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Understanding Big-O Notation: A Simplified Explanation | Coding Lesson
Understanding Big-O Notation: A Simplified Explanation
If you’re a developer or someone learning algorithms, you’ve probably come across the term Big-O Notation. It is a mathematical concept that helps analyze the efficiency of algorithms. This guide
will explain Big-O in simple terms, using Python examples to clarify the concept. By the end of this article, you’ll understand why Big-O matters and how to use it in your coding journey.
What is Big-O Notation?
Big-O Notation measures how well an algorithm scales with the size of its input. It’s a way to describe the time or space complexity of an algorithm, or how fast an algorithm runs as the input grows.
Why Does Big-O Notation Matter?
As your data grows, the performance of your algorithm can change dramatically. Big-O helps you predict these changes and choose the best algorithm for the job. Imagine sorting a small list of 10
numbers—it doesn’t matter much whether you use a fast or slow sorting algorithm. But when you have millions of items, the choice of algorithm can mean the difference between a fast application and a
painfully slow one.
Common Big-O Notations and Their Meanings
Here are some common types of Big-O complexities that you’ll encounter:
1. O(1) – Constant Time
• The algorithm’s performance does not change with the size of the input.
• Example: Accessing an element in an array by its index.
arr = [1, 2, 3, 4]
print(arr[2]) # O(1) operation
2. O(log n) – Logarithmic Time
• The algorithm reduces the problem size by a factor with each step. Binary search is a good example.
• Example: Binary search in a sorted array.
def binary_search(arr, target):
low, high = 0, len(arr) - 1
while low <= high:
mid = (low + high) // 2
if arr[mid] == target:
return mid
elif arr[mid] < target:
low = mid + 1
high = mid - 1
return -1
3. O(n) – Linear Time
• The algorithm’s running time grows linearly with the input size.
• Example: Searching for an element in an unsorted list.
def linear_search(arr, target):
for i in range(len(arr)):
if arr[i] == target:
return i
return -1
4. O(n log n) – Linearithmic Time
• This is typically the time complexity of efficient sorting algorithms like Merge Sort or Quick Sort.
• Example: Merge Sort.
def merge_sort(arr):
if len(arr) > 1:
mid = len(arr) // 2
left_half = arr[:mid]
right_half = arr[mid:]
i = j = k = 0
while i < len(left_half) and j < len(right_half):
if left_half[i] < right_half[j]:
arr[k] = left_half[i]
i += 1
arr[k] = right_half[j]
j += 1
k += 1
while i < len(left_half):
arr[k] = left_half[i]
i += 1
k += 1
while j < len(right_half):
arr[k] = right_half[j]
j += 1
k += 1
5. O(n^2) – Quadratic Time
• Algorithms with this complexity become slow as the input size grows. It is common with algorithms that involve nested loops, such as Bubble Sort.
• Example: Bubble Sort.
def bubble_sort(arr):
for i in range(len(arr)):
for j in range(0, len(arr)-i-1):
if arr[j] > arr[j+1]:
arr[j], arr[j+1] = arr[j+1], arr[j]
6. O(2^n) – Exponential Time
• These algorithms double their workload with every additional input, making them impractical for large inputs.
• Example: Recursive algorithms for solving the Fibonacci sequence.
def fib(n):
if n <= 1:
return n
return fib(n-1) + fib(n-2)
Big-O Notation Visualized
The graph of Big-O shows the growth rate of algorithms as input sizes increase. Constant time, O(1), remains flat, meaning performance stays the same regardless of input size. O(log n) grows slowly,
while O(n log n) and O(n^2) grow much faster, making them less efficient for large inputs.
How to Analyze an Algorithm Using Big-O Notation
When analyzing an algorithm, follow these steps:
1. Identify the loops
□ If there are no loops, the algorithm likely runs in constant time, O(1).
□ A single loop suggests O(n), while nested loops imply O(n^2).
2. Look for recursive calls
□ If the algorithm involves recursion, consider the number of recursive calls and their impact on time complexity.
3. Combine operations
□ If an algorithm consists of multiple operations, you may need to combine their complexities. For example, if you have an O(n) operation followed by an O(log n) operation, the overall
complexity is O(n).
Real-World Applications of Big-O Notation
1. Search Engines
Google uses algorithms that need to handle large amounts of data efficiently. Big-O analysis helps determine which algorithms scale best as data size increases.
2. Social Media
Platforms like Facebook and Instagram optimize their algorithms using Big-O to ensure their applications can handle millions of users at once.
3. E-commerce
Amazon uses algorithms to recommend products based on user activity. These algorithms are optimized for speed and efficiency, using techniques analyzed with Big-O.
Understanding Big-O Notation is critical for writing efficient code. It helps you predict how an algorithm will perform as data scales, allowing you to choose the best solution for any problem. By
practicing with different algorithms and analyzing their time complexities, you can significantly improve your coding skills and develop efficient, scalable applications. | {"url":"https://coding-lesson.com/understanding-big-o-notation-a-simplified-explanation/","timestamp":"2024-11-08T12:42:07Z","content_type":"text/html","content_length":"121264","record_id":"<urn:uuid:226a6e9d-13a4-4c92-9de5-2209ab3f7b07>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00389.warc.gz"} |
Angel Number Calculator: Decode Your Spiritual Signals
This tool calculates your personalized angel number based on your birthdate and full name.
How to Use the Angel Number Calculator
To use the angel number calculator, simply enter a number into the input field and click the “Calculate” button. The calculator will then compute the “angel number” for the input number.
How It Works
The calculator takes the input number and sums its digits. If the resulting sum is a multi-digit number (i.e., 10 or greater), the digits of the sum are added together. This process is repeated until
a single digit is obtained. The final single digit is the angel number. For example:
• Input: 1234
• Summing the digits: 1 + 2 + 3 + 4 = 10
• Summing the digits of the result: 1 + 0 = 1
• Angel Number: 1
– This calculator only works with numerical inputs. Non-numeric inputs will result in an error.
– The calculator is designed for whole numbers. Decimal numbers will be treated as two separate sequences of digits: the integer part and the fractional part. | {"url":"https://madecalculators.com/angel-number-calculator/","timestamp":"2024-11-09T15:35:37Z","content_type":"text/html","content_length":"141782","record_id":"<urn:uuid:90461dac-3941-4fec-9ecf-392466e9492c>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00577.warc.gz"} |
How do you make a circle in a square?
When a circle is inscribed in a square, the length of each side of the square is equal to the diameter of the circle. That is, the diameter of the inscribed circle is 8 units and therefore the radius
is 4 units. The area of a circle of radius r units is A=πr2 . Substitute r=4 in the formula.
What is a circle inside a square called?
A squircle is a shape intermediate between a square and a circle. There are at least two definitions of “squircle” in use, the most common of which is based on the superellipse. The word “squircle”
is a portmanteau of the words “square” and “circle”. Squircles have been applied in design and optics.
How many circles can fit in a square?
Number of circles (n) Square size (side length (L)) Number density (n/L2)
1 2 0.25
2 ≈ 3.414… 0.172…
3 ≈ 3.931… 0.194…
4 4 0.25
Is there such thing as a square circle?
abcda is a circle with radius r and a square with side 2r. This proves that there are square circles, at least in the abstract, mathematical sense in which there are round circles, or square quadri-
laterals. 5 Rather than being an impossibility, the existence of a square circle implies no contradiction at all.
What does a square with a circle inside mean?
The drying laundry symbol is notated by a square. If tumble dying is OK with the item, there will be a circle inside of the square. If the circle has two small dots inside of it, that means you can
dry it in your clothes dryer without any problems.
What is the ratio of circles to squares?
So the area of the circle is πr2=10×2π. So the ratio of the area of the circle to the area of the square is 10×2π:36×2, which is equivalent to 5π:18, as required.
Why do they call it Squaredcircle?
Known as a ‘ring’ due to its history of beginning as a circle on the ground, the name ‘squared circle’ became a common term for a boxing ring after a squared ring was introduced in the 1830s under
the new London Prize Ring Rules.
Who invented the squircle?
The word “Squircle” was coined by Peter Panholzer in the summer of 1966 in Toronto , Canada . As an aspiring architect born and studying in Vienna , Austria , Peter had spent four consecutive summers
(1963-66) working for architects in Toronto .
How do I draw a square or circle in Photoshop?
In ‘Shape’, press the ‘Shift’ key to draw a square or circle. In ‘Shape’, select a square or circle and click and drag while holding down the ‘Shift’ key.
What is the area of a circle drawn inside a square?
Area of the square = s x s = 12 x 12 = 144 square inches or 144 sq.inch Hence the shaded area = Area of the square – The area of the circle = 144 – 113.04 = 30.96 sq.in Finally we wrap up the topic
of finding the area of a circle drawn inside a square of a given side length.
How to draw a square in AutoCAD?
After selecting the tool, check “Fixed aspect ratio”, and select a random spot. From the “Selected area” tab, choose “Selected boundary drawing”. Choose an optional number (for the thickness of the
outline), uncheck “Round corners, and maintain the line thickness”, and when you click “OK”,you will have a drawing of just the outline of a square.
How do I draw a circle on my computer screen?
Normally, you can do a drawing with an optional aspect ratio, but by checking “Fixed aspect ratio” on the top of the screen, you can draw a square. And by selecting “Circle”, you can draw a circle in
whatever shape you want. | {"url":"https://www.thenewsindependent.com/how-do-you-make-a-circle-in-a-square/","timestamp":"2024-11-04T04:28:53Z","content_type":"text/html","content_length":"57699","record_id":"<urn:uuid:ee060ed2-59b3-43d2-8b08-474188f0d89a>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00636.warc.gz"} |
How Slide Rules Revolutionized Mathematics and Engineering
The invention of slide rules revolutionized the fields of mathematics and engineering, providing a powerful tool for calculations and problem-solving. This article explores the origins of slide
rules, how they work, their impact on mathematics, their applications in engineering, and their eventual decline with the emergence of electronic calculators and digital computing. Here are the key
takeaways from this article:
Key Takeaways
• Slide rules were invented in the 17th century and underwent significant development before becoming popularized.
• Slide rules operate based on logarithmic scales and allow for multiplication, division, and other mathematical operations.
• They facilitated faster and more accurate calculations, enabling advancements in various fields of science and engineering.
• Slide rules were widely used in mathematical education and played a crucial role in teaching complex mathematical concepts.
• In engineering, slide rules were used in civil, mechanical, and aerospace engineering for calculations, design, and problem-solving.
The Origins of Slide Rules
Invention of the Slide Rule
The invention of the slide rule revolutionized mathematical calculations and engineering practices. It was a significant development in the field of computation, allowing for faster and more accurate
calculations than manual methods. The slide rule was invented in the 17th century by William Oughtred, an English mathematician. It consisted of two logarithmic scales that could be slid against each
other to perform multiplication and division. This innovative device greatly simplified complex calculations and became an essential tool for scientists, engineers, and mathematicians.
Early Development and Design
During the early development and design phase of slide rules, several key advancements were made to improve their functionality and usability.
One important development was the introduction of logarithmic scales, which allowed for more precise calculations and increased accuracy. These logarithmic scales were carefully calibrated and marked
on the slide rule, enabling users to perform complex mathematical operations with ease.
Another significant improvement was the addition of additional scales and markings on the slide rule. These included trigonometric scales, exponential scales, and various conversion scales, expanding
the range of calculations that could be performed using the slide rule.
To enhance the durability and portability of slide rules, early designs incorporated materials such as wood, ivory, or celluloid. These materials were chosen for their strength and resistance to wear
and tear, ensuring that slide rules could withstand frequent use.
Overall, the early development and design of slide rules laid the foundation for their widespread adoption and eventual revolution in mathematics and engineering.
Popularization of Slide Rules
The popularization of slide rules can be attributed to their simplicity and efficiency in performing mathematical calculations. As slide rules became more widely available, they were adopted by
engineers, scientists, and mathematicians as essential tools for their work. The ability to quickly and accurately perform calculations using slide rules greatly increased productivity and allowed
for more complex calculations to be performed. This widespread adoption led to the integration of slide rules into mathematical education, where students were taught how to use them effectively. The
popularity of slide rules continued to grow until the emergence of electronic calculators, which eventually replaced them as the primary tool for mathematical calculations.
How Slide Rules Work
Principles of Operation
Slide rules operate based on the principles of logarithms. Logarithms are mathematical functions that allow for the simplification of complex calculations involving multiplication, division, and
exponentiation. The slide rule consists of two logarithmic scales, one fixed and one movable. By aligning the numbers on the scales, users can perform calculations by adding or subtracting the
logarithmic values. This method enables quick and accurate estimation of results without the need for extensive manual calculations.
A key feature of slide rules is the ability to perform calculations using a linear scale. This linear scale allows for the representation of numbers in a linear fashion, making it easier to visualize
and manipulate values. The logarithmic scales on the slide rule are designed to be proportional to the logarithmic values, allowing for precise calculations. The use of logarithmic scales also
enables the slide rule to handle a wide range of values, from very small to very large, with relative ease.
To further enhance the functionality of slide rules, additional scales and markings are often included. These scales can represent trigonometric functions, exponential functions, and other
mathematical operations. By utilizing these additional scales, users can perform more complex calculations and solve a variety of mathematical problems.
Components of a Slide Rule
The components of a slide rule include the sliding scale, the fixed scale, and the cursor. The sliding scale is a movable part of the slide rule that contains the logarithmic scales. It can be moved
back and forth to perform calculations. The fixed scale, on the other hand, is a stationary part of the slide rule that provides reference values. The cursor is a transparent indicator that helps
align the scales and read the results accurately.
Here is a table summarizing the components of a slide rule:
Using these components, users can perform various mathematical operations with ease and accuracy.
Using a Slide Rule
Using a slide rule involves a series of steps to perform calculations. Here is a step-by-step guide:
1. Align the leftmost scale, called the C scale, with the number you want to multiply or divide.
2. Move the cursor, also known as the hairline, to the right to read the result on the D scale.
3. For addition and subtraction, align the leftmost scale with the first number and move the cursor to the right to find the sum or difference on the D scale.
4. To find square roots, align the leftmost scale with the number and move the cursor to the right to read the result on the D scale.
Using a slide rule requires practice and familiarity with the scales and their corresponding operations. It was a skill that engineers and mathematicians developed to perform calculations efficiently
and accurately.
Impact of Slide Rules on Mathematics
Advancements in Calculation Speed
Advancements in calculation speed were one of the key benefits of slide rules. With the ability to perform complex calculations quickly and accurately, slide rules greatly increased the efficiency of
mathematical and engineering tasks. Engineers and mathematicians could now solve equations and perform calculations in a fraction of the time it would take using manual methods.
In fact, slide rules were so efficient that they were often used in situations where speed was crucial, such as in the field of navigation. The ability to quickly calculate distances, angles, and
other important measurements made slide rules indispensable tools for sailors, pilots, and surveyors.
Additionally, slide rules allowed for the simultaneous calculation of multiple operations. By aligning different scales on the slide rule, users could perform addition, subtraction, multiplication,
and division all in one step. This feature further enhanced the speed and efficiency of calculations, making slide rules invaluable in a wide range of mathematical and engineering applications.
Facilitation of Complex Calculations
The slide rule greatly facilitated complex calculations by allowing users to perform multiple mathematical operations quickly and accurately. With the slide rule, engineers and mathematicians could
easily perform calculations involving multiplication, division, logarithms, and trigonometric functions. This eliminated the need for time-consuming manual calculations and significantly increased
In addition, the slide rule provided a visual representation of the calculations, allowing users to easily track their progress and verify the accuracy of their results. This visual feedback was
particularly useful for complex calculations that involved multiple steps or iterations.
Furthermore, the slide rule enabled engineers and mathematicians to perform calculations with a high degree of precision. The logarithmic scales on the slide rule allowed for the estimation of values
to several decimal places, providing a level of accuracy that was not easily achievable with manual calculations alone.
Overall, the facilitation of complex calculations by the slide rule revolutionized the field of mathematics and engineering, enabling faster and more accurate computations.
Integration into Mathematical Education
The integration of slide rules into mathematical education had a profound impact on students and teachers alike. Mathematics educators recognized the value of slide rules in teaching students
practical problem-solving skills and fostering a deeper understanding of mathematical concepts.
One of the key benefits of using slide rules in the classroom was their ability to facilitate quick and accurate calculations. Students could perform complex mathematical operations with ease,
allowing them to focus more on the underlying principles and less on tedious computations.
In addition to their practical use, slide rules also served as visual aids that helped students visualize mathematical relationships and patterns. The linear scales and logarithmic scales on slide
rules provided a tangible representation of mathematical concepts, making them more accessible and intuitive.
To further enhance the learning experience, some educators incorporated slide rule competitions and challenges into their curriculum. These activities not only motivated students to improve their
slide rule skills but also fostered a sense of camaraderie and friendly competition among classmates.
Slide Rules in Engineering
Applications in Civil Engineering
Slide rules were widely used in civil engineering for various calculations and measurements. They were particularly useful in tasks such as:
• Determining distances and angles for surveying land
• Calculating structural loads and stresses
• Estimating material quantities for construction projects
Slide rules provided engineers with a portable and efficient tool for performing these calculations in the field. They allowed for quick and accurate results, reducing the need for manual
calculations and minimizing errors.
Use in Mechanical Engineering
Slide rules were widely used in mechanical engineering for various calculations and design tasks. One of the key applications of slide rules in mechanical engineering was for performing quick and
accurate calculations of mechanical properties such as force, torque, and power. Engineers could use slide rules to determine the required dimensions of mechanical components, such as gears and
shafts, based on the desired performance specifications.
In addition, slide rules were also used for solving complex equations and performing mathematical operations involved in mechanical engineering analysis. Engineers could use slide rules to solve
equations related to stress and strain, fluid dynamics, and thermodynamics. The slide rule's logarithmic scales allowed engineers to perform calculations involving exponential and logarithmic
functions with ease.
Furthermore, slide rules played a crucial role in the design and analysis of mechanical systems. Engineers could use slide rules to perform calculations related to gear ratios, mechanical advantage,
and efficiency of mechanical systems. This helped in optimizing the performance and efficiency of various mechanical devices and systems.
Overall, slide rules were an indispensable tool for mechanical engineers, providing them with a quick and reliable method for performing calculations and design tasks. The use of slide rules in
mechanical engineering continued until the advent of electronic calculators and digital computing, which offered more advanced and efficient methods for performing complex calculations.
Contributions to Aerospace Engineering
Slide rules made significant contributions to the field of aerospace engineering. They were used extensively in calculations related to aerodynamics, propulsion systems, and trajectory planning. The
ability to perform complex calculations quickly and accurately using slide rules greatly enhanced the efficiency of aerospace engineers. The precision and reliability of slide rules were crucial in
the design and analysis of aircraft and spacecraft.
In addition to their use in calculations, slide rules also played a role in the education and training of aerospace engineers. They were commonly used in classrooms and engineering programs to teach
students about mathematical principles and problem-solving techniques. The hands-on experience of using slide rules helped engineers develop a deeper understanding of mathematical concepts and their
practical applications in aerospace engineering.
Overall, slide rules were an indispensable tool for aerospace engineers, enabling them to perform complex calculations, analyze data, and design innovative aircraft and spacecraft.
The Decline of Slide Rules
Emergence of Electronic Calculators
The emergence of electronic calculators marked a significant turning point in the history of mathematical and engineering tools. These compact devices, powered by transistors and integrated circuits,
revolutionized the way calculations were performed. Unlike slide rules, which relied on manual manipulation and estimation, electronic calculators provided precise and accurate results with minimal
With the introduction of electronic calculators, calculations that would have taken hours or even days to complete using slide rules could now be done in a matter of seconds. This dramatic increase
in calculation speed greatly enhanced productivity and efficiency in both mathematics and engineering.
Additionally, electronic calculators offered advanced functionalities such as memory storage, complex number calculations, and trigonometric functions. These features further expanded the
capabilities of mathematicians and engineers, allowing them to tackle more complex problems and explore new areas of research.
The widespread adoption of electronic calculators eventually led to the decline of slide rules. As these handheld devices became more affordable and accessible, they quickly replaced slide rules as
the preferred tool for mathematical and engineering calculations.
Transition to Digital Computing
As digital computing technology advanced in the mid-20th century, the slide rule faced increasing competition from electronic calculators. These calculators offered faster and more accurate
calculations, making them more convenient for engineers and mathematicians. The transition from slide rules to digital computing was driven by the desire for greater efficiency and precision in
mathematical and engineering calculations.
While slide rules were still used in some applications, such as in educational settings or by enthusiasts, their practicality diminished as electronic calculators became more affordable and
The advent of digital computing not only revolutionized the speed and accuracy of calculations but also opened up new possibilities for complex mathematical modeling and simulation. The ability to
perform calculations quickly and efficiently on computers paved the way for advancements in fields such as computer-aided design, computational physics, and data analysis.
In summary, the transition to digital computing marked a significant turning point in the history of mathematical and engineering tools. The slide rule, once a ubiquitous instrument, gradually faded
into obscurity as electronic calculators and computers took center stage.
Legacy and Collectibility
The decline of slide rules began in the 1970s with the emergence of electronic calculators. These compact devices offered faster and more accurate calculations, making slide rules obsolete in many
professional settings. The transition to digital computing further accelerated the decline, as computers became more powerful and accessible. Despite their obsolescence, slide rules hold a special
place in the hearts of collectors and enthusiasts. Today, they are sought after as historical artifacts and symbols of a bygone era in mathematics and engineering.
In conclusion, the invention of slide rules had a profound impact on the fields of mathematics and engineering. Slide rules revolutionized the way calculations were performed, allowing for faster and
more accurate results. They were widely used by scientists, engineers, and mathematicians for centuries before the advent of electronic calculators. Although slide rules are no longer in common use
today, their legacy can still be seen in the development of modern computing devices. The slide rule remains an important symbol of the ingenuity and innovation of early mathematicians and engineers.
Frequently Asked Questions
1. What is a slide rule?
A slide rule is a mechanical calculating device used for performing mathematical calculations.
2. How does a slide rule work?
A slide rule works by using logarithmic scales and sliding the movable part to perform calculations.
3. Who invented the slide rule?
The slide rule was invented by William Oughtred in the early 17th century.
4. What were the main applications of slide rules?
Slide rules were commonly used in mathematics, engineering, and scientific calculations.
5. When did slide rules become popular?
Slide rules became popular in the 17th century and remained widely used until the mid-20th century.
6. Why did slide rules decline in usage?
The decline of slide rules was mainly due to the emergence of electronic calculators and digital computing. | {"url":"https://www.iancollmceachern.com/single-post/how-slide-rules-revolutionized-mathematics-and-engineering","timestamp":"2024-11-13T19:04:12Z","content_type":"text/html","content_length":"1050488","record_id":"<urn:uuid:fdcd43e8-3970-4d78-8b35-2965802cbfcf>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00888.warc.gz"} |
Can you solve this Grade 5 Math problem?
Susie bought 5 kg of flour and 4 kg of sugar for $14.80. If 3/4 kg of flour cost as much as 3/5 kg of sugar, find the cost of 1 kg of sugar.
Step 1: Draw 4 small boxes to represent the cost of 4 quarters (or 1 whole) of 1 kg of flour.
Since 3/4 kg of flour cost as much as 3/5 kg of sugar, mark out 3 boxes (which is 3/4 of 1 kg of flour) to equate to 3 boxes (which is 3/5 of 1 kg of sugar) of sugar.
Add 2 small boxes to the right of the "sugar" model to show the cost of the rest (2/5 kg of the sugar) of the 1 kg of sugar.
Click here to see the detailed solution.
Comments? Ideas? Feedback? I'd love to hear from you. Just reply to this zine and tell me what you think! | {"url":"http://www.teach-kids-math-by-model-method.com/Modelmatics-modelmatics5.html","timestamp":"2024-11-09T03:51:48Z","content_type":"text/html","content_length":"7585","record_id":"<urn:uuid:dec87016-ff82-4251-99dc-368884e0facf>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00717.warc.gz"} |
Electric power
Electric power is the amount of work done by an electric current in a unit time. When a current flows in a circuit with resistance, it does work. Devices can be made that convert this work into:
1. heat (electric heaters), light (light bulbs and neon lamps)
2. motion, that is, kinetic energy (electric motors).
Electric power, like mechanical power, is represented by the letter P in electrical equations, and is measured in units called watts (symbol W), after Scottish engineer James Watt.
Description[ ]
In resistive circuits, instantaneous electrical power is calculated using Joule's Law, which is named for British physicist James Joule, who first showed that electrical and mechanical energy were
${\displaystyle P=IV}$
P = power in watts
I = current in amperes
V = potential difference in volts
In reactive circuits, energy storage elements such as inductance and capacitance may result in periodic reversals of the direction of energy flow. The portion of power flow that, averaged over a
complete cycle of the AC waveform, results in net transfer of energy in one direction is known as real power. That portion of power flow due to stored energy, that returns to the source in each
cycle, is known as reactive power.
The unit for reactive power is given the special name VAR, which stands for volt-amperes-reactive. In reactive circuits, the watt unit (symbol W) is generally reserved for the real power component.
The vector sum of the real power and the reactive power is called the apparent power. Apparent power is conventionally expressed in volt-amperes (VA) since it is the simple multiple of rms voltage
and current.
The relationship between real power, reactive power and apparent power can be expressed by representing the quantities as vectors. Real power is represented as a horizontal vector and reactive power
is represented as a vertical vector. The apparent power vector is the hypotenuse of a right triangle formed by connecting the real and apparent power vectors. This representation is often called the
power triangle. Using the Pythagorean Theorem, the relationship among real, reactive and apparent power is shown to be:
${\displaystyle real\ power (W)^2 + reactive\ power(VAR)^2 = apparent\ power(VA)^2. \,}$
Power factor[ ]
The ratio between real power and apparent power in a circuit is called the power factor. Where the waveforms are purely sinusoidal, the power factor is the cosine of the phase angle (φ) between the
current and voltage sinusoid waveforms. Equipment data sheets and nameplates often will abbreviate power factor as "${\displaystyle \cos \phi}$" for this reason.
Power factor equals unity (1) when the voltage and current are in phase, and is zero when the current leads or lags the voltage by 90 degrees. Power factor must be specified as leading or lagging.
For two systems transmitting the same amount of real power, the system with the lower power factor will have higher circulating currents due to energy that returns to the source from energy storage
in the load. These higher currents in a practical system may produce higher losses and reduce overall transmission efficiency. A lower power factor circuit will have a higher apparent power and
higher losses for the same amount of real power transfer.
Capacitive circuits cause reactive power with the current waveform leading the voltage wave by 90 degrees, while inductive circuits cause reactive power with the current waveform lagging the voltage
waveform by 90 degrees. The result of this is that capacitive and inductive circuit elements tend to cancel each other out. By convention, capacitors are said to generate reactive power whilst
inductors are said to consume it (this probably comes from the fact that most real-life loads are inductive and so reactive power has to be supplied to them from power factor correction capacitors).
In power transmission and distribution, significant effort is made to control the reactive power flow. This is typically done automatically by switching inductors or capacitor banks in and out, by
adjusting generator excitation, and by other means. Electricity retailers may use electricity meters which measure reactive power to financially penalize customers with low power factor loads
(especially larger customers).
Kilowatt-hour[ ]
When paired with a unit of time, the term watt is used for expressing energy consumption. For example, a kilowatt hour, is the amount of energy expended by a one kilowatt device over the course of
one hour; it equals 3.6 megajoules. A megawatt day (MWd or MW·d) is equal to 86.4 GJ.
These units are often used in the context of power plants and home energy bills. Electricity utilities bill residential customers for only real power consumed, as opposed to apparent power.
Industrial customers are more scrutinized as they are penalized for electric loads that have low power factors which create a large difference between the supplied apparent power and the power
consumed by the load (real power).
Applications[ ]
There are few places in the world today where electricity is not utilised; be it in industries, office and residential buildings, or public places.
For all these places electricity is supplied by a utility company licensed or authorised for this purpose observing the procedures laid down by law, which takes into account the electric power supply
as explained.
See also[ ]
• Electrical power industry
This page uses Creative Commons Licensed content from Wikipedia (view authors). | {"url":"https://engineering.fandom.com/wiki/Electric_power","timestamp":"2024-11-08T14:23:33Z","content_type":"text/html","content_length":"171746","record_id":"<urn:uuid:db955e60-8d89-4db8-ae14-03772e7c8e9f>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00676.warc.gz"} |
Chapter:Dave Operations Budgeting And Cost-variance-profit Analysis
Questions and Answers
• 1.
Which of the following budgeting processes ensures that plans are specifically geared to individual operations within multi-unit food service companies?
□ A.
□ B.
□ C.
□ D.
Correct Answer
B. Bottom-up budgeting
Bottom-up budgeting is a budgeting process that ensures plans are specifically geared to individual operations within multi-unit food service companies. In this approach, each unit or department
within the organization is responsible for creating its own budget based on its specific needs and goals. These individual budgets are then aggregated to create an overall budget for the entire
company. This approach allows for greater accuracy and accountability at the operational level, as each unit has a clear understanding of its own requirements and can tailor its budget
• 2.
Which of the following methods for projecting revenues in the budgeting process assumes that past trends are good predictors of future growth?
□ A.
□ B.
□ C.
□ D.
Correct Answer
A. Revenue history
Revenue history is the correct answer because it assumes that past trends in revenue can be used to predict future growth. By analyzing the patterns and trends in past revenue data, organizations
can make informed projections about their future revenue streams. This method assumes that historical data is a reliable indicator of future performance and can be used as a basis for budgeting
and forecasting.
• 3.
Costs that remain constant in the short term, even though sales volume may vary, are called ________ costs
□ A.
□ B.
□ C.
□ D.
Correct Answer
D. Fixed
Fixed costs are costs that do not change regardless of the sales volume. These costs remain constant in the short term and do not fluctuate with changes in production or sales. They are often
associated with expenses such as rent, salaries, insurance, and utilities, which do not directly depend on the level of output. Fixed costs are important for businesses to consider when analyzing
their cost structure and determining the breakeven point.
• 4.
Which of the following is the most likely to be classified as a variable cost?
□ A.
General manager’s salary
□ B.
□ C.
□ D.
Correct Answer
D. Food costs
Food costs are the most likely to be classified as a variable cost because they directly vary with the level of production or sales. As the business produces more or sells more, the cost of food
will increase. Conversely, if production or sales decrease, the cost of food will also decrease. This is in contrast to the other options: the general manager's salary, rent expense, and property
taxes, which are typically fixed costs that do not vary with production or sales levels.
• 5.
At the 120-seat Riverside Restaurant, total variable costs for September were $12,000. For October, the manager expects to sell 10 percent more meals than in September. If the increase in sales
volume occurs, the manager should expect the total fixed costs for October to be:
□ A.
□ B.
□ C.
Relatively the same as in September
□ D.
Impossible to forecast with any accuracy
Correct Answer
C. Relatively the same as in September
The correct answer is "relatively the same as in September" because fixed costs do not change with the level of production or sales. Since the increase in sales volume is only 10%, it is unlikely
to have a significant impact on fixed costs. Therefore, the manager should expect the total fixed costs for October to be relatively the same as in September.
• 6.
Using the percentage method for estimating expenses, if the current beverage cost is 20 percent and projected beverage revenue is $60,000, the estimated beverage cost in dollars for the new
budget period would be:
□ A.
□ B.
□ C.
□ D.
Correct Answer
A. $12,000
The percentage method for estimating expenses involves calculating the estimated cost based on a percentage of the projected revenue. In this case, the current beverage cost is 20 percent of the
projected beverage revenue of $60,000. To find the estimated beverage cost for the new budget period, we multiply the projected revenue by the percentage: 20% of $60,000 is $12,000. Therefore,
the estimated beverage cost in dollars for the new budget period would be $12,000.
• 7.
At the Virtual Café, the average price per meal sold is $15 with an average variable cost of $7. Fixed costs for July are expected to be $30,000. If the restaurant manager expects to sell $5,000
meals in July, the net income (or loss) for the month would be:
□ A.
□ B.
□ C.
□ D.
Correct Answer
B. $10,000 net income
The net income for the month can be calculated by subtracting the total variable costs and fixed costs from the total revenue. The total revenue can be found by multiplying the average price per
meal sold ($15) by the number of meals sold (5,000), which equals $75,000. The total variable costs can be found by multiplying the average variable cost per meal ($7) by the number of meals sold
(5,000), which equals $35,000. The fixed costs are given as $30,000. Therefore, the net income can be calculated as $75,000 - $35,000 - $30,000 = $10,000.
• 8.
The Night Owl Restaurant expects to sell 6,000 meals during the upcoming month with an average variable cost per meal sold of $6. Total fixed costs are expected to be $24,000. The average selling
price per meal sold at the breakeven point would be:
Correct Answer
D. $10
The breakeven point is the point at which total revenue equals total costs, resulting in zero profit. In this case, the fixed costs are $24,000 and the variable cost per meal is $6. To find the
breakeven selling price per meal, we need to divide the total costs by the expected number of meals. The total costs can be calculated by multiplying the variable cost per meal by the number of
meals and adding the fixed costs. So, the total costs would be (6000 * $6) + $24,000 = $36,000. Dividing this by the number of meals (6000) gives us a breakeven selling price per meal of $6.
Therefore, the correct answer is $6.
• 9.
The Daylight Diner expects to sell 6,000 meals during the upcoming month with an average variable cost per meal sold of $6, if total fixed costs are expected to be $24,000, what would the average
selling price per meal sold be if the operation is to meet its $12,000 profit goal for the month
□ A.
□ B.
□ C.
□ D.
Correct Answer
D. $12
To find the average selling price per meal sold, we need to calculate the total cost per meal and add the desired profit. The total cost per meal is the sum of the average variable cost per meal
($6) and the fixed cost per meal (total fixed costs divided by the number of meals, which is $24,000 divided by 6,000). This gives us a total cost per meal of $10. Adding the desired profit of
$12,000 to the total cost per meal, we get a selling price per meal of $12. | {"url":"https://www.proprofs.com/quiz-school/story.php?title=chapterdave-operations-budgeting-and-costvarianceprofit-analysis","timestamp":"2024-11-08T20:33:12Z","content_type":"text/html","content_length":"431826","record_id":"<urn:uuid:7fa1b6de-8d3a-49de-849c-97e31fa51e02>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00560.warc.gz"} |
Getting started with sparsegl
This package provides tools for fitting regularization paths for sparse group-lasso penalized learning problems. The model is fit for a sequence of the regularization parameters.
The strengths and improvements that this package offers relative to other sparse group-lasso packages are as follows:
• Compiled Fortran code significantly speeds up the sparse group-lasso estimation process.
• So-called “strong rules” are implemented during group wise coordinate descent steps screen out groups which are likely to be 0 at the solution.
• The design matrix X may be a sparse.
• An estimate_risk() function may be used to evaluate the quality of fitted models via information criteria, providing a means for model selection if cross-validation is too computationally costly.
• Additional exponential families may be fit (though this is typically slower).
For additional details, see Liang, Cohen, Sólon Heinsfeld, Pestilli, and McDonald (2024).
You can install the released version of sparsegl from CRAN with:
You can install the development version from GitHub with:
# install.packages("remotes")
Vignettes are not included in the package by default. If you want to include vignettes, then use this modified command:
build_vignettes = TRUE,
dependencies = TRUE
For this getting-started vignette, first, we will randomly generate X, an input matrix of predictors of dimension $n\times p$. To create y, a real-valued vector, we use either a
• Linear Regression model: $y = X\beta^* + \epsilon$.
• Logistic regression model: $y = (y_1, y_2, \cdots, y_n)$, where $y_i \sim \text{Bernoulli}\left(\frac{1}{1 + \exp(-x_i^\top \beta^*)}\right)$, $i = 1, 2, \cdots, n.$
where the coefficient vector $\beta^*$ is specified as below, and the white noise $\epsilon$ follows a standard normal distribution. Then the sparse group-lasso problem is formulated as the sum of
mean squared error (linear regression) or logistic loss (logistic regression) and a convex combination of the $\ell_1$ lasso penalty with an $\ell_2$ group lasso penalty:
• Linear regression: $\min_{\beta\in\mathbb{R}^p}\left(\frac{1}{2n} \rVert y - \sum_g X^{(g)}\beta^{(g)}\rVert_2^2 + (1-\alpha)\lambda\sum_g \sqrt{|g|}\rVert\beta^{(g)}\rVert_2 + \alpha\lambda\
rVert\beta\rVert_1 \right) \qquad (*).$
• Logistic regression: $\min_{\beta\in\mathbb{R}^p}\left(\frac{1}{2n}\sum_{i=1}^n \log\left(1 + \exp\left(-y_ix_i^\top\beta\right)\right) + (1-\alpha)\lambda\sum_g \sqrt{|g|}\rVert\beta^{(g)}\
rVert_2 + \alpha\lambda\rVert\beta\rVert_1 \right) \qquad (**).$
• $X^{(g)}$ is the submatrix of $X$ with columns corresponding to the features in group $g$.
• $\beta^{(g)}$ is the corresponding coefficients of the features in group $g$.
• $|g|$ is the number of predictors in group $g$.
• $\alpha$ adjusts the weight between lasso penalty and group-lasso penalty.
• $\lambda$ fine-tunes the size of penalty imposed on the model to control the number of nonzero coefficients.
n <- 100
p <- 200
X <- matrix(data = rnorm(n * p, mean = 0, sd = 1), nrow = n, ncol = p)
beta_star <- c(
rep(5, 5), c(5, -5, 2, 0, 0), rep(-5, 5),
c(2, -3, 8, 0, 0), rep(0, (p - 20))
groups <- rep(1:(p / 5), each = 5)
# Linear regression model
eps <- rnorm(n, mean = 0, sd = 1)
y <- X %*% beta_star + eps
# Logistic regression model
pr <- 1 / (1 + exp(-X %*% beta_star))
y_binary <- rbinom(n, 1, pr)
Given an input matrix X, and a response vector y, a sparse group-lasso regularized linear model is estimated for a sequence of penalty parameter values. The penalty is composed of lasso penalty and
group lasso penalty. The other main arguments the users might supply are:
• group: a vector with consecutive integers of length p indicating the grouping of the features. By default, each group only contains one feature if without initialization.
• family: A character string specifying the likelihood to use, could be either linear regression "gaussian" or logistic regression loss "binomial". Default is "gaussian". If other exponential
families are required, a stats::family() object may be used (e.g. poisson()). In that case, arguments providing observation weights or offset terms are allowed as well.
• pf_group: Separate penalty weights can be applied to each group $\beta_g$ to allow differential shrinkage. Can be 0 for some groups, which implies no shrinkage. The default value for each entry
is the square-root of the corresponding size of each group.
• pf_sparse: Penalty factor on $\ell_1$-norm, a vector the same length as the total number of columns in x. Each value corresponds to one predictor Can be 0 for some predictors, which implies that
predictor will be receive only the group penalty.
• asparse: changes the weight of lasso penalty, referring to $\alpha$ in $(*)$ and $(**)$ above: asparse = $1$ gives the lasso penalty only. asparse = $0$ gives the group lasso penalty only. The
default value of asparse is $0.05$.
• lower_bnd: lower bound for coefficient values, a vector in length of 1 or the number of groups including non-positive numbers only. Default value for each entry is -$\infty$.
• upper_bnd: upper bound for coefficient values, a vector in length of 1 or the number of groups including non-negative numbers only. Default value for each entry is $\infty$.
Plotting sparsegl objects
This function displays nonzero coefficient curves for each penalty parameter lambda values in the regularization path for a fitted sparsegl object. The arguments of this function are:
• y_axis: can be set with either "coef" or "group". Default is "coef".
• x_axis: can be set with either "lambda" or "penalty". Default is "lambda".
To elaborate on these arguments:
• The plot with y_axis = "group" shows the group norms against the log-lambda or the scaled group norm vector. Each group norm is defined by: $\alpha\rVert\beta^{(g)}\rVert_1 + (1 - \alpha)\sum_g\
rVert\beta^{(g)}\rVert_2$ Curves are plotted in the same color if the corresponding features are in the same group. Note that the number of curves shown on the plots may be less than the actual
number of groups since only the groups containing nonzero features for at least one $\lambda$ in the sequence are included.
• The plot with y_axis = "coef" shows the estimated coefficients against the lambda or the scaled group norm. Again, only the features with nonzero estimates for at least one $\lambda$ value in the
sequence are displayed.
• The plot with x_axis = "lambda" indicates the x_axis displays $\log(\lambda)$.
• The plot with x_axis = "penalty" indicates the x_axis displays the scaled group norm vector. Each element in this vector is defined by: $\frac{\alpha\rVert \beta\rVert_1 + (1-\alpha)\sum_g\rVert
\beta^{(g)}\rVert_2}{\max_\beta\left(\alpha \rVert \beta\rVert_1 + (1-\alpha)\sum_g\rVert \beta^{(g)}\rVert_2\right)}$
plot(fit1, y_axis = "group", x_axis = "lambda")
plot(fit1, y_axis = "coef", x_axis = "penalty", add_legend = FALSE)
This function performs k-fold cross-validation (cv). It takes the same arguments X, y, group, which are specified above, with additional argument pred.loss for the error measure. Options are
"default", "mse", "deviance", "mae", and "misclass". With family = "gaussian", "default" is equivalent to "mse" and "deviance". In general, "deviance" will give the negative log-likelihood. The
option "misclass" is only available if family = "binomial".
fit_l1 <- cv.sparsegl(X, y, group = groups, pred.loss = "mae")
A number of S3 methods are provided for both sparsegl and cv.sparsegl objects.
• coef() and predict() return a matrix of coefficients and predictions $\hat{y}$ given a matrix X at each lambda respectively. The optional s argument may provide a specific value of $\lambda$ (not
necessarily part of the original sequence), or, in the case of a cv.sparsegl object, a string specifying either "lambda.min" or "lambda.1se".
coef <- coef(fit1, s = c(0.02, 0.03))
predict(fit1, newx = X[100, ], s = fit1$lambda[2:3])
#> s1 s2
#> [1,] -4.071804 -4.091689
predict(fit_l1, newx = X[100, ], s = "lambda.1se")
#> s1
#> [1,] -15.64857
#> Call: sparsegl(x = X, y = y, group = groups)
#> Summary of Lambda sequence:
#> lambda index nnzero active_grps
#> Max. 0.62948 1 0 0
#> 3rd Qu. 0.19676 26 20 4
#> Median 0.06443 50 19 4
#> 1st Qu. 0.02014 75 25 5
#> Min. 0.00629 100 111 23
With extremely large data sets, cross validation may be to slow for tuning parameter selection. This function uses the degrees of freedom to calculate various information criteria. This function uses
the “unknown variance” version of the likelihood. Only implemented for Gaussian regression. The constant is ignored (as in stats::extractAIC()).
• object: a fitted sparsegl object.
• type: three types of penalty used for calculation:
□ AIC (Akaike information criterion): $2 df / n$
□ BIC (Bayesian information criterion): $2 df\log(n) / n$
□ GCV (Generalized cross validation): $-2\log(1 - df / n)$
where df is the degree-of-freedom, and n is the sample size.
• approx_df: indicates if an approximation to the correct degree-of-freedom at each penalty parameter $\lambda$ should used. Default is FALSE and the program will compute an unbiased estimate of
the exact degree-of-freedom.
The df component of a sparsegl object is an approximation (albeit a fairly accurate one) to the actual degrees-of-freedom. However, computing the exact value requires inverting a portion of $\mathbf
{X}^\top \mathbf{X}$. So this computation may take some time (the default computes the exact df). For more details about how this formula, see Vaiter, Deledalle, Peyré, et al., (2012).
Liang, X., Cohen, A., Sólon Heinsfeld, A., Pestilli, F., and McDonald, D.J. 2024. “sparsegl: An R Package for Estimating Sparse Group Lasso.” Journal of Statistical Software 110(6), 1–23. https://
Vaiter S, Deledalle C, Peyré G, Fadili J, and Dossal C. 2012. “The Degrees of Freedom of the Group Lasso for a General Design.” https://arxiv.org/abs/1212.6478. | {"url":"https://dajmcdon.github.io/sparsegl/articles/sparsegl.html","timestamp":"2024-11-11T04:50:54Z","content_type":"text/html","content_length":"43881","record_id":"<urn:uuid:44a7a4b8-b571-45fa-853e-a3ef17b055f7>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00096.warc.gz"} |
Problem 412 - TheMathWorld
Problem 412
A study found that 40% of the assisted reproductive technology (ART) cycles resulted in pregnancies. Twenty-three percent of the ART pregnancies resulted in multiple births.
(a) Find the probability that a random selected ART cycle resulted in a pregnancy and produced a multiple birth.
(b) Find the probability that a randomly selected ART cycle that resulted in a pregnancy did not produce a multiple birth.
(c) Would it be unusual for randomly selected ART cycle to result in a pregnancy and product a multiple birth? Explain.
(a) Let A be the event that an ART cycle pregnancy and B be the event that an ART cycle resulted in a multiple birth. Note that the probability that two events A and B will occur in sequence is as
P(A and B) = P(A).P(B│A)
Determine P(A) and P(B│A).
P(A) =
P(B│A) =
The probability that a randomly selected ART cycle resulted in a pregnancy and produced a multiple birth is P(A ns B). Substitute the value for P(A) and P(B│A) into the formula and simplify to find P
(A and B).
P(A and B) = P(A). P(B│A)
= 0.4 * 0.23
Thus, the probability that a randomly selected ART cycle resulted in a pregnancy and produced a multiple birth is 0.092.
(b) Let B ’ be the complement of B. Hence, the probability that a randomly selected ART cycle that resulted in a pregnancy did not product a multiple birth is P(B’│A). Notice that, in the sample
space of ART cycle resulting in pregnancies, the event {B’│A} is the set of all outcomes that are not included in the event {B│A}. Therefore, the event {B’│A} is the complement of the event {B│A}.
Determine P(B’│A) using the formula the complement of an event, P(E’) = 1 – P(E), where E is an event and E’ is its complement. Recall that P(B│A) = 0.23.
P(B’│A) = 1 – P(B│A)
= 1 – 0.23
= 0.77
Therefore, the probability that a randomly selected ART cycle that resulted in a pregnancy did not produce a multiple birth is 0.77.
(c) An event that occurs with a probability of 0.05 or less is typically considered unusual. Use this information to determine whether it would be unusual for a randomly selected ART cycle to
result in a pregnancy and produce a multiple birth. Recall that P(A and B) = 0.092. | {"url":"https://mymathangels.com/problem-412/","timestamp":"2024-11-11T13:50:13Z","content_type":"text/html","content_length":"61832","record_id":"<urn:uuid:2dd05e41-f5b4-483e-936e-16b4654ffd0c>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00522.warc.gz"} |
Where Form 8815
Hi there and welcome to another video from Hegarty Maths. If Mr. Haggerty here and this is our fourth video on index form. This time, we're going to be talking about multiplying indices and trying to
generate a rule for that. So, to start with, I've introduced you to index form. How would a mathematician usually write - x - x - 7 x? Well, they would write 2^7. Now, I'm going to just break up this
two in different ways. We know that when you're multiplying, multiplying is associative, so I can do it in any order I want. So let's just say I went like this: I did that on its own and then I did
those multiplied. 2 here, that's 2 to the power of 1. These six 2's here are 2 to the power of 6. Obviously, I know my answer is 2 to the 7th. What if I did it in a different order? What if I said,
"Right, I'm going to do them 2 and then I'm going to do them 5?" 2 squared multiplied by 2 to the power of 5 is, of course, 2 to the 7th. What if I broke it up in a different order again? What if I
said, "I don't know, like this?" and I said, "Right, 2 to the 4, therefore multiplied by 2 to the 3." Well, obviously, that's 2 to the 7th. And I could break it up in any way I want. So let's say I
said, "I don't know, I did these two and then I did these three and then I did these two last." So that would be 2 squared multiplied by 2 cubed multiplied by 2 squared. And I know the answer's 2 to
the 7. Now, can you spot what I'm trying to... | {"url":"https://form-8815.com/105353-where-form-8815-index","timestamp":"2024-11-14T22:06:09Z","content_type":"text/html","content_length":"32042","record_id":"<urn:uuid:9cbfa1a9-f56a-41d4-b691-1da227f44fbf>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00865.warc.gz"} |
class astropy.modeling.functional_models.Multiply(factor=1, **kwargs)[source]#
Bases: Fittable1DModel
Multiply a model by a quantity or number.
Factor by which to multiply a coordinate.
Attributes Summary
param_names Names of the parameters that describe models of this type.
Methods Summary
evaluate(x, factor) One dimensional multiply model function.
fit_deriv(x, *params) One dimensional multiply model derivative with respect to parameter.
Attributes Documentation
factor = Parameter('factor', value=1.0)#
fittable = True#
linear = True#
param_names = ('factor',)#
Names of the parameters that describe models of this type.
The parameters in this tuple are in the same order they should be passed in when initializing a model of a specific type. Some types of models, such as polynomial models, have a different
number of parameters depending on some other property of the model, such as the degree.
When defining a custom model class the value of this attribute is automatically set by the Parameter attributes defined in the class body.
Methods Documentation
static evaluate(x, factor)[source]#
One dimensional multiply model function.
static fit_deriv(x, *params)[source]#
One dimensional multiply model derivative with respect to parameter. | {"url":"https://docs.astropy.org/en/latest/api/astropy.modeling.functional_models.Multiply.html","timestamp":"2024-11-15T00:36:50Z","content_type":"text/html","content_length":"36181","record_id":"<urn:uuid:48ef4837-8725-4f48-b542-21f8bfcd9b02>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00633.warc.gz"} |
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Decimal to Fractional Odds C
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Welcome to the decimal to fractional odds website. We have built all the tools you need to make your sports betting (and specifically converting from decimal to fractional odds) experience better!
This website does exactly as the name suggests, lets you convert from decimal to fractional odds!
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How does the decimal to fractional odds converter work?
The decimal to fractional odds converter takes any given fractional odds value and converts it into it's decimal odds equivalent.
• What is the difference between decimal and fractional odds?
Decimal odds is an odds format typically used in Australia. Fractional odds is an odds format typically used in the UK. To be able to convert from Decimal to Fractional odds you first have to
enter a decimal.Converting Decimal Odds to Fractional Odds can be confusing to begin with because of the different ways they are displayed. The main difference being decimal odds consider your
original stake being returned, whereas fractional odds are a display of your profit. Decimal odds of course, are just a simple multiplier to your original stake. If an event at odds of $5.00
wins, you will receive back 5x your stake. That is, a $5 bet would net you $25.
To learn more about odds conversion, please visit our odds converter . Also feel free to visit Fractional to Decimal odds. | {"url":"https://decimaltofractionalodds.com/","timestamp":"2024-11-12T00:53:36Z","content_type":"text/html","content_length":"30974","record_id":"<urn:uuid:d7c751a8-69a6-4241-92f1-d8f1081ae85a>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00067.warc.gz"} |
The course reviews the basic concepts, terminology, and notation involved in geometry, and is designed for the student who successfully completed Algebra 1 as a freshman, though any student may
apply. Both abstract and practical aspects are covered. Conditional statements, conjectures, theorems, and written justifications are systematically brought into the course, along with the subjects
to which they pertain, in the context of problem solving as well as in the context of the preparation of formal proofs. Students construct an understanding by spending some of their class time
working in collaborative learning groups. Review of algebraic and geometric concepts is employed throughout the course. In this way, algebra skills are maintained and the students are better
prepared to enter into the geometric aspects of advanced algebra, math analysis, precalculus, and calculus courses. A Texas Instruments TI-83 or TI-84 series graphing calculator is required. | {"url":"https://curriculum.siprep.org/courses/geometry/","timestamp":"2024-11-13T19:53:24Z","content_type":"text/html","content_length":"72273","record_id":"<urn:uuid:5fa87fc5-8408-4edb-a8d0-7d77606c3533>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00819.warc.gz"} |
A Fully Row/Column-Parallel In-Memory Computing Macro in Foundry MRAM With Differential Readout for Noise Rejection
This work demonstrates two integrated 256-kb in-memory computing (IMC) macros based on foundry MRAM, implemented in a 22-nm fully depleted silicon on insulator (FD-SOI) CMOS process. Embedded
non-volatile memory (eNVM), including MRAM, resistive RAM (ReRAM), and phase-change memory (PCM), is an emerging class of technologies that have drawn interest for IMC due to their potential to
achieve high density with advanced-node scaling as well as low-power always-on/ duty-cycled operation. However, the typically low bit-cell signals (i.e., resistance contrast) necessitate
high-sensitivity readout circuitry, particularly with the high levels of IMC row parallelism desired for maximizing energy efficiency and compute density. This work analyzes power supply and coupling
noise, which arises and poses a primary limitation in recent high-sensitivity, high-efficiency architectures, preventing their integration and scale-up in systems on chip (SoCs). To address this, a
differential readout architecture is demonstrated, which retains the previous efficiency and density while overcoming power-supply interference and coupling by over <inline-formula> <tex-math
notation="LaTeX">$100 \times$</tex-math> </inline-formula> between the many parallel readout channels. The architecture is based on conductance-to-current (<inline-formula> <tex-math notation=
"LaTeX">$G$</tex-math> </inline-formula>-to-<inline-formula> <tex-math notation="LaTeX">$I$</tex-math> </inline-formula>) conversion, column-weighted combining for analog-to-digital converter (ADC)
sharing, and 6-b digitization via a successive-approximation current-to-digital converter (IDC). Enabling fully parallel operation across 128–512 rows and 512 columns, the macros achieve the
state-of-the-art energy efficiency of 68.6 1b-TOPS/W, the compute density of 5.43 1b-TOPS/<inline-formula> <tex-math notation="LaTeX">$\text{mm}^{2}$</tex-math> </inline-formula>, and the efficiency&
#x2013;throughput product (reciprocal of area-normalized energy-delay product) of <inline-formula> <tex-math notation="LaTeX">$3.72 \ttimes 10^{26}$</tex-math> </inline-formula>, for the 256
row-parallel operation. CIFAR-10 classification is demonstrated by mapping a six-layer convolutional neural network (NN), achieving iso-software accuracy of 90.25%.
All Science Journal Classification (ASJC) codes
• Electrical and Electronic Engineering
• Computer architecture
• Edge computing
• embedded non-volatile memory (eNVM)
• Energy efficiency
• Foundries
• in-memory computing (IMC)
• MRAM
• Parallel processing
• Phase change materials
• Resistance
• scalable architecture
• Signal to noise ratio
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The Application of ENG (Extended Newtonian Gravitation) to Wide Binary Systems
The Application of ENG (Extended Newtonian Gravitation) to Wide Binary Systems ()
1. Introduction
Zwicky published a paper at the beginning of the 1930s [1] where an inconsistency between the galaxy circular speeds and the mass of the Coma galaxy cluster was noticed (using the virial theorem).
One of his interpretations of the problem was the potential existence of dark matter.
In the 1970s, the need for the so coined dark matter in the outer part of galaxies and well beyond their optical edge was noticed [2]. Later on, missing mass problems were also found in galaxy
clusters [3].
Up to this point, the need for dark matter was noticed only in large-scale cosmic structures. However, in 2011, Hernandez et al. [4] found a non-Newtonian behavior of the velocity difference between
binary stars with large separation distances (greater than about 0.1 pc) in the Hipparcos and Gaia catalogues. This finding implies a missing mass problem that is incompatible with the current dark
matter models because the dark matter amount will be very little in such a small distance scale [5] [6].
Since current alternatives to dark matter that modify Newton’s gravitation have not solved completely the missing mass problem, reference [7] extended the Newtonian gravitation (ENG) to reproduce the
non-Newtonian behavior of rotation curves in galaxies which yields results similar to MOND (Modified Newtonian Dynamics) [8] [9] for galaxies and larger circular speeds than MOND results in a
simulated hypothetic compact cluster of galaxies. Note that it is in galaxy clusters where MOND still has a missing mass problem [3].
