The Excel POWER function returns a number raised to a given power. The POWER function is an alternative to the exponent operator (^). It can be used as a worksheet function (WS) in Excel. The syntax for the POWER function is =POWER(number, power), where number is the base number and power is the exponent to which the base number is raised.

The POWER function is a built-in function in Excel that is categorized as a Math/Trig Function. It returns the result of a number raised to a given power. In Excel, the caret symbol (^) is used to indicate exponents. For example, to calculate the volume of a room, you can use the formula =B2^3, where B2 is the length of the room.

To lock a cell in Excel, select the cell, go to the Home tab, click the small arrow to open the Format Cells popup window, and select the Locked check box. You can also press Control + 1 to open the Format Cells dialog box and select the Protection tab.

To calculate the average in Excel, click a cell below the column or to the right of the row of the numbers for which you want to find the average, then click the arrow next to AutoSum > Average, and press Enter.

There are three horizontal alignments available in Excel: left, center, and right. There are also three vertical alignments available: top, middle, and bottom.

To merge cells in Excel, select the cells, click Home > Merge & Center. Make sure only one of the cells in the range has data.

To create a chart in Excel, select the data, click INSERT > Recommended Charts, and scroll through the list of charts that Excel recommends for your data.

To align text in a cell, select the cell, go to the Home tab, and choose one of the following alignment options: Top Align, Middle Align, or Bottom Align for vertical alignment, and Align Text Left, Center, or Align Text Right for horizontal alignment.

To remove protection and unhide formulas in Excel, go to the Home tab, click the Format button, and select Unprotect Sheet from the drop-down menu. Alternatively, go to the Review tab > Changes group, and click the Unprotect Sheet button.

To protect a sheet or range in Google Sheets, open a spreadsheet, click Data, select Protected sheets and ranges, and click Add a sheet or range or click an existing protection to edit it. Choose how you want to limit editing and click Save or Done.

To create a bar chart in Microsoft Word, open Microsoft Word, go to the insert tab on the ribbon, and select the chart icon. Select column and then select clustered bar type, then select OK.


Evaluate the given problem by ensuring a full understanding of what is being asked. Note any specific conditions or constraints and determine how to verify the correctness of the answer. Consider rephrasing the question to facilitate comprehension.

If difficulties arise, refer to the provided hints. The first hint suggests utilizing partial fractions to approach the problem. If this method yields no progress, proceed to the next hint, which advises using a polynomial of degree one in the numerator since the denominator cannot be simplified further.

To write the integrand using partial fractions, compare coefficients to obtain equations. Solve these equations to find the values of the variables. This process involves adding twice one equation to another, yielding a specific value. This value then determines the coefficients, resulting in the partial fraction decomposition.

The key steps involve:
- Understanding the problem and its constraints
- Applying partial fractions as hinted
- Comparing coefficients to form equations
- Solving these equations to find variable values
- Using these values to express the integrand in partial fractions.


**Introduction to Trigonometry**

Trigonometry is a part of math science that deals with the specific functions of angles, degrees, and their application. The name "trigonometry" comes from the Greek words **trigon**, meaning triangle, and **metron**, meaning measure. Trigonometry can be divided into planar (angles and distances in the plane) and spherical (angles and distances in space) trigonometry.

**What is Trigonometry?**

Trigonometry refers to studying the properties of a right triangle, lengths, and angles. It involves learning the relationships between the lengths of the sides of a right triangle. Additionally, trigonometry deals with waves and oscillations. It is not just a study of plane figures, as it has a branch called spherical trigonometry that studies triangles in three-dimensional spaces.

**History of Trigonometry**

Trigonometry has its roots in ancient Babylon, Egypt, India, and China. The founder of modern trigonometry is considered to be Hipparchus of Greece (around 190-120 BC), who was the first to calculate the values of trigonometric functions. In the 2nd century, Ptolemy contributed to the development of trigonometry in his work Almagest. The invention of logarithms by John Napier in 1614 also greatly facilitated numerical computation, including compiling tables of trigonometric functions.

**Trigonometric Identities**

Trigonometric identities are equalities of two expressions that remain accurate even when any values are included instead of general numbers. Basic trigonometric identities include:

- Connection of sine and cosine: sin²α + cos²α = 1
- Sine and tan connection: sin²α = tg²α / (1 + tg²α)
- Cosine and tan connection: cos²α = 1 / (1 + tg²α)

**Trigonometry Table**

The trigonometry table is used to remember and easily find trigonometric numbers of basic angles (in degrees). The table includes values for sine, cosine, tangent, and cotangent for angles 0°, 30°, 45°, 60°, and 90°.

**Trigonometry vs. Geometry**

Mathematics has three main branches: Arithmetic, Algebra, and Geometry. Geometry is the study of the shapes, sizes, and properties of a space of a certain number of dimensions. Trigonometry is a branch of geometry that studies the properties of a right triangle.

**Using a Trigonometry Calculator**

To use a trigonometry calculator, enter the values you know, such as one side and one angle, two sides, area, and one side. The calculator will help you find the missing sides or angles of a right triangle.

**Real-world Applications**

Trigonometry has numerous real-world applications, including:

- Surveying
- Military and engineering
- Optics and statics
- Physical chemistry and physics
- Music and architecture
- Aerial control and air traffic control
- Image and video analysis

**Frequently Asked Questions**

- **What are the 6 basic trigonometric functions?**: Sine, cosine, tangent, cotangent, secant, and cosecant.
- **Is trigonometry hard to learn?**: With effort and professional help, it is possible to explore this area of science at a basic level.
- **How to learn Trigonometry?**: Lectures at school and independent work, understanding basic ideas and methods, and learning algebra and geometry beforehand.
- **Who invented trigonometry?**: Trigonometry occurs in ancient Babylon, Egypt, India, and China, with Hipparchus of Greece being the founder of modern trigonometry.
- **When do you learn trigonometry?**: At the age of 14-15, with advanced trigonometry learned at 17-18.
- **What grade do you learn trigonometry?**: In 9th grade.


#### Square Feet Equation

## What is the formula of square feet?

To calculate the area of a square or rectangular room, measure the length and width, then multiply them. The formula is: Length x Width = Area. For example, a room measuring 11 feet wide x 15 feet long has an area of 165 square feet.

## Calculating Square Feet for Various Room Sizes

- A 10×10 room has an area of 100 square feet.
- A 12×16 room has an area of 192 square feet.
- A 20×40 room has an area of 800 square feet, calculated by multiplying the width (20 feet) by the length (40 feet).

