It has been well known that the profile of a PSD of levitated particles is given by a Lorentzian function, with spectral widths being the damping rate determined by the background pressure <cit.>. Because the relation between the pressure and the damping rate depends on the mass of particles, the spectral widths extracted from PSDs have served as a crucial means to estimate the mass (and thus the radius) of trapped nanoparticles. In this article, we show that the spectral widths of the PSDs at pressures lower than tens of Pa are nearly constant and determined dominantly by the thermal broadening. Accordingly, the spectral profiles at low pressures significantly deviate from the Lorentzian function. The observed spectral widths at low pressures strongly depends on the direction of the motion as well as on the size of the nanoparticles. These behaviors are caused by the nonlinearities in the optical dipole force, which lower the frequency of the center-of-mass motion (trap frequency) under finite oscillation amplitudes. We identify two physical mechanisms for broadening: the anharmonicity and the cross-dimensional couplings. The former effect means a deviation of an optical potential from a harmonic potential, while the latter effect means that finite oscillation amplitudes in orthogonal directions effectively lowers the trap depth and hence the trap frequency. We derive a theoretical model accounting for both effects and show that the spectral width normalized by the trap frequency is directly connected to the ratio of the thermal energy to the trap depth. Our model explains the observed features well. With a slight modification on numerical coefficients representing nonlinearities, our model can be readily extended to a more general case with a single-beam optical trap.The present study is closely related to the study on thermal nonlinearities <cit.>, where they investigated the shift of the trap frequency introduced by the nonlinearities of the optical force. In most part of their study, they applied feedback cooling to the orthogonal motions, which means canceling the cross-dimensional coupling, and focused on the one-dimensional problem of how the anharmonicity of the trap shifted the trap frequency. They did not explicitly explore the spectral widths. By contrast, the present study provides detailed investigations of the widths without applying feedback cooling and therefore deals with the three-dimensional problem. We also provide a new perspective on the relation between the width and the particle size.§ EXPERIMENTAL SETUP Our setup is based on previous experimental studies on feedback cooling the center-of-mass motion of nanoparticles optically levitated in vacuum <cit.>. Here we briefly describe our setup. The schematic of our experimental setup is shown in Fig. <ref>. A single-frequency fiber laser at a wavelength of [1550]nm with a power of about [300]mW is used for both trapping and detecting the motion of trapped nanoparticles. A high-numerical-aperture lens focuses the trapping beam to beam waists of about [1.7]μ m and [2.0]μ m in x and y directions, respectively. Another lens placed after the focusing lens is used for both collimating the trapping beam and collecting the scattered light from nanoparticles. The trapping beam is retro-reflected to form a standing-wave trap, namely, a one-dimensional optical lattice commonly used in cold atom experiments <cit.>. As a result, the trap frequency in the z direction is much higher than in the other directions; the trap frequencies are about 36, 54, and [230]kHz× 2π in the x, y, and z directions, respectively.The three-dimensional motion of trapped nanoparticles is observed via a balanced photodetector, where the signal of the light without nanoparticles is subtracted from that of nanoparticles. For each measurement, we record the signal for [1]s. The PSDs are obtained by performing a Fourier transform to the recorded data on a computer. The time duration for the Fourier transform is chosen to be [200]ms, which is sufficiently longer than the Fourier limit of the observed narrowest spectrum (about 2π×[50]Hz). From a single data set with a duration of [1]s, we obtain five independent PSDs and extract the spectral widths from each. The standard deviation of the five values allows us to estimate errors in fitting. The Rayleigh-scattered light from trapped nanoparticles is collected also through the upper viewport. The scattering of an imaging beam at [532]nm, overlapped with the trapping beam, is shone on a CMOS camera for acquiring the information on the number and the positions of trapped nanoparticles (Fig. <ref>). The magnitude of the scattering of the trapping beam is measured with an additional photodetector. This signal, which hereafter we call a fluorescence signal, provides a reliable means to estimate the relative size of trapped nanoparticles. In the present study, the observed fluorescence signals are ranged between 0.04 and [50]μ W. For loading of nanoparticles into the trap region, we introduce a mist of ethanol including silica nanoparticles with radii of about [60]nm into a vacuum chamber at atmospheric pressure. After we observe trapped nanoparticles on the camera, we evacuate the chamber to arbitrary pressures. Below [0.6]Pa, nanoparticles escape from the trap. Hence, we investigate the spectra at pressures above [0.6]Pa.The main feature of our setup lies in the optical lattice structure of the trap region. As a result, we are able to trap multiple nanoparticles at the same time as shown in Fig. <ref>. Trapping multiple nanoparticles in a standing-wave trap, formed by two counter-propagating beams emitted from two optical fibers, has been reported <cit.>. In our setup, we are able to adjust the position of the trapped nanoparticles by controlling the position of the retro-reflecting mirror. Such a feature is necessary for the two following reasons. First, during the evacuation process, the position of trapped nanoparticles has to be adjusted to compensate a slight variation of the refractive index from that of air to that of vacuum, because such a variation in the refractive index shifts the standing wave by about [50]μ m in our setup. Second, by pushing nanoparticles away from the focus of the beam, we are able to release them. In this way, we control number of remaining nanoparticles relevant to experiments. In the present study, we prepare a situation with only single nanoparticles trapped at the focus of the beam. § THEORETICAL DESCRIPTION OF THE THERMAL BROADENING A previous study on thermal nonlinearities in the motion of trapped nanoparticles mainly focuses on a Duffing nonlinearity arising from the anharmonicity of the Gaussian distribution of a laser beam <cit.>. Although their theoretical framework starts with a general three dimensional problem, they simplify it to a one-dimensional problem because in their study the motions in the orthogonal two directions are suppressed by feedback cooling. On the contrary, we are interested in a situation where no feedback cooling is applied. Here we extend the formalism in ref. <cit.> such that both the anharmonicity and the cross-dimensional couplings are considered.The purpose of this section is to find expressions of the thermal spectral widths on the basis of three dimensional equations of motion.We first derive an expression of the optical dipole force taking into account nonlinearities to the first order. The spatial distribution of the optical potential created by a one-dimensional optical lattice beam is approximately given by <cit.>U(x,y,z) ≈ -U_0 cos^2(2π z/λ) exp(-2x^2/w_x^2-2y^2/w_y^2)where U_0 is the depth of the potential, λ is the wavelength of the beam, w_x and w_y are beam waists in x and y directions, respectively. Here, we assumed that the confinement in the z direction dominantly arises from the standing wave andignored the intensity variation due to a beam divergence. The trap frequencies, without taking into account nonlinearities, are written as follows: Ω_x = √(4U_0mw_x^2), Ω_y = √(4U_0mw_y^2),Ω_z = √(8π^2U_0mλ^2) The optical dipole force exerted by the trapping beam is obtained by differentiating eq. (<ref>) and can be approximated to the first order byF(x,y,z) ≈ U_0([ -4xw_x^2( 1-2x^2w_x^2-2y^2w_y^2-4π^2z^2λ^2); -4yw_y^2( 1-2x^2w_x^2-2y^2w_y^2-4π^2z^2λ^2); -8π^2zλ^2( 1-2x^2w_x^2-2y^2w_y^2-8π^2z^23λ^2) ])We define nonlinear coefficients ξ_k^(j) such that the j-th component of the force (<ref>) is expressed by them as follows: F_j(x,y,z)= -l_j ( 1+ ∑^_kξ_k^(j) x_k^2 ) x_jwhere l_j is a spring constant in the j-th direction and {k,j}∈{x,y,z}. The nonlinear coefficients are then related to the beam waists and the wavelength as given in Table <ref>.Note that, for dealing with the case of a single-beam trap, only the nonlinear coefficients in the z direction have to be modified. With modified coefficients, the following arguments on the spectral profiles and the spectral widths of PSDs hold also for a single-beam trap.Using the nonlinear coefficients in Table <ref>, we can write down the three dimensional equations of motion as follows:q_j +Γ_0 q_j+Ω_j^2(1+ϵ_j cosω_j t + ∑_kξ_k^(j) q_k^2 )q_j ≈ 0where ϵ_j and ω_j are the amplitude of parametric modulation and the modulation frequency of parametric driving in the j-th direction, respectively. The terms of parametric driving are necessary to find actual trap frequencies in the presence of thermal motions <cit.>, whereas terms relevant to feedback included in ref. <cit.> are omitted here. The damping coefficient Γ_0 is proportional to the background pressure P <cit.>: Γ_0 = 0.6199πϕ d_m^2√(2)aρ k_BT_ airPwhere a is the radius of nanoparticles, d_m is the diameter of air molecules, ρ is the density of nanoparticles, ϕ is the viscosity of air, k_B is the Boltzmann constant, and T_ air is the temperature of the surrounding gas. Following the approach of ref. <cit.>, based on the secular perturbation theory, we introduce a dimensionless slow time scale τ given byτ= κ_jΩ_j tκ_j≡ Γ_0Ω_jwhere κ_j corresponds to the inverse of the quality factor of the center-of-mass oscillation in the j-th direction and is smaller than unity at low pressures relevant to this study. We take the ansatz that the solutions of eqs. (<ref>) are written in the following form: q_j = q_j02A_j e^iΩ_j t + c.c. q_j0= √(-κ_j/ξ_j^(j))where q_j0 is a time-independent scale factor and A_j is a time-dependent complex amplitude of the dynamics. The time derivative of A_j is written asA_j=Ω_j κ_j dA_jdτThe terms relevant to eqs. (<ref>) are then given byq_j= q_j0Ω_j2( κ_j dA_jdτ+iA_j)e^iΩ_jt+c.c.q_j= q_j0Ω_j^22( κ_j^2d^2A_jdτ^2+2iκ_jdA_jdτ-A_j)e^iΩ_jt+c.c. q_j^3= 3q_j0^38|A_j|^2A_je^iΩ_jt+c.c. ϵ_jq_jcosω_jt = ϵ_jq_j04A_j^*e^iΩ_jte^i(ω_j-2Ω_j)t+c.c.q_j^2q_k= q_j0^2q_k08(A_j^2A_k^*e^i(2Ω_j-Ω_k)t+2|A_j|^2A_ke^iΩ_kt) +c.c.where c.c. denotes complex conjugates and we ignored fast oscillating terms. The last term (<ref>) represents the cross-dimensional coupling that is introduced in the present study. Similarly to ref. <cit.>, we define rescaled parameters for Γ_0 and ϵ_j:γ_0 ≡Γ_0Ω_jκ_j ϵ_j ≡ϵ_jκ_jSubstituting eqs. (<ref>–<ref>) into eq. (<ref>) and considering the terms proportional to exp(iΩ_j t) yields q_j0Ω_j^22(κ_j^2d^2A_jdτ^2+2iκ_jdA_jdτ-A_j) +q_j0κ_jΩ_j^22(κ_jdA_jdτ+iA_j)γ_0+q_j0Ω_j^24(2A_j+ϵ_jκ_jA_j^*e^i(ω_j-2Ω_j)t) +q_j0Ω_j^24∑_k≠ jq_k0^2ξ_k^(j)|A_k|^2A_j+38q_j0^3Ω_j^2ξ_j^(j)|A_j|^2A_j =0where we assumed that 2Ω_j is sufficiently separated from Ω_k for any combination of {j,k}. This is the case with the present study.Considering that κ_j is smaller than unity at high vacuum, we ignore higher order terms of O(κ_j^2) and obtain dA_jdτ= -γ_02A_j+iϵ_j4A_j^*e^i(ω_j-2Ω_j)t+i∑_k≠ jq_k0^2ξ_k^(j)4κ_j|A_k|^2A_j-i38|A_j|^2A_j Here we rewrite A_j asA_j=p_jexp(iϕ_j-iδ_jτ/2)with δ_j=(2-ω_j/Ω_j)/κ_j. Substituting eqs. (<ref>) into eqs. (<ref>), we obtain dp_jdτ+ip_j(dϕ_jdτ-δ_j2)= -γ_02p_j+iϵ_jp_j4e^-2iϕ_j+i∑_k≠ jq_k0^2ξ_k^(j)4κ_jp_k^2p_j-i38p_j^3 The real and imaginary parts of these equations can be separately written asdp_jdτ= -γ_02p_j+ϵ_j4p_jsin2ϕ_jdϕ_jdτ= δ_j2+ϵ_j4cos2ϕ_j+∑_k≠ jq_k0^2ξ_k^(j)4κ_jp_k^2-38p_j^2 For recovering equations without rescaling, we apply the inverse of the rescaling (<ref>) to eqs. (<ref>,<ref>) and arrive at equations for the amplitude Q_j=q_j0p_j and the phase ϕ_j: dQ_jdt= -Γ_02Q_j+ϵ_jΩ_j4Q_jsin2ϕ_jdϕ_jdt= Ω_j-ω_j2+ϵ_jΩ_j4cos2ϕ_j+∑_k≠ jξ_k^(j)Ω_j4Q_k^2+38Ω_jξ_j^(j)Q_j^2 The actual trap frequency ω_j/2, which takes nonlinearities into account, is obtained by considering the steady state solution of eqs. (<ref>), namely, dϕ_j/dt=0 at weak parametric driving ϵ_j≈ 0: ω_j2=Ω_j+∑_k≠ jξ_k^(j)Ω_j Q_k^24+38ξ_j^(j)Ω_j Q_j^2 These equations indicate that the actual trap frequency is lowered by finite oscillation amplitudes in any direction. The second terms in the right hand side of eqs. (<ref>) represent the cross-dimensional couplings, whereas the third term represents the anharmonicity of the trap. Due to collisions with background gases, the oscillation amplitudes fluctuate (Fig. <ref>). Amplitude fluctuations in a specific direction broadens the widths in the orthogonal two directions through the cross-dimensional couplings as well as the width in this direction through the anharmonicity. In reality, the oscillation amplitude fluctuates in all directions; therefore, the spectral widths are influenced by the motion in all directions. Our goal is to relate the temperature of the center-of-mass oscillation T_NP to the spectral broadening via nonlinear frequency shifts in eqs. (<ref>). The oscillation amplitude in the j-th direction is written by Q_j=√(2K_jmΩ_j^2)where K _j denotes the temporal center-of-mass energy of nanoparticles in the j-th direction.The energies K_j are expected to follow the Boltzmann distribution, which we experimentally confirm by taking the histogram of the observed signal (Fig. <ref>). Substituting eq. (<ref>) and eqs. (<ref>) into eqs. (<ref>), with nonlinear coefficients of Table <ref>, we obtain expressions for the temporal trap frequency in the j-th direction: ω_j2= Ω_j[1-14U_0(∑_k≠ jK_k+32c_jK_j)]= Ω_j[1-E4U_0(3c_j+46)]where E denotes the total center-of-mass energy and numerical factors c_j is 2/3 for j=z and is 1 for other cases. Here we assumed that the energy is equally distributed among each direction. The probability distribution of E follows the Maxwell-Boltzmann distribution in three dimensions: f_ MB(E) ∝√(E)exp( -Ek_BT_NP)From eqs. (<ref>,<ref>), we obtain the spectral profile of the thermal fluctuation for ΔΩ_j = ω_j/2-Ω_j (<0) f_ th(ΔΩ_j)∝√(-ΔΩ_j)exp(24U_0ΔΩ_j(3c_j+4)k_BT_NPΩ_j)The peak of this function lies atΔΩ_j0 = -(3c_j+4)k_BT_NP48U_0Ω_jFor a small displacement near the peak δ = ΔΩ_j - ΔΩ_j0, eq. (<ref>) is approximated by a Gaussian function: f_ th(δ) ∝exp[ -( 24U_0δ(3c_j+4)k_BT_NPΩ_j)^2] Thus, we arrive at a representation for the spectral width: ΔΩ_j ≈ (3c_j+4)k_BT_NP24U_0Ω_jWe anticipate that actual spectral profiles are given by the Lorentzianfunction <cit.> S_ Lor(ω)=2k_BT_airπ mΓ_0(ω^2-Ω_j^2)^2+ω^2Γ_0^2weighted by the thermal distribution of eq. (<ref>). § COMPARISON OF EXPERIMENTS AND THEORYWe investigate the PSD of laser-trapped nanoparticles in a pressure range between 0.6 and [600]Pa. The observed PSDs for a fluorescence signal of [0.49]μ W are shown in Fig. <ref>. At high pressures above tens of Pa, the spectra fit very well with the Lorentzian function for any directions regardless of the size of nanoparticles. However, at lower pressures and for small nanoparticles, the observed profiles deviate from the Lorentzian function. The deviation is the most pronounced in the z direction (Fig. <ref>), where the profile is nearly a Gaussian function. For even smaller nanoparticles, we observe asymmetric near-Gaussian profiles for three directions (Fig. <ref>). Profiles for very large nanoparticles (with fluorescence signals of the order of [10]μ W) fit well with the Lorentzian function for any directions. The dependence of the observed width on the pressure is shown in Fig. <ref>. At pressures above tens of Pa, where the spectra are well described by the Lorentzian function, the widths are nearly identical among three directions and are proportional to the pressure. Such a behavior indicates the damping of nanoparticles' motion by collisions with background gases and is consistent with eq. (<ref>). At low pressures, where the profiles deviate from the Lorentzian function and are closer to Gaussian functions, the widths settle to values that are nearly independent from the pressure. Here the widths in the z direction are much broader than those in the x and y directions.Our observations are understood as follows. At high pressures, broadenings dominantly arise from the damping (<ref>) and therefore the profiles agree well with the Lorentzian function. However, at low pressures the contribution of the damping (<ref>) becomes smaller and that of the thermal fluctuation (<ref>) is dominant. As a result, the profiles are closer to the Gaussian function (<ref>). Because the broadening due to the thermal fluctuations (<ref>) is proportional to the trap frequency, the widths in the z direction, in which the trap frequency is much higher, are much broader than those in x and y directions.A previous work reported that taking a short duration for Fourier transform revealed an original narrow Lorentzian profile, while a long duration for Fourier transform resulted in a broad spectrum <cit.>. However, in our experiments, the spectral widths are nearly independent from the duration for Fourier transform. In the previous work, they focused on the thermal behavior in a specific direction and applied feedback cooling in the other two directions. By contrast, we investigate the situation without feedback cooling, implying that the cross-dimensional couplings broaden the spectral width even with a short duration for Fourier transform. In the following discussions, we focus on how the thermal broadening depends on the size of nanoparticles. We first find a correspondence between the nanoparticles' radii and the fluorescence signal. We convert the slope of the pressure dependence at high pressures into nanoparticles' radii by using eq. (<ref>). Fig. <ref> plots the derived radii with respect to the fluorescence signal. The observed radii range between 20 and [120]nm and their mean value is in consistent with the specification of our sample ([60]nm). The observed radii have large scatters, which we interpret as follows. The fluorescence signal is based only on radii, whereas the radii obtained from the slope depends also on the density of nanoparticles. Because we are interested in the interaction of nanoparticles with the trapping beam, hereafter we employ the fluorescence signal as a measure of the size of nanoparticles.The spectral profiles at low pressures can be nearly Gaussian functions, for which fitting with the Lorentzian function may result in wrong fit values. For the systematic comparison of the observed widths among nanoparticles with largely different sizes, we fit the spectra at [0.6]Pa with Gaussian functions and extract the thermal widths for each. Although the extracted widths largely differ among three directions, we find that the widths normalized by the trap frequencies nearly fall on a single line in a wide range of the fluorescence signal (Fig. <ref>). Such a behavior is in good agreement with our model in Sec. <ref>. Although our model predicts a slight difference in widths of about [17]% between x,y and z directions, we are unable to find a systematic difference among the three directions within the error of our measurements. Our model implies that the values of k_BT_NP/U_0 are directly obtained from the observed normalized widths. From Fig. <ref>, we find that the ratio k_BT_NP/U_0 is ranged between 4× 10^-3 and 1× 10^-1. Assuming that the temperature of the center-of-mass motion T_NP is [300]K, we estimate that the trap depth U_0 is ranged between k_B×[3× 10^3]K and k_B×[7× 10^4]K. For testing the validity of our model quantitatively, we compare the value of U_0 estimated from the spectral widths with that from a calculation based on the polarizability. On the one hand, at a mean radius of [60]nm, we estimate that U_0 ≈ k_B×[1.5× 10^4]K from the width. On the other hand, the trap depth is related to the intensity of the trap beam I_0 as <cit.> U_0 = 2π a^3c Re(ϵ_r-1ϵ_r+2)I_0where c is speed of light and ϵ_r = 2.1 is the dielectric constant of silica at [1550]nm. With our beam intensity of I_0=[21]MW/cm^2, the trap depth is calculated to be k_B×[1.7× 10^4]K. Thus, our estimation from the spectral width is in good agreement with the purely calculated value.The power-laws obtained from the fit in Fig. <ref> are -0.44(1), -0.39(1), and -0.44(2) in x, y, and z directions, respectively. According to our model, the thermal broadenings are proportional to k_BT_NP/U_0, which scales as a^-3 because of the scaling of U_0 ∝ a^3 as shown in eq. (<ref>). The fluorescence signal is expected to scale as a^6 because the scattering of the trapping beam by nanoparticles is in the regime of Rayleigh scattering (a≪λ). Thus, theoretically we expect that the power-law of the normalized widths with respect to the fluorescence signal is -0.5. This value is in reasonable agreement with our observation. The slight deviation in the power-laws can, in principle, arise from laser absorption for the following reason. The larger the nanoparticles, the higher the internal temperature of laser-trapped nanoparticles, because the absorption scales with a^3 while cooling by surrounding gases scales with a^2. As reported in ref. <cit.>, when the internal temperature of a nanoparticle is higher than the temperature of the surrounding gases, it heats the colliding molecules and eventually heats the center-of-mass motion of nanoparticles as well via the back-action. Therefore, the larger the nanoparticles, the higher the center-of-mass temperature. This means k_BT_NP∝ a^n with n being positive and alters the power-laws in Fig. <ref>.§ CONCLUSIONWe investigate the PSD of nanoparticles trapped in one-dimensional optical lattice in vacuum. At high pressures, the spectra are in good agreement with the profile expected from the model based on the damping with background gases. In this pressure regime, the widths linearly scales with the pressure. However, at pressures lower than tens of Pa, the spectra deviate from such a profile and become closer to a Gaussian profile. Here the widths do not scale with the pressure and nearly saturate to values specific to the orientation and the size of nanoparticles. For the z direction, in which a standing wave is formed, the widths are much broader than those in the other two directions. We develop a theory to explain the widths at low pressures as broadenings originating from the thermal motions in three directions. We experimentally confirm the two important aspects of our model. First, the widths normalized by the trap frequency are nearly identical among three directions. Second, the normalized widths directly reveals the ratio of the thermal energy of nanoparticles to the trap depth. Our results are readily extended to the general case of a single-beam optical trap only with a slight modification on nonlinear coefficients. Furthermore, we believe that our results are generally the case with smaller particles such as atoms and molecules. The agreement between theory and experiments implies that the trap frequency is shifted by eq. (<ref>) due to the thermal motion. Such a shift has been an important issue in the precision measurement with ultracold atomic gases <cit.>.In the present study, the PSD is fit either by the Lorentzian function or by the Gaussian function for clarity. Investigating the combined profile of the Lorentzian and Gaussian functions in the intermediate situation will be important future work. The detailed analysis of the spectral profile itself can provide the information on the nanoparticles' size. We expect that parametric feedback cooling of the center-of-mass motion <cit.>, which can lower the temperature to a millikelvin regime, allows more elaborate studies on the relation between the widths and the thermal energy. Once the temperature is lowered to such a regime, the contribution of the thermal broadening to the widths is expected to be of the order of [1]mHz and is negligibly smaller than the broadening due to the damping. The method of studying the widths with respect to the size of nanoparticles is generally applicable to nanoparticles of other species. The widths at low pressures provide the important information on the ratio k_BT_NP/U_0, which is useful not only to estimate the trap depth but also to observe if T_NP is dependent on the size of nanoparticles. Indirectly, the information on T_NP tell us the extent of heating due to laser absorption. We thank S. Inouye for lending us a fiber laser and M. Kozuma for fruitful discussions and also for lending us a diode-pumped solid state laser at the early stage of the experiments. We are grateful to Y. Matsuda for careful reading of the manuscript and R. Nagayama for experimental assistance. 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"authors": [
"M. Yoneda",
"K. Aikawa"
],
"categories": [
"cond-mat.mes-hall"
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"primary_category": "cond-mat.mes-hall",
"published": "20170824225303",
"title": "Thermal broadening of the power spectra of laser-trapped particles in vacuum"
} |
1Department of Astrophysics, Princeton University, Princeton, NJ 08540, USA2Department of Astronomy and Astrophysics, University of California, Santa Cruz, 1156 High Street, Santa Cruz, CA 95064, USA3European Southern Observatory, Karl-Schwarzschild-Str. 2, D-85741 Garching, Germany4National Astronomical Observatories, Chinese Academy of Sciences, 20A Datun Road, Chaoyang District, Beijing 100012, China5School of Physics and Astronomy, The University of Nottingham, University Park, Nottingham, NG7 2RD, UK6Institute of Astronomy and Astrophysics, Academia Sinica, Taipei 10617, Taiwan7Physics Department and Tsinghua Centre for Astrophysics, Tsinghua University, Beijing 100084, China 8Jodrell Bank Centre for Astrophysics, School of Physics and Astronomy, The University of Manchester, Oxford Road, Manchester M13 9PL, UK9Institute of Cosmology and Gravitation, University of Portsmouth, Dennis Sciama Building, Burnaby Road, Portsmouth PO1 3FX, UK; South East Physics Network10Kavli Institute for the Physics and Mathematics of the Universe (WPI), Tokyo Institutes for Advanced Study, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa-shi, Chiba, 277-8583, Japan11Department of Physical Science, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8526, Japan 12Hiroshima Astrophysical Science Center, Hiroshima University, Higashi-Hiroshima, Kagamiyama 1-3-1, 739-8526, Japan 13Core Research for Energetic Universe, Hiroshima University, 1-3-1, Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8526, Japan14Department of Astronomy and Astrophysics, The Pennsylvania State University,University Park, PA 16802 15Institute for Gravitation and the Cosmos, The Pennsylvania State University, University Park, PA 16802 16School of Physical Sciences, The Open University, Milton Keynes,MK7 6AA, UK 17Department of Physics, University of North Carolina, Asheville, NC 28804, USA18Apache Point Observatory and New Mexico State University, P.O. Box 59, Sunspot, NM, 88349-0059, USA 19Sternberg Astronomical Institute, Moscow State University, Moscow19Department of Physics and Astronomy, University of Kentucky, 505 Rose Street, Lexington, KY 40506-0057, USA20Department of Astronomy, Yale University, PO. Box 208101,New Haven, CT 06520-8101 SDSS-IV MaNGA: Uncovering the Angular Momentum Content of Central and Satellite Early-type Galaxies J. E. Greene1, A. Leauthaud2, E. Emsellem3, J. Ge4, A. Aragón-Salamanca5, J. Greco1, Y.-T. Lin6, S. Mao4,7,8, K. Masters9, M. Merrifield5, S. More10, N. Okabe11,12,13, D. P. Schneider14,15, D. Thomas9, D. A. Wake16,17, K. Pan18, D. Bizyaev18,19, D. Oravetz18, A. Simmons18, R. Yan19, F. van den Bosch20 December 30, 2023 ================================================================================================================================================================================================================================================================================================================== We study 379 central and 159 satellite early-type galaxies with two-dimensional kinematics from the integral-field survey Mapping Nearby Galaxies at APO (MaNGA) to determine how their angular momentum content depends on stellar and halo mass. Using the <cit.> group catalog, we identify central and satellite galaxies in groups with halo masses in the range 10^12.5 h^-1 M_⊙< M_200b < 10^15 h^-1 M_⊙. As in previous work, we see a sharp dependence on stellar mass, in the sense that ∼ 70% of galaxies with stellar mass M_* > 10^11 h^-2 M_⊙ tend to have very little rotation, while nearly all galaxies at lower mass show some net rotation.The ∼ 30% of high-mass galaxies that have significant rotation do not stand out in other galaxy properties except for a higher incidence of ionized gas emission.Our data are consistent with recent simulation results suggesting that major merging and gas accretion have more impact on the rotational support of lower-mass galaxies. When carefully matching the stellar mass distributions, we find no residual differences in angular momentum content between satellite and central galaxies at the 20% level. Similarly, at fixed mass, galaxies have consistent rotation properties across a wide range of halo mass. However, we find that errors in classification of centrals and satellites with group finders systematically lowers differences between satellite and central galaxies at a level that is comparable to current measurement uncertainties. To improve constraints, the impact of group finding methods will have to be forward modeled via mock catalogs. § INTRODUCTION Integral field spectroscopic (IFS) surveys have provided an exciting new window into the formation histories of massive galaxies <cit.>. We have long understood that rotational support V/σ (with V the maximum radial velocity, and σ the stellar velocity dispersion) is a function of internal galaxy properties, with lower-mass ellipticals and S0s (early-type galaxies) having more rotation than the most massive elliptical galaxies <cit.>. IFS studies have definitively shown that ∼ L^* early-type galaxies typically have some net rotation <cit.>, while the most massive elliptical galaxies tend to have little to no net rotation <cit.>. In the era of IFS surveys, the classic ratio of V/σ that quantifies the level of rotation in galaxies has been replaced by a light-weighted two-dimensional analog, λ. We adopt the definition of λ_R from <cit.>, inspired by the two-dimensional modeling of V / σ from first principles as described in <cit.>, with V and σ the locally measured values, R the projected galactocentric distance, and brackets representing flux weighting:λ_R = ⟨ R | V | ⟩/⟨ R √(V^2 + σ^2)⟩. Note that λ_R is a cumulative measurement out to radius R, which has been shown to be a robust proxy for the spin parameter <cit.>. Early surveys did not cover sufficient volume to study the importance of extrinsic factors (such as local galaxy density) on the galaxy angular momentum content. The ATLAS^ 3D survey <cit.>,volume-limited to 42 Mpc, includes few slowly rotating galaxies and contains only one dense environment (the Virgo Cluster). In principle, there are many reasons to believe that the location of a galaxy within its halo may impact its angular momentum. For instance, if a series of minor mergers can lead to loss of net angular momentum <cit.> then central galaxies, that likely experience enhanced minor merging due to their location at the center of the potential, may be expected to preferentially lack rotation. The halo mass could also matter, since galaxies in more massive halos may have experienced more merging, even at fixed M_*. Ongoing large-scale IFS surveys, including CALIFA <cit.>, SAMI <cit.>, MASSIVE <cit.>, and MaNGA <cit.> provide the larger volumes that are needed to simultaneously control for galaxy properties, such as stellar mass, along with large-scale environment. They also provide a larger baseline in halo mass, providing the leverage to examine trends in stellar mass M_* and halo mass M_200b simultaneously, as well as separate central and satellite galaxies. Thanks to the Sloan Digital Sky Survey <cit.>, we now have the sample size to investigate all of these important parameters in setting the angular momentum content of galaxies. In this paper, we will use spatially resolved spectroscopy from the MaNGA survey <cit.> to address the relationship between internal galaxy properties, large-scale environment, and angular momentum, λ_R. We address the possible correlation between λ_R and local overdensity in a companion paper.This paper proceeds as follows. In <ref>, we discuss the MaNGA survey and our galaxy sample. We present the kinematic measurements in <ref>, and examine the results in <ref>. We conclude in <ref>. For consistency with our group catalog from <cit.> (2), we assume a flat ΛCDM cosmology with Ω_ m=0.238, Ω_Λ=0.762, H_0=100 h^-1 km s^-1 Mpc^-1. Halo masses are defined as M_200b≡ M(<r_200b)=200ρ̅4/3π R_200b^3 where R_200b is the radius at which the mean interior density is equal to 200 times the mean matter density (ρ̅) and the `b' indicates background (rather than critical) density. Stellar mass is denoted M_* and has been derived using a <cit.> Initial Mass Function (IMF). For consistency with our adopted stellar mass measurements, we assume h=1.§ DATA AND SAMPLE §.§ The MaNGA SurveyThe aim of MaNGA, one of three core SDSS IV projects, is to obtain integral-field spectroscopy of 10,000 nearby galaxies. Like previous SDSS surveys, MaNGA utilizes the 2.5m Sloan Foundation Telescope <cit.> and the BOSS spectrographs <cit.>. Unlike previous SDSS surveys, the MaNGA survey groups individual fibers into 17 hexagonal fiber bundles to perform a multi-object IFS survey <cit.>. Each fiber has a diameter of 2, while the bundles range in diameter from 12 to 32 with a 56% filling factor. The two dual BOSS spectrographs cover the wide wavelength range of 3600-10,300Å while maintaining a spectral resolution of σ_r ≈ 70 , appropriate for galaxy studies. Careful spectrophotometry yields a relative calibration accurate to a few percent <cit.>. The survey design is described in <cit.>, the observing strategy in <cit.>, and the data reduction pipeline in <cit.>. The MaNGA team has also developed useful tools for data visualization and vetting <cit.>.The MaNGA sample is selected from an enhanced version of the NASA-Sloan Atlas (NSA; Blanton M.; http://www.nsatlas.org) with redshifts primarily taken from the SDSS DR7 MAIN galaxy sample <cit.> and photometry from reprocessed SDSS imaging. The sample is built in i-band absolute magnitude (M_i)-complete shells, with more luminous galaxies observed in more distant M_i shells such that the spatial resolution (in terms of R_e) is roughly constant across the sample <cit.>. The MaNGA Primary sample, which forms ∼ 50% of the total sample, is selected such that 80% of the galaxies in each M_i shell can be covered to 1.5 R_e by the largest MaNGA IFU. There is also a Secondary sample, accounting for ∼ 40% of the targets, selected such that 80% of these galaxies are covered to 2.5 R_e. The remainder of the MaNGA main sample is the Color-enhanced supplement, which fills in poorly-covered regions of the color-magnitude diagram (e.g., faint red galaxies or the most luminous blue galaxies) and, like the Primary sample, is covered to 1.5 R_e. Given this sample construction, one must re-weight the galaxy distribution according to the volume in each shell to construct volume-limited samples. Whenever possible, we apply these weights before drawing conclusions. In this work we take a conservative approach and consider the Primary and Secondary samples only, excluding the Color-enhanced supplement. We make this choice since the addition of the color term in the Color-enhanced selection potentially increases the uncertainty in volume weights for these galaxies.§.§ Ancillary ProgramThe MaNGA survey awarded a small fraction of fiber-bundles to “Ancillary” Programs that can help to boost survey efficiency by serving as filler targets and also enhance the science output of MaNGA with a small time investment. We were awarded an Ancillary program to augment the number of central galaxies in the most massive halos available within the local volume targeted by MaNGA. Here we describe our target selection, along with a brief description of the group catalog that our selection is based upon.While there are several samples of central galaxies in the literature, building an uncontaminated sample of central galaxies spanning a range of halo masses is a non-trivial exercise. Different cluster finders and central galaxy selections, automated and visual, often disagree. We base our target selection on the <cit.> cluster catalog updated to DR7, created from the SDSS DR7 New York University Value Added Catalog <cit.>, a spectroscopic galaxy catalog.Y07 use an iterative, adaptive group finder to assign galaxies to halos. In short, they first use a friends-of-friends algorithm <cit.> to identify potential groups. Each group is assigned a characteristic luminosity, defined as the combined r-band luminosity of all group members with M_r-5 log(h) ≤-19.5 mag where M_r is the absolute Galactic-extinction corrected r-band luminosity, K-corrected to z = 0.1. Roughly speaking, the stellar mass comprises 1% of the total mass. This characteristic luminosity is used to assign halo masses to groups and to refine the group identification using an iterative method. Using a mock catalog, Y07 estimate the scatter in the assigned halo masses to be of order σ_Q∼0.35 dex for groups with 10^13h^-1M_⊙<M_200b<10^14.5h^-1M_⊙, where σ_Q is the standard deviation of Q=[log(M_L)-log(M_200b)]/√(2), with M_L the halo mass inferred from the group luminosity. They report that the overall scatter is dominated by intrinsic scatter between halo mass and M_L, while the details of the group finder (interlopers, incompleteness effects), are a relatively small component. <cit.> demonstrate that in general it is possible to extract meaningful physical correlations from Y07 as a function of color, stellar, and halo mass despite misidentifications and errors in halo mass <cit.>. We adopt the Y07 modelC catalog, which uses the SDSS model magnitudes and includes redshifts from SDSS and the 2dF Galaxy Redshift Survey <cit.> and nearest neighbors from the VAGC. We identify central galaxies as the most luminous (in r-band) galaxy, as noted in the Y07 imodelC_1 catalog. We adopt the Y07 group halo mass based on the total luminosity ranking of the groups, which is in the modelC_group file in the Y07 catalog. To build our sample, we avoid edges of the catalog by applying grp_f_edge>0.6 as recommended by Y07, and 0.02<z_ group<0.15, with the lower limit reflecting the Y07 limits and the upper limit set to match the MaNGA sample.Because massive halos are rare, the default Primary and Secondary MaNGA catalogs do not contain many central galaxies in high-mass halos (Figure <ref>). To address this lack, in our ancillary program we select central galaxies in halos more massive than M_200b=10^13.75 h^-1 M_⊙, dividing our sample into four halo mass bins. We construct the ancillary sample such that in each halo bin, we add sufficient galaxies to enable stacked stellar population gradient measurements in each halo bin for two bins in stellar velocity dispersion. In practice, we insist on a stacked S/N of 50 at a radius of 1.25-1.75 R_e. In total, the ancillary program aims to observe ∼ 50 additional central galaxies to add to the Primary and Secondary MaNGA observations, of which seven are included in this paper. +0mm-1mm< g r a p h i c s > 5mm§.§ Galaxy Sample We now turn to the properties of the entire sample of galaxies considered here. We are working with the MaNGA Data Reduction Pipeline version 2.0.1 sample (MaNGA Product Launch 5; MPL5; K. Westfall et al. in preparation). An initial set of central galaxies are selected from the Y07 catalog and are defined as the most luminous galaxy among the group members. Two coauthors performed visual inspection in the r-band of all of the Y07 groups with M_200b>10^14 h^-1 M_, the halo-mass range that we focus on for the ancillary sample. In doing these visual checks, we both ensure that the chosen overdensity exists and check the validity of the choice of central galaxy. Based on the visual inspection, we judge that the Y07 algorithm overall selected visually reasonable clusters and central galaxy candidates. We apply a stellar mass cut M_* > 10^10 h^-2 M_⊙ (in practice, 95% of central galaxies have stellar masses M_*> 10^10.5 h^-2 M_⊙) and a halo mass cut M_200b> 10^12.5 h^-1 M_⊙. As shown in <cit.>, the groups are quite incomplete below this halo mass in the redshift range of interest. As a result, the majority of our central galaxies have stellar masses M_*>10^10.5 h^-2 M_⊙. Satellite galaxies are those in the Y07 catalog that are not central galaxies. Of course, the satellite galaxies extend to much lower stellar masses. However, our primary goal here is to compare the satellite and central galaxy populations. Furthermore, at low stellar mass, the early-type galaxies in MaNGA are overwhelmingly unresolved (see also Appendix B). Therefore, we apply a stellar mass cut to the satellite galaxies of M_*> 10^10 h^-2 M_⊙. We adopt measured properties (e.g., stellar mass, galaxy radius, redshift) from the MaNGA source catalog, which in turn is based on version v1_0_1 of the NSA. The galaxy magnitudes are based on elliptical Petrosian apertures, measured as Petrosian magnitudes <cit.> but using elliptical apertures[http://www.sdss.org/dr13/manga/manga-target-selection/nsa/]. The stellar masses are derived using the k-correct code <cit.>, which fits spectral energy distributions to the elliptical Petrosian magnitudes to derive the mass-to-light ratio. A <cit.> Initial Mass Function is assumed.Typically λ_R is measured in a fixed aperture (often R_e), so that all galaxies can roughly be on the same footing. There are two complications to this approach for our sample. One is that the galaxies are typically poorly resolved spatially at R_e, which compromises the λ_R measurements (see Appendix B). The other issue is that it is notoriously difficult to measure a uniform effective radius <cit.>. In the end, we therefore adopt a measurement of λ_R that is not directly tied to R_e. Nevertheless, we must adopt some measure of size.There are two sizes tabulated by the NSA that we consider here. The one used by the MaNGA team to define the sample is the elliptical Petrosian radius. The other possibility within the NSA is the R_e provided by a single- fit. We adopt the aspect ratio (B/A) and position angle (PA) derived from the parametricfit because they are PSF-corrected. Indeed, when we compare B/A derived from the Petrosian andfits we find clear evidence that the PSF-correction makes a difference, since theB/A is typically 10% smaller than the Petrosian value. To be consistent between the size measurement R_e and the ellipticity and PA measurements, we adopt the circularized R_e from thefit. All R_e measurements use thefits. This measurement is roughly 30% larger than the elliptical Petrosian measurements used to define the MaNGA targets. Again, we emphasize that this decision does not impact the final λ_R measurements, but the circularized -derived R_e is used as a benchmark throughout the paper.§.§.§ Galaxy Morphologies Many of the central galaxies are late-type (spiral) galaxies. Late-type galaxies tend to have high λ_R values, and our main goal is to investigate the distribution in λ_R for the early-type galaxies. We have visually classified galaxies into those with or without spiral structure (early and late-type galaxies). Visual inspection was performed by the first author using the three-color SDSS images. Of the 475 central galaxies, there are 379 early-type central galaxies; of the 241 satellite galaxies, 159 are of early type. Of the 379 central galaxies, there are 217 Primary and 162 Secondary galaxies, while the 241 satellite galaxies comprise 90 and 69 Primary and Secondary galaxies (Table 1). There are 15 central and 13 satellite galaxies that we classify as ambiguous, most of which are edge-on galaxies that may be S0 or later spiral types. We exclude these ambiguous cases, although the numbers are too small to impact our conclusions. Throughout the paper we focus on the sample of early-type galaxies unless explicitly noted otherwise. Galaxy classification grows harder at higher redshifts. <cit.> has shown that above z ≈ 0.08, the fraction of galaxies classified as elliptical rises unphysically as detail is lost in imaging. We could introduce a redshift-dependent fraction of fast-rotating spiral galaxies into our early-type sample if this effect is at play. The morphological bias would cause us to measure a higher early-type fraction at z>0.08 relative to the lower-redshift bin, caused entirely by spirals appearing as early-types. To search for this effect, we take all the galaxies with stellar masses 10^10.75 < M_*/M_⊙ < 10^11 h^-2, and examine the early-type fraction with redshifts above and below z=0.08. There are 69 (63) objects in the low (high) redshift bin. We find consistent early-type fractions of 0.85 ± 0.12 and 0.80 ± 0.13 in the two redshift bins, suggesting that this morphology bias is not impacting our results.Of the central, early-type galaxies, seven belong to our ancillary program. Our sample contains 30 central galaxies in halos more massive than M_200b>10^14 h^-1 M_⊙ and six central galaxies in halos more massive than M_200b>10^14.5 h^-1 M_⊙. Many central galaxies are known to have a large extended halo, sometimes known as cD galaxies <cit.>; as the MaNGA sample grows it will become possible to examine trends between cD halo and λ_R. § ANALYSIS §.§ Kinematic Measurements We use the kinematic measurements provided by the MaNGA Data Analysis Pipeline (DAP; K. Westfall et al. in preparation). The individual spaxels are combined using Voronoi binning <cit.> to maintain a signal-to-noise ratio of at least 10 per spectral pixel of 70 . The number of combined spectra at ∼ R_e varies considerably from object to object depending on R_e, with a mean value of 12 spectra, a median value of 3 spectra, and a maximum of 200. Spaxels with individual S/N<1 are excluded from the binning. In calculating λ_R, we use the distance to the center of each spaxel.The kinematics are measured using the penalized pixel-fitting code pPXF <cit.>, with emission lines masked. Stellar templates from the MILES library<cit.>, which cover the spectral range 3525-7500Å at 2.5 Å (FWHM) spectral resolution <cit.>, are convolved with a Gaussian line-of-sight velocity distribution to derive the velocity and velocity dispersion of the stars (values are not corrected for instrumental resolution at this stage). An eighth-order additive polynomial is included to account for flux calibration and stellar population mismatch. When we extract the σ measurements from the MaNGA catalog, we correct them for instrumental resolution using the measured σ_r, which is the unweighted average of the difference in resolution between the templates and the data as measured over all wavelengths and all spectra in the cube (K. Westfall et al. in prep). The MILES templates are used at their native resolution of 2.5Å, which is higher than the MaNGA data over the full spectral range for all cubes <cit.>. With this approach, the pipeline is able to reliably recover intrinsic dispersions of ∼ 35 or more. The MaNGA team also tried to employ the wavelength-dependent kernel convolution available within pPXF, but could not recover such low velocity dispersions with that functionality enabled. As shown by <cit.>, even for intrinsic dispersions of 40 , it is possible to recover the dispersion to within 10% for high S/N spectra, while at our limiting S/N=10, it is possible to recover the dispersions above the nominal limit of 70 to within ∼ 20%. The DAP then supplies a single resolution correction that is the effective difference between the MILES templates and the MaNGA spectra calculated for each cube. The resolution as a function of wavelength is calculated from arcs taken before and after each exposure and then corrected using strong sky lines <cit.>. Tests indicate that these corrections are better than 5% for dispersions of 70or higher (Westfall et al. in prep). In cases that σ is below the spectral resolution of the instrument (a condition that does occur in the outer parts of some galaxies) we mask these values. Example maps for interesting subsets of the population are presented in Appendix A.While in general these kinematic measurements are robust, there are some known systematic failures. There are foreground stars, which have been successfully masked by MaNGA. At lower signal-to-noise ratios, particularly in the outer regions of the galaxies, the velocity dispersion measurements can peg at the unphysical value of 1000 km s^-1. Finally, the superposition of two companion galaxies in the IFU field of view can lead to unphysical velocity and velocity dispersion measurements if the two components are not jointly modeled. The MPL5 catalog has identified these unphysical values; we adopt their “DONOTUSE” flags, as well as flagging all σ >500 and V>400 values. We only analyze objects for which at least 50% of the fibers within R_e are not flagged. Visual inspection verifies that we remove most of the clear merger cases through these cuts, and no galaxies are removed by visual inspection. Excluding these problematic cases does not lead to any systematic bias in redshift or stellar mass distributions of the galaxies. We perform two checks on our measurements. We explore a different analysis of the MPL4 data cubes performed by J. Ge <cit.>. The kinematics are also measured with pPXF, but the binning and masking prescriptions are different. Themeasurements agree well in general, with ⟨λ_ MPL4 - λ_ MPL5⟩ = 0.02 ± 0.09. This provides confidence that the measurements are robust to detailed choices about masking and binning. We also compare the central σ measurements with the original SDSS measurements that were made on different spectra using a different fitting technique. We find decent agreement, with ⟨ (σ_ MaNGA-σ_ SDSS)/σ_ SDSS⟩ = -0.05 ± 0.1.§.§.§ Radial Coverage Keeping only galaxies with at least 50% of the fibers unmasked leaves 357 central galaxies. We have verified that no bias in stellar or halo mass is incurred when we remove the galaxies with problematic measurements. Among these 357 centrals, 272 have coverage at or beyond R_e, 136 have coverage at or beyond 1.5 R_e, and 73 have coverage at or beyond 2 R_e (Figure <ref>; Table 1). The median effective radius of the Primary sample is 5.2, while the Secondary sample has a slightly smaller median of R_e = 4.7. Turning to the satellites, there are 159 early-type galaxies with >50% of their fibers unmasked. Of these, 137 reach R_e and are spatially resolved by that point, 94 reach 1.5R_e, and 34 galaxies reach 2 R_e. Only 50 of the early-type satellites have R_e > 4 and radial coverage out to 1.5 R_e. In 3.2.1 we will return to the issue of spatial resolution and angular momentum measurements.§.§ λ_R Measurements The V and σ measurements on the binned data are used to calculate λ_R (Equation 1) in elliptical apertures, as defined from the single-component <cit.> fit from the NSA. We adopt the ellipticity [ϵ = 1- (B/A)] and position angle from this fit because the model is corrected for seeing and thus should be the most robust measurement available of these parameters, roughly measured at the effective radius (see also 2.3). The R_e value is the circularized half-light radius from thefit, and in the following figures and calculations, we use the radial coordinate R_ Area, which is calculated as the radius of the circle that would have the equivalent area as the enclosed spaxels (R_ Area = √(R_e,a× R_e,b) = √(N_ pixels× A_ pixel/π)), where A_ pixel is the area of a pixel <cit.>. Examples of λ_R are shown in Figure <ref>. In internal comparisons between different MaNGA teams, there is good agreement betweenvalues calculated with different prescriptions (M. Graham et al. in preparation).§.§.§ Spatial Resolution Constraints It is common in the literature to report λ_R measured at the effective radius of the galaxy: . The typical galaxy in our sample has R_e ≈ 5. Assuming typical seeing of FWHM≈ 2, there are only four to five resolution elements across a galaxy, meaning that the λ_R measurements are not well resolved at R_e. We have performed simulations (Appendix B) using the most resolved cubes in the MaNGA sample. Objects with low <0.2 can be highly biased towards lowervalues, by ∼ 40%. However, this bias can be mitigated by measuring λ_R at larger radius. As demonstrated quantitatively in Appendix B, a decent compromise uses the outer value of λ_R, measured in the outer 10% of the profile, . In practice, because of the typical radial extent of our data,matches λ(1.5 R_e) with no bias and a scatter of ∼ 20% (see Appendix B). Therefore, in what follows we will report values of , but our results do not change on average if we use the smaller sub-sample with λ(1.5 R_e) directly available.There is another challenging regime, those galaxies with incomplete coverage at large radius. In some cases, the λ_R curves may not reach an asymptotic value. To quantify how often this occurs, we extrapolate each profile to 2 R_e using a slope fitted to the outer 20% of the λ_R curve. We then ask whether the fast/slow designation would change at large radius, taking into account the error in ϵ as well as in the extrapolation.We find that 10% of the galaxies would change designation, with these galaxies having a very similar mass distribution to the overall sample.Thus, while limited spatial coverage is problematic, it should not change our basic results. §.§.§vs. ϵ The distribution ofas a function of ϵ is presented in Figure <ref>. In addition to the galaxies, we show an analytic model of an edge-on galaxy in which anisotropy is proportional to ellipticity <cit.>. The majority of the fast rotators lie above this magenta line as expected from prior work <cit.>. Galaxies scatter above the magenta line because of inclination and internal variations in ellipticity. The ATLAS^ 3D survey also defines an empirical division between “slow” and “fast” rotators. We will adopt a similar prescription to separate the two (black line), but in <cit.> the division between slow and fast rotator is defined at r=R_e. An empirical value of /√(ϵ)=0.31 best divides the populations. However, we are not adopting R_e as our aperture. We determine the revised value of /√(ϵ) empirically using our data. By comparingwith , we find that the former is 14% larger than the latter. Therefore, we scale the division between slow and fast rotators to /√(ϵ)=0.35. A number of different divisions into fast and slow rotator have been proposed in the literature <cit.>. If we were to adopt the Cappellari (2016) definition instead, only <10% of galaxies would change designation.We draw attention to the lower-right region of the figure, galaxies that apparently have relatively high ϵ > 0.4 and low <0.2. <cit.> do not have galaxies that populate this region <cit.>. Some of these outliers appear round visually, suggesting that they have poorly measured ϵ values. A related issue is that we utilize an effective ϵ value, but in general these massive galaxies grow more flattened at larger radius <cit.>, which may contribute to the scatter. However, the majority of these galaxies are quite elongated and display low levels of rotation along their major axis, with a high central dispersion (see example maps in Appendix A). Objects in this region may be galaxies with high angular momentum that masquerade as slow rotators. In particular, galaxies known as “double σ” galaxies have two well-separated peaks in their σ distributions, and have been shown to have counter-rotating disks <cit.>. We visually examine the low-, high-ϵ outliers and show a possible candidate for a double-σ galaxy in Appendix A, although our ability to distinguish such features is limited by spatial resolution <cit.>. Otherwise, there are no obvious differences between these galaxies and those with lowbut correspondingly low ϵ. We conclude that double-sigma galaxies are unlikely to dominate this outlier population, and since they constitute such a small fraction of the sample, we do not remove them from consideration in what follows.§.§.§ Notes on the Ancillary Galaxies There are seven massive central galaxies in this work that were added as part of our ongoing Ancillary Program (2.2). These galaxies preferentially live in the richest environments within the MaNGA footprint by design. As a result, most of them contain companions within the MaNGA IFS footprint. In two of the seven galaxies, more than 50% of the spaxels are masked due to contamination from this substructure.We are currently working to jointly model the kinematics from all the different substructures to build clean kinematic maps for this sample. Our work on decomposing galaxies into distinct components follows similar analysis by <cit.>. In the meantime, we include only five of the seven ancillary galaxies in our analysis (Table 1). § ANGULAR MOMENTUM CONTENT AS A FUNCTION OF STELLAR AND HALO MASS §.§ Slow Rotator Fraction as a Function of Stellar Mass Withand slow/fast rotator determinations in hand, we investigate trends in the standard slow rotator fraction using the Emsellem et al. definition (Figure <ref>; Table 2). Our stellar mass coverage extends only to M_*>10^10.5 h^-2 M_⊙ for the central galaxies due to the halo mass cut, while our satellite sample extends a bit lower and is limited by spatial resolution.+10mm-1mm< g r a p h i c s > 5mm Also, this figure includes only the early-type galaxies, but there are likely to be some biases in these by-eye determinations <cit.>. As a sanity check, we recalculate the slow-rotator fraction with spiral galaxies included. The slow-rotator fraction changes by ∼ 50% at the lowest stellar masses, but is unaffected for M_*>10^11 h^-2 M_⊙. Our conclusions are unchanged if we include the spiral galaxies in the sample. Consistent with previous work, we see a steep increase in slow rotator fraction with stellar mass. At M_* < 3 × 10^10 h^-2 M_⊙, galaxies are overwhelmingly fast rotators, with only 10-15% slow-rotator fraction. Galaxies with stellar mass M_* > 10^11 h^-2 M_⊙ are mostly slowly rotating, and this mass scale is consistent with the quoted transition mass from <cit.> once our h^-2 and IMF scales are matched. Our results are in good agreement with the literature. After correcting the ATLAS^ 3D masses to be in h^-2 units and shifting them to match our assumed Chabrier IMF from Salpeter <cit.> there is good agreement <cit.>. At the highest masses, our ancillary sample results agree well with the results from both <cit.> and <cit.>, as well as a number of studies of individual massive clusters <cit.>. Our results are also in qualitative agreement with the study by <cit.> based on the mass and luminosity dependence of the disky vs. boxy fraction of early-type galaxies, if one associates disky (boxy) galaxies with fast (slow) rotators, as suggested by the seminal work of <cit.> and <cit.>. §.§ Central vs. Satellite M_* clearly correlates strongly with λ_R. We now address whether there is an additional dependence on large-scale environment. One approach to evaluate the impact of environment is to compare central and satellite galaxies at fixed stellar mass. Figure <ref> displays a trend whereby satellite galaxies have a slightly higher slow-rotator fraction than the central galaxies at fixed stellar mass.+0mm +15mm< g r a p h i c s >This behavior is mirrored in Figure <ref> (top left; Table 3), where there is a difference in the median /√(ϵ), and also possibly a larger scatter towards highin the central relative to the satellite population. One possible driver of this difference could be spatial resolution. However, because of the design of the MaNGA survey, within the mass range of M_* = 10^10.5-10^11 h^-2 M_⊙, the central and satellite galaxy samples have similar median apparent sizes of 5 and comparable median redshifts of ⟨ z ⟩ = 0.065, suggesting that spatial resolution is not obviously to blame for the systematic difference between the two populations. Given that the central sample is much larger than the satellite sample, and given that the detailed mass distributions do not match between the two samples, we perform an additional test to compare the satellite and central galaxies. We focus on the mass range where the two populations overlap: M_*=10^10.5-10^11 h^-2 M_⊙ (203 central and 54 satellite galaxies). The observed mass distribution of the satellite sample is sharply falling towards the higher mass end of this bin, while the reverse is true for the central galaxy mass distribution. We thus assign weights to the central galaxies to force the mass distribution between the two populations to match. We then build the weighted distribution in /√(ϵ) for both the central and satellite galaxies (Figure <ref>) and compare the two distributions using an Anderson-Darling test <cit.>. There is a probability of P=33% that the two samples are drawn from the same underlying distribution. There is no compelling evidence for a difference between the central and satellite galaxies in their distributions of . However, we caution that some subtle differences may still exist due to the different radial coverage between the central and satellite samples seen in Figure <ref>. Furthermore, central galaxy samples are not pure, with the level of contamination depending on halo mass <cit.>. In 4.5 we revisit the impact of contamination in the group catalog using the mock catalogs from <cit.>.+1mm -5mm< g r a p h i c s > -0mm [] Distribution of /√(ϵ) in the mass range M_*=10^10.5-10^11 h^-2 M_⊙. We apply the MaNGA weights to both samples and also re-weight the central galaxies to match the mass distribution of the satellite galaxies. When the mass distributions are carefully matched, the difference in /√(ϵ) is not significant, with an Anderson-Darling test returning a probability P=33% that the two distributions match. 5mm We conclude that the central and satellite galaxies have statistically consistent distributions in /√(ϵ) when their mass distributions are carefully matched. We do not detect any significant difference between the central and satellite galaxies in their slow rotator fraction. §.§ Dependence on Halo Mass As an additional probe of the large-scale environment, we attempt to disentangle the stellar from the halo mass dependence (Figure <ref>; upper-right). We treat the central and satellite populations separately. We divide each into two groups based on their host halo mass, and examine the weighted mean /√(ϵ) of that subpopulation. We divide the central galaxies with a halo mass above and below 10^13.3 h^-2 M_⊙, which is the median halo mass. There are 172 (185) central galaxies in the higher (lower)-mass halo bin. The satellites are divided at their median halo mass of 10^13.8 h^-2 M_⊙, There are 83 (76) satellites in the higher (lower)-mass bin respectively. We re-weight the distributions such that the stellar mass distributions match, over the mass range of overlap (M_*=10^10.8-10^11.1 h^-2 M_⊙) for the ∼ 100 galaxies in each mass-limited sample. Although this comparison is limited to a narrow range in stellar mass, the distributions of central and satellite λ/√(ϵ) are consistent with each other, with an Anderson-Darling test returning a P=30% chance that the two samples were drawn from the same distribution. Similar results are found for the satellite galaxies. This group is divided at a higher halo mass of M_200b=10^13.8 h^-1 M_⊙. In a mass range of M_*=10^10.0-10^10.7 h^-2 M_⊙ (∼ 60 galaxies in each bin) we see that satellites in lower-mass halos tend to have a higher /√(ϵ). After forcing the mass distributions to match, we find only a marginally significant difference between the high and low halo masses, with an Anderson-Darling test returning a probability P=4% that the two samples are drawn from the same distribution. A larger sample is needed to investigate whether there is a real difference in the satellite population as a function of halo mass, but in the centrals our finding of no halo mass dependence is consistent with prior work <cit.>. Finally, we check for a correlation between /√(ϵ) and the magnitude difference between the central galaxy and the next brightest galaxy, and find no correlation.In a companion paper <cit.>, we further compare the fast and slow rotator fractions as a function of local overdensity to compare with the recent literature <cit.>.§.§ Galaxy Properties of Slow vs Fast Rotators While the majority of galaxies at high stellar mass are slowly rotating, there is a tail of fast-rotating galaxies even in the highest stellar mass bin <cit.>. Maps of massive fast and slow rotators are shown in Appendix A. Nearby small companions add some contamination to this class of objects (at the ∼ 15% level) but the majority are single objects with real rotation. We now investigate whetherproperties of the fastest and slowest rotating galaxies differ in any other interesting ways.We select the 32 fast-rotating central galaxies with > 0.3 and M_* > 10^11 h^-2 M_⊙ and compare with the 43 galaxies of the same stellar mass that have < 0.1. The two samples have similar median galaxy sizes of ⟨ R_ petro⟩ =16 and 15 kpc, respectively, and an Anderson-Darling test shows that they have indistinguishable distributions in size. They are at similar median redshift of ⟨ z ⟩ = 0.1, and have similar flattening of ⟨ϵ⟩ = 0.3 and 0.2, respectively, again with distributions consistent with arising from the same distribution. The galaxies have similar median color, ⟨ g-r⟩ =0.94 mag. The velocity dispersions in the fast rotators are lower (mean of 230 and 260 km s^-1 respectively; P=6 × 10^-4 that the distributions are the same). This difference is not surprising sincedepends inversely on σ.The one independent difference between the two samples appears to be in their emission line properties. Focusing on the Hα equivalent width within 3, and including only systems where the emission line within that radius is measured with >3 σ significance, there is a measurable difference in the equivalent width distribution (P=0.003 of belonging to the same distribution), with 47% of the fast rotators having Hα EW>1Å, while only 16% of the slow rotators have Hα EW>1Å (Fig. <ref>). This difference in emission-line properties suggests that the fast rotators do typically have higher gas content, as we might expect for galaxies with a disk component, perhaps associated with the event that increased their spin (see <ref> below). Unfortunately, the S/N ratios of the other strong lines are not high enough to examine the sources of photoionization (e.g., star formation, active nuclei) in these objects without more work. We perform a similar test with the satellite galaxies, but to construct a decent sample we must shift the mass limit down to log (M_*) = 10.5 h^-2 M_⊙. We see similar trends, although the difference in emission properties is less significant for the satellites. In the future, it will be interesting to investigate additional galaxy properties, such as luminosity-weighted mean age, which seems to correlate with the outer ellipticity of halos <cit.>, disky or boxy isophotes <cit.>, stellar population gradients <cit.>, or even cD envelope fraction, as <cit.> have argued that the cD envelope is an alternate tracer of merger history. §.§ Impact of Imperfect Group CatalogsGroup finders cannot perfectly identify dark matter halos. For our purposes, there are two main concerns. There is scatter in the assignment of halo mass to groups, leading to uncertainties in the halo masses. Also, there are errors in the identification of central galaxies <cit.>. To explore how these two errors impact our results, we employ a mock catalog built by <cit.>. Campbell et al. use N-body simulations to create a “true” galaxy catalog matched to the SDSS with perfect knowledge of the groups, halo masses, and central galaxies. They then run the Y07 algorithm on their mock galaxy catalog to create a mock Y07 group catalog. This catalog contains the same scatter in halo mass and in central designation as Y07, being constructed in the same manner.+1mm -5mm< g r a p h i c s > -0mm [] Distribution of Hα equivalent widths measured at the galaxy center for central galaxies with masses M_*>10^11 h^-2 M_⊙ and >0.3 (blue) or <0.1 (red). The fast rotators have a broader distribution towards high equivalent widths. 5mm We assign every mock galaxy a value of /√(ϵ) based on its stellar mass, with a linear dependence designed to roughly reproduce our measurements. We model the /√(ϵ) distribution as a sum of two Gaussians. The primary Gaussian (containing 80% of the galaxies) has a narrow width of 0.2, and a central value that varies with stellar mass as /√(ϵ) = -0.7log M_* - 0.14. The secondary Gaussian has no mass dependence and a large scatter and is added in an ad hoc way to match the scatter that we see at all masses. The center of the distribution is fixed at /√(ϵ)=0.8 with a dispersion of 0.5, and this component comprises 20% of the total. In the fiducial assignment, satellite and central galaxies are treated the same way at fixed stellar mass.First, we use these catalogs to test our sensitivity to differences between central and satellite populations. We create a suite of simulated satellite galaxies in which /√(ϵ) is boosted relative to the default values for central galaxies at a given stellar mass by δ/√(ϵ) = 0.1, 0.2, 0.3, 0.35 on average, with a scatter of 0.05. These differences are introduced in the true catalog, and then we ask whether we can recover this difference in the Y07 mock catalog. We then create 100 data sets with statistics matched to our true sample by selecting a mock galaxy with stellar mass within 0.05 dex and halo mass within 0.1 dex of each sample galaxy, for both central and satellite galaxies. We then calculate the dependence of /√(ϵ) and slow-rotator fraction on M_* as with the real data (adopting the appropriate MaNGA weight for each mock galaxy). We find that the difference between satellite and central becomes measurable when the offset is δ/√(ϵ) = 0.2 and significant when the offset is δ/√(ϵ) = 0.35 (Figure <ref>). As is apparent from the figure, there is a bias introduced by the mixing between satellites and centrals.Second, we test whether we could uncover a secondary trend in /√(ϵ) with M_200b at fixed stellar mass. We scatter the default values in the catalog by an amount that depends on halo mass. Specifically, we perturb the values according to: δ /√(ϵ) = m (log M_200b - log M_ 200b, median), for slope m, with m=0.1, 0.2, 0.3 and a scatter in m of 0.05. In Figure <ref> we see that there is minimal bias introduced by scatter in halo masses, and the two halo-mass bins become measurably different for a slope of >0.3, corresponding to changes in /√(ϵ)∼ 0.2 at M^* ≈ 10^11 . We see that the errors in the satellite/central designation suppress the input difference between central and satellite. The amount of suppression (λ/√(ϵ)∼ 0.2) is of the same order as the current uncertainty in the mean value. Thus, simply increasing the total number of objects will not help uncover subtle differences between central and satellite galaxies; rather we are limited by the systematic errors in the designation of centrals and satellites by group finders. While it is possible to uncover trends in halo mass for central galaxies (test 2 above), to distinguish differences in satellite vs. central galaxies, the impact of group finding methods will have to be forward modeled via mock catalogs. §.§ Linking λ_R with formation historySeveral papers have used simulations to investigate the primary mechanisms that impact λ_R and galaxy flattening (ϵ) in galaxies. We compare our results with their predictions here.Several studies <cit.> have argued that the dichotomy of early-type galaxies can be explained in a scenario whereby boxy, slowly rotating ellipticals have their origin in a merger that is both major (i.e., mass ratio of progenitors close to unity) and dry (i.e., progenitors have small gas mass fractions). In particular, <cit.> conclude that the observed stellar mass dependence of the boxy fraction requires that slow rotators result from mergers with a progenitor mass ratio < 2 and with a combined cold gas mass fraction < 0.1. <cit.> also find that major merging is a primary driver of angular momentum evolution. Interestingly, <cit.> find that the cluster galaxies in their simulations with no major merging are the ones with the most rapid decrease in λ_R, but they are not certain what physical process drives this decline. <cit.> examine the distribution of λ_R with stellar mass using the Illustris simulation <cit.>. They report that major mergers and gas accretion have the strongest impact on λ_R, with the former typically spinning down and the latter spinning up galaxies, although they do not yet consider black hole feedback or large-scale intrinsic alignments of galaxies as possible factors <cit.>. They find that lower mass galaxies can be spun up by accretion of gas, while at higher mass the accretion rates are not high enough to change λ_R. They suggest that high-mass galaxies above M_* ≈ 10^11 M_⊙ have uniformly low λ_R because at later times mergers and accretion are unable to significantly change their angular momentum content. In contrast, lower-mass galaxies can be spun up at late times by accretion and star formation. <cit.> also find that faster-rotating galaxies are more gas rich (and more metal-enhanced) and they do not find a discernible difference between satellite and central galaxies at fixed mass. <cit.> use cosmological simulations of 44 massive galaxies to examine the relationship between merger history and angular momentum content. In addition to λ_R, they compare their simulations with observations of the flattening of the merged remnant (ϵ), and higher-order moments of the line-of-sight velocity distribution parametrized with Gauss-Hermite polynomials; h_3 is the asymmetry parameter <cit.>. They identify three pathways that form fast rotators: galaxies with little merging but some gas (e.g., a faded disk galaxy), late gas-rich merging that spins up the remnant (these have anti-correlated h_3 and V/σ), and late dissipationless merging that spins up the remnant <cit.>. <cit.> have identified SAMI galaxies without an h_3-V/σ anticorrelation that may indeed be the remnants of late dissipationless merging. There are also three main pathways to make slow rotators in their simulations: galaxies that form early and only experience minor merging since, galaxies with a gas-rich major merger that spins down the remnant, and galaxies with a gas-poor major merger that spins down the remnant. Apparently these last two events produce highly flattened configurations, and may correspond to our small tail of high , high ϵ galaxies. § SUMMARYWe present an unprecedented sample of 503 (379 early-type) central and 241 (159 early-type) satellite galaxies observed with the SDSS-IV MaNGA IFU survey. We leverage this sample to study the dependence of the specific stellar angular momentum λ_R on stellar and halo mass. We define a new measure of λ, the asymptotic value , that allows us to compare cubes with different spatial resolution and spatial coverage. We investigate the slow rotator fraction along with thedistributions as a function of stellar (M_*) and halo (M_200b) mass. Overall, the observed distribution of galaxies in thevs ϵ plane matches expectations from previous work, with most fast rotators well-described by oblate rotator models in which the anisotropy correlates with ϵ. However, there is a small but interesting tail of galaxies with lowand high ϵ.Aligned with all previous work on this topic, we find a clear and strong dependence of /√(ϵ) on stellar mass M_*. There is a tail of high angular momentum galaxies (∼ 30%) even at the highest masses. These galaxies tend to contain more ionized gas emission, but otherwise show no other differences with the slowly-rotating systems. Central and satellite galaxies have similar slow-rotator fractions and distributions in /√(ϵ) at fixed stellar mass. There is no evidence for a residual dependence of /√(ϵ) on M_200b for central or satellite galaxies once the mass distributions are matched. As the MaNGA survey continues, the number of massive central galaxies in the most massive halos will increase, producing a wider baseline for study. We will also investigate secondary trends with stellar populations and gradients therein <cit.>, the spatial extent of the ionized gas, and the presence or absence of an extended stellar halo.We thank the anonymous referee for carefully reading the manuscript multiple times. We thank P. Hopkins and K. Bundy for useful conversations. We especially thank Z. Penoyre and S. Genel for providing their manuscript before submission. J.E.G. is partially supported by NSF AST-1411642. SM is supported by the Japan Society for Promotion of Science grants JP15K17600 and JP16H01089. FB acknowledges support from the Klaus Tschira Foundation, through the HITS-Yale Program in Astrophysics (HYPA). Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS-IV acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. The SDSS web site is www.sdss.org.SDSS-IV is managed by the Astrophysical Research Consortium for theParticipating Institutions of the SDSS Collaboration including theBrazilian Participation Group, the Carnegie Institution for Science,Carnegie Mellon University, the Chilean Participation Group, the French Participation Group, Harvard-Smithsonian Center for Astrophysics,Instituto de Astrofísica de Canarias, The Johns Hopkins University,Kavli Institute for the Physics and Mathematics of the Universe (IPMU) /University of Tokyo, Lawrence Berkeley National Laboratory,Leibniz Institut für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA Heidelberg),Max-Planck-Institut für Astrophysik (MPA Garching),Max-Planck-Institut für Extraterrestrische Physik (MPE),National Astronomical Observatories of China, New Mexico State University,New York University, University of Notre Dame,Observatário Nacional / MCTI, The Ohio State University,Pennsylvania State University, Shanghai Astronomical Observatory,United Kingdom Participation Group, Universidad Nacional Autónoma de México, University of Arizona,University of Colorado Boulder, University of Oxford, University of Portsmouth,University of Utah, University of Virginia, University of Washington, University of Wisconsin,Vanderbilt University, and Yale University.This research made use of Marvin, a core Python package and web framework for MaNGA data, developed by Brian Cherinka, Jose Sanchez-Gallego, and Brett Andrews (MaNGA Collaboration, 2017). 94 natexlab#1#1[Abazajian et al.(2009)]abazajianetal2009 Abazajian, K. 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R., Drory, N., Roman Lopes, A., & Malanushenko, O. 2017, , 465, 4572 § APPENDIX AThree figures displaying examples of galaxies with high stellar mass (M_*>10^11 h^-2 M_⊙) and <0.1 (Figure A8), galaxies in the same mass range but with >0.4, and finally galaxies with <0.15 and ϵ>0.45.5mm5mm 5mm § APPENDIX B§.§ Binning and Signal-to-Noise Voronoi binning is done to keep the signal-to-noise (S/N) ratio above a baseline level of 10. However, some of the outermost bins have lower S/N in practice. They also can cover large regions of the galaxy. It is natural to ask whether the λ_R measurements are sensitive either to degraded S/N in the outermost parts of the galaxy or to irregular bin shapes. We briefly address each concern here.First, we ask whether including noisy measurements in the outer bins has a strong impact on the value ofthat we infer. We perform a simple test. We simply perturb each velocity measurement with a random Gaussian variate using twice the reported error in the measurement, and then recompute the perturbed _,p. The distribution in δ=-_,p is strongly peaked at zero, with a negative tail. That is, adding noise tends to increaseslightly, with a median δ = -0.001, with 30% of objects having δ < -0.05 and 5% of objects having δ < -0.1. We do the same test with σ rather than V. This change has even less effect, presumably because σ is an even quantity, and so the average σ is the same everywhere at a given radius, unlike V. The distribution in δ is peaked at zero, with a FWHM of 0.01. Only 8% of systems have |δ| >0.02 and only one object has |δ| >0.04. Finally, we repeat the same test but introduce scatter in both V and σ at the same time. In this case the systematic bias is slightly larger, with a median δ = -0.02 ± 0.01 (Figure <ref>). Only 6% of systems have |δ| >0.04 and 1% of systems have |δ| >0.05.The punchline is that scatter in the outer measurements does not influencestrongly. While we experimented with excluding measurements based on large uncertainties, this test suggests that it is worth keeping all the measurements, even those that are noisy. If we exclude lower S/N measurements, we simply restrict the radial range of the data unnecessarily.Second, we do the following test to see how the large and irregular bin sizes impact . We run our same λ_R measuring apparatus on the unbinned spaxels as we do on the Voronoi-binned data. Since the unbinned data can get very noisy in the outer parts, we restrict attention in this test to . Taking only galaxies that are binned (with at least five or more spaxels per bin at R_e), we again find that the majority of objects have δ = 0, and again we find a bias towards largerin the binned versus unbinned data. Again, the bias is quite small, with only 11% of galaxies having |δ| > 0.01 and only 5% having |δ| > 0.02. §.§ Spatial Resolution Because of the design of the MaNGA survey, most galaxies are moderately resolved, with typical angular radii (in our early-type, central galaxy sample) of 5. We cannot afford to exclude galaxies that are only marginally resolved at R_e (Figure 2). In this Appendix, we explore creative ways to use the available information to derive robust λ_R measurements that can be compared across the entire sample.To understand how the resolution impacts our measurements, we define a subset of 50 galaxies that have sizes R_e > 5.5 (more than four beams across R_e, for a median seeing FWHM of 2.5) and radial coverage to at least 1.5 R_e. While some previous work <cit.> has performed simulations starting with the ATLAS^ 3D sample, which has very high sampling and spatial resolution, working directly with the most spatially extended MaNGA cubes provides two key advantages. Firstly, because of the different sized MaNGA bundles, we are able to find galaxies that are both well-resolved and have coverage beyond 1.5 R_e. Very few ATLAS^ 3D cubes extend to such radii, but this is the regime where the MaNGA data are best.Secondly, because of the small volume of ATLAS^ 3D, the sample is heavily weighted towards low-mass galaxies. Our well-resolved sample spans the same range in stellar mass as our parent sample (Figure <ref>).Taking this sample, we degrade the spatial resolution such that the final FWHM=R_e/3.6 (which is typical for the sample as a whole). In degrading the data, we assume a Gaussian PSF and that the line-of-sight velocity distribution (LOSVD) for each fiber is a Gaussian, simply adopting the measured velocity and velocity dispersion as measured by pPXF for each fiber. A new LOSVD is constructed at each position by weighting each spectrum with the new PSF and combining the weighted spectra, while the new velocity and velocity dispersion are derived from a fit to the new LOSVD. We then recompute λ(R), and compare the values of λ_ R_e between the input and smoothed cube.In general, λ_ R_e is sensitive to the resolution, and the situation is most severe for systems with <0.2. For galaxies with inputbetween 0-0.2,values from the simulations are typically lower by 45% compared to the true values (assuming the typical MaNGA galaxy ratio of seeing to R_e of ∼ 2.5). Galaxies with >0.2 are impacted at the 15% level on average (Figure <ref>, left). If we measure λ at 1.5 R_e instead, the bias decreases. Galaxies with intrinsic <0.2 decline by only 35%, while galaxies with higherbiased low by 10% (with an absolute offset of -0.02 in λ_R). Using λ at 1.5R_e, however, excludes 5% of the galaxies whose rotation curves do not quite reach this radius. Thus, we introduce a final measurement, the mean λ as measured in the outermost 10% of the λ_R curve, . The quantitymatches λ(1.5 R_e) with no bias, and a scatter of 20%. To maximize the sample size, we compare objects using , but we repeat all of our analysis for λ at 1.5 R_e and none of our conclusions change. We apply one more correction to mitigate the final 10% bias. For obvious reasons, within the central beam (∼ 2 arcsec), all the rotation is measured as dispersion, naturally lowering λ everywhere. If we excise this inner region, then bothand λ at 1.5R_e increase by ∼ 10%. We demonstrate this in two ways. First, we take our smoothed models, remove the central spaxels that are within the smoothed FWHM, and remeasure λ(1.5 R_e; Fig. <ref>). We find that λ(1.5 R_e) increases on average, such that Δλ = (λ_1.5 R_e, m-λ_1.5 R_e)/λ_1.5 R_e = 0.04. There is quite a bit of scatter introduced in these models because of their relatively limited radial coverage. As a second test, we go back to the full sample and experiment with removing the central 2 , in Figure <ref>. Interestingly, the slowest rotators have theirvalues drop slightly when the center is excluded (by ∼ 0.01-0.02, which is in the noise of ourmeasurements) but at >0.2,systematically increases by ∼ 10%. Thus we are able to fully recover the input λ(1.5R_e) values in our simulations, on average. This correctedis the value utilized here. §.§ Impact of resolution on slow-rotator fraction We can use these simulations to calculate an error bar on the slow-rotator fraction, by asking how many galaxies might change designation based on residual bias in ourvalues. For >0.2, there is effectively no bias. At lower <0.2, we tend to systematically underestimateby 30% (Figure <ref>). Here we estimate the resulting uncertainty in slow-rotator fraction that ensues from this uncertainty, using two methods. First, we take all the objects in the sample with a measured <0.14 (corresponding to an intrinsic <0.2). If we assume that these all have an intrinsicthat is 30% higher, we can ask what fraction of these would be considered fast rotators if we did not have a biasedmeasurement for them. Of the 72 sources with <0.14, 16 are mis-identified as slow rotators, constituting a 10% error in the slow-rotator fraction. The mass distribution of the lowsources is comparable to that of the full sample, with a median ⟨log M_* ⟩ = 10.9 . Thus, we feel confident that the residual errors incurred from our lower-than-optimal spatial resolution do not impact the main conclusions of this paper.Second, we do an actual Monte Carlo experiment. In three bins of(<0.1,0.1-0.2,>0.2), we measure the mean offset and scatter from Figure <ref>. In each Monte Carlo run, we perturb λ/√(ϵ) based on this mean offset and scatter. Based on these perturbed values, we calculate a new slow rotator fraction. We then calculate the mean and scatter in the slow rotator fraction from these runs at each mass. In Figure <ref>, we show the perturbed values in the dotted lines; there is little difference between the dotted lines and our calculated values in solid. | http://arxiv.org/abs/1708.07845v2 | {
"authors": [
"Jenny E. Greene",
"Alexie Leauthaud",
"Eric Emsellem",
"J. Ge",
"A. Arag'on-Salamanca",
"J. P. Greco",
"Y. -T. Lin",
"S. Mao",
"K. Masters",
"M. Merrifield",
"S. More",
"N. Okabe",
"D. P. Schneider",
"D. Thomas",
"D. A. Wake",
"K. Pan",
"D. Bizyaev",
"D. Oravetz",
"A. Simmons",
"R. Yan",
"F. van den Bosch"
],
"categories": [
"astro-ph.GA"
],
"primary_category": "astro-ph.GA",
"published": "20170825180036",
"title": "SDSS-IV MaNGA: Uncovering the Angular Momentum Content of Central and Satellite Early-type Galaxies"
} |
KU Leuven, Instituut voor Theoretische Fysica, Celestijnenlaan 200D, B-3001 Leuven, Belgium KU Leuven, BIOSYST-MeBioS, Kasteelpark Arenberg 30, B-3001 Leuven, Belgium05.70.Ln, 87.16.A-, 87.16.Nn Recent experiments demonstrate that molecular motors from the Myosin II family serve as cross-links inducing active tension in the cytoskeletal network. Here we revise the Brownian ratchet model,previously studied in the context of active transport along polymer tracks, in setups resembling a motor in a polymer network, also taking into account the effect of electrostatic changes in the motor heads.We explore important mechanical quantities and show that such a model is also capable of mechanosensing.Finally, we introduce a novel efficiency based on excess heat production by the chemical cycle which is directly related to the active tension the motor exerts.The chemical efficiencies differ considerably for motors with a different number of heads, while their mechanical properties remain qualitatively similar.For motors with a small number of heads, the chemical efficiency is maximal when they are frustrated, a trait that is not found in larger motors.Tension and chemical efficiency of Myosin-II motors Herman Ramon December 30, 2023 ===================================================§ INTRODUCTIONThe cytoskeleton of a cell mainly consists of long semi-flexible actin polymers and myosin II molecular motors <cit.>.These motors contain several heads that are able to bind to polymer filaments and walk along them <cit.>.The mechanism for walking relies both on a chemical cycle in the motor's head and on the asymmetry of a periodic electrostatic potential along such a polymer <cit.>.The cytoskeleton serves multiple purposes:transporting large vesicles, providing the cell's stiffness, driving the separation of the cell during cytokinesis and governing the cell's motility, i.e. the ability of the cell to move and change its shape <cit.>. The part of the cytoskeleton that plays a crucial role in the stiffness and motility of the cell is the actin-myosin cortex <cit.>, which will serve as the exemplary system in this paper.Recently, it has been argued that the main role of myosin II motors inside the cytoskeleton is to generate tension in the actin polymer network <cit.>. Therefore, the main goal of this work is to capture the behavior of myosin II motors in such a networkand to introduce an efficiency of the motor that is not related to transport, but rather to tension in the motor. We show that this efficiency is directly related to the net supply of energy provided by the chemical cycleand we investigate how this chemical efficiency depends on ATP concentrations and on the external force on the system. However, before we do so, we revisit certain features of molecular motors in a simple Brownian ratchet model <cit.> with constant chemical rates and relate the force-velocity relation to the mean force generated by the ratchet potential and to the tension in the motor.Cytoskeletal polymers exhibit a sequence of monomers that cause a periodic structure <cit.>.These monomers are polarized, so that when chained together in a polymer, they give rise to an asymmetric, ratchet-like, electrostatic potential along the length of the filament <cit.>.A simplified version of such a potential, used in this work, is shown in Fig. <ref>.The heads of a myosin II motor also carry an electric charge <cit.> andtherefore feel the sawtooth potential of the filaments. The chemical cycle that powers the molecular motor is driven by ATP molecules that can attach to a head of the motor, causing it to change the charge distribution of the head <cit.>. As a consequence, the head loosens its bond with the polymer filament.When bound to a motor head, the ATP molecule will hydrolyze to ADP and release a phosphate <cit.>.It this state, the head can attach to the polymer again, ADP will be released from the motor head and the chemical cycle can start over.Hence, the chemical state of a motor head in our model will determine how strongly it can be coupled to a cytoskeletal filament <cit.>.Therefore, the head will experience a sawtooth potential that is stochastically flashing in time, changing the amplitude of the potential from high to low.This mechanism provides a motivation for using ratchet models,where the motor heads interact with a periodic asymmetric potential that changes depending on the chemical state of the head.The basic cycle of the ratchet model works as follows: Whenever a motor is in a state where no ATP is bound to heads, i.e., the state in which the motor is tightly bound to the filament,the head will most probably be very close to a minimum of the ratchet potential.On the other hand, while in the state where the ATP is bound to heads, it is easier for the motor to diffuse over a larger distance.Due to the asymmetry of the ratchet potential, the motor will be biased to explore one side of the ratchet potential rather than the other.Therefore, the motor will more likely jump to an adjacent period of the sawtooth in the preferred direction, indicated by the asymmetry of the ratchet potential <cit.>. This will eventually lead to directed motion of the motor on long timescales <cit.>. Ratchet models have already been studied extensively in the context of molecular motors <cit.>. The models, used in these studies, are very similar to each other but often differ in some basic assumptions. The specific model that we have studied is presented in section <ref>.It has the important trait that the chemical rates do not depend on spatial degrees of freedom,the sawtooth potential has a non-zero amplitude in all statesand motor heads can attach and detach anywhere along the polymer filament. Furthermore, the rates in our model do not depend on external forces acting on the motor but solely on concentrations of chemicals involved in ATP hydrolysis.We assume that constant concentrations are maintained in the environment at all times. While for discrete models force depended rates are deemed necessary to reproduce experimental results <cit.>,in the scope of this study, constant rates will prove to suffice.The type of motor that we study is non-muscle myosin II, which has a small number of heads <cit.>.These non-muscle myosin II motors are crucial to the motility of most eukaryotic cells <cit.>.In muscle cells and in experimental studies with synthetic myosin motors,the number of heads per motor is also much higher than in non-muscle cells <cit.>. For those motors, mean-field calculations have led to analytic results for the force-velocity relation <cit.>, which play an important role in interpreting and understanding those in-vitro experiments. For motors with fewer heads, mean-field methods are not suitable and numerical simulations prove to be useful.In section <ref> we describe the dynamics of this motor-polymer system in detail by means of numerical methods. Specifically we study the mean velocity, force – velocity relation and stall force of myosin II motors in order to verify our proposed model.As the main role of the myosin II motor is to generate tension in the actin polymer networks <cit.>,we investigate the tension in the motor in section <ref> for multiple setups,mimicking the typical polymer environment,e.g., the coupling of the motor to an elastic environment. Earlier studies have also shown that myosin II motors are able to sense the stiffness of their environment and that they can adapt the active forces that they generate accordingly <cit.>. In those studies, the chemical rates did depend on the load on the motor.Weshow that our simple ratchet model with constant rates is not only capable of capturing this important biological trait, it naturally emerges from the dynamics of the system without any additional assumptions.Another important aspect of molecular motors is their ability to convert chemical energy into mechanical work <cit.>. The chemical cycle of ATP hydrolysis consists of many consecutive reactions and after one full cycle a net amount of energy is provided to the motor head <cit.>. However, often this cycle is represented by only two states <cit.>, e.g. ATP bound and ADP bound,in which case a stochastic process on these states always satisfies detailed balance and no chemical energy can be extracted.In other words, the energetic gain of one reaction is canceled by the cost of the following reaction. Nevertheless, since the chemical process is coupled to a diffusion process, we do break detailed balance in our model,as the energy gain per transition varies while the rates remain constant.In section <ref>, we look at the energy balances in various setups of a myosin II motor interacting with actin filaments. We link the excess heat for the chemical cycle to the ratchet force and consequently to the active tension produced by the motor. This consideration leads to a novel way of defining a chemical efficiency, which is presented in section <ref>.This efficiency compares the net amount of energy that is extracted from the ATP cycle to the total energy that is supplied to the motor. Simultaneously, we show that the chemical efficiency relates the chemical input, measured by the differences in potential energy in each jump,to its active tension, rather than to its displacement. We investigate how this efficiency depends on the external load, number of heads of the motor and ATP concentration.Consequently, we find that motors with a low number of heads, contrary to those with a high number of heads, achieve maximal chemical efficiency when they are frustrated, i.e., close to maximal tension. That contrasts with the observation that from the mechanical perspective there is no qualitative difference between them. § THE RATCHET MODEL To investigate the movement of myosin along the actin filament and the forces it generates, we consider the following one–dimensional model. At first, we assume that the rigidity of the actin polymer and myosin backbone is high enough so that their elongations can be neglected. Secondly, we assume that all myosin heads are attached directly to the myosin backbone at fixed equidistant positions; see Fig. <ref>.These simplifications allow us to model both the motor and the filament as point particles undergoing overdamped diffusion in the inter-cellular environment <cit.>. Moreover, each myosin head has internal (chemical) states, e.g. tellingwhether or not there is an ATP bounded to the myosin head. The dynamics of the positions is described by coupled overdamped Langevin equations [b]x_M =- 1/γ_M[ ∇_M V_t(x_M(t) - x_A(t)) + F_load] t + √(2 k_B T/γ_M)W_M , [b][.42]x_A =- 1/γ_A∇_A V_t(x_M(t) - x_A(t)) t+ √(2 k_B T/γ_A)W_A ,where x_M denotes the position of the motor's first head,x_A denotes the position of an arbitrary reference point on the actin filament,γ is the Stokes drag, T is the absolute temperature of the environment, W_M and W_A denote independent Wiener processes, F_load is an external load on the motor,and the interaction potential V_t represents the overall ratchet interaction,which is the sum over individual contributions from all motor heads,V_t(x) = ∑_i=0^N-1ζ_t(i) V_r (x - i d ),where d is the distance between myosin heads,N is the number of heads per motor, ζ_t(i) represents the internal state,and V_r is the periodical ratchet potential; see Fig. <ref>, V_r(x) = V_r(x+ℓ) x < 0 V x / a ℓ0 ≤ x < a ℓV (ℓ-x) / (1-a) ℓa ℓ≤ x < ℓV_r(x-ℓ)ℓ≤ x ,with total height V, period ℓand skewness parameter a ∈ [0,1]. The dynamics of the internal state of heads is governed by a continuous time Markov proces on two states with constant rates,ζ_t = 1 λ_bλ_u⇄ζ_t = c ,where c∈[0,1] determines the amplitude of the ratchet potential when the motor is weakly interacting with the filament,which we use to account for changes in the free energy landscape <cit.>.This choice is justified by multiple considerations:First, in our model the internal state represents whether ATP (or one of its products) is bound or unbound to the myosin head, rather than the binding of the myosin motor to the actin filament itself. In other words, the internal state characterizes the electrochemical properties of the head,which further determine the magnitude of the interaction potential with the actin filament.Secondly, we assume that whether the head itself is bound or not has no influence on the binding site of the ATP molecule.In other wordsthere is no effective dependence of the dynamics of the chemical cycle on the relative position of myosin and actin. Similarly, we assume that an external load does not influence the binding site of ATP on the motor head.It only acts on the motor backbone and opposes forces originating from the ratchet interaction. Finally, we simplify the full chemical network governing the internal states of myosin II <cit.> to only two states, which represent the long lasting states that significantly change the physical properties of the motor head. Namely, we look at states where either ATP is bound or ADPis bound to the myosin head,as escape rates from those states are small with respect to other transitions <cit.>.For these states, the estimated free energy landscapes <cit.> very well reflect the changes in conformation and electric charge of the myosin head <cit.>.Note that while the rates λ_b, λ_u are independent of the load,they depend on the concentrations of the relevant chemical species involved in ATP hydrolysis.More specifically λ_b is proportional to [ATP] while λ_u is inversely proportional to [ADP] and [P_i].In order to obtain transition rates between these two states, we start from the three state model introduced by Albert et al. <cit.>, which we further simplify by unifying the states corresponding to the hydrolized ATP (i.e. ADP+P) and ADP with P released,as this process occurs on a much shorter time-scale than other transitions.While both the diffusion and the jump process are in equilibrium separately,the mechanism of the flashing ratchet brings the combined system out of equilibrium, leading to movement of the motor in a preferred direction, given by the asymmetry of the filament <cit.>.Fig. <ref> depicts the whole setup of the system and Fig. <ref> depicts the dynamics involved in flashing ratchet.Table <ref> lists the values of the parameters used in this model.Some of them were taken from the literature while others were fitted, leading to realistic mean relative velocities <cit.> and stall force <cit.> for myosin II. The friction coefficients have been calculated with Stokes' law <cit.>, using the known dimensions of the molecules <cit.> and the viscosity of the intracellular medium <cit.>. §.§ Alternative setups The previously introduced setup, shown in Fig. <ref>, is most commonly studied <cit.>, with the aim to characterize the transport induced by molecular motors. However, recent experiments show that the main role of myosin II motors is to cross-link the cytoskeleton <cit.> and consequently produce an active tension inside the network <cit.>. As we want to investigate also the validity of our model under these conditions,we present a couple of natural extensions to this setup, more faithfully capturing the behavior of myosin II motors inside these networks. The original setup will be further referred to as “load on the motor”.§.§.§ Load on the polymer The first alternative is very similar to the scenario of “load on the motor”,but now with the external load applied to the polymer instead of the molecular motor.This corresponds to a situation where the motor is attached to the polymer, which itself is under the tension due to the action of other motors and complex interactions in a larger network. As weshow later, in this setup the tension on the motor vanishes for large loads, which will be relevant for our discussion of the chemical efficiency.The basic scheme of this setup is depicted in Fig. <ref>. As was advertised, the equations of motion are very similar to those of the “load on the motor”, i.e. (<ref>) and (<ref>).[b][.42]x_M =- 1/γ_M∇_M V_t(x_M(t) - x_A(t)) t + √(2 k_B T/γ_M)W_M , [b][.42]x_A =1/γ_A[ F_load - ∇_A V_t(x_M(t) - x_A(t)) ] t+ √(2 k_B T/γ_A)W_A .In these equations, the argument of the ratchet potential is preserved,while the load F_load is applied on the polymer and its direction is reversed with respect to the “load on the motor” setup.This ensures that the mean relative velocity of the motor with respect to the polymer is directly comparable in both setups.The equations of motion of this setup can be mapped into the “load on the motor” setup by interchanging the label of myosin and actinalong with inverting the x-axis,i.e. γ_M↔γ_A and x_M↔ - x_A. §.§.§ Tug of war A third setup that we investigate is a natural extension of the previous setup, but now the myosin motor is interacting with two, rather than one,anti-parallel actin filaments.The external load is again applied directly to the filaments instead of to the motor, as demonstrated in Fig. <ref>. This corresponds to a situation where the motor is actively cross-linking two polymers, which are subjected to the tension caused by other motors and other external influences. Hence the equations of motion read[b][.45]x_M =- 1/γ_M∇_M[V_t(x_A_1(t) - x_M(t))+ V_t(x_M(t) - x_A_2(t))] t+ √(2 k_B T/γ_M)W_M ,[b][.45]x_A_1 =- 1/γ_A[ F_load + ∇_A_1 V_t(x_A_1(t) - x_M(t)) ] t+ √(2 k_B T/γ_A)W_A_1 ,[b][.45]x_A_2 =1/γ_A[ F_load - ∇_A_2 V_t(x_M(t) - x_A_2(t)) ] t+ √(2 k_B T/γ_A)W_A_2 ,where subscripts A_1 and A_2 respectively denote the first and second actin filamentand where the anti-alignment of polymers is represented by reversing the sign in the argument of the ratchet potential (<ref>) for polymer A_1.§.§.§ Elastic environment As polymers inside the cytoskeleton are usually thoroughly interconnected rather than freely diffusing <cit.>, we can opt for an alternative approach. Rather than applying an external load we attach filaments to fixed points by a spring with constant stiffness k_sp,similarly to the approach discussed in <cit.>.These represent points of interaction with the rest of the network while the spring stiffness k_sp represents the elastic properties of the full network.The myosin motor is again interacting with two filaments that are arranged in an anti-parallel fashion.Hence, similar to the previous case, it no longer has a preferred direction of motionand it eventually reaches a steady state where its mean velocity will become zero.The observable of interest is now the mean force that the motor applies to the filaments,which is obtained via the extension of the springs attached to the filaments, and which represent the active tension in the network produced by the single myosin minifilament.The corresponding setup is sketched in Fig. <ref>.The equations of motion are just a modification of (<ref>), i.e., [b][.45]x_M =- 1/γ_M∇_M[V_t(x_A_1(t) - x_M(t))+ V_t(x_M(t) - x_A_2(t))] t+ √(2 k_B T/γ_M)W_M ,[b][.45]x_A_1 =- 1/γ_A[ ∇_A_1 V_t( x_A_1(t) - x_M(t) )+ k_sp( x_A_1(t) - x_A_1(0) ) ] t+ √(2 k_B T/γ_A)W_A_1 ,[b][.45]x_A_2 =- 1/γ_A[ ∇_A_2 V_t( x_M(t) - x_A_2(t) ) + k_sp( x_A_2(t) - x_A_2(0) ) ] t+ √(2 k_B T/γ_A)W_A_2 ,where the load is replaced by the force caused by springs and where the reference point for the springs is chosen as their position at some initial time.§.§ Numerical methodWe use an Euler-Maruyama integration scheme to stochastically evolve the systems from a randomized initial configuration <cit.>.Waiting times between chemical jumps are generated by a Gillespie algorithm <cit.>. Most results are obtained after evolving the system up to a time of 1 s with time step Δ t = 0.01 ns and averaging over 100 independent copies. An exception is the data in Fig. <ref>,obtained after a time-average every 1 ms over 10 s with time step Δ t = 0.01 ns and ensemble average over 1000 independent copies. Finally, to obtain the empirical distribution in Fig. <ref>,we sampled every 0.5 ms for a total time of 1 s with time step Δ t = 0.001 ns and from 100 independent simulations.The data were converted into a histogram with a bin width of 0.1 nm.§ MODEL VERIFICATION As our model deviates from the traditionally used ratchet model, as discussed in section <ref>, we need to verify that we can reproduce the behavior reported in the literature for flashing ratchet models and myosin II motors.The mean velocity, force–velocity relation and the stall force are numerically obtained for our model. §.§ Mean velocity We start with a study of the mean velocity of a four-headed motor in the “load on the motor” setup,⟨ v ⟩= ⟨ x_M⟩/ t - ⟨ x_A⟩/ t,for different values of the scaling factor c and without any external load F_load = 0. Since the scaling factor c modifies the interaction strength between the motor and the polymer, it effectively models the electrostatic screening of the charges in the ionic solution inside a cell <cit.>. Note that in this particular case, where there is no external load, setups “load on the motor” and “load on the polymer” coincide. From Fig. <ref>we conclude that the mean velocity has a maximum at c=0 and goes to zero when c approaches 1.This result is expected since c=1 corresponds to equilibrium, where there is no difference between the two internal states, cf. (<ref>). On the other hand, for c=0 we have the biggest discrepancy between the two internal states of each head.Secondly, the mean velocity (<ref>) also depends on the concentration of ATP through the binding rate λ_b.In this setup, we fix the ATP unbinding rate λ_u and scaling factor c, cf. table <ref>, and vary the ATP binding rate λ_b.The resulting dependency is shown in Fig. <ref>. We observe that the maximal velocity is reached for rates close to λ_b = λ_u.Deviating far from this ratio means that one of the states become dominant and thus the system effectively approaches equilibrium. The main difference between λ_b/λ_u being low or high is the mean amplitude of the interaction with actin.In the ATP bound state the interaction is weaker,which is reflected in the fact that the system is trapped for shorter times close to the local minima and diffuses more.Hence, the variance of the relative velocity is higher when compared to the ATP unbound state. §.§ Force–velocity relation In the previous section, we reported on the mean relative velocity for a motor in the “load on the motor” setup without any external load.Here we focus on the dependence of the mean relative velocity (<ref>) on an external load force F_load applied on the motor anti-parallel to the preferred direction,cf. Fig. <ref>.Fig. <ref> shows a typical force – mean velocity relations for various ATP binding rates λ_u, which represents the dependency on ATP concentration.For small load forces, the interaction with an actin polymer provides an additional friction force countering the load force,which results in small response of the relative velocity ⟨ v ⟩. For larger forces the system looses its resilience and the contribution from the load will dominate.The larger the load, the more the velocity will approach the velocity of a non-interacting motor, ⟨ v ⟩≍ - 1/γ_M F_load.We prove this dependency in appendix <ref>.Curves for higher ATP binding rates are closer to the curve for the completely unbound motor. As the motor is more loosely bound to the polymer, it is more submissive to the external load. Such a force-velocity relation was already presented in <cit.> for a flashing ratchet, representing a motor with a single head.A similar plot is shown in Fig. <ref>, where instead of varying ATP binding rate λ_b we look at motors with different numbers of heads N. Motors with only one head are very susceptible to external loads and the force from the ratchet offers little resistance.On the other hand, the motors with four and sixteen heads have almost identical force–velocity relations.However, it would be naive to consider these motors identical as they behave very differently regarding their energetics, which we will discuss in section <ref>.A comparison of the force–velocity relation for alternative setups with the “load on the motor” setup is provided in appendix <ref>. §.§ Stall forceFrom Fig. <ref> it is apparent that when applying a force against the preferred direction of the motor,the motor first slows down until it stops before it reverses its direction.The force magnitude that is needed to stop the motor's movement relative to the actin filament is called the stall force F_stall,⟨ v ( F_load≡ F_stall ) ⟩ = 0 .Fig. <ref> shows a zoomed in version of Fig. <ref> for small loads, opposite to the motor's preferred direction of motion. The intersections with the horizontal axis correspond to stall forces, while intersections of the curves with the vertical axis correspond to the mean velocities for system without any external load, compare Fig. <ref>.We see that for small loads the mean velocity decreases linearly with increasing load.Moreover, the slope of the force – mean velocity relation increases with the binding rate λ_b, that is responsible for the monotonous decrease of the stall force F_stall with respect to the ATP binding rate λ_b, see Fig. <ref>.When the ATP binding rate increases, so does the fraction of time in which the motor is weakly interacting with the actin filament.Hence, on average it takes a smaller load to halt a motor since the amplitude of its confining potential is typically lower. Ultimately, when the system is found only in the ATP unbound state the relative mean velocity is zero, ⟨ v ⟩ = 0, as the system reached an equilibriumand thus the stall force reaches zero as well, F_stall→ 0. § TENSION While the force–velocity relation captures the non-linear nature of the motor's transport,it alone does not truly capture its behavior inside polymer networks. There, the myosin II minifilament typically crosslinks two polymers being not necessarily displaced on average. This is demonstrated in setups “tug of war” and “elastic environment”.However, as we show in this section, the motor exerts an active tension in the system. This tension is further propagated through actin polymers over the full network up to the cell membrane and largely contributes to the overall stiffness of the cell <cit.>. §.§ Ratchet force In order to define a tension generated by the motor, we first need to evaluate all forces applied on the motor in the steady state.In the equation of motion for the “load on the motor” setup (<ref>) we identify the force generated by the ratchet potential asF_r = - ∇_M V_t( x_M - x_A ) = ∇_A V_t(x_M - x_A ) .Without the ratchet interaction the motion of the myosin, relative to the actin filament, would satisfy γ_M⟨ v ⟩ = -F_load.The discrepancy in the relation for the free motor, up to a pre-factor, equals the mean force exerted via the flashing ratchet ⟨ F_r⟩,see also equation (<ref>) in appendix <ref>. By taking the mean value of equation (<ref>),we can extract the mean ratchet force in terms of external load and mean relative velocity as ⟨ F_r⟩ = γ_Aγ_M/γ_A + γ_M(⟨ v ⟩ + F_load/γ_M) .Note that this is the same, up to a scaling factor, as the distance between the force-velocity curves and the ratchet-free curve in Fig. <ref>.Also note that, using the re-scaled parameters (<ref>) introduced in appendix <ref>,we can write the mean ratchet force as ⟨ F_r⟩ = γ^* ⟨ v ⟩ - F^*. We will discuss the ratchet force in other setups later on in the context of the tension for those setups. The typical behavior of the mean ratchet force in the “load on the polymer” setup is discussed in appendix <ref>.§.§ Tension in the motor Since both the load and ratchet force are directly applied on the respective ends of the motor in the “load on the motor” setup,the tension applied to the motor τ_M is proportional to their difference. Here we consider the contribution from the thermal force and from the drag negligible in the steady state as they are typically applied at the center of the mass of the motor.We choose the sign of the tension in such a waythat a positive value corresponds to a pair of forces applied on each end of the motor in the direction away from the motor, i.e.,⟨τ_M⟩ = ⟨ F_r⟩ + F_load= γ_Aγ_M/γ_A+γ_M⟨ v⟩+ (1 + γ_A/γ_A+γ_M) F_load,where we will consider the contribution from the ratchet potential as an active component of the tension, i.e. τ_M^(A)≡ F_r, in this particular setup.Note, that we omit the normalization by the length of the motor as it is just a division by a constant due to the infinite rigidity of the motor in our model.In a similar fashion we can define the tension on the polymer as⟨τ_A⟩ = - ⟨ F_r⟩= - γ_Aγ_M/γ_A+γ_M⟨ v⟩- γ_A/γ_A+γ_M F_load. We plot the tension on the motor as a function of the load F_load in Fig. <ref> for various ATP binding rates and in Fig. <ref> as a function of the amount of heads on a motor N. We observe that for small loads the tension is linear in load, in good agreement with the observation of the plateau in the force velocity relation; see Fig. <ref>. In that regime, the velocity is small compared to the initial load as most of the load is countered by the ratchet interaction.Hence only the second term in (<ref>) contributes to the tension, ⟨τ_M⟩≈(1 + γ_A/γ_A+γ_M) F_load. For large forces, the motor slips over the potential and thus the mean ratchet force ⟨ F_r⟩ approaches zero, as can be seen in Figs. <ref> and <ref>. Hence, the large force asymptotic is given solely by the load⟨τ_M⟩≍ F_loadfor F_load→±∞.That is further supported by calculations of the mean velocity for a motor with a single head, provided in appendix <ref>, cf. equation (<ref>). §.§.§ Load on the polymerA similar analysis can be made for alternative setups presented in Section <ref>.In the case where the load is applied directly to the polymer instead of the motor, cf. the setup “load on the polymer,”the motor only experiences a force from the ratchet interaction with the polymer and not from a load.Therefore, the tension on the motor is given only by the mean ratchet force, cf. equation (<ref>), where the load must be divided by γ_A since the load is exerted on the actin filament instead of the motor, i.e.⟨τ_M⟩= ⟨ F_r⟩= γ_Aγ_M/γ_A + γ_M(⟨ v ⟩ + F_load/γ_A) .Following the same argumentation as in the “load on the motor” setup, for small loads, the tension is given by⟨τ_M⟩≈γ_M/γ_A+γ_M F_load.This is similar to (<ref>), where the motor experiences the load directly, up to an exchange of friction coefficients of the motor and actin filament. Furthermore, since the load is not directly applied to the motor in this setup, the overall term F_load is not present.Therefore, the tension vanishes for large load in the same way that the relative velocity approaches -F_load / γ_A; see appendix <ref>,⟨τ_M⟩∝ F_load^-1for F_load→∞.§.§.§ Tug of warIn the case of Section <ref>, the motor is aligned anti-parallel between two actin polymers. We again start from the mean velocities for all components of the system in this setup (<ref>),⟨ v_A_1⟩ = - 1/γ_A( ⟨ F_r_1⟩ + F_load) ,⟨ v_M⟩ = 1/γ_M( ⟨ F_r_2⟩ + ⟨ F_r_1⟩) ,⟨ v_A_2⟩ = 1/γ_A( F_load - ⟨ F_r_2⟩) ,where A_1 and A_2 denote the two different actin filamentsand F_r_1 and F_r_2 are the ratchet forces (<ref>) between the motor and respectively filament A_1 and A_2,F_r_1 = - ∇_M V( x_A_1 - x_M ) ,F_r_2 = - ∇_M V( x_M - x_A_2 ) .Note, that due to the symmetry of the setup in the steady state the mean velocity of the myosin is inherently zero,⟨ v_M⟩= - 1/γ_M[⟨∇_M V_t( x_A_1(t) - x_M(t) ) ⟩ + ⟨∇_M V_t( x_M(t) - x_A_2(t) ) ⟩]= - γ_A/γ_M[⟨ v_A_1⟩+ ⟨ v_A_2⟩] ≡ 0 ,which is demonstrated in Fig. <ref>. Although the load is not applied to the motor directly,the motor is subjected to a tension due to interaction with both actin filaments.This tension is again given by the difference in the forces applied to the respective ends of the motor,⟨τ_M⟩ = ⟨ F_r_2⟩ - ⟨ F_r_1⟩= 2 F_load - γ_A(⟨ v_A_2⟩ - ⟨ v_A_1⟩) .The tension in the motor for the different setups is shown in Fig. <ref>, compared to the tension in the setups with only one actin filament.For small loads, ratchet forces balance the load applied on both filaments.The tension is thus approximately proportional to twice the load ⟨τ_M⟩≈ 2 F_load . This behavior is similar to the tension in the previously discussed setups“load on the motor” and “load on the polymer” setup; cf. equations (<ref>) and (<ref>),where the proportionality constants are 1 + γ_A / (γ_A + γ_M ) and γ_M / (γ_A + γ_M ) respectively.Note that the proportionality constant for the “load on the motor” setup, which lies in the closed interval [1,2],is always bigger than or equal to the one in the “load on the polymer”, which is in the interval [0,1], and smaller than or equal to the one in the “tug of war” setup.In the “load on the polymer” setup, the tension in the motor does not build up as fast with increasing load because the motor experiences a ratchet force only from one actin filament. The initial tension there is less than half of the tension in the setup with two loaded filaments. This suggests that the ratchet force, per motor head, is better transmitted to the motor in the “tug of war” setup. That is confirmed by Fig. <ref> and represents our first result.As described earlier, the motor starts to slip when the load is increased. Consequently, the mean ratchet force decreases, cf. Fig. <ref> and <ref>, and the increase in tension diminishes,as shown in Fig. <ref>. This behavior underlines the main difference between the “tug of war” and “load on the polymer” setups and the “load on the motor” setup,where the load is applied directly to the motor.Furthermore, as the ratchet force vanishes for large load, a residual tension from the load remains. However, in the “tug of war” and “load on the polymer” setup, this residual tension is not transmitted further to the motor due to the vanishing ratchet interaction.Finally, the variance of the motor velocity, cf. Fig. <ref>, seems to behave similarly to the tension on the motor in Fig. <ref>.While the mean velocity stays close to zero, the variance increases as the tension builds up in the motor.As the motor begins to slip over the filaments, the variance appears to decrease again. The asymmetry with respect to reversal of the load comes again from the inherent asymmetry of the ratchet potential. §.§ Mechanosensing The last setup, introduced in Section <ref>, is the case where the motor is situated between two anti-parallel actin filaments, which are attached to a wall via springs, see sketch in Fig. <ref>.Similarly as in the previous setup, the tension in the motor is given by the difference of the ratchet forces applied on it. However, in this setup, there is no external load on actin polymers and thus in the steady state we have a simple balance between the ratchet interaction and springs.That means that the tension on the motor is given byτ_M= ⟨ F_r_2⟩ - ⟨ F_r_1⟩= k_sp( ⟨Δ x_2 ⟩ - ⟨Δ x_1 ⟩) ,where k_sp is the spring stiffness and Δ x_1 = x_1(t) - x_1(0) (Δ x_2 = x_2(t) - x_2(0)) is the extension of the spring attached to first (second) actin.The tension is plotted in Fig. <ref> for different stiffnesses k_sp that represent the varying elasticity of the surrounding network,over a biologically relevant range <cit.>.The tension that the motor generates depends strongly on this elasticity k_sp and saturates for large k_sp.This suggests that the motor is not only able to respond to external forces, but also to sense mechanical properties of its environment.Similar results were obtained for different motor models <cit.>.However, the models used in those studies exhibit rates that depend on the strain or tension in the myosin motor.In our study, we show that this behavior also occurs in a simple Brownian ratchet model with constant chemical rates. It is important to include the fluctuations in this stochastic model to preserve the mechanosensing feature. In mean field models, where the phenomenological force-velocity relation would be implemented deterministically,the motor would essentially stall in this given setup and cease to exert a net force on the filaments.Therefore mechanosensing will not occur spontaneously in those models. This concludes the main result of the first part of our study. It has been shown that mechanosensing takes place on the cellular scale,where the cytoskeleton is coupled to the environment through focal adhesions <cit.>.§ ENERGY BALANCE Up to this point we have discussed mechanical properties of myosin II inside various setups mimicking the acto-myosin cortex.Most of these mechanical properties emerge from the motor activity,which is driven by a chemical energy turnover into active tension and active movement of polymers and the motor itself inside the network.This leads to a natural question: “How efficient is the motor?”. Presently we formulate a relation for the excess heat produced by the ATP cycle, that demonstrates its dependency on tension in the system. This relation will be important in our interpretation of the efficiency in section <ref>.We start our discussion from the energy balance on the level of individual trajectories <cit.>, which can be symbolically summarized as E = 𝒬_M^hb + 𝒬_A^hb + 𝒲^ext_M + 𝒬^chem,where the energy of the full system consists of the interaction with the ratchet potential (<ref>) and the energy stored in the ATP molecules that are bound to the motor heads,E( x_M, x_A, {ζ_i }) = V_t( x_M - x_A ) + Δ E ∑_i=0^N-1χ( ζ_i = c ) ,where Δ E represents the actual internal energy stored by in a single ATP and χ is a characteristic function [ χ(condition) = 1 if condition is fulfilled otherwise 0. ].The energy of the system is subject to changes driven by the interaction with the environment. In the case of overdamped diffusion, the environment represents a heath bath and thus every interaction is associated with a heat exchange <cit.>𝒬_M^hb = ∫ x_M∘[ - γ_M x_M/ t + √( 2 k_B T γ_M)W_M/ t ] , 𝒬_A^hb = ∫ x_A∘[ - γ_A x_A/ t + √( 2 k_B T γ_A)W_A/ t ] , where the first term is the dissipation of energy due to the friction and the second term is the energy absorbed from the thermal activity of the environment. Note that those integrals are evaluated in the Stratonovich sense. The energy is also changed by the work of the external load 𝒲^ext_M = - ∫ x_M∘ F_load = - Δ x_M F_load,where we use the fact that the external load F_load is constant. The last term in the energy balance (<ref>) is the contribution from the chemical cycle.The chemical cycle of a motor head, in the most basic form <cit.>, is a sequence of binding an ATP molecule, its hydrolysis, release of phosphor and finally release of ADP. The first step provides the chemical energy to the system, stored in ATP, while the next steps are associated with dissipation of the remaining energy to the surroundings. Thus the total chemical energy consists of two parts 𝒬^chem = 𝒬^chem_in + 𝒬^chem_out,𝒬_in^chem = ∑_t_U≤ t_fin[ Δ E + (c-1) V_r(x_t_U) ] , 𝒬_out^chem = -∑_t_B≤ t_fin[ Δ E + (c-1) V_r(x_t_B) ]where we sum over the jump times t_B(t_U) of all ATP binding (ADP unbinding) events. t_fin denotes the duration of the process. Fig. <ref> depicts a time evolution of the internal energy (<ref>) for a single trajectory of the system without any external load, F_load = 0.Note, that while we can identify the chemical input and output by the direction of the jump in the total energy,without any additional information it is impossible to distinguish which part of the heat dissipated to the heat bath corresponds to myosin or to actin. For non-zero load, the heat dissipated to the heat bath will contain the work of the external forces as well, which makes the separation in individual components even more difficult.Also note that while expressions (<ref>), (<ref>), (<ref>) and (<ref>) are independent of the setup,the total energy balance (<ref>) and the internal energy itself (<ref>) are not.§.§ Load on the motorIn the “load on the motor” setup, we can further simplify the expressions (<ref>) and (<ref>)by using the equation of motion (<ref>),which will allow us to interpret the excess chemical heat in terms of the ratchet force (<ref>) and thus link it to the tension. We start with the heat exchange of the molecular motor with the surrounding, i.e. (<ref>) expressed as 𝒬_M^hb= ∫ x_M∘[ ∇_M V_t(x_M - x_A ) + F_load]= ∫ x_M∘∇_M V_t(x_M - x_A ) - 𝒲_M^ext ,where we recognize two terms.The first one is the work done by the ratchet potential, while the second one is the work done by an external load (<ref>).Similarly for the heat dissipated by the polymer's interaction with the environment we get 𝒬_A^hb= ∫ x_A∘∇_A V_t(x_M - x_A ) = - ∫ x_A∘∇_M V_t(x_M - x_A ) .Consequently, the total energy balance equation (<ref>) simplifies to E=x_M∘∇_M V_t +x_A∘∇_A V_t + 𝒬_in^chem + 𝒬_out^chem ,which does not explicitly depend on the load F_load anymore.Note that the total differential here is taken in the Stratonovich sense. Finally, in the steady state,we can see that the excess of the chemical cycle in the steady state is dissipated as the work of ratchet potential (<ref>) over the separation of motor and filament ⟨𝒬_in^chem + 𝒬_out^chem⟩=⟨∫( x_M - x_A) ∘[ - ∇_M V_t( x_M - x_A ) ] ⟩ , which can be further interpreted in terms of tension on the polymer τ_A, cf. (<ref>), and the relative velocity v; see (<ref>), ⟨𝒬_in^chem + 𝒬_out^chem⟩= ⟨∫ t vτ_A⟩ .§.§ Heat fluxes The internal energy of the system (<ref>) is bounded. Hence, if we interpret the energy balance equation (<ref>) in terms of mean dissipation rates defined as q = lim_Δ t →∞1/Δ t⟨𝒬(Δ t) ⟩ ,we obtain an equation representing the conservation of fluxes 0 = q_M^hb + q_A^hb + w^ext_M + q^chemfor arbitrary initial condition.Such equality is trivially true for arbitrarily small times when the initial condition corresponds to the steady state.§.§ Alternative setups In section <ref> we introduced three alternative models,more suitable for capturing the behavior of myosin motors inside the polymer network known as the actomyosin cortex.§.§.§ Load on the polymerAs the “load on the polymer” setup is similar to the “load on the motor” setup, cf. section <ref>,the energy balance (<ref>) is preservedwith the for heat dissipation (<ref>) and (<ref>) kept intactand external work on the motor 𝒲_M^ext being replaced by the external work on the polymer𝒲_A^ext = ∫ x_A∘ F_load = Δ x_AF_load .Note, that in this particular setup the load is applied in the opposite direction, as can be seen from the equations of motion (<ref>). Similarly expressions (<ref>) and (<ref>) are modified to𝒬_M^hb = ∫ x_M∘∇_M V_t(x_M - x_A ) ,𝒬_A^hb = - ∫ x_A∘∇_M V_t(x_M - x_A ) - 𝒲_A^ext . Consequently the excess of the chemical cycle (<ref>) changes to, cf. (<ref>), ⟨𝒬_in^chem + 𝒬_out^chem⟩=⟨∫( x_M - x_A) ∘[ - ∇_M V_t( x_M - x_A ) ] ⟩ , which can be again interpreted in terms of the relative velocity v, cf. (<ref>),and the tension on the motor τ_M, cf. (<ref>),⟨𝒬_in^chem + 𝒬_out^chem⟩= ⟨∫ t vτ_M⟩ .An argumentation similar to the one used in the previous setup can be also made here. §.§.§ Tug of warIn the “tug of war” setup, see section <ref> and Fig. <ref>, the motor is placed between two polymers on which the load is applied. Therefore the energy balance (<ref>) is modified toE = 𝒬_M^hb + 𝒬_A_1^hb + 𝒬_A_2^hb+ 𝒲^ext_A_1 + 𝒲^ext_A_2 + 𝒬^chem,where the energy now contains contributions from both polymersE( x_M, x_A_1, x_A_2, {ζ_i })= V_t( x_M - x_A_2 )+ V_t( x_A_1 - x_M )+ Δ E ∑_i=0^N-1χ( ζ_i = c ) .Note, that the last sum is over all heads of the motor, N = N_A_1 + N_A_2. Contributions from the heat exchange with the environment (<ref>) remain the same, even though in this case we have two contributions from both filaments𝒬_A_1^hb = ∫ x_A_1∘[ - γ_A x_A_1/ t + √( 2 k_B T γ_A)W_A_1/ t ] ,𝒬_A_2^hb = ∫ x_A_2∘[ - γ_A x_A_2/ t + √( 2 k_B T γ_A)W_A_2/ t ] ,and now the external load is not applied on the motor but on each actin polymer. Hence, equation (<ref>) is replaced by𝒲^ext_A_1 = - ∫ x_A_1∘ F_load = - Δ x_A_1 F_load, 𝒲^ext_A_2 = ∫ x_A_2∘ F_load = Δ x_A_2 F_load. Note, that the interpretation of the energy balance (<ref>) in terms of the heat fluxes (<ref>) while following the same arguments as for (<ref>) is possible and leads to 0 = q_M^hb + q_A_1^hb + q_A_2^hb + w^ext_A_1 + w^ext_A_2 + q^chem . Similarly to the “load on the motor” setup, we can express the excess from the chemical cycle in the steady state as sum of work by the ratchet forces with respect to their respective displacements,⟨𝒬_in^chem + 𝒬_out^chem⟩=⟨∫( x_M - x_A_1) ∘[ - ∇_M V_t( x_A_1 - x_M ) ] ⟩+ ⟨∫( x_M - x_A_2) ∘[ - ∇_M V_t( x_M - x_A_2 ) ] ⟩ .If we naively assume that relative velocities and ratchet force are well localized random Gaussian variables,we can further approximate the excess chemical heat as ⟨𝒬_in^chem + 𝒬_out^chem⟩≈ T [⟨ v_1 ⟩⟨ F_r_1⟩ + ⟨ v_2 ⟩⟨ F_r_2⟩] = T/2⟨ v ⟩⟨τ_M⟩ ,where the last equality follows from the anti-symmetry of the mean relative velocities of the motor with respect to individual polymers in the steady state. As the previous assumption is far from reality, such straightforward interpretation of the excess heat is invalid.However, if we assume that the displacements in the respective integrals are typically anti-symmetric, due to the anti-symmetry of the setup,we can interpret the excess chemical heat as a covariance of the tension on the motor (<ref>) with the relative velocity of the motor with respect to a single polymer (<ref>) integrated over time,as in equation (<ref>).Note, that in this particular setup the interpretation in terms of the tension is less straightforward in comparison with the “load on the motor” or the “load on the polymer” setup,because the individual displacements x_M - x_A_1 and x_M - x_A_2 are subjected to thermal fluctuations on short time scale.Only on the long time scale they are anti-symmetric and the tension relaxes to a steady value. §.§.§ Elastic environment For the model described in paragraph <ref> and depicted in Fig. <ref>,the energy balance (<ref>), resp. (<ref>) is further modified,E = 𝒬_M^hb + 𝒬_A_1^hb + 𝒬_A_2^hb + 𝒬^chem,as there is no external load but all the accumulated energy is now stored in harmonic potentials E( x_M, x_A_1, x_A_2, {ζ_i })= V_t( x_M - x_A_2 )+ V_t( x_A_1 - x_M )+ 1/2 k_sp[ x_A_1 - x_A_1(0) ]^2+ 1/2 k_sp[ x_A_2 - x_A_2(0) ]^2+ Δ E ∑_i=0^N-1χ( ζ_i = c ) .Contrary to the previous setup, the total energy of the system is no longer bound, hence the heat fluxes (<ref>) are only balanced in the steady state.Finally we look again at the excess output from the chemical cycle⟨𝒬_in^chem + 𝒬_out^chem⟩=⟨∫( x_M - x_A_1) ∘[ - ∇_M V_t( x_A_1 - x_M ) ] ⟩+ ⟨∫( x_M - x_A_2) ∘[ - ∇_M V_t( x_M - x_A_2 ) ]⟩+ k_sp⟨∫ x_A_1∘( x_A_1 - x_A_1(0) ) ⟩ + k_sp⟨∫ x_A_2∘( x_A_2 - x_A_2(0) ) ⟩ , which further simplifies to ⟨𝒬_in^chem + 𝒬_out^chem⟩=⟨∫( x_M - x_A_1) ∘[ - ∇_M V_t( x_A_1 - x_M ) ] ⟩+ ⟨∫( x_M - x_A_2) ∘[ - ∇_M V_t( x_M - x_A_2 ) ] ⟩+ ⟨Δ E^sp_1⟩ + ⟨Δ E^sp_2⟩ , where E^sp denotes the energy stored in the spring;, i.e.,⟨Δ E^sp_α⟩=1/2 k_sp⟨(x_A_α(t)-x_A_α(0) )^2⟩ .The interpretation of this excess chemical energy in terms of the tension is even more difficult than in the “tug of war” setup (<ref>) due to multiple factors. First, there are additional terms related to the change of the energy of the springs. Secondly, the mean velocity of all elements in the steady state is zero due to the anti-symmetry of the setup and no additional external force breaking this symmetry.§ EFFICIENCYIn previous years, a lot of effort was directed towards the assessment of the efficiency of molecular motors <cit.>.The main focus of these works was towards the transport properties of the aforementioned motors.However the main role of myosin II is to generate tension in a cytoskeletal network <cit.>.Efficiency, in general, relates a “useful” output to the input. As what is considered “useful” heavily depends on the considered setup, there is a plethora of possible choices. Here we discuss a common choice as well as one particular choice which is to the best of our knowledge not discussed previously in the literature.First, we briefly discuss the most common definition of efficiency, well discussed in the literature <cit.>, which is related to the transport properties of molecular motors.Then we provide an alternative definition of the chemical efficiency, which can be related to the tension generated by myosin II in networks.§.§ Transport efficiency As we have stated, the first efficiency we discuss is related to transport properties.In this respect, we identify the heat dissipated to the environment by the motor alone as the output work 𝒲^out_M = - 𝒬^hb_M and chemical work 𝒬^chem and the work done by the load 𝒲^ext_M. Hence the transport efficiency can be defined as η_tr = ⟨𝒲^out_M⟩/⟨𝒬^chem⟩ + ⟨𝒲^ext_M⟩ .Note that for a large load (F_load→∞) such efficiency approaches η_tr→ 1,while in absence of a load F_load = 0 it captures how much of the chemical energy is dissipated by the motor's motionη_tr( F_load = 0 ) = ⟨∫ x_M∘[ - ∇_M V_t ] ⟩/⟨𝒬^chem⟩ ,i.e., how much of the chemical energy provided to the system is dissipated by the motor and not by the polymer. Note that this particular definition of the efficiency has several shortcomings. First, the output work has two components, one from the chemical source and the second from the external driving,thus in principle the efficiency can be greater than one.Secondly,only for long time scales, in situations where the motor displacement and ratchet force are to a great extent uncorrelated,one can relate the output work to the heat dissipated by the drag force to the medium. This means that in systems where the motor is frustrated and does not move on average,like in the “tug of war” and “elastic environment” setups, one cannot distinguish between various sources of the output work.Finally, the output work depends on the details of the ratchet interaction, which renders it practically immeasurable.§.§ Chemical efficiencyAn alternative approach is to identify the “useful” output by looking at the chemical energy, supplied by ATP, that is transformed to other forms of energy,i.e. either it is dissipated by myosin and actin through their diffusion or it is stored in the interaction potential and thus increases the tension in the system.This consideration leads to a novel chemical efficiency defined as η_chem = ⟨𝒬^chem_in⟩+⟨𝒬^chem_out⟩/⟨𝒬^chem_in⟩ .Note that in the steady state all excess chemical energy is dissipated to the environment, see section <ref>,and thus can in principle be measured by calorimetric techniques <cit.>.§.§.§ Load on the motorWe start our discussion again with the “load on the motor” setup with a four headed motor, see Fig. <ref>.We observe that the chemical efficiency has two local maxima.We show later that these two maxima correspond to a frustrated system,in which the load is close to the point where it is able to overcome any force induced by the ratchet potential.Another observation, which we discuss later in more detail, is that the efficiency reaches negative values for large loads,which we interpret as work done by the load F_load that is converted back to chemical energy. In order to properly understand these features,we try to evaluate the chemical efficiency (<ref>) in a steady state.We start by evaluating the mean heat (<ref>) and (<ref>) produced over a time interval of length Δ tby transitions of a single head i displaced with respect to the end of the motor by i d,[b][.42] ⟨𝒬^chem_in(i) ⟩= Δ tλ_u ×∑_{ζ_j} : ζ_i = 1 ρ({ζ_j } )[ Δ E - (1-c) ⟨ V_r(x - i d ) | {ζ_j }⟩] [b][.42] ⟨𝒬^chem_out(i) ⟩= - Δ tλ_b ×∑_{ζ_j} : ζ_i = c ρ({ζ_j } )[ Δ E - (1-c) ⟨ V_r(x - i d ) | {ζ_j }⟩]where ⟨ V | X ⟩ is the steady state conditional mean value with respect to the fixed internal state Xand ρ(X) ∈ [0,1] denotes the probability of the occupation of the corresponding internal state X in the steady state,which does not depend on the position due to the independence of the internal state dynamics (<ref>) on position.Moreover, due to the independence of individual heads the probability occupation of a given internal state can be written as ρ( {ζ_i } ) = ∏_i=0^N-1ρ_1( ζ_i ),where ρ_1(ζ) = λ_b/λ_b + λ_u ζ = 1 ,λ_u/λ_b + λ_u ζ = c . Another consequence of the independence of the individual headsis that we can express the total mean heat, produced in the chemical cycle, as a sum of contributions from individual heads⟨𝒬_in^chem⟩= ∑_i=0^N-1⟨𝒬_in^chem(i) ⟩= Δ t λ_bλ_u/λ_b + λ_u×[ NΔ E- (1-c) ∑_i=0^N-1⟨ V_r(x - i d ) | ζ_i = 1 ⟩] ,⟨𝒬^chem_out⟩= ∑_i=0^N-1⟨𝒬_out^chem(i) ⟩= - Δ t λ_bλ_u/λ_b + λ_u×[ NΔ E- (1-c) ∑_i=0^N-1⟨ V_r(x - i d ) | ζ_i = c ⟩] ,which leads to a more explicit expression for the efficiency (<ref>) η_chem = 1-c/N ×∑_i=0^N-1⟨ V_r(x-id) | ζ_i = c ⟩ - ⟨ V_r( x -id ) | ζ_i = 1 ⟩/Δ E - 1-c/N∑_j=0^N-1⟨ V_r(x-jd) | ζ_j = 1 ⟩.We can further introduce an average probability density over all heads ρ̅(x,ζ) = 1/N∑_i=0^N-1∑_{ζ_j } : ζ_i = ζρ( x + i d, {ζ_j } ) ,and consequently the conditional average probability densityρ̅(x|ζ) = ρ̅(x,ζ) [ ∫_0^ℓ yρ̅(y,ζ) ]^-1which allow us to further simplify the expression for the chemical efficiency to η_chem = (1-c) ⟨ V_r(x) | ζ = c ⟩_ρ̅ - ⟨ V_r(x) | ζ = 1 ⟩_ρ̅/Δ E - (1-c) ⟨ V_r(x) | ζ = 1 ⟩_ρ̅.We notice that the numerator is given by the average difference between mean potentials in the ATP attached and detached states over all heads.As the potential is known in our model, the only missing information is thus the steady state distributionor to be more precise the average steady state density for all heads (<ref>). These distributions are empirically obtained by collecting the positions of the heads over all times in which the head was in the ATP bound/unbound state. They are shown in Fig. <ref> for different load forces.The difference between the average distribution (<ref>) in the ATP bound state and ATP unbound state is directly translated to the mean heat flux (<ref>) out of the chemical systemq_in^chem + q_out^chem= (1-c) λ_bλ_u/λ_b + λ_u ×[ ⟨ V_r(x) | ζ = c ⟩_ρ̅ - ⟨ V_r(x) | ζ = 1 ⟩_ρ̅] , which is depicted in Fig. <ref>, where we show the spatial dependency of the heat fluxes. We can see that a major flux is close to the ratchet potential's minimum (<ref>) where both the probability density is localized and the gap is the largest.On the other hand, close to the ratchet potential's maximum the heat flux is close to zero. However, if we compare it to the numerator of the chemical efficiency (<ref>); see Fig. <ref>, we can see that a major contribution to the chemical efficiency actually comes from the region around the ratchet potential's maximum. This discrepancy can be interpreted in the following sense:While most transitions occur close to the ratchet potential's minimum due to the large probability density, thus providing a large flux,those particles do not diffuse far before the next transition,hence the contribution to the chemical efficiency, associated with the discrepancy between chemical input and output, is relatively small.On the other hand, a particle that switched its internal state close to the potential maximum relaxes relatively fast, compared to the transition rates,to the vicinity of the ratchet potential's minimum and thus creates a larger discrepancy leading to high contribution to the efficiency.However these transitions are rare due to the low probability density and hence the total flux density is relatively small.In absence of an external load, F_load=0,the maximum of the distribution, see Fig. <ref>, is close to the minimum of the potential, where the discrepancy in the position comes from the averaging over multiple heads with a fixed distance between them,which does not match with the period of the ratchet potential.Also the fact that the distributions seem to exhibit multiple peaks, can be attributed to the motor having four heads.Both in the bound and unbound state, the head can be found predominantly on the “soft” slope of the ratchet potential.When the load is increased to F_load = - 25pN, the average steady state distribution shifts to the right in both the ATP bound and unbound state.While the distribution for the ATP unbound state remains peaked around the potential minimum, cf. Fig. <ref>, the distribution in the ATP bound state flattens out and increases around the potential maximum, which is a trademark of a frustrated system. This is due to the potential being less steep, meaning that heads in that particular state are more susceptible to external forces, which in this particular case leads to an increase of η due to the increase in discrepancy between the average distributions,see Figs <ref> and <ref>. For a load in the opposite direction F_load= 25pN the effect on average densities is similar but less pronounced,see Fig. <ref>,since the slope of the ratchet potential on the side is less steep and the distributions get more spread out, cf. Fig. <ref>. For larger loads, distributions shift and flatten even more, Fig. <ref>.In the extreme situation, it leads to completely flat distribution and thus zero efficiency, Fig. <ref>. For high loads we also observe a population inversion; see Fig. <ref>,where the potential maximum is more occupied than the potential minimum.This is due to the effective friction caused by the potential.As the motor slows down when dragged along the potential gradient, it spends more time on the up-slope than on the down-slope of the potential. Such behavior is associated with a negative efficiency.In Appendix <ref> we have proven this hypothesis on a system with a single head and we have shown that the efficiency decays with the force η≍ - 1/ F_load^2 for F_load→±∞ . §.§.§ Dependency on number of headsIf we interpret the dependency of the chemical efficiency (<ref>) on the load as a probabilistic distribution with finite variance and assume that the main contribution to the shape, Fig. <ref>, is from the numerator,by the central limit theorem the efficiency-load dependency approached a Gaussian distribution with increasing number of heads.This is confirmed by Fig. <ref>, where indeed the distribution is broadening with an increasing number of heads and has only a single maximum.It is also apparent that there is no qualitative difference in chemical efficiency between the system with four heads and one head. Moreover we can see that the empirical efficiency has a smaller variance with an increasing number of heads.From these, we can conclude that there is a significant qualitative difference between motors with low and high numbers of heads. While the motors with a high number of heads perform better without any external load, motors with a low number of heads perform the best when they are frustrated. Thus in-vitro experiments with large myosin II motors <cit.> do not necessarily capture the behavior of the motors inside a cytoskeleton. That is the first main result of the second part of this study. §.§.§ Load on the polymerAs was already discussed, the situation with the load on the polymer is almost identical to the “load on the motor” setup. The main difference is in the notion of what we perceive as motor or as polymer, cf. section <ref>. Hence, it is not surprising that the chemical efficiency (<ref>) as a function of the load F_loadresembles the situation with the load on the motor, see Fig. <ref>. The main difference between the curve for the “load on the motor” and “load on the polymer” setup is the width of the distribution.This can be understood from equations (<ref>) and (<ref>),where we see that the excess of the chemical energy entering the system is proportional to the relative velocity. In Fig. <ref> we observe a different asymptotic behavior of the relative velocity by a factor γ_M / γ_A, Hence we expect that the same factor applies in the steady state for the rate of the excess chemical heat production.That hypothesis is further confirmed in Fig. <ref>, where we plot the chemical efficiency η_chem with respect to re-scaled load. §.§.§ Tug of warFig. <ref> and <ref> also depict the chemical efficiency η dependency on the load F_load for the “tug of war” setup, cf. Section <ref>. The general shape is preserved. From Fig. <ref>, we see that the large load asymptotic behavior corresponds to the “load on the polymer” setup. That can be explained by the same asymptotic behavior for the relative velocity ⟨ v⟩, cf. Fig. <ref>,and for the tension on the motor τ_M, cf. Fig <ref>, while having equation (<ref>) in mind.For small loads, see Fig. <ref>, the chemical efficiency for the “tug of war” setup corresponds to the “load on polymer” setup only when the load is scaled by a factor 2.This points to the fact that, rather than the external load F_load, the mean tension on the motor ⟨τ_M⟩ generated by the ratchet interaction is important, which for small loads, when the polymer does not slip, is close to twice the external load.We discuss this connection later, see Fig <ref>.§.§.§ Elastic environmentIn the case of the“elastic environment” setup; see Fig. <ref>, we observe that the chemical efficiency (<ref>) is practically independent of the stiffness of the environment, k_sp. Unfortunately, here we cannot use the same reasoning as for the “tug of war” setup, as the mean velocity of the motor in the steady state is zero. §.§.§ Tension and chemical efficiencyDuring the investigation of the chemical efficiency dependency on the external load F_load for the “tug of war” setup,we discovered that rather than the load, the tension on the motor τ_M might be the intrinsic parameter.However, if we take the results of section <ref> into account, namely equations (<ref>), (<ref>) and (<ref>),the mean ratchet force (<ref>) might be a better candidate. Indeed, as seen in Fig. <ref>, the chemical efficiency has a universal dependency on the mean ratchet force across all setups.For a small mean ratchet force, caused either by a small external load F_load or a small stiffness of the surrounding network k_sp,the chemical efficiency has a relatively flat profile with weak positive response to an increase in tension. This can be explained by perturbations in the steady state average probability density for the motor's headsand was discussed in more details insection <ref>; see equations (<ref>) and (<ref>), and Figs. <ref> and <ref>.With increasing load, both positive and negative, the mean ratchet force on the motor in the “load on the polymer”, “load on the motor” and “tug of war” setups increase up to a certain threshold. There, the polymers start to slip with respect to the motor and a further increase in the driving leads only to a decrease in the mean ratchet force and thus to a decrease in the tension (<ref>) and (<ref>) as well. This creates a second branch of the dependency for the slipping motor.For the “tug of war” setup the threshold value is lower than for the “load on the polymer” setup due to the interaction mediated by two ratchet potentials,as for the complete motor to slip it is sufficient if only one of the ratchet interactions on either side of the motor fails. Note that a similar discussion for the “elastic environment” setup is highly non-trivial, as shown in section <ref>. However, the dependency of the efficiency on the mean ratchet force follows a common pattern and agrees with the results of the other setups. As the mean ratchet force is not directly measurable in comparison with the tension on the motor, we plot the chemical efficiency as a function of the tension in the motor for all the setups mentioned above; see Fig. <ref>. Here we can see a deviation from a universal behavior,most notably for the “load on the motor” setup, where there is always a residual tension from the load alone (<ref>). Also note that in the “tug of war” setup the threshold value of the tension for motor to slip is larger than in other setups due to two contributions from the mean ratchet force to the tension, one at each respective side of the motor. These considerations complement the discussion that related the excess chemical energy with the build up tension in the motor,or more precisely with the additional tension created by the ratchet interaction, see discussion related to equations (<ref>), (<ref>) and (<ref>). In the end, we can conclude that the additional tension caused by the ratchet potential, or in other setups representing the tension in the motor itself, is the true intrinsic quantity that determines the chemical efficiency of the molecular motor. § CONCLUSIONSIn this paper we introduced a scaled-ratchet model, which is a variant of a Brownian ratchet. With this model, we were able to reproduce experimental values forvelocities and stall-forces of the myosin II molecular motor.We inspected the dependency of these quantities on the ATP binding rate λ_b, which can be linked to the ATP concentration itself,and on the scaling factor c, that models the screening of the interactions by ionic concentrations in the inter-cellular environment. We also revisited the force-velocity relation for small non-muscle myosin II motorsand investigated how it depends on λ_b and on the number of heads of the motor N. From that, we inferred that, from a mechanical point of view, there is hardly any difference between the myosin motor with as little as four heads and a motor with a large number of heads. Recently it was found that the main role of myosin II motors is not in active transport, but rather in the generation of tension inside large networks.In order to capture the behavior of the motor in such an environment, we studied mechanical quantities like the mean ratchet force and the tension in the motor for multiple setups mimicking to various degrees the motor inside such a network. We observed that for large loads, the motor will slip over the filament track.Consequently, the force from the ratchet potential, and the tension generated by it, will vanish. The decay of the ratchet force corresponds to the theoretical result, obtained by a perturbative expansion around infinitely large loads.In addition, the motor's ability of mechanosensing naturally emerges from the model without imposing a load-dependence on the chemical rates. The main result is that all dynamical aspects follow solely from the continuous, mechanical ratchet interaction between the motor and actin filament.The traditional way of how to evaluate the efficiency of molecular motors is with respect to the efficiency of the transport along the actin filament. However, such a quantity is unsuitable for motors inside large polymer networks where there is little to no mean displacement of the motor.Here we propose an alternative which we call chemical efficiency, that quantifies the fraction of energy, supplied by ATP hydrolysis,that is consumed in the diffusion processes of the myosin II motor and actin filament. This energy is partially used in the movement of the motor along the actin filament or in the build-up of tension in the motor, depending on the setup.We have shown that, contrary to the mechanical study, a significant qualitative difference in the behavior of the motor with a high or low number of heads is present.The main conclusion is that motors with a small amount of heads perform most efficiently when they are frustrated, a treat which is missing in motors with large number of heads.We further observed, that for large loads, the chemical efficiency becomes negative,which can be interpreted as the energy that is being drained from the external driving to the chemical cycle. That behavior was confirmed by perturbative calculations, that also predict that the efficiency decays to zero for even bigger loads.Finally, it seems that this chemical efficiency has a universal dependency on the mean ratchet force, independently of the considered setup, which is directly related to the tension the motor experiences. In summary, we have shown that the scaled ratchet model, introduced in this paper, is naturally capable of covering various biologically relevant treats like the non-linear force velocity relation,its mechanosensing ability and the fact that the motor's chemical efficiency reaches a maximal value under load.The authors are grateful to Bart Smeets and Maxim Cuvelier for proofreading of the manuscript. § FORCE – VELOCITY RELATION FOR ALTERNATIVE SETUPS In Fig. <ref> we compare the force-velocity relation of the “load on the motor” setup with alternative setups, namely with the “load on the polymer”, cf. Section <ref> and the “tug of war”, cf. Section <ref>.Note that for the “tug of war” setup we have two equivalent choices for the relative velocity, however due to the symmetry of the setup, cf. Fig. <ref>, in the steady state these options differ only by sign,hence we define the relative velocity (<ref>) as ⟨ v ⟩≡⟨ x_M⟩/ t- ⟨ x_A_2⟩/ t= - ( ⟨ x_M⟩/ t- ⟨ x_A_1⟩/ t ) .For small loads, the mean relative velocity ⟨ v ⟩ of the motor with respect to the polymer remains close to each other in all setups. However, for high loads a clear difference in their asymptotic behavior becomes apparent. When the load is applied to the motor the relative velocity goes to - F_load / γ_M. For the setups where the load is applied to the filaments, the relative velocity goes to - F_load / γ_A with increasing load.In the inset of Fig. <ref>,the force-velocity relations are plotted on rescaled x-axes for the different setups,which takes the difference in their asymptotic behavior into account.Thus, the “load on the motor” setup is plotted with respect to - F_load / γ_M instead of F_load, while the other setups are plotted with respect to - F_load / γ_A. The curve for the simplest setups, with one motor and one polymer, collapse in this plot,confirming the fact that they can be mapped onto each other by switching friction coefficients γ_M↔γ_Aand reversing the relative velocity.However, the “tug of war” setup manifests a clear difference for mid-range loads due to the more complex interaction with two polymers.§ VELOCITY WITH RESPECT TO MEDIUM We have investigated the relative velocity of the myosin and actin polymer (<ref>) in Section <ref>.Here we provide a study whether it is the motor that is moving with respect to the medium or rather the actin polymer that is being pushed back through the medium. In order to find the dependency between individual velocities and relative velocity in the “load on the motor” setup,we start by taking expectation values of their equations of motion, cf. equations (<ref>) and (<ref>),γ_M⟨ v_M⟩ = ⟨ x_M⟩/ t = ⟨ F_r⟩ - F_load , γ_A⟨ v_A⟩ = ⟨ x_A⟩/ t = -⟨ F_r⟩ ,where F_r denote ratchet force (<ref>).Next, we express the mean relative velocity (<ref>) in terms of the mean ratchet force ⟨ F_r⟩ and the external load F_load⟨ v ⟩= ⟨ v_M - v_A⟩= γ_A + γ_M/γ_Aγ_M⟨ F_r⟩ - 1/γ_M F_load ,which leads to individual velocities expressed in terms of the mean relative velocity and load ⟨ v_M⟩ = 1/γ_A + γ_M( γ_A⟨ v ⟩ - F_load) ,⟨ v_A⟩ = -1/γ_A + γ_M( γ_M⟨ v ⟩ + F_load) . Note, that for the load equal the stall force, F_load = F_stall, both velocities are equal and non-zero. This is related to the fact that the full system is moving with respect to the liquid, as can be seen from the mean velocity of the geometric center ⟨ v_center⟩ = ⟨ v_M + v_A/2⟩= = γ_A - γ_M/ 2 ( γ_A + γ_M ) ⟨ v ⟩ - 1/γ_A + γ_M F_load ,which is proportional to both the load and the relative velocity and is zero only when ⟨ v ⟩ = 2 F_load / ( γ_A - γ_M ).Also note, that the ratchet force is bounded by -N V /ℓ (1-a) ≤F_r≤ N V /ℓ a,cf. equations (<ref>) and (<ref>),which means that there is a maximal portion of the load that the ratchet interaction can transmit from the motor to the polymerand thus there exists an upper and lower bound for the polymer's velocity (<ref>).That is seen in Fig. <ref> depicting the individual velocities as a function of the external load F_load,where the polymer's velocity is clearly bounded. For small loads, the motor and actin filament are moving together, as the coupling by the ratchet interaction proves to be dominant.This hypothesis can be further supported by the linear dependency of equations (<ref>) and (<ref>) on the load,which is due to the fact that for small loads also the mean relative velocity (<ref>) is linear in the load; see Fig. <ref>.Moreover as the mean velocity's response to the external force is typically very small compared to the direct contribution from the external load to the individual velocities, we can further approximate them by⟨ v_M⟩ ≈ -F_load/γ_A + γ_M + γ_A/γ_A + γ_M⟨ v ⟩_ F_load = 0 ,⟨ v_A⟩ ≈ -F_load/γ_A + γ_M - γ_M/γ_A + γ_M⟨ v ⟩_ F_load = 0.Such a response is depicted in the insert of Fig. <ref>, where the shift in individual velocities caused by the relative velocity for very small loads is constant.Since the load is only applied to the motor in this setup, for very high loads, the motor will start slipping over the ratchet potential along the actin filament.Consequently, the actin filament's mean velocity will stop increasing and eventually approaches zero,while the myosin motor's mean velocity goes to ⟨ v_M⟩≍ F_load/γ_M as can be seen in Fig. <ref>.This is due to the fact that the asymptotic behavior of the relative velocityis⟨ v ⟩≍ -F_load/γ_M,which can be obtained directly from equation (<ref>) as the contribution from the ratchet force is bounded. The asymptotic behavior for the motor and the polymer follows from equations (<ref>) and (<ref>) for ⟨ v ⟩ = - F_load / γ_M.From there it is also apparent that the relative mean velocity in the limit of infinitely large load F_load→±∞ will be equal to the motor's velocity ⟨ v_M⟩. That agrees well with the data previously shown in Fig. <ref>.§ MEAN RATCHET FORCE In Fig. <ref> we show,that the mean ratchet force in the “load on the polymer” setup for small loads is linear and independent of the chemical rates and hence of the ATP concentration. In this regime the motor hardly moves along the filament and the load is almost completely countered by the mean ratchet force, i.e. it acts as a friction force. With increasing load, the force generated by the ratchet interaction will reach a maximal value and eventually decay for larger external load forces.In appendix <ref>, cf. equation (<ref>), we show that the friction provided by the ratchet will go to zero as the load goes to infinity as | ⟨ F_r⟩_F_load→∞| ≍1/F_load ,which is depicted in Fig. <ref>.Note that while there are natural bounds on the ratchet force due to the shape of the ratchet potential,see equation (<ref>) in appendix <ref>,here we have shown that the minimal and the maximal ratchet force are comparable to these theoretical bounds.Compare this to the maximal tension the motor is able to withstand before it start to slip in section <ref>.§ LOCAL MINIMUM OF THE CHEMICAL EFFICIENCY Another trait of the chemical efficiency curves are the local minima around F_load=0.This can be understood from equation (<ref>), which is determined by the ratchet potential averaged with the heads' distribution in the attached and detached state.For a one headed motor, these distribution will both be peaked around the minimum of the ratchet potential.In fact, these distributions will be given by the Boltzmann distributionρ_0(x,ζ) = 1/Z(ζ) e^-βζ V_r(x) in the lowest order of approximation, as the binding and unbinding rates operate on much longer time scale than the relaxation governed by the diffusion.From this point of view, ATP binding is effectively equivalent to an increasing of the temperature of the heat bath T→T/c.Hence the distributions for the ATP bound state are broader and the expectation value of V_r will be bigger than in the attached state and η will be positive. Now if we apply a small external force, those distributions will be slightly perturbed.Following the McLennan formula <cit.> we obtainρ^*(x) ∼ e^-β[V_r(x) + F_load( x - x_min) ]≈ρ_0(x) [1 - β F_load( x - x_min) ]close to the potential minimum x_min, i.e. V(x) ≥ V(x_min).This leads, for small F_load, to a shift of the distribution ρ in the opposite direction of the load as can be seen from ⟨ x ⟩_ρ^*= x_min + ⟨ x - x_min⟩_ρ^* ≈ x_min + ⟨ x - x_min⟩_ρ_0- β F_load⟨( x - x_min)^2 ⟩_ρ_0= ⟨ x ⟩_ρ_0 - β F_load⟨( x - x_min)^2 ⟩_ρ_0 .From the previous equation it follows that a more spread-out distribution will be affected more by the force. This means that in our case the ATP bound state will be more affected. A larger displacement of the distribution, away from the potential's minimum,means that the mean ratchet potential in the ATP bound state ⟨ V_r| ζ = c ⟩ will increase more than the mean ratchet potential in the ATP unbound state ⟨ V_r| ζ = 1 ⟩ and hence the chemical efficiency (<ref>) has to increase. This argument can be further supported by Jensen's inequality. The ratchet potential is convex close to its minimum.Moreover, we have seen in Fig. <ref> that the probability distribution for small loads is well localized around it's mean value. Hence we can concur that the Jensen's inequality will hold for sufficiently small loadsV_r(⟨ x | ζ⟩) ≤⟨ V_r(x) | ζ⟩ , where the lower bound on the expectation value of the ratchet potential (<ref>) is given by the value of the ratchet potential at the mean position of the heads in a given state.Consequently, any shift in the mean position of the heads due to the external force shifts the lower bound for the mean value of the ratchet potential itself. Furthermore, in the ATP bound state the mean position will be further away from the minimum of the ratchet potential (<ref>) so the lower bound is higher than in the ATP unbound state. Therefore, as the distribution is well localized around the minimum, the shift in the lower bound for the potential represents a shift in the mean value as well. For a four headed motor, the distributions of the heads at zero load are not exactly peaked around the minimum of the potential.This is due to the fixed distance between each head, which does not correspond to the period of the ratchet potential.Therefore the local minimum of η_chem lies not exactly at, but close to, F_load =0; see Fig. <ref>.§ PERTURBATIVE TREATMENT Here we show that the chemical cycle must extract energy from the ratchet potential in the system with one head for large load forces,i.e. the chemical efficiency (<ref>) must be negative.We show this by calculating the perturbative corrections to the probability density of the position of the motor with respect to the actin filament in the regime of large load F_load→∞, both in the ATP bound and ATP unbound state. First, we assume that the relaxation time of the position in a given potential is much shorter than the typical time between jumps in the chemical state of the heads.This effectively leads to time scale separations, thus the conditional probability distribution (<ref>) is given as a steady state solution of the corresponding Fokker-Planck equation in a given state ζ0 = - 1/γ_M∂_M[ ( F_load + ζ∂_M V_r( x_M - x_A ) ) μ( x_M, x_A) .. + k_B T ∂_Mμ( x_M, x_A) ]+ 1/γ_A∂_A[ζ∂_M V_r μ( x_M, x_A )- k_B T ∂_Aμ( x_M, x_A)] .If we change the variables to describe the geometric center x_C and the distance between motor and actin Δ xx_C= 1/2 ( x_M + x_A ) ,Δ x= x_M - x_A,a more convenient version of the Fokker-Planck is obtained0= [t][.4] - ∂_Δ[( 1/γ_M F_load + ( 1/γ_M + 1/γ_A) ζ∂ V_r( Δ x ) ) μ + ( 1/γ_M + 1/γ_A) k_B T ∂_Δμ]- [t][.4] 1/2∂_C [ ( 1/γ_M F_load + ( 1/γ_M - 1/γ_A) ζ∂ V_r( Δ x ) ) μ+ 1/2( 1/γ_M + 1/γ_A) k_B T ∂_C μ] .If we integrate over all possible position of the center as our focus in this appendix is on the steady state distribution of the relative position of the motor and actin, which completely determines the mean value of both the ratchet potential and the relative velocity,we obtain a reduced Fokker-Planck equation just for the aforementioned displacement (x ≡Δ x further on)0 = 1/γ^*∂_x [ ( F^* - ζ∂_x V_r(x) ) ρ(x) - k_B T ∂_x ρ(x) ] ,where V_r is the ratchet potential (<ref>) and [.4]γ^*= γ_Mγ_A/γ_M + γ_A ,F^*= - γ_A/γ_M + γ_A F_load . The first step is expressing equation (<ref>) in terms of ϵ = ( F^* )^-1 instead of load force f_load, which makes it more convenient for the expansion∂_x [ ρ(x) - ϵ( ζ∂_x V_r(x) ρ(x) + k_B T ∂_x ρ(x) ) ] = 0Due to the periodical boundary conditions, the zeroth order approximation correspond to the flat distribution, i.e. ρ_0 = 1/ℓ.Note that at the leading order the conditional probability density is independent of the internal state ζ, thus the chemical efficiency (<ref>) is zero. For the first order solution we start with an ansatz ρ_1 = ρ_0 + δρ_1.When inserted into (<ref>) we getδρ_1 = ϵ/ℓ∂_x V_r + Cwhere C is an integration constant. Due to the normalization 1 = ∫_0^ℓ xρ_1(x),and periodicity of the ratchet potential (<ref>) the aforementioned integration constant has to be equal to 0.Note, that the first order correction does not contribute to calculations of ⟨ V_r | ζ⟩_ρ ⟨ V_r | ζ⟩_δρ_1= ∫_0^ℓ V_r(x)δρ_1(x) x∝∫_0^ℓ V_r(x)∂_x V_r(x) x= 1/2∫_0^ℓ∂_x V_r^2(x) x= 0where we again used the fact that the ratchet potential V_r(x) is periodic.Consequently, we need to calculate the second order correction δρ_2.Starting from the ansatz ρ_2 = ρ_1 + δρ_2 in (<ref>) and keeping up the terms up to quadratic order in ϵ, we findδρ_2 = ϵ^2 /ℓ[ ( ζ∂_x V_r)^2 - k_B T ζ∂_x^2 V_r] + C^' .Again by requiring the normalization (<ref>), the integration constant C^' can be determined1 = ∫_0^ℓρ_2(x) x= 1 + ϵ^2/ℓ∫_0^ℓ[ ( ζ∂_x V_r(x) )^2 - k_B T ζ∂_x^2 V_r(x) ] x+ ℓ C^' ,which leads toC^' = - ϵ^2 ζ^2/ℓ^2 ∫_0^ℓ( ∂_x V_r(x) )^2 x .Now the density up to second order is given byρ_2(x) = 1/ℓ{ 1+ ϵζ∂_x V_r(x) + ϵ^2 ζ^2 [ ( ∂_x V_r(x) )^2 - 1/ℓ∫_0^ℓ( ∂_y V_r(y) )^2 y ]- ϵ^2 ζ k_B T ∂_x^2 V_r(x) } .Note that up to the quadratic order in ϵ the conditional probability density depends on the ratchet potential only through a first or second order derivatives.Therefore the density will appear as a piece-wise constant function on x ∈ [0,ℓ], with different values for x < a ℓ and x > a ℓ.This trend can already be seen in the empirical density obtained from simulations for large load forces F^* = ± 250pN in Fig. <ref>, and is further demonstrated on motor with a single head in Fig. <ref>. With the second order correction, we can now calculate the change in the expected value of the potential energy⟨ V_r | ζ⟩_δρ_2= ∫_0^ℓ V_r(x)δρ_2 x = ϵ^2 ζ^2/ℓ∫_0^ℓ V_r(x) (∂_x V_r(x) )^2 x- ϵ^2 ζ^2/ℓ^2∫_0^ℓ V_r(x) x ∫_0^ℓ( ∂_y V_r(y) )^2 y - ϵ^2 ζ k_B T/ℓ∫_0^ℓ V_r(x)∂_x^2 V_r(x) x .After tedious but straightforward calculation we found that the first two terms will cancel each other.Hence only the last term remains, which after an explicit calculation leads to⟨ V_r | ζ = 1 ⟩_ρ_2 = V/2 + k_B T/(F^*)^2V^2/a (1-a) ℓ^2 ,⟨ V_r | ζ = c ⟩_ρ_2 = V/2 + k_B T/(F^*)^2c V^2/a (1-a) ℓ^2.By using these expression we get the chemical efficiency (<ref>) up to the second order in ϵη_chem = - (1-c)^2 / a (1-a)k_B T/Δ E - (1-c) V/2( V/ℓ F^*)^2= - (1-c)^2 / a (1-a) k_B T/Δ E - (1-c) V/2[( γ_A + γ_M ) V /γ_Aℓ F_load]^2 .As was advertised, the zeroth and linear order in ϵ do not contribute,the chemical efficiency decays towards zero as F_load^-2 and it is negative for large loads independently of the direction of the load. The symmetry for of the chemical efficiency η_chem is for large loads confirmed in Fig. <ref>.The fact that the chemical efficiency is negative in this regime,means the driven system builds up a chemical energy in the environment, while the external force is performing work on the system.Also note that in the expression for the chemical efficiency,the ratio between the amplitude of the ratchet potential V and the work done by the external force over a period of the potential ℓ F^* appears in the formula,as well as the ratio between the thermal energy k_B T and the mean energy gap between the attached and detached state. In addition we can compute the steady state probabilistic current in each respective internal state. The probabilistic current is given by 𝒥(x) = 1/γ^*[ ( F^* - ∂_x V_r(x) ) ρ(x) - k_B T ∂_x ρ(x) ]Which simplifies in the second order in the large load regime to𝒥(x) = F^*/γ^* ℓ - ζ^2/F^* γ^* ℓ^2∫^ℓ_0 ( ∂_x V_r(x) )^2 x = F^*/γ^* ℓ - ζ^2 V^2/γ^* F^* ℓ^3 a (1-a),which is independent of the position.Note, that the current integrated over the period of the ratchet potential ℓ, represents the mean velocity of the motor in the given state⟨ v | ζ⟩= ∫_0^ℓ x𝒥(x)= F^*/γ^* - ζ^2 V^2/γ^* F^* ℓ^2 a (1-a) = - F_load/γ_M -( γ_M + γ_A ) ζ^2 V^2/γ_Mγ_A F_loadℓ^2 a (1-a).Indeed, the dominant term for large external forces is ⟨ v ⟩_ρ_0 = - F_load / γ_M independently of the internal state,which was already demonstrated in Fig. <ref>.Furthermore, the force-independent term (i.e. term proportional to F_load^0) vanishes leading to an anti-symmetric form with respect to the load F_load.The first correction decay linearly in the external load, which was clearly seen in Fig. <ref>,which exposes the first correction to the mean velocity, cf. equation (<ref>).As there is no fundamental difference in the Fokker-Planck equation for one head and for four heads regarding the currents for large driving forces,the result obtained for a one-headed motor also holds for the simulations of motors with four heads. | http://arxiv.org/abs/1708.07454v2 | {
"authors": [
"Pieter Baerts",
"Christian Maes",
"Jiří Pešek",
"Herman Ramon"
],
"categories": [
"physics.bio-ph",
"cond-mat.stat-mech"
],
"primary_category": "physics.bio-ph",
"published": "20170824152304",
"title": "Tension and chemical efficiency of Myosin-II motors"
} |
http://arxiv.org/abs/1708.07963v1 | {
"authors": [
"A. Kuiroukidis"
],
"categories": [
"hep-th"
],
"primary_category": "hep-th",
"published": "20170826125002",
"title": "Inflationary $α-$attractors and $F(R)$ gravity"
} |
|
Persistent Sinai type diffusion in Gaussian random potentials with decayingspatial correlations Ralf Metzler December 30, 2023 ================================================================================================= Spaces of (subsets of) phylogenetic networks are known to be connected by several types of local moves,for example the rNNI and rSPR moves defined by Gambette et al. <cit.>, or the SNPR moves defined by Bordewich, Linz and Semple <cit.>. Building on the concept of rSPR moves, we here propose a more restricted type of move: the tail move.Whereas rSPR moves can change either the starting point or the end point of an arc, a tail move can only move the starting point. Gambette et al. prove that the space of all phylogenetic networks on a set X with k reticulations, which we call the k-th tier of X, is connected by rSPR moves. Here we prove that, unless k=1 and |X|=2, the k-th tier is already connected when we only use tail moves. Our proof also gives us a bound on the diameter of this space in terms of the rNNI-distance for networks with at least two leaves. § INTRODUCTION§ DEFINITIONS AND PROPERTIES §.§ Phylogenetic networksPhylogenetic networks are a generalisation of phylogenetic trees. The difference between the two is that in phylogenetic networks branches may recombine. In the network, such an event occurs at a particular node which we call a reticulation. The following definition gives names to all types of nodes we encounter in a phylogenetic network. Let G be a directed acyclic graph, then we use the following terminology for the nodes of G.* A node with in-degree 0 and out-degree 1 is called a root.* A node with in-degree 1 and out-degree ≥ 2 is called a tree node.* A node with in-degree ≥ 2 and out-degree 1 is called a reticulation.* A node with in-degree 1 and out-degree 0 is called a leaf. We have now named all different kinds of nodes that can be found in a phylogenetic network. We now define such a network using this new terminology. A phylogenetic network on a finite set X consists of the following: * A (connected) directed acyclic graph with no parallel edges and one root node, in which all other nodes are one of the following: a tree node, a reticulation, or a leaf.* a bijection between X and the leaf nodes of G.A phylogenetic network is called binary if the total degree of each node is at most 3. Note that this definition excludes internal nodes with in-degree 1 and out-degree 1. Suppressing all these nodes does not change the shape of the network, so their exclusion does not change the topological generality of the definition. Another topological note about this definition, regards the addition of connected to the restrictions on the network. This addition is not necessary, as having exactly one root and no directed cycles already implies connectedness of the network.Phylogenetic networks are considered more complex if they are less tree-like. Therefore we need a definition of the tree-likeness of a network. A standard measure for phylogenetic networks is the reticulation number: the number of times branches recombine. Let N be a phylogenetic network. The reticulation number of N is ∑_v∈ R(N)(v)-1,where R(N) denotes the set of reticulation nodes of N. The reticulation number of binary networks is especially easy to compute, since it is equal to the number of reticulation nodes. The reticulation number is one way to define the complexity of a network. It is hence natural to study all networks with the same reticulation number.Let X be a (finite) set of taxa. The k-th tier on X is the set of all binary phylogenetic networks with label set X containing k reticulation nodes.Although this definition could easily be given for arbitrary (non-binary) networks, we restrict to the binary case because our results apply to binary networks only. From this point on, we will only consider binary networks.§.§.§ Orders and AncestorsIn rooted trees and rooted networks, all edges are directed away from the root. We depict all phylogenetic networks and trees with the root at the top and the leaves at the bottom. In this representation all arcs are directed downward. In line with these conventions, we use the following terminology for `directions'. Let u and v be nodes in a phylogenetic network, then we say u is above v and v is below u if u ≤ v in the order induced by the directed graph underlying the network.Similarly, if e=(x,y) is an edge of the network and u a node, then we say that e is above u if y is above u and e is below u if x is below u.This definition is about order over possibly large distances in the network, the next is about directional relations over a short distance. Let e=(u,v) be an arc of a phylogenetic network, then we say that u is the tail of e and v is the head of e. In this situation, we also say that u is a parent of v, or u is directly above v and v is a child of u or v is directly below u. Note that in a rooted tree, there is always a unique (shortest) path between two nodes, and such a path always goes up, and then down. The node at which it switches from going up to going down is called the lowest common ancestor. In a phylogenetic network there is no unique node through which such paths go. Hence we will define a lowest common ancestor of two nodes, and directly after we prove that there is at least one such lowest common ancestor for any pair of vertices. Let u and v be nodes of a phylogenetic network. A lowest common ancestor (LCA) of u and v is a node x (often denoted (u,v)) such that x ≤ ux ≤ v,and for any other x' with the same property we have x≰x'.For every pair of vertices u,v of a phylogenetic network there is at least one LCA. Consider the set of nodes which are above u and vV={x∈ V(N): x≤ u , x≤ v },then the LCAs of u and v are the maximal elements of this set under the induced partial order. The set V is non-empty because it contains the root. It is finite because the network consists of a finite number of nodes. Hence the set of maximal elements is non-empty.Let u and v be distinct nodes of a phylogenetic network. Then every LCA of these nodes is a tree node, and any node and edge under this LCA is above at most one of u and v.§.§ Local movesTo search the space of phylogenetic networks in an orderly way, we want to have a concept of locality. Several analogues of local tree moves have been defined for phylogenetic networks.<cit.>Such a local move typically changes the head, the tail, or both head and tail of one edge. Let e=(u,v) and f be edges of a phylogenetic network. A head move of e to f consists of the following steps: * removal of edge e,* subdividing f with a new node v',* suppressing the in-degree 1, out-degree 1 node v,* adding the edge (u,v').Head moves are only allowed if the resulting digraph is a phylogenetic network. We say that the move is over distance d if there is a path from v to v' of graph length d+1 in the underlying undirected graph where both nodes exist (after step 2).Let e=(u,v) and f be edges of a phylogenetic network. A tail move of e to f consists of the following steps: * removal of edge e,* subdividing f with a new node u',* suppressing the in-degree 1, out-degree 1 node u,* adding the edge (u',v).Tail moves are only allowed if the resulting digraph is a phylogenetic network. We say that the move is over distance d if there is a path from u to u' of graph length d+1 in the underlying undirected graph where both nodes exist (after step 2). Note that connectedness after a move implies that all allowed moves are over a well defined finite distance. Because if this distance were undefined, i.e. no path between v and v' for head moves or u and u' in the case of tail moves, the resulting network would be disconnected.It should also be noted that head moves are only possible for reticulation arcs, otherwise we create a node which does not fit any of the descriptions of Definition <ref>. This means that head moves are actually not part of the natural generalisation of SPR moves on trees, which consist of only tail moves. Here, we study the connectivity of the tiers of phylogenetic network space using only tail moves. We initially prove our connectivity result using the following theorem by Gambette et al.<cit.>, who study the rNNI move which is equivalent to either a head move or a tail move over distance 1. Let N and N' be two rooted binary phylogenetic networks on tier k of X. Then there exists a sequence of rNNI moves turning N into N'.The fact that not all edges of a network can be moved by a tail move (e.g. edges with a reticulation node as tail) is quite important. Hence we want to have a notion of movability of edges. Let e=(u,v) be an edge in a phylogenetic network. Then e is called non-movable if u is the root, if u is a reticulation, or if removal of e followed by subduing u creates parallel edges. Otherwise, it is called movable. It is clear that there is only one situation in which removing an edge can result in parallel edges. This is when one would move the tail of an edge (u,v) and there is an arc from the parent of u to the child of u other than v. The situation is characterized in the following definition. Let N be a phylogenetic network and let x,u,y be nodes of N. We say x,u and y form a triangle if there are arcs (x,u), (u,y) and (x,y). The edge (x,y) is called the long edge and (u,y) is called the bottom edge. The interesting case in Definition <ref> is when the node on the side of the triangle is a tree node. Let x,u and y form a triangle in a network, and let v be the other child of u. The edge (u,v) is not movable because it creates parallel edges from x to y.The edge (u,y) is movable, however,and if it is moved sufficiently far up, the new edge (x,v) is also movable. The following remark is an important part of the arguments we use in the next section. Let u be a tree node, then at least one of its child edges is movable. This is a direct consequence of Remark <ref>. Note that movability does not consider whether there actually is a valid tail move of e: it only considers whether we can remove the tail without problems, not if we can also reattach it anywhere else. Of course a tail can always be attached to the root edge, but it is not certain that this results in a non-isomorphic network. The following remark characterizes valid moves. The tail of an edge e=(u,v) can be moved to another edge f=(s,t) iff all the following hold: * e is movable, * v is not above f, * and t≠ v.The first condition assures that the tail can be removed, the second that we do not create cycles, and the third that we do not create parallel edges. The following lemma is related to the previous remarks, and will be used to find a tail that can be moved down sufficiently far. Let x,y be nodes of a phylogenetic network N. Then there exists a movable edge in N that is not both above x and above y. Consider an arbitrary LCA u of x and y. This LCA is a tree node, and both child edges are above at most one of x and y. Because at least one of the child edges of a tree node is movable (Remark <ref>), at least one of the child edges of the LCA has the desired properties. § HEAD MOVES REWRITTENThis section presents the main result: the connectedness of tier k of phylogenetic networks on a fixed leaf set X using tail moves. Theorem <ref> tells us that it is connected by rNNI moves, i.e. head and tail moves over distance one. This means that, to achieve our goal, it suffices to prove that any distance-one head move can be replaced with a sequence of tail moves. To this end, we distinguish all different situations in which we can do such a head move. Each distance one head move belongs to exactly one of the cases (a)-(f) in Figure <ref>. Note that the figure does not indicate whether u,w,x,y are distinct. It turns out that there are only a few cases in which u,w,x,y are not all distinct, and the head move is valid. Identification of x and y in move (a) and of u and y in move (d and f) are valid, but result in isomorphic networks. Identification of u and w in move (a) is the only identification which results in non-isomorphic networks. All other identifications make a head move invalid by introducing a cycle or parallel edges. It is easy to see that moves (c) and (e) as well as moves (d) and (f) are each others reversions. Because all tail moves are also reversible, we only have to show that head moves (a)-(d) can be rewritten as a sequence of tail moves. We now treat all cases separately in the following lemma. We also take care of all the subcases given by the location of the tail of the edge whose head we move. All head moves over distance one can be substituted by a sequence of tail moves, except for the head move in the network depicted in Figure <ref>.In light of Remark <ref>, we distinguish two cases: u≠ w and u=w. All other nodes in Figure <ref> are distinct. *u≠ w.In this case we can use the sequence of tail moves depicted in Figure <ref>. The sequence in the figure only depicts part of the network. In this part of the network, we do not find any directed cycles or parallel edges, but this does not imply the validity of the tail moves in the whole network. We have to check that validity of the head move implies the intermediate network does not contain directed cycles or parallel edges.If the intermediate network were cyclic, then the (directed) cycle must involve edge (z',y), and there is a path from y to z'. This implies there is a path from y to v, which then implies there is a path from y to either u or to w. Because the tail move does not change any other part of the network, this path must also exist in the networks before and after the head move. This leads to a contradiction, because then one of these networks already contains a cycle, making the head move invalid. It is immediately clear that the moving tails do not connect to triangles by inspection of Figure <ref>. The only other possibility for creating parallel edges, is if x=y. But in that case the networks before and after the head move are isomorphic. We conclude that the intermediate network is valid, and that therefore the sequence of tail moves is valid.*u=w.Note that u and v form a triangle together with the tree node z. In this situation, we cannot directly use the same sequence as before, since this sequence would create parallel edges. We can solve this problem in two ways. *Using an extra tail.Instead of moving edge e directly, we first subdivide it by moving a tail to e. Then we can do the sequence of moves depicted in Figure <ref>. Barring the addition of the extra tail, the sequence of moves is quite similar to the moves in case <ref>. *Destroying the triangle. The bottom edge of the triangle, edge (z,v) is movable. If it is moved elsewhere, the situation changes to that of case <ref>. Do the sequence of moves for that case, and move the bottom edge of the triangle back.Option <ref> is possible if there is at least one vertex above the triangle in addition to the root: move the bottom edge to the root. And, option <ref> can be used if there is a tree node somewhere not above the triangle.The two networks to which neither of these conditions apply are shown in Figure <ref>. The first is the network on one leaf with two reticulations. In this network no head move leads to a different (non-isomorphic) network. The only non-trivial case is the excluded network with two leaves and one reticulation. The head move cannot be substituted by a sequence of tail moves, because there is no valid tail move.The idea of this substitution is to use an extra tail again. The proposed sequence is given in Figure <ref>.The main problem is to find a usable tail: to use the sequence of moves, we need a tail that can be moved below either x or y. This is sufficient because of the symmetry of the situation and the reversibility of tail moves: if the movable edge is not above y, we do the sequence of moves in reverse order, where we switch the labels x and y.Consider an LCA of x and y in Figure <ref>. At least one of the child edges of this LCA can be moved (Lemma <ref>), and this edge is not above both x and y. Hence we can move this tail down below one of these nodes (Remark <ref>).Without loss of generality, let the movable edge (s,t) be above y and not above x. There are two situations in which the sequence of moves invalid: x is a tree node with children v and t, then one of the networks has parallel edges from x to t; and x=s then the sequence cannot be used as show, because it shows s' and x as distinct nodes. We now give solutions for both these problems.*The children of x are v and t. Note that we have (s,t) with s=(x,y) above y movable. Because there is an edge from x to t, there is a path from x to y. Hence x is an LCA of x and y, but then x must be the unique such LCA. We conclude s=x, which is the next case we treat. *x=LCA(x,y). Because x is the LCA of x and y and there is no path from v to y, x must be a tree node. Let the children of x be v and t, and let the parent of x be p. *(x,v) is movable. In this case we can use a similar sequence as before, but without the addition of a tail: (x,v) to (y,z), (x',z) to (p,t).* (x,v) isnot movable. In this case p,x and t form a triangle. We employ a strategy to break the triangle similar to the strategy for Case <ref> of head move (a). * There is a node above the triangle besides the root.Move the bottom edge of the triangle to the root, we have reduced the problem to Case <ref> with p instead of x in the situation. Do the sequence of tail moves and move x back to the original position below p.* The node above the triangle is the root. If there is no node above the triangle, then there is a tree node not above x (e.g. LCA(u,y)) and hence an edge e that can be moved to (x,t). With this move, we make (x,v) movable, and we can do the moves as in Case <ref>, after which we move e back to its original position.In all sequences we gave, no directed cycles occur, because such cycles imply directed cycles in one of the networks before and after the head move. We conclude that any allowed head move of type (b) can be substituted by a sequence of tail moves. This is the easiest case, as we can substitute the head move by exactly one tail move: (z,y) to (w,v), with labelling as in Figure <ref>. Parallel edges and directed cycles cannot occur because there are no intermediate networks. This head move is in some sense the most involved, as there are easy cases, and a very complicated cases. We exploit the following symmetry of this case: relabelling u↔ y and v↔ z transforms the network before the head move into the network after the head move. Let us start with the easiest case: *Either (u,v) or (y,z) is movable. Assume w.l.o.g. that (u,v) is movable and that the other child and parent of u are p and t, we move (u,v) to (y,z) and then (u',z) to (p,t).*Either u or y is a tree node.If one of them is movable, we are in the previous situation, otherwise we do something similar: we ignore the triangle and restore it later as shown in Figure <ref>. * both u and y are reticulations. In this case we try to recreate the situation of Case <ref> by adding a tail on one of the edges (u,v) and (y,z). *The network contains at least two leaves. We can pick two distinct leaves, at least one of which is below w. Suppose first that both of these leaves are below w, then an LCA of the leaves is also below the reticulation and one of its child edges can be used as the extra tail. If only one of the leaves is below the reticulation, then any LCA of the leaves has one edge that is not above w. If this edge can be moved, we can directly use it, otherwise, we first move the lower part of the triangle to the root, and then still use this edge. * The network contains one leaf, and w is not this leaf. The only leaf is below the lowest reticulation r of the network, which is not z. By removing the lowest reticulation and the leaf below it, and then adding leaves to the loosened edges, we reduce to Case <ref>. The associated sequence of moves where we treat the parents of r as leaves substitutes the head move.* The only leaf in the network is w. We have not found a way to solve this case `locally' as we did before. Go up from these reticulations to some nearest tree nodes. The idea, then, is to use one of these tree edges to do the switch as in the case we discussed previously (Figure <ref>). The result is that all reticulations on the path to the nearest tree node move with the tree edge. These reticulations can be moved back to the other side using the previously discussed moves. In particular we move the heads sideways using head move (b), and then we move them up using head move (f) where the main reticulation (z) is not the lowest reticulation in the network.Using Theorem <ref> and the above lemma, we directly get the connectivity result for tail moves.Let N and N' be two rooted binary networks on the k-th tier of X. Then there exists a sequence of tail moves turning N into N', except if k=1 and |X|=2. § DIAMETER BOUNDS Let N, N' be two networks with leaves labeled with X.Then an isomorphism between N and N' is a bijection ϕ: V(N) → V(N') such that * Two nodes u,v ∈ V(N) are adjacent in N if and only if ϕ(u) and ϕ(v) are adjacent in N';* For any leaf u ∈ L(N), ϕ(u) is the leaf in L(N') that has the same label as u. We say N and N' are isomorphic if there exists an isomorphism between N and N'.Given a binary phylogenetic network N and a set of nodes Y in N, we say Y is downward-closed if for any u ∈ Y, every child of u is in Y.Let N and N' be binary networks on X with the same number of reticulations, such that N and N' are not the networks depicted in Figure <ref>.Let Y ⊆ V(N), Y' ⊆ V(N') be downward-closed sets of nodes such thatL(N)⊆ Y, L(N') ⊆ Y', and N[Y] is isomorphic to N[Y'].Then there is a sequence of at most 4|N∖ Y| tail moves turning N into N'. By setting Y = L(N) and Y' = L(N'), we note that such Y and Y' always exist.We first observe that any isomorphism between N[Y] and N[Y'] maps reticulations (tree nodes) of N to reticulations (tree nodes) of N'. Indeed, every node in N is mapped to a node in N' of the same out-degree, and the tree nodes are exactly those with out-degree 2. It follows that Y and Y' contain the same number of reticulations and the same number of tree nodes.Furthermore, as N and N' have the same number of reticulations (and thus the same number of tree nodes), it also follows that V(N)∖ Y and V(N')∖ Y' contain the same number of reticulations and the same number of tree nodes. We prove the claim by induction on |N ∖ Y|.If|N ∖ Y| = 0, then N = N[Y], which is isomorphic to N'[Y'] = N', and so there is a sequence of 0 moves turning N into N'.If|N ∖ Y| = 1, then as Y is downward-closed, N ∖ Y consists of ρ_N, the root of N, and by a similar argument N '∖ Y' consists of ρ_N'.Let x be the only child of ρ_N, and note that in N[Y], x is the only node of in-degree 0, out-degree 2.It follows that in the isomorphism between N[Y] and N'[Y'], x is mapped to the only node in N'[Y'] of in-degree 0, out-degree 2, and this node is necessarily the child of ρ_N'.Thus we can extend the isomorphism between N[Y] and N'[Y'] to an isomorphism between N and N' by letting ρ_N be mapped to ρ_N'.Thus again there is a sequence of 0 moves turning N into N'.So now assume that |N ∖ Y| > 1.We consider three cases, which split into further subcases.In what follows, a lowest node of N ∖ Y (N' ∖ Y') is a node u in V(N)∖ Y (V(N')∖ Y') such that all descendants of u are in Y (Y').Note that such a node always exists, as Y (Y') is downward-closed.* There exists a lowest node u' of N' ∖ Y' such that u' is a reticulation: In this case, let x' be the single child of u'.Then x' is in Y', and therefore there exists a node x ∈ Y such that x is mapped to x' by the isomorphism between N[Y] and N'[Y'].Furthermore, x has the same number of parents in N as x' does in N', and the same number of parents in Y as x' has in Y'.Thus x has at least one parent z such that z is not in Y.We now split into two subcases: *z is a reticulation: in this case, let Y_1 = Y ∪{z} and Y_1' = Y' ∪{u'}, and extend the isomorphism between N[Y] and N[Y'] to an isomorphism between N[Y_1] and N[Y_1'], by letting z be mapped to u' (see Figure <ref>). We now have that Y_1 and Y_1' are downward-closed sets of nodes such that N[Y_1] is isomorphic to N[Y_1'], and L(N)⊆ Y_1, L(N') ⊆ Y_1'. Furthermore |N ∖ Y_1| = |N ∖ Y|-1. Thus by the inductive hypothesis, there is a sequence of 4|N ∖ Y_1| =4|N ∖ Y|-4 tail moves turning N into N'.* z is not a reticulation: then z cannot be the root of N (as this would imply|N ∖ Y| = 1), so z is a tree node. It follows that the edge (z,x) is movable, unless the removal of (z,x) followed by suppressing z creates parallel edges. *(z,x) is movable: In this case, let u be any reticulation in N ∖ Y (such a node must exist, as u' exists and N ∖ Y,N' ∖ Y' have the same number of reticulations). Let v be the child of u (which may be in Y), and observe that the edge (u,v) is not below x (as x ∈ Y and u ∉ Y). If v = x, then u is a reticulation parent of x that is not in Y, and by substituting v for z, we have case <ref>. So we may assume v ≠ x. Then it follows from Remark <ref> that the tail of (z,x) can be moved to (u,v). Let N_1 be the network derived from N by applying this tail move, and let z_1 be the new node created by subdividing (u,v) during the tail move (see Figure <ref>). Thus, N_1 contains the edges (u,z_1), (z_1,v), (z_1,x).(Note that if z is immediately below u in N i.e. z=v, then in fact N_1 = N. In this case we may skip the move from N to N_1, and in what follows substitute z for z_1.)Note that (z_1,v) is movable in N_1 (since the parent uof z is a reticulation node, and therefore deleting u and suppressing z_1 cannot create parallel edges). Let a be one of the parents of u in N_1. Then the tail of (z_1,v) can be moved to (a,u) (as u ≠ v, and (a,u) is not below vas this would imply a cycle in N_1). So now let N_2 be the network derived from N_1 by applying this tail move (again see Figure <ref>). In N_2, the reticulation u is the parent of x (as z_1 was suppressed), and thus case <ref> applies to N_2 and N'. Therefore there exists a sequence of 4|N ∖ Y|-4 tail moves turning N_2 into N'. As N_2 is derived from N by two tail moves, there exists a sequence of 4|N ∖ Y|-2 tail moves turning N into N'. * The removal of (z,x) followed by suppressing z creates parallel edges: Then there exists nodes c ≠ x, d ≠ x such that c,d,z form a triangle with long edge (c,d). As c has out-degree 2 it is not the root of N, so let b denote the parent of c. * b is not the root of N: In this case, let a be a parent of b in N. Observe that the edge (a,b) is not below d and that b ≠ d. Furthermore, (c,d) is movable since c is not a reticulation or the root, and there is no edge (b,d) (such an edge would mean b has in-degree 3, as the edges (c,d) and (z,d) exist). It follows that the tail of (c,d) can be moved to (a,b)(again using Remark <ref>). Let N_1 be the network derived from N by applying this tail move (see figure <ref>). Observe that in N_1 we now have the edge (b,z) (as c was suppressed), and still have the edge (z,d) but still do not have the edge (b,d). Thus, deleting (z,x) and suppressing z will not create parallel edges, and so (z,x) is movable in N_1. Thus case <ref> applies to N_1 and N', and so there exists a sequence of 4|N ∖ Y|-2 tail moves turning N_1 into N'. As N_1 is derived from N by a single tail move, there exists a sequence of 4|N ∖ Y|-1 tail moves turning N into N'. *b is the root of N: In this case, we observe that if d ∈ Y then every reticulation in N is in Y. This contradicts the fact that N∖ Y and N'∖ Y' contain the same number of reticulations. Therefore we may assume that d ∉ Y. Then we may proceed as follows. Let N_1 be the network derived from Y by moving the head of (c,d) to (z,x) (see Figure <ref>). We observe that this move cannot create a cycle (such a cycle is only possible if there is a path in Nfrom x to c, a contradiction as x ∈ Y, c ∉ Y), and cannot create parallel edges (such parallel edges would have to be out-edges of z, and this does not happen as one of the out-neighbors of z is the newly created node subdividing (z,x)). Thus, this is a valid head move. Furthermore, this is a head move of type (a), and therefore can be replaced with a sequence of 4 tail moves. Observe that in N_1,x has a parent not in Y which is a reticulation, and that N_1[Y] = N[Y]. Then case <ref> applies to N_1 and N', and so there exists a sequence of 4|N ∖ Y|-4 tail moves turning N_1 into N'. As N_1 can be derived from N by at most 4 tail moves, it follows that there exists a sequence of 4|N ∖ Y| tail moves turning N into N'. *There exists a lowest node u of N ∖ Y such that u is a reticulation: By symmetric arguments to case <ref> , we have that there is a sequence of at most 4|N∖ Y| tail moves turning N' into N. As all tail moves are reversible, there is also a sequence of at most 4|N∖ Y| tail moves turning N into N'. * Every lowest node of N ∖ Y and every lowest node of N'∖ Y' is not a reticulation: As |N ∖ Y| > 1, we have that in fact every lowest node of N ∖ Y and every lowest node of N'∖ Y' is a tree node. Then we proceed as follows. Let u' be an arbitrary lowest node of N' ∖ Y', with x' and y' its children.Then x',y' are in Y', and therefore there exist nodes x,y ∈ Y such that x (y) is mapped to x' (y') by the isomorphism between N[Y] and N'[Y'].Furthermore, x has the same number of parents in N as x' does in N', and the same number of parents in Y as x' has in Y'.Thus x has at least one parent not in Y. Similarly, y has at least one parent not in Y.*x and y have a common parent u not in Y: In this case, let Y_1 = Y ∪{u} and Y_1' = Y' ∪{u'}, and extend the isomorphism between N[Y] and N[Y'] to an isomorphism between N[Y_1] and N[Y_1'], by letting u be mapped to u' (see figure <ref>). We now have that Y_1 and Y_1' are downward-closed sets of nodes such that N[Y_1] is isomorphic to N[Y_1'], and L(N)⊆ Y_1, L(N') ⊆ Y_1'. Furthermore |N ∖ Y_1| < |N ∖ Y|. Thus by the inductive hypothesis, there is a sequence of 4|N ∖ Y_1| =4|N ∖ Y|-4 tail moves turning N into N'.* x and y do not have a common parent not in Y: In this case, let z_x be a parent of x not in Y, and let z_y be a parent of y not in Y. Recall that z_x and z_y are both tree nodes. It follows that either one of (z_x,x), (z_y,y) is movable, or deleting (z_x,x) and suppressing x(deleting (z_y,y) and suppressing y) would create parallel edges. * (z_x,x), is movable:in this case, observe that the edge (z_y,y) is not below x (as x ∈ Y and z_Y ∉ Y), and that x ≠ y.Then by Remark <ref>, the tail of (z_x,x) can be moved to (z_y,y). Let N_1 be the network derived from N by applying this tail move (see Figure <ref>). Then as x and y have a common parent in N_1 not in Y, and as N_1[Y] = N[Y], we may apply the arguments of case <ref>to show that there exists a sequence of 4|N ∖ Y|-4 tail moves turning N_1 into N'. As N_1 is derived from N by a single tail move, there exists a sequence of 4|N ∖ Y|-3 tail moves turning N into N'. * (z_y,y) is movable: By symmetric arguments to case <ref>, we have that there is a sequence of at most4|N ∖ Y|-3tail moves turning N into N'. * Neither (z_x,x) nor (z_y,y) is movable: In this case, there must exist nodes d_x,c_x, d_y, c_y such that c_x,d_x,z_x form a triangle with long edge (c_x,d_x), andc_y,d_y,z_y form a triangle with long edge (c_y,d_y). Moreover, as z_x,z_y are different nodes with one parent each, c_x ≠ c_y. It follows that one of c_x,c_y is not the child of the root of N. Suppose without loss of generality that c_x is not the child of the root. Then there exist nodes a_x,b_x and edges (a_x,b_x), (b_x,c_x). By similar arguments to those used in Case <ref>, the tail of (c_x,d_x) can be moved to (a_x,b_x), and in the resulting network N_1, (z_x,x) is movable (see Figre <ref>). Thus case <ref> applies to N_1 and N', and so there exists a sequence of 4|N∖ Y|-2 tail moves turning N_1 into N'. As N_1 is derived from N by a single tail move, there exists a sequence of 4|N∖ Y|-1 tail moves turning N into N'. By setting Y = L(N) and Y' = L(N'), we have the following:Let N and N' be binary networks on X with the same number of reticulations, such that N and N' are not the networks depicted in Figure <ref>.Then there is a sequence of at most 4(|N|-|X|) tail moves turning N into N'.§ TAIL DISTANCE §.§ Relation with other distances §.§ Tail1 distanceComparing rNNI and tail moves directly might be hard because rNNI moves are more local. To make this comparison easier, we would like to consider local tail moves separately. The following lemma indicates how restricting to distance-1 tail moves influences our results. Let e=(u,v) to f=(s,t) be a valid tail move, then there is a sequence of distance-1 tail moves resulting in the same network. Note that there exist (possibly non-unique) directed paths u←(u,s)→ s for any choice of (u,s). We prove that a tail move of e to any edge on either path is valid, this gives a sequence of distance-1 tail moves.Let g=(x,y) be an edge of either path. We first use a proof by contradiction to show that the move to g cannot create cycles, then we prove that we do not create parallel edges.Suppose moving e to g creates a cycle, then this cycle must involve the new edge e' from g to v. This means there is a path from v to x. However, x is above u or above s, which means that either the begin situation is not a phylogenetic network, or the move e to f is not valid. From this contradiction, we conclude that the move of e to g does not create cycles.Note that e is movable, because the move of e to f is valid. Hence the only way to create parallel edges, is by moving e=(u,v) to an edge g=(x,y) with y=v. It is clear that g is not in the path u←(u,s), as this would imply there is a cycle in the original network. Hence g must be on the other path. If f=g, then the original move of e to f would create parallel edges, and if g is above f, the original move moves e to below e creating a cycle. We conclude that there cannot be an edge g=(x,y) on either path such that y=v, hence we do not create parallel edges.Lemma <ref> gives an upper bound on the number of distance-1 tail moves needed to get the same result as one general tail move. §.§ Computationspmpsci | http://arxiv.org/abs/1708.07656v1 | {
"authors": [
"Remie Janssen",
"Mark Jones",
"Péter L. Erdős",
"Leo van Iersel",
"Celine Scornavacca"
],
"categories": [
"math.CO",
"cs.DM",
"92D15, 68R10, 68R05, 05C20"
],
"primary_category": "math.CO",
"published": "20170825090230",
"title": "Exploring the tiers of rooted phylogenetic network space using tail moves"
} |
Subspace Approximation for Approximate Nearest Neighbor Search in NLP Jing Wang [email protected] December 30, 2023 ===================================================================== Functional equations satisfied by additive functions have a special interest not only in the theoryof functional equations, but also in the theory of (commutative) algebra because the fundamental notions such as derivations and automorphisms are additive functions satisfying some further functional equations as well. It is an important question that how these morphisms can be characterized among additive mappings in general. The paper contains some multivariate characterizations of higher order derivations.The univariate characterizations are given as consequences by the diagonalization of the multivariate formulas. This method allows us to refine the process of computing the solutions of univariate functional equations of the form ∑_k=1^nx^p_kf_k(x^q_k)=0,where p_k and q_k (k=1, …, n) are given nonnegative integers and the unknown functions f_1, …, f_n R→ R are supposed to be additive on the ring R. It is illustrated by some explicit examples too. As another application of the multivariate setting we use spectral analysis and spectral synthesis in the space of the additive solutionsto prove that it is spanned by differential operators.The results are uniformly based on the investigation of themultivariate version of the functional equations. § INTRODUCTION Functional equations satisfied by additive functions have a rather extensive literature <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>. They appear not only in the theoryof functional equations, but also in the theory of (commutative) algebra <cit.>, <cit.>, <cit.>, <cit.>, <cit.>.It is an important question that how special morphisms can be characterized among additive mappings in general.This paper is devoted to the case of functional equations characterizing derivations. It is motivated by some recent results <cit.>, <cit.>, <cit.> due to B. Ebanks. We are looking for solutions of functional equations of the form ∑_k=1^nx^p_kf_k(x^q_k)=0,where p_k and q_k (k=1, …, n) are given nonnegative integers and the unknown functions f_1, …, f_n R→ R are supposed to be additive on the ring R. According to the homogeneity of additive functions <cit.> it is enough to investigate equations with constant pairwise sum of the powers, i.e. p_1+q_1=…=p_n+q_n.The general form of the solutions is formulated as a conjecture in <cit.>. The proof can be found in <cit.>by using special substitutions of the variable and an inductive argument. We have asignificantly different starting point by following the method of free variables. The paper contains some multivariate characterizations of higher order derivations in Section 2.The basic results of <cit.>, <cit.> (univariate characterizations of higher order derivations) are given as consequences by the diagonalization of the multivariate formulas, see Section 3.The method allows us to refine the process of computing the solutions of functional equations of the form (<ref>). The examples (Examples I, Examples II) illustrate that the multivariate version of the functional equations provides a more effective and subtle way to determine the structure of the unknown functions. Especially, functional equations with missing powers (Examples II) can be investigated in this way to avoidformal (identically zero) terms in the solution. As a refinement of Ebanks' method we follow the main steps such as * The formulation of the multivariate version of the functional equation. * The substitution of value 1 as many times as the number of the missing powers (due to the symmetry of the variables there is no need to specify the positions for the substitution).* The application of Theorem <ref>/Corollary <ref>. The second step decreases the homogeneity degree of the functional equation to keep only non-identically zero terms in the solutions. Therefore it can be easily seen how the number of the nonzero coefficients is related to the maximal order of the solution <cit.>. In Section 4 we present another approach to the problem. The application of the spectral analysis and the spectral synthesis is ageneral method to find the additive solutions of a functional equation, see <cit.>.It is a new and important trend in the theory of functional equations; see e.g. <cit.> and <cit.>.Although the domain should be specified as a finitely generated subfield over the rationals,the multivariate version of the functional equation generates a system of functional equations concerning the translates of the original solutionsin the space of complex valued additive functions restricted to the multiplicative subgroup of the given subfield.Taking into account the fundamental result <cit.> such a closed translation invariant linearsubspace of additive functions contains automorphism solutions (spectral analysis) and the space of the solutions isspanned by the compositions of automorphisms and differential operators (spectral synthesis).All functional equations in the paper are also discussed by the help of the spectral analysis and the spectral synthesis.We prove that the automorphism solutions must be trivial (identity) and, consequently,the space of the solutions is spanned by differential operators. According to the results presented in Section 2 and Section 3,the investigation of the detailed form of the differential operator solutions is omitted.However, Subsection 4.4 contains an alternative way to prove Theorem <ref>/Corollary <ref>in the special case n=3.The proof uses a descending process instead of the inductive argument.In what follows we summarize some basic theoretical facts, terminology and notations. §.§ Derivations For the general theory of derivations we can refer to Kuczma <cit.>, see also Zariski–Samuel <cit.> and Kharchenko <cit.>.Let Q be a ring and consider a subring P⊂ Q. A function f:P→ Q is called a derivationderivation if it is additive, i.e.f(x+y)=f(x)+f(y) (x, y∈ P)and also satisfies the so-called Leibniz ruleLeibniz rulef(xy)=f(x)y+xf(y) (x, y∈ P).Let 𝔽 be a field, and 𝔽[x] be the ring of polynomials with coefficients from 𝔽. For a polynomial p∈𝔽[x], p(x)=∑_k=0^na_kx^k, define the function f𝔽[x]→𝔽[x] asf(p)=p',where p'(x)=∑_k=1^nka_kx^k-1 is the derivative of the polynomial p. Then the function f clearly fulfillsf(p+q)=f(p)+f(q)andf(pq)=pf(q)+qf(p)for all p, q∈𝔽[x]. Hence f is a derivation. Let 𝔽 be a field, and suppose that we are given a derivation f𝔽→𝔽. We define the mapping f_0𝔽[x]→𝔽[x] in the following way.If p∈𝔽[x] has the form p(x)=∑_k=0^na_kx^k,then let f_0(p)= p^f(x)=∑_k=0^nf(a_k)x^k.Then f_0𝔽[x]→𝔽[x] is a derivation.The following lemma says that the above two examples have rather fundamental importance. Let (𝕂, +, ·) be a field and let (𝔽, +, ·) be a subfield of 𝕂. If f𝔽→𝕂 is a derivation, then for any a∈𝔽 and for arbitrary polynomial p∈𝔽[x] we havef(p(a))=p^f(a)+f(a)p'(a).As the following theorem shows, in fields with characteristic zero, with the aid of the notion of algebraic base,we are able to construct non-identically zero derivations. Let 𝕂 be a field of characteristic zero, let 𝔽 be a subfield of 𝕂, let S be an algebraic base of 𝕂 over 𝔽, if it exists, and let S=∅ otherwise. Let f𝔽→𝕂 be a derivation. Then, for every function u S→𝕂, there exists a unique derivation g𝕂→𝕂 such that g |_𝔽=f and g |_S=u.If R is a commutative ring and f R→ R is a derivation, thenf(x^2)=2xf(x)(x∈ R)is a direct consequence of the Leibniz rule. In general this identity does not characterize derivations among additive functions as the following argument shows. Substituting z=x+y in place of xf(z^2)=2(x+y)f(x+y)= 2(xf(x)+xf(y)+yf(x)+yf(y)).On the other hand f(z^2)=f(x^2+2xy+y^2)= 2xf(x)+2f(xy)+2yf(y).Therefore 2f(xy)=2xf(y)+2yf(x) (x∈ R),i.e. 2f is a derivation. Unfortunately the division by 2 is not allowed without any further assumption on the ring R.Therefore some additional restrictions on the ring appear typically in the results.The notion of derivation can be extended in several ways. We will employ the concept of higher order derivations according to Reich <cit.> and Unger–Reich <cit.>. Let R be a ring. The identically zero map is the only derivation of order zero. For each n∈ℕ, an additive mapping f R→ R is termed to be a derivation of order n, if there exists B R× R→ R such that B is a bi-derivation of order n-1 (that is, B is a derivation of order n-1 in each variable) andf(xy)-xf(y)-f(x)y=B(x, y)(x, y∈ R). The set of derivations of order n of the ring R will be denoted by 𝒟_n(R).Since 𝒟_0(R)={0}, the only bi-derivation of order zero is the identically zero function, thus f∈𝒟_1(R) if and only iff(xy)=xf(y)+f(x)ythat is, the notions of first order derivations and derivations coincide.Let R be a commutative ring and d R → R be a derivation. Let further n∈ℕ be arbitrary andd^0= id,andd^n= d∘ d^n-1(n∈ℕ).Then the mapping d^n R→ R is a derivation of order n. To see this, we will use the following formula d^k(xy)= ∑_i=0^kkid^i(x)d^k-i(y) (x, y∈ R),which is valid for any k∈ℕ, and can be proved by induction on k. For n=1 the above statement automatically holds, since the notion of first order derivations and that of derivations coincide. Let us assume that there exists an n∈ℕ so that the statement is fulfilled for any k≤ n, that is,d^k∈𝒟_k(R) holds. Then d^n+1(xy)=∑_i=0^n+1n+1id^i(x)d^n+1-i(y) (x, y∈ R),yielding that d^n+1(xy)-xd^n+1(y)-yd^n+1(x)= ∑_i=1^nn+1id^i(x)d^n+1-i(y)(x, y∈ R).Thus, the only thing that has to be clarified is that the mapping B R× R→ R defined by B(x, y)= ∑_i=1^nn+1id^i(x)d^n+1-i(y) (x, y∈ R)is a bi-derivation of order n.Due to the induction hypothesis, for any k≤ n, the function d^k is a derivation of order k.Further, we also have 𝒟_k-1(R)⊂𝒟_k(R) for all k∈ℕ and due to the fact that𝒟_k(R) is an R-module, we obtain that the function B is a derivation of order n in each of its variables.Of course, 𝒟_n(R)∖𝒟_n-1(R)≠∅ does not hold in general. To see this, it isenough to bear in mind that 𝒟_n(ℤ)={ 0} for any n∈ℕ.At the same time, in case R is an integral domain with char(R)>n or char(R)=0and there is a non-identically zero derivation d R→ R, thenfor any n∈ℕ we have d^n∈𝒟_n(R) and d^n∉𝒟_n-1(R),yielding immediately that 𝒟_n(R)∖𝒟_n-1(R)≠∅. This can also be proved by induction on n. For n=1 this is automatically true, since d R→ R is a non-identically zeroderivation. Assume now that there is an n∈ℕ so that the statement holds for any k≤ n-1 and suppose to the contrary thatd^n∈𝒟_n-1(R). Then due to the definition of higher order derivations, the mapping B R× R→ R defined by B(x, y)= ∑_i=1^n-1ni d^i(x)d^n-i(y) (x, y∈ R)has to be a bi-derivation of order n-2, i.e., R ∋ x ⟼ B(x, y^∗)= ∑_i=1^n-1ni d^i(x)d^n-i(y^∗)has to be a derivation of order n-2, where y^∗∈ R is kept fixed. Clearly, d^i∈𝒟_n-2(R) holds, if i≤ n-2.Since 𝒟_n-2(R) is an R-module, from this nd(y^∗)d^n-1∈𝒟_n-2(R) would follow, which is a contradiction. §.§ Multiadditive functions Concerning multiadditive functions we follow the terminology and notations of L. Székelyhidi <cit.>, <cit.>.Székelyhidi, L.Let G, S be commutative semigroups, n∈ℕ and let A G^n→ S be a function. We say that A is n-additive if it is a homomorphism of G into S in each variable. If n=1 or n=2 then the function A is simply termed to be additiveadditive function or biadditivebiadditive function, respectively.The diagonalizationmultiadditive function!— diagonalization or trace of an n-additive function A G^n→ S is defined asA^∗(x)=A(x, …, x)(x∈ G).As a direct consequence of the definition each n-additive function A G^n→ S satisfies A(x_1, …, x_i-1, kx_i, x_i+1, …, x_n) =kA(x_1, …, x_i-1, x_i, x_i+1, …, x_n)(x_1, …, x_n∈ G)for all i=1, …, n, where k∈ℕ is arbitrary.The same identity holds for any k∈ℤ provided that G and S are groups, andfor k∈ℚ, provided that G and S are linear spaces over the rationals.For the diagonalization of A we have A^∗(kx)=k^nA^∗(x)(x∈ G). One of the most important theoretical results concerning multiadditive functions is the so-called Polarization formula, that briefly expresses that every n-additive symmetric function is uniquely determined by its diagonalization under some conditions on the domain as well as on the range. Suppose that G is a commutative semigroup and S is a commutative group. The action of the difference operator Δ on a functionf G→ S is defined by the formulaΔ_y f(x)=f(x+y)-f(x);note that the addition in the argument of the function is the operation of the semigroup G and the subtraction means the inverse of the operation of the group S. polarization formulaSuppose that G is a commutative semigroup, S is a commutative group, n∈ℕ and n≥ 1.If A G^n→ S is a symmetric, n-additive function, then for all x, y_1, …, y_m∈ G we haveΔ_y_1, …, y_mA^∗(x)= {[0 ifm>n; n!A(y_1, …, y_m) if m=n. ]. Suppose that G is a commutative semigroup, S is a commutative group, n∈ℕ and n≥ 1.If A G^n→ S is a symmetric, n-additive function, then for all x, y∈ G Δ^n_yA^∗(x)=n!A^∗(y).Let n∈ℕ, n≥ 1 and suppose that the multiplication by n! is surjective in the commutative semigroup G or injective in the commutative group S. Then for any symmetric, n-additive function A G^n→ S, A^∗≡ 0 implies thatA is identically zero, as well.The polarization formula plays the central role in the investigations of functional equations characterizing higher order derivations on a ring.This is another reason (see also Remark <ref>)why some additional restrictions on the the ring appear in the results.They essentially correspond to the conditions for the multiplication by n! in the domain as well as in the range in Lemma <ref>. § MULTIVARIATE CHARACTERIZATIONS OF HIGHER ORDER DERIVATIONS In what follows we frequently use summation with respect to the cardinality of I⊂{1, …, n+1} as I runs through the elements of the power set 2^{1, …, n+1}, where n∈ℕ.As another technical notation, we introduce the hat operator v to delete arguments from multivariate expressions.Let R be a commutative ring and consider the action of a second order derivation A∈𝒟_2(R)on the product of three independent variables as a motivation of the forthcoming results: A(x_1x_2x_3)-x_1A(x_2x_3)-A(x_1)x_2x_3=B(x_1, x_2x_3)=x_2B(x_1, x_3)+B(x_1, x_2)x_3 =x_2(A(x_1x_3)-x_1A(x_3)-x_3A(x_1))+(A(x_1x_2)-x_1A(x_2)-A(x_1)x_2 )x_3. In general∑_i=0^n(-1)^i∑_card(I)=i(∏_j∈ Ix_j)· A(∏_k∈{1, …, n+1}∖ Ix_k)=0(x_1, …, x_n+1∈ R)as a simple inductive argument shows. Conversely suppose that equation (<ref>) is satisfied and let us define the (symmetric) biadditive mapping byB(x,y)=A(xy)-A(x)y-xA(y).An easy direct computation shows that A satisfies equation (<ref>) with n∈ℕif and only if B satisfies equation (<ref>) for each variablewith n-1∈ℕ, where n≥ 1.By a simple inductive argument we can formulate the following result.Let A R→ R be an additive mapping, where R is a commutative ring,n∈ℕ and n≥ 1. A∈𝒟_n(R) if and only if∑_i=0^n(-1)^i∑_card(I)=i(∏_j∈ Ix_j)· A(∏_k∈{1, …, n+1}∖ Ix_k)=0(x_1, …, x_n+1∈ R).As a generalization of the previous result we admit more general coefficients in equation (<ref>). Some additional requirements for the ring R should be also formulated. A commutative unitary ring R is called linear if it is a linear space over the field of rationals. It can be easily seen that a linear ring admits the multiplicationof the ring elements by fractions such as 1/2, …, 1/n, ….Using the linear space properties the field of the rationals ℚ can be isomorphically embedded into R as a subring. Therefore each element1, 2=1+1, …, 𝐧=1+…+1_n-timesis invertible, where 1 is the unity of the ring, n∈ℕ and n≥ 1.Hence the multiplication by n! is injective in R as the domain of the higher order derivations (cf. Lemma <ref>).Some typical examples for linear rings: fields, rings formed by matrices over a field, polynomial rings over a field.Another natural candidates are the integral domains.Since each cancellative semigroup G can be isomorphically embedded into a group we have that each integration domain Rcan be isomorphically embedded into a field 𝔽 as a subring. In the sense of Definition <ref> we can take 𝔽as the range of derivations on the subring R.Such an extension of R provides the multiplication by n! to be obviously surjective in the range 𝔽,assuming that the characteristic is zero (cf. Lemma <ref>).In what follows we do not use special notation for the unity of the ring. The meaning of 1 depends on the context as usual.Let A R→ R⊂𝔽 be an additive mapping satisfying A(1)=0, where R is an integral domain with unity, 𝔽 is its embedding field, assuming that char(𝔽)=0 andn∈ℕ. A∈𝒟_n(R) if and only if there exist constants a_1, …, a_n+1∈𝔽, not all zero, such that∑_i=0^na_n+1-in+1i∑_card(I)=i(∏_j∈ Ix_j)· A(∏_k∈{1, …, n+1}∖ Ix_k)=0,where x_1, …, x_n+1∈ R.Especially, ∑_i=1^n+1ia_i=0 provided that A is a not identically zero solution. If A∈𝒟_n(R) then we have equation (<ref>) with a_n+1-i=(-1)^in+1i(i=0, …, n);none of the constants a_1, …, a_n+1 is zero and ∑_i=1^n+1ia_i=0.Conversely, let n∈ℕ, n≥ 1 and A R→ R⊂𝔽 be an additive function such that A(1)=0.Suppose that equation (<ref>) holds for all x_1, …, x_n+1∈ R. Substituting x_1=… =x_n=1andx_n+1=x∈ Rit follows that (∑_i=1^n+1ia_i)· A(x)=0(x∈ R),i.e. ∑_i=1^n+1ia_i=0 provided that A is a not identically zero solution of equation (<ref>). If n=1, then it takes the form2 a_2A(xy)+a_1(xA(y)+yA(x))=0 (x, y∈ R).Since 2a_2+a_1=0, it follows that a_2≠ 0. Otherwise 0=a_2=a_1 which is a contradiction. Thereforea_2A(xy)-a_2(xA(y)+yA(x))=0 (x, y∈ R)and A(xy)=xA(y)+yA(x)(x, y∈ R).This means that A∈𝒟_1(R), i.e. the statement holds for n=1. The inductive argument can be completed as follows. Taking x_n+1=1 equation (<ref>) gives that 0=∑_i=1^na_n+1-in+1i∑_n+1∈ I∑_card(I)=i(∏_j∈ I∖{n+1}x_j)· A(∏_k∈{1, …, n}∖ (I∖{n+1})x_k) +∑_i=0^n-1a_n+1-in+1i∑_n+1 ∉ I∑_card(I)=i(∏_j∈ Ix_j)· A(∏_k∈{1, …, n}∖ Ix_k)+a_1n+1n(∏_j∈{1, …, n}x_j) · A(1) =∑_i=0^n-1( a_n+1-(i+1)n+1i+1+ a_n+1-in+1i) ∑_card(I)=i(∏_j∈ Ix_j)· A(∏_k∈{1, …, n}∖ Ix_k) =∑_i=0^n-1ã_(n-1)+1-in-1+1i∑_card(I)=i(∏_j∈ Ix_j)· A(∏_k∈{1, …, (n-1)+1}∖ Ix_k)(x_1, …, x_n∈ R)because A(1)=0. We have that A∈𝒟_n-1(R) ⊂𝒟_n(R) (inductive hypothesis) or all the coefficients are zero,that is, ã_(n-1)+1-i=n-1+1i(a_n+1-(i+1)n+1i+1+a_n+1-in+1i)=0 (0≤ i≤ n-1)and, consequently,(n+1-i)a_n+1-i+(i+1)a_n+1-(i+1)=0(0≤ i≤ n-1).Therefore a_n+1-(i+1)=(-1)^i+1n+1i+1a_n+1(0≤ i≤ n-1).Substituting into (<ref>) a_n+1·∑_i=0^n(-1)^i∑_card(I)=i(∏_j∈ Ix_j)· A(∏_k∈{1, …, n+1}∖ Ix_k)=0 (x∈ R).This means that A∈𝒟_n(R) since a_n+1 can not be zero due to the recursive formula (<ref>) and the condition to provide the existence of a nonzero element among the constants a_1, …, a_n+1. The condition for R to be an integral domain with unity embeddable into a field 𝔽 with characteristic zero, is used only in the last step of the proof because of thedivision by a_n+1. If the coefficients are supposed to be rationals, then the previous statement holds for a linear commutative ring with unity, see the next theorem.Let f_1, …, f_n+1 R→ R be additive functions such that f_i(1)=0 for all i=1, …, n+1, where R is a linear commutative ring with unity, n∈ℕ and n≥ 1. Functional equation∑_i=0^n1n+1i∑_card(I)=i(∏_j∈ Ix_j)· f_n+1-i(∏_k∈{1, …, n+1}∖ Ix_k) =0 (x_1, …, x_n+1∈ R)holds if and only if f_n+1-i=(-1)^i∑_k=0^in+1-i+kkD_n-i+k(i=0, …, n),where for all possible indices i, we have D_i∈𝒟_i(R).If n=1 then the equation takes the form2f_2(x_1x_2)+x_1f_1(x_2)+x_2f_1(x_1)=0.Substituting x_1=x and x_2=1 we have that2f_2(x)+f_1(x)=0and, consequently,f_2(x_1x_2)-x_1f_2(x_2)-x_2f_2(x_1)=0.This means that f_2=D_1∈𝒟_1(R) and f_1=-D_0-2D_1, where D_0∈𝒟_0(R) is the identically zero function. The converse of the statement is clear under the choice f_2=D_1 and f_1=-D_0-2D_1. The inductive argument is the same as in the proof of the previous theorem by the formal identification f_n+1-i=a_n+1-iA. We have 0=∑_i=0^n-11n-1+1i∑_card(I)=i(∏_j∈ Ix_j)·f̃_(n-1)+1-i(∏_k∈{1, …, n}∖ Ix_k),wheref̃_(n-1)+1-i=(i+1)f_n+1-(i+1)+ (n+1-i)f_n+1-i (0≤ i ≤ n-1).Using the inductive hypothesis (n+1-i)f_n+1-i+(i+1)f_n+1-(i+1)=(-1)^i∑_k=0^in-i+kkD_(n-1)-i+k,where D_i∈𝒟_i(R) for all possible indices. Especially, all the unknown functions f_1, …, f_n can beexpressed in terms of f_n+1 and D_0, …, D_n-1 in a recursive way:f_n+1-(i+1)= (-1)^i{(∑_k=0^in-ki-kD_n-(k+1)k+1)-n+1i+1f_n+1},where i=0, …, n-1. Rescaling the indices, f_n+1-i= f_n+1-(i-1+1) =(-1)^i-1{(∑_k=0^i-1n-ki-1-kD_n-(k+1)k+1)-n+1if_n+1},where i=1, …, n. Substituting into equation (<ref>) ∑_i=0^n(-1)^i∑_card(I)=i(∏_j∈ Ix_j)· f_n+1(∏_k∈{1, …, n+1}∖ Ix_k)=0,because the term containing D_l is of the form ∑_i=n-l^n(-1)^i-1/n+1il+1i-n+l∑_card(I)=i(∏_j∈ Ix_j) ·D_ln-l(∏_k∈{1, …, n+1}∖ Ix_k) = 1n+1n-l∑_i=n-l^n(-1)^i-1in-l∑_card(I)=i(∏_j∈ Ix_j) ·D_ln-l(∏_k∈{1, …, n+1}∖ Ix_k) = (-1)^n-l-1n+1n-l∑_m=0^l(-1)^mm+n-ln-l ·∑_card(I)=n-l+m(∏_j∈ Ix_j)·D_ln-l(∏_k∈{1, …, n+1}∖ Ix_k) =(-1)^n-l-1n+1n-l∑_1≤ j_1 < … < j_l+1≤ n+1(∏_k∈{ 1, …, n+1}∖{j_1, …, j_l+1}x_k)·∑_m=0^l(-1)^m∑_card(J)=m(∏_j∈ Jx_j)·D_ln-l(∏_k∈{j_1, …, j_l+1}∖ Jx_k)=0,due to Theorem <ref>. Finally, let D_n= f_n+1and D_n-(k+1)=-D_n-(k+1)k+1(k=0, …, n-1).Taking m=i-1-k in formula (<ref>) it follows thatf_n+1-i=(-1)^i-1{(∑_m=0^i-1n+1-i+mmD_n-i+mi-m)-n+1if_n+1}=(-1)^i{(∑_m=0^i-1n+1-i+mmD_n-i+m)+n+1if_n+1}=(-1)^i∑_m=0^in+1-i+mmD_n-i+mas had to be proved.The converse statement is a straightforward calculation. Theorem <ref> can be also stated in case of integral domains provided that the range of the functionsf_1, …, f_n+1 is extended to a field of characteristic zero, containing R as a subring; see Remark <ref>.The proof works without any essential modification.§ UNIVARIATE CHARACTERIZATIONS OF HIGHER ORDER DERIVATIONS Each multivariate characterization implies an univariate characterization ofthe higher order derivations under some mild conditions on the ring R due to Lemma <ref>.The ring R is supposed to be a linear commutative, unitary ring or an integral domain embeddable to a field of characteristic zero;see also Remark <ref>.The following results can be found in <cit.>, <cit.> but the proofs are essentially different.We present them as direct consequences of the multivariate characterizations by taking the diagonalization of the formulas.This provides unified arguments of specific results but we can also use the idea as ageneral method to solve functional equations of the form (<ref>); see Examples I and Examples II. Let A R→ R be an additive mapping, where R is a linear commutative, unitary ring or an integral domain with unity, embeddable to a fieldof characteristic zero and n∈ℕ and n≥ 1. Then A∈𝒟_n(R) if and only if∑_i=0^n(-1)^in+1ix^iA(x^n+1-i)=0(x∈ R).Let A R→ R⊂𝔽 be an additive mapping satisfying A(1)=0, where R is an integral domain with unity,𝔽 is its embedding field with char(𝔽)=0,n∈ℕ and n≥ 1.A∈𝒟_n(R) if and only if there exist constants a_1, …, a_n+1∈ F, not all zero, such that∑_i=0^na_n+1-ix^iA(x^n+1-i)=0 for any x∈ R. Especially, ∑_i=1^n+1ia_i=0 provided that A is a not identically zero solution.As an application of Theorem <ref> we are going to give the general form of the solutions of functional equations of the form ∑_k=1^nx^p_kf_k(x^q_k)=0(x∈ R),where n∈ℕ, n≥ 1, p_k, q_k are given nonnegative integers for all k=1, …, n and f_1, …, f_n R→ R are additive functions on the ring R.The problem is motivated by B. Ebanks' paper <cit.>, <cit.> although there are some earlier relevant results.For example the case n=2 is an easy consequence of <cit.>. Using the homogeneity of additive functions it can be easily seen that equation (<ref>) can be assumed to be of constant degree of homogeneity:p_k+q_k=l(k=1, …, n);see <cit.>. In other words collecting the addends of equation (<ref>)of the same degree of homogeneity we discuss the vanishing of the left hand side term by term.To formulate the multivariate version of the functional equation let us define the mappingΦ(x_1, …, x_l)=∑_k=1^n1lp_k∑_card(I)=p_k(∏_j∈{1, …, l}∖ Ix_j)·f_k(∏_i∈ Ix_i)(x_1, …, x_l∈ R),where the summation is taken for all subsets I of cardinality p_k of the index set{1, 2,…, l}. Due to the additivity of the functions f_1, …, f_n, the mapping Φ R^l→ Ris a symmetric l-additive function with vanishing traceΦ^∗(x)=Φ(x, …, x)=∑_k=1^nx^p_kf_k(x^q_k) =0(x∈ R)and we can conclude the equivalence of the following statements by Lemma <ref>:* The functions f_1, …, f_n fulfill equation (<ref>).* For any x_1, …, x_l∈ R ∑_k=1^n1lp_k∑_card(I)=p_k(∏_j∈{1, …, l}∖ Ix_j) · f_k(∏_i∈ Ix_i)=0.In what follows we summarize all the simplifications we use to solve equation (<ref>).(𝒞0) R is a linear commutative, unitary ring or an integral domain with unity embeddable to a field of characteristic zero. (𝒞1) Each addend has the same degree of homogeneity, i.e. p_k+q_k=l for all k=1, …, n. If p_i=p_j for some different indices i≠ j∈{1, …, n} then q_i=q_j by the constancy of the degree of homogeneity and we can write that x^p_if_i(x^q_i)+x^p_jf_j(x^q_j)=x^p_if(x^q_i),where f=f_i+f_j. Therefore the number of the unknown functions has been reduced. If q_i=0 then, by choosing j≠ i, we can write equation (<ref>) into the form ∑_ν∈{1, …, n}∖{ i, j}x^p_νf_ν(x^q_ν)+x^p_jf(x^q_j)=0,where f(x)= f_i(1) x+f_j(x) and the number of the unknown functions has been reduced again. Without loss of the generality we can suppose that(𝒞2) p_1, …, p_n are pairwise different nonnegative integers,q_1, …, q_n are positive, pairwise different integers. Especially, l≥ n because of (𝒞1). By multiplying equation (<ref>) with x if necessary we have that l≥ n+1.(𝒞3) Moreover, for any k=1, …, n, condition f_k(1)=0 can be assumed. Otherwise, let us introduce the functionsf_k(x)=f_k(x)-f_k(1) x(x∈ R).They are obviously additive and, for any x∈ R∑_k=1^nx^p_kf_k(x^q_k)=∑_k=1^nx^p_k[f_k(x^q_k)-f_k(1) x^q_k]=∑_k=1^nx^p_kf_k(x^q_k)-x^l∑_k=1^nf_k(1)=0because ∑_k=1^nf_k(1)=0 due to (<ref>).Assuming conditions (𝒞0), (𝒞1), (𝒞2) and (𝒞3) it is enough to investigate functional equation ∑_i=0^nx^if_n+1-i(x^n+1-i)=0(x∈ℝ).For those exponents that do not appear in the corresponding homogeneous term of the original equation we assign the identically zero function (that is clearly additive).Let f_1, …, f_n+1 R→ R be additive functions such that f_i(1)=0 for all i=1, …, n+1, where R is a linear commutative ring with unity, n∈ℕ and n≥ 1. Functional equation∑_i=0^nx^if_n+1-i(x^n+1-i)=0(x∈ R)holds, if and only if f_n+1-i=(-1)^i∑_k=0^in+1-i+kkD_n-i+k(i=0, …, n),where for all possible indices i, we have D_i∈𝒟_i(R). The statement is a direct consequence of Theorem <ref> and Lemma <ref>. Corollary <ref> can be also stated in case of integral domains provided that the range of the functions f_1, …, f_n+1is extended to a field having characteristic zero,containing R as a subring; see Remark <ref>.The proof is working without any essential modification. §.§ Examples I: equations without missing powers As an application of the results presented above, we will show some examples. Let f R→ R be a non-identically zero additive function and assume thatf(x^3)+xf(x^2)-2x^2f(x)=0is fulfilled for any x∈ R. Then the function A(x)=f(x)-f(1) x satisfies the conditions of Corollary <ref>. Therefore there exists a derivation D_2∈𝒟_2(R) such thatf(x)=D_2(x)+f(1) x (x∈ R).Moreover, if we define a_3=1, a_2= 1 and a_1=-2, then 3a_3+2a_2+a_1≠ 0, i.e. D_2≡ 0in the sense of Corollary <ref>. The solution is f(x)=f(1)x, where f(1)≠ 0 and x∈ R. Let f R→ R be an additive function and assume thatf(x^5)-5xf(x^4)+10x^2f(x^3)-10x^3f(x^2)+5x^4f(x)=0(x∈ R).Let a_5-i=(-1)^i5i(i=0, 1, 2, 3, 4).Then we have ∑_i=0^4a_5-ix^if(x^5-i)=0 (x∈ R).Thus there exists a fourth order derivation D_4∈𝒟_4(R) such that f(x)=D_4(x)+f(1)x (x∈ R).Since f(1)=0, the solution is f(x)=D_4(x), where x∈ R.§.§ Examples II: equations with missing powers The following examples illustrate how to use the method of free variables in the solution offunctional equations with missing powers instead of the direct application of Theorem <ref>/Corollary <ref>.The key step is the substitution of value 1 as many times as the number of the missing powers(due to the symmetry of the variables there is no need to specify the positions for the substitution).This provides the reduction of the homogeneity degree of the functional equation to avoid formal (identically zero) terms in the solution.It is given in a more effective and subtle way. The method obviously shows how the number of the nonzero coefficients isrelated to the maximal order of the solution.In <cit.> it was shown that if the number of nonzero coefficients a_i ∈ R is m,then the solution of ∑_i=1^n a_i x^p_if(x^q_i)=0is a derivation of order at most m-1.Assume that we are given two additive functionsf, g R→ R such that f(1)=g(1)=0 and f(x^3)+x^2g(x)=0(x∈ R).If we define the functions f_1, f_2, f_3 R→ R as f_3=f, f_2=0 and f_1=g then, by using Corollary <ref> with n=2, it follows that [f_3=D_2;f_2=-3D_2-D_1;f_1= 3D_2+2D_1+D_0, ]where D_i∈𝒟_i(R), if i=0, 1, 2. It is a direct application of Corollary <ref> used by B. Ebanks in <cit.>, <cit.>.Another way of the solution is to formulate the multivariate version of the equation in the first step: 3f(x_1x_2x_3)+x_1x_2g(x_3)+x_1x_3g(x_2)+x_2x_3g(x_1)=0(x_1, x_2, x_3∈ R).If x_1=x_2=x and x_3=1, then we get that 3f(x^2)+2xg(x)=0 (x∈ R).Applying Corollary <ref> with n=1,f_2=3f and f_1=2g, it follows that[f_2=D_1; 2D_1+D_0+f_1= 0, ]where D_i∈𝒟_i(R) for all i=0, 1. Therefore 3f=D_1 and g=-D_1, where D_1∈𝒟_1(R). Let f, g, h R→ R be additive functions so that f(1)=g(1)=h(1)=0. Furthermore, assume thatf(x^5)+xg(x^4)+x^4h(x)=0is fulfilled for all x∈ R. To determine the functions f, g, h, we will show two ways. The first is a direct application of the results above used byB. Ebanks in <cit.>, <cit.>.Let us define the functions f_1, f_2, f_3, f_4, f_5 R→ R as f_1= h, f_2=0, f_3=0, f_4=g and f_5=f.Then the equation takes the form ∑_i=0^4x^if_5-i(x^5-i)=0 (x∈ R).and, by Corollary <ref>, for any i=0, 1, 2, 3, 4f_5-i= (-1)^i∑_k=0^i5-i+kkD_4-i+kholds, that is [f_5-D_4=0;5 D_4+f_4+D_3=0;-10 D_4-4 D_3+f_3-D_2=0; 10 D_4+6 D_3+3 D_2+f_2+D_1=0; -5 D_4-4 D_3-3 D_2-2 D_1+f_1-D_0= 0, ]where for all i=0, 1, 2, 3, 4 we have D_i∈𝒟_i(R).Bearing in mind the above notations, for the functions f, g and h this yields that [f-D_4=0;5 D_4+g+D_3=0;-10 D_4-4 D_3-D_2=0; 10 D_4+6 D_3+3 D_2+D_1=0; -5 D_4-4 D_3-3 D_2-2 D_1+h= 0. ] The second way gives a much more precise form of the unknown functions by formulating the multivariate version of the equation in the first step: 5 f(x_1 x_2 x_3 x_4 x_5) +g(x_2 x_3 x_4 x_5) x_1 + g(x_1 x_3 x_4 x_5) x_2+ g(x_2 x_1 x_4 x_5) x_3 + g(x_2 x_3 x_1 x_5) x_4+ g(x_2 x_3 x_4 x_1) x_5+x_2 x_3 x_4 x_5 h(x_1)+ x_1 x_3 x_4 x_5 h(x_2) + x_2 x_1 x_4 x_5 h(x_3)+ x_2 x_3 x_1 x_5 h(x_4) + x_2 x_3 x_4 x_1 h(x_5) =0,where x_1, x_2, x_3, x_4, x_5∈ R. Substituting x_1=x_2=x_3= x and x_4=x_5=1 we get that 5f(x^3)+2g(x^3)+3xg(x^2)+3x^2h(x)=0.Taking f_3= 5f+2g, f_2= 3g and f_1= 3h it follows thatf_3(x^3)+xf_2(x^2)+x^2f_1(x)=0 (x∈ R)and, by Corollary <ref>, f_3= D_2, f_2= -D_1-3D_2, f_1=2D_1+3D_2,where D_i∈𝒟_i(R) for all i=1, 2. This means that [ 15 f=2D_1+9D_2; 3g=-D_1-3D_2; 3h= 2D_1+3D_2, ]where D_i∈𝒟_i(R) for all i=1, 2. § THE APPLICATION OF THE SPECTRAL SYNTHESIS IN THE SOLUTION OF LINEAR FUNCTIONAL EQUATIONS In what follows we present another approach to the problem of linear functional equations characterizing derivations among additive mappings in the special case of a finitely generated field K over the field ℚ of rationals as the domain R of the equation, i.e. ℚ⊂ K=ℚ(x_1, …, x_m) ⊂ℂ, where m∈ℕ and ℂ denotes the field of complex numbers. The linearity of the functional equation means that the solutions form a vector space over ℂ. The idea of using spectral synthesis to find additive solutions of linear functional equationsis natural due to the fundamental work <cit.>.The key result says that spectral synthesis holds in any translation invariant closed linear subspaceformed by additive mappings on a finitely generated subfield K⊂ℂ.Therefore such a subspace is spanned by so-called exponential monomials which can be given in terms of automorphisms ofℂ and differential operators (higher order derivations), see also <cit.> and <cit.>. §.§ Basic theoretical factsLet (G, *) be an Abelian group. By a variety V on G we mean a translation invariant closed linear subspace of ℂ^G, where ℂ^G denotes the space of complex valued functions defined on G.The space of functions is equipped with the product topology. The translation invariance of the linear subspace provides that f_gG →ℂ, f_g(x)=f(g*x) is an element of V for any f∈ V and g∈ G. An additive mapping is a homomorphism of G into the additive group of ℂ. The so-called polynomials are the elements of the algebra generated by the additive and constant functions. An exponential mapping is a nonzero (and, consequently, injective) homomorphism of G into the multiplicative group of ℂ, i .e. m∈ℂ^G such that m(x*y)=m(x)· m(y). An exponential monomial is the product of an exponential and a polynomial function. The finite sums of exponential monomials are called polynomial-exponentials. If a variety V is spanned by exponential monomials then we say that spectral synthesis holds in V. Ifspectral synthesis holds in every variety on G then spectral synthesis holds on G.Especially, spectral analysis holds on G, i.e. every nontrivial variety contains an exponential;see Lemma 2.1 in <cit.>.To formulate the key result of <cit.> let G:=K^* (the multiplicative group of K) and consider the variety V_a on K^* consisting of the restriction of additive functions on K (as an additive group) to K^*, i.e. V_a={A |_K^* |A(x+y)=A(x)+A(y),where x and y∈ K}.It can be easily seen that if A is an additive function then its translate A_c(x)=A(cx) with respect to the multiplication by c∈ K^* is also additive. Theorem 4.3 in<cit.> states that if the transcendence degree of K over ℚ isfinite[If K is finitely generated over ℚ then it is automatically satisfied.]then spectral synthesis holds in every variety V contained in V_a.By Theorem 3.4 in <cit.>, the polynomials in Vcorrespond to mappings of the form D(x)/x (x∈ K^*),where D is a differential operator on K. A differential operator means the (complex) linear combination offinitely many mappings of the form d_1 ∘…∘ d_k, where d_1, …, d_k are derivations on K.If k=0, then this expression is by convention the identity function.The exponentials in V satisfy both m(x+y)=m(x)+m(y) and m(x· y)=m(x)· m(y), where x, y∈ K and m(0)=0.Extending m to an automorphism of ℂ (see Lemma 4.1 in <cit.>),a special expression for the elements (exponential monomials) spanning a variety V contained in V_a can be given:they are of the form φ∘ D, where φ is the extension of an exponentialm∈ V to an automorphism of ℂ and D is a differential operator on K; see Theorem 4.2 in <cit.>.§.§ Spectral synthesis in the variety generated by the solutions of equation (<ref>)Let n∈ℕ and consider an additive mapping A K→ℂ satisfying equation∑_i=0^n(-1)^in+1ix^iA(x^n+1-i)=0(x∈ K).First of all observe that the solutions of functional equation (<ref>) form a linear subspace.To get some information about the translates of the form A_c(x)=A(cx) of the solutions we have to set the variable x free by using a symmetrization process:Φ(x_1, …, x_n+1) =A(x_1⋯ x_n+1)-∑_1≤ i≤ n+1x_iA(x_1⋯x̂_i⋯ x_n+1)+∑_i≤ i< j≤ n+1x_ix_jA(x_1⋯x̂_i⋯x̂_j⋯ x_n+1)-⋯ +(-1)^n∑_1≤ i≤ n+1x_1⋯x̂_i⋯ x_n+1A(x_i) (x_1, …, x_n+1∈ K).Due to equation (<ref>), the diagonal Φ^∗(x)=Φ(x,…,x) is identically zero. In the sense of Lemma <ref>, the mapping Φ is also identically zero. By some direct computations Φ(x, …, x, cx)=A(c x^n+1)-n1xA(c x^n)-c xA(x^n)+n2x^2A(c x^n-1)+n1c x^2A(x^n-1)- n3x^3A(c x^n-2)-n2c x^3A(x^n-2) +…+(-1)^n x^nA(c x)+(-1)^nnn-1c x^nA(x), Φ(x, …, x, c)=A(c x^n)-n1x A(c x^n-1)-c A(x^n)+n2x^2A(c x^n-2)+n1c xA(x^n-1)- n3x^3A(c x^n-3)-n2c x^2A(x^n-2) +…+(-1)^n x^nA(c)+(-1)^nnn-1c x^n-1A(x)and, consequently, for any c∈ K^*Φ(x, …, x, cx)-xΦ(x, …, x, c)= ∑_i=0^n+1(-1)^in+1ix^iA(c x^n+1-i) (x∈ K), i.e. the translation invariant linear subspace generated by the solutions of equation (<ref>) can be described by the family of equations ∑_i=0^n+1(-1)^in+1ix^iA(c x^n+1-i)=0(x∈ K, c∈ K^* ). Let c∈ K^* and x∈ K be given and suppose that f K→ℂ is the limit function[Since K is countable, the limit can be taken in a pointwise sense.] of the sequence A_l of solutions of equation (<ref>), that is for any ε>0 we have that|A_l(c x^n+1-i)-f(c x^n+1-i) | < ε (i=0, …, n+1)provided that l is large enough. Then | ∑_i=0^n+1(-1)^in+1ix^if(c x^n+1-i)| = |∑_i=0^n+1(-1)^in+1ix^if(c x^n+1-i)-∑_i=0^n+1(-1)^in+1ix^iA_l(c x^n+1-i)| ≤∑_i=0^n+1n+1i|x|^i |f(c x^n+1-i)-A_l(c x^n+1-i) |=ε(1+|x|)^n+1. Therefore the space of solutions is a translation invariant closed linear subspace, i.e. it is a variety in V_a.Using the exponential element m we have that 0=∑_i=0^n+1(-1)^in+1ix^im(c x^n+1-i)=m(c)∑_i=0^n+1(-1)^in+1ix^im^n+1-i(x)= m(c)(m(x)-x)^n+1and, consequently, the exponential element must be the identity on the finitely generated field over ℚ.This means that the space of the solutions is spanned by differential operators.§.§ Spectral synthesis in the variety generated by the solutions of equation (<ref>) Let n∈ℕ and consider an additive mapping A K→ℂ satisfying equation∑_i=0^na_n+1-ix^iA(x^n+1-i)=0(x∈ K).First of all observe that the solutions of functional equation (<ref>) form a linear subspace.To get some information about the translates of the form A_c(x)=A(c x) of the solutions we have to set the variable xfree by using a symmetrization process:Φ(x_1, …, x_n+1)=∑_i=0^na_n+1-i1n+1i∑_card(I)=i(∏_j∈ Ix_j)· A(∏_k∈{1, …, n+1}∖ Ix_k)(x_1, …, x_n+1∈ K).Then Φ^∗(x)=Φ(x, …, x)= ∑_i=0^na_n+1-ix^iA(x^n+1-i)=0(x∈ K)because of equation (<ref>). In the sense of Lemma <ref>, Φ is also identically zero. By some direct computations Φ(x, …, x, cx)-xΦ(x, …, x, c) =1n+1∑_i=0^n (n+1-i) a_n+1-i(x^iA(c x^n+1-i)-x^i+1A(c x^n-i)),that is the translation invariant linear subspace generated by the solutions of equation (<ref>) can be described by the family of equations ∑_i=0^n (n+1-i) a_n+1-i(x^i A(c x^n+1-i)-x^i+1A(c x^n-i))=0 (x∈ K, c∈ K^* ).By the same way as above we can prove that the space of the solutions of equation (<ref>) is a variety in V_a. Using the exponential element m we have that 0=m(c)(m(x)-x)∑_j=1^n+1 ja_j m^j-1(x)x^n-(j-1),where j=n+1-i, i=0, …, n. If m(x)≠ x for some x∈ K^* then we can write that0=∑_j=1^n+1 ja_j (m(x)/x)^j-1.Therefore m(x)/x is the root of the polynomial ∑_j=1^n+1 ja_j t^j-1 and it has only finitely many different values. This is obviously a contradiction because for any x∈ K^*, the functionr∈ℚ⟼m(x+r)/x+r=m(x)+r/x+rprovides infinitely many different values unless m(x)=x; note that m(x)≠ x is equivalent to m(x+r)≠ x+r for any rational number r∈ℚ. Therefore we can conclude that the exponential element must be the identity on any finitely generated field over ℚ and the space of the solutions is spanned by differential operators. §.§ The solutions of equation (<ref>) in case of n=3 Let n∈ℕ, n≥ 1 be arbitrary and assume that the additive functionsf_1, …, f_n+1 K→ℂ satisfy equation ∑_i=0^nx^if_n+1-i(x^n+1-i)=0(x∈ K)under the initial conditions f_i(1)=0 for all i=1, …, n+1. In the first step we set the variable x free by using a symmetrization process:Φ(x_1, …, x_n+1)=∑_i=0^n1n+1i∑_card(I)=i(∏_j∈ Ix_j) · f_n+1-i(∏_k∈{1, …, n+1}∖ Ix_k) (x_1, …, x_n+1∈ K).ThenΦ^∗(x)=Φ(x, …, x)= ∑_i=0^nx^if_n+1-i(x^n+1-i)=0 (x∈ K)because of equation (<ref>). In the sense of Lemma <ref>, Φ is also identically zero.We are going to investigate the explicit case of n=3, that is, equation∑_i=0^3x^if_4-i(x^4-i)=0(x∈ K).By some direct computations Φ(x,x,x,cx) =f_4(cx^4)+34xf_3(cx^3)+14cxf_3(x^3)+12x^2f_2(cx^2) +12cx^2f_2(x^2)+14 x^3f_1(cx)+34cx^3f_1(x), Φ(x,x,x,c) =f_4(cx^3)+34xf_3(cx^2)+14cf_3(x^3)+12x^2f_2(cx) +12cxf_2(x^2)+14 x^3f_1(c)+34cx^2f_1(x)and, consequently, for any c∈ K^*Φ(x, x,x, cx)-xΦ(x, x,x, c) =f_4(cx^4)+x(34f_3-f_4 )(cx^3) +x^2(12f_2-34f_3 )(cx^2) +x^3(14f_1-12f_2 )(cx)-14x^4f_1(c), where x∈ K, i.e. we can formulate the family of equations g_4(cx^4)+xg_3(cx^3)+x^2g_2(cx^2)+x^3g_1(cx)=x^4(g_1+g_2+g_3+g_4)(c) (x∈ K),where c runs through the elements of K^* and[ [ g_1; g_2; g_3; g_4; ]]=[ [1/4 -1/200;01/2 -3/40;003/4 -1;0001;]][ [ f_1; f_2; f_3; f_4 ]].The inverse formulas are f_4=g_4, f_3=43(g_3+g_4),f_2=2(g_2+g_3+g_4), f_1=4( g_1+g_2+g_3+g_4).Taking c=1 we have that g_4(x^4)+xg_3(x^3)+x^2g_2(x^2)+x^3g_1(x)=0 (x∈ K)because of the initial conditions f_1(1)=f_2(1)=f_3(1)=f_4(1)=0. Therefore the space of the solutions is invariant under the action of the linear transformation represented by the matrixM:=[ [1/4 -1/200;01/2 -3/40;003/4 -1;0001;]]=1/4[ [1 -200;02 -30;003 -4;0004;]].As a MAPLE computation shows [ [ 4^-n-2^1-n+2 4^-n3^1+n4^-n-6 2^-n+3 4^-n -4+12 3^n4^-n-12 2^-n+ 2^2-2 n;0 2^-n -3^1+n4^-n+3 2^ -n6-12 3^n4^-n+6 2^-n;00 3^n 4^-n-4+2^2-2 n3^n;0001 ]]and, consequently, lim_n→∞ M^n=[ [000 -4;0006;000 -4;0001;]]. Therefore f_4(x^4)-4xf_4(x^3)+6x^2f_4(x^2)-4x^3f_4(x)=0 (x∈ K) and, by Theorem <ref> (Corollary <ref>), we can conclude that f_4 is a differential operator on any finitely generated field K. By taking the difference of equations (<ref>) and (<ref>) the number of the unknown functions can be reduced: f̃_3(x^3)+xf̃_2(x^2)+x^2f̃_1(x)=0 (x∈ K), where f̃_3=f_3+4f_4, f̃_2=f_2-6f_4, f̃_1=f_1+4f_4. Repeating the process above we can conclude that f̃_3 is a differential operator on any finitely generated field K and so on. Note that it is an alternative way to prove Theorem <ref>/Corollary <ref> by using a descending process instead of the inductive argument. Acknowledgement. This paper is dedicated to the 65th birthday of Professor László Székelyhidi.The research of the first author has been supported by the Hungarian Scientific Research Fund (OTKA) Grant K 111651 and by the ÚNKP-4 New National Excellence Program of the Ministry of Human Capacities. The work of the first and the third author is also supported by the EFOP-3.6.1-16-2016-00022 project. The project is co-financed by the European Union and the European Social Fund.The second author was supported by the internal research project R-AGR-0500 ofthe University of Luxembourg and by the Hungarian Scientific Research Fund (OTKA) K 104178. plain | http://arxiv.org/abs/1709.03038v2 | {
"authors": [
"Eszter Gselmann",
"Gergely Kiss",
"Csaba Vincze"
],
"categories": [
"math.CA",
"39B50, 13N15, 43A45"
],
"primary_category": "math.CA",
"published": "20170825152133",
"title": "On functional equations characterizing derivations: methods and examples"
} |
STAT on words]Mahonian STAT on rearrangement class of wordsS. Fu]Shishuo Fu College of Mathematics and Statistics, Chongqing University, Huxi Campus LD 506, Chongqing 401331, P.R. China. [email protected]. Hua]Ting Hua College of Mathematics and Statistics, Chongqing University, Huxi Campus, Chongqing 401331, P.R. China. [email protected]. Vajnovszki]Vincent Vajnovszki LE2I, Université de Bourgogne Franche-Comté, BP 47870, 21078 Dijon Cedex, France [email protected] 2000, Babson and Steingrímsson generalized the notion of permutation patterns to the so-called vincular patterns, and they showed that many Mahonian statistics can be expressed as sums of vincular pattern occurrence statistics. STAT is one of such Mahonian statistics discoverd by them. In 2016, Kitaev and the third author introduced a words analogue of STAT and proved a joint equidistribution result involving two sextuple statistics on the whole set of words with fixed length and alphabet. Moreover, their computer experiments hinted at a finer involution on R(w),the rearrangement class of a given word w. We construct such an involution in this paper, which yields a comparable joint equidistribution between two sextuple statistics over R(w). Our involution builds on Burstein's involution and Foata-Schützenberger's involution that utilizes the celebrated RSK algorithm. [ [ December 30, 2023 ===================== § INTRODUCTION An occurrence of a classical pattern p in a permutation π is a subsequence of π that is order-isomorphic to p, i.e., that has the same pairwise comparisons as p. For example, 41253 has two occurrences of the pattern 3142 in its subsequences 4153 and 4253. The study of (classical) permutation patterns have mostly been with respect to avoidance, i.e., on the enumeration of permutations in the symmetric group _n that have no occurrences of the pattern(s) in question. The tone is generally believed to be set in 1968, when Knuth published Volume one of The Art of Computer Programming <cit.>. It was observed in <cit.> that the number of 231-avoiding permutations are enumerated by the Catalan numbers. See <cit.> and the references therein for a comprehensive introduction to patterns in permutations.In 2000, Babson and Steingrímsson <cit.> traced out a new line of research by generalizing classical patterns to what are now known as vincular patterns. In an occurrence of a vincular pattern, some letters of that order-isomorphic subsequence may be required to be adjacent in the permutation, and this is done by underscoring the adjacent letters in the expression for the pattern. For our previous example, only 4153 represents an occurrence of the vincular pattern 3142 in 41253. As was investigated thoroughly in <cit.>, these vincular patterns turned out to be the “building blocks” in their classification of Mahonian statistics. Recall that a permutation statistic is called Mahonian precisely when it has the same distribution, on _n, as , the number of inversions. Given a vincular pattern τ and a permutation π, we denote by (τ)π the number of occurrences of the pattern τ in π. When it causes no confusion, we will even drop the parentheses. For instance, it was observed/defined in <cit.>:=21=21+312+321+231,=21+132+231+321,=21+132+231+321,=21+132+213+321.Note that the first three Mahonian statistics are well-known ones putting on a new look, while the last onewas one of those introduced in <cit.> and shown to be Mahonian.Now we state the joint equidistribution results involving STAT obtained by Burstein <cit.> on _n and by Kitaev and the third author <cit.> on [m]^n, the set of length n words over the alphabet [m]:={1,2,…,m}. These two results largely motivated this work. All undefined statistics will be introduced in the next section. For the spelling of the statistics, we follow the convention of <cit.>, i.e., all Mahonian statistics are spelled with uppercase letters, all Eulerian statistics are spelled with lowercase letters, while all the remaining ones are merely capitalized. Moreover, the italic letters are used for set-valued statistics, such as Id in Theorem <ref> and Corollary <ref>. Statistics (, , , , ) and (, , , , ) have the same joint distribution on _n for all n.Statistics (, , , , , ) and (, , , , , ) have the same joint distribution on [m]^n for all m and n. A quick look at Table <ref> reveals that when one wants to extend the above results to the rearrangement class R(w), the statisticmust be dropped. In the meantime, the involution we are going to construct actually preserves the inverse descent set Id (see Definition <ref>). The following are the main results of this paper.Statistics (, Id, , , ) and (, Id, , , ) have the same joint distribution on _n for all n. It is worth mentioning that in <cit.> it has been shown that (Id, , ) and (Id, , ) are equidistributed on _n for all n. In view of this theorem and the standard “coding/decoding” between words and permutations <cit.>, we derive the following version for words, where the statistics have to be redefined on words accordingly. We defer the details to the next section. Statistics (, Id, , , ) and (, Id, , , ) have the same joint distribution on R(w) for any word w. Since Id uniquely determinates bothand(see Definition <ref>), the next corollary is a consequence of the previous one, and it is similar to Theorem <ref>, exceptis replaced byand the equidistribution is stated on the rearrangement classes. Statistics (, , , , , ) and (, , , , , ) have the same joint distribution on R(w) for any word w.In the next section, we will first present the formal definitions of all the statistics that concern us here, and then explain the transition from Theorem <ref> to Corollary <ref>. Theorem <ref> will be proved in Section <ref>, where Foata-Schützenberger's involution via RSK correspondence will be recalled. We conclude with some further questions suggested by this work.§ PRELIMINARIES Since all the statistics that concern us here can be defined on both permutations and words, and that we have two versions of joint equidistribution results, some care needs to be taken to avoid unnecessary repetitions. We decide to define all the statistics on the rearrangement class R(w) for a word w of length n, making sure that when w is composed of distinct letters, and thus R(w) becomes essentially _n, all the statistics reduce to the original ones defined on _n. On the other hand, for the two equidistribution results we first explain how to derive Corollary <ref> from Theorem <ref> and then construct, in the next section, the key involution on _n.We consider words w=w_1w_2⋯ w_n on a totally ordered alphabet 𝒜. Without loss of generality, we can always take 𝒜 to be the interval [m]={1,2,…,m}. By rearrangement class R(w), we mean the set of all words that can be obtained by permuting the letters of w. Within each class R(w), there is a unique non-decreasing word that we denote by w. If the letters of w are distinct, then R(w) is in bijection with the symmetric group _n via the following “coding” map <cit.>.Given a word w=w_1w_2⋯ w_n, we code it by the permutation c(w)=c(w)_1c(w)_2⋯ c(w)_n∈_n uniquely defined byc(w)_i < c(w)_j if and only ifw_i<w_j,orw_i=w_j,for i<j.This gives rise to a map c:[m]^n→_n, which we call the coding map, and clearly c is asurjection if and only if m≥ n.For example, the word w=2 1 2 2 3 1 is coded by c(w)=3 1 4 5 6 2. Now we are ready to define all seven statistics encountered in the introduction. The true or false function χ is used in the definition offor ourconvenience: χ(S)=1 (resp. χ(S)=0) if the statement S is true (resp. false). Let w=w_1w_2⋯ w_n be a word. The descent set of w, D(w), the inverse descent set of w, Id(w), and a peculiar index set (needed in Section <ref>) that we call the shuffle set of w, Sh(w) are set-valued statistics defined as follows:D(w) ={i: w_i>w_i+1, 1≤ i<n},Id(w) ={c(w)_i: ∃ j<i,s.t.c(w)_j=c(w)_i+1},Sh(w) ={i: w_i≥ w_1>w_i+1 or w_i< w_1≤ w_i+1, 1≤ i<n}.We define the following seven statistics on w:w= w_1, w = ∑_j∈ D(w)1, w = ∑_j∈ Id(w)1, w= ∑_j=1^nχ(c(w)_j=c(w)_j+1+1),take c(w)_n+1=0, w=∑_j∈ D(w)j=(132+121+231+221+321+21)w, w= ∑_j∈ Id(w)j, w= (213+212+132+121+321+21)w. We note that , , , all have been extended to words in <cit.>. Some of the definitions given here may look different but essentially are equivalent to those used in <cit.>, and our formulation ofmakes it more convenient to defineon words. When the word w happens to be a permutation in _n, then c(w)=w and all the above statistics agree with their original counterparts defined on permutations. This hopefully justifies our abuse of notation, i.e., we use the same name for these statistics, both on words and permutations. Moreover, it should be routine to verify that the coding map c preserves six statistics defined above, withbeing the only exception.(, , Id, , )w=(, , Id, , )c(w).For the previous example with w=2 1 2 2 3 1 and c(w)=3 1 4 5 6 2, we see w=2= c(w), Id w={2}=Id c(w), w=6= c(w), etc. In order to deduce Corollary <ref> from Theorem <ref>, we need to pay some attention to the “decoding” map from _n back to [m]^n. In general, the preimage of a given permutation under the coding map c is not unique and therefore this decoding is not well defined. For instance, for both w=2 1 2 2 3 1 and v=2 1 2 3 3 1, we see c(w)=c(v)=3 1 4 5 6 2, with R(w)≠ R(v). This should be somewhat expected since when w contains repeated letters, we have |R(w)|<|_n|. To deal with this, we introduce the notion of compatible permutations. Given the rearrangement class R(w) for a word w of length n, we denote _R(w) the subset of _n that is the image of R(w) under the coding map c, and we call it the set of compatible permutations with respect to R(w). More precisely,_R(w)={π∈_n: π=c(v),for some v∈ R(w)}. It is clear from Definition <ref> that if we restrict c on a rearrangement class R(w), it is injective, hence it induces a bijection between R(w) and _R(w), with a unique inverse c^-1 defined on _R(w):c=c|_R(w):R(w)→_R(w),c^-1: _R(w)→ R(w), c^-1∘ c(v) =v, ∀ v∈ R(w), c∘ c^-1(π) =π, ∀π∈_R(w). The next proposition characterizes _R(w) using w and the inverse descent set. Given R(w) and suppose w=u_1⋯ u_b_1u_b_1+1⋯ u_b_2⋯ u_b_k+1⋯ u_n, where 1≤ b_1<b_2<⋯<b_k≤ n andu_1=⋯=u_b_1<u_b_1+1=⋯=u_b_2<⋯<u_b_k+1=⋯=u_n.Then π∈_R(w) if and only if Id(π)⊆{b_1, b_2,…, b_k}. Take a permutation π∈_R(w), then π=c(v) for some v∈ R(w), and consequently Id(π)⊆{b_1, b_2,…, b_k}, this takes care of the “only if” part. We finish the proof by simply noting that both _R(w) and {π∈_n:Id(π)⊆{b_1, b_2,…, b_k}} are enumerated by |R(w)|=nb_1,b_2-b_1,…,n-b_k. In the next section, we are going to construct an involution ϕ on _n such that for any π∈_n,(, Id, , , )π =(, Id, , , )ϕ(π).This proves Theorem <ref>. Now consider the function compositionϕ_R(w) =c^-1∘ϕ∘ c.Note that for any v∈ R(w), c(v)∈_R(w), and since Id(ϕ(c(v)))=Id(c(v)), we have ϕ(c(v))∈_R(w) thanks to Proposition <ref>, thus ϕ_R(w): R(w)→ R(w) is well-defined. Now we show that ϕ_R(w) is an involution on R(w) that proves Corollary <ref>.We first prove that ϕ_R(w) is indeed an involution. Let v∈ R(w), we haveϕ_R(w)^2(v) =(c^-1∘ϕ∘ c)∘(c^-1∘ϕ∘ c)v=(c^-1∘ϕ)∘(c∘ c^-1)∘(ϕ∘ c)v (<ref>) =(c^-1∘ϕ)∘(ϕ∘ c)v=(c^-1∘(ϕ∘ϕ)∘ c)v ϕ is an involution=(c^-1∘ c)v (<ref>) =v.Next we apply (<ref>) (twice) and (<ref>) to get(, Id, , )v=(, Id, , )ϕ_R(w)(v).Finally it is straightforward to check that v= ϕ_R(w)(v) since π= ϕ(π) by relation (<ref>). This immediately establishes Corollary <ref> and ends the proof.§ INVOLUTION Φ In this section we aim to construct the aforementioned map ϕ from _n into itself, and we show that i) it is an involution and ii) it satisfies (<ref>). First off, we recall an involutiondefined on _n that was first considered by Schützenberger <cit.> and further explored by Foata-Schützenberger <cit.>. It builds on two permutation symmetries, namely reversal r and complement c, as well as the famous Robinson-Schensted-Knuth (RSK) algorithm <cit.>. We collect the relevant definitions and notations below. Given a permutation π=π_1π_2⋯π_n∈_n,π^r=π_1^rπ_2^r⋯π_n^r,with π^r_i=π_n+1-i,for 1≤ i≤ n,π^c=π_1^cπ_2^c⋯π_n^c,with π^c_i=n+1-π_i,for 1≤ i≤ n,π^rc=π_1^rcπ_2^rc⋯π_n^rc,with π^rc_i=n+1-π_n+1-i,for 1≤ i≤ n,πRSK⟶ (P_π,Q_π) RSK^-1⟶π,where RSK is a bijection between _n and the set of ordered pairs of n-cell standard Young tableaux (SYT) with the same shape. Note that π↦π^r, π↦π^c and π↦π^rc=π^cr are involutions on _n.We usually call P_π the insertion tableau and Q_π the recording tableau of π. In general, for two unrelated permutations π,σ∈_n, P_π and Q_σ are unlikely to be of the same shape. However, it was noted in <cit.> that P_π and Q_π^rc are indeed of the same shape, so the following mapis a well-defined involution on _n.:_n→_nπ↦ RSK^-1(P_π,Q_π^rc)The next property ofis of great importance for our later use and is the main reason that urged us to involvein our construction of ϕ.For any π∈_n, the mapas defined above preserves the inverse descent set Id(π) and exchanges the descent set D(π) with its complement to n. In other words one has simultaneously:Id((π)) =Id(π),and D((π)) ={n-k: k∈ D(π)}. Now given a permutation π∈_n with π=k, let us denote as π^t=π_t_1⋯π_t_n-k (resp. π^b=π_b_1⋯π_b_k-1) the top (resp. bottom) subword that is composed of all the letters larger (resp. smaller) than k. We also need the shuffle set Sh(π) defined in Definition <ref>. For example, let π=5 4 6 7 3 1 9 8 2, then π^t=6 7 9 8, π^b=4 3 1 2 and Sh(π)={1,2,4,6,8}. Conversely, it should be clear how to recover the unique permutation π that corresponds to an appropriate triple (π^t,π^b,Sh(π)), so we omit the details. We are ready to describe our key map ϕ using this defining triple: (top subword, bottom subword, shuffle set).Given π∈_n, let ϕ(π) be the unique permutation that corresponds to (ϕ(π)^t,ϕ(π)^b,Sh(ϕ(π))), whereϕ(π)^t:= c^-1∘∘ c(π^t), ϕ(π)^b:= (π^b), Sh(ϕ(π)):= {1}∪{n+1-k:k≥ 2, k∈ Sh(π)}if |Sh(π)| is odd, {n+1-k:k≥ 2, k∈ Sh(π)}if |Sh(π)| is even.It should be noted that the idea to associate π with the triple (π^t,π^b,Sh(π)) is motivated by Burstein's involution p introduced in <cit.> to prove Theorem <ref>. Actually in our setting, Burstern's p can be defined using the same (<ref>), together with (<ref>) and (<ref>) replaced byp(π)^t= c^-1((c(π^t))^rc),and p(π)^b= (π^b)^rc.Moreover, in view of (<ref>) and Proposition <ref>, (<ref>) is well-defined. Thirdly, by the above definition, it follows that ϕ(π)= π. Lastly, using the fact thatis an involution, it is routine (using the associated triple) to check that ϕ^2(π)=π for any π∈_n, i.e., ϕ is an involution on _n as claimed. All remained to be done is to prove (<ref>). Namely, we need to show for every π∈_n:ϕ(π) = π,Id(ϕ(π)) =Id(π), ϕ(π) = π, ϕ(π) = π, ϕ(π) = π.We break it up into the following three lemmas and one corollary.Id(ϕ(π))=Id(π). First note that when the first letter π is fixed, Id(π) only depends on Id(π^t) and Id(π^b), and is not affected by Sh(π) at all. Then we simply use (<ref>) to finish the proof.π+ π=(n+1) π-( π-1). This is precisely Lemma 2.7 in <cit.>, hence we omit the proof.π+ ϕ(π)=(n+1) π-( π-1). The basic idea is to pair the descents of π with the descents of ϕ(π). We discuss by two cases. * If i∈ D(π)\ Sh(π), then either π_i>π_i+1>π_1 or π_1>π_i>π_i+1. Using (<ref>) and (<ref>) we see that either ϕ(π)_n+1-i>ϕ(π)_n+2-i>ϕ(π)_1 or ϕ(π)_1>ϕ(π)_n+1-i>ϕ(π)_n+2-i, both of which lead to (n+1-i)∈ D(ϕ(π))\ Sh(ϕ(π)). So this pair contributesi+(n+1-i)=n+1to the sum π+ ϕ(π).* If i∈ D(π)∩ Sh(π), then π_i≥π_1>π_i+1, and according to (<ref>), ϕ(π)_n+1-i-s_i≥ϕ(π)_1>ϕ(π)_n+2-i-s_i, where s_i is the length of the maximal block of consecutive entries of ϕ(π) that ends with ϕ(π)_n+1-i and is contained in ϕ(π)^b. Therefore this pair contributesi+(n+1-i-s_i)=n+1-s_ito the sum π+ ϕ(π).All such s_i as defined above clearly sum up to the total length of ϕ(π)^b:∑_i∈ D(π)∩ Sh(π)s_i=|ϕ(π)^b|=|π^b|= π-1.We add the contributions from both cases to arrive at the desired identity.ϕ(π)= π, ϕ(π)= π, ϕ(π)= π. It immediately follows from Lemma <ref> and <ref> that ϕ(π)= π, and since ϕ is an involution we also get ϕ(π)= π. Finally we have:(n+1) ϕ(π)-( ϕ(π)-1) Lemma <ref>= ϕ(π)+ ϕ^2(π)aaaϕ^2=1aa= π+ π Lemma <ref>= (n+1) π-( π-1).This directly leads to ϕ(π)= π since we already noted that ϕ(π)= π.This involution ϕ is admittedly a bit involved, so we include one complete example here to illustrate it.Given w=1 1 2 3 4 4 4 5 6, take v=4 3 4 4 2 1 6 5 1 ∈ R(w). We first compute the statistics on v:(, Id, , , )v=(5, {2,3,4,8}, 4, 25, 21).Now let π=c(v)=5 4 6 7 3 1 9 8 2∈_9, we proceed to find ϕ(π). First we get the defining triple associated with π:(π^t,π^b,Sh(π))=(6 7 9 8,4 3 1 2, {1,2,4,6,8}).Applying ϕ and according to (<ref>), we get immediately Sh(ϕ(π))={1,2,4,6,8}. But for ϕ(π)^t and ϕ(π)^b, which involve the map , we need to invoke the RSK algorithm. We first apply RSK on c(π^t)=1 2 4 3 and c(π^t)^rc=2 1 3 4 to get the following four SYT.3+1 111 122 133 214 0.8-1.6P_c(π^t) 3+1 111 122 133 214 0.8-1.6Q_c(π^t) 3+1 111 123 134 212 0.8-1.6P_c(π^t)^rc 3+1 111 123 134 212 0.8-1.6Q_c(π^t)^rcThen we apply inverse RSK on the pair (P_c(π^t), Q_c(π^t)^rc) to get (c(π^t))=4 1 2 3, henceϕ(π)^t=c^-1∘∘ c(π^t)=9 6 7 8. Similarly for ϕ(π)^b, we apply RSK on π^b=4 3 1 2 and (π^b)^rc=3 4 2 1 to get the following four SYT.2+1+1 111 122 213 314 0.6-2P_π^b 2+1+1 111 124 212 313 0.6-2Q_π^b 2+1+1 111 124 212 313 0.6-2P_(π^b)^rc 2+1+1 111 122 213 314 0.6-2Q_(π^b)^rcReversing RSK on (P_π^b, Q_(π^b)^rc) then produces ϕ(π)^b=(π^b)=1 4 3 2. Together we have(ϕ(π)^t,ϕ(π)^b,Sh(ϕ(π)))=(9 6 7 8,1 4 3 2, {1,2,4,6,8}),which corresponds uniquely to ϕ(π)=5 1 9 6 4 3 7 8 2∈_R(w). Therefore ϕ_R(w)(v)=c^-1(ϕ(π))=4 1 6 4 3 2 4 5 1. Finally we check the statistics match up indeed (withandswitched).(, Id, , , )ϕ_R(w)(v)=(5, {2,3,4,8}, 4, 25, 21).§ FURTHER QUESTIONS Seeing our equidistribution result in Corollary <ref>, a natural next step is to consider the so-called “k-extension” (see <cit.>). Namely, for our prototype of the alphabet, [m]={1,2,…,m}, we take two non-negative integers l and k such that l+k=m. Then we call the letters 1,2,…,l small and the letters l+1,l+2,…,l+k=m large. Now consider any total ordering, say Ø, on [m] that is compatible with k, in the sense that for any small letter x and any large letter y, we have (x,y)∈Ø. Bothandhave their k-extensions defined with respect to Ø, so one may ask if there exists a k-extension for , such that the joint distribution of (, , ) are still the same as that of (, , ). Finally, Corollary <ref> justifies (, ) as a new Euler-Mahonian (over rearrangement class of words) pair in the family consisting of (, ), (, ) and (, ) (see <cit.> for undefined statistics and further details). It would be interesting to pair other Mahonian statistics discovered in <cit.> with eitheror , and then pigeonhole the resulting pairs into the existing (or even new) families of Euler-Mahonian statistics. Another possible direction for furthering the study onis to consider the equidistribution problems on pattern avoiding permutations, see the recent work in <cit.>. § ACKNOWLEDGEMENTThe first two authors were supported by the Fundamental Research Funds for the Central Universities (No. CQDXWL-2014-Z004) and the National Natural Science Foundation of China (No. 11501061). 99 AmN. Amini, Equidistributions of Mahonian statistics over pattern avoiding permutations, https://arxiv.org/abs/1705.05298arXiv:1705.05298.BSE. Babson and E. Steingrímsson, Generalized permutation patterns and a classification of the Mahonian statistics, Sém. Lothar. Combin, B44b (2000): 18 pp.BurA. Burstein, On joint distribution of adjacencies, descents and some Mahonian statistics, in: Proc. AN, in: Discrete Math. Theor. Comp. Sci, 2010, pp. 601-612 (also in FPSAC 2010, San Francisco, USA).ChJ. N. Chen, Equidistributions of MAJ and STAT over pattern avoiding permutations, https://arxiv.org/abs/1707.07195arXiv:1707.07195.CFR. J. Clarke and D. Foata, Eulerian calculus, II: an extension of Han's fundamental transformation, European Journal of Combinatorics, 16(3) (1995): 221-252.CSZR. J. Clarke, E. Steingrímsson and J. Zeng, New Euler-Mahonian statistics on permutations and words, Advances in Applied Mathematics, 18(3) (1997): 237-270. FSD. Foata and M. P. Schützenberger, Major index and inversion number of permutations, Mathematische Nachrichten, 83(1) (1978): 143-159. Han1G. N. Han, Une transformation fondamentale sur les réarrangements de mots, Advances in Mathematics, 105(1) (1994): 26-41.Han2G. N. Han, The k-extension of a Mahonian statistic, Advances in Applied Mathematics, 16(3) (1995): 297-305.KitS. Kitaev, Patterns in permutations and words, Springer Science & Business Media, 2011.KVS. Kitaev and V. Vajnovszki, Mahonian STAT on words, Information Processing Letters, 116(2) (2016): 157-162.KnuD. E. Knuth, The Art of Computer Programming. Volume 1. Addison-Wesley Publishing Co., Reading, Mass., 1968.SchuM. P. Schützenberger, Quelques remarques sur une construction de Schensted, Séminaire Dubreil. Algèbre et théorie des nombres, 16(2) (1963): 1–12.StanR. P. Stanley, Enumerative combinatorics, vol. 2, Cambridge University Press, Cambridge, United Kingdom, 1999. | http://arxiv.org/abs/1708.07928v1 | {
"authors": [
"Shishuo Fu",
"Ting Hua",
"Vincent Vajnovszki"
],
"categories": [
"math.CO",
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"primary_category": "math.CO",
"published": "20170826042854",
"title": "Mahonian STAT on rearrangement class of words"
} |
Cosmogenic activation of materials Susana Cebrián December 30, 2023 ==================================December 30, 2023§ ABSTRACTThe Yangtze River has been subject to heavy flooding throughout history, and in recent times severe floods such as those in 1998 have resulted inheavy loss of life and livelihoods. Dams along the river help to manage flood waters, and are important sources of electricity for the region. Being able to forecast high-impact events at long lead times therefore has enormous potential benefit. Recent improvements in seasonal forecasting mean that dynamical climate models can start to be used directly for operational services.The teleconnection from El Niño to Yangtze River basin rainfall meant that the strong El Niño in winter 2015/2016 provided a valuable opportunity to test the application of a dynamical forecast system. This paper therefore presents a case study of a real time seasonal forecast for the Yangtze River basin, building on previous work demonstrating the retrospective skill of such a forecast.A simple forecasting methodology is presented, in which the forecast probabilities are derived from the historical relationship between hindcast and observations.Its performance for 2016 is discussed. The heavy rainfall in the May–June–July period was correctly forecast well in advance. August saw anomalously low rainfall, and the forecasts for the June–July–August period correctly showed closer to average levels.The forecasts contributed to the confidence of decision-makers across the Yangtze River basin. Trials of climate services such as this help to promote appropriate use of seasonal forecasts, and highlight areas for future improvements. This document isCrown CopyrightMet OfficeKey words: Seasonal forecasting, flood forecasting, Yangtze basin rainfall, ENSO,hydroelectricity§ INTRODUCTIONThe Yangtze River basin cuts across central China, providing water, hydroelectricity and agricultural land for millions of people. The Yangtze has been subject to flooding throughout history <cit.>, linked to variations in the East Asian monsoon that are sometimes driven by factors such as the El Niño–Southern Oscillation (ENSO; e.g. ).Large hydroelectric dams along the river and its tributaries, such as the Three Gorges Dam <cit.>, have flood defence as their primary responsibility. However, by lowering the water level behind the dam to protect against flooding, less electricity will be produced. There are therefore clear benefits of forecasting impactful rainfall events at long lead times, allowing mitigation planning for flooding and electricity production. The relationship between ENSO and the East Asian monsoon is complex and not fully understood. However, it has long been clear that a strong El Niño peaking in winter is likely to be followed by above-average rainfall in China the following summer <cit.>.The extreme El Niño event of 1997/1998 was followed by devastating floods in the Yangtze River basin <cit.>: thousands of people died, millions of people were made homeless, and the economic losses ran into billions of CN.In the subsequent years, much work has gone into better water management and flood prevention, and into improving both the accuracy and communication of climate forecasts, to prevent such a disaster happening again. Statistical relationships between large-scale climate phenomena and precipitation at more local scales have long been used to produce seasonal forecasts across China, and for the Yangtze in particular <cit.>. Recent advances in the dynamical seasonal forecast system developed at the UK Met Office, GloSea5 <cit.>, have resulted in the development of operational and prototype climate services for the UK in many sectors<cit.>. Recent work has shown that GloSea5 also has useful levels of skill for various processes in China <cit.>, including for summer precipitation over the Yangtze River basin <cit.>, without having to use statistical models based on larger-scale drivers.In parallel to these findings, <cit.> demonstrated that there was a clear demand from users for improved seasonal forecasts for the Yangtze, both from the flood risk and hydropower production communities.The very strong El Niño that developed during the winter of 2015–2016 <cit.> provided a perfect opportunity to develop a trial operational seasonal forecast using GloSea5 for the subsequent summer of 2016.We focused on forecasting the mean precipitation for the June–July–August (JJA) period, as that is where <cit.>demonstratedskill.However, we alsoproduced forecasts for the upcoming 3-month period each week, from February to the end of July 2016.In the last week of each month, a forecast was issued by the Met Office to the Chinese Meteorological Administration (CMA). In this paper, we describe the observed rainfall in the Yangtze region in summer 2016, and assess how the real time forecasts performed. We describe the data sets used in section <ref>, and our forecast production methodology in section <ref>.In section <ref> we compare the forecasts to the observed behaviour, and discuss possible future developments in section <ref>.§ DATA SETSThe current operational version of GloSea5 <cit.> is based on the Global Coupled 2 (GC2) configurationof the HadGEM3 global climate model, described in detail in <cit.> and references therein. Within HadGEM3-GC2, the atmospheric component (the Met Office Unified Model, UM, )is coupled to the JULES land surface model <cit.>, the NEMO ocean model <cit.> and the CICE sea ice model <cit.>.The atmosphere is modelled on a grid of 0.83° in longitude and 0.55° in latitude with 85 levels vertically, including a well-resolved stratosphere; the ocean model is modelled on a0.25° horizontal grid with 75 levels vertically.Using this configuration, GloSea5 runs operationally, producing both forecasts and corresponding hindcasts(used to bias-correct the forecasts).Each day, two initialised forecasts are produced, running out to 7 months.To produce a complete forecast ensemble for a given date, the last three weeks of individual forecasts are collected together to form a42-member lagged forecast ensemble.At the same time, an ensemble of hindcasts is produced each week. As described by <cit.>, 3 members are run from each of 4 fixed initialisation dates in each month, for each of the 14 years covering 1996–2009.The full hindcast ensemble is made by collecting together the four hindcast datesnearest to the forecast initialisation date, yielding a 12-member, 14-year hindcast.Note that the hindcast was extended in May 2016 to cover 23 years (1993–2015). This operational hindcast is not intended to be used for skill assessments: with only 12 members, skill estimates would be biased low <cit.>. However, a separate, dedicatedhindcast was produced for skill assessment, with 24 members and 20 years.Using that hindcast, we find a correlation skill of 0.56 for summer Yangtze rainfall, statistically indistinguishable from the previous value of 0.55 found by <cit.>. We use precipitation data from the Global Precipitation Climatology Project (GPCP) as our observational data set. This is derived from both satellite data and surface rain gauges, covering the period from 1979 to the present at 2.5° spatial resolution <cit.>. The verification we present here uses version 2.3 of the data <cit.>. Only version 2.2 was available when we started our operational trial, although we have confirmed that the choice of v2.2 or v2.3 makes negligible difference to our forecasts or results.§ FORECAST PRODUCTIONTypically when producing a seasonal forecast, the distribution of forecast ensemble members is used to represent the forecast probability distribution directly. However, experience has shown that the GloSea5 ensemble members may contain anomalously small signals, such that the predictable signal only emerges through averaging a large ensemble <cit.>. While this effect is less pronounced in subtropical regions like the Yangtze Basin, it is still present <cit.>. We therefore implemented a simple precipitation forecasting methodology, based entirely on the historical relationship between the hindcast ensemble means and the observed precipitation, averaged over the Yangtze River basin region (91°–122°E,25°–35°N, following ), for the season in question.The prediction intervals, derived from the linear regression of the hindcasts to the observations <cit.>, provide a calibrated forecast probability distribution.This is illustrated in Figure <ref>, where we show the precipitation forecasts issued in late April for MJJ, and in late May for JJA.The distribution of hindcasts and observations is shown as a scatter plot, with the ensemble mean forecast also included as a green circle. The uncertainty in the linear regression (grey) determines the forecast probabilities (green bars).The GloSea5 data is shown in standardised units, that is, the anomaly of each year from the mean, as a fraction of the standard deviation of hindcast ensemble means.The observations on the vertical axis are presented as seasonal means of monthly precipitation totals. The relationship with ENSO is indicated though colour-coding of the hindcast points: Years are labelled as El Niño (red) or La Niña (blue) according to whether their Oceanic Niño Index,[<http://www.cpc.noaa.gov/products/analysis_monitoring/ensostuff/ensoyears.shtml>.] based on observed sea surface temperature anomalies in the Niño 3.4 region, is above 0.5 K or below -0.5 K,respectively.Forecasts like those shown in Figure <ref> are produced each week, using the forecast model runs initialised each day of the preceding 3 weeks to generate the 42-member ensemble. We only issued the forecast produced near the end of each month. The full release schedule is described in Table <ref>.It is important to note that, due to the linear regression method we employ, our forecast probabilities are explicitly linked to both the hindcasts and the observations.The correlations between hindcasts and observations are biased low due to the smaller size of the hindcast ensemble compared to the forecast ensemble – a larger hindcast ensemble would not necessarily alter the gradient of the linear regression, but would reduce its uncertainty. Our forecast probabilities are therefore conservative (likely to be too small).The forecast information provided was designed to show very clearly and explicitly the uncertainties in the forecast system, to prevent over-confidence on the part of potential decision-makers.In addition to the scatter plot showing the forecast and the historical relationship (Figure <ref>), we also provided the probability of above-average precipitation as a `headline message'. This was accompanied by a contingency table showing the hit rate and false alarm rate for above-average forecasts over the hindcast period. For the MJJ and JJA forecasts, these are shown in Tables <ref> and <ref>.§ RESULTSThe observed precipitation in May, June, July and August 2016 is shown in Figure <ref>.We use standardised units here to show the precipitation anomaly relative to the historical variability over the hindcast period (1993–2015). It is clear that the most anomalously high rainfall was in May and June, and largely in the eastern half of the basin. July was close to normal overall when considering the box we were forecasting for, although there were disastrous floods further north. August had anomalously low rainfall across most of the region.<cit.> have examined the observedsummer 2016 rainfall in China and the Yangtze River basin in detail, including its relationship to larger-scale drivers: the anomalously low rainfall in August 2016 is in marked contrast to the situation in 1998, and is related to the behaviour of Indian Ocean temperatures and the Madden–Julian Oscillation during the summer.Figures <ref> and <ref> show the 3-month mean precipitation anomalies for MJJ and JJA respectively, for both GPCP and the forecast averages from the GloSea5 model output. While we do not expect the spatial patterns to match in detail (considering the skill maps of ), the overall signal is similar to the observations, with stronger anomalous precipitation in the eastern region in MJJ, and closer-to-average precipitation in JJA.We examine our forecasts for the Yangtze basin box more quantitatively in Figures <ref> and <ref>, where we show the variation with lead time of the hindcast–observation correlation, the 2016 forecast signal, and the probability of above-average precipitation, for MJJ and JJA respectively.Neither the hindcast–observation correlation nor the forecast signal vary significantly with lead time; indeed, they are remarkably consistent back to3 months before the forecast season, and when the 23-year hindcast is introduced in May.The forecasts did a good job of giving an indication of precipitation in the coming season. For MJJ, the forecast gave a high probability of above-average precipitation (80%), and it was observed to be above average. In JJA, the mean precipitation was observed to be slightly below average, due to the strong drier-than-average signal in August, although it was within a standard deviation of the interannual variability. While our forecast marginally favoured wetter than average conditions (65% probability of above-average rainfall), it was correctly near to the long-term mean, and the observed value was well within the forecast uncertainties. § DISCUSSION AND CONCLUSIONSThe heavy rainfall in the Yangtze River region in early summer 2016 was at a similar level tothat of 1998, and caused heavy flooding <cit.>.While deaths due to the flooding were roughly an order of magnitude fewer than those caused by the 1998 floods (i.e. hundreds rather than thousands of lives), the economic losses nevertheless ran into tens of billions of CN.Furthermore, it was reported that insurance claims, mostly from agricultural losses, amounted to less than 2% of the total economic loss, suggesting significant levels of underinsurance <cit.>.The prior experience of the 1998 El Niño-enhanced flooding, and the high levels of awareness of the strong El Niño in winter 2015/2016, meant that dams along the Yangtze were prepared in anticipation of high levels of rainfall.Our forecasts from GloSea5, produced using the simple methodology described here,contributed to the confidence of users adapting to the impending rainfall (Golding et al., in prep).Our verification has shown that our forecasts gave a good indication of the observed levels of precipitation for both MJJ and JJA averages over the large Yangtze Basin region.A greater degree of both spatial and temporal resolution – splitting the basin into upper and lower sections, and producing additional forecasts at a monthly timescale – would of course be preferable to users.However, smaller regions and shorter time periods may well be less skillful, so further work is needed to assess how best to achieve skillful forecasts in these cases.One significant improvement would be to increase the ensemble size of the hindcast. During 2017 the GloSea5 system was changed from 3 hindcast members per start date to 7. This could result in noticeable improvements in forecasts like those described here, as the hindcast–observations relationship will be less uncertain, especially when a predictable signal is present such as from El Niño.We will be issuing forecasts again in 2017. However, unlike 2016, in 2017 there are no strong drivers such as El Niño.Nevertheless, understanding the behaviour of the forecast system under such conditions will be informative, for both the users and the producers of the forecasts.Ultimately, trial climate servives such as this help to drive forecast development, improve understanding of forecast uncertainties, and promote careful use by stakeholders in affected areas. § ACKNOWLEDGEMENTSThis work and its contributors (PB, AS, ND, DS, CH, NG) were supported by the UK–China Research & Innovation Partnership Fund through the Met Office Climate Science for Service Partnership (CSSP) China as part of the Newton Fund. CL and RL were supported by the National Natural Science Foundation of China (Grant No. 41320104007).HR was supported by the Project for Development of Key Techniques in Meteorological Operation Forecasting (YBGJXM201705).The trial forecast service was first suggested by AS in 2015.GPCP precipitation data provided by the NOAA/OAR/ESRL PSD, Boulder, Colorado, USA, from their web site at <http://www.esrl.noaa.gov/psd/>. The Yangtze river basin shapefile used in the maps was obtained from <http://worldmap.harvard.edu/data/geonode:ch_wtrshed_30mar11> and is based on the watersheds shown in the China Environmental Atlas (2000),Chinese Academy of Science, Environmental Data Center.plainnat | http://arxiv.org/abs/1708.07448v2 | {
"authors": [
"Philip E. Bett",
"Adam A. Scaife",
"Chaofan Li",
"Chris Hewitt",
"Nicola Golding",
"Peiqun Zhang",
"Nick Dunstone",
"Doug M. Smith",
"Hazel E. Thornton",
"Riyu Lu",
"Hong-Li Ren"
],
"categories": [
"physics.ao-ph"
],
"primary_category": "physics.ao-ph",
"published": "20170824151132",
"title": "Seasonal forecasts of the summer 2016 Yangtze River basin rainfall"
} |
[pages=1-last]AHC_v0.pdf | http://arxiv.org/abs/1708.07452v1 | {
"authors": [
"Ariel H. Curiale",
"Flavio D. Colavecchia",
"Pablo Kaluza",
"Roberto A. Isoardi",
"German Mato"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170824151904",
"title": "Automatic Myocardial Segmentation by Using A Deep Learning Network in Cardiac MRI"
} |
Learning the Enigma with Recurrent Neural Networks Sam GreydanusDepartment of Physics and Astronomy, Dartmouth CollegeHanover, New Hampshire 03755 December 30, 2023 =========================================================================================================== Recurrent neural networks (RNNs) represent the state of the art in translation, image captioning, and speech recognition. They are also capable of learning algorithmic tasks such as long addition, copying, and sorting from a set of training examples. We demonstrate that RNNs can learn decryption algorithms – the mappings from plaintext to ciphertext – for three polyalphabetic ciphers (Vigenere, Autokey, and Enigma). Most notably, we demonstrate that an RNN with a 3000-unit Long Short-Term Memory (LSTM) cell can learn the decryption function of the Enigma machine. We argue that our model learns efficient internal representations of these ciphers 1) by exploring activations of individual memory neurons and 2) by comparing memory usage across the three ciphers. To be clear, our work is not aimed at 'cracking' the Enigma cipher. However, we do show that our model can perform elementary cryptanalysis by running known-plaintext attacks on the Vigenere and Autokey ciphers. Our results indicate that RNNs can learn algorithmic representations of black box polyalphabetic ciphers and that these representations are useful for cryptanalysis. Learning the Enigma with Recurrent Neural Networks Sam GreydanusDepartment of Physics and Astronomy, Dartmouth CollegeHanover, New Hampshire 03755 December 30, 2023 =========================================================================================================== § INTRODUCTION Given an encrypted sequence, a key and a decryption of that sequence, can we reconstruct the decryption function? We might begin by looking for small patterns shared by the two sequences. When we find these patterns, we can piece them together to create a rough model of the unknown function. Next, we might use this model to predict the translations of other sequences. Finally, we can refine our model based on whether or not these guesses are correct. During WWII, Allied cryptographers used this process to make sense of the Nazi Enigma cipher, eventually reconstructing the machine almost entirely from its inputs and outputs <cit.>. This achievement was the product of continuous effort by dozens of engineers and mathematicians. Cryptography has improved in the past century, but piecing together the decryption function of a black box cipher such as the Enigma is still a problem that requires expert domain knowledge and days of labor. The process of reconstructing a cipher's inner workings is the first step of cryptanalysis. Several past works have sought to automate – and thereby accelerate – this process, but they generally suffer from a lack of generality (see Related Work). To address this issue, several works have discussed the connection between machine learning and cryptography <cit.> <cit.>. Early work at the confluence of these two fields has been either theoretical or limited to toy examples. We improve upon this work by introducing a general-purpose model for learning and characterizing polyalphabetic ciphers. Our approach is to frame the decryption process as a sequence-to-sequence translation task and use a Recurrent Neural Network (RNN) -based model to learn the translation function. Unlike previous works, our model can be applied to any polyalphabetic cipher. We demonstrate its effectiveness by using the same model and hyperparameters (except for memory size) to learn three different ciphers: the Vigenere, Autokey, and Enigma. Once trained, our model performs well on 1) unseen keys and 2) ciphertext sequences much longer than those of the training set. All code is available online[]. By visualizing the activations of our model's memory vector, we argue that it can learn efficient internal representations of ciphers. To confirm this theory, we show that the amount of memory our model needs to master each cipher scales with the cipher's degree of time-dependence. Finally, we train our model to perform known-plaintext attacks on the Vigenere and Autokey ciphers, demonstrating that these internal representations are useful tools for cryptanalysis. To be clear, our objective was not to crack the Enigma. The Enigma was cracked nearly a century ago using fewer computations than that of a single forward pass through our model. The best techniques for cracking ciphers are hand-crafted approaches that capitalize on weaknesses of specific ciphers. This project is meant to showcase the impressive ability of RNNs to uncover information about unknown ciphers in a fully automated way. In summary, our key contribution is that RNNs can learn algorithmic representations of complex polyalphabetic ciphers and that these representations are useful for cryptanalysis. § RELATED WORKRecurrent models, in particular those that use Long Short-Term Memory (LSTM), are a powerful tool for manipulating sequential data. Notable applications include state-of-the art results in handwriting recognition and generation <cit.>, speech recognition <cit.>, machine translation <cit.>, deep reinforcement learning <cit.>, and image captioning <cit.>. Like these works, we frame our application of RNNs as a sequence-to-sequence translation task. Unlike these works, our translation task requires 1) a keyphrase and 2) reconstructing a deterministic algorithm. Fortunately, a wealth of previous work has focused on using RNNs to learn algorithms. Past work has shown that RNNs can find general solutions to algorithmic tasks. Zaremba and Sutskever <cit.> trained LSTMs to add 9-digit numbers with 99% accuracy using a variant of curriculum learning. Graves et. al. <cit.> compared the performance of LSTMs to Neural Turing Machines (NTMs) on a range of algorithmic tasks including sort and repeat-copy. More recently, Graves et al. <cit.> introduced the Differentiable Neural Computer (DNC) and used it to solve more challenging tasks such as relational reasoning over graphs. As with these works, our work shows that RNNs can master simple algorithmic tasks. However, unlike tasks such as long addition and repeat-copy, learning the Enigma is a task that humans once found difficult.In the 1930s, Allied nations had not yet captured a physical Enigma machine. Cryptographers such as the Polish Marian Rejewski were forced to compare plaintext, keyphrase, and ciphertext messages with each other to infer the mechanics of the machine. After several years of carefully analyzing these messages, Rejewski and the Polish Cipher Bureau were able to construct 'Enigma doubles' without ever having seen an actual Enigma machine <cit.>. This is the same problem we trained our model to solve. Like Rejewski, our model uncovers the logic of the Enigma by looking for statistical patterns in a large number of plaintext, keyphrase, and ciphertext examples. We should note, however, that Rejewski needed far less data to make the same generalizations. Later in World War II, British cryptographers led by Alan Turing helped to actually crack the Enigma. As Turing's approach capitalized on operator error and expert knowledge about the Enigma, we consider it beyond the scope of this work <cit.>. Characterizing unknown ciphers is a central problem in cryptography. The comprehensive Applied Cryptography <cit.> lists a wealth of methods, from frequency analysis to chosen-plaintext attacks. Papers such as <cit.> offer additional methods.While these methods can be effective under the right conditions, they do not generalize well outside certain classes of ciphers. This has led researchers to propose several machine learning-based approaches. One work <cit.> used genetic algorithms to recover the secret key of a simple substitution cipher. A review paper by Prajapat et al. <cit.> proposes cipher classification with machine learning.Alallayah et al. <cit.> succeeded in using a small feedforward neural network to decode the Vigenere cipher. Unfortunately, they phrased the task in a cipher-specific manner that dramatically simplified the learning objective. For example, at each step, they gave the model the single keyphrase character that was necessary to perform decryption. One of the things that makes learning the Vigenere cipher difficult is choosing that keyphrase character for a particular time step. We avoid this pitfall by using an approach that generalizes well across several ciphers.There are other interesting connections between machine learning and cryptography. Abadi and Andersen trained two convolutional neural networks (CNNs) to communicate with each other while hiding information from from a third. The authors argue that the two CNNs learned to use a simple form of encryption to selectively protect information from the eavesdropper. Another work by Ramamurthy et al. <cit.> embedded images directly into the trainable parameters of a neural network. These messages could be recovered using a second neural network. Like our work, these two works use neural networks to encrypt information. Unlike our work, the models were neither recurrent nor were they trained on existing ciphers. § PROBLEM SETUPWe consider the decryption function m = P(k, c) of a generic polyalphabetic cipher where c is the ciphertext, k is a key, and m is the plaintext message. Here, m, k, and c are sequences of symbols a drawn from alphabet A^l (which has length l). Our objective is to train a neural network with parameters θ to make the approximation P̂_NN(k, m, θ) ≈ P(k, m) such that θ̂= min_θℒ(P̂_NN(k, m, θ) - P(k, c)) where ℒ is L2 loss. We chose this loss function because it penalizes outliers more severely than other loss functions such as L1. Minimizing outliers is important as the model converges to high accuracies (e.g. 95^+%) and errors become infrequent. In this equation, P(k, c) is a one-hot vector and P̂_NN is a real-valued softmax distribution over the same space.Representing ciphers. In this work, we chose A^27 as the uppercase Roman alphabet plus the null symbol, . We encode each symbol a_i as a one-hot vector. The sequences m, k, and c then become matrices with a time dimension and a one-hot dimension (see Figure <ref>). We allow the key length to vary between 1 and 6 but choose a standard length of 6 for k and pad extra indices with the null symbol. If N is the number of time steps to unroll the LSTM during training, then m and c are N-6 × 27 matrices and k is a 6 × 27 matrix. To construct training example (x_i, y_i), we concatenate the key, plaintext, and ciphertext matrices according to Equations <ref> and <ref>. x_i = (k, c) y_i = (k, m) Concretely, for the Autokey cipher, we might obtain the following sequences: input : target: We could also have fed the model the entire key at each time step. However, this would have increased the size of the input layer (and, as a result, total size of θ) introducing an unnecessary computational cost. We found that the LSTM could store the keyphrase in its memory cell without difficulty so we chose the simple concatenation method of Equations <ref> and <ref> instead. We found empirical benefit in appending the keyphrase to the target sequence; loss decreased more rapidly early in training. § OUR MODEL Recurrent neural networks (RNNs). The simplest RNN cell takes as input two hidden state vectors: one from the previous time step and one from the previous layer of the network. Using indices t=1 … T for time and l=1 … L for depth, we label them h^l_t-1 and h^l - 1_t respectively. Using the notation of Karpathy et al. <cit.>, the RNN update rule ish^l_t = tanh W^l [ h^l - 1_t; h^l_t-1 ]where h ∈ℝ^n, W^l is a n × 2n parameter matrix, and the tanh is applied elementwise.The Long Short-Term Memory (LSTM) cell is a variation of the RNN cell which is easier to train in practice <cit.>. In addition to the hidden state vector, LSTMs maintain a memory vector, c_t^l. At each time step, the LSTM can choose to read from, write to, or reset the cell using three gating mechanisms. The LSTM update takes the form: .5[ i; f; o; g ] = [ sigm; sigm; sigm; tanh ] W^l [ h^l - 1_t; h^l_t-1 ] .5c^l_t = f ⊙ c^l_t-1 + i ⊙ gh^l_t = o ⊙tanh(c^l_t) The sigmoid (sigm) and tanh functions are applied element-wise, and W^l is a 4n × 2n matrix. The three gate vectors i,f,o ∈ℝ^n control whether the memory is updated, reset to zero, or its local state is revealed in the hidden vector, respectively. The entire cell is differentiable, and the three gating functions reduce the problem of vanishing gradients <cit.> <cit.>. We used a single LSTM cell capped with a fully-connected softmax layer for all experiments (Figure <ref>). We also experimented with two and three stacked LSTM layers and additional fully connected layers between the input and LSTM layers, but these architectures learned too slowly or not at all. In our experience, the simplest architecture worked best. § EXPERIMENTS We considered three types of polyalphabetic substitution ciphers: the Vigenere, Autokey, and Enigma. Each of these ciphers is composed of various rotations of A, called Caesar shifts. A one-unit Caesar shift would perform the rotation A[i] ↦ A[(i+1) N].The Vigenere cipher performs Caesar shifts with distances that correspond to the indices of a repeated keyphrase. For a keyphrase k of length z, the Vigenere cipher decrypts a plaintext message m according to: P(k^i_t, m^j_t) = A[(j+(i z)) l] where the lower indices of k and m correspond to time and the upper indices correspond to the index of the symbol in alphabet A^l.The Autokey cipher is a variant of this idea. Instead of repeating the key, it concatenates the plaintext message to the end of the key, effectively creating a new and non-repetitive key: i as in k^i_t,ift≤ z m^i_t-z,otherwise The Enigma also performs rotations over A^l, but with a function P that is far more complex. The version used in World War II contained 3 wheels (each with 26 ring settings), up to 10 plugs, and a reflector, giving it over 1.07 × 10^23 possible configurations <cit.>. We selected a constant rotor configuration of , ring configuration ofand no plugs. For an explanation of these settings, see <cit.>. We set the rotors randomly according to a 3-character key, giving a subspace of settings that contained 26^3 possible mappings. With these settings, our model required several days of compute time on a Tesla k80 GPU. The rotor and ring configurations could also be allowed to vary by appending them to the keyphrase. We tried this, but convergence was too slow given our limited computational resources.Synthesizing data. The runtime of P(k,m) was small for all ciphers so we chose to synthesize training data on-the-fly, eliminating the need to synthesize and store large datasets. This also reduced the likelihood of overfitting. We implemented our own Vigenere and Autokey ciphers and used the historically accurate [] Python package as our Enigma simulator.The symbols of the input sequences k and m were drawn randomly (with uniform probability) from the Roman alphabet, A^26. We chose characters with uniform distribution to make the task more difficult. Had we used grammatical (e.g. English) text, our model could have improved its performance on the task simply by memorizing common n-grams. Differentiating between gains in accuracy due to learning the Enigma versus those due to learning the statistical distributions of English would have been difficult.Optimization. Following previous work by Xavier et al. <cit.>, we use the 'Xavier' initialization for all parameters. We use mini-batch stochastic gradient descent with batch size 50 and select parameter-wise learning rates using Adam <cit.> set to a base learning rate of 5 × 10^-4 and β=(0.9, 0.999). We trained on ciphertext sequences of length 14 and keyphrase sequences of length 6 for all tasks. As our Enigma configuration accepted only 3-character keys, we padded the last three entries with the null character, .On the Enigma task, we found that our model's LSTM required a memory size of at least 2048 units. The model converged too slowly or not at all for smaller memory sizes and multi-cell architectures. We performed experiments on a model with 3000 hidden units because it converged approximately twice as quickly as the model with 2048 units. For the Vigenere and and Autokey ciphers, we explored hidden memory sizes of 32, 64, 128, 256, and 512. For each cipher, we halted training after our model achieved 99^+% accuracy; this occurred around 5 × 10^5 train steps on the Enigma task and 1 × 10^5 train steps on the others.The number of possible training examples, ≈ 10^19, far exceeded the total number of examples the model encountered during training, ≈ 10^7 (each of these examples were generated on-the-fly). We were, however, concerned with another type of overfitting. On the Enigma task, our model contained over thirty million learnable parameters and encountered each three-letter key hundreds of times during training. Hence, it was possible that the model might memorize the mappings from ciphertext to plaintext for all possible (26^3) keys. To check for this sort of overfitting, we withheld a single key, , from training and inspected our trained model's performance on a message encrypted with this key.Generalization. We evaluated out model on two metrics for generalization. First, we tested its ability to decrypt 20 randomly-generated test messages using an unseen keyphrase (''). It passed this test on all three ciphers. Second, we measured its ability to decrypt messages longer than those of the training set. On this test, our model exhibited some generalization for all three ciphers but performed particularly well on the Vigenere task, decoding sequences an order of magnitude longer than those of the trainign set (see Figure <ref>). We observed that the norm of our model's memory vector increased linearly with the number of time steps, leading us to hypothesize that the magnitudes of some hidden activations increase linearly over time. This phenomenon is probably responsible for reduced decryption accuracy on very long sequences. Memory usage. Based on our model's ability to generalize over unseen keyphrases and message lengths, we hypothesized that it learns an efficient internal representation of each cipher. To explore this idea, we first examined how activations of various units in the LSTM's memory vector changed over time. We found, as shown in Figure <ref>, that 1) these activations mirrored qualitative properties of the ciphers and 2) they varied considerably between the three ciphers.We were also interested in how much memory our model required to learn each cipher. Early in this work, we observed that the model required a very large memory vector (at least 2048 hidden units) to master the Enigma task. In subsequent experiments (Figure <ref>), we found that the size of the LSTM's memory vector was more important when training on the Autokey task than the Vigenere task. We hypothesize that this is because the model must continually update its internal representation of the keyphrase during the Autokey task, whereas on the Vigenere task it needs only store a static representation of the keyphrase. The Enigma, of course, requires dramatically more memory because it must store the configurations of three 26-character wheels, each of which may rotate at a given time step.Based on these observations, we claim that the amount of memory our model requires to learn a given cipher can serve as an informal measure of how much each encryption step depends on previous encryption steps. When characterizing a black box cipher, this information may be of interest. Reconstructing keyphrases. Having verified that our model learned internal representations of the three ciphers, we decided to take this property one step further. We trained our model to predict keyphrases as a function of plaintext and ciphertext inputs. We used the same model architecture as described earlier, but the input at each timestep became two concatenated one-hot vectors (one corresponding to the plaintext and one to the ciphertext). Reconstructing the keyphrase for a Vigenere cipher with known keylength is trivial: the task reduces to measuring the shifts between the plaintext and ciphertext characters. We made this task more difficult by training the model on target keyphrases of unknown length (1-6 characters long). In most real-world cryptanalysis applications, the length of the keyphrase is unknown which also motivated our choice. Our model obtained 99^+% accuracy on the task.Reconstructing the keyphrase of the Autokey was a more difficult task, as the keyphrase is used only once. This happens during the first 1-6 steps of encryption. On this task, accuracy exceeded 95%. In future work, we hope to reconstruct the keyphrase of the Engima from plaintext and ciphertext inputs. Limitations. Our model is data inefficient; it requires at least a million training examples to learn a cipher. If we were interested in characterizing an unknown cipher in the real world, we would not generally have access to unlimited examples of plaintext and ciphertext pairs.Most modern encryption techniques rely on public-key algorithms such as RSA. Learning these functions requires multiplying and taking the modulus of large numbers. These algorithmic capabilities are well beyond the scope of our model. Better machine learning models may be able to learn simplified versions of RSA, but even these would probably be data- and computation-inefficient. § CONCLUSIONS This work proposes a fully-differentiable RNN model for learning the decoding functions of polyalphabetic ciphers. We show that the model can achieve high accuracy on several ciphers, including the the challenging 3-wheel Enigma cipher. Furthermore, we show that it learns a general algorithmic representation of these ciphers and can perform well on 1) unseen keyphrases and 2) messages of variable length. Our work represents the first general method for reconstructing the functions of polyalphabetic ciphers. The process is fully automated and an inspection of trained models offers a wealth of information about the unknown cipher. Our findings are generally applicable to analyzing any black box sequence-to-sequence translation task where the translation function is a deterministic algorithm. Finally, decoding Enigma messages is a complicated algorithmic task, and thus we suspect learning this task with an RNN will be of general interest to the machine learning community. § ACKNOWLEDGEMENTSWe are grateful to Dartmouth College for providing access to its cluster of Tesla K80 GPUs. We thank Jason Yosinski and Alan Fern for insightful feedback on preliminary drafts. aaai | http://arxiv.org/abs/1708.07576v2 | {