The application of the ENG model to wide binary systems yielded results compatible with reported experimental results as will be seen later.
MOND (as an inertial modification of Newton’s 2^nd law) failed to yield its non-Newtonian behavior (where the Newtonian acceleration is much smaller than MOND’s characteristic acceleration) when the
external field effect was considered. Without the external field effect, the non-Newtonian behavior is restored but with an asymptotic velocity difference significantly smaller than the experimental
The main purpose of this manuscript is to determine if the ENG model (developed previously by the author for galaxies and galaxy clusters) could explain the missing mass problem detected in wide
binary systems. The results of this research confirm its applicability.
This paper has the following structure: Section 2 describes the equations for the velocity difference of binary stars for the Newtonian, Mondian, and ENG models. Section 3 shows a comparison of the
results of the models in question. In Section 4, the summary and concluding remarks are presented.
2. ENG in Wide Binary Systems
Before describing the ENG model the Newtonian and Mondian models are described for a better illustration of the problem and models in question following similar approach as the description presented
in [5] [6].
The magnitude of the velocity of a component of a binary star system of equal masses using Newtonian theory (gravitational and inertial force) can be written as
The magnitude of the velocity difference between the components is then written as
$\Delta v=\sqrt{\frac{2GM}{r}}$(1)
G: Newton gravitational constant;
M: Mass of the stars;
r: Separation between the stars.
Note that the velocity direction of one star is the opposite of the other one and therefore $\Delta v=2v$.
The balance between the gravitational and the Mondian inertial acceleration for a component of the binary system is written as
a: Newtonian acceleration;
a[0]: MOND characteristic acceleration ${a}_{0}\approx \frac{c{H}_{0}}{6}~1.2\text{E}\text{ }-\text{ }10\text{\hspace{0.17em}}\text{m}/{\text{s}}^{\text{2}}$ ;
a[e]: Acceleration to take into account MOND external field effect [5] [6].
Note that the standard interpolation function is used.
That balance can be express as a quadratic equation:
Using circular acceleration and equal mass the following Quartic equation is obtained:
Making the substitution ${v}^{\prime }={v}^{2}$ that equation is converted to a quadratic one the solution of which is
${v}^{\prime }=\frac{1}{4}\left(\frac{GM}{r}-{a}_{e}r±\sqrt{{\left({a}_{e}r-\frac{GM}{r}\right)}^{2}+4GM\left({a}_{e}+{a}_{0}\right)}\right)$
from which
The velocity difference is $\Delta v=2v$ where the positive sign is taken in the inner square root of v.
In the ENG model [7] the balance between the inertial and gravitational acceleration for a non-relativistic star is written as
$a=G\frac{M}{{r}^{2}}+{G}_{1}\frac{M}{r}$, ${G}_{1}=f\left(M\right)\ll G$
where the following expression was obtained for disk galaxies:
${G}_{1}=\frac{\pi G}{2\sqrt{2M}}⇒{v}_{a}^{4}={\left(\frac{\pi G}{2}\right)}^{2}\frac{M}{2}$
The Baryonic Tully-Fisher relation for the asymptotic speed.
The values of ${v}_{a}$ calculated in this way are close to the binned experimental data shown in the next section.
Notice that even though G[1] has a mass dependency it is not a free parameter. It is supposed to be obtained from experimental data and/or experimentally verified correlations.
The value of G[1] for the wide binary systems was obtained from an extrapolation of a power fit to the binned experimental data of galaxies and galaxy clusters as will be seen in the next section.
For convenience, the explicit mass dependency of G[1] will not be incorporated into the equation of the velocity difference that follows.
For circular acceleration and equal masses:
The velocity difference is then
$\Delta v=2v=\sqrt{2M\left(\frac{G}{r}+{G}_{1}\right)}$(3)
For large r $\Delta v=\sqrt{2M{G}_{1}}$ (an asymptotic velocity difference).
3. Computational Results and Analysis
Table1 shows the binned experimental data of the baryonic mass and the asymptotic circular speeds for galaxies and galaxy clusters reported in [10]. That Tablealso shows the value of G[1] (m^2・s^−2
・kg^−1) corresponding to the binned data ( ${G}_{1}={v}_{c}^{2}/{M}_{b}$ ) along with a power law fit to G[1] (see Figure 1).The evaluation of G[1] fit for the mass of the sun yields a value of
1.7103 × 10^−25.
Table 2 and Figure 2 show the asymptotic circular velocity difference for the 3 models described in the previous section.
MOND external field effect was considered taking ${a}_{e}=1×{10}^{-10}\text{m}/{\text{s}}^{\text{2}}$ corresponding to the acceleration of the sun around the center of the Milky Way [5] [6]. Note
that the external field effect makes the asymptotic behavior of MOND disappear (it is almost Newtonian). So MOND behavior (using the cited acceleration (a[e]) in the solar neighborhood) is
incompatible with the experimental trend results for the binary system reported in [4] [11]. This MOND behavior was previously noticed by [5] [6]. Note that without the external field effect MOND
yields an asymptotic speed of 0.5 km/s with a monotonous decreasing speed profile. Figure 3 shows the Newtonian acceleration for comparison with a[0].
The ENG model shows a distinctive asymptotic behavior. Its velocity profile is very different from the Newtonian model even for relatively smaller separations. Notice that ENG yields an asymptotic
circular speed of 0.8 km/s, which is about the same value obtained in [11] and shown in [5] [6].
Figure 1. G[1] (m^2・s^−2・kg^−1) vs. M (solar mass). Power fit to the binned (G[1]) data. Both axes are in Log10 scale.
Figure 2. Asymptotic circular velocity difference (km/s) vs. separation between binary stars (pc, Log10 scale). From top to bottom: ENG (red), MOND (no external field, green), MOND (magenta), Newton
Figure 3. Newtonian acceleration (m/s^2) vs. Separation (pc). Log 10 scale for both axes.
Table 1. Binned experimental data: Baryonic mass and Asymptotic circular speed [10]. G[1] (m^2・s^−2・kg^−1): Calculated from the binned data along with its fit to a power law.
Table 2. Asymptotic circular velocity difference vs. separation between binary stars.
Asymptotic speed of future precise experiments concerning the wide binary systems could be used to extend the binned data of Table 1 to an even lower mass range from which ${G}_{1}\left({v}^{2}/M\
right)$ could be obtained. In this case the speed profile (before it levels off) calculated using the ENG model will be the prediction to be tested by precise measurements.
It is noted that the MNG [6] model has an asymptotic behavior yielding also an asymptotic speed about 0.8 km/s however it has a non-monotonous profile in the region between ~0.01 and ~0.1 pc and it
has 3 free parameters.
4. Summary and Concluding Remarks
Three models (Newtonian, MOND, ENG) were applied to wide binary systems with an individual star mass of a solar mass to calculate the stars’ velocity difference.
The Newtonian and MOND (considering its external field effect) models showed similar results which significantly deviate from the experimental values. The non-Newtonian behavior of MOND for very low
acceleration is restored if the external field effect is not considered but its asymptotic velocity difference deviates significantly from the reported asymptotic experimental values.
The ENG model yielded an asymptotic velocity difference close to the reported experimental values.
It is important to reduce the uncertainty of the experiments concerning the velocity difference of the wide binary system because it could provide a solid way to confirm (or falsify) physical
hypotheses concerning dark matter and dark physics (non-Newtonian behavior). Note that it has passed about 11 years since the apparent missing mass problem was first noticed in wide binary systems. | {"url":"https://scirp.org/journal/paperinformation?paperid=118116","timestamp":"2024-11-04T15:14:14Z","content_type":"application/xhtml+xml","content_length":"110306","record_id":"<urn:uuid:fc47f8bf-b9d3-4ff9-9f06-aaff68c35e60>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00785.warc.gz"} |
Equaffine --- Introduction ---
A line in the plane can be described under one of 4 following forms: an explicit function, an implicit equation, a pair of parametric equations, or two points on the line. A situation that
generalises over affine subspaces of higher dimension (such as a line or a plane in the space).
This exercise can present on such subspace under one of the forms of the description, and ask you to describe it under another form. With the variation of dimension, it can be used either in very
elementary levels (line in the plane), or right up to situations requiring complicated computations of linear algebra.
Note that the server does not give standard solutions, as the solutions are in general not unique.
Attention. The condition on integer coordinates/coefficients may considerably increase the difficulty of the problem.
This page is not in its usual appearance because WIMS is unable to recognize your web browser.
Please take note that WIMS pages are interactively generated; they are not ordinary HTML files. They must be used interactively ONLINE. It is useless for you to gather them through a robot program.
• Description: rewrite a line/plane/hyperplane: by points or by explicit/implicit/parametric equations. This is the main site of WIMS (WWW Interactive Multipurpose Server): interactive exercises,
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Quantum Algorithms for Lattice Problems
Paper 2024/555
Quantum Algorithms for Lattice Problems
We show a polynomial time quantum algorithm for solving the learning with errors problem (LWE) with certain polynomial modulus-noise ratios. Combining with the reductions from lattice problems to LWE
shown by Regev [J.ACM 2009], we obtain polynomial time quantum algorithms for solving the decisional shortest vector problem (GapSVP) and the shortest independent vector problem (SIVP) for all
$n$-dimensional lattices within approximation factors of $\tilde{\Omega}(n^{4.5})$. Previously, no polynomial or even subexponential time quantum algorithms were known for solving GapSVP or SIVP for
all lattices within any polynomial approximation factors. To develop a quantum algorithm for solving LWE, we mainly introduce two new techniques. First, we introduce Gaussian functions with complex
variances in the design of quantum algorithms. In particular, we exploit the feature of the Karst wave in the discrete Fourier transform of complex Gaussian functions. Second, we use windowed quantum
Fourier transform with complex Gaussian windows, which allows us to combine the information from both time and frequency domains. Using those techniques, we first convert the LWE instance into
quantum states with purely imaginary Gaussian amplitudes, then convert purely imaginary Gaussian states into classical linear equations over the LWE secret and error terms, and finally solve the
linear system of equations using Gaussian elimination. This gives a polynomial time quantum algorithm for solving LWE.
Note: Update on April 18: Step 9 of the algorithm contains a bug, which I don’t know how to fix. See Section 3.5.9 (Page 37) for details. I sincerely thank Hongxun Wu and (independently) Thomas
Vidick for finding the bug today. Now the claim of showing a polynomial time quantum algorithm for solving LWE with polynomial modulus-noise ratios does not hold. I leave the rest of the paper as it
is (added a clarification of an operation in Step 8) as a hope that ideas like Complex Gaussian and windowed QFT may find other applications in quantum computation, or tackle LWE in other ways.
Available format(s)
Publication info
Contact author(s)
chenyilei ra @ gmail com
2024-04-19: revised
2024-04-10: received
Short URL
author = {Yilei Chen},
title = {Quantum Algorithms for Lattice Problems},
howpublished = {Cryptology {ePrint} Archive, Paper 2024/555},
year = {2024},
url = {https://eprint.iacr.org/2024/555} | {"url":"https://eprint.iacr.org/2024/555","timestamp":"2024-11-14T14:10:56Z","content_type":"text/html","content_length":"15666","record_id":"<urn:uuid:44a823a7-935c-44a9-a049-817cb77316b7>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00614.warc.gz"} |
Approximating pstar and ustar
The Godunov method relies on the solution of pressure and velocity inside of the star region set up within local Riemann problems, which before now, has been done using an iterative approach inside
of the Exact Riemann Solver. The iterative approach is computationally costly, and so if we could devise schemes which approximate pstar and ustar, we can make a more efficient time marching
Linearization of the Euler equations
There are 2 methods which make use of approximating the Euler equations themselves in linearized form.
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energy stored and not stored
Two capacitors are in parallel and the energy stored is 45J, when the combination is raised to potential of 3000 V. with the same two capacitors in series, t
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About energy stored and not stored
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Seven Flipped Poster
Poster based on Seven Flipped task.
The poster is available as a
, or the image below can be clicked on to enlarge it.
Student Solutions
In this problem it doesn't matter where you start on the picture.
1) Pick any 3 hexagons (they will turn blue)
2) Pick 2 more hexagons and then pick one that you chose in the first go. You should now have 4 hexagons blue.
3) Finally pick the three remaining red hexagons.
So, it will take three moves.
If the number of tiles is a multiple of 3, the number of moves is the number of tiles divided by 3.
If the number of tiles is one more than a multiple of 3, you add 1 to the number of moves for the number of tiles immediately below it. For example, for 6 tiles it is 3 moves, for 7 tiles it is 4
If the number of tiles is two more than a multiple of 3, you add 2 to the number of moves for the number of tiles immediately below it. For example, for 6 tiles it is 3 moves, for 8 tiles it is 5 | {"url":"https://nrich.maths.org/problems/seven-flipped-poster","timestamp":"2024-11-11T01:18:14Z","content_type":"text/html","content_length":"37991","record_id":"<urn:uuid:0f9c1d2f-5abb-4aa0-8910-0fc0b95861f7>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00093.warc.gz"} |
2023 UPDATE: California Lottery - Scratchers - picking the winning tickets using math - Ivan Kuznetsov
2023 UPDATE: Full analysis in this spreadsheet has been updated to reflect the latest products offered by California Lottery.
I’ve always been fascinated by how the lottery works. From a mathematical point of view spending money on lottery tickets is a complete waste of time and money. There are a few exceptions – when a
lottery is poorly designed, it is possible to game the system and actually earn money – here’s a famous example of how MIT Students Won $8 Million in the Massachusetts Lottery.
The North American lottery system is a $70 billion-a-year business, an industry bigger than movie tickets, music, and porn combined. So general public seems to ignore the probability theory and
continues to spend money in pursuit of the elusive multi-million grand prizes of the numerous lotteries.
I decided to take a closer look at California State Lottery and came across a very nice analysis done by the Wizard of Odds. I do wholeheartedly agree with the advice given by the Wizard: “Don’t play
in the first place. Every state lottery offers terrible odds. With few exceptions, it is the worst bet you can make.”
But I was intrigued by the following passage in the article: “The California Lottery is nice enough to indicate how many tickets for each win have already been cashed. If there is a game that is
almost sold out, as evidenced by the small wins, with a high ratio of large wins still unclaimed, then it may mean the remaining unsold tickets are rich in big winners. The same principle as card
counting in blackjack.”
I decided to take a close look at the odds of winning and the average return for each of the scratch card games CA Lottery offers and see if there are substantial changes in probability of winning
and return on investment with the additional data provided by state lottery.
Here is a summary of my findings:
1. More expensive scratchers have higher odds of winning and higher return:
│Scratch Card Average Returns │
│Bet │Average Return │
│$1 │52.49% │
│$2 │56.06% │
│$3 │57.57% │
│$5 │62.37% │
│$10 │67.75% │
│$20 │69.02% │
│$30 │72.63% │
2. There is indeed a potential advantage play in scratch card games, as with the information about the number of claimed prizes that CA Lottery publishes it is possible to calculate updated winning
odds and returns – see the spreadsheet for details.
3. Over time changes in the odds of winning and expected return can go up or down ~20%, so if you gamble, it definitely makes sense to run the numbers first
4. As of February 10, 2018 Set For Life scratchers have the highest estimated return of 75.65%
5. As of February 10, 2018 $10 Million Dazzler scratchers have the highest estimated chance of winning of 36.5%
6. In the unlikely event you win a grand prize – be aware of the fine print – the lottery is not going to pay it to you straight away. You’ll have a chance of taking home ~1/2 of it before taxes
immediately or getting it paid out as installments over 25 years or so.
My full analysis is in this spreadsheet – it automatically downloads the latest stats on claimed prizes and winning odds from calottery.com, so results you see there might differ from the examples I
provided above.
And as a closing note – my general advice is “do not gamble”. If you do, then California Lottery is a good choice – its mission is to maximize supplemental funding for the state’s public schools,
which is very similar to what we have as a mission for Veikkaus – the Finnish national betting agency. At least the money you lose will go to a good cause.
15 responses to “2023 UPDATE: California Lottery – Scratchers – picking the winning tickets using math”
Your analysis here seems to assume that tickets are claimed at an equal rate. Small prizes (which can be claimed at any lottery retailer) can be claimed in a shorter amount of time than large
prizes (which have to be mailed in). So your data will make it seem like there are a higher proportion of large prizes available than is true in reality.
It’s a f****** jip joint.
I’m not sure when, but as of current, the CA Lottery website changed dramatically. The details it gives on prizes claimed and unclaimed are superficial at best now.
this world we live in is outrageous high to survive these days you would think lottery instead having one winner of 200 million why couldn’t be 500 people or even 200 split pot that way wealth
gets spread out more
I agree with David Tolbert 100%
This analyses is useless. Winning tickets are NOT distributed evenly or proportionally to all the counties, cities, stores, gas stations, etc.
ValS i completely agree!!
What David d Tolbert said is something I’ve thought about for such a long time. Does one person really need 200 million? If you split that spread it out so that 500 people could receive a share
of that would make everyone happy. But! This is America and everyone has to be greedy.
California lottery hides important information from the customer. I don’t see the total number of dollars spent on each game or the complete odds per game level. They will print the overall odds
like 1-2.97 but not the odds of winning the big prize and others where the odds could be 1-2000000 or higher. What a rip off.
I’m not sure with the percentage of return that is shown above. Is there any article or study that can support the data? It’s somewhat out of reality.
Ca lotto scratcher odds are totally BOGUS!!! Such a rip! Their odds are sooo off its not even funny but an actual RIP!! I have over 8,000 yes 8,000 loser tickets since 2017 and most ive ever won
was $500 just a few times!! As far as 2nd chance goes…. IT DOESN’T!!! Biggest joke of all jus to hold on to disappointed memories and keep ur trash instead of gettin angry n tossin em carelessly
into tha street!
Best advise is still …best advise, don’t gamble. You win EVERYTIME !
That being said, I think we all want to take a shot at the big enchilada at least once.
Might as well go big. Two items on your list of things to do will be checked.
Throwing caution to the wind and burning money. ….but, the possibilities. Good luck !
There is no need to trick consumers by offering tickets that all 10 prizes can be won so if you have a 5 dollar ticket and you happen to win all 10 prizes then how much is the minimum amount you
will win?
Lets see, If the back of card reads PRIZES RANGE FROM A FREE TICKET TO 30,000 DOLLARS then it appears you could win 10 free tickets at a minimum.worth $50. Hell no! they lie and make each prize $
1.00 so really it should read PRIZES RANGE FROM $ 1.00 TO 30,000 DOLLARS. there is no need to trick us ! So they brag how many tickets they produce with 10 prize winners on one ticket and really
its all a scam to get more to pay the lottery top employees vacatiopns
I’ve played powerball and mega many times noticed my numbers I picked off by 2 , like the Computer knows and it gives you false hopes to keep you playing. All of a sudden 1 winning ticket in
which person probably doesn’t exist and pot goes back to restart , poof the millions of dollars gone. They’re probably laughing about it cause there’s no way to prove it’s a scam.
Can you kindly provide an updated chart for 2024 I find your chart within a proper percent of accuracy Thanks
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Calculating Continuously Compounded Rates of Returns
Abdulla Javeri
30 years: Financial markets trader
Continuously compounded rates of return are widely used in financial markets, especially in the world of derivatives. In this video, Abdulla outlines the concept and provides some examples framed as
investment returns.
Continuously compounded rates of return are widely used in financial markets, especially in the world of derivatives. In this video, Abdulla outlines the concept and provides some examples framed as
investment returns.
Access this and all of the content on our platform by signing up for a 7-day free trial.
Calculating Continuously Compounded Rates of Returns
Key learning objectives:
• Define continuous compounding
• Calculate both the continuously compounded rate of return, and the future value of an investment
Continuously compounded rates of return are widely used in financial markets, especially in the world of derivatives. They are typically useful when dealing with returns on assets whose price cannot
fall below zero.
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What are investment returns?
• Usually measured as a percentage rate
• Can be calculated for the holding period, or on an annualised basis
• Can be calculated on a discrete basis, or on a continuously compounded basis
What is a discrete return?
If we start with an initial investment of 100, which a year later is worth 110, we’ve earned a discrete return of 10% a year. A profit of 10, on our original starting point of 100.
What is continuous compounding?
Continuously compounding is essentially investing for an immeasurably small period, and re-investing both principal and return for an infinite number of immeasurably small periods until the end of
the year, and we need to find the rate that gets us from 100 to 110.
What is the generic compounding formula, and why won’t it work?
P x (1+r/n)^n x t = FV
Can be re-arranged to solve for r
r = ((FV/P)^(1/(n x t))-1) x n
With continuous compounding, n approaches infinity - so this formula won’t work.
What formula can we use that fixes the n = infinity issue?
P x E^r x t = FV
Can be re-arranged to solve for r
r = In(FV/P)/t
Using the information below, how do we calculate the compounded rate of return and the future value?
• P = Principal (100)
• r = Annual rate of return
• n = Number of compounding periods in a year
• r/n = Rate for the period - the periodic rate
• t = Time in years (1)
• n x t = Number of compounding periods
• FV = Future value of principal (110)
1. Continuously compounded rate of return: ln(110/100)/1 = 0.953102. Hence, if we invest at about 9.53% a year, on a continuous basis, we will move from 100 at the beginning of the year to 100 at
the end of the year.
2. Future Value (FV): 100(e^0.953102) = 110
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Abdulla Javeri
Abdulla’s career in the financial markets started in 1990 when he entered the trading floor of the London International Financial Futures Exchange, LIFFE, and qualified as a pit trader in equity and
equity index options. In 1996, Abdulla became a trainer for regulatory qualifications and then for non-exam courses, primarily covering all major financial products.
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Macaulay Duration Calculator: Optimize Bond Investment Strategies
Unlock the power of bond investing with our Macaulay Duration Calculator. Discover how this essential tool can help you assess interest rate risk, compare bonds, and optimize your portfolio. From
beginners to seasoned investors, elevate your fixed-income strategy today. Ready to make smarter bond decisions?
Macaulay Duration Calculator
Thanks for your feedback!
Macaulay Duration Calculator: Optimize Your Bond Investment Strategy
How to Use the Macaulay Duration Calculator Effectively
Our Macaulay Duration Calculator is designed to help investors, financial analysts, and bond enthusiasts accurately determine the weighted average time to receive the present value of a bond’s cash
flows. Follow these simple steps to use the calculator effectively:
1. Enter the Face Value of the Bond: Input the par value or face value of the bond in USD.
2. Input the Coupon Rate: Enter the annual coupon rate as a decimal (e.g., 0.05 for 5%).
3. Select the Maturity Date: Choose the date when the bond will mature.
4. Choose the Coupon Frequency: Select how often coupon payments are made (annually, semi-annually, or quarterly).
5. Enter the Discount Rate: Input the market yield or discount rate as a decimal.
6. Calculate: Click the “Calculate” button to obtain the Macaulay Duration result.
Understanding Macaulay Duration: Definition, Purpose, and Benefits
Macaulay Duration, named after Frederick Macaulay who developed the concept in 1938, is a crucial metric in fixed-income investing. It measures the weighted average time until all cash flows from a
bond are received, providing investors with valuable insights into bond price sensitivity and investment risk.
Macaulay Duration is defined as the weighted average term to maturity of the cash flows from a bond. It is expressed in years and represents the time it takes to recover the true cost of a bond,
considering both the present value of future coupon payments and the principal.
The primary purpose of calculating Macaulay Duration is to assess the interest rate risk of a bond or fixed-income portfolio. It helps investors understand how changes in interest rates might affect
bond prices, enabling them to make informed decisions about their investment strategies.
• Provides a single number to compare bonds with different maturities and coupon rates
• Helps in immunizing bond portfolios against interest rate risk
• Assists in yield curve analysis and bond pricing
• Facilitates better risk management in fixed-income investments
• Aids in the construction of bond ladders and portfolio optimization
The Mathematical Formula Behind Macaulay Duration
The Macaulay Duration is calculated using the following formula:
$$ D = \frac{\sum_{t=1}^{n} \frac{t \cdot C_t}{(1+r)^t}}{\sum_{t=1}^{n} \frac{C_t}{(1+r)^t}} $$
• D = Macaulay Duration
• t = Time period
• C[t] = Cash flow at time t
• r = Yield to maturity (discount rate)
• n = Total number of periods
Benefits of Using the Macaulay Duration Calculator
Our Macaulay Duration Calculator offers numerous advantages for investors and financial professionals:
• Time-saving: Quickly compute Macaulay Duration without complex manual calculations.
• Accuracy: Minimize human error and ensure precise results for better decision-making.
• Flexibility: Easily adjust input parameters to analyze various bond scenarios.
• Educational tool: Understand the relationship between bond characteristics and duration.
• Risk assessment: Evaluate interest rate sensitivity of bonds or bond portfolios.
• Portfolio management: Facilitate bond selection and portfolio construction strategies.
• Comparative analysis: Quickly compare durations of different bonds or portfolios.
Addressing User Needs: How the Macaulay Duration Calculator Solves Specific Problems
Our Macaulay Duration Calculator addresses several key challenges faced by investors and financial analysts:
1. Assessing Interest Rate Risk
Problem: Investors need to understand how changes in interest rates might affect their bond investments.
Solution: The calculator provides a clear measure of a bond’s sensitivity to interest rate changes. A higher Macaulay Duration indicates greater sensitivity, allowing investors to adjust their
portfolios accordingly.
2. Comparing Bonds with Different Characteristics
Problem: It’s challenging to compare bonds with varying coupon rates, face values, and maturities.
Solution: Macaulay Duration offers a standardized metric for comparison, enabling investors to evaluate different bonds on a level playing field.
3. Portfolio Immunization
Problem: Investors need to protect their portfolios against interest rate fluctuations.
Solution: By calculating the Macaulay Duration of individual bonds, investors can construct portfolios with a target duration that matches their investment horizon, effectively immunizing against
interest rate risk.
4. Yield Curve Analysis
Problem: Understanding how changes in the yield curve affect bond prices can be complex.
Solution: Macaulay Duration helps investors analyze the impact of yield curve shifts on bond prices, facilitating more informed investment decisions.
Practical Applications: Examples and Use Cases
Let’s explore some practical applications of the Macaulay Duration Calculator through examples:
Example 1: Comparing Two Bonds
Suppose an investor is considering two bonds:
• Bond A: $1,000 face value, 4% coupon rate, 5-year maturity, semi-annual payments
• Bond B: $1,000 face value, 6% coupon rate, 3-year maturity, annual payments
Assuming a discount rate of 5% for both bonds, we can use the calculator to determine their Macaulay Durations:
• Bond A Macaulay Duration: 4.55 years
• Bond B Macaulay Duration: 2.83 years
Interpretation: Bond A has a higher duration, indicating it’s more sensitive to interest rate changes. If the investor expects interest rates to fall, Bond A might be preferable as its price would
increase more than Bond B’s. Conversely, if rates are expected to rise, Bond B might be a safer choice.
Example 2: Portfolio Immunization
An institutional investor has a liability due in 5 years and wants to immunize their portfolio against interest rate risk. They can use the Macaulay Duration Calculator to select bonds that, when
combined, have an average duration of 5 years.
For instance, they might choose:
• 60% allocation to a bond with a duration of 4 years
• 40% allocation to a bond with a duration of 6.5 years
The weighted average duration would be: (0.6 * 4) + (0.4 * 6.5) = 5 years, matching their investment horizon and providing immunization against interest rate changes.
Example 3: Yield Curve Analysis
An analyst is studying the impact of a potential parallel shift in the yield curve. They calculate the Macaulay Duration for a 10-year bond with a 5% coupon rate, semi-annual payments, and a current
yield of 4.5%:
Macaulay Duration: 8.36 years
If the yield curve shifts up by 0.5%, the analyst can estimate the bond’s price change using the approximation:
$$ \text{Price Change} \approx -\text{Duration} \times \text{Yield Change} $$
Estimated price change: -8.36 * 0.005 = -0.0418 or -4.18%
This quick estimation helps the analyst understand the potential impact of yield curve shifts on bond prices without complex calculations.
Frequently Asked Questions (FAQ)
1. What is the difference between Macaulay Duration and Modified Duration?
Macaulay Duration measures the weighted average time to receive cash flows, while Modified Duration measures the price sensitivity of a bond to changes in interest rates. Modified Duration is derived
from Macaulay Duration by dividing it by (1 + yield to maturity).
2. How does coupon rate affect Macaulay Duration?
Generally, bonds with higher coupon rates have shorter Macaulay Durations because a larger portion of the bond’s value is received earlier through coupon payments. Conversely, lower coupon rates
result in longer durations.
3. Why is Macaulay Duration important for bond investors?
Macaulay Duration helps investors assess interest rate risk, compare bonds with different characteristics, and construct portfolios that match their investment horizons. It’s a crucial tool for risk
management in fixed-income investing.
4. Can Macaulay Duration be negative?
No, Macaulay Duration cannot be negative. It represents a weighted average time, which is always positive for conventional bonds. However, for certain complex financial instruments, duration can
theoretically be negative.
5. How often should I recalculate Macaulay Duration for my bond portfolio?
It’s advisable to recalculate Macaulay Duration periodically, especially when market conditions change significantly or when you make changes to your portfolio. Many investors recalculate quarterly
or semi-annually, but more frequent calculations may be necessary in volatile markets.
6. Is a higher or lower Macaulay Duration better?
Neither is inherently better; it depends on your investment strategy and market expectations. Higher duration bonds are more sensitive to interest rate changes, offering greater potential returns but
also higher risk. Lower duration bonds are less sensitive, providing more stability but potentially lower returns.
7. Can Macaulay Duration be used for stocks or other non-fixed income securities?
Macaulay Duration is primarily used for fixed-income securities like bonds. While similar concepts can be applied to other securities, such as dividend-paying stocks, the traditional Macaulay
Duration calculation is not directly applicable to non-fixed income investments.
Please note that while we strive for accuracy and reliability, we cannot guarantee that our webtool or the results it provides are always correct, complete, or reliable. Our content and tools may
contain errors, biases, or inconsistencies. Always consult with a qualified financial professional before making investment decisions.
Conclusion: Empowering Your Bond Investment Strategy
The Macaulay Duration Calculator is an invaluable tool for investors, financial analysts, and anyone involved in fixed-income markets. By providing a clear measure of a bond’s sensitivity to interest
rate changes, it empowers users to make more informed investment decisions, manage risk effectively, and optimize their bond portfolios.
Key benefits of using our Macaulay Duration Calculator include:
• Quick and accurate calculations
• Enhanced understanding of bond characteristics
• Improved risk assessment and management
• Facilitation of portfolio immunization strategies
• Better comparison of bonds with different features
• Support for yield curve analysis and bond pricing
By incorporating Macaulay Duration into your investment analysis toolkit, you’ll be better equipped to navigate the complex world of fixed-income investing, make data-driven decisions, and
potentially enhance your investment returns.
Take advantage of our user-friendly Macaulay Duration Calculator today and elevate your bond investment strategy to new heights. Whether you’re a seasoned professional or a novice investor, this
powerful tool will provide valuable insights to support your financial goals.
Ready to optimize your bond investments? Start using our Macaulay Duration Calculator now and gain a competitive edge in the fixed-income market!
Important Disclaimer
The calculations, results, and content provided by our tools are not guaranteed to be accurate, complete, or reliable. Users are responsible for verifying and interpreting the results. Our content
and tools may contain errors, biases, or inconsistencies. We reserve the right to save inputs and outputs from our tools for the purposes of error debugging, bias identification, and performance
improvement. External companies providing AI models used in our tools may also save and process data in accordance with their own policies. By using our tools, you consent to this data collection and
processing. We reserve the right to limit the usage of our tools based on current usability factors. By using our tools, you acknowledge that you have read, understood, and agreed to this disclaimer.
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Lesson 18
Surface Area of a Cube
18.1: Exponent Review (5 minutes)
In this warm-up, students compare pairs of numerical expressions and identify the expression with the greater value. The task allows students to review what they learned about exponents and prompts
them to look for and make use of structure in numerical expressions (MP7).
Students should do these without calculators and without calculating, although it is fine for them to check their answers with a calculator.
Give students 1–2 minutes of quiet think time. Ask them to answer the questions without multiplying anything or using a calculator, and to give a signal when they have an answer for each question and
can explain their reasoning.
Student Facing
Select the greater expression of each pair without calculating the value of each expression. Be prepared to explain your choices.
• \(10 \boldcdot 3\) or \(10^3\)
• \(13^2\) or \(12 \boldcdot 12\)
• \(97+97+97+97+97+97\) or \(5 \boldcdot 97\)
Anticipated Misconceptions
When given an expression with an exponent, students may misinterpret the base and the exponent as factors and multiply the two numbers. Remind them about the meaning of the exponent notation. For
example, show that \(5 \boldcdot 3\) = 15, which is much smaller than \(5 \boldcdot 5 \boldcdot 5\), which equals 125.
Activity Synthesis
Ask one or more students to explain their reasoning for each choice. If not mentioned in students’ explanations, highlight the structures in the expressions that enable us to evaluate each one
without performing any calculations.
Point out, for example, that since we know that \(10^3\) means \(10 \boldcdot 10 \boldcdot 10\), we can tell that it is much larger than \(10 \boldcdot 3\).
For the last question, remind students that we can think of repeated addition in terms of multiple groups (i.e., that the sum of six 97s can be seen as six groups of 97 or \(6 \boldcdot 97\)). The
idea of using groups to write equivalent expressions will support students as they write expressions for the surface area of a cube later in the lesson (i.e., writing the areas of all square faces of
a cube as \(6s^2\)) .
18.2: The Net of a Cube (20 minutes)
This activity contains two sets of problems. The first set involves computations with simple numbers and should be solved numerically. Use students’ work here to check that they are drawing a net
The second set encourages students to write expressions rather than to simplify them through calculations. The goal is to prepare students for the general rules \(s^3\) and \(6s^2\), which are more
easily understood through an intermediate step involving numbers.
Note that students will be introduced to the idea that \(5 \boldcdot x\) means the same as \(5x\) in a later unit, so expect them to write \(6 \boldcdot 17^2\) instead of \(6 (17^2)\). It is not
critical that they understand that a number and a variable (or a number and an expression in parentheses) placed next to each other means they are being multiplied.
As students work on the second set, monitor the ways in which they write their expressions for surface area and volume. Identify those whose expressions include :
• products (e.g., \(17 \boldcdot 17\) or \(17 \boldcdot 17 \boldcdot 17\)),
• sums of products (e.g., \((17 \boldcdot 17)+(17 \boldcdot 17)+…\)),
• combination of like terms (e.g., \(6 \boldcdot(17 \boldcdot 17)\)),
• exponents (e.g., \(17^2 + 17^2 +…\)) or \(17^3\)), and
• completed calculation (e.g., \(289\)).
Select these students to share their work later. Notice the lengths of the expressions and sequence their explanations in order—from the longest expression to the most succinct.
Arrange students in groups of 2. Give students access to their geometry toolkits and 8-10 minutes of quiet work time. Tell students to try to answer the questions without using a calculator. Ask them
to share their responses with their partner afterwards.
Representation: Develop Language and Symbols. Activate or supply background knowledge about calculating surface area and volume. Share examples of expressions for a cube in a few different forms to
illustrate how surface area and volume can be expressed. Allow continued access to concrete manipulatives such as snap cubes for students to view or manipulate.
Supports accessibility for: Visual-spatial processing; Conceptual processing
Student Facing
1. A cube has edge length 5 inches.
1. Draw a net for this cube, and label its sides with measurements.
2. What is the shape of each face?
3. What is the area of each face?
4. What is the surface area of this cube?
5. What is the volume of this cube?
2. A second cube has edge length 17 units.
1. Draw a net for this cube, and label its sides with measurements.
2. Explain why the area of each face of this cube is \(17^2\) square units.
3. Write an expression for the surface area, in square units.
4. Write an expression for the volume, in cubic units.
Anticipated Misconceptions
Students might think the surface area is \((17 \boldcdot 17)^6\). Prompt students to write down how they would compute surface area step by step, before trying to encapsulate their steps in an
expression. Dissuade students from using calculators in the last two problems and assure them that building an expression does not require extensive computation.
Students may think that refraining from using a calculator meant performing all calculations—including those of larger numbers—on paper or mentally, especially if they are unclear about the meaning
of the term “expression.” Ask them to refer to the expressions in the warm-up, or share examples of expressions in a few different forms, to help them see how surface area and volume can be expressed
without computation.
Activity Synthesis
After partner discussions, select a couple of students to present the solutions to the first set of questions, which should be straightforward.
Then, invite previously identified students to share their expressions for the last two questions. If possible, sequence their presentation in the following order. If any expressions are missing but
needed to illustrate the idea of writing succinct expressions, add them to the lists.
Surface area:
• \((17 \boldcdot 17)+(17 \boldcdot 17)+(17 \boldcdot 17)+(17 \boldcdot 17)+(17 \boldcdot 17)+(17 \boldcdot 17)\)
• \(17^2+17^2+17^2+17^2+17^2+17^2\)
• \(6 \boldcdot(17 \boldcdot 17)\)
• \(6 \boldcdot (17^2)\)
• \(6 \boldcdot (289)\)
• 1,734
• \(17 \boldcdot 17 \boldcdot 17\)
• \(17^3\)
• 4,913
Discuss how multiplication can simplify expressions involving repeated addition and exponents can do the same for repeated multiplication. While the last expression in each set above is the simplest
to write, getting there requires quite a bit of computation. Highlight \(6\boldcdot 17^2\) and \(17^3\) as efficient ways to express the surface area and volume of the cube.
As the class discusses the different expressions, consider directing students’ attention to the units of measurements. Remind students that, rather than writing \(6 \boldcdot (17^2)\) square units,
we can write \(6 \boldcdot (17^2)\) units^2, and instead of \(17^3\) cubic units, we can write \(17^3\) units^2. Unit notations will appear again later in the course, so it can also be reinforced
If students are not yet ready for the general formula, which comes next, offer another example. For instance, say: “A cube has edge length 38 cm. How can we express its surface area and volume?”
Help students see that its surface area is \(6 \boldcdot (38^2)\) cm^2 and its volume is \(38^3\) cm^3. The large number will discourage calculation and focus students on the form of the expressions
they are building and the use of exponents.
Representing, Conversing: MLR7 Compare and Connect. Use this routine to prepare students for the whole-class discussion. At the appropriate time, invite groups to create a visual display showing
their strategy and calculations for the surface area and volume of a cube with an edge length of 17 units. Allow students time to quietly circulate and analyze the strategies in at least 2 other
displays in the room. Give students quiet think time to consider what is the same and what is different. Next, ask students to return to their original group to discuss what they noticed. Listen for
and amplify observations that highlight the advantages and disadvantages to each method and their level of succinctness. This will help students make connections between calculations of cubes,
regardless of the edge length.
Design Principle(s): Optimize output; Cultivate conversation
18.3: Every Cube in the Whole World (10 minutes)
In this activity, students build on what they learned earlier and develop the formulas for the surface area and the volume of a cube in terms of a variable edge length \(s\).
Encourage students to refer to their work in the preceding activity as much as possible and to generalize from it. As before, monitor for different ways of writing expressions for surface area and
volume. Identify students whose work includes the following:
• products (e.g., \(s \boldcdot s\), or \(s \boldcdot s \boldcdot s\)),
• sums of products (e.g., \((s \boldcdot s)+(s \boldcdot s)+…\)),
• combination of like terms (e.g., \(6 \boldcdot(s \boldcdot s)\)), and
• exponents (e.g., \(s^2 + s^2 +…\), or \(s^3\)).
Select these students to share their work later. Again, notice the lengths of the expressions and sequence their explanations in order—from the longest expression to the most succinct.
Give students access to their geometry toolkits and 7–8 minutes of quiet think time. Tell students they will be answering on the same questions as before, but with a variable for the side length.
Encourage them to use the work they did earlier to help them here.
Student Facing
A cube has edge length \(s\).
1. Draw a net for the cube.
2. Write an expression for the area of each face. Label each face with its area.
3. Write an expression for the surface area.
4. Write an expression for the volume.
Anticipated Misconceptions
If students are unclear or unsure about using the variable \(s\), explain that we are looking for an expression that would work for any edge length, and that a variable, such as \(s\), can represent
any number. The \(s\) could be replaced with any edge length in finding surface area and volume.
To connect students’ work to earlier examples, point to the cube with edge length 17 units from the previous activity. Ask: “If you wrote the surface area as \(6 \boldcdot 17^2\) before, what should
it be now?”
As students work, encourage those who may be more comfortable using multiplication symbols to instead use exponents whenever possible.
Activity Synthesis
Discuss the problems in as similar a fashion as was done in the earlier activity involving a cube with edge length 17 units. Doing so enables students to see structure in the expressions (MP7) and
to generalize through repeated reasoning (MP8).
Select previously identified students to share their responses with the class. If possible, sequence their presentation in the following order to help students see how the expressions \(6 \boldcdot s
^2\) and \(s^3\) come about. If any expressions are missing but needed to illustrate the idea of writing succinct expressions, add them to the lists.
Surface area:
• \((s \boldcdot s)+(s \boldcdot s)+(s \boldcdot s)+(s \boldcdot s)+(s \boldcdot s)+(s \boldcdot s)\)
• \(s^2+s^2+s^2+s^2+s^2+s^2\)
• \(6(s \boldcdot s)\)
• \(6 \boldcdot (s^2)\) or \(6 \boldcdot s^2\)
• \(s \boldcdot s \boldcdot s\)
• \(s^3\)
Refer back to the example involving numerical side length (a cube with edge length 17 units) if students have trouble understanding where the most concise expression of surface area comes from.
Present the surface area as \(6 \boldcdot s^2\). You can choose to also write it as \(6s^2\).
Lesson Synthesis
Review the formulas for volume and surface area of a cube.
• The volume of a cube with edge length \(s\) is \(s^3\).
• A cube has 6 faces that are all identical squares. The surface area of a cube with edge length \(s\) is \(6 \boldcdot s^2\).
18.4: Cool-down - From Volume to Surface Area (5 minutes)
Student Facing
The volume of a cube with edge length \(s\) is \(s^3\).
A cube has 6 faces that are all identical squares. The surface area of a cube with edge length \(s\) is \(6 \boldcdot s^2\). | {"url":"https://curriculum.illustrativemathematics.org/MS/teachers/1/1/18/index.html","timestamp":"2024-11-04T10:43:54Z","content_type":"text/html","content_length":"101721","record_id":"<urn:uuid:07cb5424-4aa2-4832-9868-7712ba5d62ef>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00148.warc.gz"} |
Algebra 1 Unit 1 Interactive Notebook Pages – The Foundations of Algebra (2024)
Starting the year off right is SO important for any class, but especially in Algebra 1, in particular. Everything that is done in the first unit lays the foundation for everything to come throughout
the rest of the year, so there is is a lot riding on starting the year strong.
Here’s what to include in your first unit of Algebra 1 to start the year off right…
Students NEED to have a strong foundation, or else they’ll be fighting an uphill battle all year, which is no good. I’ve spent a lot of time thinking about what topics are most important for students
to know (from vocabulary to skills), so that each following unit has a strong foundation. Here are all of the notes I used with my students during the 1st unit of Algebra 1.
If you want to look inside any of the pages included in this unit, you can take a look at these topic-specific posts for a more detailed look!
• 1.0 – Notebook Setup
• 1.1 – The Real Number System, Classifying Real Numbers, & Closure
• 1.2 – Properties of Real Numbers
• 1.3 – Order of Operations
• 1.4 – Evaluating Algebraic Expressions
• 1.5 – Combining Like Terms
• 1.6 – The Distributive Property
• 1.7 – Translating Algebraic Expressions, Equations & Inequalities
• 1.8 – Solving 1-Step & 2-Step Equations
• 1.9 – Solving 2-Step Inequalities
How do I set up an interactive notebook?
If you need suggestions about what to put on the first few pages, check out this post.
1.1 – The Real Number System, Classifying Real Numbers & Closure
Classifying real numbers is so much more important that we often give it credit. Math is very much a foreign language class as much it is a class about patterns, persevering, and problem solving.
Without knowing the language of the land, math class can be so much more difficult than it needs to be and needlessly can lose students who are otherwise capable.
I like to start by explicitly teaching vocabulary and helping students look at distinguishing the similarities and differences between sets of numbers. This also embeds several opportunities to
review representations of numbers from middle school.
I’m all about making my notebooks highly visual for students, so including flowcharts are a MUST. I often put them before the notes for a topic as a visual introduction and reference sheet for
students. You can get this classifying real numbers flowchart here.
I always start the next class with a recap warm-up of the prior day’s lesson. I use this as an opportunity to encourage students to start using their notes to refer back to, and I can use it as an
opportunity to catch misconceptions and iron out any lingering questions students might have from the day before.
1.2 – Properties of Real Numbers
Properties of real numbers may seem like a bit of a snooze fest for notes, but I have this foldable set up where students can help build examples to demonstrate each property which makes it much more
lively. This is a great way to help students start thinking mathematically after being on vacation for the summer. Again, the next day we do a recap warm-up where students apply their learning from
the day before and are encouraged to use their notes. This helps to start building the notion that their notebook is meant to be used and referred back to, over and over, day after day.
It’s one thing to say that students should use their notebook as a tool. It’s another to explicitly teach it and integrate it into our daily class routine. Studying and referring back to notes
actually needs to be explicitly taught, and Turing it into a daily warm-up habit goes a long way.
1.3 – Order of Operations
PEMDAS, GEMDAS, GEMS, BEDMAS…whatever you call it, I offer this order of operations “tips for success” page with that acronym. This is a quick reminder for students that emphasizes the different
grouping symbols and the need to work from left to right.
The notes include common mistakes students could make as well as an opportunity to write some “notes to self” after completing each problem that had a potential common mistake in it.
1.4 – Evaluating Algebraic Expressions
Evaluating algebraic expressions is a great topic to include in the first unit of the year. Not only does it reinforce order of operations, but the substitution property that students will need to
use countless times. As always, we start the next day’s class with a recap warm-up to encourage students to refer back to their notes, and to allow me to iron out any misconceptions that may have
1.5 – Combining Like Terms
Combining like terms can seem like such a simple concept to us math teachers, but it is actually pretty tricky for students. This set of notes encourages students to think about what it is that they
are doing, and forces them to face some common misconceptions along the way! Application problems like finding an expression for the perimeter of a shape are included!
1.6 – The Distributive Property
This set of notes on the distributive property is sure to touch on common mistakes students might make…like having a 4+3(x+2) and students wanting to make it a 7(x+2) or what to do when there’s just
a negative sign in front of parentheses like -(5x-4). These all trip students up, so we intentionally practice them to make it less scary.
1.7 – Translating Algebraic Expressions, Equations & Inequalities
Color code. Color code. Color code. Please just read this post where I go into my method for teaching translating expressions, but, trust me, when I say: color-coding is ESSENTIAL.
I start off this topic by going over key words and phrases, and then our notes focus on applying our color-coding strategy to translate written expressions into algebraic expressions, equations, and
inequalities. This is such an important lesson to start off the year, because I like to incorporate equation writing into each unit.
1.8 – Solving 1-Step & 2-Step Equations
Before we start solving equations, it’s of the utmost important that students know what a solution actually is. How can they check their work if they don’t even know what it is that they did?
We start by looking at the definition of a solution, which is any value that makes an equation true. We use replacement sets to help locate the solution(s) to two equations. One of them is even
quadratic! It’s as simple as substituting and evaluating the expression (see how that skill always comes back!).
We then move onto actually solving 1-step and 2-step equations. Lots of common stuck-points are included and students are encouraged to check their solutions on every 2-step equation. It takes quite
a bit of practice for students to be comfortable checking their work, so constantly relating it back to that definition of a solution being a value that makes an equation true is a must!
1.9 – Solving 2-Step Inequalities
For our last lesson of the unit, we do solving 2-step inequalities. Before doing any solving, we do a sign-flipping investigation that is meant to really build up the understanding of why it is
necessary to flip the inequality symbol after multiplying or dividing both sides of an inequality by a negative number.
We then move onto a foldable that covers the different types of endpoints, includes a reminder about flipping the sign, and lots of practice of solving, graphing, and writing the solution set.
Like your own copy of this download-and-done unit? Get this complete unit in one easy download!
What do all those symbols mean?
Pages marked with the “understanding” symbol provide special opportunities to slow down a topic and build conceptual understanding so you can speed up later and in the long run. These will be
essential in developing your students’ understanding of what it is that they are doing and why.
Pages marked with the “flowchart” symbol are the perfect companion differentiation tool to help ensure that all students can find success. They are great study tools that students can refer back to
over and over again and to turn to when they need help getting “un-stuck.” Use them to introduce a topic!
Pages marked with the “bonus file” symbol provide extra utility to the interactive notebook. Starting an interactive notebook can feel really daunting, but it doesn’t need to be. These pages are
those “extras” that make an interactive notebook function all the better, and make your life easier, too!
These understanding builders are included in all Algebra 1 interactive notebook kits.
Flowcharts are not included in notebook kits and are sold separately here.
Bonus files are special to the mega bundle of all Algebra 1 notebook kits, sold here. | {"url":"https://ctekproducttool.com/article/algebra-1-unit-1-interactive-notebook-pages-the-foundations-of-algebra","timestamp":"2024-11-09T14:11:40Z","content_type":"text/html","content_length":"133051","record_id":"<urn:uuid:22e3c715-1dee-4427-b845-bd555b7f518f>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00708.warc.gz"} |
Probability is the chance of something to happen.
A very good example for is when you toss a coin. When a coin is tossed, you either get a tail or a head. In short, you have two possible outcomes. So the possibility of you tossing a tail is 1 in 2
or ½. The same is true with tossing a head that is also 1 in 2 or 1/2.
It would then be safe to say that the formula of probability is this:
= Number of ways it can happen
Total number of outcomes
1. There are 6 bars of chocolate in a box. 4 are white chocolates and 2 are dark chocolates. What is the
probability of picking a white chocolate?
There are 6 total bars of chocolates.
There are only 4 white chocolates.
= 4/6
= 0.67
There are 3 plates in the dishwasher. 1 is a round shaped and the other 2 are square shaped. What is the
probability of getting a round plate?
There are 3 plates.
There is only 1 round plate.
= 1/3
= 0.33
3. You have 5 pairs of socks in the drawer. 3 of the pairs are knee length and 2 are ankle length. What is the
probability that a knee length pair will be picked?
There are 5 pairs of socks.
There are 3 pairs of knee length socks.
= 3/5
= 0.6
4. In the kitchen drawer, there are 10 utensils. 4 spoons, 3 forks and 3 knives. What is the probability that a spoon is picked?
There are 10 utensils.
There are 4 spoons.
= 4/10
= 0.4
5. In an animal adoption center, a pet dog just gave birth to 10 cute little puppies. 6 with black spots and 4 without spots. What is the probability that a puppy with black spots will be selected?
There are 10 puppies.
6 puppies have black spots
= 6/10
= 0.6 | {"url":"https://athometuition.com/simple-probabilities.aspx","timestamp":"2024-11-11T11:32:07Z","content_type":"application/xhtml+xml","content_length":"28315","record_id":"<urn:uuid:1010377f-0409-4536-9338-4fbbf147c73e>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00430.warc.gz"} |
IFAS -- Structural Analysis and Design
integrated system for 3D frame structural analysis and design.
• fully interactive graphical interfaces
• highly efficient asynchronous parallel solvers
IFAS: • partially rigid connections
• theoretical (FEM) analysis of effective length factors
• Visual analysis
• Visual design
IFAS has many distinctive features. For example,
Effective length factors play an important role in design. There are three methods of determining effective length factors:
1. alignment charts
2. approximate methods
3. theoretical (FE) analysis
The alignment charts were introduced when structural professionals didn't have a computer to analyze the effective length factors. Those charts are for estimate. It is simple, but accuracy is always
a problem.
Approximate methods assume a member only depends on the adjacent members. Each adjacent member has a rotational stiffness. Approximate methods replace the adjacent members with rotational springs and
then calculate the effective length factors from the isolated member. This approach can provide a good approximation for small and regular structures; for other situations, this approach may fail to
provide an acceptable accuracy. The problem is that the assumption made in the beginning is not true for large-scaled or irregular structures. In fact, every pair of effective length factors depend
on all the joints, members, and supports.
IFAS uses the theoretical (FE) method to analyze the effective length factors. Each pair of effective length factors are analyzed in the entire structure, not only on a local member as the one
assumed in approximate methods. IFAS uses the most accurate finite element formulation, that considers support conditions (settlements, spring constants, or inclined along a line or on a plane), and
released joints. Partially rigid effective-length analysis is also available in IFAS.
Analysis of effective length factors is in a nonlinear process. IFAS has an efficient parallel algorithm that not only run highly efficient but also can speed up on multiprocessors. IFAS
automatically calculates the theoretical values for designs. With IFAS, structural professionals don't need to assume a length factor and don't need to determine if member is braced or not. IFAS can
provide more accurate data from FE analysis and reduces the design cost.
IFAS uses parallel solvers and procedures for static, dynamic, and effective length analyses. The parallel solvers are programmed in the analysis kernel. Parallel computing is a trend for scientific
and engineering computing, that may distribute the computing streams onto employed processors so as to speed up computations. IFAS not only can run on uniprocessor (or single core) computers but also
can speed up on multiprocessors (or multicores).
IFAS is an open system without a predefined limitation on problem size. IFAS has asynchronous parallel solvers for static, dynamic, and effective-length factor analyses. The preprocessor and
postprocessor are also capable of parallel visualization.
IFAS is the first structural package that is capable of partially rigid analysis. The difference between the "traditional analysis" and partially rigid analysis is on the connector. The traditional
analysis assumes the connector is either perfect rigid with an infinite stiffness or perfect pin-connected. In most real structures, the connectors are deformable. It is no way to design a connector
with infinite stiffness. It can be understood that the traditional analysis makes an assumption that does not exist in most real structures. This is the reason to consider partially rigid
connections. Partially rigid analysis takes the connector strength into considerations, and can determine the stiffness a connector requires. Certainly, partially rigid analysis can provide more
accurate data to design a more safe structure. | {"url":"http://www.equation.com/servlet/equation.cmd?fa=ifas","timestamp":"2024-11-13T07:29:36Z","content_type":"text/html","content_length":"16586","record_id":"<urn:uuid:e3f1e799-8bee-4f75-ac69-4343cd300693>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00368.warc.gz"} |
Perturbation problems for extremal Kähler metrics
主讲人 Speaker：Lars Martin Sektnan (University of Gothenburg)
时间 Time： 3:30 - 5:00 pm. 2021-12-6/7/8
地点 Venue：Zoom Meeting ID:859 5212 7925； Passcode: Kahler
The aim of this lecture series is to present some results on perturbation problems for extremal Kähler metrics, which are a generalisation of constant scalar curvature Kähler (cscK) metrics. The goal
in these problems is to create new such metrics from old ones, via analytic techniques where the linearisation of the equation plays a central role. The first lecture will be on background on cscK
and extremal metrics (following [13]). To prepare for the perturbation problems I will discuss, a particular emphasis will be on the linear theory, where automorphisms play a key role. I will then
highlight the strategy and method of proofs in these problems by going through the proof of a classical such theorem -- the LeBrun-Simanca openness theorem ([10]). I will also discuss the statements
of some other perturbation problems (without proofs). In the second lecture, I will discuss the results of joint work with Spotti ([11]). This falls under the general theme of constructing extremal
metrics on the total space of fibrations, which has a rich history of study ([9, 8, 4, 5, 7]). I will explain how the result fits into this story, and the new elements. In the third and final
lecture, I will discuss recent joint work with Dervan ([6]). This result concerns the construction of extremal metrics on blowups. Again, this is a problem with a long history of study ([1, 2, 3, 12,
14]), which have given sufficient conditions for the blowup of an extremal manifold to admit extremal metrics. We also are able to deal with blowing up certain semistable manifolds, a case which has
not been considered before. I will explain how the result compares to previous constructions, and the novelty of our approach.