## Converting Square Feet to Feet

The conversion from square feet to feet involves understanding that square feet measure area, not length. However, for a square:
- 1 square foot is equivalent to a square with sides of 1 foot.
- 3 square feet would be a square with sides of approximately 1.7321 feet.
- 4 square feet would be a square with sides of 2 feet.

## Understanding Square Feet

1 square foot is a unit of area equal to a square measuring one foot on each side, or 0.0929 square meters. Abbreviated as ft² or sq.

## Calculating Square Feet for Specific Rooms

- A 5×5 room has an area of 25 square feet.
- A 10×10 room has an area of 100 square feet, calculated by multiplying the width (10 feet) by the length (10 feet).

## Understanding 1000 Square Feet

A 1000 square foot area can be achieved with various dimensions, such as 50 feet long by 20 feet wide, since 50 x 20 = 1000 square feet.

## Converting Area to Dimensions

- 500 square feet can be achieved with dimensions such as 25 feet x 20 feet, but it cannot be directly converted into a length or distance.
- 17 feet is a length, not an area, but if a room is 17 feet wide and 7 feet long, its area would be 119 square feet.

## Calculating Square Feet for Larger Areas

- A 12×24 feet area is 288 square feet.
- A 20×20 feet area is 400 square feet.

## Multiple Choice Questions

1. What is the formula for calculating square feet?
a) Length + Width
b) Length x Width
c) Length - Width
d) Width / Length

Answer: b) Length x Width

2. How many square feet is a 10×10 room?
a) 50 square feet
b) 100 square feet
c) 200 square feet
d) 500 square feet

Answer: b) 100 square feet

3. What does 1 square foot measure?
a) Length
b) Width
c) Area
d) Volume

Answer: c) Area


## Re-Teach Idea
The Algebra 1 class struggled with graphing linear equations, scoring under 50% on a basic quiz. Despite covering the material, students did not report difficulties during independent practice.

## Homework
Homework plays a crucial role in promoting learning, allowing students to work independently, make mistakes, and explore concepts. Its purpose is not to earn grades or keep students busy, but to reinforce understanding.

## Mathematical Literacy
Integrating literacy development into math instruction is essential. A professional development seminar highlighted the importance of best practices and strategies for student success in math programs like Cambridge IGCSE.

## Favorite Technology
The evolution of technology, from radio to the internet, has led to claims of revolutionizing education. However, the true impact of technology on learning remains to be seen.

## Teaching Square Roots Conceptually
Teaching square roots conceptually is challenging, as students often rely on procedural methods. For example, 24−−√=62–√ or 4–√=2–√24=62. To address this, instructors should focus on conceptual understanding rather than just procedural corrections.

## The Question
Students often ask, "When will I use this in real life?" The honest answer may be "never," but this question can be an opportunity to discuss the value and relevance of mathematical concepts.

## Problem-Solving without Algebra
Given the equations 3a = k and ax = 4k, can you solve for x without using algebraic manipulation? This problem requires critical thinking and an understanding of mathematical relationships.

## Exponents Part 2
Exponents can be tricky, especially when determining the base and its relationship to the exponent. Understanding the concept of repeated multiplication is crucial for working with exponents, particularly in division.

## Exponents Part 1
Exponents represent repeated multiplication, and their syntax is essential to understanding mathematical notation. The spatial arrangement of characters conveys meaning, and recognizing this syntax is vital for reading and working with exponents.

## The Purpose of Homework
The primary purpose of homework is to promote learning, allowing students to try concepts independently, make mistakes, and explore mathematical ideas. It is not solely for earning grades or completing tasks, but for deepening understanding and fostering a growth mindset.


## Counting Vertices of a Star

To count the vertices of a star, start with a circle of 5 points, labeled 0, 1, 2, 3, and 4. The second number indicates which points to connect, counting 2 places each time.

## Types of Stars

A 4-pointed star is called a Diamond star or Compass star. A star polygon is a non-convex polygon that resembles a star, with regular star polygons being the most studied.

## Regular Polygons

A star is considered a regular polygon if it has equal sides and angles. By definition, a polygon is a two-dimensional shape formed with straight lines and is closed. A regular polygon has equal sides and angles.

## Star Polygons

In geometry, a star polygon is a polygon with a star-like shape. Examples of regular star polygons include the pentagram and hexagram. Regular star polygons were first studied systematically in the 14th century.

## Vertices of Stars

A regular star pentagon has 5 corner vertices and intersecting edges, while a concave decagon has 10 edges and 2 sets of 5 vertices. The number of vertices of a star depends on its type and structure.

## Polygons and Vertices

A hexagon is a 6-sided shape with 6 vertices and 6 angles. A quadrilateral is a polygon with 4 sides and 4 vertices. A circle has 0 vertices.

## Vertices of Specific Shapes

A 5-pointed star has 5 vertices, while a hexagon has 6 vertices. A cross has 4 vertices, and a 2-dimensional cross has 12 vertices.

## Definition of Vertices

Vertices are the points where the sides of a shape meet. Sides are lines, and vertices are the points that connect them.

## Shapes and Vertices

A shape with 4 vertices is a quadrilateral, and a shape with 6 angles is a hexagon. The number of vertices of a shape depends on its type and structure.

## Conclusion

In conclusion, counting the vertices of a star involves understanding the structure and type of the star. Regular polygons, including star polygons, have equal sides and angles. The number of vertices of a shape depends on its type and structure, and can range from 0 for a circle to 12 for a 2-dimensional cross.


To find the focal point of a parabolic cooker, you need to determine where the sun's rays will be strongest. The formula for a parabola is f = x^{2}/4a. Here's a step-by-step guide to find the focal point:

1. Measure the longest diameter (width) of the parabola at its rim.
2. Divide the diameter by two to determine the radius (x) and square the result (x^{2}).
3. Measure the depth of the parabola (a) at its vertex and multiply it by 4 (4a).
4. Divide the answer from Step 2 by the answer to Step 3 (x^{2}/4a).

The result is the distance from the vertex of the parabola to its focal point, where you should place your water container for maximum sun exposure. For further exploration, visit www.ies.co.jp/math/java/conics/focus/focus.html.


It is proposed to build a single circular park equal in area to the sum of areas of two circular parks of diameters 16 m and 12 m. The radius of the new park would be determined as follows:

Let the diameters of the two circular parks be D_{1} = 16 m and D_{2} = 12 m, with corresponding radii R_{1} = 8 m and R_{2} = 6 m.

The areas of the two parks are:
- Area of first circular park = π(8)^{2} = 64 π m^{2}
- Area of second circular park = π(6)^{2} = 36 π m^{2}

The area of the single circular park is the sum of the areas of the two parks:
πR^{2} = 64 π + 36π = 100π

Solving for R:
πR^{2} = 100π
R^{2} = 100
R = 10

Therefore, the radius of the single circular park is 10 m.

