Problem: Solving a System of Nonlinear Equations

In applied mathematics, solving nonlinear systems of equations is a common challenge.

These systems appear in engineering, physics, and optimization problems.

Problem Statement

We need to solve the following system of nonlinear equations:
$$
f_1(x, y) = x^2 + y^2 - 1 = 0
$$
$$
f_2(x, y) = x^3 - y = 0
$$

Here, $ f_1(x, y) $ represents a unit circle, and $ f_2(x, y) $ represents a cubic curve.

We aim to find the intersection points of these two curves.

Approach

1. Use a numerical solver to find solutions to the equations.
2. Visualize the results by plotting both curves and highlighting the intersection points.

Python Implementation

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import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import fsolve

# Define the system of equations
def equations(vars):
    x, y = vars
    f1 = x**2 + y**2 - 1  # Equation of the unit circle
    f2 = x**3 - y         # Equation of the cubic curve
    return [f1, f2]

# Initial guesses for solutions
initial_guesses = [
    (0.5, 0.5),   # Near the first intersection
    (-0.5, 0.5),  # Near the second intersection
    (-0.5, -0.5), # Near the third intersection
    (0.5, -0.5)   # Near the fourth intersection
]

# Solve the system for each initial guess
solutions = [fsolve(equations, guess) for guess in initial_guesses]

# Unique solutions (rounded to avoid numerical artifacts)
unique_solutions = np.unique(np.round(solutions, decimals=5), axis=0)

# Print the solutions
print("Intersection points:")
for sol in unique_solutions:
    print(f"x = {sol[0]:.5f}, y = {sol[1]:.5f}")

# Visualization
x = np.linspace(-1.5, 1.5, 400)
y_circle = np.sqrt(1 - x**2)  # y-values for the unit circle (top half)
y_cubic = x**3                # y-values for the cubic curve

plt.figure(figsize=(8, 8))
plt.plot(x, y_circle, label="Unit Circle (top half)", color="blue")
plt.plot(x, -y_circle, label="Unit Circle (bottom half)", color="blue", linestyle="--")
plt.plot(x, y_cubic, label="Cubic Curve", color="green")

# Mark the solutions
for sol in unique_solutions:
    plt.scatter(sol[0], sol[1], color="red", zorder=5)
    plt.text(sol[0], sol[1], f"({sol[0]:.2f}, {sol[1]:.2f})", fontsize=10, color="red")

# Formatting the plot
plt.title("Intersection of Unit Circle and Cubic Curve", fontsize=14)
plt.xlabel("x", fontsize=12)
plt.ylabel("y", fontsize=12)
plt.axhline(0, color="black", linewidth=0.7, linestyle="--", alpha=0.7)
plt.axvline(0, color="black", linewidth=0.7, linestyle="--", alpha=0.7)
plt.grid(alpha=0.3)
plt.legend()
plt.axis("equal")  # Ensure aspect ratio is square for proper visualization
plt.show()

Explanation of Code

1. System of Equations:

   • $ f_1(x, y) = x^2 + y^2 - 1 $: Represents the unit circle.
   • $ f_2(x, y) = x^3 - y $: Represents the cubic curve.

2. Numerical Solver:

   • scipy.optimize.fsolve is used to find approximate solutions to the nonlinear equations starting from initial guesses.

3. Visualization:

   • The unit circle and cubic curve are plotted.
   • The intersection points are marked and labeled on the graph.

Expected Output

1. Intersection Points:

   1 2 3 4

   x = -0.82603, y = -0.56362 x = -0.48198, y = 0.48869 x = 0.48198, y = -0.48869 x = 0.82603, y = 0.56362

2. Graph:

   • A plot showing the unit circle and cubic curve with red dots highlighting their intersection points.

Here’s a realistic integration problem with a $Python$ solution and graphical representation:

Problem

A car is accelerating from rest along a straight road. The acceleration $ a(t) $ (in $ \text{m/s}^2 $) as a function of time $ t $ (in seconds) is given by:
$$
a(t) = 3t - 0.2t^2 \quad \text{for } 0 \leq t \leq 15.
$$

1. Find the velocity $ v(t) $ of the car as a function of time by integrating the acceleration.
2. Calculate the total distance traveled by the car in the $15$-second interval by integrating the velocity.
3. Visualize $ a(t) $, $ v(t) $, and the distance $ s(t) $ on a graph.

