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Customer Service - Understanding your water meter
If you have an AMI meter click this link to the Customer Service Portal (My Account) to see your usage analytics.
Otherwise information on your analog meter follows.
Understanding and Reading Your Analog Meter
There are two basic types of analog water meters – straight reading and circular reading.
The straight reading meter records cubic feet of water used in much the same way that a car’s odometer records miles. The dial with a single hand measures tenth of a cubic foot.
The circular reading meter uses a series of circular dials to record cubic feet of water used. To read this type of meter, start with the 100,000th circle, then read the 10,000 circle, and so forth
on down to the circle that reads in 1 cubic foot increments. If a hand is between two numbers, always read the lower number.
This meter, for example, shows a reading of 2,425.92 cubic feet (the “6” in the last dial position has not quite rolled over). One cubic foot is 7.48 gallons of water. This meter has registered
almost 18,146 gallons since it was new (2,425.92 x 7.48).
When the Water Authority reads your meter, only the white dials with the black letters are read to measure the number of “units” that have been used. One unit equals 100 cubic feet, or 748 gallons.
In this example, the meter reading would be 24. This reading appears on the left side of your bill, in the box labeled “Metered Usage.” If the meter reading the month prior was 20, this customer
would be billed for four units.
The triangle is a leak detector. The triangle will rotate if water is passing through the meter. If no one is using water, but the triangle is turning, you may have an undiscovered leak in your
plumbing system.
For more information on your water usage: Understanding and Measuring Water Usage & Checking for Leaks
Understanding and Measuring Water Usage
To measure the amount of water used for any activity, follow these instructions:
1. Before you begin the measurement, write down the meter reading to two decimal places. (see the photo example above for help).
2. Perform the activity you want to measure, but be sure that no other water is being used during the test. Some examples include:
• Washing a load of laundry or dishes
• Taking a shower or bath
• Washing your car
• Watering your lawn
• Filling your swimming pool
3. If you are doing an activity that may have a variable duration, such as taking a shower or running your sprinklers, you should measure the number of minutes the activity required. This information
will allow you to determine the number of gallons per minute the activity requires.
4. After the activity is complete, read the meter again. Subtract the first reading from the second reading, and then multiply the remainder by 7.48 to convert to gallons.
5. To get gallons per minute, divide the number of gallons by the number of minutes the activity required. For example, if you ran your sprinkler system for 10 minutes and used 110 gallons of water,
your system uses 11 gallons per minute!
Keep in mind that the “units” shown on your water bill are equal to 100 cubic feet. The Water Authority water meter readers only record complete units for billing purposes, so the last two digits
(the “tens” and the “ones” figures) are omitted. Remember: one unit equals 748 gallons, so one cubic foot equals about 7.5 gallons.
Using your Meter to Check for Leaks
If you suspect you have a leak, you can measure the volume:
• Write down the meter reading and the time of day to the minute.
• Don’t use any water during the test. Usually it is best to do this when you will be away from home for an hour or more. Make sure devices such as evaporative coolers and ice makers are turned
• Read the meter again when you return and note the time of day.
• Subtract the second reading from the first. Multiply the remainder by 7.48. This is the number of gallons that passed through the meter during the test period.
• Divide the amount of water by the number of minutes in the test. For example, if 17 gallons leaked out during a 180 minute period, you have a leak of 0.094 gallons per minute.
• Multiply the gallons per minute by 1,440 to calculate gallons per day. Multiply gallons per minute by 43,920 to calculate gallons per month. In this example, just 0.094 gallons per minute equates
to over 4,128 gallons each month! | {"url":"https://www.abcwua.org/customer-service-understanding-your-water-meter/","timestamp":"2024-11-10T08:06:21Z","content_type":"text/html","content_length":"90723","record_id":"<urn:uuid:69fcfb21-631a-4e6e-b477-290cfd74f0ef>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00375.warc.gz"} |
6 GMAT Function Practice Questions With Explanations // Ambitio
2 September 2024
6 minutes read
6 GMAT Function Practice Questions With Explanations
Key Takeaways
• Focus on understanding functions, their compositions, and how to solve them.
• Regular practice with real-world examples is crucial for mastering GMAT function questions.
• Employ strategy like visualization, and the two-pass system to enhance your learnings.
• Teach the material to others and engage actively to solidify your knowledge.
• Practice under real test conditions to build stamina and familiarity with the test environment.
Preparing for the GMAT test? We understand your headache. GMAT exams can be challenging, especially when it comes to mastering functions and equations. In this last-minute guide, we will cover 6 GMAT
function practice questions that will enhance your preparation. Understanding the value of each variable, how to plug numbers into equations, and defining the domain are crucial skills for excelling
in the GMAT.
We provide detailed explanations and expert solutions to ensure you grasp each concept thoroughly. Whether you’re working with integers, sequences, or complex functions, these practice questions are
designed to help you succeed.
So, stick to the guide till the end – we gonna cover the essential elements of GMAT functions to boost your score and confidence.
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What are the function questions in the GMAT exam?
GMAT function questions are a key part of the quantitative section that requires a solid understanding of mathematical expressions and their applications. These questions often involve interpreting
functions, using parentheses correctly, and solving for the exact value of variables like “x”.
Practice tests are invaluable for mastering these concepts because they provide real-world examples and challenges similar to those found on the actual GMAT. In addition to the GMAT test prep,
getting ready for other standardized tests like the SAT, ACT test prep, and SSAT also benefits from a strong grasp of functions. Whether you’re enrolled in ISEE courses, SSAT test prep, or even MCAT
courses, understanding functions is crucial.
For those seeking personalized guidance, options abound from tutors to classes in various locations. By honing your skills through targeted practice, you can achieve precise output and improve your
overall performance on test day.
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6 mostly asked GMAT function practice questions with explanations
The following questions cover various concepts related to functions, including function composition, domain and range, inverse functions, and properties of exponential and logarithmic functions.
Here are 6 mostly-asked GMAT function practice questions related to functions:
1. If f(x) = 2x + 3 and g(x) = x^2 – 2, what is f(g(2))?
Evaluate π (2)g(2): The function π (π ₯)g(x) is defined as:
π (π ₯)=π ₯2β 2g(x)=x2β 2
We substitute π ₯=2x=2 into the function π (π ₯)g(x):
π (2)=22β 2g(2)=22β 2
Calculate the exponent and subtraction:
π (2)=4β 2=2g(2)=4β 2=2
Evaluate π (π (2))f(g(2)): We now have π (2)=2g(2)=2. Next, we use this result as the input for the function π (π ₯)f(x). The function π (π ₯)f(x) is defined as:
π (π ₯)=2π ₯+3f(x)=2x+3
We substitute π ₯=2x=2 (which is the result of π (2)g(2)) into the function π (π ₯)f(x):
π (2)=2(2)+3f(2)=2(2)+3
Perform the multiplication and addition:
π (2)=4+3=7f(2)=4+3=7
Therefore, π (π (2))=7f(g(2))=7.
2. Let h(x) = (x^2 + 3x – 2) / (x – 1). Find the value of h(2).
To find the value of β (2)h(2) for the function β (π ₯)=π ₯2+3π ₯β 2π ₯β 1h(x)=xβ 1×2+3xβ 2β , we need to substitute π ₯=2x=2 into the function and simplify.
• Substitute π ₯=2x=2 into β (π ₯)h(x): β (2)=22+3(2)β 22β 1h(2)=2β 122+3(2)β 2β
• Simplify the numerator: 22=422=4 3(2)=63(2)=6 4+6β 2=84+6β 2=8
• Simplify the denominator: 2β 1=12β 1=1
• Combine the simplified numerator and denominator: β (2)=81=8h(2)=18β =8
Therefore, the value of β (2)h(2) is 8.
3. If f(x) = 3x^2 – 2x + 5 and g(x) = 2x – 1, find (f β g)(x).
To find (π β π )(π ₯)(fβ g)(x), which is the composition of the functions π (π ₯)f(x) and π (π ₯)g(x), we need to substitute π (π ₯)g(x) into π (π ₯)f(x). This means we will
replace every π ₯x in π (π ₯)f(x) with π (π ₯)g(x).
Given: π (π ₯)=3π ₯2β 2π ₯+5f(x)=3×2β 2x+5 π (π ₯)=2π ₯β 1g(x)=2xβ 1
We want to find π (π (π ₯))f(g(x)).
• Substitute π (π ₯)g(x) into π (π ₯)f(x): π (π (π ₯))=π (2π ₯β 1)f(g(x))=f(2xβ 1)
• Replace every π ₯x in π (π ₯)f(x) with 2π ₯β 12xβ 1: π (2π ₯β 1)=3(2π ₯β 1)2β 2(2π ₯β 1)+5f(2xβ 1)=3(2xβ 1)2β 2(2xβ 1)+5
• Expand and simplify (2π ₯β 1)2(2xβ 1)2: (2π ₯β 1)2=(2π ₯β 1)(2π ₯β 1)=4π ₯2β 4π ₯+1(2xβ 1)2=(2xβ 1)(2xβ 1)=4×2β 4x+1
• Substitute back into the function: π (2π ₯β 1)=3(4π ₯2β 4π ₯+1)β 2(2π ₯β 1)+5f(2xβ 1)=3(4×2β 4x+1)β 2(2xβ 1)+5
• Distribute the constants: π (2π ₯β 1)=12π ₯2β 12π ₯+3β 4π ₯+2+5f(2xβ 1)=12×2β 12x+3β 4x+2+5
• Combine like terms: π (2π ₯β 1)=12π ₯2β 16π ₯+10f(2xβ 1)=12×2β 16x+10
Therefore, (π β π )(π ₯)=12π ₯2β 16π ₯+10(fβ g)(x)=12×2β 16x+10.
4. The function f is defined by f(x) = 2^x for all real numbers x. Find f(log2 8).
To find π (logβ ‘28)f(log2β 8) for the function π (π ₯)=2π ₯f(x)=2x, follow these steps:
• Evaluate the inner expression logβ ‘28log2β 8: The logarithm logβ ‘28log2β 8 asks the question: “To what power must 2 be raised to get 8?” Since 23=823=8: logβ ‘28=3log2β 8=3
• Substitute the value of logβ ‘28log2β 8 into π (π ₯)f(x): Given π (π ₯)=2π ₯f(x)=2x, we need to find π (3)f(3): π (3)=23f(3)=23
• Calculate 2323: 23=823=8
Therefore, π (logβ ‘28)=8f(log2β 8)=8.
5. Let f(x) = |x – 3| and g(x) = 2x – 1. Find the value(s) of x for which f(x) = g(x).
To find the value(s) of π ₯x for which π (π ₯)=π (π ₯)f(x)=g(x), given π (π ₯)=β £π ₯β 3β £f(x)=β £xβ 3β £ and π (π ₯)=2π ₯β 1g(x)=2xβ 1, we need to solve the equation β £π ₯β 3β
£=2π ₯β 1β £xβ 3β £=2xβ 1.
The absolute value equation β £π ₯β 3β £=2π ₯β 1β £xβ 3β £=2xβ 1 can be split into two separate equations:
1. Case 1: π ₯β 3=2π ₯β 1xβ 3=2xβ 1
2. Case 2: β (π ₯β 3)=2π ₯β 1β (xβ 3)=2xβ 1
Let’s solve each case separately.
Case 1: π ₯β 3=2π ₯β 1xβ 3=2xβ 1
• Subtract π ₯x from both sides: β 3=π ₯β 1β 3=xβ 1
• Add 1 to both sides: β 2=π ₯β 2=x
So, π ₯=β 2x=β 2.Case 2: β (π ₯β 3)=2π ₯β 1β (xβ 3)=2xβ 1
• Distribute the negative sign: β π ₯+3=2π ₯β 1β x+3=2xβ 1
• Add π ₯x to both sides: 3=3π ₯β 13=3xβ 1
• Add 1 to both sides: 4=3π ₯4=3x
• Divide by 3: π ₯=43x=34β
• Check π ₯=β 2x=β 2: π (β 2)=β £β 2β 3β £=β £β 5β £=5f(β 2)=β £β 2β 3β £=β £β 5β £=5 π (β 2)=2(β 2)β 1=β 4β 1=β 5g(β 2)=2(β 2)β 1=β 4β 1=β 5 Since π (β 2)β π (β
2)f(β 2)ξ =g(β 2), π ₯=β 2x=β 2 is not a solution.
• Check π ₯=43x=34β : π (43)=β £43β 3β £=β £43β 93β £=β £β 53β £=53f(34β )=β £β £β 34β β 3β £β £β =β £β £β 34β β 39β β £β £β =β £β £β β 35β β £β £β =35β π (43)=2(43)β 1=
83β 33=53g(34β )=2(34β )β 1=38β β 33β =35β Since π (43)=π (43)f(34β )=g(34β ), π ₯=43x=34β is a solution.
Therefore, the value of π ₯x for which π (π ₯)=π (π ₯)f(x)=g(x) is 4334β .
6. If f(x) = 3^(2x) and g(x) = log3(x^2 – 1), find f(g(8)).
To find π (π (8))f(g(8)) for the functions π (π ₯)=32π ₯f(x)=32x and π (π ₯)=logβ ‘3(π ₯2β 1)g(x)=log3β (x2β 1), we need to follow these steps:
• Evaluate π (8)g(8): The function π (π ₯)g(x) is defined as: π (π ₯)=logβ ‘3(π ₯2β 1)g(x)=log3β (x2β 1) Substitute π ₯=8x=8 into π (π ₯)g(x): π (8)=logβ ‘3(82β 1)g(8)=log3β
(82β 1) Calculate the value inside the logarithm: 82=6482=64 64β 1=6364β 1=63 So, π (8)=logβ ‘3(63)g(8)=log3β (63)
• Evaluate π (π (8))f(g(8)): Now we need to use this result as the input for the function π (π ₯)f(x). The function π (π ₯)f(x) is defined as: π (π ₯)=32π ₯f(x)=32x Substitute π ₯=
logβ ‘3(63)x=log3β (63) into π (π ₯)f(x): π (logβ ‘3(63))=32β logβ ‘3(63)f(log3β (63))=32β log3β (63)
• Simplify the expression: Using the property of logarithms and exponents: 32β logβ ‘3(63)=3logβ ‘3(632)32β log3β (63)=3log3β (632) Since 3logβ ‘3(π )=π 3log3β (a)=a: 3logβ ‘3(632)=
6323log3β (632)=632
• Calculate 632632: 632=3969632=3969
Therefore, π (π (8))=3969f(g(8))=3969.
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Tips to answer better in GMAT exams
Look – we know that the GMAT is hard – especially if you are from a non-math background, but winning test-taking strategies can make a significant difference in your performance. Beyond the standard
advice of regular practice and time management, unique and insightful tips can give you an edge. Here are seven uncommon strategies to help you excel on exam day.
Visualize Success
Before diving into your study session or the exam itself, take a few minutes to close your eyes and visualize yourself successfully answering questions. This mental rehearsal can boost your
confidence and reduce anxiety, making you more focused and effective during the test.
Use the Elimination Method
Instead of looking for the correct answer right away, start by eliminating the obviously wrong choices. This strategy can increase your chances of selecting the right answer by narrowing down your
options, especially in tricky quantitative and verbal questions.
Practice Mindfulness and Breathing Techniques
Incorporate mindfulness exercises and deep breathing into your study routine. These techniques can help you stay calm and maintain concentration during the exam, particularly during challenging
sections or when facing time pressure.
Teach the Material
Another way of learning and enhancing our knowledge retention is through facilitated learning where one is encouraged to pass the knowledge being studied. Student-Directed Activities: Look for a
study buddy or practice explaining what you learned to an imaginary audience. This approach helps to consolidate information within you and show a focus on aspects that require more focus.
Use the Two-Pass System
During the exam, go through the entire section quickly first, answering the questions you find easiest. On the second pass, tackle the more difficult questions. This approach ensures you secure easy
points and manage your time more effectively.
Develop a Question Identification System
Create a personal system to quickly identify the type of question you’re facing (e.g., algebra, geometry, critical reasoning). This system allows you to switch mental gears efficiently and apply the
most appropriate strategies for each question type.
Simulate Test Conditions
Regularly practice under actual test conditions. Use a timer, sit in a quiet environment, and take full-length practice tests. Simulating the test environment helps you build stamina and get
accustomed to the pressure and timing constraints of the GMAT.
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Remember to practice regularly, use strategic approaches, and stay calm under pressure. These practice questions and tips are designed to help you excel and achieve your desired GMAT score.
To enhance your understanding, actively engage with the material. Instead of passively reading, try solving problems without looking at the solutions first. Discuss questions with peers, and donβ t
hesitate to teach others. Active learning solidifies your knowledge and exposes areas needing improvement, ultimately leading to better performance on test day.
Transform your GMAT preparation with Ambitio’s expert guidance. Our comprehensive approach includes personalized study plans, adaptive practice tests, and strategic insights, all designed to enhance
your understanding and performance across the exam’s quantitative and verbal sections.
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How do I Register for the GMAT?
How Much Does the GMAT Cost?
How Often Can I Take the GMAT?
You can take the GMAT up to five times every 12 months, with a limit of not more than once in a 16-day period or more than eight times in total
Can I Reschedule a GMAT Appointment?
You can reschedule your GMAT appointment by logging into your personal GMAT account at mba.com, but there are fees involved depending on the timing of the rescheduling
Can I Retake the GMAT?
What Material Is Tested on the GMAT?
The GMAT tests essential skills needed in business school and subsequent business careers. It includes sections like analytical writing assessment, integrated reasoning, quantitative, and verbal,
each assessing different skills and concepts
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Teaching the Types of Triangles with Math Wheel Magic! - Cognitive Cardio MathCognitive Cardio Math
Have you noticed how triangles can be a bit of a math maze for our students? I mean, equilateral, isosceles, scalene – it’s like they’ve entered a geometric wonderland without a compass! I have felt
this way time and again, so I set out to create a tool to help break down the different types of triangles in a memorable way for my students. I’m excited to share with you my student-approved Types
of Triangles Math Wheel Guided Notes! It’s the treasure map your students need to navigate the triangle jungle. With color-coded notes, practice problems scattered around the wheel, and a graphic
organizer that breaks down and organizes the different types, you’ll be turning confusion into a full-blown math adventure. Get ready for those “Aha!” moments that turn the triangle saga from
puzzling to downright fun!
What Are Math Doodle Wheels?
I’m standing at the front of my class and all I see are blank stares, math nerves, and a whole bunch of “I’d-rather-be-anywhere-else” vibes. My students were going through the motions of note-taking,
but it was clear, the info wasn’t sticking. That’s when the lightbulb moment happened, and Math Doodle Wheels were born!
I wanted notes to be more than just a checkmark on the to-do list. I wanted them to be a trusty sidekick, a go-to guide during units and review sessions.
So, I crafted these wheels where math concepts are broken down into bite-size sections of the wheel, vocab gets its VIP definition moment, and each step is practiced with examples. Plus, there’s
space for some extra practice. The secret? We’re color-coding and doodling symbols as we go to help them remember the steps and vocabulary. Suddenly, note-taking became an art, and my kiddos enjoyed
it and used these notes as tools.
To learn more about Doodle Wheels and their versatility, check out Transform Your Upper Elementary or Middle School Math Class with this Unique Note-Taking Method!
How I Teach Types of Triangles
We’re diving headfirst into the world of triangles, and it all begins right at the heart of our Types of Triangles Math Wheel. I gather my students around, and we start chatting about the six main
players in the triangle game – equilateral, scalene, acute, obtuse, right, and isosceles. Now, here’s the math magic. No matter how different these triangles are, their angles always add up to 180°.
I have my students write that 180° right in the center of our wheel to be a constant reminder.
With the center complete, I kick off the grand tour of the wheel. The first thing we discuss is that there are two ways we can classify triangles. We can classify them by their sides and we can
classify them by their angles. To keep this in mind, we tackle the three types of triangles classified by their sides first, then we move on to the three types of triangles classified by angles. With
that explanation, we have now taken 6 types of triangles and grouped them into two groups of three and made them a little easier to understand.
Sections of the Types of Triangles
Equilateral Triangle
I explain that this triangle is known for having three equal sides, which we jot down in that section. I draw three hash marks, one on each leg of the triangle, to show that they are equal.
To give them an example of number measurements, I write 6 meters on all three sides. I like starting with equilateral because it’s the least intimidating since all three sides are the same
Scalene Triangle
Now, onto the scalene triangle.
I kind of think of it as the rebel of the triangle crew. It’s the complete opposite of our equilateral triangle because it plays by its own rules with no equal sides.
We fill in the blank in our notes section. Then, it’s example time. I draw my trusty triangle and label it with the following measurements: 5 cm on the short leg, 7 cm on the top leg, and 10 cm on
the bottom leg. Why? To show my students how each side has its unique length.
Isosceles Triangle
Now, we wrap up our note-taking adventure with the isosceles triangle.
An isosceles triangle is all about the buddy system, boasting two equal sides. In our triangle example, I draw single hash marks on the left and right legs because they’re equal measurements. I label
both legs with 4 feet, emphasizing their equal status. Now, I draw two hash marks on the bottom leg, signaling that its measurements are playing solo. I label 6 feet for the bottom leg, driving home
the point that it’s not on the equal side with the left and right legs.
Acute Triangle
Next up is the acute triangle. I walk my students through jotting down the key detail that all three angles inside an acute triangle are less than 90°.
I draw two examples to show my students a couple of different scenarios they may encounter. In the first triangle, I sketch a curved line at each corner, representing an angle, and drop 60° in each.
We add each of the three angles, each at 60°, sum up to 180°, sealing the deal that triangles have a total of 180° inside.
I draw another triangle to explain that acute triangles won’t always have the same angle measurements. In this second one, I draw those curved lines again and jot down 70°, 60°, and 50° in the
corners. Quick adding by my eager students confirms the angles total to 180°.
Obtuse Triangle
In our next section, we dive into the world of the obtuse triangle.
For a triangle to earn the “obtuse” badge, it needs to boast one obtuse angle, meaning one angle must be greater than 90°.
In my example, I draw an obtuse triangle.
I draw the curved line to represent the angle in the top angle inside the triangle and write 120°.
We talk about how we can check that this is obtuse by noting how it’s a number larger than 90.
Right Triangle
We move to the right triangle section.
We review what makes a right angle, which is when a 90° angle is formed at the intersection of two perpendicular lines. I have my students write down that one angle must measure 90°. Then, in the
triangle, we practice drawing the square symbol where the two lines intersect to represent 90°.
What’s Next?
Once we wrap up our note-taking session, it’s time to put our knowledge to the test by tackling the examples scattered around the math doodle wheel.
I pre-select a few examples, modeling to my students my thought process and reasonings for classifying which triangles to which measurements. After a couple of examples, I give the steering wheel to
my students. They become the teacher and start leading me through the steps for a few examples. Then, I partner them up to complete the remaining practice examples. I set a timer so we have time to
check their answers and talk through their reasoning.
After conquering those practice problems, I give my math enthusiasts a little artistic freedom. They go back into their notes, adding splashes of color and extra symbols to etch the differences
between the triangles into their memory. It’s a personal touch that not only triggers those “aha” moments but also makes math feel like a comfortable space. The math doodle wheel isn’t about
bombarding them with gibberish and 50 problems. It’s all about breaking the skill into bite-sized chunks, creating a handy reference tool that’ll be their math companion throughout the year!
More Resources for Types of Triangles
Once I complete notes with my students, I want them to be applying what they have learned. If you don’t use it, you lose it, right? When I am planning practice activities, I want to make sure they
are interactive and allow everyone to have a seat at the table. Math anxiety is real! It’s important to be aware of all the emotions your kiddos may be feeling when it comes to math. With that in
mind, I have included some types of triangle activities that have students practicing the skills in a collaborative, low-intensity manner.
Classifying Triangles Footloose Math Task Cards Activity and Math Wheel
An awesome classroom activity I use with my students to review types of triangles and to get them up and moving is my Types of Triangles Footloose cards.
Footloose isn’t just your typical sit-and-listen routine. I hand out these cards, and suddenly, we’re not just classifying triangles by sides and angles.
We take it a step further by finding those missing angles and sides like geometry detectives. Sometimes, alarms go off in my students’ heads when they discover a missing angle.
We pull out our math doodle wheels and remind ourselves that all three angles within a triangle equal the grand number of 180!
I tend to shake up the delivery of the task cards because sometimes I have them posted around the classroom ahead of time. Then they travel around to each card with a partner to solve and record
answers on their paper.
With that being said, Footloose isn’t a one-size-fits-all deal. You can roll it out for the whole class, turn your class into an interactive experience, use it in small groups, or throw it into
center rotations. It’s flexible, it’s fun, and it’s got the whole class moving!
Triangles Color by Number Activity Missing Angles and Sides
My Triangles Color by Number activity is a low-intensity math activity for different types of angles and missing angles with a dash of color. Students solve problems while turning their answers into
My students tackle 20 problems that dive deep into triangle types. Problems range from finding missing side lengths and angle measures to calculating the perimeter of triangles. Instead of the usual
pencil and paper routine, they find their solutions on a coloring sheet/section.
I love using this resource for many reasons, but one of my favorites is that it’s self-checking! If their answer isn’t on the coloring sheet, they know it’s time for a quick math check. It’s instant
feedback that we know and love!
Plus, there are both print and digital versions available. I like flexibility, which is the name of the game for us teachers, so I wanted this resource to be flexible for your teaching style.
Classifying Triangles Math Game | Math Activity | Truth or Dare
Truth or Dare? I choose truth! I’ve got a game-changer for your next class or review session for identifying types of triangles that’s as fun as it is educational.
I’ve crafted 36 question cards that’ll turn your class into a buzz of excitement. It’s not your average Q&A session because this one comes with a “truth” or “dare” twist!
I pull this Truth or Dare activity out for center time when I am working with small groups, will be out of the classroom, or for review sessions. My students love it when they see this activity on
the schedule.
Your students dive into a world where they get to choose their fate.
Questions focus on the different triangle types (acute, obtuse, right, isosceles, equilateral, scalene) and essential concepts like the 180 degrees in a triangle.
They earn a 1-point “truth” for those classic true or false questions. If they’re feeling daring, there are “dare” cards worth 2 or 3 points, throwing in some challenges. Think along the lines of
finding missing angles or sides, explaining why a triangle is classified a certain way, or even determining if a shape can make the triangle cut based on angle measures or side lengths.
The beauty of this is it’s not just a solo gig. Gather your kiddos in groups of 3-4, and let the game begin. Keep in mind the smaller groups mean more questions tackled by each student, meaning more
Time to Teach Types of Triangles
Exploring types of triangles with the math doodle wheel makes note-taking into a masterpiece, it’s all about transforming math anxiety into math excitement.
These resources are tools that keep on giving throughout the unit and year. The adventure continues with activities that have your mathematicians applying their skills through interactive activities
like Footloose, Color by Number, and Truth or Dare. Get ready to witness “Aha!” moments, foster collaborative learning, and make math a destination worth exploring.
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Unknown in Multiplication (examples, solutions, videos, worksheets, homework, lesson plans)
Related Topics:
Lesson Plans and Worksheets for Grade 3 Lesson Plans and Worksheets for all Grades Common Core For Grade 3 More Lessons for Grade 3 Math
Examples, videos, and solutions to help Grade 3 students learn how to interpret the unknown in multiplication and division to model and solve problems.
Common Core Standards: 3.OA.3, 3.OA.4, 3.OA.5, 3.OA.7, 3.OA.1, 3.OA.2, 3.OA.6, 3.OA.8
New York State Common Core Math Grade 3, Module 3, Lesson 7
Worksheets for Grade 3 Concept Development
Thad sees 7 beetles when he weeds his garden. Each beetle has 6 legs. How many legs are there on all 7 beetles?
Lesson 7 Homework
1. Match the words on the arrow to the correct equation on the target.
2. Ari sells 6 boxes of pens at the school store.
a. Each box of pens sells for $7. Draw a tape diagram and label the total amount of money he makes as m. Write an equation and solve for m.
b. Each box contains 6 pens. Draw a tape diagram and label the total number of pens as p. Write an equation and solve for p.
3. Mr. Lucas divides 28 students into 7 equal groups for a project. Draw a tape diagram and label the number of students in each group as n. Write an equation and solve for n.
Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.
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Cayley-Bacharach applications
The Cayley-Bacharach theorem in projective geometry states that if two cubics share nine points (in sufficiently general position) and a third cubic passes through eight of those points, then it also
passes through the ninth. Here, ‘sufficiently general position’ means that as long as you’re not deliberately stupid with applying this theorem, it will work.
I’ve heard this referred to as a ‘super-theorem’, because its main use is to prove simpler theorems. Firstly, note this very important fact:
• The union of a degree-m and a degree-n algebraic curve is a degree-(n+m) algebraic curve.
So, for cubics, we can have either an irreducible cubic such as an elliptic curve, or the union of a conic and a line, or indeed the union of three lines. In most applications, we don’t use the full
force of Cayley-Bacharach, and concentrate instead on these ‘degenerate’ cases.
Pascal’s theorem (and converse)
Pascal’s theorem states that if we inscribe a hexagon in a conic, the extensions of opposite sides intersect at three collinear points. The hexagon can self-intersect, such as the following example:
I’ve coloured each cubic in a different colour, such that the nine points are triple intersections. If we lack a single piece of information (such as a concurrency or collinearity) but know that all
the other pieces of information hold, we can deduce the final piece of information through Cayley-Bacharach. For instance, the statement of Pascal’s theorem says that the red line exists, and the
converse states that the red conic exists.
To determine on which branch of the cubic the point lies (either the conic or the line), we can just apply Bezout’s theorem.
Projective dual: Brianchon’s theorem
Once we have Pascal’s theorem, we can take poles and polars about an arbitrary conic to obtain the projective dual theorem. In projective duality, we interchange lines and points, whereas conics
remain conics. Concurrency of lines corresponds to collinearity of points, and so on.
The projective dual of Pascal’s theorem is called Brianchon’s theorem, which states that a hexagon circumscribed about a conic has concurrent diagonals. I’ve colour-coordinated this diagram to
correspond with the diagram of Pascal’s theorem; the black lines were originally black points, and the coloured points were originally lines of that colour.
We can tell that this is a projective dual of a C-B corollary rather than a C-B corollary itself, because the lines are black and the points are colourful.
Both Pascal’s theorem and Brianchon’s theorem have special ‘degenerate’ cases. If the conic in Pascal’s theorem degenerates into a pair of lines, we obtain Pappus’s theorem instead. This is also a
direct corollary of Cayley-Bacharach, for obvious reasons.
In the same way that Pappus is a degenerate case of Pascal, we can view Pascal as a degenerate case of the theorem obtained by replacing the red conic and line with a single elliptic curve. This
associativity law is used in elliptic curve cryptography, and more recreationally in my elliptic curve calculator. It is the most difficult part of showing that the points on an elliptic curve form a
group under a geometric operation.
Radical-axis theorem and pivot theorem
When we apply Cayley-Bacharach, we actually use the complex projective plane rather than the ordinary Euclidean plane. There are two invisible ‘circular points at infinity’, through which all circles
pass. Using these, we can derive some theorems involving circles from Cayley-Bacharach. Only seven of the points are visible here; the other two are the circular points.
With three circles and three lines, there are two distinct theorems. Firstly, the pivot theorem:
We also obtain the radical-axis theorem in this way:
Circle inversion
When we’re dealing with circles and lines rather than arbitrary conics, we can concentrate on the real Euclidean plane and extend it to give the Riemann sphere, or complex projective line. This is
quite a sneaky trick to switch seamlessly between these two universes, but we can get away with it here. Inverting the pivot theorem yields Miquel’s theorem. This states that if we have eight points
and ‘faces’ correspond to concyclic points, then five faces of a cube determine the sixth face:
The diagram above was obtained by stereographic projection, so that the cubic nature can easily be appreciated. The radical axis theorem treats the six ‘diagonal’ planes inside a cube instead of the
six faces. In the very rare instances where both apply simultaneously, we get the famous orthocentric configuration.
Quartic Cayley-Bacharach
It turns out that there is a generalisation of Miquel’s theorem to eight conics and sixteen points, but this requires the quartic analogue of Cayley-Bacharach to prove. I’ll leave this particular one
as an exercise to the reader, and instead use QCB to derive some other beautiful results. There was a particularly elegant problem posted on Art of Problem Solving, the solution to which I’ll
describe here.
Firstly, it’s necessary to know the statement of quartic Cayley-Bacharch. If we have 16 sufficiently-general-position points lying on two quartics, and a third quartic passes through 13 of those
points, then it also passes through the other 3. In particular, if we need to prove that eight points to lie on a conic, five of them automatically do and we can get the other three through
Cayley-Bacharach. Let’s have a look at the AoPS problem:
We want to show that the centres of the eight pink/purple circles lie on a conic. We’ll apply the fact that they lie on the angle bisectors of the triangles. Indeed, once we’ve drawn the angle
bisectors, we can forget about the grey lines and pink/purple circles completely:
The red and blue angle bisectors intersect to give the eight incentres. However, they also appear to converge on the circumference of the circle to give another four points. This is not a
coincidence, and can be deduced from the converse of ‘equal arcs subtend equal angles’. These sixteen points lie on the blue quartic and the red quartic, and we can also draw a green conic through
five of the remaining eight points. Quartic Cayley-Bacharach then reveals that the last three points also lie on this conic:
This diagram is a generalisation of Pascal’s theorem, which is a corollary of generalised Cayley-Bacharach. In particular, if we have a 2n-gon inscribed in a conic, and colour alternate edges red and
blue, the remaining n^2 − 2n points lie on an algebraic curve of degree n − 2. This is explored in a paper by Gabriel Katz.
Solving an AMS problem
We’ll solve a non-trivial olympiad-style problem using as much pure projective geometry as possible. This differs completely from any ordinary way of solving the problem, although it’s somewhat
neater (the standard solution has horrible Menelaus bashes and so forth).
Notice that Q and R are defined in quite a contrived way, so we’ll see if we can learn more about these points. It’s always good to draw a diagram in these contexts; I’ll leave it relatively
uncluttered and just include F, E, B, C, Q and R (these are the only ones we’ll require at this stage).
FE and QR meet at a point on the line at infinity, which is thus collinear with the two circular points. Colour the line at infinity red. We already have that the green circle exists (and has
diameter BC, by Thales’ theorem), so we can conclude that BCRQ lie on a blue circle by Cayley-Bacharach. You could alternatively use an angle chase, but that’s too boring for cp4space.
The orthocentre exists; hence, (P,D;B,C) is a harmonic range. Consequently, the circle on diameter PM is the inverse of the line AD (polar of P) with respect to the circle on diameter BC (centre M).
So, AD is the radical axis of circles on diameters PM and BC. This means we can just apply a degenerate case of the radical axis theorem to the points {D, Q, R, B, C, P, M} to deduce that {M, P, Q,
R} are indeed concyclic.
7 Responses to Cayley-Bacharach applications
1. That last paragraph was hard to follow – it would have been nice if you’d included another diagram.
2. Heello mate great blog post
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Does Sex-Differential Gene Expression Drive Sex-Differential Selection in Humans?
Sex differences in human transcriptomes have been argued to drive sex-differential natural selection (SDS). Here, we show that previous evidence supporting this hypothesis has been largely unfounded.
We develop a new method to test for a genome-wide relationship between sex differences in expression and selection on expression-influencing alleles (eQTLs). We apply it across 39 human tissues and
find no evidence for a general relationship. We offer possible explanations for the lack of evidence, including that it is due in part to eQTL ascertainment bias towards sites under weak selection.
We conclude that the drivers of ongoing SDS in humans remain to be identified.
Sex-differences in gene expression have been theorized to be the result of long-term sex-differential selection (SDS), where allelic fitness effects differ between males and females^1,2. When a gene
product is only beneficial in one sex, it is expected that expression modifiers will evolve to increase expression for that sex and decrease expression in the other ^3. The reverse causality has also
been proposed: sex differences in gene expression might be the driver of SDS on expression modifiers^1,4.
SDS acting on viability within the current generation can generate between-sex differences in allele frequency^5,6 at expression modifiers (expression Quantitative Trait Loci, or eQTLs)^3. Sexually
dimorphic traits (diseases being one example with strong repercussions on survival^6,7,8,9) may be subject to SDS acting through viability. Previous work has proposed and tested theoretical models
for relating divergence in eQTL allele frequencies with differential expression between sexes^1,10,11. Some of the empirical results have, however, been called into question^12,13,14.
A controversial result by Cheng and Kirkpatrick (2016)^1 (hereafter “CK16”) is a characteristic pattern relating between-sex F[ST]^1,5,15 (henceforth “F[ST]”) to sex differences in gene expression
(henceforth “Δ”). They observed high F[ST] values at intermediate values of Δ and near-zero values of F[ST] when a gene is expressed evenly between the sexes (Δ = 0) or is only expressed in one sex
(Δ = ±1). They nicknamed this bimodal pattern “Twin Peaks”. The Twin Peaks pattern has been used as a signal to detect SDS in several species^1,4,16,17.
Here, we revisit the CK16 model and statistical approach. We find that their interpretation was based on previously unappreciated statistical artifacts. We then refine their model and apply it to
new, more extensive data on gene expression and allele frequencies. Across 39 human tissues, we find no evidence for a genome-wide relationship between viability SDS and Δ. We discuss how a bias in
eQTL discovery towards eQTLs under weaker selection can explain the lack of signal for a relationship between SDS and Δ, and how the drivers of SDS may still be investigated.
Twin Peaks is a Statistical Artifact
We begin by revisiting the Twin Peaks pattern with a critical eye towards caveats in its application and interpretation. Most importantly, we reconsider the model that CK16 proposed to explain the
pattern. It is based on two key assumptions. First, the relationship between a gene’s expression levels and its effect on fitness in each sex is linear. Second, sexually antagonistic selection is
symmetric, meaning selection coefficients are equal in magnitude and opposite in sign between sexes. This yields a quadratic relationship between F[ST] and Δ at a biallelic site affecting expression.
At small absolute values of Δ, where p is the allele frequency in zygotes and q = 1 − p. F[ST] is the between-sex fixation index^18 which is used to quantify allele frequency differences between
males and females. In the model, these differences are due solely to the sex differences in post-zygotic fitness effects of alleles. The quantity Δ is the sex difference in gene expression (Methods).
Finally, A is a compound parameter involving the within-sex effect of gene expression level on fitness.
The expectation in CK16 for F[ST] at extreme expression differences (|Δ| → 1), however, is based on intuition rather than a model. The authors suggest that if a gene is not expressed in one sex (Δ =
±1), then selection will not act on it. Selection on the other sex should optimize expression levels, so under the symmetrical selection assumption there will be no ongoing directional selection in
that sex either. As neither sex experiences selection, there will be no force driving increased F[ST]. CK16 then interpolate a bimodal shape by joining the quadratic relationship at low |Δ| and the
expectation for F[ST] = 0 at Δ = ±1 (Fig. 1).
However, the assumption of symmetric selection used at low values of Δ may be inappropriate for large Δ values. In particular, when Δ = ±1, the lack of expression in one sex plausibly suggests
different selection between males and females. It is therefore not intuitive that F[ST] should simply go to zero at sites regulating expression in these genes. Additionally, although the quantity 2pq
appears in Eq. 1, CK16 did not include that term in fitting the model, effectively assuming it to only contribute random noise to the relationship between F[ST] and Δ.
Because of these caveats to the model, we decided to revisit the support for Twin Peaks. We first ask whether the Twin Peaks pattern is due to SDS by applying the statistical tests of CK16 to data
generated under a null hypothesis of no relationship between Δ and SDS. We replicated the pattern shown in CK16 by using the same data and methods. Namely, we performed a 4^th degree polynomial
regression of F[ST] on Δ, using allele count data from 1000 Genomes^19 and expression data from the gonads (ovaries and testes) in GTEx v3^20. The curve and associations generated using these
datasets we refer to as the “real data” (blue line in Fig. 2a).
We then generated an empirical null by permuting ovary and testes tissue labels in the expression data but retaining sex labels associated with F[ST] values (Methods), then recomputed Δ values, and
again fit a 4^th-degree polynomial. We find 21% of permutations qualify as Twin Peaks according to the criteria used by CK16, and that the polynomial fit to the real data is not visually distinct
from those fits to the null (Fig. 2a). Higher order polynomial regressions can also yield spurious fits because distant points have an oversized impact^21,22. We therefore binned genes by Δ values
and examined the relationship with mean F[ST] in each bin. Again, the real data shows no unusual relationship between F[ST] and Δ compared to null data (Fig. 2b). From these results, we conclude the
Twin Peaks pattern is not statistically significant.
Importantly, this permutation method differs from the method used in CK16, which permuted Δ values across genes. Both methods break the associations between F[ST] and Δ as desired. Our sex -label
permutation method, however, preserves F[ST] associations with the gene’s overall expression, maintaining gene features such as expression variance which the CK16 method does not. This explains why
their reported p-value for Twin Peaks curves (0.016) is lower than our (0.21).
We hypothesized that one reason for the spurious Twin Peaks pattern is confounding. In particular, variance in expression (regardless of sex) is positively correlated with both Δ and F[ST]. Previous
work has shown that genes subject to weaker stabilizing selection show higher variance in expression^23. Higher variance means larger differences between randomly selected subgroups, and therefore it
should translate to larger values of Δ even in the absence of SDS. In turn, stronger (sex-agnostic) selection can lead to stronger drift at linked sites^24,25,26,27. Taken together, the confounding
with variance in gene expression could generate a relationship between SDS and |Δ| in the absence of SDS. Indeed, expression variance and F[ST] are positively correlated (Pearson p = 0.011; Fig. 2c),
consistent with this hypothesis. In sum, we do not find support for Cheng and Kirkpatrick’s conclusion that there is a genome-wide relationship between F[ST] and Δ.
No evidence for genome-wide SDS on eQTLs
Although we do not find support for a relationship between sex differences in gene expression and selection using CK16’s methodology for generating Twin Peaks, one may still exist. We test this
hypothesis across many tissues using improved statistical modeling, data, and methods. Despite the caveats to portions of the model discussed above, we believe that CK16’s theoretical expectation of
a quadratic relationship between F[ST] and small values of Δ is valid. We therefore built on that model by introducing the compound parameter δ^2 = 4qpΔ^2 and rewriting Eq. 1 as (Methods). To
estimate A, for each gene-tissue pair, we used the single cis-eQTL from GTEx v8^28 with the strongest association with its expression. Our estimator of A is then the inverse-variance-weighted linear
regression of F[ST] on δ^2 (Eq. 2). The advantages of this formulation over CK16’s are that it allows direct estimation of A, the strength of SDS on sex differences in expression. This model also
accounts for variation in allele frequencies across sites. Further, by using a single eQTL to calculate F[ST] for the whole gene, we circumvent biases which can arise when using a simple mean to
estimate gene-wide F[ST]^29,30.
To accompany the updated regression methodology, we also updated the datasets for both allele frequencies and gene expression. For allele frequencies, we used the Non-Finnish European subset in
gnomAD v3 (averaging over 60,000 allele samples per site)^31. We used expression data from 49 tissues from the GTEx v8 dataset^28 (averaging over 200 male samples and 150 female samples). Both
datasets greatly expand our sample size compared to CK16, and GTEx v8 provides sample-specific sex labels for calculating Δ across multiple tissues instead of just the gonads—ovaries and testes—as in
the original Twin Peaks paper.
Using the updated statistical framework method, we find no evidence that A differs significantly from zero in any tissue (Fig. 3; Methods). One explanation for the absence of a pattern may be found
in recent work by Mostafavi et al. (2023). The authors contrasted how selection impacts the discovery of genome-wide association study (GWAS) hits with the discovery of eQTLs. Variants with large
effects on phenotypes are expected to segregate at low frequencies, reducing discovery power. In GWAS, this is counterbalanced by increased power due to their large effect sizes. Consequently, in
GWAS, low frequency variants can still be detected if their effect is large enough. In contrast, eQTL discovery is based only on the effect of genotype on gene expression which does not necessarily
translate to fitness-relevant trait variation. Detection of strongly selected sites is therefore less likely^32. This ascertainment bias can weaken the relationship between Δ and F[ST] at eQTLs, as
sites with high F[ST] are less likely to be eQTLs.
In “Twin Peaks is a Statistical Artifact”, we suggested that large sex differences in expression may be entirely unrelated to SDS—for instance, merely tagging genes with high expression variance.
There are other potential explanations for the lack of a genome-wide relationship. While some sex differences in expression may be driven by or drive SDS, these are the exception rather than the
rule. The majority of sex differences in expression may be a regulatory side effect of SDS on different genes. Lastly, sex-differential gene expression may be due to past SDS, but not drivers of
current SDS^33. Regardless of the reason, if a causal relationship between Δ and FST is rare in the genome, it would be difficult to detect using models that assume a pervasive, persistent
relationship between the two.
Previous work suggested a genome-wide relationship between sex differences in expression and SDS. However, this work was based on statistical artifacts and confounded effects, such as stabilizing
selection and variance in gene expression. Even when using newer data and improved statistical methods, we found no evidence for a genome-wide relationship between sex-differential expression and
contemporary SDS. In contrast, studies that measured SDS irrespective of expression or trait variation have reported pervasive, genome-wide signals of SDS in the human genome^5,10,34. While causal
relationships, past and present, between SDS and sex-differential gene expression in humans remain plausible, they are yet to be fully elucidated.
Materials and Methods
Permuting Twin Peaks sex-labels
In the section “Twin Peaks is a Statistical Artifact”, we demonstrate that the Twin Peaks curve presented in CK16 is not statistically significant compared to a permuted null. To do so, we used the
methods and datasets described in CK16 and Eq. 1. We computed allele count using the 1000 Genomes dataset^19 and calculating F[ST]. We filtered out any sites with only a single alternative allele in
either males or females (i.e., singletons). We used the Transcripts Per Million (TPM) normalized GTEx v3 (referred to as “pilot” on the download page) dataset^20 for expression levels for calculating
Δ. Because GTEx v3 does not have individual sample labels, this analysis only compared expression in the gonads (ovaries and testes) where expression level summaries represent a single biological
sex. We used the Ensembl GRCh38 v77^35 annotation file for gene annotations. Following CK16, we limited our analysis to protein-coding genes.
To estimate F[ST], we used Hudson’s estimator^36 based on the R package used in CK16^37 as presented by Bhatia et al. (2013)^29, Here, pm and pf are the allele frequencies in males and females
respectively, and nm and nf are the number of males and females respectively. As a measure of sex-differential expression we used Δ as defined in CK16: Here, x[m] and x[f] are sex-averaged
TPM-normalized expression levels in males and females respectively. Because in Eq. 3 is a site-specific estimator while Δ is gene-wide, we generated a gene-wide estimate of F[ST] by taking a simple
average of all sites within a gene body plus 1000bp upstream and downstream. To generate the Twin Peaks curve, we used a 4^th-degree polynomial regression between and Δ. We note that here and in
CK16, this regression therefore ignores variation in heterozygosity across sites (Eq. 1). In our improved model (Results: “No evidence for SDS on eQTLs across all genes in multiple tissues” and
Methods: “Applying a new model and method for SDS-expression regression”) we correct this omission.
To compare the original Twin Peaks curve to curves generated under the null of no SDS, we generated an empirical null distribution of 4^th-degree regression curves using sex-label permutations of
gene expression data. We permuted the ovary and testes labels for each GTEx sample, then recalculated Δ for all genes. We then re-performed the 4^th-degree polynomial regression on the new Δ values
(the F[ST] values remain unchanged). This was repeated 500 times. By permuting sex labels, we break the association between sex-differential expression and sex-differential expression, while
preserving other gene-level features. To quantitatively evaluate the significance of Twin Peaks in the null distribution, we used the three criteria laid out by Cheng and Kirkpatrick (2016) for
classifying a curve as Twin Peaks. Namely, a 4^th-degree polynomial must 1) be significant (p < 0.05) for the 4^th-degree term by ANOVA, 2) have a negative coefficient for the Δ^4 term, and 3) have
three real roots. Any 4^th-degree regression passing all three criteria is classified as Twin Peaks.
Applying a new model and method for SDS-expression regression
In the section “No evidence for SDS on eQTLs across all genes in multiple tissues”, we described a revised method for testing a genome-wide relationship between SDS and Δ. For this analysis, we
revised the inference model and data. We used the Non-Finnish European subset in the gnomAD V3 dataset^31 to calculate between-sex F[ST] and heterozygosity at each eQTL. This set of samples has an
order of magnitude more individuals (average of 31,470 samples per site) compared to 1000 Genomes data (average of 2,300 samples per site when combining all ancestry groups) originally used in CK16.
Additionally, we used the normalized TPM from the current GTEx v8 for expression level data, which includes many more samples (average of 236 male and 168 female samples per tissue) than the pilot (8
male samples and 13 female samples in the gonads)^28. This version also provides sample-specific sex labels, enabling us to use additional tissues beyond gonads.
The equation we base our regression on is the model relating F[ST] with δ^2 shown in Eq. 2. Now, rather than performing a 4^th-degree polynomial regression as in the analysis for “Twin Peaks is a
Statistical Artifact”, we perform a weighted linear regression of F[ST] to δ^2 = 4pqΔ^2, isolating A as the coefficient of regression. Note that by including 4pq in the independent variable, we allow
information about heterozygosity to affect the SDS-expression relationship. We weighted each point of the regression by 1/Var(expression), where Var(expression) is the averaged variance in expression
of each sex. This should decrease the leverage of points with large |Δ|. To get a gene-wide estimate of F[ST], we used eQTLs from GTEx as mapped in the v8 study^31. For each gene, we chose the single
cis-eQTL (within 1Mbp of the gene’s midpoint) with lowest p-value for association with the gene and use that site for calculating F[ST], p, and q. By using an eQTL selected this way, we tried to
isolate the effect of selection on gene expression to the site presumed to be contributing most to expression changes in that gene.
To determine the significance of our A estimates, we generated 90% null acceptance regions by permuting sex labels. We permuted sex labels as in “Twin Peaks sex-label permutation” such that for each
tissue, sex labels are permuted in GTEx expression samples and Δ is recalculated across all genes. Then, for each iteration A is recalculated by linear regression using the new Δ values (F[ST] and 4
pq remain unchanged). The 90% null acceptance region was obtained by the 5^th and 95^th percentiles of 1,000 A values calculated on permuted expression data.
We thank Jared Cole for comments on the manuscript and other members of the Harpak Lab for helpful conversations. The work was funded by NIH grants GM11685307 to M.K., and R35GM151108 and a Pew
Scholarship to A.H. | {"url":"https://www.biorxiv.org/content/10.1101/2024.07.23.604672v1.full","timestamp":"2024-11-14T13:27:39Z","content_type":"application/xhtml+xml","content_length":"195011","record_id":"<urn:uuid:3235cffc-feef-4078-920d-3d9faa42e861>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00501.warc.gz"} |
Energy Equation
Next: Polytropic Relation Up: Basic Equation of Fluid Previous: Expression for Momentum Density   Contents
The above basic equations (A.3) and (A.10) or equations (A.7) and (A.11) contain three dependent variables v. The number of the variables, 3, is larger than the number of equations, 2. Therefore, an
extra equation is needed to close the basic equations.
Kohji Tomisaka 2007-07-08 | {"url":"http://th.nao.ac.jp/MEMBER/tomisaka/Lecture_Notes/StarFormation/3/node105.html","timestamp":"2024-11-02T14:06:33Z","content_type":"text/html","content_length":"3385","record_id":"<urn:uuid:2e820935-157b-4447-924b-8fc22f428c2d>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00607.warc.gz"} |
While going through my puzzle files, I came across the Slovak Championships folder. I realised I had never shared these puzzles on my blog. So I figured I might as well share them now.
I was contacted by Zuzanna Hromcova to write puzzles for their championship. We were given a number of categories to write puzzles in. One of the categories was non-grid puzzles, for which I provided
three different genres; namely ABC Decoder, Dice and Mastermind. Dice and ABC Decoder are types I enjoy writing; Mastermind I didn't have that much experience with. But it was something I'd like to
give a go.
The other categories I picked were Latin Squares and Division puzzles. For each type we had to write a standard genre and a variant on the genre. I picked Skyscrapers, with Haido as the variant. I
like Skyscrapers and I thought Haido still had part of the Skyscraper logic, but used differently enough to make it not like solving four skyscraper puzzles.
For the Division set I picked ABCD Division, with Sum Division as the variant. It's a type I have seen a lot when I first started puzzling, but I haven't really seen it much since. I thought sums was
an obvious variant, but I haven't really seen it this way much. I have seen a similar variant where the grid has to be divided into a complete set of pentominoes, but not really without this
I tried to put a bit of theming in the non-grid puzzles. I wrote a few nine digit ABC Decoders for the 2014 24 hour championships, and I thought that was a good size to use in a championship. The
letters spell out THE SLOVAK, which was the nicest way I could use nine different letters to write something Slovakia related. I found some words with opposite meanings in the letter set, so I used
those. I think it turned out well.
I used a similar opposites theme for the Dice puzzle, with an addition of 5 words to make it unique. I think not all words are necessary for uniqueness, but it solves pretty well this way.
The first Mastermind puzzle looks really nice, with a sequence of numbers and only white circles it solves really nicely. The second one was merely an attempt to construct a nice logical 5 digit
I thought both Skyscrapers puzzles turned out nicely. The first puzzle uses three 4s and three 5s. The second puzzle has a trio of the same digit on each side. Of course I couldn't use four different
digits as these are the only three digits you can have three of the same clue on the same side in this size.
I find it hard to theme Haido puzzles as the clues are a bit limited, but they both have nice logical paths.
The first time I saw an ABCD puzzle this way was at a Dutch championship. It was a bit of a surprise then. I wrote a similar puzzle for puzzlepicnic once and I thought it would be fun to include one
for the championship. The ABCDE puzzle is a standard layout and I think it solves well.
The sum puzzles were a bit hard to work out openings at first as there are so many ways to reach the sums. So I went with obvious opening digits for both puzzles to then work back to more ambiguous
digits towards the end. I think they both turned out well.
Puzzles can be found below.
Read more »Posted in Dice, Division, Haido, Mastermind, Puzzle, Puzzle Championship, Skyscrapers, Word | {"url":"https://logicmastersindia.com/forum/forums/thread-view.asp?pid=15826","timestamp":"2024-11-06T19:13:02Z","content_type":"text/html","content_length":"20559","record_id":"<urn:uuid:318326af-b6eb-4f4b-9a22-fc19a43a850b>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00208.warc.gz"} |
Cubic Inches to Liters Conversion (in³ to L) - Inch Calculator
Cubic Inches to Liters Converter
Enter the volume in cubic inches below to convert it to liters.
Do you want to convert liters to cubic inches?
How to Convert Cubic Inches to Liters
To convert a measurement in cubic inches to a measurement in liters, multiply the volume by the following conversion ratio: 0.016387 liters/cubic inch.
Since one cubic inch is equal to 0.016387 liters, you can use this simple formula to convert:
liters = cubic inches × 0.016387
The volume in liters is equal to the volume in cubic inches multiplied by 0.016387.
For example,
here's how to convert 5 cubic inches to liters using the formula above.
liters = (5 in³ × 0.016387) = 0.081935 L
Cubic inches and liters are both units used to measure volume. Keep reading to learn more about each unit of measure.
What Is a Cubic Inch?
A cubic inch is a unit of volume equal to the space consumed by a cube with sides that are one inch in all directions. One cubic inch is equivalent to about 16.387 cubic centimeters or 0.554 fluid
The cubic inch is a US customary and imperial unit of volume. A cubic inch is sometimes also referred to as a cubic in. Cubic inches can be abbreviated as in³, and are also sometimes abbreviated as
cu inch, cu in, or CI. For example, 1 cubic inch can be written as 1 in³, 1 cu inch, 1 cu in, or 1 CI.
Learn more about cubic inches.
What Is a Liter?
A liter is a unit of volume equal to 1,000 cubic centimeters or 0.264172 US gallons.^[2] The liter is a special name defined for the cubic decimeter and is exactly equal to the volume of one cubic
decimeter (1 decimeter is 1/10 of a meter, or 10 centimeters).
The liter is an SI accepted unit for volume for use with the metric system. A liter is sometimes also referred to as a litre. Liters can be abbreviated as L, and are also sometimes abbreviated as l
or ℓ. For example, 1 liter can be written as 1 L, 1 l, or 1 ℓ.
Learn more about liters.
Cubic Inch to Liter Conversion Table
Table showing various
cubic inch measurements
converted to liters.
Cubic Inches Liters
1 in³ 0.016387 L
2 in³ 0.032774 L
3 in³ 0.049161 L
4 in³ 0.065548 L
5 in³ 0.081935 L
6 in³ 0.098322 L
7 in³ 0.114709 L
8 in³ 0.131097 L
9 in³ 0.147484 L
10 in³ 0.163871 L
11 in³ 0.180258 L
12 in³ 0.196645 L
13 in³ 0.213032 L
14 in³ 0.229419 L
15 in³ 0.245806 L
16 in³ 0.262193 L
17 in³ 0.27858 L
18 in³ 0.294967 L
19 in³ 0.311354 L
20 in³ 0.327741 L
21 in³ 0.344128 L
22 in³ 0.360515 L
23 in³ 0.376902 L
24 in³ 0.39329 L
25 in³ 0.409677 L
26 in³ 0.426064 L
27 in³ 0.442451 L
28 in³ 0.458838 L
29 in³ 0.475225 L
30 in³ 0.491612 L
31 in³ 0.507999 L
32 in³ 0.524386 L
33 in³ 0.540773 L
34 in³ 0.55716 L
35 in³ 0.573547 L
36 in³ 0.589934 L
37 in³ 0.606321 L
38 in³ 0.622708 L
39 in³ 0.639095 L
40 in³ 0.655483 L
1. National Institute of Standards and Technology, Specifications, Tolerances, and Other Technical Requirements for Weighing and Measuring Devices, Handbook 44 - 2019 Edition, https://
2. National Institute of Standards and Technology, Units outside the SI, https://physics.nist.gov/cuu/Units/outside.html
More Cubic Inch & Liter Conversions | {"url":"https://www.inchcalculator.com/convert/cubic-inch-to-liter/","timestamp":"2024-11-07T12:12:14Z","content_type":"text/html","content_length":"71455","record_id":"<urn:uuid:70d2f042-530e-44ee-9a42-821bb280619c>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00230.warc.gz"} |
"IT Organization" name in WinPE
Can someone please point me in the right direction so I can set the "IT Organization" name in the Task Sequence progress window. I've set this in the custom client settings which is deployed to the
same collection as the task sequence.
When I run a task sequence that captures User State (requires task sequence to be launched from Windows) the "IT Organization" works fine and it displays our company name. However, when I PXE boot a
task sequence for a task sequence with no USMT capture, it only displays "IT Organization" in the task sequence progress window.
This is when I launch from Software Center:
it gets the name from the Computer Agent part of the Default Client Settings defined for the hierarchy,
• 1
I face the same, I am 100 % sure that I have custom client settings and deployed this to target collection (not been deployed to UNKNOWN).
But still it says running IT Organization.
Created a client settings again, deployed it to a different collection, same results.
Hi Peter,
Thanks for reply.
I can update the DP (which I have tested already) but cant Re-Distribute since there is no DP available if I try to distribute the contents again.
Is it something I am missing?
Hi Niall
Yes, I am updating DP.
I followed your famous sccm guides in LAB. Created a couple of custom client device settings, deployed them to specific test collections, created TS and checked, all gives running IT Organization.
Today, I changed computer agent settings in DEFAULT client settings, now I can see that it appears the settings I provided in DEFAULT client settings.
Note: I have multiple boot images, one set is default x64 , x86 and other is copy of default boot images but I have changed background /PE wallpaper.
Any thoughts?
do as follows, change your organization name in Default Client settings,
once done, update the boot image that is linked in your task sequence to it's distribution points, that should solve it. to identify the boot image, right click on the task sequence and you'll see it
listed under advanced. | {"url":"https://www.windows-noob.com/forums/topic/5613-it-organization-name-in-winpe/","timestamp":"2024-11-12T18:09:12Z","content_type":"text/html","content_length":"187024","record_id":"<urn:uuid:73e9e69e-9b82-4fbb-9853-83e8a5d7e553>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00624.warc.gz"} |
5 Corrected constraint logic programming exercises - Complex systems and AI
5 Corrected Constraint Logic Programming Exercises
Consider the following puzzle:
Each region of the grid must be filled with a number between 0 and 9 so that
• the numbers in two adjacent regions (vertically or horizontally) are different
• whenever there are four regions that meet at a point (indicated by a small circle), the sum of their numbers is equal to 20.
Solve this problem by constraint logic programming.
Let's name from top left to bottom right (by line) the empty squares by a letter. Each domain has a value between 0 and 9. Thereafter, the two constraints must be filled in manually.
For each square, its score must be different from its neighbours.
For each circle, the sum must be equal to 20.
We therefore have the following constraint logic programming:
L = [A,B,C,D,E,F,G,H,I,J,K,M],
At #\= 7, At #\= 8, At #\= 6,
B #\= 8, B #\= 4,
C #\= 7, C #\= 6, C #\= 9,
D #\= 6, D #\= 8,
E #\= 6, E #\= 8, E #\= 4, E #\= 5,
F #\= 9, F #\= 6, F #\= G, F #\= Y,
G #\= 6, G #\= 5, G #\= K,
H #\= 5, H #\= 4, H #\= M,
I #\= 9, I #\= J, J #\= K,
K #\= 5, K #\= M,
7+6+A+C #= 20, A+6+8+D #= 20, 6+8+D+E #= 20, 4+8+B+E #= 20,
9+6+C+F #= 20, 5+6+E+G #= 20, 4+5+E+H #= 20,
9+F+I+J #= 20, F+G+J+K #= 20, 5+H+K+M #= 20,
Consider the game Bokkusu. The object of the game is to mark certain squares of the given grid in black.
Each box has two values: an A value and a B value. The numbers on the right denote the A values of the boxes in each corresponding row and the numbers at the bottom denote the B values of the boxes
in each corresponding column (These values always range from 1 to the size of the grid by increasing by 1 from left to right or from top to bottom).
The values on the left indicate the sum of the B values of the black boxes for each row and the values on the top indicate the sum of the A values of the black boxes for each column. An example of a
grid and its solution is given in the figure. In the solution of this example we have for example 7 = 1 + 2 + 4 (the boxes with B value of 1, 2 and 4 are marked black) and 6 = 2 + 4 (the boxes with A
values of 2 and 4 are marked).
Solve this problem by constraint logic programming.
Let's name each square in the grid in a matrix fashion. For the constraints, the sum of the binary variables (white=0 or black=1) multiplied by the coefficients on the left or on the top must be
equal to the value opposite.
L = [X11,X12,X13,X14,
X11 + 2*X12 + 3*X13 + 4*X14 #= 3,
X21 + 2*X22 + 3*X23 + 4*X24 #= 7,
X31 + 2*X32 + 3*X33 + 4*X34 #= 1,
X41 + 2*X42 + 3*X43 + 4*X44 #= 2,
X11 + 2*X21 + 3*X31 + 4*X41 #= 5,
X12 + 2*X22 + 3*X32 + 4*X42 #= 6,
X13 + 2*X23 + 3*X33 + 4*X43 #= 1,
X14 + 2*X24 + 3*X34 + 4*X44 #=2
We consider the problem of generating a timetable. There are 7 classes and 4 slots. A slot can accommodate a maximum of 2 lessons. There are the following constraints:
– Course 4 must be before course 6.
– Course 5 must be before course 7.
– Course 6 must be before course 2.
– Course 1 cannot be in parallel with courses 2, 3, 4 and 7.
– Course 2 cannot be in parallel with courses 3 and 6.
– Course 3 cannot be in parallel with courses 4, 5 and 6.
– Course 4 cannot be in parallel with courses 5 and 6.
– Course 5 cannot be in parallel with course 7.
Solve this problem by constraint logic programming.
We will consider that the courses are domains, and that they can take a value from 1 to 4 corresponding to the niche that will be chosen. Constraint logic programming is as follows:
L = [C1,C2,C3,C4,C5,C6,C7],
C4 #< C6, C5 #< C7, C6 #< C2,
C1 #\= C2, C1 #\= C3, C1 #\= C4, C1 #\= C7,
C2 #\= C3, C2 #\= C6,
C3 #\= C4, C3 #\= C5, C3 #\= C6,
C4 #\= C5, C4 #\= C6,
C5 #\= C7,
Inshi no heya is a Japanese logic game. It is played on a rectangular grid of cells which are separated into rectangles. One dimension of each rectangle is 1 while the other dimension varies.
Each rectangle is placed horizontally or vertically and contains a number. The goal is to
fill all cells with numbers from 1 to 9 so that:
– If we multiply all the digits of each rectangle we obtain the number indicated in the rectangle
– No number appears more than once in a column.
– No number appears more than once in a line.
An example of an original grid and a solution are given below:
Solve this problem by constraint logic programming.
Each box of the matrix is considered as a domain, each box can take a value from 1 to 5. The constraints are simple since the sum of the boxes of a certain perimeter must be equal to a given number.
L = [X11,X21,X31,X41,X51,
X11 #= 5, X31*X41 #= 2, X32*X42 #= 4, X33 #= 4,
X43*X53 #= 15, X34*X44 #= 15, X15*X25*X35*X45 #= 120,
X21*X22 #= 15, X51*X52 #= 12, X12*X13*X14 #= 8,
X23*X24 #= 2, X54*X55 #= 2,
Skyscrapers are built on an NxN grid. Each skyscraper has a number of floors corresponding to its size (from 1 to N). In each column and each row each size appears exactly once. The numbers on the
edges of the grid indicate how many skyscrapers can be seen when looking from that point towards the corresponding row or column. A skyscraper is visible if all the skyscrapers in front of it are
smaller. Here is an example of a grid and its solution:
For example, in the second line from the bottom looking from the left we can see 2 skyscrapers: the one of height 3 and the one of height 4. Here is a second problem of a grid (5×5) to solve:
Solve this problem by constraint logic programming.
You can use the predefined predicate fd_cardinality(+List, ?Count)
where List is a list of constraints and Count is the number of List constraints satisfied.
fd_cardinality([A5 #>A4 #/\A5 #>A3 #/\A5 #>A2 #/\A5 #>A1,
A4 #>A3 #/\A4 #>A2 #/\A4 #>A1,
A3 #> A2 #/\A3 #> A1, A2 #> A1],C).
L = [Line1,Line2,Line3,Line4,Line5],
Line1 = [X11,X12,X13,X14,X15],
Line2 = [X21,X22,X23,X24,X25],
Line3 = [X31,X32,X33,X34,X35],
Line4 = [X41,X42,X43,X44,X45],
Line5 = [X51,X52,X53,X54,X55],
fd_all_different(Line3), fd_all_different(Line4),
% The first skyscraper is still visible. So you have to remove 1
% of each number of visible skyscrapers.
constraint(Line1,0), constraint(Line2,3), constraint(Line3,2), | {"url":"https://complex-systems-ai.com/en/combinatorial-optimization-2/corrected-constraint-logic-programming-exercises/","timestamp":"2024-11-11T19:55:20Z","content_type":"text/html","content_length":"157918","record_id":"<urn:uuid:d3f2ec5a-d9c5-4b64-996f-c22cb2299bc4>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00023.warc.gz"} |
Elements of Geometry
Because the polygon BECFD is greater than the circle LIM, the prism BCFKHG is greater than the cylinder LMNO, for they have the same altitude, but the prism has the greater base. But the pyramid
ABECFD is the third part of the prism (15. 3. Sup.) BCFKHG, therefore it is greater than the third part of the cylinder LMNO. Now, the cone ABECFD is, by hypothesis, the third part of the cylinder
LMNO, therefore the pyramid ABECFD is greater than the cone ABCD, and it is also less, because it is inscribed in the cone, which is impossible. Therefore, the cone ABCD is not less than the third
part of the cylinder BFKG: And in the same manner, by circumscribing a polygon about the circle BCD, it may be shewn that the cone ABCD is not greater than the third part of the cylinder BFKG;
therefore. it is equal to the third part of that cylinder.
PROP. XIX. THEOR.
If a hemisphere and a cone have equal bases and altitudes, a series of cylinders may be inscribed in the hemisphere, and another series may be described about the cone, having all the same altitudes
with one another, and such that their sum shall differ from the sum of the hemisphere, and the cone, by a solid less than any given solid.
Let ADB be a semicircle of which the centre is C, and let CD be at right angles to AB; let DB and DA be squares described on DC, draw CE, and let the figure thus constructed revolve about DC: then,
the sector BCD, which is the half of the semicircle ADB, will describe a hemisphere having C for its centre (7 def. 3. Sup.), and the triangle CDE will describe a cone, having its vertex to C, and
having for its base the circle (11. def. 3. Sup.) described by I)E, equal to that described by BC, which is the base of the hemisphere. Let W be any given solid. A series of cylinders may be
inscribed in the hemisphere ADBĚ, and another described about the cone ECI, so that their sum shall differ from the sum of the hemisphere and the cone, by a solid less than the solid W.
Upon the base of the hemisphere let a cylinder be constituted equal to W, and let its altitude be CX. Divide CD into such a number of equal parts, that each of them shall be less than CX; let these
be CH, HG, GF, and FD. Through the points F, G, H, draw FN, GO, HP parallel to CB, meeting the circle in the points K, L and M; and the straight line CE in the points Q, R and S. From the points K,
L, M draw Kf, Lg, Mh, perpendicular to GO, HP and CB; and from Q, R, and S, draw Qq, Rr, Ss, perpendicular to the same lines. It is evident, that the figure being thus constructed, if the whole
revolve about CD, the rectangles Ff, Gg, Hh will describe cylinders (14. def. 3. Sup.) that will be circumscribed by the hemispheres BDA; and the rectangles DN, Fq, Gr, Hs, will also describe
cylinders that will circumscribe the cone ICE. Now, it may be demonstrated, as was done of the prisms inscribed in a pyramid (13. 3. Sup.), that the sum of all the cylinders described within the
hemisphere, is exceeded by the hemisphere by a solid less than the cylinder generated by the rectangle HB, that is, by a solid less than W, for the cylinder generated by HB is less than W. In the
same manner, it may be demonstrated, that the sum of the cylinders circumscribing the cone ICE is greater than
the cone by a solid less than the cylinder generated by the rectangle DN, that is, by a solid less than W. Therefore, since the sum of the cylinders inscribed in the hemisphere, together with a solid
less than W, is equal to the hemisphere; and, since the sum of the cylinders described about the cone is equal to the cone together with a solid less than W; adding equals to equals, the sum of all
these cylinders, together with a solid less than W, is equal to the sum of the hemisphere and the cone together with a solid less than W. Therefore, the difference between the whole of the cylinders
and the sum of the hemisphere and the cone, is equal to the difference of two solids, which are each of them less than W; but this difference must also be less than W, therefore the difference
between the two series of cylinders and the sum of the hemisphere and cone is less than the given solid W.
PROP. XX. THEOR.
The same things being supposed as in the last proposition, the sum of all the cylinders inscribed in the hemisphere, and described about the cone, is equal to a cylinder, having the same base and
altitude with the hemisphere.
Let the figure BCD be constructed as before, and supposed to revolve about CD; the cylinders inscribed in the hemisphere, that is, the cylinders described by the revolution of the rectangles Hh, Gg,
Ff, together with those described about the cone, that is, the cylinders described by the revolution of the rectangles Hs, Gr, Fq, and DN are equal to the cylinder described by the revolution of the
rectangle BD.
Let L be the point in which GO meets the circle ABD, then, because CGL is a right angle if CL be joined, the circles described with the distances CG and GL are equal to the circle described with the
distance CL (2. Cor. 6.1 Sup.) or GO; now, CG is equal to GR, because CD is equal to DE, and therefore also, the circles described with the distance GR and GL are together equal to the circle
described with the distance GO, that is, the circles described by the revolution of GR and GL about the point G, are together equal to the circle described by the revolution of GO about the same
point G; therefore also, the cylinders that stand upon the two first of these circles, having the common altitudes GH, are equal to the
cylinder which stands on the remaining circle, and which has the same altitude GH. The cylinders described by the revolution of the rectangles Gg, and Gr are therefore equal to the cylinder described
by the rectangle GP. And as the same may be shewn of all the rest, therefore the cylin ders described by the rectangles Hh, Gg, Ff, and by the rectangles Hs, Gr, Fq, DN, are together equal to the
cylinder described by BD, that is, to the cylinder having the same base and altitude with the hemisphere.
PROP. XXI. THEOR.
Every sphere is two-thirds of the circumscribing cylinder.
Let the figure be constructed as in the two last propositions, and if the hemisphere described by EDC be not equal to two-thirds of the cylinder described by BD, let it be greater by the solid W.
Then, as the cone described by CDE is one-third of the cylinder (18. 3. Sup.) described by BD, the cone and the hemisphere together will exceed the cylinder by W. But
that cylinder is equal to the sum of all the cylinders described by the rectangles Hh, Gg, Ff, Hs, Gr, Fq, DN (20. 3. Sup.); therefore the hemisphere and the cone added together exceed the sum of all
these cylinders by the given solid W, which is absurd; for it has been shewn (19. 3. Sup.), that the hemisphere and the cone together differ from the sum of the cylinders by a solid less than W. The
hemisphere is therefore equal to two-thirds of the cylinder described by the rectangle BD; and therefore the whole sphere is equal to two-thirds of the cylinder described by twice the rectangle BD,
that is, to two-thirds of the circumscribing cylinder.
END OF THE SUPPLEMENT TO THE ELEMENTS.
PLANE TRIGONOMETRY.
TRIGONOMETRY is the application of Arithmetic to Geometry: or, more precisely, it is the application of number to express the relations of the sides and angles of triangles to one another. It
therefore necessarily supposes the elementary operations of arithmetic to be understood, and it borrows from that science several of the signs or characters which peculiarly belong to it.
The elements of Plane Trigonometry, as laid down here, are divided into three sections: the first explains the principles; the second delivers the rules of calculation; the third contains the
construction of trigonometrical tables, together with the investigation of some theorems, useful for extending trigonometry to the solution of the more difficult problems.
SECTION I.
LEMMA I.
An angle at the centre of a circle is to four right angles as the arc on which it stands is to the whole circumference.
Let ABC be an angle at the centre of the circle ACF, standing on the circumference AC: the angle ABC is to four right angles as the arc AC to the whole circumference ACF.
Produce AB till it meet the circle in E, and draw DBF perpendicular to AE.
Then, because ABC, ABD are two angles at the centre of the circle ACF, the angle ABC is to the angle ABD as the arc AC to the arc AD, (33. 6.); and therefore also, the angle ABC is to four times the
angle ABD as the arc AC to four times the arc AD (4. 5.).
But ABD is a right angle, and therefore four times the arc AD is equal to
the whole circumference ACF; therefore the angle ABC is to four right angles as the arc AC to the whole circumference ACF.
COR. Equal angles at the centres of different circles stand on arcs which have the same ratio to their circumferences. For, if the angle ABC, at the centre of the circles ACE, GHK, stand on the arcs
AC, GH, AC is to the whole circumference of the circle ACE, as the angle ABC to four right angles; and the arc HG is to the whole circumference of the circle GHK in the same ratio.
1. IF two straight lines intersect one another in the centre of a circle, the arc of the circumference intercepted between them is called the Measure of the angle which they contain. Thus the arc AC
is the measure of the angle ABC.
2. If the circumference of a circle be divided into 360 equal parts, each of these parts is called a Degree; and if a degree be divided into 60 equal parts, each of these is called a Minute; and if a
minute be divided into 60 equal parts, each of them is called a Second, and so on. And as many degrees, minutes, seconds, &c. as are in any arc, so many degrees, minutes, seconds, &c. are said to be
in the angle measured by that arc.
COR. 1. Any arc is to the whole circumference of which it is a part, as the number of degrees, and parts of a degree contained in it is to the number 360. And any angle is to four right angles as the
number of degrees and parts of a degree in the arc, which is the measure of that angle, is to 360.
COR. 2. Hence also, the arcs which measure the same angle, whatever be the radii with which they are described, contain the same number of degrees, and parts of a degree. For the number of degrees
and parts of a degree contained in each of these arcs has the same ratio to the number 360, that the angle which they measure has to four right angles (Cor. Lem. 1.).
The degrees, minutes, seconds, &c. contained in any arc or angle, are usually written as in this example, 49°. 36′. 24′′. 42""; that is, 49 degrees, 36 minutes, 24 seconds, and 42 thirds.
3. Two angles, which are together equal to two right angles, or two arcs which are together equal to a semicircle, are called the Supplements of one another.
4. A straight line CD drawn through C, one of the extremities of the arc
« ΠροηγούμενηΣυνέχεια » | {"url":"https://books.google.gr/books?id=5rcUAAAAYAAJ&pg=PA217&vq=triangle+ABC&dq=editions:OCLC25489442&lr=&hl=el&output=html_text&source=gbs_toc_r&cad=3","timestamp":"2024-11-07T08:50:44Z","content_type":"text/html","content_length":"33033","record_id":"<urn:uuid:a85fac9c-6af3-45d7-bdb3-9b4111f7d1d7>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00496.warc.gz"} |
Vortices shed from a row of square bars
Interactions between the wakes in a flow past a row of square bars are investigated by numerical simulations, the linear stability analysis and the bifurcation analysis. It is assumed that the row of
square bars is placed across a uniform flow. Two-dimensional and incompressible flow field is also assumed. The flow is steady and symmetric along a streamwise centerline through the center of each
square bar at low Reynolds numbers. However, it becomes unsteady and periodic in time at the Reynolds numbers larger than a critical value, and then the wakes behind the square bars become
oscillatory. It is found by numerical simulations that vortices are shed synchronously from every couple of adjacent square bars in the same phase or in the anti-phase depending upon the distance
between the bars. The synchronous shedding of vortices is clarified to occur due to an instability of the steady symmetric flow by the linear stability analysis. The bifurcation diagram of the flow
is obtained and the critical Reynolds number of the instability is evaluated numerically.
Translated title of the contribution Vortices shed from a row of square bars
Original language Other
Title of host publication 境界層遷移の解明と制御研究会講演論文集
Editors Takeshi Akinaga, Takao Kadoishi, Jiro Mizushima
Pages 19-22
Number of pages 4
Publication status Published - Dec 2002
Publication series
Name 航空宇宙技術研究所特別資料
Number NAL-SP-56
ISSN (Print) 1347-457X
Dive into the research topics of 'Vortices shed from a row of square bars'. Together they form a unique fingerprint. | {"url":"https://research-test.aston.ac.uk/en/publications/%E8%A7%92%E6%9F%B1%E5%88%97%E3%82%92%E9%81%8E%E3%81%8E%E3%82%8B%E6%B5%81%E3%82%8C%E3%81%AE%E5%90%88%E6%B5%81%E7%8F%BE%E8%B1%A1%E3%81%A8%E4%B9%B1%E6%B5%81%E3%81%B8%E3%81%AE%E9%81%B7%E7%A7%BB","timestamp":"2024-11-11T11:39:39Z","content_type":"text/html","content_length":"54927","record_id":"<urn:uuid:95f8c70c-9359-4b9d-a28b-8cfb79384b4b>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00097.warc.gz"} |
Write Your Own map Method
Now that you’ve seen how to write your own higher-order functions, let’s take a quick look at a more real-world example.
Imagine for a moment that the List class doesn’t have its own map method, and you want to write your own. A good first step when creating functions is to accurately state the problem. Focusing only
on a List[Int], you state:
I want to write a map method that can be used to apply a function to each element in a List[Int] that it’s given, returning the transformed elements as a new list.
Given that statement, you start to write the method signature. First, you know that you want to accept a function as a parameter, and that function should transform an Int into some generic type A,
so you write:
The syntax for using a generic type requires declaring that type symbol before the parameter list, so you add that:
def map[A](f: (Int) => A)
Next, you know that map should also accept a List[Int]:
def map[A](f: (Int) => A, xs: List[Int])
Finally, you also know that map returns a transformed List that contains elements of the generic type A:
def map[A](f: (Int) => A, xs: List[Int]): List[A] = ???
That takes care of the method signature. Now all you have to do is write the method body. A map method applies the function it’s given to every element in the list it’s given to produce a new,
transformed list. One way to do this is with a for expression:
for expressions often make code surprisingly simple, and for our purposes, that ends up being the entire method body.
Putting it together with the method signature, you now have a standalone map method that works with a List[Int]:
def map[A](f: (Int) => A, xs: List[Int]): List[A] =
for (x <- xs) yield f(x)
def map[A](f: (Int) => A, xs: List[Int]): List[A] =
for x <- xs yield f(x)
Make it generic
As a bonus, notice that the for expression doesn’t do anything that depends on the type inside the List being Int. Therefore, you can replace Int in the type signature with the generic type parameter
def map[A, B](f: (B) => A, xs: List[B]): List[A] =
for (x <- xs) yield f(x)
def map[A, B](f: (B) => A, xs: List[B]): List[A] =
for x <- xs yield f(x)
Now you have a map method that works with any List.
These examples demonstrate that map works as desired:
def double(i : Int): Int = i * 2
map(double, List(1, 2, 3)) // List(2, 4, 6)
def strlen(s: String): Int = s.length
map(strlen, List("a", "bb", "ccc")) // List(1, 2, 3)
Now that you’ve seen how to write methods that accept functions as input parameters, let’s look at methods that return functions.
Contributors to this page: | {"url":"https://docs.scala-lang.org/scala3/book/fun-write-map-function.html","timestamp":"2024-11-02T15:29:57Z","content_type":"text/html","content_length":"87196","record_id":"<urn:uuid:b3344049-3754-466f-915f-21533ffcb683>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00323.warc.gz"} |
efficiency of ball mill graph
The graphs show that for a courser feed a three ball mixture (50 + 20 + 10 mm) will provide the much required largest amount of size class of interest. For a finer feed a binary mixture 20 mm + 10 mm
will satisfy the objective function which is to get more of the 150 + 75 μm size class material. | {"url":"https://www.obiadypracownicze.pl/2022_Mar_17-7959.html","timestamp":"2024-11-02T06:03:20Z","content_type":"text/html","content_length":"47270","record_id":"<urn:uuid:516e3a0a-e4df-4703-a758-45cf8707cafe>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00413.warc.gz"} |
Whats the meaning of whole numbers?
Whole numbers are a set of numbers including all positive integers and 0. Whole numbers are a part of real numbers that do not include fractions, decimals, or negative numbers. Counting numbers are
also considered as whole numbers.
What does whole numbers mean in math term?
Whole Numbers The numbers that include natural numbers and zero. Not a fraction or decimal.
What are whole numbers in short definition?
A whole number is an exact number such as 1, 7, and 24, as opposed to a number with fractions or decimals.
What’s another word for whole number?
positive integers
The whole numbers are also called the positive integers (or the nonnegative integers, if zero is included).
How do you know if a number is a whole number?
a more advanced example would be. You can multiply it by 10 and then do a “modulo” operation/divison with 10, and check if result of that two operations is zero. Result of that two operations will
give you first digit after the decimal point. If result is equal to zero then the number is a whole number.
How are whole numbers used in everyday life?
Whole numbers like 0, 1, and 2 are the building blocks to understanding more complex number identifiers like real numbers, rational numbers, and irrational numbers. Rounding to the nearest whole
number can also help you make calculations or do mental math more quickly.
What is the difference between natural numbers and whole numbers?
By definition, natural numbers are a part of the number system that contains all positive integers starting from the number 1 to infinity. Whereas, a whole number includes all positive numbers
starting from the number 0 to infinity. The whole numbers are 121, 4, 0, 30.
Which is the greatest whole number?
So, 0, 1, 2, 3, 4…… are the whole numbers. We can clearly say that 1 is the smallest natural number and 0 is the smallest whole number. But there is no largest whole number because each number has
its successor. Thus, there is no largest whole number.
How many whole numbers are there in?
My Standard
Name Numbers Examples
Whole Numbers { 0, 1, 2, 3, 4, } 0, 27,398, 2345
Counting Numbers { 1, 2, 3, 4, } 1, 18, 27, 2061
Integers { −4, −3, −2, −1, 0, 1, 2, 3, 4, } −15, 0, 27, 1102
What does whole numbers mean?
Whole numbers are the same thing as counting numbers. Also known as natural numbers. They are used to count the number of physical objects. Example 101 students.
What makes a whole number?
A whole number is any number you can make by adding any number of 1s to 0 — including 0 itself. Some examples of whole numbers include 2, 5, 17, and 12,000.
What are whole numbers examples?
In mathematics, whole numbers are the basic counting numbers 0, 1, 2, 3, 4, 5, 6, … and so on. 17, 99, 267, 8107 and 999999999 are examples of whole numbers.
What are some examples of a whole number?
A whole number is any number that does not contain a fraction, decimal, or negative value. For example, 1, 25, and 365 are whole numbers. | {"url":"https://sage-advices.com/whats-the-meaning-of-whole-numbers/","timestamp":"2024-11-08T15:22:33Z","content_type":"text/html","content_length":"145522","record_id":"<urn:uuid:4ce6c668-0ec8-4d42-b96f-d95a9b40cfb5>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00021.warc.gz"} |
30000 BC - 500 BC - Chronology
30000 BC - 500 BC
• Palaeolithic peoples in central Europe and France record numbers on bones.
• Early geometric designs used.
• A decimal number system is in use in Egypt.
• Babylonian and Egyptian calendars in use.
• The first symbols for numbers, simple straight lines, are used in Egypt.
• The abacus is developed in the Middle East and in areas around the Mediterranean.
• Hieroglyphic numerals in use in Egypt. (See this History Topic.)
• Babylonians begin to use a sexagesimal number system for recording financial transactions. It is a place-value system without a zero place value. (See this History Topic.)
• Harappans adopt a uniform decimal system of weights and measures.
• The Moscow papyrus (also called the Golenishev papyrus) is written. It gives details of Egyptian geometry. (See this History Topic.)
• Babylonians use multiplication tables.
• The Babylonians solve linear and quadratic algebraic equations, compile tables of square and cube roots. They use Pythagoras's theorem and use mathematics to extend knowledge of astronomy. (See
this History Topic.)
• The Rhind papyrus (sometimes called the Ahmes papyrus) is written. It shows that Egyptian mathematics has developed many techniques to solve problems. Multiplication is based on repeated
doubling, and division uses successive halving. (See this History Topic.)
• About this date a decimal number system with no zero starts to be used in China. (See this History Topic.)
• Apastamba writes the most interesting Indian Sulbasutra from a mathematical point of view. (See this History Topic.)
• Thales brings Babylonian mathematical knowledge to Greece. He uses geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore.
• Pythagoras of Samos moves to Croton in Italy and teaches mathematics, geometry, music, and reincarnation. | {"url":"https://mathshistory.st-andrews.ac.uk/Chronology/1/","timestamp":"2024-11-14T10:58:43Z","content_type":"text/html","content_length":"15038","record_id":"<urn:uuid:2d5c6d2b-9d67-43eb-bfe4-64451cac6dfb>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00773.warc.gz"} |
Cite as
Ruben Becker, Arnaud Casteigts, Pierluigi Crescenzi, Bojana Kodric, Malte Renken, Michael Raskin, and Viktor Zamaraev. Giant Components in Random Temporal Graphs. In Approximation, Randomization, and
Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 29:1-29:17, Schloss Dagstuhl – Leibniz-Zentrum
für Informatik (2023)
Copy BibTex To Clipboard
author = {Becker, Ruben and Casteigts, Arnaud and Crescenzi, Pierluigi and Kodric, Bojana and Renken, Malte and Raskin, Michael and Zamaraev, Viktor},
title = {{Giant Components in Random Temporal Graphs}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)},
pages = {29:1--29:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-296-9},
ISSN = {1868-8969},
year = {2023},
volume = {275},
editor = {Megow, Nicole and Smith, Adam},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.29},
URN = {urn:nbn:de:0030-drops-188542},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2023.29},
annote = {Keywords: random temporal graph, Erd\H{o}s–R\'{e}nyi random graph, sharp threshold, temporal connectivity, temporal connected component, edge-ordered graph} | {"url":"https://drops.dagstuhl.de/search/documents?author=Casteigts,%20Arnaud","timestamp":"2024-11-05T00:02:32Z","content_type":"text/html","content_length":"118795","record_id":"<urn:uuid:a41d21ac-a45f-4366-8953-6c445fdc947e>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00817.warc.gz"} |
Math 116
Course Objectives: This is a fundamental course in numerical analysis, with a focus on numerical linear algebra and optimzation methods, which are the building blocks for a variety of algorithms used
in data science, machine learning, signal and image processing, and evolutionary dynamics. The primary goal is for students to gain a complete understanding of how to evaluate the efficacy of
(general) numerical methods, specifically their accuracy, stability, and convergence properties. This is not a survey course. Students will write simple computer programs in MATLAB or python to gain
understanding of how methods work, in particular to compare advantages and disadvantages of various algorithms. This course will not emphasize programming or the use of numerical software.
Numerical analysis is a fundamental topic in applied mathematics. Many practical problems that scientists try to solve are based on mathematical models, but few can be solved analytically, either due
to their complexity, large number of variables, or lack of information. Computational algorithms are therefore needed for approximating these solutions. It is critically important to maintain the
important mathematical properties of the underlying system when developing these computational algorithms. Numerical analysis is about developing good computational techniques for broad based
problems and analyzing their properties. Numerical analysts seek to demonstrate when computational algorithms are trustworthy so that domain scientists can be confident in the results of their
experiments. Numerical analysts also answer the question, "what assumptions of the underlying problem are necessary for this computational method to succeed?" | {"url":"https://math.dartmouth.edu/~m116w21/","timestamp":"2024-11-02T11:30:34Z","content_type":"text/html","content_length":"8561","record_id":"<urn:uuid:cde2f206-8c65-4737-a56a-adf1dc7df660>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00235.warc.gz"} |
Binning data by quantiles? Beware of rounded data
In my article about how to create a quantile plot, I chose not to discuss a theoretical issue that occasionally occurs. The issue is that for discrete data (which includes rounded values), it might
be impossible to use quantile values to split the data into k groups where each group has approximately the same number of elements. To put it bluntly, the algorithm that I proposed for creating a
quantile bin plot can fail for large values of k.
The problem can occur for data that are rounded to the nearest unit, such as age, height, weight, and blood pressure. The algorithm assumes that the data quantiles are (mostly) unique and that they
divide the empirical cumulative distribution function (ECDF) of N observations into k groups that each have approximately N/k observations. However, if there are more than N/k repeated values, the
repeated value can occupy more than one quantile value. In fact, this will always happen if a particular value is repeated more than 2N/k times.
For example, suppose that you want to divide the diastolic blood pressure data in the Sashelp.Heart data set into 10 groups that each have approximately 10% of the data. The following statements draw
the ECDF for the diastolic measurements and mark the deciles:
proc univariate data=Sashelp.Heart;
var diastolic;
cdfplot diastolic / vref=(10 to 90 by 10);
ods select CDFPlot Quantiles;
The value 80 appears in 711 of the 5209 observations, which is about 14% of the data. Notice that the value 80 intersects two reference lines, one at 30 and the other at 40. This means that 80 is
both the 30th percentile and the 40th percentile. Assuming that you do not want to split the value 80 across two bins, the deciles of the data produce at most nine bins.
Another way to see this graphically is to use the RANK procedure to try to group the data into 10 groups, as described in the article "Grouping observations based on quantiles." The following
statements create a new variable called Group, which for continuous data would have the values 0–9. However, for this rounded data only nine groups exist:
proc sort data=Sashelp.Heart out=Heart;
by Diastolic;
proc rank data=Heart out=Heart groups=10 ties=high;
var Diastolic; /* variable on which to group */
ranks Group; /* name of variable to contain groups 0,1,...,k-1 */
proc sgplot data=Heart;
scatter x=Group y=Diastolic / group=Group;
refline 80 / axis=y;
xaxis values=(0 to 9);
As shown by the legend on the scatter plot, only nine groups were created, instead of the 10 that were asked for. Group 3 was not created.
I've described a problem, but what can you do about it? Not a lot. The problem is in the data. If you are flexible about the number of groups, you can try decreasing the value of k. Because each
group contains approximately N/k observations, decreasing k might circumvent the problem. Another possibility is to jitter the data by adding a small amount of random noise. That will eliminate the
problem of repeated values.
So I'll leave you with this warning: beware of using quantiles to bin rounded data into groups. Although the technique works great when almost all of the data values are distinct, you can run into
problems if you ask for many bins and your data contain many repeated values.
2 Comments
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division Archives - Page 2 of 4 - EducationUnboxed.com - Free Help for Homeschool
This is part three in the series of videos about long division. This free math tutoring video has divisors and quotients that are 2-digit numbers and shows how you can use drawings to aid in the
conceptual understanding of these types of problems. This method works especially well for visual, right-brained, or global learners.
For more practice, click here to download a worksheet.
Long Division – Part Two – Free Math Tutoring Video
This is the second free math tutoring video in the long division series. These problems have divisors and quotients that are between 10 and 19. Doing multiplication problems with Cuisenaire Rods
where they multiply teen numbers would be an excellent intro to this topic. This method enables children to truly understand what the long division algorithm is all about instead of simply memorizing
the steps. It is great for visual and kinesthetic learners!
For more practice, click here to download a worksheet.
Long Division – Part One – Free Math Tutoring Video
This is the first of several free math tutoring videos that show how to do long division using Cuisenaire Rods to aid in conceptual understanding. Most people have no idea WHY the long division
algorithm works. They’ve just memorized the steps without understanding. This may work for many people, but there are a whole lot of children in school today who have sequencing issues, who struggle
with memorizing the steps to math formulas that have no meaning to them, who simply want to understand WHY all this stuff works. Those children will be better served if their teachers understand why
and can teach from a conceptual framework instead of a formula-driven approach. Even for those students who are willing to just “memorize the formula,” this method will be beneficial because it
builds a clearer understanding of the decimal system and the distributive property which will give them confidence that they really do understand and are “good at” math.
I discovered this way of teaching long division through Crewton Ramone’s House of Math. I’ve changed his method very slightly to what makes more sense to me, but it is essentially the same idea. This
method is ingenious. I don’t know why more people don’t know about it!
For more practice, click here to download a worksheet. | {"url":"http://www.educationunboxed.com/tag/division/page/2/","timestamp":"2024-11-09T00:16:02Z","content_type":"text/html","content_length":"49518","record_id":"<urn:uuid:121aa971-2d20-4402-820a-5c763ba1e6ec>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00539.warc.gz"} |
• A Sharp Theory of Hardy Spaces on Ahlfors-Regular Quasi-Metric Spaces. With M.Mitrea, Springer Lecture Notes in Mathematics, Vol.2142, (2015), 486 pages. Introduction Springerlink
• Borel regularity is equivalent to Lusin's theorem and the existence of Borel representatives. With P.Górka and A.Słabuszewski. (submitted) arXiv
• Compact embeddings of Sobolev, Besov, and Triebel-Lizorkin spaces. With P.Górka and A.Słabuszewski. (submitted) arXiv
• Optimal Embeddings for Triebel-Lizorkin and Besov Spaces on Quasi-Metric Measure Spaces. With D.Yang and W.Yuan, Math. Z. 307, 50 (2024). arXiv
• A simple proof of reflexivity and separability of N^{1,p} Sobolev spaces. With P.Hajłasz and Lukáš Malý. Ann. Fenn. Math. 48 (2023), no. 1, 255-275. arXiv
• Pointwise Characterization of Besov and Triebel–Lizorkin Spaces on Spaces of Homogeneous Type. With F.Wang, D.Yang and W.Yuan. Studia Math. 268 (2023), no. 2, 121–166. PDF
• A Measure Characterization of Embedding and Extension Domains for Sobolev, Triebel-Lizorkin, and Besov Spaces on Spaces of Homogeneous Type. With D.Yang and W.Yuan. J. Funct. Anal. 283 (2022),
no. 12, Paper No. 109687, 71 pp. arXiv
• The game of cycles. With M.Averett, B.Gaines, C.Jackson, M.L.Karker, M.A.Marciniak, F.E.Su, and S.Walker. Amer. Math. Monthly 128 (2021), no. 10, 868-887. arXiv
• Sobolev embedding for M^{1,p} spaces is equivalent to a lower bound of the measure. With P.Górka and P.Hajłasz. J. Funct. Anal. 279 (2020), no. 7, 108628 arXiv
• A note on metric-measure spaces supporting Poincaré inequalities. With P.Hajłasz. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 31 (2020), no.1, 15-23. arXiv
• Characterizing Lusin's Theorem and the Density of Continuous Functions in Lebesgue Spaces via the Regularity of the Measure. With M.Mitrea and B.Schmutzler. (preprint)
• Whitney-type extensions with control of the modulus of continuity in geometrically doubling quasi-metric spaces. With I.Mitrea and M.Mitrea. Commun. Pure Appl. Anal., 12 (2013), No. 1, pp. 59-88.
• Sharp geometric maximum principles for semi-elliptic operators with singular drift. With D.Brigham, V.Maz'ya, M.Mitrea and E.Ziadé, Math. Res. Lett., Vol.18 (2011), No. 04, pp. 613-620. PDF
• On the regularity of domains satisfying a uniform hour-glass condition and a sharp version of the Hopf-Oleinik Boundary Point Principle. With D.Brigham, V.Maz'ya, M.Mitrea and E.Ziadé, Journal of
Mathematical Sciences, Vol. 176 (2011), No. 3, pp. 281-360.
• Topics in Harmonic Analysis and Partial Differential Equations: Extension Theorems and Geometric Maximum Principles. Masters Thesis. (2011). PDF | {"url":"https://rjalvarado.people.amherst.edu/publications.php","timestamp":"2024-11-11T03:34:54Z","content_type":"text/html","content_length":"31287","record_id":"<urn:uuid:38a2b567-fdda-4c6c-9561-56edf64ebf41>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00063.warc.gz"} |
Automata Theory Questions for Campus Interviews - Sanfoundry
Automata Theory Questions and Answers – Pumping Lemma for Context Free Language
This set of Automata Theory Questions and Answers for Campus interviews focuses on “Pumping Lemma for Context Free Language”.
1. Which of the following is called Bar-Hillel lemma?
a) Pumping lemma for regular language
b) Pumping lemma for context free languages
c) Pumping lemma for context sensitive languages
d) None of the mentioned
View Answer
Answer: b
Explanation: In automata theory, the pumping lemma for context free languages, also kmown as the Bar-Hillel lemma, represents a property of all context free languages.
2. Which of the expressions correctly is an requirement of the pumping lemma for the context free languages?
a) uv^nwx^ny
b) uv^nw^nx^ny
c) uv^2nwx^2ny
d) All of the mentioned
View Answer
Answer: b
Explanation: Let L be a CFL. Then there is an integer n so that for any u that belong to language L satisfying |t| >=n, there are strings u, v, w, x, y and z satisfying
|vwx|<=n For any m>=0, uv^nwx^ny ∈ L
3.Let L be a CFL. Then there is an integer n so that for any u that belong to language L satisfying
|t|>=n, there are strings u, v, w, x, y and z satisfying
Let p be the number of variables in CNF form of the context free grammar. The value of n in terms of p is
a) 2p
b) 2p
c) 2p+1
d) p^2
View Answer
Answer: c
Explanation: This inequation has been derived from derivation tree for t which must have height at least p+2(It has more than 2^p leaf nodes, and therefore its height is >p+1).
4. Which of the following gives a positive result to the pumping lemma restrictions and requirements?
a) {a^ib^ic^i|i>=0}
b) {0^i1^i|i>=0}
c) {ss|s∈{a,b}*}
d) None of the mentioned
View Answer
Answer: b
Explanation: A positive result to the pumping lemma shows that the language is a CFL and ist contradiction or negative result shows that the given language is not a Context Free language.
5. Using pumping lemma, which of the following cannot be proved as ‘not a CFL’?
a) {a^ib^ic^i|i>=0}
b) {ss|s∈{a,b}*}
c) The set legal C programs
d) None of the mentioned
View Answer
Answer: d
Explanation: There are few rules in C that are context dependent. For example, declaration of a variable before it can be used.
6. State true or false:
Statement: We cannot use Ogden’s lemma when pumping lemma fails.
a) true
b) false
View Answer
Answer: b
Explanation: Although the pumping lemma provides some information about v and x that are pumped, it says little about the location of these substrings in the string t. It can be used whenever the
pumping lemma fails. Example: {a^pb^qc^rd^s|p=0 or q=r=s}, etc.
7. Which of the following cannot be filled in the blank below?
Statement: There are CFLs L1 nad L2 so that ___________is not a CFL.
a) L1∩L2
b) L1′
c) L1*
d) None of the mentioned
View Answer
Answer: c
Explanation: A set of context free language is closed under the following operations:
a) Union
b) Concatenation
c) Kleene
8. The pumping lemma is often used to prove that a language is:
a) Context free
b) Not context free
c) Regular
d) None of the mentioned
View Answer
Answer: b
Explanation: The pumping lemma is often used to prove that a given language L is non-context-free, by showing that arbitrarily long strings s are in L that cannot be “pumped” without producing
strings outside L.
9. What is the pumping length of string of length x?
a) x+1
b) x
c) x-1
d) x2
View Answer
Answer: a
Explanation: There exists a property of all strings in the language that are of length p, where p is the constant-called the pumping length .For a finite language L, p is equal to the maximum string
length in L plus 1.
10. Which of the following does not obey pumping lemma for context free languages ?
a) Finite languages
b) Context free languages
c) Unrestricted languages
d) None of the mentioned
View Answer
Answer: c
Explanation: Finite languages (which are regular hence context free ) obey pumping lemma where as unrestricted languages like recursive languages do not obey pumping lemma for context free languages.
Sanfoundry Global Education & Learning Series – Automata Theory.
To practice all areas of Automata Theory for Campus Interviews, here is complete set of 1000+ Multiple Choice Questions and Answers. | {"url":"https://www.sanfoundry.com/automata-theory-questions-answers-campus-interviews/","timestamp":"2024-11-07T22:29:04Z","content_type":"text/html","content_length":"143187","record_id":"<urn:uuid:d5008179-23e4-4aff-9546-785953acff42>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00786.warc.gz"} |
Cite as
Mikołaj Bojańczyk, Lê Thành Dũng (Tito) Nguyễn, and Rafał Stefański. Function Spaces for Orbit-Finite Sets. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024).
Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 130:1-130:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
Copy BibTex To Clipboard
author = {Boja\'{n}czyk, Miko{\l}aj and Nguy\~{ê}n, L\^{e} Th\`{a}nh D\~{u}ng (Tito) and Stefa\'{n}ski, Rafa{\l}},
title = {{Function Spaces for Orbit-Finite Sets}},
booktitle = {51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
pages = {130:1--130:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-322-5},
ISSN = {1868-8969},
year = {2024},
volume = {297},
editor = {Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.130},
URN = {urn:nbn:de:0030-drops-202730},
doi = {10.4230/LIPIcs.ICALP.2024.130},
annote = {Keywords: Orbit-finite sets, automata, linear types, game semantics} | {"url":"https://drops.dagstuhl.de/search?term=Lasota%2C%20Slawomir","timestamp":"2024-11-07T20:03:38Z","content_type":"text/html","content_length":"168284","record_id":"<urn:uuid:5733bf5c-8c73-4384-9a40-de5fe4514572>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00635.warc.gz"} |
mathematics Archives • Turton School
Induction to KS4 A demonstration of Sparx Maths and Corbett Maths with Mr. Hodgson. Induction to KS4 A demonstration of Sparx Maths and Corbett Maths with Mr. Hodgson. Please find additional guides
for other Year 10 subjects here… English Science Faith & Ethics
Year 11 Science & Maths Exams (Mocks /OCT)
Please find attached a copy of the exam timetable for the upcoming Year 11 Science and Maths exams. All year 11 students have been issued with this timetable and have a copy on their school email
account. Your son / daughter’s Maths and Science teachers will have discussed these exams with them and all Read More …
Year 11 Revision Outline Maths Higher (Mathematics)
Year 11 Revision Outline for Maths Higher (Mathematics)
Year 11 Revision Outline for Maths Foundation (Mathematics)
Year 11 Revision Outline for Maths Foundation (Mathematics)
Maths Learning Walk
Maths Learning WalkOn 5th January 2016 fellow Governor, Dave Miller and I met with Deputy Head Teacher, Mrs Bach and Ms Gorse the Head Teacher. Initially we observed their book scrutiny of a
selection of pupils’ work books from various classes and years. Their very thorough scrutiny was to ensure consistent teaching and marking was Read More … | {"url":"https://www.turton.uk.com/tag/mathematics/","timestamp":"2024-11-09T15:37:46Z","content_type":"text/html","content_length":"64217","record_id":"<urn:uuid:a7516c35-33c1-4271-aad8-92b284db76ee>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00147.warc.gz"} |
Suggestion: support graphing families of functions
Topic: Suggestion: support graphing families of functions
A friend just showed me this package last night to illustrate some functions we were looking at. Quite liked it. But when I asked her if she could plot a family of functions, no
can do.
Pretty important for looking at trends.
To clarify, imagine the very simple function:
y = x^n
it would be nice to ask Graph to plot this for n=0 to 5 step 1 say.
Thus a family of functions varying n.
The same applies to more complex functions indeed is where it becomes truly useful, to look at how a given coefficient alters the plot.
As it was she had to create a separate line manually for each value we wanted to look at, time consuming.
Re: Suggestion: support graphing families of functions
Animations can do something a little similar. But instead of adding more functions it will change the existing one. That is you can add the function f(x)=x^n and animate n from 0 to 5.
A more advanced possibility is to use the scripting engine, which requires you to install Python 3.2 32 bit.
When you have Python installed you can press F11 inside Graph to get a Python interpreter and enter the following code to add the family of functions:
for n in range(6):
F=Graph.TStdFunc("x^%d" % n)
I will consider creating an easier way to do this in a future version. | {"url":"https://forum.padowan.dk/viewtopic.php?pid=2592","timestamp":"2024-11-09T07:34:24Z","content_type":"text/html","content_length":"10896","record_id":"<urn:uuid:515b97b8-f083-4003-8583-665b37d50a38>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00386.warc.gz"} |
Roots third order quadratic
roots third order quadratic Related topics: math power 8
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Least Common Denominator In Algebra
Author Message
Calnicta Posted: Wednesday 07th of Dec 07:13
I have trouble with roots third order quadratic. I tried a lot to get somebody who can assist me with this. I also searched for a coach to teach me and crack my problems on quadratic
formula, simplifying expressions and radicals. Though I found a few who could perhaps crack my problem, I recognized that I cannot afford them. I do not have much time too. My exam is
coming up in a little while. I am worried . Can someone assist me with this situation? I would really value any help or any information.
From: UK
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IlbendF Posted: Wednesday 07th of Dec 13:12
Algebrator is one of the most powerful resources that can offer help to a person like you. When I was a beginner, I took support from Algebrator. Algebrator offers all the basics of
Remedial Algebra. Rather than using the Algebrator as a step-by-step guide to solve all your homework assignments, you can use it as a coach that can offer the basics of radicals, powers
and trinomials. Once you assimilate the principles, you can go ahead and work out any tough assignments on Algebra 1 in no time .
