A long-run manufacturing system can experience machine breakdown at any time for various reasons such as unskilled labor or outdated machinery technology. In an integrated green inventory model, the produced green products cannot all be perfect throughout a cycle, particularly when machines malfunction. Therefore, an inspection policy is introduced to clean the production process from unusable defect products, the correctness of which depends on the discussion of the inspected errors. The perfect products detected via the inspection process are delivered to the retailer as well as the market. To transport green products, it is essential to control the capacity of the containers and the quantities of green products transported per batch. In this study, the greenhouse gas equivalence factor of CO2 emissions is calculated for all green products' manufacturing and transportation mediums. These types of energies are used in the manufacturing process: electricity, natural gas, and coal. Whereas within transportation, four transportation modes are considered: railways, roadways, airways, and waterways. The retailer can agree to transport their inventories to the customers' house according to their requirement by requiring a third-party local agency via outsourcing criteria. The model solves the problem of CO2 emissions through production and transportation within the machine breakdown.
Citation: Bijoy Kumar Shaw, Isha Sangal, Biswajit Sarkar. Reduction of greenhouse gas emissions in an imperfect production process under breakdown consideration[J]. AIMS Environmental Science, 2022, 9(5): 658-691. doi: 10.3934/environsci.2022038
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A long-run manufacturing system can experience machine breakdown at any time for various reasons such as unskilled labor or outdated machinery technology. In an integrated green inventory model, the produced green products cannot all be perfect throughout a cycle, particularly when machines malfunction. Therefore, an inspection policy is introduced to clean the production process from unusable defect products, the correctness of which depends on the discussion of the inspected errors. The perfect products detected via the inspection process are delivered to the retailer as well as the market. To transport green products, it is essential to control the capacity of the containers and the quantities of green products transported per batch. In this study, the greenhouse gas equivalence factor of CO2 emissions is calculated for all green products' manufacturing and transportation mediums. These types of energies are used in the manufacturing process: electricity, natural gas, and coal. Whereas within transportation, four transportation modes are considered: railways, roadways, airways, and waterways. The retailer can agree to transport their inventories to the customers' house according to their requirement by requiring a third-party local agency via outsourcing criteria. The model solves the problem of CO2 emissions through production and transportation within the machine breakdown.
In this paper, we consider the Schrödinger operators
L=−△+V(x),x∈Rn,n≥3, |
where Δ=∑ni=1∂2∂2xi and V(x) is a nonnegative potential belonging to the reverse Hölder class RHq for some q≥n2. Assume that f is a nonnegative locally Lq(Rn) integrable function on Rn, then we say that f belongs to RHq (1<q≤∞) if there exists a positive constant C such that the reverse Hölder's inequality
(1|B(x,r)|∫B(x,r)|f(y)|qdy)1q≤C|B(x,r)|∫B(x,r)|f(y)|dy |
holds for x in Rn, where B(x,r) denotes the ball centered at x with radius r<∞ [1]. For example, the nonnegative polynomial V∈RH∞, in particular, |x|2∈RH∞.
Let the potential V∈RHq with q≥n2, and the critical radius function ρ(x) is defined as
ρ(x)=supr>0{r:1rn−2∫B(x,r)V(y)dy≤1},x∈Rn. | (1.1) |
We also write ρ(x)=1mV(x),x∈Rn. Clearly, 0<mV(x)<∞ when V≠0, and mV(x)=1 when V=1. For the harmonic oscillator operator (Hermite operator) H=−Δ+|x|2, we have mV(x)∼(1+|x|).
Thanks to the heat diffusion semigroup e−tL for enough good function f, the negative powers L−α2(α>0) related to the Schrödinger operators L can be written as
Iαf(x)=L−α2f(x)=∫∞0e−tLf(x)tα2−1dt,0<α<n. | (1.2) |
Applying Lemma 3.3 in [2] for enough good function f holds that
Iαf(x)=∫RnKα(x,y)f(y)dy,0<α<n, |
and the kernel Kα(x,y) satisfies the following inequality
Kα(x,y)≤Ck(1+|x−y|(mV(x)+mV(y)))k1|x−y|n−α. | (1.3) |
Moreover, we have Kα(x,y)≤C|x−y|n−α,0<α<n.
