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Research article Special Issues

Solar Power: A new mathematical definition and theoretical proof it is a Green Public Good

  • Received: 08 September 2019 Accepted: 27 September 2019 Published: 30 September 2019
  • JEL Codes: C6, D2, D4, D6, H2, H4, L5, P1, Q2, Q4, Q5

  • Solar Power is redefined in a new mathematical framework which describes its pertinent properties. It is shown that it is an economic good and that there are Property Rights and Economic Ownership associated with it. Its externalities have been examined with a view of including economic obligations created during the post emission control periods which correspond to growth without climate sustainability obligations. Also its valuation has been examined in states of both non-use and use in electricity production. A new definition of a Green Public Good is described which retains Samuelson's classic definition and has an climate externality component which under Pigouvian tax/subsidy does not satisfy Samuelson's second condition for Pareto optimality.

    Citation: Kalomoira Zisopoulou, Dionysia Panagoulia. Solar Power: A new mathematical definition and theoretical proof it is a Green Public Good[J]. Green Finance, 2019, 1(3): 312-327. doi: 10.3934/GF.2019.3.312

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  • Solar Power is redefined in a new mathematical framework which describes its pertinent properties. It is shown that it is an economic good and that there are Property Rights and Economic Ownership associated with it. Its externalities have been examined with a view of including economic obligations created during the post emission control periods which correspond to growth without climate sustainability obligations. Also its valuation has been examined in states of both non-use and use in electricity production. A new definition of a Green Public Good is described which retains Samuelson's classic definition and has an climate externality component which under Pigouvian tax/subsidy does not satisfy Samuelson's second condition for Pareto optimality.


    Fractional derivatives, which have attracted considerable attention during the last few decades, can be defined according to their type. These include the Caputo [1,2,3,4,5,6,8,7,9,10,11,12,13], Riemann-Liouville [14,15,16,17,18,19,20,21], Riesz [22,23,24,25,26], and Caputo-Fabrizio (CF) [27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46] types. Especially for the CF derivative, there were many related reports based on some discussions of different aspects, see the above references, and the Refs. [43,44,45]. In order to better grasp the fractional order problem, one can refer to the related works [47,48] and other fractional books.

    Based on these fractional derivatives, numerous models have been developed. However, these models are difficult to solve directly by applying the general analytical methods because of the existence of fractional derivatives. This problem has inspired scholars to develop numerical algorithms to derive numerical solutions efficiently. In [5,49,50,51], a few high-order approximation formulas for the Riemann-Liouville, Caputo, and Riesz fractional derivatives were proposed and developed using different techniques or ideas. Recently, high-order discrete formulas for the CF fractional derivatives were designed and discussed in Refs. [32,33,34,35,36,37,38].

    Another difficulty for simulating the models with fractional derivatives is the non-locality which greatly reduces the efficiency of the algorithm and requires much more memory storage compared with the traditional local models. Specifically, to obtain the approximation solutions {Uk}Mk=1 with M a positive integer, for the fractional models its computing complexity is O(M2), and the memory storage is O(M), in contrast to the local models with O(M) and O(1), respectively. For fast algorithms aimed at the Riemann-Liouville, Caputo, and Riesz fractional derivatives, see Refs. [14,52,53,54,55]. However, few scholars studied the fast algorithm for the CF fractional derivative. To the best of our knowledge, authors in [39] proposed numerically a fast method for the CF fractional derivative without further analysing the error accuracy.

    In this study, our aim is to construct a novel efficient approximation formula for the following CF fractional derivative [31]

    CF0αtu(t)=11αt0u(s)exp[α1α(ts)]ds,0<α<1, (1.1)

    where t[0,T], 0<T<. Our contributions in this study mainly focus on

    Propose a novel second-order approximation formula for the CF fractional derivative with detailed theoretical analysis for the truncation error.

    Develop a fast algorithm based on the novel discretization technique which reduces the computing complexity from O(M2) to O(M) and the memory storage from O(M) to O(1). Moreover, we theoretically show that the fast algorithm maintains the optimal convergence rate.