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Ashe Posted: Thursday 08th of Dec 19:58
Algebrator is one beneficial tool. I don’t have much interest in math and have found it to be complicated all my life. Yet one cannot always leave math because it sometimes becomes a
compulsory part of one’s course work. My friend is a math wiz and I found this software in his palmtop . It was only then I understood why he finds this subject to be so simple .
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vromolx Posted: Saturday 10th of Dec 08:19
I'm so happy I got these answers so fast, I can't wait to buy Algebrator. Can you tell me one more thing, where could I find this program? I'm not so good at searching for things like
this, so it would be good if you could give me a link . Thanks a lot!
From: Alken
- Belgium
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Vild Posted: Saturday 10th of Dec 14:31
Here is the link https://softmath.com/comparison-algebra-homework.html
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Bet Posted: Saturday 10th of Dec 16:45
Algebrator is the program that I have used through several math classes - College Algebra, College Algebra and Algebra 1. It is a really a great piece of math software. I remember of
going through difficulties with graphing, trinomials and radical expressions. I would simply type in a problem from the workbook , click on Solve – and step by step solution to my math
homework. I highly recommend the program.
From: kµlt
øƒ Ø™
Back to top | {"url":"https://softmath.com/algebra-software/radical-equations/roots-third-order-quadratic.html","timestamp":"2024-11-11T04:39:43Z","content_type":"text/html","content_length":"43479","record_id":"<urn:uuid:27f9def9-bf09-4385-acd6-6c4c6920e034>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00866.warc.gz"} |
Graphical Representation of Motion along a Straight Line
Plotting the distance/displacement or speed/velocity on a graph helps us visually understand certain things about time and position.
1. The distance – time graph for uniform motion
The following Table shows the distance walked by Surya at different times.
A graph is drawn by taking time along X-axis and distance along Y-axis. The graph is known as distance – time graph. When we look at the distance – time graph of Surya’s walk, we notice certain
things. First, it is a straight line. We also notice that Surya covers equal distances in equal intervals of time.
We can therefore conclude that Surya walked at a constant speed. Can you find the speed at which Surya walked, from the graph? Yes, you can. The parameter is referred to as the slope of the line.
Speed at which Surya walked = distance covered / time taken = BC/AC (From the graph)
= slope of the straight line
= 500 / 5 = 100 ms^-1
Steeper the slope (in other words the larger value) the greater is the speed.
Let us take a look at the distance–time graphs of three different people – Surya walking, Monica cycling and Hari going in a car, along the same path. We know that cycling can be faster than walking
and a car can go faster than a cycle. The distance – time graph of the three would be as given in the following graph. The slope of the line on the distance – time graph becomes steeper and steeper
as the speed increases.
2. The distance time graph for non uniform motion
We can also plot the distance – time graph for accelerated motion (non uniform motion). Table given below shows the distance travelled by a car in a time interval of two second.
Note that the graph is not a straight line as we got in the case of uniform motion. is nature of the graph shows non – linear variation of the distance travelled by the car with time. us, the graph
represents motion with non uniform speed.
3. Velocity – Time graph
The variation in velocity of an object with time can be represented by velocity – time graph. In the graph, time is represented along the X – axis and the velocity is represented along the Y – axis.
If the object moves at uniform velocity, a straight line parallel to X-axis is obtained. is Graph shows the velocity – time graph for a car moving with uniform velocity of 40 km/hour.
We know that the product of velocity and time gives displacement of an object moving with uniform velocity.
The area under the velocity – time graph is equal to the magnitude of the displacement.
So the distance (displacement) S covered by the car in a time interval of t can be expressed as
S = AC × CD
S = Area of the rectangle ABCD (shaded portion in the graph)
We can also study about uniformly accelerated motion by plotting its velocity – time graph. Consider a car being driven along a straight road for testing its engine. Suppose a person sitting next to
the driver records its velocity for every 5 seconds from the speedometer of the car. The velocity of the car in ms-1 at different instants of time is shown in the Table below.
In this case, the velocity – time graph for the motion of the car is shown in graph (straight line). The nature of the graph shows that the velocity changes by equal amounts in equal intervals of
time. Thus, for all uniformly accelerated motion, the velocity – time graph is a straight line.
One can also determine the distance moved by the car from its velocity – time graph. The area under the velocity – time graph gives the distance (magnitude of displacement) moved by the car in a
given interval of time.
Since the magnitude of the velocity of the car is changing due to acceleration, the distance S travelled by the car will be given by the area ABCDE under the velocity – time graph. That is
S = area ABCDE
= area of the rectangle ABCD + area of the triangle ADE
S = (AB × BC) + ½ (AD × DE)
The area ABCDE can also be calculated by considering the shape as trapezium. Area of the quadrangle ABCDE can also be calculated by calculating the area of trapezium ABCDE. It means
S = area of trapezium ABCDE
= ½ × sum of length of parallel sides × distance between parallel sides
S = ½ × (AB + CE) × BC
In the case of non uniformly accelerated motion, distance – time graph, velocity – time graphs can have any shape as shown in Figure below: | {"url":"https://www.brainkart.com/article/Graphical-Representation-of-Motion-along-a-Straight-Line_35767/","timestamp":"2024-11-04T20:16:18Z","content_type":"text/html","content_length":"42376","record_id":"<urn:uuid:34508c73-5a36-486d-b262-d2b62457a63e>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00174.warc.gz"} |
Ch 7-8 Review Questions
Review Questions
Ch 7 & 8 – Work and Energy
Concept Questions
c1. Sign of work done
In the following scenarios, a force pushes on an object. Determine if the work done by the force is positive, negative, or zero.
a. A box is pushed toward the left, and slides along the floor at a constant speed toward the left.
b. The same box as in (a). Consider the work done by gravity and work done by the normal force.
c. A car turns a corner on a level road at constant speed. Consider the work done by static friction, which causes the centripetal acceleration.
d. A crane lowers a mass at constant speed. Consider the work by the tension in the cable.
e. Force \(\vec F\) and displacement \(\Delta \vec r\), as shown below.
a. Work done by pushing force is positive since force and displacement are in the same (leftward) direction.
b. Work done by gravity and the normal force are zero, because these forces are perpendicular to the motion.
c. Work done by static friction is zero because \(\vec f\) is always perpendicular to \(\vec v\).
d. Work done by the tension is negative because it points up while the motion is down.
e. Work done is postive because \(\vec F \cdot \Delta \vec r > 0\).
c2. Energy Transformations
For the following scenarios, describe the energy transformation taking place. Use \(K\), \(U\), and \(E_{th}\) to refer to kinetic, potential and thermal energy, respectively.
a. A diver falls downward, before hitting the water.
b. The diver hits the water and rapidly slows.
c. A skier is sliding down a slope at a constant speed.
d. A box slides up a gentle but slightly rough incline until stopping at the top.
a. \(U \rightarrow K\)
b. \(K \rightarrow E_{th}\)
c. \(U \rightarrow E_{th}\)
d. \(K \rightarrow U\) and \(E_{th}\)
a. As the diver falls, they pick up speed. Elevation loss means \(U\) decreases. Speed increase means \(K\) increases.
b. The speed decrease means \(K\) decreases. There is no substantial elevation change, so the energy has been lost to thermal energy in the water.
c. The skier’s speed is constant, so \(K\) does not change. There is a loss of elevation, meaning \(U\) is lost. If the skier does not speed up, it must be a friction force that does work. Meaning
thermal energy is generated.
d. Losing speed means \(K\) is lost. Gaining elevation means \(U\) increases. The “rough” surface implies that friction does work, so thermal energy is also created.
c3. Work and kinetic energy
Object \(A\) has mass \(M\), and \(B\) has mass \(2M\). Both are at rest.
Compare their kinetic energies (\(K_A\) and \(K_B\)) after
a. They are each pushed with the same force for the same distance \(d\).
b. They are each pushed with the same force for the same time \(t\).
a. \(K_A = K_B\)
b. \(K_A \gt K_B\)
In each case, \(K\) depends on the work done, which is force times distance.
a. Work done is the same, so \(\Delta K\) is the same.
b. Over time \(t\), \(A\) will accelerate more and move a greater distance. So work done on \(A\) is greater.
c4. Height vs speed
A roller coaster rolls from rest down a frictionless track, reaching speed \(v\) at the bottom. If you want the car to go twice as fast at the bottom, by what factor must you increase the height of
the track?
The track must be \(4\) times higher.
Energy conservation gives \(mgh = \frac{1}{2} m v^2\), so \(h = \frac{v^2}{2g}\).
So if \(v\) is doubled, the new height will be \(4h\).
c5. Spring energy
Rank the elastic potential enegies stored in the following springs.
\(U_1 = \frac{1}{2} k d^2\)
\(U_2 = \frac{1}{2} k d^2 = U_1\)
\(U_3 = \frac{1}{2} (2k) d^2 = 2 U_1\)
\(U_4 = \frac{1}{2} k (2d)^2 = 4 U_1\)
\(U_4 > U_3 > U_2 = U_1\)
c6. Spring gun muzzle velocity
A spring-loaded gun shoots out a plastic ball at speed \(v_0\). The spring is then compressed twice the distance it was on the first shot.
a. By what factor is the spring’s potential energy increased?
b. By what factor is the ball’s launch speed increased?
a. \(U = \frac{1}{2} k x^2\), so double \(x\) means four times more energy.
b. \(U \rightarrow \frac{1}{2} m v^2\), so four times more energy means double \(v\).
c7. Energy diagram
Below are a set of axes on which you are going to draw a potential-energy curve. By doing experiments, you find the following information:
• A particle with energy \(E_1\) oscillates between positions D and E.
• A particle with energy \(E_2\) oscillates between positions C and F.
• A particle with energy \(E_3\) oscillates between positions B and G.
• A particle with energy \(E_4\) enters from the right, turns around at A, then never returns.
Draw a potential-energy curve that is consistent with this information.
c8. Features of energy diagram
The graph shows the potential energy curve of a particle moving along the \(x\)-axis under the influence of a conservative force.
a. In which intervals of \(x\) is the force on the particle to the right?
b. In which intervals of \(x\) is the force on the particle to the left?
c. At what values of \(x\) is the magnitude of the force maximum?
d. What values of \(x\) are positions of stable equilibrium?
e. What values of \(x\) are positions of unstable equilibrium?
f. If the particle is released from rest at \(x=0\) m, will it reach \(x=10\) m?
a. \(F_x= - (dU)/dx\) (negative of slope). So \(\vec F_x\) is to the right from \(0\) m to \(2\) m, and from \(5\) m to \(8\) m.
b. \(\vec F_x\) is to the left from \(2\) m to \(5\) m, and from \(8\) m to \(10\) m.
c. Slope is steepest at \(x=0\) m, \(x=3.5\) m, \(x=6.5\) m, and \(x=10\) m.
d. At the minima, \(x=2\) m and \(x=8\) m.
e. At the maxima, \(x=5\) m.
f. No. The particle will move right until it turns around at about \(x=4\) m.
c9. Compare projectiles
Two projectiles (with the same mass) are launched over flat ground with the same intial speed. Projectile \(A\) is launched at angle \(10\)°, and projectile \(B\) at angle \(50\)° (above horizontal).
You may ignore air resistance.
a. Which projectile has greater speed at its peak height?
b. Which projectile has greater speed upon hitting the ground?
Note the equal initial speed means that they have the same total energy at all times throughout their flights: \(K+U =\) constant.
a. \(A\) is moving faster. \(K_A \gt K_B\), since \(U_A \lt U_B\) at their peak height.
b. Same speed. \(K_A = K_B\), since \(U_A=U_B=0\) at the ground.
Long Answer Questions
1. Work done by given force
Consider a particle on which several forces act, one of which is known to be constant in time: \(\vec{F}_1 = 3 \hat{\imath} + 4 \hat{\jmath}\) N
As a result, the particle moves along a straight path from a Cartesian coordinate of (\(-1\) m, \(2\) m) to (\(5\) m, \(6\) m). What is the work done by \(\vec F_1\) ?
Since the force is constant and the path is straight, \(W = \vec F \cdot \Delta \vec s\).
\(W= (3 \hat{\imath} + 4 \hat{\jmath}) \cdot (6\hat{\imath} + 4 \hat{\jmath}) = 18+16 =34\) N·m
2. Accelerated electron
An electron in a television tube is accelerated uniformly from rest to a speed of \(8.4 \times 10^7\) m/s over a distance of \(2.5\) cm. What is the power delivered to the electron at the instant
that its displacement is \(1.0\) cm?
The electron has a mass of \(9.11 \times 10^{-31}\) kg.
Work done is the gain in kinetic energy: \(W=Fd= \Delta K =\frac{1}{2} m v^2\).
So with \(d=2.5\) cm, you can find the force on the electron. From this, acceleration is
\(a = F/m = 1.41 \times 10^{17}\) m/s²
Now with \(d= 1.0\) cm, find velocity with \(v^2 = 2 a \Delta s\).
\(v = 5.31 \times 10^{7}\) m/s
Then \(P = F \cdot v\).
\(P = 6.83\) μW
3. Work done by different forces
A \(40\) kg crate is pushed uphill at constant velocity a distance \(8.0\) m along a \(30\)° incline by the horizontal force \(\vec F\). The coefficient of kinetic friction between the crate and the
incline is \(\mu_k\) = \(0.40\).
Calculate the work done by
a. the applied force
b. the normal force
c. the frictional force
d. the gravitational force
e. the net force
a. \(3.46\) kJ
b. \(0\)
c. \(–1.89\) kJ
d. \(–1.57\) kJ
e. \(0\)
Consider the free-body diagram:
Note that the normal force is perpendicular to displacement, and therefore does zero work.
\(\vec{F}=m\vec{a}\) with the axes shown gives the following:
\(\Sigma F_x = 0 = F - N \sin \theta - \mu N \cos \theta\)
\(\Sigma F_y = 0 = N \cos \theta - \mu N \sin \theta - mg\)
where \(\theta = 30^{\circ}\).
The second equation yeilds \(N = \frac{mg}{\cos \theta - \mu \sin \theta} = 588.57\) N.
With this, the first equation yeilds \(F = 498.17\) N.
With these you can find work done with \(W = \vec F \cdot \Delta \vec s\).
Since \(\vec{a} = 0\), \(\vec{F}_{\textup{net}} = 0\). So the net force does zero work.
4. Kinetic Energy after work is done
A 10-kg box slides along a track at 8 m/s. As the box slides up a ramp, it gains \(1.0\) m of elevation, while friction does \(-100\) J of work on it. How fast is it moving afterward?
The change in energy \(\Delta (K+U)\) equals work done by friction (which is negative because energy is lost).
\(K_f + U_f - (K_i +U_i) = W\)
\(K_f = \frac{1}{2} m v_f^2\)
\(U_f = mgh\)
\(K_i = \frac{1}{2} m v_i^2\)
\(U_i = 0\)
Plug these in and solve for \(v_f\).
\(v_f^2 = v_i^2 - 2gh + 2\frac{W}{m} = 8^2 - 2(9.8)(1.0) - 2(100)/10 = 24.4\) m²/s²
\(v_f = 4.9\) m/s
Last modified: Tue October 22 2024, 09:25 PM. | {"url":"http://madisoncollegephysics.net/phys1fall24/model_questions/07-8.html","timestamp":"2024-11-14T23:47:06Z","content_type":"application/xhtml+xml","content_length":"26636","record_id":"<urn:uuid:e9107164-0a17-429c-80ee-4afab7910740>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00612.warc.gz"} |
Femtometer/Second Squared Converter | Kody Tools
Conversion Description
1 Femtometer/Second Squared = 1e-15 Meter/Second Squared 1 Femtometer/Second Squared in Meter/Second Squared is equal to 1e-15
1 Femtometer/Second Squared = 1000 Attometer/Second Squared 1 Femtometer/Second Squared in Attometer/Second Squared is equal to 1000
1 Femtometer/Second Squared = 1e-13 Centimeter/Second Squared 1 Femtometer/Second Squared in Centimeter/Second Squared is equal to 1e-13
1 Femtometer/Second Squared = 1e-14 Decimeter/Second Squared 1 Femtometer/Second Squared in Decimeter/Second Squared is equal to 1e-14
1 Femtometer/Second Squared = 1e-16 Dekameter/Second Squared 1 Femtometer/Second Squared in Dekameter/Second Squared is equal to 1e-16
1 Femtometer/Second Squared = 1e-17 Hectometer/Second Squared 1 Femtometer/Second Squared in Hectometer/Second Squared is equal to 1e-17
1 Femtometer/Second Squared = 1e-18 Kilometer/Second Squared 1 Femtometer/Second Squared in Kilometer/Second Squared is equal to 1e-18
1 Femtometer/Second Squared = 1e-9 Micrometer/Second Squared 1 Femtometer/Second Squared in Micrometer/Second Squared is equal to 1e-9
1 Femtometer/Second Squared = 1e-12 Millimeter/Second Squared 1 Femtometer/Second Squared in Millimeter/Second Squared is equal to 1e-12
1 Femtometer/Second Squared = 0.000001 Nanometer/Second Squared 1 Femtometer/Second Squared in Nanometer/Second Squared is equal to 0.000001
1 Femtometer/Second Squared = 0.001 Picometer/Second Squared 1 Femtometer/Second Squared in Picometer/Second Squared is equal to 0.001
1 Femtometer/Second Squared = 1.296e-8 Meter/Hour Squared 1 Femtometer/Second Squared in Meter/Hour Squared is equal to 1.296e-8
1 Femtometer/Second Squared = 0.00001296 Millimeter/Hour Squared 1 Femtometer/Second Squared in Millimeter/Hour Squared is equal to 0.00001296
1 Femtometer/Second Squared = 0.000001296 Centimeter/Hour Squared 1 Femtometer/Second Squared in Centimeter/Hour Squared is equal to 0.000001296
1 Femtometer/Second Squared = 1.296e-11 Kilometer/Hour Squared 1 Femtometer/Second Squared in Kilometer/Hour Squared is equal to 1.296e-11
1 Femtometer/Second Squared = 3.6e-12 Meter/Minute Squared 1 Femtometer/Second Squared in Meter/Minute Squared is equal to 3.6e-12
1 Femtometer/Second Squared = 3.6e-9 Millimeter/Minute Squared 1 Femtometer/Second Squared in Millimeter/Minute Squared is equal to 3.6e-9
1 Femtometer/Second Squared = 3.6e-10 Centimeter/Minute Squared 1 Femtometer/Second Squared in Centimeter/Minute Squared is equal to 3.6e-10
1 Femtometer/Second Squared = 3.6e-15 Kilometer/Minute Squared 1 Femtometer/Second Squared in Kilometer/Minute Squared is equal to 3.6e-15
1 Femtometer/Second Squared = 3.6e-15 Kilometer/Hour/Second 1 Femtometer/Second Squared in Kilometer/Hour/Second is equal to 3.6e-15
1 Femtometer/Second Squared = 8.503937007874e-9 Inch/Hour/Minute 1 Femtometer/Second Squared in Inch/Hour/Minute is equal to 8.503937007874e-9
1 Femtometer/Second Squared = 1.4173228346457e-10 Inch/Hour/Second 1 Femtometer/Second Squared in Inch/Hour/Second is equal to 1.4173228346457e-10
1 Femtometer/Second Squared = 2.3622047244094e-12 Inch/Minute/Second 1 Femtometer/Second Squared in Inch/Minute/Second is equal to 2.3622047244094e-12
1 Femtometer/Second Squared = 5.1023622047244e-7 Inch/Hour Squared 1 Femtometer/Second Squared in Inch/Hour Squared is equal to 5.1023622047244e-7
1 Femtometer/Second Squared = 1.4173228346457e-10 Inch/Minute Squared 1 Femtometer/Second Squared in Inch/Minute Squared is equal to 1.4173228346457e-10
1 Femtometer/Second Squared = 3.9370078740157e-14 Inch/Second Squared 1 Femtometer/Second Squared in Inch/Second Squared is equal to 3.9370078740157e-14
1 Femtometer/Second Squared = 7.0866141732283e-10 Feet/Hour/Minute 1 Femtometer/Second Squared in Feet/Hour/Minute is equal to 7.0866141732283e-10
1 Femtometer/Second Squared = 1.1811023622047e-11 Feet/Hour/Second 1 Femtometer/Second Squared in Feet/Hour/Second is equal to 1.1811023622047e-11
1 Femtometer/Second Squared = 1.9685039370079e-13 Feet/Minute/Second 1 Femtometer/Second Squared in Feet/Minute/Second is equal to 1.9685039370079e-13
1 Femtometer/Second Squared = 4.251968503937e-8 Feet/Hour Squared 1 Femtometer/Second Squared in Feet/Hour Squared is equal to 4.251968503937e-8
1 Femtometer/Second Squared = 1.1811023622047e-11 Feet/Minute Squared 1 Femtometer/Second Squared in Feet/Minute Squared is equal to 1.1811023622047e-11
1 Femtometer/Second Squared = 3.2808398950131e-15 Feet/Second Squared 1 Femtometer/Second Squared in Feet/Second Squared is equal to 3.2808398950131e-15
1 Femtometer/Second Squared = 6.9978402e-12 Knot/Hour 1 Femtometer/Second Squared in Knot/Hour is equal to 6.9978402e-12
1 Femtometer/Second Squared = 1.1663067e-13 Knot/Minute 1 Femtometer/Second Squared in Knot/Minute is equal to 1.1663067e-13
1 Femtometer/Second Squared = 1.9438445e-15 Knot/Second 1 Femtometer/Second Squared in Knot/Second is equal to 1.9438445e-15
1 Femtometer/Second Squared = 1.9438445e-18 Knot/Millisecond 1 Femtometer/Second Squared in Knot/Millisecond is equal to 1.9438445e-18
1 Femtometer/Second Squared = 1.3421617752326e-13 Mile/Hour/Minute 1 Femtometer/Second Squared in Mile/Hour/Minute is equal to 1.3421617752326e-13
1 Femtometer/Second Squared = 2.2369362920544e-15 Mile/Hour/Second 1 Femtometer/Second Squared in Mile/Hour/Second is equal to 2.2369362920544e-15
1 Femtometer/Second Squared = 8.0529706513958e-12 Mile/Hour Squared 1 Femtometer/Second Squared in Mile/Hour Squared is equal to 8.0529706513958e-12
1 Femtometer/Second Squared = 2.2369362920544e-15 Mile/Minute Squared 1 Femtometer/Second Squared in Mile/Minute Squared is equal to 2.2369362920544e-15
1 Femtometer/Second Squared = 6.2137119223733e-19 Mile/Second Squared 1 Femtometer/Second Squared in Mile/Second Squared is equal to 6.2137119223733e-19
1 Femtometer/Second Squared = 1.0936132983377e-15 Yard/Second Squared 1 Femtometer/Second Squared in Yard/Second Squared is equal to 1.0936132983377e-15
1 Femtometer/Second Squared = 1e-13 Gal 1 Femtometer/Second Squared in Gal is equal to 1e-13
1 Femtometer/Second Squared = 1e-13 Galileo 1 Femtometer/Second Squared in Galileo is equal to 1e-13
1 Femtometer/Second Squared = 1e-11 Centigal 1 Femtometer/Second Squared in Centigal is equal to 1e-11
1 Femtometer/Second Squared = 1e-12 Decigal 1 Femtometer/Second Squared in Decigal is equal to 1e-12
1 Femtometer/Second Squared = 1.0197162129779e-16 G-unit 1 Femtometer/Second Squared in G-unit is equal to 1.0197162129779e-16
1 Femtometer/Second Squared = 1.0197162129779e-16 Gn 1 Femtometer/Second Squared in Gn is equal to 1.0197162129779e-16
1 Femtometer/Second Squared = 1.0197162129779e-16 Gravity 1 Femtometer/Second Squared in Gravity is equal to 1.0197162129779e-16
1 Femtometer/Second Squared = 1e-10 Milligal 1 Femtometer/Second Squared in Milligal is equal to 1e-10
1 Femtometer/Second Squared = 1e-16 Kilogal 1 Femtometer/Second Squared in Kilogal is equal to 1e-16 | {"url":"https://www.kodytools.com/units/acceleration/from/fms2","timestamp":"2024-11-09T07:16:26Z","content_type":"text/html","content_length":"105955","record_id":"<urn:uuid:017e22b3-fb37-4f8b-9d0b-69ad81041199>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00741.warc.gz"} |
KEAM (Engineering) 2010 Questions (MCQ) on Electronics
"You may never know what results come of your actions, but if you do nothing, there will be no results."
–Mahatma Gandhi
Questions on electronics will be generally interesting to most of you. Today we will discuss questions in this section which appeared in Kerala engineering entrance (KEAM - Engineering) 2010 question
paper. Here are the questions with their solution:
(1) A full wave rectifier with an a.c. input is shown:
The output voltage across R[L] is represented as
The rectified output voltage will be a direct voltage but there will be very large amount of ripples. The capacitor C acts as a filter to remove the ripples; but there will still be a small amount of
ripples in the output. Therefore the correct option is (e).
(2) In the given circuit the current through the battery is
(b) 1 A
(c) 1.5 A
(d) 2 A
(e) 2.5 A
Since the diode D[1] is reverse biased, no current will flow through the D[1] branch. Diodes D[2] and D[3] are forward biased and hence the battery drives currents through the 20 Ω resistor and the
series combination of the two 5 Ω resistors.
The current driven through the 20 Ω resistor is 10 V/20 Ω = 0.5 A.
The current driven through the 10 Ω resistor is 10 V/10 Ω = 1 A.
Therefore, total current through the battery is 0.5 A + 1 A = 1.5 A
(3) The collector supply voltage is 6 V and the voltage drop across a resistor of 600 Ω in the collector circuit is 0.6 V, in a transistor connected in common emitter mode. If the current gain is 20,
the base current is
(a) 0.25 mA
(b) 0.05 mA
(c) 0.12 mA
(d) 0.02 mA
(e) 0.07 mA
We have I[C]R[C] = 0.6 V where I[C ]is the collector current and R[C] is the resistance in the collector circuit.
Therefore, I[C]×600 Ω = 0.6 V from which I[C] = 0.6/600 A = 10^–3 A = 1 mA.
Since the current gain β is given by
β = I[C]/I[B] where I[B] is the base current, we have
I[B] = I[C]/β = 1 mA/20 = 0.05 mA.
(4) A pure semiconductor has equal electron and hole concentration of 10^16 m^–3. Doping by indium increases n[h] to 5×10^22 m^–3. Then the value of n[e] in the doped semiconductor is
(a) 10^6 m^–3
(b) 10^22 m^–3
(c) 2×10^6 m^–3
(d) 10^19 m^–3
(e) 2×10^9 m^–3
According to the law of mass action we have
n[i]^2 = n[e]n[h] where n[i] is the electron concentration as well as the hole concentration in the intrinsic (pure) semiconductor, n[e] is the electron concentration in the doped semiconductor and n
[h] is the hole concentration in the doped semiconductor.
Therefore n[e] = n[i]^2/n[h] = (10^16)^2/(5×10^22) = 2×10^9 m^–3 | {"url":"http://www.physicsplus.in/2012/04/keam-engineering-2010-questions-mcq-on.html","timestamp":"2024-11-13T15:38:20Z","content_type":"application/xhtml+xml","content_length":"101184","record_id":"<urn:uuid:662fe61c-9c44-4677-963a-69e574a5985a>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00725.warc.gz"} |
About this project
Biodose Tools is an open source project that aims to be a tool to perform all different tests and calculations needed by biological dosimetry laboratories. The app is developed using the R
programming language and Shiny as a framework to offer an online, easy-to-use solution. Although the intention is to provide the application as a website, all R routines are available as an R
package, which can be downloaded for improvement or personal use.
We also aim to clarify and explain the tests used and to propose those considered most appropriate. Each laboratory in its routine work should choose the most suitable method, but the project aims to
reach a consensus that will help us in case of mutual assistance or intercomparisons.
The project is initially developed by RENEB association, but contributions are always welcome.
Dicentrics: Dose-effect fitting
Dicentrics: Dose estimation
Curve fitting data options
Variance-covariance matrix
Translocations: Dose-effect fitting
Genomic conversion factor
Translocations: Dose estimation
Curve fitting data options
Variance-covariance matrix
Genomic conversion factor
Micronuclei: Dose-effect fitting
Micronuclei: Dose estimation
Curve fitting data options
Variance-covariance matrix | {"url":"https://biodosetools.reneb.bfs.de/","timestamp":"2024-11-03T01:12:59Z","content_type":"text/html","content_length":"455066","record_id":"<urn:uuid:24af4148-a161-4ce3-80de-328619a1a348>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00833.warc.gz"} |
The algebraic structure of a trading stop-loss system
I was once an undergraduate student in a joint Mathematics & Physics program. Some of the math courses, namely group theory and algebra, remained very abstract to me throughout my education. There is
some group theory in the description of symmetries of physical systems; but being an experimentalist, I didn’t use more than 5% of what I learned in my undergrad during my PhD.
However, in the course of my work now in finance, I had the pleasure of discovering that I was actually working with an algebraic structure. This post describes how that happened.
The small trading firm for which I work is focusing a bit more on automated performance monitoring these days. With detailed trading performance data streaming in, it is now a good time to implement
a stop-loss system.
A stop-loss system is a system which receives trading performance data, and emits three categories of signal:
• an all-clear signal, meaning that nothing in recent trading performance indicates a problem;
• a warning signal, meaning that recent trading performance is degraded – but not yet concerning – and a human should take a look under the hood;
• a halt signal, meaning that there is most probably something wrong, trading should be halted at once.
Of course, we’re trading different products in different markets and even jurisdictions, and therefore the trading performance of every product is monitored independently. Moreover, our risk
tolerance or expectations may be different for every product, and so a stop-loss system is really a framework in which to express multiple stop-loss rules, with different products being supervised by
completely different stop-loss rules.
Let us consider examples: assume that we’re trading a particular stock like AAPL^1. Sensible stop-loss rules might be:
• If our current position has lost >10% in value over the last month, emit a warning; if the position has lost >25% over the last month, emit a halt signal.
• If we’re expecting market volatility in the next hour to be high (for example, due to expected high-impact news), emit a halt signal.
• If our forecast of the ticker price is way off – perhaps due to a problem in the forecasting model –, emit a halt signal.
Here is what a rule framework might looks like^2:
from enum import Enum, auto, unique
from typing import Callable
class Signal(Enum):
AllClear = auto()
Warn = auto()
Halt = auto()
class Context:
Rule = Callable[[Context], Signal]
# Example rule
def rule(context: Context) -> Signal:
A Rule is a function from some Context object to a Signal. We’re packing all information required to make decisions in a single data structure for reasons which will become obvious shortly. In this
framework, we may express one of the stop loss rule examples as:
def rule(context: Context) -> Signal:
recent_loss = loss_percent( context.recent_performance(period="30d") )
if recent_loss > 0.25:
return Signal.Halt
elif recent_loss > 0.10
return Signal.Warn
return Signal.AllClear
For the remainder of this post, I don’t care anymore about the domain-specific content of a rule.
My colleagues and I are expecting that, in practice, we will have pretty complex rules. In order to build complex rules from smaller, simpler rules, I wanted to be able to compose Rules together.
This is straightforward because all rules have the same input and output types. Consider two rules, rule1 and rule2. If I want a new rule to halt if both rule1 and rule2 emit Signal.Halt, I could
write it like this:
def rule1(context: Context) -> Signal:
def rule2(context: Context) -> Signal:
def rule_lax(context: Context) -> Signal:
sig1 = rule1(context)
sig2 = rule2(context)
if sig1 == sig2 == Signal.Halt:
return Signal.Halt
elif sig1 == sig2 == Signal.Warn:
return Signal.Warn
return Signal.AllClear
That is an acceptable definition of rule composition. Since rule_lax will emit a Halt signal if both sub-rules emit a Halt signal, we’ll call this type of composition conjunction. In order to make it
more ergonomic to write, let us wrap all rules in an object and re-use the & (overloaded and) operator:
from dataclasses import dataclass
from enum import Enum
from operator import attrgetter
class Signal(Enum):
Signals can be composed using (&):
>>> Signal.AllClear & Signal.AllClear
< Signal.AllClear: 1 >
>>> Signal.Warn & Signal.Halt
< Signal.Warn: 2 >
>>> Signal.Halt & Signal.Halt
< Signal.Halt: 3 >
AllClear = 1
Warn = 2
Halt = 3
def __and__(self, other: "Signal") -> "Signal":
return min(self, other, key=attrgetter('value'))
class rule(Callable):
_inner: Callable[[Context], Signal]
def __call__(self, context: Context) -> Signal:
return self._inner.__call__(context=context)
def __and__(self, other: "rule"):
def newinner(context: Context) -> Signal:
return rule1(context) & rule2(context)
return self.__class__(newinner)
and now we can re-write rule_lax like so:
# The @rule decorator is required in order to lift rule1 from a regular function
# to the `rule` object
def rule1(context: Context) -> Signal:
def rule2(context: Context) -> Signal:
rule_lax = rule1 & rule2
Now, rule_lax is defined such that it’ll emit Signal.Halt if both rule1 and rule2 emit Signal.Halt. The same is true of warnings; if both rules emit a warning, then rule_lax will emit Signal.Warning.
Here is a table which summarizes this composition:
$A$ $B$ $A ~ \& ~ B$
$C$ $C$ $C$
$C$ $W$ $C$
$C$ $H$ $C$
$W$ $C$ $C$
$W$ $W$ $W$
$W$ $H$ $W$
$H$ $C$ $C$
$H$ $W$ $W$
$H$ $H$ $H$
where $C$ is Signal.AllClear, $W$ is Signal.Warning, and $H$ is Signal.Halt. Therefore, & is a binary function from Rules to Rule.
This is not the only natural way to compose rules. What about this?
def rule_strict(context: Context) -> Signal:
sig1 = rule1(context)
sig2 = rule2(context)
if (sig1 == Signal.Halt) or (sig2 == Signal.Halt):
return Signal.Halt
elif (sig1 == Signal.Warning) or (sig2 == Signal.Warning):
return Signal.Warning
return Signal.AllClear
In this case, rule_strict is more, uh, strict than rule_lax; it emits Signal.Halt if either rule1 or rule2 emits a stop signal. We’ll call this composition disjunction and re-use the | (overloaded
or) operator to make it more ergonomic to write:
class Signal(Enum):
Signals can be composed using (&) and (|):
>>> Signal.AllClear & Signal.AllClear
< Signal.AllClear: 1 >
>>> Signal.Warn & Signal.Halt
< Signal.Warn: 2 >
>>> Signal.Warn | Signal.Halt
< Signal.Halt: 3 >
AllClear = 1
Warn = 2
Halt = 3
def __and__(self, other: "Signal") -> "Signal":
return min(self, other, key=attrgetter('value'))
def __or__(self, other: "Signal") -> "Signal":
return max(self, other, key=attrgetter('value'))
class rule(Callable):
_inner: Callable[[Context], Signal]
def __call__(self, context: Context) -> Signal:
return self._inner.__call__(context=context)
def __and__(self, other: "rule"):
def newinner(context: Context) -> Signal:
return rule1(context) & rule2(context)
return self.__class__(newinner)
def __or__(self, other: "rule"):
def newinner(context: Context) -> Signal:
return rule1(context) | rule2(context)
return self.__class__(newinner)
With this implementation, we can express rule_lax and rule_strict as:
# The @rule decorator is required in order to lift rule1 from a regular function
# to the `rule` object
def rule1(context: Context) -> Signal:
def rule2(context: Context) -> Signal:
rule_lax = rule1 & rule2
rule_strict = rule1 | rule2
We can update the table for the definition of & and |:
$A$ $B$ $A ~ \& ~ B$ $A ~ | ~ B$
$C$ $C$ $C$ $C$
$C$ $W$ $C$ $W$
$C$ $H$ $C$ $H$
$W$ $C$ $C$ $W$
$W$ $W$ $W$ $W$
$W$ $H$ $W$ $H$
$H$ $C$ $C$ $H$
$H$ $W$ $W$ $H$
$H$ $H$ $H$ $H$
So for a given a given Context, which is fixed when the trading stop-loss system is running, we have:
• A set of rule outcomes of type Signal;
• A binary operation called conjunction (the & operator);
□ & is associative;
□ & is commutative;
□ & has an identity, Signal.Halt;
□ & does NOT have an inverse element.
• A binary operation called disjunction (the | operator).
□ | is associative;
□ | is commutative;
□ | has an identity, Signal.AllClear;
□ | does NOT have an inverse element.
That looks like a commutative semiring to me! Just a few more things to check:
• | distributes from both sides over &:
□ $a ~|~ (b ~\&~ c)=(a ~|~ b) ~\&~ (a ~\&~ c)$ for all $a$, $b$, and $c$;
□ $(a ~ \& ~ b) ~|~ c = (a ~|~ c) ~\&~ (b ~\&~ c)$ for all $a$, $b$, and $c$.
• The identity element of & (called $0$, in this case Signal.Halt) annihilates the | operation, i.e. $0 ~ | ~ a = 0$ for all $a$.
Don’t take my word for it, we can check exhaustively:
from itertools import product
zero = Signal.Halt
one = Signal.AllClear
# Assert & is associative
assert all( (a & b) & c == a & (b & c) for (a, b, c) in product(Signal, repeat=3) )
# Assert & is commutative
assert all( a & b == b & a for (a, b) in product(Signal, repeat=2) )
# Assert & has an identity
assert all( a & zero == a for a in Signal )
# Assert | is associative
assert all( (a | b) | c == a | (b | c) for (a, b, c) in product(Signal, repeat=3) )
# Assert | has an identity
assert all( a | one == a for a in Signal )
# Assert | distributes over & on both sides
assert all( a | (b & c) == (a | b) & (a | c) for (a, b, c) in product(Signal, repeat=3) )
assert all( (a & b) | c == (a | c) & (b | c) for (a, b, c) in product(Signal, repeat=3) )
# Assert identity of & annihilates with respect to |
assert all( (zero | a) == zero for a in Signal)
and there we have it! This design of a trading stop-loss system is an example of commutative semirings. This fact does absolutely nothing in the practical sense; I’m just happy to have spotted this
structure more than 10 years after seeing it in undergrad.
1. I’m actually not involved in trading securities at all, but I think intuition about stock markets is more common↩︎
2. I’ll be using Python in this post because it was a requirement of the implementation, but know that I’m doing this under protest.↩︎ | {"url":"https://laurentrdc.xyz/posts/algebra-stoploss.html","timestamp":"2024-11-12T16:21:06Z","content_type":"text/html","content_length":"62975","record_id":"<urn:uuid:68903cc5-7c18-4d5b-b72e-2c3aad2b451d>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00363.warc.gz"} |
testing ibpdv and ibpu
05-12-2015, 01:57 PM
Post: #1
salvomic Posts: 1,396
Senior Member Joined: Jan 2015
testing ibpdv and ibpu
hi all,
I'm testing the "integration by parts" in the Prime.
Inputting ibpu(x*sin(x),x) I get [-x*cos(x) cos(x)], ok
with ibpdv(x*sin(x),sin(x) I get [x*sin(x)^2/cos(x) (-x*sin(x)^3-cos(x)*sin(x)^2-x*cos(x)^2)/cos(x)^2]
I'm a bit confused: shouldn't I get the same result as above? Am I using wrong syntax? :-)
Then another question: there is also a "integration by substitution" method in the Prime?
thank you
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05-12-2015, 02:44 PM
Post: #2
parisse Posts: 1,337
Senior Member Joined: Dec 2013
RE: testing ibpdv and ibpu
The second argument of ibpdv should be the antiderivative of one term in the product.
05-12-2015, 02:45 PM
(This post was last modified: 05-12-2015 02:46 PM by salvomic.)
Post: #3
salvomic Posts: 1,396
Senior Member Joined: Jan 2015
RE: testing ibpdv and ibpu
(05-12-2015 02:44 PM)parisse Wrote: The second argument of ibpdv should be the antiderivative of one term in the product.
thank you!
in fact, doing so it works well!
ibpdv(x*sin(x),-cos(x) -> [-x*cos(x) cos(x)]
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05-12-2015, 07:45 PM
Post: #4
Aries Posts: 159
Member Joined: Oct 2014
RE: testing ibpdv and ibpu
Hey Salvo,
as to the integration by substitution, take a look here:
05-12-2015, 07:59 PM
(This post was last modified: 05-12-2015 08:05 PM by salvomic.)
Post: #5
salvomic Posts: 1,396
Senior Member Joined: Jan 2015
RE: testing ibpdv and ibpu
(05-12-2015 07:45 PM)Aries Wrote: Hey Salvo,
as to the integration by substitution, take a look here:
thank you, Aries!
I'm reading it and trying
\[ \int_0^1{\sqrt{1-x^{2}}}dx \]
assume(u>0); a:=subst('integrate(sqrt(1-x^2)),0,1,x)',x=sin(u)); a;
and I get [u, π/4 π/4]
Manually I have solution = (½)arcsin1 = π/4 (only?)
Is it right?
I thought to have this series: ∫cos(u)^2du, then (u/2)+(sin(2u))/4+c and finally (½)(arcsin(x)+x*sqrt(1-x^2)+c (changing also the integral bounds) ...
However it's interesting this approach.
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05-12-2015, 10:18 PM
(This post was last modified: 05-12-2015 10:29 PM by Arno K.)
Post: #6
Arno K Posts: 472
Senior Member Joined: Mar 2015
RE: testing ibpdv and ibpu
Hello Salvo,
as I am fairly new to the Prime I spend a lot of time trying it, so I entered your Integration example and got the list :[u, int(sqrt(1-sin(u)^2)*cos(u),u,0,PI/2),PI/4].
This is: [substitution var, new integral, result], your solution is only shown after a final hit on simplify. The list is produced because you are entering three commands in one line. So :simplify
(subst('int(sqrt(1-x^2),x,0,1)',x=sin(u))) directly produces nothing but PI/4.
Now I tried int(sqrt(1-x^2),x) to see what the Prime would deliver: the answer is too long to be typed here,containing i and LN aside the first part, I thought of getting:0.5*(x*sqrt(1-x^2)+arcsin
So I entered: assume(x,float) and gave it a try, no change, I added: additionally(x>=0), still no change. By the way, Complex is switched off.
Perhaps you or someone else can give it a try.
edit: I now took the real Prime and entered the integral which clearly produced the same result, xcas on my Computer, xcas pad on my mobile and my formerly preferred cas (MathStudio for Android)
delivered the simplified answer, which you provided and I had expected. I am a bit disappointed...
05-13-2015, 05:14 AM
Post: #7
parisse Posts: 1,337
Senior Member Joined: Dec 2013
RE: testing ibpdv and ibpu
(05-12-2015 07:59 PM)salvomic Wrote: I'm reading it and trying
\[ \int_0^1{\sqrt{1-x^{2}}}dx \]
assume(u>0); a:=subst('integrate(sqrt(1-x^2)),0,1,x)',x=sin(u)); a;
It should be
assume(u>0); a:=subst('integrate(sqrt(1-x^2),x,0,1)',x=sin(u)); a
05-13-2015, 07:18 AM
Post: #8
salvomic Posts: 1,396
Senior Member Joined: Jan 2015
RE: testing ibpdv and ibpu
(05-12-2015 10:18 PM)Arno K Wrote: ...
So :simplify(subst('int(sqrt(1-x^2),x,0,1)',x=sin(u))) directly produces nothing but PI/4.
Now I tried int(sqrt(1-x^2),x) to see what the Prime would deliver: the answer is too long to be typed here,containing i and LN aside the first part,
I am a bit disappointed...
(05-13-2015 05:14 AM)parisse Wrote: It should be
assume(u>0); a:=subst('integrate(sqrt(1-x^2),x,0,1)',x=sin(u)); a
thank you Parisse!
iw would be interesting also to have a method to get "step by step" solution, here and for the "integration by parts" functions, but anyway still already this is ok...
please, control your CAS Settings: I've in CAS: Exact, "Use √" and Principal checked, Complex not checked (with firmware 6975)
with the integral you should get (½)*ASIN(x)+(½)*x*√(-x^2-1) from 0 to 1, that's π/4 (in Home you gets 0.785398...)
try, and let me know...
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05-13-2015, 07:34 AM
Post: #9
Arno K Posts: 472
Senior Member Joined: Mar 2015
RE: testing ibpdv and ibpu
Yes, the bounded integral is no problem, the one I tried later , int(sqrt(1-x^2),x) without boundaries, does not work.
05-13-2015, 07:57 AM
Post: #10
salvomic Posts: 1,396
Senior Member Joined: Jan 2015
RE: testing ibpdv and ibpu
(05-13-2015 07:34 AM)Arno K Wrote: Yes, the bounded integral is no problem, the one I tried later , int(sqrt(1-x^2),x) without boundaries, does not work.
but have you already controlled the settings?
perhaps it's that...
I'm using textbook.
Otherwise, I don't know
I've the correct result also in emulator (firmware 6975)
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05-13-2015, 08:16 AM
Post: #11
Arno K Posts: 472
Senior Member Joined: Mar 2015
RE: testing ibpdv and ibpu
Hi Salvo,
that are the same cas-Settings as mine, , Software Version is 6975 on the emulator as well as on my calculator, I am not able to get the symbolic integral, that is
PI/4 is no problem
05-13-2015, 08:33 AM
Post: #12
salvomic Posts: 1,396
Senior Member Joined: Jan 2015
RE: testing ibpdv and ibpu
(05-13-2015 08:16 AM)Arno K Wrote: Hi Salvo,
that are the same cas-Settings as mine, , Software Version is 6975 on the emulator as well as on my calculator, I am not able to get the symbolic integral, that is
PI/4 is no problem
I think however it's only a settings' problem.
Let's see...
CAS settings: simplify "minimum"; Home settings Entry "textbook", complex "a+b*i", allow complex output... ok (but those shouldn't change much)...
Use helper on key C: insert ∫ and so on (without boundaries), I can see the graphic (textbook) integral, and after Enter I get the correct result.
I'm in CAS mode, the "x variable" is x and it isn't "valued" (check that it doesn't have value or use use purge(x)).
It *should* work by you
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05-13-2015, 08:52 AM
(This post was last modified: 05-13-2015 09:17 AM by Arno K.)
Post: #13
Arno K Posts: 472
Senior Member Joined: Mar 2015
RE: testing ibpdv and ibpu
Yes, you are right, I reset the Emulator, entered the integral and it worked fine. Now I reinstalled my backup, reentered the integral and got the same weird result than before. Your CAS Settings
don't work either, the helper from key C doesn´t change anything. So it is up to me, to look for a solution, the Problem seems to be caused by one of the installed programs which I keep on the same
level on both the emu and the calc, I now start thinking of having the emu empty for testing and then using these things on the calc
edit: after playing around with various settings I again reset the Emulator, then looked at the settings there, reinstalled the backup, set everything like in the fresh emu and it worked. The whole
problem is caused by: use i, which I had checked in the cas-settings. I hadn't awaited that, this flag was, in my opinion, for the calc to be used for factorization of polynomials
(I don't see the point where here a polynomial is factored but perhaps can M. Parisse help and make things clearer)
and perhaps for output purposes of complex numbers , i.e. (3,5) versus 3+5*i, but it seems to influence other calculations as well.
It would be nice if things like that, which are rather annoying, were clearly mentioned in the manual.
05-13-2015, 08:57 AM
Post: #14
salvomic Posts: 1,396
Senior Member Joined: Jan 2015
RE: testing ibpdv and ibpu
(05-13-2015 08:52 AM)Arno K Wrote: Yes, you are right, I reset the Emulator, entered the integral and it worked fine. Now I reinstalled my backup, reentered the integral and got the same weird
result than before. Your CAS Settings don't work either, the helper from key C doesn´t change anything. So it is up to me, to look for a solution, the Problem seems to be caused by one of the
installed programs which I keep on the same level on both the emu and the calc, I now start thinking of having the emu empty for testing and then using these things on the calc
Have already compared the Settings (both CAS and Home) in Emulator (after reset) and the Prime?
Maybe there is one different...
I don't know if a custom program could change the result of the integral, I think this is a bit strange...
Try again and then tell us, I'm curious about this weird result.
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05-13-2015, 09:32 AM
Post: #15
Arno K Posts: 472
Senior Member Joined: Mar 2015
RE: testing ibpdv and ibpu
(05-13-2015 08:57 AM)salvomic Wrote:
(05-13-2015 08:52 AM)Arno K Wrote: Yes, you are right, I reset the Emulator, entered the integral and it worked fine. Now I reinstalled my backup, reentered the integral and got the same
weird result than before. Your CAS Settings don't work either, the helper from key C doesn´t change anything. So it is up to me, to look for a solution, the Problem seems to be caused by one
of the installed programs which I keep on the same level on both the emu and the calc, I now start thinking of having the emu empty for testing and then using these things on the calc
Have already compared the Settings (both CAS and Home) in Emulator (after reset) and the Prime?
Maybe there is one different...
I don't know if a custom program could change the result of the integral, I think this is a bit strange...
Try again and then tell us, I'm curious about this weird result.
see my edit above
05-13-2015, 09:43 AM
Post: #16
salvomic Posts: 1,396
Senior Member Joined: Jan 2015
RE: testing ibpdv and ibpu
(05-13-2015 09:32 AM)Arno K Wrote: see my edit above
I read!
yes, I tried here: "use i" produce your same output, using "i" evaluating the integral, unchecking it, it works fine.
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05-13-2015, 02:43 PM
Post: #17
parisse Posts: 1,337
Senior Member Joined: Dec 2013
RE: testing ibpdv and ibpu
If you check use i (Xcas complex mode), you will get complex logarithms instead of inverse trigonometric functions.
05-13-2015, 05:22 PM
Post: #18
Arno K Posts: 472
Senior Member Joined: Mar 2015
RE: testing ibpdv and ibpu
Hello parisse, yes, I noticed this behaviour as you can see above, but the help on this flag which can be seen in CAS-Settings claims something like: Use i to factor polynomials and not that it
influences production on various kinds of results, so I thought it would not matter if switched on or off. Now I know better.
Thanks Arno
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Generate Untargeted and Targeted Adversarial Examples for Image Classification- MATLAB & Simulink- MathWorks 日本 (2024)
This example shows how to use the fast gradient sign method (FGSM) and the basic iterative method (BIM) to generate adversarial examples for a pretrained neural network.
Neural networks can be susceptible to a phenomenon known as adversarial examples [1], where very small changes to an input can cause the input to be misclassified. These changes are often
imperceptible to humans.
In this example, you create two types of adversarial examples:
• Untargeted — Modify an image so that it is misclassified as any incorrect class.
• Targeted — Modify an image so that it is misclassified as a specific class.
Load Network and Image
Load a network that has been trained on the ImageNet [2] data set and convert it to a dlnetwork.
net = squeezenet;lgraph = layerGraph(net);lgraph = removeLayers(lgraph,lgraph.Layers(end).Name);dlnet = dlnetwork(lgraph);
Extract the class labels.
classes = categories(net.Layers(end).Classes);
Load an image to use to generate an adversarial example. The image is a picture of a golden retriever.
img = imread('sherlock.jpg');T = "golden retriever";
Resize the image to match the input size of the network.
inputSize = dlnet.Layers(1).InputSize;img = imresize(img,inputSize(1:2));figureimshow(img)title("Ground Truth: " + T)
Prepare the image by converting it to a dlarray.
X = dlarray(single(img),"SSCB");
Prepare the label by one-hot encoding it.
T = onehotencode(T,1,'ClassNames',classes);T = dlarray(single(T),"CB");
Untargeted Fast Gradient Sign Method
Create an adversarial example using the untargeted FGSM [3]. This method calculates the gradient ${abla }_{X}L\left(X,T\right)$ of the loss function $L$, with respect to the image $X$ you want to
find an adversarial example for, and the class label $T$. This gradient describes the direction to "push" the image in to increase the chance it is misclassified. You can then add or subtract a small
error from each pixel to increase the likelihood the image is misclassified.
The adversarial example is calculated as follows:
${\mathit{X}}_{\mathrm{adv}}=\mathit{X}+ϵ.\mathrm{sign}\left({abla }_{\mathit{X}}\mathit{L}\left(\mathit{X},\mathit{T}\right)\right)$.
Parameter $ϵ$ controls the size of the push. A larger $ϵ$ value increases the chance of generating a misclassified image, but makes the change in the image more visible. This method is untargeted, as
the aim is to get the image misclassified, regardless of which class.
Calculate the gradient of the image with respect to the golden retriever class.
gradient = dlfeval(@untargetedGradients,dlnet,X,T);
Set epsilon to 1 and generate the adversarial example.
epsilon = 1;XAdv = X + epsilon*sign(gradient);
Predict the class of the original image and the adversarial image.
YPred = predict(dlnet,X);YPred = onehotdecode(squeeze(YPred),classes,1)
YPred = categorical golden retriever
YPredAdv = predict(dlnet,XAdv);YPredAdv = onehotdecode(squeeze(YPredAdv),classes,1)
YPredAdv = categorical Labrador retriever
Display the original image, the perturbation added to the image, and the adversarial image. If the epsilon value is large enough, the adversarial image has a different class label from the original
The network correctly classifies the unaltered image as a golden retriever. However, because of perturbation, the network misclassifies the adversarial image as a labrador retriever. Once added to
the image, the perturbation is imperceptible, demonstrating how adversarial examples can exploit robustness issues within a network.
Targeted Adversarial Examples
A simple improvement to FGSM is to perform multiple iterations. This approach is known as the basic iterative method (BIM) [4] or projected gradient descent [5]. For the BIM, the size of the
perturbation is controlled by parameter $\alpha$ representing the step size in each iteration. This is as the BIM usually takes many, smaller, FGSM steps in the direction of the gradient. After each
iteration, clip the perturbation to ensure the magnitude does not exceed $ϵ$. This method can yield adversarial examples with less distortion than FGSM.
When you use untargeted FGSM, the predicted label of the adversarial example can be very similar to the label of the original image. For example, a dog might be misclassified as a different kind of
dog. However, you can easily modify these methods to misclassify an image as a specific class. Instead of maximizing the cross-entropy loss, you can minimize the mean squared error between the output
of the network and the desired target output.
Generate a targeted adversarial example using the BIM and the great white shark target class.
targetClass = "great white shark";targetClass = onehotencode(targetClass,1,'ClassNames',classes);
Increase the epsilon value to 5, set the step size alpha to 0.2, and perform 25 iterations. Note that you may have to adjust these settings for other networks.
epsilon = 5;alpha = 0.2;numIterations = 25;
Keep track of the perturbation and clip any values that exceed epsilon.
delta = zeros(size(X),'like',X);for i = 1:numIterations gradient = dlfeval(@targetedGradients,dlnet,X+delta,targetClass); delta = delta - alpha*sign(gradient); delta(delta > epsilon) = epsilon; delta(delta < -epsilon) = -epsilon;endXAdvTarget = X + delta;
Predict the class of the targeted adversarial example.
YPredAdvTarget = predict(dlnet,XAdvTarget);YPredAdvTarget = onehotdecode(squeeze(YPredAdvTarget),classes,1)
YPredAdvTarget = categorical great white shark
Display the original image, the perturbation added to the image, and the targeted adversarial image.
Because of imperceptible perturbation, the network classifies the adversarial image as a great white shark.
To make the network more robust against adversarial examples, you can use adversarial training. For an example showing how to train a network robust to adversarial examples, see Train Image
Classification Network Robust to Adversarial Examples.
Supporting Functions
Untargeted Input Gradient Function
Calculate the gradient used to create an untargeted adversarial example. This gradient is the gradient of the cross-entropy loss.
function gradient = untargetedGradients(dlnet,X,target)Y = predict(dlnet,X);Y = stripdims(squeeze(Y));loss = crossentropy(Y,target,'DataFormat','CB');gradient = dlgradient(loss,X);end
Targeted Input Gradient Function
Calculate the gradient used to create a targeted adversarial example. This gradient is the gradient of the mean squared error.
function gradient = targetedGradients(dlnet,X,target)Y = predict(dlnet,X);Y = stripdims(squeeze(Y));loss = mse(Y,target,'DataFormat','CB');gradient = dlgradient(loss,X);end
Show Adversarial Image
Show an image, the corresponding adversarial image, and the difference between the two (perturbation).
function showAdversarialImage(image,label,imageAdv,labelAdv,epsilon)figuresubplot(1,3,1)imgTrue = uint8(extractdata(image));imshow(imgTrue)title("Original Image" + newline + "Class: " + string(label))subplot(1,3,2)perturbation = uint8(extractdata(imageAdv-image+127.5));imshow(perturbation)title("Perturbation")subplot(1,3,3)advImg = uint8(extractdata(imageAdv));imshow(advImg)title("Adversarial Image (Epsilon = " + string(epsilon) + ")" + newline + ... "Class: " + string(labelAdv))end
[1] Goodfellow, Ian J., Jonathon Shlens, and Christian Szegedy. “Explaining and Harnessing Adversarial Examples.” Preprint, submitted March 20, 2015. https://arxiv.org/abs/1412.6572.
[2] ImageNet. http://www.image-net.org.
[3] Szegedy, Christian, Wojciech Zaremba, Ilya Sutskever, Joan Bruna, Dumitru Erhan, Ian Goodfellow, and Rob Fergus. “Intriguing Properties of Neural Networks.” Preprint, submitted February 19, 2014.
[4] Kurakin, Alexey, Ian Goodfellow, and Samy Bengio. “Adversarial Examples in the Physical World.” Preprint, submitted February 10, 2017. https://arxiv.org/abs/1607.02533.
[5] Madry, Aleksander, Aleksandar Makelov, Ludwig Schmidt, Dimitris Tsipras, and Adrian Vladu. “Towards Deep Learning Models Resistant to Adversarial Attacks.” Preprint, submitted September 4, 2019.
See Also
dlnetwork | onehotdecode | onehotencode | predict | dlfeval | dlgradient | estimateNetworkOutputBounds | verifyNetworkRobustness
Related Topics
• Verification of Neural Networks
• Train Image Classification Network Robust to Adversarial Examples
• Generate Adversarial Examples for Semantic Segmentation
• Grad-CAM Reveals the Why Behind Deep Learning Decisions
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Pasteur and parachutes: when statistical process control is better than a randomized controlled trial
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Pasteur and parachutes: when statistical process control is better than a randomized controlled trial
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On 6 July 1885 Joseph Meister, a 9 year old boy who had been severely bitten 2 days before by a rabid dog, was treated in Paris with the rabies vaccine developed in Louis Pasteur’s (1822–95)
laboratory after years of brilliant scientific research and experimentation on animals.^1,^2 Before this, no one who had developed the symptoms of rabies had survived. Meister was at the highest risk
of developing symptoms but, after 10 days of vaccinations, he was fine and lived for many years. The results of this and a second case were so dramatic compared with previous experience that, by
October, speakers at the French Academy of Sciences stated that it was “necessary to organize this treatment for everyone” and “This is a memorable day in the history of medicine”. Philanthropic
contributions poured in and by 1888 the Pasteur Institute was founded. By then about 1200 patients had been vaccinated with a mortality rate of 1%.^3
Transmitted to humans by animal bites, rabies has always been a rare event. A biting animal may not have rabies. Verification that an animal is rabid is not always possible if the animal cannot be
caught and watched. A rabid animal that bites does not always transmit the rabies virus. If the person is infected, the incubation period is about 20–60 days before symptoms develop leading to a
painful and certain death.^4 Today there are occasional reports of survival but these are so rare as to make headlines.^5 The rarity of rabies and the long incubation period led to Pasteur’s novel
approach of not vaccinating everyone preventively. The incubation period allowed time for his 10 days of vaccination to build immunity.
Would you—having been infected by a rabid dog—be willing to participate in a randomized controlled trial (RCT) when being in the control group had a certainty of a “most awful death”? If volunteers
could be found, the trial would have to be small and would therefore have low statistical power. If one wished to show that vaccination was effective for men and women—young, middle aged, and old—the
sample size would have to be 264, condemning half those people to a certain death. In this example the application of statistical process control (SPC) makes more sense. SPC avoids the ethical
issues, saves lives, builds on prior experience to control confounding variables, gives an answer more rapidly, and has much more statistical power.
In order to demonstrate the application of statistical process control (SPC) by the use of control charts, we have simulated pre and post 1885 rabies mortality using data from the literature. This
simulation assumes a mean (SD) survival of 20 (10) days before 1885. Patients were grouped into blocks of five so that each point on the control chart represents five patients recorded sequentially
over time. This would be equivalent to an average of five patients bitten in a day. The survival in days is plotted for 500 groups of five patients sequentially. Survival in days after receiving
Pasteur’s treatment is simulated for a similar number of patients based on a mean survival of 45 years combined with a mortality rate of 1% from the treatment (figs 1 and 2).
The results are so dramatic that we have presented them in two formats. In fig 1 the vertical axis is measured in number of days of survival. In this presentation the variation before 1885 is so
small as to be unobservable. Even the 3-sigma upper and lower control limits are too close together to be seen. The shorter survival points in the post 1885 treatment experience reflect one death in
that group of five patients. In order to show more clearly the pre 1885 variation, fig 2 presents exactly the same data but with the vertical axis transformed into a logarithmic scale. The pre 1885
data show a very stable process without special cause variation. Every one with rabies symptoms soon died. The serious scholar may disagree with our simple approach and assumptions about survival,
but we think that the before and after differences were so great that this simulation is plausible.
The statistical power of this evidence is overwhelming. Based on these pre 1885 control limits, the probability of living 4000 days is above the 894-sigma level yet the results were even more
dramatic than that. When little Joseph Meister had lived to October (or 90 days), his survival was at the 20-sigma level (p=6.4×10^−13) and the French academicians were right in declaring Pasteur’s
treatment a great victory. Thus, SPC methods can demonstrate a dramatic difference as a result of the outcome from one patient and the new treatment can be started immediately, rather than waiting
for the results of a prospective controlled trial.
Gordon Smith and Jill Pell wrote a fine satirical article in 2003 pointing out that the use of parachutes has never been subject to a randomized controlled trial.^6 A careful literature review found
rare examples of people falling from great heights and living. Olympic ski jumpers survive their falls. The authors report that parachutes can fail to open resulting in death. We all learn at an
early age the unvarying power of gravity and do not need to be convinced that falling from a great height is likely to be fatal. SPC evaluation using past experience which we applied to rabies
vaccination may be relevant here. Even more dramatic control charts could be simulated, particularly if ski jumpers are excluded.
There are other examples. The progress made in the treatment of acute lymphoblastic leukemia (ALL) of childhood is one of the true success stories of modern medicine.^7 Incremental advances over 50
years mean that ALL has gone from a uniformly fatal disease to one with an overall cure rate of more than 75%.
Another example was the dramatic introduction of ether as an anaesthetic during surgery. In this case the outcome measure would be pain rather than mortality.
For the four dramatic improvements described here, we propose that information from prior experience using SPC is to be preferred to RCTs for six reasons (table 1). In these circumstances SPC has
greater statistical power to exclude chance as an explanation. The RCT is designed to control for unknown confounding variables. Perhaps the treatment only works for young boys and not for older
women. In the case of symptomatic rabies before 1885, men and women (young and old) all died without variation. If there were any unknown confounding variables they would appear as special cause
variation in a long series of prior observations.
SPC can give a very rapid answer in these circumstances. There needs to be a plausible process (treatment) change associated with the astonishing outcome that is replicable for the next patients.
Without this scientifically based replicable treatment, Joseph Meister’s survival could be declared a miracle and his cure associated with the intervention of a saint—the statistical analysis of
miracles. One test of an ethical randomized trial is whether you yourself and others would volunteer for random assignment to control or experimental groups.
When the expected differences are small, when unknown confounding variables are likely to overwhelm the treatment effect, where the casual model is weak, where prior information is thin, when there
is a single intervention and single end point, and results are not urgent—then RCTs are more useful.
Appendix: Technical Note
This section describes the simulation we performed to demonstrate the applicability of control charts to detect extraordinary causes in the rabies case study.
We generated two samples to represent the survival of two populations of individuals after onset of rabies symptoms. They are before and after July 1885, the date in which Pasteur incidentally
initiated the test of the rabies vaccine in humans. We modeled each population’s survival assuming that the survival time presents a Weibull distribution, a common probability distribution used to
model time-to-an-event variables.^8 This distribution allows for a dependence of the hazard on time. In this case, this would represent a potential change in the risk of death with time since
presentation of rabies symptoms. Also, in practical terms, this distribution has more flexibility for modeling than a simpler exponential distribution, also commonly used in survival analysis.
Following is the survival function of a random variable exhibiting a Weibull distribution with parameters λ and γ representing the scale and shape parameters of this distribution:
We selected for the first sample, before July 1885, the parameters λ and γ such that the mean and standard deviation of the survival time were close to 20 and 10 days, respectively. For the second
sample, after July 1885, we chose the parameters λ and γ in order to represent survival time with mean and standard deviation of 45 and 5 years, respectively. This was the average survival time in
France for 1885.^9 For this second sample we incorporated individuals in whom the vaccine did not work with a frequency of 1/100 individuals. We assumed for these particular patients a survival time
identical to those for people before July 1885. For both samples we assumed a period of 15 months each. This was the interval reported in the study by Moulin^10 in which 2500 people had received the
rabies vaccine. We assumed this rate of approximately five bitten people/day for both the before and after July 1885 samples. Table 2 shows the summary of the parameter values we used to generate
both samples.
Control charts are a key component of the area of statistical process control (SPC) initially developed and more disseminated in quality control of industrial processes. Several types of control
charts are available depending on the nature of the measurement to control and the particular statistical control scheme. One of the purposes of control charts is to identify instances in which a
particular process monitored through a particular measurement goes “out of control” and the potential causes for such situation. Another purpose is to improve the process itself. One of the most
popular ones is the mean control chart developed by Shewart.^11 As its name indicates, the mean control chart displays the time evolution of the mean of a continuous variable of interest (fig 3). The
chart has a distinctive pattern marked by three reference lines. One in the center, designated the center line (CL), set at the average value of the characteristic to monitor while “in control”. This
value can be obtained from previous experience or assigned as a milestone. The other two reference lines assigned upper and lower control limits (UCL and LCL respectively) corresponding to the
boundaries beyond which the process will be signaled as out of control. For mean control charts these limits are commonly set at three standard error of the process mean of the process in control or
what is known as the 3-sigma limit in the SPC jargon.
We performed all the computations for the simulation in SAS for Windows Version 9.1. | {"url":"https://qualitysafety.bmj.com/content/14/2/140?ijkey=a0c95de465dee4c73c0f520f4b59a3e25c447f48&keytype2=tf_ipsecsha","timestamp":"2024-11-07T22:45:43Z","content_type":"text/html","content_length":"121616","record_id":"<urn:uuid:1ab4bda2-1be7-4c4c-8281-d7e6d0d27432>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00118.warc.gz"} |
Source code for dclab.lme4.wrapr
"""R lme4 wrapper"""
import logging
import numbers
import pathlib
import tempfile
import importlib_resources
import numpy as np
from .. import definitions as dfn
from ..rtdc_dataset.core import RTDCBase
from . import rsetup
logger = logging.getLogger(__name__)
class Rlme4(object):
def __init__(self, model="lmer", feature="deform"):
"""Perform an R-lme4 analysis with RT-DC data
model: str
One of:
- "lmer": linear mixed model using lme4's ``lmer``
- "glmer+loglink": generalized linear mixed model using
lme4's ``glmer`` with an additional a log-link function
via the ``family=Gamma(link='log'))`` keyword.
feature: str
Dclab feature for which to compute the model
#: modeling method to use (e.g. "lmer")
self.model = None
#: dclab feature for which to perform the analysis
self.feature = None
#: list of [RTDCBase, column, repetition, chip_region]
self.data = []
self.set_options(model=model, feature=feature)
# Make sure that lme4 is available
if not rsetup.has_lme4():
logger.info("Installing lme4, this may take a while!")
def add_dataset(self, ds, group, repetition):
"""Add a dataset to the analysis list
ds: RTDCBase
group: str
The group the measurement belongs to ("control" or
repetition: int
Repetition of the measurement
- For each repetition, there must be a "treatment" (``1``) and a
"control" (``0``) group.
- If you would like to perform a differential feature analysis,
then you need to pass at least a reservoir and a channel
dataset (with same parameters for `group` and `repetition`).
assert group in ["treatment", "control"]
assert isinstance(ds, RTDCBase)
assert isinstance(repetition, numbers.Integral)
region = ds.config["setup"]["chip region"]
# make sure there are no doublets
for ii, dd in enumerate(self.data):
if dd[1] == group and dd[2] == repetition and dd[3] == region:
raise ValueError("A dataset with group '{}', ".format(group)
+ "repetition '{}', and ".format(repetition)
+ "'{}' region has already ".format(region)
+ "been added (index {})!".format(ii))
self.data.append([ds, group, repetition, region])
def check_data(self):
"""Perform sanity checks on ``self.data``"""
# Check that we have enough data
if len(self.data) < 3:
msg = "Linear mixed effects models require repeated "
+ "measurements. Please add more repetitions."
raise ValueError(msg)
def fit(self, model=None, feature=None):
"""Perform (generalized) linear mixed-effects model fit
The response variable is modeled using two linear mixed effect
- model: "feature ~ group + (1 + group | repetition)"
(random intercept + random slope model)
- the null model: "feature ~ (1 + group | repetition)"
(without the fixed effect introduced by the "treatment" group).
Both models are compared in R using "anova" (from the
R-package "stats" :cite:`Everitt1992`) which performs a
likelihood ratio test to obtain the p-Value for the
significance of the fixed effect (treatment).
If the input datasets contain data from the "reservoir"
region, then the analysis is performed for the differential
model: str (optional)
One of:
- "lmer": linear mixed model using lme4's ``lmer``
- "glmer+loglink": generalized linear mixed model using
lme4's ``glmer`` with an additional log-link function
via ``family=Gamma(link='log'))`` :cite:`lme4`
feature: str (optional)
dclab feature for which to compute the model
results: dict
Dictionary with the results of the fitting process:
- "anova p-value": Anova likelihood ratio test (significance)
- "feature": name of the feature used for the analysis
- "fixed effects intercept": Mean of ``self.feature`` for all
controls; In the case of the "glmer+loglink" model, the intercept
is already back transformed from log space.
- "fixed effects treatment": The fixed effect size between the mean
of the controls and the mean of the treatments relative to
"fixed effects intercept"; In the case of the "glmer+loglink"
model, the fixed effect is already back transformed from log
- "fixed effects repetitions": The effects (intercept and
treatment) for each repetition. The first axis defines
intercept/treatment; the second axis enumerates the repetitions;
thus the shape is (2, number of repetitions) and
``np.mean(results["fixed effects repetitions"], axis=1)`` is
equivalent to the tuple (``results["fixed effects intercept"]``,
``results["fixed effects treatment"]``) for the "lmer" model.
This does not hold for the "glmer+loglink" model, because
of the non-linear inverse transform back from log space.
- "is differential": Boolean indicating whether or not
the analysis was performed for the differential (bootstrapped
and subtracted reservoir from channel data) feature
- "model": model name used for the analysis ``self.model``
- "model converged": boolean indicating whether the model
- "r model summary": Summary of the model
- "r model coefficients": Model coefficient table
- "r script": the R script used
- "r output": full output of the R script
self.set_options(model=model, feature=feature)
# Assemble dataset
if self.is_differential():
# bootstrap and compute differential features using reservoir
features, groups, repetitions = self.get_differential_dataset()
# regular feature analysis
features = []
groups = []
repetitions = []
for dd in self.data:
features.append(self.get_feature_data(dd[1], dd[2]))
# concatenate and populate arrays for R
features_c = np.concatenate(features)
groups_c = np.zeros(len(features_c), dtype=str)
repetitions_c = np.zeros(len(features_c), dtype=int)
pos = 0
for ii in range(len(features)):
size = len(features[ii])
groups_c[pos:pos+size] = groups[ii][0]
repetitions_c[pos:pos+size] = repetitions[ii]
pos += size
# Run R with the given template script
rscript = importlib_resources.read_text("dclab.lme4",
_, script_path = tempfile.mkstemp(prefix="dclab_lme4_", suffix=".R",
script_path = pathlib.Path(script_path)
rscript = rscript.replace("<MODEL_NAME>", self.model)
rscript = rscript.replace("<FEATURES>", arr2str(features_c))
rscript = rscript.replace("<REPETITIONS>", arr2str(repetitions_c))
rscript = rscript.replace("<GROUPS>", arr2str(groups_c))
script_path.write_text(rscript, encoding="utf-8")
result = rsetup.run_command((rsetup.get_r_script_path(), script_path))
ret_dict = self.parse_result(result)
ret_dict["is differential"] = self.is_differential()
ret_dict["feature"] = self.feature
ret_dict["r script"] = rscript
ret_dict["r output"] = result
assert ret_dict["model"] == self.model
return ret_dict
def get_differential_dataset(self):
"""Return the differential dataset for channel/reservoir data
The most famous use case is differential deformation. The idea
is that you cannot tell what the difference in deformation
from channel to reservoir, because you never measure the
same object in the reservoir and the channel. You usually just
have two distributions. Comparing distributions is possible
via bootstrapping. And then, instead of running the lme4
analysis with the channel deformation data, it is run with
the differential deformation (subtraction of the bootstrapped
deformation distributions for channel and reservoir).
features = []
groups = []
repetitions = []
# compute differential features
for grp in sorted(set([dd[1] for dd in self.data])):
# repetitions per groups
grp_rep = sorted(set([dd[2] for dd in self.data if dd[1] == grp]))
for rep in grp_rep:
feat_cha = self.get_feature_data(grp, rep, region="channel")
feat_res = self.get_feature_data(grp, rep, region="reservoir")
bs_cha, bs_res = bootstrapped_median_distributions(feat_cha,
# differential feature
features.append(bs_cha - bs_res)
return features, groups, repetitions
def get_feature_data(self, group, repetition, region="channel"):
"""Return array containing feature data
group: str
Measurement group ("control" or "treatment")
repetition: int
Measurement repetition
region: str
Either "channel" or "reservoir"
fdata: 1d ndarray
Feature data (Nans and Infs removed)
assert group in ["control", "treatment"]
assert isinstance(repetition, numbers.Integral)
assert region in ["reservoir", "channel"]
for dd in self.data:
if dd[1] == group and dd[2] == repetition and dd[3] == region:
ds = dd[0]
raise ValueError("Dataset for group '{}', repetition".format(group)
+ " '{}', and region".format(repetition)
+ " '{}' not found!".format(region))
fdata = ds[self.feature][ds.filter.all]
fdata_valid = fdata[~np.logical_or(np.isnan(fdata), np.isinf(fdata))]
return fdata_valid
def is_differential(self):
"""Return True if the differential feature is computed for analysis
This effectively just checks the regions of the datasets
and returns True if any one of the regions is "reservoir".
See Also
get_differential_features: for an explanation
for dd in self.data:
if dd[3] == "reservoir":
return True
return False
def parse_result(self, result):
resd = result.split("OUTPUT")
ret_dict = {}
for item in resd:
string = item.split("#*#")[0]
key, value = string.split(":", 1)
key = key.strip()
value = value.strip().replace("\n\n", "\n")
if key == "fixed effects repetitions":
rows = value.split("\n")[1:]
reps = []
for row in rows:
reps.append([float(vv) for vv in row.split()[1:]])
value = np.array(reps).transpose()
elif key == "model converged":
value = value == "TRUE"
elif value == "NA":
value = np.nan
value = float(value)
except ValueError:
ret_dict[key] = value
return ret_dict
def set_options(self, model=None, feature=None):
"""Set analysis options"""
if model is not None:
assert model in ["lmer", "glmer+loglink"]
self.model = model
if feature is not None:
assert dfn.scalar_feature_exists(feature)
self.feature = feature
def arr2str(a):
"""Convert an array to a string"""
if isinstance(a.dtype.type, np.integer):
return ",".join(str(dd) for dd in a.tolist())
elif a.dtype.type == np.str_:
return ",".join(f"'{dd}'" for dd in a.tolist())
return ",".join(f"{dd:.16g}" for dd in a.tolist())
def bootstrapped_median_distributions(a, b, bs_iter=1000, rs=117):
"""Compute the bootstrapped distributions for two arrays.
a, b: 1d ndarray of length N
Input data
bs_iter: int
Number of bootstrapping iterations to perform
(output size).
rs: int
Random state seed for random number generator
median_dist_a, median_dist_b: 1d arrays of length bs_iter
Boostrap distribution of medians for ``a`` and ``b``.
See Also
From a programmatic point of view, it would have been better
to implement this method for just one input array (because of
redundant code). However, due to historical reasons (testing
and comparability to Shape-Out 1), bootstrapping is done
interleaved for the two arrays.
# Seed random numbers that are reproducible on different machines
prng_object = np.random.RandomState(rs)
# Initialize median arrays
median_a = np.zeros(bs_iter)
median_b = np.zeros(bs_iter)
# If this loop is still too slow, we could get rid of it and
# do everything with arrays. Depends on whether we will
# eventually run into memory problems with array sizes
# of y*bs_iter and yR*bs_iter.
lena = len(a)
lenb = len(b)
for q in range(bs_iter):
# Compute random indices and draw from a, b
draw_a_idx = prng_object.randint(0, lena, lena)
median_a[q] = np.median(a[draw_a_idx])
draw_b_idx = prng_object.randint(0, lenb, lenb)
median_b[q] = np.median(b[draw_b_idx])
return median_a, median_b | {"url":"https://dclab.readthedocs.io/en/latest/_modules/dclab/lme4/wrapr.html","timestamp":"2024-11-02T19:06:47Z","content_type":"text/html","content_length":"59395","record_id":"<urn:uuid:44ad4e97-d466-4eab-aab3-0fd513cc8bc2>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00199.warc.gz"} |
seminars - Transference theorems, parametric geometry of numbers, and spectra II
We define classical exponents of Diophantine approximation attached to real matrices and state various transference theorems between them. We show how the parametric geometry of numbers, introduced
by Schmidt and Summerer and further developed by Roy, allows us to give an easy proof of these transference theorems. Furthermore, we explain how this recent theory can be applied to determine the
spectra of several exponents of Diophantine approximation (that is, the set of values they take on real entries). | {"url":"https://www.math.snu.ac.kr/board/index.php?mid=seminars&sort_index=date&order_type=asc&page=83&document_srl=1199970","timestamp":"2024-11-06T17:18:55Z","content_type":"text/html","content_length":"47140","record_id":"<urn:uuid:e9c77750-a020-4bef-a461-9969cf4a86e6>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00762.warc.gz"} |
The largest subsemilattices of the endomorphism monoid of an independence algebra
Araújo, João; Bentz, Wolfram; Konieczny, Janusz
Linear Algebra and its Applications, 458 (2014), 50-79
http://dx.doi.org/10.1016/j.laa.2014.05.041 (preprint - http://arxiv.org/pdf/1007.4845)
An algebra A is said to be an independence algebra if it is a matroid algebra and every map ?:X?A, defined on a basis X of A, can be extended to an endomorphism of A. These algebras are particularly
well-behaved generalizations of vector spaces, and hence they naturally appear in several branches of mathematics such as model theory, group theory, and semigroup theory.
It is well known that matroid algebras have a well-defined notion of dimension. Let A be any independence algebra of finite dimension n , with at least two elements. Denote by End(A) the monoid of
endomorphisms of A. We prove that a largest subsemilattice of End(A) has either 2n?1 elements (if the clone of A does not contain any constant operations) or 2n elements (if the clone of A contains
constant operations). As corollaries, we obtain formulas for the size of the largest subsemilattices of: some variants of the monoid of linear operators of a finite-dimensional vector space, the
monoid of full transformations on a finite set X, the monoid of partial transformations on X, the monoid of endomorphisms of a free G-set with a finite set of free generators, among others.
The paper ends with a relatively large number of problems that might attract the attention of experts in linear algebra, ring theory, extremal combinatorics, group theory, semigroup theory, universal
algebraic geometry, and universal algebra. | {"url":"https://cemat.tecnico.ulisboa.pt/document.php?project_id=7&member_id=10&doc_id=426","timestamp":"2024-11-11T00:21:44Z","content_type":"text/html","content_length":"9594","record_id":"<urn:uuid:31508f09-c42d-49b0-b530-a5959b27cdce>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00359.warc.gz"} |
Stokes Radius Calculator - Savvy Calculator
Stokes Radius Calculator
The Stokes radius is a measure of the size of a spherical particle in a fluid, based on its terminal velocity in the fluid. It is particularly useful in fields such as biology, chemistry, and fluid
dynamics for understanding particle behavior and interactions.
The Stokes radius (rrr) can be calculated using the formula:
r=9⋅η⋅v2⋅g⋅(ρp−ρf)r = \sqrt{\frac{9 \cdot \eta \cdot v}{2 \cdot g \cdot (\rho_p – \rho_f)}}r=2⋅g⋅(ρp−ρf)9⋅η⋅v
• rrr is the Stokes radius (meters)
• η\etaη is the viscosity of the fluid (Pa·s)
• vvv is the terminal velocity of the particle (m/s)
• ggg is the acceleration due to gravity (9.81 m/s²)
• ρp\rho_pρp is the density of the particle (kg/m³)
• ρf\rho_fρf is the density of the fluid (kg/m³)
How to Use
To use the Stokes Radius Calculator:
1. Enter the viscosity of the fluid in Pascal-seconds (Pa·s).
2. Enter the terminal velocity of the particle in meters per second (m/s).
3. Enter the density difference between the particle and the fluid in kilograms per cubic meter (kg/m³).
4. Click the “Calculate” button.
5. The Stokes radius of the particle will be calculated and displayed in meters (m).
Suppose we have a particle in a fluid with the following properties:
• Viscosity = 0.002 Pa·s
• Terminal velocity = 0.1 m/s
• Density difference = 50 kg/m³
Using the calculator:
1. Enter 0.002 for viscosity.
2. Enter 0.1 for terminal velocity.
3. Enter 50 for density difference.
4. Click “Calculate.”
5. The Stokes radius will be calculated as approximately 0.018 meters.
1. What is the Stokes radius?
□ The Stokes radius is the radius of a spherical particle in a fluid, calculated based on its terminal velocity and fluid properties.
2. Why is the Stokes radius important?
□ It helps in determining the size of particles in suspension and understanding their behavior in fluids.
3. Can the Stokes radius be used for non-spherical particles?
□ The Stokes radius formula assumes spherical particles. For non-spherical particles, different calculations or models may be required.
4. What are the units of Stokes radius?
□ Stokes radius is measured in meters (m).
5. How does density difference affect the Stokes radius?
□ A greater density difference between the particle and the fluid results in a larger Stokes radius.
6. Is the Stokes radius the same as hydrodynamic radius?
□ No, they are different. Hydrodynamic radius considers the overall size of a particle’s diffusion in a fluid, whereas Stokes radius specifically calculates size based on terminal velocity.
7. Can Stokes radius be used in biological sciences?
□ Yes, it is commonly used to determine the size of macromolecules and particles in biological fluids.
8. What is the significance of the terminal velocity in Stokes radius calculation?
□ Terminal velocity indicates the speed at which a particle falls through a fluid under gravity, crucial for determining its Stokes radius.
9. Does the calculator account for all factors influencing particle size in fluids?
□ The calculator focuses on the basic parameters required for Stokes radius calculation. For specific applications, additional factors may need consideration.
10. Can Stokes radius be experimentally measured?
□ Yes, experimental methods involve observing the behavior of particles in fluid flow and relating it to their properties and terminal velocities.
The Stokes Radius Calculator provides a straightforward method for determining the size of particles in fluids based on their terminal velocities and fluid properties. By using the formula and
inputting the relevant data, users can quickly obtain the Stokes radius, facilitating research and analysis in various scientific disciplines. | {"url":"https://savvycalculator.com/stokes-radius-calculator","timestamp":"2024-11-08T11:25:15Z","content_type":"text/html","content_length":"143308","record_id":"<urn:uuid:01c6da64-a502-46cb-bef2-ccc270f383f2>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00126.warc.gz"} |
Subtractor | BimStudies.Com
A subtractor is a digital circuit that performs subtraction of binary numbers.
It’s a fundamental component in digital logic systems and is used to calculate the difference between two binary values.
Key components of a subtractor include:
Minuend: The number from which subtraction is performed.
Subtrahend: The number that is subtracted from the minuend.
Borrow (Borrow-in): A signal indicating whether a borrow needs to be taken into account in subtraction.
Difference: The result of the subtraction operation.
Borrow-out: A signal indicating whether a borrow occurred in the subtraction operation.
There are different types of subtractors, including half subtractors, full subtractors, and n-bit parallel subtractors.
Half Subractor:
A half subtractor is a digital circuit that performs subtraction of two single binary digits (bits).
It produces two outputs: a difference (D) and a borrow (B) output.
Unlike a full subtractor, a half subtractor doesn’t take into account any borrow from a previous stage, making it suitable for subtracting two individual bits.
Block Diagram:
Truth Table:
Logic Circuit:
Full Subtractor:
A full subtractor is a digital circuit that performs subtraction of three single binary digits: the minuend bit (A), the subtrahend bit (B), and a borrow-in bit (Bin) from a previous stage.
It produces two outputs: a difference (D) and a borrow-out (Bout) output.
Unlike a half subtractor, a full subtractor takes into account both the subtrahend and the potential borrow from the previous stage.
Block Diagram:
Truth Table:
Logic Circuit: | {"url":"https://bimstudies.com/docs/digital-logic/combinational-logic/subtractor/","timestamp":"2024-11-13T12:05:36Z","content_type":"text/html","content_length":"476677","record_id":"<urn:uuid:6b52ac03-4017-450c-bf3a-1056f1001d10>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00361.warc.gz"} |
Williams, Llewelyn January 1960 (has links)
Thesis (Ph.D.)--Boston University / One of the more powerful and consistent tools available to nature is the phenomenon of alternate freezing and thawing. Mechanically, extraordinary pressures may
be involved because of the density differential existing between the liquid and the solid phases of water. Physiologically, there is the availability or non-availability of water to sustain
growth. Despite this, catastrophic changes are not to be expected. On the other hand, such a powerful tool must leave its imprint in one manner or another upon the natural landscape. In most
arctic and highland areas the imprint is directly discernible. In more moderate climes the imprint is indirectly applied principally as a limiting parameter within an aggregate of generally
1 favorable conditions. The phenomenon of freeze-thaw is a climatic parameter but not a climatic element. Unlike the elements, there is a definite threshold involved; that is, 32 degrees Fahrenheit
or 0 degrees Centigrade. At this threshold water may exist in either the liquid or solid state but by the addition or subtraction of heat it can change from one state to the other without a gain
or loss in temperature. In the natural environment the terms are not quite so precise. Time for the process to take place, impurities in the water, and the variation of temperature regimes among
the many nooks and crannies of the landscape point to the necessity of relaxing the temperature threshold. In this study the zone of 34 degrees F and 28 degrees F is used. Conditions favorable for
a thaw are thought to occur when the temperature rises through the zone and conditions favorable for a freeze when the temperature drops through this zone [TRUNCATED]
Panturat, Suwanna, January 1987 (has links)
2 Thesis (Ph.D.)--University of Oklahoma, 1987. / Bibliography: leaves 213-219.
Kaczmarczyk, Edward Bruce. January 1978 (has links)
3 Thesis (M.S.)--Wisconsin. / Includes bibliographical references (leaves 87-91).
Wang, Xiaoxue, 王霄雪 January 2015 (has links)
4 abstract / Mechanical Engineering / Doctoral / Doctor of Philosophy
Urban climatology
LaJoie, Mark R. 03 1900 (has links)
Department of Defense (DoD) climatology products rely mainly on long term means (LTMs) of climate system variables. In this project, we have demonstrated that climatologies based on LTMs can be
substantially improved using modern data and methods, especially by accounting for climate variations. We analyzed, and identified mechanisms for, enhanced (suppressed) autumn precipitation in the
Horn of Africa (HOA) during El Nino (La Nina) events. El Nino (La Nina) precipitation anomalies were associated with anomalously warm (cool) western Indian Ocean sea surface temperatures, and with
5 anomalously onshore (offshore) moisture transports in the HOA. These transport anomalies supported anomalously strong (weak) precipitation over the HOA. To improve climatological support for DoD
operations, we developed and tested a six-step smart climatology process. We applied this process in the context of a notional, unclassified non-combatant evacuation operation (NEO) set in the HOA
during autumn of an El Nino year. Using this process, we translated our scientific and operational findings into warfighter impacts. The smart climatology process we have developed is readily
adaptable to other regions, seasons, climate variations, and military operations. We have provided a detailed description of our smart climatology process to facilitate its use by DoD agencies.
Cannon, Alex Jason 05 1900 (has links)
This dissertation develops multivariate statistical models for seasonal forecasting and downscaling of climate variables. In the case of seasonal climate forecasting, where record lengths are
typically short and signal-to-noise ratios are low, particularly at long lead-times, forecast models must be robust against noise. To this end, two models are developed. Robust nonlinear canonical
correlation analysis, which introduces robust cost functions to an existing model architecture, is outlined in Chapter 2. Nonlinear principal predictor analysis, the nonlinear extension of
principal predictor analysis, a linear model of intermediate complexity between multivariate regression and canonical correlation analysis, is developed in Chapter 3. In the case of climate
downscaling, the goal is to predict values of weather elements observed at local or regional scales from the synoptic-scale atmospheric circulation, usually for the purpose of generating climate
scenarios from Global Climate Models. In this context, models must not only be accurate in terms of traditional model verification statistics, but they must also be able to replicate statistical
6 properties of the historical observations. When downscaling series observed at multiple sites, correctly specifying relationships between sites is of key concern. Three models are developed for
multi-site downscaling. Chapter 4 introduces nonlinear analog predictor analysis, a hybrid model that couples a neural network to an analog model. The neural network maps the original predictors
to a lower-dimensional space such that predictions from the analog model are improved. Multivariate ridge regression with negative values of the ridge parameters is introduced in Chapter 5 as a
means of performing expanded downscaling, which is a linear model that constrains the covariance matrix of model predictions to match that of observations. The expanded Bernoulli-gamma density
network, a nonlinear probabilistic extension of expanded downscaling, is introduced in Chapter 6 for multi-site precipitation downscaling. The single-site model is extended by allowing multiple
predictands and by adopting the expanded downscaling covariance constraint.
Cannon, Alex Jason 05 1900 (has links)
This dissertation develops multivariate statistical models for seasonal forecasting and downscaling of climate variables. In the case of seasonal climate forecasting, where record lengths are
typically short and signal-to-noise ratios are low, particularly at long lead-times, forecast models must be robust against noise. To this end, two models are developed. Robust nonlinear canonical
correlation analysis, which introduces robust cost functions to an existing model architecture, is outlined in Chapter 2. Nonlinear principal predictor analysis, the nonlinear extension of
principal predictor analysis, a linear model of intermediate complexity between multivariate regression and canonical correlation analysis, is developed in Chapter 3. In the case of climate
downscaling, the goal is to predict values of weather elements observed at local or regional scales from the synoptic-scale atmospheric circulation, usually for the purpose of generating climate
scenarios from Global Climate Models. In this context, models must not only be accurate in terms of traditional model verification statistics, but they must also be able to replicate statistical
7 properties of the historical observations. When downscaling series observed at multiple sites, correctly specifying relationships between sites is of key concern. Three models are developed for
multi-site downscaling. Chapter 4 introduces nonlinear analog predictor analysis, a hybrid model that couples a neural network to an analog model. The neural network maps the original predictors
to a lower-dimensional space such that predictions from the analog model are improved. Multivariate ridge regression with negative values of the ridge parameters is introduced in Chapter 5 as a
means of performing expanded downscaling, which is a linear model that constrains the covariance matrix of model predictions to match that of observations. The expanded Bernoulli-gamma density
network, a nonlinear probabilistic extension of expanded downscaling, is introduced in Chapter 6 for multi-site precipitation downscaling. The single-site model is extended by allowing multiple
predictands and by adopting the expanded downscaling covariance constraint.
Trevaskis, Graham Arthur. January 1963 (has links)
8 Thesis (Ed.D.)--Teachers College, Columbia University, 1963. / Typescript; issued also on microfilm. Sponsor: James P. Matthai. Dissertation Committee: Margaret Lindsey. Includes bibliographical
references (leaves 206-219).
Climatology. Geography
Hall, Leonard Frazier, January 1977 (has links)
9 Thesis--Wisconsin. / Vita. Includes bibliographical references (leaves 162-166).
Plains Climatology
Lettau, Bernhard. January 1961 (has links)
10 Thesis (M.S.)--University of Wisconsin--Madison, 1961. / Typescript. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references
(leaf 19).
Bogs. Climatology. | {"url":"http://search.ndltd.org/search.php?q=subject%3A%22Climatology%22","timestamp":"2024-11-08T18:35:45Z","content_type":"text/html","content_length":"72279","record_id":"<urn:uuid:ec334c6f-fa83-4128-92be-febcb5758dc9>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00706.warc.gz"} |
Algorithm for reducing computational costs in problems of calculation of asymmetrically loaded shells of rotation
Algorithm for reducing computational costs in problems of calculation of asymmetrically loaded shells of rotation
rotation shells, variable stiffness, asymmetric loading, coefficient prediction in the Fourier method, reduction in computing costs
The problem of calculating the shells of rotation of a variable along the meridian of rigidity under asymmetric loading is reduced o a set of systems of one-dimensional boundary value problems with
respect to the amplitudes of decomposition of the required functions into trigonometric Fourier series.
A method for reducing the number of one-dimensional boundary value problems required to achieve a given accuracy in determining the stress-strain state of the shells of rotation with a variable along
the meridian wall thickness under asymmetric load. The idea of the proposed approach is to apply periodic extrapolation (prediction) of the values of the decomposition coefficients of the required
functions using the results of calculations of previous coefficients of the corresponding trigonometric series, thus replacing them with some prediction values calculated by simple formulas.
To solve this problem, we propose the joint use of Aitken-Steffens extrapolation dependences and Adams method in the form of incremental component, which is quite effective in solving the Cauchy
problem for systems of ordinary differential equations and is based on Lagrange and Newton extrapolation dependences.
The validity of the proposed approach was verified b the results of a systematic numerical experiment by predicting the values of the expansion coefficients in the Fourier series of known functions
of one variable.
The approach is quite effective in the calculation of asymmetrically loaded shells of rotation with variable along the meridian thickness, when the coefficients of decomposition of the required
functions into Fourier series are functions of the longitudina lcoordinate and are calculated by solving the corresponding boundary value problem. In this case, the approach allows solving solutions
of differential equations for the amplitudes of decompositionin to trigonometric series only for individual "reference" harmonics, and the amplitudes for every third harmonic can be calculated by
interpolating their values for all node integration points of the corresponding boundary value problem. This significantly reduces the computational cost of obtaining the solution as a whole.
As an example, the results of the calculation of the stress-strain state of a steel annular plate under asymmetric transverse loading are given.
Biderman V. L. Mechanics of thin-walled structures / V. L. Biderman. – Mashinostroenie. –Мosscow, 1977. – 488 p. (in Russian).
Grigorenko Ya. M. Methods for calculating shells. Theory of shells of variable rigidity. Vol. 4 / Ya. M. Grigorenko, A. T. Vasilenko. – Naukova Dunka, Kiev, 1981. – 544 p. (in Russian).
Grigorenko A. Ya. Stress-strain state of shallow rectangular shells of variable thickness under various boundary conditions / A. Ya Grigorenko, N. P Yaremchenko., C. N. Yaremchenko // Bul. of NAS
Ukraine. – Vol. 6. – 2016. – P. 31-37 (in Ukrainian).
Dzyuba A. P. Calculation algorithm on the basis of a discrete-continuous approach for cylindrical shell of variable rigidity in circular direction / A. P. Dzyuba, I. A. Safronova, L. D. Levitina //
Problems of computational mechanics and strength of structures, Collection of scientific articles. –Vol. 30. – 2019. – P. 53-67 (in Ukrainian).
Myachenkov V. I. Calculation of composite shell structures on a computer: Reference / V. I. Myachenkov I. V. Grigoryev. – Mashinostroenie. – Moscow, 1981. – 212 p. (in Russian).
Sineva N. F. Calculation of a cylindrical shell of variable stiffness interacting with a nonlinear elastic base / N. F. Sineva, F. S. Selivanov, D. V. Nikityuk // Bull. Saratov State Techn. Un-ty.
Ser.: Construction and architecture. – Vol. 4(60). – Iss 2. – 2011. – P. 15-21 (in Russian).
Models and algorithms for optimization of elements of nonuniform shell structures, in N. V. Pоlyakov (Eds.) / A. P. Dzyuba, V. N. Sirenko, A. A. Dzyuba and I. A. Safronova // Actual problems of
mechanics: Monograph, Lira, Dnipro, 2018. – P. 225-243 (in Ukrainian).
Ovchinnikov I. G. Thin-walled structures under conditions of corrosion wear: Calculation and optimization / I. G. Ovchinnikov, Yu. M. Pochtman. – DNU. – Dnepropetrovsk, 1995. – 190 p. (in Russian).
Dashchenko A. F. ANSYS in the problems of mechanical engineering: monograph. second ed. / A. F. Dashchenko, D. V. Lazareva, N. G. Suryaninov. – BURUN and K0. – Kharkiv, 2011. – 504 p. (in Russian).
Alyamovsky A. A. SolidWorks/COSMOSWorks. Finite Element Engineering Analysis. / A. A. Alyamovsky. – DMK Press. – Moscow, 2004. 432 p. (in Russian).
Grigorenko Ya. M. Solving the problems of shell theory on a computer / Ya. M. Grigorenko, A. P. Mukoed. – High School. – Kiev, 1979. – 280 p. (in Russian).
Mossakovsky V. I. Contact Interactions of Elements of Shell Structures / V. I. Mossakovsky, V. S. Hudramovich, E. M. Makeev. – Naukova Dunka. – Kiev, 1988. – 288 p. (in Russian).
Emel’yanov I. G. Application of discrete Fourier series to the stress analysis of shell structures / I. G. Emel’yanov // Computational Continuum Mechanics. – Vol 8(3). – 2015. – P. 245-253. (in
Hudramovich V. S. Contact interaction and optimization of locally loaded shell structures / V. S. Hudramovich, A. P. Dzyuba // Journal of mathematical Science - Springer Science + Business media. –
2009. – P. 231-245.
Strength. Sustainability. Fluctuations: Handbook, vol.1. / (Eds.) I. A. Birger, Ya. G. Panovko. – Mashinostroenie. – Moscow, 1968. – 821 p. (in Russian).
Tolstov G. P. Fourier Series / G. P. Tolstov. – Fizmatgiz. – Moscow, 1960. – 392 p. (in Russian).
Godunov S. K. On the numerical solution of boundary value problems for systems of linear ordinary differential equations / S. K. Godunov // Advances in Mathematical Sciences. – Vol. 16. – Iss.3 (99).
–1961. – P. 171-174 (in Russian).
Bulakajev P. I. An algorithm for the prediction of search trajectory in nonlinear programming problems optimum design / P. I. Bulakajev, A. P. Dzjuba // Structural Optimization: Research Journal of
Intern. Society of Struct. and Multidisciplinary Optimiz. – Springer – Verlag. –Vol. 13(2, 3). – 1997. –P. 199-202.
Dzyuba A. P. An algorithm for reducing the computational cost of using the Fourier method in the problems of shell structural mechanics / A. P. Dzyuba, O. O. Bobilev, P. I. Bulakaev // Bull. of
Dnepropetrovsk state university. Vol. 2(2). – 1999. – P. 47-57 (in Russian).
Shamansky V. E. Methods of numerical solution of boundary value problems on a computer / V. E. Shamansky. – Academia Science USSR. – Kiev. – Part. 1, 1963. – 196 p. – Part. 2, 1966. – 242 p. (in
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with
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Flip a Column of Data Vertically - Free Excel Tutorial
This post will guide you how to flip a column of data vertically preserving the original formatting and formulas in Excel. How do I reverse a column of data order in a column arranged alphabetically
or from largest to smallest. How to use the Excel Sort feature to reverse the data order? Or how to use a VBA macro code to flip a column of data vertically in Excel.
If you want to flip or reverse a column of data that is already in numeric or alphabetical order, and you can easily to flip it with Sort feature. And if the column data is not sorted and you can try
to create a helper column to achieve the result.
1. Flip a Column of Data with Sort Feature
#1 select the column that you want to flip, right click on it, and select Insert to add helper column. Choose Entire column radio button. Then insert 1,2,3,4,…in helper column.
#2 select your helper column, then go to the Data tab, click the Sort Largest to Smallest command under Sort&Filter group.
#3 you will see that the numbers in help column will be sorted and the data of column also be reversed. Now you can delete the help column. Or you can also hide the help column in case you want to
restore the data of column.
#4 let’s see the result
2. Flip a Column of Data Vertically with Function
You can also create an excel formula based on the INDEX function and the ROWS function to flip a column of data vertically. Type the following formula in the formula box of cell C1, then press enter
Let’s see the result as below:
3. Flip a Column of Data Vertically with VBA Macro
You can also write an Excel VBA Macro to flip a column of data vertically quickly. Just do the following steps:
#1 click on “Visual Basic” command under DEVELOPER Tab.
#2 then the “Visual Basic Editor” window will appear.
#3 click “Insert” ->”Module” to create a new module.
#4 paste the below VBA code into the code window. Then clicking “Save” button.
Sub FlipCol()
Dim R As Range
Dim W As Range
Dim A As Variant
Dim i As Integer
Dim j As Integer
Dim k As Integer
On Error Resume Next
titleS = "Flip data in a column"
Set W = Application.Selection
Set W = Application.InputBox("select a range that you want to reverse", titleS, W.Address, Type:=8)
A = W.Formula
For j = 1 To UBound(A, 2)
k = UBound(A, 1)
For i = 1 To UBound(A, 1) / 2
xTemp = A(i, j)
A(i, j) = A(k, j)
A(k, j) = xTemp
k = k - 1
W.Formula = A
End Sub
#5 back to the current worksheet, then run the above excel macro. Click Run button.
#6 select a range that you want to flip.
#7 click OK button, then check the result.
4. Video: Flip a Column of Data Vertically
This Excel video tutorial, we’ll explore three methods to flip a column of data vertically. We’ll start by using a formula with the INDEX and ROWS functions, followed by leveraging the Sort feature,
and finally, we’ll delve into a VBA Macro.
5. Related Functions
• Excel INDEX function
The Excel INDEX function returns a value from a table based on the index (row number and column number)The INDEX function is a build-in function in Microsoft Excel and it is categorized as a
Lookup and Reference Function.The syntax of the INDEX function is as below:= INDEX (array, row_num,[column_num])…
• Excel ROWS function
The Excel ROWS function returns the number of rows in a cell reference.The syntax of the ROWS function is as below:= ROWS(array)… | {"url":"https://www.excelhow.net/flip-a-column-of-data-vertically.html","timestamp":"2024-11-05T07:32:59Z","content_type":"text/html","content_length":"94363","record_id":"<urn:uuid:c55ef31a-b474-4934-bd5e-9c97f3b68a5f>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00750.warc.gz"} |
For homework submissions, I require that all work be completed - Essay Writing
For homework submissions, I require that all work be completed in excel, which means that you will use the excel functions to arrive at the answer. I should be able to click on the cell that holds
your answer, and the formula to arrive at this answer will be shown in the function ribbon. I will not accept answers that were simply re-typed into the excel workbook.
Chapter 1.
Year Stock T Stock B
1 0.19 0.08
2 0.08 0.03
3 -0.12 -0.09
4 -0.03 0.02
5 0.15 0.04
a. Compute the arithmetic mean annual rate of return for each stock. Which stock is most desirable by this measure?
b. Compute the standard deviation of the annual rate of return for each stock. (Use Chapter 1 Appendix if necessary.) By this measure, which is the preferable stock?
c. Compute the coefficient of variation for each stock. (Use the Chapter 1 Appendix if necessary.) By this relative measure of risk, which stock is preferable?
d. Compute the geometric mean rate of return for each stock. Discuss the difference between the arithmetic mean return and the geometric mean return for each stock. Discuss the differences in the
mean returns relative to the standard deviation of the return for each stock.
7. A stockbroker calls you and suggests that you invest in the Lauren Computer Company. After analyzing the firm’s annual report and other material, you believe that the distribution of expected
rates of return is as follows:
Possible Rate of Return Probability
-0.60 0.05
-0.30 0.20
-0.10 0.10
0.20 0.30
0.40 0.20
0.80 0.15
Compute the expected return [E(Ri)] on Lauren Computer stock.
9. During the past year, you had a portfolio that contained U.S. government T-bills, long-term government bonds, and common stocks. The rates of return on each of them were as follows:
U.S. government T-bills 5.50%
U.S. government long-term bonds 7.50
U.S. common stocks 11.60
During the year, the consumer price index, which measures the rate of inflation, went from 160 to 172 (1982 – 1984 = 100). Compute the rate of inflation during this year. Compute the real rates of
return on each of the investments in your portfolio based on the inflation rate.
12. Assume that the consensus required rate of return on common stocks is 14 percent. In addition, you read in Fortune that the expected rate of inflation is 5 percent and the estimated long-term
real growth rate of the economy is 3 percent. What interest rate would you expect on U.S. government T-bills? What is the approximate risk premium for common stocks implied by these data?
Chapter 2.
4. a. Someone in the 36 percent tax bracket can earn 9 percent annually on her investments in a tax-exempt IRA account. What will be the value of a one-time $10,000 investment in 5 years? 10 years?
20 years?
b. Suppose the preceding 9 percent return is taxable rather than tax-deferred and the taxes are paid annually. What will be the after-tax value of her $10,000 investment after 5, 10, and 20 years?
5. a. Someone in the 15 percent tax bracket can earn 10 percent on his investments in a tax-exempt IRA account. What will be the value of a $10,000 investment in 5 years? 10 years? 20 years?
b. Suppose the preceding 10 percent return is taxable rather than tax-deferred. What will be the after-tax value of his $10,000 investment after 5, 10, and 20 years?
6. Assume that the rate of inflation during all these periods was 3 percent a year. Compute the real value of the two tax-deferred portfolios in problems 4a and 5a. | {"url":"https://essaywriting.blog/for-homework-submissions-i-require-that-all-work-be-completed/","timestamp":"2024-11-03T18:38:27Z","content_type":"text/html","content_length":"156298","record_id":"<urn:uuid:1e1a7a42-4bc1-44ff-b67f-32c21400bedc>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00447.warc.gz"} |
Speed Conversion
Speed Conversion
Conversion between nearly 75 different Metric, English, Imperial, Indian Speed (and Velocity) measurement units. Supports conversions between Kmph, Mph, Knot, Mach, Benz, Yard per hour, Foot per
second, Meter per second, Megameter per hour, Speed of Light, Speed of Sound and many more Metric, English and Imperial units.
