🧠:<think>Alright, so I've got this problem about a city planner looking into economic disparities across three neighborhoods: A, B, and C. The planner is using two different methods to analyze the data: z-scores and the Gini coefficient. I need to figure out how to approach both sub-problems.Starting with Sub-problem 1: Calculating z-scores for a household income of 50,000 in each neighborhood. I remember that a z-score tells you how many standard deviations an element is from the mean. The formula is z = (X - μ) / σ, where X is the value, μ is the mean, and σ is the standard deviation.So for each neighborhood, I'll plug in the numbers:- Neighborhood A: Mean = 40,000, SD = 8,000. So z = (50,000 - 40,000) / 8,000 = 10,000 / 8,000 = 1.25- Neighborhood B: Mean = 55,000, SD = 10,000. So z = (50,000 - 55,000) / 10,000 = (-5,000) / 10,000 = -0.5- Neighborhood C: Mean = 70,000, SD = 12,000. So z = (50,000 - 70,000) / 12,000 = (-20,000) / 12,000 ≈ -1.67Now, interpreting these z-scores. A positive z-score means the income is above the mean, negative means below. So in A, 50k is above average, in B, it's below average, and in C, it's significantly below average. This tells us about the relative standing within each neighborhood.Moving to Sub-problem 2: The Gini coefficients are given as 0.25, 0.35, and 0.40 for A, B, and C respectively. The planner wants to reduce each by 10%. So I need to calculate 10% of each Gini coefficient and subtract that from the original.For Neighborhood A: 10% of 0.25 is 0.025. So new Gini = 0.25 - 0.025 = 0.225For Neighborhood B: 10% of 0.35 is 0.035. So new Gini = 0.35 - 0.035 = 0.315For Neighborhood C: 10% of 0.40 is 0.04. So new Gini = 0.40 - 0.04 = 0.36Now, the question is which neighborhood should be prioritized for interventions. I think this depends on how much the Gini coefficient can be reduced and perhaps the current level of inequality. Neighborhood C has the highest Gini, so reducing it might have the most significant impact. Alternatively, maybe the one where the reduction is the largest in absolute terms, but since all are reduced by 10%, the absolute reduction is proportional to their original Gini. So C would have the largest absolute reduction (0.04), followed by B (0.035), then A (0.025). So maybe C should be prioritized because it has the highest inequality and the largest reduction in absolute terms.But wait, I should also consider the z-scores from Sub-problem 1. Maybe the neighborhoods with higher z-scores for a given income have more disparity? Or perhaps the combination of both measures. Hmm, not sure if the z-scores directly influence the prioritization based on Gini reduction. Maybe the Gini alone is sufficient for this part.I think the main point is that reducing the Gini coefficient by 10% in each neighborhood, and since C had the highest Gini, the impact there might be more noticeable. So the planner should prioritize Neighborhood C for interventions to achieve the most significant reduction in economic disparities.</think>

🧠:<think>Alright, so I've got this problem about Alex, a personal care assistant, who helps clients with exercises. There are two parts to the problem. Let me try to tackle them step by step.Starting with the first question: Calculate the total effort exerted by Alex and the client together over a 10-minute session. They want the result as a definite integral.Okay, so Alex's effort is given by A(t) = 5t² + 2t, and the client's effort is C(t) = 3t² + 4t + 1. The total effort together would be the sum of these two functions, right? So, I should add A(t) and C(t) together.Let me write that out:Total effort E(t) = A(t) + C(t) = (5t² + 2t) + (3t² + 4t + 1)Combine like terms:5t² + 3t² = 8t²2t + 4t = 6tAnd then the constant term is just +1.So, E(t) = 8t² + 6t + 1.Now, to find the total effort over a 10-minute session, I need to integrate E(t) from t=0 to t=10. That makes sense because integrating the effort over time gives the total effort exerted.So, the definite integral would