Shen [1] obtained Lp estimates of the Schrödinger type operators when the potential V∈RHq with q≥n2. For Schrödinger operators L=−Δ+V with V∈RHq for some q≥n2, Harboure et al. [3] established the necessary and sufficient conditions to ensure that the operators L−α2(α>0) are bounded from weighted strong and weak Lp spaces into suitable weighted BMOL(w) space and Lipschitz spaces when p≥nα. Bongioanni Harboure and Salinas proved that the fractional integral operator L−α/2 is bounded form Lp,∞(w) into BMOβL(w) under suitable conditions for weighted w [4]. For more backgrounds and recent progress, we refer to [5,6,7] and references therein.
Ramseyer, Salinas and Viviani in [8] studied the fractional integral operator and obtained the boundedness from strong and weak Lp(⋅) spaces into the suitable Lipschitz spaces under some conditions on p(⋅). In this article, our main interest lies in considering the properties of fractional integrals operator L−α2(α>0), related to L=−Δ+V with V∈RHq for some q≥n2 in variable exponential spaces.
We now introduce some basic properties of variable exponent Lebsegue spaces, which are used frequently later on.
Let p(⋅):Ω→[1,∞) be a measurable function. For a measurable function f on Rn, the variable exponent Lebesgue space Lp(⋅)(Ω) is defined by
Lp(⋅)(Ω)={f:∫Ω|f(x)s|p(x)dx<∞}, |
where s is a positive constant. Then Lp(⋅)(Ω) is a Banach space equipped with the follow norm
‖f‖Lp(⋅)(Ω):=inf{s>0:∫Ω|f(x)s|p(x)dx≤1}. |
We denote
p−:=essinfx∈Ωp(x) and p+:=esssupx∈Ωp(x). |
Let P(Rn) denote the set of all measurable functions p on Rn that take value in [1,∞), such that 1<p−(Rn)≤p(⋅)≤p+(Rn)<∞.
Assume that p is a real value measurable function p on Rn. We say that p is locally log-Hölder continuous if there exists a constant C such that
|p(x)−p(y)|≤Clog(e+1/|x−y|),x,y∈Rn, |
and we say p is log-Hölder continuous at infinity if there exists a positive constant C such that
|p(x)−p(∞)|≤Clog(e+|x|),x∈Rn, |
where p(∞):=lim|x|→∞p(x)∈R.
The notation Plog(Rn) denotes all measurable functions p in P(Rn), which states p is locally log-Hölder continuous and log-Hölder continuous at infinity. Moreover, we have that p(⋅)∈Plog(Rn), which implies that p′(⋅)∈Plog(Rn).
Definition 1.1. [8] Assume that p(⋅) is an exponent function on Rn. We say that a measurable function f belongs to Lp(⋅),∞(Rn), if there exists a constant C such that for t>0,
∫Rntp(x)χ{|f|>t}(x)dx≤C. |
It is easy to check that Lp(⋅),∞(Rn) is a quasi-norm space equipped with the following quasi-norm
‖f‖p(⋅),∞=inf{s>0:supt>0∫Rn(ts)p(x)χ{|f|>t}(x)dx≤1}. |
Next, we define LipLα,p(⋅) spaces related to the nonnegative potential V.
Definition 1.2. Let p(⋅) be an exponent function with 1<p−≤p+<∞ and 0<α<n. We say that a locally integrable function f∈LipLα,p(⋅)(Rn) if there exist constants C1,C2 such that for every ball B⊂Rn,
1|B|αn‖χB‖p′(⋅)∫B|f(x)−mBf|dx≤C1, | (1.4) |
and for R≥ρ(x),
1|B|αn‖χB‖p′(⋅)∫B|f(x)|dx≤C2, | (1.5) |
where mBf=1|B|∫Bf. The norm of space LipLα,p(⋅)(Rn) is defined as the maximum value of two infimum of constants C1 and C2 in (1.4) and (1.5).
Remark 1.1. LipLα,p(⋅)(Rn)⊂Lα,p(⋅)(Rn) is introduced in [8]. In particular, when p(⋅)=C for some constant, then LipLα,p(⋅)(Rn) is the usual weighted BMO space BMOβL(w), with w=1 and β=α−np [4].