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1) Center for Nonlinear Science, University of North Texas, P.O. Box 311427, Denton, Texas 76203-1427, USA2) Army Research Office, Research Triangle Park, NC, United StatesWe present a technique to search for the presence of crucial events in music, based on the analysis of the music volume. Earlier work on this issue was based on the assumption that crucial events correspond to the change of music notes, with the interesting result that the complexity index of the crucial events is μ≈ 2, which is the same inverse power-law index of the dynamics of the brain. The search technique analyzes music volume and confirms the results of the earlier work, thereby contributing to the explaination as to why the brain is sensitive to music, through the phenomenon of complexity matching. Complexity matching has recently been interpreted as the transfer of multifractality from one complex network to another. For this reason we also examine the mulifractality of music, with the observation that the multifractal spectrum of a computer performance is significantly narrower than the multifractal spectrum of a human performance of the same musical score. We conjecture that although crucial events are demonstrably important for information transmission, they alone are not sufficient to define musicality, which is more adequately measured by the multifractality spectrum.Keywords: Crucial events, Multifractality, Complexity matching, Musicality, 1/f noiseComplexity Measures of Music Bruce J. West^2 December 30, 2023 ============================§ INTRODUCTION One of the outstanding mathematicians of the twentieth century, George Birkhoff, argued that the aesthetics of art have mathematical, which is to say a quantitative, measure. The structure in various art forms, music in particular, that he discussed in his book <cit.> was largely overlooked by other scientists until the last quarter of the twentieth century, when Mandelbrot introduced the scientific community to fractals<cit.> and his protégée Voss applied these ideas to the mathematical analysis of music. Voss and Clark <cit.> used stochastic, or 1/f, music, in which notes are selected at random and the frequency with which a particular note is used is determined by a prescribed distribution function, to gain insight into the structure of more conventional music. They determined that a variety of musical forms, jazz, blues, classical, have a blend of regularity and spontaneous change characteristic of 1/f-music. Aesthetically pleasing music was found to have a 1/f^α spectrum, with an inverse power-law index in the interval 0.5<α <1.5, thereby connecting the structure of music to the physical phenomena of 1/f-noise <cit.>. In 1987 the newly developed concept of self-organized criticality (SOC) was used by Bak et al. <cit.> to explain the source of 1/f -noise. Subsequently, 1/f-noise has been found to be a ubiquitous property of complex networks near criticality, such as the brain. This suggests an exciting connection with the problem of cognition <cit.> because 1/f noise may represe nt the brain self-organizing through a vertical collation of the body's spontaneous physiological events. Somaet al. <cit.> have shown that the brain is more sensitive to1/f-fluctuations than to other forms of noise, resulting in higher information transfer rates in the visual cortex <cit.>, pain-relief efficiency by electrical stimulation <cit.> and enhanced efficiency by biological ventilators <cit.>. West et al.<cit.> speculate that there is a complexity matching between Mozart's music (1/f-composition), the brain's organization (1/f -complex network) and the hearbeat (another 1/f-process), to explain the result of Tsuruoka et al. <cit.> that listening to Mozart has the effect of inducing 1/f-noise on heart beating. This also supports the conjecture that music mirrors the mind<cit.> in that its complexity is a reflection of the 1/f-complexity of brain cognition.The observation that listening to Mozart's music enhances the reasoning skills of students <cit.> contributed to the ever-expanding circle of research interest centered on the possible complexity matching between Mozart's music and brain function. This is a throny problem having aspects of a number of fundamental human issues, including but not limited to creativity, free will, determinism and randomness <cit.>. Our purpose here is to present a mathematical theory that explains these interesting aspects of music, which picks up where the above mentioned popular works leave off. The approach presented herein uses the concept of a crucial event as a fundamental building block for the underlying time series, resulting in1/f-variability being the signature of complexity. The theory is an application of the recent work of Mahmoodi et al <cit.>, which contains an intuitive description of crucial events and develops a generalized form of SOC, self-organized temporal criticality (SOTC), based on the dynamics of complex networks. Experimental observation shows that moving from physics to biology is signaled by the emergence of the breakdown of ergodic behavior with increasing complexity. Ergodic behavior is one of the foundational assumptions of statistical physics, that being that time averages of system vaiables produce results equivalent to those obtained from ensemble averages of those variables <cit.>. Its almost ubiqutous breakdown in complex system came as a surprise.Ergodicity breakdown is caused by complex fluctuations being driven by crucial events. The time interval between consecutive crucial events are statistically independent and described by a markedly non-exponential waiting-time probability density function (PDF). To realize the temporal complexity of crucial events requires the concept of an intermediate asymptotic region, characterized by an inverse power law (IPL) with index μ <3. These events are renewal <cit.> in the sense that their occurrence invokes a total rejuvenation of the system implying that sequential renewal events occur at times having no correlation with the times of occurrence of preceeding events. SOTC shows that beyond the intermediate asymptotic region an exponential time region appears that entails the system recovering the normal ergodic condition in the long-time limit. This exponential truncation, generated by the same cooperative interaction responsible for the IPL nature of the intermediate asymptotic region, is often confused with the effects produced by the finite size of the observed time series.We conjecture that music, being the mirror of mind, naturally reflects the brain's dynamics, which is a generator of 1/f -fluctuations <cit.>, thereby confirming the early observations of Voss and Clarke<cit.>. However, the arguments adopted by these pioneers are based on the assumption that 1/f-noise is generated by fluctuations with very slow, but stationary correlation functions. Whereas, the crucial events emerging from the statistical analysis of the time series generated by the brain <cit.>, on the contrary, have non-stationary correlation functions. The significance of SOTC modeling is that the crucial events generated are the same as the 1/f-noise produced by the brain, that is, the fluctuations have non-stationary correlation functions.The importance of crucial events for music composition was recognized in two earlier publications of our group <cit.>; the first paper illustrates an algorithm for composing music based on crucial events. The present paper is closer to the main goal of the second publication, which was the detection of crucial events in existing music composition. Vanni and Grigolini <cit.> assummed that the time at which a note change occurs is a crucial event and found that the IPL index was μ≈ 2. Herein we adopt a different criterion for detecting crucial events, one based on the observation of music volume. This technique was inspired by a method developed and used by Kello <cit.>, who, however, did not evaluate the time interval between two consecutive events. Consequently, the question of whether or not the events are crucial events was left unanswered. Herein we confirm that music is driven by crucial events. Our analysis establishes a significant difference between computer and human performance of the same music score, which is not surprising. The computer plays the notes as written by the composer, without interpretation. Humans, on the other hand, bring all their knowledge, experience and feeling for the music to their performance. The computer can provide the heart of the music, but only a human can make the heart beat. We also make a preliminary attempt at establishing a connection between crucial events and multifractality. A time series without a characteristic time scale can be characterized by a scaling exponent, the fractal dimension. An even more complex time series can have a time-dependent fractal dimension, resulting in a spectrum of fractal dimensions. This spectrum defines a multifractal time series and the width of the multifractal spectrum is a measure of the varability of the time series scaling behavior.§ IN SEARCH OF CRUCIAL EVENTS According to the theoretical perspective established in earlier work <cit.> we define crucial events, as events for which the time interval between two consecutive events is described by a waiting-time PDF ψ (τ ) with the asymptotic IPL structure: ψ (τ )∝1/τ ^μ,with an IPL index μ <3. The time intervals between two different pairs of consecutive events are not correlated ⟨τ _iτ _j⟩∝δ _ij,where the bracket indicates an average over the waiting-time PDF. The occurrence of crucial events establishes a new kind of fluctuation-dissipation process <cit.> and the transport of information from one complex system P to another complex system S is determined by the influence that the crucial events of P exert on the time occurrence of the crucial events of S <cit.> (complexity management).To search for renewal events in music we adopt a method that Kello used at the 2016 Denton Workshop <cit.> to record music amplitude and turn it into a sequence of events significantly departing from a homogeneous Poisson sequence of events. We assume that the intersections of the music signal with a suitably selected threshold line may correspond to the occurrence of a sequence of renewal events. We apply this analysis technique to the time series resulting from both computer and human performances of a music composition. The computer performance consisted of programmed MIDI (Musical Instrument Digital Interface) file and a FLAC recording (Free Lossless Audio Codec) provided a human performance. In Fig. <ref> we depict the music selection for human performance of Mozart'sConcerto for Flute, Harp, and Orchestra, Allegro. The music signal was sampled at 44100 samples per second.The IPL in Fig. <ref> have slopes ofμ = 2.07 and 2.2 for the human and computer performances, respectively (shown in black). The red and blue curves were evaluated using the aging experiment <cit.>. Red is the waiting time PDF of the time series after being aged. Blue has the τ's of the time series that are first shuffled and then aged. As can be seen in the figure, both red and blue more or less overlap one another. We interpret this overlap to mean the events defined by the crossings are renewal and are therefore crucial events.§ POWER SPECTRUM Another way to establish that the events detected are renewal is to evaluate the spectrum S(ω ) to detemine if it is IPL. This is so because a signal hosting crucial events may give the impression of being random. Actually, that signal, as a consequence of hosting crucial events becomes a fluctuating time series, characterized by a non-stationary correlation function. The lack of stationarity is a consequence of ergodicity breakdown becoming perennial when μ <3. <cit.>.According to <cit.>, the spectrum of fluctuations in that case cannot be derived from the Wiener-Khintchine theorem, relying as it does, on the stationarity assumption. It is necessary to take into account that in both caces (μ =2.07 and μ =2.2), the average time interval between two consecutive events diverges, thereby making non-stationary the process driven by the crucial events. This anomalous condition leads to a spectrum that is dependent on the length of the time series L <cit.>:S(ω )∝1/L^2-μ1/ω ^β,with the IPL indexβ =3-μ . This result was obtained by going beyond the Wiener-Khintchine theorem adopted by Voss and Clarke in their analysis, but which cannot be applied to our condition if we make the reasonable assumption, basedon the results depicted in Fig. <ref>,that the events detected using the adaptation of Kello's method, are renewal. If they are renewal and they drive the signal ξ (t), namely the music intensity, then the spectrum S(ω ) is expected to follow the prescription of Eq. (<ref> ). In the case where the process yields a slow, but stationary correlation function, we would have β <1 <cit.>. Evaluating the power spectrum in this case becomes computationally challenging because, as shown by Eq. (<ref>), the noise intensity decreases with increasing L , the length of the time series. Nevertheless, the results depicted in Fig.<ref> yield the IPL index β≈ 1, and Eq.(<ref>) yields β = 3-2=1, and the agreement between the results is very encouraging.Both results satisfactorily support the claim that the events revealed using the threshold method are crucial events. To clarify this point Fig. <ref> illustrates the waiting-time PDF of the intervals between two consecutive crossings of the threshold line, when the threshold is set equal to 0.002. This threshold is not large enough to filter out the events that are not crucial. We see that these non-crucial events produce a well pronounced exponential shoulder in the waiting-time PDF. The results of Fig.<ref> have been obtained by filtering out these non-crucial events. We therefore conclude that the crucial events, which are the mechanism for information transport <cit.>, also have the significant effect of determining the behavior of the spectrum for ω→ 0.§ MULTIFRACTALITY The discovery of 1/f noise in music by Voss and Clarke <cit.> was interpreted assuming the music time series is stationary, which is consistent with Fractional Brownian Motion (FBM), and yields a mono-fractal<cit.>. However, the present work goes beyond <cit.> and the ergodic assumption, by taking a non-stationary approach consistent with multifractality. Using the method of Multifractal Detrended Fluctuation Analysis (MF-DFA)<cit.> to analyze each time series of the two performances gives the results shown in the Fig. <ref>.The computer performance yields the narrow multifractal distribution, whereas the multifractal distribution of the human performance is significantlybroader. This notable difference between the multifractal spectra indicates different levels of complexity in the two performances. The narrowness of the computer performance suggests a strict adherence to the single fractal dimension and consequenty less complexity then in the human performance. In fact, the difference between the two performances may be better described by what the computer performance lacks compared to the human performance. This extra information, or musicality, contained in the human performance includes specific techniques that add to the complexity of the music through subtle variation in timing, intonation, articulation, dynamics, etc., which are likely a better match to the brain's complexity. The human performance is largely more aestheticly pleasing to the listener than is the computer performance.Suppose the brain of Mozart contains a certain complexity, which is well described by SOTC as a generator of 1/f-noise. Then Mozart transcribes his complexity, albeit incompletely, into the music score of the chosen selection. To recover this lost musicality, the human performer injects their own interpretation of the lost complexity using their specific performance techniques. Conversely, the computer performance is unable to interpret this lost component and delivers exactly what was transcribed, resulting in less variability in complexity. The computer performance is a record of the brain of Mozart even if Mozart himself would have produced a broader multifractal distribution when the piece was performed.§ CONCLUDING REMARKS Crucial events exist in the changing of notes in music, as found by Vanniet al <cit.>. Similarly, this analysis done differently by analyzing the music signal, the change in volume, leads to the same conclusion (through the analysis of a different aspect of the music). This difference in analysis is very important because the statistical analysis of the dynamics of the brain <cit.> shows that the brain is a generator of crucial events with the same IPL index. Additionally, we found that there is a noticeable difference in the fractal measures between human and computer performances.Music is aesthetically pleasing to the brain <cit.> because of the crucial events described by μ = 2. Multifractality may provided a clearer picture of which performance of Mozart was more pleasing. The difference in musicality is obvious to the listener's ear and this difference can be quantified through the narrower and broader multifractal spectra. The increasing aesthetics of music favors a broader multifractal spectrum. Indeed, multifractality describes an additional measure of complexity. Crucial events measure complexity of the intermediate asymptotics, whereas multifractality contains additional information, beyond the intermediate asymptotics, regarding the transient region and exponential truncation. All this subtlety in composition is experienced by the brain through the transfer of the music time series' multifracality.Acknowledgments. AP and KM warmly thank ARO and Welch for support through Grant No. W911NF-15-1-0245 and Grant No. B-1577, respectively. birkhoff33Birkhoff GD. Aesthetic Measure. Cambridge: Harvard University Press; 1933. p. 1-225. mandelbrot77 Mandelbrot BB. Fractals, form, chance and dimension. San Francisco: WH Freeman & Co.; 1977. p. 1-365.voss1Voss RV, Clarke J.1/f noise in music and speech. Nature 1975;258:317-8.voss2Voss RV,Clarke J. 1/f noisein music: music from 1/f noise. J. Acoust. Soc. Am 1978;63:258-63.voss3 Voss RV,Clarke J. 1/f noise from systems in thermal equilibrium.Phys. Rev. Lett. 1976;36:42-5.bak87 Bak P, Tang C, Wiesenfeld K. Self-organized Criticality: An Explanation of q/f-noise. Phys. Rev. Lett. 1987;59:381-4.anderson Anderson CM. From molecules to mindfulness: How vertically convergent fractal time fluctuations unify cognition and emotion.Consciousness & Emotion 2000;1:193-226.soma03 Soma R, Nozaki D, Kwak S, Yamamoto Y. 1/f Noise Outperforms White Noise in Sensitizing Baroreflex Function in the Human Brain. Phys. Rev. Lett. 2003;91:078101-4.yu05 Yu Y,Romero R,Lee TS. Preference of Sensory Neural Coding for 1/f Signals. Phys. Rev. Lett. 2005;94:108103-4.takakura79 Takakura K,Sano K, Kosugy Y, Ikebe J. Pain Control by Transcutaneous Electrical Nerve Stimulating Using Irregular Pulse of 1/f Fluctuation. Appl. NeuroPhysiol. 1979;42:314-315.mutch05 Mutch WAC. Fluctuations ans Noisees in Biological, Biophysical and Biomedical Systems III. Proc. of SPIE 2005;5841:1-250. west08 West BJ, Geneston EL, Grigolini P. Maximizing information exchange between complex networks. Phys. Rep. 2008;468:1-99.tsuruoka07 Tsuruoka M, Tsuruoka Y, Shibasaki R, Yasuoka Y. Proc 29th Ann. Int. Conf. IEEE EMBS Cite Int., Lyon, France (2007).bianco Bianco S, Ignaccolo M, Rider MS,Ross MJ, Winsor P, Grigolini P. Brain, music, and non-Poisson renewal processes. Phys.Rev. E 2007;75:061911-10. nature Rauscher FH, Shaw GL,Ky KN. Music and spatial task performance. Nature 1993;365:611-11.wang Hong FT. Free will: A case study in reconciling phenomenological philosophy with reductionist sciences. Progress in Biophysics and Molecular Biology 2015;119:670-726.korosh Mahmoodi K, West BJ, Grigolini P. Self-organizing Complex Networks: individual versus global rules. Front. Physiol. 2017;8:00478.grigolini Grigolini P. Emergence of biological complexity: Criticality, renewal and memory. Chaos, Solitons and Fractals 2015;81:575-88. allegrini1/f Allegrini P, Menicucci D, Bedini R, Fronzoni L, Gemignani A, Grigolini P, West BJ, Paradisi P. Spontaneous brain activity as a source of ideal 1/f noise. Phys. Rev. E2009;80:061914-13. davidadams Adams D, Grigolini P. Music, New Aesthetic and Complexity.J. Zhou (Ed.): First International Conference, Complex 2009 Part II, LNICST 2009 5, 2212. lestocart Vanni F, Grigolini P. Music as a mirror of mind. “Esthétique de la complexité".Hermann, Paris: 2017 Chapter 6.kello C. Kello, The Sound of Cooperation: Complexity Matching in Spoken Interaction, UNT Workshop 2016. crucialevents Allegrini P, Barbi F, Grigolini P, Paradisi P. Renewal, modulation, and superstatistics in time series. Phys. Rev. E 2006;73:046136-13.crucialevents2 Allegrini P, Bologna M, Grigolin P, West BJ. Fluctuation-Dissipation Theorem for Event-Dominated Processes. Phys. Rev. Lett. 2007;99:010603-4. complexitymanagement Aquino G, Bologna M, Grigolini P, West BJ.Beyond the Death of Linear Response: 1/f Optimal Information Transport. Phys. Rev. Lett. 2010;105:040601-4.mirko Lukovic M, Grigolini P. Power spectra for both interrupted and perennial aging processes J. Chem. Phys. 2008;129:184102-11.Kantelhardt Kantelhardt JW, Zschiegner SA, Koscielny-Bunde E, Havlin S, Bunde A, Stanley HE. Multifractal detrended Fluctuation analysis of nonstationary time series. Physica A 2002;316:87-114. | http://arxiv.org/abs/1708.08041v1 | {
"authors": [
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[pages=1-last]Optimized_Observing_Final.pdf | http://arxiv.org/abs/1708.07589v1 | {
"authors": [
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conicet,cab,ib]M. N. Kuperman [email protected] conicet,cab]M. F. Laguna [email protected] conicet,cab,ib]G. Abramson [email protected] conicet,fb]A. Monjeau [email protected] iidypca]J. L. Lanata [email protected][conicet]Consejo Nacional de Investigaciones Científicas y Técnicas [cab]Centro Atómico Bariloche (CNEA), R8402AGP Bariloche, Argentina [ib]Instituto Balseiro, Universidad Nacional de Cuyo, Argentina [fb]Fundación Bariloche, R8402AGP Bariloche, Argentina [iidypca]Instituto de Investigaciones en Diversidad Cultural y Procesos de Cambio, CONICET-UNRN, R8400AHL Bariloche, Argentina We develop a mathematical model of extinction and coexistence in a generic predator-prey ecosystemcomposed of two herbivores in asymmetrical competition and a hunter exerting a predatory pressure onboth species. With the aim of representing the satiety of hunters when preys are overabundant, weintroduce for the predation behavior a dependence on preys density. Specifically, predation ismodeled as growing proportionally to the presence of herbivores at low density, and saturating when the total population of prey is sufficiently large. The model predicts theexistence of different regimes depending on the parameters considered: survival of a single species,coexistence of two species and extinction of the third one, and coexistence of the three species.But more interestingly, in some regions parameters space the solutions oscillate in time, both as a transient phenomena and as persistent oscillations ofconstant amplitude. The phenomenon is not present for the more idealized linear predation model, suggesting that it can be the source of real ecosystems oscillations. hierarchical competition predation bifurcation analysis ecological cycles§ INTRODUCTION The use of mathematical models in biology in general and in ecology in particular has grownsignificantly in the last decade. This is due in part to their predictive capacity, but also dueto their power to order and systematize assumptions and thus contribute toelucidate the behavior of complex biological systems. In fact, the interrelation of factors asdiverse as climate, access to resources, predators and human activity, makes it necessary todevelop mathematical models that allow predicting the effect of each of them on the speciesinvolved, showing possible scenarios of coexistence or extinction. A large number ofpublications on topics such as predator-prey models <cit.>, intra- and inter-specificcompetition <cit.>, orhabitat fragmentation <cit.> can be found, but more research isstill needed on how to integrate all these mechanisms together.In previous workswe developed a mathematical model of extinction and coexistence in a genericpredator (or hunter)-prey ecosystem. In order to characterize the general behaviors we focused on a trophicnetwork of three species: two herbivores in asymmetric competition and one predator <cit.>.This problem was studied by means of ordinary differential equations and stochastic simulations.Both approaches provided similar and interesting results. The model predicts the existence ofdifferent regimes depending on the parameters considered: survival of one species, coexistence oftwo and extinction of the third (in the three possible combinations), and coexistence of the threespecies involved <cit.>. Moreover, the results presented in <cit.> indicate that the superiorcompetitor of the hierarchy is driven to extinction after the introduction of hunters in the model. Thishappens even in pristine habitats (with no environmental degradation) and, more relevantly, even ifthe predatory pressure is higher on the inferior herbivore. In the original model we proposed that predation grew proportionally to the density of herbivores.While this approach is valid for ecosystems with low density of preys, it introduces an unrealisticbehavior of the predator population when the density of available prey ishigh. The model implicitly assumes that the predator or hunter never quench, even when there is anoverabundance of prey. In the present work, and in search of a better representation of predation,we analyze a variation of this model. Satiety of the predator or hunter is incorporated in themathematical description as an asymptotic saturation in the term of predation.The analysis of this model shows new results. The most interesting aspect of the solutions is thetemporal oscillation of the populations. Under certain conditions these oscillations are transientand decay to a stable equilibrium. But in other situations oscillations are maintainedindefinitely. In fact, we found regions of coexistence of the three species with persistentoscillations of constant amplitude. These dynamic regimes enrich the predictive properties of themodel, so we expect our results to drive the search for evidence of oscillations in populations ofcurrent and extinct species.In the Section <ref> we introduce the mathematical model. Section <ref> is devotedresults, whereas in Section <ref> we discuss the main implications of the resultsandpossible future directions.§ MODEL WITH SATURATION IN PREDATION Our dynamical model requires a set of rules determining the temporal evolution of the system. Theserules are inspired by the life history and the ecological interactions of the species involved,corresponding to biotic, environmental and anthropic factors <cit.>. In order to gain insightinto the possible outcomes of different scenarios of interest, we have intentionally kept our systemrelatively simple: two herbivores in a hierarchical competition and a hunter exerting a predatorypressure on both. Some details of the ecological implications are discussed below. The original model, proposed in <cit.>, can be described by the following set of equations: dx_1/dt =c_1 x_1(1-x_1)-e_1 x_1-μ_1 x_1 y, dx_2/dt =c_2 x_2(1-x_1-x_2)-e_2 x_2-μ_2x_2 y, dy/dt =c_y y(x_1+x_2-y)-e_y y. Each species is described by a dynamical variable: x_1 and x_2 are the herbivores, and y isthe predator. Eqs. (<ref>)describe the time evolution of these variables. We imagine thatboth herbivores feed on the same resource and therefore compete with each other. This is representedby the first terms of Eqs. (<ref>-<ref>). Such competition is asymmetrical, as it happens in most natural situations. This has interestingconsequences, since coexistence under these circumstances requires advantages and disadvantages ofone over the other. Consider, for example, that the individuals of each species are of differentsize, or temperament, such that species x_1 can colonize any available patch ofhabitat, and even displace x_2, while species x_2 can only occupy sites that are not alreadyoccupied by x_1. In this regard, we call x_1 the superior or dominant species of the hierarchy,and x_2 the inferior one. Since body size is often the main factor establishing this hierarchy, weimagine that x_1 is larger than x_2. This asymmetry is reflected in the logistic termsdescribing the competition in Eqs. (<ref>) and (<ref>), as x_1 limits the growth ofx_2 in Eq. (<ref>), while the reciprocal in not true. This mechanism can be considered as aweak competitive displacement. A strong version of the competitive interaction between theherbivores was also considered in our previous works, consisting of an additional term -c_1 x_1x_2 in Eq.(<ref>). In other words, we have intra-specific competition in both species, but only x_2 suffers from thecompetition with the other species, x_1. In this context, for x_2 to survive requires that theyhave some advantage other than size, typically associated with a higher reproductive rate or a lowerneed of resources.Besides, the equations for the herbivores also include a mortality term with coefficient e_ianda predation term with coefficient μ_i. The equation for the predator y is also logistic, witha few differences. The reproduction rate of the predator is limited by intra-specific competitionbut enhanced by the presence of prey. Now we analyze a variation of the previous model, seeking a more realistic representation of the predation (or hunting) term. The differential equations in the model with saturation are as follows:dx_1/dt =c_1 x_1(1-x_1)-e_1 x_1-μ_1 x_1y/x_1+x_2+d_1, dx_2/dt =c_2 x_2(1-x_1-x_2)-e_2 x_2-μ_2 x_2 y/x_1+x_2+d_2, dy/dt =c_y y(x_1+x_2-y)-e_y y. Observe that, while the predation terms undermine the population of herbivores, predation does notgrow proportionally to the presence of prey, but rather saturates if the combined prey population issufficiently large. This represents a satiation if preys are overabundant. Note that two newparameters d_1 and d_2 are included in Eqs. (<ref>), representing the departure fromproportionality.In the next section we present the main results of the model described by Eqs. (<ref>) and compare them with those of Eqs. (<ref>). § RESULTSThe study of the system with saturation of the predating pressure shows interesting and much richerresults than those of our previous models. While the model described by Eqs. (<ref>) predictsseveral different regimes, with three and twospecies coexistence, the steady state solutions arealways stable nodes. Here we show that the saturation effect induces oscillatory solutions.Without losing generality, we have restricted the values of the parameters within a range thatshows all the behaviours displayed by the model, especially those scenarios of coexistence betweentwo or all three species. The parameters are chosen in such a way that in the absence of predationpressure (μ_1 = μ_2 = 0), a coexistence of the three species is achieved. Moreover, thepredation pressure over x_1 is kept fixed at a value μ_1 < μ_2, corresponding tosituations where the inferior species is hunted more frequently than the superior one. We plot in Fig. <ref> the temporal evolution of the population densities for different values of μ_2,the predation pressure over the inferior herbivore, x_2. The first panel (Fig. <ref>a) shows the behavior of the populations when a relativelylow predation pressure is exerted on x_2, μ_2=0.33. In this case we observe damped oscillations, which converge to the extinction of the superior herbivore x_1 and to the coexistenceof the other two species, the predator y and the inferior herbivore x_2. A higher value of μ_2=0.40 is not enough to allow the survival of x_1 but produces sustained oscillations ofy and x_2 (see Fig. <ref>b). An even higher pressure on x_2(μ_2=0.50) and the equilibrium between herbivores is achieved, and the three species coexist.This is shown in Fig. <ref>c, with persistent oscillations of constant amplitude. If we increase further the predation pressure on x_2, theoscillations disappear. Still, the coexistence of the three species is possible, as shown inFig. <ref>d. As expected, a larger predation pressure on the inferior herbivore will finally produce itsextinction, as seen in Fig. <ref>e.As mentioned before, these non-oscillating behaviors were also observed in our previous model.In order to provide a visual representation of the steady state behavior of bothsystems,Eqs. (<ref>) and (<ref>), we show in Fig. <ref> the stable equilibria and limitcycles corresponding for the solutions of both models, for a range of μ_2 and the same choice ofthe values of the rest of the parameters as in Fig. <ref>. On the one hand the asymptotic solutions corresponding to the model described byEqs. (<ref>), without saturation in the predation, converge to stable nodes, showing threespecies coexistence for all the values of μ_2 displayed. These are the set of solutionsindicated as A on Fig. <ref>. On the other hand, the steady state solutions of Eqs. (<ref>) show both stable nodes and cycles. These are indicated as B on Fig. <ref>. The dynamics of the cycles is rather interesting.For μ_2≲ 0.36 we have non-oscillatory solutions, nodes located on the vertical (x_2,y)plane, that appear as an oblique line of dots in Fig. <ref> on the left of the plot. In thisregime the dominant herbivore, despite of being less predated on than the inferior one, can notpersist.At μ_2≈ 0.36 there is a Hopf bifurcation and cycles (still on the vertical (x_2,y)plane) appear.Then, at μ_2 ≈ 0.46 a new bifucartion occurs. This time it is a transcritical bifurcationof cycles, as will be shown later. The superior herbivore can now coexist with the other two speciesand the cycle detaches from the (x_2,y) plane. We can observe in Fig. <ref> how these twistin the three-dimensional phase space,displaying an oscillatorycoexistence ofthe three species.At μ_2≈ 0.56 another Hopf bifurcation occurs, this time destroying the cycle, preserving the coexistence between the three species, as shown by the three rightmost points of Fig. <ref>B. A bifurcation diagram of the phenomenon, using μ_2 as a control parameter, is shown in Fig. <ref>, where the five regimes of Fig. <ref> are indicated by the same letters, in vertical stripes in both panels. The upper paneldisplays the equilibria of the solutions. Dashed lines indicate linearly unstable equilibria, and in such circumstances sustained oscillations occur. The amplitude of these oscillations is shown in the bottom panel of Fig. <ref>. We can observe more clearly that there is a region where species x_2 and the predator coexist (that is, withextinction of the dominante herbivore), corresponding to values of μ_2≲ 0.46. This regime contains the Hopf bifurcation H1, with two-species oscillations for μ_2≳ 0.36. When the predation pressureon the inferior herbivore is increased above the transcritical bifurcation TC we observe coexistence of the threespecies, both in an oscillating regime and in a stationary equilibrium, achieved after the Hopf bifurcation H2 is crossed. Finally, if this predation is too high, it is x_2 the extinct species, allowing for the survival of the dominant species x_1. Complementing the bifurcation analysis, we show in Fig. <ref> the real part of theeigenvalues of the linearized system at the unstable equilibria in the region of cycles, around thetranscritical bifurcation TC. Thicker lines (of both colors) correspond to the pair ofcomplex-conjugate eigenvalues of each cycle. Black lines correspond to the two-species oscillation,which is stable for μ_2≲ 0.462. The eigenvalue with negative real part corresponds to thestable manifold of the cycle, which is normal to the plane (x_2,y). At the transcriticalbifurcation point TC this eigenvalue exchanges stability with the corresponding one of the othercycle (thin red line), the center manifold abandons the plane x_1=0 and three-species coexistenceensues.§ FINAL REMARKS AND CONCLUSIONSWe have presented here the main results obtained with a simple three-species model, composed of apredator and two herbivores in asymmetric competition, where the predation pressure saturates if thedensity of preys is high enough. As shown, the model predicts the existence of different regimes asthe values of the parameters change. These regimes consist of the survival of a single species (anyof them), the coexistence of two species and the extinction of the third one (the three combinationsare possible) and also the coexistence of the three species. But the most interesting aspect of thesolutions of this model is that in some cases the populations oscillate in time. Under someconditions these oscillations are transient phenomena that decay to a stable equilibrium. Yet inother situations the oscillations are maintained indefinitely. In fact, we have found regions ofcoexistence of the three species with persistent oscillations of constant amplitude. It was shown that, while in the original model without saturation in the predation the asymptoticsolutions converge to stable nodes, the steady state solutions of the model whith saciation showsboth stable nodes and cycles. Our results indicate that, for low predation pressures on the inferiorherbivore, the superior one extinguishes and non-oscillatory solutions appear for the remainingspecies, as indicated by the nodes observed in Fig. <ref>. At higher predation pressure aHopf bifurcation and cycles develop, but still the superior herbivore cannot survive. After that,for an even higher value of μ_2,a transcritical bifurcation of cycles occurs to a state of three-species coexistence.Bear in mind that the persistence of the inferior competitor requires that they have some advantage over the dominant one (in this case, a greater colonization rate). In such a context, the superior competitor is the most fragile of both with respect to predation (or to habitat destruction, as shown for example in <cit.>). For this reason an increase of the predation on x_2 releases competitive pressure, allowing x_1 to survive. Finally, at an even higher predation pressure, another Hopf bifurcation occurs which destroys the cycle. The coexistence of the three species is preserved until the pressure μ_2 is high enough to extinguish the inferior herbivore x_2.Transcritical bifurcation of cycles in the framework of population models has been found in severalsystems described by equations that include saturation <cit.>. Three-species foodchain models were extensively studied through bifurcation analysis <cit.>. A rich set ofdynamical behaviors was found, including multiple domains of attraction, quasiperiodicity, andchaos. In Ref. <cit.> the dynamics of a two-patches predator-prey system is analyzed, showingthat synchronous and asynchronous dynamics arises as a function of the migration rates. In aprevious work, the same author analyzes the influence of dispersal in a metapopulation modelcomposed of three species <cit.>. Our contribution, through the model presented here, naturallyextends those results in two directions. First, our model has three species in two trophic levels,two of them in asymmetric competition and subject to predation. Second, spatial extension andheterogeneity has been taken into account implicitlyasmean field metapopulations in the framework of Levins' model <cit.>.Of course, we have not exhausted here all the possibilities of the model defined by Eqs. (<ref>), but it is an example of the most interesting results that we have found.One can also imagine that the cyclic solutions arise from the interplay of activation and repressioninteractions, as in metabolic systems <cit.>. The same pattern could be applied toregulations in community ecology if we replace the satiation inhibitor by the addition of a secondpredator, superior competitor with respect to the other predator, inhibiting its actions. This is awell documented pattern in several ecosystems <cit.>. We believethat these behaviors are very general and will provide a thorough analysis elsewhere. Thesedynamical regimes considerably enrich the predictive properties of the model. In particular, we believe that the prediction of cyclic behavior for a range of realistic predator-prey models should motorize the search for their evidence in populations of current and extinct species. § ACKNOWLEDGMENTSThe authors gratefully acknowledge grants from CONICET (PIP 2011-310), ANPCyT (PICT-2014-1558) and UNCUYO (06/506).§ REFERENCES99 1 Swihart RK, Feng Z, Slade NA, Mason DM, Gehring TM, Effects of habitat destruction andresource supplementation in a predator–prey metapopulation model, J. Theor. Biol. 210 (2001)287-303.2 Bascompte J, Solé RV, Effects of habitat destruction in a prey–predator metapopulationmodel, J. Theor. Biol. 195 (1998) 383-393.3 Kondoh M, High reproductive rates result in high predation risks: a mechanism promotingthe coexistence of competing prey in spatially structured populations, Am. Nat. 161 (2003) 299-309.4 Nee S, May RM, Dynamics of metapopulations: habitat destruction and competitivecoexistence, J. Anim. Ecol. 61 (1992) 37-40.5 Tilman D, May RM, Lehman CL, Nowak MA, Habitat destruction and the extinction debt,Nature 371 (1994) 65-66.6 Hanski I, Coexistence of competitors in patchy environment, Ecology 64 (1983) 493–500.7 Hanski I, Ovaskainen O, The metapopulation capacity of a fragmented landscape, Nature404 (2000) 755-758.8 Hanski I, Ovaskainen O,Extinction debt at extinction threshold, Conserv. Biol. 16(2002) 666-673.9 Ovaskainen O, Sato K, Bascompte J, Hanski I, Metapopulation models for extinctionthreshold in spatially correlated landscapes, J. Theor. Biol. 215 (2002) 95-108.10 Laguna MF, Abramson G, Kuperman MN, Lanata JL, Monjeau JA, Mathematical model oflivestock and wildlife: Predation and competition under environmental disturbances, EcologicalModeling 309-310 (2015) 110-117.11 Abramson G, Laguna MF, Kuperman MN, Monjeau JA, Lanata JL, On the roles of hunting andhabitat size on the extinction of megafauna, Quaternary International 431B (2017) 205-215.12 McCann K, Yodzis P, Bifurcation structure of a three species food chain model, Theor. Pop. Biol. 48 (1995) 93-125.13 Klebanoff A, Hastings A, Chaos in three species food chains, J. Math. Biol. 32 (1994) 427-451.14 Kuznetsov YA, Rinaldi S, Remarks on Food Chain Dynamics, Math. Biosciences 134 (1996) 1-33. 15 Jansen VVA, The dynamics of two diffusively coupled predator-prey populations, Theor. Pop. Biol. 59 (2001) 119-131. 16 Jansen VVA, Effects of dispersal in a tri-trophic metapopulation model, J. Math. Biol. 34 (1994) 195-224.17 Bacelar FS, Dueri S, Hernández-García E, Zaldívar JM, Jointeffects of nutrients and contaminants on the dynamics of a food chain in marine ecosystems,Math. Biosciences 218, (2009)24-32monod1961 Monod J and Jacob F, General conclusions: Teleonomics mechanisms incellular metabolism, growth and differentiation, in Cold Spring Harbor Symposia on QuantitativeBiology 26 (1961) 389-401.terborgh2010 Terborgh J and Estes JA, Trophic cascades, predators, prey, and the changing dynamics of nature (Island Press, Washington DC, 2010). | http://arxiv.org/abs/1708.07774v1 | {
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"Marcelo N Kuperman",
"Fabiana Laguna",
"Guillermo Abramson",
"Adrian Monjeau. Jose Luis Lanata"
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"primary_category": "nlin.AO",
"published": "20170825152210",
"title": "Oscillations from satiation of predators"
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Study of Clear Sky Models for Singapore T Qureishy^1, C Laliena^2, E Martínez^2, A J Qviller^3, J I Vestgården^1 ^4, T H Johansen^1 ^5, R Navarro^2 and P Mikheenko^1 August 24, 2017 =================================================================================================================================Soumyabrata Dev^1, Shilpa Manandhar^2 ^*, Yee Hui Lee^3, and Stefan Winkler^4 ^**^1Nanyang Technological University Singapore, Singapore 639798, email:^2Nanyang Technological University Singapore, Singapore 639798, email:^3 Nanyang Technological University Singapore, Singapore 639798, email:^4Advanced Digital Sciences Center (ADSC), Singapore 138632, email: ^* Presenting author^** Corresponding author The estimation of total solar irradiance falling on the earth's surface is important in the field of solar energy generation and forecasting. Several clear-sky solar radiation models have been developed over the last few decades. Most of these models are based on empirical distribution of various geographical parameters; while a few models consider various atmospheric effects in the solar energy estimation.In this paper, we perform a comparative analysis of several popular clear-sky models, in the tropical region of Singapore. This is important in countries like Singapore, where we are primarily focused on reliable and efficient solar energy generation. We analyze and compare three popular clear-sky models that are widely used in the literature. We validate our solar estimation results using actual solar irradiance measurements obtained from collocated weather stations. We finally conclude the most reliable clear sky model for Singapore, based on all clear sky days in a year. Introduction The total solar irradiance falling on the earth's atmosphere, in the event of the clear day without any clouds, is referred as the clear-sky solar radiation. It is important, particularly in tropical countries like Singapore, that experiences most of the direct solar irradiance <cit.>, and is useful for solar energy generation. Several theoretical models have been developed that estimates the clear sky global solar irradiance (GHI). In this paper, we discuss three popular clear-sky models: Bird model <cit.>, Yang model <cit.> and CAMS McClear model <cit.>. Clear Sky Models Bird et al. uses results from radiative transfer codes, to estimate the total irradiance, by keeping the various atmospheric factors as constant throughout the year. It explains the various scattering of the solar radiation in the earth's atmosphere, and calculates the diffused- and direct- solar radiation at the earth's surface. The various atmospheric scattering from aerosols, ozone and water vapor are considered in its estimation of solar radiation. Recently, Yang et al. proposed an empirical regression model, that is specifically designed for Singapore. This model is created by considering three distinct weather station data positioned at various locations in Singapore. It modifies the clear-sky model in <cit.>, and proposes a Singapore-specific solar radiation model based on two user inputs – zenith angle of the sun at the particular location on earth's surface and day number of the year.The total clear-sky solar irradiance G_c in W/^2 is defined as: G_c = 0.8277E_0I_sc(cosθ_z)^1.3644e^-0.0013×(90-θ_z), In Eq. <ref>, E_0 is the eccentricity correction factor of earth, I_sc is a solar irradiance constant (1366.1W/m^2), θ_z is the solar zenith angle (in degrees). We compute the correction factor E_0 as follows: E_0 = 1.00011 + 0.034221cos(Γ) + 0.001280sin(Γ) +0.000719cos(2Γ) + 0.000077sin(2Γ). Γ = 2π(d_n-1)/365 is the day angle (in radians), where d_n is the day number in a year. In addition to these empirical models, the atmosphere service of Copernicus (CAMS) proposed a Copernicus McClear clear-sky solar radiation model. It provides a fully physical model, by attempting to replace earlier empirical models. It uses the recent measurement results of various aeorosol properties and water vapor content, to provide reliable estimates of solar irradiance. The clear-sky solar radiation data are available for download from its site [<http://www.soda-pro.com/web-services/radiation/cams-mcclear>.], in temporal resolution of 1 minute, 15 minute, hour, day, and month.Experiments and Results In this paper, we study these models [The source codes of these models and the associated results are available online at <https://github.com/Soumyabrata/clear-sky-models>.] for a specific location in Singapore. In our experiments, we manually select all the clear-sky days in Singapore. However, we need to note that the existence of completely clear-sky day is rare in a tropical country like Singapore. Data Collection The data is collected on a particular rooftop (1.34^∘N, 103.68^∘E) of Nanyang Technological University Singapore. The total solar radiation is measured in resolution of 1 minute, by Davis Instruments 7440 Weather Vantage Pro II. In addition to solar radiation, it also provides us other useful meteorological data, such as temperature, humidity, precipitation and wind speed. We have also designed a ground-based sky camera called WAHRSIS (Wide Angled High Resolution Sky Imaging System) <cit.> that captures images of the sky scene at regular intervals of time. These sky cameras provide us more useful insights, while manually selecting the clear-sky days in a year. Results Figure <ref> shows the various models for a sample clear sky day of 7-April-2016, along with the actual solar irradiance measurements, recorded by the weather station at this location. We observe that, for this sample day, both Bird model and Yang model over-estimates the actual solar irradiance values. However, McClear model can better estimate the actual solar irradiance measurements.In order to provide an objective evaluation of the various models, we compute the Pearson correlation co-efficient and Root Mean Square Error (RMSE) between the actual- and estimated- solar irradiance values. Table <ref> shows the results for 7-April-2016. The McClear model has the least RMSE value, and highest correlation value amongst all the models. For a larger statistical duration, we compute such objective measures for all the clear-sky days in the year of 2016. Table <ref> summarizes the results, and shows the average correlation- and RMSE- values. Table <ref> clearly concludes that the McClear model performs the best across other benchmarking clear-sky radiation models. Conclusion In this paper, we have provided a comparative analysis of various state-of-the-art clear sky models. We recorded our observations for the tropical country of Singapore. We recorded all the clear sky days in a single year, and estimated the best model amongst all benchmarking methods. We conclude that the CAMS McClear model is the most accurate clear-sky model for Singapore. In our future work, we plan to integrate images from whole-sky imager for reliable solar energy estimation and forecasting. We also plan to design hardware-efficient sky cameras with low-power for continuous usage, as it is ubiquitous in the areas of biomedical systems <cit.>. The authors would like to thank Luke Witmer for his python implementation of Bird Clear Sky Model for Solar Radiation Modeling <cit.>. 1IGARSS_solar S. Dev, F. M. Savoy, Y. H. Lee, and S. Winkler, “Estimation of solar irradiance using ground-based whole sky imagers,” in Proc. International Geoscience and Remote Sensing Symposium (IGARSS), 2016.Bird81 R. E. Bird and R. L. Hulstrom, “Simplified clear sky model for direct and diffuse insolation on horizontal surfaces,” Tech. Rep., Golden, CO: Solar Energy Research Institute, 1981.dazhi2012estimation D. Yang, P. Jirutitijaroen, and W. M. Walsh, “The estimation of clear sky global horizontal irradiance at the equator,” Energy Procedia, vol. 25, pp. 141–148, 2012.McClear13 M. Lefèvre, A. Oumbe, P. Blanc, B. Espinar, B. Gschwind, Z. Qu, L. Wald, M. Schroedter-Homscheidt, C. Hoyer-Klick, A. Arola, A. Benedetti, J. W. Kaiser, and J.-J. Morcrette, “Mcclear: a new model estimating downwelling solar radiation at ground level in clear-sky conditions,” Atmospheric Measurement Techniques, vol. 6, no. 9, pp. 2403–2418, 2013.Robledo2000 L. Robledo and A. Soler, “Luminous efficacy of global solar radiation for clear skies,” Energy Conversion and Management, vol. 41, no. 16, pp. 1769–1779, 2000.WAHRSIS S. Dev, F. M. Savoy, Y. H. Lee, and S. Winkler, “WAHRSIS: A low-cost, high-resolution whole sky imager with near-infrared capabilities,” in Proc. IS&T/SPIE Infrared Imaging Systems, 2014.IGARSS2015 S. Dev, F. M. Savoy, Y. H. Lee, and S. Winkler, “Design of low-cost, compact and weather-proof whole sky imagers for High-Dynamic-Range captures,” in Proc. International Geoscience and Remote Sensing Symposium (IGARSS), 2015, pp. 5359–5362.Deepu2016 C. J. Deepu, X. Zhang, C. H. Heng, and Y. Lian, “A 3-Lead ECG-on-chip with QRS detection and lossless compression for wireless sensors,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 63, no. 12, pp. 1151–1155, Dec. 2016.Witmer2015 L. Witmer, “Birdclearskymodel-python,” <https://github.com/lukewitmer/BirdClearSkyModel-Python>, 2015. | http://arxiv.org/abs/1708.08760v1 | {
"authors": [
"Soumyabrata Dev",
"Shilpa Manandhar",
"Yee Hui Lee",
"Stefan Winkler"
],
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"published": "20170824174038",
"title": "Study of Clear Sky Models for Singapore"
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[email protected] Physics Department, Clarkson University, Potsdam, New York 13699-5820, USAWe haveextended our analytically derived PDB-NMA formulation, ATMAN <cit.>, to include protein dimers using mixed internal and Cartesian coordinates. A test case on a 1.3Å resolution model of a small homodimer,ActVA-ORF6, consisting of two112-residue subunitsidentically folded in a compact 50Å sphere, reproduces the distinct experimental Debye-Waller motility asymmetry for the two chains, demonstrating that structure sensitively selectsvibrational signatures. The vibrational analysis of this PDB entry, together with biochemical and crystallographic data, demonstrates the cooperative nature of the dimeric interaction of the two subunits and suggests a mechanical model for subunit interconversion during the catalytic cycle. PDB-NMA of a Protein Homodimer Reproduces Distinct Experimental Motility Asymmetry Daniel ben-Avraham December 30, 2023 =================================================================================== § INTRODUCTIONAn internal force analysis of protein data bank (PDB) entries via normal mode analysis (NMA) reveals the softest coordinates inherent to the system and provides the basis of a theoretical derivation of the crystallographic Debye-Waller, or B factors. We have extended our PDB-NMA, ATMAN <cit.>, to dimeric protein systems according to the formalism presented in Levitt, Sander and Stern (LSS) <cit.>.While completely general and valid for any collection of oscillators, our original implementation of the LSS formalism to protein systems was restricted to single chain polypeptides using only“soft”torsional angle coordinates: the main chain ϕ and ψ dihedrals as well as the side chains χ dihedrals <cit.>. For the analysis of dimers, we use these torsional angle coordinates{ϕ_i^A,ψ_i^A,χ_i^A;ϕ_j^B,ψ_j^B,χ_j^B} for chain A and chain B, along with six additional coordinates, {x,y,z,α,β,γ},to account for the overall motion of chain B relative to chain A: the first three denote rigid body translations of chain B relative to A along the x-, y-, and z-direction;α is a rigid body rotation of chain B around its own x-axis (passing through B's center of mass), and β and γ are analogous rotations about B's y- and z-axes, respectively. The LSS formalism extends straightforwardly to any kind of generalized coordinates, including mixed. Briefly, given a set of coordinates {q_l} and a potential energy field V({q_l}), one produces the Hessian matrix F,F_ln=∂^2/∂ q_l∂ q_nV ,and the inertia matrix H:H_ln=∑_k=1^N m_k∂ r_k/∂ q_l·∂ r_k/∂ q_n ,where the sum runs over all the atoms in the system (m_k is the mass of the k-th atom and r_k is itslocation vector). The normal modes are then obtained by co-diagonalizing the F and H matrices:FA=Λ HA .Λ is a diagonal matrix containing the mode frequencies, Λ_ll=ω^2_l, and A_kl contains the l-th eigenmode amplitudes. (The modes are H-orthogonal: A^T HA =I.)The {q_l} in this formalism represent any kind of coordinates, including the mixed types we use in our analysis. The practical implementation, however, is far from trivial.For example, the ∂ r/∂ q derivatives required for the H-matrix must exclude overall rigid bodytranslations and rotations of the whole system, and derivatives of a complex potential energy field (needed for the computation of F) require some care as well, especiallysince the different types of coordinates, coupled with the separate domains of chain A and chain B, give rise to numerous blocks in the F-matrix, each withits own set of rules, etc. We have tested our codes for the production of F by performing all derivatives numerically (involving updates of the system's configuration by small increments δ q_i),as well as analytically, and confirming the agreement between these two very different procedures. In addition, the analysespass several“consistency" requirements: the diagonalization yields non negative eigenvalues; the emergent eigenfrequencies distribution istypical to folded proteins; and the root mean square deviations per mode i, RMS^i, are smoothly decreasing with i, with the first three modes obtaining in excess of 50% of the total RMS.One important reason to work with the complexity of dihedral angles is that these coordinates allow one to scale the computations to much largersystems without sacrificing proper stereochemical topology <cit.>. Dihedral bonds typically constitute a seventh or an eighth of the full complement of available degrees of freedom.The excluded degrees of freedom, bond lengths and bond angles,are quite stiffer than the soft dihedral coordinates and thus do not contribute to the slower modes of motion, the main focus of normal mode analyses. All-atom PDB-NMA that use Cartesian coordinates without topological constraints do not maintain proper stereochemistry and yield atypical eigenvalue spectra <cit.>. Design of proper topological constraints for another common reduction scheme, the use of only Cα coordinates, is likewise challenging <cit.>.Crystallized proteins adopt conformations that are long-lived and stable, implying that these structures already reside at a minimum of a multidimensional energy surface. PDB-NMA therefore assumes a balanced distribution of pairwise atomic interactions, with suitably assignedspring constants <cit.>. Early formulations assigned a universal spring constant to all interatomic interactions, while currentPDB-NMAs like sbNMA or ATMAN derive the pairwise interatomic spring constant strengths from the atom types and distances of separation of every interacting atom pair according to a parent potential like CHARMM (sbNMA) <cit.> or L79/ENCAD (ATMAN) <cit.>.This yields identical results to those derived from classical NMA on energy minimized structures, when both analyses proceed from the same, energy minimized structures.The PDB data bank includes many structures with nearly identical folds: differences in their vibrational signatures will be lost if such structures must be energy minimized first, motivating the use of these types of energy potentials <cit.>.To test our PDB-NMA formulation for dimeric protein systems,we analyzed the PDB entry of a small homodimer with a large hydrophobic interface: a monooxygenaseproduced by a soil-dwelling bacterium, Streptomyces coelicolor. The enzyme, ActVA-Orf6,catalyzes the oxidation of large, 3-ringed aromatic polyketides in the biosynthetic pathway leading to actinorhodin production, one of fourantibiotics produced by this gram-positive actinobacterium. Sciara et al published the 1.3Å resolution structure of the native, unliganded enzyme at 100K as PDB entry 1LQ9in 2003 <cit.>. As seen in Fig. <ref>, the enzyme consists of two identical chains of112 residues (Ala2-Ser113) with the secondary structure sequence N-(B1-H1H2-B2)-(B3-H3H4-B4)-B5-C, that together fold into a compact sphere roughly 50Å in diameter. Each amino terminus N leads intoβ strand B1 andloops back with a broken α helix H1H2 to form the parallel β strand B2.Halfway through the primary sequence,therefollows a sharp turn at Thr55-Thr58, and the same topology repeats: β strand B3, situated between and antiparallel to B1 and B2,loops back via the broken α helix pair H3H4 to form a fourth antiparallel, outer β strand B4 aligned along B1.Topologically, the β strands present in the order B4, B1, B3 and B2, with theH1H2 helical arch over and parallel to B2 and next to it the H3H4 arch over and parallel toB4. Thisfold presents as a β sheet “floor” supporting two arches to create an open space accessible from the “front” (strand B4) as well as the “rear” (B2). However, the C terminal sequence Phe103-Ile110 after B4 forms a final β strand, B5, in the dimer. This β strand extension of B4 interlaces with and extends the neighboring chain's β sheet at B2 and sealsthe rear aperture, creating an enclosed region, the active site cavity, accessible only from the front. The monomer's structureis reminiscent of a shell-shaped stage.In the dimerized complex, Fig. <ref>, the two β sheet floors are parallel, their main chains roughly 10Å apart and theirstrands juxtaposed by 90^∘, with the arched helices arrayed on the exterior surfaces, somewhat suggestive of a classic TIM barrel roll.Dimerization is mediated by the N- as well as the C-terminal regions of each chain. As mentioned, the two C terminal arms interlock with the B2 strands of the opposing chain, zipper style, with seven hydrogen-bond “teeth” extending from the Phe103 carbonyl oxygen of one chain to the Thr55 amide hydrogenof the other chain,to the final interchain hydrogen bond between the Ile110 carbonyl oxygen of one chain to the Ala49 amide nitrogen of the other chain.As reported in Sciara <cit.>, this swapped chain featureresults in a 2200Å^2 contact area between the two chains.An additional1100Å^2 contact area is created by the interface of the N termini withthe Ala2Glu3Val4 juxtaposed in an antiparallel fashion to create a knob-like packing of the N terminiat the rearof the central pseudo-barrel, between the C terminal arms. This packing arrangementblocks access of solvent to the rear of the barrel-type construct of the β sheets. Altogether the 3300Å^2surface contact between the A and B chains represents30% of the total surface area of the dimer, which is reported in the 1LQ9 header file as 11000Å^2.A rigid-body, all-atom superposition of chains A and B results in a RMSD of 0.5Å, while a main chain superpositionresults in a RMSD of 0.3Å. The mainchain traces overlap closely, withthe N termini as well as all five β strandsnearly perfectly aligned, while there exists a slight mismatch in the orientation of the loop (residues 34-38) linking H1 to H2 of the rear-most arch as well as a slight overall shift of the H3 helix and break between helices H3 and H4 that frame the entrance to the active site cavity. The C terminal arms after Pro98Pro99again align nearly perfectly to residue 109,with the final 4 C-terminal residues slightly out of register.While most sidechains are seen to adopt identical conformations in chains A and B, several sidechains situated on the surface and at the entry to the active site cavity orientdifferently, including Gln37 which either orients towards the active site cavity (chain A) or folds away from the cavity (chain B).§ RESULTS Each chain consists of 845 heavy atoms and obtains 112 ψ dihedrals, 101 ϕ dihedrals (the 10proline ϕ dihedrals as well as the N terminal ϕ_1 are fixed), and 165 χ sidechain dihedrals, or 378 dihedral coordinates per chain. Along with the 6 inter-monomer degrees of freedom therefore the dimerobtains a total of762dihedral coordinates versus 5064 Cartesian coordinates: a nearly 7-fold reduction in coordinate space that leads to a significant reduction in matrix size. Chain A obtains 4609 intra-chain NBI and chain B 4666 intra-chain NBI between nonbonded atom pairs at least 4 bonds lengths apartand less than the cutoff distance defined by the inflection point of their van der Waals curves <cit.>; the atom pair forming the longest-range interaction having a distance of separation of 4.78Å.The average number of NBI per atom for chain A is 9.74 and for chain B is 9.86. The chains have zero cysteine residues and no disulfide bridges.The number of inter-chain interactions between chains A and B, using identical selection criteria as for the intra-chain NBI, is 714 of which 37 are 2-3Å apart, 291 are 3-4Å apart, and 386 are 4-5Å apart.After matching theanalytically derivedto numerically computed terms of the dimer's Hessian matrix, we found that the resultant eigenfrequencies vary smoothly from 5-600 cm^-1, with no anomalous or negative values. The solid line in Fig. <ref> shows the distribution of these frequencies as a histogram with bin width of 5cm^-1. To ascertain the effects of dimerization on the eigenspectrum signature, we plot two additional curves. The dashed curve is the histogram of the eigenfrequencies of the isolated chain A summed with the histogram of the isolated chain B (and therefore obtains 6 fewer modes). To compare these curves to the`universal' curves obtained for singly folded protein chains <cit.>, we sought a PDB entry with similar hallmarks as the dimer: a single chain consisting ofroughly 2×112 residues folded in a TIM barrel fashion with a central eight-stranded β-barrel surrounded by a series of α helices. We identified a high resolution 1.3Å isomerase with 244 residues folded intosuch a classic TIM barrel fold:PDB entry 2Y88 <cit.>. The eigenspectrum distribution of this protein is shown in the inset of Fig. <ref>.The inset plot for PDB entry 2Y88 demonstrates the typical or universal character of vibrational eigenspectra of single-chain crystalline proteins:a sparse concentration of slow modesrapidly rising to a densest concentration of modes in the 55-60cm^-1 region and then falling off with a broad shoulderbeyond 140cm^-1. The sum of the isolated 1LQ9A and 1LQ9B spectra, shown as the dashed line in Fig. <ref>, deviates from this pattern in having an anomalous concentration of slow modes, pushing the maximum towards 20cm^-1. This is not surprising, as the isolated chains A and B represent unstable configurations with unsupportedN- and C-terminal arms creating anomalous vibrational patterns. Dimerization, it is seen in Fig. <ref>, tightens these slowest modes with a decrease in the density in this region,but still retains a greater buildup of slow modes relative to isolated chains. This is not unexpected, as interfaces generally cannot obtain a rigidity greater than their constituent bodies. We next investigated the character of the eigenmodesthat give rise to these computed frequencies. TheRMS fluctuations of all α carbon atoms at room temperature per mode i, RMS^i, are plotted in the inset of Fig. <ref> for the first two hundred modes, with the y-axis extending from 0-0.15Å. The first three modes contribute 63% to thetotal 0.24Å (summed over all modes),while the higher frequency modes contribute roughly0.02Å per mode, beyond the first 40 modes. This plot,typical for protein systems, gives a rough indication of the persistence length associated with each eigenmode: modes with longer correlation lengths obtain larger RMS^i values. For PDB-NMA using simplified Hookean formulations, it is not unusualfor this plot to display an irregular decay in amplitude, with one or more slow modes obtaining vanishingly small magnitudes. This irregularity isabsent when using standard classical potential energy formulations on energy-minimized structures. An examination of such anomalous modes typically indicates an inadequate “knitting”of the structure's motility, with some regions, such as surface side chains, moving independently of the rest of the structure, although structural pathologies, such as steric clashes, will also give rise to this anomalous decay pattern. ATMAN parameterizes every bonded and nonbonded interaction according to the parent potential, L79 <cit.>, and successfully reproduces results from classical, energy-minimized NMA on single chains <cit.>. As the dimer interface of 1LQ9 obtains an extensive, hydrophobic and core-like packing arrangement, no further adjustments to the nonbonded interaction list were required:theRMS^i plotdecays smoothly, as expected, for the slowest 40 modes, with some discontinuities appearing thereafter.As the current formulation ignores bond angle and bond length degrees of freedom that likely contribute at these higher frequencies,we report only on the slower modes.Thecomputed B factors of each C_α due to the combined effect of the dimer's 40 slowest modes, unscaled and in units of Å^2, are shown in Fig. <ref> forchain A (blue) andchain B (red).The five β strands in each molecule obtain B values under 3Å^2, reinforcing the description of these sheets as static floors (a B factor of 5Å^2 corresponds to a mean displacement from equilibrium of 0.25Å). The four α helices, the midpoint of the sequence at Thr55-Thr58,as well as the C-terminal arm and the loop connecting the N terminal “knob” to B1obtain B factor values over 5Å^2, with the break between H1 and H2 at the back of the stage and H4 plus the loop leading to B4 at the front especially mobile, with B values greater than 9Å^2. Fig. <ref> demonstrates that two identical subunits with a mainchain RMS overlap of 0.3Å do not necessarily obtain identical vibrational signatures. The computed B factors of each C_α for chains A and B obtain a correlation of 0.91, with residues 34-38 connecting H1 and H2 in subunit Bobtaining a peak B value nearly twice that of chain A: 17Å^2 versus 8Å^2. In addition, the loop connecting N to B1of chain A as well as the loop connecting H4 to B4 in chain B appear to be more mobile than their partners.Are these trends supported by the experimentally deduced B factors reported in the PDB file? Fig. <ref> shows the unscaled isotropic experimental B factors for each C_α atom for chain A (cyan) and chain B (orange).The experimental B values for chains A and B have a correlation of 0.13, with the motility of the arched helices H1H2 and H3H4 not well matched.Experimental B factors obtain contributions not merely from intra-dimer, thermal vibrations, but also from numerous other sources includingbulk solvent, inter-dimer (packing) vibrations, disorder such as mosaicity, and possible systematic errors <cit.>. Generally, the non-vibrationary sources are homogeneous and provide a uniform noise level to the atomic coordinates. Crystal vibrations span wavelengths delimited by the maximum crystal dimensionand the minimum unit cell dimension, and therefore contributea slowly varying “noise” within the unit cell. For this reason, variations in the B factors for the atoms comprising the asymmetric unit are ofinterest. Within the asymmetric unit,the relative contribution of plasticity (inter-minimum mobility) versus elasticity (intra-minimum mobility) have been examined in an effort to quantify and characterize the structural heterogeneity withinprotein crystals <cit.>. The current PDB-NMA demonstrates the degree to which a strictly elastic response of the reported structure to thermal perturbations reproduces the experimentally deduced B factors. And indeed, the experimental B factors in Fig. <ref> broadly support the theoretical, intra-dimer predictions, with the β strands obtaining smaller B values than either the helices, the midpoint Thr55-Thr58 or the N and C terminal regions, and the distinct peak at 34-38 for chain B also evident.To more clearly ascertain the overlap of the experimental and theoretical B factors,we superpose the scaled theoretical B factors onto the experimental curves for chains A (Fig. <ref>A) and B (Fig. <ref>B). A uniform scale factor of 3 applied to the computed values brings the minima of each curve into alignment in order to highlight the relevant correlations. For thisPDB entry the correlationsare reasonably high, with the 40 slowest intra-dimer modes reproducing the observed motility trends of the various secondary structure elements in each case. For chain A, the correlation of the experimental and computed B factors is 0.55, withover-estimates for the motilities of theH1H2 arch, the midchain 55-58 loop, and the H4 to B4 regionwhilethe C terminal region has a higher experimentally deduced motility. Packing interactions may partly explain these difference. 1LQ9 crystallized in an orthorhombic space group (P2_12_12_1) where each dimer obtains 4 unique packing interactions that introduce a total of 281 NBI (using identical selection criteria as for the intra-chain NBI),of which 6 are 2-3Å apart, 127 are 3-4Å apart, and 148 are 4-5Å apart. These packing interactions, relatively few and weak in comparison to the intra- and inter-monomer interactions, are typically not included in PDB-NMA. Visual inspection of the 1LQ9 model and its symmetrically related subunits shows that the region of closest approach is between the external surface of the H3H4 arch of a B subunit that wedges into thebinding site aperture of an A chain. This packing likely inhibits the vibrations in the 1LQ9 crystal of the front arch of chain A, and helps explain the lower observed motility of these regions.Chain B obtains acorrelation of 0.81 between the experimental and computed B factors,including an excellent overlap for the 34-38 loop and close matches for the midchain loop and the N and C termini. The greater motility predicted than observed for the H3H4 helical arch at the front of the ligand aperture is perhaps also due to the packing arrangement that stacks the outer arches of dimers together. The relatively high correlations of the experimental to the theoretical B factors for chains A (0.55) and B (0.81), in contrast to the lower correlation of the experimental curves comparing chains A and B (0.13), invites an examination of the character of those modes of the dimer that contribute to this asymmetry. As the slowest 3 modes contribute 63% to the total computed RMS, we examined the character of each of those modes via 3d PyMol animations. We plot the computed mean square motility, scaled to the experimental B factors, of each Cα atom for modes 1, 2 and 3 and for subunits A (blue) and B (red) in Fig. <ref>A, B and C.Mode 1 presents as a flexing or yawing of the two subunits, with the open, barrel-type construct between the two subunits (the front of the illustration in Fig <ref>) pulling apartand the C and N terminal regions at the opposing end of the barrel-constructcoordinating this flexing while maintaining, along with the Thr55-Thr58 midchain motif, thestructural integrity of the dimer. The contribution of β strands B5 to mode 1 is distinct, with the 7 hydrogen bonds linking each B5 strand with the opposing chain's B2 strand strained, but in a non-equivalent manner. AB5 remains in register with BB2 while straining the interstrand bonds laterally away from the plane of the β sheet floor, while the interstrand distance of separation remains fixed between BB5 and AB2 as those two strands display a tendency to slide past each other. The source of this variable flexibility appears to be related tothe manner in which each C terminal packs Ser113. In subunit B, Gln37 folds away from the binding cavity towards the subunit interface,which simultaneously extends Arg41 to interact with ASer113,latching this C terminal residue, along with Glu38 and Glu41, in a “ERE pocket” (Fig. <ref>).In subunit A, in contrast, Gln37 extends away from the interface toward the binding cavity, Arg41 is pulled away fromBSer113, the ERE pocket does not form and BSer113 is not similarly latched but is insteadsolvent exposed.This variable packing of the C termini results in a variable transmission of motility, with the AB5 strand, latched at ASer113 to the neigboring ERE pocket in BH2, vibrating in sync with that α helix and thence also the neighboring BH4 region. Meanwhile, the unlatched and less restrained BB5 strand swings out of tandem with AH2, hammer-style, with its Ile110 sidechain like a peen strikingthe extended AGln37 sidechain. So for example, the distance of separation between CD1 of BIle110 and OE1 of AGln37 in the crystal structure is 2.8 Å and varies from 2.6Å to 3.1Å due to mode 1, while simultaneously, at the other subunit, the distance of separation between the CD1 of AIle110 and OE1 of BGln37 remains at a steady 8.2Å.This feature, then, explains the asymmetry between the computedB factors of chains A and B of Fig. <ref>A: when yoked to the neighboring subunit's H2 motif at the ERE pocket,the vibrational motilityof the C terminal β strand adds to that of the H2 helix and thence also to H4, via the base stacking ofTrp39 with Phe76 for example, and numerous other noncovalent interactions between the H1H2 and H3H4 arches.The C terminal Ser113 that does not latch into the ERE pocketdoes not vibrate in tandem with H2 and does not contribute as effectively to that region's displacement in this mode of oscillation. Higher order modes, it will be seen, reveal various motility patterns of the arched helices, but only mode 1 reveals a high motility of the C terminal arms and the midchain peaks as well as the distinct chain A versus B asymmetry at H2 (Fig. <ref>A).In higher order modes the B5 strands oscillate in tight synchrony with the B2 strands of the opposing chain, no yawing of the barrel-construct between the chains is apparent, and the midchain peaks are likewise damped. The source of the experimental H2 B factorasymmetry, therefore, may be due to a single mode, the softest dimer mode, and the variable packing of the C termini which either effectively transmits the intrinsic vibrational propensities to H2, or not.Mode 2 presents as a sidewise tilting of the dimer that does not distort the central-barrel motif, but instead swings the active site roofs, especiallythe front arches H3H4, forward. This motility seems to be mediated by the B4B5 junction with Pro106 serving as a hinge to allow the relative reorientation of B4 and B5. While the character of the motility at the two active site cavities is identical,their amplitudes of activation are not. This asymmetry is likely due to the variable packing within the active site cavitiesof residues such as Glu37 at the rear and Arg86 at the entrance of the active site.In subunit B, for example, Arg 86 is fully extended and directed into the active site cavity, while in subunit A it folds away from the active site cavity, towards the exterior surface of the protein.The result is that the vibrational character of mode 2 isdifferently expressed in subunits A and B, with the packing arrangement within the active site of subunit B seeming to frustrate the motility expressed in subunit A.Mode 3 presents as a type of twisting of the two chains resulting once again in a high mobility of the arches, H3H4, framing the active site cavity. Rather than tilt forwards as in mode 2, in mode 3 the arches are seen to tilt sidewise in a manner that distorts the shape of the active site entrance. Interestingly,the vibrational character of mode 3 seems to favor subunit B, with BH4 and the subsequent loop to BB4 obtaining noticeably larger amplitudes of vibration than in subunit A (Fig. <ref>C).It is interesting to note that Pro80 and Pro93 bracket α helix H4 while Pro93 and Pro98Pro99 similarly bracket the loop connecting H4 to B4.Prolines disrupt mainchain hydrogen bonding patterns and also reduce the instrinsic flexiblity of the polypeptide chain by eliminating the ϕ rotational degree of freedom, and thus are well suited to isolate regions with flexibility characteristics distinct from neighboring regions, as seen in mode 3. § DISCUSSION The PDB-NMA of entry 1LQ9 contributes new insights into the structure/function relationship of ActVA-Orf6.The crystallographically determined atomic model implicates two gates for the reaction mechanism for this homodimeric enzyme <cit.>. A proton gate consisting of Arg86 at the entrance to the active site, together with Tyr72 inside the active site cavity, are thought to shuttle the proton extracted fromsubstrate to bulk solvent. Awater gate consisting of Gln37 in the mobile 34-38 loop opens a narrow passage for diffusion of molecular oxygen from the bulk solvent or of a water molecule to the bulk.As evidence, Sciara et al point out that only the extended Gln37 conformation in subunit A forms a hydrogen bond with Tyr51, a residue implicated in catalysis, whereas in subunit Bthis residue folds back and a water molecule replaces Gln37 in its interaction with the hydroxyl of Tyr51. The crystal model <cit.> together with biochemical data <cit.> also demonstrate that disruption of the dimeric interface eliminates catalytic activity: the enzyme functions only as a dimer. In addition, the crystal structures of the enzyme with different ligands demonstrate that in the crystals, subunits A seem to bind product-like analogues while subunits B seem to bind substrate-like analogues <cit.>.Crystallographically deduced B factors reveal that the two chains of this homodimer obtain distinct mobility signatures for the H1H2 and H3H4 arches, with especially the mobile 34-38 loop connecting H1 and H2in subunit B obtaining significantly higher motility than in subunit A. Without a vibrational analysis of the proposed structure, however, the source of the variation and asymmetry of the B factors cannot be discerned. We find that thePDB-NMA of the 1LQ9 model reproduces themotility patterns for chains A and B,suggesting that thedistinct B factor signatures are not an artefact of crystal packing interactions but rather an innate feature of the chains. An examination of themodes of oscillation intrinsic to this homodimerprovides a mechanistic explanation of the B factors that implicates a special role for the C terminal β strands.Taken together, the biochemical, crystallographic and vibrational analyses demonstrate the cooperative nature of the interaction of the two subunits in this homodimer. Subunit A, bound with product and with Gln37 extended to interact with Tyr51 in the active site, retracts Arg41 and does not form the ERE pocket to restrain the swinging of subunit B's C terminal arm. The resultantoscillations of the C terminal hammer hasthe Ile110 peenstrike Gln37,which serves perhaps to dislodge product and simultaneously to reorient the Gln37 residue to the retracted orientation compatible withsubstrate binding. The retraction of Gln37 simultaneously extends Arg41 that therefore snags the C terminal arm of the other subunit, whose vibrational motilitythen contributes to thevibrational propensity of the arched helical roof.This sequence of events would sequentially interconvert subunit A to subunit B. The motilities presented in mode 1 appear consistent withthe proposed Gln37 water gate, while modes 2 and 3 might serve to expedite entry and exit of ligands. Crystal structures reveal only dimers with distinct subunits, A and B, and neverwithidentical orientations of both subunits,suggesting that the chains catalyze substrate sequentially. How this higher level of cooperativity might be achieved is unclear, although it might involve the tight dynamical coupling and rich communication between the C terminal arms across the N terminal knobs. The importance to enzymatic activity of residues that confer specific dynamic signatures, such as Ser112, Arg41, Pro112 or Pro106, could be tested by site directed mutagenesis studies. § CONCLUSION The formalismfor macromolecular NMApresented in LSS <cit.> is completely general and applies to multimeric systems using mixed, internal and Cartesian, coordinates. An application of PDB-NMA to a high resolution homodimer provides a plausible representation of internal flexibility of the complex, and requires no special reformulation of the inter-protein potential energy expression presented in Tirion & ben-Avraham <cit.>. The reliability of the computation for this multimeric system is demonstrated by internal consistency checks that include matching the numeric and analytic expressions of each term in the Hessian matrix; a diagonalization of the generalized eigenvalue equation that results in no negative eigenvalues; an eigenvalue histogram distribution that adheres to the universal character of stably folded proteins; and an RMS^i curve that smoothly decreases as a function of mode number. In addition to these internal consistency checks, the computed B factors exhibit a high degree of correlation to the experimentally deduced B factors, including the presence of a distinct peak in one of the chains.The precise pattern of covalent and noncovalent bonding deduced from crystallographic diffraction data therefore selects unique vibrational signatures consistent with the independently refined Debye-Waller factors.Vibrational analyses of PDB entries that maintain stereochemically refined bond lengths and bond angles and adhere to the character ofbonded and nonbonded pairwise atomic interactions as provided by standard energy potentials are feasible, both for monomeric and multimeric systems. PDB-NMA is a quick, concise and reproduceable way to characterize the vibrational propensities of stably folded protein systems, and provide a rich source of insight to further comprehend and model enzymatic activities.99tirion14M. M. 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"authors": [
"Monique M. Tirion",
"Daniel ben-Avraham"
],
"categories": [
"q-bio.BM"
],
"primary_category": "q-bio.BM",
"published": "20170825150702",
"title": "PDB-NMA of a Protein Homodimer Reproduces Distinct Experimental Motility Asymmetry"
} |
A Conservation Law Method in Optimization Bin Shi December 30, 2023 ========================================= We build the coherent states for a family of solvable singular Schrödinger Hamiltonians obtained through supersymmetric quantum mechanics from the truncated oscillator.The main feature of such systems is the fact that their eigenfunctionsare not completely connected by their natural ladder operators.We find a definition that behaves appropriately in the complete Hilbert space of the system, through linearised ladder operators. In doing so, we study basic properties of such states like continuity in the complex parameter, resolution of the identity, probability density, time evolution and possibility of entanglement. § INTRODUCTIONWhen studying quantum systems few tools have proven to be as fruitfulas those describing them through coherent states (CS). These states were originally introduced to characterise the most classicalbehaviour for the harmonic oscillator, despite the system being quantum in nature <cit.>. Since then they became a widespread tool in the understanding of quantum phenomena <cit.>.On the other hand, supersymmetric quantum mechanics is a spectral design method that allowsobtaining solvable Hamiltonians departing from an initial one whose solution is already known. Such a technique has been used to produce plenty of new solvable quantum systems <cit.>. Although most of the work has focused on non-singular potentials,recent efforts start to consider potentials with singularities in their domains <cit.>. In particular, the truncated oscillator,a system described by a harmonic oscillator potential with an infinite barrier at the origin, has shown to be an appropriate and simple model of this kind <cit.>. Moreover, there is a natural definition of ladder operatorsfor the supersymmetric partners of the truncated oscillator,even though the energy spectra of such systems are divided into subsets, that cannot be connected through the action of these operators. In fact, the spectral manipulation achievable for such a system ishighly affected by the boundary condition at the origin.Previously, it has been possible to define CS for the supersymmetric partners of non-singular potentials <cit.>. Thus, it is natural to proceed further and study such states for some singular cases. However, earlier works have shown that a reduction process on the ladder operators is required in order to obtain appropriate CS for a kind of energy spectrum that some supersymmetric partners possess. Linearised coherent states have been found to be a successful definition in these problematic cases <cit.>. This was done for non-singular systems connected with the Painlevé IV equation. In this work we shall extend the study as well to singular systems, that can be connected with the Painlevé V equation <cit.>.This work is devoted to study several alternative definitions of coherent statesfor the truncated oscillator and its supersymmetric partners.In order to achieve this goal, the article is organised as follows: In Section 2 we introduce the definitions of coherent states we are interested in. In Section 3 we study the coherent states for the truncated oscillator, while inSection 4 we analyse these states for its supersymmetric partners.Finally, in Section 5 we present our final remarks and conclusions.§ COHERENT STATESIn this section we will introduce in general terms several definitions of coherent states (CS)starting from a given set of operators {H,l^+,l^-} characterising the system under study,where H is a Schrödinger Hamiltonian and l^± are its ladder operators, that are not necessarily hermitian conjugate to each other. We suppose that these operators can be realised in a separable Hilbert space ℋ. In the Fock representation, an element of this basis is given by |k⟩,where k∈ℕ_0={0,1,2,...,d-1}, where d is either finite or infinite, and the action of these operators on a state |k⟩ is given by:H|k⟩ = ξ(k) |k⟩ ,l^-|k⟩ = √(f(k))|k-1⟩ , l^+|k-1⟩ = √(g(k))|k⟩ ,where ξ(k) is a growing function of k that definesthe energy eigenvalue associated to |k⟩and f(k) and g(k) are functions of k such that f(0)=0 and, if d is finite, g(d)=0. §.§ l^- coherent statesThe first definition of coherent states |z⟩ to be considered,the l^--CS, is given by the conditionl^- | z ⟩ =z|z⟩ ,where z∈ℂ, i.e., |z⟩ is an eigenstateof l^- with complex eigenvalue z.Through a standard procedure <cit.>, |z⟩can be formally expanded in the Fock basis as|z⟩=C_z∑_k=0^d-1z^k/√([f(k)]!)|k⟩ ,where C_z=[∑_k=0^d-1|z|^2k/[f(k)]!]^-1/2 is a normalisation constant, and [f(k)]!={1 fork=0 ∏_k'=1^k f(k') fork>0.is a generalisation of the factorial function. Normalisability of these states is attained as long as C_z remains finite. For finite d this fulfillment is immediate, but for an infinite dimensional Hilbert space it depends on the behavior of the function f(k) when k→∞.Notice that if the space ℋ is finite-dimensional (dim(ℋ)=dwith d finite), the operator l^- has a d× d-matrixrepresentation that is nilpotent of index d.Thus, the only solution to the exact eigenvalue equation (<ref>) is z=0 <cit.>. Such a result makes the condition (<ref>) unsuccessful for definingfamilies of CS in a finite dimensional space ℋ.Work has been done in order to avoid this problem for some physical models with such a finite energy spectrum (like the Morse potential <cit.>) by considering “almost eigenstates of l^-” . §.§ D_l(z) coherent statesAn alternative definition of coherent states, the D_l(z)-CS,is given by the action of the so-called displacement operator D_l(z) on an extremal state of the system, that we shall identify with |0⟩. The extremal states of a quantum system are those non-trivial eigenstates of H that are in the kernel of the annihilation operator l^-.Recall that the displacement operator for the harmonic oscillator is D_a(z)= exp(za^+-z^*a^-)=exp(-1/2|z|^2)exp(za^+)exp(-z^*a^-) ,where the second equation (the factorised expression)is derived from the commutation relation between a^- and a^+. For systems described by more general algebraic structures, we use the following definition for the displacement operator(possibly non-unitary) <cit.>D_l(z)=C̃_z exp(zl^+)exp(-z^*l^-).It means that for the systems of interest in this work,the D_l(z)-CS are formally given by:|z⟩=D_l(z)|0⟩ =C̃_z ∑_k=0^d-1√([g(k)]!)/k! z^k |k⟩.The normalisation condition requires again that C̃_z=[∑_k=0^d-1[g(k)]!/k!^2|z|^2k]^-1/2which, as mentioned before, relies on its finiteness. If ℋ is infinite-dimensional, this cannot be guaranteed in general, since the main quantity to be controlled for obtaining an appropriate normalisation is the ratio [g(k)]!/k!^2 and its behavior for k→∞. §.§ Linearised coherent statesIn order to reconciliate both definitions of CS (l^--CS and D_l(z)-CS) it has beenproposed to use a deformed version of the ladder operators l^± <cit.> that linearises their commutator by using the associated number operator.Recently such a reduction was used to obtain a definition forD_l(z)-CS working well in both finite- and infinite-dimensional cases <cit.>.One of the reasons to look for a definition of coherent states thatworks well for both types of Hilbert spaces is to be able to generate themfor systems as those that will be described in the following sections. Indeed, let us introduce the linearised ladder operators ℓ^±through their action on the Fock basis:ℓ^-|k⟩=√(α k)| k-1⟩ ,ℓ^+|k-1⟩=√(α k)|k⟩ . In the case of an infinite-dimensional Hilbert space the generalised factorial becomes [f(k)]!=k! α^k and thus the ℓ^--CS are given by:|z⟩=C_z∑_k=0^∞1/√(k!)(z/√(α))^k|k⟩,where C_z=exp(-|z|^2/2α). As indicated in section <ref>, in the case of a finite dimensional Hilbert space such a definition of ℓ^--CS gives only the state |0⟩.On the other hand, having linearised the ladder operatorsmakes the normalisability of the D_ℓ(z)-CS straightforward.In fact, the linearised D_ℓ(z)-CS are given by|z⟩=C̃_z ∑_k=0^d-11/√(k!) (√(α)z)^k |k⟩ ,whereC̃_z=[exp(α |z|^2)-|z|^2dα^d/Γ(d+1) _2F_2(1,d+1;d+1,d+1;α|z|^2)]^-1/2 and d= dim(ℋ). In the infinite-dimensional case we have that C̃_z=exp(-α |z|^2/2).Note that, throughout this article _pF_q(a_1,...,a_p;b_1,...,b_q;x) represents the generalised hypergeometric function.From equations (<ref>) and (<ref>) we can see that when the Hilbert space is infinite dimensional (d=∞) and α=1 both definitions coincideand the standard expression for the CS for the harmonic oscillator is reproduced.§ TRUNCATED OSCILLATOR AND COHERENT STATESLet us remember that a quantum harmonic oscillator truncated by an infinite barrier at the origin is described by the Schrödinger HamiltonianH_0=-1/2 d^2/ dx^2+V_0(x), whereV_0(x) = {[x^2/2 if x > 0;∞ if x ≤ 0;].,is the potential of the system. The general solution to the stationary Schrödinger equation H_0u(x)=ϵ u(x)for x>0 is a linear combination of two solutions of definite parity, an odd and even one, given by:u(x,ϵ)=e^-x^2/2[ _1 F_1(1-2ϵ/4;1/2;x^2)+2νΓ(3-2ϵ/4)/Γ(1-2ϵ/4) x _1 F_1(3-2ϵ/4;3/2;x^2)] .Moreover, the boundary conditions of the problem, ψ_k(0)=ψ_k(∞)=0,make the admissible eigenfunctions of H_0 to be ψ_k(x)=A_k xe^-x^2/2 _1_1(-k;3/2;x^2) =B_ke^-x^2/2 H_2k+1(x) ,where A_k=[√(π) 4^k-1 (k!)^2/(2k+1)!]^-1/2,B_k=[√(π) 4^k (2k+1)!]^-1/2and H_n is the n-th Hermite polynomial, with corresponding eigenvalues E_k=2k+3/2, k=0,1,... Note that the states ψ_k of the truncated oscillator are relatedto the states ψ_n^ HO of the harmonic oscillator restrictedto the domain (0,∞) by ψ_k=√(2) ψ_2k+1^ HO.The system described by H_0 has the natural ladder operators l^±=(a^±)^2,where a^± are the ladder operators for the standard harmonic oscillator. We thus get the commutation relations: [H_0,l^±]=± 2l^± ,[l^-,l^+]=4H_0 .Their action on a Fock state |k⟩ associated to the eigenfunction ψ_k(x)=⟨ x|k⟩ of equation (<ref>) is given by l^-|k⟩=√(2k(2k+1)) |k-1⟩ ,l^+|k-1⟩=√(2k(2k+1)) |k⟩ .It means that for this system we have f(k)=g(k)=2k(2k+1). §.§ Coherent states and their propertiesSuch a system has an infinite dimensional Hilbert space ℋ_0, and we examine next the three definitions of CS presented in section <ref>.The l^--CS are found as|z⟩=C_z∑_k=0^∞ z^k/√((2k+1)!) |k⟩,where C_z=[sinh(|z|)/|z|]^-1/2.The D_l(z)-CS become:|z⟩=C̃_z ∑_k=0^∞ √((2k+1)!)/k! z^k |k⟩ .Such states are normalisable only for |z|<1/2, where the normalisation constantis C̃_z=(1-4|z|^2)^3/4, making this definition a poor choice. The third choice leads to the linearised CS already given inequations (<ref>) and (<ref>) when α=2.Let us study next some properties of the l^--CS. First of all, through a standard procedure <cit.>, we can show that these coherent states are continuous in the complex parameter z. Even more, they provide a resolution of the identity in ℋ,1=∫|z⟩⟨ z| μ(z) dz, where the integral measure isμ(z)=|z|^2/8 π C_z^2exp(-|z|) . The probability density distribution for a l^--CS |z⟩is given by P_z(x)=|⟨ x|z⟩|^2, where⟨ x|z⟩=∑_n=0^∞ ⟨ x|n⟩⟨ n|z⟩=C_z ∑_n=0^∞z^n/√((2n+1)!) ψ_n(x) . Plots for some particular examples of such a probability density can be found in figure <ref>. We could also compute the state probability, i.e., the probabilityp_n(z)=|⟨ n|z⟩|^2 of obtaining the eigenvalue ξ(n)as a result of an energy measurement when the system is in the coherent state|z⟩ (see equation (<ref>)), that is given byp_n(z)=C^2_z |z|^2n/(2n+1)! . The time evolution of a l^--CS is obtained through the expressionU(t)|z⟩=exp(-iH_0t)|z⟩ .Since the energy eigenvalues E_k are linear in k, these coherent states are temporally stable, i.e.,an arbitrary l^--CS evolves into another state that is also a l^--CS. It is worth mentioning that these properties were analysed also for the ℓ^--CSand the results do not pose great difference from those of the l^--CS.The linearised coherent states are continuous in z and temporally stable. They provide a resolution of the identity and their probability density shows well localisation similar to that of the l^--CS depicted in figure <ref>. §.§ Measures of entanglementNow, we shall turn to the question of whether these CS can be used to produce entanglement. To deal with this possibility let us study two measures of entanglement: the uncertainty relation and the linear entropy. For this we need to compute explicitly expectation values in the coherent states, obtained here as follows. For an observable O it is possible to obtain its expectation value in a l^--CS in the Schrödinger picture through the equation⟨ O⟩=⟨ z|O|z⟩=∑_m,n=0^∞ Λ_mn(z) ⟨ O_nm⟩ ,whereΛ_mn(z)=C_z^2 z^m(z^*)^n/√((2m+1)!(2n+1)!)and ⟨ O_nm⟩=∫ ψ^*_n(x) O ψ_m(x)dx. In order to simplify future calculations let us use the fact that O is hermitian to rewrite equation (<ref>) as follows⟨ O⟩=∑_n=0^∞ Λ_nn(z) ⟨ O_nn⟩+∑_n=1^∞∑_m=0^n-1 2Re(Λ_mn(z) ⟨ O_nm⟩) .For example, if the observable O is chosen as the Hamiltonian H_0, we obtain the expectation value of the energy ⟨ H_0⟩=1/2+|z|(|z|).Before moving on, let us recall that on the domain (0,∞) the momentum operator p=-i d/ dx is not an observable, as can be asserted by the fact that its deficiency indices are (1,0), and thus, by the deficiency theorem p has no self-adjoint extensions <cit.>. Nonetheless, we are still interested in studying the behavior of the expectation value of the operator p. §.§.§ Uncertainty relation.We can now compute the standard deviations of the position and momentum operators σ_x=√(⟨ x^2⟩-⟨ x⟩^2), σ_p=√(⟨ p^2⟩-⟨ p⟩^2),and ultimately their product σ_xσ_p. The lower bound of this product characterises the uncertainty relationfor the pair of operators x and p. Let us recall that, since the ladder operators are now l^±=(a^±)^2, then x and p cannot be written in a simple manner in terms of l^±.In order to compute the uncertainty relation for the operators x and p,we use the matrix elements⟨ x_nm⟩ = √((2n+1)!/(2m+1)!)(-2)^n-m-1Γ(n-m-1/2)/π (2n-2m)! × _2F_1[-2m-1,n-m-1/2;2(n-m)+1;2] , ⟨ x^2_nm⟩ = 1/2δ_nm+√( (2n+1)!(2m+1)!)/(n+m)!(1/2δ_n,m-1+δ_n,m+1/2δ_n,m+1) ,⟨ p_nm⟩ = i ⟨ x_nm⟩-i(2m+1)/π √((2n+1)!/(2m+1)!)(-2)^n-m(2n-2m-1)Γ(n-m-1/2)/(2n-2m+1)! × _2F_1[n-m+1/2,-2m;2(n-m+1);2] , ⟨ p^2_nm⟩ = 1/2δ_nm-2√(2m(2m+1))δ_m-1,n+2√((2n+1)!(2m+1)!)/(n+m)!(3/2δ_n,m-1+δ_n,m-1/2δ_n,m+1) , where n≥ m.Figure <ref> shows approximations of the corresponding standard deviationsand their product as functions of the modulus of the complex parameter z.These were obtained by truncating the infinite sums in equation (<ref>) up to the 30th term. We can see that the dispersion of the position xis always lower than that of the momentum p.However, as |z| increases both σ_x and σ_p approach the value 1/√(2) and thus their product tends to the value 1/2. For the linearised coherent states, figure <ref>shows the results of the approximations carried out up to the 30th term of (<ref>). Observe that σ_p is a decreasing function of |z| while σ_x is a growing function of |z|. For 0<|z|<1 the dispersion of the momentum is greater than that of the position and they become equal for |z|=1.For 1<|z| the dispersion of the position surpasses that of the momentum. Opposite to the results for the l^--CS,in this case we can see that the dispersions ofx and p do not approach to each other as |z| increases. However, once again the product σ_xσ_p tends to 1/2 as |z| increases. In any case, these graphs yield an uncertainty relation such thatσ_xσ_p≥1/2 ,and, since σ_x and σ_p are different from each otherin the case of linearised as well as non-linearised coherent states, we can say that there is squeezing phenomena <cit.>, that suggests the possibility of producing entangled states by means of these coherent states.§.§.§ Entanglement and linear entropy.As a measure of entanglement let us use the beam splitter tooland compute the linear entropy of the out-state <cit.>. By definition the beam splitter operator isℬ= e^θ/2(a^+be^iφ-ab^+e^-iφ), where θ is the angle of the beam splitter and φis the phase difference between the reflected and transmitted states, from which the reflection and transmission amplitudes are r=- e^-iφsin(θ/2), t=cos(θ/2), respectively.The beam splitter acts on an input that is a bipartite state| in⟩ yielding the out-state <cit.>:| out⟩=ℬ| in⟩ = e^θ/2(a^+be^iφ-ab^+e^-iφ)| in⟩ ,where the in-state is such that | in⟩∈ℋ_1⊗ℋ_2, a, a^+ and b, b^+ are the standard first-order ladder operatorsof the harmonic oscillator in each space of the tensor product. Notice that the operators K_-=ab^+, K_+=a^+b, K_0=1/2(a^+a-b^+b)generate a su(2) algebra.In our case, | in⟩=|0⟩⊗| z ⟩ where | z ⟩ is a l^--CS. We thus get| out⟩=𝒞_z∑_n 1/√((2n+1)!) z^n ℬ(|0,n⟩) . Using this last state we can define the density operatorρ_AB=| out⟩⟨ out|,and by taking a partial trace we obtain ρ_A.The linear entropy is thus S=1- Tr(ρ_A^2),and we know that S=0 corresponds to no entanglementwhile S=1 corresponds to maximal entanglement. Let us recall that |n⟩=√(2) |2n+1⟩_ HO; thusℬ|0,n⟩=2 e^θ/2(a^+b e^iφ-ab^+ e^-iφ)|1,2n+1⟩_HO=2 ∑_k=0^2n+21/k![e^iφtanθ/2]^k[(cosθ/2)^2n√((k+1)!(2n+1)!/(2n+1-k)!) |k+1,2n+1-k⟩_HO. .-tan(θ/2) e^-iφ√(2n+2)(cosθ/2)^(2n+2)√((k)!(2n+2)!/(2n+2-k)!) |k,2n+2-k⟩_HO ] where we have used the Baker-Campbell-Haussdorf formula for the su(2) algebra<cit.> generated by K_-=ab^+, K_+=a^+b, K_0=1/2(a^+a-b^+b): e^τ K_+-τ^*K_-= e^τ/|τ|tan|τ|K_+ e^-2(lncos|τ|)K_0 e^-τ^*/|τ|tan|τ|K_- . In terms of the reflection and transmission amplitudes we obtain| out⟩ = 𝒞_z∑_n=0^∞ z^n/√((2n+1)!) ∑_k=0^2n+2(-r/t)^k/k!e^i2kφ×[Q_1 |k+1,2n+1-k⟩_ HO+Q_2 |k,2n+2-k⟩_ HO] ,whereQ_1(r,t,n,k) = t^2n√((k+1)!(2n+1)!/(2n+1-k)!) , Q_2(r,t,n,k) = r t^(2n+1)√(2n+2) √((k)!(2n+2)!/(2n+2-k)!) .Remember that the bra-ket product is obtained by an integration over the domain (0,∞).In fact <cit.>(⟨α|β⟩)_ HO=∫_0^∞ e^-x^2H_α(x)H_β(x) dx =√(π)_2F_1[-α,-β;1-α+β/2;1/2]/2^1-α-βΓ(1-α+β/2) . Now the density operator ρ_AB=| out⟩⟨ out| is given byρ_AB=𝒞_z^2∑_m,n=0^∞ z^n(z^*)^m/√((2n+1)!(2m+1)!) ∑_k=0^2n+2∑_l=0^2m+2(-r/t)^k/k!(-r^*/t)^l/l! e^i2(k-l)φ ×[Q^*_2(r,t,m,l)Q_2(r,t,n,k) (|k,2n+2-k⟩⟨l,2m+2-l|)_HO. + Q^*_1(r,t,m,l)Q_2(r,t,n,k) (|k,2n+2-k⟩⟨l+1,2m+1-l|)_HO + Q^*_2(r,t,m,l)Q_1(r,t,n,k) (|k+1,2n+1-k⟩⟨l,2m+2-l|)_HO .+ Q^*_1(r,t,m,l)Q_1(r,t,n,k) (|k+1,2n+1-k⟩⟨l+1,2m+1-l|)_HO] . Computing now the partial trace ρ_A we get the linear entropy S=1- Tr(ρ_A^2). Figure <ref> shows numerical approximations for S(|z|), obtained by truncating the resulting infinite sum up to the 20th term.It is worth noticing that the plot is overall flat and it shows thatthe out-sate is entangled, although not maximally entangled. If we repeat the computations now for the linearised coherent states, the plot of S(|z|) behaves in general in the same way as with the l^--CS.§ THE SUPERSYMMETRIC TRUNCATED OSCILLATOR AND COHERENT STATESIn quantum mechanics, a supersymmetric transformation relatestwo Hamiltonians as follows <cit.>.Suppose that the Hamiltonian H=-1/2 d^2/ dx^2+V(x)is obtained through a q-th order supersymmetric transformation from aninitial Hamiltonian H_0=-1/2 d^2/ dx^2+V_0(x),i.e., there exists a differential operator Q of order q,called intertwining operator, such that the following relation holdsHQ=QH_0 . Up to an overall shift in the energy, the set of eigenvalues of H differsin a finite number, κ=κ(q), of elements from that of H_0.This means that the supersymmetric transformation has added and/or erasedlevels in the energy spectrum of H_0 to obtain the one of H.If the energy spectrum of H_0 is infinite countable, then the Hilbert space ℋspanned by the eigenfunctions of H is the direct sum of two subspaces:ℋ=ℋ_ iso⊕ℋ_ new, an infinite-dimensional one denoted by ℋ_ iso and a κ-dimensional subspace denoted by ℋ_ new. The eigenvectors of H are given by the union of those|E_n⟩∈ℋ_ iso and the ones |ℰ_j⟩∈ℋ_ new. Then, the energy spectrum of H is also given by a union of two sets. The first one is an infinite set given by {E_0,E_1,E_2,⋯} while the second one is the finite set {ℰ_0,ℰ_1,⋯,ℰ_κ-1}.If the initial system admits ladder operators l^±,then the system described by H has natural ladder operators inthe subspace ℋ_ iso given by the product Q l^± Q^†,where Q^† is the hermitian adjoint of Q. On the other hand, in ℋ_ new such a definition does not yieldin general operators that connect the eigenenergies in this subspace.However, in the following section we will use a reduction theorem that willallow us to obtain well-behaved ladder operators in ℋ_ new. §.§ Supersymmetric partners of the truncated oscillatorThe supersymmetric partners of the truncated oscillator <cit.>, obtained through a q-th order supersymmetric transformation,are described by Hamiltonians H=-1/2 d^2/ dx^2+Vwith potentials given by <cit.>V=V_0-{ln[W(u_1,...,u_q)]}” ,where W(u_1,...,u_q) is the Wronskian of q seed solutionsof the initial Schrödinger equation H_0 u_j=ϵ_j u_j,j=1,...,q, given in general by (<ref>). These u_j's do not need to satisfy any boundary conditions,although they must not introduce new singularities to the superpartner potential V.Without loss of generality, we can order the ϵ_j's as ϵ_1<...<ϵ_q. Moreover, as is usual we shall suppose that ϵ_q<1/2. Under these assumptions, the set of eigenvalues of H hasκ=[q/2] (the integer part of q/2)new elements corresponding to the eigenvalues ℰ_iadded by the non-singular supersymmetric transformation,which associated eigenfunctions indeed satisfy the boundary conditions <cit.>. Hence, the eigenvalues in ℋ_ iso form an infinite ladder{E_0,E_1,E_2,...} such that E_k=E_0+2k, where E_0=3/2,while in ℋ_ new the eigenvalues{ℰ_0,ℰ_1,...,ℰ_κ-1} are not in general equally spaced.Now, in order to define ladder operators working appropriately on both subspaces, we have to consider that the eigenvalues in ℋ_ neware related by ℰ_j=ℰ_0+2j.Thus, the partial spectrum {ℰ_0,ℰ_1,...,ℰ_κ-1} formsan equidistant set of eigenvalues, and by means of a reduction theoremapplied to the operators Q (a^±)^2 Q^†, where a^± are the standard first-order ladder operators of the harmonic oscillator, it is possible to obtain now sixth-order ladder operators L^± in the whole space ℋ that satisfy <cit.>L^+L^-=(H-1/2)(H-3/2) (H-ϵ_1)(H-ϵ_2) (H-ϵ_q-1-2 )(H-ϵ_q-2 ).Moreover, the commutation relations [H,L^±]=± 2L^± still hold, and they yield the following action of L^± on the basis vectors of ℋ.In ℋ_ iso:L^-|E_n⟩ =[(E_n-1/2)(E_n-3/2)(E_n-ϵ_1)(E_n-ϵ_2 ). ×.(E_n-ϵ_q-1-2)(E_n-ϵ_q-2 )]^1/2| E_n-1⟩ , L^+|E_n⟩ =[(E_n+1-1/2)(E_n+1-3/2) (E_n+1-ϵ_1)(E_n+1-ϵ_2). ×.(E_n+1-ϵ_q-1-2)(E_n+1-ϵ_q-2 )]^1/2| E_n+1⟩ . In ℋ_ new:L^-|ℰ_j⟩ =[(ℰ_j-1/2)(ℰ_j-3/2)(ℰ_j-ϵ_1)(ℰ_j-ϵ_2 )..×(ℰ_j-ϵ_q-1-2)(ℰ_j-ϵ_q-2 )]^1/2|ℰ_j-1⟩ ,L^+|ℰ_j⟩ =[(ℰ_j+1-1/2)(ℰ_j+1-3/2)(ℰ_j+1-ϵ_1)(ℰ_j+1-ϵ_2 )..×(ℰ_j+1-ϵ_q-1-2)(ℰ_j+1-ϵ_q-2 )]^1/2|ℰ_j+1⟩ . From these results we can see that L^- annihilates the eigenstatescorresponding to 3/2 and ℰ_0, while L^+annihilates the eigenstate corresponding to ℰ_κ-1.It is well known that the differential operators L^-, L^+ and H defining second and third degree polynomial Heisenberg algebras are determined by solutions of thePainlevé IV and Painlevé V equations respectively <cit.>. In <cit.> it was explicitly shown how different supersymmetric partners of the truncated oscillator,ruled by appropriate L^-, L^+ and H, are thus connected tothese non-linear second-order differential equations. In <cit.> coherent states were built for systems connected to the Painlevé IV equation.Here we will build coherent states for supersymmetric partners of the truncated oscillator that are ruled by both, the second and fifth degree polynomial Heisenberg algebras.When trying to define coherent states in ℋ using L^±, one realises thatthese ladder operators do not connect energy levels fromℋ_ iso with those in ℋ_ new,which justifies the direct sum decomposition of ℋ.Thus, we require a definition of coherent states appropriatefor finite- and infinite-dimensional spaces simultaneously. From section <ref> we know that the L^--CS are appropriate forℋ_ iso but failin ℋ_ new.The opposite happens with D_L(z) coherent states: while this definition is appropriate for ℋ_ newit fails in general in ℋ_ iso. Therefore, we will implement the definition of linearised coherent states from section <ref> for the supersymmetric partners of the truncated oscillator. As previously remarked, each subspace, ℋ_ iso or ℋ_ new,possesses one extremal state given by its lowest energy state, thus allowing one to apply the displacement operator in each case. Before doing this, however, we will illustrate with an example how the supersymmetry method applied to the truncated oscillator works.Let us consider the case where a fourth order (q=4)supersymmetric transformation has been implemented, with ϵ_1=-11/2, ϵ_2=-9/2,ϵ_3=-7/2, ϵ_4=-5/2. Then we get that κ=2, and the eigenstates in ℋ_ neware those associated to the eigenvalues ℰ_0=ϵ_2=-9/2 and ℰ_1=ϵ_4=-5/2. The supersymmetric partner potential thus obtained is given by V(x)=x^2/2-4(256 x^16-2560 x^14+10496 x^12-19584 x^10+27360 x^8-10080 x^6-10800 x^4+16200 x^2+2025)/(16 x^8-64 x^6+120 x^4+45)^2where x∈(0,∞).The subspace isospectral to the truncated oscillator ℋ_ iso is spanned byϕ_n =Qψ_n/√(∏_i=1^4(E_n-ϵ_i)),where ψ_n is given by equation (<ref>) and the intertwining operator Q=1/4( d^4/ dx^4+η_3(x) d^3/ dx^3+η_2(x) d^2/ dx^2+η_1(x) d/ dx+η_0(x))is a fourth-order differential operator whose coefficients are given byη_3 = -4 x (16 x^8-32 x^6+24 x^4+120 x^2+45)/16 x^8-64 x^6+120 x^4+45, η_2 = 6 (16 x^10-16 x^8+88 x^6-120 x^4+165 x^2-105)/16 x^8-64 x^6+120 x^4+45, η_1 = 4 x (-16 x^10+16 x^8-216 x^6+48 x^4+915 x^2+315)/16 x^8-64 x^6+120 x^4+45, η_0 = 16 x^12-32 x^10+360 x^8+240 x^6-795 x^4-7110 x^2+1935/16 x^8-64 x^6+120 x^4+45.The lowering operator L^- acts on the basis of this subspace as followsL^-|E_n⟩=√((E_n-1/2)(E_n-3/2)(E_n+11/2)(E_n+9/2)(E_n+3/2 )(E_n+1/2 ))| E_n-1⟩. On the other hand, the only two eigenfunctions in ℋ_ new are given byϕ_ℰ_0 = 4 √(3) e^-x^2/2 x (8 x^6-4 x^4+10 x^2+15)/√(π)(8 (2 x^4-8 x^2+15) x^4+45), ϕ_ℰ_1 = 2 e^-x^2/2 x (16 x^8+72 x^4-135)/√(3)√(π)(8 (2 x^4-8 x^2+15) x^4+45),associated to ℰ_0=-9/2, ℰ_1=-5/2, respectively. The lowering operator L^- acts on the corresponding basis vectors as followsL^-|ℰ_j⟩ =√((ℰ_j-1/2)(ℰ_j-3/2)(ℰ_j+11/2)(ℰ_j+9/2)(ℰ_j+3/2)(ℰ_j+1/2))|ℰ_j-1⟩.§.§ Linearised D_ℒ(z) coherent statesA linearisation of L^± can be obtained by definingthe following linearised ladder operatorsℒ^+:= γ(H) L^+ ,ℒ^-:= γ(H+2) L^- ,where γ(H)=[(H-1/2)(H-ϵ_1)(H-ϵ_2)(H-ϵ_q-1-2)(H-ϵ_q-2)]^-1/2. The action of ℒ^± on the basis vectors of ℋ_ iso isℒ^-|E_n⟩=√(E_n-3/2)| E_n-1⟩=√(2n)| E_n-1⟩, ℒ^+|E_n⟩=√(E_n+1-3/2)| E_n+1⟩=√(2n+2)| E_n+1⟩.Thus, the operator ℒ^- annihilates the state |E_0⟩,which is the extremal state in ℋ_ iso. Also, note that the commutators between ℒ^± and Hare those of a Heisenberg-Weyl algebra in this subspace[H,ℒ^±]=±2ℒ^±, [ℒ^-,ℒ^+]=2. Acting the displacement operator D_ℒ(z)=exp(-|z|^2)exp(zℒ^+)exp(-z^*ℒ^-) on the extremal state |E_0⟩ we obtain the desired linearised CS in ℋ_iso:|z⟩_iso=D_ℒ(z)|E_0⟩ =exp(-|z|^2)∑_n=0^∞(√(2)z)^n/√(n!)|E_n⟩. On the other hand, the action of ℒ^± onthe basis vectors of ℋ_new is given byℒ^-|ℰ_j⟩ = (1-δ_j,0)√(ℰ_j-3/2)|ℰ_j-1⟩=(1-δ_j,0)√(2j-Δ_1)|ℰ_j-1⟩, ℒ^+|ℰ_j⟩ = (1-δ_j,κ-1)√(ℰ_j+1-3/2)|ℰ_j+1⟩=(1-δ_j,κ-1)√(2j+2-Δ_1)|ℰ_j+1⟩ ,where Δ_1=3/2-ℰ_0. Note that ℒ^- annihilates the state |ℰ_0⟩, while ℒ^+ annihilates the state |ℰ_κ-1⟩ so that |ℰ_0⟩ is the extremal state in ℋ_ new. Although one needs to be cautious when these operators act on |ℰ_0⟩ and |ℰ_κ-1⟩,in general, the commutators in (<ref>) still hold when acting on the other basis vectors of ℋ_ new.Displacing now the state |ℰ_0⟩ gives usthe following CS in ℋ_new:|z⟩_new=D_ℒ(z)|ℰ_0⟩ =Ĉ_z ∑_j=0^κ-1(√(2)z)^j/j!√(Γ(-Δ_1/2 +j)/Γ(-Δ_1/2)) |ℰ_j⟩ ,whereĈ_z=_1F_1[-Δ_1/2;1;2|z|^2]-|z|^2κ 2^κΓ(2κ-Δ_1/2)/[Γ(κ+1)]^2 Γ(-Δ_1/2) _2F_2[1,κ-Δ_1/2;κ+1,κ+1;2|z|^2] . The D_ℒ(z)-CS are continuous in the complex parameter zand they lead to a resolution of the identity, where the integral measures in each subspace ℋ_ iso and ℋ_ new are given respectively byμ_ iso(z)=2/π , μ_ new(z)=2 Γ(-Δ_1/2)/πĈ_z^2 G^2,0_1,2(-(Δ_1+2/2) 0 , 0 | 2|z|^2 ) ,where G^m,n_p,q(a_1,...,a_p b_1,...,b_q | x) is the Meijer G-function.As in the non-supersymmetric case, these states are temporally stable.The mean value of the energy in each subspace reads⟨ H⟩_ iso=3/2+4|z|^2 ,⟨ H⟩_ new= ℰ_0+2|Ĉ_z|^2∑_j=0^σ-1j(2 |z|^2)^j/(j!)^2Γ(Δ_1/2-j) ,while the state probability is given byP_n(z)=(2|z|^2)^n/n!exp(-2|z|^2) ,P_j(z)=(2 |z|^2)^j-1/[(j-1)!]^2|Ĉ_z|^2/Γ(Δ_1/2+1-j) ,in ℋ_ iso and ℋ_ new, respectively.Consider again the previous example,where the levels ℰ_0=-9/2 and ℰ_0=-5/2have been added to the energy spectrum of the supersymmetric partner. The probability density P(x)=|⟨ x|z⟩|^2 in each subspace ofℋ is depicted in figure <ref>, showing fairly good localisation. We notice that for |z⟩_ iso this localisationis somewhat lost for small values of |z|.In the case of |z⟩_ new,for big values of |z| two distinct peaks occur, that however remain close to each other.Greater localisation is achieved for small values of |z| since the peaks start to merge.§.§ Measures of entanglementFollowing the results and notation in equation (<ref>) we obtain the matrix elementsΛ_m,n(z) =exp(-2|z|^2) (√(2)z)^m(√(2)z^*)^n/√(m!n!) , Λ_i,j(z) = Ĉ_z^2 (√(2)z)^i(√(2)z^*)^j/i!j! √(Γ(-Δ_1/2+i)Γ(-Δ_1/2+j))/Γ(-Δ_1/2) ,in ℋ_ iso and ℋ_ new, respectively.§.§.§ Uncertainty relation.Let us rely on the example introduced at the end of section <ref>, where a fourthorder supersymmetric transformation was carried out for the truncated oscillator. Recall that in this example ℋ_ new was two-dimensional, since the added eigenfunctions spanning it were those associated tothe eigenvalues ℰ_0=-9/2 and ℰ_1=-5/2,while the space ℋ_ iso remained infinite-dimensional, as it is isospectral to the truncated oscillator. Figure <ref> shows the behavior of σ_x,σ_p and their product σ_xσ_p,as functions of the modulus of the complex parameter z. In both subspaces we can observe squeezing phenomena: while in ℋ_ new the uncertainty in the momentum is alwaysgreater than the uncertainty in the position, in ℋ_ iso we find the squeezing of σ_x for |z|<1 and the squeezing of σ_p for |z|>1. This indicates once again that entanglement is attainable by means of these coherent states.§.§.§ Entanglement and linear entropy.Let us obtain the linear entropy in each of the two subspacesℋ_ iso and ℋ_ new.The in-states in these subspaces are given by | in⟩_ iso= e^-|z|^2 ∑_i=0^∞(√(2)z)^i/√(i!) |ℰ_0,E_i⟩ , | in⟩_ new=Ĉ_z ∑_j=0^κ-1(√(2)z)^j/j!√(Γ(-Δ_1/2 +j)/Γ(-Δ_1/2)) |ℰ_0,ℰ_j⟩ ,respectively. By means of equation (<ref>) we can obtain the out-states | out⟩_ iso, | out⟩_ new. Through a similar treatment as in section <ref>, for which φ=0,we find that, in coordinates representation the out-states are given by⟨x,y|out⟩_iso =𝒞̃_z ∑_j=0^∞ (√(2)z)^j/√(j!) ∑_l,m,n=0^∞1/l!m!n![tanθ/2]^n[ln(cosθ/2)]^m[- tanθ/2]^l ×∑_i=0^mmi (-1)^-i∂_x^n x^i ∂_x^i[x^l ⟨x|ℰ_0⟩] y^m+n-i⟨y|E_j⟩^(l+m-i) ,⟨x,y|out⟩_new =Ĉ_z ∑_j=0^κ-1 (√(2)z)^j/j!√(Γ(-Δ_1/2 +j)/Γ(-Δ_1/2)) ∑_l,m,n=0^∞1/l!m!n![tanθ/2]^n[ln(cosθ/2)]^m[- tanθ/2]^l ×∑_i=0^mmi (-1)^-i∂_x^n x^i ∂_x^i[x^l ⟨x|ℰ_0⟩] y^m+n-i⟨y|ℰ_j⟩^(l+m-i) , in ℋ_ iso and ℋ_ new, respectively, where 𝒞̃_z and Ĉ_z are normalisation constants.As before, we use the out-state to obtain the operatorρ=| out⟩⟨ out| and then we compute the linear entropy S=1- Tr(ρ_A^2).For the system described by the potential in equation (<ref>), a numerical approximation to the linear entropy in ℋ_ newas function of |z| can be found in figure <ref>.The infinite series were truncated at the 30th term.Although being flat, with values close to 0.5,this plot show evidence of entanglement, but not of a maximally entangled out-state. In the case of the subspace ℋ_ iso the behavior of the linear entropyis similar to that depicted in figure <ref>.§ CONCLUSIONSIn this work we studied several definitions of coherent statesfor the truncated oscillator and its supersymmetric partners. The main difficulty in defining coherent states for systems obtained through a supersymmetric transformation lays in the fact that the resulting energy spectrumis not completely connected by the action of the natural ladder operators, leading to a decomposition of ℋ as a direct sum of two subspaces:a finite dimensional one and other infinite dimensional.It has been found that the conventional CS definitions that work well for finitedimensional spaces fail when applied to infinite dimensional ones and vice versa. We surpassed such a difficulty by using a reduced linearised version of the natural ladder operators for the system. Thus we arrived to a common definition of coherent states, that can be applied to both subspaces. Consequently, we studied some properties of these coherent states. We found the production of squeezing in the position and momentum of these coherent states, as well as the arising of entanglement in the out-state of a beam splitter whose in-statewas one of these coherent states.§ ACKNOWLEDGMENTSThis work has been supported in part by research grants fromNatural sciences and engineering research council of Canada (NSERC). The authors acknowledge the financial support of the Spanish MINECO(project MTM2014-57129-C2-1-P) and Junta de Castilla y León (VA057U16). VS Morales-Salgado also acknowledges the Conacyt fellowship 243374 and the Department of Foreign Affairs, Trade and Development of Canada for the Emerging Leaders in the Americas Program (ELAP) scholarship provided.§ REFERENCES100 s26 Schrödinger E 1926 Naturwiss 14 664 k63a Klauder J R 1963 J. Math. Phys. 4 1955 k63b Klauder J R 1963 J. Math. Phys. 4 1958 g63a Glauber R J 1963 Phys. Rev. 130 2529 g63b Glauber R J 1963 Phys. 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A 65 032323x02 Xiang-Bin W 2002 Phys. Rev. A 66 024303 gk05 Gerry C and Knight P 2005 Introductory Quantum Optics (Cambridge University Press) | http://arxiv.org/abs/1708.08010v2 | {
"authors": [
"David J Fernández",
"Véronique Hussin",
"VS Morales-Salgado"
],
"categories": [
"math-ph",
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],
"primary_category": "math-ph",
"published": "20170826185602",
"title": "Coherent states for the supersymmetric partners of the truncated oscillator"
} |
]Fragmentation inside atomic cooling haloes exposed to Lyman-Werner radiationJ.A. Regan & T.P. Downes]John A. ReganE-mail:[email protected], Marie Skłdowska-Curie Fellow & Turlough P. Downes Centre for Astrophysics & Relativity, School of Mathematical Sciences, Dublin City University, Glasnevin, Ireland2016firstpage–lastpage [ [ December 30, 2023 ===================== Supermassive stars born in pristine environments in the early Universe hold the promise of being the seeds for the supermassive black holes observed as high redshift quasars shortly after the epoch of reionisation. suppression is thought to be crucial in order to negate normal Population III star formation and allow high accretion rates to drive the formation of supermassive stars. Only in the cases where vigorous fragmentation is avoided will a monolithic collapse be successful giving rise to a single massive central object. We investigate the number of fragmentation sites formed in collapsing atomic cooling haloes subject to various levels of background Lyman-Werner flux. The background Lyman-Werner flux manipulates the chemical properties of the gas in the collapsing halo by destroying . We find that only when the collapsing gas cloud shifts from the molecular to the atomic cooling regime is the degree of fragmentation suppressed. In our particular case we find that this occurs above a critical Lyman-Werner background of J ∼ 10 J_21. The important criterion being the transition to the atomic cooling regime rather than the actual value of J, which will vary locally. Once the temperature of the gas exceeds T ≳ 10^4 K and the gas transitions to atomic line cooling, then vigorous fragmentation is strongly suppressed. Cosmology: theory – large-scale structure – first stars, methods: numerical§ INTRODUCTION Supermassive black holes (SMBHs) populate the centres of massive galaxies <cit.> and shine as extremely luminous quasars out to high redshift <cit.>. Their progenitors are unknown, however, with competing theories for whether the progenitors of SMBHs were stellar mass black holes or supermassive stars with initial masses of M_*,init≳ 10^4 <cit.>. Stellar mass black holes formed from the remnants of the first stars (PopIII stars) must accrete at the Eddington limit for their entire history to reach the billion mass threshold by a redshift of 7 if they are to be the seeds of the first quasars. This scenario appears exceedingly difficult as the first stellar mass black holes are born “starving” <cit.> in mini haloes which have been disrupted by both the ionising radiation from PopIII stars and the subsequent supernova explosions <cit.>. As a direct result, investigation of supermassive star (SMS) formation as a viable alternative has been undertaken and appears attractive <cit.>. The larger initial seed masses of SMSs alleviate the growth requirements somewhat <cit.>. More fundamentally however, is the fact that the SMS is born in a much larger halo compared to the canonical PopIII case. SMS formation is thought to be possible only in pristine atomic cooling haloes with virial temperatures T_vir≳ 10^4 K with accretion rates exceeding Ṁ≳ 0.1 /yr <cit.>. The larger halo leads to a much deeper potential well relative to the minihalo case, the SMS is also expected to collapse directly into a black hole skipping the supernovae phase and what's more the radiation spectrum expected from a SMS peaks in the infrared rather than the UV as is the case for PopIII stars<cit.>. Taken together, SMSs appear to offer far more favourable conditions for being the original seeds for SMBHs. The conditions for achieving SMS formation are, however, not yet fully understood. It is widely suspected that high mass accretion rates combined with high temperature gas will lead to the formation of a SMS. This can be achieved in a number of ways. Lyman-Werner (LW) radiation from a nearby galaxy will disrupt abundances removing a coolant, allowing a halo to grow sufficiently without PopIII formation <cit.>. Haloes which accrete unusually rapidly through a succession of mergers provide another pathway<cit.> as do streaming velocities<cit.> left over from the decoupling of baryons and dark matter after recombination. A combination of one or more of these scenarios is also possible. We focus here on haloes which are exposed to a nearby source of LW radiation as it illustrates the necessary points and is straightforward to implement. The fragmentation of gas inside a collapsing halo could potentially disrupt the formation of a SMS if the gas were to fragment into sufficiently distant clumps to avoid the formation of a single (or small multiple) of very massive stars at the centre of a collapsing atomic cooling halo. We note that any mechanism (with or without a LW component) which suppresses PopIII star formation and induces the same kind of thermodynamic change to the gas as an external LW field will also likely impact fragmentation in the same way. While the fragmentation of primordial haloes has been thoroughly investigated for the case of PopIII formation<cit.> it has not been as systematically probed for the case of irradiated haloes (but see <cit.>, <cit.> (L14b), <cit.> and <cit.> for some preliminary work). Here we investigate this scenario more systematically, varying the intensity of the radiation and investigating the degree of fragmentation as a function of resolution and radiation intensity.The goal of this study is to identify fragmentation sites in collapsing metal-free atomic cooling haloes which are irradiated by a uniform LW background. As noted above these pristine haloes are expected to be ideal cradles for SMS formation if the abundances can be suppressed. If fragmentation levels are reduced in the face of impacting LW radiation, as expected, this would be favourable for monolithic collapse of a single massive object. We probe this scenario in this work. The paper is laid out as follows: in <ref> we describe themodel setup and the numerical approach used as well as introducing our star particle formulation; in <ref> we describe the results of our numerical simulations; in <ref> we discuss the importance of the results and present our conclusions.Throughout this paper weassume a standard ΛCDM cosmology with the following parameters<cit.>, Ω_Λ,0= 0.6817,Ω_ m,0 = 0.3183, Ω_ b,0 = 0.0463, σ_8 = 0.8347 and h = 0.6704.We further assume a spectral index for the primordial density fluctuations of n=0.9616.§ NUMERICAL FRAMEWORK In this study we have used the publicly available adaptive mesh refinement code [http://enzo-project.org/] to study the fragmentation properties of gas within haloes irradiated by a background LW field. Into we have added a new star particle type which we have dubbed . We now describe both components.§.§ Enzo[Changeset:fedb30ff370b] <cit.> is an adaptive mesh refinement code ideally suited for simulations of the high redshift universe. Gravity in is solved using a fast Fourier technique <cit.> which solves the Poisson equation on the root grid at each timestep. On subgrids, the boundary conditions are interpolated to the subgrids and the Poisson equation is then solved at each timestep. Dark matter is represented using particles, each particle is stored on the highest refinement grid available to it and thus the particle has the same timestep as the gas on that grid. The particle densities are interpolated onto the grid and solved at the same time as the gas potential. contains several hydrodynamics schemes to solve the Euler equation. We use the piecewise parabolic method which was originally developed by <cit.> and adapted to cosmological flows by <cit.>. The PPM solver is an explicit, higher order accurate version of Godunov's method for ideal gas dynamics with a spatially third accurate piecewise parabolic monotonic interpolation scheme employed. A nonlinear Riemann solver is used for shock capturing. The method is formally second order accurate in space and time and explicitly conserves mass, linear momentum and energy making the scheme extremely useful for following the collapse of dense structures. Chemistry is an important component is following the collapse of (ideal) gas. We use the [https://grackle.readthedocs.org/]^,[Changeset:482876c71f73] <cit.> library to follow the evolution of ten individual species: H,H^+,He,He^+, He^++,e^-,H_2, H_2^+ H^-and HeH^+. We adopt here the 26 reaction network determined by <cit.> as the most appropriate network for solving the chemical equations required by gas of primordial composition with no metal pollution and exposed to an external radiation source. The network includes the mostup-to-date rates as described in <cit.>. The cooling mechanisms included in the model are collisional excitation cooling, collisional ionisation cooling, recombination cooling, bremsstrahlung and Compton cooling off the CMB. §.§ Simulation SetupAll simulations are run within a box of 2 (comoving),the root grid size is 256^3 and we employ three levels of nested grids. The grid nesting and initial conditions were created using MUSIC <cit.>. Within the most refined region (i.e. level 3) the dark matter particle mass is ∼ 103 . In order to increase further the dark matter resolution of our simulations we split the dark matter particles according to the prescription of <cit.> and as described in <cit.>. We split particles centered on the position of the final collapse as found from lower resolution simulations within a region with a comoving side length of 43.75 h^-1 kpc. Each particle is split into 13 daughter particles resulting in a final high resolution region with a dark matter particle mass of ∼ 8 . The particle splitting is done at a redshift of 40 well before the collapse of the target halo. Convergence testing to study the impact of lower dark matter particle masses was discussed in <cit.>. The baryon resolution is set by the size of the grid cells, in the highest resolution regionthis corresponds to approximately 0.48comoving (before adaptive refinement). We vary the maximum refinement level (see Table <ref>) to explore the impact of resolution on our results. Refinement is triggered in when the refinement criteria are exceeded. The refinement criteria used in this work were based on three physical measurements: (1) The dark matter particle over-density, (2) The baryon over-density and (3) the Jeans length. The first two criteria introduce additional meshes when the over-density (Δρρ_mean) of a grid cell with respect to the mean density exceeds 8.0 for baryons and/or DM. Furthermore, we set the MinimumMassForRefinementExponent parameter to -0.1 making the simulation super-Lagrangian and therefore reducing the threshold for refinement as higher densities are reached. For the final criteria we set the number of cells per Jeans length to be 32 in these runs. We use 16 levels of refinement as our fiducial refinement level. This corresponds to a minimum cell size of Δ x ∼ 0.01 pc/h comoving (∼ 6 × 10^-3 pc at z ∼ 30).We set the effective temperature of the background radiation field to T_eff = 30000 K. This background temperature suitably models the spectrum of a population of young stars <cit.>. The effective temperature of the background is important as the radiation temperature determines the dominant photo-dissociation reaction set in the irradiated halo. This in turn leads to a value of J_crit - the flux above which complete isothermal collapse of the irradiated halo is observed due to the complete suppression of . The actual value of J_crit depends on the nature of the source spectrum <cit.>. <cit.> proposed that the J_crit needed from a given stellar population modelled using realistic stellar spectra can vary widely over 2-3 orders of magnitude. They argue that in an external pristine atomic cooling halo, DCBH formation is better parameterised by using a critical curve in the H_2 and H^- photo-destruction rate parameter space (further confirmed by <cit.>). While our choice of T_eff = 30000 K falls well within the range advocated by these studies, including a realistic source spectrum derived from population synthesis models is beyond the scope of the current work and remains to be explored in a future study. It should also be noted that <cit.> found that J_crit is only very weakly dependent on the nature of the source spectrum in tension with the results of both <cit.> and <cit.>, however they explored a somewhat smaller parameter space. §.§ Smart StarsAs the gas density increases in high density regions hydro codes, including , require a method to convert the high density gas into stars in many cases. This is done to allow gas which has reached themaximum allowed refinement level of the simulation and for which further collapse is being artificially suppressed through artificial pressure support to be dealt with. In this case it can often be prudent to introduce particles into the calculation to mimic the act of real star formation. It should however be noted that in many cases there is only a loose correspondence between the numerical particle and resolving actual star formation. Depending on the level of resolution, primarily, the numerical particle may represent a single star or an entire cluster of stars. In our study here the numerical particle introduced will be a proxy for star formation (fragmentation) sites and represent only regions in which we expect star formation to be likely to occur. Star (sometimes also known as sink) particles were initially introduced into grid codes by <cit.>. The implementation used here also follows this model with some modifications. We employ the same model for gas accretion as <cit.> for our star particles using a fixed accretion radius of four cells and accreting gas based on the Bondi-Hoyle prescription. However, while <cit.> use only two criteria to evaluate the formation of sink particles; that the gas is at the highest refinement level and that the cell density exceeds the Jeans density (see equation 6 in <cit.>) we also employ additional sink particle formation criteria. Similar to the prescriptions described by <cit.> we only form sink particles when the following criteria are met: * The cell is at the highest refinement level* The cell exceeds the Jeans Density* The flow around the cell is converging along each axis* The cooling time of the cell is less than the freefall time* The cell is at the minimum of the gravitational potentialWe calculate the gravitational potential in a region of twice the Jeans length around the cell. We experimented with also including the additional conditions relating to the gas boundedness and the Jeans instability test (see <cit.> for more details). However, we found that these additional tests were sub-dominant compared to the criteria noted above and so in the interest of optimisation we did not include them.Once a is formed it can accrete gas within its accretion radius (4 cells) and it can merge with other particles. In this study we are focused only on ascertaining the degree of fragmentation experienced in haloes as a function of Lyman-Werner radiation. The particle type contains algorithms for different accretion modes and feedback modes as well as the ability to differentiate between normal PopIII star formation and supermassive star formation based on accretion rates. However, we do not employ these more sophisticated accretion or feedback algorithms here and instead leave those for an upcoming study. Here as noted previously we focus only on identifying fragmentation sites in collapsing atomic cooling haloes. In Table <ref> we list the simulation parameters and some of the fundamental parameters and results.§ RESULTS §.§ The onset of fragmentation as a function of LW intensity In Figure <ref> we show the central 1 pc of the simulations with LW intensities of 0 J_21[J_21 is defined as ], 2 J_21, 4 J_21, 10 J_21, 50 J_21, 100 J_21. Each panel has an identical maximum refinement level of 16 levels and hence an equivalent minimum cell size in comoving units. The projection is made 650 kyrs after the first forms.Each realisation shows the formation of multiple fragments within the collapsing gas with the exception of the halo exposed to a radiation level of 100 J_21. In the final halo of Figure <ref> (i.e. bottom right panel) only a single fragment forms at the very centre of the halo and is surrounded by a dense accretion disk (though mild fragmentation is also observed in this realisation at higher resolution as we will see). The mass of the fragments show a general trend of increasing mass with increasing radiation level as expected since the Jeans mass increases in each case due to the suppression of .In the bottom left of each panel in Figure <ref> we give the mass of the most massive at this time. We do not directly associate this mass with the final mass of the star that forms since further fragmentation may well, and does (see Figure <ref>), occur below this maximum resolution scale. That is not the goal in this study. Instead we are focused on identifying regions within the collapsing gas cloud which are likely to experience fragmentation and likely to foster star formation. Interestingly what we see is that, initially, as the LW intensity increases, there is a noticeable increase in the number of fragmentation sites. In Figure <ref> we have plotted the number of fragments as a function of the background LW intensity for our fiducial refinement level of 16. We choose two different times at which to examine the number of fragments. We calculate the cooling time in a given region, t_cool, as the average cooling time in a region. The cooling time is the time after which the first forms that we expect an actual star to form within the fragmentation site. The coarser the resolution the longer this time will be. At the other extreme with infinite resolution the fragmentation site would be the actual star and t_cool would be zero. In order to account for our limited resolution we use the cooling time as a estimate of the collapse of the gas within the fragmentation site into stars a process which occurs well below the grid scale. We initially select each cell and calculate its Jeans length. We then calculate the average cooling time in a sphere surrounding that cell with radius its Jeans length. We then loop over all cells in this manner calculating the cooling over multiple regions to find the cooling time over a given domain. The equation for the domain cooling time, τ_cool, is then given by:t_cool =< ∑^N_cell_i = 0 t_cool, i^J_v >τ_cool =∑^N_cell_i = 0 t_coolρ_i∑^N_cell_i = 0ρ_iwhere N_cell is the number of cells in the domain and J_V is the Jeans volume (i.e. the sphere surrounding the cell i of radius the jeans length of that cell) and t_cool, i^J_V is the average cooling time of the gas within that cooling sphere. We then sum over all cooling spheres (i.e. all cells) and taking the average cooling time within each sphere as the cooling time for that sphere. We further weight the contribution of each cell in the calculation by it's density with the highest density cells (those with the shortest cooling times) having a larger contribution. Low density cells then do not have an appreciable contribution to the calculation. We choose two different domains. In the first instance we choose the Jeans length of the maximum density cell and examine all cells within a sphere of radius the Jeans length surrounding that cell (J_L ∼ 0.03 pc). That leads to an average cooling time of approximately 80 kyrs for the fiducial runs (i.e. those with a maximum refinement level of 16). The second region we choose is the entire computational domain. We found through experimentation that the cooling time of the domain saturates after the region exceeds approximately 10 kpc as outside of this region the densities become too low and so those cells do not contribute significantly to the computation of the average cooling time. We find the average cooling time in the total computational domain corresponds to approximately 650 kyrs. The cooling time will be somewhat dependent on resolution as well. As higher densities are resolvedwe are able to better resolve the actual star formation timescale and hence our proxy for the star formation timescale (here estimated using the cooling time) drops. We fix the computation to only calculate the cooling time for the fiducial refinement level as a benchmark. The first time (80 kyrs) corresponds to roughly when we expect the first actual star to form after the formation has been triggered. This will also correspond to the time at which we expect radiation feedback effects to begin to impact subsequent fragmentation. The second time is the other extreme and is the maximum time it could take the surrounding gas to cool and initiate widespread fragmentation. These two timescales should then bracket the actual fragmentation timescale - which is unknown and will depend on specific environmental conditions which are either not resolved by our simulation or not included. In determining the fragmentation scale the Jeans length of the gas must be adequately resolved. The Jeans length (and cooling time/length) depend sensitively on the temperature of the gas. In the 0 J_21 case the Jeans length is of order J_L ∼ 0.007 pc at the centre of the collapse while it increases to approximately J_L ∼ 0.04 pc for the cases with radiation. For our fiducial runs, Δ x ∼ 0.004 pc, and so we are not well resolving fragmentation sites for the 0 J_21 case. The number of fragmentation sites for the 0 J_21 case can only be therefore taken as lower limits. For the cases with radiation, which is the goal of this study, we are resolving the Jeans length reasonably well. Nonetheless we include the 0 J_21 case for the case of completeness.For the case of J = 4 J_21 the number of fragmentation sites is 5 after 80 kyrs. It is only when the background is sufficiently strong that it can effect the detailed chemistry of the gas - we then see a reduction in the number of fragmentation sites. For the highest intensity case probed here (J = 100 J_21) the number of fragmentation sites is one both after 80 kyrs and after 650 kyrs. If we focus on the trend from 2 J_21 to 100 J_21 we see the number of fragmentation sites decrease as a function of increasing radiation. While there is also a drop from 1J_21 to 0 J_21 this is likely due to a lack of resolution in the fiducial case and the fragmentation levels for the 0 J_21 are best treated as a lower limit.In Figure <ref> we plot the radial profiles of the gas as a function of radius for the realisation at the same refinement level (Ref16) and varying background intensities. The plots are made just before the first forms in each realisation. The densities reached are approximately the same in each case since the maximum refinement level is identical and the densities are close to those at which formation is triggered. Note that the times (redshifts) are different in each case as the LW background delays the onset of cooling within the halo. Note also how the enclosed mass, fraction and temperature vary according to the LW intensity. For the highest LW backgrounds the temperature is clearly higher in the central parts of the halo. These thermodynamic differences effect the fragmentation rates as we saw in Figure <ref>. To further emphasise the temperature and associated thermodynamic differences we show the phase profiles for each realisation in Figure <ref>. As the LW intensity increases we clearly see the state of the gas change reflecting the chemical changes. As the intensity increases up to and above 10 J_21 we see the gas transition from the molecular to the atomic cooling regime with a significant mass fraction of the gas above approximately 2000 K. This transition and the thermodynamic consequences of it are reflected in the decrease in the level of fragmentation we observe. The increased temperature of a significant fraction of the gas results in a larger Jeans mass and less fragmentation for the most highly irradiated haloes. §.§ The dependence of fragmentation on resolutionIn Figure <ref> we plot the fragmentation as a function of resolution for two different values of LW intensity - 4 J_21 and 100 J_21. In the left hand column the maximum refinement level is set to 16 (Δ x ∼ 6 × 10^-3 pc), in the middle column the maximum refinement level is increased to 20 (Δ x ∼ 4 × 10^-4 pc) and in the right hand column the maximum refinement level is increased to 24 (Δ x ∼ 3 × 10^-5 pc). For the left and middle column the snapshots are plotted at approximately 80 kyrs (i.e. the cooling time with a Jeans length of the gas for the Ref16 runs) while for the right hand column the short dynamical times make extending the runs to those times prohibitive and instead we plot them at 5 kyrs instead (cf. <cit.>). The projection is made using the temperature field to illustrate the changing chemical conditions as a function of LW intensity. Note that the spatial scale varies between columns from left to right. We centre on the most massive in each case and as the resolution increases we are probing fragmentation deeper and deeper inside individual fragmentation sites. As the resolution is increased we see that the 4 J_21 realisation shows increased fragmentation, with the colder gas and subsequently smaller Jeans masses likely facilitating the fragmentation. This is in contrast with the bottom row where the 100 J_21 intensities show comparatively little fragmentation for similar scales. The right hand column shows the most significant fragmentation demonstrating the most complex structures exist at the smallest scales. Again, however, the 100 J_21 intensities show over an order of magnitude less fragmentation compared to the 4 J_21 intensities at comparable times. The prohibitively short Courant times associated with the highest resolution runs (even allowing for the fact that a portion of the high density gas is tied up in particles) prevents us from evolving the highest resolution simulations for a large number of dynamical times. However, what is clear is that the level of fragmentation is greatly reduced at the smallest scales as the LW intensity is increased.§.§ Further evolution of fragmentationFinally in Figure <ref> we explore the time evolution of the realisations with LW intensity of 4 J_21 and 100 J_21 respectively. It should be noted here again that we do not include the effects of radiation or accretion feedback in these simulations and therefore we are neglecting some important physical processes which will effect the levels of fragmentation as star formation proceeds. Additionally, the formation of one , may impact physically, on the formation of a subsequent . This would be particularly important if the releases ionising radiation. For the case of SMS formation with accretion rates around 0.1 yr^-1 the bloated stellar radii of SMSs lead to effective surface temperatures of approximately 6000 K. With these, relatively low, effective temperatures the radiation is emitted predominantly in the infrared meaning that it's effect may be quite mild and not impact surrounding fragmentation sites. Therefore, for the high external flux cases where SMS formation is expected, the formation of one is not expected to negatively (or positively) impact subsequent SMS formation. If, however, smaller POPIII stars form instead then the ionisation front may become important. However, a recent study by <cit.> shows that the ionisation from around POPIII stars which form in close proximity to SMSs have small HII regions which are closely bound to the star and do not have significant negative impacts at least in the first 100,000 years.However, since we do not take these feedback effects into account here, we simply outline the evolution of fragmentation in the absence of these effects. In Figure <ref> we see that initially (left hand column) the particles are quite clustered (there is only particle for the 100 J_21 case). As the simulation proceeds up to 1 Myr (after which the first particle would be expected to go supernova) the fragmentation sites in the 4 J_21 spread out and lie approximately 1 pc apart. For the 100 J_21 case the collapse remains completely monolithic. We can also see that for the 4 J_21 case the surrounding gas remains comparatively cold compared to the 100 J_21 case, the temperature of the gas in the 4 J_21 case is likely to stimulate fragmentation and hence the increased fragmentation in the lower LW intensity case is not surprising. The main finding is that the transition to the atomic cooling regime in the 100 J_21 case leads to and sustains complete monolithic collapse. §.§ Comparison to previous studies in the literatureL14b and <cit.> (L15b) performed similar studies to that performed here. In L14b their goal was to identify the mass scale of stars formed when exposed to a moderate LW background and in L15b they investigated the accretion rates on sink particles in the case of non-isothermal collapse. We will focus our comparison with the first work - L14b - as it most closely corresponds to our study. In L14b they investigated the flux impinging on two separate haloes with LW backgrounds of 10 J_21, 100 J_21 and 500 J_21. Unlike the study performed here they did not systematically vary their resolution and did not explore lower LW background cases (i.e. below 10 J_21). Their sink particle prescription was based on that of <cit.> which is broadly similar to the prescription we employed although their accretion radius was significantly larger than ours (∼32 cells compared to our 4 cells). Nonetheless our results are broadly consistent for the relatively high flux cases of J =10 J_21 and 100 J_21 where our two studies overlap. The main differences being that we form more sinks for the very high resolution tests (δ_x ∼ 1 AU) but with smaller masses.L14b find that one to two sinks form in their very high resolution cases. We find, at comparable resolution to their study, between 3 and 7 sinks have formed. Our higher values are due to the fact that we have a much smaller accretion radius and hence less merging (since the merging radius is identical to the accretion radius in both implementations). Our mass scales therefore also differ somewhat, L14b have masses of 7337 and 2592 for their haloes, for the J = 100 J_21 after 10 kyr. In our work we see sink particles with maximum masses of 1794 (after 80 kyr) and 6 (after 5 kyr). Our decreased masses can be attributed to our significantly smaller accretion radius. Choosing the correct accretion radius is non-trivial and different prescriptions for the accretion/merging radius will lead to different results as noted recently by <cit.> in a similar context. Nonetheless, the studies give broadly the same result with differences attributable to slightly different subgrid implementations. Since their study did not investigate the lower flux cases the degree of fragmentation suppression is not available from their study.Comparing to the results of <cit.>, which is for the zero flux case,we achieve similar resolution and similar results to their very high resolution runs. While they investigate the build up of a PopIII IMF we focus on simply identifying fragmentation sites rather than individual star formation sites. Encouragingly their fragmentation levels are similar to ours (comparing our 4 J_21 case to their fiducial (0 J_21) case). After approximately 5 kyrs the ionisation front from the first star to form has slowed the accretion rates and subsequent star formation in their simulations. They find that of the order of 100 sink particles have formed after 5 kyr compared to our 81 sink particles. It is encouraging to see that given our resolution levels are comparable that we achieve similar fragmentation levels for the low flux cases when the dominant chemistry is still due to .§ SUMMARY & DISCUSSIONWe investigated here the degree of fragmentation of atomic cooling haloes as a function of both the strength of the LW background and as a function of resolution. The degree of fragmentation within irradiated atomic cooling haloes is an important consideration in understanding the zero age main sequence masses of supermassive stars <cit.> that are candidates for the supermassive black holes that exist as high redshift quasars <cit.>. Haloes that are exposed to high Lyman Werner backgrounds have delayed collapse times and larger Jeans masses enabling them to form larger objects. However, if fragmentation of the gas is prevalent the final masses of the (super)massive stars is likely to be greatly reduced. The thermodynamic conditions of the gas is therefore critical in estimating the initial and (by accounting for mass loss) the final masses of these objects.We find that haloes exposed to a LW background of J ≳ 10 J_21 experience progressively less fragmentation than haloes exposed to lower LW backgrounds. Haloes exposed to a LW background of J ≳ 10 J_21 have a significant fraction of their gas pushed onto the atomic cooling track (see Figure <ref>) and are much less prone to fragmentation. The transition to the atomic cooling track is likely then to be the most stringent indicator of fragmentation levels. The increased temperature of the gas in the centre limits the degree of fragmentation significantly and for the highest irradiation levels examined here fragmentation was eradicated almost entirely. Comparing to the results of <cit.> we achieve similar resolution and similar results to their very high resolution runs. While they investigate the build up of a PopIII IMF we focus on simply identifying fragmentation sites rather than individual star formation sites. Encouragingly their fragmentation levels are similar to ours (comparing our 4 J_21 case to their fiducial (0 J_21) case). After approximately 5 kyrs the ionisation front from the first star to form has slowed the accretion rates and subsequent star formation in their simulations. Given our resolution levels are comparable it suggests that our assertion that once the gas chemistry switches to the atomic cooling regime fragmentation is likely to be strongly suppressed which is something they did not investigate. We note also that any mechanism which suppresses formation and forces a transition to the atomic cooling track is likely to achieve a similar outcome. For example in the simulations conducted by <cit.>, collapsing haloes are exposed to streaming velocities of up to 18 at z = 200. The streaming velocities act in a similar way to a LW background by suppressing production and hence impacting the gas cooling. The thermodynamic state of the gas is therefore effected in a similar way to a LW background and hence for haloes exposed to large streaming velocities fragmentation is also likely to be suppressed. Similarly, dynamical heating effects <cit.> combined with mild LW radiation from nearby galaxies as recently identified by Wise et al. (2017)(in prep) would likely not be effected by vigorous fragmentation and would in fact be ideal candidates for a single monolithic collapse.In summary we find that haloes exposed to LW radiation experience both delayed collapse and suffer from decreased fragmentation. Once the gas in the core exceeds approximately 2000 K, and has transitioned chemically to predominantly atomic line cooling, then fragmentation is almost entirely suppressed and conditions becomes ideal for monolithic collapse. In future work we will explore the impact of feedback from the collapsing protostar and the subsequent transition to a (massive) black hole seed.§ ACKNOWLEDGEMENTSJ.A.R and T.P.D thank David Hubber, Peter Johansson and Bhaskar Agarwal for useful discussion on both the sink particle implementation and direction of the study.J.A.R. acknowledges the support of the EU Commission through the Marie Skłodowska-Curie Grant - “SMARTSTARS" - grant number 699941. Computations described in this work were performed using thepublicly-available code (http://enzo-project.org), which is the product of a collaborativeeffort of many independent scientists from numerous institutions around the world.Theircommitment to open science has helped make this work possible. The freely available astrophysicalanalysis code YT <cit.> was used to construct numerous plots within this paper. The authorswould like to extend their gratitude to Matt Turk et al. for an excellent software package. J.A.R. would like to thank Lydia Heck and all of the support staff involved with Durham's COSMA4 and DiRAC's COSMA5 systems for their technical support. This work was supported by the Science and Technology Facilities Council (grant numbers ST/L00075X/1 and RF040365). This work used the DiRAC Data Centric system at Durham University,operatedbytheInstituteforComputational Cosmology on behalf of the STFC DiRAC HPC Facility(www.dirac.ac.uk). This equipment was funded by BIS National E-infrastructure capital grant ST/K00042X/1, STFC capital grant ST/H008519/1, and STFC DiRAC Operations grant ST/K003267/1 and Durham University.DiRAC is part of the National E-Infrastructure. The authors also wish to acknowledge the SFI/HEA Irish Centre for High-End Computing (ICHEC) for the provision of computational facilities and support. 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We tested the implementation using both the Singular Isothermal Sphere test <cit.> and by analysing the results from a Rotating Cloud test <cit.>. §.§ Collapse of a Singular Isothermal SphereThe collapse of a singular isothermal sphere with a ρ(r) ∝ r^-2 density profile produces a constant flux of mass through spherical shells i.e. a constant accretion rate. We compute the collapse of a singular isothermal sphere to test whether our model of sink particle accretion reflects this collapse behaviour. We follow the work of <cit.> in conducting this numerical test. The sphere is set to have a truncation radius of R = 5 × 10^16 cm, a density at this radius of ρ(R) = 3.82 × 10^-18 g cm^-3 and therefore a total mass of 3.02 . The sphere is initialized at a temperature of 10 K, the mean molecular weight of the gas is set to 3.0 resulting in a sound speed of 0.166 . These values lead to a large instability parameter for the sphere of A = 29.3 with A = 4 π G ρ(R) R^2/c_s^2 <cit.>. We ran the test at a maximum refinement level of 10 leading to a minimal grid spacing of Δ x ∼ 3.05 × 10^-5 pc. No radiative cooling or chemical reactions are calculated during the collapse. In the upper panel of Figure <ref> we show the mass, in solar masses, as a function of time. The particle mass increases monotonically as it accretes matter eventually accreting the entire envelope (3.02 ). The slope of the line is 1.5 × 10^-4 /yr in excellent agreement with the analytical solution predicted by <cit.>. In the lower panel of Figure <ref> we show the mass accretion rate during this time. The test predicts a constant accretion rate of1.5 × 10^-4 and this is confirmed by our test. The bump at approximately 17000 yrs corresponds to the particle accreting the last of the matter and at this point the accretion rate fluctuates mildly before falling to zero. We also examine the behaviour of the gas itself and the mass that is accreted by the particle. In Figure <ref> we see the density profile, radial velocity profile and the mass accretion through spherical shells as a function of radius. Overlaid on each plot are the analytical expressions for the collapse taken from <cit.>. The agreement in all three profiles is excellent with mild deviations only appearing once the the entire envelope has been accreted at T ∼ 17000 yrs.§.§ Rotating Cloud Core Fragmentation TestFinally, we also analyse the behaviour of the particles when the gas is subject to rotation and collapse. This is the classic <cit.> test. It is a standard test for examining fragmentation in hydrodynamical simulations <cit.>. We employ a setup very similar to that of <cit.>, <cit.> and <cit.>. The cloud is initialised with the following parameters: radius R = 5 × 10^16 cm, constant densityρ_0 = 3.82 × 10^-18 g cm^-3, mass = 1 , angular velocity, Ω = 7.2 × 10^-13 rad s^-1 (ratio of rotational to gravitational energy β = 0.16), sound speed = 0.166 (ratio of thermal to gravitational energy α = 0.26), global free-fall time t_ff = 1.075 × 10^12 s =3.41 × 10^4 yr. The cloud is then subject to a density perturbation with an m = 2 mode: ρ = ρ_0[1 + 0.1 Cos(2ϕ)], where ϕ is the azimuthal angle. We use an isothermal equation of state throughout the test. The initial rotation (β = 0.16) forces the gas cloud to collapse to a disk with a central bar due to the m = 2 initial density perturbation. particles then form at the ends of the bar once the conditions for particle formation are met. The test was run at a maximum refinement level of 5 with a minimum cell size of 16.32 AU (2.44 × 10^14 cm).The evolution of the system is shown in Figure <ref>. The figures show the initial formation of the disk and bar structure face on. Two particles form at either end of the bar at the same time due to the inherent symmetry of the system. A third fragment then forms at the centre of the cloud core. Subsequently the two end fragments move along the bar and merge with the central fragment. The results of this test are in excellent agreement with similar tests carried out by <cit.>. | http://arxiv.org/abs/1708.07772v2 | {
"authors": [
"John A. Regan",
"Turlough P. Downes"
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"published": "20170825152046",
"title": "Fragmentation inside atomic cooling haloes exposed to Lyman-Werner radiation"
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ℝ ℕ ℂ ℤ ℋ ℳ≲plainthmTheorem[section] prop[thm]Proposition lem[thm]Lemmacor[thm]Corollarydefi[thm]Definitionclaim[thm]ClaimA uniform estimate]A remark on oscillatory integrals associated with fewnomials831 E. Third St., Bloomington, 47405, IN, [email protected][2010]42B20, 42B25We prove that the L^2 bound of an oscillatory integral associated with a polynomial phase depends only on the number of monomials that this polynomial consists of.[ Shaoming Guo December 30, 2023 =====================§ INTRODUCTION Let d∈. Consider the operator H_Q f(x):=∫_ f(x-t)e^i Q(t)dt/t,with Q(t)=a_1 t^α_1+… +a_d t^α_d.Here a_i∈ and α_i is a positive integer for each 1≤ i≤ d. Given d∈, we haveH_Q f_2 ≤ C_d f_2.Here C_d is a constant that depends only on d, but not on any a_i or α_i. On ^2, define the Hilbert transform along the polynomial curve (t, Q(t))_t∈ byH_Q f(x, y)=∫_ f(x-t, y-Q(t))dt/t.As a corollary of Theorem <ref>, we have Given d∈, we haveH_Q f_2 ≤ C_d f_2.Here C_d is a constant that depends only on d, but not on any a_i or α_i. Corollary <ref> follows from Theorem <ref> via applying Plancherel's theorem to the second variable of H_Q f. We leave out the details.Denote by nthe degree of the polynomial Q given by (<ref>). Then it is well-known (see Stein and Wainger <cit.>) that the estimate (<ref>) holds true if we replace C_d by C_n, a constant that is allowed to depend on the degree n. Moreover, Parissis <cit.> proved thatsup_P∈P_n|p.v.∫_ e^iP(t)dt/t| ≃log n,where P_n is the collection of all real polynomials of degree at most n. It would also be interesting to know whether the constant C_d in (<ref>) can be made to (log d)^c for some c>0.Acknowledgements. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring semester of 2017. The author thanks the anonymous reviewers for their careful reading of the manuscript and suggestions on how to improve the exposition of the paper.§ REDUCTION TO MONOMIALSWe start the proof. In this section, we will splitinto different intervals, and show that for all but finitely many of these intervals, there always exists a monomial which “dominates” our polynomial Q. In dimension one, this idea has been used extensively in the literature, for instance Folch-Gabayet and Wright <cit.>. Here we follow the formulation of Li and Xiao <cit.>.Notice that we can always let the function f absorb the linear term of Q. Hence we assume that1<α_1< … < α_d. Denote by n the degree of the polynomial Q, that is n=α_d. Let λ=2^1/n. Define b_j∈ such thatλ^b_j≤ |a_j|< λ^b_j+1.We define a few bad scales. For 1≤ j_1< j_2≤ d, defineJ^(0)_bad(Γ_0, j_1, j_2):={l∈: 2^-Γ_0 |a_j_2λ^α_j_2l|≤ |a_j_1λ^α_j_1l|≤ 2^Γ_0 |a_j_2λ^α_j_2l|}.Here Γ_0:=2^10d!. Notice that l satisfies -2-nΓ_0 + b_j_2-b_j_1≤ (α_j_1-α_j_2) l≤ nΓ_0 + b_j_2-b_j_1+2.Hence J^(0)_bad(Γ_0, j_1, j_2) is a connected set whose cardinality is smaller than 4nΓ_0. DefineJ^(0)_good:=( ⋃_j_1≠ j_2J^(0)_bad(Γ_0, j_1, j_2) )^cNotice that J^(0)_good has at most d^2 connected components. Moreover, on each component, there is exactly one monomial which is “dominating”.Similarly, we defineJ^(1)_bad(Γ_0, j_1, j_2):= {l∈: 2^-Γ_0 |α_j_2(α_j_2-1) a_j_2λ^α_j_2l|≤ |α_j_1(α_j_1-1) a_j_1λ^α_j_1l|≤ 2^Γ_0 |α_j_2(α_j_2-1) a_j_2λ^α_j_2l|}.Moreover, J^(1)_bad:= ⋃_j_1≠ j_2J^(1)_bad(Γ_0, j_1, j_2)and J_good:=J^(0)_good∖J^(1)_bad.Analogously, J_good has at most d^4 connected components. § BAD SCALES Due to the control on the cardinalities of various bad sets, the contributions from those l∉J_good can be controlled by a multiple of the Hardy-Littlewood maximal function.Let us be more precise. Suppose that we are working on the collection of bad scales J^(0)_bad(Γ_0, j_1, j_2) for some j_1 and j_2. Define H_l f(x)=∫_ f(x-t) e^i Q(t)ψ_l(t)dt/t.Here ψ_0is a non-negative smooth bump function supported on[-λ^2, -λ^-1]∪[λ^-1, λ^2] such that∑_l∈ψ_l(t)=1for everyt≠ 0,with ψ_l(t):=ψ_0(t/λ^l).By the triangle inequality, we have|∑_l∈J^(0)_bad(Γ_0, j_1, j_2) H_l f(x)| ≤∑_l∈J^(0)_bad(Γ_0, j_1, j_2)∫_ |f(x-t)| ψ_l(t)dt/|t|.Recall that the cardinality of J^(0)_bad(Γ_0, j_1, j_2) is at most 4nΓ_0. Now we partition the set J^(0)_bad(Γ_0, j_1, j_2) into subsets of consecutive elements, and such that each subsetcontains exactly n elements, with possibly one exception which can be handled in the same way. The scale that these n elements can see is about λ^n=2, in the sense that for every l_0∈, supp(∑_l=l_0^l_0+nψ_l)has Lebesgue measure about λ^n. Hence the contribution from each of these subsets can be controlled by 2 Mf(x). Here M denotes the Hardy-Littlewood maximal operator. Hence the right hand side of (<ref>) can be controlled by 8Γ_0 · Mf(x). This takes care of the contribution from bad scales.§ GOOD SCALESSuppose we are working on one connected component of J_good, and for each integer l in such a component, we assume thata_j_1 t^α_j_1 dominatesQ(t) in the sense of (<ref>), that is,|a_j_1λ^α_j_1l|≥ 2^Γ_0 |a_j'_1λ^α_j'_1l|for everyj'_1≠ j_1,and a_j_2α_j_2(α_j_2-1) t^α_j_2-2 dominatesQ”(t) in the sense of (<ref>), that is,|a_j_2α_j_2(α_j_2-1)λ^α_j_2l|≥ 2^Γ_0 |a_j'_2α_j'_2(α_j'_2-1)λ^α_j'_2l|for everyj'_2≠ j_2. Let us call such a set J_good(j_1, j_2). Under this assumption, we have the estimates |Q(t)|≤ 2 |a_j_1 t^α_j_1|and|Q”(t)|≥ |a_j_1t^α_j_1-2|,for every t∈ [λ^l-2, λ^l+1]withl∈J_good(j_1, j_2). Recall that λ=2^1/n is the smallest scale that we will work with. This scale is only visible when a_n t^n dominates. When some other monomial dominates, at such a small scale, our polynomial will not have enough room to see the oscillation. This will be reflected when we come to the stage of applying van der Corput's lemma (see (<ref>) below). Defineλ_j_1:=2^1/α_j_1. We choose this scale because the monomial a_j_1 t^α_j_1 dominates. LetΦ_j_1, j_2(t)= ∑_l∈J_good(j_1, j_2)ψ_l(t).Notice that here we join all the small scales from J_good(j_1, j_2) to form a larger scale. Next we will apply a new partition of unity to the function Φ_j_1, j_2. Define H^(j_1)_l' f(x)=∫_ f(x-t) e^i Q(t)ψ^(j_1)_l'(t)Φ_j_1, j_2(t)dt/t.Here ψ^(j_1)_0 is a non-negative smooth bump function supported on [-λ_j_1^2, -λ_j_1^-1]∪[λ_j_1^-1, λ_j_1^2] such that∑_l'∈ψ^(j_1)_l'(t)=1for everyt≠ 0,with ψ^(j_1)_l'(t):=ψ^(j_1)_0(t/λ_j_1^l'). We define B_j_1∈ such that λ_j_1^-B_j_1≤ |a_j_1|<λ_j_1^-B_j_1+1,denote γ_j_1=B_j_1/α_j_1 and split the sum in l' into two cases.∑_l'∈H^(j_1)_l f=∑_l'≤γ_j_1H^(j_1)_l' f+∑_l'> γ_j_1H^(j_1)_l' f.Thefirst summand in (<ref>) can be controlled by the maximal function and the maximal Hilbert transform. To be precise, we have a bound ∑_l'≤γ_j_1|∫_ f(x-t)ψ^(j_1)_l'(t)Φ_j_1, j_2(t)dt/t| +∑_l'≤γ_j_1|∫_ f(x-t) (e^i Q(t)-1)ψ^(j_1)_l'(t)Φ_j_1, j_2(t)dt/t| H^* f(x)+ Mf(x)+ ∑_l'≤γ_j_1|∫_ f(x-t) (e^i Q(t)-1)ψ^(j_1)_l'(t)Φ_j_1, j_2(t)dt/t|.Here H^* stands for the maximally truncated Hilbert transform. The last summand in (<ref>) can be further controlled by ∑_l'≤γ_j_1∫_ |f(x-t)| |a_j_1t^α_j_1|ψ^(j_1)_l'(t)dt/|t|≤∑_l∈∫_λ^γ_j_1-l-2_j_1^λ^γ_j_1-l+1_j_1 |f(x-t)| |a_j_1| |t|^α_j_1-1dt≤∑_l∈λ^(γ_j_1-l+1)(α_j_1-1)_j_1∫_λ^γ_j_1-l-2_j_1^λ^γ_j_1-l+1_j_1 |f(x-t)| |a_j_1|dt≤ 8Mf(x). Hence it remains to handle the latter term from (<ref>).We will prove that there exists δ>0 such thatH^(j_1)_γ_j_1+l f_2 ≤ C_d 2^-δ lf_2,for everyl≥ 0,with a constant C_d depending only on d. This amounts to proving a decay for the multiplier∫_ e^iQ(t)+itξψ^(j_1)_γ_j_1+l(t)dt/t=∫_ e^iQ(λ^γ_j_1+l_j_1t)+iλ^γ_j_1+l_j_1tξψ^(j_1)_0(t)dt/t.We calculate the second order derivative of the phase function: λ^2γ_j_1+2l_j_1 |Q”(λ^γ_j_1+l_j_1t)|≥1/2 |a_j_1|λ^B_j_1+α_j_1 l_j_1≥ 2^l-2.Hence the desired estimate follows from van der Corput's lemma, for which we refer to Proposition 2 in Page 332 <cit.> 20[FW12]FW M. Folch-Gabayet and J. Wright. Weak-type (1,1) bounds for oscillatory singular integrals with rational phases.Studia Math. 210 (2012), no. 1, 57–76.[LX16]LiXiao X. Li and L. Xiao. Uniform estimates for bilinear Hilbert transforms and bilinear maximal functions associated to polynomials. Amer. J. Math. 138 (2016), no. 4, 907–962.[Par08]Par I. Parissis. A sharp bound for the Stein-Wainger oscillatory integral.Proc. Amer. Math. Soc. 136 (2008), no. 3, 963-972. [Stein93]Stein E. Stein. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals.With the assistance of Timothy S. Murphy. Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, III. Princeton University Press, Princeton, NJ, 1993. xiv+695 pp. ISBN: 0-691-03216-5[SW70]SW E. Stein and S. Wainger. The estimation of an integral arising in multiplier transformations. Studia Math. 35 1970 101–104. | http://arxiv.org/abs/1708.08047v2 | {
"authors": [
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"published": "20170827032329",
"title": "A remark on oscillatory integrals associated with fewnomials"
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The Young Substellar Companion ROXs 12 B:Near-Infrared Spectrum, System Architecture, and Spin-Orbit Misalignment[Someof the data presented herein were obtained at the W.M. Keck Observatory, which is operated as a scientific partnershipamong the California Institute of Technology, the University of California and the National Aeronautics and Space Administration.The Observatory was made possible by the generous financial support of the W.M. Keck Foundation.] Lucas A. Cieza December 30, 2023 ========================================================================================================================================================================================================================================================================================================================================================================================================================================================================================This paper studies the expected optimal value of a mixed 0-1 programmingproblem with uncertain objective coefficients following a joint distribution. We assume that the true distribution is not known exactly, but a set of independent samples can be observed. Using the Wasserstein metric, we construct an ambiguity set centered at the empirical distribution from the observed samples and containing the true distribution witha high statistical guarantee. The problem of interest is to investigate the bound on the expected optimal value over the Wasserstein ambiguity set. Under standard assumptions, we reformulate the problem into a copositive program, which naturally leads to a tractable semidefinite-based approximation. We compareour approach with a moment-based approach from the literature on three applications. Numerical results illustrate the effectiveness of our approach.Keywords: Distributionally robust optimization; Wasserstein metric;copositive programming; semidefinite programming § INTRODUCTION We consider the following uncertain mixed 0-1 linear programming problem:v(ξ) := max{ (Fξ)^Tx : [ Ax = b, x ≥ 0; x_j ∈{0,1} ∀j ∈ ]}where A ∈^m × n, F ∈^n × k, and b ∈^m are the problem data,x ∈^n_+ is thevector of decision variables, ⊆{ 1, …, n } is an index set of binary variables, and the objective coefficients are linear in the random vector ξ∈^k via F. Problem (<ref>) entails two extreme classes of programs: if = ∅, then (<ref>) represents the regular linear program with uncertain objective coefficients; if = { 1, …, n }, then (<ref>) represents the regular binary program with uncertain coefficients. In general, problem (<ref>) is NP-hard <cit.>. The optimal value v(ξ) is a random variable as ξ is a random vector. We assumethat ξ follows a multivariate distributionsupported on a nonempty set Ξ⊆^k, which is, in particular, defined as a slice of a closed, convex,full-dimensional cone Ξ⊆_+ ×^k-1:Ξ := {ξ∈Ξ: e_1^Tξ =ξ_1 = 1 },where e_1 is the first standard basis vector in ^k. In words, Ξ is the homogenization of Ξ. We choose this homogenized version for notational convenience. Note that it, in fact, enables us to model affine effects of the uncertain parameters in (<ref>). The expected optimal value of (<ref>),denoted by v_, is defined asv_ := _ [v(ξ)] = ∫_Ξ v(ξ) d ( ξ).The problem of computing v_ has been extensively studied in the literature. Hagstrom <cit.> showed that computing v_ for the longest path problem over a directed acyclic graph is #𝒫-complete even if the arc lengths are each independently distributed and restricted to taking two possible values. Aldous <cit.> studied a linear assignment problem with random cost coefficients following either an independent uniform distribution on [0,1] or an exponential distribution with parameter 1 and proved that the asymptotic value of v_ approaches π^2/6 as the number of assignments goes to infinity. For additional studies, see <cit.>.In practice, it is difficult or impossible to knowcompletely, and computing v_ is thus not well definedin this situation. An alternative is to construct an ambiguity set, denoted by , that contains a family of distributions supported on Ξ and consistent with any known properties of . Ideally, the ambiguity set will possess some statistical guarantee,e.g., the probability that ∈ will be at least 1 - β, where β is the significance level. In analogy with v_,we define v_for any ∈. Then, we are interested in computingthe maximum expected optimal value v_ over the ambiguity set :v_^+ :=sup_∈ v_.Note that, when the probability of ∈ is at least 1 - β, the probability of v_≤ v_^+ is at least 1 - β. There are three main issues (somehow conflicting) regardingthe computation of v_^+. First, one would like an ambiguitysetwith a high statistical guarantee to contain the true distribution .In this way, the computed v_^+ will be an upper boundon v_ with a high confidence level. (We will introduce severalapproaches in the following paragraph.) Second, one would like v_^+ to be tight in the sense that it is as close to v_ as possible. Generally, ifenforces more informationabout , then v_^+ will be closer to v_. Finally, the thirdconcern is the complexity of the resulting optimization problem,i.e., whether the problem can be solved in polynomial time.Bertsimas et al. <cit.> constructed moment ambiguity sets using the first two marginal moments of each ξ_i. Denote the first and second of each uncertain parameter by μ_i and σ_i respectively. They computed v_^+ over alljoint distributions sharing the same first two marginal moments and provedpolynomial-time computability if the corresponding deterministicproblem is solvable in polynomial time. However, the computed bound may not be tight with respect to v_ since the marginal-momentmodel does not capture the dependence of the random variables.In a closely related direction, Natarajan et al. <cit.> proposed an ambiguity set that was constructed from the known marginal distributions of each random variable ξ_i, and they computed v_^+ by solving a concave maximization problem. As an extension to the marginal moment-based approach, Natarajan et al. <cit.> proposed a cross-moment model that was based on an ambiguity set constructed using both marginal and cross moments. Compared to the marginal-moment approach, the cross-moment approachhas tighter upper bounds as the model captures the dependence of the random variables. However, computing the bound requires solving a completely positive program, which itself can only be approximated in general.Thus, the authors proposed semidefinite programming (SDP)relaxations to approximate v_^+. Moment-based ambiguity sets are also used prominently in a parallel vein of research, called distributionally robust optimization (DRO); see <cit.>.The popularity of the moment-based approach is mainly due to the fact that it often leads to tractable optimization problems and relatively simple models. Its weakness, however, is that moment-based sets are not guaranteed to converge to the true distributionwhenthe sample size increases to infinity, even though the estimations of the first and second moments are themselves guaranteed to converge. As an attractive alternative to moment-based ambiguity sets,distance-based ambiguity sets haven been proposed in recent years.This approach definesas a ball in the space of probability distributions equipped with a distance measure, and the center of the ball is typically the empirical distribution derived from a series of independent realizations of the random vector ξ. The key ingredient of this approach is the distance function. Classical distance functions include the Kullback-Leibler divergence <cit.>, the ϕ-divergence <cit.>, the Prohorov metric <cit.>, empirical Burg-entropy divergence balls <cit.>, and theWasserstein metric <cit.>.In this paper, we apply the Wasserstein metric to construct a data-driven ambiguity setcentered at the empirical distribution _N derived from N independent observations of ξ. This approach has several benefits. The conservativeness of the ambiguity set can be controlled by tuning a single parameter, the radius of the Wasserstein ball; we will discuss this parameter in detail in Section <ref>. Also, under mild conditions on , the Wasserstein ambiguity provides a natural confidence set for .Specifically, the Wasserstein ball around the empirical distribution onN independent identical samples containswith confidence 1-βif its radius exceeds an explicit threshold ϵ_N(β) that can be computedvia a closed form equation <cit.>.We then formulate v_^+ in (<ref>) over the constructedWasserstein ambiguity set. That is, we model the maximum value ofv_ over the ambiguity setconstructed by the Wassersteinmetric. In Section <ref>, we reformulate problem (<ref>)into a copositive problem under some standard assumptions. As the copositivereformulation is computationally intractable, we apply a standard approachbased on semidefinite programming techniques to approximate v_^+from above. In Section <ref>, we numerically verify our approachon three applications from the literature. In particular, we compare ourapproach with the moment-based approach proposed in<cit.>. We have several importantobservations from the experimental results. First, we find that the gapsbetween the bound from our semidefinite programsand the true expected optimal value becomes narrower as the sample sizeincreases. However, the moment-based bound remains the same regardless of the increase in the sample size. Second, we observe that our boundconverges to the true expected optimal value on the first two applications where the underlying deterministic problems are linear programs. Although our bound on the third application is not able to converge to the true expected optimal value, it is tighter than the moment-based bound after the sample size increases to a certain level.We conclude our researchand discuss some future directions in Section <ref>. We point out some similarities of our paper to a recent technical report by Hanasusanto and Kuhn <cit.>. In their report, they proposed a Wasserstein-metric ambiguity set for a two-stage DRO problem. In particular, they applied copositive programming techniques to reformulate the second-stage worst-case value function, which is essentially a max-min optimization problem, while we use copositive techniques to reformulate a max-max optimization problem; see (<ref>). Furthermore,they directly used a hierarchy schema to approximate the copositive cones, while we derive natural SDP approximations based on the copositivereformulation. Note that their hierarchy of approximations lead to SDPapproximations as well. Finally, they developed an approach to derive anempirical Wasserstein radius, which is in spirit similar to our approachin this paper. §.§ Notation, terminology, and basic techniquesWe denote by ^n the n-dimensional Euclidean space and by _+^n the nonnegative orthant in ^n. For a scalar p ≥ 1, the p-norm of z ∈^n is defined z_p := (∑_i=1^n |z_i|^p)^1/p, e.g., z_1 = ∑_i=1^n |z_i|. We will drop the subscript for the 2-norm, i.e., z := z_2. For v,w ∈^n, the inner product of v and w is denoted by v^Tw := ∑_i=1^n v_iw_i. For thespecific dimensions k and n of the problem in this paper,we denote by e_i the i-th standard basis vector in ^k,and similarly, denote by f_j the j-th standard basis vectorin ^n. We will also define g_1 := e_10∈^k+n.We denote by δ_ξ the Dirac distribution concentrating unit mass at ξ∈^k. For any N ∈ℕ,we define [N] := {1, …, N}.Let ^m × n denote the space of real m × n matrices, and A∙ B := trace(A^TB) denote thetrace of the inner product of two matrices A, B ∈^m × n.We denote by ^n the space of n × n symmetric matrices,and for X ∈^n, X ≽ 0 represents that X is positivesemidefinite. In addition, we denote by (X) the vector containingthe diagonal entries of X, and denote by (v) the diagonalmatrix with vector v along its diagonal. I ∈^n denotesthe identity matrix. Finally, letting K⊆^n be a closed, convex cone, and K^* be its dual cone, we give a brief introduction to copositive programming with respect to the cone K. The copositive cone with respect to K is defined as( K) := { M ∈^n: x^T M x ≥ 0∀ x ∈ K},and its dual cone, the completely positive cone with respect to K, is given as( K) := { X ∈^n: X = ∑_i x^i (x^i)^T,x^i ∈ K},where the summation over i is finite but its cardinality is unspecified. The term copositive programming refers to linear optimization over ( K) or, via duality, linear optimization over ( K). In fact, these problems are sometimes called generalized copositive programming or set-semidefinite optimization <cit.> in contrast with the standard case K = _+^n. In this paper, we work with generalized copositive programming, although we use the shorter phrase for convenience. § A WASSERSTEIN-BASED AMBIGUITY SETIn this section, we define the Wasserstein metric and discussa standard method to construct a Wasserstein-based ambiguity set. Using this ambiguity set, we fully specify problem (<ref>). Denote by Θ_N := {ξ̂^1, …, ξ̂^N}the set of N independent samples of ξgoverned by . The uniform empiricaldistribution based on Θ_N is_N := 1N ∑_i=1^N δ_ξ̂^i where δ_ζ is the Dirac distribution concentrating unit mass at ζ∈^k. Let ^2(Ξ) be the set of all probability distributionsthat are supported on Ξ and that satisfy _[ξ- ξ'^2] = ∫_Ξξ - ξ'^2 d (ξ) < ∞ whereξ' ∈Ξ is some reference point, e.g., ξ' = ξ̂^i for some i ∈ [N].The 2-Wasserstein distance between any , ' ∈ M^2(Ξ) isW^2(, ') := inf{ (∫_Ξ^2ξ - ξ'^2Π (dξ, dξ'))^1/2 :[ Π is a joint distribution of ξ and ξ'; with marginalsand ', respectively ]}.The Wasserstein distance is essentially the minimum cost of redistributing mass fromto '. It is also called the “earth mover's distance" in the community of computer science; see <cit.>. In fact, the Wasserstein distance between two discrete distributions witha finite number of positive masses corresponds to a transportation planningproblem in finite dimensions.Example <ref> illustrates the Wasserstein distance between two discrete distributions. Consider two discrete distributions: := ∑_i=1^Mq_i δ_ξ_i and ' := ∑_j=1^M' q'_jδ_ξ'_j where q_i ≥ 0 i = 1, …, M,q'_j ≥ 0j = 1, …, M', and ∑_i=1^M q_i = ∑_j=1^M'q'_j = 1. Define c_ij = ξ_i - ξ'_j^2∀i = 1, …, M j = 1, …, M'. Then, the 2-Wasserstein distance betweenand ' equals the square root of the optimal value of the following transportation planning problem:[ min_π ∑_i=1^M ∑_j=1^M' c_ijπ_ij;∑_j=1^M'π_ij = q_i∀i = 1, …, M; ∑_i=1^Mπ_ij = q'_j∀j = 1, …, M'; π_ij≥ 0∀i = 1, …, M,j = 1, …, M', ]where π is the joint distribution of ξ and ξ' with marginals ofand ' and π is the matrix variable in this optimization problem. With this setting, our ambiguity set contains a family ofdistributions that are close to _N with respect to theWasserstein metric. In particular, we define our ambiguity set as a 2-Wasserstein ball of radius ϵ that is centered at theuniform empirical distribution _N:(_N, ϵ) := {∈^2(Ξ) :W^2(, _N) ≤ϵ}. The reader is referred to <cit.> forthe general case of ^r(Ξ) and W^r(, ') for any r ≥ 1. We use the 2-Wasserstein distance in this paper for two reasons. First, the Euclidean distance is one ofthe most popular distances considered in the relevant literature; see <cit.>.Second, we will find that problem (<ref>) with an ambiguityset based on the 2-Wasserstein distance can be reformulated intoa copositive program; see Section <ref>.Then, we replace the generic ambiguity setwith the Wassersteinball (_N, ϵ) in problem (<ref>) to compute a data-driven upper bound: [ v_(_N, ϵ)^+ =sup_Π,∈^2(Ξ)∫_Ξv(ξ)d (ξ);∫_Ξ^2ξ - ξ'^2Π(dξ, dξ') ≤ϵ^2; Π is a joint distribution of ξ and ξ'; with marginalsand _N, respectively. ] We next close this subsection by making some remarks. First, the Wasserstein ball radius in problem (<ref>) controls the conservatism of the optimal value. A larger radiusis more likely to contain the true distribution and thus a more likelyvalid upper bound on v_, but even if it is valid, it could be aweaker upper bound. Therefore, it is crucial to choose anappropriate radius for the Wasserstein ball. Second, theKullback-Leibler divergence ball is also considered in recent research;see <cit.>. However, in the case of our discrete empirical distribution, the Kullback-Leibler divergence ball is asingleton containing only the empirical distribution itself, with probabilityone. Third, the ambiguity sets constructed by goodness-of-fittests in <cit.> also possess statistical guarantees, however, they often lead to complicated and intractable optimization problems for the case ofhigh-dimensional uncertain parameters.§.§ An empirical Wasserstein radiusThe papers <cit.>present a theoretical radius ϵ_N(β) fordatasets of size N, which guarantees a desired confidencelevel 1 - β for ∈(_N, ϵ_N(β))under the following standard assumption on : [Light-tailed distribution] There exists an exponent a > 1 such that _ [exp(ξ^a)] = ∫_Ξexp(ξ^a)d (ξ) <∞. Note that ϵ_N(β) depends on Nand β. However, ϵ_N(β) is known to beconservative in practice; see <cit.> forexample. In other words, (_N, ϵ_N(β))might contain significantly more irrelevant distributions sothat the computed v_(_N,ϵ_N(β))^+is significantly larger than v_. So, we propose aprocedure to derive an empirical radius that providesa desired confidence level 1 - β but is much smallerthan ϵ_N(β). Our approach is based onthe data set Θ_N. In particular, weapply a procedure, similar to cross validation in spirit,that computes an empirical confidence level (between0 and 1) for a given radius ϵ; see detailsin the next paragraphs. Our procedure guarantees thata larger radius leads to a higher confidence level.Therefore, by iteratively testing different ϵ,we can find a radius with a desired confidence level basedon the data set Θ_N. Although the derivedϵ(Θ_N,β) depends on thedata set Θ_N, our experimental results inSection <ref> indicate that it can be usedfor other datasets ofthe same sample size. We willshow the numerical evidence in Section <ref>.Our approach is also similar in spirit to the one usedin <cit.>.Our procedure requires an oracle to compute (or approximate)v_^+. Later in Section <ref>, we will proposea specific approximation; see (<ref>). Assume also that,in addition to the dataset Θ_N, we predetermine aset ℰ containing a large,yet finite, number of candidate radii ϵ. We randomly divideΘ_N into training and validation datasets K times. We enforce the same dataset size denoted by N_T on each of theK training datasets. Next, for each ϵ∈ℰ, we derive an empiricalprobability based on the following procedure: (i) we use each ofthe K training datasets to approximatev_(_N_T,ϵ)^+ with a value calledv_WB(ϵ) by calling the oracle, where _N_T represents the empirical distribution from the training set; (ii) we then usethe corresponding K validation datasets to simulate the expectedoptimal values denoted by v_SB [This processis to solve a linear program or integer program corresponding toeach sample in the validation dataset and then to take the averageof the optimal values.]; and (iii) we finally computethe percentage of the K instances where v_WB(ϵ)≥ v_SB. Let us call this empirical probability as theempirical confidence level. Thus, the empirical confidence levelcan roughly approximate the confidence level that the underlyingdistribution is contained in the Wasserstein-based ambiguityset with the radius ϵ. Note that the percentagecomputed is non-decreasing in ϵ and equal to 1 forsome large ϵ_0. Therefore, the set containing all theempirical confidence levels is essentially an empirical cumulative distribution.Then, given a desired confidence level, we canchoose a corresponding empirical radius ϵ∈ℰ.The numerical results in Section <ref> indicate that our choices of ϵ indeed return the desired confidence levels. We specify the above procedure in Algorithm <ref>.§ PROBLEM REFORMULATION AND TRACTABLE BOUNDIn this section, we propose a copositive programming reformulation for problem (<ref>) under some mild assumptions.As copositive programs are computationally intractable, we then propose semidefinite-based relaxations for the purposes of computation. Let us first define the feasible set for x ∈^nin (<ref>)as follows::= { x ∈^n :[ Ax = b, x ≥ 0; x_j ∈{0,1} ∀j ∈ ]}.We now introduce the following standard assumptions: The set ⊆^n is nonempty and bounded.Ax = b,x ≥ 00 ≤ x_j ≤ 1∀j ∈. Assumption <ref> can be easily enforced.For example (see also <cit.>, <cit.>),if = ∅, then the assumption is redundant; if problem (<ref>) is derived from the network flow problems, for instance the longest path problem on a directed acyclic graph, then Assumption <ref> is implied from the network flow constraints; ifis a nonempty set and theassumption is not implied by the constraints, we can add constraints x_j + s_j = 1,s_j ≥ 0∀j ∈.The support set Ξ⊆^k is convex, closed,and computationally tractable. For example, Ξ could be represented using a polynomialnumber of linear, second-order-cone, and semidefinite inequalities.In particular, the set Ξ possesses a polynomial-time separationoracle <cit.>. Ξ is bounded.By Assumption <ref>, we know that v(ξ) is finite and attainable for any ξ∈Ξ.Note that under Assumptions<ref>-<ref>,v_^+ is finite and attainable and thus we canreplace sup with max in (<ref>) underthese conditions. Assumption <ref>could be merged with Assumption <ref>, butit is stated separately to highlight its role in proving theexactness of the copositive programming reformulationbelow. §.§ A copositive reformulationWe reformulate problem (<ref>) via conic programming duality theory and probability theory. We introduce a useful result from the literature as follows.v_(_N, ϵ)^+ equals the optimal value of[sup1/N∑_i=1^N ∫_Ξv(ξ)d _i(ξ);1/N∑_i=1^N ∫_Ξξ - ξ̂_i^2d _i(ξ) ≤ϵ^2;_i ∈ M^2(Ξ)∀i ∈ [N], ]where _i represents the distribution of ξ conditional on ξ' = ξ̂^i for all i ∈ [N]. As _i represents the distribution of ξ conditional onξ' = ξ̂^i, the joint probability Π in problem (<ref>)can be decomposed as Π = 1/N∑_i ∈ [N]_iby the law of total probability. Thus, the optimal value of (<ref>) coincides with v_(_N, ϵ)^+, which completes the proof.We next provide a copositive programming reformulation for problem (<ref>). As the first step, we use a standard duality argument to write the dual of (<ref>) (see also <cit.>):v_(_N, ϵ)^+ =sup__i ∈^2(Ξ)inf_λ≥ 01/N∑_i=1^N ∫_Ξ v(ξ)d _i(ξ) + λ (ϵ^2 - 1/N∑_i=1^N ∫_Ξξ - ξ̂_i^2d _i(ξ))≤ inf_λ≥ 0sup__i ∈^2(Ξ)λ ϵ^2 + 1/N∑_i=1^N ∫_Ξ (v(ξ) - λξ - ξ̂_i^2)d _i(ξ)=inf_λ≥ 0λ ϵ^2 + 1/N∑_i=1^N sup_ξ∈Ξ (v(ξ) - λξ - ξ̂_i^2),where (<ref>) follows from the max-min inequality,while equation (<ref>) follows from the fact that M^2(Ξ) contains all the Dirac distributions supported on Ξ.By Assumption <ref>, v(ξ) is finite for all ξ∈Ξ. Then, the inequality in (<ref>) becomes an equality for any ϵ > 0 due to a straightforward generalization of a strong dualityresult for moment problems in Proposition 3.4 in <cit.>; see also Theorem 1 in <cit.> and Lemma 7 in<cit.>.By introducing auxiliary variables s_i, the minimization problem in (<ref>) is equivalent to [v_(_N, ϵ)^+ = inf_λ, s_i λ ϵ^2 + 1/N∑_i=1^N s_i; sup_ξ∈Ξ (v(ξ) - λξ - ξ̂_i^2) ≤ s_i∀i ∈ [N];λ≥ 0. ]For each i ∈ [N], consider the following maximization problem corresponding to the left-hand side of the constraints in (<ref>):[ h^i(λ) := sup (Fξ)^Tx - λ(ξ^Tξ - 2ξ̂_i^Tξ + ξ̂_i^2);Ax = b,x ≥ 0;x_j ∈{0, 1} ∀j ∈;e_1^Tξ = 1,ξ∈Ξ̂, ]which is a mixed 0-1 bilinear program. Under Assumption <ref>,it holds also that the optimal value of (<ref>) equals the optimal value of an associated copositive program <cit.>,which we now describe. Definez := ξ x∈^k + n,E := [ -be_1^T A ]∈^m × (k + n), H^i(λ) := [ -λ (I - ξ̂_i e_1^T - e_1ξ̂_i^T + ξ̂_i^2e_1e_1^T)1/2 F^T;1/2 F0 ]∈^k+n,and for any j ∈, defineQ_j :=0f_j0f_j^T - 1/20f_je_10^T-1/2e_100f_j^T ∈^k+n. where f_j denotes the j-th standard basis vector in ^n. Because bothand Ξ are bounded by Assumptions <ref> and <ref>, there exists a scalar r > 0 such that the constraint z^T z = ξ^T ξ + x^T x ≤ r is redundant for (<ref>).Furthermore, it is well-known that we can use the following quadratic constraints to represent the binary variables in the description of : x_j^2 - x_j = 0⇔ Q_j∙ zz^T = 0. After adding the redundant constraint and representing the binary variables, we homogenize problem (<ref>) as follows:[maxH^i(λ) ∙ zz^T; Ez = 0,g_1^Tz = 1;I ∙ zz^T ≤ r;Q_j ∙ zz^T = 0∀j ∈;z ∈Ξ×_+^n, ]where g_1 = e_10∈^k+n and e_1 denotes the standard basisvector in ^k. The copositive representation is thus[ maxH^i(λ) ∙ Z; (EZE^T) = 0; g_1 g_1^T ∙ Z = 1; I ∙ Z ≤ r; Q_j ∙ Z = 0∀j ∈;Z ∈(Ξ×_+^n). ]Letting u^i ∈^m, ρ^i ∈_+, α^i ∈, and v^i ∈^|| be the respective dual multipliers of (EZE^T) = 0, I ∙ Z ≤ r, g_1 g_1^T ∙ Z = 1, and Q_j ∙ Z = 0, standard conic duality theory implies the dual of (<ref>) is[ min_α^i, ρ^i, u^i, v^iα^i + r ρ^i;α^i g_1 g_1^T - H^i(λ) + E^T (u^i)E + ∑_j ∈ v^i_jQ_j + ρ^i I ∈(Ξ×_+^n);ρ^i ≥ 0. ]Holding all other dual variables fixed, for ρ^i > 0 large, the matrix variable in (<ref>) is strictly copositive—in fact, positive definite—which establishes that Slater's condition is satisfied, thus ensuring strong duality: the optimal value of (<ref>) equals the optimal value of (<ref>). Therefore, we can reformulate problem (<ref>) as follows:[v_(_N, ϵ)^+ =min λϵ^2 + 1/N∑_i=1^N (α^i + rρ^i); α^i g_1 g_1^T - H^i(λ) + E^T (u^i)E + ∑_j ∈ v^i_jQ_j + ρ^i I ∈(Ξ×_+^n)∀i ∈ [N]; ρ^i ≥ 0 ∀i ∈ [N];λ≥ 0. ]Note that if Assumption <ref> fails, the constraint I ∙ Z ≤ r should be excluded from (<ref>) and thus the terms rρ^i and ρ^i I in the objective function and the constraint, respectively, should be excludedin (<ref>) as well. As such, strong dualitybetween (<ref>) and (<ref>) cannot be established in this case. However, the modified (<ref>) still provides an upper bound on h^i(λ).Accordingly, the modified problem (<ref>) still provides an upper bound on v_(_N, ϵ)^+. §.§ A semidefinite-based relaxationAs problem (<ref>) is difficult to solve in general, wepropose a tractable approximation based on semidefinite programming techniques. In particular, we propose an inner approximation of (Ξ×_+^n) in (<ref>) so that the resulting problem has an optimal value that is an upper bound on v_^+. Now, define(Ξ×_+^n) := { S + M: [ S_11∈(Ξ), Rows(S_21) ∈Ξ^*; S_22≥ 0,M ≽ 0 ]},where (Ξ) is an inner approximation of (Ξ), i.e.,(Ξ) ⊆(Ξ).Immediately, we have a relationship between (Ξ×_+^n) and (Ξ×_+^n): (Ξ×_+^n) ⊆(Ξ×_+^n). Let arbitrary pq∈Ξ×_+^n be given. We need to showpq^T ( S + M ) pq = pq^T S pq + pq^T M pq≥ 0. pq^T ( S + M ) pq = pq^T S pq + pq^T M pq = p^T S_11p + 2q^TS_21p + q^TS_22q +pq^T M pq ≥0The first term is nonnegative because p ∈Ξ and S_11∈(Ξ) ⊆(Ξ); the second term is nonnegative because p ∈Ξ, q ≥ 0, and Rows(S_21) ∈Ξ^*; the third term is nonnegative because q ≥ 0 and S_22≥ 0; the last term is nonnegative because M ≽ 0. When Ξ= {ξ∈^k: Pξ≥ 0} is a polyhedral cone based on some matrix P ∈^p × k, a typical inner approximation (Ξ) of (Ξ) is given by(Ξ) := { S_11 = P^T Y P : Y ≥ 0 },where Y ∈^p is a symmetric matrix variable. This corresponds to the RLT approach of <cit.>. When Ξ= {ξ∈^k : (ξ_2, …, ξ_k)^T≤ξ_1 } is the second-order cone, it is known <cit.> that(Ξ) = { S_11 = τ J + M_11 : τ≥ 0,M_11≽ 0 },where J = (1, -1, …, -1). Because of this simple structure, it often makes sense to take (Ξ) = (Ξ) in practice.Now consider the following problem by replacing (Ξ×_+^n) with (Ξ×_+^n) in (<ref>). [v̅_(, ϵ)^+=min λϵ^2 + 1/N∑_i=1^N (α^i + rρ^i); α^i g_1 g_1^T - H^i(λ) + E^T (u^i)E + ∑_j ∈ v^i_jQ_j + ρ^i I ∈(Ξ×_+^n)∀i ∈ [N]; ρ^i ≥ 0 ∀i ∈ [N];λ≥ 0. ] Obviously, we have the following result:v_(_N, ϵ)^+ ≤v̅_(, ϵ)^+.§ NUMERICAL EXPERIMENTSIn this section, we validate our proposed Wasserstein-ball approach (WB) on three applications. We will compare WB with the moment-based approach (MB) proposed in<cit.> where the exact values of thefirst two moments of the distributions are known.In practice, the moments of the distribution are often not knownexactly. To this end, Delage and Ye <cit.> proposed a data-driven approach to handle this case. However, in this paper,we assume that the moments are known exactly for MB. Actually, thischoice favors MB, but the goal of our experiments is to demonstrate thatour approach provides a valid upper bound that gets closer to v_ as the size of the data set increases, while the MB provides an upper bound, which does not improve with the size of the data set.All computations are conducted with Mosek version 8.0.0.28 beta <cit.> onan Intel Core i3 2.93 GHz Windows computer with 4GB of RAM and are implementedusing the modeling language YALMIP <cit.> in MATLAB(R2014a)version 8.3.0.532. In order to demonstrate the effectiveness of WB, we also implement a Monte Carlo simulation-based approach (SB) which requires a sufficiently large number of randomly generated samples. For the project management problem in Section <ref>, a linear program is solved for each sample of the Monte Carlosimulation, while for the knapsack problem in Section <ref>, an integer program is solved for each sample. We employ CPLEX 12.4 to solve theselinear programs and integer programs. §.§ Statistical sensitivity analysis of highest-order statistic The problem of finding the maximum value from a set ζ = (ζ_1, …, ζ_n) of n numbers can be formulatedas the optimization problem:max{ζ^Tx : e^T x = 1,x ≥ 0}.For example, suppose ζ_1 = max{ζ_1, …, ζ_n },then the optimal solution to (<ref>) is x_1^* = 1, x_2^* = ⋯ = x_n^* = 0. For the statistical sensitivity analysis problem, we consider a random vector ζ following a joint distribution .In the situation where the true distribution is not known exactly, our focus is to investigate the upper bound on the expected maximum value over an ambiguity set containing distributions that possess partial shared information. We consider an instance with n = 3 and the true distributionof ζ is assumed to be jointly lognormal with first andsecond moments given by μ_log∈^3 andΣ_log∈^3, respectively. In our experiments, we use the following procedure to randomly generate μ_log and Σ_log. We first sampleμ∈^3 from a uniform distribution [0, 2]^3. Then, werandomly generate a matrix Σ∈^3 as follows: we set the vector of standard deviations to σ = 14 e ∈^3, sample a random correlation matrix C ∈^3 using the MATLAB command `gallery(`randcorr',3)', and set Σ = (σ)C(σ)+μμ^T.We set μ and Σ as the first and second moments respectively of the corresponding normal distribution of . Then μ_logand Σ_log can be computed based on the followingformulae <cit.>:[ (μ_log)_i = e^μ_i + 0.5Σ_ii,; (Σ_log)_ij = e^μ_i + μ_j + 0.5(Σ_ii + Σ_jj)(e^Σ_ij-1). ]We can cast this problem into our framework by setting m = 1, k = n+1, ξ = (1, ζ_1, …, ζ_n), F = (0, I), and = ∅. Obviously, Assumptions <ref> and <ref> are satisfied.Assumption <ref> is vacuous. Although Assumption<ref> does not hold, problem (<ref>) can still provide a valid upper bound on the expected optimal value asdiscussed in Section <ref>.§.§.§ The deviation of empirical Wasserstein radii In this experiment, we consider a particular underlying distributionthat is generated by the procedure mentioned above. Also, we consider eight cases for the size of the dataset: N ∈{10, 20, 40, 80, 160, 320, 640, 1280 }. For each case, we randomly generate a dataset Θ_Ncontaining N independent samples fromand use the procedure in Section <ref> to determine a desired radius from a pre-specified setℰ = {0.001, 0.005, 0.01, 0.05, 0.1, 0.2,0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 2.0}[ From preliminary experiments, the largest element 2.0 inset E returned 1 as the empirical confidence level for all theexperiments we conducted. Thus, we believe it is sufficient to have 2.0 as the largest element here.].In particular, we set K = 100 in Algorithm <ref>. Figure <ref> shows the trend of the reliabilities over different Wasserstein radii for N ∈{ 20, 80, 320, 1280 }. Clearly,smaller Wasserstein radii tend to have lower empirical confidencelevels. Furthermore, as the sample size increases, the empirical confidence level increases as well for the same Wasserstein radius. The result of this experiment indicates that we can practically choose a Wasserstein radius with a desired statistical guarantee for each case of N.We remark that the derived radii can be used for datasets of the same sizes generated from different distributions of the same family (lognormal distributions in this application). §.§.§ Instances with the same underlying distributionOur next experiment is to focus on a particular joint lognormal distribution . We consider eight cases: N ∈{ 10, 20, 40, 80, 160, 320, 640, 1280 }. For each case, we test 100 trials and in each trial we randomly generate N independent samples from and choose the Wasserstein radius with an empirical confidence level of 0.90. We compare our approach with MB where the first two moments are directly given as μ_log and Σ_log.We also randomly generate 100000 independent samples fromto simulate the true expected optimal value.We demonstrate experimental results in Figure<ref>.Note that the solid black line represents the simulated value of the true expected optimal value, while the dashed black linerepresents the upper bound calculated by the moment-basedapproach. Furthermore, we solve an instance of (<ref>)for each of the 100 trails in each case of N. We use the blue, red, and green lines to respectively represent the 80^th quantile, the median, and the 20^th quantile of the values from the100 trials in each case. Figure <ref> shows that our approach provides weaker bounds on the expected optimal value for smaller sample sizes. However, as the size of samples increases, our approach provides stronger bounds and the bounds get relatively close to the simulated value. In contrast, the value from MB remains the sameregardless of sample size. §.§.§ Instances with different underlying distributions In this experiment, we consider eight cases N ∈{ 10, 20, 40,80, 160, 320, 640, 1280 }. In each case, we randomlygenerate 100 trials. For each trial in each case we generate N samples from a random lognormal distribution whose first and secondmoments are generated by using the procedure at the beginning of this section. For each trial in each case, we solve an instance of(<ref>) with a Wasserstein radius corresponding toan empirical confidence level of 0.90. We also simulate the true expectedoptimal values by randomly generating 100000 samples from the true distributions. For each trial in each case, we denote the optimal value from (<ref>)by v̅_WB^+ and the simulated value by v_SB.Then, we calculate the relative gap between WB and SB as gap(WB) := v̅_WB^+ - v_SBv_SB. We take the average of the relative gaps over the 100 trials for each case.Then, for each trial in each case, we solve MB with the first two moments computed by (<ref>). Denote the optimal value from MB by v̅_MB^+. Similarly, we calculate the relative gapbetween MBand SB asgap(MB) := v̅_MB^+ - v_SBv_SB. We then take the average of the relative gaps over the 100 trials in each case.Figure <ref> illustrates the average relative gaps from both WB and MB over the eight cases. Clearly, the upper bound from WB approaches the simulated value along with the increase of the size of samples, while the average relative gapbetween the bound from MB and the simulated value does not. Table <ref> showsthe percentage of the 100 trials where the optimal values from WB are greater than or equal to the corresponding simulated optimal values in the eight cases. The result demonstrates that the derived empirical Wasserstein radii indeed provide desired statistical guarantees in practice. §.§ Project management problemIn this application, we consider a project management problem, which can be formulated as a longest-path problem on a directed acyclic graph. The arcs denote activities and the nodes denote the completions of a set of activities. Arc lengths denote the time to complete the activities. Thus, the longest path from the starting node s to the ending node t gives the time needed to compete the whole project. Let ζ_ij be the length (time) of arc (activity) from node i to node j. The problem can be solved as a linear program due to the network flow structure as follows: [ max∑_(i,j) ∈ Aζ_ij x_ij; ∑_i : (i,j) ∈ A x_ij - ∑_j : (i,j) ∈ A x_ji ={[1, if i = s; 0, if i ∈ N, and i ≠ s,t; -1, if i = t ].; x_ij≥ 0,∀(i,j) ∈ A, ]where A denotes the set containing all the arcs, N denotes the set containing all nodes on the network, and x_ij denotes the number of units of flow sent from node i to node j through arc(i,j) ∈ A. For the stochastic project management problem, the activity times are random. In such cases, due to the resource allocation and management constraints, the project manager would like to quantify the worst-case expected completion time of the project, which is corresponding to the worst-case longest path of the network. We consider an instance with a network structure shown in Figure <ref>. This network consists of 7 arcs and 6 nodes. There are 3 paths from the starting node to the ending node on the network. In the experiments of this example, we consider truncated joint normal distributions. We use the following procedure to generate a truncated joint normal distribution : denoting | A| by the cardinality of set A,we generate ζ≥ 0 from a jointly normal distribution with first and second moments given by μ∈^| A| and Σ∈^| A|, respectively. Specifically, we sample μ from a uniform distribution [0, 5]^| A| while the matrix Σ is generated randomly using the following procedure: we set the vector of standard deviations to σ = e, sample a random correlation matrix C ∈^| A| using the MATLAB command `gallery(`randcorr',| A|)', and set Σ = (σ)C(σ)+μμ^T.Skipping the details, we can cast the network flow problem into our framework. It is straightforward to check that Assumptions<ref>, <ref>, and <ref> are satisfied and Assumption <ref> is vacuous. §.§.§ Instances with the same underlying distribution The first experiment of this example focuses on a particular underlying distribution . We consider seven cases: N ∈{10, 20, 40, 80, 160, 320, 640}. For each case, we run 100 trials and in each trial we randomly generate a dataset Θ_N containing N independent samples from .We use the procedure in Section <ref> to compute an empirical confidence level set for each case. Then, we use computed empirical confidence level sets to derive empirical Wasserstein radii for the following computations. For each trial in each case, we solve an instance of (<ref>) with a Wasserstein radius corresponding to an empirical confidence level of 0.90.We compare WB with MB where the first two moments are approximated by using the sample mean and variance from 100000 samples. The computed moments are close to their theoretical counterparts as thesample size is considerably large. We also simulate the expected optimalvalue over the 100000 samples. Figure <ref> shows that WB provides weaker bounds on the expected optimal value forsmaller sample sizes. However, as the size of samples increases, WB provides stronger bounds and the bounds get relatively close to the simulated value. In contrast, the bounds from MB remains the sameregardless of the change of sample sizes. §.§.§ Instances with different underlying distributions In this experiment, we consider seven cases: N ∈{ 10, 20, 40, 80,160, 320, 640}. For each case, we randomly generate 100 trials inwhich N independent samples are drawn from a randomly generatedtruncated joint normal distribution. Then, for each trial in each case, we solveand instance of (<ref>) with a Wassersteinradius corresponding to a 0.90 empirical confidence level.We solve MB where the first two moments are approximatedby computing the sample mean and variance of 100000 samples.We also simulate the expected optimal value over the 100000 samplesfor each trial in each case. We compute the relative gap between the WB and SB as well as the relative gap between MB and SB.Then, for each case, we take the average of the relative gaps from both WB and MB over the 100 trials.Figure <ref> illustrates the average relative gapsover the seven cases.Clearly, the upper bound from WB approaches to the simulated value along with the increase in the size of samples, while the gap between the bound from MB and the simulated value remains relatively the same as the sample size increases. Table <ref> showsthe percentage of the 100 trials where the optimal values from WB are greater than or equal to the corresponding simulated optimal values over the seven cases.§.§ Knapsack problemA standard knapsack problem is defined as follows: given a set of items and each with a weight and a value, the problem is to determine the numberof items to include in a knapsack such that the total weight is less than or equal to a given capacity limit and the total value is maximized; see the detail in<cit.>. Let w_i and ζ_i be the weight and valueof item i(i = 1, …, n) respectively. Let W be the maximum weightcapacity of the knapsack. Then, the knapsack problem can be formulatedas an integer program:v(ζ) := max{∑_i = 1^n ζ_ix_i : ∑_i=1^n w_i x_i ≤ W,x_i ∈{0, 1}},where x_i represents the number of item i to include in the knapsack.Assume that the values of the items are random and follow an unknownjoint distribution. Assume also that we can collect a dataset containing Nsamples with each corresponding to an observation of the n item values.In such cases, we would like to compute a data-driven distributionally robustupper bound on the expected maximum value of the knapsack. We canapproximate the upper bound by solving problem (<ref>).We consider an instance with n=4, w = (5,4,6,3 )^T, and W = 10. The true distributionof ζ is assumed to be jointly lognormal with first and second moments given by μ_log∈^4 andΣ_log∈^4, respectively. Similar to the procedure described inSection <ref>, we sample μ∈^4 from a uniform distribution [0, 2]^4.Then, we randomly generate a matrix Σ∈^4 as follows: we set the vector of standard deviations to σ = 14 e ∈^4, sample a random correlation matrix C ∈^4 using the MATLAB command `gallery(`randcorr',4)', and set Σ = (σ)C(σ)+μμ^T.Then μ_log and Σ_log can be computed based on (<ref>). We can easily cast this problem into our framework. For simplicity, we skip the details. It is also straightforward to check that the conditions in Assumptions <ref>, <ref>, and <ref> are satisfied. AlthoughAssumption <ref> is not satisfied, we still can solve (<ref>) to obtain a valid upper bound on the expected optimal value of the knapsack problem. §.§.§ Instances with the same underlying distribution In the first experiment, we focus on a particular underlying distributionand consider eight cases: N ∈{10, 20, 40,80, 160, 320, 640, 1280 }. For each case, we run 100trials and in each trial we randomly generate a dataset Θ_Ncontaining N independent samples from . Similarly, we derive empirical Wasserstein radii for each case. Then, for each trial in each case, we solve an instance of (<ref>) with an empirical Wasserstein radiuscorresponding to an empirical confidence level of 0.90. We compare our approachwith MB where the first two moments are computed by (<ref>).We simulate the expected optimal value over 100000 samples.Note that we solve an integer program for each sample in the simulation.Figure <ref> shows that WB provides weaker bounds on theexpected optimal value for smaller sample sizes. However, as the size ofsamples increases, WB provides stronger bounds and the bounds getrelatively close to the simulated value. In contrast, the bounds from MBremains the same regardless of the change of sample sizes. We remark that the upper bound computed by WB may not converge to the true expected optimal value as the sample size increases to infinity; see the trendshown in Figure <ref>. This is due to the fact that problem(<ref>) is a relaxation of problem (<ref>) and the fact the relaxation is not tight in this example. §.§.§ Instances with different underlying distributions This experiment considers eight cases: N ∈{ 10, 20, 40, 80, 160,320, 640, 1280 }. For each case, we randomly generate100 trials with each trial is drawn from a randomly generatedjoint lognormal distribution. Then, for each trial in each case, we solve an instance of (<ref>) with a Wassersteinradius corresponding to a 0.90 empirical confidence level.We simulate the expected optimal value over 100000 samplesfor each trial in each case. Next, we compute the relative gapbetween the values of WB and SB as well as the relative gap between the values of MB and SB. Then, for each case, we take the average ofthe relative gaps from both WB and MB over the 100 trials.Figure <ref> illustrates the average relative gaps overthe seven cases. Clearly, the WB gap becomes narrower as thesample size increases, while the MB gap remains relatively the same. Table <ref> showsthe percentage of the 100 trials where the optimal values from WB are greater than or equal to the corresponding simulated optimal values over the eight cases.§ CONCLUDING REMARKSIn this paper, we have studied the expected optimal value of a mixed 0-1 programming problem with uncertain objective coefficients following a joint distribution whose information is not known exactly but a setof independent samples can be collected. Using the samples, we have constructed a Wasserstein-based ambiguity set that contains the true distribution with a desired confidence level. We proposed an approach to compute the upper bound on the expected optimal value. Then under mild assumption, the problem was reformulated to a copositive program, which leads to a semidefinite-based relaxation.We have validated the effectiveness of our approach over three applications.§ ACKNOWLEDGEMENTSWe would like to thank Kurt Anstreicher, Qihang Lin, Luis Zuluaga, and Tianbao Yang for many useful suggestions, which helped us improve the results of the paper. plain | http://arxiv.org/abs/1708.07603v1 | {
"authors": [
"Guanglin Xu",
"Samuel Burer"
],
"categories": [
"math.OC",
"90B50"
],
"primary_category": "math.OC",
"published": "20170825025807",
"title": "A Data-Driven Distributionally Robust Bound on the Expected Optimal Value of Uncertain Mixed 0-1 Linear Programming"
} |
§ BACKGROUNDIn recent years, tremendous efforts have been dedicated to establish the evolution of the globalcomoving SFR density up to z∼8. The observations show that there is a peak in the comoving SFRdensity between 1<z<3, followed by an order of magnitude decrease towards z∼0 over the last 10 Gyrs, i.e. 70% of the age of the Universe <cit.>. But a clear picture of the underlying processesthat drive such a strong evolution is yet to emerge.Since stars form in molecular clouds, the observations of the molecular gas (H_2), which is the basicfuel for star formation, and the atomic gas (), which is the dominant gas component of galactic discs,can be used to understand the evolution of SFR density.The atomic gas component of the interstellarmedium (ISM) is easily observable through the21-cm emission line. But due to technicallimitations the21-cm emission line studies at present are mostly confined to the local Universe.Through the observations of21-cm emission line in the local Universe and the large samples of dampedLyα absorbers (DLAs[ column density, N()≥2×10^20 cm^-2.])at high-z, themass density (Ω_HI) of the Universe is reasonablywell-constrained at z<0.2 and 2<z<4, respectively. They reveal a very modest (factor ∼2) decreasein Ω_HI compared to an order of magnitude decrease in the SFR density over the same redshift range.This implies that the processes leading to the conversion of atomic gas to molecular gas and eventually tostars need to be understood directly via observations of cold atomic and molecular gas, rather than theevolution of Ω_HI.It is well known that21-cm absorption is an excellent tracer of the cold neutral medium (CNM; T∼100 K)in the Galaxy. Similarly, the OH radical is one of the common constituents of diffuse and dense molecularclouds, and can be observed through the twomain lines at 1665 MHz and 1667 MHz, and the two satellite lines at1612 MHz and 1720 MHz. As the volume filling factor of ISM phases and the observables associated withtheand OH absorption lines depend sensitively on in situ star formation through various stellarfeedback processes, these lines can be used to probe the physical conditions in the ISM and understandthe cosmic evolution of the SFR density <cit.>. In addition, the observed frequencies of these lines can be used to place the most stringent constraintson the variation of fundamental constants of physics <cit.> and cosmicacceleration <cit.>. Despite the scientific potential of intervening21-cm and OH 18-cm lines, and the tremendous effortsfrom the community over last three decades, mainly due to technical limitations, only ∼50intervening21-cm <cit.> and 3 intervening molecular absorbers <cit.>at radio wavelengths are known. The small number of detections has hampered attempts to systematicallymap the cold gas evolution with any statistical significance <cit.>. The availability of various SKA pathfinders in RFI-free environments offers an unique opportunityto overcome these limitations. In 2010, MALS was identified as one of the ten large surveys to be carried out with MeerKAT. The main objective of the survey is to use MeerKAT's L- and UHF-band receivers to carry out the most sensitive(N()>10^19 cm^-2) dust-unbiased search of intervening21-cm and OH 18-cm absorption lines at 0<z<2.The survey is expected to detect ∼200 intervening 21-cm absorbers.This will provide reliable measurements of the evolution of cold atomic and molecular gas cross-sections of galaxies,and unravel the processes driving the steep evolution in the SFR density. Due to the large field-of-view and excellent sensitivity of MeerKAT, the survey is also expected to detect∼500 21-cm absorbers associated with AGNs. These associated and intervening absorbers detected from the survey will enable discovery througha wide range of science themes that are described in Section 2. The survey design is presented in Section 3. A high-level plan for data products, observing plan and data releases is provided in Section 4. The relationship with other SKA pathfinder surveys and key science is presented in Section 5.§ MALS - KEY SCIENCE THEMESThe key science themes based on the survey design (see Section 3; Table <ref>) are as follows: * Evolution of cold gas in galaxies and relationship with SFR density:MALS will be sensitive to detect21-cm absorption and constrain the number per unit redshift range ofintervening21-cm absorbers, n_21, from sub-DLAs to DLAs, i.e. the CNM fraction in a wide range ofcolumn densities tracing the inner disks of galaxies, as well as the extended disks, halos, intra-group gasand the circumgalactic medium (CGM).It will measure the evolution of cold atomic and molecular gas cross-section of galaxies in a luminosity- anddust-unbiased way and constrain the ISM physics at pc scales over the entire redshift range 0<z<2.Through MALS, the detection of ∼200 intervening 21-cm absorbers is expected. With these it willconstrain the evolution of n_21(z), split in 3 redshift bins over 0<z<1.5, with ∼20% accuracy. If n_21(z) scales with Ω_HI(z), the 20% accuracy will allow us to detect the evolution inn_21(z), and hence the CNM cross-section of galaxies, at the 3σ level (for example, see <cit.>).Through OH lines, the surveywill be sensitive to the molecular gas cross-section of galaxies, and will provide the first estimate of the numberper unit redshift range of OH absorbers, n_OH(z).The n_21(z) and n_OH(z) from the surveywill be key observables to understand the process of condensation of warm gas into cold gas and eventually stars.In Table <ref>, we compare MALS with other upcomingimaging surveys with footprint and survey depthlarge enough to deliver a substantial number of 21-cm absorbers.The Westerbork Synthesis Radio Telescope (WSRT) shallow survey will search foremissionline at δ>+30^∘ with Apertif, a phased-array feed for WSRT. Simultaneously, it will deliver a 21-cmabsorption line survey called SHARP (PI: Morganti).The Australian SKA Pathfinder (ASKAP) will have a dedicated21-cm absorption line survey at 0.5<z<1 called the First Large Absorption Survey in(FLASH; PI: Sadler).WALLABY (PIs: Koribalski and Staveley-Smith) is the ASKAPemission all-sky survey at δ<30^∘that will simultaneously deliver a 21-cm absorption line survey at z<0.26. From Table <ref> it is clear that amongst various upcoming major21-cm absorption line surveys, MALS iswell distinguishable. It is the only survey that will uniformly cover 0<z<1.5, the most important redshift rangeas far as the evolution of SFR and AGN activity are concerned (cf. columns 2-3). It also provides excellent spectralline sensitivity in relatively much less observing time per pointing (cf. columns 4-5). The last column provides thenumber of sources or sight lines within the survey area towards which21-cm absorption from a 100 K gas withthe DLA column density would be detectable. The flux density cut-off corresponding to this optical depth sensitivityis also provided. Note that the last column does not take into account the redshift distribution of radio sources andany variation in gas properties with redshift or radio source type, but provides a useful metric to compare the overalleffectiveness of these surveys with different characteristics. Fig. <ref> shows the distribution of redshift path for intervening absorption for the same optical depth sensitivity. Here, the analysis takes intoaccount the redshift and spatial distribution of radio sources over the full survey area. For MALS, we haveconsidered the survey area corresponding to 20% sensitivity of the maximum of the primary beam. In summary, itis evident that except for WALLABY for which a large amount of telescope time in a narrow redshift range has beenallocated to an21-cm emission line survey, the MALS is comparable to other surveys at z>0.26.In particular, both in the L- and UHF- bands it covers more than 50% of the redshift ranges inaccessible to any ofthe other surveys.Notably, the redshift ranges of 0.3<z<0.5 and 1<z<1.5 are only accessible to MALS. The absorbers from MALS will befollowed-up with (1) VLBI to map the pc-scale structure in the ISM <cit.>,(2) HST, SALT and VLT to detect various atomic/metal absorption lines to constrain the ambient radiationfield, metallicity, density and temperature via detections, and VLA/MeerKAT to probe the magnetic fields ingalactic discs via rotation measure synthesis <cit.>. These parameters, in addition to n_21and n_OH, will be valuable inputs to test and refine the physical models of the ISM so far based primarilyon the galactic ISM <cit.> and understand the processes driving the evolution of SFR density.Further, the multi-wavelength follow-ups of the absorbers using ALMA, NOEMA, VLA, VLT and SALT will lead tothe discovery of rare atomic and molecular species which will be used to measure the evolution of CMB temperatureand constrain the effective equation of state of decaying dark energy <cit.>. * Fuelling of AGN, AGN feedback and dust-obscured AGNs:AGN activity is known to peak during the same epoch as SFR density. Both theoretical and observationalconsiderations suggest a close relationship between galaxy evolution and AGN activity <cit.>.Further, the existence of red i.e. dust-obscured AGNs is predicted in both the orientation-based AGN unificationmodels and the evolutionary scenarios but the exact fraction of such AGNs is still uncertain with the estimatesranging over 10-50% <cit.>. Through a dust-unbiased search ofand OH absorptions, MALSwill determine the (1) distribution and kinematics of atomic and molecular phases associated with circumnuclear gasand the ISM of AGN hosts, and (2) the fraction of dust obscured AGNs missed out in optical/ultraviolet surveys.Through these the survey will address several fundamental issues in the context of AGN evolution, AGN feedback andthe AGN Unification paradigm.Presently, about 100 associated21-cm absorbers and only a few OH absorbers are known <cit.>.In MALS survey area, there will be ∼2000 sources brighter than 30 mJy (∼1 GHz) and suitable for theassociated 21-cm and OH absorption line search. Adopting a detection rate of 25% <cit.>, the detectionof ∼500 associated 21-cm absorbers is expected. This will provide a large sample to study the propertiesof cold gas in powerful AGNs (>10^24 W Hz^-1) by splitting these into subsamples of different AGN typesand redshifts. For the brighter sources, the survey will also be sensitive to detect rare blue-shifted wings in the absorption profile <cit.>. Such wings representing outflows have been identifiedin ∼5% of the AGNs observed at z<0.2. The evolution of this fraction and the mass-outflow rate withredshift is a key observable to compare with the predictions from the models of galaxy evolution. MALS willextend the observations of these outflows to a largely unexplored (z>0.2) and the most relevant redshiftrange in this context.* Galaxy evolution throughand OH emission: With an integration time of 56 minutes per pointing in L-band, MALS is a substantial blind emission line survey. MALS provides a set of 740 quasi-random pointings (with a potential sub-sample pointing towards regions of high interest such as galaxy groups). With the natural rms of 0.26 mJy beam^-1over 20at the pointing centre delivered by MALS, a galaxy like Leo T with a dark matter mass that isjust enough to retain baryons, M()∼2.8×10^5 M_⊙ and radius∼240 pc<cit.>, would be detectable above a 5σ level and resolved to a distance of 4 Mpc.Over an area of 300 deg^2, MALS will be able to detect an M_* galaxy up to a redshift ofz∼0.1-0.2 and the galaxies with the largestmasses out to z > 0.5 (see Fig. 1). We estimatethe number of detections to be of the order of 5000-15000 galaxies, many of which will be resolved. In addition,tenuous gas at column densities of 0.2 M_⊙pc^-2 will be detectable over 20at the 6σ levelat 45^'' resolution. MALS will therefore probe the gas distributions and kinematics of a substantial number of galaxy disks while providingan unique opportunity to relate these to the cold atomic and molecular gas detected throughand OH absorption. The main goals of MALS as anemission line survey are: (1) to catalog basic emission line parameters of observedobjects, (2) to produceemission line maps and data cubes of resolved emission line objects, (3) to parameterizethe extended morphology and kinematics, spin characteristics, rotation curves, and surface density profiles of theresolved galaxies independently as well as in connection with the absorption line survey to understand the natureof 21-cm and OH absorbers, and (4) to study velocity and mass functions, (dark) matter distribution and quantifyaccretion properties. With these observables MALS will be a perfect emission lineprogenitor survey for already addressing major questions raised in the SKA science book: (1) studying the radial gasprofiles of galaxies in the light of galaxy evolution <cit.>, (2) studying the matterdistribution in galaxies, (3) quantifying the intrinsic and mutual spin distribution in galaxies and its dependenceon their environment <cit.>, and (4) studying the low column density environment of galaxies toquantify (AGN) feedback and accretion processes on a statistical basis <cit.>.Besidesemission, MALS will also be capable of detecting OH Megamasers (MMs). Powerful OH MMs have already beendetected up to a redshift of 0.265 (1318 MHz; <cit.>) and the emission profiles show strong outflowsand evidences of strong interactions. The population of sources will increase at least up to a redshift of 2.5 becauseof the link between ULIRG[Ultraluminous infrared galaxy] host galaxies and the increasing merger rate withredshift. OH MMs are a good tracer ofstarburst activity and merger history and MALS will establish their luminosity function at higher luminosities and redshifts.* Radio continuum and polarization science with MALS: Magnetic fields are known to be present over a wide range of physical scales in astrophysical objects.They are an important component in the energy balance of the interstellar medium (ISM) of galaxies.Besides being important in determining the conditions for the onset of star formation in protostellarclouds they are also relevant for (1) the formation and stabilisation of gas disk and spiral arms,(2) the heating of gas, especially, in the outer disks and halos, and (3) the launching of galacticoutflows and determining how the matter and energy are distributed throughout the galaxy. Overall, magnetic fieldsare tightly coupled to various stellar feedback processes and play an important role in a wide range ofphysical processes that drive galaxy evolution.MALS will provide extremely sensitive wide-band L- andUHF-band observations to observe total intensity and polarized emission from a large number of normal andactive galaxies enabling wide range of projects some of which are described below. Clusters, halos and relics: MALS will allow us to conduct a blind survey of a statistically significantnumber of galaxy clusters to discover new diffuse cluster emission and constrain their formation models throughspectral studies. We expect to detect about 770 clusters with M_500>3×10^14 M_⊙within the survey region, where M_500 is the cluster mass within a density five hundred times the criticaldensity. All MALS pointings will be within the footprint of the Advanced ACT cosmic microwave background survey<cit.>, in which MALS team members are actively involved, and which will deliver Sunyaev-Zel'dovichmasses for all clusters above this mass limit. With the ∼3 μJy flux density limit of MALS, we expect to observe diffuseradio halo emission in approximately 80 of these clusters, over a wide range of cluster mass and redshift(0<z<0.8). This will provide a large enough statistical sample to study the evolution of radio haloproperties, which significantly extends the existing heterogeneous sample of about 50 giant radio halos<cit.> beyond the current range of masses (M_500 > 5×10^14 M_⊙) and redshifts(0.2 < z < 0.4). We will place constraints on radio halo formation models, by measuring fluxes and spectralindices over the wide MeerKAT frequency bands, as well as distinguish between turbulent halos generated byre-acceleration during cluster mergers, and off-state halos of a purely hadronic origin found in more relaxedsystems <cit.>. The polarization data from MALS will be crucial for a full study of the relicemission by testing the shock- related formation theories through inferred Mach numbers and magnetic fieldorientations.Magnetic fields in galaxies and AGNs: With the sensitivity of ∼3 μJy, we expect more than 30 polarized sourcesper square degree at 1.4 GHz <cit.>. The wide-band coverage of MALS will allow us to quantify theeffects of depolarization and measure currently unknown polarized source counts at low frequencies. The strongintervening 21-cm absorbers detected through MALS will be an excellent tracer of galactic discs and ideallysuited to probe the evolution of magnetic fields in galaxies by rotation measure synthesis. Until now, suchstudies using samples of Mg ii absorbers have yielded ambiguous results <cit.> becauseof the small sample sizes and the difficulty in interpreting the origin of Mg ii absorbers. Additionally, the sizes and structures of radio sources are often found to be affected by the ambient gas in thecentral regions of host galaxies, suggesting strong dynamical interaction with the external medium. Evidence ofthis gas may also be probed via polarization measurements and will be used to understand the role of jet-modefeedback in driving the outflows detected in 21-cm absorption and the evolution of AGNs.* Constraining space- and time-variation of fundamental constants:The absorption lines seen in the spectra of distant QSOs can be used to place constraints on the space andtime variations of different dimensionless fundamental constants of physics <cit.>. For the pastdecade or so based on systematic efforts using high resolution echelle spectrographs at VLT and Keck, thecommunity has constrained the variation of α=(e^2/hc) and μ = (m_e/m_p)at the level of 1 ppm at high-redshift.The next major step is to improve these constraints by another factor of 10 i.e. 1 part in 10^7which will be comparable to local measurements based on atomic clocks. The major systematics that preventsany further improvement stems from the stability of wavelength scales in present day optical spectrographs<cit.>. Further progress is possible through radio absorption lines because thefrequency scales at radio telescopes are known to be well-defined and compared tooptical/ultraviolet lines the radio lines are more sensitive to the variation of fundamental constants.But a large sample of radio absorbers is required to achieve this so that any systematics introduced due toionisation, radiation and excitation inhomogeneities in the gas that gets reflected as the variation ofconstants is randomised. MALS will provide such a sample for the first time. Through main- andsatellite- OH absorption lines from the survey, or the joint analysis of 21-cm and OH absorption withvarious metal, atomic and molecular absorption lines detected through follow-ups with ALMA, NOEMA, VLA,VLT and SALT the survey will lead to tightest constraints on the variations of fundamental constants of physics.* Stacking line, continuum and polarization data: Stacking of undetected21-cm line spectra of sources with known redshifts has become a tested method tostudy the atomic gas in galaxies <cit.>. We will apply stacking technique to spectral line, radiocontinuum and polarization data from MALS to make measurements that would otherwise be possible only withmuch deeper observations or SKA-I. Inand OH absorption, MALS will likely detect galaxies and AGNs undetectedat other wavelengths. This will not only constrain the redshift of such sources but also investigate theirmulti-wavelength properties by stacking. § SURVEY DESIGNThe MALS survey design is summarized in Table <ref>.The total L- and UHF-band on-source timesfor 740 and 370 pointings are 691 and 746 hrs respectively.Assuming 15% overheads this corresponds to a total observing timeof 1655 hrs. The survey strategy that yields almost uniform redshift path coverage over the entire redshift range of interest(see Fig. 1) is based on the optimization of the following two parameters that drive the number of detections and thescience output from the survey: * Optical depth sensitivity (∫τdv): The target 5σ integrated optical depth sensitivityfor intervening21-cm absorption at the center of the primary beam is ∫τdv = 0.045 . Thiscorresponds to a sensitivity to detect the CNM (100 K) in N()∼10^19 cm^-2 gas via21-cm absorption,and is essential to detect the CNM in a wide range of physical environments.This will be achieved by centeringL- and UHF-band pointings on bright (>400 mJy at 1 GHz) radio sources. * Redshift coverage/ Survey redshift path (Δz): As the sensitivity to detect the CNM down to acolumn density of 10^19 cm^-2 will beachieved only towards the central source, it is essential that these be selected carefully.The MeerKAT L- and UHF-band receivers cover frequency ranges of 900-1670 MHz and 580-1015 MHz respectively. For the21-cm line, these correspond to a redshift coverage of 0<z<0.58 and 0.40<z<1.44,respectively[The corresponding redshift ranges for OH main lines are 0<z<0.85and 0.64<z<1.87, respectively.].The redshift path for central bright sources will be maximized by selecting these to be at z>0.6 andz>1.4, respectively. With this approach, based on the n_21 from <cit.>, which has ∼50% errors,we expect to detect ∼100 intervening 21-cm absorbers towards the central bright sources (we adopt n_21=0.13;Δz = 800 and assume no redshift evolution). The spectral rms of ∼0.5 mJy would also ensure that an adequate redshift path to deliver a similar number ofintervening 21-cm absorbers with N()∼10^20 cm^-2 (i.e. ∫τdv=0.5for 100 K)is achieved towards off-axis weaker sources, and the sight lines with rare high columndensity absorbers are not under-represented in the survey. Over the last few years results from several21-cm absorption line searches in thesamplesof Mg ii absorbers at 0.5<z<1.5 <cit.>, DLAs at z>2<cit.>and quasar-galaxy pairs (QGPs) at z<0.3 <cit.> have become available.From VLBA imaging and spectroscopy of a subset of these samples it has become increasingly obvious thathigher detection rates of intervening 21-cm absorption are achieved for flat-spectrum radio sources(see Fig. 6 of <cit.>). This is related to the typical correlation length of 30-100 pcexhibited by the CNM gas (see also <cit.>). The gas clouds are expected to be much morecompact (<10 pc) in the case of translucent or dense phases of the ISM <cit.>.Therefore, for a blind search of intervening absorbers the background sources have to be as compactas possible. The flat-spectrum radio quasars (FSRQs), which are known to be compact, are hence the straightforward choice for MALS.Thus, in addition to the criteria that the central sources for L- and UHF-band pointings be atz>0.6 and z>1.4, respectively, we further require that a majority of these be FSRQs.§ SURVEY STRATEGY, OBSERVING PLAN AND DATA RELEASES The survey is expected to start in 2018 and will be carried out over a period of five years.Based on the outcomesfrom the first year of observations, the proportion of L- and UHF-band pointings for the survey will be revised. Survey strategy and observing setup * Observing setup: The data will be acquired in both parallel- and cross-polarization hands using a correlator dump time of 4 s. The L (900 - 1670 MHz) and UHF (580 - 1015 MHz) bands will be split into 32768frequency channels.This would deliver a spectral resolution of ∼5 km s^-1, and will beadequate to detect and resolve the structure in intervening 21-cm absorption lines (e.g. see Fig.12 of <cit.>).* Calibration overheads: For a two-hour observing block we assume a calibration overhead of 15%. This includes observations of a standard calibrator every two hours and a 2 min scan on a phasecalibrator every 45 mins for flux density scale and bandpass calibrations.The leakage calibration will be carried out through observations of an unpolarized calibrator.We will alsorequire a polarization angle calibrator. It is quite possible that the polarization angle can be calibratedusing data from previous days, and that there will be no need to observe a separate phase calibrator. All this will be tested and confirmed during the commissioning phase.* Observing blocks and Scheduling: The time spent on each pointing ranges from 56 to 121 mins.The total time on each pointing may be split into two equal sessions, which will be executedwithin a few months with a gap that ensures a shift of 20-30in the absorption lines. This shift due to theheliocentric motion of the Earth would allow us to unambiguously distinguish between true absorptions and RFI, andis required to build a reliable absorption line catalog. We intend to execute these sessions through observing blocksof one to several hours. * Cadence: Each pointing will have two epochs separated by a few months. The pointings (∼300) that arecommon between the L- and UHF-band phases will have four epoch observations. * Strategy for the selection of pointings: For the first year of observations, we will select 70% of the pointings at -40^∘< δ < +30^∘ and 100% of these will be centered at FSRQs withspectroscopically confirmed redshifts. This will allow accurate determination of redshift path length functionstowards the central bright source and lead to the first reliable constraints on n_21(z) and n_OH(z).The availability of VLBI images for all of the central targets in the first year of observations will allow us to quantify the effect ofradio structure on the detectability of absorption and its redshift evolution. All this will allow us to usefirst year's observations as a proving ground and, if needed, alter the choice of pointings for thesubsequent phases.Bright FSRQs with known spectroscopic redshifts, especially in the southern hemisphere, are rare. Therefore, we are carrying out an ambitious optical spectroscopic campaign using SALT and NOTto increase the number of FSRQs with z>1.4.For this, using WISE IR colors we have developed a pre-selection method that allows us to select z>1.4`candidate' quasars with ∼75% efficiency (Fig. <ref>).Through this ongoing program we have already identified ∼100 new FSRQs over last two years. All the data products from the survey (i.e. calibrated visibilities and images) will be released to communitythrough regular data releases.§ RELATIONSHIP WITH OTHER SKA PATHFINDER SURVEYS Understanding galaxy evolution through21-cm line is one of the key science drivers ofthe SKA <cit.>.MALS will measure the evolution of the cold atomic and molecular gas cross-sections of galaxies in aluminosity- and dust-unbiased way and constrain the ISM physics at pc scales over the entire redshiftrange: 0<z<2 (cf. Table <ref> for complementarity with other 21-cm absorption line surveys).Through the excellent sensitivity of MeerKAT over a wide range of spatial resolutions and widebandspectro-polarimetry, MALS will simultaneously deliver a highly competitive blind / OH emissionline and radio continuum survey, and will address a wide range of science issues raised in the SKA science book. The survey will also complement21-cm emission line surveys such as(1) LADUMA and WALLABY that will be affected by luminosity biasand will probe only most luminous objects at higher redshifts, and (2) MHONGOOSE that willmeasure the sub-kpc scale properties of gas but in a sample of nearby galaxies. The conversion of gas intostars involves complex physical processes from pc to kpc and even Mpc scales. Together, these surveys will provide observational constraints on the properties of cold gas and the underlying physical processes at allphysical scales required to understand the evolution of SFR density and design the future21-cm line surveys with SKA.We thank Kenda Knowles of UKZN for useful discussions. We thank IUCAA, Rutgers University and the South African members for the SALT observing timecontributed to the survey to identify high-z quasars for MALS. NOT is operated by the Nordic Optical Telescope Scientific Association at the Observatoriodel Roque de los Muchachos, La Palma, Spain, of the Instituto de Astrofisica de Canarias. We acknowledge the support from ThoughtWorks India Pvt. Limited in developing ARTIP - a prototype of MALS dataanalysis pipeline. JHEP § AUTHOR AFFILIATIONS^1 Inter-University Centre for Astronomy and Astrophysics, India;^2 ASTRON, the Netherlands Institute for Radio Astronomy, The Netherlands; ^3 Rutgers, the State University of New Jersey,USA; ^4 University of Manchester, UK; ^5 National Radio Astronomy Observatory, USA; ^6 Laboratoire d'Astrophysique de Marseille, France;^7 Physics and Astronomy Department, University of Western Cape, South Africa;^8 Observatoire de Paris, France;^9 Centre for High Performance Computing, South Africa;^10 Dark Cosmology Center, Niels Bohr Institute, Denmark;^11 CSIRO Astronomy and Space Science, Australia; ^12 Astrophysics and Cosmology Research Unit, University of KwaZulu Natal, South Africa;^13 Astrophysics, University of Oxford, UK;^14 Square Kilometre Array South Africa, South Africa;^15 National Centre for Radio Astrophysics, India;^16 Argelander-Institut für Astronomie (AIfA), Universität Bonn, Germany;^17 Max-Planck-Institut für Radioastronomie, Germany;^18 Institut d'Astrophysique de Paris, France; ^19 University of South Carolina, Department of Physics and Astronomy, USA;^20 European Southern Observatory, Chile;^21 Astronomy Department, California Institute of Technology, USA; ^22 South African Astronomical Observatory, South Africa;^23 INAF-Osservatorio Astronomico di Cagliari, Italy;^24 Department of Physics, Royal Military College of Canada, Canada;^25 Kapteyn Astronomical Institute, University of Groningen, The Netherlands;^26 University of Utah, USA;^27 International Centre for Radio Astronomy Research, The University of Western Australia, Australia. | http://arxiv.org/abs/1708.07371v1 | {
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"R. Beswick",
"S. Bhatnagar",
"D. Bhattacharya",
"A. Bosma",
"C. Carilli",
"M. Cluver",
"F. Combes",
"C. Cress",
"R. Dutta",
"J. Fynbo",
"G. Heald",
"M. Hilton",
"T. Hussain",
"M. Jarvis",
"G. Jozsa",
"P. Kamphuis",
"A. Kembhavi",
"J. Kerp",
"H. -R. Klöckner",
"J. Krogager",
"V. Kulkarni",
"C. Ledoux",
"A. Mahabal",
"T. Mauch",
"K. Moodley",
"E. Momjian",
"R. Morganti",
"P. Noterdaeme",
"T. Oosterloo",
"P. Petitjean",
"A. Schröder",
"P. Serra",
"J. Sievers",
"K. Spekkens",
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"published": "20170824122158",
"title": "The MeerKAT Absorption Line Survey (MALS)"
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[][email protected] []www.goldsborough.org.uk Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, D-85748 Garching, Germany [][email protected] []www.glenevenbly.com Département de Physique and Institut Quantique, Université de Sherbrooke, Québec, Canada We propose a tensor network method for investigating strongly disordered systems that is based on an adaptation of entanglement renormalization [G. Vidal, Phys. Rev. Lett. 99, 220405 (2007)]. This method makes use of the strong disorder renormalization group to determine the order in which lattice sites are coarse-grained, which sets the overall structure of the corresponding tensor network ansatz, before optimization using variational energy minimization. Benchmark results from the disordered XXZ model demonstrates that this approach accurately captures ground state entanglement in disordered systems, even at long distances. This approach leads to a new class of efficiently contractible tensor network ansatz for 1D systems, which may be understood as a generalization of the multi-scale entanglement renormalization ansatz (MERA) for disordered systems. 75.10.Jm, 05.30.-d, 02.70.-cEntanglement renormalization for disordered systems Glen Evenbly December 30, 2023 =================================================== § INTRODUCTION Disordered systems are naturally of interest in quantum many-body physics as real world materials are rarely perfect. Furthermore it is known that the presence of disorder can dramatically alter the properties of a system, as evidenced in the ground-breaking work of Anderson <cit.> which that showed that the amount of disorder in a non-interacting three dimensional electron system dictates whether the states are localized or extended. More recently there has been great excitement around the idea of many-body localization, which extends Anderson localization to interacting many-body systems <cit.>.Unfortunately, disordered systems can also be challenging to study using standard approaches. For instance the density matrix renormalization group (DMRG) <cit.>, which is otherwise regarded as an extremely reliable and robust method for 1D quantum systems, may have difficulty even for a relatively simple example like a 1D disordered Heisenberg model due to the presence of long range singlets in the ground state <cit.>. On the other hand, specialist methods developed for disordered systems such as the strong disorder renormalization group (SDRG) method of Ma, Dasgupta and Hu <cit.> may capture the qualitative features of the ground state entanglement, but generally only produce quantitatively precise results when the disorder is infinitely strong. In this manuscript we propose and benchmark a new tensor network method designed for disordered systems. In most previous tensor network approaches the structure of the network, i.e. the pattern of how the tensors in the network are connected, is fixed independent of the problem under consideration whilst the parameters within the tensors are optimized variationally. Conversely, the method we introduce here is based on allowing both the structure of the tensor network as well as the parameters within the tensors to be adapted to the problem under consideration. This idea is similar to previous work <cit.>, which adapted a coarse-graining transformation, based on blocking together sites using isometries, to specific instances of a disordered system, and thus produced an approximation to the quantum ground state as a disordered tree tensor network (TTN).However, in the present work the coarse-graining transformation is based on entanglement renormalization <cit.>. This differs by including unitary disentanglers, such that the ground state is approximated as a generalized form of the multi-scale entanglement renormalization ansatz (MERA) <cit.>. Here we use SDRG to adapt the overall structure of the network to the problem under consideration, which helps ensure the qualitative features of the ground state entanglement are captured, before then applying variational sweeping to optimize the tensors and improve the quantitative accuracy significantly.The manuscript is organized as follows: Section <ref> provides an overview of some of the previous approaches to numerical strong disorder renormalization, and Sect. <ref> discusses the problems of trying to combine SDRG with a variational tensor network algorithm. Section <ref> describes the newly proposed tensor network method for disordered systems, also outlining the variational optimization algorithm that is employed. Finally, Sect. <ref> presents benchmark results for the disordered XX and Heisenberg (XXX) models, which are compared to various other numerical methods. § STRONG DISORDER RENORMALIZATION In this section we review some of previous approaches proposed for strongly disordered systems. Let us consider the disordered XXZ model on a 1D lattice of spin-12 sites, with periodic boundary conditions (PBCs). The Hamiltonian is defined asH = ∑_i J_i{1/2[ s^+_i s^-_i+1 + s^-_i s^+_i+1] + Δ_z s^z_i s^z_i+1}where s^x, s^y and s^z are spin-12 matrices, with s^± = s^x ± i s^y. Notice that the Heisenberg model when is recovered when the anisotropy is fixed at Δ_z = 1, while the XX model is when Δ_z = 0. Disorder is introduced in the couplings J_i which are allowed to vary in strength with position; here we choose these couplings randomly between 0<J_i<J_max according to some probability distribution P(J). Many previous studies of the disordered Heisenberg model have used variants of the SDRG <cit.>, the archetypal method for the antiferromagnetic case was set out by Ma, Dasgupta and Hu <cit.>. In this procedure the pair of spins with the strongest coupling J_i are approximated as a singlet, and then second order perturbation theory is used to find an effective coupling J̃_i between the spins either side, as illustrated in Fig. <ref>(a). This step is repeated to produce a random singlet phase, an example is given in Fig. <ref>(a), where singlets created later on in the renormalization scheme can potentially span large distance scales. When averaged over many realizations of randomly chosen couplings J_i, the spin-spin correlation function is known to obey an inverse-square power law decay and the entanglement entropy scales logarithmically in a manner consistent with an effective conformal field theory (CFT) of central charge c=1 <cit.>.More recent developments have included higher order components to increase the accuracy of observables such as the spin-spin correlation functions <cit.>, which is effectively an application of Wilson's numerical renormalization group (NRG) <cit.> to a system with many impurities. Later it was shown in Ref. GolR14 that this form of SDRG has an efficient tensor network representation in terms of an inhomogeneous TTN, where the structure is set by the renormalization order of the couplings, which we refer to as the tSDRG method. An example of a wavefunction produced from tSDRG is given in Fig. <ref>(b). The basic building block of these wavefunctions are isometries v, which annihilate to an identity with their conjugates v v^† = as also depicted in Fig. <ref>(a), and are responsible for coarse-graining two sites into a single effective site. In the tSDRG approach the isometries are found by diagonalizing the block Hamiltonian supported on the pair of sites with the strongest coupling. Whilst this method was shown to reproduce qualitative features of the correlation functions and entanglement entropy, the ground state energy was not given with high precision <cit.>. Although the accuracy of the energy could be improved through the subsequent use of a variational sweep to minimize the energy of the corresponding TTN, this was found to be at the expense of accuracy in the other features of the state. In particular, the energy minimization update favoured short-range correlations and ignored the long-range singlets, which are key to the proper characterization of the disordered ground states. § TENSOR NETWORKS AND DISORDER The goal of this manuscript is to design a tensor network that can properly capture the structure of entanglement in a disordered ground state, in particular the pattern of long range singlets.Whereas the tSDRG approach constructs the TTN and sets the values in the tensors once, in this section we propose a more sophisticated network that allows a variational update whilst still capturing the long range entanglement. It follows the ideas of entanglement renormalization, employing unitary disentanglers in such a way as to modify the random singlet phase of SDRG to be relevant for finite systems with finite disorder. Each step of the SDRG can be interpreted as a coarse-graining transformation that maps a lattice Ł to a new lattice Ł' of two fewer sites by fixing two neighbouring spins, specifically those which possess the strongest coupling J_i, in a singlet state as depicted in Fig. <ref>(a). Here we propose two generalizations of this transformation, each a mapping from Ł to Ł'. The first, presented in Fig. <ref>(b), enacts a unitary disentangler u across the coupling J_i to be renormalized before employing isometries v_L and v_R that each coarse-grain two sites into one. The second, presented in Fig. <ref>(c), also enacts a unitary disentangler u across the coupling J_i before employing additional unitary gates w_L and w_R with the boundary spins. The sites at the location of coupling J_i are then fixed in an entangled singlet state, similar to SDRG.One should notice that if the spins associated to J_i in the second scheme, Fig. <ref>(c), were instead fixed as a product state then the first scheme of Fig. <ref>(b) is recovered. This follows as unitary gates w_L and w_R with a fixed (product state) input index can be simplified to isometries v_L and v_R, a property that allows the MERA to be interpreted both as a tensor network and as a quantum circuit <cit.>. As with other tensor network methods, we set an upper bound to the index dimensions of the tensors in order to truncate of the Hilbert space dimension for the coarse grained sites. This is known as setting the bond dimension χ.These coarse-graining schemes satisfy two important properties. Firstly, it is easy to see that there exist choices of tensors in both schemes for which the singlet state of Fig. <ref>(a) is reproduced, thus they should at least match the effectiveness of the standard SDRG. Secondly it should be noted that these schemes preserve locality: any operator supported on two contiguous sites of the initial lattice Ł is mapped to an operator supported on two contiguous sites of the coarser lattice Ł'. Hence it also follows that a nearest-neighbour Hamiltonian on Ł is mapped to a nearest-neighbour Hamiltonian on Ł'. Preservation of locality is important in order to produce a computationally viable numerical algorithm as, in general, the cost of representing a local operator grows exponentially with the size of its support.The proposed lattice transformations of Fig. <ref>(b-c) can be seen to yield a class of tensor network ansatz, just as entanglement renormalization yields the class of tensor network known as the MERA. For a lattice Ł of N sites, we consider the class of tensor network that encompasses all possible ways of applying N/2 of the transformations from either Fig. <ref>(b) or Fig. <ref>(c) in order to reach a coarse-grained lattice of O(1) sites. An example of a tensor network from this class is depicted in Fig. <ref>(c). Importantly, one should notice that a standard MERA arises as a specific instance in this more general class. More precisely, the modified binary MERA, as introduced in <cit.>, can be understood as a particular (uniform) arrangement of the unit cell of Fig. <ref>(b). Accordingly, we refer to class of tensor network that generalizes the MERA for disordered systems as a disordered MERA or dMERA. Notice that, as with the standard MERA, all instances of dMERA possess a bounded causal structure, due to the preservation of locality in the underlying coarse-graining transformation, and are thus efficiently contractible for expectation values of local observables and for correlation functions.§ALGORITHM In this section we discuss an algorithm for how the proposed coarse-graining transformations can be implemented for the study of disordered systems. This algorithm consists of a procedure for first determining the overall geometry of the dMERA tensor network, before then applying variational energy minimization to optimize the tensors.Here we use the original SDRG algorithm <cit.> to determine the network geometry, then build the tensor network by patterning the unit cell of isometric and unitary gates, as depicted in Fig. <ref>(b-c), to match the order of singlets.An example of a random singlet state produced by SDRG and the corresponding dMERA is presented in Fig. <ref>(a,c).In general, we employ the transformation of Fig. <ref>(c) throughout, apart from when we wish to increase the bond dimension χ of the network (usually during the earlier RG steps). The reason is that, despite the block in Fig. <ref>(b) reducing the geometric distance between any two sites connected by a singlet <cit.>, the update process still tends to omit long range correlation. When the singlets are an explicit part of the coarse-graining block as in Fig. <ref>(c), the long range properties are preserved.Once the structure of the dMERA has been set, we then adapt the standard MERA energy minimization algorithm <cit.> in order to optimize the tensors. Here we provide only an outline of the algorithm, the full details of which are provided in appendix <ref>.In a similar manner to standard MERA, we optimize the tensors by means of a variational sweep over the layers of the network. Each layer is comprised of a coarse-graining block as depicted by Fig. <ref>(b-c), which transforms between an initial lattice Ł and a coarser lattice Ł'. Following the strategy proposed in Ref. EveV09 the tensors within the cell are updated by first computing their corresponding linearized environments from the coarse-grained Hamiltonian on lattice Ł and the two-site local reduced density matrices on lattice Ł', themselves obtained using the appropriate ascending and descending superoperators respectively. Each of the environments is then decomposed using a singular value decomposition (SVD) in order to find the new isometry or unitary that minimizes the (linearized) energy. The update is performed for all tensors within each layer, and sequentially over all layers, and this variational sweep is iterated until the tensors converge. All the manipulations required to optimize a dMERA scale as 𝒪(χ^7) in the bond dimension χ of the network. Once the tensor network state is converged we then calculate expectation values and other properties of interest. Calculating expectation values of one- and two-site operators is straightforward; one can simply use the ascending superoperators to coarse-grain the operator through the network as discussed in appendix <ref>. As with optimization, this evaluation has the cost 𝒪(χ^7) in general but can be simplified to 𝒪(χ^6) for operators located on particular sites. Two-point correlation functions can also be efficiently computed by initially coarse-graining each of the operators separately and then fusing them into a single operator at the scale they meet. As with standard MERA, this may incur a higher computational cost as the fusion of the two operators may result in an operator with a larger support. As shown in appendix <ref>, the full cost of calculating the correlation function can vary between 𝒪(χ^6) and 𝒪(χ^11) depending on the particular sites involves and the geometry of the network. Block entanglement entropy is one of the key quantities that will be analysed in sec. <ref>, as one of the aims of this work is to achieve the correct entanglement structure of the wavefunction. However, the calculation of entanglement entropy is generally computationally difficult. Fortunately, the form of the tensor network alleviates this to some degree, as discussed in appendix <ref>, such that the entanglement entropy can be computed from tensor network states with small χ. § RESULTS In order to benchmark the proposed method we focus on the disordered anti-ferromagnetic XX and Heisenberg models, where Δ_z = 0, 1 in Eq. (<ref>) respectively. We select coupling strengths 0 < J_i < 1 from the strong disorder distribution <cit.>, which has the probability density function P(J) = 1/Δ_J J^-1 + Δ_J^-1,where Δ_J≥ 0 is the disorder strength. When Δ_J = 1 the distribution is uniform and when Δ_J→∞ the distribution matches that of the infinite randomness fixed point.For the remainder of this paper we will concentrate on the stronger disorder regime, when Δ_J≥ 1, as this tends to be more challenging for numerical tenchniques. Throughout this section we compare the efficacy of the dMERA with various other numerical methods, including the tSDRG algorithm <cit.> and DMRG, which is performed using the ITensor libraries <cit.>.It may be possible to obtain more accurate DMRG results by, for example, changing the algorithm to take into consideration inhomogeneities of the system <cit.> or performing measurements at multiple χ and extrapolating to the χ→∞ limit. Here we concentrate on a standard DMRG implementation without extrapolating so we can compare the finite χ performance of the algorithms. Finally we note that, as the XX model can be mapped to a free fermion model via a Jordan-Wigner transformation <cit.>, we can obtain exact results to benchmark against, even for large system sizes.§.§ The disordered XX model We begin by applying a dMERA, built from composition of the unit cell depicted in Fig. <ref>(c), of bond dimension χ = 2 to find the ground state of the (disordered) XX model. We observe that after optimization with any disorder strength, the unitaries w_L,R, when viewed as a matrix between incoming and outgoing indices, may be written asw_L,R = [ 1 0 0 0; 0cosθ_L,R -sinθ_L,R 0; 0sinθ_L,Rcosθ_L,R 0; 0 0 0 1 ].with θ_L,R a free variational parameter.The disentanglers u converge to a fixed matrix,u = [1000;00 -10;0 -100;000 -1 ],that does not contain any free variational parameters. Therefore, in this setting, the χ=2 dMERA is fully specified by L - 3 parameters, with L the system size. The optimization of this network can also be simplified from the general algorithm introduced in Sect. <ref>, as discussed in appendix <ref>.Furthermore, one should notice that if all the angles are set to θ_L = θ_R = 0, then the w_L,R become identity matrices and the the random singlet phase of SDRG is recovered. The reason that this parameterization arises is not clear to us, and a proof using analytic methods is beyond the scope of this paper. In the absence of a proof, the numerical results in the rest of the section are performed using a full optimization, but the simplified update reproduces the results in all test cases.Results comparing how the average ground state entanglement entropy ⟨ S_A⟩ scales with block size L_A for disorder strengths Δ_J = 1, 2, 4 are presented in Figure <ref>. Note that we average both over all possible blocks of size L_A and 50 different disorder configurations. The results demonstrate that dMERA, even when restricted to bond dimension χ=2, reproduces the ground state entanglement accurately over the range of disorder strengths considered. In comparison the tSDRG algorithm has a tendency to over estimate the entanglement entropy, particularly when the disorder is small, yet still reproduces the expected logarithmic scaling.For small disorder, Δ_J = 1, we see that DMRG requires bond dimension χ = 50 in order to rival the accuracy of the χ=2 dMERA, while for stronger disorder DMRG is substantially worse and the entanglement quickly saturates to a constant even for large bond dimension χ.Notice also that the results from standard SDRG become more accurate as the disorder increases, which is to be expected as the approach becomes exact in the limit of infinite disorder <cit.>.The staggered two-point correlation functions ⟨⟨s⃗_x_1 . s⃗_x_2⟩⟩, computed from 50 site systems, are plotted in Fig. <ref>. These have also been averaged both over all positions and over 100 disorder configurations. For length scales smaller than half the system size, the exact results demonstrate the characteristic power law decay of the random singlet phase, with an exponent that depends on the disorder strength Δ_J. While the χ=2 dMERA does reproduce this power law decay, it tends to underestimate the correlation especially for weak disorder Δ_J=1. However, the results are still significantly more accurate than those of the standard SDRG approach or from χ=20 DMRG, where the correlations decay exponentially at larger distances.That the χ=2 dMERA tends to underestimate correlators can be understood by analysing individual disorder realizations, where it can be seen that the problem occurs when the maximum coupling J_i is not significantly stronger than one of its neighbours, J_i-1 or J_i+1. In such instances the SDRG that is used to determine the structure of the dMERA is also less accurate; although optimization of the tensors allows the wavefunction to be improved the correlations still end up systematically smaller than they should be at larger separations. In disorder instances where the maximum coupling is much greater than its neighbours (J_i≫ J_i-1, J_i+1), which is more likely to occur at stronger disorder Δ_J, we find that the correlation functions are faithfully reproduced by the χ = 2 dMERA.§.§ The disordered Heisenberg modelWe now address the disordered Heisenberg model, which has Δ_z = 1 in Eq. (<ref>). This model no longer corresponds to free fermions and thus cannot be diagonalized exactly as with the XX model. Also, a dMERA of bond dimension χ = 2 is no longer viable as the set of unitary gates that preserve the SU(2) symmetry would be trivial; thus we increase the bond dimension to χ = 4, 8.Figure <ref> shows the average entanglement ⟨ S_A⟩ for a 20 site system when averaged over 50 disorder realizations. One can infer a rough upper bound to the expected results from tSDRG, which tends to overestimate the entanglement, and a lower bound from SDRG, which tends to underestimate the entanglement. It appears that the approaches based on energy minimization, DMRG and the dMERA, underestimate the entanglement for small bond dimension, approaching from below as the bond dimension is increased. For small disorder, Δ_J = 1 as depicted in Fig. <ref>(a), the dMERA with χ = 8 and 4 appears to be slightly less accurate than DMRG with χ = 50 and 20 respectively. However in the cases with stronger disorder, Δ_J = 2,4 as depicted in Fig. <ref>(b-c), the dMERA clearly reproduces the entanglement more accurately than DMRG, where the entanglement saturates to a constant for larger blocks. As expected, SDRG also becomes more accurate with stronger disorder, almost matching the entanglement from a χ=4 dMERA, although still falling short of the χ=8 case.The two-point spin correlation functions are plotted in Fig. <ref> for a system of 50 sites, averaged over 200 disorder configurations. The dMERA is seen to reproduce the expected algebraic decay of correlations, producing a pattern similar to that from tSDRG but differing in magnitude at longer distances. However, it is not clear which approach is more accurate as both methods seem quite well converged in bond dimension χ. § SUMMARY AND CONCLUSIONSWe have proposed a generalization of entanglement renormalization for disordered systems, which yields a new class of efficiently contractible tensor network ansatz that we call disordered MERA. A key feature of the dMERA is that the geometry of the network can be adjusted to the specific problem under consideration, whereas previous tensor network ansatzes typically use a fixed network geometry only adjust the content of the tensors to the specific problem. This allows much of the entanglement structure of wavefunction to be captured before the variational optimization of the tensors even begins.Conceptually, the benefit of adjusting the network geometry can easily be understood from an example such as the concentric singlet phase also known as the rainbow state <cit.>. This state, or something close to it, could easily be represented by a dMERA, but would require exponential growth in bond dimension with system size in order to be represented by aconventional tensor network, such as an MPS or MERA, due to the linear growth of half-chain entanglement entropy.In practice, the advantage of the this approach was seen in the benchmark results of Sect. <ref>. Here a dMERA of dimension χ=2 yielded remarkably accurate results for the XX model, significantly improving over DMRG calculations with χ=50.The dMERA also seemed to capture the entanglement structure in the disordered Heisenberg model, although a larger bond dimension was required to do so adequately. Accordingly, we expect this method may require further development and refinement before it is viable for more challenging models.There are many possible extensions and modifications to our proposed approach. In the basic unit cell of the coarse-graining transformation we imposed that the truncated sites were either in a product state, as depicted in Fig. <ref>(b), or in a singlet state, as depicted in Fig. <ref>(c). In general this could be chosen as any quantum state; for instance a spin trio type state could be used to describe the baryonic states found in Ref. QuiHM15. It could also be possible to use a different metric to determine the geometry of a dMERA, rather than through use of the SDRG as was considered here, allowing the approach to be extended beyond states based on random singlet phases. For instance, one could try to use the mutual information to set the geometry, similar to the proposal in Ref. HyaGB17. Alternatively, it may be possible to design an algorithm that adjusts the network structure automatically during the variational optimization, which remains an interesting avenue for future work.AMG would like to thank Norbert Schuch and Rudolf A. Römer for fruitful discussions.§ ALGORITHMIC DETAILS §.§ The Hamiltonian Before the update can begin, the Hamiltonian needs to have its energy shifted so that the entire spectrum is negative.This is important to allow the update process to target the ground state, the details of which are discussed in sec. <ref>. The shift is performed by diagonalizing the two-site Hamiltonian operators and subtracting the identity multiplied by the largest eigenvalue λ_i,i+1^max from eachh^shifted_i,i+1 = h_i,i+1 - λ_i,i+1^max( _2⊗_2).The sum of the shifts are stored and added to the final energy after minimization to obtain the true ground state energy once more.§.§ The SDRG order The order in which the coarse-graining blocks are combined to create the full network is given by ordinary SDRG <cit.>. The algorithm coarse-grains in the order of strongest interaction, renormalizing neighbouring interactions via second order perturbation theory. For more information see <cit.> and <cit.>.The SDRG rules for the XXZ model are as follows: Find the largest coupling J_i in the set of coupling strengths. The strongest coupled pair form a singlet to first order.Second order perturbation theory is then used to find an effective coupling J̃ for the spins either sideJ̃ = J_i-1 J_i+1/(1 + Δ_z) J_i.The location i is stored, then the strongest coupled spins are removed, coarse-graining the chain by two sites. This is repeated until there are no remaining sites. The result is a list of the locations of the singlets at each level of coarse-graining as shown in Fig. <ref>(a), which can then be used to set the geometry of the tensor network.§.§ The initial tensors The order of coarse-graining and the maximum allowed χ is used to set the initial state of the tensors. If χ = 2 the network is in the most simple form; all u and w tensors are 2 × 2 × 2 × 2 and unitary, and each coarse graining block is of the form of Fig. <ref>(c). Setting these as ( _2⊗_2) means that the initial state is the random singlet state. Another option is to generate random numbers to fill the tensors.The top of the network takes a slightly different form due to the small system size and periodic boundaries.When the system consists of four sites it is only possible to put one coarse-graining block, then the remaining pair of sites will be connected by a singlet. As shown in Fig. <ref>, due to the PBCs this second singlet can be considered to be part of the same coarse-graining operation.The full tensor network for a system of 20 sites with χ = 2 is shown in Fig. <ref>(c). Notice that this is the network obtained given the order set out in Fig. <ref>(a).If χ > 2 then the network needs to be altered to allow the bond dimensions to grow. By looking at Fig. <ref>(c) it is clear that if the tensors at the lower levels of the network have χ = 2, then all of the tensors above must do also. Thus the coarse-graining block of Fig. <ref>(b) is used to increase the dimension of the effective sites. Once the bond dimension is above 2 it can grow using the standard coarse-graining block of Fig. <ref>(c). The dimensions and form of the tensors is set in the initialization stage and will then not change during the algorithm.There are many choices that can be made relating to how to implement the bond dimension increase. We choose to only use the χ-increase block [Fig. <ref>(b)] when no information is to be discarded, this is because this form has been shown to be unreliable at retaining long range interactions with lower bond dimensions. The update is performed without the ρ's so that is is just a coarse-graining of the Hamiltonian. As such, we only update it on the first sweep as the environment will not change as the algorithm progresses.For the rest of the blocks the dimensions of the top two legs of the unitary in the χ-increase block are swapped, e.g. if the bottom two legs have dimensions 2 and 4 then the top legs have 4 and 2. This is to be consistent with the simplified results of the XX model, which show that the unitary has properties similar to a swap tensor.All of these choices have been seen to be more effective as well as being conceptually preferable.§.§ The descending superoperatorsThe two-site local reduced density matrices {ρ} contain all of the tensors that are above the tensor to be updated so that the process minimizes the energy of the network as a whole. To create the set of ρ's, start at the top and contract down. The tensors of the coarse-graining block make up the descending superoperators, mapping a density matrix on level Ł' to one at level Ł. The highest level ρ's consist of the four ways of contracting the top block and its conjugate as shown in Fig. <ref>, where i is the position of maximum coupling from SDRG when the system is four sites. Here and in the following diagrams the optimal contraction order is found using the netcon function <cit.> and the contraction itself performed using ncon <cit.>.For the next level down there are six sites on the lattice and the ρ's are created by contracting the coarse-graining block to the correct ρ at the level above (labelled as ρ^' in the figures). There are three sites (i-1, i and i+1) that are below ρ^'_i-1 shown in Figs. <ref>(b,c,d). The sites either side of the main coarse-graining block, ρ_i-2 and ρ_i+2 are below ρ^'_i-2 and ρ^'_i respectively as shown in Figs. <ref>(a,e). Any sites that are not affected by the coarse-graining block have ρ's copied down from the level above.This is repeated until the bottom of the network creating the necessary ρ's for the updates of all of the tensors. §.§ The update The process is much the same as for standard MERA <cit.>. The update of the unitary and isometries is quadratic in the sense that we want to find, for example, the optimal u and u^† at the same time. Unfortunately there is no exact way to perform a general quadratic update, thus the problem is linearized. Linearization here means setting u^† as the hermitian conjugate of the current u, so the problem now is just to update u on its own. Once the new tensor has been found the u^† is replaced with the new u and the process is repeated until convergence.The update is the process of finding the tensor, here u, that minimizes the energy of the network as a whole. We call the rest of the network the environment of u, Υ_u. The fact that the Hamiltonian tensors contain all of the tensors below the coarse-graining block and the ρ's contain all of the network above makes the contraction of Υ_u simply the sum of the three diagrams containing the Hamiltonian components below the u, as shown in Fig. <ref>. Notice that the linearization is accomplished by contracting u^† into the environment.Then the optimization of u can be stated asE_min = min_u ∈𝒰Tr[ u Υ_u],where 𝒰 is the set of unitary tensors. Because the Hamiltonian is shifted, as mentioned in section <ref>, the minimum energy is the negative trace of the singular values of Υ_u <cit.>. Thus u = -VU^† where U and V are found by performing a singular value decomposition (SVD) of the environment (Υ_u = USV^†) This is shown in Fig. <ref>, where it is clear that this choice of u results in the energy being the negative trace of S.The environment is then re-contracted with the new u^† and the process is repeated until convergence. The update of the isometries is effectively the same but with different environment diagrams to be contracted. The environment of w_L is given in Fig. <ref> and the environment of w_R constructed similarly. The main difference to that of u is the i - 2 term, which does not contribute to the environment of u. The top of the network is again slightly different as there are no ρ's, just the singlets that make up the top, as shown in Fig. <ref>. This is updated by SVD in the same way as the rest of the tensors. Diagrams for the environments of w_L and w_R are omitted but are straightforward to recreate. §.§ The ascending superoperators Once the tensors of a block have been updated the Hamiltonian components are raised to the next level of coarse-graining using ascending superoperators <cit.>. This is accomplished by applying the block to each Hamiltonian tensor individually. Because the centre three components get mapped onto the same site of the coarse-grained lattice, the new tensor for that site is the sum of these three terms. The tensor network diagrams for this raising operation are shown in Fig. <ref>. § OBSERVABLES §.§ Single- and two-site operators Expectation values are calculated by applying the coarse-graining blocks on operators. Because the block maps two-site operators on one level to two-site operators on the level above, calculating the expectation values means simply applying the raising operators on the operator in the same way as the Hamiltonian operators during the update. However due to the unitary or isometric properties of the tensors, usually only a fraction of the tensors need to be contracted and the rest cancel to identities. The remaining tensors are said to be in the causal cone of the operator. An example of this is given in Fig. <ref>, which shows an operator acting on sites 11 and 12 in the network from Fig. <ref>(c). The causal cone is highlighted in blue, the remaining tensors are not involved in the expectation value. This allows expectation values to be calculated in a highly efficient manner. The diagrams for the raising operation are effectively the same as those in Fig. <ref> except the summation of terms in (a) is unnecessary as the expectation value will only include one of the three diagrams at each level. The numerical result of the expectation value is found when the top is contracted, as shown in Fig. <ref>. The leading order cost is 𝒪(χ^7), but can be 𝒪(χ^6) for operators at some sites in some network geometries. Single-site operators can be expressed as two-site operators via tensor product with an identity.The expectation values of which can then be contracted using the same diagrams as before. This is the most practical method as a single-site operator will be mapped to two sites by the coarse-graining block anyway. §.§ Two-point correlation functionsCorrelation functions are calculated by applying the coarse-graining blocks on the two operators independently just as single- and two-site expectation values. As before, only the tensors within the causal cone need to be contracted.However, at some point in the process the two sites will be joined by a single block. This is the point at which the causal cones of the two operators join, as shown in Fig. <ref>. The joining means that the calculation of two-point correlation functions can be significantly more costly than a single-site operator. This additional expense was also observed for standard MERA and was one of the reasons for the development of ternary MERA as opposed to the original binary variant <cit.>. The reason is that when the two operators meet the outcome is not necessarily another two-site operator. For the ternary MERA case however, there are choices of sites where the cost is 𝒪(χ^8). This is when the two sites are at the centre leg of the isometry, i.e. at separations r = 3^q with q = 1,2,3 ….Otherwise the cost grows beyond 𝒪(χ^8). For dMERA the problem can not be so simply worked around. When performing calculations for disordered systems, discarding individual correlations will result in errors in the averaging and therefore losing the characteristic power law of the random singlet phase. It is therefore necessary to calculate correlations between all pairs of sites or discard all data for a given separation r if one is too costly to compute.When the operators in dMERA meet the possible outcomes are two-, three- and four-site operators, as shown in the tensor network diagrams of Figs. <ref>(a-f).The raising of two-site operators is the same as in sec. <ref>, three- and four-site operators are shown in Figs. <ref>(g-l) and <ref>(m-t) respectively. The expectation value of the correlation function is finally determined by contraction with the top of the network as shown in Fig. <ref>. In full, the cost of a calculating a two-point correlation function can be between 𝒪(χ^6) and 𝒪(χ^11) depending on the sites and structure of the network. As mentioned in sec. <ref>, for algorithmic simplicity a single site-operator is encoded in a two-site operator by tensor product with an identity. This suggests that there are two ways of encoding the each operator: O_i⊗_i+1 and _i-1⊗ O_i. Due to the fact that the network is inhomogeneous, this choice of encoding can affect the cost of contracting the correlation function.It is therefore important to choose the operator encoding that minimizes the cost.§.§ Entanglement entropyThe entanglement entropy S_A of a region A given byS_A = -Trρ_Alog_2ρ_A,where ρ_A is the reduced density matrix, obtained by tracing over region B. The contraction of ρ_A in general is inefficient as the computational complexity grows exponentially with system size. The construction of the dMERA wavefunction makes the problem significantly easier as we only need to contract the tensors that join blocks A and B <cit.>, as illustrated by Fig. <ref>.This makes it possible to calculate S_A when χ and L are relatively small. We note that in all cases the entanglement is still captured by the tensor network and it is just the calculation of the entropy that is problematic. § ENERGY MINIMIZATION OF THE XX MODEL As discussed in sec. <ref>, dMERA for the χ = 2 XX model has a simplified form where the u tensors have no variables and the w_L,R have one angle each. The number of parameters in this tensor network is then L - 3. This number comes from the fact that there are L/2 - 1 coarse-graining blocks as the top one is trivial, each containing two w tensors. The final pair of angles are also correlated; the difference between the two being the only free parameter. The update process can be performed by simply differentiating to find the minimum energy. For example, we require the minimum energyE_min = min_θtr[ Υ w(θ) ].Writing the environment Υ, and w as matrices as in Eq. (<ref>), the energy can be written asE= tr[ Υ w(θ) ] = Υ_1,1 + Υ_4,4 + ( Υ_2,2 + Υ_3,3) cosθ + ( Υ_2,3 - Υ_3,2) sinθ,where Υ_i,j are elements of the environment. Differentiation shows that the stationary points are attanθ = Υ_2,3 - Υ_3,2/Υ_2,2 + Υ_3,3,and the second derivative is used to find if it is a maximum or minimum. 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Straight quantum layerwith impurities inducing resonancesSylwia Kondej============================================================= Mathematical models that couple partial differential equations (PDEs) and spatially distributed ordinary differential equations (ODEs) arise in biology, medicine, chemistry and many other fields. In this paper we discuss an extension to the FEniCS finite element software for expressing and efficiently solving such coupled systems. Given an ODE described using an augmentation of the Unified Form Language (UFL) and a discretisation described by an arbitrary Butcher tableau, efficient code is automatically generated for the parallel solution of the ODE. The high-level description of the solution algorithm also facilitates the automatic derivation of the adjoint and tangent linearization of coupled PDE-ODE solvers. We demonstrate the capabilities of the approach on examples from cardiac electrophysiology and mitochondrial swelling.finite element methods, adjoints, coupled PDE-ODE, FEniCS, dolfin-adjoint, code generation65L06, 65M60, 65M32, 35Q92§ INTRODUCTIONIn this paper we discuss solvers for systems involving spatially dependent ordinary differential equations (ODEs) of the following form: given an initial condition y_0(x), find y = y(x, t) such thaty(x, t) = f(y, x, t),y(x, t_0) = y_0(x),for all x in a point set X ⊂Ω⊆^d, where d is the spatial dimension. The subscript t refers to differentiation in time.The right-hand side f cannot depend on spatial derivatives of y; the ODE is decoupled at different points. Problems of this form often arise when discretizing time-dependent mathematical models that couple PDEs with spatially distributed systems of ODEs via operator splitting. Examples of application areas include cardiac electrophysiology in general <cit.> and cardiac ion channel modeling in particular <cit.>, mitochondrial swelling <cit.>, groundwater flow and contamination <cit.>, pulmonary gas transport <cit.>, and plasma-enhanced chemical vapor deposition <cit.>. We also discuss the automated derivation of the adjoint and tangent linearization of such models: these can be used to identify the sensitivity of the solution to model parameters, solve inverse problems for unknown parameters, and characterize the stability of trajectories.For concreteness, we present two biological examples of coupled PDE-ODE systems where problems of the form (<ref>) arise. We return to numerical results for these examples in Section <ref>.[The bidomain equations] As our first motivating example, we will consider the bidomain equations for the propagation of an electrical signal in a non-deforming domain Ω <cit.>: find the transmembrane potential v = v(x, t), the extracellular potential u_e(x, t) and additional state variables s = s(x, t) such that for t ∈ (0, T]:v -(M_iv + M_iu_e)= - I_ ion(v, s) + I_sin Ω, -(M_iv + (M_i + M_e)u_e)= 0in Ω,s = F(v, s)∀ x ∈Ω. In (<ref>), I_ ion is a given nonlinear function describing ionic currents and F defines a system of nonlinear functions, while M_i and M_e are the intracellular and extracellular conductivity tensors, respectively, and I_s is a given stimulus current. The function F cannot depend on spatial derivatives of v and s; it defines a pointwise system of ODEs. The specific form of I_ ion and F are typically prescribed by a given cardiac cell model and may vary greatly in complexity: from involving a single state variable s such as the FitzHugh-Nagumo model <cit.> to models with e.g. 41 state variables such as the model of O'Hara et al. <cit.>. The system (<ref>) is closed with appropriate initial and boundary conditions. After the application of operator splitting, the ODE step is decoupled from the PDE step and is a system of the form of (<ref>). [Mitochondrial swelling] As a second example, we consider a model proposed in <cit.> to describe the swelling of mitochondria. Mitochondrial swelling plays a key role in the process of programmed cell death (apoptosis) and thus for the life cycle of cells. Mathematically, we consider the following model <cit.>: find the calcium concentration u = u(x, t) and the densities of mitochondria states N_i = N_i(x, t) for i = 1, 2, 3 such thatu = d_1 Δ (|u|^q-2 u) + d_2 g (u) N_2in Ω,N_1 = - f(u) N_1∀ x ∈Ω,N_2 = f(u) N_1 - g(u) N_2∀ x ∈Ω,N_3 = - g(u) N_2∀ x ∈Ω. where d_1 ≥ is a diffusion coefficient, d_2 ≥ 0 is a feedback parameter, q ≥ 2 ∈ℕ determines the nonlinearity of the diffusion-type operator, and g and f are prescribed functions. The functions N_1, N_2 and N_3 describe the densities of unswollen, swelling and completely swollen mitochondria respectively. Equation (<ref>) is a spatially-coupled PDE, while equations (<ref>)-(<ref>) define pointwise ODEs. The system is closed with initial conditions and Dirichlet boundary conditions for u. Again, after operator splitting, a subproblem of the form (<ref>) results. Over the last decade, there has been a growing interest in computational frameworks for the rapid development of numerical solvers for PDEs such as the FEniCS Project <cit.>, the Firedrake Project <cit.>, and Feel++ <cit.>. The rapid development of solvers is achieved by offering high-level abstractions for the expression of such problems. Each of these projects provides a domain-specific language for specifying finite element variational formulations of PDEs and associated software for their efficient solution. The methods presented in this paper are implemented in the the FEniCS Project and dolfin-adjoint <cit.>; dolfin-adjoint automatically derives discrete adjoint and tangent linear models from a FEniCS forward model. These allow for the efficient computation of functional gradients and Hessian-vector products, and are essential ingredients in stability analyses, parameter identification and inverse problems.However, these systems do not currently efficiently extend to the kind of coupled PDE-ODE systems arising in computational biology.While monolithic finite element discretizations of coupled PDE-ODE systems such as (<ref>) or (<ref>) may easily be specified and solved using the current software features in FEniCS, this approach will involve an unreasonably large computational expense, as the monolithic system is highly nonlinear and the solver cannot exploit the fact that the ODE is spatially decoupled.Operator splitting is the method of choice for such coupled PDE-ODE systems <cit.>, and is implemented as standard in many hand-written codes, see e.g. <cit.> in the context of cardiac electrophysiology. Operator splitting decouples a PDE-ODE system into a spatially-coupled system of PDEs and a spatially-decoupled collection of ODE systems. This collection of ODE systems then typically takes the form (<ref>), which can be solved using well-known temporal discretization methods. Heretofore it has not been possible to specify spatially-decoupled ODE systems in high-level PDE frameworks such as the FEniCS or Firedrake projects.This work addresses the gap in available abstractions, algorithms and software for arbitrary PDE-ODE systems. We introduce high-level domain-specific language constructs for specifying collections of ODE systems and for specifying multistage ODE schemes via Butcher tableaux. This has several major benefits. By automatically generating the solver from a high-level description of the problem, practitioners can flexibly explore a range of models; this is especially important in biological problems where the model itself is uncertain.Another advantage is that a high-level description facilitates the automated derivation of the associated tangent linear and adjoint models.This represents a significant saving in the time taken to investigate questions of biological interest.The main new contributions of this paper are (i) the extension of the FEniCS finite element system to enable efficient the large-scale forward solution of coupled, time-dependent PDE-ODE systems via operator splitting, and (ii) the extension of dolfin-adjoint to automatically derive and solve the associated tangent linear and adjoint models, enabling efficient automated computation of functional gradients for use in e.g. optimization or adjoint-based sensitivity analysis.This paper is organized as follows. In Section <ref>, we describe the general operator splitting setting, the resulting separate PDE and ODE systems, and natural discretizations of these. We continue in Section <ref> by briefly describing the well-established ODE schemes that we consider in this work: the multistage and Rush-Larsen families. We discuss the adjoint and tangent linear discretizations of a general operator splitting scheme and derive the adjoint and tangent linear models for the multistage and Rush-Larsen schemes in Section <ref>. Key features of our new implementation are described in Section <ref>. We present numerical results for two different application examples in Section <ref>, and use these to evaluate the performance of our implementation. In Section <ref>, we provide some concluding remarks and discuss current limitations and possible future extensions. § OPERATOR SPLITTING FOR COUPLED PDE-ODE SYSTEMS §.§ Operator splittingA classical approach to the solution of the bidomain system (<ref>) <cit.> and similar systems is to apply operator splitting. This approximately solves the full system of equations by alternating between the solution of a system of PDEs and a system of ODEs defined over each time interval. The advantage of this approach is that it decouples the solution of the typically highly nonlinear ODEs from the spatially coupled PDEs at each time step. We will consider this general setting, but use (<ref>) as a concrete example.Applying a variable order operator split to (<ref>), the resulting scheme reads: given initial conditions v^0, s^0, an order parameter θ∈ [0, 1], and time points {t_0, …, t_N} with an associated timestep _n = t_n+1 - t_n, then for each time step n = 0, 1, …, N - 1: * Compute v^∗ and s^∗ by solving the ODE systemv = - I_ ion (v, s)s = F(v, s) over Ω× [t_n, t_n + θκ_n] with initial conditions v^n, s^n.* Compute v^† and u^n+1_e by solving the PDE systemv -(M_iv + M_iu_e)= I_s -(M_iv + (M_i + M_e)u_e)= 0 over Ω× [t_n, t_n+1] with initial condition v^∗.* If θ < 1, compute solutions v^n+1 and s^n+1 solving (<ref>) over Ω× [t_n + θκ_n, t_n+1] with initial conditions v^† and s^∗.The split scheme relies on the repeated solution of a nonlinear system of ODEs and (in this case) a linear system of PDEs. The main advantage of the approach is that it allows for the separate discretization and solution of the ODEs and PDEs, with the respective solutions as feedback into the other system. For θ = 1/2, the resulting Strang splitting scheme is second-order accurate; for other values of θ the resulting scheme is first-order accurate. §.§ Discretization of the separate PDE and ODE systems Suppose now that the relevant PDE system (e.g. (<ref>)) is discretized in space by a finite element method defined over a mesh _h of the domain Ω, and in time by some suitable temporal discretization. Efficient solution algorithms for such discretizations are well-established (such as <cit.>) and will not be detailed further here.At each iteration the solution of the PDE system relies on the solution of the ODE system (e.g. (<ref>)), or at least the ODE solution evaluated at some finite set of points X = {x_i}_i=1^|X| in space. For instance, the set of points X may be taken as the nodal locations of the finite element degrees of freedom, or the quadrature points of the mesh. As a consequence of this and the spatial locality of ODEs, a natural approach to discretizing the system of ODEs is to step the ODE system forward in time at this set of points X. Typically, |X| is very large and the efficient repeated solution of these systems of ODEs is key. This is the setting that we focus on next. § SOLUTION SCHEMES FOR THE ODE SYSTEMSThe initial value problem (<ref>) decouples in space, and henceforth we consider its solution for a fixed x. With a minor abuse of notation, let y = y(x) and f(y, t) = f(y, x, t) so that (<ref>) reads as the classical ODE problem: find y ∈ C^1([T_0, T_1]; ℝ^m) for a certain m ∈ℕ such thaty(t) = f(y, t),y(T_0) = y_0.for t ∈ [T_0, T_1]. There exists a wide variety of solution schemes for (<ref>) including but not limited to multistage, multistep, and IMEX schemes, see e.g. <cit.>. In this work, we focus on two classes of schemes that are widely used in computational biology: multistage schemes and so-called Rush-Larsen schemes. The Rush-Larsen schemes are commonly used in cardiac electrophysiology in general and for discretizations of the bidomain equations (<ref>) in particular. These two classes of schemes are detailed below. Keeping the general iterative operator splitting setting in mind, we present the schemes on a single time-step [T_0, T_1] with κ = T_1 - T_0 for brevity of notation. §.§ Multistage schemesAn s-stage multistage scheme for (<ref>) is defined by a set of coefficients a_ij, b_i and c_i for i, j = 1, …, s, commonly listed in a so-called Butcher tableau; for more details, see e.g. <cit.>. Given y^0 at T_0, the scheme finds the stage variables k_i for i = 1, …, s satisfyingk_i = f(y^0 + κ∑_j=1^s a_ij k_j, T_0 + c_i κ),and subsequently sets the solution y^1 at T_1 viay^1 = y^0 + κ∑_i=1^s b_i k_i.Note that (<ref>) defines a system of (non-linear) equations to solve for the stage variables k_i (i = 1, …, s) if a_ij≠ 0 for any j ≥ i. Our implementation demands that a_ij = 0 for j > i, i.e. does not allow the computation of earlier stages to depend on the values of later stages. §.§ The Rush-Larsen scheme and its generalizationRecall that y = {y_i}_i=1^m and f(y) = {f_i(y)}_i=1^m. The original Rush-Larsen scheme employs an exponential integration scheme for all linear terms for each component f_i (for i = 1, 2, …, M), and a forward Euler step for all non-linear terms <cit.>. Let J_i = ∂ f_i/∂ y_i(y^0, T_0) be the i'th diagonal component of the Jacobian at time T_0 for i = 1, …, M, and assume that J_i is non-zero. For brevity, denote f_i(y^0, T_0) = f_i^0. The Rush-Larsen scheme then computes y_i^1 at T_1 byy_i^1 = y_i^0 + {[ J_i^-1 f_i^0 (e^κ J_i-1 ) ,; κ f_i^0 . ] . For stiff systems of ODEs, the forward Euler step in (<ref>) can become unstable for large κ. This motivates a generalized version of the Rush-Larsen scheme <cit.>. In this generalization, the exponential integration step is used for all components of y reducing the scheme toy_i^1 = y_i^0 + J_i^-1 f_i^0 (e^κ J_i-1 )Both (<ref>) and (<ref>) are first order accurate in time <cit.>.Both the original and the generalized Rush-Larsen schemes can be developed into second order schemes by repeated use as follows. The solution y^1/2 at T_1/2 = T_0 + κ/2 is computed in the first step and is used in f and its linearization J to compute y^1 in the second step. Let f_i^1/2 = f_i(y^1/2, T_1/2), and let J_i, 1/2 = ∂ f_i/∂ y_i(y^1/2, T_1/2) be the diagonal component of the Jacobian at time T_1/2 for i = 1, …, M. More precisely, the second order version of (<ref>) and (<ref>) is then given byy_i^1/2 = y_i^0 + {[ J_i^-1 f_i^0 (e^κ/2J_i - 1 ) ;κ/2 f_i^0].y_i^1 = y_i^0 + {[ J_i, 1/2^-1 f_i^1/2 (e^κ J_i, 1/2 - 1 ); κ f_i^1/2 ]. for the original Rush-Larsen scheme andy_i^1/2 = y_i^0 + J_i^-1 f_i^0(e^κ/2 J_i-1 ) y_i^1 = y_i^0 + J_i, 1/2^-1 f_i^1/2( e^κ J_i, 1/2 - 1 ) for the generalized Rush-Larsen scheme. Here, (<ref>) is simplified compared to the scheme in <cit.>: instead of evaluating f and J at y̅^(i) (cf. eq. (7) in <cit.>) we use y_i^1/2. § ADJOINTS AND TANGENT LINEARIZATIONS OF COUPLED PDE-ODE SYSTEMS Our aim is to automatically derive the adjoint and tangent linear equations of operator splitting schemes in a manner that allows for efficient solution of the resulting systems. We proceed as follows: we first state the adjoint and tangent linear system for general problems, then consider the special case of mixed PDE-ODE systems and finally discuss the specific adjoint and tangent linear versions of multistage and Rush-Larson schemes.We begin by considering a general system of discretized equations in the form: find y ∈ Y ⊂ℝ^M such thatF(y) = 0,The adjoint equation of (<ref>) associated with a real-valued functional of interest J: Y → is: find the adjoint solution y̅∈ℝ^M such that∂ F/∂ y^∗(y) y̅ = ∂ J/∂ y(y)where the superscript ∗ denotes the adjoint operator. The associated tangent linear equation with respect to some auxiliary control parameter m is given by: find the tangent linear solution ẏ such that∂ F/∂ y (y)ẏ = ∂ F/∂ m(y) We now turn our attention to the case where F is a coupled PDE-ODE system. For a guiding example, we consider again the bidomain equations (<ref>)–(<ref>) and, without loss of generality restrict ourselves to a single iteration (N = 1).For brevity, we denote the ODE operator (<ref>) as O and the PDE operator (<ref>) as P. We can then rewrite the scheme in the form (<ref>): v_0 - v^0= 0, (Set initial condition)s_0 - s^0= 0, (Set initial condition)O(v^*, s^*; v^0, s^0)= 0, (Solve bidomain ODEs (<ref>))P(v^†, u_e^1; v^*)= 0, (Solve bidomain PDEs (<ref>))O(v^1, s^1; v^†, s^*)= 0.(Solve bidomain ODEs for final state)The first pair of equations in (<ref>) set the initial conditions, the following equations represents a collection of nonlinear ODEs, the third equation a system of PDEs, and the last equation again a collection of ODEs.From (<ref>), the adjoint system for (<ref>) derives as:( [2c|2*I2|c|2*∂O^*/∂(v^0, s^0)2c|2*0 2c2*0;xx;2c|2*0 2c|2*∂O^*/∂(v^*, s^*) 2c|2*∂P^*/∂(v^0, s^0) 2c2*∂O^*/∂s^*;;2c|2*02c|2*0 2c|2*∂P^*/∂(v^†, u_e^1) 2c2*∂O^*/∂v^†;;2c|2*02c|2*02c|2*02c2*∂O^*/∂(v^1, s^1); ]) [ v̅^0; s̅^0; v̅^∗; s̅^∗; v̅^†; u̅_e^1; v̅^1; s̅^1;] = [ 0; 2*⋮;;;;; ∂ J/∂ v^1; ∂ J/∂ s^1 ]For brevity, the left hand side matrix combines 2×2 blocks, and the functional of interest is assumed to only depend on the final state variables v^1, s^1. The adjoint system (<ref>) is a linear coupled PDE-ODE system with upper-triangular block structure, which can efficiently by solved by backwards substitution.The last block row represents a linear adjoint ODE system, preceded by a adjoint PDE system, and preceded by another adjoint ODE system, and finalised by a variable assignment in the first block row which represents the adjoint version of setting the initial conditions.The derivation of adjoint and tangent linear equations for finite element discretizations of PDEs, such as the third block row in (<ref>), is well-established, see e.g. <cit.>. However, the automated derivation of adjoint and tangent linear systems for the multistage and Rush-Larsen discretizations of the generic ODE system (<ref>), such as the second and fourth block row in (<ref>), is less so, and these derivations are presented below. §.§ Adjoints and tangent linearizations of multistage schemes Consider a multistage discretization of (<ref>) as described in Section <ref> with a given Butcher tableau a_ij, b_j, c_i, i, j = 1, …, s. Taking s = 3 for illustrative purposes, we can write (<ref>)–(<ref>) in form (<ref>) as:[ y^0; k_1; k_2; k_3; y^1 ] - [ y_0; f(y^0 + κ w_1, T_0 + c_1 κ); f(y^0 + κ w_2, T_0 + c_2 κ); f(y^0 + κ w_3, T_0 + c_3 κ);y^0 + κ∑_i=1^3 b_i k_i ] =0,withw_i = y^0 + κ∑_j=1^s a_ij k_j.Assuming that the functional of interest J is independent of the internal stage values k_i, we can derive the adjoint problem as:[ y̅^0; k̅_1; k̅_2; k̅_3; y̅^1 ] - [ 0 ∂ f_1/∂ w_1^* ∂ f_2/∂ w_2^* ∂ f_3/∂ w_3^* I; 0 κ a_11∂ f_1/∂ w_1^* κ a_21∂ f_2/∂ w_2^* κ a_31∂ f_3/∂ w_3^* κ b_1; 0 0 κ a_22∂ f_2/∂ w_2^* κ a_32∂ f_3/∂ w_3^* κ b_2; 0 0 0 κ a_33∂ f_3/∂ w_3^* κ b_3; 0 0 0 0 0; ][ y̅^0; k̅_1; k̅_2; k̅_3; y̅^1 ] = [ ∂ J/∂ y^0; 0; 0; 0;y̅_0 ] ,where y̅_0 is the terminal condition for the adjoint solution at T_1.Generalizing to s stages, we see that we first solve for the adjoint stage values and then compute the adjoint solution at time T_0 via(I - κ a_ii∂ f_i/∂ w_i^*) k̅_i= κ b_i y̅^1 + ∑_j=i + 1^s κ a_ji∂ f_j/∂ w_j^* k̅_j,y̅^0= y̅^1 + ∑_i=1^s ∂ f_i/∂ w_i^* k̅_i.Following a similar calculation, the tangent linearization of the multistage scheme with s stages is: given y_0 at T_0 compute k̇_i for i = 1, …, s and then ẏ^1 at T_1 via(I - κ a_ii∂ f_i/∂ w_i) k̇_i = ∂ f_i/∂ w_iẏ^n + ∑_j = 1^i-1κ a_ij∂ f_i/∂ w_ik̇_j + ∂ f_i/∂ m,ẏ^n+1 = ẏ^n + ∑_i=1^s κ b_i k̇_i. §.§ Adjoints and tangent linearizations of Rush-Larsen schemes We now turn to consider adjoints and tangent linearizations of the (generalized) Rush-Larsen schemes. We here derive the equations for the first order generalized Rush-Larsen scheme given by (<ref>):y_i^1 = y_i^0 + J_i^-1 f_i^0 (e^κ J_i-1 )for each component i = 1, …, M of the state variable.We can write (<ref>) in form (<ref>) as[y^0 - y_0; y^1 - L(y^0) - y^0;] = 0where y_0 is the given initial condition at T_0 and L(y^0) = { L_i(y^0) }_i=1^M withL_i(y^0) = J_i^-1 f_i^0 ( e^κ J_i-1) ≡∂ f_i/∂ y_i(y^0, T_0)^-1 f_i(y_0, T_0) (e^κ∂ f_i/∂ y_i(y^0, T_0) - 1) .The adjoint system of the first order generalised Rush-Larsen scheme is then given by[ y̅^0; y̅^1;] - [0 I +∂ L/∂ y^0^∗;00;][ y̅^0; y̅^1;] = [ ∂ J/∂ y^0;y̅_0; ]where y̅_0 is a given terminal condition for the adjoint at T_1. Similarly, the tangent linear system (<ref>) with respect to an auxiliary parameter m is given by[ ẏ^0; ẏ^1 ] - [ 0 0; I + ∂ L/∂ y^0 0; ][ ẏ^0; ẏ^1 ] = [ẏ_0; ∂ L(y^0)/∂ m ]where ẏ_0 is a given initial condition at T_0.The adjoint and tangent linear equations for the other Rush-Larsen schemes are derived following the same steps, and we therefore omit the details here. § FENICS AND DOLFIN-ADJOINT ABSTRACTIONS AND ALGORITHMSThe FEniCS Project defines a collection of software components targeting the automated solution of differential equations via finite element methods <cit.>. The componentsinclude the Unified Form Language (UFL) <cit.>, the FEniCS Form Compiler (FFC) <cit.> and the finite element library DOLFIN <cit.>. The separate dolfin-adjoint project and software automatically derives the discrete adjoint and tangent linear models from a forward model written in the Python interface to DOLFIN <cit.>. This section presents our extensions of the FEniCS form language and the FEniCS form compiler, and other new FEniCS and dolfin-adjoint software features targeting coupled PDE-ODE systems in general and collections of systems of ODE in particular. §.§ Variational formulation of collections of ODE systems Consider a spatial domain Ω⊂^d tessellated by a mesh _h, and a collection of general initial problems as defined by (<ref>) over a set of points X defined relative to _h. For x_i ∈ X, denote by δ_x_i the Dirac delta function centered at x_i ∈^d such thatf(x_i) = ∫_^d fδ_x_ifor all continuous, compactly supported functions f. We will also write∑_x_i ∈ X∫_^d fδ_x_i≡∑_x_i ∈ X∫ f (x_i) ≡∑_x_i ∈ Xf1_x_i≡f1_X Drawing inspiration from our context of finite element variational formulations, we can then write (<ref>) as: find y(·, t) such thaty_t(t)ψ_i_X = f(y, t)ψ_i_Xfor all (basis) functions (or distributions) ψ_i such that ψ_i(x_i) = 1 and ψ_j(x_i) = 0 for j ≠ i.The remainder of this section describes the extensions of the FEniCS and Dolfin-adjoint systems to allow for in particular abstract representation and efficient forward and reverse solution of systems of the form (<ref>). §.§ Extending UFL with vertex integrals UFL is an expressive domain-specific language for abstractly representing (finite element) variational formulations of differential equations. In particular, the language defines syntax for integration over various domains. Consider a mesh _h of geometric dimension d with cells {𝒯}, interior facets {ℱ^i} and boundary facets {ℱ}^b. Interior facets are defined as the d-1 dimensional intersections between two cells and thus for any interior facet ℱ^i we can write ℱ^i = 𝒯^+∩𝒯^-. UFL defines the sum of integrals over cells, sum of integrals over boundary facets and sum of integrals over interior facets by the , , andmeasures, respectively. For example, the following linear variational form defined in terms of a piecewise polynomial u defined relative over _h:L(u) = ∑_{𝒯}∫_𝒯 u + ∑_{ℱ^i}∫_ℱ^i u |_T^+ + ∑_{ℱ^b}∫_ℱ^b uis naturally expressed in UFL as:L = u*dx + u('+')*dS + u*ds To allow for point evaluation over the vertices of a mesh (cf. (<ref>)), we have introduced a new vertex integral (or vertex measure in UFL terms) type with default instantiation :from ufl import Measure dP = Measure('vertex') L = f*dPIn agreement with (<ref>), the vertex integral is defined by:𝚏*𝚍𝙿≡f1_𝒱(_h) = ∑_x_i ∈𝒱(_h) f (x_i),where V(_h) is the set of all vertices of the mesh _h. Vertex measures restricted to a subset of vertices can be defined as for all other UFL measure types.This basic extension of UFL crucially allows for the abstract specification of collections of ODEs such as (<ref>) in a manner that is consistent and compatible with abstract specification of finite element variational formulations of PDEs. It also allows for the native specification of e.g. point sources in finite element formulations of PDEs in FEniCS. §.§ Vertex integrals in the FEniCS Form Compiler The FEniCS Form Compiler FFC generates specialized C++ code <cit.> from the symbolic UFL representation of variational forms and finite element spaces <cit.>. To accommodate the new vertex integral type, we have extended the UFC interface with a classdefining the interface for the tabulation of the element matrix corresponding to the evaluation of an expression at a given vertex, listed below:/// This class defines the interface for the tabulation of /// an expression evaluated at exactly one point. class vertex_integral: public integralpublic:/// Constructor vertex_integral() /// Destructor virtual vertex_integral() /// Tabulate tensor for contribution from local vertex virtual void tabulate_tensor(double * A,const double * const * w,const double * coordinate_dofs,std::size_t vertex,int cell_orientation) const = 0; ;The FFC code generation pipeline has been correspondingly extended to allow for the generation of optimized code from the UFL representation of variational forms involving vertex integrals. This allows for subsequent automated assembly of variational forms involving single vertex integrals (as illustrated above) and also in combination with other (cell, interior facet, exterior facet) integrals. §.§ DOLFIN features for solving collections of ODE systems §.§.§ Assembly of vertex integrals We have extended DOLFIN with support for automated assembly of variational forms that include vertex integrals. The support is currently limited to forms defined over test and trial spaces with vertex-based degrees of freedom only. The assembly algorithm follows the standard finite element assembly pattern by iterating over the vertices of the mesh, computing the map from local to global degrees of freedom, evaluating the local element tensor based on generated code, and adding the contributions to the global tensor. The vertex integral assembler runs natively in parallel via MPI.§.§.§ Specification and generation of multistage schemes Since version , DOLFIN supports the specification of multistage schemes of the form (<ref>)–(<ref>) for the solution of collections of ODE systems of the form (<ref>), via their Butcher tableaus. The DOLFIN classtakes as input the right hand side expression f, the solution y, the Butcher tableau specified via a, b and c, and secondary variables such as a time variable, the (integer) order of the scheme. From this specification, DOLFIN automatically generates a variational formulation of each of the separate stages in the multistage scheme, with each stage in accordance with (<ref>)-(<ref>). DOLFIN can also automatically generate the variational formulations corresponding to the adjoint scheme (<ref>) and to the tangent linear scheme (<ref>) on demand. A set of common multistage schemes are predefined including Crank-Nicolson, explicit Euler, implicit Euler, 4^th-order explicit Runge-Kutta, ESDIRK3, ESDIRK4 <cit.>.Similar features have also been implemented for easy specification of Rush-Larsen schemes cf. (<ref>)–(<ref>) and the automated generation of the corresponding adjoint and tangent linear schemes cf. e.g. (<ref>) and (<ref>) through the class . Bothandsubclass theclass.§.§.§ PointIntegralSolvers To allow for the efficient solution of the multistage schemes for collections of systems of ODEs, we have introduced a targetedclass in DOLFIN . This solver class takes aas input and its main functionality is to compute the solutions over a single time step.The solver iterates over all vertices of the mesh and solves for the relevant (stage and/or final) variables at each vertex. The solution algorithm for each stage depends on whether the stage is explicit or implicit. For implicit stages, a custom Newton solver is invoked that allows for Jacobian reuse across stages, across vertices, and/or across time steps on demand. As the resulting linear systems are typically small and dense, a direct (LU) algorithm is used for the inner solves. For explicit stages, a simple vector update is performed.The point integral solver runs natively in parallel via MPI. For a cell-partitioned mesh distributed between N processes, each cell is owned by one process and that process performs the solve for each vertex on the cell (once). As little communication is required between processes, the total solve is expected to scale linearly in N. §.§ Extensions to dolfin-adjoint Dolfin-adjoint has been extended to support the new features including automatically deriving and computing adjoint and tangent linear solutions for . The symbolic derivation of the variational formulation for the adjoint and tangent linear equations for theare based on (<ref>), (<ref>) for the Butcher tableau defined schemes and (<ref>), (<ref>) and its analogies for the Rush-Larsen schemes.The overall result is that operator splitting algorithms as described in Section <ref> can be fully specified and efficiently solved within FEniCS. Moreover, the corresponding adjoint and tangent linear models and first and second order functional derivatives may be efficiently computed using dolfin-adjoint with only a few lines of additional code.§ APPLICATIONSIn this section, we demonstrate the applicability and performance of our implementation by considering the two examples presented in the introduction, originating from the computational modelling of mitochondrial swelling and cardiac electrophysiology respectively. The complete supplementary code is openly available, see <cit.>. §.§ Application: mitochondrial swelling We consider the mathematical model defined by (<ref>). Inspired by <cit.>, we let x = (x_0, x_1) ∈Ω = [0, 1]^2 and t ∈ [0, T] with T = 35, and consider the initial conditions N_1, 0 = 1, N_2, 0 = 0, N_3, 0 = 0 andu_0(x) = 30/∫_Ω M(x)M(x),M(x_0, x_1) = 1/2 π e^- 0.5 (X_0^2 + X_1^2),with X_0 = (α - β) x_0 + β, X_1 = (α - β) x_1 + β taking α = 3 and β = -1. We considerf(s) =0 s < C^-, f^∗s > C^+,f^∗/2 (1 -cos (s - C^-/C^+ - C^-π ))otherwise,andg(s) =g^∗s > C^+,g^∗/2 ( 1 - cos ( s/C^+π )) otherwise.with C^- = 20, C^+ = 200, f^∗ = 1, g^∗ = 0.1, d_1 = 2 × 10^-6, and d_2 = 30.We are interested in the total amount of completely swollen mitochondria (whose density is given by N_3) at the final time T and its sensitivity to the initial condition for u (u_0). Thus, our functional of interest J is given byJ = ∫_Ω N_3(T)and our goal is to compute the derivative of J with respect to u_0.We use a second order Strang splitting scheme for (<ref>), a Crank-Nicolson discretization in time and continuous piecewise linear finite elements in space. The scheme then reads as: given initial conditions u_0, N_1,0, N_2,0, N_3,0, and time points {t_0, t_1, …, t_N} with time step κ_n = t_n+1 - t_n, then for each n = 0, 1, …, N - 1: * Compute u^∗ and N_i, 0^∗ i = 1, 2, 3 solvingu = d_2 g(u) N_2,N_1 = - f(u) N_1,N_2 = f(u) N_1 - g(u) N_2,N_3 = g(u) N_2,over Ω× [t_n, t_n + 1/2κ_n] with initial conditions u^n, N_i^n for i = 1, 2, 3.* Compute a solution u^† of: find u_h ∈ V_h such that∫_Ω ( u_h - u^∗ ) v+ κ_n𝒜 (u_h)· v= 0for all v ∈ V_h with u = 1/2 (u + u^∗ ) and where 𝒜(u) = u or 𝒜(u) = |u|^q-2 u.* Compute solutions u^n+1 and N_i^n+1, i = 1, 2, 3, solving (<ref>) over Ω× [t_n + 1/2κ_n, t_n+1] with initial conditions u^† and N_i^∗, i = 1, 2, 3.We choose to discretize (<ref>) via the ESDIRK4 method in time and set a tolerance of 10^-10 for the inner Newton solves. Further, we take q = 3, take κ_n = 0.5, and let _h be a uniform tessellation of Ω with N_x × N_x × 2 triangles.The initial and final solutions are presented in Figure <ref> while the L^2-gradient of the objective functional J given by (<ref>) with respect to the initial calcium concentration u_0 is presented in Figure <ref>. To verify that the computed gradient is correct, we performed a Taylor test at a given spatial and temporal resolution with a given perturbation seed. The results are listed in Table <ref> and demonstrate the expected orders of convergence, indicating a correctly computed gradient. In a multi-stage scheme, a number of intermediate stage solutions are computed during the solution process. To avoid excessive memory usage, dolfin-adjoint does not store these stage solutions. Thus, in order to compute the adjoint solution, the stage solutions are recomputed for each time-step. As a consequence, the minimal ratio of adjoint runtimeto forward runtime for multistage ODE solves using this strategy is approximately 2. For a linear PDE solve, the optimal ratio of adjoint run time to forward run time, assuming all linear solves take the same amount of time, is 1. For a nonlinear PDE solve via a Newton iteration, the optimal ratio of adjoint run time to forward run time, assuming all linear solves take the same amount of time, is 1/N where N is the number of average Newton iterations in the forward solve. These optimal ratios for linear and nonlinear PDE solves have observed for typical dolfin-adjoint usage <cit.>. For this example, two Newton iterations were required on average to solve the PDEs to within a tolerance of 10^-10 both for N_x = 40 and 80, and assembly dominated the PDE solve runtime. Thus for this example with two multistage ODE solves and a nonlinear PDE solve in each timestep, the optimal ratio of adjoint runtime to forward runtime is expected to be in the range 0.5-2, and closer to the upper bound than the lower bound depending on the distribution of computational cost between the PDEs and ODEs.Experimentally observed timings for this example are listed in Tables <ref>–<ref> for N_x = 40, 80, 160. From Table <ref>, we observe that the adjoint-to-forward runtime ratio for the total solve is in the range 1.38-1.95 and decreasing with increasing problem size. The corresponding gradient-to-forward runtime is in the range 1.44 - 2.28, again decreasing with increasing problem size as expected. From Table <ref>, we observe that adjoint-to-forward runtime ratio for the ODE solves is in the range 2.35-2.41 for this set of problem sizes, also decreasing with increasing problem size. We also note that the ODE solve runtime, both for the forward and adjoint solves, appears to scale linearly with the problem size M = N_x^2 as anticipated and as is optimal. The ODE solves account for 37-40% of the total forward runtime for this example.We also conducted numerical experiments varying the multistage scheme used in the ODE solves, including the 1-stage Backward Euler (BDF1), the 2-stage Crank-Nicolson (CN2), the 3-stage ESDIRK3 scheme in addition to the previously reported 4-stage ESDIRK4 scheme. The results (using N_x = 40) are reported in Table <ref>. We observe that the adjoint-to-forward runtime ratio ranges from 2.04 (BDF1) to 2.41 (ESDIRK3, ESDIRK4) and the general trend is that this ratio increases with the number of stages as expected, but stabilizes around 2.3-2.4.§.§ Application: cardiac electrophysiology (2D)In this example, we consider the bidomain equations (<ref>) over a two-dimensional rectangular domain Ω = [0, 50] × [0, 50] (mm) with coordinates (x_0, x_1). We will consider a set of different cardiac cell models of increasing complexity: a reparametrized FitzHugh-Nagumo (FHN) model with 1 ODE state variable <cit.>, the Beeler-Reuter (BR) model with 7 ODE state variables <cit.>, the ten Tusscher and Panfilov (TTP) epicardial cell model with 18 ODE state variables <cit.>, and the Grandi et al (GPB) cell model with 38 ODE state variables <cit.>. The description of each of these cell models is available via the CellML repository, and our implementation of the ionic current I_ ion and F arising in (<ref>) was automatically generated from the CellML models for the BR, TTP and GPB models. We refer to the supplementary code <cit.> for the precise description of the models including our choice of FitzHugh-Nagumo parameters.Our parameter setup is otherwise as follows. We let χ = 140 (mm^-1), C_m = 0.01 (μF/mm^2), and let M_i = diag (g_ if, g_ if) and M_e = diag (g_ ef, g_ es) where g_ ef = 0.625/(χ C_m), g_ es = 0.236/(χ C_m) and g_ if = 0.174/(χ C_m). We let I_s = 0 and set the initial conditionv_0 = 10( x_0/50 ) ^2 + 10for v. The other state variables are initialized to the default initial conditions given by the respective CellML models.We use a second-order Strang splitting scheme (i.e. θ = 0.5 for (<ref>)–(<ref>)), a Crank-Nicolson discretization in time and continuous piecewise linear finite elements in space for the PDE system, and different Rush-Larsen type discretizations (RL1, GRL1, RL2, GRL2) for the time discretization of the ODE systems[We observed spurious wave propagation results when using a first-order splitting scheme and implicit Euler for with the BR model.]. The coupled linear systems that arise at each PDE step (<ref>) were solved with a block-preconditioned GMRES scheme with a relative tolerance of 10^-10. The complete details of the solver are detailed in the supplementary code <cit.>.We take κ_n = 0.1 (ms), and consider a uniform mesh of the computational domain with N_x × N_x × 2 triangles. The coarsest mesh considered used N_x = 160. For this case, simulations converged using any of the cell models and schemes, and gave qualitatively correct results, except the Grandi cell model discretized by the RL1 scheme for which the numerical solution scheme failed to converge.We are interested in computing the gradient with respect to the extracellular fiber conductivity g_ ef of the following objective functionalJ(v) = ∑_i=1^5 ∫_Ω ( v(t_i)^2 + s(t_i)^2)for a equidistributed set of time points t_i. The model initial condition and solution for the transmembrane potential at T = 100, together with the L^2-gradient of the objective functional with respect to the fiber conductivity are illustrated in Figure <ref>.The following numerical experiments were performed on the Abel supercomputer at the University of Oslo. All runtime experiments were repeated three times, of which the minimum timing values are reported here.§.§.§ Verifying the correctness of the discrete gradientTo verify that the computed functional gradient is correct, we performed a Taylor test in a random perturbation direction at given spatial and temporal resolutions. The results with the Fitz-Hugh Nagumo cell model and the GRL1 scheme are listed in Table <ref> and demonstrate the expected orders of convergence without a computed gradient (order 1) and when using the computed discrete gradient (order 2) in the Taylor expansion.We also performed similar experiments for the other cell models (BR, TTP, and GPB) and other Rush-Larsen solution schemes (RL1, GRL2, RL2). We used the same settings as in Table <ref>, except for the Grandi cell model with any other Rush-Larsen scheme than GRL1, where the time step had to be reduced to 0.01 in order for the forward solver to converge. We obtained the expected convergence order for all combination of cell models and schemes tested. These results indicate that the automatically derived and computed adjoints and gradient are correct for all the Rush-Larsen schemes implemented.§.§.§ Adjoint runtime performance As the bidomain PDEs (<ref>) are linear, a rough theoretical estimate of the optimal PDE adjoint to PDE forward runtime ratio is 1, assuming that the solution times in the adjoint and forward PDE solves are equal and dominated by the finite element assembly and linear solvers. Similarly, the ODEs are solved with explicit Rush-Larsen schemes with a theoretically optimal adjoint-to-forward ratio of 1 as well.In Table <ref>, we list the total runtime, and the runtime components corresponding to only the ODE solves, only the PDE solves and an intermediate variable update (merge) step, for both the forward and the adjoint solution computation for a test case with T = 10.0, κ_n = 0.1, N_x = 160, the GRL1 scheme and the four different cardiac cell models (FHN, BR, TTP, GPB).From Table <ref>, for the forward solves, we observe that the total runtime increases with cell model complexity. The forward PDE step runtime is essentially independent of the cell model, while the forward ODE step runtime increases with the cell model complexity, as expected: the ODE runtime corresponds to approximately 10 % of the PDE runtime for the simplest FitzHugh-Nagumo model, and approximately 114 % of the PDE runtime for the most complex Grandi model. The runtime of the forward merge step is insignificant in comparison to the ODE and PDE steps, but increases with cell model complexity.We observe that also the adjoint PDE runtimes are comparable for all cell models. The resulting adjoint-to-forward PDE step ratios are in the range 0.79-0.82. The decrease in compute time for the adjoint PDE solves compared to the forward PDE solves are attributable to the Krylov solvers: fewer Krylov solver iterations were necessary for convergence for the adjoint solves compared to the forward solves.For the ODE solves, we observe that the adjoint-to-forward ratio for the ODE solves is in the range 1.88-2.58 for the different cell models. Additionally, we observe that the runtime of the adjoint merge step is significantly larger than the corresponding time for the forward merge step and with an increasing adjoint-to-forward ratio with increasing cell model complexity 3.76-5.40. We suggest that the reason for the increased and increasing adjoint runtime is that the adjoint merge step involves additional assembly of variational forms of complexity comparable to that of the cell model.Overall, we observe that the adjoint-to-forward total runtime ratios are in the range 1.15-1.66 for the different cell models considered, with increasing adjoint-to-forward ratio with increasing cell model complexity. Since the PDE runtime dominates the total runtime for the simpler cell models, the total adjoint-to-forward ratio is closer to 1 for these models. On the other hand, for the more complicated cell models, the ODE runtimes dominate, and so we observe adjoint-to-forward ratios closer to 2 for these.In the last column of Table <ref>, we observe that the combined ODE and PDE solve time contributes to between 97% (for FitzHugh-Nagumo) and 86% (for Grandi) of the total forward runtime for the cell models considered. The remaining forward runtime cost is primarily caused by the initialization routines such as loading the computational mesh, and creating the required function spaces. For the corresponding adjoint runtimes, the combined ODE and PDE solve times contribute to between 86% (for FitzHugh-Nagumo) and 85% (for Grandi) of the total run time, and the percentage contribution decreases with the complexity of the cell model.The remaining runtime cost can primarily be attributed to recording the forward states during the adjoint solve (which is included in the reported total run time but not in the ODE or PDE runtimes here). We also timed the computation of the discrete gradient and noted that the cost of additionally computing the gradient was negligible (less than 0.1% of the adjoint solve runtime).§.§.§ Forward and adjoint parallel scalabilityIn this section, we discuss the weak and strong parallel scalability of the forward and adjoint ODE solvers and the scalability of the point integral solvers in particular applied to this 2D electrophysiology test case. We focus on the scalability of the point integral solver particular as this is the primary new feature described in this paper. Previous studies have discussed the parallel scalability of FEniCS in general <cit.>.We ran a series of weak scaling experiments on Abel, hosted by the Norwegian national computing infrastructure (NOTUR), for the previously considered series of cell models (FHN, BR, TTP, GPB) on an increasing number of CPUs (1, 4, 16, 64) and with an increasing mesh resolution.We timed the total runtime of the point integral solves, both for the forward solves and the adjoint solves. For each cell model and resolution, we ran three experiments and extracted the minimal run times. The results are listed in Table <ref>. For both the forward and adjoint run times, we computed the parallel efficiency (PE) as the ratio of the 1-CPU runtime to the n-CPU run time for n = 4, 16, 64, also listed in Table <ref>. The optimal PE would be 1. The last column of the Table lists the computed the adjoint-to-forward ratio for these solves.For the forward runtimes, we observe that the parallel efficiencies range from 0.66 to 0.95 for all cell models and resolutions, with efficiencies higher than 0.8 for all but the least complex cell model. In general, we observe that the parallel efficiency increases with increasing cell model complexity and that it decreases moderately with the number of CPUs. Some of the loss of parallel efficiency (from 1) is attributable to only a near-perfect distribution of mesh vertices across CPUs; we observed approximately 10% variation between CPUs in the number of local vertices.For the adjoint runtimes, we observe parallel efficiencies ranging from 0.75 (FHN on 64 cores) to 0.95, again with efficiencies higher than 0.8 for all but the least complex cell model. In general, we note the same trends as for the forward run times: increasing parallel efficiency with increasing cell model complexity and moderately decreasing parallel efficiency with increasing number of CPUs. We emphasize that we thus achieve the same parallel efficiency for the adjoint point integral solves as for the forward point integral solves.Comparing the forward and adjoint runtimes, we see that the adjoint-to-forward ratio for the point integral solvers range from 1.19 to 1.56. Comparing these numbers with the adjoint-to-forward ratios of the overall ODE solves as discussed in the previous, we observe that the adjoint-to-forward point integral solver performance is better. The adjoint-to-forward ratio is moderately increasing with cell model complexity, but is stable with respect to the number of CPUs. This latter point again illustrates that the adjoint point integral solvers demonstrate the same parallel efficiency as the forward point integral solves. We also ran a series of strong scaling experiments on Abel for the previously considered series of cell models (FHN, BR, TTP, GPB) on an increasing number of CPUs (1, 4, 16, 64) and with a fixed mesh resolution. We timed the total runtime of the point integral solver steps for both the forward and adjoint solves. Again, for each cell model and resolution, we ran three experiments and extracted the minimal run times. The results are listed in Table <ref>. For both the forward and adjoint run times, we computed the (strong scaling) parallel efficiency (PE) as the ratio of the 1-CPU runtime to n-times the n-CPU runtime for n = 4, 16, 64, also listed in Table <ref>. The optimal (strong scaling) PE would be 1. The last column of the Table lists the computed the adjoint-to-forward ratio for these solves.We observe that the parallel efficiencies are comparable to the results from the weak scaling test case with efficiencies in the range from 0.60 to 0.94. We note that the adjoint strong parallel efficiency is comparable to the forward strong parallel efficiency. The adjoint-to-forward ratios for this strong scaling test are also in the same range as for the weak scaling test. §.§ Application: biventricular cardiac electrophysiology (3D)In this example, we aim to compute the sensitivity of the squared L^2-norm of the transmembrane potential with respect to its initial condition, in order to illustrate the features discussed in this paper for a more complex test case. This example also aims to illustrate how different representations may seamlessly be used for the control variables; this is useful for instance when computing sensitivities with respect to spatially constant, regionally defined or highly resolved control variables.Inspired by <cit.>, we consider the monodomain variation of (<ref>) over a mesh of a three-dimensional bi-ventricular domain Ω with the ten Tusscher and Panfilov epicardial cell model <cit.> as detailed in Section <ref>. For the monodomain equations, we replace the bidomain equations (<ref>)–(<ref>) by: find v such thatv_t -Mv = - I_ ion(v, s) in Ω.with homogeneous Neumann boundary conditions and with v(t = 0) = v_0. As before, we let χ = 140 (mm^-1), C_m = 0.01 (μF/mm^2), and for simplicity (ignoring realistic, spatially-varying fiber directions), let M = diag (g_ f, g_ s, g_ n) where:g_f = 0.255,g_n = 0.0775,g_s = 0.0775. To solve the coupled equations (<ref>) and (<ref>), we use a second-order Strang splitting scheme (θ = 0.5), a Crank-Nicolson discretization in time and continuous piecewise linear finite elements in space for the resulting PDEs, and the first-order generalized Rush-Larsen (GRL1) scheme for the resulting ODEs. We take κ_n = 0.05 (ms). The moderately coarse mesh has mesh cell diameters in the range [0.61, 2.70] (mm) and bounding box [0.103, 40.2] × [-15.8, 36.8] × [-30, 13.9] (mm^3), with 45 625 vertices and 236 816 cells, and is illustrated in Figure <ref>.To quantify the transmembrane potential exceeding a zero threshold , we consider the following objective functional:J(v) = ∫_Ω_v v(T)^2 ,at a fixed time T = 6.0 (ms) and where Ω_v = { x ∈Ω| v(x) ≥ 0 }. We are interested in computing the gradient of J with respect to the initial condition u_0.We consider two different representations of the initial condition to illustrate computing the gradient with respect to low and high resolution fields. For both cases, we letv_0(x) =-61.1 x_2 > z_ mid, -61.3 x_2 ≤ z_ mid,with z_ mid = -8.07, but consider (i) v_0 as represented by two constants (c_0, c_1) i.e. the two-dimensional space ^2 and (ii) v_0 represented by a continuous piecewise linear defined relative to the mesh i.e. as a n_ verts-dimensional space V where n_ verts denotes the number of mesh vertices.The initial condition and computed solution at t = 6.0 (ms) are illustrated in Figure <ref>. (With reference to the chosen perspective in this Figure, we denote the subdomain where x_2 > z_ mid as the top part of the domain and the subdomain x_2 ≤ z_ mid as the bottom part.)For the case where the initial condition is represented at a low resolution by v_0 = (c_0, c_1) ∈^2, we considered a series of experiments starting with different initial conditions v_0^α = (c_0 + α, c_1 + α) ∈^2 for α∈{ -0.1, -0.05, 0.0, 0.05, 0.1 }. The resulting gradients (in ^2) with respect to v_0 ∈^2 are illustrated in Figure <ref> (left). We observe that as we increase α, the gradient with respect to the top initial condition c_0 goes from large and positive to small and negative, while the gradient with respect to the bottom initial condition c_1 varies from moderately negative to close-to-zero and rapidly to large and positive. The rapid changes in the gradients illustrate the nonlinear nature of the problem at hand.The sensitivity of the objective functional J defined by (<ref>) with respect to the initial condition v_0 ∈ V is illustrated in Figure <ref> (right). We observe that the gradient in the upper part of the domain is large and negative, that the gradient in a boundary zone is large and positive and the gradient in the lower part of the domain is close to zero. This indicates that infinitesimal increases/decreases in the upper part of the domain will lead to a large decrease/increase in the functional value, infinitesimal increases/decreases in the boundary zone domain will lead to a large increase/decrease in the functional value, and that infinitesimal increases/decreases in the lower part of the domain will have relatively small effect on the functional value.§ CONCLUDING REMARKSWe have presented a high-level framework that allows for the specification of coupled PDE-ODE systems and their efficient forward and adjoint solution with operator splitting schemes. We have illustrated the features of the framework with a series of examples originating from cell modelling and computational cardiac electrophysiology. Our numerical results indicate adjoint runtime performance near optimal (measured in terms of adjoint-to-forward runtimes), and parallel efficiency indices of 84-94% for the more realistic cardiac cell models.The main advantages of the framework are the speed of developing solvers for new systems, and the ability to automatically derive their adjoints from the high-level specification. This allows for flexible exploration of models when the relevant governing equations are uncertain, and for the identification of unknown parameters via the solution of inverse problems as demonstrated e.g. in <cit.>. Both of these capabilities are of significant importance in numerous areas of scientific computing, and in particular in computational medicine, physiology and biology. Future work will focus on further improving the parallel scalability of the solvers and its application to problems in personalized medicine. siamplain | http://arxiv.org/abs/1708.07648v1 | {
