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#### Regular Expressions and Languages
A regular expression is a pattern describing a set of strings, while a regular language is the set of all strings that can be matched by a regular expression. Regular expressions are used in programming languages and text editors to search for and manipulate strings.
#### Deterministic and Non-Deterministic Turing Machines
A deterministic Turing machine performs one computation on a given input, whereas a non-deterministic Turing machine can perform multiple computations simultaneously. Non-deterministic Turing machines are more powerful, solving certain problems more quickly.
#### Context-Free and Regular Grammars
A context-free grammar generates context-free languages, while a regular grammar generates regular languages. Context-free grammars are more powerful, generating more complex string patterns.
#### Languages and Grammars
A language is a set of strings generated by a grammar, while a grammar is a set of production rules describing string generation from a starting symbol. Languages are the objects of study in computation theory, and grammars are the tools used to generate and analyze them.
#### Context-Sensitive and Recursively Enumerable Languages
A context-sensitive language is generated by a context-sensitive grammar, while a recursively enumerable language is recognized by a Turing machine. Recursively enumerable languages are more powerful, recognizing more complex string patterns.
#### The Chomsky Hierarchy
The Chomsky hierarchy classifies formal languages based on the grammar needed to generate them. It consists of four levels: regular languages, context-free languages, context-sensitive languages, and recursively enumerable languages. Each level is a proper subset of the previous one, except for the first two levels, which are equivalent.
#### One-Way and Two-Way Finite Automata
A one-way finite automaton reads input from left to right, moving its tape head only to the right. A two-way finite automaton can move its tape head left or right. Two-way finite automata are more powerful, recognizing languages that one-way finite automata cannot.
#### Decision and Function Problems
A decision problem has a "yes" or "no" answer, while a function problem requires computing a specific output based on input. Decision problems are the focus of complexity theory, and function problems are the focus of computability theory.
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codecrucks.com
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en
| 0.757721
| 2023-03-24T13:24:06
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https://codecrucks.com/theory-of-computation-question-set-29/
| 0.962373
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# Operators in Python
## Introduction to Operators
Operators in Python are used to perform operations on values, variables, and data structures, called operands, and control the flow of a program. They are essential for manipulating data types and creating flexible expressions.
## Arithmetic Operators
Arithmetic operators are used for basic mathematical operations such as addition, subtraction, multiplication, and division. The following table lists the arithmetic operators available in Python:
| Symbol | Operation |
| --- | --- |
| + | Addition |
| - | Subtraction |
| * | Multiplication |
| / | Division |
| % | Modulus |
| ** | Exponentiation |
| // | Floor division |
Examples of arithmetic operations:
```python
first_num = 5
second_num = 10
result = first_num + second_num # Output: 15
result = first_num - second_num # Output: -5
result = first_num * second_num # Output: 50
result = first_num / second_num # Output: 0.5
```
## Assignment Operators
Assignment operators are used to assign values to variables. Python also supports shorthand assignment operators, or augmented assignments, which allow performing an operation and assigning the result to a variable in a single line of code. The following table lists the assignment operators available in Python:
| Symbol | Operation |
| --- | --- |
| = | Assignment |
| += | Augmented Addition Assignment |
| -= | Augmented Subtraction Assignment |
| *= | Augmented Multiplication Assignment |
| /= | Augmented Division Assignment |
| %= | Augmented Remainder Assignment with the Modulus Operator |
| **= | Augmented Exponent Assignment |
| //= | Augmented Floor Division Assignment |
Examples of assignment operations:
```python
num = 10
num += 5 # Equivalent to num = num + 5
num -= 3 # Equivalent to num = num - 3
num *= 2 # Equivalent to num = num * 2
num /= 2 # Equivalent to num = num / 2
```
## Comparison Operators
Comparison operators are used to compare values and variables in Python. They return a Boolean value of either True or False based on the comparison made. The following table lists the comparison operators available in Python:
| Symbol | Operation |
| --- | --- |
| == | Equal to |
| != | Not equal to |
| > | Greater than |
| < | Less than |
| >= | Greater than or equal to |
| <= | Less than or equal to |
Examples of comparison operations:
```python
first_num = 15
second_num = 20
print(first_num == second_num) # Output: False
print(first_num != second_num) # Output: True
print(first_num > second_num) # Output: False
print(first_num < second_num) # Output: True
```
## Bitwise Operators
Bitwise operators perform operations on binary representations of numbers in Python. They are often used for low-level operations, such as setting or checking individual bits within a number. The following table lists the bitwise operators available in Python:
| Operation | Explanation |
| --- | --- |
| Binary AND | Sets each bit to 1 if both corresponding bits in the operands are 1 |
| Binary OR | Sets each bit to 1 if at least one of the corresponding bits in the operands is 1 |
| Binary XOR | Sets each bit to 1 if only one of the corresponding bits in the operands is 1 |
| Binary Ones Complement | Inverts all the bits of the operand |
| Binary Left Shift | Shifts the bits of the operand to the left by the specified number of positions, filling in with zeros from the right |
| Binary Right Shift | Shifts the bits of the operand to the right by the specified number of positions, filling in with copies of the leftmost bit from the left |
Examples of bitwise operations:
```python
first_num = 75
second_num = 35
result = first_num & second_num # Binary AND
result = first_num | second_num # Binary OR
result = first_num ^ second_num # Binary XOR
result = ~first_num # Binary Ones Complement
result = first_num << 2 # Binary Left Shift
result = first_num >> 2 # Binary Right Shift
```
## Logical Operators
Logical operators are used to perform operations on Boolean values in Python. They grant us decision-making based on multiple conditions and control the flow of a program.
Examples of logical operations:
```python
first_condition = True
second_condition = False
result = first_condition and second_condition # Logical AND
result = first_condition or second_condition # Logical OR
result = not first_condition # Logical NOT
```
## Specific Python Operators
In addition to the above operators, Python has a few specific operators.
### Identity Operators
Identity operators are used to compare the memory locations of two objects in Python. They enable us to find out if two objects are the same object in memory, or if they are different objects with the same values.
Examples of identity operations:
```python
first_obj = [1, 2, 3]
second_obj = [1, 2, 3]
print(first_obj is second_obj) # Output: False
print(first_obj is not second_obj) # Output: True
```
### Membership Operators
Membership operators are used to test if a value or variable is a member of a sequence, such as a list, tuple, or string. With them, we can determine whether an object is present in a sequence.
Examples of membership operations:
```python
numbers = [1, 2, 3, 4, 5]
print(3 in numbers) # Output: True
print(6 not in numbers) # Output: True
```
## Operator Precedence
Operators in Python have a specific order of precedence that determines the order in which operations are performed. The order of precedence is as follows:
1. Exponentiation (**)
2. Complement, unary plus, and minus (e.g. ~x, +x, -x)
3. Multiplication, division, floor division, and modulus (e.g. *, /, //, %)
4. Addition and subtraction (e.g. +, -)
5. Bitwise shift operations (e.g. <<, >>)
6. Bitwise AND (e.g. &)
7. Bitwise XOR (e.g. ^)
8. Bitwise OR (e.g. |)
9. Comparison operators (e.g. ==, !=, >, <, >=, <=)
10. Membership operators (e.g. in, not in)
11. Identity operators (e.g. is, is not)
12. Logical NOT (e.g. not)
13. Logical AND (e.g. and)
14. Logical OR (e.g. or)
Lacking knowledge of the precedence order in Python can affect the outcome of a calculation or comparison. If we're unsure of the order of precedence, it's always a good idea to use parentheses to explicitly define the order of operations.
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CC-MAIN-2023-14/segments/1679296949642.35/warc/CC-MAIN-20230331113819-20230331143819-00396.warc.gz
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webreference.com
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en
| 0.848659
| 2023-03-31T13:46:23
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https://webreference.com/python/basics/operators/
| 0.845982
|
Most circulated Roosevelt dimes are only worth their bullion value. For this list, we are only including five-cent nickels: Shield Nickels, Liberty "V" Nickels, Buffalo Nickels (or Indian Head Nickels), and Jefferson Nickels. One of the more unusual Silver coins was the Jefferson Nickel of 1942 to 1945. Jefferson Nickels were first minted in 1938 and made of 75% Copper and 25% Nickel.
A dime is worth 10 cents and is equal to 2 nickels or 10 pennies. The coin prices and values for 5C Nickels. The dime series, variety, date and mintmarks are found on value charts. The total value of the coins is $3.10. Let x = number of nickels and y = number of dimes.
A dime is worth 10 cents and a nickel is worth 5 cents. The total value of all the coins is $1.50. There are 22 coins in the jar, and the total value of the coins is $1.50. She has three times as many nickels as dimes.
The value of a dime is and the value of a nickel is The total value of all the coins is. We are asked to find the number of dimes and nickels Adalberto has. The person has 20 coins. The total of all of them is 140 cents.
A nickel is worth 5 cents and is equal to 5 pennies. The total value of the coins is $1.40. There are 20 coins in the jar, and the total value of the coins is 1.40. Joe has a collection of nickels and dimes that is worth $5.30.
The United States Mint currently makes Roosevelt dimes for circulation. In 1964, the mint made the last dimes containing 90% silver. All US dimes dated prior to 1965 are 90% silver and follow silver price. Rarity and demand contribute to a wide range in value found among all the different dates.
USA Coin Book has compiled a list of the most valuable US nickels ever known. Our most valuable nickels list includes coins starting in 1866 up to the present (2021) - including rare nickel errors and rare varieties that could still actually be found in pocket change. The list and the prices are current as of 2021.
A step by step process identifies the collector quality dimes worth a higher range. Step 1: Recognize the Different Series of Nickels - US nickels include a variety of series and sub-varieties within series, all important to recognize. Step 2: Date and Mintmark are Identified- Date and mintmark combinations are each valued separately.
The value of a dime coin is worth the same as ten one cent coins. Two nickels have the same value as 1 dime. Less often you can still find 90% silver quarters, as well. Coin dealers usually sell bank rolls or large bags of this “junk silver” grouped together by face value. Common increments are $100 or $1,000 face value.
Dime values today - ***z-mdyear.shtml*** are ***zs-roos-d1.shtml*** each. The least? The more valauble of the dimes are at the top of the list and the less valuable ones are at the bottom, and all the coins in the middle follow that value system.
Kimela T. asked • 12/02/15 A jar contains n nickels and d dimes. The total value of the coins is $1.50. Melanie has $1.80 worth of nickels and dimes. Martha has some nickels and dimes worth $6.25. She has 12 more nickels than dimes.
The child may be encouraged to work out the problem on a piece of paper before entering the solution. We encourage parents and teachers to select the topics according to the needs of the child. We hope that the free math worksheets have been helpful. We welcome your feedback, comments and questions about this site or page.
Student Visa Insurance Get Quotes For visitors, travel, student and other international travel medical Insurance. Embedded content, if any, are copyrights of their respective owners. Note that you will lose points if you ask for hints or clues. Try the given examples, or type in your own problem and check your answers.
Money value (dimes, nickels, pennies) Worksheet. Count the coins and match to its value. Kids count the pennies, nickels, dimes, quarters and half dollars and match to the corresponding value expressed in cents and dollars. This printable batch of simple worksheets helps strengthen counting money skills.
A dime is worth 10 cents. A nickel is worth 5 cents. 1 penny = 1 cent. 1 nickel = 5 cents. The value of a dime is and the value of a nickel is The total value of all the coins is. The total value of the coins is $3.10. The total value of the coins is $1.50. The total value of the coins is $1.40. The total value of the coins is 1.40.
The person has 20 coins. There are 22 coins in the jar. There are 20 coins in the jar. The total of all of them is 140 cents. She has three times as many nickels as dimes. She has 12 more nickels than dimes. How many nickels and dimes are in the jar? How many nickels and how many dimes are in the jar?
The value of all the dimes in dollars:0.10 × 2y = 0.20y. The value of all the nickels in dollars: 0.05 × y = 0.05y. Since the number of dimes are 2 times more than the number of nickels, the number of dimes will be 2y. The number of nickels is y.
The total value of the coins is $4.50. The total value of the coins is $8.40. If the number of dimes was tripled and the number of nickels was increased by 14, the value of the coins would be $8.40. The total value of the coins is $1.80. The total value of the coins is $6.25. The total value of the coins is $5.30.
This money song for kids helps your children learn to identify and know the value of a penny, nickel, dime and quarter. Objective: I know the value of dimes, nickels and pennies. We hope that the kids will also love the fun stuff and puzzles.
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CC-MAIN-2021-25/segments/1623487608856.6/warc/CC-MAIN-20210613131257-20210613161257-00517.warc.gz
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kpi.ua
|
en
| 0.912319
| 2021-06-13T13:22:13
|
https://cad.kpi.ua/wildflower-seedling-vce/nickels-and-dimes-value-76cf99
| 0.758791
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The Tanzalin Method for Easier Integration by Parts
The Tanzalin Method is a technique used to perform certain integrations, providing an alternative to the traditional Integration by Parts method. This approach can be easier to follow and is particularly useful for checking work in examinations.
Example 1: Integration by Parts Method
To compare, let's first examine the traditional Integration by Parts method. We identify u, v, du, and dv as follows:
u = 2x
dv = (3x − 2)^6 dx
du = 2 dx
Integration by Parts gives us:
∫u dv = uv - ∫v du
= (2x)(3x − 2)^6 - ∫(3x − 2)^6 (2 dx)
= (2x)(3x − 2)^6 - 2∫(3x − 2)^6 dx
Now, we find the unknown integral:
∫(3x − 2)^6 dx = (1/3)(3x − 2)^7 + C
Putting it together, we have:
∫(2x)(3x − 2)^6 dx = (2x)(3x − 2)^6 - (2/3)(3x − 2)^7 + C
We can then factor and simplify this to give the final answer.
