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Numerical Computing & Linear Algebra Essentials

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Floating Point Systems & Numerical Error

A Floating Point (FP) System represents numbers as: x = ± (d0 + d1/β + d22 + ... + dt-1(t-1)). The Unit Roundoff (u) is defined as εmachine/2, where fl(1 + ε) > 1.

Rounding to Nearest

When rounding to the nearest representable number, fl(x) = x(1 + ε) where |ε|.

IEEE 754 Standard for Floating Point

Normalized Numbers

If the exponent (e) is not equal to 0, it's a normalized FP number. The value is x = (-1)sign ⋅ β(e - offset) ⋅ (1.d1 d2...dt-1).

Denormalized Numbers

If the exponent (e) is 0, the number is denormalized. The value is x = (-1)sign ⋅ β(e - offset + 1) ⋅ (0.d1 d2...dt-1). The sticky bit 0 is free because it is always determined by the value of exponent e.

Exceptional Values

  • If
... Continue reading "Numerical Computing & Linear Algebra Essentials" »

Map Symbols, Scale, and Distance/Direction

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Map Generalization

Types of Symbols

  • Line Symbols: Represent real-life objects with a linear path.
  • Point Symbols: Represent objects occurring at a single point on Earth's surface using a dot.
  • Area (Polygon) Symbols: Represent real-life objects spread over Earth's surface using geometric shapes.

Generalization Techniques

Reality contains too much information for a single 2D map. Generalized geometry and content make a map useful. A good map suppresses less important information to highlight what needs to be seen.

  • Selection: Only relevant line, point, and area features are chosen.
  • Classification: Grouping similar features and using a common symbol to represent them.
  • Simplification: Reduction of unnecessary detail.
  • Smoothing: Smoothing out abruptly joined
... Continue reading "Map Symbols, Scale, and Distance/Direction" »

Cost Accounting Essentials: Key Concepts and Calculations

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Chapter 2: Predetermined Overhead Rate

Predetermined Overhead Rate = Estimated Total Manufacturing Overhead (MOH) / Estimated Total MOH Driver (e.g., Direct Labor hours, Direct Labor costs, Machine Hours)

Prime Cost = Direct Materials + Direct Manufacturing Labor

Conversion Cost = Direct Manufacturing Labor + Indirect Manufacturing Overhead

Cost Accumulation: Data is collected in an organized way (also known as cost pools).

Cost Assignment: Systematically links an actual cost pool to a distinct cost object (e.g., Tires, engine, labor assigned to car cost).

Activity Base: Examples include kilometers driven in a car, units produced, units sold, machine hours.

Product Cost: Costs tied to creating a product (Direct Materials, Direct Labor, Manufacturing... Continue reading "Cost Accounting Essentials: Key Concepts and Calculations" »

Understanding Financial Formulas and Calculations

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Tutorial 1

If you get a positive value times a number,

You need to shift the decimal to the right as many times as the number specified.

If negative, move it to the right.

Simple interest formula = S = FV = P(1 + iK)

Compound interest formula = Sk = P(1 + i)^k

Sn = P(1 + I/T)^n
where I is interest
T is frequency of compounding per year
K is the number of years
N is the total number of periods - K T or TK

Depreciation Formula = Vo or P = Initial value,
Vk = P(1 - d)^k

Tutorial 2

1. 5 years 1 + r = (FV/PV)^(1/5)
(i) r = 10.38%
(ii) r = 10.47%
(iii) r = 10.51%
(iv) r = 10.52%
(v) r = 10.52%
2. 1 + r = (1 + 0.06/12)^8 ∙ (1 + 0.072/12)^4
1 + r = (1.005)^8 ∙ (1.006)^4
1 + r = (1.0407) ∙ (1.0242) = 1.06591
r = 6.59%

For an initial outlay of $1000, the net return is

... Continue reading "Understanding Financial Formulas and Calculations" »

Matrices: multiplication, rank, determinant, inverse and Rouche-Frobenius theorem

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System Types
The systems of equations can be
classified by the number of solutions that can arise. According to that case may have the following cases:
· Incompatible system if it has no solution.
· Compatible system if you have any solution in this case can also distinguish between:
or compatible system determined when it has a finite number of solutions.
indeterminate
or compatible system when it admits an infinite set of solutions.
Fitting and classification:
Image
Calculating the rank of a matrix for determining
Image
1. We can rule a line if:.
· All the coefficients are zeros.
· There are two equal lines.
A line is proportional to another.
A line is a linear combination of others.
Delete the third column because it is a linear... Continue reading "Matrices: multiplication, rank, determinant, inverse and Rouche-Frobenius theorem" »

English Grammar and Vocabulary Exercises

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Part I: Complete the Following Table

Sentence TypeExample
AffirmativeCarlos is in the house.
NegativeCarlos isn't in the house.
QuestionIs Carlos in the house?
AffirmativeWe are 23 years old.
NegativeWe aren't 23 years old.
QuestionAre we 23 years old?