Example: , , , , , , , , , , , and many more. Acceleration Conversion
Conversion between nearly 25 different Acceleration measurement units. Supports conversion between Centigal, Decigal, Milligal, G-Unit (G), Gal, Galileo, Gn, Grav, Meter per second-square,
Centimeter per second-square, Inch per second-square, Decimeter per second-square, Dekameter per second-square, Foot per second-square, Hectometer per second-square, Kilometer per hour-second,
Kilometer per second-square, Mile per hour-minute, Mile per hour-second, Mile per second-square, Millimeter per second-square and many more
Torque Conversion
Conversion between nearly 20 different Torque measurement units. Supports conversion between Dyne Centimeter, Gram Centimeter, Kilogram Centimeter, Kilogram Meter, Kilonewton Meter, Kilopond
Meter, Meganewton Meter, Micronewton Meter, Millinewton Meter, Newton Centimeter, Newton Meter, Ounce Foot, Ounce Inch, Pound Foot, Pound Inch, Poundal Foot and many more
Dynamic Viscosity Conversion
Conversion between nearly 25 different Dynamic Viscosity measurement units. Supports conversion between Poise, Decipoise, Centipoise, Millipoise, Poiseuille (France), Reyn, Newton second/m²,
Pascal second, Millipascal second, Dyne second/cm², Gram-Force second/cm², Gram/centimeter second, Kilogram-Force second/m², Kilogram/meter hour, Kilogram/meter second, Pound-Force second/f²,
Pound-Force second/in², Pound/foot hour, Pound/foot second, Poundal hour/feet², Poundal second/feet², Slug/foot second and many more
Kinematic Viscosity Conversion
Conversion between nearly 30 different Kinematic Viscosity measurement units. Supports conversion between Stokes, Centistokes, Lentor, Liter per Centimeter-Day, Liter per Centimeter-Hour, Liter
per Centimeter-Minute, Liter per Centimeter-Second, Poise Cubic Centimeter per Gram, Square-Centimeter per Day, Square-Centimeter per Hour, Square-Centimeter per Minute, Square-Centimeter per
Second, Square-Feet per Day, Square-Feet per Hour, Square-Feet per Minute, Square-Feet per Second, Square-Inch per Day, Square-Inch per Hour, Square-Inch per Minute, Square-Inch per Second,
Square-Meter per Day, Square-Meter per Hour, Square-Meter per Minute, Square-Meter per Second, Square-Millimeter per Day, Square-Millimeter per Hour, Square-Millimeter per Minute,
Square-Millimeter per Second and many more
Viscosity of Oil, Water and Glycerin Conversion
Conversion of viscosity at different temperature of Oil, Water, Glycerin and thick fluids between nearly 15 different measurement units. Supports conversion between Poise, Centipoise, Water at
20C, Water at 40C, Heavy Oil at 20C, Heavy Oil at 40C, Glycerin at 20C, Glycerin at 40C, SAE 5W at -18C, SAE 10W at -18C, SAE 20W at -18C, SAE 5W at 99C, SAE 10W at 99C, SAE 20W at 99C and many
Angular Velocity Conversion
Conversion between degree/day, degree/hour, degree/minute, degree/second, radian/second, radian/day, radian/hour, radian/minute, revolution/day, revolution/hour, revolution/minute, revolution/
second and more
Angular Acceleration Conversion
Conversion between radian/second^2, radian/minute^2, degree/second^2, degree/minute^2, revolution/second^2, revolution/minute^2
Moment of Inertia Conversion
Conversion between nearly 20 different Moment of Inertia measurement units. Support conversion between Kg-m^2, Kg-cm^2, Kg-mm^2, g-cm^2, g-mm^2, oz-in^2, lb-ft^2, lb-in^2, slug-ft^2 and many more
Speed of common Devices, Peripheral, Protocols
Find speed of sound, light, electricity, speed of different computer devices, peripherals, protocols, upload and download speed, bit rate and more
Escape Velocity of different Planets, Moons, Comets and Asteroids
Find escape velocity, surface gravity, mass of different solar system objects, planets, moons, comets, asteroids, centaurs, kuiper belt objects and more.
Example: Find Escape Velocity of , , , , , , , , , , , , , , , , , , , , , , , , Speed of common Animals
Find top speed of different animals, top speed of common land animals, top speed of common birds, top speed of aquatic (sea) animals, top speed of different insects, fastest animal, fastest
mammal, fastest fish, fastest bird and many more.
Also see ... | {"url":"https://www.webconversiononline.com/speed/default.aspx","timestamp":"2024-11-12T02:34:48Z","content_type":"text/html","content_length":"73918","record_id":"<urn:uuid:0ff41053-abbd-4b77-b98c-e56fab15f0c4>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00053.warc.gz"} |
Integrability and complexity in quantum
SciPost Submission Page
Integrability and complexity in quantum spin chains
by Ben Craps, Marine De Clerck, Oleg Evnin, Philip Hacker
This is not the latest submitted version.
This Submission thread is now published as
Submission summary
Authors (as registered SciPost users): Marine De Clerck
Submission information
Preprint Link: scipost_202307_00036v1 (pdf)
Code repository: https://doi.org/10.5281/zenodo.7876467
Data repository: https://doi.org/10.5281/zenodo.7876467
Date submitted: 2023-07-28 14:48
Submitted by: De Clerck, Marine
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
Specialties: • Mathematical Physics
• Quantum Physics
There is a widespread perception that dynamical evolution of integrable systems should be simpler in a quantifiable sense than the evolution of generic systems, though demonstrating this relation
between integrability and reduced complexity in practice has remained elusive. We provide a connection of this sort by constructing a specific matrix in terms of the eigenvectors of a given quantum
Hamiltonian. The null eigenvalues of this matrix are in one-to-one correspondence with conserved quantities that have simple locality properties (a hallmark of integrability). The typical magnitude
of the eigenvalues, on the other hand, controls an explicit bound on Nielsen's complexity of the quantum evolution operator, defined in terms of the same locality specifications. We demonstrate how
this connection works in a few concrete examples of quantum spin chains that possess diverse arrays of highly structured conservation laws mandated by integrability.
Current status:
Has been resubmitted
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2023-11-13 (Invited Report)
• Cite as: Anonymous, Report on arXiv:scipost_202307_00036v1, delivered 2023-11-13, doi: 10.21468/SciPost.Report.8099
1 - tackles an ambitious and timely problem: complexity of chaotic vs. integrable dynamics
2 - focuses on a computable upper bound to complexity, which exhibits different behavior in chaotic/integrable models
3 - very detailed study of a very interesting object -the 'Q-matrix'- which allows to probe the local conservation laws and has implications on the complexity bound
4 - analytical results on the Q-matrix from random matrix
5 - extensive numerical checks in non-trivial interacting integrable spin chains
6 - very carefully written
1 - complete lack of conciseness
2 - if one reads only this paper, then it is not quite clear what is new or not new here, compared to [SciPost Phys. 13, 090 (2022)] by the same authors
Motivated by the notion of complexity of unitary evolutions, the authors study an upper bound on Nielsen's complexity -Eq. (2.24)- related to a matrix $Q$ of size $\dim(\mathcal{H}) \times \dim(\
mathcal{H})$ defined for a subspace of 'easy' operators (such as local or few-body operators). The spectrum of the matrix $Q$ is claimed to be highly sensitive to integrability, in particular its
kernel is directly tied to the existence of local conservation laws. The authors make general conjectures about properties of $Q$, which they relate to the behavior of their upper bound on
complexity, especially to the plateau that the bound displays at long time.
The main claims are supported by analytical calculations for GUE random matrices, in the case of chaotic dynamics (Section 3). Extensive numerical checks are then provided both for chaotic and
integrable spin chains (Section 4).
I find that the results are very substantial and exciting. They open a new pathway towards the fundamental goal of characterizing the impact of conservation laws on the complexity of quantum
many-body dynamics. I also think they provide a synergetic link between research on complexity of quantum evolution operators, which so far has largely focused on chaotic models and/or fully
connected models like the SYK model, and interacting integrable spin chains.
In my opinion, the only problem of this paper is the fact that it is too long. It took me a very long time to go through it, and while I feel that it was very instructive and inspiring, I am also a
bit frustrated because the reading could have been faster and less painful if the manuscript were organized more clearly.
Several sections are extremely long, with no clear substructure. For instance, Section 2.1 is 10 pages long, presented as a whole block. It starts by discussing the definition of Nielsen's
complexity, then some of its properties and equivalence/differences with other notions of complexity, then its geometric interpretation, then discusses a warm-up calculation of complexity in a simple
case, then some generic properties of its short-time behavior, then its long-time behavior and plateau, in relation with typical distances to hybercube lattices, then refers to Ref. [6] for some
observation about chaotic vs. integrable dynamics, then comes back to the general discussion of the problem with a penalty factor, then drops Eq. (2.24) -which plays a central role in the whole
paper-, then discusses again distances in hypercubic lattices, then makes a long digression about some known numerical methods to solve the 'CVP', then repeats some information about distances in
hypercubic lattices, then briefly mentions with Eq. (2.28) that the kernel of the matrix Q encodes the conservation laws of the model -a central point for the rest of the paper-, then discusses
whether or not it is a good choice to declare that the identity operator is local, and then provides a loose discussion of integrability, repeating information that already appeared above, and
finally briefly hints at the results of Sections 3 and 4. Please...
It would not be difficult to break this long section (and other similar sections, such as 4.1.3) that contain digressions and repetitions, into smaller, more focused, subsections or paragraphs, each
with a clear title. In addition, it would help the reader to have a clear distinction between the results of Ref. [6] that are partially summarized in section 2.1, and the new results specific to
this new paper.
In summary, I am happy to recomment publication of these results in Scipost Physics, but I would encourage the authors to try to make the manuscript a bit easier to read, if possible.
Requested changes
See above. If the authors could give more structure to the manuscript (subsections, paragraph titles, etc), and perhaps also trim it in order to avoid unnecessary repetitions, I think it could really
make the manuscript more accessible.
Otherwise, the manuscript is very carefully written. I caught only a few typos:
- before Eq. (3.10), it seems that a word is missing: perhaps 'We simplify the computation by working to leading order in $1/D$ and we define'
- before Eq. (4.3): 'these non-trivial site' -> 'sites'
- after Eq. (4.14): 'The former has an explicit U(1) symmetry' sounds weird, because both the former and the latter have that symmetry here
Finally, in the captions of the figures, it is said many times that what is shown is 'the complexity' (see e.g. caption of Fig. 6). It would be clearer to recall that it is the 'complexity bound'
that is shown
Report #1 by Anonymous (Referee 1) on 2023-9-29 (Invited Report)
• Cite as: Anonymous, Report on arXiv:scipost_202307_00036v1, delivered 2023-09-29, doi: 10.21468/SciPost.Report.7876
1. Important and relevant results on the connection between complexity and dynamics.
2. Clearly written. Also well contrasted with previous results in the literature (especially Ref. [6] by the same authors)
3. Many practical examples for integrable and chaotic dynamics.
This manuscript proposes a new interesting technique to distinguish between chaotic and integrable models based on the concept of complexity. The topic of complexity in many body quantum systems is
very timely and of fundamental importance. The same is also the case for the quest of setting apart chaotic and integrable dynamics by means of simple and computable quantities. It is very remarkable
to establish a connection between these two subjects.
The manuscript is extremely well organized with a clear introduction where the main results are summarized. Sec. 2 contains the definitions of the quantities of interest. In Secs. 3 and 4 the main
results for chaotic and integrable dynamics are derived. Many examples are reported to corroborate the results and ideas. I have no doubts concerning the publication of this interesting manuscript in
Scipost Physcis. However, I would like that before the authors consider to comment on the point raised below.
In the literature, several other quantities to distinguish chaotic and integrable dynamics have been introduced and discussed. Among them, a very important and effective one is the local operator
entanglement. Indeed it has been shown (see V. Alba, J. Dubail, and M. Medenjak, Phys. Rev. Lett. 122, 250603 (2019) and references therein) that such quantity grows at most logarithmically in time
for integrable models while it grows linearly in chaotic ones. The authors should mention this fact and comment about possible connections between their results and the local operator entanglement. | {"url":"https://www.scipost.org/submissions/scipost_202307_00036v1/","timestamp":"2024-11-10T02:52:16Z","content_type":"text/html","content_length":"43406","record_id":"<urn:uuid:74402a29-b5af-4a31-8345-1815a06e6f2e>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00617.warc.gz"} |
Power Measurements on AC-DC Power Supplies
AC to DC power supplies are fully integrated into our daily lives. They are the heart of all our electronics, providing energy for the everyday devices that we rely on. A cell phone charger is a
power supply, but so are the power electronic circuits embedded inside almost any electronics or appliance connected to the AC line. This near universal use of power supplies means that although the
power that each individual supply draws is small, the cumulative effects are very significant. For this reason it is desirable, and often required by regulation, to control power supply
characteristics such as efficiency.
AC –DC Power supplies convert electrical energy from their AC line input to provide DC outputs that are:
• Isolated from the dangerous high-voltage, high-capacity AC line.
• Smoothed and low-noise DC voltage.
• Regulated to be largely independent of input voltage changes.
• Current controlled to avoid damage to the load (especially batteries) and to the power supply itself
Typically, an AC to DC power supply converts the AC line (110 220V 50/60Hz) into low voltage (12, 5, 3V) DC.
Power supply designers strive to improve the efficiency of their designs while maintaining specified performance over a range of input and load conditions and complying with demanding international
regulations for harmonic current content, standby power, safety and EMC.
This application note is intended to assist engineers who design and test power supplies and other equipment connected to the AC line make power related measurements accurately, quickly and safely.
2. Power Supply Measurements Essential Background
AC Power is the product of RMS input voltage, RMS input current and power factor.
The power supply input voltage is the normal AC line in most cases and its shape is near sine-wave with low distortion.
The input current may be highly distorted and rich in harmonic content.
Most power supplies have input circuits consisting of arectifier followed by a smoothing capacitor to produce a high voltage DC. The capacitor maintains the DC voltage when the AC line input voltage
is falling and is charged only when the AC voltage is higher than the DC ‘bus’ voltage. The AC input current is thus no sine wave, but only flows near the peak of the voltage sinewave.
An example of a typical switch mode power supply is shown in Figure 1. Figure 2 shows typical input waveforms of a power supply.
Figure 1. Switch-mode power supply simplified schematic.
Figure 2. Vrms and Arms waveform.
This distorted current increases the RMS current drawn from the AC line, but does no useful work and does not contribute to the real power (watts) drawn or used by the power supply.
This extra, wasted current, multiplied up by the millions of power supplies in use means that the the AC supply and distribution system must be larger than required to supply the necessary power
(watts) and that power is wasted providing the excess current by resistive losses in the transmission and distribution system.
This is one reason why the current distortion drawn by power supplies is limited by international regulations such as IEC61000-3-2.
Most modern power supply designs mitigate this problem by employing some form of wave shaping circuit on their input, often called a (PFC) Power Factor Correction circuit. An example of PFC
implementation is shown in Figure 3. These circuits help shape the input current such that the input impedance of the power supply appears fairly linear to the power line, similar to a resistive
load. By doing so, the problems of distorted input current waveforms can be significantly improved. Another key issue, most designers face today is measuring and certifying energy consumption and
efficiency on their class of device to comply with institutions like ENERGY STAR™ and CEC.
Figure 3. Simplified view of an SMPS power supply (primary side only) and its power quality measurement test points. Simultaneous input VAC and IAC readings are necessary for power quality
In order to satisfy these requirements for low input current distortion and harmonics, high efficiency and low average and standby power consumption its important to understand how to measure them in
accordance with internationally accepted practice.
Figure 4. Distorted input current sampling.
3. Critical Power Supply Measurements
It's important for a power supply designer to keep check of all these parameters that are so crucial to a power supply’s performance proper operation and compliance to specification. Some of the most
important power measurements a designer has to make are listed as follows:
4. Making Power Supply Input Measurements Using a Tektronix – PA1000
Tektronix PA1000 is a single channel high precision Power Analyzer that helps makes complex power measurements on a power supply very easy. This section will discuss a practical example of making
power supply input measurements on a typical power supply. The DUT (Device under test) for this example is a generic laptop charger.
The standard current inputs of a power analyzer will measure a large range of current, from milli-amps to 20 or 30 amps RMS. This is suitable for most power supplies up to 3kW.
A single power analyzer wattmeter input channel consists of a voltage input pair (V[HI] and V[LO]) and a current input pair (A[HI] and A[LO]).
These connections are simplified by use of a break-out box that makes the analyzer connections with 4mm safety connectors and provides a standard AC outlet for connection to the power supply. Make
the connections as shown in the Figure 5.
Figure 5. PA1000 wiring diagram.
Connect the PA1000 and breakout box using the color coded safety-banana lead set as seen in picture
For this example a generic laptop charger was plugged in the AC socket on the breakout box and allowed to charge a laptop
PA1000 will start displaying results as soon as everything is plugged in and powered on as shown in Figure 6
Figure 6. Result screen.
Figure 7
To view more parameters press the "Menu" button on the front panel and navigate to "Measurements" as shown in Figure 8
Figure 8. Measurements menu.
Options available include crest factors, power factor, current and voltage harmonics, THD, active and reactive power and many more. Please refer to the user manual for the list of all available
Figure 9 shows an example of some of these measurements on the display.
Figure 9. More measurement choices.
Voltage, current and power waveforms and harmonic bar charts can also be viewed as seen in following Figure 10 and Figure 11 respectively by pressing the Graph button or accessing the Graph Menu.
Figure 10. Graph display
Figure 11. Harmonic bar chart display.
One of the important features that most of the power supply designers and testing labs need is ability to log data efficiently over extended periods of time.
PA1000 provides a standard USB port on the front panel for data logging which can be very handy for performing extended testing.
The test results can be exported as an excel file and used for presentation or analysis as shown in the Figure 12.
Figure 12. Data logged to Excel file.
The best and the most convenient way to test power supply is to use the included PWRVIEW software.
5. PWRVIEW Software
PWRVIEW software compliments Tektronix Power Analyzers and is available as free download on Tektronix website.
PWRVIEW offers easy, wizard-driven test solutions for power supply, standby power and many other target applications
For download and installation info go to www.tek.com/power-analyzer-series/pa1000
Tektronix Power Analyzers remote operation
Transfer, view, analyze, record and export measurement data in real-time from the instrument, including waveforms, harmonic bar charts, and stand power trend charts.
Default applications and standard compliance tests
Figure 13 shows the screen shot of PWRVIEW PC software
Figure 13. PWRVIEW software.
PWRVIEW makes is very easy to monitor, record and analyze the critical power measurement that have been discussed earlier
The following section will demonstrate how to make efficiency measurements using two PA1000 using PWRVIEW software.
6. Power Supply Efficiency Measurements
Two wattmeter channels are required for performing the efficiency measurements. Either a multichannel Power Analyzer like PA4000 or two PA1000 connected with PWRVIEW can do the job. Two PA1000’s are
used for this example to demonstrate efficiency measurements on a power supply.
Connect two PA1000’s, one on input and one on output of the power supply under test making connections as demonstrated earlier.
Connect both PA’s to the computer via USB, Ethernet or GPIB and add them to PWRVIEW.
Once added, PWRVIEW will display two tabs for two Power Analyzers connected as shown in Figure 14.
Figure 14. Adding two PA1000 simultaneously
For calculating efficiency, just click on the Configure radio tab on the top ribbon.
This will open a new tab for input and output selection as shown in Figure 15.
Figure 15. Efficiency configuration.
Once proper groups are selected for input and output just click on the radio button – Enable Watts Efficiency Measurements.
Once done, click on the Measure tab on the top and then click start on the Measure page.
The Measurement grid will start making efficiency measurement based on the input and output wattage as seen in the Figure 16.
Figure 16. Measurement grid with efficiency.
PWRVIEW can also be used to view waveforms and bar charts as shown in following Figure 17 and Figure 18 respectively.
Figure 17. Waveform display.
Figure 18. Harmonic bar chart display.
7. Standby Power Compliance Tests
The power consumed by power supply while they are not loaded is called standby power.
The standby power is trivial when considered for individual power supplies but when equated as a whole over a longer period of time and considering the number of power supplies plugged in to sockets
all the time, becomes a significant waste of energy
Many programs are already in place to reduce standby power including ENERGY STAR and the EU Eco Directive. The scope of these programs continues to grow and the level of standby power in Watts
necessary to achieve compliance steadily falls.
PWRVIEW software offers engineers a straight forward and accurate tool to conduct a full-compliance standby power test and get to the market faster and cheaper.
Figure 19 shows a snapshot of a full compliance standby test in action.
Figure 19. Full compliance standby power test.
Figure 20. PWRVIEW PC software charts harmonics and compares to limits.
For more details on standby power and full compliance stand by test, please refer to Standby Power app note on
Harmonics Limits
Using PC software coupled to the power analyzer, harmonics measurements may be quickly and conveniently recorded and compared to the limits of IEC61000-3-2 and others.
Software features such as PDF report export provide complete reporting functions for power supply conformance measurements.
8. Definitions
8.1. RMS (Root Mean Squared Value)
The RMS value is the most commonly used and useful means of specifying the value of both AC voltage and current. The RMS value of an AC waveform indicates the level of power that is available from
that waveform and is equivalent DC at the same voltage. This is one of the most important attributes of any AC source. The calculation of an RMS value can best be described by considering an AC
current waveform and its associated heating effect such as that shown in Figure 21.
Figure 21. Root Mean Square waveform.
The current (Amp) is considered to be flowing through a resistance; the heating effect at any instant is given by the equation:
By dividing the current cycle into equally spaced coordinates (samples), the variation of the heating effect with time can be determined as shown in Figure 2 above. The average heating effect (power)
is given by:
To determine the equivalent value of current that would produce the average heating effect value shown above, then the following applies:
The RMS value of the current = the square root of the mean of the squares of the current. This value is often termed the effective value of the AC waveform, as it is equivalent to the direct current
that produces the same heating effect (power) in the resistive load. It is worth noting that for a sinusoidal waveform:
8.2 Average Value
The average value of a waveform such as that shown in Figure 22 is given by:
It is clear the average value can only have real meaning over one half cycle of the Waveform, for a symmetrical waveform, the mean or average value over a complete cycle is zero. Most simple
multi-meters determine AC values by full-wave rectification of the AC waveform, followed by a calculation of the mean value. Such meters; however, will be calibrated in RMS and will make use of the
known relationship between RMS and average for a sinusoidal waveform, i.e.
However, for waveforms other than a pure sine wave, the readings from such meters will be invalid. RMS values will need to be calculated using techniques shown in Figure 22.
Figure 22. Average calculated methodology.
8.3 Real and Apparent Power (W & VA)
A sinusoidal voltage source of, say, 100V RMS is connected to a resistive load of, say, 100Ω, then the voltage and current can be depicted as in Figure 23 and are said to be "in phase". The power
that flows from the supply to the load at any instant is given by the value of the product of the voltage and the current at that instant, as illustrated in Figure 23.
Figure 23. Voltage and current phase waveform.
From this, it can be seen that the power flowing into the load fluctuates (at twice the supply frequency) between 0 and 200W and that the average power delivered to the load equals 100W which is what
one might expect from 100V RMS and a resistance of 100 Ω. However, if the load is reactive (i.e. contains inductance or capacitance as well as resistance) with an impedance of 100 Ω, then the current
that flows will still be 1A RMS but will no longer be in-phase with the voltage. This is shown in Figure 24 for an inductive load with the current lagging by 60 degrees. Although the power flow
continues to fluctuate at twice the supply frequency, it now flows from the supply to the load during only a part of each half cycle—during the remaining part, it actually flows from the load to the
supply. The average net flow into the load, therefore, is much smaller than in the case of a resistive load as shown in Figure 23— with only 50W of useful powering delivered into the inductive load.
Figure 24. The power that flows from the supply to the load.
In both of the above cases the RMS voltage was equal to 100V RMS and the current was 1A RMS. The product of these two values is the apparent power delivered into the load and is measured in VA as
The real power delivered has been shown to depend on the nature of the load. It is not possible to determine the value of real power from the knowledge of RMS voltage and the use of a true AC power
meter, capable of computing the product of the instantaneous voltage and current values and displaying the mean of the result is required. VA is often measured to ensure that the ac supply has
sufficient capacity.
Figure 24. Power factor waveform.
8.4 Power Factor
It is clear that, in comparison with DC systems, the transferred AC power is not simply the product of the voltage and current values. A further element, known as the power factor must also be taken
into consideration. In the previous example (real and apparent power) with an inductive load, the power factor is 0.5 because the useful power is exactly one half of the apparent power. We can
therefore define power factor as:
In the case of sinusoidal voltage and current waveforms, the power factor is actually equal to the cosine of the phase angle (q) between the voltage and current waveforms. For example, with the
inductive load described earlier, the current lags the voltage by 60 degrees.
It is for this reason that power factor is often referred to as cosq. However, it is important to remember that this is only the case when both voltage and current are sinusoidal [Figure 24 (I1 and
I2)] and that power factor is not equal to cosq in any other case [Figure 24 (I3)]. This must be remembered when using a power factor meter that reads cosq, as the reading will not be valid except
for pure sinusoidal voltage and current waveforms. A true power factor meter will compute the ratio of real to apparent power as described in section discussing real and apparent power.
Tektronix power analyzer retains high accuracy even at very low power factor, which is very important for product characterization and development.
8.5 Crest Factor
Tektronix power analyzers can measure a high Crest Factor (~10). This is critical for characterization of switched mode power supplies, which usually have high peak current draw. It has already been
shown that for a sinusoidal waveform:
The relationship between peak and RMS is known as the crest factor and is defined as:
Thus, for a sinusoid:
Many items of modern equipment connected to the AC supply take non-sinusoidal current waveforms. These include power supplies, lamp dimmers, and even fluorescent lamps. Power supplies can often
exhibit a current crest factor of around 4 and up to 10.
8.6 Harmonic Distortion
If a load introduces distortion of the current waveform, it is useful, in addition to knowing the crest factor, to quantify the level of distortion of the wave shape. Observation on an oscilloscope
will indicate distortion but not the level of distortion. It can be shown by Fourier analysis that a nonsinusoidal current waveform consists of a fundamental component at the supply frequency plus a
series of harmonics (i.e.) components at frequencies that are integral multiples of the supply frequency). For example, a SMPS, a lamp dimmer or even a speed-controlled washing machine motor can
contain harmonics of even greater significance as shown in Figure 25.
Figure 25. An example of Harmonics Barchart.
The only useful current is the fundamental component of current, as it is only this that can generate useful power. The additional harmonic current not only flows within the power supply itself, but
in all of the distribution cables, transformers and switchgear associated with the power supply and will thus cause additional loss. There is an increasing awareness of the need to limit the level of
harmonics that equipment can produce. Controls exist in many territories to provide mandatory limits on the level of harmonic current permitted for certain types of load. Such regulatory controls are
becoming more widespread with the use of internationally recognized standards such as EN61000-3-2. Thus, there is a need for an increased awareness amongst equipment designers as to whether their
products generate harmonics and at what level. It is only the fundamental which generates power, harmonics in general do not.
8.7 Standby Power
Standby power is the power drawn by a power supply when its load is not performing its full function. This may be the power consumed just by the clock on a microwave oven or the power drawn by a
laptop charger when the battery is fully charged.
To make the measurement requires not only very low measurement ranges, but special techniques to overcome the problems of power supplies operating in burst mode.
Tektronix power analyzers have both a quick 1-button standby mode for the designer and, together with PWRVIEW pc software will perform full compliance ENERGY STAR and IEC62310 Ed.2 standby power
measurements. Please see our detailed application note on this subject.
9. Conclusion
Power and power related measurements on power supplies require sophisticated and accurate instrumentation to ensure that the power supply performs to its specification.
The Tektronix PA1000 power analyzer incorporates a wide range of advanced features that provide class-leading accuracy from mW to kW. With the PA1000, Power supply designers faced with the challenges
of ever increasing efficiency and lower standby power requirements can always be confident that their designs meet specifications. | {"url":"https://www.tek.com/vn/documents/application-note/power-measurements-ac-dc-power-supplies","timestamp":"2024-11-03T15:08:07Z","content_type":"text/html","content_length":"119072","record_id":"<urn:uuid:537ab297-5c8c-444a-a2df-ee82131ccd40>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00687.warc.gz"} |
Houghton corrector:
telescopeѲptics.net ▪ ▪ ▪ ▪ ▪▪▪▪ ▪ ▪ ▪ ▪ ▪ ▪ ▪ ▪ ▪ CONTENTS
◄ 10.2.4.3. Houghton telescopes ▐ 10.2.4.6. Plano-symmetrical HCT ►
10.2.4.5. Houghton corrector: secondary spectrum reduction
As the ray plots clearly show, chromatism of the aplanatic (symmetrical) single-glass Houghton corrector becomes unacceptably large as the mirror relative aperture approaches and exceeds ~ƒ/3. While
both, secondary spectrum and spherochromatism contribute to it, it is the former that dominates. The cause of the Houghton secondary spectrum is that its lens elements are not ideal thin lenses with
near-zero thickness and separation. Consequently, the thin lens focal length formula given by (1/ƒ)=(1/ƒ1)+(1/ƒ2) doesn't apply. Instead, the effective focal length is given by the thick lens focal
length formula for a pair of lenses,
with the individual focal lengths of the front and rear lens, as the actual thick-lens focal lengths, given by:
respectively, with the quantity Π being the separation between the second principal plane of the front lens and the first principal plane of the rear lens (FIG. 202) given by:
with ƒ1=(n-1)[(1/R1)-1/R2] and ƒ2=(n-1)[(1/R3)-1/R4] being the front and rear lens focal length, respectively, t1 and t2 the respective lens center thicknesses, s the lens separation and R1, R2, R3
and R4 the respective lens surface radii. With 1/ƒ1 and 1/ƒ2 being very close numerically, it is the value of Π that mainly determines corrector's power (Eq. 150), and hence its secondary spectrum.
FIGURE 202: Houghton corrector principal plane separation Π, between the second principal plane of the front lens (P1') and the first principal plane of the second lens (P2), is the main determinant
of the effective non-zero focal length of the doublet. According to Eq. 151, the principal plane separation increases with lens thickness and lens spacing (s), reducing further corrector (positive)
focal length, nearly in proportion to ~ƒ1ƒ2/Π. This in turn results in the increase of corrector's secondary spectrum. The principal plane separation - and secondary spectrum - for given aperture
also increase with more strongly curved lens surfaces, needed to correct spherical aberration and coma of faster mirrors. When the two lenses are in contact, the separation s is likely to be,
roughly, somewhat less than a half of the value of Π. This means that any relative increase in lens separation will result in less than half as much of a relative increase of secondary spectrum. With
the typical values of ƒ1/R1~1.3 and ƒ2/R4~0.5, change in the thickness of the front lens results in secondary spectrum larger by a factor of ~2.5 than with identical thickness increase of the rear
lens. Also, any relative change in the front lens thickness will result in ~1.3 times greater secondary spectrum than identical increase in lens separation, while thickness increase of the rear lens
will result in the increase of secondary spectrum smaller by a factor of 0.5 than identical increase in lens separation.
As can be seen from Eq. 151, the size of Π it is not affected by a change in index, or radii, but it does change in proportion to lens thickness. Since the lenses can only be so thin, there is a
practical minimum level of secondary spectrum that can not be further lessened with a single-glass Houghton corrector. It could only be cancelled by changing the sign of R4, but it would require the
rear element to be a negative out-curving meniscus, which would make corrector with acceptable coma impossible.
In the symmetrical corrector configuration, the positive element has slightly less power than the negative element as a single lens, but when combined the two have a weak positive power. Neglecting
the smaller (negative) value of (1/ƒ1)-1/ƒ2 in Eq. 150, the corrector focal length is approximated by ƒc~ƒ1ƒ2/Π (at the end of this section is explained in more details why taking somewhat smaller
value is likely to give better end result), with the variation in the focal length for non-optimized wavelengths approximated by
where n and n' are the refractive index of the optimized and non-optimized wavelength, respectively. It places the secondary spectrum of the Houghton corrector - as the axial separation of its
paraxial foci - roughly at ~ƒc/200, or some 10 times greater than in a doublet achromat. It is only due to the weak corrector's power that it induces relatively small amount of chromatism. Nominally,
the secondary spectrum is negative for shorter (than the optimized) wavelengths, and positive for longer wavelengths - a consequence of the positive effective power of the corrector. It simply means
that the former focus shorter, and the latter longer than the optimized wavelength.
Should the secondary spectrum be the only chromatic aberration present, the transverse chromatic blur diameter in units of the green e-line Airy disc diameter would be approximated by B~745(Δƒ)/Fc2,
with Fc being the corrector's focal ratio, approximated by Fc~ƒc/D~ƒ1ƒ2/ΠD. In reality, secondary spectrum is always combined with a certain amount of sphero-chromatism, in which case the actual
secondary spectrum is measured by the separation between best aberrated foci for different wavelengths, and the blur size results from the combined size of defocus (secondary spectrum) and spherical
aberration (sphero-chromatism) for the wavelength.
Houghton corrector secondary spectrum can be minimized by either slightly weakening chromatic power of the front lens, or by slightly strengthening the rear lens. The two options for reducing the
secondary spectrum of the Houghton corrector are: (1) abandoning symmetrical radii, and (2) using two different glass types for the lenses. Another option is to use plano-symmetrical corrector type,
which has generally less of the residual power, hence smaller secondary spectrum. Abandoning equal radii design is a practical disadvantage fabrication-wise (also requiring more complex
calculations), while plano-lens design puts constraints on coma-correction. Thus, the most effective way of reducing the Houghton secondary spectrum is by using two different glass types.
Starting out with a typical single-glass aplanatic Houghton corrector, minimizing the secondary spectrum requires small change in either power, or dispersion (or both) of one of the two lens
elements. Lens power changes with (n-1), and its dispersion with 1/V, n being the glass refractive index and V its nominal dispersion. The measure of needed chromatic power change can be obtained
from the general rule of achromatism for near-contact or contact doublet, requiring ƒ1/ƒ2=V2/V1. Since the aplanatic Houghton doublet acts as a thin lens pair in which the rear lens focal length is
somewhat longer than that of the front lens, resulting in a weak positive power of the doublet, the appropriate dispersion V2 for minimizing the secondary spectrum should be different in
approximately the same proportion.
Assuming thin-lens doublet, the rear lens effective focal length ƒ2e is obtained from 1/ƒc=(1/ƒ1)+1/ƒ2e, as 1/ƒ2e=1/ƒc-(1/ƒ1). The appropriate rear lens dispersion V2 can be obtained from ƒ1/ƒ2e=V2/V
0, with V0 being the dispersion of the basic single-glass corrector. However, glass of different dispersion will almost invariably also have different refractive index. Since it is also a factor
affecting chromatic power (secondary spectrum), it needs to be taken into account. With it, the achromatic relation takes the form ƒ1/ƒ2e=(n0-1)V2/(n2-1)V0, with needed properties of the
achromatizing glass obtained from:
with n0, V0 and n2, V2 being the refractive index (optimized wavelength) and Abbe number for the starting single-glass aplanatic corrector, and the achromatized negative element, respectively.
Helpful indicator of the effect of new glass type on secondary spectrum is given by 1/ç=(n2-1)V0/(n0-1)V2. This parameter can be called corrector's relative chromatic power. If it is the front lens
to be achromatized, ç will be slightly smaller, and for the rear (negative) lens slightly greater than 1 (rough average with n~1.5 and mirrors in the ƒ/2.5-ƒ/3 range is ±1%). It reflects needed
change in the relative chromatic power of one of the lens elements for near-optimal balancing of the combined secondary spectrum of the corrector.
In general, the second glass alternative for minimizing the secondary spectrum will be of very similar index and dispersion values to those of the single-glass corrector to be achromatized (as the
nominal dispersion and index change in opposite directions with significant index changes, it becomes difficult to impossible to find a glass with relative chromatism ç close to 1). Once suitable
glass is found, it will make possible significant reduction of the secondary spectrum. An integral part of the optimization is tailoring out best combination of the secondary spectrum and spherical
aberration, with lens thickness and separation also being possible factors (FIG. 203).
FIGURE 203: Effect of achromatizing on the chromatism of a HCT with symmetrical aplanatic Houghton corrector (compare with Fig. 132a and 133a). By replacing the BK7 negative lens element glass with
BK8, the wavefront error is reduced from 1.3 and 0.38 to 0.45 and 0.048 wave RMS for the h and r spectral lines, respectively (system a in the Appendix). By increasing lens separation, the chromatism
is further reduced and optimized (balanced) to 0.14 and 0.12 wave RMS for the h and r lines, respectively, as shown on the plot (system b in the Appendix). The h-line correction is now nearly twice
better than in non-achromatized plano-lens version, or at the level of a 4" ƒ/200 achromat. While still approximately double the chromatism of a comparable SCT, it is vastly improved over the
non-achromatized version (note that further reduction could be possible). An ƒ/2.5 relative aperture for this aperture size is probably near-limit for all-spherical Houghton system. At this primary
focal ratio, higher-order spherical, responsible for most of the scattered rays, cannot be corrected significantly better than 1/30 wave RMS without adding aspheric surface term. SPEC'S
Achromatizing is still worthwhile at ~ƒ/3 mirror focal ratios. By switching a rear lens from BK7 to PK3, and increasing lens spacing from 1.4mm to 3mm (to compensate for the induced spherical
aberration) in the ƒ/3/10 symmetrical aplanatic Houghton (Fig. 133c), corrector's chromatism is reduced from 0.37 and 0.1 to 0.085 and 0.043 wave RMS wavefront error for the violet h-line and red
r-line, respectively. That compares very favorably to the reduction of chromatism by compromising it with some coma (0.28 and 0.075 wave RMS for the h- and r-lines, respectively). Note that the
corrector type with all four radii different allows for still better color correction in a single-glass arrangement.
An alternative to increasing lens separation of the achromatized corrector is slightly (typically less than 1%) relaxing second radius. It is more likely to bring best foci of different wavelengths
closer together, but the choice is best made using ray-trace.
As mentioned, secondary spectrum is main, but not a lone contributor to the chromatism of the Houghton corrector. The other is sphero-chromatism. As a consequence, the secondary spectrum - and
chromatism in general - is minimized by bringing together best foci for different wavelengths, not the paraxial foci. This would certainly require more involved optimization than the one outlined
above. In general, however, considering typical symmetrical aplanatic Houghton LA properties, even this crude form of optimization alone - in particular with the change in relative chromatic power
purposely reduced to ~2/3 of the power differential indicated by the ç value - should result in a significant reduction of corrector's chromatism.
The achromatizing glass often will, to some extent, also change the spherochromatism of the basic (single-glass) corrector. It can be for better, or for worse, but the effect is, in general,
secondary to that of the change in size of secondary spectrum.
◄ 10.2.4.3. Houghton telescopes ▐ 10.2.4.6. Plano-symmetrical HCT ►
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ACO Seminar
The ACO Seminar (2014–2015)
Jan. 15, 3:30pm, Wean 8220
Alexander Barvinok, University of Michigan
Computing partition functions in hard problems of combinatorial optimization
Consider an NP-complete problem, such as finding whether a given graph
vertices contains a Hamiltonian cycle. First, we attempt to solve an even more general problem: given an
non-negative matrix A, interpreted as the matrix of weights on the edges of the complete graph
, we consider the sum (partition function) over all Hamiltonian cycles in
of the products of weights on the edges of the cycle. Thus if
the adjacency matrix of
, we obtain the number of Hamiltonian cycles in
. Next, we modify
by replacing all zeros with some positive epsilons. It turns out that the partition function becomes efficiently computable, which allows us to distinguish graphs that are far from having a
Hamiltonian cycles from graphs that have many Hamiltonian cycles (even when "many" still means that the probability to hit a Hamiltonian cycle at random is exponentially small).
In a joint work with P. Soberon, we attempt to establish the most general framework when this approach appears to work: it concerns computing the partition function of graph homomorphisms with
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10000 Mortgage Over 15 Years
10000 Mortgage Over 15 Years
For those looking to secure a Texas land mortgage, our land loan calculator Input your loan term (total years on the loan). Determine your payment. Most common terms are 15 years and 30 years.
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SciPy Programming Succinctly - Free Computer, Programming, Mathematics, Technical Books, Lecture Notes and Tutorials
SciPy Programming Succinctly
• Title SciPy Programming Succinctly
• Author(s) James McCaffrey
• Publisher: Syncfusion (2016)
• Hardcover/Paperback N/A
• eBook HTML, PDF (120 pages), ePub, adn MOBI (Kindle)
• Language: English
• ISBN-10: N/A
• ISBN-13: N/A
• Share This:
Book Description
This book offers readers a quick, thorough grounding in knowledge of the Python open source extension SciPy. The SciPy library, accompanied by its interdependent NumPy, offers Python programmers
advanced functions that work with arrays and matrices.
Each section presents a complete demo program for programmers to experiment with, carefully chosen examples to best illustrate each function, and resources for further learning. Use this e-book to
install and edit SciPy, and use arrays, matrices, and combinatorics in Python programming.
About the Authors Reviews, Ratings, and Recommendations: Related Book Categories: Read and Download Links:Similar Books:
Book Categories
Other Categories
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Eilenberg-Steenrod Axioms -- from Wolfram MathWorld
A family of functors category of pairs of topological spaces and continuous maps, to the category of Abelian groups and group homomorphisms satisfies the Eilenberg-Steenrod axioms if the following
conditions hold.
1. long exact sequence of a pair axiom. For every pair
where the map inclusion map inclusion map map boundary map.
2. homotopy axiom. If induced maps
3. excision axiom. If space with subspaces set closure of inclusion map
4. dimension axiom. Let groups. The coefficients of the homology theory
These are the axioms for a generalized homology theory. For a cohomology theory, instead of requiring that functor, it is required to be a co-functor (meaning the induced map points in the opposite
direction). With that modification, the axioms are essentially the same (except that all the induced maps point backwards). | {"url":"https://mathworld.wolfram.com/Eilenberg-SteenrodAxioms.html","timestamp":"2024-11-04T04:35:25Z","content_type":"text/html","content_length":"58128","record_id":"<urn:uuid:ca6c1d6a-5273-4598-89b6-4f6cb1877243>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00632.warc.gz"} |
Understanding the Area of a Cube
Understanding the Area of a Cube
Introduction to the Area of a Cube
Cubes are geometric marvels that we encounter in everyday life, from dice in our game nights to shipping boxes. But beyond their boxy charm lies an interesting mathematical concept: their surface
area. Calculating the area of a cube is a fundamental concept in geometry that provides valuable insights for various real world applications. Let’s dive into it!
Dissecting the Formula
The formula to find the area of a cube is simple yet powerful: A = 6s². Here:
• A represents the total surface area of the cube, expressed in square units like square meters (m²) or square feet (ft²).
• s is the length of one side of the cube, expressed in linear units like meters (m) or feet (ft).
In essence, the surface area (A) is equal to six times the square of the side length (s).
Real life Example: Packaging Design
Imagine you are designing a gift box for a new product launch. You’ve settled on a chic cube shaped box with each side measuring 0.5 meters. What’s the total surface area?
Plugging into the formula, we have:
A = 6 * (0.5)² = 6 * 0.25 = 1.5 m²
Thus, you’ll need 1.5 square meters of material to cover the entire surface of the cube.
Practical Application: Construction
Engineers and architects regularly use this formula in designing structures. For instance, if a company plans to construct cube shaped storage units, knowing the surface area helps in estimating
material costs.
Data Validation and Practical Limitations
It’s important to ensure that the side length (s) is a positive number. Negative or zero values are not physically meaningful for length and should return an error message.
Calculating the area of a cube is a straightforward yet invaluable skill in geometry. From packaging design to construction, this formula A = 6s² helps you quantify the surface area required for
various practical applications. Understanding this basic formula opens the door to numerous real world applications, making it an essential tool in both education and industry.
Q: Can the side length (s) of a cube be in different units?
A: Yes, the side length can be in any linear unit like meters, feet, inches, etc. Just ensure consistency when calculating the area.
Q: What if the side length is zero or negative?
A: The side length should be a positive number. Zero or negative values don’t make sense and should return an error message.
Example Calculations
1. s = 1 m
Surface Area: A = 6 * 1² = 6 m²
2. s = 2 ft
Surface Area: A = 6 * 2² = 24 ft²
3. s = 3 cm
Surface Area: A = 6 * 3² = 54 cm²
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Andre Weil: Founding Member of the Mathematical Bourbaki Group
ANDRE WEIL: Founding Member of the Mathematical Bourbaki Group
André Weil was a very influential French mathematician around the middle of the 20th Century. Born into a prosperous Jewish family in Paris, he was brother to the well-known philosopher and writer
Simone Weil, and both were child prodigies. He was passionately addicted to mathematics by the age of ten, but he also loved to travel and study languages (by the age of sixteen he had read the
“Bhagavad Gita” in the original Sanskrit).
He studied (and later taught) in Paris, Rome, Göttingen and elsewhere, as well as at the Aligarh Muslim University in Uttar Pradesh, India, were he further explored what would become a life-long
interest in Hinduism and Sanskrit literature.
Even as a young man, Weil made substantial contributions in many areas of mathematics, and was particularly animated by the idea of discovering profound connections between algebraic geometry and
number theory. His fascination with Diophantine equations led to his first substantial piece of mathematical research on the theory of algebraic curves. During the 1930s, he introduced the adele
ring, a topological ring in algebraic number theory and topological algebra, which is built on the field of rational numbers.
The early leader of the Bourbaki group
Weil was an early leader of the Bourbaki group who published many influential textbooks on modern mathematics
It was also at this time that he became a founding member, and the de facto early leader, of the so-called Bourbaki group of French mathematicians. This influential group published many textbooks on
advanced 20th Century mathematics under the assumed name of Nicolas Bourbaki, in an attempt to give a unified description of all mathematics founded on set theory. Bourbaki has the distinction of
having been refused membership of the American Mathematical Society for being non-existent (although he was a member of the Mathematical Society of France!)
When the Second World War broke out, Weil, a committed conscientious objector, fled to Finland, where he was mistakenly arrested as a possible spy. Having made his way back to France, he was again
arrested and imprisoned as for refusing to report for military service. In his trial, he cited the Bhagavad Gita to justify his stand, arguing that his true dharma was the pursuit of mathematics, not
assisting in the war effort, however just the cause. Given the choice of five more years in prison or joining a French combat unit, though, he chose the latter, an especially lucky decision given
that the prison was blown up shortly afterwards.
But it was in 1940, in a prison near Rouen, that Weil did the work that really made his reputation (although his full proofs had to wait until 1948, and even more rigorous proofs were supplied by
Pierre Deligne in 1973). Building on the prescient work of his countryman Évariste Galois in the previous century, Weil picked up the idea of using geometry to analyze equations, and developed
algebraic geometry, a whole new language for understanding solutions to equations.
Weil Conjectures
An illustration of the “cycle évanescent” or “vanishing cycle” described in Deligne’s proof of the Weil conjectures
The Weil conjectures on local zeta-functions effectively proved the Riemann hypothesis for curves over finite fields, by counting the number of points on algebraic varieties over finite fields. In
the process, he introduced for the first time the notion of an abstract algebraic variety and thereby laid the foundations for abstract algebraic geometry and the modern theory of abelian varieties,
as well as the theory of modular forms, automorphic functions and automorphic representations. His work on algebraic curves has influenced a wide variety of areas, including some outside of
mathematics, such as elementary particle physics and string theory.
In 1941, Weil and his wife took the opportunity to sail for the United States, where they spent the rest of the War and the rest of their lives. In the late 1950s, Weil formulated another important
conjecture, this time on Tamagawa numbers, which remained resistant to proof until 1989. He was instrumental in the formulation of the so-called Shimura-Taniyama-Weil conjecture on elliptic curves
which was used by Andrew Wiles as a link in the proof of Fermat’s Last Theorem. He also developed the Weil representation, an infinite-dimensional linear representation of theta functions which gave
a contemporary framework for understanding the classical theory of quadratic forms.
Over his lifetime, Weil received many honorary memberships, including the London Mathematical Society, the Royal Society of London, the French Academy of Sciences and the American National Academy of
Sciences. He remained active as professor emeritus at the Institute for Advanced Studies at Princeton until a few years before his death.
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Varney, Moses Varney, Carey Varney, Donny Varney, Colby Varney, Saul Varney, Trenton Varney, Marion Varney, Rory Varney, Tracey Varney, Darrel Varney, Gavin Varney, Reynaldo Varney, Erich Varney,
Bennie Varney, Fabian Varney, Luther Varney, Tomas Varney, Blaine Varney, Jerald Varney, Dave Varney, Shad Varney, Edmund Varney, Reuben Varney, Clark Varney, Kory Varney, Jarrett Varney, Sonny
Varney, Donnell Varney, Demond Varney, Robby Varney, Aron Varney, Leonardo Varney, Jamison Varney, Lionel Varney, Kurtis Varney, Deon Varney, Elliott Varney, Stefan Varney, Emanuel Varney, Nickolas
Varney, Efrain Varney, Kristian Varney, Archie Varney, Joesph Varney, Graham Varney, Lynn Varney, Jerrod Varney, Wilson Varney, Esteban Varney, Reggie Varney, Mason Varney, Sylvester Varney, Leland
Varney, Hugo Varney, Willard Varney, Vance Varney, Ken Varney, Miles Varney, Taylor Varney, Jeffry Varney, Will Varney, Jean Varney, Rodrick Varney, Willis Varney, Antwan Varney, Hans Varney, Rudolph
Varney, Daren Varney, Jefferson Varney, Roosevelt Varney, Amos Varney, Emmanuel Varney, Elmer Varney, Grady Varney, Joaquin Varney, Tommie Varney, Jeromy Varney, Dorian Varney, Dane Varney, Tobias
Varney, Anton Varney, Santiago Varney, Sterling Varney, Freddy Varney, Shon Varney, Rico Varney, Jarvis Varney, Brant Varney, Monty Varney, Collin Varney, Nolan Varney, Alexis Varney, Quinton Varney,
Bob Varney, Stevie Varney, Avery Varney, Aric Varney, Emilio Varney, Shayne Varney, Carlo Varney, Conrad Varney, Jed Varney, Denny Varney, Elliot Varney, Laurence Varney, Delbert Varney, Jennifer
Varney, Doug Varney, Landon Varney, Moises Varney, Zachariah Varney, Kristofer Varney, Buddy Varney, Ignacio Varney, Elvis Varney, Chuck Varney, Torrey Varney, Jonas Varney, Humberto Varney, Aubrey
Varney, Elton Varney, Chance Varney, Marlin Varney, Vicente Varney, Solomon Varney, Horace Varney, Kim Varney, Tory Varney, Royce Varney, Wilbert Varney, Dirk Varney, Thad Varney, Ervin Varney, Louie
Varney, Jonah Varney, Raphael Varney, Wilfredo Varney, Santos Varney, Shelby Varney, Noe Varney, Hubert Varney, Mack Varney, Van Varney, Branden Varney, Ali Varney, Marcel Varney, Jamel Varney, Jamar
Varney, Jasen Varney, Rodger Varney, Pierre Varney, Blair Varney, Harley Varney, Johnathon Varney, Winston Varney, Ellis Varney, Marquis Varney, Galen Varney, Dewey Varney, Reid Varney, Bert Varney,
Brain Varney, Bradly Varney, Dusty Varney, Darian Varney, Wyatt Varney, Alphonso Varney, Vaughn Varney, Brenton Varney, Waylon Varney, Benito Varney, Kirby Varney, Jeramy Varney, Jarod Varney, Isaiah
Varney, Lane Varney, Rufus Varney, Domingo Varney, Jackson Varney, Jerod Varney, Edmond Varney, Tad Varney, Ernie Varney, Quinn Varney, Percy Varney, Andrea Varney, Lowell Varney, Cornell Varney,
Rhett Varney, Shelton Varney, Rickie Varney, Korey Varney, Cleveland Varney, Deandre Varney, Nathanael Varney, Arron Varney, Theron Varney, Boyd Varney, Logan Varney, Ariel Varney, Brendon Varney,
Adan Varney, Donte Varney, Doyle Varney, Reed Varney, Rodrigo Varney, Josue Varney, Cole Varney, Rob Varney, Zane Varney, Jeremie Varney, Kenyatta Varney, Barrett Varney, Rocco Varney, Sebastian
Varney, Michel Varney, Denis Varney, Brooks Varney, Danial Varney, Garth Varney, Heriberto Varney, Nakia Varney, Giovanni Varney, Mauricio Varney, Agustin Varney, Cedrick Varney, Tyron Varney, Brice
Varney, Darwin Varney, Carter Varney, Kip Varney, Garland Varney, Malik Varney, Elbert Varney, Gerry Varney, Deshawn Varney, Ronny Varney, Issac Varney, Jeramie Varney, Emmett Varney, Adolfo Varney,
Daron Varney, Rashad Varney, Lincoln Varney, Homer Varney, Jacques Varney, Nicky Varney, Dino Varney, Markus Varney, Ahmad Varney, Kristoffer Varney, Wilbur Varney, Antione Varney, Jan Varney, Ezra
Varney, Antony Varney, Gino Varney, Mikel Varney, Ari Varney, Tremayne Varney, Judson Varney, Garrick Varney, Kasey Varney, Kraig Varney, Rigoberto Varney, Diego Varney, Edwardo Varney, Jarred
Varney, Chet Varney, Hunter Varney, Jude Varney, Billie Varney, Elvin Varney, Alvaro Varney, Lenny Varney, Irvin Varney, Jasper Varney, Judd Varney, Carson Varney, Kenyon Varney, Keven Varney, Sammie
Varney, Keenan Varney, Darron Varney, Russel Varney, Leif Varney, Tyree Varney, Woodrow Varney, Chase Varney, Tod Varney, Richie Varney, Randell Varney, Vito Varney, Deron Varney, Gregorio Varney,
Federico Varney, Weston Varney, Davis Varney, Torrance Varney, Ulysses Varney, Trey Varney, Jeremey Varney, Riley Varney, Vince Varney, Nigel Varney, Chauncey Varney, Davin Varney, Leonel Varney,
Tobin Varney, Michelle Varney, Bernardo Varney, Dedrick Varney, Titus Varney, Osvaldo Varney, Wilfred Varney, Errol Varney, Jade Varney, Carmen Varney, Cliff Varney, Kenya Varney, Jammie Varney,
Kelley Varney, Harrison Varney, Lonny Varney, Abdul Varney, Coy Varney, Denver Varney, Robb Varney, Telly Varney, Carroll Varney, August Varney, Tristan Varney, Fidel Varney, Octavio Varney, Dameon
Varney, Josef Varney, Eloy Varney, Dax Varney, Barton Varney, Eddy Varney, Cyrus Varney, Broderick Varney, Zachery Varney, Hiram Varney, Raymundo Varney, Giuseppe Varney, Terrill Varney, Burton
Varney, Hank Varney, Sedrick Varney, Jamil Varney, Germaine Varney, Myles Varney, Maxwell Varney, Harlan Varney, Norris Varney, Emil Varney, Kennith Varney, Deric Varney, Bobbie Varney, Levar Varney,
Francesco Varney, Hassan Varney, Jerrold Varney, Jayme Varney, Josiah Varney, Junior Varney, Laron Varney, Duncan Varney, Kenton Varney, Sandy Varney, Brennan Varney, Jamaal Varney, Brook Varney,
Coby Varney, Augustine Varney, Merle Varney, Abram Varney, Rod Varney, Jefferey Varney, Frederic Varney, Eldon Varney, Nathanial Varney, Silas Varney, Gonzalo Varney, Greggory Varney, Brannon Varney,
Demarcus Varney, Erwin Varney, Shea Varney, Lon Varney, Alec Varney, Antwon Varney, Melissa Varney, Phil Varney, Adrain Varney, Hal Varney, Mitchel Varney, Antonia Varney, Irving Varney, Derik
Varney, Tavares Varney, Edgardo Varney, Gus Varney, Paris Varney, Kimberly Varney, Jevon Varney, Carmelo Varney, Britt Varney, Efren Varney, Moshe Varney, Shanon Varney, Dominique Varney, Cordell
Varney, Kermit Varney, Whitney Varney, Lashawn Varney, Jovan Varney, Chadd Varney, Dedric Varney, Chadrick Varney, Lars Varney, Kelsey Varney, Marcellus Varney, Dejuan Varney, Tanner Varney, Braden
Varney, Cortez Varney, Ezekiel Varney, Christoper Varney, Donell Varney, Delvin Varney, Timmothy Varney, Linwood Varney, Lisa Varney, Alfonzo Varney, Tate Varney, Douglass Varney, Quintin Varney,
Bennett Varney, Pat Varney, Cruz Varney, Garret Varney, Aldo Varney, Brandy Varney, Jereme Varney, Arnulfo Varney, Taurus Varney, Dereck Varney, Ned Varney, Trever Varney, Kirt Varney, Liam Varney,
Angela Varney, Marques Varney, Amy Varney, Roscoe Varney, Aurelio Varney, Genaro Varney, Nestor Varney, Dwain Varney, German Varney, Tarik Varney, Cristopher Varney, Forest Varney, Barney Varney,
Chaim Varney, Eliseo Varney, Thurman Varney, Donta Varney, Che Varney, Mariano Varney, Lanny Varney, Donavan Varney, Jameson Varney, Andreas Varney, Isidro Varney, Anderson Varney, Jedediah Varney,
Rahsaan Varney, Baron Varney, Emery Varney, Michale Varney, Jered Varney, Charley Varney, Bronson Varney, Odell Varney, Armand Varney, Shay Varney, Grover Varney, Carmine Varney, Norberto Varney,
Deangelo Varney, Brandt Varney, Chandler Varney, Otto Varney, Dannie Varney, Torrence Varney, Buck Varney, Darby Varney, Lindsey Varney, Trinity Varney, Cale Varney, Franco Varney, Damond Varney,
Jerad Varney, Reinaldo Varney, Wiley Varney, Ivory Varney, Everette Varney, Leopoldo Varney, Andra Varney, Dario Varney, Dalton Varney, Ward Varney, Brien Varney, Paulo Varney, Lazaro Varney, Emory
Varney, Westley Varney, Marcelino Varney, Emerson Varney, Mary Varney, Kwame Varney, Toney Varney, Marcelo Varney, Rasheed Varney, Derrell Varney, Jermey Varney, Russ Varney, Vincenzo Varney, Raheem
Varney, Asa Varney, Lemuel Varney, Murray Varney, Pasquale Varney, Burt Varney, Hollis Varney, Valentin Varney, Bernie Varney, Jamin Varney, Lenard Varney, Roel Varney, Weldon Varney, Ahmed Varney,
Cortney Varney, Tito Varney, Claudio Varney, Kristin Varney, Lorne Varney, Enrico Varney, Marlo Varney, Loyd Varney, Amir Varney, Arnoldo Varney, Harris Varney, Cristobal Varney, Mac Varney, Corbin
Varney, Cullen Varney, Shedrick Varney, Maria Varney, Jermain Varney, Ramsey Varney, Torey Varney, Nicole Varney, Corry Varney, Anson Varney, Heather Varney, Parker Varney, Elizabeth Varney, Tye
Varney, Lesley Varney, Lamonte Varney, Sanford Varney, Karim Varney, Stephanie Varney, Toriano Varney, Bruno Varney, Estevan Varney, Jerel Varney, Orville Varney, Brion Varney, Demetrios Varney,
Anibal Varney, Reginal Varney, Lavar Varney, Rosendo Varney, Romeo Varney, Simeon Varney, Elisha Varney, Stan Varney, Vidal Varney, Enoch Varney, Prince Varney, Eugenio Varney, Leigh Varney, Mervin
Varney, Bartholomew Varney, Delmar Varney, Khalid Varney, Kristen Varney, Johnpaul Varney, Carnell Varney, Domenic Varney, Adalberto Varney, Jaron Varney, Luciano Varney, Kai Varney, Stanford Varney,
Uriah Varney, Demarco Varney, Jabari Varney, Zackary Varney, Ritchie Varney, Ezequiel Varney, Basil Varney, Brenden Varney, Garett Varney, Major Varney, Jerold Varney, Wes Varney, Yancy Varney,
Benjamen Varney, Fredric Varney, Jerrell Varney, Benji Varney, Barron Varney, Chip Varney, Michele Varney, Britton Varney, Milo Varney, Samir Varney, Charlton Varney, Royal Varney, Rubin Varney,
Stoney Varney, Kieth Varney, Javon Varney, Derwin Varney, Duke Varney, Antone Varney, Daryle Varney, Demetric Varney, Layne Varney, Reyes Varney, Walker Varney, Fletcher Varney, Freeman Varney,
Jarret Varney, Delano Varney, Oren Varney, Marcell Varney, Trinidad Varney, Luigi Varney, Rustin Varney, Shan Varney, Sharif Varney, Isiah Varney, Levon Varney, Brody Varney, Coleman Varney, Ollie
Varney, Thor Varney, Darell Varney, Desi Varney, Horacio Varney, Kiley Varney, Mauro Varney, Vinson Varney, Dimitrios Varney, Kevan Varney, Merlin Varney, Monroe Varney, Rebecca Varney, Rashawn
Varney, Tyrell Varney, Zackery Varney, Samson Varney, Darcy Varney, Donavon Varney, Jessica Varney, Napoleon Varney, Muhammad Varney, Trevis Varney, Berry Varney, Antwain Varney, Eliezer Varney,
Kendell Varney, Lamarr Varney, Micky Varney, Demian Varney, Seneca Varney, Tucker Varney, Cheyenne Varney, Davon Varney, Norbert Varney, Regan Varney, Adolph Varney, Jerimiah Varney, Benton Varney,
Jeb Varney, Rian Varney, Corwin Varney, Jameel Varney, Tyrus Varney, Alexandro Varney, Benson Varney, Darien Varney, Augustus Varney, Erie Varney, Gil Varney, Raymon Varney, Abelardo Varney, Terance
Varney, Tremaine Varney, Cristian Varney, Jedidiah Varney, Christina Varney, Homero Varney, Rueben Varney, Rey Varney, Jace Varney, Caesar Varney, Hakim Varney, Lydell Varney, Marcello Varney,
Parrish Varney, Isaias Varney, Johnie Varney, Tavis Varney, Eliot Varney, Fermin Varney, Nikolas Varney, Orion Varney, Alden Varney, Clement Varney, Lucio Varney, Rondell Varney, Ronnell Varney,
Demetris Varney, Juston Varney, Len Varney, Cade Varney, Detrick Varney, Sherwin Varney, Truman Varney, Reymundo Varney, Danielle Varney, Domenico Varney, Octavius Varney, Anwar Varney, Christofer
Varney, Dwaine Varney, Leander Varney, Reese Varney, Houston Varney, Conor Varney, Dillon Varney, Mitch Varney, Demario Varney, Artis Varney, Boris Varney, Samual Varney, Dustan Varney, Gorge Varney,
Jory Varney, Stanton Varney, Amit Varney, Julie Varney, Kelby Varney, Kody Varney, Christos Varney, Fritz Varney, Lyndon Varney, Constantine Varney, Erasmo Varney, Jodie Varney, Brooke Varney, Conan
Varney, Kane Varney, Merrill Varney, Bryson Varney, Isreal Varney, Woody Varney, Tadd Varney, Emiliano Varney, Cyril Varney, Davey Varney, Gabe Varney, Pernell Varney, Curtiss Varney, Jacky Varney,
Nicholaus Varney, Hasan Varney, Davy Varney, Oswaldo Varney, Tammy Varney, Felton Varney, Jarrad Varney, Rosario Varney, Yusef Varney, Anthoney Varney, Dimitri Varney, Donn Varney, Graig Varney,
Glendon Varney, Alain Varney, Alonso Varney, Dwane Varney, Laura Varney, Millard Varney, Milan Varney, Everardo Varney, Taj Varney, Kendal Varney, Prentice Varney, Hilario Varney, Hayden Varney,
Lukas Varney, Lorin Varney, Stevan Varney, Benedict Varney, Ferdinand Varney, Renard Varney, Geoff Varney, Raleigh Varney, Amanda Varney, Jules Varney, Marshal Varney, Ephraim Varney, Patricia
Varney, Sarah Varney, Kalvin Varney, Sky Varney, Vladimir Varney, Kipp Varney, Rashid Varney, Burke Varney, Derrek Varney, Margarito Varney, Rojelio Varney, Von Varney, Wayland Varney, Mohammad
Varney, Sandro Varney, Renato Varney, Shiloh Varney, Torry Varney, Kenji Varney, Del Varney, Elgin Varney, Booker Varney, Chistopher Varney, Lavell Varney, Mohammed Varney, Refugio Varney, Damen
Varney, Nicola Varney, Daryn Varney, Elwood Varney, Kenric Varney, Andrae Varney, Christine Varney, Clemente Varney, Corby Varney, Edmundo Varney, Channing Varney, Maynard Varney, Roderic Varney,
Wilton Varney, Kedrick Varney, Kieran Varney, Lucius Varney, Cliffton Varney, Geraldo Varney, Adrien Varney, Hershel Varney, Reco Varney, Dudley Varney, Jame Varney, Michal Varney, Omari Varney,
Christophe Varney, Delton Varney, Lindsay Varney, Philippe Varney, Faron Varney, Brandan Varney, Williams Varney, Adonis Varney, Jeanpaul Varney, Mckinley Varney, Bertram Varney, Randel Varney, Audie
Varney, Fransisco Varney, Gideon Varney, Lafayette Varney, Renaldo Varney, Winfred Varney, Lacy Varney, Ravi Varney, Denton Varney, Bjorn Varney, Demetrice Varney, Duwayne Varney, Lavon Varney,
Porfirio Varney, Eldridge Varney, Hosea Varney, Lupe Varney, Corbett Varney, Grayson Varney, Sanjay Varney, Emile Varney, Emmitt Varney, Olin Varney, Ramone Varney, Yusuf Varney, Leandro Varney,
Amado Varney, Leighton Varney, Malachi Varney, Stephon Varney, Wilford Varney, Keon Varney, Timmie Varney, Errick Varney, Jarad Varney, Kaleb Varney, Dayton Varney, Jelani Varney, Rance Varney,
Corrie Varney, Jerrad Varney, Yancey Varney, Jonpaul Varney, Theo Varney, Amador Varney, Jamaine Varney, Lorenza Varney, Valentino Varney, Dee Varney, Eleazar Varney, Jarett Varney, Justen Varney,
Lauren Varney, Tyronne Varney, Willy Varney, Elroy Varney, Esequiel Varney, Ike Varney, Renee Varney, Karen Varney, Herschel Varney, Konstantinos Varney, Arlo Varney, Buford Varney, Rasheen Varney,
Reagan Varney, Tobey Varney, Haywood Varney, Khristopher Varney, Patricio Varney, Zack Varney, Jamon Varney, Khary Varney, Augustin Varney, Benjiman Varney, Ishmael Varney, Wally Varney, Faustino
Varney, Hamilton Varney, Jami Varney, Vincente Varney, Griffin Varney, Jeramiah Varney, Manny Varney, Santino Varney, Vern Varney, Richardo Varney, Franklyn Varney, Wilmer Varney, Christpher Varney,
Derric Varney, Drake Varney, Lino Varney, Nathen Varney, Ryon Varney, Vernell Varney, Johann Varney, Khalil Varney, Cris Varney, Foster Varney, Gale Varney, Teodoro Varney, Creighton Varney, Farrell
Varney, Jai Varney, Chirstopher Varney, Lavelle Varney, Mikal Varney, Orin Varney, Ceasar Varney, Pietro Varney, Sedric Varney, Sydney Varney, Val Varney, Dawayne Varney, Odis Varney, Wardell Varney,
Murphy Varney, Nevin Varney, Quenton Varney, Santo Varney, Slade Varney, Dawn Varney, Marko Varney, Ricco Varney, Rogers Varney, Abe Varney, Jerimy Varney, Jessy Varney, Montrell Varney, Cassidy
Varney, Hilton Varney, Christen Varney, Kamal Varney, Leron Varney, Miquel Varney, Yosef Varney, Judah Varney, Malcom Varney, Rayford Varney, Arlen Varney, Gaetano Varney, Montez Varney, Kareen
Varney, Dajuan Varney, Marlow Varney, Uriel Varney, Glynn Varney, Bud Varney, Eben Varney, Geronimo Varney, Deshaun Varney, Hernan Varney, Zebulon Varney, Deshon Varney, Gareth Varney, Paolo Varney,
Edison Varney, Gaylon Varney, Shamus Varney, Carleton Varney, Florentino Varney, Osbaldo Varney, Brodie Varney, Jaret Varney, Johathan Varney, Tariq Varney, Seamus Varney, Devan Varney, Richmond
Varney, Cynthia Varney, Cleo Varney, Huey Varney, Talmadge Varney, Dain Varney, Dennie Varney, Paxton Varney, Peyton Varney, Darrius Varney, Jodi Varney, Ronell Varney, Jeremi Varney, Quinten Varney,
Sandra Varney, Waymon Varney, Earle Varney, Franz Varney, Christoher Varney, Micahel Varney, Sal Varney, Sharon Varney, Keegan Varney, Eusebio Varney, Geoffery Varney, Monica Varney, Dontae Varney,
Jimi Varney, Kendric Varney, Amin Varney, Cleon Varney, Kale Varney, Tina Varney, Ambrose Varney, Armond Varney, Ashton Varney, Lucien Varney, Maximo Varney, Reno Varney, Susan Varney, Chico Varney,
Jermel Varney, Lennie Varney, Rajesh Varney, Artie Varney, Kerwin Varney, Sage Varney, Dietrich Varney, Eriberto Varney, Niles Varney, Rolland Varney, Butch Varney, Courtland Varney, King Varney, Kit
Varney, Menachem Varney, Rich Varney, Florencio Varney, Lovell Varney, Jacque Varney, Antwone Varney, Darick Varney, Kennth Varney, Asher Varney, Efrem Varney, Lamarcus Varney, Terron Varney, Jacinto
Varney, Karlos Varney, Mohamed Varney, Rahman Varney, Ammon Varney, Davie Varney, Diallo Varney, Kelton Varney, Mathias Varney, Sunny Varney, Zeke Varney, Jaison Varney, Nate Varney, Demetrious
Varney, Maceo Varney, Vasilios Varney, Demetrio Varney, Lyman Varney, Angus Varney, Arvin Varney, Camron Varney, Daryll Varney, Ehren Varney, Les Varney, Nils Varney, Akil Varney, Markeith Varney,
Santana Varney, Warner Varney, Alphonse Varney, Arnaldo Varney, Art Varney, Domenick Varney, Prentiss Varney, Blas Varney, Dwan Varney, Geno Varney, Kahlil Varney, Feliciano Varney, Hayward Varney,
Ulises Varney, Valentine Varney, Darold Varney, Deven Varney, Mordechai Varney, Rodriquez Varney, Gentry Varney, Jorje Varney, Sunil Varney, Wilburn Varney, Anders Varney, Damone Varney, Sherrod
Varney, Dashawn Varney, Otha Varney, Rahim Varney, Javan Varney, Wendy Varney, Camilo Varney, Jeromie Varney, Jaysen Varney, Modesto Varney, Quincey Varney, Tyran Varney, Cleve Varney, Raynard
Varney, Tori Varney, Antonino Varney, Bartley Varney, Kristofor Varney, Sheridan Varney, Alva Varney, Correy Varney, Dandre Varney, Darek Varney, Wylie Varney, Anand Varney, Decarlos Varney, Domonic
Varney, Rachel Varney, Jamieson Varney, Nikolaos Varney, Rashaan Varney, Roque Varney, Shelly Varney, Sven Varney, Evans Varney, Jakob Varney, Lateef Varney, Noble Varney, Artemio Varney, Kameron
Varney, Kavin Varney, Dick Varney, Filiberto Varney, Henri Varney, Joby Varney, Montgomery Varney, Salomon Varney, Vashon Varney, Abdullah Varney, Cal Varney, Shain Varney, Sherwood Varney, Dewitt
Varney, Newton Varney, Nicklaus Varney, Darvin Varney, Jeron Varney, Khari Varney, Tyrome Varney, Benjie Varney, Kunta Varney, Karsten Varney, Patric Varney, Reece Varney, Olen Varney, Rowdy Varney,
Terrel Varney, Addison Varney, Blaise Varney, Fidencio Varney, Ibrahim Varney, Idris Varney, Lavern Varney, Mickel Varney, Shun Varney, Athanasios Varney, Danilo Varney, Giles Varney, Mateo Varney,
Nino Varney, Alonza Varney, Cardell Varney, Job Varney, Mahlon Varney, Anil Varney, Destry Varney, Lamon Varney, Martez Varney, Mikael Varney, Nickey Varney, Tywan Varney, Abner Varney, Jamell
Varney, Nels Varney, Raoul Varney, Braulio Varney, Randle Varney, Roddy Varney, Toni Varney, Lynwood Varney, Thurston Varney, Chrisopher Varney, Eron Varney, Fausto Varney, Keir Varney, Kelcey
Varney, Marquette Varney, Mikeal Varney, Anselmo Varney, Hipolito Varney, Neville Varney, Rodriguez Varney, Antron Varney, Walton Varney, Devlin Varney, Lucian Varney, Rudolfo Varney, Avi Varney,
Brantley Varney, Dathan Varney, Doran Varney, Erica Varney, Franky Varney, Kary Varney, Tiffany Varney, Yuri Varney, Bucky Varney, Hardy Varney, Justus Varney, Cecilio Varney, Hoyt Varney, Jasson
Varney, Jerardo Varney, Johnson Varney, Kenney Varney, Panagiotis Varney, Ajay Varney, Alvis Varney, Conrado Varney, Mel Varney, Burl Varney, Daymon Varney, Dorsey Varney, Giancarlo Varney, Riccardo
Varney, Antuan Varney, Braxton Varney, Candelario Varney, Colton Varney, Rommel Varney, Irwin Varney, Reynold Varney, Tino Varney, Alessandro Varney, Corie Varney, Dameion Varney, Dwyane Varney,
Jabbar Varney, Lajuan Varney, Mose Varney, Taiwan Varney, Yisroel Varney, Danniel Varney, Darris Varney, Jerred Varney, Lori Varney, Crispin Varney, Maximilian Varney, Skip Varney, Yaakov Varney,
Brodrick Varney, Fabio Varney, Gerrit Varney, Iran Varney, Trace Varney, Amar Varney, Aram Varney, Buster Varney, Byran Varney, Jens Varney, Joseluis Varney, Karriem Varney, Kenan Varney, Tonya
Varney, Trampas Varney, Durand Varney, Lou Varney, Regis Varney, Cain Varney, Christ Varney, Daivd Varney, Marquise Varney, Meredith Varney, Dell Varney, Matthias Varney, Alphonzo Varney, Latroy
Varney, Allison Varney, Darion Varney, Melton Varney, Tara Varney, Boyce Varney, Cletus Varney, Derron Varney, Madison Varney, Tait Varney, Yehuda Varney, Arik Varney, Donato Varney, Duston Varney,
Kimani Varney, Lucious Varney, Nader Varney, Baldemar Varney, Bertrand Varney, Blane Varney, Dujuan Varney, Ellery Varney, Kennedy Varney, Misael Varney, Tremain Varney, Veronica Varney, Christy
Varney, Columbus Varney, Little Varney, Marius Varney, Skyler Varney, Alexandre Varney, Celestino Varney, Crystal Varney, Jordon Varney, Justine Varney, Demon Varney, Derreck Varney, Flavio Varney,
Kennard Varney, Roby Varney, Schuyler Varney, Garrison Varney, Cipriano Varney, Cori Varney, Denise Varney, Keyon Varney, Laverne Varney, Obie Varney, Tige Varney, Gabino Varney, Joselito Varney,
Karlton Varney, Lashon Varney, Lucky Varney, Neill Varney, Shmuel Varney, Ara Varney, Blain Varney, Pascual Varney, Rashan Varney, Serge Varney, Thane Varney, Bradlee Varney, Connor Varney, Cooper
Varney, Jonny Varney, Merrick Varney, Porter Varney, Zeb Varney, April Varney, Edrick Varney, Flint Varney, Heith Varney, Jospeh Varney, Luiz Varney, Macario Varney, Zach Varney, Candido Varney,
Dangelo Varney, Dov Varney, Dru Varney, Eladio Varney, Rasheem Varney, Tarek Varney, Willian Varney, Ezell Varney, Konrad Varney, Mick Varney, Tarrance Varney, Yitzchok Varney, Rahul Varney, Shelley
Varney, Theodis Varney, Timonthy Varney, Connie Varney, Derrik Varney, Ennis Varney, Gail Varney, Gaston Varney, Gunnar Varney, Justice Varney, Romel Varney, Bilal Varney, Jamy Varney, Jomo Varney,
Keary Varney, Nehemiah Varney, Rashon Varney, Rolf Varney, Sameer Varney, Jere Varney, Jerone Varney, Kennon Varney, Rodd Varney, Ronaldo Varney, Teresa Varney, Coley Varney, Enos Varney, Narciso
Varney, Robinson Varney, Tobby Varney, Virgilio Varney, Christiaan Varney, Clarke Varney, Demetruis Varney, Jemal Varney, Llewellyn Varney, Miller Varney, Shilo Varney, Tarus Varney, Young Varney,
Arlie Varney, Djuan Varney, Marcial Varney, Pamela Varney, Rayshawn Varney, Robyn Varney, Tirrell Varney, Dontay Varney, Joshuah Varney, Jovon Varney, Lemar Varney, Oran Varney, Taron Varney, Jaimie
Varney, Kacey Varney, Wendall Varney, Armondo Varney, Curry Varney, Klint Varney, Lauro Varney, Maximillian Varney, Rashard Varney, Anastasios Varney, Jermiah Varney, Joao Varney, Lonnell Varney,
Matias Varney, Rollin Varney, Thimothy Varney, Toma Varney, Jahmal Varney, Jerame Varney, Skye Varney, Alford Varney, Baltazar Varney, Gianni Varney, Hayes Varney, Lamond Varney, Linda Varney,
Neftali Varney, Antwoine Varney, Arden Varney, Dionicio Varney, Ely Varney, Kacy Varney, Syed Varney, Tyrel Varney, Windell Varney, Allyn Varney, Antwane Varney, Dayne Varney, Donal Varney, Maxie
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Nikki Varney, Terell Varney, Ace Varney, Donna Varney, Gerome Varney, Jermane Varney, Joeseph Varney, Raynaldo Varney, Carrie Varney, Demetri Varney, Rupert Varney, Shadrick Varney, Sol Varney,
Wendel Varney, Zechariah Varney, Delon Varney, Demitrius Varney, Mcarthur Varney, Nickie Varney, Tavaris Varney, Baby Varney, Cornelious Varney, Deborah Varney, Haven Varney, Merritt Varney, Vijay
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Geremy Varney, Kobie Varney, Leeroy Varney, Maximino Varney, Orrin Varney, Wayman Varney, Brenda Varney, Celso Varney, Daman Varney, Denzil Varney, Geary Varney, Ladon Varney, Marland Varney,
Servando Varney, Kamau Varney, Migel Varney, Ramond Varney, Terrace Varney, Ubaldo Varney, Chaz Varney, Demetrick Varney, Freddrick Varney, Gray Varney, Ivy Varney, Shlomo Varney, Torre Varney,
Damein Varney, Deryl Varney, Dusten Varney, Katherine Varney, Kreg Varney, London Varney, Patrice Varney, Payton Varney, Ramel Varney, Rosalio Varney, Serjio Varney, Tramaine Varney, Adolphus Varney,
Ameer Varney, Antwaun Varney, Ashish Varney, Benjamine Varney, Charleston Varney, Darry Varney, Elmo Varney, Nicklas Varney, Nikolai Varney, Ricki Varney, Tai Varney, Bernardino Varney, Devaughn
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Roddrick Varney, Shaw Varney, Silvestre Varney, Tavon Varney, Unknown Varney, Daniell Varney, Andrei Varney, Hashim Varney, Hobert Varney, Terris Varney, Aundre Varney, Avrohom Varney, Elden Varney,
Guido Varney, Holly Varney, Marice Varney, Martinez Varney, Daneil Varney, Mattew Varney, Shamar Varney, Shannan Varney, Shomari Varney, Daniele Varney, Darrian Varney, Demetrus Varney, Erron Varney,
Hasani Varney, Robbin Varney, Sabino Varney, Sloan Varney, Zev Varney, Cassius Varney, Dawan Varney, Sekou Varney, Tyjuan Varney, Bentley Varney, Ean Varney, Edric Varney, Gamaliel Varney, Gerold
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Raffaele Varney, Barret Varney, Chadley Varney, Frantz Varney, Gabrial Varney, Geroge Varney, Harland Varney, Obed Varney, Sharod Varney, Silverio Varney, Vernard Varney, Alfie Varney, Darryle
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Varney, Conway Varney, Dempsey Varney, Eligio Varney, Alicia Varney, Jairo Varney, Kathleen Varney, Koby Varney, Nikia Varney, Pierce Varney, Sherard Varney, Spiro Varney, Winford Varney, Alfonza
Varney, Augusto Varney, Christin Varney, Dakota Varney, Gannon Varney, Lucus Varney, Rand Varney, Renardo Varney, Robet Varney, Atiba Varney, Dondi Varney, Jammy Varney, Jarrid Varney, Jonthan
Varney, Marlan Varney, Olando Varney, Salim Varney, Shaine Varney, Arun Varney, Broc Varney, Cash Varney, Cord Varney, Ebony Varney, Gage Varney, Jacobo Varney, Loy Varney, Orlanda Varney, Sammuel
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Varney, Theadore Varney, Yves Varney, Aidan Varney, Antjuan Varney, Decarlo Varney, Fitzgerald Varney, Frances Varney, Jhon Varney, Price Varney, Theophilus Varney, Theresa Varney, Aden Varney,
Alaric Varney, Armon Varney, Johny Varney, Kenith Varney, Mallory Varney, Rondal Varney, Samer Varney, Coty Varney, Gerrod Varney, Maximiliano Varney, Rito Varney, Robie Varney, Skipper Varney, Wali
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Varney, Jonothan Varney, Lannie Varney, Shaka Varney, Arlan Varney, Chas Varney, Conley Varney, Donzell Varney, Jamarr Varney, Karon Varney, Landis Varney, Marie Varney, Marino Varney, Michial
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Kedric Varney, Kimo Varney, Kristan Varney, Kyron Varney, Rayburn Varney, Sandor Varney, Undra Varney, Waymond Varney, Avraham Varney, Brennen Varney, Delaney Varney, Evangelos Varney, Geoffry
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Varney, Philbert Varney, Arlyn Varney, Armin Varney, Bonifacio Varney, Kari Varney, Kofi Varney, Tamara Varney, Wayde Varney, Anna Varney, Cheston Varney, Jarom Varney, Jemel Varney, Jermy Varney,
Jeronimo Varney, Ontario Varney, Roshawn Varney, Adams Varney, Barbara Varney, Darran Varney, Horatio Varney, Jedd Varney, Jeramey Varney, Rajeev Varney, Rendell Varney, Ansel Varney, Apolonio
Varney, Daric Varney, Dewan Varney, Krishna Varney, Lenwood Varney, Lex Varney, Marlowe Varney, Mickael Varney, Octavious Varney, Rashaad Varney, Catherine Varney, Chesley Varney, Natividad Varney,
Ahren Varney, Alvie Varney, Ardell Varney, Gaylen Varney, Iain Varney, Jeffrie Varney, Levy Varney, Naeem Varney, Ramey Varney, Rapheal Varney, Recardo Varney, Welton Varney, Wiliam Varney, Bakari
Varney, Diana Varney, Durrell Varney, Gaspar Varney, Jacqueline Varney, Jewell Varney, Marvell Varney, Noland Varney, Tedrick Varney, Waleed Varney, Aloysius Varney, Amber Varney, Audrey Varney,
Curley Varney, Farris Varney, Gian Varney, Jarmaine Varney, Lazarus Varney, Parish Varney, Stavros Varney, Sultan Varney, Laquan Varney, Londell Varney, Parnell Varney, Ryland Varney, Shem Varney,
Yolanda Varney, Arnie Varney, Brennon Varney, Chan Varney, Dionisio Varney, Frederico Varney, Joy Varney, Juancarlos Varney, Keno Varney, Kwasi Varney, Lael Varney, Laroy Varney, Normand Varney,
Omarr Varney, Tyre Varney, Valente Varney, Gavino Varney, Kade Varney, Kolby Varney, Linton Varney, Malvin Varney, Moishe Varney, Rhonda Varney, Sherry Varney, Verlin Varney, Antoino Varney, Benjy
Varney, Dmitri Varney, Duan Varney, Ford Varney, Hezekiah Varney, Kalani Varney, Kinte Varney, Lacey Varney, Massimo Varney, Ryder Varney, Seann Varney, Valerie Varney, Waldo Varney, Edson Varney,
Eliud Varney, Gustave Varney, Harrell Varney, Henderson Varney, Obadiah Varney, Christipher Varney, Constantinos Varney, Cort Varney, Duran Varney, Eldred Varney, Jennings Varney, Jubal Varney,
Macarthur Varney, Marvis Varney, Ric Varney, Dimas Varney, Emily Varney, Golden Varney, Jessee Varney, Juaquin Varney, Kennie Varney, Lerone Varney, Paige Varney, Shean Varney, Termaine Varney,
Antoni Varney, Connell Varney, Gustav Varney, Meir Varney, Reza Varney, Rollie Varney, Ronney Varney, Sherron Varney, Teron Varney, Theotis Varney, Vittorio Varney, Athony Varney, Burnell Varney,
Crawford Varney, Esau Varney, Gina Varney, Jamahl Varney, Kenyata Varney, Mickeal Varney, Rexford Varney, Rockey Varney, Thadeus Varney, Trevon Varney, Barclay Varney, Chuckie Varney, Early Varney,
Ernst Varney, Fredy Varney, Gildardo Varney, Hyrum Varney, Kerby Varney, Kimball Varney, Marek Varney, Ransom Varney, Shareef Varney, Sid Varney, Tillman Varney, West Varney, Wynn Varney, Danyel
Varney, Eduard Varney, Elmore Varney, Holland Varney, Johannes Varney, Langston Varney, Long Varney, Lyn Varney, Montreal Varney, Rui Varney, Shae Varney, Shondell Varney, Tor Varney, Caine Varney,
Destin Varney, Dionne Varney, Hernando Varney, Isabel Varney, Lamark Varney, Leandre Varney, Nicholus Varney, Nikola Varney, Olaf Varney, Orson Varney, Presley Varney, Randi Varney, Regina Varney,
Rufino Varney, Shariff Varney, Shaune Varney, Torri Varney, Arcadio Varney, Arlando Varney, Carlus Varney, Efraim Varney, Emmet Varney, Etienne Varney, Garvin Varney, Jermine Varney, Larnell Varney,
Michiel Varney, Pieter Varney, Wilber Varney, Christapher Varney, Derk Varney, Enrigue Varney, Isidoro Varney, Kriston Varney, Ladell Varney, Morton Varney, Nash Varney, Nickolaus Varney, Paula
Varney, Sheron Varney, Wellington Varney, Antawn Varney, Brandyn Varney, Christan Varney, Deonte Varney, Elie Varney, Hampton Varney, Jeremias Varney, Marque Varney, Weylin Varney, Antwine Varney,