Example 1: Tanzalin Method
The Tanzalin Method involves setting up a table with successive derivatives of the simplest polynomial term in the first column and the integrals of the second term in the second column.
| Derivatives | Integrals | Sign | Same-color Products |
| --- | --- | --- | --- |
| 2x | (3x − 2)^6 | + | 2x(3x − 2)^6 |
| 2 | (1/3)(3x − 2)^7 | - | -2(1/3)(3x − 2)^7 |
| 0 | (1/3)(3x − 2)^7 | + | 0 |
We multiply the terms with the same color background, and the answer for the integral is the sum of the terms in the final column.
The Tanzalin Method is somewhat less messy than the traditional Integration by Parts method.
Example 2: Tanzalin Method
We'll go straight to the Tanzalin Method for this example.
| Derivatives | Integrals | Sign | Same-color Products |
| --- | --- | --- | --- |
| x | sin x | + | x sin x |
| 1 | -cos x | - | cos x |
| 0 | -sin x | + | 0 |
We multiplied (x) by (-cos x) without changing the sign and then multiplied (1) by (-sin x) with a changed sign.
Adding the final column gives us the answer:
∫x sin x dx = x(-cos x) + cos x + C
Example 3: Tanzalin Method
Using the Tanzalin Method requires four rows in the table this time, since there is one more derivative to find.
| Derivatives | Integrals | Sign | Same-color Products |
| --- | --- | --- | --- |
| x^2 | (3x − 2)^6 | + | x^2(3x − 2)^6 |
| 2x | (1/3)(3x − 2)^7 | - | -2x(1/3)(3x − 2)^7 |
| 2 | (1/9)(3x − 2)^8 | + | 2(1/9)(3x − 2)^8 |
| 0 | (1/9)(3x − 2)^8 | - | 0 |
Our final answer is:
∫x^2(3x − 2)^6 dx = x^2(3x − 2)^6 - (2/3)x(3x − 2)^7 + (2/9)(3x − 2)^8 + C
Example 4: Tanzalin Method
We need to choose ln 4x for the first column, following the Integration by Parts priority recommendations.
| Derivatives | Integrals | Sign | Same-color Products |
| --- | --- | --- | --- |
| ln 4x | x^2 | + | ln 4x(x^2) |
| 1/x | (1/3)x^3 | - | -(1/3)x^2 |
| 0 | (1/3)x^3 | + | 0 |
When do we stop? The derivatives column will continue to grow, as will the integrals column. The Tanzalin Method requires one of the columns to "disappear" (have a value of 0) so we have somewhere to stop.
Our final answer is:
∫ln 4x(x^2) dx = x^2 ln 4x - (1/3)x^3 + C
Conclusion
While the Tanzalin Method only handles integrals involving at least one polynomial expression, it is worth considering as a simpler way of writing Integration by Parts questions. This method provides an alternative approach to integration, making it easier to organize and solve problems.
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CC-MAIN-2019-04/segments/1547583826240.93/warc/CC-MAIN-20190122034213-20190122060213-00295.warc.gz
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intmath.com
|
en
| 0.890537
| 2019-01-22T04:58:18
|
https://www.intmath.com/blog/mathematics/tanzalin-method-for-easier-integration-by-parts-4339
| 0.931644
|
To calculate the geometric mean for a set of numbers, multiply the numbers together and take the nth root of this product. This calculation is only valid when all numbers in the set are nonzero and positive.
For example, consider a set of numbers: 5, 8, 2, 8, and 10. Multiply these numbers together to get a product of 6,400. Since there are five numbers in the set, take the fifth root of 6,400 to find the geometric mean, which is approximately 5.771.
The geometric mean is useful for calculating average rates of return and other investment scenarios where numbers are typically percentages rather than counts. In contrast, the arithmetic mean (which involves adding numbers and dividing by the count) is more suitable for items involving counting rather than multiplication. Key steps for calculating the geometric mean include:
1. Multiplying the numbers in the data set together.
2. Counting the numbers in the set.
3. Taking the nth root of the product, where n is the count of numbers.
By following these steps, you can calculate the geometric mean for any set of nonzero and positive numbers, making it a valuable tool for investment analysis and other applications involving multiplication.
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CC-MAIN-2017-51/segments/1512948517181.32/warc/CC-MAIN-20171212134318-20171212154318-00478.warc.gz
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reference.com
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en
| 0.85321
| 2017-12-12T15:01:32
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https://www.reference.com/math/calculate-geometric-mean-2753eae7a9bfb7c9
| 0.999969
|
To pass in physics board exams, here are some important theory questions of paper-1 that could help you score near 50% plus marks in your 12th physics board exams.
Important questions of physics paper 1 to go with while appearing for class 12 board exams are as follows:
**1st Chapter – Circular Motion**
1. Relation between linear velocity and angular velocity.
2. Difference between centripetal force and centrifugal force.
3. Show that the angle of banking is independent of the mass of the vehicle.
4. Obtain an expression for the time period (T) of a conical pendulum.
**3rd Chapter – Rotational Motion**
1. Derive an expression for the kinetic energy of a rigid rotating body about an axis.
2. Obtain an expression for the torque acting on a rotating body about an axis.
3. State and prove the theorem/principle of parallel axes.
4. State and prove the theorem/principle of perpendicular axes.
5. Expression for the angular momentum of a rotating body.
6. Principle of conservation of angular momentum.
**4th Chapter – Oscillations**
1. Obtain the differential equation of linear S.H.M.
2. Obtain an expression for the period of a simple pendulum.
3. State the laws of a simple pendulum.
**5th Chapter – Elasticity**
All topics in this chapter are important.
**6th Chapter – Surface Tension**
1. Obtain an expression for the rise of a liquid in a capillary tube.
2. Define the angle of contact and its characteristics, including diagrams.
3. What is capillarity, and give some applications of it.
**7th Chapter – Wave Motion**
1. Explain the production of beats and deduce analytically the expression for beats frequency.
2. Explain what is Doppler Effect and its applications.
3. Derive an expression for one-dimensional simple harmonic waves traveling towards the +ve X-axis.
**8th Chapter – Stationary Waves**
1. Explain the formation of stationary waves by analytical method, including nodes and antinodes.
2. State and explain the laws of vibrating strings and end correction.
3. Explain resonance with an example, its merits, and demerits.
4. With a neat diagram, explain the fundamental mode of an air column in a pipe when the pipe is open at both ends and when the pipe is closed at one end.
**9th Chapter – Kinetic Theory of Gases & Radiation**
1. Write the assumptions of the kinetic theory of gases.
2. Explain a perfectly black body.
3. State Kirchhoff’s law of radiation and give its theoretical proof.
4. Explain Newton’s law of cooling and Stefan’s Law.
5. Derive an expression for the pressure exerted by a gas on the basis of the kinetic theory of gases.
6. Deduce Boyle’s law using the expression for pressure exerted by the gas.
These around 30 theory questions can help you score passing marks or near 50-60 percent marks, including your practical exams performance. For example, if you perform well in your practical exams and score around 20-25 out of 30 marks, and then you come across a few of these questions in the theory exam and write proper points as per the requirement of the question, you can score around 10-15 marks out of 70. This can result in a total score of 35, which can help you pass.
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CC-MAIN-2023-14/segments/1679296949694.55/warc/CC-MAIN-20230401001704-20230401031704-00339.warc.gz
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techniyojan.com
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en
| 0.865436
| 2023-04-01T00:25:25
|
https://techniyojan.com/2020/01/how-to-pass-in-physics-board-exams-class-12.html
| 0.496979
|
# STARKs, Part 3: Into the Weeds
STARKs ("Scalable Transparent ARgument of Knowledge") are a technique for creating a proof that \(f(x)=y\) where \(f\) may potentially take a very long time to calculate, but where the proof can be verified very quickly. A STARK is "doubly scalable": for a computation with \(t\) steps, it takes roughly \(O(t \cdot \log{t})\) steps to produce a proof, and it takes ~\(O(\log^2{t})\) steps to verify.
## MIMC
MIMC (see paper) is used as an example because it is simple to understand and interesting enough to be useful in real life. The function can be viewed visually as follows:
MIMC with a very large number of rounds is useful as a verifiable delay function - a function which is difficult to compute, and particularly non-parallelizable to compute, but relatively easy to verify.
## Prime field operations
A convenience class is built to perform prime field operations and polynomial operations over prime fields. The code includes trivial bits such as addition, subtraction, and multiplication, as well as the Extended Euclidean Algorithm for computing modular inverses.
## Fast Fourier Transforms
The FFT only takes \(O(n \cdot log(n))\) time, though it is more restricted in scope; the x coordinates must be a complete set of roots of unity of some order \(N = 2^{k}\).
## Thank Goodness It's FRI-day (that's "Fast Reed-Solomon Interactive Oracle Proofs of Proximity")
A low-degree proof is a (probabilistic) proof that at least some high percentage of a given set of values represent the evaluations of some specific polynomial whose degree is much lower than the number of values given.
## The STARK
The actual meat that puts all of these pieces together is the `def mk_mimc_proof(inp, steps, round_constants)` function, which generates a proof of the execution result of running the MIMC function with the given input for some number of steps.
The extension factor is the extent to which the computational trace is "stretched". The step count multiplied by the extension factor must be at most \(2^{32}\), because there are no roots of unity of order \(2^{k}\) for \(k > 32\).
The computational trace is generated, and then converted into a polynomial. The polynomial is evaluated in a larger set, of successive powers of a root of unity \(g_2\) where \((g_2)^{steps \cdot 8} = 1\).
The round constants of MIMC are converted into a polynomial. Because these round constants loop around very frequently, they form a degree-64 polynomial, and the expression and extension can be computed.
\(C(P(x))\) is calculated, which is \(C(P(x), P(g_1 \cdot x), K(x))\). The goal is that for every \(x\) that is laid down in the computational trace (except for the last step), the next value in the trace is equal to the previous value in the trace cubed, plus the round constant.
There is an algebraic theorem that proves that if \(Q(x)\) is equal to zero at all of these x coordinates, then it is a multiple of the minimal polynomial that is equal to zero at all of these x coordinates: \(Z(x) = (x - x_1) \cdot (x - x_2) \cdot ... \cdot (x - x_n)\).
The quotient \(D(x) = \frac{Q(x)}{Z(x)}\) is provided, and FRI is used to prove that it's an actual polynomial and not a fraction.
The prover wants to prove \(P(1) = input\) and \(P(last\_step) = output\). If \(I(x)\) is the interpolant - the line that crosses the two points \((1, input)\) and \((last\_step, output)\), then \(P(x) - I(x)\) would be equal to zero at those two points.
The Merkle root of \(P\), \(D\), and \(B\) is committed to. A pseudorandom linear combination of \(P\), \(D\), and \(B\) is computed, and an FRI proof is done on that.
The proof consists of a set of Merkle roots, the spot-checked branches, and a low-degree proof of the random linear combination. The largest parts of the proof are the Merkle branches and the FRI proof.
At every position that the prover provides a Merkle proof for, the verifier checks the Merkle proof and checks that \(C(P(x), P(g_1 \cdot x), K(x)) = Z(x) \cdot D(x)\) and \(B(x) \cdot Z_2(x) + I(x) = P(x)\).
The verifier also checks that the linear combination is correct and calls `verify_low_degree_proof` to verify the FRI proof.
The STARK proving overhead for MIMC is remarkably low, because MIMC is almost perfectly "arithmetizable" - its mathematical form is very simple. For "average" computations, which contain less arithmetically clean operations, the overhead is likely much higher, possibly around 10000-50000x.
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cloudflare-ipfs.com
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en
| 0.84683
| 2023-03-24T03:43:14
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https://cloudflare-ipfs.com/ipfs/bafybeigsn4u4nv4uyskxhewakk5m2j2lluzhsbsayp76zh7nbqznrxwm7e/general/2018/07/21/starks_part_3.html
| 0.997855
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## Solving the Equation: (x+1/x-2)^2 + x+1/x-4 - 3(2x-4/x-4)^2 = 0
This equation presents a challenge due to its complex structure. We will break down the steps to solve it:
### 1. Simplification
We can simplify the equation by factoring out common terms and expanding the squares:
(x+1/x-2)^2 + (x+1)/(x-4) - 3(2(x-2)/(x-4))^2 = 0
Expanding the squares gives:
[(x+1)^2/(x-2)^2] + (x+1)/(x-4) - 3[4(x-2)^2/(x-4)^2] = 0
Combining terms with the same denominator, the common denominator is (x-2)^2(x-4)^2:
[(x+1)^2(x-4)^2 + (x+1)(x-2)^2(x-4) - 12(x-2)^2(x-4)] / [(x-2)^2(x-4)^2] = 0
### 2. Solve the Numerator
Since the denominator cannot be zero, we only need to solve the numerator:
(x+1)^2(x-4)^2 + (x+1)(x-2)^2(x-4) - 12(x-2)^2(x-4) = 0
Factoring out common terms gives:
(x-2)^2(x-4)[(x+1)^2 + (x+1)(x-2) - 12] = 0
Simplifying the expression inside the brackets:
(x-2)^2(x-4)[2x^2 + x - 13] = 0
### 3. Solve for x
Now we have a simpler equation to solve. We need to find the values of x that make this equation true:
x - 2 = 0 => x = 2
x - 4 = 0 => x = 4
2x^2 + x - 13 = 0
To solve the quadratic equation 2x^2 + x - 13 = 0, we can use the quadratic formula:
x = [-b ± √(b^2 - 4ac)] / 2a
Where a = 2, b = 1, and c = -13.
Solving for x using the quadratic formula, we get two more solutions:
x = (-1 + √105) / 4
x = (-1 - √105) / 4
### 4. Verification
It's essential to verify if the solutions we found are valid. We need to check if any of the solutions make the original denominator zero.
We find that x = 2 and x = 4 make the denominator zero, so these solutions are extraneous and need to be discarded.
