The number 3600 at the end of solving the problem represents the final result.
In order to understand where the number 3600 came from at the end of solving the problem, we need to examine the steps taken during the solution process.
Let's assume we were solving a mathematical problem and arrived at the equation 120 + 3600 = 3720. The equation implies that by adding 120 and 3600 together, we obtain the sum of 3720. However, in this context, we are specifically interested in the origin of the number 3600.
To determine where this number came from, we would need to review the specific calculations or operations conducted before arriving at the final equation.
It could be the result of performing a series of mathematical operations such as multiplication, division, or exponentiation. The detailed calculations leading up to the addition of 120 and 3600 would provide the necessary context to understand the origin of the number 3600.
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If A and B are two Hermitian operators show that AB is Hermitian only if A and B commute.
The commutation of Hermitian operators has significant implications in quantum mechanics, where Hermitian operators represent observables, If two Hermitian operators commute, their product will also be Hermitian.
Let's consider two Hermitian operators, A and B. For AB to be Hermitian, it means that (AB)† = AB, where the dagger symbol (†) represents the Hermitian conjugate.
Taking the Hermitian conjugate of AB, we have (AB)† = B†A†. Since A and B are Hermitian operators, we know that A† = A and B† = B.
Substituting these values, we get (AB)† = BA.
For AB to be Hermitian, we require (AB)† = AB. Comparing this with (AB)† = BA, we can see that AB is Hermitian only if BA = AB.
In other words, AB is Hermitian if and only if A and B commute, meaning that they can be applied in any order without changing the result.
This result highlights an important property of Hermitian operators. If two Hermitian operators commute, their product will also be Hermitian.
However, if they do not commute, their product will not be Hermitian.
The commutation of Hermitian operators has significant implications in quantum mechanics, where Hermitian operators represent observables, and their commutation relation affects the simultaneous measurability of physical quantities.
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Express the following numbers in polar form (magnitude and phase). The angle is in degrees and between −180° and + 180°. (a) 19+i(27)= (b) −21+i(27)= (c) −16+i(−22)= (d) 23+i(−26)=
Complex numbers are:(a) 19+i(27)(b) −21+i(27)(c) −16+i(−22)(d) 23+i(−26)In order to express the given numbers in polar form (magnitude and phase), first we need to find the magnitude and the phase angle of the given complex number.(a)19+i(27).Polar form of 23+i(−26) is 5√219∠-48.21°
Here, x = 19 and y = 27 Magnitude, r = sqrt(x^2+y^2) = sqrt(19^2+27^2) = sqrt(1360) = 20*sqrt(17)Phase angle, θ = tan⁻¹(y/x) = tan⁻¹(27/19)≈53.13°Hence, polar form of 19+i(27) is 20√17∠53.13°.(b)−21+i(27)Here, x = -21 and y = 27 Magnitude, r = sqrt(x^2+y^2) = sqrt(21^2+27^2) = sqrt(1350) = 3*5*sqrt(6)Phase angle, θ = tan⁻¹(y/x) = tan⁻¹(-27/21)≈-53.13°Hence, polar form of −21+i(27) is 15√6∠-53.13°.(c)−16+i(−22)Here, x = -16 and y = -22 Magnitude, r = sqrt(x^2+y^2) = sqrt(16^2+22^2) = sqrt(820) = 2*2*5*sqrt(41)Phase angle, θ = tan⁻¹(y/x) = tan⁻¹(-22/-16)≈36.87°Hence, polar form of −16+i(−22) is 20√41∠36.87°.(d)23+i(−26)Here, x = 23 and y = -26Magnitude, r = sqrt(x^2+y^2) = sqrt(23^2+26^2) = sqrt(1095) = 5*sqrt(219)Phase angle, θ = tan⁻¹(y/x) = tan⁻¹(-26/23)≈-48.21°Hence, polar form of 23+i(−26) is 5√219∠-48.21°.
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Find all Nash equilibria for the following game: b. Hotelling model: Two ice-cream vendors decide where to set-up their stands on a linear beach that's broken into 100 segments. There are two customers per segment and each customer will go to the closest stand. Half will go to one and half will go to the other if they 're equal distance. The vendors make $1 profit per cone. Find all Nash equilibria for this Hotelling game. Clearly demonstrate that these equilibria must be the only ones. (I suggest you do this logically instead of writing out the entire 100×100 matrix.)
In the Hotelling model with two ice-cream vendors on a linear beach divided into 100 segments, there are two Nash equilibria: the "Corner Equilibria." These equilibria occur when each vendor sets up their stand at one of the extreme ends of the beach. These equilibria are unique and cannot be surpassed by any other configuration.
In the Hotelling model, the vendors aim to maximize their profits by attracting customers. Since customers go to the closest stand, each vendor wants to position themselves in a way that minimizes the distance to potential customers.
If one vendor sets up their stand at one end of the beach, the other vendor will position themselves at the opposite end to evenly split the customers. This creates a stable equilibrium as neither vendor can gain by moving closer to the other.
Any deviation from the corner equilibria would result in a longer distance for customers, making the deviating vendor less attractive. Therefore, the corner equilibria are the only stable outcomes in this game. Other configurations would lead to a disadvantage for the vendors and thus would not be sustainable Nash equilibria.
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