be:∫₀¹⁰ (8t² + 6t + 1) dtI think that's the answer for the first part. Let me just double-check. Yes, adding the two functions and setting up the integral from 0 to 10. That seems correct.Moving on to the second question: Alex has to distribute his efforts optimally across three clients in a single day, such that the total effort he exerts is evenly distributed among them. Each client requires a 10-minute session. I need to find the function that represents the distribution of Alex's effort for each client and verify that the total effort for each client over their respective sessions is equal.Hmm, okay. So, first, I think I need to find the total effort Alex exerts over a 10-minute session. Then, since he has three clients, each client should get an equal share of that total effort. So, the function for each client would be scaled down by a factor of 3.Wait, let me think. If Alex is distributing his effort across three clients, each getting 10 minutes, does that mean he's working with each client for 10 minutes, but splitting his effort in some way? Or is he working with all three clients simultaneously? The problem says he has to distribute his efforts optimally across three clients in a single day, such that the total effort he exerts is evenly distributed among them.So, I think it means that over the day, he works with each client for 10 minutes, but his effort during each session is scaled so that the total effort per client is the same. So, perhaps the function A(t) is scaled by 1/3 for each client? Because he has three clients, so each client gets 1/3 of his effort.Wait, but the problem says "the total effort he exerts is evenly distributed among them." So, the total effort Alex exerts in the day is the sum of his efforts across all three clients. So, if each client requires a 10-minute session, and he has to distribute his effort such that each client gets an equal portion of his total effort.So, first, let's compute the total effort Alex exerts in one 10-minute session. Then, since he has three clients, each client would get 1/3 of that total effort.But wait, actually, no. Because he is working with each client for 10 minutes, so he has three separate 10-minute sessions. So, the total effort he exerts in the day is 3 times the effort of one session. But he needs to distribute his effort such that each client gets an equal share of his total effort.Wait, I'm getting confused. Let me parse the problem again."Alex has to distribute his efforts optimally across three clients in a single day, such that the total effort he exerts is evenly distributed among them. If each client requires a session of 10 minutes, find the function that represents the distribution of Alex's effort for each client, and verify that the total effort for each client over their respective sessions is equal."So, each client requires a session of 10 minutes. So, he has three separate 10-minute sessions. He needs to distribute his effort across these three sessions such that the total effort he exerts in each session is equal.Wait, no. It says "the total effort he exerts is evenly distributed among them." So, the total effort he exerts in the day is split equally among the three clients.So, first, compute the total effort Alex exerts in one 10-minute session, then multiply by three to get the total effort for the day. Then, each client should get 1/3 of that total effort.But wait, each client already has a 10-minute session. So, does that mean that for each client, the effort function is scaled so that the integral over 10 minutes is 1/3 of the total daily effort?Alternatively, maybe Alex is splitting his time or effort across the three clients in some way, but the problem says each client requires a session of 10 minutes. So, perhaps he is working with each client for 10 minutes, but his effort during each session is scaled so that the total effort per client is equal.Wait, perhaps it's that he is working with all three clients simultaneously, but that doesn't make much