### Conclusion
Therefore, the solutions to the equation (x+1/x-2)^2 + x+1/x-4 - 3(2x-4/x-4)^2 = 0 are:
x = (-1 + √105) / 4
x = (-1 - √105) / 4
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jasonbradley.me
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en
| 0.764842
| 2024-09-08T17:01:41
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https://jasonbradley.me/page/(x%252B1%252Fx-2)%255E2%252Bx%252B1%252Fx-4-3(2x-4%252Fx-4)%255E2%253D0
| 0.999968
|
### Introduction to Gradual Release of Instruction
The gradual release of instruction is a crucial strategy in math instruction, applicable to all topics, including fractions. This approach involves a recursive and cyclical process, rather than a linear one, to layer instruction effectively.
### What is Gradual Release of Instruction?
The traditional gradual release model consists of three stages:
1. **I show the students**: The teacher demonstrates the concept or skill.
2. **We do it together**: The teacher and students work together to practice the concept or skill.
3. **They try it alone**: Students apply the concept or skill independently.
However, this model can be oversimplified. A more effective gradual release plan involves recursive cycles of instruction, practice, and assessment.
### An Alternative Gradual Release Model
In math instruction, a second type of gradual release can be applied. This model focuses on providing students with real-world applications and challenging problems to deepen their understanding. For example:
- **Word problems**: "Sue walked into a bakery to buy some cookies for her family. She noticed that each tray held 24 cookies. If half of the cookies on one of the trays had sprinkles, how many would that be? What if half of the cookies with sprinkles also had chocolate chips–how many would that be?"
- **Mathematical computations**: 3/4 + 4/4 = ?, 5 x 3/8 = ?
- **Equivalent fractions**: Generate a list of 3 fractions equivalent to 4/5.
### Assessing True Understanding
Students may appear to understand fractions by completing practice sheets and identifying given fractions. However, true understanding requires application and real-world context. Teachers must work closely with students, observing, listening, and asking questions to ensure a deep understanding of fractions and their practical applications.
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theteacherstudio.com
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en
| 0.93984
| 2023-03-23T16:55:56
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https://theteacherstudio.com/getting-students-ready-to-learn/
| 0.788911
|
## Introduction to Square Root of 64
The square root of 64 is the number that, when multiplied by itself, equals 64. In other words, this number to the power of 2 equals 64. The square root of 64 can be written as ²√64 or 64^1/2.
## Calculator
Our calculator shows that ²√64 = ±8.
## Second Root of 64
The term "square root of 64" usually refers to the positive number, which is the principal square root. Since the index 2 is even and 64 is greater than 0, 64 has two real square roots: ²√64 (positive) and -²√64 (negative).
## Inverse of Square Root of 64
Extracting the square root is the inverse operation of squaring. The inverse operation of 64 square root is raising the result to the power of 2.
## What is the Square Root of 64?
The square root of 64 is ±8. The parts of the square root symbol are:
- ²√64: square root of 64 symbol
- 2: index
- 64: radicand (the number below the radical sign)
- √: radical symbol or radical only
## Table of nth Roots of 64
The following table provides an overview of the nth roots of 64:
| Index | Radicand | Root | Symbol | Value |
| --- | --- | --- | --- | --- |
| 2 | 64 | Square Root of 64 | ²√64 | ±8 |
| 3 | 64 | Cube Root of 64 | ³√64 | 4 |
| 4 | 64 | Forth Root of 64 | ⁴√64 | ±2.8284271247 |
| 5 | 64 | Fifth Root of 64 | ⁵√64 | 2.29739671 |
| 6 | 64 | Sixth Root of 64 | ⁶√64 | ±2 |
| 7 | 64 | Seventh Root of 64 | ⁷√64 | 1.8114473285 |
| 8 | 64 | Eight Root of 64 | ⁸√64 | ±1.6817928305 |
| 9 | 64 | Nineth Root of 64 | ⁹√64 | 1.587401052 |
| 10 | 64 | Tenth Root of 64 | ¹⁰√64 | ±1.5157165665 |
## Summary
To sum up, the square roots of 64 are ±8, with the positive real value being the principal. Finding the second root of 64 is the inverse operation of rising the result to the power of 2, i.e., (±8)^2 = 64. For more information about roots, visit our page on nth Root.
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industrialdevicesindia.com
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en
| 0.877525
| 2024-09-12T05:38:00
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https://industrialdevicesindia.com/article/what-is-the-square-root-of-64-information-and-calculator/1763
| 0.977974
|
# Resources tagged with: Linear functions
### There are 9 results
Broad Topics > Functions and Graphs > Linear functions
##### Age 11 to 14 Challenge Level:
Collect as many diamonds as possible by drawing three straight lines to understand linear functions.
##### Age 14 to 16 Challenge Level:
Position lines perpendicular to each other and analyze the equations of perpendicular lines. What relationship can be observed between their slopes?
##### Age 11 to 14 Challenge Level:
Investigate how the position of a line affects its equation. Compare the equations of parallel lines and identify any patterns or relationships.
##### Age 11 to 14
Alf Coles discusses creating 'spaces for exploration' in the classroom to facilitate student learning.
##### Age 11 to 14 Challenge Level:
Translate different lines and observe the changes in their equations. Predict and explain the effects of translation on linear equations.
##### Age 11 to 14 Challenge Level:
Reflect different lines in the x or y-axis and analyze the changes in their equations. Predict and explain the effects of reflection on linear equations.
##### Age 14 to 16 Challenge Level:
Given the graph y=4x+7, determine the possible order of four transformations that could have been applied to it.
##### Age 14 to 16 Challenge Level:
Investigate the relationship between the coordinates of a line's endpoints and the number of grid squares it crosses. Is there a consistent pattern or formula?
##### Age 11 to 14 Challenge Level:
Compare different pocket money systems and choose the most preferable one. Consider factors such as weekly allowance, rewards, or penalties.
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maths.org
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en
| 0.870419
| 2020-01-27T21:33:28
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https://nrich.maths.org/public/topic.php?code=59&cl=3&cldcmpid=6982
| 0.977834
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The problem asks for the largest integer for which $n!$ is a factor of the sum $98! + 10,000$. To solve this, we can factor out $n!$ from $98!$, resulting in $\frac{98!}{n!}$. This expression has factors of $n+1, n+2, ..., 98$.
We need to find the largest $n$ for which $n!$ is a factor of $98! + 10,000$. Factoring out $n!$ from $98!$ gives us $\frac{98!}{n!}$, which has factors of $n+1, n+2, ..., 98$. The number $98!$ has $22$ factors of $5$, and $10,000$ has $4$ factors of $5$.
The total number of factors of $5$ in $98! + 10,000$ is $22 + 4 = 26$. Since $n!$ needs to be a factor of $98! + 10,000$, the largest possible value of $n$ is the largest integer for which $n!$ has $26$ or fewer factors of $5$.
This occurs when $n = 22$, since $22!$ has $22$ factors of $5$ from the multiples of $5$ up to $22$, and $23!$ would have $23$ factors of $5$, exceeding the total number of factors of $5$ in $98! + 10,000$.
Therefore, the largest integer $n$ for which $n!$ is a factor of $98! + 10,000$ is $n = 25$, but we must verify if $25!$ is indeed a factor of $98! + 10,000$. Upon closer examination, we see that $25!$ has the necessary factors to divide $98!$, and since $10,000 = 2^4 \cdot 5^4$, $25!$ also has the necessary factors of $2$ and $5$ to divide $10,000$.
Hence, $n = 25$ is indeed the largest integer for which $n!$ is a factor of $98! + 10,000$.
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artofproblemsolving.com
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en
| 0.800055
| 2023-03-25T07:27:59
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https://artofproblemsolving.com/wiki/index.php?title=2017_AMC_8_Problems/Problem_19&oldid=152053
| 0.999297
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## NCERT Solutions for Class 9 Maths Chapter 15 Probability Exercise 15.1
The correct NCERT Solutions for Chapter 15 Probability Exercise 15.1 in Class 9 Maths are provided here to help understand the chapter's basics. These solutions are useful for completing homework and achieving good exam marks. Experts have prepared detailed answers to every question, allowing for easy doubt clearance.
Exercise 15.1 is the sole exercise in this chapter, comprising 13 questions. These questions require preparing frequency distribution tables and calculating the probability of certain events.
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studyrankers.com
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en
| 0.708597
| 2023-03-31T16:23:18
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https://www.studyrankers.com/2020/03/ncert-solutions-for-class-9-maths-chapter-15-exercise-15.1.html
| 0.910289
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The largest subset of [1,2,...,n] which contains no 3-term geometric progression is being discussed. A sequence, now A208746, has been created to track this. For n = 16, omitting 1 and 4 was initially considered, but it was found that 1..16 \ {1,4} still contains 3,6,12 and 9,12,16.
A corrected and extended sequence is:
1,2,3,3,4,5,6,7,7,8,9,10,11,12,13,13,14,14,15,15,16,17,18,19,19,20,21,21,22,23,
24,24,25,26,27,27,28,29,30,31,32,33,34,34,35,36,37,38,38,38,39,39,40,41,42,43,
44,45,46,46,47,48,49,49,50,51,52,52,53,54,55,55,56,57,57,57,58,59,60,61,61,62
This computation used a floating-point IP solver for the packing subproblems. The approach was to enumerate geometric progressions using a nested loop structure and then solve the integer program of maximizing the subset of {1..N} subject to not taking all 3 of any progression.
The original question was posed by Neil Sloane, who also maintains sequence A003002, which gives the size of the largest subset of [1,2,...,n] which contains no 3-term arithmetic progression. The sequence for geometric progressions was initially calculated by hand for n >= 1, resulting in:
1,2,3,3,4,5,6,7,7,8,9,10,11,12,13,14
However, this has been corrected and extended as mentioned above.
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seqfan.eu
|
en
| 0.763072
| 2023-03-26T03:27:48
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http://list.seqfan.eu/pipermail/seqfan/2012-March/060886.html
| 0.883141
|
## Integers and Absolute Value
### Objective
Students will be able to order and compare integers, including the absolute value of integers.
### Lesson Duration
65 minutes
### Launch (10 minutes)
The lesson begins with an opener that allows students to construct viable arguments and critique the reasoning of others, which is mathematical practice 3. The learning target for the day is: "I can compare integers using <, >, and = by identifying their position on a number line. I understand that the absolute value of a number is its distance from zero."
### Explore (50 minutes)
The explore portion of the lesson starts with a video introduction to integers and absolute value. Students fill in their notes during the video and then participate in guided notes with real-world examples of integers, applying mathematical practice 4. A number line is used to demonstrate the progression from -10 to 10, utilizing mathematical practice 5. Comparison problems are discussed, and volunteers explain their answers and reasoning.
The concept of absolute value is introduced, emphasizing that it represents distance and cannot be negative. An example is used to illustrate this concept: if the supermarket is 10 miles north and Grandma's house is 10 miles south, Grandma's house is not -10 miles away, as distance cannot be negative. Students practice problems, including one that requires treating absolute value bars like parentheses in the order of operations, highlighting the importance of precision, which is mathematical practice 6.
### Instructional Strategy
A table challenge may be conducted using XP Math - Math Games Arcade, where tables participate in a task on the smartboard, and the table with the highest score wins. This challenge reminds students of the importance of paying attention and precision in their work.
### Real-World Examples of Absolute Value
Examples include distances between locations, such as the distance between home and school or the distance between two cities. These examples help students understand that absolute value represents a measure of distance without considering direction.
### Similar Lessons and Units
This lesson is part of a larger unit on integers and rational numbers, which includes lessons on adding, subtracting, multiplying, and dividing integers and rational numbers, as well as applying these concepts to real-world problems. The full unit plan includes:
- UNIT 1: Introduction to Mathematical Practices
- UNIT 2: Proportional Reasoning
- UNIT 3: Percents
- UNIT 4: Operations with Rational Numbers
- UNIT 5: Expressions
- UNIT 6: Equations
- UNIT 7: Geometric Figures
- UNIT 8: Geometric Measurement
- UNIT 9: Probability
- UNIT 10: Statistics
- UNIT 11: Culminating Unit: End of Grade Review
Specific lessons in the sequence include:
- LESSON 1: Integers and Absolute Value
- LESSON 2: Modeling Addition
- LESSON 3: Integer Addition Word Problems
- LESSON 4: Adding Integers
- LESSON 5: Multiple Addends
- LESSON 6: Adding Integers Review
- LESSON 7: Adding Integers Test
- LESSON 8: Subtracting Integers
- LESSON 9: Subtracting Integers Practice
- LESSON 10: Addition and Subtraction of Integers - DOMINOES!
- LESSON 11: Adding and Subtracting Integers - Real World Applications
- LESSON 12: Adding and Subtracting Integers - REVIEW!
- LESSON 13: Adding and Subtracting Integers Test
- LESSON 14: Adding and Subtracting Signed Fractions
- LESSON 15: Adding and Subtracting Signed Fractions Fluency Practice
- LESSON 16: Adding and Subtracting Signed Decimals
- LESSON 17: Adding and Subtracting Rational Numbers - Practice Makes Perfect!
- LESSON 18: Adding and Subtracting Rational Numbers - Test
- LESSON 19: Multiplying and Dividing Integers
- LESSON 20: Multiplying and Dividing Rational Numbers
- LESSON 21: Problem Solving with Rational Numbers
- LESSON 22: Fractions to Decimals - Terminate or Repeat?
- LESSON 23: Rational Number Unit Test
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betterlesson.com
|
en
| 0.816985
| 2017-12-16T11:04:57
|
https://betterlesson.com/lesson/443522/integers-and-absolute-value-are-two-steps-forward-and-two-steps-back-the-same-thing?from=consumer_breadcrumb_dropdown_lesson
| 0.989243
|
To determine if the function f(x) = 2x^{3} + 300 + 4 is increasing, decreasing, or not changing at x = -2, we need to find its derivative. First, rewrite the function as f(x) = 2x^{3} + 300x^{0} + 4, which simplifies to f(x) = 2x^{3} + 300 + 4, since x^{0} = 1. However, to apply the power rule for differentiation correctly to the constant term, we treat 300 as 300x^{0}. The derivative of x^{0} is 0, so the derivative of 300 is 0. Thus, we focus on differentiating 2x^{3} and the constant term 4, which also differentiates to 0.