#### Options

10 m
15 m
20 m
24 m

#### Solution
The radius of the new park would be **10 m**.


In vector calculus, the **Jacobian matrix** of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, its determinant is referred to as the **Jacobian determinant**. Both the matrix and the determinant are often referred to simply as the **Jacobian**.

Suppose **f** : **R**^{n} → **R**^{m} is a function such that each of its first-order partial derivatives exist on **R**^{n}. The Jacobian matrix of **f** is defined to be an *m*×*n* matrix, denoted by **J**, whose (*i*,*j*)th entry is .

The Jacobian matrix represents the differential of **f** at every point where **f** is differentiable. If **h** is a displacement vector, the matrix product **J**(**x**) ⋅ **h** is another displacement vector, that is the best linear approximation of the change of **f** in a neighborhood of **x**.

When *m* = *n*, the Jacobian matrix is square, so its determinant is a well-defined function of **x**, known as the **Jacobian determinant** of **f**. It carries important information about the local behavior of **f**. The function **f** has a differentiable inverse function in a neighborhood of a point **x** if and only if the Jacobian determinant is nonzero at **x**.

The Jacobian determinant also appears when changing the variables in multiple integrals. If **f** : **R**^{n} → **R**^{n} is a function, the Jacobian determinant is used to transform integrals between the two coordinate systems.

The Jacobian matrix can also be thought of as describing the amount of "stretching", "rotating" or "transforming" that the function imposes locally near a point. For example, if (*x*′, *y*′) = **f**(*x*, *y*) is used to smoothly transform an image, the Jacobian matrix **J**_{f}(*x*, *y*), describes how the image in the neighborhood of (*x*, *y*) is transformed.

If a function is differentiable at a point, its differential is given in coordinates by the Jacobian matrix. However, a function does not need to be differentiable for its Jacobian matrix to be defined, since only its first-order partial derivatives are required to exist.

The Jacobian matrix is used in various fields, including regression and least squares fitting, dynamical systems, and Newton's method. It is also used to determine the stability of equilibria for systems of differential equations.

### Examples

#### Example 1

Consider the function **f** : **R**^{2} → **R**^{2}, with (*x*, *y*) ↦ (*f*_{1}(*x*, *y*), *f*_{2}(*x*, *y*)), given by

Then we have

and

and the Jacobian matrix of **f** is

and the Jacobian determinant is

#### Example 2: polar-Cartesian transformation

The transformation from polar coordinates (*r*, *φ*) to Cartesian coordinates (*x*, *y*), is given by the function **F**: **R**^{+} × [0, 2π) → **R**^{2} with components:

The Jacobian determinant is equal to *r*. This can be used to transform integrals between the two coordinate systems:

#### Example 3: spherical-Cartesian transformation

The transformation from spherical coordinates (*ρ*, *φ*, *θ*) to Cartesian coordinates (*x*, *y*, *z*), is given by the function **F**: **R**^{+} × [0, *π*) × [0, 2*π*) → **R**^{3} with components:

The Jacobian matrix for this coordinate change is

The determinant is *ρ*^{2} sin *φ*. This can be used to transform integrals between the two coordinate systems:

#### Example 4

The Jacobian matrix of the function **F** : **R**^{3} → **R**^{4} with components

is

This example shows that the Jacobian matrix need not be a square matrix.

#### Example 5

The Jacobian determinant of the function **F** : **R**^{3} → **R**^{3} with components

is

From this we see that **F** reverses orientation near those points where *x*_{1} and *x*_{2} have the same sign; the function is locally invertible everywhere except near points where *x*_{1} = 0 or *x*_{2} = 0.

### Inverse

According to the inverse function theorem, the matrix inverse of the Jacobian matrix of an invertible function is the Jacobian matrix of the *inverse* function. That is, if the Jacobian of the function **f** : **R**^{n} → **R**^{n} is continuous and nonsingular at the point **p** in **R**^{n}, then **f** is invertible when restricted to some neighborhood of **p** and

### Critical points

If **f** : **R**^{n} → **R**^{m} is a differentiable function, a *critical point* of **f** is a point where the rank of the Jacobian matrix is not maximal. This means that the rank at the critical point is lower than the rank at some neighbour point.

### Other uses

The Jacobian serves as a linearized design matrix in statistical regression and curve fitting. It is also used in dynamical systems to determine the stability of equilibria.

### Notes

* Differentiability at **x** implies, but is not implied by, the existence of all first-order partial derivatives at **x**, and hence is a stronger condition.

### References

* "Jacobian - Definition of Jacobian in English by Oxford Dictionaries".
* "the definition of jacobian".
* "Jacobian pronunciation: How to pronounce Jacobian in English".
* "Jacobian".
* Smith? RJ (2015). "The Joys of the Jacobian".
* Joel Hass, Christopher Heil, and Maurice Weir.
* Arrowsmith, D. K.; Place, C. M. (1992). "The Linearization Theorem".
* Hirsch, Morris; Smale, Stephen (1974).
* Gandolfo, Giancarlo (1996). "Comparative Statics and the Correspondence Principle".
* Protter, Murray H.; Morrey, Charles B., Jr. (1985). "Transformations and Jacobians".

### Further reading

* Mariano Zeron and Ignacio Ruiz -

### External links

* "Jacobian",
* Mathworld A more technical explanation of Jacobians


Establishing learning goals is a crucial part of designing a lesson. Many teachers write learning goals as behavioral outcomes, such as "Students will be able to..." However, learning in an academic context is a mental activity, and learning goals should describe a desired internal change in the student, such as understanding something new, knowing something new, or feeling differently about something.

Articulating learning goals in this way helps teachers think more deeply about the content they are trying to teach. For example, consider a typical learning goal for 5th grade: "Students will be able to add fractions with unlike denominators." This goal can be rephrased as internal changes in the student, such as:

(A) The student knows the procedure for adding fractions with different denominators, including finding the least common multiple, converting each fraction, and adding the numerators.

(B) The student understands that fractions with different denominators must be converted to a common unit before they can be added, and can use their knowledge of equivalent fractions and least common multiples to solve the problem.

Goal A focuses on a new procedure, while Goal B connects the new problem to the student's existing knowledge of fractions and equivalent ratios. Lessons designed around these goals will be very different. A student who achieves Goal A may feel burdened with a new procedure to remember, while a student who achieves Goal B may feel empowered to solve new problems using their existing knowledge.

