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I'm a Science Olympiad coach trying to optimize the performance of our "Scrambler", a car which must be accelerated by only a falling mass.
Most competitors simply tie a weight to a string and route that string over a set of pulleys (using no mechanical advantage to convert the vertical falling acceleration horizontal. Even if the car
had no mass and there were no loss of energy due to friction, that would limit the final departure velocity of the car to the final velocity of the falling weight:
v(final) = SQRT(2*g*h) = SQRT(2*9.8*1) = 4.427 m/s Intuitively, we believe we can get the car to depart faster if we store much of the KE (Kinetic Energy) of the falling mass into PE of the stretched
rubber band, which would allow us to accelerate a car of much less mass at a rate considerable more than g (gravity).
Here's are target configuration, so far:
Mass to drop: 2Kg (max allowable)
Height of fall: 1 meter (max allowable) Mass of car we will accelerate:
0.5 Kg Max length of our launch device: 1 meter Useful details about our launch device:
1) Held at 1m high, our 2Kg mass is attached to a string that goes over a "top pulley'
2) Down on the floor, sits our car, which (for simplification) is 1 meter long.
3) Directly overhead and inline with the length of the car is a suspended 'slide rod' running the same 1 meter length of the car.
4) Hanging down from the rod is a sliding 'push block'. The bottom of this push block touches the 'back bumper' of our car, ready to push the car forward as it slides forward on the overhead rod.
(NOTE: the actual length of the 'push' doesn't need to be a full 1 meter, and likely won't be)
5) Connection between falling mass and car: A string is attached to the 2Kg mass, immediate goes up and over a 'top pulley' then extends down to a 'bottom pulley' right at the front of the car and
the front end of the slide rod. The string then continues horizontally to the other end of the slide rod over the back end of the car and is attached to a 'push block'
6) The 'push block' hangs from and slides along the slide rod and extends down just behind the 'back bumper' of the car. When the string gets pulled, the slide block is pulled forward, in turn,
pushing the car forward as the 'launch'.
That was all simple, but needed as base information. Here's where it gets interesting!
1) The car is LOCKED into place and is released ONLY AFTER the falling mass arrives at the BOTTOM of its fall.
2) Very important: Our connection 'string' is not ALL string. inserted in between two end lengths is a strong rubber band.
3) Also very important: The string connection is 'slack' enough that the band doesn't not begin to stretch until the mass has fallen a significant % of the falling distance (we can adjust this). in
this way we can allow the falling mass to accumulate KE (Kinetic Energy) before 'capturing' and storing that energy at PE in the rubber bank).
4) Through adjusting both the strength of the rubber bank and the % of the falling distance we use for KE-to-PE storage, we in tend to decelerate the falling weight back to near-zero by the time is
reaches the floor. In this way, we will hopefully maximize the transfer of energy into the rubber band.
5) With the slight amount of inertia left in the falling/decelerated mass, this mass will fall onto a 'trap' that a) captures it, so it can't proceed back upward in response to the rubber band's
'pull', and b) releases the trigger holding car in place, thus allowing the stored energy to now pull on the slide block, pushing and accelerating the car.
HOW WE THING WE CAN OUT-PERFORM (OUT-ACCELERATE) OUR COMPETITION: By waiting to accumulate significant KE from the fall before converting it to PE in the band, we believe we can store a significant
stretch force to accelerate the car. And, since our car is only 1/4 the mass of the falling weight, we should be able to use that force to accelerate the at a rate much higher than gravity's 9.8 m/s^
WHERE WE NEED YOUR HELP: Obviously, if we can get our heads around all the necessary formulae, we can calculate the resulting FINAL LAUNCH VELECITY OF THE CAR, and by experimenting with different
condition variables, we can select the configuration for optimal performance.
So, getting to that calculation of Final Launch Velocity is our goal, and the target of our request for help.
Here are some of the thinking we've gone through, the formulae we've used, and some of the control limits we've settled on to keep our variables from getting out of hand:
1) We know we're starting with a PE = mgh = 2x9.8x1 = 19.6 Joules
2) We know the most straightforward way to increase our potential for faster acceleration is to lighten the car, but let's assume that 1/2 Kg is as good as we can get.
3) WE THINK WE'RE OUR THINKING IS SOUND HERE: Let's assume that we could precisely adjust the appropriate MOE (modulus of elasticity) and length of the rubber band so that we could always get the
falling weight to decelerate back to zero just as it got to the floor, no matter when we allow the string to become taut and the band to begin to stretch. We're thinking that, ignoring friction and
other inefficiencies, we'd always end up storing the same 19.6 Joules of PE in the rubber band. Two examples:
a) Weight falls .5 meter before band begins to stretch:
V(final) = SQRT(2 x g x distance) = 2 x 9.8 x 0.5 = 3.13 m/s
Accumulated KE = 1/2 x m x v^2 = 1/2 x 2 x 3.13m/s = 9.8 Joules
Remaining PE = mgh = 2 x 9.8 x 0.5 = 9.8 Joules.
Assuming band strength to decelerate the mass back to 0 when
reaching the floor, the energy stored in the band = 19.6 Joules
b) Weight falls .75 meter before band begins to stretch:
V(final) = SQRT(2 x g x distance) = 2 x 9.8 x 0.75 = 3.834 m/s
Accumulated KE = 1/2 x m x v^2 = 1/2 x 2 x 3.834m/s = 14.7 Joules
Remaining PE = mgh = 2 x 9.8 x 0.25 = 4.9 Joules.
Assuming band strength to decelerate the mass back to 0 when
reaching the floor, the energy stored in the band = 19.6 Joules
So, unless we've miscalculated, we believe there's no "best time" to start capturing energy into the band, assuming, again, that we capture all the energy by perfectly decelerating the falling mass
back to zero at the bottom.
4) NOW, HERE'S A POSSIBLE AREA OF MISPERCEPTION:
While we've concluded that different 'stretch length / MOE' combinations will all result in the same 19.6 Joule energy transfer to our rubber band in our 'ideal' machine', it's also become obvious
that, the longer we wait before beginning to stretch the rubber band, the greater will be the INITIAL force available to us to use to accelerate the car, and we're not sure if that will make any
difference in how fast we can get our car moving.
1) Example 1: the 2Kg mass falls 75cm (.75m) before the string goes taut and the 'well-adjusted' band slows the mass to a stop at zero height, having
stored 19.6 PE Joules. When this energy is used to accelerate the car, all
19.6 Joules will be expended within the 25 CENTIMETERS of 'un-stretching' it take to bring the band back to a 'limp' condition
2) Example 1: the 2Kg mass falls 50cm (.5m) before the string goes taut and the 'well-adjusted' band slows the mass to a stop at zero height, having
stored 19.6 PE Joules. When this energy is used to accelerate the car, all
19.6 Joules will be expended within the 50 CENTIMETERS of 'un-stretching' it take to bring the band back to a 'limp' condition.
To us, it just stands to reason that the magnitude of pulling force available from the shorter band will be much greater than the initial force available from the longer band. Since the shorter band
will release/transfer all of it 19.6 Joules of energy in just 25cm while the longer band will take 50cm to do that,
That larger initial force pulling the 1/2 Kg car will, at first, impart a greater initial acceleration, but will dissipate more quickly.
Given that the slide rod is 1 meter long, we can keep pulling the car until for the full distance needed to "un-stretch" the band back to rest, regardless of whether we use 25cm stretch or the 50cm
stretch. So, ultimately, ignoring inefficiencies (friction, etc.), the ultimate WORK available to be performed on the car is the same, derived from the 19.6 Joules of energy. But, somehow, we're not
sure that necessarily means both stretch lengths will impart the same acceleration and highest launch velocity as one another. In the past, with similar but different car-competition events, we've
observed that the more quickly we can impart acceleration force on our car, the greater the ultimate final launch velocity we can accomplish.
SO, OUR QUESTION IS THIS: Given the greater initial pulling force available from the shorter, stronger 25cm-stretch band and the resulting quicker initial acceleration, would we be able to reach a
greater launch velocity that with the weaker 50cm-stretch band, or (ignoring inefficiencies) will the final launch velocities be the same?
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A right triangle has sides A, B, and C. Side A is the hypotenuse and side B is also a side of a rectangle. Sides A, C, and the side of the rectangle adjacent to side B have lengths of 7 , 6 , and 15 , respectively. What is the rectangle's area? | Socratic
A right triangle has sides A, B, and C. Side A is the hypotenuse and side B is also a side of a rectangle. Sides A, C, and the side of the rectangle adjacent to side B have lengths of #7 #, #6 #, and
#15 #, respectively. What is the rectangle's area?
1 Answer
The above (not to scale) picture contains the information given in the problem. In many geometry problems, drawing a picture is a good way to start.
The area of the rectangle is the product of the lengths of its sides, in this case $15 B$. To solve for $B$, we can use the Pythagorean theorem, which states that the square of the length of the
hypotenuse of a right triangle is equal to the sum of the squares of its legs. In this case, that translates to ${A}^{2} = {B}^{2} + {C}^{2}$
Substituting in the given values for $A$ and $C$, we obtain
$49 = {B}^{2} + 36$
$\implies {B}^{2} = 13$
$\implies B = \sqrt{13}$
Thus the area of the rectangle is $15 \sqrt{13}$
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How do you calculate overtime pay in Texas? | 2024
Calculating overtime pay accurately is crucial for both employers and employees to ensure fair compensation and compliance with labor laws. In Texas, the Fair Labor Standards Act (FLSA) governs the
overtime regulations of the state. Understanding the correct methods for calculating overtime helps payroll errors, and potential disputes, and ensures adherence to legal requirements.
This article provides a comprehensive guide on how to calculate overtime pay in Texas, addressing various scenarios and providing practical examples to help navigate these calculations effectively.
This Article Covers
Understanding Overtime in Texas
Which Overtime Laws Apply in Texas?
Texas state laws do not cover overtime, so federal regulations apply. The federal Fair Labor Standards Act (FLSA) governs overtime with the following key points:
• Non-exempt employees must receive overtime pay for any hours worked beyond 40 in a workweek.
• Overtime pay is required to be at one and a half times the regular pay rate.
• Working on weekends, nights, or holidays does not automatically qualify as overtime unless it involves extra hours worked during these periods.
• There are no limits on the number of hours an employer can require an employee to work.
• The FLSA operates on a weekly basis, defined as seven consecutive 24-hour periods. Weekly schedules do not necessarily align with the calendar week. Employers have the flexibility to establish
workweeks for their employees based on their business needs.
How are Overtime Laws in Texas Different from Federal laws?
Overtime laws in Texas are governed by federal regulations under the FLSA, as the state does not have specific overtime laws. Both Texas and federal laws require non-exempt employees to receive 1.5
times their regular pay rate for hours worked beyond 40 in a workweek. Texas follows the FLSA’s definition of a workweek, which can start on any day.
How Much is Overtime Pay in Texas?
In Texas, overtime pay is calculated at 1.5 times the employee’s regular rate of pay for any hours worked over 40 in a workweek. This rate is mandated by the FLSA and applies to most non-exempt
For example, the minimum wage in Texas is $7.25, and the overtime rate would be $10.88 per hour ($7.25 x 1.5) for any hours worked beyond the 40-hour threshold.
If you want to learn more, check out our guide on Texas overtime laws.
Who is Eligible for Overtime Pay in Texas?
Employees in Texas are classified as either exempt or non-exempt when it comes to overtime. Non-exempt workers, who are usually paid hourly and perform jobs involving manual labor or customer
service, are eligible for overtime pay, provided they are at least 16 years old.
Check out our guide on overtime rights in Texas.
Who is Exempt from Overtime Pay in Texas?
In Texas, overtime regulations are governed by the FLSA since there are no state-specific overtime laws. The FLSA outlines various employee categories that are exempt from overtime pay.
Exempt employees work in “white-collar” positions such as professional, administrative, or executive roles or as salespersons. However, it is not just the job title that determines exemption.
Employees must also meet specific criteria assessed through three tests:
• Salary Basis Test: Employees must receive a consistent salary, irrespective of hours worked or work completed, meaning they should be salaried rather than hourly.
• Salary Test: The employee’s salary must meet a minimum threshold. As of 2024, the FLSA has set the minimum threshold at $844 per week or $43,888 annually.
• Duties Test: The employee’s primary duties must involve administrative, professional, or executive functions requiring discretion and independent judgment.
Other job categories exempt from overtime include:
• Airline employees
• Babysitters
• Commissioned sales employees
• Computer professionals
• Drivers and loaders
• Live-in domestic employees
• Farmworkers on small farms
• Federal criminal investigators
• Fishermen
• Outside sales employees
• Railroad employees
• Salesmen and mechanics
• Switchboard operators
What is the Regular Rate of Pay in Texas?
The regular rate of pay is the amount an employee earns per hour worked and must be at least the set minimum wage. In Texas, the minimum wage is $7.25 per hour. For hourly employees, calculating this
rate is straightforward as it matches their standard hourly wage. However, for other types of employees, to determine their regular hourly rate:
• Salaried Employees: Calculate the weekly salary by the standard 40-hour workweek to find the hourly rate.
• Piecework or Commission Employees: There are three ways to calculate their regular rate: using the rate per piece or commission, dividing the total earnings for the workweek by the number of
hours worked, and for group work, calculating the group rate by dividing the total number of pieces by the number of people in the group, then multiply this rate by the hours each person worked
to determine their hourly rate.
Overtime for Hourly and Salaried Employees in Texas
How do you Calculate Overtime for Hourly Employees in Texas?
To calculate overtime hours for hourly employees in Texas:
• Determine the employee’s regular hourly rate: This is typically their base hourly wage.
• Calculate the employee’s overtime rate: Multiply the regular hourly rate by 1.5 to determine the overtime rate. For example, if an employee’s regular rate is $20 per hour, their overtime rate
would be $30 per hour ($20 x 1.5).
• Identify overtime hours: count the total number of hours worked beyond 40 in a workweek. Only these hours qualify for overtime pay.
• Calculate overtime pay: Multiply the overtime hours worked by the overtime rate. For instance, if an employee worked 5 hours of overtime in a week, and the overtime rate is $30 per hour, the
overtime pay would be $150 (5 hours x $30).
• Combine pay: Add the regular pay (for hours worked up to 40) and the overtime pay to get the total weekly pay. For example, if the employee worked 40 regular hours at $20 per hour and 5 overtime
hours at $30 per hour, their total weekly pay would be $950 ($800 regular pay + $150 overtime pay).
To learn more, you can read our guide on Your Rights as an Hourly Employee in Texas.
How is Overtime Calculated for Salaried Employees in Texas?
To calculate overtime for salaried employees in Texas:
• Determine the regular rate of pay: Divide the weekly salary by 40 to get the regular hourly rate. If an employee works 40 hours per week and is paid $1,200 weekly, then their regular rate of pay
would be $30 per additional hour.
• Calculate the regular rate of pay: Multiply the regular hourly rate by 1.5 to find the overtime rate. Using the $30 hourly rate, the overtime rate would be $45 per hour (30 x 1.5).
• Identify overtime hours: Count the total number of hours worked beyond 40 in a workweek. Only these hours qualify for overtime pay.
• Calculate overtime pay: Multiply the number of overtime hours by the overtime rate. For example, if the employee worked 10 overtime hours, the overtime pay would be $450 ($10 hours x $45).
• Combine pay: Add the regular pay and the overtime pay to get the total weekly pay. For example, if the total weekly salary is $1,200 and the overtime pay is $450, the total weekly pay would be
For more details, check out our guide on Your Rights as a Salaried Employee in Texas.
How do you Calculate Overtime for Seven Consecutive Working Days in Texas?
Texas does not have a specific provision for overtime for seven consecutive working days. The calculation of overtime typically follows a standard workweek of 40 hours rather than a rolling seven-day
period. To calculate overtime, you have to:
• Determine the employee’s regular hourly rate and calculate the overtime rate, which is one and a half (1.5) times the regular rate. For example, if the employee’s regular hourly rate is $10 per
hour, their overtime rate would be $15 per additional worked hour.
• Identify any hours worked beyond 40 within the seven consecutive working days. Only these hours qualify for overtime pay. If the employee worked 46 hours within the seven consecutive days, the
employee is entitled to receive overtime compensation for the six hours worked.
• Multiply the overtime hours by the overtime rate to determine the total of the overtime pay. For example, if the employee worked 46 hours in a seven-day period, six of which were overtime hours,
and the overtime rate is $15 per hour, the overtime pay would be $90 (6 hours x $15).
• Add the regular pay (for the first 40 hours) and the overtime pay to get the total pay for the seven days. If the employee’s regular hourly rate is $10 per hour and they worked 46 hours total,
their total pay would include both regular pay for 40 hours ($400) and overtime pay ($90), totaling $490.
Overtime for Complex Pay Structures in Texas
How do you Calculate Overtime for Piece Rate or Commission Employees in Texas?
In Texas, employees who earn commissions are entitled to overtime pay just like other non-exempt employees. When calculating overtime, commissions must be included in the employee’s total wages.
However, the overtime rate is half of the regular rate.
Let’s have this as an example: An employee works 45 hours a week. The employee is paid $10 per hour and earns $100 in commissions. Here’s how you can calculate the commissioned employee’s overtime:
• Calculate the employee’s weekly wage by multiplying the total hours worked by the hourly and adding their earned commission. The employee’s weekly wage would be $550 (45 hours x $10 = $450 + 100
= $550).
• Determine the employee’s new hourly rate by dividing the amount by the total hours worked. The employee’s new hourly rate would be $12.22 ($550 / 45 = $12.22).
• Calculate the overtime rate by multiplying the new regular hourly rate by half (0.5). The overtime rate would be $6.11 ($12.22 x 0.5 = $6.11).
• Calculate the overtime pay by multiplying the overtime rate by the number of additional hours worked. The overtime compensation would be $30.56 ($6.11 x 5 = $30.56).
How do you Calculate Overtime with Multiple Pay Rates in Texas?
To calculate overtime with multiple pay rates in Texas:
• Determine total earnings: Calculate the total earnings for each pay rate within the workweek. For example, if an employee worked 20 hours at $15 per hour (20 x 15 = $300) and 20 hours at $25 per
hour (20 x 25 = $500), the total earnings of the employee would be $800 ($300 + $500).
• Calculate the weighted average rate: To find the regular rate of pay for overtime calculations, compute the weighted average hourly rate. This is done by dividing the total earnings by the total
hours worked. If the total earnings is $800 the weighted average rate would be $20 per hour ($800 / 40 = $20).
• Compute the overtime rate: Multiply the weighted average by 1.5 to find the overtime rate. For the weighted average rate of $20, the overtime rate would be $30 per hour ($20 x 1.5).
• Calculate overtime pay: Determine the number of overtime hours worked and multiply by the overtime rate. If an employee worked 5 hours of overtime, their overtime pay would be $150 (5 x $30 =
• Combine pay: Add the regular pay and the overtime pay to calculate the total pay for the week. Using the example above, the total pay would be $950 ($800 + $150).
Additional Considerations for Texas Overtime
Are there Exceptions to the Standard Overtime Rules in Texas?
The Fluctuating Workweek (FWW) Method is a specific approach to calculating overtime pay for salaried employees whose hours vary each week. Under this method, employees are paid a fixed salary
regardless of the number of hours worked each week. If their hours fluctuate and they work over 40 hours in a week, they receive overtime pay at a rate of one-half (½) of their regular hourly rate.
To qualify, employees must have a minimum hourly wage of $7.25 and a fluctuating workweek pattern. This method provides an exception to the standard overtime rules, allowing salaried employees who
meet these criteria to earn overtime pay, unlike the typical rule where fixed-salary employees may be exempt from overtime.
In Texas, while many employees are exempt from overtime pay, the FWW method provides a way for certain salaried employees with fluctuating hours to receive overtime compensation. This is an exception
to the general rule that salaried employees may not qualify for overtime under standard regulations.
Are there Industry-Specific Overtime Rules in Texas?
Yes, there are industry-specific overtime rules in Texas that differ from standard overtime regulations, which include:
• Agricultural Workers: Agricultural workers in Texas are generally exempt from the FLSA’sbovertime requirements, meaning they do not receive overtime pay for hours worked beyond 40 in a workweek.
This exemption covers workers involved in farming activities, such as growing and harvesting crops, and related processing. Employers in agriculture or agricultural commodity processing are
included in this exemption. Special rules may apply to horticultural and dairy farm workers depending on their specific duties. Workers engaged in non-farming activities or employed by
non-agricultural businesses may not be exempt from overtime provisions.
• Transportation Workers: Under the FLSA, certain transportation workers, including drivers and mechanics, are exempt from standard overtime rules if they work for motor carriers or private
carriers engaged in interstate or foreign commerce and perform safety-affecting duties.
• Public Sector Employees: In Texas, state employees subject to the FLSA who work over 40 hours a week can receive either 1.5 hours of compensatory time for each overtime hour or overtime pay at
1.5 times their regular rate. Employees can accumulate up to 240 hours of compensatory time, or 480 hours for those in public safety or emergency roles.
• Healthcare Workers: Healthcare employees have special considerations under the FLSA, they follow a 14-day work period for calculating overtime instead of the standard 7-day workweek. This can be
advantageous for these industries, which often have varying schedules and extended shifts.
How can Employers Ensure Compliance with Texas Overtime Laws?
Employers in Texas can ensure compliance with Texas overtime laws by accurately classifying employees as exempt or non-exempt according to FLSA criteria. Exempt employees, including those in
executive, administrative, or professional roles, are not entitled to overtime pay. Implementing reliable time and attendance tracking software is crucial for accurately recording all hours worked,
this ensures precise tracking and correct calculation of overtime. In addition, maintaining comprehensive records of hours worked, wages paid, and overtime calculations is essential, as the FLSA
mandates that these records must be kept for at least three years. Lastly, regularly reviewing and updating company policies helps ensure compliance with current federal and state labor laws, and
staying informed about changes in regulations allows employers to adjust their policies accordingly.
Important Cautionary Note
This content is provided for informational purposes only. While we make every effort to ensure the accuracy of the information presented, we cannot guarantee that it is free of errors or omissions.
Users are advised to independently verify any critical information and should not solely rely on the content provided.
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How do you calculate aggregate impact test?
How do you calculate aggregate impact test?
Calculation. Aggregate impact value is the ratio of the weight of the fraction passing through 2.36 mm (weight W2) by the total weight of the sample (weight W1 + W2).
How is aggregate impact value expressed?
The aggregate impact value is expressed as the percentage of the fine passing 2.36 mm sieve formed in terms of the total weight of the sample.
What tests are done on aggregate?
Specific Gravity and Water Absorption Test of Coarse Aggregate. Specific gravity and Water Absorption Test of Aggregates are major important tests to be performed on aggregate. These two parameters
or properties of aggregate play an important role in the mix design of concrete.
Is code for aggregate impact value test?
Aggregate Impact Value Test Procedure as per IS: 2386 Part-4 (1963)
Why AIV test is done?
PURPOSE OF TEST: Aggregate Impact Value test determines the Aggregate Impact Value (AIV) of aggregates which provides a relative measure of the resistance of an aggregate to sudden shock or impact.
Resistance of the aggregates to impact is termed as toughness.
What is ACV test?
The Aggregate Crushing Value (ACV) Test Set measures how resistant an aggregate is, when being crushed under a gradually applied compressive load. Each set consists of steel cylinder, plunger, base
plate, cylindrical measure and tamping rod.
What is the main objective of aggregate impact test?
What is the principle of aggregate impact test?
The test sample shall be subjected to a total of 15 such blows each being delivered at an interval of not less than one second. The crushed aggregate shall then be removed from the cup and the whole
of it sieved on the 2.36 mm IS Sieve until no further significant amount passes in one minute.
Which test is used for coarse aggregate?
Durability of coarse aggregate is normally evaluated in the sulfate soundness test and water absorption tests, and by measuring resistance to impact in the Los Angeles abrasion and impact test. These
tests suffer from some disadvantages: poor precision and inadequate correlation with field performance.
Is 2386 a part6?
1.1 This standard (Part VI) covers the test procedure for measuring the mortar-making properties of fine aggregate for concrete by means of a compression test on specimens made from a mortar of a
plastic consistency and gauged to a definite water-cement ratio.
Is 383 an impact value?
The aggregate impact value shall not exceed 45 percent by weight for aggregates used for concrete other than for wearing surfaces and 30 percent by weight for concrete for wearing surfaces, such as
runways, roads and pavements.
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How to Read Lead Time Distribution: examples and guidance
How to Read Lead Time Distribution?
Step 1 – Building the Lead Time Histogram
Let`s start with understanding the main term:
Lead time – the time you need to deliver the service / to fulfill the request. It is calculated from the commitment point (for example, the “requested” column) to an agreed ready-for-delivery or
acceptance point which means the work is done (for example “Done” or “Deployed” column on a Kanban board). Analyzing the lead time of tickets gives us an understanding of how long before we need to
take delivery, we must place an order and have it accepted.
To build a Lead Time histogram you should remember the following:
• x-axis = time (days of the lead time of the tickets);
• y-axis = frequency (the quantity of the tickets, that took a certain lead time (x-axis) which have gone through the board over a certain period).
Let's imagine you are analyzing data from 3 months of the work. You know that the minimum lead time of a ticket on your board is 1 day. After 3 months you have 53 tickets with 1-day lead time – which
means, that your first data point will be 1 on the x-axis and 53 on the y-axis, and so on.
Note: if the minimum lead time is greater than 1 day, but 3 days, for instance, it means your chart starts from 3 as a starting point (see another example below).
Step 2 – Analyzing your lead time distribution
We believe that a widely spread concept of average is not precise enough. Let's define “mode”, “medium” and “mean” for your lead time distribution. Let'ssee how we can use it as a more precise
“average” in your communication with the client.
The mode – is the most commonly occurring lead time in the data set = the biggest quantity of tickets you have with this lead time. We tend to remember this lead time because it happens the most
often. This is the top of the hill of our lead time histogram, the peak of the curve.
The median – is the 50th percentile. This means half of the data (the quantity of the tickets) is on one side of the histogram and the other half – on another. If the median is twenty days, it means
that half of the items processed took less than twenty days, while the other half took twenty or more days. Every other one is less than the median value, and the other half is greater than the
median value.
The mean – is the arithmetic average: sum up the value of all the data points (lead time days) and divide by the number of points (the quantity of the tickets). The word “average” is usually used to
refer to this arithmetic mean.
We need to understand, that the mean tends to accelerate away from the mode and the median as the tail extends farther to the right with longer lead times (creating a fat tail). A fat tail affects
the mean much more than it affects the median and is unlikely to impact the mode at all. Understanding this is important for planning, risk management, and customer satisfaction.
Step 3 – Identify your 85%-ile and 98%-ile
We have already identified that the median – is the 50th percentile. To define the 85%-ile we need to determine what is the lead time when 85% of the data points are being processed (see the image
below). We like the 85%ile because it represents 6 out of 7. Only 1 out of 7 takes longer than the 85%ile point.
How to use this data in communication with your client
All this data will give you more trustworthiness as the averages are simply not enough to communicate the delivery capability. To be reliable use at least two data points: the average (mean) and a
high percentile (e.g. 85%). Why? Let`s have a look at our example below.
• the mode (most frequent): 5 days;
• the mean (arithmetical average): 6 days;
• 85%-ile (85% of items are done): 8 days;
• 98%-ile (98% of items are done): 12 days.
Imagine you are talking to your client and you are considering committing to 5 days delivery. You remember you often deliver in 5 days. You feel you can make it and you don`t either want to sound too
slow in delivering.
However, knowing your lead time gives you more variations to make a proper commitment and be more reliable for your client. Now you know that your mean is 6 days and 85%-ile is 8 days. The smart
solution is to say, “we will deliver it in 5-8 days”. If you anticipate that this particular request may take longer, just add “maximum, you'll have it in 12 days”. This way you will have many more
chances to deliver in time and gain a trustworthy reputation with your client.
For example, you delivered in 5 days – good for you, you`ve made your client happy. In 8 days – you delivered in time, in 12 days – the delivery is still within the SLE.
Explore more about Kanban on Kanban+
Kanban+ is one source of truth when it comes to the Kanban Method. It is one platform that gathers all possible Kanban method materials, taught and used by Kanban University. Create your free
account now and get access to a set of free content such as posters, infographics, book chapters, and more. Learn more about the Kanban Method today on kanban.plus!
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Tuesday, October 15, 14:00, 7.527
Csaba Szántó (Cluj)
Ringel-Hall polynomials and Gabriel-Roiter measure over Euclidean quivers.
Bo Chen's theorem in the Dynkin case states that if T is a Gabriel-Roiter submodule of M then Hom(T,M/T)=0 (all modules being indecomposable). Ringel proved this theorem comparing all possible
Ringel-Hall polynomials (involving only indecomposables) with the special form they take in case of a Gabriel-Roiter inclusion. We will implement Ringel's idea in the Euclidean quiver context
obtaining some new results on Gabriel-Roiter inclusions. For this purpose we have also determined a list of special Ringel-Hall polynomials in the Euclidean case which may have further applications.
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We practiced in the light of Chandra, a moon-practice like Chandra Namaskar, the salutation of the moon. The syllable "tha" of the word Hatha Yoga (more physical oriented Yoga) represents the moon
and the femine energy, one of the main energies in our body. d you know that the atomic weight of silver is 108? In astrology, the metal silver is said to represent the moon.Di
Time to shed light on the multi dimesional meaning of the number 108. For example Hindu and Buddhist prayer beads (malas) have 108 beads, or some fraction of that number.
What does 108 represent? Here are 12 answers...
1) SANSKRIT: There are 54 letters in the Sanskrit alphabet. Each has masculine and feminine, shiva and shakti. 54 times 2 is 108.
2) 9 TIMES 12: Both of these numbers have been said to have spiritual significance in many traditions. 9 times 12 is 108. Also, 1 plus 8 equals 9. That 9 times 12 equals 108.
3) CHAKRAS: The chakras are the intersections of energy lines, and there are said to be a total of 108 energy lines converging to form the heart chakra. One of them, sushumna leads to the crown
chakra, and is said to be the path to Self-realization.
4) TIME: Some say there are 108 feelings, with 36 related to the past, 36 related to the present, and 36 related to the future.
5) SUN and EARTH: The diameter of the sun is 108 times the diameter of the Earth.
6) MOON and EARTH: The average distance of the Moon from the Earth is 108 times the diameter of the Moon.
7) GODESS NAMES: There are said to be 108 Indian goddess names.
8) ASTROLOGY: In astrology, there are 12 houses and 9 planets. 12 times 9 equals 108.
9) SOUL: According to Atman, the human soul or center goes through 108 stages on the journey.
10) PYTHAGORAS: In Pythagorean, the nine is the limit of all numbers, all others existing and coming from the same. ie: 0 to 9 is all one needs to make up an infinite amount of numbers.
11) PATHS to GOD: Some suggest that there are 108 paths to God.
12) FIRST MAN in SPACE: The first manned space flight lasted 108 minutes, and was on April 12, 1961 by Yuri Gagarin, a Soviet cosmonaut.
And: There are 108 cards in an UNO deck. I´m sure there are another 12, or 12 times 9, or maybe 2 times 54 answers to contemplate on the meaning of 108. Enjoy by doing so ♥ ♡ ❥!
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SVG - The path directive - Moment For Technology
Make writing a habit together! This is my first day to participate in the “Gold Digging Day New Plan · April More text challenge”, click to see the details of the activity.
Antecedents to review
The last two we through the SVG path M/H/V/L/C/Q/Z several instructions parsing. Converted the NUGGETS LOGO SVG to the native path rendering of Flutter and added some rendering effects.
In addition to these, there are several other instructions. The purpose of this article is to take a thorough look at the path commands under the PATH tag in SVG. The list is as follows:
M/ M (x,y)+ Move current position L/ L (x,y)+ straight line H/ H (x)+ horizontal line V/ V (x)+ vertical line Z/ Z Closed path Q/ Q (x1,y1,x,y)+ quadratic Bezier curve T/ T (x,y)+ smooth draw quadratic Bezier curve C/ C (x1,y1,x2,y2,x,y)+ Cubic Bezier curve S/ S (x2,y2,x,y)+ smooth draw cubic Bezier curve A/ A (rx, RY, Xr, LAf,sf,x,y) arcCopy the code
Absolute and relative instructions
As you may see, each instruction has uppercase and lowercase letters. Capital letters indicate that the following coordinates are absolute, that is, the coordinates starting from the upper left
corner of the region. Absolute and relative coordinates are the most basic concepts in drawing and are easy to understand. The source code for the SVG sample file in this article is idRAW/
EXTRA_02_SVG /base.
1. Absolute coordinate movement example
Absolute move, use capital letters. For example, in the second paragraph M30,30 means to move the starting point to (30,30), V60 means that the abscissa remains unchanged and the ordinate reaches the
absolute point 60.
<path d="M30,20 H80 V40Z M30,30 V60" stroke="# 000082" />
Copy the code
2. Example of relative coordinate movement
For relative movement, use lowercase letters. Below is m30, where 30 indicates how much the coordinates have moved with reference to the end of the current path. Before m30, 30, the end of the path
is (30,20); So m30, 30 will move the next segment to (30+30,20+30), which is (60,50). V30 means: taking the end point of the current path as reference, how much the coordinate has moved vertically
will be straight line to (60,50+30), i.e. (60,80).
<path d="M30,20 H80 V40Z M30, 30 v30" stroke="# 000082" />
Copy the code
3. Use scenarios of absolute and relative coordinates
If the coordinates of a point can be known precisely, it is most convenient to use absolute coordinates. But A lot of times, the absolute coordinates are not directly available, so let’s say we move
A 50 to the right, 20 to the down, to B. Although you can calculate the absolute coordinates of B from the data, it’s a bit of a hassle, especially for some curvilinear paths, relative offset is very
<path d="M30,20 H80 V40Z m0,20 l50,20 V40 l-50,20Z" stroke="# 000082" />
Copy the code
2. Curve path
The curve path includes:
• Quadratic Bessel curve, the instruction isQ/q, and smooth formT/t ；
• Cubic Bezier curves, the instruction isC/c, and smooth formS/s ；
• Arc curve, the instruction isA/a ；
1. Quadratic Bessel curveQ/q
Each quadratic Bezier curve consists of two coordinates representing the control point and the end point respectively. The following two Bessel curves are formed by absolute Q and relative Q
respectively. For example, the control points above are 70 and 10, which are tangent to the lines and curves of the starting and ending points, as shown in dotted lines:
<path d="10, 80, 40 (, 20 Q70," stroke="# 000082" />
<path d="M80, 40 q - 40 30-40 10" stroke="#FF743D" />
Copy the code
2. Cubic Bezier curvesC/c
Each cubic Bezier curve consists of three coordinates representing control point 1, control point 2 and end point respectively. The following two Bessel curves are formed by absolute C and relative C
respectively. For example, the control point above is control point 1: (50,10) and control point 2: (80,20). The line between control point 1 and the starting point and the line between control point
2 and the end point is tangent to the curve, as shown in the dotted line:
<path d="(20 C50, 10, 80, 20, 80, 40." " stroke="# 000082" />
<path d="M80, 40 c -, 10 -, 30-50 10 40." " stroke="#FF743D" />
Copy the code
3. Arc curveA/a
This command has 7 parameters, looks like a little crash. In Flutter, it corresponds to the Path#arcToPoint method with the following seven parameters:
The next two arcs are formed by absolute A and relative A respectively. An arc is essentially an arc cut from an ellipse. The first two values are the lengths of two half axes of the ellipse. The
fourth value indicates whether a large arc is taken, the following solid line is taken as a large arc, dotted line is taken as a small arc; The fifth value represents whether the line is clockwise.
The solid line is clockwise and the dotted line is counterclockwise. The sixth and seventh values represent the end point coordinates.
<path d="M30,20 A20 20 0 1 1 50 50 a15 20 0 1 1 20 30" stroke="# 000082" />
Copy the code
And the third value, in particular, is the Angle of rotation, and notice that this is the Angle value not the radian value. The orange color below is the effect of rotation 45. The rotation is not
the rotation of the center of the ellipse, but the tilt Angle of the Y-axis, and it needs to meet the starting and ending point of the ellipse.
-- -- -- - > [07_Aa_rotate.svg]----
<path d="M40,50 A20 30 0 1 1 60 70" stroke="# 000082" />
<path d="M40,50 A20 30 45 1 1 60 70" stroke="#FF743D" />
Copy the code
4. Smooth cubic Bessel curvesS/s
Each S instruction is followed by two coordinates, but it is a cubic Bezier curve. The following examples show the difference between it and Q and its relationship with C. In this case, S represents
the coincidence between control point 1 and the starting point, and control point 2 is 40,70. The following S and C are the same curve:
<path d="M20, 10 C20, 10; seven 80 40, 50" stroke="#F619FF"/>
<path d="70, 80, 50 M20, 10 S40," stroke="# 000082"/>
<path d="70, 80, 50, 10 Q40 M20," stroke="#FF743D"/>
Copy the code
In addition, the most difficult point of S is:
If the upper segment of S is a cubic Bezier curve: the first control point of S is the symmetric point of the previous cubic Bezier curve [second control point] with respect to [S starting point]. Otherwise: the first control point of S is the starting point of S.Copy the code
As shown below, to understand S is to understand what the Sp1 point below is.
<path d="M10,40 C20,10 40,10 50,40 S90,70 90,20" stroke="# 000082"/>
Copy the code
Mathematically, if P0 and P1 are symmetric with respect to p, their coordinates should satisfy the following relation:
Note: p0(x0,y0) p1(x1,y1) p(x,y) if p0 and P1 are symmetric with respect to p, then the coordinate relation satisfies: (x0 + x1)/2 = x (y0 + y1)/2 = y and given the coordinates of p0 and p, it is easy to find p1: x1 = 2 x - x0 and y1 = 2 y - y0Copy the code
In addition, s is the relative coordinate, same effect.
5. Smooth quadratic Bessel curvesT/t
The T/ T instruction is similar:
If the upper segment of T is a quadratic Bessel curve: the control point of T is the symmetric point of the previous quadratic Bessel curve [control point] with respect to [S starting point]. Otherwise: the control point of T is the starting point of T.Copy the code
The following is the test effect of Q on T. Think about what Tp point is.
<path d="M10, Q30 40, 50, 60, 40 T90, 40." " stroke="# 000082" stroke-width="1"/>
Copy the code
Three, this series harvest
You can download SVG ICONS in iconFont,
SVG can be drawn directly from the Flutter drawing API by parsing SVG as follows:
Through these three articles, you have implemented an extremely rudimentary SVG parser. Although it has no practical application value, we have learned the meaning of the path directives in SVG. This
is a more basic accumulation of knowledge. By associating SVG paths with Flutter drawing, you can also practice your Flutter drawing skills. In addition, I tried to parse SVG, in which the process of
finding and solving problems is the accumulation of personal experience. I also got some training in the use of regular expressions.
In addition, for SVG path resolution, pub has a complete package path_Drawing, and based on this package, SVG files to display package Flutter_SVG. Every time I see open source libraries written by
other big names, I feel like I’m playing house. Studying, thinking and learning, I hope that one day, I can also like them, see the elegant demeanor in their eyes.
The extra_02_SVG /08
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Frontiers | The Transformative Impact of a Mathematical Mindset Experience Taught at Scale
• ^1Graduate School of Education, Stanford University, Stanford, CA, United States
• ^2Inter-American Development Bank, Washington, DC, United States
A wide range of evidence points to the need for students to have a growth mindset as they approach their learning, but recent critiques of mindset have highlighted the need to change teaching
approaches, to transfuse mindset ideas throughout teaching. This shifts the responsibility from students themselves to teachers and schools. This paper shares a mixed methods study conducted across
the US, that measured the impact of a “mathematical mindset teaching approach” shown to be effective when taught by the authors, scaled to teachers in 10 US districts. The effectiveness of this novel
mathematics approach was measured using pre and post assessments during a summer intervention followed by measures of GPA change when students returned to schools. Both measures showed that a
mathematical mindset approach to teaching significantly improves students’ mathematical achievement, and changes students’ beliefs about themselves and their approach to learning. Accompanying
analyses of teaching and of teacher interviews give insights into the ways students change, highlighting the need to bring about shifts in students’ mindsets through a changed approach to mathematics
teaching and learning.
Introduction and Literature Review
In recent years there has been considerable attention paid to the idea of mindset, a construct developed and researched by Carol Dweck and teams of other researchers (Dweck, 2007). Dweck has shown
that students with a “growth mindset”, who believe that they can learn anything and that their intelligence develops as they learn more, outperform those with a fixed mindset who believe their
intelligence is fixed (Aronson et al., 2002; Good et al., 2003; Blackwell et al., 2007). Dweck’s book summarizing mindset is an international best seller (Dweck, 2007) and her ideas have been used by
tens of thousands of schools worldwide, as well as businesses, sports teams, and parents. Despite the extensive research base showing the impact of mindset changes, critiques of the concept have
emerged. Dweck herself has now written about the dangers of “false growth mindset” work in schools, when teachers learn only to praise effort, but do not implement teaching strategies to help develop
growth mindsets. Kohn (2015) argues that teachers and administrators who urge students to change their mindset without changing the teaching environment are doing them a great disservice, merely
shifting the responsibility and perhaps blame onto students. This paper shares a teaching approach that is particularly important in light of the mindset critiques. The approach that will be examined
in this paper differs from many mindset initiatives as it infuses mindset ideas and brain science messages throughout mathematics teaching. In doing so, the approach takes account of both Dweck and
Kohn’s warnings and critiques, shifting the responsibility for mindset awareness from students to teachers and schools. The approach was enacted in ten school districts across the United States
resulting in significant mindset and mathematics achievement gains.
Multiple research studies have demonstrated the positive impact of having a ‘growth mindset’ in mathematics and other subjects. Blackwell et al. (2007), for example, followed students with a growth
and fixed mindset during seventh and eighth grade, who were taught by the same teachers, to look for impact on mathematics achievement. They found that those with a growth mindset pulled increasingly
ahead and by the end of eighth grade their achievement was at a significantly higher level than those with a fixed mindset, even though they were taught in the same classes and by the same teachers.
In an interventional study building upon their prior work, Blackwell et al. (2007) performed a growth mindset intervention with students of color (97% African American and Latinx) making the
transition to seventh grade, many of whom were already showing declining grades. The control group received eight sessions of training in study skills, while the growth mindset group received eight
sessions of study skills plus training in growth mindset. The key growth mindset message was that effortful learning changes the brain by forming new connections, and that students control this
process. The growth mindset intervention led to a 0.25 difference (on a 4.0 scale) in mathematics grades between the experimental and control conditions (adjusting for pre-intervention differences).
Good et al. (2003) also created a growth mindset intervention for seventh grade students (largely Latinx) and compared it to a control group that received an anti-drug workshop. In both groups,
mentors met with their students in person for 90 min at two separate times. The impact of the intervention on statewide end-of-year achievement test scores was assessed. The growth mindset
intervention led to significantly higher achievement in mathematics and reading test scores. Moreover, in the control group, the gender difference in mathematics was highly significant, but in the
growth mindset group the gender gap was largely eliminated.
Finally, a study with college students looked at the impact of growth mindset on overall grade point average compared to two control groups, a multiple intelligence intervention and a no-treatment
control (Aronson et al., 2002). While the control groups showed no change in achievement, the growth mindset intervention led to a clear gain in achievement, particularly for African American
students. In the following term, African American students gained one quarter of a grade point, and the grade-point gap between White and African American students was no longer significant. In
addition, the African American students in the growth mindset group showed a significant increase in their valuing and enjoyment of academics.
Taking the growth mindset message beyond the traditional boundaries of classrooms and schools, Boaler et al. (2018) developed an online course entitled ‘How to Learn Mathematics’ that shares
information about mindset and productive mathematics learning through a massive, open, online course (MOOC), taken by approximately one half of a million people. In a randomized controlled trial
investigating its impact, middle school teachers teaching two classes were recruited to give the online course to one of their classes. The students in both experimental and control groups were then
followed over a school year. At the end of the school year the students who took the online course achieved at significantly higher levels than those who did not on standardized, Smarter Balanced
state scores. The students were also 68% more engaged in work, as measured by their teachers in their mathematics classes, and they changed their mindset and ideas about mathematics significantly (
Boaler et al., 2018).
These different studies all suggest that when students change their minds about what is possible, and they are released from ideas of fixed intelligence, they achieve at higher levels, whether or not
the teaching they receive changes. Despite this, Kohn (2015) has cautioned that it is irresponsible to tell students that they need to change their ideas, without changing the school systems they
work within. We support this view and recognize that many parts of the school system communicate messages about mindset to students, such as assessment, grading, other forms of feedback, student
grouping, and even the nature of the questions used in classrooms (Kraker-Pauw et al., 2017). This study shares a teaching approach in which mindset ideas are infused throughout the teaching
practices used by teachers, as part of a summer intervention.
A key part of a mathematical mindset teaching approach (Boaler, 2016, 2022) is the use of open tasks, that are “low floor and high ceiling”—these are tasks that all students can access but that
extend to high levels, and that can be approached in multiple ways. Mathematics classrooms are typically filled with closed, narrow questions–that can contravene growth mindset messages. Students
often interpret mathematics as a fixed subject, as questions have one right answer with one valued method. If questions are, by contrast, open, with invitations to students to draw, discuss, and make
connections with prior knowledge, then they are more likely to see mathematics as a growth subject that they can learn (Boaler, 2002; Boaler, 2019a). These types of tasks also allow students to
engage in authentic mathematical thinking and reasoning in ways that more traditional problem sets do not allow (Schoenfeld, 2016).
While traditional narrow questions communicate to students that mathematics is about recalling and applying a procedure, open tasks provide opportunities for students to engage in what Stein et al.
(1996) call “doing mathematics” that is: “framing and solving problems, looking for patterns, making conjectures, examining constraints, making inferences from data, abstracting, inventing,
explaining, justifying, challenging, and so on” (p. 456). Rich mathematical tasks also support the development of autonomous learners, as students are not dependent upon reproducing the teacher’s
example to gain the correct solution, rather they are encouraged to follow their own creative thinking and ideas (Silver and Stein, 1996; Silver, 1997). Several studies have shown the connection
between the use of open tasks and the development or strengthening of students’ growth mindsets (Boaler, 1998; Stohlmann et al., 2018; Sun, 2018).
Some of the growth mindset information that is most powerful to students draws upon the evidence from neuroscience, showing the potential of the brain to grow and develop connections (Maguire et al.,
2006; Iuculano et al., 2015; Boaler, 2019a). Additional neuroscientific evidence that underpinned the teaching intervention was the evidence showing that brains are made up of ‘distributed networks’,
and when people work on mathematics problems, different areas of the brain light up and communicate with each other (Menon, 2015). In particular, brain activity is distributed between different
networks, which include two visual pathways: the ventral and dorsal visual pathways (see Figure 1). Neuroimaging has shown that even when people work on a number calculation, such as 12 x 25, with
symbolic digits (12 and 25), mathematical thinking is grounded in visual processing.
FIGURE 1
FIGURE 1. Brain network underpinning mathematics knowledge, lang chen in Boaler (2019a).
The dorsal visual pathway has reliably been shown to be involved when both children and adults work on mathematics tasks. This area of the brain particularly comes into play when students consider
visual or spatial representations of quantity, such as a number line. A number line representation of number quantity has been shown in cognitive studies to be particularly important for the
development of numerical knowledge and a precursor of children’s academic success (Siegler and Booth, 2004; Hubbard et al., 2005; Schneider et al., 2009; Kucian et al., 2011).
The different studies on mindset and on teaching with a growth mindset suggest that while mindset interventions aimed at changing students’ ideas can be powerful, the biggest improvements can be
brought about when students’ ideas change at the same time as teaching is designed to encourage a growth mindset (Anderson et al., 2018; Boaler, 2019a). The approach of changing students’ ideas and
changing teaching, with a mathematical mindset intervention, has not, before now, been studied. This paper shares research that investigates the impact of mathematical mindset teaching, implemented
by multiple teachers in ten school districts distributed across the United States.
The Mathematical Mindset teaching approach was first developed and studied in a youcubed summer camp implemented in the summer of 2015 and detailed in Boaler (2019a, b). Eighty-one students who had
just finished sixth or seventh grade attended a four-week mathematics camp, held on a university campus. The students were recruited from two local school districts, had been identified by
administrators as having had negative math experiences, and were from a range of ethnic backgrounds, with the majority of students identifying as mixed race. Mindset was infused in two ways: (1) the
teaching of mathematics through a curriculum of open tasks that can be approached in different ways and (2) explicit growth mindset messaging. Students engaged in low-floor, high-ceiling tasks, and
the four weeks of teaching were centered around four “big ideas” (California Department of Education (CD), 2021; Cabana et al., 2014; Bransford et al., 2000): number sense, pattern seeking, algebra
as a problem-solving tool, and generalizing. Additionally, teachers explicitly communicated messages about growth mindset and brain science, highlighting the importance of mistakes, struggle, and
visual thinking, and dispelling myths about the importance of speed and procedural approaches to mathematics. Analyses of students’ achievement on a standardized assessment of algebraic thinking that
was taken before and after camp revealed that students improved their performance by an average of 50 percent across the students, with an effect size of 0.91 standard deviation, equivalent to 2.8
school years of growth in school (Boaler, 2019b). Additionally, qualitative analyses of student interviews revealed that the majority of students shifted their perspectives over the course of summer
camp, changing their minds about their own potential and about the nature of mathematics (Boaler 2019b). In particular, they began to see themselves as capable, they saw mathematics as a creative set
of ideas, and they saw their role in mathematics as people who investigated ideas, explored conjectures, and reasoned about them (see also: https://www.youcubed.org/resources/solving-math-problem/)
While the results of the original youcubed camp were promising, several important questions remained. Could the mathematical mindset approach to teaching only be done by this particular teaching team
at this particular university? Could it be specified well enough to scale this approach to other summer programs? If so, would students at other programs experience increased achievement and shifts
in mindset as a result? The remainder of this paper will communicate the results of a study monitoring the impact of the youcubed summer camp, taught in ten districts across the United States,
considering any potential improvements in mathematics achievement after the camp and when the students returned to their mathematics classes in the following school year.
Over several years, workshops designed by the research team were offered for teachers to learn about the mathematical mindset teaching approach. During the workshop teachers were given the curriculum
and trained with mathematical mindset pedagogical practices, and multiple additional resources were shared with them to support their learning on this teaching approach. In 2019 a partnership between
ten school districts and youcubed enabled a study of the learning of students who participated in the camps in their districts, which is the focus of this paper.
Research Design
In the summer of 2019, ten districts in five states implemented the youcubed summer camp, agreeing to provide data on their students’ mathematics achievement at the beginning and end of the camp and
later when the students returned to school. The districts recruited students to attend the youcubed camps who were diverse in terms of ethnicity, gender, and socioeconomic status. Additionally,
district recruitment focused on students who are Black, Latinx, and/or experiencing poverty, to ensure that camp attendees reflected these groups. Overall district data is shown in Table 1. Camps
also exhibited variation in enrollment size, attendance rates, and amount of instruction delivered, which is shown in Table 2. The duration of camps analyzed for this study ranged from 10 to 28 days,
comprising 30–84 h of math instruction in the summer of 2019. The wide range in implementation characteristics provides important context for this analysis.
TABLE 1
TABLE 2
Different forms of support were offered to participating teachers before and during the study summer camps. In the spring of 2019, all participating teachers were required to take part in three 1-h
webinars and were offered additional learning opportunities, including a book detailing the approach (Boaler, 2016) and an online class that shared videos and teaching designs from the original camp
Teachers of the multiple camps were all given detailed curriculum that described the objectives and activities for each lesson during the camp. Two sequences of the curriculum were shared with
teachers to account for the variation in instructional days across the sites: one for two-week camps expected to include 30 h of instruction, and one for four-week camps expected to include 60 h of
instruction. The mathematical topics included in the curriculum were number sense, algebra as a tool for problem solving, generalization and mathematics as pattern seeking. Additionally, specific
structures and activities were provided. A typical day included a “number talk” to build number flexibility and a short video with growth mindset messages. The remaining time was dedicated to
instruction organized into “big ideas,” with students engaging in an orientation activity, open-ended mathematics tasks that encouraged then to engage with agency and authority (Gresalfi and Cobb,
2006; Sengupta-Irving, 2016), time to work in groups, and often a whole class discussion.
Research Methods
Given the goal of understanding the impact of a mathematical mindset approach taught within summer camps, scaled to ten districts, a mixed methods approach was implemented, drawing from both
quantitative and qualitative methods. A matched comparison analysis was employed to assess the effect of the approach on students’ achievement. School districts provided a variety of achievement
measures of both participant and non-participant students (GPA and MARS scores, before and after camp participation; and a baseline math standardized test score), and a battery of control variables
(race, ethnicity, gender, free and reduced-price lunch status, English learner status, and special education status). To examine the enactment of the mathematical mindset approach, a subset of
classroom videos from the camps were collected and analyzed using qualitative methods. Finally, to investigate students’ mindsets, interviews with a subset of teachers were conducted and transcripts
of these interviews were analyzed.
Student Achievement
Two data sources were used to examine student achievement: a standardized assessment of conceptual mathematics– MARS tasks–was administered at the start and end of the camps to measure changes in
students’ mathematical understanding. Participating sites administered four MARS performance tasks at the beginning of camp and on the final day, with the same tasks used for pre and post camp across
all grade levels. Each task was scored by an external partner Silicon Valley Mathematics Institute (SVMI) on a point-score analytic rubric for numerical responses and mathematical reasoning. There
was a total of 36 possible points across the four tasks. MARS scores were analyzed for all students who met the following criteria: (i) they were in a district that had submitted MARS assessment
papers by November 4, 2019; (ii) they had both pre and post camp scores available, and; (iii) they had completed at least two of the four MARS tasks.
To consider change in mathematical understanding, measured through MARS tasks, a composite score was achieved by summing students’ scores across the four tasks with the pre-score subtracted from the
post-score, to give a measure of growth and enable the calculation of main effect sizes, following the same method as the original camp study (Boaler, 2019b). This enabled analysis of gains by
district understand the variation of effects, including the relationship between effect size and hours of instruction.
To measure the program’s impact on student achievement when students returned to their school classrooms after the conclusion of the summer camps, mathematics grade point average (GPA) were collected
for the school year following the mathematics summer camp (2019–2020). A matched comparison analysis (Rosenbaum and Rubin, 1983; Stuart, 2010) was used to analyze the effects of the youcubed camp on
students’ GPA. The sample included 536 camp participants enrolled in grades 5, 6, and 7 (the camp’s focal grades) in 10 districts during the Spring of 2019. This sample included all camp participants
for whom baseline GPA, baseline math standardized test score, and the outcome GPA were able to be gathered, making up 64% of the original sample of camp participants. Students changing school
districts was a common reason for missing data.
The analysis was conducted through the creation of a uniform GPA variable across the 10 districts by mapping standards-based grades (which have four levels) onto a standard, 4-point GPA scale (i.e.,
“advanced” was coded as 4, “proficient” was coded as 3, “below proficient” was coded as 2, and “basic” was coded as 1, equivalent to a D letter grade in the standard GPA scale). Neighbor matching (
Abadie et al., 2004) was used to identify comparison students for each camp participant based on proximity in baseline GPA and mathematics standardized test score. The post-youcubed mathematics camp
GPA of that comparison student served as the estimate of the grade each camp participant would have received if they had not attended camp. When multiple comparison group students had equally similar
baseline grades and scores, this algorithm used the average Fall 2019 GPA of those students as the estimated comparison outcome. To identify the average effect of the youcubed mathematics camp among
participants, this approach calculated the difference in average GPA between camp participants and the matched comparison students.
Multiple model specifications were used to assess whether the overall impact estimates were robust. Among models that included the key baseline variables of mathematics GPA and test score, all impact
estimates were positive and of a similar magnitude, and the model with the richest set of matching variables (adding race, ethnicity, gender, free and reduced-price lunch (FRPL) status, English
language learner (ELL) status, and special education status as matching variables) yielded a very similar impact estimate (0.14 GPA points). The chosen model included prior GPA and math score, both
to avoid reducing the sample size (thus making the findings as broadly applicable to camp participants as possible) and to prioritize identifying matched comparison students with the most similar
prior academic achievement.
Studying the Enactment of the Youcubed Camp Approach
To capture the teaching that was implemented in the different camps, not only the intended teaching approach, seven classroom videos across four sites were analyzed. All teachers had been asked to
record and submit a classroom video of the same task, “Painted Cube” (shown in Figure 2). Across all districts, nine classroom videos were submitted and seven were determined to have strong enough
audio and video quality for analysis.
FIGURE 2
The teaching was analyzed in two different ways. Researchers created content logs (Derry et al., 2010) of approximately 7 hours of video, outlining the events on each video and conducting a time
analysis of how many minutes were spent on each segment of the lesson (i.e., task launch, work time, and whole-class discussion) for each teacher and across the teachers. One of the seven videos was
excluded from the time analysis because it only captured one segment of the lesson.
In the second form of analysis, researchers examined the seven videos using the Mathematical Mindset Teaching Guide (https://www.youcubed.org/
mathematical-mindset-teaching-guide-teaching-video-and-additional-resources/) as a tool for coding classroom practice. An initial video was selected to consider in depth, based on the teacher’s
implementation of several mathematical mindset teaching practices, which was determined during the content logging process. A team of three researchers then re-watched this video, independently
identifying 7–10 “critical moments” in which one of the mathematical mindset teaching practices (Growth Mindset Culture, Nature of Mathematics, Challenge and Struggle, Connections & Collaboration)
was enacted. The fifth practice, assessment, was excluded because it was not feasible to identify the teacher’s range of assessment practices in one lesson. After discussing these moments and the
extent to which each aligned with the proposed practice, the combined critical moments were developed into a list of indicators for each dimension of each practice. For example, in one critical
moment the teacher called on a student to explain their thinking, and when the student asked if they should come up to the board, the teacher said, “whatever you need to do to prove it.” This moment
was combined with another moment--in which the teacher asked the class questions like “how do you know?”--to create the following indicator for the “Reasoning & Multiple Perspectives” dimension of
the Nature of Math practice: “students are expected and explicitly invited to bring multiple ideas to the task and justify/reason through their ideas (in writing and/or verbally).” This initial draft
was then tested on two contrasting cases from two different sites, for which researchers identified evidence that either validated an indicator or suggested a need to refine an indicator (e.g.,
re-wording, clarifying, adding). These pieces of evidence were discussed until consensus was reached, which led to refinement of the indicators. For example, the previously mentioned indicator was
validated by a moment in which a different teacher asked the class for different answers following the sharing of a solution from one student. The revised indicators were then used to code the
remainder of the data set for critical moments in which mathematical mindset teaching practices were enacted.
Studying Student Mindsets and Engagement with Mathematics
To consider changes in students’ mindsets and engagement with mathematics, semi-structured interviews (Glesne, 2005) were conducted with teachers during the 2019–2020 school year, following the
implementation of the youcubed camp. Teachers were recruited for interviews and the 20 that were interviewed came from six districts, representing both the highest and lowest achieving camps amongst
the group based on MARS effect sizes. These interviews lasted approximately 45 min. The interviews were conducted and recorded via Zoom, and transcripts of the audio were analyzed. Two members of the
research team coded the transcripts to systematically identify instances in which teachers provided detail of students’ experiences in the youcubed classrooms. These excerpts were then open coded for
emergent themes around students’ mindsets and relationships with mathematics, after which analytic memos were written (Emerson et al., 2011) and a codebook was created. Two researchers then re-coded
the excerpts using this codebook. Next, a theme analysis was conducted on excerpts from the three most common codes (tasks, student engagement, and student belief). Finally, researchers calculated
the presence and co-occurrence of these three codes across the data set to quantify these themes.
Analyses revealed that students’ mathematics achievement both at the conclusion of camp and in the following school year significantly increased, as measured by MARS scores and mathematics GPA. To
better understand the mechanism for this change, teachers’ enactment of the mathematical mindset teaching practices was analyzed. This analysis of teaching revealed that students were given
significant time to grapple with open tasks in summer camp. Additional analyses of teachers’ interviews showed that students’ experiences with open tasks was a significant factor in students’ changed
mindset and engagement with mathematics.
MARS Results
Students who attended the youcubed camps achieved at significantly higher levels at the conclusion of the camps, as evidenced by a significant difference in pre/post MARS assessments. The average
gain score for participating students across all sites was 0.52 standard deviation units (SD), equivalent to 1.6 years of growth in math. On average, at baseline, camp participants received 6.6
points out of a total of 36 on the 4 MARS tasks, whereas the mean score after the camp was 8.8, a gain of 2.2 points that was statistically significant with a 99% degree of confidence. There was
variation across ten districts in the size of gains students demonstrated, with gains ranging from 0.24 SD to 0.96 SD (i.e., 1.02 to 4.16 points, respectively). In nine out of the ten camps, gains
were statistically significant with a 90% of confidence. Table 3 presents the results both in the aggregate and by district. The overall sample result of 0.52 SD is lower than the original youcubed
summer camp at Stanford of 0.91 SD.
TABLE 3
To consider the impact of the teaching time in different camps, investigation of correlations between the amount of instruction provided by a camp (in days of camp and hours of instruction) and the
growth in learning students demonstrated (the effect sizes of the learning gains) were conducted. These showed moderate, positive correlations in the total number of days of camp duration (r = 0.65)
and total number of hours (r = 0.58) each site devoted to the youcubed camp approach. Figure 3 present scatterplots of these relationships. The correlation with total days is statistically
significant at the 0.05 level.
FIGURE 3
FIGURE 3. (A) Effect size of MARS by number of days of MMSP instruction. (B) Effect size of MARS by number of hours of MMSP instruction.
The MARS gains showed that camps who offered the mathematical mindset approach for more days and hours, achieved significantly larger gains. There was little evidence of a difference in MARS gains
based on recruitment approach (see Table 5).
Post Camp Grades
When the students returned to their regular mathematics classes in their school district they experienced a variety of forms of instruction. The students who had attended the camps were compared with
students in their districts who were at similar levels of achievement but had not attended the camps. This analysis showed that at the end of the first term or semester back at school, the students
who attended the youcubed summer camp achieved a significantly higher mathematics GPA (p < 0.01, n = 2,417). On average, students who attended camp had a math GPA that was 0.16 points higher than
similar non-attendees (i.e., students from the same district and grade and who had a similar baseline math GPA and test score) (Table 4). In addition, compared to control students, camp participants
were 6 percentage points more likely to receive a grade of B or higher, and 5 percentage points less likely to receive a grade of D or lower (Table 4).
TABLE 4
Among the seven sites that shared science GPA data, a matched-comparison analysis indicated that camp participants also had slightly higher science GPAs than similar nonparticipants, but that
difference–0.11 GPA points–was not statistically significant at the 5% level.
Overall, exploratory subgroup analyses suggest that the mathematical mindset intervention had a similar impact for students with different demographic characteristics including students of different
racial groups, English Learners, and students who received low or average grades at baseline (Table 5). The only exception to the overall pattern of similar impacts among subgroups was that the large
impact on GPA for special education students (0.46 GPA points) was significantly different from the impact for non-special education students (0.10 GPA points).
TABLE 5
At camps that targeted recruitment to students with substantially lower math test scores than the district average, impacts on math GPA were larger (Table 5). The pattern in GPA impacts from these
exploratory analyses suggests that the mathematical mindset approach particularly benefits students with a lower level of baseline math knowledge, even with a modest number of hours of instruction.
Analyses of Teaching
Video analyses of teaching were conducted to consider the aspects of mathematical mindset teaching practices that brought about these positive gains in mathematics achievement, and that stand in
contrast to more typical forms of mathematics teaching (Li and Schoenfeld, 2019). The task that was the subject of analysis (see Figure 2, Painted Cube) prompts students to consider the faces of the
small 1 × 1 × one cubes that comprise a 3 × 3 × three cube--a challenging task that gives students opportunities to struggle and gives teachers opportunities to share messages of the value of
struggle as well as the value of visual thinking. The task is low floor–all students can build with cubes and think about patterns–and high ceiling, as the upper ends of the task involve forming
different expressions, linear, quadratic, and cubic. The different teachers studied all invited students to build different sized cubes using sugar cubes--beginning with a 3 × 3 × three cube and then
extending to a 4 × 4 × four cube--and to engage in three-dimensional visualization and drawing, which encouraged students to experience math physically, see it visually, and think about
generalization. To help them draw different sized cubes, students were provided notebooks with squared paper, as shown in one student’s journal in Figure 4. These critical moments in which teachers
supported students with the resources and time to deeply explore one problem through multiple approaches served as evidence of their enactment of the mathematical mindset teaching practice of nature
of math and the dimensions of “reasoning and multiple perspectives” and “depth over speed”.
FIGURE 4
To support students in finding and extending patterns during their exploration, teachers in six out of seven classrooms created a table on the whiteboard to document the number of cubes within each
type of cube that would have each amount of their faces shaded. Of these six classrooms, two teachers constructed partial tables, which documented the number of cubes with each number of faces shaded
for solely a 3 × 3 × three cube or both a 3 × 3 × three and 4 × 4 × four cube. The other four teachers constructed tables that extended to 5 × 5 × five and n x n x n cubes. An example of this table,
written by a student in their notebook, is shown in Figure 5.
FIGURE 5
Surprisingly perhaps, teachers rarely shared explicit growth mindset messages during this task, but they frequently supported a growth mindset culture in implicit ways, as evidenced by critical
moments in which teachers pushed students to justify their thinking, invited students to come up to the board to share their thinking, gave students time to grapple with the task on their own before
intervening, and praised students’ thinking and struggle.
Time analysis showed that teachers afforded students ample time to grapple with and persist through the task, supporting the students in encountering challenge and struggle–a key aspect of a
mathematical mindset teaching approach. Teachers launched the task for approximately 5 minutes on average and then allowed students to grapple with the task--building cubes, collaborating with peers,
and recording in their journals or on chart paper--for an average of 53 min. The open nature of the task meant that even as students figured out one part of the question there were still other areas
to explore. After students had sufficient work time and most students had moved beyond the original question to work on the 4 × 4 × 4 cube or generalized even further, the teacher then facilitated a
whole class discussion to synthesize the ideas from multiple student groups. This was noted in five of the six videos and lasted approximately 9 minutes on average. Analyses revealed that students
were afforded significant time to work on the task in groups and that teachers pushed students to justify their thinking and to connect to each other’s ideas in whole-class discussions.
The teaching analyses revealed that teachers offered students multiple opportunities to experience mathematical ideas in multidimensional ways--they saw a 2-D representation of the cube, built a 3-D
model, drew different sized cubes, collected and recorded patterns, organized their thinking, discussed ideas with each other, and considered generalization of different sized cubes. Painted Cube was
one of many open tasks in the summer camp curriculum, which afforded students a new mathematical experience, through which they could experience important brain connections, as they saw and
experienced mathematics in different ways. An absence of any tests or grading practices during the weeks of the camps was also an important feature designed to avoid the fixed messages associated
with such practices (Kraker-Pauw et al., 2017). Teachers chose instead to give diagnostic feedback to students as they worked on open tasks (Black and Wiliam, 1998).
Students’ Mindsets and Engagement with Mathematics
The main focus of the study reported in this paper was the relationship between a mathematical mindset teaching approach and student understanding and achievement, but teacher interviews conducted
with 20 teachers also enabled consideration of the students’ shifts in engagement and mindset, as observed by the teachers. All of the interviews were coded and the three most common codes that
emerged, as teachers discussed the students’ experiences, were: tasks, student engagement, and student beliefs. Theme analysis across these three codes showed that teachers reported that a
significant factor in the students’ engagement in the mathematics in camp came from the openness of tasks, which also helped support students’ changes in mindsets.
All 20 of the teachers interviewed shared details of how the summer camp curriculum impacted students’ engagement and mindsets positively. Sixteen of the teachers (80%) reported the importance of the
tasks allowing students to develop and share their own thinking and reasoning--rather than share a single method or answer--and the ways this shifted the dynamics of the classroom. The 16 teachers
differed in the particular aspect(s) of the tasks they foregrounded in their interviews: 30% foregrounded the opportunities for multiple approaches to the tasks, 25% foregrounded the open and
explorative nature of the tasks, 25% foregrounded the focus on students’ explaining their thinking, and 20% foregrounded the opportunities for students to experience mathematical ideas physically.
The teachers explained that when their students shared their thinking with one another, they saw that there were multiple ways to think about the same problem, shifting students’ ideas about what it
means to solve a mathematics problem. The teachers noted the multiple entry points for students to participate and engage in tasks and the multiple ways students could find success. These features of
tasks resulted in an overall increase in student excitement and engagement and a decrease in anxiety and fear of making mistakes.
Interview analyses also revealed that the majority of teachers observed shifts in students’ engagement throughout the summer camp. Fifteen of the teachers (75%) commented on two types of shifts in
student engagement: shifts at the whole-class level and shifts for particular groups of students or individual students. For both types of shifts in engagement, teachers shared stories of students
increasing their participation in groupwork, sharing and showing their thinking more readily, persisting on problems rather than shutting down, and building confidence in thinking mathematically.
Eight of the teachers (40%) shared that their students displayed excitement while doing mathematics tasks (to the point of not wanting to go to recess or lunch).
Additionally, many teachers connected the task not only to increased engagement but to students’ changed beliefs about themselves as mathematics learners. Sixteen (80%) of the teachers commented on
shifts in two types of student beliefs: beliefs about the nature of mathematics and the ways students could engage in the subject (n = 6) and beliefs about their ability to be successful (n = 10).
Teachers shared that students changed their ideas of mathematics as being a closed subject of speed, individual work, and procedures, to regarding mathematics as a subject in which they could engage
deeply, visually and collaboratively. Additionally, teachers noted that many students who had had previously negative experiences with math, built confidence in their abilities as “doers of
mathematics,” became excited about mathematics, and developed positive attitudes towards their mathematics learning.
Many of the teachers shared stories of student transformation, particularly highlighting students who had previously been unsuccessful, shedding negative ideas about the nature of mathematics and
their potential, and engaging in new ways. We close this section of the paper with one of the teacher’s reflections:
“I had a student who really, really struggled. We had the entrance exam for her in the Youcubed program and she did not do very well at all, I mean very close to zero. What I found was by the end of
the summer, she was more confident to answer those questions, she had no fear about this test, she had no remorse about this test, she put answers on paper, she thought about nontraditional ways, she
put a lot more time and energy into it and she did exceptional on it regarding her first score. She went from zero to a passing score, which for somebody like that is a really, really important
thing, it builds that confidence huge. So I can just. I’ll never forget this one student who really had no way to access that information when she came into the camp, but only five weeks later, she
could develop that into some really, really solid thinking. And she wasn’t necessarily always right, but you could see her thinking progress, and that was a beautiful thing.”
Discussion and Conclusion
The summer camp intervention built upon research from psychology, neuroscience, and mathematics education, in designing new ways for students to experience mathematics (Wittmann, 1995) as an open,
visual and creative subject, that we describe as a Mathematical Mindset approach (Boaler, 2016). From psychology, the concept of mindset and the importance of helping students believe that they can
learn anything, has become widely known in education among teachers and leaders. In a national survey of teachers in the U.S., 98% reported that growth mindsets were important for students to have,
and 90% reported that they associated students’ mindsets with increased effort and persistence. Strikingly, only 20% of the same sample believed that they could foster a growth mindset through their
teaching, and 85% said they needed professional development to learn about ways to encourage mindset through teaching (Education Week research center, 2016). The disconnect between teachers’
practices and mindset messaging is well known by leaders and becoming established in different research studies. Research that has studied teachers’ assessment practices, a key way in which mindset
ideas–fixed or growth–are communicated, has found that teachers often reveal growth mindset beliefs and ideas but then assess students with closed practices–giving students no opportunity to improve
or “grow” their learning and achievement (Kraker-Pauw et al., 2017). Other studies reveal disconnects between mindset messages and the mathematics problems and tasks used in classrooms, with closed
and narrow mathematics tasks causing students to believe that students are smart or they are not, and that speed is the most important part of mathematics success (LaMar et al., 2020).
An important feature of a growth mindset approach to learning is a comfort with struggle and the belief that struggle is good for learning. Studies of beginning college students, who were asked to
engage in complex tasks, found that students were uncomfortable with struggle and their lack of awareness of the value of struggle caused them to avoid complex tasks (Deslauriers et al., 2019). The
need to encourage student comfort with struggle, and student awareness of the value of struggle for brain development (Coyle, 2018) is why messages of struggle are centralized in the mathematical
mindset approach. Alongside the messages that teachers were trained to give, the tasks that were part of the youcubed curriculum required high level thinking and gave students multiple opportunities
to struggle–in supportive classroom environments. By designing instructional tasks and teaching strategies that were fully aligned with growth mindset, the project moved beyond what has been termed a
“false growth mindset” approach - of encouraging messages with no change in teaching (Dweck, 2015; Dweck, 2016; Sun, 2018; Sun, 2019). Students received not only growth mindset messaging, but
teaching practices that reflected and reinforced this messaging.
The mathematics tasks, that emerged clearly as pivotal in the students’ experiences, from the teacher interviews, and that we have described as low floor and high ceiling, also had another important
feature–they were mathematically interesting to students. Other studies have highlighted the value of students working on tasks that are based in realistic contexts, and that give students
opportunities to consider and tackle social injustices (Gutstein, 2016). The tasks in the youcubed camps did not do this, instead they centered the idea of mathematics being ‘the science of patterns’
(Wittmann, 1995; Devlin, 1996) and they invited students to investigate patterns in the borders of squares, in the growth of cubes, in dot cards and number talks, in displays of number visuals, in
Pascal’s triangle, and in other examples of what some describe as “pure mathematics.” In both the original youcubed camp and the camps taught in ten districts, that are the focus of this paper,
students were fascinated by these mathematical investigations and deeply engaged in the discovery of patterns, confirming what Devlin has claimed to be a natural human desire–to study and understand
patterns (Devlin, 1996). Perhaps surprisingly to some, these pattern based tasks, taught students school mathematics–including the mathematics of number sense, geometry, and algebra. They also
contributed to the pursuit of equitable outcomes, helping students who had previously under achieved, see a future for themselves in mathematics, and other STEM subjects.
This study had several limitations and some unanswered questions. Missing data, due to student turnover and the difficulties of following students who changed districts, was one challenge. One
unanswered question is why some camps raised students’ achievement significantly more than others, a question that could be investigated in a further study.
Despite these limitations and unanswered questions, the data reported shows that a mathematics approach that is based on mindset and neuroscience, that enables students to embrace struggle and to
encounter mathematics in multiple ways, can have a transformative impact on students. This approach is not one that is typically used in schools, partly because of the pressure teachers feel to
“cover” the curriculum, and to prepare students for narrow tests, as well as the textbooks on offer to teachers, usually filled with narrow questions. For these reasons the teachers who took part in
the study believed that a summer camp is needed, free from these constraints, to unlock students’ potential (Boaler, 2019a), and to help them approach school mathematics differently. Some teachers
have learned about the mathematical mindset approach and have infused it into their regular classroom teaching, with resulting achievement gains for students (Anderson et al., 2018).
The study reported in this paper adds to this important evidence–showing that a two-to-four-week summer camp sharing a mathematical mindset approach can have a transformative impact on student
mathematics achievement. We hope that these different forms of evidence, from summer camp and from school teaching, will prompt policy makers to reconsider the mathematics approaches they encourage
in schools–that contravene mindset messages and have resulted in widespread under achievement across the US. When students are released from negative ideas about mathematics and themselves, they
learn and approach mathematics differently, and begin a changed, mindset infused, pathway.
Data Availability Statement
The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.
Ethics Statement
The studies involving human participants were reviewed and approved by Institutional Review Board, GSE, Stanford University. Written informed consent to participate in this study was provided by the
participants’ legal guardian/next of kin. Written informed consent was obtained from the individual(s), and minor(s)’ legal guardian/next of kin, for the publication of any potentially identifiable
images or data included in this article.
Author Contributions
JB designed and directed the study, GP-N, JD, and MS-A ran statistical analyses, TL and ML led the analysis of teaching. The full team interpreted results and contributed the writing and revising
This report is based on research funded in part by the Bill and Melinda Gates Foundation. The findings and conclusions contained within are those of the authors and do not necessarily reflect
positions or policies of the Bill and Melinda Gates Foundation.
Conflict of Interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s Note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the
reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
The authors wish to recognize Cathy Williams, Executive Director of youcubed at Stanford as a co-designer of the summer camp curriculum and member of the research team. The team also would like to
acknowledge Gregory Chojnacki, senior researcher at Mathematica, for his invaluable assistance with this research.
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Keywords: Mindset, Mathematics, teaching, learning, beliefs, student math Learning, student math Achievement
Citation: Boaler J, Dieckmann JA, LaMar T, Leshin M, Selbach-Allen M and Pérez-Núñez G (2021) The Transformative Impact of a Mathematical Mindset Experience Taught at Scale. Front. Educ. 6:784393.
doi: 10.3389/feduc.2021.784393
Received: 27 September 2021; Accepted: 23 November 2021;
Published: 10 December 2021.
Copyright © 2021 Boaler, Dieckmann, LaMar, Leshin, Selbach-Allen and Pérez-Núñez. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The
use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in
accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Jack A. Dieckmann, jackd1@stanford.edu
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CPM Homework Help
The diagram at right shows the region bounded by the $x$-axis, $f(x) = 0.5x^2$, $x = 1$, and $x = 3$. The region is revolved about the $y$-axis to create the solid shown with dotted lines. .
a. Describe a method you can use to determine the volume of the solid.
Will you use Washers or Disks? Will the bounds and integrand be written in terms of $x$ or $y$?
b. Set up and evaluate the integrals needed to calculate the volume. (Using washers, the solution will require two integrals.)
The outside of this solid will be a cylinder with radius $3$.
The inside of the solid will have a hole. But what shape is that hole? Is it cylindrical?
$\text{Or is it determined by }f(y) = \sqrt{(2y)}?$
Notice that the bottom of the hole is cylindrical (with radius $1$), while the top of the hole is determined by
Consequently, you will need to use the Washer Method twice... both of which are rotated about the $y$-axis.
Use the eTool below to help solve the problem.
Click on the link to the right to view the full version of the eTool. Calc 8-63 HW eTool
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OpenStax College Physics for AP® Courses, Chapter 1, Problem 4 (Problems & Exercises)
American football is played on a 100-yd-long field, excluding the end zones. How long is the field in meters? (Assume that 1 meter equals 3.281 feet.)
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Solution video
OpenStax College Physics for AP® Courses, Chapter 1, Problem 4 (Problems & Exercises)
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Video Transcript
This is College Physics Answers with Shaun Dychko. We are going to express the 100 yard length of a football field in meters and we multiply by 3 feet for every yard to get rid of the yards units
leaving us with feet and we want to do that because the conversion factor of the question tells us that there is 1 meter for every 3.281 feet and I write it this way so that the feet are on the
bottom and they will cancel with these feet on the top here leaving us with meters behind. And our answer is 91.4 meters and the question is, how many significant figures should it have? Strictly
speaking, when you have the number 100 written like this, this has only one significant figure however because we know this is the length of a football field, that detail is important because
football fields are well standardized and in order to have fair gameplay, one should expect quite a precise measurement of a 100 yards for all the different football fields that the teams can play
on. And so we can assume that this actually has three significant figures and so that's why 91.4 meters is the correct way to express this answer.
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Help: fdtd in Sullivan's book
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Jul 31, 2009
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I am a newer to fdtd. I am reading Sullivan's book. In the FDTD_2.2.c program, they caculate the Fourier transform of the input pulse.
/* Fourier transform of the input pulse*/ ---> why have to caculate the Fourier transform of the input pulse.
if (T<100) ---->why T have to be smller than 100
for (m=0;m<=2;m++)
{ real_in[m]=real_in[m]+cos(arg[m]*T)*ex[10] --->why have to time ex[10]
thanks a lot
May 2, 2005
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Hi joywwj
As you know FDTD equations are in time domain but some of our interesting parameters are in frequency domain such as reflections coefficients and other S parameters so we have to use Fourier
transform of the input pulse and the other parameters for FT convergence
I hope it'll be helpful
Jul 20, 2009
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I've been wondering about this, too. I think it's got to do with the fact that he's just collecting data points for that incident field - the Ex(10) is actually a location 10 cells into the
simulation space. The T<100 has to do with the propagation of that wave from one side of the simulation space to the other - he wants to stop after 100 iterations of the field calculation, which I
think would be about where the signal would hit that dielectric medium if you look at Fig. 1.5.
If somebody has a good explanation of how this works with the Fourier transform, that would be awesome!
Apr 12, 2008
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I don't have the book available right now but I think skysearcher is right.
1) T<=100 probably means that by that time the pulse has weakened to a negligible amount.
2) EX[10] is just the location (if I remember correctly chapter 2 is for 1d-simulations then
this should mean the 10th or 11th cell from the boundary
3) The reason for using the FT is that you want to know the spectral content (= the strength of
each frequency) of the pulse.
If you do this at some other point (e.g. after an interface between two materials) you can compute
the transmission and reflection coefficients for several frequencies in a single simulation
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Limiting floats to two decimal points
I want a to be rounded to 13.95. I tried using round:
>>> a
>>> round(a, 2)
You are running into the old problem with floating point numbers that not all numbers can be represented exactly. The command line is just showing you the full floating point form from memory.
With floating point representation, your rounded version is the same number. Since computers are binary, they store floating point numbers as an integer and then divide it by a power of two so 13.95
will be represented in a similar fashion to 125650429603636838/(2**53).
Double precision numbers have 53 bits (16 digits) of precision and regular floats have 24 bits (8 digits) of precision. The floating point type in Python uses double precision to store the values.