    The remainder of this paper is structured as follows. In Section 2, we derive a novel approximation formula with second-order convergence rate for the CF fractional derivative. In Section 3, we develop a fast algorithm by splitting the CF fractional derivative into two parts, the history part and local part, and then rewrite the history part by a recursive formula. Further we prove the truncation error for the fast algorithm. In Section 4, two numerical examples are provided to verify the approximation results and the efficiency of our fast algorithm. In Section 5, we provide a conclusion and offer suggestions for future studies.

    Throughout this article, we denote C as a positive constant, which is free of the step size Δt.

    To derive a novel approximation formula, we choose a uniform time step size Δt=TM=tktk1 with nodes tk=kΔt,k=0,1,,M, where M is a positive constant. We denote uk=u(tk) on [0,T].

    We next give a discrete approximation of CF fractional derivative CF0αtu(t) at tk+12(k1)

    CF0αtu(tk+12)=11αkj=1tj+12tj12u(s)exp[α1α(tk+12s)]ds+11αt12t0u(s)exp[α1α(tk+12s)]ds=11αkj=1tj+12tj12[u(tj)+u(tj)(stj)+12u(ξj)(stj)2]exp[α1α(tk+12s)]ds+11αt12t0[u(t12)+u(t12)(st12)+12u(ξ0)(st12)2]exp[α1α(tk+12s)]ds=1αkj=1uj+1uj12Δt(Mk+12j+12Mk+12j12)+1αu1u0Δt(Mk+1212Mk+120)+Rk+121+Rk+122CF0Dαtu(tk+12)+Rk+12, (2.1)

    where ξj generally depending on s satisfies ξj(tj12,tj+12) for j1 and ξ0(t0,t12). The coefficients Mkj and error Rk+12 are defined as follows

    Mkj=exp[α1α(tktj)],     Rk+12=Rk+121+Rk+122, (2.2)
    Rk+121=11αkj=1tj+12tj12[u(tj)(stj)+12u(ξj)(stj)2]exp[α1α(tk+12s)]ds+11αt12t0[u(t12)(st12)+12u(ξ0)(st12)2]exp[α1α(tk+12s)]ds,Rk+122=11αO(Δ2t)tk+12t0exp[α1α(tk+12s)]ds. (2.3)

    From (2.1), we obtain the following approximation formula for CF0αtu(tk+12) with k1.

    CF0Dαtu(tk+12)=1αkj=1uj+1uj12Δt(Mk+12j+12Mk+12j12)+1αu1u0Δt(Mk+1212Mk+120). (2.4)

    Based on this discussion, we obtain the novel approximation formula (2.4). We next discuss the truncation error of the novel approximation formula.

    Theorem 1. For u(t)C3[0,T], the truncation error Rk+12(k0) satisfies the following estimate

    |Rk+12|CΔ2t, (2.5)

    where the constant C is independent of k and Δt.

    Proof. According to formula (2.3), we can obtain:

    |Rk+12||Rk+121|+|Rk+122||11αkj=1u(tj)tj+12tj12(stj)exp[α1α(tk+12s)]ds|+|11αu(t12)t12t0(st12)exp[α1α(tk+12s)]ds|+|12(1α)kj=1tj+12tj12u(ξj)(stj)2exp[α1α(tk+12s)]ds|+|12(1α)t12t0u(ξ0)(st12)2exp[α1α(tk+12s)]ds|+|11αtk+12t0O(Δ2t)exp[α1α(tk+12s)]ds|=I1+I2+I3+I4+I5. (2.6)

    For the term I1, using integration by parts, we can arrive at:

    I1maxt[0,T]|u(t)|Δ2t8(1α)kj=1{exp[α1α(tk+12tj+12)]exp[α1α(tk+12tj12)]}+maxt[0,T]|u(t)|αΔ2t8(1α)2tk+12t12exp[α1α(tk+12s)]dsmaxt[0,T]|u(t)|Δ2t4(1α)(1Mk+1212)CΔ2t. (2.7)

    Next, for the term I2, by using the mean value theorem of integrals, we obtain:

    I2=|Δt2(1α)u(t12)(tεt12)exp[α1α(tk+12tε)]|maxt[0,T]|u(t)|Δ2t4(1α)CΔ2t, (2.8)

    where t0tεt12.