Assessing students' understanding is also different for these two goals. A worksheet or quiz that asks students to add fractions with unlike denominators may assess their ability to follow a procedure, but it will not assess their understanding of why the procedure works. To design effective lessons and assessments, teachers must think beyond observable behaviors and identify the internal mental changes they wish to make in their students.

When designing lessons, it is essential to consider how students will feel after achieving the learning goal. Will they feel confident and empowered, or overwhelmed and burdened? The answer to this question can inform the design of the lesson and the assessment strategy. By focusing on internal mental changes and deep understanding, teachers can create lessons that promote meaningful learning and help students develop a sense of power and agency over their learning.


# Evaluating Boundary Integrals

James Lai inquired about integrating a solution over a given boundary edge using DOLFIN through Python. After obtaining the solution with `problem = VariationalProb` and `U = problem.solve()`, he attempted to integrate `n dot u` over subdomain 1.

To achieve this, he set up subdomains and tried the following code:
```python
ds_in = Measure()
n_in = Constant((-1,0))
val = Integral()
```
However, `val` resulted in an `Integral` type instead of a numerical value. He sought a way to evaluate this integral and obtain a numerical result.

The question was solved by Johan Hake, providing a solution for evaluating boundary integrals in DOLFIN. The key concept involves using the `Measure` and `Integral` functions correctly to compute the desired integral over a specific boundary edge. By refining the code and understanding the underlying principles, one can successfully integrate the solution over a given boundary edge and obtain a numerical answer.


**College Physics Course Planner PHYS103**

**Week 1: Sept 6-9**

* Day 1-2: Course overview, expectations, goal setting, metacognition, math skills assessment, and problem-solving strategies
* Topics: variable lists, formulas, dimensional analysis, algebra, and scientific notation
* Readings: Text P. 5-11, 16-18, MCQ #3, 4, & 11, P. 20 #9 & 10
* Concepts & Skills:
	1. Decipher word problems to identify variables and quantities
	2. Rearrange formulas to solve for different variables using algebra
	3. Represent powers of ten using scientific notation
	4. Verify relationships based on unit consistency

**Weeks 2 & 3: Sept 12-23**

* Day 3-5: One-dimensional linear motion, speed vs. velocity, distance vs. displacement, average velocity, and motion graphs
* Topics: scalars, vectors, factor-label unit conversions, slopes, and acceleration
* Readings: Text P. 19, 24-30, Quick Quiz 2.1, P. 49 #1 & 2
* Concepts & Skills:
	1. Use motion equations to solve word problems
	2. Determine average velocity and differentiate it from initial, final, and instantaneous velocities
	3. Convert units using the F-L approach
	4. Represent motion on graphs and determine slope
	5. Calculate acceleration using formulas and graph slopes

**Weeks 4 & 5: Sept 26-Oct 7**

* Day 8-12: Vector components of velocity, projectile motion, and distance formula
* Topics: trig functions, max height and range, sketching projectile paths, and motion graphs
* Readings: Text P. 52-58, 62-69, Quick Quiz 3.5 & 3.6
* Concepts & Skills:
	1. Use trig functions to find x and y velocity components of projectiles
	2. Find max height and range of a projectile launched at an angle
	3. Sketch the path of projectiles and draw motion dot diagrams
	4. Use motion equations in x and y to solve 2D projectile problems

**Week 6: Oct 11-14**

* Day 13-14: Newton's 1st and 2nd Laws of Motion, mass, inertia, force vectors, and free body diagrams
* Topics: concurrent forces, forces in equilibrium, and practice force vector drawings
* Readings: Text P. 83-89, Quick Quiz 4.1, P. 111 #1, 3, 5, & 7
* Concepts & Skills:
	1. Acceleration of an object is directly proportional to the net force acting on it
	2. Find Fnet using vector diagrams and identify all forces acting on an object
	3. Find resultant of two or more vectors acting concurrently

**Labs**

* Due Friday, Oct 14th
* Requirements:
	+ Well-written abstract
	+ Purpose or objective
	+ Procedure (if required)
	+ Data and observations
	+ Calculations and graphs
	+ Questions
	+ Well-written complete conclusion
* Note: Labs must be word-processed


A scientist creates a chemical believed to change the temperature of water. The scientist applies this chemical to water and takes measurements, then measures temperature in samples without the chemical.

**Summary Data:**

* **First Set of Measurements (with chemical):** N = 8, SD = 2.167124, Mean = 72.875
* **Second Set of Measurements (without chemical):** N = 8, SD = 1.669046, Mean = 75.25

To analyze the data, select "Compare Means" from the "Analyze" dropdown, then choose "Summary Independent-Samples T Test". Enter the provided information and click "OK", using the default 95% confidence level.

The output shows a p-value of 0.028 and a t-value of -2.456. Since the p-value (0.028) is less than 0.05, the scientist can conclude that, at a 95% confidence interval, the chemical is altering the temperature of the water.


# Geometric Concepts

A circular arc is a segment of a circle. A circular sector, or circle sector, is the portion of a disk enclosed by two radii and an arc, where the smaller area is the sector. A circular segment is a region of a circle "cut off" from the rest of the circle by a secant or a chord.

An oloid is a three-dimensional curved geometric object discovered by Paul Schatz in 1929. It is the convex hull of a skeletal frame made by placing two linked congruent circles in perpendicular planes.

A spherical polygon on the surface of a sphere is defined by a number of great circle arcs. A deltoid, also known as a tricuspoid or Steiner curve, is a hypocycloid of three cusps.

Tortuosity is a property of a curve being tortuous, with several attempts to quantify this property. The general equation for a circle gives the length of the radius of the circle.

In Euclidean geometry, a tangential quadrilateral is a convex quadrilateral whose sides are all tangent to a single circle. Spirograph is a geometric drawing toy that produces mathematical roulette curves, including hypotrochoids and epitrochoids.

An epitrochoid is a roulette traced by a point attached to an external circle rolling around the outside of a fixed circle. A hypotrochoid is a roulette traced by a point attached to a circle rolling around the inside of a fixed circle.

A hyperbola is a type of smooth curve defined by its geometric properties or by equations. The law of cotangents is a relationship among the lengths of the sides of a triangle and the cotangents of the halves of the three angles.

An annulus is a ring-shaped object, especially a region bounded by two concentric circles. A circumscribed circle or circumcircle of a triangle is a circle that passes through all the vertices of the triangle. The center of this circle is called the circumcenter.


The normal form of a lambda term is crucial in understanding its behavior. However, the evaluation order can significantly impact the outcome. Consider the term `(\x.y)(ww)`, which can be reduced to normal form using a specific strategy. But if we use a different strategy, the reduction may not terminate.