For example,
>>> 125650429603636838/(2**53)
>>> 234042163/(2**24)
>>> a = 13.946
>>> print(a)
>>> print("%.2f" % a)
>>> round(a,2)
>>> print("%.2f" % round(a, 2))
>>> print("{:.2f}".format(a))
>>> print("{:.2f}".format(round(a, 2)))
>>> print("{:.15f}".format(round(a, 2)))
If you are after only two decimal places (to display a currency value, for example), then you have a couple of better choices:
1. Use integers and store values in cents, not dollars and then divide by 100 to convert to dollars.
2. Or use a fixed point number like decimal.
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Data-First Development with gurobipy-pandas: Speed, Best Practices and Other Considerations
Monday, September 9, 2024
In a Gurobi webinar (view the recording here) on Data-First Optimization Development, Irv discussed the transformative benefits of pandas and best practices using the gurobipy-pandas library, and
walked through an example. Following is a lightly edited excerpt from Irv’s presentation and the Q&A with practitioners from around the world.
In emphasizing Data-First optimization development, I have drawn on the Job Task Analysis (JTA) created by INFORMS that serves as the basis of the Certified Analytics Professional exam. The JTA also
serves as an outline for a solution development process that prescribes preparing and working with the data before creating a model; unfortunately, this is not the way optimization is typically
taught in the Operations Research community, which focuses on models. Based on the JTA and the methodology at Princeton Consultants which is proven to deliver high quality optimization applications,
I strongly recommend you start with the data.
I discovered pandas (https://pandas.pydata.org/) about nine years ago and realized that it was going to solve a lot of problems for me in developing optimization applications. After making a number
of contributions to pandas , I was invited to join the core team, https://pandas.pydata.org/about/team.html and I participate in its ongoing improvements.
Inspired in part by some of our work in tying together pandas and Gurobi together, Gurobi built the gurobipy-pandas library, https://www.gurobi.com/features/gurobipy-pandas/, which takes advantage of
pandas to manage data, allowing Gurobi objects to be placed into pandas DataFrames and Series. The library provides the capability to write code that executes quickly when creating models. Good
knowledge of pandas is required to map mathematical formulations into gurobipy-pandas.
We have written about this topic in previous blog posts:
Question: Have you compared the performance of gurobipy-pandas against the more traditional interface or alternatives? Can you note the performance in terms of both model-building as well as how it
might affect the solution time?
Irv: We were internally using pandas to create models before gurobipy‑pandas existed. Back in 2018, we had models we’d written with gurobipy with and without pandas that were larger scale, and we
were seeing faster model building times using pandas. One of the real issues that comes into play has to do with slicing: pandas more naturally slices than gurobipy does because it has some very
efficient ways of doing groupby operations in an efficient way that I think is better than what's under the hood in gurobipy.
I don't know how that would compare today. In fact, one of the biggest pieces of overhead at the time was the naming of the constraints, which was taking the longest amount of time. The actual
creation of the model was flying by in both cases. It is useful to have names when you name your constraints and variables because it makes debugging your models a lot easier. There is an overhead
for creating names that might sometimes be the most expensive part of creating a model.
Question: Are there any advantages to this method versus maybe using Pyomo with Gurobi?
Irv: In my experience, using gurobipy instead of Pyomo results in better performance. We have evaluated Pyomo and believe its design was based on AMPL and a model‑first way of thinking, which we
avoid. In my earlier example, we inferred the sets of nodes and commodities from the tables, but in Pyomo you have to explicitly say what those sets are and read them in separately. A real advantage
working with gurobipy-pandas is that you are thinking about and understanding your data. As you develop a model in a notebook, you can look at the data, plot it, generate different descriptive
statistics, and see where missing values are. When you achieve an understanding of the data, it then plugs into the model you write. In the Pyomo approach, you are not really making that data
analysis and hooking it in. And when it comes to slicing and groupby operations, the gurobipy‑pandas is going to be a lot faster than using Pyomo.
Question: When dealing with large-scale issues, do you find it more effective to define variables within a single DataFrame, maybe allocating a column for each variable, or should you opt for a
DataFrame dedicated to each variable?
Irv: It very much depends. In my simple example earlier with one set of decision variables, I used a Series. In the commercial models that we create, there is typically a mix of variables that have
different index sets. If you consider the classical facility location problem in which you are deciding which facilities to open, you will have a variable that is indexed just on the facilities that
you might open, and then you might have other variables that are going to be indexed on things like supply and demand, and then connections in a network. In that case, you are going to place them in
separate Series or DataFrames. In the larger models we have developed at Princeton, we typically keep each set of decision variables in its own pandas Series, which allows us to keep things straight
from a development perspective: when we are sharing code and collaborating, we all know that is our best practice. We can see that all the variables are in a collection of different Series that are
named, however we want to name them—that is typically what we do here at Princeton.
To discuss this topic with Irv, email us to set up a call.
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A multivector Lagrangian for Maxwell’s equation, w/ electric and magnetic current density four-vector sources - Peeter Joot's BlogJune 2022 – Peeter Joot's Blog
A multivector Lagrangian for Maxwell’s equation, w/ electric and magnetic current density four-vector sources
[Click here for a PDF version of this and previous related posts .]
Initially I had trouble generalizing the multivector Lagrangian to include both the electric and magnetic sources without using two independent potentials. However, this can be done, provided one is
careful enough. Recall that we found that a useful formulation for the field in terms of two potentials is
F = F_{\mathrm{e}} + I F_{\mathrm{m}},
F_{\mathrm{e}} = \grad \wedge A \\
F_{\mathrm{m}} = \grad \wedge K,
and where \( A, K \) are arbitrary four-vector potentials.
Use of two potentials allowed us to decouple Maxwell’s equations into two separate gradient equations. We don’t want to do that now, but let’s see how we can combine the two fields into a single
multivector potential. Letting the gradient act bidirectionally, and introducing a dummy grade-two selection into the mix, we have
&= \rgrad \wedge A + I \lr{ \rgrad \wedge K } \\
&= – A \wedge \lgrad – I \lr{ K \wedge \lgrad } \\
&= -\gpgradetwo{ A \wedge \lgrad + I \lr{ K \wedge \lgrad } } \\
&= -\gpgradetwo{ A \lgrad + I K \lgrad } \\
&= -\gpgradetwo{ \lr{ A + I K } \lgrad }.
Now, we call
N = A + I K,
(a 1,3 multivector), the multivector potential, and write the electromagnetic field not in terms of curls explicitly, but using a grade-2 selection filter
F = -\gpgradetwo{ N \lgrad }.
We can now form the following multivector Lagrangian
\LL = \inv{2} F^2 – \gpgrade{ N \lr{ J – I M } }{0,4},
and vary the action to (eventually) find our multivector Maxwell’s equation, without ever resorting to coordinates. We have
\delta S
&= \int d^4 x \inv{2} \lr{ \lr{ \delta F } F + F \lr{ \delta F } } – \gpgrade{ \delta N \lr{ J – I M } }{0,4} \\
&= \int d^4 x \gpgrade{ \lr{ \delta F } F – \lr{ \delta N } \lr{ J – I M } }{0,4} \\
&= \int d^4 x \gpgrade{ -\gpgradetwo{ \lr{ \delta N} \lgrad } F – \lr{ \delta N } \lr{ J – I M } }{0,4} \\
&= \int d^4 x \gpgrade{ -\gpgradetwo{ \lr{ \delta N} \lrgrad } F +\gpgradetwo{ \lr{ \delta N} \rgrad } F – \lr{ \delta N } \lr{ J – I M } }{0,4}.
The \( \lrgrad \) term can be evaluated using the fundamential theorem of GC, and will be zero, as \( \delta N = 0 \) on the boundary. Let’s look at the next integrand term a bit more carefully
\gpgrade{ \gpgradetwo{ \lr{ \delta N} \rgrad } F }{0,4}
\gpgrade{ \gpgradetwo{ \lr{ \lr{ \delta A } + I \lr{ \delta K } } \rgrad } F }{0,4} \\
\gpgrade{ \lr{ \lr{\delta A} \wedge \rgrad + I \lr{ \lr{ \delta K } \wedge \rgrad }} F }{0,4} \\
\gpgrade{ \lr{\delta A} \rgrad F – \lr{ \lr{\delta A} \cdot \rgrad} F + I \lr{ \delta K } \rgrad F – I \lr{ \lr{ \delta K } \cdot \rgrad} F }{0,4} \\
\gpgrade{ \lr{\delta A} \rgrad F + I \lr{ \delta K } \rgrad F }{0,4} \\
\gpgrade{ \lr{ \lr{\delta A} + I \lr{ \delta K} } \rgrad F }{0,4} \\
\gpgrade{ \lr{ \delta N} \rgrad F }{0,4},
\delta S
&= \int d^4 x \gpgrade{ \lr{ \delta N} \rgrad F – \lr{ \delta N } \lr{ J – I M } }{0,4} \\
&= \int d^4 x \gpgrade{ \lr{ \delta N} \lr{ \rgrad F – \lr{ J – I M } } }{0,4}.
for this to be zero for all variations \( \delta N \) of the 1,3-multivector potential \( N \), we must have
\grad F = J – I M.
This is Maxwell’s equation, as desired, including both electric and (if desired) magnetic sources.
This summarizes the significant parts of the last 8 blog posts.
[Click here for a PDF version of this post]
STA form of Maxwell’s equation.
Maxwell’s equations, with electric and fictional magnetic sources (useful for antenna theory and other engineering applications), are
\spacegrad \cdot \BE &= \frac{\rho}{\epsilon} \\
\spacegrad \cross \BE &= – \BM – \mu \PD{t}{\BH} \\
\spacegrad \cdot \BH &= \frac{\rho_\txtm}{\mu} \\
\spacegrad \cross \BH &= \BJ + \epsilon \PD{t}{\BE}.
We can assemble these into a single geometric algebra equation,
\lr{ \spacegrad + \inv{c} \PD{t}{} } F = \eta \lr{ c \rho – \BJ } + I \lr{ c \rho_{\mathrm{m}} – \BM },
where \( F = \BE + \eta I \BH = \BE + I c \BB \), \( c = 1/\sqrt{\mu\epsilon}, \eta = \sqrt{(\mu/\epsilon)} \).
By multiplying through by \( \gamma_0 \), making the identification \( \Be_k = \gamma_k \gamma_0 \), and
J^0 &= \frac{\rho}{\epsilon}, \quad J^k = \eta \lr{ \BJ \cdot \Be_k }, \quad J = J^\mu \gamma_\mu \\
M^0 &= c \rho_{\mathrm{m}}, \quad M^k = \BM \cdot \Be_k, \quad M = M^\mu \gamma_\mu \\
\grad &= \gamma^\mu \partial_\mu,
we find the STA form of Maxwell’s equation, including magnetic sources
\grad F = J – I M.
Decoupling the electric and magnetic fields and sources.
We can utilize two separate four-vector potential fields to split Maxwell’s equation into two parts. Let
F = F_{\mathrm{e}} + I F_{\mathrm{m}},
F_{\mathrm{e}} &= \grad \wedge A \\
F_{\mathrm{m}} &= \grad \wedge K,
and \( A, K \) are independent four-vector potential fields. Plugging this into Maxwell’s equation, and employing a duality transformation, gives us two coupled vector grade equations
\grad \cdot F_{\mathrm{e}} – I \lr{ \grad \wedge F_{\mathrm{m}} } &= J \\
\grad \cdot F_{\mathrm{m}} + I \lr{ \grad \wedge F_{\mathrm{e}} } &= M.
However, since \( \grad \wedge F_{\mathrm{m}} = \grad \wedge F_{\mathrm{e}} = 0 \), by construction, the curls above are killed. We may also add in \( \grad \wedge F_{\mathrm{e}} = 0 \) and \( \grad
\wedge F_{\mathrm{m}} = 0 \) respectively, yielding two independent gradient equations
\grad F_{\mathrm{e}} &= J \\
\grad F_{\mathrm{m}} &= M,
one for each of the electric and magnetic sources and their associated fields.
Tensor formulation.
The electromagnetic field \( F \), is a vector-bivector multivector in the multivector representation of Maxwell’s equation, but is a bivector in the STA representation. The split of \( F \) into
it’s electric and magnetic field components is observer dependent, but we may write it without reference to a specific observer frame as
F = \inv{2} \gamma_\mu \wedge \gamma_\nu F^{\mu\nu},
where \( F^{\mu\nu} \) is an arbitrary antisymmetric 2nd rank tensor. Maxwell’s equation has a vector and trivector component, which may be split out explicitly using grade selection, to find
\grad \cdot F &= J \\
\grad \wedge F &= -I M.
Further dotting and wedging these equations with \( \gamma^\mu \) allows for extraction of scalar relations
\partial_\nu F^{\nu\mu} = J^{\mu}, \quad \partial_\nu G^{\nu\mu} = M^{\mu},
where \( G^{\mu\nu} = -(1/2) \epsilon^{\mu\nu\alpha\beta} F_{\alpha\beta} \) is also an antisymmetric 2nd rank tensor.
If we treat \( F^{\mu\nu} \) and \( G^{\mu\nu} \) as independent fields, this pair of equations is the coordinate equivalent to \ref{eqn:maxwellLagrangian:1760}, also decoupling the electric and
magnetic source contributions to Maxwell’s equation.
Coordinate representation of the Lagrangian.
As observed above, we may choose to express the decoupled fields as curls \( F_{\mathrm{e}} = \grad \wedge A \) or \( F_{\mathrm{m}} = \grad \wedge K \). The coordinate expansion of either field
component, given such a representation, is straight forward. For example
&= \lr{ \gamma_\mu \partial^\mu } \wedge \lr{ \gamma_\nu A^\nu } \\
&= \inv{2} \lr{ \gamma_\mu \wedge \gamma_\nu } \lr{ \partial^\mu A^\nu – \partial^\nu A^\mu }.
We make the identification \( F^{\mu\nu} = \partial^\mu A^\nu – \partial^\nu A^\mu \), the usual definition of \( F^{\mu\nu} \) in the tensor formalism. In that tensor formalism, the Maxwell
Lagrangian is
\LL = – \inv{4} F_{\mu\nu} F^{\mu\nu} – A_\mu J^\mu.
We may show this though application of the Euler-Lagrange equations
\PD{A_\mu}{\LL} = \partial_\nu \PD{(\partial_\nu A_\mu)}{\LL}.
\PD{(\partial_\nu A_\mu)}{\LL}
&= -\inv{4} (2) \lr{ \PD{(\partial_\nu A_\mu)}{F_{\alpha\beta}} } F^{\alpha\beta} \\
&= -\inv{2} \delta^{[\nu\mu]}_{\alpha\beta} F^{\alpha\beta} \\
&= -\inv{2} \lr{ F^{\nu\mu} – F^{\mu\nu} } \\
&= F^{\mu\nu}.
So \( \partial_\nu F^{\nu\mu} = J^\mu \), the equivalent of \( \grad \cdot F = J \), as expected.
Coordinate-free representation and variation of the Lagrangian.
F^2 =
F^{\mu\nu} F_{\mu\nu}
\lr{ \gamma_\alpha \wedge \gamma^\beta }
\epsilon_{\mu\nu\alpha\beta} F^{\mu\nu} F^{\alpha\beta},
we may express the Lagrangian \ref{eqn:maxwellLagrangian:1870} in a coordinate free representation
\LL = \inv{2} F \cdot F – A \cdot J,
where \( F = \grad \wedge A \).
We will now show that it is also possible to apply the variational principle to the following multivector Lagrangian
\LL = \inv{2} F^2 – A \cdot J,
and recover the geometric algebra form \( \grad F = J \) of Maxwell’s equation in it’s entirety, including both vector and trivector components in one shot.
We will need a few geometric algebra tools to do this.
The first such tool is the notational freedom to let the gradient act bidirectionally on multivectors to the left and right. We will designate such action with over-arrows, sometimes also using
braces to limit the scope of the action in question. If \( Q, R \) are multivectors, then the bidirectional action of the gradient in a \( Q, R \) sandwich is
Q \lrgrad R
&= Q \lgrad R + Q \rgrad R \\
&= \lr{ Q \gamma^\mu \lpartial_\mu } R + Q \lr{ \gamma^\mu \rpartial_\mu R } \\
&= \lr{ \partial_\mu Q } \gamma^\mu R + Q \gamma^\mu \lr{ \partial_\mu R }.
In the final statement, the partials are acting exclusively on \( Q \) and \( R \) respectively, but the \( \gamma^\mu \) factors must remain in place, as they do not necessarily commute with any of
the multivector factors.
This bidirectional action is a critical aspect of the Fundamental Theorem of Geometric calculus, another tool that we will require. The specific form of that theorem that we will utilize here is
\int_V Q d^4 \Bx \lrgrad R = \int_{\partial V} Q d^3 \Bx R,
where \( d^4 \Bx = I d^4 x \) is the pseudoscalar four-volume element associated with a parameterization of space time. For our purposes, we may assume that parameterization are standard basis
coordinates associated with the basis \( \setlr{ \gamma_0, \gamma_1, \gamma_2, \gamma_3 } \). The surface differential form \( d^3 \Bx \) can be given specific meaning, but we do not actually care
what that form is here, as all our surface integrals will be zero due to the boundary constraints of the variational principle.
Finally, we will utilize the fact that bivector products can be split into grade \(0,4\) and \( 2 \) components using anticommutator and commutator products, namely, given two bivectors \( F, G \),
we have
\gpgrade{ F G }{0,4} &= \inv{2} \lr{ F G + G F } \\
\gpgrade{ F G }{2} &= \inv{2} \lr{ F G – G F }.
We may now proceed to evaluate the variation of the action for our presumed Lagrangian
S = \int d^4 x \lr{ \inv{2} F^2 – A \cdot J }.
We seek solutions of the variational equation \( \delta S = 0 \), that are satisfied for all variations \( \delta A \), where the four-potential variations \( \delta A \) are zero on the boundaries
of this action volume (i.e. an infinite spherical surface.)
We may start our variation in terms of \( F \) and \( A \)
\delta S
\int d^4 x \lr{ \inv{2} \lr{ \delta F } F + F \lr{ \delta F } } – \lr{ \delta A } \cdot J \\
\int d^4 x \gpgrade{ \lr{ \delta F } F – \lr{ \delta A } J }{0,4} \\
\int d^4 x \gpgrade{ \lr{ \grad \wedge \lr{\delta A} } F – \lr{ \delta A } J }{0,4} \\
-\int d^4 x \gpgrade{ \lr{ \lr{\delta A} \lgrad } F – \lr{ \lr{ \delta A } \cdot \lgrad } F + \lr{ \delta A } J }{0,4} \\
-\int d^4 x \gpgrade{ \lr{ \lr{\delta A} \lgrad } F + \lr{ \delta A } J }{0,4} \\
-\int d^4 x \gpgrade{ \lr{\delta A} \lrgrad F – \lr{\delta A} \rgrad F + \lr{ \delta A } J }{0,4},
where we have used arrows, when required, to indicate the directional action of the gradient.
Writing \( d^4 x = -I d^4 \Bx \), we have
\delta S
-\int_V d^4 x \gpgrade{ \lr{\delta A} \lrgrad F – \lr{\delta A} \rgrad F + \lr{ \delta A } J }{0,4} \\
-\int_V \gpgrade{ -\lr{\delta A} I d^4 \Bx \lrgrad F – d^4 x \lr{\delta A} \rgrad F + d^4 x \lr{ \delta A } J }{0,4} \\
\int_{\partial V} \gpgrade{ \lr{\delta A} I d^3 \Bx F }{0,4}
+ \int_V d^4 x \gpgrade{ \lr{\delta A} \lr{ \rgrad F – J } }{0,4}.
The first integral is killed since \( \delta A = 0 \) on the boundary. The remaining integrand can be simplified to
\gpgrade{ \lr{\delta A} \lr{ \rgrad F – J } }{0,4} =
\gpgrade{ \lr{\delta A} \lr{ \grad F – J } }{0},
where the grade-4 filter has also been discarded since \( \grad F = \grad \cdot F + \grad \wedge F = \grad \cdot F \) since \( \grad \wedge F = \grad \wedge \grad \wedge A = 0 \) by construction,
which implies that the only non-zero grades in the multivector \( \grad F – J \) are vector grades. Also, the directional indicator on the gradient has been dropped, since there is no longer any
ambiguity. We seek solutions of \( \gpgrade{ \lr{\delta A} \lr{ \grad F – J } }{0} = 0 \) for all variations \( \delta A \), namely
\grad F = J.
This is Maxwell’s equation in it’s coordinate free STA form, found using the variational principle from a coordinate free multivector Maxwell Lagrangian, without having to resort to a coordinate
expansion of that Lagrangian.
Lagrangian for fictitious magnetic sources.
The generalization of the Lagrangian to include magnetic charge and current densities can be as simple as utilizing two independent four-potential fields
\LL = \inv{2} \lr{ \grad \wedge A }^2 – A \cdot J + \alpha \lr{ \inv{2} \lr{ \grad \wedge K }^2 – K \cdot M },
where \( \alpha \) is an arbitrary multivector constant.
Variation of this Lagrangian provides two independent equations
\grad \lr{ \grad \wedge A } &= J \\
\grad \lr{ \grad \wedge K } &= M.
We may add these, scaling the second by \( -I \) (recall that \( I, \grad \) anticommute), to find
\grad \lr{ F_{\mathrm{e}} + I F_{\mathrm{m}} } = J – I M,
which is \( \grad F = J – I M \), as desired.
It would be interesting to explore whether it is possible find Lagrangian that is dependent on a multivector potential, that would yield \( \grad F = J – I M \) directly, instead of requiring a
superposition operation from the two independent solutions. One such possible potential is \( \tilde{A} = A – I K \), for which \( F = \gpgradetwo{ \grad \tilde{A} } = \grad \wedge A + I \lr{ \grad \
wedge K } \). The author was not successful constructing such a Lagrangian.
This is the 8th part of a series on finding Maxwell’s equations (including the fictitious magnetic sources that are useful in engineering) from a multivector Lagrangian representation.
[Click here for a PDF version of this series of posts, up to and including this one.] The first, second, third, fourth, fifth, sixth, and
seventh parts are also available here on this blog.
There’s an aspect of the previous treatment that has bugged me. We’ve used a Lagrangian
\LL = \inv{2} F^2 – \gpgrade{A \lr{ J – I M } }{0,4}, \end{equation}
where \( F = \grad \wedge A \), and found Maxwell’s equation by varying the Lagrangian
\grad F = J – I M.
However, if we decompose this into vector and trivector parts we have
\grad \cdot F &= J \\
\grad \wedge F &= -I M,
and then put our original \( F = \grad \wedge A \) back in the magnetic term of this equation, we have a contradiction
0 = -I M,
\grad \wedge \lr{ \grad \wedge A } = 0,
provided we have equality of mixed partials for \( A \). The resolution to this contradiction appears to be a requirement to define the field differently. In particular, we can utilize two separate
four-vector potential fields to split Maxwell’s equation into two parts. Let
F = F_{\mathrm{e}} + I F_{\mathrm{m}},
F_{\mathrm{e}} &= \grad \wedge A \\
F_{\mathrm{m}} &= \grad \wedge K,
and \( A, K \) are independent four-vector potential fields. Plugging this into Maxwell’s equation, and employing a duality transformation, gives us two coupled vector grade equations
\grad \cdot F_{\mathrm{e}} – I \lr{ \grad \wedge F_{\mathrm{m}} } &= J \\
\grad \cdot F_{\mathrm{m}} + I \lr{ \grad \wedge F_{\mathrm{e}} } &= M.
However, since \( \grad \wedge F_{\mathrm{m}} = \grad \wedge F_{\mathrm{e}} = 0 \), these decouple trivially, leaving
\grad \cdot F_{\mathrm{e}} &= J \\
\grad \cdot F_{\mathrm{m}} &= M.
In fact, again, since \( \grad \wedge F_{\mathrm{m}} = \grad \wedge F_{\mathrm{e}} = 0 \), these are equivalent to two independent gradient equations
\grad F_{\mathrm{e}} &= J \\
\grad F_{\mathrm{m}} &= M,
one for each of the electric and magnetic sources and their associated fields.
Should we wish to recover these two equations from a Lagrangian, we form a multivector Lagrangian that uses two independent four-vector fields
\LL = \inv{2} \lr{ \grad \wedge A }^2 – A \cdot J + \alpha \lr{ \inv{2} \lr{ \grad \wedge K }^2 – K \cdot M },
where \( \alpha \) is an arbitrary multivector constant. Variation of this Lagrangian provides two independent equations
\grad \lr{ \grad \wedge A } &= J \\
\grad \lr{ \grad \wedge K } &= M.
We may add these, scaling the second by \( -I \) (recall that \( I, \grad \) anticommute), to find
\grad \lr{ F_{\mathrm{e}} + I F_{\mathrm{m}} } = J – I M,
which is \( \grad F = J – I M \), as desired. This resolves the eq \ref{eqn:fsquared:1720} conundrum, but the cost is that we essentially have an independent Lagrangian for each of the electric and
magnetic sources. I think that is the cost of correctness. Perhaps there is an alternative Lagrangian for the electric+magnetic case that yields all of Maxwell’s equation in one shot. My attempts to
formulate one in terms of the total field \( F = F_{\mathrm{e}} + I F_{\mathrm{m}} \) have not been successful.
On the positive side, for non-fictitious electric sources, the case that we care about in physics, we still have the pleasantry of being able to use a simple multivector (coordinate-free) Lagrangian,
and vary that in a coordinate free fashion to find Maxwell’s equation. This has an aesthetic quality that is arguably superior to the usual procedure of using the Euler-Lagrange equations and lots of
index gymnastics to find the tensor form of Maxwell’s equation (i.e. the vector part of Maxwell’s) and applying the Bianchi identity to fill in the pieces (i.e. the trivector component of Maxwell’s.)
This is the 7th part of a series on finding Maxwell’s equations (including the fictitious magnetic sources that are useful in engineering) from a multivector Lagrangian representation.
[Click here for a PDF version of this series of posts, up to and including this one.] The first, second, third, fourth, fifth, and sixth parts are also available here on this blog.
For what is now (probably) the final step in this exploration, we now wish to evaluate the variation of the multivector Maxwell Lagrangian
\LL = \inv{2} F^2 – \gpgrade{A \lr{ J – I M } }{0,4},
without resorting to coordinate expansion of any part of \( F = \grad \wedge A \). We’d initially evaluated this, expanding both \( \grad \) and \( A \) in coordinates, and then just \( \grad \), but
we can avoid both.
In particular, given a coordinate free Lagrangian, and a coordinate free form of Maxwell’s equation as the final destination, there must be a way to get there directly.
It is clear how to work through the first part of the action variation argument, without resorting to any sort of coordinate expansion
\delta S
\int d^4 x \lr{ \inv{2} \lr{ \delta F } F + F \lr{ \delta F } } – \gpgrade{ \lr{ \delta F } \lr{ J – I M } }{0,4} \\
\int d^4 x \gpgrade{ \lr{ \delta F } F – \lr{ \delta A } \lr{ J – I M } }{0,4} \\
\int d^4 x \gpgrade{ \lr{ \grad \wedge \lr{\delta A} } F – \lr{ \delta A } \lr{ J – I M } }{0,4} \\
-\int d^4 x \gpgrade{ \lr{ \lr{\delta A} \grad } F – \lr{ \lr{ \delta A } \cdot \grad } F + \lr{ \delta A } \lr{ J – I M } }{0,4} \\
-\int d^4 x \gpgrade{ \lr{ \lr{\delta A} \grad } F + \lr{ \delta A } \lr{ J – I M } }{0,4}.
In the last three lines, it is important to note that \( \grad \) acts bidirectionally, but on \( \delta A \), but not \( F \).
In particular, if \( B, C \) are multivectors, we interpret the bidirectional action of the gradient as
B \lrgrad C &=
B \gamma^\mu \lrpartial_\mu C \\
(\partial_\mu B) \gamma^\mu C
B \gamma^\mu (\partial_\mu C),
where the partial operators on the first line are bidirectionally acting, and braces have been used in the last line to indicate the scope of the operators in the chain rule expansion.
Let’s also use arrows to clarify the directionality of this first part of the action variation, writing
\delta S
-\int d^4 x \gpgrade{ \lr{\delta A} \lgrad F + \lr{ \delta A } \lr{ J – I M } }{0,4} \\
-\int d^4 x \gpgrade{ \lr{\delta A} \lrgrad F – \lr{\delta A} \rgrad F + \lr{ \delta A } \lr{ J – I M } }{0,4}.
We can cast the first term into an integrand that can be evaluated using the Fundamental Theorem of Geometric Calculus, by introducing a
a parameterization \( x = x(a_\mu) \), for which the tangent space basis vectors are \( \Bx_{a_\mu} = \PDi{a_\mu}{x} \), and the pseudoscalar volume element is
d^4 \Bx = \lr{ \Bx_{a_0} \wedge \Bx_{a_1} \wedge \Bx_{a_2} \wedge \Bx_{a_3} } da_0 da_1 da_2 da_3 = I d^4 x.
Writing \( d^4 x = -I d^4 \Bx \), we have
\delta S
-\int_V d^4 x \gpgrade{ \lr{\delta A} \lrgrad F – \lr{\delta A} \rgrad F + \lr{ \delta A } \lr{ J – I M } }{0,4} \\
-\int_V \gpgrade{ -\lr{\delta A} I d^4 \Bx \lrgrad F – d^4 x \lr{\delta A} \rgrad F + d^4 x \lr{ \delta A } \lr{ J – I M } }{0,4} \\
\int_{\partial V} \gpgrade{ \lr{\delta A} I d^3 \Bx F }{0,4}
+ \int_V d^4 x \gpgrade{ \lr{\delta A} \lr{ \rgrad F – J + I M } }{0,4}.
The first integral is killed since \( \delta A = 0 \) on the boundary. For the second integral to be zero for all variations \( \delta A \), we must have
\gpgrade{ \lr{\delta A} \lr{ \rgrad F – J + I M } }{0,4} = 0,
but have argued previously that we can drop the grade selection, leaving
\grad F = J – I M
where the directional indicator on our gradient has been dropped, since there is no longer any ambiguity. This is Maxwell’s equation in it’s coordinate free STA form, found using the variational
principle from a coordinate free multivector Maxwell Lagrangian, without having to resort to a coordinate expansion of that Lagrangian.
This is the 6th part of a series on finding Maxwell’s equations (including the fictitious magnetic sources that are useful in engineering) from a multivector Lagrangian representation.
[Click here for a PDF version of this series of posts, up to and including this one.] The first, second, third, fourth, and fifth parts are also available here on this blog.
We managed to find Maxwell’s equation in it’s STA form by variation of a multivector Lagrangian, with respect to a four-vector field (the potential). That approach differed from the usual variation
with respect to the coordinates of that four-vector, or the use of the Euler-Lagrange equations with respect to those coordinates.
Euler-Lagrange equations.
Having done so, an immediate question is whether we can express the Euler-Lagrange equations with respect to the four-potential in it’s entirety, instead of the coordinates of that vector. I have
some intuition about how to completely avoid that use of coordinates, but first we can get part way there.
Consider a general Lagrangian, dependent on a field \( A \) and all it’s derivatives \( \partial_\mu A \)
\LL = \LL( A, \partial_\mu A ).
The variational principle requires
0 = \delta S = \int d^4 x \delta \LL( A, \partial_\mu A ).
That variation can be expressed as a limiting parametric operation as follows
\delta S
= \int d^4 x
\lim_{t \rightarrow 0} \ddt{} \LL( A + t \delta A )
\lim_{t \rightarrow 0} \ddt{} \LL( \partial_\mu A + t \delta \partial_\mu A )
We eventually want a coordinate free expression for the variation, but we’ll use them to get there. We can expand the first derivative by chain rule as
\lim_{t \rightarrow 0} \ddt{} \LL( A + t \delta A )
\lim_{t \rightarrow 0} \PD{(A^\alpha + t \delta A^\alpha)}{\LL} \PD{t}{}(A^\alpha + t \delta A^\alpha) \\
\PD{A^\alpha}{\LL} \delta A^\alpha.
This has the structure of a directional derivative \( A \). In particular, let
\grad_A = \gamma^\alpha \PD{A^\alpha}{},
so we have
\lim_{t \rightarrow 0} \ddt{} \LL( A + t \delta A )
= \delta A \cdot \grad_A.
\lim_{t \rightarrow 0} \ddt{} \LL( \partial_\mu A + t \delta \partial_\mu A )
\PD{(\partial_\mu A^\alpha)}{\LL} \delta \partial_\mu A^\alpha,
so we can define a gradient with respect to each of the derivatives of \(A \) as
\grad_{\partial_\mu A} = \gamma^\alpha \PD{(\partial_\mu A^\alpha)}{}.
Our variation can now be expressed in a somewhat coordinate free form
\delta S = \int d^4 x \lr{
\lr{\delta A \cdot \grad_A} \LL + \lr{ \lr{\delta \partial_\mu A} \cdot \grad_{\partial_\mu A} } \LL
We now sum implicitly over pairs of indexes \( \mu \) (i.e. we are treating \( \grad_{\partial_\mu A} \) as an upper index entity). We can now proceed with our chain rule expansion
\delta S
&= \int d^4 x \lr{
\lr{\delta A \cdot \grad_A} \LL + \lr{ \lr{\delta \partial_\mu A} \cdot \grad_{\partial_\mu A} } \LL
} \\
&= \int d^4 x \lr{
\lr{\delta A \cdot \grad_A} \LL + \lr{ \lr{\partial_\mu \delta A} \cdot \grad_{\partial_\mu A} } \LL
} \\
&= \int d^4 x \lr{
\lr{\delta A \cdot \grad_A} \LL
+ \partial_\mu \lr{ \lr{ \delta A \cdot \grad_{\partial_\mu A} } \LL }
– \lr{\PD{x^\mu}{} \delta A \cdot \grad_{\partial_\mu A} \LL}_{\delta A}
As usual, we kill off the boundary term, by insisting that \( \delta A = 0 \) on the boundary, leaving us with a four-vector form of the field Euler-Lagrange equations
\lr{\delta A \cdot \grad_A} \LL = \lr{\PD{x^\mu}{} \delta A \cdot \grad_{\partial_\mu A} \LL}_{\delta A},
where the RHS derivatives are taken with \(\delta A \) held fixed. We seek solutions of this equation that hold for all variations \( \delta A \).
Application to the Maxwell Lagrangian.
For the Maxwell application we need a few helper calculations. The first, given a multivector \( B \), is
\lr{ \delta A \cdot \grad_A } A B
\delta A^\alpha \PD{A^\alpha}{} \gamma_\beta A^\beta B \\
\delta A^\alpha \gamma_\alpha B \\
\lr{ \delta A } B.
Now let’s compute, for multivector \( B \)
\lr{ \delta A \cdot \grad_{\partial_\mu A} } B F
\delta A^\alpha \PD{(\partial_\mu A^\alpha)} B \lr{ \gamma^\beta \wedge \partial_\beta \lr{ \gamma_\pi A^\pi } } \\
\delta A^\alpha B \lr{ \gamma^\mu \wedge \gamma_\alpha } \\
B \lr{ \gamma^\mu \wedge \delta A }.
Our Lagrangian is
\LL = \inv{2} F^2 – \gpgrade{A \lr{ J – I M } }{0,4},
\lr{\delta A \cdot \grad_A} \LL
-\gpgrade{ \lr{ \delta A } \lr{ J – I M } }{0,4},
\lr{ \delta A \cdot \grad_{\partial_\mu A} } \inv{2} F^2
\inv{2} \lr{ F \lr{ \gamma^\mu \wedge \delta A } + \lr{ \gamma^\mu \wedge \delta A } F } \\
\lr{ \gamma^\mu \wedge \delta A } F
}{0,4} \\
\lr{ \delta A \wedge \gamma^\mu } F
}{0,4} \\
\delta A \gamma^\mu F
\lr{ \delta A \cdot \gamma^\mu } F
}{0,4} \\
\delta A \gamma^\mu F
Taking derivatives (holding \( \delta A \) fixed), we have
-\gpgrade{ \lr{ \delta A } \lr{ J – I M } }{0,4}
\delta A \partial_\mu \gamma^\mu F
}{0,4} \\
\delta A \grad F
We’ve already seen that the solution can be expressed without grade selection as
\grad F = \lr{ J – I M },
which is Maxwell’s equation in it’s STA form. It’s not clear that this is really any less work, but it’s a step towards a coordinate free evaluation of the Maxwell Lagrangian (at least not having to
use the coordinates \( A^\mu \) as we have to do in the tensor formalism.)
A multivector Lagrangian for Maxwell’s equation: A summary of previous exploration.
June 21, 2022 math and physics play action, bivector, Dirac basis, dot product, electric sources, four vector, Fundamental Theorem of Geometric Calculus, Geometric Algebra, grade selection, gradient,
Lagrangian, magnetic sources, Maxwell's equation, pseudoscalar, space time algebra, spatial basis, STA, tensor formalism, variational principle, vector, wedge product
Curl of F revisited.
June 20, 2022 math and physics play bivector, electric current and charge density, electric potential, four vector, Geometric Algebra, Lagrangian density, magnetic current and charge density,
magnetic potential, Maxwell's equation, vector
A coordinate free variation of the Maxwell equation multivector Lagrangian.
June 18, 2022 math and physics play bivector, coordinate expansion, Geometric Algebra, Lagrangian density, Maxwell's equation, multivector, multivector Lagrangian, space time algebra, STA,
variational principle
Progressing towards coordinate free form of the Euler-Lagrange equations for Maxwell’s equation
June 17, 2022 math and physics play bivector, coordinates, Dirac basis, Euler-Lagrange equations, four vector, Lagrangian density, Maxwell's equation, multivector, space time algebra, STA,
variational principle
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Analytical Optimal Load Calculation of RF Energy Rectifiers Based on a Simplified Rectifying Model
Analytical Optimal Load Calculation of RF Energy Rectifiers Based on a Simplified Rectifying Model^ †
Electronics System, Department of Electrical Engineering, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands
Holst-Centre, IMEC-NL, 5656 AE Eindhoven, The Netherlands
Author to whom correspondence should be addressed.
This paper is an extended version of our paper published in The 2021 IEEE Wireless Power Transfer Conference, San Diego, CA, USA, 1–4 June 2021, titled “On the Analytical Optimal Load Resistance of
RF Energy Rectifier”.
Submission received: 29 September 2021 / Revised: 26 November 2021 / Accepted: 28 November 2021 / Published: 1 December 2021
Wireless power transfer (WPT) is an essential enabler for novel sensor networks such as the wireless powered communication network (WPCN). The efficiency of an energy rectifier is dependent on both
input power and loading condition. In this work, to maximize the rectifier efficiency, we present a low-complexity numerical method based on an analytical rectifier model to calculate the optimal
load for different rectifier topologies, including half-wave and voltage-multipliers, without needing time-consuming simulations. The method is based on a simplified analytical rectifier model based
on the diode equivalent circuit including parasitic parameters. Furthermore, by using Lambert-W function and the perturbation method, closed-form solutions are given for low-input power cases. The
method is validated by means of both simulations and measurements. Extensive transient simulation results using different diodes (Skyworks SMS7630 and Avago HSMS285x) and frequency bands (400 MHz,
900 MHz, and $2.4$ GHz) are provided for validation of the method. A 400 MHz 1- and 2-stage voltage multiplier are designed and fabricated, and measurements are conducted. Different input signals are
used when validating the proposed methods, including the single sinewave signal and the multisine signal. The proposed numerical method shows excellent accuracy with both signal types, as long as the
output voltage ripple is sufficiently low.
1. Introduction
Wireless power transfer (WPT) is an emerging technology that removes the traditional charging cables. Due to its convenience, WPT can be found or foreseen in many applications such as electric
vehicles, consumer electronics, and new communication networks [
]. Specifically, far-field WPT based on RF signals can deliver wireless power over a long distance up to kilometers, which enables new communication and sensing networks in the IoT domain such as
wirelessly powered communication network (WPCN). These networks consist of low-power sensor nodes whose power is provided by either dedicated RF sources or ambient RF energy, which prolongs the
sensors’ lifetime and reduces the maintenance cost [
The receiving side of an RF WPT system is called an energy rectifier. The rectifier converts the RF signal into a DC voltage that either directly powers electronics or is stored in storage units such
as batteries or super-capacitors. It often consists of an antenna that captures the wireless signal; a matching and rectifying network for RF-to-DC conversion; and a power management unit (PMU),
which is the rectifier’s load. The power conversion efficiency (PCE) of a rectifier has been shown to depend on both received RF power and its load; thus, the optimal load for a rectifier needs to be
understood [
]. Additionally, novel excitation waveforms such as multisine waveform featuring high peak-to-average-power ratio (PAPR) and multiple frequency components [
] have been proposed to boost the rectification efficiency. To analyze the optimal load with general excitation, transient simulations are often conducted [
], which are time-consuming and computationally intensive. Harmonic balance (HB) is another option, but its complexity poorly scales with the number of frequency components in the excitation waveform
so that it soon becomes impractical [
Instead of numerical solvers, many efforts were put into the analytical modeling of rectification. The works [
] used the time-domain method to analytically analyze the shunt-diode rectifier. Afterwards, ref [
] the model was extended with class-f harmonic termination. In [
], Bessel functions are used to separate DC component and the first harmonic of the diode voltage and the optimal load using the developed model is calculated. A limitation to the aforementioned
works is that they all assume the applied excitation to be a sine wave. Works in [
] extensively analyzed the incurred losses in the complete rectification chain and pointed out the optimal load resistance for the overall efficiency in the low input power range is equal to the
diode junction resistance and series resistance combined. The junction resistance, however, depends on the junction bias voltage, which in turn depends on the load; thus, additional steps are needed
to calculate or measure this quantity.
There have also been works focusing on developing analytical rectification models for general multisine signals. In [
], a simplified analytical model was developed to mathematically prove the efficiency gain of the multisine excitation. Later, this model was used in [
] to optimize the transmission waveform with frequency-selective fading channels because of the tractability of this rectifier model and its ability to capture the non-linearity of the rectifier
circuitry. For the same reason, this model was also used in system performance analysis and optimization of WPCNs and shows superior accuracy to the conventional linear rectifier model in [
]. Despite the successful applications of this model, the key assumptions in [
] when developing it are the ideal diode and the half-wave rectifier topology.
In our previous work [
], the model with diode parasitics in the simplest half-wave rectifier was discussed; then, the model was extended for the voltage-doubler. We also showed the low-complexity method to derive the
optimal load. The method works with general multisine input signals, provided that the output voltage ripple is small enough. In the current work, we further extend the model to generic N-stage
voltage-multipliers. More extensive transient simulations are conducted to validate the result. Two different Schottky diodes are considered in the simulation, and three different frequency bands:
400 MHz, 900 MHz, and 2.4 GHz, which are simulated as well to investigate the impact of frequency. Finally, rectifier prototypes are designed and fabricated and a measurement campaign is conducted to
provide experimental data to further support the results.
The paper is organized as follows:
Section 2
introduces the simplified analytical rectifier model for both half-wave and N-stage voltage multiplier with a realistic diode equivalent model;
Section 3
describes the calculation of the optimal load and its closed-form asymptotic solutions; and numerical and experimental validations are discussed in
Section 4
, including the simulation setup, PCB design considerations, measurement setup, results, interpretation, and discussion. Finally,
Section 5
summarizes the paper and discusses the implications of applications.
2. Analytical Rectification Model
In this section, we will summarize the rectification model for the half-wave topology and analyze the effects of diode parasitic parameters. Then, we will extend the model to a generic N-stage
voltage multiplier.
2.1. Half-Wave Rectification Model
The schematic of a half-wave rectifier is shown in
Figure 1
a. The diode is modeled by the equivalent circuit shown in
Figure 2
, where there is the ideal diode junction
$D j$
, junction capacitance
$C j$
, series resistance
$R S$
, parallel capacitance
$C P$
, and series inductance
$L S$
. Assume a multisine input voltage to the rectifier circuit being:
$v i n ( t ) = ∑ n = 0 N f − 1 V A cos ( 2 π f n t + ϕ n )$
$N f$
is the number of sub-carriers or tones,
$V A$
is the amplitude of each tone,
$f n$
$ϕ n$
are the frequency and phase of the
-th tone, respectively. The tones are assumed to follow a uniform frequency grid, such that
$f n = f 0 + n Δ f$
, with
$Δ f$
being the frequency separation. As a result,
$v i n ( t )$
is a periodic signal with period
$T = 1 / Δ f$
$N f > 1$
. The CW signal can be viewed as a special case with
$N f = 1$
and a period of
$T = 1 / f 0$
According to the Kirchhoff’s voltage and current law, and
Figure 1
a and
Figure 2
, we have the following relationships:
$C d v o u t ( t ) d t + i o u t ( t ) = C P d v C P ( t ) d t + i D j ( t ) + C j d v C j ( t ) d t$
$i D j ( t ) = i s ( e α v D j ( t ) − 1 )$
$v D j ( t ) = v i n ( t ) − v L S ( t ) − v R S ( t ) − v o u t ( t )$
$i R S ( t ) = i D ( t ) − C P d v C P ( t ) d t$
$i s$
is the diode saturation current, and
$α = 1 / ( n v t )$
$v t$
being ideality factor and thermal voltage, respectively. Equation (3) is the Shockley equation of the diode junction. Because we are interested in the DC output voltage rather than its transient, we
average both sides of Equation (2) over a signal period after the system reaches steady state:
$i d c = E { i D j ( t ) } = i s T ∫ 0 T e α v D j ( t ) d t − i s$
$i d c$
is the DC component of output current
$i o u t$
$E { . }$
denotes time averaging, and the juntion current
$i D j ( t )$
is substituted by Equation (3). Note that during the time averaging, the current terms related to capacitors vanishe. This is because in the steady state, the amount of electronic charges on a
capacitor remains the same in the beginning and the end of a period. As a result, the average current has to be zero.
Next, we assume the output capacitor
Figure 1
a is sufficiently large, such that the output voltage ripple is negligible. Hence, the output voltage over the load is effectively a DC signal, such that we can write
$i o u t ( t ) ≈ i d c$
$v o u t ( t ) ≈ i d c R L$
. This is a reasonable assumption since a steady state voltage source is essential for the proper functionality of the circuitry behind the rectifier. Following this assumption, we can further
approximate the current through diode
$i D ( t ) = C d v o u t ( t ) d t + i o u t ( t ) ≈ i d c$
. Substituting these approximations in Equation (4), we obtain
$v D j ( t ) ≈ v i n ( t ) − i R S ( t ) R S − i d c R L$
The series inductance term is dropped because
$L S$
is typically very small so naturally
$v L S ( t ) = L s d i D ( t ) d t ≈ 0$
. Similarly, the parallel capacitance term in Equation (5) is also dropped due to small
$C P$
value. As a result, Equation (5) can be rewritten by
$i R S ( t ) ≈ i d c$
, and Equation (7) is now:
$v D j ( t ) ≈ v i n ( t ) − i d c ( R S + R L )$
Substitute it back to Equation (6):
$i d c = e − α i d c ( R S + R L ) i s T ∫ 0 T e α v i n ( t ) d t − i s$
Note now that
$i d c$
is still on both sides of the equation. Move
$i s$
to the left hand side and multiply
$α R h e α ( i d c + i s ) R h$
to both sides:
$α ( i d c + i s ) R h e α ( i d c + i s ) R h = α R h e α i s R h i s T ∫ 0 T e α v i n ( t ) d t$
$R h = R S + R L$
. Equation (10) can be solved for
$i d c$
by using the principle branch of the Lambert W-function [
$i d c ( v i n , R L ) = − i s + 1 α R h W α R h ( i s + z d c ) e α i s R h$
$z d c = i s T ∫ 0 T e α v i n ( t ) d t − i s$
, which is a monotonic function with the amplitude of input voltage, and
$W ( x )$
is the Lambert W-function whose value is the solution of
to the equation
$w e w = x$
. The Lambert W-function does not have an explicit formula but can be evaluated by simple numerical methods described in [
2.2. N-Stage Voltage-Multiplier Rectification Model
In this section, we will generalize the analytical half-wave rectification model developed in the previous section to
-stage voltage-multiplier.
Figure 1
b shows the schematic of a
-stage voltage-multiplier. A voltage-multiplier is often used to boost the output DC voltage by cascading voltage-doublers. The capacitors
$C n$
with even
are used to provide DC offset to each stage so the output voltage is stepped up gradually. Assume all diodes used in
Figure 1
b are the same. According to the Kirchhoff’s current law, for the upper diode of the last stage
$D 2 N$
$C 2 N − 1 d v o u t ( t ) d t + i o u t ( t ) = C 2 N d v C 2 N ( t ) d t + C P 2 N d v C P ( 2 N ) ( t ) d t + i D j ( 2 N ) ( t ) + C j ( 2 N ) d v C j ( 2 N ) ( t ) d t$
$C P ( 2 N )$
$D j ( 2 N )$
denote the parallel capacitance and junction of the
$2 N$
-th diode. By time-averaging the above equation in the steady state like we did with Equation (2), we get:
$i d c = E { i D j ( 2 N ) ( t ) } = i s T ∫ 0 T e α v D j ( 2 N ) ( t ) d t − i s$
According to the Kirchhoff’s voltage law for the diode
$D 2 N$
, the junction voltage is:
$v D j ( 2 N ) ( t ) = v i n ( t ) − v C 2 N ( t ) − v L S ( 2 N ) ( t ) − v R S ( 2 N ) ( t ) − v o u t ( t )$
$L S ( 2 N )$
$R S ( 2 N )$
are series inductance and series resistance of the
$2 N$
-th diode. The same treatment with the series inductance and resistance can be done as when analyzing the half-wave rectifier, to approximate
$v L S ( 2 N ) ≈ 0$
$v R S ( 2 N ) ≈ i d c R S$
. To ensure a small output ripple, all capacitors in a voltage-multiplier need to be large enough so the time constant is larger than the signal period. This means the capacitors can be considered
short-circuits at high frequency so that their voltage drop has only DC component [
]. At DC, the capacitors are open circuit and the input is shorted because the input voltage does not have DC component. As a result, the voltage drop across
$C 2 N$
$v C 2 N ( t ) = − 2 N − 1 2 N i d c R L$
which equals to the voltage drop across the first
$2 N − 1$
cascaded diodes. Using it in Equations (13) and (14), we get:
$i d c = e α i d c ( R S + R L 2 N ) i s T ∫ 0 T e α v i n ( t ) d t − i s$
Again, solve it for
$i d c$
using the principle branch of the Lambert W-function:
$i d c ( v i n , R L ) = − i s + 1 α R W α R ( i s + z d c ) e α i s R$
$R = R S + R L 2 N$
is the number of stages of a voltage-multiplier. Given the similarity between Equation (11) for half-wave and Equation (17) for voltage-multiplier, the half-wave model can be viewed as a special case
of the multiplier model with number of stages
$N = 0.5$
3. Calculation of Optimal Load Resistance
3.1. Problem Formulation
We have so far developed the output DC current in the last chapter. By definition, the output DC power
$P d c$
The optimal load that maximizes $P d c$ can be found by numerically evaluating Equation (18) based on Equation (17) with a scanned $R L$. This solution is called numerical solution of the analytical
To find the closed-form solution to the optimal load, the first derivative of
$P d c$
needs to be formulated. We first write the
$i d c$
’s first derivative with respect to the load using (17):
$∂ i d c ∂ R L = − 1 2 α N R 2 W ( E ) + 1 α R ∂ W ( E ) ∂ R L$
$E = α R ( i s + z d c ) e α i s R$
. To simplify the notation, we omit the dependency of both
$i d c$
$P d c$
$v i n$
$R L$
in equations from here on. The derivative of
$P d c$
with respect to the load resistance:
$∂ P d c ∂ R L = i d c ( i d c + 2 R L ∂ i d c ∂ R L )$
Use Equations (17) and (19) in (20) we further have:
$∂ P d c ∂ R L = i d c − i s + N R − R L α N R 2 W ( E ) + 2 R L α R ∂ W ( E ) ∂ R L ≜ i d c I 0$
Since $i d c$ is by definition a non-negative number, solving for $I 0 = 0$ is equivalent to solving $∂ P d c ∂ R L = 0$. However, finding the closed-form solution can be a challenge due to the lack
of explicit formula of the W-function.
3.2. Closed-Form Approximations for Low Input Power
The W-function can be approximated in closed-form under some assumptions. By definition, the value of the W-function in Equation (21) is the solution of the equation:
$W ( E ) e W ( E ) = α i s R e α i s R + α z d c R e α i s R$
An easy solution would be obtained if the second term on the right hand side was absent, which is
$W ( E ) = α i s R$
. This situation is similar to solving a nonlinear ordinary differential equation (ODE). When the ODE is constructed in a way that there is a simple part added by a complex nonlinear term, often the
perturbation method can be applied if the nonlinear term is small [
]. Here, we apply the perturbation method to solve Equation (22) under the condition that
$z d c$
is small compared with
$i s$
. Because
$z d c$
is a monotonic function of input voltage amplitude, the condition is equivalent to a small input power.
The exact solution
$W ( E )$
is obviously a function of
$z d c$
; thus, a power series about
$z d c$
exists that approximates
$W ( E )$
$W ( E ) ≈ W a ( K ) ( E ) = ∑ k = 0 K z d c k W k$
$W a ( K ) ( E )$
is the approximation to
$W ( E )$
with order
and the coefficients
$W k$
$∀ k = 0 , 1 , … , K$
are the generating solutions. Naturally, the smaller
$z d c$
is, the less order
is needed before the approximation converges. After substituting (23) into (22) and taking logarithm on both sides, we obtain
$ln ∑ k = 0 K z d c k W k + ∑ k = 0 K z d c k W k = ln α ( i s + z d c ) R + α i s R$
Taking the derivative of this equation from 0 to
times and equating
$z d c$
to zero each time gives us
$K + 1$
generating equations. We list them with
$K = 2$
$ln ( W 0 ) + W 0 = ln ( α i s R ) + α i s R$
$1 W 0 W 1 + W 1 = 1 i s$
$− W 1 W 0 2 + 2 W 2 W 0 + 2 W 2 = − 1 i s 2$
Then, it is straightforward to get the generating solutions:
$W 0 = α i s R , W 1 = α R 1 + α i s R , W 2 = − α 2 R 2 ( 2 + α i s R ) 2 ( 1 + α i s R ) 3$
Using the first two generating solutions and (23), the
$W ( E )$
is approximated in order
$K = 1$
$W a ( 1 ) ( E ) = α i s R + α R z d c 1 + α i s R$
Substitute this into
$I 0 = 0$
and solve for
$R L$
, the optimal resistance based on the 1st order approximation is:
$R L , 1 * = 2 N ( R S + 1 α i s )$
This solution is then the closed-form approximation with the first-order truncation for extremely low input power. Note that what is inside the bracket is the diode’s resistance at low power [
], which suggests the load should match the resistance of all diodes in series to obtain maximum output power.
Furthermore, using all three generating solutions in (28), the
$W ( E )$
is approximated in order
$K = 2$
by the following:
$W a ( 2 ) ( E ) = W a ( 1 ) ( E ) − α 2 R 2 ( 2 + α i s R ) z d c 2 2 ( 1 + α i s R ) 3$
This approximation of higher order is accurate over a wider
$z d c$
range than the 1st order approximation in (29). By substituting this into
$I 0 = 0$
, multiplying a positive term
$α ( α − 1 + i s R ) 4 / ( i s 2 z d c )$
to both sides of the equation, and simplifying, we get a cubic equation about
$μ R 3 + μ α i s R 2 − μ + 5 z d c / 2 α 2 i s 2 R + 1 α 3 i s 2 = 0$
$μ = z d c / 2 − i s$
. Since only positive multipliers are used during the derivation of (32), the Equation (32) is equivalent to
$∂ P d c ∂ R L = 0$
. During the simplification of the above equation, we used approximation
$R ≈ R L / 2 N$
in order to simplify the derivation. This is supported by the fact that the diode series resistance
$R S$
is normally no more than a few tens of Ohm while the optimal load is typically in the order of kilo-Ohm. As the optimal load decreases to lower magnitudes with higher input power, the approximation
(31) will become inaccurate, as we will show in
Section 4
for validation results. The solution to the above cubic function is found by using Cardano’s general cubic formula. The roots of a cubic equation
$a x 3 + b x 2 + c x + d = 0$
are given by:
$x k = − 1 3 a ( b + ξ k B + Δ 0 ξ k B ) , k ∈ { 0 , 1 , 2 }$
$x k$
is the
-th root,
$B = Δ 1 ± Δ 1 2 − 4 Δ 0 3 2 3$
$Δ 0 = b 2 − 3 a c$
$Δ 1 = 2 b 3 − 9 a b c + 27 a 2 d$
$ξ = − 1 + − 3 2$
. The choice of plus or minus in
is arbitrary as long as it does not lead to
$B = 0$
. We then choose the smallest positive real root out of the three, i.e.,
$R L , 2 * = min ∀ k ∈ Θ ( 2 N x k )$
$Θ = { k ∈ { 0 , 1 , 2 } | x k is real and positive }$
Theorem 1.
The smallest positive root of (32) is the optimal load resistance that maximizes $P d c$ when $z d c < 2 i s$.
Proof of Theorem 1.
Denote the left hand side of (32) by $C I 0$. $C I 0$ has a positive y-intercept $1 α 3 i s 2$, so its positive before R increases to its minimum positive root and becomes negative after that. $C I
0$ has the same polarity as partial derivative $∂ P d c ∂ R L$ because there is only a positive term multiplied to $I 0$, which means the first positive root is a local maximizer of $P d c$. It can
be easily proven that $C I 0$ either monotonically decreases or increases first then decreases when $z d c < i s$ by inspecting $C I 0$’s derivative. This means $C I 0$ always has a single positive
root when $z d c ≤ 2 i s$; thus, the local maximizer is also a global maximizer. When $z d c ≥ 2 i s$, there may be more than one positive root, but the small $z d c$ assumption is violated so the
perturbation approximation is inaccurate anyway. □
4. Validation and Discussions
The validation consists of two parts: simulation and measurement, whose details will be explained in this section. The results and insights obtained from the validation will also be discussed.
4.1. Simulation Setup and Results
To verify the accuracy of the proposed methods, a set of transient simulations are carried out with MATLAB Simscape Electrical [
] by sweeping the load
$R L$
. The load sweep starts from 10
to several tens of k
. Two low-barrier Schottky diodes Skyworks SMS7630 and Avago HSMS285x are used for comparison. Key parameters of the two considered types of diode are taken from their data sheets and summarized in
Table 1
. In Simscape, the junction capacitance
$C J$
is modeled as a voltage dependent parameter calculated based on zero-bias junction capacitance
$C J 0$
, junction potential
$V J$
, and grading coefficient
. All capacitors in
Figure 1
are set to 500 pF. Besides, the simulation is carried out in three different frequency bands: 400 MHz, 900 MHz, and 2.4 GHz, due to their availability of license-free bands.
Figure 3
(a1) shows the calculated optimal load of three different rectifier topologies, i.e., half-wave, 1-, and 2-stage voltage-multipliers with Skyworks SMS7630 diode at 400 MHz frequency band. It can be
seen that the numerical solution to the analytical model (red) has good accuracy compared with simulated results (blue) for all topologies. This proves that our proposed simplified analytical model
is sufficiently accurate and that no numerical simulation is needed. Besides, the optimal load of 1- and 2-stages voltage-multipliers are roughly 2 and 4 times larger than that of the half-wave
rectifier, which corresponds to 2 and 4 times more series diodes from the load’s view point.
Figure 3
(a1) also shows the closed-form solutions with truncation order
$K = 1$
(yellow) and
$K = 2$
(purple). When
$K = 1$
, the closed-form result is accurate only when input power is extremely low and is an upper bound of optimal load. This is helpful when determining the specification of an adaptive optimal load
system. For
$K = 2$
, the valid input power region is wider until approximately 30 mV of input voltage amplitude, after which it becomes inaccurate. This completely closed-form solution is helpful in very low power
applications due to its extremely low computational complexity.
Figure 3
(a2,a3) also shows results at 900 MHz and 2.4 GHz bands. No clear difference can be observed when frequency is increased to 2.4 GHz. This is because the Skyworks diode has very small parasitic
parameters, see
Table 1
, which means the reactance part of the diode impedance remains negligible within the frequency spectrum that we considered.
Figure 3
(b1–b3) shows the calculated optimal load at three different frequency bands, with multisine signals consisting of four subcarriers that are separated by 1 MHz. The numerical solution to the
analytical model (red) still shows high accuracy in low input power region, while larger discrepancy is observed as input amplitude increases than when
$N f = 1$
. This is because a much larger signal period and thus a much larger ripple is created by the multisine signal. As a result, to minimize the output ripple, an output R-C section with a much larger
time-constant is needed than when single-sinusoid signal is used. In the simulation, the output capacitor is 500 pF all the time, so when the load decreases, it will come to a point when the output
ripple becomes significant, which is also when the low ripple assumption of the rectification model fails. In practice, this can be avoided by using a large enough capacitor in the output R-C section
based on the applied signal.
Figure 4
shows the optimal load calculated by the proposed methods and simulation when using Avago HSMS285x. Similarly to the results with the Skyworks diode, the proposed analytical solution (red) shows very
good accuracy compared with simulated results (blue). However, at 2.4 GHz, a slightly larger discrepancy can be observed between the analytical and the simulated results, due to the fact that the
Avago diode has considerably larger parasitic capacitance and inductance as can be seen from
Table 1
. This means at higher frequency, the effect of parasitics becomes more significant while the simplified model neglects it. Nevertheless, the error is still minor within the frequency range that we
Another observation is that the optimal load with the Avago diode is almost twice as large as the Skywork’s. This can be interpreted from the order 1 closed-form solution. The saturation current $i
s$ of the Avago diode is almost half of the Skyworks one. According to (30), this corresponds to an approximate twice as large optimal load. This observation shows the optimal load of a rectifier is
highly dependent on the diode’s parameter.
4.2. Measurement Setup
To further validate the results, the one-, and two-stage voltage multiplier PCBs are designed in Altium Designer at 400 MHz and fabricated, see
Figure 5
. Only 400 MHz is chosen for fabrication because of the lack of high-frequency probes for debugging purposes in our lab. An IS400 substrate with dielectric constant
$ϵ r = 4.3$
and thickness
$h = 0.119$
mm is used. Grounded co-planar waveguide (GCPW) with vias is used as transmission line with track and slot width of 0.225 mm and 0.17 mm to ensure 50
characteristic impedance. An edge-mount SMA connector is used for RF input, and a pin header is used to connect external variable load. All capacitors used on the PCBs are 500 pF. The used Schottky
diode is SMS7630-040LF from Skyworks. Measurements have been conducted to obtain experimental data to validate the model presented in
Section 2
The variable load is achieved by a resistor bank on a bread board, see
Figure 6
. In total, there are 14 resistors, and 16 resistance values are used in the measurement. The used resistance values are listed in
Table 2
A Rohde & Schwarz SMW200A signal generator is used as RF source to generate the input signal. The RF source is fed to the SMA connector on the PCB using a coaxial RF cable. The average output voltage
of the rectifier is measured by a Keysight MSO7104B digital oscilloscope. The acquisition of measured average voltage is controlled by a windows PC through SCPI remote commands via a USB connection.
The remote control session is established by MATLAB using Instrument Control Toolbox. The output DC voltage is measured 20 times during each measurement with
s interval between consecutive acquisitions. The 20 acquired samples are then averaged to obtain a final measurement result. A picture of the measurement setup is shown in
Figure 7
The measured output DC voltage and power with one- and two-stage multipliers are shown in
Figure 8
Figure 9
, respectively. The voltage and power are plotted against the load resistance with different tone amplitude
$V A$
. During measurement,
$V A$
is measured at the central pin of the SMA connector on the PCB using a Teledyne LeCroy SDA816Zi serial data analyzer with a ZS1000 active probe with 1 GHz bandwidth. The fluctuation with the measured
results with low input
$V A$
and especially low load resistance is because of the low output voltage, which is close to the digital oscilloscope’s noise floor. Despite this, the measured and simulated results are consistently in
very good agreement.
Figure 8
Figure 9
also show the output DC voltage and power calculated by the analytical model given by (17) and (18). The accuracy of the analytical model is confirmed with respect to both measured and simulated
results when
$N f = 1$
for all input levels. When
$N f = 4$
, however, the model fails to predict the rectifier output with small loads when the input level is high. This is because high PAPR signals in general (in this case the multisine signal) need an R-C
section with a higher time constant than the conventional single-sine signal to eliminate the output voltage ripple, and a negligible ripple is a prerequisite for our simplified model, as we
explained in
Section 4.1
. Indeed from
Figure 8
Figure 9
, as the load increases, which leads to a larger time-constant, the model becomes more and more accurate.
Moreover, the optimal load based on the measured data as a function of
$V A$
is shown in
Figure 10
. The simulated results and the numerical solution of the analytical model are also shown as solid and dashed lines, respectively, for comparison. Note that the resolution of the measured optimal
load is limited by the step size of the variable load listed in
Table 2
. Also note that the optimal load associated with the lowest input level tends to be an outlier since the rectifier’s output voltage is close to the oscilloscope’s noise floor, so more randomness is
observed on the left-most measured point in
Figure 10
a,b. It can be seen that the simulated data is in close agreement with the measured data when
$N f = 1$
$N f = 4$
. The numerical solution of the analytical model also shows great agreement with the measured data except in the high input
$V A$
region with four-tone multisine signal, which has already been explained before.
5. Conclusions
In this paper, we analyzed simplified analytical rectification models for the half-wave rectifier and the N-stage voltage-multiplier. The targeted rectifier topologies are generic, and the models
consider the diode as its realistic equivalent circuit. Based on the models, a set of methods that calculate the optimal loading condition for the rectifiers are given, including a low-complexity
numerical method, and closed-form approximations for low input power scenarios. The proposed methods are validated by both simulation and measurement.
The simulation results show that the parameters of the diode, namely, saturation current $i s$ and ideality factor n, significantly influence the optimal loading condition. The simulation results
also show that the carrier frequency does not influence the optimal loading condition with Skyworks SMS7630 diode. The effect of frequency gets larger only when the frequency and the diode parasitics
get larger. In our simulation, the Avago HSMS285x diode with much higher parasitics exhibits more discrepancy between simulated and analytical optimal loads at $2.4$ GHz than 400 MHz and 900 MHz.
However, the discrepancy is still negligible. This means the frequency impact is negligible at least below $2.4$ GHz.
The proposed numerical and closed-form methods have low computational complexity, which can provide a head start when designing a rectifier system. It provides very good accuracy without the need for
either harmonic balance or transient simulation provided that the output voltage ripple is eliminated. Moreover, the proposed methods are also valid for general signals, for example, novel input
signals such as the multisine waveform, with which the problem can quickly become infeasibly large for harmonic balance as the number of tones increases [
]. Another possible application is adaptive load control in an actual rectifier to ensure optimal efficiency. Implementation of such a control scheme constitute future work.
Author Contributions
Conceptulization, L.Y. and G.D.; methodology, L.Y. and J.R.; validation, L.Y.; formal analysis, L.Y., G.D. and J.R.; writing—original draft preparation, L.Y.; writing—review and editing, G.D. and
J.R.; visualization, L.Y.; project administration, G.D.; funding acquisition, G.D. All authors have read and agreed to the published version of the manuscript.
This research was funded by EIT Digital with project number 17199.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
The data can be made available upon request to the correspondence author.
The authors gratefully acknowledge the advice on PCB design and fabrication received from Erwin Allebes and Chengyao Shi, and laboratory support by Sherwin Gatchalian.
Conflicts of Interest
The authors declare no conflict of interest.
Figure 2.
Diode equivalent circuit [
Figure 3. Optimal load calculated by transient simulation (blue), numerical solution of the analytical model (red), and closed-form solutions (yellow and purple), when (a) $N f = 1$, and (b) $N f =
4$ with $Δ f = 1$ MHz, with half-wave, 1-, and 2-stage voltage-multipliers using Skyworks SMS7630. The results under different frequency bands (a1,b1) 400 MHz, (a2,b2) 900 MHz, and (a3,b3) 2.4 GHz
are shown, too.
Figure 4. Optimal load calculated by transient simulation (blue), numerical solution of the analytical model (red), and closed-form solutions (yellow and purple) when (a) $N f = 1$, and (b) $N f = 4$
with $Δ f = 1$ MHz, simulated with half-wave, 1-, and 2-stages voltage-multipliers using Avago HSMS285x. The results under different frequency bands (a1,b1) 400 MHz, (a2,b2) 900 MHz, and (a3,b3) 2.4
GHz are shown, too.
Figure 8. Comparison between measurement, simulation, and analytical model of the one-stage multiplier. Output DC voltage with (a) $N f = 1$ and (c) $N f = 4$; output DC power with (b) $N f = 1$ and
(d) $N f = 4$.
Figure 9. Comparison between measurement, simulation, and analytical model of the two-stage multiplier. Output DC voltage with (a) $N f = 1$ and (c) $N f = 4$; output DC power with (b) $N f = 1$ and
(d) $N f = 4$.
Figure 10. Optimal loads calculated by simulation, numerical solution of the analytical model, and measurement with (a) $N f = 1$ and (b) $N f = 4$. Note that the measured results with the lowest
input tone amplitude $v A$ tend to be outliers since the rectifier’s output voltage is close to the noise floor of the oscilloscope.
$i s$ $R S$ N $C J 0$ M $V J$ $L S$ $C P$
SMS7630 5 $μ$A 20 $Ω$ $1.05$ 0.14 pF $0.4$ 0.51 V 0.05 nH 0.005 pF
HSMS285x 3 $μ$A 25 $Ω$ $1.06$ 0.18 pF $0.5$ 0.35 V 2 nH 0.08 pF
R1 R2 R3 R4 R5 R6 R7$/$R8 R7
Value [$Ω$] 16.2 100.3 328 558 822 1.2 k 1.99 k 3.29 k
R8 R9 R11 $/$ R14 R10 R11 R12 R13 R14
Value [$Ω$] 5.1 k 7.49 k 9.63 k 11.97 k 14.96 k 18.01 k 22 k 26 k
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© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:
Share and Cite
MDPI and ACS Style
Yao, L.; Dolmans, G.; Romme, J. Analytical Optimal Load Calculation of RF Energy Rectifiers Based on a Simplified Rectifying Model. Sensors 2021, 21, 8038. https://doi.org/10.3390/s21238038
AMA Style
Yao L, Dolmans G, Romme J. Analytical Optimal Load Calculation of RF Energy Rectifiers Based on a Simplified Rectifying Model. Sensors. 2021; 21(23):8038. https://doi.org/10.3390/s21238038
Chicago/Turabian Style
Yao, Lichen, Guido Dolmans, and Jac Romme. 2021. "Analytical Optimal Load Calculation of RF Energy Rectifiers Based on a Simplified Rectifying Model" Sensors 21, no. 23: 8038. https://doi.org/10.3390
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Honours Actuarial and Financial Mathematics Co-op (B.Sc.)
Nov 09,
Undergraduate Calendar 2023-2024 [-ARCHIVED CALENDAR-]
Honours Actuarial and Financial Mathematics Co-op (B.Sc.)
Enrolment in this program is limited. Admission is by selection, and possession of the published minimum requirements does not guarantee admission. Selection is based on academic
achievement and an interview but requires, as a minimum, submission of the on-line application by the stated deadline, and completion of Level II Honours Actuarial and Financial Mathematics
(B.Sc.) with a Grade Point Average of at least 5.0.
Information about this program and the selection procedure can be obtained from the Science Career and Cooperative Education Office.
Program Notes
1. This is a five-level (year) co-op program which includes two eight-month work terms which must be spent in actuarial and financial mathematics related placements.
2. Students must be registered full-time and take a full academic workload, as prescribed by Level and Term.
3. Students are required to complete SCIENCE 2C00 prior to the Fall Term of Level III. Students are required to complete SCIENCE 3C00 before the first work placement.
4. Students interested in focusing on financial mathematics are strongly encouraged to complete MATH 2XX3 , 3FF3 and one of COMPSCI 1MD3 , MATH 1MP3 or 3NA3 . Students should note that
MATH 2X03 is a prerequisite for MATH 2XX3 and that MATH 2R03 is a prerequisite for MATH 3FF3 .
5. Students must complete STATS 4A03 or COMMERCE 2FB3 (or COMMERCE 3FA3).
6. Students should take COMMERCE 1AA3 and COMMERCE 2FA3 by the end of Level II, to enable completion of COMMERCE 2FB3 (or COMMERCE 3FA3) in a Fall Term of Level III or IV.
7. Alternatives for meeting the requirement of three units of STATS 4A03 or COMMERCE 2FB3 (or COMMERCE 3FA3) would include distance learning courses accredited by the actuarial agencies
for fulfillment of either the Applied Statistical Methods VEE, or the Corporate Finance VEE, respectively. Students considering this alternative must speak with a faculty advisor from
the Department of Mathematics and Statistics .
8. Students who entered the program prior to September 2023 may use COMMERCE 4FP3 towards the Course List.
Course List
120 units total (Levels I to IV), of which no more than 48 units may be Level I
Level I: 30 Units
Completed prior to admission to the program
Level II: 30 Units
30 units
Completion of Level II Honours Actuarial and Financial Mathematics program including:
Level III
Consists of academic studies (Fall Term), Co-op Work Term (Winter Term), and Co-op Work Term (Spring/Summer Term)
Fall Term: 15 units:
0-3 units
0-3 units
• Electives
Winter Term:
Work Term
1 course
Spring/Summer Term:
Work Term
1 course
Level IV
Consists of academic studies (Fall and Winter Terms) and Co-op Work Term (Spring/Summer Term)
Fall and Winter Terms: 30 units:
0-3 units
9 units
• Course List (See Program Notes 4 and 8 above.)
12-15 units
• Electives
Spring/Summer Term:
Work Term
1 course
Level V
Consists of Co-op Work Term (Fall Term) and academic studies (Winter Term)
Fall Term:
Work Term
1 course
Winter Term: 15 units:
6 units
3 units
6 units
• Electives
Requirements For Students Who Entered in September 2020 or Prior
120 units total (Levels I to IV), of which no more than 48 units may be Level I
Level I: 30 Units
Completed prior to admission to the program
Level II: 30 Units
30 units
Completion of Level II Honours Actuarial and Financial Mathematics program including:
Level III
Consists of academic studies (Fall Term), Co-op Work Term (Winter Term), and Co-op Work Term (Spring/Summer Term)
Fall Term: 15 units:
0-3 units
0-6 units
• Electives
Winter Term:
Work Term
1 course
Spring/Summer Term:
Work Term
1 course
Level IV
Consists of academic studies (Fall and Winter Terms) and Co-op Work Term (Spring/Summer Term)
Fall and Winter Terms: 30 units:
3 units
3 units
0-3 units
0-3 units
9 units
• Course List (See Program Notes 4 and 8 above.)
9-15 units
• Electives
Spring/Summer Term:
Work Term
1 course
Level V
Consists of Co-op Work Term (Fall Term) and academic studies (Winter Term)
Fall Term:
Work Term
1 course
Winter Term: 15 units:
3 units
3 units
3 units
• Course List (See Program Notes 4 and 8 above.)
6 units
• Electives
Co-op Program Chart
│ │FALL TERM │WINTER TERM │SPRING/SUMMER TERM│
│ │ │ │ │
│ │(September to December) │(January to April) │(May to August) │
│ │ │Work Term │Work Term │
│ │15 units from Academic Level III │ │ │
│Level III│+ │SCIENCE 3WT0 │SCIENCE 3WT0 │
│ │SCIENCE 3C00 │ │ │
│ │ │ │ │
│ │ │ │Work Term │
│ │ │ │ │
│Level IV │15 units from Academic Levels III, IV│15 units from Academic Levels III, IV│SCIENCE 4WT0 │
│ │ │ │ │
│ │ │ │ │
│ │Work Term │ │ │
│ │ │ │ │
│Level V │SCIENCE 5WT0 │15 units from Academic Level IV │ │
│ │ │ │ │
│ │ │ │ │
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27908 calculation for speed of ball mill of 21 rpm
WEBIn some situations, the months and day result of this age calculator may be confusing, especially when the starting date is the end of a month. For example, we count Feb. 20 to Mar. 20 to be
one month. However, there are two ways to calculate the age from Feb. 28, 2022 to Mar. 31, 2022. If we consider Feb. 28 to Mar. 28 to be one month, then ...
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WEBJun 1, 2015 · Such significant effect of grinding media size has already been observed earlier for the planetary mill [15, 43] and for ball mills [21]. The median impact energy losses
presented in Table 8 are ...
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WEBGeneral: This calculator is designed to coincide with standard feed and speed charts for various materials and carbide end mills. In the future we will be adding data for HSS and cobalt end
mills. Therefore the diameters that are presented are the most common diameters in the range of 1/8" 1" and 3mm 25mm. The Green Boxes are calculated.
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WEBWith our RPM calculator, you can calculate your engine's RPM based off of your vehicle's speed, tire diameter, rear gear ratio, and transmission gear ratio. Keep in mind that our calculator
does NOT take into account drive train loss, road and weather conditions, or the skill of the driver. Engine RPM, or revolutions per minute, is defined as ...
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WEBThis set of Mechanical Operations Multiple Choice Questions Answers (MCQs) focuses on "Ball Mill". 1. What is the average particle size of ultrafine grinders? a) 1 to 20 µm b) 4 to 10 µm c) 5
to 200 µm ... At what speed stirred mill operates? a) 50 to 500 rpm b) 5 to 10 rpm c) 10 to 500 rpm d) 100 to 1500 rpm
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WEBDec 16, 2023 · The goal of this calculator is to enable you to calculate the swing speed by using the ball speed. Using historical data from various other players and tournaments, a formula is
created to give you an idea of how fast you are swinging depending on the speed of the golf ball off the golf club head. Keep in mind that the faster the ball comes off ...
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WEBCrushed ore is fed to the ball mill through the inlet; a scoop (small screw conveyor) ensures the feed is constant. For both wet and dry ball mills, the ball mill is charged to approximately
33% with balls (range 3045%). Pulp (crushed ore and water) fills another 15% of the drum's volume so that the total volume of the drum is 50% charged.
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WEBNov 16, 2020 · If the observed capacity of a mill at speed n1 is = T1 tph, the capacity T2 of the same mill at speed n2 should be. The acceleration factor of the ball or rod mass is a function
of the peripheral speed of the mill. Thus. P .
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WEBOperating Speed of Rotation of a Ball MillIntroductionA ball mill is a cylindrical device used to grind or mix materials like ores, chemicals, ceramic raw materials, and paints. It rotates
around a horizontal axis and partially filled with the material to be ground plus the grinding medium (balls). The diameter of the ball mill and the size of the balls can significantly .
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WEBJul 26, 2023 · Speed of the outermost edge of a rotating blade. Formula. Blade Tip Speed (ft/min) = (π * Diameter * RPM) / 12. Unit. Feet per minute (ft/min) or Meters per second (m/s)
Importance. Determines efficiency, performance, and safety of blades. Appliion. Wind turbines, propellers, cutting tools, etc.
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WEBFeb 20, 2023 · The RPM (rotations per minute) of a ball mill depends on the diameter of the mill and the desired particle size and grinding efficiency. However, as a general rule of thumb, the
optimal RPM for a ...
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WEBto ball filling variation in the mill. The results obtained from this work show, the ball filling percentage variation is between – % which is lower than mill ball filling percentage,
according to the designed conditions (15%). In addition, acquired load samplings result for mill ball filling was %.
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WEBThe critical speed of a ball mill in rpm whose diameter is 12 inches with grinding balls diameter of 1⁄2 in is approximately _____ rpm. Your solution's ready to go! Our expert help has broken
down your problem into an easytolearn solution you can count on.
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WEBFigure Grate discharge mill. ... (1988) have developed a method to calculate ball mill charge by using a grinding circuit simulator with a model of ball wear in a tumbling mill. ... The work
input to a mill increases in proportion to the speed, and ball mills are run at as high a speed as is possible without centrifuging. Normally this is ...
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WEBOct 1, 2015 · Recommended Mill Operating Speed RPM. Here is a table of typically recommended ball mill speed or rod mill speed as a % of critical will operate at. In summary, the larger the
mill, the slower you will want the RPM to .
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WEBThe proper speed and feed help improve tool life and remove material at the optimal rate. Find SFM, IPM, RPM, and more here.
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WEBMay 7, 2024 · Start with writing down the known values. Let's say that you know the diameter and RPM of the driver pulley (d₁ = m and n₁ = 1000 RPM), the diameter of the driven pulley (d₂ =
m), and the transmitting power (P = 1500 W).You have also measured the distance between the pulley centers to be equal to D = 1 m.. Determine .
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WEBDuration (Time) formula. The time, or more precisely, the duration of the trip, can be calculated knowing the distance and the average speed using the formula: t = d / v. where d is the
distance travelled, v is the speed (velocity) and t is the time, so you can read it as Time = Distance / Speed. Make sure you convert the units so both their ...
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WEBAug 14, 2023 · RPM Calculator RPM Calculator Speed (m/s): Diameter (meters): Calculate RPM FAQs GEGCalculatorsGEG Calculators is a comprehensive online platform that offers a wide range of
calculators to er to various needs. With over 300 calculators covering finance, health, science, mathematics, and more, GEG Calculators provides .
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WEBAug 1, 2013 · Sepúlveda (2004) has done calculations on ball breakage based on impact, showing that the speed (v) in metres per second at which a ball could be moving, can be estimated by (3)
v = π N c D mill where N c (rad/s) is the critical mill speed, and D mill the mill diameter (m).
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WEBCutting Speed = 75 (fpm)Diameter of Cutter = for a drill. Onethird the speed for countersinking would be = 789/ 3 = 263 RPM. Center Drill RPM Calculations. A center drill or combination drill
and countersink (Figure 6) is used for spotting holes in workpieces or for making center holes for turning work.
