Egyptian Civilization

Work Done by a Force

• Work is the dot product of the force and displacement vectors: (W = \overrightarrow{F} \cdot \overrightarrow{d} = Fd\cos\theta)
• (\overrightarrow{F}) is the force vector,
• (\overrightarrow{d}) is the displacement vector, and
• (\theta) is the angle between them.
• The unit for work is the Joule (J), where 1 J = 1 N·m.
• Work is a scalar quantity.
• Work can be positive (0 \le \theta < 90^\circ), negative (90^\circ < \theta \le 180^\circ), or zero (\theta = 90^\circ), depending on the angle ( \theta ) between the force and displacement.
• Work equals the component of force along the displacement direction multiplied by the displacement magnitude.

Example Work Calculation

• Given a force (\overrightarrow{F} = (4\hat{i} + 5\hat{j})) N and a displacement (\overrightarrow{d} = (3\hat{i} + 7\hat{j})) m, the work done is: (W = (4\hat{i} + 5\hat{j}) \cdot (3\hat{i} + 7\hat{j}) = 47) J.

Work Done by a Varying Force

• Applied forces that aren't constand across a ditance can be integrated to find the total amount of work: (W = \int_{x_i}^{x_f} F_x dx)

Work-Kinetic Energy Theorem

• The net work done on an object equals the change in its kinetic energy: (W_{net} = K_f - K_i = \Delta K).
• (K_f) and (K_i) represent the final and initial kinetic energies.

Kinetic Energy

• Object's kinetic energy is defined as: (K = \frac{1}{2}mv^2), where (m) is mass and (v) is speed.

Potential Energy

• Potential energy is the energy related to the system's configuration of objects interacting with forces.
• Gravitational potential energy is (U_g = mgy).
• Elastic potential energy is (U_s = \frac{1}{2}kx^2).

Conservative vs Nonconservative Forces

• Conservative forces perform the same work between two points regardless of the path taken. Examples are gravitational and elastic forces.
• Nonconservative forces perform different work depending on the path. Examples include friction, air resistance, tension, and motor forces.

Conservation of Mechanical Energy

• In a closed system with only conservative forces, the total mechanical energy (kinetic plus potential) is conserved: (\Delta K + \Delta U = 0) or (K_f + U_f = K_i + U_i).

Work Done by Nonconservative Forces

• The work done by nonconservative forces equals the change in kinetic and potential energies: (W_{nc} = \Delta K + \Delta U = (K_f - K_i) + (U_f - U_i)).

Power

• Power is the rate of energy transfer.
• Average power is (\overline{P} = \frac{W}{\Delta t}).
• Instantaneous power is (P = \frac{dW}{dt} = \overrightarrow{F} \cdot \overrightarrow{v}).
• The SI unit for power is the Watt (W), where 1 W = 1 J/s.

Energy Loss Example due to Friction

• For a 70 kg base-runner sliding to rest from 4.0 m/s with a friction coefficient of 0.70:
  • Mechanical energy lost due to friction: (\Delta E = -560) J.
  • Sliding distance: (d = 1.2) m.

Independent Random Variables

• Knowing the value of one random variable doesn't change the probability mass function (PMF) or probability density function (PDF) of the other.
• (p_{X, Y}(x, y) = p_{X}(x) p_{Y}(y)) for discrete variables.
• (f_{X, Y}(x, y)=f_{X}(x) f_{Y}(y)) for continuous variables.

Dice Roll Example

• If rolling a dice twice the probability that (X_1 + X_2 = 8):
• (P(X_{1}+X_{2}=8) = \frac{5}{36})

Functions of Multiple Random Variables

• For (Z=g(X, Y)), the distribution of (Z) is:
  • (p_{Z}(z)=\sum_{{(x, y): g(x, y)=z}} p_{X, Y}(x, y)) if (X) and (Y) are discrete.
  • (f_{Z}(z)=\int_{-\infty}^{\infty} f_{X, Y}(x, y) d y) if (X) and (Y) are continuous, where (S={(x, y): g(x, y) \leq z}).

Sum of Independent Random Variables

• If (Z=X+Y) and (X) and (Y) are independent, the PDF of (Z) is the convolution of the PDFs of (X) and (Y): (f_{Z}(z)=\int_{-\infty}^{\infty} f_{X}(x) f_{Y}(z-x) d x).