To illustrate this, let's examine the use of the fixed point combinator `Y = \f.(\h.f(hh))(\h.f(hh))` to define a recursive function. The reduction of `Y (\f n. blah)` would result in an infinite sequence of terms, never reaching the base case.

Restricting evaluation order is essential to gain control over the reduction process. This can be achieved by adjusting the notion of beta reduction, which is part of the untyped calculus. To discuss this precisely, we need to refine our vocabulary.

There are three kinds of lambda terms: variables (e.g., `x`, `y`, `z`), abstractions (e.g., `\x.x`, `\x.y`, `\x.\y.y`), and applications (e.g., `(x x)`, `((\x.x) y)`, `((\x.x)(\y.y))`). A redex is a lambda term of the form `((\aM) N)`, where beta reduction is defined. In an application `(M N)`, `M` is the functor, and `N` is the argument.

We can distinguish four types of normal forms. To understand the reduction process, consider Plotkin's CBN CPS, which defines the reduction of terms as follows:
- `[x] ~~> x`
- `[\xM] ~~> \k.k(\x[M])`
- `[MN] ~~> \k.[M](\m.m[N]k)`

Using this system, let's reduce the term `[(\xy)(ww)]`:
- `[(\xy)(ww)] ~~> \k.[\xy](\m.m[ww]k)`
- `~~> \k.[\xy](\m.m(\k.[w](\m.m[w]k))k)`
- `~~> \k.[\xy](\m.m(\k.(\k.k(\x[xx]))(\m.m[w]k))k)`
- `~~> \k.[\xy](\m.m(\k.(\x[xx])[w]k)k)`
- `~~> \k.[\xy](\m.m(\k.(\x(\k.[x](\m.m[x]k)))[w]k)k)`
- `~~> \k.[\xy](\m.m(\k.(\x(\k.x(\m.mxk)))[w]k)k)`
- `~~> \k.[\xy](\m.m(\k.[w](\m.m[w]k))k)`

This reduction process demonstrates the importance of controlling the evaluation order to achieve the desired outcome.


Chapter 3: Boolean Algebra and Digital Logic

**Objectives:**
- Understand the relationship between Boolean logic and digital computer circuits.
- Learn how to design simple logic circuits.
- Understand how digital circuits work together to form complex computer systems.

**3.1 Introduction:**
George Boole introduced the concept of representing logical thought through mathematical equations in the 19th century. This idea is the foundation of modern computers, as realized by John Atanasoff and Claude Shannon.

**3.2 Boolean Algebra:**
- A Boolean operator can be described using a truth table.
- The AND operator is the Boolean product, and the OR operator is the Boolean sum.
- The NOT operator is often designated by an overbar and has the highest priority in Boolean operations.
- A Boolean function has at least one Boolean variable, one Boolean operator, and one input from the set {0,1}, producing an output in the same set.

**Boolean Functions and Operators:**
- The truth table for a Boolean function can include extra columns to evaluate subparts of the function.
- Boolean operations follow rules of precedence: NOT, then AND, then OR.
- Most Boolean identities have both AND (product) and OR (sum) forms.
- DeMorgan’s law provides a way to find the complement of a Boolean function by replacing each variable with its complement and changing ANDs to ORs and vice versa.

**Canonical Forms:**
- There are two canonical forms for Boolean expressions: sum-of-products and product-of-sums.
- In sum-of-products form, ANDed variables are ORed together.
- In product-of-sums form, ORed variables are ANDed together.
- Converting a function to sum-of-products form can be done using its truth table by listing variable values that result in a true function value and then ORing these groups together.

**3.3 Logic Gates:**
- Boolean functions are implemented in digital circuits called gates.
- A gate is an electronic device producing a result based on two or more input values.
- The simplest gates are AND, OR, and NOT gates, corresponding directly to their respective Boolean operations.
- Other useful gates include the exclusive OR (XOR) gate, where the output is true only when input values differ, and NAND and NOR gates, known as universal gates because any Boolean function can be constructed using only one type of these gates.
- Gates can have multiple inputs and outputs, and combinations of gates implement Boolean functions, with simpler circuits often being the goal of simplifying Boolean expressions.


## Quadratic Equations

A quadratic equation in the variable x is of the form ax^2 + bx + c = 0, where a, b, c are real numbers and a ≠ 0. This is a polynomial equation of degree 2 in one variable, with 'a' as the leading coefficient and 'c' as the absolute term. The values of x that satisfy the equation are called the roots (α, β).

A quadratic polynomial becomes a quadratic equation when equated to zero. The roots of the equation are the values of x that satisfy it.

### Quadratic Equation Formula

The solution or roots of a quadratic equation are given by the quadratic formula:
(α, β) = [-b ± √(b^2 – 4ac)]/2a

### Roots of Quadratic Equation

The roots of a quadratic equation are the values of variables that satisfy the given equation. In other words, x = α is a root of f(x) if f(α) = 0. The real roots of an equation f(x) = 0 are the x-coordinates of the points where the curve y = f(x) intersects the x-axis.

Key properties of roots:
- If c = 0, one root is zero and the other is -b/a.
- If b = c = 0, both roots are zero.
- If a = c, the roots are reciprocal to each other.


Largest Palindromic Number by Rearranging Digits
Given a very large number N, the goal is to find the largest palindromic number by rearranging its digits. If it's not possible to form a palindrome, print "Palindrome not found".

Examples:
Input: N = 113223
Output: 321123
Input: N = 333
Output: 333
Input: N = 123
Output: Palindrome not found

Naive Approach:
The naive approach involves finding all palindromic permutations of the digits in N and printing the maximum value. However, this approach is too slow for very large numbers.

Efficient Approach:
To efficiently solve this problem, follow these steps:
1. Store the occurrences of every digit in the number.
2. Count the number of digits that occur an odd number of times. If the count is 0 or 1, a palindrome can be formed by rearranging the digits.
3. If a palindrome can be formed, apply the greedy approach to obtain the largest palindromic number.

The Greedy Approach:
1. Start checking the occurrences of every digit from 9 to 0.
2. If the occurrences of a digit are even, put half of the occurrences at the front and subtract that half from the actual occurrences.
3. After completing the traversal, add the digit that occurs an odd number of times and make its occurrence zero.
4. Traverse the digits again from 0 to 9 and add the remaining occurrences to the number.

Important Points:
- The number can be very large, so store it in a string.
- Use a count array of size 10 to store the occurrences of the digits.
- Include a function to check if the given number can form a palindrome.