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WEBFeb 14, 2020 · About the Author. Photo Credits. Performing a conversion from RPM to speed in a linear direction involves two steps: first convert the RPM to a standard angular velocity, and
then use the formula v = ωr to convert to linear velocity. You divide the figure in RPM by 60, multiply by 2π and then multiply by the radius of the circle.
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WEBMay 20, 2023 · Solved a face milling operation on machine steel plate isEnd mill information and tips End mill rpm and feed rate rule of thumb ?Feed rpm mill end rule thumb rate metric tool
sizes mig welding forum edit added. Proper Speed For End Mills The Home Machinist! Check Details. Aluminum end mill 3 flute standard length
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WEBMar 31, 2021 · The critical speed of ball mill is given by, where R = radius of ball mill; r = radius of ball. But the mill is operated at a speed of 15 rpm. Therefore, the mill is operated at
100 x 15/ = % of critical speed.
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WEBOct 21, 2023 · distance = speed x time. Rate and speed are similar since they both represent some distance per unit time like miles per hour or kilometers per hour. If rate r is the same as
speed s, r = s = d/t. You can use the equivalent formula d = rt which means distance equals rate times time. distance = rate x time. To solve for speed or rate use .
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WEBThe formula for machine RPM is mentioned below: Machine RPM= (Motor × Pulley 1 Diameter)/Pulley 2 Diameter. Example: RMP= (1465 x 200)/250 = 1172 (rpm) Motor RPM is the rotational speed of the
motor. Pulley 1 diameter refers to the diameter of the motor (drive) pulley. Pulley 2 diameter represents the diameter of the machine (driven) pulley.
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WEBThis free percentage calculator computes a number of values involving percentages, including the percentage difference between two given values.
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WEBOct 19, 2006 · For instance, if your jar had in inside diameter of 90 mm and your milling media was mm diameter lead balls, the optimum rotation would be 98 RPM. Optimum RPM= .65 x Critical
speed (cascading action of the media stops) Critical speed = /sqrt (Jar Media ) with dimensions in inches.
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WEBFixed fixed = . Characteristic speed. DN = d o n max. and. n max = DN / d o. DN = characteristic speed (rpm) d o = nominal diameter of the screw (mm) n max = maximum allowable rotational speed
(typical 60,000 150,000 mm/min) Typical DN values for ball screw return designs are: 76,200 mmRPM (3,000 inrpm) for Standard external return ...
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WEBFeb 15, 2019 · 3. Results and discussion. Fig. 1 shows as an example the pressuretime record corresponding to a milling experiment performed using the following experimental conditions: ω d =
250 rpm, k = and BPR = 24, from which a t ig value of 110 min and 22 s was determined. As can be seen, the pressure spike at ignition is very intense and the .
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WEBAug 3, 1999 · 2. Experiment. To examine the dependence of critical rotation speed on ballcontaining fraction, we measured critical speeds at various ballcontaining fractions from to stepped by
Since at lower fraction than we could not observe the centrifugal motion, we chose this fraction range. A jar of ballmill consists of a cylinder ...
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WEB1 day ago · In a nutshell, with this RPM calculator, you can compute any one of the RPM, vehicle speed, transmission ratio, or tire diameter by providing the other three!. For example, let's
say you are driving a car at 60 mph, you're in 3 rd gear, and your tachometer indies 3,500 know that your tire code is: 185/55R15. First, you can look up a .
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WEBResult #1: This mill would need to spin at RPM to be at critical speed. Result #2: This mill's measured RPM is % of critical speed. Calculation Backup: the formula used for Critical Speed is:
N c = (D ) where, Nc is the critical speed,in revolutions per minute, D is the mill effective inside diameter, in feet.
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Exponential Functions with Base e
Learning Outcome
• Evaluate exponential functions with base [latex]e[/latex]
As we saw earlier, the amount earned on an account increases as the compounding frequency increases. The table below shows that the increase from annual to semi-annual compounding is larger than the
increase from monthly to daily compounding. This might lead us to ask whether this pattern will continue.
Examine the value of [latex]$1[/latex] invested at [latex]100\%[/latex] interest for [latex]1[/latex] year, compounded at various frequencies.
Frequency [latex]A\left(t\right)={\left(1+\frac{1}{n}\right)}^{n}[/latex] Value
Annually [latex]{\left(1+\frac{1}{1}\right)}^{1}[/latex] [latex]$2[/latex]
Semiannually [latex]{\left(1+\frac{1}{2}\right)}^{2}[/latex] [latex]$2.25[/latex]
Quarterly [latex]{\left(1+\frac{1}{4}\right)}^{4}[/latex] [latex]$2.441406[/latex]
Monthly [latex]{\left(1+\frac{1}{12}\right)}^{12}[/latex] [latex]$2.613035[/latex]
Daily [latex]{\left(1+\frac{1}{365}\right)}^{365}[/latex] [latex]$2.714567[/latex]
Hourly [latex]{\left(1+\frac{1}{\text{8766}}\right)}^{\text{8766}}[/latex] [latex]$2.718127[/latex]
Once per minute [latex]{\left(1+\frac{1}{\text{525960}}\right)}^{\text{525960}}[/latex] [latex]$2.718279[/latex]
Once per second [latex]{\left(1+\frac{1}{31557600}\right)}^{31557600}[/latex] [latex]$2.718282[/latex]
These values appear to be reaching a limit as n increases. In fact, as n gets larger and larger, the expression [latex]{\left(1+\frac{1}{n}\right)}^{n}[/latex] approaches a number used so frequently
in mathematics that it has its own name: the letter [latex]e[/latex]. This value is an irrational number, which means that its decimal expansion goes on forever without repeating. Its approximation
to six decimal places is shown below.
A General Note: The Number [latex]e[/latex]
The letter e represents the irrational number
as n increases without bound
The letter e is used as a base for many real-world exponential models. To work with base e, we use the approximation, [latex]e\approx 2.718282[/latex]. The constant was named by the Swiss
mathematician Leonhard Euler (1707–1783) who first investigated and discovered many of its properties.
In our first example, we will use a calculator to find powers of e.
Calculate [latex]{e}^{3.14}[/latex]. Round to five decimal places.
Show Solution
Investigating Continuous Growth
So far we have worked with rational bases for exponential functions. For most real-world phenomena, however, e is used as the base for exponential functions. Exponential models that use e as the base
are called continuous growth or decay models. We see these models in finance, computer science, and most of the sciences, such as physics, toxicology, and fluid dynamics.
The Continuous Growth/Decay Formula
For all real numbers r, t, and all positive numbers a, continuous growth or decay is represented by the formula
• a is the initial value,
• r is the continuous growth or decay rate per unit time,
• and t is the elapsed time.
If r >[latex]0[/latex], then the formula represents continuous growth. If r < [latex]0[/latex], then the formula represents continuous decay.
For business applications, the continuous growth formula is called the continuous compounding formula and takes the form
• P is the principal or the initial amount invested,
• r is the growth or interest rate per unit time,
• and t is the period or term of the investment.
In our next example, we will calculate continuous growth of an investment. It is important to note the language that is used in the instructions for interest rate problems. You will know to use the
continuous growth or decay formula when you are asked to find an amount based on continuous compounding. In previous examples we asked that you find an amount based on quarterly or monthly
compounding where, in that case, you used the compound interest formula.
A person invested [latex]$1,000[/latex] in an account earning a nominal [latex]10\%[/latex] per year compounded continuously. How much was in the account at the end of one year?
Show Solution
In the following video, we show another example of interest compounded continuously.
How To: Given the initial value, rate of growth or decay, and time [latex]t[/latex], solve a continuous growth or decay function
1. Use the information in the problem to determine a, the initial value of the function.
2. Use the information in the problem to determine the growth rate r.
1. If the problem refers to continuous growth, then r > [latex]0[/latex].
2. If the problem refers to continuous decay, then r < [latex]0[/latex].
3. Use the information in the problem to determine the time t.
4. Substitute the given information into the continuous growth formula and solve for A(t).
In our next example, we will calculate continuous decay. Pay attention to the rate – it is negative which means we are considering a situation where an amount decreases or decays.
Radon-222 decays at a continuous rate of [latex]17.3\%[/latex] per day. How much will [latex]100[/latex] mg of Radon-[latex]222[/latex] decay to in [latex]3[/latex] days?
Show Solution
In the following video, we show an example of calculating the remaining amount of a radioactive substance after it decays for a length of time.
Continuous growth or decay functions are of the form [latex]A\left(t\right)=a{e}^{rt}[/latex]. If r > [latex]0[/latex], then the formula represents continuous growth. If r < [latex]0[/latex], then
the formula represents continuous decay. For business applications, the continuous growth formula is called the continuous compounding formula and takes the form [latex]A\left(t\right)=P{e}^{rt}[/
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Fraction City
CONTRIBUTOR: Dr. Mavis Kelley, mkelley@ badlands.nodak.edu
Materials: Poster board for each student (or pairs); 8 colors of 9×12 construction paper, cut into strips for each student (1 of each color – extra for practicing); glue; markers; ruler; plastic cars
or trucks;
Fractional Concepts: Continuous Fractions (partitioned wholes); Fraction Sense; Addition and Subtraction of Fractions; Equivalent Fractions Procedure
1. Have the students practice folding paper strips into equal parts. Thirds, sixths, ninths, and twelfths are more difficult for children to fold. Discuss how many parts/folds are in each strip and
how the fractions are named.
2. Mark the large piece of construction paper with a vertical line about 2 cm from the left margin.
3. Glue on unfolded strip (one whole) lengthwise on the sheet by placing it flush against the line. This is First Street. (One whole=First Street).
4. Show the students how to fold a strip into two equal parts by working with the fold until the strip lies flat. Darken the crease with a pencil or thin marker.
5. Place this strip below the first one, leaving 2-3 cm between the strips.
6. Do not glue the strips until they are all done. That way the strips can still be moved.
7. Work with the students to fold another strip in half and then in half again. Open the strip and fold it back and forth so it will stay flat. Darken the folds and place this strip 2-3 cm below the
second strip. Continue the same process for eighths.
8. Help children fold another strip in thirds. Take the time needed to make the folds as accurate as possible. You can use a ruler. Prepare this strip as before and place it on the large sheet,
leaving about 2-3 cm space between all the strips.
9. Fold the next strip in thirds and then in half to get sixths. Prepare and position the strip in the same manner as previously described. Discuss how the same result can be obtained by folding a
strip in half and then by folding those sections into thirds. Have the children try it. Continue for ninths and twelfths.
10. Throughout the process, ask the students to look for patterns. How many folds does a strip have? How many parts? Use this discussion to rearrange the strips so that the strip with 2 parts is
followed by the strips with 3 parts, 4 parts, etc. Glue the strips to the poster board.
11. With older students, make the connection to multiplying fractions while folding. Example: if you are folding the fourth strip, you fold in half and half again. 1/2 of 1/2 is 1/4.
12. Each strip of paper is a street and all streets are the same length.
13. Each part of a street is a block, and all the blocks on one street are the same length. Therefore, First Street has one block, Second Street has two blocks, and so on.
14. Have the children drive their cars on a street. Help them discuss on which street they are located and how many blocks they have driven. Help them to become aware that there is a different name
for the total distance on each street because each street has a different number of blocks.
15. Talk about how hard it is to tell where they are at without the strteets being labeled. Lead them into identifying their location by street signs in which the bottom number tells how many blocks
a street has, and the top number tells how many blocks on that street their car has driven.
1 The number of blocks driven
2 The number of blocks on the street
16. Have the students put the street signs on all the blocks on their map of Fraction City.
17. Use the Fraction City map for teaching the following concepts:
Comparing Fractions – Have the students use 2-3 toy cars to drive in Fraction City. Ask them to drive two blocks on Third Street and park. Ask them to drive another car two blocks on Fourth Street
and park. Encourage them to discuss how far the two cars are from the beginning of each street. Encourage them to explore the fact that even though each car traveled two blocks, the cars did not
travel the same distance. Discuss the same number of blocks on other streets. Encourage further exploration and discussion.
Equivalent Fractions – Drive 1 block on Second Street. Drive 2 blocks on Fourth Street. What do you notice?
Addition of Fractions of Like and Unlike Denominators –
Travel 4 blocks on Sixth Street; travel 2 more blocks on Sixth Street. How far have you traveled? Make up additional problems such as this. 4/6 = 2/6 = 6/6 (you went the entire street)
Travel one block on Third Street. Travel one block on Fourth Street. How could you determine the distance traveled? (Find the block with equivalent fractions.)
Subtraction of Fractions – Travel three blocks on Ninth Street. Go back one block. How far have you traveled?
18. When ready, students can write number sentences to describe their movement on their Fraction City map.
19. To explore mixed numerals, place two Fraction Cities together.
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Alice Deanin
Dr. Alice Deanin
This is the homepage of Dr. Alice Deanin.(picture): (official picture)
Department of Mathematical Sciences, Villanova University,
800 Lancaster Avenue, Villanova, PA 19085-1699 USA
e-mail: alice.deanin@villanova.edu
Spring 2015 was my last semester at Villanova. I am now retired from the faculty. I no longer have an office or a phone at Villanova, but my email is still active, and my primary method for contact.
I look forward to hearing from you!
Research interests:
My research training is in Number Theory, and my primary interests in this area are Computational Number Theory and Diophantine Approximation. My published research has been about p-adic continued
fraction algorithms, but my recent endeavors have been directed at continued fractions in power series fields, particularly those with bounded partial quotients. This rather solitary investigation
has been neglected of late in favor of more socially interactive endeavors.
I have strong interest in and commitment to teaching. My axiomatic premise as a teacher is that all students have talents to develop, use, and enjoy mathematics. Instruction should be directed at
ensuring that all students succeed in the classroom, and value their success there. I have used this basic premise in dealing with students at many levels and in different programs. I have done
classroom enrichment in primary grade and middle school classrooms, written curriculum materials and conducted workshops supporting integrative teaching of Mathematics and Science at the Middle and
High School level, in suburban and city Regional School districts in NJ and PA. I have supervised independent study projects for undegraduate math majors and education majors interested in
mathematics education at all levels, as well as education and curriculum development projects with graduate mathematics students.
I like teaching the freshman Calculus for Life Sciences, (which is really intro to mathematical models in biology). I work every semester with junior and senior mathematics majors, usually teaching
modern algebra or linear algebra. I also have seized the geometry course, offered as an elective for math majors and math grad students, required for math education majors. I took my inspiration for
this course from a geometry topics course given by Thurston and Conway in Princeton and later at the Geometry Center. These courses all provided opportunities for interactive communication about
mathematics and cooperative and project based work.
I conduct and direct seminars, as capstone courses for our graduate program leading to M.A. in Mathematics. In these seminars, each student selects a topic to investigate for the semester, and gives
four presentations and writes four reports on the topic, at increasing levels of sophistication. I heartily recommend this colloquium type of format; it requires that the instructor develop some
project management skills and sufficient chutzpah to offer to direct projects in anything. But it has had a tremendous payoff in study and research skills for students, and more importantly, a
tangible evidence of the interconnectedness of different, seemingly distant, branches in mathematics. Graduate Math Seminar History of Topics
1999 Grad Seminar 2002 Grad Seminar 2003 Grad Seminar 2004 Grad Seminar 2005 Tu Grad Seminar
2005 Th Grad Seminar 2006 Grad Seminar 2007 Tu Grad Seminar 2007 Th Grad Seminar 2008 Grad Seminar
2009 Grad Seminar 2010 Grad Seminar 2011 Grad Seminar 2012 Grad Seminar 2013 Grad Seminar 2014 Grad Seminar
Mathematical Links
21 May 2015 alice.deanin@villanova.edu
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Use systuneOptions to create an option set for the systune function.
options = systuneOptions returns the default option set for the systune command.
options = systuneOptions(Name,Value) creates an option set and sets Properties using one or more name-value arguments
Display — Information to display
'final' (default) | 'off' | 'iter'
Amount of information to display during systune runs, specified as one of these values:
• 'final' — Display a one-line summary at the end of each optimization run. The display includes the best achieved values for the soft and hard constraints, fSoft and gHard. The display also
includes the number of iterations for each run.
Final: Soft = 1.09, Hard = 0.68927, Iterations = 58
• 'sub' — Display the result of each optimization subproblem.
When you use both soft and hard tuning goals, the software solves the optimization as a sequence of subproblems of the form:
$\underset{x}{\mathrm{min}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{max}\left(\alpha f\left(x\right),g\left(x\right)\right).$
Here, x is the vector of tunable parameters, f(x) is the largest normalized soft-constraint value, and g(x) is the largest normalized hard-constraint value. (See the “Algorithms” section of the
systune reference page for more information.) The software adjusts the multiplier α so that the solution of the subproblems converges to the solution of the original constrained optimization
problem. When you select 'sub', the report includes the results of each of these subproblems.
alpha=0.1: Soft = 3.97, Hard = 0.68927, Iterations = 8
alpha=0.5036: Soft = 1.36, Hard = 0.68927, Iterations = 8
alpha=1.47: Soft = 1.09, Hard = 0.68927, Iterations = 42
Final: Soft = 1.09, Hard = 0.68927, Iterations = 58
• 'iter' — Display optimization progress after each iteration. The display includes the value after each iteration of the objective parameter being minimized. The objective parameter is whichever
is larger of αf(x) and g(x). The display also includes a progress value that indicates the percent change in the constraints from the previous iteration.
Iter 1: Objective = 4.664, Progress = 93%
Iter 2: Objective = 2.265, Progress = 51.4%
Iter 3: Objective = 0.7936, Progress = 65%
Iter 4: Objective = 0.7183, Progress = 9.48%
Iter 5: Objective = 0.6893, Progress = 4.04%
Iter 6: Objective = 0.6893, Progress = 0%
Iter 7: Objective = 0.6893, Progress = 0%
Iter 8: Objective = 0.6893, Progress = 0%
alpha=0.1: Soft = 3.97, Hard = 0.68927, Iterations = 8
Iter 1: Objective = 1.146, Progress = 42.7%
Iter 2: Objective = 1.01, Progress = 11.9%
alpha=1.47: Soft = 1.09, Hard = 0.68927, Iterations = 42
Final: Soft = 1.09, Hard = 0.68927, Iterations = 58
• 'off' — Run in silent mode, displaying no information during or after the run.
MaxIter — Maximum number of iterations in each optimization run
300 (default) | positive scalar
Maximum number of iterations in each optimization run, when the run does not converge to within tolerance, specified as a positive scalar.
RandomStart — Number of additional optimizations starting from random values
0 (default) | nonnegative scalar
Number of additional optimizations starting from random values of the free parameters in the controller, specified as a nonnegative scalar.
If RandomStart = 0, systune performs a single optimization run starting from the initial values of the tunable parameters. Setting RandomStart = N > 0 runs N additional optimizations starting from N
randomly generated parameter values.
systune tunes by finding a local minimum of a gain minimization problem. To increase the likelihood of finding parameter values that meet your design requirements, set RandomStart > 0. You can then
use the best design that results from the multiple optimization runs.
Use with UseParallel = true to distribute independent optimization runs among MATLAB^® workers (requires Parallel Computing Toolbox™ software).
UseParallel — Option to enable parallel computing
false (default) | true
Option to enable parallel computing, specified as the comma-separated pair consisting of 'UseParallel' and false or true.
When you use the RandomStart option to run multiple randomized optimization starts when tuning a structured controller, you can also use parallel computing to distribute the optimization runs among
workers in a parallel pool. When you set this option to true, if there is an available parallel pool, then the software performs independent optimization runs concurrently among workers in that pool.
If no parallel pool is available, one of the following occurs:
• If you select Automatically create a parallel pool in your Parallel Computing Toolbox preferences (Parallel Computing Toolbox), then the software starts a parallel pool using the settings in
those preferences.
• If you do not select Automatically create a parallel pool in your preferences, then the software performs the optimization runs successively, without parallel processing.
Using parallel computing requires Parallel Computing Toolbox software.
SkipModels — Models or design points to ignores
[] (default) | array of linear indices
Models or design points to ignore, specified as an array of linear indices.
Use this option to skip specific models or ignore portions of the design space when tuning gain-scheduled control systems. For example, you might want to skip grid points outside the flight envelope
of an airplane model, or points outside the operating range for tuning. Identify the models to skip by absolute index in the array of models to tune. Using SkipModels lets you narrow the scope of
tuning without reconfiguring each tuning goal. For more information, see Change Requirements with Operating Condition.
SoftTarget — Target value for soft constraints
0 (default) | scalar
Target value for soft constraints, specified as a scalar.
The optimization stops when the largest soft constraint value falls below the specified SoftTarget value. The default value SoftTarget = 0 minimizes the soft constrains subject to satisfying the hard
SoftTol — Relative tolerance for termination
0.001 (default) | scalar
Relative tolerance for termination, specified as a scalar.
The optimization terminates when the relative decrease in the soft constraint value decreases by less than SoftTol over 10 consecutive iterations. Increasing SoftTol speeds up termination, and
decreasing SoftTol yields tighter final values.
SoftScale — A-priori estimate of best soft constraint value
1 (default) | scalar
A-priori estimate of best soft constraint value, specified as a scalar.
For problems that mix soft and hard constraints, providing a rough estimate of the optimal value of the soft constraints (subject to the hard constraints) helps to speed up the optimization.
MinDecay — Minimum decay rate for closed-loop poles
1e-7 (default) | positive scalar
Minimum decay rate for stabilized dynamics, specified as a positive scalar.
Most tuning goals carry an implicit closed-loop stability or minimum-phase constraint. Stabilized dynamics refers to the poles and zeros affected by these constraints. The MinDecay option constrains
all stabilized poles and zeros to satisfy:
• Re(s) < -MinDecay (continuous time).
• log(|z|) < -MinDecay (discrete time).
Adjust the minimum value if the optimization fails to meet the default value, or if the default value conflicts with other requirements. Alternatively, use TuningGoal.Poles to control the decay rate
of a specific feedback loop.
For more information about implicit constraints for a particular tuning goal, see the reference page for that tuning goal.
MaxRadius — Maximum spectral radius for stabilized dynamics
1e8 (default) | scalar
Maximum spectral radius for stabilized dynamics, specified as a scalar.
This option constrains all stabilized poles and zeros to satisfy |s| < MaxRadius. Stabilized dynamics are those poles and zeros affected by implicit stability or minimum-phase constraints of the
tuning goals. The MaxRadius constraint is useful to prevent these poles and zeros from going to infinity as a result of algebraic loops becoming singular or control effort growing unbounded. Adjust
the maximum radius if the optimization fails to meet the default value, or if the default value conflicts with other requirements.
MaxRadius is ignored for discrete-time tuning, where stability constraints already impose |z| < 1.
For more information about implicit constraints for a particular tuning goal, see the reference page for that tuning goal.
Create Options Set for systune
Create an options set for a systune run using five random restarts. Also, set the display level to show the progress of each iteration, and increase the relative tolerance of the soft constraint
value to 0.01.
options = systuneOptions('RandomStart',5,'Display','iter',...
Alternatively, use dot notation to set the values of options.
options = systuneOptions;
options.RandomStart = 5;
options.Display = 'iter';
options.SoftTol = 0.01;
Configure Option Set for Parallel Optimization Runs
Configure an option set for a systune run using 20 random restarts. Execute these independent optimization runs concurrently on multiple workers in a parallel pool.
If you have the Parallel Computing Toolbox software installed, you can use parallel computing to speed up systune tuning of fixed-structure control systems. When you run multiple randomized systune
optimization starts, parallel computing speeds up tuning by distributing the optimization runs among workers.
If Automatically create a parallel pool is not selected in your Parallel Computing Toolbox preferences (Parallel Computing Toolbox), manually start a parallel pool using parpool (Parallel Computing
Toolbox). For example:
If Automatically create a parallel pool is selected in your preferences, you do not need to manually start a pool.
Create a systuneOptions set that specifies 20 random restarts to run in parallel.
options = systuneOptions('RandomStart',20,'UseParallel',true);
Setting UseParallel to true enables parallel processing by distributing the randomized starts among available workers in the parallel pool.
Use the systuneOptions set when you call systune. For example, suppose you have already created a tunable control system model, CLO. For tuning this system, you have created vectors SoftReqs and
HardReqs of TuningGoal requirements objects. These vectors represent your soft and hard constraints, respectively. In that case, the following command uses parallel computing to tune the control
system of CL0.
[CL,fSoft,gHard] = systune(CL0,SoftReqs,HardReqs,options);
Version History
Introduced in R2016a
R2016a: Functionality moved from Robust Control Toolbox
Prior to R2016a, this functionality required a Robust Control Toolbox™ license.
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Mathematics: essential learning?
Are there things everyone should be required to learn? If so, what are they?
A page of logarithms from the Handbook of Chemistry and Physics, 44th edition, 1962-1963
There are lots of things that are useful to know or be able to do. Reading and writing are fundamental. Knowing how to count, add and subtract. Grammar can be useful, and spelling too. So is
recognising street signs. The list could go on.
These are things that are useful to know, but they are not identical to things students have to study. In high school in the US, I had to take two years of a foreign language in order to get into a
good university. French was my worst subject. Then, at Rice University, I had to take two years of a language to graduate, even though my major was physics. I chose German this time around, and
despite studying hard, was lucky to pass. For me, studying foreign languages was challenging, and I retained little of what I learned.
I vaguely remember some of the things learned in school mathematics classes, like interpolating in a table of logarithms. To multiply or divide numbers, we would look up the logarithm of each number,
add or subtract the logarithms and then find the number corresponding to the result. For greater accuracy, we would interpolate in the tables, namely estimate the number between two entries in the
I learned how to use a slide rule, which is basically two rulers with logarithmic scales that can be used to multiply and divide. I remember in year 8 daring to use my slide rule in an exam, and then
checking the result by calculating it longhand.
These skills became outdated decades ago, after the introduction of pocket calculators. No one says today that anyone should have to learn how to interpolate in tables of logarithms or to use a slide
rule. Most young people have never heard of a slide rule.
Some knowledge becomes obsolete and other knowledge is never used. So is there anything that everyone must study and learn?
The math myth
These reflections are stimulated by Andrew Hacker’s new book The Math Myth. He is greatly disturbed by the requirement that all US students must study math (or maths as we say in Australia) to a
level far beyond what is required in most people’s lives and jobs.
Hacker, a political scientist at Queens College in New York City, actually loves maths, and shows his knowledge of the field by dropping references to polynomials and Kolmogorov equations. He is
ardent in his support of learning maths, primarily arithmetic (requiring addition, subtraction, multiplication and division) and practical understanding of real world problems. His target for
criticism is requirements for learning algebra, trigonometry and calculus that damage the morale and careers of many otherwise capable students.
In the US, according to Hacker, the most common reason students fail to complete high school or university is a maths requirement. Everyone has to pass maths courses, and learn how to solve quadratic
equations, whether they are going to become a hairdresser, truck driver or ballet dancer. His argument is that many people have talents they are prevented from fully developing because of an absurd
requirement to pass courses in mathematics. Even when students pass, many of them quickly forget what they learned because they never use it.
Hacker makes a bolder claim. He says that in many professions in which maths might seem essential, actually most practitioners use only arithmetic. This includes engineering. Hacker interviewed many
engineers who told him that they never needed to solve algebraic equations or use trigonometric functions.
On the flip side, Hacker cites studies of some occupations, like carpet laying, in which workers in essence solve difficult equations, but they do it in a way passed down from experienced workers.
The irony is that many of these workers never passed the maths classes mandated for finishing high school.
The resulting picture is damning. Millions of students struggle through maths classes, some of them falling to the wayside, others developing maths anxiety, yet few of them ever use the knowledge
presented in these classes.
Why maths requirements?
How has this situation arisen? Hacker puts the blame on leaders of the mathematics profession, mostly elite pure mathematicians, who sit on panels that advise on high school and university
syllabuses. Few of these research stars have any expertise in teaching, and indeed few of them spend much time with beginning students. Not only do they seldom visit a high school classroom, but most
avoid teaching large first-year university maths classes. Educational administrators defer to these gurus rather than consulting with teachers who actually know what is happening with students.
It might be argued that being able to do well in maths is a good indicator of doing well in other subjects. Perhaps so, but this is not a good argument for imposing maths on all students. Research on
expert performance shows that years of dedicated practice are required to become extremely good at just about any skill, including music, sports, chess and maths. The sort of practice required,
called deliberate practice, involves focused attention on challenges at the limits of one’s ability. This sort of practice can compensate for and indeed supersede many shortcomings in so-called
general intelligence. In other words, you don’t need to be good at maths to become highly talented in other fields.
Hacker argues that the test most commonly used for entry to US universities, the SAT, is unfairly biased towards maths, to the detriment of students with other capabilities. Not only do maths classes
screen out many students with talents in other areas, but selection mechanisms for the most prestigious universities, whose degrees are tickets to lucrative careers, unfairly discriminate against
those whose interests and aptitudes are in other areas.
Education as screening
Hacker’s analysis of maths is compatible with a wider critique of education as a screening mechanism. Randall Collins in his classic book The Credential Society argued that US higher education served
more to justify social stratification than to stimulate learning. In other words, students go through the ritual of courses, and those with privileged backgrounds have the advantage in obtaining
degrees that give them access to restricted professions.
In another classic critique, Samuel Bowles and Herbert Gintis in Schooling in Capitalist America argued that schooling reproduces the class structure. Their Marxist analysis gives the same general
conclusion as Collins’ approach. Then there is The Diploma Disease by Ronald Dore, who described education systems worldwide, but especially in developing countries, as irrelevant in terms of
producing skills that can be applied in jobs.
Schooling, up to teenage years, remains one of the few compulsory activities in contemporary societies, along with taxation. (In some countries, military service, jury duty and voting are
compulsory.) There is no doubt that education can be a liberating process in the right circumstances, but for many it is drudgery with little compensating benefit, aside from obtaining a certificate
needed for obtaining a job, while what is learned has little practical relevance.
A different system would be to set up entry processes to occupations, ones closely related to actual skills used in practice. Exams and apprenticeships are examples. Attendance at schools and
universities then would be optional, chosen for their value in learning. There is one big problem: attendance would plummet.
Some teachers set themselves the task of stimulating a love of learning. Rather than trying to convey particular facts and frameworks, they see that learning facts and frameworks is a way of learning
how to learn. The ideal in this picture is lifelong learning.
The trouble with schooling systems is that they undermine a love of learning by imposing syllabi and assessments. Students, rather than studying a topic because they are fascinated by it, instead
learn that studying is tedious and to be avoided, and only undertaken under the whip of assessment.
How many students do you know who keep studying after the final exam? On the other hand, people who are passionate about a topic will put in hours of concentrated effort day after day in a quest for
improvement and in the engaged mental state called flow.
The paradox of educational systems is that they are designed to foster learning yet, by subjecting students to arbitrary requirements, can actually hinder learning and create feelings of inadequacy.
The more that everyone is put through exactly the same hoops — the same learning tasks at the same time — the more acute the paradox.
A different sort of education
Taking this argument a step further leads to a double implication. Education should be designed around the needs of individual students, as attempted in free schools and in some forms of home
schooling. The second implication is that work should be designed around the jointly articulated needs of workers and consumers. Rather than students having to compete for fixed job slots, instead
work would be reorganised around the freely expressed needs and capacities of workers and local communities.
Whether this ideal could ever be reached is unknown, but it nonetheless provides a useful goal for restructuring education — including maths education. This brings us back to Hacker’s The Math Myth.
There are two sides to his argument. The first, as I’ve described it, is that US maths requirements are damaging because few people ever need maths beyond arithmetic and the requirements screen
talented people out of careers where they could make valuable contributions.
The second element in Hacker’s argument is that for the bulk of the population, there are useful things to learn about maths and that these can be made accessible using a practical problem-solving
approach. To show what’s involved, Hacker describes a course he taught in which students tackled everyday challenges.
Hacker’s course shows his capacity for innovative thinking. The Math Myth is not an attack on mathematics. Quite the contrary. Hacker wants everyone to engage with maths by designing tasks that
relate to their lives.
Whether Hacker’s powerful critique will lead to changes in US educational requirements remains to be seen. Although Hacker talks only about pointless maths requirements, his arguments challenge the
usual basis for screening that helps maintain social inequality. If maths cannot be used to legitimise inequality in educational outcomes, what will be the substitute?
Whether you respond to maths with affection or anxiety, it’s worth reading The Math Myth and thinking about its implications.
Brian Martin
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Research On Improved Hybrid Recommendation Algorithm Based On Clustering
Posted on:2019-06-17 Degree:Master Type:Thesis
Country:China Candidate:X Pan Full Text:PDF
GTID:2417330575450435 Subject:Applied statistics
With the rapid development of Internet technology and information technology,information and data resources have grown exponentially,human society has entered into the big data era of information
overload.Personalized recommendation is an effective way to solve information overload.Collaborative filtering method is the most popular personalized recommendation method in the actual
recommendation system.And the method mainly makes recommendations for users based on the preferences of the group.However,the traditional collaborative filtering algorithm have data
sparsity,cold-start,scalability and other problems.If we can effectively overcome the above defects,not only can we improve the user’s satisfaction,but also increase sale profits.In recent years,some
scholars have introduced clustering algorithms into collaborative filtering algorithms to alleviate their shortcomings in the research on recommendation systems.In this paper,we analyze the
collaborative filtering algorithm and the clustering algorithm.Then we propose three improvements for the cluster-based collaborative filtering algorithm.For the data sparsity problem,we use the
weighted Slope One algorithm to fill the user scoring matrix,which increases the data density and effectively reduces the sparseness of the data.For the problem that the initial clustering center of
K-means clustering algorithm is difficult to select,this paper uses the particle swarm optimization algorithm having the characteristics of strong global search and optimization ability to search the
initial clustering center.In the particle swarm optimization algorithm,the inertia weight determines the degree of influence of the current velocity and position of the particle on the next
iteration.The learning factor determines the information interaction between different particles of the entire particle swarm and the ability to transmit information.The improvement and optimization
of the whole recommendation algorithm are mainly reflected in the following aspects:(1)According to the characteristics of data,we use the inertia weight reduction strategy based on the sin
function.In the initial stage of the particle swarm algorithm,the inertia weight is larger to make the algorithm converges to the global optimal solution faster.Then with the rapid decrement of the
inertia weight,particle swarm algorithm quickly enters the local search state.In the later stage of the algorithm’s iteration,as the deceleration slows down,the algorithm can perform a more detailed
search on the optimal solution and obtain a high-precision solution.(2)In order to improve the efficiency of the particle swarm algorithm,we make some dynamic adjustments to learning factors.In the
early stage of the particle swarm optimization algorithm,the value of the learning factor c1 should be large,so as to enhance the population expansion ability of the particle group,the value of c2
should be small to avoid premature convergence.In the later stage of the particle swarm optimization algorithm,the value of the learning factor c,should be small to speed up the convergence speed of
the population,and the value of c2 should be larger to increase the probability of population convergence.The traditional Pearson similarity only considers the items of common scoring between
users,and the accuracy is not high enough when the scoring matrix is sparse.Therefore,we consider the product popularity and the factors of the user’s common rating when calculating the user
similarity,and weight it to reduce the error of the score prediction.Finally,the paper verifies the effectiveness of the improved clustering algorithm by calculating the clustering accuracy and
fitness values on the Wine dataset and the BreastCancer dataset.Then the absolute recommendation error is calculated on the Movielens 1M dataset to verify the superiority of the improved
recommendation algorithm and the hybrid recommendation algorithm of the fusion implicit semantic model.
Keywords/Search Tags: Recommendation System, Clustering, Particle Swarm, Slope One Algorithm, Latent Factor Model
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Daylight, Durlston Castle, and Where is Hamburg?
“Is Hamburg that much further north than London?”
I furrowed my brow. Hamburg, to the best of my knowledge, is not that much further north than London. But here it was, written in stone (on the side of Durlston Castle in Swanage.)
(I’ve transcribed the sign at the bottom of this post).
Things were about to get geometric.
Colin’s model Earth
My model of Earth is not as large as the Large Globe, a short walk from where we were, but that makes it much more portable. I can carry it in my head.
It consists of a sphere of unit radius (because “Earth radius” is a perfectly good unit to measure the Earth in), with a plane cutting it at an angle of (up to) 23.5º ((Yes, degrees are the standard
unit for measuring the planet. I don’t like it, but I have to go along with it.)) to the vertical, passing through the centre of the sphere.
Each line of latitude is a horizontal circle, with radius $\cos(\lambda)$, where $\lambda$ is the latitude.
The Longest Day
In fact, we’re looking at the longest day of the year, when the angle is 23.5º. Now, I happen to know the latitude of London, it’s about 51.5º north of the equator, which makes the radius of its
circle of latitude $\cos(51.5º) \approx 0.623$.
Where does the day/night plane cut the circle? A diagram will help here.
We’re trying to find the shortest leg of the red triangle. Its longer leg is $\sin(51.5º)$, and the angle at its base is 23.5º, so the leg is $\sin(51.5º)\tan(23.5º)$.
So, how much of the line of latitude is in shadow at solstice? Another diagram will help.
Now we’re looking down on London, and not just because Dorset is so much nicer. The line is where the day/night plane intersects the circle, so if we know the green angle, we can work out the number
of hours of darkness we have.
Splitting it into two right-angled triangles, we know they each have a hypotenuse of $\cos(51.5º)$ and a base of $\sin(51.5º)\tan(23.5º)$ - so we can work out the angle of each as $\arccos\br{\tan
(51.5º)\tan(23.5º)} \approx 58.6º$. The green angle is about 117.2º.
We’re not particularly interested in the angle itself, but rather the fraction of a circle it represents: here, that’s a bit less than a third; dividing by 360º, multiplying by 24 and converting to
hours gives 7 hours and 49 minutes – meaning 16 hours and 11 minutes of daylight, which isn’t too far removed from the answer written in stone.
How about Hamburg?
We now have a model: at summer solstice, the number of hours of darkness is:
\(H = \\arccos\\br{\\tan(\\lambda)\\tan(23.5º)} \\times \\frac{24}{180º}\).
I’d generally rewrite that, as I have a mild allergy to arccosines:
\[\\cos(H \\times 7.5º) = \\tan(\\lambda)\\tan(23.5º)\]
This also means we can easily work from either side to find $\lambda$ given $H$ or vice-versa.
Now, the writing on the wall says that Hamburg’s longest day is 19 hours - so just five hours of darkness. Does that check out?
$\cos(37.5º) = \tan(\lambda)\tan(23.5º)$
That give $\tan(\lambda) \approx 1.824$, and $\lambda \approx 61.3º$, putting Hamburg just south of the Arctic Circle. That seems… slightly off.
That’s not my Hamburg! Its latitude is too northerly!
According to worldatlas.com, Hamburg’s latitude is more like 53.6º - roughly level with Manchester.
We can work out the maximum day length there, too: if $ \cos(H \times 7.5º) = \tan(53.6º)\tan(23.5º)$, we get 7 hours and 10 minutes of darkness, which is 16 hours and 50 minutes.
According to The US Navy, the true answer is 17 hours - again, off by about 10 minutes ((I believe this is due to how the sunlight refracts through the atmosphere, but that’s physics.)).
I don’t know how long the stone has been up at Durlston Castle, but if I were them, I’d be taking it back for a refund.
* After a request from Quantum Mechanic, I’ve transcribed the writing:
Duration of Longest Day: At London: 16hrs 30mins Hamburg: 19 0 Spitzbergen: 3 1/2 months The Poles: 6 [months]
Clock Times of the World
These differ from Greenwich 4 mins every degree When 12 o’clock noon at Greenwich, it is
At Paris 12 9
At Swanage 11 52
Rome 1 50
Edinburgh 11 47
Vienna 1 6
Dublin 11 35
Calcutta 5 54
New York 7 4? ((Cut off in the photo, sorry))
* Edited 2018-02-14 to fix some LaTeX.
A selection of other posts
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Comparison of Deflection of Functionally Gradient Material Plate Under Mechanical, Thermal and Thermomechanical Loading
Volume 02, Issue 07 (July 2013)
Comparison of Deflection of Functionally Gradient Material Plate Under Mechanical, Thermal and Thermomechanical Loading
DOI : 10.17577/IJERTV2IS70403
Download Full-Text PDF Cite this Publication
Manoj Sharma, Manish Bhandari, Dr. Kamlesh Purohit, 2013, Comparison of Deflection of Functionally Gradient Material Plate Under Mechanical, Thermal and Thermomechanical Loading, INTERNATIONAL
JOURNAL OF ENGINEERING RESEARCH & TECHNOLOGY (IJERT) Volume 02, Issue 07 (July 2013),
• Open Access
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• Authors : Manoj Sharma, Manish Bhandari, Dr. Kamlesh Purohit
• Paper ID : IJERTV2IS70403
• Volume & Issue : Volume 02, Issue 07 (July 2013)
• Published (First Online): 18-07-2013
• ISSN (Online) : 2278-0181
• Publisher Name : IJERT
• License: This work is licensed under a Creative Commons Attribution 4.0 International License
Text Only Version
Comparison of Deflection of Functionally Gradient Material Plate Under Mechanical, Thermal and Thermomechanical Loading
Manoj Sharma1* Manish Bhandari2* , Dr. Kamlesh Purohit3 Rajasthan Technical University, Kota1,2 , Prof. & HOD, JNVU, Jodhpur3
Assistant Professor1 , Associate Professor2,Jodhpur Institute of Technology, Jodhpur
ABSTRACT Functionally gradient materials are one of the most widely used materials. The objective of this research work is to perform a thermo-mechanical analysis of functionally gradient material
square laminated plate made of Aluminum / Zirconia and compare with pure metal and ceramic. The plates are assumed to have isotropic, two-constituent material distribution through the thickness, and
the modulus of elasticity of the plate is assumed to vary according to a power-law distribution in terms of the volume fractions of the constituents. To achieve this objective, we shall use first
shear deformation theory of plates and numerical analysis will be accomplished using finite element model prepared in ANSYS software. The laminated Functionally Gradient Material plate is divided in
to layers and their associated properties are then layered together to establish the through-the-thickness variation of material properties. The displacement fields for functionally gradient material
plate structures under mechanical, thermal and thermo mechanical loads are analyzed under simply supported boundary condition
Keywords FGM, Computational techniques, Thermo mechanical properties in FGM.
History is often marked by the materials and technology that reflect human capability and understanding. Many times scales begins with the stone age, which led to the Bronze, Iron, Steel, Aluminum
and Alloy ages as improvements in refining, smelting took place and science made all these possible to
move towards finding more advance materials possible. It has become possible to develop new composite materials with improved physical and mechanical properties. Functionally gradient materials (FGM)
are a class of composites that have a gradual variation of material properties from one surface to another. These novel materials were proposed by the Japanese in 1984 and are projected as thermal
barrier materials for applications in space planes, space structures and nuclear reactors, to name only a few. In general, all the multi- phase materials, in which the material properties are varied
gradually in a predetermined manner, fall into the category of functionally gradient materials. The gradients can be continuous on a microscopic level, or they can be laminates comprised of gradients
of metals, ceramics, polymers, or variations of porosity/density as shown in figure 1.
Figure 1: Gradient of FGMs; (a) continuously graded and (b) discretely layered FGMs.
A huge amount of published literature observed for evaluation of
thermomechanical behavior of functionally gradient material plate using finite element techniques. It includes both linearity and non linearity in various areas. Few of published literature highlight
the importance of topic. A laminated theory for a desired degree of approximation of the displacements through the laminate thickness, allowing for piecewise approximation of the inplane deformation
through individual laminae reported by reddy [1]. S. Suresh and A. Mortensen (1997) focus a review of the processing of functionally graded metal- ceramic composites and their thermo mechanical
behavior. They discussed various approximations for determination of properties and their limitations are highlighted. They have focused on various issues related to functionally gradient material
manufacturing [2]. G. N. Praveen and reddy (1997) reported the static and dynamic response of the functionally graded material plates by varying the volume fraction of the ceramic and metallic
constituents using a simple power law distribution. [3]. J. N. Reddy (1998) reported theoretical formulations and finite element analyses of the thermomechanical, transient response of functionally
graded cylinders and plates with Nonlinearity. [4]. J. N. Reddy (2000) gives Navier's solutions of rectangular plates, and Finite element models based on the third-order shear deformation plate
theory for functionally graded plates. [5]. J.N. Reddy et. al. (2001) reported three-dimensional thermomechanical deformations of simply supported, functionally graded rectangular plates. The
temperature, displacements and stresses of the plate are computed for different volume fractions of the ceramic and metallic constituents. [6]. Bhavani V. Sankar (2002) solved the thermoelastic
equilibrium equations for a functionally graded beam in closed-form [7]. Senthil S. Vel and R.C. Batra (2003) calculate an analytical solution for three-dimensional thermomechanical deformations of a
simply supported functionally graded rectangular plate subjected to time-dependent thermal
loads [8]. M. Tahani1, M. A. Torabizadeh and A. Fereidoon (2006), have reported analytical method is developed to analyze analytically displacements and stresses in a functionally graded composite
beam subjected to transverse load and the results obtained from this method are compared with the finite element solution done by ANSYS [9]. Ki-Hoon Shin (2006) suggests that the Finite Element
Analysis (FEA) is an important step for the design of structures or components formed by heterogeneous objects such as multi-materials, Functionally Graded Materials (FGMs), etc [10]. Fatemeh
Farhatnia, Gholam-Ali Sharifi and Saeid Rasouli (2009), determined the thermo- mechanical stress distribution has been determined for a three layered composite beam having a middle layer of
functionally graded material (FGM), by analytical and numerical methods. They found that there is no practically considerable difference, between stress profiles obtained analytically and from FEM
model and ANSYS [11].
M.K. Singha, T.Prakash and M.Ganapathi (2011) reported The nonlinear behaviors of functionally graded material (FGM) plates under transverse distributed load. [12].
D.K. Jha, Tarun Kant and R.K. Singh (2012) reported a critical review of the reported studies in the area of thermo-elastic and vibration analyses of functionally graded (FG) plates since 1998. They
have presented various areas of work for FGM and their application. [13]. Srinivas.G and Shiva Prasad.U focused on analysis of FGM flat plates under pressure i.e. mechanical loading in order to
understand the effect variation of material properties has on structural response. [14].
1. Modeling Introduction
With the advent of powerful computers and robust software, computational modeling has emerged as a very informative and cost effective tool for materials design and analysis. Modeling often
can both eliminate
costly experiments and provide more information than can be obtained experimentally. A wide variety of software, for e.g. ABAQUS, ANSYS etc., are commercially available and can be used to
model and analyze FGMs. In this report Ansys 13.0 is used as a tool for anaylsis and the element SHELL 181 is used.
2. MATERIAL PROPERTIES
Volume fraction and material properties of FGMs may vary in the thickness direction or in the plane of a plate. The FGM modeled usually is done with one side of the material as ceramic and
the other side as metal. A mixture of the two materials composes the through-the-thickness characteristics. This material variation is dictated by a parameter, n. At n = 0 the plate is a
fully ceraic plate while at n = the plate is fully metal. Material properties are dependent on the n value and the position in the plate and vary according to a power law.
Here we assume that the material property gradation is through the thickness and we represent the profile for volume fraction variation by the expression of power law, i.e.
P(z) = (Pt Pb)V +Pb
z n
Table: 1 Material property
Property Aluminum Zirconia
1 Young's modulus 70 GPa 151 GPa
2 Poisson's ratio 0.3 0.3
3 Thermal conductivity 204 W/mK 2.09 W/mK
4 Thermal expansion 23×10-6 /°C 10×10-6 /°C
The study of the behaviour of an FGM plate under mechanical loads is done for a square plate whose constituent materials are taken to be Aluminum and zirconia. The top surface of the plate is
ceramic (zirconia) rich and the bottom surface is metal (Aluminium) rich. Variation of effective youngs modulus, Thermal conductivity and Thermal expansion with respect to parameter z/h for
various material index as shown in figure 2,3 and 4 respectively.
Variation of youngs modulus in FGM
Variation of youngs modulus in FGM
For the material index (n)=2;
At bottom layer, (z/h)=0 and so V-=0 hence P(z) = Pb
At top layer , (z/h)=1 and so V=1
hence P(z)= Pt
where P denotes a generic material property like modulus, Pt and Pb denote the property of the top and bottom faces of the plate, respectively, h is the total thickness of the plate, and n is
a parameter that dictates the material variation profile through the thickness.
Ypoungs Modulus (Ef)
Ypoungs Modulus (Ef)
Parameter (z/h)
Parameter (z/h)
Figure 2: Variation of effective youngs
modulus with respect to parameter z/h for various material index
Variation of Thermal conductivity in FGM
Variation of Thermal conductivity in FGM
results are presented in terms of non- dimensional stress and deflection. The various non dimensional parameters used are
Thermal conductivity (kf)
Thermal conductivity (kf)
non dimensional center deflection w = w0Etp/(q0a4) and non dimensional shear
0 0.2 0.4 0.6 0.8 1
Parameter (z/h)
0 0.2 0.4 0.6 0.8 1
Parameter (z/h)
Figure 3: Variation of effective Thermal conductivity with respect to
parameter z/h for various material index
Figure 4: Variation of effective Thermal expansion with respect to
parameter z/h for various material index
3. ANALYSIS
The static analysis was performed on a square plate of side length a=b = 0.2m and thickness h = 0.01 m. The plate is assumed to be simply supported on all its edges. A regular 8 by 8 mesh of
linear elements in a full size plate was chosen after convergence studies. The value of the uniformly distributed loading chosen was equal to q0= 0.01×106 N/m2. The results were plotted. The
analysis is performed for fix values of the volume fraction exponent i.e n=2. The
stress xz =xzh /(q0a )
In the present analysis, in addition to the uniform loading, the plate is subjected to a temperature field where the uniform temperature up to 300°C is given and the reference surface
temperature is held at 20°
C. The materials are assumed to be perfectly elastic throughout the deformation. A simply supported FG plate subjected to a uniformly distributed mechanical load and thermal loading as sown
in figure 5.
Figure 5: A simply supported FG plate subjected to a uniformly distributed mechanical load and thermal loading
4. BOUNDARY CONDITIONS AND MESHING
The square plate modeled is meshed using the mesh tool. The mesh tool provides a convenient path to many of the most common mesh controls, as well as to the most frequently performed meshing
operations. The plate modeled throughout this project is subjected to simply supported Boundary condition i.e. along the X direction, Uy=UZ=0 and along the Y direction Ux=UZ=0. It is
illustrated in Figure 6.
gradient material plate as shown in figure 7, 8 and 9 respectively.
Mechanical Loading
Metal plate
Metal plate
Non dimensional deflection
Non dimensional deflection
Figure 6: Square plate with 8 layers, 8×8 mesh and Simply supported boundary condition.
Using the APDL tool, the layer of the model along the thickness are divided into the number of layers desired; the other layers are then selected and divided
Ceramic plate
FGM plate
Ceramic plate
FGM plate
0 0.2 0.4 0.6 0.8 1
Non dimensional parameter (z/h)
depending on the mesh size required. The following figure 6 shows an FGM plate modeled with 8 layers and a mesh count of size 8×8 along the x-y plane. Once the model is meshed; the model is
modified in order to create layers with different material properties. This is done with the help of shell section. The material properties are then assigned to the respective layers defined
along the thickness. It is to be noted that each layer is isotropic in nature.
5. Result
In this section we present several numerical simulations, in order to assess the behavior of functionally graded plates subjected to mechanical, thermal and thermo-mechanical loads. A simple
supported plate is considered for the investigation. The plate is made up of a ceramic material at the top, a metallic at the bottom. The simple power law with different values of n = 2 is
used for the through-the-thickness variation. Following trends obtained as shown in various graphs
1. Non dimensional deflection:
Figure 7: Variation of Non dimensional deflection for mechanical loading with z/h for n=2
Thermal Loading
Thermal Loading
Metal plate
Metal plate
Non dimensional parameter (z/h)
Non dimensional parameter (z/h)
Ceramic plate
FGM plate
Ceramic plate
FGM plate
Non dimensional deflection
Non dimensional deflection
Figure 8: Variation of Non dimensional deflection for thermal loading with z/h for
Thermomechanical Loading
1. E+02
Metal plate
Metal plate
Ceramic plate
Ceramic plate
FGM plate
FGM plate
Non dimensional deflection
Non dimensional deflection
Here the non dimensional deflection parameter is plotted against non dimensional parameter (z/h) for mechanical loading, thermal loading and thermo mechanical loading for metal plate,
ceramic plate and functionally
0 0.2 0.4 0.6 0.8 1
Non dimensional parameter (z/h)
Figure 9: Variation of Non dimensional deflection for thermo mechanicalloading with z/h for n=2
2. ANSYS DIAGRAM
In this section we present several numerical simulations diagram of deflection in figure 10,11 and 12 for mechanical , thermal and thermomechanical loading respectively to assess the behavior of
functionally graded plates.
Figure 10: Variation of Non dimensional deflection for mechanical loading in FGM
Figure 11: Variation of Non dimensional deflection for thermal loading in FGM plate
Figure 12: Variation of Non dimensional deflection for thermo mechanical loading in FGM plate
1. CONCLUSION
In this report analysis is carried out on a functionally gradient material square plate made of Aluminium/Zirconia. The plate considered is thick plate with a/h=20 and a/b=1. The structural
response of this plate is studied with respect to mechanical, thermal and thermomechancial loads. The structural response of functionally gradient material plate is also compared with pure metal
and pure ceramic plate under mechanical, thermal and thermomechancial loading. The properties of functionally gradient material are calculated for each layer according to power law. The material
index, number of layers and mesh size is kept constant. The following points are summarized:
1. The modeling of functionally gradient material plate in step wise variation in properties is successfully developed.
2. It is observed that the response of plates depends upon the intermediate properties of the metal and the ceramic.
3. In case of pure mechanical loading the non dimensional deflection of functionally gradient material plate is in
between pure metal and ceramic plate.
4. In case of pure thermal and thermomechanical loading the non dimensional deflection is same nature. Ceramic plate having minimum deflection in mechanical, thermal and thermomechanical
5. As for as review of literature it is also concluded that fine the number of mesh better the results. Also ANSYS gives faster approximate results and degree of accuracy depends on the mesh
size, layers and solver.
6. In this report first order shear deformation theory has been used for formulation of the problem, it is concluded from the review that higher order theory can give approximate better results.
The plate modeled here was a step wise graded structure, with each layer being isotropic with specific material properties. The material properties for each layer have been calculated by any other
methods like Mori-Tanka etc.which may give better estimation of properties. Also we can go for the coding of the material to get the continous variation of the properties. The material index, no of
layers and meshing size can also be changed to get the better results. The position of natural axis and its eccentricity can also be considered for perfect analysis.
1. E. J. Barbero and J. N. Reddy, An accurate determination of stresses in thick laminates using a generalized plate theory International Journal for Numerical
Methods in Engineering, 29, p.p .1-
14 ,1990.
2. S. Suresh and A. Mortensen, Functionally graded metals and metal-ceramic composites: Part 2, Thermomechanical behavior, International Materials Reviews, Vol. 42 No.3 85, 1997.
3. G. N. Praveen and J. N. Reddy, Nonlinear transient thermoelastic analysis of functionally graded ceramic-metal plates, Int. J. Solids Structure. Vol. 35. No. 33. pp. 4457- 4476, 1997.
4. J. N. Reddy, Thermomechanical behavior of functionally graded materials, Final Report for AFOSR Grant F49620-95-1-0342, CML Report 98-01, August 1998.
5. J. N. Reddy, Analysis of functionally graded plates, International journal for numerical methods in engineering, Int. J. Numer. Meth. Engg. 47, pp 663-684, 2000.
6. J.N. Reddy and Zhen-Qiang Cheng, Three-dimensional thermomechanical deformations of functionally graded rectangular plates, Eur. J. Mech. A/Solids 20, pp. 841855, 2001.
7. Bhavani V. Sankar and Jerome T. Tzeng, Thermal stresses in functionally graded beams, AIAA journal, Vol. 40, No. 6, June 2002.
8. Senthil S. Vel and R.C. Batra, Three-dimensional analysis of transient thermal stresses in functionally graded plates, International Journal of Solids and Structures 40, pp 71817196, 2003.
9. M. Tahani1, M. A. Torabizadeh and
1. Fereidoon, Non-Linear Response of Functionally graded beams under transverse loads 14th
Annual (International) Techanical Engineering Conference ,Isfahan University of Technology, Isfahan, Iran, May 2006.
10. Ki-Hoon Shin, FEA Based Design of Heterogeneous Objects International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, Philadelphia, Pennsylvania,
USA, September 10- 13, 2006,
11. Fatemeh Farhatnia, Gholam-Ali Sharifi and Saeid Rasouli, Numerical and Analytical Approach of Thermo-Mechanical Stresses in FGM Beams Proceedings of the World Congress on Engineering, London,
U.K, Vol II
, July 1 – 3, 2009.
12. M.K. Singha, T.Prakash and M.Ganapathi , Finite element analysis of functionally graded plates under transverse load Finite Elements in Analysis and Design 47 ,453460 , 2011.
13. D.K. Jha, Tarun Kant and R.K. Singh, A critical review of recent research on functionally graded plates , Composite Structures, (2012).
14. Srinivas.G and Shiva Prasad.U, Simulation of Traditional Composites Under Mechanical Loads, International Journal of Systems , Algorithms & Applications, Volume 2, Issue ICASE 2012, ISSN Online:
2277- 2677, August 2012.
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Graphing A Parabola From Vertex Form Worksheet Answers
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Kimberly Fessel's BlogAccuracy, Precision, and Recall — Never Forget Again!Python for Data Science: An Interview with Course ReportDelorean for Datetime ManipulationMeasuring Statistical Dispersion with the Gini CoefficientWeb Scraping in Python: Real Python PodcastLet's Scrape the Web: PyCon 2020 Video TutorialLevel Up: spaCy NLP for the WinMath for Data Science: An Interview with Course ReportDown and Up: A Puzzle Illustrated with D3.jsHow to Gather Data from YouTube
https://kimfetti.github.io/ 2024-02-18T02:48:19+00:00 Kimberly Fessel is a data science consultant. Her enthusiasm for data storytelling often leads her toward better math, better visuals, and better
science! Kimberly Fessel Jekyll https://kimfetti.github.io/mathematics/data/accuracy-precision-recall/ 2022-04-03T00:00:00+00:00 KFessel <em>Designing an effective classification model requires an
upfront selection of an appropriate classification metric. This posts walks you through an example of three possible metrics (accuracy, precision, and recall) while teaching you how to easily
remember the definition of each one.</em> <center> <iframe width="818" height="500" src="//www.youtube.com/embed/qWfzIYCvBqo" frameborder="0" allowfullscreen=""></iframe> </center> <hr /> <p>To
design an effective supervised machine learning model, data scientists must first select appropriate metrics to judge their model’s success. But choosing a useful metric often proves more challenging
than anticipated, especially for classification models that have a slew of different metric options.</p> <p>Accuracy remains the most popular classification metric because it’s easy to compute and
easy to understand. Accuracy comes with some serious drawbacks, however, particularly for imbalanced classification problems where one class dominates the accuracy calculation.</p> <p>In this post,
let’s review accuracy but also define two other classification metrics: precision and recall. I’ll share an easy way to remember precision and recall along with an explanation of the precision-recall
tradeoff, which can help you build a robust classification model.</p> <h2 id="model-and-data-setup">Model and Data Setup</h2> <p>To make this study of classification metrics more relatable, consider
building a model to classify apples and oranges on a flat surface such as the table shown in the image below.</p> <center> <img src="https://kimfetti.github.io/images/OA_training_data.jpg" alt=
"Apples and oranges arranged on a table with most of the apples on the right side" width="700" /> </center> <p><br /></p> <p>Most of the oranges appear on the left side of the table, while the apples
mostly show up on the right. We could, therefore, create a classification model that divides the table down its middle. Everything on the left side of the table will be considered an orange by the
model, while everything on the right side will be considered an apple.</p> <center> <img src="https://kimfetti.github.io/images/OA_precision_recall_header.png" alt="Left side identified as the orange
side and right side as the apple side of the model" width="700" /> </center> <h2 id="what-is-accuracy">What is accuracy?</h2> <p>Once we’ve built a classification model, how can we determine if it’s
doing a good job? Accuracy provides one way to judge a classification model. To calculate accuracy just count up all of the correctly classified observations and divide by the total number of
observations. This classification model correctly classified 4 oranges along with 3 apples for a total of 7 correct observations, but there are 10 fruits overall. This model’s accuracy is 7 over 10,
or 70%.</p> <center> <img src="https://kimfetti.github.io/images/OA_accuracy.png" alt="Accuracy calculated from example apple-orange model as 70%" width="700" /> </center> <p><br /></p> <p>While
accuracy proves to be one of the most popular classification metrics because of its simplicity, it has a few major flaws. Imagine a situation where we have an imbalanced dataset; that is, what if we
have 990 oranges and only 10 apples? One classification model that achieves a very high accuracy predicts that all observations are oranges. The accuracy would be 990 out of 1000, or 99%, but this
model completely misses all of the apple observations.</p> <p>Furthermore, accuracy treats all observations equally. Sometimes certain kinds of errors should be penalized more heavily than others;
that is, certain types of errors may be more costly or pose more risk than others. Take predicting fraud for example. Many customers would likely prefer that their bank call them to check up on a
questionable charge that is actually legitimate (a so-called “false positive” error) than allow a fraudulent purchase to go through (a “false negative”). Precision and recall are two metrics that can
help differentiate between error types and can still prove useful for problems with class imbalance.</p> <h2 id="precision-and-recall">Precision and Recall</h2> <p>Both precision and recall are
defined in terms of just one class, oftentimes the positive—or minority—class. Let’s return to classifying apples and oranges. Here we will calculate precision and recall specifically for the apple
class.</p> <p>Precision measures the quality of model predictions for one particular class, so for the precision calculation, zoom in on just the apple side of the model. You can forget about the
orange side for now.</p> <p>Precision equals the number of correct apple observations divided by all observations on the apple side of the model. In the example depicted below, the model correctly
identified 3 apples, but it classified 5 total fruits as apples. The apple precision is 3 out of 5, or 60%. To remember the definition of precision, note that preci<strong>SI</strong>on focuses on
only the apple <strong>SI</strong>de of the model.</p> <center> <img src="https://kimfetti.github.io/images/OA_precision.jpg" alt="Precision calculated as 60% for the apple class from example
apple-orange model" width="700" /> </center> <p><br /></p> <p>Recall, on the other hand, measures how well the model did for the actual observations of a particular class. Now check how the model did
specifically for all the actual apples. For this, you can pretend like all of the oranges don’t exist. This model correctly identified 3 out of 4 actual apples; recall is 3 over 4, or 75%. Remember
this simple mnemonic: rec<strong>ALL</strong> focuses on <strong>ALL</strong> the actual apples.</p> <center> <img src="https://kimfetti.github.io/images/OA_recall.jpg" alt="Recall calculated as 75%
for the apple class from example apple-orange model" width="700" /> </center> <h2 id="precision-recall-tradeoff">Precision-Recall Tradeoff</h2> <p>So what are the benefits of measuring precision and
recall instead of sticking with accuracy? These metrics certainly allow you to emphasize one specific class since they are defined for one class at a time. That means that even if you have imbalanced
classes, you can measure precision and recall for your minority class, and these calculations won’t get dominated by the majority class observations. But it turns out that there’s also a nice
tradeoff between precision and recall.</p> <p>Some classification models, such as logistic regression, not only predict which class each observation belongs to but also predict the probability of
being in a particular class. For example, the model may determine that a specific fruit has 80% probability of being an apple and 20% probability of being an orange. Models like these come with a
decision threshold that we can adjust to divide the classes.</p> <p>Let’s say you’d like to improve the precision of your model because it’s very important to avoid falsely claiming that an actual
orange is an apple (false positive). You can just move the decision threshold up, and precision gets better. For our apple-orange model, that means shifting the model line to the right. In the
example image, the updated model boundary yields perfect precision of 100% since all predicted apples are actually apples. When we do this, however, recall will likely decrease because moving the
threshold up leaves out actual apples in addition to the erroneous oranges. Here, recall dropped to 50%.</p> <div class="row"> <div class="large-6 columns"> <img src="https://kimfetti.github.io/
images/OA_precision_boundaryRight.jpg" alt="With the decision threshold increased, precision increased to 100% for the apple class" width="450" /> </div> <div class="large-6 columns"> <img src=
"https://kimfetti.github.io/images/OA_recall_boundaryRight.jpg" alt="With the decision threshold increased, recall decreased to 50% for the apple class" width="450" /> </div> </div> <p><br /></p> <p>
Okay, what if we want to improve recall? We could make our decision threshold lower by moving our model line to the left. We now capture more actual apples on the apple side of our model, but as we
do this, our precision likely decreases since more oranges sneak into the apple side as well. With this update, recall improved to 100% but recall declined to 50%.</p> <div class="row"> <div class=
"large-6 columns"> <img src="https://kimfetti.github.io/images/OA_recall_boundaryLeft.jpg" alt="With the decision threshold decreased, recall increased to 100% for the apple class" width="450" /> </
div> <div class="large-6 columns"> <img src="https://kimfetti.github.io/images/OA_precision_boundaryLeft.jpg" alt="With the decision threshold decreased, precision decreased to 50% for the apple
class" width="450" /> </div> </div> <p><br /></p> <p>Monitoring and selecting an appropriate precision-recall tradeoff allows us to prioritize certain types of errors, either false positives or false
negatives, as we adjust the decision threshold of our model.</p> <h2 id="conclusion">Conclusion</h2> <p>Precision and recall offer new ways to judge classification model predictions as opposed to the
standard accuracy computation. With apple precision and recall, we focus in on the apple class. High precision assures that what our model says is an apple actually is an apple (preci<strong>SI</
strong>on = apple <strong>SI</strong>de), but recall prioritizes correctly identifying all of the actual apples (rec<strong>ALL</strong> = <strong>ALL</strong> apples).</p> <p>Precision and recall
allow us to distinguish between different types of errors, and there’s also a great tradeoff between precision and recall because we can’t blindly improve one without often sacrificing the other. The
balance between precision and recall can also help us build more robust classification models. In fact, practitioners often measure and try to improve something called the F1-score, which is the
harmonic average between precision and recall, when building a classification model. This ensures that both metrics stay healthy and that the dominant class doesn’t overwhelm the metric like it
generally does with accuracy.</p> <p>Choosing an appropriate classification metric is a critical early step in the data science design process. For example, if you want to be sure not to miss a
fraudulent transaction, you’ll likely prioritize recall for cases of fraud. Though in other situations, accuracy, precision, or F1-score may be more appropriate. Ultimately, your choice of metric
should be intimately linked to the goal of your project, and once it’s determined, that metric of choice should drive your model development and selection process.</p> 2022-04-03T00:00:00+00:00
https://kimfetti.github.io/python/course-report-python-ds/ 2020-09-15T00:00:00+00:00 KFessel <em>Python is one of the most popular computer programming languages in the world. Find out how Python is
used for data science in interview with Course Report.</em> <!--more--> <p>In a recent interview with <a href="https://www.coursereport.com/">Course Report</a>, I discussed the basics of Python and
how Python is used for data science. Python serves as an all-purpose programming language, so data scientists, engineers, analysts, and web developers alike utilize Python to build end-to-end
projects, ready for launch into production. Python also has incredibly simple syntax, which makes it a great first programming language for beginners. We chat about these topics and many more in the
video!</p> <p>You can also check out a write up of our interview on the <a href="https://www.coursereport.com/blog/how-is-python-used-for-data-science-metis">Course Report blog</a>.</p>
2020-09-15T00:00:00+00:00 https://kimfetti.github.io/python/datetime/delorean-datetime-manipulation/ 2020-07-25T00:00:00+00:00 KFessel <em>Working with dates and times in Python can lead to
frustration, heartache, and, ironically, lost time – but it doesn’t have to! This brief demo introduces Delorean, a library constructed to make datetime manipulation in Python easier. </em> <!
--more--> <p>This year’s pandemic necessitated different conference formats for data science professionals. The organizers of PyOhio decided to ask speakers to create 5- or 10-minute pre-recorded
talks to be streamed continuously while participants discussed the content in a live chat session. The format was a success! And I am proud to have created this video all about the Python library
Delorean.</p> <p>Delorean makes working with datetimes in Python much less of a burden. Its simple syntax allows users: to do datetime arithmetic, to handle time zone shifts, to convert datetimes
into human language like “3 days ago,” and to generate equally spaced datetime intervals.</p> <p>Check out my video for a look at Delorean (along with many, many <em>Back to the Future</em>
references) or watch the <a href="https://www.youtube.com/watch?v=OGmzRIgDgOY&list=PL2k6bbM_wgjtGSzPXzUzP3AfVO-o4imbB">full PyOhio 2020 conference playlist on YouTube</a>.</p>
2020-07-25T00:00:00+00:00 https://kimfetti.github.io/mathematics/applications/gini-use-cases/ 2020-06-05T00:00:00+00:00 KFessel <em>The Gini coefficient is a good general-purpose measure of
statistical dispersion. Long since popular in the field of economics, this metric can be leveraged much more broadly to explore data from nearly any discipline. The following post includes a thorough
mathematical explanation of the Gini coefficient as well as a few non-standard use cases.</em> <center> <iframe width="900" height="550" src="//www.youtube.com/embed/nFbAnwIYle4" frameborder="0"
allowfullscreen=""></iframe> </center> <hr /> <p>If you work with data long enough, you are bound to discover that a dataset’s mean rarely–if ever–tells you the full data story. As a simple example,
each of the following groups of people have the same <strong>average pay</strong> of $100:</p> <ul> <li>100 people who make $100 each</li> <li>50 people who make $150 each and 50 people who make $50
</li> <li>1 person who makes $10,000 and 99 people who make nothing</li> </ul> <p>The primary difference, of course, is the way that money is distributed among the people, also known as the <a href=
"https://en.wikipedia.org/wiki/Statistical_dispersion">statistical dispersion</a>. Perhaps the most popular measurement of statistical dispersion is standard deviation or variance; however, you can
leverage other metrics, such as the Gini coefficient, to obtain a new perspective.</p> <p><a href="https://en.wikipedia.org/wiki/Gini_coefficient">The Gini coefficient</a>, also known as the Gini
index or the Gini ratio, was introduced in 1912 by Italian statistician and sociologist Corrado Gini. Analysts have historically used this value to study income or wealth distributions; in fact,
despite being developed over 100 years ago, <a href="https://www.bbc.com/news/blogs-magazine-monitor-31847943">the United Nations still uses the Gini coefficient</a> to understand monetary inequities
in their annual ranking of nations. But the Gini coefficient may be utilized much more broadly! After a more thorough mathematical explanation, let’s apply the Gini coefficient to a few non-standard
use cases that do not involve international economies: baby names and healthcare pricing.</p> <h2 id="defining-gini">Defining Gini</h2> <p>The first step in understanding the Gini coefficient
requires a discussion about the Lorenz curve, a graph developed by Max Lorenz for visualizing income or wealth distribution. To trace out the Lorenz curve, begin by taking the incomes of a population
and sorting them from smallest to largest. Then build a line plot where the \(x\)-values represent the percentage of people seen thus far and the \(y\)-values represent the cumulative proportion of
wealth attributed to this percentage of people. For example, if the poorest 30% of the population holds 10% of a population’s wealth, the curve should pass through the scaled \(x,y\) coordinates
(0.3, 0.1). Note also that if wealth is distributed evenly among all members of a population, the Lorenz curve follows a straight line, \(x=y\). See the figure below for an illustration of a
hypothetical Lorenz curve along with the line of equality.</p> <div class="row"> <div class="large-6 columns"> <img src="https://kimfetti.github.io/images/gini_explanation.png" alt="The areas
surrounding the Lorenz curve define the Gini coefficient: A/(A+B)" width="350" /> </div> <div class="large-6 columns"> <img src="https://kimfetti.github.io/images/gini_animation.gif" alt="The Gini
coefficient increases as the inequality gap widens." width="350" /> </div> </div> <p><br /></p> <p>The Gini coefficient measures how much a population’s Lorenz curve deviates from perfect equality or
how much a set of data diverges from equal values. The Gini coefficient typically ranges from zero to one<sup id="fnref:1" role="doc-noteref"><a href="#fn:1" class="footnote" rel="footnote">1</a></
sup>, where</p> <ul> <li>zero represents perfect equality <em>(e.g. everyone has an equal amount)</em> and</li> <li>one represents near perfect inequality <em>(e.g. one person has all the money)</
em>.</li> </ul> <p>For all situations in between, the Gini coefficient \(G\) is defined as \[G = \frac{A}{A + B}\] where \(A\) signifies the region enclosed between the line of perfect equality and
the Lorenz curve, as indicated in the figure above, while \(A + B\) represents the total triangular area.</p> <p>Each of the three situations discussed in the introduction produce an average of $100
per person. The Gini coefficient, however, varies greatly for each scenario as seen in the figure below.</p> <p><img src="https://kimfetti.github.io/images/gini_compare.png" alt="Gini coefficient
increases with wealth inequality." width="1000" /></p> <h2 id="gini-in-python">Gini in Python</h2> <p>To calculate a dataset’s Gini coefficient with Python, you have the option of computing the
shaded area \(A\) with something like <code class="language-plaintext highlighter-rouge">scipy</code>’s <a href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.quad.html">
quadrature</a> routine. If this style of numerical integration proves slow or too complicated for applications at scale, you can utilize an alternative, <a href="https://en.wikipedia.org/wiki/
Gini_coefficient#Definition">equivalent definition of the Gini coefficient</a>.</p> <blockquote> <p>The Gini coefficient may also be expressed as half of the data’s <a href="https://en.wikipedia.org/
wiki/Mean_absolute_difference#Relative_mean_absolute_difference">relative mean absolute difference</a>, a normalized form of the average absolute difference among all pairs of observations in the
dataset. \[ G = \frac{\sum\limits_i \sum\limits_j |x_i - x_j|}{2\sum\limits_i\sum\limits_j x_j}\]</p> </blockquote> <p>The calculation simplifies further if the data consist of only positive values
as it becomes <a href="https://www.statsdirect.com/help/default.htm#nonparametric_methods/gini.htm">unnecessary to evaluate all possible pairs</a>. Sorting the datapoints in ascending order and
assigning a positional index \(i\) yields \[G = \frac{\sum\limits_i (2i - n - 1)x_i}{n\sum\limits_i x_i}, \] which is even speedier to compute.</p> <p>The best Python implementation of the Gini
coefficient that I’ve found comes from <a href="https://github.com/oliviaguest/gini/blob/master/gini.py">Olivia Guest</a>. I will subsequently leverage her vectorized <code class="language-plaintext
highlighter-rouge">numpy</code> routine to calculate Gini in the case studies that follow.</p> <h2 id="case-1-baby-names">Case #1: Baby Names</h2> <p>So far we have mostly addressed the Gini
coefficient in the context of its original field of economics. This metric generalizes, however, to provide insight whenever statistical dispersion plays a critical role. I will now illustrate two
atypical applications to demonstrate how using the Gini coefficient augments the workflow of exploratory data analysis.</p> <p>The Social Security Administration of the United States (SSA) <a href=
"https://www.ssa.gov/oact/babynames/limits.html">hosts public records</a> on the names given to US babies for research purposes. Aggregating these data for children born since 1950, I discovered that
18 out of the top 20 most popular names more commonly associate with male children. So where are the females?</p> <center> <img src="https://kimfetti.github.io/images/popular_names.png" alt="Most
popular names given to US babies since 1950" width="500" /> </center> <p>Slightly <a href="https://www.npr.org/sections/health-shots/2015/03/30/396384911/why-are-more-baby-boys-born-than-girls">more
male babies are actually born each year</a>, and certainly more male babies have been registered with the SSA (53% male vs 47% female); nonetheless, I was still surprised to see such a large
proportion of male names in my quick popularity chart. Digging into the data further, I found that even though fewer females appear in the data, there have been consistently more unique female names
each year.</p> <center> <img src="https://kimfetti.github.io/images/unique_names.png" alt="Number of unique names for male and female babies since 1950" width="700" /> </center> <p><br /></p> <p>
Statistical dispersion appears to play a significant role. To put it back in financial terms, some male names like the ones on my top 20 list are just extremely “wealthy.” (The most popular name,
“Michael,” accounts for over 3% of all male children born since 1950.) These ultra-popular masculine names likely pass down from generation to generation. Females babies, on the other hand, are
distributed more widely across a variety of names, so extra names share in the “wealth” of female children. We can verify this theory by returning to the Gini coefficient.</p> <p>Consider how female
children disperse across each name. Some names in the dataset account for only 5 babies<sup id="fnref:2" role="doc-noteref"><a href="#fn:2" class="footnote" rel="footnote">2</a></sup> since 1950,
while “Jennifer” represents nearly 1.5 million individuals. Tallying up all females born with each name since 1950 and sorting the names from least to most popular, we find the Gini coefficient to be
0.96, implying a huge disparity in the most popular versus the most unique names.</p> <p>Male names exhibit a very similar Lorenz curve but with a little more skew, registering a Gini coefficient of
0.97. The difference between male and female coefficients appears insignificant, but consider an alternative viewpoint. Instead of aggregating across time, calculate a yearly Gini coefficient for
each gender. Plotting both the female and male Gini coefficients for each year since 1950 demonstrates a clear and persistent pattern where the male coefficient presents consistently higher.<sup id=
"fnref:3" role="doc-noteref"><a href="#fn:3" class="footnote" rel="footnote">3</a></sup> Thus male names experience more statistical dispersion than female monikers. Also of note, the Gini values for
both genders have ticked downward since the 1990s, indicating a trending preference toward more diverse naming conventions.</p> <center> <img src="https://kimfetti.github.io/images/gini_names.png"
alt="The male Gini coefficient tracks consistently higher throughout time" width="700" /> </center> <p><br /></p> <p>In a final look at this dataset, let’s examine popularity trends for individual
names over time. Now utilize Gini by grouping the female data by name and calculating the Gini coefficient as it pertains to yearly frequencies; that is, for any given name, sort each year of the
dataset by that name’s least to most popular year in order to compute Gini. Names with lower Gini coefficients demonstrate similar levels of popularity throughout the entire time span, while higher
coefficients imply uneven popularity levels. The figure below compares popularity trends for the names “Scarlett” and “Miriam.” Both names represent about 60,000 female babies in the dataset;
however, the sharp increase in babies named “Scarlett” generates a large Gini coefficient while “Miriam” sees a low Gini value since the name has consistently been given to roughly 1,000 babies every
year since 1950.</p> <center> <img src="https://kimfetti.github.io/images/scarlett_vs_miriam.png" alt="The popularity of female names Miriam and Scarlett over time with Gini coeffients" width="900"
/> </center> <p><br /></p> <h2 id="case-2-healthcare-prices">Case #2: Healthcare Prices</h2> <p>Now shift to <a href="https://www.cms.gov/Research-Statistics-Data-and-Systems/
Statistics-Trends-and-Reports/Medicare-Provider-Charge-Data/Inpatient2017">this 2017 healthcare pricing dataset</a> hosted by the Centers for Medicare and Medicaid Services, a federal agency of the
United States. These data, aggregated as procedural averages for individual hospitals, include the charges and eventual payments for over 500 separate inpatient procedures for Medicare patients. I
applied Gini coefficient calculations to determine which, if any, procedures require better billing standardization. The underlying basis for my analysis boils down to this: the higher the Gini
coefficient, the greater the disparity in what different hospitals charge for a given procedure. Procedures with large Gini values could then necessitate regulation or more transparent cost details.
