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

Markowitz Portfolio Selection

The Markowitz model focuses on the relationship between portfolio risk, expected returns, and the covariance and correlation between assets. Diversification reduces the variability of expected returns. The model identifies combinations that provide the maximum return for a given level of risk. A portfolio is considered efficient when it offers the highest return for a specific risk level or the minimum risk for a target return.

Core Investment Hypotheses

• Rational Investor Behavior: Investors prefer higher returns for a given risk level.
• Risk Aversion: Investors prefer lower risk for a given return level.

The Three-Step Optimization Process

1. Determine the Efficient Frontier

The efficient frontier consists of all efficient... Continue reading "Markowitz Portfolio Theory and Sharpe Optimization" »

Relative Positions in 3D Space

Position of Three Planes

The relative position of three planes is determined by analyzing the rank of the system of equations.

• Intersect at one point: The rank of the coefficient matrix and the augmented matrix is 3.
• Intersect in a line: The rank of both matrices is 2.
• Parallel or Coincident: If the ranks are 1 or 2, the planes can be parallel, coincident, or form a prismatic surface.

Position of Two Lines

Given two lines with direction vectors v₁, v₂ and points P₁, P₂:

• Intersecting Lines: The lines are coplanar. The determinant of the matrix formed by vectors v₁, v₂, and P₁P₂ is zero. To find the intersection point, solve the system of their parametric equations.
• Parallel or Coincident Lines: The direction

Measures of Central Tendency:

Arithmetic Mean

Used for:

Intervals and pooled data (xi = class mark)

Data are not grouped (tables).

Data are grouped (tables) but no intervals.

Median

The middle value in a sorted dataset.

For an odd number of observations, it's the middle number.

Sort data from lowest to highest before finding the median.

Mode

The value that appears most frequently in a dataset.

Mid-Range

RM = (Maximum Value + Minimum Value) / 2

Geometric Mean

G = ⁿ√(x₁ * x₂ * ... * xn)

Harmonic Mean

H = n / ( (1/x₁) + (1/x₂) + ... + (1/xn) )

Quadratic Mean (Root Mean Square)

Q = √[ (x₁² + x₂² + ... + xn²) / n ]

Percentile

A measure indicating the value below which a given percentage of observations in a group falls.

Kth Percentile

Measures of... Continue reading "Statistical Measures: Central Tendency, Dispersion, and Form" »

Bolzano's Theorem

If f(x) is continuous on [a, b], there exists at least one value c in (a, b) such that f(c) = 0.

Consequences

If f(a) > g(a) and f(b) < g(b), there exists c in (a, b) such that f(c) = g(c).

Mean Value Theorem

If f(x) is continuously differentiable on [a, b], then there exists a point where the derivative f'(c) equals the average rate of change.

Linear Combinations and Geometry

A linear combination is defined as: a(x₁, y₁, z₁) + b(x₂, y₂, z₂) + c(x₃, y₃, z₃) = (x₄, y₄, z₄).

Distance Formulas

• Between two points: The magnitude of the vector.
• Point to line: dist(P, r) = |AP × Vd| / |Vd|, where A is a point on the line and Vd is the direction vector.
• Point to plane: |ax + by + cz + d| / √(a² + b² + c²).

The Warehouse

The warehouse is essential for managing product timing and consumption. The approver must assess the risk of goods that necessitate storage. This facility serves as the destination for purchased goods, rejected items from clients, and leftover stock. Note: Storage is a phase of the production process that does not add value to the product.

Receipt of Orders

In procurement, receiving expected orders is critical. Warehouse personnel must be aware of all realized orders. Upon delivery, the manager must:

• Contrast delivery vouchers with original order data.
• Oversee the unloading process.
• Recount packages.
• Inspect for external damage.
• File timely claims if discrepancies exist.

This verification process is known as quality control. Technological... Continue reading "Warehouse Management: Best Practices for Inventory Control" »

Derivatives of Algebraic and Transcendental Functions

• f(x) = (x³ - 3x)³: f'(x) = 3(x³ - 3x)² · (3x² - 3) = 9x⁸ - 63x⁶ + 135x⁴ - 81x²
• f(x) = 2e^x · (3x - 1): f'(x) = 2e^x · (3x - 1) + 2e^x · 3 = 6xe^x - 2e^x + 6e^x = 6xe^x + 4e^x
• f(x) = ln(x² + 6x): f'(x) = (1 / (x² + 6x)) · (2x + 6)
• f(x) = 1 / ln x: f'(x) = -1(ln x)^-2 · (1 / x) = -1 / (x ln² x)
• f(x) = 2 cos x · sin x: f'(x) = 2(-sin x) · sin x + 2 cos x · cos x = 2(cos² x - sin² x)
• f(x) = 2^{x2 - 3}: f'(x) = 2^{x2 - 3} · 2x · ln 2
• f(x) = cos³(2x): f'(x) = 3 · (cos 2x)² · (-sin 2x) · 2 = -6 cos²(2x) · sin(2x)
• f(x) = (2x³ - 3x) / (x² - 1): f'(x) = [(6x² - 3)(x² - 1) - (2x³ - 3x)(2x)] / (x² - 1)²
• f(x) = (x² - 1) / (x + 1): f'(x) = [(2x)(x + 1) - (x² - 1) · 1] / (x + 1)²
• f(x) = e^x2 / (1 - e^x): f'(x) =