Time Complexity: O(N)

Example:
N = 1122334
Step 1: Count the digits with odd occurrences.
Frequency array:
occurrences: 0 2 2 2 1 0 0 0 0 0
index: 0 1 2 3 4 5 6 7 8 9
There is 1 digit with an odd occurrence, so a palindrome can be formed.

Step 2: Traverse from 9 to 0.
Initialize Answer = 0.
Frequency array:
occurrences: 0 2 2 2 1 0 0 0 0 0
index: 0 1 2 3 4 5 6 7 8 9
Answer = 0
Frequency array:
occurrences: 0 2 2 2 1 0 0 0 0 0
index: 0 1 2 3 4 5 6 7 8 9
Take half and subtract half: Answer = 3
Frequency array:
occurrences: 0 2 2 1 1 0 0 0 0 0
index: 0 1 2 3 4 5 6 7 8 9
Take half and subtract half: Answer = 32
Frequency array:
occurrences: 0 2 1 1 1 0 0 0 0 0
index: 0 1 2 3 4 5 6 7 8 9
Take half and subtract half: Answer = 321
Frequency array:
occurrences: 0 1 1 1 1 0 0 0 0 0
index: 0 1 2 3 4 5 6 7 8 9

Step 3: Add the digit with an odd occurrence (4).
Answer = 3214

Step 4: Traverse from 0 to 9.
Frequency array:
occurrences: 0 1 1 1 0 0 0 0 0 0
index: 0 1 2 3 4 5 6 7 8 9
Add remaining occurrences: Answer = 32141
Frequency array:
occurrences: 0 0 1 1 0 0 0 0 0 0
index: 0 1 2 3 4 5 6 7 8 9
Add remaining occurrences: Answer = 321412
Frequency array:
occurrences: 0 0 0 1 0 0 0 0 0 0
index: 0 1 2 3 4 5 6 7 8 9
Add remaining occurrences: Answer = 3214123

The largest palindromic number by rearranging the digits is 3214123.


A matching network can be built using a quarterwave transformer to match a load of impedance \(Z_L\) to a source having impedance \(Z_S\). This is accomplished by using a transmission line with an electrical length of 90 degrees and characteristic impedance \(Z_0 = \sqrt{Z_S\,Z_L}\). Alternatively, lumped elements can be used, where \(Z_0 = \sqrt{\frac{L}{C}}\).

To achieve a desired characteristic impedance, \(L\) and \(C\) can be chosen as \(C = \frac{1}{2\,\pi\,f_m\,Z_0}\) and \(L = \frac{Z_0}{2\,\pi\,f_m}\). For example, to match a 75-ohm load to a 50-ohm source at 50 MHz, \(Z_0 = 61.2\), \(C = 52\,\mathrm{pF}\), and \(L = 195\,\mathrm{nH}\). Simulation results show a good match around 50 MHz, with a 3 MHz bandwidth for a return loss better than 30 dB and a 10 MHz bandwidth for a 20 dB return loss.

To increase the bandwidth, multiple matching networks can be cascaded. For instance, matching 50 to 75 ohms can be done in two stages: 50 to 62.5 ohms and then 62.5 to 75 ohms. This approach is like an arithmetic series, where the difference between impedances is constant. The bandwidth for a return loss better than 30 dB is around 10 MHz, and for a 20 dB return loss, it is around 16 MHz.

In general network analysis, consider a network with source and load impedances labeled \(R_1\) and \(R_2\). Using the mesh rule and the impedance matrix \(\mathbf{Z}\), the currents in each loop can be found. The impedance matrix is given by:

$$
\mathbf{Z} =
\begin{pmatrix}
R_1 + \frac{1}{s\,C_1} & -\frac{1}{s\,C_1} & 0 & 0 \\
-\frac{1}{s\,C_1} & \frac{1}{s\,C_1} + s\,L_1 + \frac{1}{s\,C_2} & -\frac{1}{s\,C_2} & 0 \\
0 & -\frac{1}{s\,C_2} & \frac{1}{s\,C_2} + s\,L_2 + \frac{1}{s\,C_3} & -\frac{1}{s\,C_3} \\
0 & 0 & -\frac{1}{s\,C_3} & \frac{1}{s\,C_3} + R_2
\end{pmatrix}
$$

The structure of the impedance matrix can be exploited to construct matrices for networks of arbitrary size. The main diagonal contains series impedances plus shunt impedances, while the off-diagonals contain negative shunt impedances. The first and last elements also contain the terminal impedances. With a Matlab script, matching networks of this type can be calculated, and \(S_{11}\) and \(S_{21}\) can be plotted.


NumPy's broadcasting functionality is a set of rules for applying binary operations on arrays of different sizes. Broadcasting allows binary operations to be performed on arrays of different sizes, for example, adding a scalar to an array. This can be thought of as an operation that stretches or duplicates the value into the array, and adds the results. The advantage of NumPy's broadcasting is that this duplication of values does not actually take place.

Broadcasting can be extended to arrays of higher dimension. When adding a one-dimensional array to a two-dimensional array, the one-dimensional array is stretched or broadcast across the second dimension to match the shape of the two-dimensional array. In more complicated cases, broadcasting can involve both arrays, where both arrays are stretched to match a common shape.

The geometry of broadcasting is visualized in a figure, where the light boxes represent the broadcasted values. This extra memory is not actually allocated in the course of the operation, but it can be useful conceptually to imagine that it is.

Broadcasting in NumPy follows a strict set of rules to determine the interaction between the two arrays:

1. If the two arrays differ in their number of dimensions, the shape of the one with fewer dimensions is padded with ones on its leading (left) side.
2. If the shape of the two arrays does not match in any dimension, the array with shape equal to 1 in that dimension is stretched to match the other shape.
3. If in any dimension the sizes disagree and neither is equal to 1, an error is raised.

To illustrate these rules, consider adding a two-dimensional array to a one-dimensional array. The shape of the arrays are `M.shape = (2, 3)` and `a.shape = (3,)`. By rule 1, the array `a` has fewer dimensions, so we pad it on the left with ones: `M.shape -> (2, 3)` and `a.shape -> (1, 3)`. By rule 2, we now see that the first dimension disagrees, so we stretch this dimension to match: `M.shape -> (2, 3)` and `a.shape -> (2, 3)`. The shapes match, and we see that the final shape will be `(2, 3)`.

Another example involves adding a one-dimensional array to a two-dimensional array, where the matrix is transposed. The shape of the arrays are `M.shape = (3, 2)` and `a.shape = (3,)`. Again, rule 1 tells us that we must pad the shape of `a` with ones: `M.shape -> (3, 2)` and `a.shape -> (1, 3)`. By rule 2, the first dimension of `a` is stretched to match that of `M`: `M.shape -> (3, 2)` and `a.shape -> (3, 3)`. Now we hit rule 3 – the final shapes do not match, so these two arrays are incompatible.