The derivative f'(x) is found by applying the power rule to each term:
- The derivative of 2x^{3} is 2 * 3x^{3-1} = 6x^{2}.
- The derivative of 300 is 0, since the derivative of any constant is 0.
- The derivative of 4 is also 0.
So, f'(x) = 6x^{2}.
To evaluate if the function is increasing or decreasing at x = -2, we substitute x = -2 into f'(x):
f'(-2) = 6(-2)^{2} = 6 * 4 = 24.
However, the original solution provided an incorrect derivative and evaluation. Let's correct that and follow the original function given: f(x) = 2x^{3} + 300 + 4. The correct derivative, considering the function as is, should directly apply the power rule:
- The derivative of 2x^{3} is 6x^{2}.
- The derivatives of 300 and 4 are 0.
Thus, the correct derivative is f'(x) = 6x^{2}. Evaluating at x = -2:
f'(-2) = 6(-2)^{2} = 24.
This indicates the function is increasing at x = -2 because the derivative is positive. The confusion arose from an incorrect manipulation of the function and its derivative in the original solution. The key concept here is applying the power rule correctly to find the derivative and then evaluating it at the specified point to determine the function's behavior.
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expertsmind.com
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en
| 0.833064
| 2017-12-13T17:05:40
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http://www.expertsmind.com/questions/some-interpretations-of-the-derivative-30153374.aspx
| 0.998761
|
## Learning Objectives
By the end of this section, you will be able to:
- Describe the law of conservation of linear momentum.
- Derive an expression for the conservation of momentum.
- Explain conservation of momentum with examples.
- Explain the law of conservation of momentum as it relates to atomic and subatomic particles.
The information presented in this section supports the following AP learning objectives and science practices:
**5.A.2.1** The student is able to define open and closed systems for everyday situations and apply conservation concepts for energy, charge, and linear momentum to those situations.
**5.D.1.4** The student is able to design an experimental test of an application of the principle of the conservation of linear momentum, predict an outcome of the experiment using the principle, analyze data generated by that experiment whose uncertainties are expressed numerically, and evaluate the match between the prediction and the outcome.
**5.D.2.1** The student is able to qualitatively predict, in terms of linear momentum and kinetic energy, how the outcome of a collision between two objects changes depending on whether the collision is elastic or inelastic.
**5.D.2.2** The student is able to plan data collection strategies to test the law of conservation of momentum in a two-object collision that is elastic or inelastic and analyze the resulting data graphically.
**5.D.3.1** The student is able to predict the velocity of the center of mass of a system when there is no interaction outside of the system but there is an interaction within the system.
Momentum is conserved when the net external force on a system is zero. This means that the total momentum of a closed system remains constant over time. The law of conservation of momentum can be expressed mathematically as:
p_tot = p'_tot
where p_tot is the initial total momentum and p'_tot is the final total momentum.
Consider a two-car collision. The momentum of the first car is p1 = m1v1, and the momentum of the second car is p2 = m2v2. After the collision, the momentum of the first car is p'1 = m1v'1, and the momentum of the second car is p'2 = m2v'2. The total momentum before the collision is equal to the total momentum after the collision:
p1 + p2 = p'1 + p'2
Using the definition of impulse, the change in momentum of the first car is given by:
Δp1 = F1Δt
where F1 is the force on the first car and Δt is the time over which the force acts. Similarly, the change in momentum of the second car is given by:
Δp2 = F2Δt
where F2 is the force on the second car. Since the forces are equal and opposite (F1 = -F2), the changes in momentum are also equal and opposite:
Δp1 = -Δp2
The total momentum of the two-car system remains constant:
p1 + p2 = p'1 + p'2
This result can be generalized to any isolated system, with any number of objects. The law of conservation of momentum states that the total momentum of an isolated system remains constant over time.
## Conservation of Momentum Principle
The conservation of momentum principle can be applied to systems as different as a comet striking Earth and a gas containing huge numbers of atoms and molecules. Conservation of momentum is violated only when the net external force is not zero. But another larger system can always be considered in which momentum is conserved by simply including the source of the external force.
## Isolated System
An isolated system is defined as one for which the net external force is zero. In an isolated system, the total momentum remains constant over time.
## Making Connections: Cart Collisions
Consider two air carts with equal mass (m) on a linear track. The first cart moves with a speed v towards the second cart, which is initially at rest. If the collision is elastic, the first cart will stop after the collision, and the second cart will have a final velocity v after the collision. The momentum of the system will be conserved in the collision.
If the two carts stick together after the collision, the final velocity of the two-cart system will be half the initial velocity of the first cart. The kinetic energy of the system will not be conserved in this case.
## Making Connections: Take-Home Investigation—Drop of Tennis Ball and a Basketball
Hold a tennis ball side by side and in contact with a basketball. Drop the balls together. The tennis ball will bounce off the basketball, and the basketball will remain largely stationary. This is because the momentum of the tennis ball is conserved, and the basketball has a much larger mass.
## Making Connections: Take-Home Investigation—Two Tennis Balls in a Ballistic Trajectory
Tie two tennis balls together with a string about a foot long. Hold one ball and let the other hang down, then throw it in a ballistic trajectory. The center of the string will remain stationary, and the two balls will move in opposite directions. This is because the momentum of the system is conserved.
## Applying Science Practices: Verifying the Conservation of Linear Momentum
Design an experiment to verify the conservation of linear momentum in a one-dimensional collision, both elastic and inelastic. Measure the momentum of each object before and after the collision, and compare the results to your prediction.
## Making Connections: Conservation of Momentum and Collision
Conservation of momentum is crucial to our understanding of atomic and subatomic particles. The conservation of momentum principle is used to analyze the masses and other properties of previously undetected particles, such as the nucleus of an atom and the existence of quarks that make up particles of nuclei.
## Subatomic Collisions and Momentum
The conservation of momentum principle is valid when considering systems of particles. Experiments seeking evidence that quarks make up protons scattered high-energy electrons off of protons. The analysis was based partly on the same conservation of momentum principle that works so well on the large scale.
The law of conservation of momentum is a fundamental principle in physics that applies to all systems, from the smallest subatomic particles to the largest galaxies. It states that the total momentum of a closed system remains constant over time, and it is a powerful tool for analyzing and understanding the behavior of physical systems.
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openstax.org
|
en
| 0.811771
| 2024-09-13T15:47:47
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https://openstax.org/books/college-physics-ap-courses/pages/8-3-conservation-of-momentum
| 0.687777
|
**Optimal Control: Direct Solution Methods**
This lecture covers the basics of numerical optimal control, focusing on direct solution methods such as direct single shooting, direct multiple shooting, and collocation with Legendre polynomials. Key concepts include problem formulation, scaling of input and state variables, and the use of toolboxes like CASADI for solving optimal control problems. The lecture also explains time transformation, input rate constraints, and scaling dynamics, providing practical tips for successful problem solving.
**Course Information**
The lecture is part of the doctoral course EE-715: Optimal Control, which introduces optimal control theory, numerical implementation, and problem formulation for applications.
**Related Concepts**
1. Numerical analysis: the study of algorithms using numerical approximation for mathematical analysis problems.
2. Numerical methods for ordinary differential equations: methods for finding numerical approximations to solutions of ordinary differential equations (ODEs).
3. Iterative method: a mathematical procedure using an initial value to generate a sequence of improving approximate solutions.
4. Direct multiple shooting method: a numerical method for solving boundary value problems by dividing the interval into smaller intervals and imposing matching conditions.
5. Numerical methods for partial differential equations: the branch of numerical analysis studying numerical solutions of partial differential equations (PDEs).
**Instructors and Related Lectures**
The lecture is taught by two instructors and is related to 258 other lectures, including:
- Finite Difference Grids (MATH-351)
- Finite Elements: Elasticity and Variational Formulation (MATH-212)
- Nonlinear Problem Solving (MOOC: Numerical Analysis for Engineers)
- Finite Elements: Problem with Limits (MATH-251(b))
- Direct Methods for Linear Systems of Equations (ChE-312)
**Note**: This page is automatically generated and may contain incorrect or outdated information. Please verify the information with EPFL's official sources.
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epfl.ch
|
en
| 0.886774
| 2024-09-11T23:16:55
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https://graphsearch.epfl.ch/en/lecture/0_1816fy63
| 0.999941
|
# How to Calculate Motor Starting Time
Calculating motor starting time can be complex due to various influencing factors. To understand how to calculate this, it's essential to consider the motor characteristic, specifically the torque curve, and how it interacts with the load torque curve.
## Understanding Motor Starting Time
The torque available to accelerate the motor up to speed is the difference between motor torque (CM) and load torque (CL). This difference, denoted as Ca, is crucial for understanding the starting dynamics. The formula for Ca is:
Ca = CM - CL
Where:
- Ca is the torque to accelerate the motor, in N.m
- CM is the motor torque, in N.m
- CL is the load torque, in N.m
Both motor and load torque vary with speed, and the motor torque characteristic depends on the motor's design and construction. Starting methods can also affect the available motor torque and the shape of its curve.
## Deriving the Equation for Starting Time
To derive an equation for the time to accelerate from zero to the running speed, we consider the inertia of both the motor and the load. The equation involves the following parameters:
- t: time to accelerate to running speed, in seconds
- n: motor running speed, in rpm
- C: motor torque, in N.m
- CL: load torque, in N.m
- J: inertia of the motor, in kg.m^2
- JL: inertia of the load, in kg.m^2
The equation for starting time can be complex due to the variability of torque with speed. For precise calculations, especially with complex torque curves or starting arrangements, numerical solutions or software tools are recommended.
## Simplified Approximation for Starting Time
For a more straightforward calculation, simplifications can be introduced:
1. Use an average value of motor torque, considering the inrush torque (CS) and the maximum torque (Cmax).
2. Apply an adjustment factor KL to account for varying load torque due to speed changes.
The load factor (KL) values for different types of loads are:
- Lift: 1
- Fans: 0.33
- Piston: 0.5
- Flywheel: 0
Using these simplifications, the approximate starting time can be calculated with the formula involving the effective acceleration torque (Cacc).
## Example Calculation
For a 90 kW motor driving a fan:
- Motor Rated Speed (n): 1500 rpm
- Motor Full Load Speed: 1486 rpm
- Motor Inertia (J): 1.4 kg.m^2
- Motor Rated Torque: 549 Nm
- Motor Inrush Torque (CS): 1563 Nm
- Motor Maximum Torque (Cmax): 1679 Nm
- Load Inertia (JL): 30 kg.m^2
- Load Torque (CL): 620 Nm
- Load Factor (KL): 0.33
This example demonstrates how to apply the simplified formula to estimate the starting time.
## Conclusion
While accurately calculating motor starting time can be complex, using simplifications and understanding the key factors involved can provide realistic estimates for common starting scenarios. For more precise calculations, especially in critical applications, consulting detailed references or using specialized software is advisable.
## References
- Three-phase asynchronous motors. Generalities and ABB proposals for the coordination of protective devices. ABB, 2008.
For specific load types like centrifugal and positive displacement pumps, the load factor can be approximated as follows:
- Centrifugal pump: similar to a fan, use KL = 0.33
- Positive displacement pump: similar to a piston, use KL = 0.5
These approximations can be used in the simplified formula to estimate the starting time for these types of loads.
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|
myelectrical.com
|
en
| 0.88879
| 2023-03-22T06:24:10
|
https://myelectrical.com/notes/entryid/107/how-to-calculate-motor-starting-time
| 0.875781
|
## Understanding Entropy in Information Theory
Entropy is a measure of uncertainty or randomness in a dataset. It quantifies the level of disorder in the data and helps in decision-making algorithms. In information theory, entropy is related to the reduction of uncertainty. When new knowledge is obtained from data, uncertainty decreases, and information increases.
### A. Information and Uncertainty
Information refers to the reduction of uncertainty. When uncertainty increases, it becomes harder to make predictions. Entropy, at its core, quantifies this uncertainty.
### B. Shannon’s Information Theory and Entropy
Claude Shannon introduced the concept of entropy as a measure of uncertainty. Shannon’s entropy, symbolized as H, quantifies the amount of information contained in a random variable or a probability distribution. Higher entropy implies higher unpredictability.
### C. Calculating Entropy for Discrete and Continuous Probability Distributions
To calculate entropy, we sum up the probabilities of all outcomes, each multiplied by the logarithm of its inverse probability, for discrete distributions. For continuous distributions, we integrate a similar expression.
### D. Interpretation of Entropy Values
Entropy values range from 0 to log(n), where n is the number of distinct outcomes. An entropy of 0 indicates certainty, while maximum entropy indicates maximum uncertainty.
## Entropy as a Measure of Uncertainty in Machine Learning
### A. Entropy as a Metric for Evaluating Decision Trees
Entropy serves as a metric to guide decision tree construction. The information gain achieved by a split is the reduction in entropy. By selecting splits that yield the most significant information gain, decision trees become more accurate.
### B. Information Gain and Its Relationship with Entropy
Information gain measures how much an attribute contributes to reducing overall entropy. By selecting attributes with high information gain, decision trees make better predictions.
### C. Using Entropy to Measure the Purity of a Dataset
Entropy provides a measure of purity, indicating how well-separated classes are. Minimizing entropy leads to higher accuracy and better generalization.
### D. Relationship between Entropy and Gini Impurity
Gini impurity is another criterion for evaluating decision tree splits. Both entropy and Gini impurity aim to minimize uncertainty but can yield different results.
## Applications of Entropy in Machine Learning
### A. Decision Trees and Random Forests
Entropy-based methods help Random Forests select the best attributes for splitting, leading to robust and accurate ensemble models.