Some basic knowledge of differential geometry, complex manifolds and Kähler geometry.
[1] Claudio Arezzo and Frank Pacard, Blowing up and desingularizing constant scalar curvature Kähler manifolds, Acta Math. 196 (2006), no. 2, 179-228. MR 2275832 (2007i:32018)
[2] -------, Blowing up Kähler manifolds with constant scalar curvature. II, Ann. of Math. (2) 170 (2009), no. 2, 685-738. MR 2552105 (2010m:32025)
[3] Claudio Arezzo, Frank Pacard, and Michael Singer, Extremal metrics on blowups, Duke Math. J. 157 (2011), no. 1, 1-51. MR 2783927 (2012k:32024)
[4] Till Brönnle, Extremal Kähler metrics on projectivized vector bundles, Duke Math. J. 164(2015), no. 2, 195-233. MR 3306554
[5] Ruadhaí Dervan and Lars Martin Sektnan, Extremal metrics of fibrations, Proc. Lond. Math. Soc. (3) 120 (2020), no. 4, 587-616. MR 4008378
[6] -------, Extremal Kähler metrics on blowups, (2021), arXiv:2110.13579.
[7] -------, Optimal symplectic connections on holomorphic submersions, Comm. Pure Appl. Math. 74 (2021), no. 10, 2132-2184. MR 4303016
[8] Joel Fine, Constant scalar curvature Kähler metrics on bred complex surfaces, J. Differential Geom. 68 (2004), no. 3, 397-432. MR 2144537
[9] Ying-Ji Hong, Constant Hermitian scalar curvature equations on ruled manifolds, J. Differential Geom. 53 (1999), no. 3, 465-516. MR 1806068
[10] Claude LeBrun and Santiago R. Simanca, On the Kähler classes of extremal metrics, Geometry and global analysis (Sendai, 1993), Tohoku Univ., Sendai, 1993, pp. 255-271. MR 1361191
[11] Lars Martin Sektnan and Cristiano Spotti, Extremal metrics on the total space of destabilising test configurations, (2021), arXiv:2110.07496.
[12] Gábor Székelyhidi, On blowing up extremal Kähler manifolds, Duke Math. J. 161 (2012), no. 8, 1411-1453. MR 2931272
[13] -------, An introduction to extremal Kähler metrics, Graduate Studies in Mathematics, vol.152, American Mathematical Society, Providence, RI, 2014. MR 3186384
[14] -------, Blowing up extremal Kähler manifolds II, Invent. Math. 200 (2015), no. 3, 925-977. MR 3348141
Notes download：
Note: Notes.pdf
Talk1: Talk 1.pdf
Talk2: Talk 2.pdf
Talk3: Talk 3.pdf | {"url":"https://ymsc.tsinghua.edu.cn/en/info/1050/1897.htm","timestamp":"2024-11-06T23:28:36Z","content_type":"text/html","content_length":"40357","record_id":"<urn:uuid:1cafb0ce-0512-42b5-a5e8-1d6f71ad6628>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00329.warc.gz"} |
Part 5 Review | Wilderness Labs Developer Portal
Important Concepts
This was a big chapter of the tutorial. Along the way, we learned about Kirchhoff's laws, which are as foundational as Ohm's law and are key to analyzing, understanding, and designing circuits. We
also built our first practical circuits, learned to solder, and integrated those circuits with Netduino.
Kirchoff's Laws
• Kirchhoff's current law states that the current flowing into a junction is the same as the current flowing out.
• Kirchhoff's voltage law states that voltage drops in proportion to the resistance at any point in a circuit, and the sum of all voltage drops is equal to the voltage source.
Voltage Division
• Voltage division can be calculated using the equation Vout = Vs * (R2 / (R1 + R2)), where R2 represents the bottom half (closest to ground) and R1 represents the top half of the voltage divider.
• Voltage divider circuits are useful for using resistive sensors and level shifting from a higher voltage domain to a lower voltage domain. Potentiometers also use voltage division internally to
control their output voltage.
• Voltage dividers should never be used as a voltage regulator.
• Level shifting is the process of changing the the voltage of a signal.
Analog to Digital Conversion
• Meadow can measure a voltage signal from 0V to 3.3V using the onboard Analog to Digital Converter (ADC) and reports that voltage as a digital value from 0 to 1023.
Circuit Analysis
• Complex circuits can be understood by considering them as collection of smaller circuits and analyzing the smaller sub-circuits.
• Datasheets are published for nearly every component and contain important information about how they work, how they should be wired up, etc.
• Soldering is the process of melting solder to make permanent connections.
Circuit Software
• There are many software applications to help design and analyze circuits. Some of the most common and accessible apps are Fritzing, iCircuit, Eagle CAD, and KiCad. | {"url":"https://developer.wildernesslabs.co/Hardware/Tutorials/Electronics/Part5/Review/","timestamp":"2024-11-09T13:16:38Z","content_type":"text/html","content_length":"29055","record_id":"<urn:uuid:5c3cb81b-1516-41c2-89a0-d07967a8c1e6>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00618.warc.gz"} |
Electric Charge
Electric Charge
Electric Charge is a Transcendent Descendant module in The First Descendant. Here’s everything you need to know about its effects and how to get it.
Module Details :
Rarity : Transcendent
Class : Descendant
Character : Ultimate Bunny
Electric Charge Skill Effect per Level
Level Effect Capacity Mastery Kuiper Gold
Level Shard
0 Increases the landing damage after Rabbit Foot's Double Jump. Double Jump deals significantly increased damage after hitting enemies with the skill a 13 0 - -
certain number of times.
1 Increases the landing damage after Rabbit Foot's Double Jump. Double Jump deals significantly increased damage after hitting enemies with the skill a 12 1 1,500 15,000
certain number of times.
2 Increases the landing damage after Rabbit Foot's Double Jump. Double Jump deals significantly increased damage after hitting enemies with the skill a 11 2 3,000 30,000
certain number of times.
3 Increases the landing damage after Rabbit Foot's Double Jump. Double Jump deals significantly increased damage after hitting enemies with the skill a 10 3 5,500 55,000
certain number of times.
4 Increases the landing damage after Rabbit Foot's Double Jump. Double Jump deals significantly increased damage after hitting enemies with the skill a 9 4 10,000 100,000
certain number of times.
5 Increases the landing damage after Rabbit Foot's Double Jump. Double Jump deals significantly increased damage after hitting enemies with the skill a 8 5 17,000 170,000
certain number of times.
6 Increases the landing damage after Rabbit Foot's Double Jump. Double Jump deals significantly increased damage after hitting enemies with the skill a 7 6 28,500 285,000
certain number of times.
7 Increases the landing damage after Rabbit Foot's Double Jump. Double Jump deals significantly increased damage after hitting enemies with the skill a 6 7 47,500 475,000
certain number of times.
8 Increases the landing damage after Rabbit Foot's Double Jump. Double Jump deals significantly increased damage after hitting enemies with the skill a 5 8 77,500 775,000
certain number of times.
9 Increases the landing damage after Rabbit Foot's Double Jump. Double Jump deals significantly increased damage after hitting enemies with the skill a 4 9 125,000 1,250,000
certain number of times.
10 Increases the landing damage after Rabbit Foot's Double Jump. Double Jump deals significantly increased damage after hitting enemies with the skill a 3 10 200,000 2,000,000
certain number of times.
Electric Charge Decompose Data
Level Kuiper Shard
0 4,500
1 5,700
2 7,625
3 10,893
4 16,557
5 25,812
6 40,849
7 65,267
8 104,189
9 165,679
10 262,250
How to Get Electric Charge
Electric Charge cannot be obtained through missions or activities. You can acquire Electric Charge by combining other Modules at Silion in Albion. | {"url":"https://tfdtools.com/modules/electric-charge","timestamp":"2024-11-13T01:43:04Z","content_type":"text/html","content_length":"159831","record_id":"<urn:uuid:cf2b68d9-8a23-4ebf-a97a-8e6330573c85>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00133.warc.gz"} |
Gennadiy Averkov
For a finite set $X \subset \Z^d$ that can be represented as $X = Q \cap \Z^d$ for some polyhedron $Q$, we call $Q$ a relaxation of $X$ and define the relaxation complexity $\rc(X)$ of $X$ as the
least number of facets among all possible relaxations $Q$ of $X$. The rational relaxation complexity $\rc_\Q(X)$ restricts … Read more | {"url":"https://optimization-online.org/author/averkov/","timestamp":"2024-11-07T06:47:46Z","content_type":"text/html","content_length":"89983","record_id":"<urn:uuid:eb565524-fbdf-4ea3-a935-e017da5a6923>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00593.warc.gz"} |
Enhancing HNSW Efficiency: Cutting Index Build Time by 85% - GSI Technology
Enhancing HNSW Efficiency: Cutting Index Build Time by 85%
Optimizing Graph Tradeoffs with Parameter Tuning
Consequences of Slow Index Build Times
Limited Developer Application Experimentation
Slow Update of Dynamic Datasets
Ways to Reduce Index Build Time
Compute-In-Memory (CiM) Associative Processing – Flexible Massive Parallel Processing
The vector database market is growing quickly, with applications like Retrieval Augmented Generation (RAG) for Generative AI (GenAI) and vector similarity search for ecommerce leading the way. Many
leading vector database providers use the HNSW (Hierarchical Navigable Small Worlds) algorithm for vector similarity search because it provides fast search with good recall.
This fast search and good recall come at a price, however—it takes a long time to build an HNSW index. Long index build time can lead to issues such as limited scalability, increased operational
costs, reduced developer application experimentation, and slow update of dynamic datasets.
Fortunately, there are ways to reduce HNSW index build time. One key approach is to parallelize the nearest neighbor distance calculations needed to build the index. Those calculations represent the
longest portion of the index build, so speeding them up is important.
This paper will review the HNSW algorithm, how an HNSW index is built, and why the build time using traditional solutions, like CPU, takes a long time. It will also present a solution that leverages
a high degree of parallelism to reduce index build time by roughly 85% compared to CPU.
Market Drivers
A report from MarketsandMarkets™ ^1 forecasts the vector database market to grow from $1.5 billion in 2023 to $4.3 billion by 2028, representing a CAGR of 23.3%.
RAG is one of the main drivers of this growth. RAG retrieves relevant documents from a large corpus and uses those documents in GenAI applications to generate a response. By incorporating this
external information, it allows for more accurate and reliable responses and reduces hallucinations, where inaccurate responses are given that are not based on factual information.
RAG leverages vector databases to efficiently store and retrieve those relevant documents, which are in the form of vector embeddings. The need for effective and efficient retrieval of vector
embeddings has made vector databases integral to RAG.
Additionally, RAG is contributing to the trend of billion-scale vector databases. That is because larger datasets provide a wider range of information to retrieve relevant information from. Cohere,^2
a leading provider of LLMs, states that some of their customers scale to tens of billions of vector embeddings.
Another application that is driving the need for billion-scale vector databases is ecommerce. For example, eBay^3 stated that their “marketplace has more than 1.9 billion listings from millions of
different sellers.” Those listings are in the form of vector embeddings that the vector database stores and retrieves as part of the recommendation process.
Within the vector database market, HNSW is one of the most popular Approximate Nearest Neighbor (ANN) algorithms used to search and retrieve vector embeddings. It has been adopted by many vector
database providers like Vespa^4 and Weaviate^5 and by companies like AWS^6 and eBay.^7
HNSW Overview
HNSW is a graph-based algorithm that efficiently searches for the approximate nearest neighbors of a query. The graph is hierarchical, meaning it has multiple layers of nodes (vector embeddings),
where each layer contains a subset of the previous layer. The nodes are connected by edges, which represent the distance between them. Distance is measured by a metric, such as cosine similarity.
The higher layers have fewer nodes and longer edges between the nodes, which allows for bridging distant regions of the graph. The bottom layer includes all the vectors and has short-range edges
(connecting to nearby nodes).
Figure 1 provides a simple example of an HNSW graph structure.
Figure 1: Example HNSW graph showing the hierarchical layers structure. The top layer has the fewest nodes and longer connections, while the bottom layer contains all the nodes and has short-range
This multi-layered graph structure with fewer, long-range connections on the top layers and denser, short-range connections on the bottom layers allows for a coarse-to-fine search approach. This
provides faster pruning of irrelevant portions of the graph during nearest neighbor search, resulting in reduced search time.
The fewer nodes and longer edges in the top layers allow fast exploration of the vector space, while the denser nodes and shorter edges in the lower layers allow for more refined searches.
The HNSW search algorithm starts by taking big jumps across the top layers and progressively moving down the hierarchy of layers to refine the search. This minimizes the number of distance
calculations needed, speeding up similarity searches.
This approach is analogous to the steps one might take in searching for a specific house. You start at the city level, then move to a specific neighborhood, a particular street, and finally, you find
the house you are looking for.
The long-range connections in the top layers act like shortcuts, and they ensure the “small world” property of the graph. This means most nodes can be reached in a few hops. This concept is shown in
Figure 2—a long-range connection is used to help find the nearest neighbor for the query in two hops from the entry point.
Figure 2: “Small world” concept demonstrated where a long-range connection allows for an efficient 2-hop graph traversal from the Entry Point to the Query’s nearest neighbor.
Optimizing Graph Tradeoffs with Parameter Tuning
Parameters M and ef_construction, which are both related to the number of links a node has in the HNSW graph, help optimize the trade-off between index build time, memory usage, and accuracy.
During index build or update, new nodes are inserted into the graph based on their distance from existing nodes. At each layer, a dynamic list of the nearest neighbors seen so far is kept. The
parameter ef_construction determines the size of this list.
The index build algorithm iterates over the nodes in the list and performs nearest neighbor distance calculations to see if the nodes’ neighbors are closer to the query than they are, and if so,
considers adding the nodes’ neighbors to the dynamic list.
A larger ef_construction means that more candidates are tracked and evaluated, increasing the chance of finding the true nearest neighbors for each node in the graph. This improves accuracy, but
increases the index build time since more distance calculations are needed.
Similarly, increasing M, which is the number of bi-directional links created for each node in the graph, improves the accuracy of the graph but also increases the build time since more distance
calculations and updates are needed.
By tuning the values of M and ef_construction, developers can experiment with different indexes to optimize the specific requirements of their application. For example, they can tune the values to
prioritize index build time over accuracy or vice versa.
Consequences of Slow Index Build Times
As seen in the previous sections, building an HNSW index is a compute-intensive process that requires a lot of distance calculations to find the nearest neighbors for each vector in a hierarchy of
graph layers. While this results in a graph with low search latency and good recall, it also comes with the tradeoff of a graph that is slow to build and update.
For example, as seen in Figure 3, eBay found that building an HNSW graph for an index size of 160M vectors can take about 3 hours to 6 hours based on the different parameters chosen.
Figure 3: Index build time vs. index size. Different quantization, M, and ef_construction values are used.^7
The figure also shows that build time increases rapidly as the number of vectors in the index grows. This can be problematic for a company such as eBay that, as mentioned earlier, has billions of
live listings.
Slow index build times can lead to many challenges, such as: scalability issues, increased operational costs, limited developer application experimentation, and slow update of dynamic datasets.
Scalability Issues
Slow index builds can limit a company’s ability to scale. Two examples of this can be seen in ecommerce and RAG:
• Ecommerce—As a company’s customer base and product catalog grow, so does the time needed to build an index to search that catalog to provide relevant and timely product recommendations. This can
limit their ability to take on more customers and products, impacting growth potential.
• RAG—RAG retrieves contextual information to help generate better GenAI responses. The vector databases they use to retrieve that information are getting bigger. That is because larger datasets
provide more relevant information to draw from to generate high-quality responses and to reduce hallucinations. Slow index build times can limit how much of this valuable information can be added
to the database without severely impacting system performance.
Increased Operational Costs
With vector search applications scaling to billions of items, slow index build time can lead to increased operational costs, which lowers profit margins. For example:
• Even as applications become billion-scale, customers expect the system to maintain its performance. More computational resources are needed to keep index build times down to maintain that
performance, resulting in higher costs.
• Slow index build times means computational resources like CPUs and GPUs are in use longer. In cloud computing environments, where resources are billed based on usage (CPU hours, GPU hours), this
translates to higher costs.
• The extended use of computational resources not only increases direct operational costs but also raises energy consumption. This means higher energy bills.
Limited Developer Application Experimentation
As mentioned earlier, developers experiment with different parameters, such as M and ef_construction, to test different indexes and tune the performance of their application.
In a post, eBay commented on the importance of being able to experiment quickly — “Our ML engineers develop many ML embedding models as they iterate on their hypotheses for improved features. Any
time there is a change in the ML model, even when it’s a small feature or a new version, the vector space of embeddings changes and therefore a new vector search index needs to be created…. Rapid ML
model iteration is ultimately what brings value to the end user by an improved recommendations experience.”^3
Longer index build time impacts this ability to experiment in a few ways:
• It limits the number of experiments developers can run, which could lead to suboptimal application performance. Reduced experimentation can also lead to less innovation, potentially causing a
company to fall behind the competition.
• It increases the time it takes a developer to evaluate the impact of their experiments, which could delay improvements and new releases.
Slow Update of Dynamic Datasets
Applications like ecommerce and RAG have dynamic datasets where new products or information is frequently added to the database. Long index build time delays updates to this dynamic data, affecting
the system’s relevancy and accuracy of search results.
• Ecommerce—If new products are added but the index build time is slow, this could cause the new products not to be included in search results. This can lead to outdated recommendations, which can
negatively impact customer satisfaction and lead to lost revenue opportunities. It can also damage the company’s brand.
• RAG—In a customer support scenario, new information is frequently added, or existing information is updated. Slow index build and update can affect the system’s ability to keep up with this
latest information. That could lead to incorrect or irrelevant responses, eroding customer trust and satisfaction.
Ways to Reduce Index Build Time
Two effective ways to reduce index build time are parallel processing and compressing the vectors through quantization.
Parallel Processing
A few ways to use parallel processing to reduce index build time are to:
• Split the dataset into clusters of vectors and search multiple clusters in parallel to find the nearest neighbors for a vertex. The nearest neighbors from multiple clusters can be merged to
provide a final set of neighbors.
• Perform the nearest neighbor distance calculations within the clusters in parallel. These distance calculations account for the longest portion of the index build time.
Unfortunately, CPUs, which most HNSW index build solutions use, have limited parallel processing capabilities. While some high-end multi-core CPUs have 64 cores that can perform parallel processing,
this is still a small number of cores. This limits the number of parallel operations they can perform, so they can only process a minimal amount of vectors at a time.
What is needed is a solution with massive parallel processing capability.
Vector Quantization
Compressing the vectors in the index through quantization speeds up the build time by:
• Packing more vectors per data transfer. This reduces the number of accesses from slower external memory to faster internal memory.
• Speeding up the nearest neighbor distance calculations since the computation is simplified by performing it on fewer bits.
What is needed is a flexible solution that can work with quantized vectors of varied bit lengths. This allows for experimenting with different compression algorithms to see which bit length is
optimal for achieving a particular set of goals, such as build time or accuracy.
Compute-In-Memory (CIM) Associative Processing—Flexible Massive Parallel Processing
GSI Technology’s compute-in-memory Associative Processing Unit (APU) is a technology that allows for massive parallel processing and flexible quantization. It is based on bit-level processing, and
computation is done in place, directly in memory, which avoids the traditional bottleneck between processors and memory.
Building an index with the APU involves the following steps:
1. Quantize the dataset to 4 bits per feature (or any bit value of choice). For the results seen in Figure 6, 4-bit quantization was used.
2. Cluster the dataset into N[c] clusters of approximately the same size by using K-means clustering.
3. Assign each vertex to its N[p ]≥ 2 closest clusters based on distance to the cluster centroids created in step 2.
4. Load multiple clusters into the APU.
5. Check a list to see which vertices are assigned to the clusters loaded in the APU.
6. Batch search the vertices assigned to the clusters currently loaded in the APU to find their K nearest neighbors in those clusters. The loaded clusters are searched in parallel.
7. Repeat steps 4–6 until all clusters have been processed and all vertices inserted into the graph.
8. A vertex might have nearest neighbors from multiple clusters. The K nearest neighbors for that vertex are the union of the nearest neighbors from the multiple clusters.
9. After the union of clusters from step 8, the edges are made bidirectional and redundant edges are pruned to ensure that each vertex has <= K neighbors.
The APU has millions of bit processors that perform computation at the bit level. This allows for massive parallel processing on any sized data element.
The APU’s bit-level processing allows each data bit to be processed independently. For example, for a 16-bit data item, each of the 16 bits can be processed individually.
The APU architecture stores the corresponding bits of multiple data elements together. For the 16-bit example mentioned above, this means the first bits of a set of 16-bit numbers are stored
together, all the second bits together, etc. This can be seen in Figure 4, where Bit Slice 0 holds all the 0 bits for each 16-bit data item, Bit Slice 1 holds all the 1 bits, etc.
All the data elements for each bit slice are accessed in parallel, which coupled with the millions of bit processors, allows for massive parallel processing.
Figure 4: The APU’s bit-parallel architecture stores the corresponding bits of multiple data elements together. For example, Bit Slice 0 holds the 0 bits for each data item in a set of items.^8
The APU takes advantage of this bit-level processing and massive parallel processing to speed up the index build process detailed earlier:
1. Bit-level processing
1. Allows for flexible quantization (e.g., 4-bit quantization as in step 1 above). Quantization compresses the vectors to allow more vectors to be loaded into the APU’s memory, where they can be
processed in parallel directly in place, speeding up the build process.
2. Quantized values use fewer bits, which simplifies and speeds up the many nearest neighbor distance calculations needed in step 6. Also, quantization minimizes the time spent loading the
clusters from external memory into the APU since fewer bits are transferred.
2. Parallel Processing—The APU uses its millions of bit processors to perform the nearest neighbor distance calculations from step 6 in parallel. This massive parallel processing significantly
reduces the time needed to compute the distance calculations. Performing nearest neighbor distance calculations is the most time-consuming part of building an index, so reducing it will have the
greatest impact on speeding up the index build.
Figure 5, from Nvidia, shows that an Intel Xeon Platinum 8480CL CPU takes 5,636 seconds (about 1 and a half hours) to build an HNSW index for 100M vectors.
Figure 5: CPU HNSW index build time for 100M vectors^9
Figure 6 presents the HNSW index build times using an APU system for datasets ranging from 100 million vectors to 1 billion vectors. It shows an APU system can build a 100 million vector index in 864
seconds (about 15 minutes). This represents nearly an 85% reduction in build time compared to the 5,636 seconds (about 1 and a half hours) for an Intel Xeon Platinum 8480CL CPU.
Figure 6 also shows that an APU system can build a 1 billion vector HNSW index in 7800 seconds (about 2 hours), which is a significant improvement compared to Figure 3 from eBay if the eBay build
times were extrapolated to 1 billion vectors.
Figure 6: HNSW index build time for 100 million–1 billion vectors.
This fast index build time has many benefits:
• Scale—It provides timely access to contextual information for RAG applications or product recommendations for ecommerce applications—even as those applications scale to billions of vectors.
• Reduced operational costs—computational resources are in use for less time, which for cloud computing, where resources are billed by the hour, means lower costs.
• Improved Developer Application Experimentation—It increases the number of experiments a developer can run and reduces the time to evaluate those experiments. This leads to more innovation and
faster product releases.
• Faster Update of Dynamic Datasets—In ecommerce applications, new products can be added to the index more quickly. This means they can be included in product recommendations promptly, leading to
increased revenue opportunities.
In RAG applications, it allows the latest information to be accessed sooner, leading to more relevant and accurate responses.
HNSW is a leading algorithm used for vector similarity search in applications like GenAI and ecommerce. It provides high recall and fast search, but with the tradeoff of long index build time.
Long index build time limits scalability, increases costs, leads to less experimentation, and delays updates to dynamic data. This lowers a company’s growth potential, reduces profits, leads to less
innovation, and erodes customer confidence if incorrect or irrelevant responses are generated.
One of the main ways to reduce index build time is through parallelization. Unfortunately, most current solutions use CPUs, which have limited parallel processing capabilities.
GSI’s APU addresses this issue by providing massive parallel processing. The APU’s millions of bit processors allow for massive parallel computation of the many nearest neighbor distance calculations
needed to build an index. The nearest neighbor distance calculations are the longest step in building an index, so performing them in parallel significantly reduces index build time.
The result—roughly an 85% reduction in index build time compared to traditional CPU-based solutions.
To find out more about how GSI Technology’s on-prem and cloud-based offerings can significantly reduce your HNSW index build times, contact us at [email protected]. | {"url":"https://gsitechnology.com/enhancing-hnsw-efficiency-cutting-index-build-time-by-85-percent/","timestamp":"2024-11-04T02:11:07Z","content_type":"text/html","content_length":"142310","record_id":"<urn:uuid:030f9067-6bea-4851-a40a-ece07098f72a>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00163.warc.gz"} |
Convert cm to km | Easy Online Length Conversion Tool | ORCHIDS
Example 1:
Given: Length in centimeters = 250,000
Using the formula:
${\text{Length in kilometers}}_{}=\frac{{\text{Length In Centimeters}}_{}}{{\text{100,000}}_{}}$
${\text{Length in kilometers}}_{}=\frac{{\text{250,000}}_{}}{{\text{100,000}}_{}}$
Result: 2.5 Kilometers
Example 2:
Given: Length in centimeters = 600,000
Using the formula:
${\text{Length in kilometers}}_{}=\frac{{\text{Length In Centimeters}}_{}}{{\text{100,000}}_{}}$
${\text{Length in kilometers}}_{}=\frac{{\text{600,000}}_{}}{{\text{100,000}}_{}}$
Result: 6 Kilometers
Example 3:
Given: Length in centimeters = 1,500,000
Using the formula:
${\text{Length in kilometers}}_{}=\frac{{\text{Length In Centimeters}}_{}}{{\text{100,000}}_{}}$
${\text{Length in kilometers}}_{}=\frac{{\text{1,500,000}}_{}}{{\text{100,000}}_{}}$
Result: 15 Kilometers | {"url":"https://www.orchidsinternationalschool.com/converters/centimeter-kilometer-converter","timestamp":"2024-11-06T17:24:04Z","content_type":"text/html","content_length":"34328","record_id":"<urn:uuid:690fe1e6-d967-420e-afa1-fd7def1efb2b>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00679.warc.gz"} |
Redfield Consulting's blog
MCT has just published a power curve for its Seagen device in Strangford Narrows (see figure above and link
). The curve shows a period of output at the design capacity of 1.2 MW, in what is described as a “medium tide”. This tide appears to have peaked at 3.1 m/s, which may be medium for Strangford, but
is pretty impressive for most sites.
We realised that it’s possible to drill into this curve to come up with some (very) theoretical ideas of the capacity factor which might be achieved by the technology. First we constructed a velocity
lookup table by taking a ruler to the graph, which shows the power output (kW) at various stream speeds (m/s).
m/s - Power
0 - 0
1 - 20
1.25 - 100
1.5 - 180
1.75 - 400
2 - 600
2.25 - 900
2.4 - 1200
We then constructed a model which characterises a simplified tidal environment, with stream speed varying according to a diurnal cycle (sinusoidal variation over 24 hours, in 2 flood, 2 ebb tides)
and a 28 day lunar cycle (again simple sinusoidal variation).
We entered a maximum stream speed (peak rate achieved at spring tide) and a minimum stream speed (peak rate achieved at neap tide) and constructed a lookup on an hour by hour basis to estimate the
power output over a month.
Based on a maximum stream speed of 3.2 m/s and a minimum stream speed of 1.6 m/s (ie neap maximum is half as fast as spring maximum), we find that the average theoretical power output (assuming no
outages) to be 450 kW, making the capacity factor 38%.
The model shows that output is sensitive to both maximum stream speed and the ratio between spring and neap peak rates. The table below shows the relationship between capacity factor and the maximum
stream speed) assuming that neaps are limited to 50% of the maximum stream speed in springs. The month-average capacity factor for various maximum spring stream speeds is:
m/s - CF (%)
2.8 - 28%
3 - 34%
3.2 - 38%
3.5 - 45%
4 - 53%
The table below shows how the capacity factor is influenced by the ratio between the maximum neap speed and the maximum spring speed based on a maximum spring stream speed of 3.2 m/s. The table shows
maximum neap speed and month-average capacity factor.
m/s - CF (%)
0.8 - 31%
1.2 - 33%
1.6 - 38%
2.1 - 45%
2.4 - 48%
All of these power output estimates are wildly theoretical – and should be treated with extreme caution. Next we’re going to combine this power curve with some actual tidal data from tidal diamonds
on charts to see how that looks. | {"url":"https://redfieldconsulting.blogspot.com/2009/11/","timestamp":"2024-11-03T13:19:44Z","content_type":"application/xhtml+xml","content_length":"42365","record_id":"<urn:uuid:e560841d-3d5f-424d-8647-90947bd9013d>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00208.warc.gz"} |
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Exactly how can moms and dads support their children's multiplication method in the house?
Motivating consistent method, offering assistance, and producing a favorable learning setting are advantageous actions. | {"url":"https://crown-darts.com/en/two-digit-multiplication-worksheets-with-grid.html","timestamp":"2024-11-04T08:05:52Z","content_type":"text/html","content_length":"28840","record_id":"<urn:uuid:18cad58f-8cf3-4072-8b04-d54d6cfe32e7>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00178.warc.gz"} |
First digit of Radix number
+ General Questions (11)
The radix-10 number <[33]>10 is 33 in decimal and it is the radix-4 number <[201]>4. Radix-B numbers never need leading zeros unless you have to extend them to a certain number of digits. Leading
zeros don't change the value of Radix-B numbers.
However, 33 in decimal is the 4-complement number <[0201]>4. Here, we need a leading zero, since otherwise, the number <[201]>4 would be -31. With 4-complement numbers leading digits 0,1 denote
non-negative numbers, and leading digits 2,3 denote negative numbers. | {"url":"https://q2a.cs.uni-kl.de/1628/first-digit-of-radix-number?show=1631","timestamp":"2024-11-12T03:48:03Z","content_type":"text/html","content_length":"47914","record_id":"<urn:uuid:f056f311-b413-4ab6-94f6-fa7162ae4f2d>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00493.warc.gz"} |
Kind ¶
Every Term has a kind which represents its type, for example whether it is an equality ( Equal ), a conjunction ( And ), or a bit-vector addtion ( BVAdd ). The kinds below directly correspond to the
enum values of the C++ Kind enum.
class cvc5. Kind ( value ) ¶
The Kind enum | {"url":"https://cvc5.github.io/docs/cvc5-1.0.2/api/python/base/kind.html","timestamp":"2024-11-13T07:56:49Z","content_type":"text/html","content_length":"783996","record_id":"<urn:uuid:19b67fdc-cb78-4653-ad77-c2379140a04e>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00422.warc.gz"} |
Issues With Translating Conditionals: "If A then B"
The most notorious mismatches between the meaning of a logical connective and the meaning of its natural language counterpart are found with conditional statements. The English language expression
“if A then B” is used in a variety of ways, and not all of them map onto the meaning of the conditional as defined in propositional logic.
In the lecture on the logic of conditional statements I said that a conditional is any statement that is logically equivalent to a statement of the form “If A then B”, and I defined the truth
conditions for such statements as follows:
“If A then B” is FALSE when the antecedent “A” is true and the consequent “B” is false; for all other combinations of truth values, the conditional is TRUE.
So, for example, given the conditional statement
“If I shoot you with this gun, you will die”
the only scenario in which this statement is FALSE is the case where I shoot you with this gun but you don’t die, i.e. the case where the antecedent is true but the consequent is false.
Now, here’s the odd bit: for all other assignments of truth values, the logical conditional is assigned the value true.
That includes cases where the antecedent is false. On this definition of the conditional, BOTH of the following statements are TRUE:
(1) “If I don’t shoot you with this gun, you will die.”
(2) “If I don’t shoot you with this gun, you won’t die.”
Most of us, looking at these two statements, will wonder how either of these follows from the original conditional claim. It’s not like we have strong intuitions the other way. It’s just … we don’t
know why we should feel any particular way about them.
So, right off the bat, we can say that there is a mismatch between the semantics of the logical conditional as defined in propositional logic, and our intuitions about natural language conditionals.
To talk about this further, let’s clarify our language a bit.
The logical conditional, the conditional as defined by the truth table definition in propositional logic, is known as the MATERIAL CONDITIONAL.
The material conditional is what is represented by the various conventional symbolizations used in logic textbooks, usually one of these:
The material conditional, by definition, is false is when A is true and B is false, and true otherwise.
This interpretation allows us to write various logical equivalents:
(A → B) (“if A is true then B is true”) is logically equivalent to ~(A & ~B) (“it is not the case that A is true and B is false”).
Also, (A → B) (“if A is true then B is true”) is logically equivalent to (~A ∨ B) (“either A is false or B is true”).
It’s a standard logic exercise to write the truth tables for these expressions to demonstrate that they are indeed logically equivalent.
Now, the question that thousands of logic students have asked their logic instructors, and hundreds of philosophers and logicians have written about, is this:
What is the relationship between the material conditional and conditional statements expressed in natural language?
If you push this question deeply enough it opens up vast areas of research in analytic philosophy.
I’ll give you a little tour of the issues, but I can’t possibly do the topic justice. A little tour is enough to appreciate the general point, which is that there are many uses of “if-then” in
natural language that are well described by the material conditional, and there are many other uses that are not.
1. Indicative vs Counterfactual Conditionals
In English there are at least two different kinds of conditional. Consider the following part of English conditionals:
(1) “If humans did not build Stonehenge, then nonhumans did.”
(2) “If humans had not built Stonehenge, then nonhumans would have.”
Conditional (1) seems clearly true.
Conditional (2) seems clearly false. There is no reason to think, for example, that if humans had not built Stonehenge, then aliens would have.
Yet it is natural to regard each of them as composed of two component propositions — “humans did not build Stonehenge” and “nonhumans built Stonehenge” — using a two-place “if-then” connective.
But if one is true and the other is false, then the conditional connective cannot be the same in both cases.
We call the conditional connective in (1) an indicative conditional. Here are some other examples of indicative conditionals:
(3) “If it rains tonight, we shall get wet.”
(4) “If the roof leaked last night, there will be water on the kitchen floor.”
We call the conditional in (2) a subjunctive or counterfactual conditional. Here are some other examples of counterfactual conditionals:
(5) “If it were to rain tonight, we would get wet.”
(6) “If the roof had leaked last night, there would be water on the kitchen floor.”
2. Counterfactual Conditionals and the Material Conditional
It is clear that the counterfactual conditional is NOT correctly translated into propositional logic as the material conditional A → B.
This is because counterfactual conditionals are not truth functional.
Consider the claim “if you had put your sandwich down, a dog would have eaten it.”
Suppose we regard this statement as a compound made up from the counterfactual conditional connective and the two propositions “you put your sandwich down” and “a dog ate your sandwich”.
In a situation in which you ate your sandwich quickly, without putting it down, while surrounded by hungry dogs, both components are false, and the counterfactual conditional is true.
But in a situation in which you ate your sandwich quickly, without putting it down, but there are no hungry dogs present, both components are again false, but the counterfactual conditional is false.
From this it follows that the counterfactual conditional is not a truth functional connective. What determines the truth of the counterfactual conditional isn’t JUST the truth of the component
statements. We need to know MORE than just the truth values of the components in that situation.
To handle the semantics of counterfactual, or subjunctive, conditionals, we need more machinery than propositional logic gives us. The truth table for the material conditional won’t do the job.
3. Indicative Conditionals and the Material Conditional
The material conditional connective in propositional logic is intended to model the semantics of indicative conditionals, not counterfactual conditionals.
But even the claim that the indicative conditional is correctly translated as A → B is controversial in logic and philosophy.
On the one hand, one can argue that if the indicative conditional is truth functional at all, it has to be the material conditional, since no other truth table is a serious candidate.
And there are arguments that the semantics of the indicative conditional really does follow the truth table rules.
Consider this conditional statement: “if it rains, then the match will be cancelled”. It seems clear that this implies that “either it will not rain, or it will rain and the match will be cancelled”.
But this statement is true if and only if either “it will rain” is false or “the match will be cancelled” is true. That is, if and only if the material conditional is true.
So what’s the problem with translating indicative conditionals using →?
Well, recall that the material conditional A → B is true whenever the antecedent A is false or the consequent B is true (or both). However, the following conditionals all seem quite wrong, even
though the first two have false antecedents and the last has a true consequent.
(1) “If my cat has fleas, then my cat’s name is Sam.”
(false antecedent, true consequent)
(2) “If my cat has fleas, then the Earth is the largest planet in the solar system.”
(False antecedent, false consequent)
(3) “If carbon is an element, then the computer I’m typing on is an iMac.”
(true antecedent, true consequent)
If we’re following the truth table for the material conditional, all of these are TRUE statements. Yet they don’t strike us as true, at least at first glance.
The problem seems to arise from the fact that in each case the antecedent has nothing to do with the consequent: believing the antecedent is true gives us no reason to think that the consequent is
One might conclude that for an indicative conditional to be true, there must be some connection between the antecedent and the consequent, such that the truth of the former is relevant to the truth
of the latter.
If we pursue this line of thought, we’ll be lead to the conclusion that the indicative conditional is not truth functional — whether “if A then B” is true will depend not just on whether A and B are
true or false, but also on what they actually say, and in particular, on whether there is the right kind of connection between what A says and what B says.
The alternative is to defend the view that indicative conditionals have the same truth conditions as material conditionals. How? By demonstrating how the problematic examples only seem to get the
semantics wrong. There are several well known defenses of this view in the literature but I won’t bother to go into them here.
The take-away point of this discussion is that conditional relationships are expressed in many different ways in natural language, and only some of those ways are well described by the material
conditional of propositional logic. Counterfactual conditionals clearly do not follow the semantics of the material conditional. Whether, and what sorts of, indicative conditionals can be interpreted
as material conditionals, is a subject of ongoing debate.
However, we should note that when we’re using conditional argument forms like modus ponens and modus tollens in argument analysis, we’re assuming that the conditional follows the logic of the
material conditional. Otherwise the following argument forms would not be valid:
1. If A then B
2. A
Therefore, B (modus ponens)
1. If A then B
2. Not-B
Therefore, not-A (modus tollens)
This is an important point to keep in mind. | {"url":"https://criticalthinkeracademy.com/courses/propositional-logic/lectures/761263","timestamp":"2024-11-02T03:20:00Z","content_type":"text/html","content_length":"154322","record_id":"<urn:uuid:a27c832b-1c40-4812-8527-cacfb95ba3f0>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00467.warc.gz"} |
Application of real-world modulation schemes to advanced spatial modulation systems.
This dissertation does not contain the writings of other persons, unless specifically acknowledged as a source by other researchers. Thus, the second contribution of this dissertation is the
application of circular constellations, in particular amplitude phase shift keying (APSK) modulation to existing SM, GSM and GSM-CR systems.
Motivation and Context
In these conventional MIMO systems, diversity techniques are used to improve the overall link reliability of a wireless communication system. In this scheme, the Alamouti structure was incorporated
to improve the error performance over traditional GSM systems.
Figure 1.1: Block Diagram for the Alamouti System [12]
Research Aim and Objectives
As hypothesized, the M-APSK GSM-CR system outperforms the corresponding M-APSK GSM and M-APSK SM systems. Derive a closed form expression for the average BER for the M-APSK SM, M-APSK GSM and M-APSK
GSM-CR systems.
Figure 1.3: Block Diagram for the Proposed Systems
Organization of Dissertation
Zhang, “Dual learning-based channel and signal estimation in massive MIMO with generalized spatial modulation,” IEEE Transactions on Communications, vol. Cheng, “Low-complexity ML detection for MIMO
spatial modulation with APSK constellation,” IEEE Transactions on Vehicle Technology, vol.
Previous bit error rate (BER) performance studies of M-APSK SM systems have only been presented using simulation results and have not been verified by an analytical framework. The theoretical average
BER expressions are shown to have a tight bound in the high signal-to-noise ratio (SNR) region compared to Monte Carlo simulation results. The theory expression verified the simulation results of a 6
bit/s/Hz system configuration presented in a previous study.
The performance study in this paper is then extended by presenting theory and simulation results for 7 bit/s/Hz and 8 bit/s/Hz system configurations. Performance analysis of APSK in Spatial
Modulation 12 M-APSK SM system and compare it with the simulation results. The theoretical BER expressions for the 16-APSK and 32-APSK constellations in SM are derived in Section IV.
System Model
The entries of Hand are independent and identically distributed (i.i.d) according to the complex Gaussian distribution CN(0,1). The constellation diagrams and the corresponding bit allocation for the
two modulation schemes are shown in Fig. In the 16-APSK constellation, the ratio of the outer and inner rays is indicated by β0 = R2/R1, while the ratios in the 32-APSK constellation are defined as
β1 =R2/R1 and β2=R3/R1.
Performance Analysis of M -APSK SM System
Analytical BER of Symbol Estimation in AWGN (P d )
An upper bound is obtained by considering only the nearest neighbors for each PEP in (2.11). The process for deriving the 32-APSK BER is the same as in the 16-APSK case.
Analytical BER of Symbol Estimation in Rayleigh Fading (P d )
Analytical BER of Transmit Antenna Index Estimation (P a )
Results and Discussion
Sinanovic, Chang Wook Ahn, and Sangboh Yun, “Spatial Modulation,” IEEE Transactions on Vehicular Technology, vol. Haas, “Energy evaluation of spatial modulation in a multi-antenna base station,” in
2013 IEEE 78th Vehicular Technology Conference (VTC Fall), 2014, p. ,” IET Communications, vol.
Hanzo, "Star-QAM Signaling Constellations for Spatial Modulation," IEEE Transactions on Vehicular Technology, vol. Al-Mumit Quazi, "Spatial modulation: Optimal Detector Asymptotic Performance and
multiple-stage detection," IET Communications, vol. Generalized Spatial Modulation ( GSM) is a recently developed multiple-input multiple-output (MIMO) technique that aims to improve data rates over
conventional Spatial Modulation (SM) systems.
Context of Research
An analytical bound on the average BER of the proposed M-APSK GSM and M-APSK GSM-CR systems over fading channels is derived. The overall spectral efficiency in GSM is improved by the base-two
logarithm of the number of transmit antennas compared to SM. Several schemes have been developed to improve the reliability of traditional GSM systems [6-11, 13].
In their paper Zhou et al designed APSK constellations based on the theoretical symbol error probability (SER) of the NCSM system. The design objective is to ensure that adjacent symbols are spaced
further apart in the secondary mapper than the primary mapper. The first approach is to use geometric heuristics, but to the best of the authors' knowledge, heuristics for generalized APSK
constellations have not yet been presented.
However, the resulting constellations in these works [19–22] have certain limitations that make them unsuitable for the M-APSK GSM and GSM-CR systems proposed in this chapter: a) they are
specifically designed for coded systems b) they deviate from those recommended by the DVB-S2 standard. The challenge of developing an M-APSK GSM-CR system is the design of a secondary mapper for a
given M-APSK mapper. Even after reducing the search space, Samra et al [15] report that the algorithm is still too complex for constellations where M¿16.
Recently, a new approach for mapping based on GA was proposed by Patel et al [23, 24]. This algorithm allows the design of higher modulation scheme designers, with possible computational complexity.
Thus, this algorithm is applied in this chapter to design secondary mappers for the proposed M-APSK GSM-CR system.
Structure and Notation
System Model
The signal domain in the GSM-CR system consists of two symbols mapped by bs = log2M bits, where M indicates the order of the APSK modulation scheme used. The same process was applied for the proposed
M-APSK GSM and M-APSK GSM-CR. Alternatively, the received vector for the APSK GSM-CR can be represented as: . ejθk is the transmitted symbol pair and hk = h. is a Nr ×2 dimensional channel matrix
corresponding to the active antenna pair index k. The receiver uses the ML detection rule for estimating the transmit antenna pair index and the transmitted symbol as shown in Eq. Performance
analysis of M-APSK generalized spatial modulation with constellation.
These schemes are referred to as n1 + n2+..+nl APSK where l is the total number of rings and nl is the number of points on the 1th ring. Furthermore, it is worth noting that 4+12 APSK and 4+12+16
APSK are the chosen modulation schemes in the latest DVB-S2 standard for satellite communication over non-linear channels [18]. The constellation diagrams for 16-APSK and 32-APSK and the
corresponding bit allocation for mappers ω1 and ω2 in decimal are shown in Fig.
BER Performance Analysis
Analytical BER of Transmit Antenna Index Estimation (P a )
Analytical BER of Symbol Pair Estimation (P d )
The average BER for the transmit antenna index is calculated assuming that the transmitted signal is correctly detected. Quazi [34] showed that choosing n greater than 6 has sufficient accuracy of
numerical integration. The PEP conditioned on H defined in Eq. 3.13) is averaged by integrating the attenuation distribution.
Hence, the average BER expression of symbol estimation can be obtained by the unconditional probability expressed in Eq.
Constellation Reassignment Mapper Design
Description of Genetic Algorithm
The genetic algorithm described by Patel et al [23] deals with the case where ω1 is known and ω2 is desired. The block diagram in Figure 3.3 provides a high-level illustration of the genetic
algorithm designed by Patel et al[23]. The set of chromosomes considered at each iteration of the algorithm is called a "population".
In the remainder of this section, the authors provide a summarized discussion of each phase of the genetic algorithm. Generating a population of chromosomes The population of chromosomes refers to
the set of candidate secondary mappers evaluated by the genetic algorithm at each iteration. In the genetic algorithm for mapping design, 2× z2. chromosomes are discarded at the end of each
Results and Discussion
The transmit antenna pairs used for M-APSK GSM and M-APSK GSM-CR were obtained from Basar et al. for a 4×Nr and 6×Nr and are shown in Table 3.2 and Table 3.3 [9], respectively. To ensure a fair
comparison, identical spatial mappings were used for both GSM and GSM-CR systems. It is also clear that the GSM-CR outperforms the GSM and SM systems in all cases.
As the graphs in Fig. 3.5, the 16-APSK GSM-CR system achieves a gain of 2.5 dB over its equivalent GSM system and 1.5 dB over its equivalent SM system at BER of 10−5. It can be seen in Fig. 3.7 that
the 16-APSK GSM-CR system achieves a gain of 2.5 dB over its equivalent GSM system and 1.5 dB over its equivalent SM system at BER of 10−5. 3.8, 32-APSK GSM-CR systems achieve gains of 2 dB and 1.2
dB at BER of 10−5 compared to its equivalent GSM and SM systems, respectively.
Figure 3.6: Average BER - 7 bits/s/Hz Configuration
-CR systems achieve gains of 2.2 dB and 2 dB at BER of 10−5 compared to its equivalent GSM and SM systems respectively. It should be noted that similar performance improvements were observed in M-QAM
GSM-CR over SM and GSM in the original work done by Naidoo et al[13]. The performance benefits for the M-APSK GSM-CR system over the M-APSK GSM system at different AASCs are attributed to the
improved symbol pair estimation error performance, Pd.
Since Pd was significantly improved using Eq. 3.8), we can conclude that the overall probability of the system is now bounded by Pa. Therefore, for future works, the next logical step would be to
improve the average BER estimate of the pair of transmit antennas, Pa, in order to further improve the connection reliability of GSM systems.
Future Works
Second, this chapter shows that the GSM-CR system is bound by the error probability of antenna par estimation. Future work should focus on improving Pa to further improve the link reliability of GSM
and GSM-CR systems. Finally, there are more advanced SM systems that further improve performance compared to GSM-CR, such as Space-Time Quadrature Spatial Modulation (ST-QSM), Generalized QSM (GQSM),
and Extended QSM (EQSM) [36–38].
Appendix: Derivation of the Q-function Derivation
Haas, “General Spatial Modulation,” in Conference Proceedings of the 2010 Asilomar Forty-Fourth Conference on Signals, Systems, and Computers, Nov. Yuan, “Incoherent spatial modulation and optimal
design of APSK multi-ring constellation,” IEEE Communications Letters, vol. . Martinez, “Performance Analysis of Turbo Coded APSK Modulations over Nonlinear Satellite Channels,” IEEE Transactions on
Wireless Communications, vol.
Ba¸sar, "Space-time Quadrature Spatial Modulation," in 2017 IEEE International Black Sea Conference on Communications and Networking (BlackSeaCom), 2018, pp. Therefore, an M-APSK GSM-CR system was
developed to improve the overall link reliability of the conventional GSM system. The presented results show that the M-APSK GSM-CR system outperforms its equivalent GSM and SM systems for various
Block Diagram for the Alamouti System [12]
Block Diagram for the Labelling Diversity System [15]
Block Diagram for the Proposed Systems
System Model for M -APSK SM
APSK Constellations
Average BER - 6 bits/s/Hz Configuration
Average BER - 6 bits/s/Hz and 7 bits/s/Hz Configurations
Average BER - 7 bits/s/Hz and 8 bits/s/Hz Configurations
System Model for GSM-CR
GSM-CR Constellations, Key=ω 1 /ω 2
Block Diagram of Genetic Algorithm for CR Mapper Design
Illustrating the position of Genes in M -APSK Constellations
Average BER - 6 bits/s/Hz Configuration
Average BER - 7 bits/s/Hz Configuration
Average BER - 7 bits/s/Hz Configuration
Average BER - 8 bits/s/Hz Configurations | {"url":"https://pubpdf.net/za/docs/application-world-modulation-schemes-advanced-spatial-modulation-systems.10329463","timestamp":"2024-11-11T10:39:20Z","content_type":"text/html","content_length":"156404","record_id":"<urn:uuid:6307aa6a-d828-4b17-a48d-ed3c9486f411>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00250.warc.gz"} |
Updated. Animation added.
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on 02/06/2013(UTC)
Good work uni
I've noted the "Export" button was added (maybe it was here but I missed it). This can be used to save the plot in the GIF format, animated or not, like the
Rank: Administration SMath's base plots.
Groups: Registered, Advanced Member I suppose copying the plots from one worksheet to another will be fixed as well as soon as the plot saving fix will work.
Joined: 23/06/2009(UTC) Regards,
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Thank you for the new version of MapleWrapper plugin. This is really impressive to have so beautiful tool like your plugin in SMath Studio. Once again thank you
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I know how to dissolve one differential equation of the any degree using MapleWrapper plugin. But how to dissolve the system of ODE using MapleWrapper, this
Rank: Advanced Member unfortunately I do not know. Please for the help.
Groups: Registered I tried in different ways to reach the result, however ineffective.
Joined: 19/04/2013(UTC) From above heartily thank you for your help.
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Originally Posted by: sija
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I know how to dissolve one differential equation of the any degree using MapleWrapper plugin. But how to dissolve the system of ODE using MapleWrapper, this
Groups: Registered, Advanced unfortunately I do not know. Please for the help.
I tried in different ways to reach the result, however ineffective.
Joined: 23/06/2009(UTC)
From above heartily thank you for your help.
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I am not quite sure this was correctly solved using Maple
Edited by user 03 June 2013 00:03:23(UTC) | Reason: Not specified
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omorr attached the following image(s):
When Sisyphus climbed to the top of a hill, they said: "Wrong boulder!"
Hello Radovan,
Thank you for these examples. I think that I know already where I made mistakes. Thank you also for your kind - already several times you helped me.
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You are welcome Janusz
And do not hesitate to ask, as I always do
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When Sisyphus climbed to the top of a hill, they said: "Wrong boulder!"
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Make calculations in MapleWrapper must be record the tangent function as: "tan" what it is visible on the first picture. To closing of the file and his repeated opening SMath Studio
Rank: Advanced changes the function tanges from the record: "tan" - required through MapleWrapper, on "tg", and in the this moment MapleWrapper cannot make calculations because it has implemented
Member the tangent function as "tan" and not "tg" as in SMath Studio in the this moment - look the second picture. Well would be to standardize this mark of the tangent function in
MapleWrapper and SMath Studio.
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I think this is a bug (Smath Studio, not mine). The same works and for cot(). You can define the function tg():=tan() and it should work.
The names of the trigonometric functions depends on the settings of the program. I consider them.
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Viacheslav N. Mezentsev
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Hello Uni,
Thank you for explanations and the council.
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There still need to add that maple() function doesn't support nested expressions fully. I had not planned to use such complex commands.
uni attached the following image(s):
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This is very instructive and surprising
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Please for the explanation why I receive two different results as this is visible on the example below:
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There is no error. This behavior depends on the relative positions of the expressions on the sheet. When you put the definition of x above of a function call, the
variable x in the expression solver becomes zero.
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769 post(s)
Hello Uni,
Thank you for the explanation. The first time I meet with such by chance.
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I cannot obtain such result as you. Can I ask you for the source file of your results.
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I Installed MapleWrapper from the Online Gallery
It seems Maple is working fine, but, unfortunately, MaplePlot can not produce a plot for me. There are still "empty" plots there, in spite of inserting the
Rank: Administration plotting text inside them
Groups: Registered, Advanced Member I hope I did not do something wrong.
Joined: 23/06/2009(UTC) Regards,
Posts: 1,740 Radovan
Was thanked: 318 time(s) in 268 Edited by user 06 June 2013 21:37:31(UTC) | Reason: Not specified
When Sisyphus climbed to the top of a hill, they said: "Wrong boulder!"
Not so fast
Rank: Advanced Member
Groups: Registered, Advanced Member
Joined: 10/11/2010(UTC)
Russia ☭ forever
Posts: 1,563 Viacheslav N. Mezentsev
Was thanked: 1311 time(s) in 769 post(s)
on 06/06/2013(UTC)
Originally Posted by: uni
There is no error. This behavior depends on the relative positions of the expressions on the sheet. When you put the definition of x above of a function call, the variable x
Rank: Advanced Member in the expression solver becomes zero.
Groups: Registered
Joined: 19/04/2013(UTC) Hello,
Posts: 46 Right now I noticed, that the power in the calculation of the value t (0) at the end of your sheet is positive. From here the conclusion: there is the error because the value
t(0) have to be zero. The value x equal zero is given at the end because here I pass to numerical calculations.
Location: Podkarpackie,
Poland If I am wrong please for the explanation.
Was thanked: 7 time(s) Regards,
in 7 post(s)
Edited by user 07 June 2013 19:48:19(UTC) | Reason: Not specified
When you are trying to calculate the last expression the maple() function takes the following expression (see mvr5\mvr5.txt):
As you can see: solve({(
Rank: Advanced Member
Groups: Registered, Advanced Member
, but you expected solve({(
Joined: 10/11/2010(UTC)
Posts: 1,563
. That's the difference. The solve() function solves another equation.
Was thanked: 1311 time(s) in 769 post(s)
Russia ☭ forever
Viacheslav N. Mezentsev
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Regularity and stability analysis of discrete-time Markov jump linear singular systems
In this paper, the regularity and stability analysis of discrete-time Markov jump linear singular systems (MJLSS) is performed. When dealing with singular systems, a primary concern is related to the
existence and uniqueness of a solution to the system. This problem, that is called regularity problem, has a known solution when the linear singular system is not subject to jumps (LSS). It turns out
that when the pair of matrices that describes the dynamics of the LSS satisfies a certain condition, then it is regular. By extending this condition to MJLSS, a unique solution for this class of
systems is derived. Indeed through the idea of mode-to-mode regularity, which is stronger than the mode-by-mode notion that can be found in the literature, the existence of a unique solution of an
MJLSS is shown. Furthermore, for systems that are mode-to-mode regular, it is shown how to obtain recursive second moment models whose complexity depends on how anticipative the systems is. The
derived results on regularity and second moment modeling enable us to study stability of an MJLSS. By following the literature on Markov jump linear systems (MJLS), four different concepts of
stability are introduced and their relationship is established. The results presented in this paper generalize well known results concerning the stability given in the MJLS theory. For instance, it
is shown that the stability of the system is equivalent to the spectral radius of an augmented matrix being less than one, as happens in the theory of MJLS.