Bonnie Varney, Chucky Varney, Matthews Varney, Mikell Varney, Padraic Varney, Tarence Varney, Christphor Varney, Cindy Varney, Daven Varney, Dock Varney, Enzo Varney, Gerrick Varney, Jeanpierre
Varney, Jericho Varney, Joan Varney, Kenley Varney, Kentrell Varney, Sherod Varney, Carlous Varney, Deandra Varney, Demar Varney, Dieter Varney, Frazier Varney, Garon Varney, Howell Varney, Joh
Varney, Librado Varney, Melinda Varney, Micha Varney, Steffan Varney, Stevens Varney, Victoriano Varney, Zak Varney, Addam Varney, Arick Varney, Cheryl Varney, Cornel Varney, Cosme Varney, Damani
Varney, Dillard Varney, Edsel Varney, Garner Varney, Gerad Varney, Giacomo Varney, Ismail Varney, Link Varney, Rohit Varney, Alexandros Varney, Brandin Varney, Dolan Varney, Dustyn Varney, Eran
Varney, Imran Varney, Johnthan Varney, Lashun Varney, Linus Varney, Misty Varney, Shaheed Varney, Ann Varney, Antionio Varney, Arie Varney, Aris Varney, Armen Varney, Clancy Varney, Colt Varney,
Dione Varney, Garren Varney, Janson Varney, Justyn Varney, Megan Varney, Niel Varney, Niels Varney, Rockie Varney, Thornton Varney, Turner Varney, Viet Varney, Albino Varney, Alma Varney, Artemus
Varney, Cam Varney, Claud Varney, Daimon Varney, Dequan Varney, Fransico Varney, Ilan Varney, Laurent Varney, Sigmund Varney, Tal Varney, Tray Varney, Apollo Varney, Barak Varney, Brentley Varney,
Cleavon Varney, Delvon Varney, Dyron Varney, Jaye Varney, Jermell Varney, Ladd Varney, Leamon Varney, Nico Varney, Ovidio Varney, Peder Varney, Raven Varney, Teofilo Varney, Trae Varney, Alon Varney,
Charle Varney, Erek Varney, Evin Varney, Graeme Varney, Jerimie Varney, Juanito Varney, Keoni Varney, Kwan Varney, Lathan Varney, Leonidas Varney, Marrio Varney, Mayer Varney, Ozzie Varney, Rajiv
Varney, Rami Varney, Shann Varney, Teague Varney, Terran Varney, Branson Varney, Briant Varney, Christohper Varney, Delmas Varney, Demone Varney, Duffy Varney, Grabiel Varney, Hudson Varney, Jae
Varney, Jefrey Varney, Kali Varney, Klinton Varney, Markel Varney, Saverio Varney, Steffen Varney, Akbar Varney, Brandi Varney, Desmon Varney, Detric Varney, Doron Varney, Georgie Varney, Hilary
Varney, Isadore Varney, Kawika Varney, Keefe Varney, Montell Varney, Prentis Varney, Rishi Varney, Zacharia Varney, Adriano Varney, Arash Varney, Chancy Varney, Deshun Varney, Donnel Varney,
Johndavid Varney, Jonn Varney, Kerrick Varney, Ruperto Varney, Silvio Varney, Smith Varney, Alen Varney, Damarcus Varney, Dixon Varney, Isacc Varney, Landry Varney, Laurie Varney, Lennard Varney,
Mckenzie Varney, Rakesh Varney, Selwyn Varney, Torris Varney, Yohance Varney, Zakary Varney, Alison Varney, Baxter Varney, Bienvenido Varney, Christon Varney, Cosmo Varney, Cuauhtemoc Varney, Drue
Varney, Ferris Varney, Jensen Varney, Jerime Varney, Kathryn Varney, Kingsley Varney, Prescott Varney, Rayfield Varney, Sanders Varney, Tarvis Varney, Bernell Varney, Calvert Varney, Channon Varney,
Clovis Varney, Duron Varney, Dushawn Varney, Elam Varney, Elwin Varney, Johnmichael Varney, Lawton Varney, Linden Varney, Meyer Varney, Raheen Varney, Santonio Varney, Shonn Varney, Vikram Varney,
Wesly Varney, Yoel Varney, Ysidro Varney, Asim Varney, Authur Varney, Cyle Varney, General Varney, Page Varney, Tab Varney, Tamar Varney, Thaddaeus Varney, Tyshawn Varney, Akiva Varney, Alexi Varney,
Dannon Varney, Draper Varney, Garet Varney, Immanuel Varney, Julia Varney, Manfred Varney, Monique Varney, Monta Varney, Nile Varney, Rome Varney, Tywon Varney, Verne Varney, Amando Varney, Askia
Varney, Cannon Varney, Chavis Varney, Davide Varney, Deepak Varney, Garold Varney, Jabar Varney, Japheth Varney, Klaus Varney, Naftali Varney, Ora Varney, Ronn Varney, Shandon Varney, Toussaint
Varney, Travers Varney, Verlon Varney, Antwann Varney, Barnaby Varney, Bretton Varney, Cesario Varney, Georgios Varney, Jajuan Varney, Lisandro Varney, Maurizio Varney, Shakir Varney, Tavarus Varney,
Waverly Varney, Chaun Varney, Diane Varney, Dontrell Varney, Jamiel Varney, Jashua Varney, Leopold Varney, Lynell Varney, Montie Varney, Rafe Varney, Reginold Varney, Romulo Varney, Thayne Varney,
Tierre Varney, Wilhelm Varney, Akim Varney, Baldomero Varney, Darrien Varney, Durell Varney, Eden Varney, Gergory Varney, Kerri Varney, Lennon Varney, Margaret Varney, Nikita Varney, Noam Varney,
Rajan Varney, Reynard Varney, Sachin Varney, Seferino Varney, Ventura Varney, Warrick Varney, Washington Varney, Antonius Varney, Barrington Varney, Billyjoe Varney, Christo Varney, Derald Varney,
Dewight Varney, Ferrell Varney, Filippo Varney, Garin Varney, Herbie Varney, Joachim Varney, Khalif Varney, Mikey Varney, Severo Varney, Tahir Varney, Taras Varney, Vinay Varney, Aharon Varney,
Antiono Varney, Apolinar Varney, Arsenio Varney, Furman Varney, Gayle Varney, Girard Varney, Grey Varney, Hillary Varney, Martel Varney, Murry Varney, Nabil Varney, Neel Varney, Ozell Varney, Rylan
Varney, Samantha Varney, Tellis Varney, Thompson Varney, Varian Varney, Antoinne Varney, Derrel Varney, Erika Varney, Faisal Varney, Kristina Varney, Laird Varney, Lancer Varney, Marcelle Varney,
Melvyn Varney, Sabin Varney, Tajuan Varney, Terrick Varney, Thadius Varney, Timoteo Varney, Treavor Varney, Tyrese Varney, Victoria Varney, Vishal Varney, Aran Varney, Christoffer Varney, Devron
Varney, Durwin Varney, Elon Varney, Hansel Varney, Haskell Varney, Jerell Varney, Jonathen Varney, Lanier Varney, Marcellous Varney, Marquez Varney, Regino Varney, Skylar Varney, Christobal Varney,
Claudia Varney, Dartagnan Varney, Errin Varney, Hung Varney, Juvenal Varney, Kaseem Varney, Keri Varney, Marwan Varney, Matteo Varney, Raymone Varney, Rennie Varney, Rondale Varney, Shadd Varney, Yul
Varney, Arnel Varney, Concepcion Varney, Daran Varney, Flynn Varney, Jawara Varney, Jeremia Varney, Karry Varney, Rodric Varney, Romon Varney, Aren Varney, Bevan Varney, Casper Varney, Collis Varney,
Delane Varney, Erric Varney, Gerhard Varney, Hillel Varney, Jansen Varney, Jenny Varney, Keelan Varney, Michell Varney, Petros Varney, Saeed Varney, Shamon Varney, Starsky Varney, Tamir Varney, Umar
Varney, Verdell Varney, Vicent Varney, Alpha Varney, Arther Varney, Case Varney, Cavin Varney, Deacon Varney, Franciso Varney, Freedom Varney, Gregary Varney, Harlen Varney, Ichael Varney, Jacy
Varney, Jeremaine Varney, Jud Varney, Judge Varney, Monico Varney, Raffi Varney, Ricci Varney, Sacha Varney, Adelbert Varney, Andree Varney, Bron Varney, Cisco Varney, Creed Varney, Criag Varney,
Edilberto Varney, Gillermo Varney, Hobart Varney, Justino Varney, Kodi Varney, Martino Varney, Mickie Varney, Newell Varney, Odie Varney, Robbi Varney, Thaddius Varney, Tonny Varney, Tully Varney,
Werner Varney, Zoltan Varney, Carrol Varney, Cesare Varney, Clarance Varney, Clem Varney, Davion Varney, Dvid Varney, Graydon Varney, Latrell Varney, Manual Varney, Pervis Varney, Randolf Varney,
Sinclair Varney, Stephane Varney, Torian Varney, Tylor Varney, Tzvi Varney, Umberto Varney, Baruch Varney, Benard Varney, Glenwood Varney, Jaun Varney, Jermie Varney, Johnney Varney, Kenard Varney,
Kyler Varney, Luca Varney, Martha Varney, Nephi Varney, Nicholos Varney, Nikhil Varney, Olan Varney, Sir Varney, Stevon Varney, Teon Varney, Tryone Varney, Vanessa Varney, Ames Varney, Breon Varney,
Gregroy Varney, Heyward Varney, Jebediah Varney, Kye Varney, Larue Varney, Lonell Varney, Nevada Varney, Norma Varney, Renny Varney, Rosa Varney, Ana Varney, Blue Varney, Cresencio Varney, Danon
Varney, Debra Varney, Denard Varney, Deondre Varney, Elzie Varney, Gamal Varney, Gerrard Varney, Heber Varney, Jerremy Varney, Jobe Varney, Jun Varney, Karey Varney, Marquel Varney, Martell Varney,
Marvel Varney, Ocie Varney, Raudel Varney, Romaine Varney, Rush Varney, Ruston Varney, Talib Varney, Therman Varney, Vann Varney, Vaughan Varney, Wilberto Varney, Albin Varney, Antwuan Varney, Aramis
Varney, Bodie Varney, Darrion Varney, Diron Varney, Issa Varney, Izaak Varney, Javis Varney, Jerami Varney, Keane Varney, Kjell Varney, Laurance Varney, Marico Varney, Philipp Varney, Placido Varney,
Remy Varney, Rion Varney, Sanchez Varney, Shahid Varney, Valdez Varney, Acie Varney, Aundrey Varney, Buckley Varney, Eliyahu Varney, Erroll Varney, Leotis Varney, Levern Varney, Mace Varney, Perrin
Varney, Phillipe Varney, Rama Varney, Steward Varney, Tawan Varney, Zvi Varney, Carlon Varney, Daris Varney, Deandrea Varney, Dominque Varney, Gill Varney, Keola Varney, Larone Varney, Lew Varney,
Luz Varney, Montel Varney, Sylvan Varney, Tobi Varney, Twan Varney, Ashraf Varney, Avram Varney, Benzion Varney, Casimiro Varney, Chane Varney, Christie Varney, Chritopher Varney, Cirilo Varney,
Constantino Varney, Dupree Varney, Gregor Varney, Jathan Varney, Keaton Varney, Kostas Varney, Larenzo Varney, Parag Varney, Rudi Varney, Sameul Varney, Sanjeev Varney, Soloman Varney, Tarrell
Varney, Traci Varney, Tracie Varney, Watson Varney, Zoran Varney, Aries Varney, Aristotle Varney, Avelino Varney, Bram Varney, Camden Varney, Dagan Varney, Damin Varney, Danta Varney, Eliott Varney,
Felicia Varney, Juwan Varney, Kemp Varney, Kenn Varney, Kenyatte Varney, Michail Varney, Orval Varney, Salvadore Varney, Shirley Varney, Boone Varney, Dionte Varney, Durwood Varney, Ellison Varney,
Garron Varney, Gerson Varney, Jerrald Varney, Lavaughn Varney, Musa Varney, Rayvon Varney, Richrd Varney, Sharrod Varney, Stony Varney, Vikas Varney, Ector Varney, Elia Varney, Jerret Varney, Josua
Varney, Marlen Varney, Mical Varney, Niall Varney, Nicolaus Varney, Niki Varney, Raymund Varney, Simone Varney, Stormy Varney, Suzanne Varney, Teddie Varney, Tron Varney, Wolfgang Varney, Anthonio
Varney, Berton Varney, Beverly Varney, Clive Varney, Corky Varney, Danell Varney, Darl Varney, Ellsworth Varney, Gloria Varney, Gregery Varney, Ivey Varney, Kam Varney, Kenderick Varney, Magdaleno
Varney, Markee Varney, San Varney, Stacie Varney, Valdemar Varney, Zac Varney, Albaro Varney, Berkley Varney, Boe Varney, Casimir Varney, Delroy Varney, Janet Varney, Jimmey Varney, Kalen Varney,
Kregg Varney, Larkin Varney, Lenell Varney, Natalie Varney, Ruel Varney, Sabrina Varney, Thom Varney, Yale Varney, Adnan Varney, Ankur Varney, Arnett Varney, Bishop Varney, Brently Varney, Carla
Varney, Courtenay Varney, Dashon Varney, Ebon Varney, Edd Varney, Ezzard Varney, Gifford Varney, Jermon Varney, Jeryl Varney, Jock Varney, Kiven Varney, Mandrell Varney, Nichols Varney, Norwood
Varney, Olivier Varney, Ramin Varney, Rikki Varney, Aleksandar Varney, Burley Varney, Kawan Varney, Kourtney Varney, Lary Varney, Laszlo Varney, Loyal Varney, Marin Varney, Richey Varney, Rio Varney,
Rogerio Varney, Salem Varney, Shalom Varney, Sharron Varney, Tamer Varney, Torie Varney, Antar Varney, Barrie Varney, Cletis Varney, Colter Varney, Duaine Varney, Elpidio Varney, Erice Varney, Farid
Varney, Fortunato Varney, Gerrad Varney, Jad Varney, Jenaro Varney, Jones Varney, Kamran Varney, Kelli Varney, Keneth Varney, Kortney Varney, Levell Varney, Minh Varney, Monti Varney, Rodell Varney,
Ronal Varney, Sheddrick Varney, Sonia Varney, Terryl Varney, York Varney, Andrian Varney, Anne Varney, Attila Varney, Avis Varney, Blaze Varney, Chi Varney, Corin Varney, Dann Varney, Darik Varney,
Deleon Varney, Demarko Varney, Dennison Varney, Dovid Varney, Dutch Varney, Emmit Varney, Evert Varney, Iman Varney, Joshue Varney, Lindy Varney, Mohamad Varney, Purnell Varney, Rayshon Varney,
Tyreese Varney, Alistair Varney, Arman Varney, Blakely Varney, Brit Varney, Carlis Varney, Doniel Varney, Earnie Varney, Hershell Varney, Jasin Varney, Kendale Varney, Laine Varney, Leandrew Varney,
Lenin Varney, Macon Varney, Merrell Varney, Mychal Varney, Nat Varney, Nicanor Varney, Nirav Varney, Waldemar Varney, Yakov Varney, Adrion Varney, Bowen Varney, Carolyn Varney, Creg Varney, Farley
Varney, Favian Varney, Ferman Varney, Hurley Varney, Jairus Varney, Kevyn Varney, Korry Varney, Mandell Varney, Melchor Varney, Osualdo Varney, Shanti Varney, Sheila Varney, Sotirios Varney, Taft
Varney, Tijuan Varney, Antonyo Varney, Antwoin Varney, Arley Varney, Carlisle Varney, Christerpher Varney, Derell Varney, Dimitrius Varney, Fabrizio Varney, Gibran Varney, Jasn Varney, June Varney,
Kenta Varney, Kenyetta Varney, Lantz Varney, Mart Varney, Naaman Varney, Nasser Varney, Norvell Varney, Ole Varney, Perfecto Varney, Philemon Varney, Primitivo Varney, Quint Varney, Rony Varney,
Shanta Varney, Tighe Varney, Trini Varney, Antionne Varney, Antwaine Varney, Bradon Varney, Campbell Varney, Chao Varney, Donel Varney, Isa Varney, Kendra Varney, Kiel Varney, Lejon Varney, Lonn
Varney, Micael Varney, Namon Varney, Perez Varney, Ramses Varney, Renwick Varney, Rigo Varney, Rober Varney, Rondall Varney, Terral Varney, Tobe Varney, Akili Varney, Dalen Varney, Deanthony Varney,
Elson Varney, Fulton Varney, Hillard Varney, Jedadiah Varney, Jihad Varney, Keevin Varney, Kile Varney, Kito Varney, Kwane Varney, Lamount Varney, Lanell Varney, Leith Varney, Marshawn Varney, Marwin
Varney, Raymont Varney, Rumaldo Varney, Tabari Varney, Tex Varney, Yechiel Varney, Chirag Varney, Crist Varney, Damain Varney, Damar Varney, Deke Varney, Earlie Varney, Eleuterio Varney, Garen
Varney, Joanthan Varney, Josephus Varney, Marisol Varney, Phillips Varney, Priest Varney, Romie Varney, Sonya Varney, Sung Varney, Tandy Varney, Traves Varney, Augusta Varney, Avon Varney, Daemon
Varney, Danie Varney, Delwyn Varney, Eldrick Varney, Gerod Varney, Hannibal Varney, Harper Varney, Javiel Varney, Jawan Varney, Kathy Varney, Kiran Varney, Linn Varney, Manolito Varney, Remus Varney,
Salome Varney, Savalas Varney, Shamel Varney, Sigfredo Varney, Tylon Varney, Yehoshua Varney, Damean Varney, Gualberto Varney, Guiseppe Varney, Huy Varney, Jaymes Varney, Jaymie Varney, Jeshua
Varney, Maher Varney, Mahmoud Varney, Otoniel Varney, Radford Varney, Rodman Varney, Tevis Varney, Timohty Varney, Tripp Varney, Wells Varney, Allie Varney, Ason Varney, Beth Varney, Carlas Varney,
Cederick Varney, Colleen Varney, Cortland Varney, Deanna Varney, Farron Varney, Gayland Varney, Jaeson Varney, Jamerson Varney, Kenon Varney, Keoki Varney, Kevon Varney, Kristy Varney, Ladale Varney,
Mare Varney, Maurilio Varney, Niko Varney, Nikos Varney, Orland Varney, Remigio Varney, Rhys Varney, Ruby Varney, Simcha Varney, Thatcher Varney, Travas Varney, Zachry Varney, Zenon Varney, Amish
Varney, Anthon Varney, Aswad Varney, Erickson Varney, Fernandez Varney, Finley Varney, Gorden Varney, Ignatius Varney, Illya Varney, Jerico Varney, Jermayne Varney, Kamron Varney, Keita Varney,
Kwabena Varney, Lonzell Varney, Manley Varney, Markanthony Varney, Mika Varney, Muhammed Varney, Nicholes Varney, Percival Varney, Quin Varney, Radley Varney, Ruth Varney, Sabas Varney, Shaft Varney,
Shante Varney, Sullivan Varney, Waylan Varney, Westly Varney, Woodie Varney, Algie Varney, Arvel Varney, Benedetto Varney, Bradrick Varney, Chancellor Varney, Chawn Varney, Daylon Varney, Devonne
Varney, Galvin Varney, Herb Varney, Jacen Varney, Jadon Varney, Janice Varney, Kristi Varney, Leeland Varney, Merton Varney, Ossie Varney, Patsy Varney, Reynolds Varney, Sebastiano Varney, Seymour
Varney, Shanan Varney, Shawndell Varney, Terrie Varney, Toure Varney, Zakee Varney, Zebulun Varney, Alexei Varney, Branon Varney, Derrin Varney, Deryck Varney, Edelmiro Varney, Felice Varney, Hamid
Varney, Higinio Varney, Jule Varney, Kellie Varney, Kern Varney, Khaled Varney, Marciano Varney, Morrell Varney, Nazario Varney, Oneil Varney, Said Varney, Silvester Varney, Sotero Varney, Yvonne
Varney, Adron Varney, Alando Varney, Anothony Varney, Brando Varney, Brenten Varney, Canaan Varney, Colon Varney, Dany Varney, Fenton Varney, Griffith Varney, Halbert Varney, Johney Varney, Karlo
Varney, Keone Varney, Leviticus Varney, Maurio Varney, Murad Varney, Nahum Varney, Vester Varney, Anita Varney, Archibald Varney, Brinton Varney, Cayce Varney, Craige Varney, Damont Varney, Heinz
Varney, Joaquim Varney, Lajuane Varney, Lakendrick Varney, Lanard Varney, Lesly Varney, Nam Varney, Remi Varney, Semaj Varney, Shadrach Varney, Sholom Varney, Silvano Varney, Timothey Varney, Vernal
Varney, Vernie Varney, Alix Varney, Aristeo Varney, Bubba Varney, Carlie Varney, Ceaser Varney, Dallin Varney, Darrow Varney, Deondra Varney, Deonta Varney, Engelbert Varney, Jabier Varney, Jarron
Varney, Kekoa Varney, Lasean Varney, Manolo Varney, Marqus Varney, Marshell Varney, Menno Varney, Ram Varney, Raynell Varney, Rohan Varney, Amadeo Varney, Arch Varney, Arlis Varney, Ayinde Varney,
Barnard Varney, Darriel Varney, Darvis Varney, Derryl Varney, Dewain Varney, Dietrick Varney, Glyn Varney, Jasmine Varney, Kenyan Varney, Michae Varney, Montoya Varney, Oracio Varney, Roney Varney,
Sascha Varney, Shermaine Varney, Sigifredo Varney, Theodor Varney, Virginia Varney, Waco Varney, Yaphet Varney, Zebadiah Varney, Alika Varney, Aundrea Varney, Brendt Varney, Carvin Varney,
Christophere Varney, Clevon Varney, Drayton Varney, Evelio Varney, Gerasimos Varney, Jemaine Varney, Kian Varney, Lucan Varney, Moise Varney, Richy Varney, Tyshon Varney, Zebedee Varney, Alcides
Varney, Allister Varney, Altonio Varney, Bengie Varney, Cavan Varney, Desiderio Varney, Dewon Varney, Eulogio Varney, Faheem Varney, Georges Varney, Hasson Varney, Hollie Varney, Izell Varney, Jamee
Varney, Jestin Varney, Keldrick Varney, Kier Varney, Lorn Varney, Mister Varney, Niraj Varney, Omega Varney, Osman Varney, Ragan Varney, Raja Varney, Talbert Varney, Thanh Varney, Triston Varney,
Vashawn Varney, Victorio Varney, Aubry Varney, Audwin Varney, Babatunde Varney, Bejamin Varney, Cantrell Varney, Carman Varney, Chaddrick Varney, Claiborne Varney, Creston Varney, Deshone Varney,
Dewane Varney, Gabriele Varney, Hason Varney, Jessi Varney, Keenon Varney, Keithan Varney, Kostantinos Varney, Leah Varney, Ludwig Varney, Maverick Varney, Neilson Varney, Octaviano Varney, Orestes
Varney, Otho Varney, Parry Varney, Percell Varney, Rashod Varney, Rochelle Varney, Rondel Varney, Rondey Varney, Rowan Varney, Sang Varney, Shawnta Varney, Sulaiman Varney, Suresh Varney, Taran
Varney, Tatum Varney, Venson Varney, Agustine Varney, Babak Varney, Biagio Varney, Cedar Varney, Christorpher Varney, Cloyd Varney, Corinthian Varney, Curran Varney, Dani Varney, Dannell Varney,
Dayle Varney, Dermot Varney, Encarnacion Varney, Hendrick Varney, Imani Varney, Jackey Varney, Keion Varney, Kennis Varney, Langdon Varney, Lawayne Varney, Leticia Varney, Levan Varney, Liberty
Varney, Mardy Varney, Nicolo Varney, Nolen Varney, Octavian Varney, Purvis Varney, Reuven Varney, Ronrico Varney, Tharon Varney, Turhan Varney, Tyren Varney, Wess Varney, Aman Varney, Amiel Varney,
Apostolos Varney, Clayborn Varney, Devone Varney, Genesis Varney, Giorgio Varney, Jaleel Varney, Jarmar Varney, Jasan Varney, Jermar Varney, Jojo Varney, Jondavid Varney, Martyn Varney, Melecio
Varney, Michelangelo Varney, Mikkel Varney, Montague Varney, Nicki Varney, Quenten Varney, Ramie Varney, Rudolpho Varney, Shenandoah Varney, Tilden Varney, Tosh Varney, Zedrick Varney, Able Varney,
Aldon Varney, Alejo Varney, Atif Varney, Cason Varney, Daiel Varney, Davian Varney, Donaciano Varney, Dywane Varney, Eliberto Varney, Ephriam Varney, Galo Varney, Giulio Varney, Hanif Varney, Harlon
Varney, Hermon Varney, Homar Varney, Jhonny Varney, Jonnathan Varney, Kara Varney, Keath Varney, Kennieth Varney, Kirsten Varney, Kriss Varney, Lennis Varney, Leonides Varney, Levelle Varney, Marlyn
Varney, Merced Varney, Montrel Varney, Naveen Varney, Newman Varney, Osiris Varney, Raymondo Varney, Renzo Varney, Royden Varney, Shawnn Varney, Shuan Varney, Thurmond Varney, Treg Varney, Tremell
Varney, Trenten Varney, Aldric Varney, Alexandra Varney, Alston Varney, Anish Varney, Anuj Varney, Arif Varney, Ashby Varney, Ashok Varney, Chavez Varney, Cinque Varney, Derris Varney, Donovon
Varney, Doroteo Varney, Fontaine Varney, Hadley Varney, Jeanclaude Varney, Josha Varney, Kenyada Varney, Laban Varney, Neeraj Varney, Nilesh Varney, Oral Varney, Quran Varney, Raheim Varney, Rockford
Varney, Rondy Varney, Sesar Varney, Thayer Varney, Vincient Varney, Amilcar Varney, Antion Varney, Antrone Varney, Ardis Varney, Arvil Varney, Avid Varney, Branton Varney, Carmon Varney, Claudell
Varney, Deion Varney, Dejon Varney, Delonte Varney, Delvecchio Varney, Dickson Varney, Elwyn Varney, Fredie Varney, Garnet Varney, Gunther Varney, Jawanza Varney, Jermone Varney, Katrina Varney,
Krista Varney, Landy Varney, Levester Varney, Levin Varney, Rahiem Varney, Rashun Varney, Rinaldo Varney, Ronson Varney, Terril Varney, Ambrosio Varney, Arvid Varney, Atlee Varney, Calogero Varney,
Cobey Varney, Donathan Varney, Donelle Varney, Dorman Varney, Dung Varney, Garcia Varney, Jase Varney, Jeanette Varney, Kolin Varney, Kurtiss Varney, Lorenz Varney, Mathis Varney, Natale Varney,
Patton Varney, Petar Varney, Richar Varney, Thadd Varney, Timithy Varney, Tomislav Varney, Tommaso Varney, Tyris Varney, Uvaldo Varney, Anothy Varney, Aristides Varney, Arno Varney, Deshannon Varney,
Eamonn Varney, Hiawatha Varney, Jamol Varney, Jasun Varney, Joedy Varney, Lenn Varney, Lord Varney, Manoj Varney, Merl Varney, Michaelangelo Varney, Navin Varney, Per Varney, Peterson Varney, Sherri
Varney, Simmie Varney, Socrates Varney, Tramell Varney, Trevino Varney, Winslow Varney, Andrell Varney, Arlington Varney, Austen Varney, Aziz Varney, Benuel Varney, Cristofer Varney, Delrico Varney,
Dereke Varney, Edmon Varney, Finis Varney, Frederich Varney, Hussein Varney, Jerman Varney, Johnathen Varney, Joni Varney, Kariem Varney, Kasim Varney, Klayton Varney, Lafe Varney, Lamario Varney,
Manning Varney, Markos Varney, Natasha Varney, Nicholis Varney, Nickolaos Varney, Nima Varney, Nitin Varney, Piero Varney, Robertson Varney, Sabastian Varney, Sandon Varney, Shandy Varney, Sylvia
Varney, Tacuma Varney, Tarance Varney, Tarl Varney, Traver Varney, Ander Varney, Bartolo Varney, Brentt Varney, Chace Varney, Chay Varney, Clete Varney, Colbert Varney, Domonick Varney, Dondre
Varney, Emanuele Varney, Enoc Varney, Hercules Varney, Inocencio Varney, Ivery Varney, Jaja Varney, Jalal Varney, Jamelle Varney, Jin Varney, Jumaane Varney, Kendel Varney, Kurk Varney, Lendell
Varney, Linc Varney, Louise Varney, Luc Varney, Mackie Varney, Mehul Varney, Myrick Varney, Nasir Varney, Omero Varney, Osiel Varney, Rhyan Varney, Roni Varney, Roshan Varney, Sander Varney, Thai
Varney, Toy Varney, Willia Varney, Zacharias Varney, Zenas Varney, Abelino Varney, Ameen Varney, Bridget Varney, Claudius Varney, Creig Varney, Danton Varney, Danuel Varney, Durward Varney, Duval
Varney, Fredrico Varney, Gibson Varney, Hoang Varney, Huston Varney, Ildefonso Varney, Jamaar Varney, Jerrard Varney, Jsaon Varney, Korie Varney, Lang Varney, Latron Varney, Lawernce Varney, Lief
Varney, Manu Varney, Mervyn Varney, Mingo Varney, Nadeem Varney, Nunzio Varney, Odin Varney, Pinchas Varney, Quoc Varney, Ramy Varney, Randol Varney, Reginaldo Varney, Tam Varney, Theodoros Varney,
Vinnie Varney, Arland Varney, Atul Varney, Baudelio Varney, Bernhard Varney, Chadric Varney, Charon Varney, Damaris Varney, Derin Varney, Donavin Varney, Feliz Varney, Ferdinando Varney, Fortino
Varney, Geron Varney, Gershon Varney, Hanson Varney, Helen Varney, Hodari Varney, Jamen Varney, Jayce Varney, Kalonji Varney, Kayle Varney, Lamorris Varney, Lennox Varney, Mannix Varney, Minor
Varney, Nathaneal Varney, Olegario Varney, Pantelis Varney, Princeton Varney, Quince Varney, Rashied Varney, Rose Varney, Shaye Varney, Son Varney, Stephone Varney, Trenell Varney, Ulric Varney,
Arion Varney, Burgess Varney, Chadwin Varney, Collier Varney, Derico Varney, Egan Varney, Foy Varney, Jacobi Varney, Jemar Varney, Kay Varney, Lake Varney, Markey Varney, Micaiah Varney, Nasario
Varney, Oakley Varney, Remo Varney, Richad Varney, Ringo Varney, Romano Varney, Thierry Varney, Vergil Varney, Vinton Varney, Ajamu Varney, Andrzej Varney, Annette Varney, Asif Varney, Aundray
Varney, Benn Varney, Bernice Varney, Byrant Varney, Clemon Varney, Cleotha Varney, Cully Varney, Darious Varney, Diondre Varney, Donivan Varney, Fard Varney, Gianfranco Varney, Gyasi Varney, Hosie
Varney, Inez Varney, Jaden Varney, Julious Varney, Junious Varney, Ketan Varney, Kieron Varney, Loring Varney, Noal Varney, Pepe Varney, Pharoah Varney, Pilar Varney, Raquel Varney, Remon Varney,
Sederick Varney, Severiano Varney, Shawna Varney, Shawnee Varney, Sim Varney, Spurgeon Varney, Stace Varney, Taber Varney, Tarron Varney, Thorin Varney, Wil Varney, Zaire Varney, Andray Varney,
Berlin Varney, Betty Varney, Biran Varney, Chaney Varney, Chon Varney, Dearl Varney, Demetria Varney, Dewaine Varney, Edan Varney, Ediberto Varney, Froilan Varney, Henrique Varney, Jamille Varney,
Jessey Varney, Kaine Varney, Kendricks Varney, Kynan Varney, Laray Varney, Laren Varney, Mandy Varney, Marilyn Varney, Neale Varney, Nyle Varney, Okey Varney, Ramesh Varney, Ricard Varney, Saladin
Varney, Sherrill Varney, Shonta Varney, Sione Varney, Steaven Varney, Stefon Varney, Taji Varney, Toris Varney, Travell Varney, Urbano Varney, Wolf Varney, Alvah Varney, Ante Varney, Arlon Varney,
Auther Varney, Bran Varney, Chevy Varney, Cormac Varney, Damione Varney, Delando Varney, Deno Varney, Dow Varney, Earvin Varney, Erinn Varney, Florian Varney, Georg Varney, Gerren Varney, Henery
Varney, Isac Varney, Jamond Varney, Jaramie Varney, Kinta Varney, Kylan Varney, Lashan Varney, Mckay Varney, Molly Varney, Ngai Varney, Nolberto Varney, Quang Varney, Solon Varney, Tobiah Varney,
Tricia Varney, Wai Varney, Willem Varney, Yaron Varney, Agostino Varney, Alfonse Varney, Antonious Varney, Antwion Varney, Bryen Varney, Camille Varney, Carsten Varney, Cheo Varney, Chevis Varney,
Colen Varney, Corneilus Varney, Davidson Varney, Dawon Varney, Deldrick Varney, Dia Varney, Dimitris Varney, Donnelle Varney, Dory Varney, Ewell Varney, Hai Varney, Heron Varney, Hope Varney, Imari
Varney, Jaremy Varney, Lindon Varney, Loreto Varney, Nichole Varney, Nieves Varney, Phelan Varney, Randale Varney, Rik Varney, Saturnino Varney, Schawn Varney, Sterlin Varney, Talmage Varney, Tavoris
Varney, Toribio Varney, Trebor Varney, Vick Varney, Worth Varney, Zaid Varney, Adin Varney, Alim Varney, Artez Varney, Bashir Varney, Dameian Varney, Demorris Varney, Deryk Varney, Estill Varney,
Gautam Varney, Hyman Varney, Isom Varney, Jeral Varney, Juanita Varney, Kalman Varney, Kee Varney, Keithen Varney, Kipling Varney, Lesean Varney, Love Varney, Nadir Varney, Norton Varney, Philander
Varney, Sebastien Varney, Sherif Varney, Tabitha Varney, Webb Varney, Akin Varney, Blu Varney, Calbert Varney, Cassandra Varney, Cephus Varney, Dalvin Varney, Daxton Varney, Delfin Varney, Drexel
Varney, Elijio Varney, Fareed Varney, Geffrey Varney, Jabin Varney, Jodey Varney, Jomar Varney, Judy Varney, Jujuan Varney, Klay Varney, Kyriakos Varney, Laderrick Varney, Landen Varney, Latoya
Varney, Lebron Varney, Mat Varney, Rashi Varney, Roberta Varney, Rodderick Varney, Shant Varney, Summer Varney, Viktor Varney, Abayomi Varney, Adrienne Varney, Akeem Varney, Amaury Varney, Andrez
Varney, Antuane Varney, Barnabas Varney, Corneluis Varney, Delray Varney, Demarlo Varney, Deshan Varney, Dev Varney, Diarra Varney, Duriel Varney, Emigdio Varney, Eon Varney, Evelyn Varney, Hasaan
Varney, Janmichael Varney, Jeston Varney, Jobie Varney, Kalon Varney, Kumar Varney, Lemond Varney, Raimundo Varney, Rainier Varney, Robbert Varney, Rulon Varney, Skeeter Varney, Starling Varney, Sun
Varney, Tammie Varney, Terrelle Varney, Tiwan Varney, Toddrick Varney, Varick Varney, Zavier Varney, Zion Varney, Aniceto Varney, Asmar Varney, Boaz Varney, Cayetano Varney, Dakarai Varney, Deadrick
Varney, Debbie Varney, Dominik Varney, Dugan Varney, Emad Varney, Glennon Varney, Haralambos Varney, Harlin Varney, Hilliard Varney, Hristopher Varney, Humphrey Varney, Javar Varney, Javin Varney,
Jeffre Varney, Josuha Varney, Kashif Varney, Keino Varney, Kemper Varney, Kendrix Varney, Keshawn Varney, Lamone Varney, Lashone Varney, Lekeith Varney, Markham Varney, Maruice Varney, Miguelangel
Varney, Nichalos Varney, Olanda Varney, Rodrigues Varney, Tarig Varney, Tarius Varney, Tonnie Varney, Velton Varney, Walden Varney, Yasin Varney, Alexie Varney, Autry Varney, Burnett Varney, Burnis
Varney, Cephas Varney, Crandall Varney, Curvin Varney, Dashaun Varney, Donyel Varney, Emir Varney, Eston Varney, Farrel Varney, Giovani Varney, Jamaul Varney, Jarmel Varney, Jarold Varney, Jeorge
Varney, Joshwa Varney, Jousha Varney, Joyce Varney, Justan Varney, Kem Varney, Lavance Varney, Ledell Varney, Manuelito Varney, Michaell Varney, Mischa Varney, Osborne Varney, Park Varney, Partick
Varney, Rasaan Varney, Ravon Varney, Romone Varney, Stefen Varney, Veron Varney, Vondell Varney, Airrion Varney, Aki Varney, Alegandro Varney, Author Varney, Autumn Varney, Brannan Varney, Caliph
Varney, Carles Varney, Cartez Varney, Codey Varney, Dammon Varney, Daunte Varney, Demonte Varney, Devell Varney, Domonique Varney, Donyale Varney, Dornell Varney, Duwan Varney, Edvardo Varney,
Emeterio Varney, Gerado Varney, Gordan Varney, Jahi Varney, Jair Varney, Jasmin Varney, Jaydee Varney, Judas Varney, Kennan Varney, Lehman Varney, Macy Varney, Mayo Varney, Mazen Varney, Mikhail
Varney, Osama Varney, Rainer Varney, Raman Varney, Rocio Varney, Ryen Varney, Shayn Varney, Tee Varney, Temple Varney, Timathy Varney, Vinh Varney, Adarryl Varney, Adrin Varney, Alastair Varney, Anh
Varney, Braun Varney, Burrell Varney, Dalon Varney, Demetrias Varney, Derrill Varney, Donnis Varney, Dwon Varney, Fredick Varney, Gasper Varney, Glade Varney, Hulon Varney, Jonatha Varney, Jorma
Varney, Kaipo Varney, Karel Varney, Kejuan Varney, Kristine Varney, Larrie Varney, Majid Varney, Mar Varney, Mareo Varney, Nahshon Varney, Ondre Varney, Rasool Varney, Rayn Varney, Reade Varney,
Roshon Varney, Rozell Varney, Ryann Varney, Salbador Varney, Seon Varney, Sharone Varney, Shown Varney, Taggart Varney, Tarell Varney, Tymon Varney, Akira Varney, Alice Varney, Arend Varney, Baretta
Varney, Billyjack Varney, Breton Varney, Cleophas Varney, Dejan Varney, Delante Varney, Delmus Varney, Derren Varney, Doni Varney, Ezekial Varney, Fernand Varney, Gable Varney, Gilford Varney, Irfan
Varney, Jamarcus Varney, Jamason Varney, Jermond Varney, Jozef Varney, Kal Varney, Kalin Varney, Kayne Varney, Kosta Varney, Lancelot Varney, Lois Varney, Marlos Varney, Penny Varney, Rodriques
Varney, Rogerick Varney, Roldan Varney, Ronel Varney, Sha Varney, Shana Varney, Taris Varney, Torres Varney, Trapper Varney, Venancio Varney, Vinod Varney, Whit Varney, Allah Varney, Angelos Varney,
Antawan Varney, Bobbi Varney, Declan Varney, Eliasar Varney, Emanual Varney, Ericson Varney, Erol Varney, Faris Varney, Filemon Varney, Guthrie Varney, Helder Varney, Jahn Varney, Jamine Varney,
Jeoffrey Varney, Jervis Varney, Ketih Varney, Kimble Varney, Kole Varney, Krzysztof Varney, Lorena Varney, Martice Varney, Orie Varney, Oziel Varney, Ralphael Varney, Randie Varney, Seldon Varney,
Shanna Varney, Shawndale Varney, Shawnte Varney, Smokey Varney, Teri Varney, Tyrrell Varney, Wilburt Varney, Winton Varney, Yonatan Varney, Brand Varney, Briton Varney, Budd Varney, Charlene Varney,
Chetan Varney, Chivas Varney, Chriss Varney, Daryel Varney, Davi Varney, Dewarren Varney, Elston Varney, Garo Varney, Hilberto Varney, Hughie Varney, Jarrel Varney, Juma Varney, Kala Varney, Kamel
Varney, Lasalle Varney, Leoncio Varney, Linnie Varney, Markis Varney, Melville Varney, Nekia Varney, Obert Varney, Otilio Varney, Quintus Varney, Romualdo Varney, Romulus Varney, Sanjuan Varney,
Shadi Varney, Shonte Varney, Stelios Varney, Tavaras Varney, Tennyson Varney, Timon Varney, Torrell Varney, Tramel Varney, Tyrece Varney, Vertis Varney, Vipul Varney, Wenceslao Varney, Zuri Varney,
Amer Varney, Aquil Varney, Ardie Varney, Brayden Varney, Cederic Varney, Claudie Varney, Colm Varney, Cristen Varney, Crosby Varney, Darlene Varney, Darrol Varney, Darroll Varney, Dequincy Varney,
Donold Varney, Eddrick Varney, Elder Varney, Gaines Varney, Giovanny Varney, Hermilo Varney, Ivo Varney, Jermale Varney, Jibri Varney, Joshawa Varney, Kendrell Varney, Kerin Varney, Kert Varney, Kojo
Varney, Koy Varney, Lawerance Varney, Letroy Varney, Lumumba Varney, Machael Varney, Marshon Varney, Napolean Varney, Nimrod Varney, Phong Varney, Ramondo Varney, Ricahrd Varney, Sanjiv Varney,
Scotti Varney, Shaunn Varney, Shedric Varney, Sundance Varney, Tally Varney, Tiant Varney, Tysen Varney, Ulyses Varney, Vivian Varney, Ajani Varney, Anthany Varney, Antowan Varney, Brack Varney,
Chason Varney, Damany Varney, Danney Varney, Demitri Varney, Diangelo Varney, Dontaye Varney, Dragan Varney, Einar Varney, Gaspare Varney, Gerrell Varney, Gery Varney, Heston Varney, Isai Varney,
Jamale Varney, Jamone Varney, Jane Varney, Jerri Varney, Karla Varney, Kelan Varney, Kerrie Varney, Kobi Varney, Krist Varney, Lalo Varney, Latif Varney, Leman Varney, Levert Varney, Lorenso Varney,
Marcy Varney, Markese Varney, Merlyn Varney, Milburn Varney, Nicolai Varney, Norval Varney, Oris Varney, Ranjit Varney, Rashidi Varney, Richardson Varney, Rohn Varney, Roshaun Varney, Shepherd
Varney, Sonnie Varney, Stephens Varney, Stirling Varney, Sundiata Varney, Troyce Varney, Vander Varney, Waddell Varney, Yarnell Varney, Yitzchak Varney, Zerrick Varney, Adair Varney, Amani Varney,
Arvind Varney, Ascencion Varney, Athan Varney, Biff Varney, Burnice Varney, Carlito Varney, Carlitos Varney, Carlson Varney, Carry Varney, Christepher Varney, Clemmie Varney, Corbet Varney, Cristoval
Varney, Damaso Varney, Dickey Varney, Doral Varney, Eliu Varney, Erving Varney, Favio Varney, Helmut Varney, Hisham Varney, Ignazio Varney, Jamail Varney, Jarius Varney, Jermanie Varney, Jonte
Varney, Juluis Varney, Jushua Varney, Keron Varney, Kevis Varney, Kollin Varney, Labron Varney, Lavan Varney, Lavoris Varney, Lazar Varney, Nafis Varney, Nathanel Varney, Othello Varney, Quantrell
Varney, Ramzi Varney, Rossi Varney, Rye Varney, Siegfried Varney, Stanly Varney, Takashi Varney, Taro Varney, Tarvares Varney, Tehran Varney, Therron Varney, Thoms Varney, Tico Varney, Timm Varney,
Tylan Varney, Wheeler Varney, Zalman Varney, Aleksander Varney, Alok Varney, Alvan Varney, Angelica Varney, Ato Varney, Aureliano Varney, Ayman Varney, Berkeley Varney, Bracken Varney, Bren Varney,
Brittany Varney, Caley Varney, Camara Varney, Christifer Varney, Clenton Varney, Corley Varney, Deante Varney, Denorris Varney, Donielle Varney, Eleftherios Varney, Eleno Varney, Eliah Varney,
Enrrique Varney, Frederik Varney, French Varney, Hani Varney, Herby Varney, Howie Varney, Joab Varney, Kylie Varney, Leshon Varney, Maribel Varney, Marquese Varney, Martine Varney, Maurico Varney,
Melford Varney, Mikie Varney, Myreon Varney, Nabeel Varney, Nabor Varney, Orlandus Varney, Rafi Varney, Rahn Varney, Ramont Varney, Romain Varney, Ronold Varney, Rosco Varney, Shabazz Varney,
Shandell Varney, Shaughn Varney, Sumner Varney, Tamika Varney, Tarry Varney, Tedric Varney, Tellas Varney, Theordore Varney, Tolbert Varney, Wanda Varney, Winthrop Varney, Wyndell Varney, Abrahm
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Regginald Varney, Schon Varney, Secundino Varney, Slyvester Varney, Socorro Varney, Sundeep Varney, Tao Varney, Tasha Varney, Tivon Varney, Twain Varney, Verl Varney, Yousef Varney, Adem Varney,
Akash Varney, Archer Varney, Barnett Varney, Binh Varney, Darrly Varney, Detrich Varney, Diandre Varney, Dinesh Varney, Donya Varney, Eirik Varney, Fuquan Varney, Garrin Varney, Goerge Varney, Griff
Varney, Haneef Varney, Jamian Varney, Janos Varney, Jervon Varney, Josie Varney, Jumar Varney, Keithon Varney, Kelii Varney, Konstantine Varney, Latonya Varney, Laurel Varney, Liborio Varney, Lyon
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Trung Varney, Zeferino Varney, Abu Varney, Adaryll Varney, Adolf Varney, Allon Varney, Almon Varney, Alphonza Varney, Andras Varney, Angelito Varney, Aurelius Varney, Banjamin Varney, Bartlett
Varney, Caprice Varney, Cheskel Varney, Corrado Varney, Damieon Varney, Dariel Varney, Darral Varney, Demetreus Varney, Fleming Varney, Fotios Varney, Graylin Varney, Hari Varney, Hilbert Varney,
Holden Varney, Iram Varney, Jalon Varney, Jams Varney, Jaquan Varney, Jerimey Varney, Johnthomas Varney, Kain Varney, Lonney Varney, Lukus Varney, Marquet Varney, Melody Varney, Michall Varney,
Nicodemus Varney, Octavia Varney, Petro Varney, Prashant Varney, Primo Varney, Rahmel Varney, Rita Varney, Shepard Varney, Siddhartha Varney, Talbot Varney, Tomy Varney, Tyon Varney, Tyrie Varney,
Victory Varney, Aaren Varney, Ade Varney, Adel Varney, Akida Varney, Anjel Varney, Arrick Varney, Becky Varney, Bekim Varney, Camillo Varney, Cerrone Varney, Chett Varney, Cicero Varney, Collie
Varney, Cregg Varney, Danel Varney, Demitrios Varney, Desiree Varney, Dontez Varney, Doris Varney, Dorrell Varney, Edouard Varney, Emmette Varney, Everton Varney, Eward Varney, Geofrey Varney,
Gregrey Varney, Greig Varney, Gust Varney, Jaycee Varney, Jes Varney, Jocob Varney, Judith Varney, Karlin Varney, Kaveh Varney, Keil Varney, Kino Varney, Kwaku Varney, Lamel Varney, Lea Varney,
Leodis Varney, Loni Varney, Lonie Varney, Miriam Varney, Nnamdi Varney, Olufemi Varney, Patrik Varney, Raun Varney, Refujio Varney, Rossie Varney, Rydell Varney, Severin Varney, Shadeed Varney,
Shannen Varney, Shellie Varney, Sheri Varney, Sumit Varney, Tamiko Varney, Tien Varney, Tirso Varney, Tonie Varney, Trino Varney, Tyra Varney, Wright Varney, Zalmen Varney, Abdon Varney, Abrahan
Varney, Adrean Varney, Aeron Varney, Ajene Varney, Atthew Varney, Augie Varney, Carmel Varney, Cezar Varney, Chioke Varney, Chung Varney, Delonta Varney, Derrich Varney, Deyon Varney, Dicky Varney,
Elihu Varney, Eustacio Varney, Fernado Varney, Filbert Varney, Free Varney, Goran Varney, Habib Varney, Haile Varney, Han Varney, Heraclio Varney, Jarin Varney, Javaris Varney, Jeramine Varney,
Joanna Varney, Johnel Varney, Joie Varney, Juddson Varney, Kason Varney, Leverne Varney, Liston Varney, Marck Varney, Marcoantonio Varney, Margarita Varney, Messiah Varney, Nicholaos Varney, Onofre
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Varney, Zeno Varney, Abimael Varney, Adell Varney, Adren Varney, Agron Varney, Brick Varney, Bridger Varney, Buddie Varney, Cosimo Varney, Cristin Varney, Deitrich Varney, Demetre Varney, Dolphus
Varney, Dorell Varney, Dylon Varney, Fadi Varney, Fredrich Varney, Gerron Varney, Jacobe Varney, Jarell Varney, Karson Varney, Kegan Varney, Latasha Varney, Laureano Varney, Lemonte Varney, Lev
Varney, Marcellino Varney, Marsha Varney, Maureen Varney, Ori Varney, Prudencio Varney, Rockwell Varney, Selby Varney, Shah Varney, Stefanos Varney, Tarris Varney, Taryll Varney, Telley Varney, Teryl
Varney, Thoams Varney, Tiko Varney, Tilford Varney, Toraino Varney, Tyce Varney, Undrea Varney, Walid Varney, Williard Varney, Yasir Varney, Yvette Varney, Abiel Varney, Aimee Varney, Ainsley Varney,
Akram Varney, Antino Varney, Bengy Varney, Benjimen Varney, Bohdan Varney, Canyon Varney, Chandra Varney, Cheron Varney, Christhoper Varney, Cordney Varney, Covey Varney, Danyale Varney, Daquan
Varney, Darnelle Varney, Dong Varney, Donjuan Varney, Elieser Varney, Feliberto Varney, Fernie Varney, Gaylan Varney, Gerritt Varney, Glennis Varney, Goldie Varney, Hermes Varney, Hernandez Varney,
Herny Varney, Hervey Varney, Italo Varney, Jeno Varney, Joell Varney, Kaiser Varney, Kalem Varney, Kijana Varney, Knute Varney, Korby Varney, Leaf Varney, Lejuan Varney, Lin Varney, Mahdi Varney,
Mindy Varney, Mithcell Varney, Mylon Varney, Nichlas Varney, Nickolus Varney, Quanah Varney, Rachael Varney, Rafiq Varney, Ranier Varney, Rashaud Varney, Rebel Varney, Reuel Varney, Rodgerick Varney,
Ryun Varney, Samad Varney, Steffon Varney, Tan Varney, Tonio Varney, Trevell Varney, Truett Varney, Woodley Varney, Ziad Varney, Alisha Varney, Amory Varney, Ashford Varney, Bard Varney, Brenan
Varney, Burney Varney, Carnel Varney, Cevin Varney, Clare Varney, Corban Varney, Dashan Varney, Deddrick Varney, Dennise Varney, Dimetrius Varney, Diogenes Varney, Dodd Varney, Dominico Varney,
Dorion Varney, Eberardo Varney, Ethen Varney, Eyal Varney, Fedrick Varney, Gonsalo Varney, Gorje Varney, Hart Varney, Irby Varney, Jarmal Varney, Jaworski Varney, Johnston Varney, Joshuwa Varney,
Json Varney, Kenzie Varney, Kristie Varney, Labaron Varney, Lan Varney, Leroi Varney, Lysander Varney, Marten Varney, Matthieu Varney, Meghan Varney, Misha Varney, Norvel Varney, Ohn Varney, Olivia
Varney, Omoro Varney, Powell Varney, Qunicy Varney, Shaheen Varney, Sirron Varney, Suraj Varney, Teran Varney, Theopolis Varney, Tion Varney, Toddy Varney, Toronto Varney, Tyrin Varney, Urban Varney,
Vencent Varney, Vida Varney, Virgle Varney, Weslee Varney, Wilfrido Varney, Ahron Varney, Alanzo Varney, Amol Varney, Asad Varney, Avel Varney, Bao Varney, Benancio Varney, Britten Varney, Cassie
Varney, Choya Varney, Christien Varney, Chrles Varney, Cochise Varney, Criss Varney, Culley Varney, Danyelle Varney, Deforest Varney, Deland Varney, Deshane Varney, Dynell Varney, Eluterio Varney,
Emeka Varney, Eyad Varney, Fabricio Varney, Graviel Varney, Gunner Varney, Harun Varney, Hiroshi Varney, Ihsan Varney, Irineo Varney, Jacon Varney, Jamile Varney, Jawad Varney, Jeramia Varney, Joann
Varney, Johnn Varney, Josph Varney, Kaleo Varney, Katie Varney, Kelwin Varney, Kwanza Varney, Laddie Varney, Lanorris Varney, Larron Varney, Maron Varney, Mehdi Varney, Monolito Varney, Mont Varney,
Munir Varney, Murrell Varney, Nachman Varney, Natan Varney, Nguyen Varney, Nickalus Varney, Nikitas Varney, Orlin Varney, Phillippe Varney, Rasean Varney, Rino Varney, Robi Varney, Rodgers Varney,
Rolan Varney, Roswell Varney, Sagar Varney, Schaun Varney, Segundo Varney, Senaca Varney, Severino Varney, Sloane Varney, Spenser Varney, Talvin Varney, Teddrick Varney, Theran Varney, Tiron Varney,
Tshombe Varney, Zephaniah Varney, Abbas Varney, Adil Varney, Alireza Varney, Ancil Varney, Antoan Varney, Atanacio Varney, Benedicto Varney, Benjimin Varney, Blandon Varney, Bradey Varney, Brannen
Varney, Chalmer Varney, Charleton Varney, Chrsitopher Varney, Cleatus Varney, Corwyn Varney, Costas Varney, Cranston Varney, Damario Varney, Delmon Varney, Dermaine Varney, Duc Varney, Duff Varney,
Farhad Varney, Gabor Varney, Garreth Varney, Heshimu Varney, Ines Varney, Ivar Varney, Jahan Varney, Jamael Varney, Janes Varney, Jaycen Varney, Jett Varney, Johnhenry Varney, Jona Varney, Josey
Varney, Khristian Varney, Kilian Varney, Lander Varney, Laval Varney, Lebaron Varney, Leocadio Varney, Lonzie Varney, Mackey Varney, Marios Varney, Maxim Varney, Naquan Varney, Nickalaus Varney,
Nicolaos Varney, Nkosi Varney, Ozie Varney, Ponciano Varney, Praveen Varney, Ramar Varney, Rizwan Varney, Ronen Varney, Saad Varney, Shalin Varney, Shalon Varney, Shaunte Varney, Steele Varney, Sten
Varney, Terrin Varney, Tiran Varney, Valdis Varney, Virgel Varney, Yamil Varney, Adriana Varney, Ajit Varney, Alger Varney, Amil Varney, Ananda Varney, Angie Varney, Anup Varney, Arjun Varney, Ark
Varney, Bayard Varney, Bolivar Varney, Burnie Varney, Chae Varney, Dallan Varney, Danile Varney, Deward Varney, Didier Varney, Dusan Varney, Elaine Varney, Franchot Varney, Gavriel Varney, Heliodoro
Varney, Igor Varney, Irene Varney, Iven Varney, Jamien Varney, Jaquay Varney, Kaylon Varney, Larico Varney, Lenord Varney, Lenton Varney, Lynden Varney, Mahesh Varney, Malo Varney, Mercedes Varney,
Nicklos Varney, Raiford Varney, Rashed Varney, Rayshaun Varney, Reubin Varney, Senica Varney, Shaan Varney, Shermon Varney, Sherrell Varney, Sierra Varney, Tamarcus Varney, Tavius Varney, Thurmon
Varney, Tonia Varney, Torino Varney, Urian Varney, Verner Varney, Zacarias Varney, Abdel Varney, Aniel Varney, Arjuna Varney, Ashon Varney, Candy Varney, Caroline Varney, Chukwuemeka Varney, Cid
Varney, Court Varney, Cristino Varney, Darshan Varney, Delshawn Varney, Demetrie Varney, Deondray Varney, Deone Varney, Diante Varney, Dolores Varney, Donahue Varney, Eligah Varney, Essex Varney,
Eutimio Varney, Gustavus Varney, Hebert Varney, Hyun Varney, Janssen Varney, Jaxon Varney, Jerre Varney, Joanne Varney, Jorden Varney, Joseantonio Varney, Kainoa Varney, Kalif Varney, Karin Varney,
Katina Varney, Keisha Varney, Khalfani Varney, Leanthony Varney, Michaelpaul Varney, Murl Varney, Nazareth Varney, Pearl Varney, Purcell Varney, Quay Varney, Raylon Varney, Ren Varney, Renell Varney,
Reshard Varney, Rodel Varney, Shaen Varney, Talon Varney, Torsten Varney, Victorino Varney, Vonzell Varney, Yong Varney, Zaki Varney, Abdiel Varney, Adon Varney, Alferd Varney, Amjad Varney, Arlester
Varney, Bassam Varney, Bela Varney, Belton Varney, Branko Varney, Burk Varney, Calixto Varney, Cassey Varney, Cayle Varney, Cedrie Varney, Cuong Varney, Damu Varney, Danelle Varney, Darcey Varney,
Dayna Varney, Domanic Varney, Dougles Varney, Elex Varney, Elrico Varney, Estaban Varney, Ferlando Varney, Filipe Varney, Geovanni Varney, Glenford Varney, Gumaro Varney, Gustabo Varney, Haley
Varney, Herberto Varney, Jabez Varney, Jaman Varney, Joshau Varney, Jullian Varney, Kairi Varney, Karnell Varney, Kin Varney, Kinsey Varney, Kipper Varney, Laquincy Varney, Laterrance Varney, Laurice
Varney, Leshaun Varney, Marcia Varney, Marcio Varney, Melquiades Varney, Miklos Varney, Ming Varney, Mylo Varney, Omid Varney, Oskar Varney, Paco Varney, Rafel Varney, Ralf Varney, Ravindra Varney,
Rivers Varney, Rody Varney, Saint Varney, Santee Varney, Shai Varney, Shelvin Varney, Shyam Varney, Silvino Varney, Squire Varney, Star Varney, Steen Varney, Stein Varney, Tae Varney, Taryn Varney,
Tavare Varney, Thedore Varney, Yehudah Varney, Zacchaeus Varney, Zachory Varney, Zan Varney, Adewale Varney, Aleksandr Varney, Alf Varney, Alvester Varney, Amiri Varney, Araceli Varney, Arcenio
Varney, Arvell Varney, Arvis Varney, Atlas Varney, Binyomin Varney, Chasen Varney, Cheyne Varney, Coray Varney, Dak Varney, Davone Varney, Egbert Varney, Ferlin Varney, Gilles Varney, Glendell
Varney, Granger Varney, Jafar Varney, Jalil Varney, Jamain Varney, Jarrette Varney, Jeret Varney, Jocelyn Varney, Joenathan Varney, Jurgen Varney, Keene Varney, Keller Varney, Kenneith Varney, Kinney
Varney, Labarron Varney, Levie Varney, Linford Varney, Marcas Varney, Mickle Varney, Morrison Varney, Oba Varney, Qasim Varney, Rahsan Varney, Rajah Varney, Raphel Varney, Rehan Varney, Rodregus
Varney, Ronan Varney, Salman Varney, Seddrick Varney, Shauna Varney, Simpson Varney, Sunshine Varney, Terrall Varney, Tiawan Varney, Waylen Varney, Weslie Varney, Abdur Varney, Alric Varney, Arien
Varney, Arren Varney, Bari Varney, Briar Varney, Burdette Varney, Carver Varney, Charvis Varney, Contrell Varney, Costa Varney, Dara Varney, Darly Varney, Darwyn Varney, Davied Varney, Dayon Varney,
Delmont Varney, Dena Varney, Devlon Varney, Dimitry Varney, Duwane Varney, Esteven Varney, Fergus Varney, Harles Varney, Hutch Varney, Ion Varney, Isidore Varney, Jamus Varney, Jerett Varney, Jerrit
Varney, Jontue Varney, Lemarcus Varney, Lequan Varney, Leslee Varney, Linell Varney, Lugene Varney, Makoto Varney, Malek Varney, Marquell Varney, Osei Varney, Peggy Varney, Ralphie Varney, Ramona
Varney, Rickard Varney, Ronnald Varney, Sabian Varney, Saleh Varney, Salih Varney, Samule Varney, Sarkis Varney, Sharrieff Varney, Snehal Varney, Sony Varney, Timotheus Varney, Tood Varney, Tyreece
Varney, Wei Varney, Zackariah Varney, Aleem Varney, Antwione Varney, Ayodele Varney, Barkley Varney, Bartt Varney, Bethany Varney, Bing Varney, Braheem Varney, Brek Varney, Caron Varney, Cathy
Varney, Cecilia Varney, Cristina Varney, Dainel Varney, Dal Varney, Darrek Varney, Darrelle Varney, Dashun Varney, Delrick Varney, Demitrus Varney, Dishon Varney, Donnald Varney, Ehab Varney, Erico
Varney, Erskin Varney, Faraji Varney, Gillis Varney, Hamed Varney, Hermann Varney, Hondo Varney, Jamis Varney, Jeric Varney, Jermyn Varney, Jessup Varney, Jonatan Varney, Joshaua Varney, Jotham
Varney, Khaalis Varney, Kion Varney, Lam Varney, Larence Varney, Lavonne Varney, Marti Varney, Mclean Varney, Mercury Varney, Miki Varney, Norvin Varney, Onnie Varney, Paulmichael Varney, Pavan
Varney, Paz Varney, Phuong Varney, Race Varney, Ralston Varney, Ramadan Varney, Rashee Varney, Roan Varney, Sally Varney, Shanga Varney, Slater Varney, Tarvaris Varney, Tevin Varney, Travus Varney,
Viviano Varney, Zerick Varney, Abdula Varney, Achilles Varney, Adarryll Varney, Ananias Varney, Antinio Varney, Antrell Varney, Anzio Varney, Aristidis Varney, Ashante Varney, Atom Varney, Audley
Varney, Azim Varney, Bandy Varney, Bary Varney, Britain Varney, Bryne Varney, Calen Varney, Ceddrick Varney, Chalmers Varney, Chanc Varney, Chap Varney, Charly Varney, Charron Varney, Cornellius
Varney, Darth Varney, Dayan Varney, Deran Varney, Dewand Varney, Fran Varney, Girolamo Varney, Harvie Varney, Haskel Varney, Hoy Varney, Islam Varney, Ivon Varney, Jafari Varney, Jamario Varney, Jehu
Varney, Josemanuel Varney, Jung Varney, Kabir Varney, Kael Varney, Khan Varney, Khayree Varney, Kimber Varney, Koran Varney, Lamonta Varney, Lavalle Varney, Lavel Varney, Lenardo Varney, Lydia
Varney, Mandrill Varney, Mannie Varney, Marguis Varney, Marisa Varney, Marissa Varney, Marshaun Varney, Medardo Varney, Naomi Varney, Naser Varney, Natanael Varney, Oji Varney, Orvil Varney, Pasqual
Varney, Previn Varney, Reo Varney, Ruddy Varney, Shahram Varney, Sharad Varney, Shaunta Varney, Shuron Varney, Teo Varney, Thalmus Varney, Theodoric Varney, Tibor Varney, Tres Varney, Ulrich Varney,
Winter Varney, Yasser Varney, Ygnacio Varney, Abasi Varney, Alwin Varney, Alwyn Varney, Amalio Varney, Arty Varney, Baker Varney, Bishara Varney, Briane Varney, Catalino Varney, Darrall Varney,
Dartanyon Varney, Deano Varney, Demico Varney, Deontae Varney, Deveron Varney, Devonn Varney, Donnovan Varney, Durant Varney, Duy Varney, Evander Varney, Ever Varney, Ewing Varney, Fouad Varney,
Froylan Varney, Gabriela Varney, Garnell Varney, Gerlad Varney, Ginger Varney, Glendale Varney, Grafton Varney, Haig Varney, Hale Varney, Iver Varney, Ivin Varney, Jarel Varney, Javen Varney, Jeran
Varney, Jereld Varney, Jermichael Varney, Jimel Varney, Karreem Varney, Keiron Varney, Kester Varney, Laronn Varney, Lavor Varney, Lehi Varney, Lemon Varney, Lourdes Varney, Marqui Varney, Marsh
Varney, Marx Varney, Montego Varney, Murice Varney, Mychael Varney, Nashawn Varney, Nichalas Varney, Nickalas Varney, Nuri Varney, Obinna Varney, Onaje Varney, Patrich Varney, Pavel Varney, Priscilla
Varney, Rakeem Varney, Rande Varney, Rashann Varney, Rayan Varney, Raynold Varney, Ricko Varney, Riki Varney, Rooney Varney, Saulo Varney, Schad Varney, Seiji Varney, Shahn Varney, Shantel Varney,
Shanton Varney, Sisto Varney, Taiwo Varney, Travion Varney, Tristen Varney, Truong Varney, Tywann Varney, Tywayne Varney, Uhuru Varney, Welby Varney, Wesam Varney, Alexius Varney, Ammar Varney,
Andrews Varney, Atticus Varney, Audy Varney, Belinda Varney, Bennet Varney, Bomani Varney, Bulmaro Varney, Charlies Varney, Chato Varney, Chelsea Varney, Chipper Varney, Clearence Varney, Cleotis
Varney, Cleven Varney, Corydon Varney, Dadrian Varney, Damiano Varney, Darrold Varney, Davell Varney, Derone Varney, Derral Varney, Derwood Varney, Django Varney, Doc Varney, Dontell Varney, Doren
Varney, Elza Varney, Esmeralda Varney, Estil Varney, Finn Varney, Fredrik Varney, Garrie Varney, Greogory Varney, Harlow Varney, Iris Varney, Jascha Varney, Jediah Varney, Jef Varney, Jefferie
Varney, Jerimah Varney, Jerrie Varney, Joal Varney, Johsua Varney, Joon Varney, Joson Varney, Kamar Varney, Kellen Varney, Kelvis Varney, Ketrick Varney, Kimberley Varney, Kingston Varney, Meldon
Varney, Mendell Varney, Michaeljohn Varney, Moss Varney, Nuno Varney, Onofrio Varney, Phineas Varney, Ramell Varney, Raydell Varney, Regie Varney, Ronnel Varney, Sadat Varney, Sajid Varney, Shanard
Varney, Shaul Varney, Shondale Varney, Shyrone Varney, Simuel Varney, Sjon Varney, Stephenson Varney, Tadeusz Varney, Takeshi Varney, Tilman Varney, Tomasz Varney, Travares Varney, Truitt Varney,
Ulrick Varney, Vandy Varney, Yvon Varney, Zebulan Varney, Adolpho Varney, Alek Varney, Amery Varney, Andrej Varney, Aniello Varney, Anurag Varney, Audra Varney, Boruch Varney, Burr Varney, Cable
Varney, Cheney Varney, Cimarron Varney, Clell Varney, Clinten Varney, Daimen Varney, Dameyon Varney, Dason Varney, Deanglo Varney, Delaine Varney, Demeco Varney, Denim Varney, Derryck Varney, Domnick
Varney, Doy Varney, Dwyne Varney, Edwards Varney, Elisa Varney, Esmond Varney, Gawain Varney, Green Varney, Hallie Varney, Hameed Varney, Heinrich Varney, Jeri Varney, Kaj Varney, Kalan Varney, Kash
Varney, Kente Varney, Koji Varney, Kunte Varney, Laurin Varney, Livingston Varney, Meliton Varney, Michaeal Varney, Myran Varney, Nephtali Varney, Nicasio Varney, Nyles Varney, Obediah Varney, Pride
Varney, Primus Varney, Rae Varney, Rahmaan Varney, Rhoderick Varney, Rip Varney, River Varney, Roberts Varney, Rodricus Varney, Rommie Varney, Romney Varney, Roverto Varney, Samar Varney, Sergei
Varney, Shadrack Varney, Shavon Varney, Stevin Varney, Tarone Varney, Tashawn Varney, Tora Varney, Tri Varney, Vinny Varney, Yesenia Varney, Yvan Varney, Zubin Varney, Absalom Varney, Aldrick Varney,
Alif Varney, Amad Varney, Anacleto Varney, Andru Varney, Argelio Varney, Arrington Varney, Ashely Varney, Auston Varney, Baird Varney, Beryl Varney, Celester Varney, Chadron Varney, Chante Varney,
Dack Varney, Darain Varney, Darvell Varney, Daune Varney, Daylan Varney, Dedan Varney, Deitrick Varney, Dejaun Varney, Denzel Varney, Deverick Varney, Eris Varney, Flor Varney, Frankey Varney,
Gaberial Varney, Gianluca Varney, Hesham Varney, Iban Varney, Ilya Varney, Isham Varney, Issiah Varney, Jacey Varney, Jahmel Varney, Jaimey Varney, Jamane Varney, Jammey Varney, Jareb Varney, Jarek
Varney, Jemery Varney, Jeremaih Varney, Jernard Varney, Jerramy Varney, Jese Varney, Johnaton Varney, Jontae Varney, Kayode Varney, Kaz Varney, Kedar Varney, Kell Varney, Kewan Varney, Kiyoshi
Varney, Ladarryl Varney, Lamart Varney, Lataurus Varney, Lavarr Varney, Lavert Varney, Lawan Varney, Loney Varney, Lonnel Varney, Luan Varney, Macio Varney, Mansa Varney, Miachel Varney, Neftaly
Varney, Nichlos Varney, Nicklous Varney, Nik Varney, Orlander Varney, Quadir Varney, Raeford Varney, Rajohn Varney, Ranson Varney, Rasul Varney, Renald Varney, Reynald Varney, Roben Varney,
Rutherford Varney, Salathiel Varney, Savoy Varney, Sharief Varney, Sherrick Varney, Siddharth Varney, Staci Varney, Steed Varney, Tauheed Varney, Thaddus Varney, Toland Varney, Toran Varney, Torriano
Varney, Turon Varney, Vashaun Varney, Zebediah Varney, Aasim Varney, Abron Varney, Ahman Varney, Aiden Varney, Alexsander Varney, Alias Varney, Aly Varney, Ameet Varney, Anothny Varney, Ansley
Varney, Ballard Varney, Bedford Varney, Beecher Varney, Benaiah Varney, Clent Varney, Colvin Varney, Dakari Varney, Darric Varney, Daved Varney, Demichael Varney, Deray Varney, Deren Varney,
Dionisios Varney, Dionta Varney, Dorothy Varney, Dorrian Varney, Dorris Varney, Dyke Varney, Elmon Varney, Erle Varney, Eunice Varney, Eustace Varney, Evon Varney, Fay Varney, Federick Varney, Feras
Varney, Gabrielle Varney, Gaven Varney, Gay Varney, Gjon Varney, Grier Varney, Guillaume Varney, Hansen Varney, Harrel Varney, Hazel Varney, Heathe Varney, Herschell Varney, Ingram Varney, Isauro
Varney, Jaimy Varney, Jaymar Varney, Jerin Varney, Jermall Varney, Jevin Varney, Jillian Varney, Johnmark Varney, Jovaughn Varney, Kamuela Varney, Kendon Varney, Khoa Varney, Lamberto Varney, Latham
Varney, Luster Varney, Malon Varney, Micholas Varney, Mitul Varney, Montrice Varney, Moroni Varney, Mylan Varney, Nason Varney, Nic Varney, Nowell Varney, Oryan Varney, Owens Varney, Phill Varney,
Raed Varney, Raji Varney, Ramzy Varney, Rebekah Varney, Reeve Varney, Regenald Varney, Roddie Varney, Roshun Varney, Salvator Varney, Seanpaul Varney, Shanne Varney, Shawntay Varney, Sheldrick
Varney, Sheppard Varney, Snapper Varney, Tarick Varney, Terelle Varney, Tery Varney, Thelonious Varney, Trampis Varney, Tushar Varney, Vernice Varney, Whalen Varney, Willim Varney, Alban Varney,
Alexy Varney, Almalik Varney, Amedeo Varney, Antoney Varney, Asia Varney, Aurthur Varney, Avelardo Varney, Barth Varney, Cartrell Varney, Chaderick Varney, Champ Varney, Chano Varney, Charity Varney,
Christhopher Varney, Christino Varney, Clavin Varney, Cuyler Varney, Dalbert Varney, Darcel Varney, Darcell Varney, Darrie Varney, Denney Varney, Denson Varney, Dezmond Varney, Donyea Varney, Dorien
Varney, Emidio Varney, Etan Varney, Evagelos Varney, Everado Varney, Farren Varney, Fate Varney, Filomeno Varney, Fonzie Varney, Franko Varney, Garrod Varney, Graciano Varney, Gustin Varney, Isrrael
Varney, Jac Varney, Jakie Varney, Jamas Varney, Jamiyl Varney, Jaylon Varney, Jeanmarc Varney, Jebadiah Varney, Jeffey Varney, Jiles Varney, Johnnell Varney, Josip Varney, Juanmanuel Varney, Kemo
Varney, Kemuel Varney, Kindrick Varney, Kirtis Varney, Kyley Varney, Lacorey Varney, Lansing Varney, Lawyer Varney, Lenzie Varney, Lessie Varney, Lindley Varney, Lynette Varney, Lyonel Varney, Magnus
Varney, Marki Varney, Marrell Varney, Mccoy Varney, Mcdonald Varney, Meade Varney, Mickell Varney, Mikah Varney, Miranda Varney, Mitchael Varney, Mitcheal Varney, Montee Varney, Morio Varney, Murat
Varney, Nashon Varney, Nova Varney, Olusegun Varney, Ondra Varney, Orian Varney, Orren Varney, Reggy Varney, Rodrico Varney, Ronzell Varney, Runako Varney, Ryszard Varney, Shady Varney, Shango
Varney, Shawan Varney, Terez Varney, Thang Varney, Tomothy Varney, Toron Varney, Torrin Varney, Toya Varney, Trayon Varney, Trellis Varney, Tremel Varney, Tryon Varney, Tyrek Varney, Wessley Varney,
Yan Varney, Zed Varney, Ako Varney, Aldrich Varney, Alesandro Varney, Alrick Varney, Ayo Varney, Bobbyjoe Varney, Broadus Varney, Brockton Varney, Callie Varney, Ched Varney, Conn Varney, Cristo
Varney, Dallen Varney, Dannel Varney, Dasan Varney, Datron Varney, Dawaun Varney, Delance Varney, Delawrence Varney, Demarrio Varney, Denell Varney, Derak Varney, Derrie Varney, Deundra Varney, Dorin
Varney, Dyon Varney, Elizardo Varney, Filimon Varney, Fitzroy Varney, Gari Varney, Glover Varney, Gretchen Varney, Hartley Varney, Holt Varney, Jabe Varney, Jaclyn Varney, Jadrien Varney, Jamah
Varney, Jarman Varney, Jarreau Varney, Jearld Varney, Jerimi Varney, Jerris Varney, Joeph Varney, Jorel Varney, Joshus Varney, Jovany Varney, Kaven Varney, Kawaski Varney, Kiron Varney, Korrey
Varney, Kort Varney, Lavale Varney, Lenorris Varney, Lizandro Varney, Lynard Varney, Manson Varney, Marky Varney, Marlando Varney, Maxime Varney, Mcgarrett Varney, Merwin Varney, Miko Varney, Niklas
Varney, Nkrumah Varney, Orman Varney, Osmar Varney, Paresh Varney, Ponce Varney, Prem Varney, Rachelle Varney, Rajendra Varney, Rashone Varney, Rawn Varney, Raynor Varney, Rechard Varney, Rogan
Varney, Rollo Varney, Romando Varney, Royd Varney, Salah Varney, Scoey Varney, Selvin Varney, Swain Varney, Tarun Varney, Tawn Varney, Teal Varney, Telesforo Varney, Thaddeaus Varney, Timoth Varney,
Toren Varney, Vicki Varney, Wael Varney, Whitfield Varney, Winifred Varney, Wyeth Varney, Abdu Varney, Abdulla Varney, Akio Varney, Aleck Varney, Antwian Varney, Aquiles Varney, Armad Varney, Averill
Varney, Bijan Varney, Billyjo Varney, Bladimir Varney, Bray Varney, Brigido Varney, Candice Varney, Changa Varney, Chanon Varney, Charlotte Varney, Colie Varney, Colonel Varney, Crespin Varney,
Dabney Varney, Dace Varney, Damaine Varney, Danna Varney, Danya Varney, Darelle Varney, Darrnell Varney, Deaundre Varney, Dijon Varney, Dina Varney, Dontel Varney, Dracy Varney, Duayne Varney, Easton
Varney, Eban Varney, Elric Varney, Ervey Varney, Festus Varney, Filip Varney, Furnell Varney, Gardiner Varney, Gleen Varney, Herve Varney, Hogan Varney, Hossein Varney, Ikaika Varney, Imad Varney,
Jacek Varney, Jamaica Varney, Jamichael Varney, Jammal Varney, Jarmon Varney, Jerren Varney, Joshu Varney, Juanjose Varney, Kanoa Varney, Kia Varney, Kiernan Varney, Kirtus Varney, Lacedric Varney,
Legrand Varney, Lekendrick Varney, Lemoyne Varney, Lige Varney, Loranzo Varney, Loyce Varney, Masai Varney, Master Varney, Michah Varney, Min Varney, Mir Varney, Mohan Varney, Montrail Varney,
Montray Varney, Nacoma Varney, Nakoa Varney, Navid Varney, Nole Varney, Opie Varney, Orpheus Varney, Payam Varney, Pearce Varney, Ramal Varney, Recco Varney, Renn Varney, Rigel Varney, Roe Varney,
Rolondo Varney, Romaldo Varney, Severn Varney, Shakim Varney, Shulem Varney, Silviano Varney, Strider Varney, Tayari Varney, Tiburcio Varney, Tilmon Varney, Tomar Varney, Tran Varney, Travaris
Varney, Trayvon Varney, Trev Varney, Trevar Varney, Tyreek Varney, Windle Varney, Yaw Varney, Yechezkel Varney, Zeth Varney, Zollie Varney, Acey Varney, Adisa Varney, Arlanda Varney, Artavius Varney,
Avin Varney, Beauford Varney, Bernadette Varney, Birch Varney, Brittan Varney, Brown Varney, Cabe Varney, Calton Varney, Caribe Varney, Carrington Varney, Chadwich Varney, Chrstopher Varney, Chun
Varney, Crescencio Varney, Damyon Varney, Dantrell Varney, Dariusz Varney, Dartanion Varney, Delio Varney, Demarius Varney, Deniz Varney, Dennard Varney, Domingos Varney, Dominie Varney, Doyal
Varney, Dwanye Varney, Earon Varney, Efstathios Varney, Elkin Varney, Ellen Varney, Fabain Varney, Frans Varney, Gopal Varney, Haleem Varney, Iverson Varney, Japeth Varney, Johanna Varney, Jorgen
Varney, Jourdan Varney, Kassim Varney, Keidrick Varney, Keif Varney, Knox Varney, Lacharles Varney, Ladarius Varney, Ladarrell Varney, Leonce Varney, Lipa Varney, Lorren Varney, Luqman Varney,
Mcihael Varney, Mercer Varney, Micharl Varney, Mikle Varney, Montre Varney, Morad Varney, Nichael Varney, Phi Varney, Quantez Varney, Quency Varney, Quinlan Varney, Quintrell Varney, Redell Varney,
Regnald Varney, Renauld Varney, Ronelle Varney, Rudell Varney, Ruffin Varney, Salil Varney, Seanmichael Varney, Shaheem Varney, Sharieff Varney, Shawndel Varney, Stamatis Varney, Tadashi Varney,
Tiras Varney, Toan Varney, Toben Varney, Tyrik Varney, Tyshaun Varney, Ulisses Varney, Visente Varney, Wadell Varney, Wilfrid Varney, Won Varney, Aashish Varney, Adama Varney, Aja Varney, Alcindor
Varney, Alejandra Varney, Alyn Varney, Ami Varney, Andrel Varney, Antonine Varney, Artemas Varney, Artimus Varney, Ary Varney, Atiim Varney, Avinash Varney, Brittain Varney, Callan Varney, Cari
Varney, Claire Varney, Conard Varney, Danan Varney, Darol Varney, Day Varney, Delma Varney, Demetricus Varney, Dewone Varney, Dougals Varney, Elester Varney, Espiridion Varney, Estanislado Varney,
Esther Varney, Fabien Varney, Famous Varney, Fredderick Varney, Friedrich Varney, Garrette Varney, Greogry Varney, Gwendolyn Varney, Hardin Varney, Hendrik Varney, Hendrix Varney, Jabulani Varney,
Jacin Varney, Jacquelyn Varney, Jantzen Varney, Jarren Varney, Jasyn Varney, Jayde Varney, Jeris Varney, Jery Varney, Jolyon Varney, Karem Varney, Key Varney, Kibwe Varney, Kyon Varney, Lamin Varney,
Lasaro Varney, Ledon Varney, Malcomb Varney, Mando Varney, Marcelus Varney, Margus Varney, Marzell Varney, Mearl Varney, Miron Varney, Mitesh Varney, Montay Varney, Montrey Varney, Morey Varney,
Morrie Varney, Mozell Varney, Narada Varney, Naveed Varney, Nenad Varney, Nicolaas Varney, Nilton Varney, Ojay Varney, Olga Varney, Oman Varney, Pancho Varney, Pepper Varney, Poul Varney, Rajon
Varney, Rawley Varney, Rayon Varney, Read Varney, Rees Varney, Reeves Varney, Romond Varney, Rondrick Varney, Sancho Varney, Santi Varney, Selso Varney, Sevan Varney, Shondel Varney, Tavarius Varney,
Thelbert Varney, Theodoro Varney, Theus Varney, Thien Varney, Timoty Varney, Timur Varney, Travius Varney, Trisha Varney, Tyquan Varney, Usbaldo Varney, Varnell Varney, Videl Varney, Willi Varney,
Zacariah Varney, Zain Varney, Aamir Varney, Abdallah Varney, Almando Varney, Almond Varney, Alpheus Varney, Alvarez Varney, Alverto Varney, Andrick Varney, Anthonie Varney, Anthoy Varney, Antiona
Varney, Antwand Varney, Arrie Varney, Artur Varney, Baldwin Varney, Bardo Varney, Carols Varney, Chau Varney, Clayborne Varney, Conell Varney, Confesor Varney, Cottrell Varney, Craven Varney, Curits
Varney, Damiel Varney, Deandrae Varney, Derrius Varney, Dirrick Varney, Dora Varney, Dorain Varney, Dushaun Varney, Eliodoro Varney, Elvie Varney, Faraz Varney, Fernanda Varney, Ferron Varney,
Frankin Varney, Fraser Varney, Giuliano Varney, Godwin Varney, Hakan Varney, Hershal Varney, Jacobie Varney, Jada Varney, Jaimeson Varney, Jamill Varney, Jarrard Varney, Jeromi Varney, Jerral Varney,
Jilberto Varney, Jonel Varney, Joseangel Varney, Josedejesus Varney, Joshva Varney, Jovani Varney, Juana Varney, Juvencio Varney, Karol Varney, Keanon Varney, Keli Varney, Kelven Varney, Kemal
Varney, Khanh Varney, Kiah Varney, Kimmy Varney, Kitt Varney, Laith Varney, Latravis Varney, Lebarron Varney, Lenon Varney, Leondre Varney, Lue Varney, Luka Varney, Lyndal Varney, Macedonio Varney,
Mahdee Varney, Marcellas Varney, Mardell Varney, Monnie Varney, Naron Varney, Nehal Varney, Nicholai Varney, Nicol Varney, Nimesh Varney, Orlan Varney, Pal Varney, Paras Varney, Pasha Varney, Pio
Varney, Piotr Varney, Rahmon Varney, Reilly Varney, Rommell Varney, Romondo Varney, Ronie Varney, Rontae Varney, Roth Varney, Ruven Varney, Senecca Varney, Sequoia Varney, Shadwick Varney, Shaquan
Varney, Shaundell Varney, Shaya Varney, Shloma Varney, Shonnon Varney, Shunta Varney, Silver Varney, Stanislaw Varney, Sumeet Varney, Sylvain Varney, Tajiddin Varney, Tejas Varney, Tereso Varney,
Tilton Varney, Tjay Varney, Tramon Varney, Troi Varney, Troye Varney, Tyrelle Varney, Usher Varney, Valeriano Varney, Verlyn Varney, Vickie Varney, Wallis Varney, Yahya Varney, Yair Varney, Zacharie
Varney, Zakery Varney, Zoe Varney, Abdual Varney, Adeyemi Varney, Alisa Varney, Alois Varney, Alondo Varney, Alprentice Varney, Altariq Varney, Amaro Varney, Amen Varney, Andria Varney, Anthoni
Varney, Apurva Varney, Arrow Varney, Aryn Varney, Atilano Varney, Auburn Varney, Auturo Varney, Aven Varney, Averil Varney, Bernerd Varney, Bethel Varney, Biju Varney, Bogdan Varney, Boston Varney,
Bree Varney, Camaron Varney, Candace Varney, Cara Varney, Cato Varney, Ceferino Varney, Chantry Varney, Chappell Varney, Charels Varney, Ciaran Varney, Codi Varney, Colan Varney, Crayton Varney,
Dakar Varney, Dallis Varney, Danen Varney, Daoud Varney, Davien Varney, Deny Varney, Derius Varney, Derrico Varney, Devere Varney, Dmarcus Varney, Dwright Varney, Dyami Varney, Eileen Varney, Eitan
Varney, Elgie Varney, Elimelech Varney, Ellie Varney, Ellwood Varney, Emmanual Varney, Eryk Varney, Esdras Varney, Estanislao Varney, Euell Varney, Everard Varney, Hagen Varney, Happy Varney, Harald
Varney, Harrington Varney, Hays Varney, Henrik Varney, Hillery Varney, Honorio Varney, Hristos Varney, Ilario Varney, Indalecio Varney, Izaac Varney, Jahmar Varney, Jarome Varney, Jarone Varney,
Javaughn Varney, Jeanne Varney, Jeannie Varney, Jearl Varney, Jenner Varney, Jennie Varney, Jerit Varney, Jet Varney, Jonath Varney, Jumal Varney, Kahil Varney, Kealii Varney, Keeley Varney, Kehinde
Varney, Kene Varney, Kermitt Varney, Keshon Varney, Kore Varney, Lapatrick Varney, Larmar Varney, Laroyce Varney, Larren Varney, Latisha Varney, Latonio Varney, Lewayne Varney, Malachy Varney,
Manford Varney, Marcanthony Varney, Marks Varney, Marley Varney, Marlone Varney, Marquies Varney, Mihir Varney, Minas Varney, Montae Varney, Mortimer Varney, Mykel Varney, Nadia Varney, Nadim Varney,
Naftoli Varney, Nakai Varney, Nashid Varney, Nello Varney, Nichola Varney, Nilo Varney, Nthony Varney, Orenthial Varney, Palani Varney, Pericles Varney, Quentine Varney, Rafiel Varney, Rainey Varney,
Remberto Varney, Renan Varney, Riyad Varney, Robey Varney, Rochell Varney, Romaro Varney, Roosvelt Varney, Satish Varney, Scotte Varney, Sentell Varney, Serapio Varney, Shabaka Varney, Srinivas
Varney, Stanely Varney, Taraus Varney, Tehron Varney, Temujin Varney, Todrick Varney, Torrian Varney, Treven Varney, Trinidy Varney, Tung Varney, Uchenna Varney, Vineet Varney, Waldon Varney, Wilkins
Varney, Xzavier Varney, Yaser Varney, Zahir Varney, Zeus Varney, Abdurrahim Varney, Ahmet Varney, Albion Varney, Andi Varney, Andri Varney, Angello Varney, Angle Varney, Aquilla Varney, Arlee Varney,
Arliss Varney, Asante Varney, Ashwin Varney, Benford Varney, Bernd Varney, Bevin Varney, Blanton Varney, Brandie Varney, Brint Varney, Carle Varney, Carvel Varney, Carvell Varney, Chang Varney,
Charls Varney, Christa Varney, Deidrick Varney, Derrion Varney, Dervin Varney, Devine Varney, Dixie Varney, Dom Varney, Dwone Varney, Edman Varney, Efthimios Varney, Eian Varney, Elazar Varney,
Elridge Varney, Emma Varney, Eoin Varney, Ericka Varney, Everick Varney, Everitt Varney, Franck Varney, Fredi Varney, Ganesh Varney, Gaurav Varney, Geza Varney, Gianpaolo Varney, Grahm Varney, Haden
Varney, Haitham Varney, Hansford Varney, Heberto Varney, Hieu Varney, Jabali Varney, Jabir Varney, Jamilah Varney, Jarriel Varney, Javares Varney, Jawaun Varney, Jessen Varney, Jonanthan Varney,
Joshoa Varney, Joslyn Varney, Jovian Varney, Jozsef Varney, Juanpablo Varney, Justun Varney, Kalief Varney, Kaseen Varney, Kato Varney, Keagan Varney, Keefer Varney, Kishan Varney, Kitwana Varney,
Konata Varney, Lajohn Varney, Lakeisha Varney, Lamichael Varney, Latonia Varney, Lelan Varney, Lendon Varney, Leovardo Varney, Leray Varney, Linh Varney, Lovie Varney, Manasseh Varney, Marcella
Varney, Marguette Varney, Marke Varney, Marlene Varney, Martie Varney, Martrell Varney, Marus Varney, Masaki Varney, Mathhew Varney, Maurisio Varney, Mayur Varney, Meco Varney, Meguel Varney, Meko
Varney, Mensah Varney, Micki Varney, Mikele Varney, Milas Varney, Mirko Varney, Mondell Varney, Mondo Varney, Mondrell Varney, Murdock Varney, Nairobi Varney, Nathaiel Varney, Neco Varney, Neely
Varney, Nelvin Varney, Nichol Varney, Nicholous Varney, Norm Varney, Olander Varney, Pace Varney, Paschal Varney, Patick Varney, Pearson Varney, Phuc Varney, Ramil Varney, Ravinder Varney, Rayman
Varney, Remond Varney, Rendall Varney, Reshawn Varney, Rhian Varney, Romy Varney, Roozbeh Varney, Rutledge Varney, Ryker Varney, Satoshi Varney, Shafton Varney, Shahab Varney, Shantell Varney,
Shontez Varney, Standley Varney, Taha Varney, Tarren Varney, Tavio Varney, Terah Varney, Texas Varney, Thabiti Varney, Theodric Varney, Timtohy Varney, Torence Varney, Torrick Varney, Towan Varney,
Travor Varney, Tyge Varney, Vandell Varney, Virginio Varney, Wilkie Varney, Windsor Varney, Yarrow Varney, Yuji Varney, Zedric Varney, Ziyad Varney, Abie Varney, Abundio Varney, Adi Varney, Aisha
Varney, Akintunde Varney, Alfio Varney, Anival Varney, Antonnio Varney, Asencion Varney, Ashly Varney, Beaux Varney, Bodhi Varney, Bowie Varney, Brandal Varney, Bristol Varney, Carrick Varney, Chales
Varney, Chantz Varney, Chenier Varney, Chima Varney, Chrystopher Varney, Cian Varney, Clearance Varney, Cohen Varney, Colman Varney, Corneil Varney, Dannis Varney, Dantonio Varney, Danyal Varney,
Darus Varney, Daudi Varney, Dejohn Varney, Demeko Varney, Dewaun Varney, Dondrick Varney, Donley Varney, Donney Varney, Donyelle Varney, Dorjan Varney, Drummond Varney, Dwayn Varney, Dywayne Varney,
Euclides Varney, Farhan Varney, Fritzgerald Varney, Gabreil Varney, Gerrald Varney, Greer Varney, Halim Varney, Haroon Varney, Hatim Varney, Herald Varney, Ismeal Varney, Jamesmichael Varney, Jamir
Varney, Janel Varney, Janelle Varney, Jerrol Varney, Jerryl Varney, Jonta Varney, Jumoke Varney, Karanja Varney, Karina Varney, Karis Varney, Karrem Varney, Kavan Varney, Keddrick Varney, Kedrin
Varney, Keifer Varney, Keldric Varney, Kenichi Varney, Kery Varney, Khris Varney, Kiko Varney, Kimothy Varney, Kippy Varney, Kivin Varney, Kojak Varney, Kortez Varney, Lambros Varney, Lealand Varney,
Lillian Varney, Llewelyn Varney, Loron Varney, Maneesh Varney, Mansoor Varney, Mariah Varney, Maricus Varney, Marta Varney, Marte Varney, Martese Varney, Maston Varney, Maylon Varney, Mcclain Varney,
Mikko Varney, Montrez Varney, Najee Varney, Najeeb Varney, Nakoma Varney, Nakuma Varney, Oliverio Varney, Orlondo Varney, Osceola Varney, Osric Varney, Othoniel Varney, Ovid Varney, Peterjohn Varney,
Rad Varney, Rogue Varney, Rolly Varney, Roshad Varney, Sahwn Varney, Santosh Varney, Shakeem Varney, Shanen Varney, Shey Varney, Sina Varney, Sonja Varney, Stone Varney, Stphen Varney, Sue Varney,
Sylvanus Varney, Tameka Varney, Tauris Varney, Tawayne Varney, Tennille Varney, Terrice Varney, Thalamus Varney, Tore Varney, Tranell Varney, Travone Varney, Trevan Varney, Tristram Varney, Tywaun
Varney, Valton Varney, Venice Varney, Wagner Varney, Walberto Varney, Weyman Varney, Willliam Varney, Wisam Varney, Yisrael Varney, Zachari Varney, Zon Varney, Amal Varney, Amie Varney, Antwun
Varney, Audre Varney, Bertrum Varney, Bond Varney, Canon Varney, Careem Varney, Carly Varney, Charmaine Varney, Corian Varney, Coye Varney, Daisuke Varney, Damacio Varney, Damarius Varney, Damiane
Varney, Darrik Varney, Dartanian Varney, Dary Varney, Davan Varney, Deana Varney, Deaundra Varney, Delos Varney, Develle Varney, Dontray Varney, Dorsett Varney, Dyrell Varney, Gered Varney, Gilad
Varney, Hadrian Varney, Haji Varney, Harl Varney, Hideki Varney, Husani Varney, Jacson Varney, Jaimes Varney, Jama Varney, Janathan Varney, Jaran Varney, Jauron Varney, Jeffie Varney, Jejuan Varney,
Jerick Varney, Jermery Varney, Jerramie Varney, Kalei Varney, Karam Varney, Keevan Varney, Kei Varney, Kel Varney, Keo Varney, Khaleel Varney, Klye Varney, Korrie Varney, Kunal Varney, Kyrone Varney,
Ladaryl Varney, Laman Varney, Larome Varney, Latwan Varney, Lavaris Varney, Lavoy Varney, Lenoard Varney, Levarr Varney, Matilde Varney, Mazin Varney, Nakie Varney, Nikolos Varney, Oak Varney, Orrie
Varney, Pleas Varney, Quanta Varney, Rayshun Varney, Rhasaan Varney, Ruark Varney, Saxon Varney, Seith Varney, Sham Varney, Shamont Varney, Shontel Varney, Terre Varney, Thoma Varney, Thunder Varney,
Torance Varney, Travelle Varney, Tulio Varney, Tyrance Varney, Ustin Varney, Verle Varney, Vicky Varney, Wlliam Varney, Xerxes Varney, Yero Varney, Adetokunbo Varney, Adika Varney, Adriane Varney,
Afshin Varney, Alto Varney, Alvon Varney, Anant Varney, Anatole Varney, Aston Varney, Aul Varney, Aza Varney, Azriel Varney, Bay Varney, Bee Varney, Benoit Varney, Bradden Varney, Brayton Varney,
Buren Varney, Caton Varney, Cesareo Varney, Chandar Varney, Charistopher Varney, Chike Varney, Chong Varney, Cline Varney, Dabid Varney, Damascus Varney, Delron Varney, Delvis Varney, Demeterius
Varney, Demetries Varney, Demorio Varney, Demtrius Varney, Derrian Varney, Dilanjan Varney, Donti Varney, Doulgas Varney, Doyce Varney, Eber Varney, Elena Varney, Ellington Varney, Eros Varney,
Esiquio Varney, Esley Varney, Ewan Varney, Firas Varney, Freeland Varney, Frenchie Varney, Garlan Varney, Gaylin Varney, Hamin Varney, Hoyle Varney, Hubbard Varney, Ikechukwu Varney, Issam Varney,
Jameil Varney, Janard Varney, Jann Varney, Jarmain Varney, Jayjay Varney, Jemell Varney, Jerem Varney, Jeremian Varney, Jerritt Varney, Juda Varney, Kahn Varney, Kartik Varney, Katrell Varney, Kavon
Varney, Kelsie Varney, Kelsy Varney, Kenna Varney, Kenner Varney, Kentaro Varney, Kenyotta Varney, Keonte Varney, Kindell Varney, Kodie Varney, Kurtus Varney, Lacory Varney, Latanya Varney, Lathaniel
Varney, Lauriano Varney, Lavares Varney, Lemarr Varney, Lomont Varney, Lyndel Varney, Mandeep Varney, Mansur Varney, Marrion Varney, Mathan Varney, Maurie Varney, Melburn Varney, Mia Varney, Micheil
Varney, Milledge Varney, Mourice Varney, Mousa Varney, Muneer Varney, Naphtali Varney, Navarro Varney, Nehemias Varney, Nickalous Varney, Nickolous Varney, Ola Varney, Oshua Varney, Osmond Varney,
Quinta Varney, Quintel Varney, Rahshawn Varney, Raney Varney, Raye Varney, Rayne Varney, Rockne Varney, Romar Varney, Sebastain Varney, Shang Varney, Sherrard Varney, Smauel Varney, Stamatios Varney,
Talal Varney, Taurin Varney, Terdell Varney, Thong Varney, Tishawn Varney, Tizoc Varney, Trad Varney, Trafton Varney, Trenity Varney, Tyray Varney, Ulices Varney, Vareck Varney, Videll Varney,
Voltaire Varney, Wilmot Varney, Zacary Varney, Aaran Varney, Abdalla Varney, Abdias Varney, Abiodun Varney, Adlai Varney, Ahamad Varney, Alexandar Varney, Alie Varney, Alissa Varney, Alterick Varney,
Amitabh Varney, Andros Varney, Angelia Varney, Antwonne Varney, Arnez Varney, Arrion Varney, Asaf Varney, Auden Varney, Bacilio Varney, Brynn Varney, Byan Varney, Can Varney, Carlen Varney, Carliss
Varney, Carlose Varney, Cazzie Varney, Chadney Varney, Chapman Varney, Chedrick Varney, Cherokee Varney, Christophen Varney, Clearnce Varney, Cleto Varney, Cliford Varney, Colburn Varney, Cordy
Varney, Craigory Varney, Dalan Varney, Dam Varney, Damonn Varney, Dandrea Varney, Danford Varney, Danieal Varney, Dano Varney, Daril Varney, Darrence Varney, Dartanyan Varney, Dawone Varney, Deatrick
Varney, Delayne Varney, Delfred Varney, Delshon Varney, Demark Varney, Demea Varney, Demetra Varney, Demetress Varney, Demeturis Varney, Dene Varney, Derran Varney, Dewann Varney, Dezi Varney,
Dondrell Varney, Donsha Varney, Dontai Varney, Durelle Varney, Ebay Varney, Edin Varney, Efran Varney, Emelio Varney, Erasto Varney, Erby Varney, Eriks Varney, Erine Varney, Ester Varney, Everet
Varney, Excell Varney, Falando Varney, Gaius Varney, Garald Varney, Garlon Varney, Germane Varney, Gernard Varney, Glenden Varney, Graciela Varney, Gram Varney, Grayling Varney, Gunter Varney, Gustaf
Varney, Gwyn Varney, Hamp Varney, Hanley Varney, Harrold Varney, Harvest Varney, Hermenegildo Varney, Hooman Varney, Husam Varney, Idrees Varney, Jaaron Varney, Jaben Varney, Jacent Varney, Jacyn
Varney, Jadrian Varney, Jaisen Varney, Jamard Varney, Jamare Varney, Jarl Varney, Jarvin Varney, Jary Varney, Jashon Varney, Jaso Varney, Jaton Varney, Jawann Varney, Jebidiah Varney, Jedadia Varney,
Jenard Varney, Jeramaine Varney, Jerrimy Varney, Jerrin Varney, Jibreel Varney, Jigar Varney, Joah Varney, Jolene Varney, Jonell Varney, Jordy Varney, Josuah Varney, Jwan Varney, Kantrell Varney,
Keithrick Varney, Kennet Varney, Keshaun Varney, Kief Varney, Kisha Varney, Kiwan Varney, Kizzy Varney, Koury Varney, Kylon Varney, Lakisha Varney, Lamarco Varney, Lavonte Varney, Lemanuel Varney,
Lennell Varney, Letcher Varney, Lethaniel Varney, Levone Varney, Lilton Varney, Lindel Varney, Linzy Varney, Lora Varney, Lorie Varney, Lovelle Varney, Lowry Varney, Lucia Varney, Luisa Varney,
Macheal Varney, Magdiel Varney, Mahmood Varney, Manrique Varney, Manus Varney, Maricela Varney, Marjoe Varney, Marquett Varney, Mars Varney, Mathews Varney, Mattthew Varney, Mehmet Varney, Mehran
Varney, Menashe Varney, Merril Varney, Moe Varney, Morgen Varney, Natalio Varney, Nero Varney, Nickolos Varney, Nidal Varney, Nishant Varney, Nixon Varney, Oreste Varney, Osie Varney, Parks Varney,
Pax Varney, Phillipp Varney, Qaadir Varney, Quantas Varney, Rameen Varney, Ranell Varney, Rani Varney, Rasheim Varney, Rawlin Varney, Rena Varney, Rett Varney, Riko Varney, Rimas Varney, Roark
Varney, Roma Varney, Romale Varney, Rosemary Varney, Roshell Varney, Sadiki Varney, Saquan Varney, Sargon Varney, Shantez Varney, Shawntel Varney, Shelia Varney, Sheryl Varney, Sholem Varney, Sly
Varney, Sohail Varney, Spero Varney, Stepehn Varney, Tabor Varney, Taki Varney, Talley Varney, Tamboura Varney, Tarrus Varney, Tau Varney, Taz Varney, Tejuan Varney, Terik Varney, Terone Varney,
Terreance Varney, Terriel Varney, Terriss Varney, Theodus Varney, Tobian Varney, Tolan Varney, Tollie Varney, Townsend Varney, Tramain Varney, Trennis Varney, Tshaka Varney, Tynell Varney, Tyone
Varney, Tywain Varney, Uganda Varney, Unique Varney, Vahe Varney, Vashone Varney, Verna Varney, Welsey Varney, Woodson Varney, Wydell Varney, Youssef Varney, Zebulin Varney, Zeyad Varney, Aaryn
Varney, Ademola Varney, Adom Varney, Alandus Varney, Alcide Varney, Aldridge Varney, Alfonsa Varney, Alfonzia Varney, Allyson Varney, Alter Varney, Amari Varney, Amr Varney, Ancel Varney, Andie
Varney, Andrus Varney, Antiwan Varney, Antoinette Varney, Antonis Varney, Antwyne Varney, Anupam Varney, Aquila Varney, Arville Varney, Asael Varney, Assad Varney, Aurohom Varney, Autrey Varney, Ayal
Varney, Azikiwe Varney, Bashar Varney, Basim Varney, Beauregard Varney, Belal Varney, Benyamin Varney, Bertis Varney, Bertran Varney, Boban Varney, Boby Varney, Bracy Varney, Branndon Varney, Burtis
Varney, Cameo Varney, Carley Varney, Carnelius Varney, Caroll Varney, Carolos Varney, Carvis Varney, Cashawn Varney, Cean Varney, Charod Varney, Chevelle Varney, Chrisotpher Varney, Christerfer
Varney, Christi Varney, Clois Varney, Cloyce Varney, Constancio Varney, Corkey Varney, Corvin Varney, Corye Varney, Costello Varney, Coury Varney, Crecencio Varney, Cuahutemoc Varney, Curby Varney,
Curly Varney, Dade Varney, Dag Varney, Daimeon Varney, Dakin Varney, Damiean Varney, Dandy Varney, Danian Varney, Davinder Varney, Daylin Varney, Deljuan Varney, Delone Varney, Delta Varney, Demaris
Varney, Demel Varney, Demerick Varney, Demetrous Varney, Demiko Varney, Demorrio Varney, Deonne Varney, Derian Varney, Detron Varney, Deva Varney, Dillion Varney, Divine Varney, Dniel Varney, Doanld
Varney, Donaldo Varney, Donshay Varney, Dontee Varney, Doyne Varney, Dubois Varney, Dulani Varney, Duong Varney, Dupre Varney, Durante Varney, Edith Varney, Edris Varney, Edwyn Varney, Efrin Varney,
Ehrin Varney, Ehsan Varney, Eldwin Varney, Elic Varney, Elis Varney, Elven Varney, Eren Varney, Errik Varney, Erubey Varney, Evertt Varney, Exavier Varney, Fahad Varney, Fares Varney, Fatmir Varney,
Ferando Varney, Fonta Varney, Fotis Varney, Gaither Varney, Gar Varney, Garvey Varney, Gilmer Varney, Godofredo Varney, Greggery Varney, Gregoire Varney, Gregorey Varney, Hagan Varney, Haim Varney,
Hanz Varney, Hartford Varney, Hassen Varney, Haynes Varney, Herrick Varney, Herry Varney, Hoa Varney, Hud Varney, Hurbert Varney, Ilir Varney, Ilyas Varney, Ivica Varney, Jabarr Varney, Jamara
Varney, Jamarco Varney, Jana Varney, Janis Varney, Jarry Varney, Javad Varney, Javas Varney, Javed Varney, Jayden Varney, Jebb Varney, Jeptha Varney, Jerimia Varney, Joffrey Varney, Johanthan Varney,
Johari Varney, Johnanthony Varney, Johnnathan Varney, Johnta Varney, Jolly Varney, Jonahtan Varney, Jong Varney, Jonh Varney, Josep Varney, Joss Varney, Juergen Varney, Juliocesar Varney, Junichi
Varney, Jusitn Varney, Kalid Varney, Kamali Varney, Kannon Varney, Karlon Varney, Kashawn Varney, Keilan Varney, Kelcy Varney, Kellis Varney, Kelon Varney, Kendrich Varney, Kentrel Varney, Kerwyn
Varney, Kesley Varney, Kimon Varney, Kindu Varney, Kinneth Varney, Kondwani Varney, Krystal Varney, Kyran Varney, Lacarlos Varney, Laney Varney, Lelon Varney, Lem Varney, Lena Varney, Leno Varney,
Leshan Varney, Lexie Varney, Lisle Varney, Lonne Varney, Lorence Varney, Lowery Varney, Lundy Varney, Lynford Varney, Maleek Varney, Maliek Varney, Mano Varney, Marcellis Varney, Marcellius Varney,
Marci Varney, Maris Varney, Marl Varney, Marston Varney, Matthe Varney, Matther Varney, Maximiano Varney, Mecca Varney, Melesio Varney, Mic Varney, Mican Varney, Mickal Varney, Mitchum Varney, Mohit
Varney, Myrl Varney, Nael Varney, Najja Varney, Nedal Varney, Nektarios Varney, Nemiah Varney, Niclas Varney, Nilay Varney, Nina Varney, Noberto Varney, Noell Varney, Obrian Varney, Orbie Varney,
Ormond Varney, Orvin Varney, Osborn Varney, Pasco Varney, Pele Varney, Penn Varney, Perryn Varney, Philipe Varney, Prakash Varney, Quarterrio Varney, Quiency Varney, Rahaman Varney, Rahshan Varney,
Rahson Varney, Raimond Varney, Ramsay Varney, Ranard Varney, Ranulfo Varney, Rasha Varney, Rashine Varney, Rason Varney, Ravis Varney, Raylan Varney, Rayme Varney, Raymel Varney, Redrick Varney, Regi
Varney, Renier Varney, Rhea Varney, Riad Varney, Richerd Varney, Rishard Varney, Rockland Varney, Rolfe Varney, Roudy Varney, Saed Varney, Salaam Varney, Samie Varney, Savino Varney, Selim Varney,
Senneca Varney, Seung Varney, Shadric Varney, Shaman Varney, Sharn Varney, Shawen Varney, Shiron Varney, Shohn Varney, Shunn Varney, Sims Varney, Skeet Varney, Soctt Varney, Sohn Varney, Standly
Varney, Starbuck Varney, Stratton Varney, Sulton Varney, Sunday Varney, Sylas Varney, Taff Varney, Tallis Varney, Tamal Varney, Tarrant Varney, Taurice Varney, Teko Varney, Telvis Varney, Terek
Varney, Terrius Varney, Thermon Varney, Thorsten Varney, Timothee Varney, Tirus Varney, Tomi Varney, Torren Varney, Travolta Varney, Travoris Varney, Tyan Varney, Vasco Varney, Vashion Varney, Verron
Varney, Vinicio Varney, Vyron Varney, Wilkin Varney, Winson Varney, Worthy Varney, Wynne Varney, Yance Varney, Yates Varney, Yecheskel Varney, Yonah Varney, Yuseff Varney, Abby Varney, Adaryl Varney,
Addis Varney, Afrim Varney, Ahmand Varney, Ahmon Varney, Alejos Varney, Alphonsa Varney, Alphonsus Varney, Altie Varney, Anastasio Varney, Andris Varney, Ankit Varney, Antario Varney, Antero Varney,
Arafat Varney, Armard Varney, Armel Varney, Arrin Varney, Artavious Varney, Ashlee Varney, Atari Varney, Aundrae Varney, Avan Varney, Avion Varney, Azad Varney, Azariah Varney, Babe Varney, Banks
Varney, Bentura Varney, Bily Varney, Blakeley Varney, Bonner Varney, Bradshaw Varney, Braeden Varney, Brainard Varney, Brianne Varney, Brigg Varney, Broch Varney, Bryian Varney, Buell Varney,
Burchell Varney, Butler Varney, Callen Varney, Cardale Varney, Carlester Varney, Carwin Varney, Cassell Varney, Casy Varney, Chancelor Varney, Chapin Varney, Chee Varney, Chey Varney, Chianti Varney,
Clarenc Varney, Clayburn Varney, Cleofas Varney, Clevester Varney, Clydell Varney, Constance Varney, Coran Varney, Correll Varney, Cotton Varney, Currie Varney, Dae Varney, Dai Varney, Daisy Varney,
Dametrius Varney, Damico Varney, Danal Varney, Danien Varney, Danis Varney, Dar Varney, Darivs Varney, Darmon Varney, Deejay Varney, Delia Varney, Demarquis Varney, Demetrik Varney, Demonta Varney,
Denys Varney, Derec Varney, Derl Varney, Deshay Varney, Destiny Varney, Detroy Varney, Devar Varney, Dewyane Varney, Dierre Varney, Dillan Varney, Din Varney, Dionysios Varney, Dishawn Varney, Doak
Varney, Donaldson Varney, Donaven Varney, Donzel Varney, Doremus Varney, Dorn Varney, Doss Varney, Duglas Varney, Durk Varney, Dushon Varney, Duvon Varney, Dwaylon Varney, Dywan Varney, Edjuan
Varney, Effrem Varney, Eliel Varney, Elizah Varney, Elray Varney, Elsa Varney, Emerick Varney, Epigmenio Varney, Epimenio Varney, Erasmus Varney, Erez Varney, Erman Varney, Esgar Varney, Everrett
Varney, Fateen Varney, Felimon Varney, Felis Varney, Ferguson Varney, Firman Varney, Fitz Varney, France Varney, Freddick Varney, Frederi Varney, Gadiel Varney, Galdino Varney, Gant Varney, Gared
Varney, Garic Varney, Garrell Varney, Gavan Varney, Gemini Varney, Genard Varney, Geoge Varney, Germany Varney, Gioacchino Varney, Gladys Varney, Glennie Varney, Glenroy Varney, Gordy Varney, Graylon
Varney, Griselda Varney, Gussie Varney, Hall Varney, Hammad Varney, Hamza Varney, Handy Varney, Hardie Varney, Hassel Varney, Heiko Varney, Henley Varney, Hersh Varney, Hewitt Varney, Hiran Varney,
Hoke Varney, Holmes Varney, Hunt Varney, Ikenna Varney, Imre Varney, Irma Varney, Isaul Varney, Ishmail Varney, Ishmel Varney, Iyad Varney, Jaamal Varney, Jabaar Varney, Jacquin Varney, Jameison
Varney, Jasom Varney, Jefery Varney, Jeffy Varney, Jermanine Varney, Jerol Varney, Jesses Varney, Jibril Varney, Jimy Varney, Joffre Varney, Johm Varney, Johne Varney, Johnothan Varney, Josias
Varney, Joslin Varney, Jovanny Varney, Juel Varney, Kaare Varney, Kalib Varney, Kalim Varney, Kalum Varney, Kedron Varney, Kerim Varney, Kerman Varney, Keyan Varney, Khai Varney, Khareem Varney,
Kierre Varney, Kimmie Varney, Kipton Varney, Kord Varney, Lajaune Varney, Lamor Varney, Lancy Varney, Lando Varney, Landrum Varney, Lani Varney, Larmont Varney, Laurens Varney, Lawrenc Varney, Lealon
Varney, Leeman Varney, Legrande Varney, Leone Varney, Lewie Varney, Loretta Varney, Lorrin Varney, Loukas Varney, Luisito Varney, Lutalo Varney, Machel Varney, Macklin Varney, Mahir Varney, Manpreet
Varney, Mansour Varney, Marcum Varney, Mariusz Varney, Markcus Varney, Marquest Varney, Marven Varney, Mattias Varney, Melbourne Varney, Melisa Varney, Mihcael Varney, Mile Varney, Monserrate Varney,
Montique Varney, Morice Varney, Nataniel Varney, Neiman Varney, Noa Varney, Obdulio Varney, Odilon Varney, Oma Varney, Onathan Varney, Orlandis Varney, Osamah Varney, Pastor Varney, Patterson Varney,
Pattrick Varney, Pavlos Varney, Phoenix Varney, Pierson Varney, Puneet Varney, Quartez Varney, Quetin Varney, Rahsheen Varney, Randee Varney, Ras Varney, Raymie Varney, Raymound Varney, Raysean
Varney, Regginal Varney, Reico Varney, Reino Varney, Reynardo Varney, Rodrecus Varney, Rodrickus Varney, Rolin Varney, Royston Varney, Rynell Varney, Ryo Varney, Sacramento Varney, Seaborn Varney,