### B. Clustering Algorithms
Entropy-based criteria assess the quality of clustering and enhance accuracy. K-means and hierarchical clustering can benefit from entropy-based approaches.
### C. Reinforcement Learning
Entropy plays a crucial role in balancing exploration and exploitation. Entropy regularization encourages exploration, leading to better decision-making.
## Entropy in Deep Learning
### A. Entropy in Neural Network Loss Functions
Cross-entropy loss is a widely-used loss function for classification tasks. It measures the dissimilarity between predicted and actual probabilities.
### B. Regularization Techniques Based on Entropy
Dropout regularization adds noise and randomness, akin to entropy, leading to improved generalization. Maximum entropy regularization aims to maximize the entropy of a model’s predictions.
## Challenges and Considerations
### A. Overfitting and Underfitting
Entropy can lead to overfitting or underfitting. It is essential to address these issues to ensure model performance.
### B. Bias in Data
Data bias influences entropy-based metrics, leading to biased predictions. Addressing bias is crucial for fair and ethical model use.
### C. Handling Imbalanced Datasets
Entropy-based methods can help alleviate the challenges of imbalanced datasets, ensuring fair and accurate model performance.
## FAQs About Machine Learning Entropy
### What is Entropy in Machine Learning?
Entropy is a measure of uncertainty or randomness in a dataset.
### What are the Different Types of Entropy?
There are mainly two types: Shannon entropy and Gini impurity.
### What Does High Entropy Mean?
High entropy indicates a higher level of disorder or uncertainty.
### What is the Entropy of a Model?
The entropy of a model refers to the uncertainty in its predictions.
### How Do You Explain Entropy?
Entropy is a measure of unpredictability or randomness in a dataset.
### What is Entropy Used to Explain?
Entropy explains the amount of disorder or randomness in data.
### What Does the Entropy of 1 Mean?
An entropy value of 1 signifies maximum disorder or uncertainty.
### What is a Good Entropy Value?
A good entropy value depends on the context and algorithm.
### What Does Entropy of 0 Mean?
An entropy value of 0 indicates a perfectly ordered dataset.
### What is Gini and Entropy in Machine Learning?
Gini and entropy are measures of impurity used in decision tree algorithms.
### How is Entropy Calculated?
Entropy is calculated by summing the negative of the probability of each class multiplied by the logarithm of that probability.
## Final Thoughts About Machine Learning Entropy
Machine learning entropy is a powerful concept that measures uncertainty or randomness within a dataset. It plays a crucial role in various machine learning algorithms and helps in building effective models. By embracing entropy, we can make better-informed decisions and improve model performance.
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CC-MAIN-2024-38/segments/1725700651722.42/warc/CC-MAIN-20240917004428-20240917034428-00351.warc.gz
|
kingpassive.com
|
en
| 0.878602
| 2024-09-17T02:04:44
|
https://kingpassive.com/machine-learning-entropy/
| 0.872358
|
The circumference or perimeter of a circle is the distance around the edge of the circle. To calculate the circumference of a circle, use the formula:
C = π × d
* C represents the Circumference.
* d represents the diameter of the circle, which is the distance all the way across the center of the circle.
* π stands for Pi, a number equal to 3.142 rounded to 3 decimal places.
To find the circumference, multiply the diameter of the circle by π.
To find the area of the circle, use the formula:
A = π × r²
* A represents the area.
* r represents the radius of the circle, which is the distance halfway across the center, so it is half the value of the diameter.
Note that the formula for circumference uses the diameter, and the formula for the area uses the radius.
**Example 1**
Find the circumference and area of a circle with a diameter of 38cm. The radius is 19cm.
First, calculate the circumference:
C = π × d
C = π × 38
C = 119.4 cm to 1 decimal place.
Next, calculate the area:
A = π × r²
A = π × 19²
A = 1134 cm² rounded to the nearest whole number.
**Example 2**
Calculate the circumference and area of a circle with a radius of 4.2m. The diameter is 8.4m.
First, calculate the circumference:
C = π × d
C = π × 8.4
C = 26.4 m to 1 decimal place.
Next, calculate the area:
A = π × r²
A = π × 4.2²
A = 55.4 m² rounded to the nearest whole number.
**Example 3**
A bicycle wheel has a diameter of 33cm. To find the distance the bike travels in one turn, calculate the circumference:
C = π × d
C = π × 33
C = 104 cm to the nearest centimeter.
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infobarrel.com
|
en
| 0.847919
| 2016-12-09T23:27:19
|
http://www.infobarrel.com/Calculating_the_area_and_circumference_of_a_circle
| 0.999734
|
Hall's conjecture states that for positive integers x and y, if k = x^3 - y^2 is nonzero, then |k| > x^(1/2-o(1)). This conjecture is a special case of the Masser-Oesterlé "ABC conjecture." Despite limited theoretical progress, significant experimental work has been done, starting with Hall's original paper.
A new algorithm has been developed to find all solutions of |k| << x^(1/2) with x < N in time O(N^(1/2+o(1))). Implementing this algorithm in 64-bit C for N=10^18 and running it for almost a month yielded 10 new cases of 0 < |k| < x^(1/2) in that range. Two of these cases improved on the previous record for x^(1/2) / |k|, with one breaking the old record by a factor of nearly 10.
The following table lists the 24 solutions of 0 < |k| < x^(1/2) with x < 10^18, including x, k, and r = x^(1/2) / |k|. The table does not include y, which is always the integer closest to x^(3/2).
1. |k| = 1641843, x = 5853886516781223, r = 46.60
2. |k| = 30032270, x = 38115991067861271, r = 6.50
3. |k| = 1090, x = 28187351, r = 4.87 (GPZ)
4. |k| = 193234265, x = 810574762403977064, r = 4.66
5. |k| = 17, x = 5234, r = 4.26 (GPZ, P(-3))
6. |k| = 225, x = 720114, r = 3.77 (GPZ)
7. |k| = 24, x = 8158, r = 3.76 (GPZ, P(3))
8. |k| = 307, x = 939787, r = 3.16 (GPZ)
9. |k| = 207, x = 367806, r = 2.93 (GPZ)
10. |k| = 28024, x = 3790689201, r = 2.20 (GPZ)
11. |k| = 117073, x = 65589428378, r = 2.19
12. |k| = 4401169, x = 53197086958290, r = 1.66
13. |k| = 105077952, x = 23415546067124892, r = 1.46
14. |k| = 1, x = 2, r = 1.41
15. |k| = 497218657, x = 471477085999389882, r = 1.38
16. |k| = 14668, x = 384242766, r = 1.34 (GPZ, P(-9))
17. |k| = 14857, x = 390620082, r = 1.33 (GPZ, P(9))
18. |k| = 87002345, x = 12813608766102806, r = 1.30
19. |k| = 2767769, x = 12438517260105, r = 1.27
20. |k| = 8569, x = 110781386, r = 1.23 (GPZ)
21. |k| = 5190544, x = 35495694227489, r = 1.15
22. |k| = 11492, x = 154319269, r = 1.08 (GPZ)
23. |k| = 618, x = 421351, r = 1.05 (GPZ)
24. |k| = 548147655, x = 322001299796379844, r = 1.04 (D)
Notes:
- GPZ: Solutions found by J.Gebel, A.Pethö, and H.G.Zimmer with 1 < |k| < 10^5.
- D: Danilov's infinite family, with r approaching 5^(5/2)/54 = 1.035+.
- P(t): Polynomial family, where t is an integer congruent to 3 mod 6.
- *: Obtained from the record solution by multiplying x, y, k by 4, 8, 64, reducing r by a factor of 32.
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|
harvard.edu
|
en
| 0.787713
| 2024-09-17T02:42:31
|
https://people.math.harvard.edu/~elkies/hall.html
| 0.969047
|
Introduction to Recursion
A binary tree is a tree data structure in which each node has at most two children, referred to as the left child and the right child. The structure of a binary tree can be represented by the following class:
```java
public class Tree {
Object value;
Tree right;
Tree left;
}
```
Not every binary tree is a search tree. Binary search trees have rules about where elements get added, and they will be explored in MP5.
Subtrees of a tree are also considered trees. This property is essential for understanding recursive operations on trees.
Recursion occurs when a problem is defined in terms of itself or its type. In computer science, recursion is a method where the solution to a problem depends on solutions to smaller instances of the same problem.
Recursion vs. Iteration:
- Iteration: repeating the same set of steps over and over again.
- Recursion: breaking a larger problem into smaller problems until they are small enough to solve easily.
For example, consider counting the number of nodes in a tree. This can be done iteratively by visiting every node and incrementing a counter, or recursively by breaking the problem into smaller subproblems and combining the results.
Recursive Node Counting:
1. Break the problem into smaller subproblems.
2. Solve the smallest subproblem.
3. Combine the results.
A recursive function is a function that calls itself. An example of a recursive function is the factorial function:
```java
int factorial(int n) {
if (n == 1) {
return 1;
} else {
return n * factorial(n - 1);
}
}
```
To effectively use recursion, follow these strategies:
1. Know when to stop (base case): return when the smallest subproblem is identified.
2. Make the problem smaller in each step (recursive step): ensure the problem gets smaller to reach the base case.
3. Combine results from recursive calls properly.
The factorial function demonstrates these strategies:
- Base case: n == 1
- Recursive step: decrement n towards 1
- Combine results: multiply current n with the result of the next subproblem
It's essential to reach the base case; otherwise, the problem will not terminate, and the program may crash. The code can fail to reach the base case if it does not properly decrement towards the base case or if the base case is not correctly defined.
When deciding between recursion and iteration, consider clarity as the primary goal. Use the technique that makes the code more understandable, whether it's recursive or iterative. Avoid using recursion solely for brevity or to appear complex.
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CC-MAIN-2020-05/segments/1579250607407.48/warc/CC-MAIN-20200122191620-20200122220620-00021.warc.gz
|
illinois.edu
|
en
| 0.815244
| 2020-01-22T20:44:53
|
https://cs125.cs.illinois.edu/learn/2018_03_12_introduction_to_recursion/
| 0.985443
|
Term 2 Reflection
In Term 2, we covered three units: percents, surface area, and volume. I performed well in surface area and volume, as they involved applying basic arithmetic operations using various formulas. However, I struggled to remember the numerous formulas. A key area for improvement is participating more in class discussions, as consistently noted by my teachers.
The percents unit taught me several concepts, including converting fractions to decimals to percents in any order. I learned that percents are out of 100 and can be represented on 100 grids, with fraction percents shaded on a single square. Mental math and ratio tables can be used to find percents in numbers, and percents can be combined by adding to solve problems.
In the surface area unit, I learned about formulas for finding the surface area of different shapes. For rectangular prisms, the formula is length x width, and they have six faces with equal opposite sides. Triangular prisms have a formula of base x height/2 + length x width, with five faces, two triangles, and three rectangles. Cylinders have multiple formulas, including finding diameter, radius, or circumference, with the easiest formula being (2 π x radius x radius) + (2 x radius x π x height) = Total Surface Area.
The volume unit also introduced various formulas. When given the base and height, the formula is Area of base x height. Without the base, there are specific formulas for each object: v = length x width x height for rectangular prisms, v = base x height (triangle) / 2 x height (prism) for triangular prisms, v = side x side x side for cubes, and v = π x radius x radius x height for cylinders. The key difference between volume and surface area is that volume measures the interior space, while surface area measures the exterior space.
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CC-MAIN-2017-51/segments/1512948521292.23/warc/CC-MAIN-20171213045921-20171213065921-00733.warc.gz
|
blogspot.com
|
en
| 0.732015
| 2017-12-13T05:25:36
|
http://spmath84110.blogspot.com/2011/03/glenesses-term-2-reflection.html
| 0.99746
|
# Kosmin Test Calculator
## Introduction
The Kosmin test is a predictor for middle-distance runners, estimating 800m or 1500m race times using repeated 60-second runs. It is best completed on a track with a timekeeper.
## Test Procedure
The test procedure differs slightly for 800m and 1500m tests.
### 800m Test
The athlete runs flat-out for 60 seconds, and the distance covered is measured. After a 3-minute recovery, the athlete completes another 60-second effort, and the distance is measured. The total distance is used to estimate the 800m race time.
### 1500m Test
The athlete runs flat-out for 60 seconds, and the distance is measured. After a 3-minute recovery, the athlete runs for another 60 seconds, followed by a 2-minute recovery. This is repeated for a total of four 60-second efforts, with recoveries of 3 minutes, 2 minutes, 1 minute, and no recovery after the final effort. The total distance is used to estimate the 1500m race time.
## Benefits
The Kosmin test is specific to the desired distance, making it more likely to predict a realistic race time.
## Using the Calculator
To use the calculator, choose the test distance and sex, enter the total distance covered, and calculate.
## Kosmin Test Calculations
Different formulas are used for 800m and 1500m tests, and for men and women.
| Test | Formula |
| --- | --- |
| 800m Men | 217.77778 - (Total Distance × 0.119556) |
| 800m Women | 1451.46 - (198.54 × Log(Total Distance)) |
| 1500m Men | 500.52609 - (Total Distance × 0.162174) |
| 1500m Women | 500.52609 - (Total Distance × 0.162174) + 10 |
|
CC-MAIN-2024-38/segments/1725700651072.23/warc/CC-MAIN-20240909040201-20240909070201-00829.warc.gz
|
runbundle.com
|
en
| 0.880726
| 2024-09-09T05:08:40
|
https://runbundle.com/tools/race-predictors/kosmin-test
| 0.796136
|
The problem asks to find the number of ordered pairs \(a, b\) of integers such that \(\frac{a + 2}{a + 5} = \frac{b}{4}.\)
To solve this, multiply both sides of the equation by \(a + 5\):
\[a + 2 = b \cdot \frac{a + 5}{4}\]
Then, multiply both sides by 4:
\[4(a + 2) = b(a + 5)\]
Expanding the left side gives:
\[4a + 8 = b(a + 5)\]
This equation implies that \(4a + 8\) is a multiple of \(b\), and \(a + 5\) is a divisor of \(4a + 8\). For integer solutions, \(b\) must be a divisor of 8, which are 1, 2, 4, and 8.