Original language Spanish
Pages (from-to) 32-40
Number of pages 9
Journal Automatica
Volume 76
State Published - 1 Feb 2017 | {"url":"https://cris.pucp.edu.pe/en/publications/regularity-and-stability-analysis-of-discrete-time-markov-jump-li-2","timestamp":"2024-11-13T18:35:56Z","content_type":"text/html","content_length":"39053","record_id":"<urn:uuid:f3dffa0d-653e-4aad-bfc3-83a90f9ec113>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00777.warc.gz"} |
Lesson 31: Multiple Regression Analysis
Lesson 31: Multiple Regression Analysis
In this lesson, we'll explore using the REG procedure for performing regression analyses with a continuous response variable and more than one predictor variable. We'll also explore using the
LOGISTIC procedure for performing logistic regression analyses with a binary response variable and one or more predictor variables.
Upon completion of this lesson, you should be able to:
• use the REG procedure to analyze data arising from a designed experiment
• use the REG procedure to analyze data arising from nonexperimental research
• use the variable selection methods available in the REG procedure to find the "best" model (or models!) for a set of data
• create and use dummy variables in a regression model
• use the variance inflation factor to look for multicollinearity
• use the LOGISTIC procedure to analyze regression data having a response variable with just two levels (such as yes/no or dead/alive)
• capture the necessary output from the LOGISTIC procedure in order to create a receiver operating characteristic (ROC) curve
Textbook Reference
Chapter 9 of the textbook.
31.1 - Lesson Notes
31.1 - Lesson Notes
B. Designed Regression
It is worth reinforcing the comment on page 287, that the adjusted r-square is the more appropriate measure of the amount of variation explained by a regression with more than one explanatory
D. Stepwise and Other Variable Selection Methods
To drive home the point about how difficult it might be to select the best model when you have a number of predictor variables, let's make it concrete, and suppose we have five possible predictor
variables, a, b, c, d, and e. In that case, there'd be as many as 31 different regression models we'd have to try out on our data:
• 5 models with just one predictor variable: a, b, c, d, and e
• 10 models with two predictor variables: ab, bc, bd, be, ac, ad, ae, cd, ce, and de
• 10 models with three predictor variables: abc, abd, abe, acd, ace, ade, bcd, bce, bde, and cde
• 5 models with four predictor variables: abcd, abce, abde, acde, and bcde
• 1 model with all five predictor variables: abcde
Page 295. Did we learn anything from this study? The best predictor of reading achievement at the end of the sixth grade is reading achievement at the end of the fifth grade. Hmmm.
G. Logistic Regression
Page 309. Sensitivity and specificity rates are typically used in quantifying the value of a diagnostic test. Sensitivity is defined as ... given that a person has a disease, what is the probability
that the diagnostic test will detect the disease? Specificity is defined as ... given that a person is healthy, what is the probability that the diagnostic test will indicate that the person is
healthy? Based on these definitions, it becomes clear that we desire the highest sensitivity and specificity rates that we can get. As you can see, though, on the classification table on page 307,
the two values play off of each other. That is, as sensitivity increases, specificity generally decreases. The goal is to find the point at which we can live with the sensitivity and specificity (or
find another diagnostic test!). In the example here, the authors are suggesting making the cutoff 0.3, so that the sensitivity is high (92%), but the specificity is not too low (45%).
31.2 - Summary
31.2 - Summary
In this lesson, we learned how to use the REG procedure to analyze regression data having a continuous response variable and more than one predictor variable. We also learned how to use the LOGISTIC
procedure to analyze regression data having a binary response variable and one or more predictor variables.
The homework for this lesson will give you more practice with multiple regression and logistic regression analyses. | {"url":"https://online.stat.psu.edu/stat482/book/export/html/649","timestamp":"2024-11-11T19:25:18Z","content_type":"text/html","content_length":"11781","record_id":"<urn:uuid:0a329d83-c40a-4503-81ac-05dde8a3907f>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00045.warc.gz"} |
Q Is a Ratio
1-24 and useful to designers. Technicians should have some knowledge of the factor because it affects so many
things. The factor is known as Q. Some say it stands for quality (or merit). The higher the Q, the better the circuit; the lower the losses (I^2R), the closer the circuit is to being perfect.
Having studied the first part of this chapter, you should not be surprised to learn that resistance (R) has a great effect on this figure of merit or quality. Q Is a Ratio
Q is really very simple to understand if you think back to the tuned-circuit principles just covered.
Inductance and capacitance are in all tuners. Resistance is an impurity that causes losses. Therefore,
components that provide the reactance with a minimum of resistance are "purer" (more perfect) than those
with higher resistance. The actual measure of this purity, merit, or quality must include the two basic quantities, X and R. The ratio
does the job for us. Let's take a look at it and see just why it measures quality. First, if a perfect circuit has zero resistance, then our ratio should give a very high value of Q to
reflect the high quality of the circuit. Does it? Assume any value for X and a zero value for R. Then: Remember, any value divided by zero equals infinity. Thus, our ratio is infinitely high for a
theoretically perfect circuit. With components of higher resistance, the Q is reduced. Dividing by a larger number always yields a
smaller quantity. Thus, lower quality components produce a lower Q. Q, then, is a direct and accurate measure of the quality of an LC circuit.
Q is just a ratio. It is always just a number no units. The higher the number, the "better" the
circuit. Later as you get into more practical circuits, you may find that low Q may be desirable to provide certain characteristics. For now, consider that higher is better.
Because capacitors have much, much less resistance in them than inductors, the Q of a circuit is very often expressed as the Q of the coil or:
The answer you get from using this formula is very near correct for most purposes. Basically, the Q
of a capacitor is so high that it does not limit the Q of the circuit in any practical way. For that reason, the technician may ignore it. | {"url":"https://electriciantraining.tpub.com/14181/css/Q-Is-A-Ratio-36.htm","timestamp":"2024-11-08T15:26:44Z","content_type":"text/html","content_length":"20589","record_id":"<urn:uuid:7152066c-f1d1-4227-8265-988410b18155>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00491.warc.gz"} |
Cosmic concordance and quintessence
We present a comprehensive study of the observational constraints on spatially flat cosmological models containing a mixture of matter and quintessence - a time-varying, spatially inhomogeneous
component of the energy density of the universe with negative pressure. Our study also includes the limiting case of a cosmological constant. We classify the observational constraints by redshift:
low-redshift constraints include the Hubble parameter, baryon fraction, cluster abundance, the age of the universe, bulk velocity and the shape of the mass power spectrum; intermediate-redshift
constraints are due to probes of the redshift-luminosity distance based on Type Ia supernovae, gravitational lensing, the Lyα forest, and the evolution of large-scale structure; high-redshift
constraints are based on measurements of the cosmic microwave background temperature anisotropy. Mindful of systematic errors, we adopt a conservative approach in applying these observational
constraints. We determine that the range of quintessence models in which the ratio of the matter density to the critical density is 0.2 ≲; Ω[m] ≲ 0.5, and the effective, density-averaged equation of
state is -1 ≤ w ≲ -0.2, is consistent with the most reliable, current lowredshift and microwave background observations at the 2 σ level. Factoring in the constraint due to the recent measurements of
Type Ia supernovae, the range for the equation of state is reduced to -1 ≤ w ≲ -0.4, where this range represents models consistent with each observational constraint at the 2 σ level or better
(concordance analysis). A combined maximum likelihood analysis suggests a smaller range, -1 ≤ w ≲ -0.6. We find that the best-fit and best-motivated quintessence models lie near Ω[m] ≈ 0.33, h ≈
0.65, and spectral index n[s] = 1, with an effective equation of state w ≈ 0.65 for "tracker" quintessence and w = -1 for "creeper" quintessence.
All Science Journal Classification (ASJC) codes
• Astronomy and Astrophysics
• Space and Planetary Science
Dive into the research topics of 'Cosmic concordance and quintessence'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/cosmic-concordance-and-quintessence","timestamp":"2024-11-03T10:58:53Z","content_type":"text/html","content_length":"54113","record_id":"<urn:uuid:14eca68d-22be-4902-9886-04744b0c3175>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00051.warc.gz"} |
Numerical Solution to the Time-Dependent Maxwell Equations in Two-Dimensional Singular Domains: The Singular Complement Method
In this paper, we present a method to solve numerically the time-dependent Maxwell equations in nonsmooth and nonconvex domains. Indeed, the solution is not of regularity H^1 (in space) in general.
Moreover, the space of H^1-regular fields is not dense in the space of solutions. Thus an H^1-conforming Finite Element Method can fail, even with mesh refinement. The situation is different than in
the case of the Laplace problem or of the Lamé system, for which mesh refinement or the addition of conforming singular functions work. To cope with this difficulty, the Singular Complement Method is
introduced. This method consists of adding some well-chosen test functions. These functions are derived from the singular solutions of the Laplace problem. Also, the SCM preserves the interesting
features of the original method: easiness of implementation, low memory requirements, small cost in terms of the CPU time. To ascertain its validity, some concrete problems are solved numerically.
• Conforming finite elements
• Maxwell's equation
• Reentrant corners
• Singularities
Dive into the research topics of 'Numerical Solution to the Time-Dependent Maxwell Equations in Two-Dimensional Singular Domains: The Singular Complement Method'. Together they form a unique | {"url":"https://cris.ariel.ac.il/en/publications/numerical-solution-to-the-time-dependent-maxwell-equations-in-two-3","timestamp":"2024-11-12T03:52:43Z","content_type":"text/html","content_length":"56393","record_id":"<urn:uuid:1ffab0a6-79cc-4909-9945-84f082cec898>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00176.warc.gz"} |
How do you use the half-angle identity to find the exact value of sin (-pi/12)? | HIX Tutor
How do you use the half-angle identity to find the exact value of sin (-pi/12)?
Answer 1
Find $\sin \left(- \frac{\pi}{12}\right)$
Answer: - 0.259
Call# sin (-pi/12) = sin t -> cos 2t = cos (- pi/6) = cos (pi/6) = sqrt3/2# Use trig identity: #cos 2t = sqrt3/2 = 1 - 2sin^2 t# #2sin^2 t = (2 - sqrt3)/2# #sin^2 t = (2 -sqrt3)/4#
#sin t = sin (-pi/12) = +- sqrt(2 - sqrt3)/2# = +- 0.259. Only the negative answer is accepted: - 0.259
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Answer 2
You can use the half-angle identity for sine to find the exact value of sin(-π/12) as follows:
sin(θ/2) = ±√((1 - cos(θ)) / 2)
sin(-π/12) = sin((-π/6)/2)
sin((-π/6)/2) = ±√((1 - cos(-π/6)) / 2)
First, find the value of cos(-π/6):
cos(-π/6) = cos(π/6) = √3/2
Now, substitute cos(-π/6) into the half-angle identity:
sin(-π/12) = ±√((1 - √3/2) / 2)
To determine the sign, consider the quadrant. Since -π/12 is in the fourth quadrant where sine is negative:
sin(-π/12) = -√((1 - √3/2) / 2)
Then, simplify to find the exact value of sin(-π/12).
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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• Readily available 24/7 | {"url":"https://tutor.hix.ai/question/how-do-you-use-the-half-angle-identity-to-find-the-exact-value-of-sin-pi-12-8caa474a68","timestamp":"2024-11-04T17:57:22Z","content_type":"text/html","content_length":"569766","record_id":"<urn:uuid:8eb3b7c4-fbd0-457a-8752-254d07422dba>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00097.warc.gz"} |
LaTeX Block
There really needs to be a block that supports LaTeX. It would likely take LaTeX code in a way that looks similar to the code block, but then when unselected would output it as an image of the Math
• Fully agree. It'd be good to the latex functionality.
• Please can you add the ability to write equations. For example in latex if i paste something like this $L = \frac{1}{n} \sum_{i=1}^{n} (y_i - \hat{y}_i)²$ it will convert into a correctly format
Thank you
• latex editor to be able to write equations
• Hey! I merged these same ideas under one so that the upvotes are added together 😁.
• Hey all!
I thought @Adrian Goleby's tip here might be helpful to you as well: | {"url":"https://community.meister.co/discussion/897/latex-block","timestamp":"2024-11-13T06:20:13Z","content_type":"text/html","content_length":"300851","record_id":"<urn:uuid:111cf6f5-52b0-4bab-ac2e-4b30f2bac71e>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00777.warc.gz"} |
CAMB: matter power spectrum from interpolator and transfer function
Hi, I am using CAMB version 1.5.8 and seeing small differences in the matter power spectrum obtained from transfer function `get_matter_transfer_data` (using the formula P_k = A_s*(k/pivot_scalar)^
ns*transfer_fn^2)) and `get_matter_power_interpolator`. The difference is on the order of ~1% regardless of the wavenumber. Is this expected or did I miss something?
The code snippet can be found here:
Re: CAMB: matter power spectrum from interpolator and transfer function
My guess is you want ns-1 not ns, and factor of h somewhere
Re: CAMB: matter power spectrum from interpolator and transfer function
Thanks very much.
I actually tried adjusting the power-law index and including a factor h without success. I see (with nrun=0 and nrun,run=0) and in the CAMB Notes. (I am not sure if in the CAMB notes is different
from in Eq (1) in Hahn & Abel (2011)?)
I don't see how changing ns to ns-1 would help in this situation because the 1% relative difference is independent of k and shows up consistently for different h and other parameter values. Maybe
it's my lack of understanding of the equations in the notes. I would appreciate it if you could point me to useful references or example notebooks.
Re: CAMB: matter power spectrum from interpolator and transfer function
Everything checks out now. Thank you very much! | {"url":"https://cosmocoffee.info/viewtopic.php?t=3886&sid=335c2960e91537cd63f4eca744aa2740","timestamp":"2024-11-02T03:16:09Z","content_type":"text/html","content_length":"38719","record_id":"<urn:uuid:b8dba6a2-3856-4446-8f9c-43a63b741c75>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00754.warc.gz"} |
Mathematics at the University of Virginia
We have started a newsletter for the Department, the Virginia Math Bulletin. We intend to produce a new edition every year. If you would like to be added to our mailing list to receive an electronic
version of the newsletter, please write to us at | {"url":"https://math.virginia.edu/newsletter/","timestamp":"2024-11-06T02:31:23Z","content_type":"text/html","content_length":"98866","record_id":"<urn:uuid:b889488a-41a8-4d38-ace4-83c31c4e5108>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00242.warc.gz"} |
Interior-Point Methods - NEOS Guide
Interior-Point Methods
See Also: Constrained Optimization Linear Programming
The announcement by Karmarkar in 1984 that he had developed a fast algorithm that generated iterates that lie in the interior of the feasible set (rather than on the boundary, as simplex methods do)
opened up exciting new avenues for research in both the computational complexity and mathematical programming communities. Since then, there has been intense research into a variety of methods that
maintain strict feasibility of all iterates, at least with respect to the inequality constraints. Although dwarfed in volume by simplex-based packages, interior-point products have emerged and have
proven to be competitive with, and often superior to, the best simplex packages, especially on large problems.
The NEOS Server offers several solvers that implement interior-point methods, including bpmpd, MOSEK, and OOQP.
Here, we discuss only primal-dual interior point algorithms, which are effective from a computational perspective as well as the most amenable to theoretical analysis. To begin, the dual of the
standard form linear programming problem can be written as
\[\max\left\{b^T y \; : \; s = c – A^T y \geq 0 \right\}.\] Then, the optimality conditions for \((x,y,s)\) to be a primal-dual solution triplet are that
\[Ax = b, \quad A^T y + s = c, \quad S X e = 0, \quad x \geq 0, \quad s \geq 0,\] where
\[S = \mbox{diag}(s_1, s_2, \ldots, s_n), \quad X = \mbox{diag}(x_1, x_2, \ldots, x_n)\] and \(e\) is the vector of all ones.
Interior-point algorithms generate iterates \((x_k,y_k,s_k)\) such that \(x_k > 0\) and \(s_k > 0\). As \(k \to \infty\), the equality-constraint violations \(\|A x_k – b \|\) and \(\| a^T y_k + s_k
– c\|\) and the duality gap \(x_k^T s_k\) are driven to zero, yielding a limiting point that solves the primal and dual linear programs.
Primal-dual methods can be thought of as a variant of Newton's method applied to the system of equations formed by the first three optimality conditions. Given the current iterate \((x_k,y_k,s_k)\)
and the damping parameter \(\sigma_k \in [0,1]\), the search direction \((w_k,z_k,t_k)\) is generated by solving the linear system
\[A w = b – A x_k, \; A^T z + t = c – A^T y_k – s_k, \; S_k w + X_k t = – S_k X_k e + \sigma_k \mu_k e\] where
\[\mu_k = x_k^T s_k / n\]
The new point is then obtained by setting
\[(x_{k+1},y_{k+1},s_{k+1}) \leftarrow (x_k,y_k,s_k) + (\alpha_k^P w_k , \alpha_k^D z_k, \alpha_k^D t_k)\] where \(\alpha_k^D\) and \(\alpha_k^P\) are chosen to ensure that \(x_{k+1} > 0\) and \(s_
{k+1} > 0\).
When \(\sigma_k=0\), the search direction is the pure Newton search direction for the nonlinear system,
\(A x = b, \; A^T y + s = c, \; S X e = 0\)
and the resulting method is an ''affine scaling'' algorithm. The effect of choosing positive values of \(\sigma_k\) is to orient the step away from the boundary of the nonnegative orthant defined by
\(x \geq 0, \; s \geq 0\), thus allowing longer step lengths \(\alpha_k^P, \; \alpha_k^D\) to be taken. ''Path-following'' methods require \(\alpha_k^P, \; \alpha_k^D\) to be chosen so that \(x_k\)
and \(s_k\) are not merely positive but also satisfy the centrality condition\[(x_k,s_k) \in C_k\] where
\[C_k = \left\{ (x,s): x_i s_i \geq \gamma \mu_k, \; i=1,\ldots,n \right\}\] for some \(\gamma \in (0,1)\). The other requirement on \(\alpha_k^P\) and \(\alpha_k^D\) is that the decrease in \(\mu_k
\) should not outpace the improvement in feasibility (that is, the decrease in \(\| A x_k – b \|\) and \(\| A^T y_k + s_k – c \|\)). Greater priority is placed on attaining feasibility than on
closing the duality gap. It is possible to design path-following algorithms satisfying these requirements for which the sequence \(\{ \mu_k \}\) converges to zero at a linear rate. Further, the
number of iterates required for convergence is a polynomial function of the size of the problem (typically, order \(n\) or order \(n^{3/2}\)). By allowing the damping parameter \(\sigma_k\) to become
small as the solution is approached, the method behaves more and more like a pure Newton method, and superlinear convergence can be achieved.
The current generation of interior-point software is quite sophisticated. Most current codes are based on a predictor-corrector algorithm outlined in a 1993 paper by Mehrotra. Mehrotra's algorithm
consists of the basic infeasible-interior-point approach described above, with a corrector component added to each step to increase the order of approximation to the nonlinear (complementarity) term
in the KKT conditions. Implementations of the algorithm also incorporate other heuristics that are essential for robust behavior on many practical problems, including the choice of step length and
centering parameter, handling of free variables (i.e. variables without explicit bounds), preprocessing to remove redundant information in the problem specification, determination of a starting
point, and so on.
Another useful option in practical software is the ability to cross over to a simplex-like method once a good approximation to the solution has been found. This feature is present in the CPLEX
Barrier code.
Most of the computational cost of an interior-point method is associated with the solution of the linear system that defines the search direction. Typically, block elimination is used to obtain a
single linear system in \(z\) alone, specifically,
: \(A X_k S_k^{-1} A^T z = A X_k S_k^{-1}\left[ c – A^T y_k – s_k \right] + b – \sigma_k \mu_k A S_k^{-1} e\).
The coefficient matrix is formed explicitly, except that dense columns of \(A\) (which would cause \(A X_k S_k^{-1} A^T\) to have many more non zeros than \(A\) itself) are handled separately, and
columns that correspond to apparently nonbasic primal variables are eventually dropped.
N. Karmarkar,A new polynomial-time algorithm for linear programming, Combinatorica, 4 (1984), pp. 373-395.
Please see the home page for the book Primal-Dual Interior-Point Methods for more information. | {"url":"https://neos-guide.org/guide/algorithms/ipm/","timestamp":"2024-11-03T23:13:38Z","content_type":"text/html","content_length":"77570","record_id":"<urn:uuid:99ffbf6f-d701-4aab-8463-480b82e5c9fc>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00526.warc.gz"} |
The Stacks project
Lemma 8.11.3. Let $\mathcal{C}$ be a site. Let $p : \mathcal{X} \to \mathcal{C}$ and $q : \mathcal{Y} \to \mathcal{C}$ be stacks in groupoids. Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism
of categories over $\mathcal{C}$. The following are equivalent
1. For some (equivalently any) factorization $F = F' \circ a$ where $a : \mathcal{X} \to \mathcal{X}'$ is an equivalence of categories over $\mathcal{C}$ and $F'$ is fibred in groupoids, the map $F'
: \mathcal{X}' \to \mathcal{Y}$ is a gerbe (with the topology on $\mathcal{Y}$ inherited from $\mathcal{C}$).
2. The following two conditions are satisfied
1. for $y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{Y})$ lying over $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ there exists a covering $\{ U_ i \to U\} $ in $\mathcal{C}$ and objects
$x_ i$ of $\mathcal{X}$ over $U_ i$ such that $F(x_ i) \cong y|_{U_ i}$ in $\mathcal{Y}_{U_ i}$, and
2. for $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, $x, x' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X}_ U)$, and $b : F(x) \to F(x')$ in $\mathcal{Y}_ U$ there exists a covering $\{
U_ i \to U\} $ in $\mathcal{C}$ and morphisms $a_ i : x|_{U_ i} \to x'|_{U_ i}$ in $\mathcal{X}_{U_ i}$ with $F(a_ i) = b|_{U_ i}$.
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Sciencemadness Discussion Board - Vanillin... - Powered by XMB 1.9.11 (Debug Mode)
Nick F posted on 15-7-2004 at 14:19
Hazard to Others Vanillin...
Posts: 439
I guess this goes in bioactive...
7-9-2002 Does anyone know the % of vanillin in a typical vanilla pod? I've read that with good vanilla it can crystalise on the surface, although I'm a bit sceptical of this... I can't
believe it's pure vanillin. Prove me wrong..?
Member Is Offline I thought a fun project this summer would be to try to make some 3,4(,5)-trisubstituted-beta-nitrostyrenes and/or similarly substituted phenylnitroethanes, from OTC stuff. I also
want to get tryptophan from albumin, but I know all about that already.
Of course I would not reduce them any further
No Mood
OK, so you all know what I'm doing, and I know drug preparation isn't discussed here, but a) I'm only asking for a percentage and b) I'm not making drugs, I'm making
nitrostyrenes (
Thanks for any help!
[Edited on 15-7-2004 by Nick F]
[Edited on 15-7-2004 by Nick F]
JohnWW posted on 9-8-2004 at 13:38
Posts: 2849
Registered: I read somewhere that vanillin, which is the sole constituent of artificial vanilla essence, is not the only active constituent of natural vanilla, although it is the major
27-7-2004 constituent. Natural vanilla contains other compounds related to or derived from vanillin; and has a slightly different bouquet and taste as the result.
Location: New Vanillin is a substituted phenol and benzaldehyde, being 3-methoy-4-hydroxy-benzaldehyde, or 2-methoxy-4-aldehyde-phenol. The other consituents of natural vanilla are probably
Zealand isomers of this with different placements of the three functional groups on the benzene ring, and possibly an extra methoxy group, or ethoxy in place of methoxy.
Member Is Offline John W.
No Mood
S.C. Wack posted on 9-8-2004 at 13:59
Posts: 2419
7-5-2004 vanillin content
Cornworld, Central 2%.
Member Is Offline
thefips posted on 25-8-2004 at 03:36
Posts: 33
Why donĀ“t you buy vanillin?I have bought 200g to ...well... make some reactions.It is very pure and no problem to get here in Germany.
Location: Germany But it is a difficult way to nitrostyrenes,if you even do not have the possibility to get the starting material in pure form...
Member Is Offline
self destructive
chemoleo posted on 26-8-2004 at 18:34
Posts: 3005
Nick F - you mentioned having details about the isolation of tryptophan from albumin (the protein)
Registered: Would you care sharing this with us ? I had a look around, but couldnt find much on it
23-7-2003 Damn, even espacenet doesnt have decent patents.
If you post this, please create a sep. thread as this is somewhat unrelated to vanillin.
Location: England
Member Is Offline
Never Stop to Begin, and Never Begin to Stop...
Mood: Tolerance is good. But not with the intolerant! (Wilhelm Busch)
Nick F posted on 27-8-2004 at 04:00
Hazard to Others
Posts: 439
Registered: Actually, the procedure I was thinking of used casein, I got mixed up
7-9-2002 Although, now that I think about it, I'm pretty sure HalfaPint from the Hive (god rest his soul
Member Is Offline
No Mood
jarynth posted on 17-10-2008 at 23:52
Hazard to Self Vanillin from wood
Posts: 76
I've been looking into the synthesis of vanillin from lignin. This is how synthetic vanillin was made until the process was deemed polluting.
12-8-2008 Lignosulfonates are produced by a chemical wood pulping process: wood chips are digested for one day in sulfurous acid. Does anyone know the necessary concentration of the acid?
A winemaking supply store here sells it about 5%. The sulfite process requires a pH between 1.5 and 5.
Member Is Offline
Calcium lignosulfonates are precipitated with Ca(OH)2 by the Howard process (US1981176). Vanillin is formed by oxidation of the lignoslfonates, and it can be separated from them
Mood: according to US4277626. I couldn't find quantitative details about the oxidation step either.
No Mood This slide reports the higher ratio of polymethoxy substituted rings in lignin from deciduous timber. Would this then by the same procedure give a possible route to (e.g.)
Sauron posted on 18-10-2008 at 01:05
Posts: 5351
Vanillin is almost always synthetic or semisynthetic these days. Semisynthetic product is produced from other essential oil components (like isoeugenol).
22-12-2006 The forum library has the two part THE CHEMISTRY OF ESSENTIAL OILS for free download. Better to make use of it than to start new threads without having done any research on your
own first.
Barad-Dur, Mordor
Member Is Offline
Sic gorgeamus a los subjectatus nunc.
jarynth posted on 18-10-2008 at 01:16
Hazard to Self
Posts: 76 Quote:
Originally posted by Sauron
Registered: Better to make use of it than to start new threads without having done any research on your own first.
I agree, that's why I posted in the old thread.
Member Is Offline
Mood: This gives some insight into the sulfite pulping.
No Mood The Chem. of Ess. Oils has something about the coniferin and eugenol processes, nothing about lignin though. The eugenol route was discussed in other threads already; the educt
is not particularly affordable compared to wood.
[Edited on 18-10-2008 by jarynth]
Sauron posted on 18-10-2008 at 01:58
Posts: 5351
Registered: True, but you are not the author of this thread, are you? My remark was directed at him.
Barad-Dur, Mordor
Sic gorgeamus a los subjectatus nunc.
Member Is Offline
jarynth posted on 18-10-2008 at 03:49
Hazard to Self
Posts: 76
Registered: Quote:
12-8-2008 Originally posted by Sauron
True, but you are not the author of this thread, are you? My remark was directed at him.
Member Is Offline
Mood: Isn't there a statute of expiration against flaming thread creators?
No Mood
Sauron posted on 18-10-2008 at 03:51
Posts: 5351
Registered: Ah, the thread author has probably expired in a geriatric ward by now. Oh well, it's the principle that matters.
[Edited on 19-10-2008 by Sauron]
Barad-Dur, Mordor
Member Is Offline
Sic gorgeamus a los subjectatus nunc.
S.C. Wack posted on 18-10-2008 at 15:50
Posts: 2419 Which one - the principle of 10 posts a day?
Registered: I happen to have 6 JACS articles from '36-41 on the subject of benzaldehydes from woody material. No one is going to like the methods used, but no one said it was really
7-5-2004 convenient. No doubt that industrial chemistry encyclopedias, patent databases, and the journal literature have much, much more.
Location: Unfortunately the zip uploaded to the board returns a "seek error" on downloading. No such problem with the same zip uploaded elsewhere.
Cornworld, Central
USA http://mihd.net/96lutjr
Member Is Offline EDIT: response to below
Mood: Originally posted by Sauron
Hey, S.C., best bone up on your arithmetic.
Stats: Averages:
18.21 posts per member
8,180.56 posts per forum
12.11 replies per thread
55.86 posts per day
3.05 new members per day
45.46% of all members have posted.
Member of the Day is Sauron with 11 posts
Would have been 10 (so far today) if you hadn't contributed yet another useless post to this thread.
EDIT 2, again in response to below post:
Originally posted by Sauron
Speaking of useless posts, have you stopped beating your wife? (Y/N)
?????? Is that supposed to be an invitation for me to ask WTF so that you can make yet another worthless post?
[Edited on 18-10-2008 by S.C. Wack]
Sauron posted on 18-10-2008 at 16:17
Hey, S.C., best bone up on your arithmetic.
Posts: 5351
My 2nd anniversary on the forum is fast approaching (12/'06), so my overall thread count works out to more like 5 posts a day. Not 10.
22-12-2006 Honi soi qui mal y pens, mon ami.
Location: The thread author opened this thread to ask the essential oil yield from vanilla pods. A datum readily available from the forum library, from books that may well have been
Barad-Dur, Mordor uploaded by a guy named S.C.Wack. So I would have thought that you would have approved of encouraging people to at least try to crack a book for an answer once in a while.
Member Is Offline That I did not note the date of the thread, mostly because it is essentially invisible to me, is not material.
Mood: If you want to call that post whoring, go ahead. I don't think I am more than mildly post-promiscuous.
metastable [Edited on 19-10-2008 by Sauron]
Sic gorgeamus a los subjectatus nunc.
Sauron posted on 18-10-2008 at 18:10
Posts: 5351 Oh, I see. You are jealous because some silly-ass forum stats function named me Member of the Day?
Registered: Well, I was (a) blissfully unaware of the dubious honor, and (b) don't give a damn for it.
Speaking of useless posts, have you stopped beating your wife? (Y/N)
Barad-Dur, Mordor If I post more than you perhaps it is because I have more to say. In any case, I do not take instructions on when, what, or how to post from you, S.C. so kindly don't forget it.
Member Is Offline
Sic gorgeamus a los subjectatus nunc.
Sauron posted on 18-10-2008 at 21:23
Posts: 5351 Jarynth, the lignin based process is a useful one if you happen to be operating a lumber mill, paper miss, sulfite type wood pulping plant, which are ebvironmentally monstrous,
notorious sources of dimethylmercury etc.
22-12-2006 On a bench scale it's a joke.
Location: You'd be better off buying coniferol and going from there or starting from guiacol (thus skipping the hazardous methylation of catechol) to get wherever it is that you want to
Barad-Dur, Mordor go.
Member Is Offline Of course if you sre just studying vanillin/eugenol production by this process out of idle curiosity, no worries.
Sic gorgeamus a los subjectatus nunc.
nitro-genes posted on 24-10-2008 at 19:14
Posts: 1048 Just found this interesting article about armotherapy for elderly people with obvious psychiatric problems:
Registered: http://lancashirecare.wordpress.com/2008/05/19/dementia-the-...
Supposedly vanilin can help those people to relax and escape their boring daily routines...
Member Is Offline
[Edited on by nitro-genes]
Fleaker posted on 28-10-2008 at 15:39
Vanillin is commercially available; it's cheap (about 55 USD/kg), comes in high purity, and has many uses, whether it be in baking cookies or as a stain for TLC.
Posts: 1252
If anyone wants it, I have a trusted supplier for USP grade vanillin (more than adequate for TLC sulfuric stain), just ask!
Member Is Offline
Neither flask nor beaker.
nucleophilic "Kid, you don't even know just what you don't know. "
--The Dark Lord Sauron
Random posted on 26-4-2011 at 02:43
Posts: 1120
Did someone have success in creating vanillin from wood? Maybe precipitated calcium lignosulfonates could be oxidized using mild oxidizing agents which oxidize alcohols just to
Location: In ur aldehydes and form vanillin?
Member Is Offline | {"url":"https://www.sciencemadness.org/whisper/viewthread.php?tid=2299#pid138789","timestamp":"2024-11-12T02:38:26Z","content_type":"application/xhtml+xml","content_length":"67719","record_id":"<urn:uuid:2089721e-84e1-42a2-9ab1-796c3ca982b3>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00190.warc.gz"} |
: Data Se
Data Set Information
DATA_SET_NAME AMES MARS GENERAL CIRCULATION MODEL 5 LAT PRES VARIABLE V1.0
DATA_SET_ID MODEL-M-AMES-GCM-5-LAT-PRES-V1.0
DATA_SET_DESCRIPTION Data Set Overview : The Ames Mars General Circulation Model is a three dimensional model based on the primitive equations of meteorology. It includes the radiative effects of dust and carbon dioxide as well as other features such as large-scale topography (see [POLLACKETAL1990]; [BARNESETAL1993]; and [HABERLEETAL1993]). The model has 25 latitude bins (7.5 degree resolution), 40 longitude bins (9.0 degree resolution) and 13 vertical layers. The spacing between each layer varies. The following array defines the layer spacing from the tropopause to the surface in mbars: 0.005779, 0.009511, 0.01568, 0.02585, 0.04263, 0.07025, 0.1159, 0.1544, 0.1586, 0.1364, 0.1223, 0.09270, 0.05000. The model spins up from a resting isothermal state with the global temperature equal to 200K. The spin up starts with hour equal to 0. The model saves data every 1.5 hours for a total of 16 hour bins per day. Some other initial conditions that are set when the model spins up: INITIAL SURFACE PRESSURE : 7.60 mbars TROPOPAUSE PRESSURE : 0.06699 mbars DUST OPTICAL DEPTH : 0.3 (for periods where Ls < 200) DUST OPTICAL DEPTH : 1.0 (for periods where Ls > 200) ICE CLOUD OPTICAL DEPTH : 0.00 TIME STEP : 9.25 minutes RICHARDSON NUMBER TIME SCALE : 200000000. seconds SCALE HEIGHT FOR DUST PROFILE : 0.03000 TIME CONSTANT FOR RAYLEIGH FRICTION : 2.0000 days This data set is composed of 20 to 30 day averages of the data resulting from model runs simulating martian conditions for four Ls periods during 1977 (Ls 10-24, 94-103, 202-220, and 267-286). Native start time : 10.15 AREOCENTRIC LON, EARTH YEAR 1977 Native stop time : 286.09 AREOCENTRIC LON, EARTH YEAR 1977 The latitude-pressure data set contains the time and zonally averaged values for several first order, heating, eddy, phase and amplitude variables. The data are given as a function of latitude and vertical pressure. The calculations were done in the pi or uv coordinate system. Pi coordinates are located at the intersection points of a set of evenly spaced latitude circles and a set of evenly spaced longitude circles. UV grid points lie midway between PI grid points. All data are given in the pi coordinate system. The data set is in the following configuration (listed as field followed by description): LATITUDE Latitude, degrees PRESSURE Vertical pressure coordinate, mbars WINDS_MER Meridional wind velocity, m/sec WINDS_VERT Vertical wind velocity, m/sec WINDS_ZON Zonal wind velocity, m/sec TEMP_VERT Vertical temperature profile, K MASS_STRM Mass stream function, kg/sec RES_MASS_STRM Residual mass stream function, kg/sec TOT_NET_HEAT Total net heating, K/day TOT_NET_ACC Total net acceleration, m/sec/day Diabatic heating parameters --------------------------- RAD_HEAT Radiative heating, K/day RF_HEAT Rayleigh friction heating, K/day SOL_HEAT Solar heating, K/day IR_HEAT Infrared heating, K/day 15_MIC_HEAT 15 micron band heating, K/day OUT_15_MIC_HEAT Outside 15 micron band heating, K/day Eddy variables -------------- GEO_HT_VAR RMS geopotential height variance, m GEO_TEMP_VAR RMS geopotential temperature variance, K Phase and amplitude parameters ------------------------------ TEMP_AMP_D Diurnal temperature amplitudes, K TEMP_AMP_SD Semi-diurnal temperature amplitudes, K TEMP_PHASE_D Diurnal temperature phase, hours TEMP_PHASE_SD Semi-diurnal temperature phase, hours
DATA_SET_RELEASE_DATE 1995-01-01T00:00:00.000Z
START_TIME 1994-11-01T12:00:00.000Z
STOP_TIME 1994-12-01T12:00:00.000Z
TARGET_NAME MARS
TARGET_TYPE PLANET
INSTRUMENT_HOST_ID MODEL
INSTRUMENT_NAME UNKNOWN
INSTRUMENT_ID AMES-GCM
INSTRUMENT_TYPE UNKNOWN
NODE_NAME Planetary Atmospheres
ARCHIVE_STATUS ARCHIVED
CONFIDENCE_LEVEL_NOTE Confidence Level Overview : The Ames Mars General Circulation Model is a three dimensional computer model based upon the primitive equations of meteorology. It includes the radiative effects of dust and carbon dioxide as well as other features such as large-scale topography. Model results are dependent upon the selected initial conditions and spatial and temporal resolution of the model, and may not fully incorporate the effects of some physical processes represented by paramaterizations made by the model.
CITATION_DESCRIPTION Citation TBD
ABSTRACT_TEXT The Ames Mars General Circulation Model is a three dimensional model based on the primitive equations of meteorology. It includes the radiative effects of dust and carbon dioxide as well as other
features such as large-scale topography (see [POLLACKETAL1990]
PRODUCER_FULL_NAME MAUREEN BELL
SEARCH/ACCESS DATA Atmospheres Online Archives | {"url":"https://pds.nasa.gov/ds-view/pds/viewProfile.jsp?dsid=MODEL-M-AMES-GCM-5-LAT-PRES-V1.0","timestamp":"2024-11-11T01:28:01Z","content_type":"text/html","content_length":"16423","record_id":"<urn:uuid:b73ab6a8-3be9-4d76-bb9e-254b093bdccd>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00199.warc.gz"} |
pita.pl -- pita
This module performs reasoning over Logic Programs with Annotated Disjunctions and CP-Logic programs. It reads probabilistic program and computes the probability of queries.
See https://friguzzi.github.io/cplint/ for details.
Reexports cplint_util and bddem
- Fabrizio Riguzzi
- Fabrizio Riguzzi
- Artistic License 2.0 https://opensource.org/licenses/Artistic-2.0
Loads File.lpad if it exists, otherwise loads File.cpl if it exists.
Loads FileWithExtension.
The predicate computes the best solution for the decision theory problem. It returns the best strategy in Strategy and it cost in Cost. Complete solution without pruning.
To be used in place of prob/2 for meta calls (doesn't abolish tables)
The predicate computes the most probable abductive explanation of the ground query Query. It returns the explanation in Delta together with its Probability
The predicate builds the BDD for Query and writes its dot representation to file FileName and a list in LV with the association of variables to rules. LV is a list of list, each sublist has three
elements: the multivalued variable number, the rule number and the grounding substitution.
The predicate builds the BDD for Query and returns its dot representation in DotString and a list in LV with the association of variables to rules. LV is a list of list, each sublist has three
elements: the multivalued variable number, the rule number and the grounding substitution.
The predicate builds the BDD for the abductive explanations for Query and returns its dot representation in DotString and lists LV and LAV, the association of variables to rules and to abductive
variables to rules respectively. LV and LAV are lists of list, each sublist has three elements: the multivalued variable number, the rule number and the grounding substitution.
The predicate builds the BDD for the abductive explanations for Query It returns the explanation in Delta together with its Probability. The predicate builds the BDD for Query and returns its dot
representation in DotString and lists LV and LAV, the association of variables to rules and to abductive variables to rules respectively. LV and LAV are lists of list, each sublist has three
elements: the multivalued variable number, the rule number and the grounding substitution.
The predicate computes the explanation of the ground query Query with Maximum A Posteriori (MAP) probability. It returns the explanation in Delta together with its Probability
The predicate computes the explanation of the ground query Query with Maximum A Posteriori (MAP) probability. It returns the explanation in Delta together with its Probability The predicate
builds the BDD for Query and returns its dot representation in DotString and lists LV and LAV, the association of variables to rules and of query variables to rules respectively. LV and LAV are
lists of list, each sublist has three elements: the multivalued variable number, the rule number and the grounding substitution.
The predicate computes the probability of Query If Query is not ground, it returns in backtracking all ground instantiations of Query together with their probabilities
Equivalent to prob/4 with an empty option list.
To be used in place of prob/3 for meta calls (doesn't abolish tables)
Returns the index Variable of the random variable associated to rule with index Rule, grounding substitution Substitution and head distribution Probabilities in environment Environment.
Returns the index Variable of the random variable associated to rule with index Rule in environment Environment.
Returns the index Variable of the random variable associated to rule with index Rule, grounding substitution Substitution and head distribution Probabilities in environment Environment.
Returns a BDD representing Var=Value. This is a predicate for programs in the PRISM syntax
Returns a BDD representing Var=Value when there is a depth bound on derivations. This is a predicate for programs in the PRISM syntax
The predicate sets the value of a parameter For a list of parameters see https://friguzzi.github.io/cplint/
The predicate returns the value of a parameter For a list of parameters see https://friguzzi.github.io/cplint/
Re-exported predicates
The following predicates are exported from this file while their implementation is defined in imported modules or non-module files loaded by this module.
Values is a list of pairs V-N where V is the value and N is the number of samples returning that value. The predicate returns a dict for rendering with c3 as a bar chart with a bar for each value
V. The size of the bar is given by N.
Makes Variable belonging to Environment a query random variable for MAP inference. Returns in BDD the diagram of the formula encoding the required constraints among the Boolean random variable
that represent Variable.
The predicate writes the BDD in dot format to to file FileName.
EBDD is a couple (Environment,BDD) Returns the Probability of BDD belonging to environment Environment Uses
Returns in Variable the index of a new random variable to be queried in MAP inference with NumberOHeads values and probability distribution ProbabilityDistribution. The variable belongs to
A and B are couples (Environment, BDDA) and (Environment, BDDB) respectively Returns in AandB a couple (Environment, BDDAandB) where BDDAandB is pointer to a BDD belonging to environment
Environment representing the conjunction of BDDs BDDA and BDDB. fails if BDDB represents the zero function
Computes the standard deviation of Values. Values can be
□ a list of numbers
□ a list of pairs number-weight, in which case each number is multiplied by the weight before being considered
□ a list of pairs list-weight, in which case list is considered as a matrix of numbers. The matrix in each element of List must have the same dimension and are aggregated element- wise
Prints the debug information which is the result of the call of Cudd_ReadDead, Cudd_CheckZeroRef, Cudd_CheckKeys and Cudd_DebugCheck(env->mgr));
Computes the variance the average of Values. Values can be
□ a list of numbers
□ a list of pairs number-weight, in which case each number is multiplied by the weight before being considered
□ a list of pairs list-weight, in which case list is considered as a matrix of numbers. The matrix in each element of List must have the same dimension and are aggregated element- wise
Returns the Probability of BDD belonging to environment Environment.
Given a list of numeric Values, a Lower value and BinWidth, returns in Couples a list of N pairs V-Freq where V is the midpoint of a bin and Freq is the number of values that are inside the bin
interval [V-BinWidth/2,V+BinWidth/2) starting with the bin where V-BinWidth/2=Lower
Returns in Zero a pointer to a BDD belonging to environment Environment representing the zero Boolean function.
Returns in Variable the index of a new decision variable in Environment
Computes the optimal strategy given a pointer to the ADD belonging to environment Environment. Decision is a list of selected facts, Cost is the total cost.
Returns a Value sampled from a Dirichlet distribution with parameters Alpha. Alpha and Value are lists of floating point numbers of the same length.
Returns a set of 3-dimensional points representing the plot of the density of a sets of 2-dimensional samples. The samples are in List as pairs [X,Y]-W where (X,Y) is a point and W its weigth.
Options is a list of options, the following are recognised by density2d/3:
the minimum value of the X domain, default value the minimum in List
the maximum value of the X domain, default value the maximum in List
the minimum value of the Y domain, default value the minimum in List
the maximum value of the Y domain, default value the maximum in List
the number of bins for dividing the X and Y domains, default value 40
Draws a line chart of the density of two sets of samples, usually prior and post observations. The samples from the prior are in PriorList while the samples from the posterior are in PostList.
PriorList and PostList must be lists of pairs of the form [V]-W or V-W where V is a sampled value and W is its weight, or lists of values V. Options is a list of options, the following are
recognised by histogram/3:
the number of bins for dividing the domain, default value 40 */
Initializes a data structure for storing a single BDD. Returns an integer Environment that is a pointer to a data structure for storing a single BDD to be used for inference only (no learning).
The pseudo-random number generator is initialized using the argument passed as Seed. It calls the C function srand.
Returns in BDD a couple (Env,B) with B a pointer to a BDD belonging to environment Environment representing the disjunction of all the BDDs in ListOfBDDs (a list of couples (Env,BDD))
The predicate returns a dict for rendering with c3 as a bar chart with a bar for the probability
A and B are couples (Environment, BDDA) and (Environment, BDDB) respectively Returns in AorB a couple (Environment, BDDAorB) where BDDAorB is pointer to a BDD belonging to environment Environment
representing the disjunction of BDDs BDDA and BDDB.
Returns in Zero a couple (Environment,BDD) where BDD is pointer to a BDD belonging to environment Environment representing the zero Boolean function
Returns the abductive Explanation of BDD and its Probability. BDD belongs to environment Environment.
Computes the standard deviation and the average of Values. Values can be
□ a list of numbers
□ a list of pairs number-weight, in which case each number is multiplied by the weight before being considered
□ a list of pairs list-weight, in which case list is considered as a matrix of numbers. The matrix in each element of List must have the same dimension and are aggregated element- wise
Given In=A0-N, to_atom/2 returns Out=A-N where A is an atom representing A0
Computes the variance of Values. Values can be
□ a list of numbers
□ a list of pairs number-weight, in which case each number is multiplied by the weight before being considered
□ a list of pairs list-weight, in which case list is considered as a matrix of numbers. The matrix in each element of List must have the same dimension and are aggregated element- wise
Returns in AandB a pointer to a BDD belonging to environment Environment representing the conjunction of BDDs A and B.
Given a pair Key-Vaule, returns its second element Value
Sets the type of parameter initialization for EM on Environment: if Alpha is 0.0, it uses a truncated Dirichlet process if Alpha is a float > 0.0, it uses a symmetric Dirichlet distribution with
that value as parameter
Returns in BDD the BDD belonging to environment Environment that represents the equation Variable=Value.
NumberOfHeads is a list of terms, one for each rule. Each term is either an integer, indicating the number of head atoms in the rule, or a list [N] where N is the number of head atoms. In the
first case, the parameters of the rule are tunable, in the latter they are fixed.
Performs EM learning. Takes as input the Context, information on the rules, a list of BDDs each representing one example, the minimum absolute difference EA and relative difference ER between the
log likelihood of examples in two different iterations and the maximum number of iterations Iterations. RuleInfo is a list of elements, one for each rule, with are either
□ an integer, indicating the number of heads, in which case the parameters of the corresponding rule should be randomized,
□ a list of floats, in which case the parameters should be set to those indicated in the list and not changed during learning (fixed parameters)
□ [a list of floats], in which case the initial values of the parameters should be set to those indicated in the list and changed during learning (initial values of the parameters) Returns the
final log likelihood of examples LL, the list of new Parameters and a list with the final probabilities of each example. Parameters is a list whose elements are of the form [N,P] where N is
the rule number and P is a list of probabilities, one for each head atom of rule N, in reverse order.
Equivalent to density2d/3 with an empty option list.
Computes the sum of the two ADDs ADD1 ADD2 belonging to environment Environment. The result in saved in ADDOut.
Equivalent to densities/4 with an empty option list.
Returns a Value sampled from a uniform distribution in [0,1]
Terminates the context data structure for performing parameter learning. Context is a pointer to a context data structure for performing the EM algorithm. Context must have been returned by a
call to init_em/1. It frees the memory occupied by Context.
Terminates the environment data structure for storing a single BDD. Environment is a pointer to a data structure returned by a call to init/1.
The predicate returns a dict for rendering with c3 as a bar chart with a bar for the number of successes and a bar for the number of failures
Returns in BDD a pointer to a BDD belonging to environment Environment representing the disjunction of all the BDDs in ListOfBDDs
EBDD is a couple (Environment,A) Returns in NotEBDD a couple (Environment,NotA) where NotA is pointer to a BDD belonging to environment Environment representing the negation of BDD A
Returns in Variable the index of a new abducible random variable in Environment with NumberOHeads values and probability distribution ProbabilityDistribution.
Returns in One a couple (Environment,BDD) where BDD is pointer to a BDD belonging to environment Environment representing the one Boolean function
Aggregate values by summation. The first argument is a couple _-N with N the new value to sum to PartialSum
Computes the average of Values. Values can be
□ a list of numbers
□ a list of pairs number-weight, in which case each number is multiplied by the weight before being summed
□ a list of lists, in which case lists are considered as matrices of numbers and averaged element-wise
□ a list of pairs list-weight, in which case the list is considered as a matrix of numbers. The matrix in each element of List must have the same dimension and are aggregated element- wise
Terminates the evnironment data structure for storing a BDD. Environment is a pointer to a data structure returned by init_ex/2. It frees the memory occupied by the BDD.
Returns in AorB a pointer to a BDD belonging to environment Environment representing the disjunction of BDDs A and B.
Given a pair Key-Vaule, returns its first element Key
Returns a Value sampled from a discrete distribution with parameters Theta. Theta is a list of floating point numbers in [0,1] that sum to 1. Value is in 0..(length(Theta)-1)
Multiplies the ADD belonging to environment Environment with the value Utility and stores the result in ADDOut.
Initializes a data structure for performing parameter learning. It returns an integer in Context that is a pointer to a context data structure for performing the EM algorithm.
Draws a line chart of the density of a sets of samples. The samples are in List as pairs [V]-W or V-W where V is a value and W its weigth.
Options is a list of options, the following are recognised by density/3:
the minimum value of domain, default value the minimum in List
the maximum value of domain, default value the maximum in List
the number of bins for dividing the domain, default value 40
Returns a Value sampled from a Gaussian distribution with parameters Mean and Variance
Draws a histogram of the samples in List. List must be a list of pairs of the form [V]-W or V-W where V is a sampled value and W is its weight, or a list of values.
Options is a list of options, the following are recognised by histogram/3:
the minimum value of domain, default value the minimum in List
the maximum value of domain, default value the maximum in List
the number of bins for dividing the domain, default value 40
The predicate returns the BDD in dot format.
The predicate returns a dict for rendering with c3 as a bar chart with a bar for the probability and a bar for one minus the probability.
A and B are couples (Environment, BDDA) and (Environment, BDDB) respectively Returns in AandB a couple (Environment, BDDAandB) where BDDAandB is pointer to a BDD belonging to environment
Environment representing the conjunction of BDDs BDDA and BDDB.
Returns in EBDD a couple (Environment,BDD) where BDD belongs to environment Environment and represents the equation Variable=Value.
Returns in Variable the index of a new random variable in Environment with NumberOHeads values and probability distribution ProbabilityDistribution.
Returns the MAP state MPAState of BDD and its Probability. BDD belongs to environment Environment.
Succeeds if Goal is an atom whose predicate is defined in Prolog (either builtin or defined in a standard library).
Converts the BDD belonging to environment Environment into an ADD.
Computes the value of the multivariate beta function for vector Alphas https://en.wikipedia.org/wiki/Beta_function#Multivariate_beta_function Alphas is a list of floats
Returns in NotA a pointer to a BDD belonging to environment Environment representing the negation of BDD A.
Returns in One a pointer to a BDD belonging to environment Environment representing the one Boolean function.
Given a pair E-W, returns a pair Ep-W where Ep=EE if E=[EE], otherwise Ep=E
Returns a Value sampled from a symmetric Dirichlet distribution with parameter Alpha. K is the number of dimensions of the result.
Initializes an enviroment data structure for storing a BDD. Context is an integer that is a pointer to a context data structure created using init_em/1. Returns an integer Environment that is a
pointer to a data structure for storing a single BDD to be used for the EM algorithm.
Equivalent to density/3 with an empty option list.
Returns a Value sampled from a gamma distribution with parameters Shape and Scale
Equivalent to histogram/3 with an empty option list. | {"url":"https://eu.swi-prolog.org/pack/file_details/cplint/docs/pldoc/pita.html","timestamp":"2024-11-02T11:12:00Z","content_type":"text/html","content_length":"46976","record_id":"<urn:uuid:b76353bd-ae2c-46aa-bd83-f10d2c4eaf9c>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00569.warc.gz"} |
Fast Simulation of Simple Temporal Exponential Random Graph Models
This package provides functions for the computationally efficient simulation and resimulation of dynamic networks estimated with the statistical framework of temporal exponential random graph models
(TERGMs), implemented in the tergm package within the Statnet suite of R software. Networks are represented within an edgelist format only, with nodal attributes stored separately. Also includes
efficient functions for the deletion and addition of nodes within that network representation.
The statistical framework of temporal exponential random graph models (TERGMs) provides a rigorous, flexible approach to estimating generative models for dynamic networks and simulating from them for
the purposes of modeling infectious disease transmission dynamics. TERGMs are used within the EpiModel software package to do just that. While estimation of these models is relatively fast, the
resimulation of them using the tools of the tergm package is computationally burdensome, requiring hours to days to iteratively resimulate networks with co-evolving demographic and epidemiological
dynamics. The primary reason for the computational burden is the use of the network class of object (designed within the package of the same name); these objects have tremendous flexibility in the
types of networks they represent but at the expense of object size. Continually reading and writing larger-than-necessary data objects has the effect of slowing the iterative dynamic simulations.
The tergmLite package reduces that computational burden by representing networks less flexibly, but much more efficiently. For epidemic models, the only types of networks that we typically estimate
and simulate from are undirected, binary edge networks with no missing data (as it is simulated). Furthermore, the network history (edges or node attributes) does not need to be stored for
research-level applications in which summary epidemiological statistics (e.g., disease prevalence, incidence, and variations on those) at the population-level are the standard output metrics for
epidemic models. Therefore, the network may be stored as a cross-sectional edgelist, which is a two-column matrix of current edges between one node (in column one) and another node (in column two).
Attributes of the edges that are called within ERGMs may be stored separately in vector format, as they are in EpiModel. With this approach, the simulation time is sped up by a factor of 25-50 fold,
depending on the specific research application.
Version 2.0 Installation Notes
Versions >= 2.0 implement the new networkLite API that is implemented across the tergmLite, network, and EpiModel packages. If you would like to use the version before the implementation of this new
API, you should install version 1.2.0 with: | {"url":"https://cran.uni-muenster.de/web/packages/tergmLite/readme/README.html","timestamp":"2024-11-11T14:22:20Z","content_type":"application/xhtml+xml","content_length":"8250","record_id":"<urn:uuid:2e317753-0cf7-4aa8-b9d5-6b33a175abb8>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00605.warc.gz"} |
Atomic Functions
This module provides a set of functions to do atomic operations towards mutable atomic variables. The implementation utilizes only atomic hardware instructions without any software level locking,
which makes it very efficient for concurrent access. The atomics are organized into arrays with the following semantics:
Atomics are 64 bit integers.
Atomics can be represented as either signed or unsigned.
Atomics wrap around at overflow and underflow operations.
All operations guarantee atomicity. No intermediate results can be seen. The result of one mutation can only be the input to one following mutation.
All atomic operations are mutually ordered. If atomic B is updated after atomic A, then that is how it will appear to any concurrent readers. No one can read the new value of B and then read the old
value of A.