Sedgwick Varney, Shaddrick Varney, Shaffer Varney, Shamal Varney, Shamone Varney, Sharman Varney, Sharrief Varney, Shawnell Varney, Shawntae Varney, Shimshon Varney, Shontay Varney, Shuaib Varney,
Steveland Varney, Stonewall Varney, Sujal Varney, Tage Varney, Tarin Varney, Tarrick Varney, Tarryl Varney, Tauras Varney, Tayvon Varney, Terrol Varney, Thearon Varney, Thinh Varney, Tirell Varney,
Tobius Varney, Tobyn Varney, Toderick Varney, Torben Varney, Torran Varney, Torray Varney, Traven Varney, Trek Varney, Trina Varney, Twayne Varney, Tyee Varney, Tyrick Varney, Tyvon Varney, Ubong
Varney, Ulysess Varney, Valerian Varney, Vennie Varney, Venus Varney, Vincen Varney, Vonn Varney, Wandell Varney, Warden Varney, Warnell Varney, Wille Varney, Wolfram Varney, Worley Varney, Wren
Varney, Yoseph Varney, Zacharian Varney, Zacheriah Varney, Anoop Varney, Antohny Varney, Bianco Varney, Billey Varney, Blanca Varney, Bradely Varney, Brison Varney, Chanson Varney, Cheng Varney,
Choice Varney, Cosmos Varney, Cymande Varney, Cyprian Varney, Dang Varney, Derrus Varney, Dorwin Varney, Everest Varney, Farzad Varney, Gralin Varney, Idi Varney, Jakari Varney, Jezreel Varney,
Jorell Varney, Kong Varney, Liza Varney, Mendy Varney, Onix Varney, Rajaee Varney, Random Varney, Rashean Varney, Revis Varney, Saman Varney, Santini Varney, Shahin Varney, Talton Varney, Teco
Varney, Temitope Varney, Vimal Varney, Wacey Varney, Aeric Varney, Alhaji Varney, Anas Varney, Anselm Varney, Bamidele Varney, Behrang Varney, Binu Varney, Brenon Varney, Brittney Varney, Catina
Varney, Chaunce Varney, Christerphor Varney, Claxton Varney, Colley Varney, Courtnee Varney, Darrio Varney, Darweshi Varney, Dashiell Varney, Daymeon Varney, Deontay Varney, Devlyn Varney, Effrey
Varney, Eldric Varney, Faith Varney, Garris Varney, Holley Varney, Hombre Varney, Hussain Varney, Jamari Varney, Jaramy Varney, Jaydon Varney, Jeramee Varney, Jerediah Varney, Jermiane Varney,
Jermont Varney, Jerud Varney, Johncharles Varney, Jolan Varney, Kairaba Varney, Kawon Varney, Kemon Varney, Kennen Varney, Kenyun Varney, Khevin Varney, Kiowa Varney, Kwabene Varney, Ladrick Varney,
Lamarc Varney, Laramy Varney, Larance Varney, Lebrone Varney, Loc Varney, Lynda Varney, Marcius Varney, Mariel Varney, Marisela Varney, Meshach Varney, Milon Varney, Monterio Varney, Najib Varney,
Natha Varney, Nery Varney, Niklaus Varney, Pinchus Varney, Quaid Varney, Rael Varney, Rain Varney, Ralpheal Varney, Ramiah Varney, Redmond Varney, Regniald Varney, Rennard Varney, Rhet Varney, Ritesh
Varney, Rodolpho Varney, Rontrell Varney, Rossano Varney, Shale Varney, Shevin Varney, Shiraz Varney, Sirica Varney, Sridhar Varney, Staton Varney, Suliman Varney, Tchalla Varney, Telvin Varney,
Terren Varney, Traun Varney, Trong Varney, Trygve Varney, Unborn Varney, Vaden Varney, Valery Varney, Vuong Varney, Waseem Varney, Zebulen Varney, Abed Varney, Abhay Varney, Adebayo Varney, Ahmid
Varney, Akai Varney, Aldwin Varney, Altonia Varney, Alvertis Varney, Alvey Varney, Amondo Varney, Amrom Varney, Andera Varney, Andrico Varney, Anterio Varney, Antolin Varney, Antorio Varney, Arkim
Varney, Arran Varney, Aslan Varney, Ausencio Varney, Aviel Varney, Avrum Varney, Ayron Varney, Azeem Varney, Azel Varney, Azell Varney, Basel Varney, Bengi Varney, Benjamyn Varney, Berish Varney,
Bianca Varney, Bonny Varney, Boy Varney, Brach Varney, Brishen Varney, Britney Varney, Burlin Varney, Buzz Varney, Carolina Varney, Casimer Varney, Cedeno Varney, Cemal Varney, Chadly Varney, Chen
Varney, Chirs Varney, Christion Varney, Christohpher Varney, Christpoher Varney, Clate Varney, Cleaven Varney, Clif Varney, Coburn Varney, Conroy Varney, Corbitt Varney, Crispus Varney, Daiman
Varney, Daimion Varney, Damarco Varney, Dammian Varney, Darr Varney, Darreyl Varney, Davar Varney, Dawain Varney, Dayron Varney, Dectrick Varney, Dederick Varney, Demarkis Varney, Denman Varney,
Derril Varney, Devanand Varney, Dharma Varney, Dionysus Varney, Domanick Varney, Dorey Varney, Duel Varney, Ebenezer Varney, Edon Varney, Effie Varney, Ehron Varney, Eliceo Varney, Elija Varney,
Elijha Varney, Elizandro Varney, Enriquez Varney, Eryn Varney, Essie Varney, Etoyi Varney, Fabrice Varney, Farouk Varney, Garick Varney, Gean Varney, Geovani Varney, Germar Varney, Gillian Varney,
Gilmore Varney, Giuseppi Varney, Governor Varney, Graden Varney, Gresham Varney, Haashim Varney, Hagop Varney, Hamzah Varney, Harlie Varney, Hatem Varney, Hien Varney, Hose Varney, Ibe Varney, Ingo
Varney, Irven Varney, Isam Varney, Ishi Varney, Istvan Varney, Jabriel Varney, Jamont Varney, Janette Varney, Jaroslaw Varney, Jashawn Varney, Java Varney, Javid Varney, Jaysun Varney, Jemario
Varney, Jemiah Varney, Jerek Varney, Jeren Varney, Jeriah Varney, Jerode Varney, Jimmel Varney, Jitu Varney, Jmichael Varney, Johnas Varney, Johnwilliam Varney, Joran Varney, Jori Varney, Josehua
Varney, Joshlin Varney, Joushua Varney, Juane Varney, Juba Varney, Juvon Varney, Kamaal Varney, Kasib Varney, Kate Varney, Kayin Varney, Keffer Varney, Kentay Varney, Kentrail Varney, Kenyell Varney,
Kerr Varney, Khalik Varney, Khurram Varney, Kipchoge Varney, Koa Varney, Kowan Varney, Kreig Varney, Kristien Varney, Krystopher Varney, Kurry Varney, Kylee Varney, Lamaine Varney, Lamell Varney,
Lamour Varney, Lanoris Varney, Laquann Varney, Laquinn Varney, Lavone Varney, Leanord Varney, Leeandrew Varney, Lenzy Varney, Lequient Varney, Leshun Varney, Lopaka Varney, Lopez Varney, Lyall
Varney, Malcoln Varney, Mali Varney, March Varney, Marchant Varney, Matthen Varney, Medrick Varney, Mildred Varney, Mitchelle Varney, Moriah Varney, Mykal Varney, Neema Varney, Nii Varney, Nikkia
Varney, Okechukwu Varney, Olatunde Varney, Olu Varney, Oluseyi Varney, Orazio Varney, Orel Varney, Ortiz Varney, Orvel Varney, Oseph Varney, Overton Varney, Peer Varney, Phyllis Varney, Pratik
Varney, Quante Varney, Quindell Varney, Raeshawn Varney, Rahshon Varney, Ran Varney, Real Varney, Rhodes Varney, Rhodney Varney, Rodeny Varney, Rondi Varney, Rozelle Varney, Rutilio Varney, Saif
Varney, Sandford Varney, Seab Varney, Sevag Varney, Shahied Varney, Shamari Varney, Shammah Varney, Sharmon Varney, Shervin Varney, Skippy Varney, Stefanie Varney, Stylianos Varney, Tafari Varney,
Tag Varney, Tamon Varney, Tanisha Varney, Tearle Varney, Tellys Varney, Terriance Varney, Timber Varney, Timthoy Varney, Tomiko Varney, Torr Varney, Torrez Varney, Toshiro Varney, Trammell Varney,
Trance Varney, Tregg Varney, Treston Varney, Tristian Varney, Tymaine Varney, Tyne Varney, Tyrane Varney, Vada Varney, Valerio Varney, Varion Varney, Vasean Varney, Vaugh Varney, Viviana Varney,
Welford Varney, Windel Varney, Yaacov Varney, Yaniv Varney, Yusuke Varney, Zebulah Varney, Zuberi Varney, Aarin Varney, Abbott Varney, Abdule Varney, Abdulrahman Varney, Abrahim Varney, Abrian
Varney, Adham Varney, Aditya Varney, Adre Varney, Akito Varney, Ala Varney, Alberico Varney, Alexsandro Varney, Aljandro Varney, Allin Varney, Alyssa Varney, Amond Varney, Angelique Varney, Ankoma
Varney, Antonne Varney, Antroine Varney, Antwanne Varney, Antwayne Varney, Arben Varney, Arby Varney, Arcangelo Varney, Arek Varney, Arel Varney, Artavis Varney, Arvon Varney, Arye Varney, Ascension
Varney, Ashanta Varney, Atilla Varney, Atrick Varney, Austreberto Varney, Avant Varney, Avian Varney, Avier Varney, Ayatollah Varney, Baltasar Varney, Banning Varney, Barin Varney, Barlow Varney,
Barnie Varney, Barrick Varney, Baruti Varney, Bascom Varney, Basheer Varney, Bear Varney, Beatriz Varney, Bengt Varney, Bennette Varney, Berl Varney, Berley Varney, Berwin Varney, Biren Varney, Blase
Varney, Bon Varney, Bowe Varney, Bowman Varney, Brance Varney, Bravlio Varney, Brette Varney, Brewster Varney, Bria Varney, Brookes Varney, Buel Varney, Buffy Varney, Cairo Varney, Caldwell Varney,
Cali Varney, Carlan Varney, Carmello Varney, Carmichael Varney, Carney Varney, Carr Varney, Carrell Varney, Carzell Varney, Castro Varney, Caswell Varney, Catrell Varney, Cedrice Varney, Celeste
Varney, Celina Varney, Cesear Varney, Chade Varney, Chadwell Varney, Chalres Varney, Chamar Varney, Chanan Varney, Chanda Varney, Chanin Varney, Charo Varney, Chayim Varney, Chelton Varney, Chevez
Varney, Chih Varney, Chin Varney, Chino Varney, Christaphor Varney, Ciprian Varney, Conny Varney, Corbit Varney, Corinthians Varney, Cosby Varney, Curlee Varney, Cyrano Varney, Daimian Varney, Daks
Varney, Dalin Varney, Damir Varney, Damisi Varney, Damiso Varney, Dantae Varney, Danthony Varney, Dantoni Varney, Darcus Varney, Darence Varney, Darrious Varney, Darylle Varney, Datril Varney,
Davidlee Varney, Dawane Varney, Daylen Varney, Deandrew Varney, Deloy Varney, Delson Varney, Demaine Varney, Dementrius Varney, Demetrise Varney, Demien Varney, Demingo Varney, Demitris Varney, Deni
Varney, Dennys Varney, Denon Varney, Dent Varney, Deondrick Varney, Derelle Varney, Derike Varney, Derrall Varney, Des Varney, Deshea Varney, Deston Varney, Deval Varney, Devery Varney, Deville
Varney, Devrin Varney, Dewell Varney, Diondray Varney, Dishan Varney, Dkwon Varney, Dmario Varney, Dolph Varney, Dominador Varney, Donae Varney, Donni Varney, Dontea Varney, Donye Varney, Dossie
Varney, Dray Varney, Dwann Varney, Earic Varney, Earley Varney, Eaton Varney, Edwar Varney, Eiad Varney, Eirc Varney, Eith Varney, Elba Varney, Elefterios Varney, Eleodoro Varney, Elisabeth Varney,
Elrod Varney, Elzy Varney, Emilo Varney, Emmanouel Varney, Enis Varney, Ephrain Varney, Erec Varney, Erique Varney, Eriverto Varney, Erven Varney, Esco Varney, Esquiel Varney, Essa Varney, Ettore
Varney, Eulis Varney, Evay Varney, Fabius Varney, Faiz Varney, Felando Varney, Fenwick Varney, Fernell Varney, Francesca Varney, Francisca Varney, Franke Varney, Gabiel Varney, Garrit Varney, Garyn
Varney, Geffery Varney, Generoso Varney, Genghis Varney, Georgia Varney, Geovanny Varney, Gerell Varney, Gerred Varney, Gerrett Varney, Ghassan Varney, Gilliam Varney, Gilson Varney, Givon Varney,
Goeffrey Varney, Gorgonio Varney, Grahame Varney, Grayland Varney, Grzegorz Varney, Guan Varney, Gumercindo Varney, Gwen Varney, Hannah Varney, Harden Varney, Harsha Varney, Harvy Varney, Hassain
Varney, Helio Varney, Hemant Varney, Herchel Varney, Herold Varney, Hershy Varney, Hillis Varney, Hinton Varney, Hommy Varney, Hong Varney, Huberto Varney, Hughes Varney, Husain Varney, Hussien
Varney, Ibis Varney, Ilias Varney, Irl Varney, Iron Varney, Isak Varney, Ishaq Varney, Issaac Varney, Iva Varney, Jabon Varney, Jachin Varney, Jacobus Varney, Janeiro Varney, Japhet Varney, Jarion
Varney, Jaris Varney, Jarmell Varney, Jarnell Varney, Jarvon Varney, Jaryl Varney, Jasonn Varney, Jauan Varney, Javian Varney, Javoris Varney, Jawhar Varney, Jawwaad Varney, Jayesh Varney, Jeffree
Varney, Jeffri Varney, Jerauld Varney, Jereny Varney, Jereomy Varney, Jerom Varney, Jerrall Varney, Jerran Varney, Jerrico Varney, Jessiah Varney, Jesten Varney, Jevan Varney, Jimbob Varney, Jiro
Varney, Johnpatrick Varney, Jolon Varney, Jonothon Varney, Jonthomas Varney, Josephine Varney, Jovonne Varney, Juanantonio Varney, Julis Varney, Jumah Varney, Jwyanza Varney, Kail Varney, Kaleem
Varney, Kalil Varney, Kamari Varney, Kamien Varney, Kanon Varney, Kanta Varney, Karac Varney, Karega Varney, Karras Varney, Kealoha Varney, Kearston Varney, Kedryn Varney, Keiven Varney, Keland
Varney, Kelechi Varney, Kelson Varney, Kendle Varney, Kenjuan Varney, Kennell Varney, Kenson Varney, Kerick Varney, Ketric Varney, Kevine Varney, Keylon Varney, Keynan Varney, Khaaliq Varney, Khali
Varney, Khang Varney, Khoi Varney, Kinshasa Varney, Kono Varney, Konstandinos Varney, Kontar Varney, Kou Varney, Kreston Varney, Kristapher Varney, Kristophor Varney, Kumasi Varney, Kuntakinte
Varney, Kwami Varney, Kwon Varney, Laderick Varney, Lajon Varney, Lakesha Varney, Lakieth Varney, Lakota Varney, Lamarkus Varney, Lameen Varney, Lamondo Varney, Lamonica Varney, Landers Varney,
Landrick Varney, Lanis Varney, Lapriest Varney, Laquane Varney, Larvell Varney, Larwence Varney, Latavius Varney, Latimer Varney, Laton Varney, Latrelle Varney, Latwon Varney, Laurencio Varney,
Lavelton Varney, Laver Varney, Lavester Varney, Lavonn Varney, Lawanda Varney, Lawarren Varney, Lazerick Varney, Lecedric Varney, Leconte Varney, Leib Varney, Lemel Varney, Lenis Varney, Lenox
Varney, Lenward Varney, Leonid Varney, Leory Varney, Lerin Varney, Leroyal Varney, Linnell Varney, Linville Varney, Lional Varney, Loel Varney, Lofton Varney, Londale Varney, Lonza Varney, Lovis
Varney, Lucy Varney, Luie Varney, Lyndall Varney, Macker Varney, Makarios Varney, Malcon Varney, Manly Varney, Marell Varney, Margo Varney, Markco Varney, Markeis Varney, Markian Varney, Marsden
Varney, Martina Varney, Martinus Varney, Mashon Varney, Masud Varney, Mathaniel Varney, Mazi Varney, Merel Varney, Merwyn Varney, Michaelanthony Varney, Michaeljames Varney, Mico Varney, Mihran
Varney, Mikki Varney, Milik Varney, Milos Varney, Miran Varney, Mirza Varney, Mishael Varney, Mondale Varney, Mondre Varney, Montsho Varney, Morse Varney, Mostafa Varney, Mumin Varney, Mynor Varney,
Nantambu Varney, Narayana Varney, Naseem Varney, Nathane Varney, Navada Varney, Nazar Varney, Nectarios Varney, Nell Varney, Nevelle Varney, Nichalaus Varney, Nichlous Varney, Nicholson Varney,
Nicklus Varney, Nickolai Varney, Nicolino Varney, Nihar Varney, Nishan Varney, Nizar Varney, Nocholas Varney, Noelle Varney, Novel Varney, Nyerere Varney, Obbie Varney, Ocean Varney, Oded Varney,
Oden Varney, Olaniyan Varney, Ondray Varney, Orentha Varney, Osby Varney, Othell Varney, Othniel Varney, Ottie Varney, Panayiotis Varney, Parren Varney, Pedram Varney, Petey Varney, Petter Varney,
Pharaoh Varney, Precious Varney, Pressley Varney, Procopio Varney, Pryor Varney, Quaashie Varney, Quinte Varney, Quintez Varney, Quinzell Varney, Quirino Varney, Rabon Varney, Rachard Varney, Rade
Varney, Raju Varney, Ramee Varney, Ranjan Varney, Ransford Varney, Rashawd Varney, Raynald Varney, Regenal Varney, Reiko Varney, Reinhold Varney, Remijio Varney, Renaud Varney, Ritchard Varney,
Roanld Varney, Roarke Varney, Robley Varney, Rochester Varney, Rodolph Varney, Roi Varney, Romelle Varney, Romey Varney, Romolo Varney, Rondo Varney, Ronni Varney, Ronta Varney, Roselio Varney,
Rosheen Varney, Ruban Varney, Rustan Varney, Sajan Varney, Salik Varney, Samel Varney, Santonia Varney, Sargent Varney, Saud Varney, Saxton Varney, Schane Varney, Sebron Varney, Shadley Varney,
Shadron Varney, Shafi Varney, Shalako Varney, Shanell Varney, Shani Varney, Shantae Varney, Shari Varney, Shauntay Varney, Shauwn Varney, Shawkat Varney, Shell Varney, Shephen Varney, Sherief Varney,
Shin Varney, Shneur Varney, Shontell Varney, Shoun Varney, Shundell Varney, Sigmond Varney, Spanky Varney, Stpehen Varney, Suhail Varney, Sulieman Varney, Sundown Varney, Taboris Varney, Taijuan
Varney, Talion Varney, Tamario Varney, Tamel Varney, Tana Varney, Tari Varney, Tarnell Varney, Tashon Varney, Tasos Varney, Tavar Varney, Tawon Varney, Tay Varney, Tederick Varney, Tedman Varney,
Teejay Varney, Tephen Varney, Terranc Varney, Terric Varney, Tevita Varney, Thao Varney, Theoplis Varney, Thuan Varney, Tiger Varney, Tigh Varney, Timmon Varney, Tin Varney, Tionne Varney, Tipton
Varney, Todo Varney, Tomatra Varney, Tomeka Varney, Tomie Varney, Toriono Varney, Torivio Varney, Toye Varney, Tramine Varney, Tre Varney, Treaver Varney, Tredell Varney, Treva Varney, Trinton
Varney, TRUE Varney, Tydus Varney, Tyeson Varney, Tyle Varney, Tyrae Varney, Tyrand Varney, Tyresse Varney, Tyri Varney, Tysean Varney, Tyus Varney, Uland Varney, Usman Varney, Vahid Varney, Vang
Varney, Vassilios Varney, Vernis Varney, Vidale Varney, Vikash Varney, Vishnu Varney, Volney Varney, Vondale Varney, Warees Varney, Webber Varney, Whitman Varney, Willilam Varney, Windy Varney, Winn
Varney, Wynell Varney, Xan Varney, Yeshaya Varney, Yuma Varney, Zackory Varney, Zahid Varney, Zarak Varney, Zayd Varney, Zebula Varney, Zef Varney, Zeljko Varney, Zephyr Varney, Zeshan Varney, Aakash
Varney, Aalon Varney, Aaraon Varney, Aaric Varney, Aarn Varney, Abduel Varney, Abid Varney, Abigail Varney, Abijah Varney, Achille Varney, Ada Varney, Adali Varney, Adante Varney, Adarsh Varney,
Adedayo Varney, Adeyinka Varney, Adib Varney, Adran Varney, Adrell Varney, Adrew Varney, Adriaan Varney, Adriene Varney, Afton Varney, Agim Varney, Ahmond Varney, Aiman Varney, Akiba Varney, Alamin
Varney, Aland Varney, Alanson Varney, Albertico Varney, Albertus Varney, Alcario Varney, Aldrin Varney, Alejando Varney, Alexey Varney, Alfreda Varney, Alfreddie Varney, Allateef Varney, Almer
Varney, Almondo Varney, Alpesh Varney, Alrahman Varney, Alson Varney, Altarik Varney, Alven Varney, Amadi Varney, Amancio Varney, Amandeep Varney, Amauris Varney, Amdrew Varney, Amel Varney, Amelia
Varney, Amhad Varney, Amrit Varney, Ande Varney, Anderw Varney, Andon Varney, Andren Varney, Andretti Varney, Andrius Varney, Andrue Varney, Andrw Varney, Aneesh Varney, Aneudy Varney, Anglo Varney,
Anissa Varney, Annie Varney, Anquan Varney, Ansar Varney, Ansen Varney, Anslem Varney, Anthone Varney, Anthory Varney, Antoinio Varney, Antojuan Varney, Antonial Varney, Antowain Varney, Antowine
Varney, Antrione Varney, Antroy Varney, Anttwan Varney, Antwin Varney, Antwonn Varney, Antyon Varney, Anwon Varney, Aquan Varney, Aquarius Varney, Aracely Varney, Aragorn Varney, Arbie Varney, Arhtur
Varney, Aristede Varney, Aristotelis Varney, Arkee Varney, Arlandus Varney, Arlondo Varney, Armstead Varney, Arnab Varney, Aroldo Varney, Arric Varney, Arshad Varney, Artice Varney, Arvelle Varney,
Arvie Varney, Aryan Varney, Arzell Varney, Asbury Varney, Aseem Varney, Ash Varney, Ashan Varney, Ashkan Varney, Ashlin Varney, Ashur Varney, Astor Varney, Asuncion Varney, Ata Varney, Atsushi
Varney, Aubery Varney, Audi Varney, Augustino Varney, Aurther Varney, Aviv Varney, Ayan Varney, Azael Varney, Azar Varney, Azhar Varney, Azizi Varney, Bai Varney, Bain Varney, Baldo Varney, Balentin
Varney, Banyon Varney, Barbaro Varney, Barett Varney, Barnell Varney, Barren Varney, Bartolomeo Varney, Basem Varney, Basilios Varney, Bassey Varney, Bayete Varney, Baylen Varney, Bayne Varney,
Bayron Varney, Beaufort Varney, Bejan Varney, Belvin Varney, Benajmin Varney, Bengamin Varney, Bently Varney, Berel Varney, Berman Varney, Bernadino Varney, Bernis Varney, Bernon Varney, Bertie
Varney, Bertin Varney, Bibiano Varney, Biko Varney, Binyamin Varney, Birche Varney, Blakley Varney, Bleu Varney, Blythe Varney, Bobak Varney, Bowden Varney, Boz Varney, Brahim Varney, Brahin Varney,
Brandell Varney, Brandley Varney, Brann Varney, Brannigan Varney, Brantly Varney, Brayan Varney, Braydon Varney, Breandan Varney, Breh Varney, Brence Varney, Brenner Varney, Brentwood Varney, Bric
Varney, Bridgette Varney, Brig Varney, Briggs Varney, Brionne Varney, Bronc Varney, Brunson Varney, Brycen Varney, Jennifer Varney, Amy Varney, Melissa Varney, Michelle Varney, Kimberly Varney, Lisa
Varney, Angela Varney, Heather Varney, Stephanie Varney, Nicole Varney, Jessica Varney, Elizabeth Varney, Rebecca Varney, Kelly Varney, Mary Varney, Christina Varney, Amanda Varney, Julie Varney,
Sarah Varney, Laura Varney, Shannon Varney, Christine Varney, Tammy Varney, Tracy Varney, Karen Varney, Dawn Varney, Susan Varney, Andrea Varney, Tina Varney, Patricia Varney, Cynthia Varney, Lori
Varney, Rachel Varney, April Varney, Maria Varney, Wendy Varney, Crystal Varney, Stacy Varney, Erin Varney, Jamie Varney, Carrie Varney, Tiffany Varney, Tara Varney, Sandra Varney, Monica Varney,
Danielle Varney, Stacey Varney, Pamela Varney, Tonya Varney, Sara Varney, Michele Varney, Teresa Varney, Denise Varney, Jill Varney, Katherine Varney, Melanie Varney, Dana Varney, Holly Varney, Erica
Varney, Brenda Varney, Deborah Varney, Tanya Varney, Sharon Varney, Donna Varney, Amber Varney, Emily Varney, Linda Varney, Robin Varney, Kathleen Varney, Leslie Varney, Christy Varney, Kristen
Varney, Catherine Varney, Kristin Varney, Misty Varney, Barbara Varney, Heidi Varney, Nancy Varney, Cheryl Varney, Theresa Varney, Brandy Varney, Alicia Varney, Veronica Varney, Gina Varney,
Jacqueline Varney, Rhonda Varney, Anna Varney, Renee Varney, Megan Varney, Tamara Varney, Kathryn Varney, Melinda Varney, Debra Varney, Sherry Varney, Allison Varney, Valerie Varney, Diana Varney,
Paula Varney, Kristina Varney, Ann Varney, Margaret Varney, Cindy Varney, Victoria Varney, Jodi Varney, Natalie Varney, Brandi Varney, Kristi Varney, Suzanne Varney, Samantha Varney, Beth Varney,
Tracey Varney, Regina Varney, Vanessa Varney, Kristy Varney, Carolyn Varney, Yolanda Varney, Deanna Varney, Carla Varney, Sheila Varney, Laurie Varney, Anne Varney, Shelly Varney, Diane Varney,
Sabrina Varney, Janet Varney, Katrina Varney, Erika Varney, Courtney Varney, Colleen Varney, Carol Varney, Julia Varney, Jenny Varney, Jaime Varney, Kathy Varney, Felicia Varney, Alison Varney,
Lauren Varney, Kelli Varney, Leah Varney, Ashley Varney, Kim Varney, Traci Varney, Kristine Varney, Tricia Varney, Joy Varney, Krista Varney, Kara Varney, Terri Varney, Sonya Varney, Aimee Varney,
Natasha Varney, Cassandra Varney, Bridget Varney, Anita Varney, Kari Varney, Nichole Varney, Christie Varney, Marie Varney, Virginia Varney, Connie Varney, Martha Varney, Carmen Varney, Stacie
Varney, Lynn Varney, Monique Varney, Katie Varney, Kristie Varney, Shelley Varney, Sherri Varney, Angel Varney, Bonnie Varney, Mandy Varney, Jody Varney, Shawna Varney, Kerry Varney, Annette Varney,
Yvonne Varney, Toni Varney, Meredith Varney, Molly Varney, Kendra Varney, Joanna Varney, Sonia Varney, Janice Varney, Robyn Varney, Brooke Varney, Kerri Varney, Sheri Varney, Becky Varney, Gloria
Varney, Mindy Varney, Tracie Varney, Angie Varney, Kellie Varney, Claudia Varney, Ruth Varney, Wanda Varney, Jeanette Varney, Cathy Varney, Adrienne Varney, Kelley Varney, Rachael Varney, Beverly
Varney, Candace Varney, Sylvia Varney, Penny Varney, Charlene Varney, Trisha Varney, Charlotte Varney, Belinda Varney, Candice Varney, Yvette Varney, Lindsay Varney, Keri Varney, Melody Varney,
Caroline Varney, Rosa Varney, Bethany Varney, Trina Varney, Joyce Varney, Joanne Varney, Tabitha Varney, Vicki Varney, Tamika Varney, Gretchen Varney, Ginger Varney, Betty Varney, Dorothy Varney,
Debbie Varney, Darlene Varney, Helen Varney, Latoya Varney, Ellen Varney, Leigh Varney, Rose Varney, Shawn Varney, Karla Varney, Frances Varney, Ana Varney, Alice Varney, Jean Varney, Hope Varney,
Tasha Varney, Latonya Varney, Latasha Varney, Nikki Varney, Maureen Varney, Bobbie Varney, Rita Varney, Peggy Varney, Shirley Varney, Marsha Varney, Cara Varney, Christa Varney, Audrey Varney, Norma
Varney, Shana Varney, Juanita Varney, Leticia Varney, Kimberley Varney, Billie Varney, Evelyn Varney, Tonia Varney, Desiree Varney, Judith Varney, Joann Varney, Shanna Varney, Elaine Varney, Angelica
Varney, Charity Varney, Staci Varney, Tami Varney, Judy Varney, Rebekah Varney, Raquel Varney, Tammie Varney, Jane Varney, Meghan Varney, Jackie Varney, Sally Varney, Jana Varney, Priscilla Varney,
Sandy Varney, Rochelle Varney, Summer Varney, Jodie Varney, Marcia Varney, Alisha Varney, Keisha Varney, Stefanie Varney, Casey Varney, Loretta Varney, Eva Varney, Dena Varney, Cristina Varney,
Janelle Varney, Christi Varney, Naomi Varney, Rachelle Varney, Marisa Varney, Irene Varney, Kirsten Varney, Jeanne Varney, Jenifer Varney, Roxanne Varney, Miranda Varney, Roberta Varney, Dina Varney,
Shauna Varney, Sonja Varney, Marilyn Varney, Eileen Varney, Gwendolyn Varney, Vickie Varney, Katina Varney, Teri Varney, Olivia Varney, Amie Varney, Lora Varney, Cheri Varney, Jennie Varney, Lindsey
Varney, Lesley Varney, Lara Varney, Lydia Varney, Alisa Varney, Sheryl Varney, Autumn Varney, Jacquelyn Varney, Lynette Varney, Esther Varney, Nina Varney, Antoinette Varney, Gail Varney, Deana
Varney, Jaclyn Varney, Lorraine Varney, Alexandra Varney, Jami Varney, Ronda Varney, Candy Varney, Bobbi Varney, Ericka Varney, Abigail Varney, Chandra Varney, Ebony Varney, Latisha Varney, Cherie
Varney, Marla Varney, Lee Varney, Angelia Varney, Kenya Varney, Joan Varney, Jeannie Varney, Shari Varney, Ruby Varney, Miriam Varney, Lakisha Varney, Tameka Varney, Marcy Varney, Angelina Varney,
Faith Varney, Marisol Varney, Krystal Varney, Dianna Varney, Adriana Varney, Audra Varney, Angelique Varney, Marissa Varney, Grace Varney, Alyssa Varney, Janine Varney, Alexis Varney, Elisa Varney,
Jolene Varney, Karin Varney, Shelby Varney, Wendi Varney, Tania Varney, Melisa Varney, Bernadette Varney, Lorena Varney, Shelia Varney, Rosemary Varney, Ramona Varney, Terry Varney, Nora Varney,
Marlene Varney, Camille Varney, Tanisha Varney, Carey Varney, Alma Varney, Sherrie Varney, Cecilia Varney, Celeste Varney, Kate Varney, Betsy Varney, Brandie Varney, Annie Varney, Darla Varney,
Lillian Varney, Maribel Varney, Glenda Varney, Chasity Varney, Patrice Varney, Amelia Varney, Lorie Varney, Tabatha Varney, Elena Varney, Lana Varney, Jocelyn Varney, Jeannette Varney, Guadalupe
Varney, Marcie Varney, Johanna Varney, Josephine Varney, Tisha Varney, Michael Varney, Leanne Varney, Vivian Varney, Whitney Varney, Darcy Varney, Hilary Varney, Marci Varney, Doris Varney, Selena
Varney, Constance Varney, Serena Varney, Daphne Varney, Hollie Varney, Bridgette Varney, Tammi Varney, Jeanine Varney, Terra Varney, Margarita Varney, Vicky Varney, Allyson Varney, Marianne Varney,
Elisabeth Varney, Paige Varney, Iris Varney, Lena Varney, Lakeisha Varney, Jillian Varney, Arlene Varney, Latanya Varney, Emma Varney, Aisha Varney, Tia Varney, Alissa Varney, Sophia Varney, Hannah
Varney, Lawanda Varney, Chrystal Varney, Lea Varney, Cari Varney, Jessie Varney, Mia Varney, Chastity Varney, Lynda Varney, Cassie Varney, Marjorie Varney, Liza Varney, Edith Varney, Jenna Varney,
Joni Varney, Misti Varney, Blanca Varney, Dionne Varney, Lashonda Varney, Leann Varney, Yesenia Varney, Corinne Varney, Jasmine Varney, Malinda Varney, Gabriela Varney, Shonda Varney, Laurel Varney,
Hillary Varney, Kimberlee Varney, Karrie Varney, Irma Varney, Pauline Varney, Claire Varney, Dora Varney, Isabel Varney, Georgia Varney, Esmeralda Varney, Nadine Varney, Lakesha Varney, Marcella
Varney, Luz Varney, Ladonna Varney, Katharine Varney, Phyllis Varney, Sue Varney, Janie Varney, Kisha Varney, Dianne Varney, Myra Varney, Gabrielle Varney, Ingrid Varney, Maryann Varney, Rene Varney,
Lucy Varney, Bridgett Varney, Karyn Varney, Janel Varney, Mandi Varney, Paulette Varney, Lucinda Varney, Carissa Varney, Genevieve Varney, Olga Varney, Dolores Varney, Catina Varney, Randi Varney,
Tera Varney, Jeannine Varney, Nakia Varney, Janette Varney, June Varney, Alana Varney, Annmarie Varney, Lashawn Varney, Noelle Varney, Chelsea Varney, Brittany Varney, Clarissa Varney, Shanda Varney,
Cathleen Varney, Susana Varney, Susanne Varney, Chanda Varney, Edna Varney, Catrina Varney, Kerrie Varney, Francine Varney, Mona Varney, Alyson Varney, Doreen Varney, Maggie Varney, Lynne Varney,
Tori Varney, Clara Varney, Demetria Varney, Beatrice Varney, Valarie Varney, Jeri Varney, Kasey Varney, Silvia Varney, Gena Varney, Janna Varney, Abby Varney, Cristy Varney, Deirdre Varney, Deidre
Varney, Tawana Varney, Deena Varney, Jan Varney, Maricela Varney, Tamra Varney, Daisy Varney, Sondra Varney, Caryn Varney, Nadia Varney, Lorrie Varney, Patty Varney, Anissa Varney, Farrah Varney,
Bianca Varney, Tiffani Varney, Carly Varney, Delia Varney, Susie Varney, Tessa Varney, Latrice Varney, Shellie Varney, Renae Varney, Corina Varney, Deanne Varney, Rena Varney, Louise Varney, Larissa
Varney, Latosha Varney, Lois Varney, Maritza Varney, Julianne Varney, Mildred Varney, Janell Varney, Karina Varney, Christal Varney, Antonia Varney, James Varney, Ida Varney, Marina Varney, Lacey
Varney, Kristal Varney, Ami Varney, Christopher Varney, Gladys Varney, Cori Varney, Gwen Varney, Eleanor Varney, Elisha Varney, Justine Varney, Madeline Varney, Ursula Varney, Charmaine Varney,
Rosemarie Varney, Suzette Varney, Elise Varney, Bertha Varney, Gayle Varney, Lynnette Varney, Sheree Varney, Araceli Varney, Chantel Varney, Kesha Varney, Dayna Varney, Malissa Varney, Dara Varney,
Latonia Varney, Jayme Varney, Robert Varney, Jerri Varney, Carolina Varney, Athena Varney, Hilda Varney, Ivy Varney, Mara Varney, Tosha Varney, Marion Varney, Elsa Varney, Stella Varney, Devon
Varney, Marlo Varney, Danette Varney, Mayra Varney, Kay Varney, Bernice Varney, Adrianne Varney, Geneva Varney, Colette Varney, David Varney, Anitra Varney, Letitia Varney, Marian Varney, Windy
Varney, Jason Varney, Katy Varney, Felecia Varney, Elissa Varney, Susanna Varney, Lucia Varney, Celia Varney, Margie Varney, Kami Varney, Adrian Varney, Holli Varney, Delores Varney, Sasha Varney,
Tamera Varney, Tonja Varney, Taryn Varney, Stacia Varney, John Varney, Tomeka Varney, Anastasia Varney, Marta Varney, Patti Varney, Jeanna Varney, Georgina Varney, Eugenia Varney, Tawnya Varney,
Corey Varney, Leeann Varney, Juliet Varney, Chris Varney, Dominique Varney, Mitzi Varney, Margo Varney, Selina Varney, Marnie Varney, Casandra Varney, Deann Varney, Leanna Varney, Vera Varney,
Consuelo Varney, Buffy Varney, Lourdes Varney, Florence Varney, Geraldine Varney, Tyra Varney, Alecia Varney, Greta Varney, Carole Varney, Francesca Varney, Meagan Varney, Jeanie Varney, Rosalinda
Varney, Trudy Varney, Haley Varney, Sandi Varney, Corrie Varney, Maura Varney, Sharonda Varney, Graciela Varney, Michell Varney, Terrie Varney, Eve Varney, Tiffanie Varney, Rosalind Varney, Marlena
Varney, Shannan Varney, Thelma Varney, Carie Varney, Lashanda Varney, Shelli Varney, Karie Varney, Danelle Varney, Tawanda Varney, Ryan Varney, Simone Varney, Maya Varney, Mollie Varney, Deidra
Varney, Josie Varney, Noemi Varney, Briana Varney, Venus Varney, Lauri Varney, Richelle Varney, Candi Varney, Ella Varney, Cora Varney, Latricia Varney, Marisela Varney, Beatriz Varney, Shanon
Varney, Alejandra Varney, Aubrey Varney, Jeana Varney, Nanette Varney, Brianne Varney, Leona Varney, Helena Varney, Kris Varney, Christian Varney, Annemarie Varney, Andria Varney, Mindi Varney,
Rosalyn Varney, Shanta Varney, Mercedes Varney, Polly Varney, Juana Varney, Rosie Varney, Martina Varney, Therese Varney, Cory Varney, Kayla Varney, Shantel Varney, Joey Varney, Shalonda Varney,
Mellissa Varney, Ada Varney, Brandee Varney, Sunshine Varney, Tamiko Varney, Danita Varney, Niki Varney, Rosanna Varney, Joelle Varney, Monika Varney, Trista Varney, Chiquita Varney, William Varney,
Brook Varney, Janis Varney, Rae Varney, Kizzy Varney, Cortney Varney, Cecelia Varney, Jaimie Varney, Christin Varney, Christen Varney, Tanika Varney, Darci Varney, Claudine Varney, Sharla Varney,
Toya Varney, Racheal Varney, Noel Varney, Jada Varney, Davina Varney, Della Varney, Kori Varney, Tomika Varney, Callie Varney, Jamila Varney, Aileen Varney, Dusty Varney, Rocio Varney, Kia Varney,
Benita Varney, Jena Varney, Aurora Varney, Penelope Varney, Sadie Varney, Patsy Varney, Valencia Varney, Lisette Varney, Tamela Varney, Mari Varney, Tana Varney, Ivette Varney, Eliza Varney, Darcie
Varney, Tawanna Varney, Bonita Varney, Debora Varney, Lacy Varney, Tarsha Varney, Stephenie Varney, Johnna Varney, Kathrine Varney, Lakeshia Varney, Griselda Varney, Charla Varney, Lesa Varney, Sunny
Varney, Celina Varney, Karri Varney, Corinna Varney, Dixie Varney, Rosalie Varney, Rhiannon Varney, Eunice Varney, Mariah Varney, Angeline Varney, Danna Varney, Shayla Varney, Brenna Varney, Juliana
Varney, Brianna Varney, Octavia Varney, Leila Varney, Liliana Varney, Shameka Varney, Lissette Varney, Cary Varney, Chrissy Varney, Corrine Varney, Destiny Varney, India Varney, Adriane Varney, Kandi
Varney, Coleen Varney, Angelita Varney, Denice Varney, Janeen Varney, Suzanna Varney, Dee Varney, Magdalena Varney, Lola Varney, Brian Varney, Nikole Varney, Sofia Varney, Nicolle Varney, Lucretia
Varney, Mellisa Varney, Morgan Varney, Alexa Varney, Faye Varney, Daniel Varney, Antionette Varney, Tennille Varney, Ayanna Varney, Maxine Varney, Christel Varney, Alysia Varney, Alexandria Varney,
Lenora Varney, Aida Varney, Kyla Varney, Joseph Varney, Bree Varney, Kira Varney, Renita Varney, Sharron Varney, Jenni Varney, Ethel Varney, Fatima Varney, Hazel Varney, Nellie Varney, Juli Varney,
Delilah Varney, Pearl Varney, Caren Varney, Melonie Varney, Robbie Varney, Lindy Varney, Tangela Varney, Trena Varney, Daniela Varney, Michaela Varney, Carri Varney, Sommer Varney, Alanna Varney,
Tressa Varney, Georgette Varney, Nikita Varney, Chanel Varney, Aretha Varney, Charisse Varney, Laquita Varney, Natalia Varney, Josette Varney, Roxanna Varney, Kirstin Varney, Shani Varney, Nichol
Varney, Regan Varney, Salina Varney, Anjanette Varney, Susannah Varney, Jammie Varney, Kenyatta Varney, Jasmin Varney, Anthony Varney, Ginny Varney, Lillie Varney, Missy Varney, Starla Varney,
Lucille Varney, Lorna Varney, Tiana Varney, Melodie Varney, Fawn Varney, Candida Varney, Carin Varney, Elvira Varney, Shonna Varney, Carmela Varney, Spring Varney, Shara Varney, Yolonda Varney,
Josefina Varney, Evette Varney, Carisa Varney, Sherita Varney, Roseann Varney, Saundra Varney, Myrna Varney, Bambi Varney, Layla Varney, Lilia Varney, Mechelle Varney, Cathryn Varney, Jayne Varney,
Lolita Varney, Rosario Varney, Eric Varney, Valeria Varney, Jade Varney, Nicki Varney, Maranda Varney, Felisha Varney, Raina Varney, Tracee Varney, Cindi Varney, Freda Varney, Gillian Varney, Damaris
Varney, Ava Varney, Brigitte Varney, Nicola Varney, Esperanza Varney, Leilani Varney, Sharlene Varney, Marni Varney, Claudette Varney, Keely Varney, Hayley Varney, Chantelle Varney, Joleen Varney,
Elsie Varney, Rebeca Varney, Mendy Varney, Adria Varney, Harmony Varney, Francisca Varney, Jolie Varney, Vikki Varney, Nikia Varney, Tamatha Varney, Keli Varney, Keshia Varney, Charissa Varney,
Imelda Varney, Francis Varney, Richard Varney, Shantell Varney, Shawanda Varney, Caitlin Varney, Frankie Varney, Shea Varney, Cherry Varney, Sheena Varney, Brigette Varney, Keesha Varney, Cristi
Varney, Justina Varney, Rashida Varney, Emilie Varney, Ashlee Varney, Shane Varney, Cherise Varney, Qiana Varney, Latarsha Varney, Crissy Varney, Celena Varney, Kylie Varney, Rolanda Varney, Crista
Varney, Roslyn Varney, Angella Varney, Shamika Varney, Wilma Varney, Violet Varney, Kyra Varney, Kevin Varney, Marybeth Varney, Alesha Varney, Matthew Varney, Shanika Varney, Shawnda Varney, Cristal
Varney, Julissa Varney, Lila Varney, Alycia Varney, Camilla Varney, Johnnie Varney, Dori Varney, Nichelle Varney, Venessa Varney, Anika Varney, Lily Varney, Tamica Varney, Rhoda Varney, Alethea
Varney, Carina Varney, Roxann Varney, Latrina Varney, Charles Varney, Bessie Varney, Julianna Varney, Marquita Varney, Gayla Varney, Kandy Varney, Estella Varney, Marguerite Varney, Chana Varney,
Robbin Varney, Cristin Varney, Raven Varney, Tanesha Varney, Alina Varney, Juliette Varney, Milagros Varney, Inez Varney, Vonda Varney, Jenelle Varney, Elaina Varney, Dedra Varney, Verna Varney,
Starr Varney, Bobby Varney, Lizette Varney, Dawna Varney, Danyelle Varney, Petra Varney, Stephaine Varney, Angelic Varney, Brittney Varney, Corrina Varney, Britt Varney, Stephany Varney, Lorri
Varney, Shayna Varney, Flora Varney, Jesse Varney, Velma Varney, Racquel Varney, Stefani Varney, Phoebe Varney, Yadira Varney, Liana Varney, Milissa Varney, Pam Varney, Zoe Varney, Libby Varney,
Shasta Varney, Luisa Varney, Barbra Varney, Janae Varney, Shandra Varney, Thomas Varney, Kelsey Varney, Shante Varney, Bettina Varney, Lashunda Varney, Melina Varney, Mattie Varney, Meaghan Varney,
Timothy Varney, Rosetta Varney, Mistie Varney, Steven Varney, Patrica Varney, Shay Varney, Takisha Varney, Geri Varney, Teressa Varney, Aaron Varney, Clare Varney, Noreen Varney, Adina Varney,
Rosanne Varney, Luciana Varney, Nakisha Varney, Trinity Varney, Cicely Varney, Minerva Varney, Devin Varney, Gabriella Varney, Lela Varney, Kendall Varney, Treva Varney, Hallie Varney, Kristan
Varney, Lia Varney, Viola Varney, Portia Varney, Jonna Varney, Kellee Varney, Danica Varney, Kristyn Varney, Lina Varney, Jewel Varney, Tamala Varney, Jenniffer Varney, Melony Varney, Kacey Varney,
Quiana Varney, Shawnna Varney, Rana Varney, Dannielle Varney, Andra Varney, Lenore Varney, Carlene Varney, Malia Varney, Lanette Varney, Jacquline Varney, Willie Varney, Precious Varney, Collette
Varney, Mariana Varney, Ayana Varney, Daniella Varney, Thea Varney, Viviana Varney, Kindra Varney, Gracie Varney, Louisa Varney, Adele Varney, Annamarie Varney, Kiesha Varney, Sherie Varney, Althea
Varney, Catalina Varney, January Varney, Ernestine Varney, Corie Varney, Lesli Varney, Shanita Varney, Tresa Varney, Maryanne Varney, Tamar Varney, Rayna Varney, Siobhan Varney, Demetra Varney,
Kimberlie Varney, Rhea Varney, Elva Varney, Elvia Varney, Karissa Varney, Maryellen Varney, Valorie Varney, Adrianna Varney, Nicolette Varney, Daniele Varney, Trish Varney, Twila Varney, Abbie
Varney, Towanda Varney, Lorinda Varney, Marcela Varney, Kyle Varney, Carmelita Varney, Helene Varney, Mark Varney, Jeffrey Varney, Lidia Varney, Kenyetta Varney, Lupe Varney, Cami Varney, Cher
Varney, Minnie Varney, Toby Varney, Karena Varney, Katheryn Varney, Renea Varney, Kathie Varney, Laverne Varney, Shona Varney, Linette Varney, Kecia Varney, Elana Varney, Eboni Varney, Loren Varney,
Tabetha Varney, Dione Varney, Glenna Varney, Marti Varney, Gia Varney, Tena Varney, Cameron Varney, Roxana Varney, Alberta Varney, Dolly Varney, Pamala Varney, Talia Varney, Tobi Varney, Latesha
Varney, Larhonda Varney, Sybil Varney, Kandice Varney, Yasmin Varney, Christiana Varney, Sydney Varney, Chantal Varney, Estela Varney, Jaimee Varney, Salena Varney, Tamekia Varney, Dalia Varney,
Twana Varney, Jacklyn Varney, Scarlett Varney, Tanja Varney, Kenisha Varney, Daria Varney, Agnes Varney, Mandie Varney, Lani Varney, May Varney, Sharee Varney, Tenisha Varney, Twyla Varney, Chantell
Varney, Shiela Varney, Isela Varney, Karey Varney, Alaina Varney, Loriann Varney, Star Varney, Samara Varney, Janene Varney, Gerri Varney, Lasonya Varney, Lavonne Varney, Rikki Varney, Meridith
Varney, Lorene Varney, Scott Varney, Stephani Varney, Kimberli Varney, Zandra Varney, Sallie Varney, Cecily Varney, Rachele Varney, Melynda Varney, Reagan Varney, Carmella Varney, Danyell Varney,
Ivonne Varney, Jamey Varney, Shawana Varney, Aja Varney, Casie Varney, Alena Varney, Barbie Varney, Clarice Varney, Kassandra Varney, Tiffiny Varney, Madelyn Varney, Cherish Varney, Lavonda Varney,
Malisa Varney, Stormy Varney, Amalia Varney, Machelle Varney, Abbey Varney, Evangelina Varney, Serina Varney, Fannie Varney, Joi Varney, Tarra Varney, Dale Varney, Shaunna Varney, Jeremy Varney,
Sarina Varney, Harriet Varney, Holley Varney, Reyna Varney, Ayesha Varney, Natosha Varney, Princess Varney, Shavon Varney, Ashleigh Varney, Laticia Varney, Cristine Varney, Asia Varney, Antonette
Varney, Dinah Varney, Jordan Varney, Cherilyn Varney, Mackenzie Varney, Giselle Varney, Edwina Varney, Mae Varney, Kathi Varney, Shalanda Varney, Donita Varney, Melissia Varney, Cristie Varney, Mirna
Varney, Cheyenne Varney, Leslee Varney, Jeni Varney, Jacinda Varney, Consuela Varney, Evangeline Varney, Joanie Varney, Nickie Varney, Jolynn Varney, Malika Varney, Nathalie Varney, Mamie Varney,
Paul Varney, Davida Varney, Erinn Varney, Laronda Varney, Latina Varney, Liberty Varney, Nicol Varney, Danyel Varney, Tommie Varney, Joshua Varney, Shaun Varney, Jodee Varney, Teena Varney, Shanell
Varney, Calandra Varney, Iesha Varney, Devona Varney, Aracely Varney, Brandon Varney, Dona Varney, Micah Varney, Adela Varney, Lashon Varney, Reina Varney, Sierra Varney, Addie Varney, Carlotta
Varney, Nia Varney, Ariana Varney, Chante Varney, Tamisha Varney, Ali Varney, Deedee Varney, Sarita Varney, Jeniffer Varney, Kiley Varney, Ronnie Varney, Cherri Varney, Nickole Varney, Hattie Varney,
Syreeta Varney, Chaya Varney, Dani Varney, Catharine Varney, Tamie Varney, Perla Varney, Billiejo Varney, Deloris Varney, Lashawnda Varney, Meggan Varney, Temeka Varney, Dorothea Varney, Marianna
Varney, Lou Varney, Tesha Varney, Karmen Varney, Luann Varney, Rona Varney, Elyse Varney, Latashia Varney, Dacia Varney, Nita Varney, Tashia Varney, Kenneth Varney, Mireya Varney, Carleen Varney,
Alfreda Varney, Candie Varney, Delicia Varney, Lakiesha Varney, Renata Varney, Juliann Varney, Patience Varney, Alexia Varney, Lakeesha Varney, Petrina Varney, Liane Varney, Ester Varney, Kimberely
Varney, Millie Varney, Roni Varney, Reba Varney, Tory Varney, Maryjo Varney, Aundrea Varney, Kasandra Varney, Melba Varney, Tiffaney Varney, Maisha Varney, Kenna Varney, Detra Varney, Jerry Varney,
Meghann Varney, Sanjuanita Varney, Emilia Varney, Denita Varney, Shaunda Varney, Shavonne Varney, Jose Varney, Roseanne Varney, Kiersten Varney, Sophie Varney, Maren Varney, Giovanna Varney, Bethann
Varney, Ilene Varney, Dorian Varney, Lanita Varney, Ariel Varney, Piper Varney, Shawnee Varney, Cammie Varney, Joellen Varney, Millicent Varney, Rachell Varney, Billy Varney, Felica Varney, Jacinta
Varney, Krissy Varney, Letisha Varney, Pennie Varney, Delisa Varney, Evonne Varney, Keysha Varney, Migdalia Varney, Opal Varney, Taunya Varney, Farah Varney, Ina Varney, Samatha Varney, Linnea
Varney, Lyn Varney, Merry Varney, Patrina Varney, Terese Varney, Leandra Varney, Cornelia Varney, Manda Varney, Lissa Varney, Coretta Varney, Gidget Varney, Demetrice Varney, Mabel Varney, Necole
Varney, Torrie Varney, Kandace Varney, Loraine Varney, Tarah Varney, Cody Varney, Cinnamon Varney, Shonta Varney, Rosalia Varney, Delana Varney, Nannette Varney, Amee Varney, Ofelia Varney, Shakira
Varney, Aletha Varney, Kacy Varney, Rosita Varney, Talisha Varney, Moira Varney, Sebrina Varney, Katrice Varney, Eden Varney, Jaqueline Varney, Kirstie Varney, Gregory Varney, Letha Varney, Apryl
Varney, Kristel Varney, Fabiola Varney, Steffanie Varney, Elicia Varney, Nedra Varney, Maia Varney, Sharita Varney, Tammara Varney, Charise Varney, Cheree Varney, Andrew Varney, Honey Varney, Velvet
Varney, Anglea Varney, Brett Varney, Henrietta Varney, Jenine Varney, Meg Varney, Khalilah Varney, Sheronda Varney, Dyan Varney, Marty Varney, Suzan Varney, Liz Varney, Brande Varney, Flor Varney,
Tameika Varney, Manuela Varney, Stephen Varney, Kamilah Varney, Karolyn Varney, Demetrius Varney, Lacie Varney, Lakeysha Varney, Jannette Varney, Vonetta Varney, Gale Varney, Shiloh Varney, Shira
Varney, Gisela Varney, Jacquelin Varney, Natisha Varney, Billi Varney, Charleen Varney, Brigid Varney, Dorinda Varney, Jacki Varney, Keena Varney, Lamonica Varney, Shena Varney, Michel Varney, Nanci
Varney, Donya Varney, Leisa Varney, Rosalba Varney, Anabel Varney, Annalisa Varney, Amity Varney, Mimi Varney, Erma Varney, Lyndsey Varney, Marlana Varney, Skye Varney, Chloe Varney, Inga Varney,
Jonathan Varney, Melaine Varney, Lorelei Varney, Marivel Varney, Nelly Varney, Amiee Varney, Cathrine Varney, Jonelle Varney, Sacha Varney, Jennette Varney, Lilly Varney, Shawnta Varney, Tisa Varney,
Judi Varney, Cherrie Varney, Annetta Varney, Charlette Varney, Christiane Varney, Jesica Varney, Pilar Varney, Elida Varney, Twanna Varney, Madonna Varney, Tausha Varney, Carletta Varney, Leesa
Varney, Matilda Varney, Nakita Varney, Natashia Varney, Kaci Varney, Mica Varney, Dionna Varney, Margot Varney, Maribeth Varney, Darby Varney, Shondra Varney, Socorro Varney, Misha Varney, Taylor
Varney, Becki Varney, Kiana Varney, Valentina Varney, Ashlie Varney, Desirae Varney, Lizabeth Varney, Alesia Varney, Roxane Varney, Donielle Varney, Kelle Varney, Larisa Varney, Paola Varney, Aleta
Varney, Dorene Varney, Kitty Varney, Shenita Varney, Tatiana Varney, Felisa Varney, Yashica Varney, Jeanetta Varney, Tony Varney, Latrisha Varney, Nyree Varney, Ronald Varney, Romona Varney, Shawanna
Varney, Edward Varney, Justin Varney, Tuesday Varney, Avis Varney, Edie Varney, Aurelia Varney, Nova Varney, Ruthie Varney, Estelle Varney, Mikki Varney, Jamee Varney, Micaela Varney, Phaedra Varney,
Sean Varney, Fiona Varney, Shilo Varney, Trudi Varney, Blythe Varney, Julee Varney, Alia Varney, Lachelle Varney, Roseanna Varney, Donald Varney, Nydia Varney, Effie Varney, Jenell Varney, Carolee
Varney, Karma Varney, Violeta Varney, Ilana Varney, Leonor Varney, Talitha Varney, Deedra Varney, Jimmie Varney, Buffie Varney, Iva Varney, Joycelyn Varney, Randa Varney, Tawny Varney, Bonny Varney,
Dallas Varney, Felice Varney, Lawana Varney, Corine Varney, Kizzie Varney, Ladawn Varney, Torri Varney, Marva Varney, Sabina Varney, Taneka Varney, Deandra Varney, Kary Varney, Juliane Varney, Asha
Varney, Mika Varney, Twanda Varney, Dawne Varney, Ellie Varney, Lawanna Varney, Rowena Varney, Caron Varney, Chad Varney, Ronna Varney, Tatum Varney, Caprice Varney, Cinthia Varney, Debby Varney,
Deeann Varney, Shalon Varney, Ernestina Varney, Andre Varney, Annamaria Varney, Latoshia Varney, Shemeka Varney, Stacee Varney, Tyesha Varney, Annabelle Varney, Germaine Varney, Risa Varney, Latoria
Varney, Patrick Varney, Wendie Varney, Laureen Varney, Cristen Varney, Kati Varney, Niccole Varney, Sabra Varney, Shoshana Varney, Tequila Varney, Melva Varney, Micki Varney, Darcey Varney, Coral
Varney, Fonda Varney, Lashaunda Varney, Pepper Varney, Trinette Varney, Eloisa Varney, Goldie Varney, Tijuana Varney, Venita Varney, Danae Varney, Tessie Varney, Rebecka Varney, Taisha Varney, Heide
Varney, Joslyn Varney, Rebbecca Varney, Magda Varney, Roxie Varney, Daniell Varney, Kellye Varney, Timika Varney, Dannette Varney, Lakita Varney, Shawnte Varney, Dusti Varney, Shenika Varney, Amberly
Varney, Koren Varney, Carolynn Varney, Thomasina Varney, Bobbiejo Varney, Candance Varney, Jacque Varney, Nikisha Varney, Torie Varney, Cyndi Varney, Diann Varney, Marylou Varney, Nelda Varney,
Prudence Varney, Sharyn Varney, Carman Varney, Coreen Varney, Amina Varney, Neva Varney, Scarlet Varney, Ammie Varney, Haydee Varney, Jaymie Varney, Rina Varney, Lyndsay Varney, Shantelle Varney,
Alyce Varney, Danika Varney, Leeanne Varney, Tamesha Varney, Iliana Varney, Kylee Varney, Lisha Varney, Mercy Varney, Nona Varney, Teresita Varney, Xochitl Varney, Carmel Varney, Jessi Varney,
Kourtney Varney, Elia Varney, Lasandra Varney, Alisia Varney, Candis Varney, Nicholle Varney, Nissa Varney, Micheal Varney, Kali Varney, Tajuana Varney, Aleshia Varney, Daina Varney, Fay Varney,
Rivka Varney, Jennefer Varney, Kena Varney, Cassidy Varney, Toshia Varney, Larry Varney, Travis Varney, Chaka Varney, Anya Varney, Leia Varney, Nola Varney, Rosalva Varney, Arica Varney, Hanna
Varney, Kenda Varney, Tiesha Varney, Jewell Varney, Roshanda Varney, Arianne Varney, Eryn Varney, Denna Varney, George Varney, Jerilyn Varney, Lakia Varney, Melani Varney, Raeann Varney, Season
Varney, Sherilyn Varney, Lizbeth Varney, Myesha Varney, Suzy Varney, Emilee Varney, Kristian Varney, Leighann Varney, Akilah Varney, Britney Varney, Juan Varney, Bobbijo Varney, Dalila Varney, Ileana
Varney, Contessa Varney, Darleen Varney, Deshawn Varney, Audrea Varney, Kawana Varney, Lona Varney, Margret Varney, Melany Varney, Michal Varney, Odessa Varney, Reva Varney, Lula Varney, Mariann
Varney, Kanika Varney, Paris Varney, Shaneka Varney, Sherrell Varney, Suzann Varney, Chelsey Varney, Laquisha Varney, Willow Varney, Kaye Varney, Muriel Varney, Britta Varney, Dania Varney, Jesenia
Varney, Nilda Varney, Dottie Varney, Nereida Varney, Kellyann Varney, Alishia Varney, Denisha Varney, Marietta Varney, Merideth Varney, Enid Varney, Isis Varney, Rosio Varney, America Varney, Belen
Varney, Randy Varney, Karol Varney, Kasie Varney, Julieann Varney, Sari Varney, Bronwyn Varney, Concepcion Varney, Criselda Varney, Natacha Varney, Ariane Varney, Breanna Varney, Shayne Varney,
Alysha Varney, Gary Varney, Rasheeda Varney, Sherice Varney, Tonisha Varney, Katrena Varney, Shemika Varney, Sirena Varney, Breanne Varney, Simona Varney, Lakenya Varney, Winona Varney, Adam Varney,
Caridad Varney, Deonna Varney, Kimber Varney, Lashaun Varney, Maile Varney, Tomekia Varney, Bernadine Varney, Miesha Varney, Adelina Varney, Tiara Varney, Selene Varney, Shae Varney, Tomi Varney,
Laila Varney, Keith Varney, Lilian Varney, Sherika Varney, Mariela Varney, Moriah Varney, Page Varney, Raegan Varney, Timeka Varney, Tonda Varney, Danya Varney, Kacie Varney, Sabrena Varney, Shala
Varney, Yalonda Varney, Jermaine Varney, Ranee Varney, Sherron Varney, Tameca Varney, Winter Varney, Yulanda Varney, Gabriel Varney, Johna Varney, Keya Varney, Tobie Varney, Johnetta Varney, Rochell
Varney, Celestine Varney, Christinia Varney, Kymberly Varney, Vernita Varney, Jina Varney, Lashawna Varney, Malena Varney, Gilda Varney, Somer Varney, Aliza Varney, Christene Varney, Marilee Varney,
Ricki Varney, Tiffiney Varney, Winifred Varney, Antonio Varney, Cassondra Varney, Suzie Varney, Etta Varney, Toi Varney, Ashli Varney, Charis Varney, Isabelle Varney, Jayna Varney, Nena Varney, Nisha
Varney, Jenise Varney, Lizzie Varney, Maegan Varney, Contina Varney, Joe Varney, Sunday Varney, Tanna Varney, Eleni Varney, Mirella Varney, Concetta Varney, Jeneen Varney, Nerissa Varney, Bethanie
Varney, Latoyia Varney, Maryjane Varney, Treena Varney, Benjamin Varney, Keturah Varney, Paulina Varney, Rasheda Varney, Shaunta Varney, Vickey Varney, Ashanti Varney, Georgiana Varney, Noelia
Varney, Georgianna Varney, Troy Varney, Yulonda Varney, Carley Varney, Stephine Varney, Oralia Varney, Pricilla Varney, Ranae Varney, Lory Varney, Africa Varney, Lekisha Varney, Rosaura Varney,
Shelita Varney, Louann Varney, Shanti Varney, Takesha Varney, Tonie Varney, Santa Varney, Zulema Varney, Dahlia Varney, Jacquelynn Varney, Karon Varney, Teisha Varney, Carlos Varney, Jerrie Varney,
Shyla Varney, Jacqulyn Varney, Janee Varney, Lupita Varney, Ora Varney, Pia Varney, Dodie Varney, Mariam Varney, Meegan Varney, Shanelle Varney, Chanelle Varney, Janetta Varney, Chanell Varney, Hiedi
Varney, Lesia Varney, Nikkia Varney, Ivory Varney, Karry Varney, Marcey Varney, Theodora Varney, Lonnie Varney, Kai Varney, Mina Varney, Seana Varney, Kathlene Varney, Melvina Varney, Temika Varney,
Kala Varney, Leta Varney, Shaunte Varney, Sheron Varney, Tameeka Varney, Eloise Varney, Tatanisha Varney, Catrice Varney, Malaika Varney, Raymond Varney, Sharhonda Varney, Tandra Varney, Tess Varney,
Linsey Varney, Lottie Varney, Miki Varney, Ronni Varney, Tiffini Varney, Jenee Varney, Alta Varney, Elba Varney, Jamillah Varney, Jannie Varney, Loree Varney, Roselyn Varney, Winnie Varney, Alvina
Varney, Dorcas Varney, Laci Varney, Merri Varney, Mickie Varney, Penni Varney, Allegra Varney, Dreama Varney, Nidia Varney, Pattie Varney, Gianna Varney, Kimberlyn Varney, Roshonda Varney, Joel
Varney, Nettie Varney, Blair Varney, Terina Varney, Ebonie Varney, Kera Varney, Ruthann Varney, Randee Varney, Bryan Varney, Jolyn Varney, Stephannie Varney, Delta Varney, Janella Varney, Laquanda
Varney, Lonna Varney, Marcelle Varney, Tavia Varney, Terrell Varney, Chasidy Varney, Danell Varney, Khadijah Varney, Neely Varney, Nicholas Varney, Nicky Varney, Jennine Varney, Tomica Varney,
Xiomara Varney, Deneen Varney, Magaly Varney, Zoraida Varney, Marquetta Varney, Tianna Varney, Annabel Varney, Brie Varney, Caterina Varney, Dennise Varney, Madeleine Varney, Marika Varney, Angelena
Varney, Doretha Varney, Quintina Varney, Rashonda Varney, Carlie Varney, Krysta Varney, Lianne Varney, Martine Varney, Nacole Varney, Regena Varney, Jeanene Varney, Kassie Varney, Larae Varney, Leana
Varney, Ronica Varney, Tenesha Varney, Deeanna Varney, Echo Varney, Melita Varney, Johnny Varney, Lorianne Varney, Makeba Varney, Nakesha Varney, Zenaida Varney, Ophelia Varney, Gennifer Varney,
Rasheedah Varney, Jenette Varney, Lashana Varney, Lashandra Varney, Blanche Varney, Kina Varney, Dia Varney, Emmy Varney, Janina Varney, Krishna Varney, Shawnette Varney, Wenona Varney, Hailey
Varney, Marita Varney, Charita Varney, Diedra Varney, Annika Varney, Gay Varney, Karima Varney, Keila Varney, Lajuana Varney, Marya Varney, Omayra Varney, Carl Varney, Gricelda Varney, Lynsey Varney,
Jama Varney, Lynelle Varney, Telisha Varney, Todd Varney, Camie Varney, Luanne Varney, Mira Varney, Raylene Varney, Rosana Varney, Nakeisha Varney, Brea Varney, Leisha Varney, Maira Varney, Shera
Varney, Tangie Varney, Tari Varney, Treasa Varney, Angle Varney, Jaimi Varney, Joie Varney, Ryann Varney, Sueann Varney, Tiffney Varney, Alida Varney, Jackeline Varney, Tonika Varney, Lakeitha
Varney, Lezlie Varney, Stephania Varney, Zulma Varney, Candra Varney, Charlie Varney, Aleisha Varney, Andreana Varney, Berta Varney, Deon Varney, Gigi Varney, Meri Varney, Cherice Varney, Mya Varney,
Rani Varney, Sunni Varney, Felicity Varney, Marcus Varney, Myisha Varney, Eureka Varney, Gertrude Varney, Lavette Varney, Karisa Varney, Lacresha Varney, Olympia Varney, Bryn Varney, Frank Varney,
Jovita Varney, Leola Varney, Augusta Varney, Karlene Varney, Shaina Varney, Tenille Varney, Fanny Varney, Katerina Varney, Lainie Varney, Natalee Varney, Tandy Varney, Thalia Varney, Elda Varney,
Jacalyn Varney, Beckie Varney, Chenoa Varney, Leeanna Varney, Sharie Varney, Tyisha Varney, Zenobia Varney, Diedre Varney, Charline Varney, Janey Varney, Shanan Varney, Keva Varney, Samuel Varney,
Sandie Varney, Zabrina Varney, Earlene Varney, Jacy Varney, Janessa Varney, Sarai Varney, Sharice Varney, Tambra Varney, Cammy Varney, Lashun Varney, Leonora Varney, Dennis Varney, Lelia Varney,
Cherita Varney, Courtenay Varney, Keira Varney, Tommy Varney, Genia Varney, Jordana Varney, Kaylene Varney, Loni Varney, Mickey Varney, Sheba Varney, Sherell Varney, Anisa Varney, Camelia Varney,
Domenica Varney, Ines Varney, Nathan Varney, Sherrill Varney, Delma Varney, Essie Varney, Lavon Varney, Marilynn Varney, Myrtle Varney, Takiyah Varney, Alba Varney, Delena Varney, Leyla Varney,
Lynell Varney, Tayna Varney, Jazmin Varney, Vannessa Varney, Awilda Varney, Carma Varney, Derrick Varney, Kasi Varney, Malka Varney, Carli Varney, Cordelia Varney, Jesus Varney, Kristeen Varney,
Chrissie Varney, Guillermina Varney, Joana Varney, Lauralee Varney, Serenity Varney, Alethia Varney, Dulce Varney, Lavinia Varney, Nilsa Varney, Tinisha Varney, Veda Varney, Demetris Varney, Jamilah
Varney, Janean Varney, Soledad Varney, Talisa Varney, Yael Varney, Adeline Varney, Amparo Varney, Shannah Varney, Anel Varney, Corissa Varney, Myriam Varney, September Varney, Shawntel Varney,
Camisha Varney, Cathie Varney, Reanna Varney, Shavonda Varney, Azure Varney, Darnell Varney, Delphine Varney, Keren Varney, Lashondra Varney, Yajaira Varney, Yessenia Varney, Adelaida Varney, Aura
Varney, Dorthy Varney, Karly Varney, Latonja Varney, Mendi Varney, Wende Varney, Adriene Varney, Arin Varney, Cinda Varney, Jeanmarie Varney, Khristina Varney, Rory Varney, Shanel Varney, Zena
Varney, Astrid Varney, Darice Varney, Melia Varney, Nell Varney, Sage Varney, Tedra Varney, Tyler Varney, Inger Varney, Lekesha Varney, Celine Varney, Marchelle Varney, Shaundra Varney, Bethanne
Varney, Kathryne Varney, Yanira Varney, Jacquetta Varney, Jessika Varney, Niesha Varney, Shantae Varney, Tira Varney, Elke Varney, Freida Varney, Gretta Varney, Kerstin Varney, San Varney, Suzannah
Varney, Tamira Varney, Zina Varney, Charmain Varney, Cherisse Varney, Kylene Varney, Lacretia Varney, Mavis Varney, Telisa Varney, Alysa Varney, Carlita Varney, Christianne Varney, Franchesca Varney,
Joetta Varney, Lashelle Varney, Rosamaria Varney, Seema Varney, Youlanda Varney, Clarisa Varney, Donella Varney, Markita Varney, Mesha Varney, Poppy Varney, Arleen Varney, Cecile Varney, Krisha
Varney, Mckenzie Varney, Sharmaine Varney, Willa Varney, Anessa Varney, Donetta Varney, Donnetta Varney, Leora Varney, Christena Varney, Dorie Varney, Eddie Varney, Herlinda Varney, Lise Varney,
Rosalee Varney, Temple Varney, Tonette Varney, Cheryle Varney, Cleo Varney, Elma Varney, Erlinda Varney, Peter Varney, Sparkle Varney, Evie Varney, Lavina Varney, Marna Varney, Meranda Varney,
Tynisha Varney, Alyse Varney, Anisha Varney, Bevin Varney, Glory Varney, Jonnie Varney, Marinda Varney, Natascha Varney, Shanetta Varney, Angelle Varney, Dion Varney, Quinn Varney, Shanette Varney,
Shaniqua Varney, Shantay Varney, Tamarra Varney, Diona Varney, Hortencia Varney, Latara Varney, Lisamarie Varney, Sidney Varney, Alessandra Varney, Idalia Varney, Malina Varney, Sindy Varney,
Antonella Varney, Kasha Varney, Laquinta Varney, Tysha Varney, Adena Varney, Aide Varney, Bernita Varney, Bettie Varney, Delinda Varney, Lashundra Varney, Sanjuana Varney, Susann Varney, Vilma
Varney, Deandrea Varney, Devorah Varney, Marisha Varney, Reena Varney, Takeisha Varney, Tora Varney, Carlyn Varney, Carrieann Varney, Chanta Varney, China Varney, Rhianna Varney, Sharleen Varney,
Jessenia Varney, Kemberly Varney, Rosalina Varney, Rosella Varney, Latrese Varney, Mecca Varney, Nneka Varney, Alva Varney, Douglas Varney, Mandee Varney, Sherese Varney, Taya Varney, Tenika Varney,
Donelle Varney, Frieda Varney, Joye Varney, Kayce Varney, Sammie Varney, Shereen Varney, Tomiko Varney, Tommi Varney, Marleen Varney, Shonte Varney, Avery Varney, Delfina Varney, Dollie Varney,
Kaycee Varney, Lakesia Varney, Penney Varney, Tinika Varney, Tywanda Varney, Betina Varney, Debrah Varney, Latia Varney, Lynetta Varney, Michaelle Varney, Shunta Varney, Soraya Varney, Anastacia
Varney, Charisma Varney, Jonie Varney, Shunda Varney, Terah Varney, Trinidad Varney, Valeri Varney, Charlena Varney, Elizebeth Varney, Joella Varney, Johnette Varney, Nikol Varney, Rosalynn Varney,
Aime Varney, Aviva Varney, Joselyn Varney, Lashanna Varney, Tarrah Varney, Candyce Varney, Estrella Varney, Mable Varney, Renay Varney, Shontel Varney, Wendee Varney, Angelene Varney, Karli Varney,
Maricruz Varney, Minda Varney, Robbi Varney, Towana Varney, Charolette Varney, Kandis Varney, Keeley Varney, Kimbley Varney, Kristene Varney, Lashell Varney, Merissa Varney, Taneshia Varney, Tristan
Varney, Vida Varney, Zelda Varney, Angele Varney, Beronica Varney, Damita Varney, Debbra Varney, Lanetta Varney, Priya Varney, Tamia Varney, Arlinda Varney, Armida Varney, Chere Varney, Errin Varney,
Kamala Varney, Lashan Varney, Luana Varney, Yalanda Varney, Antonietta Varney, Colby Varney, Jona Varney, Keitha Varney, Dava Varney, Herminia Varney, Allie Varney, Dawnmarie Varney, Eula Varney,
Monet Varney, Shirlene Varney, Tarina Varney, Yasmine Varney, Cassaundra Varney, Chelsie Varney, Fern Varney, Rosina Varney, Dyana Varney, Lavern Varney, Mario Varney, Ollie Varney, Rodney Varney,
Bridgit Varney, Julieta Varney, Korie Varney, Sumer Varney, Arianna Varney, Carmon Varney, Chassidy Varney, Ivana Varney, Jerusha Varney, Marne Varney, Melisha Varney, Regenia Varney, Sarena Varney,
Shanae Varney, Tona Varney, Cherese Varney, Deitra Varney, Dustin Varney, Kory Varney, Suzana Varney, Synthia Varney, Zakia Varney, Lisbeth Varney, Matilde Varney, Terresa Varney, Tiffeny Varney,
Toinette Varney, Charmin Varney, Julienne Varney, Marcelina Varney, Kama Varney, Katisha Varney, Tiffiany Varney, Tobey Varney, Adelita Varney, Atiya Varney, Bradley Varney, Cathi Varney, Dominica
Varney, Jeanelle Varney, Jeffery Varney, Rachal Varney, Sammi Varney, Anamaria Varney, Arcelia Varney, Isabella Varney, Janay Varney, Janett Varney, Jonell Varney, Lakeya Varney, Leigha Varney,
Alexander Varney, Denae Varney, Dustie Varney, Felicita Varney, Freedom Varney, Harriett Varney, Katia Varney, Londa Varney, Marlyn Varney, Tereasa Varney, Analisa Varney, Annissa Varney, August
Varney, Jenae Varney, Jenice Varney, Kamie Varney, Shanee Varney, Sherise Varney, Tamieka Varney, Karianne Varney, Naima Varney, Natarsha Varney, Vita Varney, Anjali Varney, Ena Varney, Jaci Varney,
Jolanda Varney, Kimi Varney, Celestina Varney, Devonna Varney, Latrece Varney, Leatrice Varney, Lucila Varney, Mignon Varney, Stefany Varney, Tamyra Varney, Tracye Varney, Beulah Varney, Danetta
Varney, Dede Varney, Dwana Varney, Julisa Varney, Kanisha Varney, Lanae Varney, Ronette Varney, Sandee Varney, Veronique Varney, Deirdra Varney, Ember Varney, Kerin Varney, Lekeisha Varney, Letisia
Varney, Pandora Varney, Zaneta Varney, Earline Varney, Evon Varney, Lisandra Varney, Lura Varney, Raelene Varney, Shontell Varney, Ila Varney, Kamisha Varney, Tinamarie Varney, Tish Varney, Jay
Varney, Kayleen Varney, Maryam Varney, Phillip Varney, Santina Varney, Taneisha Varney, Zakiya Varney, Brandice Varney, Jennifier Varney, Mariaelena Varney, Mirian Varney, Shalene Varney, Tammera
Varney, Tesa Varney, Zaida Varney, Anjelica Varney, Arika Varney, Babette Varney, Cameo Varney, Dwan Varney, Kiera Varney, Michella Varney, Nekia Varney, Rashelle Varney, Robynn Varney, Sheilah
Varney, Velia Varney, Betsey Varney, Chiara Varney, Kortney Varney, Tressie Varney, Tywana Varney, Valery Varney, Arnetta Varney, Carlee Varney, Craig Varney, Debbi Varney, Genesis Varney, Genny
Varney, Jori Varney, Kandie Varney, Lorin Varney, Melodee Varney, Cherlyn Varney, Karleen Varney, Windi Varney, Beverley Varney, Loreen Varney, Malynda Varney, Nada Varney, Nichola Varney, Sakinah
Varney, Steffany Varney, Tunisia Varney, Vanita Varney, Gemma Varney, Jennell Varney, Lakecia Varney, Shameeka Varney, Towanna Varney, Tywanna Varney, Vanesa Varney, Dea Varney, Desire Varney, Donnie
Varney, Kiki Varney, Makesha Varney, Rebeccah Varney, Anglia Varney, Branda Varney, Kawanna Varney, Tawnia Varney, Debi Varney, Elysia Varney, Lorretta Varney, Renetta Varney, Tamu Varney, Anneliese
Varney, Devan Varney, Rochel Varney, Vernetta Varney, Anetra Varney, Brynn Varney, Claribel Varney, Denielle Varney, Eugena Varney, Fara Varney, Genie Varney, Joya Varney, Keisa Varney, Loria Varney,
Tahira Varney, Tawna Varney, Angila Varney, Chara Varney, Cybil Varney, Jimmy Varney, Kendal Varney, Serita Varney, Tova Varney, Vernice Varney, Cruz Varney, Jameelah Varney, Karan Varney, Kyna
Varney, Larita Varney, Maricella Varney, Mariel Varney, Elyssa Varney, Kathyrn Varney, Raechel Varney, Jolee Varney, Earnestine Varney, Livia Varney, Nechama Varney, Shawndra Varney, Antonina Varney,
Chassity Varney, Codi Varney, Georgetta Varney, Gwyn Varney, Jacqualine Varney, Latania Varney, Tawni Varney, Tiwana Varney, Toyia Varney, Willette Varney, Anette Varney, Brandye Varney, Danny
Varney, Enedina Varney, Korina Varney, Latisa Varney, Vasiliki Varney, Victor Varney, Daneen Varney, Elnora Varney, Shareen Varney, Amey Varney, Carroll Varney, Channon Varney, Charletta Varney,
Deserie Varney, Jovan Varney, Latrica Varney, Lorina Varney, Love Varney, Carolann Varney, Fran Varney, Ilona Varney, Kandra Varney, Keela Varney, Lasonia Varney, Leatha Varney, Lizzette Varney, Macy
Varney, Magan Varney, Rebekka Varney, Vanetta Varney, Azucena Varney, Brigit Varney, Jamaica Varney, Jameka Varney, Sheridan Varney, Aliya Varney, Chantay Varney, Giuseppina Varney, Lyndi Varney,
Ragan Varney, Retha Varney, Bari Varney, Daryl Varney, Elonda Varney, Grisel Varney, Joline Varney, Korey Varney, Milinda Varney, Rubi Varney, Sabine Varney, Silvana Varney, Turkessa Varney, Myla
Varney, Sharese Varney, Cheron Varney, Joell Varney, Lachandra Varney, Lateefah Varney, Sanya Varney, Wesley Varney, Amira Varney, Imani Varney, Joette Varney, Kimmie Varney, Lynna Varney, Sheneka
Varney, Sunnie Varney, Andi Varney, Angelyn Varney, Maurice Varney, Melodi Varney, Nelida Varney, Sherida Varney, Shronda Varney, Takia Varney, Toshiba Varney, Trishia Varney, Alonda Varney, Araseli
Varney, Laketa Varney, Lesha Varney, Magen Varney, Neisha Varney, Selma Varney, Shannen Varney, Toria Varney, Aline Varney, Aubree Varney, Delaina Varney, Joannie Varney, Mai Varney, Niya Varney,
Dimitra Varney, Karl Varney, Kimberle Varney, Quanda Varney, Alica Varney, Chaundra Varney, Cherly Varney, Donnell Varney, Ellyn Varney, Lanell Varney, Lawrence Varney, Tonita Varney, Vania Varney,
Genea Varney, Kathaleen Varney, Laquetta Varney, Lorien Varney, Mallory Varney, Margery Varney, Porsha Varney, Agatha Varney, Alayna Varney, Barbi Varney, Fatimah Varney, Jaclynn Varney, Kalisha
Varney, Liesl Varney, Alita Varney, Janita Varney, Ola Varney, Shawne Varney, Talena Varney, Angla Varney, Autum Varney, Bertina Varney, Samira Varney, Shalena Varney, Consuella Varney, Corin Varney,
Delora Varney, Lanna Varney, Pat Varney, Roshunda Varney, Jonette Varney, Latishia Varney, Reginia Varney, Ronnette Varney, Sherree Varney, Tamicka Varney, Alejandrina Varney, Carlisa Varney, Jene
Varney, Loralee Varney, Lorenza Varney, Luis Varney, Yasmeen Varney, Zakiyyah Varney, Ashly Varney, Cassy Varney, Chari Varney, Indira Varney, Ivelisse Varney, Jaymi Varney, Karalee Varney, Lauretta
Varney, Raelynn Varney, Tifany Varney, Torrey Varney, Amyjo Varney, Annalee Varney, Hester Varney, Josefa Varney, Latresa Varney, Lita Varney, Marysol Varney, Mindie Varney, Shan Varney, Susy Varney,
Teneka Varney, Arnita Varney, Charice Varney, Cleopatra Varney, Cotina Varney, Dawnette Varney, Julene Varney, Lashay Varney, Latrell Varney, Lucrecia Varney, Ricky Varney, Shamekia Varney, Camellia
Varney, Cassi Varney, Curtis Varney, Hana Varney, Isha Varney, Jacquelyne Varney, Jenica Varney, Norah Varney, Oneida Varney, Subrina Varney, Tyrone Varney, Angi Varney, Anjeanette Varney, Bella
Varney, Donica Varney, Kareen Varney, Laina Varney, Luci Varney, Mischelle Varney, Tanita Varney, Heaven Varney, Madelin Varney, Michaele Varney, Rashell Varney, Tamarah Varney, Viki Varney, Candee
Varney, Danyale Varney, Karis Varney, Mishelle Varney, Shanequa Varney, Shellee Varney, Despina Varney, Dodi Varney, Katherin Varney, Mechele Varney, Pebbles Varney, Roshelle Varney, Carry Varney,
Denine Varney, Dory Varney, Jolinda Varney, Julieanne Varney, Karlyn Varney, Kathern Varney, Kerensa Varney, Kerrin Varney, Lafonda Varney, Launa Varney, Marin Varney, Ona Varney, Sheli Varney,
Trixie Varney, Yahaira Varney, Casi Varney, Ciara Varney, Darline Varney, Drema Varney, Francie Varney, Gerald Varney, Khara Varney, Lashone Varney, Meloney Varney, Melyssa Varney, Merrie Varney,
Neysa Varney, Queen Varney, Rashawn Varney, Robbyn Varney, Sharilyn Varney, Sharyl Varney, Tamitha Varney, Tresha Varney, Amaris Varney, Bernetta Varney, Channel Varney, Corrin Varney, Deadra Varney,
Filomena Varney, Jaye Varney, Kerriann Varney, Kysha Varney, Lanie Varney, Lashauna Varney, Lashona Varney, Laurinda Varney, Monifa Varney, Sharri Varney, Shontae Varney, Terrica Varney, Thresa
Varney, Tondra Varney, Geralyn Varney, Ginamarie Varney, Jesseca Varney, Lasonja Varney, Marica Varney, Marlen Varney, Virgie Varney, Genna Varney, Kriste Varney, Kyndra Varney, Tequilla Varney,
Vernell Varney, Zoila Varney, Aiesha Varney, Andera Varney, Blake Varney, Carlena Varney, Deangela Varney, Lavonia Varney, Lesly Varney, Savannah Varney, Danisha Varney, Dierdre Varney, Hermelinda
Varney, Kam Varney, Milena Varney, Omega Varney, Rosaria Varney, Russell Varney, Shaquita Varney, Tashika Varney, Arlena Varney, Delisha Varney, Demetric Varney, Lakendra Varney, Marilu Varney, Neha
Varney, Terria Varney, Carita Varney, Kallie Varney, Makisha Varney, Martie Varney, Mitzie Varney, Stephane Varney, Donette Varney, Joli Varney, Kathlyn Varney, Keona Varney, Lakishia Varney, Laurice
Varney, Melena Varney, Nan Varney, Rafaela Varney, Taura Varney, Wynette Varney, Arielle Varney, Carlette Varney, Cyrstal Varney, Dewanda Varney, Kimmy Varney, Pauletta Varney, Shannel Varney, Shelbi
Varney, Camillia Varney, Charlyn Varney, Christianna Varney, Gerry Varney, Jamy Varney, Karoline Varney, Katrinia Varney, Lorissa Varney, Magdalene Varney, Mariza Varney, Markisha Varney, Mindee
Varney, Terica Varney, Terisa Varney, Tyronda Varney, Aracelis Varney, Betzaida Varney, Charon Varney, Chastidy Varney, Henry Varney, Jamika Varney, Janiece Varney, Kariann Varney, Kawanda Varney,
Saprina Varney, Chera Varney, Genoveva Varney, Jacey Varney, Jannifer Varney, Jaylene Varney, Meisha Varney, Daphney Varney, Delois Varney, Denelle Varney, Diamond Varney, Felita Varney, Hilarie
Varney, Jamica Varney, Joelene Varney, Kisa Varney, Lilliana Varney, Sulema Varney, Wendolyn Varney, Calista Varney, Darrell Varney, Elinor Varney, Faviola Varney, Jenean Varney, Lalena Varney,
Mischa Varney, Tamikia Varney, Alanda Varney, Ashia Varney, Bryna Varney, Serene Varney, Shauntel Varney, Sheria Varney, Talya Varney, Thersa Varney, Walter Varney, Yevette Varney, Yolande Varney,
Ashlea Varney, Gaylene Varney, Khadija Varney, Latitia Varney, Micheline Varney, Monalisa Varney, Sakina Varney, Stacye Varney, Taffy Varney, Tifani Varney, Wilhelmina Varney, Amara Varney, Gypsy
Varney, Joely Varney, Kameron Varney, Keasha Varney, Louella Varney, Riki Varney, Torey Varney, Adrien Varney, Arwen Varney, Crysta Varney, Delaine Varney, Denyse Varney, Georgie Varney, Janea
Varney, Medina Varney, Meka Varney, Nika Varney, Rashanda Varney, Corena Varney, Denee Varney, Kimya Varney, Lanisha Varney, Luella Varney, Marylin Varney, Merritt Varney, Mysti Varney, Shawntay
Varney, Shirelle Varney, Siri Varney, Cayce Varney, Chava Varney, Jovanna Varney, Lorine Varney, Maryelizabeth Varney, Sonji Varney, Titania Varney, Brent Varney, Jamye Varney, Kela Varney, Korin
Varney, Monette Varney, Rebbeca Varney, Rebeka Varney, Stormie Varney, Angelo Varney, Camela Varney, Darlena Varney, Dietra Varney, Gwendolen Varney, Kenia Varney, Kesia Varney, Kolleen Varney,
Leighanne Varney, Lindi Varney, Marcee Varney, Miya Varney, Sami Varney, Tameko Varney, Tamora Varney, Tanaya Varney, Tiki Varney, Toniann Varney, Jeaneen Varney, Lizeth Varney, Manuel Varney,
Marlina Varney, Monisha Varney, Nichele Varney, Rechelle Varney, Richele Varney, Shenna Varney, Sky Varney, Tahnee Varney, Brita Varney, Erikka Varney, Jani Varney, Kattie Varney, Kimbra Varney,
Quincy Varney, Shaquana Varney, Sherelle Varney, Sherryl Varney, Shireen Varney, Theressa Varney, Yecenia Varney, Albert Varney, Ansley Varney, Clair Varney, Danille Varney, Evelia Varney, Jacob
Varney, Jon Varney, Katrin Varney, Kiva Varney, Mirta Varney, Pearlie Varney, Roger Varney, Shawntell Varney, Terrance Varney, Vincent Varney, Aimie Varney, Alise Varney, Caryl Varney, Devora Varney,
Dian Varney, Genelle Varney, Johannah Varney, Lanora Varney, Memory Varney, Mila Varney, Shandy Varney, Shella Varney, Shila Varney, Tahisha Varney, Channa Varney, Christeen Varney, Janise Varney,
Kristee Varney, Melaney Varney, Tanis Varney, Teasha Varney, Arthur Varney, Derek Varney, Ginette Varney, Kamesha Varney, Keriann Varney, Lateisha Varney, Merrilee Varney, Nikesha Varney, Sherrita
Varney, Shontay Varney, Bess Varney, Candelaria Varney, Charo Varney, Dominga Varney, Kateri Varney, Kelsie Varney, Kenyata Varney, Lane Varney, Lateshia Varney, Lili Varney, Linnette Varney, Odette
Varney, Sharika Varney, Sherra Varney, Tama Varney, Cicily Varney, Clarinda Varney, Corri Varney, Donnita Varney, Duana Varney, Gisele Varney, Laquitta Varney, Mahogany Varney, Seanna Varney, Sena
Varney, Shelbie Varney, Steffani Varney, Tauna Varney, Zara Varney, Aminah Varney, Amye Varney, Damon Varney, Doria Varney, Frederica Varney, Hollis Varney, Katey Varney, Nila Varney, Pamella Varney,
Ray Varney, Taina Varney, Zenia Varney, Ananda Varney, Becca Varney, Dessie Varney, Donnette Varney, Marney Varney, Nekisha Varney, Sharell Varney, Stefania Varney, Tandi Varney, Allana Varney,
Andree Varney, Arlette Varney, Denisa Varney, Ieshia Varney, Latrecia Varney, Meryl Varney, Shakia Varney, Shaye Varney, Clover Varney, Cresta Varney, Francina Varney, Fredericka Varney, Kimbery
Varney, Shameika Varney, Sherene Varney, Armanda Varney, Cally Varney, Cydney Varney, Dasha Varney, Gwynne Varney, Josephina Varney, Lavonna Varney, Melessa Varney, Nycole Varney, Nyoka Varney, Pansy
Varney, Shakita Varney, Sonjia Varney, Wanita Varney, Xenia Varney, Any Varney, Bobi Varney, Bonni Varney, Chevelle Varney, Chrystie Varney, Genine Varney, Iona Varney, Kezia Varney, Latresha Varney,
Lucie Varney, Malikah Varney, Micha Varney, Reginald Varney, Ronita Varney, Sharina Varney, Sharise Varney, Stormi Varney, Tammey Varney, Angelika Varney, Anja Varney, Carianne Varney, Dalene Varney,
Florinda Varney, Irasema Varney, Karren Varney, Kedra Varney, Koreen Varney, Leena Varney, Libra Varney, Moncia Varney, Rosaline Varney, Roshawn Varney, Rusty Varney, Sonda Varney, Talina Varney,
Trudie Varney, Audry Varney, Celinda Varney, Deona Varney, Jennipher Varney, Jolena Varney, Kimerly Varney, Kym Varney, Laquana Varney, Madalyn Varney, Renate Varney, Una Varney, Barrie Varney,
Chanin Varney, Deven Varney, Hadley Varney, Hether Varney, Ninfa Varney, Sha Varney, Sloane Varney, Tamla Varney, Amada Varney, Belynda Varney, Celesta Varney, Chelsa Varney, Corene Varney, Dawana
Varney, Elly Varney, Laure Varney, Pascha Varney, Shenell Varney, Tashana Varney, Trecia Varney, Angelea Varney, Dominque Varney, Frederick Varney, Ira Varney, Karine Varney, Lachanda Varney, Lyla
Varney, Melanee Varney, Nikkole Varney, Ronetta Varney, Shanise Varney, Shiree Varney, Vincenza Varney, Yolando Varney, Adelle Varney, Ara Varney, Bria Varney, Cheryll Varney, Delila Varney, Delina
Varney, Jonetta Varney, Kamela Varney, Lacinda Varney, Lashanta Varney, Laurene Varney, Lenita Varney, Martin Varney, Neda Varney, Sharen Varney, Tanishia Varney, Bronwen Varney, Jamilla Varney,
Jennelle Varney, Lavetta Varney, Makeda Varney, Omaira Varney, Randie Varney, Ricci Varney, Romy Varney, Sherlyn Varney, Sonal Varney, Tierra Varney, Tonyia Varney, Caressa Varney, Carmina Varney,
Dann Varney, Dawnelle Varney, Isa Varney, Lameka Varney, Lavada Varney, Lavita Varney, Lessie Varney, Miracle Varney, Mylinda Varney, Nancie Varney, Raeanne Varney, Shanique Varney, Sharolyn Varney,
Alex Varney, Cheyanne Varney, Christan Varney, Halima Varney, Karlee Varney, Katherina Varney, Lisabeth Varney, Marylynn Varney, Naimah Varney, Ria Varney, Tierney Varney, Velda Varney, Abbe Varney,
Alane Varney, Belia Varney, Falisha Varney, Kaylee Varney, Latausha Varney, Lynnea Varney, Myranda Varney, Nyla Varney, Rima Varney, Seneca Varney, Aprille Varney, Carlina Varney, Christle Varney,
Corliss Varney, Crystle Varney, Desha Varney, Kea Varney, Kristena Varney, Kristiana Varney, Lari Varney, Lorita Varney, Marlaina Varney, Marquitta Varney, Ondrea Varney, Tashara Varney, Devonne
Varney, Domonique Varney, Jannet Varney, Kila Varney, Krysten Varney, Leslye Varney, Lynnae Varney, Shela Varney, Sherica Varney, Timberly Varney, Arminda Varney, Essence Varney, Gala Varney, Hellen
Varney, Jannine Varney, Kamille Varney, Lalita Varney, Layne Varney, Letecia Varney, Panagiota Varney, Philana Varney, Trisa Varney, Akisha Varney, Audria Varney, Berenice Varney, Bette Varney,
Denese Varney, Donia Varney, Johana Varney, Kamika Varney, Katelyn Varney, Laine Varney, Lajuan Varney, Lasondra Varney, Marketta Varney, Patrizia Varney, Sascha Varney, Shamara Varney, Shamica
Varney, Tirzah Varney, Trinia Varney, Tyla Varney, Venice Varney, Carline Varney, Deyanira Varney, Isabell Varney, Jackqueline Varney, Kendria Varney, Marlee Varney, Sharisse Varney, Shawntae Varney,
Sholanda Varney, Thomasine Varney, Tikisha Varney, Tyna Varney, Aria Varney, Colene Varney, Daphine Varney, Dorthea Varney, Genise Varney, Keia Varney, Leasa Varney, Nohemi Varney, Rosann Varney,
Ruthanne Varney, Shundra Varney, Taria Varney, Valinda Varney, Adrain Varney, Brunilda Varney, Caralee Varney, Charlee Varney, Katasha Varney, Krystina Varney, Krystyna Varney, Lovie Varney,
Marquette Varney, Marybel Varney, Melvin Varney, Nathaniel Varney, Rickie Varney, Romana Varney, Tene Varney, Tinesha Varney, Tyann Varney, Wednesday Varney, Abra Varney, Adia Varney, Casondra
Varney, Charmane Varney, Chevonne Varney, Gwenn Varney, Laketha Varney, Louis Varney, Marvin Varney, Micole Varney, Novella Varney, Oriana Varney, Raena Varney, Sharone Varney, Suellen Varney, Thais
Varney, Trenda Varney, Zuleika Varney, Aundria Varney, Ayisha Varney, Catrena Varney, Corinthia Varney, Dorine Varney, Dulcie Varney, Dyanna Varney, Kimyatta Varney, Lekeshia Varney, Lenette Varney,
Nesha Varney, Reta Varney, Rian Varney, Rori Varney, Samia Varney, Shelonda Varney, Sima Varney, Adriann Varney, Beata Varney, Dewanna Varney, Elisia Varney, Lida Varney, Myriah Varney, Porsche
Varney, Regine Varney, Roma Varney, Shandi Varney, Sharae Varney, Brady Varney, Denitra Varney, Erik Varney, Jeane Varney, Jinny Varney, Karah Varney, Mechell Varney, Mikel Varney, Nikkie Varney,
Nisa Varney, Nubia Varney, Peaches Varney, Philip Varney, Ramie Varney, Romelia Varney, Roshell Varney, Tai Varney, Tracia Varney, Wonda Varney, Alix Varney, Aprile Varney, Ilda Varney, Kacee Varney,
Kathrin Varney, Lettie Varney, Michaelene Varney, Rania Varney, Shilpa Varney, Teddi Varney, Tegan Varney, Terrilyn Varney, Albertina Varney, Aryn Varney, Augustina Varney, Belle Varney, Britton
Varney, Irena Varney, Jacquie Varney, Katrine Varney, Lorenda Varney, Magali Varney, Mikaela Varney, Remona Varney, Rossana Varney, Shree Varney, Adrea Varney, Ashaki Varney, Beatris Varney, Elisabet
Varney, Gene Varney, Greer Varney, Hillery Varney, Jillene Varney, Keyona Varney, Latifa Varney, Marshell Varney, Miguel Varney, Resa Varney, Roy Varney, Tarnisha Varney, Tekisha Varney, Teshia
Varney, Tomara Varney, Vashti Varney, Brienne Varney, Calvin Varney, Corrinne Varney, Dorina Varney, Errica Varney, Eulalia Varney, Jacelyn Varney, Jimi Varney, Kavita Varney, Keyana Varney, Kiara
Varney, Lahoma Varney, Merilee Varney, Milisa Varney, Pollyanna Varney, Rabecca Varney, Shalisa Varney, Tyree Varney, Baby Varney, Cali Varney, Cherron Varney, Conswella Varney, Dayle Varney, Deniece
Varney, Elesha Varney, Eugene Varney, Fredricka Varney, Letty Varney, Marlisa Varney, Meadow Varney, Nana Varney, Ricardo Varney, Sunita Varney, Tahirah Varney, Aleasha Varney, Alene Varney, Jerica
Varney, Jonni Varney, Jorie Varney, Keenya Varney, Phillis Varney, Scottie Varney, Tamy Varney, Adelaide Varney, Akia Varney, Alona Varney, Brandilyn Varney, Davita Varney, Deondra Varney, Guinevere
Varney, Jaquetta Varney, Joyelle Varney, Kaitlin Varney, Kammy Varney, Lashonna Varney, Manisha Varney, Nyesha Varney, Phebe Varney, Shadonna Varney, Tamila Varney, Treasure Varney, Wayne Varney,
Charmayne Varney, Corry Varney, Doni Varney, Jannell Varney, Kaia Varney, Karna Varney, Kinya Varney, Lasha Varney, Lecia Varney, Letricia Varney, Paulita Varney, Sharnell Varney, Sheniqua Varney,
Toy Varney, Unique Varney, Wynona Varney, Audrie Varney, Cambria Varney, Crissie Varney, Deseree Varney, Deva Varney, Jenney Varney, Jessamyn Varney, Leda Varney, Lorry Varney, Marjory Varney, Marnee
Varney, Monigue Varney, Nakeya Varney, Passion Varney, Ryanne Varney, Sharona Varney, Shelina Varney, Tabbatha Varney, Tameshia Varney, Adelia Varney, Anica Varney, Calli Varney, Carolyne Varney,
Chiffon Varney, Elizabet Varney, Ermelinda Varney, Freddie Varney, Jamia Varney, Korrie Varney, Latrenda Varney, Makia Varney, Marcel Varney, Meta Varney, Noni Varney, Sigrid Varney, Zita Varney,
Angelee Varney, Balinda Varney, Brandyn Varney, Christia Varney, Eneida Varney, Irish Varney, Jackelyn Varney, Jacqui Varney, Kammie Varney, Kendell Varney, Kiya Varney, Mellanie Varney, Sherina
Varney, Syretta Varney, Teia Varney, Teneshia Varney, Tomasa Varney, Tonna Varney, Virgina Varney, Zelma Varney, Adella Varney, Alea Varney, Annita Varney, Denetra Varney, Georgeann Varney, Gwyneth
Varney, Jenea Varney, Kalli Varney, Khristine Varney, Krissie Varney, Lanelle Varney, Letrice Varney, Meshell Varney, Michon Varney, Nekita Varney, Raelyn Varney, Shaleen Varney, Sharanda Varney, Sol
Varney, Suzi Varney, Ulanda Varney, Ame Varney, Aprill Varney, Brooks Varney, Carlye Varney, Francisco Varney, Gaynell Varney, Gwenda Varney, Heidy Varney, Helga Varney, Janese Varney, Kathrina
Varney, Katonya Varney, Kodi Varney, Leshawn Varney, London Varney, Lyndee Varney, Marc Varney, Miyoshi Varney, Onika Varney, Peyton Varney, Shandell Varney, Terena Varney, Acacia Varney, Aleida
Varney, Antoine Varney, Arletha Varney, Carlin Varney, Danni Varney, Davette Varney, December Varney, Jenene Varney, Karalyn Varney, Kimbely Varney, Lakisa Varney, Lynae Varney, Margaux Varney,
Michela Varney, Renne Varney, Tanzania Varney, Tenia Varney, Toyna Varney, Veronika Varney, Amal Varney, Camila Varney, Chi Varney, Dominic Varney, Macie Varney, Maida Varney, Malisha Varney, Milagro
Varney, Mitchell Varney, Shondell Varney, Taira Varney, Argelia Varney, Casaundra Varney, Courtnie Varney, Dalena Varney, Donyell Varney, Ericia Varney, Kelsi Varney, Madelene Varney, Mikelle Varney,
Moria Varney, Salome Varney, Starlet Varney, Stepanie Varney, Stephnie Varney, Teesha Varney, Allisa Varney, Aneesah Varney, Breann Varney, Chevon Varney, Deatrice Varney, Delaney Varney, Jodene
Varney, Julian Varney, Lorilee Varney, Margarette Varney, Marshall Varney, Meleah Varney, Nadene Varney, Nefertiti Varney, Rebekkah Varney, Shakina Varney, Sibyl Varney, Teria Varney, Briget Varney,
Cortina Varney, Gennie Varney, Georgeanna Varney, Jael Varney, Katharina Varney, Kennetha Varney, Kerrianne Varney, Meggin Varney, Rochele Varney, Santos Varney, Senta Varney, Shanie Varney, Sona
Varney, Yana Varney, Alicea Varney, Capri Varney, Cassey Varney, Daun Varney, Denette Varney, Harold Varney, Jacie Varney, Jennica Varney, Krissa Varney, Lawonda Varney, Lenna Varney, Maija Varney,
Ninette Varney, Tomeika Varney, Trenna Varney, Amberlee Varney, Avril Varney, Charnell Varney, Chenita Varney, Crystel Varney, Eugenie Varney, Ginnie Varney, Jerome Varney, Karee Varney, Latunya
Varney, Lianna Varney, Lisaann Varney, Maribell Varney, Marylyn Varney, Molli Varney, Raye Varney, Risha Varney, Sherrice Varney, Taralyn Varney, Tika Varney, Tyanna Varney, Wilda Varney, Brandalyn
Varney, Camesha Varney, Ericca Varney, Ernest Varney, Glynis Varney, Larena Varney, Latecia Varney, Lindsy Varney, Melea Varney, Monic Varney, Suanne Varney, Tammra Varney, Tyeisha Varney, Allen
Varney, Ambra Varney, Cindee Varney, Deja Varney, Denika Varney, Easter Varney, Emerald Varney, Jann Varney, Jeannetta Varney, Keyla Varney, Lataya Varney, Natividad Varney, Nikola Varney, Quanita
Varney, Shalyn Varney, Temekia Varney, Tishia Varney, Allena Varney, Andrena Varney, Aurea Varney, Buffi Varney, Darren Varney, Dovie Varney, Evelina Varney, Fatina Varney, Ivey Varney, Jennafer
Varney, Kimberlea Varney, Kishia Varney, Lacrecia Varney, Laney Varney, Latora Varney, Robinette Varney, Shalita Varney, Shanin Varney, Terasa Varney, Tiphanie Varney, Tiwanna Varney, Ashlyn Varney,
Breana Varney, Emely Varney, Evan Varney, Evangelia Varney, Halle Varney, Lynnetta Varney, Mandisa Varney, Mardi Varney, Neomi Varney, Peggie Varney, Senaida Varney, Shantee Varney, Alondra Varney,
Bettyjo Varney, Dawnell Varney, Jaunita Varney, Jaymee Varney, Jeneane Varney, Kamara Varney, Korinne Varney, Lacosta Varney, Latrena Varney, Letticia Varney, Luanna Varney, Meighan Varney, Merinda
Varney, Raechelle Varney, Randall Varney, Shermaine Varney, Tacy Varney, Telicia Varney, Trica Varney, Unknown Varney, Venetia Varney, Virgen Varney, Chanita Varney, Christe Varney, Collene Varney,
Deni Varney, Eliana Varney, Evelin Varney, Jessa Varney, Katarina Varney, Kersten Varney, Krysti Varney, Lanise Varney, Lannette Varney, Latacha Varney, Lavenia Varney, Magnolia Varney, Mayte Varney,
Missie Varney, Renda Varney, Sapna Varney, Shawan Varney, Starlene Varney, Stasia Varney, Tamaria Varney, Tiffanee Varney, Yazmin Varney, Ammy Varney, Blossom Varney, Cherokee Varney, Chrissa Varney,
Dean Varney, Donell Varney, Jannelle Varney, Jenessa Varney, Jenne Varney, Kimiko Varney, Lenise Varney, Lotoya Varney, Mariko Varney, Myeshia Varney, Nadja Varney, Shalana Varney, Tangee Varney,
Waynette Varney, Alan Varney, Bennie Varney, Berlinda Varney, Brinda Varney, Brooklyn Varney, Bruce Varney, Daisha Varney, Devina Varney, Donisha Varney, Judie Varney, Khristy Varney, Margarett
Varney, Markesha Varney, Melannie Varney, Nickol Varney, Pamelia Varney, Roselle Varney, Shirleen Varney, Terrence Varney, Trenise Varney, Alda Varney, Ariella Varney, Aubrie Varney, Aziza Varney,
Carrol Varney, Charnita Varney, Deshanna Varney, Donyale Varney, Dorrie Varney, Elayne Varney, Feather Varney, Gregoria Varney, Kanesha Varney, Kirsti Varney, Maryalice Varney, Maude Varney, Millissa
Varney, Monya Varney, Shareese Varney, Shereka Varney, Tenaya Varney, Vonnie Varney, Bunny Varney, Carmin Varney, Cissy Varney, Damika Varney, Dayla Varney, Dewana Varney, Jorge Varney, Kaila Varney,
Kaisha Varney, Kaylynn Varney, Kemba Varney, Lael Varney, Lysa Varney, Roberto Varney, Tanasha Varney, Tynisa Varney, Vina Varney, Angelette Varney, Angenette Varney, Aquila Varney, Carrianne Varney,
Charlynn Varney, Denean Varney, Dondi Varney, Enriqueta Varney, Georganna Varney, Glinda Varney, Jenel Varney, Larinda Varney, Latika Varney, Lequita Varney, Licia Varney, Marilou Varney, Sylvie
Varney, Aaliyah Varney, Cozette Varney, Damian Varney, Ginna Varney, Janalee Varney, Jenniefer Varney, Kierstin Varney, Lexie Varney, Maja Varney, Michelene Varney, Norine Varney, Sana Varney, Sarrah
Varney, Sharna Varney, Sherre Varney, Shoshanna Varney, Tangi Varney, Tillie Varney, Aishia Varney, Allene Varney, Antoniette Varney, Brittani Varney, Candise Varney, Carlota Varney, Debroah Varney,
Demeka Varney, Dondra Varney, Franca Varney, Gaye Varney, Jeanell Varney, Lark Varney, Laurin Varney, Lourie Varney, Machell Varney, Pasha Varney, Shelena Varney, Tangelia Varney, Tani Varney, Twilla
Varney, Tynesha Varney, Vena Varney, Vernessa Varney, Aleesha Varney, Cassia Varney, Catheryn Varney, Chavon Varney, Codie Varney, Conchita Varney, Felipa Varney, Jeanice Varney, Kelleen Varney,
Latona Varney, Loreal Varney, Lutricia Varney, Natanya Varney, Perry Varney, Samaria Varney, Sharene Varney, Shavone Varney, Stavroula Varney, Tatia Varney, Terrina Varney, Veroncia Varney, Alaine
Varney, Carmalita Varney, Chirstina Varney, Dawanna Varney, Delonda Varney, Ebonee Varney, Jamela Varney, Kerie Varney, Lanika Varney, Latressa Varney, Priscella Varney, Sharifa Varney, Shenequa
Varney, Shon Varney, Taresa Varney, Viva Varney, Annabell Varney, Arelis Varney, Carra Varney, Char Varney, Chekesha Varney, Cherylann Varney, Crystall Varney, Diandra Varney, Dwayne Varney, Elishia
Varney, Emi Varney, Jennife Varney, Kassi Varney, Keyonna Varney, Kresta Varney, Kriston Varney, Laree Varney, Lashea Varney, Lauryn Varney, Lennie Varney, Nailah Varney, Neena Varney, Nelia Varney,
Reiko Varney, Sejal Varney, Shanea Varney, Sharman Varney, Talesha Varney, Teresia Varney, Tunya Varney, Annett Varney, Catherina Varney, Charlita Varney, Dandrea Varney, Darcel Varney, Eleanore
Varney, Falicia Varney, Glynda Varney, Karra Varney, Kerra Varney, Ladona Varney, Loretha Varney, Lus Varney, Marketa Varney, Megen Varney, Shalini Varney, Sheela Varney, Tally Varney, Thuy Varney,
Tristen Varney, Yashika Varney, Aleah Varney, Arline Varney, Catarina Varney, Cris Varney, Dawnita Varney, Kelliann Varney, Kristle Varney, Lanesha Varney, Malanie Varney, Nekesha Varney, Rhona
Varney, Roslynn Varney, Sariah Varney, Takeshia Varney, Theda Varney, Altagracia Varney, Anneke Varney, Aspen Varney, Azalea Varney, Jamelle Varney, Kalyn Varney, Lavonya Varney, Malea Varney,
Marykate Varney, Meliss Varney, Noelani Varney, Patina Varney, Riann Varney, Rolonda Varney, Shaney Varney, Shanice Varney, Wynne Varney, Yocheved Varney, Aysha Varney, Cindie Varney, Daphanie
Varney, Darcee Varney, Demitra Varney, Francene Varney, Ian Varney, Janaya Varney, Jera Varney, Jeremiah Varney, Karlie Varney, Kiwana Varney, Krystin Varney, Larina Varney, Marylee Varney, Maurine
Varney, Mistee Varney, Rainey Varney, Raynette Varney, Robyne Varney, Ronelle Varney, Sharma Varney, Shelene Varney, Stevie Varney, Wynter Varney, Ylonda Varney, Zondra Varney, Aesha Varney, Alpha
Varney, Angala Varney, Che Varney, Courtnay Varney, Deatra Varney, Genell Varney, Jerrica Varney, Jozette Varney, Kerianne Varney, Krissi Varney, Millisa Varney, Minna Varney, Naeemah Varney,
Shantrell Varney, Theresia Varney, Tomeko Varney, Vanda Varney, Annelise Varney, Bettye Varney, Brina Varney, Carmelina Varney, Charlotta Varney, Charmine Varney, Darbi Varney, Devra Varney,
Donnamarie Varney, Enza Varney, Jack Varney, Lamika Varney, Lashaundra Varney, Latavia Varney, Mauri Varney, Nevada Varney, Patrisha Varney, Rashel Varney, Rayann Varney, Shanay Varney, Shareka
Varney, Sharlyn Varney, Sharra Varney, Suzzanne Varney, Tascha Varney, Tomeca Varney, Annisa Varney, Anthea Varney, Bethel Varney, Charlesetta Varney, Davetta Varney, Deonne Varney, Emmie Varney,
Fanta Varney, Felicitas Varney, Flavia Varney, Glenn Varney, Gwendolynn Varney, Hollye Varney, Jalene Varney, Jamel Varney, Jenita Varney, Jonda Varney, Kameka Varney, Lagina Varney, Lateasha Varney,