Part II: Complete the Following Sentences

  1. She is from the United States. She is American.
  2. She is from Nigeria. She is Nigerian.
  3. She is from Germany. She is German.
  4. They are from Egypt. They are Egyptian.
  5. We are from Canada. We are Canadian.

Part III: Select the Correct Sentence

  1. Select the correct sentence.
    • a) She lives with Carlos, and she works on Saturdays.
  2. Select the correct sentence.
    • c) They are running in the park.
  3. Select the correct sentence.
    • b) She is 20 years old.
  4. Select the correct sentence.
    • a) We always
... Continue reading "English Grammar and Vocabulary Exercises" »

Understanding Bonds: Key Features and Market Dynamics

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Bond Characteristics

  • Coupon: The interest payment made by the bond issuer, usually expressed as an annual percentage of the bond's face value.
  • Par (Face Value): The amount the bondholder receives when the bond matures, typically $1,000.
  • Term to Maturity: The time remaining until the bond's maturity date when the issuer must repay the bond's par value.
  • Denomination: The face value of the bond, usually in increments of $1,000.
  • Quotation: Bonds are quoted as a percentage of their face value (e.g., a bond quoted at 95 is selling for 95% of $1,000, or $950).

Bond Prices, Yield to Maturity (YTM), Current Yield, and Rate of Return (HPR)

  • Bond Prices: The market price of a bond depends on interest rates. Prices and interest rates have an inverse relationship.
... Continue reading "Understanding Bonds: Key Features and Market Dynamics" »

Introduction to Statistics: Discrete and Continuous Random Variables, Probability Distributions, and Sampling Techniques

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Discrete Random Variables

Discrete random variables are variables that can take on a finite number of distinct values. In simpler terms, a discrete random variable is a set of possible outcomes that is countable.

Continuous Random Variables

Continuous random variables are random variables that take an infinitely uncountable number of potential values, typically measurable amounts.

Example

  1. List the sample space in the given experiment. How many outcomes are possible?

The sample space is: S = {NNN, NND, NDN, NDD, DNN, DND, DDN, DDD}

  1. Count the number of defective keyboards in each outcome in the sample space and assign this number to the outcome. For instance, if you list NND, then the number of defective keyboards is 1.

The possible values of X are 0,... Continue reading "Introduction to Statistics: Discrete and Continuous Random Variables, Probability Distributions, and Sampling Techniques" »

Inventory Management Principles and Practices

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Inventory Fundamentals

One use of inventory is to provide a hedge against inflation. ABC analysis divides an organization's on-hand inventory into three classes based upon annual dollar volume. Cycle counting is a process by which inventory records are verified. The difference(s) between the basic EOQ model and the production order quantity model is that the production order quantity model does not require the assumption of instantaneous delivery. Extra units that are held in inventory to reduce stockouts are called safety stock. Inventory record accuracy would be decreased by increasing stockroom accessibility. The two most important inventory-based questions answered by the typical inventory model are when to place an order and how many of

... Continue reading "Inventory Management Principles and Practices" »

Statistical Relationships: Scatter, Correlation, Regression

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What is a Scatter Diagram?

Definition

A scatter diagram (or scatter plot) is a graphical representation of two variables where each point represents an observation consisting of paired values from two datasets. The horizontal axis (X-axis) represents one variable, and the vertical axis (Y-axis) represents the other.

Construction

Each point (x_i, y_i) is plotted on the graph for the corresponding values of the two variables.

Utility in Correlation Analysis

Scatter diagrams are essential for:

  • Visualizing relationships: Helps identify if a linear or non-linear relationship exists.
  • Direction of correlation:
    • Positive correlation: As X increases, Y increases (points slope upwards).
    • Negative correlation: As X increases, Y decreases (points slope downwards).
... Continue reading "Statistical Relationships: Scatter, Correlation, Regression" »