sense because he can't be in three places at once. So, more likely, he works with each client for 10 minutes, one after another, and during each session, he exerts effort, but he wants the total effort per client to be equal.But in that case, if he exerts the same effort function for each client, then each client would have the same total effort. But the problem says he has to distribute his efforts optimally across three clients such that the total effort he exerts is evenly distributed among them.Wait, maybe the idea is that instead of doing one 10-minute session with maximum effort, he splits his effort across three sessions, so each session has less effort, but the total effort over the day is the same as one session, but spread out.Wait, no, the problem says "the total effort he exerts is evenly distributed among them." So, the total effort he exerts in the day is split equally among the three clients.So, first, compute the total effort Alex exerts in one 10-minute session, which is the integral of A(t) from 0 to 10. Then, since he has three clients, each client should get 1/3 of that total effort.But each client requires a session of 10 minutes. So, he needs to adjust his effort function during each session so that the integral over 10 minutes is 1/3 of the original total effort.So, the original total effort for one client is ∫₀¹⁰ A(t) dt. For three clients, the total effort would be 3 * ∫₀¹⁰ A(t) dt. But he wants each client to receive 1/3 of that total effort. Wait, no, that would mean each client gets ∫₀¹⁰ A(t) dt, which is the same as before. That doesn't make sense.Wait, perhaps the total effort he exerts in the day is the same as one session, but spread across three clients. So, each client gets 1/3 of the effort.But the problem says "the total effort he exerts is evenly distributed among them." So, the total effort he exerts in the day is split equally among the three clients.So, first, compute the total effort he would exert in one 10-minute session: ∫₀¹⁰ A(t) dt.Then, since he has three clients, the total effort he exerts in the day is 3 * ∫₀¹⁰ A(t) dt. But he wants each client to receive an equal share of his total effort, so each client should get (1/3) * 3 * ∫₀¹⁰ A(t) dt = ∫₀¹⁰ A(t) dt. Wait, that brings us back to each client getting the same as before.I think I'm overcomplicating this. Maybe the idea is that instead of doing one session with full effort, he does three sessions, each with 1/3 of the effort, so that the total effort is the same as one session. But the problem says "the total effort he exerts is evenly distributed among them," which suggests that the total effort is split equally among the three clients.Wait, perhaps the total effort he exerts in the day is the same as one session, but he splits it across three clients, so each client gets 1/3 of the effort. But each client still requires a 10-minute session. So, he needs to adjust his effort function during each session so that the integral over 10 minutes is 1/3 of the original total effort.So, the original total effort for one client is ∫₀¹⁰ A(t) dt. Let's compute that first.Compute ∫₀¹⁰ (5t² + 2t) dt.The integral of 5t² is (5/3)t³, and the integral of 2t is t². So, the integral from 0 to 10 is:[(5/3)(10)^3 + (10)^2] - [0 + 0] = (5/3)(1000) + 100 = (5000/3) + 100 = (5000 + 300)/3 = 5300/3 ≈ 1766.666...So, the total effort for one client is 5300/3.If he has three clients, the total effort he exerts in the day is 3 * (5300/3) = 5300.But he wants this total effort to be evenly distributed among the three clients, meaning each client should get 5300 / 3 ≈ 1766.666... which is the same as the original. So, that doesn't change anything.Wait, that can't be. Maybe I'm misunderstanding.Alternatively, perhaps he is working with all three clients simultaneously, but that's not practical. So, maybe he works with each client for 10 minutes, but during each session, he exerts 1/3 of his usual effort, so that the total effort over the day is the same as one session.Wait, but