Solution Explanation

1. Integration of Acceleration:
   Velocity is the integral of acceleration:
   $$
   v(t) = \int a(t) , dt + C,
   $$
   where $ C $ is the integration constant. Since the car starts from rest $( v(0) = 0 $), $ C = 0 $.

2. Integration of Velocity:
   Distance is the integral of velocity:
   $$
   s(t) = \int v(t) , dt.
   $$

3. Visualization:
   Plot $ a(t) $, $ v(t) $, and $ s(t) $ on the same graph to understand how acceleration, velocity, and distance evolve over time.

Python Implementation

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import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import quad

# Acceleration function
def acceleration(t):
    return 3 * t - 0.2 * t**2

# Velocity function (integrating acceleration)
def velocity(t):
    v, _ = quad(acceleration, 0, t)  # Integrate acceleration from 0 to t
    return v

# Distance function (integrating velocity)
def distance(t):
    s, _ = quad(velocity, 0, t)  # Integrate velocity from 0 to t
    return s

# Generate data
time = np.linspace(0, 15, 500)
acc_values = acceleration(time)
vel_values = [velocity(t) for t in time]
dist_values = [distance(t) for t in time]

# Plotting
plt.figure(figsize=(10, 6))

# Acceleration
plt.plot(time, acc_values, label='Acceleration $a(t)$', color='red')
# Velocity
plt.plot(time, vel_values, label='Velocity $v(t)$', color='blue')
# Distance
plt.plot(time, dist_values, label='Distance $s(t)$', color='green')

# Annotations and labels
plt.title("Car's Acceleration, Velocity, and Distance over Time", fontsize=14)
plt.xlabel("Time (s)", fontsize=12)
plt.ylabel("Value", fontsize=12)
plt.axhline(0, color='black', linewidth=0.7, linestyle='--', alpha=0.7)
plt.legend()
plt.grid(alpha=0.3)

# Display the plot
plt.show()

# Total distance traveled
total_distance = distance(15)
print(f"Total distance traveled in 15 seconds: {total_distance:.2f} meters")

Explanation of Code

1. Acceleration Function:
   $ a(t) = 3t - 0.2t^2 $ defines the instantaneous acceleration.
2. Integration:
   • scipy.integrate.quad is used for numerical integration of $ a(t) $ and $ v(t) $.
3. Visualization:
   • All three functions $ a(t), v(t), s(t) $ are plotted together for easy comparison.
4. Total Distance:
   • The result of $ s(15) $ gives the total distance traveled by the car in $15$ seconds.

Expected Output

This graph illustrates the relationship between acceleration, velocity, and distance over time for a car accelerating along a straight road:

1. Red Curve (Acceleration, $ a(t) $):

   • The car’s acceleration starts at a high value, increases initially, and then decreases to zero at $ t = 15 $ seconds. This indicates the car is initially speeding up but stops accelerating as $ t $ approaches $15$ seconds.

2. Blue Curve (Velocity, $ v(t) $):

   • Velocity increases over time, but the rate of increase slows down as acceleration decreases. This reflects that while the car continues to gain speed, it does so more gradually as $ t $ approaches $15$ seconds.

3. Green Curve (Distance, $ s(t) $):

   • The distance traveled by the car increases monotonically and steeply. The curvature reflects the combined effect of velocity and acceleration, resulting in greater distances covered as time progresses.

4. Total Distance Traveled:

   • The text at the bottom indicates the total distance traveled by the car in $15$ seconds, which is approximately 843.75 meters.

This visualization effectively demonstrates how changes in acceleration impact velocity and how these two factors contribute to the total distance covered by the car.

Here’s a challenging probability problem with a $Python$ solution and a graphical visualization:

Problem

A factory produces items that pass through two quality control tests: Test $A$ and Test $B$.

The probabilities are as follows:

1. The probability of passing Test $A$ is $( P(A) = 0.7 )$.
2. The probability of passing Test $B$ is $( P(B) = 0.8 )$.
3. The probability of passing both tests is $( P(A \cap B) = 0.6 )$.

An item is selected at random.

Let $( X )$ denote the number of tests the item passes.

Find and visualize the probability mass function ($PMF$) of $( X )$, i.e., the probabilities of passing $0$, $1$, or $2$ tests.

Solution Explanation

1. Definitions and Relationships:

• $ P(X = 0) $: Probability of failing both tests.
  $ P(X = 0) = 1 - P(A \cup B) $, where $ P(A \cup B) = P(A) + P(B) - P(A \cap B) $.
• $ P(X = 1) $: Probability of passing exactly one test.
  $ P(X = 1) = P(A) + P(B) - 2P(A \cap B) $.
• $ P(X = 2) $: Probability of passing both tests.
  $ P(X = 2) = P(A \cap B) $.

