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

a) Divide and Conquer Idea (≤ 8 sentences)

Split the array into two halves around a middle index. Recursively compute the maximum subarray sum entirely in the left half and entirely in the right half. Also, compute the best “crossing” subarray that ends in the left half (best suffix of left) and continues into the right half (best prefix of right). The maximum subarray for the whole array is the maximum of these three values. The crossing sum can be found in linear time by scanning leftward from the middle to get a max suffix and scanning rightward from the middle + 1 to get a max prefix. Use this recursively until the base case of a single element is reached.

b) Recurrence and Asymptotic Time

Let T(n) be the running time on n items. We... Continue reading "Divide and Conquer Algorithms and Asymptotic Analysis" »

Digital Differential Analyzer (DDA) Algorithm

• Input: The starting point (x1, y1) and ending point (x2, y2).
• Calculate:
  • dx = x2 - x1
  • dy = y2 - y1
• Find:
  • Steps = max(|dx|, |dy|)
• Compute:
  • xinc = dx / Steps
  • yinc = dy / Steps
• Plot: The initial point.
• Iterate: Add xinc and yinc repeatedly until the endpoint is reached.

Bresenham's Line Drawing Algorithm

• Calculate dx and dy.
• Initialize the decision parameter.
• Plot the starting pixel.
• According to the decision parameter, choose the next pixel.
• Update the decision parameter.
• Repeat until the endpoint is reached.

Midpoint Circle Drawing Algorithm

• If P < 0, choose the East pixel.
• If P ≥ 0, choose the South-East pixel.
• Update the decision parameter.
• Continue until x > y.
• Use 8-way symmetry to complete the circle.

Sutherland-

Functions and Objectives

(a) f(z) = |z|, where z is a complex number.

(b) f(x, y) = x²y / (x² + y²)

Proof Objective

• (a) Prove that f(z) = |z| is continuous everywhere but nowhere differentiable except at the origin.
• (b) Find the iterative limit and simultaneous limit of f(x, y) = x²y / (x² + y²) as (x, y) → (0, 0).

Proof Process

(a) Continuity of f(z) = |z|

[Step 1]: Show that f(z) = |z| is continuous everywhere.

Let z₀ be an arbitrary complex number. We want to show that for any ε > 0, there exists a δ > 0 such that if |z - z₀| < δ, then |f(z) - f(z₀)| < ε.

We have f(z) = |z| and f(z₀) = |z₀|. Then |f(z) - f(z₀)| = ||z| - |z₀||.

By the reverse triangle inequality, we know that ||z| - |z₀|| ≤ |z - z₀|.

So, if... Continue reading "Complex Analysis: Continuity, Differentiability, and Limits" »

Matrices (Question 6a)

Verify that A · (\text{adj } A) = (\text{adj } A) · A = |A| · I_3 for
A = \begin{bmatrix} 2 & 3 & 4 \\ 3 & 0 & 1 \\ 2 & 1 & 5 \end{bmatrix}.

Tasks:

• Find the determinant |A|:
• Find the Adjoint (\text{adj } A): This involves finding the cofactor of each element and then transposing the resulting matrix.
• Cofactors: C₁₁ = -1, C₁₂ = -13, C₁₃ = 3, C₂₁ = -11, C₂₂ = 2, C₂₃ = 4, C₃₁ = 3, C₃₂ = 10, C₃₃ = -9
• Multiply A · (\text{adj } A)

4. Financial Arithmetic (Question 2g)

Find the compound interest on Rs. 8,000 for 1 1/2 years at 10% per annum, compounded annually.

• Amount for the first year:
• Interest for the next half year: Use simple interest on the new principal.
• Total Compound Interest:

Answers use standard... Continue reading "Matrix Determinant and Adjoint Verification with AP/GP and CI" »

Calculating the Future Value of an Annuity Due

Step 1: Determine the Variables

The problem provides the following details:

• Annual payment: Rs. 200. Therefore, the half-yearly payment (Pmt) is:
  
  Rs. 200 / 2 = Rs. 100
  Rs. 200 / 2 = Rs. 100
• Annual interest rate (r): 4% or 0.04. Since the interest is compounded half-yearly, the interest rate per period (i) is:
  
  0.04 / 2 = 0.02
  0.04 / 2 = 0.02
• Term: 20 years. Payments are made half-yearly, so the total number of periods (n) is:
  
  20 × 2 = 40
  20 × 2 = 40
• The annuity type is an annuity due, meaning payments are made at the beginning of each period.

Step 2: Apply the Future Value Formula

The formula for the Future Value (FV) of an annuity due is given by:

FV = Pmt × [((1 + i)^n - 1) / i] × (1 + i)

FV = Pmt × [((1

Solving Polynomials Using the Remainder Theorem

We are given the function:

f(x) = mx³ − 3x² + nx + 2

• When divided by (x + 3), the remainder is −3.
• When divided by (x − 2), the remainder is −4.

By the Remainder Theorem, if f(x) is divided by (x − a), the remainder is f(a).

Step 1: Substitute x = −3

f(−3) = m(−3)³ − 3(−3)² + n(−3) + 2

= −27m − 27 − 3n + 2

= −27m − 3n − 25

Given remainder −3:

−27m − 3n − 25 = −3

−27m − 3n = 22

27m + 3n = −22 (Equation 1)

Step 2: Substitute x = 2

f(2) = m(2)³ − 3(2)² + n(2) + 2

= 8m − 12 + 2n + 2

= 8m + 2n − 10

Given remainder −4:

8m + 2n − 10 = −4

8m + 2n = 6 (Equation 2)

Step 3: Solve the System of Equations

Multiply Equation 2 by 3 → 24m + 6n = 18

Multiply Equation... Continue reading "Solving Polynomial Remainder Theorem and Transformations" »

Customer Value: (Quality, Time, Flexibility, Customer Experience, Innovation) / Price

Sustainability Paradigms: Economic (Viable, Equitable), Environment (Viable, Bearable), Social (Bearable, Equitable)

- Efficiency (Optimization), Differentiator (Innovation), Driver (Motivation) | - Audits (Assess Sustainability Performance)

Table 7.3: Factors for Calculating Three-Sigma Limits for the X (Bar) Chart and R-Chart

*A sample is out of control if its value falls below the LCL or above the UCL*

Corporate Distribution Layers

Distributions are applied in the following order: Accumulated E&P + Current E&P, then basis, then the remainder is treated as capital gain.

• No Dividend: If there is no E&P, reduce current E&P first. Negative AEP still distributes E&P.
• Offset: Half of current E&P is lost to offset AEP before distribution.

Property Distributions

Recognize gains, but not losses.

• Dividend: FMV - Liabilities
• Basis: FMV
• Gains: FMV - Tax Basis

Distribution Table

The amount from CEP is calculated as: (Period distribution / Total distribution) * CEP. AEP is applied to the remainder, then to stock basis.

Stock Distributions and Redemptions

• Stock Distribution: Proportionate distributions are nontaxable and do not reduce E&