    For the term I3, we can easily obtain:

    I3maxt[0,T]|u(t)|Δ2t8(1α)tk+12t12exp[α1α(tk+12s)]dsCΔ2t. (2.9)

    Similarly, we can estimate the term I4 as follows:

    I4maxt[0,T]|u(t)|Δ2t8(1α)t12t0exp[α1α(tk+12s)]dsCΔ2t. (2.10)

    Finally, for the term I5, we can derive:

    I511α|O(Δ2t)|(1Mk+120)CΔ2t. (2.11)

    Based on the aforementioned estimates for the terms I1, ,I5, we can complete the proof of the Theorem.

    It is obvious that the approximation formula (2.4) is nonlocal since the value at node tk+12 for the CF fractional derivative is concerned with all the values of uj, j=0,1,,k,k+1, which means the computing complexity when apply the formula (2.4) to ODEs is of O(M2) and the memory requirement is O(M). In the following analysis, inspired by the work [14], we develop a fast algorithm based on the new discretization technique used in this paper, with which the computing complexity is reduced from O(M2) to O(M) and the memory requirement is O(1) instead of O(M).

    We split the derivative CF0Dαtu(tk+12) for k1 into two parts: the history part denoted by Ch(tk+12) and the local part denoted by Cl(tk+12), respectively, as follows

    CF0αtu(tk+12)=Ch(tk+12)+Cl(tk+12)=11αtk12t0u(s)exp[α1α(tk+12s)]ds+11αtk+12tk12u(s)exp[α1α(tk+12s)]ds. (3.1)

    For the local part Cl(tk+12), we have

    Cl(tk+12)=uk+1uk12αΔt(1Mk+12k12)+Rk+12l, (3.2)

    where Mkj is defined by (2.2), and the truncation error Rk+12l is

    Rk+12l=11αtk+12tk12[u(tk)(stk)+12u(ξk)(stk)2]exp[α1α(tk+12s)]ds+11α[u(tk)uk+1uk12Δt]tk+12tk12exp[α1α(tk+12s)]ds,k1. (3.3)

    For the history part Ch(tk+12), we rewrite it into a recursive formula when k2 in the following way

    Ch(tk+12)=11αtk32t0u(s)exp[α1α(tk+12s)]ds+11αtk12tk32u(s)exp[α1α(tk+12s)]dsC(1)h(tk+12)+C(2)h(tk+12), (3.4)

    and when k=1,

    Ch(t32)=11αt12t0u(s)exp[α1α(t32s)]dsC(2)h(t32). (3.5)

    Careful calculations show that

    C(1)h(tk+12)=exp(αΔtα1)Ch(tk12),k2. (3.6)

    For the term C(2)h(tk+12), by similar analysis for the Theorem 1, we have

    C(2)h(tk+12)={ukuk22αΔt(Mk+12k12Mk+12k32)+Rk+12h,if k2u1u0αΔt(M3212M320)+R32h,if k=1, (3.7)

    where, for k2,

    Rk+12h=11αtk12tk32[u(tk1)(stk1)+12u(ξk1)(stk1)2]exp[α1α(tk+12s)]ds+11α[u(tk1)ukuk22Δt]tk12tk32exp[α1α(tk+12s)]ds, (3.8)

    and, for k=1,

    R32h=11αt12t0[u(t12)(st12)+12u(ξ0)(st12)2]exp[α1α(tk+12s)]ds+11α[u(t12)u1u0Δt]t12t0exp[α1α(t32s)]ds. (3.9)

    For the truncation error Rk+12l and Rk+12h defined respectively by (3.3) and (3.8)-(3.9), we have the estimates that

    Lemma 1. Suppose that u(t)C3[0,T], then for any k1, Rk+12l and Rk+12h satisfy

    |Rk+12l|CΔ3t,|Rk+12h|CΔ3t, (3.10)

    where the constant C is free of k and Δt.