</p> <p>The procedure, or diagnosis related group (DRG), with the highest Gini coefficient in this dataset<sup id="fnref:4" role="doc-noteref"><a href="#fn:4" class="footnote" rel="footnote">4</a></
sup> is labeled as, “Alcohol/Drug Abuse or Dependency w Rehabilitation Therapy.” This perhaps elicits little surprise given that rehabilitation therapies vary widely both in terms of treatment length
and illness severity; we probably expect a wide range in what assorted hospitals charge. In fact, all diagnoses with the largest Gini coefficients, such as coagulation disorders and psychoses, can
vary in severity. Procedural charges that show the most uniformity among the hospitals, on the other hand, mostly describe one-time cardiac events such as value replacement, percutaneous surgeries,
or observation for chest pain.</p> <center> <table width="800"> <caption>Gini coefficients among average hospital charges per diagnosis related group (DRG)</caption> <colgroup> <col span="1" style=
"width: 50%;" /> <col span="1" style="width: 50%;" /> </colgroup> <thead> <tr> <th><center>Highest Gini</center></th> <th><center>Lowest Gini</center></th> </tr> </thead> <tbody> <tr> <td>Alcohol/
Drug Abuse or Dependence w Rehabilitation Therapy</td> <td>Aortic and Heart Assist Procedures except Pulsation Balloon w MCC</td> </tr> <tr> <td>Coagulation Disorders</td> <td>Angina Pectoris</td> </
tr> <tr> <td>Alcohol/Drug Abuse or Dependence, Left AMA</td> <td>Cardiac Valve & Oth Maj Cardiothoracic Proc w/o Card Cath w/o CC/MCC</td> </tr> <tr> <td>Psychoses</td> <td>Heart Transplant or
Implant of Heart Assist System w MCC</td> </tr> <tr> <td>Other Respiratory System Diagnoses w MCC</td> <td>Perc Cardiovasc Proc w/o Coronary Artery Stent w/o MCC</td> </tr> </tbody> </table> </
center> <p>So what about billing regulation? Do we need more safeguards in place to be sure hospitals are charging similar amounts for similar procedures? Well, more cost transparency certainly
doesn’t hurt, especially for treatments that range in duration or intensity, but let’s go back to the dataset. In addition to the information about the amounts hospitals charge, the data also contain
<a href="https://www.cms.gov/Research-Statistics-Data-and-Systems/Statistics-Trends-and-Reports/Medicare-Provider-Charge-Data/Downloads/Inpatient_Outpatient_FAQ.pdf">the total payments that the
hospitals actually received</a>. Applying the same type of analysis to the payments received yields much lower Gini values. In fact, the Gini coefficient is lower for the average payments received
than the hospital charges, for <em>every single procedure</em>. This curious insight signals that the contracts in place for Medicare payments <em>already</em> do quite a lot to moderate and
regularize procedural costs.<sup id="fnref:5" role="doc-noteref"><a href="#fn:5" class="footnote" rel="footnote">5</a></sup></p> <center> <img src="https://kimfetti.github.io/images/gini_health.png"
alt="Comparison of Gini coeffients for total payments vs hospital charges" width="600" /> </center> <h2 id="conclusion">Conclusion</h2> <p>The Gini coefficient continues to provide insight over 100
years after its inception. As a good general-purpose measure of statistical dispersion, Gini can be used broadly to explore and understand data from nearly any discipline. Currently, the most popular
metric for understanding data spread is likely standard deviation; however, there are <a href="https://stats.stackexchange.com/questions/210829/
difference-is-summary-statistics-gini-coefficient-and-standard-deviation/211595">several key differences</a> between standard deviation and the Gini coefficient. Firstly, standard deviation retains
the scale of your data. You report the standard deviation of US incomes in dollars, while you might give the standard deviation of temperatures in degrees Celsius. The Gini coefficient, however, has
no measurement unit, also called scale invariance. Secondly, standard deviation is unbounded in that it can be any non-negative value, but Gini typically ranges between zero and one. Gini’s scale
invariance and strict bounds make comparing statistical dispersion between two dissimilar data sources much easier. Lastly, standard deviation and the Gini coefficient judge statistical dispersion
through different lenses. Gini reaches its maximum value for a non-negative dataset if it contains one positive and the rest zeros. Standard deviation reaches its maximum if half the data live at the
extreme maximum and the other half register at the extreme minimum.</p> <p><a href="https://www.scientificamerican.com/article/ask-gini/">Certain limitations</a> apply to the Gini coefficient despite
its many benefits. Like other summary statistics, Gini condenses information thereby losing the granularity of the original dataset. Gini is also many-to-one, which means various different
distributions map to the same coefficient. The Gini coefficient proves to be quite sensitive to outliers such that a singular extreme datapoint (large or small) can increase Gini dramatically. Yet,
economists have also criticized the Gini coefficient for being <a href="https://www.bbc.com/news/blogs-magazine-monitor-31847943">undersensitive to wealth changes in upper and lower echelons</a>.
Researchers have go on to introduce several alternative metrics to study different aspects of income inequality, such as the <a href="https://en.wikipedia.org/wiki/Income_inequality_metrics#
Palma_ratio">Palma ratio</a>, which explicitly captures financial fluctuations for the richest 10% and the poorest 40% of a population.</p> <p>No matter which metric you choose to understand
statistical dispersion, building data intuition certainly goes beyond simple estimates of the mean or median. The Gini coefficient, long since popular in the field of economics, provides excellent
insight about the spread of data regardless of your chosen subject area. As demonstrated in this post, Gini could be tracked over time, calculated for specific segments of your data, or used to
detect processes requiring better price standardization. Its applications are limitless, and it might just be the missing component of your EDA toolkit.</p> <hr /> <p><a href="https://github.com/
kimfetti/Blog/blob/master/gini_coefficient.ipynb">Check out this code on GitHub!</a></p> <div class="footnotes" role="doc-endnotes"> <ol> <li id="fn:1" role="doc-endnote"> <p>The Gini coefficient is
strictly non-negative, \(G \geq 0\), as long as the mean of the data is assumed positive. Gini can theoretically be greater than one if some data values are negative, which occurs in the context of
wealth if some people contribute negatively in the form of debts owed. <a href="#fnref:1" class="reversefootnote" role="doc-backlink">↩</a></p> </li> <li id="fn:2" role="doc-endnote"> <p>The
Social Security Administration does not include names that are given to fewer than 5 babies per gender per state due to privacy reasons; therefore, five children for one given female name since 1950
signifies the absolute minimum allowed. <a href="#fnref:2" class="reversefootnote" role="doc-backlink">↩</a></p> </li> <li id="fn:3" role="doc-endnote"> <p>The Gini values displayed in the
yearly figure are less than the aggregate because popular names tend to stay popular year after year thus bolstering naming inequality and increasing the Gini coefficient. <a href="#fnref:3" class=
"reversefootnote" role="doc-backlink">↩</a></p> </li> <li id="fn:4" role="doc-endnote"> <p>Some diagnosis related groups (DRGs) occur at as few as one hospital for the entire year. I have
filtered the dataset down to procedures that are documented by at least 50 hospitals to avoid high variance issues. <a href="#fnref:4" class="reversefootnote" role="doc-backlink">↩</a></p> </
li> <li id="fn:5" role="doc-endnote"> <p>The payments hospitals receive are strictly less than the amounts they charge. Decreasing a dataset’s mean while holding its standard deviation fixed <a href=
"https://repository.upenn.edu/gse_grad_pubs/6/">actually <em>increases</em> the Gini coefficient</a>. Here we observe just the opposite effect so statistical dispersion must be lessened in the
payments received. <a href="#fnref:5" class="reversefootnote" role="doc-backlink">↩</a></p> </li> </ol> </div> 2020-06-05T00:00:00+00:00 https://kimfetti.github.io/data/web%20scraping/
realpython-podcast-web-scraping/ 2020-06-05T00:00:00+00:00 KFessel <em>Do you want to get started with web scraping using Python? Find out more in this Real Python podcast.</em> <!--more--> <p>I
recently sat down with Christopher Bailey at the Real Python Podcast to discuss web scraping as well as my <a href="https://www.youtube.com/watch?v=RUQWPJ1T6Zc">PyCon 2020 tutorial</a>: “It’s
Officially Legal so Let’s Scrape the Web.” In this podcast we talk about web scraping tools and techniques, HTML basics and data cleaning, as well as a recent change to the legal landscape regarding
scraping.</p> <p>Check out the YouTube video above or listen to the podcast at <a href="https://realpython.com/podcasts/rpp/12/">Real Python</a>.</p> 2020-06-05T00:00:00+00:00 https://
kimfetti.github.io/data/web%20scraping/python-web-scraping/ 2020-05-04T00:00:00+00:00 KFessel <em>Developing web scraping skills allows you to save time and to broaden your access to data. This
tutorial covers web scraping with Python from the basics of HTML to the full scraping pipeline.</em> <!--more--> <p>Web scraping empowers you to write computer programs to collect data from websites
automatically and recent legal rulings support your right to do so. This tutorial covers the breadth and depth of web scraping: from HTML basics through pipeline methods to compile entire datasets.
My video provides step-by-step instructions on utilizing Python libraries like <code class="language-plaintext highlighter-rouge">requests</code> and <code class="language-plaintext
highlighter-rouge">BeautifulSoup</code> as well as links to supplementary tutorial resources in the form of Google Colab or Jupyter notebooks.</p> <p>Check out the supplementary materials via Google
Colab (<a href="https://bit.ly/pycon2020_scrapingbasics">Scraping Basics</a> and <a href="https://bit.ly/pycon2020_scrapingwiki">Scraping Wikipedia</a>) or on <a href="https://github.com/kimfetti/
Conferences/tree/master/PyCon_2020">GitHub</a>.</p> 2020-05-04T00:00:00+00:00 https://kimfetti.github.io/nlp/spacy-for-the-win/ 2020-02-21T00:00:00+00:00 KFessel <em>spaCy provides an easy-to-use
framework for getting started with NLP. This post covers the basics of spaCy and highlights its functionality on a small corpus of restaurant reviews..</em> <!--more--> <p>Natural language processing
(NLP) is a branch of artificial intelligence in which computers extract information from written or spoken human language. This field has experienced a massive rise in popularity over the years, not
only among academic communities but also in industry settings. Because unstructured text makes up so much of the data we collect today (e.g. emails, text messages, and even this blog post), many
practitioners regularly use NLP at the workplace and require straightforward tools to reliably parse through substantial amounts of documents. The open-source library spaCy meets these exact demands
by processing text quickly and accurately, all within a simplified framework.</p> <p><a href="https://explosion.ai/blog/introducing-spacy">Released in 2015</a>, spaCy was initially created to help
small businesses better leverage NLP. Its practical design offers users a streamlined approach for accomplishing necessary NLP tasks, and it assumes a more pragmatic stance toward NLP than
traditional libraries like NLTK, which were developed with a more research-focused, exploratory intention. spaCy can be quite flexible, however, as it allows more experienced users the option of
customizing just about any of its tools. spaCy is considered a Python package, but the “Cy” in spaCy indicates that Cython powers many of the underlining computations. This makes spaCy incredibly
fast, even for more complicated processes. I will illustrate a selection of spaCy’s core functionality in this post and will end by implementing these techniques on sample restaurant reviews.</p> <p>
Please continue to the <a href="https://opendatascience.com/level-up-spacy-nlp-for-the-win/">ODSC blog</a> to read my full post covering this introduction to spaCy.</p> 2020-02-21T00:00:00+00:00
https://kimfetti.github.io/mathematics/course-report-math-ds/ 2020-02-17T00:00:00+00:00 KFessel <em>Math skills are critical for a successful career in data science. Find out why in this interview
with Course Report.</em> <!--more--> <p>I recently sat down with <a href="https://www.coursereport.com/">Course Report</a> to discuss the math needed to become a data scientist. Blending coding
skills with mathematics lies at the heart of data science, so understanding fundamental math concepts is critical for a successful career within the field. Linear algebra, calculus, probability, and
statistics are the four math disciplines that fuel the bulk of data science. In this interview, I discuss the role each topic plays in data science; I also work through an example problem from all
four subjects.</p> <p>Please continue to the <a href="https://www.coursereport.com/blog/math-for-data-science-with-metis">Course Report blog</a> for a write-up of the interview.</p> <!-- image:
thumb: CourseReport_Thumb.png homepage: CourseReport_Feb2020.png title: CourseReport_Feb2020.png caption: "Photo by CP. Image constructed by Course Report." caption_url: "https://www.coursereport.com
/" --> 2020-02-17T00:00:00+00:00 https://kimfetti.github.io/visualizations/puzzles/down-and-up/ 2020-01-05T00:00:00+00:00 KFessel <em>Math puzzles provide great amusement for some people, but many
others approach them with dread--especially during interviews. Such trepidation may be unwarranted, however, because a simple visual--like the ones illustrated in this post--could be all you need to
find a solution.</em> <head> <script src="https://d3js.org/d3.v4.min.js"></script> <!--Multiple button functions--> <script src="https://cdnjs.cloudflare.com/ajax/libs/d3/4.3.0/d3.js"></script>
<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.6.1/jquery.min.js"></script> <style> input { border: none; color: white; padding: 8px 16px; margin: 4px 2px; cursor: pointer; } input[name=
paintButton] { background-color: #271B77; font-weight: bold; } input[name=danceButton] { background-color: #6BA450; font-weight: bold; } input[name=resetButton] { background-color: #ADADB0;
margin-top: 15px; } </style> </head> <!-- Begin Post --> <p>On a recent vacation my husband and I happened upon an entertainment shop that was well stocked with board games, dice, playing cards, etc.
We quickly found an item that both of us, absolute nerds that we are, deemed an essential purchase: a book by Boris A. Kordemsky called <a href="https://www.amazon.com/
Moscow-Puzzles-Mathematical-Recreations-Recreational/dp/0486270785/">The Moscow Puzzles: 359 Mathematical Recreations</a>. No, we didn’t spend our entire vacation solving all 359, but we did bring
the book home with us and have continued working through them–often over a glass of wine in the evenings.</p> <p>One particular puzzle recently caught my attention for several reasons. I’ll come back
to those reasons in a bit, but for now, the problem is called “Down and Up” and it goes like this:</p> <blockquote> <p>Suppose you have two pencils pressed together and held vertically. One inch of
the pencil on the left, measuring from its lower end, is smeared with paint. The right pencil is held steady while you slide the left pencil down 1 inch, continuing to press the two pencils together.
You then move the left pencil back up and return it to its former position, all while keeping the two pencils touching. You continue these actions until you have moved the left pencil down and up 5
times each. Assume the paint does not dry or run out during this process. <b>How many inches of each pencil are smeared with paint after your final movement?</b></p> </blockquote> <p>Take a minute to
solve this problem before proceeding if you’d like–spoilers ahead!</p> <h2 id="first-thoughts">First Thoughts</h2> <p>When I first heard this problem, I initially thought that perhaps the paint is
not smeared to the right pencil at all and perhaps only one inch of paint appears on the left pencil throughout the entire process. (Did you also expect this?) But the <em>second</em> time I read
through the problem I started to visualize what might actually be happening. The solution became much more clear as soon as I tried to make a mental picture of the process. Since my husband was
solving the problem with me, I made him this sketch to share what I was thinking:</p> <center> <img src="https://kimfetti.github.io/images/pencil_sketch.png" alt="Initial ideas as a sketch" width=
"550" /> </center> <p><br /></p> <p>I managed to distinctly envision the situation, arrive at a solution, and communicate my thought process just with this simple sketch. For many math puzzles a
rough picture provides all you need find the answer, but if my crude drawing hasn’t fully conveyed the solution to you, no worries. Let’s dive in a bit more methodically with a much nicer
illustration.</p> <p><img style="float: right; padding: 30px;" src="https://kimfetti.github.io/images/pencil_initial.gif" alt="Paint is spread to both pencils immediately" width="500" /></p> <h2 id=
"problem-setup">Problem Setup</h2> <p>From the problem directions, we know that initially only the left pencil is smeared with paint. Recall though that the left pencil presses directly against the
right. This means paint immediately transfers to the right pencil as they are squeezed together. So both pencils are smeared with one inch of paint even before any of the five down-up movements
occur.</p> <p><br /> <br /> <br /></p> <h2 id="solving-and-illustrating-the-full-problem">Solving and Illustrating the Full Problem</h2> <p>The problem gets a little more complicated as the left
pencil moves down and up, but returning to a visual interpretation once again helps immensely. Also feel free to reread the problem statement at any point to regain your bearings.</p> <p>Both pencils
are currently smeared with one inch of paint. Then the left pencil moves down one inch while both pencils continue pressing together. Can you envision what happens when the left pencil moves down?
Yes! A clean portion of the left pencil makes contact with the bottom of the right pencil; therefore, another inch of paint transfers over to the left.</p> <p>The left pencil now lingers one inch
lower than the right. One inch of the right pencil is smeared with paint, but paint covers <em>two inches</em> of the left pencil. The left pencil moves up in the next step of the problem, coming
back to its original position. So the two pencils realign, but what happens to the paint? Since the left pencil continually makes contact with the right, paint smears over to the right pencil and
coats two inches of both pencils at the end of the first down-and-up cycle.</p> <p>The four remaining cycles proceed similarly, with paint transferring first to the left pencil and then to the right.
<b>Finally after five rounds of movements, both pencils are smeared with a total of six inches of paint: an initial inch plus five more inches, one for each of the down-up cycles.</b></p> <p>This
problem ultimately hinges on the ability to translate the problem statement into an explanatory visual. To further contextualize this solution, I created an interactive figure with D3.js. Below both
pencils start with one inch of paint as described in the problem setup. Use the “Move Pencil” button to convince yourself of the answer I provided.</p> <p><em>Note: these pencils are six fictitious
inches long. After the fifth movement, the pencils reach equilibrium in that paint completely covers them. Hit the “Reset” button at any time to start over.</em></p> <p><br /></p> <div style="width:
100%; padding-bottom: 15px" id="pencilContainer"> <div style="float: left; width: 10%; height: 400; padding-left: 5%;"> <input name="paintButton" type="button" value="Move Pencil" onclick="movePencil
(); addPaint(1,800); addPaint(2,2000); incrUnits();" /> <br /> <input name="resetButton" type="button" value="Reset" onclick="removePaint()" /> </div> </div> <script> var pencilColor = "#F0C446"; var
paintColor = "#271B77"; var pencilData = [1, 2]; var width = $("div#pencilContainer").width(); var height = 400; var svg = d3.select("div#pencilContainer").append("svg") .attr("width", width*.6)
.attr("height", height) .style('transform', 'translate(40%, 0%)'); var objects = svg.append("g"); var pencils = objects.selectAll("g") .data(pencilData) .enter() .append("g") .attr("id", function(d,
i) { return i; }) .attr("transform", function(d, i) {return "translate(" + i*50 + ",0)"; }); var rects = pencils.append("rect") .attr("x", 50) .attr("y", 50) .attr("width", 50) .attr("height", 300)
.attr("fill", pencilColor) .style("fill-opacity", .7) .style("stroke-width",".2em") .style("stroke", pencilColor); var triangles = pencils.append("path") .attr('d', function(d, i) { var x = 0, y =
50; return 'M ' + (50+x) + ' ' + y + ' l ' + y/2 + ' ' + -y + ' l ' + y/2 + ' ' + y + ' z'; }) .attr("fill", pencilColor) .style("fill-opacity", .4) .style("stroke-width",".2em") .style("stroke",
pencilColor); var tips = pencils.append("path") .attr('d', function(d, i) { var x = 12.5, y = 25; return 'M ' + (50+x) + ' ' + y + ' l ' + y/2 + ' ' + -y + ' l ' + y/2 + ' ' + y + ' z'; }) .style
("fill-opacity", .7) .style("stroke-width",".2em") .style("stroke", "#393731"); var paint = pencils.append("rect") .attr("x", 50) .attr("y", 300) .attr("width", 50) .attr("height", 50) .attr("fill",
paintColor) .style("fill-opacity", 0.9) .style("stroke-width",".2em") .style("stroke", paintColor); var paintUnits = 1; var text = svg.append("text"); text .attr("x", 225) .attr("y", 50) .attr
("font-size",22); text.append("tspan") .text("Paint:"); var paintText = text.append("tspan") .attr("dx", 10) .style("fill", paintColor) .attr("font-weight", "bold") .text(paintUnits + " Inch");
function movePencil() { d3.select("g").selectAll("*") .filter(function (d) { return d == 1; }) .transition() .duration(750) .attr("transform", "translate(0,25)") .on("end",function() { d3.select
(this) .transition() .delay(750) .attr("transform", "translate(0,0)") }); } function addPaint(pencilNumber, delay) { d3.select("g").selectAll("*") .filter(function(d) { return d == pencilNumber; })
.filter(function(d,i) { return i == 4; }) .transition() .delay(delay) .attr("height", function(d) { return Math.min(paintUnits*50 + 50, 300); }) .attr("y", function(d) { return Math.max(300 -
50*paintUnits, 50); }); } function incrUnits() { paintUnits++; paintText.transition() .delay(2400) .text( Math.min(paintUnits, 6) + " Inches"); } function removePaint() { paint .transition()
.duration(500) .attr("y", 300) .attr("height", 50); paintUnits = 1; paintText.transition() .delay(250) .text( paintUnits + " Inch"); } </script> <h2 id="backstory-and-problem-extensions">Backstory
and Problem Extensions</h2> <p>Earlier I mentioned this problem caught my eye for several reasons. The first reason is exactly what we have been discussing. I marveled at how tricky the problem
sounds initially as opposed to how simple it becomes as soon as you construct an appropriate mental image of the situation.</p> <p>The second reason this puzzle piqued my interest is its history. As
explained in Kordemsky’s book, Leonid Mikhailovich Rybakov, a Soviet mathematician who lived in the early 20th Century, created this “Down and Up” problem. I deeply appreciate math problems that
pervade through many time periods and geographies. Solving such puzzles allows me to feel more connected to the past and to other mathematicians around the globe.</p> <p>Finally, this problem sparked
my curiosity because Rybakov first thought it up when returning home from a successful duck hunt. Kordemsky encourages readers to contemplate why this could be the case but goes on to explain in his
“Answers” section. From <em>The Moscow Puzzles</em> book:</p> <blockquote> <p>Looking at his boots, Leonid Mikhailovich noticed that their entire lengths were muddied where they usually rub each
other while he walks.<br /> “How puzzling,” he thought, “I didn’t walk in any deep mud, yet my boots are muddied up to the knees.” <br /> Now you understand the origin of the puzzle.</p> </
blockquote> <p>Just as the paint smeared the entire length of both pencils, Rybakov’s boots were covered from tip to top because mud had transferred from one boot to the other as he walked.</p> <p>I
continued to think about how this concept might apply to other situations, and I came up with one amusing but slightly unpleasant example. Consider two lines of contra dancers in which the first
dancer in the first line unfortunately feels unwell. If this dancer’s sickness is highly communicable, she will, of course, pass along her malady to her dance partner who is positioned across from
her. Sometimes in contra dancing participants exchange dance partners by shifting the two lines laterally. Regrettably, when this happens the newly infected dancer will pass the disease back across
the line, and eventually the entire group of dancers become ill. Try out my widget below to see this application in action.</p> <p><br /></p> <div style="width: 100%; padding-bottom: 15px" id=
"contraContainer"> <div style="float: left; width: 10%; height: 400; padding-left: 5%;"> <input name="danceButton" type="button" value="Dance!" onclick="moveBlushers('left', 0); sickBlusher();
moveBlushers('center', 2500); sickGrinner();" /> <br /> <input name="resetButton" type="button" value="Reset" onclick="makeWell()" /> </div> </div> <script> var blushEmoji = "https://
kimfetti.github.io/images/emoji_blush.png"; var grinEmoji = "https://kimfetti.github.io/images/emoji_grin.png"; var sickEmoji = "https://kimfetti.github.io/images/emoji_sick.png"; var contraData =
[1, 2, 3, 4, 5]; var w = $("div#contraContainer").width(); var h = 200; var canvas = d3.select("div#contraContainer").append("svg") .attr("width", w*.75) .attr("height", h) .style('transform',
'translate(30%, 0%)'); var blushGroup = canvas.append("g") .attr("id", "blushers"); var blushers = blushGroup.selectAll("image") .data(contraData) .enter() .append("image") .attr('xlink:href',
function (d, i) { if (i == 0) { return sickEmoji; } else { return blushEmoji; } }) .attr("x", function (d, i) { return w/75+d*(w/10); }) .attr("y", 0) .attr('width', w/13) .attr('height', w/13); var
grinGroup = canvas.append("g") .attr("id", "grinners"); var grinners = grinGroup.selectAll("image") .data(contraData) .enter() .append("image") .attr('xlink:href', grinEmoji) .attr("x", function (d,
i) { return w/75+d*(w/10); }) .attr("y", 100) .attr('width', w/14) .attr('height', w/14); var sickNum = 1; function moveBlushers(pos, delay) { if (sickNum == 1) { return; } else { d3.select("#
blushers") .selectAll("image") .transition() .delay(delay) .duration(1000) .attr("transform", function(d) { if (pos=="left") { return "translate(" + -w/10 +", 0)"; } else if (pos=="center") { return
"translate(0, 0)"; } }); }; } function sickBlusher() { { if (sickNum == 1) { return; } else { delay = 1200; } } d3.select("#blushers") .selectAll("image") .filter( function (d) { return d == sickNum;
}) .transition() .delay(delay) .style("opacity", 0) .attr("xlink:href", sickEmoji) .transition() .duration(800) .ease(d3.easeLinear) .style("opacity", 1); } function sickGrinner() { { if (sickNum ==
1) { delay = 300; } else { delay = 3300; } } d3.select("#grinners") .selectAll("image") .filter( function (d) { return d == sickNum; }) .transition() .delay(delay) .style("opacity", 0) .on("end",
function() { d3.select(this) .transition() .duration(800) .ease(d3.easeLinear) .style("opacity", 1) .attr("xlink:href", sickEmoji) .attr("width", w/13) .attr("height", w/13) }); sickNum++; } function
makeWell() { d3.select("#blushers") .selectAll("image") .attr('xlink:href', function (d, i) { if (i == 0) { return sickEmoji; } else { return blushEmoji; } }) .attr("width", w/13) .attr("height", w/
13); d3.select("#grinners") .selectAll("image") .attr("xlink:href", grinEmoji) .attr("width", w/14) .attr("height", w/14); sickNum = 1; } </script> <h2 id="conclusion">Conclusion</h2> <p>I hope you
have enjoyed this discussion on one of my new favorite math puzzles along with these illustrative D3 visuals. Making a mental image of a math puzzle is not always easy, but it can be invaluable when
solving problems like these–especially if you are a visual learner like myself. The next time you feel stuck on an interview question, check to see if sketching or imagining the physical setup of the
problem helps. For me it often does.</p> <p>I also hope you have enjoyed learning a little about the backstory behind this puzzle. Some of the world’s best math puzzles were created long ago, so I
believe looking to the past when attempting to sharpen our minds benefits us greatly. Furthermore, expanding this kind of problem to new applications, like I did with the contra dancers, helps
solidify core concepts and builds intuition for future brainteasers. It also makes math problems more enjoyable because you relate them to your own life. So now it’s your turn – can you think of any
other “Down and Up” scenarios?</p> <table> <tbody> <tr> <td>Check out my D3 code on GitHub!</td> <td> </td> <td><a href="https://github.com/kimfetti/Blog/blob/master/pencil_paint.html">Pencils and
Paint</a></td> <td> </td> <td><a href="https://github.com/kimfetti/Blog/blob/master/contra.html">Contra Dancers</a></td> </tr> </tbody> </table> 2020-01-05T00:00:00+00:00 https://kimfetti.github.io/
data/web%20scraping/gather-youtube-data/ 2019-11-12T00:00:00+00:00 KFessel <em>You can mine YouTube's massive content library for many different types of data. This post provides instructions for
obtaining the videos themselves, the video transcripts, as well as YouTube search results.</em> <p>Since its 2005 inception, YouTube has entertained, educated, and inspired more than <a href="https:/
/biographon.com/youtube-stats/">one billion people</a>. It now ranks as the <a href="https://www.alexa.com/siteinfo/youtube.com">2nd most visited website</a> on the planet, and its users upload 300
hours of video content every minute. YouTube clearly dominates as the world’s premier source of <a href="https://www.youtube.com/watch?v=_OBlgSz8sSM">cute baby moments</a>, <a href="https://
www.youtube.com/watch?v=vq8G81oOHhY">epic sports fails</a>, and <a href="https://www.youtube.com/watch?v=AS7_6Uv_Bn0">hilarious cat videos</a>, but its vast troves of content can also be leverage to
strengthen a wide variety of data science projects. In this post, I share how you can gain access to three types of YouTube data: the videos themselves for use in computer vision tasks, the video
transcripts for natural language processing (NLP), and video search results for hybrid machine learning efforts.</p> <p>Please continue to the <a href="https://www.thisismetis.com/blog/
how-to-gather-data-from-youtube">Metis blog</a> to read my full post covering data collection from YouTube.</p> 2019-11-12T00:00:00+00:00
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Fraction Worksheets For 4th Graders
Fraction worksheets for 4th graders provide practice and reinforcement of fraction concepts and skills. They are an essential tool for helping students develop a strong understanding of fractions,
which are a fundamental part of mathematics.
Fraction worksheets can cover a variety of topics, including:
• Identifying and comparing fractions
• Adding and subtracting fractions
• Multiplying and dividing fractions
• Solving fraction word problems
Fraction worksheets can be used in a variety of ways, including:
• As a supplement to classroom instruction
• For homework practice
• For review and remediation
Fraction worksheets are an important tool for helping 4th graders develop a strong understanding of fractions. They provide practice and reinforcement of fraction concepts and skills, and can be used
in a variety of ways to meet the needs of individual students.
Fraction Worksheets for 4th Graders
Fraction worksheets for 4th graders are an essential tool for helping students develop a strong understanding of fractions. They provide practice and reinforcement of fraction concepts and skills,
and can be used in a variety of ways to meet the needs of individual students.
• Practice: Fraction worksheets provide students with opportunities to practice fraction skills, such as identifying fractions, comparing fractions, adding and subtracting fractions, and
multiplying and dividing fractions.
• Assessment: Fraction worksheets can also be used as a formative assessment tool to help teachers identify students who need additional support with fraction concepts.
Fraction worksheets can be used in a variety of ways to support instruction. They can be used as a supplement to classroom instruction, for homework practice, or for review and remediation. Fraction
worksheets can also be differentiated to meet the needs of individual students. For example, students who are struggling with fraction concepts may need to use worksheets that focus on basic fraction
skills, such as identifying fractions and comparing fractions. Students who are more proficient with fraction concepts may be able to use worksheets that focus on more challenging fraction skills,
such as multiplying and dividing fractions.
Fraction worksheets are an important tool for helping 4th graders develop a strong understanding of fractions. They provide practice, reinforcement, and assessment of fraction concepts and skills.
Fraction worksheets can be used in a variety of ways to meet the needs of individual students.
Fraction worksheets for 4th graders provide ample opportunities for students to practice and reinforce essential fraction skills. Through these worksheets, students can engage in various activities
that strengthen their understanding of fractions and build a strong foundation for more complex mathematical concepts.
• Identifying Fractions: Fraction worksheets introduce students to different representations of fractions, such as shaded regions, fraction circles, and number lines. By identifying and labeling
fractions, students develop a visual understanding of fraction concepts.
• Comparing Fractions: Worksheets provide practice in comparing fractions using symbols (<, >, =) and common denominators. This helps students develop their number sense and understand the relative
sizes of fractions.
• Adding and Subtracting Fractions: Fraction worksheets present problems involving adding and subtracting fractions with like and unlike denominators. Through practice, students learn strategies
for finding common denominators and performing operations on fractions.
• Multiplying and Dividing Fractions: Worksheets introduce the concepts of multiplying and dividing fractions. Students practice finding the product or quotient of fractions, using visual models
and algorithms to solve problems.
By incorporating fraction worksheets into their learning, 4th graders gain valuable practice in applying fraction concepts to real-world situations. These worksheets contribute to their overall
mathematical development and prepare them for success in future math courses.
Fraction worksheets for 4th graders are valuable not only for providing practice and reinforcement of fraction concepts but also for assessing students’ understanding and identifying areas where they
may need additional support. As a formative assessment tool, fraction worksheets offer several benefits:
• Monitoring Student Progress: Fraction worksheets allow teachers to assess students’ progress towards mastery of fraction concepts. By reviewing completed worksheets, teachers can identify
students who have a strong grasp of the material and those who may need further instruction or support.
• Identifying Specific Areas of Difficulty: Fraction worksheets can help teachers pinpoint specific areas of difficulty that students may have with fractions. For example, a student who struggles
with adding fractions with unlike denominators may need additional practice with finding common denominators.
• Informing Instructional Decisions: The information gathered from fraction worksheets can inform instructional decisions and help teachers tailor their teaching to meet the individual needs of
students. For instance, if a worksheet reveals that a significant number of students are struggling with a particular concept, the teacher may decide to reteach that concept or provide additional
practice opportunities.
By utilizing fraction worksheets as a formative assessment tool, teachers can gain valuable insights into their students’ understanding of fractions and make informed decisions to support their
FAQs about Fraction Worksheets for 4th Graders
Fraction worksheets for 4th graders are a valuable resource for students, parents, and teachers alike. Here are some frequently asked questions about fraction worksheets:
Question 1: Why are fraction worksheets important for 4th graders?
Fraction worksheets provide essential practice and reinforcement of fraction concepts and skills, strengthening students’ understanding and preparing them for more complex mathematical concepts.
Question 2: What types of fraction skills do worksheets cover?
Fraction worksheets cover a wide range of fraction skills, including identifying fractions, comparing fractions, adding and subtracting fractions, multiplying and dividing fractions, and solving
fraction word problems.
Question 3: How can parents use fraction worksheets at home?
Parents can use fraction worksheets at home to supplement classroom learning, provide extra practice, and assess their child’s understanding of fraction concepts.
Question 4: How can teachers use fraction worksheets in the classroom?
Teachers can use fraction worksheets in the classroom to introduce new concepts, provide practice opportunities, assess student learning, and differentiate instruction based on individual student
Question 5: Where can I find free and printable fraction worksheets?
There are many websites and educational resources that offer free and printable fraction worksheets. A quick online search can lead to a variety of options.
Question 6: What are some tips for helping students who struggle with fractions?
Students who struggle with fractions may benefit from using visual aids, such as fraction circles or number lines. Breaking down fraction concepts into smaller steps and providing ample opportunities
for practice can also be helpful.
Fraction worksheets are an essential tool for helping 4th graders develop a strong understanding of fractions. By providing practice, reinforcement, and assessment opportunities, fraction worksheets
support students’ learning and prepare them for success in mathematics.
Transition to the next article section…
Tips for Using Fraction Worksheets for 4th Graders
Fraction worksheets can be a valuable tool for helping 4th graders develop a strong understanding of fractions. Here are five tips for using fraction worksheets effectively:
Tip 1: Start with the basics. Before students can tackle more complex fraction problems, they need to have a solid understanding of the basics. This includes being able to identify fractions, compare
fractions, and add and subtract fractions with like denominators.
Tip 2: Use visual aids. Visual aids can help students to understand abstract concepts like fractions. Fraction circles, fraction bars, and number lines can all be helpful tools for visualizing
fractions and understanding how they relate to each other.
Tip 3: Provide plenty of practice. The best way to learn fractions is through practice. Fraction worksheets provide students with the opportunity to practice identifying, comparing, adding,
subtracting, multiplying, and dividing fractions.
Tip 4: Make it fun. Learning fractions doesn’t have to be boring! There are many ways to make fraction worksheets more engaging for students. For example, you could use fraction worksheets with
real-world contexts, or you could turn fraction practice into a game.
Tip 5: Be patient. Learning fractions takes time and practice. Don’t get discouraged if your students don’t understand everything right away. With patience and perseverance, they will eventually
develop a strong understanding of fractions.
By following these tips, you can help your 4th graders to succeed with fraction worksheets and develop a strong understanding of fractions.
Fraction worksheets for 4th graders are an essential tool for helping students develop a strong understanding of fractions. They provide practice and reinforcement of fraction concepts and skills,
and can be used in a variety of ways to meet the needs of individual students.
By using fraction worksheets effectively, you can help your students to succeed with fractions and develop a strong foundation for future math learning.
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The Econbrowser Recession Indicator Index
James D. Hamilton
Dept. of Economics, University of California at San Diego
Declarations by the Business Cycle Dating Committee of the National Bureau of Economic Research (NBER) are regarded as highly authoritative by academic researchers, policy makers, and the public at
large. The NBER’s dates as to when U.S. recessions began and ended are based on the subjective judgment of the committee members, which raises two potential concerns. First, the announcements often
come long after the event. For example, NBER waited until July 17, 2003 to announce that the 2001 recession ended in November, 2001. Second, outsiders might wonder (perhaps without justification)
whether the dates of announcements are entirely independent of political considerations. For example, there might be some benefit to the presidential incumbent of delaying a declaration that a
recession had started or accelerating a declaration that a recession had ended. For these reasons, it is worth exploring whether one could perform a similar function using purely objective summaries
of the data.
Any such effort faces a tradeoff between two objectives. On the one hand, we might hope to use as much information in as much detail as possible. On the other hand, the more simple and parsimonious
the approach, the more likely it is to prove to be robust as the economy changes and data get revised. The approach described here is based on the second philosophy.
What sort of GDP growth do we typically see during a recession? It is easy enough to answer this question just by selecting those postwar quarters that the NBER has determined were characterized by
economic recession and summarizing the probability distribution of those quarters. A plot of this density, estimated using nonparametric kernel methods, is provided in the following figure; (figures
here are similar to those in a paper written in 2005 with UC Riverside Professor Marcelle Chauvet, which was published in Nonlinear Time Series Analysis of Business Cycles). The horizontal axis on
this figure corresponds to a possible rate of GDP growth (quoted at an annual rate) for a given quarter, while the height of the curve on the vertical axis corresponds to the probability of observing
GDP growth of that magnitude when the economy is in a recession. You can see from the graph that the quarters in which the NBER says that the U.S. was in a recession are often, though far from
always, characterized by negative real GDP growth. Of the 45 quarters up to the date that paper was written for which the NBER said the U.S. was in recession, 19 were actually characterized by at
least some growth of real GDP.
One can also calculate, as in the blue curve below, the corresponding characterization of expansion quarters. Again, these usually show positive GDP growth, though 10 of the postwar quarters that are
characterized by NBER as part of an expansion exhibited negative real GDP growth.
The observed data on GDP growth can be thought of as a mixture of these two distributions. Historically, about 20% of the postwar U.S. quarters are characterized as recession and 80% as expansion. If
one multiplies the recession density in the first figure by 0.2, one arrives at the red curve in the figure below. Multiplying the expansion density (second figure above) by 0.8, one arrives at the
blue curve in the figure below. If the two products (red and blue curves) are added together, the result is the overall density for GDP growth coming from the combined contribution of expansion and
recession observations. This mixture is represented by the yellow curve in the figure below.
It is clear that if in a particular quarter one observes a very low value of GDP growth such as -6%, that suggests very strongly that the economy was in recession that quarter, because for such a
value of GDP growth, the recession distribution (red curve) is the most important part of the mixture distribution (yellow curve). Likewise, a very high value such as +6% almost surely came from the
contribution of expansions to the distribution. Intuitively, one would think that the ratio of the height of the recession contribution (the red curve) to the height of the mixture distribution (the
yellow curve) corresponds to the probability that a quarter with that value of GDP growth would have been characterized by the NBER as being in a recession. Actually, this is not just intuitively
sensible, it in fact turns out to be an exact application of Bayes’ Law. The height of the red curve measures the joint probability of observing GDP growth of a certain magnitude and the occurrence
of a recession, whereas the height of the yellow curve measures the unconditional probability of observing the indicated level of GDP growth. The ratio between the two is therefore the conditional
probability of a recession given an observed value of GDP growth. This ratio is plotted as the red curve in the figure below.
Probability of recession if all we observe is one quarter’s GDP growth, as a function of the observed rate of rate of GDP. Adapted from Chauvet and Hamilton (2005)
Such an inference strategy seems quite reasonable and robust, but unfortunately it is not particularly useful– for most of the values one would be interested in, the implication from Bayes’ Law is
that it’s hard to say from just one quarter’s value for GDP growth what is going on. However, there is a second feature of recessions that is extremely useful to exploit– if the economy was in an
expansion last quarter, there is a 95% chance it will continue to be in expansion this quarter, whereas if it was in a recession last quarter, there is a 75% chance the recession will persist this
quarter. Thus suppose for example that we had observed -10% GDP growth last quarter, which would have convinced us that the economy was almost surely in a recession last quarter. Before we saw this
quarter’s GDP number, we would have thought in that case that there’s a 0.75 probability of the recession continuing into the current quarter. In this situation, to use Bayes’ Law to form an
inference about the current quarter given both the current and previous quarters’ GDP, we would weight the mixtures not by 0.2 and 0.8 (the unconditional probabilities of this quarter being in
recession and expansion, respectively), but rather by magnitudes closer to 0.75 and 0.25 (the probabilities of being in recession this period conditional on being in recession the previous period).
The ratio of the height of the resulting new red curve to the resulting new yellow curve could then be used to calculate the conditional probability of a recession in quarter t based on observations
of the values of GDP for both quarters t and t – 1. Starting from a position of complete ignorance at the start of the sample, we could apply this method sequentially to each observation to form a
guess about whether the economy was in a recession at each date given not just that quarter’s GDP growth, but all the data observed up to that point.
One can also use the same principle, which again is nothing more than Bayes’ Law, working backwards in time– if this quarter we see GDP growth of -6%, that means we’re very likely in a recession this
quarter, and given the persistence of recessions, that raises the likelihood that a recession actually began the period before. The farther back one looks in time, the better inference one can arrive
at. Seeing this quarter’s GDP numbers helps me make a much better guess about whether the economy might have been in recession the previous quarter. We then work through the data iteratively in both
directions– start with a state of complete ignorance about the sample, work through each date to form an inference about the current quarter given all the data up to that date, and then use the final
value to work backwards to form an inference about each quarter based on GDP for the entire sample.
All this has been described here as if we took the properties of recessions and expansions as determined by the NBER as given. However, another thing one can do with this approach is to calculate the
probability law for observed GDP growth itself, not conditioning at all on the NBER dates. Once we’ve done that calculation, we could infer the parameters such as how long recessions usually last and
how severe they are in terms of GDP growth directly from GDP data alone, using the principle of maximum likelihood estimation. It is interesting that when we do this, we arrive at estimates of the
parameters that are in fact very similar to the ones obtained using the NBER dates directly, and implied dates for recessions that are very close to those assigned by the NBER.
In that 2005 paper, Chauvet and I explored the potential use of this algorithm as an objective alternative to the declarations of the NBER Business Cycle Dating Committee, taking into account the
fact that the data are often revised substantially. For each quarter between 1967:Q1 and 2004:Q2 we assembled from the real-time database at the Federal Reserve Bank of Philadelphia a time series for
GDP growth as it would have been reported at that time, fit the parameters of the model to data available at the time, and calculated the implied probability of being in a recession for the
next-to-most-recent quarter. The reason for lagging the calculation by one quarter in this way is that data revisions and the extra insight from observing the subsequent quarter’s advance GDP release
are necessary in order to form a reliable inference. But if one allows for this extra quarter of smoothing, the inferences seem to be very useful. The following figure plots these real-time
probabilities, and is automatically updated using the most recent GDP statistics as described at Econbrowser.
Shaded areas in the above figure denote the dates of NBER recessions, which were not used in any way in constructing the index. Note moreover that this series is entirely real-time in construction–
the value for any date is always based solely on information as it was reported in the advance GDP estimates available one quarter after the indicated date, and the series by definition is never
The paper with Chauvet also explored how well an inference based on data as reported in real time would have performed based on the following rule. When the index rises above 67%, we declare the
economy to have been in a recession the preceding quarter, and use the full sample of information available as of that point to assign a probable date for the beginning of the recession, defined on
the biggest recent value of j for which Prob(S[t–j]=1|Y[t]) > 1/2 where S[t]=1 indicates a recession. Once a recession has been declared, that announcement remains in effect until the index falls
below 33%, at which point an end date for the recession is assigned based on the biggest recent j for which Prob(S[t–j]=1|Y[t]) < 1/2.
Since July 2005, I have been reporting the value for this index and making these calls on the website Econbrowser each time a new advance GDP figure gets released. A spreadsheet containing these
entries can be downloaded here. Note that any individual row of this spreadsheet is by construction never revised, but a new row is added with each new quarter’s data. For rows added since 2005, the
last column of the spreadsheet contains hyperlinks to the original release of that row’s numbers with discussion.
Prior to July 2020, we would recalculate the maximum-likelihood estimates of parameters using each new GDP report. The 2020:Q2 observation associated with the COVID recession was such an outlier that
it severely distorts maximum-likelihood estimates. One might try to describe the data using three regimes, where the third regime is just the 2020:Q2 observation. What we have done instead (as
here) is to keep parameter estimates fixed at their values estimated through 2020:Q2 data. Those parameter estimates are mean annualized growth rates for expansion and recession given by 3.87902 and
-1.51768, continuation probabilities for expansion and recession given by 0.943698 and 0.696427, and a variance for each regime of 10.1800.
Simulated (prior to 2005) and actual (since 2005) announcements of when recessions began and ended are provided in the following table.
Announcements based on recession indicator index
(last updated: April 2010)
Date of announcement Announcement
Simulated (through June 2005)
May 1970 recession began 1969:Q2
Aug 1971 recession ended 1970:Q4
May 1974 recession began 1973:Q4
Feb 1976 recession ended 1975:Q1
Nov 1979 recession began 1979:Q2
May 1981 recession ended 1980:Q2
Feb 1982 recession began 1981:Q2
Aug 1983 recession ended 1982:Q4
Feb 1991 recession began 1989:Q4
Feb 1993 recession ended 1991:Q4
Feb 2002 recession began 2001:Q1
Aug 2002 recession ended 2001:Q3
Actual real time (since July 2005)
Jan 30, 2009 recession began 2007:Q4
Apr 30, 2010 recession ended 2009:Q2
Jul 30, 2020 recession began 2020:Q1
Jan 28, 2021 recession ended 2020:Q2
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Muse of Mathematics
the art and science of teaching mathematics blog
June 24, 2021 Incubator Daniel and Mason presented on Smileys. That's a puzzle that has you sparsely filling a grid with smiley faces. Then you start the app running. At each time-step a grumpy face
turns into a smiley face if it touches two or more smiley faces. The...
July 8, 2021 Incubator Asmita Sodhi presented 5x5 Pentomino Sudoku puzzles. These are problems that her father, Amar worked on so they have a special place. You can read more about Amar and these
puzzles on pages 10-12 of the 2018 CMS Notes:...
Is there a tendency in our education culture to avoid teaching the core of science? Do science centres cheapen science by making it into entertainment - with all the slow discovery and math removed?
Do our schools do a disservice to our children by giving them all of...
When presented with a tough challenge a majority of our students charge right in. They see a computation and charge! We'll call this majority the Quixotic Problem Solvers. The worst offenders solve
one thing and instead of trying to generalize or to take steps to...
Standard curricula have children jump through a progression of hoops. True - they stop every so often to "review" content, but this is usually time wasted for top students and it is only tolerated in
order to get struggling students "caught-up" with the rest of the...
I want to raise a red flag about a practice that may have value, but is being pushed too much. Asking children to reflect and articulate how they think is not as important as thinking. Math class
should be spent thinking - not thinking about thinking. If I see my...
Molly Crocker contacted me to ask for my opinion on her finger counting ideas 1 to 99. I decided to take the opportunity and review different techniques to teach counting. The questions you should
ask in selecting a technique for your classroom: Does the...
I have honed some techniques for introducing new games and puzzles into the elementary classroom. From my previous blog postings (and the video below) you will know that I do NOT recommend teaching
the rules at the start, but rather to engage students immediately by...
I have been using your puzzles for a long time on my classrooms... Honestly, they are the best mathematics learning I have found to date- all the students have an entry point, and everyone is
successful and challenged... I'm wondering how you come up with questions...
I failed yesterday. Background: I encourage students to work in pairs or occasionally triples with a single puzzle-sheet shared between them. In my classes paper is a scarce resource. I love to see
co-operative math as in the photo...
Mathematics is usually taught brick on brick - each brick resting solidly on the ones underneath. This sounds good. Brick on brick mathematics education is capable of building an impressive edifice.
Look at the power of the average calculus student after a dozen years...
To teach the Scientific Method the natural impulse is to turn to the natural world. This needs rethinking. Mini-Mathethatical Universes can be created which students can poke and prod with precision.
These universes should be given to students without explanation. My...
I am most impressed with the advice of the Julia Robinson Mathematics Festival to the volunteers who come to people its tables: Be as unhelpful as possible. This is marvellous advice for
educators and parents to follow most of the time. As a parent I struggle...
Why do we allow students to work in pencil? Sometimes we do it because we want them to erase their mistakes. If done to excess, this is wrong. Mistakes are there to learn from - not to be erased or
scribbled out. They should be artfully identified so that teacher and...
Speed is essential, but some curricula value it too much. The core of every mathematics classroom should be problem solving. Ponderous problem solvers need to be protected. I am not making the case
against memorizing basic facts. Students absolutely need...
We should abolish the subject of mathematics in elementary school. Why? Because "mathematics" has become synonymous with arithmetic for many educators and parents. Problem solving, which should be at
the heart of the classroom experience of mathematics is only given...
I'm agnostic when it comes to computer games. On the negative side: 1) Too many students already have too much screen time at home - the last thing this subset of students need is to have screen time
in school. 2) Quality control is lacking. On the positive side: 1)...
I’ve already put up some magic tricks on MathPickle, but I’d like your input. What place does magic have in the classroom? Mathemagic is great. Some teachers use it to increase their coolness
factor, but the primary reason to use mathemagic should be to generate...
What about the use of games in the classroom? The heart of mathematics education is problem solving. Thinking games pose one problem after another problem - a whole sequence of problems that end
with victory or defeat. They fit naturally into a mathematics classroom...
In your last post, you ended with a recommendation that teachers maintain a level of classroom chaos so that students didn’t know if they are slow or fast. Yes - I don’t emphasize speed in my
assessment of a student, and I want to protect slow students from a negative...
Please use MathPickle in your classrooms. If you have improvements to make, please contact me. I'll give you credit and kudos 😉 For a free poster of MathPickle's ideas on elementary math education go
Gordon Hamilton
(MMath, PhD)
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How to Add Days in MS Excel? [5 Examples] - QuickExcel
How to Add Days in MS Excel? [5 Examples]
Appearances are deceptive! That is especially true when it comes to the data that is in Date format in MS Excel. Though a date may appear in the DD-MM-YYYY format or fancies any other format for that
matter, deep underneath MS Excel assigns a value to this date. To put it in simple terms, dates are just continuous serial numbers with 1 representing the 1^st of January 1900.
Observe & Believe!
In the above image, it is evident that when tried to look into the numerical equivalent for the 1^st January 1900 in the format dropdown it shows the number ‘1’. Having established this conundrum let
us now move on to finding out the different ways of adding days in MS Excel.
Adding Days in Excel:
Listed below are 5 examples to demonstrate the different ways to add the days. One can choose either of these based on whichever best suits the requirements at hand.
• Using Operand
• Using DATE Formula
• The Accelerator Way
• Using EOMONTH Formula
• Using TODAY Formula
Example 1 – Adding Days in Excel Using Operand
Beneath the skin, it is just a number so a simple addition operand shall serve the purpose of adding the days. Let us try adding 20 days to the 1^st of January 1900. It could be done by starting with
an equals (=) sign & clicking the cell which contains the date. Then include a plus (+) sign followed by the number 20 as shown below.
Adding 20 Days
Hit ENTER & the new date 20 days after the selected date shall appear.
20 Days Added!
Example 2 – Adding Days in Excel Using the DATE Formula
Another way of constructing a date is using the DATE formula in which references would be given to each of the cells containing the value of the day, month & year – the basic constituents of the
date. Have a look at the following example to get a fair idea.
DATE Formula
Syntax for DATE formula is,
=DATE( year, month, date)
So, to add the days for any given date in this formula one can put a plus sign (+) following the cell referred to in the date portion of the formula.
Adding Days within DATE Formula
Hit ENTER & the new date shall appear.
Days Added within DATE Formula
Example 3 – Adding Days in Excel Using The Accelerator Way
Using the infamous accelerator let’s add 20 days to the date in the below image.
Sample Date
Copy the count of days to be added using CTRL+C & move the active cell to J7 which has the date as shown below.
J8 Copied
Hit the following keys in the same sequence for the Paste Values Dialog box to appear.
ALT – E – S – V – D
Paste Special Dialog Box
Hit ENTER & the new date shall appear!
Days Added!
Example 4 – Adding Days in Excel Using EOMONTH Formula
EOMONTH formula is used to get the last date of a month whose syntax is as follows.
=EOMONTH( start_date, months)
• start_date – reference date to which end of the month is to be determined
• months – count of months after which end of the month is to be determined
EOMONTH Formula
If one wants to add a few days to the result of the EOMONTH, here’s what is needed to be done.
Days Added to EOMONTH
Example 5 – Adding Days in Excel Using TODAY Formula
MS Excel has this fascinating feature of displaying today’s date when the following syntax is put into use.
What if one wants to add a certain number of days to the current date? Just add a plus (+) sign followed by the required number at the end of the formula.
Adding Days to Current Date
Now that we have reached the end of this article, here’s another that details on how to add months to a date in MS Excel. There are numerous other articles too in QuickExcel that can come in handy
for those who are in looking to level up their skills in MS Excel. Adios!
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Maximization Bias
This is an idea that even though each estimate of the state-action values is unbiased, the estimate of ’s value of can be biased.
See video
“All the control algorithms that we have discussed so far involve maximization in the construction of their target policies. For example, in Q-learning the target policy is the greedy policy given
the current action values, which is defined with a max, and in Sarsa the policy is often -greedy, which also involves a maximization operation. In these algorithms, a maximum over estimated values is
used implicitly as an estimate of the maximum value, which can lead to a significant positive bias.
Consider a single state where there are many actions whose true values, , are all zero but whose estimated values, , are uncertain and thus distributed some above and some below zero. The maximum of
the true values is zero, but the maximum of the estimates is positive, a positive bias. We call this maximization bias“.
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Adaptive Regularization Minimization Algorithms with Non-Smooth Norms and Euclidean Curvature
A regularization algorithm (AR1pGN) for unconstrained nonlinear minimization is considered, which uses a model consisting of a Taylor expansion of arbitrary degree and regularization term involving a
possibly non-smooth norm. It is shown that the non-smoothness of the norm does not affect the O(epsilon_1^{-(p+1)/p})$ upper bound on evaluation complexity for finding first-order
epsilon_1-approximate minimizers using p derivatives, and that this result does not hinge on the equivalence of norms in Re^n. It is also shown that, if p=2, the bound of O(epsilon_2^{-3})
evaluations for finding second-order
\epsilon_2-approximate minimizers still holds for a variant of AR1pGN named AR2GN, despite the possibly non-smooth nature of the regularization term. Moreover, the adaptation of the existing theory
for handling the non-smoothness results in an interesting modification of the subproblem termination rules, leading to an even more compact complexity analysis. In particular, it is shown when the
Newton's step is acceptable for an adaptive regularization method. The approximate minimization of quadratic polynomials regularized with non-smooth norms is then discussed, and a new approximate
second-order necessary
optimality condition is derived for this case. An specialized algorithm is then
proposed to enforce the first- and second-order conditions that are strong enough to ensure the existence of a suitable step in AR1pGN (when p=2) and
in AR2GN, and its iteration complexity is analyzed.
Original language English
Publisher Arxiv
Volume 2105.07765
Publication status Published - May 2021
Dive into the research topics of 'Adaptive Regularization Minimization Algorithms with Non-Smooth Norms and Euclidean Curvature'. Together they form a unique fingerprint.
• Bellavia, S., Gurioli, G., Morini, B. &
TOINT, P.
Feb 2023
In: Journal of Optimization Theory and Applications. 196
p. 700-729 30 p.
Research output: Contribution to journal › Article › peer-review
• Cartis, C., Gould, N. I. M. &
TOINT, P.
Jul 2022
600 p.
(SIAM-MOS Series on Optimization)
Research output: Book/Report/Journal › Book
• Toint, P. (CoI), Gould, N. I. M. (CoI) & Cartis, C. (CoI)
1/11/08 → …
Project: Research
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dst: Using the Theory of Belief Functions
Using the Theory of Belief Functions for evidence calculus. Basic probability assignments, or mass functions, can be defined on the subsets of a set of possible values and combined. A mass function
can be extended to a larger frame. Marginalization, i.e. reduction to a smaller frame can also be done. These features can be combined to analyze small belief networks and take into account
situations where information cannot be satisfactorily described by probability distributions.
Version: 1.8.0
Depends: R (≥ 3.5.0)
Imports: dplyr, ggplot2, tidyr, Matrix, methods, parallel, rlang, utils
Suggests: igraph, knitr, rmarkdown, tidyverse, testthat
Published: 2024-09-03
DOI: 10.32614/CRAN.package.dst
Author: Peiyuan Zhu [aut, cre], Claude Boivin [aut]
Maintainer: Peiyuan Zhu <garyzhubc at gmail.com>
BugReports: https://github.com/RAPLER/dst-1/issues
License: GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]
NeedsCompilation: no
Materials: README NEWS
CRAN checks: dst results
Reference manual: dst.pdf
Bayes_Rule (source, R code)
Captain_Example (source, R code)
Crime_Scene (source, R code)
Crime_Scene_Commonality (source, R code)
Evidential_Modelling (source, R code)
Holmes_Burglary (source, R code)
Introduction to Belief Functions (source, R code)
Vignettes: PJM_example_DSC (source, R code)
PJM_example_DSC_Multivalued_Map (source, R code)
PJM_example_DSC_Simplified (source, R code)
Reliability_Proof_Machinery (source, R code)
Simple_Implication (source, R code)
Template (source, R code)
The Monty Hall Game (source, R code)
The original peter, John and Mary example (source, R code)
Peeling algorithm on Zadeh's Example (source, R code)
Package source: dst_1.8.0.tar.gz
Windows binaries: r-devel: dst_1.8.0.zip, r-release: dst_1.8.0.zip, r-oldrel: dst_1.8.0.zip
macOS binaries: r-release (arm64): dst_1.8.0.tgz, r-oldrel (arm64): dst_1.8.0.tgz, r-release (x86_64): dst_1.8.0.tgz, r-oldrel (x86_64): dst_1.8.0.tgz
Old sources: dst archive
Please use the canonical form https://CRAN.R-project.org/package=dst to link to this page.
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Walking Times
1. How many minutes does it take to walk from Oxford Circus to Tottenham Court Road?
2. How many minutes longer does it take to walk from Lancaster Gate to Marble Arch than to walk from Oxford Circus to Tottenham Court Road?
3. If I set out from Westminster at 4:40pm and walked directly to St James's Park Tube station, what time would I arrive?
4. If I set out from South Kensington at 2:57pm and walked directly to Sloane Square, what time would I arrive?
5. I want to arrive at Westminster at exactly 5:57pm. What time should I leave the coffee shop at Green Park Tube station?
6. I want to arrive at Regent's Park Tube station at exactly 4:05pm. What time should I leave Oxford Circus Tube station?
7. I walk from Covent Garden to Russell Square via Holborn the back to Covent Garden along the same route. If I leave Covent Garden at 7:50pm, what time will my walk end?
8. I set out from Oxford Circus at 11:15am and walk to the next Tube station. I get there at 11:33am. What is the name of that station?
9. What is the average walking time (in minutes) between pairs of adjacent stations on the Piccadilly line between South Kensington and Covent Garden?
10. If the distance from Angel to Old Street is one mile. What is the presumed walking speed (in mph) used in the making of this map?
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[Solved] An equilateral triangle PQR is circumscribed about a g... | Filo
An equilateral triangle is circumscribed about a given . Prove that the maximum area of is . Where are the sides of and is its area.
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Let the angle between and be
In , from the sine rule
In , from the sine rule.
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Question Text An equilateral triangle is circumscribed about a given . Prove that the maximum area of is . Where are the sides of and is its area.
Topic Trigonometric Functions
Subject Mathematics
Class Class 11
Answer Type Text solution:1
Upvotes 134
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Polar Spline Chart
Polar charts use polar coordinates to plot data. The X-Axis is a circle and the values are normally fixed in degrees (0-360) or radians typical polar charts, but the chart will allow you to set any
range desired. Data is plotted in terms of values and angles, where "x" is the angle/rotation and "y" is distance from the center of the circle.
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What Do You Need to Know for SAT Math? - Studyville
The SAT is a standardized test that covers critical reading, writing, and mathematics concepts,
and it is commonly used during the college admissions process to measure a student’s
compatibility with a specific university’s academic standards. When it comes to preparing for an
exam of such importance, understanding the breakdown of each section is essential, especially
for the subjects you find most difficult. If you are one of the many students who see math as
their blind spot, you’re in luck; I’m happy to clue you in on all the important details of the SAT
Math section and how you can use this information to your advantage.
Calculator vs. Non-Calculator
The math section of the exam is split into a calculator section and a non-calculator section. The
calculator portion has 38 questions to be completed in 55 minutes, while the non-calculator
portion has 20 questions to be completed in 25 minutes. Now, calculators are incredibly useful
tools, but as I often heard in my high school math courses, they are only as capable as the
person using them. Speaking from experience, a handful of the calculator questions do not
actually require the use of a calculator. Being able to decide whether or not you need to bust out
your fancy graphing calculator for a question will play a key role in time management.
Additionally, I cannot stress just how important it is for you to know the functions of the
calculator you intend to use on the test. All scientific calculators are allowed and more than
capable of helping you through, and some graphing calculators are permitted, as well.
Familiarize yourself with how to use your calculator to change decimals into fractions, switch
between degrees and radians, etc.
Four Content Areas of the SAT
With a total of 58 questions to answer, you may be wondering exactly which concepts from your
courses will show up on the SAT. College Board says all their Math questions fall under one of
these four categories: heart of algebra, problem solving and data analysis, passport to
advanced math, and additional topics in math. I don’t know about you, but this doesn’t tell me a
ton. To clarify for you, the SAT will cover various math topics from algebra to geometry, and
even some pre-calculus. I recommend sifting through practice exams online to gauge the
difficulty of questions. For problem solving and data analysis, many questions will simply ask
you to look at graphs or tables and interpret the information; therefore, sharpening your
mathematical reasoning skills is a must!
Multiple-Choice vs. Grid-In SAT Math Questions
Exam test school or university concept : Hand student holding pencil writing standardized answer multiple carbon paper form with gray black answers sheet bubbled of question in examination
The majority of questions on the SAT Math test are multiple-choice. This means we have
process of elimination on our side when we are not sure about how to answer a question.
Unfortunately, though, 13 grid-in questions will be asked; the odds of successfully guessing on
these are incredibly slim. As a tip, when you’re studying for SAT Math, treat all the questions
that need a numerical answer as a grid-in. Refrain from looking at the answer choices, and try to
come up with the right answer just by working it out. This will dull your reliance on multiple-choice problems without hindering your ability to answer them correctly, and it will get you
comfortable with attempting grid-in questions without feeling intimidated.
Formulas Provided
If you are stressed about what kind of concepts do or do not need to be committed to memory
prior to your test day, you might be relieved to know that College Board provides some
reference material for you. Most of these formulas are geometric (volume of a cone,
circumference of a circle, pythagorean theorem, etc.). Aside from these, other formulas you
need to commit to memory include the quadratic formula and the slope formula. Everything else
you’ll need to know will be more second-nature. You should know how to find simple probability,
change fractions into decimals or percentages and vice versa, and find the mean or average of
a set of numbers. Check out PrepScholar for more information on what kind of formulas will be
provided and which formulas you may need to memorize.
Extra Tidbits to Help You Ace SAT® Math
When I was in high school, I took the PSAT twice per school requirements at the time and the
SAT twice per my own self-competitiveness, and while math is my strongest subject,
standardized testing can often be more about understanding the objective of the test writers
than content mastery. The SAT Math and Reading/Writing sections are each scored between
200 and 800 points. When attempting to raise my SAT score, my main strategy was to practice,
practice, practice! Any student I’ve tutored in Math has probably had to endure my passionate
“practice makes perfect” speech, and I stand by it! Math is procedural, and you can build your
math ability like it is a muscle. By doing practice exams (and I mean really doing, like timing
myself and everything), I was able to significantly raise my SAT Math score. Not only are you
sharpening your math skills, you are also familiarizing yourself with how the test wants to test
you, challenge you, and trick you. Two birds with one stone.
The SAT Math section is absolutely a code that can be cracked, and hopefully, this abundance
of information will set you on a path to do exactly that. Nowadays, you can take the exam as
many times you want (with the proper finances, of course) without having to report all your
testing attempts to colleges. I encourage you to use this to your advantage, as well. Here at
Studyville, we offer SAT Private Sessions that help students reach their goals for both the SAT
and the PSAT. If you are preparing for either of these tests, keep calm, keep practicing, and
keep us in mind!
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Largest of Three numbers program in Python - CodeKyro
Largest of Three numbers program in Python
In the previous article we explored a problem involving a python game utilizing Even or Odd Logic, now we need to write a program to accept three numbers from the user and then automatically provide
to us the largest of the three numbers. Suppose the user provides the program the numbers: a=10, b= 15, c=7 it should return to us the variable b along with it’s value. If the numbers provided are a=
23, b=-1 and c= -28. The variable a and the value 23 is returned. Suppose it is a=5, b=5. c=5, the program returns the value of 5.
Implementation of Largest of Three numbers program in Python
Let us see a simple working of a python program that accepts three values from the user: a, b and c . And then it prints out the largest of the three numbers. It figures out the largest of the three
by utilizing conditional if-else statements and comparison operators.
a=int(input("Enter First Number: "))
b=int(input("Enter Second Number: "))
c=int(input("Enter Third Number: "))
if(a>=b) and (a>=c):
if(b>=a) and (b>=c):
if(c>=a) and (c>=b):
Enter First Number: 200
Enter Second Number: 400
Enter Third Number: 6000
The largest of the three numbers is 6000
The program returns the largest number found through the use of comparison operators. However there is a similar but way more simple and effective way of finding out the largest of given numbers
through the use of the in-built function ‘max( )‘.
max() in python
a=int(input("Enter First Number: "))
b=int(input("Enter Second Number: "))
c=int(input("Enter Third Number: "))
Enter First Number: 200
Enter Second Number: 7000
Enter Third Number: 6000
The max( ) function can take in any number of inputs for finding out max value.
Hence, we have seen two approaches for finding out the maximum or largest of a series of given input numbers. In the next article we will look at Leap Year program in Python logic.
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AP Board 9th Class Maths Notes Chapter 6 Linear Equation in Two Variables
Students can go through AP Board 9th Class Maths Notes Chapter 6 Linear Equation in Two Variables to understand and remember the concepts easily.
AP State Board Syllabus 9th Class Maths Notes Chapter 6 Linear Equation in Two Variables
→ Equations like x + 7 = 10; y + √3 = 8 are examples of linear equations in one variable.
→ If a linear equation has two variables then it is called a linear equation in two variables. Eg.: 3x – 5y = 8; 5x + 7y = 6 ….
→ The general form of a linear equation in two variables x and y is ax + by + c = 0; where a, b, c are real numbers and a, b are not simultaneously zero.
→ Any pair of values of x and y which satisfy ax + by + c = 0 is called the solution of linear equation.
→ An easy way of getting two solutions is put x = 0 and get the corresponding value of y. Similarly put y = 0 and get the value for x.
→ The line obtained by joining all points which are solutions of a linear equation is called graph of linear equation.
→ Equation of a line parallel to X-axis is y = k. (at a distance ‘k’ units)
→ Equation of a line parallel to Y-axis at a distance of k – units is x = k.
→ Equation of X-axis is y = 0 and Y-axis is x = 0.
→ The graph of x = k is a line parallel to Y-axis at a distance of ‘k’ units and passing through the point (k, 0).
→ The graph of y = k is a line parallel to X-axis at a distance of k – units and passing through the point (0, k).
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CPM Homework Help
Solve for the missing side lengths and angles in the triangle at right.
This triangle has known values for $\text{SAS}$ (side-angle-side) so will need to be solved using the Law of Cosines.
Refer to the Math Notes box in Lesson 5.3.3 if you need help remembering the Law of Cosines.
$c^2=10^2+17^2-2(10)(17)\text{cos }45º$
After finding the length of $c$, the Law of Sines can be used to to find the measure of $∠A$.
Refer to the Math Notes box in Lesson 5.3.2 for the Law of Sines.
$\frac{\text{sin }A}{10}=\frac{\text{sin }45}{c}$
Use the Triangle Angle Sum Theorem to find the measure of $∠B$.
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A Simple Introduction to Random Forests | Online Statistics library | StatisticalPoint.com
A Simple Introduction to Random Forests
by Erma Khan
When the relationship between a set of predictor variables and a response variable is highly complex, we often use non-linear methods to model the relationship between them.
One such method is classification and regression trees (often abbreviated CART), which use a set of predictor variables to build decision trees that predict the value of a response variable.
Example of a regression tree that uses years of experience and average home runs to predict the salary of a professional baseball player.
The benefit of decision trees is that they’re easy to interpret and visualize. The downsize is that they tend to suffer from high variance. That is, if we split a dataset into two halves and apply a
decision tree to both halves, the results could be quite different.
One way to reduce the variance of decision trees is to use a method known as bagging, which works as follows:
1. Take b bootstrapped samples from the original dataset.
2. Build a decision tree for each bootstrapped sample.
3. Average the predictions of each tree to come up with a final model.
The benefit of this approach is that a bagged model typically offers an improvement in test error rate compared to a single decision tree.
The downside is that the predictions from the collection of bagged trees can be highly correlated if there happens to be a very strong predictor in the dataset. In this case, most or all of the
bagged trees will use this predictor for the first split, which will result in trees that are similar to each other and have highly correlated predictions.
Thus, when we average the predictions of each tree to come up with a final bagged model, it’s possible that this model doesn’t actually reduce the variance by much compared to a single decision tree.
One way to get around this issue is to use a method known as random forests.
What Are Random Forests?
Similar to bagging, random forests also take b bootstrapped samples from an original dataset.
However, when building a decision tree for each bootstrapped sample, each time a split in a tree is considered, only a random sample of m predictors is considered as split candidates from the full
set of p predictors.
So, here’s the full method that random forests use to build a model:
1. Take b bootstrapped samples from the original dataset.
2. Build a decision tree for each bootstrapped sample.
• When building the tree, each time a split is considered, only a random sample of m predictors is considered as split candidates from the full set of p predictors.
3. Average the predictions of each tree to come up with a final model.
By using this method, the collection of trees in a random forest is decorrelated compared to the trees produced by bagging.
Thus, when we take the average predictions of each tree to come up with a final model it tends to have less variability and results in a lower test error rate compared to a bagged model.
When using random forests, we typically consider m = √p predictors as split candidates each time we split a decision tree.
For example, if we have p = 16 total predictors in a dataset then we typically only consider m = √16 = 4 predictors as potential split candidates at each split.
Technical Note:
It’s interesting to note that if we choose m = p (i.e. we consider all predictors as split candidates at each split) then this is equivalent to simply using bagging.
Out-of-Bag Error Estimation
Similar to bagging, we can calculate the test error of a random forest model by using out-of-bag estimation.
It can be shown that each bootstrapped sample contains about 2/3 of the observations from the original dataset. The remaining 1/3 of the observations not used to fit the tree are referred to as
out-of-bag (OOB) observations.
We can predict the value for the ith observation in the original dataset by taking the average prediction from each of the trees in which that observation was OOB.
We can use this approach to make a prediction for all n observations in the original dataset and thus calculate an error rate, which is a valid estimate of the test error.
The benefit of using this approach to estimate the test error is that it’s much quicker than k-fold cross-validation, especially when the dataset is large.
The Pros & Cons of Random Forests
Random forests offer the following benefits:
• In most cases, random forests will offer an improvement in accuracy compared to bagged models and especially compared to single decision trees.
• Random forests are robust to outliers.
• No pre-processing is required to use random forests.
However, random forests come with the following potential drawbacks:
• They’re difficult to interpret.
• They can be computationally intensive (i.e. slow) to build on large datasets.
In practice, data scientists typically use random forests to maximize predictive accuracy so the fact that they’re not easily interpretable is usually not an issue.
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A design for an electromagnetic filter for precision energy measurements at the tritium endpoint
We present a detailed description of the electromagnetic filter for the PTOLEMY project to directly detect the Cosmic Neutrino Background (CNB). Starting with an initial estimate for the orbital
magnetic moment, the higher-order drift process of E×B is configured to balance the gradient-B drift motion of the electron in such a way as to guide the trajectory into the standing voltage
potential along the mid-plane of the filter. As a function of drift distance along the length of the filter, the filter zooms in with exponentially increasing precision on the transverse velocity
component of the electron kinetic energy. This yields a linear dimension for the total filter length that is exceptionally compact compared to previous techniques for electromagnetic filtering. The
parallel velocity component of the electron kinetic energy oscillates in an electrostatic harmonic trap as the electron drifts along the length of the filter. An analysis of the phase-space volume
conservation validates the expected behavior of the filter from the adiabatic invariance of the orbital magnetic moment and energy conservation following Liouville's theorem for Hamiltonian systems.
Bibliographical note
Publisher Copyright:
© 2019 Elsevier B.V.
• CNB
• Cosmic Neutrino Background
• Neutrino mass
• PTOLEMY
• Relic neutrino
• Transverse drift filter
Dive into the research topics of 'A design for an electromagnetic filter for precision energy measurements at the tritium endpoint'. Together they form a unique fingerprint.
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JTS Overlay - the Next Generation
In the
JTS Topology Suite
is the general term used for the
binary set-theoretic
symmetric difference
. These operations accept two geometry inputs and construct a geometry representing the operation result. Along with
spatial predicates
they are the most important functions in the JTS API.
Intersection of MultiPolygons
Overlay operations are used in many kinds of spatial processes. Any system aspiring to provide full-featured geometry processing simply has to provide overlay operations. In fact, many geometry
libraries exist solely to provide implementations of overlay. Notable libraries include the
ESRI Java API
. Some of these provide overlay only for polygons, which is the most difficult case to compute.
Overlay in JTS
The JTS overlay algorithm supports the full OGC SFS geometry model, allowing any combination of
as input. In addition, JTS provides an explicit precision model, to allow constraining output to a desired precision. The overlay algorithm is also available in C++ in
. There it provides overlay operations in numerous systems, including
, and
. This codebase has had an long lifespan; it was developed back in 2001 for the very first release of JTS, and while there have been improvements over the years the core of the design has remained
However, there are some long-standing issues with JTS overlay. The most serious one is that in spite of much valiant effort over the years, overlay is not fully robust. The constructive nature of
overlay operations makes them particularly susceptible to the robustness issues which are notorious in geometric algorithms using floating-point numerics. It can happen that running an overlay
operation on seemingly innocuous, valid inputs results in the dreaded
being thrown. There is a steady trickle of issue reports about this in JTS, and even more for GEOS (such as
Another issue is that the codebase is complex, and thus hard to debug and modify. Partly this is because of the diversity of inputs and the explicit precision model. To support this the JTS overlay
algorithm has a rich and detailed semantics. But some of the complexity is due to the original design of the code. This makes it difficult to incorporate new ideas for improvements in performance and
Next Generation Overlay
So for many years it's been on my mind that JTS overlay needs a thorough overhaul. I chipped away at the problem over time, but it was clear that it was going to be a major effort. Now, thanks to the
support of my employer
Crunchy Data
, I've at last been able to focus on a complete rewrite of the JTS overlay module. It's called
The basic algorithm remains the same:
1. Extract the input linework, and node it together
2. Build a topology graph from the noded linework
3. Compute a full topological labelling of the graph
4. Extract the resultant polygons, lines and points from the graph
This algorithm is time-tested and is able to handle the complexities of multiple geometry types and topology collapse. The new codebase benefits from 20 years of experience to become simpler and more
modular, with increased testability, and potential for reuse.
OverlayNG has the following improvements:
• A snap-rounding noder is available to make overlay fully robust. This eliminates the possibility of TopologyExceptions (when an appropriate precision model is used).
• Snap-rounding allows full support for specifying the output precision model. The precision model can be specified independently for each overlay call, which is more flexible and easier to use.
The use of snap-rounding also provides fully valid precision reduction for geometries. This makes it feasible for the first time to fully operate in a fixed-precision regime.
Precision Reduction turned all the way up to 11
• Significant performance optimizations are included (notably, one which makes polygon intersection much faster in many cases)
• Pluggable noding allows providing different noding strategies. One use is to run OverlayNG with the original floating-point noder, which is faster than snap-rounding (but of course has the
robustness issues noted above). Another is to use a special-purpose noder to provide very fast polygonal coverage union.
Union of a polygonal coverage (10x faster with OverlayNG)
• A modular and cleaner codebase allows easier testability, maintenance, enhancement and reuse. A winged-edge graph model is used for the topology graph. This is simpler and less memory intensive.
• The rebuild gives an opportunity to make some semantic improvements:
□ Empty results are returned as empty atomic geometries of appropriate type, rather than awkward-to-handle empty GeometryCollections
□ Linear output is merged node-to-node. This gives union a more natural and useful semantic
A benefit of the new codebase is that it is easier to enhance and extend. For example, it should be straightforward to finally provide a SplitPolygon function for JTS. Another potential extension is
overlay for Polygonal Coverages.
Code that is so widely used needs to be thoroughly tested against real-world workloads. Initially OverlayNG will be released as a separate API in JTS. This allows it to be used along with the
original overlay. It can be used as a fallback for cases which fail in the original overlay process. Once the new code has been proved out in real world use, it is likely to become the standard
overlay code path. Also, the code will be ported to GEOS soon, where we're hoping it will provide
significant benefits
to the many systems that use GEOS.
I'll be posting more articles about aspects of OverlayNG soon. The code is almost ready to release, after some final testing. In the meantime, the pre-release code is available in a
Git branch
. It would be great to get as much beta-testing as possible before final release, so try it out and log some feedback!
4 comments:
Do you have any information regarding when this will be integrated into GEOS and QGIS? I have come across an error that apparently will not be solved until this new version is added to GEOS.
This is in GEOS as of 3.9. It's the default overlay mode, so should be in QGIS as well if it's using at least that version.
Hm, that is strange then! I get the error "GEOS geoprocessing error: intersection failed" that I thought was related to the limitations you mentioned in this post. I guess it's something else
Feel free to submit an issue on the GEOS Github repo.
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CFA Level 1 - Alternative Investments Session 13 - Reading 47
CFA Level 1 - Alternative Investments Session 13 - Reading 47 (Notes, Practice Questions, Sample Questions) 1. Compared to a mutual fund, a hedge fund is most likely to have lower levels of: A)
disclosure. B)usage of derivatives.C)leverage. {Explanation}: Hedge funds are relatively unregulated, and have minimal disclosure requirements. Unlike mutual funds, hedge funds arenot subject to
limits on the use of leverage and derivatives 2. Compared to a mutual fund, a hedge fund is most likely to have lower: A)disclosure requirements. B)lockup periods.C)fees. {Explanation}: Due to the
unregulated nature of hedge funds, hedge funds are required to provide only minimal disclosure to investors (andeven less disclosure to non-investors). Hedge funds often have a one,
two, or three year lockup periods, while mutual funds generally havedaily liquidity. Hedge fund fees are generally higher than mutual fundfees, because on top of a management fee (typically 2%),
hedge fundsalso charge a performance fee (typically 20% of profits.) 3. Alpha hedge fund limits withdrawals by investors during the first three years by imposing a redemption fee of 3%. Such provisions
by hedgefunds are called: A)hard lockup.B)regulatory disclosure. C)soft lockup. {Explanation}: Lockups which allow for redemption on payment of penalty are called as soft lockup. Under hard lockup,
withdrawals arenot permitted. 4. A hedge fund with a fixed-income arbitrage strategy is most likely to suffer a loss when: A)credit spreads widen quickly. B)leverage becomes less expensive.C)markets
for lower quality debt become more liquid {Explanation}: Fixed-income arbitrage generally involves buying high-yielding (low quality) bonds while selling low-yielding (highquality) bonds. Leverage
can be used in place of selling high quality
bonds. This strategy can suffer great losses when credit spreads widenquickly, when leverage becomes more expensive, or when the marketsfor low-quality debt become less liquid 5. A “risk arbitrage”
(or “merger arbitrage”) strategy: A)is considered a “long volatility” strategy. B)experiences losses when a planned merger is cancelled. C)involves purchasing the stock of an acquiring company
{Explanation}: A merger arbitrage strategy is considered a “short volatility” strategy: this strategy will experience a loss if the expectedmerger is cancelled. This strategy involves purchasing the
stock of thetarget company and shorting the stock of the acquiring company 6. Which of the following is most accurate in describing the problems of survivorship bias and backfill bias in the
performance evaluation ofhedge funds? A)Survivorship bias and backfill bias both result in downwardly biasedhedge fund index returns. B)Survivorship bias and backfill bias both result in upwardly
biasedhedge fund index returns. C)Survivorship bias results in upwardly biased hedge fund index returns,but backfill bias results in downwardly biased hedge fund index returns {Explanation}: The
problem in survivorship bias is that only the returns for survivors will be reported and the index return will be biased
upwards. Backfill bias results when a new hedge fund is added to anindex and the fund's historical performance is added to the index'shistorical performance. The problem is that only funds that
survivedwill have their performance added to the index, resulting in an upwardbias in index returns 7. A hedge fund investor is most likely to express a preference for returns distribution that has:
A)a negative skew. B)low kurtosis. C)high variance {Explanation}: Investors prefer a return distribution with low kurtosis, low variance, a high mean and positive skewness 8. Adding long volatility
hedge fund strategies to a portfolio of short volatility hedge fund strategies is most likely to increase theattractiveness of the portfolio return’s: A)Sharpe ratio. B)skewness and kurtosis
exposures. C)reported volatility {Explanation}: Adding long volatility strategies to a portfolio of short volatility strategies would increase the volatility of portfolio returnsand decrease the
portfolio’s Sharpe ratio. However, the resulting
portfolio returns distribution will be more normally distributed andskewness and kurtosis characteristics of the return distribution will bemore attractive to investors 9. Studies using factor models
have generally found the largest contributor to hedge fund returns to be: A)traditional market factor exposures. B)manager skill.C)exotic beta exposures {Explanation}: Studies have found that the
majority of hedge fund styles can be relatively closely replicated using traditional marketexposures such as stock and bond indices, currency, and commoditymarket returns. These traditional market
risk factors have been foundto explain 50%–80% of hedge fund returns 10. A misspecification of a hedge fund factor model that omits relevant risk factors is most likely to cause alpha to be: A)
underestimated. B)overestimated. C)negative. {Explanation}: Hedge fund factor models generally attribute hedge fund return to the sum of alpha, the risk-free rate, and the sum of theimpacts of the
relevant risk factors. If some of the relevant risk factors
are omitted from the model, the alpha (return due to manager skill) islikely to be over-estimated 11. Which of the following most accurately describes the distribution of hedge fund returns? Hedge
fund returns: A)are lognormally distributed. B)have fat tails in the distribution. C)are normally distributed {Explanation}: Investors should be concerned about hedge fund risk because hedge fund
returns have fat tails on the left hand side of theirdistribution. In other words, the probability of large losses is greaterthan that expected from a normal distribution. For this reason, it
isimperative that investors evaluate a downside measure of risk, such asmaximum drawdown and/or value at risk 12. The return distribution of a merger arbitrage strategy, in which the fund manager
purchases the target company and shorts the acquiringcompany stock, is best described as: A)normally distributed.B)positively skewed. C)highly kurtotic.