"authors": [
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"Johan E. Hake",
"Simon W. Funke",
"Marie E. Rognes"
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"published": "20170825083809",
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mn2e ();a;firstpage–lastpage2018 Maximum redshift ofgravitational wave merger events Abraham Loeb===================================================== This work presents AutoLens, the first entirely automated modeling suite for the analysis of galaxy-scale strong gravitational lenses. AutoLens simultaneously models the lens galaxy's light and mass whilst reconstructing the extended source galaxy on an adaptive pixel-grid. The method's approach to source-plane discretization is amorphous, adapting its clustering and regularization to the intrinsic properties of the lensed source. The lens's light is fitted using a superposition of Sersic functions, allowing AutoLens to cleanly deblend its light from the source. Single component mass models representing the lens's total mass density profile are demonstrated, which in conjunction with light modeling can detect central images using a centrally cored profile. Decomposed mass modeling is also shown, which can fully decouple a lens's light and dark matter and determine whether the two component are geometrically aligned. The complexity of the light and mass models are automatically chosen via Bayesian model comparison. These steps form AutoLens's automated analysis pipeline, such that all results in this work are generated without any user-intervention. This is rigorously tested on a large suite of simulated images, assessing its performance on a broad range of lens profiles, source morphologies and lensing geometries. The method's performance is excellent, with accurate light, mass and source profiles inferred for data sets representative of both existing Hubble imaging and future Euclid wide-field observations.gravitational lensing - galaxies: structure 1_Introduction.tex 2a_Overview.tex 2_Simulations.tex 3_AutoLens.tex 3a_Demonstration.tex 4_Pipeline.tex 5_Results.tex 6_Discussion.tex§ ACKNOWLEDGEMENTS JN acknowledges support from STFC grant ST/N00149/1 and the University of Nottingham. SD acknowledges support from the Midlands Physics Alliance and STFC Ernest Rutherford Fellowship Scheme. RJM acknowledges support from the Royal Society University Research Fellowship. We are grateful for access to the University of Nottingham High Performance Computing Facility. This paper used the DiRAC Data Centric system at Durham University, operated by the Institute for Computational Cosmology on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). This equipment was funded by BIS National E-infrastructure capital grant ST/K00042X/1, STFC capital grant ST/H008519/1, and STFC DiRAC Operations grant ST/K003267/1 and Durham University. DiRAC is part of the National e-Infrastructure. We acknowledge use of the software program (getdist.readthedocs.io/en/latest/).Appendices.tex | http://arxiv.org/abs/1708.07377v2 | {
"authors": [
"James Nightingale",
"Simon Dye",
"Richard Massey"
],
"categories": [
"astro-ph.CO",
"astro-ph.GA",
"astro-ph.IM"
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"primary_category": "astro-ph.CO",
"published": "20170824124146",
"title": "AutoLens: Automated Modeling of a Strong Lens's Light, Mass and Source"
} |
=1./Images/ | http://arxiv.org/abs/1708.07982v2 | {
"authors": [
"Fiona McCarthy",
"David Kubiznak",
"Robert B. Mann"
],
"categories": [
"hep-th",
"gr-qc"
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"primary_category": "hep-th",
"published": "20170826151230",
"title": "Breakdown of the Equal Area Law for Holographic Entanglement Entropy"
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Bayesian Compressive Sensing Using Normal Product Priors Zhou Zhou, Kaihui Liu, and Jun Fang, Member, IEEE Zhou Zhou, Kaihui Liu, and Jun Fang are with the National Key Laboratory on Communications, University of Electronic Science and Technology of China, Chengdu 611731, China, Email: [email protected]; [email protected]; [email protected] This work was supported in part by the National Science Foundation of China under Grant 61172114. Received: date / Accepted: date ======================================================================================================================================================================================================================================================================================================================================================================================================================= In this paper, we introduce a new sparsity-promoting prior, namely, the “normal product” prior, and develop an efficient algorithm for sparse signal recovery under the Bayesian framework. The normal product distribution is the distribution of a product of two normally distributed variables with zero means and possibly different variances. Like other sparsity-encouraging distributions such as the Student's t-distribution, the normal product distribution has a sharp peak at origin, which makes it a suitable prior to encourage sparse solutions. A two-stage normal product-based hierarchical model is proposed. We resort to the variational Bayesian (VB) method to perform the inference. Simulations are conducted to illustrate the effectiveness of our proposed algorithm as compared with other state-of-the-art compressed sensing algorithms. Compressed Sensing, sparse Bayesian learning, normal product prior.§ INTRODUCTION Compressed sensing <cit.> is a new data acquisition technique that has attracted much attention over the past decade. Existing methods for compressed sensing can generally be classified into the following categories, i.e. the greedy pursuit approach <cit.>, the convex relaxation-type approach <cit.> and the nonconvex optimization method <cit.>. Another class of compressed sensing techniques that have received increasing attention are Bayesian methods. In the Bayesian framework, the signal is usually assigned a sparsity-encouraging prior, such as the Laplace prior and the Gaussian-inverse Gamma prior <cit.>, to encourage sparse solutions. It has been shown in a series of experiments that Bayesian compressed sensing techniques <cit.> demonstrate superiority for sparse signal recovery as compared with the greedy methods and the basis pursuit method. One of the most popular prior model for Bayesian compressed sensing is the Gaussian-inverse Gamma prior proposed in <cit.>. The Gaussian-inverse Gamma prior is a two-layer hierarchical model in which the first layer specifies a Gaussian prior to the sparse signal and an inverse Gamma priori is assigned to the parameters characterizing the Gaussian prior. As discussed in <cit.>, this two-stage hierarchical model is equivalent to imposing a Student's t-distribution on the sparse signal. Besides the Gaussian-inverse Gamma prior, authors in <cit.> employ Laplace priors for the sparse signal to promote sparse solutions.In this paper, we introduce a new sparsity-encouraging prior, namely, the normal product (NP) prior, for sparse signal recovery. A two-stage normal product-based hierarchical model is established, and we resort to the variational Bayesian (VB) method to perform the inference for the hierarchical model. Our experiments show that the proposed algorithm achieves similar performance as the sparse Bayesian learning method, while with less computational complexity.§ THE BAYESIAN NETWORKIn the context of compressed sensing, we are given a noise corrupted linear measurements of a vector x_0y = Ax_0 + nHere A∈R^M × N is an underdetermined measurement matrix with low coherence of columns, n∈R^M represents the acquisition noise vector and x_0 a sparse signal. For the inverse problem, both x_0 and n are unknown, and our goal is to recover x_0 from y. We formulate the observation noise as zero mean independent Gaussian variable with variance ξ ^2 i.e. y∼ N(A x_0,ξ ^2I) and seek a sparse signal xΔ = (x_1,x_2, …x_N) for x_0.In this section, we utilize a two-stage hierarchical bayesian prior for the signal model. In the first layer of signal model, we use the Normal Product (NP) distribution as sparseness prior:x∼ NP(0,Σ)where 0∈R^N stands for its mean and Σ given as a diagonal matrix diag(σ _1^2,σ _2^2, … ,σ _N^2) ∈R^N × N represents its variance. For each element x_i of x , its probability density distribution is K_0( |x_i|/σ _i)/.-πσ _i, which exhibits a sharp peak at the origin and heavy tails, where K_0( · ) is the zero order modified Bessel function of the second kind<cit.> thus the probability density function (PDF) of x is 1/π ^N∏_i = 1^N 1/σ _iK_0(| x_i|/σ _i) .A NP distributed scaler variate x_i ∈R can be decomposed as the product of two independent normally distributed variables i.e. if x_i ∼ NP(0,σ_i^2 ), then there are a_i ∼ N(0,κ_i^2 ) and b_i ∼ N(0,γ_i^2 )satisfying x_i = a_ib_i with moment relationship σ_i= γ_i κ_i<cit.>. We call this property as the generating rule of Normal Product distribution. Similarly for the vector x,we can decompose it into the Hadamard product of two independent virtual normally distributed vector variables a and b whose variance are diagonal matrix κ^2Δ =diag(κ _1^2,κ _2^2, … ,κ _N^2) and γ^2Δ = diag(γ _1^2,γ _2^2, … ,γ _N^2) respectively i.e. x= a ∘b and Σ = κ^2∘γ^2, where ∘ denote the Hadamard product. Finally, we set σ _i^ - 2 as a realization of Gamma hyperprior Γ (α ,β ) and choose α→ 0,β→ 0 to construct a sharper distribution of x during the variational procedure <cit.> for the second layer of signal.According to the generating rule of Normal Product mentioned above, we can add two parallel nodes a and b to construct one latentlayer. Then the posterior distribution can be expressed as:P_x(x,Σ|y;α ,β ) = ∫P(x,a,b,Σ|y;α ,β )dadbwhere[ P(x,a,b,Σ|y;α ,β ); ∝ P(y|x)P(x|a,b)P(a|κ^2)P(b|γ^2)P(Σ|α ,β ); = P(y - Ax)δ (x - a∘b)P(a|κ^2)P(b|γ^2)P(Σ|α ,β ) ] and δ(·) denotes the Dirac Delta function. In this expression we know that the value of x must be equal to a∘b while keeping the value of P(x,a,b,Σ|y;α ,β ) nonzero. Consider the Bayesian Risk function:[ E(L(x̂,x)) = ∫L(x̂,x)P_x(x,Σ|y;α ,β )dxdΣ; ∝∫L(x̂,a∘b)P_a,b(a,b,Σ|y;α ,β )dadbdΣ ]where[P_a,b(a,b,Σ|y;α ,β ); = P(y - A(a∘b))P(a|κ^2)P(b|γ^2)P(Σ|α ,β ) ]and L(x̂,x) represents the loss function. Thus, we can replace P_x(x,Σ|y;α ,β ) with P_a,b(a,b,Σ|y;α ,β )in the inference procedure while maintaining the value of the Bayesian risk function at a consistent level. So the modified Bayesian Network is: y∼ N(A(a∘b),ξ ^2I) a∼ N(0,κ^2) b∼ N(0,γ^2) σ _i^ - 2 = γ _i^ - 2κ _i^ - 2∼Γ (α ,β )as depicted in Fig. <ref>. § THE VARATIONAL BAYESIAN INFERENCEIn order to infer the bayesian network, we use the mean-field variational Bayesian method to analytically approximate the entire posterior distribution of latent variable. In this theorem, it was assumed that the approximate variational distribution q(z|y) where z stands for the latent variable in the model could be factorized into the product ∏_i = 1^l q_i(z_i|y) for the partition z_1,…, z_l of z. It could be shown that the solution of the variational method for each factor q_i(z_i|y) could be written as:q_i^*(z_i|y) = e^ < ln p(z|y)) > _z_i/∫e^ < ln p(z|y)) > _z_idz_iwhere < ln p(z|y)) > _z_i means taking the expectation over the set of z using the variational distribution except z_i<cit.> . Applying the variational Bayesian method upon the bayesian model mentioned in secetion <ref>, the posterior approximation of Σ,a,b are respectively:ln (q^*(a)) ∝< ln P(a,b,Σ|y;α,β) > _q^*(b),q^*(Σ) ln (q^*(b)) ∝< ln P(a,b,Σ|y;α,β) > _q^*(a),q^*(Σ) ln (q^*(Σ)) ∝< ln P(a,b,Σ|y;α,β) > _q^*(b),q^*(a)§.§ posterior approximation of a and bSubstituting (<ref>), (<ref>), (<ref>) and (<ref>) into (<ref>), we can arrive at:q^*(b)∼ N(Γ'^ - 1c',Γ'^ - 1)whereΓ' = 1/ξ ^2(A^TA) ∘< aa^T > _q^*(a) +< γ^ - 2 > _q^*(γ)andc' = 1/ξ ^2diag(< a > _q^*(a))A^Ty . From the principle of variational inference, we know that q( b) is an approximation of P(b|y) i.e. P(b|y) ≈ q( b) and in order to get a more concise iteration formula, we choose the relaxation of the secondary moment as: < aa^T > _q^*(a)≈< a > _q^**(a) < a^T > _q^**(a) < γ^ - 2 > _q^*(γ)≈< γ > _q^**(γ)^ - 2 This relaxation can be interpreted as ignoring the posteriori variance during the updating, which is inspired by the fact that after the learning procedurethe posteriori variance always approaches zero to ensure the posteriori mean's concentration on the estimated value.Then the corrected posterior approximation is q^**(b)∼ N(Γ^ - 1c,Γ^ - 1) where[ Γ^ - 1 = {1/ξ ^2( Adiag(< a > _q^**(a)))^TAdiag(< a > _q^**(a)); +< γ > _q^**(γ)^ - 2} ^ - 1 ]andc = 1/ξ ^2diag(< a > _q^**(a))A^TyIn the noise free case, using Woodbury identity we have:[ lim_ξ→ 0Γ^ - 1c; =< γ > _q^**(γ)(Adiag( < a > _q^**(a)) < γ > _q^**(γ))^ + y ]and[lim _ξ→ 0Γ^ - 1; = (I -< γ > _q^**(γ)( Adiag(< b > _q^**(b)) < γ > _q^**(γ))^ + );×Adiag(< a > _q^**(a)) < γ > _q^**(γ)^2 ]Similarly, for a, substituting (<ref>), (<ref>), (<ref>) and (<ref>) into (<ref>), we have:q^*(a)∼ N( H'^ - 1f', H'^ - 1)whereH' = 1/ξ ^2(A^TA) ∘< bb^T > _q^*(b) +< κ^ - 2 > _q^*(κ)andf' = 1/ξ ^2diag(< b > _q^*(b))A^TySo the corrected approximation is:q^**(a)∼ N( H^ - 1f, H^ - 1)Then again, in the noiseless case, we have[ lim _ξ→ 0 H^ - 1f; =< κ > _q^**(κ)(Adiag( < b > _q^**(b)) < κ > _q^**(κ))^ + y ]and[lim _ξ→ 0H^ - 1; = (I -< κ > _q^**(κ)( Adiag(< b > _q^**(b)) < κ > _q^**(κ))^ + );×Adiag(< b > _q^**(b)) < κ > _q^**(κ)^2 ]§.§ posterior approximation of ΣAs discussed in section <ref>, we have:[ ln (q^*(Σ)) = ln (q^*(κ^2∘γ^2)) = ln (q^*(κ^2)) + ln (q^*(γ^2)); ∝< ln P(a,b,κ^2∘γ^2|y;α,β) > _q^**(b),q^**(a) ]Substituting (<ref>), (<ref>), (<ref>) into (<ref>) and using the separability of ln (q^*(κ^2)), it can be shown that:[ lnq( κ _i^2) ∝- 1/2(< a_i^2 > _q^**(a) + 2β< γ _i^ - 2 > _q^*(γ _i^2))κ _i^ - 2;+ (α+ 1/2 - 1)ln( κ _i^ - 2) ]which meansκ _i^ - 2∼Γ (1/2 + α ,1/2( < a_i^2 > _q^**(a) + 2β< γ _i^ - 2 > _q^*(γ^2)))and< κ _i^ - 2 > _q^*(κ ^2) = 1/2 + α/1/2( < a_i^2 > _q^**(a) + 2β< γ _i^ - 2 > _q^*(γ^2)) Similarly, we have:γ _i^ - 2∼Γ (1/2 + α ,1/2( < b_i^2 > _q^**(b) + 2β< κ _i^ - 2 > _q^*(κ^2)))and< γ _i^ - 2 > _q^*(γ^2 ) = 1/2 + α/1/2( < b_i^2 > _q^**(b) + 2β< κ _i^ - 2 > _q^*(κ^2)) §.§ The Proposed Algorithm As a result, we summarize the procedure above as two algorithms named “NP-0” and “NP-1” which represent the inference results for the one layer signal model and two layer signal model respectively. The difference between Np-0 and NP-1 is whether the learning process updates the precision of Normal Product. The algorithm NP-0 is similar to the FOCUSS algorithm<cit.>. The difference between them is that NP-0 uses < a>^(t) and < b>^(t), the decomposed component of x^(t) in an interleaved way to update x^(t+1) while FOCUSS uses the whole x^(t) to directly update x^(t+1). Furthermore, we set the initial value of the algorithm as a constant to avoid the local minimum results being returned.The algorithm NP-1 looks like coupling two sparse Bayesian learning (SBL) <cit.> procedures together and we will experientially prove that the MSE via NP-1 descends faster than SBL in section <ref>.§ NUMERICAL RESULTSIn this section, we compare our proposed algorithms among SBL, iterative reweighted least squares (IRLS)<cit.> and basis pursuit (BP)<cit.>. In the following results, we set the dimension of original signal to be 100 and in each experiment every entries of the sensing matrix A is uniformly distributed as standard normal random variable. The stop criterion parameters ε and t_max in both algorithm NP-0 and NP-1 are set as 10^-3 and 300 respectively.The success rate in the Fig. <ref> is defined by the average probability of successful restoration, which is counted as the ratio between the number of successful trials and the total number of the independent experiments. It collects the result of 300 Monte Carlo trials and we assume successful trial if x̂ - x_2/.-x_2 < 10^ - 3, where x̂ is the estimate of the original signal x. The mean squared error (MSE) in Fig. <ref> which measures the average of the squares of the difference between the estimator and estimated signal is calculated as follow:MSE(dB) = 20log (x̂ - x_2/.-x_2) Fig. <ref> demonstrats the superior performance of the NP-1 algorithm in a few measurements. It shows the success rate of respective algorithms versus the number of measurement M and sparse level K of the signal. We can see that the reconstruction performance of the one layer normal product algorithm(NP-0) is almost the same as BP, which is the MAP estimation of unknowns in the one layer Laplace prior framework. The comparisons also show that the two layers normal product algorithm(NP-1) requires as few number of CS measurements as SBL while inheriting the similar reconstruction precision.In Fig. <ref>, the comparison among the computational cost of the three algorithms on the reconstruction performance are performed on a personal PC with dual-core 3.1 GHz CPU and 4GB RAM. It is interesting to note that the MSE of NP-1 descends faster than SBL in Fig. 4(a) while it has the same sparsity-undersampling tradeoff performance as SBL. It should also be noticed that we haven't use any basis pruning technology as in <cit.> to reduce the computational complexity. To understand the reason of low computational time from Fig. 4(b), we observe that the NP-1 requires only a few numbers of iterations to arrive at stop condition. In summary, these experimental results confirm that the NP-1 is a fast Sparse Bayesian algorithm.§ CONCULSION In this letter, we formulated a Normal Product prior based Bayesian framework to solve the compressed sensing problem in noise free case. Using this framework, we derived two algorithms named NP-0 and NP-1 and compared them with different algorithms. We have shown that our algorithm NP-1 has the similar reconstruction performance with SBL while the interleaved updating procedure provides improved performance in computational times.IEEEtran | http://arxiv.org/abs/1708.07450v1 | {
"authors": [
"Zhou Zhou",
"Kaihui Liu",
"Jun Fang"
],
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"published": "20170824151522",
"title": "Bayesian Compressive Sensing Using Normal Product Priors"
} |
^1Infrared Processing and Analysis Center, California Institute of Technology, Pasadena, CA 91125, USA ^2Cahill Center for Astronomy and Astrophysics, California Institute of Technology, Pasadena, CA 91125, USA ^3Department of Astronomy, Cornell University, Space Sciences Building, Ithaca, NY 14853, USA ^4Max-Planck Institut für Astronomie, Königstuhl 17, D-69117, Heidelberg, Germany ^5School of Physics and Astronomy, Rochester Institute of Technology, 84 Lomb Memorial Drive, Rochester, NY 14623, USA [email protected]; Twitter: @astrofaisst Recent studies have found a significant evolution and scatter in the relation at z>4, suggesting different dust properties of these galaxies. The total far-infrared (FIR) luminosity is key for this analysis but poorly constrained in normal (main-sequence) star-forming z>5 galaxies where often only one single FIR point is available. To better inform estimates of the FIR luminosity, we construct a sample of local galaxies and three low-redshift analogs of z>5 systems. The trends in this sample suggest that normal high-redshift galaxies have a warmer infrared (IR) SED compared to average z<4 galaxies that are used as prior in these studies. The blue-shifted peak and mid-IR excess emission could be explained by a combination of a larger fraction of the metal-poor inter-stellar medium (ISM) being optically thin to ultra-violet (UV) light and a stronger UV radiation field due to high star formation densities. Assuming a maximally warm IR SED suggests 0.6dex increased total FIR luminosities, which removes some tension between dust attenuation models and observations of the relation at z>5. Despite this, some galaxies still fall below the minimum relation derived with standard dust cloud models. We propose that radiation pressure in these highly star-forming galaxies causes a spatial offset between dust clouds and young star-forming regions within the lifetime of O/B stars.These offsets change the radiation balance and create viewing-angle effects that can change UV colors at fixed IRX.We provide a modified model that can explain the location of these galaxies on the diagram. § INTRODUCTION Modern galaxy surveys have provided large and robust samples of z>5 galaxies that span a wide range of properties. These rich datasets are beginning to give us insights into the physical properties of early galaxies only one billion year after the Big Bang. In particular, the specific star formation rate (sSFR; the rate of mass build-up) is found to increase rapidly by more than a factor of 50 with increasing redshift at z<2 <cit.> and is expected to keep rising continuously out to z∼6 <cit.>.While the average low-redshift star-forming galaxy is represented by a smooth bulge/disk profile, galaxies in the early Universe are irregular with turbulent clumps in the ultra-violet (UV) light representing regions of on-going vigorous star formation <cit.>.Furthermore, the amount of dust-obscured star formation increases significantly between 0<z<4 where it can be measured directly and appears to begin declining at z>4 based on UV colors, indicating a change in the inter-stellar medium (ISM) properties of galaxies in the early Universe <cit.>. However, the far-infrared (FIR) properties of normal[As opposed to sub-millimeter galaxies (SMGs) and dusty star-forming galaxies (DSFGs) for which rich data sets in the FIR are available out to high redshifts.] main-sequence z>5 galaxies have been difficult to study even with the Atacama Large Millimeter Array (ALMA), mostly providing only one single data point at these wavelengths. The diagram <cit.> is one of the few probes available to-date to study the ISM properties for large samples of normal z>5 galaxies within reasonable observation times on the facilities that are currently available.This diagram connects the ratio of total FIR to UV luminosity (/≡ IRX)[The UV luminosity () is defined as the monochromatic luminosity measured at rest-frame 1600 Å. The total FIR luminosity (, sometimes also called “TIR”) is integrated at 3-1100 μ m.] with the UV continuum spectral slope (β)[The UV continuum slope (defined as f_λ∝λ^β) is derived for 1600 < λ < 2600 <cit.>.]. While the former is a proxy of the total dust mass, the latter depends on the column density of dust along the line-of-sight to the observer that is attenuating the UV light of stars and hence creating a red UV color. The relation is therefore sensitive to a range of ISM properties including dust geometries, dust-to-gas ratios, dust grain properties, and the spatial distribution of dust.Furthermore, the scatter and trends in this diagram are strongly related to evolutionary trends in the ISM. It is further to note here, that the UV slope β also depends on other galaxy properties such as star-formation history (SFH), age of the stellar population, and metallicity. However, in comparison to dust, these effects are shown to contribute little to β in the case of young galaxies in the early Universe <cit.>.Studies of low-redshift galaxies have shown that galaxies of different types populate different regions on the diagram. Specifically, young metal-poor galaxies similar to the Small Magellanic cloud (SMC) show a flatter relation between IRX and β compared to local starburst galaxies, which occupy regions of higher IRX at similar β <cit.>. The differences between these populations of galaxies on the diagram can be attributed to differences in the shape of the internal dust attenuation curves <cit.>, different star-formation histories <cit.>, and/or different dust geometries <cit.>. Studies at z<3 suggest relations similar to local starbursts with no significant evolution <cit.>, however, stacking analyses at z>3 suggest significant deviations from this relation for luminous and young Lyman Break galaxies <cit.>. While the Hubble Space Telescope (HST) and ground based facilities provide us with accurate measurements on the rest-frame UV properties of high-redshift galaxies, the high sensitivity of millimeter-wave interferometers such as ALMA, the Plateau de Bure Interferometer (PdBI), and its successor the Northern Extended Millimeter Array (NOEMA) enable us to push the measurement of the FIR properties of individual galaxies to higher and higher redshifts <cit.>. <cit.> (hereafter ) provided the first study of the diagram at z>5 with a diverse sample of 5.1 < z < 5.7 galaxies observed in ALMA band 7 at rest-frame ∼150 μ m. Together with deep HST near-IR imaging providing accurate UV colors for this sample <cit.>, these studies suggest a significant evolution of the relation and its scatter at z>5 <cit.>. Although some galaxies are found to be consistent with the relation of local starbursts, more than half of the galaxies show a substantial deficit in IRX at a range of UV colors compared to the samples at z<3. Such galaxies are curious because their location on the diagram is difficult to be explained with current models for dust attenuation <cit.> even for very low dust opacities and steep internal dust attenuation curves such as observed in the metal-poor SMC. Because of the poor constraints in the FIR, a common (partial) solution to this discrepancy is to assume increasing dust temperatures towards high redshifts <cit.>. However, also a significantly altered geometric distribution of stars and dust in these galaxies could explain their low IRX values. In this paper we try to understand the causes of such low IRX values in this high-redshift sample by exploring the infrared (IR) spectral energy distribution (SED) and the distribution of dust and stars. We begin by investigating the FIR luminosities of these galaxies which are derived from only one continuum data point at ∼150 μ m and are therefore very poorly constrained. Importantly, the shape of the IR SED is assumed from models that are fit at z<4. In particular, the luminosity weighted temperature T <cit.> that is accounting for mid-IR excess emission is crucial for defining the shape of the IR SED and therefore the FIR luminosity. Hence, it has a significant impact on the relation. Several studies suggest and expect a higher temperature in highly star-forming high-redshift galaxies due to a stronger UV radiation field in low-metallicity environments <cit.> and as a function of various galaxy properties <cit.>. Such an evolving temperature could bring these galaxies in agreement with local relations <cit.>. A direct measurement of this temperature is, however, very difficult to obtain for main-sequence star-forming galaxies at z>5 (see Appendix <ref>). Hence, our best chance to make progress is to investigate statistically correlations between temperature and other physical properties in large samples of low-redshift galaxies together with the study low-redshift analogs of such z>5 galaxies. We stress that this is by no means a bulletproof approach, but it allows us to gain apicture of galaxies in the early Universe and will lead and define follow-up explorations of these galaxies with future facilities that can refine the conclusions of this work. Furthermore, the geometry of the dust distribution in high-redshift galaxies could be substantially different due to their turbulent nature – something that is mostly not considered in recent studies. As we will show in this paper, this can lead naturally to a low IRX value and a large range of UV colors an thus can explain extreme cases in the galaxy population at high redshifts. This work is organized as follows: In Section <ref>, we present FIR measurements and their correlation with other physical properties for local galaxies from the literature along with three z∼0.3 emitting galaxies which appear to be good analogs of high-redshift (z>5) galaxies. We then investigate the impact of different galaxy properties on the relations from a model and observational point-of-view, which will allow us to explain empirically a possible evolution of the relation to high redshifts (Section <ref>). In Section <ref>, we revisit thegalaxies, derive FIR luminosities using the priors on IR SED shape from the low-redshift samples, and present an updated diagram at z∼5.5.Finally, we propose a simple analytical model for the dust distribution in high-redshift galaxies that can describe the observed deficit in IRX by taking into account a non-uniform distribution of dust and stars (Section <ref>). Throughout this work, we assume a flat cosmology with Ω_Λ,0 = 0.7, Ω_m,0 = 0.3, and h = 0.7. Stellar masses and SFRs are scaled to a <cit.> initial mass function (IMF) and magnitudes are quoted in AB <cit.>. Metallicities are quoted in the <cit.> calibration unless specified differently. § FIR PROPERTIES OF LOCAL AND LOW-REDSHIFT GALAXIES One way of understanding the high-redshift universe is to assemble a sample of low-redshift objects which are “analogs" to the high-redshift population.These analogs typically can provide much more detailed data, and can be used to construct priors on the physical properties of high-redshift galaxies.However, a key shortcoming of this technique is that the low-redshift sample is often an analog in only a few parameters, and so extrapolations must be made.As such it is essential to carefully study the low-redshift sample and determine how well it matches the higher redshift one.Here we start with several samples of local galaxies that have a wealth of FIR data from the Herschel observatory and other sources.We explorecorrelations between their FIR properties and other physical parameters which are measurable in our z>5 sample and distinguish them the most from galaxies at lower redshifts.We then study three z∼0.3 emitters that are analogs to the z>5 sample in rest-frame UV properties and explore their FIR properties in comparison to the Herschel sample.To ensure consistency we re-analyze the low-redshift samples using the same techniques we apply at higher redshift.§.§ The sample of Herschel observed local galaxiesIn the following, we consider local galaxies from the KINGFISH sample <cit.>, the GOALS sample <cit.>, and the Dwarf Galaxy Survey <cit.>. Combining these samples allows us to cover a wide range in galaxy properties. Here, we briefly summarize the main properties of these samples. We note that we have removed all galaxies with a known Active Galactic Nucleus (AGN) from these three samples, however, we cannot rule out the presence of heavily obscured AGNs (see discussion in Section <ref>). * The KINGFISH sample consists of 61 nearby galaxies within 30Mpc with 9 << 10 but also contains a handful of less massive dwarf galaxies at 10^7 <cit.>. Their sSFRs are predominantly 0.1-1.0Gyr^-1 and the metallicities in the range of ∼ 8.5-8.8 <cit.>. All KINGFISH galaxies are observed with Spitzer/IRAC (3.6 μ m, 4.5 μ m, 5.8 μ m, and 8.0 μ m) and Spitzer/MIPS (24 μ m and 70 μ m) as well as Herschel/PACS (70 μ m, 110 μ m, and 160 μ m) and Herschel/SPIRE (250 μ m, 350 μ m, and 500 μ m) <cit.>. * The GOALS sample consists of ∼200 of the most luminous infrared-selected galaxies including merging systems in the nearby Universe (LIRGs) out to 400Mpc <cit.>. In particular, about 20 of these are ultra-luminous Infrared Galaxies (ULIRGs). The GOALS galaxies have stellar masses mostly between 10 << 11.5 and are therefore the most massive galaxies of these three samples <cit.>. Furthermore, the galaxies span a wide range in sSFR from 0.1Gyr^-1 up to 10Gyr^-1 for the lowest masses (∼9.5). There are only a handful of metallicity measurements for these galaxies and they are in general around =8.5 <cit.>. Most of the GOALS galaxies are observed by Spitzer/IRAC and Spitzer/MIPS as well asHerschel/PACS and Herschel/SPIRE <cit.>. * The DGS sample consists of 50 low-mass (7 << 10) dwarf galaxies in the nearby Universe out to 200Mpc <cit.>. The galaxies populate mostly low metallicities <cit.> and are therefore similar to high-z galaxies and our low-redshift analogs, with the exception that the DGS galaxies have a lower sSFR (0.1-1.0Gyr^-1) similar to the KINGFISH sample <cit.>. All of the DGS galaxies are observed with Spitzer/IRAC andSpitzer/MIPS as well as Herschel/PACS and Herschel/SPIRE <cit.>. §.§.§ Parameterization of the IR SED and fitting of temperature We measure the FIR properties of the local galaxies using the parameterization introduced by <cit.> and their public IR photometry from 24 μ m to 500 μ m. We will use this parameterization throughout this work to ensure consistent measurements for all the galaxy samples discussed in the following. The <cit.> parameterization combines a single graybody (accounting for the reprocessed, cold to warm emission) with a mid-IR power-law (approximating the warm to hot dust components from AGN heating, hot star-forming regions, and optically thin dust). As shown in their study, this method results in a better fit to the rest-frame mid-IR part of the spectrum by marginalizing over multiple warm to hot dust components of a galaxy (see their figure 1 for an illustration) compared to a single graybody. Mathematically, the parametrization depends on the luminosity weighted temperature (T), the slope of the mid-IR power-law component (α), the emissivity (β_ IR), and a normalization. Note that T represents the average temperature marginalized over all warm temperature components combined in the IR SED and is therefore a luminosity weighted temperature (in the following referred to as just “temperature” for convenience). This temperature should not be confused with the peak temperature T_ peak[E.g., computed using Wien's displacement law T_ peak = b/λ_ peak with b=2898 μ m K.] that is quoted in some other studies and is only equivalent to T only in the case of a single blackbody shaped IR SED <cit.>. In this work we study the luminosity weighted temperature T, however, for convenience, we quote equivalent T_ peak temperatures whenever possible using the relation shown in <cit.>. Importantly, T defines the shape of the IR SED by taking into account the mid-IR emission. It is not only affected by the temperature of the optically thick dust, but also by the opacity of the dust in the ISM of a galaxy as well as dust grain properties, which can significantly change the mid-IR slope α as shown by <cit.> and further discussed in Section <ref>. Therefore changes in temperature T are not only limited to changes in the dust temperature, but also include changes in the opacity of the dust or its geometry. Hence, we refer with T to all these effects describing the shape of the IR SED rather than solely the dust temperature. For the fitting of the IR SED using the above parameters, we fix the wavelength at which the optical depth is unity to λ_0=200 μ m <cit.>. This is somewhat arbitrary as λ_0 is a priori unknown and is likely to change for different samples of galaxies, although <cit.> find a consistent value close to 200 μ m for z>4 SMGs. Using λ_0=100 μ m would result in ∼5K cooler temperatures, but does not change our results for the total FIR luminosity, so the choice is irrelevant as long as it is treated consistently. For all fits, we assume a freely varying β_ IR. However, because β_ IR and T can be degenerate, we also fit the photometry with a fixed β_ IR of 1.6 and 2.0 but we do not find any significant differences in the following results and conclusions.§.§ Low redshift analogs of high-z galaxies In addition to the Herschel samples we consider three galaxies at low redshift (z∼0.3) that were selected as UV analogs to z>5 systems with archival FIR and ALMA observations. Such galaxies are extremes amongst low-redshift galaxies but show very similar properties as high-redshift galaxies in terms of sSFRs, optical emission lines, metallicity, and morphology. Low-redshift analogs, such as “Green Peas” <cit.> or ultra strong emission line galaxies <cit.>, are therefore used by many studies to understand in detail the population of high-redshift galaxies, for which much less observational data exist <cit.>. A very common property of these galaxies are their high equivalent-width (equivalent to sSFR). In fact, as demonstrated by <cit.>, the equivalent-width allows a very clean selection of low-redshift analogs with very similar properties of high-redshift galaxies. The three analogs of high-redshift galaxies (hereafter named as , , and ) were originally selected from the Galaxy Evolution Explorer <cit.> emitter sample described in <cit.> <cit.>. This emitter sample contains galaxies that are spectroscopically preselected to have rest-frame > 15 Å in the GALEX far-UV or near-UV grism.Galaxies with a clear AGN signature were removed based on a cut in FWHM and another cut in the high-ionization optical emission lines (see also Figure <ref>). All three galaxies are in the main Cosmic Evolution Survey <cit.>[<http://cosmos.astro.caltech.edu>] area and thus are imaged by ground and space based facilities in more than 30 photometric bands ranging from UV to radio <cit.>. These are the only galaxies in COSMOS that are analogs of z>5 systems with a wealth of ancillary data in the UV, optical, and FIR. Rest-frame optical spectroscopy is available from the zCOSMOS-bright spectroscopic survey <cit.>[<http://archive.eso.org/cms/eso-data/data-packages/zcosmos-data-release-dr1.html>] and two of the analogs ( and ) are observed in the UV with the Cosmic Origin Spectrograph <cit.> on HST (Scarlata et al., in prep). The three galaxies appear to be good analogs of high redshift systems. Specifically, they show optical emission line properties and properties similar to galaxies at z∼5. The equivalent-widths (= 200-400 Å) are comparable to those measured in z∼5-6 systems using Spitzer colors <cit.> and their specific SFRs based on combined UV and FIR luminosities are between 1.6Gyr^-1 and 8.8Gyr^-1. This is 2-15 times higher than average galaxies on the star-forming main-sequence at z∼0.3 at stellar masses ∼9.0-10.0 <cit.>, but similar to those of z∼5-6 galaxies <cit.>. Furthermore, the /ratios are high (4.5 to 6.0) and /ratios are low (0.025 to 0.060) compared to z∼0.3 galaxies at similar stellar masses, but again comparable to z∼5-6 systems <cit.>. Finally, these line ratios suggest gas-phase metallicities of <8.3 in any calibration, which is 0.3dex below the average metallicity of z∼0.3 galaxies at similar stellar mass but comparable to galaxies at z>3.5 <cit.>. The detailed spectroscopic properties of the three analogs are listed in Table <ref>. §.§.§ HST morphology Figure <ref> shows the optical images by HST's Advanced Camera for Survey (ACS) in F814W (I-band) at 200pc resolution. The bulk of the optical light is emitted from a compact nucleus of less than 0.8kpc in diameter but diffuse components extend out to 6kpc in all of the galaxies. Several clumps of UV light in the diffuse component with same or smaller size than the nucleus suggest regions of vigorous star formation. §.§.§ UV to mid-IR data The UV to mid-IR photometry of the galaxies is extracted from the COSMOS2015 catalog <cit.>[<ftp://ftp.iap.fr/pub/from_users/hjmcc/COSMOS2015/>]. Next to deep imaging in the UV and optical, the catalog includes deep Spitzer/IRAC mid-IR data at 3.6 μ m and 4.5 μ m from the Spitzer Large Area Survey with Hyper-Suprime-Cam <cit.>[<http://splash.caltech.edu>] as well as Spitzer/MIPS 24 μ m (all three galaxies are detected). The photometry has been extracted using the positional priors from the optical bands and reliable de-blending algorithms <cit.>. The UV to optical SEDs are shown in the top and middle panels of Figure <ref>. Each of the galaxies has been observed with Spitzer/MIPS at 70 μ m as well as Herschel/PACS (110 μ m and 170 μ m) and Herschel/SPIRE (250 μ m, 350 μ m, and 500 μ m). Due to the large point spread functions (PSF) and confusion, we choose to measure the IR photometry on a galaxy-by-galaxy basis. In brief, we perform aperture photometry with an appropriate correction to total magnitudes. The uncertainties are based on random parts of background. For galaxies with close neighbors, we apply either Spitzer/IRAC or Spitzer/MIPS positions as priors and use<cit.> to carefully subtract the flux of the neighboring galaxies before performing aperture photometry on the main object. The detailed IR photometry of the three analogs is listed in Table <ref>. §.§.§ Physical properties from optical SED fitting We fit the optical SED of our low-redshift analogs to obtain their stellar masses and rest-frame 1600 Å monochromatic luminosities (). We use templates based on <cit.> and include constant, exponentially declining and increasing star formation histories (SFH) with variable metallicities from 1/20 of solar to solar. The stellar population ages range from 100Myrs to a few Gyrs and dust is parametrized by extinction laws based on local starbursts <cit.> and the SMC <cit.>. However, we found the different extinction laws do not significantly change the measured physical parameters. The redshift is fixed to the spectroscopic redshift during the fitting. We add strong optical emission lines coupled to with variable ratios /Hα, Hα/[We compute via case B recombination and the fitted value assuming a stellar-to-gas dust ratio of unity as suggested by recent studies <cit.>.], and /. Weak emission lines (such as N, S, or He) are added with a constant ratio with respect to for a sub-solar metallicity <cit.>. The GALEX far-UV photometry is corrected for line emission derived from the spectra prior to fitting as we do not include the line in our SED models. The fit is performed using a Levenberg-Marquardt algorithm, as part of thepackage[<https://cran.r- project.org/web/packages/minpack.lm/ index.html>] and errors are obtained from a Monte Carlo sampling taking into account the error in the photometry and models. The UV luminosity () is measured at rest-frame 1600 Å and the UV continuum slope is fitted in the window between 1600 Å and 2600 Å. We perform two fits for accessing the uncertainties in the fitted parameters. First, we fix , /and /to the spectroscopic value and in a second run we leave these as free parameters. The fits are shown in Figure <ref> on the top and middle panels, respectively, and we find an agreement on the order of 25% or better between the photometrically and spectroscopically measured EW and line ratios. We find stellar masses of = 9-9.5, moderate to low dust attenuation (<0.3), and UV luminosities on the order of log()∼10.2. The photometrically derived / ratios (ranging from 1.5 to 3.5) are factors 3-6 higher compared to z∼0.3 galaxies on the main-sequence but fit well the expected / ratios for z∼5 galaxies <cit.>, which is an additional verification of their validity as analogs of high-redshift galaxies. We compare the UV slope β measured from SED fitting with the direct measurement from the observed GALEX and CFHT/u^* fluxes and find a good agreement within the uncertainties. Since we want to study the location of these galaxies on the diagram, we want the measurement of β to be independent from any assumed dust attenuation law. This is not the case for β measurements from SED fitting. We therefore prefer to use the β measurement obtained from the observed fluxes. The physical properties obtained from the optical SED fits are listed in Table <ref>. §.§.§ ALMA measurements The three analogs of high-redshift galaxies have been observed with ALMA in Band 3 (ID: 2012.1.00919.S, PI: Y. Kakazu) centered at an observed frequency of 92GHz, which corresponds to the rest-frame wavelength of emission (2.6mm). The integration times vary between 1255s and 9852s and the number of antennae between 24 and 36 (see Table <ref>). The line observations are performed in “frequency division mode” at a bandwidth of 0.9375GHz in one spectral window and the continuum observations use the “time division mode” with a bandwidth of 2.0GHz in each of the three line-free spectral windows. The synthesized beam sizes are on average 3.5× 2.5. The data are reduced using the standardALMA calibration pipeline. All three galaxies are undetected in the continuum around rest-frame 2.6mm at ≲50 μ Jy at 3σ but we note a tentative detection at the redshift of at a 1-2σ level with a peak emission spatially offset by 1.8 or 7kpc. §.§.§ IR SEDs and temperature of the low-z analogs We estimate the FIR properties of the analogs in the same way as for the Herschel sample (see Section <ref>). The bottom panels of Figure <ref> show the best-fit IR SEDs to the three analogs (best fit parameters are listed in Table <ref>). We find temperatures ranging from 70K to 100K with uncertainties of ∼ 10K (corresponding to 40K to 60K with uncertainties of ∼ 5K in T_ peak). The temperature for is only poorly constrained because only upper limits for the FIR photometry are available. §.§ Correlations of FIR properties with metallicity and sSFR in low-redshift galaxies Metallicity and sSFR are the two most prominent ways high-redshift galaxies differ from lower redshift ones. We therefore begin by investigating the dependency of the shape of the IR SED (parameterized by T) on these physical quantities. Figure <ref> shows thevs. T and vs. T_ peak relation for our local samples with open symbols color-coded by the metallicities of the galaxies. The different symbols indicate the KINGFISH (circles), DGS (squares), and GOALS (diamonds) samples. The metallicities are quoted in the <cit.> calibration for a consistent comparison of the different samples. At intermediate to high total FIR luminosities (≳9) we recover the positive correlation betweenand T (and T_ peak) similar to what was found in studies of luminous FIR galaxies <cit.>. However, contrarily to expected low temperatures at very low FIR luminosities, the FIR faint and metal-poor dwarf galaxies from the DGS sample suggest increasing temperatures at ≲9. This has not be seen in previous studies as these do not go this low in total FIR luminosity. It indicates that galaxies at lowand low metallicity as well as galaxies at highand high metallicity are characterized by a warm IR SED based on our local samples. Figure <ref> shows the (peak) temperature probability densities for our local samples. The KINGFISH sample, containing the most mature and metal-rich galaxies, shows the lowest distribution in T and T_ peak of all the samples <cit.>. The dwarf galaxies in DGS, having significantly lower metallicities, show a T distribution similar to the (U)LIRGs in the GOALS sample with sSFR<2Gyr^-1. Notably, galaxies with the highest sSFR (>2Gyr^-1 in the GOALS sample) show an overall warmer IR SED, i.e., higher T. Taken at face value, this suggests that, in addition to a trend in metallicity, galaxies with higher sSFR have warmer IR SEDs. This is not entirely unexpected since sSFR is correlated with the total FIR luminosity via the SFR, which can introduce a positive correlation with T as shown above. However, strictly speaking, sSFR is more similar to a FIR luminosity density and is therefore the preferred tool over the absoluteto compare galaxies at different redshifts. The three low-metallicity analogs at z∼0.3 show 10-30K (10-20K in T_ peak) higher temperatures compared to the average of galaxies at similar metallicity, such as the dwarf galaxies (Figure <ref>). Hence, they seem to be outliers in the nearby galaxy samples. However, note that the analogs have relatively high sSFR (between 1.6Gyr^-1 and 8.8Gyr^-1) compared to these local samples which is expected to boost their temperature further. Indeed, they show similar temperatures as the GOALS (U)LIRGs at similar sSFRs (Figure <ref>). We also note that the total FIR luminosities of the analogs do not match up with the local samples. Specifically, for their given metallicity, the analogs show ∼1-2 orders of magnitude highercompared to the DGS galaxies. The same trends have been seen in other samples of analog galaxies and are also indicated in our z∼5.5 sample (∼10 at metallicities of ∼7.5-8.0 in the <cit.> calibration similar to the analogs – by definition). This simply means that galaxies with the samebut at different redshifts are not comparable, which is reasonable given their vastly different structure and stellar mass. Instead, the sSFR (a proxy for the FIR luminosity density) or equivalently surface density are better parameters to compare low and high redshift galaxies as well as to provide better analog galaxies <cit.>. Summarizing, the local samples provide evidence that galaxies at low metallicity and high sSFRs are characterized by a warmer IR SED. The UV analog galaxies (extreme cases of low-redshift galaxies but with similar properties as high-redshift galaxies) strengthen these conclusions and suggest that high-redshift galaxies (characterized by low metallicity and high sSFRs) have a warmer IR SED compared to average galaxies at lower redshifts. §.§ Possible reasons for a “warm” IR SED In the previous section, we have outlined evidence that galaxies with low metallicity and high sSFR show a “warmer” IR SED (i.e., higher T andT_ peak) compared to the average population. Ultimately, this may suggests that the average population of high-redshift galaxies (characterized by low metallicity and high sSFRs) have increased temperatures compared to typical galaxies at lower redshifts, which will be important to characterize their FIR properties as it is shown later. In the following, we list possible reasons that can cause a warm IR SED. First of all, the presence of an AGN is one possible cause of a mid-IR excess, but known AGN have been removed from the samples discussed above and the location of our three analogs on the BPT diagram suggest no AGN component. A detailed analysis ofX-ray emission from a dusty star forming galaxy at z∼5.6 (SPT0346-52; T=74K, T_ peak=43K) using Chandra also suggests that its FIR emission originates from vigorous star formation activity instead of an AGN <cit.>. This is strengthened by a recent analysis of a sample of four “Green Peas” (low-redshift analogs of z∼3 galaxies) that arrives with the same conclusion <cit.>. However, the presence of an obscured AGN cannot be ruled out and in fact it can be easily missed by traditional selections as shown in the case of GN20 at z∼4 <cit.>, or other examples at lower redshifts <cit.>. This obscured AGN can even dominate the mid-IR continuum but it is shown by these studies that it has a minor impact on the FIR part of the SED and therefore only a modest effect on the temperature. We have found evidence that galaxies with low metallicity as well as high sSFR have increased T. This suggests that there might be at least two mechanisms that cause such a warm IR SED. These might be working at the same time, especially in low-metallicity and high sSFR analogs and likely also high-redshift galaxies. On one hand, a high sSFR is indicative of a high density of intense UV radiation fields originating from young star-forming regions that are heating up the surrounding ISM. In the local Universe, this might be the case for compact FIR luminous galaxies such as the ULIRGS in GOALS (Figure <ref>). At high redshifts, however, this could generally be the case as these galaxies are very compact and reveal a higher population averaged sSFR. This picture is generally supported by the detection of extended emission from the warm ISM around local LIRGs <cit.> and in samples of high-redshift galaxies (Section <ref>). On the other hand, a likely explanation for the warm IR SEDs could be a lower optical depth in the ISM due to lower dust column densities at lower metallicities. As shown by <cit.>, the mid-IR shape of the SED is not only affected by the temperature but also by the mean optical depth of the ISM and the dust mass <cit.>. In detail, for optically thin dust, the exponential mid-IR black-body tail is modified to a power-law and the peak wavelength is shifted blue-wards, both causing an increase in T as well as T_ peak. The same trend but weaker is expected for a decreasing dust mass. At the same time, harder UV radiation might be prevalent in low-metallicity environments with massive stars caused by less efficient ISM cooling. Interestingly, <cit.> suggest also a higher number of X-ray sources at lower metallicity that could be responsible for heating the ISM. In the local Universe, these effects are expected in metal-poor and FIR faint dwarf galaxies (Figure <ref>). At high redshifts, this could be commonly the case as these galaxies have metallicities significantly below solar. § THE DIAGNOSTIC TOOL The general idea behind the correlation is that dust in the ISM and circum-galactic medium around galaxies absorbs the blue light from young O and B stars and re-emits it in the FIR. Therefore, the ratio of FIR to UV luminosity (/= IRX) is sensitive to the total dust mass and the geometry of dust grains. On the other hand, the UV color (measured by the UV continuum spectral slope β) is sensitive to the line-of sight opacity <cit.>[Note that β also depends on the age of the stellar population, the star-formation history, and metallicity. However, for young stellar populations as it is the case in young star-forming galaxies especially at high redshifts, the intrinsic UV slope is expected to be invariant <cit.> and hence we can assume β to be a good proxy for line-of-sight opacity.]. The diagram combines these two measures and can therefore be used for studying the dust and gas properties and distribution in galaxies with a minimal amount of input. §.§ Analytical models to explain in typical local to intermediate redshift galaxies <cit.> <cit.> describe a simple analytical model for the absorption of starlight in the ISM of galaxies, which can explain the location of typical galaxies on the diagram up to z∼3-4 where detailed data exist. The model assumes dust in thermal equilibrium that is distributed in the ambient ISM in one of the following three geometrical configurations: (i) a uniform foreground screen characterized by a single optical depth τ_λ^sc, (ii) a “mixed slab” characterized by a uniform mixture of dust and stars and optical depth τ_λ^sl, and (iii) a Poisson distribution of discrete clouds characterized by an optical depth per cloud (τ_λ^c) and the average number of clouds mixed with stars along the line-of-sight to the observer (n̅). Moreover, the models assume that the young stars are enshrouded in birth-clouds that dissolve after a finite lifetime of t_ bc=10Myrs <cit.>. The total transmission for UV light is therefore time-varying, namely the product of the transmission function through the birth clouds and the ambient ISM for t<t_ bc and the transmission function alone through the ISM otherwise. These different models successfully explain the typical galaxy population on the diagram. Specifically, <cit.> find that the model featuring discrete clouds in the ISM represents best the IRX and UV color distribution of typical local starburst galaxies by assuming a simple power-law relation τ_λ^c= τ_V^c (λ/5500 Å)^-1.3 for the wavelength dependence of the optical depth of the ambient ISM. Such simple forms for τ_λ with a logarithmic slope of -1.3 are expected from various detailed measurements of the dust extinction curves in the Milky Way galaxy <cit.>, the Large Magellanic Cloud <cit.>, local starburst galaxies <cit.>, and even normal star-forming galaxies at z=2-4 <cit.>. For the birth-clouds, a similar optical depth is assumed with τ_λ^bc= 0.7 (λ/5500 Å)^-0.7, but we note that the exact relation does not impact the following conclusions. We show such a model in Figure <ref> for a stellar population of 300Myrs and n̅ running from 1 to 10 with τ_V^c=0.5 (orange circles) and τ_V^c=0.1 (green squares)[We assume a simple stellar population published with<cit.> for half-solar metallicity, a constant star-formation history, and a <cit.> IMF.]. This simple model does successfully explain galaxies with similar dust properties as the local starbursts but it fails to reproduce galaxies with lower IRX values at a fixed UV color even for the lowest cloud optical depths (i.e., τ_V^c). This part of the parameter space can be covered within this model by steepening the dust extinction curve, i.e., change the form of the wavelength dependent optical depth τ_λ^c. The blue triangles in Figure <ref> show the cloud model with the more complex τ_λ^c of the metal-poor SMC parameterized by a 3-fold broken power-law <cit.> and assuming n̅=3 and τ_V^c up to 0.05. With this modification, the model can reproduce galaxies with lower IRX values at a given UV color for SMC-like dust. At the same time, this shows how we can study the dust properties of galaxies with this very simple model of dust absorption and a minimal amount of data; the location of galaxies on the diagram tells us about their total amount of dust, the optical depth of their clouds in the ISM, and the wavelength dependence of their dust attenuation curve. These properties ultimately depend on other physical properties of the galaxies, for example their metallicity as shown in the case of SMC-like dust. §.§ Correlation between location on the diagram and physical properties The diagram has also been studied thoroughly from an observational point of view and it is found that the galaxy properties change significantly as a function of position on this diagram <cit.>. For example, more mature, metal-rich, and dust-rich galaxies are expected to show high values of IRX, thus are found in the upper part of the diagram. Such galaxies are pictured as compact, dust-enshrouded, IR luminous star-forming systems that have likely experienced a recent starburst after a merger event <cit.>. The blue colors of some of these galaxies can be explained by a patchy dust screen <cit.> or tidally stripped unobscured young stars or a small spatially unresolved and unobscured star-forming satellite as suggested by some IR luminous (>12) local galaxies <cit.>. An alternative scenario that does not involve starbursts or mergers is predicted by hydrodynamical simulations. These suggest that isolated normal main-sequence galaxies with <12 can populate the upper left part of the diagram naturally due to their high gas fractions and high metal enrichment <cit.>. The observed increase in gas fraction towards higher redshifts <cit.> would make such a configuration likely for the most metal rich galaxies in the early Universe. On the other hand, younger, metal-poor, and dust-poor galaxies are expected to show low values of IRX and blue UV colors and thus populate the lower parts on the diagram (see Figure <ref>). They could be characterized by a steeper dust attenuation curve as it is seen in the metal-poor SMC, which causes a flatter relation for these galaxies <cit.>. Galaxies in the early Universe are expected to be young, dust-poor, and less metal enriched on average <cit.>, thus are expected to reside on the lower parts of the diagram as predicted by the <cit.> model with a SMC-like dust attenuation curve. This has been directly observed in some FIR detected z∼3 galaxies <cit.> and there are also hints of this in our small sample of z∼0.3 analogs; the analogs and (both sitting on the relation of local starbursts) have ∼ 0.2dex higher metallicities than , which is located on the relation of the SMC. From the correlation with metallicity, we would also expect a correlation between position on the diagram and IR SED shape (Section <ref>). As detailed in Section <ref>, a warm IR SED can be due to heating of the ISM by a high amount of star-formation or due to a low dust optical depth in the ISM as shown by <cit.>. The latter is supported by the simple dust model discussed in Section <ref>, which predicts a very low optical depth (τ_V^c<0.05) for the clouds in galaxies close to the SMC relation. Extrapolating our knowledge gained from our low-redshift samples, we expect galaxies in the early Universe to show warm SEDs as well as high rates of star formation, which will make them appear close to the SMC relation. § THE DIAGRAM AT Z∼5.5The advent of ALMA has allowed us to begin populating the diagram at z>5 <cit.>. presents today's largest and most representative sample of normal main-sequence z∼5-6 galaxies with detected emission and continuum detections or limits at 158 μ m along with robust UV measurements <cit.>.From the reasoning in Sections <ref> and <ref> we expect that most of the high-redshift galaxies reside in the lower part of the diagram, close to the relation of the SMC. However, we also expect a large variation in the metallicity and star-forming properties of high-z galaxies <cit.>. We therefore also expect a significant scatter in . Thesample consists of 9 star-forming Lyman Break galaxies at 1-4 L^*_ UV (the characteristic knee of the galaxy luminosity function) and one low luminosity quasar (HZ5) at redshifts 5.1 < z < 5.7. All galaxies are located in the 2 square-degree COSMOS field and therefore benefit from a multitude of ground and space-based photometric and spectroscopic data. The galaxies are spectroscopically selected via their rest-frame UV absorption features from spectra obtained with the Deep Extragalactic Imaging Multi-Object Spectrograph <cit.> and represent a large range in stellar masses, SFRs, and UV luminosities at z∼5.5.A detailed analysis of their DEIMOS optical spectra and Spitzer photometry properties suggests gas-phase metallicities around ∼ 8.5, which is representative of the average metallicity of ∼10 star-forming galaxies at z∼5 <cit.>. All galaxies have been observed with ALMA at rest-frame ∼150 μ m and are detected in emission at 158 μ m <cit.>. Four of them (HZ4, HZ6, HZ9, and HZ10) are detected in continuum. Furthermore, HZ6 and HZ10 have been detected in line emission and continuum at 205 μ m <cit.>. Both the low luminosity quasar (HZ5) as well as HZ8 have a detected companion (HZ5a and HZ8W) at their corresponding redshifts without significant detection in ground-based near-IR imaging data. HZ6 is a system of three galaxies (HZ6a, HZ6b, HZ6c)[These are called LBG-1a through LBG-1c in <cit.> and <cit.>.] with detected emission and 158 μ m continuum between HZ6a and HZ6b. HZ10 has a low luminosity companion (HZ10W) that is, however, not spectroscopically confirmed in either optical or FIR lines. All UV properties (morphology, luminosity, UV color) have been derived accurately from deep near-IR imaging that has been recently obtained by HST and is explained in detail in <cit.>. Table <ref> provides a summary of the UV and IR properties of thesample. §.§ FIR luminosity of high-z galaxies The IR part of the SED is essential for studying the dust properties of galaxies. Although ALMA provides us with the necessary tools to observe the FIR part of theelectro-magnetic spectrum in high-redshift galaxies, it is very poorly constrained in normal (i.e., typical main-sequence) high-redshift galaxies due to their faintness, which makes such observations time consuming (see Appendix <ref>). While good wavelength coverage in the FIR exists for a handful of bright sub-mm galaxies or FIR luminous lensed systems at z>5 <cit.>, only one continuum data point at rest-frame 150 μ m (close to the line) is commonly measured for typical main-sequence galaxies at these redshifts. Particularly, the total FIR luminosity (), one of the most important parameters to constrain the location of galaxies on the diagram, is uncertain by a factor >3 because of the poorly constrained IR SED <cit.>. The current lack of accurate studies of the IR SEDs at high redshifts makes it necessary to assume an SED shape for these galaxies. The temperature T is a primary parameter for characterizing the IR SED shape in our parameterization (see Section <ref>). The total FIR luminosity at z>5 is usually obtained by integrating over an IR SED assuming priors on temperature and other parameters from the z<4 universe, which is then normalized to the 158 μ m continuum measurement of the galaxies. These parameters are typically T=25-45K (T_ peak∼ 18-28K) and a range of α and β_ IR corresponding to the average temperature of local (U)LIRGs (see ).However, as shown in Section <ref> by correlations in our local galaxy samples and analogs, there is evidence that this assumption might be incorrect for the average population of star forming galaxies at z>5 which have lower metallicities and higher sSFRs[Note that the constant background temperature emitted by the Cosmic Microwave Background (CMB) is increased by a factor of seven to ∼19K at z=6. However, we expect this effect to increase the temperatures by less than 5K on average at these redshifts <cit.>, which is small given the uncertainties of our measurements.]. Figure <ref> illustrates the change in total FIR luminosity (Δlog()) if the temperature T or T_ peak is underestimated by the typical difference between average z<4 (U)LIRGs and the low-metallicity and high sSFR analogs in our local samples.This difference can be up to ∼40K (T=35K instead of T=75K)[This corresponds to a difference between T_ peak∼ 23K and T_ peak∼ 45K, i.e., Δ T_ peak∼ 22K.], which would lead to a significant underestimation ofof ∼0.6dex as shown by the color gradient and black contours. The figure also shows that T is the main parameter of uncertainty as Δlog() only changes by ± 0.04dex over a large and conservatively estimated range of mid-IR slopes α and emissivities β_ IR. The assumption of a warmer IR SED of high-redshift galaxies is supported by our study of local galaxies and low-redshift analogs, but also by several direct measurements in the literature. * <cit.> stack FIR SED of z∼3 galaxies and find peak temperatures T_ peak∼ 30-40K (T ∼ 45-65K), which are at the warm end of the distribution of temperatures of local galaxies. * <cit.> measure the temperatures of Herschel detected star-forming galaxies at 1.5 < z < 3.0 and find peak temperatures of 35K < T_ peak < 50K (55K < T < 85K) in good agreement with our analogs. Also, <cit.> <cit.> and <cit.> <cit.> find temperatures of T_ peak∼ 40K (T ∼ 65K) for average z∼4 and one galaxy at z∼7.5, respectively. * <cit.> <cit.> present several strongly lensed galaxies at z∼5-6 that are detected and selected in the sub-mm by the South Pole Telescope (SPT) survey <cit.> and observed between 100 μ m and 3000 μ m by SPT, ALMA, APEX, Herschel, and Spitzer.We re-fit the FIR SEDs of these galaxies using the same method as our local samples and analogs and find temperatures of 60K < T < 80K (37K < T_ peak < 48K), which is in good agreement with our local analogs (see Table <ref> and black arrows in Figure <ref>). * The analysis of the starburst AzTEC-3 at z=5.3 <cit.> also results in a temperature of 88^+10_-10K (T_ peak∼51K) and an SED similar to the low-redshift analogs. * Finally, <cit.> extended the FIR wavelength coverage of HZ6 and HZ10 from thesample by measuring fluxes in the two sidebands at 158 μ m and 205 μ m independently with ALMA. Assuming a uniform prior for temperature (10-100K, i.e., 5-60K in T_ peak) and β_ IR=1.7±0.5, they derive temperatures of 60^+35_-27K (T_ peak∼35K) and 36^+25_-10K (T_ peak∼25K) for HZ6 and HZ10, respectively. As the authors discuss, these temperature measurements are very uncertain as the current data does not cover wavelengths near the IR peak that are sensitive to warm dust. An additional measurement at bluer wavelengths (e.g., rest-frame 122 μ m, see Appendix <ref>) would increase the confidence in the estimated temperature. Following the above reasoning, we re-compute the total FIR luminosities of the 12 z∼5.5 systems fromassuming a temperature prior of 60K < T < 90K (T_ peak∼ 35-50K) for the IR SED. As in , we create IR SEDs using the <cit.> parameterization that we normalize to the observed 158 μ m continuum emission of the galaxies, which is the only data-point available in the IR. The result is marginalized over a grid of mid-IR slopes and emissivities (1.5 < α < 2.5, 1.0 < β_ IR < 2.0).The total FIR luminosity is derived by integration between 3 μ m and 1100 μ m and the errors are derived from the uncertainty in the photometric measurement at 158 μ m and the marginalization over the assumed range in α and β_ IR. We list the updatedvalues in Table <ref>, which, as expected, are increased by ∼0.6dex compared to the previous results assuming a temperature prior with 25K < T < 45K (T_ peak∼ 18-28K) as in . §.§ Constraints on metallicity and sSFR as derived from Spitzer colors The strong emission lines in high redshift galaxies can alter their observed photometry that in turn allows us to study them. In particular, Spitzer observations allow us to constrain the fluxes of strong optical lines such as and of galaxies at z>3 and thus can put strong constraints on their (specific) SFR, metal content, hence evolutionary stage of these galaxies <cit.>. Figure <ref> shows the expected [3.6 μ m]-[4.5 μ m] and [4.5 μ m]-[5.8 μ m] color in the redshift range 5.1 < z < 5.8 for different EWs and /(∝ /Hβ) emission line ratios[We assume templates from the <cit.> library with half-solar (stellar) metallicity and a 500Myr old constant SFH. As shown in <cit.>, the choice of these parameters, as long as reasonable, has little impact on the intrinsic model color (< 0.15mag for z > 5).]. The black symbols show the dust corrected[For dust correction we apply the maximum of thecalculated directly from / using the relation between IRX and A_1600 <cit.> and from β using the relation between A_1600 and β <cit.>.] colors of the galaxies in thesample with their names indicated. Note that while all galaxies have reliable measurements at 3.6 μ m and 4.5 μ m, only three galaxies have measured 5.8 μ m fluxes due to the only shallow coverage at 5.8 μ m. Furthermore, the latter measurements have large uncertainties and are therefore are only meaningful for HZ1 and HZ6. For galaxies at 5.1<z<5.3, the [3.6 μ m]-[4.5 μ m] color is a good estimator of the equivalent-width <cit.>, , while at higher redshifts it becomes degenerated with the / ratio. Both Spitzer [3.6 μ m]-[4.5 μ m] and [4.5 μ m]-[5.8 μ m] colors suggest > 500 Å integrated over the three sub-components of HZ6, which could indicate a recent starburst event maybe induced by the tidal interaction between these three galaxies. In comparison, these colors suggest < 500 Å for HZ1, HZ7, and HZ8. In addition, we estimate /≲1 (or /<2.86) for HZ9 and HZ10 and /≳1 (or />2.86) for HZ1, HZ2, and HZ4 from their [3.6 μ m]-[4.5 μ m] colors. Assuming all these galaxies sit on the BPT ”main-sequence“ locus of high-redshift galaxies <cit.>, we can relate the /line ratio to /that is a proxy for the gas-phase metallicity. Hence, we would expect /> -0.6 for HZ9 and HZ10, which translates into a gas-phase metallicity of ≳ 8.5 according to <cit.>. The same reasoning leads to ≲ 8.5 for HZ1, HZ2, and HZ4. Note that these metallicity estimates are in good agreement with the strong UV absorption features seen in these galaxies <cit.>.Overall this suggests that low-metallicity, high sSFR low-redshift galaxies are indeed good analogs for high-redshift systems and that we should be adopting higher temperatures for the FIR SEDs.Interestingly, the exception is HZ10, which red Spitzer colors and rest-UV spectrum (Section <ref>) indicate a more mature system with higher metallicity and lower sSFR in line with the cooler temperature estimated by <cit.> and being the strongest FIR continuum emitter in thesample. §.§ Comparison of UV, FIR, and morphology The galaxies inare all resolved in the ALMA FIR line and continuum observations at a resolution of 0.5, which allows, together with their high-resolution near-IR HST images, a spatial study of rest-frame UV (a proxy for unobscured star formation) and FIR continuum (a proxy for the dust distribution and obscured star formation) in typical z>5 galaxies for the first time (Figure <ref>). The FIR continuum detected galaxy HZ9 shows no offset between UV and FIR emission, indicative of a dust-obscured compact central star-formation region. In contrast, the FIR emission of HZ4 and HZ10 is offset by up to by up to 5kpc from the location of the unobscured star formation, which could indicate a substantial re-distribution of dust clouds due to UV radiation pressure, tidal interactions by a recent or on-going merger event or the production and growth of dust caused by a recent starburst <cit.>. Such an offset of FIR continuum and unobscured star formation is also seen in the two local analogs (, ) with the lowest metallicities and lowest dust content (i.e., IRX) as shown in the upper panels of Figure <ref>. Emission from singly ionized carbon is commonly originating from the photo-dissociation regions (PDRs) on the surfaces of molecular clouds. The luminosity () is therefore used as a proxy for the star formation in galaxies, therefore it should coincide well the locations of star formation <cit.>. The sample of , however, shows offsets between and UV emission of up to 5kpc as well as a very extended emission over 5-10kpc, a factor of ∼5 larger than the typical UV size the galaxies (Figure <ref>). The extended emission could indicate that a non-negligible fraction of is originating from the warm neutral gas in the diffuse ISM that is heated by the strong UV radiation of young massive stars in high-redshift galaxies <cit.>. Furthermore, recent hydrodynamical simulations suggest that offsets of up to 10kpc between emission and regions of on-going star formation are expected in young primeval galaxies at high redshift due to their strong stellar radiation pressure <cit.>. Overall, the FIR, , and UV morphologies would support a picture of a turbulent, low dust-column density ISM with a greater optically thin fraction which would cause a higher observed temperature.This is consistent with our analysis of the low metallicities and high sSFRs local galaxies (Section <ref>) and could suggest a change in the relation between SFR andfound in galaxies at lower redshifts <cit.>. Alternatively, the offsets described above could be due to differential dust obscuration as it is commonly observed in SMGs and DSFGs <cit.>. For example, is much less affected by dust obscuration and can therefore be detected at places where the UV light is completely absorbed, i.e., places of obscured star formation. On the other hand, the must then be destroyed or expelled at locations of unobscured star formation. §.§ Updated IRX-β diagram of z∼5.5 galaxies Figure <ref> shows the diagram at z∼5.5 using the HST-based β measurements <cit.> andderived for a maximally warm IR SED prior (60K < T < 90K, T_ peak∼ 35-50K) for the galaxies in . The FIR continuum detected high-redshift galaxies are shown as solid red symbols, while the non-detections are shown as empty red symbols with arrows. FIR detected companion galaxies are shown as empty red symbols. Previous measurements (if available) that assume a cooler IR SED prior of T=25-45K (T_ peak∼ 18-28K) are shown as light red points. If the warmer IR SED priors are correct, this would push the galaxies up by ∼0.6dex in IRX. This brings the galaxies at low IRX in better agreement with known dust properties measured in local starbursts <cit.> <cit.> and models adopted for the metal-poor SMC <cit.>. We note that the applied warm IR SED prior is motivated by the extrapolation of trends found in local galaxy samples as well as three low-redshift analogs which we think are reasonable representatives of a large fraction of high-redshift galaxies. However, the true temperature distribution of typical high-redshift galaxies is still unknown and we therefore argue that our temperature prior is a very reasonable upper limit and so are the updated IRX values. Taking the early measurements by <cit.> as well as the trends in the local sample at face value, a temperature gradient across the diagram is not unexpected. For example, lower temperatures for HZ9 and HZ10, which would reduce their IRX and bring them in better agreement with the local starburst and the <cit.> relation, are entirely possible. More direct measurements of the IR SED at high redshifts and larger samples of low redshift analogs are necessary to study such effects. We find that galaxies at z∼5.5 occupy a large fraction of the diagram with significant scatter, which indicates a large variety in their physical properties already 1 billion years after the Big Bang. This scatter has to be understood in order to use therelation to predict the dust properties of high-redshift galaxies from UV colors only as it is commonly done. The current sample of galaxies is too small to make such statistical predictions as well as point out the precise trends on thediagram as a function of galaxy properties. In the following, we can therefore only derive a patchy understanding of how galaxies behave in relation to β and IRX. We find galaxies with blue UV colors (β< -1.5) and large IRX values on or above the curve for local starbursts (HZ9, HZ10). As discussed in Section <ref>, these are likely amongst the most metal-rich and maturemain-sequence galaxies at these redshifts. The blue UV color of these galaxies could be explained by a recent perturbation of the dust clouds by tidal interactions or a patchy optically thick dust screen (Section <ref>). The former could be the case for HZ10 showing a large offset between UV and FIR emission, while the latter might be the case for HZ9 for which UV and FIR emission coincide (Section <ref>). We note that HZ4, which is also elevated somewhat above the location of local galaxies but not as much as the others, is expected to be less metal enriched. Also, its morphology (Figure <ref>) shows that its FIR emission is offset in a tail pointing to north-east. The large fraction of unobscured UV emission along the line-of-sight causes its blue UV color (β≲ -2) but its substantial dust mass causes a high IRX value. Next to galaxies showing increased IRX values, we find galaxies with low in IRX values (even upper limits) and red UV colors (e.g., HZ6a, HZ7, HZ8, or HZ8W). This is even the case for a maximally warm IR SED prior that is assumed here and would be even more significant if no temperature evolution is assumed. Such galaxies are expected to have a significant line-of-sight dust attenuation but only little dust FIR emission, i.e., a low total dust mass. The recent HST observations provide very accurate β measurements of these galaxies as shown by the simulations in <cit.>. Therefore we conclude that these galaxies have a deficit in IRX and are not biased in β.Also, the UV spectra and Spitzer colors strongly suggest that these galaxies are young and not dominated by old stellar populations that could cause a red UV color <cit.>. These galaxies are curious because such a configuration of IRX and β is difficult to explain with current models of dust attenuation in galaxies. As shown in Section <ref>, the simple analytical model of <cit.> cannot cover that part of the parameter space even assuming low dust opacities and a steep SMC-like dust attenuation curve. From the Spitzer colors (Section <ref>) as well as UV spectra <cit.>, we would expect these galaxies to be primeval and metal-poor. The large offsets between emission and star forming regions and the extent of emission could be indicative of vigorous star formation, strong stellar radiation pressure, and substantial heating of the diffuse ISM. We suggest that the enhanced turbulence in these galaxies paired with merger events significantly alters their spatial dust distribution. In the next section, we propose an updated model for dust absorption in the ISM of these galaxies that can explain the location of these peculiar galaxies on the diagram by taking into account geometrical effects of the dust and star distribution. §.§ Dust model for FIR faint high-z galaxies: Separating clouds from stars As discussed in Section <ref>, a simple analytical model for dust absorption in galaxies with well mixed dust and stars in thermal equilibrium <cit.> is able to explain the majority of galaxies on the diagram. However, it is not able to explain the FIR faint galaxies at very low IRX at a given UV color found at z>5 even for low dust opacities and a steep SMC-like dust attenuation curve. In order to explain the location of these galaxies, the dust clouds have to be in a configuration that allows them to attenuate the UV light enough to cause the red β but at the same time the FIR emission has to be decreased by 0.5-1.0dex with respect to SMC-like galaxies. We propose that this can be achieved by a spatial separation of the clouds and the star-forming regions that naturally occurs as the birth clouds are disrupted by the radiation pressure of the stars and turbulences in the ISM. Instead of letting the birth-clouds disappear after 10Myrs (the typical life time of O and B stars), we modify the <cit.> prescription such that the birth-clouds are separated from the star-forming regions by δ s(t) = t×δ v with t<10Myrs.The total FIR luminosity is decreased roughly by a factor of ∝ 1/δ s^2 (where δ s is the distance between star-forming region and cloud) due to the smaller angle along the line-of-sight covered by the cloud, while the UV color from the obscured regions is not significantly changed. We assume that initially, the star clusters have the same velocity as the birth-cloud. After that, the cloud is subject to hydrodynamic forces and pressures, e.g., caused by the radiation of young stars or merging events, which causes the clouds to run into the nearby low-density warm ISM. The velocity change of the clouds can be estimated by a momentum conservation argument and depends on the column density ratio of the clouds and the surrounding ISM as well as the relative velocity v of the external medium δ v ≈( C_ ISM/C_ cloud) × v. The turbulent velocity dispersion of the ISM (σ_ ISM) provides a reasonable estimate for v. From the emission line widths of the z∼5.5 galaxies measured bywe find ∼100km/s and we take this value to approximate σ_ ISM in the following. Note that a significant part of the line width may be due to rotation as it is suggested for HZ9 and HZ10 <cit.>. Therefore, this approximation might lead to an upper limit for σ_ ISM. Our simple modified model is shown in Figure <ref> by the dashed and dotted blue lines for C_ ISM/C_ cloud of 0.01 and 0.1, respectively, assuming an SMC-like wavelength dependence of the optical depth of the ambient ISM. Our model can quantitatively explain the low IRX values of the z∼5.5 galaxies and also can naturally lead to the large observed range in β at a fixed IRX value. Depending on the geometry and the viewing angle of the observer, the dust clouds might be spatially offset from the line-of-sight and the UV light can escape the galaxy without being attenuated, which would cause blue UV colors but very similar IRX values. Qualitatively, this model is also consistent with other observed characteristics of high-redshift galaxies such as the increased turbulent motion, increased radiation pressure, merger events, and strong gas inflows. In all cases, this leads to dust that is not well mixed with the stars, with isolated ISM clouds spatially offset from the regions of star formation. At the same time, low opacity dust screens due to tidal disruption or from the diffuse ISM can be created in front of the stars, producing the observed red UV colors. This setup could create lines-of-sight with attenuated UV light of the star-forming regions as well as ones that are clear, at a fixed IRX. In fact we see hints of such behavior in local galaxies.NGC 5253, NGC 1568, and Zwicky 403 are examples of local galaxies that have similar morphological properties, although the z>5 population is likely more extreme <cit.>. All three galaxies show confined isolated clouds of gas and dust along with clear lines of sight to the star forming regions with a complex morphology. §.§ Other possible geometric configurations There are other geometric configurations that can cause the observed low IRX values and red UV spectral slopes. Radiation pressure and tidal forces can stir up the ISM in these young primeval galaxies and could result in a large fraction of dust and gas that is expelled into their circum-galactic regions. The low luminosity quasar HZ5 suggests evidence for an extended halo of dust and gas out to several kilo-parsecs around its confirmed (foreground) companion HZ5a. The spectrum of the quasar shows absorption features at the redshift of HZ5a as well as a narrow emission line possibly caused by absorption in the foreground galaxy <cit.>. Such circum-galactic dust can attenuate the UV light of a companion galaxy in the background and cause a red β. At the same time, its IRX value is determined by the foreground galaxy's dust. The galaxy HZ8W (a confirmed companion of HZ8) could be an example of such a geometrical configuration. It is characterized by a red spectral slope of -0.1 and an upper limit in IRX of only 0.2. The UV light of this galaxy could be substantially attenuated by a foreground screen of dust and gas originating from the larger and more massive galaxy HZ8 in several kpc projected distance. This is also indicated by its very extended emission. The HZ6 system could show a similar configuration. §.§ Correlation with stellar mass Several studies indicate a relation between IRX and stellar mass for galaxies at z<4 <cit.>. The growing samples at z>4 show that this relation might break down at higher redshifts <cit.>. Specifically, it is found that these galaxies show a lower IRX at a fixed stellar mass than expected from the relations at lower redshifts. Part of the reason for this discrepancy is the unknown IR SED of the galaxies <cit.>. Figure <ref> shows the IRX vs. stellar mass diagram for the z=5-6 galaxies assuming a warmer IR SED prior of T=60-90K (T_ peak∼ 35-50K) together with data and relations at lower redshift taken from the literature. Even with a warmer IR SED, the FIR undetected galaxies still fall below the expected IRX vs. stellar mass relation found at z<4. Assuming the stellar masses are robust, this suggests that the stellar mass becomes uncorrelated with IRX (i.e., total dust) at z>5 similar to the relation between IRX and β. This could be due to mergers, which would increase the stellar mass without necessarily changing the IRX, or a modified geometric distribution of dust as discussed here. Alternatively it could be that at these redshifts the production of dust is lagging behind the mass growth of the galaxies which happens on very short timescales. As mentioned in Section <ref>, such defined relations between IRX and β as well as stellar mass are important to predict the dust properties of large samples of high-redshift galaxies from quantities measured in the rest-frame UV and optical. Unfortunately, we currently lack large enough galaxy samples to define such relations accurately. Large ALMA programs targeting the FIR of high-redshift galaxies will be crucial in the future to progress. § SUMMARY AND CONCLUSION Observations of the relation across cosmic time is important for understanding the evolution of ISM and dust properties of galaxies in the early Universe. Recent studies of the relation of main-sequence galaxies at z>3 have reported a significant evolution of this relation. Because the UV colors are accurately measured by HST, this evolution is attributed to a deficit in IRX at fixed β compared to galaxy samples at lower redshifts with similar stellar masses. These low IRX values and large ranges in UV color are curious, since they cannot be explained by standard models of dust attenuation in galaxies. The total FIR luminosity of these high-redshift samples is very poorly constrained often with only one data point at these wavelengths. A common solution to this discrepancy is therefore an evolving temperature, which is the most important parameter to define the IR SED. On the other hand, a different geometric distribution of dust and stars can be responsible for the evolution of the relation to high redshifts. In this work, we have investigated both ideas and find that also with a “maximally” warm IR SED the distribution of dust and stars likely needs to be altered to explain the location of some of the z>5 galaxies on the plane. We first infer the IR SED shape of high-redshift galaxies by using correlations of the luminosity weighted temperature (in this work referred as “temperature”) with metallicity and sSFR in samples of nearby galaxies, which we extend with the extreme cases of three z∼0.3 analogs of z>5 galaxies. This provides us with good evidence that the IR SEDs of high-redshift galaxies show a warm emission component as suggested by temperatures of up to T=60-90K (T_ peak∼ 35-50K), which is significantly warmer than typical z < 3 galaxies (T ∼ 25-45K, T_ peak∼ 18-28K). We believe the warmer IR SEDs are due to a combination of lower ISM optical depth and denser and stronger UV radiation fields associated with more intense star formation and low metallicity environment.A comparison of the UV, , and FIR morphologies along with metallicities in high and low redshift systems may support this picture. Assuming such maximally warm IR SEDs would increase the FIR luminosity (and therefore IRX) by up to 0.6dex (factor of 4) with respect to previous measurements that assume the on average cooler IR SEDs of local galaxies. This reduces the deficit in IRX and brings most of the galaxies at z>5 in agreement with the relation measured at lower and local redshifts, however, with significant scatter. Even with the assumption of such warm IR SEDs is correct, a population of galaxies with red UV colors and low IRX remains. We suggest that the geometric distribution of dust and stars is altered in high-redshift galaxies. Common dust attenuation models assuming a well mixed ISM <cit.> are not able to explain the galaxies with low IRX values and large range in UV colors. The large spatial offsets between star-forming regions and emission as well as the Spitzer colors suggest that these galaxies are young, metal-poor, and likely have enhanced cloud motions due to stellar radiation pressure. Based on this, we propose an updated dust attenuation model in which the dust clouds are separated from the young stars by their radiation pressure. We show that such a separation can be easily achieved within the lifetime of massive O and B stars in young star-forming galaxies at high redshifts. Such a model can successfully reproduce the location of the FIR faint galaxies with low IRX. Furthermore, it can naturally reproduce the large scatter in UV colors due to viewing-angle effects. We note that the evidence of a warmer IR SED in high-redshift galaxies is indirectly derived from correlations in local galaxies and three exemplary analogs of z>5 galaxies and the fact that high-redshift galaxies are metal-poor and have high sSFRs. However, we stress that a direct measurement of, e.g., the temperature T is absolutely crucial to verify these trends. Unfortunately, such measurements turn out to be time consuming and only possible for some of the brightest high-redshift galaxies we know to-date (see Appendix <ref>). The first estimates of the temperature of two normal z∼5.5 galaxies suggest large variations, highlighting the importance of such direct measurements <cit.>. The two galaxies occupy two different regions on the diagram and likely are different in terms of metallicity and sSFR despite having comparable UV emission. This hints towards a large diversity in the galaxy population at z>5, only 1 billion years after the Big Bang that is waiting to be explored with further observations. The authors would like to thank Nick Scoville, Caitlin Casey, Ranga-Ram Chary, Rychard Bouwens, Kirsten Larson, Shoubaneh Hemmati, and Lee Armus for valuable discussions which improved this manuscript. Furthermore, the authors thank the anonymous referee for the valuable feedback. D.R. and R.P. acknowledge support from the National Science Foundation under grant number AST-1614213 to Cornell University. R.P. acknowledges support through award SOSPA3-008 from the NRAO. We thank the ALMA staff for facilitating the observations and aiding in the calibration and reduction process. This paper makes use of the following ALMA data: ADS/JAO.ALMA#2011.0.00064.S, ADS/JAO.ALMA#2012.1.00523.S, ADS/JAO.ALMA#2012.1.00919.S, ADS/JAO.ALMA#2015.1.00928.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada) and NSC and ASIAA (Taiwan) and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. This work is based in part on observations made with the Spitzer Space Telescope and the W.M. Keck Observatory, along with archival data from the NASA/ESA Hubble Space Telescope, the Subaru Telescope, the Canada-France-Hawaii-Telescope and the ESO Vista telescope obtained from the NASA/IPAC Infrared Science Archive. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. lcccccccccc UV and optical spectroscopic properties of the 3 analog galaxies 0ptName R.A. Decl. z / / /^a SFR_ Hα (J2000.0) (J2000.0) (optical)(Å) (Å)(yr^-1)09:59:40.34 01:51:21.30 0.2506 2.11± 0.03 6.04± 0.03 0.051± 0.001 21± 3 333± 4 8.16± 0.01 4510:00:27.85 01:57:03.60 0.2647 1.96± 0.04 5.61± 0.04 0.024± 0.002 43± 2 423± 2 7.98± 0.02 1710:00:35.76 02:01:13.50 0.2653 1.65± 0.03 4.71± 0.03 0.060± 0.001 19± 2 195± 2 8.20± 0.01 11aThis uses the <cit.> calibration based on . The <cit.> calibration yields 7.90± 0.01, 7.90± 0.01, and 8.17± 0.01, respectively, but we find that this estimate is not reliable because it is based on the location where the upper and lower branch separate.lcccccccc Photometric IR properties of the 3 low-redshift analogs 0ptName 1cSpitzer/MIPS 2cHerschel/PACS 3cHerschel/SPIRE 2cALMA^a 1c 2c———————— 3c———————————— 2c———————————– 70 μ m 110 μ m 160 μ m 250 μ m 350 μ m 500 μ m S_ cont T_ exp (#Antenna) (mJy) (mJy) (mJy) (mJy) (mJy) (mJy) (mJy) (seconds) 11.5± 2.5 12.4± 2.1 12.2± 5.4 0.5± 4.0 <2.8^† <1.1^† < 0.0793^ 1255.87 (27)<3.3^† < 2.1^† 9.4± 5.4 <4.0^† <2.8^† <1.1^† <0.0484^ 9216.43 (36)7.3± 4.2 9.9±2.1 6.2± 5.4 <4.0^† <2.8^† <1.1^† <0.0658^ 9852.10 (24)aALMA observations at the position of CO (observed frequency ∼ 90GHz). No CO emission is detected in any of the three galaxies with high significance (L^'_ CO < 5.5× 10^8K km s^-1 pc^2 at 3σ). †1σ upper limits on fluxes. ^3σ upper limits on fluxes. lcccccccccc Fitted UV and optical properties of the 3 low-redshift analogs 0ptName/ / / log(/) β ^a SFR_ UV+FIR^b (Å) (mag)(UV cont. slope) (yr^-1) 452± 14 1.95± 0.14 5.56± 0.41 3.23± 0.57 9.04± 0.17 0.16± 0.06 10.12^+0.07_-0.08 -1.80± 0.46 8.8 (333)^† (2.11)^† (6.04)^† 4.11± 0.72 9.11± 0.16 0.18± 0.07 10.11^+0.08_-0.10 -1.71± 0.552cβ from GALEX: -1.43^+0.24_-0.14402± 18 2.00± 0.11 5.73± 0.32 3.39± 0.56 8.9± 0.15 0.07± 0.04 10.33^+0.05_-0.06 -2.19± 0.23 4.9(423)^† (1.96)^† (5.61)^† 3.38± 0.52 8.89± 0.14 0.07± 0.03 10.33^+0.05_-0.06 -2.20± 0.22 2cβ from GALEX: -1.48^+0.87_-0.40[0.2cm] 150± 13 1.89± 0.23 5.41± 0.66 1.55± 0.24 9.55± 0.14 0.26± 0.08 10.24^+0.11_-0.16 -1.52± 0.70 6.5(195)^† (1.65)^† (4.71)^† 1.48± 0.09 9.53± 0.13 0.26± 0.07 10.24^+0.10_-0.14 -1.54± 0.66 2cβ from GALEX: -1.86^+0.28_-0.14aNote that β derived from SED fitting is strongly correlated to the assumed dust attenuation curve (SMC or local starburst in this case). For the final diagram, we therefore use the β independently computed from the GALEX photometry, corrected by the emission contaminating the FUV filter. bUV measured at 1600 Å and FIR from , see Table <ref>. †These parameters are fixed to the spectroscopic value during the SED fitting. lcc cccc FIR properties of the 3 low-z analogs and high-z galaxies from the literature. 0ptName z α β_ IR T ^a T_ peak ^b log(/) ^c (MIR power-law slope) (emissivity) (K) (K) 7cLow-redshift analogs (This work)0.2506 2.43^+0.07_-0.15 1.16^+3.46_-1.04 97^+5_-12 54^+7_-6 10.91^+0.04_-0.07^† 0.2647 2.42^+0.47_-0.61 1.80^+3.00_-1.47 72^+35_-46 43^+18_-26 10.32^+0.15_-0.370.2653 2.47^+0.09_-0.11 1.18^+3.16_-1.28 75^+8_-5 43^+5_-0 10.68^+0.06_-0.09Stack - 2.12^+1.25_-0.63 2.88^+12.87_-2.36 84^+9_-17 48^+6_-5 10.68^+0.03_-0.04 7cHigh-z galaxies from(This work, using warm IR SED prior)7c(parameters derived for 1.5 < α < 2.5, 1.0 < β_ IR < 2.0, 60K < T < 90K [35K < T_ peak < 50K])HZ1 5.6885 - - - - <10.96^+0.22_-0.25HZ2 5.6697 - - - - <10.95^+0.22_-0.25HZ3 5.5416 - - - - <11.18^+0.22_-0.25HZ4 5.5440 - - - - 11.78^+0.22_-0.25 HZ5^∗ 5.3089 - - - - <10.95^+0.22_-0.25·HZ5a - - - - <10.95^+0.22_-0.25 HZ6 (all)^∙ 5.2928 - - - - 11.78^+0.22_-0.25·HZ6a - - - - 10.92^+0.22_-0.25·HZ6b - - - - 11.52^+0.22_-0.25·HZ6c - - - - 11.44^+0.22_-0.25 HZ7 5.2532 - - - - <10.99^+0.22_-0.25 HZ8 5.1533 - - - - <10.90^+0.22_-0.25·HZ8W- - - - <10.90^+0.22_-0.25 HZ9 5.5410 - - - - 12.18^+0.22_-0.25HZ105.6566 - - - - 12.58^+0.22_-0.25·HZ10W^≀ - - - - (12.28^+0.22_-0.25) 7cTemperature constraints on high-z galaxies from <cit.>7c(parameters derived for 1.2 < β_ IR < 2.2, 10K < T < 100K [5K < T_ peak < 60K], α is unconstraint)HZ6 (all) 5.2928 - - 60^+35_-27 35^+20_-16 12.11^+0.48_-0.48HZ10 5.6566 - - 36^+25_-10 25^+18_-7 11.55^+0.55_-0.66 7cLensed high-z Herschel detected 1.5 < z < 3.0 galaxies <cit.> 7c(modified blackbody with fixed β_ IR=1.5, corrected for magnification)A68/C0 1.59 - - - 34.5^+1_-1 11.06 A68/h7 2.15 - - - 43.4^+1_-1 12.26 A68/HLS115 1.59 - - - 37.5^+1_-1 11.53 A68/nn4 3.19 - - - 54.9^+1_-1 12.83 MACS0451 north 2.01 - - - 49.2^+1_-1 10.87 MACS0451 full arc- - - 50-80 11.27 [0.2cm] 7cLensed high-z DSFGs <cit.> 7c(refitted by our method with photometry from <cit.>SPT2319-55^ 5.2929 1.73^+53.19_-1.17 2.46^+0.17_-0.16 59^+5_-7 34^+4_-1 <13.79^+0.40_-0.17^SPT2353-50^ 5.5760 0.91^+3.28_-0.59 2.60^+0.18_-0.18 61^+6_-6 38^+1_-4 <14.02^+0.36_-0.28^SPT0346-52^ 5.6559 2.07^+5.26_-0.72 2.52^+2.68_-0.66 74^+3_-3 43^+1_-5 13.52^+0.16_-0.23^⋆SPT2351-57^ 5.8110 3.12^+17.55_-2.50 2.41^+0.17_-0.16 80^+5_-7 48^+1_-5 <13.92^+0.41_-0.08^ 7cIntense high-z starburst AzTEC-3 <cit.>AzTEC-3 5.2988 6.17^+2.73_-2.60 2.16^+0.27_-0.27 88^+10_-10 51^+6_-6 13.34^+0.08_-0.10 7cHigh-z galaxies from <cit.>7c(parameters derived for 1.5 < α < 2.5, 1.0 < β_ IR < 2.0, 60K < T < 90K [35K < T_ peak < 50K])CLM1 6.1657 - - - - 11.18^+0.22_-0.25WMH5 6.0695 - - - - 11.87^+0.22_-0.25 7cCandidate -detected high-z galaxies from <cit.> (blind search)7c(parameters derived for 1.5 < α < 2.5, 1.0 < β_ IR < 2.0, 60K < T < 90K [35K < T_ peak < 50K])IDX25 6.357 - - - - <11.16^+0.22_-0.25IDX34 7.491 - - - - <11.66^+0.22_-0.25ID02 7.914 - - - - <11.43^+0.22_-0.25ID04 6.867 - - - - 11.53^+0.22_-0.25ID09 6.024 - - - - <11.15^+0.22_-0.25ID14 6.751 - - - - <11.22^+0.22_-0.25ID27 7.575 - - - - 11.20^+0.22_-0.25ID30 6.854 - - - - <11.42^+0.22_-0.25ID31 7.494 - - - - 11.14^+0.22_-0.25ID38 6.593 - - - - <11.20^+0.22_-0.25ID41 6.346 - - - - <11.20^+0.22_-0.25ID44 7.360 - - - - <11.25^+0.22_-0.25ID49 6.051 - - - - <11.13^+0.22_-0.25ID52 6.018 - - - - <11.37^+0.22_-0.25aLuminosity weighted temperature as in <cit.>. bTemperature measured from the wavelength of peak flux emission via Wien's displacement law. cThe total FIR luminosity is integrated between rest-frame 3-1100 μ m. If necessary, the values are converted to this integration interval. ^†This source is IR faint and therefore these estimates are mostly based on limits causing the large uncertainties. ^No constraints blue-ward of rest-frame 40 μ m are available for these sources causing the estimate for α to be very uncertain. ^The magnification for these sources is not known. Therefore thehas not been de-magnified in these cases. ^⋆ has been corrected for magnification using μ=5.4 <cit.>. ^∗This object is a low-luminosity quasar. ^∙This object is called LBG-1 in <cit.> and <cit.>. ≀Assuming half of the FIR luminosity of HZ10. l ccccc Summary of the UV and FIR properties of thegalaxy sample from <cit.> and <cit.>. 0ptName z β log(/) log(/) 158 μ m continuum flux(UV spectral slope) (μJy) HZ1 5.690 -1.92_-0.11^+0.14 11.21 ± 0.01 8.40±0.32 <30 HZ2 5.670 -1.82_-0.10^+0.10 11.15 ± 0.01 8.56±0.41 <29 HZ3 5.546 -1.72_-0.15^+0.12 11.08 ± 0.01 8.67±0.28 <51 HZ4 5.540 -2.06_-0.15^+0.13 11.28 ± 0.01 8.98±0.22 202±47HZ5 5.310 -1.01_-0.12^+0.06 11.45 ± 0.004 <7.20 <32 ·HZ5a<10.37 8.15±0.27 <32HZ6 (all) 5.290 -1.14_-0.14^+0.12 11.47 ± 0.01 9.23±0.04 220±36 ·HZ6a-0.59_-1.12^+1.05 11.11 ± 0.07 8.32±0.06 30±20 ·HZ6b-1.50_-1.22^+1.05 11.00 ± 0.07 8.81±0.02 120±20 ·HZ6c-1.30_-0.37^+0.51 10.81 ± 0.07 8.65±0.03 100±20HZ7 5.250 -1.39_-0.17^+0.15 11.05 ± 0.02 8.74±0.24 <36HZ8 5.148 -1.42_-0.18^+0.19 11.04 ± 0.028.41±0.18 <30·HZ8W-0.10_-0.29^+0.29 10.57 ± 0.04 8.31±0.23 <30HZ9 5.548 -1.59_-0.23^+0.22 10.95 ± 0.029.21±0.09 516±42 HZ10 5.659 -1.92_-0.17^+0.24 11.14 ± 0.02 9.13±0.13 1261±44·HZ10W^†-1.47_-0.44^+0.77 10.23 ± 0.05 (8.83±0.13) (630±44) †Assuming half the IR flux and line luminosity of HZ10. § QUANTIFYING THE DIFFICULTY OF MEASURING IR SEDS AT Z>5 In Section <ref> we show that the knowledge of the shape of the IR SED is crucial to measure basic IR properties like the total luminosity . Furthermore, it enables us to study in detail the ISM properties of galaxies such as their temperature and optical depth, which tell us about the current star-formation conditions in these galaxies. As pointed out earlier, IR SEDs of z>5 galaxies are very poorly constrained, most of the time by just one data-point at 158 μ m if at all. Measurement of DSFGs at z=5-6 reveal increased T and thus warm SED shapes compared to galaxies a z<4 <cit.>. The first estimates of T for two normal z∼5.5 galaxies reveal two very different temperatures, indicative of large variations in the SED shape of high-redshift galaxies <cit.>. However, these measurements are uncertain as they only use wavelength data red-ward of . One way to investigate the diversity of IR SEDs of high-redshift galaxies statistically is to use larger samples of low-redshift analogs of these galaxies. This, however, might be dangerous as we do not know the degree at which analogs are actually representing the high-redshift galaxy population as a whole. Direct measurements of galaxies at high redshifts therefore seems to be the preferred strategy. Due to the faintness of these galaxies, this is a difficult endeavor and hence an optimal and effective observing strategy is needed. In the following, we outline a possible observing strategy in order to better constrain the IR SEDs (specifically constrained by the luminosity weighted temperature T) of z∼6 galaxies. Since observations in the IR are expensive, we focus in particular on minimizing the total observing time and maximizing the science output.§.§ Theoretical calculations To measure the temperature of high-redshift galaxies via continuum measurements, we choose wavelengths close to bright FIR emission lines ( at 63μ m, at 88μ m, at 122μ m and 205μ m, and at 158μ m). With this strategy, any ALMA observations will produce line diagnostics in addition to FIR continuum measurements to further investigate the ISM of these galaxies <cit.>. Figure <ref> shows the observability of these FIR emission lines with ALMA as a function of redshift. The color and width of the bands for each FIR emission line corresponds to the atmospheric transmission in per-cent assuming a precipitable water vapor of 1mm. An efficient strategy is to look for the lowest redshift (to have a reasonable sample of spectroscopically confirmed galaxies) at which most of the FIR lines are observable with ALMA. It turns out that the redshift range 5.82 < z < 5.96 fulfills this criteria with Band 5 being operational in ALMA cycle 5. Note that z∼6.1 and z∼6.3 are other possible windows at which all five emission lines are observable. If only interested in the highest transmission wavelengths (122 μ m, 158 μ m, and 205 μ m), all redshifts above z∼5.9 are a good choice. The left-side panel in Figure <ref> shows FIR continuum flux ratios around these FIR emission lines as a function of the luminosity weighted temperaturederived from a graybody + mid-IR power-law parametrization by <cit.>. We choose to use the ratios with respect to rest-frame 158 μ m, which has a high atmospheric transmission (93% at z∼6) and is well constrained by existing data sets in terms of detections and limits. In the following, we refer to 158 μ m as the “primary wavelength” and to the other wavelengths (63 μ m, 88 μ m, 122 μ m, and 205 μ m) as the “secondary wavelengths”. The width of the bands visualizes the model uncertainty (90% confidence interval) of the continuum flux ratio for a large range in emissivities (0.5 < β_ IR < 2.5) and mid-IR power-law slopes (1 < α < 3). We note that the relation between continuum flux ratio and temperature does not significantly depend on α and β_ IR and is primarily dependent on the assumed temperature T (see also Section <ref>). The black points with error bars show simulated data points with fluxes observed at different depths for comparison. The continuum flux ratio F_158 μ m/F_63 μ m would be the ideal choice to quantify the temperature because of its steep dependence on temperature. However, the atmospheric transparency for z∼5.9 at the observed wavelength of 63 μ m is lowest with 38% compared to 40%, 82%, and 94% for 88 μ m, 122 μ m, and 205 μ m, respectively. The right-side panel in Figure <ref> shows this more qualitatively. Each sub-panel corresponds to a continuum flux ratio and shows the needed depth of the flux measurement (in σ) to reach a certain σ of significance to distinguish three different pairs of temperature. The calculation of the latter combines the model uncertainty at a given temperature with the measurement uncertainty (corresponding to the y-axis). The F_158 μ m/F_63 μ m ratio gives the best turn-out for a given depth of flux measurement. All considered temperatures can be distinguished at >2σ (>3σ) significance for a 10σ (20σ) detection at 63 μ m. For F_158 μ m/F_88 μ m temperatures of 30K and 60K can be distinguished at 2-2.5σ significance for a 6σ detection at 88 μ m, while for F_158 μ m/F_122 μ m a >15σ detection in 122 μ m is necessary to reach the same significance. Finally, the F_158 μ m/F_205 μ m vs. temperature relation is too shallow to separate temperatures at T>30K with a significance of more than 1σ for any observational depth. The latter is the reason why the temperature estimates by <cit.> have such a large quoted uncertainty. §.§ A fiducial survey at z=5.9 The final success to measure a temperature depends on (i) the ability to reach the flux ratio for a given temperature (the limiting flux ratio ξ) and (ii) the ability to split different temperatures (as discussed above). Figure <ref> demonstrates these points by combining the previous results with relative integration time calculations for a fiducial redshift of z=5.9. The panels show the reached flux ratio above the limiting flux ratio as a function of integration in the secondary wavelengths with respect to the integration time necessary for a 10σ detection in 158 μ m. Note that ξ<1 only result in an upper limit in temperature for all flux ratios except for F_158 μ m/F_205 μ m where it results in a lower temperature limit (see left panel of Figure <ref>). The horizontal panels show the four different flux ratios while the sets of panels (arranged vertically) show different depths for the observations at the secondary wavelengths (5σ, 10σ, and 15σ). The width of the curves (showing temperatures of 30K, 60K, and 90K)[Corresponding to 20K, 35K, and 50K in T_ peak.] include the model and measurement uncertainties. In summary, we note the following. * Temperatures cannot be distinguished above T=60K with more than 1σ significance using the F_158 μ m/F_122 μ m and F_158 μ m/F_205 μ m ratios at flux measurement depths corresponding to less than 15σ signifiance. * Temperatures can be confined at T < 30K or T >30K at >1.5σ for a 10σ detection in 122 μ m or 205 μ m within 1-2 times the integration time in 158 μ m (with 10σ). * The measurement of temperatures with the F_158 μ m/F_63 μ m and F_158 μ m/F_88 μ m ratios is not feasible within a reasonable amount of time (less than 10 times the integration time in 158 μ m). * Therefore, the optimal observing strategy to study the temperature of z∼6 galaxies is to target rest-frame 158 μ m and 122 μ m (or 205 μ m if 122 μ m is affected by low transmission, but with a significantly decreased success rate), thereby requiring a depth of at least 10σ. After setting up the frame-work, we now proceed to establish integration times and flux limits for which temperatures can be measured efficiently. Figure <ref> shows the estimated flux root mean square (RMS) limits at a given σ significance as a function of integration time for different wavelengths and redshifts. We assume a dual polarization set-up with a bandwidth of 7.5GHz, precipitable water vapor column density of 0.913mm, 40 antennae, and a resolution of 1 in the following. The calculations are done for objects at δ(2000) = +02:00:00 observed with ALMA and the integration times do not include overheads. The large panel shows a more detailed graph for 158 μ m at z=5.9. An integration time of 2hours would result in a 10σ 158 μ m continuum flux sensitivity of 170 μ Jy per beam, which is similar the flux observed in HZ4 (see Table <ref>). To distinguish temperatures of 30 K and 60K with F_158 μ m/F_122 μ m for such a source at z=5.9 with a significance of 1σ (for which a 10σ detection at 122 μ m is necessary), we have to double the integration time at 158 μ m, thus 4hours at 122 μ m (see Figure <ref>). Alternatively, a 2× 4 = 8hour integration (corresponding to 15σ at 122 μ m) will result in a significance to distinguish T=30 K and T=60K of 2σ. The significance increases by roughly 30-50% for temperatures of T=30 K and T=90K. §.§ Temperature measurement of existing sources detected in provides four of the IR brightest ∼ L^* galaxies at z∼5.5 that can be followed up with our method to verify our inferred average temperature of such galaxies. Table <ref> shows their expected (and measured in case of 158 μ m) fluxes. We also include the other 7 galaxies that are not detected in the FIR continuum. For the prediction of the fluxes we assumed a MIR power-law slope of 1.5 < α < 2.5, an emissivity of 1 < β_ IR < 2 and a temperature 60K < T < 90K (35K < T_ peak < 50K). For 25K < T < 45K (18K < T_ peak < 28K), as assumed in , fluxes at wavelengths blue-ward of would be lower while fluxes red-ward of would be larger because of the shift of the SED to lower temperatures. In the following, we assume the same ALMA configuration as in Section <ref>. HZ10. The temperature of HZ10 at z∼5.65 has already been estimated to be T∼30K (T_ peak∼ 20K) <cit.>. However, the uncertainty of this estimate is large since only data at >158 μ m are available. A better temperature estimate can be achieved by a 17σ measurement in122 μ m. This would uniquely differentiate a temperature of T=30K and T=60K (T=90K) with 2σ (3σ) significance (Figure <ref>). With an estimated flux of ∼2000 μ Jy at 122 μ m, a 17σ measurement can be achieved in ∼20-30minutes of observing time in Band 7 (see Figures <ref> and <ref>). Assuming a temperature prior of 25K < T < 45K instead, we would expect a flux sensitivity of ∼1400 μ Jy and a required integration time of ∼40-50minutes. HZ9. For the second brightest galaxy in FIR at z∼5.55, rest-frame 122 μ m cannot be observed as it is falling between Bands 7 and 8. Similarly, 63 μ m and 88 μ m are out of range. Instead, a 20σ continuum measurement at 205 μ m can uniquely differentiate a temperature of T=30K and T=60K/90K with 1σ significance. With an estimated flux of ∼300 μ Jy at 205 μ m, a 20σ measurement can be achieved in ∼1.5hours of observation in Band 6. Assuming a temperature prior of 25K < T < 45K instead, we would expect a flux of ∼370 μ Jy, which would decrease the integration time to ∼1hour. HZ4. HZ4 is at the same redshift as HZ9 and can therefore not be observed at 122 μ m.Also, 63 μ m and 88 μ m are out of the wavelength range. The 20σ measurement at 205 μ m that is needed to uniquely differentiate a temperature of T=30K and T=60K/90K with 1σ significance, can only be achieved at integration times of more than 5hours for the estimated 205 μ m continuum flux of ∼110 μ Jy. The same is true for the source WMH5 detected by <cit.>. HZ6. HZ6 at z∼5.3 has been observed at 205 μ m by <cit.> and a temperature of T=60^+35_-27K (T_ peak∼ 35K) has been derived with large uncertainty. Note that these measurements are integrated over all the three components. HZ6 can be re-observed at 122 μ m (expected flux is 340 μ Jy) reaching a 6 σ continuum detection in 2.5hours. This results in a splitting of T=30K and T=90K at 1 σ level. The 20σ measurement at 205 μ m that is needed to uniquely differentiate a temperature of T=30K and T=60K/90K with 1σ significance, can only be achieved at integration times of more than 5hours for the estimated 205 μ m continuum flux of ∼126 μ Jy <cit.>. FIR undetected galaxies. The 158 μ m continuum flux is not known for these galaxies and therefore only a lower limit in integration time can be inferred. Taking at face value the continuum flux limit of the stacked observations (∼35 μ Jy), an integration time of at least 12hours is needed for a continuum detection of 5σ at 158 μ m as shown in the large panel in Figure <ref>. Note that this figure was made for z=5.9, but due to the flat transmission function around rest-frame 158 μ m (Figure <ref>), the integration time estimates also hold for z∼5.5. For a 5σ detection at 158 μ m (instead of 10σ) in Figure <ref>, the resulting integration times would increase by a factor 4. Thus, for a 5σ detection at 205 μ m, Figure <ref> suggests 4-8 times the integration time in 158 μ m, so more than 50hours. Thus, estimating the temperature for these FIR undetected sources is not feasible with today's capabilities. This is also true for the 158 μ m continuum detected candidates found in ALMA blind searches <cit.> as well as for CLM1 from <cit.>. l c c cccccccccExpected photometry for high-z galaxies predicted from their 158 μ m flux and 1.5 < α < 2.5, 1 < β_ IR < 2, and 60K < T < 90K (35K < T_ peak < 50K). 0ptName z log(/) 63 μ m 88 μ m 110 μ m 122 μ m 158 μ m^a 190 μ m 205 μ m 220 μ m Reference^b(μ Jy) (μ Jy) (μ Jy) (μ Jy) (μ Jy) (μ Jy) (μ Jy) (μ Jy) HZ10 5.6566 12.58^+0.22_-0.25 3100^+583_-649 2773^+268_-319 2234^+120_-136 1947^+74_-81 (1261^+ 0_- 0) 858^+31_-28 722^+38_-37 608^+45_-39 1HZ9 5.5410 12.18^+0.22_-0.25 1269^+239_-266 1135^+110_-130 914^+49_-56 797^+30_-33 (516^+ 0_- 0) 351^+13_-11 296^+15_-15 249^+19_-16 1HZ6 (all) 5.2928 11.78^+0.22_-0.25 541^+102_-113 484^+47_-56 390^+21_-24 340^+13_-14 (220^+ 0_- 0) 150^+ 5_- 5 126^+ 7_- 6 106^+ 8_- 7 2WMH5 6.0695 11.87^+0.22_-0.25 536^+101_-112 479^+46_-55 386^+21_-24 337^+13_-14 (218^+ 0_- 0) 148^+ 5_- 5 125^+ 7_- 6 105^+ 8_- 7 3HZ4 5.5440 11.78^+0.22_-0.25 497^+93_-104 444^+43_-51 358^+19_-22 312^+12_-13 (202^+ 0_- 0) 137^+ 5_- 4 116^+ 6_- 6 97^+ 7_- 6 1HZ6b 5.2928 11.52^+0.22_-0.25 295^+56_-62 264^+25_-30 213^+11_-13 185^+ 7_- 8 (120^+ 0_- 0) 82^+ 3_- 3 69^+ 4_- 3 58^+ 4_- 4 2HZ6c 5.2928 11.44^+0.22_-0.25 246^+46_-51 220^+21_-25 177^+ 9_-11 154^+ 6_- 6 (100^+ 0_- 0) 68^+ 2_- 2 57^+ 3_- 3 48^+ 4_- 3 2IDX34 7.4910 <11.66^+0.22_-0.25 <246^+46_-51 <220^+21_-25 <177^+ 9_-11 <154^+ 6_- 6 (<100^+ 0_- 0) <68^+ 2_- 2 <57^+ 3_- 3 <48^+ 4_- 3 4ID04 6.8670 11.52^+0.22_-0.25 202^+38_-42 180^+17_-21 145^+ 8_- 9 127^+ 5_- 5 (82^+ 0_- 0) 56^+ 2_- 2 47^+ 2_- 2 40^+ 3_- 3 4ID52 6.0180 <11.37^+0.22_-0.25 <172^+32_-36 <154^+15_-18 <124^+ 7_- 8 <108^+ 4_- 4 (<70^+ 0_- 0) <48^+ 2_- 2 <40^+ 2_- 2 <34^+ 3_- 2 4ID30 6.8540 <11.42^+0.22_-0.25 <160^+30_-33 <143^+14_-16 <115^+ 6_- 7 <100^+ 4_- 4 (<65^+ 0_- 0) <44^+ 2_- 1 <37^+ 2_- 2 <31^+ 2_- 2 4ID02 7.9140 <11.43^+0.22_-0.25 <135^+25_-28 <121^+12_-14 <97^+ 5_- 6 <85^+ 3_- 4 (<55^+ 0_- 0) <37^+ 1_- 1 <32^+ 2_- 2 <27^+ 2_- 2 4HZ3 5.5416 <11.18^+0.22_-0.25 <125^+24_-26 <112^+11_-13 <90^+ 5_- 6 <79^+ 3_- 3 (<51^+ 0_- 0) <35^+ 1_- 1 <29^+ 2_- 1 <25^+ 2_- 2 1CLM1 6.1657 11.18^+0.22_-0.25 108^+20_-23 97^+ 9_-11 78^+ 4_- 5 68^+ 3_- 3 (44^+ 0_- 0) 30^+ 1_- 1 25^+ 1_- 1 21^+ 2_- 1 3ID41 6.3460 <11.20^+0.22_-0.25 <108^+20_-23 <97^+ 9_-11 <78^+ 4_- 5 <68^+ 3_- 3 (<44^+ 0_- 0) <30^+ 1_- 1 <25^+ 1_- 1 <21^+ 2_- 1 4ID09 6.0240 <11.15^+0.22_-0.25 <103^+19_-22 <92^+ 9_-11 <74^+ 4_- 5 <65^+ 2_- 3 (<42^+ 0_- 0) <29^+ 1_- 1 <24^+ 1_- 1 <20^+ 2_- 1 4ID14 6.7510 <11.22^+0.22_-0.25 <103^+19_-22 <92^+ 9_-11 <74^+ 4_- 5 <65^+ 2_- 3 (<42^+ 0_- 0) <29^+ 1_- 1 <24^+ 1_- 1 <20^+ 2_- 1 4ID38 6.5930 <11.20^+0.22_-0.25 <103^+19_-22 <92^+ 9_-11 <74^+ 4_- 5 <65^+ 2_- 3 (<42^+ 0_- 0) <29^+ 1_- 1 <24^+ 1_- 1 <20^+ 2_- 1 4IDX25 6.3570 <11.16^+0.22_-0.25 <98^+19_-21 <88^+ 8_-10 <71^+ 4_- 4 <62^+ 2_- 3 (<40^+ 0_- 0) <27^+ 1_- 1 <23^+ 1_- 1 <19^+ 1_- 1 4ID44 7.3600 <11.25^+0.22_-0.25 <98^+19_-21 <88^+ 8_-10 <71^+ 4_- 4 <62^+ 2_- 3 (<40^+ 0_- 0) <27^+ 1_- 1 <23^+ 1_- 1 <19^+ 1_- 1 4ID49 6.0510 <11.13^+0.22_-0.25 <98^+19_-21 <88^+ 8_-10 <71^+ 4_- 4 <62^+ 2_- 3 (<40^+ 0_- 0) <27^+ 1_- 1 <23^+ 1_- 1 <19^+ 1_- 1 4HZ7 5.2532 <10.99^+0.22_-0.25 <89^+17_-19 <79^+ 8_- 9 <64^+ 3_- 4 <56^+ 2_- 2 (<36^+ 0_- 0) <24^+ 1_- 1 <21^+ 1_- 1 <17^+ 1_- 1 1ID27 7.5750 11.20^+0.22_-0.25 84^+16_-17 75^+ 7_- 9 60^+ 3_- 4 53^+ 2_- 2 (34^+ 0_- 0) 23^+ 1_- 1 19^+ 1_- 1 16^+ 1_- 1 4HZ5 5.3089 <10.95^+0.22_-0.25 <79^+15_-16 <70^+ 7_- 8 <57^+ 3_- 3 <49^+ 2_- 2 (<32^+ 0_- 0) <22^+ 1_- 1 <18^+ 1_- 1 <15^+ 1_- 1 1HZ5a 5.3089 <10.95^+0.22_-0.25 <79^+15_-16 <70^+ 7_- 8 <57^+ 3_- 3 <49^+ 2_- 2 (<32^+ 0_- 0) <22^+ 1_- 1 <18^+ 1_- 1 <15^+ 1_- 1 1HZ1 5.6886 <10.96^+0.22_-0.25 <74^+14_-15 <66^+ 6_- 8 <53^+ 3_- 3 <46^+ 2_- 2 (<30^+ 0_- 0) <20^+ 1_- 1 <17^+ 1_- 1 <14^+ 1_- 1 1HZ6a 5.2928 10.92^+0.22_-0.25 74^+14_-15 66^+ 6_- 8 53^+ 3_- 3 46^+ 2_- 2 (30^+ 0_- 0) 20^+ 1_- 1 17^+ 1_- 1 14^+ 1_- 1 2HZ8 5.1533 <10.90^+0.22_-0.25 <74^+14_-15 <66^+ 6_- 8 <53^+ 3_- 3 <46^+ 2_- 2 (<30^+ 0_- 0) <20^+ 1_- 1 <17^+ 1_- 1 <14^+ 1_- 1 1HZ8W 5.1532 <10.90^+0.22_-0.25 <74^+14_-15 <66^+ 6_- 8 <53^+ 3_- 3 <46^+ 2_- 2 (<30^+ 0_- 0) <20^+ 1_- 1 <17^+ 1_- 1 <14^+ 1_- 1 1ID31 7.4940 11.14^+0.22_-0.25 74^+14_-15 66^+ 6_- 8 53^+ 3_- 3 46^+ 2_- 2 (30^+ 0_- 0) 20^+ 1_- 1 17^+ 1_- 1 14^+ 1_- 1 4HZ2 5.6697 <10.95^+0.22_-0.25 <71^+13_-15 <64^+ 6_- 7 <51^+ 3_- 3 <45^+ 2_- 2 (<29^+ 0_- 0) <20^+ 1_- 1 <17^+ 1_- 1 <14^+ 1_- 1 1aThe 158 μ m fluxes are measured used for predicting the fluxes at the other wavelengths. bList of references: 1–<cit.>; 2–<cit.>; 3–<cit.>; 4–<cit.> †Assuming a temperature of T∼30K (T_ peak∼ 20K) for HZ10 <cit.>, this would result in the following fluxes for 63 μ m, 88 μ m, 110 μ m, 122 μ m, 190 μ m, 205 μ m, 220 μ m: 618 μ Jy, 1185 μ Jy, 1411 μ Jy, 1434 μ Jy, 1012 μ Jy, 902 μ Jy, 800 μ Jy. aasjournal | http://arxiv.org/abs/1708.07842v1 | {
"authors": [
"Andreas L. Faisst",
"Peter L. Capak",
"Lin Yan",
"Riccardo Pavesi",
"Dominik A. Riechers",
"Ivana Barisic",
"Kevin C. Cooke",
"Jeyhan S. Kartaltepe",
"Daniel C. Masters"
],
"categories": [
"astro-ph.GA"
],
"primary_category": "astro-ph.GA",
"published": "20170825180005",
"title": "Are high redshift Galaxies hot? - Temperature of z > 5 Galaxies and Implications on their Dust Properties"
} |
Hierarchical Multi-scale Attention Networks for Action Recognition Shiyang Yan, Jeremy S. Smith, Wenjin Luand Bailing Zhang Shiyang Yan, Wenjin Lu and Bailing Zhang are with the Department of Computer Science and Software Engineering, Xi'an Jiaotong-liverpool University, Suzhou, China. Jeremy S. Smith is with the University of Liverpool.December 30, 2023 =============================================================================================================================================================================================================================================================================================================== Recurrent Neural Networks (RNNs) have been widely used in natural language processing and computer vision. Among them, the Hierarchical Multi-scale RNN (HM-RNN), a kind of multi-scale hierarchical RNN proposed recently, can learn the hierarchical temporal structure from data automatically. In this paper, we extend the work to solve the computer vision task of action recognition. However, in sequence-to-sequence models like RNN, it is normally very hard to discover the relationships between inputs and outputs given static inputs. As a solution, attention mechanism could be applied to extract the relevant information from input thus facilitating the modeling of input-output relationships. Based on these considerations, we propose a novel attention network, namely Hierarchical Multi-scale Attention Network (HM-AN), by combining the HM-RNN and the attention mechanism and apply it to action recognition. A newly proposed gradient estimation method for stochastic neurons, namely Gumbel-softmax, is exploited to implement the temporal boundary detectors and the stochastic hard attention mechanism. To amealiate the negative effect of sensitive temperature of the Gumbel-softmax, an adaptive temperature training method is applied to better the system performance. The experimental results demonstrate the improved effect of HM-AN over LSTM with attention on the vision task. Through visualization of what have been learnt by the networks, it can be observed that both the attention regions of images and the hierarchical temporal structure can be captured by HM-AN.Action recognition, Hierarchial multi-scale RNNs, Attention mechanism, Stochastic neurons. § INTRODUCTIONAction recognition in videos is a fundamental task in computer vision. Recently, with the rapid development of deep learning, and in particular, deep convolutional neural networks (CNNs), a number of models <cit.> <cit.> <cit.> <cit.> have been proposed for image recognition. However, for video-based action recognition, a model should accept inputs with variable length and generate the corresponding outputs. This special requirement makes the conventional CNN model that caters for one-versus-all classification unsuitable.RNNs have long been explored for sequential applications for decades, often with good results. However, a significant limitation of vanilla RNN models which strictly integrate state information over time is the vanishing gradient effect <cit.>: the ability to back propagate an error signal through a long-range temporal interval becomes increasingly impossible in practice. To mitigate this problem, a class of models with a long-range dependencies learning capability, called Long Short-Term Memory (LSTM), was introduced by Hochreiter and Schmidhuber <cit.>. Specifically, LSTM consists of memory cells, with each cell containing units to learn when to forget previous hidden states and when to update hidden states with new information.Many sequential data often have complex temporal structure which requires both hierarchical and multi-scale information to be modeled properly. In language modeling, a long sentence is often composed of many phrases which further can be decomposed by words. Meanwhile, in action recognition, an action category can be described by many sub-actions. For instance, `long jump' contains `running', `jumping' and `landing'. As stated in <cit.>, a promising approach to model such hierarchical representation is the multi-scale RNN. One popular approach of implementing multi-scale RNN is to treat the hierarchical timescales as pre-defined parameters. For example, Wang et al. <cit.> implemented a multi-scale architecture by building a multiple layers LSTM in which higher layer skip several time steps. In their paper, the skipped number of time steps is the parameter to be pre-defined. However, it is often impractical to pre-define such timescales without learning, which also leads to poor generalization capability. Chung et al. <cit.> proposed a novel RNN structure, Hierarchical Multi-scale Recurrent Neural Network (HM-RNN), to learn time boundaries from data automatically. These temporal boundaries are like rules described by discrete variables inside RNN cells. Normally, it is difficult to implement training algorithms for discrete variables. Popular approaches include unbiased estimator with the aid of REINFORCE <cit.>. In this paper, we re-implement the HM-RNN by applying a recently proposed Gumbel-sigmoid function <cit.> <cit.> to realize the training of stochastic neurons due to its efficiency in practice <cit.>.In the general RNN framework for sequence-to-sequence problems, the information of input is treated uniformly without discrimination on the different parts. This will result in fixed length of intermediate features and subsequent sub-optimal system performance. On the other hand, the practice is in sharp contrast to the way humans accomplish sequence processing tasks. Humans tend to selectively concentrate on a part of information and at the same time ignores other perceivable information. The mechanism of selectively focusing on relevant contents in the representation is called attention. The attention based RNN model in machine learning was successfully applied in natural language processing (NLP), and more specifically, in neural translation <cit.>. For many visual recognition tasks, different portions of an image or segments of a video have unequal importance, which should be selectively weighted with attention. Xu et al. <cit.> systematically analyzed the stochastic hard attention and deterministic soft attention models and applied them in image captioning tasks, with improved results compared with other RNN-like algorithms. The hard attention mechanism requires a stochastic neuron which is hard to train using conventional back propagation algorithm. They applied REINFORCE <cit.> as an estimator to implement hard attention for image captioning. The REINFORCE is an unbias gradient estimator for stochastic units, however, it is very complex to implement and often with high gradient variance during training <cit.>. In this paper, we study the applicability of Gumbel-softmax <cit.> <cit.> in hard attention because Gumbel-softmax is an efficient way to estimate discrete units during training of neural networks. To mitigate the problem of sensitive temperature in Gumbel-softmax, we apply an adaptive temperature scheme <cit.> in which the temperature value is also learnt from data. The experimental results verify that the adaptive temperature is a convenient way to avoid manual searching for the parameter. Additionally, we also test the deterministic soft attention <cit.> <cit.> and stochastic hard attention implemented by REINFORCE-like algorithms <cit.> <cit.> <cit.> in action recognition. Combined with HM-RNN and two types of attention models, we systematically evaluate the proposed Hierarchical Multi-scale Attention Networks (HM-AN) for action recognition in videos, with improved results.Our main contributions can be summarized as follows:* We propose a Hierarchical Multi-scale Attention Network (HM-AN) by combining HM-RNN and both stochastic hard attention and deterministic soft attention mechanism for vision tasks.* By re-implementing Gumbel-softmax and Gumbel-sigmoid, we make the stochastic neurons in the networks trainable by back propagation.* We test the proposed model on action recognition from video, with improved results.* Through visualization of the learnt attention regions and boundary detectors of HM-AN, we provide insights for further research.§ RELATED WORKS §.§ Hierarchical RNNsThe modeling of hierarchical temporal information has long been an important topic in many research areas. The most notable model is LSTM proposed by Hochreiter and Schmidhuber <cit.>. LSTM employs the multi-scale updating concept, where the hidden units' update can be controlled by gating such as input gates or forget gates. This mechanism enables the LSTM to deal long term dependencies in temporal domain. Despite the advantage, the maximum time steps are limited within a few hundreds because of the leaky integration which makes the memory for long-term gradually diluted <cit.>. Actually, the maximum time steps in video processing is several dozens which makes the application of LSTM in video recognition very challenging.To alleviate the problem, many researchers tried to build a hierarchical structure explicitly, for instance, Hierarchical Attention Networks (HAN) proposed in <cit.>, which is implemented by skipping several time steps in the higher layers of the stacked multi-layer LSTMs. However, the number of time steps to be skipped is a pre-defined parameter. How to choose these parameters and why to choose certain number are unclear.More recent models like clockwork RNN <cit.> partitioned the hidden states of RNN into several modules with different timescales assigned to them. The clockwork RNN is more computationally efficient than the standard RNN as the hidden states are updated only at the assigned time steps. However, finding the suitable timescales is challenging which makes the model less applicable.To mitigate the problem, Chung et al. <cit.> proposed Hierarchical Multiscale Recurrent Neural Network (HM-RNN). HM-RNN is able to learn the temporal boundaries from data, which facilitates the RNN model to build a hierarchical structure and enable long-term dependencies automatically. However, the temporal boundaries are stochastic discrete variables which are very hard to train using standard back propagation algorithm.A popular approach to train the discrete neurons is the REINFORCE-like <cit.> algorithms. It is an unbiased estimator but often with high gradient variance <cit.>. The original HM-RNN applied straight-through estimator <cit.> because of its efficiency and simplicity in practice. In this paper, instead, we applied a more recent Gumbel-sigmoid <cit.> <cit.> to estimate the stochastic neurons. It is much more efficient than other approaches and achieved state-of-the-art performance among many other gradient estimators <cit.>. §.§ Attention MechanismOne important property of human perception is that we do not tend to process a whole scene in its entirety at once. Instead humans pay attention selectively on parts of the visual scene to acquire information where it is needed <cit.>. Different attention models have been proposed and applied in object recognition and machine translation. Mnih et al. <cit.> proposed an attention mechanism to represent static images, videos or as an agent that interacts with a dynamic visual environment. Also, Ba et al. <cit.> presented an attention-based model to recognize multiple objects in images. The two above-mentioned models are all with the aid of REINFORCE-like algorithms.The soft attention model was proposed for machine translation problem in NLP <cit.>, and Xu et al. <cit.> extended it to image caption generation as the task is analogous to `translating' an image to a sentence. Specifically, they built a stochastic hard attention model with the aid of REINFORCE and a deterministic soft attention model. The two attention mechanisms were applied in image captioning task, with good results. Subsequently, Sharma et al. <cit.> built a similar model with the soft attention applied in action recognition from videos.There is a number of subsequent works on the attention mechanism. For instance, in <cit.>, the attention model is utilized for video description generation by softly weighting the visual features extracted from frames in each video. Li et al. <cit.> combined convolutional LSTM <cit.> with soft attention mechanism for video action recognition and detection. Teh et al. <cit.> extended the soft attention into CNN networks for weakly supervised object detection.One important reason for applying soft attention instead of its hard version is that the stochastic hard attention mechanism is difficult to train. Although the REINFORCE-like algorithms <cit.> are unbiased estimators to train stochastic units, their gradients have high variants. To solve the problem, recently, Jang et al. <cit.> proposed a novel categorical re-parameterization techniques using Gumbel-softmax distribution. The Gumbel-softmax is a superior estimator for categorical discrete units <cit.>. It has been proved to be efficient and has high performance in <cit.>.§.§ Action RecognitionRNNs have been popular for speech recognition <cit.>, image caption generation <cit.>, and video description generation <cit.>. There have also been efforts made for the application of LSTM RNNs in action recognition. For instance, <cit.> proposed an end-to-end training system using CNN and RNNs deep both in space and time to recognize activities in video. <cit.> also explicitly models the video as an ordered sequence of frames using LSTM. Most of the previous works treat image features extracted from CNNs as static inputs to RNNs to generate action labels at each frame. However, static features indiscriminating of important and irrelevant information will be detrimental to system performance. On the other hand, the interpretation of CNN features will be much easier if attention model can be applied for action recognition because attention mechanism automatically focuses on specific regions to facilitate the classification.In this paper, as discussed previously, we re-implement the HM-RNN to capture the hierarchical structure of temporal information from video frames. By incorporating the HM-RNN with both stochastic hard attention and deterministic soft attention, long-term dependencies of video frames can be captured. Works related to ours also include the attention model proposed by Xu et al. <cit.> and <cit.>. <cit.> first applied both stochastic hard attention and deterministic soft attention mechanisms for spatial locations of images for image captioning. <cit.> instead used weighting on image patches to implement a region-level attention. In this paper, similar to <cit.>, both of stochastic hard attention and deterministic soft attention are studied. However, when implementing hard attention, <cit.> borrowed the idea of REINFORCE whilst we also propose to apply a more recent Gumbel-softmax to estimate discrete neurons in the attention mechanism. § THE PROPOSED METHODSIn this section, we first re-visit the HM-RNN structure proposed in <cit.>, then introduce the proposed HM-AN networks, with details of Gumbel-softmax and Gumbel-sigmoid to estimate the stochastic discrete neurons in the networks.§.§ HM-RNN HM-RNN was proposed in <cit.> to better capture the hierarchical multi-scale temporal structure in sequence modeling. HM-RNN defines three operations depending on the boundary detectors: UPDATE, COPY and FLUSH. The selection of these operations is determined by boundary state z_t^l-1 and z_t-1^l, where l and t stand for the current layer and time step, respectively:[UPDATE, z_t-1^l = 0 and z_t^l-1 = 1;;COPY, z_t-1^l = 0 and z_t^l-1 = 0;; FLUSH, z_t-1^l = 1. ]The updating rules of operation UPDATE, COPY and FLUSH are defined as follows:c_t^l =f_t^l⊙ c_t-1^l + i_t^l⊙ g_t^l,UPDATEc_t-1^l, COPY i_t^l⊙ g_t^l,FLUSH The updating rules for hidden states are also determined by the pre-defined operations: h_t^l = h_t-1^l,COPY o_t^l⊙ c_t^l, otherwise The (i, f, o) indicate input, forget and output gate, respectively. g is called `cell proposal' vector. One of the advantages of HM-RNN is that the updating operation (UPDATE) is only executed at certain time steps sparsely instead of all, which largely reduce the computation cost.The COPY operation simply copies the cell memory and hidden state from previous time step to current time step in upper layers until the end of a subsequence, as shown in Fig. <ref>. Hence, upper layer is able to capture coarser temporal information. Also, the boundaries of subsequence are learnt from data which is a big improvement of other related models. To start a new subsequence, FLUSH operation needs to be executed. FLUSH operation firstly forces the summarized information from lower layers to be merged with upper layers, then re-initialize the cell memories for the next subsequence.In summary, the COPY and UPDATE operations enable the upper and lower layers to capture information on different time scales, thus realizing a multi-scale and hierarchical structure for a single subsequence. On the other hand, FLUSH operation is able to summarize the information from last subsequence and forward them to next subsequence, which guarantee the connection and coherence between parts within a long sequence.The values of gates (i, f, o, g) and boundary detector z are obtained by:(i_t^lf_t^lo_t^lg_t^lz_t^l )= (sigmsigmsigmtanhhardsigm ) f_slice(s_t^recurrent(l) + s_t^top-down(l)+ s_t^bottom-up(l) + b_l ) wheres_t^recurrent(l) = U_l^l h_t-1^l s_t^top-down(l) = U_l+1^l (z_t-1^l ⊙ h_t-1^l+1) s_t^bottom-up(l) = W_l-1^l (z_t^l-1⊙ h_t^l+1)and the hardsigm is estimated using Gumbel-sigmoid which will be explained later. §.§ HM-AN The sequential problems inherent in action recognition and image captioning in computer vision can be tackled by RNN-based framework. As explained previously, HM-RNN is able to learn the hierarchical temporal structure from data and enable long-term dependencies. This inspires our proposal of HM-AN model.On the other hand, attention has been proved very effective in action recognition <cit.>. In HM-AN, to capture the implicit relationships between the inputs and outputs in sequence to sequence problems, we apply both hard and soft attention mechanisms to explicitly learn the important and relevant image features regarding the specific outputs. More detailed explanation follows.§.§.§ Estimation of Boundary Detectors In the proposed HM-AN, the boundary detectors z_t are estimated with Gumbel-sigmoid, which is derived directly from Gumbel-softmax proposed in <cit.> and <cit.>.The Gumbel-softmax replaces the argmax in Gumbel-Max trick <cit.>, <cit.> with following Softmax function:y_i = exp(log(π_i + g_i)/τ)/∑_j=1^kexp(log(π_j + g_j)/τ)where g_1, ..., g_k are i.i.d. sampled from distribution Gumbel (0,1), and τ is the temperature parameter. k indicates the dimension of generated Softmax vector.To derive Gumbel-sigmoid, we firstly re-write Sigmoid function as a Softmax of two variables: π_i and 0. sigm(π_i)=1/(1 + exp( - π_i)) =1/(1 + exp(0 - π_i)) =1/1 + exp(0)/exp(π_i)=exp(π_i)/(exp(π_i) + exp(0)) Hence, the Gumbel-sigmoid can be written as: y_i = exp(log(π_i+g_i/τ)/exp(log(π_i+g_i)/τ)+exp(log(g^')/τ)where g_i, g^' are independently sampled from distribution Gumbel (0,1).To obtain a discrete value, we set values of z_t = y_i as: y_i =1y_i ≥ 0.5 0otherwise In our experiments, all the boundary detectors z_t are estimated using the Gumbel-sigmoid with a constant temperature 0.3.§.§.§ Deterministic Soft AttentionTo implement soft attention over image regions for action recognition task, we applied a similar strategy with the soft attention mechanism in <cit.> and <cit.>.Specifically, the model predicts a Softmax over K×K image locations. The location Softmax is defined as: l_t,i = exp(W_i h_t-1)/∑_j=1^K × Kexp(W_j h_t-1) i = 1, ..., K^2where i means the ith location corresponding to the specific regions in the original image.This Softmax can be considered as the probability with which the model learns the specific regions in the image, which is important for the task at hand. Once these probabilities are obtained, the model computes the expected value of inputs by taking expectation over image features at different regions: x_t = ∑_i=1^K^2l_t,iX_t,iwhere x_t is considered as inputs of the HM-AN networks. In our HM-AN implementations, the hidden states used to determine the region softmax is only defined for the first layer, i.e., h_t-1^1. The upper layers will automatically learn the abstract information of input features as explained previously. The soft attention mechanism can be visualized in the left side of Fig. <ref>. §.§.§ Stochastic Hard AttentionREINFORCE-like algorithm The stochastic hard attention was proposed in <cit.>. Their hard attention was realized with the aid of REINFORCE-like algorithm. In this section, we also introduce this kind of hard attention mechanism.The location variable l_t indicates where the model decides to focus attention on the t^th frame of a video. l_t,i is an indicator of one-hot representation which can be set to 1 if the i^th location contains relevant feature.Specifically, we assign a hard attentive location of {α_i}:p(l_i,t = 1 | l_j<t,a)= argmax(α_t,i) = argmax ( exp(W_i h_t-1)/∑_j=1^K × Kexp(W_j h_t-1))where a represents the input image features.We can define an objective function L_l that is a variational lower bound on the marginal log-likelihood log p(y|a) of observing the action label y given image features a. Hence, L_l can be represented as:L_l = ∑_lp(l|a)log p(y|l,a) ≤ log ∑_lp(l|a)p(y|l,a) = log p(y|a)∂ L_l/∂ W = ∑_l p(l|a) [ ∂ log p(y|l,a)/∂ W + log p(y|l,a) ∂ log p(l|a)/∂ W] Ideally, we would like to compute the gradients of Equation <ref>. However, it is not feasible to compute the gradient of expectation in Equation <ref>. Hence, a Monte Carlo approximation technique is applied to estimate the gradient of the operation of expectation.Hence, the derivatives of the objective function with respect to the network parameters can be expressed as: ∂ L_l/∂ W = 1/N∑_n=1^N [ ∂ log p(y|l̃_n,a)/∂ W + log p(y|l̃_n,a) ∂ log p(l̃_n|a)/∂ W]where l̃ is obtained based on the argmax operation as in Equation <ref>.Similar with the approaches in <cit.>, a variance reduction technique is used.With the k^th mini-batch, the moving average baseline is estimated as an accumulation of the previous log-likelihoods with exponential decay: b_k = 0.9 × b_k-1 + 0.1 × log p(y|l̃_k,a) The learning rule for this hard attention mechanism is defined as follows:∂ L_l/∂ W≈1/N∑_n=1^N [ ∂ log p(y|l̃_n,a)/∂ W +λ (log p(y|l̃_n,a) -b) ∂ log p(l̃_n|a)/∂ W]where λ is a pre-defined parameter. As being pointed out in Ba et al. <cit.>, Mnih et al. <cit.> and Xu et al. <cit.>, this is a formulation which is equivalent to the REINFORCE learning rule <cit.>. For convenience, it is abbreviated as REINFORCE-Hard Attention in the following.Gumbel Softmax In hard attention mechanism, the model selects one important region instead of taking the expectation. Hence, it is a stochastic unit which can not be trained using back propagation. <cit.> applied the REINFORCE to estimate the gradient of the stochastic neuron. Although the REINFORCE is an unbiased estimator, the variance of gradient is large and the algorithm is complex to implement. To solve these problems, we propose to apply Gumbel-softmax to estimate the gradient of the discrete units in our model. Gumbel-softmax is better in practice <cit.> and much easier to implement.We can simply replace the Softmax with Gumbel-softmax in Equation <ref> and remove the process of taking expectation to realize the hard attention. l_t,i = exp(log(W_i h_t-1+ g_i)/τ)/∑_j=1^K × Kexp(log(W_j h_t-1+g_j)/τ) i = 1 ... K^2 The Gumbel-softmax will choose a single location indicating the most important image region for the task. However, the searching space for the parameter of temperature is too large to be manually selected. And the temperature is a sensitive parameter as explained in <cit.>. Hence in this paper we applied the adaptive temperature as in <cit.>. The adaptive temperature is to determine the value of temperature adaptively depending on current hidden states. In other words, instead of being treated as pre-defined parameter, the value of temperature is learnt from data. Specifically, we can set the following mechanism to determine the temperature:τ = 1/Softplus(W_temp h_t^1+ b_temp) + 1where h_t^1 is the hidden state of first layer of our HM-AN. Equation <ref> generates a scalar for temperature. In the equation, adding 1 can enable the temperature fall into the scope of 0 and 1. The hard attention mechanism can be seen in the right side of Fig. <ref>.§.§ Application of HM-AN in Action Recognition The proposed HM-AN can be directly applied in video action recognition. In video action recognition, the dynamics exist in the inputs, i.e., the given video frames. With attention mechanism embedded in RNN, the important features of each frames can be discovered and discriminated in order to facilitate the recognition. For action recognition, the HM-AN applies the cross-entropy loss for recognition.LOSS = -∑_t=1^T∑_i=1^Cy_t,ilog(ŷ_t,i)where y_t is the label vector, ŷ_t is the classification probabilities at time step t. T is the number of time steps and C is the number of action categories. The system architecture of action recognition using HM-AN is shown in Fig. <ref>§ EXPERIMENTS In this section, we first explain our implementation details then report the experimental results on action recognition. §.§ Implementation Details We implemented the HM-AN in the platform of Theano <cit.> and all the experiments were conducted on a server embedded with a Titan X GPU.In our experiments, the HM-AN is a three layer stacked RNN. The outputs are concatenated by hidden states from three layers and forwarded to a softmax layer.In addition to the baseline approach (LSTM networks), 4 versions of HM-AN were implemented for the purpose of comparison: * LSTM with soft attention (Baseline). The baseline approach is set as LSTM networks with soft attention mechanism.* Deterministic soft attention in HM-AN (Soft Attention). This is to see how soft attention mechanism performs with the HM-AN.* Stochastic hard attention with reinforcement learning in HM-AN (REINFORCE-Hard Attention). This kind of hard attention mechanism is described in Section <ref>.* Stochastic hard attention with 0.3 temperature for Gumbel-softmax in HM-AN (Constant-Gumbel-Hard Attention). The constant temperature is applied in Gumbel-softmax to accomplish the proposed hard attention model.* Stochastic hard attention with adaptive temperature for Gumbel-softmax in HM-AN (Adaptive-Gumbel-Hard Attention). The temperature is set as a function of hidden states of RNN. For the experiments, we firstly extracted frame-level CNN features using MatConvNet <cit.> based on Residue-152 Networks <cit.> trained on the ImageNet <cit.> dataset. The images were resized to 224×224, hence the dimension of each frame-level features is 7×7×2048. For the network training, we applied mini-batch size of 64 samples at each iteration. For each video clip, the baseline approach randomly selected 30 frames for training while the proposed approaches selected 60 frames for training as the proposed HM-AN is able to capture long-term dependencies. Actually, the optimal length for LSTM with attention is 30 and increasing the number will seriously deteriorate the performance. We applied the back propagation algorithm through time and Adam optimizer <cit.> with a learning rate of 0.0001 to train the networks. The learning rate was changed to 0.00001 after 10,000 iterations. At test time, we compute class predictions for each time step and then average those predictions over 60 frames. To obtain a prediction for the entire video clip, we average the predictions from all 60 frame blocks in the video. Also, we find normally the networks converge in several epoches. §.§ Experimental Results and Analysis §.§.§ Datasets We evaluated our approach on three widely used datasets, namely UCF sports <cit.>, Olympic sports datasets <cit.> and a more difficult Human Motion Database (HMDB51) dataset <cit.>. Fig. <ref> provides some examples of the three datasets used in this paper. UCF sports dataset contains a set of actions collected from various sports which are typically featured on broadcast channels such as ESPN or BBC. This dataset consists of 150 videos with the resolution of 720 × 480 and with 10 different action categories in it. Olympic sports dataset was collected from YouTube sequences <cit.> and contains 16 different sports categories with 50 videos per class. Hence, there are total 800 videos in this dataset. The HMDB51 dataset is a more difficult dataset which provides three train-test splits each consisting of 5100 videos. This clips are labeled with 51 action categories. The training set for each split has 3570 videos and the test set has 1530 videos. For UCF sports dataset, as there is lack of training-testing split for evaluation, we manually divide the dataset into training and testing sets. We randomly pick up 75 percent for training, and leave the remaining 25 percent for testing. We then report the classification accuracy on the testing dataset.As for Olympic sports dataset, we use the original training-testing split with 649 clips for training and 134 clips for testing provided in dataset. Following the practice in <cit.>, we evaluate the Average Precision (AP) for each category on this dataset.When evaluating our method on HMDB51, we also follow the original training-testing split and report the classification accuracy of split 1 on the testing set. §.§.§ Results UCF sports datasetWe firstly tested the performance of the LSTM with soft attention proposed in <cit.> on the UCF sports dataset and obtained 70.0% accuracy. All the experimental settings are the same as these in <cit.>. Then we evaluated the proposed four approaches mentioned previously. HM-AN with stochastic hard attention which is realized with REINFORCE-like algorithm improves the results to 82.0%. HM-AN with soft attention is similar to the REINFORCE-Hard Attention, with accuracy results of 81.1%. The hard attention mechanism realized by Gumbel-softmax with adaptive temperature achieves 82.0% accuracy, similar to our REINFORCE-Hard Attention model. However, the Constant-Gumbel-Hard Attention which uses Gumbel-softmax with constant temperature value of 0.3 only yields 76.0% accuracy, which indicates the significant role of adaptive temperature in maintaining the system performance. Fig. <ref> shows the curves of training cost cross entropy for the Adaptive-Gumbel-Hard Attention approach and REINFORCE-Hard Attention approach, respectively. It can be seen from the figure that the REINFORCE-Hard Attention converges marginally slower than the approach of Adaptive-Gumbel-Hard Attention. As shown in Table <ref>, we compare our model with the methods proposed in <cit.> in which a convolutional LSTM attention network with hierarchical architecture was used for action recognition. The hierarchical architecture in <cit.> was pre-defined whilst our model is able to learn the hierarchy from data. The improvements brought by our methods are obvious as shown in Table <ref>.Olympic sports dataset Olympic sports dataset is of medium size. Experiments on this dataset are shown in Table <ref>. The mAP result of baseline approach is 73.7%. Our method HM-AN with Soft attention achieves 82.4% mAP. However, unlike UCF sports dataset, the mAP result of REINFORCE-Hard Attention is 77.1%, which is lower than the approach of Soft Attention. The Constant-Gumbel-Hard Attention, which is implemented by Gumbel-softmax with a constant temperature 0.3, obtains a mAP value of 82.3%. By making the temperature value of Gumbel-softmax adaptive, the proposed model achieves 82.7% mAP, the highest among all our experimental settings. Again, our proposed methods show superior performance compared to the hand-designed hierarchical model in <cit.> even the convolutional LSTM with attention was utilized in <cit.>.HMDB51 datasetHMDB51 is a more difficult and bigger dataset. The performance of baseline approach is reported in <cit.>, with 41.3% accuracy. Our HM-AN model with soft attention improves the accuracy to 43.8%. We then apply the REINFORCE-Hard Attention approach on this dataset. The accuracy result turns out to be lower than the HM-AN with soft attention. Moreover, the model with REINFORCE-like algorithm converges slower than the Gumbel-softmax with adaptive temperature, also with more oscillations on the training cost, which is shown in Fig. <ref>.With constant value 0.3 of temperature for hard attention, the model achieves 44.0% accuracy. Again, the improvement by adding adaptive temperature is obvious, with 44.2% accuracy on HMDB51 dataset. The accuracy results are further summarized in Table <ref>. We also compare the performance of the proposed HM-AN with some published models related to ours. Firstly, the methods not from RNN family but only with the spatial image shows poor performance as shown in Table <ref>, even with CNN model fine-tuned. Specifically, the softmax regression results from <cit.> directly extracted image feature of each frame and perform softmax regression on them, with 33.5% accuracy. The spatial convolutional net from the famous two-stream approach <cit.> can be considered as a similar method with the difference that the two-stream approach performs fine-tuning on the CNN model, with an improved accuracy of 40.5%. The LSTM without attention also achieves 40.5% accuracy <cit.>. When adding soft attention mechanism, an improved accuracy of 41.3% can be obtained. The Conv-Attention <cit.> and ConvALSTM <cit.> both use convolutional LSTM with attention. The differences are that Conv-Attention extracts features from Residual-152 Networks <cit.> without fine-tuning whilst ConvALSTM extracts image features from a fine-tuned VGG16 model. The ConvALSTM leads Conv-Attention by a small margin, with 43.3% accuracy. As explained previously, CHAM <cit.> has a hand-designed hierarchical architecture in contrast with ours in which the temporal hierarchy is formed through training. Our best setting (Adaptive-Gumbel-Hard Attention) reports the highest accuracy (44.2%) among methods from the RNN family and leads the CHAM results (43.4%) by 0.8 percent. Analysis and VisualizationWe tested four approaches (Soft Attention, REINFORCE-Hard Attention, Constant-Gumbel-Hard Attention and Adaptive-Gumbel-Hard Attention) on three different datasets: UCF sports dataset, Olympic sports dataset and HMDB51 dataset. On UCF sports dataset, The REINFORCE-Hard Attention and Adaptive-Gumbel-Hard Attention generate satisfactory results and shows better performance than the soft attention and Constant-Gumbel-Hard Attention. This indicates that the adaptive temperature is an efficient method to improve performance in implementation of Gumbel-softmax based hard attention.On both of the Olympic sports dataset and HMDB51 dataset, the best approach is the Adaptive-Gumbel-Hard Attention while the REINFORCE-Hard Attention is even worse than soft attention mechanism. On the bigger datasets, the advantages of Gumbel-softmax include small gradient variance and simplicity, which are obvious compared with the REINFORCE-like algorithms. This shows that Gumbel-softmax generalizes well on large and complex datasets. This is reflected not only by the accuracy results, but also by the training cost curves in Fig. <ref> and Fig. <ref>. This conclusion is also consistent with the findings in other recent researches <cit.> which also applied both REINFORCE-like algorithms and Gumbel-softmax as estimators for stochastic neurons.The visualization of attention maps and boundary detectors learnt by the HM-AN is given in Fig. <ref>. In the attention maps, the brighter an area is, the more important it is for the recognition. The soft attention captures multi-regions while the hard attention selects only one important region. As can be seen from the figure, in different time steps, the attention regions are different which means the model is able to select region to facilitate the recognition through time automatically. The z_1, z_2 and z_3 in the figure indicate the boundary detectors in the first layer, the second layer and the third layer, respectively. In the figure for boundary detector, the black regions indicate there exists a boundary in the time-domain whilst the grey regions show the UPDATE operation can be performed. The multi-scale properties in the time-domain can be captured by the HM-AN as different layer shows different boundaries. § CONCLUSION In this paper, we proposed a novel RNN model, HM-AN, which improves HM-RNN with attention mechanism for visual tasks. Specifically, the boundary detectors in HM-AN is implemented by recently proposed Gumbel-sigmoid. Two versions of attention mechanism are implemented and tested. Our work is the first attempt to implement hard attention in vision task with the aid of Gumbel-softmax instead of REINFORCE algorithm. To solve the problem of sensitive parameter of softmax temperature, we applied the adaptive temperature methods to improve the system performance. To validate the effectiveness of HM-AN, we conducted experiments on action recognition from videos. Through experimenting, we showed that HM-AN is more effective than LSTMs with attention. The attention regions of both hard and soft attention and boundaries detected in the networks provide visualization for the insights of what the networks have learnt. Theoretically, our model can be built based on various features, e.g., Dense Trajectory, to further improve the performance. However, our emphasis in this paper is to prove the superiority of the model itself compare with other RNN-like model given same features. Hence, we chose to use deep spatial features only. Our work would facilitate further research on the hierarchical RNNs and its applications on the vision tasks.IEEEtran | http://arxiv.org/abs/1708.07590v2 | {
"authors": [
"Shiyang Yan",
"Jeremy S. Smith",
"Wenjin Lu",
"Bailing Zhang"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170825010810",
"title": "Hierarchical Multi-scale Attention Networks for Action Recognition"
} |
firstpage–lastpage Analyticity Constraints for Hadron Amplitudes: Going High to Heal Low Energy Issues G. Fox December 30, 2023 ===================================================================================== We used the optical and near-infrared imagers located on the Liverpool, the IAC80, and the William Herschel telescopes to monitor 18 M7-L9.5 dwarfs with the objective of measuring their rotation periods. We achieved accuracies typically in the range ±1.5–28 mmag by means of differential photometry, which allowed us to detect photometric variability at the 2σ level in the 50% of the sample. We also detected periodic modulation with periods in the interval 1.5–4.4 h in 9 out of 18 dwarfs that we attribute to rotation. Our variability detections were combined with data from the literature; we found that 65 ± 18 % of M7–L3.5 dwarfs with v sin i≥30 km s^-1 exhibit photometric variability with typical amplitudes ≤20 mmag in the I-band. For those targets and field ultra-cool dwarfs with measurements of v sin i and rotation period we derived the expected inclination angle of their rotation axis, and found that those with v sin i≥30 km s^-1 are more likely to have inclinations ≳40 deg. In addition, we used these rotation periods and others from the literature to study the likely relationship between rotation and linear polarization in dusty ultra-cool dwarfs. We found a correlation between short rotation periods and large values of linear polarization at optical and near-infrared wavelengths.polarization – brown dwarfs – stars: atmospheres – stars: late-type – stars: low-mass § INTRODUCTION The presence of magnetic spots and/or dust clouds in the atmosphere of very low-mass stars and brown dwarfs (“ultra-cool dwarfs”, SpT≥M7) has been proposed to account for the spectro-photometric variability observed by different groups <cit.>. These structures can modulate the shape of the light curve as the dwarf rotates. In some cases, the shape of the observed light curve does not vary significantly with each rotation <cit.>, while in other cases, changes in the shape and amplitude are observed after a few rotations <cit.>. Infrared observations from space <cit.> and the ground <cit.> have shown that the amplitudes and occurrence rates of variability do not decrease with spectral type. The observed amplitudes of photometric variabilityare typically ≤ 1.5 % at optical and infrared wavelengths—even though amplitudes ≥ 2 % have been observed in some L/T dwarfs—with periodicities in the range ≈1.4–20 h, which have been attributed to rotation. Magnetic activity was one of the first mechanisms proposed to account for the observed photometric variability. Hα observations—a sign of magnetic activity—have revealed that nearly 100% of M7–L3.5 dwarfs are magnetically active <cit.>, so the presence of magnetic spots could lead to photometric variability as in stars <cit.>. In addition, optical and radio aurorae have been proposed as another possible origin for some of the variability observed at optical and near-infrared wavelengths <cit.>. However, strong magnetic fields are not common in ultra-cool dwarfs, especially in those cooler than spectral type L3.5 <cit.>, so magnetic activity alone cannot explain the photometric variability observed from late-M to T dwarfs. The low temperatures in the upper atmospheric layers of ultra-cool dwarfs favor the natural formation of condensates <cit.>, which can also induce photometric variability. State-of-the-art radiative transfer models have shown that the observed photometric variability in several ultra-cool dwarfs cannot be explained only by magnetic spots, unless they are combined with dust clouds <cit.>. Recently, <cit.> investigated a sample of 94 ultra-cool dwarfs, all of them with optical spectra containing the Hα region and a photometric monitoring with a duration of at least a few hours. These authors found that the detection rate of magnetic activity and photometric variability follow different dependencies as a function of spectral type, and concluded that i) both phenomena are uncorrelated, and ii) that dust clouds are the most likely origin for the observed photometric variability, even for the warmest ultra-cool dwarfs. The presence of photospheric dust can linearly polarize the output flux of the ultra-cool dwarf by means of scattering processes. This was theoretically studied by <cit.>, <cit.>, <cit.>, and <cit.>. These works predicted values of linear polarization ≲2% at optical and near-infrared wavelengths in good agreement with observations in the filters R, I, Z, J, and/or H <cit.>. Non-zero net linear polarization is detectable provided an asymmetry on the surface of the dwarf avoids the cancellation of the polarimetric signal from different areas of the visible disk, such as an oblate shape and/or an heterogeneous dust distribution. Both of these asymmetries can be present in ultra-cool dwarfs at the same time; however, their effects on the net polarization are different as the dwarf rotates. For an inhomogeneous atmosphere, the polarimetric signal is expected to modulate with rotation <cit.>, while for an asymmetry produced by a non-spherical shape, the most oblate ultra-cool dwarfs show the largest values of linear polarization, but no changes are expected in the signal of linear polarization as the dwarf rotates. <cit.> studied the J-band linear polarimetric properties of a sample of rapidly rotating ultra-cool dwarfs (v sin i≥30 km s^-1), finding that dwarfs with v sin i≥60 km s^-1 show statistically larger values of linear polarization than those with smaller v sin i.Here, we present the results of our optical (I and SDSS-i) and near-infrared (J) photometric monitoring of a sample of 18 M7-L9.5rapidly-rotating ultra-cool dwarfs by using the Liverpool, the IAC80, and the William Herschel telescopes. Our sample is presented in Section <ref>, and our data acquisition and reduction are described in Section <ref>. In Section <ref>, we describe how the differential intensity light curves of the targets were derived, and how we searched for periodic photometric variability. In Section <ref>, the rotation periods (T_ rot) found for our sample are compared with others from the literature, and we use them to derive the likely inclination angle of the rotation axis of our targets. Finally in Section <ref>, we investigate the likely relationship between short rotation periods and large values of linear polarization for some of our targets that have linear polarimetric data acquired with similar filters. Our conclusions are summarized in Section <ref>. | http://arxiv.org/abs/1708.07777v1 | {
"authors": [
"P. A. Miles-Páez",
"E. Pallé",
"M. R. Zapatero Osorio"
],
"categories": [
"astro-ph.SR"
],
"primary_category": "astro-ph.SR",
"published": "20170825152952",
"title": "Rotation periods and photometric variability of rapidly rotating ultra-cool dwarfs"
} |
These authors contributed equally to this work. Center for Nanophotonics, AMOLF, Science Park 104, 1098 XG Amsterdam, The NetherlandsThese authors contributed equally to this work. Center for Nanophotonics, AMOLF, Science Park 104, 1098 XG Amsterdam, The NetherlandsDepartment of Electrical and Computer Engineering, The University of Texas at Austin, Austin, TX 78712, USADepartment of Electrical and Computer Engineering, The University of Texas at Austin, Austin, TX 78712, [email protected] Center for Nanophotonics, AMOLF, Science Park 104, 1098 XG Amsterdam, The NetherlandsOptical circulators are important components of modern day communication technology. With their ability to route photons directionally, these nonreciprocal elements provide useful functionality in photonic circuits and offer prospects for fundamental research on information processing. Developing highly efficient optical circulators thus presents an important challenge, in particular to realize compact reconfigurable implementations that do not rely on a magnetic field bias to break reciprocity. We demonstrate optical circulation based on radiation pressure interactions in an on-chip multimode optomechanical system.We show that mechanically-mediated optical mode conversion in a silica microtoroid provides a synthetic gauge bias for light, which enables a 4-port circulator by exploiting tailored interference between appropriate light paths. We identify two sideband conditions under which ideal circulation is approached. This allows to experimentally demonstrate ∼10 dB isolation and < 3 dB insertion loss in all relevant channels. We show the possibility of actively controlling the bandwidth, isolation ratio, noise performance and circulation direction, enabling ideal opportunities for reconfigurable integrated nanophotonic circuits.Optical circulation in a multimode optomechanical resonator Ewold Verhagen December 30, 2023 ===========================================================§ INTRODUCTIONOptical circulators route photons in a unidirectional fashion among different ports, with diverse applications in advanced communication systems, including dense wavelength division multiplexing and bi-directional sensors and amplifiers. Their operation offers opportunities for routing quantum information <cit.> and to realize photonic states whose propagation in a lattice is topologically protected <cit.>. Traditionally, nonreciprocal elements such as circulators and isolators have relied on applied magnetic bias fields to break time-reversal symmetry and Lorentz reciprocity. While significant progress has been made towards introducing magneto-optic materials in photonic circuits <cit.>, realizing low-loss, linear, efficient, and compact photonic circulators on a chip remains an outstanding challenge.In recent years, coupled-mode systems that create an effective magnetic field using parametric modulations have been recognized as powerful alternatives <cit.>. Optomechanical systems, where multiple optical and mechanical modes are coupled through radiation pressure <cit.>, provide an effective platform to realize such modulation <cit.>. Nonreciprocal transmission and optical isolation were demonstrated in several optomechanical implementations, including ring resonators, photonic crystal nanobeams, and superconducting circuits <cit.>. Recently, nonreciprocal circulation for microwave signals was reported in a superconducting device exhibiting directional frequency conversion <cit.>.Here, by leveraging a synthetic gauge field created in a multimode optomechanical system, we realize a compact and highly reconfigurable on-chip circulator in the photonic (telecommunication) domain.The working principle, relying on tailored interfering paths, allows for operation at equal-frequency input and output fields and could be applied in a wide variety of platforms. We reveal the importance of control fields to regulate the nonreciprocal response and identify two distinct regimes, with and without employing optomechanical gain, where this response can approach ideal circulation.§ RESULTSOur experimental system consists of a high-Q microtoroid resonator <cit.> that is simultaneously coupled to two tapered optical fibres, forming four ports through which light can enter and exit (Fig. <ref>A).The microtoroid supports nearly degenerate odd and even optical modes — superpositions of clockwise and counterclockwise propagating waves — coupled through a mechanical breathing mode, to form a three-mode optomechanical system.A control beam incident through port 1 populates both modes, labelled by i={1,2}, with intracavity control fields α_i that exhibit π/2 phase difference <cit.>.When detuned from the cavity, these control fields induce linear couplings between the mechanical resonator and both cavity modes at rates g_i=g_0α_i, where g_0 is the vacuum optomechanical coupling rate <cit.>. For red-detuned control (Fig. <ref>B), photon transfer from mode 1 to mode 2 via the mechanical resonator then takes place at rate μ_m=2g_1^∗ g_2/, withthe mechanical linewidth.In contrast, transfer from mode 2→1 occurs at rate μ_m^∗, i.e., with opposite phase.The control field thus biases the mode-conversion process with maximally nonreciprocal phase Δϕ≡(μ_m)=π/2 <cit.>.This phase is reminiscent of a synthetic d.c. magnetic flux that breaks time-reversal symmetry <cit.> and can, under appropriate conditions, serve to create an optical circulator. Our system is fully equivalent to the general Fabry-Pérot model sketched in Fig. <ref>C, where the beamsplittersimpart a π/2 phase shift on reflections from either side. From each port the two cavities are then excited with the same π/2 phase difference as the even and odd modes in the ring resonator. In absence of optomechanical coupling this realizes a reciprocal `add-drop' filter: off-resonant light propagates from port 1(3) to port 2(4) and vice versa, while resonant light is routed from 1(2) to 4(3). However, in the presence of a control beam at a frequencydetuned from the cavity frequencies _1,2 by =-, withthe mechanical resonance frequency, a weaker probe beam at frequency =+ (Fig. <ref>B) experiences an optomechanically induced transparency (OMIT <cit.>) window when ≈: the probe is routed from port 1 to 2 rather than `dropped' in port 4. This behaviour is quantified in Fig. <ref>D, which shows scattering matrix elements s_ij, signalling transmission from port j to i, for two control powers. These are obtained from a rigorous coupled-mode model describing the optomechanical three-mode system (Materials and Methods). The response is clearly nonreciprocal: the OMIT window for s_21 (green curves, left panel) does not occur for probe light incident from port 2 (grey), which remains dropped in port 3 (purple, right panel). Likewise, light from port 3 is suppressed (grey) and routed to port 4, etc. The optomechanical interactions in this multimode system thus allow nonreciprocal add-drop functionality, which can also be related to momentum-matching of intracavity control and probe fields <cit.>. For proper bias conditions this yields a circulator, i.e., a control beam incident from port 1 or 3 allows probe transmission1→2, 2→3, 3→4, 4→1, whereas a control from port 2 or 4 reverses the circulation direction.In experiment, we measure circulation using a heterodyne technique (Materials and Methods) to obtain transmission spectra for all 16 port combinations that constitute the 4× 4 scattering matrix. The studied microtoroid has nearly degenerate optical modes atω_i/2π=196.7THz with intrinsic loss rate_0/2π=8.3MHz, and supports a mechanical breathing mode at Ω_m/2π=50.12MHz (/2π=62kHz). The two optical modes are assumed to have equal total loss rates =_0+_a+_b, where _a,b are exchange losses to waveguides a and b. Figure <ref>A shows the measured probe transmittances in the direct (1↔2) and add-drop (2↔3) channels when the control is detuned to the lower mechanical sideband of the cavity (≈-Ω_m).OMIT is observed in the copropagating direct channel (|s_21|^2), accompanied by reduced transmission in the add-drop channel (|s_23|^2).The OMIT features are absent in reverse operation of the device, which thus exhibits nonreciprocal transport.The overall effect is optical circulation in the direction 1→2→3→4→1 (see Supplementary Materials for all 16 scattering matrix elements).To understand the role of mechanically-mediated mode transfer in the observed response, we recapitulate the operation of conventional add-drop filters: when a probe signal from port 1(3) does not excite the two modes (e.g. because it is off-resonance), it interferes constructively in port 2(4) and vice versa (Fig. <ref>D).When the probe does excite both modes, their π/2 phase difference means that light constructively interferes in the drop port (e.g. 1→4, Fig. <ref>E).However, a control beam reconfigures this behaviour through nonreciprocal mode conversion. Now a probe signal incident from port 1 reaches each mode either directly or via the parametric mode conversion process. As a result of the ±π/2 phase shift associated with the mode-transfer paths, the direct and mode-transfer path destructively interfere in both cavities (Fig. <ref>F).Consequently, a probe from port 1 does not excite the modes and is transmitted to port 2 instead of 4.In contrast, a probe from port 2 excites both modes with opposite phase difference, such that interference inside the cavities is constructive and light is routed to port 3 (Fig. <ref>G).Therefore, circulation can only occur in a properly configured two-mode optical system.In particular, the presence of a direct scattering path between ports 1 and 2 (and 3↔4) is crucial to induce the appropriate phase shift and interference causing circulation. The strength of the optomechanical interactions between the optical modes and the mechanical resonance is tuned by the control laser power and can be parametrised by the cooperativities _i≡ 4|g_i|^2/.Perfect optical circulation, with infinite isolation and zero insertion loss, is approached at high cooperativities and vanishing intrinsic loss _0 (Fig. <ref>D).Stronger nonreciprocity is thus obtained for larger optomechanical coupling than currently achieved in our experiment (Fig. <ref>, limited by thermal instabilities), which reached≡_1+_2=0.31 (obtained from a global fit of the 4×4 transmittance matrix, see Supplementary Materials).Moreover, the isolation could be increased by increasing the normalized coupling rates η_a,b=κ_a,b/κ such that the exchange losses to the waveguides overcome the intrinsic cavity losses (η_a,b=(0.19,0.25), /2π=14.8 MHz in Fig. <ref>).Generally, the performance of add-drop filters is limited by a finite mode splitting μ=(_2-_1)/2, which in ring resonators results from surface inhomogeneities that directly couple clockwise and counterclockwise propagating waves <cit.>. This changes the relative phase with which both (odd and even) modes are excited, leading to small (∼-16dB) reflection (|s_33|^2) and cross-coupling (|s_31|^2) signals in our circulator (Fig. <ref>B) indicative of a normalized mode splitting δ≡ 2μ/κ=0.19. Interestingly, the optomechanical interaction suppresses these unwanted signals over the nonreciprocal frequency band(Fig. <ref>B). Assuming equal control powers in the optical modes (g_2= g_1) the resonantly reflectedsignal at port j of waveguide a,b reads |s_jj|^2=4η_a,b^2 δ^2 (1++δ^2)^-2 (see Supplementary Materials). A large cooperativity thus annihilates unwanted reflection (and likewise cross-coupling).This can be understood by recognizing that the nonreciprocal mode coupling at rate μ_m then dominates the direct optical coupling μ, making the impact of the latter on optical response negligible. One way to overcome the limitation on isolation due to intrinsic loss is to introduce gain. This naturally occurs in our system when the control beam is detuned to the blue mechanical sideband (≈) <cit.>. In this regime, the device still acts as a circulator, but with opposite handedness 1←2←3←4←1, since the mode conversion process for blue-detuned control fields has opposite phase, i.e., μ_m=2g_1 g_2^∗/.As a result, constructive and destructive intracavity interference happen opposite to the scenarios depicted in Figs. <ref>F-G.Now, a probe signal co-propagating with the control beam experiences optomechanically induced absorption.Figure <ref>A plots the probe transmittances for the direct and add-drop channels for ≈. We indeed observe a reduction of co-propagating probe signal due to induced absorption (grey data points).Moreover, and resulting from mode splitting, counter-propagating signals (colored data points) experience OMIT, which significantly enhances the nonreciprocal system response. This response is characterised by inspecting the complete 4×4 transmittance matrix (for _L-_p=) shown in Fig. <ref>B. Besides displaying a clear asymmetry, we use this matrix to calculate the per-channel isolation (∼10dB or more) and insertion loss (≲3dB) as displayed in Fig. <ref>C. The properties of the realized circulator agree well with the prediction of our model for a cooperativity of =0.63 (Fig. <ref>D). Importantly, our theory predicts (Supplementary Materials) that loss and gain can be balanced to give complete blocking(transmittance) in unwanted(preferred) directions. For =1 this condition reads δ^2=2η, resulting in a nonreciprocal bandwidth(1-/(1+δ^2)) and finite backreflections δ^2, which can be relatively small in an undercoupled scenario. § DISCUSSIONIn addition to the transport properties, the noise performance of the circulator is important for faithful signal processing, especially for applications in the quantum domain.Assuming equal control powers in both modes, the added noise N_j(ω) at port j due to thermal occupation of the mechanical mode n̅_th can be written at resonance as N_1,3(±)=4n̅_thη_a,bδ^2(1±+δ^2)^-2 and N_2,4(±)=4n̅_thη_a,b(1±+δ^2)^-2, where the upper(lower) sign corresponds to a red(blue) detuned control respectively.Interestingly, the noise added to the ports is independent of the direction of circulation and ports 1 and 3 have noise contribution only in the presence of split optical modes (δ≠ 0).Active control over δ thus presents the possibility to steer noise, choosing to protect either source or receiver.For degenerate modes, the thermal bath adds less than 1 photon (N_2,4(±)<1) in the optical ports when (1±)^-2<1/(4n̅_thη_a,b), showing that it is beneficial to use high-frequency mechanical resonators exhibiting smaller thermal populations. Moreover, this regime is more easily met for a red-detuned control, where an increased cooperativity widens the OMIT peak <cit.> and distributes the noise over a larger bandwidth. In contrast, a blue-detuned control deteriorates this bandwidth and thus requires more stringent conditions on n̅_th. As such, to achieve strong circulation at small control power, blue detuning can be beneficial, whereas to maximize bandwidth and noise performance, the red-detuned regime is preferred. It would be good to ascertain whether schemes that include more optical or mechanical modes would allow increased flexibility to mitigate noise and improve bandwidth.In conclusion, we demonstrated nonreciprocal circulation of light through radiation pressure interactions in a three-mode system, with active reconfigurability regulated via the strength, phase and detuning of control fields. Our experiments and theoretical model recognize different regimes of circulating response and reveal the general importance of destructive intracavity interference between direct and mode-conversion paths on the device properties. These principles can lead to useful functionality for signal processing and optical routing in compact, on-chip photonic devices.§ MATERIALS AND METHODSExperimental detailsThe microtoroid is fabricated using previously reported techniques, see e.g. <cit.>.The sample is placed in a vacuum chamber operated at 3× 10^-6 mbar and room temperature.Two tapered optical fibres, mounted on translation stages, are positioned near the toroid tocouple the waveguides and optical modes.Three electronically controlled 1×2 optical switches (SW) are used in tandem to launch the probe beam into one of the four ports. A schematic of the experimental set-up is provided in the Supplementary Materials. A tunable laser (New Focus, TLB-6728) atis locked to one of the mechanical sidebands of the cavity mode, with a Pound-Drever-Hall scheme, and launched in port 1. The output of a vector network analyser (VNA, R&S ZNB8) at frequency ω drives a Double-Parallel Mach-Zehnder Interferometer (DPMZI, Thorlabs LN86S-FC) to generate the probe beam at frequency =±ω.The DPMZI is operated in a single-sideband suppressed-carrier mode, allowing selection of the relevant sideband at frequency =+ω or =-ω.The power and polarisation of the control and probe beams are controlled by variable optical attenuators and fibre polarisation controllers respectively.The probe beamexiting a port is combined with the control beam on a fast, low-noise photo detector whose output is analysed by the VNA.All four detectors are connected to the VNA through a 4×1 radio frequency switch (RFSW). Scattering matrix for the optomechanical circulatorConsidering an optomechanical system that consists of two optical modes coupled to one mechanical mode, the modal amplitudes of the intracavity probe photons δ a_i (i=1,2) and the phonon annihilation operator b are governed by the following coupled dynamical equations <cit.>: d/dtδ a_i=(Δ̅_i-κ_i/2)δ a_i+ g_i(b+b^†)+∑_j=1^4d_jiδ s_j^+ d/dt b = (--/2)b+∑_i=1^2 g_i^*δ a_i+g_i δ a_i^†+√()b_inIn these relations, the loss rate _i of each optical mode i is defined as _i≡_0,i + _a,i + _b,i, where _0,i refers to the intrinsic optical loss rate of the mode, and _a,i, _b,i are the exchange losses to waveguides a and b.The D-matrix elements d_ji specify the coupling between input fields δ s_j^+, incident from port j, and the optical modes. In addition, g_1,2=g_0 α_1,2 and _1,2=Δ_1,2+(2g_0^2/)(|α_1|^2+|α_2|^2) respectively represent the enhanced optomechanical coupling and the modified frequency detunings, where Δ_1,2=_L-_1,2 is the frequency detuning of the control laser with respect to the optical resonance frequencies.Here, we consider a frequency splitting of 2μ between the resonance frequencies of the two optical modes, and without loss of generality assume _1,2=ω_0∓μ thus _1,2=±μ.Equations (1,2) completely describe the behaviour of the system in connection with the input-output relations:[ s_1^-; s_2^-; s_3^-; s_4^- ] = C[ s_1^+; s_2^+; s_3^+; s_4^+ ] + D[ a_1; a_2 ]where the matrix C defines the port-to-port direct path scattering matrix of the optical circuit, while D and its transpose D describe the mode-to-port and port-to-mode coupling processes respectively.Using the rotating wave approximation, equations (<ref> - <ref>) lead to the frequency-domain scattering matrix: S =C+ D (M+ I)^-1 DS =C+ D [ ∓ |g_1|^2/^± ∓ (g_1 g_2)/^±; ∓ (g_1 g_2)/^± ∓ |g_2|^2/^± ]^-1 D^T.Here the upper and lower signs correspond to a control laser that is red- (=-) and blue-detuned (=+) with respect to the optical resonance.In addition, ^±≡∓+/2 and Σ_o_1,2≡+_1,2+κ_1,2/2 respectively represent the inverse mechanical and optical susceptibilities, where _1,2=±μ. Note that μ_m, the mechanically-mediated mode conversion rate, is identified from the off-diagonal elements of the M matrix.The direct-path scattering matrix C for the side-coupled geometry that describes both systems in Fig. <ref>A and Fig. <ref>C isC= [ 0 1 0 0; 1 0 0 0; 0 0 0 1; 0 0 1 0 ].The D matrix is obtained from symmetry considerations, power conservation (D D = diag(_a,1+_b,1,_a,2+_b,2)) and time reversal symmetry (CD=-D) <cit.>.For the even and odd mode picture, the D matrix readsD= 1/√(2)[√(_a,1) -√(_a,2);√(_a,1)√(_a,2);√(_b,1) -√(_b,2);√(_b,1)√(_b,2);].Equation <ref>, combined with eqs. <ref> and <ref> fully specifies the scattering matrix of our 4-port circulator.We assume equal losses (_0,1=_0,2=_0, _a,1=_a,2=_a, _b,1=_b,2=_b) and equal control powers (g_2= g_1= g) in the two optical modes to fit the experimentally measured transmittances (|s_ij|^2) using a global fitting procedure (see Supplementary Materials).§ SUPPLEMENTARY MATERIALSSection S1. 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It is part of the research programme of the Netherlands Organisation for Scientific Research (NWO). E.V. acknowledges support by the European Union's Horizon 2020 research and innovation programme under grant agreement No 732894 (FET Proactive HOT). Author contributions: F.R. developed the experimental setup. F.R. and J.P.M. performed the experiments and analysed the data. M.-A.M. developed the theoretical model, with contributions fromE.V., A.A., F.R, and J.P.M. E.V. and A.A. supervised the project. All authors contributed to the writing of the manuscript. Competing interests: The authors declare no competing financial interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.Supplementary Materials for “Optical circulation in a multimode optomechanical resonator" § EXPERIMENTAL SETUPFigure <ref> presents a detailed schematic of the experimental setup used in our experiment.Each port of the device is connected to an output of the optical switch network and to a fast detector (D) via commercially available fibre optic circulators (C, Thorlabs).The electronically controllable optical (SWi) and RF (RFSW) switches, along with the circulators, allow for straightforward measurement of the full transmittance matrix.Four fibre polarisation controllers (FPCs) placed directly after optical switches 2 and 3 ensure that the polarisation of the incoming probe fields can be matched to that of the cavity mode.The polarisation of the control field is separately controlled with a fifth FPC placed directly after the Electro-Optic Modulator (EOM). The FPCs are omitted in the schematic for clarity. We use the EOM for a Pound-Drever-Hall locking schemethat locks the control field to a motional sideband of the optical cavity.Furthermore, two variable optical attenuators (not shown in the schematic) placed after the DPMZI and the EOM control the power levels of the probe and control beams respectively.In our experiments with red- and blue-detuned control fields we used incident control powers of 60 and 214, respectively. § MEASUREMENT AND FITTING PROCEDURE FOR THE TRANSMITTANCE MATRIX To provide calibrated and reliable S-matrix elements, the following experimental protocol is followed: first, careful characterization of all the lossy elements between the microtoroid and detectors is performed. This includes all detector responses and losses associated with the commercial components and the tapered fibres.Next, both fibres are gently moved towards the microtoroid until the desired coupling strength is obtained.The latter step involves continuous spectroscopy over a frequency region around _c, and is performed at higher probe power levels (∼1) to allow direct measurements of all detector output voltages on an oscilloscope. These voltages are obtained by splitting the detector output lines just before the RFSW (not shown). For the direct channels (1↔2, 3↔4) it is then possible to use the measured off-resonance voltages to normalize the on-resonance cavity response, which directly yields the on-resonance transmittance efficiency.Considering for example port 1 → port 2, the non-resonant voltage on D2, together with the response of the detector and losses of circulator C2 and tapered fibre, allows us to infer the actual power that entered port 1.Using the on-resonance voltages that are recorded by detectors D1/D3/D4 (obtained via fitting a Lorentzian lineshape) and compensating with the appropriate losses and detector response functions, we can then 1) determine the power exiting ports 1/3/4 and 2) calculate the reflection/cross-coupling/add-drop efficiencies.Raw VNA spectra are converted into transmittance spectra with the help of previously determined transmittance efficiencies. This post-processing involves the removal of a small portion of raw VNA spectroscopic data surrounding the mechanical resonance peak. Next, Lorentzian lineshapes are fitted to this cropped (and squared) VNA data, of which the fitted maxima are used to normalize the reflection, cross-coupling and add-drop spectra.For the direct channels, normalization is performed with respect to the minimum of the fitted line.All curves are then multiplied with their respective transmittance efficiencies to yield experimental data.Once the transmittances are obtained, the data around the mechanical resonance peak is included and fitted by varying the mode splitting and cooperativity as fit parameters. During fitting, , and _0 are held fixed, as they are obtained from independent spectroscopic measurements.To account for experimental drift during data acquisition, the only parameters that are allowed to vary between the different s-parameters are the control detunings . The transmittance spectra for the full 4× 4 S matrix, including global fit, are shown in Fig. <ref>(<ref>) for red(blue)-detuned control beam, respectively. § CONDITIONS FOR NEAR-IDEAL CIRCULATIONHere, we examine the scattering matrix of the circulator both in the red- and blue-detuned regimes and investigate the possibility of near-ideal circulation. For simplicity we assume that both the optical modes have equal losses (_a,1=_a,2=_a, _b,1=_b,2=_b, _0,1=_0,2=_0 and =_0+_a+_b) and are pumped with equal intensity and with π/2 phase difference (g_2= g_1 =g). In addition, we focus only at the resonance frequency where the optomechanical intercations are maximal. For a red-detuned system (=-) at resonance (ω=) the S matrix can be simplified toS= [ 0 1 0 0; 1 0 0 0; 0 0 0 1; 0 0 1 0 ] -2/1++δ^2[η_aδη_a (1+)√(η_aη_b)δ √(η_aη_b)(1+); η_aη_aδ √(η_aη_b)√(η_aη_b)δ;√(η_aη_b)δ √(η_aη_b)(1+)η_bδη_b (1+); √(η_aη_b)√(η_aη_b)δ η_bη_bδ ]where, =_1+_2 represents the total cooperativity of both modes. In addition, we have defined η_a,b=_a,b/ as the ratio of leakage losses to the total losses of each mode and δ=2μ/ as the normalized frequency splitting of the even and odd modes.Here, the aim is to maximize the clockwise port-to-port coupling coefficients, i.e., {s_21,s_32,s_43,s_14}, while minimizing the counterclockwise port-to-port couplings {s_12,s_23,s_34,s_41}.On the other hand, of interest would be to simultaneously minimize the reflection coefficients (s_11,s_22,s_33,s_44) and the cross-coupling terms (s_13,s_31,s_24,s_42).According to Eq. (<ref>), ideal circulation can be acheieved in the limit of large cooperativities (→∞) and for zero internal losses when assuming equal mode coupling to the upper and lower waveguide channels (η_a,b→ 1/2).On the other hand, another interesting observation in the S matrix of Eq. (<ref>) is that the reflection coefficients and the cross-coupling terms are all proportional with the frequency splitting 2μ (appearing as δ in the S matrix).This can be understood easily in the basis of rotating whispering gallery modes. In such picture, any coupling between the counter-rotating modes is mediated through their mutual coupling rate μ which instead results in a finite reflection and coupling of the diagonal ports.Similarly, for a blue-detuned control (=) at resonance (ω=-) the S matrix can be written asS= [ 0 1 0 0; 1 0 0 0; 0 0 0 1; 0 0 1 0 ] -2/1-+δ^2[η_aδη_a (1-)√(η_aη_b)δ √(η_aη_b)(1-); η_aη_aδ √(η_aη_b) √(η_aη_b);√(η_aη_b)δ √(η_aη_b)(1-)η_bδη_b (1-); √(η_aη_b)√(η_aη_b)δ η_bη_bδ ].which is similar to that of the red-detuned system when replacingwith -. In a similar fashion, one can show that in this case a large cooperativity results in ideal circulation, however, this is an unphysical scenario given that in the blue-detuned regime large cooperativities give rise to parametric instabilities. On the other hand, the S matrix of Eq. (<ref>) exhibits interesting properties and can become close to that of an ideal circulator for critical choices of the parameters involved. Interestingly, compared to the red-detuned case, here the circulation direction is reversed, i.e., {s_12,s_23,s_34,s_41} can be close to unity while {s_21,s_32,s_43,s_14} are minimum. In fact, for a particular set of parameters, one can show that s_21 and s_43 become zero for=1+δ^2-2η_a=1+δ^2-2η_bwhich clearly requires equal leakage rate for both channels η_a,b=η. In addition, these conditions demand a total cooperativity larger than the ratio of internal loss to total losses (η_0=1-2η). Figure <ref> depicts the cooperativity required to satisfy this condition for different values of η and δ. Interestingly, the choice of δ^2=2η demands a cooperativity of =1 which instead results in the following resonant S matrix:S= [ -δ1 -δ0;0 -δ -1 -δ; -δ0 -δ1; -1 -δ0 -δ ]According to this simple relation, near-ideal circulation can be achieved for =1 and δ^2=2η. On the other hand, one can reduce the reflection coefficients and diagonal port couplings by decreasing δ which instead requires a decrease in η. This latter scenario thus happens for a weak waveguide-cavity coupling. § PARAMETRIC INSTABILITY THRESHOLDFor a blue-detuned control laser, the optomechanical system can enter a parametric instability regime. For a single-mode optomechanical system, the threshold cooperativity associated with the onset of instabilities is found to be _th=1. Similar relation holds for the microtoroid system when assuming a degenerate pair of counter rotating modes where only one of the modes is populated by the control laser. However, assuming a finite coupling μ between the two modes, this relation is no longer valid. In this case, one would expect a higher cooperativity for bringing the system to instability threshold given that the active mode is coupled to another mode which is naturally lossy. For a blue-detuned system, the dynamical equations governing the optomechanical system in the absence of input probe signals can be written as:d/dt[ δ a_1; b^†; δ a_2 ] =[ _1+κ_1/2g_10; -g_1^*+/2 -g_2^*;0g_2 _2+κ_2/2 ][ δ a_1; b^†; δ a_2 ]Assuming _1,2=±μ and g_2= g_1= g, we consider an ansatz of (δ a_1 , b^† , δ a_2 )^T=(α_1 , β , α_2 ) e^i te^st, which results in the following cubic equation for the exponent ss^3+(κ+/2)s^2+(μ^2+κ^2/4+κ/2-2|g|^2)s+μ^2/2+κ^2/4/2-κ|g|^2=0.The stability of the system can now be investigated by finding a parameter regime for which all roots of this equation fall on the left side of the imaginary axis, i.e., Real(s)<0. This latter can be obtained through the Routh-Hurwitz stability criterion. According to the Routh-Hurwitz criterion, a generic cubic equation of the form s^3+a_2s^2+a_1s+a_0 has all roots in the negative real part plane as long as a_0>0, a_2>0 and a_1a_2>a_0. For Eq.(<ref>), the stability conditions are found to be:/2(μ^2+κ^2/4)>κ |g|^2, (κ+/2)(μ^2+κ^2/4+κ/2-2|g|^2) > /2(μ^2+κ^2/4)-κ |g|^2,Given that in practice ≪κ, the first condition imposes a lower bound on the control power required to bring the system to instability threshold. Therefore, by rewriting Eq. (<ref>) in terms of the total cooperativity, the condition of stability can be written as <1+δ^2, which results into the following expression for the instability threshold_th=1+δ^2.This relation assures that the condition of near-ideal circulation (Eq.(<ref>)) is accessible without running into instability. It is worth noting that this condition is obtained under the rotating wave approximation. A more general expression can be found when considering both mechanical sidebands.§ CIRCULATION BANDWIDTHAccording to the results presented in the main text, the bandwidth of the optomechanical circulator is dictated by the linewidth of the optomechanically induced transparency (OMIT) or absorption (OMIA) window. Considering the S-matrix derived in the Materials and Methods section, the bandwidth can be obtained through the frequency-dependent factor of [(ω I+M)]^-1,1/(ω I+M)=1/Σ_o^2(ω)-μ^2∓ 2|g|^2Σ_o(ω)/Σ_m(ω),where, to simplify the analysis, we have assumed equal losses in both modes, κ_1=κ_2, and equal intensity pumping, |g_1|=|g_2|. In this relation the up and down signs are associated with the red- and blue-detuned regimes respectively.Given that in general κ≫, the inverse optical susceptibilities are fairly constant near the resonance ω=± and over the small frequency range of OMIT/OMIA.Thus one can assume Σ_o(ω)≈κ/2 which greatly simplifies Eq.(<ref>) to1/(ω I+M)=-4/κ^2(1+δ^2)(1-±/1+δ^2/ω∓+(1±/1+δ^2)/2).According to this relation, the transparency or absorption window is found to beBW=(1±/1+δ^2),which, could be compared with a similar relation for a single-mode optomechanical cavity: BW=(1±). Recalling that δ=μ/(κ/2) is a normalized mode splitting, relation (<ref>) shows that in general the lifted degeneracy of the modes decreases the bandwidth in the red-detuned regime while it increases the bandwidth in the blue-detuned case. In addition, relation (<ref>) is in agreement with the fact that the bandwidth should approach zero at the onset of parametric instabilities.§ THERMAL NOISE Thermal phonons in the mechanical resonator interact with the control beam creating cavity photons that contribute to noise at the output ports of the device. The effect of thermal noise can be considered in the linearized mechanical equation of motion as:d/dtb = (- -/2)b + (g_1^*δ a_1+g_1δ a_1^†+g_2^*δ a_2+g_2δ a_2^†)+√()ξ(t) where ξ(t) is the noise operator with correlations ⟨ξ(t)ξ^†(t') ⟩=(n̅_th+1)δ(t-t') and ⟨ξ^†(t)ξ(t') ⟩=n̅_thδ(t-t') where n̅_th=k_B T/ħ is the thermal occupation number of the mechanical mode.The coupled optomechanical equations of motion can now be used to write the frequency-domain contribution of noise to all four ports δ s_n,i^-(ω) in terms of the noise operator ξ(ω). Under the rotating wave approximation, the output noise operators for a red-detuned system are found to be:[ δ s_n,1^-(ω); δ s_n,2^-(ω); δ s_n,3^-(ω); δ s_n,4^-(ω) ]= -/^+(ω) D(M+ω I)^-1[ g_1; g_2 ]√()ξ(ω),while, for a blue-detuned control laser, we have:[ δ s_n,1^-(ω); δ s_n,2^-(ω); δ s_n,3^-(ω); δ s_n,4^-(ω) ]= -/^-(ω) D(M+ω I)^-1[ g_1; g_2 ]√()ξ^†(ω),where,M+ω I= [∓|g_1|^2/^± ∓g_1g_2^*/^±; ∓g_1^*g_2/^±∓|g_2|^2/^± ].Here, the mechanical and optical susceptibilities are defined as ^±(ω)=ω∓+/2 and Σ_o_1,2(ω)=(ω++_1,2/2).The power spectral density of the thermal photon flux at output ports can be obtained as N_i(ω)=⟨δ s_n,i^†(ω)δ s_n,i(ω) ⟩.To simplify the analysis, we assume that the optical modes have equal losses (_a,1=_a,2=_a, _b,1=_b,2=_b, _1,2=, η_a,b=_a,b/) and are driven out of phase with equal intensities such that g_2= g_1= g. The noise contribution at each port then reads[ N_1(ω); N_2(ω); N_3(ω); N_4(ω) ] = 2 |g|^2n̅_th/|(Σ_o^2-μ^2)^±∓2|g|^2Σ_o|^2[ _aμ^2; _a|Σ_o|^2; _bμ^2; _b|Σ_o|^2 ],where k_B is the Boltzmann constant, and T= 300 is the bath temperature.On resonance (|ω_p-ω_L|=) the noise contribution can be written as[ N_1(±); N_2(±); N_3(±); N_4(±) ] = 4 n̅_th/(1±+δ^2)^2[ η_aδ^2;η_a; η_bδ^2;η_b ]where the upper and lower signs correspond to red- and blue-detuned controls respectively. Figure <ref> plots the output noise at port 2 for the case of red- and blue-detuned controls as a function of the cooperativity. | http://arxiv.org/abs/1708.07792v1 | {
"authors": [
"Freek Ruesink",
"John P. Mathew",
"Mohammad-Ali Miri",
"Andrea Alù",
"Ewold Verhagen"
],
"categories": [
"physics.optics"
],
"primary_category": "physics.optics",
"published": "20170825155908",
"title": "Optical circulation in a multimode optomechanical resonator"
} |
Semantic Foggy Scene Understanding with Synthetic DataChristos Sakaridis Dengxin Dai Luc Van Gool=========================================================This paper analyses discontinuous Galerkin finite element methods (DGFEM) to approximate a regular solution to the von Kármán equations defined on a polygonal domain. A discrete inf-sup condition sufficient for the stability of the discontinuous Galerkin discretization of a well-posed linear problem is established and this allows the proof of local existence and uniqueness of a discrete solution to the non-linear problem with a Banach fixed point theorem. The Newton scheme is locally second-order convergent and appears to bea robust solution strategy up to machine precision. A comprehensive a priori and a posteriori energy-norm error analysis relies on one sufficiently large stabilization parameter and sufficiently fine triangulations. In case the other stabilization parameter degenerates towards infinity, the DGFEM reduces to a novel C^0 interior penalty method (IPDG). In contrast to the known C^0-IPDG due to Brenner et al <cit.>, the overall discrete formulation maintains symmetry of the trilinear form in the first two components – despite the general non-symmetry of the discrete nonlinear problems. Moreover, a reliable and efficient a posteriori error analysis immediately follows for the DGFEM of this paper, while the different norms in the known C^0-IPDG lead to complications with some non-residual type remaining terms. Numerical experiments confirm the best-approximation results and the equivalence of the error and the error estimators. A related adaptive mesh-refining algorithm leads to optimal empirical convergence rates for a non convex domain.§ INTRODUCTIONThe discontinuous Galerkin finite element methods (DGFEM) have become popular for the numerical solution of a large range of problems in partial differential equations, which include linear and nonlinear problems, convected-dominated diffusion for second- and fourth-order elliptic problems.Their advantages are well-known;the flexibility offered by the discontinuous basis functions eases the global finite element assembly and the hanging nodes in mesh generation helps to handle complicated geometry. The continuity restriction for conforming FEM is relaxed, thereby making it an interesting choice for adaptive mesh refinements. On the other hand, conforming finite element methods for plate problems demand C^1 continuity and involve complicated higher-order finite elements. The simplest examples are Argyris finite element with 21 degrees of freedom in a triangle and element with 16 degrees of freedom in a rectangle. Nonconforming <cit.>, mixed and hybrid <cit.> finite element methods are also alternative approaches that have been used to relax the C^1-continuity. Discontinuous Galerkin methods are well studied for linear fourth-order elliptic problems, e.g., the hp-version of the nonsymmetric interior penalty DGFEM (NIPG) <cit.>, the hp-version of the symmetric interior penalty DGFEM (SIPG) <cit.> and a combined analysis of NIPG and SIPG in <cit.>. The literature on a posteriori error analysis for biharmonic problems with DGFEM include <cit.> and a quadratic C^0-interior penalty method <cit.>. The medius analysis in <cit.> combines ideas of a priori and a posteriori analysis to establish error estimates for DGFEM under minimal regularity assumptions on the exact solution. This paper concerns discontinuous Galerkin finite element methods for the approximation of regular solution to the von Kármán equations, which describe the deflection of very thin elastic plates. Those plates aremodeled by a semi-linear system of fourth-order partial differential equations and can be described as follows. For a given load function f∈, seek u,v ∈ H^2_0(Ω) such thatΔ^2 u =[u,v]+fand Δ^2 v =-[u,u]in Ωwith the biharmonic operator Δ^2 and the von Kármán bracket [∙,∙], Δ^2φ:=φ_xxxx+2φ_xxyy+φ_yyyy, and [η,χ]:=η_xxχ_yy+η_yyχ_xx-2η_xyχ_xy=(D^2η):D^2χ for the co-factor matrix (D^2η) of D^2η. The colon : denotes the scalar product of two 2× 2 matrices.In <cit.>, conforming finite element approximations for the von Kármán equations are analyzed and an error estimate in energy norm is derived for approximations of regular solutions. Mixed and hybrid methods reduce the system of fourth-order equations into a system of second-order equations <cit.>. Conforming finite element methods for the canonical von Kármán equations have been proposed and error estimates in energy, H^1 and L^2 norms are established in <cit.> under realistic regularity assumption on the exact solution. Nonconforming FEMs have also been analyzed for this problem <cit.>. An a priori error analysis for a C^0 interior penalty method of this problem is studied in <cit.>. Recently, an abstract framework for nonconforming discretization of a class of semilinear elliptic problems which include is analyzed in <cit.>. In this paper, discontinuous Galerkin finite element methods are applied to approximate the regular solutions of the von Kármán equations. To highlight the contribution, underminimal regularity assumption of the exact solution, optimal order a priori error estimates are obtained and a reliable and efficient a posteriori error estimator is designed. Moreover, a priori and a posteriori error estimates for a C^0 interior penalty method for the von Kármán equations are recovered as a special case. The comprehensive a priori analysis in <cit.> controls the error in the stronger norm ·_h≡∙_ and therefore requires a more involved mathematics and a trilinear form b_IP without symmetry in the first two variables, cf. Remark <ref>. The remaining parts of the paper are organized as follows. Section 2 describes some preliminary results and introducesdiscontinuous Galerkin finite element methods for von Kármán equations. Section 3 discusses some auxiliary results required for a priori and a posteriori error analysis. In Section 4, a discrete inf-sup condition is established for a linearized problem for the proof of the existence, local uniqueness and error estimates of the discrete solution of the non-linear problem. In Section 5, a reliable and efficient a posteriori error estimator is derived. Section 6 derives a priori and a posteriori error estimates for a C^0 interior penalty method.Section 7 confirms the theoretical results in various numerical experiments and establishes an adaptive mesh-refining algorithm. Throughout the paper, standard notation on Lebesgue and Sobolev spaces and their norms are employed.The standard semi-norm and norm on H^s(Ω) (resp. W^s,p (Ω)) for s>0 are denoted by |∙|_s and ∙_s (resp. |∙|_s,p and ∙_s,p ).Bold letters refer to vector valued functions and spaces, e.g. = X× X. The positive constants C appearing in the inequalities denote generic constants which do not depend on the mesh-size. The notation A≲ B means that there exists a generic constant C independent of the mesh parameters and independent of the stabilization parameters σ_1 and σ_2≥ 1 such that A ≤ CB; A≈ B abbreviates A≲ B≲ A.§ PRELIMINARIESThis section introduces weak and discontinuous Galerkin (dG) formulations for the von Kármán equations. §.§ Weak formulationThe weak formulation of von Kármán equations (<ref>) reads: Given f∈, seeku,v∈X:= such that a(u,φ_1)+ b(u,v,φ_1)+b(v,u,φ_1)=l(φ_1) φ_1∈ X a(v,φ_2)-b(u,u,φ_2) =0φ_2 ∈ X. Here and throughout the paper, for all η,χ,φ∈ X, a(η,χ):= D^2 η:D^2χ, b(η,χ,φ):=- [η,χ]φ,andl(φ):=∫_Ωfφ.Given F=(f,0)∈ L^2(Ω)× L^2(Ω), the combined vector form seeks Ψ=(u,v)∈:=X× X≡× such thatN(Ψ;Φ):=A(Ψ,Φ)+B(Ψ,Ψ,Φ)-L(Φ)=0Φ∈,where, for all Ξ=(ξ_1,ξ_2),Θ=(θ_1,θ_2), and Φ=(φ_1,φ_2)∈,A(Θ,Φ):=a(θ_1,φ_1)+a(θ_2,φ_2),B(Ξ,Θ,Φ):=b(ξ_1,θ_2,φ_1)+b(ξ_2,θ_1,φ_1)-b(ξ_1,θ_1,φ_2) andL(Φ):=l(φ_1).Let ∙_2 denote the product norm on defined by Φ_2:=(|φ_1|_2,Ω^2+|φ_2|_2,Ω^2)^1/2 for all Φ=(φ_1,φ_2)∈. It is easy to verify that the following boundedness and ellipticity properties holdA(Θ,Φ)≤Θ_2Φ_2, A(Θ,Θ) ≥Θ_2^2, B(Ξ, Θ, Φ) ≤C Ξ_2Θ_2Φ_2.Since b(∙,∙,∙) is symmetric in first two variables, the trilinear form B(∙,∙,∙) is symmetric in first two variables.For results regarding the existence of solution to (<ref>), regularity and bifurcation phenomena, we referto <cit.>. It is well known <cit.> that on a polygonal domain Ω, for given f∈ H^-1(Ω), the solutions u,v belong to ∩ H^2+α(Ω), for the index of elliptic regularity α∈ (,1] determined by the interior angles of Ω. Note that when Ω is convex; α=1; that is, the solution belongs to ∩ H^3(Ω). Unless specified otherwise, the parameter α is supposed to satisfy 1/2<α≤ 1. Throughout the paper, we consider the approximation of a regular solution <cit.> Ψ to the non-linear operator N(Ψ; Φ)=0 for all Φ∈of (<ref>) in the sense that the bounded derivative DN(Ψ) of the operator N at the solution Ψ is an isomorphism in the Banach space; this is equivalent to an inf-sup condition 0<β:=inf_Θ∈ Θ_2=1sup_Φ∈ Φ_2=1(A(Θ,Φ)+2B(Ψ,Θ,Φ)).§.§ TriangulationsLet be a shape-regular <cit.> triangulation of Ω into closed triangles. The set of all internal vertices (resp. boundary vertices) andinterior edges (resp.boundary edges)of the triangulationare denoted by (Ω) (resp.(∂Ω)) and (Ω) (resp. (∂Ω)). Define a piecewise constant mesh function h_(x)=h_K= diam (K) for all x ∈ K, K∈, and set h:=max_K∈h_K. Also define a piecewise constant edge-function on :=(Ω)∪(∂Ω) by h_|_E=h_E= diam(E) for any E∈. Set of all edges of K is denoted by (K). Note that for a shape-regular family, there exists a positive constant C independent of h such that any K∈ and any E∈∂ K satisfy Ch_K≤ h_E≤ h_K.Let P_r( K) denote the set of all polynomials of degree less than or equal to r and P_r():={φ∈ L^2(Ω): ∀ K∈,φ|_K∈ P_r(K)} and write P_r():=P_r()× P_r() for pairs of piecewise polynomials. For a nonnegative integer s, define the broken Sobolev space for the subdivision asH^s()={φ∈: φ|_K∈ H^s(K) K∈}with the broken Sobolev semi-norm|∙|_H^s() and norm ∙_H^s() defined by|φ|_H^s()=(∑_K∈ |φ|_H^s(K)^2)^1/2 and φ_H^s()=(∑_K∈φ_H^s(K)^2)^1/2.Define the jump [φ]_E=φ|_K_+-φ|_K_- and the average ⟨φ⟩_E=(φ|_K_++φ|_K_-) across the interior edge E of φ∈ H^1() of the adjacent trianglesK_+ and K_-. Extend the definition of the jump and the average to an edge lying in boundary by [φ]_E=φ|_E and ⟨φ⟩_E=φ|_E for E∈(∂Ω) owing to the homogeneous boundary conditions.For any vector function, jump and average are understood componentwise. Set Γ≡⋃_E∈E. §.§ Discrete norms and bilinear formsFor 1/2<α≤ 1, abbreviate Y_h:=(X∩ H^2+α(Ω))+ and :=Y_h× Y_h. For all η,χ∈ Y_h, φ∈ X+, introduce the bilinear, trilinear and linear forms bya_(η,χ):=∑_K∈∫_K D^2η:D^2χ-(J(η,χ)+ J(χ,η)) +J_σ_1,σ_2(η,χ), b_(η,χ,φ):=-∑_K∈∫_K [η,χ]φ, l_(φ):= ∑_K∈∫_Kfφ,J(η,χ)=∑_E∈∫_E [∇χ]_E·⟨ D^2η ν_E⟩_E ,with σ_1>0 and σ_2>0 to be suitably chosen in the jump terms across any edge E∈ with unit normal vector ν_E andJ_σ_1,σ_2(η,χ):=∑_E∈σ_1/h_E^3∫_E[η]_E[χ]_E+∑_E∈σ_2/h_E∫_E[∇η·ν_E]_E[∇χ·ν_E]_E. The discontinuous Galerkin (dG) finite element formulation of (<ref>) seeks (u_,v_)∈:=× such that, for all (φ_1, φ_2)∈,a_(u_,φ_1)+b_(u_,v_,φ_1)+b_(v_,u_,φ_1)=l_(φ_1),a_(v_,φ_2)-b_(u_,u_,φ_2)=0.The combined vector form seeks Ψ_≡ (u_,v_)∈ such that, for all Φ_∈,N_h(Ψ_;Φ_):=A_(Ψ_,Φ_)+B_(Ψ_,Ψ_,Φ_)-L_(Φ_)=0,where, Ξ_=(ξ_1, ξ_2),Θ_=(θ_1,θ_2),Φ_=(φ_1,φ_2)∈,A_(Θ_,Φ_):=a_(θ_1,φ_1)+a_(θ_2,φ_2),B_(Ξ_,Θ_,Φ_):=b_(ξ_1,θ_2,φ_1)+b_(ξ_2,θ_1,φ_1)-b_(ξ_1,θ_1,φ_2), L_(Φ_):=l_(φ_1).Note that b_(∙,∙,∙) is symmetric in the first and second variables, and so is B_(∙,∙,∙).For φ∈ H^2() and Φ=(φ_1,φ_2)∈^2()≡ H^2()× H^2(), define the mesh dependentnorms ∙_ and ∙_ by φ_^2:=|φ|_ H^2()^2+∑_E∈σ_1/h_E^3[φ]_E_L^2(E)^2+∑_E∈σ_2/h_E[∇φ·ν_E]_E_L^2(E)^2,Φ_^2:=φ_1_^2+φ_2_^2.For ξ∈ Y_h≡ (X∩ H^2+α(Ω))+ and Ξ=(ξ_1,ξ_2)∈≡ Y_h× Y_h, define the auxiliary norms ∙_h and ∙_hbyξ_h^2:=ξ_^2+∑_E∈∑_j,k=1^2h_E^1/2⟨∂^2ξ/∂ x_j ∂ x_k⟩_E_L^2(E)^2 and Ξ_h^2:=ξ_1_h^2+ξ_2_h^2. § AUXILIARY RESULTS This section discusses some auxiliary results and establishes the boundedness and ellipticity results required for the analysis.§.§ Some known operator boundsThis subsection recalls a few standard results. Throughout this subsection, the generic multiplicative constant C≈ 1 hidden in the brief notation ≲ depends on the shape regularity of the triangulationand arising parameters like the polynomial degree r∈_0 or the Lebesgue index p and the Sobolev indices ℓ,s>1/2 and 1/2<α≤ 1; C is independent of the mesh-size. <cit.> There exists a positive constant C that depends on ℓ∈ℕ, p∈ and the shape regularity of the triangulation such that, for 1≤ℓ,2≤ p≤∞, for any ξ∈ P_r(K) satisfies ξ_L^p(K) ≲ h_K^(2-p)/pξ_L^2(K) and |ξ|_H^ℓ(K)≲ h_K^-1|ξ|_H^ℓ-1(K), for any K ∈ with E⊂(K), where ξ_L^p(E)≲ h_E^1/p-1/2ξ_L^2(E).The following trace inequalities hold for K∈ and s>1/2. * <cit.> ξ_L^2(∂ K)≲ h_K^-1/2ξ_L^2(K)ξ∈ P_r(K); * <cit.>ξ_L^2(∂ K)≲h_K^s-1/2ξ_H^s(K)+h_K^-1/2ξ_L^2(K) for all ξ∈ H^s(K). <cit.> There exists a linear operator Π_h: H^s()→ P_r(), such that, for 0≤ q≤ s, m=min(r+1,s) and 1/2<α≤ 1,φ-Π_hφ_H^q(K) ≲ h_K^m-qφ_H^s(K) K∈ andφ∈ H^s(), φ-Π_hφ_ ≤φ-Π_hφ_h≲ h^αφ_2+αφ∈ H^2+α(Ω).The proof of (<ref>) follows from Lemma <ref>.b, (<ref>), and an interpolation of Sobolev spaces <cit.>.For an easy notation of vectors, we denote the componentwise interpolation of ζ∈H^s():=H^s()× H^s() by Π_hζ. <cit.> For K∈, a macro-element of degree 4 is a nodal finite element (K,P̃_4,Ñ), consisting of sub-triangles K_j, j=1,2,3 (see Figure <ref>). The local element space P̃_4 is defined by P̃_4:={φ∈ C^1(K): φ|_K_j∈ P_4(K_j), j=1,2,3}. The degrees of freedom Ñ are defined as (a) the value and the first (partial) derivatives at the vertices of K; (b) the value at the midpoint of each edge of K; (c) the normal derivative at two distinct points in the interior of each edge of K; (d) the value and the first (partial) derivatives at the common vertex ofK_1, K_2 and K_3. The corresponding finite element space consisting of the above macro-elements will be denoted by S_4()⊂.The enrichment operator of <cit.> is outlined in the sequel for a convenient reading. For each nodal point p of the C^1-conforming finite element space S_4(), define (p) to be the set of K∈ which shares the nodal point p and let |(p)| denotes its cardinality. Define the operator E_h:→ S_4() for any nodal variable N_p at p byN_p(E_h(φ_)):=1/|(p)|∑_K∈(p) N_p(φ_|_K)ifp∈(Ω),0ifp∈(∂Ω). <cit.> The enrichment operator E_h: → S_4() satisfies, for m=0,1,2,∑_K∈|φ_-E_hφ_|_H^m(K)^2≲h_^1/2-m[φ_]__L^2(Γ)^2+h_^3/2-m[∇φ_]__L^2(Γ)^2 ≲ h^2-mφ__. Moreover, for some positive constant Λ≈ 1, φ_-E_hφ__≤Λinf_φ∈ Xφ_-φ_. See <cit.> for a proof of (<ref>). For the proof of (<ref>), choose m≤ 2 in (<ref>) and obtain (with h_≲ h≲ 1) that φ_-E_hφ__ H^2()^2≲h_^-3/2[φ_]__L^2(Γ)^2+h_^-1/2[∇φ_]__L^2(Γ)^2. Since [φ_-E_hφ_]_E=[φ_]_E and [∇(φ_-E_hφ_)]_E=[∇φ_]_E,those edge terms in both sides of the above inequality lead (in the definition of ·_) to φ_-E_hφ__^2≲h_^-3/2[φ_]__L^2(Γ)^2+h_^-1/2[∇φ_]__L^2(Γ)^2. Furthermore, any φ∈ X satisfies (with (<ref>) for m=2 in the end) that φ_-E_hφ__^2≲h_^-3/2[φ_-φ]__L^2(Γ)^2+h_^-1/2[∇ (φ_-φ)]__L^2(Γ)^2≲φ_-φ_. This completes the proof of (<ref>) for some h-independent positive constant Λ. It holds h_∇φ_L^∞(Ω) ≲φ_L^∞(Ω)φ∈+S_4(), φ_W^1,4()+φ_L^∞(Ω) ≲φ_φ∈+X. This follows with the arguments of <cit.> on the enrichment and interpolation operator. Further details are omitted for brevity. §.§ Continuity and ellipticity This subsection is devoted to the boundedness and ellipticity results for the bilinear form a_(∙, ∙) and boundedness results for b_(∙, ∙,∙). Any θ_,φ_∈+S_4() satisfies a_(θ_,φ_)≲θ__φ__. Given any θ_,φ_∈, recall the definition of a_(∙,∙) a_(θ_,φ_)= D^2θ_: D^2φ_-(J(θ_,φ_)+ J(φ_,θ_))+J_σ_1,σ_2(θ_,φ_). The definition of J(∙, ∙), the Cauchy-Schwarz inequality, and Lemma <ref> imply J(θ_,φ_)= [∇φ_]_E·⟨ D^2θ_ν_E⟩_E ≤σ_2^-1/2(∑_E∈σ_2/h_E[∇φ_]_E_L^2(E)^2)^1/2(∑_E∈h_E^1/2⟨ D^2θ_⟩_E_L^2(E)^2)^1/2≲σ_2^-1/2φ__|θ_|_H^2()≤σ_2^-1/2θ__φ__. The same arguments show, J(φ_,θ_)≲σ_2^-1/2θ__φ__. The definitions of J_σ_1,σ_2(∙,∙) and ∙_, and the Cauchy-Schwarz inequality lead to J_σ_1,σ_2(θ_,φ_)=∑_E∈σ_1/h_E^3∫_E[θ_]_E[φ_]_E+∑_E∈σ_2/h_E∫_E[∇θ_·ν_E]_E[∇φ_·ν_E]_E≤(∑_E∈σ_1/h_E^3[θ_]_E_L^2(E)^2)^1/2(∑_E∈σ_1/h_E^3[θ_]_E_L^2(E)^2)^1/2+(∑_E∈σ_2/h_E[∇θ_·ν_E]_E_L^2(E)^2)^1/2(∑_E∈σ_2/h_E[∇φ_·ν_E]_E_L^2(E)^2)^1/2≲θ_φ_. The combination of all displayed formulas and σ_2≥ 1 conclude the proof. The definitions of a_(∙,∙), the auxiliary norm ∙_h, and the estimate (<ref>) imply (since σ_2≥ 1) a_(θ,φ)≲θ_hφ_hθ,φ∈ Y_h≡ (X∩ H^2+α(Ω))+. The trace inequality Lemma <ref>.a implies that ∙_≈∙_h are equivalent norms on +S_4() with equivalence constants, which do neither depend on the mesh-size nor on σ _1,σ_2>0. For any σ_1>0 and for a sufficiently large parameter σ_2, there exists some h-independent positive constant β_0 (which depends on σ_2) such that β_0 θ__^2 ≤ a_(θ_,θ_) θ_∈. Forθ_∈, the definition of a_(∙,∙) leads to a_(θ_,θ_)=θ__^2-2J(θ_,θ_).Recall (<ref>) in the formJ(θ_,θ_)≤ C_0 σ_2^-1/2θ__^2 with some constant C_0≈ 1. For any 0<β_0<1 and any choice of σ_2≥ 4C_0(1-β_0)^-2, the combination of the previous estimates concludes the proof.Recall that h denotes the maximal mesh-size of the underlying triangulation . Any ξ∈ H^2+α(Ω) with 1/2<α≤ 1 and φ_∈ satisfy a_(ξ,φ_-E_hφ_)≲ h^αξ_2+αφ__. Consequently, for ξ∈^2+α(Ω) and Φ_∈, A_(ξ,Φ_-E_hΦ_)≲ h^αξ_2+αΦ__. The method of real interpolation <cit.> of Sobolev spaces defines H^2+α(Ω):=[ H^2(Ω),H^3(Ω)]_α,2 from H^2(Ω) and H^3(Ω). Given any ξ∈ H^2(Ω) and φ_∈, the boundedness of a_(∙,∙) and Lemma <ref> imply a_(Π_hξ,φ_-E_hφ_)≲Π_hξ_φ_-E_hφ__≲ξ_2φ__. Given any ξ∈ H^3(Ω) and φ_∈, the definition of a_(∙,∙), an integration by parts, and Lemma <ref> result in a_(ξ,φ_-E_hφ_) =∑_K∈∫_K D^2ξ:D^2(φ_-E_hφ_)- J(ξ,φ_-E_hφ_)=∑_K∈∫_K(D^2ξ)·∇(φ_-E_hφ_)≤ Chξ_3φ__. This,Remark <ref>, (<ref>) with α=1, and Lemma <ref> lead to a_(Π_hξ,φ_-E_hφ_) =a_(Π_hξ-ξ,φ_-E_hφ_)+a_(ξ,φ_-E_hφ_) ≲ hξ_3φ__. The estimates (<ref>)-(<ref>) and the interpolation between Sobolev spaces show a_(Π_hξ,φ_-E_hφ_)≲ h^αξ_2+αφ__. For any α>1/2, the combination with Remark <ref>, (<ref>), and Lemma <ref> results in a_(ξ,φ_-E_hφ_)=a_(ξ-Π_hξ,φ_-E_hφ_)+a_(Π_hξ,φ_-E_hφ_) ≲ h^αξ_2+αφ__. This concludes the proof of (<ref>). The estimate (<ref>) follows from (<ref>) and the definition of A_(∙,∙). * Anyη,χ, φ∈ X+ satisfyb_(η,χ,φ)≲η_χ_φ_. * Given any α>1/2, any η∈ X∩ H^2+α(Ω) and χ∈ X+ satisfy b_(η,χ,φ)≲η_2+αχ_φ_1φ∈ H^1_0(Ω), η_2+αχ_φ_L^4(Ω)φ∈ X+. (a). For η,χ,φ∈ X+, the definition of b_(∙,∙,∙) and Lemma <ref> lead to |2b_(η,χ,φ)| =| [η,χ]φ|≲ |η|_ H^2()|χ|_ H^2()η_L^∞()≲η_χ_φ_. (b). For η∈ X∩ H^2+α(Ω), χ∈ X+, and φ∈ H^1_0(Ω)∪ (X+), the generalized inequality and the continuous imbedding H^2+α(Ω)↪ W^2,4(Ω) yields |2b_(η,χ,φ)| =| [η,χ]φ|≲η_W^2,4(Ω)χ_H^2()φ_L^4(Ω)≲η_2+αχ_φ_L^4(Ω). This implies second part of (b). For φ∈ H^1_0(Ω)↪ L^4(Ω) this proves the first. § A PRIORI ERROR CONTROL This section first establishes the discrete inf-sup condition for the linearized problem, then the existence of a discrete solution to the nonlinear problem (<ref>), and finally the convergence of a Newton method.§.§ Discrete inf-sup conditionThis subsection is devoted to discrete inf-sup condition. Throughout the paper, the statement “there exists σ_2 such that for all σ_2≥σ_2 as in Lemma <ref> on ellipticity, there exists h(σ_2) such that for all h≤ h(σ_2) ...” is abbreviated by the phrase “for sufficiently large σ_2 and sufficiently small h ...”. Let Ψ∈^2+α(Ω)∩^2_0(Ω) be a regular solution to (<ref>). For sufficiently large σ_2 and sufficiently small h, there exists β such that the following discrete inf-sup condition holds 0<β≤inf_Θ_∈ Θ__=1sup_Φ_∈ Φ__=1(A_(Θ_,Φ_)+2B_(Ψ,Θ_,Φ_)). Given any Θ_∈ with Θ__=1, let ξ∈ and η∈ solve the biharmonic problems A(ξ,Φ) =2B_(Ψ,Θ_,Φ)Φ∈, A(η,Φ) =2B(Ψ,E_hΘ_,Φ)Φ∈. Lemma <ref>.b implies that B_(Ψ,Θ̃,∙) and B (Ψ,Θ̃,∙) belong to ^-1(Ω) for Θ̃∈+. The reduced elliptic regularity for the biharmonic problem <cit.> yields ξ,η∈^2+α(Ω)∩. Since Ψ is a regular solution to (<ref>), there exists β from (<ref>) and Φ∈ with Φ_2=1 such that β E_hΘ__2≤ A(E_hΘ_,Φ)+2B(Ψ,E_hΘ_,Φ). The solution property in (<ref>), the boundedness of A(∙,∙), and the triangle inequality in the above result imply β E_hΘ__2≤ A(E_hΘ_+η,Φ)≤ E_hΘ_+η_2 ≤E_hΘ_-Θ__+Θ_+ξ_+η-ξ_2. The definition of ξ,η in (<ref>)-(<ref>) and Lemma <ref>.a lead to η-ξ_2≤ 2C_bΨ_2 E_hΘ_-Θ__ for some positive constant C_b≈ 1. The combination of the previous two displayed inequalities reads β E_hΘ__2≤Θ_+ξ_+(1+2C_bΨ_2) E_hΘ_-Θ__. This and (<ref>) result in E_hΘ__2≤1/β(1+Λ(1+2C_bΨ_2))Θ_+ξ_. The triangle inequality, (<ref>), and (<ref>) lead to 1 =Θ__≤ E_hΘ_-Θ__+ E_hΘ__2≤(Λ+1/β(1+Λ(1+2C_bΨ_2)))Θ_+ξ_. In other words, β_1:=β/(1+Λ(1+β+2C_bΨ_2)) satisfies β_1≤Θ_+ξ_≤Θ_+Π_hξ_+ξ-Π_hξ_. For any given Θ_+Π_hξ∈, the ellipticity of A_(∙,∙) from Lemma <ref> implies the existence of some Φ_∈ with Φ__=1 and β_0Θ_+Π_hξ_≤ A_(Θ_+Π_hξ,Φ_) =A_(Θ_,Φ_)+A_(Π_hξ-ξ,Φ_)+A_(ξ,Φ_-E_hΦ_)+A(ξ,E_hΦ_). The choice of Φ=E_hΦ_ in (<ref>) plus straightforward calculations result in β_0Θ_+Π_hξ_ ≤ A_(Θ_,Φ_)+ A_(Π_hξ-ξ,Φ_) +A_(ξ,Φ_-E_hΦ_)+ 2B_(Ψ,Θ_,Φ_) +2B_(Ψ,Θ_,E_hΦ_-Φ_). Remark <ref>, Lemma <ref> and <ref> for the second and third term, plus Lemma <ref>.b and <ref> for the last term in (<ref>) lead to β_0Θ_+Π_hξ_≤ A_(Θ_,Φ_)+2B_(Ψ,Θ_,Φ_)+Ch^α. The combination of (<ref>) and (<ref>) with Lemma <ref> shows β_0β_1-C_*h^α≤ A_(Θ_,Φ_)+2B_(Ψ,Θ_,Φ_) for some h-independent positive constant C_*. Hence,for allh≤ h_0:=(β_0β_1/2C_*)^1/α, the discrete inf-sup condition (<ref>) follows. The next lemma establishes that the perturbed bilinear form_(Θ_,Φ_):= A_(Θ_,Φ_)+2B_(Π_hΨ,Θ_,Φ_)satisfies a discrete inf-sup condition.Let Π_hΨ be the interpolation of Ψ from Lemma <ref>. Then, for sufficiently large σ_2 and sufficiently small h, the perturbed bilinear form (<ref>) satisfies the discrete inf-sup condition β/2≤inf_Θ_∈ Θ__=1sup_Φ_∈ Φ__=1_(Θ_,Φ_). Proof. Lemma <ref> leads to the existence of h_1>0, such thatΨ-Π_hΨ_≤β/4C_b holds for h≤ h_1. Given any Θ_∈, Theorem <ref> andLemma <ref>.a lead to β/2Θ__≤βΘ__-2C_bΨ-Π_hΨ_Θ_≤sup_Φ_∈ Φ__=1(A_(Θ_,Φ_)+2B_(Ψ,Θ_,Φ_)) -2sup_Φ_∈ Φ__=1 B_(Ψ-Π_hΨ,Θ_,Φ_)≤sup_Φ_∈ Φ__=1(A_(Θ_,Φ_)+2B_(Π_hΨ,Θ_,Φ_))=sup_Φ_∈ Φ__=1_(Θ_,Φ_).§.§ Existence, uniqueness and error estimateThe discrete inf-sup condition is employed to define a nonlinear map μ:→ which enables to analyze the existence and uniqueness of solution of (<ref>).For any Θ_∈, define μ(Θ_)∈ as the solution to the discrete fourth-order problem_(μ(Θ_),Φ_)=L_(Φ_)+2B_(Π_hΨ,Θ_,Φ_)-B_(Θ_,Θ_,Φ_) for all Φ_∈. Lemma <ref> guarantees that the mapping μ is well-defined and continuous. Also, any fixed point of μ is a solution to (<ref>) and vice-versa. In order to show that the mapping μ has a fixed point, define the ball_R(Π_hΨ):={Φ_∈: Φ_-Π_hΨ_≤ R }. For sufficiently large σ_2 and sufficiently small h, there exists a positive constant R(h) such that μ maps the ball _R(h)(Π_hΨ) to itself; μ(Θ_ )-Π_hΨ_≤ R(h) holds for any Θ_∈_R(h)(Π_hΨ). The discrete inf-sup condition of _(∙,∙) in Lemma <ref> implies the existence of Φ_∈ withΦ__=1 and β/2μ(Θ_)-Π_hΨ_≤_(μ(Θ_)-Π_hΨ, Φ_). Let E_hΦ_ be the enrichment of Φ_ from Lemma <ref>. The definition of _(∙,∙), the symmetry of B_(∙,∙,∙) in first and second variables, (<ref>), and (<ref>) lead to _(μ(Θ_)-Π_hΨ, Φ_) =_(μ(Θ_), Φ_)-_(Π_hΨ, Φ_)=L_(Φ_)+2B_(Π_hΨ,Θ_,Φ_)-B_(Θ_,Θ_,Φ_)-A_(Π_hΨ,Φ_)-2B_(Π_hΨ,Π_hΨ,Φ_)=L_(Φ_-E_hΦ_) +(A(Ψ, E_hΦ_)-A_(Π_hΨ,Φ_)) +(B(Ψ,Ψ, E_hΦ_)-B_(Π_hΨ,Π_hΨ,Φ_)) +B_(Π_hΨ-Θ_,Θ_-Π_hΨ,Φ_) =:T_1+T_2+T_3+T_4. The term T_1 can be estimated using the continuity of L_ and Lemma <ref>. The continuity of A_(∙, ∙), Lemma <ref> and <ref> with Φ__=1lead to T_2: =A(Ψ, E_hΦ_)-A_(Π_hΨ,Φ_) = A_(Ψ,E_hΦ_-Φ_) +A_(Ψ-Π_hΨ,Φ_) ≲ h^αΨ_2+α. Lemma <ref>, <ref>, and <ref> result in T_3 := B(Ψ,Ψ,E_hΦ_)-B_(Π_hΨ,Π_hΨ,Φ_)= B_(Ψ,Ψ-Π_hΨ,E_hΦ_)+B_(Ψ-Π_hΨ,Π_hΨ,Φ_) +B_(Ψ,Π_hΨ,E_hΦ_-Φ_) ≲ h^αΨ_2+αΨ_2. Lemma <ref> implies T_4:=B_(Π_hΨ-Θ_,Θ_-Π_hΨ,Φ_)≲Θ_-Π_hΨ_^2. A substitution of the estimates for T_1, T_2, T_3, and T_4 in (<ref>) and Ψ_2+α≈ 1≈Ψ_2 lead to C_1≈ 1 with μ(Θ_)-Π_hΨ_≤ C_1( h^α+Θ_-Π_hΨ_^2). Then h≤ h_2:=(2C_1)^-2/α and Θ_-Π_hΨ_≤ R(h):=2C_1 h^α lead to μ(Θ_)-Π_hΨ_≤ C_1 h^α(1+4C_1^2 h^α)≤ R(h). This concludes the proof.For sufficiently large σ_2 and sufficiently small h, there exists a unique solution Ψ_ to the discrete problem (<ref>) in _R(h)(Π_hΨ). First we prove the contraction result of the nonlinear map μ in the ball _R(h)(Π_hΨ) of Theorem <ref>. Given any Θ_,Θ̃_∈_R(h)(Π_hΨ) and for all Φ_∈, the solutions μ(Θ_) andμ(Θ̃_) satisfy _(μ(Θ_),Φ_) =L_(Φ_)+2B_(Π_hΨ,Θ_,Φ_)-B_(Θ_,Θ_,Φ_),_(μ(Θ̃_),Φ_) =L_(Φ_)+2B_(Π_hΨ,Θ̃_,Φ_)-B_(Θ̃_,Θ̃_,Φ_). The discrete inf-sup of _(∙,∙) from Lemma <ref> guarantees the existence of Φ_ with Φ__=1 below.With (<ref>)-(<ref>) and Lemma <ref>, it follows that β/2μ(Θ_)-μ(Θ̃_)_≤_(μ(Θ_)-μ(Θ̃_),Φ_)=2B_(Π_hΨ,Θ_-Θ̃_, Φ_)+B_(Θ̃_,Θ̃_, Φ_)-B_(Θ_,Θ_, Φ_)=B_(Θ̃_-Θ_,Θ_-Π_hΨ,Φ_)+B_(Θ̃_-Π_hΨ,Θ̃_-Θ_,Φ_)≲Θ̃_-Θ__(Θ_-Π_hΨ_+ Θ̃_-Π_hΨ_). Since Θ_,Θ̃_∈_R(h)(Π_hΨ), for a choice of R(h) as in the proof ofTheorem <ref>, for sufficiently large σ_2 and h≤min{h_0,h_1,h_2}, μ(Θ_)-μ(Θ̃_)_≲ h^αΘ̃_-Θ__. Hence, there exists positive constant h_3, such that for h≤ h_3 the contraction result holds. For h≤h:=min{h_0,h_1,h_2,h_3}, Lemma <ref> and Theorem <ref> lead to the fact that μ is a contraction map that maps the ball _R(h)(Π_hΨ) into itself. An application of the Banach fixed point theorem yields that the mapping μ has a unique fixed point in the ball _R(h)(Π_hΨ), say Ψ_, which solves (<ref>) with Ψ_-Π_hΨ_≤ R(h). LetΨ be a regular solution to (<ref>)and let Ψ_ be the solution to (<ref>). For sufficiently large σ_2 and sufficiently small h, it holds Ψ-Ψ__≤ C h^α. A triangle inequality yields Ψ-Ψ__≤Ψ-Π_hΨ_+Π_hΨ-Ψ__. For h≤h and sufficiently large σ_2, Theorem <ref> leads to Π_hΨ-Ψ__≤ Ch^α. This, Lemma <ref>, (<ref>),and (<ref>) conclude the proof.§.§ Convergence of the Newton method The discrete solution Ψ_ of (<ref>) is characterized as the fixed point of (<ref>) and so depends on the unknown Π_hΨ. The approximate solution to (<ref>) is computed with the Newton method, where the iterates Ψ_^j solve A_(Ψ_^j,Φ_)+2B_(Ψ_^j-1,Ψ_^j,Φ_)=B_(Ψ_^j-1,Ψ_^j-1,Φ_)+L_(Φ_) for all Φ_∈. The Newton method has locally quadratic convergence. Let Ψ be a regular solution to (<ref>) and letΨ_ solve (<ref>). There exists a positive constant R independent of h, such that for any initial guess Ψ_^0 with Ψ_- Ψ_^0_≤R, it follows Ψ_- Ψ_^j_≤ R j=0,1,2,… and the iterates of the Newton methodin (<ref>) are well defined and converges quadratically to Ψ_. Following the proof of Lemma <ref>, there exists a positive constant ϵ (sufficiently small) independent of h such that for each Z_∈ with Z_-Π_hΨ_≤ϵ, the bilinear form A_(∙,∙)+2B_(Z_,∙,∙) satisfies discrete inf-sup condition in ×. For sufficiently large σ_2 and sufficiently small h, the equation (<ref>) implies Π_hΨ-Ψ__≤ Ch^α. Thus h can be chosen sufficiently small so that Π_hΨ-Ψ__≤ϵ/2. Recall β from (<ref>). Lemma <ref>.a implies that there exists a positive constant C_b̃ independent of h such thatB_(Ξ_,Θ_,Φ_)≤ C_b̃Ξ__Θ__Φ__. Set R:=min{ϵ/2,β/8C_b̃}. Assume that the initial guess Ψ_^0 satisfies Ψ_-Ψ_^0_≤R. Then, Π_hΨ-Ψ_^0_≤Π_hΨ-Ψ__+Ψ_-Ψ_^0_≤ϵ. This implies Ψ_-Ψ_^j-1_≤R for j=1 and suppose for mathematical induction that this holds for some j∈. Then Z_:=Ψ_^j-1 in (<ref>) leads to an discrete inf-sup condition of A_(∙,∙)+2B_(Ψ^j-1_,∙,∙) and so to an unique solution Ψ_^j in step j of the Newton scheme. The discrete inf-sup condition (<ref>) implies the existence of Φ_∈ with Φ__=1 and β/4Ψ_-Ψ_^j_≤ A_(Ψ_-Ψ_^j, Φ_)+2B_(Ψ_^j-1,Ψ_-Ψ_^j, Φ_). The application of (<ref>), (<ref>), and Lemma <ref> result in A_(Ψ_-Ψ_^j, Φ_)+2B_(Ψ_^j-1,Ψ_-Ψ_^j, Φ_)=A_(Ψ_,Φ_)+2B_(Ψ_^j-1,Ψ_,Φ_)-B_(Ψ_^j-1,Ψ_^j-1,Φ_)-L_(Φ_)=-B_(Ψ_,Ψ_,Φ_)+2B_(Ψ_^j-1,Ψ_,Φ_)-B_(Ψ_^j-1,Ψ_^j-1,Φ_)=B_(Ψ_-Ψ_^j-1,Ψ_^j-1-Ψ_,Φ_)≤ C_b̃Ψ_-Ψ_^j-1_^2. This implies Ψ_-Ψ_^j_≤(4C_b̃/β)Ψ_-Ψ_^j-1_^2 and establishes the quadratic convergence of the Newton methodto Ψ_. The definition of R and (<ref>) guarantee Ψ_-Ψ_^j_≤Ψ_-Ψ_^j-1_<R to allow an induction step j→ j+1 to conclude the proof. § A POSTERIORI ERROR CONTROL This section establishes a reliable and efficient error estimator for the DGFEM. For K ∈ and E ∈(Ω), define the volume and edge estimators η_K and η_E by η_K^2 := h_K^4(f+[u_,v_]_L^2(K)^2+[u_,u_]_L^2(K)^2), η_E^2 :=h_E([D^2 u_ ν_E]_E_L^2(E)^2+[ D^2 v_ν_E]_E_L^2(E)^2) +h_E^-3([u_]_E_L^2(E)^2+[v_]_E_L^2(E)^2)+h_E^-1([∇ u_]_E_L^2(E)^2+[∇ v_]_E_L^2(E)^2).Let Ψ=(u,v)∈ be a regular solution to (<ref>) and let Ψ_=(u_,v_)∈ solve (<ref>). For sufficiently large σ_2 and sufficiently small h, there exists an h-independent positive constant C_ rel such that Ψ-Ψ__^2 ≤ C_ rel^2(∑_K∈η_K^2+∑_E∈(Ω)η_E^2). The Fréchet derivative of N(Ψ) at Ψ in the direction Θ∈ reads DN(Ψ;Θ,Φ):=A(Θ,Φ)+2B(Ψ,Θ,Φ)Φ∈. Since Ψ is a regular solution, for any 0<ϵ<β with β from (<ref>), there exists some Φ∈ with Φ_2=1 and (β-ϵ)Ψ-E_hΨ__2≤ DN(Ψ;Ψ-E_hΨ_,Φ). Since N is quadratic, the finite Taylor series is exact and shows N(E_hΨ_;Φ) =N(Ψ;Φ)+DN(Ψ;E_hΨ_-Ψ,Φ)+ D^2N(Ψ;E_hΨ_-Ψ, E_hΨ_-Ψ,Φ). Since N(Ψ;Φ)=0 and D^2N(Ψ;Θ,Θ,Φ)=2B(Θ,Θ,Φ) for Θ=Ψ-E_hΨ_, (<ref>)-(<ref>) plus Lemma <ref>.a with boundedness constant C_b lead to (β-ϵ)Ψ-E_hΨ__2 ≤ -N(E_hΨ_;Φ)+B(Ψ-E_hΨ_, Ψ-E_hΨ_,Φ)≤ |N(E_hΨ_;Φ)|+C_bΨ-E_hΨ__2^2. The triangle inequality, (<ref>), and Theorem <ref> imply Ψ-E_hΨ__≤Ψ-Ψ__+Ψ_-E_hΨ__ ≤ C(1+Λ)h^α. With ϵ↘ 0, (<ref>)-(<ref>) verify (β-(1+Λ)CC_bh^α)Ψ-E_hΨ__2≤ |N(E_hΨ_;Φ)|. There exists positive constant h_4, such that h≤ h_4 implies β-(1+Λ)CC_bh^α>0. Hence, for h≤ h_4, the above equation and triangle inequality lead to Ψ-Ψ__ ≤Ψ-E_hΨ__+ E_hΨ_-Ψ__≲ |N(E_hΨ_;Φ)|+ E_hΨ_-Ψ__. The definitions of N andN_h, N_h(Ψ_;Π_hΦ)=0, the boundedness of A_(∙, ∙), B_(∙, ∙, ∙), and Theorem <ref> result in N(E_hΨ_;Φ)=A(E_hΨ_,Φ)+B(E_hΨ_,E_hΨ_,Φ)-L(Φ)=N_h(Ψ_;Φ)+A_(E_hΨ_-Ψ_,Φ)+B(E_hΨ_,E_hΨ_,Φ)-B_(Ψ_,Ψ_,Φ)=N_h(Ψ_;Φ-Π_hΦ)+A_(E_hΨ_-Ψ_,Φ)+B_(E_hΨ_-Ψ_,E_hΨ_,Φ)+B_(Ψ_,E_hΨ_-Ψ_,Φ)≲ |N_h(Ψ_;Φ-Π_hΦ)|+ E_hΨ_-Ψ__. The combination of the previous displayed estimates proves Ψ-Ψ__≲ |N_h(Ψ_;Φ-Π_hΦ)|+ E_hΨ_-Ψ__. Abbreviate Φ-Π_hΦ=:χ=(χ_1,χ_2) and recall N_h(Ψ_;χ)=A_(Ψ_,χ)+B_(Ψ_,Ψ_,χ)-L_(χ). The definition of A_(∙,∙) and an integration by parts with the facts thatu_ and v_ are piecewise quadratic polynomials lead to A_(Ψ_,χ)=(D^2 u_: D^2χ_1+D^2 v_: D^2χ_2)-J(u_,χ_1)- J(χ_1,u_)-J(v_,χ_2)- J(χ_2,v_) +∑_E∈σ_1/h_E^3∫_E( [u_]_E[χ_1]_E+[v_]_E[χ_2]_E)+∑_E∈σ_2/h_E∫_E([∇ u_·ν_E]_E[∇χ_1·ν_E]_E+[∇ v_·ν_E]_E[∇χ_2·ν_E]_E)=([D^2 u_ ν_E]_E·⟨∇χ_1⟩_E+[D^2 v_ ν_E ]_E·⟨∇χ_2⟩_E)-(J(χ_1,u_)+J(χ_2,v_)) +∑_E∈σ_1/h_E^3∫_E([u_]_E[χ_1]_E+[v_]_E[χ_2]_E)+∑_E∈σ_2/h_E∫_E([∇ u_·ν_E]_E[∇χ_1·ν_E]_E+[∇ v_·ν_E]_E[∇χ_2·ν_E]_E). The trace inequality of Lemma <ref> and interpolation estimates χ_1_H^m(K)=φ_1-Π_hφ_1_H^m(K)≤ h_K^2-mφ_1_H^2(K) for m=1,2, from Lemma <ref> result in h_^-1/2⟨∇χ_1⟩__L^2(Γ)^2≲∑_K∈h_K^-1∇χ_1_L^2(∂ K)^2≲∑_K∈h_K^-1(h_K^-1χ_1_H^1(K)^2+h_Kχ_1_H^2(K)^2)≲Φ_2^2=1. This and the Cauchy-Schwarz inequality imply ([D^2 u_ ν_E]_E·⟨∇χ_1⟩_E+[D^2 v_ ν_E ]_E·⟨∇χ_2⟩_E) ≲(∑_E∈(Ω)h_E^1/2[D^2 u_ ν_E]_E_L^2(E)^2)^1/2+(∑_E∈(Ω)h_E^1/2[D^2 v_ ν_E]_E_L^2(E)^2)^1/2. The Cauchy-Schwarz inequality, the trace inequality of Lemma <ref>, and the interpolation of Lemma <ref> lead to J(χ_1,u_)+J(χ_2,v_)≲(h_^-1/2[∇ u_]_E_Γ^2+h_^-1/2[∇ v_]_E_Γ^2)^1/2Φ_2. Similar arguments for the penalty terms result in ∑_E∈σ_1/h_E^3∫_E( [u_]_E[χ_1]_E+[v_]_E[χ_2]_E)≲(h_^-3/2[u_]__L^2(Γ)^2+h_ ^-3/2[v_]__L^2(Γ)^2)^1/2,∑_E∈σ_2/h_E∫_E([∇ u_·ν_E]_E[∇χ_1·ν_E]_E+[∇ v_·ν_E]_E[∇χ_2·ν_E]_E)≲(h_^-1[∇ u_·ν_E]__L^2(Γ)^2+h_ ^-1[∇ v_·ν_E]__L^2(Γ)^2)^1/2. The definition of B_(∙,∙,∙) yields B_(Ψ_,Ψ_,χ) = -[u_,v_]χ_1+[u_,u_]χ_2. The Cauchy-Schwarz inequality and Lemma <ref> show B_(Ψ_,Ψ_,χ)-L_(χ)≲∑_K∈ h_K^4(f+[u_,v_]_L^2(K)^2+[u_,u_]_L^2(K)^2). A substitution of the previous six estimates in (<ref>) and Lemma <ref> establish (<ref>). The efficiency of the error estimator obtained in Theorem <ref> can be established using the standard bubble function techniques <cit.>; further details are therefore omitted. Let Ψ=(u,v)∈ be a regular solution to (<ref>) and let Ψ_=(u_,v_)∈ be the local solution to (<ref>). There exists a positive constant C_ eff independent of h but dependent on Ψ such that ∑_K∈η_K^2+∑_E∈(Ω)η_E^2≤ C_ eff^2(Ψ-Ψ__^2+ osc^2(f)),where osc^2(f):=∑_K∈h_K^4f-f_h_L^2(K)^2 and f_h denotes the piecewise average of f. § A C^0 INTERIOR PENALTY METHOD The analysis of this paper extends to a C^0 interior penalty method for the von Kármán equations formally for σ_1→∞ when σ_1 disappears but the trial and test functions become continuous. The novel scheme is the above dG method but with ansatz test function restricted to ∩^1_0(Ω)=:^2_0()≡ S^2_0()× S^2_0() and the norm ∙_ is ∙_ with restriction to ^2_0() excludes σ_1 (which has no meaning as it is multiplied by zero) and ∙_ is ∙_h with restriction to ^2_0().Since the discrete functions are globally continuous for this case, the bilinear forma_(∙,∙) simplifies for some positive penalty parameter σ_2, for η_,χ_∈ S^2_0(),toa_(η_,χ_):= D^2η_:D^2χ_-⟨ D^2η_ν_E⟩_E·[∇χ_]_E-⟨ D^2χ_ν_E⟩_E·[∇η_]_E+∑_E∈σ_2/h_E∫_E[∇η_·ν_E]_E[∇χ_·ν_E]_E. This novel C^0 interior penalty (C^0-IP) method for the von Kármán equations seeks u_,v_∈ S^2_0() such thata_(u_,φ_1)+b_(u_,v_,φ_1)+b_(v_,u_,φ_1)=l_(φ_1)φ_1∈ S^2_0(),a_(v_,φ_2)-b_(u_,u_,φ_2)=0φ_2∈ S^2_0().The term related to jump which is of the form [η_]_E for each η_∈ S^2_0() vanishes in the C^0-IP method and this simplifies the analysis. LetΨ be a regular solution to (<ref>)and let Ψ_=(u_,v_) be the solution to (<ref>)-(<ref>). For sufficiently large σ_2 and sufficiently small h, it holds Ψ-Ψ__≤ C h^α. The Lemmas <ref>-<ref> hold as it is and the boundedness results in Lemma <ref> for b_(∙,∙,∙) can be modified tob_(η,χ,φ)≲η_χ_φ_η,χ,φ∈ X+S^2_0(), η_2+αχ_φ_1η∈ X∩ H^2+α(Ω) and χ,φ∈ X+S^2_0().Theorems <ref>-<ref> follow in the same lines and hence, a priori error estimates in energy norm can be established without any additional difficulty. For K ∈ and E ∈(Ω), a posteriori error estimates for the C^0-interior penalty method (<ref>)-(<ref>) lead to the volume estimator η_K and the edge estimator η_E defined byη_K^2 := h_K^4(f+[u_,v_]_L^2(K)^2+[u_,u_]_L^2(K)^2), η_E^2 :=h_E([D^2 u_ ν_E]_E_L^2(E)^2+[ D^2 v_ν_E]_E_L^2(E)^2) +h_E^-1([∇ u_]_L^2(E)^2+[∇ v_]_L^2(E)^2). Let Ψ=(u,v)∈ be a regular solution to (<ref>) and Ψ_=(u_,v_)∈ S^2_0()× S^2_0() be the solution to (<ref>)-(<ref>). For sufficiently large σ_2 and sufficiently small h, there exist h-independent positive constants C_ rel and C_ eff such that C_ rel^-2Ψ-Ψ__^2 ≤∑_K∈η_K^2+∑_E∈(Ω)η_E^2≤ C_ eff^2Ψ-Ψ__^2+ osc^2(f). This follows as inTheorems <ref>-<ref>, and hence the details are omitted for brevity. The C^0-IP formulation of <cit.> chooses the trilinear form b_(∙,∙,∙) as b_(η_,χ_,φ_) :=- [η_,χ_]φ_+∑_E∈(Ω)∫_E [⟨(D^2η_)⟩_E∇χ_·ν_E]_Eφ_ for all η_,χ_,φ_∈ S^2_0(). For the C^0-IP formulation (<ref>)-(<ref>) with b_(∙,∙,∙) replacing b_(∙,∙,∙) and ∙_h≡∙_, the efficiency of the estimator is still open, for difficulties caused by the the non-residual type average term ⟨(D^2η_)⟩_E. § NUMERICAL EXPERIMENTSThis section is devoted to numerical experiments to investigate the practical parameter choice and adaptive mesh-refinements. §.§ PreliminariesThe discrete solution to (<ref>) is obtained using the Newton method defined in (<ref>) with initial guess Ψ_^0∈ computed as the solution of the biharmonic part of the von Kármán equations, i.e., Ψ_^0∈ solvesA_(Ψ_^0,Φ_)=L(Φ_)Φ_∈.Let the ℓ-th level error (for example, in the norm Ψ-Ψ__) and the number of degrees of freedom (ndof) be denoted by e_ℓ and (ℓ), respectively. The ℓ-th level empirical rate of convergence is defined by(ℓ):=2×log(e_ℓ-1/e_ℓ)/log((ℓ)/(ℓ-1) ) for ℓ=1,2,3,…In all the numerical tests, the Newton iterates converge within 4 steps with the stopping criteria Ψ_^5-Ψ_^j-1_<10^-8 for j∈, where Ψ_^5 denotes the discrete solution generated by Newton iterates at 5-th iteration. The penalty parameters for the DGFEM and C^0-IP are consistently chosenas σ_1=σ_2=20 in all numerical examples and appear as sensitive as in the case of the linear biharmonic equations.§.§ Example on unit square domainThe exact solution to (<ref>)is u(x,y)=x^2y^2(1-x)^2(1-y)^2 andv(x,y)=sin^2(π x)sin^2(π y) on the unit square Ω with elliptic regularity index α=1 and corresponding data f and g.Figure <ref> displaysthe initial mesh and its successivered-refinements lead to sequence ofDGFEM solutions on the quasi-uniform mesheswith the errors u-u__ and v-v__ and with their empirical convergence rates in Table <ref>. The empirical convergence ratewith respect to dG norm is one as predicted from Theorem <ref>.Table <ref> displays the errorsand empirical rates of convergence for C^0-IP method of Section <ref> and the linear empirical rate of convergence as expected from the theory is observed. §.§ Example on L-shaped domain In polar coordinates centered at the re-entering corner of theL-shaped domain Ω=(-1,1)^2 ∖([0,1)×(-1,0]),the slightly singular functionsu(r,θ)=v(r,θ):=(1-r^2 cos^2θ)^2 (1-r^2 sin^2θ)^2 r^1+αg_α,ω(θ) with the abbreviation g_α,ω(θ):= (1/α-1sin((α-1)ω)-1/α+1sin((α+1)ω))×(cos((α-1)θ)-cos((α+1)θ))-(1/α-1sin((α-1)θ)-1/α+1sin((α+1)θ))×(cos((α-1)ω)-cos((α+1)ω)),are defined for the angle ω=3π/2 and the parameter α= 0.5444837367 as the non-characteristic root of sin^2(αω) = α^2sin^2(ω). With the loads f and g according to (<ref>) the DGFEM solutions are computed on a sequence of quasi-uniform meshes. Tables <ref> and <ref> display the errors and the expected reduced empirical convergence rates for the DGFE and the C^0-IP method.§.§ Adaptive mesh-refinementFor theL-shaped domain of the preceding Example <ref>the constant load function f ≡ 1, the unknown solutionto the (<ref>) is approximated by an adaptive mesh-refining algorithm. Given an initialtriangulation _0 run the steps SOLVE, ESTIMATE, MARK and REFINE successivelyfor different levels ℓ=0,1,2,….SOLVE Compute the solution of DGFEM Ψ_ℓ:=Ψ_ (resp. C^0-IP Ψ_ℓ:=Ψ_ ) with respect to _ℓ and number of degrees of freedom given by . ESTIMATE Compute local contribution of the error estimator from (<ref>) (resp. from (<ref>)) η^2_ℓ(K):=η_K^2+∑_E∈(K)η_E^2for allK∈_ℓ. MARK The Dörfler marking chooses a minimal subset ℳ_ℓ⊂_ℓ such that 0.3 ∑_K∈𝒯_ℓη^2_ℓ(K)≤∑_K∈ℳ_ℓη^2_ℓ(K). REFINE Compute the closure of ℳ_ℓ and generate a new triangulation _ℓ+1 using newest vertex bisection <cit.>.Figure <ref>(a) displays theconvergence history ofthe a posteriorierror estimatoras a function of number of degrees of freedom for uniform and adaptive mesh-refinement of DGFEM and C^0-IP method. Figure <ref>(b) depicts the adaptive mesh for C^0-IP method generated by the above adaptive algorithm for level ℓ=19, and it illustrates the adaptive mesh-refinement near the re-entering corner. The suboptimal empirical convergence rate for uniform mesh-refinement is improved to an optimal empirical convergence rate 0.5 via adaptive mesh-refinement.To show the reliability and efficiency of the estimators for DGFEM and C^0-IP, another test has been performed over L-shaped domain for the example described in Example <ref>. Figure <ref>(a) displays the convergence history of the error and the a posteriorierror estimator as a function of number of degrees of freedom for uniform and adaptive mesh-refinement of DGFEM and C^0-IP method.Figure <ref>(b) displays the convergence history ofthe error and the a posteriorierror estimator for uniform andadaptive mesh-refinement of C^0-IP method.The ratio between error and estimator C_relis plotted inFigure <ref>(a)-(b) and almost constant as a numerical evidence of the reliability and efficiency of the estimators for DGFEMand C^0-IP methods of Theorem <ref>-<ref> and Theorem <ref>. § CONCLUSIONSThis paper analyzes a discontinuous Galerkin finite element method for the approximation of regular solutions of von Kármán equations. A priori error estimate in energy norm and a posteriori error control which motivates an adaptive mesh-refinement are deduced under the minimal regularity assumption on the exact solution. The analysis suggests a novel C^0-interior penalty method and provides a priori and a posteriori error control for the energy norm.Moreover, the analysis can be extended to hp discontinuous Galerkin finite element methods with additional jump terms for higher order derivatives of ansatz and trial functions under additional regularity assumptions on the exact solution. § ACKNOWLEDGEMENTSThe first and third authors acknowledge the support of National Program on Differential Equations: Theory, Computation & Applications (NPDE-TCA) and Department of Science & Technology (DST) Project No. SR/S4/MS:639/09. The second author would like to thank National Board for Higher Mathematics (NBHM) and IIT Bombay for various financial supports.amsplain | http://arxiv.org/abs/1708.07815v1 | {
"authors": [
"Carsten Carstensen",
"Gouranga Mallik",
"Neela Nataraj"
],
"categories": [
"math.NA"
],
"primary_category": "math.NA",
"published": "20170825172405",
"title": "A Priori and A Posteriori Error Control of Discontinuous Galerkin Finite Element Methods for the Von Kármán Equations"
} |
[email protected], [email protected] university Tehran, Iran In this paper, we propose a simple way to utilize stereo camera data to improve feature descriptors. Computer vision algorithms that use a stereo camera require some calculations of 3D information. We leverage this pre-calculated information to improve feature descriptor algorithms. We use the 3D feature information to estimate the scale of each feature. This way, each feature descriptor will be more robust to scale change without significant computations. In addition, we use stereo images to construct the descriptor vector. The Scale-Invariant Feature Transform (SIFT) and Fast Retina Keypoint (FREAK) descriptors are used to evaluate the proposed method. The scale normalization technique in feature tracking test improves the standard SIFT by 8.75% and improves the standard FREAK by 28.65%. Using the proposed stereo feature descriptor, a visual odometry algorithm is designed and tested on the KITTI dataset. The stereo FREAK descriptor raises the number of inlier matches by 19% and consequently improves the accuracy of visual odometry by 23%. Keywords: Stereo Camera, Feature Descriptor, Visual Odometry, Feature Matching, Depth Information, Stereo Visual Odometry.§ INTRODUCTION Feature descriptors presently constitute one of the fundamental components in most of the computer vision tasks. They have a key role in many vision algorithms such as scene reconstruction, visual odometry, object detection, and object recognition. As a result, any improvement in feature descriptors will probably increase the performance of such algorithms.In visual odometry, feature descriptors are used to match features between displacements in the camera position. Feature matching is a very delicate work where each feature is a projection of a geometric point of the 3D scene to the image plane so that it would be recognizable from different views. Here, feature descriptors are used to match features between the left and right camera images and to track the features over time. There may be some incorrect matches among the matched features called outliers. These outliers are detected and rejected in an outlier rejection step of the visual odometry algorithm, and only the inlier matches are used. Moreover, not all of the inlier matches are suitable for motion estimation. In other words, some features are matched with a small drift and the feature descriptors are not sensitive enough to detect that. Although one-pixel error causes a small drift in the estimation, such drifts are accumulated and cause a notable error in the estimated path. Considering these issues, finding the best matches is considered as one of the main challenges in visual odometry.In the recent years, many successful feature descriptors have been proposed such as: SIFT by <cit.>, Speeded Up Robust Features (SURF) by <cit.>, Binary Robust Independent Elementary Features (BRIEF) by <cit.>, Oriented FAST and Rotated BRIEF (ORB) by <cit.>, and FREAK by <cit.>. Some of them like SIFT or SURF are robust, but slow. Others like BRIEF, ORB, and FREAK are fast, but sensitive to large transforms. Although they all have been successful in many application, they generally have a large mismatch rate. For example, in visual odometry, matched features are usually contaminated with outliers by more than 40%. This happens when the feature descriptor is not discriminative at all points, especially the challenging ones. Therefore, an outlier rejection scheme (such as Random Sample Consensus (RANSAC) by <cit.>) should be used to determine which one is a correct and which one is an incorrect match. A better solution will result if these mismatches do not occur. A solution is to prevent these mismatches from happening using more robust feature descriptors. In this paper, we propose a robust stereo feature descriptor with two properties: first, it utilizes depth information extracted from the stereo images to estimate the scale. Second, it uses the two images of the stereo camera to construct the feature descriptor vector. Our purpose is to build a feature descriptor that is both robust and efficient. A feature descriptor is calculated from a selected area around the corresponding feature. We leverage the depth information to change the area adaptively. Using the left and right images of the scene points helps us to have more information about the features and to implicitly check the features matching process. With the help of the depth information of the feature, we can correctly determine the distance between the feature and the camera. The smaller the distance from the camera is, the larger the area around the feature needed to construct the descriptor is. In other words, our stereo descriptor uses the depth information to estimate the correct scale for the features and adjust the scale for the descriptor.The rest of the paper is organized as follows. The related literature will be reviewed in Section 2. In Section 3 the basic required elements to describe the proposed method are presented. Section 4 presents the stereo camera feature descriptor algorithm. Section 5 gives the experimental results followed by the conclusions given in Section 6.§ RELATED WORK A complete overview on visual odometry can be found in <cit.> and <cit.>. Feature descriptors have been used in visual odometry for many years. But descriptors became more important in this application after <cit.>. They suggested using descriptors to match features between the left and right camera images and by using them to track features over time. This technique has recently become more popular and consequently, computing feature descriptors has become one of the major steps in the success of a visual odometry algorithm. One of the well-known and robust feature descriptors is SIFT. It has been successfully used for more than one decade in many applications including visual odometry, scene reconstruction, object recognition, etc. The SIFT descriptor vector contains 128 floating point numbers. The main issue with the SIFT descriptor is its computation and matching times. In real-time applications such as visual odometry or Simultaneous Localization and Mapping (SLAM), time limitation is a serious challenge. Moreover, in SLAM, it is needed to store feature descriptor vectors in order to find loop closure which is the main step of the path optimization. Therefore, the size of the descriptor vector is very important. The SURF descriptor, which is a fast version of SIFT, constructs a 64-D descriptor vector and reduces the computation time compared to the SIFT descriptor. However, it still has high computation and matching time. <cit.> showed that using dimensionality reduction techniques such as Principal Component Analysis (PCA) or Linear Discriminant Embedding (LDE) can reduce descriptor size without any loss in recognition performance. Another way to reduce the descriptor vector size has been presented by <cit.>. They took advantage of a quantization method to use only 4 bits to store floating numbers of the descriptor.Many methods have been employed to solve the SIFT problems to reach a fast computation and matching as well as a suitable discriminative power. BRIEF descriptor is one of the successful alternatives to SIFT. It has a good performance and a low computation cost. BRIEF computes a binary descriptor where each bit is independent. Therefore, the matching process would be much easier than the SIFT-like descriptors. Indeed, the Hamming distance would be used and the matching time will decrease significantly. More specifically, BRIEF defines a test on some pairs of image patches around the feature. The pixel intensity of each smoothed patch is computed and the test is performed on them. The result of the test which might be true or false, determines one bit of the descriptor. In other words, the BRIEF descriptor is an n-dimensional bit string, where each bit is the result of a test between two patches. They consider n=128, 256 and 512 in the largest configuration, and the vector needs just 64 bytes. A significant problem with BRIEF is that it is neither rotation- nor scale-invariant.The ORB descriptor, uses the keypoint direction to steer the BRIEF descriptor. In addition, a learning method is developed to choose a good subset of binary tests. It is a rotation invariant version of the BRIEF descriptor and it is highly robust to noise. FREAK is another method which is quite similar to the ORB descriptor. FREAK, inspired by human visual system, creates a sampling pattern which is extracted from the human retina. In human eyes, the distribution of ganglion cells is not uniform. Their distribution decreases exponentially as the cells take distance from the foveal area. FREAK uses the retina sampling pattern, which means that it raises the patch size according to the distance from the feature. In this way, descriptor extracts more detailed information from the area around the feature and extracts more global information as the patch takes distance from the feature. Moreover, the patches have some overlap in the receptive fields and lead to a better performance. <cit.> combined local neighborhood difference pattern (LNDP) and local binary pattern (LBP) to extract more information from local intensity of surrounding pixels of the feature. On the other hand, <cit.> proposed a feature descriptor based on Normalized Difference Vector (NDV). It was able to takes full advantage of the local difference between the neighboring pixels. Moreover, they proposed two strategies to make it rotation invariant. There are some approaches to construct the descriptor vector by learning a convolutional neural network such as <cit.>, <cit.>, and <cit.>. Most of these works tried to train a convolutional neural network works to extract the features of an input patch using an enormous amount of training patches. Then use these features as a descriptor. These descriptors are similar to the SIFT-like descriptors and The Euclidean distance reflects the features similarity. As a result, feature matching is more time-consuming than the binary descriptors. § PRELIMINARYIn this section, the basic elements needed to describe the proposed method are presented. First, a brief description of features is provided. Second, the feature descriptor is explained, and finally, a basic visual odometry algorithm is briefly presented.§.§ Feature PointA feature point is a pixel or a small area in an image which differs from its immediate neighborhood in terms of intensity, color, or texture <cit.> which in general, is called feature. The pixels around a good feature should have meaningful information in order to discriminate the feature from other points in the image. For example, a repeated plain texture or a simple edge is not a good feature. A good feature is a point in the 3D world which is projected on the camera image and it is easily locatable from different views. There are many feature extraction algorithms such as SIFT, SURF, FAST, and Harris. Each feature has some important properties including its location, response and size. The location shows the coordinate of the point in the image and response is the score of the feature calculated by feature extraction algorithm. The other important property of a feature is its size which represents the radius of the meaningful area around the feature.§.§ Feature DescriptorFeature descriptors have been used to match features between images of a scene taken from different views. Descriptors are numerical vectors constructed based on some calculation on the area around the feature. The goal of a descriptor is to extract meaningful information from the feature point to recognize it from different views. There are many feature descriptors for extracting this information and constructing the descriptor vector.Most of the feature descriptor algorithms use a fixed circular area around the feature to construct the descriptor. Areas with different radii lead to different descriptors. Therefore, it is crucial to have reliable criteria to select the size of this area. Most of the feature descriptor algorithms use the size properties of the feature to determine the area around the feature which should be used in the descriptor calculation.Feature descriptors construct the descriptor vector with different characteristics. Some feature descriptors such as SIFT and SURF construct a descriptor vector containing floating point numbers. Another group of descriptors such as ORB or FREAK construct a binary descriptor. Binary descriptors have a very low time complexity in descriptor construction as well as distance computation between descriptors and matching. §.§ Visual OdometryThe process of finding the translation between two positions of the camera based on images taken from those positions is called visual odometry. When this process is repeated on the images of a moving camera, the trajectory of the camera movement can be calculated. Many different visual odometry models are presented. All of the visual odometry models contain three main steps: feature extraction, feature matching or feature tracking, and motion estimation. In a stereo visual odometry model, first, the features of the left and right camera images are extracted. Then in the feature matching step, these features are matched for two time steps. Matching features between the left and right camera images and using triangulation methods results in 3D points. In the motion estimation step, these 3D points are used to estimate the motion of the camera between the two time steps.§ STEREO FEATURE DESCRIPTORSome steps of the stereo visual odometry such as feature extraction and triangulation are mandatory. In the stereo visual odometry algorithms, 3D features are used to estimate the motion. We believe that these 3D features and the stereo images can be useful in constructing a more robust feature descriptor. We propose two strategies to use these data to improve the feature descriptor. First, we estimate the scale of each feature using the 3D information and feature distance from the camera. Second, we use the two images of the stereo cameras to construct the descriptor vector. These approaches can be applied to any existing feature descriptor.The proposed stereo feature descriptor, requires 3D feature points to construct the feature descriptor. In the Figure <ref>, a suggested visual odometry model is shown that calculate 3D features before the feature descriptors calculation.Each stereo visual odometry model has two feature matching step. First, between left and right camera images to construct 3D features. Second, between two consequent positions of the camera.In the suggested model, the first feature matching step is done using feature tracking technique which are accurate when there is a small changes in the camera positions. And the proposed stereo feature descriptor is used in the second feature matching step. After feature matching, the depth information of the feature points is extracted. This is possible using several stereo geometry methods which is presented in<cit.>. Since the depth information is available, the stereo feature descriptor can be constructed. Our stereo approach is divided into two steps. In the first step, the area around each feature is calculated based on its distance from the camera. The area is a circle where its radius is a function of the depth information of the feature point. The second step constructs the feature descriptor based on the left and the right camera images. The two steps are illustrated in the next subsections. §.§ Scale Normalization Pinhole cameras, like the human eye, follow the linear perspective projection rules. This means that objects are seen smaller as their distance from the camera increases. Therefore, the scale of an object in the image of the camera would change as the object changes its distance from the camera. Scale-invariant feature descriptors can solve this problem at the cost of intensive calculations. However, they are limited on the scale change in their algorithms. The time limitation in visual odometry forces the researchers to ignore the scale problem and use fast descriptors which are scale sensitive. In addition, we need to find loop closures to optimize the motions estimation. This process needs strong feature descriptors to recognize a previously visited location.Nevertheless, there is some pre-calculated information which could help us to estimate the scale of features. The 3D positions of the features are used to estimate the motion of the camera. Therefore, a question arises: why not use this information to estimate the scale of a feature?The calculation of the exact distance of the features from the camera is possible using the stereo images where a feature is viewed from two calibrated cameras. These calculations are called triangulation. In the proposed feature descriptor, we leverage the depth information of the features to determine a fixed area around the feature point in the image and construct the feature descriptor based on this area. In other words, even if a feature is seen from a different distance, the image portion which is used for the calculation of the descriptor would be the same. To do so, the feature descriptor is calculated on a circle around the feature where the radius of this circle is derived based on the depth of that feature. The relationship between the depth and the radius should be estimated. This can be done by experimentally sampling and applying a curve fitting algorithm.Figure <ref> demonstrates an example of such situation where a feature is viewed in two images with different distances from the camera. Here, a circle is drawn around the feature. This circle also shows the area used to construct the feature descriptor. Since we determine an image portion around the feature according to its depth, the same patterns would be used to construct the feature descriptor vectors and therefore, the descriptors would be the same in the two images.The aforementioned policy can be applied to any descriptor by just modifying the size property of the features. In this paper, we use FREAK and SIFT as the base descriptors to implement our stereo descriptors. The FREAK descriptor uses Gaussian kernels, distributing them around the feature to extract local information. FREAK is inspired from the distribution of visual cells in the human retina. Figure <ref> shows these Gaussian kernels distributed in the two areas shown in Figure <ref>. The two images are quite similar, but they are different just in terms of resolution and image quality. In this step, Gaussian kernels would help us, since changing resolution does not affect on its output. §.§ Stereo DescriptorFinding perfect matches between features is a crucial step of motion estimation in visual odometry. Among matched features, there are many outlier matches. Removing these outlier matches is possible by outlier rejection methods. After the rejection of outliers, motion estimation is applied to the inlier matches. However,<cit.> proved that using all of the inliers would not lead to the best estimation. There are inliers which have some small errors. These errors are caused by small drifts in the feature extraction step. In addition, the feature descriptors are not sensitive enough to notice those small errors. As a result, they construct similar descriptors for these drifted points.Our solution for this problem is to prevent matching drifted features by constructing a more strict feature descriptor. This goal is achieved using both images of the feature in the left and right cameras. These images have slight differences which are related to the angles of the two cameras. The feature is in the viewing field of both cameras, but the surrounding area of the point, in which the descriptor is made, is different.Our descriptor vector has two sections. The first section represents the descriptor of the left view of the feature and the second section represents the descriptor of the right view of it. In this way, we have a double check on feature matching and prevent the feature matcher to match drifted features.§ EXPERIMENTAL RESULTSThe main aim of this paper is presenting a more robust feature descriptor for visual odometry, where 3D data is available. Therefore, the well-known visual odometry dataset called KITTI presented by <cit.> is used to evaluate the proposed descriptor. Note that most datasets common for evaluation of feature descriptors, such as the dataset presented by <cit.>, are monocular and do not support the stereo information. Thus they are not applicable to our stereo descriptor. Three experiments are designed to evaluate the proposed descriptor comprehensively. In the first experiment, the proposed scale normalized feature descriptor is evaluated. The second experiment examines the robustness of the stereo feature descriptor and the last experiment tests the stereo feature descriptor in a usual visual odometry problem. These experiments are presented in the next three subsections. A. Feature Matching ExperimentThe goal of this test is to show the degree of improvement the proposed scale normalization technique makes in the resulting descriptor. Therefore, this would be a contest between a standard descriptor and the scale normalized version of it. SIFT and FREAK descriptors are chosen for this evaluation. SIFT is a scale invariant and robust descriptor and FREAK is a fast and robust binary descriptor. We expect that the scale normalization technique would improves both of these descriptors.In this experiment, we count the number of frames in which a feature could be found using the specified descriptor. It is obvious that a higher correctly matched score shows higher robustness of the descriptor. Here, each feature will be matched to features in the next ten following frames. In other words, this is a multi-step feature matching and the features of frame t will be matched with the features of frame (t + step), where:step∈{1,2,3,... ,10} and t=Current frame number.This test is employed on the candidate descriptors two times. First, for the scale normalized version where the Scale Estimation flag is set to True and second, for the standard version where the Scale Estimation flag is set to False. Then, the evaluation algorithm operates as shown in Algorithm <ref>. Table 1 shown that scale normalized versions of SIFT and FREAK descriptors, named SN-SIFT and SN-FREAK, both got better scores in all sequences than their standard versions. SIFT is a scale invariant descriptor, so as it was expected, the improvement in FREAK is higher than in SIFT. However, the scale normalized SIFT also had a 8.75% improvement on the average. Since Freak is not scale invariant, the proposed technique helps FREAK to improve its score by 28.65% on the average. Scale normalization helps descriptors to become scale invariant feature descriptors.Feature extraction algorithms often use a threshold on the response property of the features to prune weak features. Using a strict threshold results in a small set of features with a high response property. In another word, a higher response indicates a stronger feature. Therefore, with the use of a strict threshold, it would be easy for descriptors to find features in consecutive frames and the proposed method results in a limited improvement over standard descriptors. However, in this experiment, in order to show the advantage of the proposed method, we use a loose threshold, so that the feature set has both strong and weak features. The experiment was repeated three times. The feature set size for these experiments was 310, 1350, and 2320, respectively. Figure <ref> shows the improvement of the proposed method over the standard descriptors for different sizes of the feature set. We see that the improvement increases as the size of the feature set increase.B. Visual Odometry Robustness Experiment To evaluate the robustness of the proposed method, we design an inlier counting visual odometry experiment in which we use the simple visual odometry model presented in Figure <ref>. In this test, instead of fitting the model to the dataset to get the highest accuracy, we aim to examine the robustness of the descriptors. Therefore, a multi-step visual odometry model is used where motion estimation will be applied to the current frame t and t+step frame later for step={1,2,3,4,5}. In a standard visual odometry, the step is equal to one. When a larger step is chosen, a larger transform would occur between frames and it becomes more challenging for the descriptor to cope with the scale change of each feature. We applied our stereo version of FREAK descriptor on the KITTI dataset and compared it against the standard version of the FREAK descriptor. The results of this test are shown in Figure <ref>. Our stereo descriptor produces 19% more inliers in comparison to the standard version of the FREAK descriptor. Due to the strict criteria on matching features in our method, considering the number of inliers as the only measure cannot constitute a comprehensive and conclusive evaluation. In order to deduce a better evaluation, we also perform an accuracy based visual odometry test on the stereo feature descriptor algorithm. C. Visual Odometry Accuracy Experiment In the final experiment, the accuracy improvement is evaluated. As before, we used the simple visual odometry model illustrated in Figure <ref>. For outlier rejection we used <cit.>. This method has an important random subsection which causes the model to have different results in each run even with fixed parameters. One way to cope with the randomness is to repeat the test for several times with fixed parameters. Therefore, the test is repeated 20 times for each descriptor and the average results are reported to have a better comparison. As mentioned before, this is not a benchmarking test and we want to evaluate the general accuracy improvement of our method.Repeating the experiment for several times, resulted in 23% improvement in the average estimation of the standard FREAK by using our stereo FREAK descriptor. Table <ref> shows the result of all tests. The stereo version of FREAK descriptor has better results in both worse and best estimations, too. This shows that the stereo descriptor is more stable than the standard version and the scale normalization method is successful to improve the visual odometry accuracy and stability at the same time. § CONCLUSIONSFeature descriptors are very useful in computer vision. Many algorithms such as visual odometry or SLAM rely on these methods to match features. Stereo visual odometry requires some mandatory computations such as 3D features calculation. This information can be used to improve feature descriptors. Our stereo feature descriptor uses the 3D information of features to adjust the scale based on the features depth. In this way, the descriptor robustness is improved without a significant computation. Moreover, we use stereo images to construct the descriptor vector. It helps the descriptor to have comprehensive information about the features and construct better descriptors. Our implementation increases the number of inlier matches of the SIFT and the FREAK descriptors by 8.75% and 28.65% on the average. Moreover, this technique results in 23% improvement in visual odometry due to preventing wrong matches and adjusting the scale of the feature descriptor . | http://arxiv.org/abs/1708.07933v2 | {
"authors": [
"Ehsan Shojaedini",
"Reza Safabakhsh"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170826053502",
"title": "Robust Stereo Feature Descriptor for Visual Odometry"
} |
http://arxiv.org/abs/1708.07848v4 | {
"authors": [
"Michael Moshe",
"Mark J. Bowick",
"M. Cristina Marchetti"
],
"categories": [
"cond-mat.soft"
],
"primary_category": "cond-mat.soft",
"published": "20170825180409",
"title": "Geometric frustration and solid-solid transitions in model 2D tissue"
} |
|
GState Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, ChinaUniversity of Chinese Academy of Sciences, Beijing 100049, China.Department of Theoretical Physics, Research School of Physics and Engineering, Australian National University, Canberra ACT 0200, AustraliaState Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, ChinaUniversity of Chinese Academy of Sciences, Beijing 100049, China.Centre for Modern Physics, Chongqing University, Chongqing 400044, ChinaDepartment of Theoretical Physics, Research School of Physics and Engineering, Australian National University, Canberra ACT 0200, AustraliaMathematical Sciences Institute, Australian National University, Canberra ACT 0200, Australia[][email protected] Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, ChinaDepartment of Theoretical Physics, Research School of Physics and Engineering, Australian National University, Canberra ACT 0200, AustraliaCenter for Cold Atom Physics, Chinese Academy of Sciences, Wuhan 430071, China71.10.Fd, 75.40.Cx,02.30.IkIn this Rapid Communicationwe showthat low energy macroscopic properties of the one-dimensional (1D)attractive Hubbard model exhibit two fluids of bound pairs and of unpaired fermions.Using the thermodynamicBethe ansatz equationsof the model, wefirstdetermine the low temperature phase diagramand analytically calculatethe Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) pairing correlation function forthe partially-polarized phase. Wethenshowthat for such a FFLO-likestatein the low density regime theeffective chemical potentials of bound pairs and unpaired fermions behave liketwo free fluids. Consequently, the susceptibility, compressibility and specific heatobeysimpleadditivity rules, indicatingthe `free' particlenature of interacting fermions on a 1D lattice.In contrast to thecontinuum Fermi gases, the correlation critical exponents andthermodynamics of theattractive Hubbardmodelessentially depend on two latticeinteracting parameters. Finally, westudy scaling functions, the Wilson ratioand susceptibilitywhichprovide universalmacroscopic properties and dimensionless constants of interacting fermions at low energy. FFLO correlationandfreefluidsin the one-dimensionalattractive Hubbard modelXi-Wen Guan December 30, 2023 ======================================================================================= The notion of Landauquasiparticlesgives rise to theFermi liquidtheorysuccessfully used for describing properties of a large variety of systems, such as Fermi liquid ^3He and electrons in metals <cit.>. In contrast, it is generally accepted that Fermi liquid theory is not applicable in 1D, wherethe description of the low-energy physics of strongly correlated electrons, spins, bosonic and fermionic atoms relies on theTomonaga-Luttinger liquid (TLL) theory <cit.>. Such an understanding of the TLLin 1D is based on collective excitations which are significantly different fromLandauquasiparticles in higher dimensions. However, concerning macroscopic properties, there are many universal properties/quantities which are common for both 2D/3D and 1D systems <cit.>.The1D repulsive Fermi-Hubbard model describing interacting fermions on a lattice provides a paradigm for understanding many-body physics, includingspin-charge separation, fractional excitations, quantum dynamics of spinons,a Mott insulatingphase and magnetism <cit.>. Very recently, ultracold atoms trapped in optical lattices <cit.> offer promising opportunitiesto test such fundamental concepts <cit.>. In contrast, the 1D attractive Fermi-Hubbard model <cit.> is a notoriouslydifficult problem due to the complicatedbound states of multi-particles and multi-spinson lattices. Despite there being a mapping by Shiba transformations between the repulsive and attractive regions of the Hubbard model <cit.>,such a mapping cannot be used for a study ofthe low energy themodynamicsof the attractive Hubbard modeldue to the differentcut-offprocesses in terms ofsuch multi-spin and multi-charge bound states. Of central importance tothis attractive Hubbard modelis the understandingof quantumcorrelations of chargebound states,for example, theFulde-Ferrell-Larkin-Ovchinnikov (FFLO) likepairing <cit.>on a 1D lattice <cit.>.In the expansion dynamics of the FFLO state in 1D <cit.>,a nature of two fluids of bound pairs andfree fermions was indicated. In this Rapid Communication, building on the thermodynamic Bethe ansatz (TBA) equationsof the attractive Hubbard model, weanalytically obtain theFFLO pairing correlation andthe universal two free quantum fluids of the FFLO-like state, where the lattice effects are seen to drive the system differently to the continuous Fermi gas <cit.>, see Fig. <ref>. More detailedstudies ofthismodel will bepresentedelsewhere <cit.>. TheBethe ansatzsolution. The 1D singleband Hubbard modelis described by the Hamiltonian <cit.> H= - ∑^L_j=1, a=↑,↓( c_j,a^† c_j+1,a +h.c.)+ u ∑_j=1^L (2n_j,↑-1) (2 n_j,↓-1),where c_j,a^† and c_j,a are the creation and annihilation operators of electrons (fermionic atoms) with spin a (internal degrees of freedom)(a=↑ or a=↓) at site j ona 1Dlattice with length L.They satisfy the anticommutation relations { c_j, a, c_k,b}= { c_j, a^† , c_k,b^†}=0 and { c_j, a, c_k,b^†}=δ_jkδ_ab.Meanwhile n_j,a=c_j,a^† c_j,a is the density operator, n_e=1/L∑_j=1^L∑_a n_j,a is the total fermion number per latticesite and u is the dimensionless interaction strength between particles (u>0 forrepulsion and u<0 forattraction).In 1968Lieb and Wu<cit.> derivedthe Bethe ansatz (BA) equations forthe 1D Hubbardmodel by meansof Bethe's hypothesis <cit.>. Takahashi <cit.> discovered the solutions of the BA equations which in general are classified as real quasimomenta k, k-Λ strings and complex spin rapidities ofΛ strings, see <cit.>. Theseroots respectively count for the quasimomenta of the single fermions, bound states of different lengths of fermions and bound states of magnons with different lengths. Athigh energy or momentum, such bound states cancoexist. Building on Takahashi's string hypothesis, we obtain the TBA equations for the 1D attractive Hubbard model <cit.> ε(k) =g_0(k)-∑_n=1^∞ a_n ∗( F[ε^'_n]-F[ε_n])(k), ε_n(Λ) = 2nB-a_n^t ∗ F[ε](Λ)-∑_m=1^∞ A_n m∗ F[ε_m](Λ),ε_n^'(Λ) = g_n(Λ) -a_n^t ∗ F[ε](Λ)-∑_m=1^∞ A_n m∗ F[ε^'_m](Λ) withthe notation F[x](y)=-Tln[1+exp(-x(y)/T)] and n=1,…,∞. The kernel function a_n(x)=1/2π2n|u|/(n|u|)^2+x^2. The driving termsare g_0(y)= -2cos y-μ-2u-B and g_n(y) =-4√(1-(y+in |u|)^2)-n(2μ+4u). In the above equations we denoted the convolutions a_n ∗ F[x] (k) = ∫_-∞^∞y a_n( k-y) F[x(y)] and a_n^t ∗ F[x](Λ)=∫_-π^πycos ya_n(sin y-Λ) F[x(y)]. The functions ε,ε_m^' and ε_nstand for the dressed energiesfor unpaired fermions, bound states of 2m fermions (the k-Λ strings) and length-n spin strings of magnons, respectively. The function A_nm(x) is given in<cit.>.It is particularly important to observe that thelonger k-Λ strings areinvolved in the thermodynamicsastemperature increases <cit.>. The free energy per site is thus given byf=u+∫_-π^πk/2π F[ε](k) +∑_n=1^∞∫_-∞^∞Λ/2πξ_n(Λ) F[ε_n^'](Λ)with ξ_n(Λ)=∫_-π^π k a_n(Λ-sin k). We also observe that in the dilute limit, u→ 0, n_e→ 0 with n_e/|u| constant <cit.>, the TBA equations (<ref>)-(<ref>) reduce to those ofthe Gaudin-Yang model <cit.>.We note that the Shiba transformation between the repulsive and attractive regions of the Hubbard model does not help to obtain universal low energy physics from the TBA equations.This is mainly because the cut-off processes regarding the above spin andcharge bound states are quite different <cit.>, unlike the case ofthe ground state <cit.>.As we shall see, in the attractive regime, the low energy physics of the modelis no longer described by the spin-charge separated theory, rather it is described by the FFLO-like quantum liquids of pairs and single fermions.Quantum phase diagram and Wilson ratio.In contrast to the repulsive case, the ground stateof the attractive Hubbard model hascharge bound states, i.e.,length-1k-Λ strings,forminga lattice version of the FFLO state.The quantum phasesand phase diagram at T=0can be directly determined from theTBA equations (<ref>)-(<ref>) in the limit T→ 0, which arecalled the dressed energy equations <cit.>. The dressed energy equationsdeterminefive quantum phases in the μ-B plane: vacuum I, fully-polarized phase II, half-filled phase III, FFLO-like state IV and fully-paired state V, see Fig. <ref>.The zero temperaturephase boundaries can also bedetermined bythe Shiba transformation <cit.>.Here we show that the Wilson ratio, namely, the dimensionless ratio of thecompressibility κ and the specific heat divided by the temperature T,R_W^κ=π^2 k_B^2/3κ/C_v/T,provides a convenient way for revealing the full phase diagram at low temperatures, see Fig. <ref>. In the above k_B is Boltzmann's constant. Thisratio can be directly calculated from the finite temperature TBA equations (<ref>)-(<ref>) with a suitable spin and charge bound state cut-offprocess,see <cit.>. We find that the ratio R_W^κis capable of distinguishing all phases of quantum states, including the FFLO-like state in the phase diagramFig. <ref>. We observe that an enhancement of this ratiooccursnear a phase transition. It gives a finite valueat the critical pointunlike the divergent values of compressibility andsusceptibility for T→ 0. Indeed, the phase boundaries determinedby the Wilson ratio(<ref>) coincide with the ones determined bythedressed energy equationsat T=0.The phases IVand VinFig. <ref>revealsignificant features, namelythe quasi-long range orderand free-fermion quantum criticality.A constant Wilson ratio implies that the two types of fluctuations are on an equal footing, regardless of the microscopic details of the underlying many-body systems.Regarding the sudden change of the Wilson ratio near a phase transition,we observe thattheparticle number and energyfluctuations become temperature dependent, see Fig. <ref>(a). At the critical point,the vanishing of the Fermi points, i.e., ε_1(0)=0 and ε_1^' (0)=0, in the Fermi sea of pairs and of unpaired fermions leads to a universality class of quantum criticality. In the critical regime, the scaling functions of thermodynamicproperties can be cast intouniversal forms. From the TBA equations (<ref>)-(<ref>),we obtain the scaling functions of compressibility and susceptibilityκ(μ,B,T) = κ_0(μ,B)+T^d/z+1-2/ν zλ_κℱ( μ-μ_c/T^1/ν z) ,χ(μ,B,T) = χ_0(μ,B)+T^d/z+1-2/ν zλ_χ𝒦( μ-μ_c/T^1/ν z).Herethe scaling functions ℱ(x)=𝒦(x)= Li_-1/2(x) indicates a free-fermion criticality classified by the dynamical critical exponents z=2 and correlation critical exponent ν=1/2, see <cit.>. Theterms κ_0 and χ_0 are the regular part and the factors λ _κ,χ are phase dependent constants. Fig. <ref>(b) andFig. <ref>(c) show such universal scaling behaviour of the susceptibility and compressibility across the phase boundaries (V,IV). Similar scaling invariant behaviour occurs whenever the model parameters are driven across the phase boundaries in Fig. <ref>.FFLO correlation. For the fully paired state V,the pairing correlation length is larger than the average interparticlespacing.In this phase, the single particle Green's function decays exponentially, whereasthe singlet pair correlation function decays as a power of distance <cit.>. However,once the external field exceeds the critical line between phases IV and V, the Cooper pairs start to break apart. Thus both of these correlation functions decay as a power of distance, indicating a quasi-long range correlation.In the phase IV, Cooper pairsand excess fermions form a 1D analogue of theFFLO pairing-like state <cit.>. However,analytical result forthe FFLO pairing correlations for the Hubbard modelis still lacking.For obtaining a universalform of the FFLO-like correlation function, wefirstfocus on thecase oflow density n_e≪ 1 and low energy. In the FFLO-like phase IV, the spin wave bound states ferromagnetically couple tothe Fermi sea ofthe unpaired fermions. Thus the spin wave fluctuations can be ignored at low temperatures due to this ferromagnetic nature. Then we simplify the TBA equations(<ref>)-(<ref>)as<cit.> ε(k) ≈ k^2-μ_1 -a_1 ∗ F[ε^'_1](k), ε_1^'(Λ) ≈ α_1 ( Λ^2- μ_2)-a_1 ∗ F[ε](Λ) - a_2 ∗ F[ε^'_1](Λ).The free energy (<ref>) reduces to f≈ u+∫_-π^πk/2π F[ε](k) +∫_-∞^∞Λ/2πβ_1 F[ε_1^'](Λ)<cit.>.In this new set of TBA equations (<ref>) and (<ref>)we have introduced two effective chemical potentialsμ_1 = μ-2|u|+B+2, μ_2 = 1/α_1 [2μ+ 4(√(u^2+1)-|u|)],for understanding the FFLO correlation andfree-fermion nature ofthe attractive Hubbard model. In the above equations theparameters α_nand β_n reflect the interacting effect of the length-nk-Λ bound states on a lattice. They are given byα_n= ∫_-π^π kcos^2 k a_n(sin k)2|u| cos^2 k (n^2u^2-3 sin^2 k)/(n^2u^2+sin^2 k)^3, β_n = ∫_-π^π k a_n(sin k).At low energy physics only length-1k-Λ stringsare involved. In this region, the lattice parameters α_1 and β_1approach 2 when u tends to zero. However, for large |u|,the band ofpairs becomes flat <cit.>.The TBA equations (<ref>) and (<ref>) are reminiscentof the `feedback interaction' equation in theLandau Fermi liquid theory <cit.>. The driving term in(<ref>)can be expressedas ħ^2/2mα_1(k^2-μ_2)=p^2_0/2α_1 m-ħ^2 /2mα_1 μ_2 with 2m=ħ=1, whichis the first-order coefficient describing the excitation energy of a single bound pair. The lattice parameterα_n characterizesthe effective massof length-nk-Λ strings (bound state of 2n atoms on a lattice).In lightof the conformal field theory approach<cit.> and using the TBA equations(<ref>) and (<ref>), we calculatetheasymptotic formofthe FFLOcorrelation function of the attractive Hubbard model inthe low density region <cit.> G_p(x,t) = ⟨Ψ_↑^†(x,t) Ψ_↓^†(x,t) Ψ_↑(0,0) Ψ_↓(0,0) ⟩≈A_p,1cos( π(n_↑-n_↓)x )/|x+ iv_u t|^2θ_1 |x+iv_b t|^2θ_2 + A_p,2cos( π(n_↑-3n_↓)x )/|x+iv_u t|^2θ_3 |x+iv_b t|^2θ_4,with the exponents θ_1 ≈ 1/2, θ_2 ≈ 1/2+n_2/|u|β_1,θ_3 ≈1/2-4n_2/|u|β_1 and θ_4 ≈5/2-4 n_1/|u|-3n_2/|u|β_1. Here n_2,1=N_2,1/L arethe dimensionlessdensitiesof pairs andunpaired fermions, respectively. The sound velocities are given by v_b=√(α_1 )/β_1π n_2(1+1/|u|β_1 (2n_1+n_2)) and v_u=√(2)π n_1 (1+4/|u|n_2 ).In the above equation the coefficients A_p,1 and A_p,2 areconstant factors. In this phase IV the spatial oscillation inthe pairing correlation is a characteristicof the FFLO state, where the imbalance n_↑-n_↓ in the densities of spin-up and spin-down fermions gives rise to a mismatch in Fermi surfaces between both species of fermions. In 1D,the spatial oscillation signature in pair correlation isa consequence ofthebackscattering for bound pairs and unpaired fermions, see also the results for the Gaudin-Yang model <cit.>. Here we observe that the critical exponent θ_2 depends essentially on the latticeparameter β_1. So do the critical exponents in other types of correlation functions <cit.>. The Fourier transform ofG_P(x,0^+) gives G̃_p(k) ∼[ (k-π(n_↑-n_↓) ) ]^2s_p|k-π(n_↑-n_↓)|^ν_p with 2s_P≈ 0 and ν_p ≈ n_2/(|u|β_1).Two free fluidsand spin gapped phase. At low temperatures, we finda significantnature of two fluids in phase IV. For the ground state, the energy can be regarded as twoTLLsof unpaired fermions and of pairs due to the quasi-long range correlation. Without losing generality,we consider a physical regimeoflow density (n_e small),low temperatureand finite strong magnetic field. This region is reachable in cold atoms <cit.>. In this regime, the chemical potentials for the unpaired fermions and pairs are given explicitly byμ_1 = π n_1^2A_1^2+4π^2α_1/3β_1^3|u|n_2^3A_2^3,μ_2 = π^2n_2^2/β_1^2A_2^2+4π^2/3α_1|u|n_1^3 A_1^3 + 2π^2/3β_1^3|u|n_2^3 A_2^3 ,whereA_1=1+2n_2/|u|+(2n_2/|u|)^2 and A_2=1+2n_1+n_2/β_1 |u|+(2n_1+n_2/β_1 |u|)^2indicateinteracting effects among pairs and unpaired fermions like that of the Fermi gas<cit.>.The effective chemical potential μ_2 inthe 2D interacting Fermi gasesshows acrossover from a Bose-Einstein condensate to a Bardeen-Cooper-Schrieffer superconductor in ultracold fermions <cit.>.Moreover, from the relations (<ref>) we demonstratethe free-particle nature of two fluidsthroughtheadditivity rules in compressibility and susceptibility:κ=κ_1+2/α_1 κ_2, 1/χ=1/χ_1 +α_1 /21/χ_2,where κ_r=( ∂rn_r/∂μ_r)|_B and χ_r=( ∂ rn_r/∂μ_r )|_n with r=1,2 for unpaired fermions and pairs, respectively.We see thatthe effective binding energy e_b=-(2u+2)n_1-4(u+√(u^2+1))n_2 of a bound pair isabsorbed into the effective chemical potentials.The compressibility and susceptibility can be explicitly calculated from the chemical potentials (<ref>) and(<ref>) via therelations1/κ_1 = J/(∂μ_1/∂ n_2 -α_1/2∂μ_2/∂ n_2 ),1/κ_2=-1/α_1J/( ∂μ_1/∂ n_1 -α_1/2∂μ_2/∂ n_1), χ_1 = 1/(∂μ_1/∂ n_1 -1/2∂μ_1/∂ n_2) ,χ_2=-1/(∂μ_2/∂ n_1 -1/2∂μ_2/∂ n_2) ,where the Jacobi determinant J=-α_1/2 (∂μ_1/∂ n_1∂μ_2/∂ n_2 -∂μ_2/∂ n_1∂μ_1/∂ n_2). The explicit forms are given in<cit.>. The additivity rules in the thermodynamic properties reveal a significant free-particle feature in the phase of multiple quantum liquids on a 1D lattice.Furthermore, using the TBA equations (<ref>)-(<ref>) and the BA equations with thelength-1k-Λ strings, we show that the specific heat, i.e., a measureof the energy fluctuations, is given by C_v=π T/3( 1/v_u + 1/v_b). Here the sound velocities v_b,u are as given above.A second-orderphase transition occurs when the system is driven across the phase boundary in the μ-B plane, see Fig. <ref>. Fig. <ref>(a) and Fig. <ref>(b)showthe compressibility and susceptibilityvsmagneticfield at different temperatures.Theyare temperature independent inphase IV, whereas the specific heat depends linearly on the temperature, having thus a common feature of the Fermi liquid in higher dimensions. We observe that in phase IV the compressibility and susceptibility curves at different temperatures collapse into the zero temperature ones obeyingthe additivity rules (<ref>).Fig. <ref>(c) shows the susceptibility vstemperature for different magneticfields. ForB>B_c, the susceptibilitydisplays a flat region in the χ-T plane, the small regionto the left ofthe green dashed line, indicating the two free fluids. For B<B_c, the susceptibility illustrates the exponential decay as temperature decreases (blue lines). In this case, the susceptibility is given by χ_s=T^-1/2/4√(π)e^-Δ/T with the energy gap Δ=-R^2+4 (2π-R^3/3)/3 |u| π^2( 1+ 2|u|πμ/2π-R^3/3)^3/2, indicating the behaviour of dilute magnons. Here we have denotedR=Re√(μ+2u+B+2). In summary,for the attractive Hubbard model, we have analytically calculatedtheFFLO pair correlation and critical exponents,along with scaling functions of thermal andmagnetic properties forwhich the lattice effect becomes prominent. We have obtained the effective chemical potentialsof the bound pairs and of the unpaired fermionsand demonstratedthe additivity rules ofthe susceptibility and the compressibility in the FFLO-like state. While wehave foundthatthe susceptibility and the compressibility are temperature independent, the specific heat depends linearly on the temperature in this phase. Theseresults provide strong evidence for the existence of two free fluids of bound pairs and of unpaired fermions, which were predicted in expansion dynamics of the FFLO state in 1D <cit.>.Acknowledgments.The authors SC and YCY contributed equally to the calculations in this paper. The authorsthank M. Takahashi and R. Huletfor helpful discussions. This work is supported by Key NNSFC grant number 11534014, MOST grant number 2017YFA0304500, NNSFC grant numbers 11374331, 11174375 and ARC Discovery Projects DP130102839, DP170104934.99Hew97 A. C. 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Guan, “Universal thermodynamics of the one-dimensional attractive Hubbard model", arXiv:1708.07784.Cheng:2017-preprint-2 S. Cheng, Y.-Z. Jiang, Y.-C. Yu, M. T. Batchelor and X.-W. Guan, Nucl. Phys. B929, 353 (2018). Lieb:1968 E. H. Lieb and F. Y. Wu, Phys. Rev. Lett.20, 1445 (1968).Bethe:1931 H. A. Bethe, Z. Phys.71, 205 (1931).Takahashi:1972 M. Takahashi, Prog. Theor. Phys.47, 69 (1972).Takahashi:1974 M. Takahashi, Prog. Theor. Phys.52, 103 (1974).Tak99M.Takahashi,Thermodynamics ofOne-Dimensional Solvable Models,(Cambridge University Press, Cambridge, 2005).note-k-L-string Private communication with Professor M Takahashi. If the effective mass of the k-Λ strings is positive, they do not form bound states. The effective mass is given by d^2 E(q)/d q ^2. In the Hubbard model,the mass can be negative at high energy or momentum. Therefore k-Λ string bound states can exist at high energy scales.note1We observe thata small value of Λ makes a realistic contribution to the free energy of the pairs in the dilute limit. Therefore forlow densityonly the leading orderof ξ_n(Λ), i.e., theβ_n, is usedforthe free energy of the pairs.Coleman P. Coleman,Introduction to Many-Body Physics, (Cambridge University Press, Cambridge, 2016).Bogolyubov:1989 N. M. Bogolyubov and V. E. Korepin, Teor. i Mat. Fiz.82,331 (1990). Frahm:1990 H. Frahm and V. E. Korepin, Phys. Rev. B42, 10553 (1990).Frahm:1991 H.Frahm and V. E. Korepin, Phys. Rev. B43, 5653 (1991).note-asymptoticHere the asymptotic behavior means the long distance behavior, i.e., the distance x is large.Lee-Guan:2011 J. Y. Lee and X.-W. Guan, Nucl. Phys. B853, 125 (2011).Guan:2007 X.-W. Guan, M. T. Batchelor, C. Lee and M. Bortz, Phys. Rev. B76, 085120 (2007).Boett15 I. Boettcher, L. Bayha, D. Kedar, P. A. Murthy, M. Neidig, M. G. Ries, A. N. Wenz, G. Zürn, S. Jochim and T. Enss, Phys. Rev. Lett.116, 045303 (2016). | http://arxiv.org/abs/1708.07776v3 | {