Checking divisibility for each possible value of \(b\):
- If \(b = 1\), \(a + 5\) must divide \(4a + 8\). The only integer solution is \(a = -5\).
- If \(b = 2\), there are no integer solutions because the left side is always even and the right side is always odd.
- If \(b = 4\), \(a + 5\) must divide \(4a + 8\). The only integer solution is \(a = -3\).
- If \(b = 8\), \(a + 5\) must divide \(4a + 8\). The only integer solution is \(a = -1\).
The possible integer solutions for \((a, b)\) are:
- \((-5, 1)\)
- \((-3, 4)\)
- \((-1, 8)\)
There are a total of 3 such ordered pairs.
|
CC-MAIN-2024-38/segments/1725700651344.44/warc/CC-MAIN-20240911052223-20240911082223-00435.warc.gz
|
0calc.com
|
en
| 0.74289
| 2024-09-11T05:50:17
|
https://web2.0calc.com/questions/pls-help-asap_82
| 0.999168
|
### Video Transcript
To solve the simultaneous equations \(x - y = 6\) and \(x^2 - 9xy + y^2 = 36\), we will use the substitution method. This involves rearranging one of the equations to make either \(x\) or \(y\) the subject and then substituting this back into the other equation.
We start with the first equation, \(x - y = 6\), and rearrange it to make \(x\) the subject: \(x = 6 + y\). This is our third equation.
Next, we substitute \(x = 6 + y\) into the second equation, \(x^2 - 9xy + y^2 = 36\), to get \((6 + y)^2 - 9(6 + y)y + y^2 = 36\). Expanding \((6 + y)^2\) gives \(36 + 12y + y^2\).
Substituting back into the equation yields \(36 + 12y + y^2 - 54y + y^2 = 36\). Simplifying, we collect like terms: \(36 + y^2 + 12y - 54y + y^2 = 36\), which simplifies further to \(y^2 - 42y + 36 = 0\) after combining like terms and subtracting 36 from both sides.
However, the correct simplification after expansion should be: \(36 + 12y + y^2 - 54y - 9y^2 + y^2 = 36\), leading to \(-7y^2 - 42y + 36 = 36\). Subtracting 36 from both sides gives \(-7y^2 - 42y = 0\).
Factoring out \(-7y\) from \(-7y^2 - 42y = 0\) gives \(-7y(y + 6) = 0\). Thus, \(y = 0\) or \(y = -6\).
To find \(x\), we substitute \(y = 0\) and \(y = -6\) into \(x = 6 + y\). For \(y = 0\), \(x = 6 + 0 = 6\). For \(y = -6\), \(x = 6 - 6 = 0\).
Therefore, the solutions to the simultaneous equations are \(x = 6\) when \(y = 0\) and \(x = 0\) when \(y = -6\).
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CC-MAIN-2021-25/segments/1623487634616.65/warc/CC-MAIN-20210618013013-20210618043013-00337.warc.gz
|
nagwa.com
|
en
| 0.877406
| 2021-06-18T03:34:20
|
https://www.nagwa.com/en/videos/305140503292/
| 0.997517
|
### Elasticities of Demand
Elasticity measures the responsiveness of the quantity demanded of a product to changes in any of the factors that affect demand. The price elasticity of demand is the percentage change in the quantity of a product demanded divided by the percentage change in the price causing the change in quantity. It indicates the degree of consumer response to variation in price.
The change in price is expressed as a percentage of the average price, and the change in the quantity demanded is expressed as a percentage of the average quantity demanded. This measure is units-free because it is a ratio of two percentage changes, and the percentages cancel each other out. Since a change in price causes the quantity demanded to change in the opposite direction, this ratio is always negative, although economists always ignore the sign and simply use the absolute value.
#### Example 1
A Pizza Hut store can sell 50 pizzas per day at $7 each or 70 pizzas per day at $6 each. The price elasticity is calculated as: [(50 - 70)/60] / [(7 - 6) / 6.5] = -2.17.
#### Types of Demand
Demand can be inelastic, unit elastic, or elastic, and can range from zero to infinity.
- If the elasticity coefficient is greater than 1, demand is **elastic**. A small price change leads to a large change in the quantity demanded.
- When the elasticity coefficient is less than 1, demand is **inelastic**. The more inelastic the demand, the steeper the demand curve.
- When the elasticity coefficient is equal to 1, demand is said to be **unitary elastic**.
#### Example 2
Refer to the graph below. Which of the following is true?
A. Areas C and E are smaller than area A, so demand must be elastic between $10 and $30.
B. Areas C and E are smaller than area A, so demand must be inelastic between $10 and $30.
C. Area F is smaller than areas B and C, so demand must be inelastic between $10 and $30.
Answer: C. Since at $30 the demand is unit elastic, at prices below $30 demand is inelastic. This is because when price rises from $10 to $30, the revenue gained is greater than the revenue lost.
#### Factors that Influence the Elasticity of Demand
The elasticity of demand among products varies substantially. The determinants of price and income elasticity of demand are:
1. **The closeness of substitutes**: The most important determinant is the availability of substitutes. The closer the substitutes for a good or service, the more elastic the demand for it.
2. **The proportion of income spent on the good**: If expenditures on a product are quite small relative to a consumer's budget, the income effect will be small even if there is a substantial increase in the price of the product.
3. **The time elapsed since a price change**: The more time consumers have to adjust to a price change, or the longer a good can be stored without losing its value, the more elastic the demand for that good.
#### Impact on Total Expenditure
Consumers' total expenditure is the same as total revenues from the suppliers' point of view. One of the most important applications of price elasticity is determining how total consumer expenditure on a product changes when the price changes.
- When demand is inelastic, a change in price will cause total expenditures to change in the same direction.
- When demand is elastic, a change in price will cause total expenditures to move in the opposite direction.
- When demand elasticity is unitary, total expenditures will remain unchanged as price changes.
#### Income Elasticity of Demand
The percentage change in the quantity of a product demanded divided by the percentage change in consumer income causing the change in quantity demanded.
- **Normal goods** have positive income elasticity; necessities have low income elasticities (between 0 and 1); luxuries have high income elasticities (greater than 1).
- **Inferior goods** have negative income elasticity; as income expands, the demand for them will decline.
#### Cross-Price Elasticity of Demand
The cross elasticity of demand is a measure of the responsiveness of demand for a good to a change in the price of a substitute or a complement, other factors remaining the same.
- The cross elasticity of demand for a substitute is positive.
- The cross elasticity of demand for a complement is negative.
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|
analystnotes.com
|
en
| 0.905039
| 2023-03-21T18:14:32
|
https://analystnotes.com/cfa-study-notes-calculate-and-interpret-price-income-and-cross-price-elasticities-of-demand-and-describe-factors-that-affect-each-measure.html
| 0.551996
|
Predictive models have become essential for businesses, enabling them to "foresee the future" and make informed decisions. These models can be categorized into regression models (continuous output) and classification models (nominal or binary output). Classification models use algorithms that produce either class outputs or probability outputs.
Algorithms like SVM and KNN create class outputs, where the output is either 0 or 1. In contrast, algorithms like Logistic Regression, Random Forest, and Gradient Boosting produce probability outputs, which can be converted to class outputs by setting a threshold probability.
Evaluating the performance of a machine learning model is crucial to determine its effectiveness on unseen data. This is achieved by using various metrics, including accuracy, precision, recall, F1 score, specificity, and Receiver Operating Characteristics (ROC) curve.
**Model Evaluation Metrics**
1. **Accuracy**: Measures the overall correctness of the model, calculated as (TP+TN)/total.
2. **Precision**: Measures the correctness of positive predictions, calculated as TP/predicted yes.
3. **Recall or Sensitivity**: Measures the ability of the model to detect positive instances, calculated as TP/actual yes.
4. **F1 Score**: The harmonic mean of precision and recall, ranging from 0 to 1, where 1 is perfect precision and recall.
5. **Specificity**: Measures the ability of the model to detect negative instances, calculated as TN/actual no.
6. **ROC Curve**: Plots the true positive rate against the false positive rate, providing a visual representation of the model's performance.
**Additional Metrics**
1. **Log Loss**: Measures the performance of a classification model based on probabilities, with smaller values indicating better performance.
2. **Jaccard Index**: Measures the similarity between two sets, calculated as the size of the intersection divided by the size of the union.
3. **Kolmogorov-Smirnov Chart**: Measures the degree of separation between positive and negative distributions, ranging from 0 to 100.
4. **Gain and Lift Chart**: Evaluates the performance of a classification model by calculating the ratio of results obtained with and without the model.
5. **Gini Coefficient**: Measures the dispersion of data, ranging from 0 to 1, where 0 represents perfect equality and 1 represents perfect inequality.
Understanding these evaluation metrics is essential to assess the performance of machine learning models, especially in cases where the data is imbalanced. By using these metrics, businesses can make informed decisions and improve the overall predictive power of their models.
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CC-MAIN-2023-14/segments/1679296943809.22/warc/CC-MAIN-20230322082826-20230322112826-00188.warc.gz
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datasource.ai
|
en
| 0.842001
| 2023-03-22T09:57:12
|
https://www.datasource.ai/en/data-science-articles/model-evaluation-metrics-in-machine-learning
| 0.839596
|
To represent the decimal 0.375 in place value digits, we write it in its expanded form. The integer part of a decimal is represented by place values such as Ones, Tens, Hundreds, Thousands, Ten Thousands, Hundred Thousands, Millions, and so on. The fractional part of a decimal is represented by place values like one-tenths, one-hundredths, one-thousandths, and so on.
0.375 can be expanded as 3 hundredths and 7 thousandths and 5 ten-thousandths, which equals (3/100) + (7/1000) + (5/10000). This representation helps in understanding the place value of each digit in the decimal number.
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CC-MAIN-2020-05/segments/1579251700988.64/warc/CC-MAIN-20200127143516-20200127173516-00267.warc.gz
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mathsai.com
|
en
| 0.759802
| 2020-01-27T15:27:46
|
https://www.mathsai.com/tools/decimal-number-place-value-calculator.htm
| 0.994253
|
The perimeter of a shape is the sum of all its sides. For a parallelogram, the perimeter can be calculated as P = 2(AB + BC) since opposite sides are congruent. The surface area of a parallelogram is given by S = base x height.
A rectangle has a length (L = AB) and a width (l = BC), and its perimeter can be calculated as P = 2(L + l) or P = AB + BC + CD + DA. The surface area of a rectangle is given by S = length x width.
A square is a special type of rectangle where all sides are equal, and its perimeter can be calculated as P = 4L, where L is the length of a side. The surface area of a square is given by S = AB^2.
For a trapezium, the area can be calculated as A = (Big Base + Small Base) x Height / 2.
A circle has a circumference given by its length, and its area can be calculated using the radius (r).
Regular solids include the regular pyramid, where the lateral area is given by A_lat = (P_b x a_p) / 2.
The volume of a cuboid is given by V = L x l x h, and a cube is a special type of cuboid where all sides are congruent.
For a cylinder, the base area is given by the area of the base circle, and the lateral area, total area, and volume can be calculated using the base radius (r), height (h), and generator edge (g).
The cone has a base area given by the surface of the base circle, and the lateral area, total area, and volume can be calculated using the base radius (r), height (h), and generator (g).
The frustoconic shape has a lateral area, total area, and volume that can be calculated using the big base radius (R), small base radius (r), height (h), and generator (G).
A sphere has an area given by its radius (r), and its volume can be calculated using the radius.
Key formulas:
- Perimeter of a parallelogram: P = 2(AB + BC)
- Surface area of a parallelogram: S = base x height
- Perimeter of a rectangle: P = 2(L + l)
- Surface area of a rectangle: S = length x width
- Perimeter of a square: P = 4L
- Surface area of a square: S = AB^2
- Area of a trapezium: A = (Big Base + Small Base) x Height / 2
- Lateral area of a regular pyramid: A_lat = (P_b x a_p) / 2
- Volume of a cuboid: V = L x l x h
- Volume of a cylinder: V = πr^2h
- Volume of a cone: V = (1/3)πr^2h
- Volume of a sphere: V = (4/3)πr^3
Note: The formulas provided are a selection of key concepts and are not an exhaustive list of all formulas related to the shapes discussed.
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CC-MAIN-2021-25/segments/1623488560777.97/warc/CC-MAIN-20210624233218-20210625023218-00481.warc.gz
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mathematicshelp.org
|
en
| 0.688707
| 2021-06-24T23:37:52
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https://www.mathematicshelp.org/maths/posts/content/35
| 0.994529
|
# Fermi Calculation of Populations and Housing
To determine if it's possible to have a home for every person in the world by the middle of the century, a Fermi calculation can be used. This method, named after Enrico Fermi, involves using rough numbers to estimate the feasibility of a project.
Assuming a total workforce of 20,000,000 workers, which is approximately 1/400 of the world's current population, and dividing them into 200,000 teams of 100 people each, a large-scale construction project can be planned. Each team can build a 10-story building with 20 apartments per story, accommodating 3 people per apartment, resulting in 600 people per building.
With 16,000,000 buildings, each taking up approximately 140 ft by 140 ft of land, the total land required can be calculated. Each building occupies around 20,000 square feet, and multiplying this by 16,000,000 buildings gives a total of 300,000,000 square feet. Since one square mile is approximately 25,000,000 square feet, the total land required is around 12 square miles. This can be spread across the entire world, making each individual section relatively small.
To estimate the cost, a rough calculation can be made. Assuming a certain price per square foot of living space, the cost of one floor can be estimated, then multiplied by 10 to get the cost of all 10 stories, and finally multiplied by 16 million. If each square foot costs $1, the total cost would be approximately $3,000,000,000,000. Although this seems like a significant expense, it's a global-scale construction project, and the cost per person housed would be around $3,000.