Louanne Varney, Ludivina Varney, Ramonita Varney, Reshonda Varney, Romina Varney, Samone Varney, Shauntae Varney, Shelle Varney, Tasheka Varney, Tashima Varney, Tekesha Varney, Trini Varney, Tyese
Varney, Alesa Varney, Brigett Varney, Camella Varney, Charee Varney, Coby Varney, Constantina Varney, Contrina Varney, Danise Varney, Delanie Varney, Denia Varney, Detrice Varney, Donyelle Varney,
Estee Varney, Freya Varney, Gabriele Varney, Heidie Varney, Idella Varney, Iraida Varney, Jonica Varney, Kaley Varney, Kirsty Varney, Laurette Varney, Nataki Varney, Patrisia Varney, Ronnetta Varney,
Signe Varney, Tarita Varney, Trenia Varney, Annessa Varney, Celene Varney, Cerissa Varney, Chinita Varney, Danah Varney, Dannelle Varney, Dawnielle Varney, Doretta Varney, Dwanna Varney, Jinger
Varney, Karilyn Varney, Loida Varney, Miko Varney, Mirtha Varney, Myia Varney, Perri Varney, Ragina Varney, Rea Varney, Saralyn Varney, Saskia Varney, Sharry Varney, Tiare Varney, Tonica Varney,
Waleska Varney, Xochilt Varney, Zipporah Varney, Delynn Varney, Detria Varney, Earl Varney, Enedelia Varney, Jeanett Varney, Jonita Varney, Kenita Varney, Laural Varney, Leonard Varney, Ligia Varney,
Linnie Varney, Lowanda Varney, Ltanya Varney, Monquie Varney, Paloma Varney, Rafael Varney, Steve Varney, Torina Varney, Yolunda Varney, Cariann Varney, Carmita Varney, Cerise Varney, Dalana Varney,
Dawnya Varney, Deshannon Varney, Deshonda Varney, Desirea Varney, Eartha Varney, Emelia Varney, Francoise Varney, Gerrie Varney, Gitty Varney, Jared Varney, Kianna Varney, Lamanda Varney, Laraine
Varney, Latonda Varney, Latrease Varney, Leaann Varney, Liv Varney, Maha Varney, Meliza Varney, Shawnetta Varney, Shontelle Varney, Sloan Varney, Taja Varney, Tammye Varney, Terika Varney, Vinita
Varney, Charese Varney, Charisa Varney, Danella Varney, Halley Varney, Jara Varney, Jawana Varney, Jobina Varney, Juniper Varney, Leondra Varney, Letonya Varney, Marianela Varney, Mariella Varney,
Marisel Varney, Marlinda Varney, Maurita Varney, Merrill Varney, Nalani Varney, Natika Varney, Ralph Varney, Shajuana Varney, Sharmon Varney, Tanda Varney, Clarence Varney, Enjoli Varney, Hali
Varney, Isaura Varney, Joellyn Varney, Kaylyn Varney, Ketra Varney, Latayna Varney, Melisse Varney, Natoshia Varney, Niko Varney, Shalina Varney, Sharrie Varney, Silva Varney, Taren Varney, Tawnie
Varney, Venetta Varney, Chenelle Varney, Chonda Varney, Drucilla Varney, Genene Varney, Jaquita Varney, Kally Varney, Laury Varney, Lucina Varney, Lynde Varney, Marley Varney, Scherrie Varney, Shaila
Varney, Shalimar Varney, Shannyn Varney, Shantina Varney, Shirl Varney, Takita Varney, Tineka Varney, Valisa Varney, Adrina Varney, Ambre Varney, Arlisa Varney, Cecila Varney, Demetrica Varney, Dyann
Varney, Erina Varney, Haven Varney, Hector Varney, Janika Varney, Johnnetta Varney, Kellyanne Varney, Lacrisha Varney, Lamar Varney, Litisha Varney, Malkia Varney, Marcellina Varney, Otilia Varney,
Pascale Varney, Ramonda Varney, Safiya Varney, Sebrena Varney, Shifra Varney, Sina Varney, Tashina Varney, Teffany Varney, Trevor Varney, Veleka Varney, Zanetta Varney, Arie Varney, Azalia Varney,
Cedric Varney, Chasta Varney, Cherelle Varney, Cyndee Varney, Darnetta Varney, Dinora Varney, Dorena Varney, Howard Varney, Joyel Varney, July Varney, Lance Varney, Lore Varney, Michelina Varney,
Olive Varney, Reem Varney, Rica Varney, Shaneen Varney, Sharlotte Varney, Shavonna Varney, Shawnie Varney, Sherria Varney, Sian Varney, Stephanee Varney, Susette Varney, Terilyn Varney, Tiajuana
Varney, Tristin Varney, Vada Varney, Vesta Varney, Yara Varney, Alyssia Varney, Arian Varney, Chianti Varney, Chriselda Varney, Darian Varney, Dawnetta Varney, Desarae Varney, Dinorah Varney, Djuana
Varney, Elka Varney, Golda Varney, Honor Varney, Jamille Varney, Jesusita Varney, Junita Varney, Kenyada Varney, Konnie Varney, Kwanza Varney, Lissett Varney, Marlon Varney, Marlys Varney, Marshelle
Varney, Meagen Varney, Myka Varney, Mylissa Varney, Patrece Varney, Raguel Varney, Sharia Varney, Shawnell Varney, Teanna Varney, Adrena Varney, Brad Varney, Charlisa Varney, Dannie Varney, Fernanda
Varney, Jeania Varney, Kareema Varney, Karyl Varney, Kellianne Varney, Lilah Varney, Lorelle Varney, Lynnell Varney, Quana Varney, Rondi Varney, Rosiland Varney, Ruben Varney, Sharmin Varney,
Taniesha Varney, Tashonda Varney, Valecia Varney, Alnisa Varney, Cherity Varney, Coletta Varney, Damary Varney, Darcia Varney, Dashawn Varney, Garnet Varney, Hadassah Varney, Javon Varney, Kareem
Varney, Keara Varney, Kerryann Varney, Loleta Varney, Marijo Varney, Maritsa Varney, Marvella Varney, Miriah Varney, Necia Varney, Priscila Varney, Ramon Varney, Shalawn Varney, Sidra Varney, Sundee
Varney, Tasia Varney, Tiwanda Varney, Xavier Varney, Alayne Varney, Anesha Varney, Anndrea Varney, Astra Varney, Breezy Varney, Carletha Varney, Chantele Varney, Damara Varney, Denell Varney, Dessa
Varney, Marcell Varney, Maretta Varney, Marline Varney, Melania Varney, Modesta Varney, Montoya Varney, Rashidah Varney, Rusti Varney, Shannell Varney, Shaune Varney, Sheresa Varney, Stasha Varney,
Talaya Varney, Taletha Varney, Tashawna Varney, Terrah Varney, Yessica Varney, Yolander Varney, Ani Varney, Cyndy Varney, Dannell Varney, Edythe Varney, Elodia Varney, Felissa Varney, Jamison Varney,
Janalyn Varney, Jodell Varney, Josetta Varney, Karmin Varney, Kenesha Varney, Keyna Varney, Kimara Varney, Kiran Varney, Ladonya Varney, Latorya Varney, Marizol Varney, Nadya Varney, Pamula Varney,
Pleshette Varney, Tamaka Varney, Tifanie Varney, Tunisha Varney, Verona Varney, Veronda Varney, Allicia Varney, Allisha Varney, Allissa Varney, Allyn Varney, Andriana Varney, Anneka Varney, Bridgitte
Varney, Brigida Varney, Chaunte Varney, Christol Varney, Elita Varney, Jacquelene Varney, Jenefer Varney, Jessyca Varney, Kammi Varney, Kasondra Varney, Katricia Varney, Katrinka Varney, Lakasha
Varney, Maryrose Varney, Mirinda Varney, Natoya Varney, Omar Varney, Santana Varney, Shakima Varney, Shatina Varney, Shawnita Varney, Sherine Varney, Starlett Varney, Tanga Varney, Teal Varney, Teara
Varney, Tiffanny Varney, Vivien Varney, Altovise Varney, Conni Varney, Dawanda Varney, Ellena Varney, Janica Varney, Jeralyn Varney, Kameelah Varney, Kasia Varney, Katharyn Varney, Lamesha Varney,
Laurieann Varney, Leon Varney, Margherita Varney, Marguerita Varney, Mariama Varney, Mariea Varney, Maryfrances Varney, Mayda Varney, Meena Varney, Mikal Varney, Nicholette Varney, Nitza Varney,
Norman Varney, Raychelle Varney, Riva Varney, Ronit Varney, Rosenda Varney, Royce Varney, Saira Varney, Samona Varney, Toye Varney, Zinnia Varney, Alfred Varney, Amita Varney, Aqueelah Varney, Blenda
Varney, Charron Varney, Chrisie Varney, Ema Varney, Endia Varney, Georgine Varney, Gretel Varney, Johnita Varney, Joscelyn Varney, Kiona Varney, Lamont Varney, Lateesha Varney, Liisa Varney, Lizet
Varney, Marny Varney, Martisha Varney, Marvette Varney, Natonya Varney, Oona Varney, Philomena Varney, Sahar Varney, Shantia Varney, Starlette Varney, Talana Varney, Tereza Varney, Torsha Varney,
Tyeshia Varney, Aislinn Varney, Alexandrea Varney, Artisha Varney, Assunta Varney, Belva Varney, Britany Varney, Bryanna Varney, Chauntel Varney, Cristel Varney, Damali Varney, Daphane Varney,
Dorianne Varney, Imogene Varney, Janai Varney, Laveda Varney, Lucero Varney, Lyssa Varney, Marielle Varney, Natalya Varney, Pammy Varney, Patches Varney, Rhiana Varney, Sakeena Varney, Tanea Varney,
Tiffinie Varney, Xaviera Varney, Arla Varney, Collen Varney, Deane Varney, Don Varney, Gari Varney, Jazmine Varney, Joani Varney, Jude Varney, Lacee Varney, Lakeasha Varney, Leandrea Varney, Marvina
Varney, Minette Varney, Nico Varney, Noell Varney, Ouida Varney, Rainbow Varney, Shania Varney, Shemekia Varney, Velinda Varney, Yetta Varney, Adell Varney, Alyshia Varney, Arlana Varney, Athina
Varney, Calley Varney, Carson Varney, Cassandre Varney, Channing Varney, Cherith Varney, Darnella Varney, Deepa Varney, Eboney Varney, Florentina Varney, Fredia Varney, Jerilynn Varney, Kamila
Varney, Keeli Varney, Korena Varney, Korri Varney, Lakeia Varney, Lulu Varney, Madelaine Varney, Marja Varney, Maryhelen Varney, Mei Varney, Quintella Varney, Rainy Varney, Rashunda Varney, Rheanna
Varney, Saran Varney, Shanese Varney, Shelisa Varney, Tamecia Varney, Tanica Varney, Tawnee Varney, Van Varney, Vivienne Varney, Yoland Varney, Yvonna Varney, Abbi Varney, Afrika Varney, Anise
Varney, Aron Varney, Carrieanne Varney, Chauna Varney, Daena Varney, Elin Varney, Jerrilyn Varney, Landa Varney, Latoyna Varney, Leane Varney, Lilliam Varney, Marquisha Varney, Marry Varney, Matasha
Varney, Matina Varney, Megin Varney, Mitchelle Varney, Nakeshia Varney, Pearline Varney, Rayne Varney, Reeshemah Varney, Romaine Varney, Shameca Varney, Sterling Varney, Suzzette Varney, Tangy
Varney, Tonetta Varney, Trenice Varney, Zahra Varney, Abena Varney, Aliesha Varney, Andreia Varney, Beryl Varney, Cathey Varney, Crystalyn Varney, Davonna Varney, Delphia Varney, Denisse Varney,
Elina Varney, Elisheva Varney, Evangela Varney, Felina Varney, Gisella Varney, Ilka Varney, Janeth Varney, Katara Varney, Laquesha Varney, Lysandra Varney, Malana Varney, Morgen Varney, Natali
Varney, Niambi Varney, Niketa Varney, Pamla Varney, Rebeckah Varney, Revonda Varney, Sarabeth Varney, Sharmila Varney, Shylo Varney, Takiya Varney, Trevia Varney, Aleatha Varney, Anabelle Varney,
Blima Varney, Bliss Varney, Christyn Varney, Chrysta Varney, Cristela Varney, Deliah Varney, Ellisa Varney, Genice Varney, Isadora Varney, Jawanna Varney, Kamaria Varney, Ladena Varney, Larenda
Varney, Laurissa Varney, Leela Varney, Lovina Varney, Lyra Varney, Maryanna Varney, Marybell Varney, Meika Varney, Meshelle Varney, Moana Varney, Nicoletta Varney, Radiah Varney, Rupal Varney,
Shannin Varney, Shyra Varney, Stanley Varney, Sundae Varney, Taunia Varney, Tomorrow Varney, Abbigail Varney, Amoreena Varney, Anamarie Varney, Chani Varney, Dayanara Varney, Delane Varney, Demika
Varney, Desi Varney, Desma Varney, Eugina Varney, Florine Varney, Johanne Varney, Karman Varney, Katheryne Varney, Kema Varney, Kenitra Varney, Lareina Varney, Laryssa Varney, Lasaundra Varney,
Lorriane Varney, Lurdes Varney, Meredyth Varney, Mireille Varney, Naomie Varney, Rewa Varney, Rosette Varney, Rukiya Varney, Shanica Varney, Sherill Varney, Shonette Varney, Adonica Varney, Aquilla
Varney, Billijo Varney, Bretta Varney, Charmian Varney, Chondra Varney, Coty Varney, Danine Varney, Davena Varney, Davia Varney, Falana Varney, Fredrica Varney, Harry Varney, Jae Varney, Jawanda
Varney, Keana Varney, Krisann Varney, Laqueta Varney, Lashannon Varney, Lavelle Varney, Layna Varney, Linde Varney, Lorisa Varney, Lyndie Varney, Macey Varney, Maricel Varney, Nadina Varney, Nadiyah
Varney, Nereyda Varney, Sequoia Varney, Stephene Varney, Tahesha Varney, Takeya Varney, Tali Varney, Tiera Varney, Tomasina Varney, Verena Varney, Yvonda Varney, Aracelia Varney, Arletta Varney,
Augustine Varney, Barry Varney, Brittny Varney, Christella Varney, Courtnee Varney, Dakota Varney, Denesha Varney, Donnella Varney, Fotini Varney, Gita Varney, Holland Varney, Jacquiline Varney,
Jamilyn Varney, Kafi Varney, Kalena Varney, Katya Varney, Kit Varney, Laniece Varney, Lauree Varney, Leslea Varney, Maite Varney, Mitzy Varney, Myria Varney, Njeri Varney, Oscar Varney, Rain Varney,
Rema Varney, Ronee Varney, Shevon Varney, Shirlee Varney, Sissy Varney, Sonali Varney, Sundra Varney, Taralee Varney, Trace Varney, Tyana Varney, Agustina Varney, Akua Varney, Allyssa Varney, Andrina
Varney, Angell Varney, Antwanette Varney, Carmelia Varney, Chelsi Varney, Cheronda Varney, Clorinda Varney, Crissa Varney, Cyrena Varney, Dante Varney, Destini Varney, Emiko Varney, Era Varney,
Gentry Varney, Gini Varney, Hanan Varney, Haylee Varney, Hollee Varney, Jannett Varney, Javier Varney, Jenay Varney, Jennier Varney, Kandee Varney, Kerilyn Varney, Kissy Varney, Konstantina Varney,
Ladana Varney, Logan Varney, Lovette Varney, Manal Varney, Marykay Varney, Phoenix Varney, Rosalin Varney, Shatara Varney, Shawnya Varney, Sheralyn Varney, Sheretta Varney, Suann Varney, Tava Varney,
Bobette Varney, Carrissa Varney, Coralee Varney, Daneille Varney, Edwin Varney, Emelda Varney, Flecia Varney, Jame Varney, Janeane Varney, Jelena Varney, Katja Varney, Kelia Varney, Lanee Varney,
Laverna Varney, Lorra Varney, Marlin Varney, Mieke Varney, Miroslava Varney, Nekeisha Varney, Nikeya Varney, Noami Varney, Saida Varney, Sienna Varney, Sundi Varney, Tylene Varney, Yakima Varney,
Alicha Varney, Amani Varney, Armandina Varney, Bethani Varney, Carine Varney, Carmencita Varney, Chantee Varney, Chavonne Varney, Chinyere Varney, Clarisse Varney, Evett Varney, Felisia Varney,
Flossie Varney, Indra Varney, Inge Varney, Keiko Varney, Kristia Varney, Lady Varney, Latrelle Varney, Lean Varney, Lewanda Varney, Marit Varney, Melodye Varney, Moneka Varney, Naisha Varney, Najah
Varney, Ronisha Varney, Shamona Varney, Shantal Varney, Sharay Varney, Shawndell Varney, Shekita Varney, Shelah Varney, Shenise Varney, Soraida Varney, Sujey Varney, Suni Varney, Tiona Varney, Tuwana
Varney, Wynetta Varney, Angelie Varney, Annalise Varney, Carlen Varney, Cherryl Varney, Donnelle Varney, Florencia Varney, Gema Varney, Genita Varney, Happy Varney, Jessy Varney, Karimah Varney,
Lanitra Varney, Louanna Varney, Maisie Varney, Marco Varney, Margaretta Varney, Marguita Varney, Marice Varney, Marygrace Varney, Retta Varney, Roderick Varney, Ronny Varney, Shalom Varney, Sherena
Varney, Tamura Varney, Threasa Varney, Ulonda Varney, Zachary Varney, Aletta Varney, Amaryllis Varney, Amethyst Varney, Antoinetta Varney, Aricka Varney, Bobbyjo Varney, Camile Varney, Charyl Varney,
Elysa Varney, Jameela Varney, Jeannett Varney, Jyl Varney, Kinda Varney, Kristianne Varney, Latifah Varney, Lavera Varney, Madaline Varney, Marea Varney, Melaina Varney, Natesha Varney, Oliva Varney,
Renell Varney, Salli Varney, Satina Varney, Shanyn Varney, Sharelle Varney, Sharena Varney, Shenetta Varney, Sherlonda Varney, Sun Varney, Teah Varney, Teana Varney, Tramaine Varney, Wenda Varney,
Wyndi Varney, Yoko Varney, Adrean Varney, Aishah Varney, Aundra Varney, Betsaida Varney, Chrisann Varney, Clarita Varney, Correna Varney, Cynitha Varney, Cythia Varney, Damien Varney, Dedria Varney,
Divina Varney, Dody Varney, Elanda Varney, Elizbeth Varney, Estrellita Varney, Eydie Varney, Gerilyn Varney, Hedy Varney, Jacklynn Varney, Kimberleigh Varney, Kristinia Varney, Lakeeta Varney, Larie
Varney, Latrisa Varney, Lesleigh Varney, Marella Varney, Martinique Varney, Melessia Varney, Nicolasa Varney, Quantina Varney, Remy Varney, Robina Varney, Roya Varney, Saba Varney, Sacheen Varney,
Shalee Varney, Sharlet Varney, Shereese Varney, Sherronda Varney, Shewanda Varney, Skyla Varney, Tallie Varney, Tanganyika Varney, Taressa Varney, Tarshia Varney, Timi Varney, Zola Varney, Ainsley
Varney, Alivia Varney, Ardith Varney, Artina Varney, Ashely Varney, Athanasia Varney, Chantale Varney, Chimene Varney, Corinn Varney, Daffney Varney, Danene Varney, Daphnie Varney, Deeanne Varney,
Erynn Varney, Glennis Varney, Kalee Varney, Kandyce Varney, Kathren Varney, Kelcey Varney, Laticha Varney, Lenee Varney, Luvenia Varney, Machele Varney, Maris Varney, Marisella Varney, Nakeia Varney,
Nickey Varney, Nikkita Varney, Noella Varney, Peri Varney, Raushanah Varney, Renisha Varney, Shahidah Varney, Shakisha Varney, Shevonne Varney, Tashema Varney, Tondalaya Varney, Trinita Varney, Akiko
Varney, Alfredia Varney, Bathsheba Varney, Billye Varney, Carter Varney, Clementina Varney, Danyele Varney, Darnisha Varney, Darryl Varney, Eda Varney, Emy Varney, Faustina Varney, Hayden Varney,
Hortensia Varney, Ivon Varney, Javonna Varney, Keita Varney, Lakeisa Varney, Lan Varney, Leota Varney, Letasha Varney, Maggi Varney, Monita Varney, Nadirah Varney, Nikka Varney, Noemy Varney, Roxy
Varney, Sabrinia Varney, Sanda Varney, Sharan Varney, Sharnette Varney, Swati Varney, Tiny Varney, Tonjia Varney, Trella Varney, Venecia Varney, Zana Varney, Zorana Varney, Adrienna Varney, Ama
Varney, Angelisa Varney, Beckey Varney, Bobbye Varney, Cammi Varney, Chequita Varney, Clinton Varney, Dama Varney, Danee Varney, Dari Varney, Deandre Varney, Deetta Varney, Devonda Varney, Edelmira
Varney, Ieasha Varney, Joylynn Varney, Krisinda Varney, Kristol Varney, Kwana Varney, Lacresia Varney, Ladawna Varney, Lavanda Varney, Lavena Varney, Leaha Varney, Massiel Varney, Michiko Varney,
Naketa Varney, Persephone Varney, Raynell Varney, Ronell Varney, Shalynn Varney, Shimeka Varney, Shiquita Varney, Tanaka Varney, Tomasita Varney, Trixy Varney, Tyson Varney, Andy Varney, Arden
Varney, Carlissa Varney, Chery Varney, Duane Varney, Ereka Varney, Gracia Varney, Halona Varney, Jamal Varney, Kechia Varney, Kelda Varney, Latessa Varney, Lynett Varney, Mikala Varney, Naida Varney,
Nancee Varney, Renna Varney, Resha Varney, Rifka Varney, Rolinda Varney, Shaneika Varney, Syliva Varney, Tamikka Varney, Tamyka Varney, Tarika Varney, Tawania Varney, Trinda Varney, Trinetta Varney,
Valrie Varney, Anabela Varney, Andretta Varney, Capricia Varney, Chistina Varney, Coree Varney, Darrah Varney, Diahann Varney, Faigy Varney, Georgiann Varney, Hillarie Varney, Ieisha Varney, Joeann
Varney, Jovana Varney, Kesa Varney, Ladeana Varney, Laresa Varney, Lisset Varney, Lynnann Varney, Lysette Varney, Melenie Varney, Natina Varney, Nicolina Varney, Queena Varney, Tameaka Varney, Tansy
Varney, Terressa Varney, Topeka Varney, Alicen Varney, Aneka Varney, Arlyn Varney, Bethzaida Varney, Caralyn Varney, Cartina Varney, Cathlene Varney, Chaquita Varney, Clifford Varney, Danitra Varney,
Dayana Varney, Delania Varney, Dennie Varney, Edina Varney, Electra Varney, Elouise Varney, Ginelle Varney, Heath Varney, Kassia Varney, Katrenia Varney, Keidra Varney, Laconda Varney, Loura Varney,
Marchell Varney, Mayela Varney, Merle Varney, Merlinda Varney, Mychelle Varney, Nannie Varney, Nikko Varney, Nori Varney, Prescilla Varney, Rika Varney, Ronika Varney, Sahara Varney, Sandrea Varney,
Senora Varney, Shawntee Varney, Sheryll Varney, Shina Varney, Terre Varney, Tinna Varney, Tonnette Varney, Torre Varney, Tya Varney, Vianey Varney, Zora Varney, Adonna Varney, Alberto Varney, Almee
Varney, Ashanta Varney, Bridgid Varney, Cindia Varney, Cricket Varney, Darah Varney, Dineen Varney, Elan Varney, Iman Varney, Jacquelynne Varney, Jennilyn Varney, Jessicca Varney, Julio Varney,
Kenosha Varney, Kimyata Varney, Lanessa Varney, Leesha Varney, Letoya Varney, Nkenge Varney, Priti Varney, Raizel Varney, Rosaisela Varney, Sameerah Varney, Sera Varney, Shallon Varney, Sharmane
Varney, Shauntay Varney, Sherhonda Varney, Shonnie Varney, Sueanne Varney, Taneesha Varney, Telina Varney, Tenecia Varney, Tyanne Varney, Alexsandra Varney, Angeles Varney, Charlean Varney, Danel
Varney, Davi Varney, Daysha Varney, Demaris Varney, Dorenda Varney, Dorita Varney, Dorlisa Varney, Genee Varney, Jamella Varney, Kathyjo Varney, Kaya Varney, Latondra Varney, Leshia Varney, Mahala
Varney, Marija Varney, Maudie Varney, Megann Varney, Phuong Varney, Reesa Varney, Ronya Varney, Selinda Varney, Shama Varney, Shirell Varney, Shonita Varney, Taiwana Varney, Takenya Varney, Talonda
Varney, Tassie Varney, Telena Varney, Theodore Varney, Violetta Varney, Willetta Varney, Adora Varney, Alvin Varney, Chauncey Varney, Chrisy Varney, Clancy Varney, Claressa Varney, Corrinna Varney,
Darra Varney, Deshawna Varney, Donise Varney, Elona Varney, Gara Varney, Georgene Varney, Gila Varney, Jania Varney, Jocelynn Varney, Joei Varney, Kassy Varney, Liset Varney, Maleka Varney, Mashell
Varney, Melida Varney, Michalene Varney, Moya Varney, Nyisha Varney, Rayanne Varney, Renia Varney, Salima Varney, Samanthia Varney, Sammy Varney, Santosha Varney, Shaneeka Varney, Shawntelle Varney,
Shwanda Varney, Sita Varney, Tarin Varney, Tawn Varney, Tosca Varney, Valisha Varney, Vangie Varney, Vernon Varney, Zella Varney, Amarilis Varney, Amelie Varney, Aubri Varney, Bobbette Varney, Byron
Varney, Chance Varney, Deshon Varney, Eraina Varney, Jaycee Varney, Joyell Varney, Keirsten Varney, Kendi Varney, Kendrea Varney, Keyanna Varney, Khristie Varney, Kimberlyann Varney, Kimyetta Varney,
Kinberly Varney, Lakeita Varney, Lakina Varney, Leida Varney, Lenae Varney, Mariette Varney, Marymargaret Varney, Meera Varney, Micky Varney, Naja Varney, Oneka Varney, Pedro Varney, Rennie Varney,
Sharlee Varney, Sharonne Varney, Shonya Varney, Solange Varney, Tashawn Varney, Amorette Varney, Andrienne Varney, Ari Varney, Carolin Varney, Chalonda Varney, Cinthya Varney, Drusilla Varney,
Erendira Varney, Franki Varney, Fredrick Varney, Isaac Varney, Jacklin Varney, Jamese Varney, Jeani Varney, Josi Varney, Jullie Varney, Kadie Varney, Kennetta Varney, Keonna Varney, Krisanne Varney,
Lakrisha Varney, Lakysha Varney, Lasheka Varney, Levette Varney, Lexi Varney, Lezlee Varney, Lovely Varney, Marah Varney, Markeeta Varney, Marlow Varney, Meloni Varney, Natausha Varney, Nate Varney,
Neoma Varney, Nicci Varney, Niema Varney, Nykia Varney, Oma Varney, Quandra Varney, Rasha Varney, Shenelle Varney, Sheryle Varney, Sylena Varney, Tasheba Varney, Tennile Varney, Terriann Varney,
Tesia Varney, Tuere Varney, Tynika Varney, Valena Varney, Beverlee Varney, Carriann Varney, Chaney Varney, Chantae Varney, Charie Varney, Cherell Varney, Chrystina Varney, Cookie Varney, Delphina
Varney, Denys Varney, Digna Varney, Doriann Varney, Elecia Varney, Elmira Varney, Irina Varney, Jacky Varney, Jimmi Varney, Joslin Varney, Katty Varney, Kimberlin Varney, Laquan Varney, Latonga
Varney, Lauran Varney, Lei Varney, Maralee Varney, Marilena Varney, Markeisha Varney, Mena Varney, Mike Varney, Nadeen Varney, Ranelle Varney, Renika Varney, Ritu Varney, Roshawnda Varney, Sandria
Varney, Shann Varney, Shellene Varney, Shemica Varney, Shenandoah Varney, Sherrye Varney, Sholonda Varney, Teka Varney, Temica Varney, Thomasena Varney, Vallerie Varney, Velva Varney, Annjanette
Varney, Arnitra Varney, Asa Varney, Barri Varney, Deborha Varney, Deshaun Varney, Disa Varney, Doree Varney, Enrique Varney, Fredrika Varney, Gisel Varney, Jemima Varney, Kellene Varney, Kinsey
Varney, Lanice Varney, Leonore Varney, Liesel Varney, Loredana Varney, Lynita Varney, Maeve Varney, Michale Varney, Michalle Varney, Nadira Varney, Orlando Varney, Radonna Varney, Rivkah Varney,
Sanita Varney, Shaketa Varney, Shatika Varney, Stepheny Varney, Tashanda Varney, Tashanna Varney, Thera Varney, Vernette Varney, Virna Varney, Wanetta Varney, Andee Varney, Aparna Varney, Armando
Varney, Brynne Varney, Carolynne Varney, Danyal Varney, Drena Varney, Faydra Varney, Felishia Varney, Gwendalyn Varney, Jennyfer Varney, Jesika Varney, Johnell Varney, Kalen Varney, Keelie Varney,
Kirby Varney, Kristopher Varney, Lynnett Varney, Mikka Varney, Moraima Varney, Nasha Varney, Nechelle Varney, Nelson Varney, Rashawnda Varney, Rianna Varney, Rogina Varney, Rudy Varney, Schwanda
Varney, Serafina Varney, Seth Varney, Shantrice Varney, Shenee Varney, Sherrel Varney, Shirin Varney, Soila Varney, Sujata Varney, Tannia Varney, Tiya Varney, Tonnie Varney, Tuwanda Varney, Young
Varney, Anjela Varney, Aquanetta Varney, Ardis Varney, Brigitta Varney, Carmilla Varney, Catherin Varney, Charna Varney, Chaunda Varney, Chela Varney, Cherylyn Varney, Clifton Varney, Danice Varney,
Daylene Varney, Delecia Varney, Dell Varney, Eleonora Varney, Gesenia Varney, Giana Varney, Iisha Varney, Jacqualyn Varney, Jule Varney, Kanita Varney, Karrin Varney, Keishia Varney, Kerith Varney,
Kima Varney, Letica Varney, Letta Varney, Lucianna Varney, Mala Varney, Maresa Varney, Michaeline Varney, Mitra Varney, Norene Varney, Raul Varney, Robi Varney, Sativa Varney, Shakeema Varney,
Shalisha Varney, Sherece Varney, Sherlene Varney, Talicia Varney, Tinia Varney, Toneka Varney, Tuwanna Varney, Tymeka Varney, Vesna Varney, Yvetta Varney, Akeisha Varney, Almeta Varney, Alvita
Varney, Ameerah Varney, Amena Varney, Anntoinette Varney, Asusena Varney, Benetta Varney, Bernard Varney, Bibiana Varney, Bracha Varney, Branden Varney, Carrisa Varney, Channell Varney, Christee
Varney, Daisey Varney, Dandra Varney, Dene Varney, Destinee Varney, Emmalee Varney, Evamarie Varney, Felicha Varney, Hally Varney, Harmonie Varney, Ivone Varney, Janira Varney, Jennfier Varney,
Josalyn Varney, Karley Varney, Kellyjo Varney, Kinshasa Varney, Laurena Varney, Lewis Varney, Louvenia Varney, Lynley Varney, Maleah Varney, Marceline Varney, Maree Varney, Maxie Varney, Melora
Varney, Minta Varney, Nella Varney, Nikea Varney, Orlanda Varney, Radhika Varney, Renelle Varney, Robynne Varney, Rosland Varney, Saudia Varney, Shannette Varney, Sharetta Varney, Sharnita Varney,
Shequita Varney, Tanessa Varney, Tannis Varney, Thu Varney, Toiya Varney, Umeka Varney, Yadhira Varney, Birdie Varney, Bradi Varney, Carola Varney, Chelle Varney, Chermaine Varney, Christelle Varney,
Cortnie Varney, Dalinda Varney, Daphene Varney, Demetri Varney, Ennifer Varney, Esta Varney, Francheska Varney, Griselle Varney, Ife Varney, Jahaira Varney, Jammi Varney, Javonne Varney, Jenie
Varney, Jocelin Varney, Karole Varney, Kelee Varney, Khrista Varney, Leya Varney, Luwanda Varney, Makayla Varney, Marly Varney, Nzinga Varney, Paquita Varney, Ramanda Varney, Rayleen Varney, Rinda
Varney, Shauntell Varney, Tamecka Varney, Tanji Varney, Tela Varney, Teneisha Varney, Valicia Varney, Willona Varney, Alejandro Varney, Ameka Varney, Ashante Varney, Brisa Varney, Charli Varney,
Charmagne Varney, Charrise Varney, Crystol Varney, Darya Varney, Destiney Varney, Elidia Varney, France Varney, Galadriel Varney, Genifer Varney, Gladis Varney, Glori Varney, Harriette Varney, Ioanna
Varney, Jeanann Varney, Kittie Varney, Ladon Varney, Latesa Varney, Linn Varney, Loralie Varney, Mazie Varney, Mekisha Varney, Nivia Varney, Paulene Varney, Renette Varney, Saadia Varney, Shakena
Varney, Sheritta Varney, Stepahnie Varney, Taliah Varney, Teretha Varney, Tessy Varney, Wendelyn Varney, Zonia Varney, Ailsa Varney, Alonna Varney, Andres Varney, Asma Varney, Austin Varney, Brandis
Varney, Corinda Varney, Deetra Varney, Dyanne Varney, Eliabeth Varney, Evelynn Varney, Geana Varney, Geeta Varney, Georgeanne Varney, Glynnis Varney, Hilari Varney, Hindy Varney, Jackquline Varney,
Joaquina Varney, Jolynne Varney, Kirsta Varney, Latissa Varney, Lucile Varney, Makeisha Varney, Marli Varney, Meeghan Varney, Michaelyn Varney, Monte Varney, Omeka Varney, Patra Varney, Ragen Varney,
Reshma Varney, Ronnell Varney, Shakera Varney, Shantella Varney, Shar Varney, Shelagh Varney, Sherisse Varney, Slyvia Varney, Stephenia Varney, Tarisha Varney, Tria Varney, Tynetta Varney, Veena
Varney, Aleksandra Varney, Analia Varney, Aneshia Varney, Anesia Varney, Anissia Varney, Caroll Varney, Celisa Varney, Chariti Varney, Charnette Varney, Cheria Varney, Dabney Varney, Davon Varney,
Demita Varney, Denny Varney, Dwanda Varney, Erick Varney, Fancy Varney, Felesha Varney, Genette Varney, Graziella Varney, Helaine Varney, Huong Varney, Idania Varney, Ilsa Varney, Iola Varney,
Janifer Varney, Jaynie Varney, Jil Varney, Joyann Varney, Kalina Varney, Kana Varney, Katheleen Varney, Kelie Varney, Kimbly Varney, Ladina Varney, Lamisha Varney, Landra Varney, Laretha Varney,
Lashuna Varney, Latorsha Varney, Lehua Varney, Libbie Varney, Loris Varney, Lovetta Varney, Marilin Varney, Marquis Varney, Medea Varney, Michaella Varney, Milly Varney, Montina Varney, Muna Varney,
Nickcole Varney, Nickia Varney, Quianna Varney, Retina Varney, Rubina Varney, Sallyann Varney, Sangeeta Varney, Shalinda Varney, Shatonya Varney, Tamasha Varney, Tenise Varney, Terrilynn Varney, Teya
Varney, Toma Varney, Trang Varney, Vienna Varney, Zenda Varney, Amisha Varney, Chenell Varney, Cherene Varney, Chimere Varney, Darcell Varney, Desa Varney, Devi Varney, Dove Varney, Eilene Varney,
Jarita Varney, Jenetta Varney, Johnie Varney, Kanani Varney, Kanya Varney, Katiria Varney, Kenyon Varney, Khristi Varney, Kidada Varney, Kristien Varney, Laressa Varney, Lashonta Varney, Lataisha
Varney, Leanda Varney, Leshonda Varney, Letia Varney, Lorenzo Varney, Makenzie Varney, Manya Varney, Maribelle Varney, Maryjean Varney, Moniqua Varney, Myeisha Varney, Mystie Varney, Nakeesha Varney,
Neeley Varney, Neil Varney, Nida Varney, Oletha Varney, Rebel Varney, Rosaland Varney, Satonya Varney, Shanah Varney, Sherly Varney, Sherrilyn Varney, Shonia Varney, Shuna Varney, Shylah Varney,
Tameisha Varney, Telecia Varney, Terrill Varney, Torry Varney, Tracyann Varney, Undrea Varney, Usha Varney, Velicia Varney, Windie Varney, Yehudis Varney, Ameenah Varney, Amika Varney, Anupama
Varney, Bethaney Varney, Brookie Varney, Bryce Varney, Cherika Varney, Courtni Varney, Danyle Varney, Darilyn Varney, Darnita Varney, Eveline Varney, Gilbert Varney, Giovanni Varney, Jacquel Varney,
Jenai Varney, Jocelyne Varney, Jull Varney, Karene Varney, Karisha Varney, Kelvin Varney, Kenzie Varney, Kimblery Varney, Kyleen Varney, Laketta Varney, Larisha Varney, Lashae Varney, Laveta Varney,
Lezli Varney, Marlynn Varney, Melana Varney, Meribeth Varney, Onita Varney, Philicia Varney, Raychel Varney, Roschelle Varney, Rosilyn Varney, Sandhya Varney, Selenia Varney, Seretha Varney, Shadia
Varney, Shalise Varney, Sonnet Varney, Tacey Varney, Taurus Varney, Telly Varney, Tenita Varney, Tiawana Varney, Timica Varney, Warren Varney, Yamile Varney, Yaminah Varney, Adel Varney, Aiyana
Varney, Amybeth Varney, Angee Varney, Angeligue Varney, Areli Varney, Asheley Varney, Bina Varney, Celines Varney, Charlet Varney, Charley Varney, Charonda Varney, Dameka Varney, Danne Varney,
Deloise Varney, Denay Varney, Denene Varney, Donata Varney, Georgianne Varney, Jamelia Varney, Jennetta Varney, Jesusa Varney, Johnda Varney, Juleen Varney, Khaliah Varney, Kimala Varney, Kimble
Varney, Kimmarie Varney, Korene Varney, Lezley Varney, Lindie Varney, Lorilyn Varney, Mayme Varney, Meeka Varney, Natlie Varney, Ranita Varney, Rasheen Varney, Rayetta Varney, Savina Varney,
Scarlette Varney, Shaindy Varney, Sheana Varney, Sheldon Varney, Shoni Varney, Stephan Varney, Taneeka Varney, Tanesia Varney, Tannya Varney, Tionna Varney, Tresia Varney, Tylisha Varney, Vonita
Varney, Vonnetta Varney, Wren Varney, Akiba Varney, Blaire Varney, Buffey Varney, Candita Varney, Charnelle Varney, Cynda Varney, Danesha Varney, Darbie Varney, Deah Varney, Deette Varney, Devita
Varney, Devonia Varney, Dianah Varney, Dustina Varney, Dylan Varney, Elli Varney, Eun Varney, Geneen Varney, Geni Varney, Jenevieve Varney, Julaine Varney, Kaitlyn Varney, Lalania Varney, Latrise
Varney, Mahalia Varney, Manon Varney, Marycatherine Varney, Nyasha Varney, Orlena Varney, Petina Varney, Precilla Varney, Raenell Varney, Ramonica Varney, Reema Varney, Scotti Varney, Shakeya Varney,
Shaylene Varney, Shenique Varney, Shimika Varney, Takiesha Varney, Tambi Varney, Tammatha Varney, Tauheedah Varney, Timiko Varney, Tomisha Varney, Venesa Varney, Venise Varney, Akesha Varney, Anmarie
Varney, Arnette Varney, Brennan Varney, Cayla Varney, Cecil Varney, Coreena Varney, Danessa Varney, Day Varney, Denni Varney, Fernando Varney, Floretta Varney, Fred Varney, Gardenia Varney, Genevie
Varney, Georgann Varney, Grasiela Varney, Jaimy Varney, Jaquelyn Varney, Jerelyn Varney, Jinnifer Varney, Johnelle Varney, Jovonna Varney, Kady Varney, Keala Varney, Kesi Varney, Kyana Varney,
Laketia Varney, Laquanta Varney, Latoy Varney, Laurine Varney, Lynea Varney, Lynee Varney, Maegen Varney, Margarite Varney, Marijane Varney, Mellany Varney, Monia Varney, Monice Varney, Mylene
Varney, Nakiya Varney, Neila Varney, Oleta Varney, Queenie Varney, Quinetta Varney, Rekha Varney, Remi Varney, Rickey Varney, Roselynn Varney, Samanatha Varney, Shakila Varney, Shalandra Varney,
Shamira Varney, Shandel Varney, Shandrea Varney, Shinita Varney, Stayce Varney, Takela Varney, Talibah Varney, Tameria Varney, Taronda Varney, Telia Varney, Ticia Varney, Tovah Varney, Venissa
Varney, Verlinda Varney, Vittoria Varney, Alicyn Varney, Amylynn Varney, Anabell Varney, Anela Varney, Annice Varney, Aubry Varney, Coralie Varney, Coryn Varney, Darlyn Varney, Dejuana Varney, Eleana
Varney, Esmerelda Varney, Georganne Varney, Jacqlyn Varney, Jodine Varney, Joelyn Varney, Josey Varney, Keanna Varney, Kendy Varney, Korine Varney, Kristianna Varney, Laneka Varney, Lannie Varney,
Mathilda Varney, Myiesha Varney, Nessa Varney, Nikiya Varney, Penina Varney, Radha Varney, Ralonda Varney, Shantele Varney, Sharlette Varney, Shelanda Varney, Shelda Varney, Shere Varney, Shuntel
Varney, Stephaney Varney, Tabbitha Varney, Tamkia Varney, Teona Varney, Terrisa Varney, Teryl Varney, Trellis Varney, Vanissa Varney, Abigale Varney, Bonnita Varney, Carren Varney, Clementine Varney,
Coni Varney, Demonica Varney, Denis Varney, Diantha Varney, Edana Varney, Elene Varney, Faithe Varney, Franklin Varney, Haidee Varney, Jaima Varney, Javonda Varney, Jayma Varney, Jeffifer Varney,
Jennel Varney, Jennfer Varney, Katryna Varney, Keta Varney, Kristyne Varney, Krystle Varney, Kymberlee Varney, Kymberli Varney, Lacole Varney, Laranda Varney, Leea Varney, Loryn Varney, Lyndell
Varney, Lyndy Varney, Mande Varney, Marlissa Varney, Marrissa Varney, Meko Varney, Miryam Varney, Neida Varney, Odilia Varney, Pamila Varney, Porscha Varney, Quentina Varney, Riley Varney, Roselia
Varney, Sharifah Varney, Sonyia Varney, Sugar Varney, Tamico Varney, Tanara Varney, Tiombe Varney, Verity Varney, Zoey Varney, Abril Varney, Aine Varney, Aixa Varney, Ange Varney, Annah Varney,
Avelina Varney, Chantil Varney, Charmel Varney, Cheralyn Varney, Chesley Varney, Colin Varney, Cornelius Varney, Cortni Varney, Deserae Varney, Dru Varney, Emeline Varney, Iran Varney, Jesslyn
Varney, Jovon Varney, Kalani Varney, Kaleen Varney, Kamilla Varney, Kemi Varney, Kendrick Varney, Lachele Varney, Letesha Varney, Makita Varney, Maricia Varney, Mariya Varney, Marlise Varney, Nara
Varney, Ngina Varney, Nikkol Varney, Okema Varney, Palma Varney, Priscillia Varney, Ranell Varney, Rashmi Varney, Renessa Varney, Ricarda Varney, Rozlyn Varney, Shamia Varney, Sharissa Varney,
Shianne Varney, Takeesha Varney, Tekia Varney, Tyhesha Varney, Vanassa Varney, Vicenta Varney, Abagail Varney, Addy Varney, Akemi Varney, Analilia Varney, Andreanna Varney, Anh Varney, Annmargaret
Varney, Caitlyn Varney, Carrin Varney, Celestial Varney, Chameka Varney, Chandrika Varney, Charlsie Varney, Cherlynn Varney, Darien Varney, Darlean Varney, Devyn Varney, Dyane Varney, Eman Varney,
Erna Varney, Fontella Varney, Halie Varney, Ilyse Varney, Janece Varney, Jayla Varney, Jeannene Varney, Juwana Varney, Kachina Varney, Keleigh Varney, Kissie Varney, Krisi Varney, Kristyl Varney,
Krystie Varney, Kyesha Varney, Lizandra Varney, Lyda Varney, Majorie Varney, Marnita Varney, Marrisa Varney, Miguelina Varney, Nadege Varney, Nakiesha Varney, Neesha Varney, Raeleen Varney, Rashana
Varney, Renada Varney, Rupa Varney, Shaunette Varney, Sheetal Varney, Sherryann Varney, Tanecia Varney, Tashiba Varney, Tawney Varney, Telma Varney, Tomeeka Varney, Xochil Varney, Yuki Varney, Ahna
Varney, Aneesha Varney, Aphrodite Varney, Areatha Varney, Cam Varney, Chalene Varney, Charnetta Varney, Clarise Varney, Cleta Varney, Danniel Varney, Deshana Varney, Eduardo Varney, Erminia Varney,
Eron Varney, Hettie Varney, Janan Varney, Janda Varney, Jayda Varney, Jeanny Varney, Katanya Varney, Kaylin Varney, Keeva Varney, Kimesha Varney, Kristeena Varney, Kristyna Varney, Lakeyshia Varney,
Latesia Varney, Latoia Varney, Lavona Varney, Lonni Varney, Loukisha Varney, Manika Varney, Marbella Varney, Marcina Varney, Marialuisa Varney, Mariellen Varney, Mariluz Varney, Mikia Varney, Morning
Varney, Myndi Varney, Ronalda Varney, Shanitra Varney, Sherea Varney, Shericka Varney, Sonny Varney, Suzane Varney, Terea Varney, Tersa Varney, Yarnell Varney, Yezenia Varney, Zona Varney, Alissia
Varney, Anjenette Varney, Antwan Varney, Bernette Varney, Cedar Varney, Chandria Varney, Cherina Varney, Chrisandra Varney, Christiann Varney, Cosandra Varney, Danea Varney, Demetress Varney,
Emmeline Varney, Ernesta Varney, Fabiana Varney, Fabienne Varney, Falecia Varney, Georgena Varney, Gerardo Varney, Gwynn Varney, Heavenly Varney, Ivett Varney, Jaya Varney, Jessicia Varney, Jetaun
Varney, Joane Varney, Kalliopi Varney, Kalynn Varney, Karmon Varney, Kischa Varney, Kita Varney, Lanea Varney, Laron Varney, Latonyia Varney, Lavone Varney, Leasha Varney, Liann Varney, Madelyne
Varney, Malissia Varney, Margit Varney, Marlowe Varney, Mashonda Varney, Meggen Varney, Merari Varney, Michille Varney, Missi Varney, Nichoel Varney, Nikie Varney, Nyssa Varney, Pooja Varney, Purvi
Varney, Ryane Varney, Sakeenah Varney, Sam Varney, Shaquan Varney, Sharmel Varney, Shlonda Varney, Tamsen Varney, Tikia Varney, Tomicka Varney, Tremeka Varney, Wenonah Varney, Yavonne Varney, Adraine
Varney, Aiysha Varney, Akila Varney, Almira Varney, Alnita Varney, Ambrosia Varney, Aminta Varney, Anedra Varney, Angeliki Varney, Aymee Varney, Barrett Varney, Batina Varney, Brandey Varney, Brena
Varney, Bryant Varney, Charidy Varney, Davona Varney, Delissa Varney, Denea Varney, Denicia Varney, Deniese Varney, Desta Varney, Dimple Varney, Emelie Varney, Eustacia Varney, Farra Varney, Giulia
Varney, Jannetta Varney, Jen Varney, Jennice Varney, Kareemah Varney, Kaysie Varney, Latice Varney, Letanya Varney, Liticia Varney, Luna Varney, Madge Varney, Marielena Varney, Merrily Varney, Mitsy
Varney, Nachelle Varney, Nolita Varney, Palmira Varney, Parthenia Varney, Ramey Varney, Raymona Varney, Rhodesia Varney, Sanna Varney, Shakema Varney, Shamieka Varney, Shandria Varney, Shaniece
Varney, Sharronda Varney, Sheletha Varney, Shoshannah Varney, Suprina Varney, Tanyia Varney, Tarri Varney, Tequita Varney, Thressa Varney, Twala Varney, Tyneshia Varney, Willena Varney, Aaryn Varney,
Ailene Varney, Alfredo Varney, Alinda Varney, Avia Varney, Brandelyn Varney, Brandolyn Varney, Brieanna Varney, Brittanie Varney, Celest Varney, Charman Varney, Chenise Varney, Colandra Varney,
Damari Varney, Donni Varney, Dorice Varney, Dustine Varney, Elease Varney, Jamara Varney, Jameca Varney, Jessalyn Varney, Joyanna Varney, Kalia Varney, Kennisha Varney, Kewanna Varney, Krystel
Varney, Lashara Varney, Leslieann Varney, Letetia Varney, Letizia Varney, Lisanne Varney, Lovella Varney, Maranatha Varney, Melisia Varney, Meriah Varney, Monaca Varney, Myrtis Varney, Nadean Varney,
Naila Varney, Narissa Varney, Natale Varney, Nealy Varney, November Varney, Petrice Varney, Renie Varney, Robecca Varney, Sarra Varney, Shakeena Varney, Shalunda Varney, Sharin Varney, Shatoya
Varney, Soni Varney, Sulma Varney, Taffany Varney, Tahara Varney, Taralynn Varney, Tarena Varney, Tatyana Varney, Tekeisha Varney, Timothea Varney, Tishana Varney, Toshua Varney, Tresea Varney,
Tykisha Varney, Varonica Varney, Yolinda Varney, Adra Varney, Amia Varney, Aquarius Varney, Baila Varney, Betheny Varney, Cady Varney, Calla Varney, Catrinia Varney, Chanika Varney, Chellie Varney,
Chelsy Varney, Christana Varney, Cicley Varney, Crescent Varney, Deronda Varney, Diamantina Varney, Dyani Varney, Elvina Varney, Emiley Varney, Haneefah Varney, Ivan Varney, Jamesetta Varney, Janele
Varney, Jolean Varney, Jory Varney, Julieanna Varney, Jyoti Varney, Kaela Varney, Katena Varney, Kaycie Varney, Kismet Varney, Kobi Varney, Lakechia Varney, Lakesa Varney, Lalanya Varney, Latiesha
Varney, Lauralyn Varney, Lekita Varney, Loletha Varney, Lonetta Varney, Lukisha Varney, Mele Varney, Nadra Varney, Natia Varney, Ngozi Varney, Nija Varney, October Varney, Olinda Varney, Preeti
Varney, Raine Varney, Ronie Varney, Sharlie Varney, Sharnetta Varney, Shontia Varney, Tamakia Varney, Taquita Varney, Temeca Varney, Tenna Varney, Teresea Varney, Teryn Varney, Tessica Varney, Trayce
Varney, Tujuana Varney, Alys Varney, Aris Varney, Asya Varney, Atina Varney, Ayonna Varney, Cherl Varney, Corretta Varney, Cybill Varney, Gaylynn Varney, Heatherly Varney, Jacquelina Varney, Jeanean
Varney, Jeniece Varney, Johnathan Varney, Krystine Varney, Larua Varney, Lelah Varney, Lilla Varney, Linh Varney, Lisia Varney, Lucilla Varney, Malita Varney, Matisha Varney, Melika Varney, Merci
Varney, Mishell Varney, Monae Varney, Monee Varney, Montrice Varney, Najla Varney, Parrish Varney, Ramsey Varney, Raya Varney, Rejeana Varney, Roanna Varney, Roxan Varney, Shai Varney, Shanen Varney,
Sharalyn Varney, Shuntay Varney, Taheerah Varney, Takeysha Varney, Tamelia Varney, Tanyika Varney, Temperance Varney, Theodosia Varney, Thyra Varney, Tyria Varney, Urania Varney, Yvett Varney,
Angelynn Varney, Anitria Varney, Anny Varney, Arlisha Varney, Aya Varney, Brielle Varney, Careen Varney, Correne Varney, Dagmar Varney, Dalynn Varney, Dannetta Varney, Delight Varney, Denetta Varney,
Desaree Varney, Donyel Varney, Eisha Varney, Eleanora Varney, Felix Varney, Fransisca Varney, Jannel Varney, Jenia Varney, Jettie Varney, Jolina Varney, Jowanna Varney, Junko Varney, Kaaren Varney,
Kathey Varney, Keryn Varney, Ketina Varney, Korrine Varney, Kristalyn Varney, Laini Varney, Lamona Varney, Lanay Varney, Leza Varney, Maddalena Varney, Mellody Varney, Merica Varney, Merita Varney,
Nafeesa Varney, Nicholl Varney, Oneika Varney, Pamelyn Varney, Romonda Varney, Ronique Varney, Rosslyn Varney, Rosy Varney, Sachiko Varney, Sarajane Varney, Satara Varney, Shamra Varney, Tamberly
Varney, Tawona Varney, Tiasha Varney, Tine Varney, Trameka Varney, Tynia Varney, Valentine Varney, Venisha Varney, Veronia Varney, Vinnie Varney, Yuri Varney, Aissa Varney, Aleda Varney, Amenda
Varney, Andromeda Varney, Antigone Varney, Aren Varney, Ariadne Varney, Bambie Varney, Belkis Varney, Bunnie Varney, Claudina Varney, Darcelle Varney, Dulcinea Varney, Enola Varney, Florida Varney,
Francia Varney, Ginni Varney, Grizelda Varney, Israel Varney, Jalynn Varney, Jamil Varney, Jenika Varney, Joeanna Varney, Katesha Varney, Katreena Varney, Keilani Varney, Kiah Varney, Kjersti Varney,
Korrin Varney, Leonda Varney, Liesa Varney, Lyna Varney, Manuelita Varney, Marka Varney, Markia Varney, Maronda Varney, Melondy Varney, Melonee Varney, Micheala Varney, Misa Varney, Nikeisha Varney,
Odalys Varney, Parul Varney, Petula Varney, Rainee Varney, Rainie Varney, Romi Varney, Rubie Varney, Shamone Varney, Shawnia Varney, Sheritha Varney, Shery Varney, Tamula Varney, Tasheena Varney,
Thao Varney, Thereasa Varney, Tjuana Varney, Tomesha Varney, Turquoise Varney, Tyiesha Varney, Vernee Varney, Annastasia Varney, Ardra Varney, Bibi Varney, Blaine Varney, Bridie Varney, Carmelle
Varney, Chalice Varney, Chanie Varney, Chauntelle Varney, Dareth Varney, Demitria Varney, Denetria Varney, Didi Varney, Dimitria Varney, Dwight Varney, Elisabetta Varney, Ellisha Varney, Ileen
Varney, Jamala Varney, Jayleen Varney, Jeny Varney, Jeraldine Varney, Jowanda Varney, Joylene Varney, Kareena Varney, Karrah Varney, Kashonda Varney, Kendel Varney, Ketrina Varney, Lagena Varney,
Lashann Varney, Lasheena Varney, Latish Varney, Leroy Varney, Maryum Varney, Mende Varney, Mieka Varney, Nakea Varney, Niurka Varney, Philippa Varney, Pina Varney, Romanda Varney, Satrina Varney,
Shaletha Varney, Shandrika Varney, Shawonda Varney, Sherae Varney, Shernita Varney, Sheronica Varney, Silena Varney, Sora Varney, Stehanie Varney, Tanikka Varney, Tashi Varney, Tawyna Varney, Tenesia
Varney, Tikesha Varney, Trenell Varney, Tristina Varney, Vallie Varney, Verla Varney, Vinessa Varney, Walida Varney, Yolandra Varney, Aarti Varney, Aries Varney, Bayyinah Varney, Bertie Varney,
Blakely Varney, Cabrina Varney, Chandrea Varney, Cherrell Varney, Chiquitta Varney, Damarys Varney, Danett Varney, Dedee Varney, Demesha Varney, Denica Varney, Derinda Varney, Deshanda Varney,
Diondra Varney, Edgar Varney, Ewa Varney, Grabiela Varney, Jeanifer Varney, Jemma Varney, Jetta Varney, Jodelle Varney, Joeleen Varney, Johari Varney, Kasaundra Varney, Kewana Varney, Kieran Varney,
Kitina Varney, Kitrina Varney, Lacheryl Varney, Lakeishia Varney, Lasean Varney, Lashawne Varney, Lashica Varney, Latanga Varney, Latreece Varney, Latysha Varney, Leonna Varney, Lilli Varney, Lititia
Varney, Lorette Varney, Makala Varney, Mea Varney, Mendie Varney, Meosha Varney, Milessa Varney, Nathasha Varney, Nefertari Varney, Nieves Varney, Odelia Varney, Phylis Varney, Phyliss Varney, Rami
Varney, Shaena Varney, Shahara Varney, Shajuan Varney, Shalane Varney, Shamaine Varney, Shaquanna Varney, Shauntelle Varney, Sheril Varney, Sherley Varney, Shuronda Varney, Sicily Varney, Sindi
Varney, Sonni Varney, Tamantha Varney, Tejal Varney, Tenley Varney, Tongela Varney, Trenita Varney, Vernica Varney, Ameena Varney, Chancey Varney, Chandi Varney, Charelle Varney, Cherisa Varney,
Corby Varney, Corianne Varney, Crystale Varney, Cyd Varney, Cyndie Varney, Dagny Varney, Daine Varney, Darin Varney, Debralee Varney, Delanda Varney, Donta Varney, Estefana Varney, Falesha Varney,
Fawna Varney, Galen Varney, Geanine Varney, Grazia Varney, Hala Varney, Jadie Varney, Jalyn Varney, Jamilia Varney, Jannice Varney, Jasmina Varney, Jehan Varney, Joannah Varney, Joclyn Varney,
Julietta Varney, Kody Varney, Krisa Varney, Lanaya Varney, Laray Varney, Larra Varney, Larrissa Varney, Leanora Varney, Lelania Varney, Lenetta Varney, Lizett Varney, Lonnette Varney, Lorey Varney,
Lynann Varney, Madalena Varney, Marlita Varney, Marrianne Varney, Maylene Varney, Merida Varney, Midori Varney, Nafeesah Varney, Nekeshia Varney, Parris Varney, Payal Varney, Perlita Varney, Prairie
Varney, Raeanna Varney, Rayette Varney, Sabriya Varney, Sadia Varney, Sequita Varney, Sharine Varney, Sharlena Varney, Spencer Varney, Sugey Varney, Taheera Varney, Taiwan Varney, Tameki Varney,
Tamme Varney, Tawonda Varney, Toscha Varney, Tyronza Varney, Valeska Varney, Vannesa Varney, Vashon Varney, Viktoria Varney, Vonya Varney, Akita Varney, Andraya Varney, Anecia Varney, Annjeanette
Varney, Antonya Varney, Brette Varney, Chase Varney, Chena Varney, Cheris Varney, Chirstine Varney, Claude Varney, Clayton Varney, Clea Varney, Clorissa Varney, Consuello Varney, Cosette Varney,
Darinda Varney, Dawnn Varney, Delene Varney, Deneka Varney, Geisha Varney, Gwendelyn Varney, Hyacinth Varney, Ilia Varney, Jacqulin Varney, Jaynee Varney, Jennifr Varney, Kaysha Varney, Keiana
Varney, Kelsy Varney, Keziah Varney, Lakaisha Varney, Lashawndra Varney, Lennette Varney, Loran Varney, Margarete Varney, Marquise Varney, Marvel Varney, Maurica Varney, Mikell Varney, Nya Varney,
Odetta Varney, Raizy Varney, Samika Varney, Scotty Varney, Senita Varney, Shasha Varney, Shaton Varney, Sheilla Varney, Shital Varney, Shonika Varney, Silver Varney, Tabathia Varney, Tabby Varney,
Taniya Varney, Tawonna Varney, Teddy Varney, Tenicia Varney, Teodora Varney, Tessia Varney, Tinya Varney, Travia Varney, Trivia Varney, Verda Varney, Veta Varney, Waneta Varney, Ysenia Varney, Yumiko
Varney, Zully Varney, Adeana Varney, Adenike Varney, Afton Varney, Angeli Varney, Anicia Varney, Arron Varney, Artie Varney, Ashlei Varney, Ayme Varney, Brieanne Varney, Cira Varney, Cynthea Varney,
Damiana Varney, Darlynn Varney, Dennette Varney, Doreena Varney, Dorris Varney, Dorsey Varney, Drew Varney, Drina Varney, Enrika Varney, Genean Varney, Glendora Varney, Gwendlyn Varney, Hedi Varney,
Honesty Varney, Jackalyn Varney, Jillyn Varney, Jode Varney, Joshlyn Varney, Joylyn Varney, Jullian Varney, Kajuana Varney, Kasee Varney, Kimberla Varney, Kjersten Varney, Kortni Varney, Latronda
Varney, Leonie Varney, Lonette Varney, Lonita Varney, Lynnelle Varney, Marenda Varney, Marybelle Varney, Paisley Varney, Patrena Varney, Rheannon Varney, Rochella Varney, Saleemah Varney, Samala
Varney, Shannie Varney, Shelitha Varney, Shemeika Varney, Shenia Varney, Shinika Varney, Shunte Varney, Sia Varney, Sinead Varney, Tal Varney, Tata Varney, Teela Varney, Tita Varney, Verenice Varney,
Wade Varney, Yesica Varney, Zaira Varney, Zuri Varney, Bobie Varney, Brendalee Varney, Carolanne Varney, Chevette Varney, Debera Varney, Denessa Varney, Desiray Varney, Destinie Varney, Diania
Varney, Edra Varney, Ekaterini Varney, Elane Varney, Eudora Varney, Evelyne Varney, Franci Varney, Gussie Varney, Herbert Varney, Ilse Varney, Jaquay Varney, Jerra Varney, Juliene Varney, Kashia
Varney, Kirstan Varney, Laconya Varney, Lajoyce Varney, Larraine Varney, Lera Varney, Loan Varney, Luwana Varney, Maire Varney, Malene Varney, Marca Varney, Marcene Varney, Mathew Varney, Mayumi
Varney, Nakeysha Varney, Orpha Varney, Quinta Varney, Robie Varney, Sarha Varney, Selah Varney, Sergio Varney, Shalaine Varney, Shazia Varney, Shenae Varney, Shenica Varney, Shineka Varney, Sonnie
Varney, Tashauna Varney, Tashica Varney, Tatjana Varney, Tecia Varney, Therisa Varney, Tracina Varney, Tressia Varney, Vallery Varney, Ximena Varney, Aidee Varney, Alpa Varney, Ambur Varney,
Annaliese Varney, Atasha Varney, Ave Varney, Barbaraann Varney, Bekki Varney, Breena Varney, Catharina Varney, Chandelle Varney, Charmion Varney, Cherilynn Varney, Chesney Varney, Chirsty Varney,
Cynethia Varney, Cyntia Varney, Dane Varney, Dorethea Varney, Elyce Varney, Emery Varney, Garrett Varney, Gittel Varney, Gizelle Varney, Hinda Varney, Jaquelin Varney, Jenaya Varney, Jeneva Varney,
Jenipher Varney, Karessa Varney, Karita Varney, Kelita Varney, Kiyana Varney, Kora Varney, Laquanna Varney, Larkin Varney, Latangela Varney, Linsay Varney, Lisett Varney, Lizzy Varney, Magdalen
Varney, Marcea Varney, Mashawn Varney, Meiko Varney, Mekia Varney, Merisa Varney, Nikcole Varney, Norina Varney, Prima Varney, Rashada Varney, Rasheena Varney, Rozella Varney, Selia Varney, Shaheen
Varney, Shamon Varney, Shaya Varney, Shirly Varney, Shunna Varney, Shyanne Varney, Skylar Varney, Sukari Varney, Sylvana Varney, Tahlia Varney, Taiesha Varney, Tamiki Varney, Tangala Varney, Taniqua
Varney, Tanza Varney, Temesha Varney, Teonna Varney, Tiffane Varney, Torya Varney, Tremaine Varney, Tricha Varney, Vanesha Varney, Yetunde Varney, Zarinah Varney, Amalie Varney, Atoya Varney, Bryony
Varney, Caree Varney, Carron Varney, Celestia Varney, Charma Varney, Chelly Varney, Christon Varney, Chrystle Varney, Conchetta Varney, Dan Varney, Deandria Varney, Donika Varney, Eulanda Varney,
Eyvonne Varney, Gloriana Varney, Hyun Varney, Joslynn Varney, Kewanda Varney, Laetitia Varney, Laketra Varney, Lakshmi Varney, Latawnya Varney, Lawren Varney, Leo Varney, Letonia Varney, Lichelle
Varney, Lue Varney, Lynnmarie Varney, Mauricia Varney, Nakkia Varney, Neka Varney, Parisa Varney, Quentin Varney, Ranada Varney, Rhianon Varney, Roselee Varney, Rozanne Varney, Rozetta Varney, Rozina
Varney, Sada Varney, Sala Varney, Samar Varney, Seandra Varney, Shaela Varney, Sharrell Varney, Shawneen Varney, Shefali Varney, Shekinah Varney, Sherriann Varney, Shuree Varney, Soyini Varney, Teala
Varney, Telitha Varney, Tempie Varney, Tirza Varney, Valori Varney, Veleda Varney, Vivianne Varney, Alexi Varney, Arnetra Varney, Azizi Varney, Belkys Varney, Cailin Varney, Carmell Varney, Charlyne
Varney, Courteney Varney, Danniell Varney, Danyiel Varney, Danylle Varney, Delita Varney, Deseri Varney, Donice Varney, Ebone Varney, Eleshia Varney, Ellison Varney, Giuliana Varney, Harvest Varney,
Hillari Varney, Ijeoma Varney, Janeene Varney, Kem Varney, Kya Varney, Laquandra Varney, Lashane Varney, Latana Varney, Latoi Varney, Lenice Varney, Lethia Varney, Lindsi Varney, Lis Varney, Lorrine
Varney, Marena Varney, Marlayna Varney, Mashanda Varney, Mercedez Varney, Merlene Varney, Najwa Varney, Paraskevi Varney, Princella Varney, Raejean Varney, Raissa Varney, Ravin Varney, Reda Varney,
Reisha Varney, Roland Varney, Sakia Varney, Salma Varney, Shakela Varney, Sharica Varney, Shatisha Varney, Shauntee Varney, Sheriann Varney, Stori Varney, Sueellen Varney, Sylina Varney, Tametra
Varney, Tanieka Varney, Tanikia Varney, Tarisa Varney, Tiffony Varney, Tolanda Varney, Treana Varney, Tumeka Varney, Ursala Varney, Yumi Varney, Zoraya Varney, Alfie Varney, Aliyah Varney, Almeda
Varney, Amarilys Varney, Amberley Varney, Arabella Varney, Ardella Varney, Atonya Varney, Audree Varney, Benedetta Varney, Billiejean Varney, Carlynn Varney, Cherylynn Varney, Christabel Varney,
Clarrisa Varney, Colinda Varney, Cutina Varney, Dalyn Varney, Dametra Varney, Daphna Varney, Emili Varney, Fatema Varney, Glennda Varney, Izetta Varney, Janielle Varney, Jessamy Varney, Jonique
Varney, Karron Varney, Kathia Varney, Katrese Varney, Kendrah Varney, Kevia Varney, Khalia Varney, Kissa Varney, Kizzi Varney, Kutina Varney, Lanetra Varney, Lashonya Varney, Latanja Varney, Latisia
Varney, Latriece Varney, Lenny Varney, Leonarda Varney, Lewanna Varney, Lien Varney, Lyndia Varney, Mallissa Varney, Marlie Varney, Marqueta Varney, Marvetta Varney, Mikayla Varney, Milicent Varney,
Montana Varney, Morgana Varney, Nathania Varney, Neema Varney, Noriko Varney, Ondria Varney, Onna Varney, Patrici Varney, Rachella Varney, Reatha Varney, Rise Varney, Rosabel Varney, Secret Varney,
Shanetha Varney, Sharah Varney, Sharion Varney, Sherilynn Varney, Summar Varney, Tam Varney, Tamanika Varney, Tarhonda Varney, Teddie Varney, Tennelle Varney, Tephanie Varney, Terryl Varney, Tiawanna
Varney, Tischa Varney, Tonesha Varney, Amirah Varney, Amory Varney, Anji Varney, Arasely Varney, Argentina Varney, Aviance Varney, Batsheva Varney, Briar Varney, Cena Varney, Chalon Varney, Chancy
Varney, Chandler Varney, Chivon Varney, Chonita Varney, Chrisa Varney, Darnelle Varney, Deanie Varney, Deeana Varney, Delise Varney, Elayna Varney, Errika Varney, Eurika Varney, Farrell Varney,
Felcia Varney, Isidra Varney, Jai Varney, Jamell Varney, Jamesha Varney, Jere Varney, Jodilyn Varney, Jovonne Varney, Jowana Varney, Juline Varney, Kameshia Varney, Katrisha Varney, Keary Varney,
Keyshia Varney, Kolette Varney, Lakecha Varney, Lakiya Varney, Larrisa Varney, Lasharn Varney, Lasundra Varney, Latica Varney, Latoiya Varney, Laurisa Varney, Lucrezia Varney, Mariateresa Varney,
Maryland Varney, Melicia Varney, Mieko Varney, Molina Varney, Mozella Varney, Nataly Varney, Nayda Varney, Olanda Varney, Onica Varney, Ourania Varney, Quinette Varney, Rayma Varney, Rianne Varney,
Sarika Varney, Shalona Varney, Shawntina Varney, Shellye Varney, Sheva Varney, Shironda Varney, Sonora Varney, Sophronia Varney, Symantha Varney, Tahra Varney, Tearsa Varney, Terrika Varney, Tonyetta
Varney, Treina Varney, Tzipora Varney, Violette Varney, Vivianna Varney, Yanet Varney, Akira Varney, Amanada Varney, Atara Varney, Britten Varney, Camika Varney, Carlo Varney, Celicia Varney, Chanon
Varney, Chanti Varney, Christeena Varney, Consandra Varney, Correen Varney, Delayna Varney, Derenda Varney, Dlynn Varney, Donyetta Varney, Fawne Varney, Feliz Varney, Gisselle Varney, Glen Varney,
Hong Varney, Hyla Varney, Jacques Varney, Jasmyn Varney, Jonel Varney, Khaleelah Varney, Kista Varney, Lachell Varney, Lajoy Varney, Lakethia Varney, Lamia Varney, Lasheba Varney, Laurianne Varney,
Lelani Varney, Lonya Varney, Loucinda Varney, Maressa Varney, Mariesa Varney, Meshia Varney, Messina Varney, Paulett Varney, Rashon Varney, Rayshawn Varney, Rochanda Varney, Ronnica Varney, Rorie
Varney, Saunya Varney, Shakeitha Varney, Shalla Varney, Shalondra Varney, Shamecca Varney, Shavonn Varney, Sherril Varney, Shonell Varney, Sonita Varney, Svetlana Varney, Takila Varney, Tala Varney,
Tamecca Varney, Tiphany Varney, Torria Varney, Vanya Varney, Verdell Varney, Vonna Varney, Zsazsa Varney, Adonia Varney, Alease Varney, Aleena Varney, Alishea Varney, Anetta Varney, Ania Varney,
Anquinette Varney, Antonieta Varney, Archana Varney, Arely Varney, Artavia Varney, Arturo Varney, Batya Varney, Bernardine Varney, Betti Varney, Breck Varney, Camara Varney, Cesar Varney, Chalise
Varney, Charly Varney, Cherrise Varney, Chikita Varney, Cristiana Varney, Dayatra Varney, Ericha Varney, Ermalinda Varney, Eulonda Varney, Fayth Varney, Halina Varney, Hunter Varney, Iveliz Varney,
Ivie Varney, Jahna Varney, Jancy Varney, Janisha Varney, Jeaninne Varney, Jeannemarie Varney, Jessamine Varney, Keeya Varney, Kelisha Varney, Kenyette Varney, Kyley Varney, Kyrie Varney, Ladora
Varney, Laguana Varney, Larri Varney, Leyda Varney, Louie Varney, Marykathryn Varney, Mekesha Varney, Meredeth Varney, Merridith Varney, Mittie Varney, Nakeitha Varney, Narda Varney, Nature Varney,
Neeta Varney, Nitasha Varney, Nuvia Varney, Paticia Varney, Promise Varney, Rabia Varney, Raquelle Varney, Richetta Varney, Shawneequa Varney, Shervon Varney, Siomara Varney, Starlyn Varney, Tahirih
Varney, Taia Varney, Talita Varney, Tamina Varney, Tamya Varney, Tanicka Varney, Tanyanika Varney, Tawan Varney, Tawanya Varney, Tiarra Varney, Timisha Varney, Tinita Varney, Tonnia Varney, Toronda
Varney, Toshika Varney, Triva Varney, Trula Varney, Tykesha Varney, Vaness Varney, Vennessa Varney, Wakeelah Varney, Wilhemina Varney, Yoshiko Varney, Afia Varney, Afiya Varney, Allysa Varney, Alsha
Varney, Amberlyn Varney, Arah Varney, Audri Varney, Bernardette Varney, Bernedette Varney, Binta Varney, Breeann Varney, Britni Varney, Carnetta Varney, Cesilia Varney, Chelli Varney, Dawnna Varney,
Desirie Varney, Fleur Varney, Gianina Varney, Hanh Varney, Ivelis Varney, Jalena Varney, Jenafer Varney, Johann Varney, Joyanne Varney, Kamia Varney, Katerine Varney, Kecha Varney, Keir Varney,
Kennita Varney, Kimra Varney, Kristiann Varney, Levita Varney, Lluvia Varney, Lorian Varney, Lyndsy Varney, Mardell Varney, Maurissa Varney, Mischell Varney, Mistey Varney, Nataya Varney, Nickola
Varney, Obdulia Varney, Preston Varney, Raynetta Varney, Rondell Varney, Shakara Varney, Shalan Varney, Shanya Varney, Shawnika Varney, Shenetha Varney, Shilah Varney, Soo Varney, Tametha Varney,
Tanicia Varney, Tanny Varney, Tennie Varney, Tiffannie Varney, Torra Varney, Trinika Varney, Val Varney, Abeer Varney, Adah Varney, Aleece Varney, Andreya Varney, Annaliza Varney, Arrie Varney,
Ayasha Varney, Berit Varney, Birgit Varney, Bre Varney, Brigetta Varney, Cassundra Varney, Christyl Varney, Cristyn Varney, Danean Varney, Daralyn Varney, Davonne Varney, Deisha Varney, Denina
Varney, Dietrich Varney, Domini Varney, Erinne Varney, Evy Varney, Forrest Varney, Genetta Varney, Gioia Varney, Guenevere Varney, Gwenetta Varney, Gwenette Varney, Hesper Varney, Ignacia Varney,
Irmalinda Varney, Jacquita Varney, Jeanita Varney, Jodiann Varney, Kalin Varney, Kenny Varney, Keon Varney, Kimberlynn Varney, Kimela Varney, Kimm Varney, Kiwanna Varney, Lacreshia Varney, Lakashia
Varney, Latarshia Varney, Latrish Varney, Lauria Varney, Likisha Varney, Lindee Varney, Little Varney, Loreta Varney, Lutisha Varney, Mahasin Varney, Maki Varney, Maxi Varney, Meesha Varney, Melitta
Varney, Mery Varney, Mysty Varney, Quanisha Varney, Ranetta Varney, Rosia Varney, Sadonna Varney, Sangita Varney, Sanora Varney, Shanina Varney, Sharonna Varney, Shavona Varney, Sherrica Varney,
Storm Varney, Tabrina Varney, Tacha Varney, Taleshia Varney, Tamee Varney, Tanetta Varney, Thanh Varney, Twanya Varney, Ulrica Varney, Yaffa Varney, Aarin Varney, Afi Varney, Aliscia Varney, Aneta
Varney, Anetria Varney, Antoinett Varney, Athenia Varney, Atisha Varney, Barb Varney, Brit Varney, Caley Varney, Camala Varney, Carena Varney, Carinne Varney, Carmine Varney, Charlott Varney, Chessa
Varney, Chundra Varney, Cotrina Varney, Drenda Varney, Elanor Varney, Eleftheria Varney, Erricka Varney, Feleshia Varney, Geanna Varney, Harley Varney, Honora Varney, Jakia Varney, Jannah Varney,
Jennene Varney, Jennett Varney, Jobeth Varney, Jordanna Varney, Juanetta Varney, Junie Varney, Kaira Varney, Kashana Varney, Katee Varney, Kateena Varney, Kelleigh Varney, Kenyana Varney, Kerrilyn
Varney, Kessa Varney, Kirk Varney, Lakitha Varney, Laneshia Varney, Lanissa Varney, Lashunta Varney, Leonia Varney, Lester Varney, Letina Varney, Llesenia Varney, Loranda Varney, Lorilynn Varney,
Lotus Varney, Luke Varney, Marisal Varney, Marvis Varney, Maurie Varney, Minnette Varney, Mishawn Varney, Natsha Varney, Ngoc Varney, Nioka Varney, Particia Varney, Phillina Varney, Prisilla Varney,
Raney Varney, Raphaela Varney, Rashaunda Varney, Raynelle Varney, Ross Varney, Rufina Varney, Sarahann Varney, Shaconda Varney, Shanicka Varney, Shaunita Varney, Shekina Varney, Shelie Varney, Sherah
Varney, Shermeka Varney, Shermika Varney, Sonnia Varney, Stephonie Varney, Taffie Varney, Takima Varney, Talea Varney, Tamaya Varney, Taneika Varney, Tanina Varney, Tanisia Varney, Taysha Varney,
Tekeshia Varney, Terie Varney, Therasa Varney, Thomasa Varney, Tifini Varney, Tomie Varney, Tressy Varney, Trissa Varney, Tristy Varney, Tritia Varney, Twania Varney, Tyrene Varney, Uganda Varney,
Wandy Varney, Wileen Varney, Yumeka Varney, Zarina Varney, Adreana Varney, Alandra Varney, Alenda Varney, Amera Varney, Arlicia Varney, Artemis Varney, Avani Varney, Bena Varney, Candia Varney,
Cathyjo Varney, Chala Varney, Charitie Varney, Charlise Varney, Chasiti Varney, Chayla Varney, Chrisanne Varney, Christell Varney, Clarrissa Varney, Cortnee Varney, Dametria Varney, Daneka Varney,
Darlisa Varney, Delorse Varney, Deshone Varney, Dorette Varney, Esperansa Varney, Evagelia Varney, Feliciana Varney, Fikisha Varney, Freeda Varney, Frida Varney, Gaila Varney, Genae Varney, Georgi
Varney, Gitel Varney, Gricel Varney, Hilliary Varney, Ichelle Varney, Ishia Varney, Ivelise Varney, Jacci Varney, Jamesa Varney, Jammy Varney, Jerrilynn Varney, Jillaine Varney, Jillann Varney, Jin
Varney, Joda Varney, Jodeen Varney, Kalisa Varney, Karne Varney, Katura Varney, Kayci Varney, Keegan Varney, Keenan Varney, Khia Varney, Koleen Varney, Krishana Varney, Krislyn Varney, Kurt Varney,
Lafaye Varney, Lakeysa Varney, Lakresha Varney, Lalisa Varney, Lashonne Varney, Leighanna Varney, Lesslie Varney, Lin Varney, Madlyn Varney, Malky Varney, Melenda Varney, Melisssa Varney, Mikita
Varney, Monicia Varney, Monik Varney, Nakima Varney, Naquita Varney, Natash Varney, Raphael Varney, Rashawna Varney, Reannon Varney, Reneta Varney, Reshanda Varney, Riana Varney, Rocky Varney,
Rosamond Varney, Roshan Varney, Saleena Varney, Schuyler Varney, Shaka Varney, Shakesha Varney, Shaleta Varney, Shanekia Varney, Surina Varney, Syrita Varney, Taj Varney, Tajuanna Varney, Tasheen
Varney, Tiffine Varney, Timmie Varney, Tomikia Varney, Trease Varney, Tyrhonda Varney, Wakisha Varney, Yvone Varney, Adalia Varney, Ahuva Varney, Angelicia Varney, Anina Varney, Annitra Varney,
Betsie Varney, Cachet Varney, Callista Varney, Camey Varney, Celita Varney, Cherissa Varney, Collin Varney, Daiana Varney, Equilla Varney, Evetta Varney, Eyvette Varney, Feige Varney, Filicia Varney,
Floyd Varney, Garnett Varney, Geoffrey Varney, Harper Varney, Ione Varney, Iyana Varney, Jametta Varney, Jeanee Varney, Jessalynn Varney, Jonalyn Varney, Juanice Varney, Kaija Varney, Kalene Varney,
Kassidy Varney, Katryn Varney, Kennethia Varney, Keosha Varney, Kresha Varney, Krisandra Varney, Labrina Varney, Lachonda Varney, Lamara Varney, Lashika Varney, Lateka Varney, Laytoya Varney, Lourdez
Varney, Lyndsi Varney, Markeita Varney, Marletta Varney, Marshella Varney, Martiza Varney, Marylouise Varney, Miri Varney, Nani Varney, Payton Varney, Puja Varney, Qianna Varney, Raffaella Varney,
Roanne Varney, Sadiqa Varney, Salvador Varney, Shabnam Varney, Shamar Varney, Shamina Varney, Sharalee Varney, Shavette Varney, Sheilia Varney, Shekia Varney, Shelise Varney, Sherrilynn Varney,
Sheyla Varney, Shy Varney, Suprena Varney, Tahitia Varney, Talley Varney, Tamiya Varney, Tashena Varney, Tashona Varney, Tatisha Varney, Toia Varney, Tonji Varney, Triana Varney, Tymika Varney, Yanna
Varney, Zainab Varney, Amrita Varney, Aprel Varney, Atalie Varney, Aubre Varney, Auria Varney, Barbarita Varney, Bettyann Varney, Boni Varney, Brandace Varney, Bryanne Varney, Calleen Varney, Chakita
Varney, Chamaine Varney, Chamika Varney, Charlett Varney, Chereese Varney, Cherine Varney, Christalyn Varney, Conya Varney, Dalilah Varney, Dela Varney, Delanna Varney, Delinah Varney, Dierdra
Varney, Domitila Varney, Donnis Varney, Dore Varney, Elenor Varney, Elizabethann Varney, Elysha Varney, Erryn Varney, Evalyn Varney, Frannie Varney, Fredda Varney, Ginnifer Varney, Hilaria Varney,
Ilisa Varney, Ivanna Varney, Janeal Varney, Jeena Varney, Jeremie Varney, Jeriann Varney, Julina Varney, Karim Varney, Kenitha Varney, Kery Varney, Korinna Varney, Lachrisha Varney, Lamonda Varney,
Laneisha Varney, Lene Varney, Letita Varney, Lillia Varney, Lindsie Varney, Lissete Varney, Lynsay Varney, Magalie Varney, Makini Varney, Manette Varney, Margareta Varney, Marlette Varney, Marsi
Varney, Maryn Varney, Megon Varney, Melainie Varney, Meria Varney, Merrideth Varney, Moranda Varney, Myah Varney, Myron Varney, Odalis Varney, Rakia Varney, Renesha Varney, Rhanda Varney, Richell
Varney, Salinda Varney, Sammantha Varney, Shamonica Varney, Shaquanda Varney, Sharmain Varney, Shereece Varney, Shron Varney, Tammika Varney, Terence Varney, Teresha Varney, Tiah Varney, Treniece
Varney, Truly Varney, Venicia Varney, Virgil Varney, Winnifred Varney, Yaritza Varney, Yarrow Varney, Zandria Varney, Alacia Varney, Albertha Varney, Andie Varney, Anuradha Varney, Artesia Varney,
Beatrix Varney, Brandin Varney, Brittni Varney, Carrey Varney, Catricia Varney, Chelise Varney, Christien Varney, Dalina Varney, Damia Varney, Deletha Varney, Eldora Varney, Eli Varney, Eufemia
Varney, Fabrienne Varney, Garland Varney, Genessa Varney, Ginnette Varney, Goldy Varney, Grant Varney, Greg Varney, Hasina Varney, Hermila Varney, Ima Varney, Jacquette Varney, Jilian Varney, Jilleen
Varney, Jolita Varney, Jorja Varney, Julita Varney, Juna Varney, Keicha Varney, Keshawn Varney, Ketty Varney, Kimmi Varney, Kizmet Varney, Koni Varney, Kristiane Varney, Krystyn Varney, Laronica
Varney, Lashonia Varney, Lateefa Varney, Latresia Varney, Lequisha Varney, Levi Varney, Lorenia Varney, Lubna Varney, Lucas Varney, Marche Varney, Marlea Varney, Masha Varney, Miah Varney, Michala
Varney, Michelyn Varney, Mistina Varney, Necola Varney, Netra Varney, Niomi Varney, Nira Varney, Nonnie Varney, Phelicia Varney, Quanta Varney, Quita Varney, Raini Varney, Rayanna Varney, Roselind
Varney, Roselinda Varney, Roshana Varney, Sagrario Varney, Sameka Varney, Shaindel Varney, Sharetha Varney, Sharol Varney, Shawnese Varney, Shellia Varney, Sheray Varney, Sherian Varney, Shi Varney,
Stepheni Varney, Subrena Varney, Tabithia Varney, Taleen Varney, Taneha Varney, Tanysha Varney, Tekela Varney, Telesa Varney, Tima Varney, Tonni Varney, Ula Varney, Yitty Varney, Yona Varney, Yonna
Varney, Yuko Varney, Zalika Varney, Aarika Varney, Ajeenah Varney, Alegra Varney, Alichia Varney, Allecia Varney, Amor Varney, Aneatra Varney, Anjannette Varney, Anntionette Varney, Antonetta Varney,
Ardelia Varney, Atlanta Varney, Ayoka Varney, Breeze Varney, Casonya Varney, Cathern Varney, Chanise Varney, Charlina Varney, Charmon Varney, Chemika Varney, Cheresa Varney, Cindra Varney, Claretha
Varney, Coren Varney, Cyndia Varney, Cyntha Varney, Danialle Varney, Danyeal Varney, Davin Varney, Deanda Varney, Deaundra Varney, Dejah Varney, Demica Varney, Demisha Varney, Dinita Varney, Donalyn
Varney, Donesha Varney, Dotty Varney, Fidelia Varney, Golden Varney, Helana Varney, Hidi Varney, Ilena Varney, Jenniferann Varney, Jolleen Varney, Jolynda Varney, Juanna Varney, Karlena Varney,
Karlin Varney, Keandra Varney, Kionna Varney, Konni Varney, Lakeyta Varney, Lechelle Varney, Levon Varney, Lilith Varney, Maliaka Varney, Mallie Varney, Margeaux Varney, Mariaisabel Varney, Marixa
Varney, Memorie Varney, Meshawn Varney, Michole Varney, Myrian Varney, Neela Varney, Nivea Varney, Peggi Varney, Porchia Varney, Racine Varney, Rashaun Varney, Reynalda Varney, Rosaleen Varney,
Sabrine Varney, Samanda Varney, Shanitha Varney, Sharece Varney, Sheretha Varney, Sherwanda Varney, Shonica Varney, Solveig Varney, Soyla Varney, Taka Varney, Taquilla Varney, Tedi Varney, Tema
Varney, Terita Varney, Tippi Varney, Tiziana Varney, Tomieka Varney, Tonga Varney, Trenace Varney, Tyeasha Varney, Tyresha Varney, Valda Varney, Vandy Varney, Victora Varney, Victory Varney, Waverly
Varney, Adair Varney, Agueda Varney, Aiko Varney, Akina Varney, Amaya Varney, Anthonette Varney, Ariann Varney, Averi Varney, Ayde Varney, Belissa Varney, Bernadett Varney, Berniece Varney, Berry
Varney, Bethsaida Varney, Bindu Varney, Bonniejean Varney, Brendy Varney, Cadence Varney, Camy Varney, Canisha Varney, Carilyn Varney, Caronda Varney, Chantrell Varney, Chavela Varney, Chemeka
Varney, Crystie Varney, Dafina Varney, Dangela Varney, Daphnee Varney, Darenda Varney, Darolyn Varney, Davelyn Varney, Desarie Varney, Devone Varney, Dionisia Varney, Donnielle Varney, Ebonique
Varney, English Varney, Eran Varney, Eryka Varney, Falisa Varney, Fareeda Varney, Herman Varney, Inda Varney, Janeice Varney, Janny Varney, Jessicah Varney, Jodean Varney, Julius Varney, Kaisa
Varney, Kanda Varney, Karel Varney, Karia Varney, Kimba Varney, Konya Varney, Kylah Varney, Kyrsten Varney, Laretta Varney, Lasharon Varney, Latascha Varney, Leni Varney, Leontyne Varney, Linell
Varney, Lovena Varney, Maghan Varney, Makeshia Varney, Malorie Varney, Marce Varney, Maryetta Varney, Mechille Varney, Media Varney, Meira Varney, Miosotis Varney, Mirjana Varney, Mlissa Varney,
Nekeya Varney, Nica Varney, Nikkisha Varney, Ortencia Varney, Porche Varney, Raffaela Varney, Raleigh Varney, Reka Varney, Rotunda Varney, Sharline Varney, Sharlonda Varney, Shawnice Varney, Shone
Varney, Shuntell Varney, Sneha Varney, Stephanne Varney, Sundy Varney, Tabita Varney, Takara Varney, Takeia Varney, Talli Varney, Tanganika Varney, Taquana Varney, Thresea Varney, Tila Varney, Tim
Varney, Tyan Varney, Vaishali Varney, Xanthe Varney, Zenja Varney, Zerlina Varney, Aggie Varney, Aleathea Varney, Aleen Varney, Alexcia Varney, Aloma Varney, Angeleque Varney, Araina Varney, Asucena
Varney, Azadeh Varney, Brendan Varney, Candina Varney, Candiss Varney, Cantina Varney, Carlisha Varney, Charra Varney, Chaunta Varney, Dalonda Varney, Damica Varney, Decarla Varney, Deneice Varney,
Derhonda Varney, Derrica Varney, Dorea Varney, Elvera Varney, Estell Varney, Fae Varney, Graceann Varney, Grissel Varney, Hina Varney, Jamielynn Varney, Jesscia Varney, Joley Varney, Junell Varney,
Kalilah Varney, Kathe Varney, Katurah Varney, Kenni Varney, Keysa Varney, Kimie Varney, Lachana Varney, Ladell Varney, Lanore Varney, Laramie Varney, Larrie Varney, Latonna Varney, Latori Varney,
Livier Varney, Loletta Varney, Magdaline Varney, Malaina Varney, Mandalyn Varney, Marshawn Varney, Marticia Varney, Maurisa Varney, Mellonie Varney, Nashira Varney, Orelia Varney, Quanna Varney,
Raisa Varney, Rakisha Varney, Raymonda Varney, Remonia Varney, Robbye Varney, Ronnita Varney, Roshaunda Varney, Sabre Varney, Sabreen Varney, Sharis Varney, Shawnae Varney, Sherrin Varney, Sindia
Varney, Staphanie Varney, Stevi Varney, Tannisha Varney, Tashunda Varney, Tekla Varney, Teralyn Varney, Tomecia Varney, Tonique Varney, Trilby Varney, Unika Varney, Vanisha Varney, Voula Varney,
Waynetta Varney, Yasmina Varney, Amesha Varney, Aniko Varney, Annica Varney, Antione Varney, Astria Varney, Camilia Varney, Caprina Varney, Carah Varney, Catonya Varney, Chalanda Varney, Chasty
Varney, Cherica Varney, Coy Varney, Crystalynn Varney, Dakisha Varney, Davine Varney, Dawnyel Varney, Demetrias Varney, Derica Varney, Dinamarie Varney, Dvora Varney, Elbony Varney, Emanuela Varney,
Gayleen Varney, Genevra Varney, Hadiyah Varney, Harolyn Varney, Hidie Varney, Indy Varney, Jaki Varney, Jalana Varney, Jalila Varney, Janiel Varney, Jennah Varney, Jessia Varney, Jihan Varney, Jinnie
Varney, Jonya Varney, Justa Varney, Justice Varney, Kameela Varney, Kameko Varney, Kaneisha Varney, Karletta Varney, Keneisha Varney, Kimberl Varney, Kimyada Varney, Kirstyn Varney, Krisy Varney,
Kyisha Varney, Labreeska Varney, Lacreasha Varney, Lajeana Varney, Lakiska Varney, Lanya Varney, Laren Varney, Latash Varney, Lataunya Varney, Leiah Varney, Loralyn Varney, Lorre Varney, Malaka
Varney, Malicia Varney, Marga Varney, Margaretann Varney, Markell Varney, Meriam Varney, Micca Varney, Morningstar Varney, Neta Varney, Nigel Varney, Noah Varney, Nuria Varney, Oliver Varney, Osha
Varney, Pierrette Varney, Quinita Varney, Quintessa Varney, Rashan Varney, Ravonda Varney, Raylynn Varney, Reyes Varney, Rhondalyn Varney, Rosey Varney, Roshanna Varney, Ruthy Varney, Sasheen Varney,
Shakeisha Varney, Shakirah Varney, Shaneta Varney, Sharmeka Varney, Sharonica Varney, Shelba Varney, Shelinda Varney, Shell Varney, Sheranda Varney, Stachia Varney, Stefanee Varney, Sydnee Varney,
Tamekka Varney, Tanisa Varney, Telissa Varney, Terissa Varney, Tilda Varney, Tomoko Varney, Trinna Varney, Tulani Varney, Varina Varney, Verlene Varney, Vernisha Varney, Yasha Varney, Yvonnie Varney,
Albertine Varney, Angeletta Varney, Anni Varney, Arrica Varney, Billyjo Varney, Brocha Varney, Calie Varney, Cass Varney, Catie Varney, Chairty Varney, Chakakhan Varney, Chassie Varney, Cherae
Varney, Cheryn Varney, Chisa Varney, Chrystine Varney, Coronda Varney, Curtina Varney, Danuta Varney, Debie Varney, Delona Varney, Demia Varney, Dosha Varney, Enrica Varney, Evone Varney, Fanchon
Varney, Fina Varney, Gaetana Varney, Genevia Varney, Gera Varney, Gracy Varney, Hollyann Varney, Iasia Varney, Iyesha Varney, Jackline Varney, Joanette Varney, Jonnette Varney, Kaamilya Varney,
Karolina Varney, Kathlynn Varney, Kayle Varney, Kayse Varney, Kehaulani Varney, Kella Varney, Kenyatte Varney, Kianga Varney, Kijuana Varney, Kinisha Varney, Kiwanda Varney, Kriss Varney, Lakea
Varney, Lakeithia Varney, Lakeyia Varney, Latusha Varney, Leala Varney, Leina Varney, Lucita Varney, Madia Varney, Mairead Varney, Makiba Varney, Marchella Varney, Marquel Varney, Martita Varney,
Meco Varney, Melissaann Varney, Merilyn Varney, Minyon Varney, Monnica Varney, Monnie Varney, Mora Varney, Netasha Varney, Nkechi Varney, Ocie Varney, Oni Varney, Patricie Varney, Quina Varney,
Quintana Varney, Renu Varney, Rhetta Varney, Rissa Varney, Rozalyn Varney, Salisa Varney, Seena Varney, Shaine Varney, Shandon Varney, Shanisha Varney, Shantai Varney, Sharisa Varney, Shary Varney,
Shaylyn Varney, Sheleta Varney, Shilonda Varney, Sojourner Varney, Syrena Varney, Taleah Varney, Tanekia Varney, Tatianna Varney, Tere Varney, Terez Varney, Teriann Varney, Tewana Varney, Ticey
Varney, Vanette Varney, Vidya Varney, Wonder Varney, Zoie Varney, Aeisha Varney, Alisande Varney, Allan Varney, Amantha Varney, Analee Varney, Ane Varney, Artrice Varney, Audelia Varney, Ayelet
Varney, Benji Varney, Benny Varney, Briann Varney, Brittan Varney, Calvina Varney, Cameka Varney, Cecilie Varney, Cecille Varney, Chanette Varney, Chantia Varney, Charlese Varney, Charolett Varney,
Chenille Varney, Chivonne Varney, Chudney Varney, Clemencia Varney, Courtny Varney, Cristalle Varney, Cynde Varney, Daveda Varney, Derika Varney, Dezarae Varney, Dionicia Varney, Dunia Varney,
Elspeth Varney, Fayette Varney, Glena Varney, Gordon Varney, Gretchin Varney, Halli Varney, Ilissa Varney, Jauna Varney, Johnnette Varney, Joycelynn Varney, Junelle Varney, Kaydee Varney, Kerryn
Varney, Koryn Varney, Kosha Varney, Kyoko Varney, Lala Varney, Lametra Varney, Laresha Varney, Lauraann Varney, Lecretia Varney, Luv Varney, Lyndsie Varney, Malessa Varney, Marcine Varney, Mellissia
Varney, Nakina Varney, Nashonda Varney, Neelam Varney, Nikitia Varney, Orit Varney, Otis Varney, Pasqualina Varney, Porshia Varney, Rayshell Varney, Rico Varney, Rockell Varney, Roshon Varney,
Savanna Varney, Shallan Varney, Shashana Varney, Shawntia Varney, Sherrise Varney, Sofie Varney, Taji Varney, Talanda Varney, Tamella Varney, Tamsyn Varney, Taquisha Varney, Tashea Varney, Terisha
Varney, Tikita Varney, Timmy Varney, Trenese Varney, Valissa Varney, Valli Varney, Vergie Varney, Yancy Varney, Aina Varney, Albina Varney, Aletheia Varney, Alisson Varney, Allisyn Varney, Amorita
Varney, Anginette Varney, Anjana Varney, Anngela Varney, Ariela Varney, Arienne Varney, Arnisha Varney, Arvella Varney, Ayodele Varney, Badia Varney, Brooklynn Varney, Candus Varney, Conswello
Varney, Curtisha Varney, Cythina Varney, Dashia Varney, Dekisha Varney, Denyce Varney, Dilia Varney, Doreatha Varney, Dortha Varney, Ernesto Varney, Fatisha Varney, Fauna Varney, Fifi Varney, Gordana
Varney, Indya Varney, Jaina Varney, Janiene Varney, Jarah Varney, Jasmyne Varney, Jeananne Varney, Jeff Varney, Jeronica Varney, Jessaca Varney, Joby Varney, Jodette Varney, Jolisa Varney, Juanda
Varney, Juanette Varney, Kalie Varney, Kaneshia Varney, Karalynn Varney, Kavitha Varney, Kelcy Varney, Kerrilynn Varney, Kierston Varney, Kineta Varney, Kiri Varney, Krisie Varney, Latorria Varney,
Lazara Varney, Lekecia Varney, Lenell Varney, Lesle Varney, Levina Varney, Liat Varney, Lindley Varney, Maiya Varney, Makela Varney, Mariane Varney, Marki Varney, Mashelle Varney, Maud Varney, Medora
Varney, Megumi Varney, Milca Varney, Missey Varney, Mistelle Varney, Nakeeta Varney, Neile Varney, Nicloe Varney, Nykisha Varney, Orly Varney, Patria Varney, Rama Varney, Ramonia Varney, Ranie
Varney, Ranya Varney, Refugio Varney, Rendi Varney, Schwanna Varney, Senia Varney, Shalaunda Varney, Shalia Varney, Shanieka Varney, Shantale Varney, Shatima Varney, Shawon Varney, Shelva Varney,
Shemeca Varney, Shemeeka Varney, Sher Varney, Sinda Varney, Skyler Varney, Stephaie Varney, Suzetta Varney, Suzzane Varney, Synethia Varney, Taesha Varney, Takina Varney, Tanyetta Varney, Taran
Varney, Tarasha Varney, Tarji Varney, Tauni Varney, Temeika Varney, Temisha Varney, Thania Varney, Thembi Varney, Thora Varney, Tiffin Varney, Toney Varney, Trine Varney, Tyrina Varney, Uraina
Varney, Valleri Varney, Vandana Varney, Vernie Varney, Vinetta Varney, Viveca Varney, Vonette Varney, Wynema Varney, Yasheka Varney, Yevonne Varney, Yovana Varney, Zanita Varney, Zarah Varney, Zendre
Varney, Abigayle Varney, Abraham Varney, Acquanetta Varney, Alika Varney, Alora Varney, Alvera Varney, Amand Varney, Annabella Varney, Annelle Varney, Antia Varney, Arkisha Varney, Arlanda Varney,
Aronda Varney, Ayo Varney, Bailey Varney, Banita Varney, Becka Varney, Brannon Varney, Camden Varney, Cantrice Varney, Ceclia Varney, Chantella Varney, Charryse Varney, Cheramie Varney, Chessie
Varney, Chiquetta Varney, Chrysti Varney, Cinzia Varney, Cristian Varney, Curry Varney, Cybele Varney, Damion Varney, Darius Varney, Dawnyell Varney, Denora Varney, Dung Varney, Ebonye Varney, Elesa
Varney, Ellery Varney, Ellis Varney, Emile Varney, Eowyn Varney, Felicite Varney, Geniene Varney, Ginia Varney, Hadiya Varney, Hang Varney, Harlene Varney, Inita Varney, Jalaine Varney, Janne Varney,
Jari Varney, Jayci Varney, Jodilynn Varney, Jomarie Varney, Jyll Varney, Kabrina Varney, Karisma Varney, Kashina Varney, Katika Varney, Katye Varney, Keiona Varney, Kemisha Varney, Kyli Varney,
Lametria Varney, Lamica Varney, Larue Varney, Lasheika Varney, Latandra Varney, Launi Varney, Ligaya Varney, Lilibeth Varney, Linetta Varney, Linnet Varney, Lorrain Varney, Loyce Varney, Lucresha
Varney, Marguetta Varney, Matrice Varney, Meca Varney, Meda Varney, Meiling Varney, Melesa Varney, Michaelann Varney, Monick Varney, Nonie Varney, Nyra Varney, Pepsi Varney, Porcha Varney, Rakeisha
Varney, Reana Varney, Rebacca Varney, Rennee Varney, Roben Varney, Rondalyn Varney, Roseline Varney, Salem Varney, Samella Varney, Sandar Varney, Santia Varney, Sarada Varney, Sayward Varney, Schelly
Varney, Shamecka Varney, Sharmeen Varney, Shayleen Varney, Sheina Varney, Sheleen Varney, Shenice Varney, Shiri Varney, Shonetta Varney, Suha Varney, Sulay Varney, Swanzetta Varney, Syndi Varney,
Tanette Varney, Tarla Varney, Tiffanni Varney, Timaka Varney, Tosheba Varney, Willene Varney, Wynn Varney, Yaisa Varney, Adrinne Varney, Aleisa Varney, Aleka Varney, Alka Varney, Anais Varney, Angy
Varney, Annetra Varney, Antinette Varney, Apryle Varney, Aretta Varney, Arnell Varney, Azusena Varney, Bianka Varney, Brandan Varney, Camron Varney, Capucine Varney, Carolene Varney, Catisha Varney,
Catrece Varney, Cendy Varney, Charmelle Varney, Charrisse Varney, Chrisha Varney, Cina Varney, Dajuana Varney, Dala Varney, Danial Varney, Daphyne Varney, Daya Varney, Delorise Varney, Deneise
Varney, Deshunda Varney, Desmond Varney, Desra Varney, Dinna Varney, Duchess Varney, Earnest Varney, Edda Varney, Eliane Varney, Elliott Varney, Erline Varney, Feleica Varney, Florance Varney, Gayl
Varney, Gaynelle Varney, Grisela Varney, Huma Varney, Jaimelyn Varney, Jamile Varney, Jeannifer Varney, Jolin Varney, Jonquil Varney, Jurea Varney, Kaili Varney, Kanitra Varney, Katana Varney, Katoya
Varney, Kenja Varney, Kennette Varney, Kenyotta Varney, Kertina Varney, Kiyomi Varney, Klarissa Varney, Lakeish Varney, Lakena Varney, Lanett Varney, Lashante Varney, Lasonda Varney, Laurelle Varney,
Lenda Varney, Leonette Varney, Madalene Varney, Madhavi Varney, Mariadelcarmen Varney, Marine Varney, Marquerite Varney, Marylu Varney, Maury Varney, Merlyn Varney, Natilie Varney, Neco Varney,
Nelissa Varney, Nocole Varney, Patrese Varney, Ranette Varney, Ranisha Varney, Rhena Varney, Ronesha Varney, Royal Varney, Safiyyah Varney, Sahra Varney, Saima Varney, Sarahjane Varney, Senetra
Varney, Serra Varney, Shallyn Varney, Shawni Varney, Sherin Varney, Sherisa Varney, Sherrelle Varney, Shontina Varney, Shontrell Varney, Solana Varney, Tamalyn Varney, Tamesia Varney, Tametria
Varney, Tamicia Varney, Taneya Varney, Tango Varney, Taquila Varney, Tekelia Varney, Tianne Varney, Tonimarie Varney, Trea Varney, Tremayne Varney, Trichelle Varney, Tunesia Varney, Uma Varney, Von
Varney, Yanina Varney, Akela Varney, Aleesa Varney, Alonzo Varney, Annel Varney, Arisha Varney, Aylin Varney, Aynsley Varney, Bernina Varney, Billee Varney, Calisa Varney, Calisha Varney, Canda
Varney, Cantrell Varney, Carylon Varney, Casee Varney, Celica Varney, Chantilly Varney, Charnissa Varney, Chioma Varney, Chrystel Varney, Clotilde Varney, Coriann Varney, Corisa Varney, Currie
Varney, Cythnia Varney, Danay Varney, Danyella Varney, Darling Varney, Deloria Varney, Deone Varney, Devida Varney, Divya Varney, Dorean Varney, Dorri Varney, Earlean Varney, Eletha Varney, Elfreda
Varney, Elizabth Varney, Elna Varney, Elycia Varney, Elza Varney, Emmanuelle Varney, Felesia Varney, Galit Varney, Garnetta Varney, Gwendolin Varney, Halee Varney, Haylie Varney, Jaala Varney,
Jacilyn Varney, Jackquelyn Varney, Jamekia Varney, Jancie Varney, Jeany Varney, Jenness Varney, Johni Varney, Kadee Varney, Kalinda Varney, Karmel Varney, Kashunda Varney, Kearston Varney, Keina
Varney, Kely Varney, Kerissa Varney, Kiandra Varney, Kimley Varney, Krysia Varney, Kyanna Varney, Lajuanda Varney, Lania Varney, Lanina Varney, Latanza Varney, Lauriann Varney, Lawan Varney, Laya
Varney, Linna Varney, Lisbet Varney, Luzmaria Varney, Maeghan Varney, Mali Varney | {"url":"https://www.instantcheckspy.com/lastname/Varney.php","timestamp":"2024-11-13T22:27:26Z","content_type":"text/html","content_length":"314924","record_id":"<urn:uuid:94146c32-4854-4d29-9ca5-0e66a062193c>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00389.warc.gz"} |
How hard is it to satisfy (Almost) all roommates?
The classic Stable Roommates problem (the non-bipartite generalization of the well-known Stable Marriage problem) asks whether there is a stable matching for a given set of agents, i.e. a
partitioning of the agents into disjoint pairs such that no two agents induce a blocking pair. Herein, each agent has a preference list denoting who it prefers to have as a partner, and two agents
are blocking if they prefer to be with each other rather than with their assigned partners. Since stable matchings may not be unique, we study an NP-hard optimization variant of Stable Roommates,
called Egal Stable Roommates, which seeks to find a stable matching with a minimum egalitarian cost γ, i.e. the sum of the dissatisfaction of the agents is minimum. The dissatisfaction of an agent is
the number of agents that this agent prefers over its partner if it is matched; otherwise it is the length of its preference list. We also study almost stable matchings, called Min-Block-Pair Stable
Roommates, which seeks to find a matching with a minimum number β of blocking pairs. Our main result is that Egal Stable Roommates parameterized by γ is fixed-parameter tractable, while
Min-Block-Pair Stable Roommates parameterized by β is W[1]-hard, even if the length of each preference list is at most five.
Original language English
Title of host publication 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018
Editors Christos Kaklamanis, Daniel Marx, Ioannis Chatzigiannakis, Donald Sannella
Publisher Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic) 9783959770767
State Published - 1 Jul 2018
Event 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018 - Prague, Czech Republic
Duration: 9 Jul 2018 → 13 Jul 2018
Publication series
Name Leibniz International Proceedings in Informatics, LIPIcs
Volume 107
ISSN (Print) 1868-8969
Conference 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018
Country/Territory Czech Republic
City Prague
Period 9/07/18 → 13/07/18
• Analysis and algorithmics
• Data reduction rules
• Kernelizations
• NP-hard problems
• Parameterized complexity
ASJC Scopus subject areas
Dive into the research topics of 'How hard is it to satisfy (Almost) all roommates?'. Together they form a unique fingerprint. | {"url":"https://cris.bgu.ac.il/en/publications/how-hard-is-it-to-satisfy-almost-all-roommates-2","timestamp":"2024-11-12T22:54:59Z","content_type":"text/html","content_length":"61779","record_id":"<urn:uuid:9c6e3f60-7a61-45ae-9f28-36bfea3cd3b5>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00822.warc.gz"} |
Curriculum for a Teach-The-Teacher Course?
As a starting point, I'd first of all look at teh exisiting curriculum and see if there is something currently covered that scratch might do better.
For example, in the UK curiculum in year 3 (7-8 yrs) there is an introduction to control ssytems that, currently, almost everyone currently teaches using "logo" or a derivative therof.
That's an obvious candidate for being superceded by Scratch.
Usign scratch to replace the exisitng curriculum products can go outside of the ICT class, as well. Your need to have introduced scratch first, but you can show the teachers some of the sample book
reports that chn have produced and submitted.
I've also seen teaching units where teh chn have to create their own, short, "choose your own adventure" book, somethign that could be translated into scratch very easily.
I'm keen - should I start teaching slightly older classes in teh future - to employ a cross curricular approach where (for example) we write a script in English class, draw the characters in art
class and then animate the story using scratch in ICT.
Finally - let the teachers know that if they themselves start getting interested in scratch, they can use it to produce simple interactive resources to use in teaching, customised to their own
I've used it to teach directions (directions game 1 and 2), past tense endings (Ed the cat), RE (the cleansing of the temple) and counting in 10ps (coin counting game), and may also use it in the
future in the ICT "introduction to modelling" unit.
As soon as I get around to writing a scratch simulation of growing a plant, that is | {"url":"https://scratcharchive.asun.co/forums/viewtopic.php?id=8505","timestamp":"2024-11-02T23:18:59Z","content_type":"text/html","content_length":"54468","record_id":"<urn:uuid:f05ffb31-3d67-4c41-b8ca-5e117ccbbe18>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00602.warc.gz"} |
Reconstruction of piano hammer force from string velocity
A method is presented for reconstructing piano hammer forces through appropriate filtering of the measured string velocity. The filter design is based on the analysis of the pulses generated by the
hammer blow and propagating along the string. In the five lowest octaves, the hammer force is reconstructed by considering two waves only: the incoming wave from the hammer and its first reflection
at the front end. For the higher notes, four- or eight-wave schemes must be considered. The theory is validated on simulated string velocities by comparing imposed and reconstructed forces. The
simulations are based on a nonlinear damped stiff string model previously developed by Chabassier, Chaigne, and Joly [J. Acoust. Soc. Am. 134(1), 648–665 (2013)]. The influence of absorption,
dispersion, and amplitude of the string waves on the quality of the reconstruction is discussed. Finally, the method is applied to real piano strings. The measured string velocity is compared to the
simulated velocity excited by the reconstructed force, showing a high degree of accuracy. A number of simulations are compared to simulated strings excited by a force derived from measurements of
mass and acceleration of the hammer head. One application to an historic piano is also presented.