"authors": [
"Song Cheng",
"Yi-Cong Yu",
"Murray T Batchelor",
"Xi-Wen Guan"
],
"categories": [
"cond-mat.str-el"
],
"primary_category": "cond-mat.str-el",
"published": "20170825152721",
"title": "FFLO correlation and free fluids in the one-dimensional attractive Hubbard model"
} |
http://arxiv.org/abs/1708.07600v1 | {
"authors": [
"Zhongfei Xiong",
"Weijin Chen",
"Peng Wang",
"Yuntian Chen"
],
"categories": [
"physics.optics"
],
"primary_category": "physics.optics",
"published": "20170825022024",
"title": "Classification of symmetry properties of waveguide modes in presence of gain/losses, anisotropy/bianisotropy, or continuous/discrete rotational symmetry"
} |
|
35Q55; 35Q35In this note, we prove the profile decomposition for hyperbolic Schrödinger (or mixed signature) equations on ^2 in two cases, one mass-supercritical and one mass-critical.First, as a warm up,we show that the profile decomposition works for the Ḣ^1/2 critical problem. Then, we give the derivation of the profile decomposition in the mass-critical casebased on an estimate of Rogers-Vargas <cit.>. [ Shaoming Guo December 30, 2023 =====================§ INTRODUCTION We will consider the hyperbolic (or mixed signature) Schrödinger equation on 𝐑^2, which is given byi ∂_tu + ∂_x∂_y u = |u|^p u,u(x, y,0) = u_0(x, y).In particular, we will focus on the cases p=4 and p=2.The case p=2 arises naturally in the study of modulation of wave trains in gravity water waves, see for instance <cit.>; it is also a natural component of the Davey-Stewartson system <cit.>.As can be observed quickly from the nature of the dispersion relation, the linear problemi ∂_tu + ∂_x∂_y u = 0,u(x,y,0) = u_0(x, y).satisfies the same Strichartz estimates and rather similar local smoothing estimates[See <cit.> for a general treatment of smoothing estimates for dispersive equations with non-elliptic symbols.] as its elliptic counterpart, the standard Schrödinger equation. In particular, there exists a constant C<∞ such that‖ e^it∂_x∂_yf‖_L^4_x,y,t≤ C‖ f‖_L^2_x,y.Hence, large data local in time well-posedness and global existence for small data with p≥2 can be observed using standard methods that can be found in classical texts such as <cit.>.For quasilinear problems with mixed signature, some local well-posedness results have been developed recently, see <cit.>.Non-existence of bound states was established in <cit.> and a class of bound states that are not in L^2 were constructed in <cit.>.Long time low regularity theory for this equation at large data remains unknown however.Recently, an approach to global existence for sufficiently regular solutions was taken in <cit.>, but it is conjecturedthat (<ref>) should have global well-posedness and scattering for all initial data in L^2. Much progress has been made recently in proving global well-posedness and scattering for various critical and supercritical dispersive equations by applying concentration compactness tools, which originated with the works of Lions <cit.>.One major step in applying modern concentration compactness tools to dispersive equations is the profile decomposition, see <cit.>.The idea is that given a small data global existence result, one proves that if the large data result is false then there is a critical value of norm of the initial data at which for instance a required integral fails to be finite.Then, the profile decomposition ensures that failure occurs because of an almost periodic critical element, which may then be analyzed further and in ideal settings ruled out completely.See <cit.> and references therein for applications of this idea in the setting of focusing and defocusing Schrödinger equations for instance.A major breakthrough in profile decompositions arose in the works of Gérard <cit.>, Merle-Vega <cit.>, Bahouri-Gérard <cit.>, Gallagher <cit.> and Keraani <cit.>.Those results have then been used to understand how to prove results about scattering, blow-up and global well-posedness in many settings, see <cit.> for some examples. We also mention the recent work by Fanelli-Visciglia <cit.>, where they consider profile decompositions in mass super-critical problems for a variety of operators, including (<ref>).As can be seen in <cit.>, the profile decomposition follows from refined bilinear Strichartz estimates. Using refined Strichartz estimates from <cit.> and bilinear Strichartz estimates, Bourgain <cit.> proved concentration estimates and global well-posedness in H^3/5 + ϵ for the defocusing, cubic elliptic nonlinear Schrödinger equation in ^2. Building on this work, Merle-Vega <cit.> proved a profile decomposition for the mass-critical elliptic nonlinear Schrödinger equation in two dimensions.For the hyperbolic NLS, Lee, Vargas and Rogers-Vargas <cit.> have provided refined linear and bilinear estimates, drawingon results of Tao <cit.> for the elliptic Schrödinger equation. In particular <cit.> gives an improved Strichartz estimate similar to our Proposition <ref> and uses it to prove lower bounds on concentration of mass at blow-up. An improved Strichartz estimate is also the key element in our profile decomposition, following the standard machinery described in <cit.>. For completeness, we provide a proof of Proposition <ref>, which, although drawing on similar ideas as in <cit.>, outlines more explicitly the additional orthogonality of rectangles with skewed ratios. The major issue in following the standard proof of the profile decomposition is that the mixed signature nature of (<ref>) means that an essential bilinear interaction estimate that holds in the elliptic case fails.This is compensated for in <cit.> by making a required orthogonality assumption for the refined bilinear Strichartz to hold (see the statement in Lemma <ref> below). To overcome this difficulty, we use a double Whitney decomposition to precisely identify the right scales, which introduces many different rectangles that are controlled using the fact that functions with support on two rectangles of different aspect ratios have small bilinear interactions.We note that while we here focus on analysis in 2 dimensions to keep the technical computations focused and directed, we expect many of the calculations to be generalizable to other dimensions as in <cit.>. The paper is structured as follows: in Section <ref>, we set up the problem, discuss some basic symmetries and establish some important bilinear estimates; in Section <ref> we establish the result in the mass-supercritical case using the extra compactness that comes from the Sobolev embedding; in Section <ref>, we establish the main precise Strichartz estimate in the paper and in Section <ref>, we obtain the profile decomposition for the mass-critical problem and deduce the existence of a minimal blow-up solution. Finally, we conclude with a short Appendix in which we correct an error in the published version of this article. Acknowledgments. The first author was supported in part by U.S. NSF Grant DMS - 1500424The second author was supported in part by U.S. NSF Grants DMS–1312874 and DMS-1352353. The third author was supported in part by U.S. NSF Grant DMS–1558729 and a Sloane Research fellowship.The fourth author was supported in part by U.S. NSF Grant DMS-1516565.We wish to thank Andrea Nahmod, Klaus Widmayer, Daniel Tataru, Nathan Totz for helpful conversations during the production of this work. Part of this work was initiated when some of the authors were at the Hausdorff Research Institute for Mathematics in Bonn, then progressed during visits to the Institut des Hautes Études in Paris, the Mathematical Sciences Research Institute in Berkeley and the Institute for Mathematics and Applications in Minneapolis.The authors would like to thank these institutions for hosting subsets of them at various times in the last several years.We thank also an anonymous referee who pointed out the results of Rogers-Vargas <cit.> and its relevance for this work and Emanuel Carneiro for pointing out that error in the original appendix and sharing his manuscript on optimizers for (<ref>), <cit.>. § PROPERTIES OF (<REF>) Observe that a solution toi ∂_tu + ∂_x∂_y u = |u|^2 u,u(x, y,t) = u_0(x, y), has a number of symmetries:* Translation: for any (x_0, y_0) ∈𝐑^2,u(x,y,t) ↦ u(x - x_0, y - y_0,t),* Modulation: for any θ∈𝐑,u(x,y,t) ↦ e^i θ u(x,y,t).* Scaling: for any λ_1, λ_2 > 0,u(x,y,t) ↦√(λ_1λ_2) u(λ_1 x, λ_2 y,λ_1λ_2 t), * Galilean symmetry: for (ξ_1, ξ_2) ∈𝐑^2,u(x,y,t) ↦ e^-it ξ_1ξ_2 e^i[x ξ_1 + y ξ_2] u(x - ξ_1 t, y - ξ_2 t,t).* Pseudo-conformal symmetry:u(x,y,t) ↦e^ixy/t/itu(x/t,y/t,1/t). These symmetries all preserve the L^2_x,y norm. The first two symmetries (<ref>)-(<ref>), as well as the scaling symmetry properly redefined, also preserve the Ḣ_h^s norm for any s ∈𝐑, where ‖ f‖_Ḣ_h^s^2 =‖|∂_x|^s/2|∂_y|^s/2f‖_L^2^2. Note that this norm has similar scaling laws as the more usual Ḣ^s norm.Other examples of anisotropic equations have appeared in for instance <cit.>. For example, for the Ḣ^1/2 - critical problem i ∂_tu + ∂_x∂_y u = |u|^4 u,u(x,y,0) = u_0(x, y), the symmetries are thus: * Translation: for any (x_0, y_0) ∈𝐑^2, u(x,y,t) ↦ u(x - x_0, y - y_0,t),* Scaling: for any λ_1, λ_2 > 0, u(x,y,t) ↦ (λ_1λ_2)^1/4 u(λ_1 x, λ_2 y,λ_1λ_2 t),* Modulation: for any θ∈𝐑, u(x,y,t) ↦ e^i θ u(x,y,t).We will treat the profile decomposition for (<ref>) as a warm-up, before tackling the profile decomposition for the mass-critical problem (<ref>). §.§ Notations Let φ be a usual smooth bump function such that φ(x)=1 when | x|≤ 1 and φ(x)=0 when | x|≥ 2. We also letψ(x)=φ(x)-φ(2x). We will often consider various projections in Fourier space. Given a rectangle R=R(c,ℓ_x,ℓ_y), centered at c=(c_x,c_y) and with sides parallel to the axis of length 2ℓ_x and 2ℓ_y, we defineφ_R(x,y)=φ(ℓ_x^-1(x-c_x))φ(ℓ_y^-1(y-c_y)).We define the operatorsQ_M,Nf(ξ,η) =ψ(M^-1ξ)ψ(N^-1η)f(ξ,η), P_Rf(ξ,η) =φ_R(ξ,η)f(ξ,η).The first operator is only sensitive to the scales involved, while the second also accounts for the location in Fourier space. We also let | R|=4ℓ_xℓ_y denote its area. §.§ Some Preliminary Estimates We start with a nonisotropic version of the Sobolev embedding. There holds that‖ f‖_L^q_x,y≲‖|∂_x|^s/2|∂_y|^s/2f‖_L^p_x,ywhenever 1<p≤ q<∞, 0≤ s<1 and1/q=1/p-s/2The proof, although easy, highlights the need to treat each direction independently. Using Sobolev embedding in 1d, Minkowski inequality and Sobolev again, we obtain that‖ f‖_L^q_x(𝐑,L^q_y(𝐑)) ≲‖|∂_x|^s/2f‖_L^q_y(𝐑,L^p_x(𝐑))≲‖|∂_x|^s/2f‖_L^p_x(𝐑,L^q_y(𝐑))≲‖|∂_y|^s/2|∂_x|^s/2f‖_L^p_x,ywhich is what we wanted. We have two basic refinements of (<ref>). Note the difference in orthogonality requirements between Lemma <ref> and Lemma <ref>.Assume that f=P_R_1f and g=P_R_2g where R_i=R(c^i,ℓ_x,ℓ_y) and | c^1_x-c^2_x| = N≥ 4ℓ_x, and let u (resp. v) be a solution of (<ref>) with initial data f (resp. g). Then‖ uv‖_L^2_x,y,t≲(ℓ_x/N)^1/2‖ f‖_L^2_x,y‖ g‖_L^2_x,y.Assume that f=P_R_1f and g=P_R_2g where R_i=R(c^i,ℓ_x,ℓ_y), | c^1_x-c^2_x|≥ 4ℓ_x and | c^1_y-c^2_y|≥ 4ℓ_y, and let u (resp. v) be a solution of (<ref>) with initial data f (resp. g). Then‖ uv‖_L^q_x,y,t≲ (ℓ_xℓ_y)^1-2/q‖ f‖_L^2_x,y‖ g‖_L^2_x,ywhenever q>5/3. Lemma <ref> is the main refined bilinear estimate and appears essentially in <cit.> when dealing with cubes. The result as stated here follows by scaling rectangles to cubes.We simply write thatu^2(ξ,η,t) =I(ξ,η,t),I(ξ,η,t) =∬_𝐑e^-itωφ_R_1(ξ_1,η_1)φ_R_2(ξ-ξ_1,η-η_1)×f(ξ_1,η_1)g(ξ-ξ_1,η-η_1)dξ_1 dη_1, ω =ξ_1η_1+(ξ-ξ_1)(η-η_1)we may now change variable in the integral(ξ_1,η_1)↦ (ξ_1,ω), J:=∂(ξ_1,ω)/∂(ξ_1,η_1)=[10; 2η_1-η 2ξ_1-ξ ]and in particular, we remark that| J|=| (ξ-ξ_1)-ξ_1|≃ N,so thatI(ξ,η,t) =∬_𝐑e^-itωφ_R_1(ξ_1,η_1)φ_R_2(ξ-ξ_1,η-η_1) ×f(ξ_1,η_1)g(ξ-ξ_1,η-η_1)· J^-1 dξ_1 dω, η_1 =η_1(ξ_1,ω;ξ,η).Taking into consideration the Fourier transform in time and using Plancherel, followed by Cauchy-Schwarz, we find that‖ I(ξ,η,·)‖_L^2_t^2=∫_𝐑|∫_𝐑φ_R_1(ξ_1,η_1)φ_R_2(ξ-ξ_1,η-η_1)f(ξ_1,η_1)g(ξ-ξ_1,η-η_1)· J^-1 dξ_1|^2 dω≤sup_ξ,η,η_1∫_𝐑φ_R_1(ξ_1,η_1)φ_R_2(ξ-ξ_1,η-η_1)dξ_1×∫_𝐑∫_𝐑φ_R_1(ξ_1,η_1)φ_R_2(ξ-ξ_1,η-η_1)|f(ξ_1,η_1)g(ξ-ξ_1,η-η_1)|^2· J^-2 dξ_1 dω.Now, we use the fact that R_1 has width ℓ_x, together with (<ref>) to obtain, after undoing the change of variables, that‖ I(ξ,η,·)‖_L^2_t^2 ≲ℓ_x/N∫_𝐑∫_𝐑φ_R_1(ξ_1,η_1)φ_R_2(ξ-ξ_1,η-η_1) |f(ξ_1,η_1)g(ξ-ξ_1,η-η_1)|^2· J^-1 dξ_1 dω≲ℓ_x/N∫_𝐑∫_𝐑|f(ξ_1,η_1)g(ξ-ξ_1,η-η_1)|^2 dξ_1 dη_1.Integrating with respect to (ξ,η), we then obtain (<ref>). We will in fact use the following consequence of Lemma <ref>. Under the assumptions of Lemma <ref>, it holds that‖ uv‖_L^40/21_x,y,t≲ (ℓ_xℓ_y)^-3/20‖f‖_L^20/11_x,y‖g‖_L^20/11_x,y.Indeed, using Lemma <ref>, we find that‖ e^it ∂_x ∂_yf· e^it ∂_x ∂_y g‖_L^12/7_x,y,t≲ (ℓ_xℓ_y)^-1/6‖f‖_L^2_x,y‖g‖_L^2_x,y,while a crude estimate gives that‖ e^it∂_x ∂_yf· e^it∂_x ∂_yg‖_L^∞_x,y,t≲‖f‖_L^1_x,y‖g‖_L^1_x,y.Interpolation gives (<ref>). Another tool we will need in the profile decomposition is the following local smoothing result which is essentially equivalent to Lemma <ref>: Let ϕ∈ L^2_x,y. There holds thatsup_x‖ Q_M,Ne^it∂_x∂_yϕ(x,·)‖_L^2_y,t ≲ N^-1/2‖ϕ‖_L^2_x,y, sup_y‖ Q_M,Ne^it∂_x∂_yϕ(·,y)‖_L^2_x,t ≲ M^-1/2‖ϕ‖_L^2_x,y.The proof is similar to the one in the elliptic case and follows from Plancherel after using a change of variable similar to (<ref>).An equivalent statement with proof occurs in <cit.>.See also <cit.> for a general statement of Local Smoothing Estimates for Dispersive Equations. § MASS-SUPERCRITICAL HNLSIn this section, we observe that Ḣ_h^s has similar improved Sobolev inequalities as the Ḣ^1/2 Sobolev norm. A typical example is the following:Let f∈ C^∞_c(𝐑^2). There holds that‖ f‖_L^6_x,y ≲(sup_M,N(MN)^-1/6‖ Q_M,Nf‖_L^∞)^1/3‖ f‖_Ḣ_h^2/3^2/3≲‖ f‖_Ḣ_h^2/3,and consequently,‖ f‖_L^4_x,y≲(sup_M,N(MN)^-1/4‖ Q_M,Nf‖_L^∞)^1/6‖ f‖_Ḣ_h^1/2^5/6≲‖ f‖_Ḣ_h^1/2.This is essentially a consequence of the following simple inequalities‖ Q_M,Nf‖_L^∞_x,y ≲ N^1/2‖ Q_M,Nf‖_L^∞_xL^2_y≲ N^1/2‖ Q_M,Nf‖_L^2_yL^∞_x≲ (MN)^1/2‖ Q_M,Nf‖_L^2_x,y,and similarly after exchanging the role of x and y.Proof of Lemma <ref>: Indeed, we may simply develop‖ f‖_L^6_x,y^6 ≲∑_M_1,…, M_6,N_1,…, N_6∬_𝐑×𝐑Q_M_1,N_1f· Q_M_2,N_2f… Q_M_6,N_6f dxdywithout loss of generality, we may assume thatM_5,M_6≲μ_2=max_2{M_1,M_2,M_3,M_4},N_5,N_6≲ν_2=max_2{N_1,N_2,N_3,N_4},where max_2(S) denotes the second largest element of the set S, and then using Hölder's inequality and summing over M_5,M_6 and N_5,N_6, we obtain‖ f‖_L^6_x,y^6 ≲(sup_M,N(MN)^-1/6‖ Q_M,Nf‖_L^∞)^2×∑_M_1,…, M_4, M_5,M_6≤μ_2N_1,…, N_4, N_5,N_6≤ν_2(M_5M_6N_5N_6)^1/6∬_𝐑×𝐑| Q_M_1,N_1f|…| Q_M_4,N_4f| dxdy≲(sup_M,N(MN)^-1/6‖ Q_M,Nf‖_L^∞)^2 ×∑_M_1,…, M_4N_1,…, N_4(μ_2ν_2)^1/3∬_𝐑×𝐑| Q_M_1,N_1f|…| Q_M_4,N_4f| dxdy.Now, using (<ref>) and estimating the norms corresponding to the two lower frequencies in each direction in L^∞, and the two highest ones in L^2, one quickly finds that∑_M_1,…, M_4N_1,…, N_4(μ_2ν_2)^1/3∬_𝐑×𝐑| Q_M_1,N_1f|…| Q_M_4,N_4f|dxdy≲‖ f‖_Ḣ^2/3_h^4,which finishes the proof. Inequality (<ref>) then follows by interpolation.At this point, the usual profile decomposition follows easily from the following simple Lemma <ref> below.There exists δ>0 such that‖ e^it∂_x ∂_yf‖_L^8_x,y,t≲(sup_M,N,t,x,y(MN)^-1/4|(e^it∂_x ∂_yQ_M,Nf)(x,y)|)^δ‖ f‖_Ḣ^1/2_h^1-δ.We use Hölder's inequality, Sobolev embedding Lemma <ref>, Strichartz estimates and (<ref>) to get for u=e^it∂_x∂_yf‖ u‖_L^8_x,y,t ≲‖ u‖_L^6_tL^12_x,y^3/4‖ u‖_L^∞_tL^4_x,y^1/4≲‖|∂_x|^1/4|∂_y|^1/4u‖_L^6_tL^3_x,y^3/4·(sup_M,N,t(MN)^-1/4‖ Q_M,Nu(t)‖_L^∞_x,y)^1/24‖ f‖_Ḣ^1/2_h^5/24≲‖ f‖_Ḣ^1/2_h^23/24·(sup_M,N,t(MN)^-1/4‖ Q_M,Nu(t)‖_L^∞_x,y)^1/24. §.§ The mass-supercritical profile decomposition Let us take the group actionon functions given by g_n^j = g(x_n^j, y_n^j, λ_1, n^j, λ_2, n^j) such that(g_n^j)^-1 f = (λ_n, 1^jλ_n, 2^j)^1/4[f](λ_n, 1^j x + x_n^j, λ_n, 2^j y + y_n^j). We can now state the Ḣ^1/2_h-profile decomposition for (<ref>).Let u_n_Ḣ^1/2_h ≤ A be a sequence that is bounded Ḣ^1/2_h. Then possibly after passing to a subsequence, for any 1 ≤ j < ∞ there exist ϕ^j∈Ḣ^1/2_h, (t_n^j, x_n^j, y_n^j) ∈𝐑^3, λ_n, 1^j, λ_n, 2^j∈ (0, ∞) such that for any J, u_n = ∑_j = 1^J g_n^j e^i t_n^j∂_x∂_yϕ^j + w_n^J,lim_J →∞lim sup_n →∞ e^it ∂_x ∂_y w_n^J_L_x,y,t^8 = 0, such that for any 1 ≤ j ≤ J, e^-i t_n^j∂_x ∂_y (g_n^j)^-1 w_n^J⇀ 0, weakly in Ḣ^1/2_h,lim_n →∞( u_n_Ḣ^1/2_h^2 - ∑_j = 1^Jϕ^j_Ḣ^1/2_h^2 -w_n^J_Ḣ^1/2_h^2) = 0, and for any j ≠ k, lim sup_n→∞[|ln( λ_n, 1^j/λ_n, 1^k)| + |ln( λ_n, 2^j/λ_n, 2^k)| + |x_n^j - x_n^k |/(λ_n, 1^jλ_n, 1^k)^1/2 + |y_n^j - y_n^k |/(λ_n, 2^jλ_n, 2^k)^1/2.. + |t_n^j (λ_n,1^jλ_n,2^j) - t_n^k (λ_n, 1^kλ_n, 2^k)|/(λ_n, 1^jλ_n, 2^jλ_n, 1^kλ_n, 2^k)^1/2]=∞. The proof of Proposition <ref> of this follows by simple adaptation of the techniques in <cit.>, as originally introduced in <cit.>. We note that a similar statement also appears in works of Fanelli-Visciglia <cit.>.§ PROFILE DECOMPOSITION FOR THE MASS-CRITICAL HLSIn this section, we focus on the mass-critical case. This case is more delicate for two reasons. First we need to account for the Galilean invariance symmetry in (<ref>) and second, we cannot use a simple Sobolev estimate as in (<ref>) to fix the frequency scales. We follow closely the work in <cit.> with a small variant in the use of modulation orthogonality and an additional argument for interactions of rectangles with skewed aspect ratios.§.§ A precised Strichartz inequality The main result in this section is the following proposition from which it is not hard to obtain a good profile decomposition. We need to introduce the norm‖ϕ‖_X_p:=(∑_R∈ℛ| R|^-p/20‖ϕ1_R‖_L^20/11^p)^1/pwhere ℛ stands for the collection of all dyadic rectangles.That is, rectangles with both sides parallel to an axis, of possibly different dyadic size, whose center is a multiple of the same dyadic numbers, given by the formℛ := {R_k,n,p,m : k, n, m, p ∈ℤ},R_k,n,p,m := {(x,y) : n-1 ≤ 2^-kx ≤ n+1,m-1 ≤ 2^-py ≤ m+1}.Note in particular that these spaces are nested: X_p⊂ X_q whenever p≤ q.The motivation for the space L^20/11 in (<ref>) can be motivated by the X^q_p Strichartz estimate in Theorem 4.23 from <cit.>. Let ϕ∈ C^∞_c(𝐑^2), then, there holds that for all p>2,‖ϕ‖_X_p≲_p ‖ϕ‖_L^2and in addition, there exists p>2 such that‖ e^it∂_x ∂_yϕ‖_L^4_x,y,t^4≲(sup_R| R|^-1/2sup_x,y,t| e^it∂_x ∂_y(P_Rϕ)(x,y)|)^4/21‖ϕ‖_X_p^80/21. We refer to <cit.> for a different proof of a slightly stronger estimate. Let us first recall the Whitney decomposition.There exists a tiling of the plane minus the diagonal𝐑^2∖ D =⋓ I× J,D ={(x,x), x∈𝐑},made of dyadic intervals such that | I|=| J| and6| I|≤dist(I× J,D)≤ 24 | I|.We will consider two independent Whitney decompositions of 𝐑×𝐑:1_{𝐑^2×𝐑^2∖ D}(ξ_1,η_1,ξ_2,η_2):=∑_I_1∼ I_2, J_1∼ J_21_I_1(ξ_1)1_J_1(η_1)1_I_2(ξ_2)1_J_2(η_2),where I_i and J_j are dyadic intervals of 𝐑 and ∼ is an equivalence relation such that, for each fixed I, there are only finitely many J's such that I∼ J, uniformly in I (i.e. equivalence classes have bounded cardinality) and if I∼ J, then | I|=| J | and dist(I,J)≃| I|. We also extend the equivalence relation to rectangles in the following fashion:I× J ∼ I'× J' if and only ifI ∼ I'and J ∼ J'. We would like to follow the argument in <cit.> for the profile decomposition for the elliptic nonlinear Schrödinger equation. However, it is at this point where we reach the main technical obstruction to doing this. Recall that to estimate the L_x,y,t^2 norm of [e^it Δ f]^2, it was possible to utilize Plancherel's theorem, reducing the L_x,y,t^2 norm to an l^2 sum over pairs of Whitney squares.This was because Plancherel's theorem in frequency turned the sum over all pairs of equal area squares to an l^2 sum over squares centered at different points in frequency space, and then Plancherel's theorem in time separated out pairs of squares with different area. Because there is only one square with a given area and center in space, this is enough. However, there are infinitely many rectangles with the same area and the same center. Thus, to reduce the L_x,y,t^2 norm of [e^it ∂_x∂_y f]^2 to a l^2 sum over pairs of rectangles, that is rectangles whose sides obey the equivalence relation in both x and y, it is necessary to deal with the off - diagonal terms, that is terms of the form [e^it ∂_x∂_y P_R_1 f][e^it ∂_x∂_y P_R_2 f][e^it ∂_x∂_y P_R_1' f] [e^it ∂_x∂_y P_R_2' f] _L_x,y,t^1, where R_1∼ R_2 and R_1' ∼ R_2' are Whitney pairs of rectangles which have the same area, but very different dimensions in x and y. In this case, Lemma <ref> gives a clue with regard to how to proceed, since it leads to the generalized result that [e^it ∂_x∂_y P_R_1 f] [e^it∂_x∂_y P_R_1' f] _L_x,y,t^2≪ P_R_1 f _L^2_x,y P_R_1' f _L^2_x,y. Thus, it may be possible to sum the off diagonal terms. We will not use Lemma <ref> specifically, but we will use the idea that rectangles of the same area but very different dimensions have very weak bilinear interactions. Before we turn to the details, we first present the main orthogonality properties we will use. For simplicity of notation, given a dyadic rectangle R, letϕ_R(x,y):=ϕ(x,y)1_R(x,y) andu_R(x,y,t):=(e^it∂_x∂_yϕ_R)(x,y)and set u=e^it∂_x∂_yϕ. Also we will consider rectangles R_1=I_1× J_1, R_2=I_2× J_2, R_1'=I_1'× J_1', R_2'=I_2' × J_2'. Proceeding with the above philosophy in mind, using (<ref>), we have that ‖ u‖_L^4_x,y,t^4 =‖ u^2‖_L^2_x,y,t^2=‖∑_R_1∼ R_2u_R_1· u_R_2‖_L^2_x,y,t^2=‖∑_Ω∑_R_1∼ R_2, | R_1|=| R_2|=Ωu_R_1· u_R_2‖_L^2_x,y,t^2=‖∑_Ω I_Ω‖_L^2_x,y,t^2.Using the polarization identity for a quadratic form,Q(x_1,y_1)+Q(x_2,y_2)=1/2[Q(x_1+x_2,y_1+y_2)+Q(x_1-x_2,y_1-y_2)], we compute thate^it/2ξηI_Ω (ξ,η,t)=∑_R_1∼ R_2, | R_1|=| R_2|=Ω∫_𝐑^41_R_1(ξ_1,η_1)1_R_2(ξ_2,η_2)e^-it/2(ξ_1-ξ_2)(η_1-η_2)×f(ξ_1,η_1)f(ξ_2,η_2)δ(ξ-ξ_1-ξ_2)δ(η-η_1-η_2)dξ_1 dξ_2 dη_1 dη_2.Now we observe that since| I_1|=| I_2|≃dist(I_1,I_2),| J_1|=| J_2|≃dist(J_1,J_2),it holds that, on the support of integration,| (ξ_1-ξ_2)(η_1-η_2)|≃| I_1|·| J_1|≃Ω.Therefore, we have the following orthogonality in time‖∑_ΩI_Ω(ξ,η,·)‖_L^2_t^2 =‖ e^it/2ξη∑_ΩI_Ω(ξ,η,·)‖_L^2_t^2 ≲∑_Ω‖I_Ω(ξ,η,·)‖_L^2_t^2. To continue, we need to control I_Ω uniformly in Ω. We write thatℐ_Ω =‖∑_R_1∼ R_2, | R_1|=| R_2|=Ω u_R_1· u_R_2‖_L^2_x,y,t^2=∑_R_1∼ R_2, R_1^'∼ R_2^', | R_1|=| R_2|=| R_1^'|=| R_2^'|=Ω∫_𝐑^3_x,y,t u_R_1· u_R_2·u_R_1^'· u_R_2^' dx dy dt=∑_R_1∼ R_2, R_1^'∼ R_2^' | R_1|=| R_2|=| R_1^'|=| R_2^'|=Ωℐ_R_1∼ R_2,R_1^'∼ R_2^'.To any rectangle R=I× J, we associate its center c = (c_x, c_y) and its scales ℓ_x(R)=| I| and ℓ_y(R)=| J|=Ω/| I|. For 2 rectangles R and R^' of equal area, we define their relative discrepancy byδ(R,R^')=min{ℓ_x(R)/ℓ_x(R^'),ℓ_y(R)/ℓ_y(R^')}.We want to decompose ℐ_Ω according to the discrepancy of R_1=I_1× J_1 and R_1^'=I_1^'× J_1^'. Using scaling relation (<ref>), we may assume that Ω=1, ℓ_x(R_1)=ℓ_x(R_2)=1 and that ℓ_x(R_1^')≤ℓ_y(R_1^'), so that R_1^' is a δ×δ^-1 rectangle, where δ=δ(R_1,R_1^').We first notice that, if ℐ_R_1∼ R_2,R_1^'∼ R_2^' 0, we must have that| c_x(R_1)-c_x(R_1^')| ≲ℓ_x(R_1)+ℓ_x(R_1'), | c_y(R_1)-c_y(R_1^')| ≲ℓ_y(R_1)+ℓ_y (R_1').and therefore, for any fixed R_1 and δ≳ 1, there can be only a bounded number of choices for R_1^', so thatℐ_1 ≲∑_R_1∼ R_2, | R_1|=| R_2|=1‖ u_R_1u_R_2‖_L^2_x,y,t^2.At this stage, we are in a similar position as in the elliptic case and we may follow the proof in <cit.>. From now on, we will focus on the case δ≪1.In the case δ≪1, we may in fact strengthen (<ref>). Indeed for ℐ_R_1∼ R_2,R_1^'∼ R_2^' to be different from 0, we must have that| c_x(R_1)-c_x(R_1^')| ≃ℓ_x(R_1)+ℓ_x(R_1'), | c_y(R_1)-c_y(R_1^')| ≃ℓ_y(R_1)+ℓ_y (R_1').This follows from the fact that (say)c_x(R_1)+c_x(R_2)-c_x(R_1^')-c_x(R_2^')= 2[c_x(R_1)-c_x(R_1^')]-[c_x(R_1)-c_x(R_2)]+[c_x(R_1^')-c_x(R_2^')],and the last bracket is bounded by 24δ, while the second to last is bounded below by 6; however, for ℐ_R_1∼ R_2,R_1^'∼ R_2^' to be nonzero, there must exists (ξ_1,ξ_2,ξ_1^',ξ_2^')∈ R_1× R_2× R_1^'× R_2^' such thatξ_1+ξ_2-ξ_1^'-ξ_2^'=0and| (ξ_1+ξ_2-ξ_1^'-ξ_2^')-(c_x(R_1)+c_x(R_2)-c_x(R_1^')-c_x(R_2^'))|≤ 2+2δ.We will keep note of this by writing R_1≃ R_1^' (or sometimes c(R_1)≃ c(R_1^')) whenever (<ref>) holds for rectangles of equal area.Recall that R_1^' is a δ×δ^-1 rectangle; we can decompose all rectangles into δ× 1 rectangles. We may then partitionR_1 =⋃_a=1^δ^-1I_1,a× J_1=⋃_a=1^δ^-1R_1,a, R_2=⋃_a=1^δ^-1 I_2,a× J_2=⋃_a=1^δ^-1R_2,a,R_1^' =⋃_b=1^δ^-1I_1^'× J_1,b^'=⋃_b=1^δ^-1R_1,b^', R_2^'=⋃_b=1^δ^-1I_2^'× J_2,b^'=⋃_b=1^δ^-1R_2,b^'and by orthogonality, we see thatℐ_R_1∼ R_2,R_1^'∼ R_2^' =∑_a∼a, b∼bℐ_R_1,a∼ R_2,a,R_1,b^'∼ R_2,b^'where a∼aif and only if | c_x (R_1,a) + c_x(R_2,a)- c_x (R_1') - c_x (R_2') | ≲δand comparably in y for b∼b. Thus, for fixed R_1, R_2, R_1^' R_2^', this gives two equivalence relations with O(δ^-1) equivalence classes of (uniformly) bounded cardinality.And proceeding as in (<ref>), we can easily see that| c_x(R_1,a)-c_x(R^'_1,b)|≳ 1,| c_y(R_1,a)-c_y(R^'_1,b)|≳δ^-1, | c_x(R_2,a)-c_x(R^'_2,b)|≳ 1,| c_y(R^'_1,a)-c_y(R^'_2,b)|≳δ^-1.At this point, we have extracted all the orthogonality we need and we are ready to proceed with the proof of Proposition <ref>. §.§ Proof of (<ref>) Using rescaling, we may assume that1=sup_R| R|^-1/2‖ e^it∂_x ∂_yϕ_R‖_L^∞_x,y,t.From the considerations above, we obtain the expression‖ e^it∂_x ∂_yϕ‖_L^4_x,y,t^4 ≲∑_Ω∑_R_1∼ R_2,R_1^'∼ R_2^', | R_1|=| R_2|=| R_1^'|=| R_2^'|=Ωℐ_R_1∼ R_2,R_1^'∼ R_2^',where the rectangles satisfy the condition (<ref>). In addition, for fixed rectangles R_1∼ R_2, R_1^'∼ R_2^' of equal area Ω, let δ=δ(R_1,R_1^'). As explained above, for fixed δ=δ_0=O(1), we are in a position similar to the elliptic case and we may follow <cit.> to get∑_Ω∑_R_1∼ R_2,R_1^'∼ R_2^', | R_1|=| R_2|=| R_1^'|=| R_2^'|=Ω, δ(R_1,R_1^')=δ_0|ℐ_R_1∼ R_2,R_1^'∼ R_2^'| ≲ ∑_Ω∑_R_1∼ R_2, | R_1|=| R_2|=Ω‖ u_R_1u_R_2‖_L^2_x,y,t^2 ≲ (sup_R| R|^-1/2‖ u_R‖_L^∞_x,y,t)^4/21∑_R_1∼ R_2| R_1|^2/21‖ u_R_1u_R_2‖_L^40/21_x,y,t^40/21 ≲ ∑_R_1∼ R_2{| R_1|^-1/20‖ϕ_R_1‖_L^20/11_x,y·| R_2|^-1/20‖ϕ_R_2‖_L^20/21_x,y,t}^40/21,where we have used Cauchy-Schwarz in the first inequality, Hölder's inequality in the second and (<ref>) together with Lemma <ref> in the last inequality.This gives a bounded contribution as in (<ref>) for any p≤ 80/21. We need to adjust the above scheme when δ≪1. In the following, we letT_≪1:=∑_δ≪1∑_Ω∑_R_1∼ R_2,R_1^'∼ R_2^', | R_1|=| R_2|=| R_1^'|=| R_2^'|=Ω, δ(R_1,R_1^')=δ|ℐ_R_1∼ R_2,R_1^'∼ R_2^'|and to conclude the proof of (<ref>), we need to prove that, for some p>2,T_≪1≲‖ϕ‖_X_p^80/21.We can now use the finer decomposition (<ref>) to writeℐ_R_1∼ R_2,R_1^'∼ R_2^' =∑_a∼a,b∼bℐ_R_1,a∼ R_2,a,R_1,b^'∼ R_2,b^'where the new rectangles satisfy (<ref>). Using Cauchy-Schwartz, then Hölder's inequality with (<ref>), we have that|ℐ_R_1,a∼ R_2,a,R_1,b^'∼ R_2,b^'| ≲‖ u_R_1,a· u_R^'_1,b‖_L^2_x,y,t·‖ u_R_2,a· u_R^'_2,b‖_L^2_x,y,t≲ (δΩ)^2/21‖ u_R_1,a· u_R^'_1,b‖_L^40/21_x,y,t^20/21·‖ u_R_2,a· u_R^'_2,b‖_L^40/21_x,y,t^20/21 .Now, using Lemma <ref> with (<ref>), we obtain that|ℐ_R_1,a∼ R_2,a,R_1,b^'∼ R_2,b^'| ≲ (δΩ)^2/21· (δ^-1Ω)^-6/21‖ϕ_R_1,a‖_L^20/11_x,y^20/21‖ϕ_R_2,a‖_L^20/11_x,y^20/21‖ϕ_R^'_1,b‖_L^20/11_x,y^20/21‖ϕ_R^'_2,b‖_L^20/11_x,y^20/21.Since 20/11 <40/21 and since for fixed a, there are only a bounded number a such that a∼a, we can sum over a to get∑_a∼a‖ϕ_R_1,a‖_L^20/11_x,y^20/21‖ϕ_R_2,a‖_L^20/11_x,y^20/21 ≲‖ϕ_R_1‖_L^20/11_x,y^20/21‖ϕ_R_2‖_L^20/11_x,y^20/21and similarly for b, so that|ℐ_R_1∼ R_2,R_1^'∼ R_2^'| ≲δ^8/21Ω^-4/21‖ϕ_R_1‖_L^20/11_x,y^20/21‖ϕ_R_2‖_L^20/11_x,y^20/21‖ϕ_R_1^'‖_L^20/11_x,y^20/21‖ϕ_R_2^'‖_L^20/11_x,y^20/21.In addition, for rectangles of fixed areas and sizes | R_1|=| R_2|=| R_1^'|=| R_2^'|, ℓ_x(R_1)=ℓ_x(R_2), ℓ_x(R_1^')=ℓ_x(R_2^') also satisfying (<ref>), we may use Cauchy Schwartz in the summation over the centers to get∑_R_1∼ R_2,R_1^'∼ R_2^'‖ϕ_R_1‖_L^20/11_x,y^20/21‖ϕ_R_2‖_L^20/11_x,y^20/21‖ϕ_R_1^'‖_L^20/11_x,y^20/21‖ϕ_R_2^'‖_L^20/11_x,y^20/21 ≲∑_R_1≃ R_1^'‖ϕ_R_1‖_L^20/11_x,y^40/21‖ϕ_R_1^'‖_L^20/11_x,y^40/21where the sum is taken over all rectangles R_1≃ R_1^' of the given sizes satisfying (<ref>).We can now get back to (<ref>) and use (<ref>) and the inequality above to getT_≪1 ≲∑_Ω∑_A∑_δ≤ 1δ^8/21Ω^-4/21·∑_c_1≃ c_1^'‖ϕ_R_1‖_L^20/11_x,y^40/21‖ϕ_R_1^'‖_L^20/11_x,y^40/21,where we have parameterized the lengths of the rectangles by Ω=| R_1|=| R_1^'|, A=ℓ_x(R_1) and δ=ℓ_x(R_1^')/ℓ_x(R_1), and their centers by c_1, c_1^'.Now for any p > 2 choose 0 < θ(p) < 1 such that2 θ/p + 1 - θ/p = 21/40.and observe that θ(p) ↘1/20 as p ↘ 2. Then by interpolation,∑_Ω,A,c_1≃ c_1^'Ω^-4/21‖ϕ_R_1‖_L^20/11_x,y^40/21‖ϕ_R_1^'‖_L^20/11_x,y^40/21 ≲ (∑_Ω,A,c_1≃ c_1^'Ω^-p/10‖ϕ_R_1‖_L^20/11_x,y^p‖ϕ_R_1^'‖_L^20/11_x,y^p)^40/211-θ/p(∑_Ω,A,c_1≃ c_1^'Ω^-p/20‖ϕ_R_1‖_L^20/11_x,y^p/2‖ϕ_R_1^'‖_L^20/11_x,y^p/2)^40/212θ/p.Now, on the one hand, we observe that for a fixed choice of scales (Ω, A and δ)and for each fixed c_1, there are at most O(δ^-1) choices of c_1^' satisfying (<ref>) so we obtain that∑_Ω,A,c_1≃ c_1^'Ω^-p/20‖ϕ_R_1‖_L^20/11_x,y^p/2‖ϕ_R_1^'‖_L^20/11_x,y^p/2 ≲δ^-1∑_Ω,A,c_1Ω^-p/20‖ϕ_R_1‖_L^20/11_x,y^pand the other sum can be handled in an easier way: using Hölder's inequality and forgetting about the relationship c_1≃ c_1^', we obtain that∑_Ω∑_A∑_c_1≃ c_1^'Ω^-p/10‖ϕ_R_1‖_L^20/11_x,y^p‖ϕ_R_1^'‖_L^20/11_x,y^p ≲ ∑_Ω∑_A(∑_c_1{Ω^-1/20‖ϕ_R_1‖_L^20/11_x,y}^p)·(∑_c_1^'{Ω^-1/20‖ϕ_R_1^'‖_L^20/11_x,y}^p) ≲ (∑_R| R|^-p/20‖ϕ_R‖_L^20/11_x,y^p)^2.Recall the definition (<ref>). Combining (<ref>) and (<ref>), we obtainT_≪1 ≲∑_δ≤ 1δ^8/21·(δ^-1‖ϕ‖_X_p^p)^80/21θ/p(‖ϕ‖_X_p^2p)^40/211-θ/p≲∑_δ≤ 1δ^8/21(1-10θ/p)‖ϕ‖_X_p^80/21and this is summable in δ for 2<p<40/17 small enough. The proof of (<ref>) is thus complete and it remains to prove (<ref>) which we now turn to. §.§ Proof of (<ref>) We first state and prove the following simple result we will need in the proof. Let 𝒟 denote the set of dyadic intervals (on 𝐑) and let p>2. For any g∈ C^∞_c(𝐑), there holds that∑_I∈𝒟| I|^-p/20‖ g1_I‖_L^20/11_x^p≲‖ g‖_L^2_x^p. We may assume that ‖ g‖_L^2_x=1. For fixed A, we let 𝒟_A denote the set of dyadic intervals of length A and we decomposeg=g^++g^-, g^+(x)=g(x)1_{| g(x)|>A^-1/2}, g^-(x)=g(x)1_{| g(x)|≤ A^-1/2}.On the one hand, using that ℓ^20/11⊂ℓ^p,∑_A∑_I∈𝒟_A| I|^-p/20‖ g^+1_I‖_L^20/11_x^p ≲(∑_AA^-1/11∑_I∈𝒟_A‖ g^+1_I‖_L^20/11_x^20/11)^11 /20p≲(∑_AA^-1/11∫_𝐑| g|^20/111_{| g(x)|>A^-1/2}dx)^11 /20p≲(∫_𝐑| g|^20/11·(∑_A>| g(x)|^-2A^-1/11)dx)^11 /20p≲ 1,while, for the other sum, we use Hölder's inequality to get∑_A∑_I∈𝒟_A A^-p/20‖ g^-1_I‖_L^20/11_x^p≲∑_A∑_I∈𝒟_A A^-p/20‖ g^-1_I‖_L^p^p·| I|^(11/20-1/p)p≲∫_𝐑| g(x)|^p ·∑_{A<| g(x)|^-2}A^p-2/2dx≲∫_𝐑| g(x)|^2dx≲ 1and the proof is complete. Now, we proceed to prove (<ref>).Recall 𝒟 stand for the set of dyadic intervals and 𝒟_A for the set of dyadic intervals of length A. We want to prove that∑_I∈𝒟| I|^-p/20∑_J∈𝒟| J|^-p/20‖ f 1_I× J‖_L^20/11_x,y^p ≲‖ f‖_L^2_x,y^p.We claim that, for any fixed interval I,∑_J∈𝒟| J|^-p/20‖ f1_I× J‖_L^20/11_x,y^p ≲‖ f1_I×𝐑‖_L^20/11_xL^2_y^p.Once this is proved, we may simply apply Lemma <ref> to the functiong(x):=‖ f(x,·)‖_L^2_yto finish the proof.From now on I denotes a fixed interval and f is a function supported on {x∈ I}, i.e. f=f1_I×𝐑. The proof of (<ref>) is a small variation on the proof of Lemma <ref>. Fix a dyadic number B and letc_B=c_B(x)=B^-1/2‖ f(x,·)‖_L^2_yand decompose accordingly[Note that f(x,y)=0 whenever ‖ f(x,·)‖_L^2_y=0, so that c_B(x)>0 on the support of f^+.]f=f^++f^-, f^+=f1_{| f(x,y)|>c_B(x)}, f^-=f1_{| f(x,y)|≤ c_B(x)}.We then compute that∑_B∑_J∈𝒟_B B^-p/20‖ f^+1_I× J‖_L^20/11_x,y^p≲(∑_B∑_J∈𝒟_B B^-1/11‖ f^+1_I× J‖_L^20/11_x,y^20/11)^11 /20p≲(∑_B B^-1/11∫_I_x∫_𝐑_y| f^+|^20/11 dydx)^11 /20p≲(∫_I_x∫_𝐑_y| f^+|^20/11·∑_{B:| f(x,y)|≥ c_B(x)}B^-1/11 dydx)^11/20p≲(∫_I_x∫_𝐑_y| f(x,y)|^20/11·(| f(x,y)|/‖ f(x,·)‖_L^2_y)^2/11 dydx)^11/20p≲(∫_I_x‖ f(x,·)‖_L^2_y^-2/11∫_𝐑_y| f(x,y)|^2dydx)^11/20p≲‖ f‖_L^20/11_xL^2_y^p , in the penultimate line, we note that though there is a negative power of the L^2_y norm, the product of the two quantities is well-defined, especially as we can assume f ∈ C^∞_c.Also, we have used the embedding ℓ^1 ⊂ℓ^11/20p in the first inequality, the fact that dyadic intervals of a fixed length tile 𝐑 in the second inequality, and we have summed a geometric series in the fourth inequality. Now for the second part, we compute that∑_B∑_J∈𝒟_B B^-p/20‖ f^-1_I× J‖_L^20/11_x,y^p≲∑_B∑_J∈𝒟_B B^-p/20B^p(11/20-1/p)‖ f^-1_I× J‖_L^p_y(J:L^20/11_x(I))^p≲∑_B B^p -2 /2∫_𝐑_y(∫_I_x| f^-(x,y)|^20/11dx)^11/20pdy≲∫_𝐑_y∑_B (B^p -2 /220/111/p∫_I_x| f^-(x,y)|^20/11dx)^11/20pdy≲∫_𝐑_y(∫_I_x∑_BB^p -2 /p10/11| f^-(x,y)|^20/11dx)^11/20pdy,where we have used Hölder's inequality in the first line and the inclusion ℓ^1⊂ℓ^11/20p in the fourth line. Now, since f^- is supported whereB≤(‖ f(x,·)‖_L^2_y/| f(x,y)|)^2,summing in B gives∑_B∑_J∈𝒟_B B^-p/20‖ f^-1_I× J‖_L^20/11_x,y^p ≲ ∫_𝐑_y(∫_I_x‖ f(x,·)‖_L^2_y^20/11p-2/p| f^-(x,y)|^20/112/pdx)^11/20pdy.Using Minkowski inequality on the functionh(x,y)=‖ f(x,·)‖_L^2_y^20/11p-2/p| f^-(x,y)|^20/112/p,we obtain∑_B∑_J∈𝒟_B B^-p/20‖ f^-1_I× J‖_L^20/11_x,y^p ≲ (∫_I_x(∫_𝐑_yh^11/20pdy)^20/111/pdx)^11/20p ≲ (∫_I_x(∫_𝐑_y| f(x,y)|^2dy)^20/111/p‖ f(x,·)‖_L^2_y^20/11p-2/pdx)^11/20p ≲ (∫_I_x‖ f(x,·)‖_L^2_y^20/11dx)^11/20p,which proves (<ref>). Thus the proof is complete.§ THE PROFILE DECOMPOSITION AND APPLICATIONS The profile decomposition then follows from Proposition <ref> in the usual way following the techniques in the proof of Theorems 4.25 (the Inverse Strichartz Inequality) and 4.26 (Mass Critical Profile Decomposition) from <cit.>, for instance.We note that it is the proof of the Inverse Strichartz Inequality that requires the local smoothing estimates as in Lemma <ref> to establish pointwise a.e. convergence of profiles to an element of L^2_x,y through compactness considerations, otherwise the proof follows mutatis mutandis.Once the Inverse Strichartz Inequality is established, the proof of the Profile Decomposition follows verbatim.Suppose g_n^j = g(x_n^j, y_n^j, λ_1, n^j, λ_2, n^j, ξ_n^j) is the group whose action on functions is given by(g_n^j)^-1 f = (λ_n, 1^jλ_n, 2^j)^1/2 e^-i ξ_n, 1^j (λ_n, 1^j x + x_n^j) e^-i ξ_n, 2^j (λ_n, 2^j x + y_n^j)× [f_n](λ_n, 1^j x + x_n^j, λ_n, 2^j y + y_n^j). The profile decomposition gives the following. Let u_n_L^2_x,y(𝐑^2)≤ A be a sequence that is bounded L^2_x,y(𝐑^2). Then possibly after passing to a subsequence, for any 1 ≤ j < ∞ there exist ϕ^j∈ L^2_x,y(𝐑^2), (t_n^j, x_n^j, y_n^j) ∈𝐑^3, ξ_n^j∈𝐑^2, λ_n, 1^j, λ_n, 2^j∈ (0, ∞) such that for any J, u_n = ∑_j = 1^J g_n^j e^i t_n^j∂_x∂_yϕ^j + w_n^J,lim_J →∞lim sup_n →∞ e^it ∂_x ∂_y w_n^J_L_x,y,t^4 = 0, such that for any 1 ≤ j ≤ J, e^-i t_n^j∂_x ∂_y (g_n^j)^-1 w_n^J⇀ 0, weakly in L^2_x,y(𝐑^2),lim_n →∞( u_n_L^2_x,y^2 - ∑_j = 1^Jϕ^j_L^2_x,y^2 -w_n^J_L^2_x,y^2) = 0, and for any j ≠ k, lim_n →∞[ |ln( λ_n, 1^j/λ_n, 1^k)| + |ln( λ_n, 2^j/λ_n, 2^k)|+ |t_n^j (λ_n,1^jλ_n,2^j) - t_n^k (λ_n, 1^kλ_n, 2^k)|/(λ_n, 1^jλ_n, 2^jλ_n, 1^kλ_n, 2^k)^1/2.+ (λ_n, 1^jλ_n, 1^k)^1/2 |ξ_n, 1^j - ξ_n, 1^k|+ (λ_n, 2^jλ_n, 2^k)^1/2 |ξ_n, 2^j - ξ_n, 2^k| + |x_n^j - x_n^k - 2 t_n^j (λ_n, 1^jλ_n, 2^j)(ξ_n, 1^j - ξ_n, 1^k)|/(λ_n, 1^jλ_n, 1^k)^1/2.+ |y_n^j - y_n^k - 2 t_n^j (λ_n, 1^jλ_n, 2^j)(ξ_n, 2^j - ξ_n, 2^k)|/(λ_n, 2^jλ_n, 2^k)^1/2] = ∞.§.§ Minimal mass blow-up solutions As an application of the profile decomposition, we turn to a calculation that for instance originated in <cit.>.Namely we construct a minimal mass solution to (<ref>) which is a solution u of minimal mass such that there exists a time T^* such that∫_-T^*^T^*∫_𝐑^2_x,y |u |^4 dx dy dt= + ∞.In other words, it is a solution of least mass for which the small data global argument fails. It turns out that if u is a minimal mass blowup solution to (<ref>) then u lies in a compact subset of L^2_x,y(𝐑^2) modulo the symmetry group g; more precisely, following <cit.>, we can establish the following theorem.Suppose u is a minimal mass blowup solution to (<ref>) on a maximal time interval I that blows up in both time directions.That is, I is an open interval and for any t_0∈ I, ∫_t_0^sup(I)∫ |u(x,y,t)|^4 dx dy dt, ∫_inf(I)^t_0∫ |u(x,y,t)|^4 dx dy dt = ∞. Then there exist λ_1, λ_2 : I → (0, ∞), ξ : I →𝐑^2, x, y : I →𝐑, such that for any η > 0 there exists C(η) < ∞ such that ∫_|x - x(t)| > C(η)/λ_1(t) |u(x,y,t)|^2 dx dy + ∫_|y - y(t)| > C(η)/λ_2(t) |u(x,y,t)|^2 dx dy + ∫_|ξ_1 - ξ_1(t)| > C(η) λ_1(t) |û(ξ,t)|^2 dξ + ∫_|ξ_2 - ξ_2(t)| > C(η) λ_2(t) |û(ξ,t)|^2 dξ < η. Take a sequence t_n∈ I. Then conservation of mass implies that after passing to a subsequence we may make a profile decomposition of u(t_n) = u_n. If there exists j such that, along a subsequence, t_n^j→±∞, say t_n^j→∞, then lim_n →∞ e^it ∂_x∂_y (g_n^j e^i t_n^j∂_x ∂_yϕ^j) _L_x,y,t^4([0, ∞) ×𝐑^2) = 0, so combining perturbative arguments, (<ref>), and the fact that u is a blowup solution with minimal mass then u scatters forward in time to a free solution. Thus, we may assume that for each j, t_n^j converges to some t^j∈𝐑. Then taking e^it^j∂_x ∂_yϕ^j to be the new ϕ^j, we may assume that each t_n^j = 0.Now suppose that sup_jϕ^j_L^2_x,y(𝐑^2) <u(t) _L^2_x,y(𝐑^2). Then if v^j is the solution to (<ref>) with initial data ϕ^j, since u(t) _L^2 is the minimal mass for blowup to occur, each v^j scatters both forward and backward in time, with v^j_L_x,y,t^4(𝐑×𝐑^2)^2 ≲‖ϕ^j‖_L^2_x,y^2 < ∞,uniformly inj. Then if v_n^j is the solution to (<ref>) with initial data g_n^jϕ^j, v_n^j = g_n^j (v^j( (λ_n,1^jλ_n,2^j)^-1t)). We note that, for v either a profile v_n^ℓ or the remainder w_n^J,v_n^j v_n^k v _L_x,y,t^4/3≤‖ v_n^jv_n^k‖_L^2_x,y,t‖ v ‖_L^4_x,y,t.In addition, ‖ v‖_L^4_x,y,t remains bounded either by (<ref>) (for v_n^ℓ) or as a consequence of the small data theory and (<ref>) (for w_n^J).By approximation by compactly supported functions, it is easy to see that, if j k,‖ v_n^jv_n^k‖_L^2_x,y,t→0when n →∞ as a consequence of (<ref>).As a result, using simple perturbation theory, we obtain that, for J large enough,‖ u(t_n+t)-∑_j=1^Jv_n^j(t)‖_L^4_x,y,t≲ 1and using again (<ref>)-(<ref>), we obtain that‖∑_j=1^Jv_n^j(t)‖_L^4_x,y,t^4≲∑_j=1^J‖ v_n^j‖_L^4_x,y,t^4≲∑_j=1^J‖ϕ^j‖_L^2_x,y^2<∞.which, together with (<ref>) contradicts (<ref>).Thus, after reordering we should have ϕ^1_L^2_x,y =u(t) _L^2_x,y and ϕ^j = 0 for any j ≥ 2. But this holds if and only if u(t) lies in a set G K, where G is the group generated by g_n^j and K is a compact set in L^2. This completes the proof of the theorem.§ EXTREMIZERS FOR STRICHARTZ ESTIMATES FOR (<REF>) We end with a few considerations on extremizers for the Strichartz inequality (<ref>), that is functions f of unit L^2_x,y-norm and constant C such that‖ e^it∂_x ∂_yf‖_L^4_x,y,t=C:=sup{‖ e^it∂_x ∂_yg‖_L^4_x,y,t:‖ g‖_L^2_x,y=1}.The original version of this paper stated erroneously that Gaussians were optimizers for the Strichartz norm above, and gave a corresponding numerical value for C. In fact, it was later proved in <cit.> that Gaussians are not critical points of the Strichartz norm and thus cannot be optimizers. One can however use the profile decomposition for Theorem <ref> to prove existence of an extremizer (see <cit.> for a similar proof). There exists f∈ L^2_x,y(𝐑^2) such that‖ e^it∂_x ∂_yf‖_L^4_x,y,t = C‖ f‖_L^2_x,y.where C is given in (<ref>). The exact nature of the extremizer f above remains mysterious.Preliminary numerical investigations confirm that the optimizer should be a genuine function of x and y and suggests that it has nice decay and smoothness properties.We reproduce some plots of the amplitude of an extremizer and its Fourier transform below. These are obtained by maximizing the Strichartz norm among functions in the span of the first 25 Hermite functions with unit L^2 norm. Some care has been given to obtain accurate computations and minimize boundary effects, but we do not mean that the result is anything but suggestive. However, it seems to indicate that extremizers are indeed smooth and localized and that they can be chosen real valued, which is why we have not reproduced their phase.MA BG H. Bahouri, and P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math. 121 (1999), 131-175.Bou1 J. Bourgain. 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Visciglia.The lack of compactness in the Sobolev-Strichartz inequalities. J. Math. Pures et Appl, 99, No. 3, (2013), 309-320.Gal1 I. Gallagher. Profile decomposition for solutions of the Navier-Stokes equations, Bull. Soc. Math. France, 129 (2001), 285-316.G1 P. Gérard,Description du defaut de compacite de l'injection de Sobolev, ESAIM Control Optim. Calc. Var. 3 (1998), 213-233.GS M. Ghidaglia and J.C. Saut,Nonexistence of Travelling Wave Solutions to Nonelliptic Nonlinear Schrodinger Equation, J. Nonlinear Sci. 6 (1996), 139–145.HuZh D. Hundertmark and V. Zharnitsky,On sharp Strichartz inequalities in low dimensions, International Mathematics Research Notices 2006 (2006), 34080.Iftimie Dragos Iftimie,A Uniqueness Result for the Navier–Stokes Equations with Vanishing Vertical Viscosity, SIAM Journal on Mathematical Analysis 33, No. 6 (2002),1483–1493.KM1 C.E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Inventiones Mathematicae 166, No. 3 (2006), 645-675.KPRV1 C. E. Kenig, G. Ponce, C. Rolvung, and L. Vega. The general quasilinear ultrahyperbolic Schrödinger equation. Adv. Math. 196, No. 2 (2005), 402-433.Ker1 S. Keraani, On the Defect of Compactness for the Strichartz Estimates of the Schrödinger Equations, Journ. Diff. Eq. 175 (2001), 353-392.KV R. Killip and M. Visan,Nonlinear Schrödinger Equations at Critical Regularity, Clay Summer School Lecture Notes, (2008). LeeS. Lee,Bilinear restriction estimates for surfaces with curvatures of different signs, Transactions of the American Mathematical Society 358, No. 8 (2006), 3511-3533.LP F. Linares and G. Ponce, On the Davey-Stewartson systems, Annales de l'Institut Henri Poincare (C) Non Linear Analysis. 10, No. 5 (1993), 523-548.Lions1 P.-L. Lions. The concentration-compactness principle in the calculus of variations. The limit case, Part I,Revista Matemática Aberoamericana, 1, Issue 1 (1985), 145-201.Lions2 P.-L. Lions. The concentration-compactness principle in the calculus of variations. The limit case, Part II,Revista Matemática Aberoamericana, 1, Issue 2 (1985), 45-121.KNZ P. Kevrekidis, A. Nahmod and C. Zeng.Radial standing and self-similar waves for the hyperbolic cubic NLS in 2D, Nonlinearity 24 (2011), No. 5, 1523-1538.MMT3 J.L. Marzuola, J. Metcalfe, D. Tataru.Quasilinear Schrödinger equation I:small data and quadratic interactions.Adv. Math., 231, No. 2 (2012), 1151-1172. MerleVega F. Merle and L. Vega.Compactness at blow - up time for L^2 solutions of the critical nonlinear Schrödinger equation in 2D.International Mathematics Research Notices, No. 8 (1998), 399-425.MVV A. Moyua, A. Vargas and L. Vega, Schrödinger maximal function and restriction properties of the Fourier transform, International Mathematics Research Notices, 1996, 793–815.RV K.M. Rogers and A. Vargas.A refinement of the Strichartz inequality on the saddle and applications.J. Functional Anal., 241, No. 2 (2006), 212-231.RSM. Ruzhansky and M. Sugimoto.Smoothing properties of evolution equations via canonical transforms and comparison principle, Proceedings of the London Mathematical Society 105, No. 2 (2012), 393-423. S-T Monique Sablé-Tougeron, Régularité microlocale pour des problemes aux limites non linéaires, Ann. Inst. Fourier 36, No. 1 (1986), 39–82.Sh S. Shao, Maximizers for the Strichartz and the Sobolev-Strichartz inequalities for the Schrödinger equation, Electronic Journal of Differential Equations (EJDE), (2009) Volume: 2009, page Paper No. 03, 13 p.SS C. Sulem and P. Sulem.Nonlinear Schrödinger Equations, Springer (1999).Tao1 Terence Tao. A sharp bilinear restriction estimate for paraboloids. Geometric & Functional Analysis, 13, No. 6 (2003), 1359-1384.T2 N. Totz.A justification of the modulation approximation to the 3D full water wave problem.Communications in Mathematical Physics, 335, No. 1 (2015), 369-443.T1 N. Totz.Global Well-Posedness of 2D Non-Focusing Schrodinger Equations via Rigorous Modulation Approximation.J. Diff. Eq., 261, No. 4 (2016), 2251-2299.TW N. Totz and S. Wu.A rigorous justification of the modulation approximation to the 2D full water wave problem.Comm. Math. Phys., 310, No. 3 (2012), 817-883.V1 A. Vargas.Restriction theorems for a surface with negative curvature.Math. Z., 249, (2005), 97-111. | http://arxiv.org/abs/1708.08014v3 | {
"authors": [
"Benjamin Dodson",
"Jeremy L. Marzuola",
"Benoit Pausader",
"Daniel Spirn"
],
"categories": [
"math.AP",
"35Q55, 35Q35"
],
"primary_category": "math.AP",
"published": "20170826194445",
"title": "The profile decomposition for the hyperbolic Schrödinger equation"
} |
L. Lopez-FuentesTM-RCS Working Group, AnsuR Technologies AS, Fornebu. Norway.University of the Balearic Islands, Dept of Mathematics and Computer Science, Crta. Valldemossa, km 7.5, E-07122 Palma, Spain.Computer Vision Center. Universitat Autònoma de Barcelona. 08193-Barcelona. Spain. [email protected]. van de Weijer Computer Vision Center. Universitat Autònoma de Barcelona. 08193-Barcelona. Spain. [email protected] M. González-Hidalgo University of the Balearic Islands, Dept of Mathematics and Computer Science, Crta. Valldemossa, km 7.5, E-07122 Palma, Spain. [email protected] H. Skinnemoen TM-RCS Working Group, AnsuR Technologies AS, Fornebu. Norway. [email protected] A. D. Bagdanov University of Florence, DINFO, Via Santa Marta 3, 50139 Florence, Italy. [email protected] on Computer Vision Techniques in Emergency Situations This work was partially supported by the Spanish Grants TIN2016-75404-P AEI/FEDER, UE, TIN2014-52072-P, TIN2013-42795-P and the European Commission H2020 I-REACT project no. 700256. Laura Lopez-Fuentes benefits from the NAERINGSPHD fellowship of the Norwegian Research Council under the collaboration agreement Ref.3114 with the UIB. We thank the NVIDIA Corporation for support in the form of GPU hardware.Laura Lopez-Fuentes Joost van de WeijerManuel González-Hidalgo Harald Skinnemoen Andrew D. Bagdanov Received: date / Accepted: date ==================================================================================================================================================================================================================================================================================================================================================================================================================================================================================== In emergency situations, actions that save lives and limit the impact of hazards are crucial. In order to act, situational awareness is needed to decide what to do. Geolocalized photos and video of the situations as they evolve can be crucial in better understanding them and making decisions faster. Cameras are almost everywhere these days, either in terms of smartphones, installed CCTV cameras, UAVs or others. However, this poses challenges in big data and information overflow. Moreover, most of the time there are no disasters at any given location, so humans aiming to detect sudden situations may not be as alert as needed at any point in time. Consequently, computer vision tools can be an excellent decision support. The number of emergencies where computer vision tools has been considered or used is very wide, and there is a great overlap across related emergency research. Researchers tend to focus on state-of-the-art systems that cover the same emergency as they are studying, obviating important research in other fields. In order to unveil this overlap, the survey is divided along four main axes: the types of emergencies that have been studied in computer vision, the objective that the algorithms can address, the type of hardware needed and the algorithms used. Therefore, this review provides a broad overview of the progress of computer vision covering all sorts of emergencies. § INTRODUCTION Emergencies are a major cause of both human and economic loss. They vary in scale and magnitude, from small traffic accidents involving few people, to full-scale natural disasters which can devastate countries and harm thousands of people. Emergency management has become very important to reduce the impact of emergencies. As a consequence, using modern technology to implement innovative solutions to prevent, mitigate, and study emergencies is an active field of research. The presence of digital cameras has grown explosively over the last two decades. Camera systems perform real-time recording of visual information in hotspots where an accident or emergency is likely to occur. Examples of such systems are cameras on the road, national parks, inside banks, shops, airports, metros, streets and swimming pools. In Figure <ref> we give some examples of frames from videos that could potentially be recorded by some of these monitoring systems. It is estimated that as of 2014 there were over 245 million active, operational and professionally installed video cameras around the world <cit.>. Mounting these video cameras has become very cheap but there are insufficient human resources to observe their output. Therefore, most of the data is being processed after an emergency has already occurred, thus losing the benefit of having a real-time monitoring tool in the first place. The problem of real-time processing of the visual output frommonitoring systems in potential emergency scenarios has become a hot topic in the field of computer vision. As we will show in this review, computer vision techniques have been used to assist in various stages of emergency management: from prevention, detection, assistance of the response, to the understanding of emergencies. A myriad of algorithms has been proposed to prevent events that could evolve into disaster and emergency situations. Others have focused on fast response after an emergency has already occurred. Yet other algorithms focus on assisting during the response to an emergency situation. Finally, several methods focus on improving our understanding of emergencies with the aim of predicting the likelihood of an emergency or dangerous situation occurring.This review provides a broad overview of the progress of computer vision covering all sorts of emergencies. The review is organized in four main axes along which researchers working on the topic of computer vision for emergency situations have organized their investigations. The first axis relates to the type of emergency being addressed, and the most important division along this axis is the distinction between natural and man-made emergencies. The second axis determines the objective of a system: if it is going to detect risk factors to prevent the emergency, help detecting the emergency once it has occurred, provide information to help emergency responders to react to the emergency, or model the emergency to extract valuable information for the future. The third axis determines where the visual sensors should be placed, and also what type of sensors are most useful for the task. The last axis determines which algorithms will be used to assist in analyzing the data. This typically involves a choice of computer vision algorithms combined with machine learning techniques.When organizing emergency situations along these axes, our review shows that there is significant overlap across emergency research areas, especially in the sensory data and algorithms used. For this reason this review is useful for researchers in emergency topics, allowing them to more quickly find relevant works. Furthermore, due to its general nature, this article aims to increase the flow of ideas among the various emergency research fields.Since it is such an active field of research, there are several reviews on computer vision algorithms in emergency situations. However these reviews focus on a single, specific topic. In the field of fire detection, a recent survey was published summarizing the different computer vision algorithms proposed in the literature to detect fire in video <cit.>. Also Gade et al. <cit.> published a survey on applications of thermal cameras in which fire detection is also considered. Another related research direction is human activity recognition and detection of abnormal human actions. This field has many applications to video surveillance, as shown in a recent review on this topic <cit.>. Work in the area of human activity recognition has focused on the specific topic of falling person detection. Mubashir et al. <cit.> describe the three main approaches to falling person detection, one of them being computer vision. Another important application area of emergency management is traffic. The number of vehicles on roads has significantly increased in recent years, and simultaneously the number of traffic accidents has also increased. Therefore a large number of road safety infrastructure and control systems have been developed. Three main reviews have been recently published on the topic of road vehicles monitoring and accident detection using computer vision <cit.>. As discussed here, there are several review papers about different specific emergency situations, however there is no one work reviewing computer vision in emergency situations considering many types of emergency situations. By doing so, we aspire to draw a wider picture of the field, and to promote cross-emergency dissemination of research ideas.Computer vision is just one of several scientific fields which aims to improve emergency management. Before going into depth on computer vision techniques for emergency management, we briefly provide some pointers to other research fields for interested readers. Emergency management is a hot topic in the field of Internet of Things (IoT) <cit.>. The goal of the IoT is to create an environment in which information from any object connected to the network be efficiently shared with others in real-time. The authors of <cit.> provide an excellent study on how IoT technology can enhance emergence rescue operations. Robotics research is another field which already is changing emergency management. Robots are being studied for handling hazardous situations with radiation <cit.>, for assisting crowd evacuation <cit.>, and for search and rescue operations <cit.>. Finally, we mention reviews on human centered sensing for emergency evacuation <cit.> and research on the impact of social media on emergency management. The main advantage of computer vision compared to these other research fields is the omni-presence of cameras in modern society, both in the public space and mobile phones of users. However, for some problems computer vision might currently be still less accurate than other techniques.§ ORGANIZATION OF THIS REVIEW Before we enter into the organization of the review it is important to define what we consider as an emergency and, therefore, what situations and events will be encompassed in this definition. We consider as an emergency any sudden or unexpected situation that meets at least one of the following conditions: * it is a situation that represents an imminent risk of health or life to a person or group of persons or damage to properties or environment;* it is a situation that has already caused loss of health or life to a person or group of persons or damage to properties or environment; or* it is a situation with high probability of scaling, putting in risk the health or life of a person or group of persons or damage to properties or environment.An emergency situation, therefore, requires situational awareness, rapid attention, and remedial action to prevent degradation of the situation.In some cases emergency situations can be prevented by avoiding and minimizing risks. Also, in some situations, the occurrence of an emergency situation can help to understand risk factors and improve the prevention mechanisms. Taking this definition in consideration, any type of uncontrolled fire would be considered as an emergency situation as it is an unexpected situation that could represent a risk for persons, the environment, and property. A controlled demolition of a building would not be considered an emergency situation, as by definition if it is controlled it is not an unexpected situation any more. However, if someone is injured or something unforeseen was damaged during the demolition, that would then be considered an emergency situation.After having defined what we consider an emergency situation we briefly explain the organization of this review. We begin in Section <ref> with an overview and classification into subcategories of emergency situations. Here, we briefly introduce the three axes along which we will organize the existing literature on computer vision for emergency situations, which corresponds to Sections 4 to 6. The organization is also summarized in Figure <ref>. The Monitoring Objective Axis (Section <ref>): the management of an emergency has a life cycle which can be divided into four main phases <cit.>: * Preparedness: during this phase there are risk factors that could trigger an emergency at some point. In order to reduce the chance of an emergency happening, it is important to estimate these factors and avoid or minimize them as much as possible. Not all emergencies can be prevented, but in some cases preventive measures can help to reduce the impact of a disaster (like teaching people how to behave during a hurricane). * Response: response activities are performed after the emergency has occurred. At this point, it is very important to detect and localize the emergency and to maintain situational awareness to have a rapid and efficient reaction. Often the minutesimmediately after the occurrence of an emergency can be crucial to mitigating its effects.* Recovery: recovery activities take place until normality is recovered. In this phase actions must be taken to save lives, prevent further damaging or escalation of the situation. We consider that normality has been recovered when the immediate threat to health, life, properties or environment has subsided.* Mitigation: this phase starts when normality has been re-established. During this phase it is important to study and understand the emergency, how it could have been avoided or what responses could have helped to mitigate it.The Sensors and Acquisition System Axis (Section <ref>): An emergency has four main management phases: prevention, detection, response and understanding of an emergency. For a computer vision algorithm to help in one or more of these phases, it is important to have visual sensors to record the needed information. There are many different types of devices for this, as also many types of devices in which they may be installed depending on the type of data being captured. Some of the most common are: fixed monitoring cameras, satellites, robots and Unmanned Aerial Vehicles (UAVs). These devices can capture different types of visual data, such as stereo or monocular images, infrared or thermal. Depending on the rate of data acquisition this data can be in the form of video or still images, and the video can be either from a static or moving source. The Algorithmic Axis (Section <ref>): The visual data collected by the sensor is then subject to a feature extraction process. Finally the features extracted are used as an input to a modeling or classification algorithm. In Figure <ref> the main feature extraction and classification algorithms are listed.§ TYPES OF EMERGENCIES There are many types of emergencies and a great deal of effort has been invested in classifying and defining them <cit.>. However there are many emergencies that due to their nature have not been studied in computer vision, such as famine or pollution. These types of emergencies do not have clear visual consequences that can be easily detected by visual sensors. In this paper we focus only on emergencies relevant in the computer vision field. In Table <ref> we classify the types of emergencies that we consider in this survey. We distinguish two generic emergency groups: natural emergencies and man-made emergencies. Natural emergencies are divided into three emergency types: fire, flood and drought. Emergencies caused by humans are divided into two sub-types depending on the scope: emergencies that cause dangers to multiple persons and emergencies that cause dangers to a single person. §.§ Natural emergenciesNatural emergencies cover any adverse event resulting from natural processes of the earth. Natural emergencies can result in a loss of lives and can have huge economic costs. The severity of the emergency can be measured by the losses and the ability of the population to recuperate. Although natural emergencies are external to humans and therefore can not be prevented, society has been increasingly able to reduce the impact of these natural events by building systems for early detection or assistance during the event. Among these systems built to detect or assist natural emergencies, computer vision has played an important role in emergencies related to fire, flood and drought.Fire: electric fire detectors have been integrated into buildings since the late 1980s. These devices detect smoke, which is a typical fire indicator. Currently, two major types of electric smoke detector devices are used <cit.>: ionization smoke detectors and photoelectric smoke detectors. Although the devices achieve very good performance in closed environments, their reliability decreases in open spaces because they rely on smoke particles arriving to the sensor. For this reason, fire detection in monitoring cameras has become an important area of research in computer vision. The use of video as input in fire detection makes it possible to use in large and open spaces. Hence, most computer vision algorithms that study fire detection focus mainly on open spaces <cit.>. However, some articles <cit.> also study fire detection in closed spaces, claiming to have several advantages over conventional smoke detectors: visual fire detectors can trigger the alarm before conventional electric devices, as they do not rely on the position of fire particles, and they can give information about the position and size of the fire.Flood: flood results from water escaping its usual boundaries in rivers, lakes, oceans or man-made canals due to an accumulation of rainwater on saturated ground, a rise in the sea level or a tsunami. As a result, the surrounding lands are inundated. Floods can result in loss of life and can also have huge economic impact by damaging agricultural land and residential areas. Flood detection or modeling algorithms are still at a very early stage. In computer vision, there are two main trends: algorithms based on still cameras monitoring flood-prone areas <cit.> and systems that process satellite images <cit.>. On the one hand, algorithms based on earth still cameras cover a limited area but do not have hardware and software restrictions. On the other hand, algorithms based on satellite imagery cover a much wider area, but if the system is to be installed on board the satellite, there are many factors that have to be considered when deciding the type of hardware and software to be used on board a satellite <cit.>.Drought: droughts are significant weather-related disasters that can result in devastating agricultural, health, economic and social consequences. There are two main characteristics that determine the presence of drought in an area: a low surface soil moisture and a decrease in rainfall. Soil moisture data is rarely computed with visual data, however in <cit.> they use high-resolution remotely sensed data to determine the water balance of the soil. Another technique used to determine the beginning of drought is to compute the difference between the average precipitation, or some other climatic variable, over some time period and comparing the current situation to a historical record. Observation of rainfall levels can help prevent or mitigate drought. Although rainfall does not have a clear visual consequence that can be monitored through cameras, there have been some image processing algorithms studied to map annual rainfall records <cit.>. These algorithms help to determine the extension and severity of a drought and to determine regions likely to suffer from this emergency.Earthquake: earthquakes are sudden shakes or vibrations of the ground caused by tectonic movement. The magnitude of an earthquake can vary greatly from almost imperceptible earthquakes (of which hundreds are registered every day) up to very violent earthquakes that can be devastating. Earthquake prediction is a whole field of research called seismology. However, due to the lack of distinctive visual information about an earthquake, this emergency has received relatively little attention from the computer vision community. Several works address the robustness of motion tracking techniques under seismic-induced motions <cit.>,with the aim of assisting rescue and reconnaissance crews. The main difficulty for these systems is the fact that the camera itself is also undergoing motion. In <cit.> this problem is addressed by introducing a dynamic camera reference system which is shared between cameras and allowstracking of objects with respect to this reference coordinate system. Also, computer vision plays a role during the rescue of people affected by these natural disaster by detecting people with robots or UAVs in the damaged areas, this is tackled in the “injured civilians” section. Hurricane/tornado: hurricanes and tornadoes are violent storms characterized by heavy wind and rain. Similarly to earthquakes, depending on their magnitude they can be almost unnoticeable or devastating and at the same time the visual information derived from these natural disasters has not yet been studied in the field of computer vision. However, similarly to earthquakes, robots and UAVs can be used to find people in the areas affected by the disaster using human detection algorithms.The more general work on “injured civilians” detection in affected areas which is shared among any other emergency that involves people rescue like earthquakes and hurricanes, is again applicable to landslide and avalanche emergencies. We found one work which evaluates this in particular for avalanches, and which takes into account the particular environment where avalanches occur <cit.>.§.§ Emergencies caused by humansEmergencies caused by humans cover adverse events and disasters that result from human activity or man-made hazards. Although natural emergencies are considered to generate a greater number of deaths and have larger economic impact, emergencies caused by humans are increasing in importance and magnitude. These emergencies can be unintentionally caused, such as road accidents or drowning persons, but they can also be intentionally provoked, such as armed conflicts. In this survey, emergencies caused by humans are divided in two types depending on their scope: emergencies affecting multiple persons and emergencies affecting a single person.§.§.§ Emergencies affecting multiple personsThese emergencies are human-caused and affects several persons.Road accident emergencies: Although depending on the extent of the emergency they can affect one or multiple persons, we consider road accidents to be multiple-person emergencies. Transport emergencies result from human-made vehicles. They occur when a vehicle collides with another vehicle, a pedestrian, animal or another road object. In recent years, the number of vehicles on the road has increased dramatically and accidents have become one of the highest worldwide death factors, reaching the number one cause of death among those aged between 15 and 29 <cit.>. In order to prevent these incidents, safety infrastructure has been developed, many monitoring cameras have been installed on roads, and different sensors have been installed in vehicles to prevent dangerous situations. In computer vision there are three main fields of study in which have focused on the detection or prevention of transport emergencies. * Algorithms which focus on detecting anomalies, as in most cases they correspond to dangerous situations <cit.>.* Algorithms specialized on detecting a concrete type of road emergency, such as a collision <cit.>.* Algorithms based on on-board cameras which are designed to prevent dangerous situations, such as algorithms that detect pedestrians and warn the driver <cit.>. Crowd-related emergencies: These type of emergencies are normally initiated by another emergency when man persons are gathered in a limited area. They normally occur in multitudinous events such as sporting events, concerts or strikes. These type of events pose a great risk because when any emergency strikes it can lead to collective panic, thus aggravating the initial emergency. This is a fairly new area of research in computer vision. In computer vision for crowd-related emergencies we find different sub-topics like crowd density estimation, analysis of abnormal events <cit.>, crowd motion tracking <cit.> and crowd behaviour analysis <cit.>, among others.Weapon threat emergencies: Although detecting weapons through their visual features and characteristics has not yet been studied in computer vision, some studies have been done in the direction of detecting the heat produced by them. A dangerous weapon that threatens innocent people every year are mines. Millions of old mines are hidden under the ground of many countries which kill and injure people. These mines are not visible from the ground but they emit heat that can be detected using thermal cameras <cit.>. Another weapon that can be detected through thermal sensors are guns, due to the heat that they emit when they are fired <cit.>.§.§.§ Emergencies affecting a single personThese are emergencies are caused by or affect one person. The classic example of this is detecting human falling events for monitoring the well-being of the elderly.Drowning: Drowning happens when a person lacks air due to an obstruction of the respiratory track. These situations normally occur in swimming pools or beach areas. Although most public swimming pools and beaches are staffed with professional lifeguards, drowning or near-drowning incidents still occur due to staff reaction time. Therefore there is a need to automate the detection of these events. One way of automating detection of these events is the installation of monitoring cameras equipped with computer vision algorithms to detect drowning persons in swimming pools and beach areas. However, drowning events in the sea are difficult to monitor due to the extent of the hazardous area, so most computer vision research on this topic has been focused in drowning people in swimming pools <cit.>. Injured person: An injured person is a person that has come to some physical harm. After an accident, reaction time is crucial to minimize human losses. Emergency responders focus their work on the rescue of trapped or injured persons in the shortest time possible. A method that would help speed up the process is the use of aerial and ground robots with visual sensors that record the affected area at the same time as detect human victims. Also, in the event of an emergency situation, knowing if there are human victims in the area affected by the event may help determine the risk the responders are willing to take. For this purpose autonomous robots have been built to automatically explore the area and detect human victims without putting at risk the rescue staff <cit.>. Detecting injured or trapped humans in a disaster area is an extremely challenging problem from the computer vision perspective due to the articulated nature of the human body and the uncontrolled environment.Falling person: With the growth of the elderly population there is a growing need to create timely and reliable smart systems to help and assist seniors living alone. Falls are one of the major risks the elderly living alone, and early detection of falls is very important. The most common solution used today are call buttons, which when the senior can press to directly call for help. However these devices are only useful if the senior is conscious after the fall. This problem has been tackled from many different perspectives. On the one hand wearable devices based on accelerometers have been studied and developed <cit.>, some of these devices are in process of being commercialized <cit.>. On the other hand, falling detection algorithms have been studied from a computer vision perspective <cit.>. § MONITORING OBJECTIVE As stated in Section <ref>, an emergency has four main phases which at the same time lead to four emergency management phases: prevention, detection, response/assistance and understanding of emergencies. It is important to understand, study and identify the different phases of an emergency life cycle since the way to react to it changes consequently. In this section, we will classify computer vision algorithms for emergency situations depending on the emergency management phase in which they are used. §.§ Emergency PreventionThis phase focuses on preventing hazards and risk factors to minimize the likelihood that an emergency will occur. The study of emergency prevention on computer vision can not be applied to any emergency since not all of them are induced by risk factors that have relevant visual features that can be captured by optical sensors. For example, the only visual feature which is normally present before a fire starts is smoke. Therefore research on fire prevention in computer vision focuses on smoke detection. As an example, in Figure <ref> (a) we see a region of forest with a dense smoke that looks like the starting of a fire, however a fire detection algorithm would not detect anything until the fire is present in the video. Some early work on this topic started at the beginning of the century <cit.>. Since then, smoke detection for fire prevention or early fire detection has been an open topic in computer vision. Most of the work done on smoke detection is treated as a separate problem from fire detection and aims to detect smoke at the smoldering phase to prevent fire <cit.>. However there is also some work on smoke and flame detection when the fire has already started since smoke in many cases can be visible from a further distance <cit.>.Traffic accidents are considered as one of the main sources of death and injuries globally. Traffic accidents can be caused by three main risk factors: the environment, the vehicle and the driver. Among these risk factors, it is estimated that the driver accounts for around 90% of road accidents <cit.>. To avoid these type of accidents, a lot of security devices have been incorporated on vehicles in the past years. There are two types of security devices: “active” whose function is to decrease the risk of an accident occurring, and “passive” which try to minimize the injuries and deaths in an accident. Among the active security devices, there is work on the usage of visual sensors to detect situations that could lead to an accident and force actions that could be taken to prevent the accident <cit.>.There are many different kind of traffic accidents but one of the most common and at the same time more deadly ones are pedestrians being run over. Due to the variant and deformable nature of humans, the environment variability, the reliability, speed and robustness that a security system is required, the detection of unavoidable impact with pedestrians has become a challenge in computer vision.This field of study is well defined and with established benchmarks and it has served as a common topic in numerous review articles <cit.>.Now, with the increase of training data and computing power this investigation has had a huge peak in the past years and even the first devices using this technology have already been commercialized <cit.>. Figure <ref> (b) shows an example of the output from a pedestrian detection algorithm on board a vehicle.Another contributing factor to many accidents is the drivers fatigue, distraction or drowsiness. A good solution to detect the state of the driver in a non intrusive way is using an on board camera able to detect the drivers level of attention to the road. There are different factors that can be taken into account to determine the level of attention to the road of a driver. In <cit.> they determine if the driver is looking at the road through the head pose and gaze direction, in Figure <ref> (c) we show some examples of the images they used to train their algorithm. While in <cit.> they determine 17 feature points from the face that they consider relevant to detect the level of drowsiness of a person. In <cit.> they study the movements of the eyelid to detect sleepy drivers.Although not so widely studied, there are other type of road accidents that could also be prevented with the help of computer vision. For example, in <cit.> they propose a framework to predict unusual driving behavior which they apply to right turns at intersections. Algorithms that use on board cameras to detect the road have also been studied <cit.>, this could help to detect if a vehicle is about to slip on the road. In <cit.> they propose a system with on board stereo cameras that assist drivers to prevent collisions by detecting sudden accelerations and self-occlusions. Multiple algorithms for traffic sign detection with on board cameras have also widely proposed in the literature <cit.>, these algorithms mostly go in the direction of creating autonomous vehicles, however they can also be used to detect if the driver is not following some traffic rules and is therefore increasing the risk of provoking an accident.In order to prevent falls, in <cit.> they propose a method to detect slips arguing that a slip is likely to lead to a fall or that a person that just had a slip is likely to have balance issues that increase the risk of falling down in the near future. They consider a slip as a sequence of unbalanced postures. §.§ Emergency Detection The field of emergency detection through computer vision in contrast with the field of prevention is much wider and has been studied in more depth. In this section we will present the visual information that is being studied to detect different emergencies.Fire detection has been widely studied in computer vision since it is very useful on open spaces, where other fire detection algorithms fail. In some cases it is useful to not only detect the fire but also some properties of the fire such as the flame height, angle or width <cit.>. In order to detect fire on visual content, the algorithms studied exploit visual and temporal features of fire. In early research, flame was the main visual feature that was studied in fire detection <cit.> however, as stated in Section <ref>, recently smoke has also been studied in the field of fire detection mainly because it appears before fire and it spreads faster so in most cases smoke will appear faster in the camera field of view <cit.>. As fire has a non-rigid shape with no well defined edges, the visual content used to detect fire is related to the characteristic color of fire. This makes the algorithms in general vulnerable to lighting conditions and camera quality.Water, the main cue for flood detection, has similar visual difficulties as fire in order to be detected since it has a non-rigid shape and the edges are also not well defined. Moreover, water does not have a specific characteristic color because it might be dirty and it also reflects the surrounding colors. Due to the mentioned difficulties flood detection in computer vision has not been so widely studied. In <cit.> they use color information and background detail (because as said before the background will have a strong effect on the color of the water) and the ripple pattern of water. In <cit.> they analyse the rippling effect of water and they use the relationship between the averages for each color channel as a detection metric.Detecting person falls has become a major interest in computer vision, specially motivated to develop systems for elderly care monitoring at home. In order to detect a falling person it is important to first detect the persons which are visible in the data and then analyse the movements. Most algorithms use background subtraction to detect the person since in indoor situations, most of the movement come from persons. Then, to determine if the movement of the person corresponds to a fall some algorithms use silhouette or shape variation <cit.>, direction of the main movement <cit.>, or relative height with respect to the ground <cit.>. However, sometimes it is difficult to distinguish between a person falling down and a person sitting, lying or sleeping, to address this issue, most systems consider the duration of the event since a fall is usually a faster movement <cit.>, for example, in <cit.> they consider 10 frames as a fast movement and use this threshold to differentiate two visually similar movements such as the ones given in Figure <ref> (a) which corresponds to sitting and falling.The problem of drowning person detection is fairly similar to the falling person detection, since it is also in a closed environment and related to human motion. Similarly to fall detection algorithms, the first step to detect a drowning person is to detect the persons visible in the camera range, which is also done by background subtraction <cit.>. In Figure <ref> we give an example of a detection of the persons in a swimming pool using a background modelling algorithm. However, in this case aquatic environments have some added difficulties because it is not static <cit.>. §.§ Emergency Response/Assistance After an emergency has already occurred, one of the most important tasks of emergency responders is to help injured people. However, depending on the magnitude of the disaster, it may be difficult to effectively localize all these persons so emergency responders may find it useful to use robots or UAVs to search for these persons. In order to automatically detect injured people, some of these robots use cameras together with computer vision algorithms. When considering disasters taking place on the ground one of the main characteristic of an injured person is that they are normally lying on the floor so these algorithms are trained to detect lying people. The main difference between them is the perspective from which these algorithms expect to get the input, depending if the camera is mounted on a terrestrial robot <cit.> or on a UAV <cit.>.A fast and efficient response the in the moments after an emergency has occur ed is critical to minimize damage. For that it is essential that emergency responders have situational awareness, which can be obtained through sensors in the damaged areas, by crowd sourcing applications, and from social media. However, extracting useful information from huge amounts of data can be a slow and tedious work. Therefore, organization of video data that facilitates the browsing on the data is very important <cit.>. Since during an emergency it is common to have data on the same event from different angles and perspectives, in <cit.> the authors combine interactive spatio-temporal exploration of videos and 3D reconstruction of scenes. In <cit.> the authors propose a way of representing the relationship among clips for easy video browsing. To extract relevant information from videos, in <cit.> the authors find recurrent regions in clips which is normally a sign of relevance. In <cit.> the authors introduce a framework of consecutive object detection from general to fine that automatically detects important features during an emergency. This also minimizesbandwidth usage, which is essential in many emergency situations where critical communications infrastructure collapses.§.§ Emergency Understanding Keeping record of emergencies is very important to analyse them and extract information in order to understand the factors that have contributed to provoke the emergency, how these factors could have been prevented, how the emergency could have been better assisted to reduce the damages or the effects of the emergency. The systems developed to with this objective should feed back into the rest of the systems in order to improve them. § ACQUISITION Choosing an appropriate acquisition setup is one of the important design choices. To take this decision one must take into account the type of emergency addressed (see Section <ref>) and objectives to fulfill (see Section <ref>). One of the things to consider is the location where the sensor will be placed (e.g. in a fixed location, mounted on board a satellite, a robot, or a UAV, etc). Then, depending on the type of sensor we acquire different types of visual information: monocular, stereo, infrared, or thermal. Depending on the frequency in which the visual information is acquired we can obtain videos or still images. Finally, this section is summarized in Table <ref> where we include the advantages and disadvantages of the types of sensors studied and the locations at which they can be placed The type of sensors chosen for the system will have an important impact on the types of algorithms that can be applied, the type of information that can be extracted and the cost of the system, which is also many times a limitation. §.§ Sensor locationThe physical location where the sensor is installed determines the physical extent in the real world that will be captured by the sensor. The location of the sensor depends primarily on the expected physical extent and location of the emergency that is to be prevented, detected, assisted or understood.In a fixed location:When the type of emergency considered has a limited extent and the location where it can occur (the area of risk) is predictable, a fixed sensor is normally installed. In many cases this corresponds to a monitoring camera mounted on a wall or some other environmental structure. These are the cheapest and most commonly used sensors. They are normally mounted in a high location, where no objects obstruct their view and pointing to the area of risk.Static cameras have been used to detect drowning persons in swimming pools, as in this case the area of risk is limited to the swimming pool <cit.>. They have also been used to monitor dangerous areas on roads, inside vehicles either pointing to the driver to estimate drowsiness <cit.>, or pointing to the road to process the area in which it is being driven <cit.>. Fixed cameras are also widely used in closed areas to detect falling persons <cit.>. Finally, they can occasionally also be used to detect floods or fire in hotspots <cit.>.On board robots or UAVs: For emergencies that do not take place in a specific area, or that need a more immediate attention, sensors can be mounted on mobile devices such as robots or UAVs. The main difference between these two types of devices is the perspective from which the data can be acquired: terrestrial in the case of a robot, and aerial for UAVs. On the one hand, the main advantage of these data sources is that they are flexible and support real time interaction with emergency responders. On the other hand, their main drawback is their limited autonomy due to battery capacity. Robots and UAVs have been used extensively for human rescue. In such cases, a visual sensor is mounted on board the autonomous machine which can automatically detect human bodies <cit.>. The authors of <cit.> use a UAV with several optical sensors to capture aerial images of the ground, estimate the soil moisture, and determine if an area suffers or is close to suffering drought.On board satellites: For emergencies of a greater extent, such as drought, floods or fire, sometimes the use of sensors on board satellites is more convenient <cit.>. By using satellites, it is possible to visualize the whole globe. However, the biggest drawback of this type of sensor is the frequency at which the data can be acquired since the satellites are in continuous movement and it may be necessary to wait up to a few days to obtain information about a specific area, as an example the Sentinel 2 which is a satellite in the Copernicus program for disaster management revisits every location within the same view angle every 5 days <cit.>. Due to this, the information acquired by this type of sensor is mainly used for assistance or understanding of an emergency rather than to prevent or detect it.Multi-view cameras: In some scenarios, like in the street, it is useful to have views from different perspectives in order to increase the range or performance of the algorithm. However, using data from different cameras also entails some difficulties in combining the different views. Since the most common scenario where multiple cameras are present is in the street, this problem has been tackled in pedestrian detection <cit.> and traffic accidents <cit.>.§.§ Type of sensor Another important decision to take when working with computer vision for emergency management is the type of optical sensor that should be used. When taking this decision it is important to take into account the technical characteristics of the available sensors, the type of data that they will capture, and the cost.Monocular RGB or greyscale cameras: Monocular cameras are devices that transform light from the visible spectrum into an image. This type of visual sensor is the cheapest and therefore by far the most widely used in computer vision and monitoring systems. Although these sensors may not be the optimal in some scenarios (such as fire detection), due to their low cost and the experience of researchers on the type of images generated by such sensors, they are used in almost all emergencies <cit.>.Stereo RGB or greyscale cameras: Stereo cameras are optical instruments that, similarly to monocular cameras, transform light from the visible spectrum into an image. The difference between stereo and monocular cameras is that stereo sensors have at least two lenses with separate image sensors. The combination of the output from the different lenses can be used to generate stereo images from which depth information can be extracted. This type of sensor are especially useful on board vehicles where the distance between the objects and the vehicle is essential to estimate if a collision is likely to occur and when <cit.>. They are also used on board of vehicles to detect lanes <cit.>.Infrared/thermal cameras: Infrared or thermal cameras are devices that detect the infrared radiation emitted by all objects with a temperature above absolute zero <cit.>. Similarly to common cameras that transform visible light into images, thermal cameras convert infrared energy into an electronic signal which is then processed to generate a thermal image or video. This type of camera has the advantage that they eliminate the illumination problems that normal RGB or greyscale cameras have. They are of particular interest for emergency management scenarios such as fire detection since fire normally has a temperature higher than its environment.It is also well known that humans have a body temperature that ranges between 36^∘C and 37^∘C in normal conditions. This fact can be exploited to detect humans, and has been used for pedestrian detection to prevent vehicle collisions with pedestrians <cit.>, to detect faces and pupils <cit.>, and for fall detection for elderly monitoring <cit.>. Although not so extensively used, thermal and infrared cameras have also been used for military purposes, such as to detect firing guns <cit.> and to detect mines under ground <cit.>.One of the disadvantages of thermal cameras compared with other type of optical sensors is that their resolution is lower than many other visual sensors. For this reason some systems use thermal cameras in combination with other sensors. The authors of <cit.> use a thermal camera among other visual sensors to study the surface soil moisture and the water balance of the ground, which is in turn used to estimate, predict and monitor drought.In order to obtain depth information in addition to thermal information, it is also possible to use infrared stereo cameras such as in <cit.> where the authors use infrared cameras to detect pedestrians by detecting their corporal heat, and then estimate pedestrian position using stereo information. §.§ Acquisition rate Depending on the rate at which optical data is acquired we can either obtain video or still images. The main difference between video and a still image is that still images are taken individually while video is an ordered sequence of images taken at a specific and known rate. To be considered a video, the rate at which images are taken must be enough for the human eye to perceive movement as continuous.Video: The main advantage of videos over images is that, thanks to the known acquisition rate, they provide temporal information which is invaluable for some tasks. Also, most of the systems studied in this survey are monitoring systems meant to automatically detect risky situations or emergencies. As these situations are rare but could happen at any moment and it is important to detect them as soon as possible, so it is necessary to have a high acquisition rate.For the study of many of the emergencies considered in this survey it is necessary to have temporal information. For example, merely detecting a person lying on the ground in a single image is not sufficient to infer that that person has fallen.Many systems distinguish between a person lying on the floor and a person that has fallen by using information about the speed at which this person reached the final posture <cit.>. Also, systems that estimate likelihood of a car crash must have temporal information to know at which speed the external objects are approaching the vehicle. To estimate driver drowsiness or fatigue, many systems take information into account such as the frequency at which the driver closes his eyes and for how long they remain closed <cit.>.Moreover, many algorithms developed for the detection of specific situations also require temporal information. In order to detect fire, most algorithms exploit the movement of the fire to differentiate it from the background. Similarly, for drowning person detection, many algorithms use temporal information to easily segment the person from the water <cit.>. Also, many falling person detection systems use tracking algorithms to track the persons in the scene <cit.> or background subtraction algorithms to segment persons <cit.>.Still images: Still images, in contrast with video, do not give information about the movement of the objects in the scene and does not provide any temporal information. Although most of the algorithms that study the problem of fire detection use temporal information and therefore require video, David Van et al. <cit.> studya system that detects fire on still images to broaden the applicability of their system. In some cases due to the sensor used by the system it is not possible or convenient to obtain videos for example in systems that use satellite imagery <cit.>, also in systems that use UAVs where in many cases is more convenient to take still images instead of videos in order to effectively send the data in real time or get higher resolution imagery <cit.>.Fixed location$50 to $2000 $200 to $6000 $200 to $2000 Up to 30 MP Up to 1.2 MP Up to 30 MP Real time data acquisition Up to 0.02ºC sensitivity Real time data acquisition Limited area of visualization Real time data acquisition Limited area of visualization Limited area of visualizationRobot or UAV $100 to $2000 $250 to $2000 $250 to $2000 Up to 30 MP Up to 1.2 MP Up to 30 MPPortable Up to 0.02ºC sensitivity Portable Bad quality data acquitision in real time Portable Bad quality data acquitision in real timeLimited autonomy Bad quality data acquitision in real time Limited autonomy Limited autonomySatellite Open data access through many research programsSpatial resolution of around 10mCan cover the entire globeMay have to wait a few days until the satellite covers the entire globe §.§ Sensor costWhen implementing the final system, another important factor that should be taken into account before designing it is the cost from setting up the system. In general, the cost of the final system is going to be the cost of the visual sensor, the cost of the installation of the system and the cost of the data transmission. In this section we will do a brief study of the three sources of costs.The most common visual sensors are monocular RGB and grayscale cameras. Since these sensors are so common and have a big demand, their technical features have improved greatly over time and their price has dramatically decreased. Therefore there exists a wide range of monocular cameras in the market with different technical features and prices. Depending on the quality of the device, its price can vary between approximately $50 to $2000. Although there also exists a wide variety of infrared cameras, these devices are less common and in general more expensive. Depending on the quality of the camera, its price fluctuates between $200 and $6000. To illustrate that, in the second row from Table <ref> we give an example of a thermal camera of $300 and its specifications. Finally, stereo cameras are the less common visual sensors and there is fewer variety of these devices. Their price oscillates between $200 and $2000. In the third row from Table <ref> we present an example of an average stereo camera. However, it is fairly common to build a stereo device by attaching two monocular cameras which makes its price double the price of a monocular camera.The cost of the installation of the system and the transmission of data varies greatly depending on the localization in which the sensor is installed. In this case, the cheapest installation would be at a fixed location, for which it would only be necessary to buy a holder to fix the sensor and acquire a broadband service in order to transmit the data in real time. On the other hand, it is also possible to install the sensor on a mobile device such as a robot or UAV. In this case it is necessary to first buy the device, whose price also have big fluctuations depending on their quality and technical characteristics such as its autonomy, dimensions and stability. A low cost robot or UAV can be bought for around $50 while high quality ones can cost around $20000. Moreover, these devices require a complex transmission system since the sensor is going to be in continuous movement. In fact, since these systems are meant to acquire data over a limited period of time, it is common not to install any transmission data system but saving all the data in the device and processing it after the acquisition is over. It is also possible to install a antenna on the device that transmits these data in real time over 2.4GHz or 5.8GHz frequency. In the first row from Table <ref> we show an example of a commonly used drone with an integrated camera and with a 2.4GHz antenna for live data transmission. § ALGORITHMS In this section we review the most popular algorithms that have been applied in computer vision applications for emergency situations. We have considered the nine different types of emergencies that we introduced in Section <ref>. When selecting the articles to review in this survey we have considered a total of 114 papers that propose a computer vision system for some emergency management. However, not all the types of emergencies have bee equally studied in the literature nor in the survey, in Figure <ref> we show a bar graph of the distribution of the articles among the different types of emergencies.This section is divided in two main parts: feature extraction and machine learning algorithms. In the feature extraction part various approaches to extracting relevant information from the data are elaborated. Once that information is extracted, machine learning algorithms are applied to incorporate spatial consistency (e.g. Markov Random Fields), to model uncertainty (e.g. Fuzzy logic) and to classify the data to high-level semantic information, such as the presence of fire or an earthquake. We have focused on methods which are (or could be) applied to a wider group of applications in emergency management.§.§ Feature extractionFeature extraction in computer vision is the process of reducing the information in the data into a more compact representation which contains the relevant information for the task at hand. This is a common practice in computer vision algorithms related to emergency situations. We divide the features in three main groups: color, shape and temporal. §.§.§ Color features The understanding of emergency situations often involves the detection of materials and events which have no specific spatial extent or shape. Examples include water, fire, and smoke. However, these textures often have a characteristic color, and therefore color features can provide distinctive features. Color analysis is a well established computer vision research field and has been broadly applied in applications in emergency management. It is frequently used to detect fire and smoke since they have characteristic colors and – although not so frequently – it has also been used to detect flooded regions. Fire color may vary depending on the combustion material and the temperature. However most flames are yellow, orange and especially red. Smoke tends to be dark or white and can be more or less opaque. Water in flooded regions, according to <cit.>, presents a smooth dark surface or strong reflective spots much brighter than the surroundings.One of the principle choices in color feature extraction is the color space or color model from which to compute the features. This choice influences characteristics of the color feature, including photometric invariance properties <cit.>, device dependence, and whether it is perceptually uniform or not <cit.>. Here we will shortly review the most used color models and briefly describe some of their advantages and disadvantages.The data produced by camera devices is typically in RGB color model. Due to the effortless availability of this color model and the separation of the red values, this color model has been often used for fire, smoke and flood detection <cit.>. The main disadvantage of this color model is its lack of illumination invariance and dependence on the sensor. The HSI and HSV color models were proposed to resemble the color sensing properties of human vision and separate the luminance component from the chromatic ones.They have been applied to smoke detection <cit.>, to detect light gray water <cit.> and to detect deep water <cit.>. Several systems combine RGB and HSI color models <cit.>. The YCbCr color model has three components, one of which encodes the luminance information and two encode the chromaticity. Having the luminance component separated from chromatic components makes it robust to changes in illumination <cit.>. Similarly to HSI and YCbCr, the CIELab color model makes it possible to separate the luminance information from the chromaticity.Moreover, it is device independent and perceptually uniform. This color model has been used to detect fire <cit.> and to improve the background models for swimming pools <cit.>.§.§.§ Shape and texture featuresThere are elements which are very characteristic of some of the emergencies presented in this survey, such as smoke or flames to detect fire, pedestrians to prevent traffic collisions or persons to detect victims in disaster areas. Detecting these characteristic elements is common to many of the algorithms proposed for emergency management with computer vision. The primary goal of many of these algorithms is to prevent or detect human harm and therefore humans are a common characteristic element of these emergency scenarios. The histogram of oriented gradients (HOG) descriptor was introduced to detect pedestrians and is one of the most used shape descriptors to detect humans. Among the algorithms studied in this survey, HOG was used to detect persons in disaster areas <cit.>, to detect pedestrians for on board vehicle cameras <cit.> but also to characterize other objects such as flames, smoke and traffic signs <cit.> and as a descriptor for a tracking algorithm <cit.> in crowd event recognition. Local binary patterns is also a well known shape descriptor which is used to describe textures and was successfully applied to the task of smoke detection <cit.>.§.§.§ Temporal featuresMost of the input data used in emergency management comes in form of videos, which provide visual information as well as temporal information. Here we briefly summarize the main features which exploit the available temporal information in the data. Wavelets: Flicker is an important characteristic of smoke and flames and has been used in several systems to detect fire and smoke in video data. Fourier analysis is the standard tool for analysis of spatial frequency patterns. The extension of frequency analysis to spatial-temporal signals was first studied by Haar <cit.> and later led to the wavelet transform <cit.>.In <cit.> the authors study the periodic behaviour of the boundaries of the moving fire-colored regions. The flickering flames can be detected by finding zero crossings of the wavelet transform coefficients which are an indicator of activity within the region. In <cit.> the flicker of smoke is detected computing the temporal high-frequency activity of foreground pixels by analysing the local wavelet energy. They also use frequency information to find smoke boundaries by detecting the decrease in high frequency content since the smoke is semi-transparent at its boundaries. In <cit.> the authors present an algorithm which uses a Discrete Wavelet Transform to analyse the blockwise energy of the image. The variance over time of this energy value was found to be an indicator of the presence or absence of smoke. In <cit.> the authors perform a vertical, horizontal and diagonal frequency analysis using the Stationary Wavelet Transform and its inverse. A characterization of smoke behaviour in the wavelet domain was done in <cit.> by decomposing the image into wavelets and analyzing its frequency information at various scales.Optical flow: Optical flow estimates the motion of all the pixels of a frame with respect to the previous frame <cit.>. The objective of algorithms based on optical flow is to determine the motion of objects in consecutive frames. Many techniques have been proposed in the literature to perform optical flow <cit.>, however the most widely used technique is the one proposed by Lucas-Kanade <cit.> with some extensions <cit.>.Optical flow algorithms have been used in several computer vision systems for emergency management, such as for the detection of fire and smoke and anomalous situations on roads or in crowds. In the case of fire and smoke detection several approaches use the pyramidal Lucas-Kanade optical flow algorithm to determine the flow of candidate regions which likely to contain fire or smoke and to discard regions that do not present a high variation on the flow rate characteristic of the fire turbulence <cit.>. The pyramidal Lucas-Kanade optical flow algorithm assumes that the objects in the video are rigid with a Lambertian surface and negligible changes in brightness but the drawback is that these assumptions do not hold for fire and smoke. Addressing this problem, Kolesov et al. <cit.> proposed a modification to Lucas-Kanade in which they assume brightness and mass conservation. Optical flow has been also used to analyse the motion of crowds in order to determine abnormal and therefore potentially dangerous situations <cit.>. In <cit.> they use an optical flow technique to model the traffic dynamics of the road, however instead of doing a pixel by pixel matching they consider each individual car as a particle for the optical flow model.Background modeling and subtraction:The separation of background from foreground is an important step in many of the applications for emergency management. Background modeling is an often applied technique for applications with static cameras (see Section <ref>) and static background. The aim of this method is to compute a representation of the video scene with no moving objects, a comprehensive review can be seen in <cit.>.In the context of emergency management, the method of Stauffer and Grimson <cit.> has been used to segment fire, smoke and persons from the background <cit.>. Although not as popular, other background subtraction algorithms have been proposed in the literature <cit.> and used in systems for emergency management <cit.>. An interesting application of background modeling was described in <cit.> in the context of abnormal event detection in emergency situations. They assume that vehicles involved in an accident or an abnormal situation remain still for a considerable period of time. Therefore an abnormal situation or accident can be detected in a surveillance video by comparing consecutive background models.Tracking:Tracking in computer vision is the problem of locating moving objects of interest in consecutive frames of a video. The shape of tracked objects can be represented by points, primitive geometric shapes or the object silhouette <cit.>. Representing the object to be tracked as points is particularly useful when object boundaries are difficult to determine, for example when tracking crowds <cit.> where detecting and tracking each individual person may be infeasible. It is also possible to represent the tracked objects as primitive objective shapes, for example in <cit.> the authors use ellipses to represent swimmers for the detection of drowning incidents. It is also common to represent the tracked objects as rectangles, e.g. in swimmer detection <cit.> and traffic accident detection <cit.>. Finally, tracked objects can also be represented with the object's silhouette or contour <cit.>.Feature selection is crucial for tracking accuracy. The features chosen must be characteristic of the objects to be tracked and as unique as possible. In the context of fall detection, the authors of <cit.> use skin color to detect the head of the tracked person. Many tracking algorithms use motion as the main feature to detect the object of interest, for example applications in traffic emergencies <cit.>, fall detection <cit.>, and drowning person detection <cit.>. Haar-like features <cit.> have been widely used in computer vision for object detection, and have been applied to swimmer detection <cit.>. Finally, descriptors such as Harris <cit.> or HOG <cit.> can also be useful for crowd analysis <cit.>. §.§.§ Convolutional features Recently, convolutional features have led to a very good performance in analyzing visual imagery. Convolutional Neural Networks (CNN) were first proposed in 1990 by LeCun et al. <cit.>. However, due to the lack of training data and computational power, it was not until 2012 <cit.> that these networks started to be extensively used in the computer vision community.CNNs have also been used in the recent years for some emergency challenges, mainly fire and smoke detection in images or video and anomalous event detection in videos. For the task of smoke detection, some researchers proposed novel networks to perform the task <cit.>. In <cit.> the authors propose a 9-layer convolutional network and they use the output of the last layer as a feature map of the regions containing fire, while in <cit.> the authors construct a CNN and use domain adaptation to alleviate the gap between their synthetic fire and smoke training data and their real test data.Other researchers have concentrated on solutions built using existing networks <cit.>. In <cit.> the authors retrain some existing architectures, while in <cit.> the authors use already trained architectures to give labels to the images and then classify those labels into fire or not fire. For the task of anomaly detection,algorithms need to take temporal and spatial information into account. To handle this problem the authors in <cit.> propose to use a Long Short-Term Memory (LSTM) to predict the evolution of the video based on previous frames. If the reconstruction error between subsequent frames and the predictions is very high it is considered to be an anomaly. In <cit.> the authors consider only the regions of the video which have changed over the last period of time and send those regions through a spatial-temporal CNN. In <cit.> the authors pass a pixel-wise average of every pair of frames through a 4-layered fully connected network and an autoencoder trained on normal regions. §.§ Machine learning In the vast majority of the applications in emergency management machine learning algorithms are applied. They are used for classification and regression to high-level semantic information, to impose spatial coherence, as well as to estimate uncertainty of the models. Here we briefly discuss the most used algorithms. §.§.§ Artificial Neural NetworksArtificial Neural Networks (ANN) have received considerable interest due to their wide range of applications and ability to handle problems involving imprecise and complex nonlinear data. Moreover, their learning and prediction capabilities combine with impressive accuracy values for a wide range of problems. A neural network is composed of a large number of interconnected nonlinear computing elements or neurons organized in an input layer, a variable number of hidden layers and an output layer. The design of a neural network model requires three steps: selection of the statistical variables or features, selection of the number of layers (really the hidden layers) and nodes and selection of transfer function. So, the different statistical variables or features, the number of nodes and the selected transfer function, make the difference between the papers found showing applications of the ANN to prevention and detection of emergencies using computer vision and image processing techniques. Applications of ANNs to fire and smoke detection and prevention can be seen in papers <cit.>. They mainly differ in the characteristics of the network. In <cit.> the authors extract features of the moving target which are normalized and fed to a single-layer artificial neural network with five inputs, one hidden layer with four processing elements, and one output layer with one processing element to recognize fire smoke. In <cit.> they propose a neural network to classify smoke features.In <cit.> they implement an ANN fire/smoke pixel classifier. The classifier is implemented as a single-hidden-layer neural network with twenty hidden units, with a softmax non-linearity in the hidden layer. The method proposed in <cit.> describes a neural network classifier with a hidden layer is trained and used for discrimination of smoke and non-smoke objects. The output layer uses a sigmoid transfer function. In the system presented in <cit.> a neural network of perceptrons was used to classify falls against the set of predefined situations. Here, the neural network analyses the binary map image of the person and identifies which plausible situation the person is at any particular instant in time.In general, the usage of ANNs provide a performance improvement compared to other learning methods <cit.>. The results rely highly on the selected statistical values for training. If the selection of statistical values is not appropriate, the results on the accuracy might be lower and simpler machine learning algorithm may be preferred.Deep Learning ANN algorithms that consist of more than one hidden layer are usually called deep learning algorithms. Adding more layers to the network increases the complexity of the algorithm making it more suitable to learn more complex tasks. However, increasing the layers of the network comes at a higher computational cost and a higher probability to overfit or converge to a local minima. Nevertheless, the advances in computers and the increase in training data has facilitated the research towards deeper models which have proven to improve the results.In <cit.> the authors propose a multi-layered ANN for smoke features classification. In the works <cit.> we find applications of ANNs to prevent and detect single-person emergencies. A neural network is used to classify falls in <cit.> using vertical velocity estimates derived either directly from the sensor data or from the tracker. The network described in <cit.>, with four hidden layers, is used in the classifier design to do posture classification that can be used to determine if a person has fallen. In <cit.> the authors monitor human activities using a multilayer perceptron with focus on detecting three types of fall.§.§.§ Support Vector MachinesSupport vector machines (SVMs) are a set of supervised machine learning algorithms based on statistical learning and risk minimization which are related to classification and regression problems. Given a set of labeled samples a trained SVM creates an hyperplane or a set of them, in a high dimensional space, that optimally separates the samples between classes maximizing the separation margin among classes. Many computer vision systems related to emergency situations can be considered as a binary classification problems: fire or no fire, fall or no fall, crash or no crash. To solve this classification problem many researches make use of SVMs <cit.> based on the classification performance that they can reach, their effectiveness to work on high dimensional spaces and their ability to achieve good performance with few data.In more complex systems where it is difficult to predict the type of emergencies that may occur, for example in systems that supervise crowded areas or traffic, algorithms that consider abnormal behaviours as potential emergencies have been proposed <cit.>. A difficulty of these kind of systems is that they can only be trained with samples of one class (normal scenes). For these kind of problems one-class SVMs, which find a region of a high dimensional space that contains the majority of the samples, are well suited.§.§.§ Hidden Markov Models Hidden Markov Models (HMMs), which are especially known for their application in temporal pattern recognition such as speech, handwriting, gesture and action recognition <cit.>, are a type of stochastic state transition model <cit.>. HMMs make it possible to deal with time-sequence data and can provide time-scale invariability in recognition. Therefore it is one of the preferred algorithms to analyze emergency situations subject to time variability. To detect smoke regions from video clips, a dynamic texture descriptor is proposed in <cit.>. The authors propose a texture descriptor with Surfacelet transformation (<cit.>) and Hidden Markov Tree model (HMT). Examples of use of Markov models to study human-caused emergencies that affect a single person can be seen in the following works <cit.>,covering fall detection and detection behavior of a swimmer and drownings. Hidden Markov Models have been applied on monitoring emergency situations in crowds by learning patterns of normal crowd behaviour in order to identify unusual or emergency events, as can be seen in <cit.>. In these two works Andrade and his team presents similar algorithms in order to detect emergency events in crowded scenarios, and based on optical flow to extract information about the crowd behaviour and the Hidden markov Models to detect abnormal events in the crowd.Töreyin and his team in <cit.> useHidden Markov Models to recognize possible human motions in video. Also they use audio track with a three-state HMM to distinguish a person sitting on a floor from a person stumbling and falling. In <cit.>the motion analysis is performed by means of a layered hidden markov model (LHMM). A fall detection system based on motion and posture analysis with surveillance video is presented in <cit.>.The authors propose a context-based fall detection system by analyzing human motion and posture using a discrete hidden Markov model. The problem of recognition of several specific behaviours of a swimmer (including backstroke, breaststroke, freestyle and butterfly, and abnormal behaviour)in a swimming pool using Hidden Markov Models is addressed in <cit.>.The algorithm presented in <cit.> detects water crises within highly dynamic aquatic environments where partial occlusions are handled using a Markov Random Field framework.Hidden Markov Models have been applied on monitoring emergency situations in crowds by learning patterns of normal crowd behavior in order to identify unusual or emergency events, as can be seen in <cit.>. In these two works Andrade and his team presents similar algorithms in order to detect emergency events in crowded scenarios, and based on optical flow to extract information about the crowd behavior and the Hidden Markov Models to detect abnormal events in the crowd.§.§.§ Fuzzy logicFuzzy sets theory and fuzzy logic are powerful frameworks for performing automated reasoning, and provide a principled approach to address the inherent uncertainty related to modeling; an aspect which relevant in many emergency management application. Fuzzy set theory, originally introduced by Lofti A. Zadeh in 1965 <cit.>, is an extension of classical set theory where to each element x from a set A is associated with a value between 0 and 1, representing its degree of belonging to the set A. One important branch of the fuzzy set theory is fuzzy logic <cit.>. An inference engine operators on rules that are structured in an IF-THEN rules, are presented. Rules are constructed from linguistic variables. A fuzzy approach has the advantage that the rules and linguistic variables are understandable and simplify addition, removal, and modification of the system's knowledge.In the papers <cit.>and <cit.> the authors proposed a multi-camera video-based methods for monitoring human activities, with a particular interest to the problem of fall detection. They use methods and techniques from fuzzy logic and fuzzy sets theory.Thome et al. <cit.> apply a fuzzy logic fusion process that merges featuresof each camera to provide a multi-view pose classifier efficient in unspecified conditions (viewpoints).In <cit.>, they construct a three-dimensional representation of the human from silhouettesacquired from multiple cameras monitoring the same scene, called voxel person. Fuzzy logic is used to determine the membership degree of the person to a pre-determined number of states at each image. A method, based on temporal fuzzy inference curves representing the states of the voxel person, is presented for generating a significantly smaller number of rich linguistic summaries of the humans state over time with the aim to detect of fall detection.Another problem addressed with methods and techniques from theory of fuzzy sets is fire prevention by smoke detection. It has been observed that the bottom part of a smoke region is less mobile than top part over time; this is called swaying motion of smoke. In <cit.> a method to detect early smoke in video is presented, using swaying motion and a diffusion feature.In order to realized data classification based on information fusion,Choquet fuzzy integral (see <cit.>) was adopted to extract dynamic regions from video frames, and then, a swaying identification algorithm based on centroid calculation was used to distinguish candidate smoke region from other dynamic regions. Finally, Truong et al. <cit.> presented a four-stage smoke detection algorithm, where they use a fuzzy c-means algorithm method to cluster candidate smoke regions from the moving regions.§.§ Algorithmic trends in emergency management To analyze the usage of the discussed algorithms across emergency types, we provide a more in depth analysis of the three most studied emergencies: fire, road accidents, and human fall. In Figure <ref>(a-c) we break down the algorithms that have been used for these three types of emergencies. As can be seen in the figures, for the three types of emergencies, the most commonly applied algorithms are the ones that use temporal features. For fire and human fall management background subtraction is the most popular algorithm since most times the most discriminative feature of the objects of interest in this scenarios move on top of a static background. In the case of road accident management it is common to use non-static cameras (such as cameras on board a vehicle) and therefore the background is not static and background subtraction algorithms are not useful. However, temporal features are used to perform optical flow or tracking in order to determine the trajectory of the objects in the road or close by. For fire color features are very distinctive and therefore more researched.§ DISCUSSIONThere are many video monitoring systems whose aim is to monitor various types of emergency scenarios. Since continuous human vigilance of these recordings is infeasible, most of this data is used a posteriori. Although this information is very useful for the later understanding of the event or as an incriminatory proof, many researchers study the possibility of exploiting these huge amounts of data and analysing them in real time with the hope of preventing some of these emergencies or to facilitate a faster or more efficient response. In this survey we reviewed and analyzed the progress of computer vision covering many types of emergencies with the goal of unveiling the overlap across emergency research in computer vision, especially in the hardware and algorithms developed, making it easier to researchers to find relevant works. In order to restrict the scope of this survey we first gave a description of what we consider emergency situations. Taking into account the description provided, we restrict ourselves to emergency scenarios that have been studied in the context of computer vision. Taking all these considerations into account we identified eleven different emergency scenarios, studied in computer vision, which we classified into natural emergencies and emergencies caused by humans.After a thorough analysis of the state-of-the-art algorithms on these topics, we inferred that there are four main axes that have to be taken into account when creating computer vision systems to help in emergency scenarios: the type of emergency that is going to be addressed, the monitoring objective, the hardware and sensors needed, and the algorithms used.When talking about the monitoring objective, we have seen that prevention through computer vision techniques can only be done when there is visual evidence of risk factors such as for fire detection using smoke features, car accidents by detecting pedestrians, driver drowsiness and unusual behaviours, and finally fall prevention falls by detecting slips. The most common objective is emergency detection, which has been studied in all the emergencies studied in this survey. It is also common to use computer vision to detect humans in disaster areas to assist emergency responders. Finally, computer vision has not played a role from the emergency understanding point of view.From all the acquisition methods studied in Section <ref> we have concluded that the most common sensors are monocular RGB or grayscale cameras – mainly for their low cost – and that they are normally situated in fixed locations. However, we see a trend in the use of UAVs which are becoming cheaper and provide great flexibility through their mobility.Finally, as discussed in Section <ref>, it is very difficult to draw conclusions on algorithmic trends for every emergency since some of them have little impact in computer vision. However, there is a clear tendency towards using color and temporal features for fire emergencies, a clear preference for temporal features in road accidents, and tracking algorithms and background subtraction techniques are the most common when detecting falling person events.This up-to-date survey will be important for helping readers obtain a global view. At the same time they can find novel ideas of how computer vision can be helpful to prevent, respond and recover from emergency scenarios, and learn about some required hardware and relevant algorithms that are developed to solve these problems.It is extremely difficult to compare different algorithms proposed in the literature for solving similar problems due to the lack of unified benchmarks. From all the papers studied during this review, over 60% used non-publicly available datasets. In order to facilitate a common benchmark to compare algorithms, in Table <ref> we report some of the most important publicly available datasets used among the emergencies studied in this survey. Note that some pedestrian datasets are shared among emergencies since detecting humans is considered useful in more than one emergency.In general, we have seen a tendency to create more complex systems that combine different algorithms or algorithms of a higher computational complexity which in many cases could not have been feasible with the previous generations of computers. Among these computationally complex algorithms, the emerging techniques that stand out are those based on Convolutional Neural Networks (CNN) and Deep Learning <cit.>. The field of computer vision is experiencing rapid change since the rediscovery of CNNs. These algorithms have already been introduced in some emergency scenarios studied in this survey and we believe that the impact of CNNs will grow over the years in the field of emergency management with computer vision. Moreover, with the decreasing cost of visual acquisition devices such as infrared cameras or stereo cameras, there is a trend of combining different information sources which can help to increase accuracy rates and decrease false alarms. We also believe that, with the help of this survey and further study of the overlap that exists between the algorithms developed to detect and study different types of emergency situations, it will be possible in the future to create system that can cover more than one type of emergency. spmpsci | http://arxiv.org/abs/1708.07455v2 | {
"authors": [
"Laura Lopez-Fuentes",
"Joost van de Weijer",
"Manuel Gonzalez-Hidalgo",
"Harald Skinnemoen",
"Andrew D. Bagdanov"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170824152447",
"title": "Review on Computer Vision Techniques in Emergency Situation"
} |
Department of Mathematics and Applied Physics Institute, Western Kentucky University, Bowling Green, KY 42101 Morphology evolution and coarsening of metal nanoislands is computedwithin the framework of a surface diffusion-type model that includes the effects of the electron energy confinement within the film, the charge spillage at the film/substrate interface,the energy anisotropy of the film surface and the surface stress. The conditions that result in large islands with flat tops, steep edges, and strongly preferred heights are determined. A strong influence of the film height on the coarsening dynamics and final morphologies is found; the conditions leading to interrupted coarsening are highlighted. The dependence of the geometric parameters of the equilibrium island on the film height and on the island initial volume is computed.Keywords: Heteroepitaxial ultra-thin metal films, quantum size effect, surface diffusion, coarsening, solid-on-solid wettingHeight transitions, shape evolution, and coarsening of equilibrating quantum nanoislands Mikhail Khenner December 30, 2023 ======================================================================================== § INTRODUCTIONFormation of a nanoscale islands on the surfaces of an ultrathin metal films was studied extensively, especially since it was discovered that a low temperature (<140K) deposition of a nonwetting Ag film on GaAs(110), followed by prolonged annealing at moderate temperature (300-400K), results in the atomically flat film above a certain critical film height (thickness) <cit.>.Soon after this discovery it was shown <cit.> that the quantum size effect (QSE), which is rooted in the quantization of the electronic energy in the direction across themetal film, is mainly responsible for this new “electronic" or “quantum" growth mode. Subsequent exploration of the quantum growth in heteroepitaxial Ag/Si(111) and Pb/Si(111) systems “revealed the presence of large islands with flat tops, steep edges, and strongly preferred heights". (Quoted from Ref. <cit.>; also see the review <cit.> and references therein.) Resolved Scanning Tunneling Microscopy (STM) images of a 3D island morphologies can be seen, for instance,in Fig. 2 of Ref. <cit.> and in Figures 2 and 17 of Ref. <cit.>. A detailed understanding of the kinetic pathways to form such “magic"island morphologies remains incomplete <cit.>. Different models were proposed <cit.> to help explain the experimental findings in some metal/substrate systems (Pb/Si, Ag/AlPdMn, and Ag/NiAl, respectively).Ref. <cit.> presents the rate equation-type model. The islands are approximated as circular mesas with radii R_i and heights h_i, and the evolution equation for the number of atoms in the i-th island is constructed and solved numerically, assuming the diffusion of adatoms on the substrate. The QSE is incorporated through a term in the island chemical potential, in a fashion very similar to this paper. That model, despite its apparent success at predicting the time-evolution of quantities such as the area fraction of the islands of a particular height, is not capable of tracking neither the surface morphology as a whole, nor the morphology of the individual islands. Besides, since it is limited by its starting assumption of the circular islands, it is unclear how the deviations from this shape would affect the model results. In Ref. <cit.> a step dynamics model is developed of the growth of asingle multilayer island by incorporation of atoms deposited within its capture zone. The island is assumed to have a “wedding cake" morphology, i.e. i circular layers withthe radii r_i(t). The conditions for creation of new top layers are prescribed, and the evolution of the radii are determined by the net attachment fluxes of diffusing adatoms to the edges of the circular layers. The fluxes, in turn, are obtained by solving the appropriate deposition-diffusion equations for adatom density on each terrace. The QSE enters rather indirectly through prescribing the enhanced adsorption energy on the top of the second layer, with the result that the growth of the 3rd layer is enhanced and the growth of the 4th layer is inhibited. The model thus emphasizes the growth of three-layer circular islands, with such height selectivity indeed dominating in the Ag/AlPdMn system. In Ref.<cit.> the Kinetic Monte Carlo simulations are performed of the atomistic lattice-gas model of film growth. The Density Functional Theory (DFT) calculations provide adatom adsorption energies (which reflect QSE promoting a bilayer island formation), interaction energies and key diffusion barriers. Lastly, Monte Carlo simulations of the post-deposition evolution of an ensemble of Pb islands are presented in Ref. <cit.>. Here, the (i,j) island is assumed in the form of a rectangle of height h_i and width j. The growth is driven bythe competition of probabilities π_1 and π_2, where the former probability is one of an atom jump from a wetting layer onto island's sidewall (and thus the corresponding flux increases the island's width), and the Arrhenius probability π_2 is one of an atom jump from a sidewall onto the top of the island (thus the corresponding flux increases the island's height).The cited models of the islands ensembles <cit.> are most closely related to the study in this paper. However, these models do not allow studies of the dynamics of the morphology evolution and the islands coarsening,since the individual island shape is pre-determined and is a constant of the model (i.e., a rectangle). Nonetheless, these models (see also Refs. <cit.>) are quite successful in predicting theequilibrium distributions of the islands heights and widths as functions of the parameters, such as various diffusion and attachment barriers on the island'ssidewalls and the top surface. Other models are even more restrictive by the assumption of a single island of a pre-determined shape <cit.>. In order to enable the studies of the morphology evolution of a metal film surface without the above-mentioned constraints,the author recently introduced a new model <cit.> that may be seen as partially complementary to Ref. <cit.> The new model is a specification to material systems with strong QSE of the classical surface diffusion model that was very successfully used to study the morphology evolution in asemiconductor-on-semiconductor systems <cit.>.Such specification is primarily achieved by using the form of the film surface energy that very roughly mimics its oscillationwith the film height due to QSE in a real metal/substrate system <cit.>. Precisely, the oscillation of the surface energy is taken as simply periodicand with a constant period. This assumption does not hold for many ultrathin metal films (Pb is the most extensively studied case), since in such films the surface energy often displays a complicated multi-layer periodicity (sometimes with a superimposed beating pattern) <cit.>. Thus the model at this stage is generic and phenomenological. Still, the numerical studiesof the (nonlinear) islands dynamics that we present in this paper reveal many minutiae details of the islands formation, equilibration, and coarsening thatare seen in experiment <cit.>. It is important that the new model predicts the final quantized island heights and associated island morphologies based (at large) on the form of the dominant QSEsurface energy that directly follows from the quantum theories <cit.>, with the physical parameters in thisexpression (Eq. (<ref>) below) that have clear physical meaning, and without any fitting or otherwise completely or partially ad-hoc treatment. Some other models that we are aware of <cit.> assume, tacitly or explicitly, that the diffusion along the island sidewalls is negligible in comparison with the peripheral diffusion along the edge of the island.This assumption helps in obtaining (or preserving) certain “magic" island heights that are detected in the experiment. The new model does not employ this assumption or similar assumptions, thusthe “magic" island heights emerge spontaneously through the standard surface diffusion-driven dynamics - given only the initial surface morphology(which often does not manifest islands, rather it is composed of bumps and pits). We caution however, that the numerical values of these heights that we calculate and compute in this paper cannot be taken literally as the approximations to their counterparts in a real system (due to the above-mentioned approximate nature of the QSE oscillation employed by the model and values of most parameters that are at best typical for most metal films).In fact, the “magic" island heights measured in experiment are the integer-multiples of a monolayer (ML) height, while such heights in this paperare estimated as the non-integer numbers of monolayers. Nonetheless, the equilibration and coarsening pathways that we compute and that lead to these heights are generally valid and persist in any quantum system, irrespective of its particular height-selection protocol imposed by a special QSE oscillation and the material parameters.Another remark is that the annealing temperature of <400K is sufficiently low topreserve QSE as the major effect <cit.>, and also sufficiently high to provide the necessary adatom mobility for the morphology evolution on a very large time scale of 24 hours or longer. In fact, in Pb films on Ni(111) the pronounced QSE was observed at even higher temperatures (up to 474K) <cit.>. This justifies the use of a continuum model. A caveat is that it is universally accepted that continuum models are useful for thick films (i.e., with heights h>∼ 50 ML); it is less known and appreciated that such models are very successful in thedescription of the dynamics of very thin (liquid or solid) films (the dynamics in either case is governed by a fourth-order PDE). For example, in Ref. <cit.> the dewetting of a sub-4nm thick liquid film is captured by a thin-film model (which, it is worth mentioning, is designed using the lubrication approximation; this is the analog of thin-film scaling employed in this paper), and in Ref. <cit.> the extended Mullins-type model is used for solid films with heights as small as 1 ML to model cluster growth during dewetting. More generally, in the mature field of solid-state dewetting the continuum models based on the Mullins-type approach deliver results that are in the semi-quantitative agreement with experiment <cit.>; these models typically consider a retracting edge of the film, with one of the boundary conditions h=0 there.Other examples include modeling of films rupture <cit.>. Thus even the infinitesimally small filmthickness is not, per se, a factor that invalidates the continuum model approach.To frame our study in the context of current and future applications, we notice that self-organized metal nanoisland arrays are important for emerging technologies based on nonlinear optics,plasmonics, photovoltaics, and photocatalysis. Such arrays hold large promise in the design of spectrally selective absorbers <cit.>, enhanced fluorescence and infrared spectroscopic devices <cit.>, polarizers and spectral filters <cit.>, and molecular detectors and biosensors <cit.>. They are also of interest for manufacture of the next-generation solar cells <cit.>. Our modeling study is a step toward design of the film growth protocols that would result in regular arrays that are in demand by these applications.§ THE MODEL FOR SURFACE MORPHOLOGY EVOLUTIONIn this section we briefly describe the model that enables the computation of the post-deposition dynamics of metal nanoislands. Also we provide the summary of key results of the linear stability analysis (LSA) for the trivial equilibrium (the planar film surface); for more details, see Ref. <cit.>.The model is based on the Mullins' surface diffusion equation: h_t=√(1+h_x^2+h_y^2) 𝒟∇_s^2 μ,where h(x,y) is the height of the surface ofthe thin film above the substrate, x, y the coordinates in thesubstrate plane, z be the coordinate normal to the substrate plane, ∇_s^2the surface Laplacian, 𝒟=Ω^2 D ν/k T the diffusion constant (Ω is the adatom volume, D the adatom diffusivity,ν the surface density of the adatoms, k T the Boltzmann's factor), and μ=δ/δ h∫γdS the chemical potential; here δ/δ h is the functional derivative, the integral is over the film surface, andγ = γ_0 +γ^(QSE) + γ^(S) +γ^(Anis)is the surface energy. In Eq. (<ref>)γ_0 is the nominal (constant) surface energy, and the contributions labeled “QSE", “S" and “Anis" are due to quantum size effect, the stress and the anisotropy, respectively.Due to crystalline anisotropy, the surface energy depends on the surface orientation n, and QSE and the stress introduce a dependence on h, i.e. γ = γ( n,h). Then the calculation of the functional derivative leads to a four-term expression <cit.> (see also Refs. <cit.>)μ = μ_κ+μ_wet+μ_anis+μ_h.o.t.,withμ_κ = γκ,μ_wet = n_3 ∂_h γ,μ_anis =-2n_3[(h_x h_xx+h_yh_xy)∂_h_xγ+(h_y h_yy+h_xh_xy)∂_h_yγ]-∂_x∂_h_xγ+ ∂_y∂_h_yγ/n_3,μ_h.o.t. = ∂_xx∂_h_xxγ/n_3+∂_xx∂_h_yyγ/n_3+ ∂_yy∂_h_xxγ/n_3+∂_yy∂_h_yyγ/n_3,where κ is the curvature, n_3 the z-component of n, and “wet", “anis", and “h.o.t." are the short-hand notations for wetting, anisotropy and higher order terms, respectively.The non-interacting electron gas model <cit.> and the DFT calculations <cit.> suggest <cit.> the following generic form of γ^(QSE) in Eq. (<ref>) for many metal films, including Ag, Pb,Al,Na, and Be <cit.>: γ^(QSE)(h) = γ_0^(QSE) + g_0 s^2/(h+s)^2cosη h-g_1 s/h+s, g_0, g_1 > 0,where the first (constant) term and the second term are due to quantum confinement of the electrons in a thin film, and the third term is due to the electrons spilling out to the film/substrate interface (the charge spilling effect).Eq. (<ref>) results in the limit of thick film of a quantum model, meaning that N≫ 1, where N is the number of energy bands occupied by the electronsin the z-direction. γ_0^(QSE) = k_F^2 E_F/(80 π) ∼ 400-1000 erg/cm^2 and η = 2k_F=4π/λ_F∼ 20 nm^-1 for Ag, Au, Mg, Al and Pb.(Here E_F is the energy at the bulk Fermi level.) The energies g_0 and g_1 also are proportional to k_F^2 E_F (see Ref. <cit.>).A small wetting length s = 0.4 ∼ 1.2 ML(where the estimate of the ML value is based on the atomic diameter of 0.35 nm for Pb)prevents divergence as h→ 0 <cit.>. The continuous form (<ref>) is the result of the interpolation of the set of data points produced by the quantum models <cit.>, whose z-coordinates are theinteger multiples of the monolayer height.Thus the characteristic amplitudes decays as ∼ 1/h^2 and ∼ 1/h also are the features of the cited quantum models; see Eq. (25) in section 3.3.5,and section 3 in Appendix D of Ref. <cit.>, and also section IIC and Fig. 5 in Ref. <cit.>. The decay of the dimensionless second derivative of γ^(QSE)(h)with increasing h can be seen in Fig. <ref>(d), and the surface energy itself decays with the same rate. We remark here that QSE oscillation is persistent only in relativelythin metal films, as seen in Fig. <ref>(d); Ref. <cit.> cites the experimental evidence of QSE oscillation in films as thick as 30 ML.With the choice of s and γ_0 for the height and energy scales, the dimensionless form of γ^(QSE)(h) reads:γ^(QSE)(H) = R_γ_0+G_0/(1+H)^2cosρ H-G_1/1+H,where R_γ_0=γ_0^(QSE)/γ_0,G_0=g_0/γ_0,G_1=g_1/γ_0, H=h/s and ρ=sη. Notice that 0 < R_γ_0, G_0, G_1 < 1, and ρ∼ 10 for cited values of s and η.We assume that the lattice mismatch between the film and the substrate (and more generally,the interaction energy of a film and substrate atoms)has a negligible effect on the stress at the surface; in other words, a thin metal film on a substrate is considered free-standing for the purpose of calculating the stress at the surface. Thus the only source of stress in the model is theintrinsic surface stress. Clearly, this is justified only for such nearly perfectly lattice-matched quantum growth systems asAg/NiAl(110) <cit.>, Pb/Ni(111) <cit.> and Ag/Fe(100) <cit.> (also see the discussion in Ref. <cit.>). But using large values of the strain parameter (up to the limit of the numerical method) allows the computations to mimic strongly stressed filmsdue to the lattice mismatch, i.e. Ag/GaAs(110) <cit.>, Ag/Si(111) <cit.>, Pb/Si(111) <cit.> and Pb/Cu(111) <cit.>. Following the seminal paper by Hamilton & Wolfer <cit.>, in Ref. <cit.> we derived the closed-form analytical expression for the surface stress energy. The dimensionless form of this expression isγ^(S)(H)=-ΣΓ[Γ + M (H-2)]/[2 Γ + M (H-2)]^2,where Σ=4 Γ P ϵ_*^2/(γ_0 ℓ^2). Also Γ (M) is the sum of the surface (bulk) elastic moduli, P the total number of atoms in the film, ϵ_*the strain, and ℓ the (large) lateral dimension of the film (the same along the x and y axes). The anisotropic part of the surface energy (<ref>) is derived naturally using a thin film scaling <cit.>.Thus the small parameter α=s/ℓ is introduced, and then the dimensionless anisotropic surface energy reads <cit.>γ^(Anis)(H_X, H_Y, ...)= ϵ_γ[a_00 +α^2(a_20 H_X^2 + α a_30H_X^3 + α^2 a_40 H_X^4 + a_02 H_Y^2 +α a_12 H_X H_Y^2 .. + .. α^2 a_22 H_X^2 H_Y^2 +α^2 a_04 H_Y^4)],where ϵ_γ < 0 is the anisotropy strength, a_ij's are the anisotropy coefficients, and X=x/ℓ, Y=y/ℓ are the dimensionless in-plane coordinates. In this paper we limit the discussion of the anisotropy to [111] surface; for that surfacea_00=1/3, a_20=4/3, a_30=-2√(2)/3, a_40=-5/2, a_02=4/3, a_04=-5/2, a_12=2√(2), a_22=-5. <cit.>Finally, the evolution PDE for the time-dependent film surface morphology is derived by first substituting Eqs. (<ref>), (<ref>) and (<ref>)into Eq. (<ref>). Then the total energy γ is substituted into the system (<ref>), (<ref>) - (<ref>), where the adimensionalization is completed usingthe time scaling t=(ℓ^2/D)t̂ and the height and in-plane coordinates scalings as shown above. Lastly, a thin film scaling is introduced and the equation is expanded in the powers of α, with the dominantcontributions retained <cit.>. This results in the final PDE that is shown in the Appendix.Primarily for reference, in Fig. <ref>(a-c) the perturbation growth rate ω from the one-dimensional (1D) linear stability analysis of the PDE is shown for the case of the isotropic surface energy and zero stress (ϵ_γ=Σ=0); in the Figure, k is the perturbation wavenumber and H_0 the height of the planar film immediately after the low-temperature deposition (the base height). (The expressions for ω in 1D and 2D cases are listed in Ref. <cit.>) FromFig. <ref>(a,b)it can be seen that as H_0 increases, the planar film surface z=H_0 is alternately unstable or stable.Such multi-height stability is unusual in a continuum, PDE-based model of a solid film, and it is noted here for the first time. Larger unstable heights are less unstable than smaller unstable heights, since the interval of the unstable wavenumbers decreases as H_0 increases.Manifestly, the stable (unstable) H_0 values belong to H-intervals where Δγ^(QSE)(H)≡ d^2 γ^(QSE)/d H^2 > 0 (d^2 γ^(QSE)/d H^2 < 0), see Fig. <ref>(d) (the thermodynamic stability/instability criterion <cit.>). Thus we emphasize here the agreement between the linear stability analysis and the thermodynamic analysis.§ NONLINEAR DYNAMICS OF NANOISLANDSIn this section we numerically study the 1D island dynamics and their transient and equilibrium morphologies.We compare the dynamics for the situations when only the QSE is operative; the QSE and anisotropy are operative; and all three effects, i.e. the QSE, the anisotropy, and the stress are operative. This way, the understanding of how each physical mechanism contributes to the morphology evolution is obtained. In our model, switching the effect on or off is accomplished simply by taking a non-zero or zero value of the associated governing parameter,i.e. ϵ_γ or Σ. The 1D setup allows to perform a large number of simulations, and as the result we present, for the first time in a modeling paper, the complete rationalization of the height and morphology evolution in a small, generic, and idealized quantum system.This rationalization is expected to fully carry over to a more realistic 2D system. The parameters set that is chosen for computations is typical. Moderate changes of major parameters would yield correspondingly moderate, quantitative-only changes of the results, without affecting the overall conclusions and rationalizations. For instance, slightly changing the amplitudes G_0 and G_1 of the confinement and spilling terms in the QSE surface energy, Eq. (<ref>), would result in the correspondingly moderateshifts of the base heights H_0, as seen in Fig. <ref>, but the computed island morphologies, heights, volumes, and the coarsening scenarios will not be greatly affected.§.§ The effects of the variation of the base height H_0To elucidate the differences in the morphology evolution at varying H_0, here we employ the simple and unique initial condition in all simulations: two Gaussian-shaped islands on the base surface z=H_0, that have the same constant width at half-maximum and the different (but also constant) heights. Specifically, the initial height (measured from the base surface z=H_0) and the width-at-half-maximumof the left island are 5 and 8, respectively; these parameters for the right island are 7 and 8. Thus it can be noticed that the initial island heightsextend across several stable and unstable height intervals as plotted in Figures <ref>(a,b). The horizontal extent of the base surface (along the X-axis) is also kept constant at 20λ_max, where λ_max=2π/k_max is the wavelength of the most unstable perturbation from the LSA; here k_max is the wavenumber at which the growth rate ω attains a maximum value. We used value of k_max from Fig. <ref>(c). The base film height H_0 is varied in a wide range (from 1.5 to 6).For most simulations we chose the linearly stable H_0 values listed in the caption to Fig. <ref>. The boundary conditions for H(x,t) are periodic at X=0 and X=20λ_max.In Fig. <ref> the typical coarsening process can be seen, until the smaller island disappears and the larger one evolves into the equilibrium shape with the flat top surface. The height of this surface is H=6.62, which is one of the stable heights according to Fig. <ref>; the corresponding equilibrium island height is thus H_e=6.62-5.3=1.32, which translates into roughly 1.6 ML (for Pb). Notice that this computation was done without the anisotropy and stress.Only for comparison, dewetting of islands is shown in panel (c) at ρ=0, i.e. in the case without QSE oscillation. Notice that the total initial volume (the one of the bulk film plus the islands) is conserved in Fig. <ref> as well as in other simulations.When the anisotropy and stress are not present, the typical features of the evolution toward a single equilibrium island seen in Fig. <ref> were also observed in the computations withall stable H_0 values larger than 2.8. These computations, in particular, show that “if the lateral size of an unstable island ... is large enough, it will not decay, but rather evolve to a stable height" <cit.>; indeed, in Fig. <ref>(a) the larger island develops a larger base when it evolves. However, when H_0=2.1 was chosen, the flat top surface disappeared from the island equilibrium shape, i.e. the top became rounded, and when H_0=1.5 was chosen the “reverse" coarsening was observed - that is, the small island grew at the expense of the large island, and the final equilibrium shapewas also rounded. Due to the absence of a wetting layer for the chosen parameters (see Fig. <ref>(e)), the atypical reverse coarsening at smaller film heights can be attributed only to a complex nonlinear interaction of the oscillatory QSE terms in the governing PDE (Eq. (<ref>) in Appendix). We confirmed that, as expected, the dynamics without QSE oscillationeither in the absence of a wetting layer (Fig. <ref>(c)), or when it ispresent <cit.> does not show a tendency to reverse coarsening.When the anisotropy and/or stress effects are turned on, the likelihood of an interrupted coarsening or a reversecoarsening increases <cit.>. In particular, when the moderate anisotropy is activated,the reverse coarsening was found not only at H_0=1.5, but also at H_0=2.1; with the stress activated, the reverse coarsening is seen at H_0=2.8, and the interrupted coarsening at H_0=2.1 and 1.5; and with both anisotropy and stress activated, the interrupted coarsening is seen at H_0=1.5, 2.1 and 2.8, and the reverse coarsening is seen atH_0=3.1 and 3.4. Fig. <ref> shows the reverse coarsening sequence for QSE+anisotropy+stress case at H_0=3.1 (the unstable base surface).Movie 1 (see the Supplemental Information) shows the sequence of shape transitions leading to interrupted coarsening for QSE+anisotropy+stress case at H_0=2.8. In the movie, the small island first reaches the stable height H=5.3 (measured from the substrate) and the flat top surface emerges, but because the large island is not stable, the small island increases its height to H=8 at the expense of the lateral shrinking; next, the large island decreases its height until it reaches the stable value H=4.1 (and simultaneously the height of the small island increases); the large island then starts to grow again while the small islandstarts getting smaller, etc. These cycles of the slight height growth or incomplete decay seem to repeat, and the equilibrium state of a single island is not reached in a reasonable computation time. For the cases where the single island equilibrium was achieved, Fig. <ref> shows the equilibrium island height, the width of the top facet, and the width at thebase as a function of H_0. In the QSE case the height of the island rapidly decreases and reaches the constant value 1.3 (∼1.56 ML for Pb) at H_0=4.1; simultaneously, W_top and W_base also reach constant values 125 and 200, respectively. Notice that the flat top does not emerge until H_0=2.8, and then W_top first grows then decays.From H_0=2.8 to H_0=6 the dynamics of W_base is “in phase" with W_top. In the QSE+anisotropy+stress case the equilibrium island does not emerge until H_0=3.1 (for this H_0 the island is shown in Fig. <ref>), and its height monotonically reaches the same constant value as in the QSE case; the flat top emerges simultaneously with the equilibrium island and its width tends to the same constant value as in the QSE case, but slower and in the oscillatory “tail" fashion. A similar trend can be conjectured for W_base. Information on the geometric parameters of the islands is not presented in the published experimental papers, but the just discussed predictions of the model may be useful for future experiments. Notice that in Fig. <ref> the base surface recedes from its initial unstable height at H_0=3.1 to a nearest lower stable height H=2.8;this also happens without the anisotropy and the stress. (Similar decomposition of an initial unstable blanket coverage into a stable blanket coverage and stable islands on top is observed in the experiment <cit.>.) Obviously, in the computation such surface retraction does not prevent the islands from evolving and coarsening. This is attributed to either of the two factors.First, if a small deformation at the base of the island emerges on a horizontal length scale which ismuch smaller than λ_max, then such short-wavelength perturbation quickly decays since ω<0 for small perturbation wavelengths, see Fig. <ref>(c). Second, if such deformation is of an unstable wavelength, then it stops growing (ω becomes zero) as soon as its amplitude reachesthe stable film height immediately below H_0. In this case the film surface will fairly quickly recede to this new stable height, as seen in Fig. <ref>. Since deformations of a second kind are characteristic of a nonlinear island dynamics even when H_0 is stable, the surface retraction was noted for several suchH_0 values (H_0 = 1.5, 2.1, 2.8 and 3.4). Movie 2 (see the Supplemental Information) shows the details of the retraction process for H_0=3.4; in a process similarto the coarsening sequence shown in Fig. <ref>, the smaller left island in the movie later is absorbed into the right island.At larger H_0 the retraction is arrested, which seems to indicate that the QSE again plays a role. The small satellite islands that are seen on the receded surface (Fig. <ref>) on a path to equilibrium are another hallmark of the two-islands dynamics on initial unstable base surfaces.In fact, in the beginning of the coarsening the size of these islands may be only slightly less than the size of either left or right island. For comparison, such satellite islands do not emerge on the receding stable base surface, as seen in the Movie 2. §.§ The effects of the variation of the island volume In this section we study the height transitions of the equilibrium island as a function of the increasing island volume, at fixed H_0 (and only for QSE case).Real islands have various volumes at different stages on a path to equilibrium; to our knowledge, the island height transitions due to volume variations were not systematically reported in the experimental publications. We chose H_0=5.3 to eliminate the base surface retraction; for the example see Fig. <ref>. The initial condition is again two Gaussian-shaped islands, which mimics the real-life situation - since in the experiment there is always multiple islands or none.The initial height (measured from the base surface z=5.3) and width-at-half-maximumof the left island are 1.3 and 8, respectively, and the height of the right island is 3.2; its width at half maximum is increased from 8 to produce the increasing volume V.The smaller left island disappears fast, and the equilibrium height of the right island is plotted vs. V.From Fig. <ref> it can be concluded that the new equilibrium island height, if measured from the substrate, is selected from the set of stable heights between 6 and 7.8 (see the caption to Fig. <ref>) every time V reaches a new critical value; notice that the transitions are sharp. It takes 83% volume growth to increase H_e from 6 to 6.6 (or from 0.7 to 1.3, as in the Figure); 71% volume growth to increase H_e from 6.6 to 7.2 (from 1.3 to 1.9); and 51% volume growth to increase H_e from 7.2 to 7.8 (from 1.9 to 2.5).The QSE again exerts a strong influence through value of the film height H_0. Decreasing H_0 from 5.3 to 1.5 while keeping all other parameters constant(including the initial heights of both islands) resulted in thedata points arrangement that is visually similar to Fig. <ref>, however, it takes only 60% volume growth to increase H_e from 2.8 to 3.4 (if measured from the substrate; or from 1.3 to 1.9, if measured from the base surface); 26% volume growth to increase H_e from 3.4 to 4.1 (from 1.9 to 2.6); and 36% volume growth to increase H_e from 4.1 to 4.7 (from 2.6 to 3.2). The slope of the linear fit to the data points for this case is 0.012, and thus the conclusion is that at smaller H_0 the height of the equilibrium island increases faster with the increase of the island volume. §.§ Coarsening of a pitted surface into the “magic" islands In this section we discuss the results of the computations of the surface coarsening, assuming that the starting base surface is “pockmarked" by irregularly shaped pits. The depth of all pits is chosen randomly, such that in a given run some pits initially may be shallow and other pits may be deep. Also, the pits number density along the base surface is a constant for all runs, but their positioning and shapes are random. This is a rather common initial condition in the experiments <cit.>. Our primary goal is to understand qualitatively the major factors that govern the outcomes of the coarsening process. We will show that there is two such factors:first, the stability or instability of the base surface, and second, the presence of deep pits.In Fig. <ref> we collected the results of four representative computations. In the panels (a)-(c) the base surface is stable at H_0=5.3,and a few deepest pits are of increasing depth from (a) to (c).In the panel (a), where the pits are shallow, the coarsening was computed until two large flat-top islands were formed. They “sit" on the stable surface H=4.1 and their top surface is at the stable level H=5.4. Thus the islands extend vertically across one stable domain with the center at H=4.7.When the depth of the deepest pit increased slightly from d=2.3 (panel (a)) to d=3.3 in panel (b), the coarsening rate sharply increased - notice that the single flat-top island formed by t=6× 10^7.The height of this island also reacted to the increase of the pit depth by stabilizing its base at a lower stable height H=2.8, making the island extend across three stable domains withthe centers at H=3.4, 4.1, 4.7. When the pits are even deeper (d=4.3) as in panel (c), the outcome of the coarsening changes dramatically. The islands now form on a stable surface whose position roughly coincides with the tip of the deepest pit, and most important, the flat-top “magic" islands do not emerge. Instead a fairly narrow and very high islands with a curved top and sidewalls emerge and coarsen.In panel (d) the base surface is at the unstable height H_0=3.1, and the pits are shallow. The flat-top islands are formed similar to the scenarios seen in panels (a) and (b). As in panel (a), two flat-top islandsextend vertically across one stable domain. Characteristically, the island's top surface is at the nearest (to H_0=3.1) larger stable height H=3.5 (despite that another, smallerstable height H=2.8 is also available; thus it is easier to increase the island height rather than to decrease it and increase the width). Lastly, the anisotropy and the stress accelerate coarsening; indeed, the single large island shown by the dashed line in panel (a) has been formed by t_f=4.2× 10^8, vs. t_f=4.9× 10^9 for the QSE case. However, these factors do not affect the height of the island and the vertical positioning of its base and top surfaces.In Fig. <ref> the quantitative measures of coarsening in Figs. <ref>(a-c) are provided. It can be seen that in these three casesthe root-mean-square roughness first decreases fast (linearly in time) for a very short period, resulting in the formation of pre-“magic" islands with the curved sidewalls and the top surface, as shown by the green line in Figs. <ref>(a,b). For the cases of initially shallow and moderately deep pits (Figs. <ref>(a,b)) this is followed by the power laws dynamics until roughness reaches a constant value, which coincides with theemergence of four flat-top “magic" islands (Fig. <ref>(c)). Further coarsening of these islands into the single equilibrium islandshown in Figs. <ref>(a,b) proceeds with islands not changing their heights, thus the roughness is unchanged.This scenario is contrasted to the case of initially deep pits in Fig. <ref>(c). In that case, after the initial decrease, the roughness grows linearly in time (Fig. <ref>(b)). This growth is capped only by the termination of the coarsening, i.e. when a single, large “non-magic" island emerges, as seen in Fig. <ref>(c).The results in this section point out again that the pits in the film surface serve as the initiation sites for the morphological transition, stronglymediated by the QSE, toward the flat-top islands <cit.>. Also we remark that these results and the results in Sec. <ref> re-emphasize and cast some new light on the importance of the QSE strength (through the film height) for the formation and dynamicsof the metal nanoislands <cit.>. § CONCLUSIONS A model that incorporates the QSE, anisotropy, and surfaces stress has been used to compute evolution to equilibrium ofthe pre-existing nanoscale metal islands and the coarsening of a severely rough (pitted) surface. Some of the most common qualitativefeatures of the morphological evolution of metal nanoislandsreported in the experiments on various metal/substrate systems are reproduced. The kinetic pathways in a formation and coarsening of the quantum islands are clarified.We showed that the nonlinear dynamics of the morphology evolution is complicated and that it depends on factors such asthe initial (base) film height H_0 after the deposition, the levels of anisotropy and stress and, for a pitted surface, whether the pits are deep. The dependence on the base height is not only on a height value itself, which sets the strength of the QSE,but also whether H_0 belongs to oneof the stable or unstable height intervals imposed by the QSE. In most situations the surface evolves into the “magic" islands that have the shape of a truncated pyramidwith a wide, flat top surface and a well-defined height, which is a difference of two stable base heights (again as imposed by the QSE). Interruptedcoarsening is often found at a small base height, and/or active anisotropy and strong stress.The dependence of the geometric parameters of the equilibrium island on the base height and on the island initial volume was computed. The equilibrium island height, width of the base, and width of the flat top surface reach constant values as the base height increases. The gradual increase of the island initial volume at a constant base height leads to the abrupt increases of the equilibrium island height. It should be again noted that we employed approximations or assumptions in order to obtain a concise and numerically tractable model of the complicated multi-physics phenomena. They are: a small aspect ratio of the film (height/lateral dimension), small lattice mismatch, and 1D modeling. The first assumption holds with overwhelming accuracy in the thin film systems such as discussed in the paper, since the film height and the lateral dimension typically are of the order of 10 ML and 10-100 microns, respectively. Our mathematical treatment merely formalizes this vast difference of the spatial scales by expanding the extended Mullins equation (<ref>) (after the adimensionalization) in powers of a small parameter, which is the above ratio, and retaining the dominant contributions. The result is the PDE that contains all physics of the original equation, but unlike the original equation, it allows not only the direct computation, but also the stability analysis of the equilibria. As we show in the paper, that analysis, when it goes hand-in-hand with computation, is instrumental and indispensablefor understanding the formation of “magic" islands (see the discussions of Figures <ref>, <ref> and <ref>). <cit.>In regard to strain/stress effects, we chose in this paper to limit the consideration to the intrinsic surface stress. This is because this effect is important and was cited as the primary source of stress in many systems of interest <cit.> and also, because a theoretical treatment so far has been insufficient. Our model of the surface stress paves way to continuum models of surface stress-dominated morphology evolution of thin films (either metal or semiconductor). In a comprehensive model that is presently under development, this treatment is augmented by the traditional model of the lattice mismatch stress <cit.>, thus both sources ofstress are considered and their combined effects on nanoscale metal islands can be computed. Since QSE strongly dominates and defines (at large) how such islands form and evolve, the augmentation by the lattice mismatch stress is expected to only provide a numerical correction and not affect the qualitative features of the dynamics of islands' morphology.Lastly, 1D modeling is useful to distill the key features of the 3D model, extract the dynamics that is at least semi-quantitatively correct, and perform a parametric analysis in order to explore a range of experimental conditions, such as the varying levels of stress and anisotropy (as is done in this paper). Fine features of the islands' shape and the corrections to coarsening rates will be in future computed with the help of a 3D model, which is unavoidably computationally intensive. Thus only a single set of the material parameters for a particular metal/substrate system can be employed by such model.By accounting for the lattice mismatch stress, growth, and the complicated, non-trivial QSE oscillation (which is unique to a chosen metal/substrate combination), the comprehensive model will allow computation of the morphology evolution and coarsening of several tens of 3D metal nanoislands.We expect that this will further advance our understanding of these technologically important systems and help design the film growth protocols that would result in better height,shape and position selectivity of the nanoislands.§ APPENDIX The 1D version of the governing PDE readsH_t=B[G_1/1+H(H_XXXX-2/(1+H)^2H_XX-2/1+HH_XH_XXX -((1+H) H_XX-2H_X^2)H_XX/(1+H)^2+6 /(1+H)^3H_X^2). - (1+R_γ_0)H_XXXX+G_0/(1+H)^2(6-ρ ^2(1+H)^2/(1+H)^2H_XX-H_XXXX+4 /1+HH_XH_XXX.- .(6 H_X^2 -2(1+H)H_XX -(1+H)^2H_X^2 ρ ^2 )H_XX/(1+H)^2+( 6(1+H)^2ρ ^2-24 )H_X^2/(1+H)^3) cosρH+ G_0ρ/(1+H)^2(4/1+HH_XX+2 H_XH_XXX-(4H_X^2 -(1+H) H_XX)H_XX/1+H+((1+H)^2ρ ^2-18 )H_X^2/(1+H)^2) sinρH- ϵ _γ(a_00+2a_20)H_XXXX-4ϵ _γa_20(2H_XXX^3+6H_XH_XXH_XXX+H_X^2H_XXXX)+ ΣΓ(Γ+(H-2) M)/(2 Γ+(H-2) M)^2( H_XXXX -2M/2 Γ+(H-2) MH_XX^2-4M /2Γ+(H-2) M H_XH_XXX.- .6M^2 /(2 Γ+(H-2) M)^2H_X^2H_XX)+3 ΣΓ(H-2)M^3/(2 Γ+(H-2) M)^4(4M/2 Γ+(H-2) MH_X^2- H_XX)+ ΣΓ M/(2 Γ+(H-2) M)^2(H_XX^2+2H_XH_XXX-6 M^2/(2 Γ+(H-2) M)^2H_X^2 +M/2 Γ+(H-2) MH_XX.- ..4M/2Γ+(H-2) MH_X^2H_XX)].Here B=Ω^2 νγ_0/ℓ^2 α kT is Mullins number; in the computations B is set equal to one. Also, t stands for the dimensionless time (that is, the hat over t̂ is dropped). The terms proportional to cosρ H and sinρ H originate in the oscillatory contribution to the surface energy due to QSE (Eq. <ref>). Also we remark that in the typical case Γ > M there is no singularity of the stress terms, as 2Γ + (H-2)M>0 at all H≥ 0.200Smith Smith A R, Chao K J, Qiu N and Shih C K 1996 Formation of atomically flat silver films on GaAs with a "silver mean" quasi periodicity Science 273 226Yu Yu H, Jiang C S, Ebert Ph, Wang X D, White J M, Niu Q, Zhang Z and Shih C K 2002 Quantitative determination of the metastability of flat Ag overlayers on GaAs(110) Phys. Rev. Lett. 88 016102Zhang Zhang Z, Niu Q and Shih C K 1998 “Electronic growth" of metallic overlayers on semiconductor substrates Phys. Rev. Lett. 80 5381Ozer Ozer M M, Wang C Z, Zhang Z and Weitering H H 2009Quantum size effects in the growth, coarsening, and properties of ultra-thin metal films and related nanostructures J. Low Temp. Phys. 157 221-251Han1 Han Y, Unal B, Jing D, Thiel P A, Evans J W and Liu D J 2010Nanoscale “quantum" islands on metal substrates: Microscopy studies and electronic structure analyzes Materials 3 3965UFTE Unal B, Fournee V, Thiel P A and Evans J W 2009 Structure and Growth of Height-Selected Ag Islands on Fivefold i-AlPdMn Quasicrystalline Surfaces: STM Analysis and Step Dynamics Modeling Phys. Rev. Lett 102 196103HUQJJLTE Han Y, Unal B, Qin F, Jing D, Jenks C J, Liu D-J, Thiel P A and Evans J W 2008 Kinetics of Facile Bilayer Island Formation at Low Temperature: Ag/NiAl(110) Phys. Rev. Lett. 100 116105YJEWWNZS Yu H, Jiang C S, Ebert Ph, Wang X D, White J M, Niu Q, Zhang Z and Shih C K 2002 Quantitative Determination of the Metastability of Flat Ag Overlayers on GaAs(110)Phys. Rev. Lett. 88 016102 Li Li M, Evans J W, Wang C Z, Hupalo M, Tringides M C, Chan T L and Ho K M 2007 Strongly-driven coarsening of height-selected Pb islands on Si(111) Surface Science 601 L140-L144Kuntova Kuntova Z, Chvoj Z, Tringides M C and Yakes M (2008) Monte Carlo simulations of growth modes of Pb nanoislands on Si(111) surface Eur. Phys. J. B 64 61–66Kuntova1 Kuntova Z, Hupalo M, Chvoj Z and Tringides M C (2007) Bilayer-ring second-layer nucleation morphology in Pb/Si(111)-7x7 Phys. Rev. B 75 205436Kuntova2 Kuntova Z, Tringides M C and Chvoj Z (2008) Height-dependent barriers and nucleation in quantum size effect growth Phys. Rev. B 78 155431JEM_Khenner Khenner M 2017 Interplay of quantum size effect, anisotropy and surface stress shapes the instability of thin metal films J. Eng. Math. 104 77-92 SDV Spencer B J, Davis S H and Voorhees P W 1993 Morphological instability in epitaxially strained dislocation-free solid films: Nonlinear evolution Phys. Rev. B 47 9760-77 Chiu Chiu C H 2004 Stable and uniform arrays of self-assembled nanocrystalline islands Phys. Rev. B69 165413GolovinPRB2004 Golovin A A, Levine M S, Savina T V and Davis S H 2004 Faceting instability in the presence of wetting interactions: A mechanism for the formation of quantum dots Phys. Rev. B 70 235342Korzec Korzec M D, Münch A and Wagner B 2012 Anisotropic surface energy formulations and their effect on stability of a growing thin film Interfaces and Free Boundaries 14 545-567Korzec1 Korzec M D andEvans P L 2010 From bell shapes to pyramids: A reduced continuum model for self-assembled quantum dot growth Physica D 239 465-474KTL Khenner M, Tekalign W T and Levine M 2011Stability of a strongly anisotropic thin epitaxial film in a wetting interaction with elastic substrate Eur. Phys. Lett. 93 26001LLRV Li B, Lowengrub J, Ratz A andVoigt A 2009 Geometric evolution laws for thin crystalline films: modeling and numerics Commun. Comput. Phys. 6433-482Aqua Aqua J-N and Frisch T 2010 Influence of surface energy anisotropy on the dynamics of quantum dot growth Phys. Rev. B 82 085322Hirayama Hirayama H 2009 Growth of atomically flat ultra-thin Ag films on Si surfaces Surf. Sci. 603 1492-1497Han Han Y and Liu D J 2009Quantum size effects in metal nanofilms: Comparison of an electron-gas model and density functional theory calculations Phys. Rev. B 80 155404Wu Wu B and Zhang Z 2008 Stability of metallic thin films studied with a free electron model Phys. Rev. B 77 035410OPL Pierre-Louis O 2016 Solid-state wetting at the nanoscale Progress in Crystal Growth and Characterization of Materials 62 177-202CHPPM Calleja F, Hinarejos J J, Passeggi M C G Jr, Vazquez de Parga A L and Miranda R 2007 Thermal stability of atomically flat metal nanofilms on metallic substrates Appl. Surf. Sci. 254 12-15Golubovic Constantinescu A, Golubovic L and Levandovsky A 2013 Beyond the Young-Laplace model for cluster growth during dewettingof thin films: Effective coarsening exponents and the role of long range dewetting interactions Phys. Rev. E 88 032113BGZP Bollmann T R J, van Gastel R, Zandvliet H J W and Poelsema B 2011 Anomalous decay of electronically stabilized lead mesas on Ni(111) Phys. Rev. Lett. 107 136103Becker Becker J, Grun G, Seemann R, Mantz H, Jacobs K, Mecke K R and Blossey R 2003 Complex dewetting scenarios captured by thin-film models Nature Materials 2 59-63Cheynis Cheynis F, Leroy F and Muller P2013 Dynamics and instability of solid-state dewettingC. R. Physique 14 578-589SrolovitzWang Y, Jiang W, Bao W Z and Srolovitz D J (2015) Sharp interface model for solid-state dewetting problems with weakly anisotropic surface energies Phys. Rev. B 91 045303Vaynblat Vaynblat D, Lister J R and Witelski T P 2001 Rupture of thin viscous films by van der Waals forces: Evolution and self-similarityPhys. Fluids 13 1130-1140 Held Held M, Stenzel O, Wilbrandt S, Kaiser N and Tunnermann A 2012 Manufacture and characterization of optical coatings with incorporated copper island films Appl. Optics 51 4436-4447Aslan Aslan K, Malyn S N and Geddes C D 2008 Angular-dependent metal-enhanced fluorescence from silver island films Chem. Phys. Lett. 453 222-228Jensen Jensen T R, van Duyne R P, Johnson S A and Maroni V A 2000 Surface-enhanced infrared spectroscopy: A comparison of metal island films with discrete and nondiscrete surface plasmons Appl. Spectroscopy 54 371-377Heger Heger P, Stenzel O and Kaiser N 2004 Metal island films for optics Proc. SPIE 5250 21Anker Anker J N, Hall W P, Lyandres O, Shah N C, Zhao J and Van Duyne R P 2008 Biosensing with plasmonic nanosensorsNature Materials 7 442–453Liao Liao H , Nehl C L, Hafner J H 2006 Biomedical applications of plasmon resonant metal nanoparticles Nanomedicine 1 201-208SantbergenSantbergen R, Temple T L, Liang R, Smets A H M, van Swaaij R A C M M and Zeman M 2012 Application of plasmonic silver island films in thin-film silicon solar cells J. Optics 14 024010MQA Man K L, Qiu Z Q and Altman M S 2010 Quantum size effect driven thermal decomposition of Ag films on Fe(100) in the presence of pinhole-growth morphological defects Phys. Rev. B 81 045426HW Hamilton J C and Wolfer W G 2009 Theories of surface elasticity for nanoscale objects Surf. Sci. 603 1284-1291WettingL A wetting layer can be introduced by increasing G_0 while keeping G_1=0.1; say, G_0=1/15 results in a wetting layer of the dimensionless height one. This follows from the expression for the critical film height <cit.>,H_0c=3G_0/G_1-1, above which all heights are linearly unstable and below which all heights are stable. At the chosen value G_0=0.01 for Fig. <ref> and other figures in the paper, H_0c<0 and thus all film heights are unstable - as shown in Fig. <ref>(e). FTN This approach, which bears the names longwave expansion, thin film scaling, or lubrication approximation has been fruitfully used in the theoretical/modeling thin film community for decades; see Refs. <cit.> for some of the first applications to solid and liquid films, respectively.WD Williams M B and Davis S H 1982 Nonlinear theory of film rupture J. Colloid and Interface Sci. 90 220-228 | http://arxiv.org/abs/1708.07418v1 | {
"authors": [
"Mikhail Khenner"
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"cond-mat.mes-hall",
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"primary_category": "cond-mat.mtrl-sci",
"published": "20170824135355",
"title": "Height transitions, shape evolution, and coarsening of equilibrating quantum nanoislands"
} |
Effective Blog Pages Extractor for Better UGC AccessingKui Zhao1, Yi Wang1, Xia Hu2,Can Wang11 Zhejiang Provincial Key Laboratory of Service RobotCollege of Computer Science and TechnologyZhejiang UniversityHangzhou, China, 310027{zhaokui,wangyi,wcan}@zju.edu.cn2Hangzhou S&T Information Research InstituteHangzhou, China, [email protected] 30, 2023 =========================================================================================================================================================================================================================================================================================================================Blog is becoming an increasingly popular media for information publishing. Besides the main content, most of blog pages nowadays also contain noisy information such as advertisements etc. Removing these unrelated elements can improves user experience, but also can better adapt the content to various devices such as mobile phones. Though template-based extractors arehighly accurate, they may incur expensive cost in that a large number of template need to be developed and they will fail once the template is updated. To address these issues, we present a novel template-independent content extractor for blog pages. First, we convert a blog page into a DOM-Tree, where all elements including the title and body blocks in a page correspond to subtrees. Then we construct subtree candidate set for the title and the body blocks respectively, and extract both spatial and content features for elements contained in the subtree. SVM classifiers for the title and the body blocks are trained using these features. Finally, the classifiers are used to extract the main content from blog pages. We test our extractor on 2,250 blog pages crawled from nine blog sites with obviously different styles and templates. Experimental results verify the effectiveness of our extractor. UGC; Blog; Content Extraction; DOM-Tree; § INTRODUCTIONAs a platform for people to publish contents they are interested in,blog is becoming an important part of online culture.In fact,the number of blog users has been increasingrapidly. Taking China as an example,according to the survey published by CNNIC,the number of blog users in China is 109 million in 2014 with a 21 million increase from the user number in 2013.[http://www.cnnic.net.cn/hlwfzyj/hlwxzbg/, in Chinese.]However, as shown by the example in Fig.<ref>,most blog pages contain irrelevant contents such as pop-up ads, decorative images, navigation links etc. [http://blogs.msdn.com/b/ie/.]Eliminating these unrelated contents can help users better browse information they are interested in, especially for people with visual impairment, who access web contents via screen readers. Significant improvements in accessibility can be made if the extracted contents instead of the raw HTML is fed to screen readers. What's more, it is much easier to adapt the extracted content to diversified terminals such as mobile phones, PDAs etc. Recent years have witnessed many successful applications using the extracted content to better serve their users, such as KnowLife <cit.>, Odin <cit.> etc.These applications need to extract, store and process the useful information from webpages by removing irrelevant information.As blog is gaining popularity and becoming a major source of User Generated Contents(UGC), extracting contents from blog pages becomes an important research topic. Most of the traditional webpage content extraction methods are template-based, such as <cit.>,<cit.> and many others.Although these template-based methods are highly accurate, they may incur expensive cost in that different extractors need to be developed for webpages of different templates. Moreover, template-based extractors will fail once the template of a site is updated. As blog sites nowadays are using greatly diversified templates, traditional extraction methods are expensive and inconvenient solutions. Moreover, many blog sites provide users with the capability to configure templates,which makes template-based solutions more difficult. To overcome the limitation of template-based extraction, many researchers in recent years attempt to developtemplate-independent extractors for webpages. Wang et.al developed a template-independent extractor for newspages by first identifying the title block and then extracting the body block<cit.>.Sun et. al extracted the webpage content by exploiting the text density in DOM-Tree nodes <cit.>.These extractors either only work on webpages with simple layout<cit.>,or only target at the main body of the webpage content <cit.>.In contrast, effectively extracting contents from blog pages is a challenging issue in that:(1) Blog pages have more complex structures than news pages;(2) Multiple titles and bodies might be contained in a single blog page;(3) Blog posts are usually written in a less rigorous way than news articles,thus the relevance between the title and the body in blog pagesis less certain than that in news pages. In this paper, we propose a novel method to extract titles and bodies from blog pages in a template-independent way.By first parsing a blog page into a DOM-Tree, all elements including titles and bodies in this page are represented by subtrees.Before extracting subtrees corresponding to titles and bodies using supervised learning,two subtree candidate sets are constructed, one for titles and another for bodies.Then the training set is formed by manually labelling the title subtrees and body subtrees, both spatial features and contentfeatures are extracted from the elements contained in subtrees.These features are first used to train the SVM classifier for title subtrees.The title subtrees are further used to construct new features for identifying body subtrees.Encouraging experimental results are achieved on 2,250 blog pages from nine very different blog sites.The rest of the paper is organized as follows:we briefly review related work in section 2and describe our method with details in section 3 and 4.Then we show our experimental setup and results,and discuss the results in section 5. Finally conclusions come out andwe plan the future research in section 6.§ RELATED WORKEarly work in content extraction from HTML documents can be traced back to <cit.> by Rahman et al. In last decade, with the explosive growth of the information on the web, content extraction from webpages has attracted attention from research and industrial communities. Among all methods, template-based ones are dominant because they are effective and easy to implement. The most simplistic ones are handcrafted web scrapers,including NoDoSE <cit.> and XWRAP <cit.>.They extracted contents from webpageswith a common template by looking for special HTML cues using regular expressions. A different category of template-based methods areusing Template Detection,such as Bar-Yossef's <cit.>, Chen's <cit.> andYi's <cit.>,in which webpages with the same template are used to learn the common structure.These methods will first identify the template and then extract contents by removing all parts found in every webpage with the same template. The major problem with template-based extractors is that different extractors must be developed for different templates, making the solution expensive. What's more, once the template is updated, as happens frequently in many web sites, the corresponding extractor will be invalidated.To overcome the limitation of the template-based extractors, many researchers attempted to extract webpage content in a template-independent way in recent years. Cai et al. <cit.> proposed a vision-based page segmentation algorithm named VIPS, extracting the semantic structure of a webpage based on its DOM-Tree and visual presentation.Song et al. <cit.> thought the rule-based method in VIPS was too simple and developed a model that assigns importance scores to blocks in a webpage according to their spatial features and content features.Zheng et al. <cit.> proposed a template-independent news page extraction method based on visual consistency. Wang et al. exploited more features and achieved a higher accuracy in the news title and body extraction <cit.>.In their method, a template-independent extractor was developed for news pages by first identifying the title blockand then extracting the body block. Some more recent methods exploited statistical information on webpages, such asCETR <cit.> and CETD<cit.>.These methods extract contents by identifying the regions with high text density,i.e., regions including many words and few tags are more likely to be contents.Qureshi et al. proposed a hybrid model <cit.> by considering not only statistical information but also formatting characteristics. Although the above methods have achieved some success in extracting webpage content in template-independent ways,developing template-independent extractors for blog pages still poses particular challenges not well addressed in any previouswork. The complicated page structure, existence of multiple titles and bodies, random style of writing etc.make the template-independent extraction for blog pages a difficult task.§ CONTENT EXTRACTION §.§ DOM-Tree We use P to denote the blog page aimed to extract titles and bodies.We use T_DOM to denote the DOM-Tree(an example shown in Fig.<ref>) built from P.We use T to denote the set of subtrees in T_DOM. A subtree s_i∈ T is called a title subtree if s_i but not any subtree of s_i contains the title.Similar definition is made for a body subtree b_i ∈ T. Our purpose is to find out the set of title subtrees S⊂ T and the set of body subtrees B⊂ T.More precisely, given a subtree t_i∈ T, our task is to tell whether 𝐭_i is in the set S or in the set Bor neither.§.§ Extraction ProcessWe parse a blog page into a DOM-Tree and try to find title and body subtrees.In order to improve the efficiency of extraction, the subtree candidate set is generated for title and body subtrees respectively. Then novel features describedin section 4 are constructed and we use them to learn extraction models.As titles in a blog page usually share the same style and bodies are always sandwichedbetween two titles in the vertical direction, title subtrees are extracted first and then used to help construct new features for identifying body subtrees.Finally, we use SVM classifiers <cit.> withGaussian RBF kernel <cit.> to identify title and body subtrees in their subtree candidate sets respectively.The whole process is illustrated in Fig.<ref>.§.§ Candidate SetA DOM-Tree always contains thousands of nodes.To make it more efficient to extract titles and bodies,we construct the candidate set for title and body subtrees by selecting proper subtrees respectively: §.§.§ Title Subtree Candidate. All titles have some textual contents, so we choose subtrees containingat least one textual leaf node as title subtree candidates. §.§.§ Body Subtree Candidate. Unlike titles, bodies may contain only images without any textual contents.So the rule for selecting title subtree candidates cannot be applied to selecting body subtree candidates.By analyzing many blog pages from various sites,we find that root nodes of all body subtrees contain<div> or <span> or <p> tags. So we choose those subtrees as body subtree candidates.§.§ SVM ClassifierSupport Vector Machines(SVM) <cit.> are famous supervised learning models,with associated learning algorithms that analyze data and recognize patterns.We can use it for the purpose of classification and regression analysis.Using kernel tricks, SVM can perform not only linear classification but also non-linear classification. We first represent every node in DOM-Tree by a feature vector and thenbuild two distinguished SVM classifiers with Gaussian RBF kernel <cit.> to extract titles and bodies from blog pages respectively.Features used to learn SVM classifiers will be described with details in the next section.§ FEATURE CONSTRUCTION §.§ Title SubtreePeople can recognize the title at first glance because of its outstanding positionand size. According to this fact, spatial features in titles can be exploited to identify the set S of title subtrees. Moreover, some content features are also very helpful to identify title subtrees.Spatial and content features used in title extraction include: §.§.§ Spatial features.* TRectCenter_N. Let TRect denote the bounding rectangle of a subtree 𝐭_i and TRectCenter denote the displaying center coordinates (x, y) of TRect in the browser.As different blog sites having different layouts,we need to normalize TRectCenter with BrowserCenter,which is the center coordinates (x, y) of the browser: TRectCenter_N=TRectCenter-BrowserCenter/||BrowserCenter|| * FontSize. FontSize is about the font size of words in a subtree t_i.It helps us to distinguish title subtrees from other subtrees in two aspects. One is that titles are usually displaying in font sizes large enough to be differentiated from other contents. The other one is that in most blog pages, all titles are displaying in the same font size.§.§.§ Content features. * TitleLen. TitleLen is the length of texts measured bythe number of words contained in a subtree t_i.This feature helps us to distinguish titles from long passages,based on the fact that the title seldom has too many words in it. * Link. Link is the state about the hyperlink in a subtree 𝐭_i.There are three situations for this feature: nolink, internal and external.Because the title usually contains an internal link to its corresponding post page,so the state of Link is usually internal,sometimes no link and seldom external.* EndWith. EndWith describes whether the text contained in a subtree 𝐭_i ends up with a colon, or a semicolon or a full stop. Our observation is that a title seldom ends with these punctuation marks. §.§ Body SubtreeBodies in a blog page consist of discrete passages separated by titles. The extraction of them poses great challenges in that their contents may varygreatly (e.g. no texts, only pictures) and they are displayed in styles of great diversity.There are several title blocks and body blocks in a single blog page andthey are in alternating arrangement, which can be seen from Fig.<ref> easily.We can use titles having been extracted to identify bodies better.Spatial and content features used in body extraction include:§.§.§ Spatial features.* Widthper. Widthper is the width of a subtree 𝐭_i in percentage of the screen width.This feature helps us to eliminate many unrelated contents such as publishing time, clicked times,which are usually of short width. * RelV. RelV is about relative vertical positions of a subtree t_i compared to extracted title blocks.This feature helps distinguish the body block from other blocks at the top ofthe blog page, such as introduction of the blogger, which are too hard to eliminate if we onlyconsider the content features. * RelH. RelH is about relative horizontal positions of a subtree t_i with respect toextracted title blocks. According to our observation,the title block and the body block are close in the horizontal position.Moreover, the central of the title block always lies between the left and right boundary of the body block,as shown in Fig.<ref>.This feature can help us to eliminate noises located far from the title, such as Ads, related links etc. * BRectCenter_N. Let BRect denote the bounding rectangle of a subtree 𝐭_i andBRectCenter denote the center coordinates (x, y) of BRect displaying in the browser.Like the TRectCenter, BRectCenter also need to be normalized using BrowserCenter, which has been mentioned in the context of title subtree features:BRectCenter_N=BRectCenter-BrowserCenter/||BrowserCenter||§.§.§ Content features. * BodyLen. BodyLen is the number of words contained in a subtree 𝐭_i.This feature is based on the fact thatthe body block often contains more words than other blocks.* MarksNum. MarksNum is thenumber of punctuation marks in a subtree 𝐭_i.Bodies often contain more punctuation marks, so this feature helps us to distinguish the body block from other blocks such as Ads, related links etc.* EndWith. EndWith describes whether the text of a subtree 𝐭_i ends with an ellipsis (...). The bodies listed in the blog page are usually not complete and end with ellipses. §.§ StandardizationWe have used the RBF kernel function for SVM, so all features should be centered around 0 andhave variance in the same order by removing the mean and scaling to unit variance.What's more, it can also reduce the time to find support vectors in SVM <cit.>.§ EXPERIMENTS AND RESULTS §.§ Experimental SetupIn our experiments, we crawled 2,250 blog pages from nine very different blog sites and labeled them manually.More precisely, 250 page are crawled and labeled for each site.The blog sites from which we crawled blog pages are listed in Table <ref>.For all experiments in the following sections, the results are averaged over multiple runs of the experiments.§.§ Single Blog Site ExperimentIn this experiment, we aim to learn the extraction accuracy of our method by training and testing the extraction model on a single blog site.For each blog site, we used 10, 20, 30, 40 blog pages as training set and trained SVM classifiers respectively.Then we tested each SVM classifier on the remaining blog pages of corresponding site.The results are shown in Table <ref>.Note that the accuracy in this table is considering the title and the body together.The results show that our method achieves a high accuracy for extracting titles and bodies in a single blog site.We can also see from the table that the number of training samples has little effect in the extraction accuracy. The reason for this high accuracy using only a small number of training pages is that the layout and structure for pages in a single blog site tend to be consistent. §.§ Multiple Blog Sites ExperimentIn this experiment, we used 360 blog pages from 9 blog sites (40 blog pages from each blog site) to construct the training set.The trained SVM classifiers were then tested on the remaining blog pages of corresponding site.The experimental results are shown in Table <ref>. Though some drop in the extraction accuracy can be observed in this experiment compared with the previous one on a single blog site, the extraction accuracy in general is still satisfactory. This experiment shows our method has the capability to accommodate the difference in page layout and structure, which is an important characteristic for template-independent extractors. From the experimental results in table <ref>, we can also seethat body extractionis more difficult than title extraction. §.§ Extraction Accuracy under Varying Training SetThis experiment was performed with the number of training pagesfrom each site varying from 2 to 20 with a step of 2. Fig.<ref> shows the change of average accuracy for title and body extraction.From the experimental results, we can see the extraction accuracy for both titles and bodies improves when the number of training pages increases. When more than 10 training pages for each blog site are used, the extraction accuracy is stabilized at about 97% and 95% for title and body respectively.§.§ Generalization Capability of the ExtractorIn this experiment, we tested the generalization capability of our method.We randomly chose 100 blog pages from all blog pages as the training set and the rest 2,150 blog pages asthe testing set. The results are shown in Table <ref>.As can be seen, the extractor achieves an average accuracy 88.6% and 86.7% for title and body extraction respectively,which demonstrates that our method has a good generalization capability.§ CONCLUSION AND FUTURE WORKIn this paper, we present a novel and effective method to constructtemplate-independent content extractor for blog pages.This extractor is learned from a set of blog pages automatically.After parsing a blog page into a DOM-Tree,the subtree candidate set is constructed for titles and bodies respectively.Then we train SVM classifiers using both spatial and content features.Moreover, we use the relation between body blocks and title blocks when we deal with body extraction.Finally, experimental results show that our template-independent extractor have a high accuracyand it is very stable when dealing with different blog sites. As future work, we first consider to develop methods to construct better subtree candidate sets.Then we consider to find more efficient features as the input of SVM classifiers,in order to improve the generalization of our method.What's more, we will also try to apply the whole framework to other applications,such as microblog extraction. § ACKNOWLEDGMENTThis work is supported by Zhejiang Provincial Natural Science Foundation of China(Grant no.LZ13F020001),National Science Foundation of China(Grant nos.61173185,61173186). IEEEtran | http://arxiv.org/abs/1708.07935v1 | {
"authors": [
"Kui Zhao",
"Yi Wang",
"Xia Hu",
"Can Wang"
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"categories": [
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"primary_category": "cs.IR",
"published": "20170826055632",
"title": "Effective Blog Pages Extractor for Better UGC Accessing"
} |
x[1] >p#1 shapes,arrows*§0pt*0*0 * §.§0pt*0*0 *§.§.§0pt*0*0 0pt*0*1 | http://arxiv.org/abs/1708.08135v1 | {
"authors": [
"Zixuan Cang",
"Lin Mu",
"Guowei Wei"
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"categories": [
"q-bio.QM",
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"primary_category": "q-bio.QM",
"published": "20170827204114",
"title": "Representability of algebraic topology for biomolecules in machine learning based scoring and virtual screening"
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Planet Four: Terrains - Discovery of Araneiforms Outside of the South Polar Layered Deposits [ December 30, 2023 ============================================================================================ Recently, a physical derivation of the Alday-Gaiotto-Tachikawa correspondence has been put forward. A crucial role is played by the complex Chern-Simons theory arising in the 3d-3d correspondence, whose boundary modes lead to Toda theory on a Riemann surface. We explore several features of this derivation and subsequently argue that it can be extended to a generalization of the AGT correspondence. The latter involves codimension two defects in six dimensions that wrap the Riemann surface. We use a purely geometrical description of these defects and find that the generalized AGT setup can be modeled in a pole region using generalized conifolds. Furthermore, we argue that the ordinary conifold clarifies several features of the derivation of the original AGT correspondence. § INTRODUCTIONThe Alday-Gaiotto-Tachikawa (AGT) correspondence is a remarkable relation between BPS sectors of four-dimensional supersymmetric gauge theories and two-dimensional non-supersymmetric conformal field theories <cit.>. The correspondence states that S^4 partition functions <cit.> of class 𝒮 theories of type A_N-1 <cit.> can be expressed as correlation functions in A_N-1 Toda theory. In particular, the conformal blocks of the Toda theory were shown to be equivalent to the instanton partition functions, computed in the Ω background <cit.>, whereas the three-point functions reproduce the one-loop determinants.The correspondence can arguably be viewed as the culmination of a long effort towards the understanding of the non-perturbative structure of 𝒩=2 Yang-Mills theories <cit.>. In particular, the systematic construction of the class 𝒮 theories provided great insight into the strong coupling limits of these super Yang-Mills theories <cit.>. In this construction, gauge couplings are identified with the complex structure parameters of a Riemann surface and strong-weak dualities are interpreted as a change of `pairs of pants'-decomposition of the Riemann surface. The AGT correspondence then explicitly brings (non-perturbative) four-dimensional Yang-Mills into the realm of two-dimensional CFT. This connection is fruitful since the latter class of theories is in general much better understood. For example, S-duality invariance of the S^4 partition function of 𝒩=2 SU(2) Yang-Mills with N_f=4 corresponds to crossing symmetry of the Liouville four-point function, which was rigorously proven some time ago<cit.>.Increasing the rank, however, the AGT correspondence maps unsolved problems in the gauge theory to other unsolved problems in Toda theory. For example, the computation of partition functions of non-Lagrangian theories is mapped onto the determination of a general three-point function. However, the correspondence allows these problems to be phrased in very distinct settings, leading to new insights and progress <cit.>. Moreover, a complete solution to either problem would kill two birds with one stone.A physical interpretation of the AGT correspondence and its generalizations to higher rank and inclusion of defects seems to rely on a six-dimensional perspective.[ Relevant references will be given in the main body of the paper. ] Indeed, the construction of class 𝒮 theories already hints at this since it assigns a class 𝒮 theory of type A_N-1 to a punctured Riemann surface Σ, by compactifying N M5 branes on Σ <cit.>. It is precisely this Riemann surface on which the Toda theory lives. The number of punctures denotes the number of primary insertions in the Toda correlation function.To be precise, the six-dimensional interpretation of the AGT correspondence is that the supersymmetric partition function of the 6d (2,0) theory 𝒯 of type A_N-1 on S^4×Σ has a four- and two-dimensional incarnation, which are equal. This is illustrated by the following diagram.[ld, black,"Σ→0" above left] Z_𝒯(S^4×Σ)[rd,black,"S^4 →0" above right] Z_𝒮(S^4) ⟺Z_Toda(Σ)The arrows denote a supersymmetric zero-mode reduction to the gauge theory 𝒮 and Toda theory respectively. The equivalence of the lower two partition functions is explained through a topological twist performed on Σ and the Weyl invariance of 𝒯. These features enable us to send the size of either manifold to zero without affecting the value of the partition function, as long as we restrict to the supersymmetric sector. However, the lack of a Lagrangian description of 𝒯 blocks a straightforward implementation of this strategy.Over the past few years, many different approaches have been taken to overcome this difficulty. See for an incomplete list of references <cit.>. A constructive derivation of the correspondence is desirable as it could provide an idea of the scope of AGT-like correspondences between supersymmetric sectors of gauge theories and exactly solvable models. Moreover, due to its six-dimensional origin, such a derivation may also shed light on the worldvolume theory of multiple M5 branes. In this paper we will build on a recent derivation by Córdova and Jafferis <cit.>. Using the relation between the type A_N-1 6d (2,0) theory on a circle and five-dimensional 𝒩=2 SU(N) Yang-Mills theory <cit.>, one performs a Kaluza-Klein reduction on S^4 to obtain A_N-1 Toda theory on a Riemann surface Σ. The Toda fields are understood as boundary fluctuations of SL(N,) Chern-Simons theory on a manifold with asymptotically hyperbolic boundary. This is understood in the following way. Near the boundary, the Chern-Simons connection satisfies the boundary conditions𝒜→dσ/σ H + du/σ T_+ + σ^0.Here, H is an element of the Cartan of sl_N, which sits together with a raising operator T_+ in an sl_2⊂ sl_N subalgebra. In a type IIA frame, these boundary conditions arise from a Nahm pole on the scalars of D4 branes ending on D6 branes <cit.>.Boundary conditions such as (<ref>) are well known to provide a reduction of the sl_N WZW theory induced by Chern-Simons on the boundary of asymptotically hyperbolic space, see for example <cit.>. For the principal sl_2 embedding found in <cit.>, such constraints give Toda theory <cit.>. Consequently, one of the building blocks in establishing the AGT correspondence is obtained.However, the residual symmetries of the constrained WZW theory strongly depend on the embedding of sl_2 into sl_N. For example, for N=3, the reduced boundary theory has 𝒲_3 symmetry if the embedding is the principal one, but it has Polyakov-Bershadsky 𝒲_3^(2) symmetries for the diagonal embedding. More generally, sl_2 embeddings into sl_N are labeled by the integer partitions λ of N. Each choice leads to a reduced boundary theory with different symmetries, which we will denote by 𝒲_λ. These generalized Toda theories play a role in extensions of the AGT correspondence. In <cit.> a relation was proposed between instanton partition functions of 𝒩=2 SU(2) quiver gauge theories with an insertion of a surface operator, which arises from a codimension two defect in the 6d theory, and conformal blocks of sl_2 WZW theories. This was generalized in <cit.> to a relation between SU(N) gauge theories and sl_N WZW theories. These cases dealt with the so-called full surface operators.It was conjectured in <cit.> that the SU(N) instanton partition functions with more general surface operators, labeled by a partition λ of N, would be equivalent to the conformal blocks of theories with 𝒲_λ symmetry. The standard AGT and full surface operator setup are now special cases of this more general setup, corresponding to the partitions N=N and N=1+… +1 respectively. The 𝒲_λ algebra, which is also labeled by a partition of N, is obtained by quantum Drinfeld-Sokolov reduction of sl_N. An explicit check was performed for the Polyakov-Bershadsky algebra 𝒲_3^(2), whose conformal blocks were shown to agree with instanton partition functions in the presence of a simple surface defect, with partition 3=2+1. Further checks of the proposal have appeared in <cit.>.Then, based on mathematical results in instanton moduli spaces, it was realized in <cit.> that the instanton partition function in the presence of a general surface operator on ^2 could be conveniently computed as an ordinary instanton partition function on /_m×, where m corresponds to the maximum number of parts of the partition λ. This technique was further used in <cit.> to compute the S^4 partition functions of 𝒩=2^* SU(N) theories in the presence of a full surface operator, and was shown in the case of SU(2) to reproduce the full sl_2 WZW correlation function. For SU(N) results were obtained as well, but could not be compared due to lack of results on the WZW side.In the following, we will denote the generalized Toda theory resulting from an sl_N reduction with partition λ by Toda_λ. The corresponding generalized AGT correspondence will be referred to as the AGT_λ correspondence.In the present paper, we propose a setup to derive these AGT_λ correspondences using the path laid out by Córdova and Jafferis. This approach is very natural for the problem at hand, since the general quantum Drinfeld-Sokolov reduction of sl_N can be understood from a Chern-Simons perspective as well, by imposing the boundary conditions (<ref>) for a general sl_2⊂ sl_N embedding. Therefore, we wish to show that upon including the appropriate codimension two defects in the six-dimensional setup, one finds these more general boundary conditions. Along the way, we will also be able to clarify some aspects of the analysis in the original paper <cit.>. §.§ Overview and summary of results Since the story is rather intricate and hinges on some important assumptions, we will briefly sketch the main logic and possible pitfalls of our arguments here.The original derivation, which we review in section <ref>, connects the 4d-2d correspondence to the 3d-3d correspondence through a Weyl rescaling. One of the main virtues of this connection is that a full supergravity background was already derived in <cit.> for the 3d-3d correspondence, which can then be put to use in the 4d-2d setting. The three-manifold M_3 on which the resulting Chern-Simons theory lives has nontrivial boundary. With specific boundary conditions, its boundary excitations lead to Toda theory.These boundary conditions manifest themselves in a IIA frame in the form of a Nahm pole on the worldvolume scalars of a D4 brane ending on a D6 brane. The original derivation attributes the Nahm pole to the D6 branes that are also related to a non-zero Chern-Simons level. We point out that the Nahm pole should instead be attributed to a distinct set of branes, which we refer to as D6' branes. The original branes will always be referred to as D6 branes, and will still be related to the Chern-Simons coupling. A crucial element in the original derivation is that the Nahm pole on the scalars transforms under Weyl rescaling to the relevant Drinfeld-Sokolov boundary condition on the Chern-Simons connection. It is argued that this boundary condition is a natural way to combine Nahm data into a flat connection, but the Drinfeld-Sokolov form is not the unique combination that achieves this. However, we have not been able to obtain a better understanding of this point and our construction still relies on this assumption. We expect that carefully examining the Weyl rescaling of the full supergravity background and the corresponding worldvolume supersymmetry equations should allow one to translate the Nahm pole arising in the 4d-2d frame to the Drinfeld-Sokolov boundary condition in the 3d-3d frame. However, a direct implementation of this procedure is ruled out by the lack of a Lagrangian description of multiple M5 branes.The Weyl rescaling of the full supergravity background should also allow one to further explain the claim in <cit.> that the Killing spinors as obtained in <cit.> for the 3d-3d background become the usual 4d Killing spinors of <cit.> after Weyl rescaling and an R-gauge transformation. This argument is not completely satisfactory, since the spinors in the 3d-3d frame are related to a squashed sphere geometry that preserves an SU(2)× U(1) isometry, whereas the Killing spinors in <cit.> are related to a squashed sphere with U(1)× U(1) isometry. We note that this slight discrepancy may in fact be immaterial at the level of partition functions, as was indeed originally found in <cit.> in the context of 3d partition functions and properly understood in <cit.>.The uplift to M-theory of the setup we propose leads to M5 branes on a holomorphic divisor in a generalized conifold, which we discuss in section <ref>. Here, we crucially use the orbifold description of codimension two defects that was advocated in <cit.>.[ The gravity duals of class 𝒮 theories similarly treat such codimension two defects geometrically <cit.>. ] This enables us to treat the defects purely geometrically, so that we do not have to worry about coupling the worldvolume theory to additional degrees of freedom on the defect.We propose to use the conifold geometry as an approximation to the pole region of a full supergravity background that would be needed to account for a defect in a squashed S^4 background.[See also <cit.> for a like-minded approach to the 3d-3d correspondence.] Although this approximation suffices for our purposes, it comes with a particular value of the squashing parameter that leads to a curvature singularity corresponding to the conifold point. In principle, such a singularity could couple to the M5 worldvolume theory. It would therefore be very interesting to obtain a class of supergravity backgrounds for arbitrary parameter values where this singularity can be avoided.The radial slices of the divisor of the generalized conifold have a U(1)× U(1) isometry. Furthermore, it supports two supercharges, in agreement with the four-dimensional Ω background. In the special case where only a single D6' brane is present, corresponding to a trivial surface operator, the isometry enhances to SU(2)× U(1), but still only two supercharges are present. This may seem strange, since one expects a nontrivial surface operator to break part of the supersymmetries. However, placing a surface operator on a fully squashed S^4 does not break any additional isometries, hence the number of preserved supercharges on a fully squashed background is the same with or without a surface operator.An important assumption in our derivation is that the connection to the 3d-3d correspondence still stands. Even though additional defects are present we claim that these only manifest themselves in the boundary conditions of the Chern-Simons theory. Since these defects are located at the asymptotic boundary of M_3, we believe that this claim is justified.Finally, it is known that at k=1 the Hilbert spaces of SL(N,) and SL(N,) Chern-Simons theories agree <cit.>. Therefore, at k=1 the reduction to generalized real Toda theories proceeds as usual. For higher k, one obtains complex Toda_λ theories. In the principal case, the original derivation puts forward a duality between complex Toda and real paraToda with a decoupled coset. It would be interesting to formulate a similar correspondence for complex Toda_λ theories.§ REVIEWIn this section we review the derivation by Córdova and Jafferis of both the 3d-3d and AGT correspondence <cit.>. Subsequently, we give an overview of the relation between Chern-Simons theory and Wess-Zumino-Witten models and their Drinfeld-Sokolov reduction to Toda_λ theories.§.§ Principal Toda theory from six dimensions Consider the 6d (2,0) CFT of type A_N-1 on two geometries which are related by a Weyl transformation <cit.>S_ℓ^4/_k×Σ Weyl S_ℓ^3/_k × M_3.We think of the S^4/_k as the Lens space S^3/_k fibered over an interval, shrinking to zero size at the endpoints. The three-dimensional manifold M_3 is a warped product of a Riemann surface Σ andand ℓ is a squashing parameter which controls the ratio between the Hopf fiber and base radius of the S^3. We will refer to these geometries as the 4d-2d and 3d-3d geometries respectively. See figure <ref> for an illustration.In older work <cit.> it was shown how to couple 5d 𝒩=2 SYM to 5d 𝒩=2 off-shell supergravity. Using the equivalence between the A_N-1 (2,0) theory on a circle and 5d 𝒩=2 SU(N) Yang-Mills theory <cit.>, these general results allow one to preserve four supercharges from the (2,0) theory on the geometry <cit.>S_ℓ^3/_k× M_3⊂ S_ℓ^3/_k× T^*M_3×^2.In the original derivation, the (2,0) theory is reduced on the Hopf fiber. This translates in 5d to a flux for the graviphoton, which is compatible with the 5d supergravity background. For general squashing, it is required to turn on all bosonic fields in the off-shell supergravity multiplet. The resulting background allows for a supersymmetric zero mode reduction on the S^3 which gives rise to SL(N,) Chern-Simons on M_3 with coupling q=k+i√(ℓ^2-1).[ This provides a derivation of the 3d-3d correspondence as formulated in <cit.>.]The complex Chern-Simons coupling consists of an integer k and a continuous parameter ℓ. The former arises from the graviphoton flux that couples to the D4 gauge fields through the 5d Chern-Simons coupling1/8π^2∫_S^2× M_3( C ∧ F ∧ F) k/4π∫_M_3( A∧ dA + 2/3 A ∧ A∧ A ).The continuous parameter ℓ arises from the squashing parameter of the three-sphere.A salient detail of the reduction is that the fermions of the (2,0) theory come to be interpreted as Faddeev-Popov ghosts for the gauge fixing of the non-compact part of the gauge algebrasl(N,)≅ su(N)⊕ isu(N).This provides a concrete explanation for the puzzle that the supersymmetric reduction of 5d supersymmetric Yang-Mills with compact gauge group SU(N) becomes a non-supersymmetric Chern-Simons theory with non-compact gauge group. The ghost and gauge fixing terms in the effective action are subleading in R_S^3, so that in the far IR the gauge fixing is undone and the final result for the effective theory on M_3 is the full SL(N,) Chern-Simons theory. The complex connection𝒜=A+iXis built out of the original Yang-Mills connection together with three of the five worldvolume scalars X_i. The latter combine into a one-form on M_3 due to the topological twist. We denote the other two scalars by Y_a. They correspond to movement in the remaining ^2 directions of (<ref>).We now return to the particular M_3 that arises from the Weyl rescaling of the 4d-2d background. Note that it has a nontrivial boundary consisting of two components Σ∪Σ. So we need to specify boundary conditions, which ultimately lead to non-chiral complex Toda theory on Σ, as we will review in section <ref>.To understand what type of boundary conditions have to be imposed, let us first look at the following table that summarizes the 4d-2d setup:The theory is topologically twisted along Σ. An ^2⊂^3 provides the fibers of its cotangent bundle T^*Σ. In the 3d-3d frame the entire ^3 is used for the topological twist on M_3.The setup is reduced on the Hopf fiber of the S^3/_k⊂ S^4/_k. Equivalently, thinking of the S^4 as two k-centered Taub-NUTs glued along their asymptotic boundary, one reduces on the Taub-NUT circle fiber. It is well known that the M-theory reduction on the circle fiber of a multi-Taub-NUT yields D6 branes at the Taub-NUT centers. The IIA setup[ Note that the resulting three-sphere has curvature singularities at the poles even for k=1.] is then given by Table <ref>.The D6 and D6 branes sit at the north and south pole of the S^3 respectively and the D4 branes end on them. The boundary conditions on the D4 worldvolume fields are then claimed to be similar to those studied in <cit.> for D3 branes ending on D5 branes. That would imply that the D4 gauge field satisfies Dirichlet boundary conditions, while the triplet of scalars X_i satisfy the Nahm pole boundary conditionsX_i→T_i/σ.Here, the T_i constitute an N dimensional representation of su(2) and σ parametrizes the interval over which the S^3/_k is fibered. This can be understood by thinking of the N D4 branes as comprising a charge N monopole on the D6 worldvolume. Indeed, the Nahm pole boundary conditions were originally discovered in a similar context <cit.>. We want to pause here for a moment to note that it is not quite clear why the present setup is related to the analyses of <cit.>. The latter deal with (the T-dual of) a D4-D6 brane system with different codimensions, as described in table <ref>.Here, the ⊢ denotes the fact that the D4 branes end on the D6 branes. The D6 branes in table <ref> are instead similar to the ones studied in <cit.> (see also <cit.>). The corresponding orbifold singularities in the 4d-2d frame reduce to a graviphoton flux in the 3d-3d frame, which is responsible for the Chern-Simons coupling through (<ref>). However, they cannot give rise to a Nahm pole. In section <ref>, we propose an alternative perspective that simultaneously allows for a non-zero Chern-Simons coupling and correct codimensions between D4 and D6 branes for a Nahm pole to arise. Leaving these comments aside for the moment, we must understand precisely what a Nahm pole in the topologically twisted scalars X_i would translate to in the 3d-3d picture. As remarked in section <ref>, transforming the supersymmetry equations that lead to a Nahm pole under the Weyl transformation is a dificult problem. However, we know that the resulting connection 𝒜=A+iX will have to be flat. Furthermore, we expect that the leading behavior of 𝒜 towards the boundary should still be fixed.As we will review in the following section, the relation between Chern-Simons theory and Wess-Zumino-Witten models requires 𝒜 to be chiral on the boundary.[ Nonchiral boundary conditions lead to reduced theories with nonzero chemical potentials <cit.>. It would be interesting to see if they have a role to play in further generalizations of the AGT correspondence. ] If (z,z̅) denote (anti)holomorphic coordinates on Σ, we should demand that 𝒜_z̅ vanishes. Thus, a natural equivalent of the boundary conditions (<ref>) would be𝒜 = _0 dσ/σ + _+dz/σ + σ^0.This is a flat connection. We have defined the sl_2 generators _0 = iT_1 and _± = T_2 ∓ i T_3. They satisfy the standard commutation relations[_a, _b] = (a-b) _a+b.These boundary conditions are precisely the ones that correspond to the reduction of the boundary sl_N algebra to the 𝒲_N algebra. The antiholomorphic connection of the complex Chern-Simons theory behaves in the same way. Adding the contributions from the two components of ∂ M_3 then gives rise to a full (non-chiral) complex Toda theory. It would be interesting to directly verify the transformation of the Nahm pole (<ref>) to the connection boundary condition (<ref>) under the Weyl transformation, as we already pointed out in section <ref>. §.§ Partitions of N and Drinfeld-Sokolov reductionOn a three-dimensional manifold M_3 with boundary, Chern-Simons theory with gauge algebra sl_N induces an sl_N Wess-Zumino-Witten model on ∂ M_3. Boundary conditions on the connection such as those we encountered in (<ref>) translate to constraints in the WZW model. Many of the results we discuss are well known in the literature on WZW models and three-dimensional gravity, see <cit.> for reviews. We simply wish to point out how they can be used in deriving the AGT_λ correspondence.We first recall how one obtains a (Brown-Henneaux) Virasoro algebra in the sl_2 case <cit.>. Here we restrict to the holomorphic sector of the complex Chern-Simons theory. The following holds similarly for the antiholomorphic sector. The variation of the Chern-Simons action isδ S_CS = k/2π∫_M_3[δ𝒜∧ℱ] + k/4π∫_∂ M_3[𝒜∧δ𝒜].The bulk term would lead us to identify the vanishing of the curvature ℱ= d𝒜 + 𝒜∧𝒜 as the equations of motion. However, this is not justified unless the boundary term vanishes. It is most commonly dealt with by requiring one of the boundary components to vanish. Using coordinates (z,z̅) on Σ, one can set𝒜_z̅ = 0 on ∂ M_3.With these boundary conditions, Chern-Simons theory describes a Wess-Zumino-Witten model on ∂ M. In particular, we can use the bulk gauge freedom to fix the radial component to beA_ρ = _0 ∈ sl_2.The most general flat connection satisfying (<ref>) and (<ref>) is then𝒜 = _0 dρ + e^ρ J^+(z)_+ dz + J^0(z)_0 dz + e^-ρ J^-(z)_- dz = e^-ρ_0( d + J^a(z) _a dz ) e^ρ_0.The remaining chiral degrees of freedom J^a(z) are the currents of the chiral Wess-Zumino-Witten model. One can consider the reduction of this model using certain constraints. In particular, using the radial coordinate e^-ρ = 1/σ, the leading order of the transformed Nahm pole boundary conditions (<ref>) can be written as𝒜_0 = e^-ρ_0( d + _+ dz ) e^ρ_0 = _0 dρ + e^ρ_+ dz.Comparing this to the WZW current components in (<ref>) up to leading order in ρ leads to a first class constraint J^+ ≡ 1. We can use the resulting gauge symmetry to fix J^0 ≡ 0, leading to a second class set of constraints. The reduced on-shell phase space consists of𝒜 = e^- ρ_0( d + _+ dz + J^-(z)dz ) e^ρ_0.Its residual symmetries form a Virasoro algebra with current T(z)=J^-(z) and central charge c = 6k.[ From the perspective of three-dimensional Einstein gravity, which can be described by two chiral sl_2 Chern-Simons actions with k=l/4G_N, the reference connection 𝒜_0 is empty AdS_3. The reduced symmetry algebra is a chiral half of the Brown-Henneaux asymptotic Virasoro symmetries. Constraining the leading-order radial falloff corresponds to imposing Dirichlet constraints on the boundary metric of asymptotically AdS_3 geometries. ] On the level of the action, the reduction outlined above produces Liouville theory from the sl_2 WZW model. Now let us return to sl_N, where the corresponding situation has been studied in the context of current algebras <cit.> and higher spin gravity <cit.>. Again, we can impose chiral boundary conditions (<ref>) and gauge fix the radial component as in (<ref>),.𝒜_z̅|_∂ M_3 = 0, 𝒜_ρ = _0 ∈ sl_N.We denote the sl_N generators by T_a. The chiral connection of (<ref>) describing the WZW model becomes𝒜 = e^-ρ_0( d + J^a(z) T_a dz ) e^ρ_0.Its asymptotic behavior is constrained by the leading order behavior of the Weyl transformed Nahm pole,𝒜_0 = e^-ρ_0( d + ^+ dz ) e^ρ_0 = _0 dρ + e^ρ_+ dz.The latter dictates a particular choice of sl_2⊂ sl_N embedding through the generators {_0, _+} appearing in it.[ From the point of view of three-dimensional gravity, this choice of embedding corresponds to choosing an Einstein sector within higher spin gravity. ] It is therefore useful to organize the sl_N basis in multiplets of the sl_2 subalgebra corresponding to _a.In particular, the radial falloff of a current component J^a in the connection is then determined by the weight of the corresponding generator T^a under _0. To be precise, if [_0, T_a] = w_(a) T_a, we see that the T_a component of 𝒜_z is.𝒜_z|_T_a T_a = J^a(z)e^-ρ_0 T_ae^ρ_0 = e^ - w_(a)ρ J^a(z) T_a.Thus, if we fix the non-normalizable part of the connection in terms of 𝒜_0,𝒜 - 𝒜_0 ≡1asρ→∞,we constrain all the current components J^a of w_(a)<0 generators,J^_+≡ 1 and J^T_a≡ 0for all other negative weight generatorsT_a.These constraints generate additional gauge freedom, which can be used to fix all but the highest weight currents of each multiplet to zero. This brings us to what is usually known as highest-weight or Drinfeld-Sokolov gauge, with a single current for each sl_2 multiplet. We shortly review sl_2⊂ sl_N embeddings and the corresponding multiplet structure. The possible decompositions of the sl_N fundamental representation are labeled by partitions λ of N,N_N = ⊕_k=1^N n_kk_2 ⟷λ: N = ∑ n_k .Here, we use k_M to denote a k-dimensional fundamental representation of sl_M. We are interested in the multiplet structure of sl_N under the adjoint action of its sl_2 subalgebra. Through the corresponding decomposition N=∑ n_k in (<ref>), the choice of partition λ determines the number of sl_2 multiplets in the adjoint representation of sl_N. For example, if N=3 we can choose 3=3, 3=2+1 or 3=1+1+1, corresponding to 3_3= 3_2 3_3 ⊗3̅_3 - 1_2 = 5_2 ⊕3_2 , 3_3= 2_2 ⊕1_2 3_3 ⊗3̅_3 - 1_2 = 3_2 ⊕2 2_2 ⊕1_2, 3_3= 1_2 ⊕1_2 ⊕1_2 3_3 ⊗3̅_3 - 1_2 = 8 1_2.These partitions correspond to the principal, diagonal and trivial embedding of sl_2 in sl_3, respectively.The residual symmetries of the sl_N WZW model constrained by (<ref>) for the first two decompositions are the 𝒲_3 algebra <cit.> and 𝒲_3^(2) Polyakov-Bershadsky algebra<cit.>, respectively. In addition to the Virasoro current, the former contains a spin three current, while the latter comes with two spin 3/2 and a spin one current. In the final decomposition, no positive radial weights appear so no constraints are imposed and we are still left with the full affine sl_2 current algebra.More generally, we denote the reduced theory obtained from a general partition λ by Toda_λ. Its corresponding 𝒲_λ algebra contains a current for each sl_2 multiplet appearing in the decomposition of the adjoint of sl_N <cit.>. As we will see in the following section, Toda_λ can be obtained from six dimensions using the generalized conifold. So far, we have been working with complex sl_N models and their reductions, whereas the original AGT correspondence involves a real version of Toda theory. To mediate this, the following relation is suggested for the principal embedding <cit.>complexToda(N,k,s) ⇔realparaToda(N,k,b) + 𝔰𝔲(k)_N/𝔲(1)^k-1.Here, N-1 gives the rank of the Toda theory. The parameters k and s are coupling constants in the complex Toda theory. On the right hand side, k describes the conformal dimension Δ=1-1/k of the parafermions in paraToda. The real Toda coupling isb=√(k-is/k+is). At k=1 the right hand side reduces to real Toda theory <cit.>. Both real and complex generalized Toda theories can be obtained as Drinfeld-Sokolov reductions of SL(N,) or SL(N,) Chern-Simons theories. Geometric quantization of the latter two theories yields identical Hilbert spaces <cit.> for k=1. After reduction, the complex and real Toda theory therefore agree at this particular level. Likewise, using a general sl_2⊂ sl_N embedding, complex and real Toda_λ theory at k=1 are identified.§ ORBIFOLD DEFECTS AND THE GENERALIZED CONIFOLDIn section <ref> we briefly summarize the setup pertaining to the AGT_λ correspondence. This leads us to consider generalized conifolds, denoted by 𝒦^k,m, whose geometry we review in section <ref>. §.§ Codimension two defects and their geometric realizationWe now consider the generalization of AGT that includes surface operators in the gauge theory partition function, which we refer to as AGT_λ. Under the correspondence, the ramified instanton partition functions are mapped to conformal blocks of Toda_λ theories <cit.>. Similarly, it is expected that one-loop determinants in the gauge theory map to three-point functions. This has been checked for the case of a full surface operator <cit.>.A six-dimensional perspective on this correspondence is provided by including codimension two defects in the 6d (2,0) theory. These defects wrap Σ and lie along a two-dimensional surface in the gauge theory. Therefore, they represent a surface defect in the gauge theory, and change the theory on the Riemann surface.There exists a natural class of codimension two defects that are labeled by partitions of N <cit.>, as we will discuss in more detail below. The following table summarizes the M-theory background for the particular instance of AGT_λ that we are interested in.For the moment, we zoom in on the region near the north pole of the S^4, where the geometry locally looks like ^2. In this region, we consider the realization of the defect as a ^2/_m orbifold singularity that spans the 01910 directions of table <ref> <cit.> (see also <cit.>). Note that these codimension two defects are usually described using an additional set of intersecting M5 branes together with an orbifold singularity. We want to emphasize here that we describe the defects using only the orbifold singularity. This interpretation is also supported by mathematical results on the equivalence between ramified instantons and instantons on orbifolds. See <cit.> and references therein.This means that from the gauge theory perspective, i.e. the 0123 directions, that the geometry locally looks like /_m×. A partition λ is then naturally associated to the M5 branesλ: N=n_1+… +n_m.It specifies the number of M5 branes with a particular charge under the orbifold group. Alternatively, when ^2/_m is thought of as a limit of an m-centered Taub-NUT space, it specifies how the M5 branes are distributed among the m centers. The M5 branes wrap the `cigars' in the second relative homology of TN_m. Upon reduction on the Taub-NUT circle fiber, the partition specifies how the N D4 branes are distributed among the m D6 branes. Another generalization of the original AGT correspondence <cit.> that was already covered in the original derivation <cit.> concerns instanton partition functions on ^2/_k, an orbifold singularity that spans the 0123 directions of table <ref> <cit.>. This generalization also naturally arises from a 3d-3d perspective since the S^3/_k in (<ref>) is mapped to S^4/_k after the Weyl transformation. The geometry near the north pole of this quotiented four-sphere is precisely ^2/_k. We now observe that there exists a simple (local) Calabi-Yau threefold that provides a particular realization of two ALE spaces ^2/_k and ^2/_m, intersecting along a two-dimensional subspace. This is the (partially resolved) generalized singular conifold𝒦^k,m: xy=z^k w^m,where we identify x or y with the 01 directions, z with the 23 and w with the 910 directions in the table <ref>. For earlier occurrences of this space, see <cit.>. More recently, it has also appeared in <cit.>. Note that 𝒦^1,m reflects the AGT_λ setup described above. This will therefore be the geometry we focus on in the following, although we will also make a brief comment on general k and m in section <ref>.We will use the generalized conifold as an approximation in the pole region of a squashed S^4 with defect included. This is a considerable simplification to the full supergravity background that would be needed to preserve supersymmetry on the S^4 with defect. One might worry that much information is lost by refraining from a similarly detailed and rigorous analysis as in <cit.>. However, we can still obtain a better understanding of the emergence of a general Nahm pole and the ensuing AGT_λ correspondence. This will be the main result of this paper. §.§ Intersecting D6s from the generalized conifoldIn this section, we provide more detail on the geometry of our proposal. First, we recall the relation between M-theory on a _k ALE space and k D6 branes in IIA. We then introduce the generalized conifold 𝒦^k,m and show how it effectively glues two such ALEs together into a single six-dimensional manifold, leading to two sets of k and m D6 branes upon reduction. Consider the _k ALE space as a surface in ^3 described byxy = z^k.Equivalently, we can think of this four-dimensional space as a ^2/_k orbifold with1∈_k: (x,y)∈^2 → (e^2π i/kx, e^-2π i/ky).The latter makes it clear that the resulting space is singular. In particular, we see that there is a k-fold angular deficit at the origin in the circleC := { ( e^iα x, e^-iα y)| e^iα∈ U(1) }≃ S^1.Reducing M-theory on the _k ALE along C gives rise in IIA to k D6 branes located at the origin and stretched along the transverse directions. The generalized conifold 𝒦^k,m in equation (<ref>) describes a _k and _m ALE for fixed w_0≠0 and z_0≠0, respectively. We will make these considerations more precise in the following.In its most common form, which we will denote by 𝒦^1,1, the standard conifold is a hypersurface in ^4 given byxy = zw.The space 𝒦^1,1 is a cone. We denote its base by T, so that its metric isds^2_𝒦^1,1 = dρ^2 + ρ^2 ds^2_T .It can easily be seen from (<ref>) that T is homeomorphic to S^2× S^3. For 𝒦^1,1 to be Kähler, the base has to have the metric <cit.>[ Note that this reproduces the standard conifold metric upon redefining ψ=(ψ'-_1-_2)/2. ]ds^2_T = 4/9( dψ + cos^2(θ_1/2) d_1 + cos^2(θ_2/2) d_2 )^2+ 1/6[ ( dθ_1^2 + sin^2θ_1 d_1^2 ) + ( dθ_2^2 + sin^2θ_2 d_2^2 ) ].This describes two two-spheres, each with one unit of magnetic charge with respect to the shared Hopf fiber parametrized by ψ. In other words, T can be described by SU(2)× SU(2)/U(1). The quotient by U(1) serves to identify the Hopf fibers of the SU(2)≃ S^3 factors. More details can be found in appendix <ref>.Now we want to choose a circle which leads to intersecting D6 branes upon reduction to IIA. Following the circle (<ref>) in the ALE case, we are led to consider the action(x,y,z,w) ↦ (e^iαx, e^-iαy, z,w), α∈ [0,2π).As can be seen from appendix <ref>, in terms of the Hopf coordinates in (<ref>) describing the bulk of the base of the conifold, the circle is the orbit of(θ_1, θ_2, _1, _2, ψ) ↦ (θ_1, θ_2, _1 + α, _2 +α, ψ - α).Thus the circle consists of equal θ_i orbits on the base two-spheres of T, together with a rotation in the Hopf fiber. At θ_i=0 or π these Hopf coordinates are no longer valid and the circle described by (<ref>) can shrink to a point. In terms of the embedding ^4 coordinates, these loci are hypersurfaces z=0 and w=0, as we can see in (<ref>). This is illustrated in figure <ref>. Reduction of M-theory along the circle generated by (<ref>) leads to two D6-branes stretched along the z and w directions, as illustrated in figure <ref>:D6 from x=y=z=0alongw,D6' from x=y=w=0alongz.Now let us look at the w=0 divisor, which we denote by 𝔇_w. Setting w=0 in the conifold equation (<ref>) implies that x=0 or y=0. These are two branches meeting along the z axis. We will choose the latter one, so that the metric (<ref>) restricts tods^2_𝔇_w = dρ^2 + ρ^2/6(dθ_2^2 + sin^2θ_2 d_2^2 ) + 4ρ^2/9( dψ + cos^2 (θ_2 /2) d_2 )^2.This is a radially fibered S^3 with a particular squashing. Note that it preserves SU(2)× U(1) isometries. At ρ=1 it is parametrized by (see appendix <ref>)x = cosθ_2/2 e^i(ψ +_2), z = sinθ_2/2 e^iψ.In these coordinates, the action (<ref>) whose orbit defines the M-theory circle is(θ_2, _2, ψ) → (θ_2, _2 + α, ψ).We see that the corresponding circles are just the equal θ_2 circles of the second SU(2) factor of the conifold, sitting at the north pole θ_1=0 of the first SU(2) factor.Where does this circle shrink? Again, we have to be careful about the range of our coordinates. At the north pole θ_2=0, the action (<ref>) shifts to the Hopf fiber,θ_2 = 0: (x,z) = (e^iβ,0), β→β + α.That is a circle of finite size unless ρ=0. In contrast, the orbit of (<ref>) shrinks to a point at the south pole θ_2=π for all ρ,θ_2 = π: (x,z) = (0, e^iδ), δ→δ.Therefore, from the perspective of the divisor, the D6' brane stretches along z at x=y=0 and the D6 brane is pointlike at x=y=z=0. To obtain m D6 and D6' branes, the M-theory circle should shrink with an m-fold angular deficit. We can achieve this by quotienting the action (<ref>) by _m⊂ U(1). We denote the resulting generalized conifold by 𝒦^m,m.[ Note that the labels on 𝒦^m,n are unrelated to the labels (p,q) that are sometimes used to describe possible base spaces of the conifold. ] It is given byxy = z^m w^m.We are mainly interested in the w=0 divisor 𝔇_w of this space. From the point of view of the divisor 𝔇_w, an m-fold angular deficit stretches along the z axis. Reducing to IIA leads to m D6' branes that stretch along z and are located at x=w=0.A similar analysis for z=0, w≠0 leads to a m-fold angular defect along w at x=y=z=0. This defect is pointlike in 𝔇_w, intersecting only at x=y=w=z=0. Upon reduction to IIA, it leads to m D6 branes. The conifold point at the origin corresponds to the location where the two orbifold singularities intersect. The generalized conifold 𝒦^k,m is described by equation (<ref>),xy = z^k w^m.It can be obtained by partially resolving the singularity along the w axis of (<ref>). In a IIA frame, such a resolution corresponds to moving out (m-k) D6 branes to infinity along the x or y axis. The resulting geometry is similar to that of 𝒦^m,m, except that it has a k-fold angular deficit intersecting at the origin with an m-fold one. Consequently, reducing to IIA produces k D6 branes and m D6' branes.By resolving all the way to k=1, 𝔇_w is equivalent to the /_m× background studied in <cit.>. In terms of the coordinates (<ref>), it is described byx = cosθ_2/2 e^i(ψ +_2/m), z = sinθ_2/2 e^iψ.The metric (<ref>) then becomesds^2_𝔇_w = dρ^2 + ρ^2/6(dθ_2^2 + 1/m^2sin^2θ_2 d_2^2 ) + 4ρ^2/9( dψ + 1/mcos^2 (θ_2 /2) d_2^2 )^2.In this case, the sphere isometries are broken to U(1)× U(1). The M-theory circle shrinks with a _m angular deficit along the z axis. Note that the three-spheres at fixed radius are in fact also squashed. This leads to an additional `squashing' singularity at the origin, corresponding to the conifold point. Finally, we should comment on how the M5 branes are placed in this geometry. The 𝔇_w divisor has two components ending on the x=y=w=0 defect, and we can place the M5s together along either one. This setup preserves supersymmetry since the divisor is holomorphic. See for instance <cit.> for a similar setup in IIB.Note that the M5 branes are in a sense fractional: a brane along the x axis needs to pair up with a brane along the y axis to be able to move off the defect. Upon reduction, these fractional M5 branes correspond to D4 branes that end on the D6' branes, as we illustrate in figure <ref>.As we will show in the next section, three of the scalars on the D4 branes will obtain a Nahm pole boundary condition dictated by how they are partitioned among the m D6' branes. On the other hand, the flux coming from the k D6 branes gives rise to the Chern-Simons coupling in the 3d-3d frame. Thus, both sets of D6 branes play distinct but crucial roles in our construction.§ TODA(LAMBDA) THEORY FROM GENERALIZED CONIFOLDSIn this section, we outline a derivation of the AGT_λ correspondence in the spirit of Córdova-Jafferis <cit.>. We will see that our proposal also sheds some light on the derivation of the original AGT correspondence. The reason for this is that the surface operator associated with the trivial partition N=N is decoupled from the field theory. In our description, this is reflected by the fact that for m=1 there is no orbifold singularity. Thus, we can view the original AGT correspondence as a special case of the AGT_λ correspondence. This will be discussed first.Moving on to the general AGT_λ correspondence, a crucial role is played by the generalized conifolds 𝒦^1,m. In the presence of our defect, the pole region of a squashed S^4 can be identified with an appropriate divisor in 𝒦^1,m. In this limit, the chirality and amount of the S^4 Killing spinors agree with those of the divisor in 𝒦^1,m, which we take as further evidence for our proposal.It should be noted that we use the (generalized) conifold merely as a technical simplification. We expect a general set of supergravity backgrounds exists that allows for a squashed S^4/_k with defect included. However, since the supersymmetry analysis is particularly easy for the conifold, we specialize to the parameter values it dictates. Several subtleties that arise are expected to be resolved in the general set of supergravity backgrounds. §.§ Compatibility of K1,1 with Córdova-Jafferis As explained in the previous section, the standard conifold 𝒦^1,1 produces a D6 and D6' brane if we reduce to type IIA. In our setup, the D4 branes end properly on the D6' brane. The latter is not present in the original derivation <cit.> where only the D6 brane is considered. The main reason for this discrepancy is that we choose a different circle fiber in (<ref>). In contrast to the Hopf fiber of the three-sphere, which only shrinks at the poles of the S^4, our circle degenerates along the entire z-plane in the pole region.To check that we can still build on the results of <cit.> we note that the D6 brane in <cit.> translates to a flux for the graviphoton field in the 3d-3d frame, which is ultimately responsible for a non-zero Chern-Simons level in three dimensions.[See also <cit.> for related discussions.] This feature is not lost in our construction, since reduction on the conifold still produces a similar D6 brane located at the north pole.Our brane configuration is illustrated in the following table.By the analysis of <cit.>, the D4-D6' system leads to a (principal) Nahm pole boundary condition on three of the D4 scalars, denoted by X_i in section <ref>. This provides a reinterpretation of the results in <cit.>, where it is claimed that the D6 brane is responsible for the Nahm pole.We will now turn to consistency checks of our identification of the conifold as an approximation of the supergravity background relevant to the AGT correspondence. From the Nahm pole onwards, the original analysis of <cit.> then goes through. Namely, after Weyl rescaling to the 3d-3d frame, the Nahm pole translates into the reduction of the WZW model to Toda theory as reviewed in section <ref>. Geometry Recall that the metric on the w=0 divisor is given by (<ref>),ds^2_𝔇_w = dρ^2 + ρ^2/6(dθ_2^2 + sin^2θ_2 d_2^2 ) + 4ρ^2/9( dψ + cos^2 θ_2 /2 d_2 )^2.This is the pole region of the metric of a squashed four-sphere,ds^2 =dσ^2 + (f(σ)^2ℓ^2/4(dθ^2 + sin^2(θ) d^2 ) +f(σ)^2 ( dψ + cos^2(θ/2) d)^2).The original derivation <cit.> of the AGT correspondence considers the class of geometries (<ref>) for general ℓ and any f(σ) that vanishes linearly near σ=0,π. As we can see, the divisor of the conifold imposes the values ℓ=3√(6) and f(σ)=23σ + σ^2.Note that the conifold singularity translates to a squashing singularity of the metric (<ref>) for these particular choices of f(σ) and ℓ. We will not be too concerned about these curvature singularities. As argued in <cit.>, the (2,0) theory cannot couple to curvature scalars of dimension four or higher such as R_μνR^μν. In principle, it could couple to the Ricci scalar, but this can be resolved by an appropriate choice of the function f(σ).For the singular conifold, which dictates f(σ) and ℓ as above, the Ricci scalar singularity is present. However, we expect the general set of supergravity backgrounds to contain solutions that exclude singularities in the Ricci scalar. The curvature singularity should be merely an artifact of the parameter values imposed by the conifold. Supersymmetries Here, we will show that the amount and chirality of supercharges preserved by the M5 brane on the conifold divisor match with what one expects for the 6d (2,0) theory in the AGT setup. Subsequently, we will relate them to the supercharges in the 3d-3d frame.It is well known that an M5 brane wrapped on a holomorphic divisor inside a Calabi-Yau three-fold has at most (0,4) supersymmetry in the remaining two dimensions <cit.>. However, since the latter lie along the Riemann surface Σ, on which the theory is topologically twisted, only two Killing spinors survive.[ This follows from the fact that the four supercharges form two doublets under the SU(2) R-symmetry whose U(1)⊂ SU(2) subgroup is used to perform the twist. ] Since the Killing spinors are chiral from both a six-dimensional and a two-dimensional perspective, they must be chiral in four dimensions as well,ξ^chiral_6d=ξ^chiral_2d⊗ξ^chiral_4d. Now let us turn to the usual AGT setup. A 4d 𝒩=2 theory on a squashed S^4 with U(1)× U(1) isometries has an SU(2)_R doublet of Killing spinors <cit.>,ε^1=(ξ_1,ξ̅_1)=e^1/2i(ϕ_1+ϕ_2)(e^-iθ/2sin(σ2),-e^iθ/2sin(σ2),i e^-iθ/2cos(σ2),-ie^iθ/2cos(σ2)) ε^2=(ξ_2,ξ̅_2)=e^-1/2i(ϕ_1+ϕ_2)(e^-iθ/2sin(σ2),e^iθ/2sin(σ2),-i e^-iθ/2cos(σ2),-ie^iθ/2cos(σ2)).Near the north pole these reduce, up to a local Lorentz and SU(2)_R gauge transformation, to the Ω backgroundξ̅^α̇_A=δ^α̇_A ,ξ_α A=-1/2v_m(σ^m)_αα̇ξ̅^α̇_A.Here, v_m is a Killing vector that generates a linear combination of the U(1)^2 isometry of the Ω background. It descends from the U(1)^2 isometry of the squashed sphere. Since v_m vanishes linearly with σ one sees that the Killing spinors are indeed chiral to zeroth order in σ. Hence, the amount and chirality of the supercharges preserved by the divisor of the conifold are consistent with the ordinary AGT setup.As noted in <cit.>, after Weyl rescaling to the 3d-3d frame, in a suitable R-symmetry gauge, the Killing spinors in (<ref>) become independent of σ. This is required to make contact with the Killing spinors in the 3d-3d correspondence <cit.> which are independent of σ due to the topological twist on M_3.A final subtlety is to be mentioned here. In the above, we have made contact with the Killing spinors corresponding to the 𝒩=2 theory on a squashed S^4 which preserves U(1)× U(1) isometries. However, the derivation of the 3d-3d correspondence in <cit.> makes use of a squashed sphere with SU(2)× U(1) isometries. As was first observed in <cit.> and then properly understood in <cit.>, the three-dimensional supersymmetric partition function is in fact insensitive to these extra symmetries. This should provide a justification for the proposed relation between the partition functions evaluated in the 4d-2d and 3d-3d frame. Twist Another perspective on the equivalence of the preserved supersymmetries in the conifold case and the AGT setup lies in their relation to topological twists. The worldvolume theory on the M5 branes is automatically topologically twisted, since it wraps a Kähler cycle inside a Calabi-Yau threefold <cit.>.[The normal bundle to the divisor is its canonical bundle. Then, the two scalars corresponding to transverse movement inside 𝒦^1,1 become holomorphic two-forms on the divisor <cit.>.] In terms of groups, the R-symmetry is broken by the setup to U(2). The U(1) R-symmetry given by the embeddingU(1)⊂ U(2)⊂ SO(4)⊂ SO(5)is used to twist the U(1)⊂ U(2) holonomy on the divisor (see e.g. appendix A in <cit.>).This feature is also reflected in the standard AGT setup. Indeed, at zeroth order in σ, the Killing spinors (<ref>) precisely reflect the ordinary (Donaldson-Witten) topological twist: the SU(2)_R index is identified with the dotted spinors index. The twist implemented by the conifold is a special version of this twist when the holonomy is reduced to U(2). §.§ K1,m and AGT(lambda) We now want to explain the relevance of the generalized conifold for the AGT_λ correspondence. First, a partition is associated to the divisor that specifies the charges of the fractional M5 branes in the orbifold backgroundλ:N=n_1+…+n_m.After reduction on the circle fiber, this partition encodes the number n_i of D4 branes ending on the i^th D6' brane.This imposes a Nahm pole on the three X^a D4 worldvolume scalars in terms of the sl_2⊂ sl_N embedding associated to λ. In the 3d-3d frame, the D6 brane is solely reflected as a graviphoton flux which leads to a k=1 Chern-Simons level <cit.>.In the M-theory frame, Weyl rescaling from 4d-2d to 3d-3d results in an asymptotically hyperbolic three-manifold M_3 corresponding to the directions 145 in table <ref>. The half-line along the 1 direction is stretched to a line with an asymptotic boundary. Since the D6' branes are located at the edge of this half-line, the Nahm pole they induce becomes a constraint at the asymptotic boundary of M_3 in the Weyl rescaled frame.The bulk of M_3 is therefore unaffected by the presence of the additional D6' branes introduced by the generalized conifold. The analysis of <cit.>, which obtains complex Chern-Simons theory from reduction of the M5 worldvolume theory, should then still hold in the bulk of M_3. We thus claim that only the constraint on the boundary behavior of the Chern-Simons theory is different.To be precise, the partition of the N D4 branes on m D6' branes translates in the 3d-3d frame to a block diagonal form of the connection at the boundary. For example,λ:3 = 2 +1 ⟷𝒜 = [ ∗ ∗; ∗ ∗; ∗ ].The Nahm pole then maps to a constraint in this block diagonal form,𝒜 = dρ + e^ρ^+ dz + ⋯, ^+ = [ 0 1; 0 0; 0 ].As we have outlined in section <ref>, such a constraint precisely reduces a SL(N) WZW model to the Toda theory associated to the partition λ. Recalling the equivalence between complex and real SL(N) Chern-Simons at level k=1 <cit.>, this leads to a derivation of the AGT_λ correspondence. Geometry The w=0 divisor of 𝒦^1,m is equivalent to /_m×, as we showed in section <ref>. The non-trivial Ω background that is manifested by the squashing of a radially fibered three-sphere, just as in the k=m=1 case, is also visible there. The squashed three-sphere in this geometry preserves U(1)× U(1) isometries since the base of the Hopf fibration is now orbifolded. This shows that the divisor in 𝒦^1,m reproduces the setup in which the AGT_λ correspondence was studied <cit.>.[The superconformal index of the 6d (2,0) theory in the presence of these orbifold singularities was computed in <cit.>.] Supersymmetries The generalized conifolds preserve the same amount of supersymmetry as the m=1 conifold. This is particularly clear from the IIA perspective, where instead of a single D6 and D6' brane, we now have one D6 intersecting with m coincident D6' branes.Likewise, the general AGT setup concerns a squashed S^4 with U(1)× U(1) isometries,t^2 + |z|^2/ℓ^2 + |x|^2/ℓ̃^2 =1It is clear that including our defect at x=0 does not break these isometries any further. Therefore, just as in our conifold construction, including such defects in the general AGT setup does not break any additional supersymmetry.Furthermore, the divisor of the generalized conifold is still Kähler, so only chiral supersymmetries survive. This agrees with the chirality of the Killing spinors of an S^4 in the pole region. §.§ General k and m Finally, we comment on a conjecture arising from the general conifold 𝒦^k,m. Here, we expect to obtain SL(N,) Chern-Simons theory at level k together with a boundary condition determined by the partitionλ:N=n_1+…+n_m.According to this partition one should obtain a quantum Drinfeld-Sokolov reduction of complex Toda theory. To arrive at a duality with a real paraToda theory, in the spirit of <cit.>, one could naively ask ifComplexToda_λ(n,k,s) ?⇔realparaToda_λ(n,k,b) +𝔰𝔲(k)_n/𝔲(1)^k-1.However, such a statement requires one to understand how parafermions couple to generalized Toda theories. We are not aware of the existence of any such constructions. An obvious first step would be to figure out how parafermions could couple to affine subsectors.§ CONCLUSIONS AND OUTLOOK We have argued that the derivation of the AGT correspondence proposed in <cit.> can be understood by replacing the north pole region of the S^4, where the excitations of the four-dimensional gauge theory are localized, with a holomorphic divisor inside the singular conifold 𝒦^1,1. This interpretation has two main virtues. Firstly, it provides a clear perspective on the origin of the Nahm pole. Secondly, the generalized conifolds 𝒦^1,m allow us to outline a generalization of the derivation of the original AGT correspondence, which we denote by AGT_λ, involving the inclusion of surface operators on the gauge theory side and a generalization of Toda theory to Toda_λ on the two-dimensional side. We used an equivalent description of these surface operators as orbifold defects, as advocated in <cit.>.Let us now turn to the possible pitfalls of our analysis and their potential resolutions. First of all, we make a number of assumptions and simplifications due to a lack of a full supergravity background and supersymmetry equations of the worldvolume theory in the presence our additional defects. Even in the original case, a full supergravity background and supersymmetry equations have only been written down for the 3d-3d frame <cit.>. For a precise understanding of the origin of the Nahm pole, one should furthermore obtain the supergravity background and supersymmetry equations of the 4d-2d frame. In the presence of our additional defects, such a background should include a geometry which resembles a conifold near its pole regions. Transforming the supersymmetry equations to the 3d-3d frame should then give rise to the Drinfeld-Sokolov boundary conditions on the Chern-Simons connection.An obstruction to performing this transformation is that these equations can only be written down in a five-dimensional setting, since the Lagrangian formulation of the 6d A_N-1 theory is unknown. In other words, one cannot directly transform the supersymmetry equations leading to a Nahm pole in 4d-2d to the 3d-3d equivalent which should give the correct Chern-Simons boundary conditions.Furthermore, the derivation of the 3d-3d correspondence <cit.> is in a sense only concerned with the bulk of the Chern-Simons theory. Even there, ghosts appear, which gauge fix the noncompact part of the gauge group at finite S^3_ℓ/_k size. On an M_3 with boundary, one should similarly impose boundary conditions on these ghosts, which are related to the boundary constraints of the Chern-Simons connection. It would be interesting to see if such ghost terms can be obtained in the 3d-3d frame, if they correspond to the proper Drinfeld-Sokolov constraints in the setting we propose, and how they translate to the 4d-2d Nahm pole setting.At level k=1, using the equality between the Hilbert spaces of complex and real SL(N) Chern-Simons theory, the complex Toda_λ theory corresponds to a real Toda_λ theory. For higher k, it would be interesting to understand the equivalent of the parafermions that were necessary to make contact with real Toda in the original derivation <cit.>.In the main body of this paper, we have not touched upon the relation between (a limit of) the superconformal index of the 6d (2,0) theory of type A_N-1 and vacuum characters of 𝒲_N algebras discovered in <cit.>. This relation was also derived in <cit.> following similar arguments to their derivation of the AGT correspondence. The geometry relevant to the superconformal index, S^5× S^1, can be Weyl rescaled to S^3 × EAdS_3. One can understand this by thinking of the S^5 as an S^3 fibration over a disc, where the S^3 shrinks at the boundary of the disc. The Weyl rescaling stretches the radial direction of the disk to infinite length and produces the EAdS_3 geometry.The boundary conditions on the Chern-Simons connection are again argued to be of Drinfeld-Sokolov type. The D6 brane that arises from reduction on the Hopf fiber wraps the boundary circle of the disc and the S^1.As in the derivation of the AGT correspondence, this D6 brane does not have the correct codimensions for a D4 brane to end on it, and for its scalars to acquire a Nahm pole. Again, we claim that the analogous identification of the divisor of a conifold in the S^5 reproduces the D6 brane and additionally produces the D6' on which the D4s can end. This construction generalizes to the inclusion of codimension two defects as orbifold singularities, as studied in <cit.>. This leads to a derivation of the conjecture, appearing before in <cit.>, that the vacuum character of a general 𝒲_λ algebra is equal to the 6d superconformal index. Further possibly interesting directions of research include the following. In the 3d-3d frame, the additional defects we introduced affect the boundary conditions of Chern-Simons theory. On the other hand, they also have two directions along the three-dimensional 𝒩=2 theory T[M_3]. It would be interesting to interpret the role these defects play on this side of the correspondence.We can also include other types of defects. Codimension two defects that are pointlike on the Riemann surface translate to operator insertions in the Toda theory. They are similarly labeled by a partition of N, which we can associate to the choice of a (possibly semi-degenerate) Toda primary. In six dimensions, these defects wrap an S^4 that maps under Weyl rescaling to an S^3 times the radial direction of M_3. One can then couple such a codimension two defect to the five-dimensional Yang-Mills theory. Reducing to M_3 should produce a Wilson line in complex Chern-Simons theory.Finally, the central charge of generalized Toda theories is known for any embedding, see for example <cit.>. It would be interesting to reproduce this central charge from six dimensions. This has been done for principal Toda in <cit.> by equivariantly integrating the anomaly eight-form over the ^4 Ω background. Following the geometric description of the codimension two defects, one could integrate a suitable generalization of the anomaly polynomial on the orbifolded ×/_m Ω background. Reproducing the generalized Toda central charge from such a computation would provide a convincing check on the validity of a geometric description of the codimension two defects.§ ACKNOWLEDGMENTSIt is a pleasure to thank Jan de Boer, Clay Córdova and Erik Verlinde for useful discussions and encouragement, and Yuji Tachikawa for correspondence on the geometric realization of codimension two defects. This work is partially supported by the Δ-ITP consortium and the Foundation for Fundamental Research on Matter (FOM), which are funded by the Dutch Ministry of Education, Culture and Science (OCW).§ THE CONIFOLDLet us review some facts on the conifold. We mainly follow <cit.> but choose slightly different coordinates in places. The conifold 𝒦^1,1 is a hypersurface in ^4 defined byzw = xy.This equation defines a six-dimensional cone. By intersecting 𝒦^1,1 with a seven-sphere of radius r, we can study its base, which we denote by T. The base T is topologically equivalent to S^2× S^3 and can be conveniently parametrized in the following way,Z := 1/r[ z x; y w ] T:Z = 0,Z^† Z = 1.We can write down the most general solution to these equations by taking a particular solution Z_0 and conjugating it with a pair (L,R) of SU(2) matrices,Z = L Z_0 R^†,Z_0 = [ 0 1; 0 0 ],L,R ∈ SU(2).Each SU(2) factor can be described using two complex coordinates,L:= [ a -b̅; ba̅ ]∈ SU(2),(a,b)∈^2,|a|^2 + |b|^2 = 1, R:= [ k -l̅; lk̅ ]∈ SU(2),(k,l)∈^2,|k|^2 + |l|^2 = 1.Now introduce Hopf coordinates on each SU(2)≃ S^3,a = cos(θ_1/2) e^i(ψ_1+_1),k = cos(θ_2/2) e^i(ψ_2+_2), b = sin(θ_1/2) e^iψ_1,l = sin(θ_2/2) e^iψ_2,Note that the parametrization of T in (<ref>) is overcomplete. Two pairs of SU(2) matrices (L,R) describe the same solution if and only if they are related by the U(1) action(L,R) ↦ (L Θ, R Θ^†), Θ = [e^iθ 0; 0 e^-iθ ]∈ U(1) ⊂ SU(2).This degeneracy should be quotiented out of the SU(2)× SU(2) parametrization. The U(1) acts on the S^3 coordinates by(a,b)→ (e^iθa, e^iθb),(k,l)→ (e^-iθk, e^-iθl).The resulting SU(2)× SU(2)/U(1) quotient is the conifold. Indeed, the invariant coordinates under this U(1) action correspond to the ones used in (<ref>). Setting r=1,x = ak,y = -bl,z = -al,w = bk.They are related by the defining equation (<ref>) of the conifold. In terms of the Hopf coordinates (<ref>), the U(1) quotient (<ref>) joins the two Hopf fiber coordinates in the invariant combination ψ:= ψ_1 + ψ_2. Then T is parametrized by x= cosθ_1/2cosθ_2/2 e^i(ψ + _1 + _2), y= - sinθ_1/2sinθ_2/2 e^iψ, z= - cosθ_1/2sinθ_2/2 e^i(ψ + _1), w= sinθ_1/2cosθ_2/2 e^i(ψ + _2). Demanding that 𝒦^1,1 is Kähler implies that the metric on T is given by <cit.>ds^2_T = 2/3( dZ^† dZ ) - 2/9|( Z^† dZ )|^2 = 4/9( dψ + cos^2(θ_1/2) d_1 + cos^2(θ_2/2) d_2 )^2+ 1/6[ ( dθ_1^2 + sin^2θ_1 d_1^2 ) + ( dθ_2^2 + sin^2θ_2 d_2^2 ) ].This is the metric we wrote down in (<ref>). It describes two three-spheres with a shared Hopf fiber. If we think of this fibration as an electromagnetic U(1) bundle, both spheres feel one unit of magnetic charge. Note that this metric is equivalent to the usual one under the coordinate redefinition ψ = (ψ' - _1 - _2)/2. The divisor In the main text, we make extensive use of the w=0 divisor of the conifold. Setting w to zero in (<ref>) implies that either x or y vanishes. In terms of the coordinates in (<ref>), these choices corresponds to setting either θ_1=0 or θ_2=π. Thus we are at the north (or south) pole of one of the S^2 base factors of T. The remaining sphere, together with the fiber, now describes an ordinary S^3. Setting θ_1=0, the parametrization in (<ref>) reduces to Hopf coordinatesx = cosθ_2/2 e^i(ψ +_2), z = sinθ_2/2 e^iψ. JHEP | http://arxiv.org/abs/1708.07840v1 | {
"authors": [
"Sam van Leuven",
"Gerben Oling"
],
"categories": [
"hep-th"
],
"primary_category": "hep-th",
"published": "20170825180002",
"title": "Generalized Toda Theory from Six Dimensions and the Conifold"
} |
Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany Max-Planck-Institut für Physik komplexer Systeme, 01187 Dresden, Germany Indian Institute for Science Education and Research, Pune 411008, IndiaRenmin University of China, Beijing 100876, China The low-energy physics of graphene is described by relativistic Dirac fermions with spin and valley degrees of freedom. Mechanical strain can be used to create a pseudo magnetic field pointing to opposite directions in the two valleys. We study interacting electrons in graphene exposed to both an external real magnetic field and a strain-induced pseudo magnetic field. For a certain ratio between these two fields, it is proposed that a fermionic symmetry-protected topological state can be realized. The state is characterized in detail using model wave functions, Chern-Simons field theory, and numerical calculations. Our paper suggests that graphene with artificial gauge fields may host a rich set of topological states. Fermionic symmetry-protected topological state in strained graphene Zheng-Xin Liu December 30, 2023 ===================================================================§ INTRODUCTION The study of topological phases has been a central topic of condensed matter physics since the observations of the quantum Hall effect <cit.>. The discovery of topological insulators signifies that the interplay between topology and symmetry can lead to a variety of exotic phenomena <cit.>. The concept of symmetry-protected topological (SPT) states has been introduced to describe gapped quantum states which are non-trivial only if certain symmetries are enforced. The SPT states are different from topologically ordered states in that they do not exhibit fractionalization or long-range entanglement, while they are distinguished from trivial product states by their non-trivial edge states if the protecting symmetries are not broken explicitly or spontaneously.For non-interacting fermions, the possible SPT states have been fully classified for various symmetries <cit.>. It is natural to ask what happens if interactions are introduced. On one hand, distinct SPT states in free fermion systems may be adiabatically connected if interactions are allowed <cit.>. On the other hand, new SPT states with no free fermion counterparts may emerge because of interactions. The SPT states in one- and two-dimensional interacting bosonic systems can be constructed and classified by the group cohomology theory <cit.>, while SPT states beyond group cohomology in three dimensions have been reported <cit.>. For two-dimensional systems, the Chern-Simons field theory turns out to be very useful for classifying the SPT states and studying their physical properties <cit.>. As an example of bosonic SPT states, the integer quantum Hall (IQH) states of bosons have been identified in several microscopic models <cit.>. The theory of fermionic SPT states is less complete except for in one dimension <cit.>. One exotic possibility is that topological orders may be found on the surfaces of three-dimensional SPT states in some ways that are impossible in strictly two-dimensional systems <cit.>.In this work, we construct a fermionic SPT state in two dimensions and analyze its properties in detail. This state depends essentially on interactions because its physical responses cannot appear in the same setup without interactions. The occurrence of this state requires two types of fermions coupled to two different magnetic fields. One possible platform fulfilling such a condition is strained graphene. The band structure of graphene contains two inequvialent valleys 𝐊^± in the Brillouin zone where the physics is described by Dirac fermions with linear dispersion. By applying suitably designed strain, a pseudo magnetic field pointing to opposite directions in the two valleys can be generated <cit.>. If an external magnetic field is also applied, the electrons in the two valleys would experience different total magnetic fields. This paper is organized as follows: the model of our interest is defined in Sec. <ref>, the fermionic SPT state is characterized using wave function and field theory in Sec. <ref>, the relevance of this state in strained graphene is corroborated by numerical calculations in Sec. <ref>, and we conclude in Sec. <ref>.§ MODELWe define the graphene sheet as the xy plane, the unit vector perpendicular to the xy plane as e_z, and the electron charge as -e. The system experience both a real magnetic field and a strain induced pseudo magnetic field as shown in Fig. <ref> (a). The real (pseudo) magnetic field is denoted as B^r (B^p) and the corresponding vector potential is 𝐀^r (𝐀^p). It is assumed that B^r>B^p and the total magnetic fields B^±=B^r±B^p in the two valleys point to the -e_z direction. The single-particle Hamiltonians in the two valleys areℋ_𝐊^± =v_ F[ [0 π^±_x±iπ^±_y; π^±_x∓iπ^±_y0 ]]where v_ F is the Fermi velocity and π^±_i=p_i-e(A^r_i±A^p_i)/c are the canonical momentum operators. The solutions in each valley contain a set of zero energy states, whose spatial components are the same as the non-relativistic lowest Landau level (LL) wave functions.For an infinite disk with symmetric gauge, the non-relativistic lowest LL wave functions have the simple forms ψ_m(x,y) ∼ z^mexp[-|z|^2/(4ℓ^2_B)], where ℓ_B=√(hc/(eB)) is the magnetic length, z=(x+iy) is the complex coordinate, and m is the angular momentum. If the direction of the magnetic field is reversed, the solutions are complex conjugates ψ^*_m(x,y). The electrons interact with each other via the Coulomb potential e^2/(ε|𝐫_j-𝐫_k|) (ε is the dielectric constant of the system). The magnetic field is taken to be sufficiently strong so the electrons are confined to the zero energy states. The many-body problem can be studied on compact surfaces such as sphere and torus <cit.> to avoid edge effects (see Appendix for some technical details). The low-energy states are assumed to be spin-polarized due to the Zeeman splitting and/or quantum Hall ferromagnetism <cit.> (this will be corroborated by numerical results later). The numbers of electrons in the two valleys are denoted as N^±_e. The system respects a U(1)_r×U(1)_p symmetry, where U(1)_r [U(1)_p] corresponds to the conservation of N^+_e+N^-_e (N^+_e-N^-_e). The numbers of magnetic fluxes through the surface of sphere or torus in the two valleys are denoted as N^±_ϕ. The magnetic length ℓ^+_B=√(hc/(eB^+)) associated with B^+ is used as the length scale and e^2/(εℓ^+_B) is used as the energy scale. For electrons on the sphere, an angular momentum quantum number L labels the many-body eigenstates. For electrons on the torus, special momentum quantum number Y labels the many-body eigenstates. § TOPOLOGICAL STATE §.§ Wave Function To construct a many-body state, we start from a non-interacting system with N^+_e=N^-_e and choose B^p appropriately such that the two valleys form two decoupled IQH states at filling factor 1 with opposite chiralities (B^r is zero at this stage). This system has the wave function∏_j<k (z^+_j-z^+_k) ∏_j<k (z^-_j-z^-_k)^*where the superscripts ± are used to label the two valleys. As we turn on the real magnetic field and the interaction between electrons, the composite fermion theory <cit.> suggests that the interaction energy can be efficiently minimized if the electrons each absorb two magnetic fluxes as shown in Fig. <ref> (b). This transformation is implemented by the factor ∏_j<k (z_j-z_k)^2, where the coordinates without superscripts are for all electrons. If we choose B^r=4B^p, the real magnetic field is completely absorbed by the electrons and the resulting composite fermions form two decoupled IQH states. The wave function for this system is𝒫_ LLL∏_j<k (z^+_j-z^+_k) ∏_j<k (z^-_j-z^-_k)^* ∏_j<k (z_j-z_k)^2where 𝒫_ LLL projects wave functions to the lowest LLs (i.e. the zero energy single-particle states in the two valleys). The filling factors in the two valleys are ν^+=1/5 and ν^-=1/3 respectively.The mapping to composite fermions also helps us to understand the excitations because the composite fermions can be taken as non-interacting objects and form their effective LLs. For the ground state, the two types of composite fermions completely fill their respective lowest effective LLs. A neutral excitation is created if we promote one composite fermion from its lowest LL to second LL. There are four types of charged minimal excitations: one type I (II) quasihole is present if the composite fermions occupy all orbitals of the lowest LLs except for one orbital in the 𝐊^+ (𝐊^-) valley; one type I (II) quasiparticle is present if the composite fermions occupy all orbitals of the lowest LLs and one orbital of the second LL in the 𝐊^+ (𝐊^-) valley. The local charges of the type I (II) quasihole/quasiparticle with respect to 𝐀^r are denoted as ±Q^r_ I (±Q^r_ II). The local charges with respect to 𝐀^p are denoted similarly with the superscript r replaced by p. Let us consider what happens if one electron is removed from the ground state [Fig. <ref> (c)]. The effective magnetic fluxes for the composite fermions in the 𝐊^+ (𝐊^-) valley increase (decrease) by two units. If the electron is from the 𝐊^+ valley, the real and pseudo charges increase by e and three type I quasiholes and two type II quasiparticles are created. If the electron is from the 𝐊^- valley, the real (pseudo) charge increases (decreases) by e and two type I quasiholes and one type II quasiparticles are created. This analysis yields the equations3Q^r_ I-2Q^r_ II=e 2Q^r_ I-Q^r_ II=e 3Q^p_ I-2Q^p_ II=e 2Q^p_ I-Q^p_ II=-eso we have Q^r_ I=e, Q^r_ II=e, Q^p_ I=-3e, and Q^p_ II=-5e, which demonstrates that the charges of electron do not fractionalize in our system.The Hall conductances can be obtained using the Laughlin flux insertion argument <cit.>. We place the system on a disk and insert one flux at the center. If the inserted flux is for the real magnetic field (which increases the flux values in both valleys), one type I quasihole and one type II quasiparticle are created at the center. The change of real charge at the center is Q^r_ I-Q^r_ II=0 so the real Hall conductance σ^r_xy=0. If the inserted flux is for the pseudo magnetic field [which increases (decreases) the flux value in the 𝐊^+ (𝐊^-) valley], one type I quasihole and one type II quasihole are created at the center. The change of pseudo charge at the center is Q^p_ I+Q^p_ II=-8e so the pseudo Hall conductance σ^p_xy=-8e^2/h. To measure the change of real (pseudo) charge due to the insertion of a pseudo (real) flux, we define a mutual Hall conductance σ^rp_xy whose value turns out to be 2e^2/h. It should be emphasized that these three Hall conductance values cannot appear simultaneously in a system of non-interacting two-component electrons, where two decoupled IQH states with the same or opposite chiralities at any filling factors can in principle be realized using real plus pseudo magnetic fields.The system is expected to possess two counterpropagating edge modes corresponding to the edge modes of the IQH states of composite fermions. These edge modes will not be gapped out if there is no tunneling between the valleys. Another possibility is that the composite fermion edge states will remain gapless so long as an emergent time-reversal symmetry of composite fermions is preserved. This is intuitively plausible because the ν=±1 IQH states can be viewed as the simplest two-dimensional topological insulator. This will be formulated in a more precise way using effective field theory below. However, if electrons can tunnel between the valleys, N^±_e, σ^p_xy, and σ^rp_xy are no longer well-defined. §.§ Field Theory The wave function Eq. <ref> can be described by the Chern-Simons theory with Lagrangian density <cit.>ℒ_1 = 1/4πħϵ^λμν K_IJ a_Iλ∂_μ a_Jν - j_Iλ a_Iλwhere a_Iλ (I=1,2, λ=0,x,y) are internal gauge fields, j_Iλ is the excitation current, andK = ( [ 3 2; 2 1 ])One important signature of topological phases is the number of degenerate ground states on torus. The existence of multiple degenerate ground states implies the presence of fractionalization. For the Chern-Simons Lagrangian ℒ_1, the ground state degeneracy on torus is |K|=1, which is consistent with our conclusion that there are no fractionally charged excitations.An excitation in the Chern-Simons field theory can be labeled by an integer vector 𝐥, which is (±1,0)^T for type I quasiparticle/quasihole and (0,∓1)^T for type II quasiparticle/quasihole. The statistical angle of an excitation labeled by 𝐥 with itself is θ=π l^T K^-1 l, so all the minimal excitations have fermionic self braid statistics. The statistical angle of two excitations labeled by 𝐥_1 and 𝐥_2 is θ_12 = 2π l_1^T K^-1 l_2, so one type I quasihole/quasiparticle and one type II quasihole/quasiparticle have bosonic mutual braid statistics. If two probing U(1) gauge fields A^r_μ and A^p_μ coupled to the excitation current are introduced, we need to add an extra termℒ_2 = e/2πħϵ^λμν ( t^r_I A^r_λ∂_μ a_Iν + t^p_I A^p_λ∂_μ a_Iν)to ℒ_1, where the charge vectors 𝐭^r=(1,1)^T and 𝐭^p=(1,-1)^T. The U(1) charges of an excitation labeled by 𝐥 are -e[ t^r]^T K^-1𝐥 and -e[𝐭^p]^T K^-1𝐥, which yield the same results for the minimal excitations as our previous analysis. The Hall conductance with respect to the real gauge field is σ^r_xy = e^2 [𝐭^r]^TK^-1𝐭^r/h=0, the one with respect to the pseudo gauge field is σ^p_xy = e^2 [𝐭^p]^TK^-1𝐭^p/h = -8e^2/h, and the mutual Hall conductance is σ^rp_xy = e^2 [𝐭^r]^TK^-1𝐭^p/h = 2e^2/h, which reproduce the results derived using the Laughlin argument.For a system described by ℒ_1+ℒ_2 in the bulk, it has gapless edge states on an open mainfold as captured by ℒ_ edge = 1/4πħ( K_IJ∂_0ϕ_I∂_xϕ_J - V_IJ∂_xϕ_I∂_xϕ_J)where ϕ_I are chiral boson fields that satisfy the Kac-Moody algebra[ ∂_xϕ_I(x) , ∂_yϕ_J(y) ] = 2πi K^-1_IJ∂_xδ(x-y)and V_IJ depends on the microscopic details at the edge <cit.>. The electron annihilation operators for the two valleys areC_1 = e^-i(3ϕ_1+2ϕ_2)C_2 = e^-i(2ϕ_1+ϕ_2)The number of edge modes is the dimension of K and their chiralities are given by the sign of the eigenvalues of K. This means that our system has two counterpropagating edge modes and is consistent with our previous analysis based on wave functions. These edge modes can be gapped out by some perturbations, but one can rule out such perturbations by imposing certain symmetries on the system <cit.>. If the system has a symmetry group G and is acted upon by an element g of G, the K matrix transforms asK → W^T_g K W_g = s_g Kand the gauge fields ϕ transform asϕ→ W^-1_gϕ + δ_gϕwhere W_g is an integer matrix, s_g is 1 for unitary symmetry and -1 for anti-unitary symmetry, and δ_gϕ is a constant. If we choose the symmetries to be U(1)_r×U(1)_p, the elements of the symmetry groups can be labeled by θ_r,θ_p∈[0,2π) and the symmetry operators are𝒰^r_θ_r=exp(-iθ_r∑_I t^r_I C^†_I C_I)and𝒰^p_θ_p=exp(-iθ_p∑_I t^p_I C^†_I C_I)The quantities W_g and δ_gϕ in Eqs. <ref> and <ref> can be written asW^r_θ_r = ( [ 1 0; 0 1 ])δ^r_θ_rϕ = θ_r∑_I t^r_I K^-1_IJfor the U(1)_r group andW^p_θ_p = ( [ 1 0; 0 1 ])δ^p_θ_pϕ = θ_p∑_I t^p_I K^-1_IJfor the U(1)_p group. The edge modes can also be protected by another combination of symmetries. To understand how this works, we convert the K matrix to the diagonal form K= Diag(1,-1)=X^TKX usingX=( [0 -1;12 ])The physics of our system would remain the same if the fields a_Iλ,ϕ_I and the vectors 𝐥, 𝐭 are also transformed properly (the transformed ones will be denoted by symbols with a tilde). In the transformed basis, the system is a quantum spin Hall insulator with edge modes that are protected by particle number conservation and time-reversal symmetry. Its symmetry group can be written as G^-_-[U(1)_r,𝒯], where 𝒯 is an emergent time-reversal symmetry, the subscript - means 𝒯^2=P_f (the fermion parity operator), and the superscript - means 𝒰^r_θ𝒯 = 𝒯𝒰^r_-θ. The time-reversal symmetry operates on K and ϕ_I as𝒯K𝒯^-1 = σ_xKσ_x = -K 𝒯ϕ_1𝒯^-1 = ϕ_2 𝒯ϕ_2𝒯^-1 = ϕ_1 + πAs one goes back to the original basis, an emergent time-reversal symmetry can be defined and it operates on K and ϕ_I as𝒯 K 𝒯^-1 = W^T_𝒯 K W_𝒯 = -K𝒯ϕ_1 𝒯^-1 = -2ϕ_1 - ϕ_2 + π 𝒯ϕ_2 𝒯^-1 = 3ϕ_1 + 2ϕ_2The quantities W_g and δ_gϕ in Eqs. <ref> and <ref> can be written asW_𝒯 = ([ -2 -1;32 ])δ_𝒯ϕ = ( [ π; 0 ]) § NUMERICAL RESULTSExact diagonalization can be used to check the validity of our previous analysis. Let us consider the cases where the electrons are spin-polarized and N^+_e=N^-_e. For electrons on the sphere, the relation between N^±_e and N^±_ϕ is N^±_ϕ=N^±_e/ν^±-𝒮^±, where 𝒮^+=3 and 𝒮^-=1 are the shift quantum numbers. There is no shift quantum number on the torus. Fig. <ref> shows the energy spectra of the N^+_e=N^-_e=5 system on sphere and torus. The existence of a unique ground state on torus is confirmed in all the systems that can be studied. The energy gap Δ (the energy difference between the first excited state and the ground state) is found to be ≈0.018e^2/(εℓ^+_B) in the thermodynamic limit based on the finite size scaling analysis in Fig. <ref> (c). For ε=3 and B^r=20 T, the numerical value is about 17 K. Based on previous experiences <cit.>, we expect that disorder and other imperfections in experimental systems would reduce the actual value to 30%∼50% of the ideal value.The spherical version of Eq. <ref> can be constructed explicitly and its high overlap with the exact ground state (0.9797 for N^+_e=N^-_e=5 and similar values for smaller systems) corroborates the accuracy of our ansatz. Besides the ground state, there are several low-energy neutral excitations in Fig. <ref> (a), which can be modeled by exciting one composite fermion in the 𝐊^+ valley to an originally unoccupied state. This is an interesting feature that distinguishes our state from fractional quantum Hall states in graphene, for which composite fermions in both valleys contribute to the low-energy neutral excitations <cit.>. Fig. <ref> shows the energy spectra of systems that contain one charged minimal excitation, where the trial wave functions also provide excellent approximations of the exact eigenstates. The edge states can be seen from entanglement spectrum <cit.>. For this calculation, the sphere is cut along its equator such that two hemispheres are separated by a virtual edge <cit.>. The ground state |Ψ⟩ is decomposed as |Ψ⟩ = ∑_ij F_ij |Ψ^S_i⟩⊗ |Ψ^R_j⟩, where S (R) is the southern hemisphere and its basis states are |Ψ^S_i⟩ (|Ψ^R_j⟩). The Schmidt decomposition of F_ij gives |Ψ⟩ = ∑_μ e^-ξ_μ/2 |Ψ^S_μ⟩⊗ |Ψ^R_μ⟩ and the ξ_μ levels comprise the entanglement spectrum. The entanglement levels can be labeled by the good quantum numbers N^S±_e (the numbers of electrons) and L^S_z (the z-component angular momentum) of the southern hemisphere. Fig. <ref> (d) presents the entanglement spectrum of the N^+_e=N^-_e=6 system in the N^S+_e=N^S-_e=3 subspace with the levels organized according to L^S_z. There is a foward-moving branch with counting 1,1,2,3,5 and a backward-moving branch with counting 1,1,2,3, which are consistent with the presence of two counterpropagating edge modes each described by a single boson field.The assumption of spin polarization can also be tested numerically. To this end, we denote the numbers of electrons with spin-valley indices as N^↑+_e,N^↓+_e,N^↑-_e,N^↓-_e. The electrons in the same valley with different spin experience the same magnetic field. For the two valleys, we define total spin operators 𝐒^2_± and z-component spin opertors S^z_±. Because the Hamiltonian is SU(2) symmetric in the spin space, the energy eigenstates are also eigenstates of 𝐒^2_± [with eigenvalues S_±(S_±+1)] and S^z_± [with eigenvalues S^z_±=(N^↑±_e-N^↓±_e)/2]. To check for all possible spin polarizations, we should study the subspace with minimal z-component spin values S^z_±. The ground states for the two systems with N^+_e=N^-_e=3 and N^+_e=N^-_e=4 are found to be spin-polarized.For a fixed N_e=N^+_e+N^-_e, there are many possible choices of N^±_e but only the special cases with N^+_e=N^-_e were studied above. It is useful to compare the ground state energy E_N^-_e at different N^-_e. Fig. <ref> shows that the lowest one appears at N^-_e=0 in the N_e=8 and 10 systems. If a pseudo Zeeman term α(N^+_e-N^-_e) is added to the Hamiltonian <cit.>, E_N^-_e increases (decreases) for N^-_e<N_e/2 (N^-_e>N_e/2) and E_N_e/2 will become the global minimum in a certain range of α (estimated to be 0.0344≲α≲0.0860). This can be seen easily from the spectra because the lines connecting all the pairs of data points at N^-_e and N_e-N^-_e lie above E_N_e/2 (one line is shown explicitly for each case).The K matrix in Eq. <ref> suggests another possible trial wave function∏_j<k (z^+_j-z^+_k)^3 ∏_j<k (z^-_j-z^-_k) ∏_j,k (z^+_j-z^-_k)^2for the ground state. It is the zero energy ground state of the Hamiltonian g_1∑_j<k∇^2 δ^(2)(𝐫^+_j - 𝐫^+_k) + g_2∑_j<kδ^(2)(𝐫^+_j - 𝐫^-_k) + g_3∑_j<k∇^2 δ^(2)(𝐫^+_j - 𝐫^-_k), where the positive coefficients g_1,2,3 determine the interaction strength <cit.>. However, if there is no interaction within the 𝐊^- valley but repulsion between the two valleys as required by this parent Hamiltonian, the system would be unstable to phase separation. We have computed the overlap between Eq. <ref> and the exact ground state but find that it is much worse than Eq. <ref> (0.6972 for the N^+_e=N^-_e=5 system on sphere).§ CONCLUSIONIn conclusion, we have studied the properties of a fermionic SPT state in detail and demonstrated that it can be realized in strained graphene. The combination of real and pseudo magnetic fields proposed here allows us to explore a broad range of gauge field configurations for multi-component electrons in graphene. We expect that many other quantum phases in such systems will be revealed. It would also very interesting if one can find some methods to engineer non-Abelian gauge fields and study the quantum phases in such systems.§ ACKNOWLEDGEMENT Exact diagonalization calculations are performed using the DiagHam package for which we are grateful to all the authors. YHW and TS were supported by the DFG within the Cluster of Excellence NIM. ZXL was supported by the Research Funds of Remin University of China (No. 15XNFL19), NSF of China (No. 11574392), and the Major State Research Development Program of China (2016YFA0300500).*§ HAMILTONIAN MATRIX ELEMENTS The two valleys are labeled using σ,τ=± and the creation (annihilation) operator for the single-particle state with quantum number m is denoted as C^†_σ,m (C_σ,m). The electrons interact via the Coulomb potential V(𝐫_1-𝐫_2)=e^2/(ε|𝐫_1-𝐫_2|). The second quantized form of the many-body Hamiltonian is1/2∑_στ∑_{m_i} F^σττσ_m_1m_2m_4m_3 C^†_σ,m_1 C^†_τ,m_2 C_τ,m_4 C_σ,m_3§.§ Sphere The particles on a sphere experience a radial magnetic field generated by a magnetic monopole at the center. If the magnetic flux through the sphere is N^σ_ϕ, the LLL single-particle wave functions are <cit.>ψ^N^σ_ϕ_m(θ,ξ) = [ N^σ_ϕ+1/4πN^σ_ϕN^σ_ϕ-m]^1/2 u^N^σ_ϕ/2+m v^N^σ_ϕ/2-mwhere u=cos(θ/2)e^iξ/2,v=sin(θ/2)e^-iξ/2 are spinor coordinates (θ and ξ are the azimuthal and radial angles in the spherical coordinate system) and m is the z component of the angular momentum. The magnetic length is related to the radius of the sphere as R=ℓ^+_B√(N^+_ϕ/2)=ℓ^-_B√(N^-_ϕ/2). The coefficients F^σττσ_m_1m_2m_4m_3 are∫ dΩ_1 dΩ_2[ ψ^N^σ_ϕ_m_1(Ω_1) ]^* [ ψ^N^τ_ϕ_m_2(Ω_2) ]^* V(𝐫_1-𝐫_2) ψ^N^τ_ϕ_m_4(Ω_2) ψ^N^σ_ϕ_m_3(Ω_1)where 𝐫=R(sinθcosϕ,sinθsinϕ,cosθ) and Ω=𝐫/R. The product of two wave functions can be expressed as ψ^N^σ_ϕ_m_1ψ^N^τ_ϕ_m_2 = (-1)^N^σ_ϕ-N^τ_ϕ[ (N^σ_ϕ+1)(N^τ_ϕ+1)/4π(N^σ_ϕ+N^τ_ϕ+1)]^1/2⟨N^σ_ϕ/2,-m_1;N^τ_ϕ/2,-m_2 | N^σ_ϕ/2+N^τ_ϕ/2,-m_1-m_2 ⟩ψ^N^σ_ϕ+N^τ_ϕ_m_1+m_2The Coulomb potential can be expanded ase^2/ε|𝐫_1-𝐫_2| = 4πe^2/εR∑^∞_L=0∑^L_M=-L1/2L+1[ ψ^0_LM(Ω_1) ]^* ψ^0_LM(Ω_2)These relations help us to obtainF^σττσ_m_1m_2m_4m_3 = δ_m_1+m_2,m_3+m_44πe^2/εR∑^ min(N^σ_ϕ,N^τ_ϕ)_L=01/2L+1 (-1)^(N^σ_ϕ+N^τ_ϕ)/2-m_1-m_4 S^1_L S^2_Lwhere the two coefficients S^1,2_L are defined by[ ψ^N^σ_ϕ_m_1(Ω_1) ]^* [ ψ^0_LM(Ω_1) ]^* ψ^N^σ_ϕ_m_3(Ω_1) = ∑^N^σ_ϕ_L_1=0 (-1)^N^σ_ϕ/2-m_1 S^1_L_1[ ψ^0_LM(Ω_1) ]^* ψ^0_L_1,m_3-m_1(Ω_1)[ ψ^N^τ_ϕ_m_2(Ω_2) ]^* ψ^0_LM(Ω_2) ψ^N^τ_ϕ_m_4(Ω_2) = ∑^N^τ_ϕ_L_2=0 (-1)^N^τ_ϕ/2-m_4 S^2_L_2[ ψ^0_L_2,m_2-m_4(Ω_2) ]^* ψ^0_LM(Ω_2)§.§ Torus The torus is spanned by the vectors 𝐋_1=L_1e_x,𝐋_2=L_2e_y and we choose the Landau gauge 𝐀^σ=(0,B^σx,0). If the magnetic flux through the torus is N^σ_ϕ, the LLL single-particle wave functions are <cit.>ψ^N^σ_ϕ_m(x,y) = 1/(√(π) L_2ℓ^σ_B)^1/2∑^ℤ_kexp{ - 1/2[ x/ℓ^σ_B - 2πℓ^σ_B/L_2( m+kN^σ_ϕ) ]^2 + i 2πy/L_2( m+kN^σ_ϕ) }where the magnetic length ℓ^σ_B=√(L_1L_2/(2πN^σ_ϕ)). By defining the reciprocal lattice vectors 𝐆_1=2πe_x/L_1,𝐆_2=2πe_y/L_2, we transform the interaction potential to momentum space as V(𝐫_1 - 𝐫_2)= 1/L_1L_2∑_𝐪 V(𝐪) e^i𝐪·(𝐫_1 - 𝐫_2)where 𝐪=q_1𝐆_1+q_2𝐆_2. The coefficients F^σττσ_m_1m_2m_4m_3 are∫ d^2 𝐫_1 d^2 𝐫_2[ ψ^N^σ_ϕ_m_1(𝐫_1) ]^* [ ψ^N^τ_ϕ_m_2(𝐫_2) ]^* V(𝐫_1-𝐫_2) ψ^N^τ_ϕ_m_4(𝐫_2) ψ^N^σ_ϕ_m_3(𝐫_1) = 1/L_1L_2∑^N^σ_ϕ_m_1∑^N^τ_ϕ_m_2∑_q_1,q_2 V(𝐪) exp{ -𝐪^2/4 (ℓ^σ2_B+ℓ^τ2_B) + i 2πq_1 [ (m_1-q_2/2)/N^σ_ϕ - (m_2+q_2/2)/N^τ_ϕ] }δ^N^ G_ϕ_m_1+m_2,m_3+m_4where N^ G_ϕ is the greatest common divisor of N^+_ϕ and N^-_ϕ and δ^N_ϕ_i,j is a generalized Kronecker delta defined asδ^N_ϕ_i,j=1iff i modN_ϕ = j modN_ϕThe many-body eigenstates are labeled by a special momentum quantum number Y ≡ (∑_σ=±∑^N^σ_ϕ_i=1 m^σ_i) modN^ G_ϕ. | http://arxiv.org/abs/1708.08040v1 | {
"authors": [
"Ying-Hai Wu",
"Tao Shi",
"G. J. Sreejith",
"Zheng-Xin Liu"
],
"categories": [
"cond-mat.str-el",
"cond-mat.mes-hall"
],
"primary_category": "cond-mat.str-el",
"published": "20170827021147",
"title": "Fermionic symmetry-protected topological state in strained graphene"
} |