Remark 1.2. It is easy to see that for some ball B, the inequality (1.5) leads to inequality (1.4) holding, and the average mBf in (1.4) can be replaced by a constant c in following sense
12‖f‖LipLα,p(⋅)≤supB∈Rninfc∈R1|B|αn‖χB‖p′(⋅)∫B|f(x)−c|dx≤‖f‖LipLα,p(⋅). |
In 2013, Ramseyer et al. in [8] studied the Lipschitz-type smoothness of fractional integral operators Iα on variable exponent spaces when p+>αn. Hence, when p+>αn, it will be an interesting problem to see whether or not we can establish the boundedness of fractional integral operators L−α2(α>0) related to Schrödinger operators from Lebesgue spaces Lp(⋅) into Lipschitz-type spaces with variable exponents. The main aim of this article is to answer the problem above.
We now state our results as the following two theorems.
Theorem 1.3. Let potential V∈RHq for some q≥n/2 and p(⋅)∈Plog(Rn). Assume that 1<p−≤p+<n(α−δ0)+ where δ0=min{1,2−n/q}, then the fractional integral operator Iα defined in (1.2) is bounded from Lp(⋅)(Rn) into LipLα,p(⋅)(Rn).
Theorem 1.4. Let the potential V∈RHq with q≥n/2 and p(⋅)∈Plog(Rn). Assume that 1<p−≤p+<n(α−δ0)+ where δ0=min{1,2−n/q}. If there exists a positive number r0 such that p(x)≤p∞ when |x|>r0, then the fractional integral operator Iα defined in (1.2) is bounded from Lp(⋅),∞(Rn) into LipLα,p(⋅)(Rn).
To prove Theorem 1.3, we first need to decompose Rn into the union of some disjoint ball B(xk,ρ(xk))(k≥1) according to the critical radius function ρ(x) defined in (1.1). According to Lemma 2.6, we establish the necessary and sufficient conditions to ensure f∈LipLα,p(⋅)(Rn). In order to prove Theorem 1.3, by applying Corollary 1 and Remark 1.2, we only need to prove that the following two conditions hold:
(ⅰ) For every ball B=B(x0,r) with r<ρ(x0), then
∫B|Iαf(x)−c|dx≤C|B|αn‖χB‖p′(⋅)‖f‖p(⋅); |
(ⅱ) For any x0∈Rn, then
∫B(x0,ρ(x0))Iα(|f|)(x)dx≤C|B(x0,ρ(x0))|αn‖χB(x0,ρ(x0))‖p′(⋅)‖f‖p(⋅). |
In order to check the conditions (ⅰ) and (ⅱ) above, we need to find the accurate estimate of kernel Kα(x,y) of fractional integral operator Iα (see Lemmas 2.8 and 2.9, then use them to obtain the proof of this theorem; the proof of the Theorem 1.4 proceeds identically).
The paper is organized as follows. In Section 2, we give some important lemmas. In Section 3, we are devoted to proving Theorems 1.3 and 1.4.
Throughout this article, C always means a positive constant independent of the main parameters, which may not be the same in each occurrence. B(x,r)={y∈Rn:|x−y|<r}, Bk=B(x0,2kR) and χBk are the characteristic functions of the set Bk for k∈Z. |S| denotes the Lebesgue measure of S. f∼g means C−1g≤f≤Cg.
In this section, we give several useful lemmas that are used frequently later on.
Lemma 2.1. [9] Assume that the exponent function p(⋅)∈P(Rn). If f∈Lp(⋅)(Rn) and g∈Lp′(⋅)(Rn), then
∫Rn|f(x)g(x)|dx≤rp‖f‖Lp(⋅)(Rn)‖g‖Lp′(⋅)(Rn), |
where rp=1+1/p−−1/p+.
Lemma 2.2. [8] Assume that p(⋅)∈Plog(Rn) and 1<p−≤p+<∞, and p(x)≤p(∞) when |x|>r0>1. For every ball B and f∈Lp(⋅),∞ we have
∫B|f(x)|dx≤C‖f‖Lp(⋅),∞‖χB‖Lp′(⋅), |
where the constant C only depends on r0.
Fo the following lemma see Corollary 4.5.9 in [10].