Conditional Expectation

• (E[X \mid Y=y]=\sum_{X} x P(X=x \mid Y=y)) if (X) is discrete.
• (E[X \mid Y=y]=\int x f_{X \mid Y}(x \mid y) d x) if (X) is continuous.
• (E[X \mid Y]) is a random variable that is a function of (Y).

Example Calculation

• (X \sim \operatorname{Unif}(0,1)) and (Y = \begin{cases}1 & \text { if } X \geq \frac{1}{2} \ 0 & \text { otherwise }\end{cases}), then (E[X \mid Y=1] = \frac{3}{4}).

10 Principles of Economics

• The following 10 principles are subdivided into three sections:
  • how people make decisions,
  • how people interact, and
  • how the economy as a whole works.
• All principles are equally weighted and important.

How People Make Decisions

• People Face Trade-offs
• The Cost of Something Is What You Give Up to Get It
• Rational People Think at the Margin
• People Respond to Incentives

How People Interact

• Trade Can Make Everyone Better Off
• Markets Are Usually a Good Way to Organize Economic Activity
• Governments Can Sometimes Improve Market Outcomes

How the Economy as a Whole Works

• A Country's Standard of Living Depends on Its Ability to Produce Goods and Services
• Prices Rise When the Government Prints Too Much Money
• Society Faces a Short-Run Trade-off between Inflation and Unemployment

Definition of the State

• A set of institutions possessing the authority to make rules governing people within a defined territory.

Institutions

• The different parts of the government.

Authority

• The legal right to exercise power.

Legitimacy

• Acceptance by citizens of the state's authority.

Elements of the State

• Population: All people in a particular area.
• Territory: Area where the state has authority.
• Government: Body within the state authorized to make and enforce laws.
• Sovereignty: Supreme and ultimate power within a territory.

Types of Government

• Democracy: Political system where people choose their rulers.
• Autocracy: Political system ruled by a single person with unlimited power.
• Oligarchy: Political system where a small group holds power.

Unitary State

• Governed as a single entity with a supreme central government.

Federal State

• Power divided between a central government and several regional governments.

Confederation

• An association of independent states.

Information Channels

• Focus on discrete memoryless channels (DMC)
• Input alphabet denoted by $X$.
• Output alphabet denoted by $Y$.
• Transition probabilities (p(y|x)) (x \in X), (y \in Y) where (p(y|x)) is the probability of observing output (y) given input (x).

Channel Capacity Formula

• Maximize the mutual information (I(X; Y)) over all possible input distributions (p(x)). $$C = \max_{p(x)} I(X; Y)$$
• Mutual information is given by (I(X; Y) = H(Y) - H(Y|X))
• Conditional entropy (H(Y|X) = \sum_{x \in X} p(x) H(Y|X = x))
• (H(Y|X = x) = - \sum_{y \in Y} p(y|x) \log p(y|x))

Noiseless Channel

• Output is identical to the input (Y = X).
• The conditional probabilities (p(y|x)) is equal to $1$ when when $y=x$ else $0$ in all other cases.
• Channel capacity: (C = \log |X|) = (\log |Y|).

Noisy Channel

• Each input leads to a unique, non-overlapping output.
• (H(Y|X) = 0)
• Channel capacity is (C = \log |X|).

Noisy Typewriter

• The output is a shifted version of the input.
• (p(y|x) = 1) if (y = x \oplus 1 \pmod{k}), otherwise (p(y|x) = 0).
• Channel capacity (C = \log k).

Binary Symmetric Channel (BSC)

• Two-symbol channel with (X = {0, 1}) and (Y = {0, 1}), bit flips occur with probability (p).
• Channel is defined by (p(y|x) = p), for ( y \ne x ) and (p(y|x) = 1-p) if ( y=x )
• The channel Capacity is (C = 1 - H(p)).

Binary Erasure Channel (BEC)

• Channel with (X = {0, 1}) and potential erasure (Y = {0, 1, e}), erasure occurs with probability (\alpha).
• (p(y|x) = 1 - \alpha) if (y = x), (\alpha) if (y = e), and (0) otherwise.
• The channel Capacity is (C = 1 - \alpha).

Channel Capacity

• Channel Capacity is upper and lower bounded by
  • (0 \le C \le \min {\log |X|, \log |Y|}) .
• The channel capacity C is an achievable rate where (\frac{1}{\log(2)}).
• Shannon's channel coding theorem states (R < C) or the coding rate (R).