[email protected] Air Force Research Laboratory, Information Directorate, Rome, New York, 13441, USA Department of Physics, Florida Atlantic University, Boca Raton, Florida, 33431, USA Quanterion Solutions Incorporated, Utica, New York, 13502, [email protected] Air Force Research Laboratory, Information Directorate, Rome, New York, 13441, [email protected] Air Force Research Laboratory, Information Directorate, Rome, New York, 13441, USATwo-mode squeezed states in the limit of small squeezing, Hong-Ou-Mandel interference and post selection on coincidence counts are some of the staples of linear quantum optics. We show that by using classical expectations on central moments of intensities, we can remove the requirement of small squeezing necessary for high fidelity coincidence detection. Utilizing existing techniques to probabilistically generate a cluster state, we construct a statistical analogue with deterministic generation at the cost of losing the ability to feed forward and requiring statistical averaging. 42.50.Ex, 42.50.Ar, 03.67.LxCentral Moment Analogues to Linear Optically Generated Cluster States Paul M. Alsing December 30, 2023 ===================================================================== Cluster states are a paradigm of quantum computing by which conditional measurements are utilized to deterministically enact quantum logic on an entangled state <cit.>. Nonlinear interaction in the form of a controlled phase gate is the canonical form of generating such an entangled state. Sadly, the nonlinearity necessary to deterministically enact a controlled phase gate on single photons has proven elusive. Accordingly, the toolbox for generating entangled single photon states hinges on linear optics and post selection <cit.>. Post selection acts as an effective nonlinearity at the cost of probabilistic success.Much work has been put into preventing this probabilistic approach from becoming intractable for arbitrarily large clusters <cit.>. On the other end of the spectrum, single photon generation is inherently stochastic and in the case of spontaneous parametric down conversion, it has an upper limit where multi-photon generation becomes non-negligible and contaminates the post selection conditions. This is mitigable through off-line preparation of photon pairs and photon number resolved heralding, now at the cost of needing to store photons <cit.>.We address the problem of single photon generation by mapping single photon statistics to intensity fluctuations on bright beams <cit.>. This approach is distinct from previous efforts, which also use bright beams, but focus on the quadrature uncertainties of squeezed fields <cit.>. Both approaches with respect to single photons trade probability of occurrence with certainty of outcome. In particular, we change from photon pair generation via spontaneous parametric down-conversion to two-mode squeezed states and map coincidences to cross central moment expectations. Using this mapping we find analogous statistics to Hong-Ou Mandel interference <cit.>, Bell states <cit.>, GHZ states <cit.>, and finally cluster states <cit.>. Along the way we find loss and amplification can be used to manipulate the central moments. Lastly, we find that the nature of using only linear optics invokes an exponential increase in the uncertainty of our measures as more modes are added to the state. Conversely, we do not have the up front cost of probabilistically generating our photons and do not require single photon detectors.The simplest example of our mapping is in observing an analogy to Hong-Ou-Mandel interference.We generalize single photon pairs to the small squeezing regime of a two-mode squeezer described by the unitary operator <cit.>Ŝ_ab(r,ϕ)=exp r e^iϕâ^†b̂^†- r e^-iϕâb̂,for positive real values r and ϕ which determine the amount of squeezing and relative phase of the generated modes a and b. The two-mode squeezer acts on the vacuum by Ŝ_ab(r,ϕ)|vac⟩ = ∑_n=0^∞[tanh(r) e^iϕ]^n/cosh(r)|n⟩_a|n⟩_b ≈|vac⟩ + r e^iϕ|1⟩_a|1⟩_b,r≪ 1. Normalization is omitted in (<ref>) to show the small r limit of two-mode squeezing is approximated by photon pair generation. Measuring the Pearson correlation of their intensities n̂_i=â_i^†â_i, shows that the two-mode squeezed vacuum exhibits perfect correlation. To see this, we note their central moment intensity operators are defined by n̅_i=n̂_i-n̂_i. The modes see equal average photon number n̂_a=n̂_b=sinh^2(r) and their variances are equal to their covariance asn̅_an̅_a = n̅_bn̅_b = n̅_an̅_b = cosh^2(r)sinh^2(r),leaving the Pearson correlation to be oneρ_ab=n̅_an̅_b/√(n̅_an̅_an̅_bn̅_b)=1,independent of squeezing, r. Pearson correlation is a measure of linear correlation between two random variables. It is bounded between plus and minus one corresponding to correlation and anti-correlation, respectively. Therefore, (<ref>) shows perfect linear correlation is present in the two mode squeezed state.We can remove all Pearson correlation by using the Hadamard operation Ĥ_ab. The Hadamard operator transforms the modes a and b asâ→ (â + b̂)/√(2) = â^', b̂→ (â - b̂)/√(2) = b̂^',and describes the operation of a 50/50 beamsplitter on two spatial modes. Measuring the state (<ref>) in this new basis shows there are no correlations between a^' and b^' as their covariance n̅_a^'n̅_b^'=0. The Hadamard operation transforms between maximal and zero intensity correlation on the two-mode squeezed state <cit.>. This is in analogy with Hong-Ou-Mandel interference which sees the Hadamard operation transform two photons from perfect coincidence to zero coincidence.Intensity detection can also be used to measure Bell state statistics. In the limit of small squeezing, the |HH⟩ + exp(iϕ)|VV⟩/√(2) state can be approximated by the application of two squeezing operations |ψ_Bell⟩ = Ŝ_a_hb_h(r,0)Ŝ_a_vb_v(r,ϕ)|vac⟩ ≈ |vac⟩+ r â^†_hb̂^†_h + e^iϕâ^†_vb̂^†_v |vac⟩, where in (<ref>) r≪ 1 and we ignored normalization. The subscripts h and v are merely markers to denote the additional modes; therefore all four Bell states can be generated in a similar way. In the case where these subscripts denote polarizations, one can use the Stokes parameters <cit.>a_1 = â^†_h â_h - â^†_v â_v,Σ̂_a_2 = â^†_h â_v +â^†_v â_h,Σ̂_a_3 =i â^†_v â_h-â^†_h â_v , and total photon number Σ̂_a_0 = â^†_h â_h + â^†_v â_v to perform polarization measurements. These operators correspond to photon number resolution in the low photon flux regime, and photocurrent measurement of p-i-n diode detectors in the high flux regime <cit.>. Using these operators, we can measure the two-mode squeezed Bell state (TMS-Bell) (<ref>) and find the a mode is not polarized a_0 = 2sinh^2(r), a_i = 0,i=1,2,3,where we make no assumption on the magnitude of r and a similar relation holds for the b mode. For concreteness we restrict to the case where ϕ=0. The nonzero joint expectations of Stokes parameters (and total intensity) between modes a and b are a_0b_0 = 2sinh^2(r)cosh^2(r) + 4sinh^4(r),a_1b_1 = 2sinh^2(r)cosh^2(r),a_2b_2 = 2sinh^2(r)cosh^2(r),a_3b_3 = -2sinh^2(r)cosh^2(r). Here, we want to explicitly mention all cross terms (a_ib_j=0, i j) are zero. Using the convention of normalizing by the joint total photon number (<ref>), it has been shown that this state can violate a CHSH inequality for small values of r but approaches the classical value of 2√(2)/3 as r is increased <cit.>. It should be clear that this is due to the contribution of 4sinh^4(r) which is only present in the joint total photon number measurement (<ref>). Following similar footsteps to the single two-mode squeezed state,we propose the measurement of central moment statistics. Accordingly, a_0b_0≡a_0b_0-a_0b_0,and the TMS-Bell state, (<ref>) has balanced (all equal) statistics for central moments. Additionally, the variances a_ia_i=2sinh^2(r)cosh^2(r) (similarly for b modes) are the same magnitude, revealing perfect Pearson correlation for each of these measurements. We have to note, the derivation of the CHSH inequality does not work with central moment statistics. Therefore, we are not claiming a CHSH inequality violation despite the TMS-Bell state producing Bell like statistics.As further proof that our state cannot violate a CHSH inequality, we find that in the presence of linear loss or misbalanced squeezing, the central moment statistics of (<ref>) can always be rebalanced through the introduction of more loss or amplification. This comes at the cost of reducing Pearson correlation. Nevertheless, it is in stark contrast to quadrature based continuous variable entanglement which is sensitive to loss and single photon states which cannot be amplified.Linear loss can be modeled as a beamsplitter with an unmeasured Bosonic mode γ̂_l which transforms annihilation operators asâ→cos(θ) â + i sin(θ) γ̂_land 0≤θ≤π/2 indicates zero to total loss, respectively. In the case where there are multiple sources of loss, we assume their unmeasured modes are different such that they commute as [γ̂_l_i,γ̂^†_l_j]=0.Conversely, a two-mode squeezer acts as an ideal linear amplifier. It adds vacuum fluctuations to the mode â asâ→cosh(g) â + sinh(g) γ̂_g^†where, g≥ 0 is the gain and γ̂_g is added noise (which is uncorrelated as in the linear loss case.)In an attempt to better represent the statistics we adopt a correspondence between the central moment spin measures and Pauli spin matrices as a_ib_j→a_ib_jσ_i⊗σ_j(where σ_0 is the identity and σ_1,2,3↔σ_z,x,y).Accordingly, we represent the unequally squeezed state Ŝ_a_hb_h(r_1,0)Ŝ_a_vb_v(r_2,0)|vac⟩ asρ_Bell = ∑_i,j=0^3 a_ib_jσ_i⊗σ_j = η0 0μ0 0 0 0 0 0 0 0μ0 0ν ,{η = sinh^2(r_1)cosh^2(r_1)μ = sinh(r_1)cosh(r_1) = ×sinh(r_2)cosh(r_2)ν = sinh^2(r_2)cosh^2(r_2). revealing the suggestive Bell statistics but we must note the usage of central moments prevents this from being a valid density matrix. In the case of r_1>r_2, the introduction of the same loss to both a_h and b_h takes the coefficient η→cos^4(θ) η and the coefficient μ→cos^2(θ) μ. Therefore, we can pick the loss to be such that cos^2(θ)sinh(r_1)cosh(r_1)=sinh(r_2)cosh(r_2), rebalancing the statistics.In a similar approach, one can find that applying equal and independent gain to the a_v and b_v modes transforms μ→cosh^2(g) μ and ν→cosh^4(g) ν. In this manner, a choice of gain where sinh(r_1)cosh(r_1)=cosh^2(g)sinh(r_2)cosh(r_2) is appropriate.As mentioned earlier, these actions harm the Pearson correlations of the measurements. In fact, the variances no longer equal each other and in order to characterize the state; the sensible thing to do is normalize by the trace such that ρ_Bell=1 or equivalently divide through by σ_0⊗σ_0a_0b_0. We must keep in mind the variances are indicators of the amount of time necessary to bound the statistics as ultimately, one needs to be capable of discriminating nonzero measures from ones that are zero.Moving forward, one way to generate a four photon two-mode squeezed GHZ (TMS-GHZ) state is to generate two TMS-Bell states and use a polarizing beam splitter to mix their information. A polarizing beam splitter acts as a perfect transmitter for one polarization and a perfect reflector for the orthogonal one giving â_h,b̂_h→â_h,b̂_h, â_v,b̂_v→b̂_v,â_v,where for simplicity, we ignore the phase gained upon reflection. We can incorporate the action of a polarizing beamsplitter (PBS) into the labeling of the squeezers giving |ψ_GHZ⟩=Ŝ_a_h b_hŜ_a_v d_vŜ_c_h d_hŜ_c_v b_v|vac⟩with the recognition that (r,0) are implicit arguments to the squeezing operators and a swap between b_v and d_v was applied to the initial TMS-Bell states on modes a, b and c, d. We find the measurement outcomes to be ρ_GHZ = ∑_i,j,k,l=0^3 a_ib_jc_kd_lσ_i⊗σ_j⊗σ_k⊗σ_l = δ0⋯0δ0 0⋯0 0⋮ ⋮ ⋱ ⋮ ⋮0 0⋯0 0δ0⋯0δ ,δ = sinh^4(r)cosh^4(r)where ρ_GHZ is of dimension 2^4×2^4 and normalization is given by dividing through with ρ_GHZ=2δ. Generalizing the two mode case of Pearson correlations one can regularize the central moments viaa_i = a_i/√(a_ia_i).We denote expectations across modes of these regularized operators as a measure of the co-fluctuation of the state. Just as in the TMS-Bell state (<ref>), the individual modes of the TMS-GHZ state have variances of a_ia_i=2√(δ) while nonzero central moment measures of the TMS-GHZ state (i.e. a_0b_0c_0d_0) all have magnitude 2δ. Therefore, the co-fluctuation of the TMS-GHZ state ends up being 1/2. In contrast, the two TMS-Bell states before application of the PBS see a co-fluctuation of 1. In the terminology of single photon statistics, we are seeing the well known result of 50% success of post selecting coincidences after mixing on a polarizing beamsplitter <cit.>. Waiting for a coincidence event to occur translates to a longer wait time necessary to bound the measurement statistics.We would like to note that this procedure only works for combining an even number of modes. Additionally, if one was so inclined, they could find other ways to assign squeezing such that they reproduce the statistics. For instance, instead of swapping b_v and d_v, one could have swapped a_v and c_v. Finally, one could generalize to an n mode TMS-GHZ state as|ψ_GHZ⟩_n = ∏_i=0^n-1Ŝ_a^(i)_h a^((i+1)modn)_v|vac⟩where a^(i) are distinct modes, and n= 2k for k underlying TMS-Bell states. In the case of k=1, we recognize the |HV⟩+|VH⟩/√(2) Bell state statistics which for higher k generalize accordingly and the magnitude of co-fluctuations drop off as 2^1-k.Combining Hadamard operations with polarizing beam splitters, one can extend beyond generation of GHZ states to cluster states. Cluster states are special states which allow a graphical representation to denote a vertex connectivity structure according to the eigenvalue relation σ_2^(i)⊗_j∈ngbh(i)σ_1^(j)ψ = ±ψ,where ngbh(i) denotes the set of vertices that share an edge with the vertex i and there is a correspondence between the Pauli spin operators as σ_1≡σ_z and σ_2≡σ_x. These eigenequations are known as stabilizer generators and they uniquely define the state<cit.>. Analogously, we define the TMS-cluster state to be one which obeys the TMS-stabilizer relationa_2^(i)∏_j∈ngbh(i)a_1^(j)∏_k ∉ngbh(i)a_0^(k) = β∀ i,where ∉ngbh(i) is not inclusive of i and β>0. Note, we have gone from an eigenvalue relation to one of nonzero central moment expectations. This is because our central moment operators are not eigenoperators of the squeezed states. Additionally, we have to include the a_0 measures for all other vertices, even those which are not connected to the ith vertex. As the analogy goes, we are post selecting on coincidences and the measure a_0^(i) is no longer simply the identity. We form a 2D TMS-cluster state by following the footsteps of Bodiya and Duan <cit.>. All that is necessary are TMS-Bell states, Hadamard rotations, and polarizing beamsplitters. The action of a polarizing beamsplitter followed by measuring central moment expectation of the whole state means that components of the two beams incident on the PBS which go the same direction will be neglected. They do not share central moment expectation with the vacuum which occupies the other mode. Consequentially, the action of using a PBS on the state restricts one to only the correlated horizontal polarization statistics or correlated vertical polarization statistics. This manifests itself as an enforcement of the measures a_0b_0 and a_1b_1 being nonzero. Starting from the TMS-Bell state (<ref>) we set the relative phase ϕ=0, and a Hadamard operation on either the a or b mode changes the nonzero central moment expectations froma_0b_0=a_1b_1=a_2b_2=-a_3b_3→a_0b_0=a_1b_2=a_2b_1=a_3b_3.The magnitude of a_0b_0=2√(δ) is preserved under this operation and the latter represents a TMS-cluster state with nonzero TMS-stabilizer relations a_2b_1=b_2a_1=2√(δ).Starting with 2 such TMS-cluster states on modes a,b and c,d, a PBS on modes a and c transforms the TMS-stabilizer relation asa_2b_1c_0d_0=b_2a_1c_0d_0=c_2d_1a_0b_0=d_2c_1a_0b_0 → a_2b_1c_2d_1=a_1b_0c_1d_0=a_0b_2c_1d_0=a_0b_0c_1d_2,where the latter results are calculated analytically and in general do not have a simple relation to the preceding TMS-stabilizers. Additionally, the transformations show a mimicry of the involution property of Pauli matrices (σ_i σ_i = σ_0). For instance, the outcome a_0b_2c_1d_0 is similar to a product of the stabilizer a1b2 (≡σ_1 ⊗σ_2) and the post selection on the PBS subspace of a1c1 such that a1b0c1d0·a1b2c0d0 = a0b2c1d0.Following with a Hadamard operation on the a mode we graphically represent this operation in Fig. <ref>(a). In the syntax of Bodiya and Duan <cit.> we denote the usage of a PBS followed by Hadamard on one mode as a PBS operation. The resulting TMS-stabilizer is a star cluster with center c as indicated in figure <ref>b and TMS-stabilizersa_2c_1b_0d_0 =b_2c_1b_0d_0=c_2a_1b_1d_1 =d_2c_1a_0b_0.The graphical procedure gives an effective means to see how one could continue to combine 4 mode TMS-star clusters to form a linear TMS-cluster with additional vertices which only contain one connection. These vertices are known as leaves in the graph community. Using the leaves, one could then combine several linear clusters to form a 2D cluster, combine many 2D clusters to form a 3D cluster and so on.With respect to noise, the effect on co-fluctuation due to Bodiya and Duan's procedure falls in line with the results we have presented thus far. In particular, each usage of a PBS operation sees a drop in co-fluctuation by a factor of 2.Although the co-fluctuation decreases exponentially with the size of the state, the central moment expectations can grow. For instance, starting with k Bell states, there are n=2k spatial modes and the magnitude of central moment expectation is ∏_i=1^na^(i)_0=√(2)coshrsinhr^n. If a loss of 0<γ≤1 was introduced per spatial mode, the expectation would drop. For scalability, we requireγ√(2)coshrsinhr^n≥1sinh2r≥√(2)/γ,such that the nonzero measures of the state will be distinguishable from those that are. As an example, a modest value of r=2.3 (which correlates to an average of ∼ 49 photons per spatial mode) can tolerate a modal loss of γ=0.028≈-15.5 dB and independent of the size of the state, the central moment expectation would be nonzero. We can consume the loss tolerance to build a TMS-cluster via PBS operations. As mentioned previously, each PBS operation is equivalent to roughly -3 dB loss therefore, by applying it to each spatial mode 3 times, one can generate a 2D TMS-cluster and still have a loss budget of -6.5 dB per mode.In order for single photon cluster states to deterministically perform computation, they require the capability of altering upcoming measurement settings dependent on previous measurement outcomes. This is known as feed forward. The decision making protocol relies on the binary nature of the measurements (a single photon will only trigger one detector) and modulo 2 addition. Both the reliance on expectation values instead of eigenvalue relations and real valued intensity measures prevent an extension of these squeezed states to the feed forward protocol. Instead, we have to invoke another factor of 2 hit in efficiency per spatial mode by post selecting on the measurement projections which do not require feed forward; dropping the loss budget of our r=2.3 example down to -3.5 dB. We believe that this approach will allow one to get in the order of 20 to 50 modes before the statistics become too noisy to bound in a reasonable time. Additionally, the reliability of state generation and connections to the recently proposed measurement based classical computing protocol <cit.> may lead to a non-trivial experiment which leverages the power of these quantum type statistics to perform a calculation which is not efficiently computable by a classical computer.In conclusion, we have explored the route of using central moment statistics on number correlated classical intensity states in lieu of post-selected coincidences on photon pair generated states. We show by using only linear optics to generate the state, central moment statistics are capable of exhibiting cluster state statistics that are non-vanishing for arbitrarily large states and finite loss. The tradeoff in using linear optics to build these TMS-cluster state statistics is an exponential increase in uncertainty of the measurement as a function of the size of the state. This approach is in contrast to post-selected single photons and linear optics which sees exponential wait times in coincidence detection both due to post-selection and the stochastic nature of their generation process.CCT, JS and PMA would like to acknowledge support of this work fromOffice of the Secretary of Defense ARAP QSEP program.JS would like to acknowledge support from the National Science Foundation. CCT would also like to thank Gregory A. Howland and Michael L. Fanto for helpful discussions. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of Air Force Research Laboratory.apsrev4-1 | http://arxiv.org/abs/1708.07588v1 | {
"authors": [
"Christopher C. Tison",
"James Schneeloch",
"Paul M. Alsing"
],
"categories": [
"quant-ph"
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"primary_category": "quant-ph",
"published": "20170825005732",
"title": "Central Moment Analogues to Linear Optically Generated Cluster States"
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<≪ >≫ ∼ #1to 0pt#1 3pt"2182.0pt"13C 3pt"2182.0pt"13E | http://arxiv.org/abs/1708.07790v1 | {
"authors": [
"Stanley P. Owocki",
"Richard H. D. Townsend",
"Eliot Quataert"
],
"categories": [
"astro-ph.SR"
],
"primary_category": "astro-ph.SR",
"published": "20170825155643",
"title": "Super-Eddington stellar winds: unifying radiative-enthalpy vs. flux-driven models"
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Subspace Approximation for Approximate Nearest Neighbor Search in NLP Jing Wang [email protected] December 30, 2023 ===================================================================== Most natural language processing tasks can be formulated as the approximated nearest neighbor search problem, such as word analogy, document similarity, machine translation. Take the question-answering task as an example, given a question as the query, the goal is to search its nearest neighbor in the training dataset as the answer. However, existing methods for approximate nearest neighbor search problem may not perform well owing to the following practical challenges: 1) there are noise in the data; 2) the large scale dataset yields a huge retrieval space and high search time complexity. In order to solve these problems, we propose a novel approximate nearest neighbor search framework which i) projects the data to a subspace based spectral analysis which eliminates the influence of noise; ii) partitions the training dataset to different groups in order to reduce the search space. Specifically, the retrieval space is reduced from O(n) to O(log n) (where n is the number of data points in the training dataset). We prove that the retrieved nearest neighbor in the projected subspace is the same as the one in the original feature space. We demonstrate the outstanding performance of our framework on real-world natural language processing tasks.§ INTRODUCTIONArtificial intelligence (AI) is a thriving field with active research topics and practical products, such as Amazon's Echo, Goolge's home smart speakers and Apple's Siri. The world becomes enthusiastic to communicate with these intelligent products. Take Amazon Echo for example,you can ask Echo the calories of every food in your plate if you are on diet. Whenever you need to check your calendar, just ask “Alexa, what's on my calendar today?” It boosts the development of natural language processing which refers to the AI technology that makes the communication between AI products and humans with human language possible. It is shown that the communication between human and AI products are mainly in the format of question answering (QA). QA is a complex and general natural language task. Most of natural language processing tasks can be treated as question answering problem, such as word analogy task <cit.>, machine translation <cit.>, named entity recognition (NER) <cit.>, part-of-speech tagging (POS) <cit.>, sentiment analysis <cit.>.There are many works designed for the question answering task, such as deep learning models <cit.> , information extraction systems <cit.>. In this work, we propose to solve the question answering task by the approximate nearest neighbor search method.Formally, given the question as a query q∈ R^d, the training data set P={p_1,⋯,p_n}∈ R^d× n, the nearest neighbor search aims to retrieve the nearest neighbor of the query q, denoted as p^∗ from P as the answer. We assume that p^∗ is within distance 1 from the query q, and all other points are at distance at least 1+ ϵ (ϵ∈ (0,1)) from the query q. The nearest neighbor p^∗ is called a (1+ϵ)-approximate nearest neighbor to q which can be expressed as:∃ p^*∈ P, q-p^*≤ 1and ∀ p∈ P∖{p^*}, q-p≥ 1+However, in real-world natural language processing applications, there are usually noise in the data, such as spelling errors, non-standard words in newsgroups, pause filling words in speech. Hence, we assume the data set P with arbitrary noise t to create P̃ ={p̃_1,⋯,p̃_n}, where p̃_i = p_i+t_i. The query q is perturbed similarly to get q̃=q+t_q. We assume that the noise is bounded, that is t_i≤ϵ/25 and t_q≤ϵ/25.There are many approaches proposed to solve the approximate nearest neighbor search problem. Existing methods can be classified as two groups: the data-independent methods and the data-dependent methods. The data-independent approaches are mainly based on random projection to get data partitions, such as Local Sensitive Search, Minwise Hashing. Recently,the data-dependent methods received interest for its outstanding performance. They utilize spectral decomposition to map the data to different subspace, such as Spectral hashing. However, theoretical guarantee about the performance is not provided. Existing methods can not handle natural language processing problems well for the following reasons. First the data set in natural language processing is usually in large scale which yields a huge search space. Moreover, the singular value decomposition which is widely used to obtain the low-rank subspace is too expensive here.Second, the data is with noise. Existing data-aware projection is not robust to noisy data which cannot lead to correct partitions. To solve the above mentioned problem, we propose a novel iterated spectral based approximate nearest neighbor search framework for general question answering tasks (Random Subspace based Spectral Hashing (RSSH)). Our framework consists of the following major steps: * As the data is with noise, we first project the data to the clean low-rank subspace. We obtain a low-rank approximation within (1+δ) of optimal for spectral norm error by the randomized block Krylov methods which enjoys the time complexity O(nnz(X)) <cit.>. * To eliminate the search space, we partition data to different clusters. With the low-rank subspace approximation, data points with are clustered corresponding to their distance to the subspace. * Given the query, we first locate its nearest subspace and then search the nearest neighbor in the data partition set corresponding to the nearest subspace. With our framework, we provide theoretical guarantees in the following ways: * With the low-rank approximation, we prove that the noise in the projected subspace is small. * With the data partition strategy, all data will fall to certain partition within O(log n) iterations. * We prove that our method can return the nearest neighbor of the query in low-rank subspace which is the nearest neighbor in the clean space.To the best of our knowledge, it is the first attempt of spectral nearest neighbor search for question answering problem with theory justification. Generally, our framework can solve word similarity task, text classification problems (sentiment analysis), word analogy task and named entity recognition problem.The theoretical analysis in this work is mainly inspired by the work in <cit.>. The difference is that the subspace of data sets is computed directly in <cit.>, in our work, we approximate the subspace by a randomized variant of the Block Lanczos method <cit.>. In this way, our method enjoys higher time efficiency and returns a (1+ϵ/5)-approximate nearest neighbor.§ NOTATIONIn this work, we let s_j(M) denote the j-th largest singular value of a real matrix M. M is used to denote the Frobenius norm of M. All vector norms, i.e. v for v∈^d, refer to the ℓ_2-norm.The spectral norm of a matrix X∈^n× d is defined as X = sup_y∈^d: y=1X y,where all vector norms · refer throughout to the ℓ_2-norm. It is clear that M=s_1(M) equals the spectral norm of M. The Frobenius norm of X is defined as X=(∑_ij X_ij^2)^1/2, and let X^ denote the transpose of X.A singular vector of X is a unit vector v ∈^dassociated with a singular value s∈ and a unit vectoru ∈^n such that X v = s u and u^ X = s v^.(We may also refer to v and u as a pair of right-singular and left-singular vectors associated with s.)Let p_Ũ denote the projection of a point p onto Ũ. Then the distance between a point x and a set (possibly a subspace) S is defined as d(x,S) = inf_y∈ Sx-y.§ PROBLEM Given an n-point dataset P and a query point q, both lying in a k-dimensional space U ⊂ ^d, we aim to find its nearest neighbor q^* which satisfying that:∃ p^*∈ Psuch that q-p^*≤ 1and ∀ p∈ P∖{p^*}, q-p≥ 1+Assume that the data points are corrupted by arbitrary small noise t_i which is bounded t_i≤/16 for all i (∈(0,1)). The observed set P̃ consists of points p̃_i=p_i+t_i for all p_i∈ P and the noisy query points q̃ = q+t_q with t_q≤/16.§ ALGORITHM§.§ Subspace ApproximationWe utilize a randomized variant of the Block Lanczos method proposed in <cit.> to approximate the low-rank subspace of the data set. <cit.> For any A∈ R^m× d, the k-dimensional low-rank subspace obtained by singular value decomposition is denoted as A_k. Algorithm <ref> returns Z_k which forms the low-rank approximation _k = Z_k Z_k^⊤ A, then the following bounds hold with probability at least 9/10: A - _k_2 ≤ (1+η) A-A_k_2≤ (1+η) σ_k ∀ i ∈ [k], |z_i^⊤ AA^⊤ z_i - u_i^⊤ AA^⊤ u_i|=|_i^2-σ_i^2 | ≤ησ_k+1^2 . where _i is the i-th singular value of A. The runtime of the algorithm is O(md k log(m)/√(η)+ k^2(m+d)/η).Algorithm <ref> returns the matrix Z_i which isthe approximation to the left singular vectors of data matrix A. We use Z^TA to approximate theright singular vectors of data matrix A. §.§ Data Partition<cit.> The nearest neighbor of q̃ in P̃ is p̃^*. Algorithm <ref> terminates within O(log n) iterations. Let U be the k-dimensional subspace of P with projection matrix V, let Ũ be the k-dimensional subspace of P̃ with projection matrix , let S be the low-rank approximation returned by Algorithm <ref> with projection matrix Z_k. The distance between data points and subspace is computed as: ∑_p̃∈P̃ d(,)^2= ∑_p̃∈P̃inf_y∈||-y||_2^2 = inf||-_k_k^T||_2^2 . ∑_p̃∈P̃ d(,S)^2= ∑inf_y∈ S||-y||_2^2 = inf||-Z_kZ_k^T||_2^2. According to Theorem <ref>, we can have: ||-Z_kZ_k^⊤||_2 ≤ (1+η)||-U_kU_k^⊤||_2 , We can get ∑_p̃∈P̃ d(,S) ≤ (1+η)^2 ∑_p̃∈P̃ d(,). Since Ũ minimizes the sum of squared distances from all p̃∈P̃ to Ũ, ∑_p̃∈P̃ d(p̃,Ũ)^2 ≤∑_p̃∈P̃ d(p̃,U)^2 ≤∑_p̃∈P̃p̃-p^2 ≤α^2 n. Then, we can get: ∑_p̃∈P̃ d(p̃,S)^2 ≤ (1+η)^2α^2n. Hence, there are at most half of the points in P̃ with distance to Sgreater than √(2)(1+η)α. The set M captures at least a half fraction of points. The algorithm then proceeds on the remaining set. After O(log n) iterations all points of P̃ must be captured. The approximated subspace S that captures p̃^* returns this as the (1+/5)-approximate nearest neighbor of q̃ (in Ũ). Fix p≠ p^* that is captured by the same S, and use the triangle inequality to write p - p̃_S ≤p-p̃ + p̃ - p̃_S≤α + √(2)(1+η) α≤ 4α. Similarly for p^*, p^* - p̃^*_S≤ 4α, and by our assumption q - q̃≤α. By the triangle inequality, we get q̃ - p̃^*_S ≤-p +p-^*_S≤-q+q-p+p-^*_S≤q-p+5α, q̃ - p̃^*_S ≥-p-^*_S-p≥q-p -q--^*_S-p≥q-p-5α, Similarly, we boundq̃ - p̃^*_S q̃ - p̃_S/q̃ - p̃^*_S =q-p ± 5α/ q-p^* ± 5α = q-p ±15 / q-p^* ±15 ≥q-p - 15 /q-p^* + 15 ≥ q-p^*+ 45 / q-p^*+ 15 > 1+15 . By using Pythagoras' Theorem (recall both p̃_S,p̃^*_S∈ S), q̃_S - p̃_S/q̃_S - p̃^*_S = q̃ - p̃_S - q̃ - q̃_S/q̃ - p̃^*_S - q̃ - q̃_S > (1+15 )^2. Hence, p̃^* is reported by the k-dimensional subspace it is assigned to. § EXPERIMENT In this experiment, we compare our algorithm with existing hashing algorithms. §.§ Baseline algorithms Our comparative algorithms include state-of-the-art learning to hashing algorithm such as * Anchor Graph Hashing (AGH) <cit.> * Circulant Binary Embedding (CBE) <cit.> * Multidimensional Spectral Hashing (MDSH) <cit.> * Iterative quantization (ITQ) <cit.> * Spectral Hashing <cit.> * Sparse Projection (SP) <cit.>We refer our algorithm as Random Subspace based Spectral Hashing (RSSH). §.§ Datasets. Our experiment datasets include MNIST [http://yann.lecun.com/exdb/mnist/], CIFAR-10[https://www.cs.toronto.edu/ kriz/cifar.html/], COIL-20 [http://www.cs.columbia.edu/CAVE/software/softlib/coil-20.php] and the 2007 PASCAL VOC challenge dataset. MNIST. It is a well-known handwritten digits dataset from “0” to “9”. The dataset consists of 70,000 samples in feature space of dimension 784. We split the samples to a training and a query set which containing 69,000 and 1,000 samples respectively. CIFAR-10. There are 60,000 image in 10 classes, such as “horse” and “truck”. We use the default 59,000 training set and 1,000 testing as query set. The image is with 512 GIST feature. COIL-20. It is from the Columbia University Image Library which contains 20 objects. Each image is represented by a feature space 1024 dimension. For each object, we choose 60% images for training and the others are querying. VOC2007. The VOC2007 dataset consists of three subsets as training, validation and testing. We use thefirst two subsets as training containing 5,011 samples and the other as query containing 4,096 samples. We set each image to the size of [80, 100] and extract the HOG feature with cell size 10 as their feature space [http://www.vlfeat.org/]. All the images in VOC2007 are defined into 20 subjects, such as “aeroplane” and “dining tale”. For the classification task on each subject, there are 200 to 500 positive samples and the following 4,000 are negative. Thus, the label distribution of the query set is unbalanced. A brief description of the datasets are presented in Table <ref>.§.§ Evaluation MetricsAll the experiment datasets are fully annotated. We report the classification result based on the groundtruth label information. That is, the label of the query is assigned by its nearest neighbor. For the first three datasets in Table <ref>, we report the classification accuracy. For the VOC2007 dataset, we report the precision as the label distribution is highly unbalanced. The criteria are defined in terms of true positive (TP), true negative (TN), false positive (FP) and false negative (FN) as,Precision = TP/TP+FP, Accuracy = TP+TN/TP+TN+FP+FN. For the retrieval task, we report the recall with top [1, 10, 25] retrieved samples. The true neighbors are defined by the Euclidean distance. We report the aforementioned evaluation criteria with varying hash bits (r) in the range of [2, 128].§.§ Classification ResultsThe classification accuracy on CIFAR-10, MNIST and COIL-20 are reported in Figure <ref>a, <ref>b and <ref>c. We can see that our algorithm achieves the best accuracy in terms of the number of hash bits in the three datasets. For example, on CIFAR-10 with r=48, the accuracy of our algorithm RSSN reaches 53.20% while the comparative algorithms are all less than 40.00%. The increase of hash bit promotes the accuracy of all algorithms, our algorithm remains the leading place. For example, on MNIST with r=96, ITH and SH reach the accuracy of 93.00%, but our algorithm still enjoys 4.00% advantage with 97.30%.Moreover, our algorithm obtains significant good performance even with limited information, that is, the r is small. For instance, in terms of r=8, RSSN reaches the accuracy of 87.70%, much better than the comparative algorithms. For the classification results on VOC2007, we report the accuracy on 12 of 20 classes as representation in Figure <ref>. We can see that it is a tough task for all the methods, but our algorithm still obtains satisfying performance. For example, on the classification task of “horse”, our algorithm obtains around 10% advantage over the all the comparative algorithms. The average precision on all the 20 classes are presented on Figure <ref>. We can see that our algorithm obtains the overall best result. §.§ Retrieval ResultsThe retrieval results on MNIST, CIFAR-10 and COIL-20 are presented in Figure <ref>d to Figure <ref>l. Our algorithm obtains the best recall with varying number of retrieved samples. For example, on CIFAR-10 with Top 10 retrieval and r=96, our algorithm reaches the recall over 70%, while the others are less than 20%. On MNIST with Top 25 retrieved samples and r=128, the recall of RSSN reaches 90%, while the comparatives algorithms are around 40%.apalike | http://arxiv.org/abs/1708.07775v1 | {
"authors": [
"Jing Wang"
],
"categories": [
"cs.AI"
],
"primary_category": "cs.AI",
"published": "20170825152615",
"title": "Subspace Approximation for Approximate Nearest Neighbor Search in NLP"
} |
Ö. D. AkyildizUniversity of WarwickThe Alan Turing [email protected]. Míguez Universidad Carlos III de Madrid & Instituto de Investigación Sanitaria Gregorio MarañónNudging the particle filterThis work was partially supported by Ministerio de Economía y Competitividad of Spain (TEC2015-69868-C2-1-R ADVENTURE), the Office of Naval Research Global (N62909-15-1-2011), and the regional government of Madrid (program CASICAM-CM S2013/ICE-2845).Ömer Deniz Akyildiz Joaquín Míguez Received: date / Accepted: date ====================================================================================================================================================================================================================================================================================== We investigate a new sampling scheme aimed at improving the performance of particle filters whenever (a) there is a significant mismatch between the assumed model dynamics and the actual system, or (b) the posterior probability tends to concentrate in relatively small regions of the state space. The proposed scheme pushes some particles towards specific regions where the likelihood is expected to be high, an operation known as nudging in the geophysics literature. We re-interpret nudging in a form applicable to any particle filtering scheme, as it does not involve any changes in the rest of the algorithm. Since the particles are modified, but the importance weights do not account for this modification, the use of nudging leads to additional bias in the resulting estimators. However, we prove analytically that nudged particle filters can still attain asymptotic convergence with the same error rates as conventional particle methods. Simple analysis also yields an alternative interpretation of the nudging operation that explains its robustness to model errors. Finally, we show numerical results that illustrate the improvements that can be attained using the proposed scheme. In particular, we present nonlinear tracking examples with synthetic data and a model inference example using real-world financial data.§ INTRODUCTION §.§ BackgroundState-space models (SSMs) are ubiquitous in many fields of science and engineering, including weather forecasting, mathematical finance, target tracking, machine learning, population dynamics, etc., where inferring the states of dynamical systems from data plays a key role.A SSM comprises a pair of stochastic processes (x_t)_t≥ 0 and (y_t)_t≥ 1 called signal process and observation process, respectively. The conditional relations between these processes are defined with a transition and an observation model (also called likelihood model) where observations are conditionally independent given the signal process, and the latter is itself a Markov process. Given an observation sequence, y_1:t, the filtering problem in SSMs consists in the estimation of expectations with respect to the posterior probability distribution of the hidden states, conditional on y_1:t, which is also referred to as the filtering distribution.Apart from a few special cases, neither the filtering distribution nor the integrals (or expectations) with respect to it can be computed exactly; hence, one needs to resort to numerical approximations of these quantities. Particle filters (PFs) have been a classical choice for this task since their introduction by <cit.>; see also <cit.>. The PF constructs an empirical approximation of the posterior probability distribution via a set of Monte Carlo samples (usually termed particles) which are modified or killed sequentially as more data are taken into account. These samples are then used to estimate the relevant expectations. The original form of the PF, often referred to as the bootstrap particle filter (BPF), has received significant attention due to its efficiency in a variety of problems, its intuitive appeal and its straightforward implementation. A large body of theoretical work concerning the BPF has also been compiled. For example, it has been proved that the expectations with respect to the empirical measures constructed by the BPF converge to the expectations with respect to the true posterior distributions when the number of particles is large enough <cit.> or that they converge uniformly over time under additional assumptions related to the stability of the true distributions <cit.>. Despite the success of PFs in relatively low dimensional settings, their use has been regarded impractical in models where (x_t)_t≥ 0 and (y_t)_t≥ 1 are sequences of high-dimensional random variables.In such scenarios, standard PFs have been shown to collapse <cit.>. This problem has received significant attention from the data assimilation community. The high-dimensional models which are common in meteorology and other fields of Geophysics are often dealt with via an operation called nudging <cit.>. Within the particle filtering context, nudging can be defined as a transformation of the particles, which are pushed towards the observations using some observation-dependent map <cit.>. If the dimensions of the observations and the hidden states are different, which is often the case, a gain matrix is computed in order to perform the nudging operation. In <cit.>, nudging is performed after the sampling step of the particle filter. The importance weights are then computed accordingly, so that they remain proper. Hence, nudging in this version amounts to a sophisticated choice of the importance function that generates the particles. It has been shown (numerically) that the schemes proposed by <cit.> can track high-dimensional systems with a low number of particles. However, generating samples from the nudged proposal requires costly computations for each particle and the evaluation of weights becomes heavier as well. It is also unclear how to apply existing nudging schemes when non-Gaussianity and nontrivial nonlinearities are present in the observation model.A related class of algorithms includes the so-called implicit particle filters (IPFs) <cit.>. Similar to nudging schemes, IPFs rely on the principle of pushing particles to high-probability regions in order to prevent the collapse of the filter in high-dimensional state spaces. In a typical IPF, the region where particles should be generated is determined by solving an algebraic equation. This equation is model dependent, yet it can be solved for a variety of different cases (general procedures for finding solutions are given by <cit.> and <cit.>). The fundamental principle underlying IPFs, moving the particles towards high-probability regions, is similar to nudging. Note, however, that unlike IPFs, nudging-based methods are not designed to guarantee that the resulting particles land on high-probability regions; it can be the case that nudged particles are moved to relatively low probability regions (at least occasionally). Since an IPF requires the solution of a model-dependent algebraic equation for every particle, it can be computationally costly, similar to the nudging methods by <cit.>.Moreover, it is not straightforward to derive the map for the translation of particles in general models, hence the applicability of IPFs depends heavily on the specific model at hand. §.§ Contribution In this work, we propose a modification of the PF, termed the nudged particle filter (NuPF) and assess its performance in high dimensional settings and with misspecified models. Although we use the same idea for nudging that is presented in the literature, our algorithm has subtle but crucial differences, as summarized below.* First, we define the nudging step not just as a relaxation step towards observations but as a step that strictly increases the likelihood of a subset of particles. This definition paves the way for different nudging schemes, such as using the gradients of likelihoods or employing random search schemes to move around the state-space. In particular, classical nudging (relaxation) operations arise as a special case of nudging using gradients when the likelihood is assumed to be Gaussian. Compared to IPFs, the nudging operation we propose is easier to implement as we only demand the likelihood to increase (rather than the posterior density). Indeed, nudging operators can be implemented in relatively straightforward forms, without the need to solve model-dependent equations.* Second, unlike the other nudging based PFs, we do not correct the bias induced by the nudging operation during the weighting step. Instead, we compute the weights in the same way they would be computed in a conventional (non-nudged) PF and the nudging step is devised to preserve the convergence rate of the PF, under mild standard assumptions, despite the bias.Moreover, computing biased weightsis usually faster than computing proper (unbiased) weights. Depending on the choice of nudging scheme, the proposed algorithm can have an almost negligible computational overhead compared to the conventional PF from which it is derived. * Finally, we show that a nudged PF for a given SSM (say _0) is equivalent to a standard BPF running on a modified dynamical model (denoted _1). In particular, model _1 is endowed with the same likelihood function as _0 but the transition kernel is observation-driven in order to match the nudging operation. As a consequence, the implicit model _1 is “adapted to the data” and we have empirically found that, for any sufficiently long sequence y_1, …, y_t, the evidence[Given a data set { y_1, …, y_t }, the evidence in favour of a modelis the joint probability density of y_1, ⋯, y_t conditional on , denoted (y_1:t|).] <cit.> in favour of _1 is greater than the evidence in favour of _0. We can show, for several examples, that this implicit adaptation to the data makes the NuPF robust to mismatches in the state equation of the SSM compared to conventional PFs. In particular, provided that the likelihoods are specified or calibrated reliably, we have found that NuPFs perform reliably under a certain amount of mismatch in the transition kernel of the SSM, while standard PFs degrade clearly in the same scenario.In order to illustrate the contributions outlined above, we present the results of several computer experiments with both synthetic and real data. In the first example, we assess the performance of the NuPF when applied to a linear-Gaussian SSM. The aim of these computer simulations is to compare the estimation accuracy and the computational cost of the proposed scheme with several other competing algorithms, namely a standard BPF, a PF with optimal proposal function and a NuPF with proper weights. The fact that the underlying SSM is linear-Gaussian enables the computation of the optimal importance function (intractable in a general setting) and proper weights for the NuPF. We implement the latter scheme because of its similarity to standard nudging filters in the literature. This example shows that the NuPF suffers just from a slight performance degradation compared to the PF with optimal importance function or the NuPF with proper weights, while the latter two algorithms are computationally more demanding.The second and third examples are aimed at testing the robustness of the NuPF when there is a significant misspecification in the state equation of the SSM. This is helpful in real-world applications because practitioners often have more control over measurement systems, which determine the likelihood, than they have over the state dynamics. We present computer simulation results for a stochastic Lorenz 63 model and a maneuvering target tracking problem. In the fourth example, we present numerical results for astochastic Lorenz 96 model, in order to show how a relatively high-dimensional system can be tracked without a major increase of the computational effort compared to the standard BPF. For this set of computer simulations we have also compared the NuPF with the Ensemble Kalman filter (EnKF), which is the de facto choice for tackling this type of systems.Let us remark that, for the two stochastic Lorenz systems, the Markov kernel in the SSM can be sampled in a relatively straightforward way, yet transition probability densities cannot be computed (as they involve a sequence of noise variables mapped by a composition of nonlinear functions). Therefore, computing proper weights for proposal functions other than the Markov kernel itself is, in general, not possible for these examples.Finally, we demonstrate the practical use of the NuPF on a problem where a real dataset is used to fit a stochastic volatility model using either particle Markov chain Monte Carlo (pMCMC) <cit.> or nested particle filters <cit.>.§.§ Organisation The paper is structured as follows. After a brief note about notation, we describe the SSMs of interest and the BPF in Section <ref>. Then in Section <ref>, we outline the general algorithm and the specific nudging schemes we propose to use within the PF. We prove a convergence result in Section <ref> which shows that the new algorithm has the same asymptotic convergence rate as the BPF. We also provide an alternative interpretation of the nudging operation that explains its robustness in scenarios where there is a mismatch between the observed data and the assumed SSM. We discuss the computer simulation experiments in Section <ref> and present results for real data in Section <ref>. Finally, we make some concluding remarks in Section <ref>.§.§ NotationWe denote the set of real numbers as , while ^d = ×d⋯× is the space of d-dimensional real vectors. We denote the set of positive integers withand the set of positive reals with _+. We represent the state space with 𝖷⊂^d_x and the observation space with 𝖸⊂^d_y.In order to denote sequences, we use the shorthand notation x_i_1:i_2 = {x_i_1,…,x_i_2}. For sets of integers, we use [n] = {1,…,n}. Thep-norm of a vector x∈^d is defined by x_p = (x_1^p + ⋯ + x_d^p)^1/p. The L_p norm of a random variable z with probability density function (pdf) p(z) is denoted z_p = (∫ |z|^p p(z)z)^1/p, for p≥ 1. The Gaussian (normal) probability distribution with mean m and covariance matrix C is denoted (m,C). We denote the identity matrix of dimension d with I_d.The supremum norm of a real function φ:𝖷→ is denoted φ_∞ = sup_x ∈𝖷 |φ(x)|. A function is bounded if φ_∞ < ∞ and we indicate the space of real bounded functions 𝖷 as B(𝖷). The set of probability measures onis denoted (), the Borel σ-algebra of subsets of 𝖷 is denoted (𝖷) and the integral of a function φ:𝖷 with respect to a measure μ on the measurable space (𝖷,(𝖷)) is denoted (φ,μ):=∫φμ. The unit Dirac delta measure located at x ∈^d is denoted δ_x( x). The Monte Carlo approximation of a measure μ constructed using N samples is denoted as μ^N. Given a Markov kernel τ(x'|x) and a measure π(x), we define the notation ξ(x') = τπ≜∫τ(x'|x) π(x).§ BACKGROUND§.§ State space models We consider SSMs of the formx_0∼π_0( x_0), x_t | x_t-1 ∼τ_t( x_t|x_t-1), y_t | x_t∼ g_t(y_t|x_t),t ∈,where x_t ∈𝖷 is the system state at time t, y_t ∈𝖸 is the t-th observation, the measure π_0 describes the prior probability distribution of the initial state, τ_t is a Markov transition kernel on 𝖷, and g_t(y_t|x_t) is the (possibly non-normalised) pdf of the observation y_t conditional on the state x_t. We assume the observation sequence { y_t }_t ∈_+ is arbitrary but fixed. Hence, it is convenient to think of the conditional pdf g_t as a likelihood function and we write g_t(x_t) := g_t(y_t|x_t) for conciseness.We are interested in the sequence of posterior probability distributions of the states generated by the SSM. To be specific, at each time t=1, 2, ... we aim at computing (or, at least, approximating) the probability measure π_t which describes the probability distribution of the state x_t conditional on the observation of the sequence y_1:t. When it exists, we use π(x_t|y_1:t) to denote the pdf of x_t given y_1:t with respect to the Lebesgue measure, i.e., π_t( x_t) = π(x_t|y_1:t) x_t.The measure π_t is often termed the optimal filter at time t. It is closely related to the probability measure ξ_t, which describes the probability distribution of the state x_t conditional on y_1:t-1 and it is, therefore, termed the predictive measure at time t. As for the case of the optimal filter, we use ξ(x_t|y_1:t-1) to denote the pdf, with respect to the Lebesgue measure, of x_t given y_1:t-1.§.§ Bootstrap particle filter The BPF <cit.> is a recursive algorithm that produces successive Monte Carlo approximations of ξ_t and π_t for t=1, 2, .... The method can be outlined as shown in Algorithm <ref>. After an initialization stage, where a set of independent and identically distributed (i.i.d.) samples from the prior are drawn, it consists of three recursive steps which can be depicted as,π_t-1^N →_samplingξ_t^N →_weightingπ̃_t^N →_resamplingπ_t^N.Given a Monte Carlo approximation π_t-1^N = 1/N∑_i=1^N δ_x_t-1^(i) computed at time t-1, the sampling step yields an approximation of the predictive measure ξ_t of the form ξ_t^N = 1/N∑_i=1^N δ_x̅_t^(i)by propagating the particles { x_t-1^(i)}_i=1^N via the Markov kernel τ_t(·|x_t-1^(i)). The observation y_t is assimilated via the importance weights w_t^(i)∝ g_t(x_t^(i)), to obtain the approximate filterπ̃_t^N = ∑_i=1^N w_t^(i)δ_x̅_t^(i),and the resampling step produces a set of un-weighted particles that completes the recursive loop and yields the approximationπ_t^N = 1/N∑_i=1^N δ_x_t^(i).The random measures ξ_t^N, π̃_t^N and π_t^N are commonly used to estimate a posteriori expectations conditional on the available observations. For example, if φ is a function 𝖷, then the expectation of the random variable φ(x_t) conditional on y_1:t-1 is [ φ(x_t) | y_1:t-1] = (φ,ξ_t). The latter integral can be approximated using ξ_t^N, namely,(φ,ξ_t)= ∫φ(x_t) ξ_t( x_t) ≈ (φ,ξ_t^N) = ∫φ(x_t)ξ_t^N( x_t) = 1/N∑_i=1^N φ(x̅_t^(i)).Similarly, we can have estimators (φ,π̃_t^N) ≈ (φ,π_t) and (φ,π_t^N) ≈ (φ,π_t). Classical convergence results are usually proved for real bounded functions, e.g., if φ∈ B(𝖷) then lim_N∞ | (φ,π_t) - (φ,π_t^N) | = 0under mild assumptions; see <cit.> and references therein.The BPF can be generalized by using arbitrary proposal pdf's q_t(x_t|x_t-1^(i),y_t), possibly observation-dependent, instead of the Markov kernel τ_t(·|x_t-1^(i)) in order to generate the particles {x̅_t^(i)}_i=1^N in the sampling step. This can lead to more efficient algorithms, but the weight computation has to account for the new proposal and we obtain <cit.>w_t^(i)∝g_t(x̅_t^(i)) τ_t(x̅_t^(i) | x_t^(i))/q_t(x̅_t^(i) | x_t-1^(i), y_t),which can be more costly to evaluate. This issue is related to the nudged PF to be introduced in Section <ref> below, which can be interpreted as a scheme to choose a certain observation-dependent proposal q_t(x_t|x_t-1^(i),y_t). However, the new method does not require that the weights be computed as in (<ref>) in order to ensure convergence of the estimators. § NUDGED PARTICLE FILTER §.§ General algorithm Compared to the standard BPF, the nudged particle filter (NuPF) incorporates one additional step right after the sampling of the particles {x̅_t^(i)}_i=1^N at time t. The schematic depiction of the BPF in (<ref>) now becomesπ_t-1^N →_samplingξ_t^N →_nudgingξ̃_t^N →_weightingπ̃_t^N →_resamplingπ_t^N,where the new nudging step intuitively consists in pushing a subset of the generated particles {x̅_t^(i)}_i=1^N towards regions of the state space 𝖷 where the likelihood function g_t(x) takes higher values.When considered jointly, the sampling and nudging steps in (<ref>) can be seen as sampling from a proposal distribution which is obtained by modifying the kernel τ_t(·|x_t-1) in a way that depends on the observation y_t. Indeed, this is the classical view of nudging in the literature <cit.>. However, unlike in this classical approach, here the weighting step does not account for the effect of nudging. In the proposed NuPF, the weights are kept the same as in the original filter, w_t^(i)∝ g_t(x_t^(i)). In doing so, we save computations but, at the same time, introduce bias in the Monte Carlo estimators. One of the contributions of this paper is to show that this bias can be controlled using simple design rules for the nudging step, while practical performance can be improved at the same time.In order to provide an explicit description of the NuPF, let us first state a definition for the nudging step.A nudging operator α_t^y_t: 𝖷→𝖷 associated with the likelihood function g_t(x) is a map such thatifx' = α_t^y_t(x)then g_t(x') ≥ g_t(x)for every x,x'∈𝖷. Intuitively, we define nudging herein as an operation that increases the likelihood. There are several ways in which this can be achieved and we discuss some examples in Sections <ref> and <ref>. The NuPF with nudging operator α_t^y_t:𝖷𝖷 is outlined in Algorithm <ref>. It can be seen that the nudging operation is implemented in two stages. * First, we choose a set of indices ℐ_t ⊂ [N] that identifies the particles to be nudged. Let M=|ℐ_t| denote the number of elements in ℐ_t. We prove in Section <ref> that keeping M ≤𝒪(√(N)) allows the NuPF to converge with the same error rates 𝒪(1/√(N)) as the BPF. In Section <ref> we discuss two simple methods to build ℐ_t in practice.* Second, we choose an operator α_t^y_t that guarantees an increase of the likelihood of any particle. We discuss different implementations of α_t^y_t in Section <ref>.We devote the rest of this section to a discussion of how these two steps can be implemented (in several ways).§.§ Selection of particles to be nudgedThe set of indices ℐ_t, that identifies the particles to be nudged in Algorithm <ref>, can be constructed in several different ways, either random or deterministic. In this paper, we describe two simple random procedures with little computational overhead. * Batch nudging: Let the number of nudged particles M be fixed. A simple way to construct ℐ_t is to draw indices i_1, i_2, …, i_M uniformly from [N] without replacement, and then let ℐ_t = i_1:M. We refer to this scheme as batch nudging, referring to selection of the indices at once. One advantage of this scheme is that the number of particles to be nudged, M, is deterministic and can be set a priori. * Independent nudging: The size and the elements of ℐ_t can also be selected randomly in a number of ways. Here, we have studied a procedure in which, for each index i = 1, ..., N, we assign i ∈ℐ_t with probability M/N. In this way, the actual cardinality |ℐ_t| is random, but its expected value is exactly M. This procedure is particularly suitable for parallel implementations, since each index can be assigned to ℐ_t (or not) at the same time as all others. §.§ How to nudgeThe nudging step is aimed at increasing the likelihood of a subset of individual particles, namely those with indices contained in ℐ_t. Therefore, any map α_t^y_t:𝖷𝖷 such that (g_t ∘α_t^y_t)(x) ≥ g_t(x) when x∈𝖷 is a valid nudging operator. Typical procedures used for optimisation, such as gradient moves or random search schemes, can be easily adapted to implement (relatively) inexpensive nudging steps. Here we briefly describe a few of such techniques.* Gradient nudging: If g_t(x_t) is a differentiable function of x_t, one straightforward way to nudge particles is to take gradient steps. In Algorithm <ref> we show a simple procedure with one gradient step alone, and where γ_t>0 is a step-size parameter and ∇_x g_t(x) denotes the vector of partial derivatives of g_t with respect to the state variables, i.e., ∇_x_t g_t = [ [ ∂ g_t/∂ x_1,t; ∂ g_t/∂ x_2,t; ⋮; ∂ g_t/∂ x_d_x,t; ]]x_t = [ [ x_1,t; x_2,t; ⋮; x_d_x,t; ]] ∈𝖷.Algorithms can obviously be designed where nudging involves several gradient steps. In this work we limit our study to the single-step case, which is shown to be effective and keeps the computational overhead to a minimum. We also note that the performance of gradient nudging can be sensitive to the choice of the step-size parameters γ_t>0, which are, in turn, model dependent[We have found, nevertheless, that fixed step-sizes (i.e., γ_t=γ for all t) work well in practice for the examples of Sections <ref> and <ref>.].* Random nudging: Gradient-free techniques inherited from the field of global optimisation can also be employed in order to push particles towards regions where they have higher likelihoods. A simple stochastic-search technique adapted to the nudging framework is shown in Algorithm <ref>. We hereafter refer to the latter scheme as random-search nudging. * Model specific nudging: Particles can also be nudged using the specific model information. For instance, in some applications the state vector x_t can be split into two subvectors, x_t^obs and x_t^unobs (observed and unobserved, respectively), such that g_t(x_t) = g_t(x_t^obs), i.e., the likelihood depends only on x_t^obs and not on x_t^unobs. If the relationship between x_t^obs and x_t^unobs is tractable, one can first nudge x_t^obs in order to increase the likelihood and then modify x_t^unobs in order to keep it coherent with x_t^obs. A typical example of this kind arises in object tracking problems, where positions and velocities have a special and simple physical relationship but usually only position variables are observed through a linear or nonlinear transformation. In this case, nudging would only affect the position variables. However, using these position variables, one can also nudge velocity variables with simple rules. We discuss this idea and show numerical results in Section <ref>. §.§ Nudging general particle filters In this paper we limit our presentation to BPFs in order to focus on the key concepts of nudging and to ease presentation. It should be apparent, however, that nudging steps can be plugged into general PFs. More specifically, since the nudging step is algorithmically detached from the sampling and weighting steps, it can be easily used within any PF, even if it relies on different proposals and different weighting schemes. We leave for future work the investigation of the performance of nudging within widely used PFs, such as auxiliary particle filters (APFs) <cit.>. § ANALYSIS The nudging step modifies the random generation of particles in a way that is not compensated by the importance weights. Therefore, we can expect nudging to introduce bias in the resulting estimators in general. However, in Section <ref> we prove that, as long as some basic guidelines are followed, the estimators of integrals with respect to the filtering measure π_t and the predictive measure ξ_t converge in L_p as N∞ with the usual Monte Carlo rate 𝒪(1/√(N)). The analysis is based on a simple induction argument and ensures the consistency of a broad class of estimators. In Section <ref> we briefly comment on the conditions needed to guarantee that convergence is attained uniformly over time. We do not provide a full proof, but this can be done by extending the classical arguments in <cit.> or <cit.> and using the same treatment of the nudging step as in the induction proof of Section <ref>. Finally, in Section <ref>, we provide an interpretation of nudging in a scenario with modelling errors. In particular, we show that the NuPF can be seen as a standard BPF for a modified dynamical model which is “a better fit” for the available data than the original SSM. §.§ Convergence in L_pThe goal in this section is to provide theoretical guarantees of convergence for the NuPF under mild assumptions. First, we analyze a general NuPF (with arbitrary nudging operator α_t^y_t and an upper bound on the size M of the index set ℐ_t) and then we provide a result for a NuPF with gradient nudging.Before proceeding with the analysis, let us note that the NuPF produces several approximate measures, depending on the set of particles (and weights) used to construct them. After the sampling step, we have the random probability measure ξ_t^N = 1/N∑_i=1^N δ_x̅_t^(i),which converts intoξ̃_t^N = 1/N∑_i=1^N δ_x̃_t^(i)after nudging. Once the weights w_t^(i) are computed, we obtain the approximate filter π̃_t^N = ∑_i=1^N w_t^(i)δ_x̃_t^(i),which finally yieldsπ_t^N =1/N∑_i=1^N δ_x_t^(i) after the resampling step.Similar to the BPF, the simple Assumption <ref> stated next is sufficient for consistency and to obtain explicit error rates <cit.> for the NuPF, as stated in Theorem <ref> below.The likelihood function is positive and bounded, i.e.,g_t(x_t) > 0 andg_t_∞ = sup_x_t∈𝖷 |g_t(x_t)| < ∞for t = 1,…,T. Let y_1:T be an arbitrary but fixed sequence of observations, with T<∞, and choose any M ≤√(N) and any map α_t^y_t:→. If Assumption <ref> is satisfied and |ℐ_t|=M, then (φ,π_t^N) - (φ, π_t) _p ≤c_t,pφ_∞/√(N)for every t=1, 2, ..., T, any φ∈ B(𝖷), any p ≥ 1 and some constant c_t,p < ∞ independent of N.See Appendix <ref> for a proof.Theorem <ref> is very general; it actually holds for any map α_t^y_t: →, i.e., not necessarily a nudging operator. We can also obtain error rates for specific choices of the nudging scheme. A simple, yet practically appealing, setup is the combination of batch and gradient nudging, as described in Sections <ref> and <ref>, respectively. The gradient of the likelihood is bounded. In particular, there are constants G_t<∞ such that∇_x g_t(x)_2 ≤ G_t< ∞for every x ∈𝖷 and t=1, 2, …, T. Choose the number of nudged particles, M >0, and a sequence of step-sizes, γ_t>0, in such a way that sup_1 ≤ t ≤ Tγ_t M ≤√(N) for some T<0. If Assumption <ref> holds and φ is a Lipschitz test function, then the error introduced by the batch gradient nudging step with |ℐ_t|=M can be bounded as,(φ, ξ_t^N) - (φ, ξ̃_t^N) _p ≤LG_t/√(N),where L is the Lipschitz constant of φ, for every t=1, …, T. See Appendix <ref> for a proof. It is straightforward to apply Lemma <ref> to prove convergence of the NuPF with a batch gradient-nudging step. Specifically, we have the following result. Let y_1:T be an arbitrary but fixed sequence of observations, with T<∞, and choose a sequence of step sizes γ_t>0 and an integer M such thatsup_1≤ t≤ Tγ_t M ≤√(N).Let π_t^N denote the filter approximation obtained with a NuPF with batch gradient nudging. If Assumptions <ref> and <ref> are satisfied and |ℐ_t|=M, then (φ,π_t^N) - (φ, π_t) _p ≤c_t,pφ_∞/√(N)for every t=1, 2, ..., T, any bounded Lipschitz function φ, some constant c_t,p < ∞ independent of N for any integer p ≥ 1.The proof is straightforward (using the same argument as in the proof of Theorem <ref> combined with Lemma <ref>) and we omit it here. We note that Lemma <ref> provides a guideline for the choice of M and γ_t. In particular, one can select M = N^β, where 0 < β < 1, together with γ_t ≤ N^1/2 - β in order to ensure that γ_t M ≤√(N). Actually, it would be sufficient to set γ_t ≤ C N^1/2 - β for some constant C<∞ in order to keep the same error rate (albeit with a different constant in the numerator of the bound). Therefore,Lemma <ref> provides a heuristic to balance the step size with the number of nudged particles[Note that the step sizes may have to be kept small enough to ensure that g_t(x̅_t^(i) + γ_t ∇_x g_t(x̅_t^(i)) ) ≥ g_t(x̅_t^(i)), so that proper nudging, according to Definition <ref>, is performed.]. We can increase the number of nudged particles but in that case we need to shrink the step size accordingly, so as to keep γ_t M ≤√(N). Similar results can be obtained using the gradient of the log-likelihood, log g_t, if the g_t comes from the exponential family of densities. §.§ Uniform convergence Uniform convergence can be proved for the NuPF under the same standard assumptions as for the conventional BPF; see, e.g., <cit.>. The latter can be summarised as follows <cit.>:(i) The likelihood function is bounded and bounded away from zero, i.e., g_t ∈ B(𝖷) and there is some constant a>0 such that inf_t>0, x∈𝖷 g_t(x) ≥ a.(ii) The kernel mixes sufficiently well, namely,for any given integer m there is a constant 0<ε<1 such thatinf_t>0;(x,x')∈𝖷^2τ_t+m|t(A|x) /τ_t+m|t(A|x')> εfor any Borel set A, where τ_t+m|t is the composition of the kernels τ_t+m∘τ_t+m-1∘⋯∘τ_t.When (i) and (ii) above hold, the sequence of optimal filters {π_t }_t≥ 0 is stable and it can be proved that sup_t>0 (φ,π_t) - (φ,π_t^N) _p ≤c_p/√(N)for any bounded function φ∈ B(𝖷), where c_p<∞ is constant with respect to N and t and π_t^N is the particle approximation produced by either the NuPF (as in Theorem <ref> or, provided sup_t>0 G_t < ∞, as in Theorem <ref>) or the BPF algorithms. We skip a formal proof as, again, it is straightforward combination of the standard argument by <cit.> (see also, e.g., <cit.> and <cit.>) with the same handling of the nudging operator as in the proofs of Theorem <ref> or Lemma <ref> . §.§ Nudging as a modified dynamical model We have found in computer simulation experiments that the NuPF is consistently more robust to model errors than the conventional BPF. In order to obtain some analytical insight of this scenario, in this section we reinterpret the NuPF as a standard BPF for a modified, observation-driven dynamical model and discuss why this modified model can be expected to be a better fit for the given data than the original SSM. In this way, the NuPF can be seen as an automatic adaptation of the underlying model to the available data.The dynamic models of interest in stochastic filtering can be defined by a prior measure τ_0, the transition kernels τ_t and the likelihood functions g_t(x)=g_t(y_t|x), for t≥ 1. In this section we write the latter as g_t^y_t(x) = g_t(y_t|x), in order to emphasize that g_t is parametrised by the observation y_t, and we also assume that every g_t^y_t is a normalised pdf in y_t for the sake of clarity. Hence, we can formally represent the SSM defined by (<ref>), (<ref>) and (<ref>) as _0={τ_0,τ_t,g_t^y_t}.Now, let us assume y_1:T to be fixed and construct the alternative dynamical model _1 = {τ_0, τ̃_t^y_t, g_t^y_t}, whereτ̃_t^y_t( x_t | x_t-1 ) :=(1-ε_M)τ_t( x_t|x_t-1) + ε_M ∫δ_α_t^y_t(x̅_t) ( x_t) τ_t(x̅_t | x_t-1)is an observation-driven transition kernel, ε_M = M/N and the nudging operator α_t^y_t is a one-to-one map that depends on the (fixed) observation y_t. We note that the kernel τ̃_t^y_t jointly represents the Markov transition induced by the original kernel τ_t followed by an independent nudging transformation (namely, each particle is independently nudged with probability ε_M). As a consequence, the standard BPF for model _1 coincides exactly with a NuPF for model _0 with independent nudging and operator α_t^y_t. Indeed, according to the definition of τ̃_t^y_t in (<ref>), generating a sample x̃_t^(i) from τ̃_t^y_t( x_t | x_t-1^(i)) is a three-step process where* we first draw x̅_t^(i) from τ_t( x_t|x_t-1^(i)), * then generate a sample u_t^(i) from the uniform distribution (0,1), and * if u_t^(i) < ε_M then we set x̃_t^(i) = α_t^y_t(x̅_t^(i)), else we set x̃_t^(i) = x̅_t^(i).After sampling, the importance weight for the BPF applied to model _1 is w_t^(i)∝ g_t^y_t(x̃_t^(i)). This is exactly the same procedure as in the NuPF applied to the original SSM _0 (see Algorithm <ref>).Intuitively, one can expect that the observation-driven _1 is a better fit for the data sequence y_1:T than the original model _0. Within the Bayesian methodology, a common approach to compare two competing probabilistic models (_0 and _1 in this case) for a given data set y_1:t is to evaluate the so-called model evidence <cit.> for both _0 and _1. The evidence (or likelihood) of a probabilistic modelfor a given data set y_1:t is the probability density of the data conditional on the model, that we denote as (y_1:t|).We say that _1 is a better fit than _0 for the data set y_1:t when (y_1:t|_1) > (y_1:t|_0). Since(y_1:t|_0) = ∫⋯∫∏_l=1^t g_l(x_l)τ_l( x_l | x_l-1) τ_0( x_0),and(y_1:t|_1) = ∫⋯∫∏_l=1^t g_l(x_l)τ̃_l^y_l( x_l | x_l-1) τ_0( x_0),the difference between the evidence of _0 and the evidence of _1 depends on the difference between the transition kernels τ̃_t and τ̃_t^y_t.We have empirically observed in several computer experiments that (y_1:t|_1) > (y_1:t|_0) and we argue that the observation-driven kernel τ̃_t^y_t implicit to the NuPF is the reason why the latter filter is robust to modelling errors in the state equation, compared to standard PFs. This claim is supported by the numerical results in Sections <ref> and <ref>, which show how the NuPF attains a significant better performance than the standard BPF, the auxiliary PF <cit.> or the extended Kalman filter <cit.> in scenarios where the filters are built upon a transition kernel different from the one used to generate the actual observations. While it is hard to show that (y_1:t|_1) > (y_1:t|_0) for every NuPF, it is indeed possible to guarantee that the latter inequality holds for specific nudging schemes. An example is provided in Appendix <ref>, where we describe a certain nudging operator α_t^y_t and then proceed to prove that (y_1:t|_1) > (y_1:t|_0), for that particular scheme, under some regularity conditions on the likelihoods and transition kernels.§ COMPUTER SIMULATIONS In this section, we present the results of several computer experiments. In the first one, we address the tracking of a linear-Gaussian system. This is a very simple model which enables a clearcut comparison of the NuPF and other competing schemes, including a conventional PF with optimal importance function (which is intractable for all other examples) and a PF with nudging and proper importance weights. Then, we study three nonlinear tracking problems:* a stochastic Lorenz 63 model with misspecified parameters, * a maneuvering target monitored by a network of sensors collecting nonlinear observations corrupted with heavy-tailed noise, * and, finally,a high-dimensional stochastic Lorenz 96 model[For the experiments involving Lorenz 96 model, simulation from the model is implemented in C++ and integrated into Matlab. The rest of the simulations are fully implemented in Matlab.]. We have used gradient nudging in all experiments, with either M ≤√(N) (deterministically, with batch nudging) or [M]≤√(N) (with independent nudging). This ensures that the assumptions of Theorem <ref> hold. For simplicity, the gradient steps are computed with fixed step sizes, i.e., γ_t=γ for all t. For the object tracking experiment, we have used a large step-size, but this choice does not affect the convergence rate of the NuPF algorithm either.§.§ A high-dimensional, inhomogeneous Linear-Gaussian state-space model In this experiment we compare different PFs implemented to track a high-dimensional linear Gaussian SSM. In particular, the model under consideration isx_0∼(0,I_d_x), x_t|x_t-1 ∼(x_t-1,Q),y_t|x_t∼(C_t x_t, R),where {x_t}_t≥ 0 are hidden states, {y_t}_t≥ 1 are observations, and Q and R are the process and the observation noise covariance matrices, respectively. The latter are diagonal matrices, namely Q = q I_d_x and R = I_d_y, where q = 0.1, d_x = 100 and d_y = 20. The sequence {C_t}_t≥ 1 defines a time-varying observation model. The elements of this sequence are chosen as random binary matrices, i.e., C_t ∈{0,1}^d_y × d_x where each entry is generated as an independent Bernoulli random variable with p = 0.5. Once generated, they are fixed and fed into all algorithms we describe below for each independent Monte Carlo run.We compare the NuPF with three alternative PFs. The first method we implement is the PF with the optimal proposal pdf p(x_t|x_t-1,y_t) ∝ g_t(y_t|x_t) τ_t(x_t|x_t-1), abbreviated as Optimal PF. The pdf p(x_t|x_t-1,y_t) leads to an analytically tractable Gaussian density for the model (<ref>)–(<ref>) <cit.> but not in the nonlinear tracking examples below. Note, however, that at each time step, the mean and covariance matrix of this proposal have to be explicitly evaluated in order to compute the importance weights. The second filter is a nudged PF with proper importance weights (NuPF-PW). In this case, we treat the generation of the nudged particles as a proposal function to be accounted for during the weighting step. To be specific, the proposal distribution resulting from the NuPF has the formτ̃_t(x_t | x_t-1) = (1-ϵ_N) τ_t(x_t|x_t-1) + ϵ_N τ̅_t(x_t|x_t-1),where ϵ_N = 1/√(N) andτ̅_t(x_t|x_t-1) = ∫δ_α_t^y_t(x̅_t) (x_t) τ_t(x̅_t|x_t-1).The latter conditional distribution admits an explicit representation as a Gaussian for model (<ref>)-(<ref>) when the α_t operator is designed as a gradient step, but this approach is intractable for the examples in Section <ref> and Section <ref>. Note that τ̃_t is a mixture of two time-varying Gaussians and this fact adds to the cost of the sampling and weighting steps. Specifically, computing weights for the NuPF-PW is significantly more costly, compared to the BPF or the NuPF, because the mixture (<ref>) has to be evaluated together with the likelihood and the transition pdf. The third tracking algorithm implemented for model (<ref>)–(<ref>) is the conventional BPF.For all filters, we have set the number of particles as[When N is increased the results are similar for the NuPF, the optimal PF and the NuPF-PW larger number particles, as they already perform close to optimally for N=100, and only the BPF improves significantly.] N = 100 . In order to implement the NuPF and NuPF-PW schemes, we have selected the step size γ = 2 × 10^-2. We have run 1,000 independent Monte Carlo runs for this experiment. To evaluate different methods, we have computed the empirical normalised mean squared errors (NMSEs). Specifically, the NMSE for the j-th simulation is (j) =∑_t=1^t_fx̅_t - x̂_t(j) _2^2/∑_t=1^t_f x_t _2^2,where x̅_t = 𝔼[x_t|y_1:t] is the exact posterior mean of the state x_t conditioned on the observations up to time t and x̂_t(j) is the estimate of the state vector in the j-th simulation run. Therefore, the notationimplies the normalised mean squared error is computed with respect to x̅_t. In the figures, we usually plot the mean and the standard deviation of the sample of errors, (1), …, (1000).The resultsare shown in Fig. <ref>. In particular, in Fig. <ref>(a), we observe that theperformance of the NuPF compared to the optimal PF and NuPF-PW (which is similar to a classical PF with nudging) is comparable. However, Fig. <ref>(b) reveals that the NuPF is significantly less demanding compared to the optimal PF and the NuPF-PW method. Indeed, the run-times of the NuPF are almost identical to the those of the plain BPF. As a result, the plot of the s multiplied by the running times displayed in Fig. <ref>(b) reveals that the proposed algorithm is as favorable as the optimal PF, which can be implemented for this model, but not for general models unlike the NuPF.§.§ Stochastic Lorenz 63 model with misspecified parametersIn this experiment, we demonstrate the performance of the NuPF when tracking a misspecified stochastic Lorenz 63 model. The dynamics of the system is described by a stochastic differential equation (SDE) in three dimensions,x_1= - 𝖺(x_1 - x_2) s +w_1, x_2= ( 𝗋x_1 - x_2 - x_1 x_3 )s +w_2, x_3= ( x_1 x_2 - 𝖻x_3 )s +w_3,where s denotes continuous time, {w_i(s)}_s∈(0,∞) for i = 1,2,3 are 1-dimensional independent Wiener processes and 𝖺,𝗋,𝖻∈ are fixed model parameters. We discretise the model using the Euler-Maruyama scheme with integration step 𝖳 > 0 and obtain the system of difference equationsx_1,t =x_1,t-1 - 𝖳𝖺 (x_1,t-1 - x_2,t-1) + √(𝖳) u_1,t, x_2,t =x_2,t-1 + 𝖳 (𝗋 x_1,t-1 - x_2,t-1 - x_1,t-1 x_3,t-1) + √(𝖳) u_2,t, x_3,t =x_3,t-1 + 𝖳 (x_1,t-1 x_2,t-1 - 𝖻x_3,t-1) + √(𝖳) u_3,t,where {u_i,t}_t∈, i = 1,2,3 are i.i.d. Gaussian random variables with zero mean and unit variance. We assume that we can only observe the variable x_1,t, contaminated by additive noise, every t_s>1 discrete time steps. To be specific, we collect the sequence of observationsy_n = k_o x_1,n t_s + v_n,n = 1, 2, ...,where {v_n}_n∈ is a sequence of i.i.d. Gaussian random variables with zero mean and unit variance and the scale parameter k_o = 0.8 is assumed known. In order to simulate both the state signal and the synthetic observations from this model, we choose the so-called standard parameter values (𝖺,𝗋,𝖻) = (10,28,8/3),which make the system dynamics chaotic. The initial condition is set as x_0 = [-5.91652, -5.52332, 24.5723]^⊤.The latter value has been chosen from a deterministic trajectory of the system (i.e., with no state noise) with the same parameter set(𝖺,𝗋,𝖻) = (10,28,8/3) to ensure that the model is started at a sensible point. We assume that the system is observed every t_s = 40 discrete time steps and for each simulation we simulate the system for t=0, 1, …, t_f, with t_f = 20,000. Since t_s = 40, we have a sequence of t_f/t_s=500 observations overall. Let us note here that the Markov kernel which takes the state from time n-1 to time n (i.e., from the time of one observation to the time of the next observation) is straightforward to simulate using the Euler-Maruyama scheme (<ref>), however the associated transition probability density cannot be evaluated because it involves the mapping of both the state and a sequence of t_s noise samples through a composition of nonlinear functions. This precludes the use of importance sampling schemes that require the evaluation of this density when computing the weights.We run the BPF and NuPF algorithms for the model described above, except that the parameter 𝖻 is replaced by 𝖻_ϵ = 𝖻+ϵ, with ϵ = 0.75 (hence 𝖻_ϵ≈ 3.417 versus 𝖻≈ 2.667 for the actual system). As the system underlying dynamics is chaotic, this mismatch affects the predictability of the system significantly.We have implemented the NuPF with independent gradient nudging. Each particle is nudged with probability 1/√(N), where N is the number of particles (hence [M]=√(N)), and the size of the gradient steps is set to γ = 0.75 (see Algorithm <ref>). As a figure of merit, we evaluate the NMSE for the 3-dimensional state vector, averaged over 1,000 independent Monte Carlo simulations. For this example (as well as in the rest of this section), it is not possible to compute the exact posterior mean of the state variables. Therefore, the NMSE values are computed with respect to the ground truth, i.e.,(j) =∑_t=1^t_fx_t - x̂_t(j) _2^2/∑_t=1^t_f x_t _2^2,where (x_t)_t≥ 1 is the ground truth signal.Fig. <ref> (a) displays the , attained for varying number of particles N, for the standard BPF and the NuPF. It is seen that the NuPF outperforms the BPF for the whole range of values of N in the experiment, both in terms of the mean and the standard deviation of the errors, although the NMSE values become closer for larger N. The plot on the right displays the values of x_2,t and its estimates for a typical simulation. In general, the experiment shows that the NuPF can track the actual system using the misspecified model and a small number of particles, whereas the BPF requires a higher computational effort to attain a similar performance.As a final experiment with this model, we have tested the robustness of the algorithms with respect to the choice of parameters in the nudging step. In particular, we have tested the NuPF with independent gradient nudging for a wide range of step-sizes γ. Also, we have tested the NuPF with random search nudging using a wide range of covariances of the form C = σ^2 I by varying σ^2.The results can be seen in Fig. <ref>. This figure shows that the algorithm is robust to the choice of parameters for a range of step-sizes and variances of the random search step. As expected, random search nudging takes longer running time compared to gradient steps. This difference in run-times is expected to be larger in higher-dimensional models since random search is expected to be harder in such scenarios. §.§ Object tracking with a misspecified modelIn this experiment, we consider a tracking scenario where a target is observed through sensors collecting radio signal strength (RSS) measurements contaminated with additive heavy-tailed noise. The target dynamics are described by the model,x_t= A x_t-1 + B L (x_t-1 - x_∙) + u_t,where x_t∈^4 denotes the target state, consisting of its position r_t ∈^2 and its velocity, v_t ∈^2, hence x_t = [ [ r_t; v_t; ]]∈^4. The vector x_∙ is a deterministic, pre-chosen state to be attained by the target. Each element in the sequence {u_t}_t∈ is a zero-mean Gaussian random vector with covariance matrix Q. The parameters A,B,Q are selected asA = [I_2κ I_2;0 0.99 I_2 ],B = [ 0 I_2 ]^⊤,andQ = [ κ^3/3 I_2 κ^2/2 I_2; κ^2/2 I_2 κ I_2 ],where I_2 is the 2× 2 identity matrix and κ = 0.04. The policy matrix L ∈^2×4 determines the trajectory of the target from an initial position x_0 = [140,140,50,0]^⊤ to a final state x_T = [140,-140,0,0]^⊤ and it is computed by solving a Riccati equation (see <cit.> for details), which yieldsL = [ [ -0.0134 0 -0.0381 0; 0 -0.0134 0 -0.0381; ]].This policy results in a highly maneuvering trajectory. In order to design the NuPF, however, we assume the simpler dynamical model x_t= A x_t-1 + u_t,hence there is a considerable model mismatch.The observations are nonlinear and coming from 10 sensors placed in the region where the target moves. The measurement collected at the i-th sensor, time t, is modelled as y_t,i = 10 log_10(P_0/r_t - s_i^2 + η) + w_t,i,where r_t ∈^2 is the location vector of the target, s_i is the position of the ith sensor and w_t,i∼𝒯(0,1,ν) is an independent t-distributed random variable for each i = 1,…,10. Intuitively, the closer the parameter ν to 1, the more explosive the observations become. In particular, we set ν = 1.01 to make the observations explosive and heavy-tailed. As for the sensor parameters, we set the transmitted RSS as P_0 = 1 and the sensitivity parameter as η = 10^-9. The latter yields a soft lower bound of -90 decibels (dB) for the RSS measurements.We have implemented the NuPF with batch gradient nudging, with a large-step size γ = 5.5 and M = ⌊√(N)⌋. Since the observations depend on the position vector r_t only, an additional model-specific nudging step is needed for the velocity vector v_t. In particular, after nudging the r_t^(i) = [ x^(i)_1,t, x^(i)_2,t]^⊤, we update the velocity variables asv_t^(i) = 1/κ (r^(i)_t - r^(i)_t-1),v_t^(i) = [ x^(i)_3,t, x^(i)_4,t]^⊤,where κ= 0.04 as defined for the model. The motivation for this additional transformation comes from the physical relationship between position and velocity. We note, however, that the NuPF also works without nudging the velocities.We have run 10,000 Monte Carlo runs with N = 500 particles in the auxiliary particle filter (APF) <cit.>, the BPF <cit.> and the NuPF. We have also implemented the extended Kalman filter (EKF), which uses the gradient of the observation model. Fig. <ref> shows a typical simulation run with each one of the four algorithms (on the left side, plots (a)–(d)) and a box-plot of the NMSEs obtained for the 10,000 simulations (on the right, plot (e)). Plots (a)–(d) show that, while the EKF also uses the gradient of the observation model, it fails to handle the heavy-tailed noise, as it relies on Gaussian approximations. The BPF and the APF collapse due to the model mismatch in the state equation. Plot (d) shows that the NMSE of the NuPF is just slightly smaller in the mean than the NMSE of the EKF, but much more stable.§.§ High-dimensional stochastic Lorenz 96 modelIn this computer experiment, we compare the NuPF with the ensemble Kalman filter (EnKF) for the tracking of a stochastic Lorenz 96 system. The latter is described by the set of stochastic differential equations (SDEs)x_i= [ (x_i+1 - x_i-2) x_i-1 - x_i + F ]s +w_i, i = 1, …, d,where s denotes continuous time, { w_i(s) }_s ∈ (0,∞), 1 ≤ i ≤ d, are independent Wiener processes, d is the system dimension and the forcing parameter is set to F = 8, which ensures a chaotic regime. The model is assumed to have a circular spatial structure, so that x_-1 = x_d-1, x_0 = x_d, and x_d+1 = x_1. Note that each x_i, i=1, ..., d, denotes a time-varying state associated to a different space location. In order to simulate data from this model, we apply the Euler-Maruyama discretisation scheme and obtain the difference equations,x_i,t =x_i,t-1 + 𝖳[ (x_i+1,t-1 - x_i-2,t-1) x_i-1,t-1.. - x_i,t-1 + F ] + √(𝖳) u_i,t,where u_i,t are zero-mean, unit-variance Gaussian random variables. We initialise this system by generating a vector from the uniform distribution on (0,1)^d and running the system for a small number of iterations and set x_0 as the output of this short run. We assume that the system is only partially observed. In particular, half of the state variables are observed, in Gaussian noise, every t_s = 10 time steps, namely,y_j,n = x_2j-1,nt_s + u_j,n,where n = 1, 2, …, j = 1, 2, …, ⌊ d/2 ⌋, and u_j,n is a normal random variable with zero mean and unit variance. The same as in the stochastic Lorenz 63 example of Section <ref>, the transition pdf that takes the state from time (n-1)t_s to time nt_s is simple to simulate but hard to evaluate, since it involves mapping a sequence of noise variables through a composition of nonlinearities.In all the simulations for this system we run the NuPF with batch gradient nudging (with M = ⌊√(N)⌋ nudged particles and step-size γ = 0.075). In the first computer experiment, we fixed the dimension d = 40 and run the BPF and the NuPF with increasing number of particles. The results can be seen in Fig. <ref>, which shows how the NuPF performs better than the BPF in terms of NMSE (plot (a)). Since the run-times of both algorithms are nearly identical, it can be seen that, when considered jointly with NMSEs, the NuPF attains a significantly better performance (plot (b)).In a second computer experiment, we compared the NuPF with the EnKF. Fig. <ref>(a) shows how the NMSE of the two algorithms grows as the model dimension d increases and the number of particles N is kept fixed. In particular, the EnKF attains a better performance for smaller dimensions (up to d=10^3), however its NMSE blows up for d > 10^3 while the performance of the NuPF remains stable. The running time of the EnKF was also higher than the running time of the NuPF in the range of higher dimensions (d≥ 10^3).§.§ Assessment of bias In this section, we numerically quantify the bias of the proposed algorithm on a low-dimensional linear-Gaussian state-space model. To assess the bias, we compute the marginal likelihood estimates given by the BPF and the NuPF. The reason for this choice is that the BPF is known to yield unbiased estimates of the marginal likelihood[Note that the estimates of integrals (φ,π_t) computed using the self-normalised importance sampling approximations (i.e., (φ,π_t^N) ≈ (φ,π_t)) produced by the BPF and the NuPF methods are biased and the bias vanishes with the same rate for both algorithms as a result of Theorem <ref>. The same is true for the approximate predictive measures ξ_t^N.] <cit.>. The NuPF leads to biased (typically overestimated) marginal likelihood estimates, hence it is of interest to compare them with those of the BPF. To this end, we choose a simple linear-Gaussian state space model for which the marginal likelihood can be exactly computed as a byproduct of the Kalman filter. We then compare this exact marginal likelihood to the estimates given by the BPF and the NuPF.Particularly, we define the state-space model,x_0∼(x_0;μ_0,P_0),x_t | x_t-1 ∼(x_t; x_t-1,Q), y_t | x_t∼(y_t; C_t x_t,R),where (C_t)_t≥ 0∈ [0,1]^1 × 2 is a sequence of observation matrices where each entry is generated as a realisation of a Bernoulli random variable with p = 0.5, μ_0 is a zero vector, and x_t ∈^2 and y_t ∈. The state variables are cross-correlated, namely,Q = [ 2.7 -0.48; -0.482.05 ],and R = 1. We have chosen the prior covariance as P_0 = I_d_x. We have simulated the system for T=100 time steps. Given a fixed observation sequence y_1:T, the marginal likelihood for the system given in Eqs. (<ref>)-(<ref>) isZ^⋆ = 𝗉(y_1:T),which can be exactly computed via the Kalman filter. We denote the estimate of Z^⋆ given by the BPF and the NuPF as Z^N_BPF and Z^N_NuPF, respectively. It is well-known that the BPF estimator is unbiased <cit.>,[Z^N_BPF] = Z^⋆,where [·] denotes the expectation with respect to the randomness of the particles. Numerically, this suggests that as one runs identical, independent Monte Carlo simulations to obtain {Z^N,k_BPF}_k=1^K and compute the averageZ̅^N_BPF = 1/K∑_k=1^K Z^N,k_BPF,then it follows from the unbiasedness property (<ref>) that the ratio of the average in (<ref>) and the true value Z^⋆ should satisfyZ̅^N_BPF/Z^⋆→ 1 as K →∞.Since the marginal likelihood estimates provided by the NuPF are not unbiased for the original SSM (and tend to attain higher values), if we defineZ̅^N_NuPF = 1/K∑_k=1^K Z^N,k_NuPF,then as K→∞, we should seeZ̅^N_NuPF/Z^⋆→ 1+ϵas K →∞,for some ϵ > 0.We have conducted an experiment aimed at quantifying the bias ϵ>0 above. In particular, we have run 20,000 independent simulations for the BPF and the NuPF with N=100, N=1,000 and N=10,000. For each value of N, we have computed running empirical means as in (<ref>) and (<ref>) for K=1, …, 20,000. The variance of Z̅_BPF^N increases with T, hence the estimators for small K display a relatively large variance and we need K>>1 to clearly observe the bias. The NuPF filter performs independent gradient nudging with step size γ = 0.1. The results of the experiment are displayed in Figure <ref>, which shows how, as expected, the NuPF overestimates Z^⋆. We can also see how the bias becomes smaller as N increases (because only and average of √(N) particles are nudged per time step). § EXPERIMENTAL RESULTS ON MODEL INFERENCEIn this section, we illustrate the application of the NuPF to estimate the parameters of a financial time-series model. In particular, we adopt a stochastic-volatility SSM and we aim at estimating its unknown parameters (and track its state variables) using the EURUSD log-return data from 2014-12-31 to 2016-12-31 (obtained from ). For this task, we apply two recently proposed Monte Carlo schemes: the nested particle filter (NPF) <cit.> (a purely recursive, particle-filter style Monte Carlo method) and the particle Metropolis-Hastings (pMH) algorithm <cit.> (a batch Markov chain Monte Carlo procedure). In their original forms, both algorithms use the marginal likelihood estimators given by the BPF to construct a Monte Carlo approximation of the posterior distribution of the unknown model parameters. Here, we compare the performance of these algorithms when the marginal likelihoods are computed using either the BPF or the proposed NuPF. We assume the stochastic volatility SSM <cit.>,x_0∼(μ, σ_v^2/1-ϕ^2), x_t|x_t-1 ∼(μ + ϕ (x_t-1 - μ), σ_v^2), y_t|x_t∼(0,exp(x_t)),where μ∈, σ_v ∈_+, and ϕ∈ [-1,1] are fixed but unknown parameters. The states {x_t}_1 ≤ t ≤ T are log-volatilities and the observations {y_t}_1≤ t ≤ T are log-returns. We follow the same procedure as <cit.> to pre-process the observations. Given the historical price sequence s_0, …, s_T, the log-return at time t is calculated asy_t = 100 log(s_t/s_t-1)for 1≤ t ≤ T. Then, given y_1:T, we tackle the joint Bayesian estimation of x_1:T and the unknown parameters θ = (μ,σ_v,ϕ). In the next two subsections we compare the conventional BPF and the NuPF as building blocks of the NPF and the pMH algorithms. §.§ Nudging the nested particle filter The NPF in <cit.> consists of two layers of particle filters which are used to jointly approximate the posterior distributions of the parameters and the states. The filter in the first layer builds a particle approximation of the marginal posterior distribution of the parameters. Then, for each particle in the parameter space, say θ^(i), there is an inner filter that approximates the posterior distribution of the states conditional on the parameter vector θ^(i). The inner filters are classical particle filters, which are essentially used to compute the importance weights (marginal likelihoods) of the particles in the parameter space. In the implementation of <cit.>, the inner filters are conventional BPFs. We have compared this conventional implementation with an alternative one where the BPFs are replaced by the NuPFs. For a detailed description of the NPF, see <cit.>.In order to assess the performances of the nudged and classical versions of the NPF, we compute the model evidence estimate given by the nested filter by integrating out both the parameters and the states. In particular, if the set of particles in the parameter space at time t is {θ_t^(i)}_i=1^K and for each particle θ_t^(i) we have a set of particles in the state space {x_t^(i,j)}_j=1^N, we computep(y_1:T) = ∏_t=1^T [ 1/KN∑_i=1^K ∑_j=1^N g_t(x_t^(i,j)) ].The model evidence quantifies the fitness of the stochastic volatility model for the given dataset, hence we expect to see a higher value when a method attains a better performance (the intuition is that if we have better estimates of the parameters and the states, then the model will fit better). For this experiment, we compute the model evidence for the nudged NPF before the nudging step, so as to make the comparison with the conventional algorithm fair.We have conducted 1,000 independent Monte Carlo runs for each algorithm and computed the model evidence estimates. We have used the same parameters and the same setup for the two versions of the NPF (nudged and conventional). In particular, each unknown parameter is jittered independently. The parameter μ is jittered with a zero-mean Gaussian kernel variance σ_μ^2 = 10^-3, the parameter σ_v is jittered with a truncated Gaussian kernel on (0,∞) with variance σ_σ_v^2 = 10^-4, and the parameter ϕ is jittered with a zero-mean truncated Gaussian kernel on [-1,1], with variance σ_ϕ^2 = 10^-4. We have chosen a large step-size for the nudging step, γ = 4,and we have used batch nudging with M = ⌊√(N)⌋.The results in Fig. <ref> demonstrate empirically that the use of the nudging step within the NPF reduces the variance of the model evidence estimators, hence it improves the numerical stability of the NPF.§.§ Nudging the particle Metropolis-Hastings The pMH algorithm is a Markov chain Monte Carlo (MCMC) method for inferring parameters of general SSMs <cit.>. The pMH uses PFs as auxiliary devices to estimate parameter likelihoods in a similar way as the NPF uses them to compute importance weights. In the case of the pMH, these estimates should be unbiased and they are needed to determine the acceptance probability for each element of the Markov chain. For the details of the algorithm, see <cit.> (or <cit.> for a tutorial-style introduction). Let us note that the use of NuPF does not lead to an unbiased estimate of the likelihood with respect to the assumed SSM. However, as discussed in Section <ref>, one can view the use of nudging in this context as an implementation of pMH with an implicit dynamical model _1 derived from the original SSM _0.We have carried out a computer experiment to compare the performance of the pMH scheme using either BPFs or NuPFs to compute acceptance probabilities. The two algorithms are labeled pMH-BPF and pMH-NuPF, respectively, hereafter. The parameter priors in the experiment arep(μ) = (0,1),p(σ_v) = 𝒢(2,0.1),p(ϕ) = ℬ(120,2),where 𝒢(a,θ) denotes the Gamma pdf with shape parameter a and scale parameter θ, and ℬ(α,β) denotes the Beta pdf with shape parameters (α,β). Unlike <cit.>, who use a truncated Gaussian prior centered on 0.95 with a small variance for ϕ, we use the Beta pdf, which is defined on [0,1], with mean α/(α + β) = 0.9836, which puts a significant probability mass on the interval [0.9,1].We have compared the pMH-BPF algorithm and the pMH-NuPF scheme (using a batch nudging procedure with γ = 0.1 and M = ⌊√(N)⌋) by running 1,000 independent Monte Carlo trials. We have computed the marginal likelihood estimates in the NuPF after the nudging step. First, in order to illustrate the impact of the nudging on the parameter posteriors, we have run the pMH-NuPF and the pMH-BPF and obtained a long Markov chain (2× 10^6 iterations) from both algorithms. Figure <ref> displays the two-dimensional marginals of the resulting posterior distribution. It can be seen from Fig. <ref> that the bias of the NuPF yields a perturbation compared to the posterior distribution approximated with the pMH-BPF. The discrepancy is small but noticeable for small N (see Fig. <ref>(a) for N=100) and vanishes as we increase N (see Fig. <ref>(b) and (c), for N=500 and N=1,000, respectively). We observe that for a moderate number of particles, such as N=500 in Fig. <ref>(b), the error in the posterior distribution due to the bias in the NuPF is very slight.Two common figures of merit for MCMC algorithms are the acceptance rate of the Markov kernel (desirably high) and the autocorrelation function of the chain (desirably low). Figure <ref> shows the acceptance rates for the pMH-NuPF and the pMH-BPF algorithms with N=100, N=500 and N=1,000 particles in both PFs. It is observed that the use of nudging leads to noticeably higher acceptance rates, although the difference becomes smaller as N increases.Figure <ref> displays the average autocorrelation functions (ACFs) of the chains obtained in the 1,000 independent simulations.We see that the autocorrelation of the chains produced by the pMH-NuPF method decays more quickly than the autocorrelation of the chains output by the conventional pMH-BPF, especially for lower values of N. Even for N=1,000 (which ensures an almost negligible perturbation of the posterior distribution, as shown in Figure <ref>(c)) there is an improvement in the ACFs of the parameters ϕ and σ_v using the NuPF. Less correlation can be expected to translate into better estimates as well for a fixed length of the chain. § CONCLUSIONS We have proposed a simple modification of the particle filter which, according to our computer experiments, can improve the performance of the algorithm (e.g., when tracking high-dimensional systems) or enhance its robustness to model mismatches in the state equation of a SSM. The modification of the standard particle filtering scheme consists of an additional step, which we term nudging, in which a subset of particles are pushed towards regions of the state space with a higher likelihood. In this way, the state space can be explored more efficiently while keeping the computational effort at nearly the same level as in a standard particle filter. We refer to the new algorithm as the “nudged particle filter” (NuPF). While, for clarity and simplicity, we have kept the discussion and the numerical comparisons restricted to the modification (nudging) of the conventional BPF, the new step can be naturally incorporated to most known particle filtering methods. We have presented a basic analysis of the NuPF which indicates that the algorithm converges (in L_p) with the same error rate as the standard particle filter. In addition, we have also provided a simple reinterpretation of nudging that illustrates why the NuPF tends to outperform the BPF when there is some mismatch in the state equation of the SSM. To be specific, we have shown that, given a fixed sequence of observations, the NuPF amounts to a standard PF for a modified dynamical model which empirically leads to a higher model evidence (i.e., a higher likelihood) compared to the original SSM.The analytical results have been supplemented with a number of computer experiments, both with synthetic and real data. In the latter case, we have tackled the fitting of a stochastic volatility SSM using Bayesian methods for model inference and a time-series dataset consisting of euro-to-US-dollar exchange rates over a period of two years. We have shown how different figures of merit (model evidence, acceptance probabilities or autocorrelation functions) improve when using the NuPF, instead of a standard BPF, in order to implement a nested particle filter <cit.> and a particle Metropolis-Hastings <cit.> algorithm. Since the nudging step is fairly general, it can be used with a wide range of differentiable or non-differentiable likelihoods. Besides, the new operation does not require any modification of the well-defined steps of the PF so it can be plugged into a variety of common particle filtering methods. Therefore, it can be adopted by a practitioner with hardly any additional effort. In particular, gradient nudging steps (for differentiable log-likelihoods) can be implemented using automatic differentiation tools, currently available in many software packages, hence relieving the user from explicitly calculating the gradient of the likelihood. Similar to the resampling step, which is routinely employed for numerical stability, we believe the nudging step can be systematically used for improving the performance and robustness of particle filters.§ PROOF OF THEOREM <REF>In order to prove Theorem <ref>, we need a preliminary lemma, which can be found, e.g., in <cit.>.Let α,β,α̅,β̅∈() be probability measures and f,h ∈ B() be two real bounded functions onsuch that (h,α̅) > 0 and (h,β̅)>0. If the identities,(f,α) = (fh,α̅)/(h,α̅)and (f,β) = (fh,β̅)/(h,β̅)hold, then we have,| (f,α)-(f,β) | ≤ 1 / (h,α̅) | (fh,α̅) - (fh,β̅) | +f _∞/ (h,α̅) | (h,α̅) - (h,β̅) |.Now we proceed with the proof of Theorem <ref>. We follow the same kind of induction argument as in, e.g., <cit.> and <cit.>.For the base case, i.e. t = 0, we draw x_0^(i), i=1, ..., N, i.i.d. from π_0 and obtain,(φ,π_0^N) - (φ,π_0)_p = [|1/N∑_i=1^N (φ(x_0^(i)) - (φ,π_0))|^p]^1/p.We define S_0^(i) = φ(x_0^(i)) - (φ,π_0) and note that S_0^(i), i = 1,…,N are zero-mean and independent random variables. Using the Marcinkiewicz-Zygmund inequality <cit.>, we arrive at,[|1/N∑_i=1^N S_0^(i)|^p]≤B_0,p/N^p[(∑_i=1^N | S_0^(i)|^2)^p/2] ≤B_0,p/N^p(N 4 φ_∞^2)^p/2,where B_0,p is a constant independent of N and the last inequality follows from |S_0^(i)| = | φ(x_0^(i)) - (φ,π_0)| ≤ 2 φ_∞. Therefore, we have proved that Eq. (<ref>) holds for the base case,(φ,π_0^N) - (φ,π_0)_p ≤c_0,pφ_∞/√(N),where c_0,p = 2 B_0,p^1/p is a constant independent of N.The induction hypothesis is that, at time t-1, (φ,π_t-1^N) - (φ,π_t-1) _p ≤c_t-1,pφ_∞/√(N)for some constant c_t-1,p < ∞ independent of N. We start analyzing the predictive measure ξ_t^N,ξ_t^N(x) = 1/N∑_i = 1^N δ_x̅_t^(i)(x),where x̅_t^(i), i = 1,…,N are the particles sampled from the transition kernels τ_t^x_t-1^(i)(x_t) ≜τ_t( x_t |x_t-1^(i)). Since we have ξ_t = τ_t π_t-1 (see Sec. 1.4), a simple triangle inequality yields,(φ,ξ_t^N) - (φ,ξ_t)_p= (φ,ξ_t^N) - (φ,τ_t π_t-1)_p ≤(φ,ξ_t^N) - (φ,τ_t π_t-1^N)_p + (φ,τ_t π_t-1^N)-(φ,τ_t π_t-1)_p,where,(φ,τ_t π_t-1^N) = 1/N∑_i=1^N (φ,τ_t^x_t-1^(i)).For the sampling step, we aim at bounding the two terms on the rhs of (<ref>).For the first term, we introduce the σ-algebra generated by the random variables x_0:t^(i) and x̅_1:t^(i), i = 1,…,N, denoted _t = σ(x_0:t^(i),x̅_1:t^(i),i = 1,…,N). Since π_t-1^N is measurable w.r.t. _t-1, we can write[(φ,ξ_t^N) | _t-1] = 1/N∑_i=1^N (φ,τ_t^x_t-1^(i)) = (φ,τ_t π_t-1^N).Next, we define the random variables S_t^(i) = φ(x̅_t^(i)) - (φ,τ_t π_t-1^N) and note that, conditional on _t-1, S_t^(i), i = 1,…,N are zero-mean and independent. Then, the approximation error of ξ_t^N can be written as,[| (φ,ξ^N_t. .) - (φ,τ_t π_t-1^N)|^p | _t-1]= [ | 1/N∑_i=1^N S_t^(i)|^p | _t-1].Resorting again to the Marcinkiewicz-Zygmund inequality, we can write,[ | 1/N∑_i=1^N S_t^(i)|^p | _t-1] ≤B_t,p/N^p[ (∑_i=1^N |S_t^(i)|^2)^p/2| _t-1],where B_t,p< ∞ is a constant independent of N. Moreover, since |S_t^(i)| = |φ(x̅_t^(i)) - (φ,τ_t π_t-1^N)| ≤ 2 φ_∞, we have,[ | 1/N∑_i=1^N S_t^(i)|^p | _t-1] ≤B_t,p/N^p( N 4 φ_∞^2)^p/2 = B_t,p/N^p/2 2^p φ_∞^p.If we take unconditional expectations on both sides of the equation above, then we arrive at(φ,ξ^N_t ) - (φ,τ_t π_t-1^N)_p ≤c_1,pφ_∞/√(N),where c_1,p = 2 B_t,p^1/p < ∞ is a constant independent of N.To handle the second term in the rhs of (<ref>), we define (φ̅,π_t-1) = (φ,τ_t π_t-1) where φ̅∈ B() and given by,φ̅(x) = (φ,τ_t^x).We also write (φ̅,π_t-1^N) = (φ,τ_tπ_t-1^N). Since φ̅_∞≤φ_∞, the induction hypothesis leads,(φ,τ_tπ_t-1^N) - (φ,τ_tπ_t-1)_p= (φ̅,π_t-1^N) - (φ̅,π_t-1)_p ≤c_t-1,pφ_∞/√(N),where c_t-1,p is a constant independent of N. Combining (<ref>) and (<ref>) yields,(φ, ξ_t^N) - (φ,ξ_t) _p ≤c_1,t,pφ_∞/√(N)where c_1,t,p = c_t-1,p + c_1,p < ∞ is a constant independent of N.Next, we have to bound the error between the predictive measure ξ_t^N and the nudged measure ξ̃_t^N. As the sets of samples {x̅_t^(i)}_i=1^N, used to construct ξ_t^N, and {x̃_t^(i)}_i=1^N, used to construct ξ̃_t^N as shown in (<ref>), differ exactly in M particles, namely x̃_t^(j_1), …, x̃_t^(j_M), where { j_1, …, j_M } = ℐ_t, we readily obtain the relationship(φ, ξ_t^N) - (φ,ξ̃_t^N) _p= 1/N∑_i ∈ℐ_t( φ(x̅_t^(i)) - φ(x̃_t^(i)) ) _p ≤ 2φ_∞ M/N≤ 2 φ_∞/√(N)where the first inequality holds trivially (since |φ(x) - φ(x')| ≤ 2 φ_∞ for every (x,x') ∈𝖷^2) and the second inequality follows from the assumption M ≤√(N). Combining (<ref>) and (<ref>) we arrive at(φ,ξ_t) - (φ,ξ̃_t^N) _p ≤c̃_1,tφ_∞/√(N),where the constant c̃_1,t,p = 2 + c_1,t,p < ∞ is independent of N. Next, we aim at bounding (φ,π_t) - (φ,π̃_t^N)_p using (<ref>). We note that, after the computation of weights, we define the weighted random measure,π̃_t^N = ∑_i=1^N w_t^(i)δ_x̃_t^(i)where w_t^(i) = g_t(x̃_t^(i))/∑_i=1^N g_t(x̃_t^(i)).The integrals computed with respect to the weighted measure π̃_t^N takes the form,(φ,π̃_t^N) = (φ g_t,ξ̃^N)/(g_t,ξ̃_t^N).On the other hand, using Bayes theorem, integrals with respect to the optimal filter can also be written in a similar form as,(φ,π_t) = (φ g_t,ξ_t)/(g_t,ξ_t).Using Lemma <ref> together with (<ref>) and (<ref>), we can readily obtain,|(φ,π̃_t^N) - (φ,π_t)| ≤ 1/(g_t,ξ_t)( φ_∞| (g_t,ξ_t) - (g_t,ξ_t^N)| . . + |(φ g_t,ξ_t) - (φ g_t,ξ_t^N)|),where (g_t,ξ_t) > 0 by assumption. Using Minkowski's inequality, we can deduce from (<ref>) that(φ,π̃_t^N) - (φ,π_t)_p ≤ 1/(g_t,ξ_t)( φ_∞ (g_t,ξ_t) - (g_t,ξ_t^N)_p . . + (φ g_t,ξ_t) - (φ g_t,ξ_t^N)_p).Noting that we have φ g_t_∞≤φ_∞g_t_∞, (<ref>) and (<ref>) together yield,(φ,π_t) - (φ, π̃_t^N) _p ≤c_2,t,pφ_∞/√(N),where c_2,t,p= 2 g_t_∞c̃_1,t,p/(g_t, ξ_t) < ∞is a finite constant independent of N (the denominator is positive and the numerator is finite as a consequence of Assumption <ref>).Finally, the analysis of the multinomial resampling step is also standard. We denote the resampled measure as π_t^N. Since the random variables which are used to construct π_t^N are sampled i.i.d from π̃_t^N, the argument for the base case can also be applied here to yield,(φ, π̃_t^N) - (φ, π_t^N) _p ≤c_3,t,pφ_∞/√(N),where c_3,t,p < ∞ is a constant independent of N. Combining bounds (<ref>) and (<ref>) to obtain the inequality (<ref>), with c_t,p = c_2,t,p + c_3,t,p < ∞, concludes the proof. § PROOF OF LEMMA <REF>Since x̃_t^(i) = x̅_t^(i) + γ_t ∇_x_t g_t(x̅^(i)_t), we readily obtain the relationships| φ(x̃_t^(i)) - φ(x̅_t^(i))|≤ Lx̃_t^(i) - x̅_t^(i)_2 = L γ_t ‖∇_x_t g(x̅^(i)_t)‖_2 ≤γ_t L G_twhere the first inequality follows from the Lipschitz assumption, the identity is due to the implementation of the gradient-nudging step and the second inequality follows from Assumption <ref>. Then we bound the error (φ, ξ_t^N) - (φ, ξ̃_t^N) _p as (φ, ξ_t^N) - (φ,ξ̃_t^N) _p= 1/N∑_i∈ℐ_t( φ(x̅_t^(i)) - φ(x̃_t^(i)) ) _p ≤ 1/N∑_i∈ℐφ(x̅_t^(i)) - φ(x̃_t^(i)) _p ≤ M/Nγ_t L G_twhere the identity is a consequence of the construction of ℐ_t and we apply Minkowski's inequality, (<ref>) and the assumption |ℐ_t|=M to obtain (<ref>). However, we have assumed that sup_1≤ t ≤ Tγ_t M ≤√(N), hence(φ, ξ_t^N) - (φ, ξ̃_t^N) _p ≤LG_t/√(N).§ NUDGING SCHEME THAT INCREASES THE MODEL EVIDENCEConsider the SSM _0={τ_0,τ_t,g_t} where the likelihoods (g_t)_t≥ 1 and the Markov kernels (τ_t)_t≥ 1 satisfy the regularity assumptions below.The functions log g_t(x), t =1, 2, ..., are differentiable and the gradients ∇log g_t(x) are Lipschitz with constant L_t^g < ∞. To be specific,∇log g_t(x) - ∇log g_t (x') _2 ≤𝖫_t^gx - x'_2.The Markov kernels τ_t( x|x') are absolutely continuous with respect to the Lebesgue measure, hence there are conditional pdf's m_t(x|x') such that τ_t( x|x') = m_t(x|x')x for any x'∈. Moreover, the log-pdf's log m_t(x|x') are uniformly Lipschitz in x', i.e., there are non-negative bounded functions L_t(x) such that| log m_t(x|x') - log m_t(x|x”) | ≤ L_t(x)x' - x”_2and 𝖫^τ_t = sup_x∈ L_t(x) < ∞. For any subset A ⊆, let us introduce the indicator function 1_A(x) := {[ 1,,; 0, ;].We construct the nudging operator α_t^y_t : ↦ of the formα_t^y_t(x) := ( x + γ_t ∇log g_t(x) )1_S_y_t(x) + x1_S_y_t(x),where S_y_t := { x ∈: ∇log g_t(x_t) ≥ 2 𝖫_t^τ} (recall that g_t(x)=g_t(y_t|x)), S_y_t = \ S_y_t is the complement of the set S_y_t and γ_t>0 is small enough to guarantee that g_t(x+γ_t∇log g_t(x)) ≥ g_t(x). Intuitively, this nudging scheme only takes a gradient step when the slope of the likelihood g_t is sufficient to insure an improvement of the likelihood with a small move of the state x.Assume, for simplicity, that we apply the nudging operator (<ref>) to every particle at every time step. The recursive step of the resulting NuPF can be outlined as follows: * For i=1, ..., N, * draw x̅_t^(i)∼τ_t( x_t|x_t-1^(i)), * nudge every particle, i.e., x̃_t^(i) = α_t^y_t(x̅_t^(i)), * and compute weights w_t^(i)∝ g_t(x̃_t^(i)). * Resample to obtain { x_t^(i)}_i=1, ..., N.The asymptotic convergence of this algorithm can be insured whenever the step sizes γ_t are selected small enough to guarantee that sup_xx - α_t^y_t(x)≤1/√(N) (this is a consequence of Corollary 1 in <cit.>). The implicit model for this NuPF is _1={τ_0,τ̃_t,g_t} where the transition kernel isτ̃_t^y_t( x_t|x_t-1) = ∫δ_α_t^y_t(x̅)( x_t) τ_t(x̅|x_t-1).Note that for any integrable function f:↦ we have(f,τ̃_t^y_t)(x_t-1)= ∫∫ f(x_t) δ_α_t^y_t(x̅)( x_t) τ_t(x̅ | x_t-1) = ∫ (f ∘α_t^y_t)(x̅) τ_t(x̅ | x_t-1) =(f ∘α_t^y_t, τ_t)(x_t-1),where (f ∘α_t^y_t)(x)=f(α_t^y_t(x)) is the composition of f and α_t^y_t. In particular, we note that (g_t,τ̃_t^y_t)(x)= (g_t∘α_t^y_t,τ_t)(x) and ( 1_,τ̃_t^y_t)(x) = ∫ 1_(α_t^y_t(x)) τ_t(x̅ | x) = 1 because α_t^y_t is ↦ and, therefore, 1_∘α_t^g_t = 1.Let β_t(x) := x + γ_t ∇log g_t(x).Assumptions <ref> and <ref> entail the following result.If Assumptions <ref> and <ref> hold and the inequalitiesγ_t ≤1/𝖫_t^g∇log g_t(x_t) _2 ≥ 2 𝖫_t^τ, are satisfied, then (g_t ∘β_t)(x_t) / g_t(x_t) ≥sup_x_t+1∈ m_t+1( x_t+1 | x_t) / m_t+1(x_t+1 | β_t(x_t)) .Proof. Assumption <ref> implies that, for any pair x,x'∈,log g_t(x) ≥log g_t(x') + ⟨∇log g_t(x'), x - x'⟩ - 𝖫_t^g/2x - x'^2_2see, e.g., <cit.> for a proof. We can readily use (<ref>)to obtain a lower bound for log g_t(β_t(x_t)). Indeed,log g_t(β_t(x_t))≥ log g_t(x_t) + ⟨∇log g_t(x_t), γ_t ∇log g_t(x_t)⟩- 𝖫_t^gγ_t^2/2∇log g_t(x_t)^2_2= log g_t(x_t) + (γ_t -𝖫_t^gγ_t^2/2) ∇log g_t(x_t)^2_2 ≥ log g_t(x_t) + γ_t/2∇log g_t(x_t)^2_2where the last inequality follows from the assumption γ_t ≤1/𝖫_t^g. In turn, (<ref>) impliesg_t(β_t(x_t))/g_t(x_t)≥exp{γ_t/2∇log g_t(x_t)^2_2}. We now turn to the problem of upper bounding the ratio of transition pdf's. From Assumption <ref> and the definition of β_t(x_t) we readily obtain thatlog m_t+1(x_t+1 | x_t) - log m_t+1(x_t+1 | β_t(x_t)) ≤𝖫_t^τγ_t ∇log g_t(x_t)_2holds for any x_t+1∈. Taking exponentials on both sides of (<ref>) we arrive atsup_x_t+1∈m_t+1(x_t+1 | x_t)/m_t+1(x_t+1 | β_t(x_t))≤exp{𝖫_t^τγ_t ∇log g_t(x_t)_2 }. If ∇log g_t(x_t) _2=0 then expressions (<ref>) and (<ref>) together readily yield the desired relationship (<ref>).If ∇log g_t(x_t) _2 > 0, then the assumption ∇log g_t(x_t) ≥ 2𝖫_t^τ implies thatγ_t/2∇log g_t(x_t)_2^2 ≥γ_t 𝖫_t^τ∇log g_t(x_t)_2which, together with (<ref>) and (<ref>), again yield the desired inequality (<ref>). Finally, we prove that the evidence in favour of _1 is greater than the evidence in favour of_0.Let the nudging scheme be defined as in (<ref>). If Assumptions <ref> and <ref> hold and the inequalityγ_t ≤1/𝖫_t^g is satisfied for t=1, …,T<∞, then (y_1:T|_1) ≥(y_1:T|_0).Proof. From the definition of τ̃_t in (<ref>) and the ensuing relationship (<ref>), the evidence of model _1 can be readily written down as(y_1:T|_1)= ∫⋯∫ g_T(α_t^y_T(x_T)) × ×∏_t=1^T-1 m_t+1(x_t+1|α_t^y_t(x_t)) g_t(α_t^y_t(x_t))× m_1(x_1|x_0)m_0(x_0)x_0 ⋯ x_T.It is apparent that g_T(α_T^y_T(x_T)) ≥ g_T(x_T) for every x_T. Moreover, for any x_t ∈, if x_t ∈ S_y_t then ∇log g_t(x_t) ≥ 2 𝖫_t^τα_t^y_t(x_t) = β_t(x_t), hence we can apply Lemma <ref>, which yields m_t+1(x_t+1|α_t^y_t(x_t)) g_t(α_t^y_t(x_t)) ≥ m_t+1(x_t+1|x_t) g_t(x_t)for every x_t+1. Alternatively, if x_t ∉ S_y_t, then α_t^y_t(x_t)=x_t and, trivially,m_t+1(x_t+1|α_t^y_t(x_t)) g_t(α_t^y_t(x_t)) = m_t+1(x_t+1|x_t) g_t(x_t).Therefore,(y_1:T|_1)≥ ∫⋯∫ g_T(x_T) ∏_t=1^T-1 m_t+1(x_t+1|x_t) g_t(x_t)× m_1(x_1|x_0) m_0(x_0)x_0 ⋯ x_T = (y_1:T|_0). spbasic | http://arxiv.org/abs/1708.07801v4 | {
"authors": [
"Ömer Deniz Akyıldız",
"Joaquín Míguez"
],
"categories": [
"stat.CO",
"math.PR"
],
"primary_category": "stat.CO",
"published": "20170825163149",
"title": "Nudging the particle filter"
} |
Genuinely Multipartite Noncausality Cyril Branciard December 7, 2017 =================================== This paper presents global tracking strategies for the attitude dynamics of a rigid body. It is well known that global attractivity is prohibited for continuous attitude control systems on the special orthogonal group. Such topological restriction has been dealt with either by constructing smooth attitude control systems that exclude a set ofmeasure zero in the region of attraction, or by introducing hybrid control systems to obtain global asymptotic stability. This paper proposes alternative attitude control systems that are continuous in time to achieve exponential stability, where the region of attraction covers the entire special orthogonal group. The main contribution of this paper is providing a new framework to overcome the topological restriction in attitude controls without relying on discontinuities through the controlled maneuvers. The efficacy of the proposed methods is illustrated by numerical simulations and experiments.§ INTRODUCTION The attitude dynamicsand control of a rigid body encountersthe unique challenge that the configuration space ofattitudes cannot be globally identified with a Euclidean space. Theyevolve on the compact nonlinear manifold, referred to as the three-dimensional special orthogonal group or , that is composed of 3× 3 orthogonal matrices with the determinant of one.Traditionally, the special orthogonal group has been parameterized via local coordinates, such as Euler angles or Rodriguez parameters. It is well known that such minimal, three-parameter representations suffer from singularities <cit.>.Quaternions have been regarded as an ideal alternative to minimal attitude representations, as they are defined by four parameters and they do not exhibit singularities. However, the configuration space of quaternions, namely the three-sphere double covers the special orthogonal group, and consequently, there are two antipodal quaternions corresponding to the same attitude. In fact, it has been shown that 5 parameters are required at least to represent the special orthogonal group globally in a one-to-one manner <cit.>.This ambiguity inherent to quaternions should be carefully resolved. Otherwise, there could occur unwinding phenomena, where the rigid body rotates unnecessarily through a large angle even if the initial attitude error is small <cit.>. This has been handled by constructing a control input such that the two antipodal quaternions yield the same input <cit.>, which is equivalent to designing an attitude controller in terms of rotation matrices. Another approach is to define an exogenous mechanism to lift attitude measurements on the special orthogonal group into the three-sphere in a robust fashion <cit.>, which may cause additional complexities.Alternatively, attitude control systems have been developed directly on the special orthogonal group to avoid the singularities of minimal parameterizations and the ambiguity of quaternions concurrently <cit.>. More specifically, a configuration error function that measures the discrepancy between the desired attitude and the current attitude is formulated via a matrix norm, and control systemsare designed such that controlled trajectories are attracted to the minimum of the error function, thereby accomplishing asymptotic stability.However, regardless of the choice of attitude representations, attitude control systems are constrained by the topological restriction on the special orthogonal group that prohibits achieving global attractivity via any continuous feedback control <cit.>. This is because the domain of attraction for those systems is homeomorphic to a Euclidean space, which cannot be identified with the tangent bundle of the special orthogonal group globally. For example, in the design of attitude control systems based on the aforementioned configuration error function, there are at least four critical points of the error function, according to the Lusternik-Schnirelmann category <cit.>. This implies that there will be at least three undesired equilibria in the controlled dynamics, and the region of attraction to the desired attitude excludes the union of the stable manifolds of those undesired equilibria. One can show that the resulting reduced region of attraction almost covers the special orthogonal group <cit.>, while precluding only a set of zero measure. But, the existence of the stable manifolds of the undesired equilibria may affect the controlled dynamics strongly <cit.>.Recently, the topological restriction has been tackled by introducing discontinuities in the controlled attitude dynamics. In particular, a set of attitude configuration error functions, referred to as synergistic potential functions, has been proposed <cit.>. The family of synergistic potential functions is constructed such that at each undesired critical point of a potential function, there is another member in the family with a lower value of the error. By consistently switching to the controller derived from the minimal potential function, robust global asymptotic stability is achieved for the attitude dynamics. While this approach avoids chattering behaviors by introducing hysteresis, discontinuities in the control input may excite unmodeled dynamics and cause undesired behaviors in practice. Interestingly, it has been unclear if a more general class of feedback control systems could accomplish the global stabilization task without introducing such disruptions in the control input <cit.>.The objective of this paper is to present an alternative framework to overcome the topological restriction on the special orthogonal group with control inputs that are continuous in time. This is achieved by shifting the desired attitude temporarily, instead of modifying the attitude error functions as in <cit.>. More explicitly, when the initial attitude and the initial angular velocity do not belong to the estimated region of attraction of a smooth attitude controller, the desired trajectory is shifted to a trajectory that is sufficiently close to the initial condition to guarantee convergence. While the initial value of the shifted reference trajectory is distinct from that of the true reference trajectory, it is constructed as a time-varying function such that the shifted reference trajectory exponentially converges to the true reference trajectory astime tends to infinity. Consequently, the corresponding continuous-time controlled trajectory, which is designed to follow the shifted reference trajectory, will converge to the true reference trajectory. The resulting time-varying attitude control system is discontinuous with respect to the initial condition, thereby bypassing the topological restriction. But, it is continuous in time so as to avoid the aforementioned issues of switching in hybrid attitude controls.All of these are rigorously examined and analyzedso as to show exponential convergence to the true reference trajectory and to verify that the region of attraction covers the special orthogonal group completely. Furthermore, the shifted reference trajectory is formulated insuch an explicit manner, using a conjugacy class in the special orthogonal group, that no complicated inequality conditions are needed.Later, this approach is also extended to adaptive controls to handleunknown constant disturbances in the attitude dynamics. In short, the unique contribution of the proposed approach is that global attractivity is accomplished on the special orthogonal group with control inputs that are continuous in time, overcoming the topological restriction.This paper is organized as follows. Mathematical preliminaries are presented and the attitude control problem is formulated in Section <ref>. With the assumption that there is no disturbance, two types of attitude control strategies are proposed in Section <ref>. These are extended to adaptive controls in Section <ref>, followed by numerical examples and experimental results.§ PROBLEM FORMULATION§.§ Mathematical PreliminariesThe inner product ⟨ A, B⟩ of two matrices or vectors A and B of the same size denotes the usual Euclidean inner product, i.e., ⟨ A, B⟩ = tr(A^TB). The normA for a matrix or vector A denotes the Euclidean norm, i.e., A^2 = ⟨ A, A⟩ = tr(A^TA). The minimum eigenvalue of a symmetric matrix A is denoted by λ_ min (A) and the maximum eigenvalue by λ_ max (A).The attitude dynamics of a rigid body evolve on the three-dimensional special orthogonal group, ={R∈^3× 3 |R^T R=I_3× 3, det[R]=1}. For any R, R_1, R_2 ∈,RR_1 - RR_2 =R_1- R_2= R_1R - R_2R. Let𝔰𝔬(3)denote the set of all 3× 3 skew symmetric matrices. The hat map : ℝ^3 →𝔰𝔬(3) is defined byv = (v_1, v_2,v_3) ↦v̂ = [0 -v_3v_2;v_30 -v_1; -v_2v_10 ],and its inverse map is denoted by ∨ and called the vee map. For any v and w in ℝ^3, v̂ w = v × w and⟨v̂, ŵ⟩= 2⟨ v, w⟩,wherethe left side is the inner producton ℝ^3× 3 and theright on ℝ^3. For θ∈ [0,2π] and a unit vector v∈ℝ^3,the matrix exponentialexp(θv̂) is computed as follows:exp(θv̂) = I +sinθv̂ + (1-cosθ)v̂^2. Next, we recall conjugacy classes in <cit.>. Let Z_θ∈ be the rotation about the axis e_3=(0,0,1) by an angle θ∈:Z_θ = exp(θê_3) = [cosθ -sinθ 0;sinθcosθ 0; 0 0 1 ].It is straightforward to showZ_θ_1 - Z_θ_2 =2 √(1-cos(θ_1 - θ_2))for any θ_1, θ_2∈ℝ. For θ∈ℝ, define theconjugacy class ofZ_θ inasC_θ = { R ∈| R = UZ_θ U^T,U∈},which is the set of all rotations through angle θ. The groupis partitioned into conjugacy classes. More explicitly,= ⋃_θ∈ [0,π]C_θandC_θ_1⋂ C_θ_2 = ∅for any 0≤θ_1 < θ_2 ≤π.Therefore, for any X∈, there exist a unique angleθ∈ [0,π] and some U∈ such thatX = U Z_θ U^T,where θ∈ [0,π] is determined byθ = arccos ( tr(X)-1/2 ).The rotation matrix U satisfying (<ref>) is not unique, but one can be obtained as follows. Let v be a unit eigenvector of X corresponding to eigenvalue 1. This vector v satisfies exp(θv̂) = X or X^T. If exp(θv̂) =X then set u_3 = v. Otherwise, set u_3 = -v. Choose a unit vector u_1 perpendicular to u_3 and let u_2 = u_3 × u_1. Then, the rotation matrix Udefined byU = [ u_1 u_2 u_3 ]satisfies (<ref>). Alternatively, for 0< θ <π the vector u_3 can becomputed as follows:u_3 = ( X-X^T/2sinθ )^∨,and the remaining columns u_1 and u_2 are constructed as discussed above.It is easy to show thatmax_R_1, R_2 ∈R_1 - R_2 = 2√(2)and the maximum value 2√(2) is attained if and only ifR_1R_2^T ∈ C_π, where C_π is the conjugacy class of Z_π.§.§ Attitude Dynamics and Control Objective The equations of motion for the attitude dynamics of a rigid body are given byṘ= RΩ̂, Ω̇ = (Ω) ×Ω + τ+Δ,with the rotation matrixR ∈ representing the linear transformation of the representation of a vector from the body-fixed frame to the inertial frame,and the angular velocity Ω∈ℝ^3 of the rigid body resolved in the body-fixed frame. The moment of inertia matrix is denoted by ∈ℝ^3× 3, which is symmetric and positive-definite, and the control torque resolved in the body-fixed frame is denoted by τ∈ℝ^3.The above equations include a constant but unknown disturbance torque Δ∈^3, which satisfies the following assumption.The magnitude of the disturbance is bounded by a known constant δ>0, i.e. Δ≤δ.Let (R_d(t), Ω_d(t)) ∈×ℝ^3 be a smooth reference trajectory such thatṘ_d(t) = R _d(t) Ω̂_d (t),for all t≥ 0. We wish to design a control torque τ such that the reference trajectory becomes asymptotically stable. § ATTITUDE TRACKING CONTROLS Throughout this section, it is assumed that there is no disturbance in the dynamics, i.e., Δ=0. A smooth attitude controller that yields almost global exponential stability is first presented, and it is extended for global attractivity. §.§ Almost Global Tracking Strategy The attitude tracking error E_R∈^3× 3 and the angular velocity tracking error e_Ω∈^3 are defined asE_R = R - R_d,e_Ω = Ω - Ω_d.Notice that our definition of e_Ω is distinct from that in <cit.>, where the desired angular velocity is multiplied by R^TR_d.Define an auxiliary vector e_R∈^3 ase_R= 1/2 (R^T_dR - R^TR_d)^∨.The relations between E_R and e_R are summarized as follows.1. For any R and R_d ∈,e_R^2 = 1/2R - R_d^2( 1 - 1/8R -R_d^2 ). 2.For any R andR_d ∈,e_R^2≤1/2R -R_d^2. 3. For any number a satisfying 0 < a < 1 andfor any R andR_d ∈ satisfying R -R_d≤ 2√(2a),(1-a)/2R -R_d^2 ≤e_R^2.Let R^TR_d=exp(θv̂) for θ∈[0,π]and v∈^3 with v=1. Using Rodrigues' formula, one can show E_R^2=4(1-cosθ),e_R= sinθ.Substituting these, it is straightforward to show the first identity, which implies the next two inequalities. Along the trajectory of therigid bodysystem,ė_R = C(R^TR_d) e_Ω + e_R ×Ω_dwhereC(R^TR_d) = 1/2(tr (R^TR_d) I - R^TR_d).From R_dot and Rd_dot,ė_R = 1/2{Ω̂R^T R_d + R_d^T R Ω̂-Ω̂_d R_d^T R - R^T R_dΩ̂_d}.Substitute Ω= e_Ω +Ω_d, and then the two terms dependent on e_Ω reduce to C(R^TR_d) e_Ω by the identity(x̂ A + A^T x̂)^∨ = (tr(A)I - A)xfor all x∈ℝ^3 and A∈ℝ^3× 3.The remaining terms, which depend on Ω_d, simplify to e_R ×Ω_d bythe definition of e_R in eR and the identity, x̂ŷ - ŷx̂ = x× y for any x,y∈^3.Consider a Lyapunovfunction (candidate) V: ×ℝ^3 ×ℝ→ℝ defined byV(R, Ω,t)= k_R/4 E_R^2 + 1/2e_Ω^2+ μ⟨ e_R, e_Ω⟩,where k_R >0 and μ>0. Define an auxiliary function V_0(R, Ω,t) as follows:V_0 (R, Ω,t) = k_R/4 E_R^2 + 1/2 e_Ω^2,which coincides with V when μ = 0.The following lemma discusses positive-definiteness of the function V and its relationship with its auxiliary V_0. Suppose0 < μ < √(k_R).Then, the symmetric matricesW_1 = [1/4k_R -1/2√(2)μ; -1/2√(2)μ 1/2 ], W_2 = [ 1/4k_R 1/2√(2)μ; 1/2√(2)μ1/2 ]are positive-definite and satisfyz^T W_1z ≤ V(R,Ω, t)≤ z^T W_2 zfor all (R,Ω) ∈×ℝ^3 and t≥ 0, where z = (E_R,e_Ω ) ∈ℝ^2. Moreover,√(k_R) - μ/√(k_R)V_0(R,Ω,t) ≤ V(R,Ω, t)≤√(k_R) + μ/√(k_R)V_0(R,Ω,t)for all (R,Ω) ∈×ℝ^3 and t≥ 0. The inequality (<ref>) follows fromLemma <ref> and the Cauchy-Schwarz inequality. Next, let z̃ = (√(k_R)/2E_R, 1/√(2)e_Ω)^T∈^2. Then, z^T W_1 z can be rewritten asz^T W_1 z = (z̃)^T [ 1 -μ/√(k_R); -μ/√(k_R) 1 ]z̃≥(1-μ/√(k_R)) z̃^2,which, together with (<ref>), shows the first inequality in(<ref>)as z̃^2=V_0(R,Ω,t). The second inequality in(<ref>) can be shown similarly. We propose the following tracking controller:τ =- (Ω)×Ω +( -k_R e_R - k_Ω e_Ω + Ω×Ω_d + Ω̇_d),where k_R>0 is the same constant as that used for the function V,and k_Ω >0. The following lemma computes the rate of change of V_0 and V along the trajectory of the closed-loop system. 1. Along the trajectory of the closed-loop system with the control (<ref>), the time-derivative of the auxiliary V_0 is given byV̇_0(R,Ω,t) =-k_Ωe_Ω^2.2. Let a be any number satisfying 0 < a <1 and choose any μ such that0 < μ < 4(1-a)k_R k_Ω/4(1-a) k_R + k_Ω^2.IfR -R_d≤ 2√(2a), then the time-derivative of the Lyapunov function along the controlled trajectories satisfiesV̇(R,Ω,t) ≤ -z^TW_3zwithz = ( E_R ,e_Ω) ∈ℝ^2 and the matrix W_3∈^3× 3 defined asW_3 = [ (1-a)/2μ k_R - 1/2√(2)μ k_Ω; - 1/2√(2)μ k_Ω k_Ω -μ ],which is positive-definite. Let Q=R_d^T R∈. From the attitude kinematics equations and the definition of e_Ω,Q̇ = Ṙ_d^T R + R_d^TṘ = -Ω̂_d Q + QΩ̂= Qê_Ω + QΩ̂_d-Ω̂_d Q.We have E_R^2 = (R-R_d)^T(R-R_d) = 2I_3× 3-Q. Therefore,d/dtk_R/4E_R^2 = -k_R/2Q̇= -k_R/2Qê_Ω + QΩ̂_d-Ω̂_d Q=-k_R/2Qê_Ω,where the last equality is obtained using AB-BA=0 for any square matrices A,B.From the identity Ax̂ = - x· (A-A^T)^∨ for any x∈^3 and A∈^3× 3, the above is rewritten ask_R/2 e_Ω·(Q-Q^T)^∨ = k_R e_R· e_Ω.Using this, along the trajectory of R_dot and Omega:dot,V̇(R,Ω,t)= ⟨ e_Ω, k_R e_R - Ω̇_d + ^-1( τ + (Ω) ×Ω ) ⟩+ μ⟨ C(R^TR_d) e_Ω, e_Ω⟩+ μ⟨ e_R, Ω_d × e_Ω - Ω̇_d + ^-1( τ+ (Ω) ×Ω ) ⟩,where the matrix C(R^TR_d) is defined in (<ref>). Substituting the control (<ref>), and using the fact that e_Ω×Ω_d = (Ω - Ω_d) ×Ω_d = Ω×Ω_d, this reduces toV̇(R,Ω,t)= - k_Ωe_Ω^2 -μ k_Re_R^2 - μ k_Ω⟨ e_R, e_Ω⟩+ μ⟨ C(R^TR_d)e_Ω, e_Ω⟩.Setting μ=0 yield V0_dot. According to <cit.>, the matrix C(R^TR_d) defined in (<ref>) satisfies C(R^TR_d)_2 ≤ 1, where ·_2 is the operator 2-norm. Using this fact, the Cauchy-Schwarz inequality and Lemma <ref>, one can easily prove (<ref>). The given bound of μ guarantees the positive-definiteness of W_3.Next, we show that the proposed control system yields exponential stability.Choose any positive numbers k_R, k_Ω, aand μ such that0 < a<1 and0 < μ < 4(1-a)k_R k_Ω/4(1-a) k_R + k_Ω^2.Letσ = λ_min(W_3)/λ_max(W_2) >0,where the matrices W_2 and W_3 aredefined in (<ref>) and (<ref>), respectively.Then, the zero equilibrium of the tracking errors (E_R,e_Ω)=(0,0) is locally exponentially stable, and for any initial state (R(0), Ω(0)) ∈×ℝ^3 satisfyingV_0(R(0), Ω(0), 0) ≤ 2 a k_R,the closed-loop trajectory (R(t),Ω(t)) for the control (<ref>) satisfiesV_0(R(t), Ω(t), t) ≤ V_0(R(0), Ω(0), 0) ≤ 2 a k_R, V(R(t), Ω(t), t) ≤ V(R(0), Ω(0), 0) e^-σ tfor all t≥ 0. Furthermore, both the attitude tracking error R(t) -R_d(t) and the body angular velocity tracking error Ω(t) - Ω_d(t) converge exponentially to zero at the exponential rate of (-σ/2) as t tends to infinity, i.e., there exists aconstant c>0 such thatE_R(t) + e_Ω(t) ≤ c(E_R(0) +e_Ω(0) )e^-σ/2 tforall t≥ 0 andall initial state (R(0), Ω(0)) satisfying (<ref>). Take any positive numbers k_R, k_Ω, a and μ that satisfy (<ref>) and (<ref>). Since4(1-a)k_R k_Ω/4(1-a) k_R + k_Ω^2≤4(1-a)k_R k_Ω/2√(4(1-a) k_R k_Ω^2) < √(k_R),we have0 < μ < √(k_R) by (<ref>).According to Lemma <ref>, the matrix W_1 defined in (<ref>) is positive-definite and both (<ref>) and (<ref>) hold for all (R,Ω) ∈×ℝ^3 and t≥ 0. It follows that the functionV is positive-definite and decrescent.Choose any initial state (R(0),Ω(0)) satisfying (<ref>). By (<ref>)in Lemma <ref>, V_0(R(t),Ω(t),t) is a non-increasing function of time along the closed-loop trajectory and (<ref>) holds for all t≥ 0, which impliesR(t) - R_d(t)^2 ≤4/k_RV_0(R(t), Ω(t), t) ≤ 8afor all t≥ 0. Since (<ref>) holds, by Lemma <ref> we have (<ref>) withW_3 being positive-definite. The region of attraction to an asymptotically stable equilibrium is often estimated by a sub-level set of the Lyapunov function <cit.>. While the estimate of the region of attraction given by (<ref>) is not a sub-level set of the Lyapunov function V(R,Ω,t), it is positively invariant as V_0(R(t),Ω(t),t) is non-increasing in t. Therefore, for any initial condition satisfying (<ref>), both of (<ref>) and (<ref>) hold true for all t≥ 0, and the exponential convergence is guaranteed as follows.From (<ref>) and (<ref>), it followsV̇(R(t), Ω (t), t)≤ -σ V(R(t),Ω(t),t)for all t≥ 0, where σ is defined in (<ref>). This shows (<ref>). Next, we show exp0. From (<ref>) and (<ref>),V_0(R(t), Ω(t), t) ) ≤√(k_R)+μ/√(k_R)-μ V_0(R(0), Ω (0), 0) e^-σ t,which impliesk_R/4E_R(t)^2 + 1/2 e_Ω(t) ^2 ≤√(k_R)+μ/√(k_R)-μ( k_R/4 E_R(0)^2 + 1/2 e_Ω(0) ^2)e^-σ t.We can view √(k_R/4A^2 + 1/2x^2) for (A,x)∈^3× 3×^3 as a norm on ^3× 3×^3. As all norms on a finite-dimensional vector space are equivalent <cit.>,there are positive constants c_1 and c_2 such thatc_1 (A + x) ≤√(k_R/4A^2 + 1/2x^2)≤ c_2 (A + x)for all (A,x)∈^3× 3×^3 . Hence, lettingc = c_2/c_1√(√(k_R)+μ/√(k_R)-μ),we haveexp0 for all t≥ 0 and all initial states satisfying (<ref>). The following corollary characterizes the region of attraction that guarantees exponential stability, estimated by (<ref>), and it discusses how to choose the values of the control parameters k_R and k_Ω for a given initial state.Given an arbitrary initial state (R(0), Ω(0)) ∈×ℝ^3 satisfyingR(0) - R_d(0) < 2√(2),take any k_R such that2 Ω (0) - Ω_d(0)^2/8 - R(0) - R_d(0)^2 < k_R.Then, V_0(R(0), Ω (0), 0) < 2k_R, and the conclusion of Theorem <ref> holds true for any k_Ω, a and μ satisfying (<ref>) andV_0(R(0), Ω (0),0)/2k_R≤ a <1.Straightforward. The inequalites condR0 and (<ref>) imply that the proposed control system can handle any initial attitude excluding the set { R∈| R - R_d = 2√(2)}, which is equal to C_π R_d, where C_π is the conjugacy class containing rotations through angle π. Since (C_πR_d) =C_π = 2 while = 3,one canclaim that for a given R_d ∈, the set {R ∈|R-R_d < 2√(2)} almost covers the entire space .Therefore,Corollary <ref> implies that our tracking control law can handle a large set of initial attitude tracking errors, excluding a set ofmeasure zero only. This property is referred to as almost global exponential stability <cit.>, and it is considered as the strongest stability property for smooth attitude controls, due to the topological obstruction that prohibits global attractivity with smooth vector fields on <cit.>.While there have been attitude controllers achieving almost global exponential stability <cit.>, they showed the exponential convergence for the auxiliary attitude error vector e_R, which is not necessarily proportional to the attitude error <cit.>. Instead, the stability analysis presented in this section guarantees the exponential convergence of the attitude error E_R satisfying E_R=R-R_d=I_3× 3-R^T R_d. §.§ Global Tracking Strategy In practice, there is a limited chance that the initial attitude is placed in the low-dimensional set of C_πR_d that does not guarantee the exponential convergence. However, it is shown that the controlled trajectories may be strongly affected by the existence of the stable manifolds of points inC_πR_d, and the rate of convergence can be reduced significantly <cit.>.To avoid these issues, hybrid attitude control systems have been introduced to achieve global asymptotic stability. These are based on a class of attitude error functions, referred to as synergistic potential functions, that are constructed by stretching and scaling the popular trace form of the attitude error function <cit.>, or by expelling the controlled attitude trajectories away from the undesired equilibria <cit.>. In these approaches, the topological obstruction to global attractivity is avoided by using discontinuities of the control input with respect to time. However, attitude actuators are constrained by limited bandwidth, and abrupt changes in the control torque may excite the unmodeled dynamics and cause undesired behaviors, such as the vibrations of solar panels in satellites. Interestingly, it has not been known whether a more general class of time-varying feedback could accomplish the global stabilization task without introducing such discontinuities <cit.>.In this section, we present an alternative approach to achieve global attractivity. The key idea is to introduce a shifted desired attitude, and design an attitude control system to follow the shifted trajectories instead of the original attitude command. The shifted desired trajectory is carefully constructed with the conjugacy class discussed in Section <ref> to guarantee the convergence to the desired attitude trajectory from any initial attitude. In contrast to hybrid attitude controls where the configuration error function defining the controlled dynamics is switched instantaneously, the proposed approach adjusts the desired attitude trajectory continuously in time, thereby avoiding any jump in the control input.Consider the attitude control system presented in Theorem <ref>. Suppose the initial condition (R(0),Ω(0)) satisfies (<ref>). Then, the exponential convergence is guaranteed, and there is no need for modification. As such, this section focuses on the other case when the initial condition does not satisfy the given estimate of the region of attraction, i.e.,V_0(R(0), Ω(0), 0) > 2 a k_R. Now, we introduce the shifted desired attitude. By (<ref>), (<ref>), and (<ref>), there exists a unique θ_0 ∈ [0, π] such that R(0)R_d(0)^T ∈ C_θ_0, i.e.R(0) = U_0 Z_θ_0U_0^T R_d(0)for some U_0 ∈. In other words, the initial attitude and the initial desired attitude is related by the fixed-axis rotation by the angle θ_0 about the third column of U_0.Next, for a constant ϵ∈(0,1), choose an angle θ_b_0∈ (0, θ_0) such that1-cos(θ_0 - θ_b_0) ≤ 2ϵ a.Note that such an angle θ_b_0 always exists for anyϵ,a∈(0,1) since one has only to make |θ_0 - θ_b_0| sufficiently small. Then, define a time-varying angle θ_b(t) asθ_b(t) = θ_b_0e^-γ/2 t,where γ >0 is a positive constant satisfyingγ < 4/θ_b_0√(a k_R (1-ϵ)).Using θ_b(t), define the shifted desired attitude asR̃_d(t) = U_0 Z_θ_b(t)U_0^T R_d(t).Therefore, R_d(t)^T R̃_d(t) belongs to the conjugacy class C_θ_b(t). The properties of the shifted desired attitude are summarized as follows. Consider the shifted desired attitude trajectory given by Rd:shifted:varying. (i) The initial attitude error from the shifted desired trajectory satisfiesR(0)-R̃_d(0) = 2√(1-cos(θ_0-θ_b_0))< 2√(2a). (ii) The difference between the shifted desired attitude and the true desired attitude exponentially converges to zero asR̃_d(t) - R_d(t)= 2√(1-cosθ_b(t))≤√(2)θ_b_0 e^-γ/2 t. (iii) The time-derivative of the shifted desired attitude trajectory is given byṘ̃̇_d(t) = R̃_d(t) Ω̂̃̂_d(t),where the shifted desired angular velocity Ω̃_d(t) isΩ̃_d(t) = Ω_d(t) + θ̇_b(t) R̃_d(t)^T U_0 e_3.(iv) The difference between the shifted desired angular velocity and the true desired angular velocity is given byΩ̃_d(t)-Ω_d(t) = |θ̇_b(t)| =γ/2θ_b_0 e^-γ/2 t.From (<ref>) and (<ref>),R(0)-R̃_d(0)=U_0 Z_θ_0 U_0^T R_d(0)-U_0 Z_θ_b_0 U_0^T R_d(0)= Z_θ_0-Z_θ_b_0 = √(1-cos(θ_0-θ_b_0)),which shows (<ref>) from newly:added:theta. Similarly,R̃_d(t) - R_d(t) = Z_θ_b(t) - I_3× 3 = 2√(1-cosθ_b(t)).From the fact that 2/π^2x^2 ≤ 1-cos x ≤1/2x^2 for any x∈[0,π],the last inequality of Rdtilde:Rd follows.Next, the time-derivative of the shifted reference attitude isd/dtR̃_d(t) = θ̇_b(t) U_0 ê_3 Z_θ_b(t)U_0^T R_d(t)+R̃_d(t)Ω̂_d(t),which can berewritten asd/dtR̃_d(t) = θ̇_b(t) U_0 ê_3 U_0^T R̃_d(t)+R̃_d(t)Ω̂_d(t)by Rd:shifted:varying. Using the property Rx=Rx̂ R^T for any R∈ and x∈^3 repeatedly,d/dtR̃_d(t)= θ̇_b(t)U_0e_3R̃_d(t)+R̃_d(t)Ω̂_d(t) = θ̇_b(t) R̃_d(t) (R̃_d(t)^TU_0e_3)^∧+R̃_d(t)Ω̂_d(t),and this shows Omega:tilde:varying. It is straightforward to show Wdtilde:Wd as R̃_d(t)^T U_0 e_3=1 for any t≥ 0.The motivation for the proposed shifted desired attitude is that the initial attitude error, namelyE_R(0)=R(0)-R_d(0) =Z_θ_0 - I_3× 3 = 2√(1-cosθ_0),is replaced by the shifted errorR(0)-R̃_d(0)=2√(1-cos(θ_0-θ_b_0)),that is strictly less than 2√(2a) from (<ref>). In other words, the inequality newly:added:theta ensures that the initial value of the shifted desired attitude, namely R̃_d(0) is sufficiently close to the initial attitude R(0). As such, even when the initial attitude error R(0)-R_d(0) is close or equal to 2√(2), we can replaceit with the shifted desired attitude so as to satisfy condR0. Furthermore, as shown by Rdtilde:Rd and Wdtilde:Wd, the shifted desired trajectories (R̃_d(t),Ω̃_d(t)) exponentially converge to their true desired trajectories (R_d(t),Ω_d(t)) as t tends to infinity. This is because θ_b(t) exponentially converges to zero from theta:b:varying. Therefore, we can design a control system to follow the shifted desired trajectories, while ensuring asymptotic convergence to the true desired trajectories.More explicitly, the shifted attitude error variables Ẽ_R∈^3× 3 and ẽ_R∈^3 are defined asẼ_R= R - R̃_d, ẽ_R= 1/2 (R̃_d^T R - R^TR̃_d)^∨.Also, the shifted angular velocity error is defined asẽ_Ω = Ω -Ω̃_d∈^3.As with (<ref>), the control input for the shifted desired reference can be designed asτ̃= - (Ω)×Ω +( - k_R ẽ_R -k_Ωẽ_Ω + Ω×Ω̃_d + Ω̇̃̇_d).From Theorem <ref>, for any initial condition satisfyingk_R/4R(0)-R̃_d(0)^2 + 1/2Ω(0)-Ω̃_d(0)^2 ≤ 2a k_R,which is equivalent to (<ref>) for the shifted desired trajectory, the trajectory of the controlled system exponentially converges to the shifted desired trajectories (R̃_d(t),Ω̃_d(t)) that tends to the true desired trajectories (R_d(t),Ω_d(t)). The resulting stability properties are summarized as follows. Choose any positive constants k_R, k_Ω, a, μ, σ, ϵ, θ_b_0, and γ such that (<ref>), (<ref>), (<ref>), newly:added:theta and gamma are satisfied. The control input is defined asτ=- (Ω)×Ω+ ( - k_R e_R -k_Ωe_Ω+ Ω×Ω_d + Ω̇_d)whenV_0(R(0),Ω(0),0)≤2ak_R,- (Ω)×Ω+ ( - k_R ẽ_R -k_Ωẽ_Ω+ Ω×Ω̃_d + Ω̇̃̇_d)whenV_0(R(0),Ω(0),0)> 2ak_R,where ẽ_R and ẽ_Ω are constructed by eR:tilde and eW:tilde, respectively.Then, the zero equilibrium of the tracking errors (E_R,e_Ω)=(0,0) is exponentially stable. More specifically, when V_0(R(0),Ω(0),0)≤ 2ak_R, the tracking errors (E_R(t),e_Ω(t)) converge to zero exponentially according to exp0. Otherwise, when V_0(R(0),Ω(0),0)> 2ak_R, there exists a positive constant c>0 such that, for any initial state (R(0), Ω(0)) ∈×ℝ^3 satisfying ROA_2,E_R(t)+e_Ω(t)≤ c(E_R(0)+e_Ω(0))exp^-1/2min{σ,γ}t,for all t≥ 0. When V_0(R(0),Ω(0),0)≤ 2ak_R, the results of Theorem <ref> are directly applied, i.e., the tracking errors (E_R(t),e_Ω(t)) converge to zero exponentially according to exp0.Next, when V_0(R(0),Ω(0),0)> 2ak_R, for any initial condition satisfying ROA_2, according to Theorem <ref>, there exists c̃>0 such thatẼ_R(t)+ẽ_Ω(t)≤c̃ (Ẽ_R(0)+ẽ_Ω(0))e^-σ/2tfor all t≥ 0. From the triangle inequality,E_R(t) +e_Ω(t)≤Ẽ_R(t)+ẽ_Ω(t) + R̃_d(t)-R_d(t)+Ω̃_d(t)-Ω_d(t).Substituting Rdtilde:Rd, Wdtilde:Wd, and tildez,E_R(t)+e_Ω(t) ≤c̃ (Ẽ_R(0)+ẽ_Ω(0))e^-σ/2t+√(2)+γ/2θ_b_0 e^-γ/2 t. Now, we derive several inequalities to show exp2. The initial shifted attitude error satisfiesẼ_R(0) ≤E_R(0)+R_d(0)-R̃_d(0)= E_R(0) + 2√(1-cosθ_b(0)).Since θ_b(0)=θ_b_0 < θ_0 from the definition of θ_b_0,Ẽ_R(0)< E_R(0) + 2√(1-cosθ_0) = 2E_R(0). Also, from Wdtilde:Wd,ẽ_Ω(0) ≤e_Ω(0) + Ω_d(0)-Ω̃_d(0)≤e_Ω(0) + γ/2θ_b_0.Next, from the fact that 2/π^2x^2 ≤ 1-cos x ≤1/2x^2 for any x∈[0,π] and θ_b_0<θ_0, we haveθ_b_0 ≤π/√(2)√(1-cosθ_b_0)<π/√(2)√(1-cosθ_0)=π/2√(2)E_R(0),which is substituted into z0 together with ER:tilde:0 to obtainE_R(t)+e_Ω(t)≤2c̃ + π/2√(2)γ/2(1+c̃)+√(2)E_R(0)+c̃e_Ω(0)× e^-1/2min{σ,γ}t.This shows exp2, which guarantees exponential stability.Next, we characterize the region of attraction of the proposed control system as follows.For the control system presented in Theorem <ref>, the region of attraction guaranteeing the exponential convergence encloses the following set,ℛ={(R(0),Ω(0))∈×^3 | e_Ω(0) <max{√(2k_R(2a-1+cosθ_0)), 2√(ak_R(1-ϵ))-γ/2θ_b_0}},where θ_0 is constructed from R(0) by R0:U:thetab.Furthermore, ℛ⊂×^3 coverscompletely, and when projected onto ^3, it is enlarged into ^3 as k_R is increased in a semi-global sense.Define three subsets of ×^3 for the initial condition asℛ_1 =k_R(1-cosθ_0)+1/2e_Ω(0)^2≤ 2ak_R , ℛ_2 =k_R(1-cosθ_0)+1/2e_Ω(0)^2> 2ak_R , ℛ_3 =k_R(1-cos(θ_0-θ_b_0))+1/2ẽ_Ω(0)^2≤ 2ak_R .The sets ℛ_1 and ℛ_2 represent the initial conditions corresponding to the two cases of the control inputs, namely tau_3a and tau_3b, respectively. The set ℛ_3 represents the set of initial conditions ROA_2, guaranteeing the exponential convergence for the second case of the control input. Therefore, the combined region of attraction ℛ̅ guaranteeing exponential stability is given by ℛ̅≡ℛ_1∪(ℛ_2∩ℛ_3). Since ℛ_1∪ℛ_2=×^3, this reduces to ℛ̅=(ℛ_1∪ℛ_2)∩ (ℛ_1∪ℛ_3)=ℛ_1∪ℛ_3.Now, we show the set ℛ defined by ROA is contained in ℛ̅, i.e., ℛ⊂ℛ̅. For any (R(0),Ω(0))∈ℛ,e_Ω(0) < √(2k_R(2a-1+cosθ_0)),ore_Ω(0) < 2√(ak_R(1-ϵ))-γ/2θ_b_0,where the right-hand side is positive due to gamma. For the former case, it is straightforward to show (R(0),Ω(0))∈ℛ_1⊂ℛ̅. For the latter case, from tilde:eW:bound,ẽ_Ω(0)≤ 2√(ak_R(1-ϵ)).Therefore,k_R(1-cos(θ_0-θ_b_0))+1/2ẽ_Ω(0)^2 ≤ k_R(1-cos(θ_0-θ_b_0)) + 2a(1-ϵ)k_R ≤ 2ak_R,where the last inequality is obtained by newly:added:theta. This follows (R(0),e_Ω(0))∈ℛ_3⊂ℛ̅. In short, any initial condition in ℛ belongs to the estimated region of attraction ℛ̅=ℛ_1∪ℛ_3 for the controlled dynamics, and the corresponding trajectory converges to zero exponentially according to exp0 or exp2.Next, for any θ_0∈[0,π], the set ℛ is non-empty due to gamma. As such, ℛ containsvia (<ref>). At ROA, the upper bound of e_Ω(0) tends to be infinite, as k_R→∞. Therefore, ℛ covers ^3 in a semi-global sense, as k_R→∞. The exceptionalproperty of the proposed control system is that the region of attraction covers the special orthogonal group completely, but the control input formulated in tau_3 is continuous in the time, i.e., there is no switching through the controlled attitude dynamics. The essential idea is that when the initial errors are large, the desired attitude is altered to an attitude that is closer to the initial attitude along the same conjugacy class, which is gradually varied back to the true desired attitude.Taking the advantage of the global attractivity in attitude controls with non-switching controls has been unprecedented, and it has been considered largely impossible to achieve that. As discussed above, it has been uncertain if a more general class of time-varying feedback could accomplish the global stabilization task without introducing switching <cit.>. The proposed control system overcomes the topological restriction withthe discontinuity of the control input with respect to the initial condition. The proposed framework of modifying the desired trajectory to achieve global attractivity has been unprecedented, and it is readily generalized to abstract Lie groups and homogeneous manifolds. § ADAPTIVE ATTITUDE TRACKING CONTROLS In this section, a constant disturbance moment Δ that has been introduced in Omega:dot is considered. Throughout this section, it is assumed that the reference trajectory is such that both Ω_d(t) and Ω̇_d(t) are bounded. The organization of this section is parallel to the preceding section: two types of attitude tracking strategies are presented with an adaptive law to eliminate the effects of the disturbance. §.§ Almost Global Adaptive Tracking Strategy The overall controller structure and the definition of the error variables and parameters are identical to those defined in Section <ref>. The adaptive control law presented in this section includes an estimate of the disturbance, denoted by Δ̅∈^3 in the control torque asτ =- (Ω)×Ω +( -k_R e_R - k_Ω e_Ω + Ω×Ω_d + Ω̇_d)-Δ̅,where Δ̅ is updated according toΔ̇̅̇ = k_Δ^-1 (e_Ω + μ e_R)for k_Δ > 0 with the initial estimate Δ̅(0)=0.Let the estimation error bee_Δ = Δ -Δ̅∈^3.From Assumption <ref>, we have the bound for the initial estimation error as e_Δ(0)≤δ.Define a Lyapunov function, augmented with an additional term for the estimation error e_Δ asV̅(R(t),Ω(t),Δ̅(t),t) = V(R(t),Ω(t),t) + 1/2k_Δ e_Δ(t)^2.Along the trajectory of the controlled system with tracking:control:delta,V̇̅̇(t) =- k_Ωe_Ω^2 -μ k_Re_R^2 - μ k_Ω⟨ e_R, e_Ω⟩ + μ⟨ C(R^TR_d)e_Ω, e_Ω⟩ +⟨ e_Δ, ^-1 (e_Ω+μ e_R) -1/k_ΔΔ̇̅̇⟩,where V̅(t) is a shorthand for V̅(R(t),Ω(t),Δ̅(t),t). Substituting bar:delta:dot, it reduces toV̇̅̇(t) = - k_Ωe_Ω^2 -μ k_Re_R^2 - μ k_Ω⟨ e_R, e_Ω⟩ + μ⟨ C(R^TR_d)e_Ω, e_Ω⟩.Thus, the proposed control torque tracking:control:delta and the adaptive law bar:delta:dot ensure that the time-derivative of the augmented Lyapunov function is identical to (<ref>) that is developed for the ideal case when Δ=0. The corresponding stability properties are summarized as follows. Consider the control torque defined in tracking:control:delta with the adaptive law bar:delta:dot. Choose positive constants k_R, k_Ω, k_Δ, a, and μ such that (<ref>), (<ref>), and the following inequality are satisfied.0 < 2a√(k_R)-μ/√(k_R)+μk_R-1/2k_Δδ^2.Then, the desired reference trajectory (R(t),Ω(t),Δ̅(t))=(R_d(t),Ω_d(t),Δ) is asymptotically stable. Furthermore, for any initial state (R(0), Ω(0)) ∈×ℝ^3 satisfyingV_0(R(0),Ω(0),0)≤ 2a√(k_R)-μ/√(k_R)+μk_R-1/2k_Δδ^2,all of the attitude tracking error R(t)-R_d(t), the angular velocity tracking error Ω(t)-Ω_d(t) and the estimation error Δ-Δ̅(t) converge to zero as t tends to infinity.According to the proof of Theorem <ref>, the augmented Lyapunov function V:bar satisfiesz^T W_1 z + 1/2k_Δe_Δ^2 ≤V̅(t) ≤ z^T W_2 z + 1/2k_Δe_Δ^2,for all t≥ 0, where the matrices W_1 and W_2 defined in (<ref>) are positive-definite as μ < √(k_R) from (<ref>) and (<ref>).Since C(R^T R_d)_2≤ 1, from Vbar_dot0,V̇̅̇(t) ≤ - (k_Ω-μ) e_Ω^2 -μ k_Re_R^2 + μ k_Ωe_Re_Ω,and it is negative-semidefinite, i.e., V̇̅̇≤ 0 as μ < 4k_Rk_Ω/4k_R+k_Ω^2 from (<ref>). Hence V̅(t) is non-increasing.The condition kdelta ensures that the set of initial conditions satisfying ineq:var:V0 is a non-empty neighborhood of the desired reference trajectory. From (<ref>) and the fact that e_Δ(0)≤δ,V̅(R(0),Ω(0),Δ̅(0),0)≤√(k_R)+μ/√(k_R) V_0(R(0),Ω(0),0)+1/2k_Δδ^2.Therefore, for any initial condition satisfying ineq:var:V0,V̅(R(0),Ω(0),Δ̅(0),0) ≤ 2(k_R-μ√(k_R))a.Since V̅(t) is a non-increasing function of time, using the lower bound of (<ref>), the above inequality impliesR(t)-R_d(t)^2 ≤4/k_RV_0(t)≤4/k_R -μ√(k_R) V(t)≤4/k_R -μ√(k_R)V̅(t) ≤4/k_R -μ√(k_R)V̅(0) ≤ 8a.Therefore, ER2a is satisfied. As discussed above, the expression for V̇̅̇(t) is identical to (<ref>), and therefore, we can apply Lemma <ref> to obtainV̇̅̇(t) ≤ - z^T W_3 z,for all t≥ 0 with z=(E_R,e_Ω), and W_3 is positive-definite due to (<ref>).In short, for any initial state (R(0), Ω(0)) ∈×ℝ^3 satisfying ineq:var:V0, the Lyapunov function is positive-definite and decrescent as V4, and its time-derivative is negative-semidefinite as V4_dot, for all t≥ 0. This implies that the reference trajectory is stable in the sense of Lyapunov, and all of the error variables,namely E_R(t), e_Ω(t), and e_Δ(t) arebounded. Also, from the LaSalle-Yoshizawa theorem <cit.>, z = ( E_R ,e_Ω)→ 0 as t→∞. Using the boundedness of the error variables, one can easily show that ë_Ω is bounded as well. Then, according to Barbalat's lemma, ė_Ω→ 0 as t→∞.Substituting these into Omega:dot and tracking:control:delta guarantees Δ̅→Δ as t→∞. Therefore, the zero equilibrium of the tracking errors and the estimation error is asymptotically stable. In contrast to Theorem <ref>, there is no guarantee of exponential convergence as thegiven ultimate bound of V̇̅̇ does not depend on the estimation error e_Δ. Alternatively, one could achieve exponential stability by redefining the error variables as shown in <cit.>, but we do not pursue it in this paper.An alternative estimate of the region of attraction is given by the sub-level set of the Lyapunov function as ROA_3a. The estimate provided by ineq:var:V0 in Theorem <ref> is more conservative, as it is a subset of ROA_3a. However, we use ineq:var:V0 as an estimate of the region of attraction in the subsequent development throughout this section for simplicity. As with Corollary <ref>, we discuss how to choosevalues of the control parameters k_R and k_Ω for a given initial state.Given an arbitrary initial state (R(0), Ω(0),Δ̅(0)) ∈×ℝ^3×^3 satisfyingR(0) - R_d(0) < 2√(2),take any k_R, k_Δ, k_Ω, and μ such thatR(0) - R_d(0)^2 < 8√(k_R)-μ/√(k_R)+μ, 2 Ω (0) - Ω_d(0)^2+2/k_Δδ^2/8√(k_R)-μ/√(k_R)+μ - R(0) - R_d(0)^2 < k_Rwith (<ref>) and kdelta.Then, V_0(R(0),Ω(0),0)< 2a √(k_R)-μ/√(k_R)+μ-1/2k_Δδ^2, and the conclusions of Theorem <ref> hold true for any a satisfyingV_0(R(0),Ω(0),0)+1/2k_Δδ^2/2√(k_R)-μ/√(k_R)+μ≤ a <1.Straightforward. This corollary implies that the region of attraction for asymptotic convergence almost coversas discussed in Section <ref>, and the region of attraction for Ω increases in a semi-global sense by increasing k_R.While these results are developed for the constant disturbance, it can be readily generalized to any other disturbance models where the unknown parameter appears linearly, such as ϕ(R,Ω,t)Δ with a known function ϕ(R,Ω,t), as commonly studied in the literature of adaptive controls with a weaker convergence property that ϕ(R,Ω,t)e_Δ asymptotically converges to zero <cit.>. Theresult presented in this paper may be considered as a special case when ϕ(R,Ω,t)=I_3× 3.§.§ Adaptive Global Tracking Strategy As in Section <ref>, we construct an adaptive control law that is continuous in time, while guaranteeing global attractivity. Consider the adaptive control system presented in Theorem <ref>. If a given initial condition satisfies ineq:var:V0, the error variables asymptotically converge to zero. Therefore, we focus on the case whereV_0(R(0),Ω(0),0)> 2a√(k_R)-μ/√(k_R)+μk_R-1/2k_Δδ^2. Define θ_0∈[0,π] and U_0 as in R0:U:thetab. For a constant ϵ∈(0,1), choose an angle θ_b_0∈(0,θ_0) such that1-cos(θ_0-θ_b_0) ≤2a√(k_R)-μ/√(k_R)+μ-1/2k_Δ k_Rδ^2ϵ.Then, define the time-varying angle θ_b(t) as theta:b:varying with a positive constant γ satisfyingγ < 2/θ_b_0√(2(1-ϵ)2a√(k_R)-μ/√(k_R)+μk_R-1/2k_Δδ^2).The shifted desired attitude is defined as Rd:shifted:varying. While the shifted desired attitude in this section is constructed with different bounds on θ_b_0 and γ, it is straightforward to show that it satisfies all of the properties summarized in Lemma <ref>.We use (R̃_d(t),Ω̃_d(t)) as a reference trajectory for the construction of the control system. The error variables Ẽ_R, ẽ_R, and ẽ_Ω are defined using the shifted reference trajectories. The control torque and the adaptive law are defined asτ̃=- (Ω)×Ω+( -k_R ẽ_R - k_Ωẽ_Ω + Ω×Ω̃_d + Ω̇̃̇_d)-Δ̅, Δ̇̅̇= k_Δ^-1 (ẽ_Ω + μẽ_R),with Δ̅(0)=0.According to Theorem <ref>, for any initial condition satisfyingk_R/4R(0)-R̃_d(0)^2+ 1/2Ω(0)-Ω̃_d(0)^2≤ 2a√(k_R)-μ/√(k_R)+μk_R-1/2k_Δδ^2,the tracking errors for the shifted desired trajectory, namely R(t)-R̃_d(t), Ω(t)-Ω̃_d(t), and Δ-Δ̅(t) asymptotically converge to zero. As the shifted reference trajectories are designed such that R_d(t)-R̃_d(t) and Ω_d(t)-Ω̃_d(t) exponentially converge to zero, these imply that all of the tracking errors R(t)- R_d(t), Ω(t)-Ω_d(t) from the original reference trajectories and the estimation error Δ-Δ̅(t) asymptotically converge to zero as t→∞. These are summarized as follows. Choose any positive numbers k_R, k_Ω, k_Δ, a, μ, ϵ, θ_b_0, and γ such that (<ref>), (<ref>), kdelta, theta_b0:4, and gamma:4 are satisfied. The control input and the adaptive law are defined as follows.τ=- (Ω)×Ω+ ( - k_Re_R -k_Ωe_Ω+ Ω×Ω_d + Ω̇_d),Δ̇̅̇= k_Δ^-1(e_Ω+μe_R),whenV_0(R(0),Ω(0),0)≤2a√(k_R)-μ√(k_R)+μk_R-12k_Δδ^2,τ=- (Ω)×Ω+ ( - k_R ẽ_R -k_Ωẽ_Ω+ Ω×Ω_d + Ω̇_d),Δ̇̅̇= k_Δ^-1(ẽ_Ω+μẽ_R),whenV_0(R(0),Ω(0),0)> 2a√(k_R)-μ√(k_R)+μk_R-12k_Δδ^2, with Δ̅(0)=0 for both cases. Then, the zero equilibrium of the tracking errors (E_R,e_Ω,e_Δ)=(0,0,0) is asymptotically stable. The corresponding region of attraction is characterized in the following corollary. For the control system presented in Theorem <ref>, the region of attraction guaranteeing asymptotic convergence encloses the following set,ℛ={(R(0),Ω(0))∈×^3 | e_Ω(0) <max{√(B-k_R(1-cosθ_0)), √(2(1-ϵ)B)-γ/2θ_b_0}},where θ_0 is constructed from R(0) by R0:U:thetab, and the positive constant B is defined asB = 2a√(k_R)-μ/√(k_R)+μk_R -1/2k_Δδ^2. Furthermore, ℛ⊂×^3 coverscompletely, and when projected onto ^3, it is enlarged into ^3 as k_R is increased in a semi-global sense. Define three subsets of ×^3×^3 for the initial condition asℛ_1 ={k_R(1-cosθ_0)+1/2e_Ω(0)^2≤ B }, ℛ_2 ={k_R(1-cosθ_0)+1/2e_Ω(0)^2 > B }, ℛ_3 ={k_R(1-cos(θ_0-θ_b_0))+1/2ẽ_Ω(0)^2≤ B }.The sets ℛ_1 and ℛ_2 represent the initial conditions corresponding to the two cases of the control inputs, namely tau_4a and tau_4b, respectively. The set ℛ_3 represents the set of initial conditions ineq:var:V0:a:varying, guaranteeing the asymptotic convergence for the second case of the control input. Therefore, the combined region of attraction ℛ̅ is given by ℛ̅≡ℛ_1∪(ℛ_2∩ℛ_3). Since ℛ_1∪ℛ_2=×^3, this reduces to ℛ̅=(ℛ_1∪ℛ_2)∩ (ℛ_1∪ℛ_3)=ℛ_1∪ℛ_3.Next, we show that ℛ⊂ℛ̅. For any (R(0),Ω(0))∈ℛ,e_Ω(0)≤√(B-k_R(1-cosθ_0)),ore_Ω(0)≤√(2(1-ϵ)B)-γ/2θ_b_0,where the right-hand side is positive due to gamma:4. For the former case, (R(0),Ω(0))∈ℛ_1 ⊂ℛ̅. For the latter case, from tilde:eW:bound, ẽ_Ω(0)≤√(2(1-ϵ)B). Hence,k_R (1-cos(θ_0-θ_b_0))+1/2ẽ_Ω(0)^2 ≤ k_R (1-cos(θ_0-θ_b_0)) + (1-ϵ)B ≤ B,from theta_b0:4. Therefore, (R(0),Ω(0))∈ℛ_3. As such, any initial condition in ℛ also belongs to the estimated region of attraction ℛ̅, and the resulting controlled trajectory asymptotically converges to the desired one.Also, for any θ_0∈[0,π], the set ℛ is non-empty due to gamma:4. At ROA:4, the upper bound on e_Ω(0) tends to be infinite as k_R→∞. As in the previous section, the region attraction covers the entire , and when projected to ^3 for the angular velocity, it is enlarged in a semi-global sense as k_R increases. But, such global attractivity onis achieved with the control input and the adaptive law that are formulated as continuous functions of t. As discussed in Section <ref>, overcoming the topological restriction on the attitude control with a non-switching control input has been unprecedented. The adaptive law presented in this section additionally allows a constant disturbance to be present in the dynamicsat the expense of sacrificing the exponential convergence. § NUMERICAL EXAMPLES §.§ Attitude Tracking Controls We first show the numerical results for the attitude tracking controls presented in Section <ref>. Throughout this section, two control strategies presented in Section <ref> are denoted by AGTS (almost global tracking strategy), and GTS (global tracking strategy), respectively. They are also compared with the hybrid attitude control presented in <cit.> that guarantees global exponential stability with discontinuities of the control input with respect to t, and it is denoted by HYB.Assume that the moment of inertia matrix of the system is= diag[3, 2, 1] kgm^2.Consider the following reference trajectory:R_d(t)= [ cos t-cos t sin t sin^2 t; cos t sin tcos^3 t -sin^2 t -cos t sin t - cos^2t sin t;sin^2tcos t sin t +cos^2 t sin t cos^2t - cos t sin^2t ]which impliesΩ_d(t)= [ 1+cos t; sin t - sin t cos t; cos t + sin^2 t ],Ω̇_d (t)= [-sin t; cos t -cos^2t + sin^2 t; -sin t + 2sin t cos t ].Notice thatR_d(0) =diag[1, 1, 1], Ω_d(0) = (2,0, 1).The initial state of the system is given byR(0) = R_d(0)exp (θ_0ê_2), Ω(0) = (2,0,1),where θ_0=0.999π and e_2 = (0,1,0). The controller parameters are chosen ask_R = 9, k_Ω=4.2, a=ϵ,μ = 4(1-a)k_R k_Ω/4(1-a) k_R + k_Ω^2ϵ, θ_b_0 = min{θ_0ϵ,θ_0-cos^-1(1-2aϵ)}, γ=4/θ_b_0√(a k_R (1-ϵ))ϵ,with a scaling parameter ϵ=0.9<1 selected to satisfy the inequality constraints. These expressions allow that all of the controller parameters can be determined by tuning only the proportional gain k_R and the derivative gain k_Ω.The corresponding simulation results are plotted in Figure <ref>. For the given initial conditions and the controller gains, 18=V_0(0) > 2ak_R=16.2. Therefore, the initial condition does not belong to the region of attraction of AGTS estimated conservatively by (<ref>) in spite of which the tracking errors for AGTS still converge to zero asymptotically.The convergence rate is, however, quite low, and there is no noticeable change of the attitude tracking error for the first three seconds. The slow initial convergence is common for controllers with almost global asymptotic stability, especially when the initial state is close to the stable manifold of the undesired equilibrium point <cit.>.Next, for HYB, the convergence rate for the attitude tracking error is improved. But the tracking error for the angular velocity is increased over the first four seconds, and there are two abruptchanges in the control input at t=1.95 and t=3.31. Compared with AGTS, the convergence rate is substantially improved at the expense of discontinuities in the control input.Finally, the proposed GTS exhibits the fastest convergence rate for both of the attitude tracking errors and the angular velocity tracking errors. This is most desirable, as excellent tracking performances are achieved without discontinuities in control input for the large initial attitude error. §.§ Adaptive Attitude Controls Next, we present the simulation results for the adaptive attitude control strategies presented in Section <ref>. Throughout this section, two control strategies of Section <ref> are denoted by aAGTS (adaptive almost global tracking strategy), and aGTS (adaptive global tracking strategy), respectively. They are compared with the extension of the hybrid attitude control with an adaptive term in <cit.>, and it is denoted by aHYB.The fixed disturbance is chosen asΔ = (1,-2,0.5),with the estimated bound of δ = 3. The values of the controller parameters k_R, k_Ω, a, and μ are identical to those used in the previous subsection. The other parameters are chosen asθ_b_0 = min{θ_0ϵ,θ_0-cos^-1(1-Bϵ/k_R)}, γ=2/θ_b_0√(2(1-ϵ)B)ϵ, k_Δ=25,with a scaling parameter ϵ=0.9<1 such that the inequality constraints are satisfied. One can easily verify that the chosen value of k_Δ satisfies kdelta. In short, all of the controller parameters can be selected by tuning k_R, k_Ω, and k_Δ with consideration of kdelta for k_Δ.Simulation results are plotted in Figure <ref>. All the three controllers successfully estimate the unknown disturbance, as the estimation error Δ̅-Δ asymptotically converges to zero, and the effects of the disturbance are mitigated. The overall performance characteristics for each method are similar to those in the previous subsection.For aAGTS, the given initial condition does not belong to the region of attraction estimated by ineq:var:V0, as 18=V_0(0)> B=10.31. However, both the tracking errors and the estimation error for aAGTS asymptotically converge to zero, although the initial convergence rate is low.With aHYB, the attitude tracking performance is substantially improved, but the initial angular velocity tracking error is increased. There exists a discontinuity in the control input at t=2.74, where the magnitude ofcontrol moment change at the jump exceeds 26.8 Nm.The proposed aGTS exhibits the best tracking performance for the attitude and the angular velocity, and it does not cause any discontinuity of the control input. § EXPERIMENTAL RESULTSThe twoadaptive attitude tracking control strategies presented in Section <ref> have been implemented on the attitude dynamics of a hexrotor unmanned aerial vehicle, to illustrate the efficacy of the proposed approaches through hardware experiments. §.§ Hexrotor Development The hardware configuration of the hexrotor is as follows. Six brushless DC motors (Robbe Roxxy) are used along with electric speed controllers (Mikrokopter BL-Ctrl 2.0). An inertial measurement unit (VectorNav VN-100) provides the angular velocity and the attitude of the hexrotor. A linux-based computing module (Odroid XU-3) handles onboard data processing, sensor fusion, control input computation, and communication with a host computer (Macbook Pro). A custom made, printed circuit board supplies power to each part from a battery after switching the voltage level appropriately.A flight software program is developed by utilizing multithread programming in gcc such that the tasks of communication, sensor fusion, and control are performed in a parallel fashion. In particular, the control input is computed at the rate of 120 Hz approximately.The hexrotor is attached to a spherical joint that provides unlimited rotation in the yaw direction, and ± 45^∘ rotations along the pitch and the roll.As the center of gravity is above the center of the spherical joint, it resembles an inverted rigid body pendulum <cit.>. §.§ Experimental Results Two adaptive attitude tracking control strategies that provide smooth control inputs, namely aAGTS and aGTS are implemented. The desired attitude corresponds to the inverted equilibrium, where the center of gravity of the hexrotor is directly above the spherical joint, and the first body-fixed axis of the inertial measurement unit points towards the magnetic north, i.e., R_d(t)=I_3× 3 for all t≥ 0. Note that the desired attitude is unstable due to the gravity.The initial condition is chosen such that the pitch angles is decreased until the limit of the spherical joint, and the first body-fixed axis points towards the magnetic south. The resulting initial attitude error is close to 180^∘, i.e., E_R(0)≃ 2√(2). The initial angular velocity is chosen as zero. The controller parameters are selected as k_R=1.45, k_Ω=0.4, k_Δ=0.2, and δ=1. Other parameters are identical to those presented in Section <ref>.The corresponding experimental results are illustrated in Figure <ref>. The overall behaviors of adaptive controllers are similar to the numerical simulation results. The unstable desired attitude is asymptotically stabilized by both adaptive attitude controllers.For aAGTS, the initial convergence rate, particularly for the attitude tracking error, is quite slow. For example, the attitude tracking error remains close to its initial value for the first few seconds. However, aGTS exhibits a satisfactory convergence rate from the beginning, and it shows most desirable results. § CONCLUSIONS We have proposed global tracking strategies for the attitude dynamics of a rigid body. The topological restriction on the special orthogonal group is circumvented by introducing a shifted reference trajectory with a conjugacy class. As a result, global attractivity is achieved without causing discontinuities incontrol input with respect to time. These are constructed on the special orthogonal group to avoid singularities and ambiguities of other attitude representations. The desirable properties of the proposed methods are demonstrated by numerical examples and experimental results.The proposed approaches are fundamentally distinctive from the current efforts to achieve global attractivity via modifying attitude configuration error functions along with the hybrid system framework. This paper shows that the desired trajectory can be adjusted instead, and global attractivity can be achieved without introducing undesired abrupt changes in control input.For future work, the idea of shifting reference trajectories can be applied to feedback control on other compact manifolds and Lie groups. Also, the results presented in this paper are readily generalized to various other attitude control problems such as velocity-free attitude controls or deterministic attitude observers.§ ACKNOWLEDGEMENTThis work was supported in part by NSF under the grants CMMI-1243000, CMMI-1335008, and CNS-1337722, and by KAIST under grant G04170001. It was also supported in part by DGIST Research and Development Program (CPS Global Center) funded by the Ministry of Science, ICT & Future Planning, Global Research Laboratory Program (2013K1A1A2A02078326) through NRF, and Institute for Information & Communications Technology Promotion (IITP) grant funded by the Korean government (MSIP) (No. 2014-0-00065, Resilient Cyber-Physical Systems Research). IEEEtran | http://arxiv.org/abs/1708.07649v1 | {
"authors": [
"Taeyoung Lee",
"Dong-Eui Chang",
"Yongsoon Eun"
],
"categories": [
"math.OC"
],
"primary_category": "math.OC",
"published": "20170825084018",
"title": "Semi-Global Attitude Controls Bypassing the Topological Obstruction on SO(3)"
} |
Height transitions, shape evolution, and coarsening of equilibrating quantum nanoislands Mikhail Khenner December 30, 2023 ======================================================================================== Given a graph, does there exist an orientation of the edges such that the resulting directed graph is strongly connected?Robbins' theorem [Robbins, Am. Math. Monthly, 1939] states thatsuch an orientation exists if and only if the graph is 2-edge connected. A natural extension of this problem is the following: Suppose that the edges of the graph is partitioned into trails. Can we orient the trails such that the resulting directed graph is strongly connected?We show that 2-edge connectivity is again a sufficient condition and we provide a linear time algorithm for finding such an orientation, which is both optimal and the first polynomial time algorithm for deciding this problem.The generalised Robbins' theorem [Boesch, Am. Math. Monthly, 1980] for mixed multigraphs states that the undirected edges of a mixed multigraph can be oriented making the resulting directed graph strongly connected exactly when the mixed graph is connected and the underlying graph is bridgeless. We show that as long as all cuts have at least 2 undirected edges or directed edges both ways, then there exists an orientation making the resulting directed graph strongly connected. This provides the first polynomial time algorithm for this problem and a very simple polynomial time algorithm to the previous problem. § INTRODUCTION AND MOTIVATION Suppose that the mayor of a small town decides to make all the streets one-way in such a way that it is possible to get from any place to any other place without violating the orientations of the streets[The motivation for doing so is that the streets of the town are very narrow and thus it is a great hassle when two cars unexpectedly meet.]. If initially all the streets are two-way then Robbins' theorem <cit.> asserts that this can be done exactly when the corresponding graph is 2-edge connected. If, on the other hand some of the streets were already one-way in the beginning then the generalised Robbins' theorem <cit.> states that it can be done exactly when the corresponding graph is strongly connected and the underlying graph is 2-edge connected.However, the proofs of both of these results assume that every street of the city corresponds to exactly one edge in the graph. This assumption hardly holds in any city in the world and therefore a much more natural assumption is that every street corresponds to a trail in the graph andthat the edges of each trail must be oriented consistently[This version of the problem was given to us through personal communication with Professor Robert E. Tarjan].In this paper we prove that Robbins' Theorem continues to hold even when the set of edges is partitioned into trails. In other words a necessary and sufficient condition for an orientation to exist is that the graph is 2-edge connected. We also provide a linear time algorithm for finding such an orientation. Finally we will consider the generalised Robbins' theorem in this new setting i.e. we allow some edges to be oriented initially and suppose that the remaining edges are partitioned into trails. We will show that if any cut (V_1,V_2) in the graph has either at least 2 undirected edges going between V_1 to V_2or a directed edge in each direction then it is possible to orient the trails making the resulting graph strongly connected. Although this condition is not necessary it does give a simple algorithm for deciding the problem. Indeed, the only cuts containing an undirected edge which we allow are the ones where this edge (and hence its trail) is forced in one direction. Hence for deciding the problem we can start by orienting all the forced trails until there are no more forced trails. Then the trails can be oriented making the graph strongly connected exactly if the resulting graph satisfies our condition. Note that when some edges are initially oriented the answer to the problem depends on the trail decomposition which is not the case for the other results. That the condition from the generalised Robbins' theorem is not sufficient can be seen from figure <ref>. Earlier methods Several methods have already been applied for solving orientation problems in graphs where the goal is to make the resulting graph strongly connected. One approach used by Robbins <cit.> is to use that a 2-edge connected graph has an ear-decomposition. An ear decomposition of a graph is a partition of the set of edges into a cycle C and paths P_1,…,P_t such that P_i has its two endpoints but none of its internal vertices on C∪(⋃_j=1^i-1P_j). Assuming the existence of an ear decomposition of 2-edge connected graphs it is easy to prove Robbins' theorem. Indeed, it is easy to see by induction that any consistent orientations of the paths and the cycle give a strongly connected graph.A second approach introduced by Tarjan <cit.> gives another simple proof of Robbins' theorem. One can make a DFS tree in the graph rooted at a vertex v and orient all edges in the DFS tree away from v. The remaining edges are oriented towards v and if the graph is 2-edge connected it is easily verified that this gives a strong orientation.A similar approach was used by Chung et al. <cit.> in the context of the generalized Robbins theorem for mixed multigraphs.The above methods not only prove Robbins' theorem, they also provide linear time algorithms for finding strong orientations of undirected or mixed multigraphs.However, none of the above methods have proven fruitful in our case. In case of the ear decomposition one needs a such which is somehow compatible with the partitioning into trails and this seems hard to guarantee. The original proof by Roberts is essentially similar to using the ear decomposition.Similar problems appear when trying a DFS-approach. Neither does the proof by Boesch <cit.> of Robbins' theorem for mixed multigraphs generalise to prove our result. Most importantly the corresponding theorem is no longer true for trail orientations as is shown by the example above.Since the classical linear time algorithms rely on ear-decompositions and DFS searches, and since these approaches do not immediately work for trail partitions, our linear time algorithm will be a completely new approach to solving orientation problems.§ PRELIMINARIES Let us briefly review the concepts from graph theory that we will need. Recall that a walk in a graph is an alternating sequence of vertices and edges v_0,e_1,v_1,e_2,…,v_k, such that for 1≤ i ≤ k the edge e_i has v_i-1 and v_i as its two endpoints. In a directed or mixed graph the ordering of the endpoints of each edge in the sequence must be consistent with the direction of the edge in case it is oriented. A trail is a walk without repeated edges. A path is a trail without repeated vertices (except possibly v_0=v_k). Finally a cycle is a path for which v_0=v_kNext, recall that a mixed multigraph G=(V,E) is called strongly connected if for any vertices u,v∈ V there exists a walk from u to v. In case that the graph contains no directed edges this is equivalent to saying that it consists of exactly one connected component. We also recall the definition of k-edge connectivity. A graph G=(V,E) is said to be k-edge connected if and only ifG'=(V,E-X) is connected for all X ⊆ E where |X|<k. A trivially equivalent condition is that any edge-cut (V_1,V_2) in the graph has at least k edges going between V_1 and V_2. Finally, if G=(V,E) is a mixed multigraph and A⊆ V we define G/A to be that graph obtained by contracting A to a single vertex and G[A] to be the subgraph of G induced by A. The following simple observation will be used repeatedly in this paper.If G=(V,E) is k-edge connected and A⊆ V then G/A is k-edge connected. Also if G is a strongly connected mixed multigraph then G/A is too.The structure of this paper is as follows. In section <ref> we prove our generalisation of Robbins' theorem for undirected graphs partitioned into trails. In section <ref> we study what happens in the case of mixed graphs. Finally in section <ref> we provide our linear time algorithm for trail orientation in an undirected graph. § ROBBINS THEOREM REVISITEDWe are now ready to state our generalisation of Robbins' theorem. Let G=(V,E) be a multigraph with E partitioned into trails. An orientation of each trail such that the resulting directed graph is strongly connected exists if and only if G is 2-edge connected. IfG is not 2-edge connected, such an orientation obviously doesn't exist so we need to prove the converse.Suppose therefore that G is 2-edge connected.Our proof is by induction on the number of edges in G.If there are no edges, the graph is a single vertex, and the statement is obviously true.Assume now the statement holds for all graphs with strictly fewer edges than G.Pick an arbitrary edge e that is at the end of its corresponding trail.If G-e is 2-edge connected, then by the induction hypothesis there is a strong orientation of G-e that respects the trails of G.Such an orientation clearly extends to the required orientation of G.If G-e is not 2-edge connected, there exists a bridge b in G-e (see figure <ref>).Let V_1, V_2 be the two connected components of G-e,b, and let e=(u_1,u_2) and b=(w_1,w_2) such that for i∈1,2, u_i,w_i∈ V_i (note that we don't necessarily have that u_i and w_i are distinct for i∈{1,2}).Now for i∈1,2 construct the graph G_i = G[V_i]∪(u_i,w_i), and define the trails in G_i to be the trails of G that are completely contained in G_i, together with a single trail combined from the (possibly empty) partial trail of e contained in G_i and ending at e_i, followed by the edge (u_i,w_i), followed by the (possibly empty) partial trail of b contained in G_i starting at b_i.Both G_1 and G_2 are 2-edge connected since they can each be obtained as a contraction of G. Furthermore, they each have strictly fewer edges than G, so inductively each has a strong orientation that respects the given trails.Further, we can assume that the orientations are such that the new edges are oriented as (u_1,w_1) and (w_2,u_2) by flipping the orientation of all edges in either graph if necessary.We claim that this orientation, together with e oriented as (u_1,u_2) and b oriented as (w_2,w_1), is the required orientation of G.To see this first note that (by construction) this orientation respects the trails. Secondly suppose v_1∈ V_1 and v_2∈ V_2 are arbitrary. Since G_1 is strongly connected G[V_1] contains a directed path from v_1 to u_1. Similarly, G[V_2] contains a directed path from u_2 to v_2. Thus G contains a directed path from v_1 to v_2. A similar argument gives a directed path from v_2 to v_1 and since v_1 and v_2 were arbitrary this proves that G is strongly connected and our induction is complete.The construction in the proof can be interpreted as a naive algorithm for finding the required orientation when it exists.The one-way trail orientation problem on a graph with n vertices and m edges can be solved in (n+m· f(m,n)) time, where f(m,n) is the time per operation for fully dynamic bridge finding (a.k.a. 2-edge connectivity).At the time of this writing[Separate paper submitted to SODA'18 by Holm, Rotenberg and Thorup.], this is (n+m(log nloglog n)^2).In Section <ref> we will show a less naive algorithm that runs in linear time. § EXTENSION TO MIXED GRAPHSNow we will extend our result to the case of mixed graphs. We are going to prove the following.Let G=(V,E) be a strongly connected mixed multigraph. Then G-e is strongly connected for all undirected e∈ E if and only if for any partition 𝒫 of the undirected edges of G into trails, and any T∈𝒫, any orientation of T can be extended to a strong trail orientation of (G,𝒫).Suppose G=(V,E) is as in the theorem. We will say that e∈ E is forced if it is undirected and satisfies that G-e is not strongly connected[This terminology is natural since it is equivalent to saying that there exists a cut (V_1,V_2) in G such that e is the only undirected edge in this cut and such that all the directed edges go from V_1 to V_2. If one wants an orientation of the trails making the graph strongly connected we are clearly forced to orient e from V_2 to V_1.]. Note that this is a proper extension of Theorem <ref> since if G is undirected and 2-edge connected then no e∈ E is forced.For proving the result we'll need the following lemma.Let G be a directed graph, and let (A,B) be a cut with exactly one edge crossing from A to B and at least one edge crossing from B to A.Then G is strongly connected if and only if G/A and G/B are. Strong connectivity is preserved by contractions, so if G is strongly connected then G/A and G/B both are.For the other direction, let (a_1,b_1) be the edge going from A to B, and let (b_2,a_2) be any edge from B to A.Since G/A is strongly connected and (a_1,b_1) is the only edge from A to B, G/A contains a path from b_1 to b_2 that stays in B.Since this holds for any edge going from B to A, and since G/B is strongly connected, A is strongly connected in G.By a symmetric argument, B is also strongly connected in G and since the cut has edges in both directions, G must be strongly connected.Now we provide the proof of Theorem <ref>. If G-e is not strongly connected, the trail T containing e can at most be directed one way since e is forced, so there is an orientation of T which not extend to a strong trail orientation of (G,𝒫). To prove the converse suppose G-e is strongly connected for all undirected e∈ E.The proof is by induction on 𝒫.If 𝒫≤ 1 the result is trivial.So suppose 𝒫>1 and that the theorem holds for all (G',𝒫') with 𝒫'<𝒫.Consider a trail T∈𝒫.Suppose there is no cut (A,B) that T crosses exactly once, which has exactly one other undirected edge crossing it, and has every directed edge crossing it going from A to B.Then regardless of the orientation of T, the resulting graph G' has no undirected edge e such that G-e is not strongly connected.Thus, by induction (G',𝒫∖T) has a strong trail orientation, which is also a strong trail orientation of (G,𝒫), as desired. Now suppose there is such a cut (A,B) (see figure <ref>).Construct a graph G/B by contracting every vertex in B into a single new vertex b.Let 𝒫_A consist of all trails in 𝒫 that are completely contained in A, together with a single trail T_A combined from the (possibly empty) fragments of the two trails that crossed the cut, joined at b.Since any cut in G/B corresponds to a cut in G, G/B is strongly connected and remains so after deletion of any single undirected edge. By induction any orientation of T_A in G/B extends to a strong orientation of (G/B,𝒫_A). Let G/A, a, 𝒫_ℬ and T_B be defined symmetrically, then by the same argument any orientation of T_B in G/A extends to a strong orientation of (G/A,𝒫_B). Now for any orientation of T, we can choose orientations of T_A and T_B that are compatible.The result follows by Lemma <ref>. Notice that the partitioning of edges into trails does not matter in the case when no edge is forced. Since any undirected graph has no forced edges if it is 2-edge connected, the theorem implies that the most naive algorithm: "directing trails that are forced and if none are forced direct an arbitrary trail" works for undirected graphs. In general for mixed graphs algorithm <ref> below can clearly be implemented in polynomial time and does solve the trail orientation problem for mixed graphs.Theorem <ref> gives a sufficient condition for when a strong orientation exists and we deal with the other cases by dealing with the forced edges first. However, the generalised Robbins' Theorem provides a simple equivalent condition, which we lack. Finding such an equivalent condition when you have trail decomposition is an essential open problem for strong graph orientations. Due to figure <ref> in this setting one has to take into account the structure of the trail partition. § LINEAR TIME ALGORITHMIn this section we provide our linear time algorithm for solving the trail orientation problem in undirected graphs. For this, we make two crucial observations.First, we show that there is an easy linear time reduction from general graphs or multigraphs to cubic multigraphs.Second, we show that in a cubic multigraph with n vertices, we can in linear time find and delete a set of edges that are at the end of their trails, such that the resulting graph has Ω(n) 3-edge connected components.We further show that we can compute the required orientation recursively from an orientation of each 3-edge connected component together with the cactus of 3-edge connected components. Since the average size of these components is constant, we can compute the orientations of most of them in linear time.The rest contains at most a constant fraction of the vertices, and so a simple geometric sum argment tells us that the total time is also linear.We start out by making the following reduction. The one-way trail problem on a 2-edge connected graph or multigraph with n vertices and m edges, reduces in (m+n) time to the same problem on a 2-edge connected cubic multigraph with 2m vertices and 3m edges.Order the edges adjacent to each vertex such that two edges that are adjacent on the same trail are consecutive in the order.Replace each single vertex v with a cycle of length (v), with each vertex of the new cycle inheriting a corresponding neighbour of v.Note that for a vertex of degree 2, this creates a pair of parallel edges, so the result may be a multigraph.Since edges on the same trail are neighbours, we can make the cycle-edge between the two corresponding vertices belong to the same trail.The rest of the cycle edges form new length 1 trails.This graph has exactly 2m vertices and 3m edges, andany one-way trail orientation on this graph translates to a one-way trail orientation of the original graph.The graph is constructed in (m+n) time.Recall now that a graph C is called a cactus if it is connected and each edge is contained in at most one cycle. If G is any connected graph we let C_1,…,C_k be its 3-edge connected components. It is well known that if we contract each of these we obtain a cactus graph. For a proof of this result see section 2.3.5 of <cit.>. As the cuts in a contracted graph are also cuts in the original graph we have that if G is 2-edge connected then the cactus graph is 2-edge connected. The edges of the cactus are exactly the edges of G which are part of a 2-edge cut. We will call these edges 2-edge critical.It is easy to check that if a cactus has m edges and n vertices then m≤ 2(n-1). We will be using this result in the proof of the following lemma.Let G=(V,E) be a cubic 2-edge connected multigraph, let X⊆ E, and let F⊇ E∖ X be minimal such that H=(V,F) is2-edge connected.Then H has at least 2/5X distinct 3-edge connected components. Let ⊆ X, be the set of edges deleted from G to obtain H, and let =X∖ be the remaining edges in X.If ≥4/5X, then by minimality of H there are at least2-edge-critical edges in H i.e. edges of the corresponding cactus, and thus at least 1/2+1≥2/5X+1 distinct 3-edge connected components.If ≤4/5X then ≥1/5X, and since G is cubic and the removal of each edge creates two vertices of degree 2 we must have that H has at least 2≥2/5Xdistinct 3-edge connected components. Let G=(V,E) be a connected cubic multigraph with E partitioned into trails.Then G has a spanning tree that contains all edges that are not at the end of their trail. Let F be the set of edges that are not at the end of their trail. Since G is cubic, the graph (V,F) is a collection of vertex-disjoint paths, and in particular it is acyclic. Since G is connected F can be extended to a spanning tree. Note that we can find this spanning tree in linear time. Indeed, we may assign weight 0 to edges in F and 1 to the remaining edges and use the so-called[Originally discovered by Jarník <cit.>, later rediscovered by Prim <cit.>] Prim's minimal spanning tree algorithm with a suitable priority queue to find the tree. Let G=(V,E) be a cubic 2-edge connected multigraph with E partitioned into trails. Let T be a spanning tree of G containing all edges that are not at the end of their trail. Let H be a minimal subgraph of G that contains T and is 2-edge connected. Then for any k≥5, less than 4/5k/k-1V of the vertices in H are in a 3-edge connected component with at least k vertices. Let X be the set of edges that are not in T.Since G is cubic, X=1/2V+1.By Lemma <ref> H has at least 2/5X>1/5V 3-edge connected components.Each such component contains at least one vertex, so the total number of vertices in components of size at least k is less than k/k-1(V-1/5V)=4/5k/k-1V.Let C be a 3-edge connected component in some graph H, whose edges is partitioned into trails.Define Γ_H(C) to be the 3-edge connected graph obtained by replacing each min-cut e,f where e=(e_1,e_2) and f=(f_1,f_2) and e_1,f_1∈ C with a single new edge (e_1,f_1).Define the corresponding partition of the edges of Γ_H(C) into trails by taking every trail that is completely contained in C, together with new trails combined from the fragments of the trails that were broken by the min-cuts together with thenew edges that replaced them. See figure <ref>.At this point the idea of the algorithm can be explained. We remove as many of the edges, at the end of their trails, as we can still maintaining that the graph is 2-edge connected. Lemma <ref> guarantees that we obtain a graph H with Ω(|V|) many 3-edge connected components of size O(1). We solve the problem for each Γ_H(C) for every 3-edge connected component. Finally, we combine the solutions for the different components like in the proof of theorem <ref>.The one-way trail orientation problem can be solved in (m+n) time on any 2-edge connected graph or multigraph with n vertices and m edges. By Lemma <ref>, we can assume the graph is cubic. For the algorithm we will use two subroutines. First of all when we have found the minimum spanning tree T containing the edges that are not on the end of their trail we can use the algorithm of Kelsen <cit.> to, in linear time, find a minimal (w.r.t. inclusion) subgraph H of G that contains T and is 2-edge connected. Secondly we will use the algorithm of Melhorn <cit.> to, in linear time, build the cactus graph of 3-edge connected components. The algorithm runs as follows: * Construct a spanning tree T of G that contains all edges that are not at the end of their trail.* Construct a minimal subgraph H of G that contains T and is 2-edge connected[See Kelsen <cit.>].* Find the cactus of 3-edge connected components of[See Melhorn <cit.>] H.* For each 3-edge connected component C_i, construct Γ_H(C_i).* Recursively compute an orientation for each[Note that Γ_H(C_i) is cubic unless it consists of exactly one node. In this case however we don't need to do anything.] Γ_H(C_i).* Combine the orientations from each component.First we will show correctness and then we will determine the running time.Recall that we can flip the orientation in each Γ_H(C_i) and still obtain a strongly connected graph respecting the trails in Γ_H(C_i). The way we construct the orientation of the edges of G is by flipping the orientation of each Γ_H(C_i) in such a way that each cycle in the cactus graph becomes a directed cycle[In practise this is done by making a DFS (or any other search tree one likes) of the cactus and repeatedly orienting each component in a way consistent with the previous ones.]. This can be done exactly because no edge of the cactus is contained in two cycles. By construction this orientation respects the trails so we need to argue that it gives a strongly connected graph.For showing that the resulting graph is strongly connected, first let every 3-edge connected component be contracted, then the graph is strongly connected since the cycles of the cactus graph have become directed cycles. Now assume inductively that we have uncontracted some of the components and call this graph G_1. Now we uncontract another component C (see figure <ref>) and obtain a new graph G_2 which we will show is also strongly connected. If u,v ∈ C, then since Γ_H(C) is strongly connected there is a path from u to v in Γ_H(C). If this path only contains edges which are edges in C clearly this path also exists in G_2 so we are done. If the path uses one of the added edges (e_1,f_1) (without loss of generality oriented from e_1 to f_1), it is because there are edges (e_1,e_2) and (f_1,f_2) forming a cut and thus being part of a cycle in the cactus. In this case we use edge (e_1,e_2) to leave component C and then go from e_2 back to component C which is possible since G_1 was strongly connected. When we get back to the component C we must arrive at f_1 since otherwise there would be two cycles in the cactus containing the edge (e_1,e_2). Hence the edge (e_1,f_1) was not needed. This argument can be used for any of the edges of Γ_H(C) that are not in C and thus we can move between any two nodes in C. Since G_1 was strongly connected this suffices to show that G_2 is strongly connected. By induction this implies that after uncontracting all components the resulting graph is strongly connected.Now for the running time. By Lemma <ref> each level of recursion reduces the number of vertices in “large” components by a constant fraction, for instance for k=10 we reduce the number of vertices in large components by a factor of 1/9. Let f(n) be the worst case running time with n nodes for a cubic graph, and pick c large enough such that cn is larger than the time it takes to go through steps 1-4 and 6 as well as computing the orientations in the “small” components. Let a_1,…,a_k be the number of vertices in the “large” 3-edge connected components. Then ∑_i a_i ≤8n/9 andf(n) ≤ cn+ ∑_i f(a_i)Inductively we may assume that f(a_i)≤ 9cn and thus obtainf(n) ≤ cn+ ∑_i f(a_i)≤ cn+∑_i 9ca_i = cn+8cn=9cnproving that f(n)≤ 9cn for all n. § OPEN PROBLEMSWe here mention two problems concerning trail orientations which remain open. First of all, our linear time algorithm for finding trail orientations only works for undirected graphs and it doesn't seem to generalise to the trail orientation problem for mixed graphs. It would be interesting to know whether there also exists a linear time algorithm working for mixed graphs. If so it would complete the picture of how fast an algorithm we can obtain for any variant of the trail orientation problem.Secondly, our sufficient condition for when it is possible to solve the trail orientation problem for mixed multigraphs is clearly not necessary. It would be interesting to know whether there is a simple necessary and sufficient condition like there is in the undirected case. Since in the mixed case the answer to the problem actually depends on the given trail decomposition and not just on the connectivity of the graph it is harder to provide such a condition. One can give the following condition. It is possible to orient the trails making the resulting graph strongly connected if and only if when we repeatedly direct the forced trails end up with a graph satisfying our condition in theorem <ref>. This condition is not simple and is not easy to check directly. Is there a more natural condition? plain | http://arxiv.org/abs/1708.07389v1 | {
"authors": [
"Anders Aamand",
"Niklas Hjuler",
"Jacob Holm",
"Eva Rotenberg"
],
"categories": [
"cs.DS",
"cs.DM"
],
"primary_category": "cs.DS",
"published": "20170824131854",
"title": "One-Way Trail Orientations"
} |
Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 226, 69120 Heidelberg, GermanyHefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 226, 69120 Heidelberg, GermanyHefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 226, 69120 Heidelberg, GermanyPhysikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 226, 69120 Heidelberg, Germany[e-mail:][email protected] Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 226, 69120 Heidelberg, Germany CAS-Alibaba Quantum Computing Laboratory, Shanghai 201315, China CAS Centre for Excellence and Synergetic Innovation Centre in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China [e-mail:][email protected] Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 226, 69120 Heidelberg, Germany CAS-Alibaba Quantum Computing Laboratory, Shanghai 201315, China CAS Centre for Excellence and Synergetic Innovation Centre in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, ChinaWe propose and implement a lattice scheme for coherently manipulating atomic spins. Using the vector light shift and a superlattice structure, we demonstrate experimentally the capability on parallel spin addressing in double-wells and square plaquettes with subwavelength resolution. Quantum coherence of spin manipulations is verified through measuring atom tunneling and spin exchange dynamics. Our experiment presents a building block for engineering many-body quantum states in optical lattices for realizing quantum simulation and computation tasks.Spin-dependent Optical Superlattice Jian-Wei Pan December 30, 2023 =================================== Ultracold atoms in optical lattices constitute a promising system for creating multipartite entangled states <cit.>, which is an essential resource for quantum information processing <cit.>. As the neutral atoms prepared in the Mott insulating state consist of highly ordered quantum registers <cit.>, multipartite entanglement can be generated via parallelly addressing single atoms together with two-body interactions <cit.>. Following this route, sub-lattice addressing and √(SWAP) operations in double-wells (DWs) were demonstrated <cit.>, where the atomic spins in decoupled DW arrays were addressed by ultilizing the spin-dependent effect <cit.>. However, extending these entangled pairs to a one-dimensional (1D) chain or a two-dimensional (2D) cluster remains challenging due to the lack of control over inter-well couplings<cit.>. In this context, a bichromatic lattice referred as “superlattice" provides an alternative degree of freedom to connect the entangled pairs by tuning the relative lattice phase <cit.>. Besides the √(SWAP) operation in such superlattices, site-selective single-qubit addressing is further required to create multipartite cluster states for measurement-based quantum computation <cit.>. In this Letter, we demonstrate a spin-dependent optical superlattice for coherently addressing and manipulating atomic spins. Such a lattice configuration allows one to first address even/odd rows of spins in parallel and then create entangled pairs, as well as to enable further connection of the pairs to form a multipartite entangled state with high fidelity. This configuration offers an efficient way for spin addressing in higher dimensions <cit.> and meanwhile becomes a powerful tool in detecting the quantum correlations of entangled states <cit.>.The optical lattice consists of two far-detuned lasers, one generates a local effective magnetic gradient, and the other one isolates the system into DWs and forms imbalanced structures for different spin states. To illustrate the spin-dependent optical potential, we consider an alkali atom placed inside a far-detuned laser field <cit.>. The monochromatic light field has a complex notation E(x,t) = E⃗(x)exp(-iω t) + c.c, where E⃗ represents the positive-frequency part with the driving frequency ω. The optical potential for the atom in the ground state reads V(x) = - E⃗^∗(x) ·α·E⃗(x). Here α is the polarizability tensor with the irreducible scalar α_s and the vector components α_v that contribute to the potential herein <cit.>. For ^87Rb with states indexed by the hyperfine F and magnetic m_F quantum numbers, the optical potential is a combination of scalar and vector light shift, V = V_s +V_v. The scalar part V_s = - α_s |E⃗|^2 is state independent and proportional to the laser intensity. The vector part V_v = iα_v (E⃗^∗×E⃗) ·F⃗ is state-dependent and can be regarded as an effective Zeeman shift with B⃗_eff∝i (E⃗^∗×E⃗). This vector potential depends on the laser polarization, quantization axis and projection of the angular momentum F⃗. It vanishes for linearly polarized light or for m_F = 0 magnetic sublevels.Our one-dimensional superlattice is formed by superimposing two optical standing waves differing in the period by a factor of two (see Fig. <ref>). The lattices are respectively marked as “short-lattice" and “long-lattice" by their wavelengths λ_s and λ_l. Without any vector light shift, the optical dipole potential can be written as, V(x) = V_scos^2( kx ) - V_lcos^2(kx/2 + φ), with k = 2π/λ_s the wave number and φ the relative phase between the lattices. The relative phase φ is controlled by tuning the laser frequency of the long-lattice. The spin dependence arising from the laser polarization is controlled by an Electro-optical modulator (EOM). Fig. 1(a) shows the setup of a superlattice that consists of a blue- (λ_s = 767 nm) and a red-detuned (λ_l = 1534 nm) lattice, where the polarization of short-lattice is denoted as “lin-θ-lin" configuration <cit.>. The local polarization can be decomposed into σ^± and π components referring to the orientation of the magnetic axis. When the magnetic axis is along x, the optical potential has a spin-dependent short-lattice term and a scalar long-lattice term,V_j(x) =V_s,j[ A^+_jcos^2(kx + θ/2) +A^-_jcos^2(kx - θ/2) ]- V_lcos^2(kx/2 + φ).For a certain spin state j, the parameters A^+ and A^- are mainly determined by the laser detuning of the short-lattice. We can define the spin states of ^87Rb as ↓≡F=1, m_F=-1 and ↑≡F=2, m_F=-2. The parameters for ↓ state are A^+_↓ = 0.55 and A^-_↓ = 0.45, while for spin ↑ are A^+_↑ = 0.40, A^-_↑ = 0.60.The spin-dependent term of Eq. <ref> equals a periodical potential V_s,j[A_eff·cos^2(kx+θ_eff/2) + (1-A_eff)/2], with effective depth A_eff = √(cos^2 θ + (A^+ - A^-)^2sin^2 θ) and phase shift θ_eff = tan^-1[(A^+ - A^-)tanθ]. For θ = π/2, the effective depth acquires a minimum A_eff =|A^+ - A^-| and the trap bottom shifts for λ_s/4. Until now, the coupling frequencies between ↓ and ↑ for each well in the spin-dependent short-lattice are the same (see Fig. <ref>(b)-left). Interestingly, the left-right symmetry breaks as the long-lattice adds a local potential to the DW unit and creates two different coupling frequencies ω_L and ω_R as in Fig. <ref>(b)-right. The DWs have imbalanced structures and tilt along opposite directions for spin ↓ and ↑. The superlattice with spin dependence therefore provides another degree of freedom for manipulating the atoms in the left or right wells of the DWs. Meanwhile, it creates a strong effective magnetic gradient field on the order of ∼ 250 G/cm, which can be switched on/off with a fast speed (the EOM ramping time is 500 μs) while does not induce any unwanted eddy current. The lattice residual potential at small angles (1-cosθ) is proportional to the second-order of the bias ∼θ^2, inducing fairly small increase of the ground-band heating compared with that in the unbiased potential. Since the EOM locates before the combining of the bichromatic lasers, it does not cause an intensity imbalance of incident and reflected beams <cit.>. Such a structure neither affects the long-lattice, therefore circumventing unwanted phase fluctuations. We implement such spin-dependent superlattice to realize the spin addressing. The experiment starts with a Bose-Einstein condensate of ^87Rb with around 2 × 10^5 atoms in the F=1, m_F=-1 hyperfine state <cit.>. Then the condensate is adiabatically loaded into a single layer of a pancake-shaped trap, which is generated by interfering two laser beams with a wavelength of λ_s and an intersection angle of 11^∘. Subsequently, we ramp up the 2D short-lattice along x and y. As the lattice depths rise to 25 E_r, atoms are thereby localized into individual sites and enter an insulator state. Here E_r = h^2/(2 m λ_s^2) denotes the recoil energy of the blue-detuned laser, with h the Planck constant and m the mass of atom. The x and y short-lattices have a frequency difference of 160 MHz to avoid the interference of these two dimensions. The filling number per lattice site and the strength of the on-site interaction are mainly controlled by adjusting the depth of the 4-μm-period “pancake" lattice.For the one-dimensional spin addressing in the DWs, we prepare atoms in a spin-dependent superlattice and then apply a Rabi flopping pulse on the atoms. After initialization of the insulating state, the short- and long-lattice along x are ramped up to 60 E_r and 21 E_r, respectively. The long-lattice divides the atoms into balanced DW units with φ=0^∘. The quantization axis is set to x and the phase of EOM is tuned to θ = 45^∘, forming an energy shift between the left and right wells as in Fig. <ref>. Subsequently, we apply a 167 μs microwave π-pulse to couple the spin ↓ and ↑. After the state addressing, the spin dependence is turned off by returning the EOM to θ =0^∘. Two alternative methods, site-resolved band mapping <cit.> and in situ imaging, are adapted to detect the spin populations (see Appendix). Fig. <ref>(a) shows the band mapping patterns of different spin states and site occupations, the efficiency for generating spin features ↑,↓, ↓, ↑ are 92(7)%, 91(5) %. However, we notice that spin-exchange dynamics during the band mapping could reduce the detection fidelity <cit.>. Since only ↑ react with the imaging cycling transitions, we also use in situ absorption imaging to detect the quantum states. Fig. <ref>(b) shows the spectroscopy of site-selective addressing, where the transition peaks of the left and right DWs are separated for 18.7(1) kHz. This spin addressing technique is extended to a two-dimensional system by implementing the superlattices on both directions. The long-lattice light along y is generated by another laser source with ∼11GHz detuning from the frequency of x long-lattice laser. Therefore, the x and y lattices have no crosstalk and their frequencies can be controlled individually. Atoms are initially prepared in the insulating state in the short-lattices, then the long-lattices are ramped up to form arrays of isolated square plaquettes. We perform the 2D spin addressing along each dimension in sequence, showing the capability to create a Néel antiferromagnetic state inside the plaquettes. For this, the quantization axis is first set along y, the atoms in two sites (marked as C and D in Fig. <ref>(a)) of a plaquette are flipped by a MW π-pulse in the spin-dependent potential. Then we switch the magnetic axis to x and tune the energy splitting to the desired value. A second MW π pulse flips the spin states of sites A and D. After these two operations, atoms in the diagonal sites A and C are transferred to ↑, achieving a ↑,↓,↑,↓ spin configuration with the site notations from A to D. For further calibrations, we then apply a third and a fourth pulse to recover the initial state by addressing x and y in the same fashion. Fig. <ref> shows the addressing sequences and the corresponding signals, where a 2D band mapping and Stern-Gerlach gradient analogous to the 1D case are applied to resolve the plaquette spins. The same 2D spin addressing can also be realized with a single MW pulse by setting the same transitions frequency for site A and C <cit.>. From the in situ imaging, we calibrate the addressing efficiency of spin state ↑,↓,↑,↓ to be 92(2)%.To demonstrate the coherent control of spin dynamics, we study the single atom tunneling and spin-exchange process in DW systems. Atom tunneling J represents the nearest-neighbor hopping term in the Bose-Hubbard model <cit.>. The spin-exchange dynamics is driven by a second-order interaction described with the Heisenberg spin model Ĥ = -2J_exŜ_L ·Ŝ_R <cit.>. By isolating the atoms into DW units, we reduce the Hilbert space close to two-level systems and thereby observe the evolutions.The starting point of the experiment is a Mott insulator state with near unity filling. To observe the single atom tunneling, the atoms in the left sites are flipped and removed from the lattices, resulting in single fillings in the right sites and zero fillings in the left sites. We then lower down the barrier of DWs and let the system evolve. After a time t the spin states are frozen by ramping up the DW barriers and we address the left sites again. The signal of the following absorption imaging has only contributions from atoms which have tunnelled to the left sites, P_L(t) =⟨n̂_↓, L⟩ (here n̂_↓, L represents the number operator of the left site). For short- and long-lattice depths of 11.9(1)E_r and 10.1(1)E_r during the evolution, the theoretical tunneling strength J/h = 377(8) Hz matches the experimental results, as can be seen in Fig. <ref>(a). The atoms are well isolated in DWs and can oscillate for 6 cycles without any discernible decay, indicating the excellent coherence of the system. Since the frequency is sensitive to the difference of the left and right well energy levels, the atom tunneling constitutes a sensitive tool to calibrate the superlattice phase φ.In the limit of J ≪ U (U is the on-site interaction of the Bose-Hubbard model), the spin-exchange interaction can be well described by a two-level system with interaction strength J_ex = 2 J^2/U <cit.>. To maximize the spin-exchange amplitude, the spin-dependent splitting should be minimized to keep the state ↑,↓ and ↓, ↑ degenerate. Thus besides setting the EOM to θ = 0^∘, we also set the quantization axis to z during spin-exchange evolutions. The lattices parameters for spin-exchange are V_s=17.0(1) E_r, V_l=10.0(1) E_r, resulting in an superexchange interaction strength of J_ex/h = 17.8(8) Hz. We monitor the dynamics using the Néel order parameter N_z = (n_↑ L + n_↓ R - n_↑ R - n_↓ L)/2, where n_↑,↓; L,R= ⟨n̂_↑,↓; L,R ⟩ denotes the corresponding quantum mechanical expectation values. Fig. <ref>(b) shows such spin-exchange oscillation with a high contrast. On the other hand, the spin-dependent effect can be used to suppress the spin-exchange process by breaking the degeneracy of spin states ↑,↓ and ↓, ↑. When the quantisation axis is set to the lattice direction, controlling the EOM can induce an energy shift δ between these two spin states. As shown in Fig. <ref>(b), the oscillations between states ↑,↓ and ↓, ↑ are dramatically suppressed as δ > J_ex. Such phenomena can be explained by a detuned Rabi oscillation in a two-level system, where the frequency becomes larger √(4J_ex^2 + δ^2) and the amplitude becomes smaller 4J_ex^2/(4J_ex^2 + δ^2). The suppression of lower order dynamics can be used to explore some high-order spin interactions, like the four-body ring-exchange interactions <cit.>. Another important feature is the flexibility to tune the phase of entangled Bell states via controlling this energy bias δ <cit.>.In summary, we developed a spin-dependent optical superlattice for tailoring the atomic states and spin interactions. Such a lattice provides a platform for engineering quantum states in two dimensions and detecting spin correlations with in situ imaging, e.g., generating Bell states and observing four-body ring-exchange interactions <cit.>. With the capability of spin addressing and manipulation, one can explore various quantum many-body models, such as spin interactions <cit.>, artificial gauge fields <cit.> and out-of-equilibrium dynamics <cit.> with an novel approach in quantum state initializations and detections. Moreover, our spin-dependent lattice could also be used in atom cooling <cit.>, which offers intriguing prospects for future researches on spin models and quantum magnetism <cit.>.This work has been supported by the MOST (2016YFA0301600), the NNSFC (91221204, 91421305) and the Chinese Academy of Sciences.§ APPENDIX: SITE-RESOLVED BAND MAPPING IN SUPERLATTICETo detect the atom and spin population on different lattice sites, we utilize a site-resolved band mapping sequence <cit.> in the optical superlattice. After the state preparation, the barriers between the intra-DWs are ramped up to freeze the quantum states.We then map the atoms in the left and right DWs onto different Bloch bands and measure the occupation via absorption imaging after a time-of-flight. The superlattice phase φ is first tuned adiabatically to 70^∘, matching the energy of the right ground band with the highly excited band of the left site. Then the DWs are merged by lowering down the short-lattice barrier in 300 μs, whereafter the x long-lattice and y short-lattice are ramped down in 600 μs. Finally, we apply a magnetic gradient during the time-of-flight to separate the spins (Stern-Gerlach separation), mapping out the spin and site populations into different Brillouin zones. Fig. <ref>(a) shows the band mapping patterns of different spin states and site occupations. The size of the Brillouin zones along x is half of the size along y-direction, reflecting the double lattice constant of the long-lattice.apsrev4-1 | http://arxiv.org/abs/1708.07406v1 | {
"authors": [
"Bing Yang",
"Han-Ning Dai",
"Hui Sun",
"Andreas Reingruber",
"Zhen-Sheng Yuan",
"Jian-Wei Pan"
],
"categories": [
"cond-mat.quant-gas"
],
"primary_category": "cond-mat.quant-gas",
"published": "20170824134159",
"title": "Spin-dependent Optical Superlattice"
} |
Applied Statistics Unit, Indian Statistical InstituteKolkata-700108, [email protected] Institute of Mathematics and ApplicationsBhubaneswar-751029,[email protected] Department of Mathematics, Pingla Thana MahavidyalayaPaschim Medinipur-721140,[email protected] Applied Statistics Unit, Indian Statistical InstituteKolkata-700108, [email protected] Dimensional Discrete Dynamics of Integral Value Transformations Pabitra Pal Choudhury December 30, 2023 =================================================================== (to be inserted by publisher)A notion of dimension preservative map, Integral Value Transformations (IVTs) is defined over ℕ^k using the set of p-adic functions. Thereafter, two dimensional Integral Value Transformations (IVTs) is systematically analyzed over ℕ×ℕ using pair of two variable Boolean functions. The dynamics of IVTs over ℕ×ℕ=ℕ^2 is studied from algebraic perspective. It is seen that the dynamics of the IVTs solely depends on the dynamics (state transition diagram) of the pair of two variable Boolean functions. A set of sixteen Collatz-like IVTs are identified in two dimensions. Also, the dynamical system of IVTs having attractor with one, two, three and four cycles are studied. Additionally, some quantitative information of Integral Value Transformations (IVTs) in different bases and dimensions are also discussed. § INTRODUCTIONClassically, discrete dynamics refers to the study of the iteration of self-maps of the complex plane or real line <cit.>. There are plenty of works carried out over last couple of decades in discrete dynamical systems as well as in arithmetic dynamics <cit.>. What is not very well studied yet is the discrete dynamics over spaces like natural numbers. In 2010, the notion of Integral Value Transformations (IVTs) which is a new paradigm in arithmetic dynamics <cit.> has been introduced. Later in 2014, a special class of one dimensional Integral Value Transformations (called Collatz-like) is studied in <cit.>. A couple of other works on IVTs and their applications in arithmetic logical circuit design and in Biology are also studied <cit.>. Applications of such transformations are noteworthy in the area of Cryptology and other field of discrete mathematics <cit.>. In this present article, an attempt has been initiated to understand the discrete dynamics of Collatz-like IVTs including other two dimensional Integral Value Transformations. It is seen that the dynamics of the IVTs are depending on the state transition diagram of two dimensional IVTs of the pair of two variable Boolean functions <cit.>. There are four types of dynamics seen with regards to attractors with different cycle length(s) 1, 2, 3 and 4. All these dynamical systems end with attractors (global and/or local) with a maximum of four length cycles. In addition, from the algebraic aspects the dynamical system of IVTs are investigated. Discrete dynamics is itself very appealing from its application viewpoint. The present study of the dynamical systems over two dimensions can be as pathways for network designing converging to the desired attractor(s). The rest of paper is organized as follows. In Section 2, the definitions of IVT in higher dimensions and in particular in two dimensions over the set of natural numbers with zero are given. The various algebraic properties of the dynamical system of IVTs are described in this section. The Section 3 deals with the state transition diagram for the IVTs of various algebraic classes. Some important theorems including all possible type of attractors are presented in this section. The Section 4 deals with the dynamics of IVTs using their profound trajectories. The Section 4 also deals with the classifications of 256 possible IVTs in two dimensions based on their dynamics over ℕ×ℕ. The quantitative measure for the various classes in different bases and dimensions are also incorporated in this section. This Section 4 also includes the dynamics of some special class of IVTs. The Section 5 concludes the key findings of current article and future research endeavors. § DEFINITIONS & BASIC ALGEBRAIC STRUCTURES IN 𝔹_2 AND 𝕋 §.§ Definitions and basic algebraic structures Integral Value Transformation in the base p and dimension k (IVT^p, k_i_1,i_2,⋯ ,i_k) over ℕ^k (ℕ is set of all natural numbers including zero) is defined as IVT^p,k_i_1,i_2,⋯ ,i_k(m_1,m_2,⋯, m_k)=(IVT^p,k_i_1 (m_1,m_2,⋯, m_k), IVT^p,k_i_2 (m_1,m_2,⋯, m_k),⋯,IVT^p,k_i_k (m_1,m_2,⋯, m_k)) Here (m_1,m_2,⋯, m_k) is an element of ℕ^k. The function IVT^p,k_i is defined using f_i in <cit.>. f_i is a p-adic function. Integral Value Transformation in two dimension (k=2) with base (p=2): (IVT^2,2_i,j) over ℕ×ℕ is defined as IVT^2,2_i,j(m,n)=(IVT^2,2_i (m,n), IVT^2,2_j (m,n)) Here (m,n) is an element of ℕ×ℕ and the function f_i or f_j is the 2-variable Boolean functions and pair-wise in combining they are represented as f_i,j which is formally defined in the Definition 3.The map f_i,j is defined on the set 𝔹_2={00,01,10,11} as f_i,j(x,y)=(f_i(x,y), f_j(x,y)) Here (x,y) belongs to 𝔹_2. In the rest of article, we shall use IVT_i,j instead of IVT^2,2_i,j for notational simplicity. Here we define a discrete dynamical system over ℕ×ℕ asx_n+1=IVT_i,j(x_n)where x_n ∈ℕ×ℕ. For every x_0 ∈ℕ×ℕ, if trajectory of the discrete dynamical system x_n+1=IVT_i,j(x_n) eventually converges to a point (0,0) ∈ℕ×ℕ, then we call such dynamical system x_n+1=IVT_i,j(x_n) as Collatz-like IVT.Let 𝕊 be the set of all possible two variable Boolean functions. Clearly, there are exactly 16 such functions.Let 𝕋 be the set of all possible IVT_i,j defined over the set ℕ×ℕ. Clearly, there are exactly 256 such IVTs. Our aim is to comprehend the dynamics of these maps. The following properties and remarks in two dimensional integral value transformation from algebraic perspective are given.* Properties:* The space (𝔹_2, +_2, *_2) forms a vector space over 𝔽_2={0, 1}. Here +_2 and *_2 are modulo 2 addition and scalar multiplication. The dimension of the vector space is 2. * The basis of the vector space (𝔹_2, +_2, *_2) consists of two elements (0,1) and (1,0). The dimension of the vector space is 2.* The space (𝕊, ⊕_2, ⊗_2) forms a vector space over 𝔽_2={0, 1}. Here ⊕_2, and ⊗_2 are modulo 2 point wise addition and scalar point-wise multiplication.* The basis of the vector space (𝕊, ⊕_2, ⊗_2) consists of 4 elements f_1, f_2, f_4, f_8. The dimension of the vector space is 4. * The space (𝕋, ⊕_2, ⊗_2) forms a vector space over 𝔽_2={0, 1}. Here ⊕_2, and ⊗_2 are modulo 2 function addition and scalar function multiplication. * The basis of the vector space (𝕋, ⊕_2, ⊗_2) consists of 16 elements IVT_1,1, IVT_1,2, IVT_1,4, IVT_1,8, IVT_2,1, IVT_2,2, IVT_2,4, IVT_2,8, IVT_4,1, IVT_4,2, IVT_4,4, IVT_4,8, IVT_8,1, IVT_8,2, IVT_8,4, IVT_8,8. The dimension of the vector space is 16. * There are 4 linear Boolean functions viz. f_0, f_6, f_10 and f_12 in 𝕊.* There are 16 linear maps in 𝕋 and they are IVT_0,0, IVT_0,6, IVT_0,10, IVT_0,12, IVT_6,0, IVT_6,6, IVT_6,10, IVT_6,12, IVT_10,0, IVT_10,6, IVT_10,10, IVT_10,12, IVT_12,0, IVT_12,6, IVT_12,10 and IVT_12,12.* Sum and Composition to two linear maps in 𝕋 is linear. * There are 24 bijective f_i,j in 𝕊. The twelve of the bijective functions are f_3,5, f_3,6, f_3,9, f_3,10, f_5,6, f_5,9, f_5,12, f_6,10, f_6,12, f_9,10, f_9,12, and f_10,12. * There are only 6 isomorphisms (linear and bijective) exist in 𝕋. They are f_6,10, f_6,12,, f_10,6, f_10,12, f_12,6, and f_12,10. * Remarks: * IVT_i,j∈𝕋 is a basis element of 𝕋 if and only if f_i and f_j are also basis elements in 𝕊. * IVT_i,j∈𝕋 is linear if and only if f_i and f_j are also linear. * If f_i,j is linear then f_j,i is also linear. * None of the basis function in 𝕋 is linear. * If f_i,j is bijective then f_j,i is also bijective. §.§ IVTs through STDsIn graph theory, a directed graph G is defined as G(V,E,F); where F is a function such that every edge in E an oder pair of vertices in V <cit.>. In our context, we have designed the state transition diagram (STD) of f_i,j as directed graph. Illustration: For an example, consider the domain of 2-variable Boolean functions which have four unique possibilities i.e. 00, 01, 10 and 11 for its constituent variables as shown in Table 1.From the truth table representation of 2-variable Boolean function f_6,13, we find that (0,0) → (0,1), (0,1)→ (1,0), (1,0)→ (1,1) and (1,1)→ (0,1) are the required transitions. Similarly, the transitions are shown for the function f_0,12. Based on these four transitions a directed graph called STD (G=(V,E)) can be drawn where the vertices or nodes in the graph are four possible bit pairs i.e. V= 𝔹_2 and the directed edges (E) are the transitions or mapping from the set 𝔹_2 to 𝔹_2. So, STD identifies the local transformations for all possible combinations of domain set to the rules, given the truth table of one or more rules of n-variable. The STDs of function f_6,13 and f_0,12 are shown in Fig. 1 and it is noted that there will be always four nodes each node with an out degree edge/transition, but the in degree edge for a particular node can vary from zero to maximum of four. From Fig. 1 (f_6,13), the in-degree of node (0,1) is 2, (1,1) is 1, (1,0) is 1, and (0,0) is 0. Total in-degrees are the summand of all in-degrees of four nodes that must be equal to 4. The STDs of linear functions are given in Fig. 2. The STDs of only linear IVT_i,j are given, we omit other STDs of linear f_j,i. The STDs of basis functions are given in Fig. 3. The STDs of only basis IVT_i,j are given, we omit other STDs of basis f_j,i. The STDs of bijective functions are given in Fig. 4. The STDs of only bijective IVT_i,j are given, we skip STDs of other bijective f_j,i. The STDs of isomorphisms are given in Fig. 5. The STDs of only IVT_i,j which are isomorphic are given, we skip other STDs of isomorphism f_j,i. § DYNAMICS OF IVT_I,J USING STATE TRANSITION DIAGRAM OF F_I,J §.§ Dynamics of STDs when p=2, k=2For i, j ∈{0, 1,…,15} in two dimensions, there are 2^4× 2^4=256 functions f_i,j. From the STDs of those 256 functions f_i,j, their dynamics are pretty clear. There are specifically two kinds of dynamics possible; one is attracting to a single attractor and other kind is periodic attractor of period (pr), 2 ≤ pr ≤ 4. For examples, the single attractor function is f_0,12 and period 3 attractor function is f_6,13 (Fig. 1).The dynamical system x_n+1=IVT_i,j(x_n) is Collatz-like if and only if STD of f^k_i,j((x,y))=(0,0) for all (x,y) ∈𝔹_2, for some k ∈ℕ. Here f^k_i,j denotes the k-composition of the function defined as f^k_i,j=f_i,j(f^k-1_i,j). There are exactly 16 Collatz-like x_n+1=IVT_i,j(x_n) in 𝕋.As per definition of Collatz-like function, we are looking for functions f_i,j such that for some k,f^k_i,j((x,y))=(0,0)for all (x,y) ∈𝔹_2. We will check and count through the possible variety of STDs structure of f_i,j.Suppose f_i,j has the required property and let G_f_i,j be the directed graph (STD) with four vertices- a is (0,0), b is (0,1), c is (1,0), d is (1,1) and an edge (x,y) → (m,n) whenever f_i,j(x,y) = (m,n).Each vertex of G_f_i,j has out degree 1. We make the following claims, all of which are simple.1. G_f_i,j must have at least one cycle, say C.2. G_f_i,j must be connected, and it has only one component.3. The cycle C is the only cycle in G_f_i,j, since every vertex must be in the same component as (0,0).4. The cycle C is a sink: for each (x,y), f^k_i,j(x,y) is in C for all sufficiently large k and (0,0) ∈ C.5. If (0,0)f^k_i,j((0,0)), then f^k_i,j(a) f^k_i,j(f((0,0))) for every k. And no other vertex is in C. So we have (0,0) = f_i,j((0,0)). Therefore, we could have only 4 topologies (STDs) for G_f_i,j as shown below. The vertex a is fixed and by interchanging b, c and d in different levels/positions we can easily enumerate how many elements that have f^k_i,j((x,y)) = (0,0). Topology: [ > = stealth,shorten > = .4pt,auto, node distance = 1cm,semithick] every state=[ draw = black, thick, fill = white, minimum size = 1mm ] [state] (1) a; [state] (2) [below of=1] c; [state] (3) [left of=2] b; [state] (4) [right of=2] d;[->] (1) edge [loop above] node(1); [->] (2) edge node(1); [->] (3) edge node(1); [->] (4) edge node(1);[ > = stealth,shorten > = .5pt,auto, node distance = 1cm,semithick] every state=[ draw = black, thick, fill = white, minimum size = 2mm ] [state] (1) a; [state] (2) [below left of=1] b; [state] (3) [below right of=1] c; [state] (4) [below of=2] d;[->] (1) edge [loop above] node(1); [->] (2) edge node(1); [->] (3) edge node(1); [->] (4) edge node(2);[ > = stealth,shorten > = .5pt,auto, node distance = 1cm,semithick] every state=[ draw = black, thick, fill = white, minimum size = 2mm ] [state] (1) a; [state] (2) [below of=1] b; [state] (3) [below left of=2] c; [state] (4) [below right of=2] d;[->] (1) edge [loop above] node(1); [->] (2) edge node(1); [->] (3) edge node(2); [->] (4) edge node(2); [ > = stealth,shorten > = .3pt,auto, node distance = 1cm,semithick] every state=[ draw = black, thick, fill = white, minimum size = 1mm ] [state] (1) a; [state] (2) [below of=1] b; [state] (3) [below of=2] c; [state] (4) [below of=3] d;[->] (1) edge [loop above] node(1); [->] (2) edge node(1); [->] (3) edge node(2); [->] (4) edge node(3);Number of functions: 1 6 3 6 Functions (f_i,j:(i,j)): (0,0)(0,4), (0,8), (2,0), (2,2), (4,4), (8,0) (0,12), (6,6), (10,0) (2,6), (2,8), (4,12), (6,4), (8,4), (10,2)Hence, the total number of Collatz-like functions in two dimensions is 16.It is noted that the attractor (0,0) is a global attractor for the Collatz-like IVTs, that is for every initial value taken from ℕ×ℕ, the dynamical system takes all the trajectories to (0,0). The dynamical system x_n+1=IVT_i,j(x_n) possesses multiple attractor(s) of the form (2^s-1,0), s∈ℕ if and only if f^k_i,j((x,y))=(1,0), for all (x,y) ∈𝔹_2, for some k ∈ℕ. There are countably infinite number of local attractors of the form (2^s-1,0), s∈ℕ. For any x_0 ∈ℕ×ℕ, the dynamical system eventually proceeds to a point of the form (2^s-1,0), according to definition of the dynamical system map IVT_i,j. It is obvious that the set of all possible points of the form (2^s-1,0), is a subset of the ℕ×ℕ. Since the set ℕ×ℕ is countable and so the subset is countable. Hence the theorem is followed. The dynamical system x_n+1=IVT_i,j(x_n) possesses multiple attractor of the form (0, 2^s-1) if and only if f^k_i,j((x,y))=(0,1), for all (x,y) ∈𝔹_2, s∈ℕ for some k ∈ℕ. There are countably infinite number of local attractors of the form (0,2^s-1), s∈ℕ for the dynamical system x_n+1=IVT_i,j(x_n). The dynamical system x_n+1=IVT_i,j(x_n) possesses multiple attractors of the form (2^s-1,2^t-1), s, t∈ℕ if and only if f^k_i,j=(1,1) for some k ∈ℕ. There are countably infinite number of local attractors of the form (2^s-1,2^t-1) s, t∈ℕ for the dynamical system x_n+1=IVT_i,j(x_n). It is noted that there are STDs of some function f_i,j such that there are periodic attractor of period p ≥ 2. In such cases, the dynamical system x_n+1=IVT_i,j(x_n) will possess to multiple attractors of form (u,2^t-1), (2^t-1,u), (0,u), (u,0) and(u,v). There are few dynamical systems which are sensitive to the initial conditions. In the following section, we shall classify all such types of dynamical systems based on their dynamical behavior. §.§ STD dynamics when p≥ 2, k≥3In p-adic and k-variable functions (or k-dimension), we have p^p^k number of functions f_i where i∈{0,1,⋯,2^k-1}. Depending on k value, we select the number of functions in k dimensions and in combining we represent as f_i_1,i_2,⋯ ,i_k and length of each function is p^k. In general, for any i, j ∈{0, 1,…2^k-1} in k dimensions, there are p^p^k× p^p^k=p^2p^k functions f_i_1,i_2,⋯ ,i_k. In that case, we could have the attractor of period (pr), 1 ≤ pr ≤ p^k. The number of functions is exponentially increasing with the increment of p and k. In principle, the discrete dynamics of such IVTs can also be obtained using STDs as we did it in two dimensions (p=2,k=2). The vertex of the STD in p and k dimension is (a_1,a_2,⋯ a_k); where a_i∈{0,1,⋯ p^k}. For an example, the STDs of two Collatz-like functions f_128,72,160 (p=2,k=3) and f_2187,12690 (p=3,k=2) and a STD of function f_41,19230 (p=3,k=2) with attractors 5 and 2 is shown in Fig. 6. The number of global attractors for different p and k value is shown in Table 2.Based on the computational evidence, the total number of Collatz-like IVTs in the base p with k dimension is conjectured. The total number of Collatz-like IVTs in the base p with k dimension is p^k(p^k-2). If l_k defines the path length from a particular point to the global attractor in the dynamics of IVT^p,k_i,j, then 1≤ l_k≤ (p^k-1). § DYNAMICS OF X_N+1=IVT_I,J(X_N)Before getting into the detailed classifications of dynamics, a vivid example of a dynamical system x_n+1=IVT_i,j(x_n), x_n ∈ℕ×ℕ is adumbrated with different initial positive natural integers and their trajectories as shown in the Fig. 7. For an example, IVT_13,3(0,2)=(1,3) i.e. (0,2)→(1,3). The trajectories (path) can be changed with the changes of p value even if the taken function is fixed. §.§ Classifications of IVT_i,j based on their dynamics when p=2, k=2In this section, we have classified the 256 of IVT_i,j into four disjoint classes (Class I-IV) according to the attractors of four different cycle length(s). For every class, several subclasses are identified according to the dynamic behavior of IVT_i,j over ℕ×ℕ. It can be seen that there are many (x,y) components in the dynamics ofIVT_i,jwhere (x,y)∈ℕ×ℕ. For the simplicity of graph visualization, we have taken certain limit for both x and y i.e. x,y∈{0,1⋯ 7}. For some of the IVTs, all the components are connected (e.g. all Collatz-like IVTs) and other cases there could be of different variety.For understanding the complete dynamics over the set of positive natural numbers in base p and dimension k, x,y∈{0,1,⋯, p^k-1} that covering all possible transitions in their p-adic representations. Here, p^k-1 is the minimum value that can be extended to desire limit.§.§.§ Class-I: Dynamics of IVT_i,j: attractor with a fixed pointA total of 125 IVT_i,j are classified into the Class-I as shown in Table 3. It is noted that each of the IVTs in the Class-I is having attractor with one length cycle (singleton attractor). There are 16 IVTs which are Collatz-like and rest are non-Collatz-like. For the non-Collatz-like dynamical systems, there are countable many local attractors with one cycle.A dynamical systemIVT_10,0 having an attractor with one length cycle is given in the Fig. 8.There are other 16 out of 125 which are sensitive to the initial conditions that is their trajectories are depending on initial values. Six possible IVT_i,j are taken and for all the cases five integer pairs trajectories are shown in Fig. 9. It is noted that each of these initial condition (taken pair) provides sensitive dynamical system ending with different cyclic attractors of length one.§.§.§ Class-II: Dynamics of IVT_i,j: attractor with two length cycles A total of 93 IVT_i,j are classified into the Class-II as shown in Table 4. All these IVTs in the Class-II are having attractors with two length cycles. Some of these dynamical systems are having global attractor of two length cycles. There are 42 out of 93 which are sensitive to the initial conditions that is their trajectories are depending on initial values. Thus, each of these initial conditions provides sensitive dynamical system ending with two length cycle attractor. A dynamical systemIVT_4,5 having an attractor with two length cycle (global attractor) is given in the Fig. 10. §.§.§ Class-III: Dynamics of IVT_i,j: attractor with three length cycles A total of 32 IVT_i,j are classified into the Class-III as shown in Table 5. All these IVTs in the Class-III are having attractor with three length cycles. There are some dynamical systems with global three length cycle attractors. There are 8 out of 32 which are sensitive to the initial conditions that is their trajectories are depending on initial values. Each of these initial conditions provides sensitive dynamical systems ending three length cycle attractor.A dynamical systemIVT_9,1 having an attractor with three length cycle is shown in the Fig. 11.§.§.§ Class-IV: Dynamics of IVT_i,j: attractor with four length cycles A total of 6 IVT_i,j are classified into the Class-IV as shown in Table 6. All these IVTs in the Class-IV are having attractors with four length cycles. These dynamical systems do possess global attractors with four length cycles. A dynamical systemIVT_9,5 having an attractor with four length cycle is shown in the Fig. 12. §.§ Classifications of IVT^p,k_i,j based on their dynamics when p≥2, k≥ 3In case of p=2, k=2, the cycle length of an attractor observed in the dynamics of IVT_i,j does not exceed 4 which is the maximum cycle length of an STD of f_i,j. It is quite impossible to observe the dynamics of individual IVT^p,k_i,j when p and k are sufficiently large. From the dynamics of STDs of some f^p,k_i,j, one can classify the corresponding IVT^p,k_i,j based on their dynamics ends up with cycle length exclusively 1,2,…, p^k (Fig. 6) into p^k number of classes. The cardinality of different classes based on dynamics of IVT^p,k_i,j having attractors of different cycle lengths (upto p^k) including an exclusive outer class are presented for different p and k in Table 7. The exclusive outer class contains IVT^p,k_i,j such that the length of the cycle of the attractors exceed the maximum cycle length (p^k) of the STD of f^p,k_i,j. Such a discrete dynamical systems is called here as chaotic dynamical system.An example of chaotic dynamical system of IVT^3,2_41,19230 using the function f_41,19230 (p=3, k=2) is shown in Fig. 13, whose STD is given in Fig. 6 (C). From the STD of f_41,19230, we could see two types of cycles (2 and 5) is present. But, the dynamical system shows an additional cycle of length 10 (l.c.m of 2 and 5). The reason can be easily seen from the ternary representation (here, p=3) of any pair (x,y)∈ℕ×ℕ. Take a pair (5,6) from the cycle of length 10 from Fig. 13. The ternary representation of (5,6) is (12,20). The pair wise component are 12 (first component from left) and 20 (second component from left). These two components are from two different cycles 2 and 5 respectively from STD of f_41,19230 (Fig. 6(C)). All the STD components are distinct. So, while calculating the IVT one by one iteration, the components 12 and 20 will come back/repeat to the same initial stage afterl.c.m (5,2) iterations or cycle of length l.c.m (5,2). This is true for arbitrary pair in any base p and also true if we increase the dimension k.Number of bijective functions (f^p,k_i_1,i_2,⋯ i_k) in p-adic, k dimension is (p^k-1)!. §.§ Dynamics of two dimensional IVT_i,j from algebraic perspective Here we go with classification of two dimensional IVT_i,j from their algebraic perspectives such as basis, linearity, bijective and isomorphism which are shown in Table 8. It is to be noted that the dynamics of the IVT_i,j is not always same as the dynamics of IVT_j,i although they belong to same algebraic class as previously pointed out.§ CONCLUSIONS AND FUTURE ENDEAVORSIn this manuscript, notion of dimension preservative Integral Value Transformations (IVTs) is defined over ℕ^k using the set of p-adic functions. The main focus is given in order to understand the two dimensional integral value transformation (p=2 and k=2, which are computationally feasible) and studied the dynamics of all possible pair functions in four different kinds of attractors. Among them, a special class of IVTs (Collatz-like) is computationally identified along with some others global and local dynamical systems. In addition, from the algebraic properties of IVT_i,j, the dynamics are critically observed. From the current analysis, it is evident that as p and k go high, the total numbers of Collatz like IVTs are increasing very fast. Consequently, the dynamical behavior of these IVTs is computationally difficult to simulate and apprehend. But, some of their dynamics can be vivid from our current analysis in two dimensions and some other cases. In this concern, we have shown some quantitative informations for other two special cases of IVTs (p=2,k=3 and p=3, k=2). Further, the analysis also clearly evident that as p and k go high, we come across with some chaotic dynamical behavior. In future endeavor, one can explore with some higher p and k considering the limitation of simulation.§.§ AcknowledgmentWe thank to anonymous reviewers for their constructive suggestions and valuable directions that help to improve the article quality.ws-ijbc | http://arxiv.org/abs/1709.05205v2 | {
"authors": [
"Jayanta Kumar Das",
"Sudhakar Sahoo",
"Sk. Sarif Hassan",
"Pabitra Pal Choudhury"
],
"categories": [
"math.DS"
],
"primary_category": "math.DS",
"published": "20170825054112",
"title": "Two Dimensional Discrete Dynamics of Integral Value Transformations"
} |
Height transitions, shape evolution, and coarsening of equilibrating quantum nanoislands Mikhail Khenner December 30, 2023 ======================================================================================== In this article we introduce a simple dynamical system called symplectic billiards. As opposed to usual/Birkhoff billiards, where length is the generating function, for symplectic billiards symplectic area is the generating function. We explore basic properties and exhibit several similarities, but also differences of symplectic billiards to Birkhoff billiards. § INTRODUCTIONBirkhoff billiard describes the motion of a free particle in a domain: when the particle hits the boundary, it reflects elastically so that the tangential component of its velocity remains the same and the normal component changes the sign. In the plane, this is the familiar law of geometric optics: the angle of incidence equals the angle of reflection, see Figure <ref>. This reflection law has a variational formulation: if the points x and z are fixed, the position of the point y on the billiard curve is determined by the condition that the length |xy|+|yz| is extremal. In particular, periodic billiard trajectories are inscribed polygons of extremal perimeter length.Another well-studied dynamical system is outer billiard, one such transformation in the exterior of a convex curve also depicted in Figure <ref>. Outer billiard admits a variational formulation as well: periodic outer billiard orbits are circumscribed polygons of extremal area. It is natural to consider two other planar billiards: inner with area, and outer with the length as the generating functions. The variational formulation of the reflection depicted in Figure <ref> on the right is as follows: if the points x and z are fixed, the position of the point y on the billiard curve is determined by the condition that the area of the triangle xyz is extremal. It would be natural to call this area billiard [A competing name would be affine billiard since this system commutes with affine transformations of the plane.]; due to higher-dimensional considerations, we prefer the term symplectic billiard.Analytically, the generating function of the symplectic billiard map is ω(x,y), where ω is the area form, as opposed to |xy|, the generating function of the usual billiard map. The aim of this paper is to introduce symplectic billiards and to make the first steps in their study.We believe that a system that has such a simple and natural definition should have interesting properties; in particular, one wantsto learn which familiar properties of the usual inner (Birkhoff) and outer billiards are specific to them, and which extend to the symplectic billiards and, perhaps, other similar systems. One can also define symplectic billiard in a convex domain with smooth boundary in a linear symplectic space (^2n,ω). Let M be its boundary. To define a reflection similarly to the planar case, one needs to choose a tangent direction at every point of M. This is canonically provided by the symplectic structure: the tangent line at x∈ M is the characteristic direction ω|_T_xM× T_xM, that is, the kernel of the restriction of the symplectic form ω on the tangent hyperplane T_x M. (A similar idea is used in the definition of the multi-dimensional outer billiard, see, e.g., the survey <cit.>.) The generating function of this map is again ω(x,y), where x,y ∈ M.This paper consists of three parts, the two-dimensional smooth and polygonal symplectic billiards, and multi-dimensional ones. Let us describe our main results.In the planar case, we show that symplectic billiard is a monotone, area preserving twist map. In Theorem <ref> we calculate the area of the phase space: it equals four times the area of the central symmetrization of the billiard table. Theorem <ref> is a version of the well known theorem by Mather <cit.>: if the billiard curve has a point of zero curvature, then the symplectic billiard possesses no caustics. Theorem <ref> is a KAM theory result, an analog of Lazutkin's theorem <cit.>: if the curve is sufficiently smooth and the curvature is everywhere positive, then the symplectic billiard has smooth caustics arbitrary close to the boundary. The coordinates in which the symplectic billiard map is a small perturbation of an integrable map are provided by the affine parameterization of the billiard curve. In Section <ref>, we apply the approach via exterior differential systems, introduced and implemented in the study of Birkhoff and outer billiards in <cit.>. It makes it possible to prove the existence of billiard tables that possess caustics consisting of periodic points. We remark that, for period 4, the billiard bounded by a Radon curve has this property. Another application of exterior differential systems is to small-period cases of the Ivrii conjecture which states that the set of periodic billiard orbits has zero measure (a weaker version says that this set has empty interior). Theorem <ref> asserts thatfor planar symplectic billiards the set of 3-periodic points and that of 4-periodic points has empty interior.In Section <ref>, we consider the area spectrum, that is, the set of areas of the polygons corresponding to the periodic orbits of the planar symplectic billiard. Let A_n be the maximal area of an n-gon inscribed in the billiard curve. The theory of interpolating Hamiltonians <cit.> implies an asymptotic series expansion of A_n in negative even powers of n whose first two terms are equal, up to numerical factors, to the area bounded by the billiard curve and itsaffine length. Then the affine isoperimetric inequality implies that ellipses are uniquely determined by their area spectrum – Theorem <ref> (a similar result for outer billiards was obtained in <cit.>). We also show that symplectic billiards do not possess the finite blocking property (or, in different terminology, are insecure): for every pair of distinct points x,y on the boundary curveand every finite set S inside γ, there exists a symplectic billiard trajectory from x to y that avoid S. See <cit.> for insecurity of Birkhoff billiards.Section <ref> concerns polygonal symplectic billiards. Theorems <ref> and <ref> assert that if the polygon is affine-regular or is a trapezoid, then all orbits are periodic, with explicitly described periods. In Propositions <ref> and <ref>, we describe two types of periodic orbits: those that appear in 2-parameter families (similarly to the periodic orbits of polygonal outer billiards), and the isolated, stable ones that survive small perturbations of the polygon. We do not know whether every convex polygon carries a periodic symplectic billiard orbit (this question is also open for the usual polygonal billiards,in contrast with outer polygonal billiards where such a result is known).After giving a careful definition of multi-dimensional symplectic billiard, we describe, in Section <ref>, the case of ellipsoids. As one may expect, this is a completely integrable case. Namely, in Theorem <ref>, we relate the symplectic billiard in an ellipsoid to the usual billiard in a different ellipsoid: given a trajectory of the symplectic billiard, construct a new polygonal line by connecting every second consecutive impact points (i.e, always skip one impact point); then a linear transformation (that depends only on the original ellipsoid) takes this polygonal line to a Birkhoff billiard trajectory on another ellipsoid (that again depends only on the original ellipsoid.) This result is analogous to a theorem of Moser and Veselov <cit.> that relates the discrete Neumann system with billiards in ellipsoids. We also explicitly describe the symplectic billiard dynamics inside a round sphere (Proposition <ref>).The last Section <ref> concerns periodic orbits in multi-dimensional symplectic billiards inside an arbitrary strictly convex closed smooth hypersurface. Theorem <ref> asserts that, for every k≥ 2, there exist k-periodic trajectory. This is a weak result and, in Theorem <ref>, we obtain a strongerone for small periods:the number of 3-periodic, and of 4-periodic, symplectic billiard orbits in 2n-dimensional symplectic space is no less than 2n (the dihedral group D_k acts on k-periodic trajectories by cyclic permutation of the vertices and reversing their order; what one counts are these D_k-orbits). We conclude this introduction with a remark concerning the interplay between convexity and symplectic geometry. Even though convexity is not a symplectically invariant notion it has long been know that convex domains in symplectic space enjoy special rigidity properties. This culminated in Viterbo's conjecture <cit.> which asserts an inequality between symplectic capacities and volume for convex domains. It is well known that Viterbo's conjecture fails if the convexity assumption is dropped. And even though it has been proved in special cases in general, it is considered wide open. The fundamental nature of Viterbo's conjecture is demonstrated by the fact that it implies Mahler's conjecture from convex geometry <cit.>. Recently, a renewed interested in this interplay arose, in part due to related results about systolic inequalities, see e.g., <cit.>.Of course, the definition of the symplectic billiard map crucially relies on the convexity of the domain. At this point it is not more than idle speculation that this fact is more than a coincidence. Acknowledgments. We are grateful to V. Dragovic and B. Jovanovic for a discussion of periodicbilliard orbits in ellipsoids, to R. Montgomery for a discussion of the Lexell theorem,and to R. Schwartz for writing a computer program for experiments with polygons and for his insight concerning the symplectic billiards in trapezoids. In the past, the second author had numerous discussions of the topic of four planar billiards with E. Gutkin; his untimely death made it impossible for him to participate in this project. S.T. also acknowledges stimulating discussions of these ideas with S. Troubetzkoy. The first author is grateful to the hospitality of the Pennsylvania State University. He was supported by SFB/TRR 191. The second author is grateful to the hospitality of the Heidelberg University. He was supported by the NSF grantDMS-1510055.§ SYMPLECTIC BILLIARD IN THE PLANE §.§ Symplectic billiard as an area preserving twist map §.§.§ A precise look at the definition Letbe a smooth, strictly convex, closed, positively oriented curve, the boundary of our billiard table.Sinceis strictly convex, for every point x∈ there exists a unique point x^*∈ with the propertyT_x=T_x^*⊂^2.Clearly we have x^**=x.Denote by ν the outer normal of . ThenT_x=T_y⟺ω(ν_x,ν_y)=0.Using the orientation of , we define:={(x,y)∈×| x < y < x^*}.Note that this is actually antisymmetric in x and y:={(x,y)∈×| y^* < x <y},see Figure <ref>. Indeed,={(x,y)∈×|ω(ν_x,ν_y)>0}.We think ofas the (open, positive part of) phase space. The next lemma formalizes the definition of the symplectic billiard map. Given (x,y)∈, there exists a unique point z∈ withz-x∈ T_y.Moreover, the new pair (y,z) lies again in our phase space: (y,z)∈. Since (x,y)∈, we know that T_x⋔ T_y. Using thatis convex, we obtain(x+T_y)∩={x,z}.Moreover, z≠ x since otherwise T_y=T_z=T_x, which contradicts x<y<x^*. In other words, the line through x parallel to T_y intersectsin a new point z, seeFigure <ref>. Equation (<ref>) is equivalent to z-x∈ T_y.To show that (y,z)∈, we observe that if y is close to x, then so is z, and therefore ω(ν_x,ν_y)>0 implies ω(ν_y,ν_z)>0. Now assume that x<y<x^* and ω(ν_y,ν_z)≤0. By continuity and moving y close to x, we can arrange ω(ν_y,ν_z)=0. But then equation (<ref>) implies that T_y=T_z, which implies y=x, a contradiction.Thus the mapΦ:→, (x,y)↦(y,z), with z∈ being the unique point satisfying z-x∈ T_y, is well-defined. Of course the same dynamics/map is defined on the negative part of phase space {(x,y)∈γ×γ| x^* < y < x} simply by reversing the orientation of .We extend Φ to the closure ={(x,y)∈γ×γ| x≤ y≤ x^*} by continuity. The first case is obviouslim_y→ xΦ(x,y)=(x,x),that is, the map extends as the identity. In the other case, we claimlim_y→ x^*Φ(x,y)=(x^*,x).This follows from the observation that, due to convexity, y↦ z(y) is monotone, and lim_y→ x^*T_y=T_x^*.The continuous extension Φ(x,x^*)=(x^*,x) is characterized by the 2-periodicity. That is,Φ(x,y)=(y,x) is equivalent to y∈{x,x^*}. One direction is exactly the continuous extension. Now assume that Φ(x,y)=(y,x). If (x,y)∈ then, by Lemma <ref>,(x+T_y)∩={x,y}with x≠ y. This is clearly a contradiction since T_y∩={y}. Therefore, either (x,y)=(x,x) or (x,y)=(x,x^*).The envelope of the 1-parameter family of chords x x^* is a caustic of our billiard. This envelope is called the centre symmetry set of , and it was studied in the framework of singularity theory <cit.>.Now we identify a generating function for the symplectic billiard map Φ. The function S:→, (x,y)↦ S(x,y):=ω(x,y) is a generating function for Φ, that isΦ(x,y)=(y,z) ⟺d/dy[S(x,y)+S(y,z)]=0. d/dy[S(x,y)+S(y,z)]=0⟺ω(x,v)+ω(v,z)=0∀ v∈ T_y⟺ω(x-z,v)=0∀ v∈ T_y(∗)⟺ x-z∈ T_y.In (∗) we used x≠ z which is due to (x,y)∈, compare with Lemma <ref>. The area of the triangle xyz on the right of Figure <ref> equals1/2[ ω(x,y) + ω(y,z) + ω(z,x) ],which, ignoring the factor, differs from S(x,y)+S(y,z) by a function of x and z, having no effect on the partial derivative with respect to y.The symplectic billiard map commutes with affine transformations of the plane. Obviously, circles are completely integrable: concentric circles are the caustics; by affine equivariance, ellipses are completely integrable as well.§.§.§ Invariant area form and the twist conditionLet (t) be a parameterization of the curvewith 0≤ x ≤ L. Denote by t^* the involution on the circle of parameters: the tangents at (t) and (t^*) are parallel. We have S(t_1,t_2) = ω((t_1),(t_2)), and Φ(t_1,t_2) = (t_2,t_3) ifS_2 (t_1,t_2) + S_1(t_2,t_3) =0,where S_1 and S_2 denote the first partial derivatives with respect to the first and second arguments.Following the theory of twist maps (see, e.g., <cit.>), we introduce new variables s_1 = - S_1(t_1,t_2), s_2 = S_2(t_1,t_2),and a 2-form Ω = S_12(t_1,t_2) dt_1∧ dt_2 on S^1× S^1.The variables (t,s) are coordinates on thephase space , and Ω is an area form therein. The map Φ is a monotone twist map, and the area form is Φ-invariant: Φ^*Ω = Ω. Identify ^2 with . Then-∂ s_1/∂ t_2(t_1,t_2)= S_12(t_1,t_2) = ω('(t_1),'(t_2)) = ω(-i'(t_1),-i'(t_2)) >0,since -i '(t) is an outward normal vector toat point (t) and, in , one has by definition ω(ν_ (t_1),ν_(t_2))>0. It follows that Ω is an area form on . It also follows that the Jacobian of the map (t_1,t_2) ↦ (t_1,s_1) does not vanish, therefore (t_1,s_1) are coordinates. The twist condition ∂ t_2/∂ s_1 < 0 follows as well: indeed, ∂ s_1 / ∂ t_2 = - S_12 < 0.Finally, it follows that equation (<ref>) implies that the 2-form S_12 (t_1,t_2) dt_1∧ dt_2 is Φ-invariant, indeed take the exterior derivative of(<ref>) and wedge multiply by d t_2.This completes the proof.Thus a variety of results about monotone twist maps apply to our symplectic billiard. In particular, we need to the following fact. For every period n ≥ 2 and any rotation number 1≤ k ≤ [n/2], the symplectic billiard has at least two distinct n-periodic orbits with rotation number k. §.§.§ Spherical and hyperbolic versionsUsing the same generating function, the area of a triangle, one can define the symplectic billiard map in thespherical and hyperbolic geometries. The definition is based on the Lexell theorem of spherical geometry, and its hyperbolic analog, that describes the locus of vertices of triangles with a given base and a fixed area, see <cit.>. Here is the formulation of Lexell's theorem. Let xyz be a spherical triangle, and let x^* and z^* be the points antipodal to x and z. Consider the arc of a spherical circle x^* y z^*. This arc is one half of the locus of points w such that the area of the spherical triangle xwz equals that of the triangle xyz, the other half being its reflection in the geodesic xz.Consider the hyperbolic plane in the Poincaré disk model. Let now x^* and z^* be the inversions of points x and z in the unit circle. Then the above formulation holds without change (in hyperbolic terms, the arc x^* y z^* is an equidistant curve, a curve of constant curvature less than 1).This leads to the following definition of the symplectic billiard in the spherical and hyperbolic geometries. Letbe a closed convex curve, and let x, y ∈. Let δ be the circle through point x^* that is tangent toat point y, and let δ^* be its image under theantipodal involution, respectively, inversion in the absolute. ThenΦ(x,y) = (y,z),where z is the intersection point of δ^*with , different from x (assuming that such a point is unique, otherwise the map Φ becomes multi-valued). See Figure <ref>. §.§ Total phase space areaIn this section we calculate the Ω-area of the phase space.Let D be the billiard table with the boundary .Parameterizeby the direction α of its tangent line. Choose an origin O inside D, and let p(α) be the support function, that is, the distance from O to the tangent line tohaving direction α+π/2, see Figure <ref>.The Cartesian coordinates of the point (α) are given by the formulasx(α)=p(α) cosα - p'(α) sinα, y(α)= p(α) sinα + p'(α) cosα.The perimeter length ofand the area bounded by it are given, respectively, by the integrals∫_0^2π p(α) dα and1/2∫_0^2π [p(α)+p”(α)] p(α) dα,see, e.g., <cit.>. The phase spaceconsists of pairs α_1,α_2 with α_1 < α_2 < α_1+π. Let w(α)=p(α)+p(α+π) be the width of D in the direction α.Thesymmetrization D̅ of the domain Dis the Minkowski sum with the centrally symmetric domain, scaled by 1/2. The support function p̅(α) of D̅ equals [p(α)+p(α+π)]/2. The total Ω-area of the phase spaceequals four times the area of the symmetrization D̅. Since Ω = S_12 dα_1∧ dα_2, we need to calculate S_12 = ['(α_1),'(α_2)].One has from (<ref>)'(α) = [p(α)+p”(α)] (-sinα,cosα),hence['(α_1),'(α_2)] = [p(α_1)+p”(α_1)] [p(α_2)+p”(α_2)] sin(α_2-α_1).Therefore the phase area is∫_0^2π∫_α_1^α_1+π [p(α_1)+p”(α_1)] [p(α_2)+p”(α_2)] sin(α_2-α_1) dα_2 dα_1.Consider the inner integral:∫_α_1^α_1+π p(α_2) sin(α_2-α_1) dα_2 + ∫_α_1^α_1+π p”(α_2) sin(α_2-α_1) dα_2 .We use integration by parts twice on the second summand to compute:∫_α_1^α_1+π p”(α_2) sin(α_2-α_1) dα_2=p'(α_2) sin(α_2-α_1)|_α_1^α_1+π_=0-∫_α_1^α_1+π p'(α_2) cos(α_2-α_1) dα_2=-∫_α_1^α_1+π p'(α_2) cos(α_2-α_1) dα_2=- p(α_2) cos(α_2-α_1)|_α_1^α_1+π - ∫_α_1^α_1+π p(α_2) sin(α_2-α_1) dα_2= p(α_1) + p(α_1+π)- ∫_α_1^α_1+π p(α_2) sin(α_2-α_1) dα_2 .Thus the inner integral (<ref>) evaluates top(α_1) + p(α_1+π).Now consider the outer integral and recall that the support function p̅(α)of the symmetrization D̅ equals 12[p(α)+p(α+π)]. ∫_0^2π [p(α_1)+p”(α_1)] [p(α_1) + p(α_1+π)] dα_1 =1/2∫_0^2π [p(α_1)+p”(α_1) + p(α_1+π)+p”(α_1+π)] [p(α_1) + p(α_1+π)] dα_1=2 ∫_0^2π [p̅(α) + p̅”(α)] p̅(α) dα= 4area(D̅),as claimed.It is interesting to compare this with the usual billiards: in that case, the area of the phase space (with respect to the canonical area form on the space of oriented lines) equals twice the perimeter length of the boundary curve (see, e.g., <cit.>). §.§ Existence and non-existence of caustics§.§.§ Non-existence of causticsThe next result is a symplectic billiard version of Mather's theorem: If a smooth convex billiard curve has a point of zero curvature, then the (usual) billiard inside this curve has no caustics <cit.> or <cit.>. Letbe a smooth closed convex curve whose curvature vanishes at some point. Then the symplectic billiard inhas no caustics. According to Birkhoff's theorem, an invariant curve of an area preserving twist map is a graph of a function; see, e.g., <cit.>. Assume that our billiard has a caustic. Then one has a 1-parameter family of chords x_1 x_2 of the curve , corresponding to the points of the invariant curve. The graph property implies that if x̅_1 x̅_2 is a nearby chord from the same family and x̅_1 has moved alongin the positive direction from x_1, then x̅_2 also has moved in the positive direction from x_2. It follows that the chords intersect inside the curveand, as a consequence, the caustic, which is the envelope of the lines containing these chords, lies inside the billiard table.Assume that a caustic exists, a chord x_1 x_2, tangent to the caustic, reflects to x_2 x_3, and the curvature at x_2 vanishes. Consider an infinitesimally close chord x̅_1 x̅_2, tangent to the same caustic. Since the curvature at x_2 vanishes, the tangent line at x̅_2 is, in the linear approximation, the same as the one at x_2. Therefore, in the same linear approximation, the line x̅_1 x̅_3 is parallel to x_1 x_3, hence these lines do not intersect inside the billiard table. This contradiction proves the non-existence of the caustic.Alternatively, one may use Mather's analytic condition from <cit.>. This necessary condition for the existence of a caustic is S_22 (t_1,t_2) + S_11 (t_2,t_3) < 0.Let t be an arc length parameter on . Since S(t_1,t_2) = ω((t_1),(t_2)), we haveS_22 (t_1,t_2) = ω((t_1),”(t_2)) = k(t_2) ω((t_1),i '(t_2)),and likewise for S_11 (t_2,t_3),where k is the curvature of .If k(t_2)=0, then S_22 (t_1,t_2) + S_11 (t_2,t_3) =0, violating Mather'scriterion.It would be interesting to find analogs of the results of Gutkin and Katok on the regions of the billiard tablefree of caustics <cit.> and of Bialyon the part of the phase space free of invariant curves <cit.>.§.§.§ Existence of causticsThe existence of caustics is provided by KAM theory, namely, Lazutkin's theorem <cit.>, applied to symplectic billiards.Let us make a simplifying assumption that the billiard curveis infinitely smooth (this can be replaced by sufficiently high finite smoothness) and has everywhere positive curvature. Arbitrary close to the curve , there exist smooth caustics for the symplectic billiard map; the union of these caustics has positive measure. It will be convenientto use an affine parameterization of ; let us recall the pertinent notions, e.g., <cit.>.We first give a geometric definition of the affine length of the curve followed by a more computational one.Given v∈ T_xγ, we consider a parametrization (t) with (0)=x and d/dt(0)=v.Our aim is to define a cubic form B(x,v) on . For this we denote by A(,t) the area bounded by the curve and the segment (0) (). We keep t in the notation as a reminder for the parametrization, see below. This area is of third order in , hence B(x,v):= lim_→ 0A(,t)/^3is well-defined and, of course, independent of the parametrization (as long as (0)=x and d/dt(0)=v.) That B indeed is a cubic form is basically the chain rule. Indeed, replace v by av for some a≠0. If we choose a new parametrization (τ) with t=aτ then at t=τ=0d/dτ=d/dt dt/dτ=av.We now can define two enclosed areas with respect to these two parametrizations: A(, t) and A(,τ). Of course, area is area, i.e.,A(,t)=A(a,τ),and thusB(x,v)=lim_→ 0A(, t)/^3=lim_→ 0A(a,τ)/(a)^3=a^-3lim_→ 0A(a,τ)/^3=a^-3B(x,av).This confirms that B is a cubic form. Therefore, B^1/3 is a 1-form, called an element of affine length. The integral of this form is called the affine length of the curve.More conveniently for computations, a parameterization (t) is affine if ['(t),”(t)]=1 for all t. In this case, ”'(t) = -k(t) '(t), where the positive function k is called the affine curvature. The relation between the affine length parameter t and the Euclidean arc length parameter x is dt = κ^1/3 dx, where κ is the (Euclidean) curvature.Consider an affine parameterization of the billiard curve. A chord (t_1) (t_2) is characterized by the numbers t_1 and :=t_2-t_1. We use (t,) as coordinates on the phase cylinder: t is a cyclic coordinate,is non-negative and bounded above by some function of t (since t_1 ≤ t_2 ≤ t_1^*).Let us describe the billiard map in these coordinates. Let Φ: (t-,) ↦ (t,δ), where δ(t,) is a function on the phase space. We claim that δ(t,) =+ ^3 f(t,),where f is a smooth function. To prove this claim, let δ = a_0 + a_1+ a_2 ^2 + a_3 ^3 + O(^4), where a_i are functions of t. Since the boundary =0 consists of fixed points of the billiard map, a_0=0. By definition of the billiard reflection,[(t+δ)-(t-),γ'(t)]=0,where the brackets denote the determinant made by two vectors.Expand (t+δ) and (t-) in Taylor series up to 4th derivative and substitute to (<ref>):[(δ+)' + (δ^2-^2/2)” + (δ^3+^3/6)”' + (δ^4-^4/24)^””,']=0,where we suppress the argument t from (t) and its derivatives.Since [',”]=1, the quadratic term in (<ref>) yields a_1=1. Since [',”']=0, the cubic term does not give any information. Since ”'=-k, one has ””=-k''-k”, and hence [””,']=k ≠ 0. Therefore the quartic term in (<ref>) yields a_2=0,proving (<ref>). Formula (<ref>) implies that the billiard map is given by the formula:(t,) ↦ (t+,+ ^3 g(t,)),where g is a smooth function. This is an area preserving map, a small perturbation of the integrable map (t,) ↦ (t+, ), satisfying the assumptions ofLazutkin's theorem (Theorem 2 in <cit.>). Applying this theorem concludes the proof.Figure <ref> shows a computer generated phase portrait of the symplectic billiard. It looks like a typical area preserving twist map.A useful tool in the study of billiards is the string construction that recovers a billiard table from its caustic; it produces a 1-parameter family of tables. A similar, area, construction is known for outer billiards, see, e.g., <cit.>. In the context of symplectic billiards, we failed to discover a string construction. §.§ Periodic caustics, Radon curves,and n=3,4 versions of the Ivrii conjecture§.§.§ Distribution on the space of polygonsIn this section we apply the approach to periodic billiard trajectories via exterior differential systems, developed by Baryshnikov and Zharnitsky <cit.> and, independently, by Landsberg <cit.>, and applied to outer billiards in <cit.>; see also <cit.>.Suppose that the symplectic billiard has an invariant curve consisting of n-periodic points. A periodic point is a polygon P=(z_1,z_2,…,z_n) inscribed into the billiard curveand having an extremal area. These n-gons form a 1-parameter family of inscribed polygons, connectingP=(z_1,z_2,…,z_n) to the polygon P'=(z_2,…,z_n,z_1) with cyclically permuted vertices. The area remains constant in this1-parameter family of area-extremal polygons.Assume that n≥ 3. One can reverse the situation: start with an n-gon P and try to construct a billiard curvethat has an invariant curve of n-periodic points including P. The main observation is that the directions ofat the vertices z_i, i=1,…,n are uniquely determined: they are the directions of the diagonals z_i+1 z_i-1 (here and elsewhere the indices are taken in the cyclic order, so that n+1=1).Consider the 2n-dimensional space of n-gons in ^2, and let𝒫_n be its open dense subset given by the condition that z_i+1-z_i-1 is not collinear with z_i+2-z_i (in particular, each vector z_i+1-z_i-1 is non-zero). Restricting the motion of the i-th vertex z_i to the direction of the diagonal z_i+1 z_i-1 defines an n-dimensional distribution 𝒟 on 𝒫_n (its analog for the usual inner billiards was called in <cit.> the Birkhoff distribution). An invariant curve of n-periodic points of the symplecticbilliard gives rise to a curve in 𝒫_n, tangent to 𝒟. We call curves tangent to 𝒟 horizontal curves. We point out that, by the very definition of 𝒫_n, we only consider invariant curves consisting of non-degenerate polygons, i.e., polygons in 𝒫_n.Let A: 𝒫_n → be the algebraic area of a polygon given byA = 1/2∑_i=1^n ω(z_i,z_i+1). The distribution 𝒟 is tangent to the level hypersurfaces of the area function A. The distribution 𝒟 is totally non-integrable on these level hypersurfaces: the tangent space at every point is generated by the vectors fields tangent to 𝒟 and by their first commutators.Let z_i=(p_i,q_i)∈^2n, i=1,…,n. Introduce the vector fieldsv_i = (p_i+1-p_i-1) ∂/∂ p_i + (q_i+1-q_i-1) ∂/∂ q_i,where, as always, the indices are understood cyclically. The fields v_1,…,v_n are linearly independent and they span 𝒟 at every point. Geometrically, it is obvious that the fields v_i preserve the area of a polygon. Analytically, dA = 1/2∑_i=1^n (q_i+1-q_i-1) dp_i - (p_i+1-p_i-1) dq_i.Let θ_i = (q_i+1-q_i-1) dp_i - (p_i+1-p_i-1) dq_i,so that dA = ∑θ_i.Then θ_i (v_j)=0 for all i,j, and it follows that dA vanishes on 𝒟. The common kernel of the 1-forms θ_i is the distribution 𝒟.Next, one has [v_i,v_j]=0 for |i-j|≥ 2, and [v_i+1,v_i] = (p_i+2-p_i)∂/∂ p_i + (p_i+1-p_i-1)∂/∂ p_i+1 + (q_i+2-q_i)∂/∂ q_i + (q_i+1-q_i-1)∂/∂ q_i+1.It follows that θ_i([v_i,v_i+1]) = - θ_i+1 ([v_i,v_i+1]) = (z_i+1-z_i-1,z_i+2-z_i) ≠ 0,where the last inequality is due to the definition of 𝒫_n. The distribution 𝒟 has codimension n-1 on a constant-area hypersurface, and it suffices to show that the rank of the matrix θ_i ([v_j,v_j+1]), i=1,…,n-1, j=1,… n, is n-1. But it follows from (<ref>) that the square submatrix with i,j=1,…,n-1 is upper-triangular with non-zero diagonal entries. This concludes the proof.As in <cit.>, one can draw two conclusions from Theorem <ref>. The first is that one has an extreme flexibility of deforming horizontal curves of the distribution 𝒟, and hence ofbilliard curves, still keeping an invariant curve consisting of n-periodic points (the technical statement is that there is a Hilbert manifold worth of such curves, obtained by deforming a circle and depending on functional parameters). We do not dwell on these technical issues; see <cit.> for a detailed discussion in the usual billiard set-up or <cit.> for the case of outer billiards.However, we present one concrete example for n=4.§.§.§ Radon curves A centrally symmetric convex closed planar curveis called a Radon curve if it has the property depicted in Figure <ref>. Radon curves are unit circles of a particular kind of Minkowski (normed) planes, called Radon planes, that share many features with Euclidean planes. In particular, in a Radon plane, normality (also called the Birkhoff orthogonality) is a symmetric relation. Radon curves have been extensively studied since their introduction by Radon in 1916; see <cit.> for a modern account. The relevance of Radon curves for us is that they haveinvariant circles consisting of 4-periodic points: the parallelogram xyx^*y^* is a symplectic billiard orbit for every x. Conversely, if a centrally symmetric curve has an invariant circle consisting of 4-periodic points, then it is a Radon curve.An obvious example of Radon curve is an ellipse. However Radon curves are very flexible, depending on a functional parameter. For instance, here is a construction of a C^1-smooth Radon curve, see <cit.>. Let a and b be two vectors with [a,b]=1. Connect a and b by a smooth convex curve C_1 that lies in the parallelogram spanned by a and b and is tangent to its sides at the end points. Parameterize C_1 by a parameter t∈[0,T] so that C_1(0)=a, C_1(T)=b, and [C_1(t),C_1'(t)]=1 for all t. The latter condition means that the rate of change of the sectorial area is constant. Differentiating, we obtain [C_1(t),C_1”(t)]=0, hence the acceleration of the curve C_1 is proportional to the position vector.Since C'_1(0) is proportional to b, and [a,b]=1, we have C'_1(0)=b. For the same reason, C'_1(T)=-a.Now consider the curve C_2(t)=C'_1(t). This curve lies in the second quadrant and connectsb with -a. Furthermore, C_2'(0)=C_1”(0) is proportional to C_1(0)=a, and C_2'(T)=C_1”(T) is proportional to C_1(T)=b. Therefore the union of C_1, C_2 and their reflections in the origin is a C^1-smooth curve.Since [C_1(t),C_2(t)]=1, one has [C'_1(t),C_2(t)]+[C_1(t),C'_2(t)]=0. Thus one summand vanishes if and only if so does the other. This impliesthe Radon property. For example, one can combine ℓ_p- and ℓ_q-norms with 1<p,q < ∞ and 1/p+1/q=1, taking C_1 and C_2 to be the quarters of their respective unit circles, see Figure <ref>. It is worth mentioning that, as p→∞, q→ 1, the respective Radon curve tends to an affine-regular hexagon. The above construction, in general, gives rise to C^1-smooth Radon curves. It can obviously adapted to produce C^k or C^∞-smooth examples. §.§.§ Scarcity of periodic pointsThe second consequence of Theorem <ref> is a version of the n=3 and n=4 cases of the Ivrii conjecture. This conjecture asserts that the set of periodic trajectories of the usual billiard has zero measure; a seemingly weaker (but, in fact, equivalent) version is that this set hasempty interior.For period n=3, the Ivrii conjecture is proved, by a number of authors and in different ways, in<cit.>; currently, the best known result is for n=4, see <cit.>. See also <cit.> for periods n=3,4,5,6 for outer billiards. The set of 3-periodic points of the planar symplectic billiard has empty interior.Ifis strictly convex, then the set of 4-periodic points also has empty interior.First we consider the case n=3.The proof is essentially the same as in <cit.> and <cit.>.Let M be the 5-dimensional manifold of triangles of unit area and 𝒟 be the above defined distribution on it. We point out that triangles of unit area lie automatically in the space 𝒫_3, i.e., M⊂𝒫_3, on which 𝒟 is defined. If the set of 3-periodic points of the symplectic billiards map contains an open set, then the respective triangles form a horizontal surface U ⊂ M (i.e., tangent to 𝒟.) Choosing local coordinates in this surface yields a pair of commutingvector fields on U.Since these fields are horizontal, they can be written in the formf_1 v_1+f_2v_2+f_3v_3and g_1 v_1+g_2v_2+g_3v_3, where f_i,g_i are functions and v_i are as in (<ref>). Without loss of generality, assume that f_1≠ 0 and g_2≠ 0. Taking linear combinations, we obtain vector fields w_1=v_1 + fv_3, w_2=v_2+gv_3 with the property that [w_1,w_2] is a linear combination of w_1 and w_2 with functional coefficients. In particular, [w_1,w_2] is horizontal.Let d=[z_2-z_1,z_3-z_1] be twice the area of the triangle z_1z_2z_3. One calculates[w_1,w_2] = [v_1,v_2] -f [v_2,v_3] - g [v_3,v_1] 𝒟,and hence, using (<ref>),θ_1([w_1,w_2]) = d(1+g), θ_2([w_1,w_2]) = -d (1+f).Since d 0, it follows that f=g=-1, and [w_1,w_2] = [v_1,v_2] + [v_2,v_3] + [v_3,v_1] = v_1+v_2+v_3. But this vector field is not a linear combination of w_1=v_1-v_3 and w_2=v_2-v_3. This contradiction concludes the proof of this case.Now consider the case n=4. Let x_1 x_2 x_3 x_4 be a 4-periodic orbit in the curve . Then the tangents T_x_1 and T_x_3 are parallel to the diagonal x_2 x_4, and hence x_3=x_1^*. Likewise, x_4=x_2^*.Let x̅_1 x̅_2 x̅_3 x̅_4 be another 4-periodic orbit, a perturbation of the first one. Assume that point x̅_1 has moved from point x_1 in the positive direction (with respect to the orientation of the curve ). We claim that then the other points x̅_i, i=2,3,4, have also moved in the positive direction. We shall refer to this property as the monotonicity condition.To prove the monotonicity condition, note that sinceis strictly convex, the relation x↦ x^* is an orientation preserving involution. Since x̅_3=x̅_1^*, this point has moved in the positive direction. Hence the segment x̅_1 x̅_3 has turned in the positive sense (compared to x_1 x_3). Since T_x̅_2 is parallel to (x̅_1) (x̅_3), point x̅_2 has moved in the positive direction as well, and so has x̅_4=x̅_2^*.Now we argue similarly to the n=3 case. Let M be the manifold of quadrilaterals of constant area, and let 𝒟 be the respective 4-dimensional distribution. Again we claim that 4-periodic points of the billiards map give rise to polygons which automatically lie in 𝒫_4. Indeed, by the definition of the symplectic billiard map the characteristic directions ofat x_1 and x_3 are parallel to x_2x_4 and similarly the characteristic directions at x_2 and x_4 are parallel to x_1x_3. Now, if we assume that x_1x_3 and x_2x_4 are parallel we find find four points with the same characteristic direction which directly contradicts the strict convexity ofwhich allows for exactly two such point.Therefore, we now assume that there exists a horizontal surface in M. Then one has two linearly independent commuting vector fields, ξ and η, that are lineal combinations of the vector fields v_1,…,v_4 with functional coefficients. Let θ_1,…,θ_4 be as above. It follows from (<ref>) that θ_i([v_i,v_i+1]) = - θ_i+1 ([v_i,v_i+1]) = 1, i=1,…,4,because the determinant involved is twice the oriented area of the quadrilateral.Without loss of generality, assume that the coefficient of v_1 in ξ is non-zero. In the following we distinguish two cases. If the coefficient of v_2 or of v_4 in η is non-zero, one can replace ξ and η by their linear combinations so that, perhaps after reversing the order of indices, one hasξ = v_1 + f v_3 + g v_4, η = v_2 + f̅ v_3 + g̅ v_4, and [ξ,η] is a linear combination of ξ and η with functional coefficients. We refer to this as case 1.If the coefficients of v_2 and of v_4 in η vanish then one can replace ξ and η by their linear combinations so that ξ = v_1, η = v_3. This is case 2.In case 1, [ξ,η]= [v_1,v_2]-g̅ [v_4,v_1] - f [v_2,v_3] + (f g̅ - f̅ g) [v_3,v_4] 𝒟.Evaluating θ_i, i=1,…,4 on this commutator and equating to zero yieldsf = g̅ = -1, f̅ g = 0.Without loss of generality, assume that g = 0. Then ξ = v_1 - v_3 is a vector field tangent to the disc U consisting of 4-periodic orbits. The flow of this field moves points x_1 and x_3 in the opposite directions, contradicting the monotonicity condition.Likewise, in case 2, ξ = v_1. The flow of this field moves point x_1 in the positive direction, but leaves the other points fixed, again contradicting the monotonicity condition. This completes the proof.As follows from the analysis in section <ref>, the monotonicity condition does not hold for 4-periodic orbits in a square: in fact, when such an orbit is perturbed, the points x_1 and x_3 move in the opposite directions, and so do x_2 and x_4. In particular, this shows that strict convexity is necessary. §.§ Interpolating Hamiltonians and area spectrum§.§.§ Area spectrum of symplectic billiardConsider the maximal action, that is, the maximal area of a simple n-gon, inscribed in the billiard curve . Let A_n be this area. We are interested in the asymptotics of A_n as n→∞.The theory of interpolating Hamiltonians, applied to the symplectic billiard, implies that the symplectic billiardmap equals an integrable symplectic map, the time-one map of a Hamiltonian vector field, composed with a smooth symplectic map that fixes the boundary of the phase space point-wise to all orders, see <cit.> and <cit.>; see also <cit.> for an application toouter billiards. In particular, this theory provides an asymptotic expansion of A_n in negative even powers of n:A_n ∼ a_0 + a_1/n^2 + a_2/n^4 + a_3/n^6 + … In our situation, the coefficient a_0 is, of course, the area of the billiard table. The next two coefficients, a_1 and a_2, were found in <cit.>: a_1= L^3/12, a_2=-L^4/240∫_0^L k(t) dt,where t is the affine parameter on the billiard curve, L=∫_ dt is the total affine length, and k the affine curvature of . For comparison, forouter billiards, that is for the circumscribed polygons of the least area, one has a_1 = L^3/24. In affine geometry one also has an isoperimetric inequality for all strictly convex closed curves, with equality only for ellipses, see <cit.>. It readsL^3 ≤ 8π^2 A, where A is the area. We point out that it “goes in the wrong direction” if compared to the usual isoperimetric inequality. Similarly to <cit.>, one has the following immediate consequence.The first two coefficients, a_0 and a_1, make it possible to recognize an ellipse: one always has the inequality3 a_1 ≤ 2π^2 a_0,with equality if and only ofis an ellipse. There is nothing to prove since by (<ref>) the affine isoperimetric inequality (<ref>) and (<ref>) are equivalent.Of course, we can rephrase Theorem <ref> as “one can hear the shape of an ellipse". This leads to an interesting open question. Can one interpret the sequence a_0, a_1, a_2, … really as a spectrum? That is, is there a differential operator whose spectrum is this sequence? For usual billiards this is well-known, see for instance <cit.>.Similarly it would be very interesting to determine the higher terms a_3, a_4, …, even in the case of ellipses, directly. In fact, for ellipses there is a little miracle that a_0 and a_1 determine all other terms. This is the affine isoperimetric inequality.§.§.§ Insecurity of symplectic billiardsA classical billiard is called secure (or has the finite blocking property) if, for every two points A and B in the billiard table, there exists a finite set of points S, such that every billiard trajectory from A to B visits S (so the set S blocks A from B). For example, billiard in a square is secure.It is proved in <cit.> that planar billiards with smooth boundary are not secure. In a nutshell, the argument is as follows. Let A and B be on the convex part of the boundary curve , and consider the shortest n-link billiard trajectory from A to B. For sufficiently large n, no points that are not on the boundary can block this trajectory. Using the theory of interpolating Hamiltonians, one shows that, modulo errors of order 1/n^2, the reflection points are regularly distributed on the arc AB with respect to the measure κ ^2/3 dx, where x is an arc length parameter, and κ is the curvature of . One can introduce a coordinate t so that dt = κ ^2/3 dx and normalize so that t ∈ [0,1] on the arc AB. If the reflection points were regularly distributed, and there were n reflections, then the reflection points would have coordinates m/n, 1≤ m < n. Then it is clear that for every finite set S ⊂ [0,1], there exists n such that S contains no fractions with denominator n: one can take n to be a prime number greater than all the denominators of the rational numbers contained in S.The actual argument uses some number theory (Proposition 2 or the more general Theorem 3.2 of <cit.>) to deal with the errors of order 1/n^2 in the distribution of the reflection points. To recap, no finite set of points on the arc AB can block all billiard trajectories from A to B.One can use the same approach to prove an analogous result for symplectic billiards that we formulate below without proof. The key observation is contained in <cit.>: with respect to the affine parameter t, thevertices of inscribed n-gons of maximal area, that is, the reflection points of the n-link symplectic billiard trajectory, are equidistributedmodulo errors of order 1/n^2. The symplectic billiard inside a smooth strictly convex curve is insecure. More precisely, for every pair of distinct points x,y on the boundary curveand every finite set S⊂, there exists a symplectic billiard trajectory from x to y that avoids S. § POLYGONSIn this section we collect a few simple results on polygonal symplectic billiards. In our opinion, this subject deserves a thorough study.We start with a remark concerning the definition: the symplectic billiard map is not defined for a chord xy if the sides of the polygon that contain these points are parallel. The map is also not defined if point x is a vertex of the polygon. Note that if the end points of a segment of a billiard trajectory are not on parallel sides of a polygon, then the same is true for the next segment of the trajectory. Let P be a convex polygon. The phase spaceof the symplectic billiard is the torus P× P, and it is naturally decomposed into rectangles, the products of pairs of sides of P (if P has pairs of parallel sides then the map is not defined on the respective rectangles). The symplectic billiard map Φ is a piecewise affine map of this phase space. §.§ Regular polygonsThe case of regular polygons is very interesting in both `classical' cases, the inner and the outer billiards. In thecase of inner billiards, the Veech Dichotomy holds: for every direction in a regular polygon, the billiard flow is either periodic or uniquely ergodic, see, e.g., <cit.>, just like in the well known case of a square. In the case of outer billiards, (affine) regular polygons have an intricate fractal orbit structure (except for n=3, 4, 6 when all orbits are periodic) which is analyzed only for n=5,8, see <cit.>.Let P be a regular n-gon whose sides are cyclically labeled 0,1,…, n-1. Assume that the initial segment of a billiard trajectory connects side 0 with side k; here 1≤ k ≤ [(n-1)/2]. We shall call k the rotation number of the trajectory. See Figures <ref>and <ref>. =1.5ex* The rotation number of an orbit is well defined: each link of the orbit connects side i with side i+k.* Let g(n,k) = n/ gcd(n,2k).=1ex* If g(n,k) is even, then the respective billiard orbits are 2g(n,k)-periodic.* If g(n,k) is odd, the orbits are 4g(n,k)-periodic. Let the orbit be x_0,x_1,… with x_0on the side labeled 0, and x_1 on the side labeled k. Due to the dihedral symmetry of the polygon, the projection along the kth side takesthe 0th side to the (2k)th side. Hence x_2 lies on the (2k)th side, and so on. It follows that the rotation number of the orbit is well defined. The segments connecting x_0 to x_2 to x_4, etc. are parallel to the sides of the polygon, and likewise for the segments connecting x_1 to x_3 to x_5, etc. One obtains two polygonal lines, even and odd. Due to the symmetry of a regular polygon P, both are usual billiard trajectories in P, see Figure <ref>. We refer to them as the even and odd (usual) billiard trajectories.Consider a particular case of such a billiard trajectory, the one whose initial segment connects the midpoint of side 0 to the midpoint of side 2k. This billiard trajectory is periodic with period g(n,k). A parallel trajectory is also periodic, with period g(n,k) if g(n,k) is even, and period 2g(n,k) if g(n,k) is odd. Next we observe that the even billiard trajectory depends only on the choice of the point x_0 (and the fixed number k), and the odd one only on the choice of the point x_1. This implies that, generically, these two billiard trajectories are different. Therefore the total number of vertices of the orbit x_0,x_1,… is 2g(n,k) if g(n,k) is even, and4g(n,k) if g(n,k) is odd.Since all triangles are affine equivalent, all orbits in triangles are periodic, generically, of period 12. §.§ TrapezoidsUp to affine transformations, trapezoids form a 1-parameter family; it is convenient to normalize them so that they are isosceles. We assume that the lower horizontal side AB is greater than the upper one, CD. Define the modulus of a trapezoid ABCD to be[ |AB|/|AB|-|CD|].The modulus is a positive integer. Let us call a trapezoid generic if |AB|/(|AB|-|CD|) is not an integer. For example, one may consider a triangle (with |CD|=0) as a trapezoid of modulus one, but it is not generic.All billiard orbits in a trapezoid are periodic. If the modulus of a generic trapezoid is n, then there are three periods: 16n-4, 16n+4, and 16n+12. Let x_0, x_1, x_2, … be a trajectory.Similarly to the proof of Theorem <ref>, we consider the even and odd subsequences x_0, x_2, x_4, … and x_1, x_3, x_5, … separately.Let us define a transformation F of the (boundary of the) trapezoid as follows. Let X be an interior point of a side of the trapezoid. Through X there pass two lines parallel to one of the sides of the trapezoid and intersecting its interior. Choose one of them and move the point X in this direction until it lands on the trapezoidat point Y. Through Y there again pass two lines parallel to one of the sides of the trapezoid. One of them takes Y back to X; choose the other one and move the point Y to the new point Z, etc. We have described the map F: X → Y → Z →….Since every trajectory necessarily visits all sides we take the starting point on the side BC and move horizontally left.After two moves, the point lands on thehorizontal side AB and makes an even number of `bouncing' moves between the horizontal sides, after which one more move in the direction AD takes the point to side BC. Thus we have a return map T to the side BC, see Figure <ref>.There is a break point E on the side BC such that T(E)=C, see Figure <ref> on the right. For X ∈ CE, thenumber of the bouncing moves between the horizontal sides equals 2(n-1), and for X ∈ BE, this number is 2n. Note that the map T consists of an odd number of moves, hence it reverses orientation. In addition, T is a local isometry: the bouncing between the horizontal sides does not distort the length, and the two moves, from a sideAD to AB and from AD to BC, distort the length by reciprocal factors. It follows that T is a reflection in a point, namely, the mid-point of the segment BE or EC.Thus we find the period of point X under the map T: it equals 4n+2 if X ∈ CE, and 4n+6 if X ∈ BE. In what follows we shall call the polygonal lines connecting the consecutive images of points under the map F the guides. A guide that is an 4n+2-gon is called short, and a guide that is an 4n+6-gon is called long.Now we can claim that every orbit of the symplectic billiard map Φ is periodic. Indeed, letx_0 x_1 be the initial segment of an orbit. As we have proved, the point x_0 determines a finite set of possible positions for even-numbered points x_i, and likewise, the point x_1 determines a finite set of possible odd-numbered points x_j. Therefore there are only finitely many possibilities for each segment x_i x_i+1, andhence the orbit is periodic.Now we calculate the periods. Let x_0,x_1,… be an orbit. Consider the guides of points x_0 and x_1. Each one of them can be either short or long. Respectively, one needs to consider three cases: short-short, short-long, and long-long. The combinatorics of the orbits from each case are the same, butdiffer from case to case. This is why one has three different periods.Let us describe the short-short case indetail. After that, we indicate how the other two cases differ. Consider Figure <ref> that depicts a trapezoid of modulus 2: one sees a 28-periodic orbit and the two short guides, each a decagon. Even for small n=2, this picture is already a mess. It is easier to analyze a particular case when the two guides coincide and contain the fixed point of the map T, see Figure <ref>. The effect of considering this special case is making the period four times as small (in other words, when we move the guides apart, the number of arrows quadruples). Let us analyze the combinatorics.The guide has 2n+1 vertices: two on the lateral sides, n on the side AB, and n-1 on the side CD (we use the notation in Figure <ref>). Let us label these points along the guide as shown in Figure <ref> (the odd-numbered points are on side AB and the even-numbered ones on side CD). Then the symplectic billiard orbit is4n-1-periodic and has the following code:(1 R 2 L 3 R 4 L … (2n-1) R L).Quadrupling this number yields the short-short period 16n-4. The long-long case is similar to the short-short one. The short-long case is illustrated in Figure <ref>. Once again, we place the two guides in a special position and label their vertices as shown in the figure. This time the effect of the special position is in halving the number of arrows. The symbolic orbit in the case of n=2 is as follows:(LI R I I L I I I R P 1 Q 2 P 3 Q 4 P 5 Q)and, in general, the period consists of 1+ 2(2n-1) + 2(2n+1) +1 = 8n+2 symbols. Doubling this numbers gives the short-long period 16n+4. It is an interesting problem to describe those polygons in which all symplectic billiard orbits are periodic. Another problem is to determine whether every convex polygon possesses a periodic symplectic billiard orbit. §.§ Stable and unstable periodic trajectoriesIt is known, and easy to prove, that an n-periodic point of a polygonal outer billiard transformation has a neighborhood consisting of periodic points of period n, if n is even, and period 2n, if n is odd, see for instance <cit.>. A similar property holds for odd-periodic orbits of the polygonal symplectic billiard. For the statement of the following Propositions we need a little preparation.Let (x_0,…,x_n-1) be an n-periodic orbit. Let ℓ_i be the line through point x_i parallel to x_i-1 x_i+1. One cannot reconstruct the billiard polygon P from the periodic orbit, but one knows that the side of P through point x_i lies on the line ℓ_i.Let (x̅_0 x̅_1), x̅_0 ∈ℓ_0, x̅_1 ∈ℓ_1, be the initial segment of a nearby billiard trajectory. As in the proof of Theorem <ref>, the evolutions of points x̅_0 and x̅_1 decouple: x̅_0 is projected to line ℓ_2 along ℓ_1, then to ℓ_4 along ℓ_3, etc., and likewise for x̅_1. Denote the projection of ℓ_i-1 to ℓ_i+1 along ℓ_i by π_i.Let α_i+1/2 be the angle between ℓ_i and ℓ_i+1. Let dl_ibe the length element on the line ℓ_i. Then the distortion of length under the affine map π_i is given by the formula:dl_i+1/dl_i-1=sinα_i-1/2/sinα_i+1/2.If n is odd, then the first time that point x̅_0 returns to line ℓ_0 is after n projections, that is, under the affine mapπ_n-1…π_3 π_1. This map is orientation reversing and it is an isometry because∏_i=1^n sinα_i-1/2/sinα_i+1/2 =1.Hence the second iteration of this map is the identity. A similar argument applies to x̅_1. Thus the orbits of both points, x̅_0 and x̅_1 consist of 2n points, and altogether, the billiard orbit is 4n-periodic. Moreover, since the map is the identity, an entire neighborhood of this orbit is periodic with the same period. This proves the following.An n-periodic point in phase space with n being odd has an open neighborhood in phase spave which consists of 4n-periodic orbits.If n is even, then the point x̅_0 returns to the line ℓ_0 after n/2 projections. The derivative of this affine map is∏_iodd mod nsinα_i-1/2/sinα_i+1/2.If this product is not equal to 1 then the fixed point x_0 is isolated. A similar argument applies to point x̅_1, with the derivative given by the formula∏_ieven mod nsinα_i-1/2/sinα_i+1/2,i.e., if this product is not equal to 1 then x_1 is isolated.In conclusion, if the derivatives of first return maps to ℓ_0, resp. to ℓ_1, at the fixed point are both different from 1 then the fixed points are hyperbolic. Hyperbolic fixed points do not vanish under small perturbations of the map, and hence of the polygon. This proves the following.If n≥ 6 is even then, for a generic polygon, an n-periodic point is isolated and is stable in the sense that it does not disappear under a small perturbation of the polygon. § SYMPLECTIC BILLIARD IN SYMPLECTIC SPACE §.§ Definition and continuous limit§.§.§ A precise look at the definitionWe now extend the definition of the symplectic billiard map to linear symplectic space ^2n. Consider a smooth closed hypersurface M⊂^2n bounding a strictly convex domain. On M× M we consider thefunction S:M× M →,(x_1,x_2) ↦ S(x_1,x_2):=ω(x_1,x_2).In analogy to Section <ref>, we define the (open, positive part of) phase space as:={(x_1,x_2)∈ M× M|ω(ν_x_1,ν_x_2)>0}.Here ν is again the outer normal. Since M is strictly convex, the Gauss map M∋ x↦ν_x∈ S^2n-1 is a bijection. Denote by R(x)=ω|_T_xM× T_xM the characteristic direction of M at the point x.The relation R(x_2)⊂ T_x_1M is equivalent to ω(ν_x_1,ν_x_2)=0. In particular, it is symmetric in x_1 and x_2:R(x_2)⊂ T_x_1M ⟺ R(x_1)⊂ T_x_2M.SinceR(x_2)=·(Jν_x_2), where J in the standard complex structure on ^2n≅^n, we observeR(x_2)⊂ T_x_1M ⟺⟨ Jν_x_2,ν_x_1⟩=0 ⟺ω(ν_x_2,ν_x_1)=0,as claimed. The sets ={(x_1,x_2)∈ M× M|ω(ν_x_1,ν_x_2)>0} and {(x_1,x_2)∈ M× M|ω(ν_x_1,ν_x_2)<0} are connected. Consider the Gauss map M∋ x↦ν_x∈ S^2n-1. Thenis mapped to_0:={(a,b)∈ S^2n-1× S^2n-1|ω(a,b)>0}.Clearly b≠± a. Therefore, the simple interpolationa_t:=a+t(b-a)/||a+t(b-a)||∈ S^2n-1, t∈(0,1]and ω(a,a_t)=ω(a,a+t(b-a))/||a+t(b-a)||=tω(a,b)/||a+t(b-a)||shows that we can move the second factor close to a. That is, we can move any point in_0 into a tubular neighborhood of the diagonal which is connected. The same argument works for <. Since the Gauss map is a bijection the results follows.We define the map Φ:→ geometrically similarly to the procedure from Section <ref>, see Figure <ref>. Given (x_1,x_2)∈, there exists a unique point x_3∈ M with(x_1+R(x_2))∩ M ={x_1,x_3}. Moreover, (x_2,x_3)∈ andx_3-x_1=tJν_x_2with t>0.Using the language of contact geometry one, of course, realizes that Jν is positively proportional to the Reeb vector field R⃗ of the standard contact form α:=-dr∘ J on M.Here r denotes the radial coordinate on ^2n. In particular, R⃗ orients (or trivializes) the line field R and the definition of the positive part of the phase space corresponds to conforming to this orientation. Since M is convex, (x_1+R(x_2))∩ M ={x_1,x_3} with possibly x_1=x_3. Since (x_1,x_2)∈, we know that R(x_2)⊄T_x_1M, and therefore, x_1≠ x_3. To show that (x_2,x_3)∈, we argue, as in the 2-dimensional case, by continuity. The statement is clearly true if x_2 is close to x_1. If ω(ν_x_2,ν_x_3)≤0 then, using the connectedness of {(x_1,x_2)∈ M× M|ω(ν_x_1,ν_x_2)<0}, we can move the point x_2 in order to achieve ω(ν_x_2,ν_x_3)=0. The latter is equivalent to R(x_2)⊂ T_x_3M which contradicts x_1≠ x_3.To determine the sign in x_3-x_1=tJν_x_2 consider the set M_x^+:={y∈ M|ω(ν_x,ν_y)>0} and similarly M_x^-. Both are connected since they correspond to hemispheres under the Gauss map. We can rephrase the definition ofas (x_1,x_2)∈⟺ x_1∈ M_x_2^- ⟺ x_2∈ M_x_1^+,and the above discussion asx_3∈ M_x_2^+ . Since x_1,x_2 determine x_3 we writex_3-x_1=t(x_2)Jν_x_2,where we think of x_1 as fixed and x_2 as a variable. First we claim that the sign of t(x_2) does not depend on x_2 as long as (x_1,x_2)∈, i.e., x_2∈ M_x_1^+. Assume that t(x_2) changes sign or vanishes. Since M_x_1^+ is connected, we can always reduce to the latter case, i.e., we find a point x_2 with t(x_2)=0. But this means x_3=x_1, which directly contradicts the very first assertion of this proof. Since the sign of t(x_2) is fixed, we now compute it at a convenient point. For that, let x_2 be the point where ν_x_2=aJν_x_1 with a>0. This point exists and is unique again by the Gauss map being a diffeomorphism. It follows that (x_1,x_2)∈ sinceω(ν_x_1,ν_x_2)=ω(ν_x_1,aJν_x_1)= a·|ν_x_1|^2=a>0.Finally we observe thatx_3-x_1=t(x_2)Jν_x_2=at(x_2)JJν_x_1=-at(x_2)ν_x_1which, again by convexity, forces t(x_2)>0, as ν is the outward normal.By Lemma <ref>, we can again define a mapΦ:→,(x_1,x_2)↦Φ(x_1,x_2):=(x_2,x_3)by the above rule. Now it follows that S(x_1,x_2)=ω(x_1,x_2) is a generating function, indeed:(x_2,x_3)=Φ(x_1,x_2)⟺∂/∂ x_2(S(x_1,x_2)+S(x_2,x_3))=0,by the latter we, of course, mean that ω(x_1,v)+ω(v,x_3)=0∀ v∈ T_x_2M.Since x_1≠ x_3, this is equivalent to the conditionx_3-x_1∈ R(x_2), as needed.We can again extend the map Φ continuously to :={(x_1,x_2)∈ M× M|ω(ν_x_1,ν_x_2)≥0} byΦ(x,x):=(x,x)ω(ν_x_1,ν_x_2)=0⇒Φ(x_1,x_2):=(x_2,x_1).Equation (<ref>) implies that the map Φ preserves the closed 2-form Ω_(x_1,x_2):= ∂^2 S/∂ x_1∂ x_2(x_1,x_2) dx_1∧ dx_2∈Ω^2(M× M).That is, for (u_i,v_i)∈ T_(x,y)M× M,Ω_(x,y)((u_1,v_1),(u_2,v_2))=ω(u_1,v_2)-ω(u_2,v_1). For x,y∈ M, we haveΦ(x,y)=(y,x)⟺ω(ν_x,ν_y)=0 ⟺Ω_(x,y) is degenerate.For (u_1,v_1)≠0 we haveΩ_(x,y) ((u_1,v_1),(u_2,v_2))=0∀ (u_2,v_2)∈ T_(x,y)M× M⟺ω(u_2,v_1)=0 ∀ u_2∈ T_xM and ω(u_1,v_2)=0 ∀ v_2∈ T_yM⟺ R(x)⊂ T_yMor/andR(y)⊂ T_xM⟺ω(ν_x,ν_y)=0⟺Φ(x,y)=(y,x),as claimed. We recall from Lemma <ref> that the relation R(x)⊂ T_yM is symmetric in x and y. Therefore “or” and “and” in the third line are equivalent.* As in the planar case,t strict convexity gives rise to an involution M∋ z↦ z^*∈ M characterized by R(z)=R(z^*).* Similarly to polygons in the plane, one can define symplectic billiards in convex polyhedra in symplectic space. For example, it would be interesting to study a simplex in ^2. §.§.§ Continuous limitConsider a (normal) billiard trajectory inside a convex hypersurface M that makes very small angles with the hypersurface (a grazing trajectory); as the angles tend to zero, one expects such a trajectory to have a geodesic on M as a limit. One observes this at the level of generating functions: the chord length |xy|, x,y∈ M, becomes, in the limit y → x, thefunction L(x,v) = |v| of the tangent vectors v ∈ T_x M. The extremals of the Lagrangian L are non-parameterized geodesics on M.What happens in the continuous limit with the symplectic billiard? Geometrically, one expects the following. Let Z_0,Z_1,Z_2,Z_3,… be an orbit. As the points merge together, the direction of the segment Z_0 Z_2 tends to the characteristic direction R(Z_1), and the direction of Z_1 Z_3 to R(Z_2). Therefore, in the limit, the even-numbered points lie on a characteristic line, and so do the odd-numbered points; and in the limit, the two characteristic lines merge together. Note however that the direction Z_0 Z_1 is not necessarily characteristic, andthe limiting behavior of the directions Z_j Z_j+1 is not determined.One observes the same phenomenon at the level of generating functions. The continuous limit of the generating function ω(x,y), x,y ∈ M, is the Lagrangian L(x,v)=ω(x,v) on the tangent bundle TM. The Euler-Lagrange equation with constraints readsd L_v/dt - L_x = λν,where t is time, ν is a normal, and λ is a Lagrange multiplier. Since L(x,v)=⟨ Jx,v⟩, one has L_v = Jx, L_x = -Jv, and the Euler-Lagrange equation reduces to v = μ(x,v) J(ν). This is the equation of non-parameterized characteristic line on M.Note however that, due to the degeneracy of the Lagrangian L, we end up with a first order differential equation, rather than the second order one.Perhaps a better way to take a continuous limit is to treat the even-numbered and odd-numbered vertices of a symplectic billiard trajectory separately: points Z_0,Z_2,… converge to one curve _0(t) ⊂ M, and points Z_1,Z_3,… to another curve _1(t)⊂ M. In this limit, one obtains the following relation between the curves:R(_1(t)) ∼_0'(t), R(_0(t)) ∼_1'(t),where a∼ b means that the vectors a and b are proportional. Compare to the discussion for symplectic ellipsoids in Section <ref>.The problem of finding pairs of curves on M satisfying (<ref>) is variational.A pair of closed curves (_0(t),_1(t)) on M ⊂^n, satisfying (<ref>), is critical for the functional L(_0,_1)= ∫ω(_0'(t),_1(t)) dt. Integrating by parts, we see that L(_0,_1) = L (_1,_0). Consider a variation of _1 given by the vector field v(t) on M along _1. Having∫ω(_0'(t),v(t)) dt =0for all such v is equivalent to _0'(t) being symplectically orthogonal to T__1(t)M for all t, that is, _0'(t) ∼ R(_1(t)). Due to symmetry of the functional L, the same argument yields _1'(t) ∼ R(_0(t)).It would be interesting to describe pairs of curves on M satisfying (<ref>). A particular solution is _0=_1, a characteristic curve. If M is a unit sphere, then a pair (_0,_1) of geodesic circles related by _1 = J(_0) are all other solutions. §.§ Ellipsoids§.§.§ Symplectic billiard and the usual billiardIt is well known that the billiard ball map in an ellipsoid is completely integrable, see, e.g., <cit.>. In this section we describe a close relation between the symplectic and the usual billiard in ellipsoids that, in particular, implies complete integrability of the former.Consider an ellipsoid in ^2n with Darboux coordinatesx_1,…, x_n, y_1, …, y_n and the symplectic structure ω_0 = ∑ dx_j ∧ dy_j. Applying a linear symplectic transformation and homothety, we may assume that the ellipsoid is given by the equationx_1^2+y_1^2/a_1 + x_2^2+y_2^2/a_2 + … + x_n^2+y_n^2/a_n =1, a_1,…, a_n>0.The diagonal linear transformation x_j ↦√(a_j) x_j, y_j ↦√(a_j) y_j, j=1,…,n, takes the ellipsoid to the unit sphere and transforms the symplectic form to ω = ∑ a_j dx_j ∧ dy_j. We shall consider the symplectic billiard inside the unit sphere S^2n-1 defined by this symplectic form. The characteristic direction at the pointz∈ S^2n-1 is · R(z), where the complex linear operator R:^n →^n is diagonal with the entries ia_1^-1,…, i a_n^-1.Consider the linear map R^-1: ^n →^n. It takes the unit sphere to the ellipsoid E given by the equation|w_1|^2/a_1^2 + |w_2|^2/a_2^2 + … + |w_n|^2/a_n^2 =1,where w_1,…,w_n are complex coordinates in the target space.Let (…, Z_0, Z_1, Z_2,…),be a trajectory of the symplectic billiard map in the unit sphere with respect to the symplectic form ω = ∑ a_j dx_j ∧ dy_j. Then a sequence (…, R^-1(Z_0), R^-1 (Z_2), R^-1 (Z_4), …) is a billiard trajectory in E. Conversely, to a billiard trajectory (…, W_0, W_2, W_4, …) in E there corresponds a unique symplectic billiard trajectory (…, Z_0, Z_1, Z_2,…) in S^2n-1 with Z_0 = R(W_0), Z_2 = R(W_2), etc. Consider the points Z_0, Z_2, Z_4 of a symplectic billiard trajectory. The points Z_1 and Z_3 are uniquely determined by the symplectic billiard reflection law:R(Z_1) = t_1 (Z_2-Z_0), R(Z_3) = t_3 (Z_4-Z_2), t_1 > 0, t_3 > 0.Hence Z_1 = t_1 R^-1(Z_2-Z_0), and the normalization |Z_1|^2=1 uniquely determines t_1; likewise for t_3. ThusZ_1 = R^-1(Z_2-Z_0)/|R^-1(Z_2-Z_0)|,Z_3 = R^-1(Z_4-Z_2)/|R^-1(Z_4-Z_2)|.The symplectic billiard reflection law also implies thatR(Z_2) = t (Z_3-Z_1) = t (R^-1(Z_4-Z_2)/|R^-1(Z_4-Z_2)| + R^-1(Z_0-Z_2)/|R^-1(Z_0-Z_2)|).Set R^-1 (Z_j)=W_j, j=0,2,4, and rewrite the last equation asR^2 (W_2) =t (W_0 - W_2/|W_0 - W_2| + W_4 - W_2/|W_4 - W_2|).Note that the vector R^2 (W_2) is normal to the ellipsoid E given by (<ref>). Therefore equation (<ref>) describes the billiard reflection in E at point W_2 that takesW_0 W_2 to W_2 W_4, as claimed.Conversely, given a segment of a billiard trajectory W_0,W_2,W_4 in E, one defines Z_j = R(W_j), j=0,2,4, Z_1=W_2-W_0/|W_2-W_0|, Z_3=W_4-W_2/|W_4-W_2|.Then equation (<ref>) implies that Z_0,…, Z_4 is a segment of a symplectic billiard trajectory. §.§.§ Continuous versionLet us also present a continuous version of Theorem <ref>. Let A =diag (a_1,a_2,…,a_n) be a diagonal matrix with real positive entries, and letx(t) be a characteristic curve on the ellipsoid ⟨ Ax,x⟩=1. The linear map A^-1/2: ^n →^n takes this curve to a geodesic curve on the ellipsoid ⟨ A^2 y,y⟩=1. If y=A^-1/2x and ⟨ Ax,x⟩=1, then ⟨ A^2 y,y⟩=1. Note that Ax is a normal to the ellipsoid ⟨ Ax,x⟩=1 at the point x. If x(t) is a characteristic, then x' = f JA x where f(t) is a non-vanishing function. The matricies A and J commute. Differentiate:x” =f' JA x + f JA x' = f'/f x' - f^2 A^2 x,hencey” = f'/f y' - f^2 A^2 y.Since A^2 y is a normal to the ellipsoid ⟨ A^2 y,y⟩=1 at the point y, we conclude that y”∈ Span (y',ν_y), that is, y(t) is a geodesic.One can reverse the argument:start with a geodesic on the ellipsoid ⟨ A^2 y,y⟩=1 and construct a pair of curves on the ellipsoid ⟨ Ax,x⟩=1 in the relation (<ref>). We use the same notations as before.Let y(t) be a geodesic on the ellipsoid ⟨ A^2 y,y⟩=1 satisfying y”(t) = f'(t)/f(t) y'(t) +g(t) A^2 y(t),where f and g are some functions with f(t)≠0 for all t. Set x(t)=A^1/2 y(t), x̅(t)= c/f(t) J A^-1 x'(t).Then, for a suitable choice of the constant c, the pair of curves (x,x̅) lie on the ellipsoid ⟨ Ax,x⟩=1 and satisfy (<ref>). We know from the previous proof that x(t) lies on the ellipsoid ⟨ Ax,x⟩=1. Alsox”(t) = f'(t)/f(t) x'(t) +g(t) A^2 x(t). Let h(t)=⟨ x'(t),A^-1 x'(t)⟩. Then h' = 2 ⟨ x”,A^-1 x'⟩ = 2 f'/f⟨ x',A^-1 x'⟩ + 2g ⟨ A^2 x,x'⟩ = 2f'/f h,where we used (<ref>) and the fact that Ax is orthogonal to x'. It follows that h=C f^2 where C is a constant. Then⟨ A x̅, x̅⟩ = c^2/f^2(t)⟨ x'(t),A^-1 x'(t)⟩ = c^2 C,hence, if c=C^-1/2, we have ⟨ A x̅, x̅⟩ =1.It remains to check (<ref>). That R(x̅) ∼ x is clear, since R = JA. Finally,x̅' = -cf'/f^2 J A^-1 x' + c/f(t) J A^-1 x” = -cf'/f^2 J A^-1 x' + c/f J A^-1( f'/f x' +g A^2 x ) ∼ JAx,as needed.Proposition <ref> seems to indicate that the correct continuous limit of symplectic billiard is a pair of curves satisfying (<ref>), as discussed in Section <ref>. The curve y̅ = A^-1/2x̅ is also a geodesic on the ellipsoid ⟨ A^2 y,y⟩=1. Thegeodesics y and y̅ are related by the composition of J and the skew-hodograph transformation, see section 3 in <cit.>.§.§.§ Symplectic billiards and the discrete Neumann systemThe discrete Neumann system (Z_0,Z_1) ↦ (Z_1,Z_2) is a Lagrangian map on the Cartesian square of the unit sphere given by the equationZ_0 + Z_2 = λ A(Z_1),where A is a self-adjoint linear map and λ is a factordetermined by the normalization |Z_2|=1, see <cit.>. Symplectic billiard inside the unit sphere is given by a similar equationZ_2 - Z_0 = t R(Z_1),where R is an anti self-adjoint linear map and t is a suitable factor. Our Theorem <ref> is an analog of Theorem 6 in <cit.> that relates, in a similar way, the discrete Neumann system and the billiard inside an ellipsoid. Thus, similarly to the Neumann system, the symplectic billiard is “a square root" of the billiard system in an ellipsoid (cf. <cit.>). Note however that the latter ellipsoid is not generic: its axes are equal pairwise.§.§.§ IntegralsThe symplectic billiard map possessesa collection of particularly simple integrals described in the following proposition. For a point Z_j ∈ S^2n-1⊂^2n, write Z_j=(x_1j,y_1j,…,x_nj,y_nj).The following functions are integrals of the symplectic billiard map (Z_0,Z_1) ↦ (Z_1,Z_2):I_k (Z_0,Z_1) = x_k0 x_k1 + y_k0 y_k1, k=1,…,n, andJ(Z_0,Z_1) = ⟨ R(Z_0), Z_1⟩.One hasZ_2 = Z_0 + 2 ⟨ R(Z_0),Z_1⟩/|R(Z_1)|^2 R(Z_1).Substitute this Z_2 to J(Z_1,Z_2) and simplify to obtain J(Z_0,Z_1).Likewise, to show that I_k is an integral it suffices to check that I_k(R(Z_1),Z_1) =0. This follows from the fact that R is a diagonal map of ^n that, up to a factor, multiplies eachcoordinate by i. In particular, a trajectory (…, Z_0, Z_1, Z_2, …) of the symplectic billiard in a sphere is an equilateral polygonal line.The integrals I_k occur due to the symmetry of the ellipsoid (or, equivalently, of the operator R): I_k corresponds to the rotational symmetry in kth coordinate complex line via E. Noether's theorem.Let us explain the billiard origin of the integral J.Consider the billiard system in an ellipsoid E given by the equation ⟨ Aw,w⟩ = 1. Thephase space of the billiard consists of the tangent vectors (w,v) with the foot point w∈ E and a unit inward vector v. It is known that the function ⟨ Aw,v⟩ is an integral of the billiard transformation, see, e.g., <cit.>.Consider the pull-back of the integral ⟨ Aw,v⟩ under the map R^-1: ^n →^n. This is a function of (Z_0,Z_2), and since Z_2 is determined by Z_0 and Z_1 via the symplectic billiard reflection, this is also a function of (Z_0,Z_1), a phase point of the symplectic billiard.This function equals ⟨ Z_0, R(Z_1)⟩. One has R^-1 (Z_0) = w, that is, R(w) = Z_0. Likewise, R(v) is positive-proportional to Z_2-Z_0 which, by the symplectic billiard reflection law, is positive-proportional to R(Z_1). Since both v and Z_1 are unit, v=Z_1. Since A= -R^2 and R is anti self-adjoint, we have⟨ Aw,v⟩ = ⟨ Rw,Rv⟩ = ⟨ Z_0,R(Z_1)⟩,as claimed.§.§.§ Low-period orbits Let us also mention a property of low-period orbits of the symplectic billiard in an ellipsoid. By a coordinate subspace in ^n we mean a subspace spanned by any number of the complex coordinate lines. As before, we consider symplectic billiard inside the unit sphere, with the characteristic vectorgiven by a diagonal complex linear operator R with the entries ia_1^-1,…, i a_n^-1. Assume that R is generic in the sense that a_1<a_2<… < a_n. Let Z_1,…,Z_k be a k-periodic symplectic billiard orbit with k<2n. If k is even, then the orbit is contained in a coordinate subspace of dimension at most k, and if k is odd, in a coordinate subspace of dimension at most k-1. Let L be the vector space spanned by Z_j, j=1,…,k. The law of the symplectic billiard reflection implies that R(Z_j) is proportional to Z_j+1-Z_j-1 for all j=1,…,k (the indices are understood cyclically mod k). Therefore L is an invariant subspace of the linear map R.We claim that L is a coordinate subspace. Let _1,…,_n be the complex coordinate lines, and let π_j be the projection of ^n on _j. Consider the projections π_j(L). If π_j(L)=0 for some j, then we may ignore the jth coordinate in what follows. In other words, assume that π_j(L)≠ 0 for all j. The claim now is that L is the whole space ^n.Let Z=(1,z_2, …,z_n) ∈ L. Then a_1^4m R^4m(Z) ∈ L. In the limit m→∞, we obtain the basic vector (1,0,…,0), hence _1 ⊂ L. Factorize by _1 and repeat the argument. It implies that _1 + _2 ⊂ L, and so on. Thus L = ^n, as claimed. To finish the proof, note that a coordinate subspace is always even-dimensional.Proposition <ref> is in perfect agreement with Proposition7.16 of <cit.>: “if the billiard trajectory within an ellipsoid in d-dimensional Euclidean space is periodic with period n ≤ d, then it is placed in one of the (n - 1)-dimensional planes of symmetry of the ellipsoid".§.§.§ Round sphereThe case of a_1=a_2=… =a_n=1, that is, the case when M=S^2n-1 is the unit sphere, is special. In this section, we describe this case in further detail. Let z_0 z_1 be the initial segment of a billiard trajectory, z_0,z_1 ∈ S^2n-1. The symplectic billiard map Φ is given by the formulaz_2 = z_0 + 2 ω (z_0,z_1) J(z_1),where J is multiplication by i. The quantity ω (z_0,z_1) is an integral: ω (z_0,z_1) = ω (z_1,z_2), and we denote it simply by ω.In the phase spacewe have 0 ≤ω, and also ω≤ 1 because M is the unit sphere. Set ω = sinα for 0≤α≤π/2, and let λ_1 = e^iα, λ_2 = - e^-iα. The case ω=0 is special: this is the boundary of the phase space, and the orbits are 2-periodic. The case ω=1 is also special. In this case, the orbit of Φ is 4-periodic. Indeed, if ω (z_0,z_1)=1 then z_1=J(z_0) and hence one obtains the sequence of points z_0↦z_1= J(z_0)↦z_2= -z_0↦z_3=- J(z_0)↦z_4=z_0↦ …The general case is described in the next proposition, where we assume that ω < 1.One hasz_n = λ_1^n-1 - λ_2^n-1/λ_1 - λ_2 z_0 +λ_1^n - λ_2^n/λ_1 - λ_2 z_1.The Φ-orbit of a point lies on the union of two circles. The orbit is periodic if α is π-rational and dense on the two circles otherwise. If α= 2π (p/q), where p/q is in the lowest terms, then the period equals q for even q, and 2q for odd q. Equation (<ref>) is a second order linear recurrence with constant coefficients that generates the sequence z_0,z_1,z_2,… Its solution is a linear combination of two geometric progressions whose denominators are the roots of the characteristic equation λ^2 - 2iωλ -1=0. These roots are distinct, and they are given by the formulaλ_1,2 = i ω±√(1-ω^2),coinciding with(<ref>). Choosing the coefficients of the two geometric progressions to satisfy the initial conditions, one obtains formula (<ref>). The map Φ: (z_0,z_1) ↦ (z_1,z_2) is a complex linear self-map of ^2n, it has the eigen-values λ_1 and λ_2, each with multiplicity n. Writing (z_0,z_1) as a column vector, Φ has the matrix(0 EE 2iω E ),where each entry is an n× n block.Decompose ^2n into the direct sum of the eigen-spaces. Specifically, (a b ) = 1/λ_2-λ_1(b - λ_1 aλ_2 b + a ) +1/λ_1-λ_2(b - λ_2 aλ_1 b + a ),where a and b are column vectors in ^n.Writing a vector accordingly as (u,v), one hasΦ: (u,v) ↦ (e^iα u , - e^-iα v). The orbit of (u,v) lies on the union of two circles (e^it u ,e^-it v) and (e^it u , - e^-it v), where t∈. The orbit is finite if α is π-rational, and dense on the two circles otherwise. Letα = 2πp/q,where p and q are coprime. Assume that v≠ 0. If q is even then the orbit closes up after q iterations, but if q is odd, one needs twice as many, due to the alternating sign of the second component in (<ref>). §.§ Periodic orbitsLet M ⊂^2n be a smooth, strictly convex, closed hypersurface. In this section we discuss periodic trajectories of the symplectic billiard map in M. Given a k-periodic trajectory, one can cyclically permute its vertices or reverse their order; accordingly, wecount the orbits of this dihedral group D_k action.§.§.§ Existence of periodic orbits of any periodFor every k≥ 2, the symplectic billiard map has a k-periodic trajectory. If k is not prime, we do not exclude the case that this trajectory may be multiple, that is, a lower-periodic trajectory, traversed several times. A periodic trajectory is a critical point of the symplectic area function F(z_1,…,z_k)=∑_i=1^k ω(z_i,z_i+1) on k-gons inscribed in M. Due to compactness, this function attains maximum. Let us show that the respective critical point is a genuine periodic trajectory of the symplectic billiard map.Let z_1,…,z_k be a critical polygon for F(z_1,…,z_k)=∑_i=1^k ω(z_i,z_i+1). Then ω(z_i-1,v)+ω(v,z_i+1) =0for all v∈ T_z_i M, i=1,…,n, that is, ω(v,z_i+1-z_i-1) =0.If z_i+1≠ z_i-1 then z_i+1-z_i-1∈ R(z_i), as the symplectic billiard map requires.A problem arises if z_i+1 = z_i-1.Let us show that if z_i+1 = z_i-1 then the value of F(z_1,…,z_k) is not maximal. Indeed, in this case the two terms, ω(z_i-1,z_i) and ω(z_i,z_i+1) cancel each other, and the function F is the symplectic area of the (k-2)-gon z_1,…,z_i-1,z_i+2,…,z_k. Thus it suffices to show that the maximum of the symplectic area F is attained on non-degenerate k-gons (and not on polygons with fewer sides).To do this we show that, given an inscribed polygon P, one can add to it one vertex(and thus two vertices) so that the symplectic areaincreases. Indeed, let ac be a side of P. We want to find a point b∈ M so that ω (a,b) + ω(b,c) > ω (a,c). This is equivalent to saying that the symplectic area of the triangle abc is positive. To achieve this, take an affine symplectic plane through ac and choose point b appropriately on itsintersection curve with M. §.§.§ Periods three and fourThe result of Theorem <ref> is quite weak: we believe, the actual number of periodic orbits is much larger (see, e.g., <cit.> for the usual multi-dimensional billiards). The next theorem concerns small periods.For every M as above, the number of 3-periodic symplectic billiard trajectories is not less than 2n.The same lower bound holds for the number of 4-periodic trajectories. The case k=3 is contained in <cit.> where 3-periodic trajectories of the outer billiard are studied. The function whose critical points are these 3-periodic trajectories is the same: it is the symplectic area of an inscribed triangle. This is also clear geometrically: if ABC is a 3-periodic trajectory of the outer billiard, then the midpoints of the sides of the triangle ABC form a 3-periodic trajectory of the symplectic billiard. Let us consider the case k=4. Let z_1,z_2,z_3,z_4 be a 4-periodic orbit. Then z_3-z_1 ∈ R(z_2) and z_3-z_1 ∈ R(z_4), and likewise, z_4-z_2 ∈ R(z_3) and z_4-z_2 ∈ R(z_1). It follows that R(z_2)=R(z_4) and R(z_1) = R(z_3). Using strict convexity of M, we conclude that z_4=z_2^* and z_3=z_1^*, where the involution z ↦ z^* is as before: the tangent hyperplanes to M at z and z^* are parallel.The (oriented) chords z^* z are called affine diameters of M. The observation made in the preceding paragraph suggests to consider the following function of a pair of oriented affine diametersω(z_1-z_1^*,z_2-z_2^*) = ω(z_1,z_2) + ω(z_2,z_1^*) + ω (z_1^*,z_2^*) + ω (z_2^*,z_1).We shall show that the critical points of this function are 4-periodic orbits of the symplectic billiard. We start with a technical statement.The map φ: M → S^2n-1 to the unit sphere, given by the formulaz ↦z-z^*/|z-z^*|,is a diffeomorphism.One has the following characterization of affine diameters: a chord of a convex body is its affine diameter if and only if it is a longest chord of the body in a given direction, see <cit.>. Since M is strictly convex, for every direction v ∈ S^2n-1, there is a unique affine diameter z z^*. The map v ↦ z is inverse of the map φ, and this map is smooth. Thus the smooth map φ has a smooth inverse map, that is, φ is a diffeomorphism. Next we consider critical points of the function (<ref>). The critical points of the function ω(z_1-z_1^*,z_2-z_2^*) are 4-periodic orbits of the symplectic billiard. Let z∈ M and let v∈ T_z M be a tangent vector. Denote by v^* ∈ T_z^* M the image of v under the differential of the involution z ↦ z^*.Let (z_1,z_2) be a critical point of the function ω(z_1-z_1^*,z_2-z_2^*). Then, for every v∈ T_z_1 M, one has ω(v-v^*,z_2-z_2^*)=0. Note that ψ, composed with normalization to unit vectors, is the map φ of Lemma <ref>, that is, a diffeomorphism. Hence the map dψ(z)v=v-v^*:T_zM→^2n is an injection.It follows that the vector z_2-z_2^* is symplectically orthogonal to the hyperplane T_z_1 M, that is, z_2-z_2^* ∈ R(z_1)=R(z_1^*). The same argument shows that z_1-z_1^* ∈ R(z_2)=R(z_2^*). Hence z_1 z_2 z_1^* z_2^* is an orbit of the symplectic billiard in M. Let 𝒟 be the set of pairs of oriented affine diameters of M. One has an action of the group _4 on this set:(D_1,D_2) ↦ (D_2,-D_1) ↦ (-D_1,-D_2) ↦ (-D_2, D_1) ↦ (D_1,D_2).Let F(D_1,D_2) be the function (<ref>). This function is _4-invariant. Let U ⊂𝒟 be the manifold with boundary given by the inequality F(D_1,D_2) ≥ for a sufficiently small generic positive . The gradient of the function F has the inward direction on the boundary of U, therefore the usual Morse-Lusternik-Schnirelman inequalities for the number of critical points apply. Note that if (D_1,D_2) ∈ U, then D_1 ≠± D_2, and the action of _4 on U is free.We need to describethe topology of the quotient space U/_4. We claim that it is homotopically equivalent to the lens space L = S^2n-1/_4, where _4 acts on the unit sphere in ^n by (z_1,…,z_n) ↦ (iz_1,…,iz_n).Let V be the set of pairs of unit vectors (e_1,e_2) with ω(e_1,e_2) >0. Using Lemma <ref>, we normalize D_1 and D_2 to unit vectors. Thus, at the first step, we get a _4-equivariant retraction of U to V.Next we want to retract V to the set of complex 2-frames (e_1,e_2) satisfying e_2=J e_1. To this end, consider the function ω(e_1,e_2) on V. We claim that the critical points of this function are complexframes with e_2= J e_1. Indeed, let (e_1,e_2) be a critical point. Then, for every v ∈ T_e_1 S^2n-1, one has ω(v,e_2) =0. Hence e_2 = ± J e_1. The condition ω(e_1,e_2) > 0 excludes the minus sign. Thus the gradient of the function ω(e_1,e_2) retracts V to the set of complex 2-frames. This set is S^2n-1 with the action of _4 given by J.It remains to use the Lusternik-Schnirelman lower bound for the number of critical points given by the category of the lens space L. 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"authors": [
