Notes, summaries, assignments, exams, and exercises for Mathematics for University

Introduce Research

• Study design/methods
• Appraisal
• Interpretation
• Application/utilization

Understand

• Evidence Informed Decision Making Model
• Sources of Evidence
• Types of research that inform practice (quantitative, qualitative, mixed methods)
• Why research studies should be critically appraised

1.2 Research, EBP & EIDM

• There are different ways of knowing; Empirical, Personal, Aesthetics, Ethical

1. Empirical (focus of class)

• Theories, models, facts
• Validation, confirmation
• Scientific competence

2. Personal knowledge

• Person stories, self
• Reflection, response
• Therapeutic use of self

3. Aesthetics knowledge

• Experience of nursing, health, illness
• Appreciation, grasp meaning
• Transformative act/acts

4. Ethical Knowledge

• Principles, codes
• Justification, dialogue

Evidence-Based Practice...

Financial Concepts and Calculation Practice

Expectation Theory and Two-Year Rates

Problem: If the one-year rate on an instrument is 9.0% and the expected rate for a one-year instrument one year from today is 9.89%, what is the expected two-year rate today, if the expectation theory holds?

• One-year rate: 9.00%
• One-year rate, one year forward: 9.89%

Formula: (1 + R1) * (1 + ER2) = (1 + R2)^2

Calculation:

• (1 + 0.09) * (1 + 0.0989) = (1 + R2)^2
• (1.09) * (1.0989) = (1 + R2)^2
• 1.197801 = (1 + R2)^2
• sqrt(1.197801) = 1 + R2
• 1.094449 = 1 + R2
• R2 = 0.094449

Expected Two-Year Rate: 9.44% (approximately 9.4%)

Understanding the Rule of 72

The Rule of 72 is a quick method to estimate the number of years it takes for an investment to double in value, given a fixed annual...

Consider an European call option with strike equal to 10, T = 1, r = 0.05 and σ = 0.2. Using the following time-series:

1. For every time moment t = 0, 1/360, 2/360:

• Calculate the Black-Scholes price.
• Calculate the corresponding delta.
• Calculate the price of the corresponding replicating portfolio and its composition, if we consider a dynamic hedging.

2. Calculate a static hedging. What is the final value of the portfolio? Compare with the dynamic hedging results.

Solution

• At time 0, time to maturity is 1, and then
  • The Black-Scholes price is given by S0N(d1)−Ke−rT N(d2) = 10N(0.35)−10e−0.05N(0.15) = 1.0450
  • The Delta is given by N(d1) = N(0.35) = 0.6368
  • At time t = 0, the value of the replicating portfolio coincides with the Black-Scholes price

Equations of State and Real Gases

Critical Point Conditions: (∂p/∂v)_Tc = (∂²p/∂v²)_Tc = 0

• Real Gas Law: pv = ZRT
• Van der Waals Equation: (p + a/v²)(v - b) = RT
• Reduced Variables: v_r = v/v_c; T_r = T/T_c; p_r = p/p_c
• Compressibility Factor: Z = Z(p_r, T_r)
• Semi-Reduced Volume: v_r,sem = v/v_c

Thermodynamic Differentials and State Functions

Functions of Temperature and Pressure (T, p)

• Enthalpy Differential: dh = C_pdT + [v - T(∂v/∂T)_p]dp
• Internal Energy Differential: dU = [C_p - p(∂v/∂T)_p]dT - [T(∂v/∂T)_p + p(∂v/∂p)_T]dp

Functions of Temperature and Volume (T, v)

• Entropy Differential: dS = (C_v/T)dT + (∂p/∂T)_vdv
• Enthalpy Differential: dh = [C_v + v(∂p/∂T)_v]dT + [T(∂p/∂T)_v + v(∂p/∂v)_T]dv
• Internal Energy Differential: dU = C_vdT + [

Statistics: Key Concepts and Methods

1. Statistics is a collection of methods for planning experiments, obtaining data, and then organizing, summarizing, presenting, analyzing, interpreting, and drawing conclusions.

2. Statistical methods consist of a series of procedures for managing, analyzing, and collecting qualitative and quantitative research data.

3. A population includes all objects of interest, whereas a sample is only a portion of the population.

Example: The population could be all students of a postgraduate program, and a representative sample of this population could be 400 students from different studies of the program.

Understanding Variables in Statistics

4. A variable is any characteristic, number, or quantity that can be measured...

Probability Distributions

Discrete Random Variables

1. If the outcomes of a discrete random variable follow a Poisson distribution, then which of the following is true?
   • (A) The mean equals the variance.
2. Which of the following is not a characteristic of a binomial probability distribution?
   • (A) Each trial has a finite number of possible outcomes.
3. If the probability of a machine producing a defective part is 0.05, what is the probability of finding exactly 5 defective parts from a sample of 100? (Assume that the process follows a binomial distribution and round the answer to four places.)
   • (C) 0.1800
4. The number of arrivals of delivery trucks per hour at a loading station is an example of which of the following processes?
   • (C) Poisson
5. When sampling without replacement

1. Many business owners spend years trying to get their enterprise started.

2. At the rate we are going, we are lucky to break even at year's end.

3. The world's biggest stock exchange is in New York.

4. Every employee has shares in the company.

5. The stock market rose 20% today. Brokers were euphoric!

6. The dip in the market caused some investors concern.

7. There was a dramatic rise in the share price at the end of the week.

8. Dividend is the part of the company's profits which is paid to shareholders.

9. One of the most difficult parts of running a restaurant is keeping the cashflow which can be variable.

10. House prices have increased by 5%. It is too expensive to buy now.

11. Demand for this product has fallen recently. We won't manufacture...

Quadratic Forms and Symmetric Matrices

The quadratic form is defined as w(x) = x^TAx, where M(w)_c = 1/2(A + A^T) is the associated symmetric matrix in the canonical basis.

Change of basis formula: M(w)_b = (P_BC)^T · M(w)_c · P_BC.

Rank properties: rg(w) = rg(A) = rg(M(w)_c). If rg(w) = max, the form is non-degenerate; otherwise, it is degenerate.

Diagonalization of Symmetric Matrices

For a symmetric matrix A, applying elementary row and column operations leads to (D | P^T), where D is diagonal. The goal is to obtain zeros in the off-diagonal positions.

A form w admits a non-degenerate orthonormal basis if the rank is maximum or the diagonal elements of D are non-zero.

Signature and Sylvester's Criterion

The signature sig(w) = sig(A) = (p, r_p), where rg(w)

Valuation Criteria

Historical Cost

For Assets: The purchase price or production cost. This includes the amount of cash paid or payable, plus the fair value of any other consideration given for the acquisition. All costs directly related to the acquisition and necessary to bring the asset to operating condition are included.

For Liabilities: The value corresponding to the consideration received in exchange for incurring the debt. In some cases, it is the amount of cash expected to be paid to settle the liability in the ordinary course of business.

Fair Value

The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between knowledgeable, willing market participants at the measurement date. It is determined...