the problem says "the total effort he exerts is evenly distributed among them." So, the total effort he exerts in the day is split equally among the three clients.So, if he works with each client for 10 minutes, and during each session, he exerts effort A(t), then the total effort for the day is 3 * ∫₀¹⁰ A(t) dt. But he wants each client to receive an equal share of his total effort, so each client should get (1/3) * 3 * ∫₀¹⁰ A(t) dt = ∫₀¹⁰ A(t) dt. Which again brings us back to the same thing.Wait, maybe the idea is that he is splitting his time or effort across the three clients in such a way that each client gets an equal amount of his effort. So, instead of doing one 10-minute session with full effort, he does three 10-minute sessions, each with 1/3 of the effort.But that would mean that the effort function for each client is (1/3)A(t). So, the function representing the distribution of Alex's effort for each client would be (1/3)(5t² + 2t) = (5/3)t² + (2/3)t.Then, the total effort for each client would be ∫₀¹⁰ [(5/3)t² + (2/3)t] dt.Let's compute that:Integral of (5/3)t² is (5/9)t³, and integral of (2/3)t is (1/3)t².So, evaluating from 0 to 10:(5/9)(1000) + (1/3)(100) = (5000/9) + (100/3) = (5000/9) + (300/9) = 5300/9 ≈ 588.888...But wait, the total effort for each client would be 5300/9, and since there are three clients, the total effort is 3*(5300/9) = 5300/3 ≈ 1766.666..., which is the same as the original total effort for one client.But the problem says "the total effort he exerts is evenly distributed among them." So, the total effort he exerts in the day is 5300/3, and each client gets 5300/9, which is 1/3 of the total effort.Wait, no. If the total effort he exerts in the day is 5300/3, and he has three clients, then each client gets 5300/9, which is 1/3 of the daily total. So, that seems to fit.But let me check again. The original total effort for one client is 5300/3. If he splits his effort across three clients, each client gets 1/3 of that, which is 5300/9. So, the function for each client is (1/3)A(t), and the total effort per client is 5300/9.So, the function representing the distribution of Alex's effort for each client is (5/3)t² + (2/3)t.Let me verify that the total effort for each client over their respective sessions is equal.Compute ∫₀¹⁰ [(5/3)t² + (2/3)t] dt = 5300/9 ≈ 588.888...And since there are three clients, the total effort is 3*(5300/9) = 5300/3 ≈ 1766.666..., which is the same as the original total effort for one client. So, it's evenly distributed.Therefore, the function for each client is (5/3)t² + (2/3)t.Wait, but the problem says "find the function that represents the distribution of Alex's effort for each client." So, it's (1/3)A(t) = (5/3)t² + (2/3)t.Yes, that makes sense.So, to recap:1. The total effort is the integral of (8t² + 6t + 1) from 0 to 10.2. The function for each client is (5/3)t² + (2/3)t, and the total effort per client is 5300/9, which is equal.I think that's the solution.</think>

🧠:<think>First, I need to determine how many disposable razors and safety razor blades the woman uses in a year. She shaves 3 times a week, so over 52 weeks, that's 156 shaves.Each disposable razor lasts for 2 weeks, which means she uses 1 razor every 2 weeks. Over a year, she would use 78 disposable razors.For the safety razor, each blade lasts for 5 shaves. With 156 shaves in a year, she would need 31.2 blades, which I'll round up to 32 blades.Next, I'll calculate the cost for both options. A pack of 8 disposable razors costs 16, so 78 razors would cost 1,560. For the safety razor blades, a pack of 50 costs 20, so 32 blades would cost 12.80.The cost savings from switching to a safety razor would be the difference between the two costs: 1,560 - 12.80 = 1,547.20.Now, for the environmental impact, I'll calculate the waste generated by each option. Each disposable razor weighs 20 grams, so 78 razors would generate 1,560 grams, or 1.56 kilograms of waste.Each safety razor blade weighs 2 grams, so 32 blades would generate 64 grams, or 0.064 kilograms of waste.The total reduction in waste mass would be the difference between the two: 1.56 kg - 0.064 kg = 1.496 kg, which is approximately 1.50 kilograms.</think>