"Peter Albers",
"Serge Tabachnikov"
],
"categories": [
"math.DS",
"math.SG"
],
"primary_category": "math.DS",
"published": "20170824132842",
"title": "Introducing symplectic billiards"
} |
copyrightspace 2 3D Binary Signatures Siddharth Srivastava Department of Electrical Engineering Indian Institute of Technology, Delhi, India [email protected] Brejesh Lall Department of Electrical Engineering Indian Institute of Technology, Delhi, India [email protected] ===================================================================================================================================================================================================================================================================================================================== In this paper, we propose a novel binary descriptor for 3D point clouds. The proposed descriptor termed as 3D Binary Signature (3DBS) is motivated from the matching efficiency of the binary descriptors for 2D images. 3DBS describes keypoints from point clouds with a binary vector resulting in extremely fast matching. The method uses keypoints from standard keypoint detectors. The descriptor is built by constructing a Local Reference Frame and aligning a local surface patch accordingly. The local surface patch constitutes of identifying nearest neighbours based upon an angular constraint among them. The points are ordered with respect to the distance from the keypoints. The normals of the ordered pairs of these keypoints are projected on the axes and the relative magnitude is used to assign a binary digit. The vector thus constituted is used as a signature for representing the keypoints. The matching is done by using hamming distance. We show that 3DBS outperforms state of the art descriptors on various evaluation metrics. <ccs2012> <concept> <concept_id>10010147.10010178.10010224.10010245.10010246</concept_id> <concept_desc>Computing methodologies Interest point and salient region detections</concept_desc> <concept_significance>500</concept_significance> </concept> <concept> <concept_id>10010147.10010178.10010224.10010245.10010255</concept_id> <concept_desc>Computing methodologies Matching</concept_desc> <concept_significance>500</concept_significance> </concept> <concept> <concept_id>10010147.10010371</concept_id> <concept_desc>Computing methodologies Computer graphics</concept_desc> <concept_significance>500</concept_significance> </concept> <concept> <concept_id>10010147.10010178.10010224.10010245.10010250</concept_id> <concept_desc>Computing methodologies Object detection</concept_desc> <concept_significance>300</concept_significance> </concept> <concept> <concept_id>10010147.10010178.10010224.10010245.10010251</concept_id> <concept_desc>Computing methodologies Object recognition</concept_desc> <concept_significance>300</concept_significance> </concept> </ccs2012> [500]Computing methodologies Interest point and salient region detections [500]Computing methodologies Matching [500]Computing methodologies Computer graphics [300]Computing methodologies Object detection [300]Computing methodologies Object recognition § INTRODUCTION Humans visualize a three dimensional world. Therefore, the ability to obtain and process information especially from visual senses in three dimensions has always been an exciting and potentially promising area of research. The success of various feature extraction and classification related tasks in 2D image analysis can be attributed to the availability of large scale annotated datasets such as ImageNet <cit.>. With the growing availability of 3D scanning technologies such as Kinect <cit.>, LIDAR etc., the availability of large and quality 3D datasets is also increasing. This has opened many research areas from 2D image analysis such as Object Recognition, Object Retrieval, Object Classification, Segmentation etc. for three dimensional data as well. Local features and Deep Learning based approaches especially Convolutional Neural Networks (CNNs) have been successfully used for solving many research problems involving 2D images <cit.>. Similar to 2D domain, various local featureand deep learning based models have been proposed for 3D point clouds. Although, CNN based techniques have been successfully proposed for 3D object classification <cit.> and recognition <cit.>, but 3D local features are state of the art in many tasks such as 3D object recognition and classification <cit.>, 3D scene reconstruction <cit.>, 3D model retrieval <cit.> etc. Comparisons of descriptors on various benchmarks <cit.> show that high descriptiveness with low storage requirements and reasonable computational complexity depending upon the number of points in the point cloud constitute key requirements for a descriptor. Moreover, for practical applications involving 3D object matching, recognition, classification and reconstruction, the efficiency in feature matching and lower storage requirements become important <cit.>. In order to address these issues, we propose a novel 3D binary descriptor deriving motivation from binary descriptors for 2D images <cit.>. These descriptors have reported to match or outperform traditional SIFT like descriptors <cit.> with significantly lower computational complexity and storage requirements. The proposed technique encodes the differences in the projection of normals into binary vectors. The projection is computed for nearest neighbours of a keypoint and aligned with a local reference frame. The nearest neighbours are chosen based on their angular orientation on a 2D projection plane. The binary descriptor thus generated is matched using Hamming Distance. We show that the proposed binary descriptor outperforms the state of the art on various benchmarks. The closest work to ours is B-SHOT <cit.> which generates a binary vector from the popular Signature of Histograms of Orientations (SHOT) <cit.> descriptor. It quantizes the real valued SHOT descriptor to a binary vector. The major difference in the proposed technique is that the binary vector is generated directly from the point cloud datainstead of first computing a real valued vector and quantizing it which results in an additional overhead. For highlighting the novelty of the technique, we refer to <cit.> where authors categorize 3d descriptors into two classes based on the methodology of generating histograms. First class of descriptors computes histograms either by computing a local reference frame andaccordingly divide the support region spatially. Various spatial distribution measurements are then accumulated into histograms. The second class of descriptors, encodesgeometric attributes such as normals, principal curvatures etc. of the points on the surface in the local neighbourhood the keypoint. The proposed approach is a hybrid of these two approaches. We define a local surface with the nearest neighbours of a keypoint with an angular constraint. Then similar to the first class we use a Local Reference Frame (LRF) for aligning this local surface followed by computing normals and encoding the projective difference among them which is similar to the second class of descriptors. Since the proposed technique does not require computation of histograms, therefore to highlight this difference, we term the extracted descriptors as signatures. In view of the above discussion, the main contributions of this paper are: * We propose a hybrid approach for constructing a highly distinctive yet compact 3D binary descriptor based on encoding differences among normal projections among nearest neighbours of a keypoint. To the best of our knowledge, this is the first work to directly generate 3D descriptors frompoint cloud data. * We show that the proposed 3D Binary Signature (3DBS) outperforms state of the art descriptors on common evaluation benchmarks. The paper is organized as follows: In Section <ref>, we discuss the techniques that are similar to ours and have motivated the construction of the proposed 3D Binary Signature. In Section <ref>, we detail the generation of 3D Binary Signatures (3DBS) followed by experimental results in Section <ref>. Finally, Section <ref> concludes this paper. § RELATED WORK Numerous 3D descriptors have been proposed in the past two decades. Broadly these descriptors fall into two categories. First, the descriptors that encode the geometric attributes of a spatial region as histograms. Second, the descriptors that encode as histograms the statistical properties of a region.Another way of looking at these descriptors is whether a Local Reference Frame (LRF) has been used for forming the descriptor. The descriptors without LRF utilize local statistics such as normals etc. to form the descriptors while those using LRF encode information from spatial distribution or geometric properties of neighbouring points to form the descriptors. Authors in <cit.> provide an exhaustive evaluation of various 3D binary descriptors. In the rest of this section, we discuss a few techniques which are either (i) similar to the proposed technique or (ii) use surface normals for generating the descriptors. We first discuss a few techniques which particularly focus on speed and memory efficiency, followed by robust and general purpose techniques. B-SHOT <cit.> is a 3D binary feature descriptor based on the popular SHOT descriptor. It is generated by quantizing the 352 length real valued SHOT descriptor to 352 length binary vector.The quantization begins by dividing the SHOT descriptor into sets of four values. The relative magnitudes of various combinations of values in each set are compared to assign a corresponding four-bit binary vector.There is a loss of information due to this quantization but it is compensated by significant gains in the descriptor matching efficiency and storage requirement. Signature of Histograms of Orientations (SHOT) <cit.>, which is the basis behind B-SHOT, is based upon encoding into histograms, the surface normals in a spatial distribution. It works by constructing an LRF for a keypoint and the points in the support region are aligned with this LRF. The support region is then divided into several volumes. Each volume results in a local histogram by accumulation of point counts into bins as per the angles between normal of points in the support region and the keypoint. The final descriptor is computed by concatenating all the local histograms.Spin Image (SI) <cit.> descriptor uses the surface normal at a keypoint as Local Reference Axis. Then in-plane and out-plane distances are computed for each point in the support region which is discretized into a 2D array. The final histogram is generated by binning the points from the support region to the 2D array constructed earlier. The dimension of the SI descriptor is equal to the number of bins across each dimension of the in-plane and out-plane space. Rotational Projection Statistics (RoPS) <cit.> constructs an LRF for each keypoint aligning it with the local surface to achieve invariance to transformations. The points on the local surface are rotated around x, y, and z axis and the corresponding support region are further projected onto the coordinate planes (xy, xz and yz). The planes are then divided into several bins and the number of points in each bin is counted.Various statistics are calculated on these bins and they are concatenated to form the final descriptor. 3D Shape Context (3DSC) <cit.> uses normal as Local Reference Axis. Further, it divides the support region into many bins along azimuth, elevation and radial dimensions. The descriptor is generated by weighted accumulation of the points lying in each bin. Unique Shape Context (USC) <cit.> improves the memory requirement and computational efficiency of 3DSC by avoiding computation of multiple descriptors at a keypoint. This is achieved by defining an LRF (same as that used in SHOT) at a keypointand aligns the local surface accordingly. The support region is then divided into bins and the weighted accumulation is performed for the points lying in corresponding bins. § PROPOSED METHODOLOGY In this section, we discuss the methodology for generating the 3D Binary Signatures. Figure <ref> shows the construction pipeline of the proposed binary descriptor. The proposed methodology is abstractly inspired from the descriptor construction pipeline of 2D binary descriptors. Generation of 2D binary descriptors in general involves three steps: a) Choosing a sampling pattern b) Orientation Compensation and c) Selection of Sampling pairs. The sampling pattern in the current methodology is obtained with the help of nearest neighbours while Local Reference Frame compensates for invariance to orientation and other transformation. The sampling pairs are the ordered nearest neighbour pairs which are compared for encoding the difference in projections of the normals to a binary vector. We describe these steps in detail in the following subsections. §.§ Keypoint Detection The keypoints can be detected using any of the standard keypoint detection techniques. From our experiments on various keypoint detectors, namely, Intrinsic Shape Signature (ISS) <cit.>, MeshDoG <cit.>, Keypoint Quality (KPQ) <cit.>, Harris3D <cit.>, Heat Kernel Signatures <cit.> and 3D SIFT <cit.>, it was observed that ISS and Harris3D performedbest in our experiments. Authors in <cit.> present a comprehensive survey on various keypoint detectors. They show that Intrinsic Shape Signatures(ISS) <cit.> demonstrate the highest repeatability while being the most efficient keypoint detector. Moreover, the work in <cit.> showed that selection of an appropriate keypoint detector has a reasonable impact on the performance of a descriptor. Therefore, for further experiments in this work, ISS has been chosen as the keypoint detector. §.§ Nearest Neighbours with Angular Constraint The objective of this criteria is to uniformly distribute the nearest neighbours along the surface of the point cloud. The process is pictorially shown in Figure 2(a). Instead of a spherical support region, we define the local surface for each keypoint with its N nearest neighbours. Nearest neighbours are traditionally computed based on a distance criterion. Authors in <cit.> observe that computing nearest neighbour in such a manner may not be an optimal choice for local surface representation and hence introduce an angle criterion. We therefore use the angle criterion to distribute neighbours around the keypoint p, by projecting the neighbours of the points on a plane ϕ which isbest fitting plane of neighbours of p. The plane ϕ is obtained as a least squares best fitting plane which turns the approximation into a quadratic form and therefore can be solved efficiently. The points thus obtained are sorted as per distance and for each neighbour of p, if the angle between p, its neighbour q_m and its successor q_m+1, does not exceed a threshold θ i.e. ∠q_mpq_m+1≤θ, it is considered added to the set of nearest neighbours. This process is repeated till N neighbours have been found. Angle criterion also helps in avoiding ambiguity in the order of neighbours having the same distance from a keypoint. For instance, if q_1, q_2 and q_3 are neighbours of keypoint p with the following distance relation, ||p-q_2||_2 = ||p- q_3||_2 < ||p-q_1||_2. If ∠q_1pq_2 < ∠ q_1pq_3, q_2 is listed before q_3 avoiding any ambiguity. §.§ Alignment with Local Reference Frame To infuse invariance to various transformations such as translation and rotation along with robustness to noise and clutter, the local surface formed previously is aligned with a Local Reference Frame. Authors in <cit.> show that repeatability of an LRF has a direct impact on the robustness and descriptive ability of a descriptor. Therefore, we construct an LRF using the technique of <cit.> which is the basis for the popular SHOT descriptor. It computes a weighted covariance matrix M given by Eq. <ref> around the keypoint p where the distant points are assigned smaller weights. M = 1/∑_i:d_i ≤ R(R-d_i)∑_i:d_i ≤ R (R-d_i)(p_i - p)(p_i - p)^T where d_i = ||p_i - p||_2 and R is the spherical support region. As the computation of the covariance matrix assumes a spherical support region, we set the radius of the support region, R, as the farthest neighbour of the point p, as given by Eq. <ref>. R = max_1 ≤ i ≤ N ||p_i - p||_2 The region from the previous step is then aligned with this LRF for further processing. We denote this collection of points excluding the keypoint as C. §.§ Descriptor Generation For generating the descriptor, the projection of surface normals on each of the three axes, x, y, z are computed. Let us denote the projection of a point q on axis a where a ∈{x,y,z} as q^a. These are computed for the points in C and are mathematically given by Eq. <ref>. q^x = ⟨q. î⟩, q^y = ⟨q. ĵ⟩, q^z = ⟨q. k̂⟩∀ q∈ C where ⟨.⟩ represents inner product and î, ĵ, k̂ are the unit vector in the direction of x,y and z axes i.e. (1,0,0), (0,1,0) and (0,0,1) respectively. Then the respective projections on each axis for pairs of ordered points (q_m,q_n) s.t. 1 ≤ m,n ≤ N are compared. The ordered pairs are constructed by first sorting the points in C based upon the distance from the keypoint p. Let us denote the sorted set of points by q_1, q_2...q_N such that ||q_m - p||_2 ≤ ||q_n - p||_2 ∀m<nandm,n ∈ (1,N) Then the ordered set of points consist of the pairs (q_m, q_n)s.t. m<n Each ordered pair thus obtained, results in three comparisons corresponding to projections on three axes. Each comparison is assigned a binary bit b_a ∈{0,1} for an axis a based upon the relative magnitude of the projections, as given in Eq. <ref>. b_a= 1,if q_m^a ≥ q_n^q 0,otherwise The comparison for an ordered pair, i, results in a bit vector of size 3 given by Eq. <ref>. b(i) = b_xb_yb_z Finally, the descriptor is constructed by concatenating the binary strings b(i) from all the comparisons resulting in a binary vector of size 3*N*(N-1)/2. The process is pictorially visualized as in Figure 2(b). §.§ Descriptor Matching The descriptors are matched using Hamming Distance. There are efficient algorithms to compute hamming distances especially on CPUs having hardware support to count bits in a word (for ex: POPCNT instruction). In practical applications, there are usually millions of such descriptors, say, D,which are needed to be matched. Performing a linear search over D descriptors would be of the order of O(D^2), which would be computationally very expensive. For binary descriptors from 2D images, the popular way of matching these descriptors is to hash the binary bit vectors using techniques such as Locality Sensitive Hashing (LSH)<cit.> and if needed, perform a linear search in each bucket. But, it is important to note here that the matching complexity of such techniques is affected by two important factors. Firstly, by the number of descriptors in the search space i.e. D and secondly by the length of the binary bit vectors. Typically if the length of the binary vector is greater than 32, various matching techniques use linear scan <cit.>. These limitations are aggravated in the context of 3D point clouds as the number of points could be of the order of millions resulting in a huge number of keypoints. Moreover, the size of the proposed 3D Binary Signature is O(N^2) for each keypoint, where N is the number of nearest neighbour of a keypoint. Therefore, we adopt a fast binary feature matching technique proposed in <cit.>. The technique has been chosen for its low memory footprint and the ability to scale to large datasets. The technique is briefly described below. §.§.§ Fast Feature Matching * Building search tree offeatures: The input data (descriptors) is divided into K clusters by randomly selecting K data points. The remaining data points are assigned to the closest cluster center (similar to k-medoids clustering). If the number of data points in a cluster (DP_C) is above a certain threshold i.e. maximum number of leaf nodes (S_L), then the algorithm is recursively repeated until each cluster has DP_C < S_L. * Search for nearest neighbours: The search is performed in parallel on multiple hierarchical clustering trees. The search begins by recursively exploring the node nearest to the query descriptor while the unexplored nodes are added to a priority queue. Once a node has been completely traversed, the next nearest node from the priority queue is extracted and is again explored recursively. This stopping criteria for this recursive search is based upon a search precision i.e. the fraction of exact neighbours discovered in the total number of returned neighbours. § EXPERIMENTS AND RESULTS In this section, we describe the experimental details and the results thus produced. §.§ Experimental Setup §.§.§ System Specification The experiments were performed on a system having an Intel i7 processor with 128GB RAM and Ubuntu 14.04 Operating System. The implementation has been done in C++ (g++ 4.8) using Point Cloud Library (PCL) <cit.> with OpenMP enabled. §.§.§ Datasets The experiments were performed on four publicly available datasets <cit.> with the details as shown in Table <ref>. The chosen datasets allow us to evaluate the proposed descriptor on various application contexts. The Random Views, Laser Scanner and LIDAR datasets are suited for object recognition. In these datasets the model are full 3D meshes while the scenes are 2.5D views from specific viewpoints. On the other hand, the Retrieval dataset is for 3D shape retrieval where the scenes are built by introducing rigid transformations and noise. Moreover, these datasets also allow for evaluating the descriptor on varying quality of point clouds. The Random Views and Retrieval have been derived from high resolution models. Comparatively, the Laser Scanner and LIDAR datasets can be categorized as medium and low quality respectively. §.§ Performance Evaluation We evaluate the proposed descriptor for descriptiveness, compactness and efficiency against the best performing descriptors in the comparative analysis of <cit.>. These descriptors are Fast Point Feature Histogram (FPFH) <cit.>, Signature of Histogram of Orientations (SHOT) <cit.>, Unique Shape Context (USC) <cit.> and Rotational Projection Statistics (RoPS) <cit.>. As discussed in Section <ref>,Intrinsic Shape Signature (ISS) <cit.> keypoint detector is used. We use the implementations available in Point Cloud Library (PCL) for the keypoint detector ISS and descriptors FPFH, SHOT, USC and RoPS.Unless specified, the default parameters from the corresponding implementation are used for various techniques. The computation of best-fitting plane was performed in parallel on a GPU. For performing the fast feature matching (Section <ref>), the number of parallel search trees has been fixed to 3, branching factor to 16 and maximum leaf nodes to 150. §.§.§ Descriptiveness Descriptiveness is measured using Area under the Precision-Recall curve (PRC). The PRC is generated using the following steps. Firstly, the keypoints are detected from the considered scenes and models. The keypoints are then described using various descriptors. For a fair comparison between the feature matching capability of the floating point and binary descriptors, we index the features using the technique described in Section <ref> which can be applied consistently across floating point and binary descriptors. Due to this, the results of our experiments are slightly different from those reported in<cit.>. However, the relative performance results are still valid even though the absolute numbers change by a small margin (7.2% on an average). The number of matches returned from the search tree depends upon the search precision τ. A linear scan is performed on the matches obtained from the search tree and following <cit.>, a match is considered correct if the matched features belong to the corresponding objects in the scene and model point clouds, and the matched keypoint lies within a small neighbourhood of the ground-truth. This neighbourhood is defined by a sphere of radius 2 mesh resolution (mr) <cit.> and centered at the ground-truth keypoint. The precision and recall are then computed as Precision = #Correct Matches/#Total Matches Recall = #Correct Matches/#Corresponding Matches We then vary τ from 0 to 1 and compute the Area under the PR curve (AUC_PR). The results are shown in Table <ref>. We denote the proposed binary descriptor with N neighbours as 3DBS-N. The ranking for FPFH, USC, RoPS and SHOT are consistent with those reported in <cit.>. It can be observed that 3DBS-32 outperforms other descriptors onRetrieval dataset while 3DBS-64 outperforms on all datasets except LIDAR and Random Views. Moreover, the magnitude of difference between AUC_PR of RoPS and 3DBS with other descriptors is approximately 80%. This observation is important since it shows that the proposed technique has good performance on low resolution point cloud while also performing consistently for various transformations in other datasets. Another observation that can be made is that 3DBS-64 consistently performs better than 3DBS-32. This is expected since 64 nearest neighbours span a larger local surface around a keypoint making the descriptor more robust to occlusion and clutter. This observation is in line with the performance of the other descriptors when an increase in the radius of the support region increases the performance of the descriptor <cit.> §.§.§ Compactness Compactness of a descriptor is a measure to compare descriptors when memory footprint and storage requirements become important. Compactness is given as the Compactness = Average AUC_PR/#Floats_descriptor The number of floats (length) in each of the considered descriptors is shown in Table <ref> with 32 bits per float. The results are graphically shown in Figure 3. It can be seen that 3DBS-32 is highly compact being close to FPFH. Moreover, 3DBS-64 has lower compactness than FPFH and is close to SHOT and RoPS. It would be important to note here that 3DBS can be made more compact by using bit vector compression schemes. Although the compactness measure is provided for completeness of comparative analysis with other popular 3D descriptors, it must be noted that the binary descriptors are not stored in memory as floats for computation. Therefore, compactness does not impact the matching efficiency of the proposed binary descriptor. §.§.§ Efficiency The major advantage of binary descriptors is that they can be matched extremely fast. To evaluate the descriptor matching time, the average matching time of keypoints on LIDAR dataset is reported in Figure <ref>. It can be seen that 3DBS is nearly 10 times as efficient than FPFH while almost three order of magnitude faster than other descriptors. This speed-up can be attributed to two factors. Firstly, the matching of binary vectors is by design faster than matching floating point vectors. Secondly, as discussed in Section <ref>, we leverage the built-in POPCNT instruction in GNU C Compiler providing tremendous computational efficiency in matching binary vectors. It can also be observed that the size of the proposed descriptor quadruples when the number of nearest neighbours doubles. As discussed previously, the number of nearest neighbours impacts the descriptiveness. Therefore, in Figure <ref> we show the descriptor retrieval and matching time by gradually increasing the number of nearest neighbours. As can be seen, the increase in matching time is nearly sublinear when the number of nearest neighbours are doubled. § CONCLUSION A novel 3D binary descriptor termed as 3D Binary Signature was proposed. The descriptor was based upon aligning the local surface as per a Local Reference Frame. The local surface has been identified with a nearest neigbour approach with angular constraint. This is in contrast with previous descriptors where a spherical region was used. The neighbourhood was characterized with projections of surface normals and encoding them as binary vector. We showed that the proposed descriptor outperforms the state of the art methods on various standard datasets. It is highly compact and nearly 3-10 times faster than traditional descriptors, while demonstrating comparable or better descriptiveness. abbrv | http://arxiv.org/abs/1708.07937v1 | {
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§ INTRODUCTIONPhysics has long been thought of as the leader of physical sciences and at the forefront of scientific discovery. However, when we consider the representation of groups in our society, we do not see the same, or even close representations in the field of physics. In fact, physics is one of the least diverse of all physical sciences with respect to gender, underrepresented minorities, and the LGBTQ+ community. The situation has not improved over time, in fact the number of degrees earned by African Americans has decreased in the last decade. Leaders at physics organizations and universities have recognized the demand for improved diversity. This has prompted official diversity statements from the American Physical Society (APS) and the American Association of Physics Teachers (AAPT). More recently was the creation of the APS forum on Diversity and Inclusion slated to begin in 2017. In response to the need for addressing the level of representation on URM in physics, the APS Department of Education and Diversity pioneered the APS Bridge Program, a national effort to increase the number of physics PhDs awarded to underrepresented minority (URM) students, defined by the project as African Americans, Hispanic Americans and Native Americans. There are numerous approaches targeted to improving diversity in physics but in this report, we present the best practices of Bridge Programs in physics and some initial outcomes of institutions that have adopted these programs. § MOTIVATIONAs briefly discussed, the percentage of bachelor's degrees earned by URM in physics continues to be close to the bottom for physical sciences. The number earned by Hispanic Americans has shown an increasing trend over the last ten years, but this growth seems to be proportional to the population growth and thus not a clear indicator of increased participation. For African Americans the percentage of bachelor's degrees earned has been steadily falling over the last 10-15 years as shown Figure <ref>. This has been compounded with the closing of some physics department at Historically Black Colleges and Universities (HBCUs). Hispanic-, African-, and Native -American account for roughly 10-11% and 6% of BS and PhDs degrees earned in physics <cit.>. The general trend for bachelors and doctoral degrees earned by URMs over the last 20 years is shown in Figure <ref>. § BRIDGE PROGRAMSBridge Programs are an innovative approach to addressing the underrepresentation of some groupsin physics. They aim to provide opportunities for students to be successful that may not have had such chances by traditional means. The APS Bridge Program has the goal to increase, within a decade, the percentage of physics PhDs awarded to underrepresented minority students to match the percentage of physics Bachelor's degrees granted to these groups as shown in Figure <ref>. It also aims to develop, evaluate, and document sustainable model bridging experiences that improve the access to and culture of graduate education for all students, with emphasis on those underrepresented in doctoral programs in physics.The project has established six bridge sites in Indiana, Ohio, Florida, and California that provide coursework, research experiences, and substantial mentoring for students who either did not apply to graduate school, or were not admitted through traditional graduate school admissions. These institutions receive funding to develop programs that prepare students to be accepted into a physics doctoral program. Usually, a bridge program accepts students who typically would not gain or did not gain acceptance into a doctoral program for 1-2 years, enhancing their academic and research skills before applying to a doctoral program. There are a few established Bridge Programs that include the Fisk-Vanderbilt bridge program, which was started around 2004, the Imes-Moore Bridge Program in Applied Physics at University of Michigan, Columbia university's bridge program, and the program at MIT. Presently, there are a few more programs under development at institutions such as Princeton and University of Chicago. Physics departments nationwide that participate with the APS Bridge Program can be seen in Figure <ref>. The are currently 93 Member Institutions in 36 states and 19 Partnership Institutions in 14 different states. This demonstrates the paradigm shift in physics graduate education.§.§ Best PracticesSuccessful Bridge Programs have certain key components and best practices in common, some of which include: (1)substantial tenured faculty involvement, (2) adopting a more holistic approach to performing graduate admissions, (3) securing financial support for at least one-year of bridging experience that allows students to resolve major gaps in their undergraduate education. (4) providing multiple mentoring resources, (5) being flexible in coursework, (6) sustained progress monitoring, and (7) looking for a great research match.§ OUTCOMESThe APS bridge program outcomes boast 24% female, and 93% URM participation of students that have been placed on bridge or graduate programs. Of the 93% URM participation, the representation of African-, Hispanic-, and Native - American is 24%, 64%, and 5% respectively. The national average of female and URM representation in all U.S. physics graduate departments is 20% and 6% respectively.A highlight of the APS Bridge Program has been the remarkable retention of the placed students, which is 88%, a stunning 28% higher than the 60% national average in physics graduate programs. Retention is defined as students that enter a PhD program in physics and earn their PhD. The higher retention is indicative of the outlined best practices as other programs have similar results. The Fisk-Vanderbilt bridge program has achieved 80% retention for the students that enter their program, while the University of Michigan's Imes-Moore bridge program in applied physics has surpassed both programs with a 90% retention rate.Fisk University is now recognized as the highest producer of black students with master's degrees in physics, and Vanderbilt has become a leading producer of doctoral degrees in astronomy, physics and materials science earned by URM students <cit.>.§ CONCLUSIONSThe outcomes discussed in section <ref> clearly demonstrate that Bridge Programs can be implemented as an approach to increase diversity in student enrollment, retention, and ultimately the entire physics community. The APS Bridge Program has been influential on placing over 100 students into Bridge or graduate programs in physics while retaining 88% of those placed. The project achievement since starting is shown in Figure <ref>. Leaders at physics departments that have Bridge Programs have acknowledged that the bridge programs best practices are beneficial to all student not only those enrolled as bridge students. If we care about science, then we must also care about scientists and Bridge Programs can provide the access and opportunity for students to be successful.§ ACKNOWLEDGEMENTSThis material is based upon work supported by the National Science Foundation under Grant No. 1143070. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. 99 AIP American Institute of Physics Statistical Research Center (www.aip.org/statistics) APS American Physical Society (www.aps.org/programs/education/statistics) APS_Connections APS Bridge Program Connections newsletter, Vol. 3,2016 FV K. Powell; Higher education: On the lookout for true grit Nature Vol. 504, 2013 | http://arxiv.org/abs/1708.08011v1 | {
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§ INTRODUCTION Anisotropy in the arrival directions of cosmic rays has been detected by many ground-based observatories over the past several decades <cit.>. A large-scale (dipole) anisotropy is consistently seen with an amplitude of 10^-4-10^-3 at energies from hundreds of GeV to ~5 PeV. The dipole amplitude and phase both exhibit an energy dependence which has recently been resolved up to PeV energies <cit.>. There are a number of possible explanations for the observed dipole: a nearby source of cosmic rays which dominates the large-scale diffusion gradient or the influence of a strong, local magnetic field are both expected to create a dipole anisotropy <cit.>. It has recently been shown that a combination of these effects can explain the observed dipole behavior <cit.>. A dipole due to the relative motion of the solar system through the isotropic cosmic-ray plasma known as the Compton-Getting effect is also expected, though none has been detected to date <cit.>. Small-scale anisotropy at the level of 10^-5-10^-4 has been seen at angular scales as small as 10 <cit.>, which may be due to the scattering of cosmic rays off of local turbulent magnetic fields <cit.>. In order to measure the anisotropy at these small amplitudes, data-driven methods are used to estimate the exposure, which needs to be known to an accuracy of < 10^-3. When using these data-driven analysis methods, ground-based observatories lose all sensitivity to the declination component of the anisotropy <cit.>. The measured anisotropy is thus a projection onto right ascension. The Fermi Large Area Telescope (LAT) is sensitive to anisotropy in both right ascension and declination, adding new information to the cosmic-ray anisotropy mystery. Furthermore, it sees the entire sky, which is not possible with any single ground-based instrument.§ FERMI LARGE AREA TELESCOPE The Fermi Large Area Telescope (LAT) is a pair-conversion gamma-ray telescope onboard the Fermi Gamma-ray Space Telescope. It was launched in June of 2008, entered full data taking operations in August 2008, and has accumulated > 8 years of data. The LAT is optimized to detect gamma rays via the electron-positron pairs created when they interact with nuclei inside the detector. Because it is essentially a charged-particle detector, it has the ability to study other charged cosmic rays such as protons, electrons, and positrons <cit.>.The LAT is a survey instrument with a wide, 2.4 sr instantaneous field of view that surveys the entire sky every 2 orbits or ~3 hours. It is able to see the entire sky by rocking N/S from zenith on each successive orbit. This rocking angle was 35 in the first 13 months of science operations and increased to 50 thereafter <cit.>. The increased rocking angle allows the LAT to explore larger zenith angles, but also introduces more background from the Earth limb. This is taken into account in the following sections to avoid contamination of the signal due to geomagnetic deflection.There are three main sub-systems in the LAT: the tracker (TKR), calorimeter (CAL), and anti-coincidence detector (ACD). The tracker is composed of 18 layers of alternating x and y silicon strips for direction reconstruction and tungsten foils to promote the conversion of gamma rays to e+/e-. The calorimeter consists of eight layers of CsI crystals in a hodoscopic arrangement which allows the LAT to measure a 3D profile of each shower. The anti-coincidence detector consists of segmented scintillator panels wrapped around the outside of the detector to identify charged particles <cit.>.§ EVENT SELECTIONThe data set used in this analysis contains 160 million events recorded between December 2008 and December 2016. We developed a custom event selection in order to construct a proton data set suitable for an anisotropy search. The event selection begins with the same selection used to measure the Fermi-LAT proton spectrum. A set of minimum quality cuts is imposed first to select events with accurately reconstructed variables. All events must have a track in the tracker, traverse a path length of at least four radiation lengths in the calorimeter, and deposit > 20 GeV of energy in the calorimeter. The onboard filter accepts all events satisfying this last requirement, which ensures that the filter efficiently selects high-energy protons. Additionally, there are two minimum quality requirements on the direction reconstruction; one to select events with good angular resolution and another to reject back-entering events. The tracker and anti-coincidence detector are used to remove cosmic rays with charge > 1. The average pulse height in the tracker and charge deposited in the anti-coincidence detector are strongly correlated with the charge of cosmic rays due to the Z^2 dependence of ionization loss (Figure <ref>). The minimum energy requirement in the ACD also rejects photons to well below the electron flux level <cit.>. We use a classifier developed for the Fermi LAT cosmic-ray electron/positron analyses <cit.> to separate protons from charged leptons. The use of this classifier restricts the energy range of the analysis to 78 GeV - 9.8 TeV in reconstructed energy (i.e. E_reco). The residual contamination from both heavy ions and leptons is estimated to be < 1% after applying these cuts. It is worth mentioning that the classifier will also reject residual photons since it classifies events based on the hadronic or leptonic nature of their showers.The LAT has several observing strategies in addition to standard survey mode. For example, there was modified pointing to study the Galactic center that began in December 2013. In order to remove non-standard observing modes such as this, we require the LAT to be in standard survey mode when each event was recorded. Additionally, we require the rocking angle to be < 52. This removes time periods when the earth is directly in the field of view.The calorimeter, which is 0.5 interaction lengths at normal incidence, does not fully contain hadronic showers due to its shallow depth. This results in wide energy resolution of protons measured by the LAT, which can be seen in the energy response matrix in Figure <ref>. The tail of the energy distribution extends down to tens of GeV, allowing a non-negligible population of low-energy protons into the data set when cutting on E_reco. These low-energy cosmic rays are deflected by large angles in the geomagnetic field and can create a false positive in an anisotropy search. The influence of the geomagnetic field is greatest towards the earth's horizon. In order to remove cosmic rays arriving from this region, we impose a set of energy-dependent cuts on the LAT's off-axis angle (theta angle of an event relative to the LAT's boresight). This effectively cuts on the field of view, requiring the off-axis angle to be smaller at lower energies. These cuts were developed by performing an analysis in horizontal coordinates (i.e. altitude and azimuth), while remaining blind to the full data set in the equatorial frame. The geomagnetic signal should be maximal in the horizontal frame, while any residual signal is expected to be smeared out in the equatorial frame.§ ANALYSIS METHODS§.§ Reference Maps The target sensitivity for this analysis is a dipole amplitude <10^-3, which is much smaller than the uncertainty in the detector's effective area. Therefore we cannot rely on simulation to estimate the exposure and directly calculate the intensity of cosmic rays from each direction on the sky. The standard method instead is to create a reference map which represents the detector response to an isotropic sky, i.e. the best estimate of what an isotropic sky would look like given the detector's effective area and pointing history. The anisotropy search is then performed by comparing the measured sky map to the reference map.There are a handful of data-driven methods used to create reference maps. We adopt a time-averaged method similar to direct integration <cit.> and the rate-based method used in <cit.>. The prescription for the construction of the reference map goes as follows: we first divide the data set into time bins of one year. The use of an integer number of years for the entire analysis removes any contamination from the solar dipole <cit.>, which is a Compton-Getting-like dipole created by the earth's rotation around the sun. This dipole averages to zero in the equatorial frame over the course of one year. We then calculate the time-averaged rate, R_avg, in each time bin and construct a PDF of the angular direction of events in the instrument frame, P(θ,ϕ), from the actual distribution of detected events. For each second of live time, the expected number of events is determined by R_avg and P(θ,ϕ) gives the probability of detecting an event from any particular direction. Given this information and the instrument pointing history, we then construct the expected sky map for each year. Any underlying anisotropy on the sky is contained in the instantaneous values of R(t) and P(θ,ϕ,t), but averaged out in the time-averaged quantities. This effectively smears the anisotropy over the entire sky in the reference map.We create 25 independent reference maps in eight energy bins from 78 GeV < E_reco < 9.8 TeV and average them in order to beat down statistical fluctuations in the reference map. The maps are then binned cumulatively in energy to create the final energy-integrated maps for the anisotropy search; the results of which are reported as a function of minimum energy. §.§ Angular power spectrum We perform a spherical harmonic analysis of the relative intensity between the measured sky map and reference map to search for anisotropy. The relative intensity is shown in Equation <ref> where n_i represents data map counts and μ_i represents reference map counts in the i^th pixel. δ I_i(α_i,δ_i) =n_i(α_i,δ_i) - μ_i(α_i,δ_i)/μ_i(α_i,δ_i) The relative intensity is decomposed into spherical harmonics (<ref>) and the angular power spectrum (<ref>) is calculated using the anafast algorithm in HEALPix <cit.>.In principle, the angular power spectrum is sensitive to anisostropies at all angular scales (angular scale of each multipole ≈180/l). We calculate the angular power spectrum up to l=30 which corresponds to an angular scale of ≈6/l. â_lm = 4π/N_pix∑_i=1^N_pix Y^*_lm(π-δ_i,α_i)δ I_i(α_i,δ_i)Ĉ_l = 1/2l+1∑_m=-l^l|â_lm^2| In order to determine the significance of the measured power at each multipole, detailed knowledge of the power spectrum under the null hypothesis (isotropic sky) is necessary. This can be calculated directly from the Poisson noise in the map. This white noise level due to finite statistics in the map is given by: C_N = 4 π/N_pix^2∑_i=1^N_pix (n_i/μ_i^2 + n_i^2/μ_i^3*N_maps) where N_maps is the number of independent reference maps that are created and averaged. Equation <ref> accounts for pixel-to-pixel variations in the sky maps due to non-uniform exposure. With the knowledge of C_N, one can calculate the white noise power at each l from the PDF of C_l which follows a χ^2_2l+1 distribution <cit.>. Any excess or deficit of the measured angular power compared to the isotropic expectation at a particular multipole indicates an anisotropy at that angular scale. The dipole anisotropy is typically described by the amplitude and phase of a harmonic function. The amplitude of the dipole can be calculated directly from the angular power at l=1: δ=3√(C_1/4π) § RESULTSSky maps for all events with E_reco>78 GeV in equatorial coordinates are shown in Figure <ref>. The structure seen in these maps is due to the exposure of the instrument. The exposure varies by ~60% across the sky and is greater towards the poles due to the LAT's rocking profile. It is clear to see that the exposure structure in the sky map is maintained in the reference map. Figure <ref> shows the relative intensity and Li & Ma significance maps <cit.> created from the sky maps in Figure <ref>. The relative intensity map exhibits larger fluctuations at equatorial latitudes due to the lower exposure in this region compared to the poles. These pixels are smoothed out in the significance map.The relative intensity was folded through the angular power spectrum analysis as described in Section <ref>. The angular power spectrum for E_reco>78 GeV is shown in Figure <ref>. The colored bands represent the 68%, 95%, and 99.7% central regions of expected distribution under the null hypothesis. The bands were calculated directly from the PDF of C_l under the null hypothesis (isotropic sky). The observed power at l=1 (dipole) is intriguing at ~2.5σ, but not significant enough to rule out the null hypothesis. The data point at l=2 (quadrupole) is significant, but its interpretation is under investigation. Many of the systematics of this analysis exist in the quadrupole due to the LAT's equatorial orbit and further work is being done to rule those out as the explanation of this excess.As described in Section <ref>, the analysis is performed on the data set as a function of minimum energy, which yields an angular power spectrum for each minimum energy. We calculate the amplitude of the dipole on the sky from these power spectra using <ref>. Observed dipole amplitudes as a function of minimum energy are plotted in Figure <ref>. The 68% and 99.7% bands show the expected distribution of measurements under the null hypothesis (isotropic sky). All measured amplitudes are consistent with an isotropic sky. Given the non-detection of a significant dipole, we calculate 90% CL upper limits which can be seen on the right in Figure <ref>. The upper limits were calculated using the likelihood ratio procedure described in <cit.>. Since the ground-based measurements in Figure <ref> are only sensitive to the anisotropy in right ascension, the Fermi LAT upper limits are the strongest to date on the declination dependence of the dipole. § CONCLUSION We analyzed 160 million cosmic-ray protons detected by the Fermi LAT over the course of eight years and searched for anisotropy in their arrival directions. We did not observe a significant anisotropy at any angular scale except in the quadrupole and further work is underway in order to rule out systematics as its source. We calculated upper limits on the dipole amplitude as a function of minimum energy. Due to the limited reconstruction capabilities of ground-based experiments, these are the strongest limits to date on the declination dependence of the dipole. § ACKNOWLEDGMENTSThe Fermi-LAT Collaboration acknowledges support for LAT development, operation and data analysis from NASA and DOE (United States), CEA/Irfu and IN2P3/CNRS (France), ASI and INFN (Italy), MEXT, KEK, and JAXA (Japan), and the K.A. Wallenberg Foundation, the Swedish Research Council and the National Space Board (Sweden). Science analysis support in the operations phase from INAF (Italy) and CNES (France) is also gratefully acknowledged. ICRC | http://arxiv.org/abs/1708.07796v1 | {
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"Justin Vandenbroucke"
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"published": "20170825160628",
"title": "A Search for Cosmic-ray Proton Anisotropy with the Fermi Large Area Telescope"
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=1 | http://arxiv.org/abs/1708.07769v2 | {
"authors": [
"Christian Fidler",
"Thomas Tram",
"Cornelius Rampf",
"Robert Crittenden",
"Kazuya Koyama",
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algorithm algorithm[1] #1 =1.5em ##1##2 ##1# ##2 ##12# ##1 ##1# ##1 ##1 ##1 1 2 3 4 | http://arxiv.org/abs/1708.08761v2 | {
"authors": [
"João Batista S. de Oliveira"
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"published": "20170824220500",
"title": "Problemas com modelagem algébrica: três exemplos para semestres iniciais"
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http://arxiv.org/abs/1708.07811v1 | {
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"title": "Channel reciprocity calibration in TDD hybrid beamforming massive MIMO systems"
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Self-Energy LLC, Scottsdale, Arizona 85259, USA [email protected] Superconducting amplifiers are key components of modern quantum information circuits. To minimize information loss and reduce oscillations a tapered impedance transformer of new design is needed at the input/output for compliance with other 50 Ω components. We show that an optimal tapered transformer of length ℓ, joining amplifier to input line, can be constructed using a variational principle applied to the linearized Riccati equation describing the voltage reflection coefficient of the taper. For an incident signal of frequency ω_o the variational solution results in an infinite set of equivalent optimal transformers, each with the same form for the reflection coefficient, each able to eliminate input-line reflections. For the special case of optimal lossless transformers, the group velocity v_g is shown to be constant, with characteristic impedance dependent on frequency ω_c=π v_g / ℓ. While these solutions inhibit input-line reflections only for frequency ω_o, a subset of optimal lossless transformers with ω_o significantly detuned from ω_c does exhibit a wide bandpass. Specifically, by choosing ω_o→ 0 (ω_o→∞), we obtain a subset of optimal low-pass (high-pass) lossless tapers with bandwidth (0,∼ω_c) ((∼ω_c,∞)). From the subset of solutions we derive both the wide-band low-pass and high-pass transformers, and we discuss the extent to which they can be realized given fabrication constraints. In particular, we demonstrate the superior reflection response of our high-pass transformer when compared to other taper designs. Our results have application to amplifier, transceiver, and other components sensitive to impedance mismatch. Variational theory of the tapered impedance transformer R. P. Erickson December 30, 2023 =======================================================§ INTRODUCTIONTapered impedance transformers are ubiquitous and represent important components in the design of microwave to millimeter transmissions lines, low-noise amplifiers, and transceivers. They address the impedance mismatch between input and load-bearing lines that tends to induce undesirable signal reflections, power loss, and poor signal-to-noise characteristics. Over the years different types of taper designs have been presented that reduce the size of reflections for MHz frequencies and greater. The most notable ones are discussed in standard engineering texts, such as Pozar.<cit.> For example, the Klopfenstein taper<cit.> is a particularly popular high-pass design used extensively in modern high-speed electronics because the maximum-reflection parameter of the model can be set below the threshold of reflection sensitivity of the application. A typical parameter setting is a maximum reflection of 2% within the range of frequencies of the passband.[Specifically, in the Klopfenstein model of Ref (Klopfenstein1956), as it is discussed in Ref. (Pozar2012), by a maximum reflection of 2% in the passband we mean the maximum reflection parameter Γ_m is set to Γ_m=0.02.] Since, in many cases, a reflection coefficient of 5% to 10% is tolerable, the Klopfenstein taper is more than adequate. In a few instances, such as superconducting nonlinear parametric amplifiers,<cit.> with added noise the order of 1 photon, small reflections of a few percent are readily amplified when the device is operated at a pump frequency of about 1-10 GHz, which can lead to poor signal-to-noise output.In particular, superconducting amplifiers are a key component of modern quantum information circuits, and represent the motivation for the present study. In order to obtain high performance, e.g., quantum-limited noise and wide bandwidth, it is not always possible to maintain a 50 Ω environment due to high kinetic inductance<cit.> and Josephson junction capacitances.<cit.> In order to minimize information loss and reduce oscillations in the circuit it is therefore necessary to use a tapered impedance transformer on the input and/or output of these devices in order to be compliant with other 50 Ω components. Due to absence of loss, these circuits are challenging because non-ideal behavior such as small reflections can quickly build up and cause undesirable oscillations and sharp frequency-dependent response. This is particularly applicable to the case of traveling-wave amplifiers,<cit.> which require extremely wide bandwidth and smooth response to support multiple idlers and various high-frequency pumps.Use of the Klopfenstein and other high-pass tapers to address impedance mismatch in these instances is not ideal. For example, in the case of the Klopfenstein taper the maximum ripple within the passband is designed to be constant, but cannot be made sufficiently small to reduce corresponding ripple in the signal gain of the traveling-wave amplifier, whereas tapers like the triangular and exponential designs described in Pozar<cit.> actually perform somewhat better in this regard.[D. P. Pappas, private communication.] Presumably this is because these latter designs exhibit asymptotic drop-off of ripple across the exploitable passband; while the reflection drop-off is no better than 1/ω^2 with increasing frequency ω, it is sufficient to enable these latter designs to outperform the Klopfenstein taper at the higher pump and idler frequencies encountered in the traveling-wave amplifier. Furthermore, the claim that any of these aforementioned tapers are optimal is not rigorously justified from a mathematical standpoint. An optimal solution must be determined via comparison with all other reasonable possibilities, which therefore suggests that these tapers can be improved upon.In the construction of an optimal impedance taper the input-line reflection coefficient is the measurable quantity of interest that must be engineered to zero. In fact, as we show in Appendix <ref>, a discontinuity exists between the zero reflection coefficient of the input line and the reflection coefficient just inside the taper. In the early work of Collin<cit.> a high-pass taper was derived from an N-section quarter-wave cascaded transformer structure by taking the continuum limit of N→∞. In the Klopfenstein taper design, the characteristic impedance of the taper was deduced from the Fourier transform of the reflection coefficient just inside the taper, which in turn was formed from an ansatz consistent with the results of Collin.<cit.> In both of these earlier treatments the assumption is that an optimal high-pass taper may be constructed via a procedure that minimizes reflections at every cross section along the length of the taper. A better approach is to treat the minimization of the input-line reflections via a variational principle, wherein the optimal reflection coefficient as a function of position along the length of the taper follows from the variational procedure itself. This later approach implicitly compares taper profiles and selects only those that are truly optimal with respect to input-line reflections.In the discussion that follows, we apply a variational approach to obtain a mathematically accurate definition of the optimal impedance transformer for the general case of an input line connected to a load-bearing transmission line. We have in mind a waveguide in place of the transmission line, but our method is also valid for the case when there is a terminated load after the tapered transformer. Specifically, for a signal of frequency ω_o incident to the transformer, we vary the magnitude of the input-line voltage reflections to obtain the form of the taper corresponding to the absolute minimum of reflections, i.e., zero input-line reflections. This is accomplished without a priori assumption about continuity of the reflection coefficient across the input-line/taper interface. We show that there is an infinite number of equivalent tapers, each with its own reflection coefficient within the transformer, which share the common property of having zero reflections in the input line, specifically for the input frequency ω_o. Because this set of equivalent solutions arises from the absolute minimum of a variation principle, it is therefore justified to refer to each as an optimal impedance transformer for signals of frequency ω_o.One problem with an optimal impedance transformer as defined above is that, in general, it is only applicable to the specific frequency ω_o for which it is designed. This is a consequence of the Bode-Fano criterion,<cit.> which prevents perfectly zero reflections over an extended range of frequencies. Nevertheless, any impedance transformer design must have a significant bandpass to be of practical use. To resolve this narrow-bandwidth issue we calculate the reflection response along the input line for a signal of arbitrary frequency ω incident upon an optimal lossless transformer of design frequency ω_o obtained from the variational principle.By construction the input-line reflection response will be precisely zero only when ω=ω_o. However, as we show, the reflection response is dependent on a characteristic frequency ω_c=π v_g/ℓ, where v_g is the constant transformer group velocity and ℓ is the transformer length. Only when ω≅ω_c, for which the wavelength of the incident signal is about 2ℓ, does the magnitude of the reflection response along the input line become larger. When a transformer design is considered for which ω_o is significantly detuned from ω_c then the magnitude of the reflection response becomes very small, over an extended range of frequencies ω, provided ω is closer to ω_o than ω_c.Thus, an optimal wide-bandwidth lossless impedance transformer is an optimal transformer whose design frequency ω_o is significantly detuned from its characteristic frequency ω_c. Moreover, if we take the limit ω_o→ 0 of the transformer design then the detuning establishes a bandpass 0<ω≲ω_c, corresponding to a low-pass transformer. Conversely, if we take the limit ω_o→∞ then detuning implies a bandpass ω_c≲ω<∞, corresponding to a high-pass transformer. In this way, an optimal wide-bandwidth impedance transformer, of either low-pass or high-pass character, is obtained from an optimal transformer design by taking the appropriate limit of the design frequency ω_o, which is as far from ω_c as possible.In what follows we apply our variational approach to obtain the reflection coefficient, propagation coefficient, and characteristic impedance of a set of optimal impedance transformers, assuming an incident signal of frequency ω_o. For arbitrary frequency ω we then calculate the input-line reflection response of an optimal lossless transformer of design frequency ω_o. By detuning design frequency ω_o from characteristic frequency ω_c, we derive the reflection response and characteristic impedance of both the low-pass (ω_o→ 0) and high-pass (ω_o→∞) cases. In each case, by examining the asymptotic behavior of the reflection response as ω→ω_o, we show how to obtain a lossless transformer with negligible reflections over a wide bandpass. In particular, for the high-pass case, we compare our solution to other transformer designs.<cit.> We also discuss the extent to which both of our solutions can be realized given fabrication constraints.§ THE VARIATIONAL THEORYWe consider an input line with forward-traveling wave of frequency ω_o incident upon a transmission line possessing a tapered interval 0≤ x≤ℓ, as in Fig. <ref>(a). The boundary between input and transmission lines is at x=0. The characteristic impedance of the input line is 𝒵_1(ω_o), whereas in the tapered interval it is 𝒵(x,ω_o). As x→ℓ, 𝒵(x,ω_o) smoothly transitions to 𝒵_2(ω_o) of the transmission-line interior, i.e., 𝒵(ℓ,ω_o)=𝒵_2(ω_o). The measurable reflection coefficient of the traveling wave within the input line is ρ_1(ω_o) while inside the taper it is ρ(x,ω_o). In Appendix <ref> we derive the Riccati differential equation of Walker and Wax<cit.> satisfied by ρ(x,ω_o); importantly, we include the accompanying boundary conditions. Assuming |ρ(x,ω_o)|^2 ≪ 1, the equation may be expressed in linearized form as∂/∂ xρ(x,ω_o) ≅ 2 γ(x,ω_o) ρ(x,ω_o) - 1/2∂/∂ xlog𝒵(x,ω_o) ; |ρ(x,ω_o)|^2 ≪ 1 ,with γ(x,ω_o) representing the propagation coefficient. From Eqs. (<ref>) and (<ref>), the accompanying boundary conditions at x=0 and x=ℓ are, respectively,ρ_1(ω_o) = 𝒵(0,ω_o) [ 1 + ρ(0,ω_o) ] - 𝒵_1 [ 1 - ρ(0,ω_o) ] /𝒵(0,ω_o) [ 1 + ρ(0,ω_o) ] + 𝒵_1 [ 1 - ρ(0,ω_o) ], ρ(ℓ,ω_o)=0 .To determine the optimal tapered impedance transformer we minimize |ρ_1(ω_o)|. Ultimately, we want to set |ρ_1(ω_o)|=0, but important information can be obtained through a formal minimization. From Eq. (<ref>) we see that |ρ_1(ω_o)| can be reduced if a proper choice of 𝒵(0,ω_o) and ρ(0,ω_o) is made. This is possible because, unlike boundary conditions 𝒵(ℓ,ω_o)=𝒵_2(ω_o) and ρ(ℓ,ω_o)=0, the boundary values of 𝒵(0,ω_o) and ρ(0,ω_o) are not firmly established. In Eq. (<ref>), ρ(x,ω_o) depends on both γ(x,ω_o) and 𝒵(x,ω_o), which may be expressed asγ(x,ω_o)=√(Z(x,ω_o)Y(x,ω_o)) ,𝒵(x,ω_o)=√(Z(x,ω_o)/Y(x,ω_o)) , respectively, where Z(x,ω_o)=R(x)+iω_o L(x) is the series impedance per unit length and Y(x,ω_o)=G(x)+iω_o C(x) is the shunt admittance per unit length. As described in greater detail in Appendix <ref>, R(x), L(x), G(x), and C(x) are the unit-length resistance, inductance, conductance, and capacitance, respectively, of the tapered region, as introduced via the ladder-type transmission-line model depicted in Fig. <ref>(b). Our goal of minimizing |ρ_1(ω_o)| may be described as adjusting the underlying values of independent variables Z(x,ω_o) and Y(x,ω_o) at each point x of the taper, subject to fixed boundary conditions of ρ(ℓ,ω_o)=0 and 𝒵(ℓ,ω_o)=𝒵_2(ω_o), such that the values of γ(x,ω_o) and 𝒵(x,ω_o), and thus also ρ(x,ω_o), may be altered to give boundary values ρ(0,ω_o) and 𝒵(0,ω_o) that make |ρ_1(ω_o)| as small as possible. Panels (c) and (d) of Fig. <ref> depict this idea, where the various curves of ρ(x,ω_o) and 𝒵(x,ω_o) are generated as Z(x,ω_o) and Y(x,ω_o) are varied. The solid curves have corresponding boundary values for ρ(0,ω_o) and 𝒵(0,ω_o) that render |ρ_1(ω_o)| minimal. The underlying Z(x,ω_o) and Y(x,ω_o) of these solid curves define the optimal impedance transformer for the incident traveling wave of frequency ω_o.To optimize |ρ_1(ω_o)| in the manner of variational calculus,<cit.> we vary both underlying functions Z(x,ω_o) and Y(x,ω_o), holding endpoint x=ℓ fixed but allowing endpoint x=0 to float. Specifically, we let Z(x,ω_o)→ Z(x,ω_o) + δ Z(x,ω_o) and Y(x,ω_o)→ Y(x,ω_o) + δ Y(x,ω_o) where δ Z(ℓ,ω_o)=0 and δ Y(ℓ,ω_o)=0 but δ Z(0,ω_o) 0 and δ Y(0,ω_o) 0. Since the variation of |ρ_1(ω_o)| is equivalent to varying ρ_1(ω_o), we vary ρ_1(ω_o) such that from Eq. (<ref>) we have δρ_1(ω_o) ≅𝒵_1(ω_o) 𝒵(0,ω_o)[ 4δρ(0,ω_o)+ δ Z(0,ω_o) / Z(0,ω_o) . - δ Y(0,ω_o) / Y(0,ω_o) . ]×{𝒵(0,ω_o) [ 1 + ρ(0,ω_o) ] + 𝒵_1(ω_o)[ 1 - ρ(0,ω_o) ] }^-2; |ρ(0,ω_o)|^2 ≪ 1 . To obtain δρ(0,ω_o) in Eq. (<ref>), we first integrate Eq. (<ref>) so that ρ(0,ω_o) may be expressed as a functional of Z(x,ω_o), Y(x,ω_o) and their derivatives, viz.ρ(0,ω_o)[Z,Z';Y,Y'] =∫_0^ℓ[ -2 ρ(x,ω_o) √(Z(x,ω_o) Y(x,ω_o))+ Z'(x,ω_o)/4Z(x,ω_o)- Y'(x,ω_o)/4Y(x,ω_o)] dx ,where ρ(x,ω_o) is implicitly a function of Z(x,ω_o), Y(x,ω_o) and derivatives. Then, using Eq. (<ref>) to obtain δρ(0,ω_o), and subsequently setting δρ_1(ω_o)=0 in Eq. (<ref>), we arrive at two Euler-Lagrange equations that may be expressed as√(Y/Z) ρ + 2 √(Z Y) ∂ρ/∂ Z - 2 d/d x[ √(Z Y)∂ρ/∂ Z'] = 0 , √(Z/Y) ρ + 2 √(Z Y) ∂ρ/∂ Y - 2 d/d x[ √(Z Y)∂ρ/∂ Y'] = 0 ,with boundary conditions at x=0 given by[ √(Z Y)∂ρ/∂ Z'] _x=0 = 0 , [ √(Z Y)∂ρ/∂ Y'] _x=0 = 0 . Normally, one would solve Eqs. (<ref>) through (<ref>) for Z(x,ω_o) and Y(x,ω_o). However, since ρ(x,ω_o) is unknown, a more fruitful approach is to solve for ρ(x,ω_o) in terms of Z(x,ω_o) and Y(x,ω_o). This givesρ(x,ω_o) = [ γ(0,ω_o) / γ(x,ω_o) ][ A(x,ω_o) + B(ω_o) Z'(x,ω_o) + C(ω_o) Y'(x,ω_o) ] ,where A(x,ω_o) is an arbitrary function of x and B(ω_o) and C(ω_o) are independent of x. If we apply this form to the boundary conditions of Eq. (<ref>) we further see that B(ω_o)=0 and C(ω_o)=0. Thus, the reflection coefficient within the optimal transformer is of the form ρ(x,ω_o) = [ γ(0,ω_o) / γ(x,ω_o) ] A(x,ω_o). The arbitrariness of A(x,ω_o) allows us immediately to set |ρ_1(ω_o)|=0, or equivalently, ρ_1(ω_o)=0. In this way, our optimization always results in the absolute minimum, |ρ_1(ω_o)|=0. Specifically, we set ρ_1(ω_o)=0 in Eq. (<ref>) and solve for ρ(0,ω_o), obtaining a second boundary condition on ρ(x,ω_o), viz.ρ(0,ω_o) = 𝒵_1(ω_o) - 𝒵(0,ω_o) /𝒵_1(ω_o) + 𝒵(0,ω_o),in addition to ρ(ℓ,ω_o)=0. Then, rescaling A(x,ω_o) by letting A(x,ω_o)=ρ(0,ω_o) f(x,ω_o), where f(x,ω_o) is now the arbitrary function of x, and incorporating the additional boundary condition of Eq. (<ref>), we now haveρ(x,ω_o) = [ 𝒵_1(ω_o) - 𝒵(0,ω_o) /𝒵_1(ω_o) + 𝒵(0,ω_o) ][ γ(0,ω_o) /γ(x,ω_o) ] f(x,ω_o);f(0,ω_o) = 1, f(ℓ,ω_o) = 0 ,where imposition of f(0,ω_o)=1 and f(ℓ,ω_o)=0 ensures the boundary conditions of ρ(x,ω_o) are satisfied. Assuming a physically meaningful solution for Z(x,ω_o) and Y(x,ω_o) exists, Eq. (<ref>) is the optimal form of ρ(x,ω_o) for any impedance transformer that eliminates input-line reflections, specifically for frequency ω_o.Given the optimal form of ρ(x,ω_o), we may determine the optimal taper design by substituting Eq. (<ref>) into Eq. (<ref>). This yields the constraint2 γ(0,ω_o) [ 𝒵_1(ω_o) - 𝒵(0,ω_o) ] f(x,ω_o) + γ(x,ω_o) { 4 γ(0,ω_o) [ 𝒵_1(ω_o) - 𝒵(0,ω_o) ] ∫_x^ℓ f(x') dx'+ [ 𝒵_1(ω_o) + 𝒵(0,ω_o) ] log𝒵(x,ω_o)/𝒵_2(ω_o)} = 0 . This equation determines optimal γ(x,ω_o) and 𝒵(x,ω_o), or equivalently, via Eqs. (<ref>), optimal Z(x,ω_o) and Y(x,ω_o).[Equation (<ref>) involves functions of complex numbers, so it actually represents two real-valued equations. For an optimal lossless transformer this is sufficient to determine the characteristic impedance and the propagation coefficient because the former is a real number while the later is an imaginary number. However, for the case of an optimal lossy taper, it may not be possible to specify the optimal characteristic impedance and/or propagation coefficient definitively, unless additional information about the nature of the lossiness, i.e., the transmission-line series resistance and shunt conductance, is also provided.] Since f(x,ω_o) is an arbitrary function of x, subject to f(0,ω_o)=1 and f(ℓ,ω_o)=0, there are an infinite number of optimal transformer designs, where each design is characterized by ω_o and f(x,ω_o).[Here f(x,ω_o) resembles a gauge transformation on scalar field ρ(x,ω_o) where |ρ_1(ω_o)|=0 is invariant.]Several key points of our variational approach are: * The boundary condition of Eq. (<ref>) illustrates the discontinuity of the voltage reflection coefficient across the x=0 interface.* A variation of the input line reflection coefficient, ρ_1(ω_o), with endpoint x=0 not fixed, is developed from Eq. (<ref>), applicable to a specific frequency ω_o of incident forward-traveling wave.* The variation of ρ_1(ω_o) employs the variation of Z(x,ω_o) and Y(x,ω_o) at each point x along the length of the taper, which necessitates the variation of ρ(x,ω_o), the voltage reflection coefficient within the taper, since it depends on both Z(x,ω_o) and Y(x,ω_o).* The optimization of ρ_1(ω_o) implies an optimal form for ρ(x,ω_o), as given by Eq. (<ref>), where f(x,ω_o) is subject to the stated boundary conditions; any other proposed form for ρ(x,ω_o) will not correspond to an optimal ρ_1(ω_o). * The optimal form of Z(x,ω_o) and Y(x,ω_o), or equivalently γ(x,ω_o) and 𝒵(x,ω_o), are obtained from Eq. (<ref>).In what follows we narrow our discussion to the case of a lossless optimal impedance transformer.§ THE OPTIMAL LOSSLESS IMPEDANCE TRANSFORMERFor the remainder of our discussion we assume input line, transformer, and transmission line are lossless, such that𝒵_1 = √(L_1/C_1) , 𝒵_2 = √(L_2/C_2) ,𝒵(x,ω_o) = √(L(x,ω_o) / C(x,ω_o) .) ,γ(x,ω_o)=iω_o √(L(x,ω_o) C(x,ω_o)) , where L(x,ω_o) and C(x,ω_o) are, respectively, the transformer inductance and capacitance per unit length at x. Also, L_1, C_1 and L_2, C_2 are the constant values of the input line and interior of the transmission line, respectively, with L(ℓ,ω_o)=L_2 and C(ℓ,ω_o)=C_2. In Appendix <ref> we apply Eqs. (<ref>) to Eq. (<ref>) to obtain the solution of the optimal lossless impedance transformer. An important result, which follows from Eq. (<ref>) and Eq. (<ref>), is that f(x,ω_o) of an optimal lossless transformer may be written in the formf(x,ω_o) = d/dx g(x) + 2π i ( ω_o/ω_c) g(x) / ℓ. ,where ω_c=π/( ℓ√(L_2 C_2)) is a frequency characteristic of the impedance transformer, and g(x) is a real-valued function of x satisfying the boundary conditionsg(0) = 0 ,g(ℓ) = 0 , . d/dx g(x) | _x=0 = 1 , . d/dx g(x) | _x=ℓ = 0 .In this way, any design of optimal lossless impedance transformer is characterized by both ω_o and g(x).Also in our analysis of Appendix <ref>, via Eq. (<ref>), the propagation coefficient γ(x,ω_o)=iω_o√(L(x)C(x)) of the lossless transformer is found to be a constant value, i.e., γ(x,ω_o)=iω_o√(L_2C_2). In the literature<cit.> one typically approaches the problem of finding an optimal impedance transformer by assuming γ(x,ω_o) is independent of x. Here, using our variational approach, the optimal propagation coefficient is indeed independent of x, the same value as in the interior of the transmission line. Since the dispersion frequency of the transformer is Ω(k)=k/√(L_2C_2), where k is a wavenumber, the group velocity is v_g=∂Ω(k)/∂ k=1/√(L_2C_2), which is constant for any g(x), allowing us to write ω_c=π v_g/ℓ. Similarly, for an incident signal of frequency ω=Ω(k), the phase velocity is v_p=ω/k=v_g. Thus, we may interpret τ_c=2π/ω_c=2ℓ/v_g as the time it takes for a signal of frequency ω=ω_c to traverse the length of the transformer and be reflected back to the input-line/taper interface. In this case the signal wavelength is λ=v_p τ_c=2ℓ since v_p=v_g.Via Eq. (<ref>), 𝒵(x,ω_o) of the optimal lossless transformer may be expressed in terms of g(x) as 𝒵(x,ω_o) = 𝒵_2 exp{[ log𝒵(0,ω_o) /𝒵_2 ][d g(x) / dx . - ( 2πω_o / ω_c . ) ^2 ∫_x^ℓ g(x') / ℓ^2 . dx' / 1 - ( 2πω_o / ω_c . ) ^2 ∫_0^ℓ g(x') / ℓ^2 . dx' ] } ,where, via Eq. (<ref>), 𝒵(0,ω_o) is a root of2 [ 𝒵_1 - 𝒵(0,ω_o) ][ 1 - ( 2πω_o / ω_c . ) ^2 1/ℓ^2∫_0^ℓ g(x') dx' ]+ [ 𝒵_1 + 𝒵(0,ω_o) ] log𝒵(0,ω_o)/𝒵_2 = 0 . Since γ(x,ω_o)=iω_o√(L_2C_2) is independent of x, the inductance per unit length and the capacitance per unit length can be obtained from L(x,ω_o) = √(L_2 C_2) 𝒵(x,ω_o) and C(x,ω_o) = √(L_2 C_2) / 𝒵(x,ω_o), respectively.We summarize the key points of the optimal lossless transformer as follows: * For the optimal lossless transformer f(x,ω_o) takes the particular form of Eq. (<ref>), as shown in Appendix <ref>, where g(x) is a real-valued function satisfying the boundary conditions of Eq. (<ref>).* Therefore a general result is that the design of the optimal lossless transformer is defined by the choice of g(x) and ω_o.* The frequency ω_c=π v_g/ℓ is characteristic of the geometry and material composition of the optimal lossless taper, and is therefore more or less fixed, save for some ability to change the geometry, such as via the transformer length ℓ.* An important result of our variational approach applied to the lossless transformer is that the optimal propagation coefficient of this case, γ(x,ω_o), is a constant in x, i.e., γ(x,ω_o)=iω_o√(L_2C_2), as demonstrated in Appendix <ref>. * The optimal characteristic impedance of the lossless taper, 𝒵(x,ω_o), is given by Eq. (<ref>), where the boundary value 𝒵(0,ω_o) is determined from the transcendental Eq. (<ref>).§.§ Reflection Response of the Optimal Lossless Impedance TransformerAs mentioned earlier, the optimal lossless impedance transformer of Eqs. (<ref>) and (<ref>) guarantees zero input-line reflections only for an incident signal corresponding to frequency ω_o. To determine reflection-response characteristics of the transformer at any other frequency ω we first solve Eq. (<ref>) for frequency ω, instead of frequency ω_o, but with characteristic impedance given by Eq. (<ref>). The result is the reflection coefficient of the transformer with respect to ω, which after some algebra may be expressed asρ(x;ω,ω_o) = ρ(x,ω_o)+ 2 π^2 [ log𝒵(0,ω_o)/𝒵_2]×( ω_o^2 - ω^2/ω_c^2) ∫_x^ℓ g(x') e^2π i ( ω/ ω_c . ) ( x - x' ) / ℓ. / ℓ^2 . dx' / 1 - ( 2πω_o / ω_c . ) ^2 ∫_0^ℓ g(x') / ℓ^2 . dx',whereρ(x,ω_o) = [ 𝒵_1 - 𝒵(0,ω_o) /𝒵_1 + 𝒵(0,ω_o) ][ d/dx g(x) + 2π i ( ω_o/ω_c) g(x) / ℓ] .Equation (<ref>) is just the lossless limit of the reflection coefficient of Eq. (<ref>), with f(x,ω_o) given by Eq. (<ref>) and γ(x,ω_o) independent of x. For arbitrary ω the input-line reflection coefficient is still of the form of Eq. (<ref>), but nowρ_1(ω,ω_o) =𝒵(0,ω_o) [ 1 + ρ(0;ω,ω_o) ] - 𝒵_1 [ 1 - ρ(0;ω,ω_o) ] /𝒵(0,ω_o) [ 1 + ρ(0;ω,ω_o) ] + 𝒵_1 [ 1 - ρ(0;ω,ω_o) ],where ρ(0;ω,ω_o) is Eq. (<ref>) at x=0. Also, setting x=0 in Eq. (<ref>) yields ρ(0,ω_o), the same as Eq. (<ref>). Thus, substituting the x=0 form of Eq. (<ref>) into Eq. (<ref>), and making use of Eq. (<ref>) for ρ(0,ω_o), we may write ρ_1(ω,ω_o)=[ρ(0;ω,ω_o)-ρ(0,ω_o)]/[1-ρ(0;ω_o,ω_o)ρ(0,ω_o)]. Assuming |ρ(0;ω,ω_o)ρ(0,ω_o)|≪ 1, this may be approximated as ρ_1(ω,ω_o)≅ρ(0;ω,ω_o)-ρ(0,ω_o), or from Eq. (<ref>), we haveρ_1(ω,ω_o) ≅ 2 π^2 [ log𝒵(0,ω_o)/𝒵_2] ( ω_o^2 - ω^2/ω_c^2)∫_0^ℓ g(x) e^-2π i ( ω/ ω_c . ) x / ℓ. / ℓ^2 . dx / 1 - ( 2πω_o / ω_c . ) ^2 ∫_0^ℓ g(x) / ℓ^2 . dx.This is the small-reflection response of a traveling wave of frequency ω incident upon an optimal impedance transformer of design defined by ω_o and g(x). Equation (<ref>) may be used to analyze the passband characteristics of any optimal lossless impedance transformer with characteristic impedance of form given by Eqs (<ref>) and (<ref>). As examples, we next consider the two wide-band cases delineated by the transformer characteristic frequency ω_c.[Another case of interest is that of setting the design frequency to ω_o=ω_c in Eq. (<ref>). This eliminates large reflection response attributable to ω_c and results in a V-like passband centered about frequency ω_o. We consider the definition of this type of notch filter to be a subject for follow-on research.] Several important points the reader should keep in mind as we investigate these cases are: * The optimal lossless transformer of design choice g(x) and ω_o has reflection coefficient within the taper, ρ(x,ω_o), given by Eq. (<ref>).* We may use the reflection-response function, ρ_1(ω,ω_o) of Eq. (<ref>), to determine choices for g(x) and ω_o that exhibit specific wide-bandwidth characteristics.* By construction ρ_1(ω_o,ω_o)=0 in Eq. (<ref>), and this is the only point of the ρ_1(ω,ω_o) versus ω curve where ρ_1(ω,ω_o) is precisely zero. §.§ The Wide-Band High-Pass Lossless Impedance TransformerRecall from our introductory remarks that a wide-band high-pass impedance transformer can be constructed if the design frequency ω_o is detuned from the characteristic frequency ω_c such that ω_o→∞. This statement is made more explicit by examination of the reflection response ρ_1(ω,ω_o) of Eq. (<ref>). First, note that ρ_1(ω_o,ω_o)=0 by construction; this is always the case, no matter the value of ω_o. If we let ω_o→∞ then we haveρ^(HP)_1(ω) = lim_ω_o→∞ρ_1(ω,ω_o) ≅-1/2( log𝒵_1/𝒵_2)∫_0^ℓ g(x) e^-2π i ( ω/ ω_c . ) x / ℓ. dx /∫_0^ℓ g(x) dx,where, from Eq. (<ref>), we note 𝒵(0,ω_o)→𝒵_1 as ω_o→∞. This all but eliminates ω_c from the expression of the reflection response, except for its appearance in the Fourier-like integral of the numerator, where it acts to delineate the region of high reflections, ρ^(HP)_1(0)=(1/2)log( 𝒵_2 / 𝒵_1 ), from that of low reflections, ρ^(HP)_1(∞)=0. An estimate of the passband is to describe it as the range of frequencies ω such that ω_c/2π<ω<∞. From Eq. (<ref>), the corresponding characteristic impedance is𝒵^(HP)(x) =lim_ω_o→∞𝒵(x,ω_o) =𝒵_2 exp[ ( log𝒵_1 /𝒵_2 )∫_x^ℓ g(x') dx' /∫_0^ℓ g(x') dx' ].The form of Eqs. (<ref>) and (<ref>) indicates that g(x) of the lossless high-pass transformer may also be expressed as g(x)=αd log𝒵^(HP)(x) / dx, where α is a constant. From Eq. (<ref>), we have an alternative expression of the boundary conditions of g(x), viz.. d/dxlog𝒵^(HP)(x)| _x=0 = 0 , . d/dxlog𝒵^(HP)(x)| _x=ℓ = 0 ,. d^2/dx^2log𝒵^(HP)(x)| _x=0 = 1/α , . d^2/dx^2log𝒵^(HP)(x)| _x=ℓ = 0 .As mentioned, the Fourier-transform-like integral in the numerator of Eq. (<ref>) defines the high-pass frequency regime to be ω_c/2π<ω<∞. The choice of g(x) determines the extent to which | ρ^(HP)_1(ω) | is negligible over this interval. A good choice for g(x) may be obtained by examining the asymptotic expansion of the integral for 2πω≫ω_c. After repeated integration by parts N times this expansion may be expressed as∫_0^ℓ g(x) e^-2π i ( ω/ ω_c . ) x / ℓ. dx = ( ω_c/ 2πω) ^2 +∑^N_n=3(i ω_c / 2πω) ^n[ g^(n-1)(ℓ) e^-2π i ( ω/ ω_c . )- g^(n-1)(0) ]+ (i ω_c / 2πω) ^N ∫_0^ℓ g^(N)(x)e^-2π i ( ω/ ω_c . ) x / ℓ. dx ,where g^(n)(x) refers to the n-th derivative with respect to x of g(x), and the integral on the right side of the equation is the expansion remainder. A good high-pass transformer is one with a g(x) that eliminates the second-order term in ω_c/(2πω) on the right side of Eq. (<ref>); a better one also eliminates the third-order term, and so on.In Appendix <ref> we demonstrate how a 2N-degree polynomial choice for g(x) can be used to eliminate terms of Eq. (<ref>) to order N in ω_c/(2πω). Using this 2N-degree polynomial as our g(x), we obtain a reflection response ρ^(HP)_1(ω,N) and characteristic impedance 𝒵^(HP)(x,N) expressable asρ^(HP)_1(ω,N) ≅-Γ( 3/2+N ) /√(π)( log𝒵_1/𝒵_2)( 2ω_c/πω) ^Nj_N ( πω/ω_c ) e^-π i ( ω / ω_c ) , 𝒵^(HP)(x,N) = 𝒵_2exp[ ( log𝒵_1 /𝒵_2 ) I( 1 - x/ℓ;N+1,N+1 ) ],respectively, where Γ(z) is the gamma function, j_N(z) is a spherical Bessel function, andI(z;N+1,N+1 ) = ( 2N + 1 )! /( N! ) ^2 ∫_0^z ( u - u^2 ) ^N duis a regularized incomplete beta function.<cit.> By construction we have |ρ^(HP)_1(ω,N)|∝ 1/ω^N+1 as ω→∞; conversely, as ω→ 0, we find |ρ^(HP)_1(0,N)|≅ (1/2)|log( 𝒵_1/𝒵_2 ) |.We illustrate our results with the specific example of a wide-band high-pass lossless transformer placed between a 50 Ω input line and 100 Ω load-bearing line. In this example our goal is a design with bandpass of 1-10 GHz, appropriate for a supconducting parametric amplifier, with high-frequency asymptotic reflections damped as strongly as possible. For concreteness, assume a transformer length of ℓ=50 mm, with L_1=2.5 pH/μm, C_1=0.001 pF/μm, L_2=10 pH/μm, and C_2=0.001 pF/μm. In this case the characteristic frequency of the transformer is ω_c/2π=0.5/(ℓ√(L_2 C_2))=100 MHz, but this frequency can be adjusted by changing the value of ℓ.In Fig. <ref>(a) we plot the absolute reflection response |ρ^(HP)_1(ω,N)| obtained from Eq. (<ref>) as a function of the ratio of the input-signal frequency ω to the transformer characteristic frequency ω_c, on a logarithmic scale, for design values N=2, 5, 10, 30, and 100. For comparison we also plot the absolute reflection response of the exponential, triangular, and Klopfenstein tapers that are described in Pozar.<cit.> Our example is constructed to match the example in the Pozar text as closely as possible. In particular, for the Klopfenstein-taper parameters, corresponding to a ripple of 2%, we set Γ_0=0.346574, Γ_m=0.02, and A=3.54468. For the Klopfenstein characteristic impedance we used the formula given by Pozar,<cit.> viz.𝒵^(K)(x) = √(𝒵_2𝒵_2)exp[ Γ_m A ∫_0^2x/ℓ-1I_1( A √(1-y^2))/√(1-y^2) dy ],where I_1(z) is a modified Bessel function of integer order. For the reflection response, we applied Eq (<ref>) to Eq. (<ref>) and solved for the reflection coefficient, then set x=0 to obtain the input-line reflection response. The result may be expressed asρ^(K)_1(ω) = 1/2∫_0^ℓ[ d/dx'log𝒵^(K)(x') ]e^-2π i ( ω/ ω_c . ) x' / ℓ.dx' = Γ_m e^-iπω/ ω_c . cos[ √(( πω/ω_c) ^2 - A^2)],where the last step follows from the Klopfenstein ansatz.<cit.> Note that the Klopfenstein model is not optimal because the characteristic impedance 𝒵^(K)(x) of Eq. (<ref>) does not satisfy all of the boundary conditions of Eq. (<ref>). Similarly, one can show that both the triangular and exponential models are not optimal because their respective characteristic impedances also do not satisfy Eq. (<ref>).In Fig. <ref>(b), we show the drop-off of absolute reflection response for a range of input frequencies starting from ω=ω_c. This provides a comparison of the ripple response of the different taper designs. The 2N-degree polynomial tapers have the smallest residual ripple, even for the case of N=2, owing to the built-in 1/ω^N+1 behavior as ω→∞. At ω/ω_c=100, the maximum absolute reflection responses of the exponential and triangular tapers are approximately -29 dB and -48 dB, respectively, while the Klopfenstein ripple retains a fixed maximum of 2%, as per the Γ_m=0.02 parameter setting, i.e., -17 dB. At ω/ω_c=100 the 2N-degree polynomial cases of N=2, N=5, N=10, N=30, and N=100 are all well below -60 dB. For the superconducting parametric amplifiers discussed in the introduction, in the operating range of 1-10 GHz, the highly-damped reflections of the 2N-degree polynomial design can substantially limit signal-gain ripple, as induced by impedance mismatch. A caveat of the 2N-degree polynomial design is that as N increases the lower bound of the passband tends to shift to higher frequencies, as is evident in Fig. <ref>(a). In particular, from Eq. (<ref>) of Appendix <ref>, we havelim_N→∞| ρ^(HP)_1(ω,N) | ≅1/2| log𝒵_1/𝒵_2| ,indicative of the passband diminishing to zero width as N→∞. This result is consistent with the Bode-Fano criterion<cit.> in the sense that as N→∞one might suspect the passband becoming a region of perfectly zero reflections since |ρ^(HP)_1(ω,N)|∝ 1/ω^N+1 as ω→∞; however, the order in which one takes limits matters, so as N→∞ for arbitrary ω the bandwidth instead goes to zero. For modest increases in N, the tendency for the lower bound of the passband to shift to higher frequencies can be compensated for in the taper design by increasing the length ℓ of the taper, thereby decreasing ω_c.In Fig. <ref>, we plot the characteristic impedance, corresponding to the the absolute reflection response of Fig <ref>, as a function of relative position x/ℓ within the taper. The characteristic impedance 𝒵^(HP)(x,N) of Eq. (<ref>) is shown for design values of N=2, 5, 10, 30, and 100, while that of the exponential, triangular, and Klopfenstein tapers is as in the Pozar text.<cit.> Recall from Fig. <ref>(a) and the discussion of Eq. (<ref>) that the lower bound of the passband shifts to higher frequencies as N increases. To maintain a fixed lower bound we must allow ℓ to increase accordingly, i.e., ω_c=π v_g/ℓ becomes smaller. Specifically, to compare the characteristic impedance of a 2N-degree polynomial taper (N≥ 2) to that of the other taper designs in Fig. <ref>, we fix the lowest frequency of the 2N-degree-polynomial passband, ω_1(N), such that ω_1(N)=ω_1(2) for all N>2. In this way, all of the 2N-degree polynomial tapers will have the same bandwidth as the N=2 case, comparable to the bandwidths of the exponential, triangular, and Klopfenstein tapers. Since the lower bound of the Klopfenstein passband is typically defined as the frequency corresponding to the first zero of reflections, we may define ω_1(N) similarly. Thus, from Eq. (<ref>), the first zero of |ρ^(HP)_1(ω,N)| is the first zero of the spherical Bessel function j_N(πω/ω_c), call it z_N, i.e., j_N(z_N)≡ 0 such thatω_1(N) = z_N/π ω_c(N) ,where ω_c(N)=π v_g / ℓ(N). For example, first zeros of the spherical Bessel function include z_2=5.76346, z_5=9.35581, and z_10=15.0335. Then, setting ω_1(N)=ω_1(2) impliesω_c(N) = z_2/z_N ω_c(2) , ℓ(N) = z_N/z_2 ℓ(2) .The inset of Fig. <ref> is a plot of Eq. (<ref>) as a function of increasing N, showing both ℓ(N), corresponding to the left vertical axis, and ω_c(N), corresponding to the right vertical axis. For example, in order that the N=100 case have the same passband as the N=2 case, ℓ(100) must be approximately 20 times greater than ℓ(2). In particular, as N→∞ then ℓ(N)→∞, corresponding to a taper of unattainably infinite length. Moreover, all of the curves corresponding to N≫ 1 are difficult to realize physically given the length of transformer required, as well as the material composition and other geometric factors, i.e., v_g, that contribute to ω_c(N).In Fig. <ref>, note that the characteristic impedance of the N=2 case of the 2N-polynomial taper design closely matches that of the triangular taper, but with a more strongly damped reflection oscillation, due to the 1/ω^3 behavior of the input-line reflections as ω→∞. As N increases the damping of these reflections increases as 1/ω^N+1, and the shape of the characteristic impedance approaches a sharper profile for x≈ℓ/2. The point 𝒵^(HP)(ℓ/2,N)=√(𝒵_1 𝒵_2)≅ 70.71 Ω is fixed for all N. In Fig. (<ref>), the sharpness of the characteristic impedance profile as N→∞ makes it difficult to fabricate such a taper due to limitations of line composition and geometry, as mentioned earlier. Even for the case of the N=100 taper, where from Fig. <ref>(a) the reflection response is negligible for ω/ω_c>100, there is a sharp change in the characteristic impedance for x≈ℓ/2 that would make this taper extremely challenging to fabricate, given the necessary length of the taper. A compromise is to select a value of N between the N=2 and N=100 cases. Clearly, a good choice is N=2, but a better choice might be N=5 or N=10. The best choice is to select the largest N permitted by the fabrication constraints of the application such that residual oscillations are damped to the greatest extent possible within the region of exploitable passband—this is the physical limit of the high-pass design for the application.Important points regarding the optimal wide-band high-pass transformer are: * By setting the design frequency ω_o of the transformer to infinity we realized a high-pass bandwidth the order of ω_c/2π<ω<∞.* To exploit this bandwidth to its greatest extent we expanded the Fourier-like integral of the numerator of Eq. (<ref>) in an asymptotic series, and we choose g(x) so as to eliminate the lowest N-1 terms of this series, as described in detail in Appendix <ref>; this defined a model parametrized by integer N, where the reflection response, by construction, is |ρ^(HP)_1(ω,N)|∝ 1/ω^N+1 as ω→∞.* The optimal characteristic impedance of the optimal high-pass-transformer model, 𝒵^(HP)(x,N), is given by Eq. (<ref>), as derived in Appendix <ref>.* We compared this optimal high-pass transformer model to the Klopfenstein, triangular, and exponential models, demonstrating its superior reflection response at high frequencies; this resulted from the optimal form of Eq. (<ref>), which allowed us to design the asymptotic behavior.* The Klopfenstein, triangular, and exponential models are not optimal transformers because their respective characteristic impedances do not satisfy the boundary conditions of Eq. (<ref>).* A limitation on the size of integer N of the optimal design is imposed by the physical composition and geometry of the taper; a value in the range of of 2≤ N ≤ 10 is probably reasonable for most applications.§.§ The Wide-Band Low-Pass Impedance TransformerIn a manner analogous to the high-pass transformer, recall from our introductory remarks that a wide-band low-pass impedance transformer can be constructed if the design frequency ω_o is detuned from ω_c such that ω_o→ 0. In this case Eq. (<ref>) becomesρ^(LP)_1(ω) =lim_ω_o→ 0ρ_1(ω,ω_o) ≅1/2 ℓ^2[ log𝒵(0,0)/𝒵_2] (2π i ω/ω_c ) ^2∫_0^ℓ g(x) e^-2π i ( ω/ ω_c . ) x / ℓ.dx ,where 𝒵(0,0) is determined from Eq. (<ref>), for the case ω_o=0, viz. 2 [ 𝒵_1 - 𝒵(0,0) ] + [ 𝒵_1 + 𝒵(0,0) ] log𝒵(0,0)/𝒵_2 = 0 .As in Eq. (<ref>) of the high-pass case, ω_c delineates the low-frequency and high-frequency regimes. From Eq. (<ref>), the corresponding characteristic impedance is𝒵^(LP)(x) = lim_ω_o→ 0𝒵(x,ω_o) =𝒵_2 exp{[ log𝒵(0,0) /𝒵_2 ] [d g(x) / dx ] } . As in the high-pass case, we can determine a choice for g(x) by considering the asymptotic behavior of the Fourier-transform-like integral of Eq. (<ref>) as ω→ 0. Assuming 2πω≪ω_c, and repeatedly integrating by parts N times, we obtain∫_0^ℓ g(x) e^-2π i ( ω/ ω_c . ) x / ℓ.dx =e^-2π i ( ω/ ω_c . ) ∑_n=0^N-1( 2π iω/ω_c) ^n ∫_0^ℓ G^(n)(x) dx + ( 2π iω/ω_c) ^N-1∫_0^ℓ G^(N)(x) e^-2π i ( ω/ ω_c . ) x/ℓdx ,where the last term is the expansion remainder, and we have definedG^(n)(x) = 1/ℓ∫_0^x G^(n-1)(x') dx' ;G^(0)(x) = g(x) . In Appendix <ref> we demonstrate how a (N+3)-degree polynomial choice for g(x) can be used to eliminate terms of Eq. (<ref>) to order N-1 in 2πω/ω_c. In this case the reflection response ρ^(LP)_1(ω,N) and characteristic impedance 𝒵^(LP)(x,N) areρ^(LP)_1(ω,N) ≅1/2( N + 2 )! /( 2N + 4 )! [ log𝒵(0,0)/𝒵_2]×(2π i ω/ω_c ) ^N+2 M ( N + 3, 2N + 5 ,2π i ω/ω_c )e^-2π i ( ω/ ω_c . ), 𝒵^(LP)(x,N) = 𝒵_2exp{[ log𝒵(0,0) /𝒵_2 ] P^(0,-1)_N+2(1-2x/ℓ) } ,respectively, where M(a,b,z) is Kummer's confluent hypergeometric function<cit.> and P^(α,β)_n(z) is a Jacobi polynomial of order n.<cit.> By construction |ρ^(LP)_1(ω,N)|∝ω^N+2 as ω→ 0; conversely, as ω→∞, we find |ρ^(LP)_1(ω,N)|→ (1/2)|log[ 𝒵(0,0)/𝒵_2 ] |.As in the high-pass case, we illustrate the wide-band low-pass lossless transformer with the specific example of a 50 Ω input line and 100 Ω load-bearing line. For concreteness, assume a transformer length of ℓ=5 mm, with L_1=2.5 pH/μm, C_1=0.001 pF/μm, L_2=10 pH/μm, and C_2=0.001 pF/μm. Therefore, we have a bandpass that extends up to the transformer characteristic frequency, ω_c/2π=0.5/(ℓ√(L_2 C_2))=1 GHz. Again, this frequency can be adjusted by changing the length ℓ of the transformer.In Fig. <ref>(a) we plot the absolute reflection response |ρ^(LP)_1(ω,N)|, as obtained from Eq. (<ref>), as a function of the ratio of the input-signal frequency ω to the transformer characteristic frequency ω_c, on a logarithmic scale, for design values of N=1, 5, 25, and 50. The figure shows negligible reflection response over a bandpass of up to ω≅ω_c, followed by a region ω≈ω_c characterized by large oscillating reflections. For ω≈ω_c the magnitude of reflections surpasses unity by as much as a factor of two due to the breakdown of the small-reflection approximation of Eqs. (<ref>) and (<ref>). This could be addressed by solving the full Riccati equation for the voltage reflection coefficient. However, the small-reflection approximation of Eq. (<ref>) is more than adequate to estimate the breadth of the transformer pass band. In particular, we see from Fig. <ref>(b) the smallness of the bandpass reflections as the frequency decreases from ω≈ω_c.In Fig. <ref> we plot the characteristic impedance of the wide-bandwidth low-pass lossless transformer as a function of relative position x/ℓ within the taper, for values of N=1, 5, 25, and 50. The figure illustrates the discontinuity in characteristic impedance at the x=0 interface that is particular to the low-pass transformer, independent of taper design choice. In this example the input line characteristic impedance is 50 Ω and just inside the taper we have 𝒵(0,0)≈ 500 Ω. The discontinuity is governed by the size of the impedance mismatch between input and load-bearing lines, with solution for 𝒵(0,0) obtained from Eq. (<ref>). As the impedance mismatch increases, fabrication of the transformer becomes a challenge because the taper characteristic impedance sharply increases at x=0. This is the principal difficulty in building the low-pass impedance transformer, regardless of design choice.Nevertheless, the solutions of the (N+3)-degree polynomial design indicate the direction to take in fabricating the low-pass transformer, if the impedance mismatch is not too great. In Fig. <ref>, as N increases the (N+3)-degree polynomial solution incurs greater undulation, which again presents fabrication difficulties. From Eq. (<ref>), the undulation behavior of N≫ 1 may be approximated as𝒵^(LP)(x) ≅𝒵_2exp{[ log𝒵(0,0) /𝒵_2 ]cos[ N φ(x) - π/4 ] /√(π N tan[ φ(x)/2 ] )} ,where φ(x) = arccos( 1-2x/ℓ). However, as Fig. <ref> shows, the simplest case of N=1 has negligible reflection response over frequency band 0<ω/ω_c<0.1, with corresponding characteristic impedance of Fig. <ref> exhibiting a very smooth and gradual undulation—only representing a challenge to fabrication at the x=0 end of the taper. Therefore, the N=1 case can represent a feasible design for an optimal low-pass transformer. Similar to the polynomial designs of the high-pass transformer, the largest value of N that can be accommodated by fabrication constraints represents the best physical taper design.An interesting aspect to the low-pass transformer is its ability to act as a filter of high frequencies. In Fig. <ref>(a), the black horizontal line is the asymptote for the limit of the absolute reflection response |ρ^(LP)_1(ω,N)|→ (1/2)|log[ 𝒵(0,0)/𝒵_2 ]|, as input frequency ω→∞, independent of N. When the asymptote approaches unity the low-pass transformer acts as low-pass filter, suppressing transmission of frequencies ω>ω_c. In this limit we can approximate 𝒵(0,0) from Eq. (<ref>) as 𝒵(0,0)≅𝒵_2exp(2)-4 𝒵_1, where 4𝒵_1≪𝒵_2exp(2). The absolute reflection response is then |ρ^(LP)_1(ω,N)|≅ |1 - 2 e^-2 (𝒵_1/𝒵_2)| as ω→∞. Thus, as the impedance mismatch between input line and load-bearing line increases, the filter becomes more effective. Efficacy of the device is limited by fabrication constraints imposed by the mismatch at the x=0 interface, as discussed earlier, but the device may have application to reduction of the Purcell effect (at ∼ 7 GHz) in superconducting transmon and Xmon qubit-readout measurements.<cit.>Important points regarding the optimal wide-band low-pass transformer are: * By setting the design frequency ω_o of the transformer to zero we realized a low-pass bandwidth the order of 0<ω<ω_c/2π.* Analogous to the high-pass case, we exploited this bandwidth to its greatest extent by expanding the Fourier-like integral of the numerator of Eq. (<ref>) in an asymptotic series, choosing g(x) so as to eliminate the lowest N terms of this series, as described in detail in Appendix <ref>; this again defined a model parametrized by integer N, where the reflection response, by construction, is |ρ^(LP)_1(ω,N)|∝ω^N+2 as ω→ 0.* The optimal characteristic impedance of the optimal low-pass-transformer model, 𝒵^(LP)(x,N), is given by Eq. (<ref>), as derived in Appendix <ref>.* We compared optimal low-pass transformer designs of different values of integer N, plotting reflection response versus frequency in Fig. <ref> and characteristic impedance versus position along the taper in Fig. <ref>; the small-reflection approximation employed in Fig. <ref> tends to break down at high frequencies and larger values of N. § CONCLUDING REMARKSWe presented a variational approach to determine the optimal form of the reflection coefficient of a tapered impedance transformer of length ℓ, for a specific design frequency ω_o. We used this result to construct the characteristic impedance and input-line reflection response of an optimal lossless transformer, defining the optimal transformer design in terms of ω_o and a real-valued function g(x) satisfying the boundary conditions of Eq. (<ref>). The input-line reflection response was shown to depend on a characteristic frequency ω_c=π v_g/ℓ, where v_g is the constant transformer group velocity.By construction the input-line reflection response, as a function of arbitrary frequency ω, is zero specifically at ω=ω_o, indicative of narrow pass band. However, we showed that if ω_o is detuned far from ω_c then, for an extended range of frequencies ω about ω_o, the magnitude of input-line reflections is negligible. Specifically, when we took the limit ω_o→∞ we obtained a high-pass transformer design with pass band ω_c≲ω<∞, where the input-line reflection response and characteristic impedance are given by Eqs. (<ref>) and (<ref>), respectively. Similarly, when we took the limit ω_o→ 0 we obtained a low-pass transformer design with pass band 0<ω≲ω_c, where input-line reflection response and characteristic impedance are given by Eqs. (<ref>) and (<ref>), respectively.Having derived a general form for wide-bandwidth transformers in terms of g(x), for both the high-pass and low-pass frequency regimes, we then showed, for each regime, how to choose a polynomial g(x) that produces the widest exploitable pass band possible. In the case of the high-pass transformer, we compared our results to existing optimal taper designs, specifically the exponential, triangular, and Klopfenstein tapers described in Pozar.<cit.> We showed that are our design exhibits superior pass-band characteristics. For the case of the low-pass transformer, we demonstrated the inherent difficulty of fabrication of the x=0 end of the taper, due to the discontinuity of the characteristic impedance at this interface. Nevertheless, we proposed the N=1 case of our design as the simplest to fabricate, with greater efficacy the smaller the impedance mismatch of the application.An important point to note is that our theory has focused on an isolated impedance transformer, one for which signals do not enter the transformer from the x=ℓ side. In practice, a transformer can be placed at both input and output of a load-bearing component, which means that the two transformers will be coupled, as in the example schematic of Fig. <ref>. Just as a forward-traveling wave (arrow pointing to right) can reflect from the input transformer, upon transmission through the input transformer, a forward-traveling wave can also reflect from the output transformer, allowing this reflected signal to encounter the input transformer from the right, as pictured.In the case the two coupled transformers of Fig. <ref>, the optimization of their design requires simultaneous minimization of the voltage reflection coefficients at both x=x_a and x=x_b, from the left. This can be performed in the manner of our variational approach, but also requires derivation of the corresponding coupled Riccati differential equations of the two tapers, as well as their boundary conditions. These equations can be obtained by extending the approach used in Appendix <ref>. We consider the case of coupled transformers to be an extension of our present work, and a subject of future focus.As mentioned earlier, our motivation for the present study is to improve the signal-to-noise ratio of superconducting amplifiers used in quantum-information research. We are presently engaged in fabrication and validation of the transformer designs derived here, for the coplanar waveguides that comprise our amplifiers. One aim of our experimental analysis is to ascertain the limit of fabrication techniques to capture and leverage improvements implied by these designs, i.e., whether these transformers may be fabricated to sufficiently high precision to realize their performance benefits. The present theoretical results, and the findings of our follow-on experimental studies, may have broad applicability to the new and burgeoning fields of high-speed electronics, particularly where sensitivity to small reflections at an input-line/load-bearing interface is of critical importance. This work was supported by the Army Research Office and the Laboratory for Physical Sciences under EAO221146, EAO241777 and the NIST Quantum Initiative. RPE and Self-Energy, LLC acknowledge grant 70NANB17H033 from the US Department of Commerce, NIST. We have benefited from discussions with D. Pappas, X. Wu, M. Bal, H.-S. Ku, J. Long, R. Lake, L. Ranzani, K. C. Fong, T. Ohki, and B. Abdo. § DERIVATION OF THE DIFFERENTIAL EQUATION OF THE REFLECTION COEFFICIENT OF AN IMPEDANCE TRANSFORMERConsider an input line and load-bearing transmission line with mismatched characteristic impedances 𝒵_1 and 𝒵_2, respectively. A tapered impedance transformer of length ℓ is fabricated within the transmission line to address the mismatch, as in Fig. <ref>(a). Each of these three components is modeled as a ladder-type transmission line with unit-length series inductance L(x), series resistance R(x), shunt capacitance C(x), and shunt conductance G(x), comprising a ladder rung at x extending over the infinitesimal length dx, as in Fig. <ref>(b). Within the input line (interior of the transmission line) L(x), R(x), C(x), and G(x) are all constant and denoted by subscript 1 (2), whereas in the transformer these quantities vary with position x, 0≤ x≤ℓ. Voltage V(x,t) and current I(x,t) at x satisfy transmission-line equations given by∂/∂ xI(x,t) + C(x)∂/∂ tV(x,t) + G(x) V(x,t) = 0 ,∂/∂ xV(x,t) + L(x)∂/∂ tI(x,t) + R(x) I(x,t) = 0 . These equations also apply in the input line (interior of the transmission line), except that L(x), R(x), C(x), and G(x) are replaced by L_1, R_1, C_1, and G_1 (L_2, R_2, C_2, and G_2). §.§ Region Before the TaperWithin the region before the taper, i.e., x<0 in Fig. <ref>(a), for a traveling-wave of frequency ω, the solution of the input line is of the formI(x,t) = I_1(x,ω) e^iω t + I_1(x,ω)^* e^-iω t , V(x,t) = V_1(x,ω) e^iω t + V_1(x,ω)^* e^-iω t . Substituting this into the input-line form of Eqs. (<ref>) we obtain ∂/∂ xI_1(x,ω) + Y_1(ω) V_1(x,ω) = 0 ,∂/∂ xV_1(x,ω) + Z_1(ω) I_1(x,ω) = 0 , where we have defined Z_1(ω)=R_1+iω L_1 to be a line impedance per unit length and Y_1(ω)=G_1+iω C_1 is a shunt admittance per unit length. If we assume amplitudes with spatial dependence of the form I_1(x,ω)=A_1(ω)exp[γ(ω) x] and V_1(x,ω)=B_1(ω)exp[γ(ω) x] then γ(ω) A_1(ω) + Y_1(ω) B_1(ω) = 0 , Z_1(ω) A_1(ω) + γ(ω) B_1(ω) = 0 , such that a non-trivial solution of A_1(ω) and B_1(ω) requires γ(ω)=±γ_1(ω), where γ_1(ω)=√(Z_1(ω)Y_1(ω)). So we have two solutions we may express as a superposition, viz. I_1(x,ω) = A^(+)_1(ω) e^-γ_1(ω) x + A^(-)_1(ω) e^γ_1(ω) x , V_1(x,ω) = 𝒵_1(ω) [ A^(+)_1(ω) e^-γ_1(ω) x - A^(-)_1(ω) e^γ_1(ω) x] , where the input-line characteristic impedance is 𝒵_1(ω)=√(Z_1(ω)/ Y_1(ω).). The amplitude A^(+)_1(ω) (A^(-)_1(ω)) is that of a forward (backward) traveling wave. In keeping with convention, we may define the reflection coefficient of the input line as the ratio of backward-traveling voltage amplitude to forward-traveling voltage amplitude, viz. ρ_1(ω) = [ 𝒵_1(ω) A^(-)_1(ω) ] / [ -𝒵_1(ω) A^(+)_1(ω) ] = -A^(-)_1(ω) / A^(+)_1(ω). §.§ Region After the TaperIn the interior of the transmission line, after the taper, i.e., x>ℓ, the solution follows similarly to the region before the taper, viz.I(x,t) = I_2(x,ω) e^iω t + I_2(x,ω)^* e^-iω t , V(x,t) = V_2(x,ω) e^iω t + V_2(x,ω)^* e^-iω t . However, in this case there is no backward-traveling component; one has instead I_2(x,ω) = A^(+)_2(ω) e^-γ_2(ω) ( x - ℓ) , V_2(x,ω) = 𝒵_2(ω) A^(+)_2(ω) e^-γ_2(ω) ( x - ℓ) , where Z_2(ω)=R_2+iω L_2, Y_2(ω)=G_2+iω C_2, and also 𝒵_2(ω)=√(Z_2(ω)/ Y_2(ω).), γ_2(ω)=√(Z_2(ω)Y_2(ω)). In the limit ℓ→ 0 then Eq. (<ref>) matches to Eq. (<ref>) at x=0, resulting in A^(+)_1(ω) + A^(-)_1(ω) = A^(+)_2(ω) ,𝒵_1(ω) [ A^(+)_1(ω) - A^(-)_1(ω) ] = 𝒵_2(ω) A^(+)_2(ω) . With ρ_1(ω)=-A^(-)_1(ω)/A^(+)_1(ω), the above equations implyρ_1(ω) = 𝒵_2(ω) - 𝒵_1(ω)/𝒵_2(ω) + 𝒵_1(ω) ,which is ρ_1(ω) in the absence of a transformer. §.§ Region of the TaperThe region of the taper corresponds to 0<x<ℓ. Since here the unit-length series impedance Z(x,ω)=R(x)+iω L(x) and shunt admittance Y(x,ω)=G(x)+iω C(x) vary with distance x we segment the taper into subregions small enough that these quantities may be considered constant in each subregion. We then solve for voltage and current in each subregion, as we did for the regions outside the taper, with the caveat that we have to join solutions of the subregions together to obtain the full solution across the length of the taper. Once we accomplish this we can join this full solution of 0<x<ℓ to that of x<0 and x>ℓ.Specifically, we segment the taper into N subregions where the n-th subregion corresponds to interval x_n-1<x<x_n, of length Δ x = ℓ/N, i.e., x_n=x_n-1+Δ x=ℓ n / N, with x_0≡ 0 and x_N≡ℓ. If Δ x is sufficiently narrow then constituents of impedance and admittance of the n-th subregion may be denoted by constant values L^(n), R^(n), C^(n), and G^(n), such that Z^(n)(ω)=R^(n)+iω L^(n) and Y^(n)(ω)=G^(n)+iω C^(n). Then the equations governing the voltage and current of the n-th subregion are of the same form as those before the taper, i.e., x_n-1 < x < x_n, viz.I(x,t) = I^(n)(x,ω) e^iω t + I^(n)(x,ω)^* e^-iω t , V(x,t) = V^(n)(x,ω) e^iω t + V^(n)(x,ω)^* e^-iω t , where we may write I^(n)(x,ω) = A^(n,+)(ω) e^-γ^(n)(ω) ( x - x_n-1)+ A^(n,-)(ω) e^γ^(n)(ω) ( x - x_n-1) , V^(n)(x,ω) = 𝒵^(n)(ω) [ A^(n,+)(ω) e^-γ^(n)(ω) ( x - x_n-1)- A^(n,-)(ω) e^γ^(n)(ω) ( x - x_n-1)] , where 𝒵^(n)(ω) = √(Z^(n)(ω) / Y^(n)(ω)) and γ^(n)(ω) = √(Z^(n)(ω)Y^(n)(ω)).At x=x_n we may match the solution of x_n-1 < x < x_n to that of x_n < x < x_n+1, viz.A^(n,+)(ω) e^-γ^(n)(ω) Δ x + A^(n,-)(ω) e^γ^(n)(ω) Δ x = A^(n+1,+)(ω) + A^(n+1,-)(ω) , 𝒵^(n)(ω) [ A^(n,+)(ω) e^-γ^(n)(ω) Δ x - A^(n,-)(ω) e^γ^(n)(ω) Δ x] =𝒵^(n+1)(ω) [ A^(n+1,+)(ω) - A^(n+1,-)(ω) ] .As Δ x→ 0 one passes to the continuum limit whereinA^(n,±)(ω) → A^(±)(x,ω) , A^(n+1,±)(ω) → A^(±)(x,ω) + Δ x ∂/∂ x A^(±)(x,ω) , e^±γ^(n)(ω) Δ x→ 1 ±γ(x,ω) Δ x ,𝒵^(n)(ω) →𝒵(x,ω) ,𝒵^(n+1)(ω) →𝒵(x,ω) + Δ x ∂/∂ x𝒵(x,ω) .Applying these limiting results to the two boundary equations at x=x_n, then as Δ x→ 0 we find-γ(x,ω) [ A^(+)(x,ω) - A^(-)(x,ω) ] =∂/∂ x A^(+)(x,ω) + ∂/∂ x A^(-)(x,ω) , -γ(x,ω) 𝒵(x,ω) [ A^(+)(x,ω) + A^(-)(x,ω) ] =𝒵(x,ω) [ ∂/∂ x A^(+)(x,ω) - ∂/∂ x A^(-)(x,ω) ]+ [ ∂/∂ x𝒵(x,ω) ] [ A^(+)(x,ω) - A^(-)(x,ω) ] .Solving for the derivatives of the amplitudes in these two equations yields∂/∂ x A^(+)(x,ω) = -γ(x,ω) A^(+)(x,ω)- 1/2[ ∂/∂ xlog𝒵(x,ω)] [ A^(+)(x,ω) - A^(-)(x,ω) ] , ∂/∂ x A^(-)(x,ω) = γ(x,ω) A^(-)(x,ω)+ 1/2[ ∂/∂ xlog𝒵(x,ω)] [ A^(+)(x,ω) - A^(-)(x,ω) ] .These are the first-order differential equations governing the solution of the amplitudes of the taper region, 0<x<ℓ.As we did for the region before the taper, we may define a reflection coefficient for a position x within the taper as ρ(x,ω)=-A^(-)(x, ω)/A^(+)(x, ω), which has a derivative with respect to x given by∂/∂ xρ(x,ω) = A^(-)(x, ω) ∂/∂ x A^(+)(x, ω)- A^(+)(x, ω) ∂/∂ x A^(-)(x, ω)/ A^(+)(x, ω)^2.If we substitute Eqs. (<ref>) and (<ref>) into Eq. (<ref>) we obtain∂/∂ xρ(x,ω) = 2 γ(x,ω) ρ(x,ω) - 1/2[ ∂/∂ xlog𝒵(x,ω)] [ 1 - ρ(x,ω)^2 ] .This is the expression derived by Walker and Wax.<cit.> We next consider boundary conditions applicable to this differential equation.First, at x=0, we can equate the current and voltage of Eqs. (<ref>), corresponding to the region before the taper, to the current and voltage of Eqs. (<ref>), corresponding to the subregion of n=1, just inside the taper. As Δ x → 0 the result is A^(+)_1(ω) [ 1 - ρ_1(ω) ] = A^(+)(0,ω) [ 1 - ρ(0,ω) ] ,𝒵_1(ω) A^(+)_1(ω) [ 1 + ρ_1(ω) ] = 𝒵(0,ω) A^(+)(0,ω) [ 1 + ρ(0,ω) ] , where the reflection coefficient of the input line is ρ_1(ω)=-A^(-)_1(ω)/A^(+)_1(ω) and, similarly, the reflection coefficient of the taper at x=0 is ρ(0,ω)=-A^(-)(0,ω)/A^(+)(0,ω). If we divide the first equation into the second and solve for ρ_1(ω) we findρ_1(ω) = 𝒵(0,ω) [ 1 + ρ(0,ω) ] - 𝒵_1(ω) [ 1 - ρ(0,ω) ] /𝒵(0,ω) [ 1 + ρ(0,ω) ] + 𝒵_1(ω) [ 1 - ρ(0,ω) ].The reflection coefficient is discontinuous across the physical boundary at x=0 unless the two lines on either side of the interface have the same characteristic impedance at x=0.Similarly, at x=ℓ, we can consider the continuity of current and voltage across this interface using Eqs. (<ref>), corresponding to the subregion of n=N, just inside the taper, and Eqs. (<ref>), corresponding to the region after the taper. As Δ x → 0 we find A^(+)(ℓ,ω) [ 1 - ρ(ℓ,ω) ] = A^(+)_2(ω) ,𝒵(ℓ,ω) A^(+)(ℓ,ω) [ 1 + ρ(ℓ,ω) ] = 𝒵_2(ω) A^(+)_2(ω) . Again, dividing the first equation into the second and solving for ρ(ℓ,ω), we obtainρ(ℓ,ω) = 𝒵_2(ω) - 𝒵(ℓ,ω) /𝒵_2(ω) + 𝒵(ℓ,ω).If the interface at x=ℓ is not a physical one then𝒵(ℓ,ω)=𝒵_2(ω), which implies ρ(ℓ,ω)=0.§ PHYSICALLY MEANINGFUL SOLUTIONS OF THE LOSSLESS IMPEDANCE TRANSFORMERTo obtain the solution of the lossless impedance transformer we apply Eqs. (<ref>) to Eq. (<ref>). Separating Eq. (<ref>) into real and imaginary parts, we may write2 | γ(0,ω_o) | [ 𝒵_1 - 𝒵(0,ω_o) ] f(x,ω_o)- | γ(x,ω_o) | { 4 | γ(0,ω_o) | [ 𝒵_1 - 𝒵(0,ω_o) ] ∫_x^ℓf(x',ω_o)dx'- [ 𝒵_1 + 𝒵(0,ω_o) ] log𝒵(0,ω_o)/𝒵_2} = 0 , 2 | γ(0,ω_o) | [ 𝒵_1 - 𝒵(0,ω_o) ] {f(x,ω_o) + 2 | γ(x,ω_o) | ∫_x^ℓf(x',ω_o)dx' } = 0 ,where |γ(x,ω_o)|=ω_o√(L(x)C(x)) and 𝒵(x,ω_o)=√((L(x)/C(x)). As in our earlier discussion of the more general case, for the lossless transformer a viable choice of ρ(x,ω_o), i.e., f(x,ω_o) in Eq. (<ref>), depends on the existence of a physically meaningful solution of L(x) and C(x). As we shall show, restrictions on the solution will narrow the possible choices for f(x,ω_o).For example, consider Eqs. (<ref>) and (<ref>) in the limit x→ 0, viz.| γ(0,ω_o) | { 2 [ 𝒵_1 - 𝒵(0,ω_o) ][ 1 - 2 | γ(0,ω_o) | ∫_0^ℓf(x',ω_o)dx' ]+ [ 𝒵_1 + 𝒵(0,ω_o) ] log𝒵(0,ω_o)/𝒵_2} = 0 , | γ(0,ω_o) | ^2 [ 𝒵_1 - 𝒵(0,ω_o) ] ∫_0^ℓf(x',ω_o)dx' = 0 .We see in Eq. (<ref>) that the real part of the integral of f(x,ω_o) must vanish since alternatively (i) 𝒵(0,ω_o)=𝒵_1, corresponding to a trivial solution of impedance matching, or (ii) |γ(0,ω_o)|=0, which implies an infinite group velocity at the materials interface. Thus, inclusive of the boundary conditions of f(x,ω_o) stated in Eq. (<ref>), constraints imposed on f(x,ω_o) aref(0,ω_o) = 1 ,f(ℓ,ω_o) = 0 , ∫_0^ℓf(x,ω_o)dx = 0 . As another example, since |γ(0,ω_o)| 0, we have from Eq. (<ref>) that| γ(x,ω_o) | =-f(x,ω_o)/ 2 ∫_x^ℓf(x',ω_o)dx',where, via L'Hospital's rule, we find| γ(ℓ,ω_o) | = lim_x→ℓ| γ(x,ω_o) | =1/2lim_x→ℓ∂/∂ x f(x,ω_o)/f(x,ω_o) .However, unless ∂ f(x,ω_o)/∂ x =0 as x→ℓ, the above limit will be unbounded since f(ℓ,ω_o)=0 via Eq. (<ref>). Therefore, insisting ∂ f(x,ω_o)/∂ x be zero at x=ℓ, we may use L'Hospital's rule again, viz.| γ(ℓ,ω_o) | = 1/2lim_x→ℓ∂^2/∂ x^2 f(x,ω_o)/∂/∂ x f(x,ω_o) .Thus, additional constraints imposed on our choice of f(x,ω_o) are . ∂/∂ x f(x,ω_o)| _x = ℓ = 0 , . ∂^2/∂ x^2 f(x,ω_o)| _x=ℓ =2 | γ(ℓ,ω_o) | . ∂/∂ x f(x,ω_o)| _x=ℓ . As a last point of observation, we note from Eq. (<ref>) that| γ(0,ω_o) | = lim_x→ 0| γ(x,ω_o) | =1/2lim_x→ 0∂/∂ x f(x,ω_o)/f(x,ω_o) .Since f(0,ω_o)=1 this limit requires. ∂/∂ x f(x,ω_o)| _x=0 0 ,otherwise we would again have |γ(0,ω_o)|=0. Thus, Eqs. (<ref>), (<ref>), and (<ref>) summarize the constraints on our choice of f(x,ω_o) to ensure a physically meaningful solution of L(x) and C(x), if one does indeed exist.The task of choosing f(x,ω_o), subject to the constraints of Eqs. (<ref>), (<ref>), and (<ref>), is made easier noting these constraints are satisfied if f(x,ω_o) is written in the formf(x,ω_o) = d/dx g(x) + 2 γ(ℓ,ω_o) g(x) ,where the real-valued function g(x), in turn, satisfies the boundary conditionsg(0) = 0 ,g(ℓ) = 0 , . d/dx g(x) | _x=0 = 1 , . d/dx g(x) | _x=ℓ = 0 .Thus, the definition of an optimal lossless impedance transformer reduces to choosing a real-valued g(x) that satisfies the boundary conditions of Eq. (<ref>).We may express the solution for the optimal lossless impedance transformer in terms of g(x). To start, using f(x,ω_o) of Eq. (<ref>), the solution of |γ(x,ω_o)| in Eq. (<ref>) becomes| γ(x,ω_o) | =-f(x,ω_o)/ 2 ∫_x^ℓf(x')dx'=| γ(ℓ,ω_o) | [ g(x)/g(x) - g(ℓ)] = | γ(ℓ,ω_o) | ,since g(ℓ)=0. Thus | γ(x,ω_o) | is constant in x, along the entire length of the optimal impedance transformer, for any choice of g(x).Then, making use of Eqs. (<ref>) and (<ref>), we have from Eq. (<ref>) the result2 [ 𝒵_1 - 𝒵(0,ω_o) ] [ 1 - 4 | γ(ℓ,ω_o) | ^2 ∫_0^ℓ g(x') dx' ]+ [ 𝒵_1 + 𝒵(0,ω_o) ] log𝒵(0,ω_o)/𝒵_2 = 0 ,which determines the solution of 𝒵(0,ω_o). Similarly, solving for 𝒵(x,ω_o) in Eq. (<ref>), with the aid of Eqs. (<ref>), (<ref>), and (<ref>), we arrive at𝒵(x,ω_o) =𝒵_2 exp{[ log𝒵(0,ω_o) /𝒵_2 ] [dg(x) / dx .- 4 | γ(ℓ,ω_o) | ^2 ∫_x^ℓ g(x') dx' / 1 - 4 | γ(ℓ,ω_o) | ^2 ∫_0^ℓ g(x') dx' ] } .Thus, once we determine 𝒵(0,ω_o) from Eq. (<ref>) we may obtain 𝒵(x,ω_o) via Eq. (<ref>). The solutions for the inductance and capacitance per unit length then follow asL(x,ω_o) = √(L_2 C_2) 𝒵(x,ω_o) ,C(x,ω_o) = √(L_2 C_2)/𝒵(x,ω_o). § DERIVATION OF THE WIDE-BAND HIGH-PASS LOSSLESS IMPEDANCE TRANSFORMERA 2N-degree polynomial g^(HP)(x,N) =ℓ[ x/ℓ - α∑_n=1^N a^(N)_n ( x/ℓ) ^2N-n + α( x/ℓ) ^2N]can be chosen for g(x) to eliminate all terms to order N on the right side of Eq. (<ref>). Here, polynomial coefficients a^(N)_1, …, a^(N)_N, and scaling factor α, are real-valued, and g^(HP)(x,N) satisfies the boundary conditions of Eq. (<ref>) via constraints∑_n=1^N a^(N)_n = 1 + 1/α,∑_n=1^N( 2N - n ) a^(N)_n = 2N + 1/α .The second-order term on the right of Eq. (<ref>) is removed in the limit α→ 0 when the equation is divided by∫_0^ℓ g^(HP)(x,N) dx =( 1/2 - α∑_n=1^N a^(N)_n / 2N -n + 1+ α/ 2N + 1 ) ℓ^2 .Higher terms are eliminated by requiring g^(n-1)(0)=0 and g^(n-1)(ℓ)=0 for n=3,…,N. The form of the polynomial guarantees g^(n-1)(0)=0 while g^(n-1)(ℓ)=0 is obtained if∑_n=1^N( 2N - n ) ! /( 2N - n - m ) !a^(N)_n = ( 2N ) ! /( 2N - m ) ! ; m = 2, 3, …, N-1 .The N coupled linear algebraic Eqs. (<ref>) and (<ref>) determine coefficients a^(N)_1, …, a^(N)_N of the polynomial. Substituting Eq. (<ref>) into Eq. (<ref>) for g(x), and taking the limit α→∞, we may writeρ^(HP)_1(ω,N) ≅-1/2( log𝒵_1/𝒵_2) [ ∑_n=0^N a^(N)_n / 2N-n+1 ] ^-1 ×∑_n=0^N a^(N)_n( 2N - n ) ! ( ω_c / 2π iω) ^2N-n+1[ 1 - e^-2π i ( ω / ω_c )∑_m=0^2N-n 1 / m! (2π i ω/ω_c ) ^m ] ,where we define a^(N)_0=-1 and note polynomial coefficients a^(N)_1, …, a^(N)_N now satisfy∑_n=1^N( 2N - n ) ! /( 2N - n - m ) !a^(N)_n = ( 2N ) ! /( 2N - m ) ! ; m = 0, 1, …, N-1 .Including the definition a^(N)_0=-1, the solution of Eq. (<ref>) isa^(N)_n = ( -1 ) ^n+1 N! / n! ( N -n )! ;n = 0, 1, …, N .Similarly, substituting Eq. (<ref>) into Eq. (<ref>) for g(x), and taking the limit α→∞, the corresponding characteristic impedance is𝒵^(HP)(x,N) = 𝒵_2 exp( ( log𝒵_1 /𝒵_2 ){ 1 - [ ∑_n=0^Na^(N)_n/2N-n+1] ^-1∑_n=0^Na^(N)_n/2N-n+1( x/ℓ) ^2N-n+1}).Equations (<ref>), (<ref>), and (<ref>) define the high-pass transformer design of the 2N-degree polynomial choice for g(x).Expanding exp( 2π iω/ω_c ) in an infinite sum, an alternate expression of Eq. (<ref>) is ρ^(HP)_1(ω,N) ≅-1/2( log𝒵_1/𝒵_2) [ ∑_n=0^N a^(N)_n / 2N-n+1 ] ^-1∑_n=0^N a^(N)_n ∑_m=0^∞( 2N - n ) ! /( 2N - n + m + 1 ) ! (2π i ω/ω_c ) ^m e^-2π i ( ω / ω_c ) .The coefficients of Eq. (<ref>) satisfy the summation identities ∑_n=0^N a^(N)_n /( 2N - n + m )=( -1 ) ^N+1 N! ( N + m - 1 )! /( 2N + m )! ;m > 0 ,∑_n=0^N( 2N - n ) ! /( 2N - n + m ) !a^(N)_n =( -1 ) ^N+1 N! ( N + m - 1 )! /( m - 1 )! ( 2N + m )! ;m > 0. Using Eq. (<ref>) and (<ref>) to evaluate the sums over n in Eq. (<ref>), we haveρ^(HP)_1(ω,N) ≅-1/2( log𝒵_1/𝒵_2) ( 2N+1 )! / N! ∑_m=0^∞( N + m )! / m! ( 2N + m + 1 ) ! (2π i ω/ω_c ) ^m e^-2π i ( ω / ω_c ) =-1/2( log𝒵_1/𝒵_2) M( n+1, 2n+2, 2π iω/ω_c) e^-2π i ( ω / ω_c ) ,where in the last step we recognize the resulting sum as Kummer's confluent hypergeometric function, M(a,b,z).<cit.> Alternatively, M(n+1,2n+2,,2iz)=Γ(3/2+n)e^iz(2/z)^n j_n(z)/√(π), where Γ(z) is the gamma function and j_n(z) is a spherical Bessel function.<cit.>. Thus, we haveρ^(HP)_1(ω,N) ≅-Γ( 3/2+N ) /√(π)( log𝒵_1/𝒵_2) ( 2ω_c/πω) ^Nj_N ( πω/ω_c ) e^-iπ( ω / ω_c ) .When N≫ 1 we may use the asymptotic form of the Bessel function for large order,<cit.> such thatρ^(HP)_1(ω,N) ≅-1/2( log𝒵_1/𝒵_2)e^-iπ( ω / ω_c );πω/ω_c≪ N ≫ 1 .Also, by construction, ρ^(HP)_1(ω,N)∝ 1/ω^N+1 as ω→∞.With the aid of Eq. (<ref>), we may write Eq. (<ref>) as𝒵^(HP)(x,N) = 𝒵_2exp{( log𝒵_1 /𝒵_2 )[ 1 - ( -1 ) ^N+1( 2N + 1 )! /( N! ) ^2S ( x/ℓ) ] } ,where we have introduced the sumS(z) = ∑_n=0^Na^(N)_n/2N-n+1 z^2N-n+1 .Differentiating S(z) with respect to z and making use of Eq. (<ref>), as well as the binomial theorem, we find derivative S'(z)=-(z^2-z)^N. Integrating S'(z) from 0 to z we arrive at the expression𝒵^(HP)(x,N) = 𝒵_2 exp{( log𝒵_1 /𝒵_2 )[ 1 - I( x/ℓ;N+1,N+1 ) ] } ,whereI(z;N+1,N+1 ) = ( 2N + 1 )! /( N! ) ^2 ∫_0^z ( u - u^2 ) ^N duis a regularized incomplete beta function.<cit.> Since I(z;a,b)=1-I(1-z;b,a) then I(1/2;a,a)=1/2 such that 𝒵^(HP)(ℓ/2,N)=√(𝒵_1𝒵_2) is a fixed point, the same for all N, and Eq. (<ref>) may be expressed alternatively as𝒵^(HP)(x,N) = 𝒵_2 exp[ ( log𝒵_1 /𝒵_2 )I( 1 - x/ℓ;N+1,N+1 ) ].With 0≤ z≤ 1, as N→∞ we have<cit.>lim_N→∞ I(z;N+1,N+1 ) = lim_N→∞1/√(2π)∫_-∞^3 √(N)ϕ(z) e^-u^2/2 du ; ϕ(z) =z^1/3 - ( 1 - z ) ^1/3/√( z^1/3 + ( 1 - z ) ^1/3) .Since ϕ(z)>0 if z>1/2 and ϕ(z)<0 if z<1/2 thenlim_N→∞ I(z;N+1,N+1 ) = Θ( z - 1/2 ) ,where Θ(z) is the Heaviside step function. Therefore, we may write𝒵^(HP)(x) = lim_N→∞𝒵^(HP)(x,N) =𝒵_2 exp[ ( log𝒵_1 /𝒵_2 ) Θ( ℓ/2 - x ) ]. § DERIVATION OF THE WIDE-BAND LOW-PASS LOSSLESS IMPEDANCE TRANSFORMERAnalogous to the derivation of the wide-band high-pass lossless transformer, we construct a (N+3)-degree polynomialg^(LP)(x,N) = ℓ[ x/ℓ - ∑_n=1^N+2 a^(N)_n ( x/ℓ) ^n+1] ,where g^(LP)(x,N) satisfies the boundary conditions of Eq. (<ref>) via constraints∑_n=1^N+2( n + 1 ) a^(N)_n = 1 ,∑_n=1^N+2 a^(N)_n = 1 .To eliminate terms to order N-1 in 2πω/ω_c of Eq. (<ref>) we set G^(n)(ℓ)=0 for n=1,…,N, which introduces N additional constraints involving polynomial coefficients a^(N)_1, …, a^(N)_N+2. Combining these constraints with those of Eq. (<ref>) we obtain a set of N+2 coupled linear algebraic equations in a^(N)_1, …, a^(N)_N+1 that we may express as∑_n=1^N+2( n + 1 )! /( n + m )!a^(N)_n = 1/m!;m = 0,1, …, N+1 .The solution isa^(N)_n = ( -1 ) ^n+1 N + 2 / n! ( n + 1 )!( N + n + 1 )! /( N - n + 2 )! ; n = 0,1,…,N+2 ,where we have included a^(N)_0=-1 for completeness.Substituting Eq. (<ref>) into Eq. (<ref>) for g(x), we obtainρ^(LP)_1(ω,N) ≅ -1/2[ log𝒵(0,0)/𝒵_2] ∑_n=0^N+2 a^(N)_n ( n+1 )!( ω_c / 2π i ω) ^n [ 1 - e^-2π i ( ω/ ω_c . ) ∑_m=0^n+11/m!(2π i ω/ω_c ) ^m ] .Expanding exp( 2π iω/ω_c ) in an infinite sum, an alternate expression isρ^(LP)_1(ω,N) ≅ -1/2[ log𝒵(0,0)/𝒵_2]∑_m=0^∞[ ∑_n=0^N+2 a^(N)_n ( n+1 )! /( m+n+2 )! ](2π i ω/ω_c ) ^m+2 e^-2π i ( ω/ ω_c . ),where we note that the sum over n may be written in closed form as∑_n=0^N+2 a^(N)_n ( n+1 )! /( m+n+2 )!= {[ 0 ; m<N; -( m + 2 )! /( m - N )!( m + N + 4 )! ; m ≥ N ]. .Applying this last result to Eq. (<ref>), and then shifting the sum over m by N, we obtainρ^(LP)_1(ω,N) ≅1/2[ log𝒵(0,0)/𝒵_2] ∑_m=0^∞( N + m + 2 )! / m!( 2N + m + 4 )! (2π i ω/ω_c ) ^N+m+2 e^-2π i ( ω/ ω_c . ).Then, similar to the high-pass case, we recognize this sum to be related to Kummer's confluent hypergeometric function, M(a,b,z),<cit.> viz.ρ^(LP)_1(ω,N) ≅1/2( N + 2 )! /( 2N + 4 )! [ log𝒵(0,0)/𝒵_2](2π i ω/ω_c ) ^N+2 M ( N + 3, 2N + 5 ,2π i ω/ω_c )e^-2π i ( ω/ ω_c . ).By construction, as ω→ 0 we have ρ^(LP)_1(ω,N)∝ω^N+2. Conversely, as ω→∞ then M ( N + 3, 2N + 5 ,2π i ω/ω_c )→[( 2N+4)!/( N+2 )! ] [ ω_c/(2πω) ]^N+2exp( 2π i ω/ω_c ),<cit.> so for fixed N we havelim_ω→∞ρ^(LP)_1(ω,N) ≅1/2log𝒵(0,0)/𝒵_2 . The characteristic impedance 𝒵^(LP)(x,N) corresponding to the design function g^(LP)(x,N) is obtained by substituting Eq. (<ref>) into Eq. (<ref>) for g(x). Within the resulting expression the derivative with respect to x of g^(LP)(x,N) is the Gauss series representation of the ordinary hypergeometric function F(-N-2,N+2;1;x/ℓ)=_2F_1(-N-2,N+2;1;x/ℓ).<cit.> Alternatively, _2F_1(-N-2,N+2;1;x/ℓ)=P^(0,-1)_n+2(1-2x/ℓ), where P^(0,-1)_N+2(1-2x/ℓ) is a Jacobi polynomial of order N+2.<cit.> Hence, we haved/dx g^(LP)(x,N) = -∑_n=0^N+2 a^(N)_n ( n+1 ) ( x/ℓ) ^n = P^(0,-1)_N+2(1-2x/ℓ) , where a^(N)_n is given by Eq. (<ref>). Therefore, the characteristic impedance may be written as𝒵^(LP)(x,N) = 𝒵_2 exp{[ log𝒵(0,0) /𝒵_2 ] P^(0,-1)_N+2(1-2x/ℓ) } .As N→∞ we may use the Darboux formula for the large-order expansion of Jacobi polynomials.<cit.> Letting φ(x) = arccos( 1-2x/ℓ), this gives𝒵^(LP)(x) = lim_N→∞𝒵^(LP)(x,N) =𝒵_2 lim_N→∞exp{[ log𝒵(0,0) /𝒵_2 ]cos[ N φ(x) - π/4 ] /√(π N tan[ φ(x)/2 ] )} = 0 ;x > 0 ,where 𝒵^(LP)(0)=𝒵(0,0). | http://arxiv.org/abs/1709.08808v2 | {
"authors": [
"R P Erickson"
],
"categories": [
"physics.app-ph",
"cond-mat.supr-con"
],
"primary_category": "physics.app-ph",
"published": "20170824203319",
"title": "Variational theory of the tapered impedance transformer"
} |
Computing separability probabilitiesfor fixed rank statesA.Razmadze Mathematical Institute, Iv.Javakhishvili Tbilisi State University,Tbilisi, Georgia Institute of Quantum Physics and Engineering Technologies, Georgian Technical University, Tbilisi, Georgia National Research Nuclear University,MEPhI (Moscow Engineering Physics Institute),Moscow, Russia Laboratory of Information Technologies,Joint Institute for Nuclear Research,Dubna, Russia The question of the generation of random mixed states is discussed, aiming for the computation of probabilistic characteristics ofcomposite finite dimensional quantum systems. Particularly, we consider the generation of the random Hilbert-Schmidt and Bures ensembles of qubit and qutrit pairs and compute thecorresponding probabilities to find a separablestate among the states of a fixed rank.On the generation of random ensembles of qubits and qutrits Arsen Khvedelidze1,2,3,[email protected] Ilya [email protected] 30, 2023 =====================================================================================§ INTRODUCTIONThe presentation addresses a special topic of quantum theory of finite dimensional systems that is of prime importance in the theory of quantum information and quantum communications. Namely, we study the problem of calculation of the probability for a random state of a binary composite quantum system, such as qubit-qubit or qubit-qutrit pairs, to bea separable or an entangled state[For definitions and references we quote thecomprehensive review <cit.> and a recent paper by P.Slater <cit.>.].According to the “Geometric Probability Theory” <cit.>, this «separability/entanglement probability», is determined bythe relative volume of separable states with respect to thevolume of the whole state space.In order to avoid subtle numerical calculation of the multidimensional integrals, (for a generic 2-qubit case the integrals are over the semi-algebraic domain of 15-dimensional Euclidean space), the Monte-Carlo ideology with a specific method of generation of random variables has been used (see e.g.. <cit.>-<cit.>, and references therein). Our report aims to present several results on the numeric studies of separability probabilities for random qubit-qubitand qubit-qutritHilbert-Schmidt and Bures ensembles. Particularly, the distribution properties of the “probability of entanglement” will be described with respect to the subsystem's Bloch vectors for the qubit-qubit and qubit-qutritrandomHilbert-Schmidt and Bures states of all possible ranks. § THE SEPARABILITY PROBABILITYThere are two ways to analyze the probabilistic aspects of entanglement characteristics, such as the so-called separability probability, 𝒫_sep . The latter is the probability to find a separable state among all possible states. The first approach is to consider the state space of a quantum system as the Riemannian space and, following the strategy of the theory of geometric probability, identify the separability probability with with a two volumes ratio:𝒫_sep= Vol(Separable states)/Vol(All states) .The second approach consists of dealing with the ensemble of random states, whose distribution is in correspondence with the volume measure in (<ref>). In this case the separability probability can be equivalently written as: 𝒫_sep= Vol( Number of separable states in ensemble)/Vol(Total number of states in ensemble) .The first method is suitable for analytic calculations, however it presents a big computational issue (see e.g., <cit.> and references therein). The second method allows us to use highly effective modern numerical computational techniques. Below we will give results on the separability probability of the qubit-qubit and qubit-qutrit systems obtained within the latterapproach.§ GENERATING THE HILBERT-SCHMIDT AND BURES RANDOM STATESHaving in mind calculations of the separability probability of states with different ranks we shortly present the algorithm for the generation of density matrices from the Hilbert-Schmidt and the Bures ensembles of a n-level quantum system.* The first step in the construction of both ensembles is the generation of complex n× n matrices Z from the Ginibre ensemble, i.e., the matrices whose entries have real and imaginary parts distributed as normal random variables;* Density matrices ϱ_HS describing states from the Hilbert-Schmidtensemble are: ϱ_HS = ZZ^+/tr( ZZ^+) ;* Density matrices ϱ_B from the Bures ensemble are generated with the aid of the Ginibre matrices Z and matrices U∈ SU(n) , distributed over the SU(n) group according to the Haar measure,ϱ_B = (I+U)ZZ^+(I+U^+)/tr( (I+U)ZZ^+(I+U^+)) . Since we are interested in studying the entanglement of states with a fixed rank of density matrices, the above algorithm requires a certain specification. For states of a lower rank than the maximal one we proceed as follows.∙Rank-3 states∙ We start with the generation of complex, rank-3 Ginibre matrices. It is known that any such matrix admits the following representation: Z= P_Z ([A_x_1x_2 x_3; y_1 y_2 y_3 D ])Q_Z ,In (<ref>) permutations of entries by P_Z and Q_Z are such that A is a regular element of the Ginibre ensemble of 3×3 matrices. Apart from that, if 3-tuples Y=(y_1, y_2, y_3) and X=(x_1, x_2, x_3) are composed of the normal random complex variables, whilethe entry D isD = YA^-1 X , we arrive at a rank-3 random matrix Z from the Ginibre ensemble of complex 4× 4 matrices. ∙Rank-2 states∙ Similarly, the4× 4 Ginibre matrix of rank-2can be writtenas, Z= P_Z ([ A B; C D ]) Q_Z ,where A, B and C are 2× 2 complex Ginibre matrices, while2× 2 matrix D reads, D = CA^-1 B . ∙Rank-1 states∙ Finally,a 4× 4 complex Ginibre matrix Z of rank-1 admits the representation: Z= P_Z ([ a y_1 y_2 y_3;x_1x_2 x_3D_ ])Q_Z ,where a, x_1, x_2, x_3 and y_1,y_2,y_3 are normal random complex variables, while3× 3matrixD is: D = 1/a( [ x_1y_1 x_1y_2 x_1y_3; x_2y_1 x_2y_2 x_2y_3; x_3y_1 x_3y_2 x_3y_3 ]) .Now, using the generic Ginibre matrix andrepresentations (<ref>)-(<ref>) for non-maximal rank matrices we can buildeither the Hilbert-Schmidt or Bures states of the required rank according to either (<ref>) or (<ref>). § COMPUTING THE SEPARABILITY PROBABILITYComputation of the sought-for separability probability (<ref>) requires the test of generated states on their separability. With this goal the Peres-Horodecki criterion <cit.> has been used in our calculations.§.§ Results of the computationsFirst of all, in table <ref> we collect results of the calculations of the separabilityprobability of the generic (rank-4) states from the Hilbert-Schmidt and Bures ensembles of qubit-qubit and qubit-qutrit pairs. Furthermore, results of the computations of the separability probabilities of non-maximal rank states of a qubit-qubit system, for both ensembles, are given in table <ref>. Here, it is worth making a comment on the zero separability probability for rank-2 and rank-1 states. That can be understood noting the following:If a density matrix of 2-qubits ϱ is such, thatrank(ϱ) < d_A=rank(ϱ_A),thenϱ is not separable<cit.>. Here, ϱ_A denotes the reduced density matrix obtained by taking the partial trace with respect to the second qubit. Indeed, since during the generation of rank-2 and rank-1, almost allreduced matrices are not singular, the above mentioned inequality is true.§ CONCLUDING REMARKSHaving the above described algorithms as a tool for the generation of ensembles of random states with different ranks, the diverse characteristics of entanglement in composite quantum systems can be studied. As an example, on figures <ref> and <ref>, we illustrate the separability probability in 2-qubit systems as a function of the Bloch radius of the constituent qubit for the states of maximal and sub-maximal ranks. Our analysis shows that the distribution of the separability probability with respect to the Bloch radius of the qubit is uniform for maximal rank states (see figure <ref>), while for the rank-3 states the deviation from the total separability probability is depicted on figure <ref>.BengtssonZyczkowskiBOOKI.Bengtsson andK.Zyczkowski,Geometry of Quantum States: An Introduction to Quantum Entanglement, Cambridge University Press, 2006.SlaterP.B.Slater, arXiv:1701.01973v7,(2017). KlainRota1997D.A.Klain and G.C.Rota,Introduction to Geometric Probability,Cambridge University Press, 1997.Braunstein1996S.L.Braunstein, A 219 169 (1996). ZyczkowskiSommers2001K.Zyczkowski andH-J.Sommers, J. Phys. A: Math. Theor. 34, 7111 (2001). OsipovSommersZyczkowski2010V.A.Osipov, H.J.Sommers and K.Zyczkowski,J. Phys. A: Math. Theor. 43, 055302, (2010)KhvedRogoj2015A.Khvedelidze and I.Rogojin, J. of Math. Sciences, 209, 6, 988–1004, (2015) RuskaiWener2009M.RuskaiandE.M.Werner, J.Phys.A.Math.Theor., 42, 095303, (2009) | http://arxiv.org/abs/1708.07846v1 | {
"authors": [
"Arsen Khvedelidze",
"Ilia Rogojin"
],
"categories": [
"quant-ph"
],
"primary_category": "quant-ph",
"published": "20170825180133",
"title": "On the generation of random ensembles of qubits and qutrits Computing separability probabilities for fixed rank states"
} |
SHT,WIEN]Gang Li WIEN]Anna Kauch WIEN,SHT]Petra Pudleiner WIEN]Karsten Held [SHT]School of Physical Science and Technology, ShanghaiTech University, Shanghai 200031, China [WIEN]Institute of Solid State Physics, Vienna University of Technology, A-1040 Vienna, Austria Victory, i.e. vienna computational tool depository, is a collection of numerical tools for solving the parquet equations for the Hubbard model and similar many body problems. Theparquet formalism is a self-consistent theory at both the single- and two-particle levels, andcan thus describe individual fermions as well as their collective behavior on equal footing. This is essential for the understanding of various emergent phases and their transitions in many-body systems, in particular for cases in which a single-particle description fails. Our implementation of victory is in modern Fortran and it fully respects the structure of various vertex functions in both momentum and Matsubara frequency space.We found the latter to be crucial for the convergence of the parquet equations, as well as for the correct determination of various physical observables.In this release, we thoroughly explain the program structure and the controlled approximations to efficiently solve the parquet equations, i.e. the two-level kernel approximation and the high-frequency regulation.Program summary* Program title: Victory* Programming language: Fortran 90* Parallel mode: Segments of vertex functions are distributed over different cores. MPI_BCAST is adopted for efficient data exchange. * Operating system: Unix, Linux, Windows* Version history: 1.0* Nature of the problem:The parquet equations require the knowledge of the fully irreducible vertexfrom which all one- and two-particle vertex and Green's functions are calculated. The underlying two-particle vertex functions are large memory objects that depend on threemomenta with periodicboundary conditions and three frequencies with open ones. The coupled diagrams of the parquet equations extend the frequency dependence of the reducible vertex functions to a larger frequency space where, a priori, no information is available. * Solution method: The reducible vertex functions are found to possess the simplest structure among all two-particle vertex functions and can be approximatedby functions with a reduced number of arguments, i.e. the kernel functions. The open boundary issue of the vertex functions in Matsubara-frequency space is then solved as follows: we solve the parquet equations with the reducible vertex functions whenever it is possible, otherwise we supplement these by the kernel functions.strongly correlated electron systems many body theory parquet equations Hubbard model Green's functions two-particle vertex § INTRODUCTION The understanding of materials with strong electronic correlations, such as transition metal oxides and f-electron systems, represents one of the greatest challenges to contemporary condensed-matter theory and remains a subject of intense research.The localized d- or f-electrons are subject to strong Coulomb interactions among themselves, which in turn causes a breakdown of a single-particle description in these systems. The problem arises essentially due to the presence of different energy scales which compete with each other in many-body systems <cit.>, and which is in factthe source of many intriguing phenomena.The model describing the competition of the electronic kinetic energy ϵ_𝐤 and the Coulomb repulsion U is the so-called Hubbard modelH = ∑_𝐤,σϵ_𝐤c_𝐤σ^†c_𝐤σ^†+U∑_in_i↑n_i↓ .Here,c^(†)_𝐤σ annihilates (creates) an electron with momentum 𝐤; and n_iσ≡ c_iσ^†c_iσ^† is the occupation number operator at lattice site i. The two terms in the Hubbard model describe single- and two-particle processes, respectively.The interacting electrons repel each other with interaction U, and hencetend to avoid each other to minimize the energy. As a result the charge carriers loose their mobility and become localized in a Mott insulator. As a competing effect, the kinetic energyϵ_𝐤 is minimized by increasing the itinerancy of the electrons, which drives the system away from a localized state. In addition to the two apparent energy scales, more competing energy scales can emerge in this model. For example, when the system temperature becomes lower than a certain condensation energy, the entire many-body system will behave collectively as composed particles. A well-known example for this is the high-T_c superconductor, where the composite electronic degrees of freedom are the Cooper pairs. Similarly, the condensation of particle-hole pairs can give rise to charge (CDW) or spin density wave (SDW) orders depending on the broken symmetry, respectively.Very often a many-body system displays both, the single-particle and the composed-particle characters, as well as their phase transitions. Here, an understanding of the single- and two-particle response is required on an equal footing.This is what the parquet equations <cit.> that areimplemented in the victory package provide. The parquet formulation for condensed matter physics has been discussed thoroughly in <cit.> and is one of the very few theories that maintains self-consistency at both single- and two-particle level. It requires knowing the two-particle fully irreducible vertex, or approximating it, e.g. by the bare interaction U in the parquet approximation <cit.>, by a self-consistently obtained constant <cit.> or by the local vertex of animpurity problem as in diagrammatic extensions of dynamical mean field theory<cit.>. In the present paper, we will discuss the detailed implementation of the parquet equations in thevictory package. The outline of the paper is as follows: In Section <ref>, we describe the program structure. Here, we start by recapitulating the parquet equations in Section <ref> and discuss the employed symmetries in Section <ref>.The structure of the victory program is outlinedin Section <ref>, important technical aspects are discussed in Section<ref>, and the parallelization strategy in Section <ref>. In Section <ref> we discuss —as an example— the application to the Hubbard model on a square lattice and present one- and two-particle results in Sections <ref> and <ref>, respectively. Finally, Section <ref> provides a brief conclusion and outlook.§ DESCRIPTION OF THE PROGRAM STRUCTURE§.§ Parquet equations The parquet formalism consists of a set of coupled equations, i.e. the Bethe-Salpeter, the parquet and the Schwinger-Dyson equations, which include both single-particle Green's function and two-particle vertex functions.To be self-contained and at the same time to keep the discussion simple,let us first introduce the necessary functions which contain all topologically invariant one- and two- particle Feynman diagrams. Let us start with the one-particle functions: The self-energy Σ contains all irreducible <cit.> diagrams with one incoming and one outgoing particle (leg). From it all (reducible) Green's function G diagrams are obtained through the Dyson equation G(k) = [G^-1_0(k) - Σ(k)]^-1 = [ iν + μ - ϵ_𝐤 - Σ(k)]^-1 ,with the chemical potential μ, the fermionic Matsubara frequency ν, and acombined notation of momentum 𝐤 and frequency ν, i.e. k=(𝐤, ν). The non-interacting dispersion relation ϵ_𝐤 directly yields the non-interacting Green's function G_0(k)=( iν + μ - ϵ_𝐤 )^-1.Regarding two-particle quantities, the concept of reducibility is more subtle. The full vertex functions F can be further subdivided into different classes, namely a fully irreducible class of diagrams Λ and classes of diagrams Φ_r which are reducible regarding a specific channel r. In the (2)-symmetric case, it is convenient to use some combinations of thespin indices known as the density (d), magnetic (m), singlet (s) and triplet (t) channel (in addition to the r channels regarding the irreducibility). This leads to the following parquet equations, see <cit.>: F_d^k,k^'(q)=Λ_d^k,k^'(q) + Φ^k,k^'_d(q) - 1/2Φ_d^k,k+q(k^'-k) - 3/2Φ_m^k,k+q(k^'-k) +1/2Φ_s^k,k^'(k+k^'+q) +3/2Φ_t^k,k^'(k+k^'+q) ;F_m^k,k^'(q)=Λ_m^k,k^'(q) + Φ^k,k^'_m(q) - 1/2Φ_d^k,k+q(k^'-k) + 1/2Φ_m^k,k+q(k^'-k)-1/2Φ_s^k,k^'(k+k^'+q) +1/2Φ_t^k,k^'(k+k^'+q) ;F_s^k,k^'(q)=Λ_s^k,k^'(q) + Φ^k,k^'_s(q) +1/2Φ_d^k,q-k^'(k^'-k)-3/2Φ_m^k,q-k^'(k^'-k)+1/2Φ_d^k,k^'(q-k-k^')- 3/2Φ_m^k,k^'(q-k-k^') ;F_t^k,k^'(q) = Λ_t^k,k^'(q)+ Φ^k,k^'_t(q)-1/2Φ_d^k,q-k^'(k^'-k)-1/2Φ_m^k,q-k^'(k^'-k) +1/2Φ_d^k,k^'(q-k-k^') + 1/2Φ_m^k,k^'(q-k-k^') .Here, F_d/m/s/t^k,k^'(q) are the complete vertices in the corresponding channels (combinations of spin indices).The equations (<ref>) respect an important symmetry of the two-particle vertex, i.e. the crossing symmetry <cit.>. With the above formulations, this symmetry has been guaranteed to be satisfied <cit.>. Note that, in principle,F_s/t can be obtained fromF_d/m at a different combination of frequencies and momenta. But this would take us out of the finite frequency box the vertex is stored in and break the crossing symmetry.The vertex functions Φ_d/m^k,k^'(q) and Φ_s/t^k,k^'(q) in (<ref>)denote the reducible vertex in the particle-hole (d/m) and particle-particle (s/t) channel, respectively. They can be calculated from the corresponding complete vertex F_d/m/s/t^k,k^'(q) and irreducible vertex functions Γ^k,k^'_d/m/s/t(q) = F_d/m/s/t^k,k^'(q) - Φ_d/m/s/t^k,k^'(q) through the Bethe-Salpeter equations:Φ_d/m^k,k^'(q)= T/N∑_k^''Γ^k,k^''_d/m(q)G(k^'')G(k^''+q)F^k^'',k^'_d/m(q) , Φ_t/s^k,k^'(q)= ±T/2N∑_k^''Γ^k,k^''_t/s(q)G(k^'')G(q-k^'')F^k^'',k^'_t/s(q) ,where T is temperature and N is the number of momenta in the first Brillouin zone. To clarify the used notation, Fig. (<ref>) explicitly illustrates Eq. (<ref>).In the Bethe Salpeter Eqs. (<ref>), we need the one-particle Green's function as an input. For a self-consistent scheme, we hence need to recalculate the one-particle quantities from the two-particle vertices. This is achieved via the (Heisenberg)equation of motion, also known as Dyson-Schwinger equation: Σ(k) = -UT^2/2N^2∑_k^',q[F_d^k,k^'(q)-F_m^k,k^'(q)]G(k+q)G(k^')G(k^'+q) .Fig. <ref> shows the equivalent Feynman diagrams thereof. At this stage, it is easy to see that given the fully irreducible vertex Λ_d/m/s/t as input, the other two-particle vertices and the single-particle self-energy can be consistently determined from the parquet equations (<ref>), the Bethe-Salpeter (<ref>) and the Dyson-Schwinger equation (<ref>). The Green's function is finally updated after each iteration via the Dyson Eq. (<ref>). This set of equations is solved self-consistently in victory. §.§ Symmetry consideration In addition to the particle-hole symmetry, the crossing symmetry and the spin (2) symmetry, we also apply the time-reversal symmetry to the calculation of the parquet equation in victory.For the two-particle vertex functions, taking F^k,k^'_r(q) as an example, the time-reversal symmetry implies[ F^k,k^'_r(q) ]^* = F^-k,-k^'_r(-q) ,which indicates that only the positive (negative) frequencies in either k, k^' or q are needed. The negative (positive) counterparts can be obtained straightforwardly from the above time-reversal symmetry operation.In victory, we hence only define zero and positive frequencies for q. But for k and k^' the frequencies still need to be storedfrom negative to positive as usual.Introducing the symmetry operation to the two-particle vertex has a clear advantage, as it can significantly reduce the memory requirement for storing the vertices which is the bottleneck for solving the parquet equations. This outweighstheadditional CPU operations needed for employing the symmetry. Other symmetries that connect different momentum and frequency sectors can be straightforwardly implemented in victory, which can further reduce the size of the two-particle vertex functions. This is currently work in progress.§.§ Outline of the program structure Flow diagram in Fig. <ref> demonstrates the victory implementation of the equations (<ref>)-(<ref>). For the interest of the advanced users, we also cross reference to the actual subroutines in the victory program. *Initialize the (fixed) fully irreducible vertex Λ_r^k,k^'(q), where r=d/m/s/t stands for the different channels. For example, one can set it to the bare interaction as in the parquet approximation (PA) or to the local, fullyirreducible vertex as in the dynamical vertex approximation (DΓA). The latter can be obtained by other methods (and program packages), such as exact diagonalization or quantum Monte Carlo. If Λ_r^k,k^'(q) is provided in a smaller frequency range than the planned computation range, it is augmented by its asymptotic value (channel dependent constant of the PA).*Initialize Σ_(0)(k)tozero or other initial values; It is often not of crucial importance to have the initial value close to the actual solution of the problem as victory iteratively updates the self-energy, but a smarter guess of the initial value will converge the parquet calculations much faster.At the phase transition boundary convergence may be hard to achieve and it can be helpful to initialize the self-energy with values that effectively start up the calculation far away from the difficult parameter regimes. CalculateG_(0)(k)=(ν-ϵ_ k+μ-Σ_(0))^-1 from it; andinitialize F_r,(0)^k,k^'(q), Γ_r,(0)^k,k^'(q) by setting them equal toΛ_r^k,k^'(q) or the bare interactions. *Determine the reducible vertexΦ_r, (n)^k,k^'(q)from the Bethe-Salpeter Eq. (<ref>). For a fixed value of q these two equations have a standard form of matrix multiplication w.r.t. the k, and k^' indices. Therefore, the level-3 BLAS routine zgemm can be applied straightforwardly. Here, Φ_r, (n)^k,k^'(q) at the nth-iteration is calculated from Γ_r,(n-1)^k,k^'(q) and F_r,(n-1)^k,k^'(q) from the . In order to improve convergence in some parameter regime, the calculation of Φ_r, (n)^k,k^'(q) may additionally be damped by its previous solution, i.e. Φ_r, (n),damped^k,k^'(q)= f_damping·Φ_r, (n)^k,k^'(q)+(1-f_damping)·Φ_r, (n-1)^k,k^'(q).*Due to the structure of the parquet equations, the reducible vertex function is needed in a larger frequency range than what is used in step <ref>. In <cit.>, cf. <cit.>, it was shown that the high frequency tail of the reducible vertex can be approximated bythe so-called kernel functionsin a sufficient way. Hence in step <ref>., the kernel functions are usedwhenever one of the two frequencies (ν ν^') exceeds the box [-N_ω,N_ω].The kernel functions are obtained from the reducible vertex in a level-2 approximation K^k_r,2(q) = Φ_r^k,(𝐤^',N_ω)(q) as well as K^k^'_r,2(q) = Φ_r^(𝐤,N_ω),k^'(q) in tandem with the constant background obtained in the level-1 approximation K_r,1(q) = Φ_r^(𝐤,N_ω),(𝐤^',N_ω)(q). *Now every term on the right hand side of the parquet Eq. (<ref>) is known, and a new full vertex function F_r, (n)^k,k^'(q) can be obtained. This step is computationally the most involved one due to the size of the two-particle vertex which depends on three momenta and three frequencies and therefore cannot be stored on a single core. Since segments of Φ_r, (n)^k,k^'(q) are stored on different cores and the parquet equations mix different parts of the memory space, extensive intercore communication is required. To this end, we develop an efficient parallelization scheme to broadcast the reducible vertex function across different cores. More details can be found in section <ref>. The new channel-dependent irreducible vertex Γ_r, (n)^k,k^'(q) is later obtained as Γ_r,(n)^k,k^'(q)=F_r,(n)^k,k^'(q) - Φ_r,(n)^k,k^'(q). *With the full vertex function F_r,(n)^k,k^'(q), a new self-energy is then calculated from the Schwinger-Dyson equation (<ref>).*At this point all terms in equations (<ref>) and (<ref>) have been updated once, the calculation can be iterated from step <ref> to step <ref> until convergence is achieved. §.§ Technical remarks * In victory, each vertex function is implemented as a two-dimensional array for a given value of q. Here, k, k^' and q are momentum-frequency mixed variables where each of them can be referred to in the program by a unique integer index. There are two types of indexing in victory. A combined momentum-frequency index stored as an integer in the range [1,2· N_ω· N_c], with N_c being the momentum cluster size and 2· N_ω the number of frequencies; and a derived-type index map containing integers which directly refer to components of momentum 𝐤 and Matsubara frequency ν (or in case of a bosonic frequency to ω). The three linear algebra operations needed in Eq. (<ref>), i.e. k+k^'+q, k^'-k and q-k-k^', can thus be easily defined by using the derived-type index maps [subroutine ]. There is a unique mapping between these two types of indexing [subroutine ]. * The memory consumption of the program is proportional to the cube of the range of each argument, i.e. k, k^' and q. The number of momentum points is simply determined by the size of the cluster on which the calculations are performed. The number of Matsubara frequencies is given by the user as input, but it is essentially determined by the temperature of the system, i.e. the lower the temperature the more Matsubara frequencies are required. The memory consumption for three different momentum cluster sizes is given in Table <ref>. * A practical problem arises due to the finite size of the frequency box. As a result of linear operations of the arguments mentioned before, i.e. k+k^'+q, k^'-k and q-k-k^',we stay in a bigger parameter space than the one defined individually for k, k^' and q. This fact would in the end make the loop iteration impossible as the evaluation of F_r^k,k^'(q) in the parquet Eq. (<ref>) requires Φ_r in a bigger parameter space.As a solution we developed the two-level kernel approximation which is discussed in step <ref> of the program structure and explained in detail in our previous works <cit.>. The kernel approximation respects the correct asymptotics of the reducible vertex function Φ_r and extends the calculation of Eq. (<ref>) to arbitrarily large parameter space.* The limitation to a finite frequency rangewithoutperiodicity conditions, e.g. as the ones for the momentum, affects not only the calculation of the full vertex functions F but also the determination of the self-energy in the Schwinger-Dyson Eq. (<ref>). In general, the truncation of the two sums seriously influences the high-frequency tails of Σ. However, with the help of the kernel approximations these sums can be regulated, leading to a correct behavior of the self-energy. That is,we take the bare interaction as F in Eq. (<ref>) for arbitrary large frequencies and the kernel functions for a (e.g. two or three times) larger frequency box. A more detailed discussion thereof is likewise provided in <cit.>.* Whenever possible, we evaluate the lowest order diagram (the "bubble" diagram of the Green's function) by using the Fast-Fourier-Transform between the Matsubara frequency ν and imaginary-time τ. The high-frequency asymptotic tail of the bubble diagram is explicitly evaluated. This scheme is used in the evaluation of the reducible vertex function Φ_r^k,k^'(q) to substitute the discrete frequency sum of the bubble part.§.§ ParallelizationIn the current release of victory, we provide parallelization based on Message Passing Interface (MPI). The central issue in the parallelization is the memory consumption, since the most memory-consuming objects in the program, i.e. the vertex functions, cannot be stored in the memory of one core only. They are distributed over cores according to their q-values, for illustration see Fig <ref>. Correspondingly, each core carries different segments of the vertex functions.The Bethe-Salpeter equation (<ref>) is not affected by the parallelization as, for a fixed value of q, these equations can be evaluated at each core without communication with others.However, the calculation of the parquet equation (<ref>) requires data exchange among all cores since different frequencies and momenta are mixed.In other words, each core has to send its data to all other cores. At the same time it also receives data from all other cores, which essentially requires a (possibly) non-blocking communication among different cores.This communication becomes the actual bottleneck of the calcuation.In victory we have tested two flavors of parallelization: (1) One can combine a non-blockingwith a blockingto ensure that every core receives the required data before starting to solve the parquet equation.(2)A more economical way is to usei.e. each core distributes its segment of vertex function to all other cores. The advantage of using collective communication in this case is in the decreased number of actual point-to-point communications (forwith a binomial tree algorithm it is N_cores·log N_cores as compared to the N_cores^2 needed byand ). This outweighs the potential gain in time that the non-blocking communication provides. Therefore, in the current version of victory only collective (blocking) communication is implemented. Another parallelization execution occurs when the self-energy function is calculated in step <ref>.After the evaluation of the parquet equation, the full vertex function F^k,k^'_r(q) is updated at all cores. At each core only a segment of F^k,k^'_r(q) is stored, as explained before. The calculation of the self-energy requires however F^k,k^'_r(q) at all q values, cf. Eq. (<ref>). Hence further communications are needed.Based on the parallelization strategy we established, we perform a partial sum over the q-values available at each core. Each core stores now only part of the self-energy and the sum over all these parts is the correct self-energy. The results from all cores are collected with , which yields the desired self-energy function at each core. We note that, in victory, only the two-particle vertex functions are distributed over cores, all other quantities such as the single-particle Green's function, self-energy, as well as the two-level kernel functions are stored on all cores to reduce the demand on core communications. § EXAMPLE CASE To illustrate the capability of victory, we consider here a single-band Hubbardmodel on a square lattice. The results presented in this section correspond mainly to a 4× 4 cluster with periodic boundary conditions.For the purpose of comparison, in some cases we also show the results for a 6×6 cluster.As stated before, the parquet equations guarantee self-consistency at both single- and two-particle levels. Correspondingly, in victory one gets both the single- and two-particle Green's function from the same calculation. All the results presented in this section were obtained using the parquet approximation (PA). §.§ Single-particle quantitiesIn a finite-size cluster, one has only a limited number of independent cluster momenta at which the single-particle Green's functions behave differently. In victory, we calculate the single-particle Green's function and two-particle vertex functions at all cluster momenta as it is straightforward to manipulate the internal momentum sum in the full Brillouin zone. However, we stress that this is not really necessary, i.e. a calculation with only the independent cluster momenta of the irreducible Brilluin zone can also be conducted after some further modifications of victory. In Fig. <ref>(a), the imaginary part of the single-particle Green's function for the 2D Hubbard model at different independent cluster momenta for the 4×4 cluster is shown as a function of Matsubara frequency ν for ⟨ n⟩=0.9. The current implementation of victory works equally for undoped and doped cases.The imaginary part of the Green's function in Fig. <ref>(a) displays a clear momentum selectivity, in particular at k = (π/2, π/2) it quickly increases in amplitude and develops a clear metallic character (corresponding to a 1/νshape at small ν). In contrast, at k=(0, 0) and (π,π), the Green's functions approach zero with the decrease of frequency, suggesting an insulating state at these momenta.As U/t=4 is still in the weak-correlation regime for the 2D Hubbard model (i.e. it is smaller than the critical interaction strength for the metal-insulator transition), the system essentially follows the same character asin the non-interacting limit.That is, k=(π/2, π/2) is on the Fermi surface, while the other two momenta are far away from it.The self-energy, as a measure of the effective correlations, is shown in Fig. <ref>(b).As one generally expects, the electronic correlations renormalize states at the Fermi surface more significantly thanthe states with higher binding energies.Consequently, the self-energy at k=(π/2, π/2) is somewhat bigger than at the other two momenta presented. Fig. <ref>(c) displays the real-space Green's function as a function of the imaginary-time.Here, r=0 corresponds to the local Green's function, r=√(2) andr=2√(2) to the nonlocal one between site (0,0) and (1,1) and (2,2), respectively. In victory, the average particle number ⟨ n⟩ is self-consistently determined by adjusting the chemical potential μ to satisfy ⟨ n⟩=∑_σG_r=0,σ(τ=0^-).Obtaining quantities on the real frequency axis is not implemented directly invictory. The Matsubara Green's function can however be used as input into a Padé analytic continuation script (not provided) to obtain momentum dependent spectral functions as presented in Fig. <ref>. Other analytical continuations are of course also possible.The results were obtained for the following parameters β t = 2, U/t = 4.0 and⟨ n⟩=1.0 for a 4×4 momentum cluster. A clear quasi-particle peak is visible at momenta on the Fermi surface: k=(π,0) and (π/2, π/2). As we move away from the Fermi surface, there is also a shift of some spectral weight to higher (lower) frequencies. For U/t = 4.0,only for these momenta away from the Fermi surface there isa two peak structure corresponding to the quasiparticle peak and the upper (lower) Hubbard band.§.§ Two-particle quantities A major output of victory are various two-particle quantities, which include the two-particle full vertex function F_r(k,k^';q), the channel-dependent reducible vertex functions Φ_r(k,k^';q) and the channel-dependent irreducible vertex functions Γ_r(k,k^';q).These vertex functions play important roles in many-body theory. For example, the magnetic, charge and pairing response functions are readily obtained by attaching four legs to the full vertex function F_r(k,k^';q). This yields, after summing over k and k^' and adding the bubble contribution, the physical responses which are directly observable in corresponding experiments, such as neutron scattering. In Fig. <ref>, the full vertex functions in the magnetic channel are shown as functions of the two fermionic Matsubara frequencies ν and ν^'. The transfer frequencies are taken as ω=0 and ω=20π T for the first and the second row, respectively, and the parquet equations have been solved on a 6× 6 cluster in momentum space. To illustrate the momentum dependence of the vertex function, we have chosen different transfer cluster momenta, i.e.q=(0, 0) and q=(π,π) at k= k^'=(0, 0). On the square lattice, the single-band Hubbard model shows strong antiferromagnetic spin fluctuationsat q=(π,π). This is reflectedin thestrongly peaked structure of the two-particle full vertex function F_m for q=(π,π)in Fig. <ref>. The frequency structure of the vertex functions is better seen in the intensity plots shown at the top of each vertex figure.As for transfer frequency ω=0, the major vertex structure is the cross appearing at ν=0 and ν^'=0.The cross shifts to ν=-ω andν^'=-ω when ω becomes nonzero. At the same time, the diagonal structure at ν+ν^'+ω=0 becomes more prominent. This can be clearly seen by comparing the second row of Fig. <ref> to the first row,and it essentially originates from the special momentum and frequency dependence of the parquet equation (<ref>). The full vertex F of Fig. <ref> can be separated, in each channel r, into its irreducible and reducible components F_r=Γ_r+Φ_r. Fig. <ref> shows both of these components for the magnetic channel and the biggest contribution atq=(π,π), ω=0. This reveals that the large cross-like contribution to F_m stems from the reduciblevertex Φ_m. That is, it is built up through the Bethe-Salpeter ladder in the magnetic channel, as is to be expected for strong antiferromagnetic spin fluctuations. Some other contributions such as the ν=ν^' diagonal structure that is a bit off-set from the rest of the vertex are contained in Γ_m. Indeed we know from perturbation theory that this diagonal stems from the particle-hole transversal channel <cit.> which is irreducible in the magnetic particle-hole channel shown in Fig. <ref>.The effect of using a finite frequency box is clearly visible in the irreducible vertex Γ_m in the left panel of Fig. <ref>. It manifests itself as a square-shaped structure at larger fermionic frequencies. It is also visible, though with smaller relative intensity, in the full vertex function F_m shown in Fig. <ref>. This finite-size effect disappears when a large enough frequency range is taken. For the data presented in Figs. <ref> and <ref> we used N_ω=32. We want to note that by using victory one can also obtain various susceptibilitiesafter further manipulating the vertex data. We do not provide such scripts in the standard victory package as they can be easily implemented by the users.Furthermore, the vertex functions calculated in victory can also serve as the input for various many-body methods.For example, the dual-fermion approach <cit.> and the non-local expansion scheme <cit.> both use the local full vertex function to construct the non-local self-energy diagram; the two-particle irreducible vertex Γ_ris the building block in the ladder flavor of DΓA <cit.> and the 1-particle irreducible approach <cit.>.In this respect,the victory package not only provides a practical implementation of the parquet method but also serves as a calculator of various vertex functions, for which a reliable numerical scheme is still missing (in particular with the momentum resolution).It thus can be combined with other many-body methods to study more complicated electronic systems.Another interesting quantity that is examined in victory are the eigenvalues of the Bethe-Salpeter equation in all four above-mentioned channels. These can be obtained as solutions of the following eigen-equations: -T/N∑_k^'Γ_d/m(k, k^';q)G(k^')G(k^'+q) v_q(k^') = λ_q(k) v_q(k) , ±T/2N∑_k^'Γ_t/s(k, k^';q)G(k^')G(q-k^')v_q(k^')=λ_q(k) v_q(k) .The corresponding eigenvalue, λ_q(k), provides a simple measure of the stability of the parquet equations.The Bethe-Salpeter equation, by iteratively appending the irreducible vertex Γ_r to the corresponding full vertex F_r, determines the ladder contribution to the corresponding two-particle fluctuations in the given channel. The victory parquet formalism is based on the translational symmetry, SU(2) symmetry and time-reversal symmetry, a divergence of the Bethe-Salpeter equation tells us that one of these symmetries is broken and we approach a phase transition.A phase transition occurs, more specifically, when the largest eigenvalue for transfer frequency ω = 0 approches 1. Note that we cannot calculate the vertex function in a symmetry-broken phase in the current implementation, albeit this is in principle possible.In victory, the eigenvalue λ_q(k) is given at different transfer momenta q.The largest λ typically appears at values of q commensurate to the lattice periodicity.For example, on a square lattice one expects λ at q=(π, π) in the magnetic channel to be most relevant.In the case of a triangular lattice, on the other hand, it is at q=(2π/3, 2π/3).For a possible charge density wave instability, e.g. in the presence of non-local Coulomb interactions, the leading eigenvalue in the charge channel is of particular interest.In Fig. <ref> we show the largest eigenvalue from the magnetic and, more interestingly,the pairing channel where the paring vertex Γ_↑↓ is obtained from the linear combination of the corresponding singlet and triplet components, Γ_↑↓=1/2(Γ_s - Γ_t).On the square lattice, the single-band Hubbard model at half-filling is unstable against antiferromagnetic ordering at arbitrary interaction at zero temperature. Concomitant with this,Fig. <ref> shows a monotonous increase of λ in the magnetic channel which approaches 1 at low temperatures.In the paring channel, we show the leading eigenvalues corresponding to the s-wave and d-wave paring symmetries, respectively.The paring symmetry can be easily identified by inspecting the symmetry of the corresponding wave function v_q(k) of Eq. (<ref>) in momentum space. It is as shown in the right panel of Fig. <ref> for s- and d-wave pairing symmetry.At high temperatures s-wave is dominating and gradually gives way to the d_x^2-y^2 wave at low temperatures.This is seen more clearly in the 6 × 6 cluster in the inset of the left plot in Fig. <ref>. At half-filling, however, the antiferromagnetic eigenvalueλ_m still prevails (is closer to the divergence at λ_m=1). The eigenvalues and eigenfunctions at each channel are a direct output of victory. § CONCLUSIONS AND OUTLOOKS In this work, we presented a Fortran implementation of the parquet equations for the single-band Hubbard model.For a given fully irreducible vertex Λ, the parquet equations allow us to calculate all one- and two-particle correlation functions. The computational bottleneck is the memory consumption since the vertices are large objects that depend on threefrequencies and momenta.We have implemented two-level kernel functions to correctly account for the asymptotic frequency behavior of the two-particle vertex function in Matsubara frequency space. Employing this asymptotic behavior is essential to do calculations with a finite frequency box.The program is parallelized using mostly collective communication modes (one-to-all and all-to-all) which significantly decreases the number of point-to-point communications. Without further optimization, the current program is already feasible for the study ofthe Hubbard model on a 6×6 cluster for an arbitrary number of electrons.Further improvements of this package to include non-local Coulomb interactions and to reduce the number of cluster momenta in the calculations by employing the point group symmetry are currently under development.§ ACKNOWLEDGMENTWe are grateful for valuable discussions with Partrick Tunström, Nils Wentzell, Agello Valli, Angese Tagliavini, Georg Rohringer, Alessandro Toschi, Sabine Andergassen. We also greatly acknowledge the help of Claudia Blaas-Schenner in optimizing the MPI parallelization scheme.GL acknowledges the financial support from the starting grant of ShanghaiTech University.AK and KH have been supportedby the European Research Council under the European Union's Seventh Framework Program (FP/2007-2013) throughERC grant agreement n. 306447; Part of thecomputationalresults presented has been obtained using the VSC. elsarticle-num | http://arxiv.org/abs/1708.07457v1 | {
"authors": [
"Gang Li",
"Anna Kauch",
"Petra Pudleiner",
"Karsten Held"
],
"categories": [
"cond-mat.str-el"
],
"primary_category": "cond-mat.str-el",
"published": "20170824152823",
"title": "The {\\it victory} project v1.0: an efficient parquet equations solver"
} |
School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai, China Department of Physics, The Chinese University of Hong Kong, Shatin, N.T., Hong KongSchool of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai, ChinaDepartment of Physics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong Percolation analysis has long been used to quantify the connectivity of the cosmic web. Most of the previous work is based on density fields on grids. By smoothing into fields, we lose information about galaxy properties like shape or luminosity. The lack of mathematical modelling also limits our understanding for the percolation analysis. In order to overcome these difficulties, we have studied percolation analysis based on discrete points. Using a Friends-of-Friends (FoF) algorithm, we generate the S-bb relation, between the fractional mass of the largest connected group (S) and the FoF linking length (bb). We propose a new model, the Probability Cloud Cluster Expansion Theory (PCCET) to relate the S-bb relation with correlation functions. We show that the S-bb relation reflects a combination of all orders of correlation functions. Using N-body simulation, we find that the S-bb relation is robust against redshift distortion and incompleteness in observation. From the Bolshoi simulation, with Halo Abundance Matching (HAM), we have generated a mock galaxy catalogue. Good matching of the projected two-point correlation function with observation is confirmed. However, comparing the mock catalogue with the latest galaxy catalogue from SDSS DR12, we have found significant differences in their S-bb relations. This indicates that the mock galaxy catalogue cannot accurately retain higher order correlation functions than the two-point correlation function, which reveals the limit of the HAM method. As a new measurement, S-bb relation is applicable to a wide range of data types, fast to compute, robust against redshift distortion and incompleteness, and it contains information of all orders of correlation functions. New leaves of the tree: percolation analysis for cosmic web with discrete points Ming-Chung CHU December 30, 2023 ================================================================================§ INTRODUCTION Cellular or clumpy? That is the question. In the last century, when the first generation of redshift surveys were available, people were confused by the filamentary structure of galaxy distribution <cit.>. The existence of the cosmic web was qualitatively confirmed, but a quantitative description of it was not available until Zeldovich's percolation analysis <cit.>. We classify the percolation analyse for the cosmic web into two branches. The first one, dealing with discrete points, is called continuum percolation, whereas the second one, dealing with the density field on grids, is called site percolation. Continuum percolation analysis was also called cluster analysis in 1980s, because the same technique was used to define galaxy clusters. Cluster analysis can clearly tell the difference between a clumpy structure and web-like pattern <cit.>. However, Dekel & West <cit.> concluded that cluster analysis was insensitive to cosmological parameters, and it was also non-trivially affected by volume and point density. Thereafter, the site percolation analysis became popular, and properties of site percolation in large scale density fields have been studied in detail <cit.> . By smoothing points into field, the technique has also been used to look for properties of galaxy distribution <cit.>. We would like to point out that direct comparison between the percolation properties of density field and galaxy distribution is not fair, because galaxies don't trace the density field perfectly. If we can generate a mock catalogue of galaxies and compare it with a real redshift survey, the comparison will be fair and the result will give us some useful insights in properties of the cosmic web. On the other hand, the establishment of the ΛCDM model and rapid development of N-body simulation techniques provide us a general picture of the formation of the cosmic web <cit.>. We are now confident about the success of the ΛCDM model in large scales. However the three-point, four-point or even N-point correlation functions should also be tested <cit.>, though they are all very difficult to calculate andmodel in theory. In principle, the two-point correlation function costs O(N^2) calculations, and the three-point correlation function costs O(N^3) calculations, so on and so forth. Although we have better ways to optimize the calculation, higher order correlation functionsstill require unbearable calculation resources. Thus, several methods to characterize the cosmic web have been proposed, such as Minkowski Functional, Minimum Spanning Tree, and Genus Analysis <cit.>. These methods focus on the morphology and topology of the cosmic web. The comparison between simulation and observation is not straight forward. We are still lacking of knowledge about galaxy formation and evolution, and matching galaxies with simulated dark halos heavily depends on model <cit.>. Surprisingly, Halo Abundance Matching (HAM), a newly developed technique based on simple arguments, generates mock galaxy catalogues successfully <cit.>. HAM follows thehalo mass, halo circular velocity, halo potential well, galaxy circular velocity, galaxy luminosity, galaxy stellar mass, independent of galaxy formation model, and HAM is easy to understand. Shandarin et al. <cit.> developed and studied the percolation analysis, treating voids and superclusters on an equal footing and finding the nature and nurture of the formation of cosmic web. Using a similar method, the SDSS galaxy sample has been tested with the percolation analysis <cit.>. However we are still far from understanding the percolation transition in the cosmic web. Density estimation is a prior step for site percolation analysis, but estimatingthe density field from galaxy observation is not easy. We should take into account galaxies' different properties in the percolation analysis.Based on these considerations, we used the continuum percolation analysis to calculate the S-bb relation, where bb stands for the FoF linking length and S stands for the mass fraction of the largest group for the given linking length bb <cit.>. It is quite similar to the largest cluster statistics in previous percolation analyses <cit.>. However, we find that the continuum percolation analysis can distinguish among cosmological models with high resolution simulations, whose resolution was not available in 1985. We propose the Probability Could Cluster Expansion Theory (PCCET), to calculate the S-bb relation theoretically. We measure the S-bb relation in our mock galaxy catalogue and compare it with a real galaxy catalogue. We find that HAM cannot reproduce the S-bb relation as well as the projected two-point correlation function. § METHODOLOGY §.§ S-bb Relation and Percolation TransitionFor any discrete point distribution, we can always apply the FoF algorithm to find a group of points. The points can be massive points in N-body simulations, halos in simulated halo catalogues or galaxy catalogues in redshift space. We define S_1 and S_2 as: S_1(bb)=N_largest/N_total, S_2(bb)=N_second-largest/N_total, whereN_total is the total number of points,N_largest (N_second-largest) is the number of points in the largest (second largest) group found by FoF, and normally, we refer to the S-bb relation the function of S_1(bb).It is obvious that S_1(bb) reveals quite similar information as the cluster analysis and site percolation analysis.Our simulation tests are all done using<cit.>. Fig. <ref> is the S-bb relation for a random distribution as a comparison for later figures. It is not surprising that all three sets of data corresponding to different box sizes and number of particles give the same S_1(bb) and the transition threshold is around bb=0.88.We define the transition threshold as the bb where S_2(bb) gets the maximum value. Another definition of the transition threshold was also proposed in <cit.>; in our case, it can be defined as the bb where the mass-weighted sum of the number of connected structures, μ^2=Σ_n N_nN_groups, where N_n is the number of particles in the n^th group and N_groups is the total number of connected structures, gets the maximum value. We measured the S_1, S_2 and μ^2 using the halo catalogue of the MDR1 simulation<cit.> with haloes with mass larger than 10^12M_⊙. The MDR1 simulation was performed in a box with 1h^-1Gpc sides, and so we separated the sample into 64 subsamples of boxes with 250h^-1Mpc sides to estimate the cosmic variance, shown as the errorbars in Fig. <ref>. The transition threshold bb_c is 0.74 and the power index α given by S_1=(bb-bb_c)^α was measured to be α=0.46, which is consistent with <cit.>.As shown in Fig. <ref>, both definitions of the transition threshold are good, and their FWHM are similar. We will discuss the meaning of the transition threshold in the definition of S_2 later for easier understanding. When the percolation transition happens, the largest structure grows tremendously as bb gets larger, and the second largest structure reaches its most massive point. After that, the mass of the second largest structure becomes smaller, not because the structure shrinks, but because the original second largest structure is connected to the largest one while another structure is recognized as the new second largest one. This connection and replacement reveals the nature of percolation: the largest structure becomes the one and the only one dominant structure, connecting most of the high density regions together and leaving only the far apart particles in voids alone. Thus, the definition of transition threshold based on either S_2 or μ^2 is appropriate.While the ΛCDM (cold dark matter) model is not clearly distinguishable from the ΛWDM (warm dark matter) model, the ΛHDM (hot dark matter) model is clearly ruled out by observation of large scale structures. We would like to test our percolation analysis with these three models. The S-bb relations are expected to be indistinguishable for ΛCDM and ΛWDM models, but those for ΛHDM and ΛCDM models should be clearly different. This test is the bench mark test of percolation analysis. We generate the initial conditions with<cit.> and run the simulation from redshift 49 to redshift zero using the N-body simulation code <cit.>. The transfer function and additional thermal velocity for HDM and WDM are given by <cit.>. These three simulations share the same realization. Some basic parameters for the simulations are given in Table. <ref>.As shown in Fig. <ref>, the HDM model is quite different from CDM and WDM models in their matterpower spectra and S-bb relations. The matter power spectrum is calculated by the publicly available code <cit.>. It is well accepted that the HDM model has been ruled out by measuring the power spectrum in both CMB <cit.> and large scale structures <cit.>. We have shown here that the difference between HDM model and CDM model is significant enough in the S-bb relation to be used as a supplementary. On the other hand, the unique advantage of using the S-bb relation will be discussed in the following sections.§.§ Probability Cloud Cluster Expansion TheoryThe percolation analysis has been used for a long time to study the cosmic web. For site percolation there are analogies to similar questions in many other fields <cit.>. But for discrete points, there is still no available theory to link percolation with basic cosmological parameters. Here we propose the Probability Cloud Cluster Expansion Theory (PCCET) to connect the S-bb relation with correlation functions. The idea is as follows:* Consider a box with only one particle and take it as the only member in the largest group. So the S-bb relation will be a constant line S=1.* Add in another particle randomly, the probability of linking these two particles in a single group P(bb) being monotonically related to the linking length bb. The S-bb relation will be a step function jumping from S_1=0.5 to S_1=1.0 at a certain transition value of P: S=1/2+1/2P.* Add in the third particle randomly, the probability of linking any two of them being still P and the probability of linking all three of them shall be P^2. While from a certain particle as the starting point of the group, the probability of linking the other two particles shall be 2P but deducting P^2 to avoid over counting. The S-bb relation will then be a three-step function, with steps at 1/3, 2/3, and 1: S=1/3+1/3P+1/3P(2P-P^2).* Continuing on, we consider more and more particles, taking care of over counting and over deducting, and we finally find the relation S(P) when the number of particles goes to infinity. We hide all information about bb in the function P(bb).So in general, we can write S(P) as a_1=1/N, a_n=a_n-1b_n-1 (n>1), b_n=1-(1-P)^n, S=∑_n=1^N a_n. This series expansion is the reason why we call it cluster expansion theory. Cluster expansion is often used in condensed matter physics. It can successfully account for phase transitions in some typical problems. We borrow the idea of cluster expansion because of the similarity between percolation transition and phase transition. We have shown that S(P) is monotonically increasing with P and has a limit between 0 and 1. When N→∞, S(P) will become the Euler Function. The proof in detail is given in Sec. <ref>. In the framework of this theory, a particle cannot be treated as a discrete point. In fact, it is better to take the particle as a representation of the probability cloud. For a random distribution, the probability for a particle to be closer to another particle than distance r is inversely proportional to the volume of a sphere with radius r. However, we need a volume to set as the standard for this probability. Here we take all the particles in the theory as test points. We can first put in the test center. We imagine that every test center has its own cube with periodic boundary conditions. The length of each side of this test cube can be set as the average distance between particles in a real simulation. Then the distance to the center can be simply understood as the linking length bb. If bb is larger than 1/2, the influence of this test center will reach outside the test box. In such a case, we shall remove the influence region outside the test box and only consider the volume inside the box as the real probability distribution region. Shown in Eq. <ref> is the P-bb relation for a random distribution. From the PCCET, we can find the relation between percolation and correlation function. Because of the cluster expansion, the 2-point correlation function is the correction to P as the probability of linking up 2 points, while the 3-point correlation function is the correction to P^2 as the probability of linking up 2 pairs of points, etc. In general, the n-point correlation function is the correction to P^(n-1) as the probability of linking up n-1 pairs of points. This shows that percolation actually contains the information of all orders of correlation functions. Our probability cloud cluster expansion theory also shows how every order of correlation function contributes to the S-bb relation, which represents the percolation phenomenon in the cosmic web.Unfortunately, in the non-linear regime, high order correlation functions are also quite large and not necessarily smaller than the 2-point correlation function. So based on our theory, it is still not possible to calculate the whole S-bb relation analytically. It is important to use N-body simulations to set up constraints for the S-bb relation. After all, PCCET is still the only model explaining percolation. It can be used to find the S-bb relation in the linear-regime and it is also possible that in the future, we may find a way to use it in the non-linear regime to extract high order correlation functions.§.§ Halo Abundance MatchingFrom PCCET, we can see that the S-bb relation contains information of all orders of correlation functions. Thus we would like to see whether it provides new information in comparing simulation with observation. We first need to generate a mock galaxy catalogue and check the projected 2-point correlation function to see whether it matches the real galaxy catalogue. Then we will compare the S-bb relations of the mock and real galaxy catalogues.Based on the Bolshoi simulation <cit.> BDM halo merger tree, we follow the work in <cit.> using Halo Abundance Matching to generate a mock galaxy catalogue. Here is the process: * We take the Bolshoi simulation BDM halo catalogue at redshift 0 to extract the final positions and peculiar velocities of the galaxies. We assume that every BDM halo, including the host haloes and subhaloes, can host at most one galaxy at its center. * From the merger tree of BDM haloes in the Bolshoi simulation, we find the maximum circular velocity V_max in the merger history with the halo circular velocity defined as the maximum value of the circular orbital velocity according to the density distribution of the halo.* We adopt the luminosity function from <cit.>, which is of the Schechter form. The luminosity function is:Φ(M)dM=0.4log_10Φ_*10^-0.4(M-M_*)(α+1) ×exp(-10^-0.4(M-M_*))dM, where M is the absolute magnitude and Φ_*=0.0078, M_*-5log_10h=-20.83, α=-1.24, given by <cit.> using SDSS6 data.* We sort the haloes in the Bolshoi simulation according to their V_max. Record every halo's ranking. Thus, we can easily obtain the number density of haloes with V_max larger than a certain value by the ranking.* The integral of the luminosity function is an incomplete gamma function. We apply an one-one mapping from the number density of haloes with V_max larger than a certain value V to the integral of the luminosity function. Thus we obtain an one-one mapping between V and absolute magnitude M.* Combining the data together, we get the halo catalogue with the galaxy absolute magnitude. This is also the mock galaxy catalogue with scale 250h^-1Mpc. In this catalogue, we have positions, peculiar velocities and luminosities for the mock galaxies at redshift 0.Halo Abundance Matching has been proven quite successful in recreating a galaxy catalogue which can reproduce the observed projected galaxy 2-point correlation function in the bins of absolute magnitude close to the steep transform range in the luminosity function, around Φ=Φ_*. Many works follow the idea of HAM <cit.>. There are slightly different methods to obtain a mock catalogue based on the same basic idea of HAM. Comparing to <cit.>, we did not take stochastic scattering into account. Since there are obvious bias to dimmer galaxies in stochastic scattering, we avoid this process. On the other hand, we take maximum circular velocity in the merger history of every halo as its monotonic mapping label. The physical insight of this picture is clear. Gas should fall into the center of a halo much faster than the merger rate, and once the gas has fallen into the halo center, it is so dense that the merger process cannot do much to the central dense region. So the galaxy luminosity depends more on the steepest potential well in the merger history, denoted by the maximum circular velocity V_max. We follow the paper <cit.> to calculate the projected 2-point correlation function. Our results are shown in Fig. <ref>. We have confirmed the previous results given by <cit.> that the HAM reproduced sample exhibits the same projected 2-point correlation functions as the observation in the luminosity range [-20,-19], [-21,-20] and [-22,-21], with differences mostly within 1 or 2 σ. We take these samples for further percolation analysis comparison. We shall also notice that the bias of the projected 2-point correlation function depends on the luminosity of the galaxy. We shall have a different view of the bias in the S-bb relation.§ RESULT §.§ S-bb Relation EvolutionWe are interested in the evolution of the cosmic web, which we study through an ΛCDM cosmological simulation with 512^3 particles in a box with 1h^-1Gpc sides, initial conditions generated by BBKS power spectrum <cit.> and Zel'dovich approximation <cit.>. We apply Ω_m=0.28, Ω_Λ=0.72 and h=0.7 for this test. As shown in Fig. <ref>, the S-bb relation at high redshifts looks similar to that of random distribution (Fig. <ref>), while as the density contrast increases with time, the S-bb relation at low redshift becomes rather different, which has already been known in many previous works <cit.>. §.§ Bias DeterminationThe error bars of the S-bb relation shown in fig. <ref> are determined as follows. Based on Bolshoi simulation<cit.>, we have only one realization of the HAM mock sample. So we use jackknife re-sampling to estimate the error bars. Here are the steps: * randomly choose 90% of the HAM mock sample for ten times, generating 10 different mock samples.* Measure the S-bb relations of these 10 mock samples.* Take the mean and standard variance of these 10 S-bb relations as the value and error bar. From Fig. <ref> we can see that the lower luminosity samples' transition threshold bb is lower than those of the high luminosity samples. Based on this result, we may guess that the lower luminosity samples have higher bias in the 2-point correlation function than higher luminosity samples. However it is well known that more luminous galaxies have higher bias which seems to be contradictory to our results here. Note that the definition of bb is such that it is normalized to the average distance of points. So we shall look at the 2-point correlation function with the projected distances r_p also normalized to the average distance, which is shown in Fig. <ref>. The higher the luminosity of the samples are,the lower their normalized 2-point correlation function is. Therefore, a smaller transition threshold is well expected.Galaxies' real spatial distribution is different from their redshift space distribution due to the peculiar velocities of the galaxies. Using the HAM mock sample data with peculiar velocity, we can study the redshift distortion effect in the S-bb relation. We uniformly choose 14 different directions to look at the sample, all of which are about z=0.1 away from the sample center. The distance is translated by the cosmological parameters used in the Bolshoi simulation. As shown in Fig. <ref>, we find that redshift distortion has little effects on the S-bb relation. Because the S-bb relation shows mainly the topological information of the cosmic web, it is reasonable that it is not sensitive to local distortions. Our observation data comes from SDSS DR12 <cit.>. We take the North Galactic Cap (NGC) as our galaxy zoo region because of the continuous, large coverage of the NGC. Thus we have the spectrum of galaxies covering huge volume in the red-shift space in NGC. We obtain the data of galaxies in the range of 130<RA<230, 10<Dec<50, and 0.07<z<0.14, including their coordinates, spectral redshifts and Petrosian magnitudes in the r band. The volume of this region is quite similar with the volume of the Bolshoi simulation. Then we use the same cosmological parameters as the Bolshoi simulation to calculate the luminosity distances of the galaxies using their spectral redshifts. We also calculate the Petrosian absolute magnitudes of the galaxies. But we didn't correct their absolute magnitudes. So the magnitudes of the galaxies are in fact in the r^0.1 band as described in <cit.>, since the Luminosity Function we use to generate the HAM mock galaxy catalogue is also in the r^0.1 band. The K-correction for this sample is also not important because we have chosen the red shift range to be 0.07<z<0.14. According to the discussion in <cit.>, the K-correction for r^0.1 is no more than 0.1. So we just ignore K-correction for our observation sample. Now we have generated an observation galaxy catalogue with 3-D space positions and reasonable absolute magnitudes. The volume covered by the sample is also similar to the volume covered by the HAM mock sample.Before SDSS DR12, there was no such large continuous angular coverage as well as deep redshift coverage. The public access to the SDSS DR12 allows us to play with a unprecedented, continuous, large scale, 3-D galaxy zoo. It is a good time to search for unusual statistical measurement of the large scale structures, not just the percolation analysis we proposed here but also many other ways of viewing the structures as a whole web.However, we have to point out that the observational sample is in redshift space while the HAM mock sample is in real space. So the direct comparison of these two samples is not fair. We must take redshift distortion into consideration. Fortunately, our HAM mock sample also contains information of peculiar velocity, which means we can easily transfer the HAM mock sample into the redshift space.We also need to consider the incompleteness of the observational sample. As shown in the DR12 paper <cit.>, that is mainly because of fiber collision. We can also simulate the incompleteness by excluding those haloes in the simulation that looks closer than 62" projected on the 2D plane due to fiber collision, which will exclude ∼ 5% halos mainly in high density region. From Fig. <ref> we can see that, this incompleteness has negligible effect for the S-bb relation. Therefore, the S-bb relation is very robust against the observational selection bias introduced by fiber collision. We also find that, changing from the adopted cosmological parameters in both Bolshoi and MDR1 simulations to Planck cosmology, the S-bb relation will not be changed significantly. However, due to the limited boxsize of Bolshoi simulation, the difference between the S-bb relation of MDR1 and Bolshoi simulations is observable. That will cause the misidentification of the transition threshold by 0.04, but for bb>0.8, the S-bb relation difference is negligible. Therefore, using the HAM mock sample built from the Bolshoi simulation to compare with observation is fair, but we need to be aware of a small misidentification of the transition threshold, because of the limited size of the simulation.§.§ Comparison with SDSS DR12 Here we are comparing the observational galaxy sample and the HAM mock galaxy sample in two luminosity ranges -22<M<-21 and -21<M<-20, as shown in Fig. <ref>. We choose these two ranges because, firstly, we need generally enough number of galaxies to reduce the Poisson error and secondly, we require the galaxies to be detectable even at red shift z=0.14 to get rid of Malmquist bias.Interestingly, although the projected 2-point correlation functions of the HAM mock sample and observed sample are quite similar, their S-bb relations are different by 2-3 σ, particularly the transition point in the S_2-bb relation. According to our PCCET, this is most probably because the HAM mock sample doesn't reproduce the high order correlation functions well enough. This is also supported by a recent work <cit.> comparing Illustris simulation and the data sample generated with the same two-point correlation function. The measured S-bb relations (called 'giant component-linking length' relation), for the two samples are significantly different. The transition threshold is also smaller for the real simulation sample. § CONCLUSION AND DISCUSSIONWe have proposed the S-bb relation as a branch of percolation analysis of the cosmic web. We have shown that the S-bb relation and its transition threshold reveal physical information of the cosmic web. It also has some advantages compared to the standard 2-point (n-point) correlation function: * The S-bb relation is based on the Friend-of-Friend (FoF) algorithm, which is easy to implement and costs very little computational resources.* Our theoretical model, PCCET, shows that the S-bb relation contains information of all orders of correlation functions.* The S-bb relation is robust against redshift distortion. * The S-bb relation is robust against incompleteness, which is very useful in observation. We have shown that 90% completeness is quite acceptable and this has been reached in SDSS.* The S-bb relation gives information on the bias of the correlation function.However, it also has some limitations: * We need large sample of galaxies to look at the S-bb relation, with at least 100h^-1Mpc box side length.* We need large continuous region of observation to study the S-bb relation, especially for 3-D analysis.* It is not easy to extract every order of the correlation functions from the S-bb relation.Cellular or clumpy? That's not the question any more. It is well known that the distribution of galaxies is cellular. How cellular is the cosmic web? That's the question. We suggest the S-bb relation as a new measurement, which is applicable to a wide range of data types from simulation particles to galaxy catalogue, fast to compute (as fast as 2-point correlation function measurement), robust (against redshift distortion and incompleteness), and it contains information of all orders of correlation functions. In the framework of PCCET, we have good understanding of the S-bb relation. We have compared the S-bb relation of the HAM mock galaxy catalogue with SDSS DR12 galaxy catalogue. A significant difference was found, though these two samples had similar projected 2-point correlation functions. The percolation analysis with discrete points can help us understand the cosmic web. At the minimal, it provides a quick check of whether our mock sample really represents the real distribution or not. § APPENDIX §.§ Proof for PCCETThe series in Eq. <ref> is known q-Pochhammer function or Euler function in mathematics. The q-Pochhammer symbol (a;q)_n is defined as (a;q)_n = ∏_k = 0^n-1(1-aq^k)= (1-a)(1-aq)(1-aq^2)⋯(1-aq^n-1).In particular, the Euler function ϕ(q) is defined as the infinite productϕ(q) = (q;q)_∞, q = 1 - p. The S_n(p) can be expressed in terms of the q-Pochhammer symbols S_n(p) = 1/n∑_k=0^n-1(q;q)_k,where0 < q < 1.Since 0<q<1, it is not difficult to realise the fact that1 > (q;q)_1 > (q;q)_2 > ⋯ (q;q)_n > 0.ThereforeS_n(q) > 1/nn(q;q)_n-1 = (q;q)_n-1.On the other hand,S_n(q) = 1/n∑_k = 0^n - 1(q;q)_k = 1/n∑_k = 0^m - 1(q;q)_k + 1/n∑_k = m^n - 1(q;q)_k.Here we introduce an integer m such thatm = INT(√(n)),where INT(x) is defined asINT(x) = max{n ∈ Z | n ≤ x}.Since as n increases, (q;q)_n decreases, and we have S_n(q) = 1/n∑_k = 0^m - 1(q;q)_k + 1/n∑_k = m^n - 1(q;q)_k < m/n(q;q)_0 + n - m/n(q;q)_m= m/n + n - m/n(q;q)_m.In summary,(q;q)_n-1 < S(n) < m/n + n - m/n(q;q)_m.m as a function of n has the following asymptotic propertieslim_n →∞ m = ∞, lim_n →∞m/n = 0.For simplicity, we denoteL_n(q) = (q;q)_n-1, U_n(q) = m/n + n - m/n(q;q)_m.Then lim_n →∞ L_n(q) = (q;q)_∞ = ϕ(q),lim_n →∞ U_n(q) = (q;q)_∞ = ϕ(q).Since we always haveL_n(q) < S_n(q) < U_n(q),according to the Squeeze Theorem, we concluded thatlim_n →∞ S_n(q) = ϕ(q).In the form of a step function, we find the limit of S_n(p) to be:lim_n →∞ S_n(p) = 0ifp = 0,ϕ(1-p)if0 < p < 1, 1ifp = 1.§ ACKNOWLEDGEMENTS J. Zhang thanks Prof. Y. Jing for giving quite useful advices, L. Yang and Z. Li for helping with measuring 2-point correlation function, Dylan for useful discussion, S. Liao for useful discussion and advice throughout this project, J. Chen for his great help in studying the PCCET mathematically in depth. “The CosmoSim database used in this paper is a service by the Leibniz-Institute for Astrophysics Potsdam (AIP). The MultiDark database was developed in cooperation with the Spanish MultiDark Consolider Project CSD2009-00064.”The authors gratefully acknowledge the Gauss Centre for Supercomputing e.V. (www.gauss-centre.eu) and the Partnership for Advanced Supercomputing in Europe (PRACE, www.prace-ri.eu) for funding the MultiDark simulation project by providing computing time on the GCS Supercomputer SuperMUC at Leibniz Supercomputing Centre (LRZ, www.lrz.de).The Bolshoi simulations have been performed within the Bolshoi project of the University of California High-Performance AstroComputing Center (UC-HiPACC) and were run at the NASA Ames Research Center. | http://arxiv.org/abs/1708.07602v2 | {
"authors": [
"Jiajun Zhang",
"Dalong Cheng",
"Ming-Chung Chu"
],
"categories": [
"astro-ph.CO"
],
"primary_category": "astro-ph.CO",
"published": "20170825024046",
"title": "New leaves of the tree: percolation analysis for cosmic web with discrete points"
} |
Learning a 3D descriptor for cross-source point cloud registration from synthetic data Xiaoshui HuangUniversity of Technology [email protected] December 30, 2023 ============================================================================================ As the development of 3D sensors, registration of 3D data (e.g. point cloud) coming from different kind of sensor is dispensable and shows great demanding. However, point cloud registration between different sensors is challenging because of the variant of density, missing data, different viewpoint, noise and outliers, and geometric transformation. In this paper, we propose a method to learn a 3D descriptor for finding the correspondent relations between these challenging point clouds. To train the deep learning framework, we use synthetic 3D point cloud as input. Starting from synthetic dataset, we use region-based sampling method to select reasonable, large and diverse training samples from synthetic samples. Then, we use data augmentation to extend our network be robust to rotation transformation. We focus our work on more general cases that point clouds coming from different sensors, named cross-source point cloud. The experiments show that our descriptor is not only able to generalize to new scenes, but also generalize to different sensors. The results demonstrate that the proposed method successfully aligns two 3D cross-source point clouds which outperforms state-of-the-art method.§ INTRODUCTION Point cloud is an important type of geometric data structure, which is not affected by occlusion, illumination and geometric transformation. Because of this property, point cloud is becoming prevalent as a format to store 3D data. There recently has endured high development in 3D sensors. Nowadays, there are many sensors available to produce point clouds, such as KinectFusion, LiDAR, range camera. Cross-source point clouds are data coming from different kinds of sensors. As different sensor shows different ability in sensing our environment, for example, KinectFusion shows strong ability in small-scale scene with many details and LiDAR shows strong ability in large-scale scene, registration of these point clouds shows indispensable. Therefore, cross-source point cloud registration problem emerges. It shows many applications including automatic driving, smart city and smart phone application.The problem existing in cross-source point cloud registration are mainly lies in the variations on density, missing data, different viewpoints, scale, noise and outliers <cit.>. There is some papers in dealing with cross-source point cloud registration <cit.>. One of the key procedure of the exiting methods is the similarity measurement of two components. The exiting methods are mainly relied on handcraft features, such as graph descriptors <cit.>. Recently, learning methods (e.g. convolution neural network (CNN)) show great ability in dealing with different kinds of computer vision problems. In this paper, we propose a method to utilize the ability of learning methods to deal with 3D cross-source point clouds registration.The main difficulty is that point cloud is usually organized as irregular format where each point represents a 3d position. This type of data is difficult to be directly utilized as input for neural network because of its irregular property ( examples of regular data are RGB-D data and images). Most researchers typically transform such data to regular 3D voxel grids or collections of images before feeding them to neural network architecture (e.g. <cit.>). However, the limitations are summarized as two folders: * they will render the resulting data unnecessarily voluminous while also introducing quantization artifacts that can obscure natural invariance of the data; * the existing methods are sensitive to rotation variance.The difficulties lying in learning on cross-source point cloud are that there is no stable keypoints in cross-source point cloud because of the many cross-source variations. It is difficult to find positive samples to train a neural network. Although there are many variations on cross-source point cloud, based on our observation, the structure of two point clouds remains similar. Inspired by this observation, in this paper, the function of structure similarity measurement is learned. If these two patches are describing a same structure, the similarity is very high. Otherwise, the similarity is very low. In order to fulfill our goal, we propose a region-based method to train a neural network.In this paper, we propose a region-based sampling method to obtain training data for neural network architecture (learning framework) and utilize the original point clouds as input for neural networks. We also use data augmentation method to let our network be robust for point cloud rotation. Furthermore, Our work focuses on more general case where point cloud are coming from different sensors (cross-source point cloud). We try to learn 3D descriptors for cross-source point cloud to address cross-source 3D registration (See Figure <ref>). The contributions are two folds: 1) a region-based method is proposed to deal with 3D point cloud training problem. It is robust to noise and outliers. 2) 3D rotation variant is dealt with by data augmentation.§ RELATED WORKS Learning a 3D descriptor for cross-source point cloud registration is a intersection area of computer vision and machine learning. We briefly review the related works on both domains. §.§ Handcraft 3D descriptors for point cloudMany 3D descriptors have been proposed including Fast Point Feature Histogram (FPFH) <cit.>,Signature of Histograms of Orientations (SHOT) <cit.>, and Clustered Viewpoint Feature Histogram (CVFH) <cit.>, Ensemble of Shape Functions (ESF) <cit.>. X. Huang <cit.> proposes relative geometry descriptor to describe graphs of 3D point cloud. Many of these 3D descriptors are now available in the Point Cloud Library <cit.>. Although these existing 3D descriptors have make a great progress for 3D computer vision, they stillstruggle to handle density variation, missing data, viewpoint difference variation, large proportion of noise and outliers in real-world data fromdifferent 3D sensors. Furthermore, since different sensors usually have different sensing mechanisms and they manually designed for specific applications, the captured 3D data are usually have large variance due to their different sensors and different captured environment. In this paper, we propose a method to learn a 3D descriptor that directly learns from irregular 3D point cloud. The new learned 3D descriptor shows more robust and high discriminative in a variety of 3D applications.§.§ Learned 3D descriptorsThere has also been rapid progress in learning geometric representations on 3Ddata. 3D ShapeNets <cit.> introduced 3D deep learning formodeling 3D shapes, and several recent works <cit.>also compute deep features from 3D data for the task of object retrieval and classification. While these works are inspiring, their focus is centered on extracting features fromcomplete 3D object models at a global level. In contrast,our descriptor focuses on learning geometric features for irregular 3D point cloud at a local level, to providemore robustness when dealing with partial data suffering from various occlusion patterns and viewpoint differences.Guo <cit.>, which uses a 2D ConvNet descriptor to match local geometric features for mesh labeling. <cit.> learned from RGB-D data and utilize to deal with 3D point cloud feature extraction. However, these methods are all trained or designed for same sensors, they face large challenging when confront with different sensors' data with different data modalities. In contrast, our work not onlytackles the harder problem of matching real-world partial 3D point cloud, but also generalize our methods to different sensors in a spatially coherent way. Our method has higher robustness to different sensors' variants and shows ability to deal with rotation spatial transformation. § REGION-BASED METHOD AND DATA AUGMENTATIONThere are two main works in the proposed methods: region-based method for sample selection and data augmentation for addressing geometric rotation between cross-source point clouds.Region-based method: Supposing two cross-source point clouds PC1 and PC2, we try to use region-based method to construct positive and negative samples. Firstly, features (e.g. curvature, FPFH, normal or 3D coordinate) are computed for every point in PC1 and PC2. Secondly, starting from PC1, we randomly select a point and use the features of the point to cluster similar neighbor points based on different random threshold. If the number of clustering points is more than 50, the 3D box containing these points are recorded. We aim to keep each region to contain similar shape or structure. Thirdly, using this 3D box, we crop a region from PC2. If the number of points are more than 50, we consider these two regions from PC1 and PC2 as positive samples (matched regions). We also capture a negative samples (non-matched regions) which the center of its containing box is a distance away to this positive sample. The distance ranges from 3*r to max radius of point cloud. r is the radius of the 3D containing box. For each scene, we sample 100 thousand positive and 100 thousand negative samples to construct the training dataset. Data augmentation: In order to be robust to rotation, we use data augmentation to expend the original database. We randomly rotate an angle from -90^∘ to 90^∘ for each positive and negative samples. After data augmentation, one positive sample becomes 2 positive samples and the same increment on negative samples.The advantages of the proposed method are three folds: First, region-based sampling methods delete many regions that have very sparse points. These high sparse regions often are very hard to maintain the structure similarity (see red box of Figure <ref>). Using the refined regions, the dataset are more robust to utilize the structure information. Second, because the region-based method are usually has different region size, combined with high variance of cross-source point clouds, this method provides a large and diverse correspondence training data set. Training by our method, although the keypoints are difficult to extract, the network can still train a robust descriptor and be applied to real-world 3D cross-source point cloud registration (See our experimental part).Third, due to the data augmentation, our network can be endowed with rotation ability. Therefore, the trained framework using the data augmentation samples can be robust to address the rotation of point clouds which is a common problem in many 3D applications. § NETWORK ARCHITECTUREThe architecture of this paper is depicted in Figure <ref>. The network (ConvNet) consists 10 layers with 9 convolution layers and 1 pooling layer. For each convolution layer, it is a standard 3D convolution layer, the only difference is the output size. For example, the convolution layer with parameter (3^3, 64) represents this layer consists 64 kernels of size 3×3×3. For pooling layer, it has 64 kernels with size 2×2×2. Because we use the 3D region with variant size as inputs, the network can be trained with even larger and diverse dataset which has higher generalization. § TRAININGDuring training, our objective is to optimize the local descriptors generated by the proposed network (ConvNet) such that they are similar for 3D patches corresponding to the same point, and dissimilar otherwise. To this end, we train our ConvNet with two streams in a Siamese fashion where each stream independently computes a descriptor for a different local 3D patch. The first stream takes in the local 3D patch around a surface point p1 , while the second stream takes in a second local 3D patch around a surface point p2 . Both streams share the same architecture and underlying weights. We use the L2 norm as a similarity metric between descriptors, modeled during training with the contrastive loss function <cit.>. There is a margin super parameter in this loss. The margin is used to control the minimizationon positive 3D point pairs, while the negative 3D point pairs only work when the distance below the margin. In training process, the network has an input of a same number of positive and negative samples. With the same positive and negative samples feeding for the network, it has shown high efficiency to learn discriminative descriptors <cit.>. § APPLICATIONWe take one application as 3D cross-source point cloud registration. Figure <ref> shows the pipeline of 3D cross-source point cloud registration. The inputs are two cross-source point clouds that needs to be registered. The outputs are the registration result and transformation matrix. In our pipeline, there are three main steps. Firstly, we extract the control points that robustly represent the structure of the point clouds. In this paper, we use 3D segmentation method <cit.> to segment the point cloud into many parts, the center of each segment is the control point.Secondly, we use the network in Figure <ref> to extract the features and compare the similarity of these control points. In this step, we use the containing sphere of its belonging 3D segment as input and use Truncated Distance Function (TDF) in <cit.> to convert the original point cloud into another 3D representation that suitable to do 3D convolution. Thirdly, we use simple 3D RANSAC to do the outliers detection and obtain the final transformation matrix.§ EXPERIMENTSIn this section, we conduct experiments and compare with the related work <cit.>. We do three experiments, (1) one experiment is to compare the ability in dealing with cross-source problem; (2) the other experiment is to evaluate how well the proposed method in dealing with rotation and translation combined with cross-source problem; (3) apply to the real cross-source 3D point cloud registration. §.§ Compare the ability on cross-source problemTo test the performance of the proposed method in dealing with cross-source problem, we separately evaluate and compare the ability in each variant and compare with <cit.>. We test how our method is influenced by factors of position, cross-source problem and rotation. We have done four experiments: Experiment 1 is a baseline to test the accuracy of the algorithm. Experiment 2 aims to test different position impacts the descriptor. Experiment 3 aims to test cross-source problem impacts the descriptor. Experiment 4 aims to test rotation impacts the descriptor.We use overlap ratio to evaluate the final registration accuracy. For overlap ratio, we compute nearest neighbor for each point, then count the number of point whose distance less than 0.05. We use the number divide by the total point number as the overlap ratio. So, overlapratio=N/N_t. N is the number whose nearest distance less than 0.05, N_t is the total point number.Table <ref> shows that both methods obtain 100% overlap ratio in Experiment 1 and Experiment 2. It reflect both our methods can be used to deal with 3D point cloud and invariant to translation variant. Experiment 3 shows our method is much more robust than <cit.> when facing cross-source problem, the overlapping ratio improves 94%. Experiment 4 shows our method still remain 6% overlapping ratior while <cit.> totally fails. Compared with <cit.>, this experiment shows the proposed method shows higher ability in cross-source problem and can deal with part of the rotation problem.§.§ Evaluate and compare the ability on transformation variantIn this section, we will conduct thorough experiments on transformation variants, which include experiments on rotation, shifting, rotation+shifting.Now, I will describe details about how do I conduct these experiments. Rotation: One point cloud (P1) is copied and rotated by every 5 degree to build another point cloud (P2). Totally, 72 sets of point cloud are produced. Each rotated point cloud with the original point cloud combined to build a pair with rotation variation. We sample 500 keypoints from P1 and record the index in the point cloud. In the rotated point cloud, I recall these keypoints by using these indexes. The groundtruth is the index correspondence. I test the training detector on these 72 pairs. I use the recall ratio R_1= R_0/R_N to evaluate the performance, where R_0 is the total number that the best matched descriptor are the groundtuth, R_N is the point number. The results show in Figure <ref>. It shows that the accuracy of our method shows more than 15% within 10 degree rotation while 3DMatch only detect about 0.1%. When the rotation angle goes higher, the recall ratio goes down to approximately 2% while 3DMatch goes 0%. The accuracy increases from 160^∘ and reach the peak in 180 ^∘. The next 180^∘ range goes similar cycle. Shifting on keypoints: I shift keypoint randomly from 1% to 50% of the radius of point cloud containing sphere, totally 50 sets. If the point are shifted out of the containing sphere, we use the original point as the shifted point. Figure <ref> shows the results that our method shows high accuracy than 3DMatch. Rotation and Shifting: I copy P1 to P2 and add each point in P2 a random shifting about 10% of containing sphere. Then rotate P2 by every 5 ∘, totally 72 sets. Figure <ref> shows our method obtains better results at most cases than 3DMatch. The proposed method achieves obviously better results within 20 degree rotation angle. Also, we obtain slightly better results than 3DMatch in angles larger than 20 degree rotation angle. It shows our ability in dealing with part of the rotation problem. § EXPERIMENTS ON REAL 3D CROSS-SOURCE POINT CLOUD REGISTRATIONIn this section, we will apply our network on real 3D cross-source point cloud registration and compare with 3DMatch <cit.>. We capture one source of point cloud by using KinectFusion, and another source of point cloud by using struction-from-motion <cit.>. These two sources of point clouds constitute cross-source point clouds. Following <cit.>, we firstly normalize the scale variation. Because stable keypoints are very rare in cross-source point clouds, we use <cit.> to extract the structure points of the cross-source point clouds which is the stable information we currently know. With these structure points are extracted, we use the proposed method to extract the feature of these points and compare with 3DMatch.Figure <ref> shows the registration results of both methods. From Figure <ref> we can see that the proposed method are much robust than 3DMatch in cross-source point cloud registration. § CONCLUSIONIn this paper, we propose a method to learn a 3D descriptor for cross-source point cloud registration. We propose two strategies to train a robust neural network. Firstly, we use region-based samples to train the network. Secondly, we use data augmentation to extend our network be robust to rotation transformation variant. The experiments demonstrate that the proposed method output other method in terms of robustness and accuracy in the problem of cross-source point cloud registration.ieee | http://arxiv.org/abs/1708.08997v1 | {
"authors": [
"Xiaoshui Huang"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170824223802",
"title": "Learning a 3D descriptor for cross-source point cloud registration from synthetic data"
} |
FIT]Aiwu Zhangcor1 [cor1]Corresponding author: Tel. +1(631)344-5483; Email mailto:[email protected]@bnl.gov. Now with Brookhaven National Laboratory. FIT]Marcus Hohlmann BNL]Babak Azmoun BNL]Martin L. Purschke BNL]Craig Woody[FIT]Department of Physics and Space Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA [BNL]Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA We study the position sensitivity of radial zigzag strips intended to read out large GEM detectors for tracking at future experiments. Zigzag strips can cover a readout area with fewer strips than regular straight strips while maintaining good spatial resolution. Consequently, they can reduce the number of required electronic channels and related cost for large-area GEM detector systems. A non-linear relation between incident particle position and hit position measured from charge sharing among zigzag strips was observed in a previous study. We significantly reduce this non-linearity by improving the interleaving of adjacent physical zigzag strips. Zigzag readout structures are implemented on PCBs and on a flexible foil and are tested using a 10 cm × 10 cm triple-GEM detector scanned with a strongly collimated X-ray gun on a 2D motorized stage. Angular resolutions ofare achieved with aangular strip pitch at a radius of 784 mm. On a linear scale this corresponds to resolutions below .GEM; Zigzag readout strip; Flex readout foil; Linear response; Spatial resolution § INTRODUCTION The concept of zigzag-shaped readout pads was first proposed for gaseous time projection chambers (TPC) in the 1980s in order to reduce the number of electronic channels required to read out the detector <cit.>. Later, MWPCs and GEMs were read out with parallel zigzag strips and good spatial resolutions were achieved <cit.>. This was confirmed in more recent studies which showed that the spatial resolution of a small GEM detector with parallel zigzag strip or zigzag pad readout can approach 70 μm <cit.>. We subsequently introduced radial zigzag strips to read out trapezoidal large-area GEM detectors <cit.>, which are intended for tracking systems at future experiments, e.g. forward tracking at the electron ion collider (EIC) <cit.>. Radial zigzag strips can precisely measure the ϕ coordinates of incident particles in order to track them and to determine their transverse momenta in a solenoidal field.In our previous studies <cit.>, we observed a non-linear relation between incident particle position and hit position measured from charge sharing among radial zigzag readout strips. This paper aims at quantifying the non-linear response of our previous zigzag designs and to demonstrate an improved zigzag design that has a linear response. A linear response ensures the accuracy of hit position measurements without the need for any corrections. For this purpose, six readout boards with different geometrical zigzag strip structures are produced and tested using a 10 cm × 10 cm triple-GEM detector on a 2D motorized stage. The incident particle position is defined by a highly collimated X-ray beam (140 μm × 8 mm collimator slit) in these measurements.§ ZIGZAG READOUT STRIP DESIGNS AND TEST BOARDS As shown in Fig. <ref>, zigzag strips can be designed by connecting certain points on strip center lines and reference lines in certain patterns. Four parameters are used to calculate the coordinates of these points: the starting radius of the strips, the period of the zigzag structure in the R direction (a fixed number of 0.5 mm in our studies), the ϕ-angle pitch between strips (1.37 or 4.14 mrad in our studies), and a fraction f of the angle pitch which defines the reference lines and determines the width, space, and interleaving of the zigzag strips <cit.>. There are two ways to use these points. In the first design method shown on the left in Fig. <ref>, a strip is outlined by points on a strip center line and two reference lines near it. In the second design method shown on the right in Fig. <ref>, a strip is outlined by points on the two center lines of neighboring strips and on the two reference lines. We define the “interleaving” quantitatively as the distance of overlap between two adjacent strips in the azimuthal direction divided by the strip pitch. Then this second design has a “100% interleaving” since the tips on one zigzag strip reach the centers of its two neighboring strips; this is expected to give better charge sharing and a more linear position response.Fig. <ref> (left) shows the zigzag design produced with the first design method that was used in our previous study <cit.>. The angle pitch was 1.37 mrad, the period of the zigzag structure in the R direction was 0.5 mm and the fractional strip pitch parameter was f = 0.75. Two printed circuit boards (PCBs) with this design were produced by industry (Fig. <ref> right): one with 48 strips of radii from 1420 to 1520 mm (“ZZ48 board”), the other with 30 strips of radii from 2240 to 2340 mm (“ZZ30 board”). The strip length on both boards was 10 cm except for strips near the edges of the active area and the strips on each board covered an area of approximately 10 cm × 10 cm. As can be seen in the inset in Fig. <ref> (right), the physical strips on the manufactured boards had a “spine” along the center of each strip due to low manufacturing precision in the etching of sharp points and corners. Also the space between strips turned out to be wider than what had been designed. No effort was made to further improve the quality of these two boards. Instead, we produce and test new zigzag strip boards with the “100% interleaving” design (Fig. <ref> left). For versatility, each board is designed with two different radial strip geometries. On the left side of the board, the radial strips are arranged with an angle pitch of 4.14 mrad and starting radius of 206 mm, while on the right side of the board radial strips are arranged with an angle pitch of 1.37 mrad and starting radius of 761 mm (Fig. <ref> right). We choose these two particular parameter sets because they correspond roughly to the inner and outer radial sections of a large-area trapezoidal GEM detector design for an EIC forward tracker prototype. They also have similar linear strip pitches around 1 mm. The two other design parameters, period of the zigzag structure in the R direction and fraction parameter f, are kept the same at 0.5 mm and 0.4, respectively. The strips are again about 10 cm long except for those near the edges of the active area. The capacitance between two adjacent zigzag strips is measured to be (22±2) pF and the capacitance between a strip and the readout board ground is measured to be (28±2) pF. The errors here reflect strip-to-strip variations in the measurement. We find that the “100% interleaving” design pushes the limits of industrial PCB production capabilities since the strong interleaving requires the spaces between adjacent strips to be less than 3 mils (76 μm). Three versions of the board are produced by a US PCB company (Accurate Circuit Engineering, ACE) and feature actual interleaves of 81%, 88%, and 133%, respectively (Fig. <ref>). CERN also produced a board with the same design, but with zigzag stripsimplemented on a kapton foil instead of on a PCB. The foil is then glued onto a honeycomb board for mechanical support. The main reason for producing this board is that currently CERN is the only place that can produce 1-meter-long zigzag designs on a kapton foil as required for our future R&D demands. Consequently, this small board serves as a pilot to see whether CERN can produce the zigzag strip structures on foil with sufficient accuracy.Fig. <ref> shows microscopy photos of the left sections of ACE and CERN boards with strip angle pitch 4.14 mrad. The first board (“ACE#1”) is produced using a standard PCB etching method, where copper thickness is about 18 μm (1/2 oz. PCB standard). In this board the tips are overetched, so they are not fully reaching the adjacent strip centers. The estimated interleaving between zigzag strips is 81%. Fig. <ref> shows a microscopy photo of the right section of the same board with strip angle pitch of 1.37 mrad; the interleaving here is about 80%. The second board (“ACE#2”) is produced by using half the copper thickness (9 μm, 1/4 oz. PCB std.) of ACE board #1. The resulting interleaving of 88% comes closest to the actual design. With the third board (“ACE#3”), we try to actively compensate in the design for the overetching problem. However, it turns out that in the actual board the strips become too thin (Fig. <ref>) and the interleaving comes out significantly larger than 100%. Finally, the CERN foil board (“CERNZZ”) achieves an interleaving of 89%, which is the closest to 100% interleaving that has been achieved with a chemical etching technique so far. For convenience, we refer to the various boards described above for short as ZZ48, ZZ30, ACE #1, ACE #2, ACE #3, and CERNZZ from here on. § EXPERIMENTAL CONFIGURATION AND PROCEDUREWe test all boards with the same 10 cm × 10 cm triple-GEM detector and a highly collimated X-ray gun on a 2D motorized stage. The detector is flushed with Ar/CO_2 (70:30). Fig. <ref> shows the setup and a sketch of the X-ray collimator. In order to be consistent with our previous study <cit.>, the gas gaps in the GEM detector are set to 3/1/2/1 mm for drift gap, transfer gaps 1 and 2, and induction gap, respectively. We apply high voltage settings as if there was an HV divider with a resistor chain of 1/0.5/0.5/0.45/1/0.45/0.5 MΩ so that a single voltage V_drift applied to the drift electrode will determine all the electric fields in the detector although individual HV channels are actually used to power the electrodes. The X-ray collimator is 100 mm thick with a slit that is 8 mm long and 140 μm wide. It is placed about 16 mm above the drift electrode and mounted on a motorized stage whose travel step length is set to 100 μm for our studies. The geometrical width of the area that the X-rays impinge on is about 189 μm at the center of the drift region (Fig. <ref> right); the rate of X-rays on the detector is only about 7 Hz (not including background) due to this strong collimation. The data acquisition software RCDAQ was previously developed at BNL <cit.> and a sketch of the electronics logic is shown in Fig. <ref>. Twenty-four pairs of charge-sensitive preamplifiers and shapers are connected to 24 zigzag strip channels and are read out by three 8-channel VME Flash ADCs (Struck SIS 3300/3301) with a VME controller (CAEN 1718). The DAQ trigger is formed from the signal induced on the bottom of the third GEM. An accepted trigger stops the sampling of the FADCs, which are then read out. A signal typically occupies about 200 ns. To efficiently catch signals within the sampling time window, we adjust a window of 5 μs with a sampling rate of 100 MHz, i.e. we take 500 samples per window. Each board is scanned in X direction across the strips as defined in Fig. <ref> (left) with a step size of 100 μm over a few millimeter distance. The movement of the collimated X-Ray source closely follows the azimuthal direction. Since the zigzag strips are very close to parallel and the X-ray collimator slit has an 8 mm length, no effort is made to align the slit perfectly radially. At each point, we collect data for 8000-10000 triggered signals. The X-ray collimator is positioned so that the 140 μm slit width is oriented along the desired scan direction. The GEM detector is operated at a moderate gas gain of a few thousand; the applied V_drift ranges from 3250 V to 3500 V during the tests. As a reference, Fig. <ref> shows a gain curve measured with an ^55Fe source for board ACE #1 with strip pitch angle 1.37 mrad.§ EXPERIMENTAL RESULTS§.§ Response linearity The ZZ48 board is scanned across a range corresponding to two strip pitches while the ZZ30 board, which has wider strips, is scanned over one strip pitch (Fig. <ref>). For all data, we reconstruct the X-ray hit position in the X-direction in the usual fashion using the charge sharing among strips in a strip cluster. As the hit position, we take the charge-weighted centroid of the strip positions s_c = Σ_i=1^nq_i· s_i / Σ_i=1^nq_i, where s_i and q_i are the position at the strip center and the induced charge (in ADC counts), respectively, for the i^th strip in the cluster. As radial strips are intended to measure azimuthal positions, we use azimuthal strip positions in the centroid calculation by default.Fig. <ref> shows the mean strip-cluster centroid over all hits vs. actual position of X-rays incident on the boards. We observe flat regions in the curves where the detector is basically completely insensitive to the X-ray position. Checking the mean strip multiplicity of strip clusters for hits in these regions reveals that in most of these events only a single strip shows a signal (Fig.<ref>). These flat regions are centered on the “spines” of the physical zigzag strips (see Fig. <ref>). The electron avalanche is not wide enough to induce sufficient charge on adjacent strips for a good charge sharing measurement given the actual geometry of these physical zigzag strips. This contributes to an overall non-linear response in these readout structures. The ACE and CERNZZ boards are scanned across the strips with angle pitch 4.14 mrad near a radius of 229 mm and across the strips with angle pitch 1.37 mrad near a radius of784 mm. The scans range over 5 mm, which corresponds to 4-5 strip pitches, with a step size of 100 μm and about 8000 triggered signals per point. We focus the discussion on the results for boards ACE#2 and CERNZZ since their zigzag structures are close to optimal and closest to the design.Since the collimator slit is 8 mm long in the Y-direction along the strips, it exposes about 16 zigzag periods along a strip to X-rays. Consequently, our response and resolution results intrinsically incorporate any biases in the reconstruction of the X-position due to the range of X-ray hit positions along the strip in Y-direction. This bias is being minimized by design because the 500 μm zigzag period along the strip is of comparable or even smaller size (depending on the total amount of charge induced) than the diameter of the area where charge is being induced on the readout by the GEM avalanche. This means that the area of induced charge always covers at least one zigzag period along the strip and any effects due to the X-ray hit position in the radial direction along the strip average out. As shown in Fig. <ref>, the hits are dominated by 2-strip and 3-strip clusters, which comprise > 90% of all hits, while in the remaining hits only a single strip shows a signal. The cluster strip multiplicity is correlated with the gas gain, which is adjusted by the applied HV. The higher the gain, the smaller the population of single-strip hits. The mean cluster strip multiplicity is basically a linear function of the HV applied to the drift electrode (Fig.<ref>).The mean strip multiplicities as a function of X-ray position are compared for the two boards in Fig. <ref>. We observe a comb-like pattern as the X-ray position moves across the zigzag structure as expected. When the X-rays hit between the centers of two strips, the mean multiplicity is close to two, whereas when they hit near the center of a strip, the multiplicity is closer to three which indicates that both adjacent strips share the charge with the central strip as intended. This effect is a bit more pronounced for the strips with 1.37 mrad angle pitch since their linear pitch of 1.07 mm is slightly larger than the 0.95 mm linear pitch for the strips with 4.14 mrad angle pitch.Fig. <ref> shows the mean strip-cluster centroid position measured at each X-ray position for boards ACE#2 and CERNZZ. Results are compared for strip multiplicities 2, 3, and overall, but excluding single-strip clusters. The small regions in the plots with negative slopes are due to motor backlash encountered during the scans when data taking was interrupted. Comparing these responses with those shown in Fig. <ref> clearly demonstrates that the spatial response is much more linear for the improved ACE#2 and CERNZZ zigzag strip boards than for the original ZZ48 and ZZ30 boards. §.§ Spatial resolutions A scatter plot of the residuals from the mean centroid using all cluster centroids vs. X-ray position for ZZ48is shown on the left in Fig. <ref>. It is not a horizontal band because the response is not linear. We correct this residual distribution for the non-linear response by first fitting the mean centroid vs. X-ray position plot from Fig. <ref> (left) with linear functions in the non-flat regions and then subtracting them from the residuals in the scatter plots to flatten them out. This pushes the band of residuals towards a more horizontal line as shown in the center of Fig. <ref>. Projecting this plot onto the vertical residual axis over one strip pitch produces the distribution shown on the right in Fig. <ref>. The width of this distribution, which we use as a measure for the overall residual width, is 128 μrad or 188 μm at the radius of 1468 mm for the ZZ48 board.In order to find the intrinsic spatial resolutions for the tested zigzag boards, we need to subtract the effect of the finite width of the X-ray collimator, which causes a smearing of the X-ray incidence position on the detector. This is done using a simple Geant4 simulation and the result is shown in Fig. <ref>. Through the simulation, the relation between an intrinsic detector resolution that is fed into the simulation and a measured width of the X-ray spot on the detector is determined. This mostly linear function is then used to obtain the intrinsic spatial resolution of the detector from the measured raw residual width observed in the data. The finite collimator width has an impact mainly below 100 μm. A few different X-ray beam shapes emerging from the source (uniform rectangular, cone, Gaussian) are tested in the simulation and their impact is found to be negligible. From the residual width and the GEANT4 curve, the intrinsic resolution of the ZZ48 board is measured to be (123.1±0.4) μrad or (180.7±0.6) μm after the X-ray collimator effect is subtracted. In our previous study <cit.>, a similar zigzag readout structure was tested using hadron beams and the resolution was found to be around 180 μrad at a slightly lower voltage of V_drift=3200 V. We estimate from the measurement of resolution as a function of HV in that previous study that a 140 V increase in V_drift corresponds to a 27% improvement in resolution, i.e. at V_drift=3340 V the resolution would be expected to be about 130 μrad. Ourcurrent measurement of 123 μrad with X-rays is reasonably consistent with this expectation from our previous measurement with hadrons.The same procedure is applied to the data for the other boards. Fig. <ref> shows the residual scatter plots for 2-strip and 3-strip clusters using the strips with 1.37 mrad angle pitch on the CERNZZ board. Linear functions are fitted to the strip cluster centroid vs. X-ray position plots and subtracted from the residuals to flatten them out so that overall residuals can be calculated. The residual widths are found to be 80 μrad (2-strip clusters) and 112 μrad (3-strip clusters). By again subtracting the X-ray collimator effect, the corresponding measured intrinsic resolutions are 57 μrad and 92 μrad for 2-strip and 3-strip clusters, respectively. This corresponds to linear instrinsic resolutions of 45 μm and 72 μm, respectively, at a radius of 784 mm. The overall intrinsic resolution is 71 μrad or 56 μm at a radius of 784 mm if we combine 2-strip and 3-strip clusters in the analysis. Statistical errors in these analyses are less than 0.3%.Table <ref> summarizes the final intrinsic resolution results for all the ACE boards and the CERNZZ board after X-ray width effects are subtracted. Most measured resolutions are better than 100 μm. The applied V_drift and corresponding gas gain are also listed in the table for reference. The uncertainty on the gain is estimated to be about 30% based on observed variations in signal amplitudes during the different measurements. The CERNZZ and ACE#2 boards show almost the same resolutions for the strips with 1.37 mrad angle pitch; the resolution difference for strips with 4.14 mrad is likely due to gain variations while scanning. The consistently better resolutions for board ACE#1 relative to board ACE#2 are due to the higher gain applied to ACE#1 during the test. § SUMMARY AND CONCLUSIONS The physical readout boards with improved radial zigzag strip design, which we have tested with highly collimated X-rays, feature close-to-optimal (89%) interleaving of adjacent strips. This enables robust charge sharing among adjacent strips, which in turn results in a marked improvement in spatial response linearity and spatial resolution over previous zigzag strip designs that we have studied. The spatial detector responses are almost fully linear, so there is no need for any corrections when reconstructing hit positions from straight-forward strip cluster centroids. With linear strip pitches around 1 mm, the boards achieve linear spatial resolutions of 50-90 μm at medium gas gains around 3000. The successful production of these improved radial zigzag strips on a foil rather than on a PCB and the observation that their performance is comparable to PCBs are promising for using zigzag-strip readouts on large foils to reduce overall material of a GEM detector. Consequently, this approach lends itself to GEM detector applications where material budgets can be critical, such as TPCs and forward tracking detectors at the EIC, while reducing the overall number of required readout channels and maintaining excellent spatial resolution.§ ACKNOWLEDGMENTSThis work is supported by Brookhaven National Laboratory under the EIC eRD-6 consortium. We also acknowledge Alexander Kiselev (BNL) for his help with finalizing the design idea for the optimized zigzag structure.§ REFERENCES 25 1985ZZ T. Miki, et al., Zigzag-shaped pads for cathode readout of a time projection chamber, Nuclear Instruments and Methods in Physics Research Section A, 236 (1985), p. 64. 1997ZZ J. Barrette, et al., A Study of MWPC with chevron cathode pad readout, Nuclear Instruments and Methods in Physics Research Section A, 385 (1997), p. 523. 2005ZZ B. Yu, et al., A GEM based TPC for the LEGS experiment, in 2005 IEEE Nuclear Science Symposium Conference Record, 23-29 October 2005, vol. 2, pp. 924-928, http://dx.doi.org/10.1109/NSSMIC.2005.1596405. 2012ZZ D. Abbaneo, et al., Beam Test Results of New Full-Scale Prototypes for CMS High-Eta Muon System Future Upgrade, in 2012 IEEE Nuclear Science Symposium Conference Record, Oct. 27-Nov. 3, pp. 1172-1176, http://dx.doi.org/10.1109/NSSMIC.2012.6551293. 2016ZZ B. Azmoun, et al., A study of a Mini-drift GEM tracking detector, IEEE Transactions on Nuclear Science, NS-63 (3) (2016), p. 1768.FITZZ2 A. Zhang, et al., Performance of a large-area GEM detector read out with wide radial zigzag strips, Nuclear Instruments and Methods in Physice Research Section A, 811 (2016), p. 30. EICWhite A. Accardi, et al., 2012, Electron Ion Collider: The Next QCD Frontier - Understanding the glue that binds us all, arXiv:1212.1701 (https://arxiv.org/abs/1212.1701). 2015IEEE A. Zhang, et al., R&D on GEM detectors for forward tracking at a future Electron-Ion Collider, in 2015 IEEE Nuclear Science Symposium COnference Record, Oct. 31-Nov. 7, http://dx.doi.org/10.1109/NSSMIC.2015.7581965. rcdaqref M. L. Purschke, Readout of GEM stacks with the CERN SRS system, in 2012 18^th IEEE-NPSS Real Time Conference, June 9-15, http://dx.doi.org/10.1109/RTC.2012.6418184. | http://arxiv.org/abs/1708.07931v1 | {
"authors": [
"Aiwu Zhang",
"Marcus Hohlmann",
"Babak Azmoun",
"Martin L. Purschke",
"Craig Woody"
],
"categories": [
"physics.ins-det",
"hep-ex"
],
"primary_category": "physics.ins-det",
"published": "20170826050149",
"title": "A GEM readout with radial zigzag strips and linear charge-sharing response"
} |
III. The California Molecular Cloud: A Sleeping Giant Revisited HP2 (Herschel-Planck-2MASS) survey is a continuation of the series originally entitled "Herschel-Planck dust opacity and column density maps" <cit.>. Note: the flux, opacity and temperature maps presented in this paper are available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/ A Sleeping Giant [email protected] Center for Astrophysics, Mail Stop 72, 60 Garden Street, Cambridge, MA 02138 University of Milan, Department of Physics, via Celoria 16, I-20133 Milan, ItalyUniversity of Vienna, Türkenschanzstrasse 17, 1180 Vienna, AustriaWe present new high resolution and dynamic range dust column density and temperature maps of the California Molecular Cloud derivedfrom a combination of Planck and Herschel dust-emission maps, and 2MASS NIR dust-extinction maps.We used these data to determine the ratio of the 2.2 extinction coefficient to the 850 opacity and found the valueto be close to thatfound in similar studies of the Orion B and Perseus clouds but higher than thatcharacterizing the Orion Acloud, indicating that variations in the fundamental optical properties of dust may exist between local clouds.We show that over a wide range of extinction, the column density probability distribution function (pdf) of the cloud can be well described by a simple power law (i.e., PDF_N ∝ A_K^ -n) with an index(n = 4.0 ± 0.1) that represents a steeper decline with A_K than found (n ≈ 3) in similarstudies of the Orion and Perseus clouds.Using only the protostellar population of the cloud and our extinction maps we investigate the Schmidt relation, that is, the relation between the protostellar surface density, Σ_*, and extinction, A_K, within the cloud. We show that Σ_* is directly proportional to the ratio of the protostellar and cloud pdfs, i.e., PDF_*(A_K)/PDF_N(A_K). We use the cumulative distribution of protostars to infer the functional forms for both Σ_* and PDF_*.We find that Σ_* is best described by two power-law functions. At extinctions A_K ≲2.5mag, Σ_* ∝ A_K^β with β = 3.3 while at higher extinctions β = 2.5, both values steeper than those (≈ 2) found in other local giant molecular clouds (GMCs). We find that PDF_* is a declining function of extinction also best described by two power-laws whose behavior mirrors that of Σ_*. Our observations suggest that variations both in the slope of the Schmidt relation and in the sizes of the protostellar populations between GMCs are largely driven by variations in the slope, n, of PDF_N(A_K).This confirms earlier studies suggesting that cloud structure plays a major role in setting the global star formation rates in GMCsThe HP2 survey Charles J. Lada1, John A. Lewis1, Marco Lombardi1,2, and João Alves3Received 22 May 2017; Accepted 3 August 2017 =========================================================================== § INTRODUCTIONLocated within the confines of the Perseus constellation, the California Molecular Cloud (CMC) rivals the Orion A molecular cloud as the most massive giant molecular cloud (GMC) within 0.5kpc of the Sun. However, despite its large mass, the CMC is a sleeping giant, characterized by a star formation rate (SFR) that is an order of magnitude lower than that of the Orion A cloud <cit.>.This fact along with its proximity make the CMC an ideal laboratory for investigating the physical process which regulates the global level of star formation in a cloud and sets the SFR.Near-infrared extinction mapping provided the first complete maps of the dust column density structure of the CMC <cit.>. Analysis of these observations and comparison with similar observations of the Orion A cloud suggested a direct connection between the level of star formation activity in the cloud and its structural properties. In particular, it was suggested that the global SFR of the cloud was set by the amount of high extinction (A_K ≳1.0mag) material it contained<cit.>. In a subsequent study <cit.> showed that it was generally the case that actively star forming clouds contained more high extinction material than did quiescent clouds.Shortly thereafter <cit.>demonstrated that a linear correlation exists between the measured SFRs andthe gas masses at high extinction (A_K ≳0.8mag) for a nearly complete sample of molecular clouds within 0.5kpc of the sun. The connection between the SFR and high cloud surface densities was further strengthened by detailed studies of both local low mass and more distant high mass clouds <cit.>.There is also some evidence that this relation between SFR and dense gas extends smoothly to scales of entire galaxies <cit.>. A further investigation of the Schmidt relation <cit.> within individual local molecular clouds, including the CMC, reinforced the idea that cloud structure was an important, if not the key, factor in determining the global SFR and level of star formation activity in a molecular cloud <cit.>. In light of the above considerations it would be of great interest to examine in more detail the relationship between star formation and the highest extinction gas. However, this is not feasible using existing near-infrared extinction maps because such maps are severely limited in their ability to probe the highest extinction regions, due to the lack of background stars that can be detected in the presence of such high dust opacity.More sensitive measurements of dust column density with higher dynamic range are required to investigate the nature of the star formation process at the high extinctions in which it occurs. With the dramatic improvements in the dynamic range of column density measurements provided by the Herschel mission, we are now in a position to re-examine the detailed structure of the CMC and its relation to star formation. A significant fraction of the CMC was surveyed by Herschel <cit.>. The surveyed region contained about half the cloud mass but importantly all regions of significant star formation. Here we analyze both Herschel and Planck observations of the CMC using the methodology of <cit.>. This method combines the space observations with ground-based, near-infrared extinction maps to produce extinction-calibrated dust column density maps of a cloud. Such maps help mitigate the uncertainties in dust emission measurements of column densities and temperatures introduced by the well known degeneracy between these two physical quantities along a given line-of-sight. This is the third paper in a series applying this methodology to near-by clouds <cit.>. The first two papers dealt with the more active star forming clouds Orion A, Orion B, and Perseus and demonstrated the power of the Herschel observations for probing high extinction material and providing dust column density maps of GMCs with high dynamic range and angular resolution. The results of those papers provide a very useful data set for comparison with the present study of the CMC and will help elucidate the nature star formation in this relatively quiescent GMC.The layout of this paper is as follows. Section 2 describes the data used for this study. Section 3 reviews the methodology used to produce the dust column density and temperature maps. Section 4 presents the results and in Section 5 we discuss the implications. A summary of our primary conclusions is found in Section 6. § DATA The CMC was observed by the all-skyPlanck observatory and by the Herschel Space Observatory as part of the “Auriga-California” program <cit.>.The Herschel data we used consisted of observations obtained in parallel mode simultaneously using the PACS and SPIRE <cit.> instruments. More details about the observational strategy can be found in <cit.> and <cit.>. We used the final data products of Planck <cit.>.Following the methodology of <cit.> and <cit.>, the Herschel data products were pre-processed using the Herschel Interactive ProcessingEnvironment <cit.> version 15.0.1, with the latest version of the calibration files.The final maps were then produced using the HIPE level 2.5 of the SPIRE data, and the Unimaps data for PACS.For the purposes of this study we use Herschel observations made in the PACS 160 band, and the SPIRE 250;350;500 bands.However, prior to analyzing the data we multiplied each SPIRE band by an updated correction factor C ≡ K_4e / K_4p <cit.> andthen performed an absolute calibration of the Herschel fluxes using Planck maps. Next weconvolved all Herschel data to the resolution of the SPIRE 500 band,i.e., 𝐹𝑊𝐻𝑀_500 = 36arcsec.A false color image of the reduced Herschel and Planck SPIRE band fluxes in the CMCregion is shown in Figure <ref>.The Herschel observations cover the bulk of the molecular cloud in the central regions of the image. Here the three Herschel SPIRE passbands have been convolved to the 500 resolution of 36arcsec. The fluxes in the regions outside the Herschel survey area are the predicted Planck fluxes at the threeSPIRE passbands and are characterized by a much lower angular resolution of 5arcmin. We used the Planck/IRAS dust model (i.e., T, β, τ_850;<cit.>) to derive the expected fluxes in the SPIRE passbands.The boundary between the two regions is clearly apparent in the image because of the differing angular resolutions of the two surveys. We note here that although these two regions are clearly demarked by their differing angular resolutions, the colors in the high resolution region match the colors of the rest of the image very well indicating that the absolute calibration of the Herschel observations is accurate. Finally, we used the NIR extinction maps of the CMC from<cit.> that were derived from the 2MASS all-sky survey with theNicest technique <cit.>.§ METHODOLOGYWe derived the dust column density and temperature maps of the cloud following the methodology developed by <cit.>.This method was previously used to construct similar maps of the Orion A and B clouds <cit.> as well as the Perseus cloud <cit.>.Here we briefly outline the general procedure. For more details the readercan consult the Lombardi et al. paper. §.§ Physical modelWe consider the dust emission to be optically thin and describe the specific intensity at a frequency ν as a modified blackbody:I_ν = B_ν(T) [ 1 - ^-τ_ν] ≃ B_ν(T) τ_ν,where τ_ν is the optical-depth at the frequency ν and B_ν(T) is the blackbody function at the temperature T:B_ν(T) = 2 h ν^3/c^21/^h ν / k T - 1.Following standard practice, the frequency dependence of the optical depth τ_ν can be written asτ_ν = τ_ν_0( ν/ν_0)^β_d,whereν_0 is an arbitrary reference frequency. We setν_0 = 353GHz, corresponding to λ = 850, and we indicate the corresponding optical depth as τ_850.This is also the standard adopted by the Planck collaboration. We note that when using this physical model we are implictly assuming that temperature gradients are negligible along the line-of-sight.This is an approximation because non-negligible gradients in the dust temperature are clearly observed in the plane of the sky in many regions of molecular clouds. Therefore, it is likely that in such regions temperature gradients exist along the line-of-sight as well.Because of the sensitive dependence ofEq. (<ref>) on temperature, the temperatures derived from the observed fluxes using this equation will bebiased somewhat toward higher values. This will ultimately result in slight underestimates in the opacities and corresponding column densities. Therefore T in Eq. (<ref>) should be considered an effective dust temperature for an observed dust column. §.§ SED fitWe can use our dust model to derive to derive the optical depth τ_850, the effective dust temperature T, and the exponent β_d in a given map pixel by fitting the modified blackbody of Eq. (<ref>) to the fluxes measured by Herschel for that pixel.We use the reduced observations in the PACS 100;160 bands, and the SPIRE 250;350;500 bands to construct the spectral energy distribution (SED) at each map pixel.We fit the modified blackbody dust model to the SED using a simple χ^2 minimization that takes into account the calibration errors (taken to be 15% in all bands). Because of the degeneracies present in the χ^2 minimization, we fixed β_d and fit only for τ_850 and T. For β_d we adopted local values computed by the Planck collaboration <cit.> on 35 arcmin scales across the entire sky.§ RESULTS §.§ Optical depth and temperature maps Figure 2 shows a combined optical depth and dust temperature map derived from the SED fits. Here the image intensity is proportional to the opacity and color scales with temperature with blue representing hot (T > 25K) dust and red representing cold (T < 15K) dust. The coldest regions of the cloud (red) have dust temperatures∼14K while the hottest blueish region is characterized by temperatures between 18–33K. These warmer temperatures are confined to the area around the the embedded cluster associated with the B star LkHα 101 and the emission nebula NGC 1579. The dust in the bulk of the cloud is characterized by temperatures in the range 15–16K. Optical depths range from roughly 10^-4 to 10^-3 with the highest values appearing closely associated with the coldest dust. Separate dust temperature and opacity maps for the region are shown in Figure 3.§.§ Converting opacities to extinctions and column densitiesWe can derive extinctions and ultimately useful column densities for the cloud from direct comparison of the τ_850 and K-band (NICEST) extinction (A_K) maps. To make this comparison we considered only the Herschel observations and first smoothed the Herschel opacity map to the lower resolution (80arcsec) of the extinction map. We empirically determined that between 0 ≤τ_850≤2e-4 the relation between τ_850 and A_K is linear,that is:A_K(NIR) = γτ_850 + δ.Note that since τ_ν = κ_νΣ_dust, where, Σ_dust is the dust-column density and κ_850 is the opacity coefficient (at 850), the slope γ is proportional to the ratio of κ_850 to C_2.2, the extinction coefficient at 2.2. This follows from:A_K = -2.5 log_10( I_obs/I_true) = (2.5 log_10e) C_2.2Σ_dust.Thus, to first approximation, γ≃ 1.0857 C_2.2 /κ_850. The coefficient δ in Eq.(<ref>) absorbs calibration and systematic uncertainties that arise in both the Herschel and extinction measurements.From fitting the linear relationship of Eq. (<ref>) to the data the following values for the two parameters were derived:γ =3593 ±75mag δ = -0.110 ±0.006magThe data and fit are displayed in Figure <ref>. The same figure also shows the predicted 3-σ boundaries around the fit of Eq. (<ref>). These boundaries were estimated from the statistical error on the extinction map alone (i.e., errors in the optical-depth were ignored). That the data points are predominately found to lie within 3-σ boundaries indicates that fit is accurate and confirms that the optical-depth map has a negligible relativeerror at the 80arcsec resolution of the extinction map. The value of γ we found for the CMC is higher than that obtained in a similar study of the Orion A cloud (2640; <cit.>), but similar to that derived for the Orion B cloud (3460; <cit.>) and the Perseus cloud (3931; <cit.>). As discussed in <cit.> this suggests a possible variation in the optical properties of grains between local molecular clouds. The value of |δ| in the CMC is also higher than reported for the Orion and Perseus Clouds (i.e., |δ|= 0.001–0.05). However, as pointed out by <cit.> the NIR extinction measurements used here underestimate the true values by a systematic offset of -0.04mag due the presence of an unrelated screen of foreground extinction that contaminated the 2MASS control fields used for this area of the sky. Once adjusted for that systematic error the value of δ is about -0.06mag, comparable to those derived for the other regions. For higher opacities the slope of the relation between A_K andflattens and the relation becomes non-linear.This departure from linearity and gradual flattening of the slope at higher extinctions is similar to that found previously for similar relations in Orion <cit.> and Perseus <cit.>.It indicates that the infrared extinctions, A_Ks,are more often underestimating than overestimating the dust column densities in regions of high opacity. This shift from linearity is the likely result of a breakdown in the extinction technique at high column densities due to the relative paucity of background stars within the indivdual 80arcsec pixels of the 2MASS extinction map we used.This is confirmed by examining spatial maps of the differences, A_K(𝐻𝑒𝑟𝑠𝑐ℎ𝑒𝑙)-A_K(2MASS), which show the largest values confined to the regions of highest opacity, similar to what has been observed in the Orion <cit.> and Perseus clouds <cit.>.To construct a column density map of the region we convert the Herschel submillimeter opacities to extinctions using only the coefficient γ, that is we set:A_K (𝐻𝑒𝑟𝑠𝑐ℎ𝑒𝑙)= γτ_850 In using equation <ref> we have ignored δ because of its relatively small magnitude and the presence of a systematic bias present in the NIR extinction measurements as discussed above.We applied equation <ref> to the Herschel-Planck opacity map to derive extinctions in each map pixel. We find a dynamic range in extinction of 0.04 ≤ A_K ≤6.8mag across our entire map (i.e., figure <ref>). In figures <ref>, <ref>, <ref> we present zoomed-in maps of the extinction derived from Eq. (<ref>) for three interesting sub regions (A, B, & C) ofthe CMC mapped by Herschel. These three maps were constructed at the higher spatial resolution (18arcsec) of the SPIRE 250 band following the methodology of <cit.>. Briefly, the SED of Eq. (1) was fitfor τ_250 in each SPIRE 250 map pixel by using the flux in the 250 band and fixing T_D to the value that was determined from the 36arcsec map (i.e., Figure <ref>) and fixing β to the value derived by Planck as before.The value of τ_850 was then derived from Eq. (3). In the high resolution maps our ability to resolve small, high opacity features was improved with the maximum measured infrared extinction increasing from 6.8 to 10.8mag.§.§ Cloud Mass The extinctions derived from Eq. (<ref>) can be converted to total (gas + dust) mass surface densities from: Σ_gas= μα_K m_p = 183A_KM_⊙ pc^-2 Here, μ = 1.37 is the mean molecular weight corrected for Helium, α_K = 1.67e22cm^-2.mag^-1 is the gas-to-dust ratio, [N(HI) + 2N(H_2)]/A_K <cit.> and m_p is the mass of a proton.The total mass of the cloud is then obtained from integrating over the (Herschel + Planck) surface area of the cloud:M_tot = ∫Σ_gas dSTo calculate the cloud mass we must first define the boundaries of the cloud. We consider the CMC to be entirely contained within a rectangular box on the sky given by -13 < b_II < -6.5 and 155 < l_ II < 167. We define the physical cloud boundary to be given by the A_K > 0.2mag contour within the rectangular region.This region includes both the inner area of the cloud surveyed by Herschel as well as outer regions surveyed by Planck. We applied Eq. (<ref>) to both data sets to derive the dust column densities across this expanse of the CMC. We then find M_tot = 1.1e5M_⊙. The region surveyed by Herschel contains 5.52e4M_⊙ or roughly half the cloud mass at A_K > 0.2mag. § DISCUSSION§.§ Protostellar Population and the Star Formation Rate Knowledge of the protostellar population of a cloud is instrumental in determining the relation between the star formation rate (SFR) and structure of the cloud <cit.>.Because of their relatively short lifetimes, the population of protostarsin a cloud can be considered the instantaneous yield of the star formation process in that cloud. Indeed, the number of protostars (N_p*) is an excellent proxy for the instantaneous SFR since, to good approximation, SFR = m_p*τ_p*^-1 N_p* where τ_p* and m_p* are the typical lifetime and mass of a protostellar object. Moreover, because protostellar objects are most likely to be within or very close to their original birth sites, their surface densities are the most appropriate for comparison with those of the gas or dust. Therefore, we consider here only protostars (i.e., Class 0, Class I, and flat spectrum YSOs) in our determination of the cloud SFR.IRAS observations provided the first census of young stellar objects which displayed characteristics of potential protostars in the California cloud. Seventeen candidate protostars were identified from IRAS observations by <cit.>.Deeper Herschel <cit.> and Spitzer<cit.> observations resulted in independent and more complete YSO catalogs of the cloud including greatly improved source classifications. These catalogs increased the known number of YSOs to 60 and 166 objects, respectively, including both protostars and more evolved (Class II + III) stars.Because the two catalogs were complied independently using slightly different criteria, there is some disagreement in source classifications between them. For this study we merged these two catalogs and re-examined the source classifications following procedure outlined by <cit.> and summarized in the Appendix.Table A.1 in the Appendix presents the merged catalog of YSOs we adopted. We identify 43 protostars in the CMC. The protostellar sources are largely confined to the portion of the cloud surveyed by Herschel. The locations of 42 of these protostellar sources are shown on the extinction maps in figures <ref>–<ref>. These regions account for about 80% of the cloud mass mapped by Herschel and roughly 40% of the total cloud mass in the Herschel + Planck maps.Within the cloud boundary of A_K > 0.2mag the protostellar surface densities are not uniform with Σ_* = 0.2, 0.03 and 0.02pc^-2 for regions A, B, and C, respectively. Region A is considerably more active and efficient in forming stars than either B or C. Region A contains ∼ 70% of the protostars in the CMC while occupying only a small fraction (7%) of the total (Herschel + Planck) cloud area and containing only a small fraction (10%) of its totalmass. This region also contains NGC 1579 (LK Hα101) the only substantial embedded cluster in this massive cloud. In all three regions the distributions of protostars closely follows the distribution of high extinction gas.§.§ Cloud StructureThe area distribution function is defined as S(>A_K) ≡∫_A_K^∞dS(A_K) where dS(A_K) is an element of cloud surface area at an extinction A_K. The area distribution function is a cumulative distribution that represents the total cloud area above a given threshold extinction as a function of A_K. The derivative of this function, -S'(>A_K) is proportional to the column density PDF of the cloud and both functions are useful descriptors of cloud structure. Figure <ref> and figure <ref> show the functions S(>A_K) and -S'(>A_K) respectively. (We note that as plotted here the latter function corresponds to a linearly binned PDF and consequently its slope will differ by -1 from that of a logarithmically binned PDF, such as the ones studied by <cit.>.)Although the Herschel observations clearly extend to significantly higher extinctions than the 2MASS observations, in the range where the two data sets overlap, the two functions, S(>A_K) and -S'(>A_K) are in reasonable agreement with both data sets, confirming the result of our earlier study <cit.> that both S(>A_K) and -S'(A_K) fall off relatively steeply with A_K. For example, with -S'(>A_K) ∝ A_K^-n we find, from a formal fitting of the CMC data, that n = 4.0 ± 0.1, for A_K > 0.35mag. This is relatively steep compared to the value (≈ 3) that describes the pdfs in Perseus and Orion.Similarly for S(>A_K) ∝ A_K^-q, where q = n - 1 = 3.0, the relation is relatively steep compared to that found (≈ 2) in the active star forming clouds Orion A, Orion B, and Perseus that we have also studied with Herschel.§.§ The Schmidt Relation In previous studies of the CMC <cit.> and <cit.> showed that a Schmidt relation of the form Σ_*∝ A_K^β existed for the cloud.Here Σ_*, the surface density of protostellar objects and A_K are proxies for Σ_SFR and Σ_gas, respectively in the usual Schmidt relation <cit.>.However, using different methodologies and extinction data as well as slightly differing YSO catalogs they derived different values (2 and 4, respectively) for the index β. In this paper we revisit the Schmidt relation for the CMC using a Bayesian (MCMC) methodology similar to <cit.> coupled with our extinction calibrated, Herschel 36arcsec resolution dust column density maps and our revised catalog of protostellar objects for the cloud.We assume that the protostellar surface density behaves according to a thresholded Schmidt relation:Σ_*(A_K) = κA_K^β H(A_K - A_K,0),where Σ_* is the protostellar surface density, A_K is the extinction in magnitudes in the pixel on which a source lies[It is important to note that here A_Krepresents the total extinction along the line-of-sight to the protostar averaged over a 36arcsec or 0.08pc region and is not the pencil-beam extinction to the source itself.], H(x) is a Heaviside function,A_K,0 is the extinction threshold for star formation, and κ is a normalization factor with units[star.pc^-2.mag^-β]. We do not model the diffusion of protostars from their birth location as was done in <cit.> and <cit.>. The diffusion, as previously measured, is at the sub-pixel level and is not degenerate with any parameter, so will not affect our determination of κ or β. We take the likelihood derived in <cit.> (their equation 7), (x_n|θ), and estimateθ̂=(κ,β,A_K,0) using the affine-invariant MCMC package<cit.>, with the chains initialized to a Gaussian distribution around the result of aNelder-Mead (amoeba)maximization of the likelihood.From this analysis we determined the three credible intervals for the model in Eq. (<ref>) to be β = 3.31 ± 0.23, κ = 0.36 ±0.09stars.pc^-2.mag^-3.31, and A_K,0 = 0.51mag. In figure <ref> we show the standard graphical representation of the power-law Schmidt relation we derived for the CMC plotted along with appropriately binned data for comparison.Visual inspection of the plot suggests that the fit and the model it is based on, i.e., Eq. (<ref>), are more approximate than precise representations of the observations. In particular, the data in the highest extinction bins clearly deviate from the β = 3.31 line. The departure of the highest extinction points from the fit suggests that a truncation may be present at high extinction. Such a truncation would likely result from the relative absence of high opacity material in the cloud as evidenced by the steep fall off of its PDF with A_K (see Section 5.2 and Figure <ref>).The values of these posterior parameters differ significantly from those (β∼ 2, κ∼ 2 and A_K,0∼0.6mag) derived by <cit.> using similar methodology. This difference could arise from the different extinction maps and protostellar catalogs used in our two studies. The difference in the protostellar catalogs employed was slight and likely not responsible for the differing results. Nonetheless, we performed our analysis on various subsets of the original Harvey et al. catalog, including a version that used all sources. We found the resulting posterior parameters to be the same within the errors and thus not particularly sensitive to the different catalogs used. We next performed our analysis using the 2MASS NIR extinction map instead of our Herschel map. In this case the analysis returned parameters that essentially reproduced the values derived by<cit.>.Closer inspection of the data showed that the extinctions associated with the individual protostars were almost always underestimated in the NIR map. This is not surprising since more than 80% of the protostars in the CMC are found at A_K > 1.0mag in the Herschel extinction map. At these high opacities the NIR derived extinctions are considerably less reliable than our Herschel extinctions and moreover are expected to underestimate the true opacities due to the small numbers of detectable background stars present in the individual pixels.Our Herschel derived value of β is closer to, but slightly less than, that (∼ 4) derived by <cit.> also using Herschel observations.Those authors employed a different methodology to estimate a value for β: First, <cit.> produced surface density maps of dust (A_K) and YSOs, both smoothed to a scale of 0.2. They then made a ratio of the two maps and constructed the corresponding plot ofΣ_* vs A_K from that data.<cit.> quote only their value for β derived from this plot and do not produce an estimate of its uncertainty so it is difficult to assess the significance of the difference between the two estimates. Because the value of β derived using Herschel dust column densities by two different methods is higher than the value we previously derived using 2MASS extinctions, we adopt our improved estimate for this parameter as being the more faithful measure of its true value. §.§ The Protostellar PDF We can gain some physical insight into the nature of the Schmidt relation and its connection to star formation by writing the relation in thefollowing form:Σ_*(A_K)= dN_*(A_K)/dS(A_K) =Σ_*0×PDF_*(A_K)/PDF_N (A_K) Here Σ_*0 is a constant equal to the global protostellar surface density of the cloud, that is, the ratio of N_*tot, the total number of protostellar objects, to S_tot, the total cloud area. PDF_*(A_K) is the pdf of the protostellar population[i.e., PDF_*(A_K) ≡1/N_*totdN_*(A_K)/dA_K] and PDF_N(A_K) is the cloud column density pdf. In this form we see that the Schmidt relation is proportional to the ratio of the protostellar and cloud pdfs. The column density pdf of a molecular cloud is a standard metric used to describe cloud structure. It has been shown to be well described by simplepower-law functions of extinction <cit.>. The protostellar pdf is not a well known distribution and to our knowledge has not been studied in the literature. Because N_*(A_K) is proportional to the total instantaneous SFR at any extinction, the protostellar pdfis essentially the normalized measure of the SFR as a function of extinction. Consider that we can rewrite equation <ref> to yield:PDF_*(A_K) =Σ_*(A_K)/Σ_*0×PDF_N(A_K)If Σ_*(A_K) and PDF_N(A_K)are both power-law functions of A_K, then PDF_*(A_K) must also be a power-law, i.e., PDF_*(A_K) ∝ A_K^p where p = β - n. In Table <ref> we list the values of β and n derived from similar analysis of local clouds with Herschel dust column densities. For all these cloudsn > β indicating that PDF_*(A_K) is a declining function of A_K and predicting that the number of protostars will steeply decline with extinction, despite the rapid rise of the Schmidt relation with A_K. For the CMC we find p = -0.69 and for all four clouds ⟨β - n ⟩ = -0.69 ± 0.27. A protostellar pdf of the form PDF_* (A_K) ∝ A_K^-0.7 seems counterintuitive, in part because it clearly cannot describe the behavior of the actual PDF_*(A_K) at low extinctions. This is clear from figures <ref>, <ref>, <ref>, which show an almost complete absence of protostars in regions of low extinction (i.e., A_K ≲1.0mag). Moreover, it also predicts that the number of protostars will sharply decline with increasing extinction and this appears in conflict with the fact that the protostellar positions seem to be correlated with the highest extinction regions in the maps of figures <ref>, <ref>, <ref>. The relatively small numbers of protostars in the CMC coupled with the large dynamic range of extinction they sample make it difficult to directly determine a well sampled PDF_*(A_K) and test these predictions. However, we can gain further insight into the nature of the protostellar pdf by considering the normalized cumulative distribution of protostars: N_*(>A_K)/N_*tot = ∫_A_K^∞PDF_*(A_K) dA_KThis distribution can be considered the fractional yield of protostars as a result of the star formation process in the cloud.Figure <ref> shows the normalized, cumulative distribution of protostarsas a function of extinction observed in the CMC.As predicted above, the observed data points (filled symbols) indicate that the number of protostars in the cloud in fact does sharply drop off with increasing extinction.Similar results were found in the cumulative protostellar distributions in Orion A, Orion B, and Perseus. Moreover, the above analysis supports the hypothesis that, because n > β, the steep drop off of protostars at the highest extinctions is a direct result of the relative lack of such high extinction material in the cloud <cit.>. Figure <ref> also confirms that rather than increasing, the number of protostars at lowextinctions becomes vanishingly small (i.e., N_*(>A_K)/ N_tot = 1.0, for A_K < 0.5mag).Clearly the protostellar pdf cannot be described by a single power-law function that extends unabated to the lowest extinctions. Using Eq. (<ref>) with PDF_*(A_K) ∝ A_K^p, the normalized cumulative distribution of protostars can be written as:N(>A_K)/ N_tot =(1 + p)C_p ∫_A_K^A_max A_K^p dA_K.where C_p is a normalization constant and A_max is the extinction measured in the highest extinction map pixel containing a protostar.In Figure <ref> we also plotted a series of theoretical cumulative distributions given by Eq. <ref> for various values of the power-law index, p, of the protostellar pdf and two different values of the parameter A_max.To be consistent with the observations of the CMC we have normalized the functions to equal 1.0 at A_K = 0.5mag, effectively truncating the relations at that column density. For the solid set of curves we set A_max = 6.3mag, the value derived from the data. In the dashed set of curves we set A_max = 10.0mag. The solid curves provide the best match to all the data.Inspection of the figure confirms the notion that a single power-law PDF_*(A_K) is an inadequate description of the data.However, at high extinction (≳2.5mag) thepower-law with index p = -1.4 comes closest to matching the observations. In the range 0.5 ≤ A_K ≤2.0mag, the data appear to best matched by the power-law with p = -0.5. From this simple comparison we can infer the general properties of the protostellar pdf in the CMC. At large extinctions PDF_*(A_K) does appear to fall off as a power-law with index ≈ -1.4. The function departs from this power-law at intermediate extinctions and appears to follow a shallower power-law with p ≈ -0.5 between0.5 and 2.5mag and then truncates or falls off very rapidly with decreasing extinction. This latter regime forms a broad peak in the pdf and accounts for roughly 80% of the protostars in the cloud. An instructive case is that of p = 0. Here there is equal probability of finding a protostar at any given extinction. In this case the indicies of the Schmidt relation and the cloud pdf would be the same, i.e., n = β. This is clearly not the case in the observations. That β is found to be positive suggests that the star formation process operates more efficiently in gas at high extinctions but since β < n, not efficiently enough to prevent a decrease in the relative number of protostars formed (and the SFR) at the highest extinctions.The other set of curves plotted on figure <ref> are cumulative distributions calculated from equation <ref> with A_max = 10.0mag.These curves do not come close to matching the data and illustrate the sensitivity of the predicteddistribution to the value of A_max and thus to S'(>A_K), the un-normalized cloud column density pdf. For example, in the CMC, A_max is very close to the maximum observed extinction (6.8mag) in the 36arcsec pixels. However, the A_max = 10mag curves could correspond to a cloud with a PDF_N(A_K)that likely falls off more slowly with extinction and as a result contains considerably more high extinction material than the CMC.Comparison of these models with the data again illustrates the importance of cloud structure in controlling the star formation process in the cloud. Finally, we plotted a curve (dashed-dot trace) for a power-law with index -0.7 that is normalized at an extinction of 0.2mag. This illustrates the situation of a low extinction truncation that would correspond to the edge of a molecular cloud. This curve fails to match any of the observations. This reinforces the idea that there is a definite threshold for or truncation of the protostellar pdf at modest (A_K ∼0.5mag) extinctions in the CMC. This truncation or threshold may be the result of an additional steepening (i.e., β > 4) of the Schmidt relation for A_K < 0.5mag in the CMC. If the protostellar pdf cannot be described as a single power-law function then equation <ref> requires that single power-law functions cannot describe the Schmidt relation and/or the cloud pdf. Inspection of figure <ref> indicates that PDF_N(A_K) is very close to being a single power-law relation over alarge range (i.e., A_K > 0.2mag) in extinction. On the other hand, as discussed earlier and shown in figure <ref>, the Schmidt relation in the CMC is not particularly well fit by a single power-law function over a similar extinction range. Indeed, since the power-law indicies of the three functions are related as β = p + n, a change in power-law slope p would be directly reflected by a change in β for a constant n. At high extinctions we would expect β = 4.0 - 1.4 = 2.6 suggesting a flattening of the relation at high extinction as is seen in figure <ref>. At lower extinctions we would predictβ = 4.0 - 0.5 = 3.5, not far from the value of 3.3 that appears to fit the lower portion of the relation.We can use the observed cumulative protostellar distribution function again to further constrain the Schmidt relation. Figure <ref> plots the predicted distribution of protostellar objects obtained by numerically integrating: N_*(>A_K) = ∫Σ_*(A_K)dS = ∫_A_K^∞Σ_*(A_K) |-S(>A_K)|dA_KThe integration is carried out over the observed differential area function with Σ_* ∝ A_K^β for differing values of β. The uncertainties are shown for the case of β = 3.31 that we derived from the data assuming a model of a single power function. As observed in figures <ref> and <ref> a single power-law does not appear match the observations. Allthough the low extinction data are well matched by the nominal β≈ 3.3 curve, the high extinction data in figure <ref> fall below the nominal β = 3.3 curve and are best matched by the β≈ 2.5 curve.Both of these values match the prediction above, that β = p + n with n and p independently determined from the data. These considerations imply that the Schmidt relation in the CMC is not described by a single power-law that rises with extinction but instead is a somewhat more complex function, more steeply rising in the outer regions of the cloud than in the inner high extinction regions. A qualitatively similarbehavior is observed in the cumulative distributions of protostars in the Orion A, Orion B and Perseus clouds and may be a general property of the internal Schmidt relation within a GMC. However, we note that the power-law indicies, β and n, for the CMC are both greater than the corresponding indicies of the other three sources, which are very similar to each other. Yet the value of the index p is essentially the same for all four sources. This may imply a certain similarity of PDF_*(A_K) between the local GMCs.A universal protostellar pdf would have interesting consequences for star formation theory.For example, equation <ref>would then suggest that cloud-to-cloud variations in the slope of the Schmidt relation are primarily driven by variations in the slope of the cloud column density pdf.§ CONCLUSIONSWe have constructed high-resolution, high dynamic range dust column-density and temperature maps of the California Molecular Cloud. The maps were derived by fitting Herschel fluxes in each map pixel with a modified blackbody to derive the dust opacities and effective (i.e, line-of-sight weighted) temperatures. The opacities were calibrated at low extinction by 2MASS NIR extinction measurements to produce final maps of dust column density expressed as A_K. The column-density maps span a high dynamic range covering 0.04mag < A_K < 11mag, or 6.7D20cm^-2 < N < 1.8D23cm^-2, a considerably larger range than measured in previous NIR extinction maps. This enables us to investigate cloud structure and star formation to much greater depths in the cloud than previously possible.We used these data to determine the ratio of the 2.2 extinction coefficient to the 850 opacity and found the value (≈ 3600) to be close to that (3500) found in a similar study of the Orion B cloud but higher than that (≈ 2500) characterizing the Orion A cloud, indicating that variations in the fundamental optical properties of dust may exist between local clouds. We find that the column density pdf of the cloud can be well described over a large range of extinction (0.35 ≲ A_K ≲11mag)by a simple power law (i.e., PDF_N(A_K) ∝ A_K^-n) with an index(n = 4.0 ± 0.1) that represents a steeper decline with A_K than found (n ≈ 3) in similar Herschel-Planck studies of the Orion A, Orion B, and Perseus clouds.We re-examined existing catalogs of YSOs in the cloud and produced a merged catalog with slightly revised classifications for the known YSOs. Using only the protostellar population of the cloud and our Herschel extinction maps we investigated the Schmidt relation within the cloud. If we assumed that this relation is given by a simple power-law that is,Σ_* ∝Σ_gas^β, we found β = 3.31 ± 0.23, a slope that is significantly steeper than that (≈ 2) found in other local GMCs. However modeling the cumulative distribution of protostars in the cloud indicated that Σ_* is not a simple power-law function. Instead it is better described by two power-laws. At low extinction it is a rapidly rising power-law with an index of 3.3, while at higher extinctions it is characterized by a more slowly rising power-law with an index ≈ 2.5. We showed that Σ_* is directly proportional to the ratio of the protostellar pdf, PDF_*(A_K), and PDF_N(A_K) and that if Σ_* and PDF_N are simple power-law functions,PDF_*(A_K) must be a simple power-law as well, that is, PDF_*(A_K) ∝ A_K^p where p= β - n. We used the cumulative distribution of protostars to constrain the functional form ofPDF_*(A_K). We found that it is not well described by a single power-law. At high extinctions PDF_*(A_K) is a declining power-law with p ≈ -1.4. Between 0.5 ≲ A_K ≲2.5mag the function is characterized by a shallower power-law with p ≈ -0.5. Below 0.5mag it appears to be truncated and must fall steeply with decreasing extinction. Its behavior closely mirrors that of β, exactly as expected ifPDF_N is well described by a single power-law function. Furthermore, our observations tentatively suggest that PDF_*(A_K) does not vary significantly between local clouds. If so, the variation inβbetween the CMC and other nearby GMCs is largely driven by differencesin the slope, n, of the column density pdfs of these clouds. However because the magnitude of the spectral index, n, of the cloud pdf is greater than that of the Schmidt relation, β, the size of the protostellar population and thus the global SFR in the cloud is largely controlled by the cloud's structure(i.e., PDF_N(A_K)).Finally, our observations of the CMC have provided valuable insights into the star formation process by adding to the evidence that cloud structure is critical to setting the level of star formation in a cloud. However, we have not explained why the CMC has the structure it does. Although this was likely inherited from its formation, we not do not know whether the CMC will remain a sleeping giant or awaken and evolve its structure to resemble the Orion clouds with considerably more high extinction material and the corresponding increased levels of star formation.We thank Jan Forbrich for informative discussions.Based on observations obtained with Planck (<http://www.esa.int/Planck>), an ESA science mission with instruments and contributions directly funded by ESA Member States, NASA, and Canada.J. Alves acknowledges support from the Faculty of the European Space Astronomy Centre (ESAC) and J. 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J., & Bouy, H. 2016, , 587, A106 § THE YSO CATALOG FOR THE CALIFORNIA MOLECULAR CLOUD The two most extensivecatalogs of YSOs in the CMC are based on Spitzer, Herscheland Wise data<cit.>.selected Spitzer YSOs using color-color and color-magnitude diagrams and then used Herschel, Spitzer and WISE observations to construct SEDs with sufficient wavelength coverage to provide suitable classifications of the objects. The 2-24 μ m SED slope was employed to obtain their YSO classifications, producing 58 protostellar objects including objects classified as flat-spectrum sources.used SED slopes from 3.6-160 μ m to classify the Herschel sources in the CMC, and used the bolometric temperature (T_ bol) over the same range to separate Class 0 (T_ bol≤ 50 K) and Class I (50 K≤ T_ bol≤ 70 K) sources. They found 49 sources with protostellar like SEDs Because the two analyses areindependent, there is some disagreement between the catalogs in classifications of the same sources. By combining both data sets we create better, more complete, spectral energy distributions to use for source classification. We re-examined the source classifications following the methodology of <cit.>. We first ensured proper matching by requiring that every objectmatched to thecatalog <cit.> is matched in our merged list, and we resolve conflicts by examining every match by eye.We fitted the source SEDs using the SED models developed by <cit.> to obtain estimates of the quantities M_⋆, the mass of the central star, Ṁ the mass accretion/infall rate, M_ env, the mass of the protostellar, infalling envelope, and M_ disk, the mass of any circumstellar disk. We then assign classifications using the following criteria: * P: protostar (Class 0/I), Ṁ/M_⋆≥ 10^-6 and M_ env/M_⋆≥ 0.05* D: disk(Class II),Ṁ/M_⋆ < 10^-6 and M_ disk/M_⋆≥ 10^-6* S: star (Class III)Ṁ/M_⋆ < 10^-6 and M_ disk/M_⋆ <10^-6 In this manner we determined that 42 sources in the merged list of Table A.1 were protostars. We note that the difference between this estimate and that ofis primarily due to our assignment of a disk classification for most of the flat spectrum sources in thelist. We further note that sources 1, 2, & 3 are outside Herschel footprint. Their extinctions are derived from Planck.One of these we classify as a protostar but it was not included in any of our subsequent analysis.lllllllll Young Stellar Objects in the California Molecular CloudID RA (J2000) DEC (J2000) A_ K (error) 2c 2cClass 5-6 7-8 ID Class ID Class1* 04 01 24.55 41 01 49.000.819 (0.007) 1 I 0 – P 2* 04 01 34.36 41 11 43.00 0.934 (0.020) 2 II 0 – D 3* 04 02 29.75 40 42 41.90 0.313 (0.008) 139 II 0 – D 4 04 09 02.00 40 19 13.10 1.190 (0.029) 140 I 1 I P 5 04 09 54.71 40 06 39.89 4.096 (0.054) – – 2 I P 6 04 10 00.64 40 02 36.10 2.857 (0.054) 3 II – – D 7 04 10 02.63 40 02 48.20 3.051 (0.050) 4 I 3 0 P 8 04 10 03.43 39 04 49.50 0.338 (0.004) 141 III – – S 9 04 10 04.53 40 02 37.50 2.415 (0.053) – – 4 0 P 10 04 10 05.62 40 02 38.60 3.146 (0.062) 5 II 5 0 D 11 04 10 07.08 40 02 34.58 2.717 (0.068) – – 6 0 P 12 04 10 08.41 40 02 24.40 2.621 (0.076) 6 I 7 I P 13 04 10 11.16 40 01 26.20 2.390 (0.071) 7 I 8 0 P 14 04 10 24.41 38 05 22.70 0.435 (0.008) 142 II – – D 15 04 10 40.51 38 05 00.40 1.339 (0.034) 8 II – – D 16 04 10 41.09 38 07 54.50 1.917 (0.090) 10 I 9 I P 17 04 10 41.63 38 08 05.80 3.277 (0.118) 9 II – – D 18 04 10 42.11 38 05 59.90 1.682 (0.017) 11 III – – S 19 04 10 47.61 38 03 33.80 0.383 (0.010) 12 II – – D 20 04 10 49.16 38 04 45.80 2.714 (0.054) 13 II 10 F D 21 04 12 08.47 38 01 46.60 0.341 (0.005) 143 III – – S 22 04 12 40.54 38 14 26.81 0.549 (0.007) – – 11 I/0 P 23 04 12 57.64 39 14 18.30 0.293 (0.005) 144 III – – S 24 04 13 44.57 39 04 35.70 0.414 (0.005) 145 III – – S 25 04 15 11.20 38 39 57.10 0.341 (0.005) 146 II – – D 26 04 15 54.05 38 34 13.10 0.396 (0.005) 147 III – – S 27 04 17 05.93 37 22 18.70 0.208 (0.005) 148 III – – S 28 04 19 44.67 38 11 21.90 0.501 (0.006) 14 F – – D 29 04 20 52.46 38 06 35.80 0.604 (0.007) 15 III – – S 30 04 21 37.95 37 34 41.80 3.497 (0.062) 16 II 12 F D 31 04 21 38.08 37 35 40.90 3.167 (0.050) 17 III – – S 32 04 21 40.80 37 33 59.00 4.874 (0.080) 18 I 13 I P 33 04 23 05.46 38 07 36.90 0.271 (0.005) 149 III – – S 34 04 24 49.34 37 16 46.40 1.128 (0.008) 19 III – – S 35 04 24 59.04 37 17 52.91 1.307 (0.020) – – 14 I P 36 04 25 07.83 37 15 19.30 2.052 (0.053) – – 15 0 P 37 04 25 38.48 37 07 01.20 6.295 (0.068) 20 I 16 I/0 P 38 04 25 39.79 37 07 08.20 6.452 (0.051) 21 F 17 II D 39 04 27 13.74 36 27 10.70 0.428 (0.003) 150 II – – D 40 04 27 50.80 36 31 26.40 0.523 (0.003) 22 II – – D 41 04 27 58.26 36 33 26.50 0.546 (0.003) 23 II – – D 42 04 28 02.89 36 40 58.60 0.802 (0.004) 24 II – – D 43 04 28 15.15 36 30 28.60 1.262 (0.004) 25 F 18 F D 44 04 28 21.16 36 24 47.80 0.753 (0.003) 26 II – – D 45 04 28 21.36 36 30 21.50 0.919 (0.003) 27 II – – D 46 04 28 35.09 36 25 06.40 1.961 (0.020) 28 I 19 I P 47 04 28 37.89 36 24 55.30 1.757 (0.017) 29 II 20 F D 48 04 28 38.56 36 25 28.90 2.815 (0.021) 30 I 21 I P 49 04 28 43.35 36 25 11.70 0.887 (0.005) 31 II – – D 50 04 28 43.67 36 28 39.30 1.801 (0.023) 32 I 22 I/0 P 51 04 28 44.43 36 24 45.60 0.758 (0.004) 33 F – – D 52 04 28 49.58 36 29 10.70 1.165 (0.004) 34 I – – D 53 04 28 55.30 36 31 22.50 1.965 (0.035) 35 I 23 I P 54 04 28 55.56 35 24 46.00 0.305 (0.004) 151 II – – D 55 04 28 59.11 36 23 11.20 0.442 (0.003) 36 II – – D 56 04 29 11.53 35 04 49.50 0.233 (0.005) 152 II – – D 57 04 29 14.38 35 15 24.50 0.363 (0.004) 153 II – – D 58 04 29 39.01 35 16 10.50 0.459 (0.003) 37 II – – D 59 04 29 40.01 35 21 08.90 0.696 (0.006) 38 I – – P 60 04 29 43.58 35 13 38.60 0.646 (0.004) 39 II – – D 61 04 29 44.21 35 12 30.00 0.612 (0.005) 40 F – – D 62 04 29 46.28 36 19 23.50 0.423 (0.003) 154 F – – D 63 04 29 47.28 35 10 19.20 0.406 (0.003) 41 II – – D 64 04 29 47.42 35 11 33.50 0.605 (0.006) 42 II – – D 65 04 29 48.54 35 12 12.50 0.671 (0.004) 43 II – – D 66 04 29 49.21 35 14 22.70 1.553 (0.019) 44 F – – D 67 04 29 49.61 35 14 43.80 2.186 (0.025) 45 II – – D 68 04 29 50.17 35 14 44.50 2.186 (0.025) 136 II – – D 69 04 29 50.84 35 15 57.90 2.249 (0.013) 46 F – – D 70 04 29 51.01 35 15 47.50 2.249 (0.013) 47 I – – P 71 04 29 52.54 35 22 23.60 0.292 (0.003) 155 III – – S 72 04 29 53.46 35 15 48.50 2.108 (0.013) 48 F – – D 73 04 29 54.15 35 10 21.60 0.457 (0.006) 49 F – – D 74 04 29 54.18 36 11 57.30 0.985 (0.006) 156 F 24 II D 75 04 29 54.79 35 18 02.50 0.780 (0.006) 50 II 25 F D 76 04 29 56.27 35 17 42.90 0.760 (0.009) 51 I – – P 77 04 29 59.19 36 10 16.10 0.773 (0.006) 157 II 26 II D 78 04 29 59.76 35 13 34.20 0.788 (0.006) 52 II – – D 79 04 30 00.16 36 03 22.70 0.531 (0.004) 53 II – – D 80 04 30 01.14 35 17 24.60 0.333 (0.002) 54 III – – S 81 04 30 01.52 36 07 33.30 0.449 (0.003) 158 II – – D 82 04 30 01.88 35 38 14.70 0.350 (0.005) 159 II – – D 83 04 30 02.63 35 15 14.30 0.831 (0.006) 55 II – – D 84 04 30 03.63 35 14 20.10 2.331 (0.016) 56 I – – P 85 04 30 04.23 35 09 45.90 0.523 (0.005) 57 II – – D 86 04 30 04.25 35 22 23.80 0.336 (0.002) 58 II – – D 87 04 30 07.43 35 14 57.90 0.659 (0.005) 59 II – – D 88 04 30 07.73 35 15 48.40 0.429 (0.002) 60 II – – D 89 04 30 08.25 35 14 10.00 1.528 (0.009) 61 I – – P 90 04 30 08.74 35 14 37.50 1.434 (0.010) 62 II – – D 91 04 30 09.51 35 14 40.30 1.101 (0.006) 63 I – – P 92 04 30 09.80 35 40 35.50 0.447 (0.005) 64 II – – D 93 04 30 09.80 36 13 35.40 0.444 (0.003) 160 II – – D 94 04 30 09.86 35 14 16.30 1.528 (0.010) 137 II – – D 95 04 30 09.91 35 15 53.90 0.420 (0.003) 65 II – – D 96 04 30 12.34 35 09 34.60 1.531 (0.015) 66 II – – D 97 04 30 13.09 35 13 58.60 1.165 (0.006) 67 II – – D 98 04 30 14.53 35 13 32.60 1.315 (0.008) 68 II – – D 99 04 30 14.74 35 20 14.30 0.583 (0.004) 69 II – – D 100 04 30 14.95 36 00 08.50 2.047 (0.028) 70 I 27 I D 101 04 30 15.21 35 16 39.80 1.067 (0.012) 138 F – – D 102 04 30 15.76 35 56 57.80 0.753 (0.020) 71 I 28 II D 103 04 30 16.27 35 42 42.90 0.424 (0.005) 72 II – – D 104 04 30 17.84 36 03 26.60 0.602 (0.005) 73 III – – S 105 04 30 18.08 35 45 38.90 0.546 (0.008) 74 II – – D 106 04 30 18.99 35 42 12.00 0.479 (0.006) 75 II – – D 107 04 30 19.59 35 08 21.60 1.220 (0.015) 76 F – – D 108 04 30 22.19 36 04 35.90 0.857 (0.008) 77 II – – D 109 04 30 22.68 35 19 08.10 0.617 (0.007) 78 II – – D 110 04 30 23.82 35 21 12.30 0.740 (0.007) 79 I – – P 111 04 30 24.33 34 59 16.50 0.440 (0.006) 80 III – – S 112 04 30 24.68 35 45 20.60 1.221 (0.028) 81 I 29 I P 113 04 30 25.03 35 43 17.90 1.115 (0.012) 82 II – – D 114 04 30 25.89 35 48 11.30 1.507 (0.019) 83 II – – D 115 04 30 27.02 35 20 28.40 0.811 (0.008) 84 II – – D 116 04 30 27.04 35 45 50.50 2.062 (0.023) 85 F 30 I P 117 04 30 27.41 35 09 17.80 1.919 (0.077) 86 I 31 I P 118 04 30 27.75 35 46 15.00 2.629 (0.022) 87 F 32 II D 119 04 30 28.09 35 09 16.40 2.970 (0.101) 88 I – – P 120 04 30 28.42 35 32 41.90 0.428 (0.005) 89 II – – D 121 04 30 28.44 35 49 17.60 2.590 (0.015) 90 F – – D 122 04 30 28.61 35 47 40.70 1.798 (0.015) 91 II 33 II D 123 04 30 28.71 35 47 49.80 1.826 (0.018) 92 II – – D 124 04 30 28.98 35 07 54.00 0.769 (0.009) 93 II – – D 125 04 30 29.61 35 27 17.20 0.965 (0.010) 94 II – – D 126 04 30 29.66 35 06 39.00 1.102 (0.010) 95 II – – D 127 04 30 30.14 35 06 39.20 1.154 (0.011) 96 II 34 II D 128 04 30 30.28 35 21 04.00 0.581 (0.006) 97 II – – D 129 04 30 30.43 35 18 33.70 0.335 (0.003) 98 II – – D 130 04 30 30.51 35 17 44.70 0.383 (0.004) 99 II – – D 131 04 30 30.56 35 51 44.00 3.344 (0.053) 100 I 35 I P 132 04 30 31.58 35 45 13.70 3.073 (0.039) 101 F 36 II D 133 04 30 32.35 35 36 13.40 1.062 (0.016) 102 II 37 F P 134 04 30 36.80 35 54 36.20 4.275 (0.132) 103 I 38 I P 135 04 30 37.40 36 00 18.00 0.711 (0.009) 104 II – – D 136 04 30 37.51 35 13 48.60 0.460 (0.005) 105 II – – D 137 04 30 37.51 35 50 31.70 4.321 (0.101) 106 II 39 F D 138 04 30 37.89 35 51 01.40 3.383 (0.080) 107 I 40 0 P 139 04 30 38.26 35 49 59.30 2.342 (0.077) 108 II 41 0 P 140 04 30 38.38 35 50 22.60 3.720 (0.092) – – 42 0 P 141 04 30 38.65 35 54 39.10 4.404 (0.162) 109 F 43 F D 142 04 30 39.12 35 44 49.80 0.726 (0.008) 110 II – – D 143 04 30 39.16 35 52 03.80 2.276 (0.040) 111 F 44 F P 144 04 30 39.31 35 52 00.70 2.647 (0.037) 112 F – – D 145 04 30 39.56 35 18 06.90 0.337 (0.004) 113 II – – D 146 04 30 39.58 35 11 12.80 0.403 (0.004) 114 II – – D 147 04 30 40.05 35 42 10.30 0.363 (0.006) 115 III – – S 148 04 30 40.14 35 31 34.10 1.497 (0.032) 116 II – – D 149 04 30 41.16 35 29 41.00 4.681 (0.069) 117 I 45 I P 150 04 30 44.23 35 59 51.10 2.782 (0.061) 118 I 46 F D 151 04 30 44.69 35 10 52.10 0.370 (0.005) 119 II – – D 152 04 30 45.58 34 58 08.00 0.997 (0.017) 120 II – – D 153 04 30 46.25 34 58 56.20 2.743 (0.096) 121 I 47 0 P 154 04 30 47.23 35 07 43.20 0.745 (0.008) 122 II 48 II D 155 04 30 47.57 34 58 24.20 3.860 (0.143) 123 II – – D 156 04 30 47.90 34 58 37.31 3.860 (0.143) – – 49 0 P 157 04 30 48.52 35 37 53.70 2.480 (0.051) 124 I 50 I P 158 04 30 48.61 34 58 53.50 5.167 (0.168) 125 I 51 F S 159 04 30 49.22 34 56 10.30 1.025 (0.012) 126 I 52 F D 160 04 30 49.33 34 50 46.00 0.585 (0.009) 161 II – – D 161 04 30 49.34 35 36 41.90 0.808 (0.011) 127 II – – D 162 04 30 49.48 34 50 56.20 0.559 (0.009) 162 II – – D 163 04 30 49.68 34 57 27.70 0.860 (0.021) 128 II 53 II D 164 04 30 50.57 35 33 23.50 0.864 (0.007) 129 II – – D 165 04 30 50.98 35 35 54.80 0.516 (0.007) 130 II – – D 166 04 30 52.08 34 50 08.90 0.653 (0.010) 163 F 54 F D 167 04 30 53.50 34 56 27.40 2.593 (0.041) 131 I 55 I D 168 04 30 53.90 35 30 11.00 1.097 (0.017) 132 II – – D 169 04 30 55.01 35 30 56.20 0.583 (0.006) 133 II – – D 170 04 30 55.99 34 56 47.80 2.140 (0.028) 134 I 56 I D 171 04 30 56.61 35 30 04.50 1.513 (0.029) 135 I 57 I P 172 04 30 57.19 34 53 53.59 2.799 (0.058) – – 58 I P 173 04 31 14.67 35 56 50.60 0.633 (0.008) – – 59 I P 174 04 32 05.77 36 06 37.50 0.359 (0.005) 164 III – – S 175 04 32 54.31 36 04 44.00 0.362 (0.005) 165 II – – D 176 04 33 03.15 36 02 04.50 0.360 (0.006) 166 II – – D 177 04 34 53.15 36 23 27.89 2.284 (0.041) – – 60 II D | http://arxiv.org/abs/1708.07847v3 | {
"authors": [
"Charles J. Lada",
"John A. Lewis",
"Marco Lombardi",
"João Alves"
],
"categories": [
"astro-ph.GA"
],
"primary_category": "astro-ph.GA",
"published": "20170825180345",
"title": "HP2 survey: III The California Molecular Cloud--A Sleeping Giant Revisited"
} |
The Institute for Nuclear Research and Nuclear Energy, BAS,Sofia, 1784, Bulgaria E-mail: [email protected] Department of High Energy Nuclear Physics, Institute of Modern Physics, CAS, Lanzhou, 730000, China^*E-mail: [email protected] formalized the nuclear mass problem in the inverse problem framework. This approach allows us to infer the underlying model parameters from experimental observation, rather than to predict the observations from the model parameters. The inverse problem was formulated for the numericaly generalized the semi-empirical mass formula of Bethe and von Weizsäcker. It was solved in step by step way based on the AME2012 nuclear database. The solution of the overdetermined system of nonlinear equations has been obtained with the help of the Aleksandrov's auto-regularization method of Gauss-Newton type for ill-posed problems.In the obtained generalized model the corrections to the binding energy depend on nine proton (2, 8, 14, 20, 28, 50, 82, 108, 124) and ten neutron (2, 8, 14, 20, 28, 50, 82, 124, 152, 202) magic numbers as well on the asymptotic boundaries of their influence. These results help us to evaluate the borders of the nuclear landscape and show their limit. The efficiency of the applied approach was checked by comparing relevant results with the results obtained independently.§ INTRODUCTIONThe main goal of our studies was to determine how well the existing data, and only data, determines the mapping from the proton and neutron numbers to the mass of the nuclear ground state. Another is to find presumed regularities by analysis of observed nuclei masses <cit.>. In addition is to provide reliable predictive model that can be used to forecast mass values away from the valley of stability. The aforementioned goals stimulated us to try to clarify the features and to find hidden regularities of the well known semi-empirical mass formula of Bethe-Weizsäker (BW), based exclusively on experimental data. A set of experimental nuclear masses from AME2012, the most recent evaluation database, that was published in December 2012 in <cit.>, constitutes the raw material for this work.Only measured nuclei are included into our consideration. The masses extrapolated from systematics and marked with the symbol # in the error column are not taken into account here. Therefore we use only 2564 experimental nuclear masses, including the Hydrogen atom, to provide a deep understanding of the mutual influence of terms in the semi-empirical BW mass formula. In present work we demonstrate the applicability of the inverse problem (IP) approach for solution of such nuclear physics problems. First we formalize the nuclear mass problem in the framework of the IP. Second we propose the generalized form of the BW mass formula, which helps us to discover the latent regularities in the nuclear masses.Afterwards we provide a solution of the formalized IP that has been obtained with the help of the Alexandrov dynamic auto-regularization method of the Gauss-Newton type for ill-posed problems (REGN-Dubna) <cit.>, which is a constructive development of the Tikhonov regularization method <cit.> for ill-posed problems. The formalism of the applied approach is given in Sec. <ref>.The numerical generalization of the BW mass formula is described in Sec. <ref>.The main conclusions of the paper are drawn in Sec. <ref>, which also includes a discussion of the principal results, the resulting rms deviations. § THEORY AND METHODIPs are based on the comparison of the theoretical and experimental data by solving the system of the nonlinear operator equations on the field of real numbers Rof following type:F(x) = y, wherex=(x_1,x_2,…,x_n)^T∈R^n, y=(y_1,y_2,…,y_m)^T∈R^m.R^n and R^m are the n and m dimensional real coordinate space, that corresponds to the x and y respectively.The Eq.(<ref>) typically involve the estimation of certain quantities based on indirect measurements of these quantities. The estimation process is often ill-posed in the sense that noise in the data may give rise to significant errors in the estimate. In other words, the problem Eq.(<ref>) is ill-posed <cit.> if its solution does not depend continuously on the right hand side y, which is often obtained by measurement and hence contains errors, y^δ-y ≤δ,where y^δ is the measured perturbed data, δ is the experimental uncertainty.Operator Fis a forward modeling nonlinear operator, that transforms any model x into the corresponding data y. The IP is formulated as the solution of the operator equation. Therefore, Eq.<ref> connects the unknown parameters of the model with some given quantities (variables) describing the model, in our case atomic mass number, A, and proton mass number, Z. These quantities take the form of the so-called input data. Techniques known as regularization methods <cit.> have been developed to deal with this ill-posedness, to get stable approximations of solutions of Eq.(<ref>). In the current work we apply the Alexandrov method<cit.>, which we found an appropriate choice for our IP that will be formulated in the next section.§ PARAMETERIZATION OF THE BETHE-WEIZSÄCKER MASS FORMULAA careful analysis of the previous attempts of calculation nucleus masses reveals that all models can reproduce experimental/empirical trends on the average<cit.>.In order to overcome this issue we consider a parameterized nonlinear dynamical system of equations for determining nuclei and atomic masses from the experimental bound-state energies, which can be written using matrix notation of Eq.<ref> as:FE_B,j^Th(A,Z,{a_i}) =E_B,j^Expt(A,Z),FM_a.m.,j^Th(A,Z,{a_i}) = M_a.m.,j^Expt(A,Z),FM_n.m.,j^Th(A,Z,{a_i}) = M_n.m.,j^Expt(A,Z),FΔ m_j^Th(A,Z,{a_i}) = Δ m_j^Expt(A,Z), where F is the rectangular d× m {d=1,…,𝒩_data; m=1,…,𝒩_param} Jacobian matrix composed of the Frechet derivatives with respect to the model parameters. Each system of Eqs.(<ref>) contains 2564 equations, which correspond to the number of the experimental data-points. Therefore, these systems are overdetermined because the number of considered equations exceeds the number of parameters used in the fit. The right hand side of these equations represented by the vector of the observed experimental data, j is the component of this vector, also it indicates the nonlinear equation in the system. The solution of the overdetermined system of Eqs.(<ref>) for the binding energy and its model is given by the real values of the parameters a_i (i=1,…, 𝒩_param).In order to proceed further one have to choose the initial model for description of the binding energy. The BW mass formula was chosen for this role, since it provides the baseline fit for all the rest models<cit.>.The generalization of the BW mass formula for the binding energy per nucleon in our approach has the following form: E_B(A,Z,{a_i}) = α_vol(A,Z,{a_i}_1)-α_surf(A,Z,{a_i}_2) 1/A^p_1(A,Z, {a_i}_6)-α_comb(A,Z,{a_i}_3)Z(Z-1)/A^p_2(A, Z, {a_i}_7) -α_sym(A,Z, {a_i}_4 )(N-Z)^2/A^p_3(A,Z, {a_i}_8 )+α_Wigner(A,Z,{a_i}_5)δ(A,Z)/A^p_4(A, Z, {a_i}_1) +K_MN(A,Z,{a_i}_10). The term δ(A,Z) is equal to 1 for even N,Z; -1, for odd N,Z and 0, for odd (Z+N). The detailed description of all the rest terms of Eq.(<ref>) is beyond the focus of the current note, therefore for a curious reader we recommend this Ref.<cit.> for more detailed examination. However, note, that totally in our parametrization we use 17 linearly independent variables and 241 free parameters.In order to solve all systems of Eqs.(<ref>) we adopt the following procedure. First, we searched for the appropriate solution for the binding energy for the given A and Z. The obtained solution was used as an approximation (`fake solution') to find the atomic masses, the solution for atomic masses was in turn used to find nuclear mass. Then, the fit is performed again, allowing the binding energy to vary for the given A and Z. The resultant binding energy is then taken to be a new seed, and this procedure is repeated iteratively until convergence is reached for all systems. Indeed the system converge due to applied iterative process and due to theorems proved in Ref.<cit.>. § RESULTS The presented parameterization of the binding energy allows us to solve the formulated IP, with the help of the Aleksandrov's auto-regularized method. This solution provides us description of the 2564 nucleus masses and their binding energies starting from ^2_1H with relative error -1.1924×10^-6 and 3.2197×10^-4 for atomic mass and binding energy respectively. Mean absolute error, ϵ̅_abs, is -0.0509 and 7.8488×10^-4 for atomic mass and binding energy respectively. The maximum absolute deviation is less than 2.6 MeV for the atomic mass and less than 0.82 MeV for the binding energy. The mean modified error, ϵ̅_mod , of our solution is of 1.232(1%) – 1.231(5%) MeV, the σ_mod of 1.11(1%) – 1.565(5%) MeV (assuming that the theoretical uncertainty varies between 1% and 5%) for the atomic mass. While for the binding energy, using the same assumption for band of the theoretical uncertainties, we obtain ϵ̅_mod, of fit of 0.044(1%) – -0.0057(5%) MeV, the σ_mod of 0.209(1%) – 0.223(5%) MeV,which can be compared with the latest fits for all modern mass formulas <cit.>.The obtained results and validation of our approach on the big set of the experimental data allows us to probe the outer boundary of the island of enhanced nuclear stability. The reshaping of the nuclear landscape, illustrated in Fig.<ref>, is the results of such probe. In finding both the proton and neutron drip lines, two neutron (S_2n) and two proton (S_2p) separation energies were used.The two-neutron and two proton drip lines are reached when S_2n≈ 0 and S_2p≈ 0, respectively. § SUMMARYWe found that results of the verification of the Bethe-Weizsäcker mass formula in the IP framework greatly improve the agreement between the experimental masses and the calculated ones and thus predicts the drip-lines more accurately than calculated earlier <cit.> with the modified BW mass formula alone. This in turn allows us to make a prediction for the binding energy, nuclear and atomic mass, and mass excess of the recently discovery nuclei at the DGFRS separator <cit.>, see Ref.<cit.>. Moreover, this finding together with the asymptotic behavior, and accurate predictions of the binding energies of all known isotopes allows us to obtain quite precise predictions for the location of the proton and neutron drip lines, and claim the exact number of bound nuclei in the nuclear landscape, see Fig.<ref>. We would like to comment that this result is not in a conflict with respect to the problem of the critical charge in QED, for interested reader we refer to <cit.>, where one may find all necessary details.We are of course aware that the super-heavy isotopes in Fig.<ref> cannot be produced at present and may be even at future facilities and that, so far, only theoretical studies may be carried out in such region of the nuclear chart.The concept of ill-posed problems and the associated regularization theories seem to provide a satisfactory framework to solve nuclear physics problems. This new perspective can be used for testing the applications of the liquid model in different areas, for forecasting the mass values away from the valley of stability, for calculation of kinetic and total energy of nuclear proton, alpha, cluster decays and spontaneous fission, for preliminary research of island stability problem and the possibility of creating new super-heavy elements in the future.§ ACKNOWLEDGMENTSWe are grateful to Lubomir Aleksandrov for the provided REGN program. S. Cht. Mavrodiev would like to thank Alexey Sissakian and Lubomir Aleksandrov for many years of a very constructive collaboration and friendship. Unfortunately, they passed away, but they made enormous contributions to the REGN development and application of it for solving different IPs, especially for the discovering latent regularities. This work was partially supported by the Chinese Academy of Sciences President's International Fellowship Initiative under Grant No. 2016PM043.ws-procs9x6 | http://arxiv.org/abs/1708.07966v1 | {