{Explanation}: The returns of many hedge fund strategies – including merger arbitrage trades – are not normally distributed; rather they arehighly kurtotic and negatively skewed. 13. Non-normality in
hedge fund returns is most likely to cause performance to be: A)underestimated.B)zero. C)overestimated {Explanation}: Non-normality of hedge fund returns necessitates consideration of higher order
moments of the returndistribution—specifically skewness and kurtosis. For most hedge funds,the return distribution is negatively skewed and highly kurtocic. Theseare undesirable qualities from the
perspective of an investor. Ignoringthese higher-order moments leads to overestimation of performance 14. An investor considering investing in a hedge fund, would be most likely motivated in pursuing
replicating strategy, rather than investing inthe hedge fund directly when the hedge fund: A)has a long lockup period. B)returns have a large alpha component.C)strategies are clearly disclosed.
{Explanation}: Investors may be motivated to choose hedge fund replication strategies over actual investments in hedge funds when: (1)hedge fund managers are not earning a positive alpha, (2)
investorsfeel that the fees paid to hedge fund managers are not justified, and (3)investors have objections to hedge funds’ lack of transparency orliquidity 15. A difficulty in applying traditional
portfolio analysis to hedge funds is that hedge funds have: A)high standard deviation.B)correlations with other asset classes that are static. C)non-normal return distribution {Explanation}:
Traditional portfolio analysis calculates the most efficient portfolio using return, correlation and volatility of assets.However, it is difficult to apply traditional portfolio analysis to hedgefunds
because: (1) it is difficult to develop accurate expected returns, (2)hedge fund correlation, beta exposures, and volatility can change overtime, and (3) standard deviation is not a complete measure
of hedgefund risk due to higher moment risks such as skewness and kurtosis.This is due to non-normal distribution of hedge fund returns 16. The usual result of adding hedge funds to a portfolio of
traditional (stocks and bonds) investments is a decrease in: A)standard deviation.
B)Sharpe ratio.C)skewness and kurtosis {Explanation}: The usual result of adding hedge funds to a portfolio of traditional investments is that: (1) standard deviation will decrease, (2)the Sharpe
ratio will increase, and (3) higher-moment exposures suchas skewness and kurtosis will increase 17. Compared to a single manager hedge fund, a fund of funds is most likely to have higher: A)
management and performance fees. B)return performance.C)standard deviation {Explanation}: Funds of funds generally have higher management and performance fees than single manager hedge funds because
funds offunds generally apply a second layer of fees on top of those paid to theunderlying fund managers. Fund of funds’ returns tend to be equal toaverage hedge fund index performance—before fund of
funds’ secondlayer of fees are deducted. By investing in 15 or more single managerfunds of various strategies (diversifying), funds of funds achieve lowerstandard deviation
18. Compared to a single manager hedge fund, a fund of funds is most likely to have higher: A)longevity. B)survivorship bias.C)backfill bias. {Explanation}: Funds of funds generally have lower
mortality, lower survivorship bias, and lower backfill bias than single manager hedgefunds
CFA Level 1 - Alternative Investments Session 13 - Reading 47
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TORIC VARIETIES, SPRING 2023
MATHEMATICS 595TV, TORIC VARIETIES, SPRING 2023
Instructor: Sheldon Katz
Email/Phone: katzs AT illinois.edu/217-265-6258 (email is usually quicker than phone)
Office and Hours: To be announced, 301 Altgeld Hall, or by appointment
Class Time and location: TuTh 12:30-1:45, 113 Davenport Hall.
Text: Introduction to Toric Varieties, W. Fulton, Annals of Math Studies 131, Princeton University Press, Princeton NJ 1993. ([F])
Supplementary Texts: Mirror Symmetry and Algebraic Geometry, D.A. Cox and S. Katz (Chapter 3 only) ([CK]), and Toric Varieties, D.A. Cox , J. Little, and H. Schenck ([CLS]).
Reserves: Hardcopy of all of the above texts are on reserve at the Mathematics Library. Electronic copies of [F] and [CK] are available, but not of [CLS]. However, here is a preliminary version of
[CLS], posted with permission of the authors.
Course Syllabus is here.
Weekly class plans
• Week 1 (Jan 17 and 19). Read [F] Ch. 1. See [CLS] Ch. 1 for more detail. Start homework 1.
• Week 2 (Jan 24 and 26). Read [F] Chapters 1.3,1.4,2.1. There will be a problem session in class on January 26.
• Week 3 (Jan 31 and Feb 2). Read [F] Chapters 2.2-2.5. See [CLS] Chapters 3.2, 3.4 for another perspective and more detail on some of these topics.
• Week 4 (Feb 7 and 9). Read [F] Chapters 2.5-2.6,3.1. There will be a problem session in class on February 9.
• Week 5 (Feb 14 and 16). Read [F] Chapters 3.1, 3.3, 3.4. See [CLS] Ch. 4 for more detail.
• Week 6 (Feb 21 and 23). Read [F] Chapter 3.4 and [CK] Chapter 3.1-3.2. The material from [CK] Section 3.2.1 and 3.2.2 will be supplemented with selected material from [CLS] Chapter 2. There will
be a problem session in class on February 23.
• Week 7 (Feb 28 and March 2). Read [CK] Section 3.2.3-3.2.4 and [CLS] Chapter 5. Then read [CK] 3.2.1-3.2.2.
• Week 8 (March 7 and 9). Read [CLS] Chapter 5 and [CK] 3.2.1-3.2.2.
Homeworks: [F]=Fulton; [CLS]=Cox-Little=Schenck
• Homework 1, for discussion during the Thursday, January 26 Problem Session.
• Homework 2, for discussion during the Thursday, February 9 Problem Session.
• Homework 3, for discussion during the Thursday, February 23 Problem Session.
Updates and announcements
• There will not be a problem session during the last week of class
• The last day of class will be Thursday, March 9.
Seminars and other activities of interest
• Graduate Algebraic Geometry and Commutative Algebra Seminar, Wednesdays at 3pm in AH 341. Talks will be posted on the department's seminar calendar
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When not to use scatter plot
Complete Scatter Chart Guide
How to use scatterplots to explore relationships in bivariate data. A scatterplot is a graphic tool used to display the relationship between two quantitative variables. slope, weak) is the pattern
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do remember if you've any continuous data use the scatterplot's. What is scatter plot? - Definition from WhatIs.com A scatter plot is a set of points plotted on a horizontal and vertical axes.
Scatter Your browser does not currently recognize any of the video formats available. Dr. Eugene O'Loughlin explains how to use Excel to create a scatter diagram.
Guide to Excel Scatter Plot Chart. Here we discuss how to create Scatter Plot Chart in Excel along with practical examples and downloadable excel template. Scatter Charts: Why and when to use it
Unlike other charts, that uses lines or bars this chart only uses markers or dots. This chart is between two points or variables. Scatter Plot Stats Assignment Homework Help Scatter Plot Stats
Homework, assignment and Project Help, Scatter Plot Assignment Help After mathematics came into being the way people used to handle things changed. It took mathematics centuries to evolve but th
What is a Scatter Plot? - TIBCO Docs Scatter plots are used to plot data points on a horizontal and a vertical axis in the However, even though a correlation may seem to be present, this might not
always be the case. Each product can be shown separately using trellising:. Scatter Plots and Bar Graphs | Human Anatomy and ... For scientific data, any other graph style is not useful in most
cases. Use either scatter plots or bar graphs for scientific data and avoid all other types. To decide Scatter Plots Help the World Make Sense - Infogram 2 Dec 2015 Scatter plots, a lot like line
graphs, use horizontal and vertical axes to plot Remember: Correlation does not always equal causation – other
Scatterplot | Better Evaluation
Scatter (XY) Plots Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents. Scatter Charts | Image Charts | Google Developers If you
do not use the chds parameter (custom scaling) it gives the exact encoded value; if you do use that parameter with any format type the value will be scaled to the range that you specify. Scatter Plot
in Excel | How to Create Scatter Chart in Excel
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Print version ISSN 0301-7036
Prob. Des vol.47 n.186 Ciudad de México Jul./Aug. 2016
Fundamentals, the Net Positions of Speculators, and the Exchange Rate in Brazil
^^1Institute for Economic research and the Faculty of Economics, UNAM, Mexico. E-mail addresses: armando_sanchez123@hotmail.com, memoare20@gmail.com iph@unam.mx, respectively.
This paper analyzes the role of fundamentals and the net positions of speculators in determining the Brazilian exchange rate from April 2002 to August 2012. Based on a cointegrated SVAR model, we
found empirical evidence to support our hypothesis that the microeconomic approach (Evans and Lyons, 2002) and the monetary model (Bilson, 1978) to determine the exchange rate are consistent with one
another. Unlike in other empirical studies, our analysis demonstrates that the net positions of speculators and economic fundamentals constitute two channels (one liquidity-related, the other
information-related) that can contribute to explaining the dynamics of the Brazilian exchange rate in the short and long term.
Key Words: Exchange rate; cointegration; SVAR; monetary models; speculators
En el presente artículo analizamos el papel de los fundamentales y de las posiciones netas de los especuladores en la determinación del tipo de cambio de Brasil en el periodo abril 2002-agosto 2012.
Con base en un modelo SVAR cointegrado, encontramos evidencia empírica que sustenta nuestra hipótesis de que el enfoque microeconómico (Evans y Lyons, 2002) y el modelo monetario (Bilson, 1978) de la
determinación del tipo de cambio son consistentes. A diferencia de otros estudios empíricos, nuestro análisis demuestra que las posiciones netas de los especuladores y los fundamentales constituyen
dos canales (de liquidez y de información) que contribuyen a explicar la dinámica del tipo de cambio de la moneda brasileña en el corto y en el largo plazo.
Palabras clave: tipo de cambio; cointegración; SVAR; modelos monetarios; especuladores
Dans cet article, nous analysons le rôle des fondamentaux et des positions spéculatives nettes dans la détermination du taux de change au Brésil pour la période avril 2002- août 2012. Sur la base
d’un modèle svar co-intégré, nous trouvons une preuve empi rique qui étaye notre hypothèse selon laquelle l’optique microéconomique (Evans et Lyons, 2002) et le modèle monétaire (Bilson, 1978) de la
détermination du taux de change sont cohérents. A la différence d’autres études empiriques, notre analyse démontre que les positions spéculatives nettes et les fondamentaux constituent deux canaux
(de liquidité et d’information) qui contribuent à expliquer la dynamique du taux de change de la monnaie brésilienne à court terme et à long terme.
Mots clés: taux de change; co-intégration; svar; modèles monétaires; spéculateurs
No presente artigo analisamos o papel dos fundamentais e das posições líquidas dos especuladores na determinação da taxa de câmbio do Brasil no período abril 2002 - agosto 2012. Com base num modelo
svar co-integrado, encontramos evidência empírica que sustenta nossa hipótese que o enfoque microeconómico (Evans e Lyons, 2002) e o modelo monetário (Bilson, 1978) da determinação da taxa de câmbio
são consistentes. Diferentemente de outros estudos empíricos, nossa análise demonstra que as posições líquidas dos especuladores e dos fundamentais constituem dois canais (de liquidez e de
informação) que contribuem para explicar a dinâmica da taxa de câmbio da moeda brasileira no corto e no largo prazo.
Palavras-chave: taxa de cambio; co-integração; svar; modelos monetários; especuladores
在文中我们分析了投机者的基本面和净头寸在2002年4月至2012年8月期间对巴西汇率的影响。基于某协整的SVAR模型，我们证明了如下假设：微观经济学视角（Evans y Lyons, 2002）与汇率决定的货币模型（Bilson, 1978）是
相符的。与其他基于经验的研究不同，我们的分析证明了投机者的净头寸与基本面构成了两条渠道（流动资金渠道及信息渠道）, 而这两条渠道有助于解释巴西货币在长期及短期内的汇率动态。
关键词: 汇率; 协整性; SVAR; 货币模型; 投机者
So long as the world continues to be divided amongst sovereign states, each of which regards the interest of its own citizens as its first priority, universal free trade may not be compatible with
that objective in the long run any more than in the short run – whether the world is under a regime of fixed rates or under a regime of floating rates.
Nicholas ^Kaldor, 1978: 113
The objective of this paper is to analyze the dynamics of the nominal exchange rate in the Brazilian economy as a function of its determinants, using two approaches: 1) the macroeconomic or monetary
approach, which we use to analyze the interest rate and monetary aggregate differentials between Brazil and the United States, and 2) the microstructure approach, which we use to analyze the net
positions of speculators. Both approaches consider the short and long term.
We aim to demonstrate the hypothesis that the exchange rate dynamics obey macroeconomic and microeconomic variables. On the one hand, the influence of the monetary aggregates differential and/or an
increase in the interest rate differential is positive; that means it will spur an increase (depreciation) in the nominal exchange rate (^Bilson, 1978); on the other, the influence of the net
positions of speculators on the exchange rate should be positive (^Evans and Lyons, 2002).
^Froot and Ramadorai (2005) distinguished between two forms of relationships between the exchange rate and speculators’ net positions. The first, the strong flow-centric version, holds that the net
positions of speculators are correlated with changes in the value of the fundamental variables of a currency, in other words, with the transmission of the fundamental macroeconomic information to the
market. The second, the weak flow-centric version, asserts that speculators’ net positions are correlated with a temporary deviation in the exchange rate with respect to its fundamental value. A
deviation from the exchange rate vis-à-vis its fundamental value can arise due to a change in demand, or due to liquidity effects or the overreaction of investors.
The “information channel” is the term used for the case in which the net positions of speculators have a permanent effect on the exchange rate and provide fundamental macroeconomic information to the
market, and the “liquidity channel” is the term used for the situation in which the net positions of speculators are related to temporary deviations in the exchange rate vis-à-vis its fundamental
The strong flow-centric version implies that speculators’ net positions and the exchange rate are non-stationary series, but cointegrate, such that there is a stable and long-term equilibrium
relationship between the time series. Although the weak flow-centric version does not necessarily imply cointegration (^Froot and Ramadorai, 2005).
In this paper, unlike in other studies that have analyzed speculators’ net positions, we used a cointegrated SVAR method in the presence of non-stationary variables, which allowed us to distinguish
between the two mechanisms, the informational and the liquidity mechanisms, to analyze the behavior of the Brazilian real exchange rate with respect to the United States dollar. The empirical results
suggest that including the net positions of speculators in a standard monetary model increases its explanatory power and the predictive accuracy of the model. But what is most important is that
speculators’ net positions were correlated both with the fundamental variables of the monetary model in Bilson’s version, the information channel, as well as with the monetary policy news version,
the liquidity channel. We can therefore say that the relationship between the exchange rate and speculators’ net positions is not inconsistent with the macroeconomic approach (MAER) and the effect of
these positions on prices is permanent.
Below, we analyze the role played by fundamentals and the microstructure in determining the exchange rate in Brazil in the time period 2002-2012. Following this introduction, we include: a review of
the literature relevant to these two models; an analysis of the speculators’ net positions approach and the monetary paradigm in determining the exchange rate; the SVAR method used; an empirical
scrutiny of the net positions of speculators and the exchange rate in Brazil, and, finally, our conclusions.
This section introduces literature pertaining to the behavior of the exchange rate from a macroeconomic perspective (the monetary or fundamentals model: interest rate differential, product, and money
supply) and the counterposing point of view, the microstructure approach, that explains the exchange rate based on the net positions taken by speculators.
Empirical Analysis of the Monetary Approach
The monetary model for the exchange rate assumes perfect capital mobility, perfect substitution among different financial assets, perfect integration of the goods and financial markets, and the
validity of purchasing power parity (PPP). In monetary models with flexible prices, it moreover assumes uncovered interest rate parity, with which the exchange rate responds to the fundamentals,
where the most important variable is the relative quantity of money (^Mussa, 1982). On the other hand, in monetary models with rigid prices, the exchange rate dynamics do not match with the PPP in
the short term; the financial and goods markets have different adjustment speeds, which gives rise to overshooting the exchange rate vis-à-vis the long-term equilibrium value (^Dornbusch, 1976).
Among the empirical analyses, ^Gardeazabal et al. (1997) found important evidence of cointegration between the exchange rate of the British pound sterling, the German Deutsche mark, the Italian lira,
and the Spanish peseta with respect to the United States dollar and its fundamentals; MacDonald and Taylor (1992) analyzed the long-term behavior of the exchange rate of the sterling pound and the
United States dollar using multivariate cointegration techniques in an unrestricted monetary model and were able to validate the monetary approach. In summary, these authors asserted that the
monetary model does indeed explain exchange rate dynamics in the long term, particularly when the model is complemented with the Balassa-Samuelson effect.
The Microstructure Approach: Speculators’ Net Positions
The microstructure approach highlights other variables not considered in the monetary model. Among the most relevant, the order flow that measures the net purchasing pressure (^Lyons, 2001: 7), the
conduct of the heterogeneous stabilizing and destabilizing agents that take part in the foreign exchange market with different expectations not necessarily consistent with the fundamental
macroeconomic variables, asymmetrical sharing of information, and the net positions of speculators that balance the markets and transmit information (^Torre Cepeda and Provorova Panteleyeva, 2007:
In their pioneering study on the microstructure of the exchange rate market, ^Evans and Lyons (2002), based on data from Reuters D 2000-1 on the net positions held by speculators, analyzed the daily
variation between the dollar, the mark, and the yen during a four-month period (May-August 1996) and found substantial correlation between the negotiations of the net positions held by speculators
and the published exchange rates. This conclusion was proven again by ^Danielsson, Luo, and Payne (2012), although the latter concluded that, lacking a perfect forecast of the future explanatory
variables, the net positions of speculators have little or practically no explanatory power at all when it comes to the exchange rate.
Various studies have underscored the explanatory significance of the net positions of speculators. ^Bjønnes, Rime et al. (2005) concluded that these positions could explain one third of the daily
volatility in the Swedish krona and the euro. This conclusion is consistent with others. ^Osler (2002, ^2003) and ^Bates, Dempster et al. (2003) examined the ledger of daily orders made at HSBC Bank.
^Torre Cepeda and Provorova Panteleyeva (2007) conducted an excellent and pioneering analysis of the microstructure of the influence of the net positions of speculators in the Mexican peso futures
market to determine the exchange rate with respect to the United States dollar in the time period 1999-2005; they found that due to the accelerated growth of this market, the relationship between the
two currencies has not been constant.
Based on the foregoing, we can say the following: the monetary model explains the exchange rate in the long term; the net positions of speculators determine the behavior of the exchange rate in the
short term; finally, a combination of the two approaches could better capture the dynamics of an exchange rate, given that the net positions of speculators requires prior information that could be
provided by the monetary model.
^Evans and Lyons (2002) added a few variables from the monetary model in determining the exchange rate (MAER), of which one classic version was developed by ^Bilson (1978).^^2 The equation proposed
by Evans and Lyons is:
Where ∆e [ t ] is the exchange rate, ∆m [ t ] refers to macroeconomic information innovations (the fundamentals), λ is a positive constant, and ∆x [ t ] denotes the net positions of speculators. In
this model, public information augments the association between the fundamentals, ∆m [ t ] , defined as the change in the interest rate differential (i.e., ∆m [ t ] = (i-i*)). Given that ∆m [ t ] can
be a function of other macroeconomic variables, we use the following monetary model proposed by Bilson:
Where (m - m), (i - i*), and (y - y*) are the differentials of the money supply, the interest rate, and national and international income, respectively, typically used as macroeconomic determinants
of the exchange rate.
By augmenting the equation (1) proposed by Evans and Lyons, we include the monetary model variables and the net positions of speculators (x), thereby integrating the macroeconomic approach and the
microfinancial variable that transmits important information about the exchange rate:
Equation (3) shows the transmission mechanism that keeps the money market in balance through variations in the nominal exchange rate in the following fashion: an increase in the net positions of
speculators (x), the money supply (m), or the interest rate differential will generate a depreciation of the nominal exchange rate, ∆e [ t ] > 0; contrarily, an increase in the income differential
will provoke an appreciation. As observed, a positive relationship is expected between the exchange rate and the net positions of speculators, δ > 0, given that an increase in net payments in foreign
currency will result in a higher price for the foreign currency in terms of the national currency. It is important to highlight that speculators’ net positions can affect the exchange rate directly
by way of an impact on prices, transmitting non-public information; they can also affect the informational and liquidity-related mechanisms in the market, and by extension, the nominal exchange rate.
Equation (3) could be taken to mean that the relationship between the exchange rate and speculators’ net positions is not consistent with the macroeconomic approach. However, speculators’ net
positions transmit non-public information (information that is not common knowledge) and can play a fundamental aggregate information role for uncovering prices in the currency exchange markets. The
way in which these transmission mechanisms play out is an empirical matter that we will tackle below.
The ^Sims (1987) SVAR method is useful in analyzing the relationships between the net positions held by speculators and the nominal exchange rate. The main objective of this method is to determine
the dynamic responses of economic variables distinct from independent impacts. The SVAR approach is an alternative to the traditional, non-theoretical VAR approach (^Sims, 1980; ^Juselius, 2006). The
classic VAR approach assumes that the variables are stationary and includes only lags for all of the variables. The reduced form of the model with one lag can be represented as follows:
Where y [ t ] is a vector of endogenous variables; d [ t ] is a vector of deterministic components, such as the constant, the trend, and stationary or intervention dummies; and u [ t ] is an error
Equation (4) does not appear to offer any explanation for instantaneous relationships (contemporaneous effects) between the variables, as it contains only the lags of the endogenous variables.
However, each contemporaneous component is concealed in the correlation structure of the variance and covariance matrix that comes from u [ t ] . This fact implies that the innovations in u [ t ] are
correlated. A more meticulous examination of the primitive VAR helps elucidate this difficulty (^Enders, 1995):
In Equation (5), the errors in ℇ [ t ] are not correlated due to the fact that matrix B contains the contemporaneous interactions among the variables. Matrix A captures all of the lagged interactions
between the same variables. As such, the reduced VAR model (4) can be seen as a reparameterization of the more general form of the primitive VAR model (5). Essentially, it can easily be observed that
C = B ^ -1 A and u [ t ] = B ^ -1 ℇ. This means that the errors of the reduced VAR model u [ t ] are a linear combination of the non-correlated impacts of ℇ [ t ] .
As such, the contemporaneous interactions of interest contained in matrix B can be recovered, as long as we are willing to impose different restrictions on a triangular structure given by the
Cholesky decomposition. This decomposition is used to calculate the impulse-response functions in the classic VAR analysis,^^3 which will give us the conditions needed for the identification. This
means that the number of coefficients different from zero in matrix A must be equal to or less than . However, we can impose a different decomposition, in other words, a matrix that contains
restrictions to identify the contemporaneous interactions in the errors in the reduced VAR model. This procedure is known as structural VAR (SVAR) analysis.
^Amisano and Giannini (1997) suggest a more general vision of the SVAR that admits a representation of the VAR with non-stationary series as the starting point for the specification of the SVAR
model. In the presence of cointegration, the model must take shape in two different phases: the first is to identify the long-term equilibrium relations and the second is to identify short-term
interactions. The final structure of the instantaneous equations is achieved through two matrices (A and B):
Where ℇ [ t ] is the error vector of the reduced VAR and u [ t ] is the error vector of the primitive VAR. Moreover, we know that E (u[t]) = 0 and E (u[t]u[t]) = I[t]. The identification of the
contemporaneous relationships among the variables in equation (7) requires a set of restrictions based on theoretical assumptions. Matrix B is a diagonal matrix that normalizes the variance of the
structural errors u [ t ] , and matrix A contains the relevant contemporaneous relationships. The final structure is obtained from a precisely identified model (matrix A correctly identified) and the
over-identification, imposing statistically valid theoretical constraints. The validation of these restrictions was confirmed using the likelihood ratio test.
The SVAR methodology can be implemented in three steps: first, estimate the reduced VAR and calculate the residuals matrix; second, use the residuals to estimate matrices A and B using the full
information maximum likelihood (FIML) approach, and, finally, estimate the immediate reaction of the system for the individual impacts and draw up the impulse-response graphs, combining the
information from the first two steps.
Stylized Facts
In the presence of non-stationary variables, we must look for robust empirical evidence as to the role played by the net positions of speculators in the exchange rate in a flexible regime. First, we
discuss some of the main stylized facts pertaining to the exchange rate (Brazilian real/dollar) and its cardinal determinants, including the net positions of speculators and a few fundamental
The data consisted of monthly, non-adjusted observations taken during the time period 2002.4-2012.8. The variables considered were as follows: the nominal exchange rate (Brazilian real/dollar); net
positions of speculators; the difference between the log of the interest rate and M1 of Brazil and the United States. We did not consider the product, given that we could not find monthly data on
this factor for Brazil. The following graphs display the behavior of each of the series:
In Figure 1, we see that from 2002.4 to 2002.12, the exchange rate rose, as competitiveness was sought via depreciation of the nominal exchange rate. However, starting in 2003-1 and up until 2007-12,
the Brazilian real appreciated, due to the inflow of foreign capital, as well as an improved balance of payments, in this way prompting an increase in the international reserves. The devaluation
observed in 2008 was due to the global financial crisis that year. Following the crisis, foreign capital began to flow in again and, once more, the balance of payments improved. As a result, the
nominal exchange rate continued to appreciate, exceeding pre-crisis levels (both in real and nominal terms); the Brazilian currency – together with that of Colombia and Uruguay – is the currency that
has appreciated the most since the end of 2009.
Below, we examine the presence of unit roots in the nominal exchange rate, the net positions held by speculators, the interest rate, and the M1, in both Brazil and the rest of the world. Table 1a, in
the Appendix, suggests that all of the variables are integration order I(1), which means that the estimation of the series in levels could lead to spurious conclusions, unless the series cointegrate.
Figure 2 exhibits the positive relationship between the nominal exchange and speculators’ net positions beginning in 2006; this is because when there are buying pressures (exerted by the net
positions), the exchange rate price rises. On the other hand, Figures 3 and 4 show that the trends between the various national rates and the rest of the world (m - m*, i - i*) have the same
behavior, which indicates the existence of more than one cointegration vector.
Table 1b, in the Appendix, shows that the differences between the variables for Brazil and the United States are stationary. As such, it is possible to find cointegration between the variables. In
this way, the cointegrated VAR model could be our point of departure for the structural analysis, in other words, for the SVAR. This not only would permit us to determine if the explanatory power of
the macroeconomic model is enhanced when the net positions of speculators are included, but also the structure of how the net positions of speculators transmit information to the price in a
multivariate set.
In this section, we describe the empirical method that shows the relevance of the net positions of speculators in determining the exchange rate and its role in the transmission of information to
price, using a cointegrated SVAR model. First, we show how the SVAR model is used to detail how the net positions of speculators transmit non-public information to prices in the monetary model
The analysis shows not only the estimation of the long-term cointegration equation associated with the theoretical exchange rate equation (3), but also the impulse-responses associated with the
variance and covariance matrix of the cointegrated VAR model of the structural form to calculate the contemporaneous correlations between the variations in the net positions of speculators and the
variations in the exchange rate. At the same time, we derive the short- and long-term transmission mechanisms.
In the presence of unit roots, the structure of the VAR model can be shaped in three stages: the first, specify a VAR representation that is adequate for the set of variables, including a selection
of the order of lags, the cointegration range, the type of deterministic polynomial analyzed, and a specification sensitive to the cointegration space. If one of the cointegration relationships
identified is consistent with the coefficient suggested by (3), we can conclude that the net positions of speculators plays an important role in explaining the long-term behavior of price. More
specifically, we analyze the existence of a statistically solid cointegration relationship, which includes the net positions held by speculators, which are related to the cointegration equation in
the following manner:
If speculators’ net positions have a permanent influence on determining the exchange rate, in this long-term specification, β[3] should be positive. On the other hand, the determination in the Bilson
model in the monetary approach to determining the exchange rate is given when the expected coefficients are β[1] > 0 and β[2] > 0.
The second stage is the “structural” step; the VAR model is used in its vector error correction model (VECM) version to identify the association between the exchange rate and speculators’ net
positions in the short term, which is hidden in the covariance matrix of the residuals of each multivariate model. This is the point of departure for structuring the VAR representation of the
exchange rate equation:
To recover the coefficients of the short-term model, we use the variance and covariance matrix of the VAR in their error correction form (9); to verify the statistical validity of the various
transmission channels between the net positions of speculators and the exchange rate, we work based off of equation (7). For example, to validate the short-term version of equation (3), we use the
decomposition of the following variance and covariance matrix (which will be analyzed in the next section).
Restrictions on the Short-Term SVAR
This set of restrictions corresponds to the following short-term transmission mechanism:
The structure of the matrix can be rewritten as the following equations related with the monetary model that includes the net positions of speculators:
If the monetary model plays a relevant role in uncovering the expected price in the short term, it means that the coefficients have to be: ∂ℇ [ e ] / ∂ℇ [ (m-m*) ] = a [ 41 ] > 0 and ∂ℇ [ e ] / ∂ℇ [
(i-i*) ] = a [42] > 0, according to Chin et al. (2007). Moreover, a short-term impact exerted by the net positions of speculators on the price, for example, ∂ℇ [ e ] / ∂ℇ [ (m-m*) ] = a [31] * a [43]
> 0, implies that the temporary monetary shock transmits information to the price by way of the net positions of speculators. It is important to emphasize that the monetary model must be identified
or over-identified and the short- and long-term restrictions must be validated by way of the likelihood ratio test.
Finally, the third phase is the short- and medium-term validation of the monetary model by way of plausible modeling of the instantaneous correlations via the impulse-response functions.
Empirical Results and Discussion
In the econometric analysis, we estimated a correctly specified VAR model with non-stationary variables for the time period 2002.4 to 2012.8. The data have monthly periodicity; all of the variables
are in their logarithmic form. The series used were: the nominal exchange rate Brazilian real/dollar (e), the net positions of speculators (x), and the national and external interest rate
differentials (i, i*) and M1 (m, m*). The VAR model includes a restricted constant, two lags, and dummy variables to capture the financial shocks of 2002, 2008, 2009, and 2011. The unit root tests,
with correct individual and joint specification, are shown in Tables 1a, 1b, 1c, and 1d in the Statistical Appendix. The number of lags was chosen based on the adequate model with correct
specification. Other tests were also used, such as the Schwartz information criteria test, the Godfrey Portmanteau test, and the LR test (the latter is suggested by ^Sims (1980)). All of the correct
specification tests are reported in the Appendix.
We began by analyzing the range of cointegration, using Johansen’s reduced range method. The trace statistic, in Table 1, suggests the existence of at least three cointegration vectors.
R Statistic 99%
0 147.26 54.46
1 67.71 35.65
2 28.99 20.04
3 5.08 6.65
Note: R is the cointegration range.
Source: Created by the authors based on data from IPEA.
To test further evidence, a sequential test was conducted for the joint determination of the cointegration range and the polynomial trend ^^4 (^Johansen, 1995; see Table 2). First, we report the
deterministic (constant and trend) tests, and then the joint test (the statistic).
The results suggest the existence of at least three cointegration vectors at a confidence level of 99% and indicate that a sensible model implies the existence of a linear trend in the cointegration
vector and a constant in the series of the model. To ensure a robust model for the cointegration range over time, we used the ^Hansen and Johansen (1992) iterative process. Figure 5 shows the results
of the test. The test reinforced the hypothesis of the existence of three cointegration vectors over time at a 99% confidence level. However, the test indicated that only two of them are stable over
time (R Model).
Model R Trace 99%
I(0) Intercept, I(1) Nothing 0 184.93 60.16
I(0) Intercept, I(1) Linear Trend 0 147.26 54.96
I(0) Intercept, I(1) Nothing 1 90.05 41.07
I(0) Intercept, I(1) Linear Trend 1 67.71 35.65
I(0) Intercept, I(1) Nothing 2 49.31 24.6
I(0) Intercept, I(1) Linear Trend 2 28.99 20.04
I(0) Intercept, I(1) Nothing 3 10.61 12.97
I(0) Intercept, I(1) Linear Trend 3 5.08 6.65
Note: R=range of cointegration
Source: Created by the author based on IPEA data.
The Johansen maximum likelihood procedure estimates a base for the cointegration space; the problem of identification remains open. One treatment used for identification is to impose a set of
sensible a priori restrictions in the space of the parameters. In this case, the first cointegration vector is normalized as a long-term exchange rate equation and we consider the hypothesis where
the differences between the national variables and the rest of the world are stationary; those restrictions permit a sensible identification of the space for the cointegration vector. Below, we
report the equation of the cointegration vector:
This is one of the ways to write the cointegration vector, which implies that there is a long-term relationship associated with the monetary model suggested by Bilson, but which includes the net
positions of speculators in determining price in the long term. In this particular interpretation of the cointegration space, we can see a positive relationship between net positions of speculators
and the exchange rate in the long term. Figure 6 also confirms that the long-term equation – which is a linear combination of the exchange rate and its determinants – has stationary behavior, just as
the monetary model suggests. The long-term relationship confirms that the net positions of speculators have a permanent effect and transmit fundamental macroeconomic information to the market, as the
strong flow-centric version suggests. We can conclude, therefore, that the net positions of speculators are correlated with the fundamental future variables, like in the Bilson version of the
monetary model, and that the relationship between exchange rate and speculators’ net positions is not inconsistent with the macroeconomic approach in determining the exchange rate.
To estimate the contemporaneous relationships in which we were interested, associated with the coefficients of equation (3), we used the restrictions suggested by equation (10), included in matrices
A and B, to obtain the structure suggested by equations (11), (12), (13), and (14). It is important to emphasize that to find each restriction, we begin with an exact identification structure given
by the decomposition of the lower triangle of the variance and covariance matrix of errors of the estimated VAR. Then, in matrix A, we save the coefficients to identify the monetary model (MAER), as
long as the variables are statistically significant, and restrict to zero the parameters that are not significant, with which we arrive to a situation of over-identification. Finally, we secure the
validity of the previously imposed restrictions through the LR test. The coefficients estimated for the contemporaneous interactions that represent the short-term version of the monetary model are
shown in equations 15 to 18. The final estimate supports the graphical representation of the instantaneous relationships among the variables shown in Figure 5.
The statistical validity of the mechanism in Table 4 confirms the a priori assumptions of the short-term connections between the fundamental variables and the exchange rate. The impacts of the
structural shocks of money and the interest rate on the nominal exchange rate are positive. Finally, monetary shocks also have a positive influence on the exchange rate via speculators’ net
positions. The simulation shows that the latter have a substantial contemporaneous effect on the exchange rate because the money supply shock affects the exchange rate through the net positions of
speculators. As such, speculators’ net positions are correlated with monetary policy news, the liquidity channel, which confirms that temporary monetary shocks are transmitted to prices via the net
positions held by speculators.
The evidence on the mechanisms can be proven using typical simulation techniques, such as impulse-response functions (IRF), based on the estimated VAR model and restricted to satisfy the
cointegration range restrictions. The IRFs and their asymptotically valid intervals are shown below:
Figure 7 shows the responses of the exchange rate to the shocks of the money supply and the interest rate differential, respectively, in graphs (a) and (b). As can be observed, both have a positive
effect in the short term. This confirms the search for contemporaneous shocks and allows us to say that the MAER is a good approach to understanding the behavior of the currency in Brazil. In
summary, the empirical results reveal that exchange rate movements respond to the shocks of the variables in the monetary model proposed by Bilson. On the one hand, pursuant to graph (c), we can see
that the net positions of speculators have had a positive effect on the exchange rate, as mentioned in the initial hypothesis. This reflects the effect of the net positions of speculators on the
exchange rate in the short term.
Finally, Table 5 shows the explanatory power of the net positions held by speculators. Although they are not the strongest, it is important to include them in the model, as the F test indicated that
they should not be excluded from the SVAR model, as they help the model on fit. ^^5 Table 6 shows that the precision of the forecast improves once the net positions of speculators are introduced into
the VAR model. This means that the net positions held by speculators are necessary to explain the long-term behavior of the exchange rate in Brazil.
Variable R^2
X 0.88946
(m-m*) 0.99978
(i-i*) 0.9867
E 0.98704
Source: Created by the author based on data from IPEA.
The empirical evidence offers an explicit characterization of how the net positions of speculators transmit non-public information to prices, and the channels through which the net positions of
speculators contribute to the allocation of this information.
In this study, we used an SVAR model to show an appropriate description of the relationship between speculators’ net positions and the exchange rate in Brazil. This microstructure approach is
consistent with the monetary model, and encompasses two mechanisms, information and liquidity. In other words, by adding the net positions held by speculators into the monetary model, the
specification is enhanced, and so is the explanatory power and predictive accuracy. Moreover, the long-term estimates show that the net positions of speculators have a permanent effect and transmit
fundamental macroeconomic information to the market, which is incorporated to prices via the net positions of speculators.
The simulations also demonstrate that the net positions held by speculators have a substantial contemporaneous effect on the exchange rate, because the shocks of the money supply affect the exchange
rate through the net positions held by speculators. In other words, these net positions are correlated with the variables of the monetary model known as MAER (information channel), but also with the
monetary policy news (liquidity channel). The results confirm that the relationship between the exchange rate and speculators’ net positions is not inconsistent with the macroeconomic approach to
determining the exchange rate in Brazil, and that monetary shocks are transmitted via the net positions of speculators.
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Engineering, Hong-Kong, marzo 20-23, pp. 355-362. [ Links ]
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Press, pp. 39-62. [ Links ]
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Links ]
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Juselius, Katarina (2006), The Cointegrated VAR Model: Methodology and Applications, Oxford, Oxford University Press. [ Links ]
Kaldor, Nicholas (1978), “The Effect of Devaluations on Trade in Manufactures”, en Further Essays on Applied Economics, Collected Economic Essays, vol. 6, Londres, Duckworth, pp. 99-116. [ Links ]
Lyons, Richard K. (2001), The Microstructure Approach to Exchange Rates, Cambridge, MA, The mit Press. [ Links ]
Macdonald, Ronald y Mark P. Taylor (1992), “Exchange Rate Economics: A Survey”, IMF Staff Papers, vol. 39(1), marzo, pp. 1-57. [ Links ]
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^______ (2003), “Currency Orders and Exchange Rate Dynamics: An Explanation for the Predictive Success of Technical Analysis”en Journal of Finance, núm. 58, 5, pp. 1791-1820. [ Links ]
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Época , vol. XVI, núm. 1, primer semestre. [ Links ]
^5The null hypothesis was that the net positions of speculators would be zero in each of the VAR model equations. This test suggests that the net positions of speculators must not be excluded from
the model Chi^2(23))=3.0088e+005[0.0000].
Note: The asterisks indicate series for the United States; m, i, e, and x refer to M1, the interest rate, the nominal exchange rate, and the net positions of speculators, respectively. The first
difference in the series is denoted by ∆.
Source: Created by the authors based on data from IPEA.
Note: LR [L(kmax)/L(h)]= The LR test determines the optimal number of lags for the VAR model. LR[L(h)/L(h-1) ]= The corrected LR test determines the optimal number of lags for the VAR model.
Source: Created by the authors based on data from IPEA.
Received: June 29, 2015; Accepted: January 11, 2016
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You May Need a Lot More (or Less) than you Thought to Retire
Challenging the 4% Rule Suggests Caution but also Offers Hope
As Ben Le Fort points out, the venerable 4% retirement rule isn’t as unassailable as many people think.
Definitions and Origins of the 4% Rule
The 4% retirement rule says that you can expect to safely withdraw 4% of your retirement portfolio in your first year of retirement as your initial draw amount, and then determine each subsequent
year’s draw by applying the rate of inflation over the preceding year to that preceding year’s draw.
The inverse formulation, or 25-times rule, says that you need to save 25 times your first retirement year’s draw to provide for your retirement (4% of 25 times your desired draw is exactly your
desired draw).
The 4% rule traces back, as Le Fort mentions, to a 1990s study by financial planner William Bengen, who found that historic returns starting before the Great Depression proved that a 50/50 portfolio
of stocks and bonds would allow at least a 30-year retirement with at most a 4% initial withdrawal, adjusted annually for inflation. A later study concurred, showing a 95% probability that following
this strategy would not deplete your retirement portfolio before 30 years were over.
Some Important Challenges Affecting How Well the 4% Rule Applies to You
The 4% Rule Isn’t Meant for Early Retirees
Le Fort correctly points out that adherents of the Financial Independence, Retire Early (FIRE) movement should not count on a 4% initial withdrawal, since early retirees would likely live in
retirement significantly longer than 30 years.
In fact, the Social Security Administration’s Actuarial Life Table shows that the life expectancy of an American male at age 40 is 38.59 years. This is already longer than 30 years, and worse yet
(from a financial standpoint :)), half of those 40-year-old men will live longer than that, potentially for more than 60 years!
For women, things are even more challenging, as a 40-year-old American woman can expect to live on average another 42.50 years. Even a 50-year-old woman has a life expectancy of 33.26 years, again,
with half of 50-year-old women expected to survive longer than that.
Thus, if you’re planning to retire early, you need to save up a lot more than 25 times the amount you want to spend in your first year in retirement.
Mortality Depends on Affluence
Next, more recent studies, including one by David M. Planchett, PhD, CFA, CFP® on the Financial Planning Association’s site point out that the Social Security Administration’s actuarial tables
provide averages for the entire population, while those people affluent enough to hire a financial planner, and by implication who would be able to save enough for a 3% or 4% withdrawal to fund a
comfortable retirement, tend to live longer.
Planchett thus used the Society of Actuaries 2012 Immediate Annuity Mortality Table. This implies that even at 50, an affluent American male can expect to live on average more than the SSA’s estimate
of 29.69 years.
Thus, if you plan to be affluent enough to retire early, expect to survive even longer in retirement than you’d think based on the SSA’s estimates. Perhaps a lot longer.
The 4% Rule Is Backward-Looking, Not Forward-Looking
Planchett also points out that forward-looking expectations suggest that a 3% initial withdrawal is likely more appropriate than 4%, given that market returns are expected to be significantly lower
in the next several decades than the average market returns of the past 90-odd years.
He uses Morningstar Investment Management LLC’s 2016 capital market assumptions as an estimate for forward-looking returns. For a 40%-equity portfolio, the study found a 92% success probability for a
30-year retirement with a 4% initial draw based on historic data, but only a 53% success probability using forward-looking return estimates. To achieve a 91% success probability in the
forward-looking case, the initial draw would need to be just 3%.
Thus, at best, even for a 30-year retirement, don’t expect a safe retirement on 25 times your initial-retirement-year draw. Accumulating 33.3 times that draw would be far safer. For a 40-year
retirement, the success probability for a 4% initial draw with a 40%-equity portfolio is 74%, dropping to a measly 27% using forward-looking returns.
With these forward-looking returns, a 40-year retirement starting with even a 3% initial draw has only a 66% probability of success. Thus, perhaps a 2.5% draw, or 40-times-initial-draw accumulation
would be a better target for an early retiree.
The Impacts of Dynamic Withdrawals and Partial Guaranteed Income
Perhaps the most interesting aspect of Planchett’s study is his assessment of the impacts of dynamic withdrawals and having various fractions of retirement funds in guaranteed income (he calculates
the latter by multiplying each dollar by 24.23 as the present value of guaranteed inflation-adjusted income for life, weighted by mortality; then calculating the weight of that amount as a fraction
of the total retirement fund including the available portfolio).
For example, if a retiree expects $25,000 in annual inflation-adjusted guaranteed income (say from Social Security), the lifetime value of those benefits would be estimated as 24.23 times the
$25,000, or $605,750. If that same retiree had a $1,000,000 portfolio, the guaranteed income would be assessed as 37.7% ($605,750 divided by $1,605,750 which is the total retirement wealth available
to the retiree in this example). The study ignores non-financial assets such as home equity, which could in principal be used to provide retirement income, e.g. via a reverse mortgage.
Planchett’s analysis shows that as your fraction of retirement wealth available in the form of guaranteed income grows from 5% to 95%, your safe initial draw for a 30-year retirement can increase by
more than 4% of your retirement wealth. For people with a strong preference for retirement income stability, having only 5% of wealth in the form of guaranteed income results in a safe initial draw
of at best 2.6%. Those same people could safely draw 6.8% initially if 95% of their wealth was in the form of guaranteed income.
Since very few of us can count on a guaranteed-benefit pension these days, if you expect to have a fairly large retirement portfolio, your guaranteed income fraction from Social Security would likely
be less than 50%. Assuming 50%, Planchett’s numbers show that there is also some difference in safe initial draw depending on what fraction of your retirement budget is discretionary as opposed to
fixed (e.g., travel is discretionary, while rent or mortgage payments are fixed), varying from 4.1% safe initial draw if all your expected expenses are fixed, up to 4.6% if they’re all discretionary.
This makes sense since having more discretion allows you to trim your expenses to a greater extent when market returns are bad.
The Bottom Line
While the 4% (or 25-times) rule has been around for decades and is still considered a useful guideline, it faces many challenges.
If you plan to retire early, and especially if you believe the experts’ view that market returns are expected to be far more muted in the coming decades than they were historically, you should be
more conservative in your retirement planning, and expect to have a safe initial draw of 3% or less for your portfolio to have a high probability of not declining to nothing before you die.
However, if you have a pension or other significant source of guaranteed income, you can safely start with a higher draw, perhaps as much as 7%, especially if most of your retirement budget is
This also implies that if you have little in the way of guaranteed income, you might consider purchasing an annuity as a means to increase your guaranteed income fraction, which would allow you to
increase your initial draw percentage, in extreme cases by as much as a factor of 3.
Financial strategy can help you create a plan based on your financial goals, including especially retirement, and what to do in your practice to make achieving that plan more likely. If you'd like to
learn more about this, email me and we'll coordinate a free, no-strings-attached call.
This article is intended for informational purposes only, and should not be considered financial advice. You should consult a financial professional before making any major financial decisions.
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Gaussian Plume Model
This page describes a Gaussian Plume Models in both MATLAB and Python. Gaussian plume models are used heavily in air quality modelling and environmental consultancy. The model can be used to
illustrate the following phenomena:
More info...
The governing equations describing the movement of pollutants in the atmosphere is the advection-diffusion equation: \[ \frac{\partial C}{\partial t} + \frac{\partial uC}{\partial x}+ \frac{\partial
vC}{\partial y}+ \frac{\partial wC}{\partial z}= K_x\frac{\partial ^2 C}{\partial x^2}+K_y\frac{\partial ^2 C}{\partial y^2}+K_z\frac{\partial ^2 C}{\partial z^2} \] Here, \(u,v,w\) are the 3
components of the wind; \(x,y,z\) are directions in 3-d space; \(C\) is the concentration of pollutant; \(K_x,K_y,K_z\) are eddy diffusivities that are used to describe the effects of turbulence in
smearing out the pollutant.
By making the assumption that the flow is steady and that the advective term along wind is much greater than the eddy diffusion along wind there is an analytical solution: the so called gaussian
plume model solution, which is described by the following equation: \[ C(x,y,z)=\frac{Q}{2\pi u \sigma _y \sigma _z}\exp\left(-\frac{y^2}{2\sigma _y^2} \right)\left[\exp\left(-\frac{(z-H)^2}{2\sigma
_z^2}\right) +\exp\left(-\frac{(z+H)^2}{2\sigma _z^2} \right) \right] \] Here, \(Q\) is the mass emitted from the stack per unit time and \(H\) is the height of the stack. The \(\sigma\) values
depend on the eddy diffusivities in a complex way.
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The majority of Department of Mathematics research activities is pursued through Research Centers. These include:
A small number of Department members is affiliated with other research centres or has no current affiliation.
The department is organized into eight research/scientific areas:
In particular faculty positions open in one or more of these areas.
The department has currently about 80 faculty members, making it the largest mathematics department in Portugal.
Seminars, courses and workshops
The Department research activities include work and research seminars and workshops grouped in several thematic series. The following series have been active recently (August 2021 — August 2023):
Other series are available in the seminar archive.
The best method to gauge the global publication output originating from researchers associated to the Department is to access MathSciNet for the institution with code P-TULT. Be aware that you might
get some results from Instituto Superior Técnico but not from the Mathematics Department. The link requires a MathSciNet subscription.
The Department made available online Research Reports from 1993 to 2008.
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Optimizing Your Crypto Portfolio: Diversification, Risk Management, and Optimization Techniques
Photo by Caspar Camille Rubin on Unsplash
In my experience, crypto portfolio optimization is a complex and challenging task that requires careful consideration of a variety of factors. As a quantitative analyst, I have found that the most
effective way to perform crypto portfolio optimization is to use a combination of diversification, risk management, and optimization techniques.
Portfolio optimization is the process of selecting the optimal mix of assets to achieve a desired level of return while minimizing risk. The principles of portfolio optimization apply to
cryptocurrencies, and there are several effective ways to perform a crypto portfolio optimization.
In this article, I will discuss some popular techniques used in crypto portfolio optimization. These techniques include diversification, risk management, fundamental and technical analysis, market
cap weighting, and risk parity. I will explore each of these techniques in detail and provide insights on how to apply them to the cryptocurrency market. Whether you are a novice or an experienced
trader in cryptocurrencies, this article aims to provide you with valuable insights on how to optimize your crypto portfolio.
1. Diversification: First and foremost, diversification is key. By investing or trading in a range of different cryptocurrencies, you can spread their risk and minimize the impact of any single
asset's price fluctuations on the overall portfolio. It is important to consider factors such as market capitalization, liquidity, and correlation when selecting which cryptocurrencies to include
in a portfolio. This helps minimize my exposure to any one cryptocurrency and spread my risk across different assets.
2. Risk Management: In addition to diversification, risk management is crucial in crypto portfolio optimization. This involves setting clear investment goals, establishing stop-loss orders, and
regularly reviewing and adjusting the portfolio based on market conditions. Risk management techniques such as Value at Risk (VaR) and Monte Carlo simulations can also be employed to help
mitigate risk.
3. Optimization techniques: These techniques can be used to further refine the portfolio and improve its performance. These techniques include mean-variance optimization, which aims to maximize
returns while minimizing risk, and Black-Litterman optimization, which incorporates an investor’s views and beliefs about the market into the portfolio allocation.
Mean-variance optimization is a widely used technique that aims to maximize returns while minimizing risk. The basic idea behind mean-variance optimization is to find the portfolio with the highest
expected return for a given level of risk, or the portfolio with the lowest risk for a given expected return.
The mean-variance optimization formula can be expressed in Python as follows:
Mean-Variance Optimization in Python
import numpy as np
from scipy.optimize import minimize
# Define inputs
returns = # vector of asset returns
covariance = # covariance matrix of asset returns
target_return = # expected portfolio return
# Define objective function to minimize variance
def objective(weights):
return np.dot(weights.T, np.dot(covariance, weights))
# Define constraint functions
def constraint1(weights):
return np.sum(weights) - 1
def constraint2(weights):
return np.dot(weights.T, returns) - target_return
# Define optimization problem
initial_weights = np.ones(len(returns))/len(returns)
bounds = [(0,1) for i in range(len(returns))]
constraints = [{'type': 'eq', 'fun': constraint1},
{'type': 'eq', 'fun': constraint2}]
result = minimize(objective, initial_weights, method='SLSQP',
bounds=bounds, constraints=constraints)
# Extract optimal weights
optimal_weights = result.x
Another optimization technique that is commonly used in crypto portfolio optimization is Black-Litterman optimization. This approach incorporates a trader’s views and beliefs about the market into
the portfolio allocation. The basic idea behind Black-Litterman optimization is to start with a prior estimate of expected returns and then adjust this estimate based on the trader’s views and market
Black-Litterman Optimization in R
# Define inputs
returns = # vector of asset returns
covariance = # covariance matrix of asset returns
views = # vector of investor views
view_uncertainty = # matrix of view uncertainty
tau = # scaling factor
asset_exposures = # matrix of portfolio asset exposures
# Calculate prior expected returns and covariance matrix
prior_return = # your calculation
prior_covariance = tau * covariance
# Calculate posterior expected returns
posterior_return = solve(tau * covariance) %*% prior_return + t(asset_exposures) %*% solve(view_uncertainty) %*% views
# Calculate posterior covariance matrix
posterior_covariance = solve(solve(tau * covariance) + t(asset_exposures) %*% solve(view_uncertainty) %*% asset_exposures)
# Calculate optimal weights
optimal_weights = solve(posterior_covariance) %*% posterior_return
Note that these are simplified code snippets that assume you have pre-calculated the necessary inputs, such as expected returns, covariance matrices, and view data. In practice, you would have to
perform additional calculations or data transformations to use these formulas in your own code.
Overall, optimization techniques like mean-variance and Black-Litterman can be powerful tools for crypto portfolio optimization, as they enable investors to create well-diversified and risk-managed
portfolios that are tailored to their specific investment goals and market views.
In conclusion, these techniques can help investors optimize their crypto portfolios by minimizing risk and maximizing returns. As a quant, I believe that a combination of these techniques can help me
build a diversified and well-managed crypto portfolio. However, it is important to remember that no investment strategy is foolproof, and you should always be prepared for the risks involved.
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18 questions MGF2106 – Project 2 Directions: This project contains multiple parts, and each part of the project
MGF2106 – Project 2
Directions: This project contains multiple parts, and each part of the project contains several questions. Answer each question to the best of your ability. Record your answer for each question in
the proper format and upload it to the appropriate Assignment (Projects) folder within Falcon Online by the due date posted within the Course Schedule.
Submission Format Instructions: Your submission must meet the following criteria to be accepted for grading.
· There are two options for submitting your work. Note that a requirement for each is that you must save your work as a single document.
1. Your submission can be a single, typed document saved as a Word document (.doc). Please know that every Daytona State College student has access to Microsoft Word. Refer to the Course Addendum
under Microsoft Office.
2. You can also hand write your work and scan it into a single Portable Document Format (.pdf), or in Joint Photographic Expert Group (.jpeg) format.
· Your name must be at the top of your submission. Once you have answered the questions, upload your completed work to the appropriate Assignment (Projects) folder in Falcon Online. Be sure to click
on “Submit” after using the “Add a file” link.
· Do not type the questions from the project into your submission. Only submit your answers to the questions.
· Follow the numbering system used in the project when typing your answers. Please be sure to answer all parts of a question.
MGF2106 Project 2: Conversions
Directions: Answer each question thoroughly using proper grammar and complete sentences. Record your answers in a separate document and submit your work according to the criteria provided in the
Project Instructions. You may work individually on this project or with a small group.
You have decided to spend 8 nights in England for your bucket list vacation. Any time you travel through different countries, you must be able to convert prices, distances, and speed limits. Remember
that the currency used in England is Pounds. Current conversions have 1 Pound = 1.23 US Dollars (USD) and 1 US Dollar (USD) = 0.81 Pounds. Use those conversions for this current project. Your plan is
travel from Orlando to London, England and spend four nights there before finding 4 other excursions in England.
You have saved up $5000 USD to cover all expenses, so you must decide how much the flight, meals, lodging, and transportation throughout your vacation will cost you. You will have choices to make to
plan your trip to your liking.
Make sure to convert each option and then state what you would select. Keep a tab of how much you are spending.
1. For each flight below, convert the cost from Pounds to US Dollars (USD) and complete the table below. The flight options from Orlando International Airport to Heathrow Airport in London
(roundtrip), with departure times as follows:
Flight Type Cheapest Quickest Most Reliable
5:20pm – 10:05am 6:55pm – 8:50am 2:10pm – 5:55am
Time and Length of Flight
18 hours, long layover 9 hours, direct flight 11 hours, quick layover
Cost in Pounds 486 Pounds 1024 Pounds 930 Pounds
Find the Cost in USD
2. Which flight would you choose given your budget? Why would you pick that flight?
3. After you bought that flight, how much money (USD) are you left with for your trip?
4. Now that you purchased your flight, you must now get from the airport to downtown London, where you are staying. Look at the options below and convert the prices to US Dollars. You have 3 options
to choose from:
Ride Type Uber Train Bus
Length 45 minutes 15 minutes 1 hour
Cost in Pounds 26 Pounds 9 Pounds 4 Pounds
Find the Cost in USD
5. Which mode of transportation would you pick from the airport to downtown London? Why would you pick that one?
6. After buying the flight and the transportation to downtown London, how much money do you have remaining for your trip?
7. Now you must think about lodging and food. You must decide if you want to splurge on a nice place to stay, spend your money on food, or try to save money on both. Here are the lodging/meal options
that you found. Convert each one to US Dollars and remember this is per day. You are planning on spending 4 nights in London, before spending 4 nights in other locations across England. You found a
cheap hostel, which is set up to share a room with 4 total bunkbeds and you also found a nice hotel to yourself, both in the same location you want to stay. Decide which option would best suit your
Options Hostel / Cheap food Hostel / Fine dining Hotel / Cheap food Hotel / Fine dining
Hostel price per night in Pounds 40 Pounds 40 Pounds 162 Pounds 162 Pounds
Food cost per day in Pounds 24 Pounds 81 Pounds 24 Pounds 81 Pounds
Hostel Price per night in USD
Food cost per day in USD
Total cost in USD for 4 days
8. Which option would you choose for lodging/food choices? Why did you pick that option? What is the total cost of that for four days?
9. For your 4 nights in London, you have given yourself a budget of $150 per day on miscellaneous expenses (museum cost, souvenirs, etc…). Calculate the total you gave yourself in USD and in Pounds.
10. Calculate the total amount that would be spent on a plane ticket, transportation from the airport, and lodging/food, and miscellaneous for London. What is the total amount in Pounds and US
Dollars that you will be spending? How much money is remaining in your budget in USD?
11. Now that you have budgeted your trip to get to London, sleep and eat, we must now find some fun overnight trips to do when we are in England. Let’s figure out how far away from London the top
tourist attractions are. Convert each location from kilometers (km) to miles (m). Use the conversion that 1 kilometer = 0.62 miles and 1 mile = 1.6 kilometers
Outside of London, you are considering visiting the Stonehenge ruins in Wiltshire, the Roman Baths in Somerset, the York Minister Church in York, the Windsor Castle in Berkshire, the Glastonbury
Music Festival in Glastonbury, and the Hull Fair in Kingston upon Hull. Convert the distance from in London from kilometers to miles. Round to the nearest mile.
Glastonbury Music Festival Hull Fair in
Destination Stonehenge in Wiltshire Roman Baths in Somerset York Minister in York Windsor Castle in Berkshire
In Glastonbury Hull
Distance in km 142 km 229 km 340 km 84 km 220 km 325km
Distance in m
12. For each destination given, calculate how many liters of gas you would need to get there (round to 1 decimal). Once you do this, you are curious what that amount is in gallons, so convert your
liters to gallons. Then, find the cost of gas in Pounds and USD, and find the total cost needed to rent the car for the day with the cost of gas.
Your rental car, for a cost of 104 Pounds daily will drive 8.5 km per liter of gas. The current mean cost of gas per liter in England is 1.24 Pounds. Also note that 1 liter = 0.26 gallons and that 1
gallon = 3.75 liters.
Destinations Stonehenge Roman Baths York Minister Windsor Castle Glastonbury Music Festival Hull Fair
Distance in km 142 km 229 km 340 km 84 km 220 km 325 km
Number of liters
Number of gallons
Cost in Pounds for gas
Cost in USD for gas
Total cost of car rental for one day + gas for trip in USD
13. Research all 6 of the above possible destinations and decide which 4 places you would want to visit. List them below and a brief reason why you would select those destinations.
14. Calculate the cost to drive to each one of these places, assuming that you are leaving from London. What is the total cost of driving and renting the car for the 4 days?
15. While driving, you have no idea how fast you are going because the speedometer is in kilometers per hour. Convert the following speeds into miles per hour so you know how fast you are going. Note
that 1km/h = 0.62 mi/h and 1 mi/h = 1.6 km/h.
Speed in km/h 50 85 120
Speed in mi/h
16. For each of the 4 destinations, you have allowed yourself $300 USD per day for lodging, food, and miscellaneous expenses / souvenirs. What is the total over 4 days that you allowed yourself in
USD and Pounds?
17. Add up the totals for all the money spent in London and your 4 other destinations including car rental. How much money did you spend altogether in USD and in Pounds? Were you able to stay under
the $5000 USD that you saved up for your trip?
18. If you could put together a trip like this, where would you want to go and why?
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Natural Transformation - (Programming Techniques III) - Vocab, Definition, Explanations | Fiveable
Natural Transformation
from class:
Programming Techniques III
A natural transformation is a way to relate two functors between categories, providing a bridge that transforms objects and morphisms in a coherent manner. It essentially captures the idea of mapping
one structure into another while preserving the categorical relationships, allowing for meaningful comparisons between different categories. This concept is crucial for understanding how different
structures can be interrelated in functional programming and category theory.
congrats on reading the definition of Natural Transformation. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. Natural transformations can be visualized as 'morphisms of functors' that provide a structured way to compare different functorial mappings.
2. They consist of a collection of morphisms, one for each object in the source category, ensuring that the structure is preserved under the mapping.
3. Natural transformations satisfy two key properties: they must respect identity morphisms and preserve composition, ensuring the coherence of the transformation.
4. In functional programming, natural transformations can often represent conversions between data types or structures while maintaining their inherent relationships.
5. The concept plays a vital role in defining concepts like adjunctions, limits, and colimits within category theory, linking various categorical constructs.
Review Questions
• How do natural transformations facilitate the comparison of different functors in category theory?
□ Natural transformations allow us to compare two functors by providing a systematic way to map objects and morphisms from one functor to another while preserving their structural
relationships. This means that if we have two functors between the same categories, a natural transformation gives us a method to relate their outputs consistently. It ensures that the
transformation behaves well with respect to the composition of morphisms, thereby allowing deeper insights into how different structures can interact.
• Discuss the importance of natural transformations in functional programming and how they help maintain type safety when converting data structures.
□ In functional programming, natural transformations are essential for ensuring type safety when transforming data structures from one form to another. By encapsulating the transformation
process within a natural transformation framework, programmers can guarantee that all relationships between types are preserved during conversion. This not only helps maintain type integrity
but also supports modularity and reusability in code by enabling smooth transitions between different abstractions or representations without losing coherence.
• Evaluate the implications of using natural transformations in defining limits and colimits within category theory.
□ Using natural transformations in defining limits and colimits allows for a more generalized approach to structuring categorical relationships. They provide the means to understand how objects
and morphisms can be combined or decomposed consistently across different categories. By ensuring that these transformations adhere to specific properties, such as preserving compositionality
and identity, natural transformations enable mathematicians and computer scientists to formulate universal constructions that capture essential aspects of structure across varied contexts,
ultimately enriching the framework of category theory.
"Natural Transformation" also found in:
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Currently there may be errors shown on top of a page, because of a missing Wiki update (PHP version and extension DPL3).
Topics Help • Register • News • History • How to • Sequences statistics • Template prototypes
• Contents
• Keyboard shortcuts
Helping Out • Editing
• Find or fix a stub
• Math rendering
• Known issues
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Reverse a list or range
The heart of this formula is the INDEX function, which is given the list as the array argument:
The second part of the formula is an expression that works out the correct row number as the formula is copied down:
1. COUNTA(list) returns the count of non-blank items in the list (10 in the example)
2. ROW(list) returns the starting row number of list (5 in the example)
3. ROW() returns the row number the formula resides in
The result of this expression is a single number starting at 10, and ending at 1 as the formula is copied down. The first formula returns the 10th item in the list, the second formula returns the 9th
item in the list, and so on:
=INDEX(list,10+5-5,1) // item 10
=INDEX(list,10+5-6,1) // item 9
=INDEX(list,10+5-7,1) // item 8
With Dynamic Arrays
Excel 365 supports dynamic array formulas, which can used to create a simpler and more efficient formula. The SORTBY function can perform a "reverse sort" with help from the SEQUENCE function. The
formula in D5 should be:
Inside the SEQUENCE function, the ROWS function is used twice to get a count of rows in the range. The first count is used as the rows argument in sequence, the second count is used for the start
argument. The step argument is supplied as -1, so that the array returned by sequence starts at 10 and counts down to 1. The result is delivered to SORTBY as by_array1:
With this configuration, SORTBY sorts the named range list in reverse order and the results spill into the range D5:D14.
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MEGA6: Molecular Evolutionary Genetics Analysis version 6.0.
Mol Biol Evol 30(12): 2725-2729
MEGA6: Molecular Evolutionary Genetics Analysis Version 6.0
^Research Center for Genomics and Bioinformatics, Tokyo Metropolitan University, Hachioji, Tokyo, Japan^Department of Biological Sciences, Tokyo Metropolitan University, Hachioji, Tokyo, Japan^Center
for Evolutionary Medicine and Informatics, Biodesign Institute, Arizona State University^School of Life Sciences, Arizona State University^Center of Excellence in Genomic Medicine Research, King
Abdulaziz University, Jeddah, Saudi ArabiaAssociate editor: S. Blair HedgesAssociate editor: S. Blair HedgesCopyright © The Author 2013. Published by Oxford University Press on behalf of the Society
for Molecular Biology and Evolution. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com
We announce the release of an advanced version of the Molecular Evolutionary Genetics Analysis (MEGA) software, which currently contains facilities for building sequence alignments, inferring
phylogenetic histories, and conducting molecular evolutionary analysis. In version 6.0, MEGA now enables the inference of timetrees, as it implements the RelTime method for estimating divergence
times for all branching points in a phylogeny. A new Timetree Wizard in MEGA6 facilitates this timetree inference by providing a graphical user interface (GUI) to specify the phylogeny and
calibration constraints step-by-step. This version also contains enhanced algorithms to search for the optimal trees under evolutionary criteria and implements a more advanced memory management that
can double the size of sequence data sets to which MEGA can be applied. Both GUI and command-line versions of MEGA6 can be downloaded from www.megasoftware.net free of charge.
Keywords: software, relaxed clocks, phylogeny
The Molecular Evolutionary Genetics Analysis (MEGA) software is developed for comparative analyses of DNA and protein sequences that are aimed at inferring the molecular evolutionary patterns of
genes, genomes, and species over time (Kumar et al. 1994; Tamura et al. 2011). MEGA is currently distributed in two editions: a graphical user interface (GUI) edition with visual tools for
exploration of data and analysis results (Tamura et al. 2011) and a command line edition (MEGA-CC), which is optimized for iterative and integrated pipeline analyses (Kumar et al. 2012).
In version 6.0, we have now added facilities for building molecular evolutionary trees scaled to time (timetrees), which are clearly needed by scientists as an increasing number of studies are
reporting divergence times for species, strains, and duplicated genes (e.g., Kumar and Hedges 2011; Ward et al. 2013). For this purpose, we have implemented the RelTime method, which can be used for
large numbers of sequences comprising contemporary data sets, is the fastest method among its peers, and is shown to perform well in computer simulations (Tamura et al. 2012). RelTime produces
estimates of relative times of divergence for all branching points (nodes) in any phylogenetic tree without requiring knowledge of the distribution of the lineage rate variation and without using
clock calibrations and associated distributions. Relative time estimates produced by MEGA will be useful for determining the ordering and spacing of sequence divergence events in species and gene
family trees. The (relative) branch rates produced by RelTime will also enable users to determine the statistical distribution of evolutionary rates among lineages and detect rate differences between
species and duplicated gene clades. In addition, relative times obtained using molecular data can be directly compared with the times from nonmolecular data (e.g., fossil record) to test independent
biological hypotheses. The RelTime computation in MEGA6 is highly efficient in terms of both performance and memory required. For a nucleotide alignment of 765 sequences and 2,000 bp (data from
Tamura et al. [2011]), MEGA6 required just 43 min and 1 GB memory (including the calculation steps mentioned below). Both time and memory requirements increase linearly with the number of sequences
in MEGA6 (fig. 1). Figure 2 shows a timetree produced by MEGA6 and displayed in the Tree Explorer, which has been upgraded from previous versions of MEGA to display confidence intervals and to export
relative divergence times and evolutionary rates for branches, along with absolute divergence times and confidence intervals (see below). The Tree Explorer also allows customization of the timetree
display in many ways for producing publication quality images.
Using calibrations to translate relative times to absolute times: The relative times produced by the RelTime method can be directly converted into absolute times when a single known divergence time
(calibration point) based on fossil or other information is available. This facility is incorporated in MEGA6 where a global time factor (f), which is computed from the given calibration point,
converts all estimates of relative times (NTs) to absolute times (ATs) where AT[x] = f × NT[x] for the internal node x. This approach is taken because NTs are already shown to be linearly related
with the true time (Tamura et al. 2012). However, researchers often use multiple calibration points along with information on upper and/or lower bounds on one or more calibration points. In order to
consider those constraints when estimating f, we have extended the RelTime implementation such that the estimate of f produces estimates of AT that satisfy the calibration constraints. In this case,
if there are a range of values for f that do not violate the calibration constraints, then the midpoint of that range becomes the estimate of f. If one or more of the ATs fall outside the calibration
constraints, then f is set so that their deviation from the constraints is minimized. In this case, NTs for the nodes with estimated ATs are adjusted to satisfy the calibration constraints, such that
the estimated ATs for the offending nodes will lie between the minimum and maximum constraint times specified by the user. These adjustments to NTs are followed by re-optimizing all other NTs in the
tree recursively using the standard RelTime algorithm. Figure 2 shows a timetree display with absolute times in the Tree Explorer, where 95% confidence intervals are shown for each node time (see
Confidence intervals for time estimates: MEGA6 also provides confidence intervals for relative and absolute divergence times, which are necessary to assess the uncertainty in the estimated time and
test biological hypotheses. In this formulation, variance contributed by the presence of rate variation among lineages (V[R,i]) is combined with the estimated variance of relative node time (V[NT,i]
). We compute V[R,i] using the mean of the coefficient of variation of lineage rates over all internal nodes (C[R]). It is obtained by first computing the mean (µ) and standard deviation (σ) of the
node-to-tip distance for each internal node in the original tree with branch lengths. Then, C[R] = ∑[σ[i]/µ[i]]^/(n-3), where n is the number of sequences. For node i, V[R,i] = (NT[i] × √C[R])^. The
variance of node height (V[H,i]) is estimated by the curvature method obtained during the maximum likelihood estimation of branch lengths, and thus relative NTs, for each node. Then, the variance of
NT is V(NT[i]) = V[NT,i] + V[R,i], which is used to generate a 95% confidence interval. The bounds of this interval in terms of relative time are then multiplied by the factor f to provide confidence
intervals on absolute times when calibrations are provided. It is important to note that this variance does not incorporate the uncertainty specified in the calibration times by the user through the
specifications of minimum and maximum bounds, because the statistical distribution of the calibration uncertainty is rarely known. Therefore, we only use the range of calibration bounds during the
estimation of f that converts relative times into absolute times, as described above, but this range does not affect the size of the confidence interval in any other way. In the future, we plan to
enhance the estimation of f when users provide statistical distributions specifying the calibration uncertainty (see also, Hedges and Kumar 2004).
Timetree Wizard: In practice, the estimation of timetrees can be cumbersome, as one must provide a phylogeny, a sequence data set, and calibration points with constraints. To simplify this process,
we have programmed a Timetree Wizard to enable users to provide all of these inputs through an intuitive step-by-step graphical interface. Figure 3A shows a flowchart of the Timetree Wizard, where
the user first provides a sequence alignment and a tree topology for use in building a timetree. MEGA6 validates these inputs by mapping (sequence) names in the topology to the names in the alignment
data. If the topology contains a subset of sequences present in the alignment, MEGA automatically subsets the sequence data. Additional automatic subsetting of data is provided in the Analysis
Preferences Dialog box (see fig. 3E). In the next step, the user has the option to provide calibration constraints by using a new Calibration Editor in MEGA6 where calibration points are specified by
1) point-and-click on individual nodes in the tree display (fig. 3B), 2) selecting name-pairs from dropdown lists such that their most recent common ancestor on the topology refers to the desired
node (fig. 3C), and/or 3) uploading a text file containing calibration constraints in a simple format (fig. 3D). If no calibration constraints are provided, then only relative times and related
statistical measurements will be produced by MEGA6, but users still have an option to specify them in the Tree Explorer where the timetree containing relative times is displayed.
The next step in Timetree Wizard is for the user to select various analysis options in the Analysis Preferences Dialog, including the types of substitutions to consider (e.g., nucleotide, codon, or
amino acid), evolutionary model describing the substitution pattern, distribution of substitution rates among sites (e.g., uniform or gamma-distributed rates and the presence of invariant sites),
options for excluding certain alignment positions, and stringency for merging evolutionary clock rates during timetree analysis. These options are available in a context-dependent manner based on the
type of sequence data being used in the analysis (e.g., nucleotide, coding vs. non-coding, or proteins). For coding nucleotide data, the users may subset the data based on the desired codon positions
or ask MEGA to automatically translate codons into amino acids and conduct analysis at the protein sequence level. The data subset options also allow for handling of gaps and missing data, where one
can choose to use all the data or exclude positions that contain a few or more gaps or missing data (e.g., Partial Deletion option). The stringency for merging clock rates option indicates the
statistical significance to use for deciding conditions in which the ancestor and descendant rates will be the same (rate merging), which is important to reduce the number of rate parameters
estimated and to avoid statistical over-fitting. Once these and other options are set, the RelTime computation begins.
Other enhancements in MEGA: In addition to the new timetree system in MEGA6, we have made several other useful enhancements. First, we have added the subtree-pruning-and-regrafting (SPR) algorithm to
search for the optimal tree under the maximum likelihood (ML) and maximum parsimony (MP) criteria (Swofford 1998; Nei and Kumar 2000). In addition, the tree-bisection-and-regrafting (TBR) algorithm
is now included to search for the MP trees. These algorithms replace the close-neighbor-interchange (CNI) approach and allow for a more exhaustive search of the tree space (Swofford 1998; Nei and
Kumar 2000). These algorithms were tested on simulated data sets that were analyzed in Tamura et al. (2011). The final trees produced by SPR heuristic search were, on average, more optimal than the
true tree, a phenomenon explained by Nei et al. (1998). Therefore, MEGA6 heuristic searches are expected to perform well in practical data analysis.
We have also upgraded MEGA source code to increase the amount of memory that MEGA can address in 64-bit computers, where it can now use up to 4 GB memory, which is twice its previous limit. The
source code upgrade has also increased the canvas size in Tree Explorer, which can now render trees with as many as 4,000 taxa. Finally, we have implemented a usage analytics system to assess options
and analyses that are the most used. At the time of installation, users have a choice to participate in this effort, where we wish to generate a better understanding of the needs of the user
community for prioritizing future developments. For the future, we have already planned the release of a full 64-bit version of MEGA as well as support for partitioned ML phylogenetic analyses. An
outcome of this effort is a 64-bit command-line version of MEGA6 that supports the timetree analysis, which can be downloaded from www.megasoftware.net/reltime (last accessed October 19, 2013) and
used for very large sequence data sets.
Click here to view.
We thank Oscar Murillo for extensive help in testing the RelTime computations. We would also like to thank Sayaka Miura, Anna Freydenzon, Mike Suleski, and Abediyi Banjoko for their invaluable
feedback. This work was supported from research grants from National Institutes of Health (HG002096-12; to S.K. and HG006039-03 to A.F.) and Japan Society for the Promotion of Science (JSPS)
grants-in-aid for scientific research to K.T.
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seminars - Emergence of phase-locked states for the Winfree model in a large coupling regime
<PARC Monthly Seminar>
◆ 일 시 : 2015년 2월 25일(수) 오후 4시
◆ 장 소 : 서울대학교 27동 220호
◆ 연 사 : 박 진 영
◆ 소 속 : 서울대학교 수리과학부
◆ 강연 제목 : Emergence of phase-locked states for the Winfree model in a large coupling regime.
◆ 초 록 :
We study the large-time behavior of the globally coupled Winfree model in a large coupling regime. The Winfree model is the first mathematical model for the synchronization phenomenon in an ensemble
of weakly coupled limit-cycle oscillators. For the dynamic formation of phase-locked states, we provide a sufficient framework in terms of geometric conditions on the coupling functions and coupling
strength. We show that in the proposed framework, the emergent phase-locked state is the unique equilibrium state and it is asymptotically stable in an $\ell^1$-norm; further, we investigate its
configurational structure.
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2.1: Free Electron Model of Polyenes
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The particle-in-a-box type problems provide important models for several relevant chemical situations
The particle-in-a-box model for motion in one or two dimensions discussed earlier can obviously be extended to three dimensions. For two and three dimensions, it provides a crude but useful picture
for electronic states on surfaces (i.e., when the electron can move freely on the surface but cannot escape to the vacuum or penetrate deeply into the solid) or in metallic crystals, respectively. I
say metallic crystals because it is in such systems that the outermost valence electrons are reasonably well treated as moving freely rather than being tightly bound to a valence orbital on one of
the constituent atoms or within chemical bonds localized to neighboring atoms.
Free motion within a spherical volume such as we discussed in Chapter 1 gives rise to eigenfunctions that are also used in nuclear physics to describe the motions of neutrons and protons in nuclei.
In the so-called shell model of nuclei, the neutrons and protons fill separate \(s\), \(p\), \(d\), etc. orbitals (refer back to Chapter 1 to recall how these orbitals are expressed in terms of
spherical Bessel functions and what their energies are) with each type of nucleon forced to obey the Pauli exclusion principle (i.e., to have no more than two nucleons in each orbital because protons
and neutrons are Fermions). For example, \(^4He\) has two protons in \(1s\) orbitals and 2 neutrons in \(1s\) orbitals, whereas \(^3He\) has two \(1s\) protons and one \(1s\) neutron. To remind you,
I display in Figure 2. 1 the angular shapes that characterize \(s\), \(p\), and \(d\) orbitals.
Figure 2.1. The angular shapes of \(s\), \(p\), and \(d\) functions
This same spherical box model has also been used to describe the valence electrons in quasi-spherical nano-clusters of metal atoms such as \(Cs_n\), \(Cu_n\), \(Na_n\), \(Au_n\), \(Ag_n\), and their
positive and negative ions. Because of the metallic nature of these species, their valence electrons are essentially free to roam over the entire spherical volume of the cluster, which renders this
simple model rather effective. In this model, one thinks of each valence electron being free to roam within a sphere of radius \(R\) (i.e., having a potential that is uniform within the sphere and
infinite outside the sphere).
The orbitals that solve the Schrödinger equation inside such a spherical box are not the same in their radial shapes as the \(s\), \(p\), \(d\), etc. orbitals of atoms because, in atoms, there is an
additional attractive Coulomb radial potential \(V(r) = -Ze^2/r\) present. In Chapter 1, we showed how the particle-in-a-sphere radial functions can be expressed in terms of spherical Bessel
functions. In addition, the pattern of energy levels, which was shown in Chapter 1 to be related to the values of x at which the spherical Bessel functions \(j_L(x)\) vanish, are not the same as in
atoms, again because the radial potentials differ. However, the angular shapes of the spherical box problem are the same as in atomic structure because, in both cases, the potential is independent of
\(\theta\) and \(\phi\). As the orbital plots shown above indicate, the angular shapes of s, p, and \(d\) orbitals display varying number of nodal surfaces. The \(s\) orbitals have none, \(p\)
orbitals have one, and \(d\) orbitals have two. Analogous to how the number of nodes related to the total energy of the particle constrained to the \(xy\) plane, the number of nodes in the angular
wave functions indicates the amount of angular or orbital rotational energy. Orbitals of \(s\) shape have no angular energy, those of \(p\) shape have less then do \(d\) orbitals, etc.
It turns out that the pattern of energy levels derived from this particle-in-a-spherical-box model can offer reasonably accurate descriptions of what is observed experimentally. In particular, when a
cluster (or cluster ion) has a closed-shell electronic configuration in which, for a given radial quantum number \(n\), all of the \(s\), \(p\), \(d\) orbitals associated with that \(n\) are doubly
occupied, nanoscopic metal clusters are observed to display special stability (e.g., lack of chemical reactivity, large electron detachment energy). Clusters that produce such closed-shell electronic
configurations are sometimes said to have magic-number sizes. The energy level expression given in Chapter 1
\[E_{L,n} = V_0 + (z_{L,n})^2 \dfrac{h^2}{2mR^2} \tag{2.1}\]
for an electron moving inside a sphere of radius \(R\) (and having a potential relative to the vacuum of \(V_0\)) can be used to model the energies of electron within metallic nano-clusters. Each
electron occupies an orbital having quantum numbers \(n\), \(L\), and \(M\), with the energies of the orbitals given above in terms of the zeros \(\{z_{L,n}\}\) of the spherical Bessel functions.
Spectral features of the nano-clusters are then determined by the energy gap between the highest occupied and lowest unoccupied orbital and can be tuned by changing the radius (\(R\)) of the cluster
or the charge (i.e., number of electrons) of the cluster.
Another very useful application of the model problems treated in Chapter 1 is the one-dimensional particle-in-a-box, which provides a qualitatively correct picture for \(\pi\)-electron motion along
the \(p_{\pi}\) orbitals of delocalized polyenes. The one Cartesian dimension corresponds to motion along the delocalized chain. In such a model, the box length \(L\) is related to the carbon-carbon
bond length \(R\) and the number \(N\) of carbon centers involved in the delocalized network \(L=(N-1) R\). In Figure 2.2, such a conjugated network involving nine centers is depicted. In this
example, the box length would be eight times the C-C bond length.
Figure 2.2. The \(\pi\) atomic orbitals of a conjugated chain of nine carbon atoms, so the box length \(L\) is eight times the C-C bond length.
The eigenstates \(\psi_n(x)\) and their energies \(E_n\) represent orbitals into which electrons are placed. In the example case, if nine \(\pi\) electrons are present (e.g., as in the
1,3,5,7-nonatetraene radical), the ground electronic state would be represented by a total wave function consisting of a product in which the lowest four \(\psi\)'s are doubly occupied and the fifth
\(\psi\) is singly occupied:
\[\Psi = \psi_1 \alpha\psi_1\beta \psi_2 \alpha \psi_2 \beta \psi_3 \alpha \psi_3\beta \psi_4 \alpha \psi_4 \beta \psi_5 \alpha. \tag{2.2}\]
The \(z\)-component spin angular momentum states of the electrons are labeled \(\alpha\) and \(\beta\) as discussed earlier.