Indexes into atomic arrays are one-based. An atomic array of arity N contains N atomics with index from 1 to N.
Identifies an atomic array returned from new/2.
• Arity = integer() >= 1
• Opts = [Opt]
• Opt = {signed, boolean()}
Create a new atomic array of Arity atomics.
Argument Opts is a list of the following possible options:
{signed, boolean()}
Indicate if the elements of the array will be treated as signed or unsigned integers. Default is true (signed).
The integer interval for signed atomics are from -(1 bsl 63) to (1 bsl 63)-1 and for unsigned atomics from 0 to (1 bsl 64)-1.
Atomics are not tied to the current process and are automatically garbage collected when they are no longer referenced.
put(Ref, Ix, Value) -> ok
get(Ref, Ix) -> integer()
add_get(Ref, Ix, Incr) -> integer()
Atomic addition and return of the result.
sub(Ref, Ix, Decr) -> ok
Subtract Decr from atomic.
sub_get(Ref, Ix, Decr) -> integer()
Atomic subtraction and return of the result.
exchange(Ref, Ix, Desired) -> integer()
Atomically replaces the value of the atomic with Desired and returns the value it held previously.
compare_exchange(Ref, Ix, Expected, Desired) -> ok | integer()
Atomically compares the atomic with Expected, and if those are equal, set atomic to Desired. Returns ok if Desired was written. Returns the actual atomic value if not equal to Expected.
info(Ref) -> Info
□ Ref = atomics_ref()
□ Info =
#{size := Size, max := Max, min := Min, memory := Memory}
□ Size = integer() >= 0
□ Max = Min = integer()
□ Memory = integer() >= 0
Return information about an atomic array in a map. The map has the following keys:
The number of atomics in the array.
The highest possible value an atomic in this array can hold.
The lowest possible value an atomic in this array can hold.
Approximate memory consumption for the array in bytes. | {"url":"https://www.erldocs.com/22.0.7/erts/atomics","timestamp":"2024-11-10T02:28:57Z","content_type":"text/html","content_length":"9162","record_id":"<urn:uuid:904b2a25-1378-4191-bd8d-b9d42074b14d>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00600.warc.gz"} |
Example 4.1.10: The Leaning Tower of Lire
Jillian, a diligent but overworked student, fell asleep in the library and got locked in for the night. When she awoke, the room was dimly lit and she was alone. To pass the time (and to annoy the
librarian in the morning) she decided to stack books on a table so that they would overhang the edge of the table.
Assuming she has an unlimited supply of books, all of equal width 2 and weight 1 (say), what is the biggest overhang she can produce? To make it more interesting, let's say she can use only one book
at each level.
This problem is sometimes known as the "Leaning Tower of Lire" problem and has a number of available solutions (see references at the end).
Jillian quickly has the idea of simply stacking books vertically near the edge of the table. After stacking a few books she pushes the entire stack so that part of the books overhang the edge of the
This will work, follows the rules, and (since the books have width 2) it is easy to see that while the stack can be arbitrarily high, the largest overhead that can be achieved this way is just about
1. Not much - the librarian in the morning would be annoyed but not impressed.
Jillian thinks again and comes up with a more imaginative way of stacking the books: she uses 'counter-weights' to produce a large overhang, as shown in the picture below.
This can produce a larger overhang indeed, but unfortunately uses more than one book per level. Thus, this idea is against the rules and has to be discarded.
Now Jillian remembers her math background and attacks the problem analytically. Recall that the combined center of gravity c of two objects with (point) mass M[1] and M[2], located at x[1] and x[1],
respectively, is:
To model our problem, let's imagine a number line extending to the right of the table so that the origin is at the right edge of the table.
We can assume that a stack of n-blocks will not fall off the table as long as its center of gravity c[n] . In particular, the center can be at zero and the stack will still (barely) balance on the
edge. Now let's start with an empty table. We will stack our blocks as follows:
Step 1
Put block 1 on the table so that its right edge is at zero. The block has width 2, mass 1, and center of gravity -1.
Now we will shift the block until the center of gravity is at zero so that it still barely stays on the table. Since the block has width 2 and center of gravity -1 we can shift it half its width
(1 unit) to the right and it will still remain on the table.
Stack 1:
□ Mass: M[1] = 1
□ Center of gravity: c[1] = 0
□ Overhang: D[1] = 1
Step 2
Lift the existing stack (one book) straight up. Put a new block on the table as before, so that its right edge is at zero. Put down the existing stack to get a new stack of 2 blocks. It will
remain balanced on the new block.
The center of gravity of this size-2 stack is:
Now shift the block right until its center of gravity is at zero. Since the stack's center of gravity is currently at -^1/[2] we can shift the stack right by that amount.
Stack 2:
□ Mass: M[2] = 2
□ Center of gravity: c[2] = 0
□ Overhang: D[2] = 1+^1/[2]
Step 3
Again lift the existing stack straight up. Put a new block on the table as before, so that its right edge is at zero. Put down the existing stack to get a new stack of 3 blocks. It will remain
balanced on the new block.
The center of gravity of this size-3 stack is:
Then shift the stack until the center of gravity is at zero. Since the stack's center of gravity is -^1/[3] we can shift the stack right by that amount.
Stack 3:
□ Mass: M[3] = 3
□ Center of gravity: c[3] = 0
□ Overhang: D[3] = 1+^1/[2]+^1/[3]
Step 4
Let's do it one more time: lift the existing stack straight up and put a new block on the table so that its right edge is at zero. Drop down the existing stack to get a new stack of 4 blocks. It
will remain balanced on the new block.
The center of gravity of this size-4 stack is:
Then shift the stack until its new center of gravity is at zero. Since the stack's center of gravity right now is -^1/[4] we can shift the stack right by that amount.
Stack 4:
□ Mass: M[4] = 4
□ Center of gravity: c[4] = 0
□ Overhang: D[4] = 1+^1/[2]+ ^1/[3]+ ^1/[4]
Step n
Now we have the algorithm down:
• Lift up the existing stack S[n] with n blocks, center of gravity 0, and overhang D[n] = 1+^1/[2]+...+^1/[n]
• Put down a new block b[n+1] with right edge at zero
• Drop down the stack to form a new stack S[n+1] with n+1 blocks
• Compute the center of gravity of the new stack S[n+1]:
• Shift stack S[n+1] to the right so that its new center of gravity will be zero. The overhang of stack S[n+1] will then be:
D[n+1] = 1 + +^1/[2] + ... + +^1/[n+1]
Thus, we created recursively a stack of N blocks that (just barely) balances on the table yet will overhang the edge of the table by . Since this is the N-th partial sum of the divergent harmonic
series, we can create an overhang as large as we wish - we might need a lot of blocks, though. For example:
│Desired overhang │Blocks needed (N) │
│ 2 │ N = 4 │
│ 4 │ N = 31 │
│ 10 │ N = 12,367 │
│ 22 │ N = 2,012,783,315 │
│ 40 │N = 132,159,290,357,566,703 │
This is a theoretical result but if you have actual blocks of equal size (made of wood, for example) and you work through this algorithm very carefully, you can achieve truely astonishing overhangs -
perhaps not 22 or more but certainly enough impress any librarian!
This version of the
"Leaning Tour of Lires"
has been created by with the help of Jillian Gaglione, a former Seton Hall student, as part of an independent studies project. And no, she did not
fall asleep in the libray - so she says.
• Johnson, Paul B.: “Leaning Tower of Lire.” American Journal of Physics. Occidental College: Los Angeles, California. Volume 23. Number 4. April 1955.
• Sangwin, Chris: “All a matter of balance- or a problem with dominoes.” 3 October 2001. Online Posting. University of Birmingham, Birmingham, UK. 28 July 2003. Available: http://www.mat.bham.ac.uk
• Steve Kifowit, Terra Stamps: "Serious About the Harmonic Series". November 10, 2005. Online Posting. Prairie State College. Available: http://faculty.prairiestate.edu/skifowit/htdocs/sd1.pdf | {"url":"http://pirate.shu.edu/~wachsmut/ira/numser/answers/lire_tower.html","timestamp":"2024-11-11T21:08:52Z","content_type":"text/html","content_length":"15791","record_id":"<urn:uuid:520a1b73-3cb2-4041-9468-114361a9835e>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00563.warc.gz"} |
ThmDex – An index of mathematical definitions, results, and conjectures.
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a
D1159: Probability space
such that
(i) $E_0, E_1, \ldots, E_N \in \mathcal{F}$ are each an D1716: Event in $P$
(ii) $$\mathbb{P}(E_0) = 1$$
Then $$\mathbb{P} \left( \bigcap_{n = 0}^N E_n \right) = \mathbb{P} \left( \bigcap_{n = 1}^N E_n \right)$$ | {"url":"https://theoremdex.org/r/4330","timestamp":"2024-11-09T23:15:52Z","content_type":"text/html","content_length":"6944","record_id":"<urn:uuid:05109b64-c53f-481f-aaa6-6eba7f6e36d5>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00023.warc.gz"} |
Optimal Crashing of an Activity Network with Disruptions
In this paper, we consider an optimization problem involving crashing an activity network under a single disruption. A disruption is an event whose magnitude and timing are random. When a disruption
occurs the duration of an activity, which has not yet started, can change. We formulate a two-stage stochastic mixed integer program, in which the timing of the stage is random. In our model the
recourse problem is a mixed integer program. We prove the problem is NP-hard, and using simple examples, we illustrate properties that differ from the problem's deterministic counterpart. Obtaining a
reasonably tight optimality gap can require a large number of samples in a sample average approximation, leading to large-scale instances that are computationally expensive to solve. Therefore, we
develop a branch-and-cut decomposition algorithm, in which spatial branching of the first stage continuous variables, and linear programming approximations for the recourse problem, are sequentially
tightened. We test our decomposition algorithm with multiple improvements and show it can significantly reduce solution time over solving the problem directly.
View Optimal Crashing of an Activity Network with Disruptions | {"url":"https://optimization-online.org/2019/10/7403/","timestamp":"2024-11-10T05:02:05Z","content_type":"text/html","content_length":"84705","record_id":"<urn:uuid:347eeee8-1957-4a90-98da-789d342e4521>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00046.warc.gz"} |
Video: Elasticity of Demand
The following video is a little long to watch, but it provides an excellent overview of elasticity and explains both the concept and the calculations in a simple, easy-to-follow way.
You can view the transcript for “Episode 16: Elasticity of Demand” here (opens in new window).
In review:
• Price elasticity measures the responsiveness of quantity demanded to a change in the product price
• The calculation for price elasticity is the percentage change in quantity demanded divided by the percentage change in price
• When the absolute value of the price elasticity is >1, the price is elastic and people are very sensitive to changes in price
• When the absolute value of the price elasticity is <1, the price is inelastic and people are insensitive to changes in price | {"url":"https://courses.lumenlearning.com/waymakerintromarketingxmasterfall2016/chapter/video-price-elasticity/","timestamp":"2024-11-03T06:18:01Z","content_type":"text/html","content_length":"46982","record_id":"<urn:uuid:39a5a43b-af67-4542-89d4-66b725fb5dfb>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00409.warc.gz"} |
Importance of Statistical Methods in Medical Research
RGUHS Nat. J. Pub. Heal. Sci Vol No: 9 Issue No: 3 eISSN: 2584-0460
Article Submission Guidelines
Dear Authors,
We invite you to watch this comprehensive video guide on the process of submitting your article online. This video will provide you with step-by-step instructions to ensure a smooth and successful
Thank you for your attention and cooperation.
Review Article
Year: 2018, Volume: 3, Issue: 4, Page no. 42-47,
Views: 1921, Downloads: 66
Research and Statistics are inseparable. The major steps in the research process are Research question, Aim and Objectives, Research design, Statistical hypotheses, Methodology and Analysis of data.
<p>Research and Statistics are inseparable. The major steps in the research process are Research question, Aim and Objectives, Research design, Statistical hypotheses, Methodology and Analysis of
Statistical Methods, Medical Research
Research is dynamic, more so in medical research. Research and Statistics are inseparable. As quote goes “Medicine without Statistics is like a ship without compass” – J E Park & K Park. In addition
to the above quote, it can also be said that “in research, Epidemiology is like a heart and Statistics is like a spine”. To understand how statistical methods play a significant role in medical
research the following schematic diagram on research process is to be understood.
The major steps in the research process are Research question, Aim and Objectives, Research design, Statistical hypotheses, Methodology and Analysis of data. Among these, the following are to be paid
utmost importance in the preparation of research protocol.
Research question: The role of statistics starts from here: in identifying the suitable target population from where the optimum number of representative samples will be selected to collect the
reliable and consistent data to answer the research question. The researcher should pay a careful attention towards PICO and FINGERS, which constitute 25% of planning of conduct of research.
Aim and Objectives: In any research, there will always be one Aim and many Objectives. The Aim addresses the global solution to the research question which is achieved through suitably framed
objectives. Objectives are of two types viz., Primary objective and Secondary objective. For all practical purposes, the primary objective is very important as it relates to the main outcome of the
study, in testing the research hypothesis (if any to be tested), the sample size calculation, and the power of the study. The secondary objectives can serve as axillary interest of the selected
research problem. While framing objectives, attention must be paid towards SMART, in which measurability is most important. It uses measurable action verbs like ‘to describe’, ‘to assess’, ‘to find
out’, ‘to correlate’, to estimate’, ‘to determine’, etc. The researcher should never ever use action verbs like ‘to study’, ‘to know’, ‘to see’, ‘to observe’, ‘to believe’, ‘to define’, etc which are
not measurable statistically. Further, it is also equally important to list the variables to be used generate the data to measure each of the objectives.
Research Design: The research design forms the heart of the research problem. It is said that ‘a strong research design requires the usual statistical methods to analyze the data, but if the research
design is weak, then what ever may be the sophisticated statistical methods chosen to analyze the data may not be able to take care the gap created by poor research design’. Based on the problem
statement any of the following epidemiological designs can be selected.
Besides epidemiological designs, the following are some of the statistical designs used in case of experimental studies, which should also be specified at the time of preparing research protocol.
(a) One group before-and-after design
(b) Two groups after only concurrent parallel design
(c) Two groups before-and-after concurrent parallel design
(d) More than two groups after only concurrent parallel design
(e) More than two groups before-and-after concurrent parallel design
In order to use the above designs, the researcher should also verify the following principles of experimental designs viz.,
(a) Randomization
(b) Replication
(c) Local Control
In case of not being able to go for random allocation of study subjects, then such study design will be referred as Quasi experimental design’, which will need a specific mention in the protocol. The
purpose of knowing statistical designs helps the researcher to know what statistical tests should be applied to arrive at the inference.
2. Statistical hypotheses: Another important step in research process is to specify what type of ‘research hypothesis’ the researcher is intended to test (must be specified in the protocol). Stating
research hypothesis in the protocol plays a vital role in the calculation of sample size, because one-tailed hypothesis need less number of samples, whereas two- tailed hypothesis requires more
samples. Suppose the researcher may wishes for two-tailed hypothesis, but the sample size calculated is for one-tailed, then it might lead to insufficient data to prove what the researcher is
intended to prove. This particularly is very important in clinical trials, where to prove clinical significance statistically, optimum size of the hour will be needed. Further, research hypothesis is
also directly related to ‘power of the study’.
3. Methodology: In addition to sample size calculation, importance should be given to inclusion and exclusion criteria. A properly framed inclusion and exclusion criteria takes care of the issues
like ‘confounders’, yet another issue the researcher has to address in the study. The next step is to decide what sampling techniques should be used for collecting the data. For a better
generalization of the results, it is always better to adopt ‘probability sampling techniques’. In inevitable situations, non-probability sampling techniques may be used, but they will serve only as a
baseline data and once again further studies need to be conducted to generalize such results. As for methods of data collection is concerned, one should decide between either oral interview technique
or self-administered questionnaire. Even though, the telephonic conversation, postal, and internet methods are also can be made use, they may not ensure consistency in the data and may possibly lead
to Berkosonian bias. However, in special cases, only for follow up data collection telephonic conversation method may be used ensuring free from communication barriers. For more reliable and
consistency in the data, it always better to prepare the questionnaire
using ‘Item analysis’, so that the first 27.5% ‘very easy level’ and 27.5% ‘very difficult level’ questions can be removed, resulting in the middle 45% questions which can be answered by all category
of responders.
Analysis of data: This is one of the crucial step in research process. Suitable statistical methods are to be chosen to analyze and present the results. Even though many statistical software’s are
available to analyze the collected data, but research should have good amount of knowledge on choosing right method for analysis. Some of the reputed paid software’s available are ‘SPSS’, ‘SAS’,
‘STATA’, ‘Sys Stat’. There are many open source software’s viz., ‘R’, ‘Python’, ‘Epi info’, ‘PSPP’. Of these, ‘R’ and “Python’ requires some basic training to use, because they are based on coding.
Although ‘Epi info’ requires some hands on training to work, ‘PSPP’ is menu driven like ‘SPSS’.
The first step in data analysis is to identify the type of variables measured and any possible relationship between them. The second step is based on the research design used in the research. The
analysis may be carried out as follows:
If the study design is descriptive and the variable(s) measured is/are qualitative/ categorical, then construct a frequency table, express the result in percentages and represent the results
graphically if necessary. In case, the sample proportion is estimated, then find the standard error of proportion and 100(1-α) % confidence interval for population proportion. This helps the
researcher to convey with what confidence the parameter of the population lie within the sample estimates, which is the ultimate goal of conducting studies based on samples
If the study design is descriptive and the variable(s) measured is/are quantitative, in addition to express the data categorical in the form of frequency table with percentages and graphs wherever
necessary. The emphasis should be given to calculate Mean and SD along with standard error and 100(1-α) % confidence interval for population mean or Median and inter quartile range (IQR). It is very
important to remember in case of analysis of data on quantitative variable that, the data should be checked for normality assumption. If the data is distributed near normal, better to use Mean ± SD
to describe the data, otherwise use Median and IQR for skewed data. In case of highly skewed data, the outliers are to be properly treated (either to retain or impute). The Box-and-Whisker plot helps
to a great extent to tackling the outliers.
If the study design is comparative and the variables measured is qualitative/ categorical, then there are two ways to analyze:
(a) Suppose the statistical estimate is proportions, then the difference between two proportions are to be tested using Standard normal distribution test (Z – test) for specified α-value, however,
based on the test-statistic Z, P-value has to be calculated. (No software has this facility to calculate directly Z – value).
(b) Suppose the independence (or no association) between two categorical variables are to be tested, then Pearson’s Chi-square test is the better choice. Caution: In case, the expected frequency in
any cell is < 5 in a 2 x 2 contingency table, then apply Fisher’s exact probability test (though Yate’s correction is an alternate choice, it gives only approximate value). But, for a m x n
contingency table having expected frequency < 5, modify the table by merging either rows or columns meaningfully and apply the Chisquare test. The degrees of freedom will be reduced accordingly.
If the study design is comparative and the variables measured are quantitative, then the following cases are to be considered to analyze data statistically:
(a) Suppose there are two independent groups and the variable measured is one, then by subjecting to the normality assumption verification using Kolmogorov-Smirnov test or Shapiro-Wilk test; the
Student’s independent two-sample (unpaired) t-test can be applied to test the difference between two population means. The standard error of difference between two means and 100(1-α) % confidence
interval should also be computed to make the results more meaningful and acceptable. The second assumption of equality of variances can be tested using Leven’s F – test. In case, the variances are
not equal, still continue to apply unpaired t-test (which is called as Welch t-test), but the degrees of freedom will be reduced. In case if the normality assumption do not fulfill, then MannWhitney
U – test (a non-parametric test) should applied
(b) In case of single variable but the data are recorded at two different time points in the same individuals, it forms a related (paired) observation. Here owing to normality assumption
verification, Student’s paired t-test should be applied. The standard error for difference between before and after observations as well as 100(1-α) % confidence interval should be computed. One
important note here is that, the paired observations are related (dependent) observations and hence either Pearson’s correlation or Spearman’s rank correlation has to be computed to know the extent
of relation between the two related measurements. In the eventuality, if normality assumption does not fulfill, then apply Wilcoxon signed rank sum test (non-parametric test).
(c) If one variable is measured among
(i) more than two independent groups, then subject to normality assumption verification, apply one-way analysis of variance (ANOVA) for testing equality of k (say) group means against the not equal.
In case, the null hypothesis is rejected, continue with post- hoc test to examine which two group’s mean difference has contributed to not rejecting research hypothesis, a very important step in
ANOVA. In case failure to fulfill normality assumption, apply Kruskal – Wallis non-parametric test.
(ii) more than two independent groups and blocks – apply two-way ANOVA. More than two independent groups, blocks, and treatments – apply three-way ANOVA (LSD). In both the cases not rejecting the
research hypothesis, post-hoc test should be applied.
(iii) Suppose more than two measurements are recorded on every individual forming repeated measures, then apply repeated measures of ANOVA along post-hoc test. Failure to fulfill normality assumption
apply Friedman’s nonparametric test.
If the study design is a correlational study and the variables measured are two, compute either Karl Pearson correlation coefficient or Spearman’s rank correlation coefficient depending on the nature
of data. The correlation coefficient should also be subjected for testing of hypothesis using Student’s t-test and desired level of confidence interval also to be computed.
Suppose the study objective is to fit a regression model (simple or multiple linear), and to predict the outcome, first check the model assumptions:
(a) If the dependent variable is quantitatively measured and normally distributed; error terms are also normally distributed with mean zero and variance σ2 , then fit a simple or multiple linear
regression model and go for prediction.
(b) If the dependent variable is categorical, it will not fulfill the normality assumption and the scatter plot will then be an S-type curve unlike the usual regression model. In such a case, fit a
logistic regression model (depending on the number of categories for dependent variable, it will be called as binary or multinomial logistic regression).
P – value:
Quite often it is mistaken that, every study should have the P – value to increase its credibility so that the readers will appreciate the study and even the publisher’s will accept it publication.
But what is P – value? Is it necessary to find P – value in every study?
The P – value is the strength of evidence against the null hypothesis that the true difference is zero. Corresponding to an observed value of the test – statistic, the P-value (or attained level of
significance) is the lowest level of significance at which the null hypothesis would have been rejected. In other words, it is customary to fix the level of significance α (generally at 5% but rarely
at 1%), and find out the attained level at which the null hypothesis may be rejected attributing the sampling variation and thus supporting alternative hypothesis which is supposed to be proved.
Generally, (i) if P < 0.01%, there may be an ‘overwhelming evidence’ that supports the alternative hypothesis, (ii) if P – value lies between 1% to 5%, there may be a ‘strong evidence’ that supports
the alternative hypothesis, (iii) if P – value lies between 5% to 10%, there may be a ‘weak evidence’ that supports the alternative hypothesis and (iv) ‘no evidence’ that supports the alternative
hypothesis if P > 10%.
Thus the determination of P – value depends on the study design. Usually for a descriptive study, it is not possible to find the P – value, because the objective of the study is to mainly describe
the occurrences in terms of percentages, Mean ± SD along with standard error and confidence interval.
Most often, the P – value will be a hyped number. Majority of research articles do not give teststatistic value and the degrees of freedom, but give only P – value which is incorrect. For a better
understanding by the reader, the test – statistic value, degrees of freedom and P – value has to be given.
Now the current practice is to provide P – value along with confidence interval of the parameter. There is a relationship between the P – value and confidence interval. If the P – value is
significant, the hypothesized value of the parameter(s) will not be included within the confidence interval and contrarily, if it is not significant the confidence interval includes the hypothesized
value of the parameter(s). For example, in case of testing the difference between two means, ‘zero’ will not be included in the confidence interval for difference in means if P – value is
significant, where as in case of odds ratio, ‘one’ will not be included in the confidence interval for odds ratio.
There are many more statistical methods used for analyzing the data depending on the research designs. Some of them are Survival analysis, Discriminant Analysis, Factor analysis, Classification
analysis, which are advance techniques and needs special training to use them. However, all most all statistical software’s have the facility to analyze them.
Supporting File | {"url":"https://journalgrid.com/view/article/rnjph/588","timestamp":"2024-11-03T03:43:51Z","content_type":"text/html","content_length":"53595","record_id":"<urn:uuid:49c21830-a5b9-4190-90fb-47f816721acc>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00452.warc.gz"} |
CauchyDensityExplained Archives » Data Science Tutorials
Return the corresponding value of Cauchy density in R, You will discover how to use the Cauchy functions in this R tutorial. There are four applications of dcauchy, pcauchy, qcauchy, and rcauchy on
this article. Example 1: Return the corresponding value of Cauchy density in R I’ll demonstrate how to make a density plot of…
Read More “Return the corresponding value of Cauchy density in R” » | {"url":"https://datasciencetut.com/tag/cauchydensityexplained/","timestamp":"2024-11-06T20:12:11Z","content_type":"text/html","content_length":"91000","record_id":"<urn:uuid:30856e04-4a18-4f02-a61f-c2a5510a274e>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00352.warc.gz"} |
Arithmetic Functions and the Summation of Series in context of Number Theory
30 Aug 2024
Arithmetic Functions and the Summation of Series: A Journey Through Number Theory
Number theory, a branch of mathematics that deals with the properties and behavior of integers, is a fascinating field that has captivated mathematicians for centuries. At its core lies the study of
arithmetic functions, which are mathematical operations that act on integers to produce new integers. In this article, we will delve into the world of arithmetic functions and explore their role in
the summation of series.
What are Arithmetic Functions?
Arithmetic functions are mathematical operations that take an integer as input and produce another integer as output. These functions can be simple, such as addition or multiplication, or more
complex, involving modular arithmetic or combinatorial calculations. Some examples of arithmetic functions include:
• The sum function: Σ(n) = n + (n-1) + … + 2 + 1
• The product function: Π(n) = n × (n-1) × … × 2 × 1
• The factorial function: !n = n × (n-1) × … × 2 × 1
The Summation of Series
One of the most fundamental concepts in number theory is the summation of series. A series is a sum of terms, where each term is a function of an integer variable. The goal is to find the value of
the series by summing up all the terms.
For example, consider the harmonic series:
1 + 1/2 + 1/3 + … + 1/n
This series is an arithmetic progression with first term 1 and common difference -1/(n+1). The sum of this series can be found using the formula:
Σ(1/i) = ln(n) + γ, where γ is Euler’s constant (approximately 0.5772)
Arithmetic Functions in Series Summation
Arithmetic functions play a crucial role in the summation of series. By applying arithmetic functions to the terms of a series, we can often simplify or evaluate the sum.
For instance, consider the series:
1 + 2 + 3 + … + n
This is an arithmetic progression with first term 1 and common difference 1. The sum of this series can be found using the formula:
Σ(k) = n(n+1)/2
Here, the arithmetic function is the sum function Σ(k), which adds up all the terms in the series.
Möbius Inversion Formula
One of the most powerful tools in number theory is the Möbius inversion formula. This formula allows us to invert a given arithmetic function, i.e., find its inverse.
Let f(n) be an arithmetic function and g(n) be its inverse. The Möbius inversion formula states:
f(n) = Σ(k=1 to n) g(k) μ(n/k)
where μ(n) is the Möbius function, which takes the value 0 if n has any prime factors other than 2 or 3, and -1 otherwise.
The Möbius inversion formula has numerous applications in number theory, including the evaluation of sums of arithmetic functions and the study of modular forms.
Arithmetic functions and the summation of series are fundamental concepts in number theory. By applying arithmetic functions to terms of a series, we can often simplify or evaluate the sum. The
Möbius inversion formula is a powerful tool that allows us to invert a given arithmetic function, opening up new avenues for research and discovery.
As we continue to explore the world of number theory, we will encounter more advanced concepts and techniques, such as modular forms, elliptic curves, and the Riemann Hypothesis. But for now, let us
bask in the beauty and simplicity of arithmetic functions and the summation of series.
• Hardy, G.H., & Wright, E.M. (1938). An Introduction to the Theory of Numbers.
• Apostol, T.M. (1974). Introduction to Analytic Number Theory.
• Knuth, D.E. (1997). The Art of Computer Programming: Volume 2: Seminumerical Algorithms.
1. Σ(k) = n(n+1)/2
2. f(n) = Σ(k=1 to n) g(k) μ(n/k)
3. Σ(1/i) = ln(n) + γ
Related articles for ‘Number Theory’ :
• Reading: Arithmetic Functions and the Summation of Series in context of Number Theory
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15.2.1: Using Linear Equations
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Before we start practicing calculating all of the variables in a regression line equation, let's work a little with just the equation on it's own.
Regression Line Equations
As we just learned, linear regression for two variables is based on a linear equation:
\[\widehat{\mathrm{Y}}=\mathrm{a}+(\mathrm{b}*{X}) \nonumber \]
where \(a\) and \(b\) are constant numbers. What this means is that for every sample, the intercept (a) and the slope (b) will be the same for every score. The X score will change, and that affects Y
(or predicted Y, or \(\widehat{\mathrm{Y}}\)). Some consider the predictor variable (X) as an IV and the outcome variable (Y) as the DV, but be careful that you aren't confusing prediction with
We also just learned that the graph of a linear equation of the form \(\widehat{\mathrm{Y}}=\mathrm{a}+(\mathrm{b}*{X}) \nonumber \) is a straight line.
Exercise \(\PageIndex{1}\)
Is the following an example of a linear equation? Why or why not?
Figure \(\PageIndex{1}\). Sample Plotted Line (CC-BY by Barbara Illowsky & Susan Dean (De Anza College) from OpenStax)
No, the graph is not a straight line; therefore, it is not a linear equation.
The minimum criterion for using a linear regression formula is that there be a linear relationship between the predictor and the criterion (outcome) variables.
Exercise \(\PageIndex{2}\)
What statistic shows us whether two variables are linearly related?
Pearson's r (correlation).
If two variables aren’t linearly related, then you can’t use linear regression to predict one from the other! The stronger the linear relationship (larger the Pearson’s correlation), the more
accurate will be the predictions based on linear regression.
Slope and Y-Intercept of a Linear Equation
As we learned previously, \(b =\) slope and \(a = y\)-intercept. From algebra recall that the slope is a number that describes the steepness of a line, and the \(y\)-intercept is the \(y\) coordinate
of the point \((0, a)\) where the line crosses the \(y\)-axis. Figure \(\PageIndex{2}\) shows three possible graphs of the regression equation (\(y = a + b\text{x}\)). Panel (a) shows what the
regression line looks like if the slope is positive (\(b > 0\)), the line slopes upward to the right. Panel (b) shows what the regression line looks like if there's no slope (\(b = 0\)); the line is
horizontal. Finally, Panel (c) shows what the regression line looks like if the slope is negative (\(b < 0\)), the line slopes downward to the right.
Figure \(\PageIndex{2}\): Three possible graphs of \(y = a + b\text{x}\) . (CC-BY by Barbara Illowsky & Susan Dean (De Anza College) from OpenStax)
I get it, everything has been pretty theoretical so far. So let's get practical. Let's try constructing the regression line equation even when you don't have the scores for either of the variables.
First, we'll start by identifying the variables in the examples.
Example \(\PageIndex{1}\)
Svetlana tutors to make extra money for college. For each tutoring session, she charges a one-time fee of $25 plus $15 per hour of tutoring. A linear equation that expresses the total amount of money
Svetlana earns for each session she tutors is \(y = 25 + 15\text{x}\).
What are the predictor and criterion (outcome) variables? What is the \(y\)-intercept and what is the slope? Answer using complete sentences.
The predictor variable, \(x\), is the number of hours Svetlana tutors each session. The criterion (outcome) variable, \(y\), is the amount, in dollars, Svetlana earns for each session.
The \(y\)-intercept is the constant, the one time fee of $25 (\(a = 25\)). The slope is 15 (\(b = 15\)) because Svetlana earns $15 for each hour she tutors.
Although it doesn't make sense in these examples, the y-intercept (a) is determined when \(x = 0\). I guess with Svetlana, you could say that she gets $25 for any sessions that you miss or don't
cancel ahead of time. But geometrically and mathematically, the y-intercept is based on when the predictor variable (x) has a value of zero.
Exercise \(\PageIndex{3}\)
Jamal repairs household appliances like dishwashers and refrigerators. For each visit, he charges $25 plus $20 per hour of work. A linear equation that expresses the total amount of money Jamal earns
per visit is \(y = 25 + 20\text{x}\).
What are the predictor and criterion (outcome) variables? What is the \(y\)-intercept and what is the slope? Answer using complete sentences.
The predictor variable, \(x\), is the number of hours Jamal works each visit. he criterion (outcome) variable, \(y\), is the amount, in dollars, Jamal earns for each visit.
The y-intercept is 25 (\(a = 25\)). At the start of a visit, Jamal charges a one-time fee of $25 (this is when \(x = 0\)). The slope is 20 (\(b = 20\)). For each visit, Jamal earns $20 for each
hour he works.
Now, we can start constructing the regression line equations.
Example \(\PageIndex{2}\)
Alejandra's Word Processing Service (AWPS) does word processing. The rate for services is $32 per hour plus a $31.50 one-time charge. The total cost to a customer depends on the number of hours it
takes to complete the job.
Find the equation that expresses the total cost in terms of the number of hours required to complete the job. For this example,
• \(x =\) the number of hours it takes to get the job done.
• \(y =\) the total cost to the customer.
The $31.50 is a fixed cost. This is the number that you add after calculating the rest, so it must be the intercept (a).
If it takes \(x\) hours to complete the job, then \((32)(x)\) is the cost of the word processing only.
Thus, the total cost is: \(y = 31.50 + 32\text{x}\)
Let's try another example of constructing the regression line equation.
Exercise \(\PageIndex{4}\)
Elektra's Extreme Sports hires hang-gliding instructors and pays them a fee of $50 per class as well as $20 per student in the class. The total cost Elektra pays depends on the number of students in
a class. Find the equation that expresses the total cost in terms of the number of students in a class.
For this example,
□ \(x =\) number of students in class
□ \(y =\) the total cost
The constant is $50 per class, so that must be the intercept (a).
So $20 per student is the slope (b).
The resulting regression equation is: \(y = 50 + 20\text{x}\)
You can also use the regression equation to graph the line if you input scores from your X variable and your Y variable into the equation. Let's see what that might look like in Figure \(\PageIndex
{3}\) for the equation: \(y = -1 + 2\text{x}\)
Figure \(\PageIndex{3}\): Regression Line for \(y = -1 + 2\text{x}\) . (CC-BY by Barbara Illowsky & Susan Dean (De Anza College) from OpenStax)
In the example in Figure \(\PageIndex{3}\), the intercept (a) is replaced by -1 and the slope (b) is replaced by 2 to get the regression equation (\(y = -1 + 2\text{x}\)). Right now, you are being
provided these constants. Soon, you'll be calculating them yourself!
The most basic type of association is a linear association. This type of relationship can be defined algebraically by the equations used, numerically with actual or predicted data values, or
graphically from a plotted. Algebraically, a linear equation typically takes the form \(y = mx + b\), where \(m\) and \(b\) are constants, \(x\) is the independent variable, \(y\) is the dependent
variable. In a statistical context, a linear equation is written in the form \(y = a + bx\), where \(a\) and \(b\) are the constants. This form is used to help readers distinguish the statistical
context from the algebraic context. In the equation \(y = a + b\text{x}\), the constant b that multiplies the \(x\) variable (\(b\) is called a coefficient) is called the slope. The constant a is
called the \(y\)-intercept.
The slope of a line is a value that describes the rate of change between the two quantitative variables. The slope tells us how the criterion variable (\(y\)) changes for every one unit increase in
the predictor (\(x\)) variable, on average. The \(y\)-intercept is used to describe the criterion variable when the predictor variable equals zero.
Contributors and Attributions
• Barbara Illowsky and Susan Dean (De Anza College) with many other contributing authors. Content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license.
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Aleksander Molak: Practical graph neural networks in Python with TensorFlow and Spektral
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Aleksander Molak: Practical graph neural networks in Python with TensorFlow and Spektral
Practical graph neural networks in Python with TensorFlow and Spektral: Learn how to build and train graph neural networks using Python, TensorFlow, and the Spektral library.
Key takeaways
• Graph neural networks can be used to model complex relationships between nodes in a graph.
• Graphs can be represented as adjacency matrices, where the elements of the matrix indicate whether two nodes are connected.
• Graph convolutional networks (GCNs) are a type of graph neural network that use convolutional neural networks to learn node representations.
• GCNs can be used for node classification, graph classification, and graph regression tasks.
• Graph attention networks (GATs) are another type of graph neural network that use attention mechanisms to learn node representations.
• GATs can be used for node classification, graph classification, and graph regression tasks.
• GraphSage is a type of graph neural network that uses a recursive neural network to learn node representations.
• GraphSage can be used for node classification, graph classification, and graph regression tasks.
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as node classification, graph classification, and graph
• Graph neural networks can be used to model complex relationships between nodes in a graph, and can be used for a variety of tasks such as | {"url":"https://conftalks.com/v/aleksander-molak-practical-graph-neural-networks","timestamp":"2024-11-04T12:23:44Z","content_type":"text/html","content_length":"24095","record_id":"<urn:uuid:f5709328-dad3-48aa-b0f5-48a01807b7f3>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00726.warc.gz"} |
It's Payback Time (again)
In a previous post on the value of modeling and how to think about payback, I shared some basic tools and frameworks to approach unit economics and sustainability.
Now I want to share a follow up that goes into a bit more detail about the tactical decisions that flow from there and start understanding what goes into basic ~~cohort modeling~~.
This is going to build on the content/concepts I covered previously. If you haven’t read Role Modeling or don’t feel like you’re comfortable with these concepts, I’d suggest pausing and checking that
out first. Here’s a refresher:
my goal is usually not to determine what will happen, but rather to understand what would need to be true for something to happen. To my mind, the point of modeling is to ask and answer questions
rigorously, and to be explicit about your assumptions. Putting things into numbers and breaking processes into discrete steps forces you to be specific in your thinking and with the story you’re
telling, even if the numbers and steps are themselves unspecific.
Because startups are money-losing growth machines by design, lots of traditional financial modeling just doesn’t apply. Too often that means overcompensating and looking at top-line performance
absent any more rigorous analysis of what I think of as “sustainability.” Is the growth healthy? People throw around all kinds of terms to asses the health and sustainability of startups. I think
it’s mostly bullshit and doesn’t capture or describe anything meaningful.
I’ve found myself increasingly creating models (which again are thinking frameworks rather than predictive tools) to blend together all the various top-line figures into a more-startup oriented
version of indicative health. I like to think about things in terms of payback in particular.
Once you’ve begun to understand the basic economics of a business, you’ll need to start thinking about more tactical (but no less important) questions using the same general framework. I want to
focus on one in particular. What’s the potential impact of an upfront payment versus a pay as you go model? This is obviously crucial to any business with designs on subscription or repeat revenue.
Once again, I’ll use Harry’s and an example and, once again, all these numbers are totally made up and very very wrong.
Let start with some simple assumptions and say that a full year of Harry’s blades and shaving cream costs you $48 spread across four, quarterly shipments. Let’s also use the same $35 CPA and 70%
gross margin we used in the previous payback analysis. The crucial output here is “periods to payback” because it answers what needs to be true. The 7 lifetime orders per customer is then a
reasonable assumption that shows out where things net out. Here’s what that got us last time around:
Back to the matter at hand. If you charge people upfront, you’ll probably have fewer customers (asking for more money today is a barrier to purchase). On the other hand, your customers probably won’t
churn as much because they’ve already committed to paying (even if you give them a cancellation option or risk free guarantee). Plus, maybe you can charge higher rates for pay as you go. After all,
“pay for the year and get 10% off” really just means “pay as you go and I’ll charge you an extra 10%.”
This seems complex enough for now so I’ll put aside the implications on cash flow for the moment. That’s a topic for another time but suffice to say that upfront payment is favorable to you for all
the reasons that pay as you go is favorable to your customers.
Reader beware
As I’ve said, the point of this exercise is to answer what needs to be true in order for me to meet my desired outcomes. Everything here is about being rigorous in our thinking, not trying to predict
the future. I’m illustrating a general concept, not proving a specific point.
Now let’s use those same assumptions for Harry’s and add in some more info. We’ll assume that Harry’s converts 1.5% of “quality” (non-bounce) visitors to its website into customers. Seems reasonable
enough. Some easy back of the envelope math tells us that that means Harry’s is paying $0.53 for each “lead” (person an ad pushes to its site). Finally, we’ll make some simple assumptions around
churn/cancellations after each shipment. Here’s what we get:
You might look at this and think the numbers don’t tie. I said it would take 4.17 orders to pay back the CPA, now that only seems like it happens around order 8. What gives?
Unlike the previous Harry’s payback model, this is a time series. That means that churn/retention happens in “real time” as people attrite off with each order rather than all at once at the end. So
if you sum up the cohort population percentages through shipment 7 (when net payback starts to get into the black, you’ll get ≈4 orders on average for that cohort. Orders to payback is right in line
for the whole population but it takes longer to get there because so many customers churn off far in advance.
(If you couldn’t already tell, this is getting dangerously close to the cohort analysis post I’ve promised.)
Now, putting on our operator hats, we want to know “how do I make this better?” At bare minimum, we’ll want to think through the tradeoffs of an altered model. Everyone seems to offer some kind of
“subscribe and save” or “pay now and save” option so there must be something to it. Let’s see what happens.
To be conservative, we’ll say that pay as you go will costs users nothing extra. Churn should go up because customers don’t feel like they’ve already spent the money and conversion should go up
because pay as you go is a lower barrier to purchase. We don’t know by how much either will change but we’ll say that both churn and conversion increase by 25% . That gain on conversion decreases CPA
because CPL stays the same but now more of those users are actually buying once they hit the site. Otherwise, the inputs are exactly the same. The outcomes, however, vary widely from the first case:
What we see is that even though orders per user over the two year period decreases from 4.37 (paying annually upfront) to 3.77 (pay as you go), net payback more than doubles from 5% to 13% throughout
the same timespan.
So obviously this is the right answer, right?
Not necessarily. You have to remember that I’m making some pretty wild assumptions. The devil is in the details and no matter how robust your model and how much data you have, early stage operators
need to have conviction behind their choices and a POV that goes beyond 20 minutes of excel. The “right” answer will vary based on factors this type of model couldn’t possibly capture, factors that
are intrinsic to your customers and your product and your brand and your cash flow needs and your goals.
But this is at least a good place to start.
For anyone who’s interested, I’ve updated the payback model in Google Sheets to include this exercise. Play around with it, let me know what I got wrong, and tell me what I should be thinking about | {"url":"https://99d.substack.com/p/its-payback-time-again","timestamp":"2024-11-11T16:50:39Z","content_type":"text/html","content_length":"165063","record_id":"<urn:uuid:ccad9663-94c6-419f-9409-c5e799daf380>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00000.warc.gz"} |
I Have Opinions on Books (Exploring my Goodreads Data) - Graph My Undergrad
I’ve always been a huge reader, so when I discovered Goodreads in 2013 in middle school, I’ve pretty much logged every single book I’ve read since on the website.
I had always wanted to explore my Goodreads data, because it’s a big data set (I’ve put over 500 books on the site), but hadn’t gotten around to it yet as I wasn’t sure how to best structure and
collect my data set from my Goodreads account. That being said, after finals (bye-bye sophomore year!), I was tooling around and found a way to export my data directly to a CSV, perfect for playing
with it in R!
Here are a few fun graphs about my Goodreads history.
The Graphs
To start playing with the data, I wanted to explore whether my opinions on books aligned with the public’s general opinions. On Goodreads, you can rate the book from 1-5 after you have finished it,
with 1 being a low rating and 5 being a high rating. I wanted to compare my ratings to the average value (others’ ratings) of books.
First, I wanted to correlate the two variables. Using R, the correlation coefficient was calculatedd to be .2576, which was lower than I expected. This low value means the two variables are not
strongly linked in a linear relationship.
with(gr_tbl, cor(My_Rating, Average_Rating))
## [1] 0.2575739
I graphed the ratings of all the books I’ve rated on Goodreads below. This histogram has more of a discrete scale, because readers can only rate books as integer values from 1-5.
## [1] 3.729412
ggplot(gr_tbl, aes(My_Rating)) +
geom_histogram(bins=15, col="black", fill="pink") +
labs(x="My Average Rating", y="Number of Books", title="A Histogram of My Ratings of Books I've Read on Goodreads")
Then I graphed the distribution of the average rating of all the books I’ve read on Goodreads. This histogram has more of a continuous x-scale because it’s the average of all of its readers’ ratings
and thus isn’t constrained to being 1-5.
## [1] 4.009784
ggplot(gr_tbl, aes(Average_Rating)) +
geom_histogram(bins=15, col="black", fill="skyblue") +
labs(x="Average Rating", y="Number of Books", title="A Histogram of Average Ratings of Books I've Read on Goodreads")
After that, I wanted to see what the 10 worst-rated books were that I had read, and what I had thought of them. This graph shows a comparison of the average rating (blue, following the color scheme
above) versus my own rating (pink). This graph was challenging to make because I had to use the dplyr verb gather(), which I find very confusing, but some Googling/Stack Overflow helped me out.
worst_books <- gr_tbl %>%
select(Title, My_Rating, Average_Rating) %>%
gather(Rating_Type, Rating, Average_Rating:My_Rating)
ggplot(worst_books, aes(Title, Rating, fill=Rating_Type)) +
geom_col(position="dodge") +
scale_x_discrete(labels=wrap_format(5)) +
labs(title="The 10 Lowest Rated Books I've Read on Goodreads", subtitle ="My Rating vs. the Average Reader's Rating") +
best_books <- gr_tbl %>%
arrange(desc(gr_tbl$Average_Rating)) %>%
subset(Average_Rating>4.50) %>%
select(Title, My_Rating, Average_Rating) %>%
gather(Rating_Type, Rating, Average_Rating:My_Rating)
ggplot(best_books, aes(Title, Rating, fill=Rating_Type)) +
geom_col(position="dodge") +
scale_x_discrete(labels=wrap_format(5)) +
labs(title="The 10 Highest Rated Books I've Read on Goodreads", subtitle="My Rating vs. The Average Reader's Rating") +
Sources of Error & Takeaways
As usual, several assumptions and errors are present in this data. First, it’s not always the best to use the arithmetic mean, but until I take higher level probability and statistics, which looks
like it’ll happen this upcoming Fall semester, I’m using the arithmetic means for now.
The other issue is that Goodreads only allows you to rate a book from 1-5, no half-stars and no non-integer values.
There’s also bias in my data because this is my Goodreads data, and I like to give high ratings to books. I read a lot of chick lit, teen romance, and murder mysteries, but I have broadly popular
tastes. That being said, this was a fun data set to poke around with; I practiced a bit more of my dplyr verbs, and I’m wondering what other websites I could download my data from to fool around | {"url":"http://graphmyundergrad.rbind.io/2019/05/27/opinions-on-books/","timestamp":"2024-11-01T19:03:14Z","content_type":"text/html","content_length":"8334","record_id":"<urn:uuid:f7dbb28a-d5f8-4ce6-a24b-80428e9db1b3>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00703.warc.gz"} |
Making Quantum Connections | Joint Quantum Institute
Making Quantum Connections
July 9, 2014
In quantum mechanics, interactions between particles can give rise to entanglement, which is a strange type of connection that could never be described by a non-quantum, classical theory. These
connections, called quantum correlations, are present in entangled systems even if the objects are not physically linked (with wires, for example). Entanglement is at the heart of what distinguishes
purely quantum systems from classical ones; it is why they are potentially useful, but it sometimes makes them very difficult to understand.
Physicists are pretty adept at controlling quantum systems and even making certain entangled states. Now JQI researchers*, led by theorist Alexey Gorshkov and experimentalist Christopher Monroe, are
putting these skills to work to explore the dynamics of correlated quantum systems. What does it mean for objects to interact locally versus globally? How do local and global interactions translate
into larger, increasingly connected networks? How fast can certain entanglement patterns form? These are the kinds of questions that the Monroe and Gorshkov teams are asking. Their recent results
investigating how information flows through a quantum many-body system are published this week in the journal Nature (10.1038/nature13450), and in a second paper to appear in Physical Review Letters.
Researchers can engineer a rich selection of interactions in ultracold atom experiments, allowing them to explore the behavior of complex and massively intertwined quantum systems. In the
experimental work from Monroe’s group, physicists examined how quickly quantum connections formed in a crystal of eleven ytterbium ions confined in an electromagnetic trap. The researchers used laser
beams to implement interactions between the ions. Under these conditions, the system is described by certain types of ‘spin’ models, which are a vital mathematical representation of numerous physical
phenomena including magnetism. Here, each atomic ion has isolated internal energy levels that represent the various states of spin.
In the presence of carefully chosen laser beams the ion spins can influence their neighbors, both near and far. In fact, tuning the strength and form of this spin-spin interaction is a key feature of
the design. In Monroe's lab, physicists can study different types of correlated states within a single pristine quantum environment (Click here to learn about how this is possible with a crystal of
atomic ions).
To see dynamics the researchers initially prepared the ion spin system in an uncorrelated state. Next, they abruptly turned on a global spin-spin interaction. The system is effectively pushed
off-balance by such a fast change and the spins react, evolving under the new conditions.The team took snapshots of the ion spins at different times and observed the speed at which quantum
correlations grew.
The spin models themselves do not have an explicitly built-in limit on how fast such information can propagate. The ultimate limit, in both classical and quantum systems, is given by the speed of
light. However, decades ago, physicists showed that a slower information speed limit emerges due to some types of spin-spin interactions, similar to sound propagation in mechanical systems. While the
limits are better known in the case where spins predominantly influence their closest neighbors, calculating constraints on information propagation in the presence of more extended interactions
remains challenging. Intuitively, the more an object interacts with other distant objects, the faster the correlations between distant regions of a network should form. Indeed, the experimental group
observes that long-range interactions provide a comparative speed-up for sending information across the ion-spin crystal. In the paper appearing in Physical Review Letters, Gorshkov’s team improves
existing theory to much more accurately predict the speed limits for correlation formation, in the presence of interactions ranging from nearest-neighbor to long-range.
Verifying and forming a complete understanding of quantum information propagation is certainly not the end of the story; this also has many profound implications for our understanding of quantum
systems more generally. For example, the growth of entanglement, which is a form of information that must obey the bounds described above, is intimately related to the difficulty of modeling quantum
systems on a computer. Dr. Michael Foss-Feig explains, “From a theorist’s perspective, the experiments are cool because if you want to do something with a quantum simulator that actually pushes
beyond what calculations can tell you, doing dynamics with long-range interacting systems is expected to be a pretty good way to do that. In this case, entanglement can grow to a point that our
methods for calculating things about a many-body system break down.”
Theorist Dr. Zhexuan Gong states that in the context of both works, “We are trying to put bounds on how fast correlation and entanglement can form in a generic many-body system. These bounds are very
useful because with long-range interactions, our mathematical tools and state-of-the-art computers can hardly succeed at predicting the properties of the system. We would then need to either use
these theoretical bounds or a laboratory quantum simulator to tell us what interesting properties a large and complicated network of spins possess. These bounds will also serve as a guideline on what
interaction pattern one should achieve experimentally to greatly speed up information propagation and entanglement generation, both key for building a fast quantum computer or a fast quantum
From the experimental side, Dr. Phil Richerme gives his perspective, “We are trying to build the world’s best experimental platform for evolving the Schrodinger equation [math that describes how
properties of a quantum system change in time]. We have this ability to set up the system in a known state and turn the crank and let it evolve and then make measurements at the end. For system sizes
not much larger than what we have here, doing this becomes impossible for a conventional computer.”
This news item was written by E. Edwards/JQI. | {"url":"https://jqi.umd.edu/news/making-quantum-connections","timestamp":"2024-11-12T10:39:58Z","content_type":"text/html","content_length":"477013","record_id":"<urn:uuid:83b3044a-eb46-437b-8d42-7d8a5c2ad71c>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00450.warc.gz"} |
Category: algorithms Component type: function
find_end is an overloaded name; there are actually two find_end functions.
template <class ForwardIterator1, class ForwardIterator2>
find_end(ForwardIterator1 first1, ForwardIterator1 last1,
ForwardIterator2 first2, ForwardIterator2 last2);
template <class ForwardIterator1, class ForwardIterator2,
class BinaryPredicate>
find_end(ForwardIterator1 first1, ForwardIterator1 last1,
ForwardIterator2 first2, ForwardIterator2 last2,
BinaryPredicate comp);
Find_end is misnamed: it is much more similar to search than to find, and a more accurate name would have been search_end.
Like search, find_end attempts to find a subsequence within the range [first1, last1) that is identical to [first2, last2). The difference is that while search finds the first such subsequence,
find_end finds the last such subsequence. Find_end returns an iterator pointing to the beginning of that subsequence; if no such subsequence exists, it returns last1.
The two versions of find_end differ in how they determine whether two elements are the same: the first uses operator==, and the second uses the user-supplied function object comp.
The first version of find_end returns the last iterator i in the range [first1, last1 - (last2 - first2)) such that, for every iterator j in the range [first2, last2), *(i + (j - first2)) == *j. The
second version of find_end returns the last iterator i in [first1, last1 - (last2 - first2)) such that, for every iterator j in [first2, last2), binary_pred(*(i + (j - first2)), *j) is true. These
conditions simply mean that every element in the subrange beginning with i must be the same as the corresponding element in [first2, last2).
Defined in the standard header algorithm, and in the nonstandard backward-compatibility header algo.h.
Requirements on types
For the first version:
• ForwardIterator1 is a model of Forward Iterator.
• ForwardIterator2 is a model of Forward Iterator.
• ForwardIterator1's value type is a model of EqualityComparable.
• ForwardIterator2's value type is a model of EqualityComparable.
• Objects of ForwardIterator1's value type can be compared for equality with Objects of ForwardIterator2's value type.
For the second version:
• ForwardIterator1 is a model of Forward Iterator.
• ForwardIterator2 is a model of Forward Iterator.
• BinaryPredicate is a model of Binary Predicate.
• ForwardIterator1's value type is convertible to BinaryPredicate's first argument type.
• ForwardIterator2's value type is convertible to BinaryPredicate's second argument type.
• [first1, last1) is a valid range.
• [first2, last2) is a valid range.
The number of comparisons is proportional to (last1 - first1) * (last2 - first2). If both ForwardIterator1 and ForwardIterator2 are models of Bidirectional Iterator, then the average complexity is
linear and the worst case is at most (last1 - first1) * (last2 - first2) comparisons.
int main()
char* s = "executable.exe";
char* suffix = "exe";
const int N = strlen(s);
const int N_suf = strlen(suffix);
char* location = find_end(s, s + N,
suffix, suffix + N_suf);
if (location != s + N) {
cout << "Found a match for " << suffix << " within " << s << endl;
cout << s << endl;
int i;
for (i = 0; i < (location - s); ++i)
cout << ' ';
for (i = 0; i < N_suf; ++i)
cout << '^';
cout << endl;
cout << "No match for " << suffix << " within " << s << endl;
[1] The reason that this range is [first1, last1 - (last2 - first2)), instead of simply [first1, last1), is that we are looking for a subsequence that is equal to the complete sequence [first2,
last2). An iterator i can't be the beginning of such a subsequence unless last1 - i is greater than or equal to last2 - first2. Note the implication of this: you may call find_end with arguments such
that last1 - first1 is less than last2 - first2, but such a search will always fail.
See also
STL Main Page | {"url":"http://ld2014.scusa.lsu.edu/STL_doc/find_end.html","timestamp":"2024-11-11T21:15:31Z","content_type":"text/html","content_length":"9201","record_id":"<urn:uuid:fda13af9-33ea-4831-8624-efa10bebb194>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00209.warc.gz"} |
Integers and Opposite Numbers (solutions, examples, worksheets, videos, lesson plans)
Related Pages
Lesson Plans and Worksheets for Grade 6
Lesson Plans and Worksheets for all Grades
More Lessons for Grade 6
Common Core For Grade 6
New York State Common Core Math Grade 6, Module 3, Lesson 3, Lesson 4
Grade 6, Module 3, Lesson 3 Worksheets (pdf)
Grade 6, Module 3, Lesson 4 Worksheets (pdf)
The following figure shows positive numbers, negative numbers and opposite numbers. Scroll down the page for examples and solutions.