Lemma 2.3. Let p(⋅)∈Plog(Rn), then for every ball B⊂Rn we have
‖χB‖p(⋅)∼|B|1p(x),if|B|≤2n,x∈B, |
and
‖χB‖p(⋅)∼|B|1p(∞),if|B|≥1. |
Lemma 2.4. Assume that p(⋅)∈Plog(Rn), then for all balls B and all measurable subsets S:=B(x0,r0)⊂B:=B(x1,r1) we have
‖χS‖p′(⋅)‖χB‖p′(⋅)≤C(|S||B|)1−1p−, ‖χB‖p′(⋅)‖χS‖p′(⋅)≤C(|B||S|)1−1p+. | (2.1) |
Proof. We only prove the first inequality in (2.1), and the second inequality in (2.1) proceeds identically. We consider three cases below by applying Lemma 2.3, and it holds that
1) if |S|<1<|B|, then ‖χS‖p′(⋅)‖χB‖p′(⋅)∼|S|1p′(xS)|B|1p′(∞)≤(|S||B|)1(p′)+=(|S||B|)1−1p−;
2) if 1≤|S|<|B|, then ‖χS‖p′(⋅)‖χB‖p′(⋅)∼|S|1p′(∞)|B|1p′(∞)≤(|S||B|)1(p′)+=(|S||B|)1−1p−;
3) if |S|<|B|<1, then ‖χS‖p′(⋅)‖χB‖p′(⋅)∼|S|1p′(xS)|B|1p′(xS)|B|1p′(xS)−1p′(xB)≤C(|S||B|)1(p′)+=C(|S||B|)1−1p−, where xS∈S and xB∈B.
Indeed, since |xB−xS|≤2r1, by using the local-Hölder continuity of p′(x) we have
|1p′(xS)−1p′(xB)|log1r1≤log1r1log(e+1|xS−xB|)≤log1r1log(e+12r1)≤C. |
We end the proof of this lemma.
Remark 2.1. Thanks to the second inequality in (2.1), it is easy to prove that
‖χ2B‖p′(⋅)≤C‖χB‖p′(⋅). |
Lemma 2.5. [1] Suppose that the potential V∈Bq with q≥n/2, then there exists positive constants C and k0 such that
1) ρ(x)∼ρ(y) when |x−y|≤Cρ(x);
2) C−1ρ(x)(1+|x−y|ρ(x))−k0≤ρ(y)≤Cρ(x)(1+|x−y|ρ(x))k0/(k0+1).
Lemma 2.6. [11] There exists a sequence of points {xk}∞k=1 in Rn such that Bk:=B(xk,ρ(xk)) satisfies
1) Rn=⋃kBk,
2) For every k≥1, then there exists N≥1 such that card {j:4Bj∩4Bk≠∅}≤N.
Lemma 2.7. Assume that p(⋅)∈P(Rn) and 0<α<n. Let sequence {xk}∞k=1 satisfy the propositions of Lemma 2.6. Then a function f∈LipLα,p(⋅)(Rn) if and only if f satisfies (1.4) for every ball, and
1|B(xk,ρ(xk))|αn‖χB(xk,ρ(xk))‖p′(⋅)∫B(xk,ρ(xk))|f(x)|dx≤C,forallk≥1. | (2.2) |
Proof. Let B:=B(x,R) denote a ball with center x and radius R>ρ(x). Noting that f satisfies (1.4), and thanks to Lemma 2.6 we obtain that the set G={k:B∩Bk≠∅} is finite.
Applying Lemma 2.5, if z∈Bk∩B, we get
ρ(xk)≤Cρ(z)(1+|xk−z|ρ(xk))k0≤C2k0ρ(z)≤C2k0ρ(x)(1+|x−z|ρ(x))k0k0+1≤C2k0ρ(x)(1+Rρ(x))≤C2k0R. |
Thus, for every k∈G, we have Bk⊂CB.
Thanks to Lemmas 2.4 and 2.6, it holds that
∫B|f(x)|dx=∫B⋂⋃kBk|f(x)|dx=∫⋃k∈G(B⋂Bk)|f(x)|dx≤∑k∈G∫B∩Bk|f(x)|dx≤∑k∈G∫Bk|f(x)|dx≤C∑k∈G|Bk|αn‖χBk‖p′(⋅)≤C|B|αn‖χB‖p′(⋅). |
The proof of this lemma is completed.