Pragmatic Analysis of Academic Discourse

When we analyze a text from a pragmatic point of view, we should consider several key elements: the purpose, the channel, the genre, the sender, and the receiver.

Characteristics of Academic Discourse

The primary purpose of academic discourse is to inform. In this context, the issuer acts as a transmitter, while the receiver is positioned at either a specialized level (scientific, such as a nurse) or a declarative level for general dissemination (divulgation).

Linguistic Markers and Procedures

Key markers of academic discourse include:

• Absence of deictics and an emphasis on impersonality.
• Use of the third-person axis.
• Predominance of the indicative mode and declarative statements.
• Absence of modalization procedures.

Practice 4: Salaries, Overtime, SSO, PHL

Basic Salary

• = (Additional Table at $[$D$24] * Time Worked)

Total Overtime

• Fx = IF(logical test = E10 [Overtime] > 40)
• True-Value: E10 - 40
• False-value: 0

Triple Overtime

• Fx = IF Function
• Logic Test: Total Overtime [G10 > 8]
• True-Value: G10 - 8
• False-value: 0

Payment of Extra Time Triple

• Triple Overtime (H10) * Pay Per Hour (D24) * 3

Double Overtime

• = G10 (Total Overtime) - H10 (Triple Overtime)

Payment of Extra Time Double

• = J10 (Double Extra Time) * D24 (Pay per hour) * 2

Compulsory Social Security (SSO)

• = (F10 [Basic Salary] + I10 [Payment of Triple Overtime] + K10 [Payment of Double Overtime]) * 5%

Housing Policy Act (HPL)

• = (F10 [Basic Salary] + I10 [Payment of Triple Overtime] + K10 [Payment of Double Overtime]

Limit Theorems and Convergence

• Root Test: lim ⁿ√a_n = L
• Ratio Test: lim a_n/a_n-1 = L
• Stolz Theorem: lim a_n/b_n = lim (a_n-a_n-1)/(b_n-b_n-1)
• Arithmetic Mean: lim (a₁ +...+ a_n)/n = lim a_n
• Geometric Mean: lim ⁿ√(a₁...a_n) = lim a_n

Convergence Criteria

• Raabe: lim n(a_n/a_n+1 - 1) > 1 (Convergent)
• Pringsheim: lim n^αa_n (if α > 1, Convergent)
• Root Test (n-th): lim ⁿ√a_n < 1 (Convergent)
• Logarithmic: lim log(1/a_n) / log(n) > 1 (Convergent)
• Absolute Convergence: If ∑|(-1)ⁿa_n| converges, then ∑(-1)ⁿa_n converges.
• Leibniz Criterion: 1) lim a_n = 0; 2) a_n+1 - a_n < 0

Series Types

• Hypergeometric Series: a_n+1/a_n = (an + b)/(n + c). Convergent if (c-b)/a > 1.
• Telescoping Series: ∑a_n = x_n - x_n+1. Sum = x₁ - lim x_n+1.

Asymptotic Equivalences

sin(a_n) ≈ a_n;... Continue reading "Essential Calculus Formulas and Convergence Theorems" »

Genetic Algorithm Implementation Functions

The following Python functions demonstrate key components of Genetic Algorithms, including crossover, selection, and population initialization.

Four-Point Crossover Method

This function performs a four-point crossover using four parents to generate offspring.

def four_point_cross(parent1, parent2, parent3, parent4):
    pos_alea = sorted(list(np.random.choice(range(1, parent1), 4, replace=False)))
    hijo1 = parent1[:pos_alea[0]] + parent2[pos_alea[0]:pos_alea[1]], parent1[:pos_alea[1:2]], parent2[pos_alea[2]:pos_alea[3]], parent1[:pos_alea[3:]]
    hijo2 = parent2[:pos_alea[0]] + parent3[pos_alea[0]:pos_alea[1]], parent2[:pos_alea[1:2]], parent3[pos_alea[2]:pos_alea[3]], parent2[:pos_alea[3:]]
    hijo3