Lesson 3 Student Outcomes
• Students use positive and negative numbers to indicate a change (gain or loss) in elevation with a fixed reference point, temperature, and the balance in a bank account.
• Students use vocabulary precisely when describing and representing situations involving integers; e.g., an elevation of - 10 feet is the same as 10 feet below the fixed reference point.
• Students choose an appropriate scale for the number line when given a set of positive and negative numbers to graph.
Example 1: A Look at Sea Level
The picture below shows three different people participating in activities at three different elevations. What do you think the word elevation means in this situation?
Refer back to Example 1. Use the following information to answer the questions.
The diver is 30 feet below sea level.
The sailor is at sea level.
The hiker is 2 miles (10,560 feet) above sea level.
1. Write an integer to represent each situation.
2. Use an appropriate scale to graph each of the following situations on the number line to the right.
Also, write an integer to represent both situations.
a. A hiker is 15 feet above sea level.
b. A diver is 20 feet below sea level.
3. For each statement there are two related statements: i and ii. Determine which related statement is expressed correctly (i and ii), and circle it. Then correct the other related statement so that
both parts, i and ii, are stated correctly.
a. A submarine is submerged 800 feet below sea level.
b. The elevation of a coral reef with respect to sea level is given as -250 feet.
Lesson 4 Student Outcomes
• Students understand that each nonzero integer, a, has an opposite , denoted -a; and that a and -a are opposites if they are on opposite sides of zero and are the same distance from zero on the
number line.
• Students recognize the number zero is its own opposite.
• Students understand that since all counting numbers are positive, it is not necessary to indicate such with a plus sign.
Example 1: Every Number has an Opposite
Locate the number 8 and its opposite on the number line. Explain how they are related to zero.
Exercises 2–3
2. Locate the opposites of the numbers on the number line.
9, -2, 4, -7
3. Write the integer that represents the opposite of each situation. Explain what zero means in each situation.
a. 100 feet above sea level.
b. 32 degrees below zero.
c. A withdrawal of $25
Example 2: A Real World Example
Maria decides to take a walk along Central Avenue to purchase a book at the bookstore. On her way, she passes the Furry Friends Pet Shop and goes in to look for a new leash for her dog. The Furry
Friends Pet Shop is seven blocks west of the bookstore. After she leaves the bookstore, she heads east for seven blocks and stops at Ray’s Pet Shop to see if she can find a new leash at a better
price. Which locations, if any, are the furthest from Maria while she is at the bookstore?
Determine an appropriate scale and model the situation on the number line below.
Explain your answer. What does zero represent in the situation?
Exercises 4–6
Read each situation carefully and answer the questions. 4. On a number line, locate and label a credit of $15 and a debit for the same amount from a bank account. What does zero represent in this
situation? 5. On a number line, locate and label 20 C below zero and 20 C above zero. What does zero represent in this situation? 6. A proton represents a positive charge. Write an integer to
represent protons. An electron represents a negative charge. Write an integer to represent electrons.
What is the relationship between any number and its opposite when plotted on a number line? How would you use this relationship to locate the opposite of a given number on the number line? Will this
process work when finding the opposite of zero?
Lesson 3 Problem Set
1. Write an integer to match the following descriptions.
a. A debit of $40
b. A deposit of $225
c. 14,000 feet above sea level
d. A temperature increase of 40°F
e. A withdrawal of $225
f. 14,000 feet below sea level
For Problems 2–4, read each statement about a real-world situation and the two related statements in parts (a) and (b) carefully. Circle the correct way to describe each real-world situation;
possible answers include either (a), (b), or both (a) and (b).
2. A whale is 600 feet below the surface of the ocean.
a. The depth of the whale is 600 feet from the ocean’s surface.
b. The whale is -600 feet below the surface of the ocean.
3. The elevation of the bottom of an iceberg with respect to sea level is given as -125 feet.
a. The iceberg is 125 feet above sea level.
b. The iceberg is 125 feet below sea level.
4. Alex’s body temperature decreased by 2°F.
a. Alex’s body temperature dropped 2°F.
b. The integer -2 represents the change in Alex’s body temperature in degrees Fahrenheit.
5. A credit of $35 and a debit of $40 are applied to your bank account.
a. What is an appropriate scale to graph a credit of $35 and a debit of $40? Explain your reasoning.
b. What integer represents “a credit of $35” if zero represents the original balance? Explain.
c. What integer describes “a debit of $40” if zero represents the original balance? Explain.
d. Based on your scale, describe the location of both integers on the number line.
e. What does zero represent in this situation?
Lesson 4 Problem Set
1. Find the opposite of each number, and describe its location on the number line.
a. -5
b. 10
c. -3
d. 15
2. Write the opposite of each number, and label the points on the number line.
a. Point A: the opposite of 9
b. Point B: the opposite of -4
c. Point C: the opposite of -7
d. Point D: the opposite of 0
e. Point E: the opposite of 2
3. Study the first example. Write the integer that represents the opposite of each real-world situation. In words, write the meaning of the opposite.
a. An atom’s positive charge of 7
b. A deposit of $25
c. 3,500 feet below sea level
d. A rise of 45°C
e. A loss of 13 pounds
4. On a number line, locate and label a credit of $38 and a debit for the same amount from a bank account. What does zero represent in this situation?
5. On a number line, locate and label 40℃ below zero and 40℃ above zero. What does zero represent in this situation?
Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.
We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page. | {"url":"https://www.onlinemathlearning.com/opposite-number.html","timestamp":"2024-11-07T22:03:49Z","content_type":"text/html","content_length":"46737","record_id":"<urn:uuid:b8860edf-b03c-4da6-8dc9-43c878ce97ee>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00818.warc.gz"} |
TOC | Previous | Next | Index
33.2 Finding Function Roots of Derivable Functions (.NET, C#, CSharp, VB, Visual Basic, F#)
Class NewtonRalphsonRootFinder implements the IOneVariableDRootFinder interface and finds roots of univariate functions using the Newton-Raphson Method. The Newton-Raphson algorithm finds the slope
of the function at the current point and uses the zero of the tangent line as an estimate of the root.
Like SecantRootFinder and RiddersRootFinder (Section 33.1), instances of NewtonRalphsonRootFinder are constructed by specifying an error tolerance and a maximum number of iterations, or by accepting
the defaults for these values. For example:
Code Example – C# root finding
double tol = 1e-8;
int maxIter = 100;
var finder = new NewtonRaphsonRootFinder( tol, maxIter );
Code Example – VB root finding
Dim Tol As Double = "1e-8"
Dim MaxIter As Integer = 100
Dim Finder As New NewtonRaphsonRootFinder(Tol, MaxIter)
Once you have constructed a NewtonRalphsonRootFinder instance, you can use the Find() method to find a root within a given interval. For instance, this polynomial has a root at 1:
This code finds the root in the interval (0, 3):
Code Example – C# root finding
var p = new Polynomial(
new DoubleVector( -2.0, -5.0, 9.0, -2.0 ) );
var finder = new NewtonRaphsonRootFinder();
double lower = 0;
double upper = 3;
double root = finder.Find( p, p.Derivative(), lower, upper );
Code Example – VB root finding
Dim P As New Polynomial(New DoubleVector(-2.0, -5.0, 9.0, -2.0))
Dim Finder As New NewtonRaphsonRootFinder()
Dim Lower As Double = 0
Dim Upper As Double = 3
Dim Root As Double = Finder.Find(P, P.Derivative(), Lower, Upper) | {"url":"https://www.centerspace.net/doc/NMath/user/root-finding-84907.htm","timestamp":"2024-11-03T12:40:40Z","content_type":"text/html","content_length":"15665","record_id":"<urn:uuid:31a46f51-168d-4d86-8a67-3fe0fb860253>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00518.warc.gz"} |
PPT - M atrix A lgebra on G PU and M ulticore A rchitectures PowerPoint Presentation - ID:2221900
1. Matrix Algebra on GPU andMulticoreArchitectures Stan TomovResearch DirectorInnovative Computing LaboratoryDepartment of Computer ScienceUniversity of Tennessee, KnoxvilleWorkshop on GPU-enabled
Numerical LibrariesUniversity of Basel, Switzerland May 11-13, 2011
2. Outline • PART I • Introduction to MAGMA • Methodology • Performance • PART II • Hands-on training • Using and contributing to MAGMA • Examples
3. Part I: Outline • Motivation • MAGMA – LAPACK for GPUs • Overview • Methodology • MAGMA with StarPU / PLASMA / Quark • MAGMA BLAS • Sparse iterative linear algebra • Current & future work
directions • Conclusions
4. Part I: Outline Goals • Motivation [ Hardware to Software Trends ] • MAGMA – LAPACK for GPUs • Overview [ Learn what is available, how to use it, etc. ] • Methodology [ How to develop, e.g.,
hybrid algorithms ] • MAGMA with StarPU / PLASMA / Quark [ Development tools ] • MAGMA BLAS [ Highly optimized CUDA kernels ] • Sparse iterative linear algebra [ Methodology use in sparse LA ] •
Current & future work directions • Conclusions
5. About ICL Last year ICL celebrated 20 years anniversary! • Mission – provide leading edge tools, enable technologies and software for scientific computing, develop standards for scientific
computing in general • This includes standards and efforts such asPVM, MPI, LAPACK, ScaLAPACK, BLAS, ATLAS, Netlib, Top 500, PAPI, NetSolve, and the LinpackBenchmark • ICL continues these efforts
with PLASMA, MAGMA, HPC Challenge, BlackJack, OpenMPI, and MuMI, as well as other innovative computing projects staff of more than40 researchers, students, and administrators Established by Prof.
Jack Dongarra
8. Hardware Trends • Power consumption and themove towards multicore • Hybrid architectures • GPU • Hybrid GPU-based systems • CPU and GPU to get integrated(NVIDIA to make ARM CPU cores alongside
GPUs) x86 host DMA hostmemory 7.5 GB/s PCI-e 3.0
9. Performance Development in Top500 100 Pflop/s 10 Pflop/s N=1 1 Pflop/s Gordon Bell Winners 100 Tflop/s 10 Tflop/s 1 Tflop/s N=500 100 Gflop/s 10 Gflop/s 1 Gflop/s 100 Mflop/s
12. Commodity plus Accelerators Commodity Accelerator (GPU) Intel Xeon 8 cores 3 GHz 8*4 ops/cycle 96 Gflop/s (DP) NVIDIA C2050 “Fermi” 448 “CUDA cores” 1.15 GHz 448 ops/cycle 515 Gflop/s (DP)
Interconnect PCI-X 16 lane 64 Gb/s 1 GW/s 17 systems on the TOP500 use GPUs as accelerators
13. Future Computer Systems • Most likely be a hybrid design • Think standard multicore chips and accelerator (GPUs) • Today accelerators are attached • Next generation more integrated • Intel’s MIC
architecture “Knights Ferry” and “Knights Corner” to come. • 48 x86 cores • AMD’s Fusion in 2012 - 2013 • Multicore with embedded graphics ATI • Nvidia’s Project Denver plans to develop an
integrated chip using ARM architecture in 2013.
14. Major change to Software • Must rethink the design of our software • Another disruptive technology • Similar to what happened with cluster computing • and message passing • Rethink and rewrite
the applications, algorithms, and • software • Numerical libraries for example will change • For example, both LAPACK and ScaLAPACK will • undergo major changes to accommodate this
18. A New Generation of Software Those new algorithms - have a very low granularity, they scale very well (multicore, petascale computing, … ) - removes of dependencies among the tasks, (multicore,
distributed computing) - avoid latency (distributed computing, out-of-core) - rely on fast kernels Those new algorithms need new kernels and rely on efficient scheduling algorithms.
19. A New Generation of Software MAGMA Hybrid Algorithms (heterogeneity friendly) Those new algorithms - have a very low granularity, they scale very well (multicore, petascale computing, … ) -
removes of dependencies among the tasks, (multicore, distributed computing) - avoid latency (distributed computing, out-of-core) - rely on fast kernels Those new algorithms need new kernels and
rely on efficient scheduling algorithms. Rely on - hybrid scheduler (of DAGs) - hybrid kernels (for nested parallelism) - existing software infrastructure
20. Challenges of using GPUs • High levels of parallelismMany GPU cores [ e.g. Tesla C2050 (Fermi) has 448 CUDA cores ] • Hybrid/heterogeneous architecturesMatch algorithmic requirements to
architectural strengths[ e.g. small, non-parallelizable tasks to run on CPU, large and parallelizable on GPU ] • Compute vs communication gapExponentially growing gap; persistent challenge[
Processor speed improves 59%, memory bandwidth 23%, latency 5.5% ][ on all levels, e.g. a GPU Tesla C1070 (4 x C1060) has compute power of O(1,000)Gflop/s but GPUs communicate through the CPU
using O(1) GB/s connection ]
21. Matrix Algebra on GPU and MulticoreArchitectures (MAGMA) • MAGMA:a new generation linear algebra (LA) libraries to achieve the fastest possible time to an accurate solution on hybrid/
heterogeneous architecturesHomepage:http://icl.cs.utk.edu/magma/ • MAGMA & LAPACK • MAGMA uses LAPACK and extends its functionality to hybrid systems (w/ GPUs); • MAGMA is designed to be similar
to LAPACK infunctionality, data storage and interface • MAGMA leverages years of experience in developing open source LA software packages like LAPACK, ScaLAPACK, BLAS, ATLAS, and PLASMA • MAGMA
developers/collaborators • U of Tennessee, Knoxville; U of California, Berkeley; U of Colorado, Denver • INRIA Bordeaux - SudOuest & INRIA Paris – Saclay, France; KAUST, Saudi Arabia • Community
effort [similarly to the development of LAPACK / ScaLAPACK]
22. PLASMAParallel Linear Algebra Software for Multicore Architectures
23. Asychronicity • Avoid fork-join (Bulk sync design) • Dynamic Scheduling • Out of order execution • Fine Granularity • Independent block operations • Locality of Reference • Data storage – Block
Data Layout PLASMA Parallel Linear Algebra Software for Multicore Architectures
24. LAPACK LU • fork join • bulk synchronous processing
25. Parallel tasks in LU • Idea:break into smaller tasks and remove dependencies • Objectives:high utilization of each core, scaling to large number of cores • Methodology:Arbitrary DAG scheduling,
Fine granularity / block data layout
26. PLASMA SchedulingDynamic Scheduling: Tile LU Trace • Regular trace • Factorization steps pipelined • Stalling only due to natural load imbalance 8-socket, 6-core (48 cores total) AMD Istanbul 2.8
GHz quad-socket quad-core Intel Xeon 2.4 GHz
27. Pipelining: Cholesky Inversion 48 cores POTRF, TRTRI and LAUUM. The matrix is 4000 x 4000,tile size is 200 x 200,
28. Big DAGs: No Global Critical Path DAGs get very big, very fast So windows of active tasks are used; this means no global critical path Matrix of NBxNB tiles; NB3 operation NB=100 gives 1 million
31. MAGMA Software Stack C P U H Y B R I D G P U distr. Tile & LAPACK Algorithms with DAGuE MAGNUM / Rectangular / PLASMA Tile Algorithms multi PLASMA / Quark StarPU LAPACK Algorithms and Tile
Kernels MAGMA 1.0 MAGMA SPARSE single MAGMA BLAS LAPACK BLAS BLAS CUDA Linux, Windows, Mac OS X | C/C++, Fortran| Matlab, Python
32. MAGMA 1.0 • 32 algorithms are developed (total – 122 routines) • Every algorithm is in 4 precisions (s/c/d/z, denoted by X) • There are 3 mixed precision algorithms (zc & ds, denoted by XX) •
These are hybrid algorithms • Expressed in terms of BLAS • Support is for single CUDA-enabled NVIDIA GPU, either Tesla or Fermi • MAGMA BLAS • A subset of GPU BLAS, optimized for Tesla and Fermi
33. MAGMA 1.0 One-sided factorizations
34. MAGMA 1.0 Linear solvers
35. MAGMA 1.0 Two-sided factorizations
36. MAGMA 1.0 Generating/applying orthogonal matrices
37. MAGMA 1.0 Eigen/singular-value solvers • Currently, these routines have GPU-acceleration for the • two-sided factorizations used and the • Orthogonal transformation related to them (matrix
generation/application from the previous slide)
38. MAGMA BLAS • Subset of BLAS for a single NVIDIA GPU • Optimized for MAGMA specific algorithms • To complement CUBLAS on special cases
39. MAGMA BLAS Level 2 BLAS
40. MAGMA BLAS Level 3 BLAS • CUBLAS GEMMs for Fermi are based on the MAGMA implementation • Further improvements– BACUGen - Autotuned GEMM for Fermi (J.Kurzak)– ZGEMM from 308 Gflop/s is now 341
41. MAGMA BLAS Other routines
43. Methodology overview • MAGMA uses HYBRIDIZATION methodology based on • Representing linear algebra algorithms as collections of TASKS and DATA DEPENDENCIES among them • Properly SCHEDULING tasks'
execution over multicore and GPU hardware components • Successfully applied to fundamentallinear algebra algorithms • One and two-sided factorizations and solvers • Iterative linear and
eigen-solvers • Productivity • High-level • Leveraging prior developments • Exceeding in performance homogeneous solutions Hybrid CPU+GPU algorithms(small tasks for multicores and large tasks for
44. Statically Scheduled One-Sided Factorizations(LU, QR, and Cholesky) • Hybridization • Panels (Level 2 BLAS) are factored on CPU using LAPACK • Trailing matrix updates (Level 3 BLAS) are done on
the GPU using “look-ahead” • Note • Panels are memory bound but are only O(N2) flops and can be overlapped with the O(N3) flops of the updates • In effect, the GPU is used only for the
high-performance Level 3 BLAS updates, i.e., no low performance Level 2 BLAS is scheduled on the GPU
45. A hybrid algorithm example • Left-looking hybrid Cholesky factorization in MAGMA 1.0 • The difference with LAPACK – the 3 additional lines in red • Line 10 (done on CPU) is overlapped with work
on the GPU (line 7)
46. Hybrid algorithms QR factorization in single precision arithmetic, CPU interface Performance of MAGMA vs MKL MAGMA QR time breakdown Time GFlop/s Matrix size x 1000 Matrix size x 1000 GPU :
NVIDIA GeForce GTX 280 (240 cores @ 1.30GHz) GPU BLAS : CUBLAS 2.2, sgemm peak: 375 GFlop/sCPU : Intel Xeon dual socket quad-core (8 cores @2.33 GHz) CPU BLAS : MKL 10.0 , sgemm peak: 128 GFlop/s
[ for more performance data, see http://icl.cs.utk.edu/magma ]
47. Results – one sided factorizations LU Factorization in double precision FERMI Tesla C2050: 448 CUDA cores @ 1.15GHz SP/DP peak is 1030 / 515 GFlop/s ISTANBUL AMD 8 socket 6 core (48 cores)
@2.8GHz SP/DP peak is 1075 / 538 GFlop/s Similar results for Cholesky & QR Fast solvers (several innovations) - in working precision, and - mixed-precision iter. refinement based on the one-sided
50. Mixed Precision Methods • Mixed precision, use the lowest precision required to achieve a given accuracy outcome • Improves runtime, reduce power consumption, lower data movement • Reformulate to
find correction to solution, rather than solution[ Δx rather than x ]. | {"url":"https://www.slideserve.com/colin/m-atrix-a-lgebra-on-g-pu-and-m-ulticore-a-rchitectures","timestamp":"2024-11-07T02:56:01Z","content_type":"text/html","content_length":"90599","record_id":"<urn:uuid:cb5729de-d9c8-43b4-8d38-4a423e812656>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00713.warc.gz"} |
DLASSQ - Linux Manuals (3)
DLASSQ (3) - Linux Manuals
dlassq.f -
subroutine dlassq (N, X, INCX, SCALE, SUMSQ)
DLASSQ updates a sum of squares represented in scaled form.
Function/Subroutine Documentation
subroutine dlassq (integerN, double precision, dimension( * )X, integerINCX, double precisionSCALE, double precisionSUMSQ)
DLASSQ updates a sum of squares represented in scaled form.
DLASSQ returns the values scl and smsq such that
( scl**2 )*smsq = x( 1 )**2 +...+ x( n )**2 + ( scale**2 )*sumsq,
where x( i ) = X( 1 + ( i - 1 )*INCX ). The value of sumsq is
assumed to be non-negative and scl returns the value
scl = max( scale, abs( x( i ) ) ).
scale and sumsq must be supplied in SCALE and SUMSQ and
scl and smsq are overwritten on SCALE and SUMSQ respectively.
The routine makes only one pass through the vector x.
N is INTEGER
The number of elements to be used from the vector X.
X is DOUBLE PRECISION array, dimension (N)
The vector for which a scaled sum of squares is computed.
x( i ) = X( 1 + ( i - 1 )*INCX ), 1 <= i <= n.
INCX is INTEGER
The increment between successive values of the vector X.
INCX > 0.
SCALE is DOUBLE PRECISION
On entry, the value scale in the equation above.
On exit, SCALE is overwritten with scl , the scaling factor
for the sum of squares.
SUMSQ is DOUBLE PRECISION
On entry, the value sumsq in the equation above.
On exit, SUMSQ is overwritten with smsq , the basic sum of
squares from which scl has been factored out.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
September 2012
Definition at line 104 of file dlassq.f.
Generated automatically by Doxygen for LAPACK from the source code. | {"url":"https://www.systutorials.com/docs/linux/man/3-DLASSQ/","timestamp":"2024-11-06T12:31:56Z","content_type":"text/html","content_length":"8679","record_id":"<urn:uuid:181a56a3-f602-4b2b-850c-f27b1b2bad72>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00342.warc.gz"} |
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Uncertainty Propagation in Wave Loadings
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working with nature- civil and hydraulic engineering to aspects of real world problems in water and at the waterfront - within coastal environments
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Uncertainty Propagation in Wave Loadings
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There is nothing noble in being superior to your fellow man; true nobility is being superior to your former self. Who can be a better person than Ernest Hemingway (1899 – 1961) – to write this in his
skillful way of crafting words in a lucid and attractive style? Sayings similar to this have been penned down in several pieces of WIDECANVAS in different contexts – not to advance is to fall back –
change and refinement as a show of intelligence – maturity – adaptation . . . etc. But Hemingway touched upon a very important aspect of human mind. That being taken over by superiority or
inferiority complex (see aspects of it, in Some Difficult Things) – inhibits a person’s ability to think and function normally. This piece is nothing about these complexes – but on something that
define Nature – in this case, the transmission or propagation of errors or uncertainties in wave loadings on coastal structures. Uncertainty (U), in its simplest term, is just the lack of surety or
absolute confidence in something.
Uncertainty Propagation (UP) refers to the transfer of uncertainties from the independent variables into the dependent variable – simply put, from the known to the unknown. It is transferred in an
equation or relation – from the individual variables on right hand side – into the dependent variable on the left. More commonly the propagation process is referred to as error propagation. The two –
error and uncertainty are often used interchangeably. In quantitative terms, while error refers to the difference between the measured and the true value – uncertainty refers to the deviation of an
individual measurement from the arithmetic mean of a set of measurements. As we shall see, the magnitude of propagated uncertainty is a function of the type of equation (e.g. linear, non-linear,
exponential, logarithmic, etc).
. . .
1. Uncertainty Fundamentals
Uncertainty of a parameter implies that, if it is measured repeatedly – one would find that there is no single value – rather a range of random values accrue that deviate from the arithmetic mean
(AM, µ) of the measured set. One needs a method or standardization to characterize the scattered deviations. If the deviations are distributed symmetrically about the arithmetic mean – then a
Gaussian (German mathematician Carl Freidrich Gauss, 1777 – 1855) bell-shaped curve can be fitted. One property of such a distribution is defined as the Standard Deviation (SD). This is estimated as
the square root of variance (defined as the mean of all deviations squared). If SD is normalized by dividing it with AM – the GD turns into Normal Distribution or ND. The normalized SD, σ/µ, termed
as the Coefficient of Variation (CV) – is SD relative to AM. Its distribution follows the symmetry about the mean – and as a fraction or percentage, it covers both sides of the mean. It is like the
unit of standard deviation – e.g. 1SD unit saying that 68.2% of the data are scattered on both sides of the mean. A high value of CV is the indication of a large scatter about the mean. CVs are due
to nature of the variable in their random response to different forcing functions or kinetic energy (see Turbulence) – and are therefore termed as random uncertainty or simply uncertainty (see more
on Uncertainty and Risk). It is the signature characteristic of the variable – and is due to many other factors including the applied measuring or sampling methods.
Not all variables follow the Gaussian distribution (GD), however. For example, a discrete random variable, like an episodic earthquake or tsunami event – are sparse and do not follow the rules of
continuity, and is best described by Poisson Distribution (PD, in honor of French Mathematician Simeon Denis Poisson, 1781 – 1840). An ideal example of a continuous variable that follows ND is
coastal water level. In this piece, all applied variables are assumed to follow ND. Here are some typical CVs from R Soulsby (1997): water density, ±0.2%; kinematic water viscosity, ±10%; sediment
density, ±2%; sediment grain diameter, ±20%; water depth, ±5%; current speed, ±10%; current direction, ±10o; significant wave height, ±10%; wave period, ±10%; and wave direction, ±15o.
Error or uncertainty propagation technique has been in use for long time dating back to the now known method since 1974 (G Dalquist and A Bjorck). The most recent treatment of the subject can be
found in BN Taylor and CE Kuyatt (1994) and in AIAA 1998 (The American Institute of Aeronautics and Astronautics). The propagated uncertainty has nothing to do with the scientific merit of a relation
or equation; it is rather due to the characteristic or signature uncertainties of the independent variables – which according to the UP principle must propagate or transmit onto the dependent
. . .
2. Propagation Basics
This piece is primarily based on four pieces posted earlier: Uncertainty and Risk; Wave Forces on Slender Structures; Breakwater; and The World of Numbers and Chances; and three of my papers:
• 2015: Longshore Sand Transport – An Examination of Methods and Associated Uncertainties.
• 2011: Role of Parameter Uncertainty in Design Decisions – Analytical Assessment for a Coastal Breakwater and Harbor Entrance Sedimentation. ISOPE-2011-TPC-230, Hawaii (paper not presented).
• 2008: Wave Loads on Piles – Spectral versus Monochromatic Approach.
. . .
Before moving on, let me try to demonstrate how UP principle works – by discussing a simple example. Suppose, we consider an equation, X = Y^2 * Z. Let us say, the variables Y and Z on the right hand
side of the equation have known CVs: ± y, and ± z, respectively. How to estimate the CV of X? According to the UP principle, the CV of X can be determined as the square root of x^2 = 2^2*y^2 + z^2.
As an example, suppose, y = ±10%, and z = ±5%; then x must be equal to 20.62%.
Further, a pertinent question must be answered. Why Uncertainty? or Why Uncertainty Propagation? The relevance of the questions stems from the quests to develop confidence of the relations or
equations one uses to compute and estimate parameters for everything – from the science of Nature to Social Interactions to Engineering and Technology. These relations developed by investigators
after painstaking pursuits convey theories and principles mostly on deterministic paradigm. But, things in Nature are hardly deterministic – which means the independent variables on which a relation
is based – suffer from uncertainties of some kind due to their stochastic characteristics and variability. These uncertainties associated with the independent variables must be accounted for in the
dependent variable or computed unknown parameter. Uncertainty propagation method developed over a period of many years – gives answer to the questions (see more on Uncertainty and Risk, and The World
of Numbers and Chances).
In engineering design processes, the traditional method of accounting for uncertainty is done simply by including some redundancy in the system – by the so-called factors of safety – conspicuously
described and/or inconspicuously embedded in some practices (for example, using maximum load and minimum strength; and summation of different loads together although they may not occur simultaneously
). Further elaboration on coastal design processes can be found in Oumeraci et al (1999), Burcharth (2003) and Pilarczyk (2003). They scaled the processes of design as: Level 0 – deterministic
approach; Level I – quasi-probabilistic approach; Level II – approximate probabilistic approach; and Level III – fully probabilistic approach. In the Level 0 approach, parameter uncertainties are not
accounted for, instead experience and professional judgment are relied upon to implant redundancy. This practice as a way of developing confidence or assurance – represents in reality – a process of
introducing another layer of uncertainty – partly because of heuristics associated with judgments. Or in another interpretation, it amounts to over-designing structure elements at the expense of high
cost. For the other three Levels, a load-strength reliability function is defined in different scales to account for parameter uncertainties.
A note on significant wave height uncertainty is warranted. Although a typical ±10% is recommended by Soulsby, in reality the uncertainty can be varied. The reasons can be traced to how the local
design significant wave height is estimated. Some likely methods that affect uncertainty are: (1) the duration, resolution and proximity of measurements to the structure; (2) extremal analysis of
measurements to derive design waves; (3) in absence of measurements, applied analytical hindcasting or numerical methods to estimate wave parameters; and (4) applied wave transformation routines or
modeling. Due to these diverse factors affecting uncertainty, instead of considering one uncertainty, this piece covers a range from10 to 30%.
. . .
3. Uncertainty of Wave Loading on a Vertical Pile
This portion of the piece starts with 2008 ISOPE paper and Wave Forces on Slender Structures. Unbroken waves passing across the location of a slender structure (when L/D < 1/5; L is local wave length
and D is structure dimension perpendicular to the direction of force) cause two different types of horizontal forces on it. The basis of determining them is the Morison equation (Morison and others
1950). Known as the drag force in the direction of velocity, the first is due to the difference in local horizontal velocity head or dynamic pressure between the stoss and the wake sides of
structure. The second, the inertial force is caused by the resistance of structure to the local horizontal water particle acceleration.
Both of the Morison Forces have their roots in Bernoulli Theorem (Daniel Bernoulli; 1700 – 1782) – and as one can imagine, they are a function of water density – and of course, the structure size.
The horizontal Drag Force: a function of water density, structure dimension perpendicular to the flow, water particle orbital velocity squared, and a drag coefficient. The horizontal Inertial Force:
a function of water density, structure cross-sectional area, water particle orbital acceleration, and an inertial coefficient.
To demonstrate UP of wave loadings at the water surface on a cylindrical vertical pile of 1 meter diameter – this piece relies on the same example wave discussed in Linear Waves; Nonlinear Waves;
Spectral Waves; Waves – Height, Period and Length and Characterizing Wave Asymmetry. This wave, H= 1.0 m; T = 10 second; d = 10 m; has a local wave length, L = 70.9 m and Ursell Number (Fritz Joseph
Ursell; 1923 – 2012) = 5.1; indicating that the wave can be treated as a linear wave at this depth. Other used and estimated parameters are: water density = 1025 kg/m^3; amplitude of horizontal
orbital velocity at surface = 0.56 m/s; and amplitude of horizontal orbital acceleration at surface = 0.44 m/s^2. In addition, while using most typical uncertainties proposed by Soulsby – the Us of
wave length, orbital velocity and acceleration have no typical values – therefore they are derived in the 2011 paper and in this piece applying the basic UP principle.
The results of uncertainties in wave loadings are shown in the two presented images – one for the drag force (UDF), the other for inertial force (UIF). They are shown as a function of uncertainties
in measured wave heights (U_H) for U_water density = 0.2% and U_linear dimension = 5%, with estimated U_cylindrical pile area = 10%. Since the uncertainties of coefficients (U_Cd and U_Cm) are not
known, the images show three cases of them, 10%, 20% and 30%. Here are some numbers for U_H = 10% and 30%.
• UDF for U_H = 10%. U_orbital velocity = 24.5%. UDF = 50.3% (for U_Cd = 10%); UDF = 53.2% (for U_Cd = 20%); and UDF = 57.7% (for U_Cd = 30%).
• UDF for U_H = 30%. U_orbital velocity = 37.4%. UDF = 75.7% (for U_Cd = 10%); UDF = 77.6% (for U_Cd = 20%); and UDF = 80.8% (for U_Cd = 30%).
• UIF at U_H = 10%. U_orbital acceleration = 22.4%. UIF = 26.5% (for U_Cm = 10%); UIF = 31.6% (for U_Cm = 20%); and UIF = 38.7% (for U_Cm = 30%).
• UIF at U_H = 30%. U_orbital acceleration = 36.1%. UIF = 38.7% (for U_Cm = 10%); UIF = 42.4% (for U_Cm = 20%); and UIF = 48.0% (for U_Cm = 30%).
The shown uncertainties indicate that they increase nonlinearly as the U_H increases; and that nonlinearity associated with drag force is a showcase of higher uncertainty than the corresponding
inertial forces.
. . .
4. Uncertainty of Wave Loading on Breakwater Armor Stones
This portion of the piece primarily depends on materials developed and presented in the Breakwater (BW) piece posted earlier, as wells as on my 2011 paper. The state-of-the-art techniques in
determining armor stone masses or sizes of rubble-mound breakwater and shore protection measures – rely either on Hudson Equation (RY Hudson 1958) or on VDM Formula (JW Van der Meer 1988). The
applicability and relative merits of the two methods are elaborated in the Breakwater piece.
For simplicity of analysis, I will focus on the uncertainty of Hudson Equation. This equation relates Stability Number to the product of a stability coefficient (KD) and a BW side slope factor. The
equation provides estimates of median armor stone mass as: a product of the stone density and wave height cubed – divided by the product of KD, side slope factor, and relative stone density cubed. It
is assumed that armor stone is forced by H = 1.0 m on the BW seaside slope = 1V:2H; with stone density = 2650 kg/m^3 and water density = 1025 kg/m^3 giving a relative stone density = 2.59. The
uncertainties of relative density and side slope factor are not known, they are estimated at 2.01% and 7.1% using basic UP principle.
The crux of the problem appears on defining the KD values. The recommended KDs vary from 1.6 for breaking to 4.0 for non-breaking wave forcing (USACE, 1984). Melby and Mlaker (1997) reported that the
KD values have uncertainty of some ±25%. In this piece the uncertainties median armor stone mass U_M50 for KD uncertainties ranging from ±10% to ±25% are investigated. Some estimated numbers are:
• U_M50 at U_H = 10%. For U_KD = 10%, U_M50 = 33.0%. For U_KD = 25%, U_M50 = 40.2%.
• U_M50 at U_H = 30%. For U_KD = 10%, U_M50 = 91.1%. For U_KD = 25%, U_M50 = 93.9%.
These estimates show the overwhelming influence of wave height; therefore utmost care is warranted to estimate it – such that local design wave conditions and scenarios are properly investigated and
accounted for.
. . .
The Koan of this piece on this International Jazz Day:
What seems to be perfect to an ordinary eye – is never finished, never perfect in the creator’s eye. The creative works continuously explore, experiment and search for something – that never comes to
the satisfaction of the creator.
. . . . .
- by Dr. Dilip K. Barua, 30 April 2021 | {"url":"https://widecanvas.weebly.com/science--technology/uncertainty-propagation-in-wave-loadings","timestamp":"2024-11-06T13:54:58Z","content_type":"text/html","content_length":"54366","record_id":"<urn:uuid:4981b778-7e28-4061-ab5d-d8acb3e7761e>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00779.warc.gz"} |
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Thanks m4 maths for helping to get placed in several companies. I must recommend this website for placement preparations. | {"url":"https://m4maths.com/40086-A-large-cube-is-dipped-into-a-tub-filled-with-colour-When-the-cube-is-taken-out-it-was-observed.html","timestamp":"2024-11-11T04:01:50Z","content_type":"text/html","content_length":"78065","record_id":"<urn:uuid:5c170502-569f-4366-8ca6-f7297869fc90>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00045.warc.gz"} |
Estimate parameters when identifying AR model or ARI model for scalar time series
sys = ar(y,n) estimates the parameters of an AR idpoly model sys of order n using a least-squares method. The model properties include covariances (parameter uncertainties) and estimation goodness of
fit. y can be an output-only iddata object, a numeric vector, or a timetable.
sys = ar(y,n,approach,window) uses the algorithm specified by approach and the prewindowing and postwindowing specification in window. To specify window while accepting the default value for
approach, use [] in the third position of the syntax.
sys = ar(y,n,___,Name,Value) specifies additional options using one or more name-value pair arguments. For instance, using the name-value pair argument 'IntegrateNoise',1 estimates an ARI model,
which is useful for systems with nonstationary disturbances. Specify Name,Value after any of the input argument combinations in the previous syntaxes.
sys = ar(y,n,___,opt) specifies estimation options using the option set opt.
[sys,refl] = ar(y,n,approach,___) returns an AR model along with the reflection coefficients refl when approach is the lattice-based method 'burg' or 'gl'.
AR Model
Estimate an AR model and compare its response with the measured output.
Load the data, which contains the time series tt9 with noise.
Estimate a fourth-order AR model.
sys =
Discrete-time AR model: A(z)y(t) = e(t)
A(z) = 1 - 0.8369 z^-1 - 0.4744 z^-2 - 0.06621 z^-3 + 0.4857 z^-4
Sample time: 0.0039062 seconds
Polynomial orders: na=4
Number of free coefficients: 4
Use "polydata", "getpvec", "getcov" for parameters and their uncertainties.
Estimated using AR ('fb/now') on time domain data "tt9".
Fit to estimation data: 79.38%
FPE: 0.5189, MSE: 0.5108
The output displays the polynomial containing the estimated parameters alongside other estimation details. Under Status, Fit to estimation data shows that the estimated model has 1-step-ahead
prediction accuracy above 75%.
You can find additional information about the estimation results by exploring the estimation report, sys.Report. For instance, you can retrieve the parameter covariance.
covar = sys.Report.Parameters.FreeParCovariance
covar = 4×4
0.0015 -0.0015 -0.0005 0.0007
-0.0015 0.0027 -0.0008 -0.0004
-0.0005 -0.0008 0.0028 -0.0015
0.0007 -0.0004 -0.0015 0.0014
For more information on viewing the estimation report, see Estimation Report.
Compare Burg's Method with Forward-Backward Approach
Given a sinusoidal signal with noise, compare the spectral estimates of Burg's method with those found using the forward-backward approach.
Generate an output signal and convert it into an iddata object.
y = sin([1:300]') + 0.5*randn(300,1);
y = iddata(y);
Estimate fourth-order AR models using Burg's method and using the default forward-backward approach. Plot the model spectra together.
sys_b = ar(y,4,'burg');
sys_fb = ar(y,4);
The two responses match closely throughout most of the frequency range.
ARI Model
Estimate an ARI model, which includes an integrator in the noise source.
Load the data, which contains the time series ymat9 with noise. Ts contains the sample time.
Integrate the output signal.
Estimate an AR model with 'IntegrateNoise' set to true. Use the least-squares method 'ls'.
sys = ar(y,4,'ls','Ts',Ts,'IntegrateNoise',true);
Predict the model output using 5-step prediction and compare the result with the integrated output signal y.
Modify Default Options
Modify the default options for the AR function.
Load the data, which contains a time series z9 with noise.
Modify the default options so that the function uses the 'ls' approach and does not estimate covariance.
opt = arOptions('Approach','ls','EstimateCovariance',false)
opt =
Option set for the ar command:
Approach: 'ls'
Window: 'now'
DataOffset: 0
EstimateCovariance: 0
MaxSize: 250000
Estimate a fourth-order AR model using the updated options.
Retrieve Reflection Coefficients for Burg's Method
Retrieve reflection coefficients and loss functions when using Burg's method.
Lattice-based approaches such, as Burg's method 'burg' and geometric lattice 'gl', compute reflection coefficients and corresponding loss function values as part of the estimation process. Use a
second output argument to retrieve these values.
Generate an output signal and convert it into an iddata object.
y = sin([1:300]') + 0.5*randn(300,1);
y = iddata(y);
Estimate a fourth-order AR model using Burg's method and include an output argument for the reflection coefficients.
[sys,refl] = ar(y,4,'burg');
refl = 2×5
0 -0.3562 0.4430 0.5528 0.2385
0.8494 0.7416 0.5960 0.4139 0.3904
Input Arguments
y — Time-series data
iddata object | numeric vector | timetable
Time-series data, specified as one of the following:
• An iddata object that contains a single output channel and an empty input channel.
• A numeric column vector containing output-channel data. When you specify y as a vector, you must also specify the sample time Ts.
• A single-variable timetable.
For more information about working with estimation data types, see Data Domains and Data Types in System Identification Toolbox.
n — Model order
positive integer
Model order, specified as a positive integer. The value of n determines the number of A parameters in the AR model.
Example: ar(idy,2) computes a second-order AR model from the single-channel iddata object idy
approach — Algorithm for computing AR model
'fb' (default) | 'burg' | 'gl' | 'ls' | 'yw'
Algorithm for computing the AR model, specified as one of the following values:
• 'burg': Burg's lattice-based method. Solves the lattice filter equations using the harmonic mean of forward and backward squared prediction errors.
• 'fb': (Default) Forward-backward approach. Minimizes the sum of a least-squares criterion for a forward model, and the analogous criterion for a time-reversed model.
• 'gl': Geometric lattice approach. Similar to Burg's method, but uses the geometric mean instead of the harmonic mean during minimization.
• 'ls': Least-squares approach. Minimizes the standard sum of squared forward-prediction errors.
• 'yw': Yule-Walker approach. Solves the Yule-Walker equations, formed from sample covariances.
All of these algorithms are variants of the least-squares method. For more information, see Algorithms.
Example: ar(idy,2,'ls') computes an AR model using the least-squares approach
window — Prewindowing and postwindowing
'now' | 'pow' | 'ppw' | 'prw
Prewindowing and postwindowing outside the measured time interval (past and future values), specified as one of the following values:
• 'now': No windowing. This value is the default except when you set approach to 'yw'. Only measured data is used to form regression vectors. The summation in the criteria starts at the sample
index equal to n+1.
• 'pow': Postwindowing. Missing end values are replaced with zeros and the summation is extended to time N+n (N is the number of observations).
• 'ppw': Prewindowing and postwindowing. The software uses this value whenever you select the Yule-Walker approach 'yw', regardless of your window specification.
• 'prw': Prewindowing. Missing past values are replaced with zeros so that the summation in the criteria can start at time equal to zero.
Example: ar(idy,2,'yw','ppw') computes an AR model using the Yule-Walker approach with prewindowing and postwindowing.
opt — Estimation options
arOptions option set
Estimation options for AR model identification, specified as an arOptions option set. opt specifies the following options:
• Estimation approach
• Data windowing technique
• Data offset
• Maximum number of elements in a segment of data
For more information, see arOptions. For an example, see Modify Default Options.
Name-Value Arguments
Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but
the order of the pairs does not matter.
Before R2021a, use commas to separate each name and value, and enclose Name in quotes.
Example: 'IntegrateNoise',true adds an integrator in the noise source
OutputName — Output signal names
" " (default) | character vector | string
Output channel names for timetable data, specified as a string or a character vector. By default, the software interprets the last variable in tt as the sole output channel. When you want to select a
different timetable variable for the output channel, use 'OutputName' to identify it. For example, sys = ar(tt,__,'OutputName',"y3") selects the variable y3 as the output channel for the estimation.
Ts — Sample time
1 (default) | positive scalar
Sample time, specified as the comma-separated pair consisting of 'Ts' and the sample time in seconds. If y is a numeric vector, then you must specify 'Ts'.
Example: ar(y_signal,2,'Ts',0.08) computes a second-order AR model with sample time of 0.08 seconds
IntegrateNoise — Add integrator to noise channel
false (default) | logical vector
Noise-channel integration option for estimating ARI models, specified as the comma-separated pair consisting of 'IntegrateNoise' and a logical. Noise integration is useful in cases where the
disturbance is nonstationary.
When using 'IntegrateNoise', you must also integrate the output-channel data. For an example, see ARI Model.
Output Arguments
sys — AR or ARI model
idpoly model object
AR or ARI model that fits the given estimation data, returned as a discrete-time idpoly model object. This model is created using the specified model orders, delays, and estimation options.
Information about the estimation results and options used is stored in the Report property of the model. Report has the following fields.
Report Field Description
Status Summary of the model status, which indicates whether the model was created by construction or obtained by estimation
Method Estimation command used
Quantitative assessment of the estimation, returned as a structure. See Loss Function and Model Quality Metrics for more information on these quality metrics. The structure has these
• FitPercent — Normalized root mean squared error (NRMSE) measure of how well the response of the model fits the estimation data, expressed as the percentage fitpercent = 100(1-NRMSE)
• LossFcn — Value of the loss function when the estimation completes
• MSE — Mean squared error (MSE) measure of how well the response of the model fits the estimation data
Fit • FPE — Final prediction error for the model
• AIC — Raw Akaike Information Criteria (AIC) measure of model quality
• AICc — Small-sample-size corrected AIC
• nAIC — Normalized AIC
• BIC — Bayesian Information Criteria (BIC)
Parameters Estimated values of model parameters
OptionsUsed Option set used for estimation. If no custom options were configured, this is a set of default options. See arOptions for more information.
RandState State of the random number stream at the start of estimation. Empty, [], if randomization was not used during estimation. For more information, see rng.
Attributes of the data used for estimation, returned as a structure with the following fields.
• Name — Name of the data set
• Type — Data type
• Length — Number of data samples
• Ts — Sample time
DataUsed • InterSample — Input intersample behavior, returned as one of the following values:
□ 'zoh' — A zero-order hold maintains a piecewise-constant input signal between samples.
□ 'foh' — A first-order hold maintains a piecewise-linear input signal between samples.
□ 'bl' — Band-limited behavior specifies that the continuous-time input signal has zero power above the Nyquist frequency.
• InputOffset — Offset removed from time-domain input data during estimation. For nonlinear models, it is [].
• OutputOffset — Offset removed from time-domain output data during estimation. For nonlinear models, it is [].
For more information on using Report, see Estimation Report.
refl — Reflection coefficients and loss functions
Reflection coefficients and loss functions, returned as a 2-by-2 array. For the two lattice-based approaches 'burg' and 'gl', refl stores the reflection coefficients in the first row and the
corresponding loss function values in the second row. The first column of refl is the zeroth-order model, and the (2,1) element of refl is the norm of the time series itself. For an example, see
Retrieve Reflection Coefficients for Burg's Method.
More About
AR (Autoregressive) Model
The AR model structure has no input, and is given by the following equation:
This model structure accommodates estimation for scalar time-series data, which have no input channel. The structure is a special case of the ARX structure.
ARI (Autoregressive Integrated) Model
The ARI model is an AR model with an integrator in the noise channel. The ARI model structure is given by the following equation:
AR and ARI model parameters are estimated using variants of the least-squares method. The following table summarizes the common names for methods with a specific combination of approach and window
argument values.
Method Approach and Windowing
Modified covariance method (Default) Forward-backward approach with no windowing
Correlation method Yule-Walker approach with prewindowing and postwindowing
Covariance method Least squares approach with no windowing. arx uses this routine
[1] Marple, S. L., Jr. Chapter 8. Digital Spectral Analysis with Applications. Englewood Cliffs, NJ: Prentice Hall, 1987.
Version History
Introduced in R2006a
R2022b: Time-domain estimation data is accepted in the form of timetables and matrices
Most estimation, validation, analysis, and utility functions now accept time-domain input/output data in the form of a single timetable that contains both input and output data or a pair of matrices
that contain the input and output data separately. These functions continue to accept iddata objects as a data source as well, for both time-domain and frequency-domain data. | {"url":"https://se.mathworks.com/help/ident/ref/ar.html","timestamp":"2024-11-05T19:04:10Z","content_type":"text/html","content_length":"136698","record_id":"<urn:uuid:17b3caa8-9e71-4300-9b0f-0f4b8c93eee0>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00053.warc.gz"} |
Adding Fractions - Steps, Examples: How to Add Fractions - Grade Potential Centennial, CO
How to Add Fractions: Examples and Steps
Adding fractions is a common math problem that students learn in school. It can look scary initially, but it becomes easy with a bit of practice.
This blog post will walk you through the procedure of adding two or more fractions and adding mixed fractions. We will then give examples to see how it is done. Adding fractions is essential for a
lot of subjects as you progress in science and math, so make sure to adopt these skills initially!
The Process of Adding Fractions
Adding fractions is a skill that a lot of children have a problem with. However, it is a relatively easy process once you understand the essential principles. There are three major steps to adding
fractions: determining a common denominator, adding the numerators, and streamlining the answer. Let’s take a closer look at every one of these steps, and then we’ll look into some examples.
Step 1: Determining a Common Denominator
With these useful tips, you’ll be adding fractions like a pro in no time! The initial step is to determine a common denominator for the two fractions you are adding. The least common denominator is
the lowest number that both fractions will divide equally.
If the fractions you wish to sum share the same denominator, you can avoid this step. If not, to look for the common denominator, you can determine the amount of the factors of each number as far as
you look for a common one.
For example, let’s say we want to add the fractions 1/3 and 1/6. The lowest common denominator for these two fractions is six for the reason that both denominators will divide evenly into that
Here’s a good tip: if you are not sure regarding this process, you can multiply both denominators, and you will [[also|subsequently80] get a common denominator, which would be 18.
Step Two: Adding the Numerators
Once you have the common denominator, the following step is to turn each fraction so that it has that denominator.
To convert these into an equivalent fraction with the same denominator, you will multiply both the denominator and numerator by the same number necessary to attain the common denominator.
Following the prior example, 6 will become the common denominator. To change the numerators, we will multiply 1/3 by 2 to get 2/6, while 1/6 would stay the same.
Now that both the fractions share common denominators, we can add the numerators simultaneously to get 3/6, a proper fraction that we will proceed to simplify.
Step Three: Simplifying the Results
The final step is to simplify the fraction. Consequently, it means we need to diminish the fraction to its lowest terms. To achieve this, we look for the most common factor of the numerator and
denominator and divide them by it. In our example, the greatest common factor of 3 and 6 is 3. When we divide both numbers by 3, we get the ultimate result of 1/2.
You follow the exact process to add and subtract fractions.
Examples of How to Add Fractions
Now, let’s move forward to add these two fractions:
2/4 + 6/4
By utilizing the procedures mentioned above, you will observe that they share equivalent denominators. Lucky for you, this means you can avoid the first step. At the moment, all you have to do is add
the numerators and allow it to be the same denominator as before.
2/4 + 6/4 = 8/4
Now, let’s try to simplify the fraction. We can notice that this is an improper fraction, as the numerator is higher than the denominator. This could suggest that you can simplify the fraction, but
this is not feasible when we work with proper and improper fractions.
In this instance, the numerator and denominator can be divided by 4, its most common denominator. You will get a final answer of 2 by dividing the numerator and denominator by 2.
As long as you go by these steps when dividing two or more fractions, you’ll be a professional at adding fractions in no time.
Adding Fractions with Unlike Denominators
This process will require an additional step when you add or subtract fractions with different denominators. To do this function with two or more fractions, they must have the same denominator.
The Steps to Adding Fractions with Unlike Denominators
As we stated above, to add unlike fractions, you must follow all three procedures stated prior to transform these unlike denominators into equivalent fractions
Examples of How to Add Fractions with Unlike Denominators
Here, we will put more emphasis on another example by summing up the following fractions:
As shown, the denominators are dissimilar, and the least common multiple is 12. Hence, we multiply each fraction by a value to achieve the denominator of 12.
1/6 * 2 = 2/12
2/3 * 4 = 8/12
6/4 * 3 = 18/12
Now that all the fractions have a common denominator, we will move ahead to add the numerators:
2/12 + 8/12 + 18/12 = 28/12
We simplify the fraction by dividing the numerator and denominator by 4, coming to the ultimate result of 7/3.
Adding Mixed Numbers
We have discussed like and unlike fractions, but presently we will go through mixed fractions. These are fractions followed by whole numbers.
The Steps to Adding Mixed Numbers
To work out addition sums with mixed numbers, you must start by converting the mixed number into a fraction. Here are the steps and keep reading for an example.
Step 1
Multiply the whole number by the numerator
Step 2
Add that number to the numerator.
Step 3
Write down your answer as a numerator and retain the denominator.
Now, you proceed by adding these unlike fractions as you usually would.
Examples of How to Add Mixed Numbers
As an example, we will work out 1 3/4 + 5/4.
Foremost, let’s transform the mixed number into a fraction. You are required to multiply the whole number by the denominator, which is 4. 1 = 4/4
Thereafter, add the whole number described as a fraction to the other fraction in the mixed number.
4/4 + 3/4 = 7/4
You will conclude with this operation:
7/4 + 5/4
By summing the numerators with the exact denominator, we will have a ultimate result of 12/4. We simplify the fraction by dividing both the numerator and denominator by 4, resulting in 3 as a final
Use Grade Potential to Improve Your Math Skills Today
If you're struggling to understand adding fractions, consider signing up for a tutoring session with Grade Potential. One of our expert teachers can assist you understand the material and nailcrack
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How Over-Parameterization Slows Down Gradient Descent in Matrix...
How Over-Parameterization Slows Down Gradient Descent in Matrix Sensing: The Curses of Symmetry and Initialization
Code Of Ethics: I acknowledge that I and all co-authors of this work have read and commit to adhering to the ICLR Code of Ethics.
Keywords: non-convex optimization, random initialization, global convergence, matrix recovery, matrix sensing
Submission Guidelines: I certify that this submission complies with the submission instructions as described on https://iclr.cc/Conferences/2024/AuthorGuide.
TL;DR: This paper study the different convergence behaviors of symmetric and asymmetric matrix sensing in exact and over-parameterized settings, and show the first exact convergence result of
asymmetric matrix sensing.
Abstract: This paper rigorously shows how over-parameterization dramatically changes the convergence behaviors of gradient descent (GD) for the matrix sensing problem, where the goal is to recover an
unknown low-rank ground-truth matrix from near-isotropic linear measurements. First, we consider the symmetric setting with the symmetric parameterization where $M^* \in \mathbb{R}^{n \times n}$ is a
positive semi-definite unknown matrix of rank $r \ll n$, and one uses a symmetric parameterization $XX^\top$ to learn $M^*$. Here $X \in \mathbb{R}^{n \times k}$ with $k > r$ is the factor matrix. We
give a novel $\Omega\left(1/T^2\right)$ lower bound of randomly initialized GD for the over-parameterized case ($k >r$) where $T$ is the number of iterations. This is in stark contrast to the
exact-parameterization scenario ($k=r$) where the convergence rate is $\exp\left(-\Omega\left(T\right)\right)$. Next, we study asymmetric setting where $M^* \in \mathbb{R}^{n_1 \times n_2}$ is the
unknown matrix of rank $r \ll \min\{n_1,n_2\}$, and one uses an asymmetric parameterization $FG^\top$ to learn $M^*$ where $F \in \mathbb{R}^{n_1 \times k}$ and $G \in \mathbb{R}^{n_2 \times k}$. We
give the first global exact convergence result of randomly initialized GD for the exact-parameterization case ($k=r$) with an $\exp\left(-\Omega\left(T\right)\right)$ rate. Furthermore, we give the
first global exact convergence result for the over-parameterization case ($k>r$) with an $\exp\left(-\Omega\left(\alpha^2 T\right)\right)$ rate where $\alpha$ is the initialization scale. This linear
convergence result in the over-parameterization case is especially significant because one can apply the asymmetric parameterization to the symmetric setting to speed up from $\Omega\left(1/T^2\
right)$ to linear convergence. Therefore, we identify a surprising phenomenon: asymmetric parameterization can exponentially speed up convergence. Equally surprising is our analysis that highlights
the importance of imbalance between $F$ and $G$. This is in sharp contrast to prior works which emphasize balance. We further give an example showing the dependency on $\alpha$ in the convergence
rate is unavoidable in the worst case. On the other hand, we propose a novel method that only modifies one step of GD and obtains a convergence rate independent of $\alpha$, recovering the rate in
the exact-parameterization case. We provide empirical studies to verify our theoretical findings.