Corollary 1. Assume that p(⋅)∈P(Rn) and 0<α<n, then a measurable function f∈LipLα,p(⋅) if and only if f satisfies (1.4) for every ball B(x,R) with radius R<ρ(x) and
1|B(x,ρ(x))|αn‖χB(x,ρ(x))‖p′(⋅)∫B(x,ρ(x))|f(x)|dx≤C. | (2.3) |
Let kt(x,y) denote the kernel of heat semigroup e−tL associated to L, and Kα(x,y) be the kernel of fractional integral operator Iα, then it holds that
Kα(x,y)=∫∞0kt(x,y)tα2dt. | (2.4) |
Some estimates of kt are presented below.
Lemma 2.8. [12] There exists a constant C such that for N>0,
kt(x,y)≤Ct−n/2e−|x−y|2Ct(1+√tρ(x)+√tρ(y))−N,x,y∈Rn. |
Lemma 2.9. [13] Let 0<δ<min(1,2−nq). If |x−x0|<√t, then for N>0 the kernel kt(x,y) defined in (2.4) satisfies
|kt(x,y)−kt(x0,y)|≤C(|x−x0|√t)δt−n/2e−|x−y|2Ct(1+√tρ(x)+√tρ(y))−N, |
for all x,y and x0 in Rn.
In this section, we are devoted to the proof of Theorems 1.3 and 1.4. To prove Theorem 1.3, thanks to Corollary 1 and Remark 1.2, we only need to prove that the following two conditions hold:
(ⅰ) For every ball B=B(x0,r) with r<ρ(x0), then
∫B|Iαf(x)−c|dx≤C|B|αn‖χB‖p′(⋅)‖f‖p(⋅); |
(ⅱ) For any x0∈Rn, then
∫B(x0,ρ(x0))Iα(|f|)(x)dx≤C|B(x0,ρ(x0))|αn‖χB(x0,ρ(x0))‖p′(⋅)‖f‖p(⋅). |
We now begin to check that these conditions hold. First, we prove (ⅱ).
Assume that B=B(x0,R) and R=ρ(x0). We write f=f1+f2, where f1=fχ2B and f2=fχRn∖2B. Hence, by the inequality (1.3), we have
∫BIα(|f1|)(x)dx=∫BIα(|fχ2B|)(x)dx≤C∫B∫2B|f(y)||x−y|n−αdydx. |
Applying Tonelli theorem, Lemma 2.1 and Remark 1.2, we get the following estimate
∫BIα(|f1|)(x)dx≤C∫2B|f(y)|∫Bdx|x−y|n−αdy≤CRα∫2B|f(y)|dy≤C|B|αn‖χB‖p′(⋅)‖f‖p(⋅). | (3.1) |
To deal with f2, let x∈B and we split Iαf2 as follows:
Iαf2(x)=∫R20e−tLf2(x)tα2−1dt+∫∞R2e−tLf2(x)tα2−1dt:=I1+I2. |
For I1, if x∈B and y∈Rn∖2B, we note that |x0−y|<|x0−x|+|x−y|<C|x−y|. By Lemma 2.8, it holds that
I1=|∫R20∫Rn∖2Bkt(x,y)f(y)dytα2−1dt|≤C∫R20∫Rn∖2Bt−n2e−|x−y|2t|f(y)|dytα2−1dt≤C∫R20t−n+α2−1∫Rn∖2B(t|x−y|2)M/2|f(y)|dydt≤C∫R20tM−n+α2−1dt∫Rn∖2B|f(y)||x0−y|Mdy, |
where the constant C only depends the constant M.