2. PMF:

The $PMF$ of $( X )$ is:
$$
P(X = k) \quad \text{for } k \in {0, 1, 2}.
$$

3. Visualization:

We will plot the $PMF$ as a bar chart.

Python Implementation

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import matplotlib.pyplot as plt
import numpy as np

# Given probabilities
P_A = 0.7
P_B = 0.8
P_A_and_B = 0.6

# Calculating probabilities
P_A_or_B = P_A + P_B - P_A_and_B
P_X_0 = 1 - P_A_or_B  # Probability of failing both
P_X_1 = P_A + P_B - 2 * P_A_and_B  # Probability of passing exactly one test
P_X_2 = P_A_and_B  # Probability of passing both tests

# PMF
x_values = [0, 1, 2]
pmf = [P_X_0, P_X_1, P_X_2]

# Visualization
plt.figure(figsize=(8, 5))
plt.bar(x_values, pmf, color=['red', 'blue', 'green'], alpha=0.7)
plt.xticks(x_values, labels=[f"Pass {i} Tests" for i in x_values])
plt.ylim(0, 1)
plt.title("Probability Mass Function of X (Number of Tests Passed)", fontsize=14)
plt.ylabel("Probability", fontsize=12)
plt.xlabel("Number of Tests Passed (X)", fontsize=12)
plt.grid(axis='y', alpha=0.3)
for i, p in enumerate(pmf):
    plt.text(x_values[i], p + 0.02, f"{p:.2f}", ha='center', fontsize=10)
plt.show()

# Print probabilities
print(f"PMF:")
print(f"P(X = 0) = {P_X_0:.2f}")
print(f"P(X = 1) = {P_X_1:.2f}")
print(f"P(X = 2) = {P_X_2:.2f}")

Explanation of Code

1. PMF Calculation:
   • Using set relationships to calculate $ P(X = 0) $, $ P(X = 1) $, and $ P(X = 2) $.
2. Bar Chart:
   • Bar heights represent probabilities for $ X = 0, 1, 2 $.
   • Labels and annotations make the graph interpretable.

Expected Output

• Text:

  1 2 3 4

  PMF: P(X = 0) = 0.10 P(X = 1) = 0.30 P(X = 2) = 0.60

• Graph:

  • A bar chart with $( X = 0, 1, 2 )$ on the x-axis.
  • Heights correspond to $( P(X = 0) )$, $( P(X = 1) )$, $( P(X = 2) )$.
  • $( P(X = 2) )$ is the highest due to the significant overlap between Test $A$ and Test $B$.

Here’s a challenging number theory problem solved using $Python$, with a clear explanation and visualization.

The problem is related to modular arithmetic and the Chinese Remainder Theorem (CRT):

Problem

Find all integers $( x )$ such that:
$$
x \equiv 2 \pmod{3}, \quad x \equiv 3 \pmod{5}, \quad x \equiv 2 \pmod{7}
$$
Then visualize the results within a range, highlighting the congruences visually.

Explanation

This is a system of simultaneous congruences.

The Chinese Remainder Theorem guarantees a unique solution modulo $( n = 3 \cdot 5 \cdot 7 = 105 )$ since the moduli are pairwise coprime.

1. Solve for $( x )$ satisfying the given congruences.
2. Verify solutions by computing $( x \mod 3 )$, $( x \mod 5 )$, and $( x \mod 7 )$.
3. Visualize the congruences within a given range, e.g., $( 0 \leq x < 200 )$.

Python Implementation:

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import numpy as np
import matplotlib.pyplot as plt
from sympy.ntheory.modular import solve_congruence

# Define the congruences
congruences = [(2, 3), (3, 5), (2, 7)]

# Solve using the Chinese Remainder Theorem
solution, modulus = solve_congruence(*congruences)

print(f"Solution: x ≡ {solution} (mod {modulus})")

# Generate values of x within a range
x_range = 200  # Define the range for visualization
x_values = [solution + i * modulus for i in range((x_range // modulus) + 1)]

# Verify the solution
assert all(x % m == r for r, m in congruences for x in x_values)

# Visualization
plt.figure(figsize=(10, 6))

# Plot the values of x
plt.scatter(x_values, [0] * len(x_values), color='blue', label='Solutions (x)')

# Show congruences modulo 3, 5, and 7
moduli = [3, 5, 7]
colors = ['red', 'green', 'orange']
for mod, color in zip(moduli, colors):
    congruent_vals = [x for x in range(x_range) if x % mod == solution % mod]
    plt.scatter(congruent_vals, [mod] * len(congruent_vals), color=color, alpha=0.5, label=f'x ≡ {solution % mod} (mod {mod})')