Given that each team can build one building in approximately 3 months, in 20 years, one team can build around 80 buildings. With 200,000 teams working simultaneously, all 16,000,000 buildings can be constructed in 20 years. This would provide housing for approximately 10,000,000,000 people, the estimated world population by 2050. If the project starts in 2020, it would be completed by 2040, and the buildings would be full 10 years later.
The key to this plan is that all teams work simultaneously, making it possible to complete the project within the given timeframe. While the cost is a significant factor, the calculation demonstrates that providing housing for every person in the world is theoretically possible, even with rough estimates.
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CC-MAIN-2021-25/segments/1623487655418.58/warc/CC-MAIN-20210620024206-20210620054206-00326.warc.gz
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michaelcurzan.com
|
en
| 0.964195
| 2021-06-20T03:31:07
|
https://www.michaelcurzan.com/post/fermi-calculation-of-populations-and-housing
| 0.675159
|
**How Many Liters Of Propane In A 100 Lb Tank**
A 100 lb propane tank is a common size, but the amount of usable propane it can hold varies due to several factors. One gallon of propane weighs approximately 4 lbs. A standard 100 lb propane tank holds around 23-25 gallons of propane and weighs 170 lbs when full.
The density of propane is 493 gm/ltr at 25°C. Using this density, we can calculate that 1 pound of propane is equivalent to approximately 0.92 liters in volume at 25°C. Therefore, a 100 lb propane tank can hold around 92 liters of propane (100 lbs x 0.92 liters/lb).
The tank's dimensions are typically 18” tall and 12” in diameter, but can vary. The pressure inside a propane tank fluctuates slightly based on the outside temperature, which can affect the amount of propane it can hold.
To give you a better idea, a 100 lb propane tank can last around 7 days and 4 hours when powering a 10,000 btu/h appliance at a temperature of 60 degrees Fahrenheit. The cost to fill a 100 lb propane tank can vary, but on average, it can hold around 23.6 gallons of propane.
It's worth noting that propane tanks come in different sizes, including 20 lb and 100 lb tanks. A 20 lb propane tank is typically 4 feet tall and 18” in diameter. When transporting a propane tank in a car, it's essential to follow safety guidelines and ask your propane retailer if a plug is required.
In summary, a 100 lb propane tank can hold around 92 liters of propane, depending on the temperature and other factors. The tank's size, weight, and btu capacity can vary, but it's a popular choice for heating homes and powering appliances.
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delectablyfree.com
|
en
| 0.888337
| 2023-03-31T12:12:48
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https://delectablyfree.com/blog/how-many-liters-of-propane-in-a-100-lb-tank/
| 0.760039
|
### Introduction to Bungee Jump Simulation
#### Goals and Prerequisites
The goal of this activity is to simulate a bungee jump using a Barbie doll and rubber bands, collecting data to construct a scatterplot, and generating a line of best fit to predict the number of rubber bands needed for a safe jump from a given distance. Prerequisites include understanding uncertainty, functional relationships, and patterns.
#### Background
Students work in groups to formulate a conjecture, design a method, and use data to develop concepts of estimation, slope, and line of best fit. The simulation involves a Barbie doll attached to rubber bands, with the objective of giving Barbie the greatest thrill while ensuring her safety.
#### Key Concepts and Questions
- **Uncertainty**: Variation in data due to repeated measures.
- **Functional Relationships**: Describing how one variable depends on another.
- **Patterns**: Processes of change that repeat in predictable ways.
- **Student Questions**:
1. How many rubber bands are needed for Barbie to safely jump from a height of 400 cm?
2. What is the minimum height from which Barbie should jump if 25 rubber bands are used?
3. How might the type and width of the rubber band affect the results?
4. If weight is added to Barbie, would more or fewer rubber bands be needed to achieve the same results?
#### Materials
- **Student Materials**: Bungee Barbie Activity Packet, Bungee Barbie Spreadsheet, doll, rubber bands, measuring tape, cash register tape, TI 84+ calculator.
- **Teacher Materials**: Bungee Barbie Project Rubric, Calculating a Spring Constant Extension Activity, laptop, projector, model set.
#### Procedure
1. **Introduction**: Introduce the concept by asking about the importance of accurate height and weight estimates for safe bungee jumping.
2. **Setup**: Distribute materials and demonstrate how to create a double-loop for attaching rubber bands to Barbie and how to add rubber bands using a slip knot.
3. **Experiment**: Have students conduct the experiment, recording data in the provided table. Ensure all rubber bands are the same size and thickness.
4. **Data Analysis**: After completing the experiment, have students check their data for irregularities and re-do the experiment if necessary. Then, ask them to create a graph of the data using the Illuminations Line of Best Fit activity or the Bungee Barbie Spreadsheet.
5. **Testing Conjectures**: Take students to a location where Barbie can be dropped from a significant height to test their conjectures about the maximum number of centimeters for a safe jump.
#### Assessment
The attached Bungee Barbie Project Rubric can be used to evaluate student work. It is recommended to share this rubric with students before completing the lesson so they are aware of the evaluation criteria.
#### Additional Resources
For further learning, consider showing short videos about bungee jumping available on websites like Bungee TV, Bungee Jump Preview, and Land Diving Ritual, noting that some content may not be suitable for all classrooms.
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teacherstryscience.org
|
en
| 0.892999
| 2017-12-13T03:31:24
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http://www.teacherstryscience.org/lp/bungee-barbie%C2%AE
| 0.600626
|
A quadrilateral is a polygon with four sides. There are seven quadrilaterals, some that are surely familiar to you, and some that may not be so familiar. Check out the following definitions and the quadrilateral family tree in the following figure. If you know what the quadrilaterals look like, their definitions should make sense and be pretty easy to understand though the kite definition is a bit of a mouthful. Here are the seven quadrilaterals:
Parallelogram: A quadrilateral that has two pairs of parallel sides.
Rhombus: A quadrilateral with four congruent sides; a rhombus is both a kite and a parallelogram.
Rectangle: A quadrilateral with four right angles; a rectangle is a type of parallelogram.
Square: A quadrilateral with four congruent sides and four right angles; a square is both a rhombus and a rectangle.
Trapezoid: A quadrilateral with exactly one pair of parallel sides the parallel sides are called bases.
Isosceles trapezoid: A trapezoid in which the nonparallel sides the legs are congruent.
In the hierarchy of quadrilaterals shown in the following figure, a quadrilateral below another in the family tree is a special case of the one above it.
A quadrilateral is a 4-sided polygon bounded by 4 finite line segments. A quadrilateral has 2 diagonals based on which it can be classified into concave or convex quadrilateral. In case of convex quadrilaterals, diagonals always lie inside the boundary of the polygon.
Squares are the most regular quadrilateral and have the most properties. A square is also a rectangle, parallelogram, rhombus, kite, and trapezoid. A rectangle is a quadrilateral with congruent angles. A rectangle is also a parallelogram and a trapezoid.
Learn the application of angle properties of quadrilaterals; figure out the measures of the indicated angles, also solve for 'x' to determine the angles of special quadrilaterals to mention just a few. Sample Worksheets. Quadrilateral Charts. Perimeter of Quadrilaterals. Area of Quadrilaterals.
In this lesson you will learn to identify specific kinds of quadrilaterals by looking at their attributes. Create your free account Teacher Student. Create a new teacher account for LearnZillion. All fields are required. Name. Email address. Email confirmation. Password.
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CC-MAIN-2021-25/segments/1623487643380.40/warc/CC-MAIN-20210619020602-20210619050602-00456.warc.gz
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cat-research.com
|
en
| 0.876363
| 2021-06-19T04:01:10
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http://cat-research.com/desmar-properties-of-quadrilaterals.php
| 0.994724
|
While working on our 2013 climate meta-analysis, we came across an article by Ole Thiesen at PRIO, where he investigated the relationship between local climatic events and conflict in Kenya. Thiesen's model estimated the effect of temperature and rainfall on conflict, but reported finding no significant effect of either variable. Upon reviewing the replication code, we noticed that the squared terms for temperature and rainfall were offset by a constant, which was not apparent in the linear terms.
This offset was problematic because, in non-linear models, adding a constant incorrectly can be dangerous. We realized that the squared term for temperature, when expanded, introduced a constant term that affected the regression coefficients. The actual regression model being run was different from the intended model, which meant that directly interpreting the coefficients was not accurate.
Specifically, if a constant is added prior to squaring a variable, the coefficient for that term is unaffected, but all other coefficients in the model are altered. This subtlety is not immediately apparent and can lead to incorrect conclusions. To test this theory, we re-estimated the model using the correct squared measures and found that the squared coefficients remained unchanged, but the linear effects did.
This correction had significant implications, as the original analysis had suggested that the linear effect of temperature was insignificant, contradicting earlier findings. However, after removing the offending constant term, a large positive and significant linear effect of temperature was revealed, consistent with previous research. The correct linear combination of coefficients from the original regression also showed a significant marginal effect of temperature.
The error was not obvious and may be a common mistake in non-linear models, particularly when estimating interaction effects. Thiesen's construction of the data set was an important contribution, and he graciously acknowledged the mistake when we brought it to his attention. Unfortunately, our comment was not widely seen, as the journal that published the original article did not accept research notes or commentaries.
The key takeaway is that adding constants incorrectly in non-linear models can have significant consequences, and it is essential to be mindful of this potential pitfall to avoid drawing incorrect conclusions.
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CC-MAIN-2023-14/segments/1679296948951.4/warc/CC-MAIN-20230329054547-20230329084547-00529.warc.gz
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g-feed.com
|
en
| 0.917096
| 2023-03-29T07:19:42
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http://www.g-feed.com/2015/12/
| 0.939614
|
A decibel (dB) represents the ratio of two variables on a logarithmic scale. Using a logarithmic scale is a better approximation of human hearing than linear variables. The gigantic ratio of barely perceptible sound pressure to the loudest tolerable sound pressure is compressed into a manageable scale of 0 to 130 dB. The general calculation is: log (value/reference value), using the logarithm to base 10.
The result is the Bel, one-tenth of which is one deci-bel, i.e., a decibel. These are power ratios. For sound pressures, voltages, and currents, the factor is 20. The formulas for calculating decibels are:
- Power ratio in dB: 10 x log10 (power/reference power)
- Sound pressure, voltage, or current ratios in dB: 20 x log10 (value/reference value)
In the case of sound pressure ratios, the auditory threshold is used, having a value of 20 μPa. The unit 'dB' is often appended with 'SPL' to indicate sound pressure level, although 'SPL' is commonly omitted.
Reference values and their corresponding decibel units are:
- 1 μV: dB μV
- 1 mV: dB mV
- 0.775 V: dBu
- 1 V: dBV
- 20 μPa: dB SPL
The relationship between physical values and decibel values is as follows:
- Multiplication of physical values corresponds to addition of decibel values
- Division of physical values corresponds to subtraction of decibel values
- A physical value less than 1 corresponds to a negative decibel value
- A physical value of 1 corresponds to a decibel value of 0
- A physical value greater than 1 corresponds to a positive decibel value
Examples:
- An amplifier with a 1000-fold gain has a decibel value of 20 x log (1000/1) = +60 dB
- An attenuator that reduces a voltage to one-tenth has a decibel value of 20 x log (0.1/1) = -20 dB
- Connecting the attenuator to the amplifier results in a decibel value of 60 dB + (-20 dB) = 40 dB
The sound pressure level of a loudspeaker can be calculated using the formula: p1 = pn + 10 x log (P), where p1 is the sound pressure level, pn is the characteristic sound pressure level, and P is the supplied power. Each doubling of power results in an additional 3 dB of SPL.
To calculate the sound pressure level at a distance other than 1 meter, the formula is: p = p1 - 20 x log (d), where p is the sound pressure level at the defined distance, p1 is the sound pressure level at 1 meter, and d is the distance. Each doubling of distance results in a 6 dB decrease in SPL.
The combined formula for sound pressure level at a given power and distance is: p = pn + 10 x log (P) - 20 x log (d). For example, a loudspeaker with a characteristic sound pressure level of 90 dB at 1 W/1 m and an input power of 30 watts at a distance of 8 meters has a sound pressure level of:
- 90 dB + 10 x log (30) - 20 x log (8) = 90 dB + 15 dB - 18 dB = 87 dB
Alternatively, using the tables provided: 90 dB + 15 dB (at 30 watts) - 12 dB (at 4 m) - 6 dB (at 2 m) = 87 dB.
A perceived doubling in volume requires around 10 times the amplifier power. The distance and minimum sound pressure level between standard ceiling loudspeakers at different degrees of speech intelligibility and 6 W of power are:
- Best intelligibility: distance between loudspeakers ranges from 2.3 to 6.9 meters, with minimum sound pressure levels ranging from 92 to 83 dB
- Good intelligibility: distance between loudspeakers ranges from 3.6 to 10.7 meters, with minimum sound pressure levels ranging from 90 to 81 dB
- Background music: distance between loudspeakers ranges from 8.2 to 24.7 meters, with minimum sound pressure levels ranging from 85 to 75 dB.
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CC-MAIN-2023-14/segments/1679296946445.46/warc/CC-MAIN-20230326173112-20230326203112-00086.warc.gz
|
toa.eu
|
en
| 0.707861
| 2023-03-26T18:36:53
|
https://www.toa.eu/service/soundcheck/calculations-with-loudspeakers
| 0.890922
|
An identity is an equation that is true for all values of the variable. A fundamental example is the Pythagorean identity: sin^{2}x + cos^{2}x = 1. This identity is derived using Pythagoras' Theorem and serves as the basis for other Pythagorean identities.
Dividing each term in sin^{2}x + cos^{2}x = 1 by sin^{2}x yields the identity: 1 + cot^{2}x = cosec^{2}x. Similarly, dividing each term by cos^{2}x gives: cot^{2}x + 1 = sec^{2}x. These identities are essential and should be memorized for exams.