We write the total wave function above as a product wave function because the total Hamiltonian involves the kinetic plus potential energies of nine electrons. To the extent that this total energy
can be represented as the sum of nine separate energies, one for each electron, the Hamiltonian allows a separation of variables
\[H \cong \sum_{j=1}^9 H(j) \tag{2.3}\]
in which each H(j) describes the kinetic and potential energy of an individual electron. Of course, the full Hamiltonian contains electron-electron Coulomb interaction potentials \(e^2/r_{i,j}\) that
cannot be written in this additive form. However, as we will treat in detail in Chapter 6, it is often possible to approximate these electron-electron interactions in a form that is additive.
Recall that when a partial differential equation has no operators that couple its different independent variables (i.e., when it is separable), one can use separation of variables methods to
decompose its solutions into products. Thus, the (approximate) additivity of \(H\) implies that solutions of \(H \psi = E \psi\) are products of solutions to
\[H (j) \psi (\textbf{r}_j) = E_j \psi(\textbf{r}_j). \tag{2.4}\]
The two lowest \(\pi\pi^*\) excited states would correspond to states of the form
\[\psi^* = \psi_1\alpha \psi_1\beta \psi_2\alpha \psi_2\beta \psi_3\alpha \psi_3\beta \psi_4\alpha \psi_5\beta \psi_5\alpha, \tag{2.5a}\]
\[\psi'^* = \psi_1\alpha \psi_1\beta \psi_2\alpha \psi_2\beta \psi_3\alpha \psi_3\beta \psi_4\alpha \psi_4\beta \psi_6\alpha,\tag{2.5b}\]
where the spin-orbitals (orbitals multiplied by \(\alpha\) or \(\beta\)) appearing in the above products depend on the coordinates of the various electrons. For example,
\[\psi_1\alpha \psi_1\beta \psi_2\alpha \psi_2\beta \psi_3\alpha \psi_3\beta \psi_4\alpha \psi_5\beta \psi_5\alpha \tag{2.6a}\]
\[ \psi_1\alpha(\textbf{r}_1) \psi_1\beta (\textbf{r}_2) \psi_2\alpha (\textbf{r}_3) \psi_2\beta (\textbf{r}_4) \psi_3\alpha (\textbf{r}_5) \psi_3\beta (\textbf{r}_6) \psi_4a (\textbf{r}_7)\psi_5\
beta (\textbf{r}_8) \psi_5\alpha (\textbf{r}_9). \tag{2.6b}\]
The electronic excitation energies from the ground state to each of the above excited states within this model would be
\[\Delta{E^*} = \dfrac{ \pi^2 \hbar^2}{2m} \left[ \dfrac{5^2}{L^2} - \dfrac{4^2}{L^2}\right] \tag{2.7a}\]
\[\Delta{E'^*} = \dfrac{ \pi^2 \hbar^2}{2m} \left[ \dfrac{6^2}{L^2} - \dfrac{5^2}{L^2}\right]. \tag{2.7b}\]
It turns out that this simple model of \(\pi\)-electron energies provides a qualitatively correct picture of such excitation energies. Its simplicity allows one, for example, to easily suggest how a
molecule’s color (as reflected in the complementary color of the light the molecule absorbs) varies as the conjugation length \(L\) of the molecule varies. That is, longer conjugated molecules have
lower-energy orbitals because \(L^2\) appears in the denominator of the energy expression. As a result, longer conjugated molecules absorb light of lower energy than do shorter molecules.
This simple particle-in-a-box model does not yield orbital energies that relate to ionization energies unless the potential inside the box is specified. Choosing the value of this potential \(V_0\)
that exists within the box such that \(V_0 + \dfrac{\pi^2 \hbar^2}{2m} \dfrac{5^2}{L^2}\) is equal to minus the lowest ionization energy of the 1,3,5,7-nonatetraene radical, gives energy levels (as \
(E = V_0 + \dfrac{\pi^2 \hbar^2}{2m} \dfrac{n^2}{L^2}\)), which can then be used as approximations to ionization energies.
The individual \(\pi\)-molecular orbitals
\[\psi_n = \sqrt{\dfrac{2}{L}} \sin\Big(\dfrac{n\pi x}{L}\Big) \tag{2.8}\]
are depicted in Figure 2.3 for a model of the 1,3,5 hexatriene \(\pi\)-orbital system for which the box length \(L\) is five times the distance \(R_{CC}\) between neighboring pairs of carbon atoms.
The magnitude of the \(k^{th}\) C-atom centered atomic orbital in the \(n^{th}\) \(\pi\)-molecular orbital is given by
\[\sqrt{\dfrac{2}{L}} \sin\Big(\dfrac{n\pi(k-1)R_{CC}}{L}\Big).\]
Figure 2.3. The phases of the six molecular orbitals of a chain containing six atoms.
In this figure, positive amplitude is denoted by the clear spheres, and negative amplitude is shown by the darkened spheres. Where two spheres of like shading overlap, the wave function has enhanced
amplitude (i.e. there is a bonding interaction); where two spheres of different shading overlap, a node occurs (i.e., there is antibonding interaction). Once again, we note that the number of nodes
increases as one ranges from the lowest-energy orbital to higher energy orbitals. The reader is once again encouraged to keep in mind this ubiquitous characteristic of quantum mechanical wave
This simple model allows one to estimate spin densities at each carbon center and provides insight into which centers should be most amenable to electrophilic or nucleophilic attack. For example,
radical attack at the \(C_5\) carbon of the nine-atom nonatetraene system described earlier would be more facile for the ground state \(\psi\) than for either \(\psi^*\) or \(\psi'^*\). In the
former, the unpaired spin density resides in \(\psi_5\) (which varies as \(\sin(5\pi x/8R_{CC}\)) so is non-zero at \(x = L/2\)), which has non-zero amplitude at the \(C_5\) site \(x= L/2 = 4R_{CC}
\). In \(\psi^*\) and \(\psi'*\), the unpaired density is in \(\psi_4\) and \(\psi_6\), respectively, both of which have zero density at \(C_5\) (because sin(npx/8RCC) vanishes for \(n = 4\) or \(6\)
at \(x = 4R_{CC}\)). Plots of the wave functions for \(n\) ranging from 1 to 7 are shown in another format in Figure 2.4 where the nodal pattern is emphasized.
Figure 2.4. The nodal pattern for a chain containing seven atoms
I hope that by now the student is not tempted to ask how the electron gets from one region of high amplitude, through a node, to another high-amplitude region. Remember, such questions are cast in
classical Newtonian language and are not appropriate when addressing the wave-like properties of quantum mechanics.
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Approximate Inclusion-Exclusion
The Inclusion-Exclusion formula expresses the size of a union of a family of sets in terms of the sizes of intersections of all subfamilies. This paper considers approximating the size of the union
when intersection sizes are known for only some of the subfamilies, or when these quantities are given to within some error, or both. In particular, we consider the case when all k-wise intersections
are given for every k≤K. It turns out that the answer changes in a significant way around K=√n: if K≤O(√n) then any approximation may err by a factor of θ(n/K^2), while if K≥ Ω(√n) it is shown how to
approximate the size of the union to within a multiplicative factor of {Mathematical expression}. When the sizes of all intersections are only given approximately, good bounds are derived on how well
the size of the union may be approximated. Several applications for Boolean function are mentioned in conclusion.
• AMS subject classification (1980): 05A20
Dive into the research topics of 'Approximate Inclusion-Exclusion'. Together they form a unique fingerprint.
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Time Totals
You may develop a fear of heights after watching this film! Mission Impossible: Ghost Protocol lasts for two hours and thirteen minutes. How many minutes is this?
I only have a couple of hours to watch Journey to the Mysterious Island. This film lasts for 1h 34mins. How many minutes less than 2 hours is that?
I started to watch The Avengers at 6:50pm and the film finished at 9:13pm. How many minutes did it last for?
Hairspray is 1 hour and 32 minutes worth of singing and energetic dancing. I was exhausted after watching only half of the film! How many minutes does the first half of this film last?
Do you have 3 hours and 21 minutes to watch The Lord of The Rings? I watched over three days, watching a third of the film each day. How many minutes is one third of 3h 21mins?
The running time for this film is 131 minutes. How many minutes less than three hours is that?
In this film elderly prison inmate Brooks Hatlen is freed on parole after 50 years inside. If you went to prison for 50 years today, in what year would you be released?
Mary Poppins is quite an old film but still very popular. It introduced the word Supercalifragilisticexpialidocious to the world. It was made in 1964. How many years ago was that?
Batman Begins opened in the US on 15th June 2005. It is estimated that by 5th July 2005 it had made $120 million at the box office. How many days did it take to earn $120 million?
The Spider-Man character first appeared in a comic book (Amazing Fantasy #15) in August 1962. I bought the Spider-Man 2 Blu-Ray disk in August 2023. How many months were there between these two
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Checking the asumption of independence in binomial trials using posterior predictive checking | Ladislas Nalborczyk
Checking the asumption of independence in binomial trials using posterior predictive checking
Posterior predictive checking
What is a posterior predictive check ? According to Gelman et al. (2013, page 151), “Bayesian predictive checking generalizes classical hypothesis testing by averaging over the posterior distibution
of the unknown parameters vector \(\theta\) rather than fixing it at some estimate \(\hat{\theta}\)”.
To explore this idea in more details, we are going to extend an example presented in Gelman et al. (2013, page 147) to a case study I have already discussed in two previous blogposts (here and here).
Let’s say I am recruiting participants for a psychology study that is lasting for approximately half an hour. If everything goes smoothly, I can manage to recruit 2 participants per hour, and doing
it between 9am and 6pm (having the first participant at 9am, the second one at 9.30am and the last one at 5.30pm) for a whole week (from Monday to Friday) would give me 90 potential participants.
Beta-Binomial model
We know that some participants won’t show up to the time slot they registered for. I am interested in knowing the mean probability of presence, that we will call \(\theta\).
This sequence of binary outcomes (presence vs. absence) \(y_{1}, \dots, y_{n}\) is modelled as a serie of independent trials with common probability of success (presence) \(\theta\), which is
attributed a conjugate Beta prior, with shape parameters \(\alpha\) and \(\beta\) (encoded in the second line of our model).
\[ \begin{aligned} y &\sim \mathrm{Binomial}(n, \theta) \\ \theta &\sim \mathrm{Beta}(\alpha, \beta) \\ \end{aligned} \]
We could choose to give \(\theta\) a uniform prior between 0 and 1 (to express our total ignorance about its value), but based on previous experiments I carried out, I know that participants tend to
be present with a probability around \(\frac{1}{2}\). Thus, we will choose a probability distribution that represents this prior knowledge (here a \(\mathrm{Beta}(2,2)\), see the first figure for an
# Checking the assumption of independence in binomial trials
# Example inspired from Gelman et al. (2013, page 147)
y <- # getting the data
0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0,
0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1,
1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0,
1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1,
0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0
The problem
Our model is assuming independent trials, i.e., it’s assuming that the presence of a participant is independent of the presence of another participant, which is akin to say that the model is assuming
no autocorrelation in the serie. Autocorrelation would be evidence that the model is flawed.
One way to estimate the degree of autocorrelation in the serie is to simply count the number of switches between presence and absence (i.e., between zeros and ones). An abnormally low number of
switches (for a particular \(n\) and \(\theta\)) would be evidence that some autocorrelation is present. Thus, the number of switches becomes a test quantity \(T(y)\), which describes the degree of
autocorrelation in the serie, and a way of testing the assumptions of our model.
# function to determine the number of switches in a numerical vector
nb_switches <- function(x) as.numeric(sum(diff(x) != 0) )
# determining the number of switches Ty in observed data y
(Ty <- nb_switches(y) )
## [1] 28
We observed 28 switches in our data. To know whether this number is surprising, given our number of observation and the mean probability of presence, we will use use posterior predictive checking.
But first, we need to compute the posterior distribution \(p(\theta | y)\).
Computing the posterior distribution
We know that the posterior density under this model is given by:
\[ p(\theta | y) \sim \mathrm{Beta}(\alpha + y, \beta + n - y) \] where \(y\) is the number of successes and \(n\) is the total number of observations. In our case, the posterior distribution of \(\
theta\) given \(y\) is then \(\mathrm{Beta}(2 + 55, 2 + 90 - 55) = \mathrm{Beta}(57, 37)\), which is plotted below.
# computing the posterior
n <- length(y) # number of trials
z <- sum(y) # number of 1s
a <- b <- 2 # parameters of the beta prior
grid <- seq(from = 0, to = 1, by = 0.01) # defines grid
# analytic derivation of the posterior
prior <- dbeta(grid, a, b)
posterior <- dbeta(grid, z + a, n - z + b)
data.frame(theta = grid, prior = prior, posterior = posterior) %>%
gather(type, value, prior:posterior) %>%
ggplot(aes(x = theta, y = value, colour = type, fill = type) ) +
geom_area(alpha = 0.8, position = "identity", size = 1) +
theme_bw(base_size = 12) +
scale_fill_grey() +
scale_colour_grey() +
The mean of the posterior distribution is given by \(\dfrac{\alpha + y}{\alpha + \beta + n}\), and is equal to (a + z) / (a + b + n) = 0.606, which can be interpreted as the mean probability of
This example allows us to define what conjugacy is. Formally, if \(\mathcal{F}\) is a class of sampling distributions \(p(y|\theta)\), and \(\mathcal{P}\) is a class of prior distributions for \(\
theta\), then the class \(\mathcal{P}\) is conjugate fo \(\mathcal{F}\) if
\[p(\theta|y) \in \mathcal{P} \text{ for all } p(\cdot | \theta) \in \mathcal{F} \text{ and } p(\cdot) \in \mathcal{P}\]
(Gelman et al., 2013, page 35). In other words, a prior is called a conjugate prior if, when converted to a posterior by being multiplied by the likelihood, it keeps the same form. In our case, the
Beta prior is a conjugate prior for the Binomial likelihood, because the posterior is a Beta distribution as well.
Posterior predictive checking
How posterior predictive checking can help us to assess whether the assumption of indendence is respected in our observed data \(y\) ? Well, our model is actually assuming independence, so we could
ask our model to generate new observations, or replications, called \(y^{rep}\), to see whether these replications differ from the observed data. If they do, it would mean that the observed data are
not well described by a model that is assuming independence.
This is done in two steps. First, we generate possible values of \(\theta\) from its posterior distribution (i.e., from a \(\mathrm{Beta}(57, 37)\) distribution). Then, for each of these \(\theta\)
values, we generate a new set of observations \(y^{rep}\) from a Binomial distribution.
# posterior predictive checks
nsims <- 1e4 # number of replicated samples
# generating nsims theta values from posterior
thetas <- rbeta(nsims, a + z, b + n - z)
# generating nsims new datasets (Yrep)
Yrep <-
# for each theta
# generating samples
function(i) sample(
c(0, 1),
# of the same length as y
replace = TRUE,
# with prob of presence equals to theta
# and prob of absence equals to 1 - theta
prob = c(thetas[i], 1 - thetas[i])
Then, we can compute the number of switches (our test quantity) in each replicated sample, to check whether the number of switches computed on datasets generated under the assumption of independence
differ from the number of switches computed on the observed dataset \(y\). We call the test quantities computed on replicated samples \(T(y^{rep})\).
# for each new Yrep sample, computing the number of switches Trep, and
# comparing it to observed number of switches Ty
Trep <- apply(Yrep, 2, nb_switches)
Trep %>%
compVal = Ty, breaks = 20,
col = "#E6E6E6", xlab = expression(T(y^rep) ) )
This histogram reveals that the mean number of switches accross the nsims replications is about 42.04, and the green vertical dotted line represents the position of \(T(y)\) in the distribution of \
(T(y^{rep})\) values.
To know whether the observed number of switches is surprising given the assumptions of our model (represented by its predictions), we can count the number of replications that lead to a greater
number of switches than the number of switches \(T(y)\) in the observed data.
sum(Trep > Ty)
## [1] 9929
Or we can compute a Bayesian p-value as (Gelman et al., 2013, page 146):
\[p_{B} = \text{Pr}(T(y^{rep}, \theta) \geq T(y, \theta) | y)\]
1 - sum(Trep > Ty) / nsims # equivalent to sum(Trep <= Ty) / nsims
## [1] 0.0071
Which gives the probability of observing this number of switches under our model. What does it mean ? Does it mean that our model is wrong ? Well, not exactly. Models are neither right or wrong (see
Crane & Martin, 2018). But our model does not seem to capture the full story, it does not seem to give a good representation of the process that generated our data (which is arguably one of the
characteristics that contribute to the soundness of a model).
More precisely, it misses the point that the probabilities of successive participants being present are not independent. This, in our case, seems to be due to temporal fluctuations of this
probability throughout the day. For instance, the probability of a participant being present seems to be the lowest early in the morning or late in the afternoon, as well as between 12am and 2pm.
This temporal dependency could be better taken into account by using gaussian process regression models, that generalise the varying effect strategy of multilevel models to continuous variables. In
other words, it would allow to take into account that participants coming to the lab at similar hours (e.g., 9am and 9.30am) are more similar (in their probability of being present) than participants
coming at very different hours (e.g., 9am and 3pm).
In this post we aimed to introduce the idea of posterior predictive checking by recycling an elegant and simple example from Gelman et al. (2013). It should be noted however that this kind of check
can be done for any test quantity of interest (e.g., the mean or the max of a distribution, or its dispersion).
As put by Gelman et al. (2013, page 148), “because a probability model can fail to reflect the process that generated the data in any number of ways, posterior predictive p-values can be computed for
a variety of test quantities in order to evaluate more than one possible model failure”.
So come on, let’s make p-values great again, they are not doomed to be used only as a point-null hypothesis test.
Crane, H., & Martin, R. (2018, January 10). Is statistics meeting the needs of science?. Retrieved from psyarxiv.com/q2s5m
Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., & Rubin, D. B. (2013). Bayesian Data Analysis, Third Edition. CRC Press.
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proportional to sides PQ and PR and median PM of another triang... | Filo
Question asked by Filo student
proportional to sides and and median of another triangle . Fig. Show that . 15. A vertical pole of length casts a shadow long on the ground and at the same time 3. tower casts a shadow long. Find the
height of the tower.
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proportional to sides and and median of another triangle . Fig. Show that . 15. A vertical pole of length casts a shadow long on the ground and at the same time 3. tower casts a
Question Text shadow long. Find the height of the tower.
Updated On Dec 22, 2022
Topic All topics
Subject Mathematics
Class Class 11
Answer Type Video solution: 1
Upvotes 136
Avg. Video 5 min
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Flashcards - AP Chemistry Flash Cards - States of Matter: Gases
1. What is the pressure exerted by 5 moles of gas in a 40 L container at 300 K?
Using the ideal gas equation, you can calculate the pressure exerted by the gas is about 3.1 atm.
2. How many atoms of a substance are in 1 mole of that substance?
In 1 mole of a given substance, there are 6.02 x 10^23 atoms.
3. On the periodic table, the masses given for the elemetns are masses for what quantity of the element?
The masses for the elements are the masses for 1 mole o the element.
4. What is the ideal gas equation?
PV = nRT
5. What is the value of the ideal gas constant, R?
□ The value of R depends on the units. The two most common values are:
□ 1. 0.0821 (L atm / mol K)
□ 2. 8.314 (J / mol K)
6. Suppose that a gas is kept at 0.3 atm in a container with a volume of 5 L. If this gas is transferred to a container with a volume of 3 L at a constant temperature, what will be the new pressure
exerted by the gas?
Using Boyle's law, you can determine that the new pressure is 0.5 atm.
7. At constant pressure, a gas that occupies 4 L is heated from 200 K to 300 K. How much volume will the gas now occupy?
Using Charles' law, you can determine that the new volume is 6 L.
8. A gas at 300 K exerts a pressure of 2 atm within a 30 L container. Approximately how many moles of gas are there?
Using the ideal gas law, you can calculate that the amount of gas present is equal to about 2.5 moles.
9. A container holds 3 moles of hydrogen gas (H[2]), 1 mole of oxygen gas (O[2]), and 4 moles of nitrogen gas (N[2]). What is the mole fraction of the hydrogen gas?
The mole fraction of hydrogen is 3/8.
10. A container holds 3 moles of hydrogen gas (H[2]), 1 mole of oxygen gas (O[2]), and 4 moles of nitrogen gas (N[2]). If the total pressure of the system is 4 atm, what is the partial pressure of
the hydrogen gas?
Using Dalton's law of partial pressure, you can calculate that the partial pressure of hydrogen is equal to 1.5 atm.
11. A container holds neon gas, hydrogen gas, and oxygen gas. If the partial pressure of the neon is 0.75 atm, the partial pressure of the hydrogen gas is 1.25 atm, and the total pressure of the
system is 4 am, what is the partial pressure of the H[2]?
Using Dalton's law of partial pressure, you can calculate that the partial pressure of H2 is 1.25 atm.
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Strictly convex vs. convex and well-behaved preferences in economics
This post is going to be a bit more technical than average and will probably be aimed towards the upper division microeconomics or perhaps even the graduate level students. When we go a letter more
in depth studying consumer theory we learned about well-behaved preferences and the associated shapes that the indifference curves take on. Below you can see a graph with three different indifference
curves where 2 are straight lines and one is bowed in. The curve that is bowed in is strictly convex, and all three of them are convex.
Lines A, B, and C all represent indifference curves, while points D and E represent points where the indifference curves touch or intersect (for discussions sake, point D is the point of tangency
between lines C and A).
Now, what does it mean to be convex or strictly convex? The difference is subtle but important. In order for a line to be convex (or express convexity) there has to be a slope to the line. For those
that have taken calculus, a strictly convex line has to have a second derivative that is greater than zero. Graphically, this means that a straight line cannot be strictly convex, but is possible to
still be convex.
Why do well-behaved preferences have a convex shape? It's because people prefer well-balanced distributions of goods and services opposed to extremes. For this to occur, the
marginal rate of substitution (MRS)
needs to change depending on where we are on the indifference curve. Remember that the MRS is the slope of the indifference curve, and if we have a straight line, it won't change.
The slope of lines B and C imply that there is a constant trade off or MRS between goods X and Y no matter the amount of X and Y we consume. However, for the strictly convex curve A, we see that a
lower total amount of goods are needed if both goods X and Y are consumed (as opposed to a very high number of X and a low number of Y).
Strict convexity isn't needed to have an indifference curve, but without it, we are assuming that the two goods are perfect substitutes, which isn't likely.
Additionally, tangency can only be achieved when preferences are well-behaved/strictly convex. This is because of the
linear nature of a budget constraint
. It is only possible for a linear indifference curve to touch a linear budget constraint at one point, and
this results in only one of the goods being consumed.
A good example of a tangent point is point D above, at the point curves A and C have the same slope (marginal rate of substitution = marginal rate of transformation). At point E, the curves cross,
but they do not share the same slope and therefor it is not tangent or optimal. The person could be better off consumer higher quantities of X (and lower quantities of Y) improving their overall
utility while staying within their budget constraint.
Since tangency only occurs when an indifference curve is strictly convex, it is a sufficient condition for convex preferences and an interior optimum.
TLDR: Well-behaved preferences (avoiding extremes) is exhibited by convexity.
Indifference curves can be either convex or strictly convex, but interior solutions generally only happen when they are strictly convex.
Presence of a tangent point (between a budget constraint and indifference curve) is a sufficient condition for strict convexity of indifference curves.
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FLOOR(3M) FLOOR(3M)
floor, floorl, ffloor, floorf, ceil, ceill, fceil, ceilf, copysign,
copysignl, drem, dreml, fmod, fmodl, fmodf, fabsl, fabs, fabsf,
remainder, rint, rintl, trunc, truncl, ftrunc, truncf - floor, ceiling,
remainder, absolute value, nearest integer, and truncation functions
#include <math.h>
double floor (double x);
long double floorl (long double x);
float ffloor (float x);
float floorf (float x);
double ceil (double x);
long double ceill (long double x);
float fceil (float x);
float ceilf (float x);
double copysign (double x, double y);
long double copysignl \
(long double x, long double y);
double drem (double x, double y);
long double dreml \
(long double x, long double y);
double remainder (double x, double y);
double trunc (double x);
long double truncl (long double x);
float ftrunc (float x);
float truncf (float x);
double fmod (double x, double y);
long double fmodl \
(long double x, long double y);
float fmodf (float x, float y);
double fabs (double x);
long double fabsl (long double x);
float fabsf (float x);
double rint (double x);
long double rintl (long double x);
The fmod, fabs, and trunc functions listed above, as well as the long
double and single-precision versions of the remaining functions, are only
available in the standard math library, -lm, and in -lmx.
Page 1
FLOOR(3M) FLOOR(3M)
The floor functions return the largest integer not greater than x. The
argument x is double for floor, long double for floorl, and float for
ffloor and its ANSI-named equivalent floorf.
copysign(x,y) returns the number with the magnitude of x and the sign of
y. copysignl is the long double counterpart of copysign.
drem(x,y) returns the remainder r := x - n*y where n is the integer
nearest the exact value of x/y; moreover if |n-x/y|=1/2 then n is even.
Consequently the remainder is computed exactly and |r| < |y|/2. But
drem(x,0) is exceptional; see below under DIAGNOSTICS. remainder is an
alternate entry point for drem. dreml is the long double counterpart of
The ceil functions return the smallest integer not less than x. The
argument x is double for ceil, long double for ceill, and float for fceil
and its ANSI-named equivalent ceilf.
The trunc functions return the integer (represented as a floating-point
number) of x with the fractional bits truncated. The argument x is
double for trunc, long double for truncl, and float for ftrunc.
fabs returns the absolute value of the double x, |x|. It also has
counterparts of type long double and float, namely fabsl, and fabsf,
rint returns the integer (represented as a double precision number)
nearest its double argument x in the direction of the prevailing rounding
mode. rintl is the long double counterpart of rint. rint has no
counterpart which accepts an argument of type float.
fmod returns the floating-point remainder of the division of its double
arguments x by y. It returns a number f with the same sign as x, such
that x = iy + f for some integer i, and |f| < |y|. Hence both the
yield 0.5, while the two invocations
yield -0.5.
fmodl is the counterpart of fmod which accepts and returns values of type
long double and fmodf is the counterpart of fmod which accepts and
returns values of type float.
In the diagnostics below, functions in the standard math library libm.a,
are referred to as -lm versions, those in math library libmx.a are
referred to as -lmx versions, and those in the the BSD math library
libm43.a are referred to as -lm43 versions. The -lm and -lmx versions
always return the default Quiet NaN and set errno to EDOM when a NaN is
used as an argument. A NaN argument usually causes the -lm43 versions to
Page 2
FLOOR(3M) FLOOR(3M)
return the same argument. The -lm43 versions never set errno. The value
of HUGE_VAL is IEEE Infinity.
If y (and, possibly, x) are zero, or if x is +/-HUGE_VAL, the fmod
functions return a quiet NaN, and set errno to EDOM.
IEEE 754 defines drem(x,0) and drem(infinity,y) to be invalid operations
that produce a NaN.
A version of the double-precision fabs function exists in the C library
as well. The C library version may not behave correctly when the input
is NaN.
Long double operations on this system are only supported in round to
nearest rounding mode (the default). The system must be in round to
nearest rounding mode when calling any of the long double functions, or
incorrect answers will result.
Users concerned with portability to other computer systems should note
that the long double and float versions of these functions are optional
according to the ANSI C Programming Language Specification ISO/IEC 9899 :
1990 (E).
Long double functions have been renamed to be compliant with the ANSI-C
standard, however to be backward compatible, they may still be called
with the double precision function name prefixed with a q. (Exceptions:
functions fabsl and fmodl may be called with names qabs and qmod, resp.)
In the default rounding mode, round to nearest, rint(x) is the integer
nearest x with the additional stipulation that if |rint(x)-x|=1/2 then
rint(x) is even. Other rounding modes can make rint act like floor, or
like ceil, or round towards zero.
Another way to obtain an integer near x is to declare (in C)
double x; int k; k = x;
The C compilers round x towards zero to get the integer k. Also note
that, if x is larger than k can accommodate, the value of k and the
presence or absence of an integer overflow are hard to detect.
IEEE 754 requires copysign(x,Nan) = _x. In this implementation of
copysign, the sign of NaN is ignored. Thus copysign(x,_NaN) = +x, and
copysign(_NaN,x) = +NaN.
abs(3C), math(3M), matherr(3M)
Page 3
FLOOR(3I) Last changed: 1-6-98
FLOOR - Returns the greatest integer less than or equal to its
FLOOR ([A=]a)
UNICOS, UNICOS/mk, and IRIX systems
Fortran 90
The FLOOR intrinsic function returns the greatest integer less than or
equal to its argument. It accepts the following argument:
a Must be of type real
FLOOR is an elemental function. The name of this intrinsic cannot be
passed as an argument.
On UNICOS systems, both execution speed and the number of bits used in
mathematical operations are affected when compiling with
f90 -O fastint, which is the default setting. For more information on
this, see the CF90 Commands and Directives Reference Manual,
publication SR-3901.
The result is a default integer. The result has value equal to the
greatest integer less than or equal to a. The result is undefined if
the target machine cannot represent this value in the default integer
FLOOR(3.7) has the value 3. FLOOR(-3.7) has the value -4.
CF90 Commands and Directives Reference Manual, publication SR-3901
Intrinsic Procedures Reference Manual, publication SR-2138, for the
printed version of this man page.
FLOOR(3I) Last changed: 1-6-98
FLOOR - Returns the greatest integer less than or equal to its
FLOOR ([A=]a)
UNICOS, UNICOS/mk, and IRIX systems
Fortran 90
The FLOOR intrinsic function returns the greatest integer less than or
equal to its argument. It accepts the following argument:
a Must be of type real
FLOOR is an elemental function. The name of this intrinsic cannot be
passed as an argument.
On UNICOS systems, both execution speed and the number of bits used in
mathematical operations are affected when compiling with
f90 -O fastint, which is the default setting. For more information on
this, see the CF90 Commands and Directives Reference Manual,
publication SR-3901.
The result is a default integer. The result has value equal to the
greatest integer less than or equal to a. The result is undefined if
the target machine cannot represent this value in the default integer
FLOOR(3.7) has the value 3. FLOOR(-3.7) has the value -4.
CF90 Commands and Directives Reference Manual, publication SR-3901
Intrinsic Procedures Reference Manual, publication SR-2138, for the
printed version of this man page.
[ Back ]
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Gnucap offers a comprehensive set of probes. This section lists system probes, node probes, and common element probes.
Probes always have the syntax name_of_probe(device_or_node). Example: vdd(m1).
You can access components in subcircuits by connecting the names with dots. For example: Xone.X67.R56 is R56 in X67 in Xone. Some built-in elements, including diodes, transistors, and mosfets,
contain subcircuits with internal elements. M12.Cgd is the gate to drain capacitor of mosfet M12.
For system probes, use the device name “0”.
iter The number of iterations needed for convergence for this printed step including any hidden steps.
Prints a code indicating whether model evaluation can be bypassed.
bypass 0=bypass turned off by user.
1=bypass enabled by user, but not active now.
11=bypass is possible.
A number indicating why the simulator chose this time to simulate at.
1=The user requested it. One of the steps in a sweep.
2=A scheduled discrete event. An element required a solution at this time.
3=The effect of the “skip” parameter.
4=The iteration count exceeded ITL4 so the last step was rejected and is being redone at a smaller time step.
5=The iteration count exceeded ITL3 so the increase in time step is limited.
control 6=Determined by local truncation error or some other device dependent approximation in hopes of controlling accuracy.
7=Determined by a movable analog event. An element required a solution at this time.
8=The step size was limited due to iteration count.
9=This is an initial step. The size was arbitrarily set to 1/100 of the user step size.
10+x=The previous step was rejected.
20+x=A zero time step was replaced by mrt.
30+x=The required step size less than mrt, so it was replaced by mrt.
damp Newton damping factor.
generator The output of the “signal generator”. In a transient analysis, it shows the output of the signal generator, as set up by the generator command. In a DC analysis, it shows the DC input
voltage (not the power supply). In an OP analysis, it shows the DC input, normally zero.
hidden The number of hidden steps.
temp The simulation temperature in degrees Celsius.
time The current time in a transient analysis. In AC analysis it shows the time at which the bias point was set, 0 if it was set in a DC or OP analysis, or -1 if it is the bias was not set
(power off).
All modes.
v Voltage.
z Impedance looking into the node.
mdy Matrix diagonal “y”.
mdz Matrix diagonal “z”. (1/mdy)
Transient, DC, OP, only.
A numeric interpretation of the logic value at the node. The value is displayed encoded in a number of the form a.bc where
a is the logic state:
0 = logic 0
1 = rising
2 = falling
3 = logic 1
4 = unknown
b is an indication of the quality of the digital signal.
0 is a fully valid logic signal.
Nonzero indicates it does not meet the criteria for logic simulation.
c indicates how the node was calculated:
0 indicates logic simulation.
1 indicates analog simulation of a logic device.
2 indicates analog simulation of analog devices.
lastchange The most recent time at which the logic state changed.
finaltime The scheduled time a pending logic state change will occur.
diter Iteration number for last digital update.
aiter Iteration number for last analog update.
AC only.
In addition to those listed here, you can add a suffix (M, P, R, I and db) for magnitude, phase, real part, imaginary part, and decibels, to any valid probe.
Each element type has several parameters that can be probed. In general, the form is Parameter(element). Wild cards are allowed in element names to allow probing the same parameter of a group of
For components in a subcircuit, the names are connected with dots. For example X13.R12 is R12 in the subcircuit X13.
Most two node elements (capacitors, inductors, resistors, sources) and four terminal elements (controlled sources) have at least the following parameters available. Others are available for some
Some of these probes do not work for all devices, or all analysis. It will print ”??” as the value when it doesn't work.
All devices:
v[n] Voltage at a port. v2(m2) is the voltage at the second port.
errortime Suggestion of next time point based on truncation or interpolation error.
eventtime Suggestion of next time point based on movable events.
timefuture Suggestion of next time point, the sooner of errortime and eventtime.
Most elements,
devices that do not have an internal subcircuit,
devices that can be defined simply by y=f(x).
v Branch voltage for two terminal devices, output voltage for four terminal devices. The first node in the net list is assumed positive.
vin Input voltage. The voltage across the “input” terminals. For two terminal elements, input and output voltages are the same.
i Branch current. It flows into the first node in the net list, out of the second.
p Power. Positive power indicates dissipation. Negative power indicates that the part is supplying power. Its value is the same as (PD - PS). In AC analysis, it is the real part only.
pd Power dissipated. The power dissipated in the part. It is always positive and does not include power sourced.
ps Branch power sourced. The power sourced by the part. It is always positive and does not consider its own dissipation.
input The “input” of the device. It is the current through a resistor or inductor, the voltage across a capacitor or admittance, etc. It is the value used to evaluate nonlinearities.
f The result of evaluating the function related to the part. It is the voltage across a resistor, the charge stored in a capacitor, the flux in an inductor, etc.
df The derivative of f with respect to input. Usually this is also the effective value of the part, in its units. If the part is ordinary, it will just show its value, but if it is time
ev variant or nonlinear, it shows what it is now.
nv Nominal value. In most cases, this is just the value which is constant, but it can vary for internal elements of complex devices.
eiv Equivalent input voltage. The voltage on which the matrix stamp is based.
y Matrix stamp admittance.
istamp Matrix stamp current.
ipassive Passive part of matrix stamp current.
ioffset Offset part of matrix stamp current.
iloss Loss part of device current.
dt Delta time. Time step for this device.
dtr dt required. Recommended dt for next step.
time Time at most recent actual calculation. It is usually the present time.
timeold Time at the previous actual calculation.
z Circuit impedance seen by this device, with this device not counted. Prints a meaningless number in transient analysis.
zraw Circuit impedance looking across this device, including this device. Prints a meaningless number in transient analysis.
AC power probes:
In addition to those listed here, you can add a suffix (M, P, R, I and DB) for magnitude, phase, real part, imaginary part, and decibels, to any valid probe. Negative phase is capacitive. Positive
phase is inductive.
p Real power. Watts.
pi Reactive (imaginary) power, VAR.
pm Volt amps. Complex power.
pp Power phase. Angle between voltage and current.
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Understanding Fractions - Elementary Education: Grade 3 Curriculum in British Columbia, Canada
Fractions are a fundamental concept in mathematics, representing parts of a whole. They are particularly useful in situations where whole numbers cannot accurately describe quantities. Understanding
fractions is crucial for students as they form the basis for more advanced mathematical concepts. In this lesson, we will explore the basics of fractions, including their components, types, and
What is a Fraction?
A fraction represents a part of a whole or, more generally, any number of equal parts. It consists of two parts: a numerator and a denominator, written as $\frac{numerator}{denominator}$. The
numerator represents how many parts of the whole are being considered, while the denominator indicates the total number of equal parts the whole is divided into.
Types of Fractions
1. Proper Fractions: The numerator is less than the denominator (e.g., $\frac{3}{4}$). It represents a quantity less than one.
2. Improper Fractions: The numerator is greater than or equal to the denominator (e.g., $\frac{5}{4}$). It represents a quantity greater than or equal to one.
3. Mixed Numbers: A combination of a whole number and a proper fraction (e.g., $1\frac{1}{4}$). It is another way to represent quantities greater than one.
4. Equivalent Fractions: Different fractions that represent the same quantity (e.g., $\frac{1}{2}$ and $\frac{2}{4}$).
Simplifying Fractions
Simplifying (or reducing) fractions means expressing them in the simplest form where the numerator and denominator are as small as possible. This is done by dividing both the numerator and the
denominator by their greatest common divisor (GCD). For example, to simplify $\frac{8}{12}$, we find the GCD of 8 and 12, which is 4, and divide both by 4 to get $\frac{2}{3}$.
Equivalent Fractions
Fractions are equivalent if they represent the same part of a whole, even if they look different. To find an equivalent fraction, you can multiply or divide both the numerator and the denominator by
the same number. For instance, multiplying the numerator and denominator of $\frac{3}{4}$ by 2 gives $\frac{6}{8}$, an equivalent fraction.
Adding and Subtracting Fractions
To add or subtract fractions, they must have the same denominator (common denominator). If they do not, you must first find equivalent fractions that do. For example, to add $\frac{1}{4}$ and $\frac
{1}{3}$, you can convert them to $\frac{3}{12}$ and $\frac{4}{12}$, respectively, and then add them to get $\frac{7}{12}$.
Multiplying Fractions
Multiplying fractions is straightforward: multiply the numerators to get the new numerator, and multiply the denominators to get the new denominator. For example, $\frac{3}{4} \times \frac{2}{5} = \
frac{6}{20}$, which can be simplified to $\frac{3}{10}$.
Dividing Fractions
To divide by a fraction, you multiply by its reciprocal. The reciprocal of a fraction $\frac{a}{b}$ is $\frac{b}{a}$. For example, to divide $\frac{3}{4}$ by $\frac{2}{5}$, you multiply $\frac{3}{4}$
by the reciprocal of $\frac{2}{5}$, which is $\frac{5}{2}$, resulting in $\frac{15}{8}$ or $1\frac{7}{8}$ as a mixed number.
Practical Applications
Understanding fractions is not just a mathematical skill but also a practical one. Fractions are used in everyday life, from cooking and dividing portions to managing finances and measuring
distances. Being comfortable with fractions allows students to tackle real-world problems more effectively.
Fractions are a key concept in mathematics that students must understand early on. They form the foundation for many topics in mathematics and are used in various real-life situations. By mastering
the basics of fractions, including their types, simplification, and operations, students can build a strong mathematical foundation that will benefit them throughout their education and beyond.
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[Solved] Given a list of noises generated by all pairs of animals, de
Given a list of noises generated by all pairs of animals, determine the least sum of noise that can be generated.
The animals have been making too much noise, and guests are complaining. Some animals antagonize nearby animals and cause them to emit screams of rage or cries of annoyance. You have experimented
with different animals next to each other, and you feel that you now have enough information to move forward with your plans.
The animals will be in a line of exhibits. Each animal will be in its own exhibit, and there are an equal number of animals as exhibits. You have found that animals only antagonize the animals in
exhibits within 2 exhibits of them. Animals that are antagonized will generate a specific level of noise. The total noise of the park is the sum of all the levels of noises generated.
To alleviate some stress from the animals and pain from the guests ear’s you will attempt to find some animal ordering that is a few decibels lower.
Given a list of noises generated by all pairs of animals, determine the least sum of noise that can be generated.
Input will begin with a line containing 1 integer, n (1 ≤ n ≤ 11), representing the number of animal exhibits. The following n lines will each contain n space separated non-negative integers,
representing the noises generated by animal pairs. The i-th value on the j-th line represents the noise level generated by animal j when antagonized by animal i. Each of the given individual noise
levels generated will be at most 1,000,000. The i-th value of the i-th line will always be 0. Animals don’t generate noise from themselves.
Expert's Answer
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How Navamsa chart is constructed?
Each Rasi of 30^0 is divided into 9 equal parts. So each part will be 3^0 20’. For Fiery Rasis (Mesha, Simha, Dhanus) Navamsa starts from Mesha. For earthy signs (Vrishabha, Kanya, Makara) Navamsa
starts from Makara. For Airy signs (Mithuna, Thula, Kumbha) the beginning is from Thula and for the watery signs (Karka, Vrischika and Meena) the order starts from Karka. An Example will clarify this
Suppose a planet say Mercury is in 15^0 25’ at Kanya. Kanya is earthy sign. Hence starting point for Navamsa will be Makara. 15^0 25’ is in the 5^th part of the rasi by dividing it with 9. Hence
counting 5 from Makara we come to Vrishabha. In the Navamsa chart Mercury will be placed in Vrishabha.
Suppose Mars is in Karka 24^0 10’ then it falls in the 8^th part of the rasi. Karka being a watery sign counting is to begin from Karka itself. 8^th from Karka is Kumbha where Mars will be placed in
Navamsa chart.
This is the way Navamsa chart is to be constructed.
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What Are An Arcsecond And Arcminute? - GEGCalculators
What Are An Arcsecond And Arcminute?
This article aims to provide a comprehensive understanding of the angular measurement units known as arcseconds and arcminutes in the field of astronomy. These units play a crucial role in celestial
calculations, enabling astronomers to precisely measure small angles and distances within the vastness of space.
By exploring their definitions, significance, and practical applications, this article will shed light on the fundamental concepts underlying these measurements and their relevance to astronomical
What Are An Arcsecond And Arcminute?
An arcsecond (arcsec) is a unit of angular measurement often used in astronomy and geometry. It’s equal to 1/3,600 of a degree, or approximately 1/1,296,000 of a circle. An arcminute (arcmin) is
larger, equal to 60 arcseconds or 1/60 of a degree. They are used to measure small angles, especially in the context of celestial objects’ apparent sizes or separations.
Measurement Definition Equivalent to
Arcsecond 1/3,600 of a degree Approximately 1/1,296,000 of a circle
Arcminute 60 arcseconds or 1/60 of a degree 1 arcminute = 1/60 degree
Key Takeaways
• Angular measurements in astronomy, expressed in degrees, are crucial for precise observations and calculations.
• Arcseconds and arcminutes are compact units that allow for more precise measurements of stellar positions and apparent sizes.
• The subdivision of a degree into 60 arcminutes achieves higher angular accuracy, particularly important for small or distant celestial objects.
• Arcminutes aid in determining angles between celestial objects, facilitating precise celestial navigation and orientation.
Understanding Angular Measurement in Astronomy
Angular measurement in astronomy is a fundamental concept that involves understanding units such as arcseconds and arcminutes. Accurate angular measurements are crucial for precise astronomical
observations and calculations. The accuracy of angular measurements determines the reliability of various astronomical parameters, including distances, sizes, and motions of celestial objects.
Angular measurements are commonly expressed in degrees, but they can also be converted to smaller units such as arcminutes and arcseconds for more precise calculations.
Converting between degrees and arcseconds involves multiplying the number of degrees by 60 to obtain the equivalent number of arcminutes, and then multiplying the number of arcminutes by another 60
to obtain the equivalent number of arcseconds.
This conversion allows astronomers to express angles with a higher level of precision necessary for their research and observations.
The Definition and Significance of Arcseconds
The unit of measurement used to quantify small angles is widely recognized in scientific research. In the field of astronomy, where measurement precision is crucial for accurate observations,
arcseconds and arcminutes play a significant role.
An arcsecond is defined as 1/3600th of a degree, while an arcminute is equivalent to 1/60th of a degree. These small units are particularly useful when measuring stellar positions, determining
apparent sizes of celestial objects, or calculating angular separations between celestial bodies.
Due to their compactness, they enable astronomers to express precise measurements in a standardized manner. This allows for consistency and comparability across different astronomical observations
and research studies.
The use of arcseconds and arcminutes ensures that the measurements obtained in the field of astronomy maintain high levels of accuracy and reliability.
Exploring the Role of Arcminutes in Celestial Calculations
When performing calculations involving celestial objects, the compact units of measurement that are smaller than degrees prove to be highly valuable. Within this context, arcminutes play a
significant role in enhancing angular accuracy and aiding in celestial navigation.
Here are three key reasons why arcminutes are essential in these calculations:
1) Precision: Arcminutes allow for finer measurements compared to degrees alone. This precision is particularly crucial when dealing with small or distant celestial objects.
2) Angular Accuracy: By subdividing a degree into 60 arcminutes, we can achieve a higher level of angular accuracy. This is especially important when determining the positions and movements of stars,
planets, and other celestial bodies.
3) Celestial Navigation: The use of arcminutes enables accurate determination of angles between celestial objects such as stars or landmarks on Earth’s surface. This aids navigators in determining
their position and plotting courses over long distances.
Practical Applications of Arcseconds and Arcminutes in Astronomy
Astronomical calculations benefit from the inclusion of smaller units of measurement, such as subdivisions of degrees, which allow for increased precision and accuracy in determining celestial
positions and movements. In addition to arcminutes, another important subdivision is the arcsecond.
An arcsecond is equal to 1/60th of an arcminute or 1/3600th of a degree. These small units are crucial in various aspects of astronomy, including calculus methods used for precise calculations and
The use of these subdivisions enables astronomers to accurately track celestial objects’ motions over time and make predictions about their future positions.
Furthermore, optical instruments like telescopes rely on these small angular measurements to precisely locate and observe astronomical phenomena. The following table illustrates the relationship
between degrees, arcminutes, and arcseconds:
Degrees Arcminutes Arcseconds
… … …
This table emphasizes how each unit is subdivided into smaller increments, allowing for finer measurements in astronomical calculations.
GEG Calculators is a comprehensive online platform that offers a wide range of calculators to cater to various needs. With over 300 calculators covering finance, health, science, mathematics, and
more, GEG Calculators provides users with accurate and convenient tools for everyday calculations. The website’s user-friendly interface ensures easy navigation and accessibility, making it suitable
for people from all walks of life. Whether it’s financial planning, health assessments, or educational purposes, GEG Calculators has a calculator to suit every requirement. With its reliable and
up-to-date calculations, GEG Calculators has become a go-to resource for individuals, professionals, and students seeking quick and precise results for their calculations.
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Math, Grade 6, Expressions, Reviewing The Greatest Common Factor
Material Type:
Lesson Plan
Middle School
Media Formats:
Interactive, Text/HTML
Rectangular Grid
Rectangular Grid
Reviewing The Greatest Common Factor
Students use a geometric model to investigate common factors and the greatest common factor of two numbers.
Key Concepts
A geometric model can be used to investigate common factors. When congruent squares fit exactly along the edge of a rectangular grid, the side length of the square is a factor of the side length of
the rectangular grid. The greatest common factor (GCF) is the largest square that fits exactly along both the length and the width of the rectangular grid. For example, given a 6-centimeter ×
8-centimeter rectangular grid, four 2-centimeter squares will fit exactly along the length without any gaps or overlaps. So, 2 is a factor of 8. Three 2-centimeter squares will fit exactly along the
width, so 2 is a factor of 6. Since the 2-centimeter square is the largest square that will fit along both the length and the width exactly, 2 is the greatest common factor of 6 and 8.
Common factors are all of the factors that are shared by two or more numbers.
The greatest common factor is the greatest number that is a factor shared by two or more numbers.
Goals and Learning Objectives
• Use a geometric model to understand greatest common factor.
• Find the greatest common factor of two whole numbers equal to or less than 100.
Which Tiles Will Exactly Cover the Grid?
Lesson Guide
Start the lesson by projecting the 12-unit × 18-unit rectangular grid from the interactive and having partners discuss the two questions.
The purpose of this discussion is to ensure that all students understand what it means to cover the grid with congruent square tiles without any gaps or overlaps. When you choose student responses to
share with the class, pick responses that make this concept clear. The 1-unit squares can cover the grid. Twelve of the square tiles fit along one side of the grid, and 18 of them fit along the other
side. The 5-unit square tiles do not work. There is a 2-unit gap along the 12-unit side of the grid and a 3-unit gap along the 18-unit side.
Which Tiles Will Exactly Cover the Grid?
Suppose you want to cover a 12-unit by 18-unit rectangular grid exactly, without any gaps or overlaps. Can you do this using square tile stamps in the following unit sizes?
• 1-unit square tiles? Explain why or why not.
• 5-unit square tiles? Explain why or why not.
INTERACTIVE: Rectangular Grid
Math Mission
Lesson Guide
Discuss the Math Mission. Students will find common factors and the greatest common factor of two numbers using a geometric model.
Investigate how to find the common factors of two numbers.
Square Tiles on the Grid
Lesson Guide
No product is required for the first problem; it is simply a chance to explore. However, students do use the results of the exploration for the second and third problems.
After a few minutes of exploration, direct students to record their findings on a separate sheet of paper. Have students work on their presentations individually or in pairs.
As students work, look for work that:
• Correctly identifies the 1-unit, 2-unit, 3-unit, and 6-unit squares as exactly covering the 12 × 18 rectangular grid.
• Misidentifies the other unit squares as exactly covering the 12 × 18 rectangular square grid.
It is not necessary to wait until all students have completed this portion of the work. As soon as about half the class is ready, start the Ways of Thinking discussion.
Mathematical Practices
Mathematical Practice 4: Model with mathematics.
Students use a geometric model to identify common factors and the greatest common factor of two numbers. Finding squares that fit along each edge of the rectangular grid is similar to dividing the
edge length of the rectangular grid by the edge length of the square tile.
Mathematical Practice 6: Attend to precision.
Watch for students who attend to precision when aligning square tiles along the length and the width of a rectangular grid and when recording their results in a table. Students who leave gaps between
tiles or have overlaps will not get the correct solution.
Mathematical Practice 8: Look for and express regularity in repeated reasoning.
Identify students who find relationships in their recordings and apply reasoning to determine the common factors and the greatest common factor of two numbers.
Student has an incorrect solution.
• Have you checked your work?
• Show me how you got [answer] as a side length of a square tile for both 12 and 18. Does your answer make sense? What happens if you try the tile along the [width, length]?
• Use your 3-unit square tile. How many tiles fit exactly along the 12-unit side? What is 12 divided by 3? Which operation are you modeling when you are finding squares that fit along the side of
the rectangular grid?
• What does the side length of the square tile that fits exactly along the side of the rectangular grid represent?
Student has a solution.
• What do the side lengths of squares that fit along the 12-unit side represent?
• What do the side lengths of squares that fit along the 18-unit side represent?
• What do the side lengths of the square tiles that exactly cover the rectangular grid represent?
• What does the square tile with the largest side length that will exactly cover the rectangular grid represent?
• How could you find the greatest common factor without drawing rectangular grids?
• Do you think a 1-unit square would fit exactly along the side of any rectangular grid with whole-number side lengths? What can you conclude about the number 1?
• These are the side lengths of square tiles that will exactly cover the 12 × 18 rectangular grid:
1 unit, 2 unit, 3 unit, and 6 unit.
• The square tiles that will exactly cover the 12 × 18 rectangular grid are factors of both 12 and 18.
• The largest square that will exactly cover the 12 × 18 rectangular gird is a 6-unit square.
Work Time
Square Tiles on the Grid
• Explore the interactive by trying square tiles of different sizes.
• Answer these questions:
□ Which square tiles will exactly cover a 12 × 18 rectangular grid?
□ What do you notice about the square tiles that will exactly cover the 12 × 18 rectangular grid?
□ What is the largest square tile that will exactly cover the 12 × 18 rectangular grid?
INTERACTIVE: Rectangular Grid
The numbers 12 and 18 are both even. Do you think 2-unit squares will exactly cover this rectangle?
Prepare a Presentation
Preparing for Ways of Thinking
As students work, look for work that:
• Correctly identifies the 1-unit, 2-unit, 3-unit, and 6-unit squares as exactly covering the 12 × 18 rectangular grid.
• Misidentifies the 4-unit, 12-unit, or 18-unit squares (or others) as exactly covering the 12 × 18 rectangular square grid.
Challenge Problem
• Squares with side lengths of 1, 2, 3, 4, 6, and 12 units will exactly cover a 36 × 12 rectangular grid. The 12-unit square is the largest possible square.
Work Time
Prepare a Presentation
List all the tile sizes you were able to use to exactly cover the 12 × 18 rectangle. Explain what your findings tell you.
Challenge Problem
• Predict what is the largest square you think will cover a 36 x 12 rectangular grid exactly. Then, test your prediction to see if it is correct.
Make Connections
Lesson Guide
During this discussion, help students understand that finding squares that fit along each edge of the rectangular grid is a geometric modeling of dividing the edge lengths of the rectangular grid by
the edge lengths of the square.
• When you are finding squares that fit along an edge of the rectangular grid, what are you doing? Think about the relationship between the edge length of the rectangular grid and the edge length
of the square.
• What are you finding when squares fit exactly along both the width and the length of the rectangular grid?
• Why does a 1-unit square fit exactly along the side of any rectangular grid with whole-number side lengths?
• What does the length of the largest square that fits exactly along both the width and the length of the rectangular grid represent? What can you reason from this model?
Explain that students are using squares to cover the rectangular grid because they are looking for factors that are common to both 12 and 18. The edge length of any square that exactly fits along
both the width and the length of the 12 × 18 rectangular grid is a factor of both 12 and 18.
The 1-unit square will exactly cover any rectangular grid with whole-number dimensions, and the number 1 is a factor of all whole numbers.
The edge length of the largest square that will cover a rectangular grid is the greatest common factor of the length and width of the rectangular grid.
If you ask students how they would find the greatest common factor without sketching rectangular grids, some students might suggest listing the factors of both numbers in a table and then finding the
greatest factor that is common to both. Other students might suggest listing the factors of one number, and then dividing the other number by those factors to see if there is a remainder. The
greatest factor that divides evenly is the greatest common factor.
Performance Task
Ways of Thinking: Make Connections
Takes notes about your classmates’ approaches to finding the tile sizes that will cover a grid exactly.
As your classmates present, ask questions such as:
• How did you decide which squares to use to try to cover the grid?
• Were there any square sizes that you knew right away would not cover the grid?
• What are you finding when squares fit exactly along both the width and the length of the rectangular grid?
• What does the largest square tile that covers the grid represent?
Greatest Common Factor
Ask questions such as the following as students are working:
• What strategy did you use to find the greatest common factor?
• Can you use a model to find the greatest common factor? Show me.
• What are other common factors of the two numbers? How do you know?
• The greatest common factor of 8 and 12 is 4.
• The greatest common factor of 18 and 24 is 6.
• The largest square tile that will cover a 27 × 63 rectangle is a 9-unit square.
• The greatest common factor of 54 and 126 is 18.
Work Time
Greatest Common Factor
Common factors are factors that are shared by two numbers. The greatest common factor of two numbers is the greatest number that is a factor of the two numbers.
• Find the greatest common factor of 8 and 12.
• Find the greatest common factor of 18 and 24.
• Find the largest square tile that will cover a 27 × 63 rectangle.
• What is the greatest common factor of 54 and 126?
• Find numbers that will divide evenly into both numbers. Which is the largest?
• Find all the factors of both numbers. Look for the common factors.
All About Factors
• Have pairs discuss factors, common factors, and greatest common factors.
• As student pairs discuss, listen for students who may still not understand how to use the model to find factors, common factors, and the greatest common factor. Make a note to clarify any
misconceptions during the class discussion.
• Then discuss the Summary as a class. Be sure to highlight these points:
□ Common factors are factors that are shared by two numbers. For example, 2 is a common factor of 4 and 8. The greatest common factor of two numbers is the greatest number that is a factor of
the two numbers. So the greatest common factor of 4 and 8 is 4, not 2.
□ When you can exactly fit congruent squares along the edge of a rectangular grid, the side length of the square is a factor of the side length of the rectangular grid.
□ Common factors are all of the factors that are shared by two or more numbers.
□ The greatest common factor is the greatest factor shared by two or more numbers.
ELL: Be prepared to assist students with explaining the subject matter in problems. Keep in mind that the content and vocabulary may be unfamiliar to some students. Make sure students know and
understand the word congruent and understand how congruent squares can be used to find the greatest common factor of two numbers in a rectangular grid.
SWD: To ensure that all students make the connection between the multiplication table, factors, and multiples, highlight the multiplication table with different colors on the outside (factors) and
inside (multiples).
Formative Assessment
All About Factors
Read and Discuss
• A factor of a number is a whole number that divides a given quantity without a remainder. For example, 5 is a factor of 10 but not a factor of 8.
• Common factors are factors that are shared by two numbers. For example, 2 is a common factor of 4 and 8.
• The greatest common factor of two numbers is the greatest number that is a factor of the two numbers. So, the greatest common factor of 4 and 8 is 4.
• When you can exactly fit congruent squares along the edge of a rectangular grid, the side length of the square is a factor of the side length of the rectangular grid.
Can you:
• Describe the relationship between the side length of the square and the side lengths of the rectangular grid?
• Identify what the largest square that exactly covers the rectangular grid represents?
• Use the terms factor ,common factor , andgreatest common factor appropriately?
Reflect On Your Work
Lesson Guide
Have each student write a brief reflection before the end of class. Review the reflections to find out what students learned about common factors and greatest common factors.
Work Time
Write a reflection about the ideas discussed in class today. Use the sentence starter below if you find it to be helpful.
What I learned about common factors and greatest common factors is …
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Control theory - Large Systems, Dynamics, Mathematics | Britannica
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More advanced and more critical applications of control concern large and complex systems the very existence of which depends on coordinated operation using numerous individual control devices
(usually directed by a computer). The launch of a spaceship, the 24-hour operation of a power plant, oil refinery, or chemical factory, and air traffic control near a large airport are examples. An
essential aspect of these systems is that human participation in the control task, although theoretically possible, would be wholly impractical; it is the feasibility of applying automatic control
that has given birth to these systems.
The advancement of technology (artificial biology) and the deeper understanding of the processes of biology (natural technology) has given reason to hope that the two can be combined; man-made
devices should be substituted for some natural functions. Examples are the artificial heart or kidney, nerve-controlled prosthetics, and control of brain functions by external electrical stimuli.
Although definitely no longer in the science-fiction stage, progress in solving such problems has been slow not only because of the need for highly advanced technology but also because of the lack of
fundamental knowledge about the details of control principles employed in the biological world.
On the most advanced level, the task of control science is the creation of robots. This is a collective term for devices exhibiting animal-like purposeful behaviour under the general command of (but
without direct help from) humans. Highly specialized industrial manufacturing robots are already common, but real breakthroughs will require fundamental scientific advances with regard to problems
related to pattern recognition and thought processes. (See artificial intelligence.)
Principles of control
The scientific formulation of a control problem must be based on two kinds of information: (A) the behaviour of the system must be described in a mathematically precise way; (B) the purpose of
control (criterion) and the environment (disturbances) must be specified, again in a mathematically precise way.
Information of type A means that the effect of any potential control action applied to the system is precisely known under all possible environmental circumstances. The choice of one or a few
appropriate control actions, among the many possibilities that may be available, is then based on information of type B. This choice is called optimization.
The task of control theory is to study the mathematical quantification of these two basic problems and then to deduce applied mathematical methods whereby a concrete answer to optimization can be
obtained. Control theory does not deal directly with physical reality but with mathematical models. Thus, the limitations of the theory depend only on the agreement between available models and the
actual behaviour of the system to be controlled. Similar comments can be made about the mathematical representation of the criteria and disturbances.
Once the appropriate control action has been deduced by mathematical methods from the information mentioned above, the implementation of control becomes a technological task, which is best treated
under the various specialized fields of engineering. The detailed manner in which a chemical plant is controlled may be quite different from that of an automobile factory, but the essential
principles will be the same. Hence further discussion of the solution of the control problem will be limited here to the mathematical level.
To obtain a solution in this sense, it is convenient to describe the system to be controlled, which is called the plant, in terms of its internal dynamical state. By this is meant a list of numbers
(called the state vector) that expresses in quantitative form the effect of all external influences on the plant before the present moment, so that the future evolution of the plant can be exactly
given from the knowledge of the present state and the future inputs. This situation implies that the control action at a given time can be specified as some function of the state at that time. Such a
function of the state, which determines the control action that is to be taken at any instant, is called a control law. This is a more general concept than the earlier idea of feedback; in fact, a
control law can incorporate both the feedback and feedforward methods of control.
In developing models to represent the control problem, it is unrealistic to assume that every component of the state vector can be measured exactly and instantaneously. Consequently, in most cases
the control problem has to be broadened to include the further problem of state determination, which may be viewed as the central task in statistical prediction and filtering theory. In principle,
any control problem can be solved in two steps: (1) building an optimal filter (a so-called Kalman filter) to determine the best estimate of the present state vector; (2) determining an optimal
control law and mechanizing it by substituting into it the estimate of the state vector obtained in step 1.
In practice, the two steps are implemented by a single unit of hardware, called the controller, which may be viewed as a special-purpose computer. The theoretical formulation given here can be shown
to include all other previous methods as a special case; the only difference is in the engineering details of the controller.
The mathematical solution of a control problem may not always exist. The determination of rigorous existence conditions, beginning in the late 1950s, has had an important effect on the evolution of
modern control, equally from the theoretical and the applied point of view. Most important is controllability; it expresses the fact that some kind of control is possible. If this condition is
satisfied, methods of optimization can pick out the right kind of control using information of type B.
The controllability condition is of great practical and philosophical importance. Because the state-vector equations accurately represent most physical systems, which only have small deviations about
their steady-state behaviour, it follows that in the natural world small-scale control is almost always possible, at least in principle. This fact of nature is the theoretical basis of practically
all the presently existing control technology. On the other hand, little is known about the ultimate limitations of control when the models in question are not linear, in which case small changes in
input can result in large deviations. In particular, it is not known under what conditions control is possible in the large, that is, for arbitrary deviations from existing conditions. This lack of
scientific knowledge should be kept in mind in assessing often-exaggerated claims by economists and sociologists in regard to a possible improvement in human society by governmental control.
Rudolf E. Kalman
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American Mathematical Society
The local spectral behavior of completely subnormal operators
HTML articles powered by AMS MathViewer
by K. F. Clancey and C. R. Putnam
Trans. Amer. Math. Soc. 163 (1972), 239-244
DOI: https://doi.org/10.1090/S0002-9947-1972-0291844-5
PDF | Request permission
For any compact set $X$, let $C(X)$ denote the continuous functions on $X$ and $R(X)$ the functions on $X$ which are uniformly approximable by rational functions with poles off $X$. Let $A$ denote a
subnormal operator having no reducing space on which it is normal. It is shown that a necessary and sufficient condition that $X$ be the spectrum of such an operator $A$ is that $R(X \cap \overline D
) \ne C(X \cap \overline D )$ whenever $D$ is an open disk intersecting $X$ in a nonempty set. Similar Articles
• Retrieve articles in Transactions of the American Mathematical Society with MSC: 47B20
• Retrieve articles in all journals with MSC: 47B20
Bibliographic Information
• © Copyright 1972 American Mathematical Society
• Journal: Trans. Amer. Math. Soc. 163 (1972), 239-244
• MSC: Primary 47B20
• DOI: https://doi.org/10.1090/S0002-9947-1972-0291844-5
• MathSciNet review: 0291844
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SUMJRNENT -- Summarize Journal Entries
SUMJRNENT SUMMARIZE JOURNAL ENTRIES TAAJRNH
The Summarize Journal Entries function allows you to 'net' the
journal entries so there is only a single entry for each relative
record number in a file/member. This reduces the number of journal
entries that must be applied if recovery is needed. A separate
command is used to apply the entries.
To use SUMJRNENT, you must first convert the journal entries to a
data base file using the i5/OS DSPJRN command. DSPJRN allows you to
control what file/entries you want to be converted, the journal
receivers to be used, the journal entry to start at etc.
* *
* Journal *
* Receiver *
* *
DSPJRN ....... ENTDTALEN(xxx)
DSPJRN OUTFILE
* 125 Bytes * Nbr of bytes per ENTDTALEN parm *
* Standard fields ***************************************
* eg File, member, * * *
* RR number ... * Physical record image * *
A typical command would be to convert all of the journal entries for
the current journal receiver such as:
DSPJRN JRN(JOURNALX) ENTDTALEN(500) JRNCDE(R)
OUTPUT(*OUTFILE) OUTFILE(LIB1/JRNOUTP)
The ENTDTALEN parameter is very important. It must be specified to
hold the largest physical record image (record length) being
journaled. If it is less than the largest physical record length,
the APYSUMJRN command will not apply changes for that file.
The use of JRNCDE(R) only converts the 'record' entries that appear
in the journal. These are the only journal entries processed by
SUMJRNENT. If other entries exist, they will be bypassed. See the
later discussion on the specific entry types processed.
A typical command to summarize the changes for a specific outfile
would be:
SUMJRNENT DSPJRNOUT(JRNOUTP) SUMENTP(SUMENTP)
This merges the contents of the latest converted journal entries in
JRNOUTP with the existing summarized information in SUMENTP. The new
'net' results are stored in SUMENTP. You must initially create this
file (See the implementation section).
In a typical environment you would keep summarizing the changes until
the file had been saved again. Each save of the file being journaled
represents a synchronization point to recover from. When the file is
saved, you would use CLRPFM to clear the SUMENTP file. If recovery
is needed, the latest saved version of the file would be restored and
the summarized journal changes would be applied using the APYSUMJRN
command. See the later section on SUMJRNENT scenario for more
A typical command to apply the changes for a specific file would be:
APYSUMJRN FILE(FILEA)
A report is printed to QPRINT which describes the number of entries
applied for each member. If an error has occurred on a member, the
problem is identified and no further updates to that member will
Because the update occurs by relative record number the function is
year 2000 ready.
SUMJRNENT scenario
The SUMENTP file will contain the summarized changes. When the file
being journaled is saved, you need to clear the SUMENTP file. Each
time the file is saved represents a new recovery or synchronization
A typical scenario would be a daily routine (or multiples times per
day) of:
- Change to a new journal receiver
- Use DSPJRN to convert the entries to an outfile
- Summarize the entries with SUMJRNENT
- Backup the SUMENTP file
- Optionally backup the journal receiver
Periodically you would establish a new synchronization point by:
- Backing up the file being journaled
- Use CLRPFM against SUMENTP
If recovery is needed, you would:
** Do the daily routine to merge in the last journal receiver.
If you are recovering because of a problem which occurred at a
specific time of day, you would only want to convert the
entries to that specific point using DSPJRN.
** Restore the last backup of the file being journaled
** Use APYSUMJRN for the file
If you reorganize or initialize the file you should then backup the
file and clear SUMENTP to establish a new synchronization point.
SUMJRNENT is dependent on consistent relative record numbers. Note
that neither the RGZPFM nor INZPFM entries in the journal will be
processed by SUMJRNENT.
If you use a 'put direct' function in your application or
REUSEDLT(*YES), see the later section.
One or multiple SUMENTP files
Normally, you would have multiple files being journaled. The SUMENTP
file can contain journal entries for more than one file. You can
have multiple SUMENTP files, but each should be in a separate
APYSUMJRN only operates on a single file per use of the command. To
recover multiple files, you would need to run the command for each
file to be recovered.
In a typical environment, you would backup all files where the
journal entries were being summarized at the same point (e.g. Sunday
night). Since you have a common synchronization point (the point of
the save), only a single SUMENTP file is needed. You could consider
multiples for your own reasons.
If you do not have a common backup point for the files where the
journal entries are being summarized, you need at least one SUMENTP
file for each backup group. For example, if you normally backup
files A, B and C on one day and D, E and F on a separate day, you do
not have a common synchronization point for all of the files. You
would need at least two SUMENTP files so that CLRPFM can be used at
the same time as the save.
Note that the SUMENTP2 file described in the implementation steps
does not have to be cleared. This occurs automatically by use of the
SUMJRNENT command.
Error conditions
Some error conditions will be found by SUMJRNENT. However, unlike
the TAA tool APYUSRCHG, there are several conditions which can only
be assumed to be valid. Because only the last journal entry is kept
for a unique relative record number, the following examples are
considered valid:
** An update entry exists for a non-existent relative record
number. It is assumed the add has occurred previously,
therefore the update is turned into an add.
** An add entry exists for an existing relative record number.
It is assumed that a delete has occurred previously, therefore
the add is turned into an update.
The following types are considered errors:
** The length of the file to be updated differs from the length
described in the JOENTL field of the journal entry.
** For any records added by PT entries, the relative record
number of the journal entry is compared with the relative
record number found in the file feedback area after the add
occurs. If a difference exists, the file restored (or copied
back), is not synchronized with when CLRPFM of the SUMENTP
file occurred.
** The maximum record length of the program has been exceeded.
See the later discussion of how to modify the program.
** The file/member could not be opened. The member must exist
and the user must be authorized.
** There are two special errors which can occur if you attempt to
delete a record that does not exist in the file. This would
be caused by the fact that an add occurred as a prior journal
entry, but the summarization function has removed the add and
kept only the delete. In this case the SUMJRNENT function
must place a deleted record into the file so that any
subsequent additions will be added to the proper relative
record number.
The INZPFM command supports the capability to initialize a
specified number of deleted records. If records exist in the
file, only the additional number of records required will be
initialized. For example, if 100 records exist in the file
and the user requests to initialize 110 records, records 101
thru 110 would be initialized. Records 1 thru 100 would be
left as is.
The SUMJRNENT command uses the relative record number of the
delete journal entry type to request INZPFM. For they typical
use of SUMJRNENT, the only valid condition is when INZPFM
causes a single record to be initialized. Therefore, two
error types are possible which indicate that the CLRPFM of the
SUMENTP file was not synchronized with the restored backup
No records were initialized which means that more records
exist in the file than are accounted for by the delete
relative record number.
If more than 1 record was initialized, it indicates that not
enough records exist in the file.
Entry types processed
SUMJRNENT will only process the JOCODE = R type journal entries from
the DSPJRN outfile. If other entries exist, they will be bypassed.
The Entry types processed are:
PT Put. Add a new record. This type of entry always
causes a record to be added at the end of the file.
UP Update. Update a record. The entire record image
exists in the journal entry.
DL Delete. Delete a record.
DR Delete at rollback. This entry only occurs if you
are using commitment control. As part of the commit
cycle, a record was added and then a rollback
occurred. This entry causes the record that was
added to be deleted.
UR Update at rollback. This entry only occurs if your
are using commitment control. As part of the commit
cycle, a record was either deleted or updated and
then rollback occurred. This entry causes the
original value of the record to be written back to
the data base.
PX Put direct. This entry is caused by the use of a
write (add) to a specific relative record number.
The record number written to must be a deleted
record. The record could have been deleted by
either a delete operation or initializing the file
to deleted records.
Most HLL usage cause PT entries to occur when adding
to a file. To request 'put direct' in RPG, you must
use the RECNO file continuation function.
When 'Put direct' is used, the apply function logic
must change because the relative record numbers of
new records will not appear in sequence. Fewer
logical errors can be checked when PX entries exist.
The most typical use of PX entries occurs when the
file is specified as REUSEDLT(*YES). PX entries are
written into the deleted record slots (if they
exist) instead of being added to the end of the
Because the process differs, APYSUMJRN requires a
special parameter to be specified to process PX
entries. You must specify PXUSD(*YES) if PX entries
exist in the journal.
The following chart describes the processing of the entries. The
'Record Exists' information describes whether the relative record
number of the journal entry exists as an active (not deleted) record
in the data base.
Record does not exist
Entry Record ---------------------------
type exists PXUSD(*NO) PXUSD(*YES)
------- -------- ------------ ------------
PT Update Add at EOF Add at EOF
UP Update Add at EOF Put direct
DL Delete Inlz 1 at EOF Note 1
DR Delete Inlz 1 at EOF Note 1
UR Update Add at EOF Note 3
PX Update Note 2 Note 3
(Note 2)
Note 1. Initialize by specifying the number of records as
described by the RR number.
Note 2. If PXUSD(*NO) is specified, an error occurs if any
PX entries exist.
Note 3. Try Put Direct. Monitor for CPF5006 which occurs if
the RR is beyond the EOF. If so, initialize as
described by the RR number. Try Put Direct again.
If it fails the second time, abort
The UP = Update Before journal entry caused by commitment control is
bypassed by SUMJRNENT.
Recovery with Put Direct journal entries
Because of the logic difference required in applying entries if the
PX (Put Direct) entries exist, you must specify PXUSD(*YES) on the
APYSUMJRN command.
Before you apply the entries, you should ensure the file is
initialized as it was at the time of the synchronization point.
There are two typical uses of 'Put direct'.
** You have a file that is specified as REUSEDLT(*YES). The
system will add new records to any existing deleted record
slots instead of at the end of the file.
** You have a file that uses a relative record number to access
records. Normally some algorithm would be used to convert a a
key like customer number to a relative record number and then
you would access the record. Your program logic would have to
handle the case of multiple customer numbers generating the
same relative record number. This type of file would normally
be initialized to deleted records. For example, you might
initialize a file with 10,000 entries even though you had only
5,000 records to be added. When you add a record to the file,
you want to write it to one of the deleted record slots.
If you save the file, the deleted records are saved.
Therefore, a restore to the synchronization point will work
properly. If you use some other technique of recovering the
file to the synchronization point, you may need to initialize
the file again to allow the PX entries to be properly applied.
SUMJRNENT command parameters *CMD
DSPJRNOUT The qualified name of the data base file that was
specified as the output file for DSPJRN. The
library defaults to *LIBL.
SUMENTP The file containing the summarized changes. The
file must be initially created by you. See the
implementation section.
APYSUMJRN command parameters *CMD
JRNLDFILE The qualified name of the data base file that was
journaled. Only a single file may be named for each
use of APYSUMJRN.
JRNLDMBR The member of the JRNLDFILE to be applied to. The
default is *ALL. A specific member may be named.
SUMENTP The qualified file name of the file containing the
summarized changes. The file contains data written
by the SUMJRNENT command. The library defaults to
STRNBR The start relative record number to begin applying
at for the named file. The default is *BEGIN which
applies any relative record number found.
STRNBR is intended for recovery purposes. If an
error occurs after some of the entries have been
applied, you may be able to make a manual change.
If you wish to restart the process at a specific
relative record number, you can use this parameter.
Since the records are in relative record number
order in the SUMENTP file, the relative record
number is used for a restart rather than the journal
entry ID.
PXUSD This is a *YES/*NO value with *NO as the default.
Specify *YES if you have made PX entries in the
journal for the file.
PX entries are caused by the 'put direct' function.
Additions to a file normally occur at the end of the
file and generate PT entries. A 'put direct'
function can be specified to write directly to a
specific relative record. The record on the data
base must contain a deleted entry. This would be
caused by either a prior operation that deleted the
record or by use of the INZPFM RECORDS(*DLT) option.
PX entries also occur if the file is specified as
REUSEDLT(*YES) and a record is added by the system
into an existing deleted record slot.
If PX entries are found in the journal and
PXUSD(*NO) is specified, an error occurs and the
remaining entries will not be applied. If
PXUSD(*YES) is specified, the logic for handling
certain types of journal entries changes. See the
earlier discussion.
** The apply function will apply to a library/file/member of the
same name as originally journaled. The file does not have to
be the same file as was originally journaled nor does it have
to be journaled at this time. The only requirement is that it
be the same name. See the earlier section on error conditions
that will be checked.
** A physical file may not exceed 2000 bytes in record length.
If you have files that exceed this length, you can change the
program to allow for it. See the later section.
** Like the i5/OS APYJRNCHG command, SUMJRNENT is totally
dependent on consistent relative record numbers to apply. If
you reorganize the file or copy it without COMPRESS(*NO), the
relative record numbers are likely to change. Although some
logical errors will be found by APYSUMJRN, you must be careful
in how you are performing the apply.
The APYJRNCHG command has a similar set of restrictions, but
also prevents the apply from crossing a boundary such as
RGZPFM. SUMJRNENT assumes that you are in control of when
RGZPFM, CLRPFM and INZPFM have occurred and are working with
the proper set of journal entries to be applied.
The following TAA Tools must be on your system:
SNDCOMPMSG Send completion message
SNDESCMSG Send escape message
SNDSTSMSG Send status message
SORTDB Sort data base
The tool is ready to use, but the journal entries must first be
converted to a data base file with the DSPJRN command. When you
specify DSPJRN, you must use an entry for the ENTDTALEN parameter
that describes the largest record length that will be in the file.
The use of JRNCDE(R) should also be used to reduce the number of
records in the outfile that will be read by SUMJRNENT.
If your largest record length is 500 bytes, a typical command would
DSPJRN JRN(JOURNALX) ENTDTALEN(500) JRNCDE(R)
OUTPUT(*OUTFILE) OUTFILE(LIB1/JRNOUTP)
The SUMJRNENT command requires two files be created. The first can
be any name, but SUMENTP is suggested. The second must be named
SUMENTP2 and must be in the same library as the SUMENTP file. The
purpose of the second file is to hold the merged output. It is then
used to copy back (automatically) to SUMENTP with the new net
It is best to create these files using CRTDUPOBJ based on the file
you named in the DSPJRN file for the OUTFILE parameter. The outfile
will have the proper description of the fixed fields and the length
you specified for the data by the ENTDTALEN parameter. Both files
should have the exact same definition.
CRTDUPOBJ OBJ(JRNOUTP) FROMLIB(xxxx) OBJTYPE(*FILE)
TOLIB(xxxx) NEWOBJ(SUMENTP)
CRTDUPOBJ OBJ(JRNOUTP) FROMLIB(xxxx) OBJTYPE(*FILE)
TOLIB(xxxx) NEWOBJ(SUMENTP2)
You should increase the record capacity size of the two files
according to how many summarized entries you will accumulate. If you
want to allow for 200,000 entries, you would specify:
CHGPF FILE(SUMENTP) SIZE(200000)
CHGPF FILE(SUMENTP2) SIZE(200000)
Handling larger than 2000 byte records
To provide better performance, SUMJRNENT is written to handle
physical record lengths up to 2000 bytes. The following describes
the changes needed to increase the size.
For the TAAJRNHR program:
** At about statements 2.00, 3.00 and 4.00 are the file
descriptions. The record size is 2125. Each journal entry
has 125 bytes of heading information. If your largest record
length is 5000 bytes, change the '2125' values to 5125.
** At about statement 33.00 is the input field description of the
JOESD field. This is the length of your largest data record.
The 'from position; should remain 126. The 'to position'
should be changed. If your largest record size is 5000 bytes,
change the '2125' value to 5125.
** At about statement 52.00 is the input field description of the
JOESD2 field. This is the length of your largest data record.
The 'from position; should remain 126. The 'to position'
should be changed. If your largest record size is 5000 bytes,
change the '2125' value to 5125.
** At about statements 54.00, 56.00 and 58.00 are the JOESD,
JOESD2 and JOESDX fields described as data structure fields to
avoid the RPG maximum field length restriction. If your
largest record length is 5000, change the '2000' values to
** At about statement 113.00 is the output location for the field
JOESDX. If your largest record length is 5000, change the
'2000' value to 5000.
For the TAAJRNHR2 program:
** At about statement 2.00 is the file descriptions for the
SUMENTP file. The record size is 2125. Each journal entry
has 125 bytes of heading information. If your largest record
length is 5000 bytes, change the '2125' value to 5125. If the
size of your SUMENTP file is less than 2000 bytes, you do not
need to change the value.
** At about statements 3.00 and 8.00 are the file descriptions
for the file being applied to. Program described files are
used since the actual file is controlled by your command
specification. A second file is needed to handle the PX
entries. The record size for both files is 2000. If your
largest record length is 5000 bytes, change the '2000' values
to 5125.
** At about statement 36.00 is the input field description of the
JOESD field. This is the length of your largest data record.
The 'from position; should remain 126. The 'to position'
should be changed. If your largest record size is 5000 bytes,
change the '2125' value to 5125.
** At about statement 46.00 is the JOESD field described as a
data structure field to avoid the RPG maximum field length
restriction. If your largest record length is 5000, change
the '2000' value to 5000.
** At about statement 136.00 is the test to ensure that the
program is not being requested to update a record greater than
the maximum record length specified in the program. If your
largest record length is 5000, change the '2000' value to
** At about statements 367.00 and 369.00 are the output locations
for the field JOESD. If your largest record length is 5000,
change the '2000' values to 5000.
Security considerations
Normal security is used. The user of the commands must be authorized
to make changes to each of the files to be applied to.
You may want to control who is authorized to use the commands and
prevent public use.
