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

Which of the following statements about the relationship between interest rates and bond prices is true?

1. There is an inverse relationship between bond prices and interest rates.
2. There is a direct relationship between bond prices and interest rates.
3. The price of short-term bonds fluctuates more than the price of long-term bonds for a given change in interest rates. (Assuming that coupon rate is the same for both)
4. The price of long-term bonds fluctuates more than the price of short-term bonds for a given change in interest rates. (Assuming that the coupon rate is the same for both)

Answer: I and IV only

Bond Duration

Consider a bond with a face value of $1,000, a coupon rate of 8%, a yield to maturity of 9%, and ten years to maturity. This bond's duration... Continue reading "Relationship between Interest Rates and Bond Prices" »

Vector Determination by Length and Angle

V = <||V|| Cosθ, ||V|| Sinθ> ---> ||V||Cosθi + ||V||Sinθj

Example:

a) Find the vector of length 2 that makes an angle of π/4 with the positive x-axis.

b) Find the angle that the vector V = -₃ i + j makes with the positive x-axis.

a) <||V||Cosθ , ||V||Sinθ> = <2cos45, 2sin45> ---> <₂, ₂>

b) Normalize... ||V|| = _{(-3)² + 1²} = ₄ = 2 -----> V/||V|| = <-_3/2 , 1/2> = <cosθ, sinθ> ----> cosθ = -_3/2, sinθ = 1/2 ---> θ = 5π/6

Dot Product

If U = <U₁, U₂> and V = <V₁, V₂>, then the dot product is U•V = U₁V₁ + U₂V₂.

Example:

a) U = <3, 5>, V = <-1, 2> -----> U•V = (3)(-1) + (5)(2) ---> U•V = -3 + 10 --> U•V = 7

b) U = &... Continue reading "Vector Operations, Dot and Cross Products, and Lines" »

Multiples consist of using the multiples of a company similar to yours to apply to your numbers and get an approximation of what yours is worth. It is not the most accurate in my opinion.

An example would be the EBITDA multiple, you multiply EBITDA by that figure obtained by dividing the value of the company you are comparing yourself to by its EBITDA.

This method is useful because although the estimated value of your company is not precise, with this method you can see the evolution over time.

250. The increase in provisions is not considered an outflow of cash, so it goes to the income statement.

– 150. The following are considered cash outflows

Add the cash paid instantly by the customer recorded in the cash flow statement as cash received... Continue reading "Valuation Methods and Cash Flow Analysis" »

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" »

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" »

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" »

Programmer Position Application

Dear Mr. Gilhooley,

I am writing to apply for the programmer position advertised on your website/in the Times Union on 25 January. As requested, I am enclosing a completed job application, my certification, my resume, and three references.

The opportunity presented in this listing is very appealing to me, and I believe that my strong technical experience and education will make me a very competitive candidate for this position and an asset to your organization. The key strengths that I possess for success in this position include:

• I have been in charge of 23 people in private banking.
• I strive for continued excellence.
• I provide exceptional contributions to customer service for all customers.
• I am a conscientious person