"authors": [
"S. Cht. Mavrodiev",
"M. A. Deliyergiyev"
],
"categories": [
"nucl-th",
"65Fxx, 65F22, 81V35, 15A29, 49N45"
],
"primary_category": "nucl-th",
"published": "20170826130940",
"title": "Computation of the Binding Energies in the Inverse Problem Framework"
} |
Imbalanced Malware Images Classification: a CNN based Approach Songqing YueUniversity of [email protected] WangAustin Peay State [email protected] 30, 2023 ============================================================================================================================Deep convolutional neural networks (CNNs) can be applied to malware binary detection via image classification. The performance, however, is degraded due to the imbalance of malware families (classes). To mitigate this issue, we propose a simple yet effective weighted softmax loss which can be employed as the final layer of deep CNNs. The original softmax loss is weighted, and the weight value can be determined according to class size. A scaling parameter is also included in computing the weight. Proper selection of this parameter is studied and an empirical option is suggested. The weighted loss aims at alleviating the impact of data imbalance in an end-to-end learning fashion. To validate the efficacy, we deploy the proposed weighted loss in a pre-trained deep CNN model and fine-tune it to achieve promising results on malware images classification. Extensive experiments also demonstrate that the new loss function can well fit other typical CNNs, yielding an improved classification performance.§ INTRODUCTIONMalware binary, usually with a file name extension of “.exe" or “.bin",is a malicious program that could harm computer operating systems. Sometimes, it may have many variations with highly reused basic patterns. This implies that malware binaries could be categorized into multiple families (classes), and each variation inherits the characteristics of its own family. Therefore, it is important to effectively detect malware binary and recognize possible variations <cit.>.This is non-trivial but challenging. A malware binary file can be visualized to a digital gray image <cit.>. After visualization, the malware binary detection turns into a multi-class image classification problem, which has been well studied in deep learning.One can manually extract features from malware images and feed them into classifiers such as SVM (support vector machine) or KNN (k-nearest neighbors algorithm) to detect malware binaries via classification. To be more discriminative, one can utilize CNN to automatically extract features as Razavian et al. did in <cit.> and perform classification in an end-to-end fashion. However, most deep CNNs are trained by properly solicited balanced data <cit.>, while malware image datasets may be highly imbalanced: some malware has many variations while some other only has few variations. For instance, the dataset <cit.> used in our paper consists of 25 classes, and some class includes more than 2000 images while some other only has 80 images or so. As a result, the reputed pre-trained CNN models <cit.> may still perform poorly in this scenario. Furthermore, pre-trained CNN models are originally designed for specific vision tasks, and they cannot be applied to malware binary detection directly. One may argue that data augmentation could be a possible approach to balance data, such as oversampling minority classes and/or down-sampling majority classes. It is, however, not suitable for our problem due to two reasons. First, down-sampling may result in a loss of many representative malware variations. Second, simply augmenting data cannot guarantee to generate images corresponding to real malware binaries. To address the challenges, we investigate how to train a CNN model with the imbalanced data on hand.Inspired by the work in <cit.> which designed new loss functions for improving CNN training, we propose a weighted softmax loss for deep CNNs on malware images classification. Based on the error rate given by the softmax loss, we weight misclassifications by different values corresponding to class size. Intuitively, misclassifications in minority classes should be amplified, and that in majority classes need to be suppressed. Our weighted loss can achieve this goal and guide the model weights updating in a proper direction. We adopt a pre-trained VGG-19 model <cit.>, and retrain it to achieve promising results on the malware image classification. Once the proposed loss has been demonstrated effective on the VGG model, it can be extended to other models, such as GoogleNet <cit.> and ResNet <cit.>. The contributions of this work can be summarized into two-fold. First, we propose a weighted softmax loss to help CNNs deal with imbalanced data. Second, we apply the proposed loss to address the challenges in malware image classification. § RELATED WORKMalware Images: Nataraj et al. <cit.> proposed a way of converting malware binaries into digital gray images. They created 25 classes of malware images that are highly imbalanced. Two sample malware images from the ‘Adialer.C’ class and ‘Skintrim.N’ class are shown in Fig. <ref>. The Microsoft Malware Classification Challenge[https://www.kaggle.com/c/malware-classification] also provides imbalanced data which includes 9 classes only. Therefore, it is not as challenging as the one in <cit.> in terms of malware diversity. Deep Learning: Deep learning recently gains a remarkable success in computer vision. Krizhevsky et al. in <cit.> opened a new era for deep CNNs and their applications on image classification. Many works have been inspired thereafter. Simonyan and Zisserman proposed a very deep CNN in <cit.>, and they significantly improved the performance by increasing the depth and using small convolutional filters with a size of 3×3. Szegedy et al. <cit.> designed a 22 layers GoogLeNet which increased the width of the network, and achieved the new state-of-the-art performance on image classification. He et al. <cit.> eased the training of a 152 layers deep CNN by presenting a residual learning scheme.Besides image classification, deep learning has also been applied on fundamental image processing tasks, such as denoising <cit.> and contour detection <cit.>. Saxe and Berlin <cit.> discussed the feasibility of using deep CNNs on malware detection. Hand-crafted features such as ‘PE import’ are extracted prior to the model training, which is not the typical manner of an end-to-end learning process.Huang et al. <cit.> proposed a quintuplet based triple header hinge loss for extracting discriminative features from imbalanced data. Nevertheless, their work requires extracting features in advance in a hand-crafted manner or using another pre-trained CNN model. Meanwhile, clustering algorithms such as k-means are needed prior to the training. These steps have not been integrated into an end-to-end fashion, employed by typical CNNs.Recently, Xu et al. <cit.> have applied deep learning in medical CT physics, and perform scatter correction via designing a new loss function.§ WEIGHTED SOFTMAX LOSSIn this section, we introduce how to weight softmax loss according to class size.§.§ Softmax Loss Softmax loss is a combination of softmax regression and entropy loss, used in multi-class classification problems. Given a K-classes training set including m images: {x^(i), y^(i)}, where x^(i) refers to an image, and y^(i) denotes the ground truth label (y^(i)∈{1,2,…,K}). Let a_j^(i) (j=1,2,…,K) be the output unit from the last fully connected layer of a CNN, then the probability that the label of x^(i) is j can be given by P_j^(i)=exp(a_j^i)/∑_l=0^K exp(a_l^(i)).Typical deep CNNs aim to minimize the entropy loss function:J_0=-1/m{∑_i=1^m ∑_j=0^K 1(y(i)=j) log p_j^(i)},where m is the batchsize, and K is the total number of the classes in a dataset. 1(.) is an indicator function such that, 1(true) gives 1, and 1(false) gives 0. In typical CNNs, convolutional filters are updated by the stochastic gradient descent (SGD) algorithm. The traditional softmax loss equally treats the misclassifications in each class, which is reasonable for balanced data, but will lead to a poor performance on imbalanced data, such as malware images. Our weighted softmax loss is proposed to address this issue.§.§ Weighted Softmax LossOur approach can be formulated as follows,J_0=-1/m{∑_i=1^m ∑_j=0^K ω_k *1(y(i)=j) log p_j^(i)},where ω_k is a weight coefficient determined byω_k=1+S_max-S_k/β*S_max,where k is the ground truth label of the i^th image. S_max is the size of the largest class in the dataset, and S_k is the size of an arbitrary class k in the dataset, and β is a parameter that controls the scaling of the weighted loss. Our empirical preference of β is 20. Extensive experiments about this parameter will be shown in section <ref>. It is worth noting that the minority classes will be assigned a larger weight coefficient whereas the majority classes will be lightly weighted. The weight coefficient will not dramatically affect the loss, thus it can be regarded as a subtle fine-tuning, that boosts the classification performance as well as avoids overfitting. In practice, the training procedure only needs to choose a ω_k according to the ground truth label of the i^th sample in a mini-batch. § CLASSIFICATION WITH CNNIn this section, we discuss how to fine-tune the VGG-19 model for malware image classification.§.§ Network Architecture Simonyan and Zisserman in <cit.> proved that classification accuracy can be improved via deepening the network. The VGG-F model <cit.> consists of 21 layers, whereas VGG-19 includes 43 layers. Convolutional filters with a size of 3×3 are used in VGG-19, and such smaller filters have been demonstrated being capable of extracting discriminative features. We fine-tune VGG-19 since it has been shown very competitive in the VGG family. Simonyan and Zisserman <cit.> argued that the local response normalization (LRN) <cit.> could be ignored in CNNs. In order to boost the classification performance, we add Batch Normalization (BN) layer <cit.> between convolutional layer and ReLU (rectified linear unit) activation. However, the very first convolutional layer is followed by a ReLU layer directly, and the fully connected layer (expect the last one) is directly followed by a ReLU layer. A big potential threat to deep learning is overfitting, especially when a small training set is used. Srivastava et al. <cit.> invented dropout, a simple yet powerful approach to avoid overfitting in CNNs. Units in layers are randomly dropped and the corresponding connections are also removed temporarily. It works as randomly training different networks and averaging the results as the final one, aiming to enhance the generalization. We place two dropout layers (with probability =0.5) between the three fully connected layers to prevent overfitting (by default, the pre-trained VGG-19 does not have dropout layers). In testing stage, the dropout layers will be removed.§.§ Weighted Softmax Loss Layer We append the proposed loss as the last layer of our model. The final structure consists of 60 layers including the added dropout layers and BN layers. For brevity, only the fully connected layers, the added dropout layers, and the weighted loss layer are illustrated in Fig. <ref>.§ EXPERIMENTSIn this section, we validate the proposed loss on malware image classification. The VGG-F,M,S models <cit.> are also employed to validate the general fit of the new loss function. We also analyze the selection of the scaling parameter and suggest an empirical option. The Top-1 validation error is adopted as the evaluation metric. We conduct the experiments with the MatConvNet[http://www.vlfeat.org/matconvnet/] framework <cit.>, which is an open source library for deep learning in matlab. One Nvidia TITAN X GPU is used to accelerate the mini-batch processing.§.§ Dataset and Experimental Settings The dataset <cit.> used in our work consists of 25 classes which are highly imbalanced. The name and the size of each class is listed in Table <ref>[http://old.vision.ece.ucsb.edu/spam/malimg.shtml]. We partition the data as follows: the first 60% images in each class are used for training, and the following 20% for validation, and the last 20% for testing. No data augmentation is applied, but we pre-process an input by subtracting its mean.§.§ Effects of the Weighted LossTo validate the general fit, we deploy the proposed loss in 3 pre-trained CNN models, namely VGG-F, VGG-M, and VGG-S, respectively. The default structures are entirely preserved but the size of the last fully connected layer is changed from 1000 to 25, corresponding to the 25 classes in the malware dataset. We retrain the three networks, and the top-1 validation errors with and without the weighted loss are illustrated in Fig. <ref>. It can be seen that our method effectively decreases the top-1 validation error, and keeps the curves stable. The test results are given in Table <ref>.§.§ Fine-tuning VGG-19 with the Weighted Loss To achieve a better performance, we fine-tune VGG-19 as discussed in section <ref>. We initialize the filter weights with MSRA <cit.>. The momentum is set to 0.9, and the training is regularized by a weight decay rate of 0.0005. The learning rate is set to 0.0001, which we find leads to the best performance. Dynamic adjusting is not considered since our data size does not require many epochs to make the training converge. Batch size is set to 80. We compare the top-1 validation error of the models with and without the weighted loss. As shown in Fig. <ref>, the weighted loss more effectively decreases the error compared to the original loss. During testing, we remove the two added dropout layers, and the results are presented in Table 2. In the dataset, the ‘Autorun.K’, ‘Malex.gen!J’, ‘Rbot!gen’, ‘VB.AT’, and ‘Yuner.A’ classes are from the same pack (UPX-Ultimate Packer for eXecutables). However, we treat them as different malware families. Otherwise, the test error can be further decreased. Moreover, CNNs can extract image features automatically. In our fine-tuned model, a 4096-dimensional feature vector for an arbitrary image can be captured at layer 41 (the second fully connected layer). We generate a feature map for each malware class by extracting the features for all the images in that class and combining all the feature vectors to form a matrix, which is finally visualized. Therefore, the dimension of any feature map is n×4096, where n is the class size. Such a feature map can reflect the characteristics of the corresponding malware class. We give the feature maps of five typical malware classes in Fig. <ref>. Since the classes are highly imbalanced, n is different for each class. To attain the best display effect, we use the `imagesc` command in matlab to visualize a feature map, automatically scaling the map. §.§ Scaling Parameter In Eq. (4), the parameter β is needed to compute the weighted loss value. Empirically, 20 is the best option for β. To show the influence of this parameter, we further investigate three β values on the fine-tuned VGG-19 and report the top-1 validation error in Fig. <ref>. As can be seen, the curve with β=20 is smoother than the others and it also converges at a lower error rate. It indicates that an appropriate selection of β is also critical to the network training.§ CONCLUSIONWe propose a weighted softmax loss for convolutional neural networks on imbalanced malware image classification. By imposing a weight, the classification error for different classes can be treated unequally. The principle of weighting the loss has a clear intuition, and our experiments demonstrate its capacity of working with existing CNN models. We also fine-tune the pre-trained VGG-19 with the proposed loss to achieve a satisfactory classification performance. In addition, we have empirically suggested an option of the scaling parameter β in computing the weighted loss. It indicates that an appropriate selection of this parameter is indispensable for the training success. unsrt | http://arxiv.org/abs/1708.08042v2 | {
"authors": [
"Songqing Yue",
"Tianyang Wang"
],
"categories": [
"cs.CV",
"cs.LG",
"stat.ML"
],
"primary_category": "cs.CV",
"published": "20170827022759",
"title": "Imbalanced Malware Images Classification: a CNN based Approach"
} |
1,2]Megan E. Schwambmycorrespondingauthor [mycorrespondingauthor]Corresponding author [][email protected]]Klaus-Michael Aye 3]Ganna Portyankina 4]Candice J. Hansen 5]Campbell Allen 6]Sarah Allen 7]Fred J. Calef III 5]Simone Duca 5] Adam McMaster 5]Grant R. M. Miller [1]Gemini Observatory,Northern Operations Center, 670 North A'ohoku Place, Hilo, HI 96720, USA [2]Institute for Astronomy and Astrophysics, Academia Sinica; 11F AS/NTU,National Taiwan University, 1 Roosevelt Rd., Sec. 4, Taipei 10617, Taiwan [3]Laboratory for Atmospheric and Space Physics, University of Colorado at Boulder, Boulder, Colorado, 80303, USA [4]Planetary Science Institute, 1700 E. Fort Lowell, Suite 106, Tucson, AZ 85719, USA [5]Oxford Astrophysics, Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, UK [6]Adler Planetarium, 1300 S. Lake Shore Drive, Chicago, IL 60605, USA [7]Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA We present the results of a systematic mapping of seasonally sculpted terrains on the South Polar region of Mars with the Planet Four: Terrains (P4T) online citizen science project. P4T enlists members of the general public to visually identify features in the publicly released Mars Reconnaissance Orbiter Context Camera (CTX) images. In particular, P4T volunteers are asked to identify:1) araneiforms (includingfeatures with a central pit and radiating channels known as `spiders'); 2) erosional depressions, troughs, mesas, ridges, and quasi-circular pits characteristic of the South Polar Residual Cap (SPRC) which we collectively refer to as `Swiss cheese terrain', and 3) craters. In this work we present the distributions of our high confidence classic spider araneiforms and Swiss cheese terrain identifications in 90 CTX images covering 11% of the South polar regions at latitudes ≤ -75^∘ N. We find no locations within our high confidence spider sample that also have confident Swiss cheese terrain identifications. Previously spiders were reported as being confined to the South Polar Layered Deposits (SPLD). Our work has provided the first identification of spiders at locations outside of the SPLD, confirmed with high resolution HiRISE (High Resolution Imaging Science Experiment) imaging. We find araneiforms on the Amazonian and Hesperian polar units and the Early Noachian highland units, with 75% of the identified araneiform locations in our high confidence sample residing on the SPLD. With our current coverage, we cannot confirm whether these are the only geologic units conducive to araneiform formation on the Martian South Polar region. Our results are consistent with the current CO_2 jet formation scenario with the process exploiting weaknesses in the surface below the seasonal CO_2 ice sheet to carve araneiform channels into the regolith over many seasons. These new regions serve as additional probes of the conditions required for channel creation in the CO_2 jet process. Mars - Mars, polar geology - Mars, polar caps - Mars, surface - ices § INTRODUCTIONThe seasonal processes sculpting the Martian South Polar region are driven by the sublimation and deposition of carbon dioxide (CO_2) ice.A significant portion of the Martian atmosphere, of which CO_2 is the predominant species, freezes or snows out on to the winter pole during the fall and winter. During the spring and summer, all or part of the deposited CO_2 returns to the atmosphere <cit.>. This is observed by the large seasonal variations in atmospheric pressure, with amplitudes up to 25%, first measured by the Viking landers <cit.>. Combined multi-season gravity field observations from Mars Global Survey (MGS),Mars Odyssey, and Mars Reconnaissance Orbiter (MRO) point to approximately 12 to 16% of the mass of the entire Martian atmosphere solidifying out on to the surface of the winter pole <cit.>. This cycle is directly linked to the current Martian climate. Thus, studying how the Martian South Pole region's inventory of CO_2 ice changes and evolves throughout the season and from Mars year to Mars year provides insight into processes driving Mars' climate and atmosphere. The Martian South Pole's CO_2 inventory can be divided into buried ice deposits and two broad surface ice caps, the temporary seasonal cap and the more permanent South Polar Residual Cap (SPRC). The buried CO_2 deposits vary from tens to thousands of meters in thickness and are topped with a 10-60 mlayer of water ice. The mass of these buried CO_2ice reservoirs if sublimated is estimated to double the planet's current atmospheric pressure <cit.>. These CO_2ice deposits are located below the South Polar Layered Deposits (SPLD) <cit.>.The SPLD is comprised mostly ofbands ofdust and water ice in addition to the buried subsurfaceCO_2 ice reservoirs <cit.>. The SPLD is thought to have formed through repeated deposition linked to Mars' orbital/obliquity variations produced by the planet'sMilankovitch cycles <cit.>.Recent modeling of Mars' climate variations is also able to produce wide-spread deposition of CO_2 on the South Polar region,<cit.>, indicating the formation of theburied CO_2reservoirs is also linked to Mars' Milankovitch cycles. The SPRC is located between -84 to -89 degrees latitude and 220 E and 50 degrees E longitude. It isprimarily made of carbon dioxide ice <cit.>. The extent of the SPRC has been observed to expand and to retreat during various Mars years, but the structure as a wholesurvives past the spring and summer season <cit.>. The temporary seasonal ice sheeton the other hand completely sublimates away by the end of the Southern summer <cit.>. The SPRC is thicker and higher albedo than most of the temporary seasonal cap, with thickness ranging between ∼0.5 and 10 meters <cit.>. The surface of the SPRC is heavily eroded with smooth edgedquasi-circular flat bottomed pits, mesas, troughs, and other depressions <cit.>. <cit.> and <cit.> provide a detailed description of morphologies of the SPRC based on orbital imagery.The SPRC pits, depressions, and troughs have been observed to change in depth and areal coverage indicative of active mass loss <cit.>.Mass balance modeling suggests these errosional features are due to the uneven sublimation and deposition of CO_2 ice on the SPRC <cit.>. The seasonal cap is a temporary CO_2 ice sheet that extends from the pole to latitudes as far north as -50^∘, and in cold protected patches to -22^∘ <cit.>. Mars Orbiter Laser Altimeter (MOLA) observations place the seasonal ice sheet thickness at ∼0.9-2.5 m <cit.>, with compaction decreasing the thickness over the winter <cit.>. A portion of the seasonal cap covers an area referred to as the cryptic terrain, areas where the albedo is low but has the temperatures of CO_2 ice (∼150 K), indicatingthe presence of semi-translucent slab ice <cit.>. Every Mars year, the springsublimationofthe seasonal polar cap results in the formation of CO_2 jets and dark seasonal fans. In the generally accepted CO_2 jet model, sunlight penetrates through the slab ofCO_2 ice to the base regolith layer,heating the ground<cit.>. This results in sublimation at the base of the ice sheet, forming a trapped layer of gas between the ice and the regolith. The trapped CO_2 gas is thought to exploit any weaknesses in the ice above, breaking through to the top of the ice sheet as a CO_2 jet. Dust and dirt from below the ice sheet are carried by the jet and expelled into the atmosphere.It is thought that the local surface winds carry the particles as they settle onto the top of the ice sheet producing the dark fan-like streaks and blotches observed from orbit during the spring and summer.When the seasonal cap disappears, the majority of the seasonal fans and blotches fade and blend into the background regolith, further supporting the idea that the fan material is the same as the regolith below the ice sheet <cit.>. Recent laboratory experiments by <cit.> were able to trigger dust eruptions from a layer of dust inside a CO_2 ice slab under Martian conditions, lending further credence to the proposed CO_2 jet and fan production model. Small pits in the surface with radiating channels a few meters deep,colloquially known as `spiders,' have also been identified in spacecraft imagery in many of the same areas as where the seasonal fans are present <cit.>. Spiders range in diameter from tens of meters to 1 km<cit.>. Many seasonal fans appear to originate from thespider `legs', but not all observed fans do <cit.>. With the arrival of MRO and the HiRISE <cit.> camera, with a pixel scale of ∼30 cm/pixel at 300 km altitude, new morphologies of spider-like channels have been found <cit.>. This includes `lace terrain', where the dendritic-like channels of spiders are connected with no visible central pit.Spiders and these other spider-like dendritic channels are now collectively referred to as araneiforms <cit.>, and they arethought to form via the CO_2 jet process <cit.>. A sample of araneiform features from high resolution imaging is shown in Figures <ref> and<ref>. Through a survey of over 5,000Mars Orbiter Camera (MOC) Narrow Angle (NA) <cit.> images, <cit.> linked the presence of CO_2 slab ice with araneiform formation. They found that araneiforms are located in regionswhere the seasonal CO_2 ice cap becomes cryptic for at least some part of the Southern spring and summer, lending further support to the idea that araneiforms are gradually carved into the ground by the trappedCO_2 gas during the formation of CO_2jets. <cit.> also found that spiders are confined to the top of the SPLD. The erosional mechanism forming araneiforms is a slow process; over many spring/summer seasonsthe trapped gas underneath the sublimating ice sheet carves these features into the top of the SPLD. Estimates from modeling by <cit.> and recent HiRISE observations of araneiform channel formation by <cit.> place araneiform ages at 10^3-10^4 years. <cit.> argue that with their areal coverage they would have seen spiders in the MOC NA images they visually inspected outside of the SPLD. Why araneiforms appear to only be constrained to the top of the SPLD is an open question. <cit.> postulate that araneiforms may be restricted to the SPLD because the SPLD is composed of more loosely consolidated material than other geologic units on the South Polar region<cit.>; perhaps making it easier to erode by the CO_2 gas than in other areas. In the past decade with the arrival of the Context Camera <cit.> aboard MRO, many areas of theSouth Polar region have been imaged multiple times each Mars Year with 6-8 m per pixel scale, better resolution than some of the MOC NA observations searched by <cit.> and also in areas that were not covered in the original <cit.> search. More widelydistributedcoverage will allow us to expand upon the previous maps of araneiform locations. With newly discoveredaraneiform locales, we can further explore what conditions (weather, types of terrain, erodibilityoftheground,latitude,and other surface and climate properties) are key for araneiform development through comparison to previous regions monitored by HiRISE for 5 Mars Years such as the informally named `Manhattan' and`Inca City' <cit.>.We can also compare the distribution of araneiforms to other features on the South Polar region produced by the sublimation ofCO_2 ice,such as the Swiss cheese terrain.In addition, finding new areas with past CO_2jets and seasonal fans activity is an important resource for future mission and target planning. We created Planet Four: Terrains[<http://terrains.planetfour.org> or <https://www.zooniverse.org/projects/mschwamb/planet-four-terrains>](P4T), an online citizen science project which enlists the general public to map the locations of1) araneiforms (includingfeatures with a central pit and radiating channels known as `spiders'); 2) erosional depressions, troughs, mesas, ridges, and quasi-circular pits characteristic of the South Polar Residual Cap (SPRC) which we collectively refer to as `Swiss cheese terrain', and 3) craters inpublicly available CTX observations. The human brain is ideally suited for this task and with very little training is easily capable of identifying these features in orbital images of Mars. Previous studies have visually identified features like araneiforms, Swiss cheese terrain, or recurring slope lineaeusing a single person or groups of researchers reviewing observations taken from orbit <cit.>.With the Internet, tens of thousands of people across the globe can be enlisted in such tasks to create a larger sample andin particular review images typically in more detail than in the time asingle researcher or group of researchers can. This citizen science or crowd-sourcing approach, where independent assessments from multiple non-expert classifiers are combined, has been applied to nearly all areas in astronomy and planetary science <cit.> (see references therein) including galaxy morphology <cit.>, exoplanet searches <cit.>, circumstellar diskidentification <cit.>, and crater counting <cit.>. In this Paper we present the first results from P4T, examining the distribution of araneiforms with spider morphology andcomparing to the Swiss cheese terrain on the Martian South Pole.In Section 2 we provide an overview of the image dataset used in this work. We describe the P4T project and web classification interface in Section 3. In Section 4 we detail the process of combining the multiple volunteer assessments to identify surface features in the Mars image data reviewed. In Section 5 we present our map of spider locations within 15^∘ of the Martian South Pole,andin Section 6 we comparethese locations to the distribution of secure Swiss cheese terrain identifications. We report the discovery of araneiforms outside of the SPLD and in Section 7 present higher resolution confirmation imaging from HiRISE. In Section 8 we discuss the implications of this result for the CO_2jet model.All place names referred to in this Paper are informal and not approved by the International Astronomical Union. All reported latitudes are areographic, and all longitudes are reported in reference to East longitude.Full machine-readable versions of the catalogs and tables presented in this Paper are also available from <https://www.zooniverse.org/projects/mschwamb/planet-four-terrains/about/results>[Note to the editor: When this manuscript is accepted we will make these online links accessible. All material has been submitted in the supplementary information as well].§ DATASETFor our analysis, we used publicly available observations from CTX <cit.> aboard MRO obtained from NASA's Planetary Data System (PDS)[<http://pds-imaging.jpl.nasa.gov/>]. CTX has widespread coverage of the Martian south polar region at a variety of solar longitudes (L_ S). The camera provides the best balance between resolutionand areal coverage with a single observation typically spanning a ∼30×60 km swath at 6 m/pixel spatial scale. <cit.> found araneiforms were constrained to the SPLD; we thus restricted our study to CTX observations with latitudes southward of -75^∘ latitude in order to encompass the majority of the SPLD defined by <cit.>[<http://pubs.usgs.gov/sim/3292/>]. We selected 90 CTX imagesfor review on P4T for this work. The area covered by our search images is shown in Figure <ref>. We have surveyed CTX observations covering 303,192 km^2 within -70^∘ latitude and 11% of the South Polar region within -75^∘ N.The areal coverage of the surveyed CTX images as a function of latitude is shown in Figure <ref>. The observing circumstances and planetographic coordinates of the selected CTX observations are summarized in Table <ref>. §.§ CTX Image Selection We provide a brief overview of the process used to select the 90 CTX images used in this analysis. Since araneiform formation takes thousands of Mars years <cit.>, we do not restrict ourselves to a single Mars year. We chose from publicly available CTX observations taken between October 2007 and October 2013, Mars years (MY) 28-32 according to the convention defined by <cit.> and <cit.>. The CO_2 jet process on the seasonal ice cap produces fans which appear as dark streaks and blotches in CTX images throughout the Southern spring and summer. We attempted to select CTX observations of locations where theCO_2 seasonal cap had already sublimated in order to reduce the number of obscuring fans. The extent of the seasonal CO_2 cap in latitude and longitude varies as as a function of L_ S.We used previous thermal measurements of the South polar region <cit.> and a brief visual inspection of the CTX images to determine latitude/L_ S ranges that were ice free. With the remaining CTX images that pass the selection cuts, we attempt to achieve as widespread coverage as possible distributed across the South Polar region. We divide the South polar region southward of -75 ^∘ latitudeinto30^∘ longitude and 5^∘latitude bins. The 90 CTX images used in this study were randomly selected from each bin with slightly more images picked with field centers between -85^∘ and -75^∘ latitude. The final area covered by our search images is shown in Figure <ref>. Figure <ref> shows the latitude distribution of the selected CTX images for this study as a function of L_ S.A visual inspection of the final 90 CTX images selected finds that majority of the images are free from clouds or obscuring seasonal fans. We do not have criteria for assessing the fraction of cloud cover, atmospheric opacity, or presence of seasonal fans ineach of the selected CTX images used in our search. Thus the lack of a positive identification of araneiforms, Swiss Cheese Terrain, or cratersby P4T does not necessarily mean the feature is not present in a given CTX image. Our analysis instead identifies locations where araneiforms, Swiss Cheese Terrain, and craters are confidently identified by P4T but does not provide a complete sample.§.§ CTX Image Processing and P4T Subject Creation The selected full frame CTX images searched by P4Tare subdivided into smaller subimages that are subsequently presented to volunteers on the P4T website. The raw CTX image products or Experiment Data Records (EDRs) were processed with Python using the United States Geological Survey's (USGS) Integrated Software for Imagers and Spectrometers (ISIS) <cit.>[<http://isis.astrogeology.usgs.gov/>] and the ISIS-3 python wrapper Pysis[<https://github.com/wtolson/Pysis>]. After radiometric calibration and noise removal, the CTX frames were divided into smaller non-overlapping 800×600 pixel (∼4.8×3.6 km) PNG (Portable Network Graphics) subimages to be presented on the P4T website. We refer to these subimages as `subjects.' The PNG exporterfrom the ISIS toolset (isis2std) was configured to output at 8-bit output resolution, meaning that the dynamic ratios of the more dynamic CTX data were visually reduced compared to scientific display systems. By default, isis2std cuts off the lowest and highest 0.5 percent of image values to exclude cold and hot pixels. After visual inspection by the science team of varying dynamic ratios, we settled on a full zero to hundred percent stretch for each subject generated to get the increased dynamic ratio which helps in identifying patterns in very dark or bright areas. The fact that we did not identify any problems with leaving the image stretch at 100 percent of the original values in the CTX subimage ISIS cube is a testament to the quality of the CTX camera and its calibration. We also experimented with over-stretching the generated subjects but did not find any improvement in image quality.A characteristic sample of P4T subjects is presented in Figure <ref>. In total 20,122 subjects were generated, and Table <ref> provides a list of the 90 CTX observations and the number of subjects associated with each full frame CTX image. A CTX observation contained a mean of224 P4T subjects with a minimum of 24 and maximum of 522 subjects. Due to the variable length and width of CTX observations, there are typically small regions on the right and bottom edges of the CTX full frame image that did not make it into a subject image, and thus not searched by P4T.Supplemental Table 1 summarizes the P4T subjects used in this work, including the center latitude, center longitude, and location within the full frame CTX observation. llllll [short]CTX Observations Examined in This StudyCTX imageLatitude Longitude L_s Observation # of(degrees) (degrees) (degrees) Time P4T Subjects3c– Continued CTX imageLatitude Longitude L_s Observation # of(degrees) (degrees) (degrees) Time P4T Subjects3lContinued on Next Page…[]The center coordinates for all CTX images searched used in the analysis presented in this paper. The table includes the latitude and longitude, UTC time and date of observation, and number of P4T subjects generated from the observation.We include the full CTX filename here; the first 15 characters are the unique CTX observation identifier. D13_032173_1031_XN_76S227W -77.02 132.51 331.71 2013-06-07T10:18:40.139 66D13_032182_1030_XN_77S112W -77.05 247.78 332.09 2013-06-08T03:08:08.698 183D13_032278_0991_XN_80S204W -81.01 155.56 336.15 2013-06-15T14:40:39.298 108D13_032298_0969_XN_83S028W -83.18 331.6 336.99 2013-06-17T04:04:24.753 66D13_032311_0999_XN_80S031W -80.19 329.04 337.54 2013-06-18T04:23:39.968 304D13_032352_0985_XN_81S063W -81.62 296.36 339.25 2013-06-21T09:04:37.135 162D14_032510_0963_XN_83S054W -83.71 305.21 345.76 2013-07-03T16:34:14.369 66D14_032511_0959_XI_84S078W -84.31 282.04 345.8 2013-07-03T18:26:12.728 48D14_032517_1000_XN_80S249W -80.04 110.55 346.04 2013-07-04T05:40:49.097 66D14_032518_0995_XN_80S281W -80.51 78.42 346.08 2013-07-04T07:32:06.155 392D14_032523_0954_XN_84S045W -84.65 314.82 346.29 2013-07-04T16:52:42.650 66D14_032530_0975_XN_82S259W -82.5 100.75 346.57 2013-07-05T05:59:04.972 78D14_032574_0969_XN_83S005W -83.17 354.99 348.35 2013-07-08T16:16:02.560 66D14_032575_0969_XN_83S028W -83.17 331.6 348.39 2013-07-08T18:08:13.419 66D14_032593_1037_XN_76S174W -76.39 185.51 349.12 2013-07-10T03:50:10.687 66D14_032600_0965_XN_83S357W -83.56 2.88 349.4 2013-07-10T16:53:27.794 66D14_032640_1003_XN_79S014W -79.69 345.3 351.01 2013-07-13T19:42:56.864 87D14_032656_0959_XI_84S078W -84.21 281.5 351.65 2013-07-15T01:36:53.707 66D14_032666_0916_XN_88S350W -88.51 9.05 352.05 2013-07-15T20:17:50.497 66D14_032675_0924_XN_87S253W -87.68 106.58 352.41 2013-07-16T13:08:05.533 66D14_032682_0925_XN_87S066W -87.53 293.1 352.69 2013-07-17T02:13:30.246 90D14_032733_1028_XN_77S036W -77.24 323.18 354.71 2013-07-21T01:38:54.660 204D14_032790_0933_XN_86S110W -86.74 248.82 356.96 2013-07-25T12:11:45.417 24G13_023338_1043_XI_75S229W -75.81 131.06 330.91 2011-07-20T00:01:53.559 336G13_023354_1032_XN_76S301W -76.82 58.54 331.6 2011-07-21T05:56:38.165 396G13_023432_1026_XI_77S262W -77.44 97.75 334.91 2011-07-27T07:49:28.555 294G13_023452_1049_XN_75S098W -75.05 261.45 335.75 2011-07-28T21:14:17.312 435G13_023456_0990_XN_81S204W -80.98 155.23 335.92 2011-07-29T04:42:00.456 108G13_023469_1046_XI_75S211W -75.5 148.2 336.47 2011-07-30T05:02:26.465 210G14_023496_1047_XI_75S213W -75.36 147.08 337.6 2011-08-01T07:32:15.443 138G14_023506_1036_XN_76S132W -76.42 227.99 338.02 2011-08-02T02:13:28.855 522G14_023507_1029_XN_77S158W -77.19 201.15 338.06 2011-08-02T04:06:06.023 120G14_023524_0999_XN_80S262W -80.16 97.56 338.77 2011-08-03T11:52:54.656 66G14_023538_1006_XN_79S285W -79.48 74.31 339.35 2011-08-04T14:03:39.719 336G14_023567_1039_XN_76S358W -76.23 2.04 340.55 2011-08-06T20:18:46.307 438G14_023577_0999_XN_80S262W -80.14 97.62 340.97 2011-08-07T15:00:08.617 66G14_023590_0975_XN_82S259W -82.54 100.99 341.51 2011-08-08T15:18:11.704 66G14_023591_0996_XN_80S284W -80.48 75.43 341.55 2011-08-08T17:10:49.067 132G14_023616_1004_XN_79S248W -79.61 111.85 342.58 2011-08-10T15:56:35.480 66G14_023634_1036_XN_76S027W -76.44 333.0 343.32 2011-08-12T01:36:48.168 522G14_023676_1023_XN_77S091W -77.75 268.09 345.04 2011-08-15T08:09:24.912 294G14_023687_1031_XN_76S036W -76.95 323.86 345.49 2011-08-16T04:44:01.680 232G14_023691_1031_XN_76S143W -76.98 216.96 345.65 2011-08-16T12:12:04.519 381G14_023718_1039_XN_76S160W -76.11 199.16 346.75 2011-08-18T14:42:24.470 348G14_023728_0958_XI_84S057W -84.31 302.44 347.15 2011-08-19T09:22:32.460 102G14_023735_1002_XN_79S262W -79.88 97.18 347.44 2011-08-19T22:29:22.283 138G14_023794_0952_XN_84S061W -84.89 298.47 349.82 2011-08-24T12:48:10.629 66G14_023807_0886_XN_88S313W -88.68 47.11 350.34 2011-08-25T13:05:07.603 66G14_023815_0970_XN_83S282W -83.05 78.09 350.66 2011-08-26T04:05:02.101 48G14_023833_0926_XN_87S017W -87.43 342.35 351.38 2011-08-27T13:43:17.984 66G14_023835_0906_XN_89S050W -89.45 310.24 351.46 2011-08-27T17:27:10.913 66G14_023851_0926_XN_87S180W -87.41 179.52 352.1 2011-08-28T23:23:15.422 66G15_023911_1021_XN_77S023W -77.9 336.45 354.49 2011-09-02T15:38:24.799 66G15_023927_0932_XI_86S057W -86.82 302.59 355.12 2011-09-03T21:30:31.425 66G15_023963_1021_XN_77S006W -77.91 353.12 356.54 2011-09-06T16:52:57.448 60P12_005705_1016_XI_78S133W -78.52 226.28 330.87 2007-10-14T23:21:44.255 294P12_005747_1035_XI_76S195W -76.65 164.52 332.66 2007-10-18T05:55:17.203 138P12_005790_0978_XI_82S284W -82.26 75.67 334.48 2007-10-21T14:18:20.245 66P12_005813_1030_XI_77S195W -77.07 164.97 335.46 2007-10-23T09:20:46.192 66P12_005839_0994_XI_80S170W -80.66 189.37 336.55 2007-10-25T09:56:39.216 120P13_005940_1035_XN_76S064W -76.51 295.91 340.76 2007-11-02T06:50:28.896 522P13_005941_0947_XI_85S065W -85.39 295.0 340.8 2007-11-02T08:39:53.181 408P13_005953_1020_XN_78S057W -78.01 302.45 341.3 2007-11-03T07:08:57.500 522P13_005958_1030_XI_77S195W -77.02 164.87 341.5 2007-11-03T16:30:56.168 150P13_006005_0947_XN_85S015W -85.32 344.59 343.44 2007-11-07T08:22:07.760 408P13_006112_0852_XN_85S304W -85.2 55.76 347.8 2007-11-15T16:27:46.904 66P13_006119_1035_XN_76S271W -76.52 88.75 348.08 2007-11-16T05:38:06.661 366P13_006123_0953_XN_84S001W -84.77 358.59 348.25 2007-11-16T13:04:04.676 522P13_006146_1045_XN_75S289W -75.53 70.6 349.17 2007-11-18T08:08:17.767 306P13_006148_1028_XN_77S342W -77.25 17.18 349.26 2007-11-18T11:51:51.830 522P13_006151_0974_XN_82S055W -82.63 305.13 349.38 2007-11-18T17:27:16.677 252P13_006161_1030_XN_77S338W -77.09 22.03 349.78 2007-11-19T12:11:05.625 324P13_006167_0931_XN_86S092W -86.96 267.2 350.02 2007-11-19T23:21:18.515 102P13_006173_0934_XN_86S263W -86.68 96.7 350.26 2007-11-20T10:34:49.912 66P13_006174_0958_XN_84S316W -84.29 43.28 350.3 2007-11-20T12:28:01.397 66P13_006176_0935_XN_86S350W -86.51 10.06 350.38 2007-11-20T16:11:29.064 204P13_006197_0930_XN_87S187W -87.04 173.3 351.22 2007-11-22T07:27:46.120 150P13_006199_1040_XN_76S296W -76.02 63.9 351.3 2007-11-22T11:15:22.194 522P13_006204_0986_XN_81S065W -81.45 295.05 351.5 2007-11-22T20:35:13.014 252P13_006206_1016_XN_78S124W -78.57 235.47 351.58 2007-11-23T00:20:04.979 504P13_006207_0956_XN_84S136W -84.49 223.45 351.62 2007-11-23T02:10:20.857 522P13_006229_0951_XN_84S014W -84.97 345.69 352.5 2007-11-24T19:19:03.593 522P13_006234_1008_XN_79S168W -79.28 191.3 352.7 2007-11-25T04:42:19.717 306P13_006239_1040_XN_76S308W -76.09 51.62 352.9 2007-11-25T14:04:14.736 408P13_006240_1030_XN_77S335W -77.04 25.01 352.94 2007-11-25T15:56:00.916 522P13_006257_1034_XN_76S079W -77.88 282.15 353.61 2007-11-26T23:43:54.535 264P13_006271_1010_XN_79S098W -79.07 261.59 354.17 2007-11-28T01:54:24.701 306P13_006282_1046_XN_75S043W -75.47 316.98 354.61 2007-11-28T22:29:37.821 522P13_006283_1003_XN_79S065W -79.75 294.76 354.65 2007-11-29T00:20:28.301 522P13_006290_1017_XN_78S258W -78.38 101.84 354.92 2007-11-29T13:26:24.488 522§ PLANET FOUR: TERRAINS (P4T) The aim of P4T is to identify features of interest in CTX observations of the South Polar region. For this endeavor we focused on three types of surface features and their distribution on the Martian South Polar region: 1) araneiforms 2) erosional depressions, troughs, mesas, ridges, and quasi-circular pits characteristic of the SPRC which we collectively refer to as `Swiss cheese terrain', and 3) craters. We aim to study the distribution of the araneiforms on the South Polar region and explore their locations compared to the locations of other CO_2 ice sublimation features The crater identifications can aide with the surface age dating of the SPLD, similarly to what has been done for the North Polar region <cit.>.Examples of each of thethree types of surfaces features (taken from the P4T site guide[<http://terrains-guide.planetfour.org/>]) are shown at the resolution of CTX in Figures <ref>, <ref>, and <ref>. Several different types of araneiform structures have been identified on the South Polar region <cit.>. Similar to <cit.>'s categories identified in HiRISE imaging, we divide araneiform terrain broadly into three araneiform morphologies distinguishable at CTX resolution: `baby spiders', `spiders', and `lace terrain'. Spiders are defined as radially converging channels that are often branching and often hosting a visible central pit (see Figure <ref>). We note for the reader, that any subsequent reference to `spiders'in the text uses this definition. Baby spiders are spiders with `short legs', where a central pit dominates with short or no radial channels (see Figure <ref>). In the help documentation, we recommend to volunteers that if the channels are shorter than the extent of the central pit, then it is a baby spider. As identification with P4T is through visual inspection, we acknowledge that there is not always a clear dividing line between spiders and baby spiders. The criteria separating the two categories is more qualitative than quantitative. With the resolution of CTX, it is difficult to distinguish patterned ground, formed by the repeating freezing and thawing of soil, from lace araneiforms, formed by the CO_2 jet process. In HiRISE images one can observe the more sinuous nature of the lace araneiforms differentiating these features from polygonal channels, but this is typically not visible in CTX observations. For P4T, we combine lace araneiforms and pattern ground together as one category, referring to them collectively as a`channel network' (see Figure <ref>).When needed for the channel network regions identified by P4T, we plan to use higher resolution imaging from HiRISE, to distinguish polygonal channels from interconnected araneiforms. The SPRC's top surface layers have been categorized into different groups of characteristic smooth-walled features including: troughs, mesas, and quasi-circular pits visible in orbital imagery <cit.>. A recent inventory of SPRC surface morphology based on HiRISE and CTX imagery is provided in <cit.>. The main priority of P4T is to identify new aranieform locations, thus we ask the P4T volunteers to sort araneiforms in more detail by distinguishing araneiforms of different morphology from each other. Given the spatial resolution of CTX and the size of each P4T subject image, we chose to not task P4 volunteers with distinguishingbetween the different categories established by<cit.>. For the sublimation features of the SPRC, we combined the morphological categories together into one for P4T which we simply refer to as `Swiss cheese terrain'. In the P4T help content (see Figure <ref>), we describe the Swiss cheese terrain as flat-floored, circular-like depressions and visual examples for P4T show the majority of the different sublimation morphologies visible on the SPRC with CTX. We note for the reader, that any subsequent reference to `Swiss cheese terrain' refers to the combined SPRC sublimation features previously identified. < g r a p h i c s > Examples of CTX surface morphologies searched for on P4T continued with the help text provided on the P4T project site guide. < g r a p h i c s > Examples of CTX surface morphologies searched for on P4T continued with the help text provided on the P4T project site guide.§.§ Web Interface The Planet Four: Terrains website[ urlhttp://terrains.planetfour.org or urlhttps://www.zooniverse.org/projects/mschwamb/planet-four-terrains] and online classification interface is built upon the Zooniverse[ <http://www.zooniverse.org>] <cit.> Project Builder platform[ <http://www.zooniverse.org/lab>. The code base for the Zooniverse Project Builder Platform is available under an open-source license at <https://github.com/zooniverse/Panoptes> and <https://github.com/zooniverse/Panoptes-Front-End>]. The Zooniverse Project Builder platform enables the rapid development of online citizen science projects by providing a set of web-tools for scientists to create and maintain their own citizen science projects. The platform and its Application Program Interface (API) is built upon Amazon Web Services which allows the P4T website to quickly andefficiently scale to handle the load from varying numbers of visitors on the site at the same time; it is capable of supporting tens of thousands of simultaneous users. When a volunteer arrives at the P4T website, the web interface (see Figure <ref>) displays a selected 800×600 pixel CTX subject. The Zooniverse API pseudo randomly selects a new set of subjects for each classifier upon request, in order to distribute the volunteer effort across the known dataset. In addition, the API algorithm also selects subjects the volunteer has not previously reviewed and that have not been viewed by enough volunteers to mark them as complete. Each subject is typically assessed independently by 20 classifiers before it is retired from review on the P4T website. Volunteers are tasked with assessing the image and determining what surface features of interest are present in the subject selecting from a choice of: `spiders', `baby spiders', `channel network', `Swiss cheese terrain', `craters' and `none of the above'. The volunteer is able to select more than one response that best describes the subject image. In this Paper, a `classification' is defined as the total amount of information collected about one subject by a single volunteer answering the question presented in the P4T classification interface. Help content and example images of each feature/answer choice can be accessed by clicking on the `Need some help with this task?' button. Additional examples and help content are provided on a linked site guide[<http://terrains-guide.planetfour.org/>].To minimize external information influencing or biasing a volunteer's response, no identifying information about the original parent CTX image including the filename or observing circumstances (such as location coordinates, time of day, or L_ s) are provided to the volunteer before they submit their classification. Thus the classifier cannot assess whether the image is from a location that previously has been identified as having araneiforms (such as the SPLD) or was previously imaged by HiRISE in previous south polar monitoring observations. To keep the multiple volunteer assessments independent for each subject, the classifier is kept blind to previous people's responses for the presented subject, and the subject's internal Zooniverse identifier is hidden from the classification interface. Once the volunteer selects the categories that best describe the presented subject and hits the `Done' button, the classification is submitted through the Zooniverse API and stored in the Zooniverse PostgreSQL database. The subject identifier, volunteer's IP (Internet Protocol) address, Zooniverse username if available, timestamp, web browser and operating system information, and user response are recorded. At this point, the volunteer cannot go back and revise their classification.P4T volunteers can classify in two modes: registered with a Zooniverse account or unregistered. The P4T classification interface is presented the same for both registered and non-registered classifiers. The only difference is that non-registered classifiers are reminded from time-to-time to log-in/register for a Zooniverse account. Registered classifications are easily linked by their associated Zooniverse account. For non-registered classifications, a unique identifier is generated and used to link the classifications completed by a given IP address. We note because of the IP tracking, a non-registered classifications from a single IP address may not necessarily equate to a single individual. Additionally, if a volunteer initially classifies non-registered and then logs-in to a Zooniverse account, the previous classifications are not linkedwith their registered account and remain attributed to an unregistered classifier.§.§ Talk Discussion Tool After submitting a classification on the P4T website, the classification interface presents the volunteer with two options: `Talk' or `Next'. `Next' will load a new subject image in the classification interface. Selecting the `Talk' button instead loads the P4T Talk discussion tool[<https://www.zooniverse.org/projects/mschwamb/planet-four-terrains/talk>]. Talk enables volunteers to further explore the P4T dataset beyond the main tasks and aims of the classification interface. The discussion tool hosts message boards that support interactions with the science team and others in the P4T volunteer community. Each subject has a dedicated page on Talk where a registered volunteer can initiate a new discussion or add commentary to an on-going discussion about the subject. Volunteers can also associate the subject with searchable Twitter-like hashtags and link multiple subjects together into groups. Reading previously posted commentary on Talk might bias a volunteer's assessment in the classification interface. To maintain the independence of the classifications, the Zooniverse identifier for the subject is not presented in the classification interface and the direct link for the Talk subject page is only revealed after a volunteer submits their classification to the Zooniverse database. For this work, we focus primarily on the results from the main classification interface. §.§ Site History and Statistics The 20,122 subjects derived from the 90 CTX full frame images used in this study were classified by 6,309 registered Zooniverse users and 8,513 non-logged-in sessions (tracked by IP address). The classifications were collected from June 24, 2015 to August 10, 2016. We plot the distribution of registered volunteers and non-logged-in sessions as a function of number of classifications in Figure <ref>. Registered volunteers classified a mean of 61 subjects with a median of 14. Non-logged-in sessions classified an average of 19 and median of 5 subjects. 20% of registered volunteers classified more than 50 subjects, while only 3% of non-logged-in sessions classified more than 50 subjects. We plot the distribution of classifications per subject in Figure <ref>. The majority of the subjects received 20 classifications or more. Due to a bug in the backend of the Zooniverse platform, some volunteers were shown the same subject to classify twice or more. To ensure the assessments for each subject remain independent, we filter the classification database and remove any duplicate classifications keeping the first response from the registered username or IP address in the case of non-logged classifications. For all the values reported in this Paper, we use the filtered classifications. In some cases, this will leave a subject with less than 20 independent assessments, but the impact is negligible with only 7% of the subjects in this study having less than 20 unique classifications and 0.07% with less than 17 unique classifications. We also note that due to a glitch in the backend of the Zooniverse platform a portion of P4T subjects were not retired after 20 classifications. 8% of the subjects in this study received more than 25 independent assessments, with only 3% of the subjects receiving more than 50 classifications.§ DATA ANALYSIS For this work, we focus solely on the P4T identification of spiders and Swiss cheese terrain. We refer the reader to Section <ref> for the specific definitions of spiders and Swiss cheese terrain we use in this analysis. We combine the multiple volunteer classifications of a given subject together, examining the number of volunteers who selected the `spiders' or `Swiss cheese terrain' buttons for each subject in order to identify spidersand Swiss cheese terrain presentin the P4T data. Some volunteers may be better at spotting these features than others. Rather than treat all volunteer assessments equality, we apply a user weighting scheme that enables us to pay more attention to those volunteers who are better at identifying spiders or Swiss cheese terrain and also reduce the influence of potentially unreliable classifiers, such as those who did not engage in or understand the task.We then apply these weights when combining the volunteer assessments for spiders or Swiss cheese to determine how likely a P4T image is to have these features of interest present. §.§ User Weighting SchemeWe use a modified version of the iterative user weighting schemes developed by <cit.> and <cit.> for the visual morphological classifications of galaxies in the Galaxy Zoo project and by <cit.> for the visual identification of planet transits in NASA Kepler data in the Planet Hunters project. The weighting scheme evaluates the ability of each classifier and assigns a weight based on their tendency to agree with the majority opinion, distinguishing those volunteers who are better at spotting Swiss cheese or spiders in order to pay attention to their responses more than others when identifying which subjects have these features present. We define a `user' for our case as either a volunteer with a registered Zooniverse account or the collective behavior of non-logged-in sessions with a unique-IP address. A unique non-logged-in IP address may not necessarily by a single individual (see <ref>), but for the weighting scheme we link the classifications together and determine a single weight. If a user excels at identifying spiders, it does notnecessarily mean they would be as good as identifying Swiss cheese terrain in the P4T data. We therefore treat the identification of spiders and Swiss cheese terrain independently as two separate classes of responses, and determine separate user weights for each class. For the analysis presented here, the volunteer classifications are effectively divided into two responses per class: `found' and `not found'. For spider identification this breaks down into`spiders found' if the volunteer selected the `spider' button while classifying the subject and `no spiders found' if the volunteer did not select the `spiders' button. For this analysis, if a volunteer clicked on the `baby spiders' or `channel network' buttons without also clicking on the 'spiders' button, this would count as a 'no spiders found' response for the subject image. We do the same thing with the responses for Swiss cheese terrain, with a vote of `Swiss cheese found' if the 'Swiss cheese terrain' button was selected when the subject image was reviewed or `no Swiss cheese found' if volunteer didn't mark the image as having Swiss cheese terrain. Each user j is assigned two weights, one for each class: w_j( spider) and w_j( Swiss). Initially, all users start out with each of those weights equal to 1. Then for each P4T subjectiscores for spiders, s_i( spider),and Swiss cheese pits,s_i( Swiss), are calculated. We define these scores per subject as follows:s_i( spider)= 1/S_i ∑_k w_k( spiders) k =users who selected `spider' for subject is_i( Swiss)= 1/C_i ∑_m w_m( Swiss)m =users who selected `Swiss cheese' for subject i where S_iandC_i are the sum of the respective user weights for all the users who classified subject i:S_i= ∑_k=j w_k( spiders) C_i= ∑_m=j w_m( Swiss) j= all users who classified subject iSubject scores vary between the values of 0 and 1, inclusive. A subject score of 1 is assigned if all volunteers who classified the subject agree and identified the same features of interest in the subject image. Once the subject scores s_i( spider) and s_i( Swiss) are calculated, we next assign new user weights for each user jby the prescription below:w_j(spiders) = {[ A/N_j∑_i=p s_i ( spider) + A/N_j∑_i=q [1- s_i ( spider)]if N_j >1;1if N_j =1 ].p= subjects volunteer classified as `spiders found' q= subjects volunteer classified as `spiders not found'w_j(Swiss) = {[ B/N_j∑_i=t s_i ( Swiss) + B/N_j∑_i=u [1- s_i ( Swiss)]if N_j >1;1if N_j =1 ]. t= subjects volunteer classified as `Swiss cheese found' u= subjects volunteer classified as `Swiss cheese not found'where N_j is the number of subjects classified by user j. The scaling factors A and B are chosen to be such that the median user weight for volunteers who classify more than one subject will be 1. We only adjust the weights of those volunteers who have classified more than one subject, which constitutes 85% of P4Tusers.With a single classification there is not much information to use to evaluate avolunteer's ability to discern spiders and Swiss cheese terrain. Thus for those users who have classified a single subject,we choose to keep the user weights static at a value of 1, where the median of the adjusted weights will lie.A volunteer who has classified more than one subject is upweighted strongly when they agree with the majority weighted vote and down weighted more harshly when their response is at odds with the majority of the volunteers who reviewed the subject image.After the user weights are first adjusted, the subject scores in each class, s_i( spider) and s_i( Swiss), are recalculated using the updated user weights with Equations 1 and 2. Then new user weights are assigned with Equations 6 and 7.The process is iterated until convergence is achieved, when the median absolute difference between the old and updated user weights is less than or equal to 1x10^-4, in this case after four iterations for both spiders and Swiss cheese. We plot the distribution of user weights for spiders in Figure <ref> and for Swiss cheese terrain in Figure <ref>. With this scheme, a user weight can never be zero. For w_j( spider),91% of users have weights greater than0.8 and 43%of user weights are greater than 1.For w_j( Swiss), 89% of users have weights greater than0.8 and 43%of user weights are greater than 1.§.§ Combining User Classifications and Gold Standard Data We then use the final subject scores, s_i( spider) and s_i( Swiss), to identify the locations of spiders and Swiss cheese terrain in the surveyed CTX images. Figure <ref> plots the cumulative distribution for the final calculated scores. Table <ref> reports the binned distribution in subject scores, andTable <ref> provides the final score values for each subject used in this study. Figures <ref>-<ref> contain a representative sample of subjects for s_i( spider) randomly selected in bins of 0.1. Figures <ref>-<ref> show the same for s_i( Swiss). It is readily apparent the closer the subject score is to 1, the more consensus amongst the weighted user vote and thus a higher likelihood of the features of interest (spiders or Swiss cheese terrain) being present ina given subject image. Of the 20,122 subjects classified by P4T volunteers, 3% (591) have s_i( spider) > 0.5 and 9% (1767) have s_i( Swiss) > 0.5. We set a detection threshold for s_i( spider) and s_i( Swiss) above which we define a clean sample of subjects identified as having spiders or Swiss cheese terrain present. We set this value such that the number of false positives is sufficiently small while retaining the largest number of true identifications. We determine this detection limit based on the expert assessment by the P4T science team for asmall fraction of the subject data used in this study. A similar validation process has been applied to crowd-sourced crater counting <cit.>. A subset comprised of 1,009 subjects (505 for Swiss cheese terrain identification and 504 for spider identification),corresponding in total to 5% of the total subjects reviewed by volunteers, were used to create the gold standard dataset. The subjects were divided into ten bins based on s_i( spider) and s_i( Swiss). The gold standard subjects were randomly selected from these binsin order to find the subject score where false positives begin to overwhelm positive identifications of spiders and Swiss cheese terrain. Table <ref> details the number of subjects per s_i( spider) and s_i( Swiss) bins randomly selected for the gold standard review. The P4T subject ids used in the expertgold standard review are provided in Supplemental Tables 2 and 3. Two members of the science team (MES and GP) each independently reviewed the gold standard subjects using the same web interface on the P4T website as used by the volunteers.The gold standard assessments are available in Supplemental Table 4. Figure <ref> plots for the gold standard dataset, the fraction of positive identification of spider and Swiss cheese detections as a function of score for each expert reviewer. The error bars represent the Poissonian 68% confidence limit on the positive identifications in each bin, as prescribed by <cit.>. There is strong agreement in expert assessments for Swiss cheese terrain, but less agreement for spider assessments. This may be due to differing levels of difficulty between the two identification tasks.§.§ Clean Spider and Swiss Cheese Terrain Sample We set our detection threshold for s_i( spider) and s_i( Swiss) at the score bin where 20% of the gold standard data subjects reviewed are false positives asdeemed by both expert reviewers. Therefore, we select our `clean' sample of spider detections as subjects with s_i( spider) ≥ 0.6, and our `clean' Swiss cheese sample is comprised of subjects with s_i( Swiss) ≥ 0.7. Applying these cuts, 390 (1.9%) of the P4T subjects have s_i( spider) ≥ 0.6, and 1,537 (7.6%) haves_i( Swiss) ≥ 0.7.We use the clean samples in the rest of our analysis.The clean spider and clean swiss cheese identifications do not represent a complete sample in our search region. Instead these are high confidence identifications where false positives in the sample are minimized. Thus a location within the searched CTX images that is not part of our clean samples, may still have spider-shaped araneiforms or Swiss cheese terrain present that arenot easily identified by P4T which includes cases where the surface may not be as visible in the CTX image (see Section <ref>). Thus our analysis only focuses on what we can glean from a sample of confident identifications with approximately less than 20% false positives. We provide a catalog of the subjects andassociated properties that comprise the clean spider and Swiss cheese samples in Tables <ref> and <ref> respectively. [h] P4T Subject Information and s_i(spider) and s_i(Swiss) Scores Center CenterSubject ID s_i(spider) s_i(Swiss) Parent CTX ImageLatitudeLongitude CTX X^* CTXY^* (degrees) (degrees)position position 83672 0.025 0.000 G14_023507_1029_XN_77S158W -77.80 200.96400300483673 0.074 0.000 G14_023507_1029_XN_77S158W -77.74 200.90 400 900 483674 0.000 0.000 G14_023507_1029_XN_77S158W -77.68 200.84400 1500 483675 0.122 0.016 G14_023507_1029_XN_77S158W -77.62 200.78400 2100483676 0.167 0.000 G14_023507_1029_XN_77S158W -77.56 200.72 400 2700 483677 0.131 0.000 G14_023507_1029_XN_77S158W -77.50 200.66400 3300 483678 0.237 0.015 G14_023507_1029_XN_77S158W -77.45 200.60 4003900 483679 0.066 0.057 G14_023507_1029_XN_77S158W -77.39 200.54400 4500483680 0.069 0.015 G14_023507_1029_XN_77S158W -77.33 200.484005100^* CTX X and CTX Y are the pixel location of the subject's center in the full frame CTX ISIS generated cube.This table in its entirety can be found in the online Supplemental and at <https://www.zooniverse.org/projects/mschwamb/planet-four-terrains/about/results>A portion is shown here for guidance regarding its form and content.We include the full CTX filename here; the first 15 characters are the unique CTX observation identifier. [h] Clean Spider Sample Subject ID s_i(spider) Parent CTX lmage CenterLatitude Center Longitude (degrees) (degrees) 489994 1.000 P12_005747_1035_XI_76S195W -76.76 165.32 491505 1.000 D14_032675_0924_XN_87S253W -87.55 110.18 1045043 1.000 P13_006151_0974_XN_82S055W-82.13 303.34 1320810 1.000 P12_005839_0994_XI_80S170W -81.10 189.69 1510306 0.953 G13_023338_1043_XI_75S229W -75.34 131.48 491434 0.953 P13_006173_0934_XN_86S263W -86.73 100.61 491859 0.951 P13_006148_1028_XN_77S342W -79.08 19.48 1491749 0.951 P13_006197_0930_XN_87S187W -86.91 172.28 1058665 0.951 D14_032593_1037_XN_76S174W -76.20 185.99This table in its entirety can be found in the online Supplemental and at <https://www.zooniverse.org/projects/mschwamb/planet-four-terrains/about/results>A portion is shown here for guidance regarding its form and content.We include the full CTX filename here; the first 15 characters are the unique CTX observation identifier.Clean Swiss Cheese Terrain Sample Subject ID s_i(Swiss) Parent CTX Image Center Latitude Center Longitude (degrees) (degrees) 1320528 1.000 P13_006005_0947_XN_85S015W-86.32 355.10 1320615 1.000 P13_006005_0947_XN_85S015W -85.39 345.54 1320728 1.000 P13_006005_0947_XN_85S015W -86.34 359.96 1491818 1.000 G14_023835_0906_XN_89S050W-89.36 342.33 1491849 1.000 G14_023835_0906_XN_89S050W -89.14 300.68 1040008 1.000 G14_023833_0926_XN_87S017W -87.75 346.66 1040047 1.000 G14_023833_0926_XN_87S017W -87.38342.37 1040089 1.000 P13_006123_0953_XN_84S001W -86.26 9.461040272 1.000 P13_006123_0953_XN_84S001W -85.75 5.97 This table in its entirety can be found in the online Supplemental and at <https://www.zooniverse.org/projects/mschwamb/planet-four-terrains/about/results>A portion is shown here for guidance regarding its form and content. We include the full CTX filename here; the first 15 characters are the unique CTX observation identifier.§ SPIDER DISTRIBUTION Our clean spider sample is comprised of 390 subjects, and their locations are plotted in Figure <ref> overlaid on a MOLA elevation map <cit.> and the geologic map from <cit.>. Using MOC Narrow Angle observations,<cit.> previously mapped the distribution of spiders in the South Polar region. They found a correlation between areneiforms and CO_2 slab ice, which is thought to be required for the CO_2 jet process. Additionally,<cit.> found that areneiformswere restricted to the SPLD. All but one grouping of their araneiform regions found in the <cit.> study were located within the definition of the SPLD at the time. The outlier region (located atapproximately latitude=-82^∘, longitude=35^∘) has similar surface characteristics to the SPLD, and the more modern <cit.> definition does include this location within the SPLD. There is no published table of <cit.>'s positive identifications, only the plotted distribution in Figure 2 from their paper. In order to compare our clean spider sample to that observed by <cit.>, in Figure <ref> we overlay our clean spider distribution on top of <cit.>'s Figure 2a. Like <cit.>, we find the spiders are concentrated on the SPLD with 75% of the clean sample located on the <cit.>defined SPLD, but we also have 96 identifications located outside of the SPLD. We list these locations and subject details in Table <ref> and display a set of representative subject images of these regions in Figures <ref>-<ref>. With the gold standard classifications providing a ∼20% value for the false positive rate, we are confident that the majority of the subjects identified in the clean sample as outside of the SPLD arevalid. In order to verify the detection of araneiform features outside of the SPLD, we have also obtained high resolution imaging with HiRISE. We present those observations in Section <ref>. With this discovery, we find for the first time that there are surfaces other than the SPLD on the Martian South Polar region that are conducive to the growth of araneiforms.We further investigated why the araneiform features outside of the SPLD in our clean spider sample were not found by <cit.>. The MOC NA camera has better resolution (∼1-2 m per pixel) but less coverage per image than CTX does. One would have expected these features to be more clearly visible if present in the MOC NA images surveyed by <cit.>. Examining the coverage of the MOC NAobservations used by <cit.>, we find that the locations listed in Table <ref> are either outside the search area or in regions with less than 20% coverage. Thus, the likely reason why these araneiforms features were first identified by P4T is that these areas were covered in the CTX observations and not in the<cit.> reviewed observations. Bolstering this argument is the araneiform identifications from <cit.> which surveyed MOC NA observations from September 1997 to March 2004, including reviewing newer observations than what was available to<cit.>. <cit.> did not compare their araneiform identifications to the outline of the SPLD or the locations of other geologic units. Although <cit.>do not resolve araneiform identificationsto subimage positions, comparing their distribution, we find that there are several identifications outside of the SPLD (E0701468, R0902433, R0902028, R0901662, R0800241, R0904104, R0801805, R0903024, R0903639, R0701390, R0801195, M1101070, R0601143, R0903607, R0901019, R0601842, R0902403, R0903025, M1102723, M0905981, M1103774,M1000442, M1301816, and M0900454). With a visual inspection, we find many of these off SPLD MOC observations identified as containing araneiforms by <cit.> are heavily covered with dark seasonal fans, making it more difficult for a positive identification. In our P4T search, the majority of the CTX observations areice free off the SPRC, allowing for clearer identification of spiders and other types of araneiforms. cccc [short]Clean Spider Sample Outside of the SPLDSubject ID Center Latitude Center Longitude(degrees)(degrees)3c– ContinuedSubject ID Center Latitude Center Longitude(degrees)(degrees)3lContinued on Next Page… 1044650 -82.4 303.03 1044654 -82.34 302.88 1044658 -82.29 302.73 1044835 -82.27 303.19 1045035 -82.24 303.64 1044662 -82.23 302.58 1058053 -82.22 304.11 1044841 -82.21 303.03 1058095 -82.20 304.56 1058054 -82.16 303.95 1058096 -82.14 304.40 1045043 -82.13 303.34 1058055 -82.11 303.79 1044851 -82.09 302.73 1058139 -82.06 304.69 1321289 -82.01 300.61 1058140 -82.00 304.53 1044676 -82.00 302.00 1044860 -81.98 302.44 1058099 -81.97 303.93 1045055 -81.96 302.89 1321290 -81.95 300.45 1058141 -81.94 304.37 1044681 -81.94 301.86 1058058 -81.93 303.34 1044865 -81.92 302.30 1058100 -81.91 303.78 1045059 -81.9 0 302.75 1321291 -81.89 300.29 1044687 -81.88 301.72 1058059 -81.88 303.19 1044868 -81.86 302.16 1058101 -81.85 303.63 1044690 -81.83 301.59 1044697 -81.77 301.45 1044878 -81.75 301.89 1321091 -81.33 297.46 1321181 -81.14 297.50 487949 -80.42 306.11 488331 -80.41 292.65 487950 -80.36 306.00 487697 -79.95 304.11 487698 -79.89 304.02 487785 -79.87 304.38 491934 -79.76 20.89 491935 -79.70 20.79 491675 -79.7 0 19.62 491849 -79.66 20.34 491850 -79.6 20.25 489241 -79.53 74.18 491679 -79.46 19.28 491766 -79.44 19.64 491594 -79.36 18.77 491768 -79.33 19.47 491595 -79.3 18.69 491682 -79.29 19.04 492029 -79.27 20.51 491769 -79.27 19.38 491856 -79.25 19.73 491596 -79.24 18.61 491943 -79.23 20.08 491683 -79.23 18.96 492030 -79.22 20.42 491857 -79.19 19.65 491944 -79.18 19.99 492031 -79.16 20.33 491771 -79.15 19.22 491858 -79.13 19.57 491945 -79.12 19.91 492032 -79.10 20.25 491859 -79.08 19.48 491946 -79.06 19.82 491947 -79.00 19.74 492034 -78.98 20.08 491040 -78.98 230.66 491774 -78.97 18.98 491127 -78.96 231.00 491948 -78.94 19.66 491863 -78.84 19.16 485155 -77.77 225.05 1510467 -77.71 323.76 485156 -77.71 225.02 1510416 -77.7 324.06 1510674 -77.69 319.85 1510468 -77.65 323.69 1510690 -77.65 319.48 485403 -77.54 226.71 490900 -77.00 227.76 1510585 -76.92 319.00 490991 -76.74 227.82 484275 -74.57 330.81 484536 -74.53 331.55 484624 -74.46 331.75 484538 -74.41 331.46 484629 -74.16 331.53 484545 -74.00 331.17 § COMPARISON TO THE SWISS CHEESE DISTRIBUTION In Figure <ref>, we compare the locations ofour clean spider sample to our clean Swiss cheese terrain distribution. There is a clear separation between the locations of the spiders and Swiss cheese terrain in our clean samples. As expected, Swiss cheese terrain is concentrated on the SPRC, the only location on the Martian South polar region that has exposed carbon dioxide ice present at all times of the Martian year. Of the 1,537 subjects identified as containing Swiss cheese terrain, only 10 are located outside of the SPRC region latitudes northward of -80^∘ (subject identifiers: 1070921, 1046056, 1058252, 1058338, 1058430, 1058511, 1058604, 1510352, 485480, and 1510512). In our clean Swiss cheese and spider samples. we find no subjects identified as containing both Swiss cheese terrain and spiders. Our coverage of the South Polar region is moderate with 11% surface area covered, but this result is consistent with the CO_2 slab ice hypothesis for the production of araneiforms, as the locations identified fall in regions that have been observed to darken in albedo at some point during the thawing of the seasonal CO_2 ice sheet. <cit.> have recently identifieddark fans emanating from exclusively on the sides of mesas in the SPRC. Given the restriction locations of seasonal fan activity mesa walls, this may suggest that araneiforms should not form on the SPRC.The dissected mesas, linear troughs, moats, and quasi-circular pits characteristic of the SPRC form where there is carbon dioxide ice that persists through the southern summer <cit.>. The 10 subject images outside of the SPRCboundary represent 0.65% of our clean Swiss cheese sample. Thus, our results are generally consistent with the previously mapped extent of the SPRC <cit.>, but indicate some newly discovered regions where carbon dioxide ice likely persists through the southern summer. Of the 10 deviant subjects from the clean Swiss cheese sample lying outside of the SPRC area (Figure<ref>), 6 are clustered around 78.487^∘ and 101.61^∘ E (CTX image: P13_006290_1017), in the periglacially-deformed thin mantling deposit topping the Hesperian polar unit<cit.>. § HIRISE OBSERVATIONS AND INTERPRETATIONS OF CANDIDATE ARANEIFORM LOCATIONS OUTSIDE OF THE SPLD To confirm the araneiform identifications outside of the SPLD (presented in Section <ref>), HiRISE targeted and imaged near these locations in the Southern spring and summer of Mars Year 33. Additionally, we visually inspected all observations outside of the SPLD withs_i(spider)greater than 0.5, and the most promising candidates were also added to the HiRISE target database during this time period. With 12-18 times finer spatial resolution than CTX, HiRISE can reveal detail not visible in the original CTX images including whether or not fans are present at the same locations of the candidate spider channels. We present these HiRISE observations obtained to date and discuss each region below. Table <ref>lists the HiRISE targets imaged so far, the relevant observations, and the informal names given to each HiRISE target. Figure <ref> plots the HiRISE pointings overlaid on a MOLA elevation map <cit.>and the geologic map from <cit.>.Overall, seasonal fans were visible in the early spring observations at these locations, confirmingCO_2jet activity consistent with other araneiform locations on the Martian South Polar region <cit.>. Although these observations are not ice free,dendritic-like channels with similar morphologies to previous HiRISE observations of araneiforms<cit.> are visible.Thus, we positively identify araneiforms outside of the SPLD.In this Section, we briefly summarize the characteristics of the araneiform morphologies, if present at each HiRISE target location and present our interpretations based on the high resolution imagery.§.§ Boulders and Fans: Hempstead and Shoram At -74.81^∘latitude Hempstead is the furthest site from the Martian South Polar region that is outside of the SPLD. Spiders are primarily centered around boulders, thebright points at the center of the spiders. The region informally nicknamed `Inca City' (latitude = -81.3^∘ N, longitude= 295.7^∘) was previously the only known location with araneiforms and boulders. Fans emanate from the boulders in Hempstead, similar to Inca City. See Figure <ref>for a comparison with ESP_029741_0985 from Inca City. Like Inca City, the boulders appear to spur the generation of the carbon dioxide jets.As proposed by <cit.>, the boulders are surrounded and covered by the seasonal ice sheet.The boulders provide a discontinuity in thermal inertia and an additional source of thermal heat that enables the surrounding ice to sublimate and crack more easily providing an outlet for the trapped carbon dioxide gas.As we see at Shoram (Figure <ref>) it is not always a simple story, however. In Shoram there are also small boulders,but they are not always associated with fans. §.§ Boundaries and Interfaces: BethpageBethpage has narrow channels radiating from broad troughs.A representative close-up is shown in Figure <ref>.Dark dust on the top of the seasonal ice surrounds the channels as highlighted in Figure <ref>. The appearance of the seasonal fans is consistent with active CO_2 jet processes on-going in this region, thus we infer that at least the narrow channels are formed by erosion from seasonal CO_2 gas escape entraining loose material from the surface under the seasonal ice.As shown in Figure <ref> the broad troughs and narrow channels are found in hummocky terrain and along the boundary of smoother terrain, consistent with the hypothesis that CO_2 jet processes exploit the weaknesses in the regolith to form sinuous araneiform channels.§.§ Short Channels and Boulders: Stony BrookAt Stony Brook the terrain is rough on the scale of meters.There are short channels but with a few exceptions they do not show the branching, dendritic characteristics or the tight sinuosity of the araneiform terrains previously reported in the literature <cit.>. HiRISE image ESP_046398_1030, shown in Figure <ref>, displays seasonal fans (some coming from boulders and some from short grooves) and a few well-developed spiders. Frosted spider outlines are also visible in the region as well.§.§ Connected Aaraneiforms: Montauk, Happauge, and ShoramMontauk has well-developed araneiforms as shown in Figure <ref>. Separated spiders are visible, but the vast majority of the araneiforms visible in these HiRISE observations are connected spiders, where dendritic channels are not emanating from a point but rather the channels form complex and interlinked networks of araneiform channels. This has a morphology similar to what<cit.> describes as Connected Araneiform Morphology. Along with having boulders, Shoram also has pockets of connected araneiform features.At L_ s=185.5^∘, fans can be seen originating from many of the dendritic channels as shown in Figure <ref>.Additionally, Happauge (see Figure <ref>) also exhibits connected araneiforms. The ground surrounding the spiders in Hauppage also has a stippled texture although this may not be related to the seasonal carbon dioxide jets. §.§ Spiders on Possible Eject Blankets: Montauk and Happauge Montauk is located on the rim of a partially eroded un-named crater (see Figure <ref>) and Hauppauge is located near a large crater named `South Crater' as shown in Figure <ref>. These two regions may reside on the crater ejecta blankets.The layering or unconsolidated nature of the surface inherent in an ejecta blanket may provide a conducive environment for spider development, similar to the SPLD, where araneiforms were originallyfound to reside <cit.>. Identification of crater ejecta is challenging. Only fresh or moderately eroded craters will have ejecta blankets surrounding the crater rim identifiable in orbital images.Following<cit.>, we examinedThermal Emission Imaging System (THEMIS )Day InfraredIR) 100m per pixel global map mosaic <cit.> of the area surrounding Montauk and Happauge (seeFigures<ref> and <ref>). Hauppage appears in the THEMIS observations to be on a darker unit, but it is not clear if this can be identified as part of South Crater's ejecta blanket. From our analysis we cannot confirm that these locations outside of the SPLD coincide with ejecta blankets. Further high resolution observations and detections of more araneiforms outside of the SPLD in close proximity to crater rims would bolster this hypothesis.§.§ Polygons and Spiders: Dix Hills Extensive polygonal cracks are visible in the surface at Dix Hills. Polygonal cracks are one of many well-documented types of patterned ground formed by the cyclic freezing and thawing of underlying water ice-cemented ground, and the subsequent cracking of ice-rich material produces shallow grooves<cit.>. The association of fans with these grooves leads us to infer that they provide routes for the flow of the trapped pressurized gas under the CO_2 seasonal cap.As shown in Figure <ref> and Figure <ref> more sinuous araneiform channels are also present at these locales. Fans are visible in later observations of Dix Hills where the polygonal channels have pits, are wider, and/or develop secondary branching channels, as shown in Figure <ref>.§.§ Sinuous Channels: Patchogue Patchogue has both sinuous channels and distinct well-separated spiders, with distinctly visible central pits and radiating channels, see Figure <ref>.The spiders' morphology is the classic spider with radially-organized channels, that have been observed by HiRISE on other locales on the SPLD <cit.>. Patchogue is near Inca City, but unlike Hempstead, Shoram, and Inca City, no boulders were visible. § IMPLICATIONS FOR THE CO_2 JET MODEL The majority of the P4T clean spider sample ( 75%) resides on the SPLD, where the carbon dioxide jet model has been previously put forth for the formation of araneiforms <cit.>. High resolution HiRISE imaging confirms the identification of several locations outside of the SPLD with araneiformsfound by P4T.All of the HiRISE observations of the locales outside of the SPLD with confirmed araneiform features visible, also exhibited seasonal fans, suggesting that activeCO_2 jets were present during the season. Additionally, all but one of these HiRISE target locations have already been identified by <cit.> as becoming cryptic (with temperatures near the CO_2 frost sublimation temperature but with an albedo lower than that of typical CO_2 frost) for a period of time between L_S = 180^∘ andL_S = 250^∘, indicative of semi-translucent CO_2 slab ice required for the CO_2 jet formation. For Hempstead, there was no albedo information available to <cit.>, but <cit.> identify the presence of the seasonal ice cap extending beyond that location to -50^∘ over multiple Mars years. Thus our clean spider distribution and new araneiform discoveries outside of the SPLD are consistent with the preferred CO_2 jet model and slab ice formation model. Our outside of the SPLD spider identifications are located inAmazonian and Hesperian polar units and the Early Noachian highland units outside of the SPLD. With our current coverage of∼11% of the South Polar region southward of 75^∘, we cannot make a definitive statement regarding whether these are the only geologic units with araneiforms within the Martian South Polar region. A larger search for araneiforms in CTX and MOC NA observations are needed. The presence araneiforms in the new areas identified in this study suggests that the regolith in these areas can be easily weakened or is loosely conglomerated like that of the SPLD, enabling the trapped carbon dioxide gas to slowly carve channels into the surface over time. Further study of the surface properties may illuminate the link between the SPLD and these regions for araneiform formation. Two of the HiRISE target locations (Hauppague and Montauk) are close to craters and may reside on associated ejecta blankets. This may indicate that crater ejecta blankets play an important role in the formation of araneiforms outside of the SPLD as an alternative source for loosely conglomerated materials to excavate. Further HiRISE observations and samples of araneiforms outside the SPLD are needed before a definitive conclusion can be made.§ CONCLUSIONS Through the effort of over 10,000 volunteers, P4T has mapped the distribution of spider araneiforms in 90 CTX observations of the Martian South Pole. This paper presents results from our analysis of 303,192 km^2 of CTX observationsbetween -75^∘and the South Pole, mapping locations of araneiforms and Swiss cheese terrain on the Martian South Pole. We present the first identification of araneiforms present atlocations outside the SPLD. This discovery is consistent with the general paradigm where the CO_2 jet process carves channels into the surface by the trapped CO_2 gas exploiting weaknesses in the conglomeration of the soil layer below<cit.>. Although the spiders are found primarily on the SPLD, the source of the non-uniformity of the spider distribution requires further study. The morphology of the aranieforms and rate of the erosion process to carve channels into the surfaces outside the SPLD is likely a combination of material strength of the surface, surface slopes,and the amount of time the seasonal CO_2 ice sheet is present at each location. We find that several of our newly identified araneiform locales are on or close to crater ejecta blankets. With our limited sample size, we can not definitively conclude that araneiform formation outside of the SPLD prefers disturbed regolith such as crater ejecta blankets. A larger survey of the South Polar region is required. Further high resolution imaging of these new spider locations coupled with additional surveys of the South Polar region will further elucidate the formation of these features and the CO_2 seasonal jet process. § FUTURE PROSPECTSThe success of P4T inproviding timely targets for follow-up high resolution imaging highlights citizen science and crowd sourcing as a potential future avenue for efficiently identifying planetary surface features in planetary mission datasets and feeding back into mission planning. The area searched by P4T presented in this Paper was constrained in order to have both the citizen science review and full analysis of the classifications completed before the beginning of the South spring Equinox in Mars Year 33 and thestart of HiRISE's seasonal processes monitoring campaign. With the completion of this first set of 90 CTX observations, additional CTX images have continued to be available on the P4T website for review; expanding the coverage wider beyond -75^∘ latitude. Classification of these images is currently underway. We expect in future works to be able to produce a sample of baby spider and channel network locations from P4T classifications. P4T will be able to further examine the abundance of spiders outside the SPLD and the correlations with surface properties that can shed light on how the CO_2 jet process is able to carve channels into the SPLD and these other regions of the South Pole.More examples of araneiform locations outside of the SPLD will enable a better analysis of the prevalence of crater ejecta blankets as suitable environments from araneiform formation. § ACKNOWLEDGMENTS The data presented in this Paper are the result of the efforts of the P4T volunteers, without whom this work would not have been possible. Their contributions are individually acknowledged at:<http://p4tauthors.planetfour.org>. We also thank our P4T Talk moderators Andy Martin and John Keegan for their time and efforts helping the P4T volunteer community. This publication uses data generated via the Zooniverse.org (<https://www.zooniverse.org>) platform, development of which is funded by generous support, including a Global Impact Award from Google, and by a grant from the Alfred P. Sloan Foundation. The authors thank the Zooniverse team for early access to their Project Builder Platform. MES was supported by Gemini Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., on behalf of the international Gemini partnership of Argentina, Brazil, Canada, Chile, and the United States of America.MES was also supported in part by an Academia Sinica Postdoctoral Fellowship. KMA and AP were supported in part by NASA Grant Nr. 14-SSW14 2-0237. The authors also thank Chris Lintott (University of Oxford), who had to decline authorship on this Paper. We thank him for his efforts contributing to the development of the Planet Four: Terrains website and for his useful discussions. The authors also thank Chris Snyder, Chris Schaller, and Serina Diniega for useful discussions. The authors also thank Peter Buhler and an anonymous reviewer for constructive reviews that improved the manuscript. The Mars Reconnaissance Orbiter mission is operated at the Jet Propulsion Laboratory, California Institute of Technology, under contracts with NASA. This research has made use of the USGS Integrated Software for Imagers and Spectrometers (ISIS) and of NASA's Astrophysics Data System. We thank the Context Camera (CTX) teamfor making their data publicly available through NASA's Planetary Data System (PDS). CTX is operated by Malin Space Science Systems. This research alsomade use of Astropy, a community-developed core Python package for Astronomy <cit.>. This work made use of the JAVA Mission-planning and Analysis for Remote Sensing (JMARS) <cit.> and the MRO Context Camera Image Explorer <http://viewer.mars.asu.edu/viewer/ctx#T=0>. § GEOLOGIC MAP LEGEND We provide in Figure <ref> a legend identifying the main geologic units of the South Polar region in the geologic map from <cit.> used in this work. We refer the reader to <cit.>for further details about the characteristics of each geologic unit. 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"authors": [
"Megan E. Schwamb",
"Klaus-Michael Aye",
"Ganna Portyankina",
"Candice J. Hansen",
"Campbell Allen",
"Sarah Allen",
"Fred J. Calef III",
"Simone Duca",
"Adam McMaster",
"Grant R. M. Miller"
],
"categories": [
"astro-ph.EP",
"astro-ph.IM"
],
"primary_category": "astro-ph.EP",
"published": "20170825184442",
"title": "Planet Four: Terrains - Discovery of Araneiforms Outside of the South Polar Layered Deposits"
} |
Department of Mathematics, Northeastern University, Boston, MA 02115, [email protected] www.math.neu.edu/ braverman/ Beijing International Center for Mathematical Research (BICMR), Beijing (Peking) University, Beijing 100871, [email protected] [2010]58J28, 58J30, 58J32, 19K56^††Partially supported by the Simons Foundation collaboration grant #G00005104. APS index with non-compact boundary]An APS index theorem for even-dimensional manifolds with non-compact boundary We study the index ofthe APS boundary value problem for a strongly Callias-type operatoron a complete Riemannian manifold M. We use this index to define the relative η-invariant η(_1,_0) of two strongly Callias-type operators, which are equal outside of a compact set. Even though in our situation the η-invariants of _1 and _0 are not defined, the relative η-invariant behaves as if it were the difference η(_1)-η(_0). We also define the spectral flow of a family of such operators and use it to compute the variation of the relative η-invariant. [ Pengshuai Shi August 24, 2017 ===================§ INTRODUCTIONIn <cit.> we studied the index of the Atiyah–Patodi–Singer (APS) boundary value problem for a stronglyCallias-type operator on a complete odd-dimensional manifold with non-compact boundary. We used this index to define the relative η-invariant η(_1,_0) of two strongly Callias-type operatorson even-dimensional manifolds, assuming that _0 and _1 coincide outside of a compact set. In this paper we discuss an even-dimensional analogue of <cit.>. Many parts of the paper are parallel to the discussion in <cit.>. However, there are two important differences. First, the Atiyah–Singer integrand was, of course, equal to 0 in <cit.>, which simplified many formulas. In particular, the relative η-invariant was an integer. As opposed to it,in the current paper the Atiyah–Singer integrand plays an important role and the relative η-invariant is a real number. More significantly, the proof of the main result in <cit.> was based on the application of the Callias index theorem, <cit.>. This theorem is not available in our current setting. Consequently, a completely different proof is proposed in Section <ref>.Our results provide a new tool to study anomalies in quantum field theory. Mathematical description of many anomalies is given by index theorems for boundary value problems, cf. <cit.>, <cit.>. However, most mathematically rigorous descriptions of anomalies in the literature only work on compact manifolds. The results of the current paper allow to extend many of these descriptions to non-compact setting, thus providing a mathematically rigorous description of anomalies in more realistic physical situations. In particular, Bär and Strohmaier, in <cit.>, gave a mathematically rigorous description of chiral anomalyby considering an APS boundary problem for Dirac operator on a Lorentzian spatially compact manifold. In a recent preprint <cit.> the first author extended the results of <cit.> to spatially non-compact case. The analysis in <cit.> depends heavily on the results of the current paper. Another applications to the anomaly considered in <cit.> will appear in <cit.>. We now briefly describe our main results.§.§ Strongly Callias-type operators A Callias-type operator on a complete Riemannian manifold M is an operator of the form = D+Ψ where D is a Dirac operator and Ψ is a self-adjoint potential which anticommutes with the Clifford multiplication and satisfies certain growth conditions at infinity. In this paper we imposeslightly stronger growth conditions on Ψ and refer to the obtained operatoras a strongly Callias-type operator. Our conditions on the growth of Ψ guarantee that the spectrum ofis discrete. The Callias-type index theorem, proven in different forms in<cit.>, computes the index of a Callias-type operator on a complete odd-dimensional manifold as the index of a certain operator induced byon a compact hypersurface. Several generalizations and applications of the Callias-type index theorem were obtained recently in <cit.>. P. Shi, <cit.>, proved a version of the Callias-type index theorem for theAPSboundary value problem for Callias-type operators on a complete odd-dimensional manifold with compact boundary. Fox and Haskell <cit.> studied Callias-type operators on manifolds with non-compact boundary.Under rather strong conditions on the geometry of the manifold and the operatorthey proved a version of the Atiyah–Patodi–Singer index theorem. In <cit.> we studiedthe index of the APS boundary value problem on an arbitrary complete odd-dimensional manifold with non-compact boundary. In the current paper, we obtain an even-dimensional analogue of <cit.>.§.§ An almost compact essential support A manifold C, whose boundary is a disjoint union of two complete manifolds N_0 and N_1, is called essentially cylindrical if outside of a compact set it is isometric to a cylinder [0,ε]× N, where N is a non-compact manifold. It follows that manifolds N_0 and N_1 are isometric outside of a compact set. We say that an essentially cylindrical manifold M_1, which contains ,is an almost compact essential support ofif the restriction of ^* to M∖ M_1 is strictly positive and the restriction ofto the cylinder [0,ε]× N is a product, cf. Definition <ref>. Every strongly Callias-type operator on M which is a product nearhas an almost compact essential support. Theorem <ref> states that the index of the APS boundary value problem for a strongly Callias-type operatoron a complete manifold M is equal to the index of the APS boundary value problem of the restriction ofto its almost compact essential support M_1.§.§ Index on an essentially cylindrical manifold Let M be an essentially cylindrical manifold and letbe a strongly Callias-type operator on M, whose restriction to the cylinder [0,ε]× M is a product. Suppose = N_0⊔ N_1 and denote the restrictions ofto N_0 and N_1 by _0 and -_1 respectively (the sign convention means that we thinkofN_0 as the “left boundary” and of N_1 as the “right boundary” of M). Let _B denote the operatorwith APS boundary conditions. Let () denote the Atiyah–Singer integrand of . This is a differential form on M which depends on the geometry of the manifold and the bundle. Since all structures are product outside of the compact set K, this form vanishes outside of K. Hence, ∫_M() is well-defined and finite. Our main result here is that _B -∫_M ()depends only on the operators _0 and _1 and doesnot dependon the interior of the manifold M and the restriction ofto the interiorof M, cf. Theorem <ref>. §.§ The relative η-invariantSuppose now that _0 and _1 are self-adjoint strongly Callias-type operators on complete manifolds N_0 and N_1. An almost compact cobordism between _0 and _1 is a pair (M,) where M is an essentially cylindrical manifold with =N_0⊔ N_1 andis a strongly Callias-type operator on M, whose restriction to the cylindrical part of M is a product and such that the restrictions ofto N_0 and N_1 are equal to _0 and -_1 respectively. We say that _0 and _1 are cobordant if there exists an almost compact cobordism between them. In particular, this implies that_0 and _1 are equal outside of a compact set. Let be an almost compact cobordism between _0 and _1. Let B_0 and B_1 be the APS boundary conditions forat N_0 and N_1 respectively. Let _B_0⊕ B_1 denote the index of the APS boundary value problem for .We define the relative η-invariant by the formulaη(_1,_0)= 2 ( _B_0⊕ B_1-∫_M () ) +_0 +_1.It follows from the result of the previous subsection that η(_1,_0) is independent of the choice of an almost compact cobordism.If M is a compact manifold, then the Atiayh-Patodi-Singer index theorem <cit.> implies that η(_1,_0)= η(_1)-η(_0). In general, for non-compact manifolds, the individual η-invariants η(_1) and η(_0) might not be defined. However, η(_1,_0)behaves like it were a difference of two individual η-invariants. In particular, cf. Propositions <ref>–<ref>,η(_1,_0) =- η(_0,_1), η(_2,_0) =η(_2,_1) +η(_1,_0). Under rather strong conditions on the manifolds N_0 and N_1 and on the operators _0, _1,Fox and Haskell <cit.> showed that theheat kernel of _j (j=0,1) has a nice asymptotic expansion similar to the onefor operators on compact manifolds. Then they were able to define theindividual η-invariants η(_j) (j=0,1). In this situation, as expected, our relative η-invariant is equal to the difference of the individual η-invariants:η(_1,_0)= η(_1)-η(_0). Under much weaker (but still quite strong) assumptions,Müller, <cit.>, suggested a definition of a relative η-invariant based on analysis of the relative heat kernel <cit.>. This invariant behaves very similar to our η(_1,_0). The precise conditions under which these two invariants are equal are not clear yet. We note that our invariant is defined on a much wider class of manifolds, where the relative heat kernel is not of trace class and can not be used to construct an η-invariant.§.§ The spectral flow Consider a family Å= {^s}_0≤ s≤ 1 of self-adjoint strongly Callias-type operatorson a complete Riemannian manifold. We assume that there is a compact set K⊂ M such that the restriction of ^s to M∖ K is independent of s. Since the spectrum of ^s is discrete for all s, the spectral flow (Å) can be defined in a more or less usual way. By Theorem <ref>, if _0 is a self-adjoint strongly Callias-type operator which is cobordant to ^0 (and hence, to all ^s), thenη(^1,_0) -η(^0,_0) -∫_0^1 (d/ds(^s,_0)) ds =2 (Å).The derivative d/ds(^s,_0) can be computed as an integral of the transgression form — a differential form canonically constructed from the symbol of ^s and its derivative with respect to s. Thus (<ref>) expresses the change of the relative η-invariant as a sum of 2(Å) and a local differential geometric expression. § BOUNDARY VALUE PROBLEMS FOR CALLIAS-TYPE OPERATORSIn this section we recall some results about boundary value problems for Callias-typeoperators on manifolds with non-compact boundary,<cit.>, keeping in mind applications to even-dimensional case. The operators considered here are slightly more general than those discussed in <cit.>, but all the definitions and most of the properties of the boundary value problems remain the same.§.§ Self-adjoint strongly Callias-type operators Let M be a complete Riemannian manifold (possibly with boundary) and let E→ M be a Dirac bundle over M, cf. <cit.>. In particular, E is a Hermitian bundle endowed with a Clifford multiplication c:T^*M→(E) and a compatible Hermitian connection ^E.Let D:C^∞(M,E)→ C^∞(M,E) be the Dirac operator defined by the connection ^E. Let Ψ∈ End(E) be a self-adjoint bundle map (called a Callias potential). Then:= D+Ψis a formally self-adjoint Dirac-type operator on E and^2 = D^2+Ψ^2+[D,Ψ]_+,where [D,Ψ]_+:=D∘Ψ+Ψ∘ D is the anticommutator of the operators D and Ψ. We calla self-adjoint strongly Callias-type operator if * [D,Ψ]_+ is a zeroth order differential operator, i.e. a bundle map;* for any R>0, there exists a compact subset K_R⊂ M such thatΨ^2(x) - |[D,Ψ]_+(x)| ≥ Rfor all x∈ M∖ K_R.In this case, the compact set K_R is called an R-essential support of , or an essential support when we do not need to stress the associated constant. Condition (i) of Definition <ref> is equivalent to the condition that Ψ anticommutes with the Clifford multiplication: [c(ξ),Ψ]_+ = 0, for all ξ∈ T^*M.§.§ Graded self-adjoint strongly Callias-type operators Suppose now that E=E^+⊕ E^- is a _2-graded Dirac bundle such that the Clifford multiplication c(ξ) is odd and the Clifford connection is even with respect to this grading. ThenD := (0 D^- D^+ 0)is the _2-graded Dirac operator, where D^± : C^∞(M,E^±)→ C^∞(M,E^∓) are formally adjoint to each other. Assume that the Callias potential Ψ has odd grading degree, i.e., Ψ = (0Ψ^-Ψ^+ 0),where Ψ^±∈(E^±,E^∓) are adjoint to each other. Then we have= D + Ψ = ( 0 D^-+Ψ^- D^++Ψ^+ 0 ) =: ( 0^-^+ 0 ). Under the same condition as in Definition <ref>,is called a graded self-adjoint strongly Callias-type operator. In this case, we also call ^+ and ^- strongly Callias-type operators. They are formally adjoint to each other. By an R-essential support (or essential support) of ^± we understand as an R-essential support (or essential support) of .When M is an oriented even-dimensional manifold there is a natural grading of E induced by the volume form. We will consider this situation in the next section. Suppose there is a skew-adjoint isomorphism γ:E^±→ E^∓, γ^*=-γ, which anticommutes with multiplication c(ξ) for all ξ∈ T^*M, satisfies γ^2=-1, and is flat with respect to the connection ^E, i.e. [^E,γ]=0. Then ξ↦γ∘ c(ξ) defines a Clifford multiplication of T^*M on E^+ and the corresponding Dirac operator is D̃^+= γ∘ D^+. Suppose also that γ commutes with Ψ. Then Φ^+=-iγ∘Ψ^+ is a self-adjoint endomorphism of E^+. In this situation, D̃^++iΦ^+ =γ∘^+: C^∞(M,E^+) → C^∞(M,E^+) is a strongly Callias-type operator in the sense of <cit.>. §.§ Restriction to the boundary Assume that the Riemannian metric g^M is product near the boundary, that is, there exists a neighborhood U⊂ M of the boundary which is isometric to the cylinderZ_r :=[0,r)×. In the following we identify U with Z_r and denote by t the coordinate along the axis of Z_r. Then the inward unit normal one-form to the boundary is given by τ = dt. Furthermore, we assume that the Dirac bundle E is product near the boundary. In other words we assume that the Clifford multiplication c:T^*M→ End(E) and the connection ∇^E have product structure on Z_r, cf. <cit.>.Let D be a _2-graded Dirac operator. In this situation the restriction of D to Z_r takes the form D=c(τ)(_t+Â)= ( 0 c(τ) c(τ) 0 ) ( _t+A 0 0_t+A^♯),whereA: C^∞(,E^+_) →C^∞(,E^+_)and A^♯= c(τ)∘ A∘ c(τ): C^∞(,E^-_) →C^∞(,E^-_)are formally self-adjoint operators acting on the restrictions of E^± to the boundary. It would be more natural to use the notation A^+ and A^- instead of A and A^♯. But since in the future we only deal with the operator A:C^∞(,E^+_)→ C^∞(,E^+_) we remove the superscript “+” to simplify the notation.Let =D+Ψ be a graded self-adjoint strongly Callias-type operator. Then the restriction ofto Z_r is given by =c(τ)(_t+) = ( 0 c(τ) c(τ) 0 ) ( _t+0 0_t+^♯),where:=A-c(τ)Ψ^+: C^∞(,E^+_) → C^∞(,E^+_).and ^♯ = A^♯-c(τ)Ψ^-.By Remark <ref>, c(τ)Ψ^±∈ End(E^±_) are self-adjoint bundle maps. Thereforeand ^♯ are formally self-adjoint operators. In fact, they are strongly Callias-type operators, cf. Lemma 3.12 of <cit.>.In particular, they have discrete spectrum. Also,^♯ = c(τ)∘∘ c(τ). We say that a graded self-adjoint strongly Callias-type operatoris product near the boundary if the Dirac bundle E isproduct near the boundary and the restriction of the Callias potential Ψ to Z_r does not depend on t. The operator(resp. ^♯)of (<ref>) is called the restriction of ^+ (resp. ^-) to the boundary.§.§ Sobolev spaces on the boundary Consider a gradedself-adjointstrongly Callias-type operator :C^∞(M,E)→ C^∞(M,E), cf. (<ref>). The restriction of ^+ to the boundary is a self-adjoint strongly Callias-type operator: C^∞(,E_^+) → C^∞(,E_^+). We recall the definition of Sobolev spaces H^s_(,E^+_) of sections overwhich depend on the boundary operator , cf. <cit.>.SetC_^∞(,E_^+) := { ∈ C^∞(,E_^+): (𝕀+^2)^s/2_L^2(,E_)^2<+∞s∈ }.For all s∈ we define the Sobolev H_^s-norm on C_^∞(,E_^+) by_H_^s(,E_)^2 := (𝕀+^2)^s/2_L^2(,E_)^2.The Sobolev space H_^s(,E_^+) is defined to be the completion of C_^∞(,E_^+) with respect to this norm.§.§ Generalized APS boundary conditions The eigensections ofbelong to H_^s(,E_^+) for all s∈, cf. <cit.>. For I⊂ we denote by H_I^s() ⊂H_^s(,E_^+) the span of the eigensections ofwhose eigenvalues belong to I.For any a∈, the subspaceB = B(a) := H_(-∞,a)^1/2().is called the the generalized Atiyah–Patodi–Singer boundary conditions for ^+. If a=0, then the space B(0)= H_(-∞,0)^1/2() is called the Atiyah–Patodi–Singer (APS) boundary condition. The spaces (a) := H_(-∞,a]^1/2()and(0) := H_(-∞,0]^1/2() are called the dual generalized APS boundary conditions and the dual APS boundary conditions respectively.The space B^=B^(a):=H_(-∞,-a]^1/2(^♯) is called the adjoint of the generalized APS boundary condition for ^+. One can see that it is a dual generalized APS boundary condition for ^-. If B is a generalized APS boundary condition for ^+, we denote by ^+_Bthe operator ^+ with domain^+_B := {u∈^+_max: u|_∈ B},where ^+_max denotes the domain of the maximal extension of ^+ (cf. <cit.>).We refer to ^+_Bas the generalized APS boundary value problem for ^+. Recall that ^- is the formal adjoint of ^+. It is shown in Example 4.9 of <cit.> that theL^2-adjoint of ^+_B(a)is given by ^- with the dual APS boundary condition B^(a):(^+_B(a))^ =^-_B^(a). Suppose that a graded strongly Callias-type operator (<ref>) is product near . Then the operator ^+_B: ^+_B→ L^2(M,E^-) is Fredholm. In particular, it has finite dimensional kernel and cokernel. For the case discussed in Remark <ref> this is proven in Theorem 5.4 of <cit.>. Exactly the same proof works in the general case.§.§ The index of generalized APS boundary value problems By (<ref>) the cokernel of ^+_B(a) is isomorphic to the kernel of ^-_B^(a).Let ^+ be a strongly Callias-type operator on a complete Riemannian manifold M which is product near the boundary. Let B= H_(-∞,a)^1/2() be a generalized APS boundary condition for ^+ and let B^= H_(-∞,-a]^1/2(^♯) be the adjoint of the generalized APS boundary condition. The integer ^+_B := ^+_B-(^-)_B^ ∈is called the index of the boundary value problem ^+_B. It follows directly from (<ref>) that (^-)_B^ =-^+_B.§.§ More general boundary value conditionsGeneralized APS and dual generalized APS boundary conditions are examples of elliptic boundary conditions, <cit.>. In this paper we don't work with general elliptic boundary conditions. However, in Section <ref> we need a slight modification of APS boundary conditions, which we define now.We say that two closed subspaces X_1, X_2 of a Hilbert space H are finite rank perturbationsof each other if there exists a finite dimensional subspace Y⊂ H such that X_2⊂ X_1⊕ Y and the quotient space (X_1⊕ Y)/X_2 has finite dimension. The relative index of X_1 and X_2 is defined by [X_1,X_2]:=(X_1⊕ Y)/X_2 - Y.One easily sees that the relative index is independent of the choice of Y. Weshall need the following analogue of <cit.>:Letbe a graded self-adjoint strongly Callias-type operator on M and let B be a generalized APS or dual generalized APS boundary condition for ^+.If B_1⊂ H^1/2_(,E^+_) is a finite rank perturbation of B, then the operator ^+_B_1 is Fredholm and^+_B -^+_B_1 =[B,B_1]. The proof of the proposition is a verbatim repetition of the proof of <cit.>.As an immediate consequence of Proposition <ref> we obtain the following Letbe the restriction of ^+ toand letB_0= H^1/2_(-∞,0)() and _0= H^1/2_(-∞,0]() be the APS and the dual APS boundary conditions respectively. Then ^+__0 =^+_B_0 +.§.§ The splitting theorem Let M be a complete manifold. Let N⊂ M be a hypersurface disjoint fromsuch that cutting M along N we obtain a manifold M' (connected or not) withand two copies of N as boundary. So we can write M'=(M∖ N)⊔ N_1⊔ N_2.Let E=E^+⊕ E^-→ M be a _2-graded Dirac bundle over M and ^±:C^∞(M,E^±)→ C^∞(M,E^∓) be strongly Callias-type operators as in Subsection <ref>. They induce _2-graded Dirac bundle E'=(E')^+⊕(E')^-→ M' and strongly Callias-type operators (')^±:C^∞(M',(E')^±) → C^∞(M',(E')^∓)on M'. We assume that all structures are product near N_1 and N_2. Letbe the restriction of (')^+ to N_1. Then - is the restriction of (')^+ to N_2 and, thus, the restriction of (')^+ to N_1⊔ N_2 is '=⊕(-). The following Splitting Theorem is an analogue of Theorem 5.11 of <cit.> with the same proof.Suppose M,^+,M',(')^+ are as above. Let B_0 be a generalized APS boundary condition on . Let B_1=H_(-∞,0)^1/2() and B_2= H_(-∞,0]^1/2(-)= H_[0,∞)^1/2() be the APS and the dual APS boundary conditions for (')^+ along N_1 and N_2, respectively. Then (')^+_B_0⊕ B_1⊕ B_2 is a Fredholm operator and^+_B_0 = (')^+_B_0⊕ B_1⊕ B_2. §.§ Reduction of the index to an essentially cylindrical manifold The study of the index of Callias-type operators on manifolds without boundary can be reduced to a computation on the essential support. For manifolds with boundary we want an analogous subset, but the one which contains the boundary. Such a set is necessarily non-compact, but we want it to be “similar to a compact set". In <cit.>, we introduce the notion of essentially cylindrical manifolds, which replace the role of compact subsets in our study of boundary value problems.An essentially cylindrical manifoldC is a complete Riemannian manifold whose boundary is a union of two disjoint manifolds, ∂ C=N_0⊔ N_1, such that* there exist a compact set K⊂ C, an open Riemannian manifold N, and an isometry C∖ K≃ [0,δ]× N; * under the above isometry N_0∖ K ={0}× N and N_1∖ K ={δ}× N. See Figure 1 for an example of essentially cylindrical manifolds. [height=5cm]Fig1.pdf (98,40)N_0 (98,8)N_1 (47,23)K Figure 1. An essentially cylindrical manifoldCLet ^+ be a strongly Callias-type operator on M. An almost compact essential support of^+ is a smooth submanifold M_1⊂ Mwith smooth boundary, which containsand such that there exist a (compact) essential support K⊂ M and ε∈ (0,r) such that M_1∖ K=(∖ K)× [0,ε] ⊂ Z_r.An almost compact essential support is a special type of essentially cylindrical manifolds. It is shown in <cit.> that for any strongly Callias-type operator which is product nearthere exists analmost compact essential support. Figure 2 illustrates the idea of building an almost compact essential support.[height=8cm]Fig2.pdf (-5,47) (34,47)N_1 (15,36)K (15,-3)M_1 (50,-4)M Figure 2. The idea to obtain an almost compact essential support M_1 §.§ The index on an almost compact essential supportSuppose M_1⊂ M is an almost compact essential support of ^+. Set N_0:= and let N_1⊂ M be such that _1=N_0⊔ N_1 as in Definition <ref>. The restriction ofto a neighborhood of N_1 need not be product. Since in this paper we only consider boundary value problems for operators which are product near the boundary, we first deformto a product form. It is shown in <cit.> that there exists a perturbation of all the structures such that* the new structures are product near N_1. In particular, the corresponding Callias-type operator' is product near N_1.* ' has a compact essential support, which is contained in M_1.* The difference -' vanishes nearand outside of a compact subset of M_1. In this situation we say that ' (or (')^±) isa compact perturbation of(or ^±).Let M_1⊂ M be an almost compact essential support of ^+. Let (')^+ be a compact perturbation of ^+ which is product near the boundary (cf. <cit.>). Letbe the restriction of ^+ to . It is also the restriction of (')^+. We denote by -_1 the restriction of (')^+ to N_1. Theorem 6.10 of <cit.> claims that the index of an elliptic boundary value problem on M can be reduced to an index on an almost compact essential support: Suppose M_1⊂ M is an almost compact essential support of ^+ and let _1=⊔ N_1.Let (')^+ be a compact perturbation of ^+ which is product near N_1 andsuch that there is a compact essential support for (')^+ which is contained in M_1. Let B_0 be a generalized APS boundary condition for ^+. View (')^+ as an operator on M_1 and let B_1=H_(-∞,0)^1/2(-_1)= H_(0,∞)^1/2(_1) be the APS boundary condition for (')^+ at N_1. Then^+_B_0 = (')^+_B_0⊕ B_1.§ THE INDEX OF OPERATORS ON ESSENTIALLY CYLINDRICAL MANIFOLDS In this section we discuss the index of strongly Callias-type operators on even-dimensional essentially cylindrical manifolds. It is parallel to <cit.>, where the odd-dimensional case was considered. From now on we assume that M is an oriented even-dimensional essentially cylindrical manifold whose boundary = N_0⊔ N_1 is a disjoint union of two non-compact manifolds N_0 and N_1. Let E be a Dirac bundle over M. As pointed out in Remark <ref>, there is a natural _2-grading E=E^+⊕ E^- on E. Let ^+: C^∞(M,E^+)→ C^∞(M,E^-) be a strongly Callias-type operator as in Definition <ref> (these data might or might not come as a restriction of another operator to its almost compact essential support. In particular, we don't assume that the restriction of ^+ to N_1 is invertible). Let _0 and -_1 be the restrictions of ^+ to N_0 and N_1 respectively. We first recall some definitions from <cit.>.§.§ Compatible data Let M be an essentially cylindrical manifold and let =N_0⊔ N_1. As usual, we identify a tubular neighborhood ofwith the product Z_r :=(N_0×[0,r) ) ⊔ (N_1×[0,r) ) ⊂ M.We say that another essentially cylindrical manifold M' is compatible withM if there is a fixed isometry between Z_r and a neighborhood Z_r'⊂ M' of the boundary of M'.Note that if M and M' are compatible then their boundaries are isometric.Let M and M' be compatible essentially cylindrical manifolds and let Z_r and Z_r' be as above. Let E→ M be a _2-graded Dirac bundle over M and let^+: C^∞(M,E^+)→ C^∞(M,E^-)be a strongly Callias-type operator whose restriction to Z_r is product and such that M is an almost compact essential support of ^+.Thismeans that there is a compact set K⊂ M such that M∖ K= [0,ε]× N and the restriction of ^+ to M∖ K is product (i.e.is given by (<ref>)). Let E'→ M' be a _2-graded Dirac bundle over M' and let (')^+: C^∞(M',(E')^+)→ C^∞(M',(E')^-) be a strongly Callias-type operator, whose restriction toZ_r' is product and such thatM' is an almost compact essential support of (')^+.In the situation discussed above we say that ^+ and (')^+ are compatible if there is an isomorphism E|_Z_r≃ E'|_Z_r' of graded Dirac bundles which identifies the restriction of ^+ to Z_r with the restriction of (')^+ to Z_r'.Let _0 and -_1 be the restrictions of ^+ to N_0 and N_1 respectively (the sign convention means that we thinkofN_0 as the “left boundary” and of N_1 as the “right boundary” of M).Let B_0=H^1/2_(-∞,0)(_0) and B_1=H^1/2_(-∞,0)(-_1)= H^1/2_(0,∞)(_1) be the APS boundary conditions for ^+ at N_0 and N_1 respectively. Since ^+ and (')^+ are equal near the boundary, B_0 and B_1 are also APS boundary conditions for (')^+.We denote by α_ AS(^+) the Atiyah–Singer integrand of ^+. It can be written asα_ AS(^+) :=(2π i)^- M Â(M)ch(E/S)where Â(M)and ch(E/S) arethe differential forms representing the Â-genus of M and the relative Chern character of E, cf.<cit.>. Note that α_ AS(^+) depends only on the metric on M and the Clifford multiplication on E and thus is independent of the potential Ψ. Since outside of a compact set K, M and E are product, the interior multiplication by / t annihilates. Hence, the top degree component ofvanishes on M∖ K. We conclude that the integral∫_Mα_ AS(^+) is well-defined and finite. Similarly, ∫_M'α_ AS((')^+) is well-defined and finite. Suppose ^+ is a strongly Callias-type operator on an oriented even-dimensional essentially cylindrical manifold M such that M is an almost compact essential support of^+. Suppose that the operator(')^+ is compatible with ^+. Let =N_0⊔ N_1 and let B_0= H^1/2_(-∞,0)(_0) and B_1= H^1/2_(-∞,0)(-_1)= H^1/2_(0,∞)(_1) be the APS boundary conditions for ^+ (and, hence, for (')^+) atN_0 and N_1 respectively. Then ^+_B_0⊕ B_1 - ∫_Mα_ AS(^+) = (')^+_B_0⊕ B_1 - ∫_M'α_ AS((')^+).In particular, ^+_B_0⊕ B_1-∫_Mα_ AS(^+) depends only on the restrictions _0 and -_1 of ^+ to the boundary. The rest of the section is devotedto theproof of this theorem. In <cit.> the odd dimensional version of Theorem <ref> was considered. Of course, in this casevanishes identically and Theorem 7.5 of <cit.> states that the indexes of compatible operators are equal. The proof in <cit.> is based on an application of a Callias-type index theorem and can not be adjusted to our current situation. Consequently, a completely different proof is proposed below.§.§ Gluing the data together We follow <cit.> to glueM with M' and ^+ with (')^+.Let -M' denotethemanifold M' with the opposite orientation. We identify a neighborhood of the boundary of -M' with the product -Z_r' :=(N_0×(-r,0] ) ⊔ (N_1×(-r,0] )and consider the union M̃ :=M∪_N_0⊔ N_1 (-M').Then Z_(-r,r):= Z_r∪ (-Z_r') is a subset of M̃ identified with the product (N_0×(-r,r) ) ⊔ (N_1×(-r,r) ).We note that M̃ is a complete Riemannian manifold without boundary. Let E_=E^+_⊕ E^-_ denote the restriction of E=E^+⊕ E^- to . The product structure on E|_Z_r gives a grading-respecting isomorphism ψ:E|_Z_r→ [0,r)× E_. Recall that we identified Z_r with Z_r' and fixed an isomorphism between the restrictions of E to Z_r and E' to Z_r'. By a slight abuse of notation we use this isomorphism to view ψ also as an isomorphism E'|_Z'_r→ [0,r)× E_. Let Ẽ→M̃ be the vector bundle over M̃ obtained by gluing E and E' using the isomorphism c(τ):E|_→ E'|_'. This means that we fix isomorphisms ϕ: Ẽ|_M→ E, ϕ': Ẽ|_M'→ E',so that ψ∘ϕ∘ψ^-1=𝕀: [0,r)× E_ → [0,r)× E_,ψ∘ϕ'∘ψ^-1= 1× c(τ): [0,r)× E_ → [0,r)× E_.Note that the grading of E is preserved while the grading of E' is reversed in this gluing process. Therefore Ẽ=Ẽ^+⊕Ẽ^- is a _2-graded bundle.We denote by c':T^*M'→(E') the Clifford multiplication on E' and set c”(ξ):= -c'(ξ). Then Ẽ is a Dirac bundle over M̃ with the Clifford multiplicationc̃(ξ) : = c(ξ), ξ∈ T^*M;c”(ξ)=-c'(ξ), ξ∈ T^*M'.One readily checks that (<ref>) defines a smooth odd-graded Clifford multiplication on Ẽ. Let D̃:C^∞(M̃,Ẽ)→ C^∞(M̃,Ẽ) be the _2-graded Dirac operator. Then the isomorphism ϕ of (<ref>) identifies the restriction of D̃^± with D^±, the isomorphism ϕ' identifies the restriction of D̃^± with -(D')^∓, and isomorphism ψ∘ϕ'∘ψ^-1 identifies the restriction of D̃^± to -Z_r' withD̃^±|_Z_r' =- c'(τ)∘ (D')^±_Z'_r∘ c'(τ)^-1.Let (Ψ')^± denote the Callias potentials of (')^±, so that (')^±=(D')^±+(Ψ')^±. Consider the bundle maps Ψ̃^±∈ (Ẽ^±,Ẽ^∓) whose restrictions to M are equal to Ψ^± and whose restrictions to M' are equal to -(Ψ')^∓. The two pieces fit well on Z_(-r,r) by Remark <ref>. To sum up the constructions presented in this subsection, we haveThe operators ^± :=D̃^± +Ψ̃^± are strongly Callias-type operators on M̃, formally adjoint to each other, whose restrictions to M are equal to ^± and whose restrictions to M' are equal to -(D')^∓-(Ψ')^∓ = -(')^∓. The operator ^+ is a strongly Callias-type operator on a complete Riemannian manifold without boundary. Hence, <cit.>, it is Fredholm. We again denote by α_ AS(^+) the Atiyah–Singer integrand of ^+. It is explained in the paragraph before Theorem <ref> that the integral ∫_M̃α_ AS(^+) is well defined.^+=∫_M̃α_ AS(^+).Since M̃ is a union of two essentially cylindrical manifolds, there exists a compact essential support K̃⊂M̃ ofsuch that M̃∖K̃ is of the form S^1× N. We can chooseK̃ to be large enough so that the restriction ofto a neighborhood W ofM̃∖K̃≃ S^1× N is a product of an operator on N and an operator on S^1. Then the restriction ofto this neighborhood vanishes. We can also assume that K̃ has a smooth boundary Σ=S^1× L. Let ^+ be a compact perturbation of ^+ (cf. Subsection <ref>) in W which is product both near Σ and on W and whose essential support is contained in K̃. Then ^+=^+.We cutalong Σ and apply the Splitting Theorem <ref>[Since Σ is compact we can also use the splitting theoremfor compact hypersurfaces, <cit.>.] to get^+ = ^+_K̃ + ^+_∖K̃,where ^+_K̃ stands for the index of the restriction of ^+ to K̃ with APS boundary condition, and ^+_∖K̃ stands for the index of the restriction of ^+ to ∖K̃ with the dual APS boundary condition.Since ^+ has an empty essential support in ∖K̃, by the vanishing theorem <cit.>, the second summand in the right hand side of (<ref>) vanishes. The first summand in the right hand side of (<ref>) is given by the Atiyah–Patodi–Singer index theorem <cit.> (Note thatΣ is outside of an essential support of ^+ and, hence, the restriction of ^+ to Σ is invertible. Hence, the kernel of the restriction of ^+ to Σ is trivial) ^+_K̃ = ∫_K̃α_ AS(^+) - 1/2η(0),where η(0) is the η-invariant of the restriction of ^+ to Σ. As (^+)= (^+)≡ 0 on W and ^+≡^+ elsewhere, we have∫_K̃α_ AS(^+)=∫_K̃α_ AS(^+)=∫_α_ AS(^+).To finish the proof of the lemma it suffices now to show η(0)=0.Let ω be the inward (with respect to K̃) unit normal one-form along Σ. Recall that Σ=S^1× L. We denote the coordinate along S^1 by θ. Suppose that {ω,dθ,e_1,⋯,e_m} forms a local orthonormal frame of T^* on Σ. Then the restriction of ^+=D̂^++Ψ̂^+ to Σ can be written as^+_Σ = -∑_i=1^mc̃(ω)c̃(e_i)∇_e_i^Ẽ-c̃(ω)c̃(dθ)∂_θ-c̃(ω)Ψ̂^+which maps C^∞(Σ,Ẽ^+|_Σ) to itself. We define a unitary isomorphism Θ on the space C^∞(Σ,Ẽ|_Σ) given byΘ u(θ,y) := -c̃(ω)c̃(dθ)u(-θ,y).One can check that Θ anticommutes with ^+_Σ. As a result, the spectrum of ^+_Σ is symmetric about 0. Therefore η(0)=0 and lemma is proved.§.§ Proof of Theorem <ref> Recall that we denote by B_0 and B_1 the APS boundary conditions for ^+=^+|_M at N_0 and N_1 respectively. Let (”)^+ denote the restriction of ^+ to -M'=M̃∖ M. Let _0 and _1 be the dual APS boundary conditions for (”)^+ at N_0 and N_1 respectively. By the Splitting Theorem <ref>,^+=^+_B_0⊕ B_1 + (”)^+__0⊕_1.By Lemma <ref>, we obtain ^+_B_0⊕ B_1 + (”)^+__0⊕_1 = ∫_Mα_ AS(^+) + ∫_M'α_ AS((”)^+),which means^+_B_0⊕ B_1 -∫_Mα_ AS(^+) = - (”)^+__0⊕_1 + ∫_M'α_ AS((”)^+). By Lemma <ref>, (”)^+= -(')^-. Thus _0⊕_1 is the adjoint of the APS boundary condition for (-')^+ (cf. Definition <ref>). Therefore,(”)^+__0⊕_1 = (-')^-__0⊕_1 =- (-')^+_B_0⊕ B_1 = -(')^+_B_0⊕ B_1,where we used (<ref>) in the middle equality. Also by the construction of local index density,α_ AS((”)^+) = α_ AS((-')^-) = α_ AS((')^-) =-α_ AS((')^+).Combining these equalities with (<ref>) we obtain (<ref>). □§ THE RELATIVE Η-INVARIANT In the previous section we proved thaton an essentially cylindrical manifold M the difference _B_0⊕ B_1- ∫_M ()depends only on the restriction ofto the boundary, i.e., on theoperators _0 and -_1. In this section we use this fact to define the relative η-invariant η(_1,_0) and show that it has properties similar to the difference of η-invariants η(_1)-η(_0) of operators on compact manifolds. For special cases, <cit.>, when the index can be computed using heat kernel asymptotics, we show thatη(_1,_0) is indeed equal to the difference of the η-invariants of _1 and _0. In the next section we discuss the connection between the relative η-invariant and the spectral flow.In the case when _0, _1 are operators on even-dimensional manifolds, an analogous constructionwas proposed in <cit.>. Even though the definition of the relative η-invariantfor operators on odd-dimensional manifolds proposed in this section is slightly more involved than the definition in <cit.>, we show that most of the properties of η(_1,_0) remain the same. §.§ Almost compact cobordismsLet N_0 and N_1 be two complete odd-dimensional Riemannian manifolds and let _0 and _1 be self-adjoint strongly Callias-type operators on N_0 and N_1 respectively, cf. Definition <ref>.An almost compact cobordism between _0 and _1 is a pair (M,), where M is an essentially cylindrical manifold with =N_0⊔ N_1 andis a graded self-adjoint strongly Callias-type operator on M such that* M is an almost compact essential support of ;*is product near ;* The restriction of ^+ to N_0 is equal to _0 and the restriction of ^+ to N_1 is equal to -_1.If there exists an almost compact cobordism between _0 and _1 we say that operator _0 is cobordant to operator _1.An almost compact cobordism is an equivalence relation on the set of self-adjoint strongly Callias-type operators, i.e., *If _0 is cobordant to _1 then _1 is cobordant to _0.*Let _0,_1 and _2 be self-adjoint strongly Callias-type operators on odd-dimensional complete Riemannian manifolds N_0, N_1 and N_2 respectively.Suppose _0 is cobordant to _1 and _1 is cobordant to _2. Then _0 is cobordant to _2.The proof is a verbatim repetition of the proof of Lemmas 8.2 and 8.3 of <cit.>. Suppose _0 and _1 are cobordant self-adjoint strongly Callias-type operators and let (M,) be an almost compact cobordism between them. Let B_0= H_(-∞,0)^1/2(_0) and B_1= H_(-∞,0)^1/2(-_1) be the APS boundary conditions for ^+. The relative η-invariant is defined as η(_1,_0)= 2 ( ^+_B_0⊕ B_1 - ∫_M (^+) ) +_0 +_1. Theorem <ref>implies thatη(_1,_0) is independent of the choice of the cobordism (M,). Sometimes it isconvenient to use the dual APS boundary conditions _0= H_(-∞,0]^1/2(_0) and _2= H_(-∞,0]^1/2(-_1) instead of B_0 and B_1. It follows from Corollary <ref> that the relative η-invariant can be written asη(_1,_0)= 2 ( ^+__0⊕_1- ∫_M (^+) ) -_0 -_1. §.§ The case when the heat kernel has an asymptotic expansionIn <cit.>, Fox and Haskell studied the index of a boundary value problem on manifolds of bounded geometry. They showed that under certain conditions (satisfied for natural operators on manifolds with conical or cylindrical ends) on M and ,the heat kernel e^-t(_B)^*_B is of trace class and its trace has an asymptotic expansion similar to the one on compact manifolds. In this case the η-function, defined by a usual formula η(s;):=∑_λ∈ spec() sign(λ) |λ|^s, Re s ≪ 0, is an analytic function of s, which has a meromorphic continuation to the whole complex plane and is regular at 0. So one can define the η-invariant ofby η()= η(0;).Suppose now thatis an operator on an essentially cylindrical manifold M which satisfies the conditions of<cit.>. We also assume thatis product near =N_0⊔ N_1 and that M is an almost compact essential support of . Let _0 and -_1 be the restrictions of ^+ to N_0 and N_1 respectively. Let η(_j) (j=0,1) be the η-invariant of _j. Then η(_1,_0)=η(_1) -η(_0).An analogue of this proposition for the case when M is odd is proven in <cit.>. This proof extends to the case when M is even without any changes.§.§ Basic properties of the relative η-invariantProposition <ref> shows that under certain conditions the η-invariants of _0 and _1 are defined and η(_1,_0) is their difference. We now show that in general case, when η(_0) and η(_1) do not necessarily exist,η(_1,_0) behaves like it were a difference of an invariant of N_1 and an invariant of N_0. Suppose _0 and _1 are cobordant self-adjoint strongly Callias-type operators. Then η(_0,_1) =- η(_1,_0).Let -M denote the manifold M with the opposite orientation and let M̃:= M∪_(-M) denote the double of M. Letbe an almost compact cobordism between _0 and _1. Using the construction ofSection <ref> (with '=) we obtain a graded self-adjoint strongly Callias-type operatoron M̃ whose restriction to M is isometric to . Let ” denote the restriction ofto -M=M̃∖M. Then the restriction of (”)^+ to N_1 is equal to _1 and the restriction of (”)^+ to N_0 is equal to -_0.Let _0 =H^1/2_[0,∞)(_0) = H^1/2_(-∞,0](-_0), _1 = H^1/2_[0,∞)(-_1)=H^1/2_(-∞,0](_1) be the dual APS boundary conditions for (”)^+.By (<ref>), (”)^+__0⊕_1 - ∫_M'α_ AS((”)^+) = -^+_B_0⊕ B_1 +∫_Mα_ AS(^+). Since ” is an almost compact cobordism between _1 and _0 we concludefrom (<ref>) thatη(_0,_1) = 2 ((”)^+__0⊕_1- ∫_M'α_ AS((”)^+) ) -_0 -_1.Combining(<ref>) and (<ref>) we obtain (<ref>). Note that (<ref>) implies that η(,) =0for every self-adjoint strongly Callias-type operator . Let _0,_1 and _2 be self-adjoint strongly Callias-type operators which are cobordant to each other. Then η(_2,_0) =η(_2,_1) +η(_1,_0).Let M_1 and M_2 be essentiallycylindrical manifolds such that _1= N_0⊔ N_1 and _2= N_1⊔ N_2. Let _1 be an operator on M_1 which is an almost compact cobordism between _0 and _1. Let _2 be an operator on M_2 which is an almost compact cobordism between _1 and _2. Then the operator _3 on M_1∪_N_1 M_2 whose restriction to M_j (j=1,2) is equal to _j is an almost compact cobordism between _0 and _2. Let B_0 and B_1 be the APS boundary conditions for _1^+ at N_0 and N_1 respectively. Then _1= H^1/2_[0,∞)(_1) is equal to the dual APS boundary condition for _2^+. Let B_2 be the APS boundary condition for _2^+ at N_2. From Corollary <ref> we obtain η(_2,_1)= 2 ((_2^+)__1⊕ B_2- ∫_M (_2^+) ) -_1 +_2.By the Splitting Theorem <ref>(_3^+)_B_0⊕ B_2 = (_1^+)_B_0⊕ B_1 + (_2^+)__1⊕ B_2.Clearly, ∫_M_1∪ M_2(_3^+) =∫_M_1 (_1^+) +∫_M_2 (_2^+).Combining (<ref>), (<ref>), and (<ref>) we obtain (<ref>). § THE SPECTRAL FLOW Suppose Å:= {^s}_0≤ s≤ 1 is a smooth family of self-adjoint elliptic operators on a closedmanifold N.Let (^s)∈/ denote the modreduction of the η-invariant η(^s).Atiyah, Patodi, and Singer, <cit.>, showed that s↦(^s) is a smooth function whose derivative d/ds(^s) is given by an explicit local formula.Further, Atiyah, Patodi and Singer, <cit.>, introduced a notion of spectral flow (Å) and showed that itcomputes the net number of integer jumps of η(^s), i.e., 2 (Å) =η(^1) -η(^0) -∫_0^1 (d/ds(^s) )ds.In this section we consider a family of self-adjoint strongly Callias-type operators Å= {^s}_0≤ s≤ 1 on a completeRiemannian manifold. Assuming that the restriction of ^s to a complement of a compact set K⊂ N is independent ofs, we show that for any operator _0 cobordant to ^0themod reduction (^s,_0) of the relative η-invariant depends smoothly on s and 2 (Å) =η(^1,_0)- η(^0,_0) -∫_0^1 (d/ds(^s,_0) )ds. §.§ A family of boundary operatorsLet E_N→ N be a Dirac bundle over a complete odd-dimensional Riemannian manifold N. We denote the Clifford multiplication of T^*N on E_N by c_N:T^*N→(E_N). Let Å= {^s}_0≤ s≤ 1 be a family of self-adjoint strongly Callias-type operators ^s:C^∞(N,E_N) → C^∞(N,E_N). The family Å= {^s}_0≤ s≤ 1 is called almost constant if there exists a compact set K⊂ N such that the restriction of ^s to N∖ K is independent of s.Consider the cylinder M:=[0,1]× Nand denote by t the coordinate along [0,1]. Set E^+= E^- :=[0,1]× E_N.Then E=E^+⊕ E^-→ M is naturally a _2-graded Dirac bundle over M with c(dt):=[0 -𝕀_E_N;𝕀_E_N0 ]and c(ξ) : =[0 c_N(ξ); c_N(ξ)0 ], forξ∈ T^*N. The family Å= {^s}_0≤ s≤ 1 is called smooth if: =c(dt) ( _t+ [^t 0; 0 -^t ] ): C^∞(M,E)→ C^∞(M,E)is a smooth differential operator on M. Fix a smooth non-decreasing function κ:[0,1]→ [0,1] such that κ(t)=0 for t≤1/3 and κ(t)=1 for t≥2/3 and consider the operator : =c(dt) ( _t+ [^κ(t)0;0 -^κ(t) ] ): C^∞(M,E)→ C^∞(M,E).Thenis product near . If Å is a smooth almost constant family of self-adjoint strongly Callias-type operators then (<ref>) is a strongly Callias-type operator for which M is an almost compact essential support. Hence it is a non-compact cobordism (cf. Definition <ref>) between ^0 and ^1. §.§ The spectral section If Å= {^s}_0≤ s≤ 1 is a smooth almost constant family of self-adjoint strongly Callias-type operators then it satisfies the conditions of the Kato Selection Theorem <cit.>, <cit.>. Thus there is a family of eigenvalues λ_j(s) (j∈)which depend continuously on s. We order the eigenvalues so that λ_j(0)≤λ_j+1(0) for all j∈ and λ_j(0)≤ 0 for j≤ 0 while λ_j(0)>0 for j>0. Atiyah, Patodi and Singer <cit.> defined the spectral flow (Å) for a family of operators satisfying the conditions of the Kato Selection Theorem(<cit.>, <cit.>) as an integer that counts the net number of eigenvalues that change sign when s changes from 0 to 1.Several other equivalent definitions of the spectral flow based on different assumptions on the family Å exist in the literature. For our purposes the most convenient is theDai and Zhang's definition <cit.> which is based on the notion of spectral section introduced by Melrose and Piazza <cit.>. A spectral section for Å is a continuous family = {P^s}_0≤ s≤ 1 of self-adjoint projections such that there exists a constant R>0 such that for all 0≤ s≤ 1, if ^su= λ u thenP^su= 0,ifλ< -R; u,ifλ> R.If Å satisfies the conditions of the Kato Selection Theorem, then the arguments of the proof of<cit.> show that Å admits a spectral section. §.§ The spectral flowLet = {P^s} be a spectral section for Å. Set B^s:=P^s. Let B_0^s:= H^1/2_(-∞,0)(^s) denote the APS boundary condition defined by the boundary operator ^s. Since the spectrum of ^s is discrete, it follows immediately from the definition of the spectral section that for every s∈[0,1] the space B^s is a finite rank perturbation of B_0^s, cf. Section <ref>. Recall thatthe relative index [B^s,B_0^s] was defined in Definition <ref>. Following Dai and Zhang <cit.> (see also <cit.>) we give the following definition. Let Å= {^s}_0≤ s≤ 1 be a smooth almost constantfamily of self-adjoint strongly Callias-type operators which admits a spectral section = {P^s}_0≤ s≤ 1. Assume that the operators ^0 and ^1 are invertible. Let B^s:=P^s and B_0^s:= H^1/2_(-∞,0)(^s). The spectral flow (Å) of the family Å is defined by the formula(Å) :=[B^1,B_0^1]-[B^0,B^0_0]. By Theorem 1.4 of <cit.> the spectral flow is independent of the choice of the spectral sectionand computes the net number of eigenvalues that change sign when s changes from 0 to 1. Let -Å denote the family {-^s}_0≤ s≤ 1. Then (-Å)=-(Å).The lemma is an immediate consequence of Lemma 5.7 of <cit.>. §.§ Deformation of the relative η-invariant Let Å= {^s}_0≤ s≤ 1 be a smooth almost constant family of self-adjoint strongly Callias-type operators on a complete odd-dimensional Riemannian manifold N_1. Let _0 be another self-adjoint strongly Callias-type operator, which is cobordant to ^0. In Section <ref> we showed that^0 is cobordant to ^s for all s∈ [0,1]. Hence, by Lemma <ref>.(ii), _0 is cobordant to ^1. In this situation we say the _0 is cobordant to the family Å.The following theorem is the main result of thissection.SupposeÅ={^s: C^∞(N_1,E_1)→ C^∞(N_1,E_1)}_0≤ s≤ 1 is a smooth almost constant family of self-adjoint strongly Callias-type operators on a complete odd-dimensional Riemannian manifold N_1. Assume that ^0 and ^1 are invertible. Let _0:C^∞(N_0,E_0)→ C^∞(N_0,E_0) be an invertible self-adjoint strongly Callias-type operator on a complete Riemannian manifold N_0 which is cobordant to the family Å. Then the modreduction (^s,_0)∈/of the relative η-invariant depends smoothly on s∈[0,1] and η(^1,_0) -η(^0,_0) -∫_0^1 (d/ds(^s,_0)) ds =2 (Å).The proof of this theorem occupies Sections <ref>–<ref>.§.§ A family of almost compact cobordisms Let M be an essentially cylindrical manifold whose boundary is the disjoint union of N_0 and N_1. First, we construct a smooth family ^r (0≤ r≤1) of graded self-adjoint strongly Callias-type operators on the manifoldM' :=M∪_N_1([0,1]× N_1 ),such that for each r∈ [0,1] the pair (M',^r) is an almost compact cobordism between _0 and ^r.Let :C^∞(M,E)→ C^∞(M,E) be an almost compact cobordism between _0 and ^0. Let E_0 and E_1 denote the restrictions of E to N_0 and N_1 respectively. Let M' be given by (<ref>) and let E'→ M' be the bundle over M' whose restriction to M is equal to E and whose restriction to the cylinder [0,1]× N_1 is equal to [0,1]× E_1. We fix a smooth function ρ:[0,1]× [0,1]→ [0,1] such that for each r∈ [0,1] * the function s↦ρ(r,s) is non-decreasing.* ρ(r,s)=0 for s≤1/3 and ρ(r,s)= r for s≥2/3.Consider the family of strongly Callias-type operators ^r:C^∞(M',E')→ C^∞(M',E') whose restriction to M is equal toand whose restriction to [0,1]× N_1 is given by ^r : =c(dt) ( _t+ [^ρ(r,t)0;0 -^ρ(r,t) ] ).Then ^r is an almost compact cobordism between _0 and ^r. In particular, the restriction of ^r to N_1 is equal to -^r. Recall that we denote by -Å the family {-^s}_0≤ s≤ 1. Let = {P^s} be a spectral section for -Å. Then for each r∈ [0,1] the spaceB^r:=P^r is a finite rank perturbation of the APS boundary conditionfor ^r at {1}× N_1. Let B_0:= H^1/2_(-∞,0)(_0) be the APS boundary condition for ^r at N_0. Then, by Proposition <ref>, the operator^r_B_0⊕ B^r is Fredholm. Recall that the domain ^r_B_0⊕ B^r consists of sections u whose restriction to '=N_0⊔ N_1 lies in B_0⊕ B^r.^r_B_0⊕ B^r= ^1_B_0⊕ B^1 for all r∈ [0,1].For r_0,r∈ [0,1], let π_r_0r: B^r_0→ B^r denote the orthogonal projection. Then for every r_0∈[0,1] there exists ε>0 such that if |r-r_0|<ε then π_r_0r is an isomorphism. As in the proofs of <cit.> and <cit.>, it induces an isomorphismΠ_r_0r : ^r_0_B_0⊕ B^r_0 → ^r_B_0⊕ B^r.Hence(^r_B_0⊕ B^r∘Π_r_0r) =^r_B_0⊕ B^r.Since for |r-r_0|<ε^r_B_0⊕ B^r∘Π_r_0r: ^r_0_B_0⊕ B^r_0 → L^2(M',E')is a continuous family of bounded operators, ^r_B_0⊕ B^r∘Π_r_0r is independent of r. The lemma follows now from (<ref>).§.§ Variation of the reduced relative η-invariant By Definition <ref>, the modreduction of the relative η-invariant is given by (^r,_0) :=-2 ∫_M' (^r). It follows that (^r,_0) depends smoothly on r and d/dr(^r,_0) =-2 ∫_M' d/dr(^r).A more explicit local expression for the right hand side of this equation is given in Section <ref>. For the moment we just note that (<ref>) implies that∫_0^1 (d/ds(^s,_0)) ds= -2 ∫_M' ((^1)- (^0) ). §.§ Proof of Theorem <ref> Since the operators _0, ^0, and ^1 are invertible, we have η(^j,_0)= 2 ( ^j_B_0⊕ B^j_0-∫_M' (^j) ),j=0,1.Thus, using (<ref>), we obtainη(^1,_0)- η(^0,_0) -∫_0^1 (d/ds(^s,_0) )ds = 2 ( ^1_B_0⊕ B^1_0 -^0_B_0⊕ B^0_0 ). Recall that, by Proposition <ref>,^r_B_0⊕ B^r=^r_B_0⊕ B^r_0 +[B^r,B^r_0],where B^r=P^r and B^r_0= H^1/2_(-∞,0)(^r) are defined in Subsection <ref>. Hence, from (<ref>) we obtain1/2( η(^1,_0)- η(^0,_0) -∫_0^1 (d/ds(^s,_0) )ds) =( ^1_B_0⊕ B^1- [B^1,B^1_0] )- ( ^0_B_0⊕ B^0- [B^0,B^0_0] ) Lemma <ref>= -[B^1,B^1_0] +[B^0,B^0_0]=-(-Å) Lemma <ref>=(Å).□ §.§ A local formula for variation of the reduced relative η-invariantIt is well known that there exists a family ofdifferential forms β_r (0≤ r≤ 1), called the transgression form such that dβ_r =d/dr(^r).The transgression form depends on the symbol of ^r and its derivatives with respect to r. For geometric Dirac operators one can write very explicit formulas for β_r. For example, if ^r is the signature operator (so that ^r is the odd signature operator) corresponding to a family ^r of flat connections on E,then β_r= L(M)∧d/dr^r, where L(M) is theL-genus of M, cf, for example, <cit.>. For general Dirac-type operators, a formula for β_r is more complicated, cf. <cit.>.We note that since the family ^r is constant outside of the compact set K, the form β_r vanishes outside of K. Hence, ∫_'β_r is well defined and finite. Thus we obtain from (<ref>) that d/dr(^r,_0)=-2∫_M' dβ_r = -2∫_' β_r= 2 (∫_{1}× N_1β_r-∫_N_0 β_r).Hence, (<ref>) expresses η(^1,_0)- η(^0,_0) as a sum of 2(Å) and a local differential geometric expression 2∫_'(∫_0^1β_r) dr. amsplain Anghel90article author=Anghel, N.,title=L^2-index formulae for perturbed Dirac operators, date=1990, ISSN=0010-3616,journal=Comm. Math. 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Phys., volume=100, number=2,pages=197229,url=http://projecteuclid.org.ezproxy.neu.edu/euclid.cmp/1103943444, | http://arxiv.org/abs/1708.08336v3 | {
"authors": [
"Maxim Braverman",
"Pengshuai Shi"
],
"categories": [
"math.DG",
"math.AP",
"58J28, 58J30, 58J32, 19K56"
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"primary_category": "math.DG",
"published": "20170824192621",
"title": "APS index theorem for even-dimensional manifolds with non-compact boundary"
} |
^1Centre for Astrophysics and Relativity, School of Mathematical SciencesDublin City University,Glasnevin, Dublin 9, Ireland^2Cornell Center for Astrophysics and Planetary Science, Cornell University, Ithaca, NY 14853^3Department of Physics, Cornell University, Ithaca, NY 14853== by 60by 60by 60 =- by :<100Bodies coupled to electromagnetic or other long-range fields are subject to radiation reaction and other effects in which their own fields can influence their motion. Self-force phenomena such as these have been poorly understood for spacetime dimensions not equal to four, despite the relevance of differing dimensionalities for holographic duals, effectively two-dimensional condensed matter and fluid systems, and so on. We remedy this by showing that forces and torques acting on extended electromagnetic charges in all dimensions d ≥ 3 have the same functional form as the usual test body expressions, except that the electromagnetic field appearing in those expressions is not the physical one; it is an effective surrogate. For arbitrary even d ≥ 4, our surrogate field locally satisfies the source-free field equations, and is conceptually very similar to what arises in the Detweiler-Whiting prescription previously established when d=4. The odd-dimensional case is different, involving effective fields which are not necessarily source-free. Moreover, we find a 1-parameter family of natural effective fields for each odd d, where the free parameter—a lengthscale—is degenerate with (finite) renormalizations of a body's stress-energy tensor. While different parameter choices can result in different forces, they do so without affecting physical observables. Having established these general results, explicit point-particle self-forces are derived in odd-dimensional Minkowski spacetimes. Simple examples are discussed for d=3 and d=5, one of which illustrates that the particularly slow decay of fields in three spacetime dimensions results in particles creating their own “preferred rest frames:” Initially-static charges which are later perturbed have a strong tendency to return to rest. While the focus here is on the electromagnetic self-force problem, all results are easily carried over to the scalar and gravitational cases. 1pcSelf-forces in arbitrary dimensions Abraham I. Harte^1, Peter Taylor^1 and Éanna É. Flanagan^2,3 December 30, 2023 ================================================================The motion of small bodies is central to some of the most enduring problems in physics. If such a body is coupled to an electromagnetic, gravitational, or other long-range field, it may be subject to net forces exerted by its own contributions to that field. This “self-force” strongly influences, for example, charged particles circulating in particle accelerators and the shrinking orbits of black hole binaries due to the emission of gravitational radiation. Despite its initial appearance of simplicity, the study of self-interaction involves a number of physical and mathematical subtleties. This has led to more than a century of literature on the electromagnetic self-force; see <cit.>.While impetuses for these works have varied considerably, the past two decades have seen a concerted effort—motivated largely by gravitational wave astronomy—to understand the gravitational self-force problem in general relativity. This has led to a number of theoretical and computational advances which considerably improve our understanding of self-interaction, in both the gravitational and electromagnetic contexts <cit.>. Separately, new aspects of the electromagnetic self-force are also beginning to be accessible to investigation via high-power laser experiments <cit.>.Here, we move beyond the existing literature to rigorously study self-interaction in different numbers of dimensions. There are three reasons for this: First, considerations in different numbers of dimensions refines our understanding of precisely what is important and what is not; lessons learned in this way may significantly inform future considerations even in four dimensions, particularly in more complicated theories which have not yet been understood. Second, considerations of theories in non-physical numbers of dimensions can, via holographic dualities, be related to ordinary four-dimensional systems; for example, the five-dimensional self-force might be used to understand jet quenching in four-dimensional dimensional quark-gluon plasmas <cit.>. Our final reason for considering different numbers of dimensions is that the self-force in three spacetime dimensions may be proportionally stronger than in four dimensions, both in terms of instantaneous magnitude <cit.> and—as argued below—in the particularly slow decay of fields which encode a kind of “memory” of a system's past. Moreover, systems in which is this is relevant may be accessible to experiment. For example, “pilot wave hydrodynamics” involves a number of striking phenomena observed to be associated with oil droplets bouncing on a vibrating bath <cit.>. Each bounce generates surface waves on the bath, but these waves also affect the horizontal motion of the droplet. This type of feedback with a long-range field (the surface waves) is reminiscent of a self-force problem in two spatial dimensions. Separately, there are a variety of condensed matter systems which act as though they are confined to one or two spatial dimensions <cit.>.We do not consider any particular fluid or condensed matter system in this Letter, but instead explore a standard electromagnetic self-force problem in different numbers of dimensions. To the best of our knowledge, the literature does not contain any rigorous derivations of self-force other than in four dimensions, except for recent work restricted to static bodies <cit.>. The dynamical case considered here is considerably different and more rich. Moreover, we find qualitative differences between charged-particle behavior in even and odd numbers of dimensions.Our strategy is not to obtain a “point particle self-force” as any kind of fundamental object, but instead to derive laws of motion for extended objects, and then to evaluate point-particle limits of those laws. Although we focus for concreteness on the electromagnetic self-force problem, analogous results are easily obtained for the scalar and (at least first order) gravitational self-force. In the electromagnetic context, a compact body in a d-dimensional spacetime (M,g_ab) is associated with a conserved current density J_a, and the electromagnetic field F_ab in a neighborhood of that body satisfies the Maxwell equations∇_[a F_bc] = 0, ∇^b F_ab = ω_d-1 J_a,where ω_D≡ 2 π^D/2 / Γ(D/2) is chosen by convention to be the area of a unit sphere in ℝ^D. We also assume that the system's stress-energy tensor is conserved, ∇_b T^ab = 0, and that it reduces immediately outside the body of interest to the standard expression T^ab_em[F_cd] for the stress-energy associated with F_ab in vacuum. Using that expression also in the interior of the body, the body's “own” stress-energy, T^ab_body≡ T^ab - T^ab_em, must satisfy ∇_b T^ab_body = F_ab J^b. Every portion of an extended charge is thus acted upon by the Lorentz force density F_ab J^b.The question we now ask is how this density “integrates up” to affect a body's overall motion. One difficulty is that F_ab J^b depends nonlinearly and nonlocally on J_a, and can be almost arbitrarily complicated. Despite this, experience suggests that there are physically-interesting regimes in which the (appropriately-defined) net force is not complicated at all, implying laws of motion which are effectively universal. Deriving this universality and making it precise is the essence of the self-force problem. The approach adopted here uses a formalism developed by one of us <cit.>, which provides a rigorous framework with which to analyze problems of motion in a wide variety of contexts. Crucially, most of that framework is agnostic to the number of dimensions. To review it briefly, one first defines “bare” linear and angular momenta for an extended body as particular integrals over the body's interior; we take these definitions to coincide with Dixon's <cit.>. Next, stress-energy conservation is used to derive forces and torques—rates of change of the momenta—which result in integrals for each force or torque component. Rather than displaying these integrals explicitly, we note that each force or torque component is a bilinear functional with the form ℱ_τ[ F_ab, J_c ], where τ provides a parametrized notion of time. Unless self-fields are negligible, these functionals are difficult to analyze as-is. The formalism we use nevertheless provides a set of tools which allows one to easily derive thatℱ_τ [ F_ab , J_c ] = ℱ_τ [ F̂_ab, J_c ] - δ𝒯_τ[ J_a ]for a wide class of nonlocal transformations F_ab↦F̂_ab. The difference term δ𝒯_τ [ J_a ] depends nonlinearly on its argument and has properties which allow it to be absorbed into a redefinition, or finite renormalization, of a body's stress-energy tensor. In flat spacetime, these renormalizations are relevant only for the body's linear and angular momenta. More generally, quadrupole and higher higher moments of the renormalized stress-energy tensor can also arise in the laws of motion, where they appear in gravitational forces and torques <cit.>. Regardless, results of this type are useful because, if F̂_ab varies slowly throughout a body's interior, ℱ_τ[ F̂_ab, J_c ] can be evaluated directly in terms of the multipole moments of J_a. Laws of motion for the renormalized momenta then arise which are structurally identical to those satisfied (instantaneously) by a test body moving in the effective field F̂_ab. All d=4 self-force results—whether in electromagnetism, scalar field theory, or general relativity—are organized today via similar statements, commonly referred to as Detweiler-Whiting principles <cit.>. We extend such principles here for all d ≥ 3. If d is even, there is no significant difference with prior work. If d is odd, a crucial difference appears in the form of F̂_ab.Before describing this, we first note that a particularly simple Detweiler-Whiting principle holds even in Newtonian gravity <cit.>, where it provides the foundation for Newtonian celestial mechanics: Letting ϕ_S denote the self-field associated with a body of mass density ρ, each force or torque component ℱ_τ^N [ϕ,ρ] may be shown to equal ℱ_τ^N [ϕ - ϕ_S,ρ]; the analog of Eq. (<ref>) holds with δ𝒯^N_τ[ρ] = 0. Self-gravitating bodies thus follow laws of motion which instantaneously coincide with those of a test body in the effective external potential ϕ̂≡ϕ - ϕ_S. We generalize this to electrodynamics by introducing a particular two-point distribution, or “propagator”G_aa'(x,x') for the vector potential, which plays a role analogous to the Green function G_N(x,x') = -|x - x'|^-1 which appears in the definition for the Newtonian self-field. In particular, our electromagnetic propagator is used to produce the “effective electromagnetic field” F̂_ab(x) ≡ F_ab (x) - 2 ∫∇_[a G_a]a'(x,x') J^a'(x') dV'.As it stands, this is merely a definition; it becomes interesting only for propagators with particular properties.If d=4, an appropriate G_aa' is known to be singled out by three constraints known as the Detweiler-Whiting axioms <cit.>. As noted also in <cit.>, these constraints require no modification for any even d ≥ 4. Eq. (<ref>) may be shown to hold and the Detweiler-Whiting principle remains valid. Explicitly, the propagator implied by this construction is a Green function for Maxwell's equations with the formG_aa' = 1/2[ U_aa'δ^(d/2-1)( σ ) +V_aa' Θ(σ) ],where σ = σ(x,x') denotes Synge's world function, one half of the squared geodesic distance between its arguments, and U_aa' and V_aa' are smooth bitensors, extensions of which also appear in the retarded Green function. The leading-order force acting on a sufficiently small body with charge q and d-velocity γ̇^a is the usual Lorentz expression q F̂_abγ̇^b, where F̂_ab is obtained from F_ab by substituting (<ref>) into (<ref>). All dipole and higher-order terms also appear in their usual forms. The remainder of this Letter focuses on odd d, cases which have not previously been understood (except in the static context <cit.>). The crux of the complication is that there is no odd-dimensional distribution which satisfies the Detweiler-Whiting axioms. Nevertheless, the formalism used to establish the suitability of those axioms in even dimensions makes it clear that there is considerable freedom to change them. Suppose in particular that a propagator may be found with the properties* G_aa' (x,x') = 0 for all timelike-separated x, x', * G_aa' (x,x') = G_a'a (x',x), * G_aa'(x,x') is constructed only from the geometry and depends quasilocally on the metric, *The source ω_d-1^-1∇^b F̂_ab for the effective field is smooth for any monopole point charge moving on a smooth timelike worldline, at least in a neighborhood of that worldline.The first two of these axioms are shared by the usual Detweiler-Whiting approach. Axiom 1 ensures that the effective momenta are physical in the sense that they depend only quasilocally on the body's state, while Axiom 2 describes a type of reciprocity in the self-field definition associated with G_aa'<cit.>. The third axiom is similar to one proposed in <cit.>, and demands more precisely that for any ψ^a, the Lie derivative ℒ_ψ G_aa'(x,x') can be written as a functional which depends only on the Lie derivative of the metric, and only in a compact region determined by x and x'. If considerations are restricted to a single flat spacetime, Axiom 3 may be simplified by demanding only that G_aa' be Poincaré-invariant. Regardless, Axioms 2 and 3 imply (<ref>).Axiom 4 is essentially a point-particle limit of the more physical requirement that simple laws of motion can result only if F̂_ab varies slowly throughout a body's interior, at least for sufficiently well-behaved initial data (as is required even for the familiar multipole expansions of Newtonian celestial mechanics). More directly, Axiom 4 ensures that ℱ_τ[F̂_ab,J_c] is generally simpler to evaluate than its bare counterpart ℱ_τ[F_ab,J_c]. Although it is satisfied if G_aa' is a parametrix for Maxwell's equations, the converse of this statement is false—a distinction which is crucial to our development.If any G_aa' can be found that satisfies the above four axioms, it may be used to write down momenta which obey simple laws of motion in the associated effective field F̂_ab. The Detweiler-Whiting propagators (<ref>) are explicit examples for all even d≥ 4. Before explaining our solution for odd d, note that the retarded Green function in those cases has the form G_aa'^ret = U_aa' (-2σ)^1-d/2Θ(-σ) for some smooth U_aa', at least within a convex normal neighborhood. Using g_aa' to denote the parallel propagator,U_aa' = α_d g_aa' , α_d ≡(-1)^1/2(d-3)Γ(d/2-1)/√(π)Γ(1/2(d-1))in odd-dimensional Minkowski spacetimes. Whether in Minkowski spacetime or not, Huygens' principle isviolated; signals travel not only along null cones, but also inside of them. Although this occurs also in curved even-dimensional spacetimes, the odd-dimensional case is different in that G_aa'^ret is unbounded even for timelike-separated events. Indeed, it is not even integrable in general. Some care is thus required to define the retarded Green function as a distribution. The correct result may be described by considering U_aa' (-2σ)^λΘ(-σ) when λ is chosen such that the singularity is integrable and then analytically continuing the associated distribution to λ→ 1-d/2<cit.>. Our odd-dimensional propagators are also defined using analytic continuation:G_aa' = β_d lim_λ→ 1-d/2ℓ^2λ∂/∂λ[ ℓ^-2λ U_aa' (2 σ)^λΘ (σ)].These actually constitute a 1-parameter family of propagators, parametrized by an arbitrary lengthscale ℓ > 0 which ensures that quantity being differentiated is dimensionless. Varying ℓ results in effective fields which differ by multiples of a vacuum solution, and although these variations genericallychange the effective field and also the force, those forces are associated, via the δ𝒯_τ in (<ref>), with different effective momenta. Physically, ℓ parametrizes different ways to describe the same physical system. As explained in more detail in <cit.>, true observables do not depend on it.Noting that U_aa' is symmetric in its arguments and quasilocally constructed from the geometry, it is immediately evident that G_aa' satisfies our first three axioms. The fourth axiom may be verified by direct computation to hold whenβ_d =(-1)^1/2 (d-1)/ 2π .Adopting this value establishes a Detweiler-Whiting principle which holds non-perturbatively for extended charges in all odd dimensions. Test body laws of motion hold to all multipole orders, and involve the field F̂_ab which is found by substituting (<ref>) into (<ref>).Our propagator G_aa' is not a Green function or even a parametrix. Some intuition for it may nevertheless be gained by noting that the derivative with respect to λ which appears in its definition evinces a procedure which “infinitesimally varies d.” This suggests that our map F_ab↦F̂_ab may reduce to dimensional regularization in a point particle limit, and may provide an underlying physical and mathematical origin for that procedure at least in the present context.Next, we obtain a point particle limit for the force in the sense described in <cit.>. The limiting effective field in this setting is described at leading order by the field of a point mass with charge q moving on a worldline γ(τ) parametrized via its proper time τ. Although there is no conceptual difficulty with considering more general cases, we focus for concreteness on Minkowski spacetime and adopt the inertial coordinates x^μ. Then g_μμ' = diag(-1,1,⋯,1), and if F_ab is assumed to equal the particle's retarded field, a rather involved calculation using (<ref>),(<ref>), and (<ref>) shows thatF̂_μν (γ(τ)) = 2 α_d q lim_ϵ→ 0^+[ (d-2) ∫^τ-ϵ_-∞ X_[μ(τ,τ') γ̇_ν] (τ')/ [-X^2(τ,τ')]^d/2 dτ' - ∑_n=0^d-4 (-1)^n / n!( ∇_[μ W_ν]^(n)/ d-3-n+ 1/ϵγ̇_[μ W_ν]^(n)) 1/ϵ^d-3-n + 1/(d-3)!( -1/ϵγ̇_[μ W_ν]^(d-3)+ ∇_[μ W_ν]^(d-3)ln (ϵ/ℓ) + 1/2∂_λ∇_[μ W_ν]^(d-3) + 1/(d-2)γ̇_[μ W_ν]^(d-2)) ]for any odd d ≥ 3, where X_μ(τ,τ') ≡γ_μ(τ) - γ_μ(τ'). If the physical field is not exactly the retarded one, any differences F_ab - F_ab^ret are simply added to the right-hand side of (<ref>). All counterterms here—which are derived from (<ref>), not put in by hand—are expressed in terms ofW_μ^(n)(x;λ) ≡. ∂^n/∂τ^n [ γ̇_μ (τ) Σ^λ (x,τ)] |_τ=1/2 (τ_++τ_-)and its gradients evaluated at (γ(τ);1-d/2), where τ_±(x) denote the advanced and retarded proper times associated with x. The smooth function Σ(x,τ) arises in the factorization 2σ(γ(τ), x) = [τ_+(x) - τ][ τ - τ_-(x)] Σ(x,τ), and reduces to unity for an unaccelerated particle. The overall limit here is well-defined, so the effective field is finite on the particle's worldline. More than this, it may be shown that F̂_μν(x) is smooth, as claimed, at least for x in a neighborhood of the particle's worldline; no infinities ever arise. Also note that even though G_aa' is not a Green function and the effective field here is not in general a solution to the source-free Maxwell equations, it is source-free for inertially-moving particles. Indeed, it vanishes in these cases.The leading-order self-force acting on a small charge is now contained in f_a = q F̂_abγ̇^b. Similarly, the lowest order self-torque acting on a particle with dipole moment q^ab = q^[ab] follows from n_ab = 2 q^c_[aF̂_b]c. Full laws of motion nevertheless require a centroid or spin supplementary condition, from which a (possibly nontrivial) momentum-velocity relation must be derived. Forces and torques involving higher multipole moments may also be more significant than those which involve the self-force or self-torque. These issues are discussed in detail in <cit.>. Here, we focus only on the Lorentz force associated with F̂_ab in odd dimensions.Evaluating (<ref>) and substituting the results into (<ref>), the leading-order d=3 self-force is explicitlyf_μ = 2 q^2 ( ∫^τ-ϵ_-∞X_[μγ̇_ν]' γ̇^ν/ (-X^2)^3/2 dτ' - 1/4ln(ϵ/e ℓ) γ̈_μ),where the limit ϵ→ 0^+ has been left implicit and e is the base of the natural logarithm. Different choices for ℓ result in different momenta, and thus different forces. It is clear in this context that the corresponding momentum differences involve only changes in magnitude, i.e., differing effective masses. The equivalent expression for d=5 is somewhat more complicated: Letting h_μν≡η_μν + γ̇_μγ̇_ν denote the spatial projection operator,f_μ = - q^2( ∫_-∞^τ-ϵ3 X_[μγ̇_ν]' γ̇^ν/ (-X^2)^3/2 dτ'+h_μν[ 3γ̈^ν/8ϵ^2 - ⃛γ^ν/2ϵ - 3/16 (⃜γ^ν - 3/2 |γ̈|^2 γ̈^ν ) ln ( ϵ/e^1/3ℓ) - |γ̈|^2 γ̈^ν/32] ).Changing ℓ thus renormalizes more than just the mass; it affects both the direction and magnitude of the 5-momentum.A force similar to (<ref>) has recently been obtained using the methods of effective field theory <cit.>, and in that context, ℓ appears as a free parameter in a dimensional regularization procedure. It is interpreted there as a lengthscale over which measurements may be performed, an interpretation which does not arise in our framework.We now consider the nonrelativistic limits of the d=3 and d=5 self-forces. The leading-order terms are purely spatial, and in three dimensions, repeatedly integrating by parts results in f(τ) = -1/2 q^2 ∫_-∞^τ⃛γ(τ')ln ( τ-τ'/ e^1/2ℓ)dτ' ,where it has been assumed that the particle's motion is inertial in the distant past. The d=5 case is similar, but with the third derivative of the particle's position replaced by the fifth derivative (and with different constants). Focusing on three dimensions for definiteness, the self-force is seen to depend on the past history of the particle's jerk, with a weighting factor for this history which increases in the increasingly distant past. If a charge is initially stationary, is subjected to a brief impulse near τ=τ_0, and moves inertially thereafter, f(τ) = -1/2 q^2 Δv/(τ-τ_0) at late times, where Δv denotes the velocity change associated with the impulse. This scenario requires a small external force to be applied after the initial impulse in order to counteract the self-force. If that force is not applied, a freely-evolving particle would slow down after the initial impulse. Using the above f(τ) to estimate that slowdown—which is analogous to the reduction-of-order approximations commonly applied in four-dimensions—the velocity change would appear to grow logarithmically at late times. Although this unphysical conclusion signals that it is inappropriate to approximate the force by the above expression, it does illustrate the remarkably strong memory of the d=3 self-force. Attempting to alternatively estimate the slowdown by cutting off the force integral before the external force is applied results in charges always returning asymptotically to rest. This approximation is also not well-controlled, although it does suggest that the self-field sourced in the distant past dissipates so slowly as to provide a kind of eternal rest frame. The actual endstate is nevertheless unclear. Faster decay rates in higher dimensions considerably simplify the situation.Another example which is worth noting is that of harmonic motion. Suppose that an external force is applied such that γ(τ) varies sinusoidally. Although the falloff conditions used to derive (<ref>) are violated in this case, applying fewer integrations by parts allows a well-defined self-force to be computed. Regardless of frequency, the phase of the d=3 self-force is advanced with respect to the phase of the particle's acceleration. Applying this for a charge in a fixed circular orbit with angular velocity ω, the nonrelativistic self-force reduces tof(τ) = 1/4q^2 [( 1 + 2 γ_E + ln (ωℓ)^2 ) γ̈(τ) - πωγ̇(τ) ] ,where γ_E denotes the Euler-Mascheroni constant. While work is performed only by the term proportional to the velocity, the term proportional to acceleration provides an effective mass which depends on lnω. The analogous force in five dimensions is essentially the same except for an additional factor of ω^2. We end by comparing the results presented here to others which have been suggested in the literature. Unlike our approach in which all results follow from first principles and in which no infinities arise, other proposals may be summarized as heuristic attempts to directly regularize point-particle self-fields <cit.> or expressions for the momentum associated with such fields <cit.>. In at least one case, the claimed force law is IR-divergent; see Eq. (4.4) in <cit.>. Other claimed force laws involve counterterms which depend on the particle's entire past history <cit.>, laws which could arise only if a particle's momenta depended on its state in a highly nonlocal manner—a physically-unacceptable option. Another result predicts a time-varying mass even at lowest order <cit.>. These conclusions are qualitatively different from ours.apsrev4-1 | http://arxiv.org/abs/1708.07813v1 | {
"authors": [
"Abraham I. Harte",
"Peter Taylor",
"Éanna É. Flanagan"
],
"categories": [
"gr-qc",
"hep-th"
],
"primary_category": "gr-qc",
"published": "20170825171128",
"title": "Self-forces in arbitrary dimensions"
} |
Tsinghua University [email protected] of Oxford OxfordUnited Kingdom [email protected] Tsinghua University Beijing China [email protected] University of California, Berkeley Berkeley United States of America [email protected] We study the problem of allocating impressions to sellers in e-commerce websites, such as Amazon, eBay or Taobao, aiming to maximize the total revenue generated by the platform.We employ a general framework of reinforcement mechanism design, which uses deep reinforcement learning to design efficient algorithms, taking the strategic behaviour of the sellers into account. Specifically, we model the impression allocation problem as a Markov decision process, where the states encode the history of impressions, prices, transactions and generated revenue and the actions are the possible impression allocations in each round. To tackle the problem of continuity and high-dimensionality of states and actions, we adopt the ideas of the DDPG algorithm to design an actor-critic policy gradient algorithm which takes advantage of the problem domain in order to achieve convergence and stability. We evaluate our proposed algorithm, coined IA(GRU), by comparing it against DDPG, as well as several natural heuristics, under different rationality models for the sellers - we assume that sellers follow well-known no-regret type strategies which may vary in their degree of sophistication. We find that IA(GRU) outperforms all algorithms in terms of the total revenue. <ccs2012> <concept> <concept_id>10003752.10010070.10010099.10010101</concept_id> <concept_desc>Theory of computation Algorithmic mechanism design</concept_desc> <concept_significance>500</concept_significance> </concept> <concept> <concept_id>10010147.10010257.10010258.10010261</concept_id> <concept_desc>Computing methodologies Reinforcement learning</concept_desc> <concept_significance>500</concept_significance> </concept> <concept> <concept_id>10010405.10003550</concept_id> <concept_desc>Applied computing Electronic commerce</concept_desc> <concept_significance>500</concept_significance> </concept> </ccs2012> [500]Theory of computation Algorithmic mechanism design [500]Computing methodologies Reinforcement learning [500]Applied computing Electronic commerceReinforcement Mechanism Design for e-commerce Yiwei Zhang December 30, 2023 =============================================§ INTRODUCTION A fundamental problem that all e-commerce websites are faced with is to decide how to allocate the buyer impressions to the potential sellers. When a buyer searches a keyword such as “iPhone 7 rose gold”, the platform will return a ranking of different sellers providing an item that fits the keyword, with different prices and different historical sale records. The goal of the platform is to come up with algorithms that will allocate the impressions to the most appropriate sellers, eventually generating more revenue from the transactions.This setting can be modeled as a resource allocation problem over a sequence of rounds, where in each round, buyers arrive, the algorithm inputs the historical records of the sellers and their prices and outputs such an allocation of impressions. The sellers and the buyers carry out their transactions and the historical records are updated.In reality, most e-commerce websites employ a class of heuristic algorithms, such as collaborative filtering or content based filtering <cit.>, many of which rank sellers in terms of “historical scores” calculated based on the transaction history of the sellers with buyers of similar characteristics. However, this approach does not typically take into account the fact that sellers strategize with the choice of prices, as certain sub-optimal prices in one round might affect the record histories of sellers in subsequent rounds, yielding more revenue for them in the long run. Even worse, since the sellers are usually not aware of the algorithm in use, they might “explore” with their pricing schemes, rendering the system uncontrollable at times. It seems natural that a more sophisticated approach that takes all these factors into account should be in place. In the presence of strategic or rational individuals, the field of mechanism design <cit.> has provided a concrete toolbox for managing or preventing the ill effects of selfish behaviour and achieving desirable objectives. Its main principle is the design of systems in such a way that the strategic behaviour of the participants will lead to outcomes that are aligned with the goals of the society, or the objectives of the designer. Cai et al. <cit.> tackle the problem of faking transactions and fraudulent seller behaviour in e-commerce using the tools from the field of mechanism design. A common denominator in most of the classical work in economics is that the participants have access to either full information or some distributional estimate of the preferences of others.However, in large and constantly evolving systems like e-commerce websites, the participants interact with the environment in various ways, and adjust their own strategies accordingly and dynamically <cit.>. In addition to that, their rationality levels are often bounded by either computational or financial constraints, or even cognitive limitations <cit.>. For the reasons mentioned above, a large recent body of work has advocated that other types of agent behaviour, based on learning and exploration, are perhaps more appropriate for such large-scale online problems encountered in reality <cit.>. In turn, this generates a requirement for new algorithmic techniques for solving those problems. Our approach is to use techniques from deep reinforcement learning for solving the problem of the impression allocation to sellers, given their selfish nature. In other words, given a rationality model for the sellers, we design reinforcement learning algorithms that take this model into account and solve the impression allocation problem efficiently. This general approach is called reinforcement mechanism design <cit.>, and we can view our contribution in this paper as an instance of this framework. §.§ No-regret learning as agent rationality As mentioned earlier, the strong informational assumptions of classical mechanism design are arguably unrealistic in complex and dynamic systems, like diverse online marketplaces. Such repeated game formulations typically require that the participants know the values of their competitors (or that they can estimate them pretty accurately based on known prior distributions) and that they can compute their payoff-maximizing strategies over a long sequence of rounds. Such tasks are usually computationally burdensome and require strong cognitive assumptions, as the participants would have to reason about the future, against all possible choices of their opponents, and in a constantly evolving environment.Given this motivation, an alternative approach in the forefront of much of the recent literature in algorithmic mechanism design is to assume that the agents follow some type of no-regret strategies; the agent picks a probability mixture over actions at each round and based on the generated payoffs, it updates the probabilities accordingly, minimizing the long-term regret. This is more easily conceivable, since the agents only have to reason about their own strategies and their interaction with the environment, and there is a plethora of no-regret algorithms at their disposal. Precisely the same argument has been made in several recent works <cit.> that study popular auction settings under the same rationality assumptions of no-regret, or similar types. In fact, there exist data from Microsoft's Ad Actions which suggest that advertisers do use no-regret algorithms for their actions <cit.>. For a more detailed discussion on related rationality models, the reader is referred to <cit.>. The seller rationality model: To model the different sophistication levels of sellers, we consider four different models of rationality, based on well-established no-regret learning approaches. The first two, ϵ-Greedy <cit.> and ϵ-First are known as semi-uniform methods, because they maintain a distinction between exploration and exploitation. The later is often referred to as “A/B testing” and is widely used in practice <cit.>. The other two approaches, UCB1 <cit.> and Exp3 <cit.> are more sophisticated algorithms that differ in their assumptions about the nature of the rewards, i.e. whether they follow unknown distributions or whether they are completely adversarial. Note that all of our rationality models employ algorithms for the multi-arm bandit setting, as in platforms like Taobao or eBay, the impression allocation algorithms are unknown to the sellers and therefore they can not calculate the payoffs of unused auctions. The update of the weights to the strategies is based solely on the observed payoffs, which is often referred to as the bandit feedback setting <cit.>.We note here that while other related rationality models can be used, the goal is to choose a model that real sellers would conceivably use in practice. The semi-uniform algorithms are quite simpler and model a lower degree of seller sophistication, whereas the other two choices correspond to sellers that perhaps put more effort and resources into optimizing their strategies - some examples of sophisticated optimization services that are being used by online agents are provided in <cit.>. Note that both UCB1 and Exp3 are very well-known <cit.> and the latter is perhaps the most popular bandit feedback implementation of the famous Hedge (or Multiplicative Weights Update) algorithm for no-regret learning in the fully informed feedback setting. §.§ The impression allocation problem We model the impression allocation problem as a Markov decision process (MDP) in which the information about the prices, past transactions, past allocations of impressions and generated revenue is stored in the states, and the actions correspond to all the different ways of allocating the impressions, with the rewards being the immediate revenue generated by each allocation. Given that the costs of the sellers (which depend on their production costs) are private information, it seems natural to employ reinforcement learning techniques for solving the MDP and obtain more sophisticated impression allocation algorithms than the heuristics that platforms currently employ.In our setting however, since we are allocating a very large number of impressions, both the state space and the action space are extremely large and high-dimensional, which renders traditional reinforcement learning techniques such as temporal difference learning <cit.> or more specifically Q-learning <cit.> not suitable for solving the MDP. In a highly influential paper, Mnih et al. <cit.> employed the use of deep neural networks as function approximators to estimate the action-value function. The resulting algorithm, coined “Deep Q Network” (DQN), can handle large (or even continuous) state spaces but crucially, it can not be used for large or continuous action domains, as it relies on finding the action that maximizes the Q-function at each step.To handle the large action space, policy gradient methods have been proposed in the literature of reinforcement learning with actor-critic algorithms rising as prominent examples <cit.>, where the critic estimates the Q-function by exploring, while the actor adjusts the parameters of the policy by stochastic gradient ascent. To handle the high-dimensionality of the action space, Silver et al. <cit.> designed a deterministic actor-critic algorithm, coined “Deterministic Policy Gradient” (DPG) which performs well in standard reinforcement-learning benchmarks such as mountain car, pendulum and 2D puddle world. As Lillicrap et al. <cit.> point out however, the algorithm falls short in large-scale problems and for that reason, they developed the “Deep-DPG” (DDPG) algorithm which uses the main idea from <cit.> and combines the deterministic policy gradient approach of DPG with deep neural networks as function approximators. To improve convergence and stability, they employ previously known techniques such as batch normalization <cit.>, target Q-networks <cit.>, and experience replay <cit.>.The IA(GRU) algorithm: We draw inspiration from the DDPG algorithm to design a new actor-critic policy gradient algorithm for the impression allocation problem, which we refer to as the IA(GRU) algorithm. IA(GRU) takes advantage of the domain properties of the impression allocation problem to counteract the shortcomings of DDPG, which basically lie in its convergence when the number of sellers increases. The modifications of IA(GRU) to the actor and critic networks reduce the policy space to improve convergence and render the algorithm robust to settings with variable sellers, which may arrive and depart in each round, for which DDPG performs poorly. We evaluate IA(GRU) against DDPG as well as several natural heuristics similar to those usually employed by the online platforms and perform comparisons in terms of the total revenue generated. We show that IA(GRU) outperforms all the other algorithms for all four rationality models, as well as a combined pool of sellers of different degrees of sophistication.§ THE SETTINGIn the impression allocation problem of e-commerce websites, there are m sellers who compete for a unit of buyer impression.[Since the buyer impressions to be allocated is a huge number, we model it as a continuous unit to be fractionally allocated. Even if we used a large integer number instead, the traditional approaches like DDPG fall short for the same reasons and furthermore all of the performance guarantees of IA(GRU) extend to that case.] In each round, a buyer[As the purchasing behavior is determined by the valuation of buyers over the item, without loss of generality we could consider only one buyer at each round.] searches for a keyword and the platform returns a ranking of sellers who provide an item that matches the keyword; for simplicity, we will assume that all sellers provide identical items that match the keyword exactly. Each seller i has a private cost c_i for the item, which can be interpreted as a production or a purchasing cost drawn from an i.i.d. distribution F_s. Typically, there are n slots (e.g. positions on a webpage) to be allocated and we let x_ij denote the probability (or the fraction of time) that seller i is allocated the impression at slot j. With each slot, there is an associated click-through-rate α_j which captures the “clicking potential” of each slot, and is independent of the seller, as all items offered are identical. We let q_i=∑_j=1^nx_ijα_j denote the probability that the buyer will click the item of seller i. Given this definition (and assuming that sellers can appear in multiple slots in each page), the usual feasibility constraints for allocations, i.e. for all i, for all j, t holds that 0≤ x_ij≤ 1 and or all j, it holds that ∑_i=1^mx_ij≤ 1 can be alternatively written asfor alli, q_i≥ 0,it holds that ∑_i=1^mq_i≤∑_j=1^nα_jand ∑_j=1^nα_j=1.That is, for any such allocation q, there is a feasible ranking x that realizes q (for ease of notation, we assume that the sum of click-through rates of all slots is 1) and therefore we can allocate the buyer impression to sellers directly instead of outputting a ranking over these items when a buyer searches a keyword.[The framework extends to cases where we need return similar but different items to a buyer, i.e, the algorithm outputs a ranking over these items. Furthermore, our approach extends trivially to the case when sellers have multiple items.] Let h_it=(v_it,p_it,n_it,ℓ_it) denote the records of seller i at round t, which is a tuple consisting of the following quantities: * v_it is the expected fraction of impressions that seller i gets, * p_it is the price that seller i sets, * n_it is the expected amount of transactions that seller i makes, * ℓ_it is the expected revenue that seller i makes at round t. Let H_t=(h_1t,h_2t,...,h_it) denote the records of all sellers at round t, and let H_it=(h_i1,h_i2,...,h_it) denote the vectors of records of seller i from round 1 to round t, which we will refer to as the records of the seller. At each round t+1, seller i chooses a price p_i(t+1) for its item and the algorithm allocates the buyer impression to sellers.MDP formulation: The setting can be defined as a Markov decision process (MDP) defined by the following components: a continuous state space 𝒮, a continuous action space 𝒜, with an initial state distribution with density p_0(s_0), and a transition distribution of states with conditional density p(s_t+1|s_t,a_t) satisfying the Markov property, i.e.p(s_t+1|s_0,a_0,...,s_t,a_t)=p(s_t+1|s_t,a_t). Furthermore, there is an associated reward function r:𝒮×𝒜→ℛ assigning payoffs to pairs of states and actions. Generally, a policy is a function π that selects stochastic actions given a state, i.e, π:𝒮→𝒫(𝒜), where 𝒫(𝒜) is the set of probability distributions on 𝒜. Let R_t denote the discounted sum of rewards from the state s_t, i.e, R_t(s_t)=∑_k=t^∞γ^k-tr(s_k,a_k), where 0<γ<1. Given a policy and a state, the value function is defined to be the expected total discounted reward, i.e. V^π(s)=E[R_t(s_t)|s_t=s;π] and the action-value function is defined as Q^π(s,a)=E[R_t(s_t)|s_t=s,a_t=a;π].For our problem, a state s_t of the MDP consists of the records of all sellers in the last T rounds, i.e. s_t=(H_t-T,...,H_t-1), that is, the state is a (T,m,4) tensor, the allocation outcome of the round is the action, and the immediate reward is the expected total revenue generated in this round. The performance of an algorithm is defined as the average expected total revenue over a sequence of T_0 rounds.Buyer Behaviour: We model the behaviour of the buyer as being dependent on a valuation that comes from a distribution with cumulative distribution function F_b.Intuitively, this captures the fact that buyers may have different spending capabilities (captured by the distribution).Specifically, the probability that the buyer purchases item i is n_it=(1-F_b(p_it))· v_it, that is, the probability of purchasing is decided by the impression allocation andthe price seller i sets.For simplicity and without loss of generality with respect to our framework, we assume that the buyer's valuation is drawn from U(0,1), i.e. the uniform distribution over [0,1]. §.§ Seller Rationality As we mentioned in the introduction, following a large body of recent literature, we will assume that the sellers employ no-regret type strategies for choosing their prices in the next round. Generally, a seller starts with a probability distribution over all the possible prices, and after each round, it observes the payoffs that these strategies generate and adjusts the probabilities accordingly. As we already explained earlier, it is most natural to assume strategies in the bandit feedback setting, where the seller does not observe the payoffs of strategies in the support of its strategy which were not actually used. The reason is that even if we assume that a seller can see the prices chosen in a round by its competitors, it typically does not have sufficient information about the allocation algorithm used by the platform to calculate the payoffs that other prices would have yielded. Therefore it is much more natural to assume that the seller updates its strategy based on the observed rewards, using a multi-arm bandit algorithm.More concretely, the payoff of a seller i that receives v_it impressions in round t when using price p_ij(t), is given by u_ij(t)=n_it(p_ij(t)-c_i)=v_it(1-F_b(p_it))(p_ij(t)-c_i). For consistency, we normalize the costs and the prices to lie in the unit interval [0,1] and we discretize the price space to a “dense enough” grid (of size 1/K, for some large enough K). This discretization can either be enforced by the platform (e.g. the sellers are required to submit bids which are multiples of 0.05) or can be carried out by the sellers themselves in order to be able to employ the multi-arm bandit algorithms which require the set of actions to be finite, and since small differences in prices are unlikely to make much difference in their payoffs.We consider the following possible strategies for the sellers, based on well-known bandit algorithms. ε-Greedy <cit.>: With probability ε, each seller selects a strategy uniformly at random and with probability 1-ε, the strategy with the best observed (empirical) mean payoff so far. The parameter ε denotes the degree of exploration of the seller, whereas 1-ε is the degree of exploitation; here ε is drawn i.i.d. from the normal distribution N(0.1,0.1/3). ε-First: For a horizon of 𝒯 rounds, this strategy consists of an exploration phase first, over ε·𝒯 rounds, followed by an exploitation phase, for the remaining period. In the exploration phase, the seller picks a strategy uniformly at random. In the remaining rounds, the sellers picks the strategy that maximizes the empirical mean of the observed rewards. For each seller, we set 𝒯=200 and ε=0.1. Exponential-weight Algorithm for Exploration and Exploitation (Exp3) <cit.>: We use the definition of the algorithm from <cit.>. Let γ∈ (0,1] be a real number and initialize w_i(1)=1 for i=1,…,K+1 to be the initial weights of the possible prices[For ease of notation, we drop the subscript referring to a specific seller, as there is no ambiguity.]. In each round t, * For i=1,…,K+1, let π_i(t)= (1-γ) w_i(t)/∑_j=1^K+1 w_j(t) + γ/K+1, where w_i(t) is the weight of price p_i in round t. * Select a price p_j(t) according to the probability distribution defined by π_1(t),…,π_K+1(t). * Receive payoff u_j(t) ∈ [0,1]. * For ℓ=1,…,K+1, let û_ℓ(t)= u_ℓ(t)/π_ℓ(t),if ℓ=j 0,otherwise and w_ℓ(t+1) = w_ℓ(t)e^γ·û_ℓ(t)/(K+1).We remark here that since the payoff of each seller in some round t actually takes values in [-1,1], we scale the payoff to [0,1] by applying the transformation f(u)=(u+1)/2 to any payoff u. Upper Confidence Bound Algorithm (UCB1) <cit.>:For each price p_j ∈ [0,1/K,2/K,…,1], initialize x_j(1)=0. At the end of each round t, update x_j(t) as: x_j(t)=x_j(t-1)/t + u_j(t)/t,ifjwas chosen in this roundt x_j(t-1),otherwiseFor any round t ∈{0,…,K}, the seller chooses a price p_j that has not been used before in any of the previous rounds (breaking ties arbitrarily). For any round t ≥ K+1, the seller chooses the price p_j with the maximum weighted value x_j, i.e, p_j(t) ∈max_j ∈{0,1/K,…,1} x_j(t)+log_2 t/∑_τ=1^tI_jτ , where I_jτ is the indicator function, i.e.I_jτ= 1,ifp_jwas chosen in round τ 0,otherwise. ε-Greedy and ε-First are simple strategies that maintain a clear distinction between exploration and exploitation and belong to the class of semi-uniform strategies. Exp3 is the most widely used bandit version of perhaps the most popular no-regret algorithm for the full information setting, the Hedge (or Multiplicative Weight updates) algorithm <cit.> and works in the adversarial bandit feedback model <cit.>, where no distributional assumptions are being made about the nature of the rewards. UCB1, as the name suggests, maintains a certain level of optimism towards less frequently played actions (given by the second part of the sum) and together with this, it uses the empirical mean of observed actions so far to choose the action in the next round. The algorithm is best suited in scenarios where the rewards do follow some distribution which is however unknown to the seller. For a more detailed exposition of all these different algorithms, <cit.> provides a concise survey. The point made here is that these choices are quite sensible as they (i) constitute choices that a relatively sophisticated seller, perhaps with a research team at its disposal could make, (ii) can model sellers with different degrees of sophistication or pricing philosophies and (iii) are consistent with the recent literature on algorithmic mechanism design, in terms of modeling agent rationality in complex dynamic environments.§ ALLOCATION ALGORITHMS In this section, we will briefly describe the algorithms that we will be comparing IA(GRU) against - two natural heuristics similar to those employed by platforms for the impression allocation problem, as well as the DDPG algorithm of Lillicrap et al. <cit.>.§.§ Heuristic Allocation AlgorithmsAs the strategies of the sellers are unknown to the platform, and the only information available is the sellers' historical records, the platform can only use that information for the allocation. Note that these heuristics do not take the rationality of the sellers into account, when deciding on the allocation of impressions. The first algorithm is a simple greedy algorithm, which allocates the impressions proportionally to the revenue contribution. Greedy Myopic Algorithm: At round 0, the algorithm allocates a 1/m-fraction of the buyer impression to each seller. At any other round τ+1 (for τ≥ 0),the algorithm allocates a fraction of ℓ_iτ/∑_j=1^mℓ_jτ of the buyer impression to each seller, i.e. proportionally to the contribution of each seller to the total revenue of the last round. The second algorithm is an algorithm for the contextual multi-arm bandit problem, proposed by <cit.>, based on the principles of the family of upper confidence bound algorithms (UCB1 is an algorithm in this family). The algorithm is among the state of the art solutions for recommender systems <cit.> and is an example of contextual bandit approaches, which are widely applied to such settings <cit.>. To prevent any confusion, we clarify here that while we also used bandit algorithms for the seller rationality models, the approach here is fundamentally different as the Linear UCB Algorithm is used for the allocation of impressions - not the choice of prices - and the arms in this case are the different sellers. Linear UCB Algorithm <cit.>: We implement the algorithm as described in <cit.> - in the interest of space, we do not provide the definition of the algorithm, but refer the reader to Algorithm 1 in <cit.>. We model each seller as an arm and set h_it as the feature of each arm i in each round t. The parameter α is set to 1. §.§ Deep Deterministic Policy GradientHere, we briefly describe the DDPG algorithm of <cit.>, which we we draw inspiration from in order to design our impression allocation algorithm. Before describing the algorithm, we briefly mention the main ingredients of its predecessor, the DPG algorithm of Silver et al. <cit.>. Deterministic Policy Gradient: The shortcoming of DQN <cit.> is that while it can handle continuous states, it can not handle continuous actions or high-dimensional action spaces. Although stochastic actor-critic algorithms could handle continuous actions, they are hard to converge in high dimensional action spaces. The DPG algorithm <cit.> aims to train a deterministic policy μ_θ:𝒮→𝒜 with parameter vector θ∈ R^n. This algorithm consists of two components: an actor, which adjusts the parameters θ of the deterministic policy μ_θ(s) by stochastic gradient ascent of the gradient of the discounted sum of rewards, and the critic, which approximates the action-value function. Deep Deterministic Policy Gradient: Directly training neural networks for the actor and the critic of the DPG algorithm fails to achieve convergence; the main reason is the high degree of temporal correlation which introduces high variance in the approximation of the Q-function by the critic. For this reason, the DDPG algorithm uses a technique known as experience replay, according to which the experiences of the agent at each time step are stored in a replay buffer and then a mini-batch is sampled uniformly at random from this set for learning, to eliminate the temporal correlation. The other modification is the employment of target networks for the regularization of the learning algorithm. The target network is used to update the values of μ and Q at a slower rate instead of updating by the gradient network; the prediction y_t will be relatively fixed and violent jitter at the beginning of training is absorbed by the target network. A similar idea appears in <cit.> with the form of double Q-value learning.§ THE IMPRESSION ALLOCATION (GRU) ALGORITHM In this section, we present our main deep reinforcement learning algorithm, termed IA(GRU) (“IA” stands for “impression allocation” and “GRU” stands for “gated recurrent unit”) which is in the center of our framework for impression allocations in e-commerce platforms and is based on the ideas of the DDPG algorithm. Before we present the algorithm, we highlight why simply applying DDPG to our problem can not work. Shortcomings of DDPG: First of all, while DDPG is designed for settings with continuous and often high-dimensional action spaces, the blow-up in the number of actions in our problem is very sharp as the number of sellers increases; this is because the action space is the set of all feasible allocations, which increases very rapidly with the number of sellers. As we will show in Section <ref>, the direct application of the algorithm fails to converge even for a moderately small number of sellers. The second problem comes from the inability of DDPG to handle variability on the set of sellers. Since the algorithm uses a two-layer fully connected network, the position of each seller plays a fundamental role; each seller is treated as a different entity according to that position. As we show in Section <ref>, if the costs of sellers at each round are randomly selected, the performance of the DDPG algorithm deteriorates rapidly. The settings in real-life e-commerce platforms however are quite dynamic, with sellers arriving and leaving or their costs varying over time, and for an allocation algorithm to be applicable, it should be able to handle such variability. We expect that each seller's features are only affected by its historical records, not some “identity” designated by the allocation algorithm; we refer to this highly desirable property as "permutation invariance". Based on time-serial techniques, our algorithm uses Recurrent Neural Networks at the dimension of the sellers and achieves the property.The IA(GRU) algorithm:Next, we explain the design of our algorithm, but we postpone some implementation details for Section <ref>. At a high level, the algorithm uses the framework of DDPG with different network structures and different inputs of networks. It maintains a sub-actor network and a sub-critic network for each seller and employs input preprocessing at each training step, to ensure permutation invariance. Input Preprocessing: In each step of training, with a state tensor of shape (T,m,4), we firstly utilize a background network to calculate a public vector containing information of all sellers: it transforms the state tensor to a (T,m× 4) tensor and performs RNN operations on the axis of rounds.At this step, it applies a permutation transformation, i.e. a technique for maintaining permutation invariance. Specifically, it first orders the sellers according to a certain metric, such as the weighted average of their past generated revenue and then inputs the (state, action) pair following this order to obtain the public vector (pv).On the other hand, for each seller i, it applies a similar RNN operation on its history, resulting in an individual temporal feature called (f_i). Combining those two features, we obtain a feature vector (pv,f_i) that we will use as input for the sellers' sub-actor and sub-critic networks. Actor network: For each seller, the input to the sub-actor network is (pv,f_i) and the output is a score. This algorithm uses a softmax function over the outputs of all sub-actor networks in order to choose an action. The structure of the policy which is shown in Figure <ref> ensures that the policy space is much smaller than that of DDPG as the space of inputs of all sub-actor networks is restricted, and allows for easier convergence, as we will show in Section <ref>.Critic network: For the critic, we make use of a domain-specific property, namely that the immediate reward of each round is the sum of revenues of all sellers and the record of each seller has the same space. Each sub-critic network inputs the expected fraction of buyer impression the seller gets (the sub-action) and (pv,f_i) (the sub-state) as input and outputs the Q-value of the corresponding seller, i.e, the expected discounted sum of revenues from the sub-state following the policy. Then, it sums up the estimated Q-value of all sub-critic networks to output the final estimated Q-value, with the assumption that the strategy of each seller is independent of the records of other sellers, which is the case in all of our rationality models. The framework of the critic network is similar to Figure <ref>.§ EXPERIMENTAL EVALUATION In this section, we present the evaluation of our algorithms in terms of convergence time and revenue performance against several benchmarks, namely the direct application of the DDPG algorithm (with a fully connected network) and the heuristic allocation algorithms that we defined in Section <ref>. We use Tensorflow and Keras as the engine for the deep learning, combining the idea of DDPG and the techniques mentioned in Section <ref>, to train the neural network. Designed experiments: First, we will compare IA(GRU) and DDPG in terms of their convergence properties in the training phase and show that the former converges while the latter does not. Next, we will compare the four different algorithms (Greedy Myopic, Linear UCB, DDPG and IA(GRU)) in terms of the generated revenue for two different settings, a setting with fixed sellers and a setting with variable sellers. The difference is that in the former case, we sample the costs c_i once in the beginning whereas in the latter case, the cost c_i of each seller is sampled again in each round. This can either model the fact that the production costs of sellers may vary based on unforeseeable factors or simply that sellers of different capabilities may enter the market in each round.For each one of these two settings, we will compare the four algorithms for each one of the four different rationality models (ϵ-Greedy, ϵ-First, UCB1 and Exp3) separately as well as in a combined manner, by assuming a mixed pool of sellers, each of which may adopt a different rationality model from the ones above. The latter comparison is meant to capture cases where the population of sellers is heterogeneous and may consist of more capable sellers that employ their R&D resources to come up with more sophisticated approaches (such as UCB1 or Exp3) but also on more basic sellers that employ simpler strategies (such as ϵ-Greedy). Another interpretation is that the distinction is not necessarily in terms of sophistication, but could also be due to different market research, goals, or general business strategies, which may lead to different decisions in terms of which strategy to adopt.Our experiments are run for 200 sellers, a case which already captures a lot of scenarios of interest in real e-commerce platforms. A straightforward application of the reinforcement learning algorithms for much larger numbers of sellers is problematic however, as the action space of the MDP increases significantly, which has drastic effects on their running time. To ensure scalability, we employ a very natural heuristic, where we divide the impression allocation problem into sub-problems and then solve each one of those in parallel. We show at the end of the section that this “scale-and-solve” version of IA(GRU) clearly outperforms the other algorithms for large instances consisting of as many as 10.000 sellers.Experimental Setup: In the implementation of DDPG, the actor network uses two full connected layers, a rectified linear unit (ReLu) as the activation function, and outputs the action by a softmax function. The critic network inputs a (state,action) pair and outputs the estimation of the Q-value using similar structure. The algorithm IA(GRU) uses the same structure, i.e. the fully connected network in the sub-actor and sub-critic networks, and uses a Recurrent Neural Network with gate recurrent units (GRU) in cyclic layers to obtain the inputs of these networks. For the experiments we set T=1, i.e, the record of all items of the last round is viewed as the state.[We found out that training our algorithms for larger values of T does not help to improve the performance.] We employ heuristic algorithms such as the Greedy Myopic Algorithm for exploration, i.e. we add these samples to the replay buffer before training. Experimental Parameters: We use 1000 episodes for both training and testing, and there are 1000 steps in each episode. The valuation of the buyer in each round is drawn from the standard uniform distribution U(0,1) and the costs of sellers follow a Gaussian distribution with mean 1/2 and variance 1/2. The size of the replay buffer is 10^5, the discount factor γ is 0.99, and the rate of update of the target network is 10^-3. The actor network and the critic network are trained via the Adam algorithm, a gradient descent algorithm presented in <cit.>, and the learning rates of these two networks are 10^-4.Following the same idea as in <cit.>, we add Gaussian noise to the action outputted by the actor network, with the mean of the noise decaying with the number of episodes in the exploration.§.§ Convergence of DDPG and IA(GRU) First, to show the difference in the convergence properties of DDPG and IA(GRU), we train the algorithms for 200 sellers using the ϵ-greedy strategy as the rationality model with variable costs for the sellers. Figure <ref> shows the comparison between the rewards of the algorithms and Figure <ref> shows the comparison in terms of the training loss with the number of steps.The gray band shows the variance of the vector of rewards near each step. From the figures, we see that DDPG does not converge, while IA(GRU) converges, as the training loss of the algorithm decreases with the number of steps. The convergence properties for the other rationality models are very similar. §.§ Performance ComparisonIn this subsection, we present the revenue guarantees of IA(GRU) in the setting with 200 sellers and how it fairs against the heuristics and DDPG, for either each rationality model separately, or for a heterogeneous pool of sellers, with a 1/4-fraction of the sellers following each strategy. As explained in the previous page, we consider both the case of fixed sellers and variable sellers. Performance Comparison for Fixed Sellers: We show the performance of DDPG, IA(GRU), Greedy Myopic and Linear UCB on sellers using * the ϵ-Greedy strategy (Figure <ref>), * the ϵ-First strategy (Figure <ref>), * the UCB1 strategy (Figure <ref>), * the Exp3 strategy (Figure <ref>).We also show the performance of the four different algorithms in the case of a heterogeneous population of sellers in Figure <ref>.Every point of the figures shows the reward at the corresponding step. We can conclude that the IA(GRU) algorithm is clearly better than the other algorithms in terms of the average reward on all rationality models. We also note that DPPG does not converge with 200 sellers and this is the reason for its poor performance. Performance Comparison for Variable Sellers: We show the performance of DDPG, IA(GRU), Greedy Myopic and Linear UCB on sellers using * the ϵ-Greedy strategy (Figure <ref>), * the ϵ-First strategy (Figure <ref>), * the UCB1 strategy (Figure <ref>), * the Exp3 strategy (Figure <ref>).We also show the performance of the four different algorithms in the case of a heterogeneous population of sellers in Figure <ref>. Again here, we can conclude that the IA(GRU) algorithm clearly outperforms all the other algorithms in terms of the average reward on all rationality models. Also, IA(GRU) fairs better in terms of stability, as the other algorithms perform worse in the setting with variable sellers, compared to the setting with fixed sellers. §.§ ScalabilityIn this subsection, we present the revenue guarantees of IA(GRU) in the setting with 10000 fixed sellers and how it fairs against the heuristics and DDPG to show the scalability properties of IA(GRU) with the number of sellers. For IA(GRU) and DDPG, we will employ a simple “scale-and-solve” variant, since applying either of them directly to the pool of 10.000 sellers is prohibitive in terms of their running time. We design 50 allocation sub-problems, consisting of 200 sellers each, and divide the total number of impressions in 50 sets of equal size, reserved for each sub-problem. We run IA(GRU) and DDPG algorithms in parallel for each sub-problem, which is feasible in reasonable time. For the heuristics, we run the algorithms directly on the large population of 10.000 sellers. The results for the case of ϵ-Greedy seller strategies are show in Figure <ref> (the results for other strategies are similar). We can see that even though we are applying a heuristic version, the performance of IA(GRU) is still clearly superior to all the other algorithms, which attests to the algorithm being employable in larger-case problems as well.§ CONCLUSION In this paper, we employed a reinforcement mechanism design framework for solving the impression allocation problem of large e-commerce websites, while taking the rationality of sellers into account. Inspired by recent advances in reinforcement learning, we designed a deep reinforcement learning algorithm which outperforms several natural heuristics under different realistic rationality assumptions for the sellers in terms of the generated revenue, as well as state-of-the-art reinforcement learning algorithms in terms of performance and convergence guarantees. Our algorithm can be applied to other dynamical settings for which the objectives are similar, i.e. there are multiple agents with evolving strategies, with the objective of maximizing a sum of payments or the generated revenue of each agent. It is an interesting future direction to identify several such concrete settings and apply our algorithm (or more generally our framework), to see if it provides improvements over the standard approaches, as it does here. Qingpeng Cai and Pingzhong Tang were supported in part by the National Natural Science Foundation of China Grant 61561146398, a Tsinghua University Initiative Scientific Research Grant, a China Youth 1000-talent program and Alibaba Innovative Research program. Aris Filos-Ratsikas was supported by the ERC Advanced Grant 321171 (ALGAME).ACM-Reference-Format | http://arxiv.org/abs/1708.07607v3 | {
"authors": [
"Qingpeng Cai",
"Aris Filos-Ratsikas",
"Pingzhong Tang",
"Yiwei Zhang"
],
"categories": [
"cs.MA",
"cs.AI"
],
"primary_category": "cs.MA",
"published": "20170825035521",
"title": "Reinforcement Mechanism Design for e-commerce"
} |
Low Cost, Open-Source Testbed to Enable Full-Sized Automated Vehicle Research Austin Costley Chase Kunz Ryan Gerdes Rajniaknt Sharma ============================================================================= An open-source vehicle testbed to enable the exploration of automation technologies for road vehicles is presented. The platform hardware and software, based on the Robot Operating System (ROS), are detailed. Two methods are discussed for enabling the remote control of a vehicle (in this case, an electric 2013 Ford Focus). The first approach used digital filtering of Controller Area Network (CAN) messages. In the case of the test vehicle, this approach allowed for the control of acceleration from a tap-point on the CAN bus and the OBD-II port. The second approach, based on the emulation of the analog output(s) of a vehicle's accelerator pedal, brake pedal, and steering torque sensors, is more generally applicable and, in the test vehicle, allowed for the full control vehicle acceleration, braking, and steering. To demonstrate the utility of the testbed for vehicle automation research, system identification was performed on the test vehicle and speed and steering controllers were designed to allow the vehicle to follow a predetermined path. The resulting system was shown to be differentially flat, and a high level path following algorithm was developed using the differentially flat properties and state feedback. The path following algorithm is experimentally validated on the automation testbed developed in the paper.§ INTRODUCTIONIn recent years, the automotive industry has been automating vehicle systems to aid drivers with features such as adaptive cruise control, lane keeping, and collision avoidance <cit.>. In more advanced systems, that are not commercially available, vehicles can drive themselves to user defined destinations <cit.>. Though these driver aid systems are just starting to emerge in production vehicles, research has been conducted for many years to help develop this technology <cit.>. The benefits of automated vehicles extends beyond convenience, and includes, safer roadways, increased highway throughput, and reduced emissions. Despite these positive effects of vehicle automation, there are possible drawbacks, and considerations should be made regarding the safety and security of these automated systems. Hackers are constantly reviewing platforms in search of exploitable vulnerabilities, and these automated features provide attack opportunities that were previously unavailable. For example, in <cit.> and <cit.>, Miller and Valasek showed that it was possible to exploit the parking assist, and lane keeping features of modern vehicles to gain limited control of acceleration and steering. In <cit.>, Koscher et al. demonstrate the ability to disable the braking system of modern vehicles. A team of undergraduate and graduate researchers at Utah State University was assembled to automate a 2013 Ford Focus Electric. To complete this challenge, the team had to find a way to control the vehicle by superseding internal control modules. The resulting testbed could then be used to further research in the fields of automated vehicle control, vehicle security, and in-motion wireless power transfer research <cit.>. The Electric Vehicle and Roadway (EVR) Research Facility and Test Track <cit.> at Utah State University is a facility designed to conduct in-motion wireless power transfer research, and provided lab space for this project. The first goal of the project was to control the vehicle without adding hardware actuators. Two approaches were discussed and prioritized in the following order: Controller Area Network (CAN) message injection, and sensor emulation. CAN message injection would attempt to take control of the vehicle by superseding modules connected to the CAN bus. Sensor emulation would supersede the user input sensors to manipulate the signals being sent to the CAN modules. A reverse engineering approach would be taken to determine if the vehicle could be controlled through either of these methods.Having enabled remote control of the vehicle, the utility of the testbed in allowing for automation research was demonstrated through the design and implementation of a path following control strategy.First, system identification steps were taken to develop an accurate model of vehicle dynamics due to user input. Low level feedback controllers were developed to allow control of meaningful vehicle states, and high level controller was developed to coordinate the low level controllers to follow a desired path. §.§ ContributionsThe main contributions of this work are as follows: * Affordable, open-source, experimentally validated automation testbed using the Robot Operating System (ROS) <cit.>. * Two ways to manipulate control input to the vehicle without adding external actuation hardware: CAN message injection and sensor emulation that enable the development of other vehicle testbeds. * Software packages, hardware descriptions, and a methodological approach that can be used to remotely control nearly any vehicle, thus allowing researchers and hobbyists to explore automation techniques and vehicle security. The software packages are available at: <https://github.com/rajnikant1010/EVAutomation>. * Proof-of-concept exploitation of a vehicle vulnerability wherein a compromised ECU or other device connected to the CAN bus (including via the OBD-II port) could control vehicle acceleration. §.§ Literature ReviewControl of Automated Ground Vehicles (AGV's) has been a subject of research since 1955 <cit.> and has naturally progressed to include commercially available passenger vehicles. In <cit.>, Rajamani details various vehicle control systems, and modeling techniques. There have been several automated vehicle competitions held to further the research in this field. The DARPA Grand Challenge, for example, was started in 2004 to encourage researchers to develop off-road autonomous vehicle technology that could be used for military applications <cit.>. The challenge was repeated in 2005 for off-road vehicles <cit.>. Then in 2007, the challenge was altered to focus on urban driving environments <cit.>. The teams in these challenges started with an existing commercial vehicle, and developed an automated system to complete the challenge. The Grand Cooperative Driving Challenge (GCDC) is another example of the advancement of automated driving through competition <cit.>. The purpose of the GCDC is to examine cooperative automated vehicle systems. Teams develop an automated vehicle to be used in cooperative challenges with other teams. In contrast to the work by the teams in the DARPA Grand Challenge and GCDC, this work is open-source, low-cost, and has been completed without vendor support. One purpose of this work is to allow other researchers to perform vehicle automation tasks similar those in these challenges without needing support from industry. Koscher et al. and Checkoway et al. in <cit.> and <cit.>showed that an attacker can gain access to Electronic Control Units (ECU's) and circumvent vehicle control and safety systems. They also demonstrated that the attacker could gain access to the CAN bus remotely, and perform similar attacks. Their results included the ability to have the car ignore a brake pedal press by the driver, a complete vehicle shut down, and complete control of the visual display. The attacks however, did not allow for arbitrary control of acceleration, braking or steering, which is required for vehicle autonomy. Miller and Valasek built on the work from Koscher and Checkoway et. al. in <cit.> and <cit.>, which detail attacks on vehicle systems in a 2010 Ford Escape, 2010 Toyota Prius, and a 2014 Jeep Cherokee. They developed an extensive platform for attacking ECU's and exploiting driver assist systems. However, the attacks had limited scope for vehicle control. For example, an attack on the parking assist module was conducted on the 2010 Ford Escape, whereby, vehicle steering could be controlled when the vehicle was traveling at under 5 mph, but this attack would not work if the vehicle was moving faster. They also demonstrated the ability to cause vehicle acceleration under very specific conditions. In contrast, this work demonstrates the ability to cause arbitrary acceleration under any condition. In addition, using the approach presented here, the acceleration can be controlled fromthe OBD-II port, any bus tap point, or by gaining access to an ECU. The current work also demonstrates the security concern that a vehicle could be examined and reverse engineered in a short amount of time, as it took a small team of students just under a year to develop the automated system for a commercial vehicle. This paper is organized as follows. Section <ref> details the generalized reverse engineering approach used to create the testbed, as well as how the methodology naturally allows for vulnerability discovery and mitigation experimentation. The efficacy of the testbed for automation research is demonstrated in subsequent sections.Longitudinal and lateral low level controls are discussed in Section <ref>, while Section <ref> discusses the theory of differential flatness and the path following controller. Section <ref> gives an overview of the testbed architecture and details the hardware and software components of the automation platform. Section <ref> shows the experimental results of the low level controllers and automation, and Section <ref> summarizes the findings and concludes the paper.§ ENABLING REMOTE CONTROLModern vehicles use a Controller Area Network (CAN) bus system for module-to-module communication <cit.>. Electronic Control Units (ECU's) are the CAN modules that connect to the bus that send and receive information. A CAN module receives data from sensors, processes the data, and generates the appropriate CAN message to be broadcast on the bus using an analog-to-digital operation. The research team for the current work used the 2013 Ford Focus Electric Wiring Guide <cit.> and the Auto Repair Reference Center Research Database from EBSCOhost <cit.> to understand signal path and critical connections. The wiring guide provided diagrams for most of the wires in the vehicle and included diagrams and pin-outs for the wiring connections. The Auto Repair Reference Center was particularly useful for reverse engineering the CAN protocols. It contains the CAN messages generated and received by each module, diagrams of the four CAN buses in the vehicle, and the module layout on each bus. The CAN message information was an incomplete list of general messages sent between CAN modules. For example, the list would indicate that a message about the acceleration pedal position is sent from one module to another, but it would not indicate the structure of the message, the arbitration ID, or a conversion to useful units.Using these resources, the team identified sensors and modules that could be used for vehicle control. Table <ref> summarizes these findings. In addition, the team took a hands-on approach to vehicle exploration and verified the location and connections of these components. The examination of this architecture led to the identification of the two possible controller insertion strategies, as shown in Fig. <ref> . First, the CAN lines between the control module and the CAN bus could be cut, and a controller could be inserted to intercept and change messages being sent from the target control module. Second, the analog signal wires from the target sensor could be cut, and the controller could be inserted between the sensor and the control module. In either strategy, the controller would insert spurious data into the system to control the vehicle. The details and results of these two approaches are discussed in the following subsections.In order to successfully implement the first controller insertion strategy and take advantage of the information on the vehicle CAN bus, the Vector CANalyzer <cit.> system was used for the initial CAN message identification process. This system provides an excellent visual tool for watching CAN messages in real time. The tool displays a table of CAN messages with the rows organized by arbitration ID. The first column of the table indicates the time since the last message with a given arbitration ID was received. The second column lists the arbitration ID, and the third column shows the value for each byte of the CAN message. If a byte is changed when a new message is received, the byte is displayed in bold. Over time the byte fades to a light gray if that value stays the same. This was useful in identifying messages such as the accelerator pedal position, brake pedal position, steering wheel angle, and vehicle speed. The CANalyzer system is a great resource, but it is expensive and closed-source. Cheaper alternatives such as the Peak Systems PCAN <cit.> device can be used that have an open platform for development. An open-source solution for monitoring CAN traffic with the PCAN device is provided with the open-source software accompanying this work. Further discussion on the use of the PCAN device can be found in Section <ref>. Another alternative is to use a microcontroller with a CAN bus interface module to monitor and report CAN traffic <cit.>. Using the resources in the previous paragraphs, it was determined that CAN messages have two main functions: status and control. A status message reports the status of a vehicle component or condition, but does not control that component or condition. For example, a module will receive input from the wheel speed sensors and send the information on the CAN bus. Changing the data in this message will not result in a change of vehicle speed. A control message, however, is sent by a module to control a component or condition of the vehicle. For example, the movement of the wing mirrors is controlled by a CAN signal, when this message is changed the mirrors will move in response.§.§ CAN Message Injection A platform for injecting CAN messages was developed using the TI TM4C129XL microcontroller evaluation kit <cit.> and TI CAN transceivers <cit.>. The platform would connect to the CAN bus using the first controller insertion point, between the target module and the bus. The microcontroller was programmed to record and playback CAN traffic. More specifically, the microcontroller would receive the output from the control module and store it in memory. Upon user request, the output from the control module could be injected on the CAN bus. This platform was used to determine which messages, organized by arbitration ID, were generated by the target control module. Using information from the Auto Repair Center Research Database, it was determined that the Powertrain Control Module (PCM) sent a CAN message to the Transmission Control Module (TCM) regarding the accelerator pedal position. In addition, the PCM was the only control module that received the sensor signal from the accelerator pedal. For these reasons, the PCM was chosen as the target module for vehicle acceleration.The connector diagrams in the Wiring Guide were used to determine the CAN bus connections to the PCM. These wires were cut and routed to the CAN injection platform. The CAN messages output from the PCM were recorded and passed through to the CAN bus for an accelerator pedal press. The recorded data was then played back on the bus and the vehicle accelerated as expected. Additional code was added to the playback function to selectively playback messages based on the message arbitration ID. This was used to search the recorded data set and isolate the acceleration control message. The messages were separated into two groups based on their arbitration ID, and each group was played back to the vehicle separately. The group that resulted in vehicle acceleration was separated again into two smaller groups and the process was repeated until one message arbitration ID was left. The acceleration control message was determined to be the message with arbitration ID . Figs. <ref> and <ref> show successful CAN injections of the acceleration control message, and the resulting speed for a ramp and step input, respectively. Due to the similarities between the acceleration control message, and a throttle signal in a gas powered car, this message is called the throttle message for the rest of this work.Another approach to identify useful CAN messages for vehicle acceleration is by correlating the CAN messages with the vehicle speed. This approach was attempted by recording an accelerator pedal press and processing the data in Matlab. The CAN modules for the 2013 Ford Focus EV broadcast messages at prescribed frequencies as opposed to broadcasting in response to another signal. The speed message for the vehicle is broadcast at 100 Hz, which is the highest frequency messages are broadcast for this vehicle. Correlation was performed on messages of the same frequency, and the speed data was downsampled to perform correlation with messages at lower frequencies. The correlation returned a value between -1 and 1 for each byte of every message to indicate how closely correlated that byte was to the speed message. If the byte did not change during the recording the correlation returned . On the EV-HS CAN bus there are 102 different messages and each message contains 8 bytes. The bytes were sorted based on the absolute value of the correlation value to identify the highest positively or negatively correlated bytes. The bytes that had avalue were rejected and the final number of bytes being ranked was 341. The highest correlated byte ofwas byte 4, which had a correlation value of 0.2359 and was ranked 57 out of 341. The low correlation value and rank indicates that the throttle message would not have been identified using this strategy. In contrast, using the approach described in the previous paragraph, the throttle message was identified and was used to control vehicle acceleration. The investigation of the braking system concluded that a CAN bus message about pedal position would not actuate the hydraulic braking system. The pedal signal is sent directly to the Automatic Braking System (ABS) CAN module, which is the only CAN module on the vehicle that is connected to the hydraulic brake lines. From this it was concluded that braking could not be fully controlledthrough the CAN bus. The 2013 Ford Focus EV has an option to include park assist <cit.>. Though our specific vehicle did not include this option, it was determined that the Electric Power Assisted Steering (EPAS) system had the same part number and motor as the EPAS system in a vehicle with the park assist feature. This meant that the power steering motor would be powerful enough to turn the wheel at low speeds, and by extension, any speed. The Power Steering Control Module (PSCM) receives inputs from the CAN bus and the steering torque sensor located at the base of the steering column. The torque sensor uses a torsion bar to determine the amount of torque being applied by the driver, which is used by the PSCM to determine how much assist the power steering motor should provide. Simply, for a given torque input, less assistance would be provided by the EPAS system at higher speeds. Similar to the brake system, the input of interest is sent from a sensor to the module that performs the desired action. This led to the conclusion that the control of steering would have to be controlled through torque sensor input, and not through the CAN bus. In addition, the work by Miller and Valasek <cit.> <cit.> identifies the short comings of exploiting the park assist feature, namely, the park assist feature will cease to control the vehicle if the speed threshold is exceeded.§.§ Sensor Emulation The CAN bus injection method was unable to control braking and steering, so the sensor emulation method was explored. The accelerator pedal position sensor was analyzed to control vehicle acceleration, the brake pedal position sensor was analyzed to control the braking system, and the steering torque sensor was analyzed to control the vehicle steering. Fig. <ref> shows these sensors and the following paragraphs discuss the analysis. The accelerator pedal position (APP) sensor is located at the top of the accelerator pedal. There are six wires connected to the APP sensor, including two 5 V power wires with corresponding ground wires, and two signal wires. The sensor power pins were connected to a voltage source, and the signal wires were connected to an oscilloscope. It was determined that the sensor outputs two DC voltages similar to the output of potentiometers. Fig. <ref> shows the voltage levels of the two output signals in response to a pedal press. The third signal on the graph is a multiplier that relates the two signals. It is seen that V_1 ≈ 2V_2.The brake pedal position (BPP) sensor is located at the top of the brake pedal. There are four wires connected to the BPP, including a 5 V power, ground, and two signal wires. The BPP was connected to the voltage source and oscilloscope in the same manner as the APP. However, the BPP outputs two PWM signals instead of DC voltage levels. When the brake pedal is not pressed the duty cycles of the signals settle at 89% for signal 1 and 11% for signal 2. During a braking event the duty cycle for signal 1 decreases and the duty cycle for signal 2 increases at the same rate. Fig. <ref> showsthe two PWM signals for the BPP. The frequency of signal 1 is 533 Hz and the frequency of signal 2 is 482 Hz.The steering torque sensor is located at the base of the steering column. The sensor connects to the Power Steering Control Module (PSCM) on the CAN bus, which determines the amount of power steering assist to provide. The assist is provided by an electric motor connected to the steering rack. In the sensor, a torsion bar is used to connect two parts of the steering shaft, where the rotational displacement can be measured to determine the torque input by the driver <cit.>. Similar to the BPP, there are 4 wires connected to the steering torque sensor, including 5 V power, ground, and two signal wires. The steering torque sensor was connected to the voltage source and oscilloscope, and it was determined that the sensor outputs two PWM signals on the signal wires, where both signals settle at 50% duty cycle when no torque is applied on the steering wheel. Both signals have a frequency of 2.15 kHz. Similar to the brake PWM signals, the duty cycles always add to 100%, and the direction that the steering wheel is being turned determines which signal's duty cycle increases and which signal's duty cycle decreases. Fig. <ref> shows the two steering PWM signals.§.§ Safety and SecurityIn 1996, the OBD-II (On-Board Diagnostics) specification was required to be implemented on any new vehicle sold in the United States <cit.>. This specification gives owners and technicians the ability to diagnose issues on the vehicle. The specification standardized connectors, message formats, and frequencies. The OBD-II port on the 2013 Ford Focus EV connects to the EV-HS CAN bus, which is the same bus that the throttle message is sent from the PCM to the TCM.An attack platform was developed to inject arbitrary throttle messages through the diagnostics port. This attack method was important because, if successful, it would demonstrate that the acceleration of the vehicle could be controlled with limited intrusion. This differed from the approach in Section <ref>, as it does not require access to the target module, or that the CAN wires be cut and re-routed. Instead, this platform could be plugged into the OBD-II port and monitor the bus for the target message arbitration ID. Also, it would show that if an attacker was able to inject messages from any module on the EV-HS CAN bus, then arbitrary vehicle acceleration could be caused. This would stand in contrast to the findings in <cit.>, <cit.>, <cit.>, <cit.>, where the acceleration of the vehicle could only be controlled under specific preconditions, and required intrusive access to the CAN bus. The platform was connected to the CAN bus through the OBD-II port (other points on the bus could be used, as well) and monitored the traffic on the bus. The user determined a target message, in this case, the throttle message, and provided that message arbitration ID to the system. In Section <ref>, it was determined that the throttle message is included in the data frame associated with arbitration ID , and is broadcast at 10 Hz. The platform waited until a message was received with the corresponding arbitration ID, and would replace throttle message data with an arbitrary throttle command value. The platform was designed to only alter the parts of the message that relate to the throttle control. The inserted message would be sent 250 μs after the actual message, leaving 9.75 ms for the inserted message to be received and processed by the TCM. This allowed the inserted message to dominate the period and cause the vehicle to accelerate. Fig. <ref> shows the successful ramp injection through the OBD-II port and the resulting vehicle speed. Thus confirming the hypothesis that vehicle acceleration can be caused by injecting CAN messages through the OBD-II port, and therefore, could be caused at any other point on the bus. These results demonstrate a CAN bus security concern. If an attacker were able to access the CAN bus, physically, or by compromising another ECU, they would be able to effect the acceleration of the vehicle without causing any errors. Remote access to the vehicle, but not necessarily the requisite CAN bus, could be effected by compromising the Telematic Control Unit (TCU) or a wireless Tire Pressure Monitoring Sensor (TPMS). The TPMS sends a signal to the Body Control Module (BCM), which in turn transmits a message on the medium speed CAN bus (MS-CAN), while the TCU is connected to the I-CAN bus (it is unlikely, however, that compromising a sensor would allow for injection of arbitrary CAN messages onto the I-CAN or MS-CAN bus). These busses are connected to the EV-HS bus through a gateway module; transmitting a message from one bus to another, which would be required for either the TCU or TPMS to impersonate the PCM by passing APP messages, was not explored in this work. Regardless of the access approach, the driver is able to stop the unwanted acceleration by pressing the brake pedal, however, other works indicate that it is possible to make the vehicle ignore braking requests <cit.> <cit.> <cit.> <cit.>. This was not investigated as part of this work. Another security concern is that of a malicious technician. Since technicians will often access the OBD-II port when a vehicle is being serviced, it would be quite simple for them to leave an OBD-II injection platform connected to the OBD-II port. The acceleration control could be initiated remotely or by a timer, causing the vehicle to accelerate at a dangerous time. We present two remediation strategies that could be employed to help protect against this vulnerability. First, a simple change in the acceleration system architecture, such that the APP sensor connectsdirectly to the TCM, which is the actuating module. This would remove the need of a throttle message to be sent from the PCM to the TCM and effectively remove the attack surface. The second approach is through device fingerprinting for both the digital and analog signals <cit.> <cit.>. This would allow the receiving module to authenticate the transmitting module, and prevent this type of attack.§ LOW LEVEL CONTROL A simple overview of the control structure for the automated vehicle platform is shown in Fig. <ref>. The high level controller plans the path and provides the desired vehicle speed, v_desired, and desired steering wheel angle, θ_desired, discussion on the high level controller can be found in Section <ref>. The low level controllers discussed in this section are the inner loops that control vehicle speed and steering wheel angle. The vehicle commands are τ_cmd, APP_cmd, and BPP_cmd, and represent, steering torque, APP, and brake pedal position, respectively.The first step in the development of the low level controllers was to determine a model of the system being controlled. A system model is expressed as a transfer function relating the input to the output of the system. Models were identified to relate the accelerator and brake pedal inputs to vehicle velocity, and steering wheel angle to vehicle heading. The following subsections review the model identification approach, and low level controller design process for the Ford Focus. §.§ Longitudinal ModelThe longitudinal characteristics of the vehicle are affected by the APP sensor, and the BPP sensor. These two systems were tested and identified separately, then implemented together as a complete longitudinal model.In <cit.>, Dias et al. perform longitudinal model identification and controller design for an autonomous vehicle. This approach was examined for the current work, however, a more straightforward classical controls technique using step responses was ultimately used. Once the acceleration and braking systems were identified, a control system was developed for each input device. The control loops for accelerator and braking were connected by switching logic to determine whether a the accelerator or brakes should be used. A similar two loop control system with a switching logic component was used for the longitudinal controller in the current work. However, this work is an open-source project that uses the Robot Operating System (ROS) <cit.>. For the APP sensor system identification, the vehicle was placed on a dynamometer <cit.> and step inputs were initiated on the APP sensor from 4% to 15% at increments of 1%. Fig. <ref> shows the step responses for some accelerator pedal inputs. It was observed that for a given APP percentage the vehicle would eventually settle at a specific speed. The relationship between APP and speed can be described by a first order transfer function. The general equation for a first order transfer function, G(s), can be represented byG(s) = K/τ s + 1.Where K is the constant or equation that relates APP to vehicle speed, and τ is the system time constant. The equation for K was derived from a linear fit of the a scatter plot of max speeds from the step input, as shown in Fig. <ref>, and given by f(x) = 3.65x - 9.7. Where f(x) is the vehicle speed and x is the APP percentage. The test track (shown in Fig. <ref>) where the vehicle was operating is an oval track with sharp corners on the north and south side. The sharp corners and the short straightaways limit the vehicle operating speeds to between 15 and 25 mph for the initial automation. The τ value that best represented the vehicle response between 15 and 25 mph was chosen as the time constant for accelerator pedal input in the longitudinal model. Fig. <ref> shows the time constants for varying APP percentages. The time constant for the accelerator pedal input was chosen to be 7 seconds, as this best represented the system response for the nominal operating conditions.For BPP system identification, the vehicle was driven in a large, flat, asphalt area at speeds ranging from 5 to 25 mph at 5 mph increments. The vehicle was accelerated to the desired speed by a driver. Once the vehicle obtained the desired speed, an input to the braking system was initiated through the ROS setup discussed in Section <ref>. Step inputs were initiated ranging from 5% to 50% of BPP percentage at increments of 5% for each speed value. The speed data seemed to show a consistent rate of change for a given BPP percentage. To confirm this, the speed data was smoothed using a 5 point moving average, the derivative of the smoothed data was taken by calculating the difference between successive data points, and dividing by the elapsed time between data points. Fig. <ref> shows the vehicle deceleration due to a braking event. It was observed that the settling value for the deceleration rate was consistent for a given BPP percentage and varying speeds, which concluded that the longitudinal model was independent of current vehicle speed. This speed independence can be seen in Fig. <ref> where each line shows the deceleration rate for a given BPP percentage. At low BPP values the lines converge meaning that deceleration is unaffected by very small brake pedal percentages. However, at higher brake pedal percentages the lines show distinct deceleration rates regardless of the vehicle speed. To show the relationship between BPP percentage and deceleration, an average was taken for each BPP value across each of the speeds. The result of this operation is shown in Fig. <ref>.Similar to the APP model, the relationship between BPP and deceleration could be described by a first order transfer function. After analyzing the deceleration curves at different BPP percentages and for different speeds the system time constant, τ, was calculated to be 0.3 seconds. The equation that relates BPP to deceleration was determined by finding a curve fit algorithm for the curve in Fig. <ref>. This would result in an equation that would provide a BPP percentage for a desired deceleration rate. The equation for K is given byf(x) = -0.0018x^2 + 0.029x - 0.3768, where f(x) is the deceleration, and x is the BPP percentage. This equation is used to describe K from the general first order transfer function equation.§.§ Lateral ModelThe lateral model of the vehicle was determined by step response analysis. The model relates an input from the steering torque sensor to changes in the steering wheel angle. As discussed in Section <ref>, the torque sensor measures the torque applied by the driver, and sends that information to the PSCM. The PSCM activates the power assist motor that connects to the steering rack, and moves the wheels. The steering wheel angle is measured by a sensor in the steering wheel and output on the CAN bus at a high level of precision. Step inputs were initiated on the steering torque duty cycle signal ranging from 50% to 63% at 1% increments. Tests were performed at a large, flat, asphalt area with vehicle speeds ranging from 5 mph to 25 mph. Fig. <ref> shows the results of the step input tests performed at 25 mph. It was observed that a general first order transfer function could be used to describe the relationship between steering torque duty cycle and steering wheel angle. However, at lower speeds and higher torque values this observation is not valid. Fig. <ref> shows the step response of the steering system at 15 mph. At the higher torque values the steering wheel angles do not settle to a consistent steering wheel angle. It was also observed that the settling angles for a given steering torque duty cycle are not consistent for varying speeds. Therefore, the lateral model identification is speed dependent and would require a speed dependent limit on the steering torque duty cycle. Providing these characteristics, the system can still be modeled as first order transfer function for a given speed.The steering data was analyzed in order to determine the gain equation, K, and the time constant, τ. Time constants were calculated for each step input response and for each speed. Fig. <ref> Bottom shows the time constants for given steering torque duty cycles. Each of the lines indicates the speed at which the test was performed. It can be seen that at low speeds and low duty cycles the time constants are not consistent. But at higher speeds the inconsistencies lessen. A time constant, τ, of 0.2 seconds was chosen to optimize for typical vehicle operation.Since the lateral system was found to be speed dependent, the gain equation K must also be speed dependent. The step input tests were performed at 5 mph increments so a gain equation K would be found for each speed value. These gain equations relate steering torque duty cycle to steering wheel angle. Fig. <ref> shows the settling angles for varying steering torque duty cycles when the vehicle was traveling at 25 mph. A curve fit approximation was completed for this data set, and a solution was determined by solving the given equation. For this data set, the given equation for K is f(x) = 59.4x^2 - 6802.7x + 195084.5, where f(x) is the steering wheel angle and x is the steering torque duty cycle.Figs. <ref> and <ref> show the step response of the vehicle due to steering torque input signals. The graphs do not include step input values below 58% because the step responses at such values had little effect on the steering wheel angle. This exposed a deadband in the response from the steering torque sensor input to the steering wheel angle. A deadband compensation algorithm was implemented to mitigate the effects of this non-linearity. As shown in Fig. <ref>, the deadband compensation code was executedjust before the signal was sent to the vehicle. If the torque input value was greater than 50%, then τ_cmd = B_max + τ - 50/τ_max - 50(τ_max - B_max)was used to compensate for the deadband. If the torque input value was less than 50%, thenτ_cmd = B_min + 50 - τ/50 - τ_min(τ_min - B_min)was used to compensate for the deadband. Where τ_cmd is the torque command sent to the vehicle, τ is the value received from the PI controller, B_max is the upper limit of the deadband, B_min is the lower limit of the deadband, τ_max is the maximum allowed value for the steering torque signal, and τ_min is the minimum allowed steering torque signal. For the deadband on the 2013 Ford Focus EV, the upper and lower limits were 55% and 45%, and the maximum and minimum values for the torque signal were 64% and 37%, respectively.§.§ PI Controller DesignLow-level control loops were designed to control vehicle speed and steering wheel angle. The desired speed and desired steering wheel angle would be input to the low-level control loops from a user or high-level controller. The low-level longitudinal controller interfaced with the accelerator and brake pedals to effect vehicle speed. A separate loop was designed for each vehicle input, and switching logic was used to choose whether the acceleration or brake loop would be used. The low-level lateral controller would receive the desired steering wheel angle and determine the appropriate input to the steering torque sensor to achieve the desired angle. A Proportional Integral (PI) Feedback Controller was implemented for longitudinal and lateral control. Fig. <ref> shows a basic PI Feedback Controller for a first order system. The transfer function block represents the vehicle and contains the system model. The 1/K block effectively cancels out the gain equation K, and helps relate the speed error to a vehicle input. For example, in the longitudinal controller, the K equation receives the APP as an input, and outputs speed. Therefore, the input to the transfer function block must be an APP value. However, the control loop is calculating a speed error, so the output of the PI block is a speed value. The 1/K block translates the speed value into appropriate APP value. Since the K and 1/K can be combined to equal 1, they can be ignored in the loop equation. The open loop transfer function of this system is then given byG_OL(s) = 1/(τ s + 1 ). Closing the feedback loop and adding the PI controller givesG_CL(s) = k_p/τ( s + k_i/k_p) /s^2+ ( 1/τ + kp/τ) s+k_i/τ. The system is stable if the real part of the closed-loop poles are negative. Solving for the closed loop poles and zero yields s = -(kp + 1) ±√((k_p+1)^2 - 4k_iτ)/2τ,ands = -k_i/k_p,respectively. From these equations it can be determined that if k_p, and k_i are positive the system will be stable.The second order transfer function obtained through closing the loop can be written, in a general form, as ω_n^2/s^2+2ζω_n s + ω_n^2. Where ζ is the damping coefficient and ω_n is the natural frequency. The damping coefficient determines whether the system will be underdamped, overdamped, or critically damped and the natural frequency helps to determine the time constant for the system. The time constant, τ, is given by the equationτ= 1/ζω_n. Values were chosen for the damping coefficient and the time constant to define the system behavior. From these values one can determine the appropriate k_p and k_i for the system. The equations for k_p and k_i are given byk_p = τ(2ζω_n - 1/τ),andk_i = τω_n^2, respectively. Table <ref> shows the calculated values for each of the control inputs, where the τ_car column shows the time constants found during model identification and are internal vehicle parameters.§ GPS-BASED HIGH LEVEL CONTROLThe high level controller was designed to take a desired trajectory or path, and provide appropriate inputs for the low-level controllers. There are a number of high level control strategies for manned and unmanned vehicle control <cit.> <cit.> <cit.>. The control strategy should be determined by the system characteristics and the system objectives. This platform was to be used on a ground vehicle to track a desired trajectory and was determined to be differentially flat <cit.> for the chosen states. A differentially flat system is one in which the output is a function of the system states, the input, and the derivatives of the input, and both the states and the input are functions of the output and the derivatives of the output. Therefore, a simple high level controller using the properties of a differentially flat system, and state feedback control were chosen <cit.>. An example control system design for a differently flat Unmanned Aerial Vehicle (UAV) system was demonstrated in <cit.>.The following paragraphs discuss differential flatness, and the high-level controller architecture shown in Fig. <ref>.§.§ Differential FlatnessA system is said to be differentially flat if there exists an output vector y in the form ofy = h ( x,u,u̇,ü,....,u^(r)),such that,x = ϕ(y,ẏ,ÿ,....,y^(r)),u = α(y,ẏ,ÿ,....,y^(r)),where h, ϕ, and α are smooth functions. The output of the differentially flat system can, therefore, be defined by a function of the system states, the control input, u, and the derivatives of u. The state vector and control input vector can each be described by a function of the flat output, y, and its derivatives.The state vector for this system was chosen to bex = [ p_n; p_e ],where p_n is the vehicle position in the north direction and p_e is the vehicle position in the east direction. The system input, u, is given by u = [ v_n; v_e ],where v_n is the velocity in the north direction and v_e is the velocity in the east direction. The system output, y, is given by y = [ p_n; p_e ].From equations <ref>–<ref> it can be concluded that this is a differentially flat system and can be defined by equations <ref> and <ref>. The dynamic model of the system is then given by ẋ = Ax + Bu= [ 0 0; 0 0 ][ p_n; p_e ] + [ 1 0; 0 1 ][ v_n; v_e ], and the output, measured by an RTK-GPS system, is given by y= Cx = [ 1 0; 0 1 ][ p_n; p_e ].§.§ Control ArchitectureA virtual target scheme was selected for path and trajectory control. The virtual target data was gathered by driving the vehicle around the test track and recording the RTK-GPS data at 10Hz with 2 cm resolution. The recorded data contained the p_n, p_e, v_n, and v_e data and was replayed in the system as a virtual target. The state, input, and output vectors of the virtual target are denoted as x_t, u_t, and y_t respectively and p_nt, p_et, v_nt, and v_et are the positions and velocities of the virtual target. The goal of the virtual target scheme is to minimize the difference between the system states and the virtual target states. The difference between the vehicle states and target states is given by x = x - x_t, and the input difference is given by u = u - u_t. Therefore, the error model is given by ẋ = Ax + Bu, where u is the commanded velocity vector. Solving for u, and using a state feedback control strategy, where u = -kx gives u = u_t - kx. The Linear Quadratic Regulator (LQR) method was used to find the optimal gain value for k. The entries in the system input vector, u, are notated by, u_1 and u_2, and are used to obtain desired velocity by v_d = √(u_1^2 + u_2^2), and desired heading by ψ_d = tan^-1(u_1/u_2). Through calculation and experimentation a relationship was determined between the desired heading, ψ_d, and the desired steering wheel angle, θ_d. Such that, θ_d = k_dψ_d. The desired steering wheel angle and the desired velocity are sent to the low level controllers discussed in Section <ref>.§ TESTBED PLATFORM OVERVIEW A versatile and robust testbed platform is required to enable full-sized autonomous vehicle research. The platform was designed to enable vehicle automation for both the CAN injection and the sensor emulation approaches discussed in Section <ref>. For the CAN injection approach the platform was able to monitor the CAN bus and inject the desired packets, whereas for the sensor emulation approach the platform required access to the output lines of the sensors to be emulated. In order to proceed to autonomy the platform had the ability to sense the environment, determine vehicle location, communicate with the vehicle, and monitor the CAN bus. A computer running Ubuntu and the Robot Operating System (ROS) was used to communicate between the platform components. The computer was connected to a microcontroller to allow communication with the vehicle. Fig. <ref> shows a diagram of the autonomous system, including the ROS software architecture, hardware connections to devices, and the vehicle interfacing hardware. A ROS-based platform was chosen for ease of use, modularity, and sensor interfacing packages. ROS is an open source framework that encourages collaboration between researchers. People can contribute to the ROS effort by creating software packages that interface with common sensors and provide tools for development. For example, an open-source software package for ROS was provided by Swift Navigation to interface with the Piksi RTK-GPS units <cit.>, which helped expedite development time for this project. The ROS architecture components used in this project are packages, nodes, and topics. A ROS package is a collection of executable files used to complete a task. Generally packages are used to compartmentalize similar parts of a project. ROS nodes are the executable files in a ROS package that can be written in C/C++ or Python. A ROS topic is a way to transfer data between nodes. Any node can publish data to a topic, and any node can subscribe to that topic to receive the data. In this sense, the ROS topic acts as a bus to transfer data. The following subsections detail the Interfacing Architecture, Sensing Architecture, and the Computational Architecture of the automation platform.§.§ Interfacing ArchitectureInterfacing devices are critical to the success of vehicle automation as they provide a way to send commands to the vehicle, and monitor the vehicle for feedback. A PCB (shown in Fig. <ref>) was designed to provide a connection between the microcontroller and the vehicle. The following subsections discuss the microcontroller and associated hardware, and the other interfacing devices used for this platform.§.§.§ Microcontroller and Associated HardwareThe TI TM4C129XL evaluation kit was the chosen microcontroller platform because it offered multiple CAN bus interfaces allowing for a combination of CAN injection and sensor emulation from the same board <cit.>. The microcontroller receives input from the computer through a UART module. After performing the appropriate computations, PWM signals are generated and appropriate DC voltage levels are determined for vehicle input. The control signals pass through a variety of circuit components to prepare the signals for vehicle injection. The signals are terminated at solid state relays that select either the original vehicle signal, or the generated signal to be sent to the vehicle. The user determines which signal is sent based on a mode switch input to the microcontroller. The sensor emulation approach requires four PWM signals to be generated by the microcontroller. The signals are passed through a level shifter to shift the amplitudes from 3.3 V to 5 V, and then sent through operational amplifiers in a voltage follower configuration to help drive the signals. The PWM signals are then terminated at the normally opened terminals of solid state relays.The accelerator pedal input is generated by a Digital to Analog Converter (DAC) that receives an I2C signal from the microcontroller. The DAC converts the digital communication to an analog voltage level, and sends it to an active filter IC to clean the signal and perform a gain two operation to provide the two output signals. The generated signals then terminate at the normally open terminals of the solid state relays. Another key feature of the PCB is the safety circuit. Next to the driver there is a mode switch and an Emergency Stop button. The mode switch allows the driver to switch between Manual Driving Mode and Autonomous Mode. The power and solid state relay control signals are routed to the front of the vehicle so the safety driver can switch between operating modes or press the Emergency stop button to prevent power from reaching the circuit. The vehicle's original sensor signals are connected to the normally closed terminals of the solid state relays, so removing the power returns the vehicle to Manual Mode. Power to the circuit is provided by a NewMar DC Uninterruptible Power Supply (UPS) which connects to the 12 V car battery, and provides safe and stable voltage levels for the circuit operations <cit.>. The NewMar UPS also has an internal backup battery. The voltage level is stepped down to ± 8 V, 5 V and 3.3 V, and distributed across the custom PCB. The UPS is shown in Fig. <ref>. §.§.§ Other Interface Devices The Peak Systems PCAN device was chosen to monitor CAN traffic <cit.>. The PCAN device can connect directly to the vehicle's OBD-II port, and provides serial output over USB. A picture of the PCAN device is shown in Fig. <ref>. Every message on the connected CAN bus is received and sent serially to the computer. The user can determine which CAN data packets are important to operation, and ignore the rest. One approach to determine necessary CAN data packets is detailed in Section <ref>. Instead of using the CANalyzer system to monitor CAN traffic, the PCAN device can be used to record CAN traffic for a desired event (e.g. vehicle acceleration). The CAN data can be replaced section by section until a message or set of messages is isolated. Additional information on the use of the PCAN device for system feedback is given in Section <ref>. A USB-to-serial device was used between the microcontroller and computer to enable communication. The control system determines the appropriate inputs to the vehicle and sends the commands to the microcontroller. The device receives the signal from the USB port and sends it to a UART module on the microcontroller. More information about the microcontroller and control system is given in Section <ref>. The TI SN65HVD230 CAN Transceiver Breakout Board <cit.> <cit.> was used to connect the microcontroller to the CAN bus. This board provided a direct connection with the vehicle CAN bus that can be used for monitoring and injection.§.§ Sensing Architecture The Swiftnav Piksi RTK-GPS unit <cit.> <cit.> was chosen to provide position and velocity estimates for the vehicle. A picture of the Swiftnav Piksi unit is shown in Fig. <ref>. Swiftnav provides an inexpensive, open-source RTK-GPS solution that provides GPS measurements at 10 Hz and ±2 cm accuracy. In order to achieve such high accuracies the system must have a base station with an RTK-GPS unit, and a second unit mounted to the vehicle. The two units communicate with radio transceivers at 955 MHz. The unit mounted on the car connects to a computer over USB and provides position and velocity data. The base station simply needs a 5 V power supply. The 2013 Ford Focus EV has an array of sensors on the vehicle that monitor everything from wheel speed to tire pressure. However, the vehicle does not have high precision wheel encoders, an RTK-GPS receiver, or inertial measurement sensors (IMU's) that could be useful for vehicle automation. The sensor data is typically received by a module and sent on the CAN bus. Using the PCAN device described in Section <ref>, the on-board sensor information can be provided to the rest of the automation platform. For autonomous driving, the accelerator pedal position sensor, brake pedal position sensor, steering wheel angle sensor, and vehicle speed data are used for feedback in the low-level controllers. The sensor data broadcast on the CAN bus does not provide the information in empirical units, and sometimes the data is masked with other signals. An important aspect to the sensing architecture is the conversion from CAN bus messages to useful units. These conversions could be found experimentally for each message found on the CAN bus, but for the purposes of this testbed the vehicle speed was the only message converted to empirical units (MPH). The vehicle speed is found on the message with arbitration IDon bytes 7 and 8, and is represented by a 16-bit value where byte 7 is the upper byte and byte 8 is the lower byte. When the vehicle was not moving the vehicle speed was represented ason the CAN bus. The decimal representation of this constant, 45,268, is subtracted from the16-bit vehicle speed to align the 0 MPH value. The vehicle was driven with the RTK-GPS units to provide a reference for the vehicle speed, and it was determined that the CAN value would then need to be divided by 54 to achieve an accurate measure of speed. This process is summarized by v_mph = (b7 << 8) + b8 - 45268/54,where b7 and b8 are the integer representations of bytes 7 and 8 from the CAN message with arbitration ID , and the << operator represents a left bit shift.§.§ Computational ArchitectureThe computational architecture includes the code required to combine sensor information, controller commands, and prepare command insertion. The two computational platforms used in this system are the microcontroller and ROS. These platforms are discussed in the following sections and code for these platforms can be found at <https://github.com/rajnikant1010/EVAutomation>.§.§.§ Microcontroller SoftwareThe code for the TI TM4C129XL was written in C and took advantage of the built in functionality of the TivaWare Peripheral Driver Library from Texas Instruments <cit.>. Table <ref> shows the peripherals used and their functionality. The following paragraphs discuss the microcontroller code. The microcontroller receives a serial packet from the computer in the form shown in Fig. <ref>. The first byte is always , the second byte gives the number of bytes in the payload, the payload contains the steering, braking and acceleration commands to be sent to the vehicle, and the last two bytes is a 16-bit Cyclic Redundancy Check (CRC) using the CRC16-CCITT algorithm to ensure data integrity. Once received, the CRC is calculated to ensure correct data, and the payload values are stored in appropriate variables. All input commands are normalized between zero and one. For PWM signals the normalized value represents the duty cycle of the signal, where 0.5 represents 50% duty cycle.§.§.§ ROS ArchitectureThe ROS architecture consists of a series of packages, nodes, and topics <cit.>. A ROS package can be used to modularize code. For example, the four packages used for this project were swiftnav_piksi (GPS), focus_serial (serial communication), pcan (CAN interface),and focus_control (high- and low-level control for GPS path following). Each of these packages has at least one node and publishes or subscribes to certain topics. A program file (either C/C++ or Python) is written for each node and when a node publishes information to a topic, other nodes can subscribe to that topic and receive the information that was published. The following paragraphs will discuss each of the ROS packages, nodes, and topics used for this system. The swiftnav_piksi package has only one node. This node initializes a connection with the GPS unit, receives the raw GPS data, and packages the data into an Odometry message. An Odometry message is a standard ROS message that contains information about position, velocity, angular positions, angular velocities, quaternions, and covariance matrices. The Odometry message is published to a topic called gps_data. The pcan package has one node called can_publisher that receives CAN data from the PCAN device and parses the requested data. The CAN data is translated to useful units for the given control strategy, and published to a ROS node called can_data. The can_data topic could provide as much CAN information as the user would like. For the purposes of automating this vehicle, the CAN data of interest is vehicle speed in MPH and steering wheel angle. The focus_control package has two nodes: gps_input and controller. The gps_input node is written in python, and it reads afile with information for a virtual target. The position and velocity data for the virtual target are published to the gps_input topic. The controller node subscribes to the can_data, gps_data, and gps_input topics. From the data in these topics it performs a path following algorithm described in Section <ref>. The path following algorithm provides a desired turn rate (steering wheel angle) and desired velocity for the vehicle.The low level PI controllers discussed in Section <ref> receive the desired values and calculate the appropriate vehicle commands. The accelerator pedal position, brake pedal position, and steering torque duty cycle commands are then published to the serial_data topic.The focus_serial package has one node called serial_transmitter. This node subscribes to the serial_data topic and sends the accelerator pedal position, brake pedal position and steering torque duty cycle information to the microcontroller. Before the data is sent, a Cyclic Redundancy Check (CRC) is performed and a checksum is added to the serial message. The microcontroller checks the CRC to verify the accuracy of the data before sending the requested commands to the vehicle.§ EXPERIMENTAL RESULTS Experiments were conducted to determine the results of the autonomous vehicle platform. A video of the system in operation and the development process can be found at <https://youtu.be/7ohWIwb6KfM>. The low level controllers were tested with given desired steering angles and velocities. The low level steering controller was improved by implementing a deadband compensation algorithm. The results for the low-level controllers are given in Section <ref>.After the low level controllers were verified through testing, experiments were conducted for full vehicle automation by including the high-level path following controller. The automation results are shown in Section <ref>.§.§ Low Level Controller The low level controllers provide speed and steering wheel angle control through the user input signal. For the steering controller, the steering wheel torque sensor signal was used to change the position of the steering wheel. As discussed in Section <ref>, the control loop was designed such that, given a desired angle, the controller would change the steering torque value until the desired angle was achieved. This controller was tested using a step input and a graph of the result can be seen in Fig. <ref>. The y-axis is the steering wheel angle as represented by a Hex value on the CAN bus. A desired steering wheel angle ofwas used as an input to the controller node of the ROS platform. The step response had a maximum value of , representing an overshoot of 9.65%. The resulting time constant of the system was 1.86 seconds. The desired behavior was for the system to be critically damped and have a time constant of 0.33 seconds. After implementing deadband compensation, the lateral controller improved. Fig. <ref> shows the step response of the lateral controller with deadband compensation. The maximum value for this response is 2,113, which represents a 5.65% overshoot, and has a time constant of 1.1 seconds. The result of the longitudinal controller during autonomous driving is shown in Fig. <ref>. The velocity error is shown in Fig. <ref>, where the average error for the autonomous test was 0.34 mph (0.151 m/s).§.§ Path and Velocity Errors of Autonomous DrivingThe high level controller was tested on the 2013 Ford Focus EV. A recorded data set of the vehicle being driven around the track was used as a virtual target for the differential flatness algorithms discussed in Section <ref>. The path error was calculated using the euclidean distance from the vehicle to the closest point on the desired path and is given byd_er = √((p_n - p_nt)^2 + (p_e - p_et)^2).A single lap autonomous test was performed with the high- and low-level controllers, and the data was recorded. The trajectory is shown in Fig. <ref>, and the velocity plot is shown in Fig. <ref>. The path and velocity error plots are shown in Fig. <ref>, where the average path error was 0.43 meters. However, if the vehicle was permitted to immediately perform a second lap of automated driving, the averagepath error for the second lap was 0.28 meters, boasting a 34.9% improvement over the first lap. § CONCLUSIONThe testbed discussed in this work provides an open source vehicle automation system that takes advantage of the vehicle's native communication and control systems to achieve vehicle control. This method was preferred to more expensive and more intrusive platforms on the market today. The overall cost of the components needed to automate a stock vehicle was $2,315, the itemize cost breakdown can be seen in Table <ref>. The automated systems could be used to further research in a variety of fields, including control systems, vehicle security, and wireless power transfer.In addition to the GPS-based path following, the low cost automation testbed is to validate a vision based path following algorithm as detailed in <cit.>. The experiment video of vision based-path following is available at <https://youtu.be/NpAUcNh4QUY>. Future work for this project include refining low-level controllers and testing new high-level controllers to improve lateral path accuracy to ±7 cm. To help improve the consistency and avoid errors introduced from GPS, a stereo-vision based controller will be implemented for lane detection. § ACKNOWLEDGMENTThe authors would like to thank the EV Automation Team from Utah State University: Cameron Sego, David Petrizze, Hunter Buxton, Tyler Travis, Aaron Kunz, Austin Goddard, Gregory Vernon, Daniel McGary, Clint Ferrin, Ishmaal Erekson, Nicolas Jugganaikloo, Ryker Shirner, and Zachary Garrard. This project would not have been possible without their hard work and dedication to the team goals. The authors would also like to thank the individuals who help manage and coordinate the research at the EVR: Ryan Bohm, Joshua Rambo, and Paul Rau. IEEEtran | http://arxiv.org/abs/1708.07771v1 | {
"authors": [
"Austin Costley",
"Chase Kunz",
"Ryan Gerdes",
"Rajnikant Sharma"
],
"categories": [
"cs.SY"
],
"primary_category": "cs.SY",
"published": "20170825151946",
"title": "Low Cost, Open-Source Testbed to Enable Full-Sized Automated Vehicle Research"
} |
College of Physics and Materials Science, Henan Normal University, Xinxiang 453007, China State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, ChinaUniversity of Chinese Academy of Sciences, Beijing 100049, China State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, ChinaUniversity of Chinese Academy of Sciences, Beijing 100049, China Hubei Key Laboratory of Pollutant Analysis and Reuse Technology, College of Chemistry and Chemical Engineering, Hubei Normal University, Huangshi 435002, China [email protected] State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, ChinaWe studied the high-order harmonic generation (HHG) from 2D solid materials in circularly and bichromatic circularly polarized laser fields numerically by simulating the dynamics of single-active-electron processes in 2D periodic potentials. Contrary to the absence of HHG in the atomic case, circular HHGs below the bandgap with different helicities are produced from intraband transitions in solids with C_4 symmetry driven by circularly polarized lasers. Harmonics above the bandgap are elliptically polarized due to the interband transitions. High-order elliptically polarized harmonics can be generated efficiently by both co-rotating and counter-rotating bicircular mid-infrared lasers. The cutoff energy, ellipticity, phase, and intensity of the harmonics can be tuned by the control of the relative phase difference between the 1ω and 2ω fields in bicircularly polarized lasers, which can be utilized to image the structure of solids.42.65.Ky, 42.65.Re, 72.20.Ht High-order harmonic generation from 2D periodic potentials in circularly and bichromatic circularly polarized laser fields Xue-Bin Bian December 30, 2023 ==========================================================================================================================§ INTRODUCTIONHigh-order harmonic generation (HHG) from atomic and molecular gases has been studied extensively <cit.>. It has been utilized to generate attosecond laser pulses. The mechanism is well described by the three-step recollision model <cit.>. Recently, more attention has been attracted to the HHG from solids <cit.> with the development of long-wavelength lasers. Solid HHG demonstrates novel characters different from the HHG from gases. For example, linear cutoff energy dependence on the amplitude of the laser field <cit.>, multi-plateau structure in the HHG spectra <cit.>, and different laser ellipticity dependence <cit.>. However, mechanisms of HHG from solids are still under debate. Inter- and intra-band transition models <cit.> are proposed. Three-step model in coordinate space <cit.> and step-by-step model in vector k space <cit.> are investigated. The drawback of the solid HHG is the low damage threshold of solid materials. Many efforts have been undertaken to enhance the yield of HHG. For example, two-color laser fields <cit.> and plasmon-enhanced inhomogeneous laser fields <cit.> are used to manipulate the HHG process, especially for the enhancement of the second plateau of the HHG spectra <cit.>.In circularly polarized laser fields, HHGs are absent in the atomic systems due to the non-recollision spin motions of electrons in the classical picture <cit.> and forbidden transition in the selection rules in the quantum picture. However, HHG occurs in molecular systems in circularly polarized lasers because of additional recollision centers <cit.>. In the solid systems, especially for 2D materials <cit.>, because of the multi-center periodic potential wells and delocalization of the wave packets, HHGs can be efficiently produced in circularly polarized driving laser fields with the polarization plane in the same plane of 2D solids <cit.>. The intensity of HHG as a function of the driving laser ellipticity has been studied recently <cit.>. In this work, we study the ellipticity of HHG from solids driven by the external circular and bicircular laser fields.In bicircular laser fields <cit.>, especially for the counter-rotating case, circular HHGs with different helicities can be efficiently generated in atomic and molecular systems <cit.>. This has been experimentally demonstrated recently <cit.>. It can also be used to illuminate chirality <cit.> andmolecular symmetries <cit.>. It is also possible to generate isolated elliptical and circular attosecond laser pulses <cit.>. However, the HHG from solids in bicircular laser fields is rarely investigated. In this work, we study the electron dynamics in 2D periodic potentials <cit.> to simulate the HHG process in solids. Atomic units are used throughout. § NUMERICAL RESULTS BY SOLVING THE TIME-DEPENDENT SCHRÖDINGER EQUATION In the single-active-electron approximation, the time-dependent Schrödinger equation(TDSE) can be written asi∂Ψ(x,y,t)∂ t=(p_x^22+p_y^22+V(x,y)+xE_x(t)+yE_y(t))Ψ(x,y,t)where E_x(t) and E_y(t) are the x and y components of the laser fields. V(x,y) are two-dimensional periodic Gaussian potentials <cit.>. The form for one unit cell isV_c(x,y)=-V_0 exp{ -[α_x(x-x_0)^2a^2+α_y(y-y_0)^2a^2]}where V_0 represents the maximum depth of the potential well, a is the length of the unit cell, (x_0, y_0) are the coordinates of the center of the potential well. In this work, a=4 a.u., V_0=3π^2/2a^2, α_x=α_y=6.5. As illustrated in Fig. <ref>, this 2D system has C_4 symmetry. The electrons in the valence band are mainly distributed in each potential well, while the the electrons in the conduction bands are more delocalized.Based on Bloch's theorem <cit.>, the band structure of the system can be obtained by diagonalization of the field-free Hamiltonian matrix in reduced Brillouin zone. The results are shown in Fig. <ref>. Γ, X and M represent the high symmetry points illustrated in Fig. <ref>(a).We solve the TDSE in the real space in the range x,y∈[-200,200] a.u. The wave functions are expanded by B-splines:Ψ(x,y,t)=∑_i,jC_i,j(t)B_i(x)B_j(y) The time-dependent wavefunction is obtained by using the Crank-Nicholson method <cit.>. An absorbing function is used at the boundary to remove artificial reflection. The laser-induced currents along x and y axes are: j_x(t)= ⟨Ψ(x,y,t)|p̂_x|Ψ(x,y,t)⟩, j_y(t)= ⟨Ψ(x,y,t)|p̂_y|Ψ(x,y,t)⟩. The HHG spectra are calculated by Fourier transforms of the above currents. The obtained complex xy components √(P_x(Ω))exp[iΦ_x(Ω) ] and √(P_y(Ω))exp[iΦ_y(Ω) ] of harmonic photon with energy Ω can be used to extract its ellipticity ε and phase δ. They are defined as <cit.>:ε = tanχ,wheresin(2χ)= sin(2γ)sinδ,tanγ = √(P_y(Ω)/P_x(Ω)),δ = Φ_y(Ω)-Φ_x(Ω).The values of ε and δ are in the range of [-1, 1] and [0, 2π], respectively. ε>0 represents right polarization, while ε<0 stands for left polarization.§ HIGH-ORDER HARMONIC GENERATION IN CIRCULARLY POLARIZED LASER FIELDSThe circularly polarized laser field is written in the following form:E_x(t)=E_0f(t)sin(ω t+ϕ), E_y(t)=E_0f(t)cos(ω t+ϕ).where the pulse envelope isf(t)=cos^2(π(t-τ/2)/τ),τ is the total pulse duration, which is set to be 10 cycles in this work. E_0=0.005 a.u. The wavelength is λ=3.2 μm. At first, the initial state in our TDSE evolution is the single state on top of the valence band, i.e., the state at M point in Fig. <ref>(b). The HHG spectra are presented in Fig. <ref>. The ground state of atoms is localized to the atomic core. Without recollision of ionized electron to the parent ion in the circularly polarized driving lasers, HHG can not be produced no matter how intense the lasers are. Different from the absence of HHG from atoms in circularly polarized driving laser fields, clear HHG signals with long plateau are generated in our simulations. Due to the multiple potential wells, the electron states in both the valence band and conduction band are delocalized. The transition probability between them is high even in the circularly polarized laser fields. In Fig. <ref>, strong odd harmonics are dominant. One can find that circularly polarized odd harmonics with order less than 19 are generated in Fig. <ref>(b). This energy is close to the minimum band gap between the valance band and the first conduction band: E_c-E_v=0.25 a.u. This is in the perturbation regime. The harmonics are mainly from the intra-band transitions from previous studies <cit.>. They reflect the global motion of the electron wavepackets along the conduction band <cit.>. The motion of electrons closely follows the spinning electric field from the circularly polarized driving lasers. Consequently, circular HHGs are produced. Different from the transition selection rules in the atomic case, circular harmonics with different helicities are allowed to be produced from solids with different symmetries <cit.>. The selection rules for 2D solids with C_4 symmetry in circular driving lasers in <cit.> explain well why the helicity of HHG changes alternately in Fig. <ref> (b). For harmonics with order N>20, they are generated from the inter-band transitions with a plateau structure in the non-perturbation regime <cit.>. The harmonics reflect the local instantaneous polarization between the conduction band and the valence band of the system <cit.>. The inter-band transitions do not closely follow the driving circular fields. Elliptically polarized harmonics are presented instead. One may also notice the weak even-order harmonics with N<20 in Fig. <ref>, which are forbidden in the transition selection rules in <cit.>. However, the rules are based on group theories, which are applicable in global intraband oscillations rather than local interband polarization. To check if it is from the asymmetry introduced by the finite laser pulses, we have increased the pulse duration to 20 cycles. The weak even harmonics remain, which suggests that they are from the influence of interband transitions, similar to the case of interband-transition-induced even-order HHG in linearly polarized laser fields <cit.>. The cutoff energy is determined by the maximum energy band gap between the first conduction band and the valence band at all possible wave vector k⃗(t) from our proposed model <cit.> in the momentum space. k⃗(t)=k⃗(0)+A⃗(t), where A⃗(t) is the vector potential of the laser fields and | k(0)_x|=| k(0)_y|=π/a (M point). From Fig. <ref>, the value of the maximum band gap between the valence band and the first conduction band is around 41ω, agreeing well with the cutoff energy in Fig. <ref>, suggesting that the HHG model for interband transition <cit.> is valid even in circularly polarized laser fields.The bandgap along X-M is also small as illustrated in Fig. <ref>. To include the contribution of all valence states, we use a normalized superposition of all valence states as the initial state to do the TDSE simulations. The results are presented in Fig. <ref>. One may find that the harmonics with order below 20 do not have perfect left or right circular polarization as shown in Fig. <ref>, but elliptical polarization. It is due to the coupling of different valence states. The similar thing occurs in the HHG from atoms in bi-circular lasers by involving the Rydberg states <cit.>. Dynamic symmetry breaking of the system introduced by the superposition state will change the polariztion of HHG. It may also explain why the experimentally measured ellipticity of HHGis not exactly 1 <cit.>. In all the following calculations, we use the above superpostion state as the initial state for the TDSE simulations.§ HIGH-ORDER HARMONIC GENERATION IN CO-ROTATING TWO-COLOR CIRCULARLY POLARIZED LASER FIELDS Two-color linearly polarized lasers have been shown to be efficient tools to generation high harmonics <cit.>. In this work, we investigate the role of two-color circularly polarized lasers in HHG. The co-rotating two-color 1ω and 2ω laser fields are written asE_x(t)=E_0f(t)(sin(ω t)+sin(2ω t+ϕ)), E_y(t)=E_0f(t)(cos(ω t)+cos(2ω t+ϕ)).The intensities of the two lasers are set to be the same. The pulse shape and the durations are the same as those in Eq. (<ref>). The phase ϕ will rotate the electric fields and vector potentials of the laser fields. The rotation angle in the Lissajous curve is equal to this phase. It plays no roles in the HHG from atoms due to the isotropy, while it plays important roles in HHG from solids. The HHG spectra with different phases ϕ are illustrated in Figs. <ref> and <ref>. For the HHG from atoms, this co-rotating scheme is not efficient <cit.> since the recollision probability is still small. However, a long plateau with clear cutoff beyond the minimum band gap can be observed from the HHG spectra in the above figures. This scheme of co-rotating two-color circular lasers is also an efficient way to generate HHGs. Different from the case of circular driving lasers, even-order harmonics are obviously generated and their intensities are comparable to those of their neighboring odd harmonics. One may find that the harmonics with order N<19 below the minimum band gap are not circularly polarized either, even though some of their ellipticities are close to 1. We also use the single state on top of the valence band to do TDSE simulations (not shown here), no circular harmonics are observed. It is mainly attributed to the different laser strengths along x and y axes as illustrated in the inset of Fig. <ref>.In the multiphoton picture <cit.>, harmonic with order N may come from different combinations of N_1 ω and N_2 2ω photons (N_1+2N_2=N) with different strengths and phases. The interference of these channels makes the phases of the harmonics complex. One may find that the harmonic with order 8 shown in Fig. <ref> is approximately linearly polarized.Orientation dependent harmonic generation in linearly polarized pulses is a good way of probing solid structure <cit.>. It can be extended to bicircular laser fields. To image the structure of the solid material, harmonic intensities along x and y axes with different phases ϕ in the driving lasers are presented in Fig. <ref>. The intensities of harmonics below the bandgap as a function of ϕ exhibit a period of π/2, reflecting well the C_4 symmetry of the system. It can be a useful tool for probing the structure of solid target. However, the harmonics contributed by the interband polarization above the bandgap with order N>19 loose the periodicity since they are very sensitive to the laser fields, especially for short pulses <cit.>. The phase in Eq. (<ref>) is equivalent to a delay of the two-color pulses. This delay between the two drivers will amplify the effect of finite pulse duration on the instantaneous interband transitions. The harmonic above the bandgap can not be used as an imaging tool. We also calculated the phase difference δ in Eq. (<ref>) of the third harmonic as a function of the phase ϕ of the lasers in Eq. (<ref>), which is shown in Fig. <ref>. One may find that it shows a half period of π/2 and sensitivity to the external laser fields, which can be used to image the symmetry group of the solid structure efficiently. From Fig. <ref>, the band structure of the system is anisotropic. As a result, the phase ϕ in the co-rotating driving lasers can be used to control the HHG process, and thus the properties of the HHG signals, such as the cutoff energy, the relative intensity, and phase difference of HHG along x and y directions.§ HIGH-ORDER HARMONIC GENERATION IN COUNTER-ROTATING TWO-COLOR CIRCULARLY POLARIZED LASER FIELDS The counter-rotating bicircular 1ω and 2ω laser fields with amplitude ratio 1:1 are written asE_x(t)=E_0f(t)(sin(ω t)+sin(2ω t+ϕ)), E_y(t)=E_0f(t)(cos(ω t)-cos(2ω t+ϕ)).The calculated HHG spectra with different ϕ are shown in Figs. <ref> and <ref>. The rotation angle in the Lissajous curve is also the same as ϕ.Circular HHGs with different helicities are generated from atoms in bicircular counter-rotating laser fields <cit.>. The harmonics with order 3N are forbidden from the transition selection rule in atomic systems. However, in Fig. <ref>, no perfect circular HHGs are produced in the case of solids. Some harmonics close to linear polarization are also generated in counter-rotating bicircular laser fields. These features suggest the different mechanisms of HHG from gases and solids. The harmonic order 3N is not restricted by transition selection rules in this C_4-symmetry potential wells. For the case of HHG from atoms, co-rotating bicircular lasers are less efficient to generate harmonics compared to the counter-rotating bicircular lasers due to the necessary condition of recollision. However, the efficiencies for generating HHG from 2D solids are comparable in the co-rotating and counter-rotating bicircular laser fields since the electron wave functions in both the valence band and conduction band are delocalized and the condition of electrons returning to its original position is unnecessary.The phase ϕ can be used to tune the ellipticity, the cutoff energies, relative intensity, and phase difference δ of the xy components of the HHGs. It can also be used to image the structure of solids. As shown in Fig. <ref>, the harmonic intensities along x and y directions below the bandgap demonstrate a period of π/2. However, to retrieve the structure of an unknown target, one should fit the intensity of a range of harmonics to achieve high sensitivity. We also calculated the phase δ of the third and fourth harmonics as a function of the phase ϕ of the lasers in Eq. (<ref>). The phase of the third harmonic demonstrates a similar -sine trend in co-rotating lasers illustrated in Fig. <ref>, while the phase of the fourth harmonic shows a sine trend in Fig. <ref>. One may find that it also demonstrates a half period of π/2 and sensitivity to the change of the external laser fields, which can be used to image the solid structure efficiently. The amplitude ratio of the bicircular lasers will also affect the HHG signals. The combinations of lasers are not restricted to the fundamental and its second harmonic (1ω+2ω), it can be generalized to any nω+mω bicircular field to control HHG in soilds. There are too many ways to combine these parameters to control the HHG processes. Their effects will not be discussed in this work. § SUMMARYIn conclusion, we have studied the HHG from 2D solids in circular and bicircular co-rotating and counter-rotating laser fields. Different from the HHG from atoms, circular HHGs from intraband transitions can be generated in solids by circular driving lasers by using a pure initial valence state. Contributions from different valence states will change the polarization of harmonics. The efficiencies to generate harmonics from co-rotating and counter-rotating bicircular lasers are comparable. Elliptically and linearly polarized harmonics can be generated. 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Gaarde, “Orientation dependence of temporal and spectral properties of high-order harmonics in solids,” Phys. Rev. A 96(6), 063412 (2017). Nagler P. Nagler, M. V. Ballottin, A. A. Mitioglu, F. Mooshammer, N. Paradiso, C. Strunk, R. Huber, A. Chernikov, P. C. M. Christianen, C. Schüller, and T. Korn, “Giant magnetic splitting inducing near-unity valley polarization in van der Waals heterostructures,” Nat. Commun. 8, 1551 (2017). | http://arxiv.org/abs/1708.07596v2 | {
"authors": [
"Guang-Rui Jia",
"Xin-Qiang Wang",
"Tao-Yuan Du",
"Xiao-Huan Huang",
"Xue-Bin Bian"
],
"categories": [
"physics.optics"
],
"primary_category": "physics.optics",
"published": "20170825015022",
"title": "High-order harmonic generation from 2D periodic potentials in circularly and bichromatic circularly polarized laser fields"
} |
