Notes, summaries, assignments, exams, and problems for Mathematics

Safety and Discipline Committees

Threat Assessment Team

TEC §37.115

Required Members:

• Principal / Assistant Principal
• Counselor
• Local Law Enforcement / School Resource Officer (SRO)

Key Responsibilities:

• Evaluates behavioral threats.
• Implements safety interventions.
• Must be trained via the Texas School Safety Center (TxSSC).

1A Panther Strategy: Cross-train your small team during the August Professional Development (PD) week so you are compliant before day one.

Placement Review Committee

TEC §37.003

Required Members:

• Two Elected Classroom Teachers
• One Alternate Teacher
• One Principal Appointee

Key Responsibilities:

• Rules on student placement if a teacher permanently removes a disruptive student from class.

1A Panther Strategy: Hold this campus election during... Continue reading "Essential Texas School Committee Compliance Strategies" »

Map Generalization

Types of Symbols

• Line Symbols: Represent real-life objects with a linear path.
• Point Symbols: Represent objects occurring at a single point on Earth's surface using a dot.
• Area (Polygon) Symbols: Represent real-life objects spread over Earth's surface using geometric shapes.

Generalization Techniques

Reality contains too much information for a single 2D map. Generalized geometry and content make a map useful. A good map suppresses less important information to highlight what needs to be seen.

• Selection: Only relevant line, point, and area features are chosen.
• Classification: Grouping similar features and using a common symbol to represent them.
• Simplification: Reduction of unnecessary detail.
• Smoothing: Smoothing out abruptly joined

Boolean Algebra Fundamentals

Formulating Expressions: SOP and POS

Sum of Products (SOP) Formulation Steps

1. Circle rows in the truth table where the output $Y = 1$.
2. Identify the minterms corresponding to the circled rows.
3. Sum (OR) the selected minterms to form the final expression.

Product of Sums (POS) Formulation Steps

1. Circle rows in the truth table where the output $Y = 0$.
2. Identify the maxterms corresponding to the circled rows.
3. Multiply (AND) the selected maxterms to form the final expression.

Order of Evaluation in Boolean Algebra

Operations are evaluated in the following sequence:

1. Parentheses
2. NOT (Complementation)
3. AND (Multiplication)
4. OR (Addition)

Fundamental Boolean Laws and Theorems

• Identity Laws

  • $A + 0 = A$
  • $A \cdot 1 = A$

• Null Laws

  • $A + 1 = 1$
  • $A \cdot

Linear Algebra and Matrix Properties

1. State one property of a square matrix that will guarantee it is singular.
   There are many possibilities, some of which are: having determinant 0, having dependent columns, having dependent rows, and having a non-trivial null-space.
2. Can a system of linear equations have exactly 2 solutions?
   No, a system of linear equations can have either 0, 1, or infinitely many solutions.
3. What advantages does the Cholesky factorization have over Gaussian elimination?
   The Cholesky factorization only requires about half the operations of Gaussian elimination, and pivoting is never required for stability.
4. Can an n × n positive-definite matrix be singular? What about an n × n symmetric positive-definite matrix?
   The condition

Chapter 1 - 

Linearity: As long as the values that vary (assuming any other values R
constant) R not themselves involved in anything more than
○addition&
○scalar multiplication

Chapter 2 -

Shallow network

y = f[x, ϕ]  
y = ϕ₀ + ϕ₁ a[θ₁₀ + θ₁₁x] + ϕ₂ a[θ₂₀ + θ₂₁x] + ϕ₃ a[θ₃₀ + θ₃₁x]

y = ϕ₀ + ϕ₁h₁ + ϕ₂h₂ + ϕ₃h₃

Terminology - The hidden units themselves are sometimes referred to as neurons. Rhe values of the inputs to the hidden layer (i.E., before the ReLU functions are applied) are termed pre-activations. The values at... Continue reading "Neural Network Fundamentals and Regularization Techniques" »

Process of Contouring and Isolines

Isolines connect points of equal measurements. Contours indicate elevation and changes in elevation, such as slopes and hills. Isotherms connect points of equal temperature and tell us the current state of the atmosphere.

Rules for Drawing Isolines

• Connect points of equal value.
• Never cross one another.
• Never touch.
• The interval for isolines is the same for the whole map.

Contours show us slopes, valleys, and hills.

Topographic Profiles and Vertical Exaggeration

A profile is a topographic profile representing the lay of the land from the side. To draw a profile, a transect (straight line) is drawn between two points to show hills, valleys, and slopes.

Vertical Exaggeration is a mathematical representation used to ensure... Continue reading "Mapping Techniques and Geographic Data Analysis" »

This step-by-step analysis covers Problem Sets 6-9, emphasizing key concepts from Problem Sets 7 and 8, essential for your final exam.

Problem Set 6: Product Differentiation & Merger Impacts

1. Why Bertrand Does Not Equal Marginal Cost in Reality

• Firms may experience:

  • Capacity constraints

  • Brand loyalty (differentiated products)

  • Reputational concerns or switching costs

2. Bertrand Competition with Differentiated Products

• Demand:

  • Q_M = 1000 - 200P_M + 100P_B

  • Q_B = 1000 - 200P_B + 100P_M

• Steps:

  1. Plug in rival's price to derive inverse demand.

  2. Derive Marginal Revenue (MR); set MR = Marginal Cost (MC) = 4.

  3. Solve for the best response price.

  4. Set both best responses equal to solve for the Nash Equilibrium (NE).

  5. Calculate quantity, profit, and price-cost margin.

Chapter 2: Predetermined Overhead Rate

Predetermined Overhead Rate = Estimated Total Manufacturing Overhead (MOH) / Estimated Total MOH Driver (e.g., Direct Labor hours, Direct Labor costs, Machine Hours)

Prime Cost = Direct Materials + Direct Manufacturing Labor

Conversion Cost = Direct Manufacturing Labor + Indirect Manufacturing Overhead

Cost Accumulation: Data is collected in an organized way (also known as cost pools).

Cost Assignment: Systematically links an actual cost pool to a distinct cost object (e.g., Tires, engine, labor assigned to car cost).

Activity Base: Examples include kilometers driven in a car, units produced, units sold, machine hours.

Product Cost: Costs tied to creating a product (Direct Materials, Direct Labor, Manufacturing... Continue reading "Cost Accounting Essentials: Key Concepts and Calculations" »

Statistical Measures and Causal Inference Concepts

Measures of Dispersion and Relationship

Variance

Variance: Estimates how far a set of numbers (random) are spread out from their mean value.

Covariance

Covariance: The relationship between two variables.

• Cov = 0: Unsure of the relationship.
• Cov > 0: Suggests Y will be above average when X is above average.
• Cov < 0: Suggests Y will be below average when X is above average.

The formula for variance is often expressed as: $\mathbb{E}[X^2] - (\mathbb{E}[X])^2$ (where $\mathbb{E}$ is the Expected Value).

The formula for covariance between two variables $X$ and $Y$ is: $\mathbb{E}[(X - \mathbb{E}[X])(Y - \mathbb{E}[Y])]$

Pearson's Correlation Coefficient

Standardizes covariance between -1 and 1:

Pearson’s