🧠:<think>Okay, so I have this problem about scattering amplitudes and their symmetry properties, and then a partial wave expansion. Let me try to work through each part step by step.Starting with part 1: Symmetry Properties. The problem states that the scattering amplitude ( mathcal{A}(s,t,u) ) must be symmetric under the exchange of any two particles. I need to show that this implies the functional equations ( mathcal{A}(s,t,u) = mathcal{A}(s,u,t) = mathcal{A}(t,s,u) ).Hmm, so first, I should recall what the Mandelstam variables s, t, u represent. They are defined in terms of the momenta of the particles involved. Specifically, ( s = (p_1 + p_2)^2 ), ( t = (p_1 - p_3)^2 ), and ( u = (p_1 - p_4)^2 ). These variables are related by the equation ( s + t + u = 2(p_1^2 + p_2^2 + p_3^2 + p_4^2) ), but I might not need that immediately.The key point is that the scattering amplitude must be symmetric under particle exchange. So, what does exchanging particles mean in terms of these variables?Let me think about exchanging particles 3 and 4. If I swap p3 and p4, what happens to s, t, u? Let's see:- ( s = (p1 + p2)^2 ) remains the same because it doesn't involve p3 or p4.- ( t = (p1 - p3)^2 ) becomes ( (p1 - p4)^2 ), which is u.- Similarly, ( u = (p1 - p4)^2 ) becomes ( (p1 - p3)^2 ), which is t.So, exchanging particles 3 and 4 swaps t and u. Therefore, the amplitude must satisfy ( mathcal{A}(s,t,u) = mathcal{A}(s,u,t) ).Similarly, if I exchange particles 2 and 3, let's see how the variables change. Let me denote the momenta after exchange as p1, p3, p2, p4. Then:- ( s = (p1 + p3)^2 )- ( t = (p1 - p2)^2 )- ( u = (p1 - p4)^2 )Wait, but originally, s was ( (p1 + p2)^2 ). So, after exchanging 2 and 3, s becomes ( (p1 + p3)^2 ), which is different. However, in terms of the Mandelstam variables, s, t, u are symmetric in a certain way.Wait, maybe I should think in terms of the process. If we have an interaction where particles 1 and 2 are incoming, and 3 and 4 are outgoing, exchanging 2 and 3 would change the roles of incoming and outgoing particles. But in terms of the Mandelstam variables, s is the square of the center-of-mass energy, t is the momentum transfer, and u is another variable.Alternatively, perhaps it's better to consider that the scattering amplitude is a function of the Mandelstam variables, which are symmetric under certain permutations. Since the amplitude must be symmetric under any particle exchange, which corresponds to permutations of the Mandelstam variables.Wait, so if I exchange particles 1 and 2, what happens? Let's see:- ( s = (p2 + p1)^2 = s ), so s remains the same.- ( t = (p2 - p3)^2 )- ( u = (p2 - p4)^2 )But originally, t was ( (p1 - p3)^2 ) and u was ( (p1 - p4)^2 ). So, exchanging 1 and 2 changes t and u in a way that might not just swap them. Hmm, maybe I'm complicating things.Alternatively, perhaps the symmetry under exchange of any two particles implies that the amplitude is symmetric under the exchange of any two Mandelstam variables. Since s, t, u are related by the conservation of energy and momentum, exchanging particles can lead to permutations of these variables.Wait, let me think about the process. If you have four particles, exchanging any two should correspond to a permutation of the Mandelstam variables. For example, exchanging particles 1 and 3 would lead to a permutation of s, t, u.But maybe a better approach is to consider that the amplitude depends only on the invariant quantities, which are s, t, u. Since the amplitude must be symmetric under any permutation of the particles, which corresponds to permutations of the Mandelstam variables. Therefore, the amplitude must be symmetric under the exchange of any two of s, t, u.But wait, the Mandelstam variables are not independent; they satisfy ( s + t + u = text{constant} ). So, the symmetry under exchange of particles would impose that the amplitude is symmetric under the exchange of any two variables, but considering the relation between them.Wait, actually, in the case of four-point functions, the Mandelstam variables s, t, u are related by ( s + t + u = 2(p_1^2 + p_2^2 + p_3^2 + p_4^2) ), but in the center-of-mass frame, perhaps this simplifies. However, regardless, the key point is that the amplitude must be symmetric under the exchange of any two particles, which translates into the exchange of the corresponding Mandelstam variables.So, for example, exchanging particles 3 and 4 swaps t and u, as I saw earlier. Therefore, the amplitude must be symmetric under t ↔ u, which gives ( mathcal{A}(s,t,u) = mathcal{A}(s,u,t) ).Similarly, exchanging particles 2 and 3 would lead to a different permutation. Let me see: if I exchange 2 and 3, then the new s would be ( (p1 + p3)^2 ), which is different from the original s. But perhaps in terms of the Mandelstam variables, this corresponds to a permutation of s, t, u.Wait, maybe it's clearer if I consider that the amplitude must be symmetric under any permutation of the particles, which can be represented as permutations of the Mandelstam variables. Since s, t, u are related by the conservation laws, the amplitude must be symmetric under the exchange of any two variables, but considering that s + t + u is constant.Wait, but in reality, s, t, u are not independent variables because of the relation ( s + t + u = 2(p_1^2 + p_2^2 + p_3^2 + p_4^2) ). So, if we exchange particles, we might not just swap two variables but also change the third in a way that maintains this relation.Alternatively, perhaps the amplitude is symmetric under the exchange of any two variables, considering the relation. So, for example, if we swap s and t, then u becomes ( u' = s + t + u - s - t = u ). Wait, that doesn't make sense. Maybe I'm overcomplicating.Let me try a different approach. The scattering amplitude is a function of the Mandelstam variables, which are s, t, u. The symmetry under particle exchange implies that the function must be invariant under certain permutations of these variables.Specifically, exchanging particles 3 and 4 swaps t and u, so ( mathcal{A}(s,t,u) = mathcal{A}(s,u,t) ).Similarly, exchanging particles 1 and 2 would leave s unchanged but swap t and u in a different way? Wait, no. If I exchange 1 and 2, then s becomes ( (p2 + p1)^2 = s ), so s remains the same. But t becomes ( (p2 - p3)^2 ), and u becomes ( (p2 - p4)^2 ). But originally, t was ( (p1 - p3)^2 ) and u was ( (p1 - p4)^2 ). So, unless p1 and p2 are the same, which they aren't, this doesn't just swap t and u.Hmm, maybe I need to think about the process in terms of the s, t, u channels. In scattering, s is the square of the center-of-mass energy, t is the momentum transfer, and u is another variable. The amplitude must be symmetric under crossing symmetry, which relates different channels.Wait, crossing symmetry is a property where the amplitude in one channel is related to the amplitude in another channel with particles exchanged. For example, the amplitude in the s-channel is related to the t-channel and u-channel amplitudes.But in this problem, it's stated that the amplitude must be symmetric under the exchange of any two particles, which is a more general symmetry. So, perhaps this implies that the amplitude is symmetric under the exchange of any two Mandelstam variables.Wait, but s, t, u are not independent, so the symmetry can't be complete. However, the problem states that the amplitude must satisfy ( mathcal{A}(s,t,u) = mathcal{A}(s,u,t) = mathcal{A}(t,s,u) ). So, let's see what these imply.First, ( mathcal{A}(s,t,u) = mathcal{A}(s,u,t) ) is the symmetry under swapping t and u, which corresponds to exchanging particles 3 and 4, as I saw earlier.Then, ( mathcal{A}(s,t,u) = mathcal{A}(t,s,u) ) would correspond to swapping s and t. What does swapping s and t mean in terms of particle exchange?Swapping s and t would correspond to exchanging the roles of the s-channel and t-channel. In terms of particles, this might correspond to a different permutation, perhaps exchanging particles 2 and 3 or 1 and 3.Wait, let me think about it. If I exchange particles 1 and 3, what happens to s, t, u?