Objects used by the tool
Object Type Attribute Src member Src file
------ ---- --------- ---------- ----------
SUMJRNENT *CMD TAAJRNH QATTCMD
APYSUMJRN *CMD TAAJRNH2 QATTCMD
TAAJRNHC *PGM CLP TAAJRNHC QATTCL
TAAJRNHC2 *PGM CLP TAAJRNHC2 QATTCL
TAAJRNHR *PGM RPG TAAJRNHR QATTRPG
TAAJRNHR2 *PGM RPG TAAJRNHR2 QATTRPG
SUMJRNENT Cmd
TAAJRNHC CL
TAAJRNHR RPG
APYSUMJRN Cmd
TAAJRNHC2 CL
TAAJRNHR2 RPG
Added to TAA Productivity tools April 1, 1995
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please do not use floating point value in Factorio
hello developper team.
thanks for publishing beautiful great interesting game to the world, and daily fix.
you gave funny days for me
am... i'm in trouble on Factorio 1.1.19. it causes Factorio using floating point value to fluid items, i guess.
attachment file(blueprint.txt) is my 'advanced-oil-processing' oil-refineries.
it works by trains like below:
input train is arriving FULL
input train departing condition is EMPTY
output train arriving EMPTY
output train departing condition is FULL
odd oil-refineries crafts 63048 times,
even oil-refineries crafts 63028 times,
(1)all fluid-wagon to output a heavy-oil has look like 25k units of a heavy-oil, but train departing condition 'FULL' does not match so this train do not departing. I guess from (2), odd fluid-wagon
has actually 24999.9 units.
(2)odd oil-refineries has 99.9 crude-oil
(3)fluid network(pipes, pumps) for input to (2) oil-refineries are empty.
i expect FULL means each fluid-wagon has 25000 fluid units.
and 'advanced-oil-processing' use a crude-oil just 100 units each time.
so it's strange that oil-refineries has 99.9 crude-oil.
oh where are you from 99.9 crude-oils? where are you gone 0.1 crude-oils?
i feel it is not intuitively.
in genellary, i think floating point value makes trouble in any software sometimes, so in the future, how about do not use floating point value in Factorio?
if you want to give a detail or accuracy by floating value, then use a some scaling value which scaled by '10^N, 10 to the power of N' instead floating point value for all fluid recipes,
fluid-wagons, storage-tanks, and barrels.
i think fluid unit value scaling flushes away like an avobe trouble and we can play as usual and happy:)
thank you.
(39.24 KiB) Downloaded 149 times
Re: please do not use floating point value in Factorio
Thanks for the report however that's already how fluid wagons work. If you use "full" they will have 100% capacity. Additionally we aren't going to be changing how fluids or any other logic works to
remove floating point values.
If you want to get ahold of me I'm almost always on Discord.
Re: please do not use floating point value in Factorio
thanks for quick reply.
now i have only one question about (2).
why oil-refinery has a fluid which amount are 99.9 units, when i send a fluids from "full" wagon to oil-refinery until wagon is empty.
at this time, fluid network that between wagon and oil-refinery are also empty.
i heard on local community, a little fluid which less than 0.1 units are stretched inside fluid network, then such a little fluid are disappear, like a water in "Terraria".
is it correct?
Re: please do not use floating point value in Factorio
That’s more or less correct. Small amounts of liquid can basically evaporate and be lost. Often it’s more a case that pipes are separated and deleted with small amounts left in them, but I think it
can happen on it’s own on rare occasions.
You might try a departure condition for your trains of item quantity heavy oil > 24k AND Idle 5 seconds OR Cargo Full.
Re: please do not use floating point value in Factorio
thanks for quick reaction.
oh it's right.
it's not your guide but i try how to lost a fluid at Map Editor.
simple case below: (fluid-lost-try.txt)
(a) full tank -> pump -> oil-refinery
no fluid lost.
(b) full tank -> pump -> pipes -> oil-refinery
0.1 fluid lost.
(c) full tank -> pump -> pipes -> pump -> oil-refinery
0.1 fluid lost.
(d) full tank -> 8 pumps -> oil-refinery
no fluid lost.
now i understand a situation when a little fluid lost.
that is, pipes can lost a little fluid.
by the way, this lost is expected or bug?
(2.26 KiB) Downloaded 126 times
Re: please do not use floating point value in Factorio
fluids not remains in pipes at final state in avobe cases (a) to (d).
Re: please do not use floating point value in Factorio
rakou1986 wrote: ↑
Tue Feb 16, 2021 11:34 am
0.1 fluid lost.
You don't actually lose 0.1, it's much much less than that, but the display rounds to the floor of the tenth's place. It would take a very long time for the error to accumulate to where you lose more
than one production cycle. It's a bug in that it is never intended for fluids to be lost, but it's known and won't be fixed.
Re: please do not use floating point value in Factorio
Fluid may be lost or created in a really tiny amounts because of floating point math. Worst enemy is when a fluid system becomes empty: if all fluidboxes in a fluidsystem have less than 0.05 of
fluid, then the whole fluid system is automatically flushed.
-- edit (27 march 2021): 97338
Re: please do not use floating point value in Factorio
boskid wrote: ↑
Tue Feb 16, 2021 8:17 pm
Fluid may be lost or created in a really tiny amounts because of floating point math. Worst enemy is when a fluid system becomes empty: if all fluidboxes in a fluidsystem have less than 0.05 of
fluid, then the whole fluid system is automatically flushed.
Is that an explicit deletion of fluid in pipes like that? As in not just an issue of floating point math?
If that is case am I wrong about how the fluid values are rounded? Or are fluids on the building recipe display rounded by the floor and other fluid piping and tanks rounded to nearest?
Re: please do not use floating point value in Factorio
Zanthra wrote: ↑
Tue Feb 16, 2021 9:30 pm
boskid wrote: ↑
Tue Feb 16, 2021 8:17 pm
Fluid may be lost or created in a really tiny amounts because of floating point math. Worst enemy is when a fluid system becomes empty: if all fluidboxes in a fluidsystem have less than 0.05
of fluid, then the whole fluid system is automatically flushed.
Is that an explicit deletion of fluid in pipes like that? As in not just an issue of floating point math?
If that is case am I wrong about how the fluid values are rounded? Or are fluids on the building recipe display rounded by the floor and other fluid piping and tanks rounded to nearest?
We don't round fluids anywhere for recipes or flow logic. The only 2 sources of fluids getting removed is floating point inaccuracy (incredibly tiny amounts) or when the systems fluids are explicitly
flushed as boskid mentions.
If you want to get ahold of me I'm almost always on Discord.
Re: please do not use floating point value in Factorio
Ah yeah, I meant for display. Rounding or truncating has to happen there at some point. I was just thinking at the moment why 0.05, and thought maybe to avoid numbers in pipes being rounded to 0.0
for display while actually having fluid in them to avoid confusion, but that would still happen if the fluid was above 0.05 and the display value was truncated, or the fluid was kept alive by another
connected fluid box with more than 0.05.
Re: please do not use floating point value in Factorio
As far as I know the reason why pipes flush when below 0.05 fluid is to prevent an incredibly tiny amount just sitting there clogging up the pipe(s).
Since all fluids in the game are infinite it's a non-issue; more will come in and the cycle continues.
If you want to get ahold of me I'm almost always on Discord.
Re: please do not use floating point value in Factorio
With fluid mixing being prevented when possible, and the ability to manually flush all connected fluid boxes, is that still a beneficial feature? Small amounts in the pipes can't clog up the movement
of the same fluid type. I don't feel it's harmful, but if it no longer serves it's purpose due to other changes should it still be there?
Of course if you were to remove it you would of course hear from at least one person who's entire factory bypasses all the protection against fluid mixing and relies on it flush pipes to change
fluids and who's factory has grinded to a halt due to the change.
Re: please do not use floating point value in Factorio
Zanthra wrote: ↑
Wed Feb 17, 2021 1:13 am
With fluid mixing being prevented when possible, and the ability to manually flush all connected fluid boxes, is that still a beneficial feature?
It still is. automated flushing vs manual flushing is an obvious choice in a game about automation. The incomplete emptying of pipes when almost empty has been often requested for years before the
current solution was implemented. Maanual flushing means that people can't rely on the system to work in its own (but are useful when a design mistake was made). When in operation mode, people won't
want to manually flush every time a fluid changes in a pipe where several fluids are supposed to flow depending on the conditions.
Koub - Please consider English is not my native language.
Re: please do not use floating point value in Factorio
Koub wrote: ↑
Wed Feb 17, 2021 7:27 am
Zanthra wrote: ↑
Wed Feb 17, 2021 1:13 am
With fluid mixing being prevented when possible, and the ability to manually flush all connected fluid boxes, is that still a beneficial feature?
It still is. automated flushing vs manual flushing is an obvious choice in a game about automation. The incomplete emptying of pipes when almost empty has been often requested for years before
the current solution was implemented. Maanual flushing means that people can't rely on the system to work in its own (but are useful when a design mistake was made). When in operation mode,
people won't want to manually flush every time a fluid changes in a pipe where several fluids are supposed to flow depending on the conditions.
Given that mixing fluids has largely been considered a mistake, it stands to reason that the reason it was an often requested feature had to do more with accidentally getting a fluid into a pipe
meant for something else, and being able to attach a pump to get it all out without picking up all the pipes and rebuilding them. That use case is superseded by manual flushing.
Re: please do not use floating point value in Factorio
Actually, the use cases I remember seeing it requested for were :
- Fluid wagons that couldn't be emptied of the last fraction of fluid unit, which prevented the trains to leave with "on empty" condition (not even trying to enter into the train reusable for
different fluids).
- Generic fluid loading/unloading stations, with pumps sending the fluid to a place or another depending on the fluid
Koub - Please consider English is not my native language.
Re: please do not use floating point value in Factorio
Koub wrote: ↑
Wed Feb 17, 2021 4:35 pm
Actually, the use cases I remember seeing it requested for were :
- Fluid wagons that couldn't be emptied of the last fraction of fluid unit, which prevented the trains to leave with "on empty" condition (not even trying to enter into the train reusable for
different fluids).
- Generic fluid loading/unloading stations, with pumps sending the fluid to a place or another depending on the fluid
Do fluid wagons currently have actual fluid boxes? The prototype does not, it only has a capacity property of 25000. They don't interact with other fluid boxes the way that other entities do, only
though pumps. The debug option for fluid box info also shows no information on fluid wagons.
As for the second one, it's not really possible to build such stations anymore. I get this when I try using pumps to control the flow of the fluid.
Re: please do not use floating point value in Factorio
thanks for many discussion.
(i learning English so i maybe writing based on misreading...)
now i want to change subject to "now we need an auto-flusing in fluid network?"
that's what i really wanted to see.
Zanthra seems to be thinking what i want to know
i understand tiny fluids that less than 0.05 are flusing automatically, it is expected on Factorio.
there is a historical reason why auto-flusing needed.
auto-flusing was helpful for mistake, and it is useful for change the usage of existing pipes.
Rseding91 wrote: ↑
Wed Feb 17, 2021 12:19 am
As far as I know the reason why pipes flush when below 0.05 fluid is to prevent an incredibly tiny amount just sitting there clogging up the pipe(s).
Since all fluids in the game are infinite it's a non-issue; more will come in and the cycle continues.
surely, pumpjack or offshore-pump is infinite source of fluid, it's a non-issue.
but wagon or tank is not infinite source in some cases, some time-scale, it's maybe a issue.
actually, i had trouble like a first post.
now, we got a manual-flusing, and mistake is protected in advance.
don't you think a manual-flusing and this protection can replace an auto-flusing?
when we really need an auto-flusing?
i think for example, we can't make a time shareing fluid network without an auto-flushing.
it shared different fluids by circuit or something in different time.
(i have'nt seen so much, such a shareing network, because it's popular, that is mistake...)
surely, auto-flusing is expected. i understand.
from the Factorio's point of view, it is explicit.
but from the player's point of view, it is implicit.
some people maybe still needs an auto-flusing, but can't it be chooseable toggle in options?
Re: please do not use floating point value in Factorio
Koub wrote: ↑
Wed Feb 17, 2021 4:35 pm
Actually, the use cases I remember seeing it requested for were :
- Fluid wagons that couldn't be emptied of the last fraction of fluid unit, which prevented the trains to leave with "on empty" condition (not even trying to enter into the train reusable for
different fluids).
- Generic fluid loading/unloading stations, with pumps sending the fluid to a place or another depending on the fluid
oh fluid wagons can't departure with "empty" condition due to tiny fluids.
it's difficult to stand each side.
if i had to choose one, other condition for departure without auto-flusing is better than lost a tiny fluid.
because there are fixed fluids 25000 per wagons in cycle of train at always.
with an auto-flusing, fluids decrease with cycle.
very complex...
my trouble is simple, it's due to different pipe length makes different decrease, so how to solve is align of pipe length,
or "idle time" condition with circuit that related other train stop's state.....
going the first time,
why fluid wagons can't be empty by pumps?
it's a limit of fluid network specifications?
if make can be, specifications be collapse?
so we need auto-flusing?
if fluid wagons can be empty by pumps, we don't worry.....
Re: please do not use floating point value in Factorio
Rseding91 wrote: ↑
Wed Feb 17, 2021 12:19 am
As far as I know the reason why pipes flush when below 0.05 fluid is to prevent an incredibly tiny amount just sitting there clogging up the pipe(s).
Since all fluids in the game are infinite it's a non-issue; more will come in and the cycle continues.
Would it be possible to override that for modded games? Specifically, Pyanodons, and the seablock mod for it? Early game, I'm slowly producing liquid glass from quartz at a rate of 10 fluid units per
10 seconds or so, and with the underground pipe length I was using, I was experiencing 75% fluid loss thanks to that 'feature'.
Even with short range transport from a producer to a consumer, I was losing liquid glass over time at a rate of 0.1 units every minute or so. I could tell since the amount in the destination went
from a whole number to .9, .8, etc.
Also, since pipes can only have one fluid type now without exploits, there is zero reason for any automatic flush to ever occur. If a manual flush is desired, it's easy enough to connect a pump to
one end of the pipe system.
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Russian Ordinal Numbers 1-100 - OrdinalNumbers.com
Russian Ordinal Numbers 1-100
Russian Ordinal Numbers 1-100 – You can enumerate an unlimited amount of sets making use of ordinal numbers as an instrument. They also can help generalize ordinal amounts.
Ordinal numbers are among the fundamental concepts in math. It’s a number that signifies the position of an object within a list. An ordinal number is usually an integer between one and twenty. While
ordinal numbers serve a variety of functions, they are most often used to represent the order in which items are placed in the list.
Ordinal numbers are represented by charts as well as words, numbers and other methods. They can also be used to explain the way in which pieces are organized.
The majority of ordinal numbers fall into one of two categories. Transfinite ordinals are represented using lowercase Greek letters, while finite ordinals are represented with Arabic numbers.
A well-organized collection should contain at least one ordinal according to the axiom. For instance, the top possible grade would be awarded to the class’s initial member. The winner of the contest
was the student who had the highest score.
Combinational ordinal number
Multiple-digit numbers are also known as compound ordinal numbers. They are generated by multiplying ordinal number by its ultimate number. These numbers are most commonly utilized for ranking and
dating purposes. They don’t have a unique ending like cardinal numbers.
To identify the order in which elements are placed within a collection, ordinal numerals can be used. The names of the elements in the collection are also indicated using the numbers. You can find
both normal and suppletive numbers to ordinal numbers.
Prefixing a number with the suffix “-u” results in regular ordinals. The number is next typed as a word, and then the hyphen is added to it. There are various suffixes that are able to be used. “-nd”
is for numbers ending in 2, is a good instance. “-th” is used for numbers ending between 4 and 9, is another.
Suppletive ordinals can be constructed by prefixing words with -u.e. or –ie. The suffix is employed for counting purposes and is typically wider than the regular one.
Limit of Ordinal
Limit ordinal numbers are ones that do not contain zero. Limit ordinal numbers have one disadvantage: they do not have an element with a maximum. You can create them by joining sets without any
maximum element.
Limits on ordinal numbers may also be utilized in the description of recursion in transfinite terms. Each infinite number of cardinals, based on the von Neumann model is also an ordinal limit.
An ordinal numbers with limits are equivalent to the sum of all ordinals that are lower than it. Limit ordinal quantities can be enumerated using mathematics and can be expressed in a series or
natural numbers.
Data is organized by ordinal number. They are used to explain the numerical location of objects. They are frequently used in set theory and arithmetic contexts. They are not in the same category as
natural numbers, despite sharing a structure with them.
In the von Neumann model, a well-ordered collection is used. Let’s say that fy, which is a subfunction in the function g’ that is given as a singular function is the case. If g’ is in line with the
specifications, g’ is a limit ordinal of fy when it is the only function (ii).
In the same way, it is similar to the Church Kleene ordinal can be described as a limit ordinal. The Church-Kleene ordinal defines a limit as a properly arranged collection of the smaller ordinals.
It also has a nonzero ordinal.
Common numbers are often used in stories
Ordinal numbers are used to indicate the hierarchy between objects or individuals. They are essential to organize, count, ranking, and other reasons. They can be utilized to show the position of
objects, in addition to indicating the sequence of things.
The letter “th” is commonly used to signify the ordinal number. Sometimes, however, the letter “nd” could be substituted. Books’ titles are often associated with ordinal numbers.
While ordinal numbers are often used in lists format, they may also be stated in terms. They can also be expressed in the form of numbers and acronyms. Comparatively, they are easier to grasp as
compared to cardinal numbers.
There are three kinds of ordinal numerals. These numbers are able to be learned through games, practice, and other activities. You can enhance your skills in arithmetic by studying more about them.
You can increase your math skills by using a coloring activity. Make use of a handy marker to record your results.
Gallery of Russian Ordinal Numbers 1-100
Russian Numbers 1 100 Chart Portuguese Language Learning Italian
Pin On Languages
Ordinal Numbers In Russian
Leave a Comment
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Theory of Combinatorial Algorithms
Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)
Mittagsseminar Talk Information
Date and Time: Thursday, June 12, 2008, 12:15 pm
Duration: This information is not available in the database
Location: OAT S15/S16/S17
Speaker: Dirk Nowotka (Univ. Stuttgart)
When the Maximum Intersection Problem Is Easy: On the Use of Threshold Phenomena in Randomized Graph Models
Given a computational problem over an input set equipped with a probability distribution. Which are suitable properties of such a distribution for the construction of fast algorithms? In particular,
we consider data mining problems where the input is modelled by random graphs. We want to discuss this problem field in the seminar starting from an example where we use that approach by showing that
the maximum intersection problem can be efficiently solved when the input follows the so called Zipf's law. The maximum intersection problem consists of the task to find for a given set q a set q'
from a given collection D of sets such that q and q' have an intersection of maximum size among all sets in D (all sets and D are finite).
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Adaptive sequential nonlinear LSE for structural damage tracking with incomplete measurements
Parameter identification is very important in structural health monitoring for structural safety and management after emergency event. In practical applications, some external inputs, such as seismic
excitations, wind loads, etc. and some structure responses may not be measured or may not be measurable. Herein detail analysis based on adaptive sequential nonlinear least-square estimation with
unknown inputs and unknown outputs (ASNLSE-UI-UO) method is developed for effective parameter identification. Simulation studies using a 3-DOF linear system and experimental studies using a 3-DOF
shear-beam model with different scenarios of unknown inputs and unknown outputs are carried out to verify the analytical results. Simulation and experimental results demonstrate that this analysis
technique for effective parameter identification of ASNLSE-UI-UO method is reasonable and accurate.
1. Introduction
An objective of structural health monitoring (SHM) is to identify structure parameters and detect damages for ensuring the safety, reliability and integrity of structures [1]. Consequently, the
structural parameter identification based on vibration data measured from the structural health monitoring system has received considerable attention. Various time-domain analysis techniques have
been developed to identify structure parameters, such as least-square estimation (LSE), extended Kalman filter (EKF), Monte Carlo filter, etc. [2-7]. Particularly, the classical LSE and EKF methods
above can be used to identify constant parameters without the requirement for accurate modal parameters. However the identified results of LSE may appear with a significant numerical drift which is
difficult to remove online due to the numerical integration in iterative process. In EKF method the analytical solution may not converge due to both the linearization in the algorithm and the
unsuitable initial guesses [8]. Recently, a new technique referred to as the sequential nonlinear least square estimation (SNLSE) method, has been proposed to eliminate the drawbacks above [9].
Further, adaptive SNLSE (ASNLSE) method has been developed to achieve the time-varying parameter tracking. In ASNLSE method, usually all the system inputs and outputs should be measured by sensors
[10]. However, in practical applications, some external inputs, such as seismic excitations, wind loads, etc., and some structure responses frequently could not be measured, or even could not be
measurable. Besides, due to practical limitations, it may not be possible to install enough sensors in health monitoring system to measure either all the external excitations or the acceleration
responses at all degrees of freedom (DOFs) [11]. Consequently, under the condition of some incomplete measurements, the improvement iterative process of ASNLSE method has been improved to achieve
parameter identification and damage tracking referred to as the adaptive sequential non-linear least-square estimation with unknown inputs and unknown outputs (ASNLSE-UI-UO) lately [12]. As known,
when the number of DOFs of the structure is greater than the total number of unknown inputs and unknown outputs, the LSE process in ASNLSE-UI-UO method can be used to minimize the objective function
for achievement of the parameter estimation [12]. However, to data, it is a fact that no reasonable analysis technique in theory based on ASNLSE-UI-UO method, the allowable scenarios of inputs and
outputs for effective parameter identification and damage tracking are discussed and verified in detail.
In this research, detail analysis in theory for effective parameter identification based on ASNLSE-UI-UO method is discussed. Besides, simulation and experimental studies are conducted to identify
structure parameters and track parameter changes online using the theoretical analysis results of ASNLSE-UI-UO method. Simulations using a 3-DOF linear system and experiments using a 3-DOF shear-beam
model with different scenarios of unknown inputs and unknown outputs are used to verify the analytical incomplete measurement condition. Simulation and experimental results demonstrate that this
analysis technique for effective parameter identification of ASNLSE-UI-UO method is reasonable and accurate.
2. Theoretical analysis for ASNLSE-UI-UO method
The equation of motion of a $m\text{-DOF}$ nonlinear structure can be expressed as:
=\mathbf{\eta }\mathbf{f}\left(t\right)+{\mathbf{\eta }}^{\mathbf{*}}{\mathbf{f}}^{\mathbf{*}}\left(t\right)-{\mathbf{M}}^{\mathbf{*}}{\stackrel{¨}{\mathbf{x}}}^{\mathbf{*}}\left(t\right),$
in which $\mathbf{x}\left(t\right)$ and $\stackrel{˙}{\mathbf{x}}\left(t\right)$ are the displacement vector and velocity vector respectively. The acceleration vector $\stackrel{¨}{\stackrel{-}{\
mathbf{x}}}\left(t\right)$ and ${\stackrel{¨}{\mathbf{x}}}^{*}\left(t\right)$ are the known and unknown responses (system outputs) respectively, which are the mass matrixes $\stackrel{-}{\mathbf{M}}$
and ${\mathbf{M}}^{\mathbf{*}}$ corresponding to. $\mathbf{f}\left(t\right)$ and ${\mathbf{f}}^{\mathbf{*}}\left(t\right)$ are the known and unknown excitations (system inputs) respectively, which
are the influence matrixes $\mathbf{\eta }$ and ${\mathbf{\eta }}^{\mathbf{*}}$ corresponding to. ${\mathbf{F}}_{C}\left[\stackrel{˙}{\mathbf{x}}\left(t\right)\right]$ and ${\mathbf{F}}_{S}\left[\
mathbf{x}\left(t\right)\right]$ are the damping force vector and stiffness force vector respectively.
The observation equation can be obtained from Eq. (1) as follows:
$\mathbf{\phi }\left(\mathbf{X}\right)\mathbf{\theta }+\mathbf{\epsilon }=\stackrel{-}{\mathbf{\eta }}\stackrel{-}{\mathbf{f}}+\mathbf{y},$
in which $\mathbf{\phi }\left(\mathbf{X}\right)$ is the observation matrix and $\mathbf{X}={\left[{\mathbf{x}}^{T},{\stackrel{˙}{\mathbf{x}}}^{T}\right]}^{T}$ is the state vector. $\mathbf{\theta }=
{\left[{\theta }_{1},{\theta }_{2},...,{\theta }_{n}\right]}^{T}$ is the unknown parameter vector of structure, involving $n$ unknown parameters ${\theta }_{i}\left(i=\text{1,2},...,n\right)$, such
as stiffness, damping and nonlinear parameters, which may be the time-varying ones needed to be tracked. $\stackrel{-}{\mathbf{f}}={\left[\begin{array}{cc}{{\mathbit{f}}^{\mathbit{*}}}^{T}& {{\
stackrel{¨}{\mathbit{x}}}^{*}}^{T}\end{array}\right]}^{T}$ is the unknown input-output vector. $\stackrel{-}{\mathbf{\eta }}=\left[\begin{array}{cc}{\mathbf{\eta }}^{*}& -{\mathbf{M}}^{*}\end{array}\
right]$ and $\mathbf{y}=\mathbf{\eta }\mathbf{f}-\stackrel{-}{\mathbf{M}}\stackrel{¨}{\stackrel{-}{\mathbf{x}}}$ are the known matrixes. $\mathbf{\epsilon }$ is a model noise vector.
In ASNLSE-UI-UO method, an extended unknown vector ${\mathbf{\theta }}_{e,k}$ at ${t}_{k}\left({t}_{k}=k\Delta t\right)$ is introduced as ${\mathbf{\theta }}_{e,k}={\text{[}{\mathbf{\theta }}_{k}{\
stackrel{-}{\mathbf{f}}}_{k}\text{]}}^{T}$, so Eq. (2) can be discretized as:
${\mathbf{\phi }}_{e,k}\left({\mathbf{X}}_{k}\right){\mathbf{\theta }}_{e,k}+{\mathbf{\epsilon }}_{k}={\mathbf{y}}_{k},$
in which ${\mathbf{\phi }}_{e,k}\left({\mathbf{X}}_{k}\right)=\left[\begin{array}{cc}{\mathbf{\phi }}_{k}\left({\mathbf{X}}_{k}\right)& -\stackrel{-}{\mathbf{\eta }}\end{array}\right]$. Herein, ${\
mathbf{X}}_{k}$ and ${\mathbf{\theta }}_{e,k}$ in Eq. (5) are unknown quantities to be identified. ASNLSE-UI-UO method solves ${\mathbf{X}}_{k}$ and ${\mathbf{\theta }}_{e,k}$ in two steps as
Step I: Suppose the state vector ${\mathbf{X}}_{k}$ is known, the recursive solutions for ${\stackrel{^}{\mathbf{\theta }}}_{k+1}$ and ${\stackrel{^}{\stackrel{-}{\mathbf{f}}}}_{k+1|k+1}$ (the
estimates of ${\mathbf{\theta }}_{k+1}$ and ${\stackrel{-}{\mathbf{f}}}_{k+1}$) are determined as follows:
${\stackrel{^}{\mathbf{\theta }}}_{k+1}\text{=}{\stackrel{^}{\mathbf{\theta }}}_{k}\text{+}{\mathbf{K}}_{\mathbf{\theta },k+1}\left({\mathbf{X}}_{k+1}\right)\left[{\mathbf{y}}_{k+1}-{\mathbf{\phi }}_
{k+1}\left({\mathbf{X}}_{k+1}\right){\stackrel{^}{\mathbf{\theta }}}_{k}\text{+}\stackrel{-}{\mathbf{\eta }}{\stackrel{^}{\stackrel{-}{\mathbf{f}}}}_{k+1|k+1}\right],$
${\stackrel{^}{\stackrel{-}{\mathbf{f}}}}_{k+1|k+1}=-{\mathbf{S}}_{k+1}\left({\mathbf{X}}_{k+1}\right){\stackrel{-}{\mathbf{\eta }}}^{T}\left[I-{\mathbf{\phi }}_{k+1}\left({\mathbf{X}}_{k+1}\right){\
mathbf{K}}_{\mathbf{\theta },k+1}\left({\mathbf{X}}_{k+1}\right)\right]\left[{\mathbf{y}}_{k+1}-{\mathbf{\phi }}_{k+1}\left({\mathbf{X}}_{k+1}\right){\stackrel{^}{\mathbf{\theta }}}_{k}\right],$
${\mathbf{K}}_{\mathbf{\theta },k+1}\left({\mathbf{X}}_{k+1}\right)=\left({\mathbf{\Lambda }}_{k+1}{\mathbf{P}}_{\mathbf{\theta },k}{{\mathbf{\Lambda }}^{T}}_{k+1}\right){{\mathbf{\phi }}_{k+1}}^{T}\
left({\mathbf{X}}_{k+1}\right)$$\bullet {\left[\mathbf{I}+{\mathbf{\phi }}_{k+1}\left({\mathbf{X}}_{k+1}\right)\left({\mathbf{\Lambda }}_{k+1}{\mathbf{P}}_{\theta ,k}{{\mathbf{\Lambda }}^{T}}_{k+1}\
right){{\mathbf{\phi }}_{k+1}}^{T}\left({\mathbf{X}}_{k+1}\right)\right]}^{-1},$
${\mathbf{S}}_{k+1}\left({\mathbf{X}}_{k+1}\right)={\left\{{\stackrel{-}{\mathbf{\eta }}}^{T}\left[\mathbf{I}-{\mathbf{\phi }}_{k+1}\left({\mathbf{X}}_{k+1}\right){\mathbf{K}}_{\mathbf{\theta },k+1}\
left({\mathbf{X}}_{k+1}\right)\right]\stackrel{-}{\mathbf{\eta }}\right\}}^{-1},$
${\mathbf{P}}_{\mathbf{\theta },k}=\left[\mathbf{I}+{\mathbf{K}}_{\mathbf{\theta },k}\left({\mathbf{X}}_{k}\right)\stackrel{-}{\mathbf{\eta }}{\mathbf{S}}_{k}\left({\mathbf{X}}_{k}\right){\stackrel
{-}{\mathbf{\eta }}}^{T}{\mathbf{\phi }}_{k}\left({\mathbf{X}}_{k}\right)\right]\left(\mathbf{I}-{\mathbf{K}}_{\mathbf{\theta },k}\left({\mathbf{X}}_{k}\right){\mathbf{\phi }}_{k}\left({\mathbf{X}}_
{k}\right)\right)\left({\mathbf{\Lambda }}_{k+1}{\mathbf{P}}_{\mathbf{\theta },k}{{\mathbf{\Lambda }}^{T}}_{k+1}\right).$
Step II: The recursive solutions for ${\stackrel{^}{\mathbf{X}}}_{k+1|k+1}$ (the estimate of ${\mathbf{X}}_{k+1}$) are derived as follows:
in which ${\stackrel{^}{\mathbf{y}}}_{k+1}\left[{\mathbf{X}}_{k+1}\left({\mathbf{X}}_{k}\right)\right]={\mathbf{\phi }}_{k+1}\left[{\mathbf{X}}_{k+1}\left({\mathbf{X}}_{k}\right)\right]{\stackrel{^}
{\mathbf{\theta }}}_{k+1}\left[{\mathbf{X}}_{k+1}\left({\mathbf{X}}_{k}\right)\right]-\stackrel{-}{\mathbf{\eta }}{\stackrel{^}{\stackrel{-}{\mathbf{f}}}}_{k+1}$, and:
${\stackrel{^}{\mathbf{X}}}_{k+1|k}={\mathbf{\Phi }}_{k+1,k}{\stackrel{^}{\mathbf{X}}}_{k|k}+{\mathbf{B}}_{1}{\stackrel{¨}{\mathbf{x}}}_{k}+{\mathbf{B}}_{2}{\stackrel{¨}{\mathbf{x}}}_{k+1},$
${\stackrel{-}{\mathbf{K}}}_{k+1}={\stackrel{-}{\mathbf{P}}}_{k+1|k}{{\mathbf{\Psi }}^{T}}_{k+1,k+1}{\left[\mathbf{I}+{\mathbf{\Psi }}_{k+1,k+1}{\stackrel{-}{\mathbf{P}}}_{k+1|k}{{\mathbf{\Psi }}^
${\stackrel{-}{\mathbf{P}}}_{k+1|k}={\mathbf{\Phi }}_{k+1,k}{\stackrel{-}{\mathbf{P}}}_{k|k}{{\mathbf{\Phi }}^{T}}_{k+1,k},\mathrm{}{\stackrel{-}{\mathbf{P}}}_{k|k}=\left(\mathbf{I}-{\stackrel{-}{\
mathbf{K}}}_{k}{\mathbf{\Psi }}_{k\text{,}k}\right){\stackrel{-}{\mathbf{P}}}_{k+1|k},$
${\mathbf{\Phi }}_{k+1,k}\left\{\begin{array}{cc}\mathbf{I}& \Delta t\mathbf{I}\\ 0& \mathbf{I}\end{array}\right\},{\mathbf{\Psi }}_{i,k+1}={\left[\frac{\partial {\stackrel{^}{\mathbf{y}}}_{i}\left
({\mathbf{X}}_{i}\right)}{\partial {\mathbf{X}}_{i}}\cdot \frac{\partial {\mathbf{X}}_{i}}{\partial {\mathbf{X}}_{k+1}}\right]}_{{\mathbf{X}}_{i}={\mathbf{X}}_{i}\left({\stackrel{^}{\mathbf{X}}}_{k+1
|k}\right)},{\mathbf{B}}_{1}=\left[\begin{array}{c}\left(0.5-\beta \right){\left(\Delta t\right)}^{2}I\\ \left(1-\gamma \right)\left(\Delta t\right)I\end{array}\right],$
${\mathbf{B}}_{2}=\left[\begin{array}{c}\beta {\left(\Delta t\right)}^{2}I\\ \gamma \left(\Delta t\right)I\end{array}\right]$
Herein $\beta$ and $\gamma$ are parameters used in the Newmark-$\beta$ method in step II (usually $\beta =$ 0.25, $\gamma =$ 0.5), and $\mathbf{I}$ represents a unit matrix.
In Eq. (6) and Eq. (8), ${\mathbf{\Lambda }}_{k+1}$ is the adaptive factor matrix for time-varying parameter tracking, and the detailed derivations are given in literature [8]. This new method
developed above is referred to as the ASNLSE-UI-UO method. Further, when the number of DOFs of the structure is greater than the total number of unknown inputs and unknown outputs, the LSE process in
step I can be used to minimize the objective function for achievement of effective parameter estimation. Herein, the incomplete measurement condition for this method has been theoretically discussed
to ensure the success of parameter identification. The objective function that is the core of this method can be expressed as:
$\mathbf{J}\left(\mathbf{\theta }\right)=\sum _{i=1}^{k+1}{\left[{\mathbf{y}}_{i}-{\mathbf{\phi }}_{i}\left({\mathbf{X}}_{i}\right){\mathbf{\theta }}_{i}\right]}^{T}\left[{\mathbf{y}}_{i}-{\mathbf{\
phi }}_{i}\left({\mathbf{X}}_{i}\right){\mathbf{\theta }}_{i}\right].$
In the iterative process of ${\stackrel{^}{\mathbf{\theta }}}_{k+1}$ and ${\stackrel{^}{\stackrel{-}{\mathbf{f}}}}_{k+1|k+1}$, ${\mathbf{K}}_{\mathbf{\theta },k+1}\left({\mathbf{X}}_{k+1}\right)$ in
Eq. (6) is the key for a successful parameter identification for ASNLSE-UI-UO method. According to Eq. (8), only when the rank of the data matrix ${\mathbf{\phi }}_{k+1}\left({\mathbf{X}}_{k+1}\
right)$ is full, leading to the fact that ${\mathbf{\phi }}_{k+1}\left({\mathbf{X}}_{k+1}\right)$ is reversible, can we obtain the solution of ${\mathbf{K}}_{\mathbf{\theta },k+1}\left({\mathbf{X}}_
{k+1}\right)$, and then ASNLSE-UI-UO method could identify all the parameters of DOFs effectively. Consequently, the analysis of matrix rank is conducted to determine the effective condition for
ASNLSE-UI-UO method by discussing the measured quantities in different scenarios. However, according to the iterative process above, if the rank of ${\mathbf{\phi }}_{k+1}\left({\mathbf{X}}_{k+1}\
right)$ is solved directly, one may not be able to judge the detailed results due to incomplete measurements, such as which parameter could not be identified, and the influence of the unidentified
parameter to others. Herein, we not only need to ensure the ASNLSE-UI-UO method of processing, but also need to ensure a successful result for all the structural parameters. Consequently, the data
matrix ${\mathbf{\phi }}_{k+1}\left({\mathbf{X}}_{k+1}\right)$ should be turned into an upper triangular matrix to clearly reveal the calculation process. The following is an example of 3-DOF linear
structure to theoretically analyze the effective condition for ASNLSE-UI-UO method to clarify the analysis ideas in this research which can provide experiences for the actual application of this
method. The motion equation can be expressed as:
$\left[\begin{array}{ccc}{m}_{1}& 0& 0\\ 0& {m}_{2}& 0\\ 0& 0& {m}_{3}\end{array}\right]\left[\begin{array}{c}{\stackrel{¨}{\mathbf{x}}}_{1}\\ {\stackrel{¨}{\mathbf{x}}}_{2}\\ {\stackrel{¨}{\mathbf
{x}}}_{3}\end{array}\right]+\left[\begin{array}{ccc}{c}_{1}+{c}_{2}& \text{-}{c}_{2}& 0\\ \text{-}{c}_{2}& {c}_{2}+{c}_{3}& \text{-}{c}_{3}\\ 0& \text{-}{c}_{3}& {c}_{3}\end{array}\right]\left[\begin
{array}{c}{\stackrel{¨}{\mathbf{x}}}_{1}\\ {\stackrel{¨}{\mathbf{x}}}_{2}\\ {\stackrel{¨}{\mathbf{x}}}_{3}\end{array}\right]+\left[\begin{array}{ccc}{k}_{1}+{k}_{2}& \text{-}{k}_{2}& 0\\ \text{-}{k}_
{2}& {k}_{2}+{k}_{3}& \text{-}{k}_{3}\\ 0& \text{-}{k}_{3}& {k}_{3}\end{array}\right]\left[\begin{array}{c}{\mathbf{x}}_{1}\\ {\mathbf{x}}_{2}\\ {\mathbf{x}}_{3}\end{array}\right]=\left[\begin{array}
{c}0\\ 0\\ f\end{array}\right].$
In this system, if acceleration ${\stackrel{¨}{\mathbf{x}}}_{1}$ is unknown and initial state vector is zero, the data matrix:
$\mathbf{\phi }\left(\mathbf{X}\right)=\left[\begin{array}{cccccc}{\mathbf{x}}_{1}& {\mathbf{x}}_{1}-{\mathbf{x}}_{2}& 0& {\stackrel{˙}{\mathbf{x}}}_{1}& {\stackrel{˙}{\mathbf{x}}}_{1}-{\stackrel{˙}
{\mathbf{x}}}_{2}& 0\\ 0& {\mathbf{x}}_{2}-{\mathbf{x}}_{1}& {\mathbf{x}}_{2}-{\mathbf{x}}_{3}& 0& {\stackrel{˙}{\mathbf{x}}}_{2}-{\stackrel{˙}{\mathbf{x}}}_{1}& {\stackrel{˙}{\mathbf{x}}}_{2}-{\
stackrel{˙}{\mathbf{x}}}_{3}\\ 0& 0& {\mathbf{x}}_{3}-{\mathbf{x}}_{2}& 0& 0& {\stackrel{˙}{\mathbf{x}}}_{3}-{\stackrel{˙}{\mathbf{x}}}_{2}\end{array}\right]$
Turns into:
$\left[\begin{array}{cccccc}0& -{\mathbf{x}}_{2}& 0& 0& -{\stackrel{˙}{\mathbf{x}}}_{2}& 0\\ 0& {\mathbf{x}}_{2}& {\mathbf{x}}_{2}-{\mathbf{x}}_{3}& 0& {\stackrel{˙}{\mathbf{x}}}_{2}& {\stackrel{˙}{\
mathbf{x}}}_{2}-{\stackrel{˙}{\mathbf{x}}}_{3}\\ 0& 0& {\mathbf{x}}_{3}-{\mathbf{x}}_{2}& 0& 0& {\stackrel{˙}{\mathbf{x}}}_{3}-{\stackrel{˙}{\mathbf{x}}}_{2}\end{array}\right]$
and the rank of this matrix is not full as $r\left(\mathbf{\phi }\left(\mathbf{X}\right)\right)=2$, leading to the fact that $\mathbf{\phi }\left(\mathbf{X}\right)$ is nonreversible, so it cannot
identify all structure parameters of DOFs. Turning the data matrix into upper triangular matrix we can find that the parameters corresponding to the all zero columns in the matrix are always zero in
iterative process, so the first DOF stiffness and damping could not be identified. Similarly, based on the same idea above, some analytical results can be obtained as follows: If acceleration ${\
stackrel{¨}{\mathbf{x}}}_{2}$ is unknown, the data matrix turns into:
$\left[\begin{array}{cccccc}{\mathbf{x}}_{1}& {\mathbf{x}}_{1}& 0& {\stackrel{˙}{\mathbf{x}}}_{1}& {\stackrel{˙}{\mathbf{x}}}_{1}& 0\\ 0& -{\mathbf{x}}_{1}& -{\mathbf{x}}_{3}& 0& -{\stackrel{˙}{\
mathbf{x}}}_{1}& -{\stackrel{˙}{\mathbf{x}}}_{3}\\ 0& 0& {\mathbf{x}}_{3}& 0& 0& {\stackrel{˙}{\mathbf{x}}}_{3}\end{array}\right]$
and the rank of this matrix is full as $r\left(\mathbf{\phi }\left(\mathbf{X}\right)\right)=3$, so all the structure parameters can be identified. If acceleration ${\stackrel{¨}{\mathbf{x}}}_{3}$ is
unknown, the data matrix turns into:
$\left[\begin{array}{cccccc}{\mathbf{x}}_{1}& {\mathbf{x}}_{1}-{\mathbf{x}}_{2}& 0& {\stackrel{˙}{\mathbf{x}}}_{1}& {\stackrel{˙}{\mathbf{x}}}_{1}-{\stackrel{˙}{\mathbf{x}}}_{2}& 0\\ 0& {\mathbf{x}}_
{2}-{\mathbf{x}}_{1}& {\mathbf{x}}_{2}& 0& {\stackrel{˙}{\mathbf{x}}}_{2}-{\stackrel{˙}{\mathbf{x}}}_{1}& {\stackrel{˙}{\mathbf{x}}}_{2}\\ 0& 0& -{\mathbf{x}}_{2}& 0& 0& -{\stackrel{˙}{\mathbf{x}}}_
and the rank of this matrix is full as $r\left(\mathbf{\phi }\left(\mathbf{X}\right)\right)=3$, so all the structure parameters can be identified. If accelerations ${\stackrel{¨}{\mathbf{x}}}_{1}$
and ${\stackrel{¨}{\mathbf{x}}}_{2}$ are unknown, the data matrix turns into:
$\left[\begin{array}{cccccc}0& 0& 0& 0& 0& 0\\ 0& 0& -{\mathbf{x}}_{3}& 0& 0& -{\stackrel{˙}{\mathbf{x}}}_{3}\\ 0& 0& {\mathbf{x}}_{3}& 0& 0& {\stackrel{˙}{\mathbf{x}}}_{3}\end{array}\right]$
and the rank of this matrix is not full as$r\left(\mathbf{\phi }\left(\mathbf{X}\right)\right)=1$, so the first and second DOF stiffness and damping could not be identified based on the analysis
above. If accelerations ${\stackrel{¨}{\mathbf{x}}}_{1}$ and ${\stackrel{¨}{\mathbf{x}}}_{3}$ are unknown, the data matrix turns into:
$\left[\begin{array}{cccccc}0& -{\mathbf{x}}_{2}& 0& 0& -{\stackrel{˙}{\mathbf{x}}}_{2}& 0\\ 0& {\mathbf{x}}_{2}& {\mathbf{x}}_{2}& 0& {\stackrel{˙}{\mathbf{x}}}_{2}& {\stackrel{˙}{\mathbf{x}}}_{2}\\
0& 0& -{\mathbf{x}}_{2}& 0& 0& -{\stackrel{˙}{\mathbf{x}}}_{2}\end{array}\right]$
and the rank of this matrix is not full as $r\left(\mathbf{\phi }\left(\mathbf{X}\right)\right)=2$, so the first DOF stiffness and damping could not be identified. If accelerations ${\stackrel{¨}{\
mathbf{x}}}_{2}$ and ${\stackrel{¨}{\mathbf{x}}}_{3}$ are unknown, the data matrix turns into:
$\left[\begin{array}{cccccc}{\mathbf{x}}_{1}& {\mathbf{x}}_{1}& 0& {\stackrel{˙}{\mathbf{x}}}_{1}& {\stackrel{˙}{\mathbf{x}}}_{1}& 0\\ 0& -{\mathbf{x}}_{1}& 0& 0& -{\stackrel{˙}{\mathbf{x}}}_{1}& 0\\
0& 0& 0& 0& 0& 0\end{array}\right]$
and the rank of this matrix is not full as $r\left(\mathbf{\phi }\left(\mathbf{X}\right)\right)=2$, so the third DOF stiffness and damping could not be identified. As known through the above
scenarios, the parameter, corresponding to the entire column of the data matrix ${\mathbf{\phi }}_{k+1}\left({\mathbf{X}}_{k+1}\right)$, cannot be identified, and the relative parameters may be
influenced in terms of accuracy. By the analysis process above, one can confirm the least number of sensors and the optimal placement for sensors. The theoretical analysis results above have been
verified by simulations and experiments in this research, and next, two typical simulation results and two typical experimental results are shown to illustrate this.
3. Simulation studies
In this section an example of 3-DOF linear structure above is analyzed to verify the theoretical analysis results for ASNLSE-UI-UO method. The structural parameters are: ${m}_{i}=$125.53 kg, ${c}_{i}
=$ 0.175 kN⋅s/m, ${k}_{i}=$ 24.5 kN/m, ($i=$ 1, 2, 3). The external excitation, modeled by a white noise process to simulate the ambient, is applied to the third DOF similar to Case 3 in [13]. The
initial stiffness and damping are 80 % of theoretical values. ${\mathbf{P}}_{k}$ is the gain matrix for parameter vector ${\mathbf{\theta }}_{k}$, and its initial value is ${\mathbf{P}}_{0}=1{0}^{10}
{\mathbf{I}}_{6}$. ${\stackrel{-}{\mathbf{P}}}_{k|k}$ is the gain matrix for state vector ${\mathbf{X}}_{k+1}$, and its initial value is ${\stackrel{-}{\mathbf{P}}}_{0|0}={\mathbf{I}}_{6}$. Suppose a
damage occurs in the first DOF, where ${k}_{\text{1}}$ reduces from 24.5 kN/m to 20 kN/m at $t=$ 20 s. The sampling frequency in simulation is 500 Hz for all the measured data. Herein, two typical
results of different measurement cases are shown in this paper to verify the analysis results for ASNLSE-UI-UO method in condition that these cases satisfy the condition: the number of DOFs of the
structure is greater than the total number of unknown inputs and unknown outputs. The two effective cases are as follows: Case 1: unknown excitation $\mathbf{f}$ and unknown acceleration ${\stackrel
{¨}{\mathbf{x}}}_{\text{2}}\left(t\right)$, known accelerations ${\stackrel{¨}{\mathbf{x}}}_{\text{1}}\left(t\right)$ and ${\stackrel{¨}{\mathbf{x}}}_{\text{3}}\left(t\right)$; Case 2: unknown
excitation $\mathbf{f}$ and unknown acceleration ${\stackrel{¨}{\mathbf{x}}}_{\text{3}}\left(t\right)$, known accelerations ${\stackrel{¨}{\mathbf{x}}}_{\text{1}}\left(t\right)$ and ${\stackrel{¨}{\
mathbf{x}}}_{\text{2}}\left(t\right)$. The following figures are parts of identified results, in which the dashed lines represent theoretical values, and the solid lines represent identified values.
Fig. 1Identified parameters in Case 1 (numerical simulation)
Fig. 2Identified parameters in Case 2 (numerical simulation)
As shown in Fig. 1 and Fig. 2, in these two cases the ASNLSE-UI-UO method is able to identify all the structure parameters and to track the parameter changes online. The differences between the
identified values and simulation values are very small, and these differences and minor fluctuations in identified curves are mainly due to the noises (5 % root mean square of white noises) that are
added into the simulations. The damage tracking capability is represented in these two cases as the stiffness changing process of the first DOF. In the first DOF the stiffness change begins at $t=$20
s, and then converges to the reference value at about $t=$22 s. The convergence rate mainly depends on both the sampling frequency and the signal-to-noise ratio. In case 1, the rank of data matrix is
full as $r\text{(}\mathbf{\phi }\text{(}\mathbf{X}\text{)}\text{)}=3$, so all the structure parameters can be identified though the excitation $f$ and acceleration ${\stackrel{¨}{\mathbf{x}}}_{\text
{2}}\left(t\right)$ are unknown, but there is a little difference between the identified k[3] and the theoretical one mainly due to the influence of ${\stackrel{¨}{\mathbf{x}}}_{\text{2}}\left(t\
right)$ as in the analysis in section 2. Similarly, in case 2, the rank of data matrix is also full leading to a successful identification with incomplete measurements, but there is some fluctuation
in the convergence process of all the parameters due to less vibration information. Comparing the results from case 1 and case 2, the unknown accelerations ${\stackrel{¨}{\mathbf{x}}}_{\text{2}}\left
(t\right)$ have heavier influence than ${\stackrel{¨}{\mathbf{x}}}_{3}\left(t\right)$, but all results can be accepted in project application. These typical results obtained from simulations are
nearly according to the analysis results of data matrix for the effective condition of ASNLSE-UI-UO method, and similar simulation results can be obtained in different scenarios of unknown inputs and
unknown outputs for the matrix rank analysis in section 2. The simulation results demonstrate that the analysis of matrix rank to determine the unknown conditions of ASNLSE-UI-UO method is efficient.
4. Experimental studies
A 400 mm × 300 mm scaled 3-DOF building model is used in the experiment, as shown in Fig. 3. The height of this building model is 885 mm and the total weight is 75.4 kg. The masses of each floor are
25.1 kg, 25.1 kg and 24.4 kg. The first three natural frequencies are 3.38 Hz, 9.47 Hz and 13.68 Hz. Based on the discretized 3-DOF shear-beam model, the stiffness of each floor is obtained as 55.5
kN/m using the finite element method (FEM) as reference values. To simulate the stiffness reduction caused by the damage online in the first floor, a stiffness element device (SED) was installed in
the first floor, the details of SED are described in literature [2]. El Centro earthquake excitation force is applied to the base of model using a vibration table, and each floor is installed with
one acceleration sensor to measure the vibration responses. During the experiment, the SED was provided an effective stiffness of 10.5 kN/m for the first floor at first, and then reduced the
effective stiffness of SED to zero at $t=$ 10 s, so that the stiffness k[1] in the first floor reduced from 66 kN/m to 55.5 kN/m abruptly at $t=\text{10 s}$ and ${k}_{2}={k}_{3}=$ 55.5 kN/m all the
time. The sampling frequency of all measurements in experimental tests is 500 Hz.
In this experiment, two typical results of different measurement cases are used to verify the analysis results for ASNLSE-UI-UO method in condition that these cases satisfy the analysis condition for
successful parameter identification. The two cases are the same as the simulation ones above: Case 1: unknown $\mathbf{f}$ and ${\stackrel{¨}{\mathbf{x}}}_{\text{2}}\left(t\right)$, known ${\stackrel
{¨}{\mathbf{x}}}_{\text{1}}\left(t\right)$ and ${\stackrel{¨}{\mathbf{x}}}_{\text{3}}\left(t\right)$; Case 2: unknown $\mathbf{f}$ and ${\stackrel{¨}{\mathbf{x}}}_{\text{3}}\left(t\right)$, known ${\
stackrel{¨}{\mathbf{x}}}_{\text{1}}\left(t\right)$ and ${\stackrel{¨}{\mathbf{x}}}_{\text{2}}\left(t\right)$. Some initial values need to be supposed to start the ASNLSE-UI-UO method to identify the
structure parameters as follows: ${k}_{i\text{,0}}=\text{100 kN/m,}$${c}_{i\text{,0}}=$0.1 kN⋅s/m, ${\mathbf{P}}_{0}=1{0}^{3}{\mathbf{I}}_{6}$ and ${\stackrel{-}{\mathbf{P}}}_{0|0}={\mathbf{I}}_{6}$.
The excitation acceleration ${a}_{d}$, which is the system unknown input, and the acceleration responses ${a}_{1}$, ${a}_{2}$ and ${a}_{3}$ are shown in Fig. 4. The identified results are shown in
Fig. 5 and Fig. 6, in which the dashed lines represent FEM values, and the solid lines represent identified ones.
Fig. 3Experimental set-up for building model on shaking table
Fig. 4Measured acceleration responses in experimental test
As shown in Fig. 5 and Fig. 6, in the two typical cases, the ASNLSE-UI-UO is able to identify all the structure parameters and to track the parameter changes online based on the experimental data.
Herein, the only stiffness values are represented due to the fact that the identified stiffness has the reference values for comparison. The identified values meet the FEM ones well, especially in
the second floor and the third floor, where there are no stiffness changes, as the signal-to-noise ratios in these two floors are higher than the signal-to-noise ratio in the first floor. In the
first floor, the identified stiffness in case 2 is slightly better than the identified one in case 1, and this phenomenon is due to the different scenarios of unknown inputs and unknown outputs. As
shown by the stiffness changing process of the first floor in Fig. 5 and Fig. 6, the damage tracking capability is represented in these two cases: stiffness change begins at $t=$10 s, and then
converges to the reference value at about $t=$12 s. Besides the convergence rate for the experimental studies mainly depends on the sampling frequency due to the Newmark-$\beta$ method in step II. In
these two cases, the rank of data matrix is full as $r\left(\mathbf{\phi }\text{(}\mathbf{X}\right)\right)=3$, so all the structure parameters can be identified for a successful identification with
incomplete measurements. In practical applications, the identified results in the experiments are acceptable in project application. Further, similar experimental results can be obtained in different
scenarios of unknown inputs and unknown outputs for verifying the matrix rank analysis. Compared to the simulation studies above, the experiment studies obtained consistent results applying the
matrix rank analysis technique. The experimental study results demonstrate that the incomplete measurement condition for ASNLSE-UI-UO method by the analysis of matrix rank to determine the unknown
conditions presented in this paper is efficient.
Fig. 5. Identified parameters in Case 1 (experimental test)
Fig. 6. Identified parameters in Case 2 (experimental test)
5. Conclusions
In this paper, detailed analysis for effective condition of parameter identification based on ASNLSE-UI-UO method is discussed in theory to help in reducing the number of vibration measurements and
providing experiences for the actual application of this method. The analysis of matrix rank to determine the unknown conditions of ASNLSE-UI-UO method is efficient. Besides, simulation and
experimental studies are conducted using vibration data, to identify parameters and track damages online based on the theoretical analysis results of the developed method. The results of simulations
using a 3-DOF linear system and experiments using a 3-DOF shear-beam model with different scenarios of unknown inputs and unknown outputs demonstrate that this analysis technique for effective
parameter identification of ASNLSE-UI-UO method is reasonable and accurate.
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About this article
structural health monitoring
parameter identification
adaptive sequential nonlinear least square estimation
incomplete measurement
experimental verification
This research is partially supported by the National Natural Science Foundation of China (Grant No. 11172128), the Funds for International Cooperation and Exchange of the National Natural Science
Foundation of China (Grant No. 61161120323), the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20123218110001), the Jiangsu Foundation for Excellent
Talent of China (Grant No. 2010-JZ-004), the Jiangsu Graduate Training Innovation Project (CXLX11_0171) and the Priority Academic Program Development of Jiangsu Higher Education Institutions.
Copyright © 2013 Vibroengineering
This is an open access article distributed under the
Creative Commons Attribution License
, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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How to Calculate Odds Ratio.
The odds ratio is defined as the ratio of the odds of A in the presence of B and the odds of A in the presence of B.
The odds ratio can also be used to determine whether a particular exposure is a risk factor for a particular outcome, and to compare the magnitude of various risk factors for that outcome.
Formula to calculate odds ratio.
If 12 smokers have lung cancer, 60 smokers do not have lung cancer, 1 non-smoker has lung cancer, and 90 non-smokers do not have lung cancer, calculate the odd ratio.
Thus, the smokers has 20 times the odds of having lung cancer than non-smokers.
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Define Logarithmic Functions
Learning Outcomes
• Convert from logarithmic to exponential form
• Convert from exponential to logarithmic form
In order to analyze the magnitude of earthquakes or compare the magnitudes of two different earthquakes, we need to be able to convert between logarithmic and exponential form. For example, suppose
the amount of energy released from one earthquake were [latex]500[/latex] times greater than the amount of energy released from another. We want to calculate the difference in magnitude. The equation
that represents this problem is [latex]{10}^{x}=500[/latex], where [latex]x[/latex] represents the difference in magnitudes on the Richter Scale. How would we solve for [latex]x[/latex]?
We have not yet learned a method for solving exponential equations. None of the algebraic tools discussed so far is sufficient to solve [latex]{10}^{x}=500[/latex]. We know that [latex]{10}^{2}=100[/
latex] and [latex]{10}^{3}=1000[/latex], so it is clear that [latex]x[/latex] must be some value between [latex]2[/latex] and [latex]3[/latex], since [latex]y={10}^{x}[/latex] is increasing. We can
examine a graph to better estimate the solution.
Estimating from a graph, however, is imprecise. To find an algebraic solution, we must introduce a new function. Observe that the graph above passes the horizontal line test. The exponential function
[latex]y={b}^{x}[/latex] is one-to-one, so its inverse, [latex]x={b}^{y}[/latex] is also a function. As is the case with all inverse functions, we simply interchange x and y and solve for y to find
the inverse function. To represent y as a function of x, we use a logarithmic function of the form [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex]. The base b logarithm of a number is the exponent
by which we must raise b to get that number.
We read a logarithmic expression as, “The logarithm with base b of x is equal to y,” or, simplified, “log base b of x is y.” We can also say, “b raised to the power of y is x,” because logs are
exponents. For example, the base [latex]2[/latex] logarithm of [latex]32[/latex] is [latex]5[/latex], because [latex]5[/latex] is the exponent we must apply to [latex]2[/latex] to get [latex]32[/
latex]. Since [latex]{2}^{5}=32[/latex], we can write [latex]{\mathrm{log}}_{2}32=5[/latex]. We read this as “log base [latex]2[/latex] of [latex]32[/latex] is [latex]5[/latex].”
We can express the relationship between logarithmic form and its corresponding exponential form as follows:
[latex]{\mathrm{log}}_{b}\left(x\right)=y\Leftrightarrow {b}^{y}=x,\text{}b>0,b\ne 1[/latex]
Note that the base b is always positive.
Because logarithms are functions, they are most correctly written as [latex]{\mathrm{log}}_{b}\left(x\right)[/latex], using parentheses to denote function evaluation, just as we would with [latex]f\
left(x\right)[/latex]. However, when the input is a single variable or number, it is common to see the parentheses dropped and the expression written without parentheses, as [latex]{\mathrm{log}}_{b}
x[/latex]. Note that many calculators require parentheses around the x.
We can illustrate the notation of logarithms as follows:
Notice that, comparing the logarithm function and the exponential function, the input and the output are switched. This means [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] and [latex]y={b}^{x}[/
latex] are inverse functions.
Definition of the Logarithmic Function
A logarithm base b of a positive number x satisfies the following definition.
For [latex]x>0,b>0,b\ne 1[/latex],
[latex]y={\mathrm{log}}_{b}\left(x\right)\text{ is the same as }{b}^{y}=x[/latex]
• we read [latex]{\mathrm{log}}_{b}\left(x\right)[/latex] as, “the logarithm with base b of x” or the “log base b of x.”
• the logarithm y is the exponent to which b must be raised to get x.
Also, since the logarithmic and exponential functions switch the x and y values, the domain and range of the exponential function are interchanged for the logarithmic function. Therefore,
• the domain of the logarithm function with base [latex]b \text{ is} \left(0,\infty \right)[/latex].
• the range of the logarithm function with base [latex]b \text{ is} \left(-\infty ,\infty \right)[/latex].
In our first example, we will convert logarithmic equations into exponential equations.
Write the following logarithmic equations in exponential form.
1. [latex]{\mathrm{log}}_{6}\left(\sqrt{6}\right)=\frac{1}{2}[/latex]
2. [latex]{\mathrm{log}}_{3}\left(9\right)=2[/latex]
Show Solution
In the following video, we present more examples of rewriting logarithmic equations as exponential equations.
How To: Given an equation in logarithmic form [latex]{\mathrm{log}}_{b}\left(x\right)=y[/latex], convert it to exponential form
1. Examine the equation [latex]y={\mathrm{log}}_{b}x[/latex] and identify b, y, and x.
2. Rewrite [latex]{\mathrm{log}}_{b}x=y[/latex] as [latex]{b}^{y}=x[/latex].
Think About It
Can we take the logarithm of a negative number? Re-read the definition of a logarithm and formulate an answer. Think about the behavior of exponents. You can use the textbox below to formulate your
ideas before you look at an answer.
Show Solution
Convert from Exponential to Logarithmic Form
To convert from exponential form to logarithmic form, we follow the same steps in reverse. We identify the base b, exponent x, and output y. Then we write [latex]x={\mathrm{log}}_{b}\left(y\right)[/
Write the following exponential equations in logarithmic form.
1. [latex]{2}^{3}=8[/latex]
2. [latex]{5}^{2}=25[/latex]
3. [latex]{10}^{-4}=\frac{1}{10,000}[/latex]
Show Solution
In our last video, we show more examples of writing logarithmic equations as exponential equations.
The base b logarithm of a number is the exponent by which we must raise b to get that number. Logarithmic functions are the inverse of exponential functions, and it is often easier to understand them
through this lens. We can never take the logarithm of a negative number, therefore [latex]{\mathrm{log}}_{b}\left(x\right)=y[/latex] is defined for [latex]b>0[/latex]
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Matter Time, Aethertime
The overwhelming success of general relativity for mainstream science's macroscopic reality of continuous space and time cannot be overstated. Likewise, quantum mechanics represents an even more
successful understanding of our microscopic reality of amplitude and phase. All of relativity’s reported successes, though, are really due to the two key notions of mass-energy equivalence (MEE) and
gravity time delay of light. Lorentz invariance, the constant of the speed of light irrespective of velocity, simply follows directly from MEE and means that the gravity deflection of light follows
from both light’s gravity MEE and the extra time delay of light.
Likewise, it is the quantum coherence of microscopic matter as amplitude and phase that is largely responsible for quantum's microscopic success stories. The quantum story is built upon space and
motion, just as is the GR story, but for GR, space and motion do not apply everywhere in the universe while quantum amplitude and phase apply everywhere. In the GR or mainstream science, velocity and
acceleration in empty space make up frames of reference from which emerge changes in inertial matter and time delays of light. Gravity affects light once as light’s MEE mass and then again as
gravity’s acceleration and red shift and so gravity deflection of light is twice that of just light’s MEE gravity deflection.
Augmenting continuous space, motion, and time with the more general notions of discrete matter and time extends the validity of gravity to all of the universe. In a sense, this means that space and
motion actually lie within the the domain of discrete changes in inertial matter and the time delay of light by gravity, not the other way around. In other words, augmenting continuous space and time
means that the basic principles of MEE and gravity time delays still apply to that part of the universe. however, the spatio-temporal tensors of GR do not apply outside of the limits of continuous
space and time and so a change in inertial matter emerges as motion in spatial frames of reference and it is from changes in gravity that space emerges from gravity time delay. Thus, space and motion
are both within the domain of changes in inertial matter and time delays and not the other way around as shown in the figure below. The total time delay for light due to gravity is after all a factor
of two greater that of just light’s gravity MEE time delay.
Any model of the universe with both gravity MEE and time delay will also be consistent with the observed gravity light deflections, but there are further notions of relativity that do not necessarily
follow from gravity MEE and time delay. For example, GR lacks an absolute frame of reference even though the CMB seems to be an absolute frame of reference and given an absolute CMB frame simply
limits the scale for GR tensor algebra.
Also, the determinate geodesic paths of GR objects in a 4D spacetime are inconsistent with the microscopic probabilistic quantum paths of the very successful quantum action. In fact, the determinate
GR geodesics in effect do away with the quantum notion of time since time becomes just a GR displacement and it is the 4D geodesic paths that then determine the futures of all objects from the
initial conditions of the universe.
In contrast, quantum mechanics shows by many different measurements that there are no determinate geodesic paths for quantum objects. In fact, there is a fundamental lack of knowledge of certain
quantum paths and a fundamental uncertainty principle limits all quantum paths. Yet despite the limitations of GR, the predictions of MEE and gravity time delay corrections allow our GPS satellites
to work and explain the deflection of starlight and the time delays of quasar radio sources by the gravity of the sun as well as the lensing of galaxies by other galaxies. All of these measurements
are consistent with gravity MEE and time delays and so any theory that incorporates MEE and gravity time delays will also be consistent with all of these observations as well.
The further notions of a lack of an absolute frame of reference in GR and GR determinate geodesics are then both open to question and neither has been verified by measurement. The CMB does seem to
represent an absolute frame of reference that then closes all motion in the universe and the well demonstrated quantum uncertainty does seem to rule out any determinate GR geodesics. Thus there are
still notable limitations embedded within general relativity despite GR’s notable successes with gravity MEE and time delays. Furthermore, as science better understands the universe, the limitations
of GR become even more apparent.
Black hole singularities are inconsistent with quantum action
Probably the most famous of all of general relativity’s limitations is the notion of a black hole singularity. Given enough mass, light’s gravity time delay will eventually be sufficient to capture
light into a singularity and therefore stop atomic time at an event horizon, two well worn predictions that simply cannot be the whole story.
Black hole event horizons are inconsistent with quantum action
A particle of matter that encounters the event horizon of a black hole is subject to two quite different predictions; gravity and quantum. According to much of the historical black hole modeling,
such a particle simply becomes part of the mass accretion and loses all information about its past.
More recent calculations find that, prior to reaching the event horizon, a particle is ripped into successively smaller pieces until the very, very small Planck limit. Those tiny pieces of matter
begin collapsing before they accrete and therefore never actually become part of the primary black hole. These eternally collapsing objects, eco’s, take the place of the primary black hole, but do
not really resolve the quantum paradox.
Quantum calculations predict something for a particle of matter at an event horizon, tearing into matter and antimatter particles, resulting in so-called Hawking radiation. The black hole event
horizon turns into a quantum firewall and just like with the eco, accretion action stops near the event horizon. There just cannot be these two very different fates for the same neutral particles.
Proper time is inconsistent with quantum time
Proper time is a key notion of GR and that proper time then becomes the fourth displacement of 4D spacetime. Ironically, time as a GR spatial displacement in effect does away with the uncertainty of
time. Because all motion in GR occurs as a result of gravity along determinate geodesic paths, the future is completely determined by the past.
Quantum time, on the other hand, is both reversible and uncertain and there is no stopping quantum time at a GR event horizon or anywhere else in the universe. However, time is simply a quantum
progress variable and there is therefore no quantum expectation value for a time duration or delay.
It is clear that the future for a given object simply cannot be both deterministic by the principles of GR and probabilistic by the principles of QM and it is likely that both GR and quantum times
will therefore need some kind of augmentation.
Dark matter and dark energy not explained
Dark matter is an extra gravity correction that explains the stabilities of galaxies and galaxy clusters while dark energy is yet another gravity correction needed to hold the universe together as
the CMB. The absence of any sign of these gravity corrections in GR is a little disconcerting and seems like a major flaw of GR to simply invent matter and energy objects.
Determinate geodesics inconsistent with quantum action
One of the basic assumptions of GR is that gravity action distorts or curves the 4D spacetime and that objects simply follow predetermined geodesics as minimum energy paths. Of course, quantum action
not only does not distort 4D spacetime, quantum action results in likely but not determinate futures. In quantum gravity, there will very likely be a number of possible futures instead of a
determinate one.
Lack of amplitude, phase coherence, interference, and entanglement
Our quantum reality depends on both the phase as well as the amplitude of matter. However, gravity force in GR only deals with the norms of quadrupole matter and time and so there is no role for
phase coherence or interference or entanglement with gravity. Since all of these notions of amplitude and phase figure prominently in quantum action, it is a major flaw in GR that there is no
corresponding quantum monopole or quadrupole gravity to complete our quantum reality of dipole charge.
Planck limit inconsistent with quantum uncertainty principle
Once a particle gets small enough, its own gravity will collapse it into a microscopic event horizon where time stops and quantum action does not apply. But quantum action functions everywhere in the
universe, even inside of black holes and there is no stopping quantum time. Quantum action limits the divisibility of matter and space to the uncertainty principle and to the quark, but there is
still something wrong with quantum time.
No absolute frame in GR
The basic relativistic tensor math of GR depends on the absence of an absolute frame of reference within continuous space and time. However, the CMB seems to represent just such an absolute frame of
reference for everywhere in the universe. In GR, the lack of an absolute frame means that we only see light in the universe within our event horizon or light cone and that there are past events that
are now beyond that event horizon. For example, the universe expansion means that the CMB will eventually move beyond our event horizon in about one billion years or so.
It would seem to be much more likely that the CMB represents an absolute frame of reference that all can seen and that necessarily closes the universe. We would not then be in an expanding universe
at all and the CMB will still be a CMB in one billion years, albeit somewhat evolved.
Quantum time is not consistent with proper time of GR
A determinate time in GR is incompatible with the uncertainty of quantum time. Quantum atomic clocks tick very precisely but their precision is limited by the uncertainty principle. Moreover gravity
clocks that tick like millisecond pulsars are also very precise and yet ms pulsar gravity clocks all decay. While that decay can be largely due to gravity and/or EM radiation, there is an average
intrinsic decay as well of 0.255 ppb/yr. That intrinsic decay means that ms pulsars tell two distinct times as their pulse periods and as their average decay.
It is therefore likely that quantum time also has both atom pulse periods and the same slow decay of atomic time as ms pulsars; 0.255 ppb/yr. This means that time actually has two dimensions; an
atomic time period and a gravity decay period and that two dimensional quantum time would then be consistent with the two dimensional gravity time of gravity ms pulsars.
Quantum space and motion are inconsistent with GR space and motion
Empty space and motion in empty space are both infinitely divisible notions that deeply underscore much of mainstream science. But while quantum space and motion are both quantized, GR space and
motion are both continuous and it is clear that notions of space and motion are simply fundamentally incompatible between QM and GR.
Many very smart people have worked very hard for nearly a century to make space and motion consistent between gravity and quantum, but to no avail. In fact, the notions of infinite divisibility for
both space and motion have actually been problematic since the time of Zeno of Elea, the Greek philosopher of 460 BCE.
The continuum of sensation of objects that fills time contrasts with the void of sensation that we presume exists as space
Unlike the void of empty space, for which we have no sensation, time is filled with a continuum of waves of sensations. There are no empty voids of time since all of light, sound, touch, smell, and
taste shine continuously onto us and our senses with a continuum of sensory information about objects and their backgrounds. Our sensation of object changes and time delays result in neural packets
of aware matter from which consciousness extracts information useful for prediction of action.
It is from this continuum of sensation that our consciousness imagines objects and also ignores or renormalizes any background time delays. Even though there are no voids of sensation in time, our
minds assign differences between object and background time delays to the lonely nothing of empty space. Space emerges to keep object sensations different from background sensations.
Objects that we sense have a different time delay from the backgrounds that we sense along with those objects. Our minds use space and motion to represent the difference in time delays as an absolute
time or Cartesian distance that separates objects from other objects and their backgrounds. Space and motion, in this sense, simply emerge as whatever they need to be in order to properly represent
the object changes and time delays of sensation, but space and motion do not exist in the same way that matter and time exist.
Therefore, the lonely nothing of empty space and motion within that space are notions that emerge from a more primitive reality of object changes and time delays. The nothing that we imagine as space
and the motion of objects in that nothing of space are both simply very useful representations of consciousness. Notions of space and motion help consciousness keep track of objects and make
predictions about the futures of those objects.
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Benchmarking multilinear Polynomial Commitment Schemes
This work would not have been possible without an outstanding effort from my team in the last 2 weeks. Huge thank you to Antonio Mejías Gil for implementing Hyrax and to Hossein Moghaddas for
developing Brakedown - you guys are true wizards!
In the previous post we gave the intuition of why efficient commitments to multilinear polynomials are important in the context of lookups in Lasso. We aim for this entry to be self-contained,
although the previous write-up provides useful context and a bigger picture.
This post
Here, we present the actual results of our technical contribution: benchmarks and implementation of 4 PC schemes:
• Ligero: native & recursive implementation; benches,
• Brakedown: native implementation; benches,
• Hyrax: native implementation; benches,
• multilinear KZG (PST13): benches.
To our knowledge, there is no single library to date that implements all the above schemes with the same backend. There exist some academic implementations of e.g. Hyrax in python; multilinear KZG
with the arkworks backend in Rust; Ligero/Brakedown with ZCash backend in Rust. As such, comparing the existing implementations for performance is hardly meaningful.
Our core contribution is implementing the first 3 schemes from scratch in arkworks - and adding benchmarks to the existing KZG module (from the amazing EspressoSystems/jellyfish library, which is
also built with the arkworks backend). We also implement a halo2 verifier of Ligero to allow for potential on-chain verification.
What should I aim for?
As hinted at in the previous post, our end goal should drive our design choices. We need to know what the constraints on our resources are. As such, some objectives we might wish to consider:
• Prover time
• Verifier time
• Proof size
• On-chain verification
• Recursion friendliness
Native benchmarks
Out of the 4 chosen schemes, 3 are by nature interactive. The standard way to render such schemes non-interactive is to apply the Fiat-Shamir transform, which in practice means instantiating these
with a cryptographic hash-based sponge. Hashing in-circuit (needed for recursive verification) is a few orders of magnitudes slower for hashes based on bitwise operations such as SHA256, than for
algebraic hash functions such as Poseidon. Therefore, for Ligero we consider 2 variants: optimized native with SHA256-based hashing, and recursion supporting (R) with Poseidon. KZG is non-interactive
to start with, so no such transformation is necessary. The results for Ligero are strong enough to let us draw some conclusions for non-interactive Brakedown and Hyrax.
Furthermore, as noted in the previous post, for the specific application to Jolt where the committed elements are “small”, we will consider additional variants of Hyrax and KZG using a modified
implementation of MSM, hackily optimized for <60 bits. We call this variant small (S) to differentiate from the case of arbitrary (A) coefficients for the commit phase.
Ligero is implemented with the rate $\rho = 0.5$, achieving the fastest prover on FFT-friendly fields. As far as Brakedown parameters go, we implemented the scheme following the 3rd parameter row
from Figure 2 of the Brakedown paper (relative distance $d = 0.04$, $\rho = 0.65$), like the benchmark presented in the original paper. Compared with Brakedown using linear codes with larger
distance, our version suffers from a larger number of column openings t (hence larger proofs), but benefits from an improved prover time. In both linear-code-based schemes, we assume that our PCS is
used within the larger context of a SNARK protocol, where the query point is an output of a random oracle. We are then able to skip the well-formedness check as described in Proximity Testing with
Logarithmic Randomness.
The above gives us a total of 7 schemes:
• KZG (S)
• KZG (A)
• Hyrax(S)
• Hyrax(A)
• Ligero
• Ligero (R)
• Brakedown
The last three are linear-code based, and they share the mode of operation up to the choice of a linear function used to encode each row of the coefficient matrix.
We start with polynomials that have their coefficients in the scalar field of the BN254 curve. We fix this particular curve to support potential verification on Ethereum.
The security of KZG and Hyrax is bound to the choice of the elliptic curve over which we perform group operations, and more specifically to the hardness of the discrete logarithm problem (DLP) in a
certain group associated to it.
Our Ligero and Brakedown implementations are parametrized with the security parameter. We aim for at least of 128 bits of security for these schemes.
We note that, although we aim for 128-bit security for the linear-code-based schemes, instead of e.g. 110 bits for a more fair comparison to the DLP on BN254, the performance difference from opening
a slightly fewer columns is only a few percentage points. We intentionally skip these to contain our parameter space.
Our contributions for Ligero, Brakedown and Hyrax are PR’ed to https://github.com/arkworks-rs/poly-commit and currently under review.
To run the all benchmarks there, checkout the relevant branch {ligero/brakedown/hyrax}-pp from https://github.com/HungryCatsStudio/poly-commit/ and execute: cargo bench.
Our contribution to Espresso Systems’ jellyfish library used for KZG has been successfully merged: https://github.com/EspressoSystems/jellyfish/pull/390.
Run the jellyfish KZG benchmarks with
• cargo bench --bench="pcs" --features="test-srs" (for time)
• cargo bench --bench="pcs-size" --features="test-srs" (for size).
Objective 1: Prover compute cost
All values are given in milliseconds. In all the tables that follow, an empty entry means that the benchmark was not run on purpose (e.g we were only interested in large $n$ for a few schemes). A
dash means that we attempted to run the test but ran out of compute resources.
Commit time (ms)
$$ \begin{array}{|l|c|c|c|c|c|} \hline \text{Scheme \ Variables} & 12 & 14 & 16 & 18 & 20 & … & 28 \\ \hline \text{Hyrax(A)} & 4.54 & 13.26 & 45.84 & 157.19 & 528.57 & \\ \hline \text{Hyrax(S)}
& 1.54 & 5.23 & 16.44 & 44.75 & 144.05 & \\ \hline \text{KZG(A)} & 4.10 & 13.62 & 43.68 & 136.87 & 474.86 & \\ \hline \text{KZG(S)} & 2.85 & 8.67 & 22.48 & 84.16 & 307.6 & \\ \hline \text{Ligero} &
1.61 & 3.63 & 10.29 & 33.18 & 128.65 & … & 25790\\ \hline \text{Ligero(R)} & 139.34 & 535.47 & 2085 & 8233 & 32704 & \\ \hline \text{Brakedown} & 3.45 & 8.60 & 18.49 & 49.09 & 131.35 & … &
23848 \\ \hline \end{array} $$
Ligero wins in terms of commit time for small-to-medium-sized polynomials. This seems to be in line with the benchmarks from the Brakedown paper. Brakedown has asymptotically linear time and indeed
we were able to show that the extra $\log(n)$ factor in Ligero prover starts to be a burden for large enough $n$. Specifically, Brakedown’s asymptotic advantage is of practical interest for $n \geq
Curiously, while Hyrax(S) takes full advantage of the small coefficients (~4x speedup for coefficients ~$\frac{1}{4}$ the original), the KZG(S) implementation seems to miss out on this optimizations.
We are investigating this further.
Ligero(R) performs significantly slower than Ligero for committing (and opening & verification), as expected.
Open time (ms)
Our current Hyrax implementation is fundamentally different from the rest of the schemes considered here in the sense that it achieves zero knowledge with respect to the evaluation. This means that
at the end of the prover-verifier interaction, the verifier does not learn the evaluation of the committed polynomial at the requested point, but is instead convinced that a certain evaluation
commitment is correct. This choice of Hyrax-PCS is motivated by the Hyrax zkSNARK and it is worth noting that there is a performance cost associated with the opening phase for the prover. A non-zk
Hyrax-PCS implementation is in progress, stay tuned.
$$ \begin{array}{|l|c|c|c|c|c|}\hline\text{Scheme \ Variables} & 12 & 14 & 16 & 18 & 20 \\ \hline \text{Hyrax} & 2.01 & 3.93 & 8.69 & 21.51 & 55.86\\ \hline \text{KZG} & 4.55 & 18.04 & 64.74 & 189.95
& 606.14 \\ \hline \text{Ligero} & 14.02 & 29.26 & 60.53 & 148.21 & 385.83 \\ \hline \text{Ligero(R)} & 147.99 & 547.19 & 2118 & 8327.3 & 32892 \\ \hline \text{Brakedown} & 39.091 & 94.077 & 206.11 &
453.70 & 1006.6 \\ \hline\end{array} $$
For opening, the competition is more even but Ligero emerges victorious here for large enough $n$, narrowly overtaking multilinear KZG (which has the edge for smaller polynomials).
We note that our implementations for linear code schemes miss out on a small optimization during the opening phase: the prover shouldn’t need to re-compute the Merkle Tree, since they’ve already done
so in the commit phase. Our engineer Hossein has already cut down the hashing work repeated by the prover by a number of $n^2$ (for $n$ columns) by avoiding column re-hashing in this PR to arkworks.
The current inability to serialize the entire Merkle Tree leaves an extra $2 \cdot n$ hashes to be optimized at a later stage.
Objective 2: Verifier compute cost
Verify time (ms)
$$ \begin{array}{|l|c|c|c|c|c|} \hline \text{Scheme \ Variables} & 12 & 14 & 16 & 18 & 20 \\ \hline \text{Hyrax} & 1.1638 & 1.3916 & 1.7535 & 2.2111 & 2.9807 \\ \hline \text{KZG} & 2.93 & 3.66 & 3.93
& 3.83 & 3.72 \\ \hline \text{Ligero} & 10.00 & 10.79 & 11.89 & 14.98 & 20.68 \\ \hline \text{Ligero(R)} & 131.09 & 231.49 & 428.38 & 808.24 & 1578.9 \\ \hline \text{Brakedown} & 70.82 & 120.85 &
140.54 & 152.45 & 179.99 \\ \hline \end{array} $$
Ligero is the expected winner in the code-based schemes due to the small number of column openings owing to its comparatively large code distance. The code distance in Brakedown requires opening many
more columns for a comparable level of security. We weren’t able to see the similar asymptotic advantage of Brakedown as for commit, at least for the values of $n$ that we benchmarked with. We
conjecture that for very large $n$ this might become apparent, but such values were impractical to bench.