# Annotations and labels
plt.title('Visualization of Solutions to Simultaneous Congruences', fontsize=14)
plt.xlabel('x', fontsize=12)
plt.ylabel('Modulo Conditions', fontsize=12)
plt.legend()
plt.grid(alpha=0.3)
plt.show()

Explanation of Code

1. Solve_congruence: Utilizes $SymPy$ to solve the system of congruences.
2. x_values: Computes the values satisfying the congruences within a specified range.
3. Visualization:
   • Scatter plot for $( x )$ values.
   • Highlights modulo congruence relationships $( x \mod 3 )$, $( x \mod 5 )$, $( x \mod 7 )$.

Expected Output

• Text Output:

  1	Solution: x ≡ 23 (mod 105)

  This means the smallest solution is $( x = 23 )$, and all solutions are of the form $( x = 23 + 105k )$ for integers $( k )$.

• Graphical Output:

  • Blue points indicate solutions $( x )$.
  • Colored scatter plots at $( y = 3, 5, 7 )$ show congruences modulo $3$, $5$, and $7$ respectively.

Problem: Partition Function and Average Energy in a 2-Level System

In statistical mechanics, the partition function is a fundamental quantity that describes the statistical properties of a system in thermal equilibrium.

Using $Python$, we can compute the partition function and derive physical quantities such as the average energy of a 2-level quantum system.

Problem Statement

Consider a system with two energy levels:

1. Ground state energy $E_0 = 0$,
2. Excited state energy $E_1 = \Delta E$.

The system is in thermal equilibrium at temperature $T$. Solve for:

1. The partition function $Z$:
   $$
   Z = \sum_{i} e^{-E_i / k_B T}
   $$
2. The average energy $ \langle E \rangle$:
   $$
   \langle E \rangle = \frac{\sum_{i} E_i e^{-E_i / k_B T}}{Z}
   $$

Python Code

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import numpy as np
import matplotlib.pyplot as plt

# Constants
k_B = 1.38e-23  # Boltzmann constant (J/K)
Delta_E = 1.5e-21  # Energy difference between levels (J)

# Functions for partition function and average energy
def partition_function(T):
    return 1 + np.exp(-Delta_E / (k_B * T))

def average_energy(T):
    Z = partition_function(T)
    numerator = (0 * np.exp(0) + Delta_E * np.exp(-Delta_E / (k_B * T)))
    return numerator / Z

# Temperature range
T = np.linspace(1, 1000, 500)  # Temperature in Kelvin

# Calculate partition function and average energy
Z = partition_function(T)
E_avg = average_energy(T)

# Plot results
plt.figure(figsize=(12, 6))

# Partition function
plt.subplot(1, 2, 1)
plt.plot(T, Z, label="Partition Function (Z)", color='blue')
plt.title("Partition Function vs Temperature")
plt.xlabel("Temperature (K)")
plt.ylabel("Z")
plt.grid(alpha=0.3)

# Average energy
plt.subplot(1, 2, 2)
plt.plot(T, E_avg, label="Average Energy ⟨E⟩", color='red')
plt.title("Average Energy vs Temperature")
plt.xlabel("Temperature (K)")
plt.ylabel("⟨E⟩ (J)")
plt.grid(alpha=0.3)

plt.tight_layout()
plt.show()

Explanation

1. Partition Function:

   • The partition function $Z$ is the sum of the $Boltzmann$ factors $e^{-E_i / k_B T}$.
   • For the 2-level system, $Z = 1 + e^{-\Delta E / k_B T}$.

2. Average Energy:

   • The average energy $\langle E \rangle$ is computed by weighting each energy level by its $Boltzmann$ factor, normalized by $Z$.

3. Temperature Dependence:

   • At low temperatures, most particles are in the ground state, and $\langle E \rangle \approx 0$.
   • At high temperatures, both levels are equally populated, and $\langle E \rangle \to \Delta E / 2$.

Results and Insights

• Partition Function:

  • $Z$ starts close to 1 at low temperatures, where only the ground state is occupied.
  • As temperature increases, $Z$ grows, reflecting the contribution of the excited state.