- s becomes ( (p3 + p2)^2 ), which is different from the original s.- t becomes ( (p3 - p1)^2 = t ) (since t was ( (p1 - p3)^2 ), which is the same as ( (p3 - p1)^2 )).- u becomes ( (p3 - p4)^2 ), which is different from the original u.Hmm, so swapping 1 and 3 doesn't just swap s and t, but changes s and u as well. So, perhaps the symmetry under swapping s and t is not directly from a simple particle exchange, but rather from the fact that the amplitude must be symmetric under any permutation of the particles, which can lead to more complex permutations of the Mandelstam variables.Alternatively, maybe the amplitude is symmetric under the exchange of any two variables, considering the relation ( s + t + u = text{constant} ). So, if we swap s and t, then u becomes ( u' = s + t + u - s - t = u ), which doesn't make sense. Wait, no, because ( s + t + u ) is a constant, so if we swap s and t, u would remain the same? That can't be right.Wait, perhaps I'm overcomplicating. The problem states that the amplitude must satisfy ( mathcal{A}(s,t,u) = mathcal{A}(s,u,t) = mathcal{A}(t,s,u) ). So, let's take it step by step.First, ( mathcal{A}(s,t,u) = mathcal{A}(s,u,t) ) is straightforward because swapping t and u corresponds to swapping particles 3 and 4, which are outgoing particles. Since the amplitude is symmetric under their exchange, this equality holds.Next, ( mathcal{A}(s,t,u) = mathcal{A}(t,s,u) ). This would mean that swapping s and t leaves the amplitude unchanged. What particle exchange would correspond to swapping s and t?Swapping s and t would correspond to exchanging the roles of the s-channel and t-channel. In terms of particles, this might involve a more complex permutation, such as exchanging particles 1 and 2 with particles 3 and 4. Let me see:If I exchange particles 1 and 3, then:- The new s would be ( (p3 + p2)^2 ), which is different from the original s.- The new t would be ( (p3 - p1)^2 = t ) (since t was ( (p1 - p3)^2 )).- The new u would be ( (p3 - p4)^2 ), which is different from the original u.So, this doesn't just swap s and t, but changes s and u as well. Therefore, perhaps the symmetry under swapping s and t is not directly from a simple particle exchange, but rather from the fact that the amplitude must be symmetric under any permutation of the particles, which can lead to more complex permutations of the Mandelstam variables.Alternatively, perhaps the amplitude is symmetric under the exchange of any two variables, considering the relation ( s + t + u = text{constant} ). So, if we swap s and t, then u becomes ( u' = s + t + u - s - t = u ), which doesn't make sense. Wait, no, because ( s + t + u ) is a constant, so if we swap s and t, u would remain the same? That can't be right.Wait, maybe I'm approaching this wrong. The key is that the amplitude must be symmetric under any permutation of the particles, which corresponds to certain permutations of the Mandelstam variables. Since s, t, u are related, the amplitude must be symmetric under the exchange of any two variables, but considering the relation.So, for example, swapping s and t would require that ( mathcal{A}(s,t,u) = mathcal{A}(t,s,u) ), which is the second equality given in the problem. Similarly, swapping t and u gives the first equality.Therefore, the amplitude must satisfy both ( mathcal{A}(s,t,u) = mathcal{A}(s,u,t) ) and ( mathcal{A}(s,t,u) = mathcal{A}(t,s,u) ), which together imply that the amplitude is symmetric under any permutation of s, t, u.Wait, but s, t, u are not independent, so the symmetry can't be complete. However, the problem states that the amplitude must satisfy these specific functional equations, so perhaps that's the answer.So, to summarize, the scattering