Objective 3.1: Proof size (bytes)
$$ \begin{array}{|l|c|c|c|c|c|} \hline \text{Scheme \textbackslash Variables} & 12 & 14 & 16 & 18 & 20 \\ \hline \text{Ligero} & 244297 & 369121 & 606329 & 1068305 & 1979817 \\ \hline \text
{Brakedown} & 1901009 & 3229073 & 4232905 & 6064001 & 9549721 \\ \hline \text{Hyrax} & 2360 & 4408 & 8504 & 16696 & 33080 \\ \hline \text{KZG} & 776 & 904 & 1032 & 1160 & 1288 \\ \hline \end{array}
Clearly schemes involving elliptic curve operations (KZG & Hyrax) beat the rest by orders of magnitude, with the clear winner being KZG. This is expected, as the proof size for linear code PCS
presented here is $\mathcal{O}({2^{n/2}})$.
Objective 3.2: Commitment size (bytes)
While not explicitly stated in the problem definition, we believe it is important to take the commitment size into account. Hyrax is the only scheme with commitment size dependent on the size of the
polynomial (one elliptic curve point per row of the coefficient matrix). For all other schemes, this size is a small constant: in the case of linear-code based PCS, it’s the size of the hash digest;
and in the case of KZG, it a single group element - which for BN254 happens to also fit into 64 bytes.
$$ \begin{array}{|l|c|c|c|c|c|c|c|}\hline \text{Scheme \textbackslash Variables} & 10 & 12 & 14 & 16 & 18 & 20 & 22 \\ \hline \text{Hyrax} & 2056 & 4104 & 8200 & 16392 & 32776 & 65544 & 131080 \\ \
hline \text{Ligero} & 64 & 64 & 64 & 64 & 64 & 64 & 64 \\ \hline \text{Brakedown} & 64 & 64 & 64 & 64 & 64 & 64 & 64 \\ \hline \text{KZG} & 64 & 64 & 64 & 64 & 64 & 64 & 64 \\ \hline \end{array} $$
Recursive benchmarks
When we say that we are using SNARK recursion or composition, we refer to having the verifier of one (inner) scheme represented as a circuit of another (outer) scheme. We first run the prover for the
inner circuit to obtain a valid (inner) proof. We then supply this proof as a (usually private) input to the outer circuit, and finally run the prover of the outer circuit, which implicitly means
running an accepting inner verifier. This might seem a bit counterproductive, but there can be good reasons for this:
• We have access to a scheme with a fast prover but large proof sizes, and yet are required to output small proofs. We can “wrap” the fast-prover scheme in a short-proof scheme.
• We have specific requirements on the verifier logic, e.g. we have an on-chain verifier that only accepts Groth16 proofs.
Now, however, in order to convince the outer verifier of some statement, we have to run both the inner and the outer prover. We must carefully consider when the balance tips in our favor when
choosing such an approach.
Note that for recursive verification, we don’t necessarily care about the “native” verifier time, as the inner-verifier is in a way subsumed into the outer prover.
For the recursive benchmarks, we currently only have the empirical results for one scheme, Ligero, which we have implemented using Axiom’s halo2-lib. At the moment, this scheme remains the IP of our
friends at Modulus Labs, for whom we’ve implemented the protocol. They will soon be open-sourcing the codebase - we ask to be trusted on the results for the time being. Modulus Labs have kindly
agreed to let us use the code for benchmarking, for which we thank them.
We extrapolate the number of columns that need to be opened in Ligero to the number opened in Brakedown. We provide very rough estimates for KZG and Hyrax based on the BN254 MSM benchmarks provided
in https://github.com/axiom-crypto/halo2-lib#bn254-msm. Here MSM refers to multi-scalar multiplication (i. e. adding together a number of group elements, each multiplied by a potentially different
scalar), which is the core opoeration in those two PCSs.
Objective 4.1: Recursive prover compute cost
This is basically the outer-prover cost described above. When implementing schemes for recursive verification, one needs to take into account the fact that the inner prover, adapted for
recursion-friendliness, pays a large overhead in foregoing the efficient hash functions like SHA256 in favor of circuit-friendly Poseidon.
Proof generation time, in seconds (halo2)
$$ \begin{array}{|l|c|c|c|c|} \hline\text{Scheme \textbackslash Variables} & 14 & 16 & 18 & 20 \\ \hline\text{Hyrax*} & 64.28 & 71.18 & 139.98 & 261.66 \\ \hline\text{KZG** (k=17)} & 88.2 & 102.24 &
115.02 & 127.8 \\ \hline\text{Ligero} & 56.76 & 93.04 & 243.10 & - \\ \hline\text{Brakedown***} & - & - & - & - \\ \hline\end{array} $$
*Hyrax costs are estimated by benchmarking the proving time for two MSMs of size $2^{n/2}$ each (for a multilinear polynomial in $n$ variables). We adapt the code from halo2-lib to perform benchmarks
over different numbers of base-scalar pairs.
**KZG costs are estimated by benchmarking a single pairing from halo2-lib, and multiplying by $n$. The actual computation of multilinear KZG verification involves a multipairing of $n$ pairs, which
in general can be optimized to be faster than $n$ individual pairings as reported here.
***Brakedown was out of reach: we were already unable to run the Ligero verifier for large polynomials with the hardware we selected for benchmarking. This is due to over 250k+ halo2 advice cells
(for $n = 18$), most of which come from the huge amount of hashing that needs to be performed in the non-interactive setting of linear-code-based PC schemes. Brakedown performs at least an order of
magnitude more column openings than Ligero (under reasonable choices of parameters for both), and since the verifier needs to hash each column, we conclude that it would be infeasible within the
given server setting.
In all estimates (Hyrax and KZG), we only focus on the key cost (MSM/pairing) and completely disregard auxiliary costs, e.g. to make the scheme non-interactive.
Objective 4.2: Recursive verifier compute cost
This refers to the outer-verifier costs. As noted above, in the recursive setting, the inner verifier doesn’t matter much on its own (aside from offline sanity checks if anything) - it is never run
natively and its computation is subsumed into the outer prover.
For all circuits we ran in halo2, we only ever encountered $<1s$ runtimes. We conclude that even when running this in an EVM-verifier, the gas costs will be acceptable and we don’t delve on the
Objective 5: Recursive verifier proof size
The halo2 proof sizes are tightly coupled with the parametrization of the halo2 circuit, namely the (logarithm of the) maximum number of rows k in the advice table. Rather than reporting the proof
size for varying k , we instead only present the size for the choice of k that results in the fastest prover, as noted in Objective 4.1: Recursive prover compute cost. The estimates for KZG, Hyrax
and Brakedown follow the same methodology.
Proof size in bytes (halo2)
$$ \begin{array}{|l|c|c|c|c|} \hline\text{Scheme \textbackslash Variables} & 14 & 16 & 18 & 20 \\ \hline\text{Hyrax} & 9856^* & 34624 & 67712 & 67264^* \\ \hline\text{KZG (k=17)} & 124096 & 141824 &
159552 & 177280 \\ \hline\text{Ligero} & 26048 & 46112 & 85536 & - \\ \hline\text{Brakedown**} & - & - & - & - \\ \hline\end{array} $$
$*$The fastest Hyrax verifier for $n=14$ and $n=20$ uses k = 20, whereas the middle values for $n$ have the fastest runtimes with k = 19. This explains the non-doubling (roughly) proof sizes as we
increase $n$.
**As for the recursive prover time, we ran out of memory for the largest instance of Ligero, and we expect the same for Brakedown.
Picking the right PCS is hard.
While implementing the above schemes, we realized that there are a plethora of code and parameter optimizations to be made for each, which depend on the specific optimization target (prover/verifier
time, etc.). The above benchmarks are just the beginning. While many of the opening proof generation instances could be batched in the case of Lasso, the commitments have to be made to each
polynomial separately.
On-chain verification of ZK proofs might be a fun application that has certainly accelerated the progress in the space of SNARKs. Nevertheless, my personal take is that the future users of this
technology will be off-chain, e.g. an AI judge or proof-of-identity. We need to open up to the possibility of not having minuscule proofs; sub-second verification times might be unnecessary; and the
real bottleneck becomes the prover.
Linear-code-based PCSs offer good tradeoffs in this space: high parallelism thanks to their matrix structure; flexibility from choosing the code distance; plus some schemes like Brakedown also are
field-agnostic: no FFTs means fewer restrictions are placed on the group of units of that field. I’ll be surprised if we don’t see an improvement of at least one order of magnitude thanks to both the
theoretical and implementation advances within the next year or two. Till then, there is no clear winner, although this PCS family seems to be a good choice for the time being.
All our benchmarks were run on the following dedicated instance:
AMD Ryzen™ 9 7950X3D
• CPU: 16 cores / 32 threads @ 4.2 GHz
• Generation: Raphael (Zen 4) with AMD 3D V-Cache™ Technology
• RAM: 128 GB ECC DDR5 RAM
Future work
• Zeromorph: transformation from any univariate scheme to multilinear. Does applying Zeromorph to known univariate schemes bring better performance?
• BaseFold: newest work on multilinear PCS, see https://eprint.iacr.org/2023/1705.pdf.
• Can we introduce any modifications to linear-code based PCS to benefit from “small” coefficients?
• In next steps, we would like to provide benchmarks over BLS12-381. This curve was proposed by the Zcash team as a potential replacement for BN254, whose security is estimated to be below the
initially assumed 128 bits. There are some projects underway to bring BLS12-381 curve operations as precompiles on Ethereum. The curve’s future on Ethereum is uncertain, but it has already found
its way into other blockchains.
• Upgrade to a newer circuit friendly hash function. Some of the new doubly-friendly constructions aim for a tradeoff between native and in-circuit efficiency, e.g. https://eprint.iacr.org/2023/
• Benchmarking batch openings for multiple polynomials and/or multiple queries. Ligero doesn’t have homomorphic commitments, so while opening multiple points for the same polynomial can be achieved
by taking linear combinations, opening multiple polynomials at the same (or multiple) points(s) doesn’t have any obvious “shortcuts” in PC Schemes using hashes, unlike in EC-based schemes (see
our write-up on batching in KZG).
• More schemes and more variants!
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A graph is called planar if it can be drawn in the plane without any crossings. In a drawing of a graph, nodes are identified with points in the plane, and edges with lines connecting the
corresponding end nodes. No edge is allowed to cross another edge or node in the drawing.
Write a program that determines whether a given undirected graph is planar or not.
Input consists of zero or more test cases. Each test case consists of a graph.
A graph is given in the following way: First, a line contains two integers n and m, where n denotes the number of vertices of the graph, and m denotes its number of edges (1 ≤ n ≤ 20 and 0≤ m). Then
follow m lines, one for every edge of the graph, each containing two integers u and v (with u≠ v) meaning that the graph contains the edge {u,v}. Vertices in the graph are labelled from 1 to n. There
are not repeated edges.
For each test case, print a line with the string YES if the graph is planar or with the string NO otherwise.
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a particularly annoying subnetting question
I was grinding away on Subnettingquestions.com like a champ, and then i got this one:
Question: How many subnets and hosts per subnet can you get from the network 172.23.0.0/23?
Answer: 128 subnets and 510 hosts
I cant even begin to follow the math on this one. I would really appreciate anyone helping me on this one. I gound away at this one for an hour!
So, you are given an IP and a subnet mask of 172.23.0.0/23
The easiest way that I do it is I break it down like this.
1. To find how many subnets, you need to know the difference between classful and classless. The classful ranges are A = 1-126 with a subnet mask of /8 (127 is in here, but it's loopback), B =
128-191 with a subnet mask of /16, and a C = 192-223 with a subnet mask of /24. They have given you a classless, so find the difference. Well, 172 is in a Class B range, so the difference of /23
and /16 is 7.
- So, I set up my decimal to binary chart of 128 64 32 16 8 4 2 1, I guess it's a per bit representation of an octet. Don't qoute me... Anyway count over from right to left starting at 2 (because
binary is multiples of 2) 7 times and you get 128. So you have 128 subnets. If you're good at math, you can do the 2 to the nth, nth being 7. I just like the graph.
2. For how many hosts, you use the leftover bits. You know there is 32 bits. So, the difference of /32 and the given /23 is 9.
- This time, your chart needs to be bigger. Binary is multiples of 2, so just go out two more times
512 256 128 64 32 16 8 4 2 1. Count over 9 times from right to left, starting at 2 and you get 512. Remember that for hosts, the math is 2 to the nth - 2. nth being 9. Minus 2, because you have a
network IP and a broadcast IP. So 512 - 2 = 510.
I hope this helped and actually didn't confuse you even further...
9 network bits 2, 4, 8, 16, 32, 64, 128, 256 next is 512 then subtract two for the network and broadcast. now for the 7 subnet bits 2, 4, 8, 16, 32, 64, 128 For questions like this you need to
know your classes to understand the subnets. Just keep trying the more practice you have the quicker you become.
Question: How many subnets and hosts per subnet can you get from the network 172.23.0.0/23?
Answer: 128 subnets and 510 hosts
The question is the issue. It should be:
Which is the maximum number of subnets for
And the maximum number of hosts?
You would better get used to this... Cisco exams have plenty of questions that are incorrectly worded like this!:)
Thank you very much for your answers. Expect to see more in the future as i begin my journey into networking....
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ball mill work index
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then it is necessarily to provide a correction. The goal of this work is to create an equation to adjust a Bond ball mill work index from one P 80 basis
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A Bond Work index mill ball charge and closing screen product size distributions for grinding crystalline grains. Int. J. Miner. Process., 165 (2017), pp. 814, / View PDF View article View in
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The laboratory procedure for running a Bond ball mill work index test requires that the operator choose a closing screen sieve size. The instruction is to choose a sieve size that results in the
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at 10 or 14 mesh, and ball mill Work Index tests are run at the desired grind if finer than 28 mesh. All the data obtained is evaluated and ...
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Dear all, Have a good day! I am trying to determine ball mill work index for a very soft . If I follow bonds procedure to do experiment, not able to determine. The following is the data of ore
sample. F80% = 2300mic Mesh of grind =300mic 300mic in feed = 47% In this case if the sample is gri ... Work index is as per plant operation data is 2 ...
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and the Bond Ball Mill Work Index Test (GMG, 2021). The sampling and surveying guideline (GMG, 2016) pro vides addi tional detail on how to collect the required data and is critical to this
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The work index can either be measured in the laboratory (the Bond ball mill work index determination is a common example) or it can be calculated from the operating performance of a milling ...
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This Table of Ball Mill Bond Work Index of Minerals is a summary as tested on 'around the world sample'. You can find the SG of each mineral samples on the other table.
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A Bond Ball Mill Work Index test is a standard test for determining the ball mill work index of a sample of ore. It was developed by Fred Bond in 1952 and modified in 1961 (JKMRC CO., 2006).
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W is the work index measured in a laboratory ball mill (kilowatthours per metric or short ton) P 80 is the mill circuit product size in micrometers; F 80 is the mill circuit feed size in
micrometers. Buhrstone mill. Another type of fine grinder commonly used is the French buhrstone mill, which is similar to oldfashioned flour mills.
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Abstract The Frederick C. Bond has several grinding indices (ball, rod, abrasion), one of which has been a significant guide for ball mills. The Bond ball mill work index is an expression of the
material's resistance to ground and a measure of the grinding efficiency.
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A Bond Ball Mill Work Index may also be used in the simulation and optimisation of existing mill(s) and the associated grinding circuit(s). Sample requirements: A minimum of 8 kg of material
crushed to nominally minus 10 mm is preferred. JKTech would stage crush the sample to minus mm, as required for the Bond Ball Mill Work Index test feed.
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For any circuit, whether a crushing circuit, a rod mill, or a closed ball mill circuit, the Work Index always means the equivalent amount of energy to reduce one ton of the ore from a very large
size to 100 um. The sample was received crushed appropriately for the ball mill test.
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The Bond ball mill grindability test is one of the most common metrics used in the mining industry for ore hardness measurements. The test is an important part of the Bond work index methodology
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Work index is the relation between the SEC and the amount of breakage in an ore. The most common form of this relationship is given as Equation 1, and is often referred to as "Bond's equation" or
"Bond's law." E=10× √P − √ F 80 80 ) ( 1 ) Where: E is the specific energy consumption in kWh/t
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The work index covering grinding of fine particles is labelled M ib. M ia values are provided as a standard output from a SMC Test® (Morrell, 2004a) whilst M ib values can be determined using the
data generated by a conventional Bond ball mill work index test (M ib is NOT the Bond ball work index). M ic and M
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Many researchers have tried to find alternative methods for determining this Index to reduce labor costs, sample weight, or to get one without standard equipment. This paper was carried out with
the purpose of reviewing, classifying and testing the existing methods for determination the Bond Ball Mill Index.
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The commonly used grindability tests included in the database are the Bond work indices for ball milling, rod milling and crushing; the drop weight test results A, b, A×b, DWi, Mia, Mic, Mih and
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Istilah BWI sendiri dapat disebut lagi menjadi Bond Ball Mill Work Index (BBWI) atau Bond Rod Mill Work Index (BRWI). Ini tergantung dari media gerus yang digunakan selama percobaan, apakah bola
baja atau batang baja. Untuk menghitung BWI digunakan formula sbb: Dimana. Wi = Nilai bond work index yang dicari (kwh/ton)
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This was displayed by a reduction in Bond ball mill work index (BBMWI) of the ore from to kWh/t after calcination. ... The ball mill had 12 grinding balls (each grinding ball had mm ...
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The work index (Wi) expresses the kWh required to reduce a short ton of material from theoretically infinite feed size to 80 % passing a square screen opening of 100 microns. The work index
values apply to ball mills grinding wet in closed circuit. For dry grinding in closed circuit the work input calculated by Bond's basic Third Theory ...
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MenéndezAguado et al. examined the possibility of determining the work index in a Denver laboratory batch ball mill with the same inner diameter as the Bond ball standard mill. The research was
performed on the size class of 100% − mm using samples of gypsum, celestite, feldspar, clinker, limestone, fluorite, and copper slag.
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The Bond ball mill work index test is thought to be additive because of its units of energy; nevertheless, experimental blending results show a nonadditive behavior. The SPI(R) value is known not
to be an additive parameter, however errors introduced by block kriging are not thought to be significant .
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Time dimension - Functions
Time dimension - Functions
A time dimension filter comes with a standard set of time calculations. Click on the plus icon next to Functions to expand the list of available calculations. The current day can be defined by an
administrator (Admin > Settings > OLAP Report > Time Dimension - Current Day) as:
1. The very last member of the level.
2. The very last member that has value in any of the measures.
3. Formula. Such as one based on the previous calendar day: Format(Now()-1, "MMMM d, yyyy") or one based on a UDF (User Defined Function) that retries the current day value from the database:
The following calculation types are supported:
│ Description │ Sample Question │
│Current [Level] │What are the sales numbers for today? │
│Previous [Level] │What were the sales numbers last year? │
│Current [Level] previous [Parent Level] │What was the profit for the current month last year? │
│Previous [Level] previous [Parent Level] │What was the profit for the previous month last year? │
│Beginning Previous [Level] until matching current [Child Level] │What was the profit for this previous month up to the matching current date? │
│Beginning Current [Level] previous [Parent Level] until matching current [Child Level]│What was the profit the same month last year up to the matching current date?│
│Trailing [Number of descendents] [Child Level] │What is the profit for the last three months? │
│Previous Trailing [Number of descendents] [Child Level] │What was the profit for the last three months three months ago? │
Below is an example of calculations created for Foodmart 2000 Time dimension. The dimension has only three levels (Year, Quarter and Month). In practice, the calculations will be different depending
on number and names of levels in your time dimension.
If your time demission is not marked as time dimension in Analysis Services you can mark it by going to Options > Dimensions tab > select your dimension > Time dimension > check.
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Science:Math Exam Resources/Courses/MATH307/December 2006/Question 01 (b)
MATH307 December 2006
Work in progress: this question page is incomplete, there might be mistakes in the material you are seeing here.
• Q1 (a) • Q1 (b) • Q1 (c) • Q2 (a) • Q2 (b) • Q2 (c) • Q2 (d) • Q3 (a) • Q3 (b) • Q3 (c) • Q4 (a) • Q4 (b) • Q4 (c) • Q4 (d) • Q4 (e) • Q5 (a) • Q5 (b) • QS101 6(a) • QS101 6(b) • QS101 6(c) • QS101
6(d) • QS101 7(a) • QS101 7(b) • QS101 7(c) • QS101 7(d) • QS101 7(e) • QS101 7(f) • QS101 7(g) • QS101 8 • QS102 6(a) • QS102 6(b) • QS102 6(c) • QS102 7(a) • QS102 7(b) • QS102 7(c) • QS102 7(d) •
QS102 7(e) • QS102 8(a) • QS102 8(b) •
Question 01 (b)
Consider the matrix ${\displaystyle \displaystyle A}$ and column vector ${\displaystyle \displaystyle {\vec {b}}}$.
${\displaystyle \displaystyle A={\begin{bmatrix}1&1&1\\0&0&1\\2&3&4\end{bmatrix}}}$ and ${\displaystyle \displaystyle {\vec {b}}={\begin{bmatrix}1\\2\\3\end{bmatrix}}}$
Recall that the ${\displaystyle \displaystyle PA=LU}$ factorization can be used to rewrite the system ${\displaystyle \displaystyle A{\vec {x}}={\vec {b}}}$ as two systems with triangular coefficient
matrix. Write down these two triangular systems.
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is
correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
• If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you
are stuck or if you want to check your work.
• If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem
and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.
Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies.
let ${\displaystyle y={\begin{vmatrix}y_{1}\\y_{2}\\y_{3}\end{vmatrix}}}$
${\displaystyle x={\begin{vmatrix}x_{1}\\x_{2}\\x_{3}\end{vmatrix}}}$
${\displaystyle {\begin{vmatrix}1&0&0\\1/2&1&0\\0&0&1\end{vmatrix}}{\begin{vmatrix}y_{1}\\y_{2}\\y_{3}\end{vmatrix}}={\begin{vmatrix}1\\2\\3\end{vmatrix}}}$
${\displaystyle {\begin{vmatrix}2&3&4\\0&-1/2&-1\\0&0&1\end{vmatrix}}{\begin{vmatrix}x_{1}\\x_{2}\\x_{3}\end{vmatrix}}={\begin{vmatrix}y_{1}\\y_{2}\\y_{3}\end{vmatrix}}}$
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Who Else Really Wants To Learn About math for kids?
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EG1989 Proceedings (Technical Papers)
Permanent URI for this collection
Accelerated Radiosity Method for Complex Environments
Pixel Selected Ray Tracing
The Macro-Regions: An Efficient Space Subdivision Structure for Ray Tracing
The Structure of Tube - A Tool for Implementing Advanced User Interfaces
A VLSI Chip for Ray Tracing Bicubic Patches
Components, Frameworks and GKS Input
Two Object-Oriented Models to Design Graphical User Interfaces
On the Software Structure of User Interface Management Systems
Visualisation of Digital Terrain Data
A Parallel Image Computer with a Distributed Frame Buffer: System Architecture and Programming
Deformation of Solids with Trivariate B-Splines
Non-Planar Polygons and Photographic Components for Naturalism in Computer Graphics
Forest of Quadtrees: An Object Representation for 3D Graphics
Adding Parallelism in Object Space to the Rendering Pipeline
Visualisation in Astrophysics
Hierarchical Texture Synthesis on 3-D Surfaces
A Topological Map-Based Kernel for Polyhedron Modelers: Algebraic Specification and Logic Prototyping
Highlight Shading: Lighting and Shading in a PHIGS+/PEX-Environment
A Model for Description and Synthesis of Heterogeneous Textures
Toward Realistic Formal Specifications for Non-Trivial Graphical Objects
Anti-Aliasing by Successive Steps with a Z-Buffer
A Reference Model for the Visualisation of Multi-Dimensional Data
Representing Tolerance Information in Feature-Based Solid Modelling
The Use of Finite Element Theory for Simulating Object and Human Body Deformations and Contacts
Magnenat-Thalmann, Nadia
Thalmann, Daniel
Graph Grammars, A New Paradigm for Implementing Visual Languages
Message-Based Object-Oriented Interaction Modeling
Recent Submissions
• Accelerated Radiosity Method for Complex Environments
(Eurographics Association, 1989) Xu, Hau; Peng, Qun-Sheng; Liang, You-Dong
As form-factor calculation costs about 90% of the computing time when applying radiosity approach for realistic image synthesis, it is of great significance to reduce the required computation, An
accelerated radiosity algorithm for general complex environments, based on environment localization and the directional form-factor concept, is presented in this paper. First we subdivide the
object space into many regions. Objects contained in each region are adjacent to each other and pose more illumination effects to their neighbours. Then form-factors are calculated in each local
environment,. The radiant light energy transfer between different regions is evaluated at their common boundaries. Directional form-factors are introduced to simulate the interaction of light
between local environments and between non-diffuse surfaces. Comparison is made to existing algorithms. Statistic results and theoretical analysis show that the new algorithm is much faster than
previous ones. The technique is especially fit for interactive design and animation sequence since modification to the shape or location of objects usually happens in local environments.
• Pixel Selected Ray Tracing
(Eurographics Association, 1989) Akimoto, Taka-aki; Mase, Kenji; Hashimoto, Akihiko; Suenaga, Yasuhito
This paper presents a new ray-tracing acceleration technique called Pixel Selected Ray-Tracing (PSRT). PSRT uses undersampling based on Iterative Central Point Selection(ICPS) along with checking
for similarities among trees in neighboring pixels. By using ICPS and trees, the largest danger of missing object borders can be drastically reduced. Although the speed increase attributable to
PSRT varies with the image generation environment, according to experiments comparing PSRT with standard ray-tracing, PSRT is 2.6 to 8.2 times faster than standard ray-tracing for 512 by 512
pixel images, maintaining the same visual image quality. It is true that images generated by this method may contain very small errors. However, such errors can be reduced and may be made
visually negligible by using ICPS and the trees of ray-object intersection to check for similarities.
• The Macro-Regions: An Efficient Space Subdivision Structure for Ray Tracing
(Eurographics Association, 1989) Devillers, Olivier
Ray tracing is the usual image synthesis technique which allows rendering of specular effects. The use of space subdivision for ray tracing optimization is studied. A new method of subdivision is
proposed : the macro-regions. This structure allows a different treatement of the regions with a low density of information, and the regions with a high density of information. A theoretical and
practical study of space subdivision methods -grid, octree- and the macro-regions structure is presented.
• MICRO-UIDT: A User Interface Development Tool
(Eurographics Association, 1989) Mao, Qijing
A user interlace development tool called Micro-UIDT is described. Micro-UIDT provides an interactive design environment fur the user interface designers. The designing language for defining user
interfaces are visual and nonprocedural. State transition diagrams are used as the notation for specifying man-machine dialogue control. Some extensions are added into the notation, so that the
user interfaces with semantic feedback can be defined. The approach of specifying the presentation of application data is bottom-up and direct operation. The designing language for this has two
major description facilities, one for describing graphics and another for describing the relationships between the graphics and the application variables. A concept of graphic object is used in
the language, which can make the design result be abstracted and reused. The user interfaces developed with Micro-UIDT are small in size and fast in speed.
• The Structure of Tube - A Tool for Implementing Advanced User Interfaces
(Eurographics Association, 1989) Hill, Ralph D.; Herrmann, Marc
Good user interfaces are very costly to implement and maintain. As user interfaces become more advanced, moving to various forms of direct manipulation, they become even more expensive and
difficult to implement. In the recent past, many user interface management systems and related tools for the rapid development of user interfaces have been developed. Some have been successful at
reducing interface development costs for some styles of interface, but none fully address the requirements of advanced direct manipulation interfaces. We claim this is because they are founded on
basic models and components that work only for simpler interaction styles. We present Tube, a tool for the rapid development of advanced direct manipulation user interfaces, describe its
structure, and show how it differs from, and is better than, traditional structures.
• Ray Tracing Polynomial Tensor Product Surfaces
(Eurographics Association, 1989) Giger, Christine
With regard to ray tracing algorithms for polynomial tensor product surfaces, the most timecritical step is to find an intersection point of a ray and a surface. In this case it proves to be very
difficult to decide whether numerical methods will converge to the correct solution. In this paper we present a new method based on numerical algorithms which is suitable to solve the
intersection problem. We mention how to force correct convergence and give some information about techniques to speed up the algorithm.
• A VLSI Chip for Ray Tracing Bicubic Patches
(Eurographics Association, 1989) Bouatouch, Kadi; Saouter, Yannick; Candela, Jean Charles
This paper deals with the integration of a VLSI chip dedicated to ray tracing bicubic patches. A recursive subdivision algorithm is embedded in this chip. The recursion stops when the termination
conditions are met. A software implementation allowed for the determination of key parameters which influenced the choice of the proposed chip' architecture. Only some modules of the chip are, at
the present time, simulated and laid out, the rest is being implemented. A detailed description of the chip' modules is given.
• Components, Frameworks and GKS Input
(Eurographics Association, 1989) Duce, D. A.; Ten Hagen, P.J.W.; Van Liere, R.
This paper was inspired by the Components/ Frameworks approach to a Reference Model for computer graphics, currently under discussion in the ISO computer graphics subject committee. The paper
shows how a formal description of the GKS input model may be given in Hoare’s CSP notation and explores some extensions in which some of the components in the GKS model are replaced by more
interesting ones. The paper thus demonstrates some of the power and flexibility inherent in the Component/ Frameworks idea. The use of a formal notation led to a deepening of the authors’
understanding of the input model and suggested some different ways of looking at the input model.
• Two Object-Oriented Models to Design Graphical User Interfaces
(Eurographics Association, 1989) Hübner, Wolfgang; Gomes, Mario Rui
Object-oriented concepts are well-suited to deal with the characteristics of user interfaces. Up to now several attempts to integrate the object-oriented paradigm in user interface models were
evolved and led to distinctive resulting models due to different requirements of the target application area. Within this paper two independently developed object-oriented interaction models are
presented which emphasize the graphical requirements to user interfaces. These are among others its hierarchical nature, dynamical topology of the user interface, strong connection between input,
output and the semantics of the application and the diversity of the graphics input devices and interaction techniques. Both approaches converge in the following aspects: Instead of having
separated user interface layers the components of an interactive graphics application's user interface are embedded locally within interaction objects. Therefore dialogue control, input, output
and the dynamical behavior are organized as a micro-cosmos within each object. Compound interaction objects can be designed. Temporal logical operators are used to specify the dialogue. Tools to
support the implementation of each model are described. By describing both approaches this paper could be a contribution toward establishing a uniform object-oriented framework for the design of
graphical user interfaces.
• On the Software Structure of User Interface Management Systems
(Eurographics Association, 1989) Burgstaller, Johann; Grollmann, Joachim; Kapsner, Franz
Specific systems for the development of user interfaces (Uls) are used today for coping with the increasing problems of human-computer communication. Some of those systems are based on
well-defined models for humancomputer interaction. Important requirements of such systems are: consideration of standards, most notably graphics functionality and windowing functionality,
openness to all interaction styles, and provision of comfortable design tools that allow UI prototyping. An evaluation of the existing systems reveals that they fulfill only some of those
requirements. We present a layered model for the interface between an application's functionality and its UI, which explicitly takes care of standards. Based on this model we implement a system
for efficient design and administration of Uls. An internal interface among all tools, namely, a PROLOG-like formalism used for the description of UI objects, is of central importance. This
formalism makes all tools independent from the dialog objects, hence our system is truly open. The core of our system consists of a comfortable graphically interactive editor for UI design and an
interpreter. The interpreter is mainly responsible for the presentation of Uls which are described according to this formalism. The output of the editor will be a description of Uls according to
the formalism. Our oal is the development of a functionally complete object-oriented set of formalism-based tools; these tools will also use artificial intelligence techniques for an adaptation
of Uls to user needs.
• Visualisation of Digital Terrain Data
(Eurographics Association, 1989) Thiemann, Rolf; Fischer, Joachim; Haschek, Guido; Kneidl, Gerald
Visualisation technology has found practical application in the field of terrain data processing. A raster data base (RDB-) concept will be introduced, i.e. a multi-dimensional concept of terrain
data using elevation data, description data and/or aerial image or satellite data. Other data sources in raster or vector form may also be considered. Methods of 2- and 3-D-imaging of terrain
data are pre- sented. Two-dimensional presentation will include grey- and colorcoding of different RDB layers. Techniques for superimposing two datasets are shown using relief data as one source.
Color-coding, color-space transformation or a slide-effect process will be used for superimposition. The mapping of aerial image or satellite data onto the relief data will be achieved by known
texture-map algorithms. The overlay technique is used for superimposing raster with vector data from geographic data bases. For superimposition, the geometry of the two datasets must not be
different. Many applications need the generation of perspective views of the terrain data. For the generation a terrainraytracer will be introduced. Color impression again will be achieved by
texture-mapping. Fore- and background can be handled separately. All algorithms presented are implemented in the GELA software.
• A Parallel Image Computer with a Distributed Frame Buffer: System Architecture and Programming
(Eurographics Association, 1989) Potmesil, Michael; McMillan, Leonard; Hoffert, Eric M.; Inman, Jennifer F.; Farah, Robert L.; Howard, Marc
We describe the system architecture and the programming environment of the Pixel Machine - a parallel image computer for 2D and 3D image synthesis and analysis. The architecture of the computer
is based on an array of asynchronous MIMD nodes with a parallel access to a large frame buffer. The system consists of a pipeline of pipe nodes which execute sequential algorithms and an array of
m x n pixel nodes which execute parallel algorithms. A pixel node accesses every m-th pixel on every n-th scan line of a distributed frame buffer. Each processing node is based on a high-speed,
floating-point programmable processor. The programmability of the computer allows all algorithms to be implemented in software. A set of mapping functions transfers image algorithms written for
conventional single-processor computers to algorithms which execute in the pixel nodes and access the distributed frame buffer. The ability to use floating-point computations in pixel operations,
such as antialiasing, ray tracing, and filtering, allows high-quality image generation and processing. The image computer provides up to 820 megaflops of peak processing power and 48 megabytes of
memory for data-visualization applications.
• Deformation of Solids with Trivariate B-Splines
(Eurographics Association, 1989) Griessmair, Josef; Purgathofer, Werner
Solid geometric models can be deformed to free-form solids by the use of trivariate B-splines. This paper describes the problems of implementing such transformations for shaded rendering. The
surfaces are subdivided into triangles adaptively so that the error in image space is limited. This adaptive triangulation ensures a smooth appearance of the resulting pictures.
• Non-Planar Polygons and Photographic Components for Naturalism in Computer Graphics
(Eurographics Association, 1989) Hofmann, Georg Rainer
The measuring of natural objects like landscapes and already existing (not simply planned!) buildings produces natural data. That data of hue geometry typically consists of Non-planar Polygons.
These may be triangulized, but the results are unfortunately: - a large increase of the number of polygons, - texture mapping becomes more complicated, - facetting effects in the rendered image.
This paper addresses methods and algorithms of the direct rendering of Non-planar Polygons. Special "texture mapping" is presented to insert especially Photographic Components in Non-planar
Polygons to obtain naturalistic images. With Photographic Components, a very simple illumination model is sufficient to obtain good results in rendering quality. Further on an application example
is presented. The images of this example are outstanding both for their naturalism and the little computer CPU time spent for their rendering. Basics on naturalism and photorealism in Computer
Graphics are discussed.
• Forest of Quadtrees: An Object Representation for 3D Graphics
(Eurographics Association, 1989) Kaufman, Arie; Bandopadhay, Amit
A forest of quadtrees is proposed as an alternative data structure for representing and manipulating 3D and 2.5D graphics. A data representation of a forest offers space savings over common
quadtrees by concentrating the vital information and discarding unused pointers. Several properties of the forest of quadtrees and the basic operations for display and elementary transformations
like rotation, reflection, enlargement, reduction, and translation are investigated. Specifically, the temporary memory requirements and duplication time of the algorithms are analyzed.
• Adding Parallelism in Object Space to the Rendering Pipeline
(Eurographics Association, 1989) Chapman, Paul A.; Lewis, Eric
This paper analyses the problem of adding parallelism to the rendering pipeline and discusses the reasons for advocating an object-space partition. Consideration of the methods of work
distribution and the rendering techniques which are desired, leads to the proposition of two algorithms for performing the partition. An architecture for their implementation is considered and
• Visualisation in Astrophysics
(Eurographics Association, 1989) Ertl, T.; Geyer, F.; Herold, H.; Kraus, U.; Niemeier, R.; Nollert, H.- P.; Rebetzky, A.; Ruder, H.; Zeller, G.
This paper reports on progress we have made in modelling cosmic X-ray sources on supercomputers. The results we present are meant to serve as an example for the fact that sophisticated
visualization techniques play a crucial role in scientific computing. Among the graphical methods we demonstrate, raytracing in curved space-time and a physically motivated 3D-volume rendering
algorithm might be of interest to the graphics community in general.
• Delauney Triangulations and the Radiosity Approach
(Eurographics Association, 1989) Schuierer, Sven
The radiosity approach requires the subdivision of complex surfaces into simple components called patches. Since we assume to have constant intensity over a patch, the generation of regular
patches is a desirable property of the subdivision algorithm. We show that constrained Delaunay triangulations produce patches that are as close to equilateral triangles as possible and thus are
well suited for the partitioning of surfaces into patches. Since a number of optimal algorithms to generate constrained Delaunay triangulations have been published, the implementation presented
here made use of the earlier work. The implementation consists of a rather simple modeling tool called POLY, a fast triangulation algorithm for arbitrary polygons and the form factor computation
combined with a z-buffer output module.
• Hierarchical Texture Synthesis on 3-D Surfaces
(Eurographics Association, 1989) Bennis, Chakib; Gagalowicz, Andre
This paper presents a new method for synthesizing hierarchical textures on 3-D surfaces. This method utilizes both a mapping technique for rendering the macroscopic structure on the surface and a
generalization of the direct 3-D microscopic synthesis algorithms (presented in earlier publications) for generating a homogeneous texture inside each pattern. To produce the macroscopic
structure on the 3-D shape a new mapping technique is proposed. With this technique patterns distortion is minimized locally. Finally a solution to the aliasing problem adapted to our mapping is
• A Topological Map-Based Kernel for Polyhedron Modelers: Algebraic Specification and Logic Prototyping
(Eurographics Association, 1989) Dufourd, Jean-Francois
This paper deals with the topology of surfaces, in the boundary representation of three dimensional objects. Orientable, not orientable, closed or open surfaces are efficiently described and
handled when considered as combinatorial generalized maps. An algebra of such maps is first described. Using this algebra, operations to build polyhedra step by step are next defined. That is the
basis of a graphical modeler presently under consideration. The presentation uses algebraic software specification techniques in an abstract way. Finally, a systematical validation of the
specification by logic prototyping is described.
• Highlight Shading: Lighting and Shading in a PHIGS+/PEX-Environment
(Eurographics Association, 1989) Poepsel, J.; Hornung, C.
Todays graphics standard for the rendering of scenes with illumination and reflection is defined by PHIGS+ . PEX is a proposal to integrate that functionality into the window environment of X.
This paper first describes the lighting and shading models of PHIGS+/- PEX . Then a comparison of the different shading methods follows. At last, a new shading method, the Highlight Shading, is
developed. The Highlight Shading combines both speed and image quality and therefore is an attractive alternative to existing shading algorithms.
• A Model for Description and Synthesis of Heterogeneous Textures
(Eurographics Association, 1989) Englert, Gabriele; Sakas, Georgios
Existing texture models either describe textures as a non-hierarchical surface property (by means of Markov chains, time series and other stochastic methods) or distinguish only between micro and
macro textures. Besides this, textures are used in general only for mapping colour-information (usually derived from digitized photographs) on the object surface or for varying the normal vector
of a given surface (bumps mapping). In addition, the different models are strongly combined with special generation algorithms and the produced textures are exclusively raster images. As a
consequence, the above models are not able to describe more than a few types of textures. In this paper a definition of the term texture is first presented and a hierarchical texture model in
accordance with the above definition is then proposed. We provide complete textures, consisting of several slices, to be mapped on geometrical objects. Each slice represents an optical surface
property. These properties are. approximated by the different parameters of an illumination model. The slices themselves are hierarchical compositions of several levels. Each "intermediate
texture" is derived by operations (transformations and combinations) performed on the textures of the next lower level. A texture is not limited in space and is described by means of a complete
texture function which affects all texture slices. Such functions can be either usual algebraical functions, or they can determine the placement of elements on the texture plane.
• GKS, Structures and Formal Specification
(Eurographics Association, 1989) Duce, D. A.
There are now three International Standards for application program interfaces for computer graphics programming, GKS, GKS-3D and PHIGS. In this paper a simplified model GKS-like system is
described and a 2D PHIGS-like system is then described in terms of this and a centralized structure store. Formal specifications of the systems are given illustrating how the specification of a
system can be built up from a hierarchy of simple components. The purpose of the paper is to illustrate one approach to the description of a compatible family of graphics standards and the use of
formal specification techniques in this process.
• Toward Realistic Formal Specifications for Non-Trivial Graphical Objects
(Eurographics Association, 1989) Fiume, Eugene
Formal specification has long been advocated in programming methodology, and is becoming increasingly popular in computer graphics to characterise the semantics of components of graphics systems.
Unfortunately, formal specifications tend to sacrifice realism for abstraction. The result is often a specification that is not as relevant to real graphics systems as it could be. This paper
suggests that the use of sharper mathematical tools, together with the use of object orientation (i.e., data abstraction with inheritance) provides a way of resolving this problem. As an example,
we attempt to specify formally classes of bitmaps and images. These are particularly interesting choices, for bitmaps and images are mutable, bitmaps can have a perceived effect on images, and
their semantics depends on context.
• On Reducing the Phong Shading Method
(Eurographics Association, 1989) Claussen, Ute
Today, the shading method of Phong plays an important role in the design of realtime image generation systems. Often, the model has been used in combination with a color interpolation,
suppressing a main property of this model, namely the visually acceptable rendering of highlights. Unfortunately, Phong’s algorithm demands a normalization, which is expensive to implement in
hardware. We will present several shading methods which are reductions of the Phong algorithm. They will be compared both visually and theoretically. The alternatives are judged concerning their
costs for a hardware implementation. The result is a hierarchy of shading methods that can be used to select the required cost-performance ratio for a given visualization task.
• Anti-Aliasing by Successive Steps with a Z-Buffer
(Eurographics Association, 1989) Ghazanfarpour, D.; Peroche, B.
We present a method allowing to solve the three problems arising when a scene is displayed with the z-buffer algorithm. The proposed algorithm only requires one extra memory bit per pixel and
delivers good quality images. It is fast because, in particular, the most expensive calculations such as antialiasing or texture mapping are made only for visible pixels of the scene.
• Algorithms for 2D Line Clipping
(Eurographics Association, 1989) Skala, Vaclav
New algorithms for 2D line clipping against convex, non-convex windows and windows that consist of linear edges and arcs are being presented. Algorithms were derived from the Cohen-Sutherland 's
and Liang-Barsky s algorithms. The general algorithm with linear edges and arcs can be used especially for engineering drafting systems. Algorithms are easy to modify in order to deal with holes
too. The presented algorithms have been verified in TURBO-PASCAL. Because of the unifying approach to the clipping problem solution all algorithms are simple, easy to understand and implement.
• GEO++ - A System for Both Modelling and Display
(Eurographics Association, 1989) Wisskirchen, Peter
We present a new concept for a graphics system which we call GEO++ . Apart from the manipulation of groups (structures in PHIGS-terminology), GEO++ permits a direct access to the tree structure
required for display. With this concept we believe to have achieved a synthesis between the requirements of modelling in the sense of manipulation of building patterns and of display in the sense
of editing individual objects (parts) on the screen.
• Subdivisions of Surfaces and Generalized Maps
(Eurographics Association, 1989) Lienhardt, Pascal
The modeling of subdivisions of surfaces is of greatest interest in Geometric Modeling (in particular for Boundary Representation) , and many works deal with the definition of models, which
enable the representation of closed, orientable subdivisions of surfaces, and with the definition of elementary operations, which can be applied to these models (Euler operators) . We study in
this paper the notion of 2-dimensional generalized map (or 2-G-map), which make possible the definition of the topology of any subdivision of surface, orientable or not orientable, opened or
closed ; reciprocally, the topology of any subdivision of any surface may be defined by a 2-G-map . Three characteristics are associated to any 2-G-map G (the most elementary being the number of
boundaries, the most known being the genus ...), and can be directly computed on G . These characteristics define the subdivision of surface modelled by G (static classification of the
subdivision) . We define also operations which can be applied to 2-G-maps . Any 2-G-map (and then any subdivision of surface) can be constructed by a sequence of operations . To these operations
correspond variations of the characteristics associated to the 2-G-maps . These variations enable the control of the effect of an operation on the modelled subdivision (dynamic classification of
the subdivision) . The notion of 2-G-map defines the different elements of a subdivision (vertex, edge, face, bound ary...) by using one unique kind of elements, in a rigorous and unambiguous
manner. Data structures may be deduced from the notion of 2-G-map . These data structures make possible the representation of any subdivision of surface , in a way near to the well-known
"windged-edge" data structure defined by B. Baumgart in [BA75] . The constraints of consistency about these data structures can be directly deduced from the definition of 2-G-maps . The set of
the properties of 2-G-maps (rigour, consistency, possibility of static or dynamic classification) makes the greatest interest of the 2-G-maps, with respect to other models of subdivisions of
surfaces used in Geometric Modeling .
• 2.5 Dimensional Graphics Systems
(Eurographics Association, 1989) Herman, Ivan
The outline of an extension of traditional 2D graphics systems is given. This extension is aimed at supporting a three dimensional application program, without incorporating full viewing into the
general graphics system itself. The resulting system might be very advantageous for large application programs which have their own three dimensional facilities.
• Blending Rational B-Spline Surfaces
(Eurographics Association, 1989) Bardis, L.; Patrikalakis, N.M.
A method for blendin non uniform rational B-spline surface patches, either open or periodic, is developed. he blending surface is expressed in terms of an integral, bicubic B-spline patch. The
blend ensures position and normal vector continuity along linkage curves to within a specified accuracy. The linkage curves are either user-defined or are obtained by offsetting the intersection
of the two patches using geodesics on each patch. An example illustrates the applicability of our method.
• A Reference Model for the Visualisation of Multi-Dimensional Data
(Eurographics Association, 1989) Bergeron, R. Daniel; Grinstein, Georges G.
This paper presents a reference model for the development of systems for the visualization of multidimensional data. The purpose of the reference model is to build a conceptual basis for thinking
about multi-dimensional visualization and for use in developing visualization environments. We describe the reference model in terms of the fundamental concepts of PHIGS (Programmer’s
Hierarchical Interactive Graphics System), but extend those concepts to the representation of objects of arbitrary dimensionality.
• Visualizing Curvature Discontinuities of Free-Form Surfaces
(Eurographics Association, 1989) Pottmann, Helmut
A new method for the visualization of curvature discontinuities of free-form surfaces is presented. It is based upon an improvement and refinement of the well-known technique of displaying
• An Analysis of Modeling Clip
(Eurographics Association, 1989) O Bara, Robert M.; Abi-Ezzi, Salim
Modeling clip gives an application the ability to remove sections of an object in order to view internal detail. The clipping volume defied by modeling clip can be concave and disjoint, and is
composed of a set of volumes that are specified in modeling coordinates. The modeling clip functionality has been included in the PHIGS specification [4], Some interesting peculiarities arise
from the fact that most graphics pipelines (such as PHIGS) are algebraically based and that modeling clip regions are specified in modeling coordinates. One such peculiarity occurs when the
transformation relating the coordinate system of the clip region to world coordinates is singular. A study on the algorithmic and architectural issues of implementing modeling clip is presented.
The resulting algorithm to implement the modeling clip mechanism represents the clip volume as a pipeline of filters with each filter representing one of the sub-volumes. The method handles all
of the sixteen possible set combinations between two regions in space. The effects of transformations on modeling clip have been examined, and has resulted in identifying when modeling clip can
be efficiently performed in device coordinates as well as the cases when it can not. When handling singular modeling transformations, it is shown that it i
• Representing Tolerance Information in Feature-Based Solid Modelling
(Eurographics Association, 1989) Falcidieno, Bianca; Fossati, Bruno
In this paper a system for defining dimensions and tolerances is presented which deals with the geometric representation of the objects in a coherent and compact way. This model is a combination
of a hierarchical boundary model to represent geometry of the object with features and a relational graph model to encode dimensions and tolerances. In this way, the proposed model can be
considered a ”product model” that, besides geometric and topological information about the feature components of a solid object, also codifies information about dimensions represented by relative
positron operators connected to faces which are the primitive geometric entities of the object model. The method can automatically control the validity of the geometric and topological model of
the object each tame that a new tolerance node is added to the structure or a tolerance constraint already existing is modified. In this case, it also translates changes in dimensional values
into corresponding changes an geometry and topology.
• The Use of Finite Element Theory for Simulating Object and Human Body Deformations and Contacts
(Eurographics Association, 1989) Gourret, Jean-Paul; Magnenat-Thalmann, Nadia; Thalmann, Daniel
This paper presents a method for combining image synthesis and modeling based on a finite element method (FEM) to get realistic intelligent images. FEM is used for modeling both elastically and
plastically deformations of objects, and impacts with or without penetration between deformable objects. The concept of deformable objects is applied to human flesh to improve the behavior of
synthetic human grasping and walking. The paper also discusses the introduction of this method in an animation system based on the concept of "intelligent" synthetic actors with automatic motion
control performed using A.I. and robotics techniques. In particular, motion is planned at a task level and computed using physical laws.
• Graph Grammars, A New Paradigm for Implementing Visual Languages
(Eurographics Association, 1989) Goettler, Herbert
This paper is a report on an ongoing work which started in 1981 and is aiming at a general method which would help to considerably reduce the time necessary to develop a syntax-directed editor
for any given diagram technique. The main idea behind the approach is to represent diagrams by (formal) graphs whose nodes are enriched with attributes. Then, any manipulation of a diagram
(typically the insertion of an arrow, a box, text, coloring, etc.) can be expressed in terms of the manipulation of its underlying attributed representation graph. The formal description of the
manipulation is done by programmed attributed graph grammars.
• Supporting Graphical Languages with Structure Editors
(Eurographics Association, 1989) Szwillus, Gerd
Graphical editors are used in numerous application fields for purposes like specification, design, modelling, or description of structures of various kinds. These editors handle the graphical
representations based on objects that are relevant to the application, rather than editing basic picture elements like line segments or rectangles. The GEGS project is concerned with the
generation of object-oriented graphical editors like these from an appropriate specification. The advantage of this is that editing a specification and generating a new editor is much quicker and
less errorprone than implementing a new graphical editor. An editor generator also allows adaption to technology changes, generation of application-dependent graphics, individualization to group
needs, and experimentation with new graphical languages.
• When is a Line a Line?
(Eurographics Association, 1989) Brodlie, Ken W.; Göbel, Martin; Roberts, Ann; Ziegler, Rolf
Conformance testing of graphics systems is a very complex and exhausting task. Years of practice with the GKS testing tools have shown a need for the automatic testing of visual output. Indeed,
with regard to graphics systems which are more precisely specified than GKS like the Computer Graphics Interface (CGI), conformance testing is not manageable at all unless a major part can be
automated. This paper discusses different strategies for the automatic testing of pictorial effect. It concentrates on the definition of lines and describes a strategy to answer the question put
in the title by the testing system. Finally, automatic testing of simple graphical operations such as segment highlighting and visibility is discussed.
• Variations on a Dither Algorithm
(Eurographics Association, 1989) Pins, Markus; Hild, Hermann
Mapping continuous-tone pictures into digital halftone pictures, i.e. 0/1-pictures, for printing purposes is a well explored technique. In this paper, one of these algorithms, the two-dimensional
error-diffusion algorithm is extended to color pictures and animated pictures. The color picture algorithm is superior to existing algorithms by considering extreme color values as well as
adjacent color values. The animation algorithm eliminates the noise created by the correct but varying pixel patterns generated by applying a single picture dithering algorithm on every frame.
The power of the algorithms is demonstrated by experiments carried out on synthetic images generated by ray tracing.
• Message-Based Object-Oriented Interaction Modeling
(Eurographics Association, 1989) Breen, David E.; Kühn, Volker
This paper describes a message-based object-oriented tool for exploring mathematically-based interactions which produce complex motions for computer animation. The tool has been implemented as an
object in the object-oriented computer animation system The Clockworks. It supports the definition of complex interactions between geometric objects through the specification of messages to the
interacting objects. Our approach is general, flexible and powerful. The tool itself is not hardcoded to a particular application. It simply sends the messages specified by the user. Messages are
specified as strings which may be stored, modified and interpreted. Since the tool is part of The Clockworks it may utilize many of the powerful features of the system, including data
structuring, mathematical, geometric modeling, and rendering objects. The tool has been used to explore a general spring and mass model, and the response of objects in a vector field.
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Many-body force
Many-body force
The many-body (or n-body) force applies mutually amongst all nodes. It can be used to simulate gravity (attraction) if the strength is positive, or electrostatic charge (repulsion) if the strength is
negative. This implementation uses a quadtree and the Barnes–Hut approximation to greatly improve performance; the accuracy can be customized using the theta parameter.
Unlike the link force, which only affect two linked nodes, the charge force is global: every node affects every other node, even if they are on disconnected subgraphs.
Source · Creates a new many-body force with the default parameters.
const manyBody = d3.forceManyBody().strength(-100);
Source · If strength is specified, sets the strength accessor to the specified number or function, re-evaluates the strength accessor for each node, and returns this force. A positive value causes
nodes to attract each other, similar to gravity, while a negative value causes nodes to repel each other, similar to electrostatic charge. If strength is not specified, returns the current strength
accessor, which defaults to:
function strength() {
return -30;
The strength accessor is invoked for each node in the simulation, being passed the node and its zero-based index. The resulting number is then stored internally, such that the strength of each node
is only recomputed when the force is initialized or when this method is called with a new strength, and not on every application of the force.
Source · If theta is specified, sets the Barnes–Hut approximation criterion to the specified number and returns this force. If theta is not specified, returns the current value, which defaults to
To accelerate computation, this force implements the Barnes–Hut approximation which takes O(n log n) per application where n is the number of nodes. For each application, a quadtree stores the
current node positions; then for each node, the combined force of all other nodes on the given node is computed. For a cluster of nodes that is far away, the charge force can be approximated by
treating the cluster as a single, larger node. The theta parameter determines the accuracy of the approximation: if the ratio w / l of the width w of the quadtree cell to the distance l from the node
to the cell’s center of mass is less than theta, all nodes in the given cell are treated as a single node rather than individually.
Source · If distance is specified, sets the minimum distance between nodes over which this force is considered. If distance is not specified, returns the current minimum distance, which defaults to
1. A minimum distance establishes an upper bound on the strength of the force between two nearby nodes, avoiding instability. In particular, it avoids an infinitely-strong force if two nodes are
exactly coincident; in this case, the direction of the force is random.
Source · If distance is specified, sets the maximum distance between nodes over which this force is considered. If distance is not specified, returns the current maximum distance, which defaults to
infinity. Specifying a finite maximum distance improves performance and produces a more localized layout.
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Reasoning Revenue Multiples
Understanding what is embedded in the high revenue multiples of today
Scott McNealy, the CEO of Sun Microsystems famously quipped the following after Sun’s stock had run up to $64/share before falling to under $10/share a year later during the dot com bubble.
At 10 times revenues, to give you a 10-year payback, I have to pay you 100% of revenues for 10 straight years in dividends. That assumes I can get that by my shareholders. That assumes I have
zero cost of goods sold, which is very hard for a computer company. That assumes zero expenses, which is really hard with 39,000 employees. That assumes I pay no taxes, which is very hard. And
that assumes you pay no taxes on your dividends, which is kind of illegal. And that assumes with zero R&D for the next 10 years, I can maintain the current revenue run rate. Now, having done
that, would any of you like to buy my stock at $64? Do you realize how ridiculous those basic assumptions are? You don’t need any transparency. You don’t need any footnotes. What were you
In light of that comment, it’s worth considering the case of public high growth SaaS companies today which tend to be valued on revenue multiples with the median one trading at ~25X EV/NTM revenue.
So what are we thinking?
Is it absurd that companies are trading at 20X+ revenue multiples? While it isn’t as simple as McNealy’s comment above might imply, it does provide us a good starting-off point to answer this
For one, stocks should be valued as the present value of all future free cash flows (FCF) of a company, since that is what can be returned to investors as dividends. While revenue is obviously a key
driver of that, as McNealy alludes to, the bottom line is more important and ultimately free cash flows and not revenue is what matters.
Now, depending on a stage in a companies life cycle, it might be difficult or even impossible to actually project out all its future cash flows, and so revenue multiples are often to triangulate
But as a company matures, over times its valuation anchors around what it actually generates in earnings and then free cash flow, as Chamath touches on in this tweet below.
Note however that McNealy’s tongue-in-cheek quip misses the importance of revenue growth and secondarily the impact interest rates have in valuations. Something valued a 20X revenue growing at 100% y
/y for two years could soon be at 5X revenue in a few years. Similarly, one might be willing to pay 10X revenue when the risk free rates is 1% but not when it is 7%, since the discount rates will be
Embedded Expectations
How do we put all this together? One way to do that is to try to understand what expectations of future performance are embedded in the current stock prices to make sense of what a 20X or even 50X
revenue multiple is actually implying.
As mentioned earlier since FCFs are what matters, companies (yes, even SaaS companies) will trade based on FCFs eventually. So let’s say in 10 years, most of these companies will trade based on free
cash flows rather than on revenues (the idea one might be willing to wait 10 years might be ludicrous in itself to some people, so one can repeat this exercise with whatever number they are
comfortable with).
Now, how much free cash flow will they generate then? That will depend on how efficiently they can convert Revenue to FCF, or their FCF margin. Its important to use what we think they’ll do
repeatably at scale / maturity, rather than what they’re doing now. This will vary sometimes quite dramatically between companies just given differences in business model, market, gross margins and
so on. This is also evident in today’s numbers in the chart below via PublicComps, although most of these businesses are not at maturity and still growing quickly. For most actual SaaS businesses, I
anticipate a 30%-40% FCF margin at scale, although their gross margins today might give a hint at whether they’ll be below that range or in the upper end of that.
Lastly, at scale, what multiple of FCF’s should they trade at? That will depend on a lot of things, such as how durable the FCF’s are, how fast FCF’s are growing (less important since we’re assuming
some sense of scale / maturity), what the interest rates are at the time, and so on.
From an earlier analysis by NYU professor Aswath Damodaran, the median company typically trade at ~20X FCF. But given lower interest rates translate to higher multiples, and a better business model
(recurring revenue) leads to more durable FCFs, I think its fair to say that let’s say that SaaS companies will trade at 30-35x FCF at scale.
So again, these are the key assumptions:
• The time period from now companies will be close to scale and trade on FCF multiples
• What percentage of their revenue is converted to free cash flow (FCF margin)
• What multiple of FCF they will trade at
Now given these assumptions, we can work backward to estimate what expectations are embedded in a companies stock price and in their revenue multiple today.
Specifically, we can estimate what kind of revenue they will need to be at 10 years from now and also what kind of revenue growth CAGR they will need to sustain to get there. Another thing one can do
is see what market share of their addressable market the company will have to capture, but I don’t particularly like this one since TAMs change and grow over time as companies enter new spaces and
markets change.
What does this mean for Zoom and Snowflake?
To illustrate what I’m talking about, we can look at Snowflake and Zoom, two companies which have some of the highest revenue multiples today.
By making assumptions on their long term FCF margin and their FCF multiple 10 years out, and assuming we want at least a 6% return between now and then, we can estimate that they need to grow their
revenues by 23X and 11X respectively between now and then which is no easy feat. That’s a CAGR of 37% and 27% between 2020 and 2030. Given that they both had triple digit growth rates in the past
year, that might not sound crazy, but we are talking about executing consistently over the decade to justify that valuation and earn a return.
We can also calculate how much market share they would need in 2030 of their TAM, which can be another data point, but not one I like a lot since TAMs change as markets change and companies enter new
We can repeat the same exercise for the median high growth SaaS company which is today growing at 40% and has a ~25X multiple. 10 years from now, such a company needs to 5X its revenue, implying an
18% CAGR between now and then. While not crazy, that’s an impressive growth rate to achieve every year over the next decade, just to give investors a 6% return.
As an investor, the best way to use this is to assess what the market believes about a stock, and then see how your own expectations line up, to make a decision on whether to buy or hold a stock.
Overall, the multiples are high, but given the low interest rates and the growth of these companies, as evident from some of the numbers above, there is a clear path to justifying these valuations,
although they do involve growing consistently over the next decade, and then generating FCFs of 30-35% of revenue.
so far i have been using p/s as a thumb rule for young growing tech companies. low p/s and high growth are key factors. p/s vs ev/fcf multiple differs. please write an article with examples on how
growth affects multiples and how to think of high growth companies for say 5 years of future growth based valuations. thanks for the excellent insights.
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Time Frame it Takes to Melt 1000kg of Cast Iron on A Induction Furnace
Power consumption at standard kW/kg ratio for Cast Iron (Ingot) is 560 kWh/ton. Means one ton of Cast Iron (Ingots) requires 560 kW for one hour (kWh). This number is varying for the metal to metal.
This number is also depends on the type of the physical charge. This number is different if the physical charge is scrap. Power consumption for the melting cast scrap is 575 kWh/ton.
Require time to melt 1000 kg (1 ton) of cast iron is depends on the usage of power consumption from the power supply unit. If the capacity of the power supply is 500 kW and you are operating it on
the full power means 500 kW than melting time can be calculated from below equation.
Here, let assume that charge is Cast Iron Ingot. So power consumption will be 560kWh/ton
500kw x time (t) 560 kW x 1 hour
______________ = ___________________
1 ton 1 ton
In order to find time, solve above equation for time (t).
Time (t) = ( 560 kW x 1 hour ) / ( 1 ton x 500 kW)
Time (t) = (560 kW x 1 hour) / 500 kW
Time (t) = 1.12 hour
Time (t) = 1.12 hour x 60 minutes = 67.2 minutes = Approx. 68 minutes
Therefore 68 minutes require to melting 1000 kg of cast Iron @ 500 kW.
If the power supply capacity is 700 kW and you are using full power (700kW) than require melting time can be determine using following method.
700 kw x time (t) 560 kW x 1 hour
______________ = ___________________
1 ton 1 ton
In order to find time, solve above equation for time (t).
Time (t) = ( 560 kW x 1 hour x 1 ton ) / ( 1 ton x 700 kW)
Time (t) = (560 kW x 1 hour) / 700 kW
Time (t) = 0.8 hour
Time (t) = 0.8 hour x 60 minutes = 48 minutes
Therefore 48 minutes take for melting 1000 kg of cast Iron @ 700 kW.
Power consumption is also depends on the manufacture of the furnace. Below is power consumption for the Electroheat furnace for different metals.
For Cast Iron melting 550-575 kWh/ton require
For SG Iron melting 550-600 kWh/ton require
For MS/SS melting 600-650 kWh/ton require
For light scrap of Aluminum melting 600-625 kWh/ton require For Solid scrap of Aluminum melting 500-575 kWh/ton require For Steel melting 625 kWh/ton require
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REU Mentor
Rutgers University
James Abello is a research professor at DIMACS. His areas of expertise include graph clustering, analysis of massive datasets, and visualization of complex data. His recent projects include Name that
Cluster and Graph Mining.
Computer Science
Rutgers University
Eric Allender is a professor of computer science and a DIMACS member who has organized and co-organized several DIMACS workshops. His primary research interest is the theory of computational
complexity. His most recent project is Small Complexity Classes.
Rutgers University
Andrew Baxter is a recently-graduated Ph.D. student in the Department of Mathematics at Rutgers. He was the co-organizer (with Doron Zeilberger) of the DIMACS-sponsored Experimental Mathematics
Semianr. He will be an instructor at Penn State starting in the fall of 2011. His research interests include permutation statistics, experimental mathematics, enumerative combinatorics, enumeration
schemes, and billiard paths. His most recent project is Billiards in Polygons.
Department of Mathematics
Rutgers University
Shabnam Beheshti Geometric Analysis, Integrable Systems, General Relativity, Soliton Theory, PDEs arising in Mathematical Physics
Mike Geis
Molecular Dynamics
RU Math 2012
Applied Research
Telcordia Technologies
Cliff Behrens is a senior research scientist at Telcordia Technologies, Applied Research, and a member of DIMACS. His areas of expertiese are telecommunications, geospatial mapping, and large-scale
discrete modeling. His most recent project is Agent Based Modeling of Crowd Behavior.
BioMaPS Institute for Quantitative Biology and Computer Science
Department of Molecular Biology and Biochemistry
Department of Physics
Rutgers University
Gyan Bhanot is professor of molecular biology & biochemistry and of physics, as well as a member of the interdisciplinary BioMaPS program. He is also a member of the Cancer Institute of New Jersey.
His areas of expertise include computational biology and evolutionary genetics. His recent projects include Cell Population Dynamics and Cancer Modeling in the Human Colon: An Extension of the
Johnston Model and Identifying patterns of migration and selection in HapMaP III data.
Rutgers University
Endre Boros is professor of operations research and director of the Rutgers Center for Operations Research (RutCOR) and a member of DIMACS. His research interests include combinatorial optimization,
integer programming, and the theory of boolean functions. Recent projects include projects related to graph theory and game theory.
BioMaPS Institute for Quantitative Biology and Computer Science
Biochemistry & Microbiology
Rutgers University
Yana Bromberg is an assistant professor of Biochemistry & Microbiology and a member of the Institute for Marine and Costal Sciences, BioMaPS, and the Cancer Institute of New Jersey. Her primary
research interest is the bioinformatic analysis of protein function prediction. Her most recent project is Protein Activity Unmasked.
Department of Mathematics
Rutgers University
Anders Buch is Professor of Mathematics at Rutgers University. His research interests include algebraic geometric and combinatorial algebra.
Mathew Palazzoto
Algebraic Geometry
RU Math 2010
Rutgers University
Tami Carpenter is a research professor and former associate director of DIMACS. Her areas of expertise include optimization and mathematical modeling. Her recent projects include Mathematical Models
of Theories of Leadership and Allocation of Monetary Resources in HIV-Infected Communities.
Computer Science
Rutgers University
Art Chaovalitwongse is an assistant professor of computer science and a member of DIMACS. His research includes topics in operations research and computer science in medical applications. His REU
projects include Optimizing Cooperative Sensors in Battlespace and Mining EEG Data to Diagnose Epilepsy.
Department of Chemistry
Rutgers University
Kevin Chen is an assistant professor in the Department of Genetics. His areas of research include population genetics and the molecular biology of regulatory RNA. His first project is Statistical
Algorithms in Population Genetics.
Department of Mathematics
Shippensburg University
Ji Young Choi is a visiting researcher at DIMACS, on sabbatical from Shippensburg. Her research interests include algebraic combinatorics and graph theory. Her ongoing REU project is The boundaries
of the minimum Pk-total weights for simple connected graphs.
AT&T Research
Graham Cormode is a researcher in AT&T Labs research division and a DIMACS member. His research intersts are in the areas of algorithms and computational complexity. His recent projects include
Outsourced Computation Verification and Standardization of Mergeable Quantile Summaries.
Rutgers University
Margaret (Midge) Cozzens is a research professor at DIMACS. She is also Education Director of CCICADA. Her research interests include mathematical modeling, game theory, operations research, and
mathematical education. Her ongoing REU project is A Game Theory Approach to Cascading Behavior in Networks.
Department of Computer Science
Rutgers University
Tina Eliassi-Rad is an associate professor of computer science and a member of DIMACS. Her research interests include data mining, machine learning, and the analysis of complex relational data. Her
first REU project was Graph Mining.
Industrial and Systems Engineering
Rutgers University
E. A. Elsayed is is professor of industrial and systems engineering and a member of DIMACS. His areas of expertise include quality control, reliability engineering, and production planning & control.
His ongoing REU project is Optimum Strategies of Container Inspection Scheduling and Sequencing at Port-of-Entry, which also included Nuclear Detection.
Ecology Evolution & Natural Resources, DIMACS
Rutgers University
Nina Fefferman is an associate professor of Ecology, Evolution, and Natural Resources, and is also a member of DIMACS's research faculty. She studies mathematical and computational models in biology,
specializing in evolutionary & behavioral ecology as well as epidemiology. Her recent projects include epidemiological modeling of infectious diseases and comparing routes of transmission in
multi-vector disease models.
Rutgers University
Gene Fiorini is associate director of DIMACS and director of the DIMACS REU. His research interests include extremal graph theory, graph pebbling, and mathematical education. His ongoing REU projects
include graph pebbling, sustainability-related projects, and projects including mathematical forensics.
Department of Computer Science
Vinod Ganapathy is an assistant professor in the Department of Computer Science and a member of DIMACS. His research areas include security & privacy, mobile systems, and virtualization. His first
REU project is Analyzing information flow in JavaScript-based browser extensions.
Department of Computer Science
Texas Southern University
Lila Ghemri is an associate professor of computer science at TSU. Her research interests include language analysis and education in computer science, and her first REU project is Accountability in
Social Networks.
Jude Ugiomoh
Texas Southern
Accountability in Networks
CCICADA 2012
Department of Mathematics
New York City College of Technology
Urmi Ghosh-Dastidar is an assistant professor at NY City Tech in the Department of Mathematics. Her research areas include optimization, signal processing, and wave propagation. She is also a DIMACS
visiting researcher. Her recent REU projects focus on modeling the food webs of the Hudson Valley.
Alma Cabral-Reynoso
NY City Tech
DIMACS 2011
Alexius Otto
Baruch College
DIMACS 2012
AT&T Research
Yifan Hu is a research at AT&T Research. His areas of expertise include graph visualization, machine learning, and numerical analysis & optimization. His recent projects include Name that Cluster and
Graph Mining.
Kevin Wong
Graph Visualization
DIMACS 2012
Department of Mathematics
Rutgers University
Yi-Zhi Huang is professor of mathematics. His research interests include vertex operator algebras and their applications to conformal field theory and to physics. His first REU project is the
Finiteness problem in the representation theory of vertex operator algebras.
Department of Computer Science
Rutgers University
Liviu Iftode is a professor in the Department of Computer Science. His research interests include distributed computing, network & mobile security, and vehicular & traffic computing. His first REU
project is Game-Theoretic Analysis of Climbing Social Ladder in Networks.
Rutgers University
Aaron Jaggard is a member of the DIMACS research faculty. His research focus includes various aspects of networking, including privacy, security, and traffic routing. He will also be starting in Fall
2011 as a visitor at the Department of Computer Science at Colgate University. His recent projects include topics in graph theory, network traffic, and permutation patterns.
Department of Mathematics
Rutgers University
Jeffry Kahn is a professor in the Department of Mathematics. His expertise includes many areas in discrete mathematics, including graph theory, combinatorial geometry, and the theory of set systems.
His first REU project investigates the Wide Partition Conjecture.
Computer Science
Rutgers University
Bahman Kalantari is a professor of computer science and a DIMACS member. His research interests include mathematical programming and optimization. His recent projects involve artistic and
mathematical aspects of root-finding algorithms and their basins of attraction.
School of Communication & Information
Rutgers University
Paul Kantor is a professor at the School of Communiation and Information and is a member of DIMACS. He is also Research Director of CCICADA. His areas of expertise include a broad range of topics in
communication, data analysis, operations research, and applications to security and defense. His first REU project is on the topic of Game Theoretic Aspects of Homeland Security.
AT&T Research
Howard Karloff is a researcher at AT&T Research Labs and a member of DIMACS. His research interests include graph theory and data analysis. His first REU project is Exploring Topics in Approximation,
Online and Randomized Algorithms.
Department of Mathematics
University of Delaware
Felix Lazebnik is professor of mathematics at the University of Delaware. His research intersts include graph theory and algebraic combinatorics.
Department of Computer Science
Rutgers University
Michael Littman is a professor of Computer Science and a member of DIMACS. His areas of expertise are broad, and include machine learning, networking, and game theory. His first REU project is In
Search of Value Equilibria.
Computer Science
Rutgers University
Qingshan Lui is a professor of computer science. His research areas focus on aspects of image and video analysis, particularly identification and analysis of faces, objects, and events. His fist
project is Identifying Objects by Incorporating RFID.
Department of Mathematics
Rutgers University
Luis Medina is an assistant professor in the Department of Mathematics. His research interests include combinatorics and number theory. His first REU project studies the p-adic valuation of Eulerian
Department of Computer Science
Rutgers University
Muthu Muthukrishnan is a professor of computer science and a member of DIMACS. He is also Director of Data Research at CCICADA. His research interests include data analysis, network analysis &
anomaly detection, and algorithms. His recent projects include Game-theoretic analysis of climbing social ladder in networks and Outsourced Computation Verification.
Roy Luo
UC Berkeley
Outsourced Verification
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Applied Research
Telcordia Technologies
Linda Ness is a researcher at Telcordia Applied Research. Her research interests focus on algorithms for representation and analysis of streaming data. Linda's first REU project is Mathematical
Models for Cognitive Systems.
Rutgers University
Alantha Newman is a postdoctoral researcher at DIMACS. Her research areas include algorithms, optimization, combinatorics, and computational biology. Her first REU project is on the topic of
Clustering Permutations.
Department of Mathematics
Rutgers University
Roger Nussbaum is a professor of mathematics. His area of expertise is in the area of functional analysis. His first REU project studies .
Department of Chemistry
Rutgers University
Wilma Olson is professor of chemistry and a member of DIMACS. Her research interests include many aspects of the chemistry of DNA, RNA, and proteins. Her ongoing REU project is Chromatin Folding &
DNA Looping.
Rutgers University
Boram Park is a postdoctoral researcher at DIMACS. Her interests include graph theory, game theory, and combinatorics. Her first REU project is Edge Clique Covers of Complete Multipartite Graphs.
Rutgers University
William Pottenger is a member of the research faculty at DIMACS and the Director of Technology Transfer for CCICADA. His research intersets include interactive automation and high-performanc
Becker Polverini
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Department of Mathematics
Alabama A&M University
Arjuna Ranasinghe is a professor of matheamtics at Alabama A&M University. His research interests include numerical analysis and differential equations.
Marene Bell
Alabama A&M
DHS 2011
Kiah Hedgman
Alabama A&M
DHS 2011
Department of Mathematics
Rutgers University
Siddhartha Sahi is a professor of mathematics. His research areas include the theory of Lie groups and Lie algebars.
Wei Chen
RU Math 2009
Computer Science
BioMaPS Institute for Quantitative Biology and Computer Science
Rutgers University
Alexander Schliep is an associate professor of computer science and a member of the BioMaPS institute. His research interests include genomics, machine learning, and algorithms for bioinformatics.
His REU projects include an ongoing project on Animations for Bioinformatics Algorithms.
Department of Mathematics
Rutgers University
Eugene Speer is a professor of mathematics. His areas of expertise include analysis and statistical mechanics.
Itai Feigenbaum
RU Math 2010
Jonathan Sloane
RU Math 2010
Computer Science
Rutgers University
Mario Szegedy is a professor in the Department of Computer Science. His areas of expertise include complexity theory, graph theory, and quantum computing. His first REU project is in the area of
graph theory.
Department of Computer Science
Princeton University
Robert Tarjan is a professor of computer science at Princeton University and the co-director of DIMACS for Princeton. His areas of expertise include graph theory, algorithms, and data structures. His
first REU project is Linear time Union-Find for Image Processing.
Department of Mathematics
Rutgers University
Roderich Tumulka is a professor of mathematics. His research interests focus on mathematical physics. His REU projects include GRW theory and The Continuum Limit of Bell's Jump Process.
Department of Operations Research and Financial Engineering
Princeton University
Robert Vanderbei is professor of operations research and financial engineering at Princeton University. He is also associated with the departments of Astrophysics, Computer Science, Mathematics, and
Mechanical & Aerospace Engineering. His areas of expertise are broad, and include linear programming, optimization, game theory, and probability. His first REU project is Climate Change Analysis.
Department of Mathematics
Tung-Hai University, Taiwan
Tao Ming Wang is a professor of mathematics at Tung-Hai University, Taiwan. His research interests lie in graph theory. His first REU project is Antimagic Labellings in Graphs.
Department of Mathematics
Rutgers University
Robert Wilson is a professor in the Department of Mathematics. His research interests include multilinear algebra and group theory. His most recent REU project is 2x2 Matrix Polynomial Equations in
Department of Mathematics
Rutgers University
Chris Woodward is a professor of mathematics. His areas of expertise include symplectic, algebraic geometry, and gauge theory.
Rutgers University
Minge Xie is a professor of statistics and a member of DIMACS. His research interests include statistical interference, confidence intervals, and nonparametric method, and applications of these
methods to medical, social, and environmental sciences. His ongoing REU project is Optimum Strategies of Container Inspection Scheduling and Sequencing at Port-of-Entry, which also included Nuclear
Department of Computer Science
Rutgers University
Danfeng Yao is assistant professor of computer science. Her projects include Detecting Drive-by-Downloads Using Human Behavior Patterns and Building Robust and Automatic Authentification Systems with
Activity-Based Personal Questions.
Rutgers University
Debbie Yuster is a posdoctoral fellow at DIMACS. Her research interests include combinatorics and geometry. Her first REU project is Computing Shift Equivalence.
Department of Mathematics
Rutgers University
Doron Zeilberger is a professor in the Department of Mathematics. His research interests include enumerative combinatorics and experimental mathematics. His first REU project studies the p-adic
valuation of Eulerian sequences.
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About the Book
Pattern Analysis is the process of finding general relations in a set of data, and forms the core of many disciplines, from neural networks, to so-called syntactical pattern recognition, from
statistical pattern recognition to machine learning and data mining. Applications of pattern analysis range from bioinformatics to document retrieval.
The kernel methodology described here provides a powerful and unified framework for all of these disciplines, motivating algorithms that can act on general types of data (e.g. strings, vectors, text,
etc.) and look for general types of relations (e.g. rankings, classifications, regressions, clusters, etc.). This book fulfils two major roles. Firstly it provides practitioners with a large toolkit
of algorithms, kernels and solutions ready to be implemented many given as Matlab code, suitable for many pattern analysis tasks in fields such as bioinformatics, text analysis, and image analysis.
Secondly it furnishes students and researchers with an easy introduction to the rapidly expanding field of kernel-based pattern analysis, demonstrating with examples how to handcraft an algorithm or
a kernel for a new specific application, while covering the required conceptual and mathematical tools necessary to do so.
The book is in three parts. The first provides the conceptual foundations of the field, both by giving an extended introductory example, and by covering the main theoretical underpinnings of the
approach. The second part contains a number of kernel-based algorithms, from the simplest to sophisticated systems such as kernel partial least squares, canonical correlation analysis, support vector
machines, principal components analysis, etc. The final part describes a number of kernel functions, from basic examples to advanced recursive kernels, kernels derived from generative models such as
HMMs and string matching kernels based on dynamic programming, as well as special kernels designed to handle text documents.
All those involved in pattern recognition, machine learning, neural networks and their applications, from computational biology to text analysis will welcome this account.
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Negative algebra
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gures Calculator
Significant figures are numbers you can add to another number that add meaning to the value. To avoid repetition of particular figures, you can also round the numbers – but carefully. If you round too high or too low, you can impact the precision of the final value. If you require help with rounding for the most precise significant figure, you may want to use a rounding calculator.
There are ways to work out what numbers you may consider to be significant, and those which are not. Use the set of rules below.
1. Any zero to the left of a non-zero number is not significant.
2. Placeholder trailing zeros are significant in numbers with a decimal point. In numbers without a decimal, this will depend on annotations and margins of error. This is often used to estimate population to the nearest thousand, for example.
3. If you have any zeros between non-zero numbers, they are significant.
4. Non-zero numbers are significant.
5. If there are fewer desired significant digits but more numbers, you round the numbers. In the case of 591,500, it becomes 592,000 with three significant figures, if trailing zeroes are considered to not be significant.
A sig fig calculator is used in two ways – number rounding and arithmetic. Sig figs can also be calculated by hand with a significant figures counter.
An example;
You have 0.005289, but you want a number with two significant figures.
As the two zeros are trailing placeholders, they do not get counted.
You can round 5289 to two figures by rounding the 289 to 3. The number becomes 0.0053.
The same rule applies for non-decimal figures;
You have 1,528,529, but only want four significant figures. You can round the value up to one thousand, making the number 1,529,000.
In the case of scientific notation, you can use the same formula. However, you will have to use a scientific notation calculator, not a significant figure calculator.
If you begin working with significant figures in the same formula as multiplication, division, addition, or subtraction, some new rules apply.
1. During addition and subtraction, multiplication and division, you can work out the calculations first then apply the standard significant figure rules at the end.
2. The final calculation shouldn’t have more significant figures than the value with the least amount.
An example:
14.26 + 3.89 + 0.39. The figure 0.49 has the least number of significant figures, which means the final result must have two as well.
14.26 + 3.89 + 0.39 = 19
3. If you mix subtraction and addition with multiplication and division, you need to round the number at each step. Doing so can ensure you get the right number of significant figures at the end.
An example:
14.26 + 3.89 x 0.39 to get 15.7771
That equals 14.26 + 1.5171
Then, round 1.5171 to two significant figures to get 14.26 + 1.50.
Add the two results – 14.26 + 1.50 to get 15.76, before rounding it to 16.
4. Defined, pure, and exact numbers do not affect the calculation’s accuracy. You can treat them as if they were significant numbers of infinite value. An excellent example of this is if you were to use a speed conversation calculator.
Let’s say you wanted to convert miles per hour to kilometers per hour. You would need to multiply the value by 1.6. The accuracy of the initial speed still determines the significant figures, such as:
14.28 x 1.6 = 22.98.
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