In pianos, precise knowledge of the hammer force in terms of amplitude, duration, and shape is essential since it fully determines the resulting free vibrations of the strings. Therefore, it is not
surprising that a vast literature exists on this matter in piano acoustics. For years, one commonly used method for deriving the force consisted in multiplying together the measured mass and
acceleration of the hammer head.^1,2 This method yields an acceptable rough estimate of the hammer pulse, though it suffers from several limitations.^3 Among these limitations, the oscillations of
the hammer shank were highlighted experimentally by several authors in the past,^4,5 and confirmed by recent simulations.^6 With the emergence of new optical techniques and, in particular, of those
using high-speed cameras, new methods are now available for measuring the hammer force with great accuracy.^3
In this paper, an alternative method is proposed for determining the hammer force. This method is based on non-contact measurements of the string velocity from which the hammer force is derived
through appropriate filtering based on the properties of the wave propagation along the string. The underlying motivation of this method follows from the observation that the hammer is not always
easily accessible on all pianos and, in addition, that sophisticated optical methods cannot always be simply implemented in all situations. This fact has been experienced by the author during
experiments on historic pianofortes, for which conservation requirements impose to find fast, portable, and non-intrusive measurements. A similar approach was used in the past for the bowed string.^7
The basic principles of the reconstruction method are presented in Sec. II for a linear undamped nondispersive string (a so-called “ideal” string) excited by an imposed hammer force pulse $fH(t)$ of
finite duration τ[H] at a given striking position x[H]. The transverse string velocity $vs(t)$ is measured at a given position x[s] along the string. The initial transient part of $vs(t)$ is
represented by the finite sum of elementary waves generated by the hammer, and traveling back and forth along the string. For piano strings, it is shown that 2 to 8 waves are sufficient for
representing this initial transient adequately for the complete piano compass, the number of waves to be considered depending on both the fundamental frequency f[1] of the note and force width τ[H].
A classical comb filter is applied to $vs(t)$ in order to reconstruct the hammer force. The appropriate choice of x[s] is imposed by the properties of the filter.
In Sec. III, the method is applied to simulated piano strings. The simulations are based on a model of a damped nonlinear stiff string recently developed by Chabassier et al.^8 The physical
parameters of the simulated strings are measured on real pianos. The string is excited by an imposed hammer force pulse. This input force has realistic amplitude, time width, and shape for each
simulated note. The pertinence of the wave filtering method is validated by comparisons between the input force and the reconstructed force. Three different types of wave filters are used in order to
cover the complete range of notes of a piano keyboard. The influence of damping, stiffness, and amplitude of the string motion on the estimation of the hammer force are examined.
In Sec. IV, the method is applied to measured piano strings. The string velocity is measured with a calibrated electromagnetic sensor. The reconstructed hammer force serves as the input force for the
corresponding simulated string, and the resulting simulated velocity is compared to the measured velocity. In some particular cases, the reconstructed hammer force is compared to the force derived
from measurements of hammer head acceleration and mass, and the influence of hammer shank motion is discussed. Finally, an example of application of the method to an historic piano with a Viennese
action is presented.
A. Ideal piano string model
In this section, it is assumed that the motion of the piano string is described by a linear wave equation without damping or stiffness terms. The string is characterized by its tension at rest T[0],
length L, and linear mass density μ. The string is rigidly fixed at both ends, agraffe (A) and bridge (B), which are considered as perfectly reflecting boundaries. Only one transverse component of
the string motion y(x,t) is considered, in the direction of the initial hammer velocity, which is here assumed to be vertical. In this ideal case, the equations of the struck string are written^9,10
$μ∂2y∂t2−T0∂2y∂x2=fH(t)δ(x−xH), ∀ x∈[0,L],y(x=0,t)=0, y(x=L,t)=0,$
where $fH(t)$ is a hammer force of finite duration τ[H] applied at the striking position x[H], and δ is the Dirac delta function. The transverse speed is $c=T0/μ$, the fundamental frequency is $f1=1
/T1=c/2L$, and $Zc=μT0$ is the characteristic impedance.^9 In pianos, the striking point is situated at a distance roughly equal to $L/10$ from the agraffe.^11 The string velocity $vs(t)$ is
measured at position x[s]. Figure 1 shows the case where x[s] is situated between agraffe and hammer. Both situations x[s]<x[H] and x[s]>x[H] will be examined in this section. At position x[H],
the hammer blow generates two pulses $vsh(t)=fH(t)/2Zc$ traveling in opposite directions: the left wave toward the agraffe (A), and the right wave toward the bridge (B). These elementary pulses reach
the sensor position at successive instants of time t[i] whose expressions are given in the Appendix.
In Sec. IIB, an intuitive analysis of the wave propagation on the string is presented, showing that the number of elementary pulses (or waves) to consider in the measured string velocity to
reconstruct the hammer force depend on the ratio between the force width τ[H] and the string period T[1]. For pianos, this leads to the elaboration of two-, four- and eight-wave schemes, depending on
the played note. In Sec. IIC, it is shown that these three schemes are particular cases of a general formulation of the transfer function between string velocity and hammer force that needs to be
inverted in order to recover the force. This inversion imposes precautions in the measurements and, in particular, in the appropriate selection of the measuring point x[s].
B. Principles of the force reconstruction method. Wave analysis
1. Bass and medium range of the piano. Two-wave scheme
Reconstructing the hammer force from the string velocity is an inverse problem. Knowing $vs(t)$, the goal is to recover the force pulse $fH(t)$ through the analysis of the wave propagation on the
string and the reflection of the pulses at the boundaries (agraffe and bridge). Figure 2 illustrates this point in the typical case of a piano note in the bass (or medium) range, where the string
velocity waveform recorded near the agraffe is clearly separated into two distinct parts. In the presented case where x[s]<x[H], the first part of $vs(t)$ is proportional to the sum of the incoming
hammer force pulse (1) and its reflection at the agraffe (2), whereas its second part is proportional to the sum of the force pulses reaching the sensor after reflection at the bridge (3) and after
further reflection at the agraffe (4). Each reflection at either end is characterized by a change of sign. Using the notations defined in Sec. IIA, we can write for the first part of $vs(t)$,
$vs(t)=12Zc[ fH(t−xH−xsc)−fH(t−xH+xsc) ].$
The calculation in the Appendix shows that Eq. (2) can be used to reconstruct the hammer force pulse as long as its width τ[H] is such that
This condition is obtained by considering that the force pulse (1) goes to zero before the third pulse reaches the sensor. In other words, Eq. (3) means that the number of elementary waves to
consider for the reconstruction is equal to the number of waves reaching the sensor during the duration τ[H] of the force pulse width. Since x[H] usually is on the order of $L/10$, the condition in
Eq. (3) corresponds to a force pulse duration slightly smaller than the period T[1] of the string's oscillation. In practice, Fig. 3 shows that this condition is fulfilled for most pianos below the
note C5. In this range, the force reconstruction method then consists in inverting the two-wave scheme expressed in Eq. (2). This inversion is achieved by applying a dedicated so-called
reconstruction filter to the string velocity, as explained in Sec. IIC. Analysis of the filter properties will show why the velocity sensor must be placed near the agraffe for this two-wave scheme.
2. Upper range of the piano. Four- and eight-wave schemes
In pianos, the width of the force pulse decreases less rapidly than the period of the string's oscillation when moving toward the treble range (see Fig. 3). As a consequence, the propagating pulses
get closer to one another and several reflections can occur during the pulse width τ[H]. It is shown in the Appendix that, under the condition
the initial force pulse goes to zero before the fifth reflected pulse reaches the sensor. Therefore, the string velocity can be represented by a sum of four waves during the initial time interval τ
[H]. An example is shown in Fig. 4. The frequency analysis of the corresponding reconstruction filter then shows that the measuring point of the string velocity x[s] now must be close to the bridge
in order to recover the force properly (see Sec. IIC). In practice, the condition in Eq. (4) required for this four-wave scheme is rather restrictive. It is fulfilled in a small range of notes for
most pianos, usually between the notes C5 and C6 (see Fig. 3).
For the remaining two upper octaves of the piano (notes C6–C8), a typical situation is shown in Fig. 5. In the example presented here, it is seen that the first six pulses are involved in the
expression of $vs(t)$ over the duration τ[H] of the initial force pulse. Following the same reasoning as previously, we could think of considering six waves over this time interval to reconstruct the
hammer force successfully, under the condition that the seventh pulse reaches the sensor after extinction of the first pulse. However, it turns out that such a six-wave scheme is unstable. The idea
used here is to consider eight waves (instead of six), because it yields again a stable filter. This procedure does not affect the result of the reconstruction: the output signal obtained is then
made of the reconstructed pulse of width τ[H] followed by a varying number of zeroes, depending on the note. Another argument for this eight-wave scheme is that it remains valid under the condition
which, in practice, is fulfilled for the remaining upper part of the piano (notes C6 to C8). As for the four-wave scheme, the properties of the corresponding filter imposes to select x[s] close to
the bridge (see Sec.IIC). In summary, the force reconstruction is governed by two important criteria. In the time-domain, the width τ[H] of the hammer force pulse imposes the number of waves to
consider in the inverse filtering. In the frequency-domain, the properties of the reconstruction filter impose the location of the measuring point x[s] along the string, as shown below in Sec. IIC.
C. Reconstruction filters
1. General formulation
The string wave propagation qualitatively presented in Sec. IIB can be formally described by a filter with the hammer force as input signal, and the string velocity at the measuring point as output
signal. For discrete signals sampled at frequency $fe=1/Te$, the appropriate tool is the z-transform where a delay of T[e] corresponds to a multiplication by $z−1$. Therefore $z−m$ corresponds to a
delay of m samples, that is, mT[e] seconds. For details on the z-transform, see, for example, Ref. 12. In what follows, $Vs(z)$ and $FH(z)$ denote the z-transforms of $vs(t)$ and $fH(t)$,
respectively. In the Appendix, the example of a plucked string with a pickup output developed by Karjalainen et al. is adapted to the present piano case.^13 For a lossless nondispersive string, it
is shown that the transfer function between string velocity and hammer force is given by
where n[1] accounts for the propagation delay between hammer and sensor, m[1] and m[2] account for the delays due to hammer and sensor positions, and m[3] is the delay for one complete loop.
Conversely, if $Vs(z)$ is known, the hammer force $FH(z)$ can be retrieved through the inverse transfer function
If the sensor is situated on the left-hand side of the hammer (on the agraffe side: x[s]<x[H]), it is shown in the Appendix that
$n1=(xH−xs)fec; m1=2xsfec=xsfeLf1; m2=(L−xH)feLf1; m3=fef1.$
Conversely, for x[s]>x[H] (sensor on the bridge side), the delay constants are obtained through permutation of x[s] and x[H]
$n1=(xs−xH)fec; m1=2xHfec=xHfeLf1; m2=(L−xs)feLf1; m3=fef1.$
2. Approximate two-, four-, and eight-wave filters
The transfer function expressed in Eq. (7) is valid for an infinite number of waves reaching the sensor. The filters corresponding to the wave analysis presented in Sec. IIB are obtained by
truncating this expression to the initial portion of the signal. Considering the first two incoming waves only, then H[FV] reduces to
${ HFV2w(z)=2Zczn11−z−m1 for xs<xH,or2Zczn11−z−m2 for xs>xH.$
Notice that the first expression in Eq. (10) is the transfer function directly obtained by taking the z-transform of Eq. (2). Similarly, a four-wave approximation is given by
Finally, for the eight-wave scheme, the corresponding filter is obtained by approximating the factor $(1−z−m3)$ by the two first terms of its expansion, which yields
Notice, in this case, that Eq. (7) can be used alternatively instead of Eq. (12). The advantage is then to deal with the feed-forward filter $1−z−m3$ rather than with the feedback version $1/(1+z−m3)
$, which adds high peaks in the spectrum.
After filtering of the velocity, the hammer force pulses are obtained by truncating the signal in time below the upper limit of validity for each of the three schemes given in Eqs. (3)–(5). Using the
general form of the inverse filter Eq. (7) theoretically yields an infinite domain of validity in time. However, in practice, due to the presence of noise and progressive departure from the
underlying model when dealing with real signals, the filtered velocity again has to be truncated to reasonable values. An upper bound of 10ms is, in general, sufficient for recovering the force.
3. Selection of the measurement point x[s]
Some precautions should be taken in using the previous filters to reconstruct the hammer force. This is due to the fact that both delays m[1] and m[2] now introduce comb filtering effects with poles
at frequencies $fkm1=kfe/m1$ and $fkm2=kfe/m2$, respectively. Because of their high gain, these pole frequencies can perturb the reconstruction of the force if they are smaller than the upper limit
f[VM] of the velocity spectrum. These perturbations are due to amplification of noise and deviations from the ideal case in real signals at these frequencies. Therefore, given the expressions of m[1]
and m[2] in Eqs. (8) and (9), the appropriate strategy for limiting these perturbations is to select x[s] so that f[VM] is smaller than the lowest poles $fe/m1$ and $fe/m2$, which yields the
following conditions:
${ for xs<xH: xsL<f1fVM and L−xHL<f1fVM,for xs>xH: xHL<f1fVM and L−xsL<f1fVM.$
For x[s]<x[H], the second condition, relative to m[2] in Eq. (13) is unacceptable since it implies that the spectrum of the string velocity should be limited to the fundamental frequency.
Therefore, only the condition on m[1] remains, which means that this configuration is valid for the two-wave scheme only. According to the wave analysis presented in Sec. IIB, this scheme is
applicable to the bass and medium range of the piano. In this range, x[s] must be small compared to L (sensor close to the agraffe) so that the poles of the filter are rejected out of the velocity
spectrum. With $xs/L=100$, for example, the lowest pole of the filter is rejected beyond the 100th harmonic of the string.
The configuration x[s]>x[H] is applicable to four- and eight-wave schemes if the two corresponding conditions in Eq. (13) are fulfilled. From the wave analysis in Sec. IIB, we know that such
schemes are applicable in the treble range between C5 and C8. The second condition on m[2] implies that x[s] is close to L, which is obtained now by placing the sensor close to the bridge. In
practice, good results are obtained for $0.05<(L−xs)/L<0.1$. The first condition related to $xH/L$ is imposed by the construction of the piano. On average $xH/L=0.1$, which means that the lowest
pole frequency of the reconstruction filter is rejected beyond the tenth harmonic. Usual piano tones show relatively little energy beyond this harmonic, between C5 and C8 (see, for example, Fig. 11),
and thus the configuration x[s]>x[H] is valid in this range. Notice also that the regularization presented in Sec. IIC4 allows to reduce the perturbating effects of the poles.
4. Further considerations: Dispersion, fractional delays and regularization
In the bass range of the piano, it becomes necessary to account for the dispersion of the wave due to string stiffness. One widely used method for this is to divide the transfer function $HVF(z)$ in
Eq. (6) by an appropriate allpass filter D(z) whose phase and delay properties are based on the physical parameters of the string. For a stiff string, the phase shift $φ(f)$ for a roundtrip is given
where $B=π2EI/T0L2$ is the stiffness coefficient. The associated group delay $γ(f)$ is given by
$γ(f)=1f11[ 1+Bf2f12 ]3/2.$
The pertinent dimensionless coefficient here is $εs=Bf2/f12$ which, for a given frequency f, continuously decreases from bass to treble. For reproducing this delay, a set of eight biquad filters in
cascade was applied, following the method proposed by Abel et al.^14,15
The delays defined in Eqs. (8) and (9) are usually not integers. Therefore one useful strategy is to use fractional delay filters to represent them. Here, Thiran allpass filters were tested
successfully.^16 Oversampling is another strategy, so that approximating n[1], m[1], m[2], and m[3] by the nearest integers yields a small error. Both methods gave comparable results.
Finally, a regularization can be made in replacing $(1−z−m1)$ and $(1−z−m2)$ by $(1−g1z−m1)$ and $(1−g2z−m2)$ in Eq. (7), with $0.99<g1,g2<0.999$, so that the gain of $HFV(z)$ does not tend to
infinity at those frequencies. The consequences on the pole frequencies are negligible. Physically, such a regularization accounts for a slight dissipation at both ends of the string. The interval of
values of g[1] and g[2] are selected somewhat arbitrarily, but the examples shown in Secs. III and IV show that this has no appreciable effect on the amplitude of the reconstructed force. However,
selecting g[1] and g[2] smaller than 0.99 (which would correspond to losses at the ends higher than 1%) generally leads to unwanted drift in the reconstructed force waveform, showing that the
corresponding damping at the string ends is then probably overestimated. Notice that, for coherence, the factor $(1−z−m3)$ must be then replaced by $(1−g1g2z−m3)$. In conclusion, after inversion,
the most general formulation of the transfer function used for reconstructing the hammer force is written
where the delays are implemented with fractional delay filters.
In this section, the method of hammer force reconstruction presented in Sec. II is tested on simulated piano strings. Using string simulations allows a perfect control on the input hammer force and
careful comparison with the reconstructed force. In addition, the influence of the main causes of departure from the ideal string case on the quality of the reconstruction can be tested and
quantified separately, which is not possible on real strings. Specifically, we take benefit of the simulation model to test the influence of absorption, dispersion, and non-linearity of the string
wave propagation on the resulting hammer force reconstruction. The waves schemes presented in Sec. II (two-, four- and eight-waves, respectively) are tested in the different note ranges of the
keyboard of various pianos: bass, medium, and treble, and for different striking forces (p,mf,f). The reconstructed hammer force is obtained by filtering the simulated string velocity at a given
sensor position x[s].
A. Model and method
The string model used for the simulations is a part of a complete numerical piano model developed by Chabassier et al.^8,17 This model will not be detailed further here, since it has been presented
in these previous papers extensively. For the string part, this model includes damping, stiffness, and nonlinear terms. The frequency-dependent damping is classically modeled by two-terms: a
so-called “fluid” term and a “viscoelastic” damping term for each component of the string motion. The stiffness term is obtained by modeling the string as a prestressed beam. The additional nonlinear
terms account for the influence of the amplitude on the string motion.^18 The transverse motion of the string is assumed here to be vertical (in the direction of the hammer force) and coupled to its
longitudinal motion. The motion of both ends (agraffe and bridge) is ignored in the description used here. The values of the parameters are derived from measurements on real strings (see Table I).
TABLE I.
Note . D♯1 . D♯1 . F♯2 . C4 . C♯5 . C♯5 . D♯6 . E6 .
Piano JBS73 JBS50 JBS73 JBS73 JBS36 JBS73 BSD BSD
L (m) 1.717 1.710 1.57 0.65 0.261 0.335 0.165 0.137
$Φ$ (mm) 1.25 1.1 1.14 0.94 0.73 0.885 0.80 0.80
T[0] (N) 781.6 520 598 552.6 267.9 593 680.5 522.8
f[1] (Hz) 36 37 87 245 547 523 1258 1328
m[H] (g) 5.9 NM 5.4 4.8 NM 4.2 7.02 6.96
x[H] (mm) 190 180 165 75 28 42 20 17
$xs$ (mm) 10 9 27 15 14 325 156 127
$μ=ρA$ (gm^−1) 51.1 32.5 8.01 5.45 3.22 4.83 3.95 3.95
$A=πΦ24, I=πΦ464, E=2.0×1011 Pa$
Note . D♯1 . D♯1 . F♯2 . C4 . C♯5 . C♯5 . D♯6 . E6 .
Piano JBS73 JBS50 JBS73 JBS73 JBS36 JBS73 BSD BSD
L (m) 1.717 1.710 1.57 0.65 0.261 0.335 0.165 0.137
$Φ$ (mm) 1.25 1.1 1.14 0.94 0.73 0.885 0.80 0.80
T[0] (N) 781.6 520 598 552.6 267.9 593 680.5 522.8
f[1] (Hz) 36 37 87 245 547 523 1258 1328
m[H] (g) 5.9 NM 5.4 4.8 NM 4.2 7.02 6.96
x[H] (mm) 190 180 165 75 28 42 20 17
$xs$ (mm) 10 9 27 15 14 325 156 127
$μ=ρA$ (gm^−1) 51.1 32.5 8.01 5.45 3.22 4.83 3.95 3.95
$A=πΦ24, I=πΦ464, E=2.0×1011 Pa$
In the simulations presented in this section, the hammer forces are given as input data. The amplitude, duration, and shape of the imposed force pulses are selected as close as possible to the
reality. In some cases, the hammer mass and acceleration were measured on real strings beforehand and, thus, the input force is obtained by multiplying together these two quantities. In the other
situations, the string-hammer coupling is modeled by a standard power law in which the coefficients are adjusted in order to obtain realistic hammer force amplitude and duration.^19,20 As it has been
already pointed out by several authors,^3,6 and as discussed further in Sec. IV of this paper, we are aware of the fact that these input hammer pulses might differ from the real forces, either
because of several causes of experimental errors (hammer misalignment, vibrations of the hammer shank,…), or because of the approximate modeling by a power law.^5 However, these differences do not
invalidate the testing of the reconstruction procedure presented here. In fact, this reconstruction is supposed to account for any hammer pulse, provided that the linear theory presented in Sec. II
is applicable, and for the appropriate wave filter (two-waves, four-waves, or eight-waves) compatible with the pulse duration τ[H].
In Sec. IIIB, it is analyzed to what extent realistic departures from the “ideal linear string,” in terms of absorption, dispersion, and nonlinear propagation, can affect the reconstruction of the
hammer force. A systematic study of the influence of each of these perturbation terms is presented. In Sec. IIIC, representative examples of hammer force reconstruction are presented in the three
main ranges of the piano: bass, medium, and treble, using the appropriate waves schemes.
B. Effects of some causes of departure from the ideal string on the reconstructed force
1. Absorption and damping
Consider a string with fundamental frequency $f1=1/T1$ and frequency-dependent time-constant $τ(f)$ due to damping. For each partial n, the attenuation due to damping during a roundtrip is
proportional to $1−exp[−T1/τ(fn)]$, which, for $T1≪τ$, reduces to $T1/τ$. An order of magnitude can be obtained by calculating this attenuation around the mid-frequency $fm$=1kHz. As an
example, we measured $τ(fm)$ =1.2, 0.7, and 0.2s, for the three strings D♯1, C4, and E6 examined here. This yields an attenuation of 2.3%, 0.6%, and 0.4%, respectively, for one roundtrip on these
three strings. In practice, the observed attenuation is smaller than these limits (<1%) since the distance traveled by the wave is less than twice the string length (see Fig. 6). As a consequence, no
correction was applied here since the attenuation error is weak compared to the other sources of error. Notice however that, if necessary, an additional filter can easily be designed for compensating
the frequency-dependent damping along the string.^21
2. Dispersion due to stiffness
At f=1kHz, the stiffness coefficient $εs$ defined in Sec. IIC takes the values 7.9×10^−2 for the string D♯1, 1.28×10^−2 for the string C4, and 2.3×10^−3 for the string E6. This results in
group delays (normalized to the period) equal to 0.12, 1.9×10^–2, and 3.4×10^–3, respectively, for strings D♯1, C4, and E6. As a consequence, dispersive effects are clearly seen on velocity
waveforms for the strings in the bass range, which results in a distortion of the wave, even for reduced propagation distances on the order of x[H] (see Fig. 6). For the two lowest octaves of most
pianos, it was thus necessary to design allpass filters in order to compensate for this dispersion (see Fig. 12). The dispersive effects decrease rapidly with the key number, and do not alter the
reconstruction of the force for the notes above A2 in all observed pianos.
3. Amplitude non-linearity
An appropriate strategy for evaluating the influence the non-linearity due to the amplitude of the string motion is to examine the nonlinear vibrating equation, which, to a first-order expansion, can
be written as^22
$∂2y∂t2≈c2∂2y∂x2[ 1+32cL2c2(∂y∂x)2 ],$
where $cL2=EA/μ$ is the longitudinal wave speed in the string of cross-sectional area A and linear density μ. An order of magnitude for the derivative $∂y/∂x$ is given by the ratio V/c between the
amplitude of the string velocity V and the transverse speed c. In total, the relevant dimensionless non-linearity coefficient can be written
This coefficient quantifies the change in transverse speed due to the amplitude of the wave, which might induce some distortion. Here, again, $εNL$ regularly decreases as the pitch of the note (or,
equivalently, the key number) increases. For V=1m/s (which, in most cases, corresponds to a mezzoforte level), $εNL$ takes the values 2.39×10^–2, 3.7×10^–3, and 2.83×10^–3 for the three
strings D♯1, C4, and E6, respectively. In practice, the effects of non-linearity on the force reconstruction were observed in the low bass range only, for V higher than 1m/s (see Fig. 6). Notice
that Eq. (17) accounts for the transverse effects of the amplitude non-linearity only, which is equivalent to a modulation of tension. The general nonlinear model used here for the simulations also
accounts for the longitudinal component of the string, which is not measured by the sensor.
In summary, the above dimensionless analysis of the three “perturbation” terms (damping, stiffness, amplitude non-linearity) in the ideal string wave equation shows that stiffness is likely to be the
main cause of errors in the force reconstruction. Nonlinear and damping effects are much weaker, although the errors due to non-linearity might rapidly increase with amplitude, since they are
proportional to the square of the string velocity. Another important result of this analysis is that the effects of these perturbation terms regularly decrease with the fundamental frequency of the
note. It is therefore expected that they are dominant in the bass range. The consequences on the reconstruction of the hammer force are shown in Sec. IIIC for some simulated notes in the bass,
medium, and treble ranges of the piano.
C. Examples of reconstructed hammer forces
1. Bass range. Two-wave scheme
Figure 6 shows an example of reconstructed hammer force for the string D♯1 of a pianoforte made by J. B. Streicher in 1873 (JBS73). The excitation corresponds to a level between forte and fortissimo,
with a maximum force of 32N, and a maximum string velocity of V=2.5m/s. A two-wave scheme is applied. Figure 6(a) shows that the reconstructed force perfectly coincides with the input force for a
simulated ideal string. The errors (in N) due to damping and nonlinear terms are shown in Fig. 6(b). The maximum discrepancy due to the damping terms is 0.02N, with corresponds to a relative error
of 6.0×10^–4, and can be ignored. The maximum error due to the presence of nonlinear terms in the string wave equation is equal to 0.36 N, or 1.13% in relative value. This error is comparable to
other potential causes of errors encountered in the application of the method to real strings (see Sec. IV). Finally, Fig. 6(c) shows the effects of stiffness on the reconstructed force. In this
case, the consequences of the dispersion are clearly seen, resulting in an underestimation of the maximum force (around 20% here), a slight broadening of the pulse, and the presence of characteristic
oscillations in the onset. The results presented here are representative of those observed in the two lowest octaves of the investigated pianos. The discrepancies due to stiffness and non-linearity
increase when approaching the lowest notes of the keyboard. They decrease when moving to the higher keys and, in general, as the amplitude of the hammer force decreases.
2. Medium range. Two-wave scheme
In this paragraph, representative examples of hammer force reconstruction in the medium range of the piano are presented. This range covers nearly three octaves (from note E2 to C5). In this interval
of notes, a two-wave reconstruction scheme is applied. The string parameters and the input force pulses are derived from measurements performed on a JBS73 piano (see Table I). In Fig. 7, the input
forces (solid lines) and the reconstructed forces (dashed lines) are shown for the notes F♯2 (fundamental 87Hz) and C4 (fundamental 245Hz). The reconstruction is made with the complete string
model, including damping, stiffness, and nonlinear terms. It can be seen that the hammer forces are very well reproduced in each case. The amplitudes, time duration, and shapes of the reconstructed
forces coincide with those of the input forces almost exactly. This means that the different causes of departure from the “ideal” string integrated in the simulation model do not affect the wave
propagation substantially. As a consequence, the effects of the additional terms (damping, stiffness, non-linearity) remain small. However, one exception should be underlined: as seen in Fig. 7, the
first pulse of the reconstructed hammer force for the note F♯2 shows a slightly more rounded profile than the input force, which results in a smaller amplitude by a few percent. This discrepancy is
due to the dispersive effects of stiffness.
3. Treble range. Four- and eight-wave schemes
Finally, reconstructed forces are presented here for the notes in the upper part of the keyboard. This range corresponds to the domain of notes where the period becomes smaller than the width of the
hammer pulse, as shown in Fig. 3. In this domain (with pitch higher than C5 for most pianos), four- and eight-wave schemes must be applied for reconstructing the hammer force. Figure 8 shows two
examples of reconstruction with the complete string model. The hammer force of the C♯5 string is reconstructed with a four-wave scheme, while the E6 force is reconstructed with an eight-wave scheme.
In both cases, no differences can be seen between input and reconstructed forces, which means that the perturbation terms have no visible effects.
In conclusion of this section, it is confirmed that the hammer forces can be reconstructed from simulated string velocities generated by a model that includes damping, stiffness, and nonlinear terms
in addition to the main inertial and tension terms. Depending on the notes, two-, four-, or eight-wave filters must be applied to the string velocity to recover the hammer force. As shown in Sec. II,
the choice of the filter depends on the ratio between the period of the string oscillation and the duration of the hammer pulse. In the low bass range, corresponding to the two lowest octaves for
most pianos, dispersion effects due to stiffness are more pronounced, which alters the quality of the reconstruction. In this case, the dispersion needs to be corrected by the use of an allpass
filter whose parameters are derived from the physical parameters of the string, as seen in Sec. IIC.
Due to the relatively small distance traveled by the waves along the string during its transient oscillation, the other causes of departure from the “ideal” “string” (damping, non-linearity) induce
only small effects in the reconstruction of the hammer force: the duration and shape of the pulse are not affected, and its magnitude is estimated within a few percent, or less, compared to the input
After validation on simulated string waveforms, this section shows some results obtained with the hammer force reconstruction method applied to real piano strings. In addition to damping, stiffness
and non-linearities, other physical phenomena, might alter the quality of the force reconstruction. The coupling between the strings of a given note, the whirling motion of the string, and lack of
accuracy in the velocity measurements, for example, might lead to wrong estimations. Therefore, it is challenging to test whether the proposed method is robust enough against these potential causes
of errors.
The general procedure is shown in Fig. 9. In the most general case (case 1), the hammer force is not known. The hammer force $fHR(t)$ is then reconstructed through filtering of the measured string
velocity $vsm(t)$, following the method presented in Sec. II. This force is used as input data in the string simulation model presented in Sec. III, which yields the simulated string velocity $vsi
(t)$, for further comparisons with $vsm(t)$, as shown in Sec. IVB. In the present study, an estimation of the measured hammer force $fHM(t)$ was also derived in some particular cases from
measurements of mass and acceleration of the hammer head. In these cases (case 2), this measured hammer force is compared to the reconstructed force, and also serves as input data in the string
simulation model, in order to compare the resulting string velocity $vSM(t)$ to both $vsm(t)$ and $vsi(t)$. This second procedure is detailed in Sec. IVC. The section ends with an example of
application of the method to the case of an historic piano with a Viennese action.
A. Measurements
Figure 10 shows a picture of the setup used for measuring the string velocity. The sensor is a standard electromagnetic transducer made of a coil wound onto a small cylindrical permanent magnet,
similar to an electric guitar pickup, and placed in the vicinity of the string. This transducer delivers a voltage proportional to the vertical component of the string velocity. The distance between
the magnet and the string is adjusted with a micrometer screw. The sensor and the screw can slide from string to string along a horizontal metallic bar placed over the instrument. A calibration
procedure is made prior to the measurements, where the signal delivered by the transducer is compared to the output of a laser vibrometer Polytec IVS-400 (Hörsching, Austria). The efficiency of the
transducer (in mV/m/s) is tested against string amplitude, frequency, and sensor-string distance. The efficiency is constant for a string amplitude between 20 and 300μm, and shows a slight increase
of 5% between 0.3 and 1.0mm. In our experiments, the sensor is placed either close to the bridge or to the agraffe, so that the string amplitude is <200μm.
B. Comparison between measured and simulated string velocity
We show here some representative results of the force reconstruction method applied to measured string velocities. This is supposed to correspond to the standard, and most useful, application of the
method. In the absence of measured hammer force, the validity of the reconstructed force is tested by comparing the measured string velocity with simulations where the input signal is the
reconstructed force (see Fig. 9).
Figure 11 shows the results of the hammer force reconstruction method applied to the string C♯5 of a J. B. Streicher piano built in 1836 (JBS36). It can be seen that the string velocity simulated
with the reconstructed force reproduces the measured waveform almost perfectly, even on a large time scale. This is confirmed by the spectral analysis performed on the first 50ms of the signal,
which shows a high degree of coincidence between 0 and 5kHz. A DC-component at −30dB below the maximum can be seen on the measured spectrum.
In the bass range, an allpass filter is applied to the measured string velocity in order to compensate for the dispersion due to stiffness. Figure 12(b) shows the effect of this filter on the hammer
pulse when the dispersion is reduced, which is comparable to the results obtained on the simulations in Sec. IIIC when the stiffness term is removed from the string equation. The string velocity
simulated with the filtered reconstructed force shows a good agreement with the measured velocity over a large time scale (35ms) despite some discrepancies in the amplitudes of some
sub-oscillations. These discrepancies could be due to some inaccuracy in the modeling of the frequency-dependent damping, always a delicate task, especially in the bass range. However, the comparison
between the two corresponding spectra shows an excellent agreement. The major differences are seen at the peaks of small amplitudes, around -20dB below the maximum (0dB), and can be attributed
either to some small errors in the estimation of the string parameters (damping) or in the measurements (precise location of the string sensor).
C. Comparison between reconstructed and measured force
Figure 13 shows an example of hammer force reconstructed from measurements of the string velocity for the note F♯2 of a J. B. Streicher piano (1873). The reconstructed force is compared to the
measured force derived from measurements of mass and acceleration of the hammer head. One can see that measured and reconstructed forces coincide during the first pulse of the waveform. However, some
differences exist in the magnitude of the second pulse. In addition, the measured force tends faster to zero and the remaining oscillations of the hammer head are clearly visible. Both forces are
used as input forces for simulating the string velocities, using the measured parameters shown in Table I. The simulated string velocities are then compared to the measured velocity. Here, again, one
can see an excellent agreement between the measured velocity and the velocity simulated with the reconstructed force, even on a long period of time. The simulations performed with the reconstructed
force as input quantity yields waveforms and spectra closer to the measurements than the simulations using measured hammer head acceleration and mass at the input. This is not surprising since the
reconstruction filter does not take the vibrations of the hammer shank into account.
In the treble range, Fig. 14 shows the results of the hammer force reconstruction method applied to the string D♯6 of a Bösendorfer piano with an eight-wave filtering scheme. The comparison between
the reconstructed force and the measured force obtained through multiplication of mass and acceleration of the hammer head again shows that this latter is affected by the oscillations of the head +
shank system, resulting in differences in duration and shape of the main pulse. Again, and for the same reasons as previously, the string velocity simulated with the reconstructed force shows a
better agreement with the measured velocity that the velocity simulated with the measured force truncated to its first positive pulse. This agreement is confirmed by observing the velocities over a
larger period of time (8ms) and on the spectra, where only small differences are visible below 5kHz.
In summary, comparing the hammer pulses derived from the present method with hammer head mass-acceleration measurements shows that the present method is insensitive to the perturbing oscillations of
the hammer shank. Its main advantage also follows from the fact that it does not need to have direct access to the hammer: this feature is of the utmost importance when measuring valuable and fragile
instruments such as historic pianofortes.
D. Application to historic pianofortes
One running application of the present method is to investigate the evolution of hammer-string interaction in the history of piano making. In this respect, Fig. 15–(a) shows an example of a
reconstructed hammer force (note D3) for a copy recently made by G. Hecher of a pianoforte built by Nanette Streicher in 1805 (GH05). This instrument uses a Viennese action,^23 and 200-yr-old leather
is glued on the hammer heads. From the qualitative comparison with the hammer acceleration measured of a modern piano in the same half-octave (Steinway D, note G3) in Fig. 15(b), one can see that
both shapes are globally similar. However, close observation shows some differences in the slopes and rounded profiles of the pulses, which are likely to be attributable to the respective behavior of
leather and felt. Also interesting is the comparison between measured and simulated velocity for the GH05-D3 string in Fig. 15(c). The first part of both waveforms are identical, but significant
differences now exist in the second part where the measured oscillations are significantly damped compared to the velocity simulated with the reconstructed force at the input. A very plausible
explanation for this difference lies in the specificity of the Viennese action, where the hammer slips along the string before leaving it, a feature that is not taken into account in the
reconstruction model.
In this paper, an original method was presented for reconstructing the piano hammer force from measurements of the string velocity. This method is based on the use of dedicated filters applied to the
measured velocity. The design of these filters is derived from the analysis of wave propagation along a lossless linear string. For the five lowest octaves of a piano, a filter based on two-wave
propagation (one incident and one reflected wave) is sufficient for reconstructing the hammer force. For the highest notes, a four- or an eight-wave scheme is necessary. The number of waves to
consider is imposed by the duration of the hammer pulse. The velocity sensor must be placed either near the agraffe (two-wave scheme) or near the bridge (four- and eight-wave schemes) in order to
ensure that the reconstructed force is not perturbed by the poles of the filter.
In a first step, the force reconstruction method has been validated on simulated piano strings, where the model includes absorbing, dispersive, and nonlinear terms. The input parameters of these
simulations were derived from measurements on a large variety of historic and modern pianos. The results show that both the damping and nonlinear terms only have small effects on the estimation of
the hammer force. On the contrary, the dispersion due to stiffness leads to significant discrepancies in the two lowest octaves, which needs to be compensated by means of a suitable allpass filter.
In a second step, the force reconstruction method is tested on velocity measurements performed on real piano strings. The measured string velocity is compared to the corresponding simulated velocity
excited by the reconstructed force. It is found that the reconstructed hammer force yields a simulated string velocity that reproduces the measured velocity with a high degree of accuracy, both in
time and frequency. This proves that the simplified string model used for designing the reconstruction filters accounts well for the observed wave propagation during the first oscillations after the
hammer stroke. During this small time interval, known complicating effects, such as the whirling motion of the strings and the coupling between adjacent strings of the same note, are not yet
initiated and do not perturb the reconstruction of the force. In this context, it would be of interest to compare the results with methods requiring the use of longer portions of velocity signal as,
for example, those based on the use of direct and inverse Fourier transforms in the frequency domain. Comparisons of the reconstructed force with the results obtained by other experimental methods
(such as optical methods and high-speed cameras) also remain to be done in order to completely assess the pertinence of the present filtering.
In some cases, the reconstructed force was compared to an estimate of the hammer force obtained by multiplying together the measured mass and acceleration of the hammer head. These comparison shows
reasonable similarities in terms of amplitude, shape, and duration, but differences are observed, which are mainly attributable to the oscillations of the hammer shank. Simulations of the string
velocity performed with this measured force at the input of the present filters show more discrepancies than with the reconstructed force when compared to the measured velocity: such a result is not
surprising if we consider that the dynamics of the shank are not included in the underlying model on which the reconstruction is based. In this context, extending the method to a filter with a
transfer function between hammer acceleration and string velocity, which would include a model of shank, could be an interesting perspective.
Finally, the reconstruction method is applied to a copy of an historic piano equipped with a particular action (the so-called Viennese action). Qualitative comparison between the reconstructed force
and the hammer acceleration measured on a neighboring note on a modern piano shows some differences that might be due to distinct compression behavior of leather and felt, respectively. The
comparison between measured and simulated string velocity of the D3 note played on the historic piano also shows interesting features that are likely attributable to the specificity of the Viennese
action, where the hammer slips along the string before leaving it. This is known to produce a softer sound than modern pianos.
This project was supported by a Lise-Meitner-Fellowship of the Austrian Science Fund (FWF; Project number M 1653-N30). The author wishes to thank Alex Mayer (University of Music and Performing Arts
Vienna), Caroline Haas and Michael Kirchweger (Technical Museum Vienna), and Gert Hecher (Das Klavier-Atelier, Vienna) for their valuable help in the measurements. The simulations presented in this
paper were carried out using the Plateforme Fédérative pour la Recherche en Informatique et Mathématiques (PLAFRIM) experimental platform, being developed under the Inria PlaFRIM development action
with support from Institut Polytechnique de Bordeaux, Laboratoire Bordelais de Recherche en Informatique, and Institut de Mathématiques de Bordeaux and other entities: Conseil Régional d'Aquitaine,
Université de Bordeaux, and Centre National de la Recherche Scientifique (and Agence Nationale de la Recherche) in accordance with the programme d'investissements d'avenir (see http://www.plafrim.fr/
). The author is indebted to Juliette Chabassier and Marc Duruflé for their assistance in using the PlaFRIM environment.
1. Wave analysis
For x[s]<x[H], the pulses generated by the hammer reach the sensor at the successive instants of time
$t1=(xH−xs)/c; t2=(xH+xs)/c; t3=(2L−xH−xs)/c; t4=(2L−xH+xs)/c.$
For a hammer pulse of width τ[H], the condition for non-overlapping with the third pulse within the time interval τ[H] is given by
Under this condition, the velocity signal is the sum of two waves between t[1] and $t1+τH$. For x[s]>x[H] and $L−xs<xH$ (sensor close to the bridge), we have
$t1=(xs−xH)/c; t2=(2L−xH−xs)/c; t3=(xH+xs)/c; t4=(2L+xH−xs)/c.$
The subsequent arrival times are such that $ti+4=ti+2L/c$. The condition of non-overlapping with the fifth pulse between t[1] and τ[H] then becomes
Under this condition, the hammer force can be reconstructed with the velocity represented by four waves. Finally, the condition of non-overlapping with the ninth pulse within the interval τ[H] is
For $T1<τH<2T1$, the force can then be reconstructed with the velocity signal represented by eight waves.
2. Reconstruction filter
The transfer function between string velocity and hammer force for a lossless nondispersive struck string expressed in Eq. (6) can be conveniently obtained with the help of the dual delay-line model
shown in Fig. 16, adapted from the plucked string model with pickup by Karjalainen et al.^13 The upper delay line accounts for the wave propagation in the direction from agraffe (A1) to bridge (B1),
through hammer (H1) and sensor (S1) positions. The lower delay line accounts for the propagation from bridge (B2) to agraffe (A2). The hammer force F[H] gives rise to two pulses $FH1$ and $FH2$ at
position x[H], and the string velocity at position x[s] is the sum of the contributions $Vs1$ and $Vs2$ from both lines. In the case x[H]<x[s], the delays (in samples) between the various points
$nH=xHfe/c; ns−nH=(xs−xH)fe/c; nL−ns=(L−xs)fe/c.$
R[a] and R[b] are the reflection coefficients at agraffe and bridge, respectively. On the right-hand side of the lines, we can write
$Vs=Vs1+Vs2=Vs1[ 1+Rb z−2(nL−ns) ].$
Denoting $VH1$ the string velocity at the hammer position traveling to the right, we can write at the sensor position
$Vs1=VH1 z−(ns−nH)+RaRb z−2nL Vs1.$
Finally, at the hammer position, we can write
$VH1=12FHZc[ 1+Ra z−2nH ].$
From Eqs. (A7)–(A9), we get
$HVF=VsFH=12Zcz−n1[ 1+Ra z−m1 ][ 1+Rb z−m2 ]1−RaRb z−m3,$
with $n1=ns−nH, m1=2nH, m2=2(nL−ns)$ and $m3=2nL$. For perfect reflecting ends, we have $Ra=Rb=−1$. For dissipative ends, we can write $Ra=−g1$ and $Rb=−g2$, with $| g1 |,| g2 |<1$.
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Design and tone in the mechanoacoustic piano. Part III. Piano strings and scale design
J. Acoust. Soc. Am.
J. O.
Introduction to Digital Filters with Audio Applications
W3K Publishing
Palo Alto, CA
), Chap. 3, pp.
. Available at
(Last viewed 3/17/2016).
, and
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Plucked-string models: From the Karplus-Strong algorithm to digital waveguides and beyond
Comput. Music J.
J. S.
, and
J. O.
, “
Robust, efficient design of allpass filters for dispersive string sound synthesis
IEEE Signal Proc. Lett.
J. S.
J. O.
, “
Robust design of very high-order allpass dispersion filters
,” in
Proceedings of the 9th Int. Conference on Digital Audio Effects
, Montreal (
), pp.
, “
Recursive digital filters with maximally flat group delay
IEEE Trans. Circuit Theory
, and
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Time domain simulation of a piano. Part 1: Model description
ESAIM: Math. Modell. Numer. Anal.
P. M.
K. U.
Theoretical Acoustics
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), Chap.
, “
Acoustics of pianos
Appl. Acoust.
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Experimental and computational studies of piano hammers
Acta Acust. Acust.
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Comput. Music J.
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Large amplitude damped free vibration of a stretched string
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The check in some early pianos and the development of piano technique around the turn of the 18th century
Early Music | {"url":"https://pubs.aip.org/asa/jasa/article/140/5/3504/679284/Reconstruction-of-piano-hammer-force-from-string","timestamp":"2024-11-03T19:06:02Z","content_type":"text/html","content_length":"430531","record_id":"<urn:uuid:9589fb45-13b8-40f5-9bac-2d94788d001a>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00098.warc.gz"} |
Automorphisms of Manifolds and Algebraic $K$-Theory: Part IIIsearch
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Automorphisms of Manifolds and Algebraic $K$-Theory: Part III
eBook ISBN: 978-1-4704-1720-8
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Automorphisms of Manifolds and Algebraic $K$-Theory: Part III
eBook ISBN: 978-1-4704-1720-8
Product Code: MEMO/231/1084.E
List Price: $71.00
MAA Member Price: $63.90
AMS Member Price: $42.60
• Memoirs of the American Mathematical Society
Volume: 231; 2014; 110 pp
MSC: Primary 57; Secondary 19
The structure space \(\mathcal{S}(M)\) of a closed topological \(m\)-manifold \(M\) classifies bundles whose fibers are closed \(m\)-manifolds equipped with a homotopy equivalence to \(M\). The
authors construct a highly connected map from \(\mathcal{S}(M)\) to a concoction of algebraic \(L\)-theory and algebraic \(K\)-theory spaces associated with \(M\). The construction refines the
well-known surgery theoretic analysis of the block structure space of \(M\) in terms of \(L\)-theory.
□ Chapters
□ 1. Introduction
□ 2. Outline of proof
□ 3. Visible $L$-theory revisited
□ 4. The hyperquadratic $L$–theory of a point
□ 5. Excision and restriction in controlled $L$–theory
□ 6. Control and visible $L$-theory
□ 7. Control, stabilization and change of decoration
□ 8. Spherical fibrations and twisted duality
□ 9. Homotopy invariant characteristics and signatures
□ 10. Excisive characteristics and signatures
□ 11. Algebraic approximations to structure spaces: Set-up
□ 12. Algebraic approximations to structure spaces: Constructions
□ 13. Algebraic models for structure spaces: Proofs
□ A. Homeomorphism groups of some stratified spaces
□ B. Controlled homeomorphism groups
□ C. $K$-theory of pairs and diagrams
□ D. Corrections and Elaborations
• Permission – for use of book, eBook, or Journal content
• Book Details
• Table of Contents
• Requests
Volume: 231; 2014; 110 pp
MSC: Primary 57; Secondary 19
The structure space \(\mathcal{S}(M)\) of a closed topological \(m\)-manifold \(M\) classifies bundles whose fibers are closed \(m\)-manifolds equipped with a homotopy equivalence to \(M\). The
authors construct a highly connected map from \(\mathcal{S}(M)\) to a concoction of algebraic \(L\)-theory and algebraic \(K\)-theory spaces associated with \(M\). The construction refines the
well-known surgery theoretic analysis of the block structure space of \(M\) in terms of \(L\)-theory.
• Chapters
• 1. Introduction
• 2. Outline of proof
• 3. Visible $L$-theory revisited
• 4. The hyperquadratic $L$–theory of a point
• 5. Excision and restriction in controlled $L$–theory
• 6. Control and visible $L$-theory
• 7. Control, stabilization and change of decoration
• 8. Spherical fibrations and twisted duality
• 9. Homotopy invariant characteristics and signatures
• 10. Excisive characteristics and signatures
• 11. Algebraic approximations to structure spaces: Set-up
• 12. Algebraic approximations to structure spaces: Constructions
• 13. Algebraic models for structure spaces: Proofs
• A. Homeomorphism groups of some stratified spaces
• B. Controlled homeomorphism groups
• C. $K$-theory of pairs and diagrams
• D. Corrections and Elaborations
Permission – for use of book, eBook, or Journal content
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Math Test #1 Solutions
(a) (3 + 2x)(1 - 3x) = (6x - 1)(1 - x),
(b) x(10x - 11) = 6
Solutions: (a) Expand both sides of the equation to obtain:
(3+2x)(1-3x) = (6x-1)(1-x) ^2 = 6x-6x^2-1+x
and so the solution written in set notation is
(b) Using the distributive property again we have
x(10x - 11) = 6 ^2 - 11x = 6 ^2 - 11x - 6 = 0:
Since the factorization method is not easy to apply here and also
the method of completing the square is still not very easy we apply
the quadratic formula a = 10, b = -11, c = -6:
(c) Eliminating the denominators we have
then after checking we see the only solution is
Solutions: (a) Squaring both sides, after isolating the radical ex-
pression, we get just an implication but good enough to see how the
solutions might look like:
or x = 30 and after checking we see that
(b) Using the same method we get
32x^2 - x - 3 = 0.
Using the quadratic formula
of which gives only one solution
relative to the water in going 8 miles upstream and then returning.
The total time for the trip was 1 hours. Use this information to find
the speed of the current.
(b) If instead of knowing that it took 1 hours for the round trip
we know that it took 20 minutes more to go upstream than to go
downstream what would be the speed of the current in this case?
Solutions: (a) If we denote by x the speed of current in miles per
hour we get that the time going upstream is
time going downstream is
hour round trip we obtain the equation:
Solving this for x by adding to the same common denominator we
which gives
(b) Since 20 minutes is 1/3 of an hour. In this case the equation
which implies
This gives only the solution
terval notation:
Solutions: (a) The inequality is equivalent to
which means
(b) The inequality becomes
(c) In this case we have similarly
(d) Since we are dealing with positive numbers the inequality is
equivalent to
which gives
x^2 + y^2 + 39x - 80y = 0.
(b) What are the coordinates of the x-intercepts ?
Solutions: (a) The equation can be written after completing the
which gives the center
(b) Making y = 0 in x^2+y^2+39x-80y = 0 we obtain x^2+39x = 0
which leads to two points of intersection:
Similarly for the y-intercept we get:
of this circle is included below:
the point of coordinates (3,-2) and parallel to the line of equation
7x + 5y - 3 = 0.
Solutions: The slope of the given line is m =
equation of the line we are looking for is y - (-2) = (-7/5)(x - 3)
Solutions: (a) We need to impose the condition:
(b) In this case we have:
1. 2x - 1 ≥ 0 or
to 2x - 1 = x^2 or 0 = x^2 - 2x + 1 (completing the square)
(x - 1)^2 = 0. We have only one non-negative solution: x = 1. | {"url":"https://mathinput.com/math-test-1-solutions.html","timestamp":"2024-11-10T21:28:53Z","content_type":"text/html","content_length":"91183","record_id":"<urn:uuid:2bb4d1dd-c9ac-4209-832d-ae4cd36190ba>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00115.warc.gz"} |
Translating and Solving Word Problems and Applications
Learning Outcomes
• Translate word phrases to equations and solve
• Translate and solve applications
In previous chapters, we translated word phrases into equations. This skill will help you when you solve word problems. Previously, you translated phrases into expressions, and now we will translate
phrases into mathematical equations so we can solve them. But before we get started, let’s define some important terminology:
• variables: Variables are symbols that stand for an unknown quantity, often represented with letters, like [latex]x, y[/latex], or [latex]z[/latex].
• coefficient: Sometimes a variable is multiplied by a number. This number is called the coefficient of the variable. For example, the coefficient of [latex]3x[/latex] is [latex]3[/latex].
• term: A single number, or variables and numbers connected by multiplication. For example, [latex]-4, 6x[/latex], and [latex]x^2[/latex] are all terms.
• expression: Groups of terms connected by addition and subtraction. For example, [latex]2x^2-5[/latex] is an expression.
• equation: An equation is a mathematical statement that two expressions are equal. An equation will always contain an equal sign with an expression on each side. Think of an equal sign as meaning
“the same as”. Some examples of equations are [latex]y = mx +b[/latex], [latex]\frac{3}{4}r = v^{3} - r[/latex], and [latex]2(6-d) + f(3 +k) = \frac{1}{4}d[/latex] .
The following figure shows how coefficients, variables, terms, and expressions all come together to make equations. In the equation [latex]2x-3^2=10x[/latex], the variable is [latex]x[/latex], a
coefficient is [latex]10[/latex], a term is [latex]10x[/latex], and an expression is [latex]2x-3^2[/latex].
Translate Phrases into Equations
The first step in translating phrases into equations is to look for the word (or words) that translate(s) to the equal sign. The table below reminds us of some of the words that translate to the
equal sign.
Equals (=)
is is equal to is the same as the result is gives was will be
Let’s review the steps we used to translate a sentence into an equation.
Translate a word sentence to an algebraic equation.
1. Locate the “equals” word(s). Translate to an equal sign.
2. Translate the words to the left of the “equals” word(s) into an algebraic expression.
3. Translate the words to the right of the “equals” word(s) into an algebraic expression.
In our first example, we will translate and solve a one-step equation.
Translate and solve: five more than [latex]x[/latex] is equal to [latex]26[/latex].
Five more than [latex]x[/latex] [latex]\Rightarrow\quad{x+5}[/latex]is equal to [latex]\Rightarrow\quad{=}[/latex]
Translate. [latex]26[/latex] [latex]\Rightarrow\quad{26}[/latex]
Subtract 5 from both sides. [latex]x+5\color{red}{-5}=26\color{red}{-5}[/latex]
Simplify. [latex]x=21[/latex]
Check:Is [latex]26[/latex] five more than [latex]21[/latex] ?
The solution checks.
Translate and solve: The difference of [latex]5p[/latex] and [latex]4p[/latex] is [latex]23[/latex].
Watch this video for more examples of how to translate a phrase into an equation, then solve it.
Translate and Solve Applications
In most of the application problems we solved earlier, we were able to find the quantity we were looking for by simplifying an algebraic expression. Now we will be using equations to solve
application problems. We’ll start by restating the problem in just one sentence, then we’ll assign a variable, and then we’ll translate the sentence into an equation to solve. When assigning a
variable, choose a letter that reminds you of what you are looking for.
The Robles family has two dogs, Buster and Chandler. Together, they weigh [latex]71[/latex] pounds.
Chandler weighs [latex]28[/latex] pounds. How much does Buster weigh?
Devise a problem-solving strategy.
1. Read the problem. Make sure you understand all the words and ideas.
2. Identify what you are looking for.
3. Name what you are looking for. Choose a variable to represent that quantity.
4. Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebra equation.
5. Solve the equation using good algebra techniques.
6. Check the answer in the problem and make sure it makes sense.
7. Answer the question with a complete sentence.
Let’s take a look at the problem-solving strategy in action.
Shayla paid $[latex]24,575[/latex] for her new car. This was $[latex]875[/latex] less than the sticker price. What was the sticker price of the car?
Now you can try translating an equation from a statement that represents subtraction.
In the following video, you will see another example of how to translate a phrase into an equation and solve. | {"url":"https://courses.lumenlearning.com/wm-developmentalemporium/chapter/applications-of-the-addition-and-subtraction-properties/","timestamp":"2024-11-11T15:10:54Z","content_type":"text/html","content_length":"60968","record_id":"<urn:uuid:720ac771-ced2-4e22-ba4f-757af6f05f99>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00693.warc.gz"} |
Post Correspondence Problem - (Theory of Recursive Functions) - Vocab, Definition, Explanations | Fiveable
Post Correspondence Problem
from class:
Theory of Recursive Functions
The Post Correspondence Problem (PCP) is a decision problem in theoretical computer science that involves finding a sequence of pairs of strings such that when the first components of these pairs are
concatenated, they equal the concatenation of the second components. This problem is significant because it was one of the first problems proven to be undecidable, meaning there is no algorithm that
can solve all instances of it. PCP is closely related to the concepts of formal languages and Turing machines, serving as a critical example in demonstrating the limits of computational solvability.
congrats on reading the definition of Post Correspondence Problem. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. The Post Correspondence Problem was introduced by Emil Post in 1946 and serves as a fundamental example in computability theory.
2. PCP is undecidable because there is no general algorithm that can solve all instances; some configurations will always lead to an infinite search without resolution.
3. The problem can be represented using pairs of strings, which can be visualized as tiles where each tile has a top and bottom string.
4. PCP has connections to other undecidable problems, such as the halting problem, illustrating broader implications in theoretical computer science.
5. A specific instance of PCP can often be constructed from finite automata or context-free grammars, demonstrating its relation to formal languages.
Review Questions
• How does the Post Correspondence Problem illustrate the concept of undecidability in computational theory?
□ The Post Correspondence Problem exemplifies undecidability by showing that no algorithm can be constructed to solve every possible instance of the problem. This means that for some
configurations, it is impossible to determine whether a solution exists, reflecting limits on what can be computed. As a result, PCP provides a concrete example used to understand the
boundaries of algorithmic solvability in computer science.
• Discuss the implications of the Post Correspondence Problem on understanding Turing machines and their limitations.
□ The Post Correspondence Problem highlights significant limitations of Turing machines by showcasing a specific scenario where these machines cannot provide an answer. Since PCP is
undecidable, it shows that even powerful computational models like Turing machines have boundaries beyond which they cannot operate. This relationship reinforces key concepts in computability
theory about what problems are solvable and which are fundamentally beyond algorithmic reach.
• Evaluate how the Post Correspondence Problem relates to other known undecidable problems like the halting problem and its broader impact on theoretical computer science.
□ The Post Correspondence Problem is crucial in connecting with other undecidable problems such as the halting problem, as both illustrate fundamental limits within computation. By analyzing
PCP alongside these problems, researchers gain insights into the nature of algorithmic computation and its boundaries. This understanding impacts theoretical computer science by establishing
criteria for recognizing undecidable problems and shaping future research directions aimed at exploring new computational theories and models.
"Post Correspondence Problem" also found in:
© 2024 Fiveable Inc. All rights reserved.
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12.8.1 Exercise: Projected Available Inventory Calculation
Intended learning outcomes: Calculate projected available inventory by completing a grid.
This exercise refers to Subsection 12.1.2.
Complete the grid in Figure 12.8.1.1.
Fig. 12.8.1.1 Projected available inventory calculation
a. What is the available inventory without any restrictions along the time axis?
b. What is the additional available inventory after order 102 9538?
c. Which receipt could be deferred?
d. Furthermore, the following orders are planned:
• Customer order ID 104 2158 of 500 units on January 20
• Stock replenishment order ID 104 3231 of 500 units on January 22
Does this situation lead to a problem? If so, how can it be solved?
a. 50
b. 300 (= 350 – 50)
c. Stock replenishment order ID 101 2897 could be deferred to Jan. 14.
d. Yes, there will not be enough available inventory on Jan. 20. Expediting order ID 104 3231 by at least two days could solve this problem.
Course section 12.8: Subsections and their intended learning outcomes | {"url":"https://opess.ethz.ch/course/section-12-8/12-8-1-exercise-projected-available-inventory-calculation/","timestamp":"2024-11-05T16:42:08Z","content_type":"text/html","content_length":"147941","record_id":"<urn:uuid:38714774-0ca6-4253-81b6-3a3de650e0ba>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00494.warc.gz"} |
Calculation of the flow rate of a rock-ramp pass
The calculation of the flow rate of a rock-ramp pass corresponds to the implementation of the algorithm and the equations present in Cassan et al. (2016)^1.
General calculation principle
After Cassan et al., 2016^1
There are three possibilities:
• the submerged case when \(h \ge 1.1 \times k\)
• the emergent case when \(h \le k\)
• the quasi-emergent case when \(k < h < 1.1 \times k\)
In the quasi-emergent case, the calculation of the flow corresponds to a transition between emergent and submerged case formulas:
\[Q = a \times Q_{submerge} + (1 - a) \times Q_{emergent}\]
with \(a = \dfrac{h / k - 1}{1.1 - 1}\)
Submerged case
The calculation is done by integrating the velocity profile in and above the macro-roughnesses. The calculated velocities are the temporal and spatial averages per plane parallel to the bottom.
In macro-roughnesses, velocities are obtained by double averaging the Navier-Stokes equations in uniform regime with a mixing length model for turbulence.
Above the macro-roughnesses, the classical turbulent boundary layer analysis is maintained. The velocity profile is continuous at the top of the macro-roughnesses and is dependent on the boundary
conditions set by the hydraulics:
• velocity at the bottom (without turbulence) in m/s:
\[u_0 = \sqrt{2 g S D (1 - \sigma C)/(C_d C)}\]
• total shear stress at the top of the roughnesses in m/s:
\[u_* = \sqrt{gS(h-k)}\]
The average bed velocity is given by integrating the flows between and above the blocks:
\[\bar{u} = \frac{Q_{inf} + Q_{sup}}{h}\]
with respectively \(Q_{inf}\) and \(Q_{sup}\) the unit flows for the part in the canopy and the part above the canopy.
Calculation of the unit flow rate Q[inf] in the canopy
The flow in the canopy is obtained by integrating the velocity profile (Eq. 9, Cassan et al., 2016):
\[Q_{inf} = \int_{0}^1 u(\tilde{z}) d \tilde{z}\]
\[u(\tilde{z}) = u_0 \sqrt{\beta \left( \frac{h}{k} -1 \right) \frac{\sinh(\beta \tilde{z})}{\cosh(\beta)} + 1}\]
\[\beta = \sqrt{(k / \alpha_t)(C_d C k / D)/(1 - \sigma C)}\]
\[C_d = C_{x} f_{h_*}(h_*)\]
and \(\alpha_t\) obtained by solving the following equation:
\[\alpha_t u(1) - l_0 u_* = 0\]
\[l_0 = \min \left( s, 0.15 k \right)\]
\[s = D \left( \frac{1}{\sqrt{C}} - 1 \right)\]
Calculation of the unit flow Q[sup] above the canopy
\[Q_{sup} = \int_k^h u(z) dz\]
with (Eq. 12, Cassan et al., 2016)
\[u(z) = \frac{u_*}{\kappa} \ln \left( \frac{z - d}{z_0} \right)\]
with (Eq. 14, Cassan et al., 2016)
\[z_0 = (k - d) \exp \left( {\frac{-\kappa u_k}{u_*}} \right)\]
and (Eq. 13, Cassan et al., 2016)
\[ d = k - \frac{\alpha_t u_k}{\kappa u_*}\]
which gives
\[Q_{sup} = \frac{u_*}{\kappa} \left( (h - d) \left( \ln \left( \frac{h-d}{z_0} \right) - 1\right) - \left( (k - d) \left( \ln \left( \frac{k-d}{z_0} \right) - 1 \right) \right) \right)\]
Emerging case
The calculation of the flow rate is done by successive iterations which consist in finding the flow rate value allowing to obtain the equality between the flow velocity \(V\) and the average velocity
of the bed given by the equilibrium of the friction forces (bottom + drag) with gravity:
\[u_0 = \sqrt{\frac{2 g S D (1 - \sigma C)}{C_d f_F(F) C (1 + N)}}\]
\[N = \frac{\alpha C_f}{C_d f_F(F) C h_*}\]
\[\alpha = 1 - (a_y / a_x \times C)\]
Formulas used
Bulk velocity V
\[V = \frac{Q}{B \times h}\]
Average speed between blocks V[g]
From Eq. 1 Cassan et al (2016)^1 and Eq. 1 Cassan et al (2014)^2:
\[V_g = \frac{V}{1 - \sqrt{(a_x/a_y)C}}\]
Drag coefficient of a single block C[d0]
\(C_{d0}\) is the drag coefficient of a block considering a single block infinitely high with \(F << 1\) (Cassan et al, 2014^2).
Block shape Cylinder "Rounded face" shape Square-based parallelepiped "Flat face" shape
Value of \(C_{d0}\) 1.0 1.2-1.3 2.0 2.2
When establishing the statistical formulae for the 2006 technical guide (Larinier et al. 2006^4), the definition of the block shapes to be tested was based on the use of quarry blocks with neither
completely round nor completely square faces. The so-called "rounded face" shape was thus not completely cylindrical, but had a trapezoidal bottom face (seen in plan). Similarly, the "flat face"
shape was not square in cross-section, but also had a trapezoidal bottom face. These differences in shape between the "rounded face" and a true cylinder on the one hand, and the "flat face" and a
true parallelepiped with a square base on the other hand, result in slight differences between them in the shape coefficients \(C_{d0}\).
Block shape coefficient σ
Cassan et al. (2014)^2, et Cassan et al. (2016)^1 define \(\sigma\) as the ratio between the block area in the \(x,y\) plane and \(D^2\). For the cylindrical form of the blocks, \(\sigma\) is equal
to \(\pi / 4\) and for a square block, \(\sigma = 1\).
Ratio between the average speed downstream of a block and the maximum speed r
The values of \(r\) depends on the block shapes (Cassan et al., 2014^2 et Tran et al. 2016 ^3):
• round : \(r_Q=1.1\)
• "rounded face" shape : \(r=1.2\)
• square-based parallelepiped : \(r=1.5\)
• "flat face" shape : \(r=1.6\)
Cassiopée implements a formula depending on \(C{d0}\):
\[ r = 0.4 C_{d0} + 0.7 \]
Froude F
\[F = \frac{V_g}{\sqrt{gh}}\]
Froude-related drag coefficient correction function f[F](F)
If \(F < 1\) (Eq. 19, Cassan et al., 2014^2):
\[f_F(F) = \min \left( \frac{r}{1- \frac{F_{g}^{2}}{4}}, \frac{1}{F^{\frac{2}{3}}} \right)^2\]
otherwise \(f_F(F) = 1\) because a torrential flow upstream of the blocks is theoretically impossible because of the hydraulic jump caused by the downstream block.
Maximum speed u[max]
According to equation 19 of Cassan et al, 2014^2 :
\[ u_{max} = V_g \sqrt{f_F(F)} \]
Drag coefficient correction function linked to relative depth f[h*](h[*])
The equation used in Cassiopeia differs slightly from equation 20 of Cassan et al. 2014^2 and equation 6 of Cassan et al. 2016^1. This formula is a fit to the experimental measurements on circular
blocks used in Cassan et al. 2016^1:
\[ f_{h_*}(h_*) = (1 + 1 / h_*^{2}) \]
Coefficient of friction of the bed Cf
If \(k_s < 10^{-6} \mathrm{m}\) then we use Blasius' formula
\[C_f = 0.3164 / 4 * Re^{-0.25}\]
\[Re = u_0 \times h / \nu\]
Else (Eq. 3, Cassan et al., 2016 d'après Rice et al., 1998^5)
\[C_f = \frac{2}{(5.1 \mathrm{log} (h/k_s)+6)^2}\]
• \(\alpha\): ratio of the area affected by the bed friction to \(a_x \times a_y\)
• \(\alpha_t\): length scale of turbulence in the block layer (m)
• \(\beta\): ratio between drag stress and turbulence stress
• \(\kappa\): Von Karman constant = 0.41
• \(\sigma\): ratio between the block area in the plane X,y et \(D^2\)
• \(a_x\): cell width (perpendicular to the flow) (m)
• \(a_y\): length of a cell (parallel to the flow) (m)
• \(B\): pass width (m)
• \(C\): blocks concentration
• \(C_d\): drag coefficient of a block under current flow conditions
• \(C_{d0}\): drag coefficient of a block considering an infinitely high block with \(F \ll 1\)
• \(C_f)\): bed friction coefficient
• \(d\): displacement in the zero plane of the logarithmic profile (m)
• \(D\): width of the block facing the flow (m)
• \(F\): Froude number based on \(h\) and \(V_g\)
• \(g\): acceleration of gravity = 9.81 m.s^-2
• \(h\): average depth (m)
• \(h_*\): dimensionless depth (\(h / D\))
• \(k\): useful block height (m)
• \(k_s\): roughness height (m)
• \(l_0\): length scale of turbulence at the top of the blocks (m)
• \(N\): ratio between bed friction and drag force
• \(Q\): flow (m^3/s)
• \(S\): pass slope (m/m)
• \(u_0\): average bed speed (m/s)
• \(u_*\): shear velocity (m/s)
• \(V\): flow velocity (m/s)
• \(V_g\): velocity between blocks (m/s)
• \(s\): minimum distance between blocks (m)
• \(z\): vertical position (m)
• \(z_0\): hydraulic roughness (m)
• \(\tilde{z}\): dimensionless stand \(\tilde{z} = z / k\) | {"url":"https://cassiopee.g-eau.fr/assets/docs/en/calculators/pam/macrorugo_theorie.html","timestamp":"2024-11-13T14:43:10Z","content_type":"text/html","content_length":"61537","record_id":"<urn:uuid:871c9514-91aa-4cfb-aeb6-3e2a25714905>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00402.warc.gz"} |
Radiation Modeling with Participating Media
In AcuSolve enclosure or surface to surface radiation models, effect of media between surfaces is ignored. This assumption is acceptable when you are dealing with lower temperature fluids.
Nonetheless, while you have semi-transparent media like glass or high temperature gases like in flames, effect of media should be considered in heat transfer analysis. Two models are available in
AcuSolve: A simple one equation P[1] model and more detailed but expensive discrete ordinates (DO) model.
Radiative Transfer Equation (RTE)
Radiative energy balance in a participating media is governed by the following integro-differential equation, known as the radiative transfer equation (RTE):
$\underset{\text{rate}\text{}\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{of}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{change}}{\underbrace{\Omega ·abla I\left(r,\Omega \
right)}}+\underset{\text{Absorption}}{\underbrace{\kappa I\left(r,\Omega \right)}}+\underset{\text{Scattering loss}}{\underbrace{{\sigma }_{s}I\left(r,\Omega \right)}}=\underset{\text{Emission}}{\
underbrace{f\left(r\right)}}+\underset{\text{Scattering addition}}{\underbrace{\frac{{\sigma }_{s}}{4\pi }\underset{4\pi }{\int }I\left(r,{\Omega }^{\prime }\right)\varphi \left({\Omega }^{\prime },\
Omega \right)d{\Omega }^{\prime }}}$
is the radiation intensity,
is the spatial vector,
is the unit directional vector,
${\Omega }^{\prime }$
is the scattering directional vector,
is the absorption coefficient, and
${\sigma }_{s}$
is the scattering coefficient,
is an emission,
the refractive index, and
is the Stefan-Boltzman constant (5.67 × 10
W m
). T is the temperature (K).
• P1 radiation model
• Discrete Ordinates (DO) model
P1 Radiation Model
The P[1] model is the lowest order P[N] (spherical harmonics) type radiation model. The method reduces the five independent variables of the Radiative Transfer Equation (RTE) into a PDE that is
relatively simple in comparison. The model is the most computationally efficient of the radiation models in AcuSolve, but it can lose accuracy, under certain conditions, for optically thin media. It
performs best in scenarios where the radiative intensity is near isotropic.
Governing Equation
P[1] approximation and assumptions
The P[1] model is derived from the general P[N] formulation (a spherical harmonic series expansion of the radiative intensity for the angular variable) by assuming that the series is limited to four
terms and integrating over all solid angles. From the first harmonic in the series approximation, the divergence of the radiative flux ( $q$) can be derived by integrating the RTE over all solid
angles as
$abla \cdot q=\kappa \left(4{n}^{2}\sigma {T}^{4}-G\right)$
where $G$ is the incident radiation $\frac{{\sigma }_{s}}{4\pi }\underset{4\pi }{\int }I\left(r,{\Omega }^{\prime }\right)\varphi \left({\Omega }^{\prime },\Omega \right)d{\Omega }^{\prime }$, $\
kappa$ is the absorption coefficient, $n$ the refractive index, and $\sigma$ is the Stefan-Boltzman constant (5.67 × 10^-8 W m^-2 K^-4), $q$ is the radiative flux.
Additionally, a second vector equation can be derived from the other three harmonic terms for the radiative flux
$q=\Gamma \text{}\text{\hspace{0.17em}}abla G$
where $\Gamma$ is the diffusion coefficient.
By taking the divergence of (2), and substituting into this the right hand side of equation (1), leads to elimination of the heat flux. The final diffusion reaction equation describing the transport
of incident radiation is given by
$\Gamma \text{\hspace{0.17em}}{abla }^{2}G-\kappa \left(4{n}^{2}\sigma {T}^{4}-G\right)=0$
where $\Gamma$ is a diffusion coefficient given by
$\Gamma =\frac{1}{3\left(\kappa +{\sigma }_{s}\right)-{A}_{1}{\sigma }_{s}}$
where $\kappa$ is the absorption coefficient, ${\sigma }_{s}$ is the scattering coefficient, and ${A}_{1}$ is the linear-anisotropic phase function.
Anisotropic scattering
The implementation in AcuSolve includes the ability to model linear anisotropic scattering
$\varphi \left({\Omega }^{\prime }\cdot \Omega \right)=1+{A}_{1}{\Omega }^{\prime }\cdot \Omega$
is the unit directional vector in the scattering direction,
${\Omega }^{\prime }$
is the unit directional vector in the incident radiation direction. This term is included in the scattering term of the RTE along with the four term spherical harmonic expansion of the radiative
intensity field (the first term being isotropic and the other three anisotropic). The final form of the P
approximation in equation (3) includes this term. The values of the phase function A
have the following meaning:
• A[1] = 1: More radiation is scattered in the forward direction
• A[1]= -1: More radiation is scattered in the backward direction
• A[1]= 0: Isotropic scattering
Coupling to energy equation
The source term in equation (1) can be substituted into the energy equation as a negative source since a local increase in radiative heat flux is due to a local decrease in thermal energy.
the default stagger sequence when the P
radiative heat transfer solver is enabled is:
1. Solve the energy equation with source term ( κ(4n^2σT^4-G) ), where G is zero for the time step
2. Pass temperature to radiation solver and solve equation (3)
3. Repeat until converged
Boundary Conditions
Marshak's boundary condition
Based on the assumption that the walls are diffused gray surfaces, that is, independent of wavelength, the appropriate wall boundary condition is
$n\cdot abla G=\frac{\epsilon }{2\left(2-\epsilon \right)}\left(4{n}^{2}\sigma {T}_{w}^{4}-{G}_{w}\right)$
where $\epsilon$ is the surface emissivity, ${T}_{w}^{}$ is the wall temperature, and ${G}_{w}$ the wall incident radiation. The boundary radiative heat flux can be calculated from the incident
radiation and the temperature at the wall:
${q}_{w}=\frac{\epsilon }{2\left(2-\epsilon \right)}\left({G}_{w}-4{n}^{2}\sigma {T}_{w}^{4}\right)$
Discrete Ordinates (DO) Model
Governing equation
The governing equation is the radiative transfer equation limited to a finite number of directions (or ordinates)
$\Omega ·abla I\left(r,\Omega \right)+\kappa I\left(r,\Omega \right)+{\sigma }_{s}I\left(r,\Omega \right)=f\left(r\right)+\frac{{\sigma }_{s}}{4\pi }\underset{4\pi }{\int }I\left(r,{\Omega }^{\prime
}\right)\varphi \left({\Omega }^{\prime },\Omega \right)d{\Omega }^{\prime }$
$f\left(r\right)=\kappa {I}_{b}\left(r\right)=\kappa \frac{\sigma {T}^{4}}{\pi }$
Scattering term (source term)
The integral over all the directions in equation (1) is replaced by a numerical quadrature for $N$ different ordinate directions ( ${\Omega }_{i}$)
$S=\frac{{\sigma }_{s}}{4\pi }\sum _{j=1}^{N}{w}_{j}I\left(r,{\Omega }_{j}\right)\varphi \left({\Omega }_{j},{\Omega }_{i}\right)$
The phase function, $\varphi \left({\Omega }_{j},{\Omega }_{i}\right)$, is given by
$\varphi \left({\Omega }^{\prime }\cdot \Omega \right)=1+{A}_{1}\left({\Omega }^{\prime }\cdot \Omega \right)$
Boundary conditions (RTE)
Diffused surface
If a surface emits and reflects diffusely, the exiting intensity is directionally independent and is given by
$I\left({r}_{w},{\Omega }_{i}\right)=\epsilon \left({r}_{w}\right){I}_{b}\left({r}_{w}\right)+\frac{1-\epsilon \left({r}_{w}\right)}{\pi }\sum _{n\cdot {\Omega }_{j}<0}{w}_{j}I\left({r}_{w},{\Omega }
_{j}\right)\left|n\cdot {\Omega }_{j}\right|,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}
\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}n\cdot {\Omega }_{j}>0\text{\hspace{0.17em}}$
Specular and diffuse surface
The diffused fraction defines the proportion of reflected radiation intensity at a surface which is diffused, that is, the reflection may also have a specular component. If the radiation intensity
reflection coefficient at the surface is defined by
$\rho ={\rho }^{S}+{\rho }^{D}=1-\epsilon$
then the diffused reflection coefficient, ${\rho }^{D}$, is defined in terms of the diffused fraction $\alpha$ and the emissivity of the surface by
${\rho }^{D}=\alpha \left(1-\epsilon \right)$
and the specular reflection coefficient
${\rho }^{S}=\left(1-\alpha \right)\left(1-\epsilon \right)$
If $\alpha$=1, then the reflection at the surface is completely diffused. If $\alpha$=0 then the reflection is specular. The outgoing intensity, I, at the surface in terms of the above two reflection
coefficients is given by
$I\left({r}_{w},{\Omega }_{i}\right)=\epsilon \left({r}_{w}\right){I}_{b}\left({r}_{w}\right)+\frac{{\rho }^{D}\left({r}_{w}\right)}{\pi }\sum _{n\cdot {\Omega }_{j}<0}^{N}{w}_{j}I\left({r}_{w},{\
Omega }_{j}\right)\left|n\cdot {\Omega }_{j}\right|+{\rho }^{S}\left({r}_{w}\right)I\left({r}_{w},{\Omega }_{j}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace
{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}n\cdot {\Omega }_{j}>0\text{\hspace
where the first terms represent emission from the surface, the second term the diffused component incoming radiation heat flux and the third the specular component. The diffused component represents
a sum over all radiation intensities along ordinates that are incident to the surface (that is, a hemisphere of incoming radiation to the surface); $n$ is the normal into the domain and ${\Omega }_
{j}$ the jth ordinate direction. The ordinate direction ( $\Omega$), the total number of ordinate directions ( $N$) and the weights ( $w$) are automatically defined by the order of the
radiation_quadrature (S2, S4, S6, S8 & S10). The specular ordinate direction ( ${\Omega }_{S}$) is the direction that the radiation intensity must strike the surface to reflect in a specular fashion
along the outgoing ordinate direction, ${\Omega }_{i}$, and is given by
${\Omega }_{s}={\Omega }_{i}-2\left({\Omega }_{i}\cdot n\right)n$
which means the angle that incident radiation intensity strikes the surface equals the angle of reflection.
Boundary conditions (Energy equation)
Interface and outflow/inflow boundary conditions
At an opaque interface between participating and non-participating media or outflows/inflows a radiative heat flux must be added to the boundaries in the energy equation. This flux is given by
${Q}_{rad}=\epsilon \left(\sum _{{\Omega }_{j}\cdot n>0}{w}_{j}I\left({r}_{w}\right)\left|{\Omega }_{j}\cdot n\right|-{n}^{2}\sigma {T}_{w}^{4}\right)\text{\hspace{0.17em}}$
For an opening, that is, outflow or inflow, the black body intensity used in the calculation of the outgoing intensity at the surface is based on the opening temperature of the surrounding:
${I}_{b}=\frac{\kappa {n}^{2}\sigma {T}_{open}^{4}}{\pi }$
while for an interface it is based on the current temperature solution.
Output metrics
Two directionally integrated output metrics can be derived from the radiative intensities: incident radiation and radiative heat flux.
Incident radiation
Incident radiation is the total intensity impinging on a point from all directions and is given by
$G=\sum _{i=1}^{N}{w}_{i}I\left(r\right)$
where $I$ is the intensity in direction i, $N$ the number of ordinates, $w$ the weights.
Interface Between two Semitransparent Media
At the interface between two semitransparent media (referred to as medium 1 and medium 2 below), radiative intensity is both transmitted and reflected. The proportion of transmitted and reflected
intensity at the interface depends on: the refractive indices (n[1], n[2]) of the two media; the incident angle that radiative intensity strikes the surface; and the diffuse fraction of the surface.
Reflection and transmission for specular interfaces
Reflection at an interface is governed by the angle of incidence of a radiative intensity and the refractive indices of the two media. The image below shows the different refracted and reflected rays
between two media.
Figure 1. Reflected and refracted directions at the interface between two participating media of different refractive indices (n[1] < n[2]). For the medium of higher refractive index if the incoming
direction is greater than the critical angle total internal reflection occurs (no transmission occurs across the interface). This is represented by the gray dashed lines in the image.
The cosine of the incident angle for the incoming ordinate is given by
$\mathrm{cos}{\theta }_{1}={\Omega }_{I}^{1}\cdot n$
where $n$ is the outward facing normal direction at the interface (towards the second medium) and ${\Omega }_{I}^{1}{}_{1}$ is the unit direction of incoming radiative intensity to the surface, given
${\Omega }_{I}^{1}{}_{1}={\Omega }_{R}^{1}-2\left({\Omega }_{R}^{1}\cdot n\right)n$
where ${\Omega }_{I}^{1}{}_{1}$ is the unit reflected ordinate direction vector and also represents the current ordinate direction being solved. The equivalent calculation can also be performed for
medium two.
Radiative intensity that is transmitted into a second medium undergoes refraction governed by Snell's law,
${n}_{1}\mathrm{sin}{\theta }_{1}={n}_{2}\mathrm{sin}{\theta }_{2}$
where n[1] and n[2] are the refractive indices of mediums. θ[1] and θ[2] are the angles of incidence and refraction of radiative intensity relative to the interface normal, respectively. This can
also be represented in vector form by
${n}_{1}\left(n×{\Omega }_{I}^{1}\right)={n}_{2}\left(n×{\Omega }_{R}^{2}\right)$
The incoming direction vector in medium two for a ray refracted from medium two to one is given by
${\Omega }_{I}^{2}=\frac{{n}_{1}}{{n}_{2}}{\Omega }_{R}^{1}+\left(\frac{{n}_{1}}{{n}_{2}}\mathrm{cos}{\theta }_{1}-\sqrt{1-{\left(\frac{{n}_{1}}{{n}_{2}}\right)}^{2}\left(1-\mathrm{cos}{\theta }_{1}\
The above expression is valid providing the expression under the radicand is greater than zero; otherwise total internal reflection occurs.
The actual reflected and refracted directions differ slightly from the calculated direction since these directions will unlikely coincide with a discrete ordinate direction. Since the number of
directions is governed by the order of radiation quadrature, higher quadrature orders are more accurate for interface problems.
Figure 2. Octant of angular quadrature with transmission direction. The calculated transmission direction (Ω[T]) is shifted to match the nearest quadrature point.
Depending on the refractive indices of the two media and the angle of incidence, θ[1], the proportion of radiation intensity that is reflected or transmitted will vary. If ${n}_{1}<{n}_{2}$, then the
radiative intensity in medium one will be partially reflected and partially transmitted into a cone defined by the critical angle, θ[c], which is given by:
${\theta }_{c}={\mathrm{sin}}^{-1}\frac{{n}_{1}}{{n}_{2}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\
and defines a cone in 3D (see the images below). The critical angle is defined by a ray that grazes the surface on the side of medium 2 and is transmitted exactly at the critical angle in medium 1.
As an example, if ${n}_{1}>{n}_{2}$ and ${\theta }_{1}>{\theta }_{c}$, the direction of incoming radiation is greater than the critical angle and total internal reflection will occur. This means the
incoming ray is reflected at the same angle of incidence and no transmission occurs.
Figure 3. Total internal reflection of intensity rays at an interface ( ${n}_{1}>{n}_{2}$). The critical angle cone defines the angle outside which total internal reflection occurs, that is, ${\theta
}_{1}>{\theta }_{c}$.
However, if ${\theta }_{1}<{\theta }_{c}$, the outgoing intensity will include transmission from medium 2 to medium 1, as shown in the image below.
Figure 4. Reflection and transmission of intensity rays at an interface ( ${n}_{1}>{n}_{2}$) for ${\theta }_{1}<{\theta }_{c}$
At the interface, the intensity is partially reflected and transmitted into the other medium if ${\theta }_{1}<{\theta }_{c}$ or the outgoing direction is in medium 2. The reflected proportion from $
{\Omega }_{I}^{1}\to {\Omega }_{R}^{1}$, or reflectance, is given by
${r}_{12}=\frac{1}{2}{\left(\frac{{n}_{1}\mathrm{cos}{\theta }_{1}-{n}_{2}\mathrm{cos}{\theta }_{2}}{{n}_{1}\mathrm{cos}{\theta }_{1}+{n}_{2}\mathrm{cos}{\theta }_{2}}\right)}^{2}+\frac{1}{2}{\left(\
frac{{n}_{2}\mathrm{cos}{\theta }_{1}-{n}_{1}\mathrm{cos}{\theta }_{2}}{{n}_{2}\mathrm{cos}{\theta }_{1}+{n}_{1}\mathrm{cos}{\theta }_{2}}\right)}^{2}$
and the transmitted proportion from the ${\Omega }_{I}^{2}\to {\Omega }_{R}^{1}$, or transmittance, is given by
${\tau }_{21}=1-{r}_{12}$
In the second medium, for the current scenario where ${n}_{2}>{n}_{1}$, if ${\theta }_{2}<{\theta }_{c}$ the radiative intensity is, as for medium one, partially reflected and partially transmitted.
The reflection coefficient is as described above since ${r}_{21}={r}_{12}$. If ${\theta }_{2}>{\theta }_{c}$, then total internal reflection occurs and ${r}_{21}=1.0$ and ${\tau }_{12}=0.0$, meaning
no transmission of radiative intensity into the second medium or from the first medium. This is shown in the image above with the gray dashed lines.
From the above, the outgoing radiative intensity on the side one of the interface for the current ordinate direction, ${\Omega }_{R}^{1}$ is given by
$I\left({\Omega }_{R}^{1}\right)={r}_{12}I\left({\Omega }_{I}^{1}\right)+{\tau }_{21}{\left(\frac{{n}_{1}}{{n}_{2}}\right)}^{2}I\left({\Omega }_{I}^{2}\right)$
where for medium one, the first term on the right-hand side represents the reflected intensity in medium one and the second term represents the transmitted intensity from medium two to one. For
medium two, if the current ordinate direction is ${\Omega }_{R}^{2}$ then the intensity outgoing radiative intensity is given by
$I\left({\Omega }_{R}^{2}\right)={r}_{21}I\left({\Omega }_{I}^{1}\right)+{\tau }_{12}{\left(\frac{{n}_{2}}{{n}_{1}}\right)}^{2}I\left({\Omega }_{I}^{1}\right)$
For ${n}_{2}<{n}_{1}$, the subscripts of the above analysis must be exchanged, and total internal reflection will now occur in medium one.
Reflection and transmission for diffuse interfaces
If the interface is diffused, for example, diffused_fraction = 1.0, the reflectivity of the interface is given by the hemispherically averaged reflectance:
is the ratio of refractive indices.
Note: ${n}_{1}$ always represents the medium with higher refractive index and ${n}_{2}$ the medium of lower refractive index.
The transmission from medium one to two is given by
${\tau }_{D,12}=1-{r}_{D,12}$
For the reverse direction the reflectance and transmittance are given by:
${\tau }_{D,21}=1-\frac{1}{{n}^{2}}\left(1-{r}_{D,12}\right)$
${\tau }_{D,21}=\frac{1}{{n}^{2}}{\tau }_{D,12}$
The incoming radiative intensity to the interface is given by the hemispherically averaged intensity for medium one and two:
${Q}_{1}=\sum _{j=1}^{N}{w}_{j}{I}_{j}\left|n\cdot {\Omega }_{j}\right|\left(n\cdot {\Omega }_{j}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\
hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{when}\text{\hspace{0.17em}}\text{\hspace{0.17em}}n\cdot {\Omega }_{j}>0$
${Q}_{2}=\sum _{j=1}^{N}{w}_{j}{I}_{j}\left|n\cdot {\Omega }_{j}\right|\left(n\cdot {\Omega }_{j}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\
hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{when}\text{\hspace{0.17em}}\text{\hspace{0.17em}}n\cdot {\Omega }_{j}\le 0$
where $n$ is the outward facing normal. From these fluxes, the outgoing radiative intensity at the wall for the current ordinate direction, $\Omega$, is given by
$I\left(\Omega \right)={r}_{D,12}\frac{{Q}_{1}}{\pi }+{\tau }_{D,21}\frac{{Q}_{2}}{\pi }$
Reflection and Transmission for partially specular and partially diffuse interfaces
For partially specular and partially diffuse interfaces 0.0 < diffused fraction < 1.0.
Interfaces between semi-transparent media are typically not 100 percent diffused or specular and the diffuse fraction lies somewhere between zero and one. In this range the outgoing radiative
intensity is treated as a linear combination of the specular and diffuse components, for example:
$I\left(\Omega \right)=\left(1-\alpha \right){I}^{S}\left(\Omega \right)+\alpha {I}^{D}\left(\Omega \right)$
where $\alpha$ is the diffuse fraction, ${I}^{S}\left(\Omega \right)$ is the outgoing specular component of radiative intensity, and ${I}^{D}\left(\Omega \right)$ is the outgoing diffuse component of
radiative intensity. For example, in medium one in the image above the components would be:
${I}^{S}\left(\Omega \right)=I\left({\Omega }_{R}^{1}\right)={r}_{12}I\left({\Omega }_{I}^{1}\right)+{\tau }_{21}{\left(\frac{{n}_{1}}{{n}_{2}}\right)}^{2}I\left({\Omega }_{I}^{2}\right)$
${I}^{D}\left(\Omega \right)={r}_{D,12}\frac{{Q}_{1}}{\pi }+{\tau }_{D,21}\frac{{Q}_{2}}{\pi }$
Specular and diffuse interfaces
For the interface equations to be applied weakly, I[w] must be applied in both mediums. If the current ordinate direction, $\Omega$, is outgoing from the interface in medium 1 then I[w] is equal to
the proportion of radiative intensity reflected and transmitted. From the analysis in the previous section, this would be:
${I}_{w}={r}_{12}I\left({\Omega }_{I}^{1}\right)+{\tau }_{21}{\left(\frac{{n}_{1}}{{n}_{2}}\right)}^{2}I\left({\Omega }_{I}^{2}\right)$
In medium 2 since the current ordinate direction, $\Omega$, is incoming to the surface no boundary flux is added to the equation, that is, $\eta {\left({I}_{w},v\right)}_{\Gamma }=0$.
Reflection and Transmission for diffuse interfaces of Type External
Exchange of radiative intensity occurs for external surfaces when the medium inside the computational domain is semitransparent. That is the medium surrounding, which is not modeled using a
computational mesh, participates in radiative transfer. For this case a mathematical model of external radiation is used. The model assumes that the surrounding medium has uniform radiative intensity
in all directions, that is, the radiative flux is isotropic. The isotropic radiative intensity is given by the following blackbody source:
${I}_{\text{ext}}=\frac{{\epsilon }_{\text{ext}}\sigma {T}_{\text{ext}}^{4}}{\pi }$
where ${\epsilon }_{\text{ext}}$ is the external emissivity and is set to one, $\sigma$ is the Stefan-Boltzmann constant, ${T}_{\text{ext}}$ is the temperature of the surrounding medium. At the
external interface ${I}_{\text{ext}}$ is transferred into the medium. This condition can only be applied to boundaries as the interface is only modeled mathematically. | {"url":"https://help.altair.com/hwcfdsolvers/acusolve/topics/acusolve/training_manual/radiation_modeling_r.htm","timestamp":"2024-11-13T11:05:31Z","content_type":"application/xhtml+xml","content_length":"340342","record_id":"<urn:uuid:c5c7d718-1ba0-447a-9cad-a219309bf83b>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00113.warc.gz"} |
factors affecting size of product from ball mill
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Longest Common Substring - Scaler Blog
Problem statement:
Given two strings string_1 and string_2 of size N and M respectively. Find the length of the longest common substring.
The problem statement says that we are given two strings and we need to return the length of the longest common substring from the given strings. For example:
string_1 = "fishes"
string_2 = "fisherman"
The longest common substring from string_1 and string_2 of lengths N = 6 and M = 9 respectively, is “fishe” and it is of length 5. Therefore the output is 5. Let’s discuss various ways for obtaining
the length of the longest common substring.
• 1<N<10^4
• 1<M<10^4
• where N and M are the lengths of the string_1 and string_2 respectively.
What is a Substring?
A substring is the contiguous sequence of characters within the given string. For example, “string” is a substring of “substring” where the order remains the same. For instance, Consider given input
string is “abce” The substrings of the given string are:
[a, b, c, e, ab, bc, ce, abc, bce, abce]
A substring is contiguous, unlike a subset. Even the whole string is also considered to be one of the substrings of the given string. If N is the length of the given string, then the number of
non-empty substrings from the given string is calculated by:
Number of non-empty substrings = (N*(N+1))/2 Now as we have a basic understanding of substring let’s discuss the longest common substring. When two strings are given we need to find the length of the
longest substring among these two strings. For instance, consider
string_1 = "acevd"
string_2 = "acenbvd"
The common substrings among these two strings are:
But the longest common substring is “ace” of length 3.
Approach 1 (Brute Force Approach):
• Generate all the substrings of string_1 and string_2 using three nested loops and store them in any data structure like an array.
• For generating substrings we use three nested loops:
□ The outer loop keeps track of starting character
□ The second loop tracks all the characters on the right of the starting character as the ending character of the substring.
□ The innermost/third loop appends the characters from the current starting index to the ending index.
□ Therefore generating substrings takes a time complexity of O(N^2) where N is the length of the strings given.
• Now from the substrings from the arrays, compare each substring of each string given and always carry forward the maximum length of the common substring formed from the given two strings.
• Return the length of the longest common substring obtained.
Since this is a naive approach we can just follow the algorithm and create all the substrings and compare for the length of the longest common substring from the given strings.
Complexity Analysis:
Time Complexity: O(N^2 * M^2) where N and M are the lengths of the string_1 and string_2 respectively.
• For generating the substrings for string_1 and string_2 and comparing the formed substring one by one.
• Here, string_1 has N^2 number of substrings and string_2 has M^2 number of strings.
• For comparing all these formed substrings of string_1 and string_2, it takes O(N^2^ * M^2^).
• Therefore the overall time complexity is O(N^2 * M^2)
Space Complexity: O(N^2) + O(M^2) where N and M are the lengths of the string_1 and string_2 respectively.
• For storing the substrings formed by each string.
• string_1 takes O(N^2^) space and string_2 takes O(M^2^) and the sum of it is the space complexity.
Approach 2: Efficient Approach using Recursion
The above approach generates all the substrings of given strings and compares iteratively. Instead of redundantly creating all the substrings, we can just use recursion for comparing the strings. If
the character at which we are checking is matched then, we increment the LCSCount by 1 else, making it 0.
This algorithm uses recursion for finding the length of the longest common substring from the given two strings. The algorithm is as follows:
• Start traversing from the end of the two strings with i and j for string_1 and string_2 respectively using recursion.
• If the last characters of both the strings match then decrement both the iterators by 1 and increment LCSCount by 1 and store it in LCSCount_1.
• Else, we need to consider the following two cases:
□ Just decrement the iterator of string_1 and call the recursive function by updating LCSCount to 0 (because we missed the contiguous character of the string) and store it in LCSCount_2.
□ The other way can be decrementing the iterator of string_2 by 1 and again calling the recursive function by updating the LCSCount to 0 and store it in LCSCount_3
• The base case of this function would be till iterator_1 and iterator_2 should be greater than zero, else return the LCSCount because once any one of the iterators becomes zero it means that we
have come to an end of the string given and there can be no further longest common substring which will be formed so, we need to return the LCSCount.
• The answer in this approach is the maximum of LCSCount1, LCSCount_2, LCSCount_3.
This recursive relations can be better explained using:
Below is the implementation of the above approach:
C++ implementation for finding the length of the longest common substring using recursion:
// C++ program for finding the length of the longest common substring using recursion
#include <iostream>
using namespace std;
string string_1, string_2;
// This is the recursive function that returns the length of the longest common substring
int LCS(int iterator_1, int iterator_2, int LCSCount){
// Base Case
if (iterator_1 == 0 || iterator_2 == 0)
return LCSCount;
// If the last character is same then the next character need to be compared to decrement the counter
// increase the LCSCount by 1
if (string_1[iterator_1 - 1] == string_2[iterator_2 - 1]) {
LCSCount = LCS(iterator_1 - 1, iterator_2 - 1, LCSCount + 1);
// if the characters are not the same then you need to decrease the iterator_1 once and update the LCSCount to 0
// The other case can be we can decrement the iterator_2 and update the LCSCount to 0
// The answer is the max of all these cases
LCSCount = max(LCSCount, max(LCS(iterator_1, iterator_2 - 1, 0),LCS(iterator_1 - 1, iterator_2, 0)));
// return the LCSCount
return LCSCount;
// Driver code
int main(){
int N, M;
string_1 = "singer";
string_2 = "singing";
// length of two strings is
N = string_1.size();
M = string_2.size();
// call for the recursive function to get the length of the longest common substring
cout << LCS(N, M, 0);
return 0;
Python implementation for finding the length of the longest common substring using recursion:
# Python program for finding the length of the longest common substring using recursion
# This is the recursive function which returns the length of the longest common substring
def LCS(iterator_1, iterator_2, LCSCount):
# Base Case
if (iterator_1 == 0 or iterator_2 == 0):
return LCSCount
# If the last character is same then the next character need to be compared to decrement the counter
# increase the LCSCount by 1
if (string_1[iterator_1 - 1] == string_2[iterator_2 - 1]):
LCSCount = LCS(iterator_1 - 1, iterator_2 - 1, LCSCount + 1)
# if the characters are not the same then you need to decrease the iterator_1 once and update the LCSCount to 0
# The other case can be we can decrement the iterator_2 and update the LCSCount to 0
# The answer is the max of all these cases
LCSCount = max(LCSCount, max(LCS(iterator_1, iterator_2 - 1, 0), LCS(iterator_1 - 1, iterator_2, 0)))
# return the LCSCount
return LCSCount
# Driver code
if __name__ == "__main__":
# Given two strings are
string_1 = "singer"
string_2 = "singing"
# length of two strings is
N = len(string_1)
M = len(string_2)
# Call the recursive function by passing the parameters
print(LCS(N, M, 0))
Java implementation for finding the length of the longest common substring using recursion:
// Java program for finding the length of the longest common substring using recursion
class Main{
static String string_1, string_2;
// This is the recursive function which returns the length of the longest common substring
static int LCS(int iterator_1, int iterator_2, int LCSCount){
// Base Case
if (iterator_1 == 0 || iterator_2 == 0){
return LCSCount;
// If the last character is same then the next character need to be compared to decrement the counter
// increase the LCSCount by 1
if (string_1.charAt(iterator_1 - 1) == string_2.charAt(iterator_2 - 1)){
LCSCount = LCS(iterator_1 - 1, iterator_2 - 1, LCSCount + 1);
// if the characters are not the same then you need to decrease the iterator_1 once and update the LCSCount to 0
// The other case can be we can decrement the iterator_2 and update the LCSCount to 0
// The answer is the max of all these cases
LCSCount = Math.max(LCSCount, Math.max(LCS(iterator_1, iterator_2 - 1, 0), LCS(iterator_1 - 1, iterator_2, 0)));
// return the LCSCount
return LCSCount;
// Driver code
public static void main(String[] args){
// length of two strings is
int N, M;
// Given two strings are
string_1 = "singer";
string_2 = "singing";
N = string_1.length();
M = string_2.length();
// call for the recursive function to get the length of the longest common substring
System.out.println(LCS(N, M, 0));
Explanation: The longest common substring from the string_1 and string_2 of lengths N=6 and M=7 respectively, is “sing” and it is of length 4. Therefore the output is 4.
Complexity Analysis:
Time Complexity: O(3^(N+M)) where N and M are the lengths of the string_1 and string_2 respectively.
• Since we are calling the recursive function three times and checking for all characters until the base case is not achieved.
• A recursive tree is formed which is: // change the values in the recursive tree put n-1, m-1 generic values.
Space Complexity: O(N+M) where N and M are the lengths of the string_1 and string_2 respectively.
• Since the recursive function takes a space of N+M for auxiliary stack space during recursion.
As we can observe in the recursive tree that the sub-problems are called, again and again, this is the situation of overlapping subproblems which can be resolved by using Dynamic programming. The
next approach is explained using Dynamic programming to find the length of the longest common substring.
Approach 3: Dynamic programming
This approach uses dynamic programming for finding the length of the longest common substring as it is considered to be more efficient in case of overlapping sub-problems for finding the length of
the longest common substring. This dynamic programming uses a bottom-up approach to fill up the table. Here, bottom-up approach first focuses on solving the smaller problems at the fundamental level
and then integrating them into a whole and giving the complete solution.
Dynamic programming is achieved by the tabulation method which uses a 2-D array and stores the length of the longest common substring. This table is filled up in a bottom-up manner.
1. Firstly initialize the 2-D array (DP table) with the size [N+1][M+1]. All the values are initialized to 0.
2. We use one-based indexing as the first row and the first column is 0.
3. This can be done using both iteratively (tabulation) or using memoisation.
4. Iterative approach is discussed below.
5. For filling the values in the table DP. Here dp[i][j] means that the length of the longest substring which ends at ithith character in string_1 and at jthjth character in string_2, and we need to
take the maximum of all of these combinations. For filling up the DP table, we must follow the below algorithm:
1. Start two nested loops:
2. First loop with iterator i and second loop with iterator j.
3. If the characters at string_1[i] matches with string_2[j] then the value at dp[i][j] will be equal to the sum of the dp[i-1][j-1] (diagonal value) + 1.
☆ This means that the previous characters of both the strings are also matched and we can increase the LCSCount in the present cell. Therefore the top left diagonal store the count of the
length of the longest common substring, until that point.
4. Else the value at dp[i][j] is zero because the string cannot be consecutive i.e, it is not a part of the matching substring. So, we make the present cell value zero.
6. Return the maximum element in the table after formation that is considered to be the length of the longest common substring for the given two strings because each cell represents the length of
the longest matching substring till that point.
For a clear understanding of this approach, let’s consider an example: Input:
string_1 = "acdefb"
string_2 = "nbgacde"
By following the algorithm mentioned above the DP table obtained is:
In the table, we can observe that the row represents string_1 and the column represents string_2. Values in the table are filled accordingly. We can observe that the maximum element in the table is
4, which is the length of the longest common substring which is “acde”.
Below is the implementation of this approach:
C++ implementation for finding the length of the longest common substring using Dynamic programming:
// C++ program for finding the length of the longest substring using Dynamic Programming
#include <iostream>
#include <string.h>
using namespace std;
// function which returns the length of longest common substring
int LCS(char* string_1, char* string_2, int M, int N){
// This table keeps track of the substrings which are common
// and contains the length of the longest substring.
// Row in this table represents the string_1 and the column as string_2.
// create a 2-D array named dp and intitialise all the values as zero.
int dp[M + 1][N + 1];
// To store length of the longest
// common substring
int LCSCount = 0;
// Following steps to fill the values in the dp array in bottom-up manner
// Iterative approach for filling the values.
// row iterates over the first string
for (int i = 0; i <= M; i++){
// coloumn iterates over the second string
for (int j = 0; j <= N; j++){
// Base Case
if (i == 0 || j == 0)
dp[i][j] = 0;
// if the characters at string_1 and string_2 matches
else if (string_1[i - 1] == string_2[j - 1]){
dp[i][j] = dp[i - 1][j - 1] + 1;
// carry forwarding the maximum element in the table
LCSCount = max(LCSCount, dp[i][j]);
// if the characters did not match then the values is zero
dp[i][j] = 0;
// return the maximum element in the dp table
return LCSCount;
// Driver code
int main(){
// Given two strings are
char string_1[] = "acdefb";
char string_2[] = "nbgacde";
// length of two strings
int M = strlen(string_1);
int N = strlen(string_2);
// call for the LCS function to get the length of the longest common substring
cout << LCS(string_1, string_2, M, N);
return 0;
Python implementation for finding the length of the longest common substring using Dynamic programming:
# Python program for finding the length of the longest common substring using dynamic programming
def LCS(string_1, string_2, M, N):
# This table keeps track of the substrings which are common and contains the length of the longest substring.
# Row in this table represents the string_1 and the column as string_2.
# create a 2-D array named dp and intitialise all the values as zero.
dp = [[0 for k in range(N+1)] for l in range(M+1)]
# To store the length of the longest common substring
LCSCount = 0
# Following steps to fill the values in the dp array in a bottom-up manner
# Iterative approach for filling the values.
# row iterates over the first string
for i in range(M + 1):
# column iterates over the second string
for j in range(N + 1):
# Base Case
if (i == 0 or j == 0):
dp[i][j] = 0
# if the characters at string_1 and string_2 matches
elif (string_1[i-1] == string_2[j-1]):
dp[i][j] = dp[i-1][j-1] + 1
# carry forwarding the maximum element in the table
LCSCount = max(LCSCount, dp[i][j])
# if the characters did not match then the values is zero
dp[i][j] = 0
# return the maximum element in the dp table
return LCSCount
# Driver code
if __name__ == "__main__":
# Given two strings are
string_1 = "acdefb"
string_2 = "nbgacde"
# length of two strings is
N = len(string_1)
M = len(string_2)
# Call the LCS function by passing the parameters
print(LCS(string_1, string_2, N, M))
Java implementation for finding the length of the longest common substring using dynamic programming:
// java program for finding the length of the longest common substring using dynamic programming
class Main {
// function which returns the length of longest common substring
static int LCS(char string_1[], char string_2[], int M, int N) {
// This table keeps track of the substrings which are common
// and contains the length of the longest substring.
// Row in this table represents the string_1 and the column as string_2.
// create a 2-D array named dp and intitialise all the values as zero.
int dp[][] = new int[M + 1][N + 1];
// To store length of the longest
// common substring
int LCSCount = 0;
// Following steps to fill the values in the dp array in bottom-up manner
// Iterative approach for filling the values.