    Proof. To avoid repetition we just prove the estimate for Rk+12l, since the estimate for Rk+12h can be derived similarly. By the definition (3.3), we have

    |Rk+12l|11αmaxt[0,T]|u(t)||tk+12tk12(stk)exp[α1α(tk+12s)]ds|+12(1α)maxt[0,T]|u(t)|tk+12tk12(stk)2exp[α1α(tk+12s)]ds+CΔ2t1αtk+12tk12exp[α1α(tk+12s)]ds.L1+L2+L3. (3.11)

    Then, for the term L1, using integration by parts and the Taylor expansion for exp(t) at zero, we have

    L1CΔ2t(Mk+12k+12Mk+12k12)+Ctk+12tk12(stk)2exp[α1α(tk+12s)]dsCΔ2t[1(1αΔt1α|O(Δ2t)|)]+CΔ3tCΔ3t. (3.12)

    For the terms L2 and L3, by the mean value theorem of integrals we can easily get L2CΔ3t and L3CΔ3t. Hence, we have proved the estimate for Rk+12l.

    Now, based on the above analysis, and for a better presentation, we can introduce an operator CF0Fαt for the fast algorithm defined by

    CF0Fαtu(tk+12)=uk+1uk12αΔt(1Mk+12k12)+Fh(tk+12),k1, (3.13)

    where the history part Fh(tk+12) satisfies

    Fh(tk+12)={exp(αΔtα1)Fh(tk12)+ukuk22αΔt(Mk+12k12Mk+12k32),if k2u1u0αΔt(M3212M320),if k=1. (3.14)

    We note that with (3.13) and (3.14), uk+1 only depends on uk, uk1 and uk2, which reduces the algorithm complexity from O(M2) to O(M) and the memory requirement from O(M) to O(1).

    The following theorem confirms the efficiency of the operator CF0Fαt, with which we can still obtain the second-order convergence rate.

    Theorem 2. Assume u(t)C3[0,T] and the operator CF0Fαt is defined by (3.13). Then

    |CF0αtu(tk+12)CF0Fαtu(tk+12)|CΔ2t, (3.15)

    where the constant C is independent of k and Δt.

    Proof. Combining (3.1), (3.2), (3.4)-(3.5) with (3.13), (3.14), we can get

    |CF0αtu(tk+12)CF0Fαtu(tk+12)||Ch(tk+12)Fh(tk+12)|+|Rk+12l|,k2. (3.16)

    Then, next we mainly analyse the estimate for |Ch(tk+12)Fh(tk+12)|. Actually, by definitions we obtain

    Ch(tk+12)Fh(tk+12)=exp(αΔtα1)[Ch(tk12)Fh(tk12)]+Rk+12h. (3.17)

    We introduce some notations to simplify the presentation. Let

    Tk+1=Ch(tk+12)Fh(tk+12),L=exp(αΔtα1). (3.18)

    Then, the recursive formula (3.17) reads that

    Tk+1=Lk1T2+Rk+1h, (3.19)

    where the term Rk+1h is defined by

    Rk+1h=Lk2R2+12h+Lk3R3+12h++Rk+12h. (3.20)

    Now, by (3.7) and (3.14) as well as the Lemma 1, we can get

    |T2|=|R32h|CΔ3t, (3.21)

    and

    |Rk+1h|CΔ3t(Lk2+Lk3++1)=CΔ3t1Lk11L. (3.22)

    Noting here that L(0,1) we have

    1L=1exp(αΔtα1)αΔt1ααΔt2(1α). (3.23)

    Combining (3.19), (3.21)-(3.23), we obtain that

    |Tk+1|CΔ2t. (3.24)

    Now, with (3.16), (3.17), (3.24) and the Lemma 1, we complete the proof for the theorem.

    To check the second-order convergence rate and the efficiency of the fast algorithm for the novel approximation formula, we choose two fractional ordinary differential equation models with the domain I=(0,T]. Let Uk be the numerical solution for the chosen models at tk, and define U0=u(0). Define the error as Err(Δt)=max1kM|Ukuk|. For the sufficiently smooth function u(t), we have the approximation formulas for u(tk+12) and its first derivative dudt|t=tk+12:

    u(tk+12)=12(uk+uk+1)+O(Δ2t),dudt|t=tk+12=uk+1ukΔt+O(Δ2t). (4.1)

    Then, combined with results (2.5) and (3.15), the second-order convergence rate in the following tests is expected.

