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

Accounting Obligations and Responsible Parties

Every employer shall keep proper accounts, appropriate to the activity of the company, to allow for the chronological tracing of all operations and the preparation of periodic reports and inventories. The records shall be maintained directly by employers or other authorized persons, without prejudice to their responsibility.

Compulsory Accounting Books

• Book of inventories and annual accounts
• Journal
• Minute book or record books (for corporations)

Legalization of Books

Legalization will take place in the Commercial Register at the address of the enterprise. It can be done a priori (before use) or afterwards in the event that the accounting entries are made on sheets which will then be bound to be correlatively... Continue reading "Accounting Regulations and Bookkeeping for Entrepreneurs" »

Hurwicz Criterion: Coefficient of Optimism

In this approach, a decision-maker may have an intermediate attitude. Since no probabilities are associated with each state of nature, this author suggests using a coefficient of optimism, simultaneously called C, and a coefficient of pessimism, (1 - C), where 0 ≤ C ≤ 1.

The coefficient C indicates the decision-maker's stance toward risk. When it is closer to 1, the decision-maker will be more optimistic, while closer to 0, the decision-maker is more pessimistic.

This criterion focuses only on the extreme scores of each alternative, weighing the result of maximum optimism with the coefficient C and the result of minimal value with the coefficient of pessimism (1 - C). The sum of these two products... Continue reading "Decision-Making Under Uncertainty: Criteria and Tables" »

Fundamental Data Mining Tasks and Concepts

Prediction: Forecasting Time Series Data

Prediction involves estimating the future value of a variable that is subject to random changes over time. This technique is strictly applied to time series, which are datasets whose domain is time.

Regression Analysis: Modeling Relationships

Regression is a generalization of classification (when the domain involves continuous classes) and prediction. The resulting model (classification or prediction) depends on the significance of the dependent and independent variables.

The goal is to find a mathematical or statistical model that properly relates the dependent variable with the independent variables. Geometrically, regression seeks a function that passes as close... Continue reading "Fundamental Data Mining Tasks and Algorithmic Approaches" »

The Concept of Materiality in Auditing

The auditor assesses the amount of errors or omissions that could affect the decisions of a reasonable user. This concept involves:

• Flexible guidelines.
• Both quantitative and qualitative elements.

Stages in the Application of Materiality

1. Establish a Preliminary Judgment on Materiality (Planning)

   This is the maximum amount of misstatement the auditor believes can exist in the financial statements without affecting the decisions of reasonable users. It is affected by the relative size of the company being audited.

   First, the auditor establishes a base (e.g., total assets, profits, turnover), which is multiplied by a percentage factor to determine an initial quantitative judgment. This initial judgment is then adjusted

Cutting speed depends on mining velocity. The cutter's speed must remain constant throughout its journey through the ore flow.

Sampling Methods

Sampling can be performed using manual or mechanical methods. Manual methods include cone and quartering, sampling with scoops or shovels, and using sampling tubes.

Manual Sampling Techniques

Cone and Quartering

1. Stack material into a conical pile by shoveling.
2. Form a cake or 'qeqe'.
3. Divide the cake into four equal fractions.
4. Take two opposite quarters as the sample; reject the other two.
5. If the sample size is adequate, combine the two retained quarters and reduce their size.
6. Repeat steps 1-4 as necessary until the desired sample size is achieved.

Sampling Scoop or Shovel

This procedure is used when ore is moved... Continue reading "Mining Ore Sampling Techniques and Procedures" »

Identities and Equations

An algebraic equality is formed by two algebraic expressions separated by an equal sign. They are of two types:

• Identity: True for any value of the letters.
• Equation: True only for specific values of the letters.

Elements of an Equation

• Member: Each of the two algebraic expressions separated by an equal sign.
• Term: Each summand in each of the members.
• Unknowns: Letters whose values are unknown.
• Degree: The greatest exponent of the unknowns.

The values of the unknowns that make the equation true are called solutions. Two equations are equivalent when they have the same solution.

2nd Degree Equations

A 2nd degree equation with one unknown is an equality of the form ax² + bx + c = 0, where a ≠ 0.

Analyzing the Number of Solutions

The... Continue reading "Algebraic Equations and Systems: Core Concepts" »

Fundamental Accounting Principles

• Basis: The van should be evaluated by its purchase price as it has been purchased for use and not for sale.
• Accrual: November 1, 20XX [2,400 / 12 = 200] [200 x 2 = 400]. For November and December: 200 + 200.
• Uniformity: To change the method, there needs to be a valid reason, such as altered circumstances that prompted the company to choose the original criteria. You should also explain the reasons for the change and its impact on the annual accounts, applying it to all elements with the same features.
• Prudence: Until the completion of the sale, it is not reflected in the accounts.
• No Compensation: Balances must be included in two different accounts: a right to receive and a payment obligation. This avoids loss of

Fundamental Concepts in Algebra

Numerical Language

Expressing mathematical information using only numbers.

Algebraic Language

Expressing mathematical information using both numbers and letters.

Algebraic Expression

A set of numbers and letters combined with mathematical operation signs.

Numerical Value of an Algebraic Expression

The number obtained by replacing the letters with their corresponding values and performing the indicated operations.

Monomial

A monomial consists of a number and one or more letters:

• The number (including its sign) is called the coefficient.
• The letter or letters that accompany it are called the literal part.

Degree of a Monomial

The sum of the exponents of the letters that form it.

Similar Monomials

Two or more monomials are similar... Continue reading "Fundamental Concepts of Algebra and Equations" »

Fortran 90: Implicit None and Vector Management

Implicit None

In Fortran 90, undeclared variables starting with I, J, K, L, or M are treated as integers, while others are treated as real numbers. However, it's best practice to avoid implicit declarations. Use IMPLICIT NONE right after the program instructions to disable implicit typing. This forces you to explicitly declare all variables, preventing potential errors due to character misuse.

Vector Management Forms

Ordering one-dimensional arrays (vectors) is a common operation. Here are three simple methods: selection, insertion, and bubble sort.

In all cases, we'll assume a one-dimensional array of N elements.

Selection Sort

1. Find the smallest element in the vector.
2. Swap it with the element in the