• Average Energy:

  • At low $T$, $(\langle E \rangle \to 0)$, as particles are mostly in the ground state.
  • At high $(T)$, the system reaches a thermal equilibrium where $(\langle E \rangle \to \Delta E / 2)$, consistent with equal population of both states.

This example demonstrates how statistical mechanics connects microscopic energy states with macroscopic thermodynamic quantities.

Problem: Numerical Solution of a Nonlinear Equation

In mathematical analysis, solving nonlinear equations is essential for understanding systems in engineering, physics, and economics.

Here, we use Newton’s Method to find the root of a nonlinear equation.

Problem Statement

Solve the nonlinear equation:
$$
f(x) = x^3 - 2x^2 - x + 2 = 0
$$
starting with an initial guess $(x_0 = 1.5)$.

Python Code

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import numpy as np
import matplotlib.pyplot as plt

# Define the function and its derivative
def f(x):
    return x**3 - 2*x**2 - x + 2

def f_prime(x):
    return 3*x**2 - 4*x - 1

# Implement Newton's Method
def newtons_method(f, f_prime, x0, tol=1e-6, max_iter=100):
    x = x0
    history = [x]
    for _ in range(max_iter):
        x_new = x - f(x) / f_prime(x)
        history.append(x_new)
        if abs(x_new - x) < tol:
            break
        x = x_new
    return x_new, history

# Solve the equation
x0 = 1.5  # Initial guess
root, history = newtons_method(f, f_prime, x0)

# Generate points for plotting
x = np.linspace(-2, 3, 500)
y = f(x)

# Plot the function and the iterations
plt.figure(figsize=(8, 5))
plt.plot(x, y, label="f(x) = $x^3 - 2x^2 - x + 2$", color='blue')
plt.axhline(0, color='black', linewidth=0.8, linestyle='--', label="y = 0")
plt.scatter(history, [f(xi) for xi in history], color='red', zorder=5, label="Iterations")
plt.plot(history, [f(xi) for xi in history], color='orange', linestyle='--', label="Convergence Path")

# Highlight the root
plt.scatter(root, 0, color='green', s=100, label=f"Root: x = {root:.6f}")

# Add labels, legend, and grid
plt.title("Numerical Solution of $x^3 - 2x^2 - x + 2 = 0$ Using Newton's Method")
plt.xlabel("x")
plt.ylabel("f(x)")
plt.legend()
plt.grid(alpha=0.3)
plt.show()

# Print the result
print(f"The root of the equation is approximately: x = {root:.6f}")

Explanation of the Code

1. Newton’s Method:

   • Newton’s Method is an iterative algorithm for finding roots of nonlinear equations.
   • The iteration formula is:
     $$
     x_{n+1} = x_n - \frac{f(x_n)}{f’(x_n)}
     $$
   • The method requires an initial guess $x_0$, the function $f(x)$, and its derivative $f’(x)$.

2. Convergence Tracking:

   • The history of $(x_n)$ values is stored and plotted to show the convergence path.

3. Graphical Representation:

   • The function $f(x)$ is plotted along with the iteration points.
   • The root is highlighted on the graph.

Results and Insights

This graph illustrates the solution of the nonlinear equation $f(x) = x^3 - 2x^2 - x + 2$ using Newton’s Method.

• The blue curve represents the function $f(x)$.
• The black dashed line is $y = 0$, showing the x-axis where the root lies.
• Red dots indicate the iteration points computed during the method.
• The orange dashed line represents the convergence path of the iterations.
• The green dot marks the final root found at $x = -1.000000$, where $f(x) = 0$.

This visualization demonstrates how Newton’s Method refines the guess iteratively, converging to the root of the equation.

Problem: Solving a Differential Equation with Boundary Conditions

In applied mathematics, differential equations model physical systems such as heat transfer, population dynamics, or mechanical vibrations.

This example demonstrates solving a second-order differential equation using boundary conditions.