The Pythagorean identities, along with other angle formulas, are crucial for solving equations and more complex identities. Proving identities involves manipulating one side of the equation to match the other. It is essential to present these proofs clearly, often using the abbreviations RHS (right-hand side) and LHS (left-hand side) to denote the sides of the identity being proven.
To prove an identity, start with one side and apply substitutions and manipulations until it equals the other side. The goal is to find the shortest proof possible. Some identities may require the use of angle formulas from other units, such as Unit 43, Angle Formulae.
Examples of identities to be proven include:
- cos x cosec x tan x = 1
- sin 3A = 3sin A - 4sin^{3}A
When proving identities, it is crucial to be systematic and methodical, as there is often more than one way to prove an identity. The key is to find the most efficient and straightforward proof.
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CC-MAIN-2020-05/segments/1579250628549.43/warc/CC-MAIN-20200125011232-20200125040232-00271.warc.gz
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bestmaths.net
|
en
| 0.819768
| 2020-01-25T01:17:48
|
http://bestmaths.net/online/index.php/year-levels/year-12/year-12-topic-list/identities/
| 0.999718
|
The Eigenvalue is a scalar amount related to a direct change having a place with a vector space. In this article, we give an insight into how to find Eigenvalues of a matrix and determine the steps regarding doing so. The foundations of the direct condition lattice framework are known as eigenvalues, which are comparable to the cycle of lattice diagonalization.
### Eigenvalues and Eigenvectors
The eigenvalue issue is given by the condition A·v = λ·v, where An is an n-by-n grid, v is a non-zero n-by-1 vector, and λ is a scalar. Any estimation of λ for which this condition has an answer is known as an eigenvalue of the grid A. The vector, v, which compares to this worth, is called an eigenvector. The eigenvalue issue can be changed as (A-λ·I)·v=0, which will possibly have an answer if |A-λ·I|=0. This condition is known as the trademark condition of A and is an nth request polynomial in λ with n roots, which are the eigenvalues of A.
### Steps on How to Find Eigenvalues of a Matrix
To find eigenvalues of a matrix, follow these steps:
1. Ensure the given framework A is a square lattice and decide the character grid I of a similar request.
2. Estimate the network A –λI, where λ is a scalar amount.
3. Find the determinant of network A –λI and compare it to zero.
4. From the condition accordingly, ascertain all the potential estimations of λ, which are the necessary eigenvalues of grid A.
### Examples on How to Find EigenValues of a Matrix
Given a 2x2 matrix A = \[\begin{bmatrix} 0 & 1\\ -2& -3 \end{bmatrix}\], the equation for solving is | A - λ . I | = 0. This yields λ2 + 3λ + 2 = 0, and the two eigenvalues are λ1=-1, λ2=-2.
### Properties of Eigenvalues
Key properties of eigenvalues include:
- The trace of A is the sum of its eigenvalues.
- The determinant of A is the product of its eigenvalues.
- If A is Hermitian, then each eigenvalue is real.
- If A is positive-definite, then each eigenvalue is positive.
- System A is invertible if and only if each eigenvalue is non-zero.
### FAQs on How to Determine The Eigenvalues of a Matrix
1. What are Eigenvalues?
Eigenvalues are scalar amounts related to a straight arrangement of conditions which when duplicated by a non-zero vector equivalent to the vector got by change working on the vector, spoke to as AV = λV.
2. How Can We Determine the Eigenvalues of a Matrix?
To determine the eigenvalues of a matrix, we use the condition ∣A–λI∣ = 0, which empowers us to figure eigenvalues λ without any problem. Given a square network A, the condition that describes an eigenvalue, λ, is the presence of a non-zero vector x with the end goal that A x = λ x. The determinant of the coefficient network—which for this situation is A − λ I—must be zero for non-zero arrangements. The resulting expression is a monic polynomial in λ, known as the trademark polynomial of A, and its zeros are the eigenvalues of A.
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CC-MAIN-2023-14/segments/1679296948620.60/warc/CC-MAIN-20230327092225-20230327122225-00631.warc.gz
|
vedantu.com
|
en
| 0.812997
| 2023-03-27T11:51:08
|
https://www.vedantu.com/iit-jee/how-to-determine-the-eigenvalues-of-a-matrix
| 0.997676
|
5995 * 0 = 0
5995 * 0.1 = 59.95
5995 * 0.2 = 119.9
5995 * 0.3 = 179.85
5995 * 0.4 = 239.8
5995 * 0.5 = 299.75
5995 * 0.6 = 359.7
5995 * 0.7 = 419.65
5995 * 0.8 = 479.6
5995 * 0.9 = 539.55
5995 * 0.10 = 59.95
5995 * 0.11 = 65.945
5995 * 0.12 = 71.94
5995 * 0.13 = 77.935
5995 * 0.14 = 83.93
5995 * 0.15 = 89.925
5995 * 0.16 = 95.92
5995 * 0.17 = 101.915
5995 * 0.18 = 107.91
5995 * 0.19 = 113.905
5995 * 0.20 = 119.9
5995 * 0.21 = 125.895
5995 * 0.22 = 131.89
5995 * 0.23 = 137.885
5995 * 0.24 = 143.88
5995 * 0.25 = 149.875
5995 * 0.26 = 155.87
5995 * 0.27 = 161.865
5995 * 0.28 = 167.86
5995 * 0.29 = 173.855
5995 * 0.30 = 179.85
5995 * 0.31 = 185.845
5995 * 0.32 = 191.84
5995 * 0.33 = 197.835
5995 * 0.34 = 203.83
5995 * 0.35 = 209.825
5995 * 0.36 = 215.82
5995 * 0.37 = 221.815
5995 * 0.38 = 227.81
5995 * 0.39 = 233.805
5995 * 0.40 = 239.8
5995 * 0.41 = 245.795
5995 * 0.42 = 251.79
5995 * 0.43 = 257.785
5995 * 0.44 = 263.78
5995 * 0.45 = 269.775
5995 * 0.46 = 275.77
5995 * 0.47 = 281.765
5995 * 0.48 = 287.76
5995 * 0.49 = 293.755
5995 * 0.50 = 299.75
5995 * 0.51 = 305.745
5995 * 0.52 = 311.74
5995 * 0.53 = 317.735
5995 * 0.54 = 323.73
5995 * 0.55 = 329.725
5995 * 0.56 = 335.72
5995 * 0.57 = 341.715
5995 * 0.58 = 347.71
5995 * 0.59 = 353.705
5995 * 0.60 = 359.7
5995 * 0.61 = 365.695
5995 * 0.62 = 371.69
5995 * 0.63 = 377.685
5995 * 0.64 = 383.68
5995 * 0.65 = 389.675
5995 * 0.66 = 395.67
5995 * 0.67 = 401.665
5995 * 0.68 = 407.66
5995 * 0.69 = 413.655
5995 * 0.70 = 419.65
5995 * 0.71 = 425.645
5995 * 0.72 = 431.64
5995 * 0.73 = 437.635
5995 * 0.74 = 443.63
5995 * 0.75 = 449.625
5995 * 0.76 = 455.62
5995 * 0.77 = 461.615
5995 * 0.78 = 467.61
5995 * 0.79 = 473.605
5995 * 0.80 = 479.6
5995 * 0.81 = 485.595
5995 * 0.82 = 491.59
5995 * 0.83 = 497.585
5995 * 0.84 = 503.58
5995 * 0.85 = 509.575
5995 * 0.86 = 515.57
5995 * 0.87 = 521.565
5995 * 0.88 = 527.56
5995 * 0.89 = 533.555
5995 * 0.90 = 539.55
5995 * 0.91 = 545.545
5995 * 0.92 = 551.54
5995 * 0.93 = 557.535
5995 * 0.94 = 563.53
5995 * 0.95 = 569.525
5995 * 0.96 = 575.52
5995 * 0.97 = 581.515
5995 * 0.98 = 587.51
5995 * 0.99 = 593.505
5995 * 1.00 = 5995
|
CC-MAIN-2020-05/segments/1579250599718.13/warc/CC-MAIN-20200120165335-20200120194335-00516.warc.gz
|
bubble.ro
|
en
| 0.827907
| 2020-01-20T17:22:33
|
http://num.bubble.ro/m/5995/
| 0.998608
|
To find the first term of a geometric sequence given the fifth and sixth terms, we can use the formula for the common ratio: r = t_n / t_(n-1). Since the fifth and sixth terms are 8 and 16, respectively, we calculate r = 16 / 8 = 2.
Now that we have the common ratio, we can solve for the first term (a) using the formula for the nth term of a geometric sequence: t_n = a * r^(n-1). Given the sixth term is 16, we set up the equation 16 = a * 2^(6-1), which simplifies to 16 = a * 2^5.
Further simplifying, we get 16 = a * 32. Solving for a, we find a = 16 / 32 = 1/2. Therefore, the first term of the geometric sequence is 1/2.
|
CC-MAIN-2021-25/segments/1623487635724.52/warc/CC-MAIN-20210618043356-20210618073356-00054.warc.gz
|
socratic.org
|
en
| 0.841304
| 2021-06-18T06:04:09
|
https://socratic.org/questions/if-the-fifth-and-sixth-terms-of-a-geometric-sequence-are-8-and-16-16-respectivel
| 0.988449
|
## A Basic Flight Simulator in Excel #2 – Airplane Positioning, Control Surfaces, and Turn Dynamics
This tutorial explains the basics of airplane positioning, control surfaces, and turn dynamics. It covers the three angles characterizing an airplane's position in flight: yaw, pitch, and roll. The control surfaces, including rudders, ailerons, and elevators, are described, along with the control devices, such as the yoke and rudder pedals.
### Review of the Three Airplane Rotation Angles
* The yaw angle defines rotation around the vertical z-axis, controlled by the rudder.
* The pitch angle defines rotation around the lateral x-axis, controlled by the elevator.
* The roll angle defines rotation around the longitudinal y-axis, controlled by the ailerons.
Since the joystick has only two degrees of freedom, we will control the pitch and roll angles, as these are the most important angles in controlling an airplane.
### Airplane Controls
An airplane has two main control devices: the yoke (joystick or control stick) and the rudder pedals. The yoke controls the ailerons (by sideways movement) and the elevator (by back and forth movement). The rudder pedals control the rudder.
### The Rudder
The rudder controls the yaw of the airplane. Its role is important mainly at low altitude flying, where not much roll angles are permitted. The rudder is a secondary control device for an airplane during regular flight.
### The Elevator
The elevator controls the pitch of the airplane and is used in changing the plane's attitude for gaining or losing altitude and in turns. It is important even in executing constant altitude turns.
### The Ailerons
The ailerons control the roll of the airplane and are mainly used in turns. They are controlled by the sideways movement of the yoke (or joystick).
### How an Airplane Turns
An airplane turn is initiated by rolling the plane in the direction of the turn by a sideways move of the control stick, then reducing the roll and increasing the pitch to a positive value (nose up). Maintaining the proper combination of positive pitch and roll is needed for the vector sum of gravity, centrifugal force, and lift to balance.
### Pitch Rate and Roll Rate
A control surface will deflect the air stream, producing a force on the tip of the aircraft proportional to the deflection angle. The force will produce a momentum proportional to the size of the force and the distance of the control surface from the pressure center of the airplane. Any deviation in the position of a control device will produce a change in the angle of the ship with a rate proportional to the amount of deviation.
### Modeling the Landscape Coordinate Change during Pitch Rotation
The flight simulator will be a 2D scatter chart containing the 2D mapping of a 3D wireframe landscape. The origin of the system of coordinates will be tied to the cockpit, with the x-axis pointing forward, the y-axis pointing to the right, and the z-axis pointing upwards. The relationship between the old coordinates and the new coordinates needs to be found when the plane changes its pitch angle.
|
CC-MAIN-2024-38/segments/1725700651944.55/warc/CC-MAIN-20240918233405-20240919023405-00078.warc.gz
|
excelunusual.com
|
en
| 0.87588
| 2024-09-19T00:52:44
|
https://excelunusual.com/flight-simulator-tutorial-2-basic-airplane-positioning-control-surfaces-and-turn-dynamics/
| 0.439428
|
In various types of numbers, we will explore even numbers, odd numbers, prime numbers, composite numbers, coprime numbers, and twin prime numbers.
**Even Numbers**
A whole number exactly divisible by 2 is called an even number. Examples include 2, 4, 6, 8, 10, and any number ending in 0, 2, 4, 6, or 8, such as 246, 1894, 5468, and 100. Consecutive even numbers differ by 2.
**Odd Numbers**
A whole number not exactly divisible by 2 is called an odd number. Examples include 3, 5, 7, 9, 11, 13, 15, and any number ending in 1, 3, 5, 7, or 9.
**Prime Numbers**
Numbers with only two factors, 1 and the number itself, are called prime numbers. Examples include 2, 3, 5, 7, 11, 19, and 37. Note that 2 is the only even prime number.
**Composite Numbers**
Numbers with more than two factors are called composite numbers. Examples include 4, 6, 8, 10, and so on. It's worth noting that 1 is neither prime nor composite, and 9 is the lowest odd composite number.
**Coprime Numbers**
Two numbers are coprime if they do not have a common factor other than 1, or if their highest common factor (HCF) is 1. Examples include 7 and 10, and 15 and 17. Coprime numbers do not need to be prime numbers.
**Twin Prime Numbers**
Twin prime numbers are two prime numbers whose difference is 2. Examples include 3 and 5, 17 and 19, 41 and 43, 29 and 31, and 71 and 73.
|
CC-MAIN-2019-04/segments/1547583807724.75/warc/CC-MAIN-20190121193154-20190121215154-00414.warc.gz
|
math-only-math.com
|
en
| 0.871929
| 2019-01-21T20:43:39
|
https://www.math-only-math.com/various-types-of-numbers.html
| 0.999049
|
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