Applying Lemma 2.1 to the last integral, we get
∫Rn∖2B|f(y)||x0−y|Mdy=∞∑i=1∫2i+1B∖2iB|f(y)||x0−y|Mdy≤∞∑i=1(2iR)−M∫2i+1B|f(y)|dy≤C∞∑i=1(2iR)−M‖χ2i+1B‖p′(⋅)‖f‖p(⋅). |
By using Lemma 2.4, we arrive at the inequality
∫Rn∖2B|f(y)||x0−y|Mdy≤C∞∑i=1(R)−M(2i)n−np+−M‖χB‖p′(⋅)‖f‖p(⋅)≤CR−M‖f‖p(⋅)‖χB‖p′(⋅). | (3.2) |
Here, the series above converges when M>n−np+. Hence, for such M,
|∫R20e−tLf2(x)tα2−1dt|≤CR−M‖f‖p(⋅)‖χB‖p′(⋅)∫R20tM−n+α2−1dt≤C|B|αn−1‖f‖p(⋅)‖χB‖p′(⋅). |
For I2, thanks to Lemma 2.8, we may choose M as above and N≥M, then it holds that
|∫∞R2e−tLf2(x)tα−22dt|=|∫∞R2∫Rn∖2Bkt(x,y)f(y)dytα−22dt|≤C∫∞R2∫Rn∖2Btα−n−N−22ρ(x)Ne−|x−y|2t|f(y)|dydt≤Cρ(x)N∫∞R2tα−n−N−22∫Rn∖2B(t|x−y|2)M/2|f(y)|dydt. |
As x∈B, thanks to Lemma 2.5, ρ(x)∼ρ(x0)=R. Hence we have
|∫∞R2e−tLf2(x)tα2−1dt|≤CRN∫∞R2tM+α−n−N2−1dt∫Rn∖2B|f(y)||x0−y|Mdy. |
Since M+α−n−N<0, the integral above for variable t converges, and by applying inequality (3.2) we have
|∫∞R2e−tLf2(x)tα2−1dt|≤C|B|αn−1‖f‖p(⋅)‖χB‖p′(⋅), |
thus we have proved (ⅱ).
We now begin to prove that the condition (ⅰ) holds. Let B=B(x0,r) and r<ρ(x0). We set f=f1+f2 with f1=fχ2B and f2=fχRn∖2B. We write
cr=∫∞r2e−tLf2(x0)tα2−1dt. | (3.3) |
Thanks to (3.1), it holds that
∫B|Iα(f(x))−cr|≤∫BIα(|f1|)(x)dx+∫B|Iα(f2)(x)−cr|dx≤C|B|αn−1‖χB‖p′(⋅)‖f‖p(⋅)+∫B|Iα(f2)(x)−cr|dx. |
Let x∈B and we split Iαf2(x) as follows:
Iαf2(x)=∫r20e−tLf2(x)tα2−1dt+∫∞r2e−tLf2(x)tα2−1dt:=I3+I4. |
For I3, by the same argument it holds that
I3=|∫r20e−tLf2(x)tα2−1dt|≤C|B|αn−1‖f‖p(⋅)‖χB‖p′(⋅). |
For I4, by Lemma 2.9 and (3.3), it follows that
|∫∞r2e−tLf2(x)tα2−1dt−cr|≤∫∞r2∫Rn∖2B|kt(x,y)−kt(x0,y)||f(y)|dytα2−1dt≤Cδ∫∞r2∫Rn∖2B(|x−x0|√t)δt−n/2e−|x−y|2Ct|f(y)|dytα2−1dt≤Cδrδ∫Rn∖2B|f(y)|∫∞r2t−(n−α+δ)/2e−|x−y|2Ctdttdy. |
Let s=|x−y|2t, then we obtain the following estimate
|∫∞r2e−tLf2(x)tα2−1dt−cr|≤Cδrδ∫Rn∖2B|f(y)||x−y|n−α+δdy∫∞0sn−α+δ2e−sCdss. |
Notice that the integral above for variable s is finite, thus we only need to compute the integral above for variable y. Thanks to inequality (3.2), it follows that
|∫∞r2e−tLf2(x)tα2−1dt−cr|≤Cδrδ∫Rn∖2B|f(y)||x−y|n−α+δdy≤C∞∑i=1Rα−n(2i)α−np+−δ‖χB‖p′(⋅)‖f‖p(⋅)≤C|B|α−nn‖f‖p(⋅)‖χB‖p′(⋅), |
so (ⅰ) is proved.
Remark 3.1. By the same argument as the proof of Theorem 1.3, thanks to Lemma 2.2 we immediately obtained that the conclusions of Theorem 1.4 hold.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
Ping Li is partially supported by NSFC (No. 12371136). The authors would like to thank the anonymous referees for carefully reading the manuscript and providing valuable suggestions, which substantially helped in improving the quality of this paper. We also thank Professor Meng Qu for his useful discussions.
The authors declare there are no conflicts of interest.
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