Problem Statement

Solve the differential equation:
$$
\frac{d^2y}{dx^2} - y = 0
$$
with the boundary conditions:
$$
y(0) = 1, \quad y(1) = 2
$$

Python Code

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import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import solve_bvp

# Define the differential equation
def equation(x, y):
    return np.vstack((y[1], y[0]))  # y'[0] = y[1], y'[1] = y[0]

# Define boundary conditions
def boundary_conditions(ya, yb):
    return np.array([ya[0] - 1, yb[0] - 2])  # y(0)=1, y(1)=2

# Initial guess for the solution
x = np.linspace(0, 1, 10)
y_guess = np.zeros((2, x.size))  # Initial guess for y and y'

# Solve the boundary value problem
solution = solve_bvp(equation, boundary_conditions, x, y_guess)

# Generate a finer grid for plotting
x_plot = np.linspace(0, 1, 100)
y_plot = solution.sol(x_plot)[0]

# Plot the solution
plt.figure(figsize=(8, 5))
plt.plot(x_plot, y_plot, label="Solution $y(x)$", color='blue')
plt.scatter([0, 1], [1, 2], color='red', label="Boundary Conditions", zorder=5)
plt.title("Solution to $\\frac{d^2y}{dx^2} - y = 0$")
plt.xlabel("x")
plt.ylabel("y(x)")
plt.axhline(0, color='black', linewidth=0.5, linestyle='--')
plt.axvline(0, color='black', linewidth=0.5, linestyle='--')
plt.legend()
plt.grid(alpha=0.3)
plt.show()

Explanation of the Code

1. Equation Definition:

   • The second-order equation is converted into a system of first-order equations:
     $$
     y_1 = y, \quad y_2 = y’
     $$
     Resulting in:
     $$
     y_1’ = y_2, \quad y_2’ = y_1
     $$

2. Boundary Conditions:

   • The boundary conditions $y(0) = 1$ and $y(1) = 2$ are applied.

3. Numerical Solution:

   • The scipy.integrate.solve_bvp function numerically solves the boundary value problem ($BVP$).

4. Visualization:

   • The computed solution is plotted as a smooth curve.
   • The boundary conditions are highlighted as red dots.

Results and Insights

The plot shows the solution $y(x)$ that satisfies the differential equation and the boundary conditions.

The method used is efficient for solving linear and nonlinear boundary value problems commonly encountered in engineering and applied sciences.

Problem: The Prisoner's Dilemma and Strategy Simulation

The Prisoner’s Dilemma is a classic example in game theory, demonstrating why two rational individuals might not cooperate, even if it appears to be in their best interest.

This example analyzes the payoff matrix and simulates multiple rounds of the dilemma using various strategies to observe long-term outcomes.

Objective

1. Define the payoff matrix for the Prisoner’s Dilemma.
2. Simulate repeated interactions using:
   • Always Defect
   • Always Cooperate
   • Tit-for-Tat (mimic the opponent’s previous move).
3. Visualize the accumulated payoffs over multiple rounds for each strategy.

Python Code

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import numpy as np
import matplotlib.pyplot as plt

# Define the payoff matrix (Player 1, Player 2)
# Rows: Player 1's actions, Columns: Player 2's actions
# Actions: 0 = Cooperate, 1 = Defect
payoff_matrix = np.array([
    [[3, 3], [0, 5]],  # Player 1 cooperates
    [[5, 0], [1, 1]]   # Player 1 defects
])

# Define strategies
def always_cooperate(history, my_history):
    return 0  # Always cooperate

def always_defect(history, my_history):
    return 1  # Always defect

def tit_for_tat(history, my_history):
    if not history:  # Cooperate in the first round
        return 0
    return history[-1]  # Mimic the opponent's last move

# Simulate the game
def simulate_game(strategy1, strategy2, rounds=10):
    history1, history2 = [], []
    score1, score2 = 0, 0

    for _ in range(rounds):
        move1 = strategy1(history2, history1)
        move2 = strategy2(history1, history2)
        
        payoff1, payoff2 = payoff_matrix[move1, move2]
        score1 += payoff1
        score2 += payoff2
        
        history1.append(move1)
        history2.append(move2)
    
    return score1, score2, history1, history2

# Run simulations
rounds = 20
strategies = [always_cooperate, always_defect, tit_for_tat]
results = {}

for i, strat1 in enumerate(strategies):
    for j, strat2 in enumerate(strategies):
        if i <= j:  # Avoid duplicating reverse matches
            label = f"{strat1.__name__} vs {strat2.__name__}"
            score1, score2, hist1, hist2 = simulate_game(strat1, strat2, rounds)
            results[label] = (score1, score2, hist1, hist2)

# Visualize results
fig, ax = plt.subplots(figsize=(10, 6))
labels = list(results.keys())
player1_scores = [results[label][0] for label in labels]
player2_scores = [results[label][1] for label in labels]

bar_width = 0.35
x_indices = np.arange(len(labels))

# Plot bars for Player 1 and Player 2
ax.bar(x_indices - bar_width/2, player1_scores, bar_width, label='Player 1')
ax.bar(x_indices + bar_width/2, player2_scores, bar_width, label='Player 2')