Broadcasting operations form the core of many examples. One commonly seen example is when centering an array of data. Imagine you have an array of 10 observations, each of which consists of 3 values, stored in a $10 \times 3$ array. You can compute the mean of each feature using the `mean` aggregate across the first dimension, and then center the array by subtracting the mean (this is a broadcasting operation).

Broadcasting is also useful in displaying images based on two-dimensional functions. If you want to define a function $z = f(x, y)$, broadcasting can be used to compute the function across the grid. You can then use Matplotlib to plot this two-dimensional array.

In summary, NumPy's broadcasting functionality is a powerful tool for performing binary operations on arrays of different sizes. By following a set of strict rules, broadcasting allows arrays to be stretched or duplicated to match a common shape, enabling efficient and flexible computation.


To subtract integers, we can use colored counters to model the sum. When subtracting a positive integer, proceed to the left on the number line, as the difference will be less than the original number. On the other hand, to subtract a negative number, do the opposite.

Key words can help us understand the concept of subtracting integers:
- Negative key words: loss, below sea level, withdraw, descend, spending
- Positive key words: profit, above sea level, deposit, ascend, saving

We can use strategies such as number lines and counters to add or subtract integers. For example, to add a positive integer, proceed to the right on the number line.

When using counters, the shaded spaces will show the signs. To subtract a negative number, change the integer after the subtraction sign to an addition sign.

Some important concepts to remember:
- Subtracting a positive number results in a difference that is less than the original number.
- Subtracting a negative number is equivalent to adding a positive number.

Practice questions can help reinforce these concepts:
1. What is the result of subtracting a positive integer?
a) The difference is greater than the original number
b) The difference is less than the original number
c) The difference is equal to the original number
d) The difference is zero

2. What is the equivalent operation when subtracting a negative number?
a) Subtracting a positive number
b) Adding a positive number
c) Multiplying a positive number
d) Dividing a positive number

Answer key:
1. b) The difference is less than the original number
2. b) Adding a positive number

By using these strategies and practicing with sample questions, we can become more comfortable with subtracting integers.


# Comparison of Biomass and Solar Energy Yields

## Basic Biomass and Solar Energy Calculations

To calculate energy yield from biomass, consider the following figures:
1 cubic meter of fresh cut timber yields 1476 kilowatt hours (kWh) of fire power, with a cellulose content of approximately 450 kg.
Converting this material to charcoal results in 135 kg, yielding 1150 kWh of fire power.
Converting it to Pyrolysis Oil yields 315 kg (25% moisture content), resulting in 1480 kWh of fire power.

For woodland growing at yield class 15, the average annual yield is 15 cubic meters of timber per hectare.
This translates to approximately 22,000 kWh of fire power per annum, or an average output of 2.5 kW per hectare.
However, when burned to make electricity, the output is reduced to 250 watts per hectare.

In comparison, peak sun power is approximately 1 kW per square meter, or 10,000 kW per hectare.
Accounting for night and low sun, the average output is around 2,500 kW per hectare.

The cost of electricity from biomass is estimated to be around $200 per hectare per annum, based on a cost of $0.10 per kWh.
In contrast, a 30 square meter solar collector can generate approximately 5 horsepower (hp) peak, or 0.125 kW per square meter peak.
This translates to an average annual output of 300 kW per hectare, valued at $250,000 per hectare per annum, based on a cost of $0.10 per kWh.

The cost of a solar concentrator system is estimated to be around $1.2 million per hectare, resulting in a 5-year payback period.
However, with an expected overall efficiency of 15%, the payback time can be reduced to 4 years, making solar energy a viable option on economic terms and an essential one on ecological terms.

Key energy yield figures:
- 1 cubic meter of fresh cut timber: 1476 kWh
- 1 cubic meter of charcoal: 1150 kWh
- 1 cubic meter of Pyrolysis Oil: 1480 kWh
- 1 hectare of woodland (yield class 15): 22,000 kWh per annum
- Peak sun power: 1 kW per square meter, or 10,000 kW per hectare
- Average sun power (accounting for night and low sun): 2,500 kW per hectare

Key economic figures:
- Cost of electricity from biomass: $200 per hectare per annum
- Cost of electricity from solar energy: $250,000 per hectare per annum
- Cost of solar concentrator system: $1.2 million per hectare
- Payback period: 5 years (initial estimate), 4 years (with 15% overall efficiency)


## Let's Make a Deal with Bayes

You're on a game show with three doors, each hiding a prize. One will make you wealthy, while the others are worthless. You pick the middle door, but then the host, who knows what's behind each door, opens the worthless door on the left and offers you a deal to either keep your first choice or switch.

The theorem behind this scenario is based on a simple yet powerful equation: P(H|E) = P(E|H) P(H) / P(E), which translates to the probability of a hypothesis being true given evidence. For those mathematically inclined, there are many resources to learn how the theorem works and applies to the game show scenario.

The path to the answer starts with Reverend Thomas Bayes, who created the equation representing the probability of an event as new information is learned. In the game show case, the probability of selecting the door with wealth is 1/3, while the probability of it being among the other unselected doors is 2/3. When the host opens one of the worthless doors, new information is added, and Bayes theorem tells us that the probability of the remaining unselected door leading to wealth is 2/3, while the original middle door only has a 1/3 probability of success.

If you want the best chance for wealth, you should switch. Many might think the odds are 1/2 for either door, but a key insight with Bayes is the need to continually update probabilities as new information is learned. This insight, combined with advancements in computing power and software tools, has made Bayesian techniques a top choice for reasoning about uncertain situations in artificial intelligence solutions.

SmartCloud is embracing Bayes theorem in the form of Bayesian belief networks as an integral part of their Situational Management solutions. For example, they are developing use cases for assessing risky situations within utility grids and for fraud detection. Bayesian networks fit well with how they represent knowledge needed to reason about a situation, particularly when evidence is missing or uncertain. The networks learn from new data by adapting probabilities assigned between related events.

With Bayesian networks, the probability of a situational state being true is modeled as the values of its contributing factors become known. The modeling is based on a combination of historical analysis and human expertise. An important advantage of this approach is the ability to explain the causes of a situation to humans, helping them decide on a response.

In a utility grid scenario, a Bayesian network analysis might start with a base probability of imbalance problems, say 3%. As more information is learned, the probability is updated. For instance, if it's a weekday with record high temperatures, the probability might increase to 20%, then 30% considering it's a high consumption day, and 40% at the peak of the day. If a major regional generator is out of service, the odds might jump to 60%, and if the weather is calm, meaning wind-generated power is unavailable, the odds might jump to 75%.