    First, we consider the following fractional ordinary differential equation with an initial value:

    {CF0αtu(t)+u(t)=g1(t), tˉI,u(0)=φ0. (4.2)

    Next, by taking the exact solution u(t)=t2 and the initial value φ0=0, we derive the source function as follows:

    g1(t)=2tα+t22(1α)α2[1exp(α1αt)]. (4.3)

    Direct scheme: Based on the novel approximation formula (2.4), we derive the following discrete system at tk+12:

    Case k=0

    (12+1M120αΔt)U1=(12+1M120αΔt)U0+g1(t12), (4.4)

    Case k1

    (12+1Mk+12k122αΔt)Uk+1=1Mk+12k122αΔtUk112UkMk+1212Mk+120αΔt(U1U0)12αΔtk1j=1(Uj+1Uj1)(Mk+12j+12Mk+12j12)+g1(tk+12). (4.5)

    Fast scheme: Applying the fast algorithm to the equation (4.2), we can get, for k1:

    (12+1Mk+12k122αΔt)Uk+1=1Mk+12k122αΔtUk112Uk+g1(tk+12)Fh(tk+12), (4.6)

    where Fh(tk+12) is defined by (3.14). For the case k=0, the formula (4.4) is used to derive U1.

    Let T=1. By calculating based on the direct scheme (4.4)–(4.5) and the fast scheme (4.6), we obtain the error results by choosing changed mesh sizes time step Δt=210,211,212,213,214 for different fractional parameters α=0.1,0.5,0.9, respectively, in Table 1. From the computed results, we can see that the convergence rate for both of the schemes is close to 2, which is in agreement with our theoretical result.

    Table 1.  Convergence results of Example 1.
    α Δt Direct scheme Fast scheme
    Err(Δt) Rate CPU(s) Err(Δt) Rate CPU(s)
    0.1 210 4.76834890E-07 0.0625 4.76834890E-07 0.0072
    211 1.19209015E-07 1.999996 0.2508 1.19209014E-07 1.999996 0.0067
    212 2.98022864E-08 1.999998 0.8776 2.98022864E-08 1.999998 0.0088
    213 7.45057174E-09 2.000000 3.3688 7.45057174E-09 2.000000 0.0100
    214 1.86264512E-09 1.999998 13.2991 1.86264512E-09 1.999998 0.0140
    0.5 210 4.76811290E-07 0.0716 4.76811290E-07 0.0067
    211 1.19206056E-07 1.999961 0.2768 1.19206056E-07 1.999961 0.0083
    212 2.98019183E-08 1.999980 0.9433 2.98019183E-08 1.999980 0.0091
    213 7.45052997E-09 1.999990 3.6143 7.45052997E-09 1.999990 0.0098
    214 1.86263893E-09 1.999995 14.2937 1.86263893E-09 1.999995 0.0135
    0.9 210 8.88400608E-07 0.0834 8.88400608E-07 0.0071
    211 2.22067159E-07 2.000214 0.2935 2.22067163E-07 2.000214 0.0074
    212 5.55126489E-08 2.000108 1.1452 5.55126587E-08 2.000107 0.0083
    213 1.38776781E-08 2.000050 4.3128 1.38775857E-08 2.000060 0.0099
    214 3.46934370E-09 2.000032 17.1595 3.46943940E-09 1.999982 0.0199

     | Show Table
    DownLoad: CSV

    Moreover, we manifest the efficiency of our fast scheme in two aspects: (i) by comparing with a published second-order scheme [35] which is denoted as Scheme I and (ii) with the direct scheme (4.5). In Figure 1, we take T=10 and plot the CPU time consumed for Scheme I and our fast scheme under the condition |Err(Δt)|107 for each α=0.1,0.2,,0.9. It is evident that our fast scheme is much more efficient. Further, to check the computing complexity of our direct and fast schemes, we depict in Figure 2 the CPU time in seconds needed with α=0.1 in the log-log coordinate system, by taking T=1, M=103×2m, m=1,2,,6. One can see that the fast scheme has reduced the computing complexity from O(M2) to O(M).

    Figure 1.  Comparison of CPU time between our fast method and the Scheme I with the error satisfying |Err(Δt)|107.
    Figure 2.  CPU time for Example 1 with α=0.1.

