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

What is a One-Way Function?

A one-way function is a function that is easy to compute in one direction but difficult to compute in the reverse. For example, given an input, a hash is easy to compute. However, given the output, it is extremely difficult (time-consuming) to determine the input.

What is a Trapdoor Function?

A trapdoor function is like a one-way function; however, a trapdoor function can reverse the one-way function if the trapdoor is known. For example, finding the two prime divisors of 6,895,601 is difficult. However, if you know that 1,931 is one of the numbers, it will be easy to divide 6,895,601 by 1,931 to determine the answer. In this case, 1,931 is the trapdoor.

What Makes RSA Difficult to Break?

The large number factorization... Continue reading "Understanding Cryptographic Functions and RSA Security" »

Essential Business and Financial Vocabulary

This list provides key English financial and business terms translated into Catalan and Spanish for professional reference.

General Business Terms

• 1. Track = Seguir
• 2. Reliability = Fiabilitat, és fiable
• 3. Shareholders = Accionistes
• 4. Invoice = Factures per cobrar
• 5. Receipt = Factures per pagar
• 6. Stakeholder = Inversor
• 7. Lender = Prestamista
• 8. Suppliers = Proveïdor
• 9. Reliability = Que no hi ha risc
• 10. Meeting = Complir amb
• 11. Showcasing = Mostrar
• 12. Often issued = Treure a la llum
• 13. Quarterly intervals = 4t trimestre

Credit and Debt Terminology

• 14. Creditworthy = Creditibilitat
• 15. Creditor = Persona que inverteix
• 16. Resources = Recursos
• 17. Balance sheet = Fulla de càlcul
• 18. Running out = Quedar-se sense
• 19.

Sample Characteristics: Measures of Location and Variation

Measures of Location

• Average

  The average is a central value, often referred to as a measure of central tendency.

• Mean

  • Also known as the arithmetic mean.
  • Found by dividing the sum of all values by the total number of values.
  • Formula: Mean = (Sum of Quantities) / (Number of Quantities)
  • Example: Find the mean of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
    • Sum of values: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55
    • Number of values = 10
    • Mean of values = 55 / 10 = 5.5

• Median

  The median of a set is the middle number when the values are arranged in order from lowest to highest.

• Mode

  • The mode is the value that occurs most frequently in a set of values, also known as the modal value.
  • Example: Find the mode of 1, 2, 2, 3,

10 reasons to use games in language teaching

1.Games create a context for meaningful communication. 2. This meaningful communication serves as a basis for comprehensible input. 3. Games add interest to what learners find boring. 4. Games can be used with all language skills. 5. Games offer a fun experience. 6. Games encourage participation from all learners. 7. Games are learner-centered activities. 8. Games work outside of class. 9. Games promote cooperative learning. 10. Games fit into multiple intelligence theory.

Using Games in Teaching English to Young Learners: How to Choose a Game

Students may wish to play games purely for fun. Teachers need more convincing reasons. The key to a successful language game is that the rules are clear. The... Continue reading "Benefits of Using Games, Songs, and Stories in Language Teaching" »

Problem 4: Bank Savings Distribution

The average savings in the clients' accounts of a certain bank follow a normal distribution N(50,000, 15,000) (mean = 50,000 €, standard deviation = 15,000 €). The bank distributes its clients into three categories:

• Category A: Those clients that have less than 20,000 € in their savings account
• Category B: Those clients that have between 20,000 € and 80,000 € in their savings account
• Category C: Those clients that have more than 80,000 € in their savings account

Calculate:

4a) What percentage of clients belong to Category A?

Use the standard normal variable Z = (X - μ) / σ.

For X = 20,000 €:

Z = (20,000 - 50,000) / 15,000 = -30,000 / 15,000 = -2.000

The cumulative probability Φ(-2.00) ≈ 0.0228,... Continue reading "Normal Distribution Problems: Z‑Scores, Percentages & Parameters" »

Fundamental Concepts in Financial Accounting

Merchandising Operations and Inventory Fundamentals

• A business whose income is derived largely from buying or selling goods, rather than rendering services, is known as a Merchandising Firm.
• The goods on hand at the beginning or the end of the accounting period are called Inventory.
• The merchandise inventory on hand at the end of the period will appear both in the Cost of Goods Sold (CGS) section of the income statement and as an asset on the balance sheet.
• The Purchase account is used only for goods intended for Re-sale. Purchases of trucks and equipment are debited to Asset accounts.
• Beginning Inventory plus Net Purchases equals Goods Available for Sale (GAS).
• In recording both beginning inventory and

Key Trends in Educational Outcomes (2009-2012)

• Equity in education outcomes deteriorated between 2009 and 2012.

Gender Differences in Academic Performance

• In 2012, girls outperformed boys in reading by an average of 29 score points.
• In 2012, boys outperformed girls by 16 score points in mathematics, a 7-point larger gender gap than seen in 2003.
• Across OECD countries, 42% of students (64% of girls) believe their failure in mathematics is due to a lack of ability. In Spain, this figure rises to 74% of students (80% of girls).

Socioeconomic Origin and Academic Performance

• In 2012, socioeconomically advantaged students outperformed less-advantaged students by 34 score points in mathematics.

Ethnicity and Academic Performance

• The immigrant student population

The Internet's Impact in the UK

Widespread Adoption and Recent History

The Internet has become an indispensable part of daily life in the UK, with widespread adoption across the population. It is now ubiquitous, found in homes, workplaces, and public spaces. Remarkably, just a decade ago, home Internet access was a rarity for many.

Demographics of UK Internet Users

In the UK, the largest percentage of Internet users falls within the 16-34 age bracket. As age increases, average Internet usage tends to decline, indicating a generational divide in digital engagement.

Popular Online Activities

Users frequently engage with various online platforms. The most commonly visited websites typically include social networking sites, email services, search engines,... Continue reading "Comprehensive Analysis: UK Digital Landscape & Car Performance" »