# Add labels and title
ax.set_xlabel("Strategy Pair", fontsize=12)
ax.set_ylabel("Accumulated Payoff", fontsize=12)
ax.set_title("Prisoner's Dilemma: Payoffs for Different Strategies", fontsize=14)
ax.set_xticks(x_indices)
ax.set_xticklabels(labels, rotation=45, ha='right', fontsize=10)
ax.legend()

# Show grid for clarity
ax.grid(axis='y', linestyle='--', alpha=0.7)

plt.tight_layout()
plt.show()

Explanation of the Code

1. Payoff Matrix:

   • Each cell in the matrix represents the payoff for Player $1$ and Player $2$, given their choices (cooperate/defect).
   • Cooperation leads to mutual benefit, while defection may yield higher short-term rewards.

2. Strategies:

   • Always Cooperate: Always chooses to cooperate.
   • Always Defect: Always chooses to defect.
   • Tit-for-Tat: Mimics the opponent’s previous move, encouraging reciprocity.

3. Simulation:

   • The game is repeated for a fixed number of rounds.
   • Accumulated scores and move histories are recorded for both players.

4. Visualization:

   • Bar plots display the total payoffs for Player $1$ and Player $2$ across all strategy pairs.
   • The results highlight how different strategies perform against each other.

Results and Insights

1. Tit-for-Tat Performance:

   • Tit-for-Tat typically performs well, fostering cooperation when paired with cooperative strategies.

2. Always Defect:

   • While it exploits cooperative players, it leads to mutual punishment when paired with itself or Tit-for-Tat.

3. Always Cooperate:

   • Vulnerable to exploitation but performs well in fully cooperative scenarios.

This simulation illustrates the dynamics of strategic decision-making in repeated games, revealing insights into the stability and outcomes of different strategies.

Problem: Numerical Integration Using the Trapezoidal Rule

Numerical integration is a key topic in computational mathematics.

In this example, we will approximate the integral of a given function using the trapezoidal rule and compare the result to the exact value.

Objective

1. Integrate $ f(x) = \sin(x) $ over the interval $([0, \pi])$ numerically using the trapezoidal rule.
2. Visualize the approximation and compare it with the actual curve of the function.

Mathematical Background

The integral of $ f(x) = \sin(x) $ over $([0, \pi])$ is:
$$
\int_0^\pi \sin(x) , dx = 2
$$

The trapezoidal rule approximates the integral by dividing the interval into $( n ) $subintervals and approximating $ f(x) $ as a straight line over each subinterval.
The formula is:
$$
\int_a^b f(x) , dx \approx \frac{h}{2} \left[ f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right]
$$
where $( h = \frac{b-a}{n} )$ is the width of each subinterval.

Python Code

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import numpy as np
import matplotlib.pyplot as plt

# Function to integrate
def f(x):
    return np.sin(x)

# Trapezoidal rule implementation
def trapezoidal_rule(func, a, b, n):
    x = np.linspace(a, b, n + 1)  # Subinterval points
    y = func(x)  # Function values at those points
    h = (b - a) / n  # Width of each subinterval
    integral = (h / 2) * (y[0] + 2 * np.sum(y[1:-1]) + y[-1])
    return integral, x, y

# Parameters
a, b = 0, np.pi  # Integration limits
n = 10  # Number of subintervals

# Calculate numerical integral
numerical_integral, x_points, y_points = trapezoidal_rule(f, a, b, n)

# Exact integral
exact_integral = 2  # From analytical solution

# Plotting
x_full = np.linspace(a, b, 1000)
y_full = f(x_full)

plt.figure(figsize=(10, 6))
plt.plot(x_full, y_full, label='f(x) = sin(x)', color='blue')
plt.fill_between(x_full, 0, y_full, color='blue', alpha=0.1, label='Exact Area')
plt.plot(x_points, y_points, 'o', color='red', label='Trapezoid Nodes')
for i in range(len(x_points) - 1):
    plt.plot([x_points[i], x_points[i + 1]], [y_points[i], y_points[i + 1]], color='orange')

plt.title("Numerical Integration Using the Trapezoidal Rule")
plt.xlabel("x")
plt.ylabel("f(x)")
plt.legend()
plt.grid(True, linestyle="--", alpha=0.7)
plt.show()

# Print results
print(f"Numerical Integral (Trapezoidal Rule): {numerical_integral:.4f}")
print(f"Exact Integral: {exact_integral}")
print(f"Error: {abs(numerical_integral - exact_integral):.4f}")

Explanation of the Code

1. Function and Interval:
   • The function $( f(x) = \sin(x) )$ is defined along with the integration interval $([0, \pi])$.
2. Trapezoidal Rule:
   • The interval is divided into $( n )$ subintervals, and the trapezoidal rule is applied to approximate the integral.
3. Exact Solution:
   • The exact integral is $( 2 )$, used here for error comparison.
4. Visualization:
   • The function is plotted alongside the trapezoids used for approximation.
   • The filled area under $ f(x) $ represents the exact integral.