Knowing the odds of an electrical imbalance situation and its major causes guides decision-making and actions for avoiding its impacts. Utility operators might request customers to curtail electric use or increase bids to purchase electricity from the energy spot market.

The likelihood of a utility's electric grid having a problematic energy imbalance situation depends on many factors, including consumption and supply. If the values for these factors are unknown, there is great uncertainty. However, by applying Bayesian analysis, the uncertainty can be reduced, and the probability of an imbalance situation can be predicted.

At SmartCloud, they are evolving state-of-the-art AI techniques for applying Bayesian analysis to real-time situations in energy, oil and gas, process manufacturing, and fraud. They combine the advantages of Bayes theorem, semantic modeling, rules, and other AI reasoning technologies to augment human ability to reach useful conclusions when facing high levels of uncertainty.


# Location Estimation - Part 2

In the previous article, we attempted to solve the location problem by using the power of each signal to roughly estimate our location. We narrowed it down to a small convex region in the plane, approximately 2500 square meters near (-181, 75).

To further determine the exact position, we take the centroid of this region as a guess at our location. This allows us to calculate the distance from each tower and use the speed of sound to determine the delay of each individual frequency in the received signal. For example, if we are 340.29 meters away from a tower emitting a 3500Hz signal, each sample we received was emitted about 1 second earlier.

We can use this distance and speed of sound to calculate the phase shift of the signal. The phase shift of the 350.0Hz tower, approximately 196.682 meters from our location, is -0.578 seconds.

To refine our location, we need to account for the initial rotation of the towers and our exact position. We can pick a random initial rotation and plot the scene, but this may not yield accurate results. Ideally, we want all the rays from each tower to intersect at one location, which would be the maximum likelihood estimate of the location given the received data.

However, due to real-world noise and imprecision, the rays may not intersect perfectly. To quantify the closeness of the rays, we can assume that all the lines should intersect at a specific point and measure the norm of the residuals of the least square solution of this linear system.

We can use this error value as an objective function and plug our initial guess at the position, as well as the range of possible tower orientations, into a BFGS non-linear solver. This should refine our unknown position to something that agrees with all the towers.

Using this method, we can estimate our most likely position to be (-179.2, -95.6), which is within 30 centimeters of the actual location (-179.4, -95.3).

The code used to achieve this includes:
```python
import math
speed_of_sound = 340.29
centroid_x = -181.666
centroid_y = -75.373
for tower_x, tower_y in towers:
    dx = centroid_x-tower_x
    dy = centroid_y-tower_y
    distance = math.hypot(dx, dy)
    phase_shift = distance / speed_of_sound
    peaks = find_peaks(times- phase_shift, powers)
```

```python
import numpy as np
points = [
    (-1, -1, 1, 1),
    (1, -1, 0, 1),
    (1, 0, -1, 1)
]
A = []
b = []
for x1, y1, x2, y2 in points:
    m = (y2 - y1) / (x2 - x1)
    c = y1 - (m*x1)
    A.append([-m, 1])
    b.append(c)
x_sol = np.linalg.lstsq(A, b)[0]
A = np.array(A)
b = np.array(b)
x_est = b - A.dot(x_sol)
R = np.linalg.norm(x_est)
```

```python
from scipy.optimize import minimize
initial_estimate = [0.0, -181, 75]
result = minimize(objective_func, initial_estimate, method='BFGS')
```
This method has implications for RADAR systems and could potentially be used to locate missing objects, such as the Malaysia flight MH370. However, it also raises concerns about revealing the capabilities and locations of RADAR systems.


Financial market participants have unique views of the market, perceiving it through different paradigms. Actors with small capital tend to focus on profitability, while those with larger capital consider risk. Risk and return are directly related, balancing the space of possible actions for each investor and creating a true reality of the financial market. These metrics are closely related to volatility, which determines the action space for each participant. Volatility is a crucial metric that should always be considered, especially when developing trading algorithms.

**Accuracy of Volatility Calculation**

Implied volatility is a fragile concept, as its calculations are based on option prices with limitations and assumptions. Accurate volatility calculation is essential, as inaccuracies can lead to the "butterfly effect," making the dynamics of the index chaotic and unreliable. Inaccurate volatility is a poor predictor, and approximations can result in a long time-lag. To accurately describe volatility, it's necessary to consider a maximum amount of information about options, including micro changes. The more options considered, the more accurate the volatility description.

**Methodology**

1. Determine the target time for calculating the index using the formula: Current time + Number of days to calculate the index [1, 2, 7, 14, 21, 28].
2. Select options that expire after the target time and those with the nearest expiration date.
3. Calculate the time difference between the target time and expiration date in minutes.
4. Set the Risk Free Rate (Rf) to 0, as the traditional calculation using the US Treasuries Index is not suitable for the cryptocurrency market.
5. Find the Strike Prices for both series to calculate the forward price, using the strike with the minimum Q-value (bid price + ask price) / 2.
6. Calculate the forward index price for each option series.
7. Determine K0, the strike price equal to or immediately below the forward index level, for near- and next-term options.
8. Select out-of-the-money put options with Strike Price <= K0 and call options with Strike Price >= K0.
9. Calculate ΔKi, half the difference between strike prices on either side of Ki.
10. Calculate the Cumulative Value for both near and long-term options.
11. Calculate squared volatility for both near and next-term options.

**Usage**

The volatility index can be used as a derivative, allowing traders to skip complex option strategies. It simplifies the hedging process and can be used for various algorithmic trading tasks, such as placing stop-loss orders. The index can also be used as a predictor, entry signal, or position filter, helping to diversify a portfolio. Additionally, it can help understand the risk associated with crypto investments or algo portfolios, allowing for adaptive changes to money and risk management.

The calculation methodology involves collecting information from several major derivatives exchanges and averaging the results. This can be done using APIs from options trading exchanges, which are relatively similar. The resulting index can be used to trade volatility as the price of the underlying asset, providing a convenient and simplified way to hedge and manage risk.


**ACT Math Tip: Vectors**

To handle difficult vector problems, review your right triangle skills.

**Step 1: Find the Magnitude**
Use the Pythagorean theorem to determine the vector's magnitude. For example, a vector's magnitude is approximately 3.5 meters per second.

**Step 2: Find the Direction**
Use a trigonometric ratio to find the vector's direction. Set up a tangent ratio to find the missing angle. The vector's direction is approximately 31 degrees south of east.