Anonymous Url: I certify that there is no URL (e.g., github page) that could be used to find authors' identity.
Supplementary Material: pdf
No Acknowledgement Section: I certify that there is no acknowledgement section in this submission for double blind review.
Primary Area: learning theory
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How To Fix Invalid Floating Point State Error in Windows 10
What is invalid floating error operation?
If you are seeing an error message that says “Invalid floating point state“, it means that there is something wrong with the Region Settings in Windows. This is a small glitch that throws this error
and prevents you to work on your system.
How do you solve a floating point error?
The IEEE standard for floating point specifies that the result of any floating point operation should be correct to within the rounding error of the resulting number. That is, it specifies that the
maximum rounding error for an individual operation (add, multiply, subtract, divide) should be 0.5 ULP.
How do you use float in Matlab?
Creating Floating-Point Data
1. x = 25.783; The whos function shows that MATLAB has created a 1-by-1 array of type double for the value you just stored in x :
2. whos x Name Size Bytes Class x 1x1 8 double. Use isfloat if you just want to verify that x is a floating-point number. ...
3. isfloat(x) ans = logical 1.
What fixed decimal?
Fixed Decimal Numbers: Have a constant number of digits after the decimal place. These are typically used to represent money, percentages, or a certain precision of the number of seconds (i.e.
limiting to milliseconds). They are mostly used in databases as a simple and efficient storage format.
Why do floating point errors occur?
Floating point numbers are limited in size, so they can theoretically only represent certain numbers. Everything that is inbetween has to be rounded to the closest possible number. This can cause
(often very small) errors in a number that is stored.
What is the main problem with floating point numbers?
The problem is that many numbers can't be represented by a sum of a finite number of those inverse powers. Using more place values (more bits) will increase the precision of the representation of
those 'problem' numbers, but never get it exactly because it only has a limited number of bits.
What causes floating point error?
It's a problem caused when the internal representation of floating-point numbers, which uses a fixed number of binary digits to represent a decimal number. It is difficult to represent some decimal
number in binary, so in many cases, it leads to small roundoff errors.
What does double mean in Matlab?
double is the default numeric data type (class) in MATLAB^®, providing sufficient precision for most computational tasks. Numeric variables are automatically stored as 64-bit (8-byte)
double-precision floating-point values.
What is double command in Matlab?
Description. double( s ) converts the symbolic value s to double precision. Converting symbolic values to double precision is useful when a MATLAB^® function does not accept symbolic values. For
differences between symbolic and double-precision numbers, see Choose Numeric or Symbolic Arithmetic.
How do I print a floating-point number in Matlab?
The input data are double-precision floating-point values rather than unsigned integers. For example, to print a double-precision value in hexadecimal, use a format like %bx . The input data are
single-precision floating-point values rather than unsigned integers.
What is fixed point binary?
Fixed point binary allows us to represent binary numbers that include a decimal point, known as real numbers. Fixed point binary numbers allow us to increase the precision of the numbers that we
What is fixed point format?
In fixed point notation, there are a fixed number of digits after the decimal point, whereas floating point number allows for a varying number of digits after the decimal point. Fixed-Point
Representation − This representation has fixed number of bits for integer part and for fractional part.
What is the difference between decimal and decimal point?
In algebra, a decimal number can be defined as a number whose whole number part and the fractional part is separated by a decimal point. The dot in a decimal number is called a decimal point. The
digits following the decimal point show a value smaller than one. | {"url":"https://en.naneedigital.com/article/how_to_fix_invalid_floating_point_state_error_in_windows_10","timestamp":"2024-11-15T04:04:40Z","content_type":"text/html","content_length":"30723","record_id":"<urn:uuid:b779a92a-cc31-47e2-9dc6-9669f34b5873>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00492.warc.gz"} |
Logic and Foundations
Research: Art, Information, And Academic Inquiry, 2024
Research: Art, Information, And Academic Inquiry, Luke D. Mckinney
Electronic Theses and Dissertations
In light of the rapidly changing landscape of knowledge production and dissemination, this paper proposes a reformation of academic research that integrates artistic methodologies, emphasizes
interdisciplinary collaboration, and prioritizes clear communication to both specialized and general audiences. By reconceptualizing research as a multidimensional, embodied practice that encompasses
both rational and irrational elements, we can create a more inclusive, adaptable, and effective approach to scholarship that bridges the rational divide between artistic and scientific inquiry that
allows for the engagement of Artistic Research from within the institution, ultimately leading to more innovative and impactful contributions to human knowledge.
Discordium Mathematica - A Symphony In Aleph Minor, 2024
Discordium Mathematica - A Symphony In Aleph Minor, Vijay Fafat
Journal of Humanistic Mathematics
How did Mathematics arise? Who created it? Why is it subject to Godel’s Incompleteness Theorems? And what does all this have to do with Coleridge’s poem, “Kubla Khan”, and “The Person from Porlock”?
Here is a complete mythology of Mathematics set in an epic poetry format, fusing thoughts and verses from Western religions and Eastern mysticism… Those with immense patience and careful reading
shall reap the fruit… (best read on a large screen or in printed form)
A Thesis, Or Digressions On Sculptural Practice: In Which, Concepts & Influences Thereof Are Explained, Set Forth, Catalogued, Or Divulged By Way Of Commentaries To A Poem, First Conceived By The
Artist, Fed Through Chatg.P.T., And Re-Edited By The Artist, To Which Are Added, Annotated References, Impressions And Ruminations Thereof, Also Including Private Thoughts & Personal Accounts Of The
Artist, 2024
A Thesis, Or Digressions On Sculptural Practice: In Which, Concepts & Influences Thereof Are Explained, Set Forth, Catalogued, Or Divulged By Way Of Commentaries To A Poem, First Conceived By The
Artist, Fed Through Chatg.P.T., And Re-Edited By The Artist, To Which Are Added, Annotated References, Impressions And Ruminations Thereof, Also Including Private Thoughts & Personal Accounts Of The
Artist, Jaimie An
Masters Theses
This thesis is an exercise in, perhaps a futile, attempt to trace just some of the ideas, stories, and musings I might meander through in my process. It’s not quite a map, nor is it a neat catalogue;
it is a haphazard collection of tickets and receipts from a travel abroad, carelessly tossed in a carry-on, only to be stashed upon returning home. These ideas are derived from much greater thinkers
and authors than myself; I am a mere collector or a translator, if that, and not a very good one, for much is lost. I do not claim comprehensive …
Canonical Extensions Of Quantale Enriched Categories, 2024
Canonical Extensions Of Quantale Enriched Categories, Alexander Kurz
MPP Research Seminar
No abstract provided.
Formalization Of A Security Framework Design For A Health Prescription Assistant In An Internet Of Things System, 2024
Formalization Of A Security Framework Design For A Health Prescription Assistant In An Internet Of Things System, Thomas Rolando Mellema
Electronic Theses and Dissertations
Security system design flaws will create greater risks and repercussions as the systems being secured further integrate into our daily life. One such application example is incorporating the powerful
potential of the concept of the Internet of Things (IoT) into software services engineered for improving the practices of monitoring and prescribing effective healthcare to patients. A study was
performed in this application area in order to specify a security system design for a Health Prescription Assistant (HPA) that operated with medical IoT (mIoT) devices in a healthcare environment.
Although the efficiency of this system was measured, little was presented to …
Multiscale Modelling Of Brain Networks And The Analysis Of Dynamic Processes In Neurodegenerative Disorders, 2024
Multiscale Modelling Of Brain Networks And The Analysis Of Dynamic Processes In Neurodegenerative Disorders, Hina Shaheen
Theses and Dissertations (Comprehensive)
The complex nature of the human brain, with its intricate organic structure and multiscale spatio-temporal characteristics ranging from synapses to the entire brain, presents a major obstacle in
brain modelling. Capturing this complexity poses a significant challenge for researchers. The complex interplay of coupled multiphysics and biochemical activities within this intricate system shapes
the brain's capacity, functioning within a structure-function relationship that necessitates a specific mathematical framework. Advanced mathematical modelling approaches that incorporate the
coupling of brain networks and the analysis of dynamic processes are essential for advancing therapeutic strategies aimed at treating neurodegenerative diseases (NDDs), which afflict millions of …
Reducing Food Scarcity: The Benefits Of Urban Farming, 2023
Reducing Food Scarcity: The Benefits Of Urban Farming, S.A. Claudell, Emilio Mejia
Journal of Nonprofit Innovation
Urban farming can enhance the lives of communities and help reduce food scarcity. This paper presents a conceptual prototype of an efficient urban farming community that can be scaled for a single
apartment building or an entire community across all global geoeconomics regions, including densely populated cities and rural, developing towns and communities. When deployed in coordination with
smart crop choices, local farm support, and efficient transportation then the result isn’t just sustainability, but also increasing fresh produce accessibility, optimizing nutritional value,
eliminating the use of ‘forever chemicals’, reducing transportation costs, and fostering global environmental benefits.
Imagine Doris, who is …
Many-Valued Coalgebraic Logic: From Boolean Algebras To Primal Varieties, 2023
Many-Valued Coalgebraic Logic: From Boolean Algebras To Primal Varieties, Alexander Kurz, Wolfgang Poiger
Engineering Faculty Articles and Research
We study many-valued coalgebraic logics with primal algebras of truth-degrees. We describe a way to lift algebraic semantics of classical coalgebraic logics, given by an endofunctor on the variety of
Boolean algebras, to this many-valued setting, and we show that many important properties of the original logic are inherited by its lifting. Then, we deal with the problem of obtaining a concrete
axiomatic presentation of the variety of algebras for this lifted logic, given that we know one for the original one. We solve this problem for a class of presentations which behaves well with
respect to a lattice structure …
Soundness And Completeness Results For The Logic Of Evidence Aggregation And Its Probability Semantics, 2023
Soundness And Completeness Results For The Logic Of Evidence Aggregation And Its Probability Semantics, Eoin Moore
Dissertations, Theses, and Capstone Projects
The Logic of Evidence Aggregation (LEA), introduced in 2020, offers a solution to the problem of evidence aggregation, but LEA is not complete with respect to the intended probability semantics. This
left open the tasks to find sound and complete semantics for LEA and a proper axiomatization for probability semantics. In this thesis we do both. We also develop the proof theory for some
LEA-related logics and show surprising connections between LEA-related logics and Lax Logic.
One Formula For Non-Prime Numbers: Motivations And Characteristics, 2023
One Formula For Non-Prime Numbers: Motivations And Characteristics, Mahmoud Mansour, Kamal Hassan Prof.
Basic Science Engineering
Primes are essential for computer encryption and cryptography, as they are fundamental units of whole numbers and are of the highest importance due to their mathematical qualities. However,
identifying a pattern of primes is not easy. Thinking in a different way may get benefits, by considering the opposite side of the problem which means focusing on non-prime numbers. Recently,
researchers introduced, the pattern of non-primes in two maximal sets while in this paper, non-primes are presented in one formula. Getting one-way formula for non-primes may pave the way for further
applications based on the idea of primes.
(R1986) Neutrosophic Soft Contra E-Continuous Maps, Contra E-Irresolute Maps And Application Using Distance Measure, 2023
(R1986) Neutrosophic Soft Contra E-Continuous Maps, Contra E-Irresolute Maps And Application Using Distance Measure, P. Revathi, K. Chitirakala, A. Vadivel
Applications and Applied Mathematics: An International Journal (AAM)
We introduce and investigate neutrosophic soft contra e-continuous maps and contra e-irresolute maps in neutrosophic soft topological spaces with examples. Also, neutrosophic soft contra econtinuous
maps are compared with neutrosophic soft continuous maps, δ-continuous maps, δ- semi continuous maps, δ-pre continuous maps and e∗ continuous maps in neutrosophic soft topological spaces. We derive
some useful results and properties related to them. An application in decision making problem using distance measure is given. An example of a candidate selection from a company interview is
formulated as neutrosophic soft model problem and the hamming distance measure is applied to calculate the distance …
(R1957) Some Types Of Continuous Function Via N-Neutrosophic Crisp Topological Spaces, 2023
(R1957) Some Types Of Continuous Function Via N-Neutrosophic Crisp Topological Spaces, A. Vadivel, C. John Sundar
Applications and Applied Mathematics: An International Journal (AAM)
The aim of this article is to introduced a new type of continuous functions such as N-neutrosophic crisp gamma continuous and weakly N-neutrosophic crisp gamma continuous functions in a
N-neutrosophic crisp topological space and also discuss a relation between them in a N-neutrosophic crisp topological spaces. We also investigate some of their properties in N-neutrosophic crisp
gamma continuous function via N-neutrosophic crisp topological spaces. Further, a contra part of continuity called N-neutrosophic crisp gamma-contra continuous map in a N-neutrosophic crisp topology
is also initiated. Finally, an application based on neutrosophic score function of medical diagnosis is examined with graphical representation.
(R1997) Distance Measures Of Complex Fermatean Fuzzy Number And Their Application To Multi-Criteria Decision-Making Problem, 2023
(R1997) Distance Measures Of Complex Fermatean Fuzzy Number And Their Application To Multi-Criteria Decision-Making Problem, V. Chinnadurai, S. Thayalan, A. Bobin
Applications and Applied Mathematics: An International Journal (AAM)
Multi-criteria decision-making (MCDM) is the most widely used decision-making method to solve many complex problems. However, classical MCDM approaches tend to make decisions when the parameters are
imprecise or uncertain. The concept of a complex fuzzy set is new in the field of fuzzy set theory. It is a set that can collect and interpret the membership grades from the unit circle in a plane
instead of the interval [0,1]. CFS cannot deal with membership and non-membership grades, while complex intuitionistic fuzzy set and complex Pythagorean fuzzy set works only for a limited range of
values. The concept of a …
(R1965) Some More Properties On Generalized Double Fuzzy Z Alpha Open Sets, 2023
(R1965) Some More Properties On Generalized Double Fuzzy Z Alpha Open Sets, K. Jayapandian, A. Saivarajan, O. Uma Maheswari, J. Sathiyaraj
Applications and Applied Mathematics: An International Journal (AAM)
In this paper, a new class of sets termed as double fuzzy generalized Z alpha closed sets and double fuzzy generalized Z alpha open sets are introduced with the help of double fuzzy Z alpha open and
double fuzzy Z alpha closed sets, respectively. Using these sets double fuzzy generalized Z alpha border, double fuzzy generalized Z alpha exterior and double fuzzy generalized Z alpha frontier of a
fuzzy set in double fuzzy topological spaces are introduced. Also, the topological properties and characterizations of these sets and operators are studied. Furthermore, suitable examples have been
provided to illustrate the theory.
Deep Learning Recommendations For The Acl2 Interactive Theorem Prover, 2023
Deep Learning Recommendations For The Acl2 Interactive Theorem Prover, Robert K. Thompson, Robert K. Thompson
Master's Theses
Due to the difficulty of obtaining formal proofs, there is increasing interest in partially or completely automating proof search in interactive theorem provers. Despite being a theorem prover with
an active community and plentiful corpus of 170,000+ theorems, no deep learning system currently exists to help automate theorem proving in ACL2. We have developed a machine learning system that
generates recommendations to automatically complete proofs. We show that our system benefits from the copy mechanism introduced in the context of program repair. We make our system directly
accessible from within ACL2 and use this interface to evaluate our system in …
On Specifications Of Positive Data Models With Effectively Separable Kernels Of Algorithmic Representations, 2023
On Specifications Of Positive Data Models With Effectively Separable Kernels Of Algorithmic Representations, Nodira R. Karimova
Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences
It is established that any effectively separable multi-sorted positively representable model with an effectively separable representation kernel has an enrichment that is the only (up to isomorphism)
model constructed from constants for a suitable computably enumerable set of sentences.
Reverse Mathematics Of Ramsey's Theorem, 2023
Reverse Mathematics Of Ramsey's Theorem, Nikolay Maslov
Electronic Theses, Projects, and Dissertations
Reverse mathematics aims to determine which set theoretic axioms are necessary to prove the theorems outside of the set theory. Since the 1970’s, there has been an interest in applying reverse
mathematics to study combinatorial principles like Ramsey’s theorem to analyze its strength and relation to other theorems. Ramsey’s theorem for pairs states that for any infinite complete graph with
a finite coloring on edges, there is an infinite subset of nodes all of whose edges share one color. In this thesis, we introduce the fundamental terminology and techniques for reverse mathematics,
and demonstrate their use in proving Kőnig's lemma …
Generations Of Reason: A Family’S Search For Meaning In Post-Newtonian England (Book Review), 2023
Generations Of Reason: A Family’S Search For Meaning In Post-Newtonian England (Book Review), Calvin Jongsma
Faculty Work Comprehensive List
Reviewed Title: Generations of Reason: A Family's Search for Meaning in Post-Newtonian England by Joan L. Richards. New Haven, CT: Yale University Press, 2021. 456 pp. ISBN: 9780300255492.
Richard Whately's Revitalization Of Syllogistic Logic, 2023
Richard Whately's Revitalization Of Syllogistic Logic, Calvin Jongsma
Faculty Work Comprehensive List
This is an expanded version of the first chapter Richard Whately’s Revitalization of Syllogistic Logic in Aristotle’s Syllogism and the Creation of Modern Logic edited by Lukas M. Verburgt and Matteo
Cosci (Bloomsbury, 2023). Drawing upon the author’s 1982 Ph. D. dissertation (https://digitalcollections.dordt.edu/faculty_work/230/ ) and more current scholarship, this essay traces the critical
historical background to Whately’s work in more detail than could be done in the published version.
Self-Reference And Diagonalisation, 2023
Self-Reference And Diagonalisation, Joël A. Doat
Journal of Humanistic Mathematics
This poem is an exercise on self-reference and diagonalisation in mathematics featuring Turing’s proof of the undecidability of the halting problem, Cantor’s cardinality argument, the Burali-Forti
paradox, and Epimenides' liar paradox. | {"url":"https://network.bepress.com/physical-sciences-and-mathematics/mathematics/logic-and-foundations/","timestamp":"2024-11-03T10:32:33Z","content_type":"text/html","content_length":"103726","record_id":"<urn:uuid:3b0090e3-2723-4f26-a269-f44f9553d317>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00716.warc.gz"} |
Question Answers | Class 11 | SaralStudy
Question Answers: NCERT Class 11 Mathematics
Welcome to the Class 11 Mathematics NCERT Solutions page. Here, we provide detailed question answers for Chapter - , designed to help students gain a thorough understanding of the concepts related to
natural resources, their classification, and sustainable development.
Our solutions explain each answer in a simple and comprehensive way, making it easier for students to grasp key topics and excel in their exams. By going through these question answers, you can
strengthen your foundation and improve your performance in Class 11 Mathematics. Whether you're revising or preparing for tests, this chapter-wise guide will serve as an invaluable resource.
• All Chapters Of Class 11 Mathematics | {"url":"https://www.saralstudy.com/study-eschool-ncertsolution/11th/mathematics/cbse-sample-papers-class-10","timestamp":"2024-11-06T23:53:07Z","content_type":"application/xhtml+xml","content_length":"51684","record_id":"<urn:uuid:ecfbce92-a594-4f7b-b6a9-d2290546a443>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00078.warc.gz"} |
Gaylord Box Dimensions Gaylord Boxes: Great Prices & In-Stock
Gaylord Box Dimensions
The Difference Between Rectangular and Octagonal Gaylord Boxes
The primary difference between rectangular and octagonal gaylord boxes is stacking ability. The four additional pressure points on octagonal gaylord boxes provide a more optimal weight distribution.
Consequently, octagonal gaylord boxes have significantly better stacking ability than their rectangular counterparts. In other words, each wall on a rectangular gaylord is forced to bear twice as
much weight as each wall on an octangular gaylord box. While this may not affect the weight capacity of the boxes, it will certainly affect the boxes’ shelf life. So, the shelf life for octagonal
gaylord boxes is typically much longer than rectangular gaylords because they will not be worn down as quickly.
The Base of Rectangular Gaylord Boxes
The bases of gaylord boxes can typically come in 5 sizes. The majority of boxes have bases that are either 36”x36”, 40” x 40”, 40” x 48”, 48” x 40” or 48” x 48” because they are designed to fit on
top of wooden pallets, which also typically share those same dimensions. The reasoning behind this is because the wooden pallets help fortify the carrying capacity, and make them much easier to
transport via forklift. It is often difficult to move a gaylord box at full carrying capacity without a pallet underneath it without damaging the bottom of the box. This is because the forks on a
forklift may puncture the bottom a box when attempting to hoist or move the boxes, which may potentially compromise the box. There are some boxes that have smaller bases, typically 36” x 36” which
can also be attached or moved on pallets. Some customers prefer these marginally smaller boxes because they typically have the same carrying capacity, but take up less space in the bed of a truck,
which means that more of them can be purchased in a single load.
The Height of Rectangular Gaylord Boxes
The height of rectangular gaylord boxes, can vary tremendously. The height of a cardboard gaylord box is almost always determined by the box’s first owner. On top of that, some people may actually
reduce the height of their boxes by cutting off a few inches on the top. The height of gaylords ranges between 24” and 48”. The most recurrent height is 36” because that is the most common height of
a new box. However, finding boxes with heights ranging between 40”-48” is fairly simple. Because the height of a box has a directly correlative relationship with its carrying capacity, taller boxes
are more expensive in both the retail and resale markets for gaylord boxes. It is important to note that height does not affect the box’s weight capacity, that is strictly determined by the number of
walls and the bottom of any particular box.
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The Base of Octagonal Gaylord Boxes
The base of octagonal boxes are typically smaller than their rectangular counterparts, which actually has a few significant advantages. The base of most octagonal gaylords is 46” x 38”, and is placed
on the same wooden pallets that are either 48” x 48” or 48” x 40”. One of the greatest advantages of the smaller base on the wooden pallets is that it evenly distributes the weight of the box’s
inventory in the center of the pallet, which optimizes their stacking ability. The second and most significant advantage of the centralized weight distribution is that it drastically increases the
box’s weight capacity. The octagonal shape itself also maximizes the box’s carrying capacity because it adds 4 additional pressure points in comparison to the rectangular gaylord boxes. While most
rectangular boxes can carry 1,000-1,500 lbs, octagonal gaylord boxes can generally carry well over 2,000 lbs in a single haul. Because of this, octagonal gaylord boxes are typically more expensive in
both retail and resale markets for gaylord boxes. While the majority of octagonal gaylords have 46” x 38” bases, there are bigger 48” x 48” options and smaller 42” x 36” options available as well.
The Height of Octagonal Gaylord Boxes
Much like the rectangular gaylord boxes, the height of octagonal gaylord boxes is variable. The height is also typically determined by a box’s first user. The height of an octagonal gaylord ranges
between 24” and 42” inches, which is marginally shorter than the average rectangular gaylord. While this may affect the overall carrying capacity, it bears no relevance to the box’s weight capacity,
which is often the most important factor for the majority of customers.
HPT Gaylord Boxes
High Performance Tote (HPT) Gaylord Boxes are the most valuable and sought after type of box. The name practically speaks for itself, all of these boxes have 5 walls, a rectangular shape, with full
flap bottoms, and almost always have full flap tops. There are 3 different types of HPT gaylord boxes – HPT-39’s, HPT-41’s, and HPT-50’s. The number of associated with the particular HPT refers to
the height of that box. HPT-41 boxes are the most common form high performance tote, but HPT-39’s are also fairly common. On the other hand, HPT-50’s are very rare and expensive. All of these boxes
have 3 parts – the rectangular outer case, a rectangular interior liner to fortify the box’s strength, and an additional octagonal insert to centralize the inventory and fortify the box’s stacking
ability. The construction of these boxes gives them the greatest carrying capacity of any particular gaylord box, which is why they are the most expensive and sought after form of gaylord box. Refer
to this website http://www.toteco.com/DShtHPT41.html for further details on the construction of an HPT box, and the HPT-41 box in particular. | {"url":"https://thegaylordboxexchange.com/gaylord-box-dimensions/","timestamp":"2024-11-12T13:53:00Z","content_type":"text/html","content_length":"138454","record_id":"<urn:uuid:0145d839-96e7-4eca-8176-c93e11bffa22>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00774.warc.gz"} |
EC8501 Digital Communication Notes MCQ - Chrome Tech
EC8501 Digital Communication Notes MCQ
EC8501 Digital Communication Notes mcq questions and answers
EC8501 Digital Communication Notes
1. Information rate is represented in average number of bits of information per second.
2. The two code was of the linear code for added by model to arithmetic then it produce 3rd code word in the code.
3. The syndrome depends only on the error pattern and not on the transmitted code word.
4. The number of ones complement are once codeword of the hamming code is called hamming weight.
5. The bandpass digital data transmission system consists of source encoder and modulator in the transmitter.
6. The career synchronization is required in core and detection method to generate component reference at the receiver.
7. The local carrier generated at the receiver is phase locked with the carrier at the transmitter.
8. Eye pattern is used to study the effect of ISI in baseband transmission.
9. The optimum linear receiver is Realised with the help of zero forcing equalizer.
10. Juman ear does not receive the noise in the given frequency band with if it is 15db below the signal level in the band.
EC8501 Digital Communication Notes
11. Adaptive quantizer changes its step size according to variance of the input signal.
12. Delta modulation includes one bit per sample hence Signalling rate is reduced in DM.
13. The sample of Contessa output and prediction error are used to derive estimates of predictor question is called order prediction with backward estimation.
14. Variable length coding is done by source encoder to get higher efficiency.
15. Shannon fano algorithm is used to encode the messages depending upon the probabilities.
16. Slope overload distortion occur mainly due to fixed step size S that cannot follow the rate of rise of input signal.
17. The minimum distance of the linear block code is equal to minimum weight of any non zero in the code.
18. Ninimum variance Huffman code can be obtained by placing combined probabilities as high as possible.
19. The mutual information is defined as the amount of information transferred when Xi it is transmitted and Yi is received.
20. Entropy will be maximum when all the messages have sample probabilities of occurrence.
21. The first as well as amplitude of the quadrature carrier is small letter is called quadrature amplitude phase shift keying QAPSK.
22. The demodulator output is quadrature into more than two level as decoder operates on soft decision made by demodulator.
OVERVIEW SYLLABUS OF EC8501 Digital Communication Notes
Unit 1
Unit 2 Name is information theory discrete memoryless source important basic problems. what is entropy and its channel capacity the basic concepts related to information contained in this unit. what
is memoryless channel, what is mutual information in digital communication all the basic concepts contains in unit 1 this is one of the most important easiest unit in digital communication.
Unit 2
Unit 2 Name is waveform coding and representation. this is all about protecting filter and DPCM data bipolar and unipolar power spectral density this all basic concepts contains in this you need to
know, what is waveform coding and representation. what is delta modulation and adaptive delta modulation basic principles contains in this unit 2.
Unit 3
Baseband transmission, what is baseband pulse shaping, coding eye pattern is one of the most important 16 mark in university exam.
Unit 4
Geometrical representation of signals this unit contains basic signal examples its like what is signal, what is error signal, what is continuous signal, what is discrete signal, what is QAM QPSK BFSK
DPSK this basic frequency contains in this unit.
Unit 5
Error control coding channel, coding theorem, linear block codes hamming codes cyclic codes conversion cost with the coding decoding is one of the most important 16 mark in in university point of
view. this is very basic questions and answers as very important question in University exam point of view if you study that question definitely you will get 10 marks in your digital communication
paper this question they will ask in both part b and part C if they ask as compulsory question if you studying this question definitely ask in exam for digital communication subject.
UNIT 1
Ik = information
Pk = probability
log 0 = math error
log 1 = 0
log 10 = 1
log 2=0.3010
log 10 =1
4/2 =2 bits
The above question is very basic log based problem. calculator that is FX 991 MS calculator because all the problems are you need to know how to use fx991ms calculator. because they using logs based
problem, so you need to know how to use scientific calculator for solving problems in digital communication.
The problematic sums is asking to find the information for the given value first you need to write the formula for information formula that is log formula. final answer you need to write the units
the each and every sums having the different units in digital communication all the units its like symbols, bits and rates.
How to pass digital communication.
First you need to study all the 5 units 2 mark questions with answers. this subject having all the information concepts. communication engineering also same information concept so you easily
understand the concept of information concepts as well as communication concetps that slightly different that is digital digital.
With the help of fx991ms calculator it’s used to calculate how to find information, how to find modulation, what is digital calculation. The all the digital based problems are very simple the
formula and the log are major roll playing in the sums. you need to the fx991mx base and mode calculations.
Digital communication MCQ contains in the above video if you go through that video that is sufficient for Anna University online examination. Remaining all the important MCQ questions update soon.
All the 5 units important multiple choice questions and answers given in above video if you go through that video you definitely clear your digital communication paper because all the important
University point of view, exam point of view that question paper and answers are very useful.
All the best
EC8501 Digital Communication Notes Video
Discrete Time Signal Processing Solutions EC8553 mcq CLICK HERE | {"url":"https://www.yakesh.in/digital-communication-mcq-ec8501-mcq-questions-ec8501-digital-communication-mcq/","timestamp":"2024-11-12T02:44:25Z","content_type":"text/html","content_length":"93660","record_id":"<urn:uuid:ae0d7781-b70a-430f-8ce1-551044ba352c>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00226.warc.gz"} |
Optimisation for Game Theory and Machine Learning
1st February 2024 to 31st January 2027
Algorithms for optimisation often in practice use local search approaches. For example, when the objective function is continuous and smooth, gradient descent is usually used (for example, in neural
networks). In game-theoretic settings, local search arises naturally in the context of multiple agents, who are attempting to improve their payoffs by best-responding to their peers’ behaviour. Of
course, there is no general guarantee about the convergence of this process, it depends on the structure of the game. Local search often works surprising well in practice (even when worst-case is
known to be poor) and it is of interest to understand why. This project aims to develop the theory of what is going on, and hopefully lead to improved algorithms. It will consider various specific
optimisation problems in more detail, as part of this agenda.
Both machine learning and game theory have given rise to diverse problems of local optimisation, and it is of interest to classify them according to their computational hardness. The project aims to
study a general issue, which is the “hard in theory, easy in practice” phenomenon of these problems (so, an aspect of the “beyond worst-case analysis” agenda). The project will include the designing
of novel algorithms with performance guarantees. In settings of continuous optimisation, where gradient descent is applicable, there are new and interesting variants of the technique, for example
‘optimistic’ gradient descent, and the issue of how to adjust the step size, or learning rate, as the algorithm runs.
Principal Investigator | {"url":"http://www.cs.ox.ac.uk/projects/OptimisationforGameTheoryandMachineLearning/","timestamp":"2024-11-13T13:04:29Z","content_type":"text/html","content_length":"30284","record_id":"<urn:uuid:bc486923-8081-408c-b397-24534a99e0d9>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00394.warc.gz"} |
s a percentage:
Learn how to turn a decimal number into a fraction and a percentage. Steps.
1. How to write the number as a percentage:
• Multiply the number by 100. Then add the percent sign, %.
2. How to write the number as a fraction:
• Write down the number divided by 1, as a fraction.
• Turn the top number into a whole number: multiply both the top and the bottom by the same number.
• Reduce (simplify) the above fraction to the lowest terms, to its simplest equivalent form, irreducible. To reduce a fraction divide the numerator and the denominator by their greatest (highest)
common factor (divisor), GCF.
• If the fraction is an improper one, rewrite it as a mixed number (mixed fraction).
• Calculate equivalent fractions. By expanding it we can build up equivalent fractions: multiply the numerator & the denominator by the same number. | {"url":"https://www.fractii.ro/decimal-number-converted-turned-into-fractions-percentage.php?number=655&repeating_decimal_places=0","timestamp":"2024-11-10T02:21:27Z","content_type":"text/html","content_length":"33822","record_id":"<urn:uuid:e59a2787-d215-4511-9264-1c4976caac1d>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00658.warc.gz"} |
Deferred annuity sample problems pdf
Math of ivestment annuity due and deferred payments slideshare. Annuity examples deferred annuity income rider illustrations. From the perspective of an investor, deferred annuities are mainly useful
for the purpose of tax deferral of earnings because of a lack of restrictions on the amount of its annual investment coupled with the guarantee of the lifelong. In annuity uncertain, the annuitant
may be paid according to certain event. An annuity is a sequence of equal payments made at regular intervals of time. An annuity is a series of equal dollar payments that are made at the end of
equidistant points in time such as monthly, quarterly, or annually over a finite period of time. The value of sallys share is really the present value of a deferred annuity, i.
Annuities due, deferred annuities, perpetuities and calculus. To calculate the present value of a perpetuity, we. Income riders are gaining in popularity as investors are better understanding these
products. If constant cash flow occur at the end of each periodyear. Tell the students that this is an example of a deferred annuity. In any problems that you see payment at the beginning of some
time period, this is the formula to use. An annuity is an investment in which the purchaser makes a sequence of periodic, equal payments. For example, if interest is compounded semiannually, then n
the number of semiannual rents paid or received, i the annual interest rate divided by 2, and r the amount of rent paid or received every six months. All the variables have the same meaning as the
original annuity formula above. Mora is 25 years old today and wants to begin making deposits to save for her retirement beginning next year. A good example of annuity certain is the monthly payments
of a car loan where the amount and number of payments are known. Deferred a deferred annuity grows, tax deferred, until the contract is annuitized put into a payment stream or surrendered paid out as
a lump sum. Deferred annuity formula how to calculate pv of deferred annuity. A tax deferred annuity tda is an annuity in which you do not pay taxes on the money deposited or on the interest earned
until you start to withdraw the money from the annuity account.
Annuity formula calculation examples with excel template. A common problem in financial management is to determine the in stallments. A common problem in financial management is to determine the
installments. At the beginning of the section, we looked at a problem in which a couple invested a set amount of money each month into a college fund for six years. Suppose the annuity problem
setting is one in which the interest rate.
Adeferred annuity is one that begins payments at some time in the future. The learner is able to investigate, analyze and solve problems involving simple. An annuity due has payments at the beginning
of each payment period, so interest accumulates for one extra period. When a sequence of payments of some fixed amount are made in an account at equal intervals of time. Annuity means a stream or
series of equal payments. An annuity is a series of payments required to be made or received over time at regular intervals. John jones recently set up a tax deferred annuity to save for his
retirement. Annuities practice problems prepared by pamela peterson drake congrats. The payments for this formula are made at the end of a period. There are 237 sample multiple choice questions in
this study note for the longterm actuarial mathematics exam. Deferred annuity due if the deferral period ends at the beginning of the first. In a nutshell, a fixed amount the principal is deposited
in either a lump sum or over time in a deferred annuity offering a guaranteed income rider.
There are two phases in the life of a deferred annuity. If the stated interest rate is eight percent, discounted quarterly, what is the present value of this annuity. Mora is 25 years old todayand
wants to begin making deposits to save for her retirement beginning nextyear, with the last deposit on her 60thbirthday. If payments are made at the end of each period, the annuity is referred to as
ordinary annuity. Test your understanding with practice problems and stepbystep solutions.
Find the present value of a deferred annuity of p500 a year for ten years that is deferred 5 years. When income payments are scheduled to begin is the determining factor as to which category an
annuity belongs. Pdf chapter 11 other types of annuties 407 ngaruiya ben. Calculate the present value of an annuityimmediate of amount. A deferred fixed annuity as an alternative to a cd in many
cases, the deferred fixed annuity may not only defer but also reduce taxes compared with a cd or other investment held in a taxable account. A deferred annuity is an insurance contract that allows
you to delay or defer your income stream indefinitely. In annuity certain, the specific amount of payments are set to begin and end at a specific length of time. A deferred annuity is one that begins
payments at some time in the future. Annuities practice problem set 2 future value of an annuity 1. Math 4 tutorial 8 annuities due, deferred annuities, perpetuities.
Present value calculations for a deferred annuity sapling. If the interest rate is 8 percent, the amount of each annuity payment is closest to which of the following. Mortgages, car payments, rent,
pension fund payments, insurance premiums. It could also be viewed as an annuitydue deferred periods j a 8j a 8j a 21j a j 319. A deferred annuity would better be defined as a category of annuities
rather than a type of annuity. Applying for deferred or postponed retirement under the. Under a 6% return assumption, the investment portfolio balance for the swp only strategy would exceed that of
the swp and deferred income annuity strategy. Let say your age is 30 years and you want to get retired at the age of 50 years and you expect that you will live for. Due ordinary deferred annuity in
the solved examples and exercise problems in.
This article explains the computation of present value of an annuity. Designed for longterm savings, a deferred annuity grows tax deferred until you withdraw the money. When the annuity reaches the
contractually agreedupon date, the investor will begin receiving several payments over a period of time or in. An annuity due has payments at the beginning of each payment period, so interest.
Longterm actuarial mathematics sample multiple choice. The questions are sorted by the society of actuaries recommended resources for this exam. To find the amount of an annuity, we need to find the
sum of all the. During the accumulation phase, the investor will deposit money into the account either periodically or all in one lumpsum. A deferred annuity is one for which the first payment starts
some time in the. Math 4 tutorial 8 annuities due, deferred annuities. Deferred annuity formula how to calculate pv of deferred.
Payments may be made annually, semiannually, quarterly, or at other periods some examples of annuities are. A deferred annuity contract is chiefly a vehicle for accumulating savings and eventually
distributing the value either as a payment stream or as a onetime, lumpsum payment. All annuities can be categorized as either deferred or immediate. With a deferred fixed annuity, all taxes on
interest are deferred until funds are withdrawn from the account. Calculating different types of annuities money instructor. In annuity problems, n is equal to the total number of rents paid or
received. You have completed at least 10 years of creditable service, including 5 years of civilian service, then you are eligible for a deferred annuity the first day of the month after you reach
the minimum retirement age mra. Home mortgage payments, car loan payments, pension payments. Math of ivestment annuity due and deferred payments. | {"url":"https://ngirtacmale.web.app/464.html","timestamp":"2024-11-06T21:24:10Z","content_type":"text/html","content_length":"12728","record_id":"<urn:uuid:bc7b3e97-83b3-4aaf-89a7-09c574b448b5>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00648.warc.gz"} |
Finite-Difference Schemes
This appendix gives some simplified definitions and results from the subject of finite-difference schemes for numerically solving partial differential equations. Excellent references on this subject
include Bilbao [53,55] and Strikwerda [481].
The simplifications adopted here are that we will exclude nonlinear and time-varying partial differential equations (PDEs). We will furthermore assume constant step-sizes (sampling intervals) when
converting PDEs to finite-difference schemes (FDSs), i.e., sampling rates along time and space will be constant. Accordingly, we will assume that all initial conditions are bandlimited to less than
half the spatial sampling rate, and that all excitations over time (such as summing input signals or ``moving boundary conditions'') will be bandlimited to less than half the temporal sampling rate.
In short, the simplifications adopted here make the subject of partial differential equations isomorphic to that of linear systems theory [449]. For a more general and traditional treatment of PDEs
and their associated finite-difference schemes, see, e.g., [481].
Finite-Difference Schemes (FDSs) aim to solve differential equations by means of finite differences. For example, as discussed in §C.2, if displacement in meters of a vibrating string at time
sampling interval
interval. Other types of finite-difference schemes were derived in Chapter
), including a look at
properties. These
finite-difference approximations
to the partial derivatives may be used to compute solutions of
differential equations
on a discrete grid:
Let us define an abbreviated notation for the grid variables
and consider the ideal
string wave equation
(cf, §
speed). Then, as derived in §
, setting
wave equation
leads to the relation
everywhere on the time-space grid (
, for all
explicit finite-difference scheme
for string displacement:
The FDS is called
because it was possible to solve for the state at time
digital filter
'') which computes a solution at time
, a non-
causal filter
is derived), the discretized differential equation is said to define an
FDS. An implicit FDS can often be converted to an explicit FDS by a rotation of coordinates [
A finite-difference scheme is said to be convergent if all of its solutions in response to initial conditions and excitations converge pointwise to the corresponding solutions of the original
differential equation as the step size(s) approach zero.
In other words, as the step-size(s) shrink, the FDS solution must improve, ultimately converging to the corresponding solution of the original differential equation at every point of the domain.
In the vibrating string example, the limit is taken as the step sizes (sampling intervals) finite-difference approximations in Eq.D.1) converge in the limit to the very definitions of the
corresponding partial derivatives, we expect the FDS in Eq.D.3) based on these approximations to be convergent (and it is).
In establishing convergence, it is necessary to provide that any initial conditions and boundary conditions in the finite-difference scheme converge to those of the continuous differential equation,
in the limit. See [481] for a more detailed discussion of this topic.
The Lax-Richtmyer equivalence theorem provides a means of showing convergence of a finite-difference scheme by showing it is both consistent and stable (and that the initial-value problem is well
posed) [481]. The following subsections give basic definitions for these terms which applicable to our simplified scenario (linear, shift-invariant, fixed sampling rates).
A finite-difference scheme is said to be consistent with the original partial differential equation if, given any sufficiently differentiable function differential equation operating on finite
difference equation operating on
Thus, in the ideal string example, to show the consistency of Eq.D.3) we must show that
for all
shift operator notation
In particular, we have
In taking the limit as FDS by
as required. Thus, the FDS is consistent. See, e.g., [481] for more examples.
In summary, consistency of a finite-difference scheme means that, in the limit as the sampling intervals approach zero, the original PDE is obtained from the FDS.
For a proper authoritative definition of ``well posed'' in the field of finite-difference schemes, see, e.g., [481]. The definition we will use here is less general in that it excludes amplitude
growth from initial conditions which is faster than polynomial in time.
We will say that an initial-value problem is well posed if the linear system defined by the PDE, together with any bounded initial conditions is marginally stable.
As discussed in [449], a system is defined to be stable when its response to bounded initial conditions approaches zero as time goes to infinity. If the response fails to approach zero but does not
exponentially grow over time (the lossless case), it is called marginally stable.
In the literature on finite-difference schemes, lossless systems are classified as stable [481]. However, in this book series, lossless systems are not considered stable, but only marginally stable.
When marginally stable systems are allowed, it is necessary to accommodate polynomial growth with respect to time. As is well known in linear systems theory, repeated poles can yield polynomial
growth [449]. A very simple example is the ordinary differential equation (ODE)
which, given the initial condition
for any constant
When all poles of the system are strictly in the left-half of the Laplace-transform stable, even when the poles are repeated. This is because exponentials are faster than polynomials, so that any
amount of exponential decay will eventually overtake polynomial growth and drag it to zero in the limit.
Marginally stable systems arise often in computational physical modeling. In particular, the ideal string is only marginally stable, since it is lossless. Even a simple unaccelerated mass, sliding on
a frictionless surface, is described by a marginally stable PDE when the position of the mass is used as a state variable (see §7.1.2). Given any nonzero initial velocity, the position of the mass
approaches either displacement as a state variable. For ideal strings and freely sliding masses, force and velocity are usually good choices.
It should perhaps be emphasized that the term ``well posed'' normally allows for more general energy growth at a rate which can be bounded over all initial conditions [481]. In this book, however,
the ``marginally stable'' case (at most polynomial growth) is what we need. The reason is simply that we wish to excluded unstable PDEs as a modeling target. Note, however, that unstable systems can
be used profitable over carefully limited time durations (see §9.7.2 for an example).
In the ideal vibrating string, energy is conserved. Therefore, it is a marginally stable system. To show mathematically that the PDE Eq.D.2) is marginally stable, we may show that
for some constants
, we can show
for all
Note that solutions on the ideal string are not bounded, since, for example, an infinitely long string (non-terminated) can be initialized with a constant positive velocity everywhere along its
length. This corresponds physically to a nonzero transverse momentum, which is conserved. Therefore, the string will depart in the positive
The well-posedness of a class of damped PDEs used in string modeling is analyzed in §D.2.2.
A Class of Well Posed Damped PDEs
A large class of well posed PDEs is given by [45]
Thus, to the ideal
string wave equation
) we may add any number of even-order partial derivatives in
well posed
, as we now show.
To show Eq.D.5) is well posed [45], we must show that the roots of the characteristic polynomial equation (§D.3) have negative real parts, i.e., that they correspond to decaying exponentials instead
of growing exponentials. To do this, we may insert the general eigensolution
into the PDE just like we did in §
to obtain the so-called
characteristic polynomial equation
Let's now set spatial frequency (called the ``wavenumber'' in acoustics) and of course Laplace transform to a spatial Fourier transform. Since there are only even powers of the spatial Laplace
transform variable real. Therefore, the roots of the characteristic polynomial equation (the natural frequencies of the time response of the system), are given by
Proof that the Third-Order Time Derivative is Ill Posed
For its tutorial value, let's also show that the PDE of Ruiz [392] is ill posed, i.e., that at least one component of the solution is a growing exponential. In this case, setting C.28), which we
restate as
yields the
characteristic polynomial equation
By the
Routh-Hurwitz theorem,
there is at least one root in the right-half
It is interesting to note that Ruiz discovered the exponentially growing solution, but simply dropped it as being non-physical. In the work of Chaigne and Askenfelt [77], it is believed that the
finite difference approximation itself provided the damping necessary to eliminate the unstable solution [45]. (See §7.3.2 for a discussion of how finite difference approximations can introduce
damping.) Since the damping effect is sampling-rate dependent, there is an upper bound to the sampling rate that can be used before an unstable mode appears.
A finite-difference scheme is said to be stable if it forms a digital filter which is at least marginally stable [449].
To distinguish between the stable and marginally stable cases, we may classify a finite-difference scheme as strictly stable, marginally stable, or unstable.
Lax-Richtmyer equivalence theorem
The Lax-Richtmyer equivalence theorem states that ``a consistent finite-difference scheme for a partial differential equation for which the initial-value problem is well posed is convergent if and
only if it is stable.'' For a proof, see [481, Ch. 10].
A condition stronger than stability as defined above is passivity. Passivity is not a traditional metric for finite-difference scheme analysis, but it arises naturally in special cases such as wave
digital filters (§F.1) and digital waveguide networks [55,35]. In such modeling frameworks, all signals have a physical interpretation as wave variables, and therefore a physical energy can be
associated with them. Moreover, each delay element can be associated with some real wave impedance. In such situations, passivity can be defined as the case in which all impedances are nonnegative.
When complex, they must be positive real (see §C.11.2).
To define passivity for all linear, shift-invariant finite difference schemes, irrespective of whether or not they are based on an impedance description, we will say that a finite-difference scheme
is passive if all of its internal modes are stable. Thus, passivity is sufficient, but not necessary, for stability. In other words, there are finite difference schemes which are stable but not
passive [55]. A stable FDS can have internal unstable modes which are not excited by initial conditions, or which always cancel out in pairs. A passive FDS cannot have such ``hidden'' unstable modes.
The absence of hidden modes can be ascertained by converting the FDS to a state-space model and checking that it is controllable (from initial conditions and/or excitations) and observable [449].
When the initial conditions can set the entire initial state of the FDS, it is then controllable from initial conditions, and only observability needs to be checked. A simple example of an
unobservable mode is the second harmonic of an ideal string (and all even-numbered harmonics) when the only output observation is the midpoint of the string.
In summary, we have defined the following terms from the analysis of finite-difference schemes for the linear shift-invariant case with constant sampling rates:
Finally, the Lax-Richtmyer equivalence theorem establishes that well posed + consistency +
implies convergence, where, as defined in §
above, convergence means that solutions of the FDS approach corresponding solutions of the PDE as
Because the range of human hearing is bounded (nominally between 20 and 20 kHz), spectral components of a signal outside this range are not audible. Therefore, when the solution to a differential
equation is to be considered an audio signal, there are frequency regions over which convergence is not a requirement.
Instead of pointwise convergence, we may ask for the following two properties:
• Superposition holds.
• Convergence occurs within the frequency band of human hearing.
Superposition holds for all linear
partial differential equations
with constant coefficients (linear, shift-invariant systems [
]). We need this condition so that errors in the inaudible bands do not affect the audible bands. Inaudible errors are fine as long as they do not grow so large that they cause numerical overflow. An
example in which this ``bandlimited design'' approach yields large practical dividends is in
bandlimited interpolator
design (see §
In many cases, such as in digital waveguide modeling of vibrating strings, we can do better than convergence. We can construct finite difference schemes which agree with the corresponding continuous
solutions exactly at the sample points. (See §C.4.1.)
Characteristic Polynomial Equation
The characteristic polynomial equation for a linear PDE with constant coefficients is obtained by taking the 2D Laplace transform of the PDE with respect to
into the PDE, where
complex variable
associated with the Laplace-transform with respect to time, and
Laplace transform.
As a simple example, the ideal-string wave equation (analyzed in §C.1) is a simple second-order PDE given by
sound speed
, as discussed in §
Substituting Eq.D.6) into Eq.D.7) results in the following characteristic polynomial equation:
Solving for
dispersion relation
or, looking only at the frequency axes (
, using
Fourier transforms
in place of Laplace transforms),
Since the
phase velocity
of a
traveling wave
is, by definition, the temporal frequency divided by
spatial frequency
, we have simply
This result can be interpreted as saying that all Fourier components of any solution of Eq.
) must
along the string with speed
may be
, in which case the phase velocity depends on frequency (see §
for an analysis of stiff
vibrating strings
, which are dispersive). Moreover, wave propagation may be
in a frequency-dependent way, in which case one or more roots of the characteristic polynomial equation will have negative real parts; if any roots have positive real parts, we say the
initial-value problem
ill posed
since is has exponentially growing solutions in response to
initial conditions
Von Neumann Analysis
Von Neumann analysis is used to verify the stability of a finite-difference scheme. We will only consider one time dimension, but any number of spatial dimensions.
The procedure, in principle, is to perform a spatial Fourier transform along all spatial dimensions, thereby reducing the finite-difference scheme to a time recursion in terms of the spatial Fourier
transform of the system. The system is then stable if this time recursion is at least marginally stable as a digital filter.
Let's apply von Neumann analysis to the finite-difference scheme for the ideal vibrating string Eq.D.3):
There is only one spatial dimension, so we only need a single 1D
Discrete Time Fourier Transform
) along
]. Using the
shift theorem
for the DTFT, we obtain
spatial frequency
(wavenumber). (On a more elementary level, the DTFT along
difference equation
) that needs its stability checked. This can be accomplished most easily using the Durbin recursion [
], or we can check that the
of the recursion do not lie outside the unit circle in the
A method equivalent to checking the pole radii, and typically used when the time recursion is first order, is to compute the amplification factor as the complex gain
The finite-difference scheme is then declared stable if
Since the finite-difference scheme of the ideal vibrating string is so simple, let's find the two poles. Taking the z transform of Eq.D.8) yields
yielding the following characteristic polynomial:
Applying the
quadratic formula
to find the roots yields
The squared pole moduli are then given by
Thus, for marginal stability, we require
Since the range of spatial frequencies is
) for the ideal vibrating string is
In summary, von Neumann analysis verifies that no spatial Fourier components in the system are growing exponentially with respect to time.
Next Section: Equivalence of Digital Waveguide and Finite Difference SchemesPrevious Section: Digital Waveguide Theory | {"url":"https://www.dsprelated.com/freebooks/pasp/Finite_Difference_Schemes.html","timestamp":"2024-11-07T05:54:38Z","content_type":"text/html","content_length":"130816","record_id":"<urn:uuid:aaffd479-8060-4e30-bf61-5f410d43a34f>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00710.warc.gz"} |
In set theory, a transitive relation on a set is a relation with the property that if x→y and y→z then x→z.
• An equivalence relation is transitive:
□ Equality is transitive: if x=y and y=z then x=z;
□ The trivial (always-true) relation is transitive;
• An order relation is transitive:
□ The usual order on the integers is transitive: if x>y and y>z then x>z;
□ Divisibility on the natural numbers is transitive: if x divides y and y divides z then x divides z;
□ Inclusion on subsets of a set is transitive: if x is a subset of y and y is a subset of z then x is a subset of z.
• The intersection of transitive relations is transitive. That is, if R and S are transitive relations on a set X, then the relation R&S, defined by x R&S y if x R y and x S y, is also transitive.
The same holds for intersections of arbitrary families of transitive relations: indeed, the transitive relations on a set form a closure system.
Transitivity may be defined in terms of relation composition. A relation R is transitive if the composite R.R implies (is contained in) R.
Transitive closure
The transitive closure of a relation R may be defined as the intersection R* of all transitive relations containing R (one always exists, namely the always-true relation): loosely the "smallest"
transitive relation containing R. The closure may also be constructed as
${\displaystyle R^{*}=R\cup (R\circ R)\cup \cdots \cup R^{{\circ }n}\cup \cdots \,}$
where ${\displaystyle R^{{\circ }n}}$ denotes the composition of R with itself n times. | {"url":"https://en.citizendium.org/wiki/Transitive_relation","timestamp":"2024-11-08T12:47:28Z","content_type":"text/html","content_length":"34209","record_id":"<urn:uuid:a8e6acbb-b095-42f2-9904-548dcdd0bc3c>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00372.warc.gz"} |
[OLD] Teen learn - Basics of programming | Reply Challenges
While the problems you need to solve will always be different, the approaches to solve them can be divided into few categories the so called "problem solving paradigms".
Let's go into more detail about the key ones.
1 _ complete search
The simplest problem-solving paradigm
Also known as "brute force", with this very generic approach, we simply list all the possible solutions among which we then search for the correct one.
To apply a complete search for solving a problem we need:
1. To generate every possible solution to a problem by using an iterative or recursive method.
2. To know how to check if a solution is valid.
A complete search always gives you the correct answer, but when?
The brute force approach works well when there’s a small amount of information, otherwise the number of possible solutions can be extremely high.
2 _ search algorithm
Two main approaches to look for a value in a list
The simplest way to search something is to look at everything.
With a linear search, however, we can just look at each possible element to find the correct one.
This approach works every time on both sorted and unsorted data, but may be slow on large data sets.
If the numbers of the list are sorted things will be easier.
With the binary search algorithm, we can take the middle element and compare it with our searched number. By repeating this process we can eliminate a lot of numbers at each step until we’ve found
our solution, without even looking at them.
Which results can I obtain with a binary search algorithm?
The result can be one of the following:
• If the middle element is the one we’re looking for, we have our solution.
• If the middle element is greater than the searched element, the solution is in the first half of the list.
• If the middle element is less than the searched element, the solution is in the second half of the list.
NOTE: Search algorithms have many other applications besides looking for elements in array. You can also use them to speed up a complete search.
3 _ greedy algorithm
Make the best choices at each step
When facing a decision, you aim to find the best solution at any given time (greedy) and hope it’s the best one among all the others.
Unfortunately, choosing the best solution isn't always the best choice. Often, choices made in a moment can result in an even more negative impact later on.
So pay attention if you have to make decisions in advance.
exercise 1
Try writing an algorithm to solve the problem of giving the minimum amount of coins in change
** example of solution **
exercise 2
Try writing an algorithm to solve the problem of maximising the profit of a given list of tasks given their deadlines and values
** code with example of solution **
4 _ divide & conquer
When problems are getting complicated... divide them
In the ‘divide and conquer’ approach, we try to simplify a problem by splitting it into smaller problems to the point where solving them becomes trivial.
With the following 3 steps, and by using recursion, you can solve problems easily. But attention, it doesn't always give the correct solution.
exercise 3
Consider an unordered array of numbers. How can you use a divide and conquer approach to sort the numbers?
** code with example solution **
exercise 4
Imagine you have two different sorted array of numbers. How you can merge them into a unique sorted array?
** code with example solution **
5 _ dynamic programming
When things are getting harder, solve the problem as a master
Of all the problem solving paradigms, "dynamic programming" is probably the most challenging to master. The idea behind dynamic programming is the same as the divide and conquer approach: solve the
bigger problem by solving smaller problems. But while in divide and conquer smaller problems are independent from each other, in dynamic programming smaller problems are used to construct the bigger
The simplest way to approach problems with this technique is to assume you’ve already solved the smaller problems, so think about how to use them to solve a bigger solution.