Results and Insights

1. Numerical Approximation:
   • The trapezoidal rule provides an approximate integral value close to the exact solution.
     For $( n = 10 )$, the numerical result is accurate within a small error margin.
2. Graph:
   • The blue curve shows $ f(x) = \sin(x) $.
   • Red points indicate the nodes where the function is evaluated.
   • Orange lines represent the trapezoids approximating the area under the curve.
3. Error Analysis:
   • Increasing the number of subintervals $( n )$ reduces the approximation error, demonstrating the method’s convergence.

Applications in Computational Mathematics

1. Integration of Complex Functions:
   • Useful when analytical integration is impossible.
2. Engineering and Physics:
   • Applied to problems involving areas, volumes, and physical quantities.
3. Data Analysis:
   • Approximation of cumulative distributions or areas under discrete data points.

This example highlights how numerical integration works and provides a visual understanding of the trapezoidal rule.

Problem: Predicting House Prices Using Linear Regression

In this example, we analyze a dataset containing house features (e.g., size, number of rooms) to predict house prices using linear regression, a core technique in data science.

Objective

1. Train a linear regression model to predict house prices based on square footage.
2. Visualize the results, including the data points and the regression line.

Dataset

We will simulate a dataset of house sizes (in square feet) and corresponding prices (in ＄1000s).

This ensures reproducibility and simplicity.

Python Code

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import numpy as np
import matplotlib.pyplot as plt
from sklearn.linear_model import LinearRegression
from sklearn.metrics import mean_squared_error

# Simulated dataset
np.random.seed(42)
sizes = np.random.uniform(500, 3500, 50)  # House sizes (square feet)
prices = sizes * 0.15 + np.random.normal(0, 50, 50)  # Prices with some noise

# Reshape data for sklearn
X = sizes.reshape(-1, 1)  # Predictor (feature)
y = prices  # Target (price)

# Train a linear regression model
model = LinearRegression()
model.fit(X, y)

# Make predictions
predicted_prices = model.predict(X)

# Evaluate the model
mse = mean_squared_error(y, predicted_prices)
print(f"Mean Squared Error: {mse:.2f}")

# Plot the results
plt.figure(figsize=(10, 6))
plt.scatter(sizes, prices, color="blue", label="Actual Prices", alpha=0.7)
plt.plot(sizes, predicted_prices, color="red", label="Regression Line", linewidth=2)
plt.title("House Prices vs. Size")
plt.xlabel("Size (Square Feet)")
plt.ylabel("Price (in $1000s)")
plt.legend()
plt.grid(True, linestyle="--", alpha=0.7)
plt.show()

Explanation of the Code

1. Simulated Data:
   • House sizes are sampled randomly between $500$ and $3500$ square feet.
   • Prices are calculated using a simple linear relationship $( \text{Price} = 0.15 \times \text{Size} + \text{Noise} )$, where the noise simulates real-world variability.
2. Model Training:
   • A LinearRegression model from sklearn is trained using house sizes as the feature and prices as the target.
3. Prediction:
   • The trained model predicts house prices, which are plotted alongside the actual prices.
4. Model Evaluation:
   • The Mean Squared Error (MSE) quantifies the model’s accuracy.

Visualization and Insights

1. Scatter Plot:
   • Blue dots represent the actual prices for different house sizes.
2. Regression Line:
   • The red line shows the best-fit line learned by the linear regression model.
3. Model Performance:
   • The MSE indicates the average squared difference between the predicted and actual prices.

Applications in Data Science

1. Regression Analysis:
   • This approach is foundational in predictive modeling for understanding relationships between variables.
2. Real Estate Analytics:
   • Real-world data would involve more features like location, number of bedrooms, and year built.
3. Scalability:
   • This method easily extends to multivariate regression with multiple predictors.

This example demonstrates the workflow of data science: data preprocessing, modeling, and visualization, providing insights into how features like size influence house prices.