amplitude must be symmetric under the exchange of any two particles, which corresponds to swapping the Mandelstam variables t and u (giving ( mathcal{A}(s,t,u) = mathcal{A}(s,u,t) )) and swapping s and t (giving ( mathcal{A}(s,t,u) = mathcal{A}(t,s,u) )). Therefore, these functional equations must hold.Moving on to part 2: Partial Wave Expansion. The amplitude is expanded in terms of Legendre polynomials as ( mathcal{A}(s,t,u) = sum_{l=0}^{infty} (2l + 1) a_l(s) P_l(cos theta) ). I need to derive an expression for the coefficients ( a_l(s) ) in terms of ( mathcal{A} ) and the Legendre polynomials.I recall that the partial wave expansion is a standard technique in scattering theory, where the amplitude is decomposed into angular momentum components. Each term in the expansion corresponds to a specific angular momentum l, and the coefficients ( a_l(s) ) are related to the scattering amplitude.The general approach is to use the orthogonality of the Legendre polynomials. The orthogonality relation is:[int_{-1}^{1} P_l(x) P_{l'}(x) dx = frac{2}{2l + 1} delta_{ll'}]So, to find ( a_l(s) ), we can multiply both sides of the expansion by ( P_{l'}(cos theta) ) and integrate over ( cos theta ) from -1 to 1.Let me write that out:[int_{-1}^{1} mathcal{A}(s,t,u) P_{l'}(cos theta) d(cos theta) = sum_{l=0}^{infty} (2l + 1) a_l(s) int_{-1}^{1} P_l(cos theta) P_{l'}(cos theta) d(cos theta)]Using the orthogonality relation, the right-hand side becomes:[sum_{l=0}^{infty} (2l + 1) a_l(s) cdot frac{2}{2l + 1} delta_{ll'} = 2 a_{l'}(s)]Therefore, we have:[2 a_{l'}(s) = int_{-1}^{1} mathcal{A}(s,t,u) P_{l'}(cos theta) d(cos theta)]Solving for ( a_{l'}(s) ):[a_{l'}(s) = frac{1}{2} int_{-1}^{1} mathcal{A}(s,t,u) P_{l'}(cos theta) d(cos theta)]Since ( l' ) is just a dummy variable, we can write:[a_l(s) = frac{1}{2} int_{-1}^{1} mathcal{A}(s,t,u) P_l(cos theta) d(cos theta)]But wait, in the original expansion, ( mathcal{A} ) is expressed in terms of ( P_l(cos theta) ), so ( theta ) is the scattering angle. However, in terms of the Mandelstam variables, ( cos theta ) can be expressed in terms of t or u. Specifically, in the center-of-mass frame, ( cos theta = 1 - frac{t}{2s} ) or something similar, but I might not need that here.The key point is that the coefficients ( a_l(s) ) are obtained by integrating the amplitude multiplied by the corresponding Legendre polynomial over the scattering angle.So, putting it all together, the expression for ( a_l(s) ) is:[a_l(s) = frac{1}{2} int_{-1}^{1} mathcal{A}(s,t,u) P_l(cos theta) d(cos theta)]Alternatively, since ( cos theta ) is a function of t (or u), we can express the integral in terms of t. But the problem asks for the expression in terms of ( mathcal{A}(s,t,u) ) and the Legendre polynomials, so the above should suffice.Wait, but in the problem statement, the expansion is given as ( mathcal{A}(s,t,u) = sum_{l=0}^{infty} (2l + 1) a_l(s) P_l(cos theta) ). So, to find ( a_l(s) ), we use the orthogonality of the Legendre polynomials, which gives us the integral expression I derived.Therefore, the coefficients ( a_l(s) ) are given by:[a_l(s) = frac{1}{2} int_{-1}^{1} mathcal{A}(s,t,u) P_l(cos theta) d(cos theta)]But I should check the normalization. The expansion has a factor of ( (2l + 1) ), and the orthogonality gives a factor of ( frac{2}{2l + 1} ). So, when I solve for ( a_l(s) ), I get ( a_l(s) = frac{1}{2} int mathcal{A} P_l d(cos theta) ). That seems correct.Alternatively, sometimes the partial wave expansion is written with a factor of ( frac{1}{2} ) included in the integral, so this matches.So, to recap, the coefficients ( a_l(s) ) are obtained by integrating the scattering amplitude multiplied by the corresponding Legendre polynomial over the scattering angle, with a factor of ( frac{1}{2} ) included.I think that's the derivation. Let me just make sure I didn't miss any steps. We started with the expansion, multiplied both sides by ( P_{l'} ), integrated, used orthogonality, solved for ( a_l ). Yes, that seems right.</think>