A technique often used with the dynamic programming approach is memoization (not to be confused with memorization): when you need the solutions of the same problems several times, you can just
memorize their value instead of reevalute them.
This simple idea can save you a lot of time.
exercise 5
Consider the same problem of sorting a array of numbers, but now all the elements are sorted except the last one. How can you use the solution of the smaller problem to solve the bigger one?
** code of example solution **
exercise 6
Go back to the greedy approach problems and try to solve them now.
** code of example solution ** | {"url":"https://challenges.reply.com/challenges/coding-teen/learn-train/basics-programming-22v3lp7g/","timestamp":"2024-11-05T16:01:55Z","content_type":"text/html","content_length":"41814","record_id":"<urn:uuid:d7dba634-da49-45e7-817f-d987c2ecb589>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00263.warc.gz"} |
What Is The Domain And Range Of A Exponential Function - Function Worksheets
Domain And Range Of Exponential Functions Worksheets – If you’re in search of an activity in math for your child to help them practice exponential … Read more
Exponential Function Domain And Range Worksheet
Exponential Function Domain And Range Worksheet – If you’re in search of an activity in math for your child to help them practice exponential functions, … Read more | {"url":"https://www.functionworksheets.com/tag/what-is-the-domain-and-range-of-a-exponential-function/","timestamp":"2024-11-11T14:09:54Z","content_type":"text/html","content_length":"69830","record_id":"<urn:uuid:a89048a3-618b-4f64-b864-5665886d68cd>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00789.warc.gz"} |
Piecewise nonlinear functions
In [1] a question is posed how we can model the piecewise nonlinear function depicted below:
Model the dashed line h(x)
We want to model that the costs are represented by the function \(h(x)\) (the dashed curve in the picture). Let's make a few assumptions:
• The function \(f(x)\) is a quadratic function while \(g(x)\) is linear.
• \(f(x)\) is convex
• We are minimizing cost (or maximizing profit)
• Update: I also assume this is part of a larger problem, with multiple items. If this was the only curve, we just can use \(\min f(x) = \min x\) due to monotonicity (see comments below). If we
want to minimize \(\displaystyle \sum_i \mathit{Cost}_i\) this is no longer applicable.
Choose cheapest
One way is to observe that for any \(x\) we choose the cheapest curve. I.e. we have \(\mathit{Cost}=\min(f(x),g(x))\). This can be modeled as:
\[\min \> & \mathit{Cost}\\ & \mathit{Cost}\ge f(x)-M\delta\\ &\mathit{Cost}\ge g(x)-M(1-\delta)\\&x \in [0,x_{max}]\\ & \delta \in \{0,1\}\] The proper value for \(M\) would be \(M=f(x_{max})-g(x_
{max})\). Basically this MIQCP (Mixed-Integer Quadratically Constrained Programming) formulation says: just choose one of the curves and ignore the other one. The objective will make sure the most
expensive curve is ignored and the cheapest curve is retained. We did not have to use \(x_0\) at all in this formulation.
The big-M value can become large in case \(x_{max}\) is large. Sometimes a SOS1 formulation is proposed. For this case this means:\[\min \> & \mathit{Cost}\\ & \mathit{Cost}\ge f(x)-s_1\\ &\mathit
{Cost}\ge g(x)-s_2\\&x \in [0,x_{max}]\\ & s_1,s_2 \ge 0\\& \{s_1,s_2\} \in SOS1\]
If the solver allows indicator constraints we can write: \[\min \> & \mathit{Cost}\\ & \delta=0\Rightarrow \mathit{Cost}\ge f(x)\\ &\delta=1 \Rightarrow \mathit{Cost}\ge g(x)\\&x \in [0,x_{max}]\\& \
delta \in \{0,1\}\]
Use Intervals
A more standard approach would be to try to formulate: \[\mathit{Cost}=\begin{cases} f(x) & \text{if $x \le x_0$} \\ g(x) &\text{if $x\gt x_0$}\end{cases}\] Instead of letting the solver decide the
best value of \(\delta\), now we use the current value of \(x\) to determine \(\delta\). The rule \[\delta = \begin{cases} 0 & \text{if $x \in [0,x_0]$} \\ 1 & \text{if $x \in [x_0,x_{max}]$}\end
{cases}\] can be rewritten as:\[ & \delta = 0 \Rightarrow x \in [0,x_0]\\ & \delta = 1 \Rightarrow x \in [x_0,x_{max}]\] (Note that we added some ambiguity for \(x=x0\). In practice that is no
problem. Int can be argued this is good thing [2].) This expression in turn can be formulated as two inequalities: \[x_0 \delta \le x \le x_0 + (x_{max}-x_0) \delta\] We can add this to any of the
earlier formulations, e.g.: \[\min \> & \mathit{Cost}\\ & \mathit{Cost}\ge f(x)-M\delta\\ &\mathit{Cost}\ge g(x)-M(1-\delta)\\ &x_0 \delta \le x \le x_0 + (x_{max}-x_0) \delta \\&x \in [0,x_{max}]\\
& \delta \in \{0,1\}\]
1. Piecewise non-linear cost function in Cplex, https://stackoverflow.com/questions/49992813/piecewise-non-linear-cost-function-in-cplex
2. Strict inequalities in optimization models, http://yetanothermathprogrammingconsultant.blogspot.com/2017/03/strict-inequalities-in-optimization.html
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Battery Single Particle
Battery model with single-particle approach
Since R2024a
Simscape / Battery / Cells
The Battery Single Particle block represents a battery by using a single-particle model. This implementation considers the ohmic and mass transport overpotentials in both the liquid electrolyte and
solid electrode phases. Additionally, it considers the reaction kinetics and the current collector resistance.
The battery comprises two electrodes, the anode and cathode, and a porous separator between the electrodes. In this block, the anode refers to the negative electrode during discharge and the cathode
refers to the positive electrode during discharge. The block models the ohmic overpotentials of the electrodes and electrolyte, as well as the concentration across the cell cross section from the
anode current collector to the cathode current collector, in a one-dimensional framework.
This figure illustrates a representative concentration in the electrolyte during discharge. The model comprises the anode (x=[0 … L^-]), the separator (x=[L^- … L^-+L^sep]) and the cathode (x=[L^-+L^
sep… L^-+L^sep+L^+]).
The block calculates the concentration in the electrodes in representative spherical particles across the radial dimension r. This figure shows the concentration gradient in the representative
particles during a continuous discharge of the battery:
Species Conservation in Solid Phase
When the block is in solid phase, the single-particle approach models the positive and negative electrodes as a single representative spherical particle.
The superscripts in these equations refer to the respective electrodes. A + superscript refers to the cathode. A - superscript refers to the anode. A sep superscript refers to the separator. A ±
superscript means that the equation applies to both anode and cathode. For example, c^+[s] is the solid-phase concentration of the cathode and c^-[s] is the solid-phase concentration of the anode.
This equation uses Fick's law to describe the concentration, c, of the cation in the negative or positive electrode. The block uses the radial coordinates only to calculate the concentration in the
electrodes. The diffusion in the spherical particle drives the mass transfer,
$\frac{\partial {c}_{s}^{±}}{\partial t}\left(r,t\right)=\frac{\partial }{\partial r}\left[{D}_{s}^{±}\frac{\partial {c}_{s}^{±}}{\partial r}\left(r,t\right)\right],$
• c[s] is the solid-phase concentration.
• D[s] is the diffusion coefficient in solid phase.
• r is the radius.
• t is the time.
At the center of the particle, the concentration gradient is equal to 0:
${D}_{s}^{±}\frac{\partial {c}_{s}^{±}}{\partial r}\left(0,t\right)=0.$
This equation calculates the ion concentration gradient at the surface of the particle:
${D}_{s}^{±}\frac{\partial {c}_{s}^{±}}{\partial r}\left({R}_{s}^{±},t\right)=\mp \frac{{J}_{s}^{±}}{{a}_{s}^{±}\text{}F}.$
In this equation, F is Faraday's constant, and J is the molar flux,
• I is the current applied to the cell.
• A is the total area of the current collector.
• L is the length of the respective electrode.
Additionally, a is the active surface area per electrode unit volume,
${a}^{±}=\frac{3{\epsilon }^{±}}{{R}^{±}},$
• ε is the active material fraction of the electrode.
• R is the total radius of the active particle.
To solve the differential equation, the Battery Single Particle block discretizes the particle with the radius R into n shells. Each shell has a radial distance equal to $\delta r=\frac{R}{n}$ from
the adjacent spheres.
For the ith sphere, this equation calculates the rate of change of concentration, δc/δt:
${\stackrel{˙}{c}}_{{s}_{i}}=\frac{{D}_{s}}{\delta {r}^{2}}\left\{\left(\frac{i-1}{i}\right){c}_{{s}_{i-1}}-2{c}_{{s}_{i}}+\left(\frac{i+1}{i}\right){c}_{{s}_{i+1}}\right\}.$
For the innermost shell in the particle, the block implements this boundary condition:
${\stackrel{˙}{c}}_{1}=\frac{2{D}_{s}}{\delta {r}^{2}}\left\{{c}_{{s}_{2}}-{c}_{{s}_{1}}\right\}.$
To implement the boundary condition at the surface of the particle, the block adds an additional node around the surface. The block does not calculate the concentration of this node because it does
not physically exist. The block uses this node to calculate the boundary condition between the outermost shell in the particle and the non-existent shell around it by using the Neumann boundary
condition. This equation describes the discretized result at the surface of the particle:
${\stackrel{˙}{c}}_{{s}_{end}}=\frac{2{D}_{s}}{\delta {r}^{2}}\left\{{c}_{{s}_{end-1}}-{c}_{{s}_{end}}\right\}-2\frac{n+1}{n}\frac{J}{F\text{}A\text{}\delta r}.$
Mass Transport Overpotential in Solid Phase
When the block is in solid phase, the open-circuit potential depends on the concentration. To calculate the mass transport overpotential at the electrodes, the Battery Single Particle block subtracts
the open-circuit potential of the average relative concentration in the particle from the open-circuit potential of the average relative concentration at the surface,
${\eta }^{±}{}_{\text{diffusion},s}={\text{ocp}}^{±}\left({c}_{s,\text{surface},\text{relative}}^{±}\right)-{\text{ocp}}^{±}\left({\overline{c}}_{s,\text{relative}}^{±}\right),$
• η[diffusion][,s] is the solid-phase mass transport overpotential.
• ocp(c[s],[surface,relative]) is the open-circuit potential for the concentration at the surface of the particle.
• ocp(c[s][relative]) is the open-circuit potential for the average concentration of the particle.
The block uses the same equation to calculate the mass transport overpotential for both the anode and the cathode.
Ohmic Overpotential in Solid Phase
To calculate the ohmic overpotential when the block is in the solid phase, the Battery Single Particle block linearly approximates the current across the electrodes and the current at the current
collector to a value equal to the electric current applied to the cell. The current at the interface between the current separator and the electrode is zero. This equation defines the ohmic
overpotential in the solid phase,
${\eta }^{±}{}_{ohmic,s}=\frac{{I}_{batt}}{2A}\ast \frac{{L}^{±}}{{\kappa }^{±}},$
• η[ohmic,s] is the solid-phase ohmic overpotential.
• I[batt]/A is the cell cross section.
• L is the length of the respective electrode and depends on the thickness of the anode or cathode.
• κ is the conductivity. The conductivity depends on the temperature of the cell. κ is equal to the Anode conductivity parameter when the block calculates the ohmic overpotential of the anode and
is equal to the Cathode conductivity parameter when the block calculates the ohmic overpotential of the cathode.
The block uses the same equation to calculate the ohmic overpotential in solid phase for both the anode and the cathode.
Species Conservation in Liquid Phase
When the block is in liquid phase, this equation describes the concentration in the electrolyte at both electrodes and at the separator. To calculate the concentration across the separator, the block
considers the diffusive flow induced by concentration gradient,
$\frac{\partial {c}_{\epsilon }^{±}}{\partial t}\left(x,t\right)=\frac{\partial }{\partial x}\left[{D}_{\epsilon }\frac{\partial {c}_{\epsilon }^{±}}{\partial x}\left(x,t\right)\right],$
• c[ε] is the concentration in the electrolyte.
• D[ε] is the diffusion coefficient in liquid phase.
• x is the location in the thickness of the battery, from the anode current collector to the cathode current collector.
• t is the time.
At the positive and negative electrodes, the block considers both the diffusive flow and the cation flux from the solid electrode into the electrolyte,
${\in }_{\epsilon }^{±}\frac{\partial {c}_{\epsilon }^{±}}{\partial t}\left(x,t\right)=\frac{\partial }{\partial x}\left[{D}_{\epsilon }^{eff}\frac{\partial {c}_{\epsilon }^{±}}{\partial x}\left(x,t\
• ∈ is the volume fraction of the electrolyte.
• D^eff[ε] is the diffusion coefficient in the liquid phase that considers the porosity of the material. The diffusivity of the liquid electrolyte depends on the properties of the surrounding solid
electrode material. The electrode comprises multiple components, such as the active material and the filler, that form a characteristic porous material.
• J is the molar flux.
• t^+ is the transference number of the cation.
• F is Faraday's constant.
Because the electrolyte is a continuous fluid, the cation concentration at the border between the negative and positive electrodes and the separator must be equal. For the concentration in the
electrolyte at the anode-separator and cathode-separator interfaces, the block must define the boundary conditions between the three sections of the battery. The block represents both electrodes and
the separator as cuboids.
This block considers the electrolyte as a continuous medium across the electrodes and the separator. Because the concentrations on both sides of the interface must be equal, a continuity boundary
condition exists for the interface between the electrodes and the separator.
This equation describes the continuity boundary condition for the concentration at the interface between the anode and the separator,
${c}_{\epsilon }^{-}\left({L}^{-},t\right)={c}_{\epsilon }^{sep}\left({L}^{-},t\right),$
• ${c}_{\epsilon }^{-}\left({L}^{-},t\right)$ is the concentration in the anode at the border between the anode and the separator.
• ${c}_{\epsilon }^{sep}\left({L}^{-},t\right)$ is the concentration in the separator at the border between the anode and the separator.
This equation describes the continuity boundary condition for the concentration at the interface between separator and cathode,
${c}_{\epsilon }^{sep}\left({L}^{-}+{L}^{sep},t\right)={c}_{\epsilon }^{+}\left({L}^{-}+{L}^{sep},t\right),$
• ${c}_{\epsilon }^{sep}\left({L}^{-}+{L}^{sep},t\right)$ is the concentration in the separator at the border between the separator and the cathode.
• ${c}_{\epsilon }^{+}\left({L}^{-}+{L}^{sep},t\right)$ is the concentration in the cathode at the border between the separator and the cathode.
The block also applies a flux boundary condition to the interfaces between the electrodes and the separator. The flux at both sides of the interface must be equal,
$\begin{array}{l}{D}_{eff}^{-}\frac{\partial {c}_{\epsilon }^{-}}{\partial x}\left({L}^{-},t\right)={D}_{eff}^{sep}\frac{\partial {c}_{\epsilon }^{sep}}{\partial x}\left({L}^{-},t\right)\\ {D}_{eff}^
{+}\frac{\partial {c}_{\epsilon }^{+}}{\partial x}\left({L}^{-}+{L}^{sep},t\right)={D}_{eff}^{sep}\frac{\partial {c}_{\epsilon }^{sep}}{\partial x}\left({L}^{-}+{L}^{sep},t\right)\end{array}$
• ${D}_{eff}^{-}$ is the diffusion coefficient of the anode.
• ${D}_{eff}^{+}$ is the diffusion coefficient of the cathode.
• ${D}_{eff}^{sep}$ is the diffusion coefficient of the separator.
• $\frac{\partial {c}_{\epsilon }^{-}}{\partial x}\left({L}^{-},t\right)$ is the concentration gradient of the anode at the border between the anode and the separator.
• $\frac{\partial {c}_{\epsilon }^{sep}}{\partial x}\left({L}^{-},t\right)$ is the concentration gradient of the separator at the border between the anode and the separator.
• $\frac{\partial {c}_{\epsilon }^{+}}{\partial x}\left({L}^{-}+{L}^{sep},t\right)$ is the concentration gradient of the cathode at the border between the cathode and the separator.
• $\frac{\partial {c}_{\epsilon }^{sep}}{\partial x}\left({L}^{-}+{L}^{sep},t\right)$ is the concentration gradient of the separator at the border between the cathode and the separator.
This equation specifies the concentration at the boundaries between the electrodes and the current collectors. The flux is proportional to the flux at the current collector, which is equal to zero
because the block does not store any ions there. Hence the resulting flux at the interface is zero,
$\frac{\partial {c}_{\epsilon }^{-}}{\partial x}\left(0,t\right)=\frac{\partial {c}_{\epsilon }^{+}}{\partial x}\left({L}^{-}+{L}^{sep}+{L}^{+},t\right)=0,$
• $\frac{\partial {c}_{\epsilon }^{-}}{\partial x}\left(0,t\right)$ is the concentration gradient in the anode at the border between the anode and the leftmost current collector.
• $\frac{\partial {c}_{\epsilon }^{+}}{\partial x}\left({L}^{-}+{L}^{sep}+{L}^{+},t\right)$ is the concentration gradient in the cathode at the border between the cathode and the rightmost current
Similar to the solid phase, the block solves the differential equation by dividing the electrolyte into n sections of equal size. This equation expresses the concentration in the ith section with a
distance δx between sections, and is valid for all [1,M[s]-1] sections:
$\frac{\partial {c}_{\epsilon }}{\partial t}\left(x,t\right)={D}_{\epsilon ,eff}^{±}\frac{{c}_{i+1}+{c}_{i-1}-2{c}_{i}}{\partial {x}^{2}}+\frac{\left(1-{t}_{+}^{0}\right)J}{F}.$
The block discretizes the separator using the equation:
$\frac{\partial {c}_{\epsilon }}{\partial t}\left(x,t\right)={D}_{\epsilon ,eff}^{sep}\frac{c{}_{i+1}+{c}_{i-1}-2{c}_{i}}{\delta {x}^{2}}.$
To calculate the concentrations at the interfaces between the electrodes and the separator, the block applies all the boundary conditions. For example, for the interface between the anode and the
separator, the block applies the equation:
${c}_{\epsilon ,i=1}^{sep}=\frac{{\epsilon }_{\epsilon }^{-}}{{\epsilon }_{\epsilon }^{sep}}\frac{{c}_{\epsilon ,i=0}-{c}_{\epsilon ,i=-1}^{sep}}{\partial x}\partial x-{c}_{\epsilon ,i=0}.$
Mass Transport Overpotential in Liquid Phase
When the block is in the liquid phase, it uses the concentrations at the interfaces between the current collector and the anode and at the interfaces between the cathode and the current collector to
calculate the mass transport overpotential in the electrolyte using the equation,
${\eta }_{diffusion,\epsilon }=\frac{2RT}{F}\left(1-{t}_{\epsilon }^{0}\right)\mathrm{ln}\frac{{c}_{\epsilon }\left({L}^{-}+{L}^{sep}+{L}^{+},t\right)}{{c}_{\epsilon }\left(0,t\right)},$
• R is the universal gas constant.
• T is the temperature.
• F is Faraday's constant.
Ohmic Overpotential in Liquid Phase
When the block is in the liquid phase, it calculates the ohmic overpotential by linearly approximating the ionic current in each section of the battery. For the electrodes, the ionic current at the
interface to the current collector is zero. At the interface to the separator, the ionic current is equal to the electric current of the battery, I[batt]. Across the separator, the block approximates
the ionic current as constant and equal to the electric current applied to the battery. The block calculates the ohmic overpotential using the equation
${\eta }_{ohmic,\epsilon }=-\frac{{I}_{batt}}{2A}\ast \left(\frac{{L}^{+}}{{\kappa }_{eff}{}^{+}}+2\frac{{L}^{sep}}{{\kappa }_{eff}{}^{sep}}+\frac{{L}^{-}}{{\kappa }_{eff}{}^{-}}\right),$
where κ[eff] is the effective conductivity that the block calculates by using the Bruggeman coefficient. For more information about effective parameters, see the Effective Electrolyte Properties
Charge Transfer Overpotential
To model the charge transfer overpotential, this block uses the Butler-Volmer equation. The Butler-Volmer equation describes the relationship between the current density, j, and the overpotential, η,
which is the difference between the actual electrode potential and the thermodynamic equilibrium potential. The Butler-Volmer equation is
${J}^{±}\left(t\right)={j}_{0,k}\left(t\right)\left[\mathrm{exp}\left(\frac{\alpha {n}_{\epsilon }F}{RT}{\eta }^{±}\left(t\right)\right)-\mathrm{exp}\left(-\frac{\left(1-\alpha \right){n}_{\epsilon }
F}{RT}{\eta }^{±}\left(t\right)\right)\right],$
• α is the charge transfer coefficient for the oxidation and reduction.
• j[0] is the exchange current density.
Solving the equation for the electrode overpotential results in these equations:
$\begin{array}{l}{\eta }_{\text{kinetic,s}}=\frac{RT}{\alpha F}\mathrm{ln}\left({\xi }^{±}+\sqrt{{\left({\xi }^{±}\right)}^{2}+1}\right)\\ {\xi }^{±}=\frac{{j}^{±}}{2{a}^{±}{i}_{0}^{±}}\end{array}$
i^±[0] is the exchange current density in the anode and in the cathode and is equal to
${i}_{0}^{±}={k}^{±}{\left[{\overline{c}}_{\epsilon }^{±}\left({c}_{s,\mathrm{max}}^{±}-{c}_{s,surf}^{±}\right){c}_{s,surf}^{±}\right]}^{\alpha },$
• k is the charge transfer rate constant and is equal to the value of the Charge transfer rate constant for Anode parameter for the anode and to the value of the Charge transfer rate constant for
Cathode parameter for the cathode.
• ${\overline{c}}_{\epsilon }^{±}$ is the average electrolyte concentration.
• c[s,max] is the maximum electrode concentration.
• c[s,surf] is the electrode surface concentration.
To calculate the kinetic overpotential of the complete cell, the block subtracts the kinetic overpotential at the anode from the kinetic overpotential at the cathode:
${\eta }_{\text{kinetic,s}}={\eta }^{+}{}_{\text{kinetic,s}}-{\eta }^{-}{}_{\text{kinetic,s}}.$
Current Collector Resistance
This block models the current collector resistance as a single resistance. You can set the current collector resistance by specifying the Current collector resistance parameter.
Cell Voltage
To model the cell voltage, this block considers the potentials at the surfaces of each electrode, the overpotentials, and the voltage loss due to the current collector resistance by using the
$V\left(t\right)={\text{ocp}}^{+}\left({c}_{\text{surface,relative}}^{+}\right)-{\text{ocp}}^{-}\left({c}_{\text{surface,relative}}^{-}\right)+{\eta }_{\text{diffusion},\epsilon }+{\eta }_{ohmic,\
epsilon }+{\eta }_{\text{kinetic},s}+{\eta }_{ohmic}^{-}+{\eta }_{ohmic}^{+}+{I}_{\text{batt}}{R}_{\text{CurrentCollector}},$
• ocp^+(c^+[surface,relative]) is the open-circuit potential for the concentration at the surface of the cathode particle.
• ocp^-(c^-[surface,relative]) is the open-circuit potential for the concentration at the surface of the anode particle.
• η[diffusion,ε] is the mass transport overpotential in the electrolyte.
• η[ohmic,ε] is the ohmic overpotential in the electrolyte.
• η[kinetic,s] is the charge transfer overpotential in the electrodes.
• η^-[ohmic] is the ohmic overpotential in the anode.
• η^+[ohmic] is the ohmic overpotential in the cathode.
• I[batt] is the battery current.
• R[CurrentCollector] is the resistance of the current collector.
You can parameterize the open-circuit potential as table data by using the relative concentration as the breakpoints by specifying the Anode open-circuit potential, Cathode open-circuit potential,
and Normalized stoichiometry breakpoints parameters.
To calculate the relative concentration, the block considers the maximum concentration and the maximum and minimum stoichiometry of each electrode. The Anode maximum ion concentration and the Cathode
maximum ion concentration parameters represent the theoretically possible maximum concentration of each electrode. To obtain the achievable maximum and minimum concentrations, the block multiplies
the values of these parameters with the value of the Anode maximum stoichiometry, Anode minimum stoichiometry, Cathode maximum stoichiometry, and Cathode minimum stoichiometry parameters,
respectively. Then, the block calculates the relative concentration by using the equation
• c[s],[max] is the maximum concentration.
• N[max] is the maximum stoichiometry.
• N[min] is the minimum stoichiometry.
Effective Electrolyte Properties
Set the values of these parameters based on the microstructure of the porous electrodes you want to model:
• Diffusion coefficient of electrolyte — Set this parameter to the value of the diffusion coefficient of the electrolyte that influences the mass transport in the electrolyte.
• Electrolyte conductivity — Set this parameter to the value of the conductivity of the electrolyte.
To model this dependency, this block uses the Bruggeman correlation,
${\text{Parameter}}_{\text{effective}}={\text{Parameter}}_{\text{block}}\ast {\phi }_{\epsilon }{}^{\alpha },$
• φ[ε] is the volume fraction of the electrolyte. This value is equal to the value of the Volume fraction of electrolyte in anode, Volume fraction of electrolyte in separator, and Volume fraction
of electrolyte in cathode parameters, accordingly.
• α is the Bruggeman exponent. This value is equal to the value of the Anode Bruggeman exponent, Separator Bruggeman exponent, and Cathode Bruggeman exponent parameters, accordingly.
The block considers the temperature constant across the cell. These block parameters depend on the temperature of the cell:
• Diffusion coefficient of anode active material and Diffusion coefficient of cathode active material— These parameters are the diffusion coefficients of electrodes that influence the mass
transport in the electrodes.
• Diffusion coefficient of electrolyte — This parameter is the diffusion coefficient of electrolyte that influences the mass transport in the electrolyte.
• Electrolyte conductivity — This parameter is the conductivity of the electrolyte.
• Anode conductivity and Cathode conductivity — These parameters are the conductivity of the electrodes.
• Charge transfer rate constant for Anode and Charge transfer rate constant for Cathode — These parameters are the charge transfer rate constants of the electrodes.
To calculate the temperature-adjusted values of these parameters, the block uses the Arrhenius equation,
${\text{Parameter}}_{\text{T-adjusted}}={\text{Parameter}}_{\text{block}}\ast {e}^{\frac{{E}_{a}}{R}\left(\frac{1}{{T}_{ref}}-\frac{1}{T}\right)},$
• Parameter[block] is the value of the temperature-dependent parameters.
• E[a] is the activation energy and is equal to the value of the activation energy parameters in the Thermal settings.
• T[ref] is the value of the Arrhenius reference temperature parameter.
• T is the battery temperature.
Heat Generation of Battery
This block models the battery as a lumped thermal mass. The single-particle model calculates the irreversible heat generation that the overpotentials cause in the battery by using this equation:
$Q={I}_{batt}\left({\eta }_{\text{diffusion},\epsilon }+{\eta }_{ohmic,\epsilon }+{\eta }_{ohmic,s}^{-}+{\eta }_{ohmic,s}^{+}+{I}_{\text{batt}}{R}_{\text{CurrentCollector}}+{\eta }_{\text{kinetic},s}
+{\eta }^{+}{}_{\text{diffusion}}+{\eta }^{-}{}_{\text{diffusion}}\right).$
Public Variables
You can use the Probe block to access these variables in the Battery Single Particle block. The units are the default values.
• anodeModel.averageStoichiometry — Average stoichiometry in the anode.
• anodeModel.massTransportOverpotential — Mass transport overpotential, in volts.
• anodeModel.normalizedAverageStoichiometry — Average stoichiometry normalized to the minimum and maximum values.
• anodeModel.normalizedSurfaceStoichiometry — Surface stoichiometry normalized to the minimum and maximum values.
• anodeModel.ohmicOverpotential — Ohmic overpotential of the anode, in volts.
• anodeModel.shellConcentration — Concentration of the modeled shells, in mol/m^3. The number of shells is equal to the value of the Anode Shell Count parameter.
• anodeModel.shellStoichiometry — Stoichiometry of the modeled shells. The number of shells is equal to the Anode Shell Count parameter.
• anodeModel.surfaceConcentration — Concentration at the surface of the particle, in mol/m^3.
• anodeModel.surfacePotential — Potential at the surface of the particle, in volts.
• anodeModel.temperatureAdjustedConductivity — Conductivity adjusted to the battery temperature, in S/m.
• anodeModel.temperatureAdjustedDiffusionCoefficient — Diffusion coefficient adjusted to the battery temperature, in m^2/s.
• averageElectrolyteConcentration — Average concentration in the particle, in mol/m^3.
• batteryCurrent — Total current measured through the battery terminals, in amperes.
• batteryTemperature — Battery average temperature that the block uses for the table lookup of resistances and open-circuit voltage. If you set the Thermal model parameter to Constant temperature,
the batteryTemperature variable is equal to the specified temperature value. If you set the Thermal model parameter to Lumped thermal mass, the batteryTemperature variable is a differential state
that varies during the simulation.
• batteryVoltage — Battery terminal voltage, or the voltage difference between the positive and the negative terminals, in volts.
• cathodeModel.averageStoichiometry — Average stoichiometry in the cathode.
• cathodeModel.massTransportOverpotential — Mass transport overpotential, in volts.
• cathodeModel.normalizedAverageStoichiometry — Average stoichiometry normalized to the minimum and maximum values.
• cathodeModel.normalizedSurfaceStoichiometry — Surface stoichiometry normalized to the minimum and maximum values.
• cathodeModel.ohmicOverpotential — Ohmic overpotential of the cathode, in volts.
• cathodeModel.shellConcentration — Concentration of the modeled shells, in mol/m^3. The number of shells is equal to the value of the Anode Shell Count parameter.
• cathodeModel.shellStoichiometry — Stoichiometry of the modeled shells. The number of shells is equal to the Anode Shell Count parameter.
• cathodeModel.surfaceConcentration — Concentration at the surface of the particle, in mol/m^3.
• cathodeModel.surfacePotential — Potential at the surface of the particle, in volts.
• cathodeModel.temperatureAdjustedConductivity — Conductivity adjusted to the battery temperature, in S/m.
• cathodeModel.temperatureAdjustedDiffusionCoefficient — Diffusion coefficient adjusted to the battery temperature, in m^2/s.
• electrolyteModel.averageConcentration — Average concentration in the electrolyte across the whole cell, in mol/m^3.
• electrolyteModel.averageConcentrationAnode — Average concentration in the electrolyte inside the anode, in mol/m^3.
• electrolyteModel.averageConcentrationCathode — Average concentration in the electrolyte inside the cathode, in mol/m^3.
• electrolyteModel.averageConcentrationSeparator — Average concentration in the electrolyte inside the separator, in mol/m^3.
• electrolyteModel.concentrationAnode — Concentration of the modeled layers of the electrolyte in the anode, in mol/m^3. The number of elements is equal to the Electrolyte layer count of anode
• electrolyteModel.concentrationCathode — Concentration of the modeled layers of the electrolyte in the cathode, in mol/m^3. The number of elements is equal to the Electrolyte layer count of
cathode parameter.
• electrolyteModel.concentrationSeparator — Concentration of the modeled layers of the electrolyte in the separator, in mol/m^3. The number of elements is equal to the Electrolyte layer count of
electrolyte parameter.
• electrolyteModel.currentDensityAnode — Current density in the anode, in A/m^3.
• electrolyteModel.currentDensityCathode — Current density in the cathode, in A/m^3.
• electrolyteModel.diffusionCoefficientAnode — Temperature-adjusted effective diffusion coefficient of the electrolyte in the anode, in m^s/s.
• electrolyteModel.diffusionCoefficientCathode — Temperature-adjusted effective diffusion coefficient of the electrolyte in the cathode, in m^s/s.
• electrolyteModel.diffusionCoefficientSeparator — Temperature-adjusted effective diffusion coefficient of the electrolyte in the separator, in m^s/s.
• electrolyteModel.conductivityAnode — Temperature-adjusted effective conductivity of the electrolyte in the anode, in S/m.
• electrolyteModel.conductivityCathode — Temperature-adjusted effective conductivity of the electrolyte in the cathode, in S/m.
• electrolyteModel.effectiveConductivitySeparator — Temperature-adjusted effective conductivity of the electrolyte in the separator, in S/m.
• electrolyteModel.massTransportOverpotential — Mass transport overpotential of the electrolyte, in volts.
• electrolyteModel.ohmicOverpotential — Ohmic overpotential of the electrolyte, in volts.
• electrolyteModel.temperatureAdjustedConductivity — Temperature-adjusted conductivity of the electrolyte, in S/m.
• electrolyteModel.temperatureAdjustedDiffusionCoefficient — Temperature adjusted diffusion coefficient of the electrolyte, in m^s/s.
• heatGenerationRate — Total battery heat generation rate, in watts. The block calculates the heat generation rate by adding the resistive losses and the reversible heating contributions.
• power_dissipated — Resistive heat generation rate or dissipated power, in watts.
• reactionKineticsModel.chargeTransferOverpotential — Charge transfer overpotential of the battery, in volts.
• reactionKineticsModel.exchangeCurrentDensityAnode — Exchange current density in the anode, in C/(m^2*s).
• reactionKineticsModel.exchangeCurrentDensityCathode — Exchange current density in the cathode, in C/(m^2*s).
• reactionKineticsModel.temperatureAdjustedChargeTransferRateAnode — Temperature-adjusted charge transfer rate constant for the anode, in m^(5/2)/(mol^(1/2) * s).
• reactionKineticsModel.temperatureAdjustedChargeTransferRateCathode — Temperature-adjusted charge transfer rate constant for the cathode, in m^(5/2)/(mol^(1/2) * s).
• stateOfCharge — Battery state of charge obtained from Coulomb counting.
• thermalModel.batteryTemperature — Temperature of the battery, in K.
• thermalModel.cellTemperature — Temperature output by the cell.
• thermalModel.heatDissipationRate — Heat dissipation rate of the battery, in watts.
• thermalModel.heatGeneration — Heat that the battery generates, in watts.
• thermalModel.thermalMass — Thermal mass of the battery, in J/K.
+ — Positive terminal
Electrical conserving port associated with the positive battery terminal.
- — Negative terminal
Electrical conserving port associated with the negative battery terminal.
H — Battery thermal mass
Thermal conserving port associated with the thermal mass of the battery.
To edit block parameters interactively, use the Property Inspector. From the Simulink^® Toolstrip, on the Simulation tab, in the Prepare gallery, select Property Inspector.
Extrapolation method for all tables — Method of extrapolation for tables
Nearest (default) | Linear | Error
Extrapolation method for the table-based parameters:
• Linear — Estimate values beyond the data by creating a tangent line at the end of the known data and extending it beyond that limit.
• Nearest — Extrapolate a value at a query point that is the value at the nearest sample grid point.
• Error — Return an error if the value goes beyond the known data. If you select this option, the block does not use extrapolation.
Programmatic Use
To set the block parameter value programmatically, use the set_param function.
Parameter: ExtrapolationMethod
Values: "nearest" (default) | "linear" | "error"
Anode thickness — Thickness of anode
3.4E-5 m (default) | positive scalar
Electrolyte layer count of anode — Electrolyte layer count of anode
10 (default) | scalar greater than 3
Electrolyte layer count of the anode.
Separator thickness — Thickness of separator
2.5E-5m (default) | positive scalar
Thickness of the separator.
Electrolyte layer count of separator — Electrolyte layer count of separator
5 (default) | scalar greater than 3
Electrolyte layer count of the separator.
Cathode thickness — Thickness of cathode
8E-5m (default) | positive scalar
Thickness of the cathode.
Electrolyte layer count of cathode — Electrolyte layer count of cathode
10 (default) | scalar greater than 3
Electrolyte layer count of the cathode.
Electrode plate area — Area of electrode plate
1.8E-1m^2 (default) | positive scalar
Area of the electrode plate.
Anode particle radius — Particle radius of anode
5E-6m (default) | positive scalar
Particle radius of the anode.
Anode shell count — Number of shells for anode
10 (default) | positive scalar greater than 3
Number of shells for the anode.
Cathode particle radius — Particle radius of cathode
5E-8m (default) | positive scalar
Particle radius of the cathode.
Cathode shell count — Number of shells for cathode
10 (default) | positive scalar greater than 3
Number of shells for the cathode.
Electrodes Properties
Volume fraction of anode active material — Volume fraction of active material in anode
0.58 (default) | scalar in the range (0,1)
Volume fraction of the active material in the anode.
Volume fraction of cathode active material — Volume fraction of active material in cathode
0.374 (default) | scalar in the range (0,1)
Volume fraction of the active material in the cathode.
Anode open-circuit potential — Open-circuit potential of anode
[.484; .335; .253; .209; .2; .181; .165; .151; .139; .131; .128; .125; .124; .124; .123; .122; .119; .117; .111; .108; .101; .098; .095; .093; .092; .088; .088; .088; .088; .087; .087; .087; .085;
.082; .079; .071; .064; .049; .026] V (default) | vector of positive elements
Open-circuit potential of the anode. The size of this parameter must be equal to the size of the Normalized stoichiometry breakpoints parameter.
Cathode open-circuit potential — Open-circuit potential of cathode
[3.81; 3.608; 3.512; 3.478; 3.459; 3.453; 3.45257; 3.45214; 3.45171; 3.45128; 3.45085; 3.45042; 3.44999; 3.44956; 3.44913; 3.4487; 3.44827; 3.44784; 3.44741; 3.44698; 3.44655; 3.44612; 3.44569;
3.44526; 3.44483; 3.4444; 3.44397; 3.44354; 3.44311; 3.44268; 3.44225; 3.44182; 3.44139; 3.44096; 3.44053; 3.44; 3.412; 3.304; 2.968] V (default) | vector of positive elements
Open-circuit potential of the cathode. The size of this parameter must be equal to the size of the Normalized stoichiometry breakpoints parameter.
Normalized stoichiometry breakpoints — Stoichiometry breakpoints for open-circuit potential of electrode
linspace(0.025, 0.975, 39) (default) | vector of elements in the range [0,1]
Stoichiometry breakpoints for the open-circuit potential of the electrode. The size of this parameter must be equal to the size of the Anode open-circuit potential and Cathode open-circuit potential
A value of 0 means that the stoichiometry of the particle is equal to the minimum stoichiometry of the electrode, as you specify in the Anode minimum stoichiometry and Cathode minimum stoichiometry
parameters. A value of 1 means that the stoichiometry of the particle is equal to the maximum stoichiometry of the electrode, as you specify in the Anode maximum stoichiometry and Cathode maximum
stoichiometry parameters.
Overall, the block calculates the values of the normalized stoichiometry by using this equation:
Anode maximum ion concentration — Maximum ion concentration in anode
30555 mol/m^3 (default) | positive scalar
Maximum ion concentration in the anode.
Anode maximum stoichiometry — Maximum stoichiometry in anode
0.811 (default) | positive scalar
Maximum stoichiometry in the anode. The value of this parameter must be greater than the value of the Anode minimum stoichiometry parameter.
Anode minimum stoichiometry — Minimum stoichiometry in anode
0.0132 (default) | positive scalar
Minimum stoichiometry in the anode. The value of this parameter must be less than the value of the Anode maximum stoichiometry parameter.
Cathode maximum ion concentration — Maximum ion concentration in cathode
22806 mol/m^3 (default) | positive scalar
Maximum ion concentration in the cathode.
Cathode maximum stoichiometry — Maximum stoichiometry in cathode
0.74 (default) | positive scalar
Maximum stoichiometry in the cathode. The value of this parameter must be greater than the value of the Cathode minimum stoichiometry parameter.
Cathode minimum stoichiometry — Minimum stoichiometry in cathode
0.035 (default) | positive scalar
Minimum stoichiometry in the cathode. The value of this parameter must be less than the value of the Cathode maximum stoichiometry parameter.
Diffusion coefficient of anode active material — Diffusion coefficient of active material in anode
3E-15 m^2/s (default) | positive scalar
Diffusion coefficient of the active material in the anode.
Diffusion coefficient of cathode active material — Diffusion coefficient of active material in cathode
5.9E-20 m^2/s (default) | positive scalar
Diffusion coefficient of the active material in the cathode.
Anode conductivity — Conductivity of anode
100 S/m (default) | positive scalar
Conductivity of the anode.
Cathode conductivity — Conductivity of cathode
0.5 S/m (default) | positive scalar
Conductivity of the cathode.
Current collector resistance — Resistance of current collector
0 Ohm (default) | nonnegative scalar
Resistance of the current collector.
Electrolyte Properties
Volume fraction of electrolyte in anode — Fraction of volume of electrolyte in anode
0.3874 (default) | scalar in the range (0,1)
Fraction of the volume of the electrolyte in the anode.
Volume fraction of electrolyte in separator — Fraction of volume of electrolyte in separator
0.45 (default) | scalar in the range (0,1)
Fraction of the volume of the electrolyte in the separator.
Volume fraction of electrolyte in cathode — Fraction of volume of electrolyte in cathode
0.5725 (default) | scalar in the range (0,1)
Fraction of the volume of the electrolyte in the cathode.
Electrolyte conductivity — Conductivity of electrolyte
0.29 S/m (default) | positive scalar
Conductivity of the electrolyte.
Anode Bruggeman exponent — Bruggeman exponent of anode
1.5 (default) | scalar greater than or equal to 1
Bruggeman exponent of the anode.
Separator Bruggeman exponent — Bruggeman exponent of separator
1.5 (default) | scalar greater than or equal to 1
Bruggeman exponent of the separator.
Cathode Bruggeman exponent — Bruggeman exponent of cathode
1.5 (default) | scalar greater than or equal to 1
Bruggeman exponent of the cathode.
Transference number — Fraction of total electric current that ion carries in electrolyte
0.363 (default) | scalar in the range [0,1]
Fraction of the total electric current that an ion carries in an electrolyte.
Diffusion coefficient of electrolyte — Diffusion coefficient of electrolyte
2E-10 m^2/s (default) | positive scalar
Diffusion coefficient of the electrolyte that the block uses inside the Fick's Law equation.
Reaction Kinetics
Charge transfer rate constant for Anode — Rate constant of charge transfer for anode
8.8E-11 m^(5/2)/(mol^(1/2)*s) (default) | positive scalar
Rate constant of the charge transfer for the anode. The block uses this value in the Butler-Volmer equation to model the electrochemical kinetics at the electrode-electrolyte interface.
Charge transfer rate constant for Cathode — Rate constant of charge transfer for cathode
2.8E-13 m^(5/2)/(mol^(1/2)*s) (default) | positive scalar
Rate constant of the charge transfer for the cathode. The block uses this value in the Butler-Volmer equation to model the electrochemical kinetics at the electrode-electrolyte interface.
Battery thermal mass — Thermal mass of battery associated with thermal port
77 J/K (default) | positive scalar
Thermal mass at the thermal port H. This parameter represents the energy required to raise the temperature of the thermal port by one kelvin.
Arrhenius reference temperature — Arrhenius reference temperature
298.15 K (default) | positive scalar
Arrhenius reference temperature.
Activation energy for diffusion in anode active material — Activation energy for diffusion inside active material of anode
39000 J/mol (default) | positive scalar
Activation energy for the diffusion inside the active material of the anode.
Activation energy for diffusion in cathode active material — Activation energy for diffusion inside active material of cathode
35000 J/mol (default) | positive scalar
Activation energy for the diffusion inside the active material of the cathode.
Activation energy for diffusion in electrolyte — Activation energy for diffusion inside electrolyte
26600 J/mol (default) | positive scalar
Activation energy for the diffusion inside the electrolyte.
Activation energy for conduction in anode — Activation energy for conduction inside anode
3000 J/mol (default) | positive scalar
Activation energy for the conduction inside the anode.
Activation energy for conduction in cathode — Activation energy for conduction inside cathode
3000 J/mol (default) | positive scalar
Activation energy for the conduction inside the cathode.
Activation energy for conduction in electrolyte — Activation energy for conduction inside electrolyte
11000 J/mol (default) | positive scalar
Activation energy for the conduction inside the electrolyte.
Activation energy for charge transfer in Anode — Activation energy for charge transfer inside anode
13000 J/mol (default) | positive scalar
Activation energy for the charge transfer inside the anode.
Activation energy for charge transfer in Cathode — Activation energy for charge transfer inside cathode
20000 J/mol (default) | positive scalar
Activation energy for the charge transfer inside the cathode.
[1] Prada, E., D. D. Domenico, Y. Creff, J. Bernard, V. Sauvant-Moynot, and F. Huet. “Simplified Electrochemical and Thermal Model of LiFePO[4]-Graphite Li-Ion Batteries for Fast Charge
Applications.” Journal of The Electrochemical Society 159, no. 9 (August 2012): A1508–A1519. https://doi.org/10.1149/2.064209jes.
[2] Kemper, P. and D. Kum. “Extended Single Particle Model of Li-Ion Batteries Towards High Current Applications”. In 2013 IEEE Vehicle Power and Propulsion Conference (VPPC), 1–6, 2013. https://
[3] Weaver, T., A. Allam, and S. Onori. “A Novel Lithium-Ion Battery Pack Modeling Framework - Series-Connected Case Study.” In 2020 American Control Conference (ACC), 365–372. Denver, CO, USA: IEEE,
2020. https://doi.org/10.23919/ACC45564.2020.9147546.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.
Version History
Introduced in R2024a
See Also | {"url":"https://it.mathworks.com/help/simscape-battery/ref/batterysingleparticle.html","timestamp":"2024-11-07T21:53:25Z","content_type":"text/html","content_length":"227627","record_id":"<urn:uuid:59c88e0b-bc45-4c9c-91c4-a25997029b40>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00201.warc.gz"} |
ctrttp - Linux Manuals (3)
ctrttp (3) - Linux Manuals
ctrttp.f -
subroutine ctrttp (UPLO, N, A, LDA, AP, INFO)
CTRTTP copies a triangular matrix from the standard full format (TR) to the standard packed format (TP).
Function/Subroutine Documentation
subroutine ctrttp (characterUPLO, integerN, complex, dimension( lda, * )A, integerLDA, complex, dimension( * )AP, integerINFO)
CTRTTP copies a triangular matrix from the standard full format (TR) to the standard packed format (TP).
CTRTTP copies a triangular matrix A from full format (TR) to standard
packed format (TP).
UPLO is CHARACTER*1
= 'U': A is upper triangular;
= 'L': A is lower triangular.
N is INTEGER
The order of the matrices AP and A. N >= 0.
A is COMPLEX array, dimension (LDA,N)
On entry, the triangular matrix A. If UPLO = 'U', the leading
N-by-N upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading N-by-N lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
AP is COMPLEX array, dimension ( N*(N+1)/2 ),
On exit, the upper or lower triangular matrix A, packed
columnwise in a linear array. The j-th column of A is stored
in the array AP as follows:
if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
September 2012
Definition at line 105 of file ctrttp.f.
Generated automatically by Doxygen for LAPACK from the source code. | {"url":"https://www.systutorials.com/docs/linux/man/3-ctrttp/","timestamp":"2024-11-15T03:39:14Z","content_type":"text/html","content_length":"9019","record_id":"<urn:uuid:f6affbde-69d3-45a2-9e21-b94c449ccc85>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00275.warc.gz"} |
Image Coding Using Zero Tree Wavelet
Image Coding Using Zero Tree Wavelet
Published on Apr 02, 2024
Image compression is very important for efficient transmission and storage of images. Embedded Zerotree Wavelet (EZW) algorithm is a simple yet powerful algorithm having the property that the bits in
the stream are generated in the order of their importance.
Image compression can improve the performance of the digital systems by reducing time and cost in image storage and transmission without significant reduction of the image quality. For image
compression it is desirable that the selection of transform should reduce the size of resultant data set as compared to source data set. EZW is computationally very fast and among the best image
compression algorithm known today.
This paper proposes a technique for image compression which uses the Wavelet-based Image Coding. A large number of experimental results are shown that this method saves a lot of bits in transmission,
further enhances the compression performance. This paper aims to determine the best threshold to compress the still image at a particular decomposition level by using Embedded Zero-tree Wavelet
encoder. Compression Ratio (CR) and Peak-Signal-to-Noise (PSNR) is determined for different threshold values ranging from 6 to 60 for decomposition level 8.
Introduction of Image Coding Using Zero Tree Wavelet
Most natural images have smooth colour variations, with the fine details being represented as sharp edges in between the smooth variations. Technically, the smooth variations in colour can be termed
as low frequency variations and the sharp variations as high frequency variations. The low frequency components (smooth variations) constitute the base of an image, and the highfrequency components
(the edges which give the detail)add upon them to refine the image, thereby giving a detailed image. Hence, the smooth variations are demanding more importance than the details. Separating the smooth
variations and details of the image can be done in many ways. One such way is the decomposition of the image using a Discrete Wavelet Transform (DWT).
Wavelets are being used in a number of different applications. The practical implementation of wavelet compression schemes is very similar to that of subband coding schemes. As in the case of subband
coding, the signal is decomposed using filter banks. In a discrete wavelet transform, an image can be analyzed by passing it through an analysis filter bank followed by a decimation operation. This
analysis filter bank, which consists of a low pass and a high pass filter at each decomposition stage, is commonly used in image compression.
When a signal passes through these filters, it is split into two bands. The low pass filter, which corresponds to an averaging operation, extracts the coarse information of the signal. The high pass
filter, which corresponds to a differencing operation, extracts the detail information of the signal. The output of the filtering operations is then decimated by two. A two-dimensional transform can
be accomplished by performing two separate one-dimensional transforms. First, the image is filtered along the x-dimension using low pass and high pass analysis filters and decimated by two.
Low pass filtered coefficients are stored on the leftpart of the matrix and high pass filtered on the right.Because of decimation, the total size of the transformed image is same as the original
image. Then, it is followed by filtering the sub-image along the y-dimension and decimated by two. Finally, the image has been split into four bands denoted by LL, HL, LH, and HH, after one level of
decomposition. The LL band is again subject to the same procedure.
Quantization :
Quantization refers to the process of approximating the continuous set of values in the image data with a finite, preferably small, set of values. The input to a quantizer is the original data and
the output is always one among a finite number of levels. The quantizer is a function whose set of output values are discrete and usually finite. Obviously, this is a process of approximation and a
good quantizer is one which represents the original signal with minimum loss or distortion.
here are two types of quantization: scalar quantization and vector quantization. In scalar quantization, each input symbol is treated in producing the output while in vector quantization the input
symbols are clubbed together in groups called vectors, and processed to give the output. This clubbing of data and treating them as a single unit,increases the optimality of the vector quantizer, but
at thecost of increased computational complexity
Image coding utilizing scalar quantization on hierarchical structures of transformed images has been a very effective and computationally simple technique. Shapiro was the first to introduce such a
technique with his EZW [13] algorithm. Different variants of this technique have appeared in the literatures which providean improvement over the initial work. Said & Pearlman[1] successively
improved the EZW algorithm byextending this coding scheme, and succeeded in presenting a different implementation based on a setpartitioning sorting algorithm. This new coding scheme, called the
SPIHT [1], provided an even better performance than the improved version of EZW.
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Micro Electronic Pill, Microwave Superconductivity, MILSTD 1553B, MOBILE IPv6, Mobile Virtual Reality Service, Magneto-Optical Current Transformer Technology MOCT, Multisensor Fusion and Integration,
Navbelt and Guidecane, Non Visible Imaging, Nuclear Batteries-Daintiest Dynamos, Optical Burst Switching, Optical Ethernet, Optical Networking and Dense Wavelength Division Multiplexing, Optical
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Computing, Poly Fuse, Polymer Memory | {"url":"https://www.seminarsonly.com/electronics/Image-Coding-Using-Zero-Tree-Wavelet.php","timestamp":"2024-11-08T18:21:58Z","content_type":"text/html","content_length":"21336","record_id":"<urn:uuid:7111e901-9963-4fbe-a797-d468b545f03b>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00367.warc.gz"} |
Implement stack using singly linked list - Quescol
Implement stack using singly linked list
Stack can be implemented by both array and linked list.
There is some limitation or you can say that drawback in stack implementation using an array is that the array must be declared with some fixed size. In case if the size of the array becomes small to
perform the required operation so this is a problem or if we declare the too larger size of an array in advance then it is also a wastage of memory. So if the array size cannot be determined in
advance, then we have an alternative solution that is linked list representation.
The storage requirement of linked representation of the stack with n elements is O(n), and the typical time required for the operations is O(1).
In a linked stack, every node has two parts—one that stores data and another that stores the address of the next node. The linked list allocates the memory dynamically. However, time complexity in
both the scenario is the same for all the operations i.e. push, pop, and peek. The START pointer of the linked list is used as TOP. All insertions and deletions are done at the node pointed by TOP.
If TOP = NULL, then it indicates that the stack is empty.
A linked stack supports all the three stack operations, that is, push, pop, and peek.
1). Push Operation
The push operation is used to insert an element into the stack. The new element is added at the topmost position of the stack.
Push Operation algorithm
Step 1 – Allocate memory to create a newNode with given value and name it as NEW_NODE.
Step 2 – Check whether TOP == NULL of Stack
Step 3 – If TOP == NULL means empty, then set newNode → next = NULL and TOP = NEW_NODE.
Step 4 – If TOP != NULL, then set newNode → next = top and TOP = NEW_NODE.
Step 5 – END
2). POP Operation
POP operation is performed on stack to remove item from the stack.
POP Operation algorithm
Step 1 – Check whether TOP == NULL of Stack.
Step 2 – If TOP == NULL then print “Stack is Empty” and terminate the function
Step 3 – If TOP != NULL, then define a Node pointer ‘temp’ and set it to ‘top’.
Step 4 – Then set ‘top = top → next’.
Step 5 – Finally, delete ‘temp’. (free(temp)).
3). PEEK Operation
Using peek operation topmost element of the stack will be retrieved without deleting it from the stack.
PEEK Operation Algorithm
Step 1 – Check whether TOP == NULL of Stack.
Step 2 – If TOP == NULL then print “Stack is Empty” and terminate the function
Step 3 – If TOP != NULL, then display top->data.
Step 4 – END
4). Display Operation
Display operation is used to print all the elements of stack.
Display Operation Algorithm
Step 1 – Check whether TOP == NULL of Stack.
Step 2 – If TOP == NULL then print “Stack is Empty” and terminate the function
Step 3 – If TOP != NULL, then define a Node pointer ‘ptr’ and initialize with top.
Step 4 – Display ‘temp → data’ util temp → next != NULL.
Step 5 – Finally Display ‘temp → data’.
Java Program for Stack implementation using linked list
public class Main {
static class StackNode {
int data;
StackNode next;
StackNode(int data) {
this.data = data;
StackNode root;
public void push(int data)
StackNode newNode = new StackNode(data);
if (root == null) {
root = newNode;
else {
StackNode temp = root;
root = newNode;
newNode.next = temp;
System.out.println("Item pushed into stack = "+data);
public int pop()
int popped;
if (root == null) {
System.out.println("Stack is Empty");
return 0;
else {
popped = root.data;
root = root.next;
return popped;
public int peek()
if (root == null) {
System.out.println("Stack is empty");
return Integer.MIN_VALUE;
else {
return root.data;
public static void main(String[] args)
Main m = new Main();
System.out.println("Item popped from stack = "+m.pop());
System.out.println(m.peek()+ " Returned by Peek operation"); | {"url":"https://quescol.com/data-structure/implement-stack-using-singly-linked-list","timestamp":"2024-11-05T15:44:15Z","content_type":"text/html","content_length":"87779","record_id":"<urn:uuid:25c8f450-4c7d-4707-b85d-36e4f0d03595>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00894.warc.gz"} |