    We next consider another initial value problem of the fractional ordinary differential equation:

    {du(t)dt+CF0αtu(t)=g2(t), tˉI,u(0)=ψ0, (4.7)

    where the exact solution is u(t)=exp(2t), the initial value is ψ0=1, and the source function is:

    g2(t)=22α[exp(2t)exp(α1αt)]+2exp(2t). (4.8)

    Direct scheme: For the model (4.7), we formulate the Crank-Nicolson scheme based on the new approximation formula (2.4) at tk+12 as follows:

    Case k=0

    (1Δt+1M120αΔt)U1=(1Δt+1M120αΔt)U0+g2(t12), (4.9)

    Case k1

    (1Δt+1Mk+12k122αΔt)Uk+1=1Mk+12k122αΔtUk1+1ΔtUkMk+1212Mk+120αΔt(U1U0)12αΔtk1j=1(Uj+1Uj1)(Mk+12j+12Mk+12j12)+g2(tk+12). (4.10)

    Fast scheme: Applying the fast algorithm to the model (4.7), we have, for k1:

    (1Δt+1Mk+12k122αΔt)Uk+1=1Mk+12k122αΔtUk1+UkΔt+g1(tk+12)Fh(tk+12). (4.11)

    Similarly, we also compute and list the convergence data in Table 2 to show further the effectiveness of the novel approximation and the fast algorithm.

    Table 2.  Convergence results of Example 2.
    α Δt Direct scheme Fast scheme
    Err(Δt) Rate CPU(s) Err(Δt) Rate CPU(s)
    0.2 210 1.85604636E-06 0.0620 1.85604607E-06 0.0066
    211 4.64007204E-07 2.000014 0.2317 4.64006418E-07 2.000016 0.0075
    212 1.16000631E-07 2.000015 0.8691 1.16003294E-07 1.999979 0.0091
    213 2.90011650E-08 1.999950 3.3423 2.89965145E-08 2.000214 0.0107
    214 7.24794447E-09 2.000467 13.4227 7.25755100E-09 1.998325 0.0154
    0.4 210 1.78086742E-06 0.0638 1.78086759E-06 0.0071
    211 4.45204337E-07 2.000041 0.2516 4.45203661E-07 2.000043 0.0080
    212 1.11298823E-07 2.000029 0.9043 1.11298892E-07 2.000026 0.0095
    213 2.78237513E-08 2.000049 3.5048 2.78246395E-08 2.000004 0.0110
    214 6.95804836E-09 1.999562 14.8677 6.95847024E-09 1.999521 0.0143
    0.8 210 3.89820152E-07 0.0785 3.89820181E-07 0.0115
    211 9.73901404E-08 2.000961 0.2962 9.73902248E-08 2.000960 0.0079
    212 2.43394851E-08 2.000477 1.0595 2.43393918E-08 2.000484 0.0095
    213 6.08529405E-09 1.999900 4.1052 6.08559336E-09 1.999823 0.0116
    214 1.51925406E-09 2.001964 16.5226 1.51877799E-09 2.002487 0.0156

     | Show Table
    DownLoad: CSV

    From the computed data summarized in Table 2, both of the schemes have a second-order convergence rate, and the fast scheme indeed improves the efficiency of the novel approximation formula without losing too much precision. Similarly as the Example 1, we compare in Figure 3 the times for both of the methods under different M=102×2m, for α=0.9 and m=1,2,,6 in the log-log coordinate system. One can see clearly that the computing complexity for the direct scheme is O(M2), and for the fast scheme it is O(M).

    Figure 3.  CPU time for Example 2 with α=0.9.

    In this study, we constructed a novel discrete formula for approximating the CF fractional derivative and proved the second-order convergence rate for the novel approximation formula. To overcome the nonlocal property of the derivative, we proposed a fast algorithm that tremendously improves the efficiency of the approximation formula. Moreover, we demonstrated the fast algorithm maintains the second-order convergence rate. In future works, this novel approximation formula and fast algorithm can be applied with the finite element, finite difference, or other numerical methods to specific fractional differential equation models with Caputo-Fabrizio derivatives.

    The authors are grateful to the three anonymous referees and editors for their valuable comments and good suggestions which greatly improved the presentation of the paper. This work is supported by the National Natural Science Fund (11661058, 11761053), the Natural Science Fund of Inner Mongolia Autonomous Region (2017MS0107), the program for Young Talents of Science, and Technology in Universities of the Inner Mongolia Autonomous Region (NJYT-17-A07).

    The authors declare no conflict of interest.



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