=5in =7.5inLemmaLemma[section] Theorem[Lemma]Theorem Proposition[Lemma]Proposition Corollary[Lemma]Corollary Remark[Lemma]Remark Definition[Lemma]Definition Question[Lemma]Question γ sH s □ Φ𝒳 𝒩 𝒴𝒰 𝒢𝒮 𝒯 𝒫 ℛ 𝒟 e + - char supp dim span sign Re Im sing supp WF diamrad Ind distdiag detΩ⟨⟨|⟩⟨⟨⟩⟩ε∂α λΛẼB̃𝒰ℱC ^∞^ -14pt ^∘ 10ptf̂ĥδℤℛ𝒮MℳetgdR_ M]Reconstruction of a compact manifold from the scattering data ofinternal sourcesM. Lassas]Matti Lassas ^□ T. Saksala]Teemu Saksala ^♢, ∗ H. Zhou]Hanming Zhou ^† [^□ Department of Mathematics and Statistics, University of Helsinki, Finland, ^♢Department of Mathematics and Statistics, University of Helsinki, Finland; Department of computational and applied mathematics, Rice University, USA^† Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, UK; Department of Mathematics, University of California Santa Barbara, Santa Barbara, CA 93106, USA, ^∗[email protected], [email protected]] Given a smooth non-trapping compactmanifold with strictly convex boundary, we consider an inverse problem of reconstructing the manifold from the scattering data initiated from internal sources. These data consist of the exit directions of geodesics that are emaneted from interior points of the manifold. We show that under certain generic assumption of themetric, the scattering datameasured on the boundary determine the Riemannian manifold up to isometry. [ [ December 30, 2023 =====================§ INTRODUCTION, PROBLEM SETTING AND MAIN RESULTIn this paper we consider an inverse problem of reconstructing a Riemannian manifold (M,g) from geodesical data that correspond to the following theoretical measurement set up. Suppose that in a domain M there is a large amount m of point sources, at points x_1,x_2,…,x_m, sending continuously light (or other high frequency waves) at different frequencies ω_j. Such point sources are observed on the boundary as bright points. Assume that on the boundary an observer records at the point z∈ M the exit direction of the light from the point source x_j, that is, an observer can see at the point z the directions of geodesics coming from x_j to z. When the observer moves along the boundary, all existing directions of geodesics coming from the point sources to the boundary are observed. We emphasize that only the directions, not the lengths (i.e., travel times) are recorded. When m becomes larger, i.e., m→∞, we can assume that the set of the point sources {x_j} form a dense set in M. Let (N,g) be an n-dimensional closed smooth Riemannian manifold, where n≥ 2. Suppose that M ⊂ N is open and has a boundary M ⊂ N that is a smooth (n-1)-dimensional submanifold of N. We also assume that M is strictly convex, meaning that the second fundamental form of M (as a submanifold) is positive definite. Let SN⊂ TN be the unit tangent bundle of the metric g. Given (p,ξ)∈ SN, it induces a unique maximal geodesic γ_p,ξ:→ N with γ_p,ξ(0)=p, γ̇_p,ξ(0)=ξ. We denoteS_pN={ξ∈ T_pN; ξ_g=1}. We will use the notation π:TN → N for the canonical projection to the base point, π(x,ξ)=x.We define the first exit time functionτ_exit(p,ξ)=inf{t>0; γ_p,ξ(t)∈ N∖M}, (p,ξ) ∈ S M,where S M is the unit tangent bundle of M. We assume that τ_exit(p,ξ)< ∞, for all (p,ξ) ∈ S M.This means that M is non-trapping.For each point p∈M we define a scattering set of the point source p [R_ M(p)= {(q,η^T)∈ TM ;there existξ∈ S_pN; andt∈ [0,τ_exit(p,ξ)]such that; q=γ_p,ξ(t), η=γ̇_p,ξ(t)}∈ 2^T M. ]Here η^T ∈ T M is the tangential component of η∈ SM and 2^S means the power set of set S. See Figure (<ref>).LetR_ M(M)={R_ M(p); p∈ M}⊂ 2 ^T M.Consider the following collection ( M ,R_ M(M)). We call this collection thescattering data of internal sources that depends on the Riemannian manifold (M,g|_M). We emphasize the connection of data (<ref>) to the scattering data (see <cit.>), that is also considered in the Section <ref> of this paper.From now on we will use a short hand notationg:=g|_M=i^∗ g, i:M↪ N.Here i is an embedding defined by i(x)=x, x ∈M. The inverse problem considered in this paper is the determination of the Riemannian manifold (M,g) from the data (<ref>). More precisely, this means the following: Let (N_j,g_j), j=1,2 andM_j be similar to (N,g) and M.We say that the scattering data of internal sources of (M_1,g_1) is equivalent to that of (M_2,g_2), ifthere exists a diffeomorphism ϕ:M_1 → M_2 such that, {Dϕ (R_ M_1(q)); q∈ M_1}= {R_ M_2(p); p∈ M_2}.Here Dϕ:TM_1 →T M_2 is the differential of ϕ, i.e., the tangential mapping of ϕ.We denote the set of all C^∞-smooth Riemannian metrics on N by ℳet(N) (we will assign the smooth Whitney topology (see <cit.>) on ℳet(N) to make it into a topological space), it turns out that there exists a generic subset (a set that contains an countable intersection of open and dense sets) G⊂ℳet(N) such that for g∈ G, the manifold (M,g) is determined by the data (<ref>) up to an isometry. Given g∈(N),p,q∈ N and ℓ>0, we denote the number of g-geodesics connecting p and q of length ℓ by I(g,p,q,ℓ). Define I(g):=sup_p,q,ℓI(g,p,q,ℓ).In Theorem 1.2 of <cit.> it is shown that there exists a generic set G ⊂(N), such that for all g∈ G, I(g)≤ 2 n+2. Next we define the collection of admissible Riemannian manifolds. Denote𝒢={(N,g); Nis a connected, closed, smooth Riemannian n-manifold;gsatisfies (<ref>)}.Our main result is as follows, Let (N_i,g_i) ∈𝒢, i=1,2 be a smooth, closed, connected Riemannian n-manifold n≥ 2, M_i ⊂ N_i be an open set with smooth strictly convex boundary with respect to g_i. Suppose that (M_i,g_i), i=1,2is non-trapping, in the sense of (<ref>).Ifproperties (<ref>)–(<ref>) hold, then (M_1,g|_M_1) is Riemannian isometric to (M_2,g_2|_M_2). The auxiliary results needed to prove the main theorem will be stated and proven in such a way that we will use the data (<ref>) to reconstruct an isometric copy of (M,g). Therefore we formulate also the following theorem.Let (N,g) ∈𝒢 be a smooth, closed, connected Riemannian n-manifold n≥ 2, M ⊂ N be an open set with smooth strictly convex boundary with respect to g. Suppose that (M,g),is non-trapping, in the sense of (<ref>). If the data (<ref>)is given we can reconstruct a isometric copy of (M,g). We want to underline that in the most partsthe proofs for Theorems <ref> and <ref> go side by side! Therefore we will formulate the lemmas and propositions, given in Section 2 and later, in a way having this dual goal in mind. §.§ Outline of this paperThe proof of Theorem <ref> is contained in Sections 2 to 5. In Section 2 we recall some properties of the strictly convex boundaries and the first exit time function τ_exit. We will also give a generic property related to the generic property (<ref>) which guarantees that the scattering set R_ M(p) is related to a unique point p ∈ M. The main result of Section 3 is to show that the data (<ref>) determines uniquely the topological structure of the manifold M. Section 4 is devoted to the smooth structure of M and the final section shows that the data (<ref>) determines the Riemannian structure of (M, g).§.§ Previous literature The research topic of this paper is related to many other inverse problems. §.§.§ Boundary and scattering rigidity problems The scattering data of internal sources (<ref>) considered in our paper takes into account the geodesic rays emitted from all the points of M, though we only know the exit positions and directions of these geodesics on the boundary. Instead of thescattering data of internal sources, thescattering relation is defined as followsL_g:_+SM→_-SM, L_g(x,ξ):=(γ_x,ξ(τ_exit(x,ξ)),γ̇_x,ξ(τ_exit(x,ξ))),where _±SM:={(x,ξ)∈ SM; ±⟨ξ,ν(x)⟩_g≤ 0}are the sets of inward and outward pointing vectors on the boundary M. Here ν is the outward pointing unit normal to M. Notice that M is non-trapping, thus L_g is well-defined. We will see later that scattering data of internal sources ( M, R_ M(M)) determines the scattering relation L_g. The scattering rigidity problem asks: if L_g=L_g̃, does it implies that g̃=ψ^* g with ψ:M→M a diffeomorphism fixing the boundary? The scattering rigidity problem is closely related to the boundary rigidity problem, which is concerned with the determination of the metric g (up to a diffeomorphism fixing the boundary) from its boundary distance function d_g:M× M→ℝ. Notice that given x,y∈ M, d_g(x,y) equals the length of the distance minimizing geodesic connecting x and y. See <cit.> for recent surveys on these topics. In particular, the scattering rigidity problem and boundary rigidity problem are equivalent on simple manifolds. A compact Riemannian manifold M is simple if the boundary M is strictly convex and any two points can be joined by a unique distance minimizing geodesic. Michel <cit.> conjectured that simple manifolds are boundary distance rigid, so far this is known for simple surfaces <cit.>. More recently, boundary rigidity results are established on manifolds of dimension 3 or larger that satisfy certain global convex foliation condition <cit.>. In the current paper, since the scattering data of internal sources contains more information than the scattering relation, we can deal with more general geometry. It is worth mentioning that the problem of recovering Riemannianmanifolds by the length data of geodesic rays emitted from internalsources was considered in <cit.>.§.§.§ The rigidity of broken geodesic flows Another related inverse problem is concerned with determining a manifold (M ,g) from the broken geodesic data, which consists of the initial and the final points and directions,and the total length, of the broken geodesics. To define the data we first set up the notations for a broken geodesicα_x,ξ_+,z,η(t)={[ γ_x,ξ_+(t), t < s, (x,ξ_+) ∈_+SM; γ_z,η(t-s), t ≥ s, (z,η) ∈ SM, ].where z=γ_x,ξ_+(s).Denote the length of the curve α_x,ξ_+,z,η by ℓ(α_x,ξ_+,z,η), the Broken scattering relation is [ ℛ ={((x,ξ_+),(y,ξ_-),t)∈_+SM×_-SM×_+;; t = ℓ(α_x,ξ_+,z,η)and(y,ξ_-)=(α_x,ξ,z,η(t),∂_t α_x,ξ,z,η(t)),; for some(z,η)∈ SM}. ]In <cit.> the authors show that the broken scattering data ( M, ℛ) determines the manifold (M,g) uniquely up to isometry, if dim M≥ 3. In particular, there is no restrictions on the geometry of the manifold. §.§.§ Inverse boundary value problem for the wave equationsNext we will present a third well known inverse problem, where the data is given in the boundary of the manifold. Consider the following initial/boundary value problem for the Riemannian wave equation[_t^2 w - Δ_g w = 0 in(0, ∞) × M,; w|_× M = f,; w|_t = 0 = _t w|_t= 0 = 0, f∈ C_0^∞((0,∞)× M) ]where Δ_g is the Laplace-Beltrami operator of metric tensor g. It is well known that for every f∈ C_0^∞((0,∞)× M) there exists a unique w^f ∈ C^∞((0,∞)×M) that solves (<ref>) (see for instance <cit.> Chapter 2.3). Thus the Dircihlet-to-Neumann operator Λ_g:C_0^∞((0,∞)× M) → C^∞((0,∞)× M),Λ_gf(t,x)=⟨ν(x), ∇_g w^f(t,x) ⟩_g is well defined and an inverse problem to (<ref>) is to reconstruct (M, g) from the data( M, Λ_g).One classical way to solve this problem is the Boundary Control method (BC). This method was first developed by Belishev for the acoustic wave equation on ^n with an isotropic wave speed <cit.>. A geometric version of the method, suitable when the wave speed is given by a Riemannian metric tensor as presented here, was introduced by Belishev and Kurylev <cit.>. The BC-method is used to determine the collection BDF(M):={d_g(p,·)|_ M; p ∈ M}of boundary distance functions. We emphasize the connection between our data R_ M(M) and BDF(M), that is for any p ∈ M andfor every z ∈ M, it holds that holds∇_ M( d_g(p,·)|_ M)|_z∈ R_ M(p), if d_g(p, ·) is smooth at z. Above ∇_ M is the gradient of a Riemannian manifold ( M, g|_ M). One significant difference is that BDF(M) contains only information about the velocities of the short geodesics as R_ M(p) contains also the information from long geodesics. We refer to <cit.> for a thorough review of the related literature.Alsothe case of partial data has been considered. Let 𝒮, ℛ⊂∂ M be open and nonempty. In this case the inverse problem is the following. Does the restriction operator Λ_g,,f:=Λ_gf|__+ ×, f∈ C^∞_0(_+ ×)with , determine (M,g)? The answer is positive if =, (see <cit.>). In <cit.> the problem has been solved in the case ∩≠∅. The general case is still an open problem.So far the sharpest results are <cit.>.§.§.§ Distance difference functions In <cit.> the collection of distance difference functions {(d_g(p,·)-d_g(p,·))|_N ∖ M; p∈ M}⊂ L^∞(N ∖ M),was studied. The authors show that (N ∖ M, g|_N ∖ M) with the collection (<ref>) determines the Riemannian manifold (N,g) up to an isometry, if N is a compact smooth manifold of dimension two or higher and M is open and has a smooth boundary. In the context of this paper one can assume that at the unknown area M there occurs an Earthquake at an unknown point p∈ M at an unknown time t≥ 0. This Earthquake emits a seismic wave. At the every point of the measurement area N ∖ M there is a device that records the time when the seismic wave hits the corresponding point. This way we obtain the function (z,w)↦ D_p(z,w):=d_g(p,z)-d_g(p,w), z,w ∈ N ∖ M that is the travel time difference of the seismic wave.Fix z ∈ M. Suppose that for every p ∈ M the corresponding distance difference function D_p(·,z): M → is smooth. Then there is a close connection between R_ M(p) andR_ M^1(p):={∇_ MD_p(·,z)|_q;q∈ M},Notice that there exists non-trapping manifolds M with strictly convex boundary such that for a given p ∈ M the mapping M∋ q ↦ d_g(q,p) is not smooth.Moreover to prove the Main theorem <ref> of this paper we will use techniques that where introduced in <cit.>. These techniques are based on the Theorem 1 of <cit.>. §.§.§ Spherical surface dataFinally we will present one more geometric inverse problem where a geodesical measurement data is considered.Let (N,g) be a complete or closed Riemannian manifold of dimension n∈ and M⊂ N be an open subset of N with smooth boundary. We denote by U:=N∖M.In <cit.> one considers the Spherical surface dataconsisting of the set Uand the collection of all pairs (Σ,r) where Σ⊂ U is a smooth (n-1) dimensional submanifold that can be written in the formΣ=Σ_x,r,W= {exp_x(rv) ∈ N; v∈ W}, where x∈ M, r>0 and W⊂ S_xN is an open and connected set. Such surfaces Σ are called spherical surfaces, or more precisely, subsets of generalised spheres of radius r. We point out that the unit normal vector field ν_x,r,W(γ_x,v(r)):=γ̇_x,v(r),v ∈ Wof the generalized sphere Σ_x,r,W determines a data that has a natural connection to our data R_ M(M). Also, in <cit.> one assumes that U is given with its C^∞-smooth coordinate atlas. Notice that in general, the spherical surface Σ may be related to many centre points and radii. For instance consider the case where N is a two dimensional sphere.In <cit.> it is shown that the Spherical surface data determine uniquely the Riemannian structure of U. However these data are not sufficient to determine (N,g) uniquely. In <cit.> a counterexample is provided. In <cit.> it is shown that the Spherical surface datadetermines the universal covering space of (N,g) up to an isometry. In <cit.> a special case of problem <cit.> is considered. The authors study a setup where M ⊂^n, n≥ 2 and the metric tensor g|_M=v^-2e, for some smooth and strictly positive function v. Let x ∈^n ∖ M and y :=γ_x,ξ(t_0) ∈ M, for some ξ∈ S_xN and t_0>0, such that y is not a conjugate point to x along γ_p,ξ. The main theorem of <cit.> is that, if the coefficient functions of the shape operators of generalized spheres Σ_y,r,W are known in a neighborhood V of γ_x,ξ([0,t_0)) and the wave speed v is known in (N ∖ M)∩ V, then the wave speed v can be determined in some neighborhood V' ⊂ V of γ_x,ξ([0,t_0]).§ STRICTLY CONVEX MANIFOLDS, ABOUT THEEXTENSION OF THE DATA AND THE GENERIC PROPERTY§.§ Analysis of strictly convex boundariesWe will write (z(p),s(p)) for the boundary normal coordinates of M, for a point p near M. (See 2.1,<cit.>, for the definitions). Here for any p ∈ dom((z,s)), the mapping p↦ z(p) stands for the closest point z(p)∈ M of p and p↦ s(p) is the signed distance to the boundary, i.e.,|s(p)|=_g(p, M)ands(p)<0ifp ∈ M, s(p)=0ifp ∈ Mands(p)>0ifp ∈ N ∖M.Thus every q ∈ M has a neighborhood U ⊂ N, such that the map U ∋ p ↦ (z(p),s(p)) is a smooth local coordinate system. We will write ν for the unit outer normal of M, therefore, ∇_g s(p)=ν for p∈ M. Therefore the distance minimizing geodesic from p to z(p) is normal to M. The second fundamental form of M is said to be positive definite, i.e., the boundary M as a submanifold is strictly convex, if the corresponding shape operatorS:T M → T M, S(X)=∇_X νis positive definite. Here ∇ is the Riemannian connection of the metric tensor g. Next we recall a few well known results (without proof) about manifolds with strictly convex boundary.Let (N,g) be a complete smooth Riemannian manifold and M⊂ N open subset with smooth boundary. Suppose that the second fundamental form of M is positive definite. Then for each (p,ξ) ∈ S M there exists ϵ>0 such that the geodesic γ_p,ξ(t)∉M for any t∈ (0,ϵ).Moreover if (p,ξ) ∈ SM∖ S M, then the maximal geodesic γ_p,ξ:[a,b] →M does not hit the boundary M tangentially. Let (N,g) be a complete smooth Riemannian manifold and M⊂ N open subset with smooth boundary. Suppose that the second fundamental form of M is positive definite. Let (p,ξ) ∈ S M. Thenγ_p,ξ((0,τ_exit(p,ξ)))⊂ M.The claim follows from Lemma <ref> and (<ref>).If M is strictly convex, then for any p ∈M there exists a neighborhood U ⊂ N of p such that for all z,q ∈ U∩M the unique distance minimizing unit speed geodesic γ from z to q is contained in U and γ(t)∈ M ∩ U for all t ∈ (0,d(z,q)). If(M,g) is compact, non-trapping and M is strictly convex, then the first exit time function τ_exit:SM→_+ given by (<ref>) is continuous and there exists L>0 such that max{τ_exit(p,ξ);(p,ξ)∈ SM}≤ L.Moreover, the exit time function τ_exit is smooth in S M∖ S M and in _+ SM. See Section 3.2. of <cit.>. §.§ Extension of the measurement dataWe start with showing that the data (<ref>) determines the boundary metric. This is formulated more precisely in the following lemma.Let (N,g) be a smooth, closed, connected Riemannian n-dimensional manifold n≥ 2, M⊂ N an open set with smooth boundary. Then (<ref>) determines the data(( M, g|_ M) ,R_ M(M)),where g|_ M= i^∗g and i: M ↪M. More precisely if (N_i,g_i) and M_i are as in Theorem <ref> and(<ref>)–(<ref>) hold, thenϕ:M_1 → M_2 is a diffeomoprhismsuch that ϕ^∗ (g_2|_ M_2)=g_1|_ M_1.We will start with showing that we can recover the metric tensor g|_ M. Choose p ∈ M, η∈ (T_p M∖{0}) and consider the set[R(η):= {ξ∈ T_pM;there existsz ∈ M such that ξ∈ R_ M(z); anda>0,that satisfiesξ=aη}. ]It is easy to see that the set R(η) is not empty and by Lemma <ref> we also have{ξ_g ∈_+; ξ∈ R(η)}=(0,1) ⊂.Therefore, η_g=inf{a>0;η/a∈ R(η)}.Since η∈ T_p M was arbitrary we have recovered the g-norm function ·_g:T_pM →. Since norm ·_g is given by the g-inner product,we recover ⟨·, ·⟩_g using the parallelogram rule, that is⟨ξ, η⟩_g=ξ_g^2+η_g^2-ξ-η_g^2/2, η, ξ∈ T_pM. Suppose then that for manifolds (M_1,g_1) and (M_2,g_2) properties (<ref>)–(<ref>) are valid. Choose p ∈ M_1 and η∈ T_pM_1 and denote R(η) as in (<ref>). Then by (<ref>)–(<ref>) we haveDϕ (R(η))=R(D ϕη)⊂ T_ϕ(p) M_2.and by the equations (<ref>)–(<ref>) we have η_g_1=D ϕη_g_2and equation (<ref>) follows from the parallelogram rule.Let p ∈M, we define the complete scattering set of the point sourcep as [R^E_ M(p)= {(q,η)∈_- SM; there is ξ∈ S_pN;andt∈ [0,τ_exit(p,ξ)]such that; q=γ_p,ξ(t), η=γ̇_p,ξ(t)}. ]We emphasize that the difference between R_ M(p) and R_ M^E(p) for p ∈M is that R_ M(p) contains the tangential components of vectors ξ∈ R^E_ M(p). We denote R^E_ M(M):={R^E_ M(p); p ∈ M}. In the next Lemma we will give an equivalent definition for (<ref>).[Corollary of Lemma <ref>]For any p ∈M R^E_ M(p)={(γ_p,ξ(τ_exit(p,ξ)),γ̇_p,ξ(τ_exit(p,ξ)));ξ∈ S_pN}.Since M is strictly convex and M is non-trapping, the claim follows from the definition of R^E_ M(p) and Corollary <ref>. Let (N_i,g_i) and M_i be as in Theorem <ref> and suppose that (<ref>)–(<ref>) hold. Since M_i is compact, there exists r>0 and a neighborhood K_i ⊂M_i of M_i such that the mapping f_i: M_i × (-r,0]→ K_i ⊂M_i, f_i(z,s):=exp_z(sν(z))is a diffeomorphism. By (<ref>) the mapΦ: M_1 × (-r,0]→ M_2 × (-r,0], Φ(z,s)=(ϕ(z),s).is a diffeomorphism. Thus the mappingΦ:K_1 → K_2,Φ=f_2 ∘Φ∘ f_1^-1,is a diffeomorphism between K_1 and K_2. Moreover the map Φ is a local representation of Φ in the boundary normal coordinates.Suppose that (N,g) and M are as in Theorem <ref>. Then the data (<ref>) determines the complete scattering data (( M, g|_ M), R^E_ M(M)). More precisely if (N_i,g_i) and M_i are as in Theorem <ref> such that (<ref>) and (<ref>) hold, then ⟨η, ξ⟩_g_1=⟨ D Φη, DΦξ⟩_g_2,for all ξ,η∈ TM_1and{DΦ (R^E_ M_1(q)); q∈M_1}= { R^E_ M_2(p); p∈M_2}.We will start with showing that for every q ∈ M we can recover R_ M^E(q). Choose q ∈ M and let (p,η)∈_- SM be such that (p,η^T) ∈ R_ M(q). Then η^T_g < 1 and in the boundary normal coordinates x ↦ (z(x),s(x)) close to pthe vector η can be written asη=η^T+(√(1-η^T_g^2))ν.Since g|_ M is known we have recovered η in the boundary normal coordinates. Since q∈ M was arbitrary, we have recovered the collection R^E_ M(M).Next we will show that for every q ∈ M we can recover R_ M^E(q). Let q ∈ M and (p,η)∈_- SM. We will define a set[ Σ(p,η) :={R^E_ M(q') ∈ R_ M^E(M);(p,η) ∈ R^E_ M(q')}; = {R^E_ M(q')∈ R_ M^E(M): q' ∈γ_p,-η(0,τ_exit(p,-η))}, ]with which we can verify if (p,η)∈ R^E_ M(q) or not. Notice that by Lemma <ref>Σ(p,η)=∅ if and only if γ_p, -η([0,τ_exit(p,-η)])∩ M is empty if and only if η is tangential to the boundary. Suppose first that Σ(p,η) ≠∅. Then (p,η)∈ R^E_ M(q) if and only if there exists (q,ξ)∈ S_qN such that Σ(p,η)=Σ(q,ξ). Suppose then that Σ(p,η)=∅ and thus η is tangential to M. By Lemma <ref> holds τ_exit(p,η)=0. Therefore, η∈ R_ M^E(q) if and only if p=q. We conclude that we can always check with data (<ref>) if (p,η)∈_- SM is included in R^E_ M(q) or not. Since q∈ M was arbitrary, we have recovered the collection R^E_ M( M). Therefore, we have recovered the set R_ M^E(M). Next we will verify (<ref>) and (<ref>). Let (N_i,g_i) and M_i be as in Theorem <ref> such that (<ref>) and (<ref>) hold.Choose p ∈ M_1 and ξ, η∈ T_pM_1. By (<ref>), (<ref>) and (<ref>) we have⟨ DΦξ, DΦη⟩_g_2=⟨ Dϕξ^T, Dϕη^T⟩_g_2+√(1-Dϕη^T_g_2^2)√(1-D ϕξ^T_g_2^2)=⟨ξ,η⟩_g_1.Thus the equation (<ref>) is valid. Next we will prove{DΦ (R^E_ M_1(q)); q∈ M_1}= { R^E_ M_2(p); p∈ M_2}. Let q ∈ M_1. By (<ref>) there exists z ∈ M_2 such that R_ M_2(z)=Dϕ R_ M_1(q).Let (p,η) ∈R^E_ M_1(q). Then η_g_1=1 and η^T∈ R_ M_1(q). Since Dϕη^T∈ R_ M_2(z) we conclude by (<ref>) that DΦ(p,η) ∈ R^E_ M_2(z). Therefore, we have DΦ (R^E_ M_1(q)) ⊂ R^E_ M_2(z). By symmetric argument we also have R^E_ M_2(z)⊂ DΦ( R^E_ M_1(q)). Thus we have proved the left hand side inclusion in(<ref>). Again by symmetric argument we prove the right hand side inclusion in (<ref>). Next we will show that{DΦ (R^E_ M_1(q)); q∈ M_1}= { R^E_ M_2(p); p∈ M_2}.Let q ∈ M_1. By the definition of the mapping Φ and the equation (<ref>) it is enough to prove thatDΦ (R^E_ M_1(q))=R^E_ M_2(ϕ(q)).Let (p,η) ∈R^E_ M_1(q) and denote Σ(p,η) as in (<ref>). By equations (<ref>) and(<ref>) it holds that the set[Σ(Φ p, DΦη) :={R_ M_2(z);z∈ M_2, DΦη∈ R_ M_2(z)};={DΦ R_ M_1(q');R_ M_1(q')∈Σ(p,η)} ]is empty if and only if Σ(p,η) is empty. Suppose first that Σ(p,η) is empty and thus p=q and η is tangential to the M_1. Therefore, by equation (<ref>) we haveDΦη=Dϕη∈(R_ M_2(ϕ(q)) ∩ R^E_ M_2(ϕ(q))). Suppose then that Σ(p,η) is not empty. This implies that there exists a vector ξ∈ S_qN_1 that satisfies Σ(p,η)=Σ(q,ξ). Thus Σ(Φ p, DΦη)=Σ(Φ q, DΦξ) and this implies that DΦ(p,η) ∈ R_ M_2(ϕ(q)). This completes the left hand inclusion of (<ref>). Replace Φ by Φ^-1 to prove the right hand side inclusion of (<ref>). Therefore, (<ref>) is valid.§.§ A generic propertyWe will now formulate a generic property in (N) that is related to the complete scattering data (<ref>). Let N be a smooth manifold and M⊂ N an open set. We say that a Riemannian metric g ∈(N) separates the points of set M if for all p,q ∈ M, p≠ q there exists ξ∈ S_pN such thatq ∉γ_p,ξ([-τ_exit(p,-ξ),τ_exit(p,ξ)]).This means that there exists a geodesic segment that starts and ends at the boundary of M and contains p, but does not containq. We emphasize that there are easy examples for N, M and g such that g does not separate the points of M. For instance let M ⊂ S^2 be a polar cap strictly larger than the half sphere. Consider any two antipodal points p,q ∈ M. Then the standard round metric does not separate the points p and q.Our next goal is to show that for every (N,g)∈𝒢, (where 𝒢 is as in(<ref>)) andM⊂ Nthat is open, non-trapping in the sense of (<ref>) and has a smooth strictly convex boundary, the metric g separates the points of M. This will be used in Section 3 to show that for two points p,q ∈ M the complete scattering sets R^E_ M(p) and R^E_ M(q) coincide if and only if p is q.Let (N,g) be a compact Riemannian manifold. Let M⊂ N be open, non-trapping in the sense of (<ref>) and have a smooth strictly convex boundary. Suppose that the metric g does not separate the points of M. Then there exist p, q∈ M,p≠ q, an interval I⊂, and a C^∞-map I ∋ s ↦(ξ(s), ℓ(s))∈ S_pN×, such that ξ̇(s)≠ 0 and q=γ_p,ξ(s)(ℓ(s)).Since g does not separate the points of M there are p,q∈ M, p≠q,such that for all ξ∈ S_pN we haveS(ξ)={s∈ (-τ_exit(p,-ξ),τ_exit(p,ξ)); exp_p(s ξ)=q }≠∅.We note that for all ξ∈ S_pN the set S(ξ) is finite due to inequality (<ref>). Fix η∈ S_pN and enumerate S(η)={s_1,…,s_K} for some K∈. Then -τ_exit(p,-η)<s_1<s_2<, …, <s_K<τ_exit(p,η). For each s_k∈{s_1,…,s_K} we choose a n-1-dimensional submanifold S_k of N such thatγ_p,η(s_k)=q∈ S_kand γ̇_p,η(s_k)⊥ T_qS_k.For instance we can defineS_k:={exp_q(tv)∈ N; v ∈ S_qN, v ⊥γ̇_p,η(s_k),t ∈ [0,inj(q))},where inj(q) is the injectivity radius at q. Choose ϵ>0 and consider a ϵ-neighborhood W of γ_p,η([-τ_exit(p,-η),τ_exit(p,η)]). We will write S_k for the component of S_k ∩ W that contains q. If ϵ is small enough, there exists δ, δ'>0 such that for any t ∈ [-τ_exit(p,-η),τ_exit(p,η)] ∖(⋃_k=1^K(s_k-δ,s_k+δ)) holds dist_g(S_k,γ_p,η(t))≥ 2δ',for everyk ∈{1,…,K}.By the continuity of the exponential mapping, we can choose a smaller ϵ>0 such that there exists an open neighborhood V⊂ S_pN of η such that for any t ∈ [-τ_exit(p,-η),τ_exit(p,η)] ∖(⋃_k=1^K(s_k-δ,s_k+δ))and ξ∈ V holds dist_g(S_k,γ_p,ξ(t))≥δ',for everyk ∈{1,…,K}.Next we define a signed distance function ϱ_k(t,ξ)={[ -dist_g(γ_p,ξ(t),S_k), (t,ξ) ∈ (s_k-δ,s_k]× V; dist_g(γ_p,ξ(t),S_k), (t,ξ) ∈ [s_k,s_k+δ)× V. ].See Figure (<ref>). Choosing a smaller δ and V, if needed, it follows that the functionϱ_k: (s_k-δ,s_k+δ)× V → is smooth. Then we have ϱ_k(s_k,η)=0 and|/ tϱ_k(t,η)|_t=s_k|=γ̇_p,η(s_k)^2=1.Therefore, by the Implicit function theorem there exists an open neighborhood V_k⊂ V of η and a smooth function f_k:V_k→ (s_k-δ,s_k+δ) that solves the equationϱ_k(f_k(ξ),ξ)=0.Define an open neighborhood U of η by U:= ⋂_k=1^K V_k and sets U_k={ξ∈ U: exp_p(f_k(ξ)ξ)=q}. Then sets U_k are closed in the relative topology of U. By (<ref>), (<ref>) and (<ref>) it must hold thatU=⋃_k=1^K U_k. We claim that for some k ∈{1, …, K} it holds that U_k^int≠∅. If this is not true, then the sets P_j:= U∖ U_j, j∈{1,…, K} are all open and dense in the relative topology of U. Moreover by (<ref>) we haveP:=⋂_j=1^KP_j=∅.This is a contradiction since U is a locally compact Hausdorff space and thus by the Baire category theorem, it should hold that the set P is dense in the relative topology of U. Thus there exists k ∈{1, …, K} for which it holds that U_k^int≠∅. Choose an open U' ⊂ U_k. In particular U' is open in S_pN and there exists ϵ'>0 and a C^∞-path ξ(s), s∈ (-ϵ',ϵ'), in U' such that ξ̇(s)≠0. Denoting ℓ(s)= f_k(ξ(s)), we haveq=γ_p,ξ(s)(ℓ(s)) fors∈ (-ϵ,ϵ). Therefore, the curve s↦ (ξ(s),ℓ(s)) satisfies the claim of this Lemma.Let (N,g) ∈𝒢 and M be as in Theorem <ref>. Then g separates the points of M.We prove the proposition by contradiction. Given g∈𝒢, assume there are p,q∈ M, p≠ q such that for all ξ∈ S_pN, q ∈γ_p,ξ([-τ_exit(p,-ξ),τ_exit(p,ξ)]).In particular (see Lemma <ref>), there exists an open interval (-ϵ, ϵ), such that for s∈ (-ϵ,ϵ), (ξ(s),ℓ(s)) is a C^1-path on S_pN× such that ξ̇(s)≠ 0 and q=γ_p,ξ(s)(ℓ(s)). This implies [0=/ s(γ_p,ξ(s)(ℓ(s))); = γ̇_p,ξ(s)(ℓ(s))·d/dsℓ(s)+ℓ(s)· D _p|_ℓ(s)ξ(s)ξ̇(s);=: T_1+T_2. ]Since ξ(s)=1, it implies ξ(s)⊥ξ̇(s). By Gauss Lemma, we have γ̇_p,ξ(s)(ℓ(s))⊥ D _p|_ℓ(s)ξ(s)ξ̇(s), thus T_1⊥ T_2. Applying equation(<ref>), we obtain T_1=T_2=0, therefore, d/ds(ℓ(s))=0. This is equivalent to saying that ℓ(s)≡ const, for s∈ (-ϵ,ϵ). This implies I(g)=∞. By equation (<ref>), we arrive a contradiction. § THE RECONSTRUCTION OF THE TOPOLOGYWe define a mapping R^E_ M: M→ 2^ SM, p ↦ R^E_ M(p).The aim of this section is to show that the map R^E_ M is a homeomorphism, with respect to some suitable subset of 2^ SM and topology of this subset. Recall that ( M, g|_ M) is known, by the Lemma <ref>. Therefore we can talk about topological properties of 2^ SM. Thus the goal is to reconstruct a homeomorphic copy of (M, g). We start with showing that the map R^E_ M is one-to-one, if the generic property <ref> holds.Suppose that (N,g) and M are as in the Theorem <ref>. Then the mapping R^E_ M:M→ 2^ SM is one-to-one.We will prove the claim by contradiction. Therefore, we divide the proof into two separate cases.Suppose first that there is p∈ M, q∈M such that R^E_ M(p)=R^E_ M(q), that is{(γ_p,ξ(τ_exit(p,ξ)),γ̇_p,ξ(τ_exit(p,ξ)));ξ∈ S_pN}= {(γ_q,η(τ_exit(q,η)),γ̇_q,η(τ_exit(q,η)));η∈ S_qN}⊂_-SM. By the definition of R^E_ M(p) it holds thatS_pM⊂ R^E_ M(p)=R^E_ M(q).Therefore, we deduce that any tangential geodesic starting at p hits q before exiting M. Since M is strictly convex, this is true only if p=q. Suppose then that there exist p ∈ M,q ∈M,p≠ q such that R^E_ M(p)=R^E_ M(q). By the first part we may assume that q ∈ M. Let ξ∈ S_pN. Then for ±ξ we have (γ_p,±ξ((τ_exit(p,±ξ)),γ̇_p,±ξ((τ_exit(p,±ξ)))∈ R^E_ M(q).Therefore, it holds thatq ∈γ_p,ξ((-τ_exit(p,-ξ),τ_exit(p,ξ))).Since ξ∈ S_pN was arbitrary, the equation (<ref>) is valid for every ξ∈ S_pN and we have proved that the metric q does not separate the points p and q. This is a contradiction with Proposition <ref> and therefore, p=q. To reconstruct the topology of M from R^E_ M(M), we need to first give a topological structure to the latter. Notice that on the unit tangent bundle S M there is a natural metric induced by the underlying metric g, namely the Sasaki metric. We denote the Sasaki metric associated with g by g_S. Then on the power set 2^S M, we assign the Hausdorff distance, i.e., given A, B∈ 2^SM d_H(A,B):=max{sup_a∈ Ainf_b∈ Bd_g_S(a,b), sup_b∈ Binf_a∈ Ad_g_S(a,b)}. However, in general the Hausdorff distance on 2^S M needs not to be metrizable. If we consider the subset 𝒞(S M):={ M}⊂ 2^S M, then (𝒞(S M), d_H) is a compact metric space by Blaschke selection theorem (see for instance <cit.>, Theorem 4.4.15). The topology on 𝒞(S M) thus is induced by the Hausdorff metric d_H.We turn to consider a subspace 𝒞( SM):={S M}⊂ 2^ SM of 𝒞(SM). Since the boundary M is compact, 𝒞( S M) is a compact metric space. For any q ∈M, the set R^E_ M(q)⊂ S M is compact.Consider a continuous mapping E_q:S_qN →_-S M given byE_q(ξ)=(γ_q,ξ(τ_exit(q,ξ)),γ̇_q,ξ(τ_exit(q,ξ))).Since S_qN is compact, and E_q is continuous it holds by Corollary <ref> thatR^E_ M(q)=E_q(S_qN)is compact.By Lemma <ref> it holds that R^E_ M(M) ⊂𝒞( S M). From now on we consider the mapping R^E_ M:M→𝒞( S M), p ↦ R^E_ M(p).Let (X,d) be a complete metric space and 𝒞(X) be the collection of all closed subsets of (X,d). We say that a sequence (A_j)_j=1^∞⊂𝒞(X) converges to A ∈𝒞(X) in the Kuratowski topology, if the following two conditions hold:(K1) Given any sequence (x_j)_j=1^∞, x_j∈ A_j with a convergent subsequence x_j_k→ x as k →∞, then the limit point x is contained in A.(K2) Given any x ∈ A, there exists a sequence (x_j)_j=1^∞, x_j ∈ A_j that converges to x.The mapping R^E_ M:M→𝒞( S M) is continuous. Let q ∈M and q_j ∈M, j∈ be a sequence that converges to q. As S M is compact it holds that the Kuratowski convergence and the Hausdorff convergence are equivalent (see e.g. <cit.> Proposition 4.4.14). Thus it suffices to show that R^E_ M(q_j) converges to R^E_ M(q) in the space 𝒞( S M) in the sense of Kuratowski.First we prove (K1). Let (p_j,η_j) ∈ R^E_ M(q_j). Changing in to subsequences, if necessary, we assume that (p_j,η_j) → (p,η) ∈ S M as j →∞. Let ξ_j∈ S_q_jN be such that for each j∈,γ_q_j,ξ_j(τ_exit(q_j,ξ_j))=p_jand γ̇_q_j,ξ_j(τ_exit(q_j,ξ_j))=η_j.As the first exit time function is continuous and S M is compact we maywith out loss of generality assume that (q_j,ξ_j) → (q',ξ) ∈ S_q'N, q' ∈M andτ_exit(q_j,ξ_j) →τ_exit(q',ξ). Since q_j → q, it holds that q'=q.By the continuity of the exponential mapping and the first exit time function the following holds(p,η)=lim_j →∞(p_j,η_j)=lim_j →∞(γ_q_j,ξ_j(τ_exit(q_j,ξ_j)),γ̇_q_j,ξ_j(τ_exit(q_j,ξ_j)))=(γ_q,ξ(τ_exit(q,ξ)), γ̇_q,ξ(τ_exit(q,ξ))).Thus (p,η)∈ R^E_ M(q). This proves (K1).Next we prove (K2). Let (p,η) ∈ R^E_ M(q). Let ξ∈ S_qNbe such thatγ_q,ξ(τ_exit(q,ξ))=pand γ̇_q,ξ(τ_exit(q,ξ))=η.Sinceq_j → q as j→∞ we can chooseξ_j ∈ S_q_jN,j ∈ such that ξ_j →ξ as j →∞. Denoteγ_q_j,ξ_j(τ_exit(q_j,ξ_j))=p_jand γ̇_q_j,ξ_j(τ_exit(q_j,ξ_j))=η_j,since the exponential map and the first exit time function are continuous, then (p_j,η_j) ∈ R^E_ M(q_j)and(p_j,η_j) → (p,η).This proves (K2).We conclude that R^E_ M(q_j) converges to R^E_ M(q) in the Kuratowski topology. The mapping R^E_ M:M→ R^E_ M(M) ⊂𝒞( S M) is a homeomorphism.Note the R^E_ M:M→ R^E_ M(M) ⊂𝒞( S M) is continuous, one-to-one and onto, and M is compact and 𝒞( SM) is a metric space and thus a topological Hausdorff space. These yield that R^E_ M:M→ R^E_ M(M) is a homeomorphism.By the Proposition <ref> the manifold topology of the data set R^E_ M(M) is determined. Thus the topological manifold R^E_ M(M)is a homeomorphic copy of M. The rest of this section is devoted to constructing a map from M_1 onto M_2 that we later show to be a Riemannian isometry. Define a map DΦ:𝒞( S M_1) →𝒞( S M_2), F ↦ DΦ(F).The map DΦ:𝒞( S M_1) →𝒞( S M_2) is a homeomorphism.We start by noticing that, if (X,d_X) and (Y,d_Y) are compact metric spaces, and f:X → Y is continuous, then the lift f:𝒞(X) →𝒞(Y),f(K)=f(K) is well defined. Let (K_i)_i=1^∞⊂𝒞(X) be a sequence that converges to K ∈𝒞(X) with respect to Kuratowski topology. We will show that f(K_i) also converges to f(K) with respect to Kuratowski topology, and by <cit.> Proposition 4.4.14 this will imply that the lift f is continuous.Let (y_i)_i=1^∞, y_i ∈ f(K_i) be a sequence with a convergent subsequence (y_i_k)_k=1^∞, y_i_k∈ f(K_i_k), we denote the limit point by y∈ Y. Thus for each i_k, there exists x_i_k∈ K_i_k such that f(x_i_k)=y_i_k. Since X is a compact metric space thesequence (x_i_k)_k=1^∞ has a convergent subsequence in X, we denote it by (x_i_k)_k=1^∞ again. Notice that K_i→ K in the sense of Kuratowski convergence, we get x_i_k→ x ∈ K. By the continuity of f, one has y=lim_k →∞y_i_k=lim_k →∞f(x_i_k)=f(x)∈ f(K). Let y ∈ f(K) and x ∈ K such that f(x)=y. Since K_i → K in the sense of Kuratowski, there exists a sequence (x_i)_i=1^∞, x_i ∈ K_i that converges to x. On the other hand, f is continuous, thus the sequence f(x_i)∈ f(K_i) converges to f(x)=y. This completes the proof of the convergence of f(K_i) to f(K). Notice that by (<ref>) the mapping D Φ: TM_1 → TM_2 is a smooth invertible bundle map. In particular D Φ is continuous with a continuous inverse. Therefore, D Φ is a well defined homeomorphism according to the first part of the proof. Next we define a mappingΨ: M_1→M_2,Ψ:= (R_ M_2^E)^-1∘DΦ∘ R^E_ M_1.By (<ref>) and Proposition <ref> the map Ψ is well defined. Now we prove the main theorem of this section.Let (N_i,g_i) and M_i be as in Theorem <ref> such that (<ref>) and (<ref>) hold. Then the map Ψ: M_1→M_2, is a homeomorphism such that Ψ|_M_1:M_1 → M_2 and Ψ|_ M_1=ϕ. By (<ref>), Proposition <ref> and Lemma <ref> the map Ψ is ahomeomorphism.Let p ∈ M_1. Suppose that q:=Ψ(p)∈ M_2, it holds that S_qM_2⊂ R^E_ M_2(q).On the other hand by the proof of Lemma <ref> there is no p' ∈ M_1 such that S_p' M_1 ⊂ R^E_ M_1(p). Since D Φ is an isomorphism, we reach a contradiction and thus q ∈ M_2. Similar argument for the inverse mapping Ψ^-1 and p ∈ M_2 proves the second claim.Let p ∈ M_1, then S_pM_1 ⊂ R^E_ M_1(p)and by (<ref>) DΦ (S_pM_1)= S_ϕ(p) M_2 ⊂ R^E _ M_2(ϕ(p)).Therefore, by the proof of Lemma <ref> we have Ψ(p)=ϕ(p).§ THE RECONSTRUCTION OF THE DIFFERENTIABLE STRUCTUREIn this section we will show that the map Ψ:M_1→M_2 is a diffeomorphism. First we will introduce a suitable coordinate system of smooth manifold M_i that is compatible with the data (<ref>). We will consider separately coordinate charts for interior and boundary points. Then we will define a smooth structure on topological manifold R^E_ M_i(M_i) such that the maps R^E_ M_i: M_i → R^E_ M_i(M_i) are diffeomorphisms. The third task is to show thatthe map D Φ:R^E_ M_1(M_1) → R^E_ M_2(M_2) is a diffeomorphism with respect to smooth structures of R^E_ M_i(M_i). By the formula (<ref>), that defines the map Ψ, the following diagram commutes [M_1R^E_ M_1⟶R^E_ M_1(M_1);; ↓Ψ ↓DΦ;;M_2R^E_ M_2⟶ R^E_ M_2(M_2), ]and the above steps prove that the map Ψ is a diffeomorphism. Also we willexplicitly construct the smooth structure for R^E_ M(M) only using the data (<ref>). In the steps below, we will often consider only one manifold and do not use the sub-indexes, M_1 and M_2, when ever it is not necessary.§.§ The recovery of self intersecting geodesics and conjugate pointsLet us choose a point p∈M for the rest of this section. The first part of the section is dedicated to finding for the point p a suitable q∈ M, a neighborhood V_q of p and to construct a mapΘ_q(z):=1/exp_q^-1(z)_gexp_q^-1(z)∈ S_qN , z∈ V_q .using our data (<ref>). We start with considering what does it require from q ∈ M to be "suitable“. Let (q,η)∈ R^E_ M(p), we say that η is a conjugate direction with respect to p, if p is a conjugate to q on the geodesic γ_q,-η. This is equivalent to the existences∈ (0,τ_exit(q,-η)) such that γ_q,-η(s)=p and a non-trivial Jacobi field J such that J(0)=0 and J(s)=0. We say that a geodesic γ is self-intersecting at the point q ∈M, if there exists t_1<t_2 such that γ(t_1)=q=γ(t_2).Notice that the geodesic γ_q,-η([0,τ_exit(q,-η)]) may be self-intersecting at p, and moreover there may be several s'∈(0,τ_exit(q,-η)) such that γ_q,-η(s')=p and non-trivial Jacobi fields that vanish at 0 and s'.In the next Lemma we show that under the assumptions of Theorem <ref> most of the geodesics that start and end at M and hit the point p, do it only one time.Let (N,g) and M be as in the Theorem <ref>. Let p ∈M. Denote by[I := {ξ∈ S_pN;the geodesic γ_p,ξ:[-τ_exit(p,-ξ),τ_exit(p,ξ)] →M;is self-intersecting atp}. ]Then the set S_pN ∖ I is open and dense.If r ∈ R_ M^E(M) is given and p∈M is the unique point for which R_ M^E(p)=r we defineK(p):={(q,η) ∈ (R^E_ M (p) ∖ S_pN) ;γ_q,-η is not self-intersecting atp}⊂∂ SM.See Figure (<ref>). Then K(p) is not empty and K(p) is determined by r and data (<ref>).More precisely if (N_i,g_i) and M_i are as in Theorem <ref> such that (<ref>) and (<ref>) hold, then for everyp ∈M_1 the set K(p)≠∅ and DΦ (K(p))=K(Ψ(p)).Suppose first that p ∈ M. We start with proving that I is closed. Let ξ∈I and choose a sequence (ξ_j)_j=1^∞⊂ I that converges to ξ in S_p N. Choose a sequence (t_j)_j=1^∞⊂ that satisfiesexp_p(t_jξ_j)=pand0<|t_j|.By (<ref>) we can without loss of generality assume thatt_j ⟶ tasj ⟶∞.Then it must hold that 0< |t| since any geodesic starting at p cannot self-intersect at p before time inj(p)>0, where inj(p) is the injectivity radius at p. Therefore, exp_p(tξ)=p and we have proved that ξ∈ I. Thus I is closed. From the proofs of Lemma <ref> and Proposition <ref> it follows that the set I is nowhere dense, this is that I does not contain any open sets. Thus S_pN ∖ I is open and dense. Moreover the setK(p)={(γ_p,ξ(τ_exit(p,ξ)),γ̇_p,ξ(τ_exit(p,ξ)))∈_-SM; ξ∈ S_pN ∖ I} is not empty. Denote r = R_ M^E(p). Next we will show that r and data (<ref>) determine the set K(p). Let (q,η) ∈ r. We write J(q,η) for the set of self-intersection points of geodesic segment γ_q,-η:(0,τ_exit(q,-η)) → M. Due to non-trapping assumption of M the set J(q,η) is finite. Since the map R_ M^E: M→ R_ M^E( M) is a homeomorphism, it holds that z ∈γ_q,-η((0,τ_exit(q,-η))) ∖J(q,η) if and only if R_ M^E(z) has a neighborhood V ⊂ R_ M^E(M) such that Σ(q,η) ∩ V is homeomorphic to interval [0,1) or (0,1), here Σ(q,η) is as in (<ref>). Therefore, for a given r=R_ M^E(p) and (q,η)∈ r the data (<ref>) determines the set R_ M^E(J(q,η)) and it holds that (q,η) ∈ K(p)if and only ifr ∉ R_ M^E(J(q,η)). Finally we will verify the equation (<ref>) in the case of p ∈ M_1. Denote r=R_ M_1^E(p). By Theorem <ref> the map Ψ is a homeomorphism. With (<ref>) this implies that for every (q, ξ)∈_-SM_1 D Φ(Σ(q,ξ))=Σ(ϕ (q),DΦξ)and the self-intersection points of Σ(y,ξ) are mapped onto self-intersection points of Σ(ϕ(y),DΦξ) under the map D Φ. Therefore, DΦ (K(p))=K(Ψ(p)). Suppose next that p ∈ M.Assume that there exists a sequence (ξ_j)_j=1^∞⊂ I that converges to a vector ξ∈ S_p M in S_pN. Choose a sequence (t_j)_j=1^∞ such thatexp_p(t_jξ_j)=pand0<t_j.Due to Lemmas <ref> and <ref> we may assume that t_j → 0 as j →∞. But then we have a contradiction with the injectivity radius of p. Therefore, I is contained in the interior of _+ S_pM. Then by a similar argument as in the case of p ∈ M we have shown that S_pN ∖ I is open and dense. Let (y,ξ) ∈_- SM.We will use a short hand notationΣ(y,ξ) := {R_ M^E(q); q ∈M, (y,ξ) ∈ R_ M^E(q))}= R^E_ M(γ_y,-ξ([0,τ_exit(y,-ξ)]))⊂ 2^C(∂ SM),for the image of the geodesic segment γ_y,-ξ([0,τ_exit(y,-ξ)]) under the map R^E_ M.Denote r = R_ M^E(p). Then for a vector (q,η)∈ r it holds that (q,η)∈ K(p) if and only if q ≠ p. Thus the set K(p)={(γ_p,ξ(τ_exit(p,ξ)),γ̇_p,ξ(τ_exit(p,ξ)))∈_-SM; ξ∈ (_+ S_pM)^int∖ I}.is not empty.Now we will verify the equation (<ref>) in the case of p ∈ M_1. Since the map Ψ is a homeomorphism we have by the data (<ref>) thatD Φ (Σ(y,ξ))=Σ(ϕ (y),D Φξ)for all(y,ξ)∈_-SM_1.Therefore, DΦ(K(p))=K(Ψ(p)) and the equation (<ref>) is proven.In the next Lemma we will show that we can find the set of conjugate directions with respect to p from data (<ref>) and there exist lots of (q,η) ∈ r:=R_ M^E(p) such that η is not a conjugate direction with respect to p. If (q,η) ∈ K(p) is not a conjugate direction with respect to p, we will later construct coordinates for p such that (n-1)-coordinates are given by η.Let (N,g) and M be as in the Theorem <ref>. If r ∈ R_ M^E(M) is given and p∈M is the unique point for which R_ M^E(p)=r, we define the set of "good“ directions[ K_G(p):= {(q,η) ∈ K(p); η is not a conjugate directionwith respect to p}⊂ SM. ] Then K_G(p) is not empty, π(K_G(p))⊂ M is open. Moreover r and the data (<ref>) determine the set K_G(p).More precisely if (N_i,g_i) and M_i are as in Theorem <ref> such that (<ref>) and (<ref>) hold, then for everyp ∈M_1 the set K_G(p)≠∅, π(K_G(p))⊂ M_1 is open and DΦ (K_G(p))=K_G(Ψ(p)).Suppose first that p ∈ M. Let δ:SN → [0,∞] be the cut distance function, i.e., δ(y,η):=sup{t>0;d_g(γ_y,η(t),y)=t}. Let ξ∈ S_pN be such a unit vector that γ_p,ξ is a shortest geodesic from p to the boundary M. By Lemma 2.13 of <cit.> it holds that δ(p,ξ)>dist_g(p,M). Therefore, (γ_p,ξ(τ_exit(p,ξ)),γ̇_p,ξ(τ_exit(p,ξ))) is not a conjugate direction with respect to p. Moreover there exists an open neighborhood V⊂ S_pN of ξ such that for any ξ' ∈ V the vector (γ_p,ξ'(τ_exit(p,ξ')),γ̇_p,ξ'(τ_exit(p,ξ'))) is not a conjugate direction. Let I ⊂ S_pN be defined as in (<ref>). DenoteV_c:={ξ∈ S_pN; Dexp_p|_τ_exit(p,ξ)ξ is not singular }that is open and non-empty, since V ⊂ V_c. Therefore, by the Lemma <ref> it holds that the set V_c ∩ (S_pN ∖ I)=V_c ∖ I is open and non-empty. ThusK_G(p)={(γ_p,ξ(τ_exit(p,ξ)),γ̇_p,ξ(τ_exit(p,ξ)))∈ (_-SM∖ S_pN);ξ∈ V_c ∖ I } is not empty and moreover π(K_G(p)) ⊂ M is open.Next we show that for given r=R_ M^E(p) the data (<ref>) determines the set K_G(p), if r∈ R^E_ M(M). Fix (q,η)∈ K(p). Let t_p ∈ (0,τ_exit(q,-η)] be such that γ_q,-η(t_p)=p. Consider the collection H(q,η) of C^∞-curves σ:(-ϵ,ϵ)→_- S M,σ(s)=(q(s),η(s)) that satisfy the following conditions(H1)σ(0)=(q,η),(H2)r∈Σ(q(s),η(s)) for all s ∈ (-ϵ,ϵ) (H3)r ∉ R_ M^E(J(q(s),η(s))).Here we use the notations from Lemma <ref> Σ(q(s),η(s))=R^E_ M(γ_q(s),-η(s)((0,τ_exit(q(s),-η(s))))),andJ(q(s),η(s)) is the collection of self-intersection points of the geodesic segmentγ_q(s),-η(s):(0,τ_exit(q(s),-η(s))) →M. Choose a C^∞-curve s ↦σ(s) on _-SM, such that (H1) holds. By (<ref>) we can check, if the property (H2) holds. By the proof of Lemma (<ref>) we can check, if property (H3) holdsfor the curve σ. Therefore, for a given r∈ R_ M^E(M) and (q,η) ∈ K(p) we can check the validity of properties (H1)–(H3) for any smooth curve σ on _-SM withthe data (<ref>).Next we show that the collection H(q,η) is not empty. Let ξ∈ S_pN be the vector that satisfies (γ_p,ξ(t_p),γ̇_p,ξ(t_p))=(q,η). Then τ_exit(p,ξ)=t_p. Let w ∈ T_pN, w ⊥ξ and ϵ>0 be small enough. We definew(s):=ξ+sw/ξ+sw_g andf(t,s):=exp_p(tτ_exit(p,w(s))w(s)), s∈ (-ϵ,ϵ), t∈ [0,1].Notice that the curve s↦ f(1,s)∈ M is smooth by Lemma <ref>. We conclude that a curve σ can be for instance defined as followsσ(s)=(q(s),η(s)):=(f(1,s),τ_exit(p,w(s))^-1/ tf(t,s)|_t=1), s∈ (-ϵ,ϵ),where ϵ>0 is small enough. Observe that the curve σ(s) satisfies conditions (H1) and (H2) automatically. The condition (H3) is valid by the Lemma <ref>, since (q,η)∈ K(p). We also note thatd/dsq(s)|_s=0=D exp_p|_t_pξ(d/dsτ_exit(p,w(s))|_s=0ξ+t_pẇ(0))andẇ(0)=w. Now we show the relationship between conjugate directions and curves σ∈ H(q,η). Let us first consider some notations we will use. Let c:(a,b) → N be a smooth path and V a smooth vector field on c. By this we mean that t ↦ V(t)∈ T_c(t)N. We will write D_tV for the covariant derivative of V along the curve c. The properties of operator D_t are considered for instance in Chapter 4 of <cit.>. We recall that locally D_t V is defined by the formulaD_t V(t)=(d/dt V^k(t)+V^j(t)ċ^i(t)Γ_ji^k(c(t)))/ x^k(c(t)),where Γ_ji^k are the Christoffel symbols of the metric tensor g. Suppose first that γ_q,-η(t_p)=p is a conjugate point of q along γ_q,-η([0,τ_exit(q,-η)]). Then there is a non-trivial Jacobi field J on γ_p,ξ that vanishes at t=0 and at t=t_p. Then D_t J(t)|_t=0≠ 0and D_t J(t)|_t=0⊥ξ.Here D_t operator is defined on γ_p,ξ(t). Define a curve σ(s)=(q(s),η(s)) ∈ H(q,η) by (<ref>) and (<ref>) with w:=D_t J(t)|_t=0. We will show that d/dsq(s)|_s=0=0and(D_sη(s)^T|_s=0)^T≠ 0.Here D_s is defined on q(s). Since the vector w is the velocity of J at 0 and J is a Jacobi field that vanishes at 0 and t_p we have by (<ref>) and (<ref>) thatd/dsq(s)|_s=0=d/dsτ_exit(p,w(s))|_s=0η.Since s ↦ q(s) is a curve on M and η≠ 0 is not tangential to the boundary, it must hold thatd/dsτ_exit(p,w(s))|_s=0=0.Thus d/dsq(s)|_s=0=0. Recall that the Jacobi field J satisfies due to (<ref>) J(t)=t Dexp_p|_tξ w=d/ds f(t',s)|_s=0, t∈ [0,t_p], t'=t/t_p. We will use the notation D_t' for the covariant derivative on the curve γ_p,t_pξ(t'). Then by (<ref>)–(<ref>), (<ref>)–(<ref>) and Symmetry Lemma (<cit.>, Lemma 6.3) we haveD_sη(s)|_s=0=t_p^-1D_s/ t'f(t',s)|_t'=1,s=0=t_p^-1D_t'/ sf(t',s)|_t'=1,s=0= t_p^-1D_t J(t)|_t=t_p≠ 0,since J is not a zero field. Therefore, we conclude that D_sη(s)|_s=0≠ 0.Moreover since η(s)_g≡ 1 we have η(s)⊥ D_sη(s). Suppose first that (D_sη(s)|_s=0)^T= 0, then D_sη(s)|_s=0 is normal to the boundary and thus η(0) is tangential to the boundary. This is a contradiction since η(0)=η, that is not tangential to the boundary. Thus we conclude that (D_sη(s)|_s=0)^T≠ 0. Write η(s)=c(s)ν(q(s))+η(s)^T,where c is some smooth function.Let x↦ (z^j(x))_j=1^n be the boundary normal coordinatesnear q. Here we consider that z^n(x) represents the distance of x to M. Then the coefficient functions of ν are V^k=δ^n_k. Therefore for every k∈{1,…,n} the observation q̇(0)=0 yields(D_s ν(s)|_s=0)^k=(d/ds V^k(s)+V^j(s)q̇^i(s)Γ_ji^k(q(s)))|_s=0=0,here Γ_ji^k are the Christoffel symbols of metric g in boundary normal coordinates. Thus(D_s η(s)|_s=0)^T= c(0)D_s ν(q(s))|_s=0+(D_s η(s)^T|_s=0)^T=(D_s η(s)^T|_s=0)^T.Therefore, (<ref>) is valid. Next we consider curves σ(s)=(q(s),η(s))∈ H(q,η). Notice that for every σ(s) there exists ϵ>0 and a smooth function s↦ a(s)∈ (0,∞), s∈ (-ϵ, ϵ) such that a(s)→ t_p as s→ 0 and Γ(s,t):=exp_q(s)(-ta(s)η(s)), t∈ [0,1], s∈ (-ϵ, ϵ)is a geodesic variation of γ_q,-η([0,t_p]) that satisfies Γ(s,1)=p. Let us consider s ↦(q(s),-ta(s)η(s)), t∈ [0,1] as a smooth curve on TN and exp:TN → N. We use a short hand notation -tW(s):=-ta(s)η(s). We use coordinates (z^j,v^j)_j=1^n for π^-1U ⊂ TM, where U is the domain of coordinates (z^i)_i=1^n. Let (y^i)_i=1^n be coordinates at exp((q(0),-tV(0).Then the variation field [V(t):= / sΓ(s,t)|_s=0=(exp^j / z^iq̇^i(0)-texp^j / v_iẆ^i(0))/ y^j; =(exp^j / z^iq̇^i(0)-tDexp_p|_-tW(0)Ẇ(0))/ y^j, ]is a Jacobi field that vanishes at t=1. By (<ref>) we haveV^j(0)=q̇^j(0). Therefore by (<ref>) the point p is a conjugate point of q along γ_q,-η([0,τ_exit(q,-η)]), if there exists a curve σ∈ H(q,η) that satisfiesq̇(0)=0and Ẇ(0)≠ 0.By the definition (<ref>)of the operator D_sand the vector field W(s) this is equivalent toq̇(0)=0andD_s(a(s)η(s))|_s=0≠ 0. Suppose that there exists such a curve σ(s)=(q(s),η(s))∈ H(q,η) for which (<ref>) is valid. Since V is a Jacobi field that vanishes at 1, we have by(<ref>),(<ref>) and the definition of σ(s) that0=V(1)=-Dexp_q|_-t_pη(D_s(a(s)η(s))|_s=0)=-ȧ(0)ξ- Dexp_q|_-t_pη(t_pD_sη(s)|_s=0).Thus ȧ(0)=0 and by (<ref>) it holds that D_sη(s)|_s=0≠ 0, which implies that (D_sη(s)|_s=0)^T≠ 0. Therefore, by similar computations as in (<ref>) we have (D_sη(s)^T|_s=0)^T≠ 0. We conclude that, if there exists a curve σ∈ H(q,η) such that (<ref>) holds then, p is a conjugate point to q on γ_q,-η and moreover, (D_s η(s)^T|_s=0)^T≠ 0. To summarize, for a given r ∈ R_ M^E(M) a vector (q,η) ∈ K(p) is a conjugate direction of p along geodesic γ_q,-η if and only if there exists σ∈ H(q,η) such that (<ref>) is valid. Moreover since s ↦ q(s) is a curve on the boundary M we can check the validity of (<ref>) for any σ∈ H(q,η) by data (<ref>). Therefore for given r∈ R_ M^E(M) the data (<ref>) determines the set K_G(p). Next we will prove equation (<ref>) in the case of p ∈ M_1. Let (q,η)∈ K(p). Suppose that D Φη∈ K(Ψ(p)) is not in K_G(Ψ(p)). Then there exists a smooth curve s ↦σ(s)=(q(s),η(s)) ∈_- SM_2 that satisfies the properties (H1)–(H3) and for which (<ref>) is valid. As it holds that Ψ is a homeomorphism and Φ is a diffeomorphism that preserves the boundary metric (in the sense of (<ref>)) the curve s ↦ (ϕ^-1(q(s)), D Φ^-1η(s)):=(q(s),η(s)) satisfies also the conditions (H1)–(H3) and moreover q̇(0)=0and (D_s η(s)^T|_s=0)^T=D Φ^-1 (D_s η(s)^T|_s=0)^T ≠ 0.Thus (<ref>) is valid for the curve (q(s),η(s)), which implies that (q,η) ∉ K_G(p). Thus (<ref>) is valid in the case of p ∈ M_1. Suppose then that p ∈ M. It follows from the Lemma <ref> that the set K_G(p) is not empty andK_G(p)={(γ_p,ξ(τ_exit(p,ξ)),γ̇_p,ξ(τ_exit(p,ξ)))∈_-SM ;ξ∈(V_c ∖ (I ∪ S M) )}. Let (q,η) ∈ K_G(p). Let ξ∈ S_pN be the unique vector that satisfies η =γ̇_p,ξ(τ_exit(p,ξ)). Since Dexp_p is not singular at τ_exit(p,ξ)ξ, we can choose an open neigborhood U of τ_exit(p,ξ)ξ such that for every v ∈ U the differential map D exp_p|_v is not singular and exp_p(U) is an open neighborhood of q. Since q ≠ p it holds that τ_exit(p,ξ) >0. Therefore we may assume that U∩ T M = ∅ and U ⊂ V_c ∖ I, since I is closed.Thus,exp_p(U)∩ M={γ_p,v(τ_exit(p,v)); v∈ h(U)}⊂π(K_G(p)), where h(v):=v/v_g and also in this case we deduce that π(K_G(p)) ⊂ M is open. Next we show that K_G(p) is determined by r and data (<ref>), if r ∈ R_ M^E( M). Fix (q,η)∈ K(p). We note that q≠ p and by Lemma (<ref>) the vector η is not tangential to the boundary. Let t_p ∈ (0,τ_exit(q,-η)] be such that γ_q,-η(t_p)=p. Consider the collection H'(q,η) of C^∞-curves σ:(-ϵ,ϵ)→_- S M,σ(s)=(q(s),η(s)) that satisfy the following conditions(H1')σ(0)=(q,η), q(s) ≠ p,for anys ∈ (-ϵ,ϵ) (H2')r∈Σ(q(s),η(s)) for all s ∈ (-ϵ,ϵ). Then by similar proof as in the case of p ∈ M we can show that H'(q,η) ≠∅, H'(q,η) is determined by r and data (<ref>) and (q,η) is a conjugate direction with respect to p if and only if there exists a curve σ∈ H'(q,η) for which equation (<ref>) is valid. The equation (<ref>) is also valid in the case of p ∈ M_1 by a similar argument as in the case of p ∈ M_1. Now we are ready to consider the map defined in (<ref>). Choose (q,η) ∈ K_G(p) and let t_p<0 be such that exp_q(t_p η)=p. Let V_q⊂ M be a neighborhood of p.Assuming that this neighborhood is small enough, there is a neighborhood U_q⊂ T_qN of t_p η such that the exponential map exp_q:U_q→ V_qis a diffeomorphism. We emphasize that the mapping Θ_q(z)=1/exp_q^-1(z)_gexp_q^-1(z)∈ S_qN , z∈ V_q .depends on the neighborhood U_q of t_p η. Let p,q ∈ N, p≠ q. Let η∈ S_qN and t>0 be such that exp_q(tη)=p. Suppose that Dexp_q is not singular at tη and denote Dexp_q|_tηη=:ξ∈ S_pN. Then ker(DΘ_q(p)) =span(ξ).Let v ∈ T_pN. Then DΘ_q(p)v=Dexp_q^-1v/t-η⟨ Dexp_q^-1v, η⟩_g/t,and the claim follows.In the next Lemma we will show that for given r=R^E_ M(p) and (q,η) ∈ K_G(p) the data (<ref>) determine the map Θ_q.Let (N,g) and M be as in the Theorem <ref>. If r ∈ R_ M^E(M) is given and p∈M is the unique point for which R_ M^E(p)=r, then for any (q,η) ∈ K_G(p) there exists a neighborhood V_q of p and a neighborhood of U_q of t_pη, where exp_q(t_pη)=p, such that exp_q^-1:V_q → U_q is well defined. Moreover, the map Θ_q:V_q → S_qN is smooth and well defined.The set R^E_ M(V_q) and the map Θ_q ∘ (R^E_ M)^-1:R^E_ M(V_q) → S_qN are determined from the data (<ref>) for given r ∈ R_ M^E(M) and (q,η)∈ K_G(p).More precisely, if (N_i,g_i) and M_i are as in Theorem <ref> such that (<ref>) and (<ref>) hold. Then for a given r=R^E_ M(p) ∈ R_ M^E(M_1) and (q,η)∈ K_G(p) it holds thatΘ_ϕ(q)(Ψ(z))= DΦ (Θ_q(z)), z ∈ V_q.Assume first that p ∈ M. Let (q,η)∈ K_G(p), then the existence of sets V_q and U_q follows. The map Θ_q is well defined and smooth since Θ_q=h∘exp_q^-1. Here h:T_qN∖{0}→ S_qN, h(v):= v/v_g is smooth and well defined since q≠ p. Next we will show that the set V_q and the map Θ_q are determined from the data (<ref>), if (q,η)∈ K_G(p) is given. Let V ⊂ M be a neighborhood of p and U⊂ T_qN a neighborhood of t_pη. Denote U:=h(U) ⊂ S_qN. Since η∈ U, we may assume that for all z∈ V the set R^E_ M(z) ∩ U⊂ S_qN is not empty.We define a set valued mapping P_q:V → 2^S_qN by formulaP_q(z)= R^E_ M(z) ∩ U.Then P_q is well defined and for given r and (q,η)∈ K_G(p) we can recover P_q from (<ref>). We claim that, if V and U are small enough, thenfor all z ∈ V the setR^E_ M(z) ∩ U⊂ S_qNhas a cardinality of1.Therefore, P_q, coincides with Θ_q in V. We will show that if (<ref>) is not valid, then we end up in acontradiction with the assumption (q,η)∈ K_G(p). We will divide the proof into two parts.Assume first that there exists a sequence (η_j)_j=1^∞⊂ R^E_ M(p) ∩ S_qN that converges to η and η_j ≠η for any j ∈. Choose t_j<0 such that exp_q(t_jη_j)=p. Due to (<ref>) we may assume that t_j converges to some t. Therefore, by the continuity of the exponential map, we have p=lim_j→∞exp_q(t_jη_j)=exp_q(t η).Since (q,η) ∈ K(p) it must hold that t=t_p. As p=exp_q(t_jη_j) for every j ∈ and t_jη_j → tη in T_qN, we end up with a contradiction to the assumption (q,η) ∈ K_G(p).Suppose next that there exist sequences (z_j)_j=1^∞⊂M and ξ^1_j,ξ_j^2 ∈ R^E_ M(z_j) ∩ S_qN,ξ_j ^1 ≠ξ_j^2 for every j ∈ such that for i∈{1,2} holdsz_j ⟶ p,and ξ_j ^i ⟶η asj ⟶∞.Choose t_j^1∈ (-τ_exit(q,-ξ_j^1),0), t_j^2∈ (-τ_exit(q,-ξ_j^2),0), j ∈ such that for i∈{1,2}, holds exp_q(t_j^iξ_j^i)=z_j. Again by (<ref>) we may assume that t_j^i converges to t^i ∈ (-τ_exit(q,-ξ),0) as j →∞ for i ∈{1,2}, respectively. Therefore, we have p=lim_j →∞z_j=lim_j→∞exp_q(t_j^iξ_j^i)=exp_q(t^iη).Since (q,η) ∈ K(p) it must hold that t^i=t_p for both i∈{1,2}. Therefore, t_j^iξ_j^i → t_pη in T_qN as j →∞ for both i∈{1,2}. Since we assumed that ξ_j^1≠ξ_j^2 for any j ∈, we have shown that mapping exp_q is not one–to–one near t_pη. Hence we end up again in a contradiction with the assumption (q,η)∈ K_G(p). Thus we have proven the existence of such sets V and U for which (<ref>) is valid. Using data (<ref>) we can verify for a given r and the neighborhoods U of η and R^E_ M(V) of r whether the property (<ref>) is valid. We will denote by U_1 ⊂ S_qN a neighborhood of η and V_q a neighborhood of p that satisfy (<ref>). The next step is to prove that for any r=R_ M_1^E(p)∈ R_ M_1^E(M_1) and (q,η)∈ K_G(p) the equation (<ref>) is valid. Choose neighborhoods U_1 ⊂ S_qN of η and V_q ⊂ M_1 of p for which (<ref>) is valid. By (<ref>) it holds that (ϕ(q),DΦη)∈ K_G(Ψ(p)). Since Ψ is a homeomorphism Ψ(V_q) is a neighborhood of Ψ(p) and due to (<ref>)DΦ(U_1)⊂ S_ϕ(q)N_2 is a neighborhood of DΦη. Therefore (<ref>) is valid for Ψ(V_q) and DΦ(U_1). Let z ∈ V_q and{ξ}=R^E_ M(z) ∩ U_1. Then {DΦξ} =R^E_ M(Ψ (z)) ∩ D Φ(U_1) and the equation (<ref>) follows.Finally we assume that p ∈ M. We notice that by Lemma (<ref>) it holds K_G(p) ∩ T M =∅ and by the definition of K_G(p) we have p ∉π(K_G(p)). Therefore, we can choosefor every (q,η) ∈ K_G(p) such a neighborhood V_q⊂M of p that q ∉ V_q. The rest of the proof is similar to the case p ∈ M.§.§ Interior coordinatesFor this subsection we assume that p ∈ M. Let (q, η) ∈ K_G(p) be such that a similar map Θ_q:V_q̃→ S_qN exists for some neighborhood V_q of p. Let v∈ T_qN and define a smooth map Θ_q,q, v(z)=(Θ_q(z),⟨v|,Θ_q̃(z)⟩_g )∈ S_qN× , z∈ V_q∩ V_q̃.See Figure (<ref>).In the next Lemma we will show that this map is a smooth coordinate map near p. By Inverse function theorem it holds that a smooth map f:N →^n is a coordinate map near p ∈ N if and only if Df(p)≠ 0. Here Df(p) is the Jacobian matrix of f at p.Let (N,g) and M be as in the Theorem <ref>. If r ∈ R_ M^E(M) is given and p∈ M is the unique point for which R_ M^E(p)=r, then there exist (q,η),(q, η) ∈ K_G(p) and v∈ T_qN such that Θ_q,q,v is a smooth coordinate mapping from a neighborhood V_q,q̃,v ofp. Moreover, for given r = R^E_ M(p) ∈ R_ M^E(M) and (q,η),(q, η) ∈ K_G(p) the data (<ref>) determines v∈ T_q̃N, a neighborhood R_ M^E(V_q,q̃,v)of r and the map Θ_q,q,v∘ (R_ M^E)^-1: R_ M^E(V_q,q̃,v) → (S_qN × (∖{0})) such that Θ_q,q,v a smooth coordinate map from a neighborhood V_q,q̃,v of p onto some open set U ⊂ (S_qN × (∖{0})). More precisely, let (N_i,g_i) and M_i be as in Theorem <ref> such that (<ref>) and (<ref>) hold. Then for any p ∈ M_1 and(q,η),(q, η) ∈ K_G(p) and v∈ T_qN the following holds: Let ξ∈ S_qN be outward pointing and s∈. ThenΘ_ϕ(q),ϕ(q),DΦ v(Ψ(z))=(DΦξ,s)if and only if Θ_q, q,v(z)=(ξ,s)andD Θ_q, q,v(p)≠ 0if and only if DΘ_ϕ(q),ϕ(q),DΦ v(Ψ(p)) ≠ 0.Choose(q,η)∈ K_G(p). By Lemma <ref> the set π(K_G(p)) is open and, there exists (q, η) ∈ K_G(p), such that q∉γ_q,-η([0,τ_exit(q,-η)]).By Lemma <ref> there exists a neighborhood V_q∩ V_q of p such that the mapping Θ_q,q,v is well defined and smooth for any v ∈ T_q N.Next we show that there exists such a vector v∈ T_q N that the Jacobian DΘ_q,q,v is invertible at p. By the choice of (q,η) and (q, η) there exist t,t >0 and ξ,ξ∈ S_pN such that ξ and ξ are not parallel and(γ_p,ξ(t),γ̇_p,ξ(t))=(q,η) and (γ_p,ξ(t),γ̇_p,ξ(t))=(q,η). Choose linearly independent vectors w_1, …, w_n-1∈ S_qN that are all perpendicular to η. Then vectors ξ_i:=Dexp_q|_-t ηw_i ∈ T_pN, i ∈{1, …, n-1}are linearly independent and perpendicular to ξ. Therefore, the vectors ξ_1, …, ξ_n-1,ξ span T_pN. Let v ∈ T_qN be such that ⟨ v , Θ_q(p)⟩_g ≠ 0 and ⟨ v , DΘ_q(p)ξ⟩_g≠ 0. Notice that by Lemma <ref> such a v exists. Choose coordinates (x^1,…, x^n) at p such that∂/ x^i(p)=ξ_i; i ∈{1,…, n-1} and ∂/ x^n(p)=ξ.In these coordinates the Jacobian matrix of Θ_q,q,v at p isDΘ_q,q,v(p)=( [ V 0; a c ]),where V is the Jacobianmatrix of Θ_q at p with respect to (n-1) first coordinates, a, 0∈^n-1 and c = / x_n⟨ v , Θ_q(x)⟩_g|_x=p. Then by the choice of basis ξ_1,…, ξ_n-1, ξ and Lemma <ref> it holds that V is invertible.Moreoverc = / x^n⟨ v , Θ_q(x)⟩_g|_x=p=d/dt⟨ v , Θ_q(γ_p,ξ(t))⟩_g|_t=0=⟨ v , DΘ_q(p)ξ⟩_g≠ 0, and we conclude thatDΘ_q,q,v(p)=(± 1) cV≠ 0.This shows that there exist (q,η),(q, η) ∈ K_G(p) and v∈ T_q N such that the corresponding map Θ_q,q,v is a coordinate map in some neighborhood of p. Since the invertibility of the Jacobian DΘ_q,q,v(p) is invariant to the choice coordinates, we conclude that the map Θ_q,q,v is a smooth coordinate map at p.Now we will check that for a given r=R^E_ M(p) and (q,η),(q, η) ∈ K_G(p) the data (<ref>) determines v∈ T_q̃N, a neighborhood R_ M^E(V_q,q̃,v) of r and the map Θ_q,q,v∘ (R_ M^E)^-1:R_ M^E(V_q,q̃,v)→ (S_qN ×). By Lemma <ref>, the data (<ref>) determines the set K_G(p) and it is not empty. Chooseany (q,η)∈ K_G(p). By equation (<ref>) we can choose (q,η) ∈ K_G(p) such that (<ref>) is valid. By Lemmas <ref> and <ref> we can construct the neighborhoods V_q, V_q of p and the map Θ_q,q,v:V_q,q→ S_qN ×, where V_q,q:=V_ q∩ V_q, for any v ∈ T_qN. Suppose now on that the points (q,η) and (q, η) are given. By (<ref>) and (<ref>) we notice that a sufficient and necessary condition for the map Θ_q,q,v to be a smooth coordinate map at p is that vector vis contained in A_q:=T_qN ∖ ((DΘ_q(p)ξ)^⊥∪Θ_q(p)^⊥) that is open and dense.Lastly we will introduce a method to test, if v ∈ T_qN is admissible, that is, DΘ_q,q,v(p)≠ 0. By the Propostion <ref> the map R_ M ^E:M→ R_ M ^E(M) is a homeomorphism. Thus we can determine the set [ B_q = {w ∈ T_qN; the map Θ_q,q,w∘ (R_ M ^E)^-1 is a homeomorphism; from a neighborhood of R_ M ^E(p)onto some open set V ⊂^n}. ]For all w,w' ∈ B_q we define a homeomorphismH_w,w':U_w,w'→ V_w,w',H_w,w':=Θ_q,q,w∘Θ_q,q,w'^-1,where U_w,w', V_w,w'⊂^n are open. Let C^2_q:={(w,w')∈ B_q× B_q; H_w,w' is smooth and(D H_w,w'(Θ_q,q,w(p))≠ 0)}.Thus A_q× A_q⊂C^2_q and the data (<ref>), with R^E_ M(p),(q,η) and (q, η) determine C^2_q. Let (w,w')∈ C^2_q. Then by the Inverse function theorem also (w',w) is included in C^2_q. Thus((T_qN ∖ A_q ) × A_q)∩ C^2_q=∅.Write C_q for the projection of C^2_q into T_qN with respect to first component. Choose v ∈ C_q. We claim that v is admissible if and only if there exists an open and dense set A ⊂ T_qN such that{v}× A ⊂ C^2_q.If v is admissible, then for every w ∈ A_q holds (v,w)∈ C^2_q. Therefore, {v}× A_q⊂C^2_q and (<ref>) follows. Suppose then that (<ref>) is valid for some open and dense set A. Then A∩ A_q is also open, dense and {v}×(A∩ A_q)⊂ C^2_q. Suppose that v ∈ T_qN ∖ A_q. Then it follows that({v}×(A∩ A_q))∩ C^2_q≠∅. But this contradicts (<ref>) and it must hold that v ∈ A_q, which means that v is admissible. If v, w ∈ T_qN, v ≠ w are chosen and they both are admissible, then the map H_v,w: U_v,w→ V_v,w is determined by r, (q,η), (q, η) ∈ K_G(p) and data (<ref>). Let U'_v,w⊂ U_v,w be an open neighborhood of Θ_q,q,w(p) such that DH_v,w(z)≠ 0 for every z ∈ U'_v,w. We conclude that a neighborhood V_q,q, v⊂ V_q,q of p such that the map Θ_q,q,v:V_q,q, v→ S_qN × is a smooth coordinate map, can be defined for instance by formulaV_q,q, v:=Θ_q,q, v^-1(U'_v,w). Finally we will prove the equations(<ref>) and(<ref>). Let p ∈ M_1. By (<ref>) and (<ref>)we have that (q,η),(q, η) ∈ K_G(p) and (<ref>) holds if and only if the same is true for (ϕ (q),D Φη),(ϕ(q), DΦη). If(q,η),(q, η) ∈ K_G(p) and (<ref>) holds then by (<ref>) and (<ref>) the equation (<ref>) follows. Lastly we notice that B_ϕ(q)=D Φ (B_q). Therefore, we conclude that for given (q,η),(q, η) ∈ K_G(p) for which (<ref>) is valid the equation (<ref>) holds for v ∈ T_qN_1 if and only if the same holds forD Φ v ∈ T_ϕ (q)N_2. Therefore, the equation (<ref>) follows from (<ref>). §.§ Boundary coordinates Next we construct the coordinates near M. Let p ∈ M. Let I be the setof self-intersection directions of p (see (<ref>)). By Lemma <ref> the set S_pN ∖ I is open and dense in S_pN. As p ∈ M the data (<ref>) determines the set I, since it holds thatI={(p,ξ) ∈_-S_pM:there exists η∈_-S_pM,η≠ξ such that Σ(p,ξ)=Σ(p,η)}.Note that the usual boundary normal coordinates might not work well with our data, since it might happen that p is a self-intersecting point of the boundary normal geodesic γ_p,ν. In this case it is difficult to reconstruct the map U ∋ w ↦ z(w) ∈ M, from the data (<ref>). Here U is the domain of boundary normal coordinates and z(w) is the closest boundary point. To remedy this issue, we will prove in the next lemma that any non-vanishing, non-tangential, inward pointing vectorfield on M determines a similar coordinate system as the boundary normal coordinates.Let (M, g) be a smooth compact manifold with strictly convexboundary and let W be a non-vanishing, non-tangential and inward pointing smooth vector field on M. Let p ∈ M.Then there exists T>0such that the mapE_W:M× [0,T) →M, E_W(z,t)=exp_z(tW(z))is well defined and moreover, there exists T_p ∈ (0,T) and ϵ>0 such that the restriction E_W: [0,T_p) × B_ M(p,ϵ) →Mis a diffeomorphism. Here B_ M(p,ϵ):={z ∈ M: d_ M(p,z) < ϵ}and d_ M(p,z) is the distance from p to z along the boundary. Since W(z) ∈ (_+ SM)^int for every z ∈ M andM is compact, the Lemma <ref>guarantees that there exists T>0 such that for every t ∈ [0,T] and z ∈ M it holds that E_W(t,p)∈M.Let (z^j)_j=1^n-1 be local coordinates at p on the boundary.Since the map E_W is smooth it suffices to show that the Jacobian matrix DE_W is invertible at (p,0).By straightforward computation we have/ tE_W(p,t)|_t=0=W(p)and / z^jE_W(z,0)|_z=p=/ z^j, j ∈{1,…, n-1}.Since W(p) is not tangential to the boundary, it holds by (<ref>) that DE_W at (p,0) is invertible. Choose a non-vanishing, non-tangential inward pointing smooth vector field W on M such that W(p)∉ I=I_p.By Lemma <ref> there exists such a neighborhood V of p such that the mapE_W^-1:V → M × [0,T_p) is a smooth coordinate map. Let w ∈ V. We denote Π_W(w):=z∈ Mif and only if E_W(z,t)=w for some uniquet∈[0,T_p).See Figure (<ref>). We note that the map Π_W is smooth.In the next Lemma we show that the data (<ref>) determines a continuous map Π_W^E that coincides with Π_W near p. Later we will use the map Π_W^E to construct (n-1)-coordinates for points near p.Let (N,g) and M be as in the Theorem <ref>. For given r ∈ R_ M^E(M) where p∈ M is the unique point for which R_ M^E(p)=r and let W be a non-vanishing, non-tangential inward pointing smooth vector field such that W(p)∉ I,the data (<ref>) with r and W,determine such a (q,η) ∈ K_G(p) and a neighborhood V_q,W⊂M of p, q ∉ V_q,W for which the map Θ_q:V_q,W→ S_qN is well defined. Moreover,γ̇_q,-Θ_q(p)(τ_exit(q,-Θ_q(p))) ∦ W(p)anda map Π^E_q,W:V_q,W→ M∩ V_q,W, is determined by (<ref>).Here Π^E_q,W is a continuous map which coincides with Π_W near p.More precisely, let (N_i,g_i) and M_i be as in Theorem <ref> and suppose that (<ref>) and (<ref>) hold.Let W be a smooth non-vanishing, non-tangential, inward pointing vector field on M_1 such that W(p)∉ I. If (i)r=R^E_ M_1(p)∈ R^E_ M_1( M_1) (ii)(q,η) ∈ K_G(p) (iii)V_q,W is neighborhood of p such that q ∉ V_q,W and for every z ∈ V_q,W the property (<ref>) holds.are given. Then [γ̇_q,-Θ_q(p)(τ_exit(q,-Θ_q(p))) ∦ W(p);if and only if; γ̇_ϕ (q),-Θ_ϕ(q)(ϕ(p)))(τ_exit(ϕ (q),-Θ_ϕ (q)(ϕ(p)))) ∦ DΦ W(ϕ(p)), ]andϕ ( Π^E_q,W(z))= Π^E_ϕ(q), D Φ W(Ψ(z)), for allz ∈ V_q,W.We start by showing how property (<ref>) can be verified. Let R_ M^E(U) ⊂ R_ M^E(M) be a neighborhood of r=R_ M^E(p). For everyr' ∈ (R_ M^E(U)∩ R_ M^E( M)), r' ≠ r we define the number N(r') to be the cardinality of the set A(r,r'):={ξ∈ (S_zN) ∩ r;Σ(z,ξ) ⊂ R^E_ M(U), (z,ξ) ∈ K_G(p)},here z=(R^E_ M)^-1(r'). By Lemma <ref> there exists a neighborhood R_ M^E(U') ⊂ R_ M^E(U) of r such that N(r')=1 for every r' ∈ (R_ M^E(U')∩ R_ M^E( M)), r'≠ r. Since the sets A(r,r') are determined by r and (<ref>) we can find the set R_ M^E(U'). We denote byU' ⊂ U the unique neighborhoods of p that are related to R_ M^E(U') andR_ M^E(U). We write z=(R^E_ M)^-1(r') ∈ U', for given r' ∈ R_ M^E(U'). Therefore, by Lemma <ref> for the vector ξ∈ A(r,r'), there exists a neighborhood V_z⊂ U' of p such that z ∉ V_z and the map Θ_z:V_z → S_zN is smooth and well defined. Moreover there exists a unique (p,v_z)∈ S_pN so that γ_p,v_z(τ_exit(p,v_z))=zand γ̇_p,v_z(τ_exit(p,v_z))=Θ_z(p)∈ A(r,r')andΣ(z,Θ_z(p))=Σ(p,v_z).Then v_z= γ̇_z,-Θ_z(p)(τ_exit(z,-Θ_z(p))) and thus the data (<ref>) determines, if v_z and W(p) are parallel. It follows from Lemma <ref> that there existsq ∈ U' such that v_q is not parallel to W(p). Therefore we can use data (<ref>) to determine a point q that satisfies the property(<ref>).Let (q,η) ∈ A(r,R_ M^E(q)). Write V_q⊂ U' for a neighborhood of p, q ∉ V_q such that the map Θ_q: V_q → S_qN is well defined. Next we will construct the map Π_W^E, see (<ref>). Since γ_p,-W is not self-intersecting at p there exists T>0 such that for any t∈ [0,T] the point γ_p,-W(t) is not a self-intersecting point on geodesic γ_p,-W. We will prove that there exists 0<T'≤ T and ϵ>0 such that for any t ∈ [0,T'] and z ∈ B_ M(p,ϵ) the point γ_z,-W(t) is not a self-intersecting point of the geodesic γ_z,-W. If that is not true, then there exists a sequence (p_j)_j=1^∞⊂ M that converges to p as j →∞ and moreover, there exist sequences (t_j)_j=1^∞, (T_j)_j=1^∞⊂ such that for every j ∈, we have T_j>t_j≥ 0, T_j≤τ_exit(p_j,-W), γ_p_j,-W(t_j)=γ_p_j,-W(T_j)andt_j → 0asj →∞.By Lemma <ref> we may assume that T_j → T_0∈_+ as j →∞. Thenp=lim_j →∞γ_p_j,-W(t_j)=lim_j →∞γ_p_j,-W(T_j)=γ_p,-W(T_0).Notice that T_0>0 since otherwice geodesic loops γ_p_j,-W([t_j,T_j]) would be short and this would contradict Lemma <ref>. Therefore due to Corollary <ref> it must hold that T_0=τ_exit(p,-W). But this contradicts the assumption that p is not a self-intersecting point of γ_p,-W.Consider the setO(p,T^(1),ϵ^(1)):={exp_z(sW(z))∈M; z ∈ B_ M(p,ϵ^(1)), s ∈(0,T^(1))}⊂ V_q,where T^(1)>0 and ϵ^(1)>0are so small thatfor any t ∈ [0,T^(1)] and z ∈ B_ M(p,ϵ^(1)) the point γ_z,-W(t) is not a self-intersecting point on the geodesic γ_z,-W.For every w ∈ M ∩ O(p,T^(1),ϵ^(1)) we denote the component of Σ(w,W) ∩ R_ M^E(O(p,T^(1),ϵ^(1))) that contains R_ M^E(w) by C(w,T^(1),ϵ^(1)). Moreover, due to the choice of T^(1) and ϵ^(1) it holds that R_ M^E(γ_w,W([0,T^(1)])) = C(w,T^(1),ϵ^(1)), for every w ∈ M ∩ O(p,T^(1),ϵ^(1)). By Lemma <ref>there exists T^(2)∈ (0, T^(1)) and ϵ^(2)∈ (0,ϵ^(1)) such that, the set O(p,T^(2),ϵ^(2)) is open and for allw,z ∈ ( M ∩ O(p,T^(2),ϵ^(2))), w≠ z it holds thatγ_w,W([0,T^(2)]) ∩γ_z,W([0,T^(2)])=∅.ThereforeC(w,T^(2),ϵ^(2)) ∩ C(z,T^(2),ϵ^(2))=∅.Finally we notice that R_ M^E(O(p,T^(2),ϵ^(2)))=⋃_w ∈ ( M ∩ O(p,T^(2),ϵ^(2)))C(w,T^(2),ϵ^(2)).Then for a point z ∈ O(p,T^(2),ϵ^(2)) there exists a unique w ∈ ( M ∩ O(p,T^(2),ϵ^(2))) such that R_ M^E(z) ∈ C(w,T,^(2), ϵ^(2)). Next we will show how we can determine a set V_q,W⊂ V_q that is similar to O(p,T^(2),ϵ^(2)) just using the data (<ref>). Let ϵ >0 be so small that B_ M(p,ϵ) ⊂ V_q. For every w ∈ B_ M(p,ϵ) we write C(w) for the maximal connected subset of Σ(w,W) that contains R_ M^E(w) and satisfies (P1) The sets C(w) are homeomorphic to [0,1).(P2) For all w,z ∈ B_ M(p,ϵ), w≠ z the sets C(w) and C(z) do not intersect.(P3) The set V:=⋃_w ∈ B_ M(p,ϵ) C(w) ⊂ R_ M^E(M) is open and contained in R_ M^E(V_q).We emphasize that properties (P1)–(P3) hold by Lemma <ref>, if ϵ>0 is small enough. Moreover since the map R_ M^E is a homeomorphism, for any given value of ϵ we can verify,by using data (<ref>), if conditions (P1)–(P3) ave valid. Thus, let us chooseϵ>0 so that (P1)–(P3) are valid and define V_q,W:=(R^E_ M)^-1(V). Finally we are ready to define the map Π^E_q,W by the equationΠ^E_q,W:V_q,W→ M,Π^E_q,W(z):={w ∈ ( M ∩ V_q,W) ; R_ M^E(z) ∈ C(w)}.By the continuity of the map E_W (as in Lemma <ref>) and the properties (P1)–(P3) it holds that the map Π^E_q,W is continuous on V_q,W. Moreover due to Lemma <ref> the map Π^E_q,W coincides with Π_W near the point p. Let p ∈ M_1 and let vector field W ∈ (_+SM)^int be such that W(p)∉ I.Choose (q,η) ∈ K_G(p) and V_q,W, a neighborhood of p such that q ∉ V_q,W for which (<ref>) holds.Finally we will verify that properties (<ref>)–(<ref>)are valid for p if and only if they are valid for Ψ(V_q,W)⊂M_2.The property (<ref>) follows from(<ref>) and (<ref>).By the definiton (<ref>) of the mapping Ψ the properties (P1)–(P3) are valid in V_q,W if and only if the same holds for Ψ(V_q,W). Therefore (<ref>) follows from (<ref>).Given p∈ M and a non-vanishing, non-tangential inward pointing smooth vector field W on M such that W(p)∉ I, we choose (q,η) ∈ K_G(p) and a neighborhood V_q,W of p such that q ∉ V_q,W as in Lemma <ref>. Let v ∈ T_qN and define Q_q,v: V_q,W→,Q_q,v(x):=⟨ v,Θ_q(x) ⟩_g. Now we are ready to construct a boundary coordinate system near p∈ M.Let (N,g) and M be as in the Theorem <ref>. Let r ∈ R_ M^E(M) and let p∈ M to be the unique point for which R_ M^E(p)=r.Let W be a non-vanishing, non-tangential inward pointing smooth vector field on M such that W(p)∉ I, we choose (q,η) ∈ K_G(p) and a neighborhood V_q,W of p such that q ∉ V_q as in Lemma <ref>. Then there exists v ∈ T_qN and a neighborhood V_q,v,W of p such that (Q_q,v, Π^E_q,W):V_q,v,W→ (× M) defines a smooth boundary coordinate system near p. HereQ_q,v(x):= ± (Q_q,v(Π^E_q, W(x))-Q_q,v(x)),where the sign is chosen such that Q_q,v≥ 0. See Figure (<ref>).Moreover, for given r=R_ M^E(p) ∈ R_ M^E(M), W such that W(p)∉ I and (q,η) ∈ K_G(p) the data (<ref>) determines v ∈ T_qN and a neighborhood R_ M^E(V_q,v,W) of r and the map (Q_q,v, Π^E_q,W)∘ (R_ M^E)^-1:R_ M^E(V_q,v,W) → (× M), such that (Q_q,v, Π^E_q,W):V_q,v,W→ (× M) is a smooth coordinate map from a neighborhood V_q,v, W of p onto some open set U ⊂ (× M).More precisely, let (N_i,g_i) and M_i be as in Theorem <ref> such that (<ref>) and (<ref>) hold. Then for a given p ∈ M_1,(q,η)∈ K_G(p), W, that is a non-vanishing, non-tangential inward pointing smooth vector field on M_1 such that W(p)∉ I, neighborhood V_q,W of p as in Lemma <ref> and v∈ T_qN_1we have that (<ref>) holds. If s∈, w ∈ M_1 and z ∈ V_q thenQ_q,v(z)=(s,w)if and only if Q_ϕ(q),DΦ v(Ψ(z))=(s,ϕ(w)),andD (Q_q,v, Π^E_q, W)(p)≠ 0,if and onlyD(Q_ϕ(q),DΦ v, Π^E_ϕ(q), D Φ W)(Ψ(p)) ≠ 0.Recall that by Lemma <ref> and (<ref>) it holds that W(p)is not contained in ker(D Θ_q). Therefore there exists v ∈ T_qN such that ⟨ v, DΘ_qW(p) ⟩_g ≠ 0.Choose the local coordinates at p with the map E_W (as in Lemma <ref>). Then with respect to these coordinates we haveD(Q_q,v, Π^E_q, W)(p)=([ 0^T -(W Q_q,v)(p);Id_n-1 0 ]),since Π^E_q,W coincides with Π_W near p. Also it holds that(W Q_q,v)(p)=⟨ v, DΘ_q W(p) ⟩_g ≠ 0.Therefore D(Q_q,v, Π^E_q, W)(p) is invertible. Moreover, (Q_q,v, Π^E_q, W)(z)=(0,z) if z ∈ M and due to(<ref>), Lemma <ref> and the Taylor expansion of Q(q,v) it holds that Q_q,v(w)≠ 0 when w ∉ M∩ V_q,W is close to p. We choose so small neighborhood V_q, v, W for p such that Q_q,v: V_q, v, W→ vanishes only in V_q, v, W∩ M. Therefore we have proven that (Q_q,v, Π^E_q, W): V_q, v, W→^n is a smooth boundary coordinate map at p.Notice that for given r ∈ R_ M^E( M), W, (q,η), ∈ K_G(p) and v∈ T_qN the data (<ref>) determines a neighborhood R^E_ M(V_q, v, W) of r and the map (Q_q,v, Π_q, W) ∘ (R^E_ M)^-1 by a similar argument as in the proof of Lemma <ref>. The equations(<ref>) and (<ref>) can be proven by a similar argument as in the proof of Lemma <ref>.Let A be such an index set that collections (V_q,q, v^α,Θ_q,q,v^α)_α∈ A and(V^β_q,v,W,(Q_q,v, Π^E_q,W)^β)_β∈ A, that are as in Lemmas <ref> and <ref>, form a smooth atlas of (M, g). We define an atlas for R^E_ M(M) using the following charts(R^E_ M(V_q,q, v^α),Θ_q,q,v^α∘ (R^E_ M)^-1)_α∈ A and(R^E_ M(V^β_q,v,W),(Q_q,v, Π^E_q,W)^β∘ (R^E_ M)^-1)_β∈ A. By Lemmas <ref> and <ref> the charts given in (<ref>) define a smooth structure for R^E_ M(M). Let (M, g) be as in Theorem <ref>. If data (<ref>) is given, we can find the smooth charts of (<ref>) using only the data(<ref>). Moreover with respect to this smooth structure the map R^E_ M:M→ R^E_ M(M)is a diffeomorphism.The claim follows from Lemmas<ref>, <ref> and the definition (<ref>). Now we are ready to present the main theorem of this section.The mapping Ψ:M_1 →M_2 is a diffeomorphism.Sincemaps R_ M_i^E, i ∈{1,2} are diffeomorphisms it suffices to prove that D Φ: R_ M_1^E (M_1)→ R_ M_2^E(M_2)is a diffeomorphism. Let p ∈ M_1. By Lemma <ref> there exist (q,η), (q, η) ∈ K_G(p), v ∈ T_qM_1 and a neighborhood V=V_q,q, v of p such that the map Θ_q,q, v:V → (S_qM_1 ×) is a smooth coordinate map. Then it follows from (<ref>) and (<ref>) that Θ_ϕ(q),ϕ(q), DΦ v:Ψ(V) → (S_qM_2 × (∖{0})) is a smooth coordinate map. Therefore the local representation of D Φ, with respect to the smooth structures as in (<ref>), on R^E_ M(V) is given by(_-S_qM_1 ×(∖{0}))∋(ξ,s) ↦ (DΦ ξ,s) ∈ (_- S_ϕ(q)M_2 ×(∖{0})).Since Φ:K_1 → K_2 (see (<ref>)) is a diffeomorphism it holds thatD Φ isdiffeomorphic on R^E_ M_1(V). Let p ∈ M_1. By Lemma <ref> there exist (q,η) ∈ K_G(p), v ∈ T_qM_1, a smooth vector field W on M_1, and a neigborhood U=V_q,v,W of p such that the map (Q_q,v, Π^E_q, W) :U →× M_1 defines smooth local boundary coondinates. Moreover by (<ref>) and (<ref>) the map (Q_ϕ(q), DΦ v, Π^E_ϕ(q), DΦ W) :Ψ(U) →× M_2 defines smooth local boundary coordinates near ϕ(p).Therefore thelocal representation of D Φ on R^E_ M_1(U), with respect to the smooth structures as in (<ref>),is given by (× M_1)∋ (s,z)↦ (s,ϕ(z)) ∈× M_2.Since ϕ: M_1 → M_2 is a diffeomorphims it holds thatD Φ on R^E_ M_1(U) is diffeomorphic to its image.Thus we have proved that Ψ is a local diffeomorphism. By Theorem <ref> the map Ψ is one-to-one and therefore it is a global diffeomorphism. § THE RECONSTRUCTION OF THE RIEMANNIAN METRICSo far we have shown that the map Ψ:M_1 →M_2 is a diffeomorphism. The aim of this section is to show that Ψ is a Riemannian isometry. We start with showing that Ψ preserves the boundary metric. This is formulated precisely in the following Lemma.The map Ψ:M_1→M_2 has a property(Ψ^∗ g_2)|_ M_1=g_1|_ M_1. Since Ψ is smooth, the metric tensor Ψ^∗ g_2 is well defined.ByTheorem <ref> and (<ref>) we have(Ψ^∗ g_2)|_ M_1 =ϕ^∗(g_2|_ M_2)=g_1|_ M_1.We denote g:=Ψ^∗g_2 the pullback metric on M_1 and g:=g_1. We will also denote from now on M_1=M. By Lemma<ref>g and g coincide on the boundary M. Next we will show that they also have the same Taylor expansion at the boundary. Suppose that N, M, g andg are as in Theorem <ref> such that data (<ref>)and (<ref>) is valid with ϕ=id.Let (x^1,…, x^n) be any coordinate system near the boundary and α∈^n any multi-index. Write g=(g_ij)_i,j=1^n and g=(g_ij)_i,j=1^n. Then for all i,j ∈{1, …,n} holds∂^α g_ij |_ M= ∂^αg_ij |_ M,^α:=∏_k=1^n (/ x^k)^α_kLet (p,η) ∈_- SM. Suppose that η is not tangential, since M is non-trapping there exists a unique vector (q,ξ) ∈_- SM such that (q,ξ)≠ (p,η) andΣ(p,η)=Σ(q,ξ), where set Σ(p,η) is defined as in (<ref>). Therefore, the data (<ref>) determines the scattering relation L_g: _+ SM→_- SM, L(p,-η)=(γ_p,-η(τ_exit(p,-η)),γ̇_p,-η(τ_exit(p,-η)))=(q,ξ).Thus by (<ref>) it holds that L_g=L_g=:L, where L_g is the scattering relation of metric tensor g.It is shown in <cit.> Proposition 2.1 that the scattering relation L_g with Lemma <ref> determine the first exit time function τ_exit for ξ close to S_p M. More precisely, for every p ∈ M there exists a neighborhood 𝒱_p ⊂_+ S_pM of S_pM such that for all (p,ξ) ∈𝒱_p τ_exit(p,ξ)=τ_exit(p,ξ),where τ_exit is the first exit time function of g. Denote 𝒱:=∪_p ∈ M𝒱_p. We call the pair (L|_𝒱, τ_exit|_𝒱) the local lens data. In Theorem 2.1. of <cit.> it is shown that the local lens data implies (<ref>).Lemma <ref> and Proposition <ref> imply that we can smoothly extend g onto N such that g=g̃ in N∖M. Next we will show that the geodesics of g and g are the same.Metrics g and g are geodesically equivalent, i.e., for every geodesic γ:→ N of metric g, there exists a reparametrisation α:→ such that the curve γ∘α is a geodesic of metric g, and vice versa.Since M is non-trapping, any geodesic arc γ of metric g that is contained in M can be parametrized with a set Σ(p,η) for some (p,η)∈_-SM =_-SM. Thus by (<ref>) any geodesic γ of g can be re-parametrized to be a geodesic γ of metric g and vice versa. This means that there exists a smooth one-to-one and onto functionα:[a',b']→ [a,b] such thatα̇>0,γ(t)=(γ∘α)(t). See Subsection 2.0.5 of <cit.> for more details. Now we are ready to prove the main result of this section. The proof is similar to the one given in <cit.> which works for more general settings. The key ingredient of the proof is the Theorem 1 of <cit.>. The metrics g and g coincide in M.Define a smooth mapping I_0:TN → asI_0(p,v)=( (g)|_p/(g)|_p)^2/n+1g(v,v),note that the function f(p):= (g)|_p/(g)|_p is coordinate invariant, since in any smooth local coordinates (U,(x^j)_j=1^n) the function f coincides with the function p ↦ (g^ik(p)g_jk(p)) in U where (g^ij)_i,j=1^n is the inverse matrix of (g_ij)_i,j=1^n and the (1,1)-tensor fieldg^ikg_jk/ x^i⊗ dx^j can be considered as a linear automorphism on TU.Let γ_g be a geodesic of metric g. Define a smooth path β in TN as β(t)=(γ_g(t), γ̇_g(t)). Then β is an integral curve of the geodesic flow of metric g. Theorem 1 of <cit.> states that, if g and g are geodesically equivalent,then function t↦ I_0(β(t)) is a constant for all geodesics γ_g of metric g.Since g and g coincide in the set N ∖ M, we have that for anypoint z ∈ N ∖ M and vector v ∈ T_zN I_0(z,v)=g_z(v,v)=g_z(v,v).Let p ∈ M. Choose γ(t):=γ_z, ξ(t), ξ∈ S_zN,z∈ N ∖ Mbe a g-geodesic such that p=γ(t_p) for some t_p> 0. Then I_0(z,ξ)=1 andby Theorem 1 of <cit.>I_0 is constant along the geodesic flow (γ(t),γ̇(t)), i.e., I_0(p,γ̇(t_p))=I_0(z,ξ)=1. Since M is non-trapping, (<ref>) implies that, in local coordinates, for any ξ∈ S_pN g_ij(p)ξ^iξ^j=1=I_0(p,ξ)=(f(p))^2/n+1g_ij(p)ξ^iξ^j.Then the equation (<ref>) holds for all ξ∈ T_pN and both sides of equation (<ref>) are smooth in ξ, we obtain g_ij(p)=(f(p))^2/n+1g_ij(p),for alli,j∈{1,…,n}. Moreover(f(p))^2n/n+1-1=1,which implies that f(p)=1. By (<ref>) we conclude that g= g at p. Since p ∈ M was arbitrary the claim follows. The proof of Theorem <ref> follows from Theorems <ref>, <ref> and Proposition <ref>.We will give a Riemannian metric G to smooth manifold R^E_ M(M) as a pullback of g, that is G:= ((R^E_ M)^-1)^∗g. A priori we don't know g, thuswe don'tknow G. However as the smooth structure is known we can recover the collection (R^E_ M(M)), that is the collection of all Riemannian metrics of the smooth manifold R^E_ M(M). Since (M,g) is non-trapping we can use the sets Σ(p,η), (p,η) ∈_-SM to recover the images of the geodesics of G. Thus by the proof of Proposition <ref> the following set contains precisely one elementthat is (G,G),[ {(h,h') ∈(R^E_ M(M))^2: ∂^α h_ij |_R^E_ M( M)= ∂^α h'_ij |_R^E_ M( M) for every α∈^n,;metricsh and h' are geodesically equivalent,; sets Σ(p,η), (p,η) ∈_-SM are the images;of their geodesics}. ] We conclude that we have proven the following Proposition. The data (<ref>) determine the metric tensor G:= ((R^E_ M)^-1)^∗g. Finally we note that the proof of Theorem <ref> follows from the Propositions <ref>, <ref> and <ref>. Acknowledgements. The research of ML, and TS was partly supported bythe Finnish Centre of Excellence in Inverse Problems Research and Academy of Finland. In particular, ML was supported by projects 284715 and 303754, TS by projects 273979 and 263235. TS would like to express his gratitude to the Department of Pure Mathematics and Mathematical Statistics of the University of Cambridge for hosting him during two research visits when part of this work was carried out. TS would like toalso thank Prof. Gunther Uhlmann for hosting him in University of Washington where part of this work was carried out. HZ is supported by EPSRC grantEP/M023842/1. He would like to thank Prof. Gunther Uhlmann and theDepartment of Mathematics and Statistics of the University of Helsinkifor the generous support of his visit to Helsinki during his PhD whenpart of the work was carried out. abbrv | http://arxiv.org/abs/1708.07573v2 | {
"authors": [
"Matti Lassas",
"Teemu Saksala",
"Hanming Zhou"
],
"categories": [
"math.DG",
"53C22, 35R30"
],
"primary_category": "math.DG",
"published": "20170824225438",
"title": "Reconstruction of a compact Riemannian manifold from the scattering data of internal sources"
} |
add1,add2]Sandro Claudio Lera [email protected],add3]Didier Sornette [email protected][add1]ETH Zurich, Singapore-ETH Centre, 1 CREATE Way, #06-01 CREATE Tower, 138602 Singapore [add2]ETH Zurich, Department of Management, Technology, and Economics, Scheuchzerstrasse 7, 8092 Zurich, Switzerland [add3]Swiss Finance Institute, c/o University of Geneva, Geneva, Switzerland A new model that combines economic growth rate fluctuations at the microscopic and macroscopic level is presented.At the microscopic level, firms are growing at different rates while also being exposed to idiosyncraticshocks at the firm and sector level.We describe such fluctuations as independent Lévy-stable fluctuations, varying over multiple orders of magnitude.These fluctuations are aggregated and measured at the macroscopic level in averaged economic output quantities such as GDP.A fundamental question is thereby to what extend individual firm size fluctuations can have a noticeable impact on the overall economy.We argue that this question can be answered by considering the Lévy fluctuations as embedded in a steep confining potential well,ensuring nonlinear mean-reversal behavior, without having to rely on microscopic details of the system.The steepness of the potential well directly controls the extend towards which idiosyncratic shocks to firms and sectors are damped at the level of the economy.Additionally, the theory naturally accounts for business cycles, represented in terms of a bimodal economic output distribution, and thus connects two so far unrelated fields in economics.By analyzing 200 years of US GDP growth rates, we find that the model is in good agreement with the data. growth rate distributions Lévy flightsidiosyncratic shocksJEL: C22 E32 E37§ INTRODUCTION As a result of the increasing interconnectedness of firms, and more general, of society <cit.>, the study of firm size fluctuations and its effect on the entire economy has become an active area of research <cit.>.A central question is how the productivity fluctuations of individual firms aggregate into productivity measures for the economy as a whole.Albeit still subject to debate <cit.>, there is now a dominant view <cit.>, that idiosyncratic shocks to individual large firms can have a significant effect on the macroeconomic output growth. Explanations for this effect range from statistical fluctuation arguments <cit.> to more intricate models taking into consideration intersectoral input-output linkages <cit.> or dynamic income-expenditure networks <cit.>. In this article, we offer a different, more coarse-grained perspective.We show how heavy-tailed idiosyncratic shocks are mitigated through a kind of renormalized economic potential to the aggregate output growth distribution. Specifically, we claim that the aggregate output growth rate distribution can be modeled as a Lévy flight in a steepconfining potential. This establishes a quantitative connection between the tail distribution of idosyncratic shocks to individual firms and the aggregate economic growth distribution.To provide empirical support for our claim, we study the gross domestic product (GDP) of the United States over that last 200 years, and find that the model is in good agreement with the data.§ STRUCTURE OF THE GDP GROWTH RATE DISTRIBUTION As a measure of overall economic growth, we analyze the real US GDP per capita(from hereon, simply GDP) over the past 200 years (figure <ref>(a)).In contrast, the unnormalized total nominal GDP contains growth contributions as a result of population growth and inflation, both of which are not true sources of increased output productivity per individual.Given the scrutiny of the GDP in economic research, the distribution of its growth rates has also received a lot of attention.It is now a broadly accepted stylized fact that the distribution of GDP growth rates has tails that are fatter than Gaussian <cit.>, and can even be power-law <cit.> (but see also ref. <cit.> for a counter argument). Another, much less studied property is the bimodal-structure of the growth rate distribution <cit.>. The two peaks of the bimodal distribution can be rationalized by the fundamental out-of-equilibrium nature of the economy that is switching between boom and bust states, i.e business cycles. Following ref. <cit.>, we first extract the annual GDP growth rates using a smoothening wavelet filter (figure <ref>(b)),from which we can then extract the growth rate distribution with a Gaussian density kernel estimate (figure <ref>, main plot).To the eye, the bimodal structure in the middle of the distribution is apparent.But one has to be careful, as this bimodal appearance is susceptible to the choice of bandwidth of the Gaussian kernel. Statistically sound evidence for this bimodal structure has been provided in ref. <cit.>. Figures <ref>(c) and (d) depict survival functions (CCDF) of the left and right tail of the distribution, respectively.The five largest positive and negative growth rates, associated with the two world wars and the great depression, are annotated explicitly. By applying likelihood ratio tests of the power-law against the family of stretched-exponentials <cit.>, we find p-values of 0.96 and 0.61 for left- and right-tail respectively. We conclude that both tails are parsimonously represented in terms of pure power laws, without the need for the more general parameterisation of the stretched exponential distribution family. Using standard procedures <cit.>, we then determine the CCDF tail exponent and its estimated standard error.The exponents of left and right tail are surprisingly symmetric, and robust with respect to removal of the smallest and largest growth rates, leading merely to slightly increased error intervals.The tail exponent ν being smaller than 2 qualfies the tails has belonging to the Levy-stable distribution regime <cit.>.We have thus shown that the GDP growth rate distribution exhibits a bimodal structure with symmetry power-law tails.In the next section, we will present a model that non-trivially combines these two apparently unrelated properties. § LÉVY FLIGHTS IN STEEP CONFINING POTENTIALS A generic representation of the Markovian dynamics of the GDP growth rate in continuous time is given by r = -V/ r t +L(μ,D)where r=r_t is the growth rate at time t and V is some function, called the `potential', whichensures a nonlinear mean-reversing dynamics. The negative derivative of the potential is then interpreted as the `force' that is guiding the growth rate.The stochastic contribution L(μ,D) denotes Lévy-stable distributed noise with tail exponent μ∈ (0,2] and scale-parameter D > 0. The most common mean-reverting model of type (<ref>) is the Ornstein-Uhlenbeck process,characterized by a square potential (V(r) = α r^2 / 2, α > 0) and Gaussian noise (μ=2).With basic stochastic methods <cit.>, it can be shown that, for the Ornstein-Uhlenbeck process, the asymptotic stationary distribution of r_t is Gaussian with standard deviation D/√(2 α).This is illustrated in figure <ref> (left). In presence of Gaussian noise, the stationary distribution remains unimodal even when the square potential is replaced by a steeper potential V(r) = α/β |r-r_0|^ β (β > 2).Here, we have also introduced a potential midpoint r_0 for more generality.Similarly, the distribution remains unimodal for a square-potential in presence of a heavy-tailed source of noise (μ < 2). However, counter-intuitively, it has been shown <cit.> that the combination of a steep potential (<ref>),with heavy-tailed noise, results in a bimodal stationary distribution, as sketched in figure <ref> (right).Furthermore, the asymptotic left and right tail of the distribution are power-laws with CCDF tail exponent ν≡β + μ - 2.In particular, this implies that, for β > 4-μ, the steep potential walls confine the Lévy noise to the extent that its tails have finite variance.§ GROWTH RATE FLUCTUATIONS AS CONFINED LÉVY FLIGHTS We propose that the Lévy flight in a steep potential is a good model for economic growth rates.Concretely, we see the Lévy noise as a representation of the firm or sector specific idiosyncratic shocks.Indeed, firm size growth rates are known to be well approximated by heavy-tail distributions with infinite variance <cit.>.A priori, it is not clear that fluctuations of individual firms (or sectors) would have a significant impact on aggregate measures for the economy.According to the central limit theorem, the average fluctuations ofthe sum of the outputs of n firms decay as ∼ 1/√(n), a negligible effect in the limit of large n.The state of the economy is then merely influenced by economy-wide shocks such as oil price shocks,currency devaluation, wars, etc. However, such arguments break down if the distribution of firm sizes is very heavy tailed <cit.>, or the interdependencies of firms are asymmetric <cit.>.In a model that takes the microscopic firm structure into account, the individual firm shocks are propagated across the economy through a network of input-output linkages.To what extent the individual shocks are averaged out upon aggregation depends on the topology of the network of firm and sector dependencies <cit.>.If a small number of firms or sectors play a disproportionately important role as input suppliers to others, the economy is more susceptible to fluctuations of these few firms, in contrast to a more balanced scenario.Empirical evidence for the US <cit.>, France <cit.> and Sweden <cit.> suggests that indeed such asymmetries in the inter-firm and sector networkslead to significant contributions of a few large companies or sectors to the overall economic performance of a country. In our model, the microscopic firm network is represented by an effective `economic potential function'.The advantage of this coarse-grained perspective is that it does not rely on any specific assumptions about the underlying generating mechanisms at the micro level.The details average out and are captured in the potential parameter β.The steeper the potential, the more the individual firm shocks average out, and the less heavy the aggregate tail.This approach is motivated by the renormalisation group approach <cit.> that provides the template to represent the collective effect of many degrees of freedom at the micro level by a few effective `renormalised' degrees of freedom at the macro level.For more quantitative support of our proposition, we now fit the GDP data to the model.The Lévy flight in the steep potential has five parameters α, β, r_0, μ and D.Using the roughly 200 annual growth rate datapoints shown in figure <ref>(b), wecalibrate these parameters with maximum likelihood, and determine the standard errors through bootstrapping.The fitted parameters are robust with respect to removal of individual data-points, with the exception of the two, in absolute value, largest growth rates, associated with World War II. To obtain statistically robust estimates, we thus remove these two largest datapoints.Beyond statistical arguments, this is justified conceptually in the sense that a world war is a global shock that affects the economy as a whole.Since we are interested in contributions of individual shocks to aggregate economic output measures,it is justified to remove such events.Technically, we should thus get rid of all economy-wide shocks.But since only the aggregate GDP growth rates are analyzed, this is technically infeasible, and not necessary insofar as our calibration results are not sensitive to individual datapoints, except for the WWII case. WWII was indeed special for the U.S, which became the factory of the world, doubling its GDP from 1940 to 1943.Details regarding the fitting procedure are found in the supporting material of this paper and the result is depicted in figure <ref>. We find that the predicted GDP tail exponent ν̂ = β̂ + μ̂ - 2 = 1.7 ± 0.2 is in excellent agreement with the actual tail exponent between 1.5 and 2 (figure <ref>(c,d)).The magnitude of the microscopic noise with μ̂ = 1.5 ± 0.2 is lighter than Cauchy noise (μ = 1), but still clearly separated from normality (μ = 2).The potential midpoint r̂_0 = 1.2 ± 0.3 is more centered towards the left of the two bimodal peaks.However, as additional simulations in the supplementary material confirm, this is well within the realms of expectations when analyzing just 200 datapoints.We also note that the potential, with an exponent of β̂ = 2.2 ± 0.1, is not much steeper than a quadratic potential with the linear mean-reverting force.It is just sufficiently steeper than quadratic to ensure a binomial structure of the distribution of growth rates in the presence of Levy-noise but close enough to the quadratic case so that the GDP growth rate distribution has diverging second moment, in line with other findings <cit.>. In conclusion, we have shown that GDP growth rates are well described as Lévy flights in a potential that is steeper than quadratic.With only minimal ingredients, this model is able to capture the aggregate effect of idiosyncratic shocks to averaged economic output growth measures.It thereby establishes a connection between the tail exponents at a micro and macro scale, while simultaneously accounting for the bimodal structure of business cycle fluctuations. These are promising results that draw novel connections, and we hope that our findings will spark future research.§ ACKNOWLEDGMENTS. Lera acknowledges stimulating discussions with D. Georgiadis,G. Britten, S. Wheatley and M. Schatz.The research performed by S. Lera was conducted at the Future Resilient Systems at the Singapore-ETH Centre (SEC).The SEC was established as a collaboration between ETH Zurich and National Research Foundation (NRF)Singapore (FI 370074011) under the auspices of the NRF's Campus for Research Excellence and Technological Enterprise (CREATE) programme.§ REFERENCESunsrt10Schweitzeretal09 F. Schweitzer, G. Fagiolo, D. Sornette, F. Vega-Redondo, A. Vespignani, and D. R. White. Economic networks: The new challenges. Science, 325:422–424, 2009.Gabaix2011 X. Gabaix. The granular origins of aggregate fluctuations. Econometrica, 79(3):733–772, 2011.Acemoglu2012 D. Acemoglu, V. M. Carvalho, A. Ozdaglar, and A. Tahbaz-Salehi. The network origins of aggregate fluctuations. Econometrica, 80(5):1977–2016, 2012.diGiovanni2014 J. Di Giovanni, A. A. Levchenko, and I. Mejean. Firms, destinations, and aggregate fluctuations. Econometrica, 82(4):1303–1340, 2014.Friberg2016 R. Friberg and M. Sanctuary. The contribution of firm-level shocks to aggregate fluctuations: The case of Sweden. Economics Letters, 147:8–11, 2016.Anthonisen2016 N. Anthonisen. Microeconomic shocks and macroeconomic fluctuations in a dynamic network economy. Journal of Macroeconomics, 47(2011):233–254, 2016.Stella2015 A. Stella. Firm dynamics and the origins of aggregate fluctuations. Journal of Economic Dynamics and Control, 55:71–88, 2015.Fagiolo2008 G. Fagiolo, M. Napoletano, and A. Roventini. Are output growth-rate distributions fat-tailed? some evidence from OECD countries. Journal of Applied Econometrics, 23(5):639–669, 2008.Fagiolo2009 G. Fagiolo, M Napoletano, M. Piazza, and A. Roventini. Detrending and the distributional properties of US output time series. Economics Bulletin, 29(4):3155–3161, 2009.Fu2005 D. Fu, F. Pammolli, S. V. Buldyrev, M. Riccaboni, K. Matia, K. Yamasaki, and H. E. Stanley. The growth of business firms: Theoretical framework and empirical evidence. Proceedings of the National Academy of Sciences of the United States of America, 102(52):18801–18806, 2005.Williams2017 M. A. Williams, G. Baek, Y. Li, L.Y. Park, and W. Zhao. Global evidence on the distribution of GDP growth rates. Physica A, 468:750–758, 2017.Franke2015 R. Franke. How fat-tailed is US output growth? Metroeconomica, 66(2):213–242, 2015.Lera2017 S. C. Lera and D. Sornette. Evidence of a bimodal US GDP growth rate distribution : a wavelet approach. Quantitative Finance and Economics, 1(1):26–43, 2017.Malevergne2005 Y. Malevergne, V. Pisarenko, and D. Sornette. Empirical distributions of stock returns: Between the stretched exponential and the power law? Quantitative Finance, 5(4):379–401, 2005.Alstott2014 J. Alstott, E. Bullmore, and D. Plenz. Powerlaw: a Python package for analysis of heavy-tailed distributions. PloS one, 9(1):e85777, 2014.GnedenkoKolmo54 B.V. Gnedenko and A. N. Kolmogorov. Limit Distributions for Sums of Independent Random Variables. Addison-Wesley, 1954.Gardiner2009 C. W. Gardiner. Stochastic methods: a handbook for the natural and social sciences. Springer, 2009.Chechkin2003 A. V. Chechkin, J. Klafter, V. Y. Gonchar, and R. Metzler. Bifurcation, bimodality, and finite variance in confined Lévy flights. Physical Review E, 67(1):010102, 2003.Chechkin2004 A. V. Chechkin, V. Y. Gonchar, J. Klafter, R. Metzler, and L. V. Tanatarov. Lévy flights in a steep potential well. Journal of Statistical Physics, 115(5-6):1505–1535, 2004.Williams2016 M. A. Williams, G. Baek, L. Y. Park, and W. Zhao. Global evidence on the distribution of economic profit rates. Physica A, 458:356–363, 2016.Wilson79 K. G. Wilson. Problems in physics with many scales of length. Scientific American, 241(2):140–157, 1979. | http://arxiv.org/abs/1709.05594v1 | {
"authors": [
"Sandro Claudio Lera",
"Didier Sornette"
],
"categories": [
"q-fin.GN",
"physics.soc-ph"
],
"primary_category": "q-fin.GN",
"published": "20170826024929",
"title": "GDP growth rates as confined Lévy flights"
} |
Analyzing Cloud Optical Properties Using Sky Cameras Uri Malamud and Hagai B. Perets Accepted ?. Received ?; in original form ?. ===========================================================Shilpa Manandhar^1 ^*, Soumyabrata Dev^2, Yee Hui Lee^3, and Yu Song Meng^4 ^**^1Nanyang Technological University Singapore, Singapore 639798, email:^2Nanyang Technological University Singapore, Singapore 639798, email:^3 Nanyang Technological University Singapore, Singapore 639798, email:^4National Metrology Centre, Agency for Science, Technology and Research (A^*STAR), Singapore 118221, email: ^* Presenting author^** Corresponding author Clouds play a significant role in the fluctuation of solar radiation received by the earth's surface. It is important to study the various cloud properties, as it impacts the total solar irradiance falling on the earth's surface. One of such important optical properties of the cloud is the Cloud Optical Thickness (COT). It is defined with the amount of light that can pass through the clouds. The COT values are generally obtained from satellite images. However, satellite images have a low temporal- and spatial- resolutions; and are not suitable for study in applications as solar energy generation and forecasting. Therefore, ground-based sky cameras are now getting popular in such fields. In this paper, we analyze the cloud optical thickness value, from the ground-based sky cameras, and provide future research directions.IntroductionCloud optical properties and cloud coverage contribute in rapid fluctuation of solar radiation during a day. A correct estimation of cloud properties is useful in the emerging field of solar energy harvesting and forecasting. Satellite images are one of the primary sources to study different cloud properties. The Moderate Resolution Imaging Spectroradiometers (MODIS) installed on National Aeronautics and Space Administration (NASA) Earth Observing System’s (EOS) Terra and Aqua satellites transmit visible and near-infrared spectral signatures <cit.> to get the information on cloud dynamics and properties like optical thickness, effective particle radius, particle phase, and liquid water content. It also provide cloud mask data, that can be correlated from ground observations <cit.>. These instruments provide such important cloud properties information. However, they have poor temporal and spatial resolutions. Therefore, ground based sky cameras are now widely used to study the cloud properties <cit.>. In this paper, we obtain COT from MODIS satellite images, and analyze its relationship with the optical properties of visible-light images obtained from sky cameras.Data CollectionWe have installed a ground-based sky camera on a particular rooftop (1.3483^∘ N, 103.6831^∘ E) of our university building at Nanyang Technological University Singapore. This sky camera is called WAHRSIS (Wide Angled High Resolution Sky Imaging System) <cit.>. It captures the images of the sky scene at an interval of 2 minutes. These images are captured in the visible-light range, and are useful to provide higher temporal and spatial information of the cloud information. Several events in the earth's atmosphere can be effectively studied using such ground-based cameras. These captured images also provide us an idea about the optical properties of the cloud, depending on the normalized luminance of the captured image. For this paper, we analyze the deviation of captured luminance from the clear-sky luminance <cit.>, and correlate with COT.The COT values are calculated from MODIS products, which has spatial resolution of 1 km^2 (1 pixel) and is available twice a day (4 UTC and 7 UTC). In this experiment, we process COT values for 3 km × 3 km area (9 pixels) with the location of sky camera as the center. The value of COT ranges from 0 to 100. A cloud optical thickness of value 100 means that almost no light can pass through the clouds whereas a value of 0 means the sky is relatively cloud free. Experiments & Results For our experiments [The source codes of these experiments are available online at <https://github.com/Soumyabrata/cloud-optical-thickness>.], we consider all the MODIS observations in the year 2015. We process the average COT values over the 9 pixels, from the extracted 3×3 cloud optical thickness matrix. This provides us an average COT value, corresponding to the MODIS timestamp.Our sky cameras capture images at temporal resolutions of 2 minutes, and therefore, has higher resolutions as compared to the MODIS satellite images. We compute the luminance of the circumsolar region, and normalize it with the camera parameters. This entire methodology of computing the luminance of the circumsolar region, from the sky cameras is described in our previous work <cit.>. Based on this linear relationship between image luminance and solar irradiance, we can now compute the clear sky luminance. The clear sky luminance is referred as the corresponding luminance of the theoretical clear-sky model. We use the model by <cit.>, to estimate the clear-sky irradiance, and subsequently the clear sky luminance. Finally, the luminance difference is the difference between the actual luminance and clear-sky luminance.Taking the corresponding MODIS COT time as reference, we compute a 15 minute average of luminance difference from sky images. We also normalize the values of the luminance difference and cloud optical thickness value. Figure <ref> shows the scatter plot of calculated average luminance difference and average COT values. This is self explanatory – high luminance difference reflects presence of clouds. And thus, cloud presence is reflected by a higher value of COT. Interestingly, we can also observe lower COT values at higher luminance difference values. This possible deviation can be explained because of the possible mismatch between the coverage of MODIS and sky cameras. Also, there can exist some time gap between MODIS time stamp and the sky imager time stamp. In the future, we plan to model the cloud optical thickness value directly from the normalized luminance difference of the sky cameras. This will help us to estimate COT values at much finer time resolutions.Conclusion In this paper, we have studied the cloud optical thickness value using the ground-based sky cameras. This is an important contribution, as ground-based sky cameras, if installed at the experiment site, can provide higher time- and spatial- resolution of cloud formation.In our future work, we will propose a best fit line between COT and normalized luminance difference; and also work on designing Internet-of-Things (IoT) enabled wireless sensors <cit.> for sky cameras. This will help the remote sensing analysts to compute COT values for finer time resolutions, as compared to the current standards.The authors would like to thank Joseph Lemaitre for automatizing the acquisition and processing of MODIS multi bands images into a user-friendly framework.1Wei_2004 H. Wei, P. Yang, J. Li, B. A. Baum, H. L. Huang, S. Platnick, Y. X. Hu, and L. Strow, “Retrieval of ice cloud optical thickness from atmospheric infrared sounder (airs) measurements,” IEEE Transactions on Geoscience and Remote Sensing, vol. 42, no. 10, pp. 2254–2267, Oct. 2004.PIERS17a S. Manandhar, S. Dev, Y. H. Lee, and Y. S. Meng, “Correlating satellite cloud cover with sky cameras,” in Proc. Progress In Electromagnetics Research Symposium (PIERS), 2017.Dev2016GRSM S. Dev, B. Wen, Y. H. Lee, and S. Winkler, “Ground-based image analysis: A tutorial on machine-learning techniques and applications,” IEEE Geoscience and Remote Sensing Magazine, vol. 4, no. 2, pp. 79–93, June 2016.WAHRSIS S. Dev, F. M. Savoy, Y. H. Lee, and S. Winkler, “WAHRSIS: A low-cost, high-resolution whole sky imager with near-infrared capabilities,” in Proc. IS&T/SPIE Infrared Imaging Systems, 2014.IGARSS2015a S. Dev, F. M. Savoy, Y. H. Lee, and S. Winkler, “Design of low-cost, compact and weather-proof whole sky imagers for High-Dynamic-Range captures,” in Proc. International Geoscience and Remote Sensing Symposium (IGARSS), 2015, pp. 5359–5362.IGARSS_solar S. Dev, F. M. Savoy, Y. H. Lee, and S. Winkler, “Estimation of solar irradiance using ground-based whole sky imagers,” in Proc. International Geoscience and Remote Sensing Symposium (IGARSS), 2016.dazhi2012estimation D. Yang, P. Jirutitijaroen, and W. M. Walsh, “The estimation of clear sky global horizontal irradiance at the equator,” Energy Procedia, vol. 25, pp. 141–148, 2012.Deepu2017 C. J. Deepu, C. H. Heng, and Y. Lian, “A hybrid data compression scheme for power reduction in wireless sensors for IoT,” IEEE Transactions on Biomedical Circuits and Systems, vol. 11, no. 2, pp. 245–254, Apr. 2017. | http://arxiv.org/abs/1708.08995v1 | {
"authors": [
"Shilpa Manandhar",
"Soumyabrata Dev",
"Yee Hui Lee",
"Yu Song Meng"
],
"categories": [
"cs.CV",
"physics.ao-ph"
],
"primary_category": "cs.CV",
"published": "20170824165054",
"title": "Analyzing Cloud Optical Properties Using Sky Cameras"
} |
Freezeout systematics due to the hadron spectrum Subhasis Samanta December 30, 2023 ================================================ We review our current software tools and theoretical methods for applying the Neural Engineering Framework to state-of-the-art neuromorphic hardware. These methods can be used to implement linear and nonlinear dynamical systems that exploit axonal transmission time-delays, and to fully account for nonideal mixed-analog-digital synapses that exhibit higher-order dynamics with heterogeneous time-constants. This summarizes earlier versions of these methods that have been discussed in a more biological context (Voelker & Eliasmith, 2017) or regarding a specific neuromorphic architecture (Voelker et al., 2017). § INTRODUCTION This report is motivated by the recent influx of neuromorphic computing architectures including SpiNNaker <cit.>, Neurogrid <cit.>, Brainstorm <cit.>, IBM's TrueNorth <cit.>, BrainScaleS <cit.>, and ROLLS <cit.>, among others. These are massively-parallel, low-power, analog, and/or digital systems (often a mix of the two) that are designed to simulate large-scale artificial neural networks rivalling the scale and complexity of real biological systems. Despite growing excitement, we believe that methods to fully unlock the computational power of neuromorphic hardware are lacking.This is primarily due to a theoretical gap between our traditional, discrete, von Neumann-like understanding of conventional algorithms and the continuous spike-based signal processing of real brains that is often emulated in silicon <cit.>.We use the term “neural compiler” to refer loosely to any systematic method of converting an algorithm, expressed in some high-level mathematical language, into synaptic connection weights between populations of spiking neurons <cit.>. To fully leverage neuromorphic hardware for real-world applications, we require neural compilers that can account for the effects of spiking neuron models and mixed-analog-digital synapse models, and, perhaps more importantly, exploit these details in useful ways when possible. There exist various approaches to neural engineering, includingthose by Denève et al. <cit.> and Memmesheimer et al. <cit.>. However, the Neural Engineering Framework (NEF; <cit.>) stands apart in terms of software implementation (Nengo; <cit.>), large-scale cognitive modeling <cit.>, and neuromorphic applications <cit.>. Competing methods consistently exclude important details, such as refractory periods and membrane voltage leaks, from their networks <cit.>. The NEF, on the other hand, embraces biological complexity whenever it proves computationally useful <cit.> and/or improves contact with neuroscience literature <cit.>. This approach to biological modeling directly mirrors a similar need to account for the details in neuromorphic hardware when building neural networks <cit.>.Nevertheless, there are many open problems in optimizing the NEF for state-of-the-art neuromorphics. In particular, we have been working to account for more detailed dynamics in heterogeneous models of the post-synaptic current (PSC) induced by each spike, as well as delays in spike propagation <cit.>. The purpose of this report is to summarize our methods, both theoretical and practical, that have progressed in this direction. There are similar challenges in extending the NEF to account for multi-compartment neuron models <cit.>, conductance-based synapses <cit.>, and to minimize the total number of spikes <cit.> – but these topics will not be addressed in this report.The remainder of this report assumes that the reader is already comfortable with Nengo and the NEF in the context of engineering dynamical systems (i.e., “Principle III”).A technical yet accessible overview of the NEF can be found in <cit.>, and a tutorial for Nengo (version 2) can be found in <cit.>.§ ACCOUNTING FOR SYNAPTIC DYNAMICSThis section provides a theoretical account of the effect of higher-order linear synapse models on the dynamics of the network, by summarizing the extensions from <cit.> and <cit.>. This yields two novel proofs of Principle III from the NEF, and generalizes the principle to include more detailed synapses, including those modeling axonal transmission delays. §.§ Linear systemsHere we focus our attention on linear time-invariant (LTI) systems:ẋ⃗̇(t)= Ax⃗(t) + Bu⃗(t)y⃗(t)= Cx⃗(t) + Du⃗(t)where the time-varying vector x⃗(t) represents the system state, y⃗(t) the output, u⃗(t) the input, and the time-invariant “state-space” matrices (A,B,C,D) fully determine the system's dynamics. We will omit the variable t when not needed.Principle III from the NEF states that in order to train the recurrent connection weights to implement (<ref>)—using a continuous-time lowpass filterto model the PSC—we use Principle II to train the decoders for the recurrent transformation (τ A + I)x⃗, input transformation τ B u⃗, output transformation Cx⃗, and passthrough transformation Du⃗ <cit.>. This drives the recurrent synapses with the signal τẋ⃗̇ + x⃗ so that their output is the signal x⃗, in effect transforming the synapses into perfect integrators with respect to ẋ⃗̇. The vector x⃗ is then represented by the population of neurons via Principle I. Thus, this provides a systematic approach for training a recurrent neural network to implement any linear dynamical system. Now we show that this approach generalizes to other synaptic models.For these purposes, the transfer function is a more useful description of the LTI system than (<ref>). The transfer function is defined as the ratio of Y(s) to U(s), given by the Laplace transforms of y⃗(t) and u⃗(t) respectively.The variable s denotes a complex value in the frequency domain, while t is non-negative in the time domain.The transfer function is related to (<ref>) by the following:F(s) = Y(s)/U(s) = C(sI - A)^-1B + D . The transfer function F(s) can be converted into the state-space model (A,B,C,D) if and only if it can be written as a proper ratio of finite polynomials in s.The ratio is proper when the degree of the numerator does not exceed that of the denominator.In this case, the output will not depend on future input, and so the system is `causal'.The order of the denominator corresponds to the dimensionality of x⃗, and therefore must be finite.Both of these conditions can be interpreted as physically realistic constraints where time may only progress forward, and neural resources are finite. Continuous IdentityIn order to account for the introduction of a synaptic filter h(t), we replace the integrator s^-1 in (<ref>) with H(s), where H(s) = ℒ{ h(t) }. This new system has the transfer function C(H(s)^-1I - A)^-1B + D = F(H(s)^-1). To compensate for this change in dynamics, we must invert the change-of-variables s ↔ H(s)^-1. This means finding the required F^H(s) such that F^H(H(s)^-1) is equal to the desired transfer function, F(s). We highlight this as the following identity:F^H ( 1/H(s)) = F(s) .Then the state-space model (A^H,B^H,C^H,D^H) satisfying (<ref>) with respect to F^H(s) will implement the desired dynamics (<ref>) given h(t). Discrete IdentityFor the discrete (i.e., digital synapse) case, we begin with F(z) and H(z) expressed as digital systems. The form of H(z) is usually determined by the hardware, and F(z) is usually found by a zero-order hold (ZOH) discretization of F(s) using the simulation time-step (dt), resulting in the discrete LTI system:x⃗[t+dt]= A̅x⃗[t] + B̅u⃗[t]y⃗[t]= C̅x⃗[t] + D̅u⃗[t] .Here, we have the same relationship as (<ref>),F(z) = Y(z)/U(z) = C̅(zI - A̅)^-1B̅ + D̅.Therefore, the previous discussion applies, and we must find an F^H(z) that satisfies:F^H ( 1/H(z)) = F(z) .Then the state-space model (A̅^H, B̅^H, C̅^H, D̅^H) satisfying (<ref>) with respect to F^H(z) will implement the desired dynamics (<ref>) given h[t]. In either case, the general problem reduces to solving this change-of-variables problem for various synaptic models. We now provide a number of results. More detailed derivations are available in <cit.>. Continuous Lowpass SynapseReplacing the integrator s^-1 with the standard continuous-time lowpass filter, so that H(s) =1/τ s + 1:F^H(τ s + 1) = F(s)F^H(s) = C(sI - (τ A + I))^-1(τ B) + Dwhich rederives the standard form of Principle III from the NEF <cit.>. Discrete Lowpass SynapseReplacing the integrator with a discrete-time lowpass filter H(z) = 1 - a/z - a in the z-domain with time-step dt, where a = e^-dt/τ:F^H ( z-a/1-a) = F(z)F^H(z) = C̅( zI - 1/1 - a(A̅ - aI) )^-1( 1/1-aB̅) + D̅.Therefore, A̅^H = 1/1 - a(A̅ - aI), B̅^H = 1/1-aB̅, C̅^H = C̅, and D̅^H = D̅. This mapping can dramatically improve the accuracy of Principle III in digital simulations (e.g., when using a desktop computer) <cit.>.Delayed Continuous Lowpass SynapseReplacing the integrator with a continuous lowpass filter containing a pure time-delay of length λ, so that H(s) = e^-λ s/τ s + 1:F^H ( τ s + 1/e^-λ s) = F(s)F^H(s) = F ( 1/λ W_0(ds) - 1/τ) ,where d = λ/τe^λ/τ and W_0(xe^x) = x is the principal branch of the Lambert-W function <cit.>.[This assumes that |η| < π and λRe[ s ] + λ/τ > - ηη, where η := λIm[ s ] <cit.>.] This synapse model can be used to model axonal transmission time-delays due to the finite-velocity propagation of action potentials, or to model feedback delays within a broader control-theoretic context.To demonstrate the case where a pure time-delay of length θ is the desired transfer function (F(s) = e^-θ s), we let c = e^θ/τ and r = θ/λ to obtain the required transfer function:F^H(s) = c ( W_0(ds)/ds)^r = cr ∑_i=0^∞(i+r)^i-1/i! (-ds)^i .We then numerically find the Padé approximants of the latter Taylor series. More details and validation may be found in <cit.> and <cit.>.To show the utility of these derivations, Fig. <ref> implements a bandpass filter,F(s) = 1/1/w_0^2 s^2 + 1/w_0Q s + 1where w_0 = 2π· 20 rad/s is the angular frequency band and Q = 5 is a constant that is inversely proportional to the cutoff width. This compares the method of (<ref>), which is the standard version of Principle III used in NEF simulations, to the method of (<ref>), which is the discretized version of Principle III that takes into account the simulation time-step. This indicates that it is important to consider the discrete nature of the synapse when using digital hardware. We remark that the standard version is still correct for analog hardware (i.e., as dt → 0).General CaseFinally, we consider any linear synapse model of the form:H(s) = 1/∑_i=0^k c_i s^i,for some polynomial coefficients ( c_i ) of arbitrary degree k. To the best of our knowledge, this class includes the majority of linear synapse models used in the literature. For synapses containing a polynomial numerator with degree q > 1 (e.g., considering the Taylor series expansion of the box filter ϵ^-1 (1 - e^-ϵ s) s^-1), we take its [0, q]-Padé approximants to transform the synapse into this form within some radius of convergence.[This is equivalent to the approach taken in <cit.>.] To map F(s) onto (<ref>), we begin by defining our solution to F^H(s) in the form of its state-space model:A^H= ∑_i=0^k c_i A^i ,C^H= C ,B^H= ( ∑_j=0^k-1 s^j ∑_i=j+1^k c_i A^i-j-1) B ,D^H= D ,which we claim satisfies (<ref>). To validate this claim, we assert the following algebraic relationship (proven in <cit.>):H(s)^-1I - A^H = ∑_i=0^k c_i ( s^i I - A^i ) = ( ∑_j=0^k-1 s^j ∑_i=j+1^k c_i A^i-j-1) (sI - A).We now verify that (<ref>) satisfies (<ref>):F^H(H(s)^-1)= C^H(H(s)^-1I - A^H)^-1 B^H + D^H = C(H(s)^-1I - A^H)^-1( ∑_j=0^k-1 s^j ∑_i=j+1^k c_i A^i-j-1) B + D = C(sI - A)^-1 B + D = F(s) .*Since s^j is the j^th-order differential operator, this form of B^H states that we must supply the j^th-order input derivatives u⃗^(j), for all j = 1 … k - 1. To be more precise, let us first define B^H_j:= ( ∑_i=j+1^k c_i A^i-j-1) B. Then (<ref>) states that the ideal state-space model must implement the input transformation as a linear combination of input derivatives, ∑_j=0^k-1 B^H_j u⃗^(j).However, if the required derivatives are not included in the neural representation, then it is natural to use a ZOH method by assuming u⃗^(j) = 0, for all j = 1 … k - 1:B^H = ( ∑_i=1^k c_i A^i-1) B ,with A^H, C^H, and D^H as in (<ref>). This is now an equivalent model to (<ref>) assuming ZOH, and in the form of the standard state-space model (<ref>).The same derivation also applies to the discrete-time domain, with respect to the discrete synapse (corresponding to some implementation in digital hardware):H(z) = 1/∑_i=0^k c̅_i z^i.Here, the only real difference (apart from notation) is the discrete version of (<ref>):B̅^H = ( ∑_j=0^k-1∑_i=j+1^k c̅_i A̅^i-j-1) B̅.These mappings are made available byandin nengolib 0.4.0 <cit.>. Additional details may again be found in <cit.>. §.§ Nonlinear systemsHere we derive two theorems for nonlinear systems, by taking a different perspective that is consistent with <ref>. This generalizes the approach taken in <cit.>, which considered the special case of a pulse-extended (i.e., time-delayed) double-exponential. We wish to implement some desired nonlinear dynamical system,ẋ⃗̇(t) = f(x⃗, u⃗) ,using (<ref>) as the synaptic filter h(t). Letting w⃗(t) = f^h(x⃗, u⃗) for some recurrent function f^h and observing that x⃗(t) = (w⃗∗ h)(t), we may express these dynamics in the Laplace domain:X⃗(s)/W⃗(s) = 1/∑_i=0^k c_i s^i W⃗(s)= X⃗(s) ∑_i=0^k c_i s^i = ∑_i=0^k c_i [ s^i X⃗(s) ]w⃗(t)= ∑_i=0^k c_i x⃗^(i)since s is the differential operator. This proves the following theorem: Let the function computed along the recurrent connection be:f^h(x⃗, u⃗) = ∑_i=0^k c_i x⃗^(i)where x⃗^(i) denotes the i^th time-derivative of x⃗(t), and c_i are given by (<ref>). Then the resulting dynamical system is precisely (<ref>). For the discrete case, we begin with some desired nonlinear dynamics expressed over discrete time-steps:x⃗[t+dt] = f̅(x⃗, u⃗) ,using (<ref>) as the synaptic filter h[t], followed by an analogous theorem: Let the function computed along the recurrent connection be:f̅^h(x⃗, u⃗) = ∑_i=0^k c̅_i x⃗^[i]where x⃗^[i] denotes the i^th discrete forwards time-shift of x⃗, and c̅_i are given by (<ref>). Then the resulting dynamical system is precisely (<ref>).The proof for the discrete case is nearly identical. For sake of completeness, let w⃗[t] = f̅^h(x⃗, u⃗) for some recurrent function f̅^h and observe that x⃗[t] = (w⃗∗ h)[t]:X⃗(z)/W⃗(z) = 1/∑_i=0^k c̅_i z^i W⃗(z)= X⃗(z) ∑_i=0^k c̅_i z^i = ∑_i=0^k c̅_i [ z^i X⃗(z) ]w⃗[t]= ∑_i=0^k c̅_i x⃗^[i]since z is the forwards time-shift operator.Continuous Lowpass SynapseFor standard Principle III, we have H(s) = 1/τ s + 1k = 1, c_0 = 1 and c_1 = τ, f^h(x⃗, u⃗) = c_0 x⃗^(0) + c_1 x⃗^(1) = x⃗ + τẋ⃗̇ = τ f(x⃗, u⃗) + x⃗.Note that (<ref>) is consistent with (<ref>) and with Principle III from the NEF. Discrete Lowpass SynapseFor the discrete case of Principle III, we have H(z) = 1 - a/z - a, where a = e^-dt / τk = 1, c̅_0 = -a(1 - a)^-1, c̅_1 = (1 - a)^-1,f̅^h(x⃗, u⃗) = c̅_0 x⃗^[0] + c̅_1 x⃗^[1] =(1 - a)^-1(f̅(x⃗, u⃗) - ax⃗) .Note that (<ref>) is consistent with (<ref>). Continuous Double Exponential SynapseFor the double exponential synapse:H(s) = 1/(τ_1 s + 1)(τ_2 s + 1) = 1/τ_1 τ_2 s^2 + (τ_1 + τ_2)s + 1 f^h(x⃗, u⃗)= x⃗ + (τ_1 + τ_2) ẋ⃗̇ + τ_1 τ_2 ẍ⃗̈= x⃗ + (τ_1 + τ_2) f(x⃗, u⃗) + τ_1 τ_2 ( ∂ f(x⃗, u⃗)/∂x⃗· f(x⃗, u⃗) + ∂ f(x⃗, u⃗)/∂u⃗·u̇⃗̇) .In the linear case, this simplifies to:f^h(x⃗, u⃗) = ( τ_1 τ_2 A^2 + (τ_1 + τ_2) A + I ) x⃗ + ( τ_1 + τ_2 + τ_1 τ_2 A ) B u⃗ + τ_1 τ_2 B u̇⃗̇.Linear SystemsAs in <ref>, Theorems <ref> and <ref> require that we differentiate the desired dynamical system. For the case of nonlinear systems, this means determining the (possibly higher-order) Jacobian(s) of f, as shown in (<ref>). For the special case of LTI systems, we can determine this analytically to obtain a closed-form expression. By induction it can be shown that:x⃗^(i) = A^ix⃗ + ∑_j=0^i-1 A^i-j-1 B u⃗^(j).Then by expanding and rewriting the summations:f^h(x⃗, u⃗)= ∑_i=0^k c_i x⃗^(i)= ∑_i=0^k c_i [ A^i x⃗ + ∑_j=0^i-1 A^i-j-1 B u⃗^(j)] = ( ∑_i=0^k c_i A^i )_4emRecurrent Matrixx⃗ + ∑_j=0^k-1( ∑_i=j+1^k c_i A^i-j-1)B_7emInput Matricesu⃗^(j).The discrete case is identical:f̅^h(x⃗, u⃗) = ( ∑_i=0^k c̅_i A̅^i )_4emRecurrent Matrixx⃗ + ∑_j=0^k-1( ∑_i=j+1^k c̅_i A̅^i-j-1)B̅_7emInput Matricesu⃗^[j].This gives a matrix form for any LTI system with a k^th order synapse, provided we can determine u⃗^(j) or u⃗^[j] for 0 ≤ j ≤ k-1.Again, (<ref>) is consistent with (<ref>) and (<ref>), as is (<ref>) with (<ref>).§ ACCOUNTING FOR SYNAPTIC HETEROGENEITYWe now show how <ref> can be applied to train efficient networks where the i^th neuron has a distinct synaptic filter h_i(t), given by:H_i(s) = 1/∑_j=0^k_i c_ij s^j.This network architecture can be modeled in Nengo using nengolib 0.4.0 <cit.>.This is particularly useful for applications to neuromorphic hardware, where transistor mismatch can change the effective time-constant(s) of each synapse. To this end, we abstract the approach taken in <cit.>. We show this specifically for Theorem <ref>, but this naturally applies to all methods in this report.Recalling the intuition behind Principle III, our approach is to separately drive each synapse h_i with the required signal f^h_i(x⃗, u⃗) such that each PSC becomes the desired representation x⃗.Thus, the connection weights to the i^th neuron should be determined by solving the decoder optimization problems for f^h_i(x⃗, u⃗) using the methods of <ref> with respect to the synapse model h_i.This can be repeated for each synapse to obtain a full set of connection weights. While correct in theory, this approach displays two shortcomings in practice: (1) we must solve n optimization problems, where n is the number of post-synaptic neurons, and (2) there are 𝒪(n^2) weights, which eliminates the space and time efficiency of using factorized weight matrices <cit.>.We can solve both issues simultaneously by taking advantage of the linear structure within f^h_i that is shared between all h_i. Considering Theorem <ref>, we need to drive the i^th synapse with the function:f^h_i(x⃗, u⃗) = ∑_j=0^k_i c_ijx⃗^(j).Let d⃗^j be the set of decoders optimized to approximate x⃗^(j), for all j = 0 … k, where k = max_i k_i. By linearity, the optimal decoders used to represent each f^h_i(x⃗, u⃗) may be decomposed as:d⃗^f^h_i = ∑_j=0^k_i c_ijd⃗^j .Next, we express our estimate of each variable x⃗^(j) using the same activity vector a⃗:x⃗̂⃗^(j) = a⃗d⃗^j.Now, putting this all together, we obtain:a⃗d⃗^f^h_i = ∑_j=0^k_i c_ija⃗d⃗^j = ∑_j=0^k_i c_ijx⃗̂⃗^(j)≈ f^h_i(x⃗, u⃗) . Therefore, we only need to solve k+1 optimization problems, decode the “matrix representation” [ x⃗̂⃗^(0), x⃗̂⃗^(1), …, x⃗̂⃗^(k)], and then linearly combine these k+1 different decodings as shown in (<ref>)—using the matrix of coefficients c_ij—to determine the input to each synapse. This approach reclaims the advantages of using factorized connection weight matrices, at the expense of a factor 𝒪(k) increase in space and time efficiency.§ DISCUSSION We have reviewed three major extensions to the NEF that appear in recent publications. These methods can be used to implement linear and nonlinear systems in spiking neurons recurrently coupled with heterogeneous higher-order mixed-analog-digital synapses. This provides us with the ability to implement NEF networks in state-of-the-art neuromorphics while accounting for, and sometimes even exploiting, their nonideal nature.While the linear and nonlinear methods can both be used to harness pure spike time-delays (due to axonal transmission) by modeling them in the synapse, the linear approach provides greater flexibility. Both extensions can first transform the time-delay into the standard form of (<ref>) via Padé approximants, which maintains the same internal representation x⃗ as the desired dynamics (within some radius of convergence). But the linear extension also allows the representation to change, since it is only concerned with maintaining the overall input-output transfer function relation. In particular, we derived an analytic solution using the Lambert-W function, which allows the neural representation x⃗, and even its dimensionality, to change according to the expansion of some Taylor series. The linear case is also much simpler to analyze in terms of the network-level transfer function that results from substituting one synapse model for another. For all other results that we have shown, the linear extension is consistent with the nonlinear extension, as they both maintain the desired representation by fully accounting for the dynamics in the synapse.§ ACKNOWLEDGEMENTS We thank Wilten Nicola for inspiring our derivation in <ref> with his own phase-space derivation of Principle III using double exponential synapses for autonomous systems (unpublished). We also thank Kwabena Boahen and Terrence C. Stewart for providing the idea used in <ref> to separately drive each h_i, and for improving this report through many helpful discussions.ieeetr § APPENDIX§.§ The Neural Engineering FrameworkFor convenience we provide a self-contained treatment of the Neural Engineering Framework (NEF) first proposed by Eliasmith and Anderson <cit.>. A simplified technical summary can be found in <cit.>.The NEF is a set of three mathematical principles for describing neural computation. This has the aim of relating low-level aspects of biology to high-level cognition. Practically speaking, it is a theoretical framework that may be used to implement algorithms using neural components as the primitive operations. This latter perspective is most relevant when engineering systems using neuromorphic hardware. The NEF is most commonly applied to building dynamic spiking neural networks, but also applies to non-spiking and feed-forward networks.Recently, the NEF has been used to build what remains the world's largest functioning brain model <cit.>, capable of performing perceptual, motor, and cognitive tasks. This model relies fundamentally on the principles described in part below.§.§.§ Principle I – Representation We use x⃗(t) ∈ℝ^k to denote a continuous time-varying vector that is represented by a population of neurons. To describe this representation, we define a nonlinear (spiking or non-spiking) encoding and a linear decoding which together determine how neural activity relates to the represented vector.Specifically, we choose encoders E ∈ℝ^n × k, gains α_i > 0, and biases β_i as parameters for the encoding, which map x⃗(t) to neural activities. These parameters can be chosen based on prior knowledge (e.g., tuning curves, firing rates, sparsity, etc.), or randomly. Most often, we select distributions from which to sample these parameters, incorporating known constraints on the system under consideration. For a spiking neuron, the encoding is:δ_i^x⃗(t) = G_i [ α_i ⟨ E_i, x⃗(t') ⟩ + β_i : 0 ≤ t' ≤ t ]where δ_i^x⃗ is the neural spike-train generated by the i^th neuron and G_i[·] is the model for a single neuron (e.g., a leaky integrate-and-fire (LIF) neuron, a conductance neuron, etc.). We then analytically determine the firing rates of each neuron under constant inputs: r_i(v⃗) = 𝔼_t ≥ 0[ δ_i^x⃗;x⃗(t) = v⃗] In the case of non-spiking neurons, we use its rate model in place of (<ref>). In biological systems, any transfer of information between layers of a network are affected by a post-synaptic response. Consequently we introduce a post-synaptic filter h(t), referred to as the synapse model, that accounts for the temporal dynamics of a receiving neuron's synapse.To determine the vector represented by the activities, we define a linear decoding: (x̂⃗̂∗ h)(t) = ∑_i=1^n (δ_i^x⃗∗ h)(t) D_i To complete our characterization of a representation, we need to determine the linear decoders D_i. This determination is the same for the first and second principles, so we consider it next.§.§.§ Principle II – Transformation The second principle of the NEF addresses the issue of computing nonlinear functions of the vector space. The encoding remains as defined in (<ref>). However, we may decode nonlinear functions of x⃗(t) by the same linear combination of temporal basis functions:(f̂(x⃗) ∗ h)(t) = ∑_i=1^n (δ_i^x⃗∗ h)(t) D^f_iIn general, we solve for the decoders D^f ∈ℝ^n × l to compute a desired nonlinear function f : S →ℝ^l, where S = {x⃗(t) : t ≥ 0} is the domain of the signal – typically the unit k-ball {v⃗∈ℝ^k : v⃗ = 1} or the k-cube [-1, 1]^k. To account for fluctuations introduced by neural spiking, or other sources of uncertainty, we introduce a noise term η∼𝒩(0, σ^2) to regularize the solution:D^f = argmin_D ∈ℝ^n, l∫_S f(v⃗) - ∑_i=1^n (r_i(v⃗) + η) D_i ^2 d^kv⃗ Note this training only depends on r_i(v⃗) for v⃗∈ S. It does not depend on x⃗(t). Performing this regularized least-squares optimization provides the decoders needed by equations (<ref>) and (<ref>) to represent the vector or compute transformations, respectively. We can use singular value decomposition (SVD) to characterize the class of functions that can be accurately approximated by the chosen basis functions [: pp. 185–217]. The accuracy of this approach for spiking neurons relies on r_i(v⃗) being a suitable proxy for δ_i^x⃗ whenever x⃗(t) = v⃗. This is clearly true for constant x⃗, and is also a suitable model in practice for low-frequency x⃗(t). In general, we can use (<ref>) and (<ref>) to derive the connection weight matrix between layers that computes the desired function f. Specifically, the weight matrix 𝒲∈ℝ^m, n which decodes from neuron i and encodes into neuron j is given by:𝒲_ji = ⟨ E_j, D^f_i ⟩ = (E(D^f)^T)_ji That is, the matrices D and E are simply a low-dimensional factorization of 𝒲. This observation implies that the process of decoding and then encoding is equivalent to the more familiar expression for mapping activities between layers of neurons in a conventional neural network:G [ α⃗𝒲δ⃗^x⃗(t) + β⃗] The crucial difference is that a constant dimensionality k means O(n) operations per time-step to implement (<ref>) as compared to O(n^2) operations for typical weight matrices.In essence, structure is imposed by the choice of representing x⃗(t) in neural activities, which in turn becomes latent within the factored weight matrix 𝒲. This additional structure gives us a way to constrain the model by describing its state, and therefore understand in detail what the network is computing.§.§.§ Principle III – Dynamics Given our ability to compute nonlinear functions with Principle II, the methods described here apply to both linear and nonlinear dynamical systems. However, for simplicity of exposition, we focus our discussion on linear time-invariant (LTI) systems:ẋ⃗̇(t)= Ax⃗(t) + Bu⃗(t)y⃗(t)= Cx⃗(t) + Du⃗(t)where the time-varying vector x⃗(t) represents the system state, y⃗(t) the output, u⃗(t) the input, and the time-invariant matrices (A, B, C, D) fully define the system dynamics. Principle III from the NEF states that in order to train the recurrent connection weights to implement (<ref>) using a continuous-time lowpass filter h(t) = 1/τ e^-t/τ, we use Principle II to train the decoders for: * the recurrent transformation f(x⃗(t)) = (τ A + I)x⃗(t);* the input transformation f(x⃗(t)) = τ B x⃗(t);* the output transformation f(x⃗(t)) = Cx⃗(t); and* the passthrough transformation f(x⃗(t) ) = Dx⃗(t) [; pp. ]. Fig. <ref> depicts this mapping in the Laplace domain. In particular, X(s) is the Laplace transform of the vector represented in neurons via Principle I. The functions on each connection implement the desired matrix transformations as described above using Principle II. In <ref> we extend this mapping to other models of the synaptic filter H(s), and provide two novel proofs of Principle III. | http://arxiv.org/abs/1708.08133v1 | {
"authors": [
"Aaron R. Voelker",
"Chris Eliasmith"
],
"categories": [
"q-bio.NC",
"cs.AI",
"cs.SY",
"math.DS"
],
"primary_category": "q-bio.NC",
"published": "20170827202701",
"title": "Methods for applying the Neural Engineering Framework to neuromorphic hardware"
} |
S. CebriánCosmogenic activation of materials Grupo de Física Nuclear y Astropartículas,Universidad de Zaragoza, Calle Pedro Cerbuna 12, 50009 Zaragoza, SpainLaboratorio Subterráneo de Canfranc, Paseo de los Ayerbe s/n 22880 Canfranc Estación, Huesca, Spain [email protected] activation of materials Susana Cebrián December 30, 2023 ================================== Day Month Year Day Month YearExperiments looking for rare events like the direct detection of dark matter particles, neutrino interactions or the nuclear double beta decay are operated deep underground to suppress the effect of cosmic rays. But the production of radioactive isotopes in materials due to previous exposure to cosmic rays is an hazard when ultra-low background conditions are required. In this context, the generation of long-lived products by cosmic nucleons has been studied for many detector media and for other materials commonly used. Here, the main results obtained on the quantification of activation yields on the Earth's surface will be summarized, considering both measurements and calculations following different approaches. The isotope production cross sections and the cosmic ray spectrum are the two main ingredients when calculating this cosmogenic activation; the different alternatives for implementing them will be discussed. Activation that can take place deep underground mainly due to cosmic muons will be briefly commented too. Presently, the experimental results for the cosmogenic production of radioisotopes are scarce and discrepancies between different calculations are important in many cases, but the increasing interest on this background source which is becoming more and more relevant can help to change this situation.PACS numbers: 13.85.Tp; 23.40.-s; 25.40.-h; 25.30.Mr; 95.35.+d § INTRODUCTIONExperiments searching for rare phenomena like the interaction of Weakly Interacting Massive Particles (WIMPs) which could be filling the galactic dark matter halo, the detection of elusive neutrinos or the nuclear double beta decay of some nuclei require detectors working in ultra-low background conditions and taking data for very long periods of time at the scale of a few years due to the extremely low counting rates expected. Operating in deep underground locations, using active and passive shields and selecting carefully radiopure materials reduce very efficiently the background for this kind of experiments <cit.>. In this context, long-lived radioactive impurities in the materials of the set-up induced by the exposure to cosmic rays at sea level (during fabrication, transport and storage) may be even more important than residual contamination from primordial radionuclides and become very problematic. For instance, the poor knowledge of cosmic ray activation in detector materials is highlighted in Ref. gondolo as one of the three main uncertain nuclear physics aspects of relevance in the direct detection approach pursued to solve the dark matter problem. Production of radioactive isotopes by exposure to cosmic rays is a well known issue <cit.> and several cosmogenic isotopes are important in extraterrestrial and terrestrial studies in many different fields, dealing with astrophysics, geophysics, paleontology and archaeology. One of the most relevant processes in the cosmogenic production of isotopes is the spallation of nuclei by high energy nucleons, but other reactions like fragmentation, induced fission or capture can be very important for some nuclei. Since protons are absorbed by the atmosphere, nuclide production is mainly dominated by neutrons at the Earth's surface, but if materials are flown at high altitudes, in addition to the fact that the cosmic flux is much greater, protons will produce significant activation as well. In principle, cosmogenic activation can be kept under control by minimizing exposure at surface and storing materials underground, avoiding flights and even using shields against the hadronic component of cosmic rays during surface detector building or operation. In addition, purification techniques could help reducing the produced background isotopes. But since these requirements usually complicate the preparation of experiments (for example, while crystal growth and detector mounting steps) it would be desirable to have reliable tools to quantify the real danger of exposing the different materials to cosmic rays. Production rates of cosmogenic isotopes in all the materials present in the experimental set-up, as well as the corresponding cosmic rays exposure history, must be both well known in order to assess the relevance of this effect in the achievable sensitivity of an experiment. Direct measurements, by screening of exposed materials in very low background conditions as those achieved in underground laboratories, and calculations of production rates and yields, following different approaches, have been made for several materials in the context of dark matter, double beta decay and neutrino experiments. Many different studies are available for germanium and interesting results have been derived in the last years also for other detector media like tellurium and tellurium oxide, sodium iodide, xenon, argon or neodymium as well as for materials commonly used in the set-ups like copper, lead, stainless steel or titanium (see for instance the summaries in Refs. cebrianlrt2013,kudrylrt2017). However, systematic studies for other targets are missed.The relevant long-lived radioactive isotopes cosmogenically induced are in general different for each target material, although there are some common dangerous products like tritium, because being a spallation product, it can be generated in any material. Tritium is specially relevant for dark matter experiments for its decay properties (it is a pure beta emitter with transition energy of 18.591 keV and a long half-life of 12.312 y <cit.>) when induced in the detector medium. Quantification of tritium cosmogenic production is not easy, neither experimentally since its beta emissions are hard to disentangle from other background contributions, nor by calculations, as tritium can be produced by different reaction channels.The aim of this work is to summarize and compare the main results obtained for estimating the activation yields of relevant long-lived radioactive isotopes due to cosmic nucleons on the Earth's surface in the materials of interest for rare event experiments, taken into consideration both measurements and calculations. The structure of the paper is as follows. First, section <ref> describes the relevant calculation tools available. Activation results derived for different detector media are summarized in the following sections, including semiconductor materials like germanium (section <ref>) and silicon (section <ref>), tellurium used in different detectors (section <ref>), scintillators like sodium iodide (section <ref>) and noble liquid-gas detectors as xenon (section <ref>) and argon (section <ref>). The production of cosmogenic isotopes in other materials typically used in experiments is also considered, in particular, for copper (section <ref>), lead (section <ref>), stainless steel (section <ref>) and titanium (section <ref>). Results for other targets are collected in section <ref>. For each material, the available estimates of production rates of the relevant radionuclides will be presented, emphasizing the comparison with experimental results whenever possible. Some results concerning the activation of materials underground are mentioned too in section <ref> before the final summary. § CALCULATION TOOLSThe activity A of a particular isotope with decay constant λ induced in a material must be evaluated taking into account the time of exposure to cosmic rays(t_exp), the cooling time corresponding to the time spent underground and sheltered from cosmic rays (t_cool) and the production rate R: A = R [1-exp(-λ t_exp)] exp(-λ t_cool). As it will be shown in the next sections, there are some direct measurements of productions rates in some materials as the saturation activity, but unfortunately they are not very common. Consequently, in many cases production rates have to be estimated from two basic energy-dependent ingredients, the flux ϕ of cosmic rays and the cross-section σ of isotope production: R=N∫σ(E)ϕ(E)dE . N being the number of target nuclei and E the particle energy. The excitation function for the production of a certain isotope by nucleons in a target over a wide range of energies (typically from some MeV up to several GeV) can be hardly obtained experimentally, since the measurements of production cross-sections with beams are long, expensive and there are not many available facilities to carry them out. The use of computational codes to complete information on the excitation functions is therefore mandatory. In addition, measurements are usually performed on targets with the natural composition of isotopes for a given element, often determining only cumulative yields of residual nuclei. Reliable calculations are required to provide independent yields for isotopically separated targets if necessary. But in any case, experimental data are essential to check the reliability of calculations. The suitability of a code to a particular activation problem depends on energy, targets and projectiles to be considered. Some systematic and extensive comparisons of calculations and available measurements have been made based on analysis of deviation factors F, defined as: F=10^√(d), d=1/n∑_i(logσ_exp,i-logσ_cal,i)^2n being the number of pairs of experimental and calculated cross sections σ_exp,i and σ_cal,i at the same energies.There are several possibilities to obtain values of production cross sections:* Experimental results can be found at EXFOR database (CSISRS in USA) <cit.>, an extensive compilation of nuclear reaction data from thousands of experiments. Available data for a particular target, projectile, energy or reaction can be easily searched for by means of a public web form. NUCLEX <cit.> is also a compilation of experimental data. * Semiempirical formulae were deduced for nucleon-nucleus cross sections for different reactions (break-up, spallation, fission,…) exploiting systematic regularities and tuning parameters to best fit available experimental results. The famous Silberberg&Tsao equations <cit.> can be applied for light and heavy targets (A≥3), for a wide range of product radionuclides (A≥6) and at energies above 100 MeV. These equations have been implemented in different codes, which offer very fast calculations in contrast to Monte Carlo simulations: * COSMO <cit.> is a FORTRAN program with three modes of calculation: excitation curve of a nuclide for a specified target, mass yield curve for given target and energy and final activities produced for a target exposed to cosmic rays. It allows a complete treatment for targets with A<210 and Z<83.* YIELDX <cit.> is a FORTRAN routine to calculate the production cross-section of a nuclide in a particular target at a certain energy. It includes the latest updates of the Silberberg & Tsao equations.* ACTIVIA <cit.> is C++ computer package to calculate target-product cross sections as well as production and decay yields from cosmic ray activation. It uses semiempirical formulae but also experimental data tables if available.The main limitation of the formulae is that they are based only on proton-induced reactions; therefore, the fact that cross sections are equal for neutrons and protons has to be assumed. * Monte Carlo simulation of the hadronic interaction between nucleons and nuclei can output also production cross sections. Modeling isotope production includes a wide range of reactions: the formation and decay of a long-lived compound nucleus at low energies, while in the GeV range the intranuclear cascade (INC) of nucleon-nucleon interactions is followed by different deexcitation processes (spallation, fragmentation, fission,…). Many different models and codes implementing them have been developed in very different contexts (studies of comic rays and astrophysics, transmutation of nuclear waste or production of medical radioisotopes for instance): BERTINI, ISABEL, LAHET, GEM, TALYS, HMS-ALICE, INUCL, LAQGSM, CEM, ABLA, CASCADE, MARS, SHIELD,…are the names of just a few of them. Some of these models have been integrated in general-purpose codes like GEANT4 <cit.>, FLUKA <cit.> and MCNP <cit.>. In Ref. aguayo, the capabilities of GEANT4 and MCNPX to simulate neutron spallation were studied; in particular, the neutron multiplicity predicted was compared with measurements finding some discrepancies for low density materials. In Ref. head2009 preferred models for each target mass range are selected, for neutron and proton-induced reactions while in Ref. titarenko it is concluded that versions of CEM03 and INCL+ABLA codes can be considered as the most accurate. A relevant advantage of Monte Carlo codes is that they are applicable not only to proton-induced but also to neutron-induced nuclear reactions. * Several libraries of production cross sections have been prepared combining calculations and experimental data, with different coverage of energies, projectiles and reactions: * RNAL (Reference Neutron Activation Library) <cit.> is restricted to 255 reactions.* LA150 <cit.> contains results up to 150 MeV, independently for neutrons and protons as projectiles, using calculations from HMS-ALICE.* MENDL-2 and MENDL-2P (Medium Energy Nuclear Data Library) <cit.> are based on calculations using codes of the ALICE family, containing excitation functions which cover a very wide range of target and product nuclides, either for neutrons or protons, for energies up to 100 MeV for neutrons (MENDL-2, using ALICE92) and 200 MeV for protons (MENDL-2P, using ALICEIPPE).* TENDL (TALYS-based Evaluated Nuclear Data Library) <cit.> offers cross sections obtained with the TALYS nuclear model code system for neutrons and protons up to 200 MeV for all the targets.* An evaluated library for neutrons and protons to 1.7 GeV was presented in Ref. study. It gives excitation functions including available experimental data and calculated results using CEM95, LAHET, and HMS-ALICE codes for selected targets and products.* HEAD-2009 (High Energy Activation Data) <cit.> is a complete compilation of data for neutrons and protons from 150 MeV up to 1 GeV. The choice of models was dictated by an extensive comparison with EXFOR data. In particular, in HEPAD-2008 (High-Energy Proton Activation Data) sub-library cross sections are obtained using a selection of models and codes like CEM 03.01, CASCADE/INPE and MCNPX 2.6.0. Only targets with Z≥12 are considered.The detailed studies of excitation functions performed in different areas, like Positron Emission Tomography (PET) nuclear medicine imaging process, for different targets and products and based on irradiation experiments and several types of calculations (see for example recent works from Refs. ef1,ef2,ef3,ef4,ef5,ef6), can serve as a reference to select the most adequate libraries or packages. Relevant activation processes on the Earth's surface are induced mainly by nucleons at MeV-GeV energy range.At sea level, the flux of neutrons and protons is virtually the same at energies of a few GeV; however, at lower energies the proton to neutron ratio decreases significantly because of the absorption of charged particles in the atmosphere. For example, at 100 MeV this ratio is about 3%<cit.>. Consequently, neutrons dominate the contribution to the cosmic ray spectrum to be considered at sea level. But it must be kept in mind that proton activation, being much smaller, is not completely negligible. The typical contribution from protons to isotope production is quoted as ∼10% of the total in Ref. lal, which agrees for instance with the results in Refs. barabanov,wei where proton activation in germanium was specifically evaluated for some isotopes. Concerning activation by other cosmic particles like muons, it is even smaller, as confirmed by calculations in Refs. mei2016,wei. Different forms of the neutron spectrum at sea level have been used in cosmogenic activation studies, like the Hess <cit.> and Lal&Peters <cit.> spectra. The ACTIVIA code uses the energy spectrum from parameterizations in Refs. armstrong,gehrels. In Ref. ziegler, a compilation of measurements was made, including the historical Hess spectrum and relevant corrections, and an analytic function valid from 10 MeV to 10 GeV was derived. In Ref. gordon a set of measurements of cosmic neutrons on the ground across the USA was accomplished using Bonner sphere spectrometers; a different analytic expression fitting data for energies above 0.4 MeV was proposed (referred hereafter as Gordon et al. spectrum). Just to appreciate the difference between different spectra, three considered parameterizations are compared in Fig. <ref>, applied for the conditions of New York City at sea level; the integral flux from 10 MeV to 10 GeV is 5.6× 10^-3cm^-2s^-1 for Ref. ziegler and 3.6× 10^-3cm^-2s^-1 for Ref. gordon. The Gordon et al. parametrization gives a lower neutron flux above 1000 MeV. To evaluate the activation at a particular location, the relative neutron flux to the sea-level flux in New York City can be determined using the calculator available in Ref. calculator. For surface protons, the energy spectrum from Ref. hagmann can be used; the total flux corresponding to the energy range from 100 MeV to 100 GeV is 1.358× 10^-4cm^-2s^-1. It is worth noting that in Ref. hagmann2, the energy spectra at sea level for different particles including neutrons, protons and muons have been generated from Monte Carlo simulation of primary protons implementing a model of the Earth's atmosphere in different codes, finding a satisfactory agreement between codes and available data. Therefore, it seems that Monte Carlo calculations could provide too quite reliable predictions of cosmic-ray distributions at sea level.§ GERMANIUMA great deal of experiments and projects looking for rare events (like the interaction of dark matter or the double beta decay of ^76Ge) have used or are using germanium crystals either as conventional diodes (like IGEX <cit.>, Heidelberg-Moscow <cit.>, GERDA <cit.>, Majorana <cit.>, CoGENT <cit.>, TEXONO or CDEX <cit.>) or as cryogenic detectors (like CDMS <cit.> or EDELWEISS <cit.>). Therefore, many different activation studies have been performed for germanium. The observation for the first time of several cosmogenic isotopes through low energy peaks produced due to their decay by electron capture in the crystals of the CoGENT experiment was reported in Ref. cogent and cosmogenic products are usually considered in the background models of experiments <cit.>. In this section, the main results obtained for quantifying the cosmogenic yields in both natural and enriched[Typical isotopic composition of enriched germanium used in double beta decay searches is 86% of ^76Ge and 14% of ^74Ge.] germanium are summarized.* Early estimates of production rates of induced isotopes were made in Refs. avignone,miley using excitation functions calculated with the spallation reaction code LAHET/ISABEL and the Hess and Lal&Peters neutron spectra. Production rates were also derived experimentally in Homestake and Canfranc laboratories from germanium detectors previously exposed <cit.>. Agreement with calculation was within a factor of 2 and in some cases within 30%. * Productions of cosmogenic isotopes was assessed in Ref. genius using a semiempirical code named Σ. * Production cross sections were measured irradiating at Los Alamos Neutron Science Center (LANSCE) a natural germanium target with a proton beam with an energy of 800 MeV <cit.>. Screening with germanium detectors was performed at Berkeley intermittently from 2 weeks to 5 years after irradiation. A quite good agreement with predictions of Silberberg&Tsao formulae was obtained. * Estimates of production rates for ^60Co and ^68Ge were made using excitation functions calculated with the SHIELD code in Ref. barabanov. It is worth noting than in this estimate rates were evaluated including not only the neutron but also the proton contribution; as mentioned in section <ref>, the latter amounts around a ∼10% of the total. * The ACTIVIA code, including the energy spectrum for cosmic ray neutrons at sea level based on the parametrization from Armstrong <cit.> and Gehrels <cit.> was used to evaluate production rates in both natural and enriched germanium for benchmark <cit.>. * Another estimate of relevant production rates can be found in Ref. mei. Excitation functions were calculated using the TALYS code and the neutron spectrum was considered from the Gordon et al. parametrization. * In Ref. elliot2010, a 11-g sample of enriched germanium was exposed at LANSCE to a wide-band pulsed neutron beam that resembles the cosmic-ray neutron flux, with energies up to about 700 MeV. After cooling, germanium gamma counting was performed underground at the Waste Isolation Pilot Plant (WIPP) for 49 days to evaluate the nuclei production. This production was also predicted by calculating cross sections with CEM03 code. Calculations seem to overestimate production in a factor of 3 depending on isotope. In addition, measured yields were converted to cosmogenic production rates considering the Gordon et al. neutron spectrum.* An estimate of production rates after a careful evaluation of excitation functions was presented in Ref. cebrian. Information on excitation functions for each relevant isotope was collected searching for experimental data (available only for protons) and available calculations (MENDL libraries <cit.> and other ones based on different codes <cit.>) and performing some new calculations (using YIELDX). Then, deviation factors were evaluated between measured cross sections and different calculations and the selected description of the excitation functions was the following: HMS-ALICE calculations for neutrons below 150 MeV and YIELDX results above this energy. Production rates were computed for both the Ziegler and Gordon et al. spectra. It was checked that, in general, estimates based on the Gordon et al. spectrum are closer to available experimental results; the uncertainty in the activation yields coming from the cosmic ray flux was for most of the analyzed products below a factor 2.* Neutron irradiation experiments have been performed on enriched germanium to determine the neutron radiative-capture cross sections on ^74Ge and ^76Ge, for neutron thermal energies <cit.> and energies of the order of a few MeV <cit.>. These results are of special interest for double beta decay experiments like GERDA and Majorana. Neutron capture results in a cascade of prompt de-excitation gamma rays from excited states of produced nucleus and the emissions from the decay of this nucleus, if it is radioactive. * Recently, a precise quantification of cosmogenic products in natural germanium including tritium has been made by the EDELWEISS III direct dark matter search experiment <cit.>, following a detailed analysis of a long measurement with many germanium detectors with different exposures to cosmic rays. The cosmogenic yields are evaluated by fitting the measured data in the low energy region to a continuum level plus several peaks at the binding energies of electrons in K and L shells, corresponding to different isotopes decaying by electron capture. The decay rates measured in detectors are converted into production rates, considering their well-known exposure history above ground during different steps of production and shipment. Estimates of the production rates calculated with the ACTIVIA code (modified to consider the Gordon et al. cosmic neutron spectrum) are also presented in Ref. edelweisscos for comparison.* Presence of tritium in the germanium detectors of the Majorana Demonstrator focused on the study of the double beta decay has been reported too <cit.>. A first estimate of the production rate of several radioisotopes in enriched germanium, including tritium, from the data of the Majorana Demonstrator has been very recently presented <cit.>. These enriched detectors (having 87% of ^76Ge and 13% of ^74Ge) have a very well-known exposure history. The fitting model to derive the abundance of cosmogenic products is comprised of a calculated tritium beta-decay spectrum, flat background, and multiple X-ray peaks. * A detailed study of the impact of cosmogenic activation in natural and enriched germanium has been carried out in Ref. wei, estimating the production rates of many isotopes including tritium not only for dominating neutrons but also for protons and muons, using Geant4 simulations and ACTIVIA calculations. In addition, the effect of a shielding against activation has been analyzed and the expected background rates in the regions of interest have been evaluated for particular exposure and cooling conditions. All results from this work considered here correspond to neutron activation using the Gordon et al. spectrum. Tables <ref> and <ref> summarize the rates of production (expressed in kg^-1 d^-1) of some of the isotopes induced in natural and enriched germanium at sea level obtained either from measurements or from calculations based on the different approaches described before. Experimental results quoted in Ref. heusser are also included. When comparing all the available results for germanium, the order of magnitude of measured values is reasonably reproduced by calculations, but there is an important dispersion in results for some isotopes which can be very relevant, like ^68Ge. In enriched germanium, cosmogenic activation is significantly suppressed for most of the isotopes.As pointed out before, germanium is being used as a target for dark matter searches for many years, either as pure ionization detectors or in cryogenic detectors measuring simultaneously ionization and heat. The production of tritium in the detector media can be specially worrisome due to its emissions. A part of the unexplained background in the low energy region of the IGEX detectors could be attributed to tritium <cit.> and tritium is highlighted as one the relevant background sources in future experiments like SuperCDMS <cit.>. Indeed, the first experimental estimates of the tritium production rate in germanium at sea level have been derived by the EDELWEISS collaboration <cit.> for natural germanium and using data from the Majorana Demonstrator <cit.> for enriched germanium; a new estimate based on CDMSlite data <cit.> has been also presented. Several calculations had been made before: a rough calculation was attempted in Ref. avignone using two different neutron spectra. This was followed by other estimates based in TALYS <cit.> and ACTIVIA and GEANT4 <cit.>, profiting from the availability of reaction codes that fully identify all the reaction products in the final state. In Ref. tritiumpaper, dedicated calculations of production rates of tritium at sea level have been performed for some of the materials typically used as targets in dark matter detectors (germanium, sodium iodide, argon and neon), based on a selection of excitation functions over the entire energy range of cosmic neutrons, mainly from TENDL for neutrons and HEAD-2009 libraries. All these results on the production rate of tritium in both natural and enriched germanium are summarized in Table <ref>. An additional estimate of the production rate in enriched germanium using the COSMO code gives 70 kg^-1 d^-1 <cit.>. For the natural material, calculations give in general lower values than the measured rates; in particular, the smallest value from <cit.> can be understood because using TALYS cross-sections only contributions from the lowest energy neutrons are considered. The range derived in Ref. tritiumpaper is well compatible with the measured rates by EDELWEISS and CDMSlite. For enriched germanium, as used in double beta decay experiments, the measured production rate is higher than in natural germanium; this can be due to the fact that cross sections increase with the mass number of the germanium isotope in all the energy range above ∼50 MeV, according to TENDL-2013 and HEAD-2009 data <cit.>.In addition to the quantification of the cosmogenic activation, the effect of this activity in germanium detectors has been studied too <cit.>. In Ref. jian, a background simulation method for cosmogenic radionuclides inside HPGe detectors for rare event experiments is presented, to quantify the expected spectrum for each internal cosmogenic isotope analyzing phenomena caused by the coincidence summing-up effect.§ SILICONSilicon is the detector medium in cryogenic detectors, like those of the CDMS experiment, or in CCD detectors, used in the DAMIC experiment <cit.>, both devoted to the direct detection of dark matter. The intrinsic isotope ^32Si, a β^- emitter with an endpoint energy of 224.5 keV and half-life of 150 y <cit.>, can be a relevant background source. It is produced as a spallation product from cosmic ray secondaries on argon in the atmosphere; ^32Si atoms can make their way into the terrestrial environment through aqueous transport and therefore its concentration can depend on the exact source and location of the silicon used in the production and fabrication of detectors. A decay rate of 80_-65^+110 kg^-1 d^-1 has been found by the DAMIC collaboration in their CCD detectors <cit.> and this isotope is considered in the estimates of SuperCDMS sensitivity <cit.>. Also tritium can be relevant for silicon detectors, as pointed out in Ref. supercdms. An estimate of the production rate based on GEANT4 and ACTIVIA (using the Gordon et al. neutron spectrum) is presented in Ref. mei2016 and results are shown in Table <ref>, together with the production rate considered by the SuperCDMS collaboration <cit.>. § TELLURIUMActivation in tellurium has been mainly studied with focus on the CUORE <cit.> and SNO+ <cit.> double beta decay experiments. The production rate of tritium in TeO_2 was also evaluated using the TALYS code <cit.> and it is reported in Table <ref>.Results for proton production cross sections obtained within the CUORE project were published in Ref. te, completing previous results on proton spallation reactions on tellurium <cit.>. Some measurements were made in USA, by irradiating a tellurium target with the 800 MeV proton beam at LANSCE and performing a gamma screening with germanium detectors in Berkeley. Other measurements were carried out in Europe, exposing TeO_2 targets to proton beams with energies of 1.4 and 23 GeV at CERN; first germanium screening was made there for several months and later in Milano 2.8 and 4.6 years after irradiation. The obtained cross sections at the three energies are in good agreement with Silberberg&Tsao predictions.In addition, as presented in Ref. ten, flux-averaged cross-sections for neutron activation in natural tellurium were measured by irradiating at LANSCE TeO_2 powder with the neutron beam containing neutrons of kinetic energies up to ∼800 MeV and having an energy spectrum similar to that of cosmic-ray neutrons at sea-level; gamma ray analysis was performed in Berkeley Low Background Facility. The cross sections obtained for ^110mAg and ^60Co, the two isotopes identified as the most relevant ones in the background model of the CUORE experiment <cit.>, were combined with results from tellurium activation measurements with 800 MeV-23 GeV protons to estimate the corresponding production rates for CUORE crystals, considering the Gordon et al. spectrum. These results are reported in Table <ref>.Within the SNO+ project, as described in Ref. telozza, production rates of cosmogenic-induced isotopes on a natural tellurium target were calculated using the ACTIVIA program for energies above 100 MeV, the neutron and proton cross sections from TENDL library in the 10-200 MeV range when available, and the cosmogenic neutron and proton flux parametrization at sea level from Armstrong <cit.> and Gehrels <cit.>. An extensive set of isotopes was considered and even expected activation underground was evaluated too. Some results are shown in Table <ref>. Although not directly comparable with the estimated rates in Ref. ten for the different targets, there is an important discrepancy between them. As shown in the excitation functions plotted in Ref. telozza for ^110mAg and ^60Co, the measured cross sections above 800 MeV agree reasonably well with predictions of ACTIVIA; the origin of the difference can be at lower energies (the region giving the dominant contribution to the production rates). The different cosmic neutron spectra considered in the two works makes difficult also the comparison. § SODIUM IODIDESodium iodide is being used for dark matter searches for a long time and in the last years activation yields in these crystals have started to be quantified in the context of various experiments. The observation of the annual modulation signal registered in the data of the DAMA/LIBRA experiment <cit.> for many years is the goal of projects at different stages of development like KIMS <cit.> and DM-Ice <cit.> (now joint in COSINE), ANAIS <cit.>, SABRE <cit.>, PICO-LON <cit.> and COSINUS <cit.>.A first direct estimate of production rates of several iodine, tellurium and sodium isotopes induced in NaI(Tl) crystals was presented in Ref. cebrianjcap. The estimate, developed in the frame of the dark matter ANAIS experiment, was based on data from two 12.5 kg NaI(Tl) detectors produced by Alpha Spectra Inc. which were installed inside a convenient shielding at the Canfranc Underground Laboratory at the end of 2012, just after finishing surface exposure to cosmic rays during production. The very fast start of data taking allowed to identify and quantify isotopes with half-lives of the order of tens of days. Initial activities underground were measured following the evolution of the identifying signatures for each isotope along several months; then, production rates at sea level were properly estimated according to the history of detectors. Production rates were also computed for comparison using the cosmic neutron flux at sea level from Gordon et al. and a description of excitation functions, carefully selected minimizing deviation factors between measured cross sections and calculations from MENDL2N, TENDL-2013, YIELDX and HEAD-2009. The ratio between calculated and measured production rates ranges from 0.7 to 1.5 for the observed isotopes. The obtained rates are reported in Table <ref>. A complete study of cosmogenic activation in NaI(Tl) crystals was also performed within the DM-Ice17 experiment <cit.>. Two crystals, with a mass of 8.5 kg each and previously operated at the dark matter NaIAD experiment in the Boulby underground laboratory in UK <cit.>, were deployed at a depth of 2450 m under the ice at the geographic South Pole in December 2010 and collected data over three-and-a-half years. The activation of detector components occurred during construction, transportation, or storage at the South Pole prior to deployment. The DM-Ice17 data provided compelling evidence of significant production of cosmogenic isotopes, which was used to derive the corresponding production rates based on simulation matching. In addition, calculations based on modified ACTIVIA using the Gordon et al. neutron spectrum were also carried out. Results are presented in Table <ref>. A very good agreement between measured production rates in ANAIS and DM-Ice17 crystals has been found. ACTIVIA estimates of metastable tellurium isotopes give rates clearly higher than measured values. Presence of cosmogenic isotopes has been also observed in NaI(Tl) crystals from other experiments <cit.>. In particular, in Ref. damanima2008, the fraction of ^129I (which can be produced by uranium spontaneous fission and by cosmic rays) in DAMA/LIBRA crystals from Saint Gobain company was determined to be ^129I/^natI=(1.7±0.1)× 10^-13; strong variability of this concentration is expected in different origin ores. The tritium content in NaI(Tl) detectors devoted to dark matter searches could be very relevant. The tritium activity in DAMA/LIBRA crystals was constrained to be <0.09 mBq/kg (95% C.L.) <cit.>. A direct identification of tritium in NaI(Tl) crystals has not been possible, but within the ANAIS experiment, the construction of a detailed background model of the operated modules in the Canfranc Underground Laboratory (based on a Geant4 simulation of quantified background components) points to the need of an additional background source contributing only in the very low energy region, which could be tritium <cit.>. The inclusion of a certain activity of ^3H homogeneously distributed in the NaI crystal provides a very good agreement between measured data and the model below 20 keV. For two Alpha Spectra detectors, both fabricated in Alpha Spectra facilities at Grand Junction, Colorado (where the cosmic neutron flux is estimated to be a factor f=3.6 times higher than at sea level <cit.>) but with different production history, the required ^3H initial activities (that is, at the moment of going underground) to reproduce the data are around 0.20 mBq/kg and 0.09 mBq/kg; the latter value is just the upper limit set for DAMA/LIBRA crystals.These results boosted the study of tritium production in NaI. Only calculations of the tritium production rate are available; results using TALYS <cit.>, GEANT4 <cit.> and ACTIVIA <cit.> are shown in Table <ref>. In the calculations of Ref. tritiumpaper, based on a selection of excitation functions mainly from TENDL for neutrons and HEAD-2009 libraries, there is a significant difference in the medium energy range between the two estimates of cross sections for I, being those from TENDL much higher; this fact is responsible of the large uncertainty in the production rate estimated for NaI. As it can be seen in Table <ref>, this rate is higher than previous estimates, but the required exposure times to get the deduced tritium activities for ANAIS detectors based on that rate roughly agree with the time lapse between sodium iodide raw material purification starting and detector shipment, according to the company <cit.>.§ XENONXenon-based detectors are being extensively used in the investigation of rare events. Following very successful predecessors, many efforts are now concentrated in the XENON1T <cit.>, LUX-ZEPLIN <cit.>, PANDAX <cit.> or XMASS <cit.> projects in the search for dark matter; EXO <cit.>, KamLAND-Zen <cit.> and NEXT <cit.> are looking for the neutrinoless double beta decay of ^136Xe. Even if in liquid xenon experiments, the purification system is presumed to suppress the concentration of non-noble radioisotopes below significance, there are several estimates of production rates of activated isotopes in xenon based on both direct measurements and calculations following different approaches. As it will be shown, the dispersion in results for most of the analyzed products in this material is important. Production of xenon radioisotopes could be problematic due to low energy deposits, which can be a relevant background in the WIMP search energy region; but these backgrounds soon reach negligible levels once xenon is moved underground <cit.>. A first dedicated study on the cosmogenic activation of xenon was presented in Ref. baudis, complemented by a study of copper activated simultaneously. Samples were exposed to cosmic rays in a controlled manner at the Jungfraujoch research station (at 3470 m above sea level in Switzerland) for 345 days. The xenon sample, with a mass of ∼2 kg, had natural isotopic composition. The samples were screened with a low-background germanium detector (named Gator) in the Gran Sasso underground laboratory before and after activation to quantify the cosmogenic products. Saturation activities were derived for several long-lived isotopes and some results are shown in Table <ref>. The measured results were compared in addition to predictions using ACTIVIA and COSMO packages, including the neutron spectrum from Refs. armstrong,gehrels. It is worth noting that from all the directly observed radionuclides, only ^125Sb is considered to be a potential relevant background for a multi-ton scale dark matter search.For natural xenon, production rates of several isotopes were estimated as made for germanium in Ref. mei, with excitation functions calculated using TALYS code, and as for different materials in Ref. mei2016, from GEANT4 simulation or ACTIVIA calculations. For GEANT4 simulations, the set of electromagnetic and hadronic physics processes included in the Shielding modular physics list were taken into account while in ACTIVIA, cross sections are obtained from data tables and semiempirical formulae. It is worth noting that in the results shown here the Gordon et al. neutron spectrum was considered for both GEANT4 and ACTIVIA, which required a modification of the latter. Some of these results are also shown in Table <ref>. In these calculations, the production rate of ^3H was also evaluated and it is presented in Table <ref>. The analysis of the radiogenic and muon-induced backgrounds in the LUX dark matter detector operating at the Sanford Underground Research Facility in US allowed to quantify the cosmogenic production of some xenon radioisotopes <cit.>. In particular, zero-field data taken 12 days after the xenon was moved underground and 70 days before the start of the WIMP search run were considered. Some of the obtained saturation activities, as reproduced in Ref. baudis, are presented in Table <ref>. In the context of neutrinoless double beta decay experiments working with xenon enriched in ^136Xe, the production of short-lived ^137Xe by neutron capture can be relevant and is considered as a background source. This isotope is the only cosmogenic radionuclide found to have a significant contribution in their region of interest <cit.>; the measured capture rate by the EXO-200 experiment is 338^+132_-93 captures on ^136Xe per year. In addition, individual production cross sections for 271 radionuclides have been determined for ^136Xe in an inverse kinematics experiment at GSI<cit.>.§ ARGONDifferent projects use liquid argon in dark matter detectors, like ArDM <cit.>, DarkSide <cit.> or DEAP/CLEAN <cit.>, and also gaseous argon is considered in the TREX-DM experiment <cit.>. The most worrisome background source in this detector medium is ^39Ar, a β^- emitter with an endpoint energy of 565 keV and half-life of 269 y <cit.>. ^39Ar is present in atmospheric argon as it is mainly produced by cosmic ray induced nuclear reactions such as ^40Ar(n,2n)^39Ar. In the context of the WARP experiment at the Laboratori Nazionali del Gran Sasso (LNGS), ^39Ar activity in natural argon was measured to be at the level of 1 Bq/kg <cit.>. But the discovery of argon from deep underground sources with significantly less ^39Ar content has allowed to improve the background prospects of experiments using argon as the active target. In Ref. darkside1, a specific ^39Ar activity of less than 0.65% of the activity in atmospheric argon corresponding to 6.6 mBq/kg was measured for underground argon from a CO_2 plant in Colorado; results from the DarkSide experiment have shown that the underground argon contains ^39Ar at a level reduced by a factor (1.4±0.2)×10^3 relative to atmospheric one <cit.>. Concerning the tritium production in argon (or neon, which is in some cases considered as an alternative target), there is no experimental information. As shown in Table <ref>, there are some estimates of the production rate in argon using TALYS code <cit.> and GEANT4 and ACTIVIA (using the Gordon et al. neutron spectrum) <cit.>. Since there was no information for neon, in Ref. tritiumpaper, the study of tritium production performed for other targets like germanium and sodium iodide was applied also to argon and neon, selecting the excitation functions and considering the Gordon et al. cosmic neutron spectrum. If saturation activity was reached for tritium, according to the production rates deduced in this study and presented in Table <ref> too, tritium could be very problematic. However, tritium is expected to be suppressed by purification of gas and minimizing exposure to cosmic rays of the purified gas should avoid any problematic tritium activation. § COPPERCopper is a material widely used in rare event experiments due to its mechanical, thermal and electrical properties, either as shield or part of the detector components. Copper is also a specially interesting material for activation studies because there are many extensive sets of measurements of production cross sections for protons and even for neutrons; this makes it very attractive to compare calculations and experimental data in order to allow a good validation of excitation functions. The most relevant results on the quantification of activation yields in copper are presented in this section.* As for germanium, the ACTIVIA code, including the energy spectrum for cosmic ray neutrons at sea level from Refs. armstrong,gehrels, was used to evaluate production rates in copper for benchmark <cit.>. * Direct measurements of production rates were made in LNGS <cit.>. Seven plates made of NOSV grade copper from Nord-deutsche Affinerie AG (Germany), with a total weight of 125 kg, were exposed for 270 days at an outside hall of the LNGS (altitude 985 m) under a roof. Screening with one GeMPI detector was carried out for 103 days. Production rates were derived as the measured saturation activity. The highest values were found for cobalt isotopes; in particular, the measured activity of ^60Co ((2100±190) μBq/kg) greatly exceeded the upper limit derived for the primordial activity. The production rates at sea level are reproduced in Table <ref>, including a correction factor estimated to be 2.1 (following Ref. ziegler) due to the altitude during exposure.* The same study of evaluation of excitation functions based of deviation factors and estimate of production rates made for germanium was also carried out for copper in Ref. cebrian. Production rates were calculated using below 100 MeV MENDL2N results for neutrons normalized to the available experimental data if possible, and above that energy experimental data for protons combined with YIELDX calculations when necessary. Differences in the production rates estimated in this work due to the different neutron spectra considered are similar to those obtained in germanium: production rates considering the Ziegler parametrization are higher than the corresponding ones using the expression from Gordon et al. Results shown in Table <ref> are the ones obtained using the Gordon spectrum. * Together with the dedicated study on the cosmogenic activation of xenon, a study on copper was simultaneously carried out in Ref. baudis. Samples were exposed to cosmic rays in a controlled manner at the Jungfraujoch research station (at 3470 m above sea level in Switzerland) for 345 days. The 10.35 kg copper sample consisted of 5 blocks of OFHC copper from Norddeutsche Affinerie (now Aurubis), from the batch used to construct inner parts of the XENON100 detector. Before each measurement, copper was properly cleaned to remove surface contaminations. The samples were screened with the Gator low-background germanium detector in the Gran Sasso underground laboratory before and after activation to quantify the cosmogenic products. Saturation activities were derived and some results are presented in Table <ref>. The measured results were compared in addition to predictions using ACTIVIA and COSMO packages, including the neutron spectrum from Refs. armstrong,gehrels.* As for different materials, production rates of several isotopes were estimated in Ref. mei using TALYS and in Ref. mei2016 from GEANT4 simulation or ACTIVIA calculations, considering the Gordon et al. parametrization for the neutron spectrum. Some of these results are also shown in Table <ref>.* In Ref. ivan, the cosmogenic activity produced in copper to be used as radiation shielding was measured, using a HPGe detector operated in the Canfranc Underground Laboratory. A sample with a mass of 18 kg, exposed at 250 m above sea level for 1 year after casting according to company records, as well as bricks exposed to cosmic rays for 41 days, were analyzed. Activities found for cobalt isotopes and ^54Mn agree with the expectations from previously measured production rates. Comparing all the information on production rates on copper available and summarized in Table <ref>, it can be seen that measured rates from the two independent activation experiments are in very good agreement for some isotopes, but there are differences for others, specially for ^60Co. In general, higher rates were found in Ref. lngsexposure. The different calculations give results with important dispersion; it must be noted that ACTIVIA calculations reported in Table <ref> were obtained using different descriptions of the cosmic neutron spectrum.§ LEADEven if tons of lead are commonly used in rare event experiments as radiation shielding, activation studies on this material are scarce. In Ref. giuseppe, results for some production rates are presented following the irradiation of a natural lead sample at LANSCE using the neutron beam resembling the cosmic neutron flux and after counting the amount of radioactive isotopes using a low background, underground germanium detector at WIPP. By scaling the LANSCE neutron flux to a cosmic neutron flux, the sea level production rates of some long-lived radionuclides were estimated and are reported in Table <ref>; calculations based on TALYS code and the Gordon et al. neutron spectrum deduced in the same work are also shown. In Ref. giuseppe it is concluded that for ordinary exposures, the cosmogenic background is less than that from the naturally occurring radioisotopes in lead. § STAINLESS STEELStainless steel is also very often used in experiments and some activation studies are available. Direct measurements of saturation activity were made in LNGS <cit.>. Samples of stainless steel (1.4571 grade) from different batches supplied by Nironit company (with masses from 53 to 61 kg) were screened with GeMPI detector at Gran Sasso for the GERDA double beta decay experiment <cit.>. One of these samples was re-exposed for 314 days in open air at the LNGS outside laboratory, after a cooling time of 327 days underground. Production rates were derived for Gran Sasso altitude and scaled down to sea level, considering a correction factor of 2.4. In this case, ^60Co is obscured by anthropogenic contamination, generally present in steel. The obtained results are reported in Table <ref>.In Ref. mei2016, calculations of production rates for stainless steel have been performed using GEANT4 and ACTIVIA and are also shown in Table <ref>. Comparing measured and calculated rates, agreement is better in some products for ACTIVIA and in others for GEANT4. None of the codes predicts the important activation yield measured for ^7Be.§ TITANIUMTitanium has been considered due to its properties as an alternative to stainless steel and copper. Indeed, a reduced cosmogenic activation is expected in this material. In the frame of the LUX experiment, radiopurity of different titanium samples was analyzed <cit.> and the activity of cosmogenic ^46Sc was quantified, ranging from 0.2 to 23 mBq/kg. Other scandium isotopes with shorter half-lives were also observed but are not relevant. One 6.7 kg control titanium sample was used to directly estimate the cosmogenic yields; the sample was screened at the Soudan Low Background Counting Facility (SOLO) after two years underground and then transported by ground to the Sanford Surface Laboratory in South Dakota. The sample was activated over a six-month period before being transported by ground back to SOLO for re-analysis. The measured activity was (4.4±0.3) mBq/kg of ^46Sc <cit.>, which agrees within a factor two with the expected result following the calculations in Ref. mei2016. In this work, production rates of ^46Sc and other isotopes were estimated using GEANT4 and ACTIVIA, as shown in Table <ref>.§ OTHER RESULTSHere, some results related to the observation and analysis of cosmogenic activation for other materials are briefly reported. Cosmogenic activation was found to play an important role in the scintillating CaWO_4 crystals operated as cryogenic detectors in the CRESST-II dark matter experiment, following the extensive background studies of a crystal (called TUM40), grown at the Technische Universitat Munchen <cit.>. Distinct gamma lines were observed in the low-energy spectrum below 80 keV, originating from activation of W isotopes: proton capture on ^182W and ^183W results in ^179Ta (after a successive decay) and ^181W respectively, which decay via electron capture. The activity of both isotopes in the crystal was quantified and, as the leaking of events from this background source into the region of interest for dark matter searches is very important, initiatives to reduce the cosmogenic activation of crystals have been undertaken. On the other hand, the production rate of tritium in CaWO_4 was evaluated using the TALYS code <cit.> and it is reported in Table <ref>. Radioactive contamination, including some cosmogenic products, measured for different inorganic crystal scintillators like CaWO_4 is presented in Ref. damaijmpa2.Excitation functions of proton-induced reactions on natural neodymium, in the 10-30 MeV energy range, were measured in Ref. nd1. Irradiation took place at the cyclotron U-120M in Rez near Prague and the radioactivity measurement was carried out using a HPGe detector. The measured production cross sections were compared with TENDL-2010 predictions finding in general good agreement in both shapes and absolute values. In addition, the corresponding contribution to the production rates of radionuclides relevant for double beta decay searches, like in the SNO+ experiment, was evaluated adopting the proton flux from Ref. barabanov. In Ref. nd2, these results were completed in 5-35 MeV energy range following an analogue experimental strategy and comparing with excitation functions from TENDL-2012 library.The same experiment performed for tellurium and germanium was also made with a natural molybdenum target, based on 800 MeV proton irradiation at LANSCE and screening with germanium detectors at Berkeley <cit.>, for determining the ^60Co production cross section. In Ref. mei2016, the production rates of a few isotopes (^7Be, ^10Be and ^14C) induced in PTFE are presented, based on analogous calculations to those performed for xenon, copper, titanium and stainless steel using GEANT4 and ACTIVIA. For quartz light guides, some production rates were calculated in Ref. pettus using ACTIVIA with the Gordon et al. neutron spectrum, including that of tritium reproduced in Table <ref>.§ UNDERGROUND ACTIVATIONAt a depth of a few tens of meter water equivalent (m.w.e.), the nucleonic component of the cosmic flux is reduced to a negligible level. The neutron fluxes in underground facilities, either radiogenic (from fission or (α,n) reactions) or induced by cosmic muons, in rock or set-up materials, are several orders of magnitude lower than at surface. The spectrum of the dominant component is concentrated at the region of a few MeV, which is below the energy threshold of most of the spallation processes activating materials at surface. In addition, to suppress the neutron activation underground it could be possible to use a neutron moderator shielding around the detector. Therefore, the cosmogenic production underground is often assumed to be negligible. The considered generation of radioactive isotopes underground is induced mainly by muons. At shallow depths, the capture of negative muons is the relevant process but deep underground interactions by fast muons dominate (direct muon spallation or the electromagnetic and nuclear reactions induced by secondary particles: nucleons, pions, photons,…). Since the muon flux and spectrum depends on depth, it is worth noting that underground activation studies are produced for particular depths; therefore, comparison of different estimates is not straightforward. Many of the results obtained for underground activation are related with experiments using large liquid scintillator detectors, as summarized in the following.* An early estimate of production rates was made in Ref. oconnel for isotopes induced in materials typically used in neutrino experiments: C, O and Ar. Inelastic scattering of muons giving electromagnetic nuclear reactions was considered and rates were evaluated at sea level and underground (2700 m.w.e.). * Production cross sections were measured in a reference experiment <cit.> performed at CERN irradiating with the SPS muon beam with energies of 100 and 190 GeV different kinds of ^12C targets placed behind concrete and water to build the muon shower like in real liquid scintillator experiments. Several detection techniques for measuring products of different half-lives were applied. Then, considering the measured cross-sections and the deduced dependence with the muon energy (σ∝ E_μ^α with α=0.73±0.10) muon induced background rates for the muon flux at Gran Sasso and BOREXINO detector were computed <cit.>. Rates for KamLAND were estimated to be a factor ∼7 higher. The most relevant contribution was that of ^11C. * In fact, the production rate of ^11C was specifically estimated in Ref. galbiatti taking into account all relevant production channels. Evaluation of cross sections from different sources combining data and calculations was made and a FLUKA simulation of monoenergetic muons in Borexino liquid scintillator (trimethilbenzene) was run to derive rates and paths of secondary particles; then, combining this information on the secondaries with the cross sections the individual and total production rates were derived for different muon energies. Good agreement was found with rates coming from measurements (at 100 and 190 GeV) <cit.> and with extrapolations for average muon energies at KamLAND, Borexino and SNOlab. * Analysis of KamLAND data (from 2002 to 2007) allowed the measurement of activation yields <cit.>. Isotopes were identified and quantified using energy and time information registered. In addition, a FLUKA simulation of monoenergetic muons (in the 10 to 350 GeV range) was performed for KamLAND liquid scintillator to estimate the same yields too. Comparing with extrapolations (based on the power-law dependance with respect to muon energy) of results from the muon beam experiment <cit.>, some inconsistencies are reported for some isotopes, indicating that estimation by extrapolation might not be sufficient. * In Ref. borexino, from data from the Borexino experiment and profiting the large sample of cosmic muons identified and tracked by a muon veto external to the liquid scintillator, not only the yield of muon-induced neutrons was characterized but also the production rates of a number of cosmogenic radioisotopes (^12N, ^12B, ^8He, ^9C, ^9Li, ^8B, ^6He, ^8Li, ^11Be, ^10C and ^11C) were measured, based on a simultaneous fit to energy and decay time distributions. Results of the corresponding analysis performed by the KamLAND collaboration for the Kamioka underground laboratory are similar. All results are compared with Monte Carlo simulation predictions using the FLUKA and Geant4 packages. General agreement between data and simulation is observed within their uncertainties with a few exceptions; the most prominent case is ^11C yield, for which both codes return about 50% lower values.* In the context of argon-based neutrino experiments, production of ^40Cl and other cosmogenically produced nuclei (isotopes of P, S, Cl, Ar and K) which can be potential sources of background was evaluated in Ref. barker. ^40Cl can be produced through stopped muon capture and the muon-induced neutrons and protons via (n,p) and (p,n) reactions; it is unwanted as it can be a background for different neutrino reactions. Geant4 simulations were carried out and analytic models were developed, using the measured muon fluxes at different levels of the Homestake Mine. Different depths were considered in the study, concluding that large backgrounds to the physics proposed are expected at a depth of less than 4 km.w.e. Studies of underground activation have been made too in other contexts, not related to large scintillator detectors. * Estimates of production rates for isotopes produced in enriched germanium detectors and set-up materials (cryogenic liquid) were made within the GERDA double beta decay experiment <cit.>, based on a GEANT4 Monte Carlo simulation for the muon spectrum at Gran Sasso. ^77mGe was identified as the most relevant product.* In Ref. romanian, the radioactivity induced by cosmic rays, including neutrinos, muons and neutrons, is analyzed for the rock salt cavern of an underground laboratory. Natural isotopes of sodium and chlorine are considered. * The activation in natural tellurium due to the neutron flux underground, in particular for Sudbury laboratory (at a depth of about 6 km.w.e.), was analyzed in Ref. telozza. For long-lived isotopes, the underground activation was estimated to be less than 1 event/(y t). The production rates for many short-lived isotopes of Sn, Sb and Te were quantified using ACTIVIA and TENLD data, for neutrons coming from (α,n) reactions as well as for muon-induced neutrons; the obtained rates are many orders of magnitude lower than the derived rates following the same approach for exposure at surface. * The production of ^127Xe in xenon detectors at the depth of the Sanford Underground Research Facility (1480 m below surface, 4.3 km.w.e.) was evaluated in Ref. mei2016 by Geant4 simulation, considering contributions from both fast and thermal neutrons. A production rate of about 3 atoms per ton per day, considered to be negligible, was found.§ SUMMARYCosmogenic activation of materials can jeopardize the sensitivity of experiments demanding ultra-low background conditions due to the production of long-lived isotopes on the Earth's surface by nucleons and, in some cases, also due to the continuous generation of short-lived radionuclides deep underground by fast muons. With the continuous increase of detector sensitivity and reduction of primordial radioactivity of materials, this background source is becoming more and more relevant. Direct measurements and estimates of production rates and yields for several materials have been made in the context of, for instance, dark matter, double beta decay and neutrino experiments. As summarized here, cosmogenic products have been analyzed for materials used as detector media and also for common materials used in ancillary components. Each material produces different relevant cosmogenic isotopes, although there are some ones generated in most of the materials; this is the case of tritium, which due to its decay properties is specially relevant for dark matter searches. In principle, sea level activation can be kept under control by minimizing exposure at surface and applying material purification techniques. But reliable tools to quantify the real danger of exposing the materials to cosmic rays are desirable. There are many options to undertake a material activation study, but unfortunately, discrepancies between different estimates are usually non-negligible. At the moment, there is no approach working better than others for all targets and products. A good recipe to attempt the calculation of the production rate of a particular isotope could be the following: * To collect all the available information on its production cross section by neutrons and protons, from both measurements (EXFOR database will help) and calculations, either from libraries or using codes (based on semiempirical formulae like YIELDX and ACTIVIA or on Monte Carlo simulation). Measurements of cross sections at different energies using nucleon beams are not straightforward, but they are essential to validate models and codes giving excitation functions. * To choose the best description of the excitation functions of products over the whole energy range, by minimizing deviation factors between measurements and calculations. * To calculate the production rates considering the preferred cosmic ray spectrum and to compare them with previous estimates or measurements if available.Uncertainties in this kind of calculations (in many cases higher than 50%) come mainly from the difficulties encountered both on the accurate description of cosmic ray spectra and on the precise evaluation of inclusive production cross-sections in all the relevant energy range; the low and medium energy regions below a few hundreds of MeV are the most problematic ones since neutron data are scarce and differences between neutron and proton cross sections may be important. 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"authors": [
"Susana Cebrian"
],
"categories": [
"nucl-ex",
"astro-ph.HE",
"hep-ex",
"physics.ins-det"
],
"primary_category": "nucl-ex",
"published": "20170824151407",
"title": "Cosmogenic activation of materials"
} |
< g r a p h i c s > The structural and electronic properties of the clathrate compounds Ba_8Al_xSi_46-x and Sr_8Al_xSi_46-x are studied from first principles, considering an Al content x between 6 and 16. Due to the large number of possible substitutional configurations we make use of a special iterative cluster-expansion approach, to predict ground states and quasi-degenerate structures in a highly efficient way. These are found from a simulated annealing technique where millions of configurations are sampled. For both compounds, we find alinear increase of the lattice constant with the number of Al substituents, confirming experimental observations for Ba. Also the calculated bond distances between high-symmetry sites agree well with experiment for the full compositional range. For x being below 16, all configurations are metallic for both materials. At the charge-balanced composition (x=16), the substitutional ordering leads to a metal-semiconductor transition, and the ground states of Ba_8Al_16Si_30 and Sr_8Al_16Si_30 exhibit indirect Kohn-Sham band gaps of 0.36 and 0.30 eV, respectively, while configurations higher in energy are metals. The finding of semiconducting behavior is a promising result in view of exploiting these materials in thermoelectric applications.§ INTRODUCTIONWaste heat is generated by all kinds of engines, from smartphones and laptops to high-performance computers, from refrigerators to power plants, from cars to airplanes, and many more. Transforming heat into electricity by exploiting the thermoelectric effect provides a possibility to reuse part of this waste heat, however, new materials with large thermoelectric efficiency are a prerequisite for profitable applications. Thus the investigation of and the search for novel thermoelectrics <cit.> concern a hot topic in materials research. Intermetallic clathrate compounds <cit.> are promising candidates. Their cage-like structure containing guest atoms is regarded as a realization of the phonon-glass electron-crystal, promising a large figure of merit. Figure <ref> (a) shows the unit cell of type-I clathrates with formula unit G_8Y_xX_46-x (cubic space group Pm3 n). The most important framework atoms X are group-IV elements, i.e. Si, Ge, or Sn. The host structure Y_xX_46-x is built from 46 atoms forming eight cages, six tetrakaidecahedra and two dodecahedra (Fig. <ref> (b)). They occupy the three symmetrically distinct Wyckoff sites 24k, 16i, and 6c, and are interconnected by tetrahedral covalent bonds as depicted in Fig. <ref>(c). This bonding, analogous to that of group-IV semiconductors, can lead to high performance in the electronic properties<cit.>. The eight guest atoms G, placed inside the cages at Wyckoff sites 6d and 2a, can induce static and dynamic disorder, thereby lowering the thermal conductivity. They are also called rattlers, as they are expected to vibrate at low frequencies in a non-directional fashion, efficiently scattering low energy phonons responsible for heat transport.Without substituents (x=0), all valence electrons of the framework atoms X fully contribute to the covalent bonds. Following the Zintl-Klemm concept, the eight guest atoms, G=Ba or Sr considered here, donate their two valence electrons to the host, making the material metallic.These 16 free charges per formula unit can be compensated by partially substituting the group-IV element X by a group-III element, e.g. Y=Al, Ga, In. Assuming a purely ionic host-guest interaction and complete absorption of the free charges by substituents, full compensation is reached at x=16 [Previous calculations <cit.> of the electron-localizability indicator suggest that, for Al-Si based clathrates, the assumption of an ionic guest-host bonding is indeed reasonable.]. This so-called charge-balanced composition <cit.> is expected to be a semiconductor, which is highly desired for thermoelectric applications <cit.>. The variety of chemical elements available for building the clathrate structure offers a wide compositional space for optimizing their electronic properties and reaching high efficiency <cit.>. Clathrates based on Ge and Ga have demonstrated good performance in their thermoelectric properties, e.g. in terms of a glass-like thermal conductivity of Sr_8Ga_16Ge_30 <cit.> and high thermoelectric efficiency of Ba_8Ga_16Ge_30 <cit.>.Al-Si-based clathrates are of technological interest in terms of price, weight, and low environmental impact <cit.>. So far, they could be synthesized in the range x≤15 for Ba <cit.>, and only for x=10 in the case of Sr[The slightly higher electronegativity of Sr as compared to Ba may lead to a lower solubility of Al in Sr-filled clathrates.]<cit.>. Thus, in both cases the charge-balanced composition has not been achieved so far. Still the question is whether the Zintl-Klemm concept holds, i.e. this composition would indeed exhibit semiconducting behavior like the guest-free, non-substituted structure. Yet, it is not clear what the ground-state configurations at this and other concentrations of the substituent for different guest elements are. Only understanding the subtle interplay between covalent bonding and ionic repulsion, tailoring of the material's structure and, hence, its electronic properties will be possible.In this work, we investigate the clathrate compounds Ba and Sr. On the search for their ground states as a function of Al substitution in the range 6 ≤ x ≤ 16, we explore the configurational space by making use of the cluster expansion (CE) technique <cit.>. This technique allows for building numerically efficient models to predict the energy E of a substitutional alloy by exploiting the unique dependence of E on the configuration. Combined with Monte Carlo (MC) simulations, it has been successfully applied to access the stable phases and thermodynamical properties of bulk and surface alloys<cit.> (for examples, see Refs.and ). Due to the large primitive cell of the clathrates, we apply a novel iterative cluster-expansion technique (iCE) <cit.>, as implemented in the package CELL<cit.>. This technique, specially designed for alloys with large parent cells, provides an efficient way to predict the energy of an arbitrary configuration (i.e. a specific arrangement of Al atoms at the lattice sites) with accuracy. Exploring the configuration space, we provide an unbiased prediction of the ground-state configurations as a function of Al content. Based on an in-depth analysis, we discuss their stability and their electronic structure. § METHODOLOGY§.§ Cluster expansionAn arbitrary configuration of the crystal can be represented by a vector s=(σ_s1,σ_s2,...), where the occupation variables σ_si take the value 1 or 0, if a site i is occupied by one or another type of atom. For a configuration s, the predicted energy E_s can be expanded in terms of 0-, 1-, 2-, ... n-body clusters α={i,j,...} as E_s = ∑_α m_α J_αX_s α,where i, j, ... indicate crystal sites. The coefficients J_α are the so-called effective cluster interactions (ECI), and the sum runs over a set of N_c symmetrically distinct clusters. The correlation functions X_s α = 1/m_α ∑_β≡αf_sβare obtained from an average of the cluster functions f_sβ over all the clusters (denoted β) which are symmetrically equivalent to the cluster α. There are m_α such clusters. The cluster functions are defined as the product of the occupation variables σ_si for the sites i belonging to the cluster, i.e., f_sβ=∏_i ∈βσ_si. For more details on the CE we refer the reader to Ref. . The coefficients J_α in Eq. <ref> are obtained by fitting the predicted energies to a set of N_t structures –the training set– whose energies are known from calculations based on density-functional theory (DFT)[Our zero-temperature calculations do not include zero-point vibrational contributions, which could affect to some extent the cluster interactions<cit.>.]. Formally, this requires the minimization of the objective function S^2=1/N_t∑_s=1^N_t(E_s- E_s)^2 +∑_αA_α|J_α|^2with respect to the ECIs, i.e. ∇_JS^2 = 0. In Eq. <ref>, the first term on the right hand side is the mean squared error (MSE) of the predicted energies E_s. The second term represents an ℓ_2 regularization which allows for finding an optimal set of ECIs even if the number of clusters N_c is larger than N_t. Both the optimal penalization strength A_α and the optimal set of clusters are obtained by minimizing the cross-validation score (CV) <cit.>, CV^2=1/N_t∑_s=1^N_t(E_s - E_(s))^2 .Here, E_(s) is the predicted value for E_s, which is obtained from a training set excluding the data point s. Other optimization procedures as the compressed sensing <cit.> consider an ℓ_1 norm as regularization term, instead of an ℓ_2 norm as used in Eq. <ref>. For the clathrate compounds of this study, such an approach did not lead to improvements, neither in the quality of the model nor in the accuracy of the predictions.In this paper, the CE is performed in an iterative manner, making use of the python package CELL [The code CELL uses the corrdump utility of ATAT for the generation of clusters and the cluster correlation matrix. For details on this utility see Ref..]<cit.>. It enables cluster expansions for large parent cells, where a full enumeration of possible structures and the corresponding calculations of their predicted values E become an impossible task in terms of computational cost. This is the case for the type-I clathrate compounds.Additional details about the iCE approach and its application to complex alloys, as well as the numerical implementation, will be published elsewhere<cit.>. The search for the ground-state (GS) structures is carried out in an iterative procedure. It starts with a set of random structures, one for each composition x (6≤ x ≤ 16). The energies of this set are calculated, and a first CE is constructed from them. The large size of the clathrates' parent cell prevents a full enumeration of symmetrically distinct structures for each composition. Instead, the configurational space is explored through a Metropolis sampling generating a Boltzmann distribution at fixed temperature. The energies of the new structures visited in this procedure are predicted by the CE. From this sampling, the lowest-energy structures visited (or the lowest non-degenerate, in case no new ground state is found) are adopted, and their energies are calculated. Based on this extended set of data, a new CE is performed. This procedure is repeated until no new GS structure is found and the CV is small enough to make predictions.§.§ Computational details The calculations are performed with the full-potential all-electron DFT package exciting, <cit.> an implementation of the (linearized) augmented planewave + local-orbital method. In order to evaluate the stability of our results, we use two different exchange-correlation functionals, namely the local-density approximation (LDA) <cit.> and the generalized-gradient approximation in terms of the PBEsol functional <cit.>.For obtaining a CE with high predictive power, well converged calculations are essential. The crucial convergence parameters, affecting the accuracy of the calculation, are the number of 𝐤-points, used for the discretization of the Brillouin zone and the basis set size. The latter is determined by the parameter R_min× G_max, where G_max is the maximum reciprocal lattice vector of the augmented planewaves used in the representation of the wavefunctions, and R_min is the smallest muffin-tin radius. Total energies are calculated with R_min× G_max=8 and a 4 × 4 × 4 k-grid, leading toan accuracy of energy differences between structures of the same composition below 0.07meV/atom. The uncertainty in the lattice parameter upon volume optimization is smaller than 0.01Å. The atomic positions are relaxed until all the forces are below a threshold of F_tol= 2.5 mRy/atom. We find that the energy differences scale roughly linearly with the number of Al-Al bonds, where each of them leads to an increase by about 7-10 meV/atom. Energy differences between structures of the same composition and no Al-Al bonds vary by a few meV/atom, with the smallest differences being around 0.2 meV per atom for structures differing only in one Al-Al second nearest-neighbor pair.§ GROUND-STATE SEARCH §.§ Results of the iterative cluster expansionThe results of the iterative CE are shown in Fig. <ref> for Ba and Sr. They are based on DFT calculations using LDA and PBEsol. The figure depicts the energy of mixing versus the number of Al substituents, e(x) per atom, defined as e(x) = 1/N{ E(x) - [ E(x=0) + ( E(x=46) - E(x=0) ) x/46]} .This expression represents the energy difference between the energy E(x) and the energy obtained from a linear interpolation between the predicted energies of the pristine structures, e.g. Ba_8Si_46 (x=0) and Ba_8Al_46 (x=46), taking the Ba clathrate as an example. N denotes the number of atoms within the unit cell (N=54). The results e(x) are shown as black circles. Starting with the LDA results, for both clathrate compounds, the GS configurations decrease almost linearly with the Al content x (red line), with an abrupt change in slope aroundx=13. This behavior can be traced back to the electronic properties as will be discussed below. To accurately model this behavior, we perform two separate CEs by dividing the composition range into two sets of data. The first one, termed CE1, is based on the data for x ∈ [6,13] (to the left of the black solid line) and the second one (CE2) for x ∈ [13,16] (to the right of the black dashed line). In both ranges, the predicted energies e(x) (black dots) match well with their energies (circles). [e(x) can be obtained from Eq. <ref> when replacing E(x) by E(x).] At x=13, predictions with CE1 or CE2 are equally good. In Fig. <ref> predictions at this composition are obtained with CE2. With a well converged CE, a large Metropolis sampling (gray dots) is performed in order to search for new GS configurations and confirm those already found. In every Metropolis run, typically 5×10^5 configurations are sampled. For these samplings, a Boltzmann distribution with a temperature of T=1000K is used. A fully independent ground-state search was performed with the PBEsol functional (bottom panels of Fig. <ref>). For both the Ba and Sr clathrates, the two functionals, LDA and PBEsol, lead to the same ground-state configurations indicating the robustness of the results. §.§ Cluster expansion models To obtain the CE models in each iteration of the iCE, we employ an optimization procedure as described in Ref. . A set of clusters with sizeN_c is defined by two parameters, the largest number of points N_p^max and the maximum radius R^max, i.e. the maximum distance between two points in the cluster. All clusters with N_p ≤ N_p^max and R ≤ R^max are thus included in the set. A nested loop with increasing R^max and N_p^max is then performed, implying increasingly larger set sizes. The set leading to the smallest CV is selected. According to this optimization procedure, a small set of relevant clusters is identified. For this set, we finally perform a combinatorial optimization where all subsets of clusters are considered. As the number of relevant clusters is small, i.e. N_c<N_t, we set the penalization strength A_α in Eq. <ref> to zero.InFig. <ref>(a) we demonstrate this combinatorial optimization for Ba in the range x ∈ [13,16] (CE2) for PBEsol. Each point represents a different cluster set. The CV (black diamonds) and the root mean-squared error (RMSE, blue diamonds) as a function of N_c are depicted. The black solid line connects the sets yielding the lowest CV for each N_c, while the blue solid line connects the corresponding RMSE. The RMSE steadily decreases with the number of clusters, indicating an improvement of the fit to the data. In contrast, the CV reaches a minimum at N_c=10, indicating the optimal CE (see Table <ref>). The increase of the CV for N_c>10 reveals overfitting, i.e. fitting to noise in the data. The optimal CE comprises the following 10 clusters: the empty cluster (α=0); the three 1-point clusters arising from the three Wyckoff sites (24k, 16i, and 6c), labelled as α=k, i, and c; and the four 2-point clusters α=kk, ki, kc, and ii (see Fig. <ref>(b)). Additionally, two 2-point clusters consisting of next-nearest neighbor sites are present. These are denoted as i-i and k-c (dashed lines in Fig. <ref>(b)). In the case of Sr, the addition of next-nearest neighbor interactions lead to overfitting and thus are not included. Tables <ref> and <ref> provide summaries of the CEs for the two clathrate compounds. Two values for the CV are given in each case. The one on the left gives the average prediction error for the whole data set, as defined in Eq.<ref>. The one on the right (bold) represents the respective value for all configurations with an energy range from the ground-state energy to 20meV/atom above. This latter quantity represents the accuracy of the predictions for low-energy configurations better.§ STRUCTURAL PROPERTIES §.§ Ground-state configurations From the CE we can deduce the site occupancies and numbers of Al-Al bonds per unit cell, which are shown in Table <ref> for the GS configurations. These are represented by the notation (n_k,n_i,n_c) N_b, where n_k, n_i, and n_c indicate the number of Al atoms sitting at the 24k, 16i, and 6c sites, respectively, and N_b the number of bonds between Al atoms in the structure. A given tuple can represent several structures. Nevertheless, this simple representation is meaningful because it captures the main interactions. In addition to the ground-state configurations, the energies of a few quasi-degenerate structures (i.e., with energy differences below ∼ 2meV/atom with respect to the GS) are present for some compositions. Overall, the results for Ba and Sr are very similar. More striking, for each of them the two functionals lead to exactly the same configurations. For both compounds, bonds between Al atoms are avoided over the entire compositional range with only one exception. This reflects the fact that all nearest-neighbor two-point interactions are positive (see Tabs. <ref> and <ref>). Moreover, the order of the one-point ECIs, J_c < J_k < J_i, indicates favored occupation of the 6c site, followed by a preference of the 24k and 16i sites, respectively. In line with this, the 6c site is fully occupied in the range x ∈ [6,12] for Ba and [6,15] for Sr. With further addition of Al atoms, the occupation of the 16i site increases while the 24k site remains empty. This seems in contradiction with the ordering of the single-point ECIs. However, adding an Al atom at a 24k site would lead to a structure with Al-Al bonds, since every 24k site has one 6c site as neighbor (see Fig. <ref>). Thus the ECIs of the two-point clusters prevent occupation of the site 24k and rather cause occupation of the site 16i. For instance, for Ba (CE1, LDA) the energy gain for occupying a 16i position instead of 24k is J_kc + J_k - J_i∼ 260.5 meV (∼ 4.8 meV/atom).In the intermediate composition range, x ∈ [9,14], we observe a puzzling competition between single-point and two-point interactions for the Ba clathrate, leading to quasi-degenerate states through an interesting reordering. The occupation of the 6c site is reduced by one, increasing the occupancy of the 24k site, a neighbor of the now empty 6c position. The three remaining neighboring 24k sites are occupied by Al atoms at the expense of the 16i site. This multiple exchange can happen up to three times. The respective energy gain can be estimated from the ECIs as 4 J_k - 3 J_i - J_c. For instance, for CE1 (LDA), this gives 9.6meV (∼ 0.2meV/atom). For x<13, the GSs are represented by configurations (0,x-6,6) 0. At x=13, the GS is (4,4,5) 0 while two quasi-degenerate states, (0,7,6) 0 and (8,1,4) 0, are present. For x=15, the configuration (12,0,3) 0 becomes the GS, which means half occupation of the 24k and 6c sites.For the Sr clathrate, GS structures with a fully occupied 6c site are present up to x=14.Avoidance of Al-Al bonds with full occupation of the 6c site is only possible up to this Al content. At this composition, the 16i site is half occupied, thus additional Al atoms at a 16i position would lead to an Al-Al bond between two 16i sites. At x=15, three degenerate structures appear. In contrast to all other cases, the GS contains one Al-Al bond. There exist, however, two more quasi-degenerate structures with N_b=0. For the charge-balanced composition (x=16), two quasi-degenerate configurations are present, with the GS being the same as for Ba.Prevention of bonds between trivalent elements (trivalent bonds, in short) has been reported from DFT calculations for other clathrate compounds with Ga and Ge atoms in the framework <cit.>. For the charge-balanced composition, the configuration (12,1,3) 0 has been found as the most stable one out of nine different configurations.From six samples of the clathrate K_8Al_8Si_38 the configuration (2,0,6) 0 has been found lowest in energy<cit.>. These results are in line with our findings for the Ba- and Sr-filled Al-Si based clathrates. It should be noted, though, that our method allows for exploring a much larger configurational and compositional space in an unbiased manner. In Ref.guidelines for the preferred occupancies have been derived postulating the avoidance of trivalent bonds. These rules are indeed confirmed by our results. An exception concerns the statement in Ref.that structures with x=16 without trivalent bonds do not exist. Conversely, we clearly show that the GS at the charge-balanced composition does not contain Al-Al bonds. Figure <ref> shows the fractional occupancy factors OF=n_w/N_w of the ground-state configurations (dots) and their degenerate states (crosses), with N_w being the multiplicity of Wyckoff site w. In a material, both the GS and the (nearly) degenerate states can be present simultaneously in different regions, therefore the average OF can take any value in between. This is indicated by the shaded areas.§.§ Lattice constantsThe lattice constants of the GS structures are presented in Fig. <ref> for Ba (left) and Sr (right). Both compounds reveal a linear increase with the number of Al substituents, as obtained with LDA (dotted lines) and PBEsol (solid lines). This is explained by the larger atomic radius of Al atoms as compared to Si <cit.>. The slope of Δ a / Δ x ≈ 0.018Å for Ba is close to the experimental value of about 0.020 Å (black dashed lines) <cit.>. Sr exhibits a similar slope of Δ a / Δ x ≈ 0.020 Å. The smaller size of Sr atoms in comparison to Ba <cit.> leads to a smaller lattice spacing by about 0.07 Å in the entire composition range. This difference agrees well with the difference in the experimental values at x=10 (the only composition synthesized so far for Sr <cit.>). As expected, the lattice parameters obtained by LDA are smaller (∼ 0.065Å) than those from PBEsol. The latter still underestimates the experimental value by about 0.5%. While for solid Al and Si the relative error is less than 0.5% for both functionals <cit.>, for alkaline-earth metals, as Sr and Ba, the lattice spacing is underestimated by LDA by more than 4% and by PBEsol by more than 2% <cit.>. This suggests for the clathrates that the slight systematic underestimation by PBEsol is caused by the guest atoms.§.§ Bond-distances Like the lattice parameters, also the average bond distances reveal a dependence on the number of Al substituents, as can be seen in Fig. <ref>. The dots indicate the PBEsol data; the LDA results (not shown) are uniformly smaller by about 0.02 Å.Focusing on the Ba clathrate, the ii and ki bonds show a monotonic, almost linear increase with Al content for 6 ≤ x≤ 12. This reflects the fact that in this composition range, additional Al atoms occupy the 16i sites, resulting in larger bond lengths when a 16i site is involved, i.e., the ii and ki distances. Since no 24k site is occupied in the range x<12, the kk bond distance remains almost constant. As shown before (Table <ref>), occupation of the 24k site appears from x=13 onwards leading to a drastic increase of the kk bond and a corresponding decrease of the ii bond from x=12 to x=13 due to simultaneous depletion of this site. This trend continues until the charge-balanced composition is reached, where the 24k site is half occupied in the GS configuration. The kc bond length is almost independent of the Al content. The reason becomes clear by recalling the Al occupations of the 6c and 24k sites. At low x, full 6c and empty 24k sites imply that one Al atom is part of every kc bond. At high x, the 6c position is less populated, but at the same time the surrounding 24k sites of an empty 6c site are occupied, thus, the number of Al atoms forming part of kc bonds does not change.For the Sr compound, the trends in the low substitutional range are analogous to those of the Ba counterpart. However, the linear increase of the ii and ki bond distances continues until the composition of x=15 is reached. This is again related to the preferred occupation of the 16i site. At x=16, the energetic ordering of the configurations changes abruptly (see above), and the bond distances now appear similar to those of the Ba clathrates. In Fig. <ref> also experimental data are depicted. For the Ba case, the kc and ki values are in good agreement with our calculations for the entire composition range (including nearly degenerate states shown as crosses). The experimental kk bond length is slightly larger than the theoretical value for x=6-7; this may point to a marginal presence of Al at the 24k site in the real sample. For x ≥ 9, good agreement between theory and experiment is observed. In this composition range containing quasi-degenerate states, the experimental values lie inside the green shaded area. This area indicates the possible values of bond lengths assuming a mixture of GS and quasi-degenerate configurations. Concerning the ii bond, almost all experimental values lie within the range of the calculated ones (blue shaded area). For kk and ii, the bond lengths of the degenerate states are very different from those of the GS. Notably, also the experimental values largely deviate from each other. Regarding the Sr clathrate, there is good agreement with experiment. The slightly larger measured size of the kk bond for x=10 may be of the same origin as in the Ba case.§ ELECTRONIC PROPERTIES The density of states at the Fermi level, DOS(E_F), is shown in Fig. <ref> for the GS configurations of Ba and Sr. Metallic behavior is observed over the whole composition range, except at x=16, where it approaches zero. Thus, the charge-balanced composition, indeed, exhibits a semiconducting GS[Due to the electronic-entropy contribution, semiconducting behavior for the lowest-energy charge-balanced composition, may represent a disadvantage in terms of stability at finite temperatures, relative to metallic compositions. This could explain the difficulties in synthesizing the charge-balanced composition for Ba-filled clathrates, which could be achieved only up to x=15.]. In all cases, a local minimum close to x=13 points to a change in the electronic configuration of conduction electrons around this concentration. This goes hand in hand with the change in slope in the GS energy of mixing in Fig. <ref>. In line with this observation is the finding that the ECIs obtained by splitting the data set at x=13 to perform two cluster expansions, resulted in much better predictions than without doing so. Band structures and densities of states of Ba and Sr for x=6 and 16 are illustrated in Fig. <ref>. There is a gap well below the Fermi level amounting to 0.25 eV for Sr and 0.45 eV for Ba, respectively, as obtained by PBEsol and in agreement with Ref. . By increasing the Al concentration, corresponding to a removal of “free” electrons, this gap moves up in energy. At the charge-balanced composition, it is centered at the Fermi level and presents an indirect Kohn-Sham band gap along the Γ-M direction of 0.30 and 0.36 eV for Sr and Ba, respectively (see Table <ref>).Since semiconducting behavior is beneficial for the thermoelectric properties, we have closer look at the electronic behavior of these materials at the charge-balanced composition. Figure <ref> illustrates the DOS around the Fermi level for Ba_8Al_16Si_30 (left) and Sr_8Al_16Si_30 (right) for different configurations appearing during the ground-state search with the PBEsol functional. Their energies of mixing and band gaps (if present) are summarized in Table <ref>, showing also the corresponding LDA results. It is interesting to note that the configurations with a large number of Al-Al bonds clearly reveal metallic behavior. There exist also structures with a few Al-Al bonds and a small Kohn-Sham band gap [tuple (8,4,4) 2]. The structures with no Al-Al bonds are, as already discussed, lower in energy, but not all of them show semiconducting behavior. However, for both materials the GS configuration has the largest Kohn-Sham band gap with respect to other configurations. The observation of a band gap in Sr_8Al_16Si_30 is in seeming contradiction with calculations reported in Ref.giving metallic behavior for the same composition. However, this difference is explained by the facts that theconfiguration considered in this reference is not the GS, and all 16 Al atoms were placed at 16i sites. This structure has 8 Al-Al bonds, since each 16i site is linked to another 16i site. As noted above, Al-Al bonds lead to high-energy structures with vanishing or very small gaps. Semiconducting behavior of Ba_8Al_16Si_30 has already been reported for the non-GS configuration (9,4,3) 0 <cit.>. From our CE we confirm this configuration to be close in energy to the GS.In general, the electronic structure of these materials strongly depends on the atomic configuration. The predicted semiconducting behavior of both compounds in some configurations at the charge-balanced composition, and predominantly those of the ground states, is an exciting finding in view of their application in thermoelectric devices.§ CONCLUSIONS AND OUTLOOKIn the present work, we have investigated the clathrate compounds Ba and Sr (6 ≤ x ≤ 16) with respect to their structural stability and electronic properties. Their large configurational space has been explored with CELL <cit.>, an iterative cluster expansion technique for large parent cells. Thanks to its numerical efficiency we have been able to predict ground-state configurations as a function of Al content without making any presuppositions. For both compounds, these GS have been verified by two independent sets of calculations based on the exchange-correlation functionals LDA and PBEsol, respectively. The CE yields total energies with an accuracy of less than 2meV/atom for low lying configurations. This accuracy has allowed us to determine site preferences and bond lengths. Detailed analysis has revealed that Al-Al bonds are energetically unfavorable. There is a clear preference for Al atoms to occupy the 6c site. In the Ba compound, a transition from a full 6c to half 6c occupation takes place between x=10 and 14, and the 6c site remains half occupied up to the charge-balanced composition. The GS configurations of the Sr clathrate reveal a constant increase in the occupation of the 16i site up to x=15 and a drastic configurational rearrangement at x=16. Both clathrate types exhibit the same GS configuration at the charge-balanced composition, with 12 atoms at the 24k site (half occupancy), one atom at the 16i site, and 3 atoms at the 6c site (half occupancy),thereby completely avoiding Al-Al bonds. The identification of such a configuration as the GS is an important finding, as it reveals semiconducting behavior. Other higher-energy configurations at charge-balanced composition are metallic, indicating thatstructural ordering results in a metal-semiconductor transition. We have confirmed the linear increase of the lattice parameter with the number of Al substituents observed for Ba, as well as the changes in the bond distances revealed by various experiments. The bond distances have been found to be strongly correlated to the Al occupation of the Wyckoff sites as reported by previous studies <cit.>. Since in X-ray diffraction experiments Al and Si atoms are not clearly distinguishable <cit.>, a reliable model for describing the correlations between the site occupancies and the bond distances is highly desired. Our findings may form a solid basis for building such a model. An extension of our work towards finite-temperature effects will be subject of a forthcoming paper. Considering the large influence of the configurational ordering on the investigated quantities, this step is promising for a quantitative analysis of experimental observations.Input and output files of our calculations can be downloaded from the NOMAD repository by following the link in Ref. . § ACKNOWLEDGEMENTSWork supported by the Einstein Foundation Berlin (project ETERNAL). The Norddeutsche Verbund für Hoch- und Höchstleistungsrechnenn (HLRN) enabled the computationally involved calculations. 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Physical Review B 2009, 79, 085104 [mt2()]mt2017data See http://dx.doi.org/10.17172/NOMAD/2017.02.16-1. | http://arxiv.org/abs/1708.08126v1 | {
"authors": [
"Maria Troppenz",
"Santiago Rigamonti",
"Claudia Draxl"
],
"categories": [
"cond-mat.mtrl-sci"
],
"primary_category": "cond-mat.mtrl-sci",
"published": "20170827193617",
"title": "Predicting ground-state configurations and electronic properties of the thermoelectric clathrates Ba$_{8}$Al$_{x}$Si$_{46-x}$ and Sr$_{8}$Al$_{x}$Si$_{46-x}$"
} |
Department of Physics, University of Virginia, Charlottesville, VA 22904, USATheoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USADepartment of Physics and Astronomy, The University of Tennessee, Knoxville, TN 37996, USADepartment of Physics and Astronomy, The University of Tennessee, Knoxville, TN 37996, USA Department of Physics and Astronomy, The University of Tennessee, Knoxville, TN 37996, USA Quantum Condensed Matter Division and Shull-Wollan Center, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USAWe present a theoretical framework for equilibrium and nonequilibrium dynamical simulation of quantum states with spin-density-wave (SDW) order. Within a semiclassical adiabatic approximation that retains electron degrees of freedom, we demonstrate that the SDW order parameter obeys a generalized Landau-Lifshitz equation. With the aid of an enhanced kernel polynomial method, our linear-scaling quantum Landau-Lifshitz dynamics (QLLD) method enables dynamical SDW simulations with N ≃ 10^5 lattice sites. Our real-space formulation can be used to compute dynamical responses, such as dynamical structure factor, of complex and even inhomogeneousSDW configurations at zero or finite temperatures. Applying the QLLD to study the relaxation of a noncoplanar topological SDW under the excitation of a short pulse, we further demonstrate the crucial role of spatial correlations and fluctuations in the SDW dynamics. Semiclassicaldynamics of spin density waves Cristian D. Batista December 30, 2023 =============================================Quantum states with unusual broken symmetries have long fascinated physicists because of their fundamental importance and potential technological applications. Of particular interest is the regular spatial modulation of electron spin known as spin-density wave (SDW) state <cit.>. SDWs are ubiquitous in strongly correlated systems and play a crucial role in several intriguing many-body phenomena. For example, the SDW state is proximate to the superconducting phase in several unconventional superconductors, including cuprates and iron pnictides. Indeed, non-Fermi liquid behavior is usually observed in the vicinity of a SDW phase transition <cit.>. Moreover, conduction electrons propagating in a noncoplanar spin texture acquire a nontrivial Berry phase and exhibit unusual transport and topological properties <cit.>.Consequently, metallic SDW with complex spin structures, such as spirals or skyrmion crystals, offers a novel route to control the charge degrees of freedom through manipulation of spins and vice versa <cit.>.While analytical techniques have yielded much insight about itinerant magnetism and SDW states <cit.>, numerical methods continue to provide valuable benchmarks and shed light on controversial issues. Among the various numerical tools <cit.>, quantum Monte Carlo (QMC) simulations provide numerically exact solutions to strongly correlated models <cit.>. However, one severe restriction of most QMC methods is the infamous sign-problem. Powerful alternativeapproaches that avoid the sign-problem include dynamical mean-field theory (DMFT) <cit.> and density-matrix renormalization group (DMRG) <cit.>.Significant developments have also been made in their nonequilibrium extension such as time-dependent (TD) DMFT <cit.> and TD-DMRG <cit.>. Both methods, however, are still very limited in their treatment of complex mesoscopic structures. In this paper, we present a different numerical approach to SDW dynamics, emphasizing the ability to simulate large-scale lattices and complex SDW orders that often occur in highly frustrated systems. Our starting point is a semi-classical treatment of Hubbard-like models, whichneglects quantum fluctuations, but retains the spatial fluctuations of the SDW field. In a way, this approach is the complement of DMFT, which includes quantum fluctuations at the expense of neglecting spatial correlations. A systematic approach is then developed to reintroduce quantum dynamics to the SDW order parameter. We show that in the leading adiabatic approximation, the SDW dynamics is described by a generalized Landau-Lifshitz (LL) equation in which the effective forces acting on the spins are generated from itinerant electrons. Our numerical scheme can be viewed as a quantum LL dynamics (QLLD), in which the electronic degrees of freedom are integrated out at each time step. By supplementing the LL equation with Ginzburg-Landau type relaxation and stochastic terms,our QLLD method can be used to simulate SDW dynamics both near and far-from equilibrium.§ SPIN-FERMION HAMILTONIAN FOR EQUILIBRIUM SDW PHASESWe consider the one-band Hubbard model with an on-site repulsion U > 0,ℋ = - ∑_ij, α t_ij c^†_i, α c^ _j, α + U ∑_i n_i,↑ n_i ↓.After performing the Hubbard-Stratonovich (HS) transformation <cit.>, we obtain the following spin-fermion Hamiltonianℋ_ SDW = - ∑_ij, α t_ij c^†_i, α c^ _j, α - 2 U ∑_i 𝐦_i ·𝐬_i + U ∑_i |𝐦_i|^2,where 𝐬_i = 1/2 c^†_i,ασ_αβ c^ _i, β is the spin operator of conduction electrons and σ is a vector of the Pauli matrices. The local HS or auxiliary field 𝐦_i is a classical O(3) vector in ℝ^3.Here we have set ħ = 1.Importantly, since ℋ_ SDW describes non-interacting electrons coupled to a magnetic background, the fermionic degrees of freedom can be integrated out either in Monte Carlo or dynamical simulations to be described below.This HS Hamiltonian is typically the starting point for determinant QMC (DQMC) simulations <cit.>. In an alternative approach, one assumes static HS variables; then the above Hamiltonian resembles the so-called spin-fermion model <cit.> and can be simulated with Markov chain Monte Carlo assuming classical “spins” 𝐦_i, while electrons are treated quantum mechanically. Applying this method to the cubic-lattice Hubbard model, the obtained Néel temperature agrees remarkably well with those from DQMC simulations <cit.>. It is worth noting that while Eq. (<ref>) is similar to the Hartree-Fock treatment of the Hubbard model, retaining spatial fluctuations of the local HS fields {𝐦_i} in this static (in imaginary time) HS-field formalism goes beyond the usual mean-field method. For instance, this approach captures the critical fluctuations, and consequently the correct universality class, of any continuous thermodynamic transition into a magnetically ordered state. Instead of Markov-Chain Monte Carlo, here we employ the stochastic Ginzburg-Landau (GL) relaxation dynamics <cit.> to sample the equilibrium SDW configurations within the static HS-field approximation,d𝐦_i/dt = - γ∂⟨ℋ_ SDW⟩/∂𝐦_i + ξ_i(t).Here γ is a damping constant, and ξ_i is a δ-correlated fluctuating force satisfying ⟨ξ_i(t) ⟩ = 0and ⟨ξ_i^μ(t) ξ_j^ν(t') ⟩ = 2 γk_B T δ_ijδ_μνδ(t - t') for vector components μ and ν. This equation is similar to the over-damped Langevin dynamics used in Ref. <cit.> for the Kondo-lattice model. We note that the magnitude |𝐦_i| of the O(3) vector 𝐦_i is not fixed in this over-damped dynamics. A fictitious inertial mass term can be added to the above dynamicsto improve the efficiency of the simulation. Unlike conventional GLsimulations, where the force is given by the derivative of a phenomenological energy functional, here the force is computed by solving the equilibrium electron liquid of ℋ_ SDW at each time-step. Or equivalently, the effective energy functional ℰ_ eff({𝐦_i}) = ⟨ℋ_ SDW⟩ is obtained by integrating out electrons on the fly. Our method is thus similar to the so-called quantum molecular dynamics (MD) simulations, in which the inter-atomic force is computed by solving the quantum electron Hamiltonian <cit.>, instead of being derived from a phenomenological classical potential. Drawing on this analogy with MD simulations, our approach can be viewed as a quantum GL method for SDW. Interestingly, the quantum MD method in conjunction with the functional integral theory has already been employed to obtain complex magnetic orderings in itinerant magnet compounds in the past <cit.>. The GL method is particularly powerful when combined with our recently developed kernel polynomial method (KPM) and its gradient transformation in which forces acting on all spins can be efficiently computed <cit.>; see Ref. <cit.> for more details. The resulting linear-scaling KPM-GL method allows us to simulate large lattices with N ≃ 10^5 to 10^6 sites. We apply the KPM-GL simulations to the triangular-lattice Hubbard model as a benchmark. The phase diagram shown in Fig. <ref> agrees very well with those obtained by holon-doublon mean-field <cit.> and the rotational-invariant slave-boson (SB) <cit.> calculations. Note that the SB method <cit.> describes both large and small U regimes, and shows quantitative agreement with QMCover a wide range of interaction and doping for the square-lattice Hubbard model <cit.>.At large U, our simulation finds the expected 120^∘-order, which is the ground state of the Heisenberg Hamiltonian arising from the strong-coupling limit of the half-filled Hubbard model. In fact, the semi-classical Hamiltonian Eq. (<ref>) reduces to the classical Heisenberg spin model in the U/t ≫ 1 limit. To see this, we note that any ground state for t_ij=0 contains exactly one electron in each site with its spin locally aligned with the SDW field 𝐦_i. The magnitude of the SDW field freezes at |𝐦_i|=1/2 in this U →∞ limit because amplitude fluctuations have an energy cost proportional to U.In analogy with the large U limit of the original Hubbard model,the ground state manifold is massively degenerate because each𝐦_i can point in an arbitrary direction.The degeneracy is removed to second order in t_ij. The low-energy effective Hamiltonian is obtained by considering virtualelectron hopping processes between two neighboring sites i and j.The Pauli exclusion principle dictates that an electron at site-i can hop to the neighboring site only when its spin is anti-aligned with that of the local moment at site j. Consequently, the effective hopping constant between the two neighboring sites is t^ eff_ij = t_ij⟨χ_i | χ_j ⟩ = t_ijsin(θ_ij/2), where |χ_i⟩ are local electron spinor eigenstate and θ_ij is the angle between the two local moments. At second-order, the energy gain through the virtual electron hopping produces the effective interactionℋ_ij = -2 t_ij^2 sin^2(θ_ij/2)/U = 4 t_ij^2/U(𝐦_i ·𝐦_j - 1/4).This result corresponds to the classical limit of the S=1/2 Heisenberg model, implying thatthe semi-classical SDW Hamiltonian (<ref>) correctly captures the classical limit of the half-filled Hubbard model in the strong-coupling regime <cit.>.As U is decreased, our simulation shows that the 120^∘ order is replaced by an interesting commensurate collinear SDW as the ground state <cit.>. The collinear SDW at this intermediate U is still gapped electronically and exhibits a zigzag structure. By computing the electron density of states (DOS), we find a metal-insulator transition at U_c2 between an incommensurate spiral and a commensurate collinear SDW phase (see Fig. <ref>). Finally, the metallic spiral SDW undergoes a continuous transition at into a paramagnetic state at U_c1. We note in passing that the metal-insulator transition, obtained with other numerical techniques (e.g. path-integral renormalization group method <cit.>), is entirely within the paramagnetic regime <cit.>, implying the existence of a paramagnetic insulator (or spin liquid) at intermediate U values. This statecannot be obtained with our semi-classical approach because it is stabilized by strong fluctuations of the HS fields along the imaginary time axis. However, the existence of this phase remains to be settled. Recent variational QMC <cit.> and DMFT <cit.> calculations show that spiral SDW is more favorable than the spin liquid phase. Nonetheless, if a magnetic phase is stabilized by, e.g. applying a magnetic field, we expect the incommensurate and collinear SDWs obtained here to be strong candidates. § SEMICLASSICAL DYNAMICS AND LANDAU-LIFSHITZ EQUATIONHaving demonstrated that the semiclassical Hamiltonian ℋ_ SDW provides a viable approach to equilibrium SDW phases, a natural question is whether we can use it to study theSDW dynamics. To this end, we need to reintroduce physical dynamics to the “static” auxiliary SDW field. We first note that the spin-fermion Hamiltonian ℋ_ SDW can also be obtained from a Hartree-Fock decoupling of the interaction term that varies from one site to another. The important difference relative to the HS approach is that the SDW field satisfies the self-consistent condition 𝐦_i = ⟨ s_i ⟩ =Tr(ρ 𝐬_i), where ρ is the density matrix characterizing the physical electron state. Consequently, the field 𝐦_i belongs to the sphere of radius 1/2(0 ≤ |𝐦_i| ≤ 1/2). To derive the time dependence of the SDW field, we start with the continuity equation associated with the total spin conservation: d 𝐬_i /dt = - ∑_j𝐉_ij, where 𝐬_i = 1/2 c^†_iασ_αβ c^ _iβ is the electron spin, and 𝐉_ij = -it_ij/2σ_αβ (c^†_iα c^ _jβ - c^†_jαc^ _iβ) is the spin current density on bond ⟨ ij ⟩. We then introduce the single-particle density matrix ρ with elements ρ_iα, jβ≡⟨ c^†_jβ c^ _iα⟩. Taking the average of the continuity equation leads to d𝐦_i/dt = -i/2∑_j t_ijσ^ _βα( ρ_iα, jβ - ρ_jα, iβ).The SDW dynamics is thus related to the time evolution of the density matrix ρ, which obeys the von Neumann equation dρ/dt = i [ρ, H^ eff]. Up to a constant, H^ eff is the effective single-electron Hamiltonian defined as ℋ_ SDW = ∑_iα, jβ H^ eff_iα, jβ c^†_iα c^ _j,β. Using Eq. (<ref>), we obtain the equation of motiondρ_iα, jβ/dt= i (t_ik ρ_kα, jβ - ρ_iα, kβt_kj) + i U(𝐦_i ·σ_αγ ρ_iγ, jβ - ρ_iα, jγ σ_γβ·𝐦_j ).The electron density matrix is partly driven by the time-varying SDW field. Eqs. (<ref>) and (<ref>) comprisea complete set of coupled ordinary differential equations for the SDW dynamics. An alternative is to dispense of Eq. (<ref>) and substitute 𝐦_i(t) =ρ_iα, iβ(t) σ_βα in Eq. (<ref>), giving rise to a set of nonlinear differential equations for ρ. In general, time dependence of physical quantities in mean-field approaches can be obtained using the Dirac-Frenkel variational principle <cit.>. Our derivation here, on the other hand, is based on the spin-density continuity equation, hence emphasizing the importance of conservation laws in physical dynamics. This physically intuitive approach can be easily generalized to obtain dynamics for other symmetry-breaking phases.Our approach here is a real-space formulation of the time-dependent Hartree-Fock (TDHF) method <cit.>, similar to the familiar time-dependent Bogoliubov-de Gennes equation for superconductors or Bose condensates <cit.>. Assuming that the order parameters are characterized by well defined momenta, e.g. 𝐦_i = ∑_r 𝐌_rexp(i 𝐐_r ·𝐫_i), the above equations can be simplified due to the translation invariance. The problem is then reduced to a set of coupled differential equations for density-matrix elements n_αβ(𝐤, t) ≡⟨ c^†_𝐤α c^ _𝐤β⟩ and g^r_αβ(𝐤, t) ≡ (𝐌^ _r ·σ_αβ) ⟨ c^†_𝐤α c^ _𝐤+𝐐_r, β ⟩ in momentum space. This mean-field approximation of TDHF has recently been applied to the out-of-equilibrium dynamics of BCS-type superconductors <cit.>, and of Néel-type SDW <cit.>. Since the order-parameter field 𝐌_r is assumed to be uniform, spatial inhomogeneity and/or fluctuations are ignored in such k-space approaches. Our formulation here does not require the prerequisite knowledge of ordering patterns, and are particularly capable of simulating complex symmetry-breaking phases, inhomogeneous configurations, and disordered phases withpreformed local moments, such as the paramagnetic state in the large U limit. The SDW dynamics Eq. (<ref>) can be simplified in the large U limit in the so-called adiabatic approximation, which assumes that electrons quickly relax to the ground state of the instantaneous SDW configuration {𝐦_i}. Using second-order perturbation theory, one readily computes the average spin current: ⟨𝐉_ij⟩ = 4 t^2_ij/U 𝐦_i ×𝐦_j. Substituting this into the right-hand side of Eq. (<ref>) gives rise to a Landau-Lifshitz (LL) equation d𝐦_i/dt = -∑_j 4 t_ij^2/U 𝐦_i ×𝐦_j,with an effective torque computed using the Heisenberg exchange of Eq. (<ref>). More details can be found in Ref. <cit.>.For intermediate and small U/t values, one needs to solve the von Neumann equation. Since the number of independent density-matrix elements is of order 𝒪(N^2) for a lattice of N spins, the computational cost of integrating the von Neumann equation is tremendous for large lattices, e.g. N∼ 10^5. To further simplify the numerical calculation, here we derive the SDW dynamics in a similar adiabatic limit for arbitrary U. Formally, we employ the multiple-time-scale method <cit.> and introduce an adiabaticity parameter ϵ∼ |d𝐦/dt| such that the fast (electronic) and slow (SDW) times are τ = ϵ t and t, respectively. The single-particle Hamiltonian varies with the slow time, i.e. H^ eff({𝐦_i }) = H^ eff(τ). Expanding the density matrix in terms of the adiabaticity parameter: ρ(t) = ρ^(0)(τ) + ϵρ^(1)(t, τ) + ϵ^2 ρ^(2)(t, τ) + ⋯, and plugging it into the von Neumann equation, we obtain [ρ^(0), H^ eff] = 0 and dρ^(ℓ)/dt - i [ρ^(ℓ), H^ eff] = -dρ^(ℓ-1)/dτ for ℓ≥ 1. This provides a systematic approach to obtain the time dependence of the density matrix. Here we use the leading adiabatic solution ρ^(0) to compute the expectation value of the spin-current density 𝐉_ij, which is the right-hand side of Eq. (<ref>). We first write H^ eff = T + Σ where T_iα, j β = - t_ijδ_αβ is the tight-binding Hamiltonian and Σ_iα, j β = -U δ_ij 𝐦_i ·σ_αβ is the spin-fermion coupling. It is then straightforward to show that Eq. (<ref>) is simply d 𝐦_i/dt = i σ_αβ [ρ, T]_i β, iα/2. Using the adiabatic equation [ρ^(0), H^ eff] = 0, we have [ρ^(0), T] = -[ρ^(0), Σ], which givesd𝐦_i/dt = -i U /4σ_αβ[ (σ^ _βγρ^(0)_iγ, iα - ρ^(0)_iβ,iγσ^ _γα) ·𝐦_i ].The right-hand side of the above equation can be further simplified using the properties of Pauli matrix multiplication: σ^a σ^b = δ_abℐ_2× 2 + i ϵ_abcσ^c, where a, b, c are x, y, z. For example,σ_αβ(σ_βγ·𝐦_i) ρ^(0)_iγ,iα = n^(0)_i 𝐦_i + i 𝐦_i ×σ^ _αβρ^(0)_iβ, iα, where n^(0)_i = ρ^(0)_iα, iα is the local electron density. After some algebra, we obtaind𝐦_i/dt = U/2𝐦_i ×σ^ _βα ρ^(0)_iα, iβ = - 𝐦_i ×∂⟨ℋ_ SDW⟩/∂𝐦_i .The second equality comes from the fact that ρ^(0) is computed from the equilibrium electron liquid described by ℋ_ SDW. The localelectron spin ⟨𝐬_i ⟩ = 1/2⟨ c^†_iασ^ _αβ c^ _iβ⟩ = 1/2σ^ _βα ρ^(0)_iα, iβ acts as an effective magnetic field and drives the slow dynamics of the SDW field. Importantly, this equation shows that the adiabatic SDW dynamics is described by the Landau-Lifshitz (LL) equation <cit.> with an effective energy functional ℰ_ eff({𝐦_i }) = ⟨ℋ_ SDW⟩, obtained from the equilibrium electronic state of the instantaneous spin-fermion Hamiltonian. §.§ Benchmark with exact diagonalization We first benchmark our semiclassical SDW dynamics, with and without the adiabatic approximation, against the exact diagonalization (ED) calculation of the original Hubbard model. To this end, we apply our formulation to the two-sublattice collinear Néel state that is obtained for the half-filled Hubbard model on a square lattice. Since we only include NN hopping, the Néel ordering is stable for any positive value of U/|t|.Specifically, as shown in Fig. <ref>, we compute the dynamical structure factor 𝒮(𝐤, ω) of a 4 × 4 Hubbard cluster with periodic boundaries forU/t=7.33. Details of the ED calculation are described in Ref. <cit.>. We compare the ED result at T=0 and the semiclassical SDW dynamics at an extremely low temperature(classical moments freeze at T=0 <cit.>). Weset temperature at T=10^-4t and verify thatthe results do not change upon decreasing the temperature to T=10^-5, indicating that our resultscapture the dynamical response of the classical moments in the T → 0 limit.For both the real-space TDHF dynamics [Eqs. (<ref>) and (<ref>)] and the adiabatic dynamics [Eq. (<ref>)], SDW states are first generated by means of GL-Langevin simulations described in Sec. <ref>. The obtained spin configurations, which are representative of the canonical ensemble, are used as the initial condition for dynamical simulations. The dynamical structure factor, 𝒮(𝐤, ω), is calculated by applying the space-time Fourier transform to the time evolution of the auxiliary field 𝐦_i(t). In the SDW dynamics,the elastic peak has a finite width for finite duration of the dynamical simulation (not shown in Fig. <ref>(b) and (c)). The area under the elastic peak is proportional to N ⟨𝐦_i ⟩^2, where N is the number of sites. In contrast, the lowest energy peak of the exact result appears at a small but finite frequency arising from quantum fluctuations neglected by the semiclassical treatment (the exact ground state is a singlet state for a finite size system). This quasi-elastic peak becomes the elastic peak of the spontaneously broken symmetry state in the thermodynamic limit. In Fig. <ref>, we normalize the spectral weights obtained from the SDW dynamics so that the total weight of inelastic peaks equals that obtained from the ED excluding the quasi-elastic peak. The low-energy spectrum of the original Hubbard model is not well described by a simple effective spin Hamiltonian for U/|t|=7.33 (charge fluctuations can strongly renormalize the spin-wave dispersion). It is then quite remarkable that all the approaches produce a rather flat magnon dispersion for the wave-vectors included in a 4× 4 square lattice. However, the excitation energies in the adiabatic approximation are roughly 25% lower than the exact result [see Fig. <ref>]. This discrepancy is attributed to two factors: the semi-classical treatment of the spin degrees of freedom and the adiabatic approximation. The excitation energies obtained from the real-space TDHF method areapproximately 15% lower than the exact result. We thus conclude that the adiabatic approximation accounts for roughly 10% of the discrepancy, while the semiclassicaltreatment accounts for the remaining 15%. In addition, the normalized spectral weights (areas) in both the semiclassical dynamics are different only approximately 30% from the exact result. A much better quantitative agreement is expected for 3D systems, but their solutions are beyond the scope of state of the art ED methods.The real-space TDHF method captures not only the transverse modes, but also the longitudinal mode arising from charge fluctuations <cit.>. We note however that the method does not capture the longitudinalspin fluctuations associated with quantum fluctuations of the magnetic moments. Unlike charge fluctuations, quantum (magnetic) fluctuations persist for arbitrarily large-U/t (they arise from fluctuations of the m-field along the imaginary time axis). These longitudinal fluctuationscorrespond totwo-magnon excitations in a 1/S expansion <cit.>. Correspondingly, they have an energy of order J ∝ t^2/U for large U/t. In contrast, the longitudinal spin fluctuations arising from charge fluctuations leadto the high-energy peaks at ω∼ U, which are observed in the real-space TDHF dynamics, as shown in the inset of Fig. <ref>(c), while they are absent in the adiabatic dynamics. Nevertheless, as expected for this value of U/t, the longitudinal mode is well separated from the transverse modes. §.§ 120^∘ SDW order in Hubbard and Anderson-Hubbard model Our benchmark study shows that both the TDHF and the adiabatic approach provide a reasonable description of the SDW dynamics. It is worth noting that the linearized TDHF equation of motion corresponds to the random phase approximation (RPA) <cit.>. Our real-space formulation of the TDHF thus provides an efficient and universal numerical approach to describe nonlinear dynamicsbeyond the RPA level. Moreover, our approach allows for computation of dynamical response functionsat any finite temperature, including the high-temperature regime in which the magnetic moments only exhibit short range correlations.Another unique feature offered by our real-space method is the capability of computing the dynamical response of inhomogeneous SDW. To demonstrate this, here we apply the above adiabatic dynamics Eq. (<ref>) to compute 𝒮(𝐤, ω) forthe 120^∘ SDWdepicted in Fig. <ref>(c) for Hubbard model with quenched disorder.Specifically, we consider the Anderson-Hubbard model by adding an on-site potential disorder ∑_i,α V^ _ic^†_iα c^ _iα to Eq. (<ref>). This model has served as a canonical platform for investigating the intriguing interplay of localization and correlations. Relevant to our study here is the effect of disorder on long-range SDW order. For Néel-type SDW on a half-filled bipartite lattice, it has been shown that increasing the disorder first closes the electron spectral gap, while the SDW remains finite <cit.>. The disappearance of the SDW order parameter occurs at a larger disorder <cit.>. This result is relevant to the non-equilibrium dynamics to be discussed below. We first compute the dynamical structure factor for the pure Hubbard model on a large lattice with N = 120 × 120 sites. The results shown in Fig. <ref>(a) resemble the linear spin wave dispersion of the Heisenbergmodel <cit.>, except for a significantly renormalized lower-energy branch.Next we include a Gaussian disorder with zero mean andstandard deviation σ_V = 0.45t_ nn, which is relatively large yet not strong enough to destroy the SDW order <cit.>. The 𝒮(𝐤, ω) computed using the adiabatic SDW dynamics is shown in Fig. <ref>(b). While the overall dispersion is similar to that of the SDW in the disorder-free Hubbard model [Fig. <ref>(a)], there are a few notable new features. Firstly, the magnon dispersion is significantly broadened by the quenched disorder, and the middle of the low-energy branch is further renormalized. Interestingly, a rather sharp dispersion remains near the zone center, indicating that these long-wavelength modes are less sensitive to disorder. Secondly, several new modes appear at low energies, especially below the original gap at the M point. Interestingly, similar disorder-induced low-energy modes are also obtained in bi-layer Heisenberg antiferromagnet using the bond-operator method <cit.>.A systematic study of the SDW dynamics with disorder will be left for future studies. § NONEQUILIBRIUM DYNAMICS AT FINITE TEMPERATURESThe adiabatic LL equation can also be used to study nonequilibrium SDW phenomena as long as the electron relaxation is much faster than the SDW dynamics. Here we first generalize the adiabatic dynamics to finite temperatures by adding dissipation and fluctuations to Eq. (<ref>). It is worth noting that the adiabatic SDW dynamics preserves the length of local moments |𝐦_i|. Longitudinal spin relaxation and fluctuations thus come from either higher order terms in the adiabatic expansion or other processes beyond the self-consistent field approach. The standard Gilbert damping also preserves the spin length <cit.>. Instead, here we combine the Ginzburg-Landau relaxation discussed in Eq. (<ref>) with the adiabatic dynamicsof Eq. (<ref>) to account for the longitudinal relaxation <cit.>. This procedure gives rise to the following generalized LL equationd𝐦_i/dt = - 𝐦_i ×∂⟨ℋ_ SDW⟩/∂𝐦_i - γ∂⟨ℋ_ SDW⟩/∂𝐦_i + ξ_i(t). ℋ_ SDW is the spin-fermion Hamiltonian defined in Eq. (<ref>), γ is a damping constant, and ξ_i is a δ-correlated fluctuating force satisfying ⟨ξ_i(t) ⟩ = 0 and ⟨ξ^μ_i(t) ξ^ν_j(t') ⟩ = 2 γk_B T δ_ijδ_μνδ(t - t'). The damping coefficient and the stochastic terms are chosen such that the dissipation-fluctuation theorem is satisfied and the above LL equation can be used to faithfully sample the equilibrium Boltzmann distribution at finite temperatures <cit.>. A microscopic calculation of the damping coefficient γ is beyond the adiabatic approximation. In the real-space TDHF method, relaxation of SDW mainly arises from the Landau damping mechanism, which describes the energy transfer from the collective SDW mode to single-particle excitations <cit.>. Electron-electron scattering, which is not captured by the TDHF, also contributes to the damping of SDW, especially in ultrafast dynamics of metals <cit.>. Moreover, for open systems as in most pump-probe experiments, coupling of electrons to other degrees of freedom <cit.>, such as phonons, also play an important role in the relaxation of SDW dynamics. Here γ is treated as as a phenomenological parameter which we chose to ensure the adiabatic approximation.That the SDW field obeys the LL dynamics can be understood intuitively from the fact that the Heisenberg equation of motion for spin operators corresponds to the classical LL equation <cit.>.Here we give a microscopic derivationstarting from the Hubbard model, which reveals the condition for the validity of the LL dynamics (adiabatic approximation of the von Neumann equation). In fact,the adiabatic approximation has been widely employed for spin dynamics in the context of time-dependent spin-density-functional theory <cit.>. Our results thus provide a theoretical foundation for the LL dynamics of SDW, and pave the way for systematic improvements beyond the adiabatic approximation. It is worth noting that, in contrast to the conventional LL method, the energy functional in our approach is obtained by solving the spin-fermion Hamiltonian ℋ_ SDW at each time-step. In analogy with the quantum MD simulations <cit.>, our numerical scheme can then be viewed as a quantum LL dynamics (QLLD) method. Although solving the electron Hamiltonian on the fly is computationally expensive, large-scale (N ∼ 10^5) QLLD simulations are enabled by our recently developed KPM algorithm with automatic differentiation, such that the “forces” can be computed along with the total energy without extra overhead <cit.>.We next apply our stochastic QLLD method to investigate the time evolution of a topological SDW on the triangular lattice that arises as a weak-coupling instability at filling fraction n = 3/4 <cit.>. The combination of a van Hove singularity and perfect Fermi surface nesting at this filling fraction gives rise to a magnetic susceptibility that diverges as χ(𝐪) ∝log^2| 𝐪 - 𝐐_η|, where 𝐐_η (η = 1,2,3) are the three nesting wavevectors <cit.>. The system thus tends to develop a triple-𝐐 SDW characterized by three vector order parameters: 𝐦_i = Δ_1 e^i𝐐_1 ·𝐫_i + Δ_2 e^i𝐐_2 ·𝐫_i + Δ_3 e^i𝐐_3 ·𝐫_i. Note that the phase factors e^i 𝐐_η·𝐫_i = ± 1. In general, there are four distinct local moments, leading to a quadrupled magnetic unit cell. The SDW instability of triangular-lattice Hubbard model at n=3/4 filling is similar to that of the half-filled Hubbard model on square lattice. However, unlike the simple Néel order in the later case, there are several possible triple-𝐐 SDWs <cit.>. At the lowest temperatures, the magnetic ordering consists of a non-coplanar SDW with |Δ_1|=|Δ_2|=|Δ_3| and Δ_1 ⊥Δ_2 ⊥Δ_3 <cit.>. This SDW is also called a tetrahedral or all-out order as spins in the unit cell point to the four corners of a regular tetrahedron <cit.>; see Fig. <ref>(a). Moreover, as the spins on each triangular plaquette are non-coplanar, the resulting nonzero scalar spin chirality 𝐦_i ·𝐦_j ×𝐦_k = ± 4Δ^3 also breaks the parity symmetry. Consequently, the tetrahedral SDW is also characterized by a discrete Z_2 chirality order parameter. More importantly, electrons propagating in this non-coplanar SDW acquire a nonzero Berry phase, which is equivalent to a uniform magnetic field. Since the Fermi surface is gapped out by the SDW, the resulting electron state exhibits a spontaneous quantum Hall effect with transverse conductance σ_xy = ± e^2/h <cit.>.Motivated bya recent pump-probe experiment on the ultrafast SDW dynamics in chromium <cit.>, we perform simulations of this topological SDW subject to a short heat pulse. For simplicity, we assume that the effect of the pump pulse is to inject energy to the electron system, which quickly equilibrates to a state characterized by temperature T_e. This is consistent with our adiabatic approximation for the SDW dynamics. The time dependence of the effective electron temperature is governed by the rate equation C dT_e/dt = -G (T_e - T_L) + Q(t) <cit.>, where C is the heat-capacity of the electron liquid, G is the coupling to the lattice, T_L is the lattice temperature, and Q(t) ∝exp[-(t - t_p)^2 / w^2] is the heat source due to the pump pulse. We further assume that T_L ≈ 0 throughout the relaxation process. The resultant T_e(t) curve is shown in Fig. <ref>(b).T_e(t)is then used for the stochastic noise ξ(t) in our QLLD simulations of Eq. (<ref>). We use the parameters, U = 3, damping γ = 0.1, G/C = 0.02, t_p = 15, and w = 5, in units of the NN hopping t_ nn. The lattice size is N=120^2. Fig. <ref>(c) shows the evolution of the magnetic order parameter at the nesting wavevectors: ℳ = √(|Δ_1|^2 + |Δ_2|^2 + |Δ_3|^2) normalized to its maximum. We also estimate the time dependence of the electron spectral gap ε_ gap from the instantaneous DOS [see Fig. <ref>(c)]. Interestingly, as the temperature rises, the decline of ℳ is rather slow compared with the closing of the energy gap. In fact, the SDW order parameters Δ_η remain finite throughout the process, while the gap closes quickly after the photoexcitation (at t ≈ 12). In equilibrium the SDW order parameters disappear along with the gap above the transition temperature <cit.>, implying that thephotoexcited SDW is in a highly non-equilibrium transient state. As the system relaxes, the gap reopens at a later time [see Fig. <ref>(c)]. This picture is further supported by our calculation of instantaneous longitudinal and transverseconductivitiesshown in Fig. <ref>(d). Here we use KPM to compute the Kubo-Bastin formula forthe conductivities <cit.>. The error barsare estimated from five independent simulations. The electrons exhibit a negligible longitudinal conductivity σ_xx≈ 0 and a quantized Hall conductivity σ_xy = e^2/h in the gapped regimes, as expected for this topological SDW. On the other hand, the longitudinal conductivity increases significantly during the period of vanishing gap, while the transverse conductance decreasesand exhibits small oscillations in the vicinity of the gap-closing transitions.The closing and subsequent re-opening of the SDW gaphave been reported in recent pump-probe experiment on chromium <cit.>. The ultrafast SDW dynamics seem to be well described by a model that assumesa thermalized electron gas. However, the closing of the gap is assumed to be always accompanied by the disappearance of the SDW order parameter in Ref. nicholson16, which is not necessarily the case.As demonstrated in our simulations, an out-of-equilibrium electron state might be gapless while the spin density remains modulated. Indeed, similar pump-probe experiments on the charge density wave (CDW) have revealed a fast collapsing of electronic gap in the time-resolved photoemission spectroscopy <cit.>, and a reduced, yet finite, modulation of charge density inferred from core-level X-ray photoemission <cit.> during the nonequilibrium melting process. Numerical simulations taking into account coupling to the lattice distortion showed that the CDW order parameter can indeed be partially decoupled from the spectral gap dynamics <cit.>. However, it should be noted that the lattice degrees of freedomintroduce a new time scale, in addition to that of the hot electron relaxation. The transient metallic SDW observed in our simulations is probably due to a different mechanism. To understand the origin of this nonequilibrium metallic SDW, we first note that the electronic gap of this topological SDW arises from the scalar spin chirality <cit.>. Indeed, the electronic gap vanishes for collinear or coplanar triple-𝐐 SDWs <cit.>. This observation leads us to investigate the temporal and spatial fluctuations of the scalar chirality. To this end, we introduce the normalized scalar spin chirality: χ_ =χ_ijk = (𝐦_i·𝐦_j×𝐦_k) /| 𝐦 |^3 for individual triangular plaquettes (here the overline indicates average over all triangles). The time dependence of the (spatial) average and the standard deviation of the scalar chirality χ_ are shown in Fig. <ref>(c). Interestingly, the average chirality remains finite and of the same sign, indicating that the chiral symmetry is still broken in this transient SDW. On the other hand, as shown in Fig. <ref>(c), the standard deviation σ_χ_ increases significantly with T.In fact, the transient gapless regime coincides roughly with the period when σ_χ_ > χ_, implying that the vanishing gap is due to thermally induced spatial fluctuations of χ_ijk.Our scenario is confirmed by the spatial distribution of the normalized plaquette chirality at the initial stage (t = 5) and the gapless regime (t= 40), shown inFig. <ref>(a) and (b), respectively. While the chirality is relatively uniform initially (χ_∼ 1), noticeable inhomogeneity develops at later times [see Fig. <ref>(b)]. Histograms of the plaquette chirality h( χ_) and the corresponding electron DOS at various simulation times are shown in Fig. <ref>(c) and (d), respectively. The chirality distribution becomes asymmetric and very broad during the period of vanishing gap. This transient gapless SDW is similar to the disorder-induced metallic antiferromagnetic state observed in the Anderson-Hubbard model <cit.>. As discussed above, the plaquette scalar chirality acts as local magnetic field and it is known that strongly disordered magnetic flux destroys the quantum Hall effect, in agreement with our simulations. Our results thus underscore the importance of thermal fluctuations and spatial inhomogeneity for the nonequilibrium dynamics of SDW, which have been overlooked in most dynamical studies of correlated systems. § SUMMARY AND OUTLOOK We have developed a new theoretical framework for the semiclassical dynamics of SDW in Hubbard-like models. Based on a real-space time-dependent Hartree-Fock (TDHF) method applied to symmetry-breaking phases, our approach provides a Hamiltonian formulation for the SDW dynamics. The time evolution of the SDW field is coupled to the von Neumann equation that describes the dynamics of single-electron density matrix. The formulation correctly reduces to the Holstein-Primarkoff dynamics of magnons (linear spin waves) in the large-U limit at half-filling. We further show that an adiabatic approximation of the von Neumann equation gives rise to a quantum Landau-Lifshitz dynamics (QLLD) for the SDW order parameter. Importantly, the energy functional of the LL equation is computed from an effective spin-fermion Hamiltonian that is obtained from a Hubbard-Stratonovich transformation of the original Hubbard model.Our benchmark study of the Néel order on a half-filled Hubbard cluster showed that the semiclassical SDW dynamics agrees reasonably well with the exact diagonalization calculation. We apply our QLLD simulations to compute the dynamical structure factor of a 120^∘ SDW at intermediate values of U/t on the triangular lattice. While the overall spectrum resembles that obtained using linear spin-wave theory for the large U/t limit of the Hubbard model (S=1/2 Heisenberg model),charge fluctuations produce a significant renormalization of the low-energy branch. We note that quantum fluctuations, not included in our approach,can also produce a significant renormalization of the spin-wave spectrum of frustrated 2D models <cit.>, like the one considered here. However, renormalization due to quantum fluctuations is much smaller in 3D models, whose dynamical structure factor is typically well described by semiclassical approaches. Importantly, our real-space approach allows us to include the effects of spatial inhomogeneities of the SDW on large lattices. We have demonstrated this unique capability by computing the dynamical structure factor of the same 120^∘ SDW on an Anderson-Hubbard model with disordered on-site potentials. Other than significant broadening of the magnon dispersion, our result shows that the disorder induces many low-energy modes, especially at the boundary of the Brillouin zone. Another important application of our QLLD method is the study of SDW-related non-equilibrium phenomena. Here we generalize the LL dynamics by including a Langevin-type damping and the corresponding stochastic noise to account for longitudinal relaxation and fluctuation. We then apply the generalized QLLD scheme to study the evolution of a topological SDW subject to a heat pulse, similar to the situation in the pump-probe setup. Our simulation shows an intriguing transient non-equilibrium SDW on the triangular lattice. While the SDW order parameter decreases with rising electron temperature, it remains finite even when the electronic gap is closed. The gap reopens at a later time as the system relaxes. Since the electronic gap in this topological SDW originates from the noncoplanar spin configuration, we show that the vanishing gap is due to strong spatial fluctuations of the scalar spin chirality, a quantity measuring the non-coplanarity of plaquette spins. Our real-space QLLD simulations thus underscores again the importance of spatial inhomogeneity and thermal fluctuation of the SDW dynamics.The theoretical framework and numerical method developed in this paper can be easily generalized to study the dynamics of other symmetry-breaking phases, notably charge-density wave and superconductivity. Compared with other phenomenological method (e.g. time-dependent Ginzburg-Landau simulation for superconductors), keeping the electron degrees of freedom allows us to also look into the instantaneous electronic structure during the evolution of the order-parameter field. Our efficient semiclassical approach can also be feasibly integrated with first-principles method such as density functional theory (DFT). Here we note the analogy with molecular dynamics (MD) simulations. While classical MD simulations use phenomenological inter-atomic potentials, the quantum MD method computes the forces by solving the electron Hamiltonian on the fly. The quantum MD methods have proven a powerful tool in many branches of physical sciences. Our method can be viewed as the quantum version of the Landau-Lifshitz dynamics. We envision that our QLLD method combined with DFT calculation will provide a novel new approach to SDW dynamics in realistic materials.It is worth pointing out that the numerical method presented here is complementary to DMFT. Both approaches are not restricted by the sign-problem that plagues the QMC methods. Conventional DMFT ignores spatial correlations from the outset and focuses on quantum effects or fluctuations along the imaginary time axis. Our semiclassical approach emphasizes the large-scale simulations in order to fully take into account the spatialcorrelations and fluctuations of the magnetic order parameter. In developing this method, we are partly motivated by several recent studies emphasizing the important role of emergent nano-scale structures in the functionality of strongly correlated materials <cit.>.Taking advantage of recent developments of efficient electronic structure method, such as KPM, our scheme is to progressively include the quantum corrections at each time step of the dynamical simulations. Our work here has laid the groundwork for systematic improvement beyond the adiabatic or TDHF approximation, which will be left for future studies. The authors would like to thank A. Chubukov for useful discussions regarding analytical approaches to SDW dynamics. Work at LANL (K. B.) was carried out under the auspices of the U.S. DOE Contract No. DE-AC52-06NA25396 through the LDRD program. 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Conventional KPM provides an efficient approach to computing the system free-energy ℱ. However, central to our QLLD simulations is the calculation of the `forces' acting on spins: ∂ℱ / ∂𝐦_i, where the effective energy functional is calculated from the quadratic fermion Hamiltonian: ℱ = ⟨ℋ_ SDW⟩. Specifically, the force is∂ℱ/∂𝐦_i = -U ⟨𝐬_i ⟩ = -U/2ρ_iα, iβ σ_βα.Computing the force is thus equivalent to evaluating the single-particle density matrix ρ_iα, jβ. We first introduce the single-particle Hamiltonian H_iα, jβ such that the quadratic spin-fermion Hamiltonian is expressed as (up to a term that is independent of fermions)ℋ_ SDW = ∑_iα,j β H_iα,jβ c^†_iα c^ _jβ = ∑_IJ H_IJ c^†_I c^ _J.Here we have introduced notation I = (i, α), J = (j, β), ⋯ for simplicity. The density matrix is then given by the derivativeρ_IJ = ⟨ c^†_Jc^ _I ⟩ = ∂ℱ/∂ H_IJ.Next we outline the KPM procedure for computing the free energy which is expressed in terms of the DOS as ℱ = ∫ρ(ϵ) f(ϵ) dϵ, where f(ϵ) = -T log [1 + e^-(ϵ-μ)/T]. KPM begins by approximating the DOS as a Chebyshev polynomial series,ρ(ϵ) = 1/π√(1 - ϵ^2)∑_m=0^M-1 (2 - δ_0, m)μ_mT_m(ϵ),where T_m(x) are Chebyshev polynomials, and μ_m are the expansion coefficients. The expansion is valid only when all eigenvalues of H_IJ have magnitude less than one. This can in general be achieved through a simple shifting and rescaling of the Hamiltonian. Moreover, damping coefficients g_m are often introduced to reduce the unwanted artificial Gibbs oscillations. Substituting ρ(ϵ) into the free energy expression givesℱ = ∑_m=0^M-1 C_m μ_m,where coefficients C_m = (2 - δ_0,m) g_m ∫_-1^1T_m(ϵ) f(ϵ)/π√(1-ϵ^2) dϵ are independent of the Hamiltonian and may be efficiently evaluated using Chebyshev-Gauss quadrature. The key step of KPM is to replace computation of the Chebyshev moments μ_m =TrT_m(H) by an ensemble average μ_m = ⟨ T_m(H) ⟩ =1/R∑_ℓ=1^R r_ℓ^† H r_ℓ over random normalized column vectors r <cit.>. Taking advantage of the recursive relation of Chebyshev polynomials: T_m(H) = 2 H · T_m-1(H) - T_m-2(H), the moments can be evaluated recursively as follows:μ_m = r^†·α_m,where r is a random vectors with complex elements drawn from the uniform distribution |r_I|^2 = 1. The random vectors α_m are given byα_m = {[r, m = 0;H · r, m=1; 2 H ·α_m-1 - α_m-2, m>1; ].The above recursion relation also indicates that evaluation of μ_m that are required for computing ℱ only involves matrix-vector products. For sparse matrix H with 𝒪(N) elements, this requires only 𝒪(M N) operations, where M is the number of Chebyshev polynomials. On the other hand, even with the efficient algorithm for ℱ, a naive calculation of the derivatives ∂ℱ/∂ H_IJ based on finite difference approximation is not only inefficient but also inaccurate. The computational cost of finite difference is similar to the KPM-based Monte Carlo method with local updates.To circumvent this difficulty, we employ the technique of automatic differentiation with reverse accumulation <cit.>. Instead of directly using Eq. (<ref>), the trick is to view ℱ as a function of vectors α_m and write∂ℱ/∂ H_IJ = ∑_m=0^M-1∂ℱ/∂α_m, K∂α_m, K/∂ H_IJ,Here α_m, K denotes the K-th component of vector α_m, and summation over the repeated index K is assumed. Using Eq. (<ref>), we have ∂α_0, K/∂ H_IJ = 0, ∂α_1, K/∂ H_IJ = δ_IK α_0, J, ∂α_m, K/∂ H_IJ = 2 δ_IK α_m-1, J (m>1)The expression of ∂ℱ/∂ H_IJ can be simplified by introducing a new set of random vectors:β_m ≡∂ℱ/∂α_m+1,From Eqs. (<ref>) and (<ref>) , we obtain∂ℱ/∂ H_IJ = β_0, I α_0, J + 2 ∑_m=1^M-2β_m, I α_m, J.Remarkably, the vectors β_m can also be computed recursively. To this end, we note that the recursion relation (<ref>) implies that ℱ depends on α_m through three paths:∂ℱ/∂α_m, K = ∂ℱ/∂μ_m∂μ_m/∂α_m, K + ∂ℱ/∂α_m+1, L∂α_m+1, L/∂α_m, K + ∂ℱ/∂α_m+2, L∂α_m+2, L/∂α_m, K.The various terms above can be straightforwardly calculated:∂ℱ/∂μ_m = C_m, ∂μ_m/∂α_m, K = r^*_K, ∂α_m+1, L/∂α_m, K = 2 H_LK, ∂α_m+2, L/∂α_m, K = -δ_LK.Consequently, β_m = C_m+1r^† + 2 β_m+1· H - β_m+2,(m < M-1).Restoring the site and spin indices, we obtain the following expression for the density matrixρ_iα, jβ = β_0, iα α_0, jβ + 2 ∑_m=1^M-2β_m, iα α_m, jβ.As in standard KPM, there are two independent sources of errors in our method <cit.>: the truncation of the Chebyshev series at order M-1, and the stochastic estimation of the moments using finite number R of random vectors.The performance of the stochastic estimation can be further improved using correlated random vectors based on the probing method <cit.>.Most simulations discussed in the main text were done on a 120× 120 triangular lattice. The number of Chebyshev polynomials used in the simulations is in the range of M = 1000 to 2000. The number of correlated random vectors used is R = 64 to 144.§ B. LARGE-U LIMIT: FORMAL DERIVATIONHere we present a formal derivation of the effective SDW Hamiltonian in the large-U limit, which is expected to be equivalent to that of the original Hubbard model. Our first step is to write the spin-fermion Hamiltonian Eq. (<ref>) in a new reference frame, such that the local quantization axis of site i coincides with the direction of the SDW field. Let 𝐦_i = |𝐦_i| (sinθ_i cosϕ_i, sinθ_i sinϕ_i, cosθ_i), the fermionic operators in the new reference frame arec̃^†_i, + =e^-i ϕ_i/2sin(θ_i/2) c^†_i, ↓ + e^i ϕ_i /2cos(θ_i/2)c^†_i, ↑, c̃^†_i, - =e^-i ϕ_i/2cos(θ_i/2) c^†_i, ↓ - e^i ϕ_i /2sin(θ_i/2)c^†_i, ↑,The inverse transformation isc^†_i, ↑ = e^-i ϕ_i/2cos(θ_i/2) c̃^†_i, + - e^-i ϕ_i /2sin(θ_i/2) c̃^†_i, -, c^†_i, ↓ = e^+i ϕ_i/2sin(θ_i/2)c̃^†_i, + + e^+ i ϕ_i/2cos(θ_i/2)c̃^†_i, -.We separate the spin-fermion Hamiltonian into two parts ℋ_ SDW = 𝒯 + 𝒰. The kinetic hopping term can be re-expressed in the new reference frame as 𝒯 = - ∑_⟨ ij ⟩∑_μν = ± t_ij(η^μν_ij c̃^†_iμ c̃^ _j ν +h.c.),whereη^++_ij =+cos(ϕ_j - ϕ_i/2) cos(θ_j - θ_i/2) + i sin(ϕ_j - ϕ_i/2) cos(θ_j + θ_i/2),η^+-_ij =-cos(ϕ_j - ϕ_i/2) cos(θ_j - θ_i/2) - i sin(ϕ_j - ϕ_i/2) cos(θ_j + θ_i/2), η^-+_ij =-cos(ϕ_j - ϕ_i/2) cos(θ_j - θ_i/2) + i sin(ϕ_j - ϕ_i/2) cos(θ_j + θ_i/2), η^–_ij =+cos(ϕ_j - ϕ_i/2) cos(θ_j - θ_i/2) - i sin(ϕ_j - ϕ_i/2) cos(θ_j + θ_i/2).To derive the effective Hamiltonian, we use the standard perturbation approach by treating the hopping 𝒯 as a perturbation to the coupling term𝒰 = - U ∑_i |𝐦_i| ( c̃^†_i,+c̃^ _i, + - c̃^†_i,-c̃^ _i, -) + U ∑_i |𝐦_i|^2.For convenience, we first introduce the resolvent of𝒰: Ĝ_0(ϵ) = 1/(ϵ - 𝒰 ). The effective Hamiltonian up to second order in t_ij is given byℋ_ eff = 𝒫 𝒯Ĝ_0(ϵ_0) 𝒯𝒫,where 𝒫 is a projector onto the lowest energy subspace with one electron per site whose spin is parallel to local moment 𝐦_i, and ϵ_0 = - NU/4 is the energy of the degenerate large-U ground state (N is the number of lattice sites). Then, in the new reference frame, the only processes contributing to ℋ_ eff are the spin-flip hoppings which annihilate electrons with spin μ=+ and create electrons in a difference site with spin μ=-. Given that each state of the lowest energy subspace is fully characterized by the field configuration {𝐦_i }, the effective Hamiltonian can be expressed in terms of the the SDW field:ℋ_ eff =-∑_⟨ ij ⟩t_ij^2/U( |η^-+_ij|^2 + |η^+-_ij|^2 )=∑_⟨ ij ⟩ t_ij^2/U[ 1 - sinθ_i sinθ_j cos(ϕ_i - ϕ_j) - cosθ_i cosθ_j ] =∑_⟨ ij ⟩4 t_ij^2/U(𝐦_i ·𝐦_j - 1/4)which is the expected Heisenberg exchange interaction for spin-1/2 in the large U limit <cit.>.We next derive the dynamics equation in the large-U limit, which is given by the Landau-Lifshitz equation. To this end, we first consider the Heisenberg equation of motion for local spin operator d 𝐬_i / dt = -i [𝐬_i, ℋ_ H]. Here 𝐬_i = 1/2 c^†_iασ^ _αβ c^ _iβ, and ℋ_ H is the Hubbard Hamiltonian. Expressing the on-site U interaction in terms of spin operators, it is given by ℋ_ H = - ∑_⟨ ij ⟩, α t_ij(c^†_iα c^ _jα +h.c.) - 2U/3∑_i 𝐬_i^2 + N_e U/2.where N_e is the number of electrons. Obviously, the U term of the Hubbard Hamiltonian commutes with the spin operator, and we have d 𝐬_i / dt =-i [𝐬_i, 𝒯]. For example, we consider the z component first. Using commutation relations [s^z_i, c^†_i, ↑] = 1/2 c^†_i, ↑ and [s^z_i, c^†_i, ↓ ] = -1/2 c^†_i,↓, we have-i [s^z_i, (c^†_i α c^ _j α + c^†_jα c^ _iα) ] = -i σ_α/2(c^†_i α c^ _j α - c^†_jα c^ _iα) = -i/2(c^†_i ασ̂^z_αβc^ _j β - c^†_jασ̂^z_αβc^ _iβ).Here σ_α = ± 1 for α = ↑, ↓, respectively. Summing over repeated indices is also implied. The equation for the x and y components of 𝐬_i can be obtained by applying π/2 rotations to this equation. We haved𝐬_i/dt = - ∑_j 𝐉_ij,where we have defined the spin current density operator 𝐉_ij. 𝐉_ij = -i σ_αβ/2(c^†_iα c^ _jβ - c^†_jα c^ _iβ).It is worth noting that Eq. (<ref>) is simply the continuity equation for the spin density. Taking the expectation value with respect to the ground state gives rise to the equation of motion for the SDW fieldd𝐦_i/dt = -∑_j⟨𝐉_ij⟩,Next we compute the expectation value of 𝐉_ij in the large U limit. In the t / U → 0 limit, obviously ⟨𝐉_ij⟩ = 0. A nonzero contribution comes from the second-order perturbation due to electron hopping. The procedure is similar to what we did to derive the effective Hamiltonian. Specifically, we project the spin current operator into the degenerate low-energy manifold of ℋ_ SDW with the electronic eigenstates corrected up to first order in the perturbation. For the z-component, we have⟨ J^z_ij⟩ = 1/2[ 𝒫 J^z_ij G_0(ϵ_0) 𝒯𝒫 + 𝒫𝒯 G_0(ϵ_0) J^z_ij𝒫].Once again, it is convenient to work in the new reference frame. For example, the current density operatorJ^z_ij = -i t_ij/2∑_μν = ±(τ^μν_ijc̃^†_i μ c̃^ _jν - (τ^μν_ij)^* c̃^†_jν c^ _iμ).Here we have introducedτ^++_ij =+cos(ϕ_j - ϕ_i/2) cos(θ_j + θ_i/2) + i sin(ϕ_j - ϕ_i/2) cos(θ_j - θ_i/2),τ^+-_ij =-cos(ϕ_j - ϕ_i/2) cos(θ_j + θ_i/2) - i sin(ϕ_j - ϕ_i/2) cos(θ_j - θ_i/2), τ^-+_ij =-cos(ϕ_j - ϕ_i/2) cos(θ_j + θ_i/2) + i sin(ϕ_j - ϕ_i/2) cos(θ_j - θ_i/2), τ^–_ij =+cos(ϕ_j - ϕ_i/2) cos(θ_j + θ_i/2) - i sin(ϕ_j - ϕ_i/2) cos(θ_j - θ_i/2).Using expressions (<ref>) and similar one for the kinetic term Eq. (<ref>), we obtain⟨ J^z_ij⟩ = -i t_ij^2/U{[ τ^-+_ij (η^-+_ij)^* - (τ^-+_ij)^* η^-+_ij] + [ τ^+-_ij (η^+-_ij)^* - (τ^+-_ij)^* η^+-_ij] }Using the definitions for τ^μν_ij and η^μν_ij, it can be shown that[ τ^-+_ij (η^-+_ij)^* - (τ^-+_ij)^* η^-+_ij]= [ τ^+-_ij (η^+-_ij)^* - (τ^+-_ij)^* η^+-_ij] =2 i sin(ϕ_j - ϕ_i) [ sin^2 (θ_j + θ_i/2) - sin^2 (θ_j - θ_i/2) ] =2 i sin(ϕ_j - ϕ_i) sinθ_i sinθ_jIn terms of the SDW field 𝐦_i, the right-hand side of the above equation is 2 i𝐦̂_i ×𝐦̂_j ·𝐳̂, where 𝐦̂_i is a unit vector along the local moment direction. Using the fact that |𝐦_i| = 1/2 in the large U limit at half filling, this result indicates the following vector identity for the spin current⟨𝐉_ij⟩ = 4 t_ij^2/U𝐦_i ×𝐦_j.Substituting this into Eq. (<ref>) gives the well known Landau-Lifshitz equation of motion (<ref>) in the main text for the Heisenberg exchange Hamiltonian.§ C. EXACT DIAGONALIZATION CALCULATION OF DYNAMICAL STRUCTURE FACTOR Here we provide details of the exact diagonalization (ED) calculation of Hubbard Model on 4×4 square lattice with periodic boundary condition (PBC). For simplicity, we consider the case where SU(2) symmetry is conserved in the model, which leads to 𝒮(q,ω)=3𝒮^zz(q,ω). To calculate 𝒮^zz(q,ω) at T=0, we first obtain the ground state |Ψ_0⟩ in the total S_z=0 sector at half-filling, by using the implicitly restarted Arnoldi method provided through the ARPACK libary <cit.>. The dynamical structure factor can be expressed through the fluctuation-dissipation theorem:S^zz(q,ω)=-2 Imχ_zz(q,ω) =-2 ⟨Ψ_0| S_q^z S_-q^z | Ψ_0⟩ Im⟨ϕ_0 | (ω + i η + E_0-H)^-1 | ϕ_0 ⟩,where | ϕ_0 ⟩≡ S^z(-q)|Ψ_0⟩ / √(⟨Ψ_0| S_q^z S_-q^z | Ψ_0⟩), and E_0 is the ground state energy: H |Ψ_0 ⟩ = E_0 | Ψ_0 ⟩.The matrix inverse (z-H)^-1 can be calculated through the Lanczos algorithm <cit.>: 1em 1emIn the new basis { |ϕ_0⟩, | ϕ_1 ⟩, | ϕ_2 ⟩,…}, the Hamiltonian is expressed by a tridiagonal matrix:H=[ a_0 b_1; b_1 a_1 b_2; b_2 a_2 ⋱; ⋱ ⋱ b_n; b_n a_n ]. With Cramer's rule, the first element of the inverse matrix can be expressed as a continued fraction:⟨ϕ_0 | (z-H)^-1 | ϕ_0 ⟩ = [ (z-a_0)-b_1^2/(z-a_1)-b_2^2/(z-a_2)-⋯]^-1,which leads to𝒮^zz(q,ω) =-2 ⟨Ψ_0| S_q^z S_-q^z | Ψ_0⟩·Im[ (z-a_0)-b_1^2/(z-a_1)-b_2^2/(z-a_2)-⋯]^-1,where z≡ω+iη+E_0. 99weisse06b A. Weiße, G. Wellein, A. Alvermann, and H. Fehske, The kernel polynomial method, Rev. Mod. Phys. 78, 275 (2006).barros13b K. Barros and Y. Kato, Efficient Langevin simulation of coupled classical fields and fermions, Phys. Rev. B 88, 235101 (2013).silver94 R. N. Silver and H. Röder, Density of states of mega-dimensional Hamiltonian matrices, Int. J. Mod. Phys. C 5, 735 (1994).griewank89 A. Griewank, in Mathematical Programming: Recent Developments and Applications, edited by M. Iri and K. Tanabe (Kluwer Academic, Dordrecht, The Netherlands, 1989), pp. 83–108.tang12 J. M. Tang and Y. Saad, A probing method for computing the diagonal of a matrix inverse, Numer. Linear Algebra Appl. 19, 485 (2012).fazekas99b P. Fazekas, Lecture notes on electron correlation and magnetism, Chapter 5 (World Scientific, Singapore, 1999).lehoucq98 R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users' Guide: Solution of Large-scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods (Society for Industrial and Applied Mathematics, 1998).lanczos90 C. Lanczos, An Iteration Method for the Solution of the Eigenvalue Problem of Linear Differential and Integral Operators, J. Res. Nat. Bur. Stand. 45, 255 (1950).dagotto94 E. Dagotto, Correlated electrons in high-temperature superconductors, Rev. Mod. Phys. 66, 763 (1994). | http://arxiv.org/abs/1708.08050v3 | {
"authors": [
"Gia-Wei Chern",
"Kipton Barros",
"Zhentao Wang",
"Hidemaro Suwa",
"Cristian D. Batista"
],
"categories": [
"cond-mat.str-el"
],
"primary_category": "cond-mat.str-el",
"published": "20170827043436",
"title": "Semiclassical dynamics of spin density waves"
} |
2017 Vol. X No. XX, 000–000Department of Physics, Assam University, Silchar 788011, India; [email protected] 2017 Feb XX; accepted 2017 XXXX We study the extinction properties of highly porous BCCA dust aggregates in a wide range of complex refractive indices (1.4 ≤ n ≤ 2.0, 0.001 ≤ k ≤ 1.0) and wavelength (0.11μ m ≤λ≤ 3.4μ m). An attempt has been made for the first time to investigate the correlation among extinction efficiency (Q_ext), the composition of dust aggregates (n, k), the wavelength of radiation (λ) and size parameter of the monomers (x). If k is fixed at any value between 0.001 and 1.0, Q_ext increases with increase of n from 1.4 to 2.0. Q_ext and n are correlatedvia linear regression when the cluster size is small whereas the correlation is quadratic at moderate and higher sizes of the cluster. This feature is observed at all wavelengths (UV to optical to infrared). We also find that the variation of Q_ext with n is very small when λ is high. When n is fixed at any value between 1.4 and 2.0, it is observed thatQ_ext and k are correlatedvia polynomial regression equation (of degree 1, 2, 3 or 4), where the degree of the equation depends on the cluster size, n and λ. The correlationis linear for small size and quadratic/cubic/quartic for moderate and higher sizes. We have also found that Q_ext and x are correlated via a polynomial regression (of degree 3,4 or 5) for all values of n. The degree of regression is found to be n and k-dependent. The set of relations obtained from our work can be used to model interstellar extinction for dust aggregates in a wide range of wavelengths and complex refractive indices. Dhar & Das Correlation among extinction efficiency and other parametersCorrelation among extinction efficiency and other parameters in an aggregate dust model Tanuj Kumar Dhar and Himadri Sekhar Das December 30, 2023 ======================================================================================= § INTRODUCTIONThe studies of cometary and interplanetary dust indicate that cosmic dust grains are likely to be fluffy, porous and composites of many small grains fused together, due to dust-gas interactions, grain-grain collisions, and various other processes <cit.>. Porous, composite aggregates are often modelled as cluster of small spheres (known as “monomers"), agglomerated under various aggregation rules. Here grain aggregates are assumed to be fluffy sub structured collections of very small particles loosely attached to one another. Each particle is assumed to consist of a single material, such as silicates or carbon, as formed in thevarious separate sources of cosmic dust. Extinction generally takes place whenever electromagnetic radiation propagates through a medium containing small particles.The spectral dependence of extinction, or extinction curve, is a function of the structure, composition, and size distribution of the particles. The study of interstellar extinction provides us useful information for understanding the properties of the dust.It is now well accepted from observation and laboratory analysis of interplanetary dust particles that cosmic dust grains are fluffy aggregates or porous with irregular shapes <cit.>. Using Discrete Dipole Approximation (DDA) technique, several investigators studied the extinction properties of the composite grains <cit.>. <cit.> studied optical properties of composite grains as grain aggregates of amorphous carbon and astronomical silicates, using the Superposition transition matrix approach. Recently, <cit.> studied the light scattering properties of aggregate particles in a wide range of complex refractive indices and wavelengths to investigate the correlation among different parameters e.g., the positive polarization maximum, the amplitude of the negative polarization, geometric albedo, refractive indices and wavelength. The simulations were performed using the Superposition T-matrix code with Ballistic ClusterCluster Aggregate (BCCA) particles of 128 monomers and Ballistic Aggre- gates (BA) particles of 512 monomers.The extinction efficiency of dust aggregates depends on aggregate size, composition and wavelength of incident radiation. The dependence of complex refractive index (n, k) on Q_ext was studied by many groups in past for spherical and irregular particles using different scattering theories (Mie theory, DDA approach, T-matrix theory etc.). But no correlation equations were reported earlier by any group. In this paper, we study the extinction properties of randomly oriented porous dust aggregates with a wide range of complex refractive indices and wavelength of incident radiation. An attempt has been made for the first time to investigate the correlation among extinction efficiency (Q_ext), complex refractive indices (m = n + ik), wavelength (λ) and the size parameter of monomer (x).§ NUMERICAL COMPUTATIONSWe have constructed the aggregates using ballistic aggregation procedure <cit.> using two different models of cluster growth. First via single-particle aggregation and then through cluster-cluster aggregation. These aggregates are built by random hitting and sticking particles together. The first one is called Ballistic Particle-Cluster Aggregate (BPCA) when the method allows only single particles to join the cluster of particles. If the method allows clusters of particles to stick together, the aggregate is called Ballistic Cluster-Cluster Aggregate (BCCA). Actually, the BPCA clusters are more compact than BCCA clusters <cit.>. The porosity of BPCA and BCCA particles of 128 monomers has the values 0.90 and 0.94, respectively. The fractal dimensions of BPCA and BCCA particles are given by D3 and 2, respectively (Meakin 1984). A systematic explanation on dust aggregate model is already discussed in our previous work <cit.>. It is to be noted that the structure of these aggregates are similar to those of Interplanetary Dust Particles (IDP) collected in the stratosphere of Earth <cit.>. It is also well understood from the laboratory diagnosis that the particle coagulation in the solar nebula grows under BCCA process <cit.>.The general extinction A_λ is given by <cit.>:A_λ = -2.5 log[F(λ)/F_0(λ)] = 1.086 N_dQ_ext σ _d,where F(λ) and F_0(λ) are the observed and expected fluxes, N_d is the dust column density, Q_ext is the extinction efficiency factor determined from Superposition T-matrix code, and σ _d is the geometrical cross-section of a single particle.The interstellar extinction curve (i.e. the variation of extinction with wavelength) is usually expressed by the ratio A_λ/E(B-V) versus 1/λ. The extinction curve covers the wavelength range 0.11 to 3.4 μ m. The entire range consists of UV (ultra violet), visible and IR (infrared) regions. The IR rangecorresponds to near infrared i.e. 0.750 to 2.5 μ m, the Visible range (0.38 to 0.76 μ m) and the UV range (the last part of violet in visible spectrum to 0.11 μ m).The radius of an aggregate particle can be described by the radius of a sphere of equal volume given by a_v = a_mN^1/3, where N is the number of monomers in the aggregate and a_m is the monomer's radius of aggregates . We have found from literature survey that most of the work related to interstellar extinction considered a normal size range of 0.001 to 0.250 micron, with a size distribution (mainly MRN distribution) (Jones 1988, Whittet 2003, Vaidya et al. 2007, Das et al. 2010). They found an `optimum' for the range of the cluster size generally used. The above size range of the monomer is more or less capable of evaluating average observed interstellar extinction curve. If we consider N = 64 and a_m in the range 0.001 to 0.065 micron (with a step size of 0.004 micron), then a_v will be 0.004 to 0.26 micron. This size range is almost comparable to the size range used by other investigators.We use JaSTA-2 (Second version of the Java Superposition T-matrix Application) <cit.>,which is an upgraded version of JaSTA <cit.>, to execute our computations which is based on <cit.>'s Superposition T-matrix code. All versions of JaSTA are freely available to download from . The computations with the T-matrix code is fast and this technique gives rigorous solutions for randomly oriented ensembles ofspheres. It is to be noted that the results obtained from the Discrete Dipole Approximation (DDA) approach and the T-matrix approach are almost same. <cit.> showed the results with aggregates using the DDA and the T-matrix code, and found almost same results with both the code. We perform the computationswith a wide range of complex refractive indices (n = 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0 and k = 0.001, 0.01, 0.05, 0.1, 0.3, 0.5, 0.7, 1.0) and wavelengths (0.11, 0.12, 0.13, 0.16, 0.175, 0.185, 0.20, 0.207, 0.22, 0.23, 0.26, 0.30, 0.365, 0.40, 0.55, 0.6, 0.7, 0.8, 0.90 and 3.4 μ m). In general, the range of n and k which is considered in our work almost covers the range of the complex refractive indices of silicate and carbon at different wavelengths. The numerical computation in the present work has been executed with BCCA cluster of 64 monomers.We present the results for a_m = 0.001μ m, 0.017μ m, 0.041μ m and 0.065μ m where a_v is given by 0.004μ m, 0.068μ m, 0.16μ m and 0.26μ m. The monomer size parameter (x = 2 π a_m/λ) is taken in a range from 0.01 to 1.6. This study is mainly concentrated on investigation of correlation among Q_ext, (n, k), λ, and x.§ RESULTS §.§ Dependence on monomer size (a_m) §.§.§ Correlation between Q_ext and nWe perform the computations with a wide range of complex refractive indices (1.4 ≤ n ≤ 2.0 and 0.001 ≤ k ≤ 1.0)and wavelength of incident radiation (0.11 ≤λ≤ 3.4μ m). To study the dependence of n and k on the extinction efficiency (Q_ext), we can either plot Q_ext versus k by keeping n fixed or plot Q_ext versus n by keeping k fixed. We first show the results for moderate size of the cluster ,i.e., at a_v = 0.16μ m. The results at other three sizes are also presented thereafter.We plot Q_ext versus n at a_v = 0.16μ m, for k = 0.001, 0.01, 0.1, 0.5 and 1.0 respectively (although we have executed the code with all eight values of k mentioned above), in a single frame, for λ = 0.11, 0.20, 0.30 and 0.90μ m, which is shown in Fig.<ref>. It is to be noted that the plots are shown for four wavelengths although the computations have been performed for all the wavelengths.It is observed from Fig.<ref> that if k is fixed at any value between 0.001 and 1.0, Q_ext increases with increase of n from 1.4 to 2.0 at all wavelengths. But a small decrease in Q_ext is also noticed at λ = 0.11μ m when n > 1.7 and k is low. The variation of Q_ext with n is small when k ≥ 0.5. The value of Q_ext is small when λ is large, i.e. Q_ext decreases when size parameter of the monomer (x = 2 π a_m/λ) decreases. We have also investigatedthat the variation of Q_ext with n is very small when λ≥ 0.70μ m.We have found that Q_ext and n can be fitted by quadratic regression where coefficient of determination[The coefficient of determination is a key output of regression analysis which is interpreted as the proportion of the variance in the dependent variable which ranges from 0 to 1. A higher coefficient is an indicator of a better goodness of fit for the observations.] (R^2) for each equation is ≈ 0.99. The best fit equation is given by 2 Q_ext = A_k n^2 + B_k n + C_k , 1.4 ≤ n ≤ 2.0, 0.001 ≤ k ≤ 1,where, A_k, B_k and C_k are k-dependent coefficients of equation (2).The coefficients obtained for different values of k (only five values of k are shown) are depicted in Table-<ref>. If we plot coefficients A_k, B_k and C_kversus k (where k = 0.001, 0.01, 0.05, 0.1, 0.3, 0.5, 0.7, 1.0), we find that the best fit curves correspond to cubic regression, which have R^2 ≈ 0.99. We donot show any figures in this case.The coefficients are correlated with k by the relations:2aA_k =D_1 k^3 + D_2 k^2 + D_3 k + D_4 , 2bB_k =E_1 k^3 + E_2 k^2 + E_3 k + E_4 , 2cC_k =F_1 k^3 + F_2 k^2 + F_3 k + F_4 , All coefficients of equation 2(a-c)are shown in Table-<ref>. Thus knowing the coefficients, the extinction efficiency (Q_ext) can be calculated for any value of n and k from the equation (2).In Fig.<ref>, we report the results for a_v = 0.004μ m at λ = 0.11, 0.20, 0.30 and 0.90 μ m. A strong linear correlation between Q_ext and n is seen at this size for all wavelengths from 0.11 to 3.4 μ m. In Fig.<ref>, we plot Q_ext versus n for a_v = 0.068μ m at λ = 0.11μ m and 0.60 μ m. We have found that the nature is quadratic at all wavelengths from 0.11 to 3.4 μ m. Finally, we show the results for a_v = 0.26μ m at λ = 0.11μ m and 0.60 μ m, shown in Fig.<ref>. We have noticed that the Q_ext and n is correlated via a cubic regression at 0.11 μ m whereas the dependence is quadratic at other higher wavelengths. We do not show any equation or table in the above three cases. In summary, we can conclude that the correlation between Q_ext and n is linear when the cluster size is small whereas the correlation is quadratic at moderate and higher sizes of cluster. §.§.§ Correlation between Q_ext and k We now plot Q_ext versus k for n = 1.4, 1.5, 1.6, 1.7, 1.8, 1.9 and 2.0 respectively, at a_v = 0.16μ m where λ is taken in between 0.11 μ m and 3.4 μ m. When λ is between 0.11 μ m and 0.26 μ m, we have found that Q_ext and k can be fitted via a polynomial regression equation where the degree of equation depends on the value of n and λ. In Fig.<ref> we show the plots for λ = 0.11, 0.16, 0.20 and 0.26μ m. At λ = 0.11μ m, we find that Q_ext and k are correlated by (i) quartic regression when n = 1.4, (ii) cubic regression when n = 1.5 and (iii) quadratic regression when n = 1.6, 1.7, 1.8, 1.9 & 2.0. Further at λ = 0.16μ m, the correlation is cubic when n = 1.5, 1.6, 1.7 & 1.8 and quadratic when n = 1.4, 1.9 & 2.0. The correlation at λ = 0.20μ m is cubic when n = 1.6, 1.7 & 1.8 and quadratic when n = 1.4, 1.5, 1.9 & 2.0. We also note that the correlation at 0.26 μ m is quadratic when n = 1.4, 1.5, 1.6, 1.7 & 1.8 and cubic when n = 1.9 & 2.0. At low values of n, Q_ext increases with increase of k whereas the trend is exactly opposite when n is high ,e.g., at λ = 0.20μ m, Q_ext increases with k when n ≤ 1.6, but it decreases when n ≥ 1.7. The vertical range of Q_ext in the plot also decreases when k increases. This range is maximum at k = 0.001and minimum at k = 1.0.In Fig.<ref>, we show the plots for 0.30 ≤λ≤ 3.4μ m and we have found that the fit is quadratic for all values of n. An increase in Q_ext with k is noticed at almost all wavelengths. The vertical range of Q_ext also decreases when k increases. Further, the plot of Q_ext with k is same at all values of n when λ is high (please see Fig.<ref>(iv)).The best fit equation in the wavelength range 0.30 μ m to 3.4 μ m is given by 3 Q_ext = A_n k^2 + B_n k + C_n , 1.4 ≤ n ≤ 2.0, 0.001 ≤ k ≤ 1where, A_n, B_n and C_n are n-dependent coefficients of equation (3).The coefficients obtained for different values of k are shown in Table-<ref>. If we plot coefficients A_n, B_n and C_nversus n (figures are not shown), we note that the best fit curves correspond to quadratic regression, which have R^2 ≈ 0.99.Thus coefficients are given by3aA_n =D'_1 k^2 + D'_2 k + D'_3 , 3bB_n =E'_1 k^2 + E'_2 k + E'_3 , 3cC_n =F'_1 k^2 + F'_2 k + F'_3 , All coefficients of equation 3(a-c)are shown in Table-<ref>. Thus, knowing the coefficients of equation 3(a-c), the extinction efficiency (Q_ext) can be also estimated for any value of n and k from the equation (3). In Fig.<ref>, we plot Q_ext against k for a_v = 0.004μ m at λ = 0.11, 0.30, 0.60 and 0.90 μ m, although the computations have been performed for wide range of wavelengths from 0.11 to 3.4 μ m. We observe the linear dependence at this size for all wavelengths. This linear nature becomes quadratic when the size of cluster is a_v = 0.068μ m. Fig.<ref> shows the results for a_v = 0.068μ m at λ = 0.11 μ m and 0.60 μ m. In Fig.<ref>, the results obtained for a_v = 0.26μ m are plotted. In this case, the nature of dependence looks similar with a_v = 0.16μ m. Q_ext and k are correlatedvia polynomial regression equation (of degree 2, 3 or 4) where the degree of equation depends on the realpart of the refractive index (n) at (λ = 0.11μ m) (please see caption of Fig.<ref>). The nature is quadratic when λ > 0.11μ m. In summary, we can conclude that the dependence of Q_ext on k depends on the cluster size. The correlation is linear for small size and quadratic/cubic for moderate and higher sizes. It is important to mention that the real part of the complex index of refraction (n) controls the effective phase speed of electromagnetic waves propagating through the medium, while the imaginary part k describes the rate of absorption of the wave. In any material, n and k are not free to vary independently of one another but rather are tightly coupled to one another via the so-called Kramer-Kronig relations. Therefore, the results presented above are quite expectable.§.§ Dependence on wavelength of radiation (λ)We now study the dependence of Q_ext on λ for a particular set of (n, k) in case of a_v = 0.16μ m only. We observe the following results: (i) For n = 1.4, 1.5 and 1.6, Q_ext versus λ can be fitted via a quartic regression for k = 0.001, 0.01, 0.05, 0.1, 0.3, 0.5, 0.7 and 1.0 (please see Fig.<ref>). (ii) For n = 1.7, 1.8, 1.9 and 2.0, Q_ext versus λ can be fitted via a quartic regression in the wavelength range 0.11 - 0.40μ m [Fig.<ref>(i,iv) and Fig.<ref>(i,iv)] and a quadratic regression in the wavelength range 0.55 - 0.90μ m [Fig.<ref>(ii,v) and Fig.<ref>(ii,v)] for k = 0.001, 0.01, 0.05, 0.1 & 0.3.Further, Q_ext versus λ can be fitted via a quartic regression in the wavelength range 0.11 - 0.90μ m for higher values of k = 0.5, 0.7 and 1 [Fig.<ref>(iii,vi) and Fig.<ref>(iii,vi)]. We did not include the plots for λ = 3.4μ m. It is noticed from Figs.<ref>, <ref> and <ref> that Q_ext decreases with increase of λ when n ≤ 1.6. When n ≥ 1.7, Q_ext initially increases with increase of λ and reaches a maximum value, then it starts decreasing if λ is increased further. We also observe that Q_ext is maximum at k = 0.001 and minimum at k = 1.0 when λ = 0.11μ m. But this trend changes at a critical value of wavelength (λ_c) where exactly opposite nature is noticed. We notice that λ_c is (i) 0.16 μ m at n = 1.4, (ii) 0.185 μ m at n = 1.5, (iii) 0.207 μ m at n = 1.6, (iv) 0.22 μ m at n = 1.7, (v) 0.26 μ m at n = 1.8, (vi) 0.26 μ m at n = 1.9 and (vii) 0.30 μ m at n = 1.4. The values of Q_ext is maximum at k = 1.0 when λ > λ_c. We do not show any equations and tables in this case.§.§ Dependence on the size parameter of monomer (x)We now study the dependence of Q_ext on the size parameter of monomer (x = 2π a_m/λ)for n = 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0 and k = 0.001, 0.01, 0.05, 0.1, 0.3, 0.5, 0.7, 1.0.A wide range of size parameter, 0.01 ≤ x ≤ 1.6 (where, N = 64), is considered to investigate the correlation between Q_ext and x. The results are plotted in Figs. <ref>, <ref>, <ref> and <ref>for k = 0.001, 0.01, 0.1 and 1.0, respectively. It can be seen from figures that if x is fixed at any value between 0.01 and 1.6, Q_ext increases with increase of n. This increase is prominent when x > 0.2. The vertical range of Q_ext also increases with increase of n from 1.4 to 2.0. This range is maximum(i) at x = 1.4 in case of k = 0.001 (where, Q_ext = [9.34, 3.45]), (ii) at x = 1.4 in case of k = 0.01 (where, Q_ext = [9.26, 3.50]), (iii) at x = 1.4 in case of k = 0.1 (where, Q_ext = [8.56, 3.90]), and (iv) at x = 1.12 in case of k = 1.0 (where, Q_ext(max) = [6.74, 5.88]). The slope of Q_ext versus x curve increases with the increase of n from 1.4 to 2.0 which is noted for all values of k. It is also interesting to notice that the vertical range of Q_ext at x = 1.6 decreases with the increase of k and is lowest at k = 1.0. Further, Q_ext value does not depend much on n for highly absorptive particles (k = 1.0) when x < 0.5. The variation is also small when x > 0.5.We have found that Q_ext and x can be fitted by a cubic regression for all values of n except 2.0, in case of k = 0.001, 0.01, and 0.1 respectively. The coefficient of determination (R^2) for each equation is ≈ 0.99. The best fit equation is given by4 Q_ext = α_1 x^3 + α_2 x^2 + α_3 x + α_4, where, α_1, α_2, α_3 and α_4 are n-dependent coefficients of equation (4). The coefficients are presented in Table-<ref>.However, the best fit equation in case of n = 2.0 corresponds to a quartic regression for k = 0.001, 0.01, and 0.1, which is given by5 Q_ext = β_1 x^4 + β_2 x^3 + β_3 x^2 + β_4 x + β_5, where, β_1, β_2, β_3, β_4 and β_5 are n-dependent coefficients, shown in Table-<ref>. The correlation between Q_ext and x is found to be quintic regression for all values of n at k = 1.0. The best fit equation is given by6 Q_ext = γ_1 x^5 + γ_2 x^4 + γ_3 x^3 + γ_4 x^2 + γ_5 x + γ_6,where, γ_1, γ_2, γ_3, γ_4, γ_5, and γ_6 are n-dependent constants of equation (6). The constants are given in Table-<ref>.Equations (4), (5) and (6) are very useful in estimating Q_ext, where one can generate a large data set for Q_ext for selected set of n, k, a_m, and λ. § RESULTS FROM CORRELATION EQUATIONSIn the previous sections, we have obtained a set of correlation equationswhich can be used to calculate the extinction efficiency of dust aggregates with a wide range of size of aggregates and wavelength of radiation. We first calculate Q_ext from relations (2) and (3) for BCCA particles with N = 64 and a_m = 0.041μ m, for selected values of n, k and λ. The calculated values are compared with the computed values obtained using the Superposition T-matrix code. The results are shown in Table-<ref> and <ref>. We also estimate Q_ext using relations (4), (5) and (6) for selected values of x, n and k, and is shown in Table-<ref>.It can be seen that the values obtained from computed values match well with the results obtained from correlation equations.In general, to model the interstellar extinction, one need to execute the light scattering code with different values of a_m (or a_v) and wavelength (λ), which is very time consuming. Using a size distribution for aggregates, it is possible to obtain the average extinction curves for silicate and graphite (and/or amorphous carbon) particles. With a suitable mixing among them, extinction curve against different wavelengths can be generated which can be fitted well with observed extinction curve. Some good pieces of work on modeling were already done for aggregate particles. Some preliminary results on modeling of interstellar extinction using aggregate dust model were already reported by <cit.>. The present study shows that it is possible to study the extinction properties of interstellar dust aggregates for a given size of the particles and wavelength using relations (4), (5) and (6). The set ofcorrelation equations can be used to estimate the general extinction A_λ using equation (1) for a given size distribution which will help to model the interstellar extinction curve. At this stage, we are not interested in modeling as this study is primarily projected to investigate the dependency of extinction efficiency on size, wavelength and composition of particles. We show how this dependency can be framed with some correlation equations to study the extinction properties of interstellar dust. <cit.> experimentally investigated the morphological effects on the extinction band in the infrared region for amorphous silica (SiO_2) agglomerates. They also compared the measured band profiles with calculations for five cluster shapes applying Mie, T-matrix and DDA codes. Our correlations will be also helpful to study the experimental data. But it is also important to check the input parameters (size parameter, composition etc.) of the experimental setup before using the correlation equations, because the relations are based on some selected set of parameters.§ SUMMARY* We have first studied the dependency of Q_ext on size of the aggregates (a_m) to investigate the correlations between Q_ext and complex refractive indices (n, k) at a particular size. Computations are performed at four different sizes (a_v = 0.004, 0.068, 0.16 and 0.26 μ m).If k is fixed at any value between 0.001 and 1.0, Q_ext increases with increase of n from 1.4 to 2.0. Q_ext and n are correlatedvia linear regression when the cluster size is small whereas the correlation is quadratic at moderate and higher sizes of cluster. This feature is observed at all wavelengths (UV to optical to infrared). We have also found that the variation of Q_ext with n is very small when λ is high.We have observed thatQ_ext and k are correlatedvia polynomial regression equation (of degree 2,3 or 4) where the degree of equation depends on the cluster size, realpart of the refractive index of the particles (n) and wavelength (λ) of incident radiation. At a_v = 0.16 μ m, Q_ext and k is found to be correlated with a polynomial regression equation (of degree 2 or 3) when λ is between 0.11 μ m and 0.26 μ m. However, when λ > 0.26μ m,we have found that the correlation between them is quadratic for all values of n. The vertical range of Q_ext in the plot also decreases when k increases. This range is maximum at k = 0.001and minimum at k = 1.0. If we include results for four different cluster sizes, we can summarize that the correlation of Q_ext and k depends mainly on cluster size. The correlation is linear for small size and quadratic/cubic/quartic for moderate and higher sizes. * We study the dependence of Q_ext on λ for a_v = 0.16μ m. Q_ext decreases with increase of λ when n ≤ 1.6. When n ≥ 1.7, Q_ext initially increases with increase of λ and reaches a maximum value, then it starts decreasing if λ is increased further. For n = 1.4, 1.5 and 1.6, Q_ext versus λ can be fitted via a quartic regression for all values of k. For other values of n, the correlation is polynomial regression where the degree of equation depends on the value of n, k and λ. * We have found that Q_ext and x are correlated via a polynomial regression (of degree 3,4 or 5) for all values of n. The degree of regression is found to be n and k-dependent. * The correlation equations can be used to model interstellar extinction for dust aggregates in a wide range of size of the aggregates, wavelengths and complex refractive indices.§ ACKNOWLEDGEMENTWe acknowledge Daniel Mackowski and Michael Mishchenko, who made their Multi-sphere T-matrix (MSTM) codepublicly available. We also acknowledge Prithish Halder for help on the execution of JaSTA-2 software package. The reviewer of this paper is highly acknowledged for useful comments and suggestions. 55 [Bhattacharjee et al.2010]b1a Bhattacharjee C., Das H. S., Sen A. K., 2010, Assam University Journal of Science & Technology : Physical Sciences and Technology, 6 (Number II), 39[Brownlee et al.1985]b1Brownlee D. E., 1985, Ann. Rev. Earth Planet. Sci., 13, 147.[Das et al.2008]b2Das H. S., Das S. R., Paul T., Suklabaidya A.,Sen A.K, 2008, MNRAS, 389,787.[Das et al.2010]b2aDas H. K., Voshchinnikov N. 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"authors": [
"Tanuj Kumar Dhar",
"Himadri Sekhar Das"
],
"categories": [
"astro-ph.EP"
],
"primary_category": "astro-ph.EP",
"published": "20170826032540",
"title": "Correlation among extinction efficiency and other parameters in an aggregate dust model"
} |
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