// row iterates over the first string
for (int i = 0; i <= M; i++){
// column iterates over the second string
for (int j = 0; j <= N; j++){
// # Base Case
if (i == 0 || j == 0){
dp[i][j] = 0;
// if the characters at string_1 and string_2 matches
else if (string_1[i - 1] == string_2[j - 1]){
dp[i][j] = dp[i - 1][j - 1] + 1;
// carry forwarding the maximum element in the table
LCSCount = Integer.max(LCSCount, dp[i][j]);
// if the characters did not match then the values is zero
dp[i][j] = 0;
// return the maximum element in the dp table
return LCSCount;
// Driver code
public static void main(String[] args){
int M,N;
// Given two strings are
String string_1 = "acdefb";
String string_2 = "nbgacde";
// length of two strings
M = string_1.length();
N = string_2.length();
// call for the LCS function to get the length of the longest common substring
System.out.println(LCS(string_1.toCharArray(),string_2.toCharArray(), M, N));
Explanation: The longest common substring from the string_1 and string_2 of lengths N=6 and M=7 respectively, is “sing” and it is of length 4. Therefore the output is 4.
Complexity Analysis:
Time Complexity: O(M*N) where M and N are the lengths of the given string_1 and string_2 respectively.
• For filling up the the DP table (2-D array) in a bottom-up manner, it requires a time of O(N*M). Each cell can be filled as zero or using the previous cell value as required.
Space Complexity: O(M*N) where M and N are the lengths of the given string_1 and string_2 respectively.
• DP table (2-D array) is of size [M+1][N+1], so the space occupied is O(M*N).
Since we are updating the values in the table using the values of the previous updates, space optimization can be done to the above approach. This algorithm is discussed in the below approach.
Approach 4: Dynamic Programming- Space Optimization
In the above approach, we are only using the last row( updated value) of the 2-D array, therefore we can optimize the space by using a 2-D array of size 2*(min(N, M)).
• Algorithm remains the same as filling up of 2-D array but instead of column size as N or M, we instead take 2 because we just need the last row.
• Generally, all the dynamic programming approaches can be optimized by using the DP state where we can use the previous state to get the present state’s value and it can also be done using the
location of the answer in the DP matrix formed.
• However, we need to traverse the two strings using two nested loops, for obtaining the length of the longest substring.
• The space optimization is done by :
□ if string_1[i – 1] is equal to string_2[j – 1], then
dp[i%2][j] = dp[(i – 1)%2][j – 1] + 1
☆ where dp[i%2][j] represents the present cell and the dp[(i-1)%2][j-1] represents the previous state.
☆ Here we have taken the value to be mod 2, because since it involves only 2 coloumns, we can just use the mod 2 operation while updating the table values.
□ Since, we need to carry forward the maximum element in the table, we can do :
☆ LCSCount = max(LCSCount, dp[i%2][j]).
□ If the characters are not matched, then the present cell value is zero.
• At last, return the length of the longest common substring.
Below is the implementation of the space-optimized dynamic programming approach for finding the length of the longest common substring from the given two strings.
C++ implementation for finding the length of the longest substring by space optimizing using DP state:
// c++ implementation using dynamic programming for space optimisation
#include <iostream>
#include <string.h>
using namespace std;
// function which returns the length of longest common substring
int LCS(string string_1, string string_2, int M, int N){
// This table keeps track of the substrings which are common
// and contains the length of the longest substring.
// Row in this table represents the string_1 and the coloum as string_2.
// create a 2-D array named dp and intitialise all the values as zero.
int dp[2][N + 1];
// To store length of the longest
// common substring
int LCSCount = 0;
// Following steps to fill the values in the dp array in bottom-up manner
// Iterative approach for filling the values.
for (int i = 1; i <= M; i++){
// column iterates over the second string
for (int j = 1; j <= N; j++){
// Base Case
if (i == 0 || j == 0)
dp[i][j] = 0;
// if the characters at string_1 and string_2 matches
else if (string_1[i - 1] == string_2[j - 1]){
dp[i%2][j] = dp[(i - 1)%2][j - 1] + 1;
// carry forwarding the maximum element in the table
LCSCount = max(LCSCount, dp[i%2][j]);
// if the characters did not match then the values is zero
dp[i%2][j] = 0;
// return the maximum element in the dp table
return LCSCount;
// Driver code
int main(){
// Given two strings are
string string_1 = "acdefb";
string string_2 = "nbgacde";
// length of two strings
int M = string_1.length();
int N = string_2.length();
// call for the LCS function to get the length of the longest common substring
cout << LCS(string_1, string_2, M, N);
return 0;
Python implementation for finding the length of the longest substring by space optimizing using DP state:
# Python program for finding the length of the longest common substring using dynamic programming
def LCS(string_1, string_2, M, N):
# This table keeps track of the substrings which are common and contains the length of the longest substring.
# create a 2-D array named dp and intitialise all the values as zero.
dp = [[0 for k in range(N+1)] for l in range(2)]
# To store the length of longest common substring
LCSCount = 0
# Following steps to fill the values in the dp array in bottom-up manner
# Iterative approach for filling the values.
for i in range(1, M + 1):
# column iterates over the second string
for j in range(1, N + 1):
# Base Case
if (i == 0 or j == 0):
dp[i][j] = 0
# if the characters at string_1 and string_2 matches
elif (string_1[i-1] == string_2[j-1]):
dp[i%2][j] = dp[(i-1)%2][j-1] + 1
# carry forwarding the maximum element in the table
LCSCount = max(LCSCount, dp[i%2][j])
# if the characters did not match then the values is zero
dp[i%2][j] = 0
# return the maximum element in the dp table
return LCSCount
# Driver code
if __name__ == "__main__":
# Given two strings are
string_1 = "acdefb"
string_2 = "nbgacde"
# length of two strings is
N = len(string_1)
M = len(string_2)
# Call the LCS function by passing the parameters
print(LCS(string_1, string_2, N, M))
Java implementation for finding the length of the longest substring by space optimizing using DP state:
// java program for finding the length of the longest common substring using dynamic programming by space optimization technique
class Main {
// function which returns the length of longest common substring
static int LCS(char string_1[], char string_2[], int M, int N) {
// This table keeps track of the substrings which are common
// and contains the length of the longest substring.
// Row in this table represents the string_1 and the coloum as string_2.
// create a 2-D array named dp and intitialise all the values as zero.
int dp[][] = new int[2][N + 1];
// To store length of the longest
// common substring
int LCSCount = 0;
// Following steps to fill the values in the dp array in bottom-up manner
// Iterative approach for filling the values.
for (int i = 1; i <= M; i++){
for (int j = 1; j <= N; j++){
// # Base Case
if (i == 0 || j == 0){
dp[i][j] = 0;
// if the characters at string_1 and string_2 matches
else if (string_1[i - 1] == string_2[j - 1]){
dp[i%2][j] = dp[(i - 1)%2][j - 1] + 1;
// carry forwarding the maximum element in the table
LCSCount = Integer.max(LCSCount, dp[i%2][j]);
// if the characters did not match then the values is zero
dp[i%2][j] = 0;
// return the maximum element in the dp table
return LCSCount;
// Driver code
public static void main(String[] args){
int M,N;
// Given two strings are
String string_1 = "acdefb";
String string_2 = "nbgacde";
// length of two strings
M = string_1.length();
N = string_2.length();
// call for the LCS function to get the length of the longest common substring
System.out.println(LCS(string_1.toCharArray(),string_2.toCharArray(), M, N));
Explanation: The longest common substring from the string_1 and string_2 of lengths N=6 and M=7 respectively, is “sing” and it is of length 4. Therefore the output is 4.
Complexity Analysis:
Time Complexity: O(M*N) where M and N are the lengths of string_1 and string_2 respectively.
• For filling up the DP table (2-D array) in a bottom-up manner, it requires a time of O(N*M).
Space Complexity: O(min(M*N)) where M and N are the lengths of string_1 and string_2 respectively.
• DP table (2-D array) is of size [2][min(N,M)+ 1], so the space occupied is min(M*N)*2.
• So the overall space complexity is O(min(M*N)).
• We have discussed various approaches for finding the length of the longest substring.
• The first approach is by generating all the substrings of the given two strings of length M and N and comparing their substrings. It takes a time complexity of O(N^2 * M^2), this can be optimized
by comparing only the same length substrings and it takes the time complexity of O(N^3) where the length of both the strings is the same N.
• The second approach uses recursion for finding the length of the longest common substring this takes a time complexity of O(3^(N+M)) and space complexity of O(M+N)
• Whenever a recursive approach is present, and a recursive tree is formed we observe that dynamic programming can be used to overcome the re-calculation of the sub-problems again and again.
• Instead, we compute the results of all sub-problems and store them. They are extracted whenever required for solving the real problem.
• So in this way, we get the third approach which uses dynamic programming for finding the length of the longest common substring from the given two strings. We write DP state and store the values
in a 2-D DP table. This takes the time complexity of O(N*M) and space complexity O(N*M) for the DP table.
• The fourth approach is the same DP approach but space optimization is done by, simply storing the previous row, therefore the time complexity remains the same i.e, O(N*M) and space complexity is
O(2*min(M, N)).
HAPPY CODING!!! | {"url":"https://www.scaler.in/longest-common-substring/","timestamp":"2024-11-08T05:51:18Z","content_type":"text/html","content_length":"119348","record_id":"<urn:uuid:9382a0dd-0301-4fe6-bef1-da78ed956fe4>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00666.warc.gz"} |
Inversion of parabolic and paraboloidal projections
The multidimensional inverse scattering problem for an acoustic medium is considered within the homogeneous background Born approximation. A constant density acoustic medium is probed by a wide-band
plane wave source, and the scattered field is observed along a receiver array located outside the medium. The inversion problem is formulated as a generalized tomographic problem. It is shown that
the observed scattered field can be appropriately filtered so as to obtain generalized projections of the scattering potential. For a 2-D experimental geometry, these projections are weighted
integrals of the scattering potential over regions of parabolic support, whereas they become surface integrals over circular paraboloids for the 2-D case. The inversion problem is therefore similar
to that of X-ray tomography, except that instead of being given projections of the object to be reconstructed along straight lines, parabolic or paraboloid projections are given. The inversion
procedure that we propose is similar to the X-ray solution, in the sense that it consists of a backprojection operation followed by a 2- or 3-D space invariant filtering. An alternative
interpretation of the backprojection operation in terms of a backpropagated field is given. A Projection-Slice Theorem is also derived relating the generalized projections and the scattering
potential in the Fourier transform domain.
Pub Date:
April 1987
□ Acoustic Scattering;
□ Born Approximation;
□ Homogeneity;
□ Inversions;
□ Parabolas;
□ Arrays;
□ Broadband;
□ Detection;
□ Fourier Transformation;
□ Integrals;
□ Inverse Scattering;
□ Parabolic Bodies;
□ Plane Waves;
□ Receivers;
□ Tomography;
□ Weighting Functions;
□ X Ray Sources;
□ Acoustics | {"url":"https://ui.adsabs.harvard.edu/abs/1987ippp.rept.....O/abstract","timestamp":"2024-11-10T16:11:31Z","content_type":"text/html","content_length":"38399","record_id":"<urn:uuid:aef4b71c-c23c-49eb-8f8d-e04969f081be>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00413.warc.gz"} |
The discovery that Gauss was most proud of – 07/27/2021 – Marcelo Viana
Draw a circle on the paper with a compass. Then, without changing the opening of the compass, draw another circle centered somewhere in the first one. Finally, use a ruler to connect the centers of
the two circles to one of the points where they intersect. The figure obtained in this way is an equilateral triangle, that is, the sides of which are all of the same length.
The ancient Greeks knew how to use a ruler and compass to build regular polygons with 3, 4, 5 and 15 sides. They also knew how to get from one regular polygon to another with double sides. So they
knew how to build the regular hexagon (6 sides) from the equilateral triangle. Can all regular polygons with any number of N sides be constructed with a ruler and compass?
The answer is no, but that wasn’t understood until the 18th century when it was proven that regular 7- and 13-sided polygons couldn’t be constructed this way. So what are the constructible values
of N, ie so that the regular polygon with N sides can only be constructed with a ruler and compass?
The problem caught the attention of none other than the great Carl Friedrich Gauss. In 1796 he showed how to build the regular heptagon (17 pages) with a ruler and compass. This was a discovery Gauss
was most proud of. In his great work “Disquisitiones Arithmeticae” he went even further and came to the conclusion that for the construction of a regular polygon it is sufficient that the number N of
sides is the product of a power of 2 divided by different Fermat prime numbers. He also stated that this condition would be sufficient, but this was only proven in 1837 by the Frenchman Pierre
Pierre de Fermat calculated the numbers of the form 1 plus 2 to 2n for n values from 0 to 4, found them to be prime numbers and held this for all n values. But a few years later Leonhard Euler
pointed out that the Fermat number with n = 5 is not a prime number and ironically, until today it has found none other than the five originals he discovered.
Since there are 31 different Fermat prime number products, the Gauss-Wantzel theorem gives 31 odd N numbers that can be constructed, and this is the best result known so far.
PRESENT LINK: Did you like this column? Subscribers can allow five free hits on any link per day. Just click the blue F below. | {"url":"https://ksusentinel.com/2021/07/28/the-discovery-that-gauss-was-most-proud-of-07-27-2021-marcelo-viana/","timestamp":"2024-11-07T04:02:26Z","content_type":"text/html","content_length":"96190","record_id":"<urn:uuid:c625b559-425a-4d04-86c0-90aa7a887ed0>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00568.warc.gz"} |
Trigonometry Assignment help
homework assignment help is most useful online help portal for the students that providing all Online Trigonometry assignment help Services .Math assignments are common mathematics problems. In most
cases, students will have to do math assignments after every math class as homework. Math assignments can be tricky, especially if a student is learning a new mathematics concept. However you can
send us Math problems from Basic Math problems like Algebra, Geometry, Trigonometry, and Arithmetic to more advanced Math Assignment Problems like Limits, Continuity, Derivatives, Pre-Calculus, and
Double angle formulae
The following are important trigonometric relationships (it is unlikely that you will need to know how to prove them and they may be given in your formula book- check!):
sin(A + B) = sinAcosB + cosAsinB
cos(A + B) = cosAcosB - sinAsinB
tan(A + B) = tanA + tanB
1 - tanAtanB
To find sin(A - B), cos(A - B) and tan(A - B), just change the + signs in the above identities to -
sin(A - B) = sinAcosB - cosAsinB
cos(A - B) = cosAcosB + sinAsinB
tan(A - B) = tanA - tanB
1 + tanAtanB
Double Angle Formulae
sin(A + B) = sinAcosB + cosAsinB
Replacing B by A in the above formula becomes:
sin(2A) = sinAcosA + cosAsinA
so sin2A = 2sinAcosA
similarly, cos2A = cos²A - sin²A
Replacing cos²A by 1 - sin²A (see Pythagorean identities) in the above formula gives:
cos2A = 1 - 2sin²A
Replacing sin²A by 1 - cos²A gives:
cos2A = 2cos²A - 1
It can also be shown that:
tan2A = 2tanA
1 - tan²A
Pythagorean Identities
This important identity can be derived as a direct result of Pythagoras's theorem, when applied to angles in trigonometry:
sin²x + cos²x = 1
By dividing each of these terms by sin²x, we can derive a second identity: 1 + cot²x = cosec²x By dividing (1) by cos²x, we arrive at the third (and final) identity: tan²x + 1 = sec²x
Radians, like degrees, are a way of measuring angles.
One radian is equal to the angle formed when the arc opposite the angle is equal to the radius of the circle. So in the above diagram, the angle ø is equal to one radian since the arc AB is the same
length as the radius of the circle.
Now, the circumference of the circle is 2pr, where r is the radius of the circle. So the circumference of a circle is 2p larger than its radius. This means that in any circle, there are 2p radians.
Therefore 360º = 2p radians.
Therefore 180º = p radians.
So one radian = 180/p degrees and one degree = p/180 radians.
Therefore to convert a certain number of degrees in to radians, multiply the number of degrees by p/180 (for example, 90º = 90 × p/180 radians = p/2). To convert a certain number of radians into
degrees, multiply the number of radians by 180/p .
Arc Length
The length of an arc of a circle is equal to rø, where ø is the angle, in radians, subtended by the arc at the centre of the circle. So in the below diagram, s = rø .
Area of Sector
The area of a sector of a circle is ½ r² ø, where r is the radius and ø the angle in radians subtended by the arc at the centre of the circle. So in the below diagram, the shaded area is equal to ½
r² ø .
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A triangle has sides A, B, and C. Sides A and B have lengths of 2 and 4, respectively. The angle between A and C is (7pi)/24 and the angle between B and C is (5pi)/8. What is the area of the triangle? | Socratic
A triangle has sides A, B, and C. Sides A and B have lengths of 2 and 4, respectively. The angle between A and C is #(7pi)/24# and the angle between B and C is # (5pi)/8#. What is the area of the
1 Answer
The area is $\setminus \sqrt{6} - \setminus \sqrt{2}$ square units, about $1.035$.
The area is one half the product of two sides times the sine of the angle between them.
Here we are given two sides but not the angle between them, we are given the other two angles instead. So first determine the missing angle by noting that the sum of all three angles is $\setminus \
pi$ radians:
$\setminus \theta = \setminus \pi - \frac{7 \setminus \pi}{24} - \frac{5 \setminus \pi}{8} = \setminus \frac{\pi}{12}$.
Then the area of the triangle is
Area $= \left(\frac{1}{2}\right) \left(2\right) \left(4\right) \setminus \sin \left(\setminus \frac{\pi}{12}\right)$.
We have to compute $\setminus \sin \left(\setminus \frac{\pi}{12}\right)$. This can be done using the formula for the sine of a difference:
$\sin \left(\setminus \frac{\pi}{12}\right) = \setminus \sin \left(\textcolor{b l u e}{\setminus \frac{\pi}{4}} - \textcolor{g o l d}{\setminus \frac{\pi}{6}}\right)$
$= \setminus \sin \left(\textcolor{b l u e}{\setminus \frac{\pi}{4}}\right) \cos \left(\textcolor{g o l d}{\setminus \frac{\pi}{6}}\right) - \setminus \cos \left(\textcolor{b l u e}{\setminus \frac{\
pi}{4}}\right) \sin \left(\textcolor{g o l d}{\setminus \frac{\pi}{6}}\right)$
$= \left(\frac{\setminus \sqrt{2}}{2}\right) \left(\frac{\setminus \sqrt{3}}{2}\right) - \left(\frac{\setminus \sqrt{2}}{2}\right) \left(\frac{1}{2}\right)$
$= \frac{\setminus \sqrt{6} - \setminus \sqrt{2}}{4}$.
Then the area is given by:
Area $= \left(\frac{1}{2}\right) \left(2\right) \left(4\right) \left(\frac{\setminus \sqrt{6} - \setminus \sqrt{2}}{4}\right)$
$= \setminus \sqrt{6} - \setminus \sqrt{2}$.
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Interpolated table lookups using SSE2 [1/2]
Something that you are very likely to encounter, when writing SSE2 assembler is the issue of using tables for looking up data. While this is a very simple operation in C, it presents a challenge with
a lot of pitfalls, when using SSE2.
When to use lookup tables?
In general, avoid them, if it is within the possibility. Let’s have a look at a simple example:
const float ushort_to_float[65535];
float convert_unsigned_short_to_float(unsigned short value) {
return ushort_to_float[value];
This is quite tempting in C – especially if you can do some transformations on the value, by modifying the lookup table, if you need something expensive like the square root, use the inverted value.
But lets look at the drawbacks.
The table is 256KB, so you will run out of level 1 cache on most CPUs. In C the penalty for a value in L1 cache this is usually a couple of cycles per value, but if it has to be fetched from L2, we
are talking about 15 cycles or more on most platforms. But if it saves you a division and a square root or something similar the trade-off may be ok for you.
In SSE2, the trade-off picture is changed. Here the penalty per pixel for doing the lookup remains the same (or even rises), while calculating the value is usually only about 25% of cost of what the
C-code. So you can see – we can do a lot more calculations in SSE2 before using a table should even be considered.
What if I have to use lookup tables?
There might be cases, where using tables are the only way, because calculating the values simply aren’t practically possible in SSE2. A practical example of this from Rawstudio is the curve
correction, that involves mapping input values to output values based on an n-point 2D curve.
Curve Correction: Nearly impossible without using lookup tables.
In the “old days”, we used a table with 65535 16 bit entries, that simply gave the output value, much as the example above. The table was 128KB, so again, we had a lot of L1 cache misses.
When we re-wrote colour correction, everything internally was converted to float, which made the table 256KB, and even slower. Furthermore it also made sure that we lost a lot of the float precision,
by having to fit the float values into 65535 slots.
So we got two problems:
• Bad precision
• Table too large for L1 cache
Using lookup tables for float data
Since most of these lookups are fairly linear, without too sudden peaks, usually we are fairly in the clear if we reduce the precision of the table, and do linear interpolation between them instead.
This avoids posterizing the image data, because all float values will have a unique output. Here is an example, where we use a curve correction with only 256 input/output values, and interpolate them
float curve_samples[257];
// Make sure we have an extra entry, so we avoid an
// "if (lookup > 255)" later.
curve_samples[256] = curve_samples[255];
for (all pixels...) {
// Look up and interpolate
float lookup = CLAMP(input * 256.0f, 0.0f, 255.9999f);
float v0 = curve_samples[(int)lookup];
float v1 = curve_samples[(int)lookup + 1];
lookup -= floorf(lookup);
output = v0 * (1.0f - lookup) + v1 * lookup;
So now the table is only 1KB in size and fits very nicely within the Level 1 cache. This approach solves both our problems, but at the expense of some calculations. Whether 256 entries are enough is
up for you to decide, but the principle applies for all sizes. The “floor” function can be quite slow on some compilers/CPUs, but it is here to demonstrate the algorithm. We will touch this later.
In part two, I will look at the actual implementation – there are still a lot of pitfalls in making this work in real life.
PS. If you enjoy technical stuff like this, please leave a comment.
7 responses to “Interpolated table lookups using SSE2 [1/2]”
1. I can’t say I know too much about the technical details of image processing, but I do appreciate the insight post like these give. Thanks!
It seems like you could reduce the size further is you used a different data type for the LUT, for instance a 16-bit or 8-bit int. With my monitor at 1024 pixels high, I couldn’t imagine being
able to enter a curve at more than 10-bits precision, and that’s if the screen is maximized. Sounds like the image data is in float, but does it take too many cycles to cast an int to a float? I
wonder if the loss of precision would be noticable. Anyways, these are just questions floating in my head, and I’m not trying to criticize your implementation, you folks know way more than I ever
will about this stuff.
□ The game is a little different when you are dealing with Raw data, because you operate in a linear colorspace. Because of this you need additional precision, since errors become very visible
in dark areas.
See this PDF on why you need at least 16 bit precision for Raw images:
2. Yes, I enjoyed this posting very much. Thanks!
3. Thanks, I enjoy this kind of blog posts! Keep them coming :)
4. […] Continued from part 1… […]
5. The problem with that approach is that decreasing the table precision leads to at least two problems.
First, the curve downscaler 65536->256 has to be chosen carefully.
And then such a big downscale can’t be reasonably inversed using a simple linear interpolation w/o introducing aliasing problems.
Do you have some PSNR numbers comparing the original results using interpolation between the 65536 sampling points of the curve, and the faster 256 entry table interpolation ?
□ Hi Edouard!
>First, the curve downscaler 65536->256 has to be chosen carefully.
The curve isn’t downscaled as such – the curve is simply sampled with 256 entries instead of 65536. So we get 256 exact values, and not some that are derived from 65536.
>And then such a big downscale can’t be reasonably inversed using a simple linear interpolation w/o introducing aliasing problems.
How do you get to that result? The curve is an input->output mapping, where neutral is input=output (straight line). You would have to have a curve that has variations where the correction is
changed at a 1/128th pixel resolution, which simply wouldn’t make sense.
So unless you have information with a frequency close to or above the nyquist frequency, a linear interpolation should not generate aliasing.
If you consider the image of the curve, you have approximately the horizontal resolution of the image (the image is ~290 pixels wide). The vertical sample point have float point precision, so
the vertical resolution is very precise. The vertical resolution is the reason there is no precision issues, even with such small tables.
>Do you have some PSNR numbers comparing the original results using interpolation between the 65536 sampling points of the curve, and the faster 256 entry table interpolation ?
I’ll make a few calculations that calculates the min, max and average error on “realistic” and some “extreme” curves.
Klaus Post on October 28, 2010 | {"url":"https://rawstudio.org/blog/?p=457","timestamp":"2024-11-01T20:40:55Z","content_type":"application/xhtml+xml","content_length":"18783","record_id":"<urn:uuid:0e4c6942-814d-48e1-bf7b-2dfe70a118df>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00776.warc.gz"} |
. Calculate the median. - Datapott Analytics
For the Week 1 Complete assignment, solve each problem and answer each question that corresponds with it. Explain how you arrived at the answer for each problem. The total word count for your
assignment should be a minimum of 1200 words.
Identify the type of data (qualitative/quantitative) and the level of measurement Explain your choice.
The average monthly temperature in degrees Fahrenheit for the city of Wilmington, Delaware, through-out the year.
The ages of the respondents in a survey.
The years in which the respondents to a survey were born.
The ethnicity of the respondents in a survey classi-fied as White, African American, Asian, or Other.
The Arizona Diamondbacks and the city of Phoenix are considering building a new baseball stadium for the Major League Baseball team. A decision needs to be made whether to enclose the new facility
with a roof. To help make this decision, data were gathered on the number of days per month it rained during the baseball season, which are shown in the following table.
a. Construct a frequency distribution for these data. b. Using the results from part a, calculate the rela-tive frequencies for each class. c. Using the results from part a, calculate the
cumulative relative frequencies for each class. d. Construct a histogram for these data. e. What is the likelihood that a month during the baseball season will experience one day or less
The following table lists the math SAT scores for 72 students.
a. Using the 2k ≥ n rule, construct a frequency distribution for these data. b. Using the results from part a, calculate the relative frequencies for each class. c. Using the results from part a,
calculate the cumulative relative frequencies for each class. d. Construct a histogram for these data.
7. The following data represent the number of days homes were on the market before being sold in New Castle County, Delaware
a. Calculate the mean. b. Calculate the median. c. Determine the mode. d. Which of these three measurements would best describe the central tendency of the data? e. Describe the shape of this
8. Internet service providers compete on services such as download speeds that are measured in megabits per second (Mbps). The following data represent the download speeds that 16 Comcast customers
experienced recently.
a. Calculate the mean. b. Calculate the median.
c. Determine the mode. d. Describe the shape of the distribution | {"url":"https://datapott.com/calculate-the-median/","timestamp":"2024-11-04T07:51:38Z","content_type":"text/html","content_length":"103771","record_id":"<urn:uuid:1853235f-93c9-415d-b1d3-3fb681a95fc7>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00567.warc.gz"} |
how to make types of angles math’s working model(TLM project) - Science Projects | Maths TLM | English TLM | Physics Projects | Computer Projects | Geography Projects | Chemistry Projects | Working Projects | Working Models | DIY for School / College Science Exhibitions or Fair
how to make types of angles math’s working model(TLM project)
In this post we written on how to make the model on types of angles math’s working project (TLM project) using cardboard and color paper
types of angles math’s working model(TLM project)
Creating a working model to demonstrate types of angles using cardboard and color paper is an excellent way to visualize and understand different angle measurements.
Here’s a step-by-step guide to making the model:
Materials you will need:
• Cardboard (for the base and angle representations)
• Color paper (different colors for different types of angles)
• Scissors
• Glue or adhesive
• Markers or pens (for adding details)
Step-by-step instructions:
1. Prepare the base:
□ Take a large piece of cardboard to serve as the base for your angle model.
2. Draw and cut out angle representations:
□ Use a ruler and a pencil to draw various angle representations on the color paper. For example:
☆ Right angle (90 degrees): Draw an “L” shape with one angle measuring 90 degrees.
☆ Acute angle (less than 90 degrees): Draw a small angle less than 90 degrees.
☆ Obtuse angle (greater than 90 degrees): Draw a larger angle greater than 90 degrees.
☆ Straight angle (180 degrees): Draw a straight line.
□ Cut out these angle representations from the color paper using scissors.
3. Glue the angles onto the cardboard:
□ Glue each angle representation onto the cardboard base, leaving some space between them.
4. Demonstrate angle relationships:
□ Use the model to demonstrate the relationships between different angle types. For example, show how two acute angles can add up to form a right angle, or how two supplementary angles form a
straight angle.
This model provides a visual representation of various angle measurements and their characteristics. It’s a fun and educational project to understand the different types of angles and how they relate
to one another in geometry.
#typesofangles #mathsworkingmodel #tlm #craftpiller #maths #workingmodel #diy
Video guide on making types of angles math’s working model(TLM project)
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The Remainders Game
If you haven't already seen Remainders, it would be worth trying that task before playing this game.
The computer will think of a number between 1 and 100. Can you work out what it is?
Choose a divisor and the computer will give you some information about the number.
The fewer divisions you require, the more points you get.
If you identify the number correctly after
1 or 2 divisions - you gain 12 points
3 divisions - you gain 11 points
4 divisions - you gain 9 points
5 divisions - you gain 6 points
6 divisions - you gain 3 points
7 or more divisions - you gain 1 point
If you guess wrongly you lose 15 points, even if your guess satisfies all the criteria.
How soon can you reach 100 points?
Challenge your friends to do better.
The modulator will help you play the game, but eventually, try to play the game without it.
Click on the purple cog to open the settings menu and change the level of the game:
In Level 1 the chosen number will be from 1 to 60 inclusive and you can divide by each of the numbers from 1 to 10.
In Level 2 (the default setting), the chosen number will be from 1 to 100 inclusive and you can divide by each of the numbers from 2 to 10.
In Level 3 the chosen number will be from 1 to 100 inclusive and you can only divide by a selection of the numbers from 1 to 10.
This page contains some examples to clarify the rules for the Remainders Game.
If you guess wrong you lose 15 points, so don't guess until you are certain that there is only one possible answer.
Suppose you have discovered that your number has a remainder of $1$ when divided by $2$, and a remainder of $0$ when divided by $5$
At this point, you know that the hidden number is both odd and a multiple of $5$. You might think of $15$ as a number that would fit with both of these criteria.
However, there are still other possibilities that the number could be, such as $5$, $25$ or even $95$. This means that you'd need to do some more divisions to work out which number it is.
Try to ensure that each division you carry out provides new information - it should rule out some numbers.
Suppose you have divided by $4$ and found that you are left with a remainder of $1$.
At this stage, you know your number is $1$ or $5$ or $9$ or..., so all the possibilities that you get are odd. Therefore, we already know that, if you choose to divide by $2$, the remainder will be
So, if you did divide by $2$ in this situation you would gain no extra information as to what the hidden number is.
If you carry out unnecessary divisions your score for getting the right answer will be lower.
You can use The Modulator tool to help you to decide whether a division will rule out any numbers. | {"url":"https://nrich.maths.org/games/remainders-game","timestamp":"2024-11-13T02:48:15Z","content_type":"text/html","content_length":"39979","record_id":"<urn:uuid:cd006202-0014-45b7-b311-c1443c91a5d0>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00305.warc.gz"} |
ThmDex – An index of mathematical definitions, results, and conjectures.
This result is a subresult to
R5185: Tight lower bound to a finite product of positive real numbers
. In that result, we see that $\prod_{n = 1}^N x_n$ attains the lowest possible value when $x_1 = x_2 = \cdots = x_N$. In this case, we have \begin{equation} \left( \frac{1}{\frac{1}{N} \sum_{n = 1}^
N \frac{1}{x_n}} \right)^N = \prod_{n = 1}^N x_n = x^N_k \end{equation} for some $k = 1, \ldots, N$. Substituting $\sum_{n = 1}^N \frac{1}{x_n} = a$ and raising both sides to power $1 / N$, we have \
begin{equation} \frac{N}{a} = \frac{1}{a / N} = x_k \end{equation} Since $k = 1, \ldots, N$ was arbitrary, the claim follows. $\square$ | {"url":"https://theoremdex.org/r/5186","timestamp":"2024-11-05T20:19:23Z","content_type":"text/html","content_length":"6444","record_id":"<urn:uuid:3e17f368-3575-45f2-aa84-835fa811c45c>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00153.warc.gz"} |
Live, Online, Honors Level, Homeschool Math Classes for Pre-Calculus
This class is designed to help homeschooling families succeed in their study of pre-calculus topics including functions, analytical geometry, probability, series, sequences, and trigonometry by
providing quality instruction and objective evaluation using Foerster’s Algebra and Trigonometry: Functions and Applications (Pearson Prentice Hall: 2005). This rigorous and highly regarded textbook
is designed to be completed in two years, so this class covers chapters 9 – 15. Students will be expected to spend roughly an hour and a half to two hours on homework daily. Class will meet two days
per week for one hour and five minutes each session for instruction. Since this is a homeschooling tutorial, parents will have the certain responsibilities as stated here.
Understand that this is an honors level class and not designed for the struggling or lazy math student. Students will be expected to complete homework assignments before the subsequent class,
participate in class with a working microphone, and proactively seek help when not understanding the material.
Students need to have completed Geometry as well as Algebra II using Foerster’s text with an A or B average or gain special permission from the instructor. It is recommended that students be at least
15 years old, entering at least 11th grade. Parents of younger students should contact the instructor before registering for the class.
The text used has recently changed publishers, but under the cover, the editions are exactly the same.
*Students may want to use a Graph Ruled Spiral Notebook for all of their work so they can draw graphs beside their calculations.
“Thank you, Mrs. Flynn, for the excellent preparation for AP Calculus that our daughter received in your Advanced Math (now named Pre-Calculus) class. After your online course, not only did she
successfully complete the AP class, but she took the AP Calculus AB/BC exam and achieved a 5. This gave her a great edge on her college applications and she has been accepted into the Honors Fellows
program at her top choice college. Thank you for your part in her math success!”
We would love to sign up [our daughter] for your Advanced Math class. I so appreciate what you have done wit…Read more | {"url":"https://www.libertytutorials.com/tutorials/pre-calculus/","timestamp":"2024-11-02T18:46:02Z","content_type":"text/html","content_length":"86338","record_id":"<urn:uuid:48b12a9b-04b4-4236-b8b7-704405bb7e23>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00718.warc.gz"} |
September 2015 - SignalSolver
This is a Berkshire Hathaway trading strategy which would have given almost ten times the return performance of buy/hold over the last 10 years with half the drawdown. The strategy is detailed in the
table below, it was straightforward, with 123 trades over the 10 year backtest period. All trading was done at the weekly close of business. This was a strategy where there was usually a buy and a
sell signal every week (398 out of 528 weeks), but selling was initiated by the presence of a sell signal and an absence of a buy signal. There was a strong buy bias, appropriate for the underlying
positive trend of the stock.
Equity curve, signals and positions are shown in the growth chart below. Notice the preponderance of white signals which indicate dual signal days.
If the sell point was set at zero percent, the algorithm gave positive results for all buy points greater than 0.25%. Below we show how return varies with buy and sell parameters.
The returns for every combination of buy and sell parameter are shown in the surface plot below.
Looking at the minimum annualized returns for each of the four 132 week periods, you can see that the algorithm was much better behaved than the underlying stock, worst case was 19.26% which happened
in the most recent quartus. The long side of the algorithm showed a min return of 15% with a stddev of only 3.37%, which is quite tight.
As always, this is not a recommendation to trade using this algorithm, just an interesting backtest result. For a list of trades, see here: BRKA.W Trades.
Please note, the above result was corrected 12/28/2015 to address a bug fix in the short side calculations.
Update Oct 21st 2016
GOOGL Trading Strategy (Weekly)
Here is a Google Inc (GOOGL) trading strategy with once a week intervention which would have performed significantly better than buy-hold over the last 10 years. Annualized return was 31.5% vs. 16.5%
(returning $149K for 10K outlay vs $36.7K, compounded), drawdown was 40.2% vs. 62.4%, so reward/risk was better.
The buy and cover signal (see table below) was present every week where the price dropped 0.03% below the last price the stock was sold at which happened 174 times over the course of the 528 weeks of
the analysis, and usually happened the week following a sell/short.
The sell and short signal happened every week the stock price rose 8.13% or more above the open price of the current week. This happened only 29 times, so there is a strong buy-side bias to this
strategy. All trading would have been done at the open of the week following the signal.
The equity curve for this GOOGL trading strategy shows that the stock was held short (the red bands in the background) for small periods, typically a week.
The scan below shows how the annualized return changed, had the buy and sell parameters changed. At 2.35% buy point, the algorithm gets stuck, resulting in a loss. This is characteristic of trading
strategies which reference buy or sell prices. At 1.5% sell point and below, the algorithm made a loss, but returns for all buy points above that were positive, for the 0.03% buy point.
One nice characteristic this algorithm had was consistent returns for each of the 132 week periods in the backtest. You can see from the table below that the annualized return was between 28% and
34.8% for every period. Compare that with buy-hold which ranged from -4.6% to 26%
You can also see this characteristic on the scans for each quartus:
You can download the list of trades in .xlsx format: GOOGL.W Trades.
As of Sept 16th 2015, the algorithm is long, awaiting a sell signal if the price hits 708.933. Last sell price was 654.34.
Please note, while this is an interesting backtest result, it is not a suggestion to trade this way. As always I don’t know how this strategy will fare in the future, but will track it from time to
This post was edited 12/28/15 to correct an error in the short-side returns.
Update Oct 21st 2016
This strategy has pretty much followed buy-hold long:
DIS Trading Strategy
Here is a Walt Disney Company (DIS) trading strategy with daily maintenance which had quite nice characteristics; 59% annualized return over the last 2 years.
The performance was better than long buy-hold with lower drawdown and about three times the reward-risk. Signal reinforcement was good, and not many dual signal days.
Below I show the return for the DIS trading strategy for each 6 month period going back 2 years. Comparing with long buy-hold you can see better consistency (higher minimum and lower StdDev).
Please note that this is simply a measurement, not an opinion or financial advice.
This post was corrected 1/8/2016 to correct a miscalculation in the short-side returns.
Update Oct 21st 2016
Strategy peaked 11/23/15, shortly after the stock price peaked.
Two GLD Trading Strategies (Daily)
GLD is the much traded SPDR Gold Trust ETF. I find these two GLD trading strategies interesting because they gave reasonable results (32.6% and 48% annualized return) for each of the four 6 month
periods of the analysis. The strategies require daily intervention.
Strategy 1: BCS AHC
This is a buy on fall, sell on rise strategy using the close price as the buy reference and the high price as the sell reference. As you can see from the life chart and the longevity analysis, the
lowest return for the last four 6 month periods was close to 30% annualized. You can easily find algorithms with over 50% annualized return for GLD, but they are not as consistent, with lowest
quartus returns of around 14%.
I would prefer to see more reinforcement on the signals, but there it is. As of Sat Sept 5th, this strategy is Short with no transactions pending.
You can view the trades in spreadsheet format here: GLD.D Trades
Strategy 2: AOO AHCI
In many ways this strategy shows better results than the BCS AHC strategy, for example there was lower drawdown, higher return, better signal reinforcement and good consistency (minimum 6 month
quartus return of 38.43%). On the other hand, signals were cluttered, with 67 dual signal days, 50 buy signal days and 75 sell signal days. Also, its not an trading strategy that makes intuitive
sense; buy on rise, sell on rise. Maybe it is one of those serendipitous occurrences, we shall see. Notice the sell strategy is almost the same as for BCS AHC but the sell signal percentage is quite
As of Sat Sept 5th, the strategy is Short with no transactions pending. For a list of trades in Excel format: GLD.D2 Trades. For a more detailed explanation of the above charts, please go here.
Algorithms were discovered by SignalSolver.
Please note, the above analysis was corrected on 12/28/2015 to reflect a bug fix in SignalSolver. Original returns were $9530 and $12753 respectively.
Update Oct 21st 2016
Both algorithms peaked 12/30/2015. | {"url":"https://www.signalsolver.com/2015/09/","timestamp":"2024-11-06T02:33:38Z","content_type":"text/html","content_length":"76194","record_id":"<urn:uuid:8f4ced75-d209-4c7f-9838-568f9f08821d>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00287.warc.gz"} |
libquadmath/printf/mul.c - gcc - Git at Google
/* mpn_mul -- Multiply two natural numbers.
Copyright (C) 1991, 1993, 1994, 1996 Free Software Foundation, Inc.
This file is part of the GNU MP Library.
The GNU MP Library is free software; you can redistribute it and/or modify
it under the terms of the GNU Lesser General Public License as published by
the Free Software Foundation; either version 2.1 of the License, or (at your
option) any later version.
The GNU MP Library is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
License for more details.
You should have received a copy of the GNU Lesser General Public License
along with the GNU MP Library; see the file COPYING.LIB. If not, write to
the Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
MA 02111-1307, USA. */
#include <config.h>
#include "gmp-impl.h"
/* Multiply the natural numbers u (pointed to by UP, with USIZE limbs)
and v (pointed to by VP, with VSIZE limbs), and store the result at
PRODP. USIZE + VSIZE limbs are always stored, but if the input
operands are normalized. Return the most significant limb of the
NOTE: The space pointed to by PRODP is overwritten before finished
with U and V, so overlap is an error.
Argument constraints:
1. USIZE >= VSIZE.
2. PRODP != UP and PRODP != VP, i.e. the destination
must be distinct from the multiplier and the multiplicand. */
/* If KARATSUBA_THRESHOLD is not already defined, define it to a
value which is good on most machines. */
#ifndef KARATSUBA_THRESHOLD
#define KARATSUBA_THRESHOLD 32
#if __STDC__
mpn_mul (mp_ptr prodp,
mp_srcptr up, mp_size_t usize,
mp_srcptr vp, mp_size_t vsize)
mpn_mul (prodp, up, usize, vp, vsize)
mp_ptr prodp;
mp_srcptr up;
mp_size_t usize;
mp_srcptr vp;
mp_size_t vsize;
mp_ptr prod_endp = prodp + usize + vsize - 1;
mp_limb_t cy;
mp_ptr tspace;
if (vsize < KARATSUBA_THRESHOLD)
/* Handle simple cases with traditional multiplication.
This is the most critical code of the entire function. All
multiplies rely on this, both small and huge. Small ones arrive
here immediately. Huge ones arrive here as this is the base case
for Karatsuba's recursive algorithm below. */
mp_size_t i;
mp_limb_t cy_limb;
mp_limb_t v_limb;
if (vsize == 0)
return 0;
/* Multiply by the first limb in V separately, as the result can be
stored (not added) to PROD. We also avoid a loop for zeroing. */
v_limb = vp[0];
if (v_limb <= 1)
if (v_limb == 1)
MPN_COPY (prodp, up, usize);
MPN_ZERO (prodp, usize);
cy_limb = 0;
cy_limb = mpn_mul_1 (prodp, up, usize, v_limb);
prodp[usize] = cy_limb;
/* For each iteration in the outer loop, multiply one limb from
U with one limb from V, and add it to PROD. */
for (i = 1; i < vsize; i++)
v_limb = vp[i];
if (v_limb <= 1)
cy_limb = 0;
if (v_limb == 1)
cy_limb = mpn_add_n (prodp, prodp, up, usize);
cy_limb = mpn_addmul_1 (prodp, up, usize, v_limb);
prodp[usize] = cy_limb;
return cy_limb;
tspace = (mp_ptr) alloca (2 * vsize * BYTES_PER_MP_LIMB);
MPN_MUL_N_RECURSE (prodp, up, vp, vsize, tspace);
prodp += vsize;
up += vsize;
usize -= vsize;
if (usize >= vsize)
mp_ptr tp = (mp_ptr) alloca (2 * vsize * BYTES_PER_MP_LIMB);
MPN_MUL_N_RECURSE (tp, up, vp, vsize, tspace);
cy = mpn_add_n (prodp, prodp, tp, vsize);
mpn_add_1 (prodp + vsize, tp + vsize, vsize, cy);
prodp += vsize;
up += vsize;
usize -= vsize;
while (usize >= vsize);
/* True: usize < vsize. */
/* Make life simple: Recurse. */
if (usize != 0)
mpn_mul (tspace, vp, vsize, up, usize);
cy = mpn_add_n (prodp, prodp, tspace, vsize);
mpn_add_1 (prodp + vsize, tspace + vsize, usize, cy);
return *prod_endp; | {"url":"https://gnu.googlesource.com/gcc/+/a787dfe21c21becea5c7180c4d6fe2c63957633f/libquadmath/printf/mul.c","timestamp":"2024-11-06T02:13:47Z","content_type":"text/html","content_length":"47751","record_id":"<urn:uuid:2b5aaa23-d540-4184-b799-2799283b5537>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00525.warc.gz"} |
Math Contest Repository
Hypatia 2024 Question 2, CEMC UWaterloo
(Hypatia 2024, Question 2, CEMC - UWaterloo)
For a positive $3$-digit integer $n$, $f(n)$ is equal to the sum of $n$ and the digits of $n$. For example, $f(351)=351+3+5+1=360$.
Note: The decimal representation of a $3$-digit number $abc$ is $a \cdot 10^2 + b \cdot 10 + c$. For example, $836 = 8 \cdot 10^2 + 3 \cdot 10 + 6$.
(a) What is the value of $f(132)$?
(b) If $f(n)=175$, what is the value of $n$?
(c) If $f(n)=204$, determine all possible values of $n$.
Answer Submission Note(s)
In part (c), sort your answers in ascending order and separate them with a comma.
Separate the answers for each part with a single space.
For example: "a b c1,c2"
Please login or sign up to submit and check if your answer is correct.
flag Report Content
You should report content if:
• It may be offensive.
• There is something wrong with it (statement or difficulty value)
• It isn't original.
Thanks for keeping the Math Contest Repository a clean and safe environment! | {"url":"https://mathcontestrepository.pythonanywhere.com/problem/hypatia24q2/","timestamp":"2024-11-12T19:03:28Z","content_type":"text/html","content_length":"10638","record_id":"<urn:uuid:9b0a3ec1-a110-42a8-a29e-6856f26e2582>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00363.warc.gz"} |
itertools.combinations_with_replacement() python | HackerRank Solutions
itertools.combinations_with_replacement() python
Problem Statement :
itertools.combinations_with_replacement(iterable, r)
This tool returns r length subsequences of elements from the input iterable allowing individual elements to be repeated more than once.
Combinations are emitted in lexicographic sorted order. So, if the input iterable is sorted, the combination tuples will be produced in sorted order.
Sample Code
>>> from itertools import combinations_with_replacement
>>> print list(combinations_with_replacement('12345',2))
[('1', '1'), ('1', '2'), ('1', '3'), ('1', '4'), ('1', '5'), ('2', '2'), ('2', '3'), ('2', '4'), ('2', '5'), ('3', '3'), ('3', '4'), ('3', '5'), ('4', '4'), ('4', '5'), ('5', '5')]
>>> A = [1,1,3,3,3]
>>> print list(combinations(A,2))
[(1, 1), (1, 3), (1, 3), (1, 3), (1, 3), (1, 3), (1, 3), (3, 3), (3, 3), (3, 3)]
You are given a string S.
Your task is to print all possible size k replacement combinations of the string in lexicographic sorted order.
Input Format
A single line containing the string S and integer value k separated by a space.
The string contains only UPPERCASE characters.
Output Format
Print the combinations with their replacements of string S on separate lines.
Solution :
Solution in C :
from itertools import combinations_with_replacement
s,k = input().split(' ')
k = int(k)
s = [x for x in s]
l = list(combinations_with_replacement(s,k))
for i in l:
s = ''
for k in i:
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There is a collection of input strings and a collection of query strings. For each query string, determine how many times it occurs in the list of input strings. Return an array of the results.
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Array Manipulation
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Print the Elements of a Linked List
This is an to practice traversing a linked list. Given a pointer to the head node of a linked list, print each node's data element, one per line. If the head pointer is null (indicating the list is
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Given a pointer to the head of a linked list, insert a new node before the head. The next value in the new node should point to head and the data value should be replaced with a given value. Return a
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View Solution → | {"url":"https://hackerranksolution.in/itertoolscombinationswithreplacementpython/","timestamp":"2024-11-15T04:37:41Z","content_type":"text/html","content_length":"39962","record_id":"<urn:uuid:6c72efe8-683e-404f-8215-8e6a1c72ebee>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00535.warc.gz"} |
Source term bricks (and Neumann condition)
Source term bricks (and Neumann condition)¶
This brick adds a source term, i.e. a term which occurs only in the right hand side of the linear (tangent) system build by the model. If \(f\) denotes the value of the source term, the weak form of
such a term is
\[\int_{\Omega} f v\ dx\]
where \(v\) is the test function. The value \(f\) can be constant or described on a finite element method.
It can also represent a Neumann condition if it is applied on a boundary of the domain.
The function to add a source term to a model is:
add_source_term_brick(md, mim,
varname, dataexpr, region = -1,
directdataname = std::string());
where md``is the model object, ``mim is the integration method, varname is the variable of the model for which the source term is added, dataexpr has to be a regular expression of GWFL, the generic
weak form language (except for the complex version where it has to be a declared data of the model). It has to be scalar or vector valued depending on the fact that the variable is scalar or vector
valued itself. region is a mesh region on which the term is added. If the region corresponds to a boundary, the source term will represent a Neumann condition. directdataname is an optional
additional data which will directly be added to the right hand side without assembly.
The brick has a working complex version.
A slightly different brick, especially dedicated to deal with a Neumann condition, is added by the following function:
add_normal_source_term_brick(md, mim,
varname, dataexpr, region);
The difference compared to the basic source term brick is that the data should be a vector field (a matrix field if the variable varname is itself vector valued) and a scalar product with the outward
unit normal is performed on it. | {"url":"https://getfem.readthedocs.io/en/latest/userdoc/model_source_term.html","timestamp":"2024-11-09T20:20:14Z","content_type":"text/html","content_length":"19098","record_id":"<urn:uuid:68024a6e-04b5-407e-8e7e-e87a808f8e15>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00186.warc.gz"} |
Tree transducer explained
In theoretical computer science and formal language theory, a tree transducer (TT) is an abstract machine taking as input a tree, and generating output – generally other trees, but models producing
words or other structures exist. Roughly speaking, tree transducers extend tree automata in the same way that word transducers extend word automata.
Manipulating tree structures instead of words enable TT to model syntax-directed transformations of formal or natural languages. However, TT are not as well-behaved as their word counterparts in
terms of algorithmic complexity, closure properties, etcetera. In particular, most of the main classes are not closed under composition.
The main classes of tree transducers are:
Top-Down Tree Transducers (TOP)
A TOP T is a tuple such that:
• is a finite set, the set of states;
• is a finite ranked alphabet, called the input alphabet;
• is a finite ranked alphabet, called the output alphabet;
• is a subset of Q, the set of initial states; and
• is a set of rules of the form
, where
is a symbol of Σ,
is the arity of
is a state, and
is a tree on Γ and
, such pairs being nullary.
Examples of rules and intuitions on semantics
For instance,
is a rule – one customarily writes
instead of the pair
– and its intuitive semantics is that, under the action of
, a tree with
at the root and three children is transformed into
where, recursively,
are replaced, respectively, with the application of
on the first child and with the application of
on the third.
The semantics of each state of the transducer T, and of T itself, is a binary relation between input trees (on Σ) and output trees (on Γ).
A way of defining the semantics formally is to see
as a term rewriting system, provided that in the right-hand sides the calls are written in the form
, where states
are unary symbols. Then the semantics
of a state
is given by
The semantics of
is then defined as the union of the semantics of its initial states:
Determinism and domain
As with tree automata, a TOP is said to be deterministic (abbreviated DTOP) if no two rules of δ share the same left-hand side, and there is at most one initial state. In that case, the semantics of
the DTOP is a partial function from input trees (on Σ) to output trees (on Γ), as are the semantics of each of the DTOP's states.
The domain of a transducer is the domain of its semantics. Likewise, the image of a transducer is the image of its semantics.
Properties of DTOP
• DTOP are not closed under union: this is already the case for deterministic word transducers.
• The domain of a DTOP is a regular tree language. Furthermore, the domain is recognisable by a deterministic top-down tree automaton (DTTA) of size at most exponential in that of the initial DTOP.
That the domain is DTTA-recognizable is not surprising, considering that the left-hand sides of DTOP rules are the same as for DTTA. As for the reason for the exponential explosion in the worst case
(that does not exist in the word case), consider the rule
. In order for the computation to succeed, it must succeed for both children. That means that the right child must be in the domain of
. As for the left child, it must be in the domain of
. Generally, since subtrees can be copied, a single subtree can be evaluated by multiple states during a run, despite the determinism, and unlike DTTA. Thus the construction of the DTTA recognising
the domain of a DTOP must account for
of states and compute the intersections of their domains, hence the exponential. In the special case of
DTOP, that is to say DTOP where each
appears at most once in the right-hand side of each rule, the construction is linear in time and space.
• The image of a DTOP is not a regular tree language.
Consider the transducer coding the transformation
; that is, duplicate the child of the input. This is easily done by a rule
, where
encodes the
. Then, absent any restrictions on the first child of the input, the image is a classical non-regular tree language.
• However, the domain of a DTOP cannot be restricted to a regular tree language. That is to say, given a DTOP T and a language L, one cannot in general build a DTOP
such that the semantics of
is that of
This property is linked to the reason deterministic top-down tree automata are less expressive than bottom-up automata: once you go down a given path, information from other paths is inaccessible.
Consider the transducer coding the transformation
; that is, output the right child of the input. This is easily done by a rule
, where
encodes the identity. Now let's say we want to restrict this transducer to the finite (and thus, in particular, regular) domain
. We must use the rules
. But in the first rule,
does not appear at all, since nothing is produced from the left child. Thus, it is not possible to test that the left child is
. In contrast, since we produce from the right child, we can test that it is
. In general, the criterion is that DTOP cannot test properties of subtrees from which they do not produce output.
• DTOP are not closed under composition. However this problem can be solved by the addition of a lookahead: a tree automaton, coupled to the transducer, that can perform tests on the domain which
the transducer is incapable of.^[2]
This follows from the point about domain restriction: composing the DTOP encoding identity on
with the one encoding
must yield a transducer with the semantics
, which we know is not expressible by a DTOP.
• The typechecking problem—testing whether the image of a regular tree language is included in another regular tree language—is decidable.
• The equivalence problem—testing whether two DTOP define the same functions—is decidable.^[3]
Bottom-Up Tree Transducers (BOT)
As in the simpler case of tree automata, bottom-up tree transducers are defined similarly to their top-down counterparts, but proceed from the leaves of the tree to the root, instead of from the root
to the leaves. Thus the main difference is in the form of the rules, which are of the form
• Book: Hubert. Comon. Max. Dauchet. Rémi. Gilleron. Florent. Jacquemard. Denis. Lugiez. Christof. Löding. Sophie. Tison. Marc. Tommasi. Tree Automata Techniques and Applications. November 2008.
Chapter 6: Tree Transducers . https://hal.inria.fr/hal-03367725/document. 11 February 2014. .
• Book: Hosoya, Haruo. Cambridge University Press. 978-1-139-49236-2. Foundations of XML Processing: The Tree-Automata Approach. 4 November 2010.
Notes and References
1. Baker, B.S.: Composition of top-down and bottom-up tree transductions. Inf. Control 41(2), 186–213 (1979)
2. Maneth. Sebastian. A Survey on Decidable Equivalence Problems for Tree Transducers. International Journal of Foundations of Computer Science. December 2015. 26. 8. 1069–1100. 10.1142/
S0129054115400134. 20.500.11820/2f1acef4-1b06-485f-bfd1-88636c9e2fe6. free.
3. Decidability results concerning tree transducers I. www.inf.u-szeged.hu. | {"url":"https://everything.explained.today/Tree_transducer/","timestamp":"2024-11-05T03:29:12Z","content_type":"text/html","content_length":"20845","record_id":"<urn:uuid:d8092637-0802-4751-9dc4-5ce1fc710329>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00472.warc.gz"} |
Objections to the Kalam Cosmological Argument for God
The Kalaam faces many issues. There are two big issues for me. First, because of premise 2, the argument is tied heavily with science and scientific discovery. This makes it dependent of scientific
trends and to the winds of change. The cutting edge of scientific discussion will change the various aspects of causality, but also the nature of time itself. In other words, unless you constantly
update the thing and hope the science continues in your favor, you are going to face a problem. Further, this makes it pragmatically useless to discuss in most cases because you need to understand
the science behind it. Without the science, you effectively are just appealing to whatever authority you like.
Second, and probably most importantly, the argument must turn into a different argument in a very short order. You will need to go back and defend another cosmological argument or similar proof of
god because the premises are highly questionable. The argument takes an observation about what occurs in our universe- "Things that begin to exist have a cause,"- and then applies this observation to
the nature of reality outside of the universe. It is currently unknown how reality outside of our universe would operate, if it exists at all. I could go on, but I think there are enough criticisms
of the argument that stick to make it problematic.
Chany OptionsShare
Need some counter-examples of causal closure.
every physical effect (i.e. caused event) has physical sufficient causes — http://philpapers.org/archive/VICOTC.pdf
Fishfry and, I expect most of the others that have responded to your thread, is/are perfectly familiar with Hilbert's Hotel, and why it is not an argument for any statement other than 'aren't
infinities interesting?' Ditto for Aristotle's notions of potential and actual infinities. — andrewk
(Y) Hilbert’s Hotel
Shandy’s Diary
, for example, are what we call veridical paradoxes, and do not imply a contradiction, but they do show some counter-intuitive implications of infinites.
Some select objections from the trenches …
• even if sound, the argument does not suggest anything “divine”, sentient, conscious, thinking, caring, loving, warranting worship or prayer, so the argument requires more to be of particular
• if gods/God can be atemporal (changeless, “outside of time”, or something), assuming that makes sense, then we might suppose any such “origin” of the universe
• if there was a definite earliest time (or “time zero”), then anything that existed at that time, began to exist at that time, and that includes any first causes, gods/God, or whatever else
• phrases like “before time” and “a cause of causation” are incoherent, violates identity (the 1^st law)
• by contemporary cosmology (e.g. Big Bang) spacetime is an aspect of the universe …
□ spatiality and temporality are aspects of the universe
□ causation is temporal
□ therefore causation is an aspect of the universe, not somehow “outside”
• dichotomistically …
□ if some God of theism could create something out of “nothing”, as it were, then nihil fit ex nihilo is already violated, and we might as well dispose of the principle, in which case said God
is an extraneous hypothesis
□ if some God of theism created the universe from something already existing, then whatever comprise the universe “always” existed, perhaps “eternally” (to the extent that’s meaningful), and we
might as well dispose of the extras, including said God
□ therefore God is neither implied nor necessary, and may be shaved off and flushed by parsimony
• there are viable alternatives to a definite earliest time, including an infinite past duration (which does not imply a contradiction however counter-intuitive), or a no-boundary, “edge-free”
universe (which is not infinite in past duration)
• the 1^st premise may be questionable or ambiguous in light of virtual particle pairs, quantum fluctuations, radioactive decay, spacetime foam/turbulence, the “pressure” of vacuum energy, the
Casimir effect, Fomin’s quantum cosmogenesis (and successors), etc
• God = not spatiotemporal incorporeal mind ⇒ such an “entity” would “be” no-where and no-when; thinking, decision making, acting, etc, would be impossible; such an entity could not be
characterized as “free”
• a supposed supernatural “beyond” is like an explanation that isn’t really much of an explanation to begin with (gaps, creative inventions)
"If you want to believe in a personal creator God, based on your personal spiritual experiences, it's perfectly reasonable for you to do so, and you can ignore the arguments as to why there is no
God, which are as flawed as the ones in favour."
To understand why this is you should google up and read the mathematician David Hibert's famous thought experiment , "Hibert's Hotel". — John Gould
Fishfry and, I expect most of the others that have responded to your thread, is/are perfectly familiar with Hilbert's Hotel, and why it is not an argument for any statement other than '
aren't infinities interesting
?' Ditto for Aristotle's notions of potential and actual infinities.
Kalam has been discussed ad nauseam on this forum and its predecessor. You are very unlikely to come up with any arguments in its favour that have not already been considered and dismissed. Don't you
think that, if it stood up to scrutiny, non-religious logicians might have noticed and written supportive papers about it?
If you want to believe in a personal creator God, based on your personal spiritual experiences, it's perfectly reasonable for you to do so, and you can ignore the arguments as to why there is no God,
which are as flawed as the ones in favour.
But try not to be tempted into the hubris of believing that the correctness of your belief can be logically proven, and the non-believers are just too silly to see that.
andrewk OptionsShare
Your arguments dont seem more solid than the those made by a logical positivist. Why do you assert that your reason is infallible and can interpret the world as it really is by itself? That logical
statements made by our fallible language can derrive at such mystical conclusions as proof for a personal creator? Why havent you ever even questioned the idea that Perhaps the whole idea of cause
and effect is an illusion etc? Would you agree if someone Said that the law of cause and effect can only be applied to things that doesnt have independent and continuing being? I guess you would.
Would you also agree that there then cant be any causal relationship between between entities that exist by their own power independent of the environment?
John Gould
What you are presuming is that actual infinities really exist. Actual infinity, however is merely an abstract notion , or, if you like, a fiction in the realm of the philosophy of mathematics which
proposes that mathematical objects like , say the infinite sequence of negative numbers you refer to above can form a complete totality or "set", I.e. a given object that is a true actual infinity.
Actual infinities, though, do not exist, they are not realities. To understand why this is you should google up and read the mathematician David Hibert's famous thought experiment , "Hibert's Hotel".
What makes anyone thing the universe "began to exist?" For all we know it's like the negative integers ..., -3, -2, -1. Every element has a predecessor (or "cause" if you like) but there is no first
What is the evidence for assumption P2?
fishfry OptionsShare
John Gould
NB: with regard to why the creator of the universe was (a la the Kalam argument) a PERSoNAL God, I'll explain that for you in detail tomorrow when I have more time. And as for your vexed query re the
notion of eternal punishment in Christian theology, please let's try to keep this discussion on track,
an invitation to present and debate substantive philosophical/scientific objections to the Kalam cosmological argument. St Augustine's position on the question of "eternal punishment" is a totally
separate issue, and at present I am not interested in explaining to you. It's something for a separate thread ( or your psychiatrist) Right?
John Gould
You say, "Why not say if the universe has a cause, that cause must be an effect of an effect of an effect" ? Here you are challenging img premise two ( P2) of the original Kalam argument : "The
Universe BEGAN to exist"; you are asking why we should not suppose that the universe NEVER began to exist, rather, there was an infinite number of past cause -and -effect events prior to today? The
reason is that an infinite number of things cannot exist. A POTENTIALLY infinite number of things can exist, but not an ACTUALLY infinite number of things. If one begins to argue than an ACTUALLY
infinite number of things can exist certain absurdities inevitably result. The best illustration of this is "Hilbert's Hotel", the brainchild of the great German Mathematician David Hibert. Google
"Hilbert's Hotel" for yourself, it a very accessible piece of reasoning written in clear and simple English ( even you Beebert will understand it and find the logic irrefutable)If you stil need
further explanations/demonstrations re why ACTUAL infinities are fictions that exist only in the domain of mathematical discourse and not actual realities, I will provide them for you.
Secondly, you ask why the universe could not have come into existence as a random phenomenon? Here you are challenging the first premise ( P1) of the traditional Kalam argument , namely : "Whatever
begins to exist must have a cause for its beginning" Your objection is easy to rebut because to claim that something can just suddenly, randomly, "pop" into existence "ex nihilo" ,as it were, without
any reasonable cause is worse than magic. When a magician pulls a rabbit out of a hat, at least you've got the hat, not to mention the magician. But if you deny P1. You are arguing that the whole
universe just appeared at some point in the past for no reason whatsoever. But NOBODY - even you - SINCERELY believes that things, say a kangaroo or a Rolls Royce car or a sky-scraper just pop into
being without a cause. Right ?
"Saint Augustine put it -" Si Comprehendis non Deus est" - i.e. if you understand Him , then he is not God. "
And yet this sadist called Augustine did Everything be could to "understand" this God with the help of a book, so that he could imagine him condemning as many People as possible to hell from before
the foundation lf the world(before the foundation of the world... What an illogical contradiction in terms in this case)
"P3. If the universe has a cause, then an uncaused, personal Creator of the universe exists, who sans the universe is beginningless, changeless, immaterial, timeless, spaceless and enormously
Well this is a grotesque leap from the former statement in the formula that the universe has a cause. Why not as well say "If the universe has a cause, that cause must be an effect of an effect of an
effect" or "If the universe has a cause, that cause must be random" or Whatever other stupid thing one might invent in one's stupid and proud head? I by the way claim that one can question that the
universe even has a cause. And Why must this creator be personal? Why not say that he must be mindless and stupid and without any power because he was FORCED to create?
Especially this is stupid. If he creates and is a cause, "he" becomes the the one who created.
There are so many other things worthy to question
"The Kalam argument provides strong evidence"
This would probably even make Immanuel Kant laugh, despite his belief in synthetic judgements apriori or whatever. For one thing, to even be able to even claim that "Whatever begins to exist has a
cause;" is true, one needs experience. But explain to me why this statement is even true?
Also, if God is all those things Kalam said he is (timeless etc), does that mean then that from God's point of view, the world has eternally and timelessly existed? If not, isnt he then changing?
John Gould
My claim is that the Kalam argument is, to date, very strongly supported by philosophical reasoning and hard experimental, empirical scientific evidence and thus it gives us powerful grounds for
believing in the EXISTENCE of a creator God. I did NOT say that the Kalam argument provides a means for us to KNOW anything about the nature of this God as He actually is in Himself. The Kalam
argument is the product of human reasoning, it logically suggests that IN OUR CRUDE, VAGUE,LIMITED, AND INADEQUATE HUMAN TERMS we might describe the God ( divine being) who created the universe we
inhabit 14 billion years ago as a beginningless, uncaused, atemporal, a -spatial, changeless, non-physical/immaterial, unconditional, enormously powerful, Personal creator.
Insofar as you are asking WHO this God is -what is His true nature as He is in Himself that is an entirely different question; a question the Kalam argument does not seek to answer. All we can
ultimately say is that this God -whom the Kalam argument gives us good reason to believe does exist is that with respect to humanity, He is transcendent and "wholly other". For us, He is utterly
unknowable, unspeakable, unthinkable and totally incomprehensible; He Himself is forever hidden from our view. If He were not, then He would not be God; or, as Saint Augustine put it -" Si
Comprehendis non Deus est" - i.e. if you understand Him , then he is not God. Briefly, there is, to be precise, no divine predicate/affirmation, no divine concept that contains in particular that
which the God who created the universe is, there is only the divine subject as such and in Him the fullness of His divine affirmation.
In short, The Kalam argument provides strong evidence for the existence of a divine creator God - God as a "known unknown".
Metaphysician Undercover
That seems unintelligible, for how can a first cause be simultaneously a final cause? — Brian A
"Final cause" refers to a type of causation, the "final" does not refer to a temporal order. So there is nothing unintelligible about the first cause being a final cause. This would just be to say
that the first cause is that type of cause, in comparison to an efficient cause for example.
I dont like any arguments that obsesses itself with thinking in causes like that. I would rather call it one of the weakest arguments. And plus, what need do we have for this kind of pseudo-proofs?
None. If one has Faith, one shall follow God. God doesnt reveal himself in "proofs" made by logical arguments.
"P3. If the universe has a cause, then an uncaused, personal Creator of the universe exists, who sans the universe is beginningless, changeless, immaterial, timeless, spaceless and enormously
Here is where Everything is lost. I would venture to say that the argument is even confused and weak right from the start. Part of learning to know God and realize he is seems to me to be to stop
reasoning about him like that.
Our common notion of causality requires the passage of time. What does it mean for something to "begin" to exist, or "have a cause" outside of time? — darthbarracuda
Without time, the cosmological argument doesn't make sense.
A cyclical universe could be posited to counter the Kalam argument. Each point in the cycle is both the end and the beginning. Since infinite regress doesn't come into the picture, we don't need to
have a first cause.
TheMadFool OptionsShare
Our common notion of causality requires the passage of time. What does it mean for something to "begin" to exist, or "have a cause" outside of time?
_db OptionsShare
Brian A
That seems unintelligible, for how can a first cause be simultaneously a final cause? Perhaps you mean that the final cause of the universe, viz. its teleology, demonstrates the hand of a Creator, as
it were. But that would be the teleological argument, not the Kalam cosmological argument mentioned in the first post.
The mere fact that the universe has a cause does not necessarily entail the view that the universe has a teleological purpose evidencing God. The latter may be so, but something beyond the poster's
argument is necessary to establish it.
Metaphysician Undercover
As much as this does not demonstrate that God is "personal", there are ways to derive the conclusion that the first cause is very likely the type of cause which is commonly referred to as "final
cause". Final cause is exemplified by freely willed actions. This is why it is often said that the existence of the universe is according to the will of God.
Brian A
Here is an objection that occurs to me: the Kalam cosmological argument does not necessarily lead to an omni-benevolent and personal God. Granting the premise that the universe must have a cause,
what prevents us from holding the view that the first cause might be impersonal and indifferent (not omni-benevolent)? And if the the first cause is impersonal and indifferent, it does not follow
that divine justice will be eventually rendered in reference to the actions of humanity. Therefore the usual view of God is not established.
P1. Whatever begins to exist has a cause;
P2. The universe began to exist;
C1. Therefore, the universe has a cause.
P3. If the universe has a cause, then an uncaused, personal Creator of the universe exists, who sans the universe is beginningless, changeless, immaterial, timeless, spaceless and enormously
C2. Therefore, an uncaused, personal Creator of the universe exists, who sans the universe is beginningless, changeless, immaterial, timeless, spaceless and enormously powerful.
One could take issue with any of the premises. Perhaps, in lieu of any actual constraints, spontaneous occurrence is possible. Perhaps the universe is beginningless. Perhaps the universe having a
cause doesn't entail the existence of an uncaused, personal Creator who sans the universe is beginningless, changeless, immaterial, timeless, spaceless and enormously powerful.
Michael OptionsShare
John Gould
I think the Kalam cosmological argument for the existence of God is one of the strongest defences for Theism that I have read.
If anyone has any material objections to the Kalam proof , I would be interested in hearing them.
What you are presuming is that actual infinities really exist. Actual infinity, however is merely an abstract notion , or, if you like, a fiction in the realm of the philosophy of mathematics
which proposes that mathematical objects like , say the infinite sequence of negative numbers you refer to above can form a complete totality or "set", I.e. a given object that is a true actual
infinity. Actual infinities, though, do not exist, they are not realities. To understand why this is you should google up and read the mathematician David Hibert's famous thought experiment ,
"Hibert's Hotel". — John Gould
Thanks for your comments.
I disagree with you on three points, summarized as follows.
Even if
I were using actual infinity, so what? After all, actual infinity is no weirder than an all-knowing, all-powerful, benevolent uncaused cause.
2. However, I am NOT using actual infinity. Your understanding of potential versus actual infinity is different than Aristotle's. My model "..., -3, -2, -1" only uses POTENTIAL infinity as defined by
Aristotle. I will expound on this point in a moment.
3. Hilbert's Hotel (HH) doesn't apply. You are correct that HH assumes actual infinity. But my example only requires potential infinity. I don't need all the numbers (or rooms) to exist all at once.
I only need that given one, I can identify the next. I never assume I have them all existing at once. That's potential infinity.
Here is more detail, especially on point #2.
1. For the moment let me grant your (false) premise. Say I did need actual infinity (which I remind you I don't). So what? Craig wants us to conclude that there must be an uncaused cause, which he
calls God. What if I call it absoulte infinity? I can't conceive of a worldview that would
grant divinity but not infinity
Cantor himself thought that his Absolute infinity was God. But Cantor's Absolute infinity is a lot bigger than the infinity of the natural numbers.
2. But #1 is irrelevant, since I don't use actual infinity, only potential. Let me outline the concept as seen by Aristotle. He said that we all have an intuition of the natural numbers 0, 1, 2, 3,
4, 5, ... Now the "dot dot dot" means something specific as defined by the
Peano axioms
Inductive axiom
: Given a number n, there is a number n' called the "successor" of n.
Another notion for the successor of n is n + 1. So if 0 exists, then 1 does. If 1 exists then 2 does. If 2 exists then 3 does.
So if you want to know, does 43242342 exist? Then you can recursively drill down all the way back to 0, the base of the induction, and you can show that any particular number exists.
There is never any claim that we have all of them together all at once
. We can imagine they don't come into existence till we need them. All I need is n + 1 given n. If I need a million, I make a million. I never have them all at once.
That is exactly
Aristotle's definition of potential infinity
. In the following quote, Aristotle is speaking of the endless regress 1/2, 1/4, 1/8, 1/16, etc. He says:
"For the fact that the process of dividing never comes to an end ensures that this activity exists potentially, but not that the infinite exists separately."
—Metaphysics, book 9, chapter 6.
Now the sequence 1/2, 1/4, 1/8, ... may be identified with the the reverse sequence I gave earlier, ..., -3, -2, -1 by the simple mathematical trick of taking the base-2 logarithm of each fraction to
get the corresponding negative integer.
1/2 maps to -1, and 1/4 maps to -2, and so forth. So these two examples are really the same example in different forms.
Aristotle calls this
infinity. Does 5 exist? Yes, if 4 exists. So we can prove that any number n exists. But we can't say that ALL the counting numbers exist all at the same time. That would be ACTUAL infinity.
It was the genius of Cantor to take the huge conceptual leap and say, What if we allow actual infinity into math? That was his brilliant history-changing leap of the imagination.
Cantor's insight was to write the following notation:
{0, 1, 2, 3, ...}
The braces symbolize the COMPLETED SET of natural numbers. The inductive axiom gives us 1 (given 0) then 2, then 3, and so forth.
Axiom of Infinity
says that there is a set containing all the natural numbers. That's actual infinity.
We can summarize all this with a table. I apologize that I could not make the right column line up no matter what I did with tabs and spaces. Advice appreciated.
Potential infinity Actual infinity
Axiom of induction Axiom of Infinity
Peano Cantor
0, 1, 2, 3, ... {0, 1, 2, 3, ...}
n+1 given n All of them at once
Negative integers Hilbert Hotel
I hope this is helpful.
3. HH is just a popularized visualization of the fact that an infinite set may be placed into bijection with one of its proper subsets. In fact this can be taken as the definining property of
infinite sets.
You are right that HH does assume actual infinity. But we do NOT need actual infinity to define the natural numbers 1, 2, 3, ... in their usual order, or ..., -3, -2, -1 in their reverse order, or 1/
2, 1/4, 1/8, ... if you use base 2 exponentials and logarithms.
I don't need the strength of the axiom of infinity to give my example. Only the Peano axioms, which define potential infinity. Given n I need n + 1. I never need to complete the process. I only need
to take the next step. So Hilbert's Hotel is not relevant here.
fishfry OptionsShare
John Gould
Fish fry,
Thank you for your detailed and very interesting response, though I do not want this thread to be diverted too far into the domain of the philosophy of mathematics; therefore let's put aside
"Hibert's Hotel" and the notions of potential and actual infinities altogether shall we, because there is, in fact, no stipulation in the Kalam argument that the cause of the origin of the universe
must be chronologically prior to that origin. Let's hypothesise instead, like Craig, that (for example) the Creator may be conceived causally, but not temporally prior to the origin of the universe
such that the act of causing the universe to exist is SIMULTANEOUSLY with its beginning to exist?
John Gould
I agree with what you say about the Kalam argument and the "cutting edge of science", it is noteworthy, however, that in a sense, the history of 20th century cosmology can be seen as one failed
attempt after another to avoid the absolute beginning of the universe predicted by the standard "Big Bang" model. That prediction is consistent with the Kalam argument and has now Stodden firm for
nearly 100 years throughout a period of enormous change and tremendous advances in observational astronomy and creative theoretical work in astrophysics (?)
You are right that the original Kalam cosmological argument refers only the space-time universe that we observe, that particular universe we human beings inhabit that is now believed by the majority
of mainstream scientists to have come into being 14 billion or so years ago with the "Big Bang", etc. Regardless of what may exist beyond it, and regardless of any unanswered metaphysical questions
about the true nature of absolute/ ultimate reality, our own universe is , in itself, no small, trifling thing , and in endeavouring to account for it the Kalam argument , in my opinion, is already
addressing a formidable challenge. In short, limited in scope as it is merely to our universe, Kalam still, in my opinion, provides a vey strong, logically sound and rationally compelling case for
the existence of a transcendent, supernatural Creator being. (And) that, for me, is quite enough food for thought just in itself ! | {"url":"https://thephilosophyforum.com/discussion/1742/objections-to-the-kalam-cosmological-argument-for-god","timestamp":"2024-11-06T14:40:31Z","content_type":"text/html","content_length":"84093","record_id":"<urn:uuid:66823769-d971-4573-9e07-b1e79e255737>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00742.warc.gz"} |
What is data structure? Definition, types, examples
Every program relies on data and algorithms. Algorithms are sets of instructions that dictate how data is processed to produce meaningful results. Data, on the other hand, represents the information
that algorithms operate on. Together, they form the foundation on which all software systems are built. Data structures play a crucial role in the interaction between algorithms and data.
This article explores the fundamentals of data structures and how they function, showing the differences between various types with examples. Whether you're just starting out or looking to enhance
your programming skills, this article will equip you with the knowledge needed to navigate the complexities of data structures.
What is data structure?
Data structure is a specialized format for organizing, processing, retrieving, updating, and storing data. Data structures serve as frameworks for arranging data for specific needs or objectives.
They not only store the actual data values but also maintain information about how those values are related to each other.
There are three main characteristics of a data structure that developers and data experts consider: correctness, time complexity, and space complexity.
Characteristics of a data structure
Correctness refers to the accuracy and reliability of the data structure's implementation. It should perform operations as expected, handle edge cases, and produce the right results.
Time complexity is the efficiency of data structure operations in terms of the time required to perform them. It helps us understand how the runtime of an operation grows as the input size increases.
Space complexity relates to the amount of memory an algorithm requires to perform its tasks efficiently on a particular data structure. Ideally, data structures should use as little memory as
possible. This is particularly important in resource-constrained environments or when dealing with large datasets.
Balancing these characteristics is essential for selecting data structures that meet the performance requirements and maintain accuracy and reliability in all scenarios.
Data structure types
This section will discuss the main data structure types, explaining their classification and functionalities. Check out the graph below to better understand how data structure types are categorized.
Data structure classification
Primitive data structures or types are the most basic data units available in programming languages. They include
• integers: whole numbers—positive, negative, or zero, for example, 1, -30, 100, etc.;
• floats: decimal numbers with a fractional part, like 3.14, -0.5, 7.0;
• characters: individual characters, for example, “A,” “d,” and “1”;
• strings: sequences of characters; and
• booleans: binary values that can be either true or false, commonly used for conditional expressions.
Primitive data structures represent simple values that cannot be broken down further into smaller components. They have a fixed size and format, and this predictability helps optimize memory usage.
Primitive data types are also consistent across various programming languages, with only slight variations in naming or specific implementations.
Non-primitive data structures are created with primitive data structures as their building blocks to efficiently organize and manage a collection of data. They can handle different data types and
complex operations like searching, sorting, insertion, deletion, and more.
Non-primitive data structures fall into two large categories: linear and non linear structures.
Linear data structures vs. non linear data structures
The following sections will explain each category in more detail.
Linear data structures
In a linear data structure, the elements are arranged sequentially in a particular order at one level. Each element has one predecessor and one successor, except for the first and last elements. This
arrangement allows for a single uninterrupted run or iteration through the structure, starting from one end and progressing to the other.
Linear data structures are typically straightforward and efficient for basic operations like adding, retrieving, or deleting. However, as the program becomes more complex, the limitations of this
category become apparent. Even though linear data structures allow for single-run traversal, the process can still be complex as you have to visit each element one by one from the beginning. This
results in a time complexity that increases linearly with the number of elements added.
Let’s look at the types of linear data structures in more detail.
Array data structure: Effective data storage and retrieval
An array stores elements at contiguous memory locations—next to each other without gaps. It’s homogeneous, meaning that all data within a structure is of the same data type (only integers,
characters, or others described earlier.) Other linear data structures can be homogeneous or heterogeneous depending on the programming language and their implementation.
In an array, each element is associated with an index, a numerical identifier that indicates the element's position. Indexing begins at 0 for the first element and increments sequentially up to the
array size minus one. For example, in an array of size 5, the indices range from 0 to 4, like in the picture below.
Compared to other linear structures, a key advantage here is that accessing any element takes the same time regardless of its position or the array’s size.
Depending on the programming language, arrays can be fixed or flexible in length. For example, in Java, you specify a constant size for an array, while in Python, you can create dynamic arrays.
Arrays are commonly used in algorithms that require random access to elements, such as searching and sorting. They are also effective for storing lists of information (like dates or addresses),
performing mathematical computations, and image processing. Additionally, arrays serve as the foundation for more complex data structures.
Stack data structure: Managing undo and accessing recent elements
A stack operates on the principle of Last In, First Out (LIFO), where inserting and retrieving data is possible from only one end. The last element added to the stack is the first one to be removed.
Stack elements have two primary operations—push and pop. Pushing adds a new element, making it the top of the stack. Popping removes the topmost element first, exposing the next one as the new top of
the stack. Regardless of the stack size, pushing or popping an element takes the same amount of time because there is always only one place to do the operation: the top of the stack.
Stacks are particularly useful for managing data when the order of operations is important and when you need to access the most recently added elements first. A clear example is undo mechanisms in
text editors, where each action performed by a user is recorded as a command and pushed onto the stack. When you initiate an undo operation, commands are popped from the stack, canceling the previous
actions and restoring the document to its previous state.
Queue data structure: Sequential processing of tasks or data
A queue operates on the principle of First In, First Out (FIFO), where the first element added to the queue is the first one to be removed. In this case, unlike with a stack, inserting and retrieving
data is done from different ends. In a queue, the elements are added at the back (enqueue) and removed from the front (dequeue). Similar to a stack, adding and removing an element takes the same time
regardless of the queue size.
Queues help process tasks or data in the order they were received. An example would be programs sending their print jobs, typically with only one printer available to process them sequentially. This
printer handles each job in order, one at a time. Additionally, in event-driven programming, where events are processed as they occur, queues preserve the correct sequence of actions in the system.
Linked list data structure: Flexibility with dynamically growing data
A linked list consists of elements called nodes, each containing both data and a pointer (reference) to the next node in the sequence. The first node is called the head, and the last one has a null
reference, indicating the end of the list. Each node can be placed at any available memory location, with the references between nodes enabling the traversal of the list.
Linked list data structure
Linked lists can grow or shrink in size dynamically, making them suitable for situations where the size of the data structure is not known in advance or may change. An example is a web browser
history, where a webpage is represented as a node containing a reference to the next webpage visited.
Non linear data structures
Non linear data structures store elements not sequentially but rather in a hierarchical or arbitrary manner at different levels, often handling multiple connections. In such structures, you cannot
traverse data in a single run.
Non linear data structures are more difficult to implement but offer better performance for complex operations. They maintain constant time and space complexity as the size of the data grows.
Non-linear structures are also more memory efficient as they allow the same connections to be shared among multiple elements. This reduces memory consumption compared to linear structures, where each
element is connected to only one predecessor and/or successor.
Let's dive into each type of non-linear data structure.
Tree data structure: Depicting hierarchical relationships with categories
A tree is a collection of elements (nodes) connected with edges that reflect parent-child relationships. Each node (except the root—the topmost node) has a single parent. Nodes without children are
known as leaves. You’ve definitely seen this structure when looking at a family tree or organizational chart.
The image above shows a binary tree where each node has at most two children. There are also trinary trees with three children per node and n-ary trees where each node can have up to n children, with
n being a fixed positive integer.
The height of a tree is the longest distance from the root to any leaf. The depth of a node is the length of the path from the root to that particular node. In other words, it is the level at which
the node resides in the tree.
Trees are commonly used in databases to represent categories and subcategories and in file systems—for hierarchically organizing files and directories.
Heap data structure: Prioritizing tasks by importance
A heap is a binary tree where the topmost node always has the highest or the lowest value in the whole tree. This property is called the heap property or heap invariant. There are two types of heaps.
In a min-heap, the value of each node is less than or equal to the values of its children. In a max-heap, the value of each node is greater than or equal to the values of its children.
With the heap data structure, we can implement priority queues, where elements have different importance. For example, job scheduling applications need to organize tasks based on urgency or deadline,
and that’s where heaps come in handy.
Trie data structure: Efficient storage and retrieval of text strings
A trie, also referred to as a prefix tree or a keyword tree, is used for storing and retrieving a set of strings. In this structure, each node contains a character—typically, a letter of the
A prefix is one or more characters that form the beginning part of one or more strings. For example, in the image above, b is a common prefix for the words ball and bee, while an is a prefix for ant
and angel.
Tries are applied in cases where efficient storage and retrieval of strings are required, particularly when dealing with large datasets or dictionaries. A common example is autocompletion, where the
trie structure allows quick suggestions based on partial input strings. In spell-checking algorithms, tries can also determine whether a given string is a valid word and can suggest corrections.
Graph data structure: Visualizing relationships in complex networks
A graph is a data structure used to represent relationships between entities. It consists of a set of nodes, also known as vertices, and each vertex connects to others through edges. One of the
fundamental distinctions between graphs and trees is that graphs can contain cycles, while trees cannot. A cycle is a path in the graph that starts and ends at the same vertex, traversing edges
without repeating any vertex.
The graph structure can depict complex network relationships, enabling efficient analysis and processing of interconnected data. For example, graphs can represent computer networks, molecular
structures, and circuit layouts.
Hash table data structure: Streamlining searches and sorting operations
A hash table or hash map stores key-value pairs in an array. The key is a unique identifier to access or retrieve the value, which is the associated data or information.
Hash table data structure
The hash function takes a key as input and produces a unique numerical value called a hash code that serves as an index for a particular data element. This mapping process is deterministic, meaning
the same key will always be hashed to the same index, facilitating rapid lookup and retrieval of values.
We use hash tables to quickly map keys to specific locations in the array. The goal of the hash function is to distribute keys appropriately across the array indices, minimizing collisions
(situations where multiple keys map to the same index).
Hash tables are common in scenarios where efficient storage and retrieval of key-value pairs are required — for example, caching systems. Additionally, we employ hash tables in data processing tasks
such as indexing and deduplication.
The evolution of data structures
Linear data structures emerged to address simple storage and retrieval needs, while non-linear structures became necessary to represent complex relationships and optimize algorithmic performance.
With the increasing volumes of data being generated and evolving domain-specific challenges, there will be a continued focus on developing data structures that offer improved efficiency, scalability,
and resilience.
Future advancements in areas like artificial intelligence, Internet of Things (IoT), blockchain, and edge computing may lead to the development of specialized data structures that cater to the unique
requirements of these domains. It might involve advancements in data structure design, algorithmic optimizations, and parallel processing techniques.
Adaptability and openness to change are essential in the ever-changing field of technology. Overlooking this may result in performance bottlenecks, limited problem-solving abilities, and suboptimal
solutions that are slower, consume more memory, and have poorer scalability. | {"url":"https://www.altexsoft.com/blog/data-structure/","timestamp":"2024-11-06T15:22:08Z","content_type":"text/html","content_length":"128191","record_id":"<urn:uuid:7358795a-7722-4540-b0dd-152ab07ee5c8>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00223.warc.gz"} |
(All citation calculations based on
Total Publications
Data From 483 Publications
This is the number of all publications for the study.
Total Citations
Data From 425 Publications
This is the number of citations to all publications for the project.
Mean Citations Per Publication
Data From 425 Publications
The total number of citations divided by the total number of publications.
Age-weighted Mean Citations Per Publication
Data From 483 Publications
Age-weighted Mean Citations Per Publication.
Median Citations
Data From 425 Publications
The median number of citations for the project.
Data From 425 Publications
h-index is the largest number h such that h publications from a study have at least h citations.
Data From 425 Publications
The number of publications that have at least 100 citations.
Data from 483 publications. (What does this mean?)
Data from 483 publications. (What does this mean?)
Data from 425 publications. (What does this mean?)
Data from 425 publications. (What does this mean?) | {"url":"https://arcnwc.github.io/publications/metrics/index.html","timestamp":"2024-11-05T16:15:53Z","content_type":"text/html","content_length":"7335","record_id":"<urn:uuid:bf3b06b7-fb77-41bd-9d9c-245eeaf3a8c6>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00527.warc.gz"} |
How do you calculate shear rate and viscosity?
The shear rate is the velocity of the upper plate (in meters per second) divided by the distance between the two plates (in meters). Its unit is [1/s] or reciprocal second [s-1]. According to
Newton’s Law, shear stress is viscosity times shear rate. Therefore, the viscosity (eta) is shear stress divided by shear rate.
How do you calculate the shear rate of a viscometer?
When we perform rheological experiments with a viscometer, usually it gives us shear stress values corresponding to particular revolution per minute (rpm). Then we convert rpm into a shear rate by
multiplying a factor 1.7.
How do you calculate extruder shear rate?
To calculate the shear rates between the screw flight and the barrel wall, use Eqn (8.1) with h = distance between the screw flight and the barrel wall. where τ = Shear stress = F/A = Force applied
per unit area. An example of the resistance factor in a rod die on the flow rate is shown in Figure 8.1.
What is shear viscosity?
1. A coefficient that characterizes the viscous properties of a fluid and is related to the absorption (loss) of energy (or else, damping) due to the presence of velocity gradients in the fluid.
What is shear measured in?
Physical quantities of shear stress are measured in force divided by area. In SI, the unit is the pascal (Pa) or newtons per square meter. In United States customary units, shear stress is also
commonly measured in pounds-force per square inch or kilopounds-force per square inch.
What are the units of shear rate?
Shear rate is the speed of deformation in the shear mode (which is typical of fluids and can be represented as layers sliding one onto another). It is expressed (SI units) in reciprocal seconds [1/
s], since it derives from radiant per second, with radiant being dimensionless (just a number).
Does viscosity depend on shear rate?
The viscosity remains constant with changing shear rate. However, most fluids are non-Newtonian, that is, their viscosities are a function of shear rate. Therefore, changing the geometry, such as the
distance between the two plates, above, and/or the speed of flow, may significantly change the viscosity.
What is the unit of shear rate?
The SI unit of measurement for shear rate is s −1, expressed as “reciprocal seconds” or “inverse seconds”. The shear rate at the inner wall of a Newtonian fluid flowing within a pipe is. where: .γ is
the shear rate, measured in reciprocal seconds; v is the linear fluid velocity; d is the inside diameter of the pipe.
How is shear strain calculated?
Shear Strain Example First, measure the original length. Measure the original length, often considered the height of a square object. Next, measure the deformation. After a shear force has been
applied to the object, measure the deformation. Finally, calculate the shear strain. Calculate the shear strain by dividing the deformation by the original length.
What is shear ratio?
Shear modulus, in materials science, is defined as the ratio of shear stress to shear strain. The shear modulus value is always a positive number and is expressed as an amount of force per unit area.
What is wall shear stress and shear strain rate?
wall shear stress is the force the ‘tear’ the fluid near the wall due to no slip boundary condition. shear strain rate is the rate of fluid being ‘teared’ . higher shear strain rate means the fluid
is being ‘teared’ more compare to lower shear strain rate when the time is constant. | {"url":"https://pegaswitch.com/popular/how-do-you-calculate-shear-rate-and-viscosity/","timestamp":"2024-11-04T21:14:18Z","content_type":"text/html","content_length":"128624","record_id":"<urn:uuid:13e8d244-d741-45ff-a446-c848b634654e>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00456.warc.gz"} |
Excel Formula Python - Select Cell with Sum of Values
In this tutorial, we will learn how to write an Excel formula in Python that selects a cell from a range based on the sum of values. This formula can be useful when you have a range of cells and you
want to find the cell that has a cumulative sum of values up to a specific number.
To achieve this, we will use the INDEX and MATCH functions in Excel, which can be implemented in Python using the openpyxl library. The formula we will use is as follows:
=INDEX(F12:F19, MATCH(TRUE, MMULT(--(F12:F19<=48958000), TRANSPOSE(COLUMN(F12:F19)^0))=1, 0))
Let's break down the formula step by step:
1. The MMULT function is used to calculate the cumulative sum of values in the range F12:F19. It converts the values to an array of 1s and 0s, where 1 represents the cells that have a cumulative sum
less than or equal to the specified number.
2. The TRANSPOSE function is used to transpose the array of 1s and 0s, so that it can be multiplied with another array.
3. The COLUMN function is used to create an array of numbers from 1 to the number of cells in the range F12:F19.
4. The ^0 operation is used to convert the array of numbers to an array of 1s.
5. The -- operation is used to convert the array of 1s and 0s to an array of TRUE and FALSE values.
6. The MMULT function multiplies the array of TRUE and FALSE values with the transposed array of 1s and 0s, resulting in an array of 1s and 0s where 1 represents the cells that have a cumulative sum
less than or equal to the specified number.
7. The MATCH function is used to find the position of the first occurrence of TRUE in the array of 1s and 0s.
8. The INDEX function is used to return the value from the range F12:F19 at the position found by the MATCH function.
To use this formula in Python, you can install the openpyxl library and use the following code:
import openpyxl
workbook = openpyxl.load_workbook('path/to/excel/file.xlsx')
worksheet = workbook['Sheet1']
cell_value = worksheet['F12'].value
Replace 'path/to/excel/file.xlsx' with the actual path to your Excel file, and 'Sheet1' with the name of your worksheet. The formula will be applied to the range F12:F19, and the selected cell value
will be printed.
In conclusion, this tutorial has shown you how to write an Excel formula in Python to select a cell from a range based on the sum of values. This can be useful in various scenarios where you need to
find a specific cell that meets certain criteria. By using the openpyxl library, you can easily implement this formula in your Python code and work with Excel files programmatically.
An Excel formula
=INDEX(F12:F19, MATCH(TRUE, MMULT(--(F12:F19<=48958000), TRANSPOSE(COLUMN(F12:F19)^0))=1, 0))
Formula Explanation
This formula uses the INDEX and MATCH functions to select the cell from F12 to F19 that has a sum of values up to 48,958,000.
Step-by-step explanation
1. The MMULT function is used to calculate the cumulative sum of values in the range F12:F19. It converts the values to an array of 1s and 0s, where 1 represents the cells that have a cumulative sum
less than or equal to 48,958,000.
2. The TRANSPOSE function is used to transpose the array of 1s and 0s, so that it can be multiplied with another array.
3. The COLUMN function is used to create an array of numbers from 1 to the number of cells in the range F12:F19.
4. The ^0 operation is used to convert the array of numbers to an array of 1s.
5. The -- operation is used to convert the array of 1s and 0s to an array of TRUE and FALSE values.
6. The MMULT function multiplies the array of TRUE and FALSE values with the transposed array of 1s and 0s, resulting in an array of 1s and 0s where 1 represents the cells that have a cumulative sum
less than or equal to 48,958,000.
7. The MATCH function is used to find the position of the first occurrence of TRUE in the array of 1s and 0s.
8. The INDEX function is used to return the value from the range F12:F19 at the position found by the MATCH function.
For example, if we have the following data in cells F12 to F19:
| F |
| |
| 100 |
| 200 |
| 300 |
| 400 |
| 500 |
| 600 |
| 700 |
| 800 |
The formula =INDEX(F12:F19, MATCH(TRUE, MMULT(--(F12:F19<=48958000), TRANSPOSE(COLUMN(F12:F19)^0))=1, 0)) would return the value 400, which is the cell from F12 to F19 that has a sum of values up to | {"url":"https://codepal.ai/excel-formula-generator/query/1m7vp7Ek/excel-formula-python-sum-values","timestamp":"2024-11-09T01:08:50Z","content_type":"text/html","content_length":"103327","record_id":"<urn:uuid:841d119b-fdc3-42e2-b85e-77d3c4e293da>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00592.warc.gz"} |
Multiple Charts In One Excel 2024 - Multiplication Chart Printable
Multiple Charts In One Excel
Multiple Charts In One Excel – You may create a multiplication graph in Excel by using a format. You will discover a number of types of web templates and learn to file format your multiplication
graph using them. Here are some tricks and tips to produce a multiplication graph. Once you have a design, all you have to do is version the formula and mixture it inside a new cell. You may then
take advantage of this formula to grow some numbers by yet another set. Multiple Charts In One Excel.
Multiplication dinner table web template
You may want to learn how to write a simple formula if you are in the need to create a multiplication table. Initially, you have to locking mechanism row one of the header column, then grow the
telephone number on row A by mobile B. A different way to develop a multiplication kitchen table is to apply mixed personal references. In this instance, you would probably enter in $A2 into line A
and B$1 into row B. The outcome is a multiplication desk having a formula that really works both for columns and rows.
If you are using an Excel program, you can use the multiplication table template to create your table. Just open the spreadsheet with the multiplication desk template and change the label for the
student’s label. You can also adjust the sheet to fit your personal needs. There is an solution to affect the shade of the cells to change the look of the multiplication desk, way too. Then, you can
change all the different multiples for your needs.
Creating a multiplication graph or chart in Stand out
When you’re utilizing multiplication desk software program, it is possible to create a easy multiplication kitchen table in Stand out. Merely build a page with columns and rows numbered in one to 35.
Where rows and columns intersect may be the respond to. For example, if a row has a digit of three, and a column has a digit of five, then the answer is three times five. The same goes for the
Initially, you may enter into the amounts that you have to flourish. For example, if you need to multiply two digits by three, you can type a formula for each number in cell A1. To create the figures
larger sized, pick the cellular material at A1 and A8, after which click on the proper arrow to decide on a variety of cellular material. You can then type the multiplication solution within the
tissue inside the other rows and columns.
Gallery of Multiple Charts In One Excel
How To Plot Multiple Data Sets In One Excel Chart YouTube
How To Quickly Make Multiple Charts In Excel YouTube
Multiple Bar Charts On One Axis In Excel Super User
Leave a Comment | {"url":"https://www.multiplicationchartprintable.com/multiple-charts-in-one-excel/","timestamp":"2024-11-06T08:00:02Z","content_type":"text/html","content_length":"53021","record_id":"<urn:uuid:2db50c2d-4e98-419f-8a7a-1d257364a233>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00538.warc.gz"} |
: .. : : 70 : : 2017 :
.. // . 70. .: , 2017. .136-167. URL: https://doi.org/10.25728/ubs.2017.70.6
, , ,
cellular automaton, autonomous agents, reflexive agents, agents' formation control
. , , , . -- .
The article deals with the algorithm for a distributed organization of the agents' formation defined by a graph and the numerical simulation of such algorithm. Agents move through terrain with many
random obstacles (``random landscape''). At first, we describe the continuous statement of the two-criteria minimization problem. The first criterion is the agent's route time. The second criterion
is the closeness of agents' formation to the desired one. Next, we introduce a cellular automaton simulating the movement of agents for obtaining quasi-optimal solutions of the problem. The cellular
automaton has one-dimensional and two-dimensional representations. Agents use reflexion to predict the motion of other agents. Then we compare obtained solutions with optimal ones for different types
of random landscapes via numerical experiment. At this point, we obtain the empirical distribution for the time of an agent's exit to finish point. Finally, we find the relation between a type of
random landscape through which agents move and the quality of agents' formation.
PDF -
: 3073, : 2815, : 23. | {"url":"http://www.mtas.ru/search/search_results_new.php?publication_id=21665&IBLOCK_ID=20","timestamp":"2024-11-13T14:56:42Z","content_type":"text/html","content_length":"12995","record_id":"<urn:uuid:63e9c725-e7b9-470c-8d70-fb7ebdf0807a>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00143.warc.gz"} |
What is 98 Celsius to Fahrenheit? - ConvertTemperatureintoCelsius.info
98 Celsius to Fahrenheit Conversion: Understanding the Relationship Between Celsius and Fahrenheit
When it comes to measurements of temperature, Celsius and Fahrenheit are two commonly used units. Each unit provides a different scale for measuring temperature, and the conversion between them can
be confusing at times. In this article, we will explore the relationship between Celsius and Fahrenheit and specifically answer the question, “What is 98 Celsius to Fahrenheit?”
Understanding Celsius and Fahrenheit
First, let’s understand the basics of Celsius and Fahrenheit. Celsius is a metric unit for measuring temperature and is commonly used worldwide. It is based on the freezing point of water at 0
degrees and the boiling point of water at 100 degrees. On the other hand, Fahrenheit is a measurement unit commonly used in the United States. It is based on a scale in which the freezing point of
water is 32 degrees and the boiling point is 212 degrees.
The Conversion Formula
To convert Celsius to Fahrenheit, you can use the following formula:
°F = (°C × 9/5) + 32
This formula allows you to easily convert temperatures from Celsius to Fahrenheit and vice versa. Now, let’s apply this formula to the specific question at hand: “What is 98 Celsius to Fahrenheit?”
Converting 98 Celsius to Fahrenheit
Using the conversion formula mentioned earlier, we can calculate the equivalent temperature in Fahrenheit for 98 degrees Celsius:
°F = (98 × 9/5) + 32
°F = (176.4) + 32
°F = 208.4
Therefore, 98 degrees Celsius is equivalent to 208.4 degrees Fahrenheit. This simple calculation provides the answer to the initial question and demonstrates the conversion between Celsius and
When using this information in everyday life, it’s important to remember that understanding temperature measurements in both Celsius and Fahrenheit can be beneficial. Whether you’re traveling to a
different country or simply using a recipe from another part of the world, having a grasp of both temperature units can be valuable.
Additionally, with the increasing globalization and interconnectedness of the world, having this knowledge can foster better communication and understanding across different cultures and regions. It
can also aid in scientific and academic pursuits, where knowledge of temperature measurements is essential.
In conclusion, understanding the conversion between Celsius and Fahrenheit is a valuable skill that can be applied in various aspects of life. By following the simple conversion formula and learning
how to convert temperatures from one unit to another, individuals can expand their knowledge and adapt to different measurement systems. So, the next time you come across the question, “What is 98
Celsius to Fahrenheit?” you’ll be equipped with the knowledge to provide a definitive answer. | {"url":"https://converttemperatureintocelsius.info/what-is-98celsius-in-fahrenheit/","timestamp":"2024-11-05T23:24:58Z","content_type":"text/html","content_length":"73124","record_id":"<urn:uuid:491a576b-727b-483c-9610-f2e7027e50ad>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00153.warc.gz"} |
ES386-15 Dynamics of Vibrating Systems
Introductory description
ES386-15 Dynamics of Vibrating Systems
Module aims
Vibrations exert a significant influence on the performance of the majority of engineering systems. All engineers should understand the basic concepts and all mechanical engineers should be familiar
with the analytical techniques for the modelling and quantitative prediction of behaviour. Thus, this module provides students with fundamental skills necessary for the analysis of the dynamics of
mechanical systems, as well as providing opportunities to apply these skills to the modelling and analysis of vibration.
This third-year module is mandatory for students pursuing a degree in Mechanical Engineering, building upon competences acquired earlier in the course. This module introduces students to the use of
Lagrange’s equations (applied to 1D and 2D systems only for this module) and to techniques for modelling both lumped and continuous vibrating systems. It includes some coverage of approximate methods
both as an aid to physical understanding of the principles and because of their continuing usefulness. The module assumes basic understanding of mechanics and linear algebra consistent with the level
of Year 2 modules.
At the end of the module students should have a sound understanding of the wide application of vibration theory and of the underlying physical principles. In particular, they should be able to use
either Newtonian or Lagrangian mechanics to analyse 2D systems, and to determine the response of simple damped and undamped multi-degrees of freedom (DOF) systems to both periodic and aperiodic
excitation. They should also be familiar with engineering solutions for measuring and influencing vibrational behaviour.
Outline syllabus
This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.
• Generalised co-ordinates, Lagrange's equation (including preliminary study of other classical methods)
• General application of the Lagrange equation to vibrating systems
• Multi-degree of freedom systems: lumped system models, continuous system models; geared and branched systems; reduction of an n-DOF system to a set of n single-DOF systems; principal co-ordinates
• Matrix methods of analysis: conservative and non-conservative (damped) systems; determination of principal co-ordinates
• Modelling of damping: hysteretic, Coulomb, viscous; measurement of damping factor
• Forced vibration: harmonic excitation of multi-DOF systems; shaft whirling; transmissibility; vibration isolation; non-harmonic and arbitrary excitation (convolution integral)
• Approximate methods e.g. Rayleigh's method, Dunkerley's method
Learning outcomes
By the end of the module, students should be able to:
• 1. Model planar mechanical systems using Newton’s and Lagrange’s equations: Determine appropriate co-ordinate systems, analyse vibrations.
• 2. Abstract more complex engineering mechanisms: analyse using lumped system models or simple distributed mass and stiffness models. Use and justify standard methods and approximations for
extended and continuous vibrating systems.
• 3. Evaluate the natural frequencies and modes of vibration of a multi-degree of freedom linear system.
• 4. Determine and analyse the free and forced response of single-degree of freedom systems to periodic and aperiodic excitation, as well as the effects of linear and non-linear damping on the
system behaviour.
• 5. Evaluate complex (multi-degree of freedom) undamped or damped systems numerically, using a systematic approach to analyse the natural frequencies and modes, and the response of the system to
periodic and aperiodic excitations.
• 6. Demonstrate a sound understanding of the application of vibration analysis to key engineering systems.
Indicative reading list
1. Theory of Vibration with Applications, by W. T. Thomson and M. D. Dahleh. Publisher: Pearson. Fifth edition, 1998. ISBN-10: 013651068X, ISBN-13: 9780136510680.
2. Principles of Vibration, by B. H. Tongue. Publisher: Oxford University Press. Second edition, 2002. ISBN-10: 0195142462.
3. Engineering Vibrations, by D. J. Inman. Publisher: Pearson. Fourth international edition, 2013. ISBN-10: 0273768441, ISBN-13: 9780273768449.
4. Mechanical vibrations, by S. S. Rao, Fook Fah Yap. Publisher: Prentice Hall. Fifth edition in SI units, 2011. ISBN-10: 9810687125, ISBN-13: 9789810687120
5. Vibrations, by B. Balachandran, E. B. Magrab. Publisher: Cengage. Second international SI edition, 2009. ISBN10: 0495411256, ISBN-13: 9780495411253.
View reading list on Talis Aspire
Subject specific skills
SSS4: Ability to apply relevant practical and laboratory skills.
SSS8: Ability to be pragmatic, taking a systematic approach and the logical and practical steps necessary for, often complex, concepts to become reality.
Transferable skills
TS1: Numeracy: apply mathematical and computational methods to communicate parameters, model and optimize solutions.
TS2: Apply problem solving skills, information retrieval, and the effective use of general IT facilities.
TS3: Communicate (written and oral; to technical and non-technical audiences) and work with others.
TS7: Overcome difficulties by employing skills, knowledge and understanding in a flexible manner.
Study time
Type Required
Lectures 30 sessions of 1 hour (25%)
Seminars 2 sessions of 1 hour (2%)
Practical classes 1 session of 2 hours (2%)
Private study 86 hours (72%)
Total 120 hours
Private study description
Guided independent learning, assignment preparation, etc 86 hours.
No further costs have been identified for this module.
You must pass all assessment components to pass the module.
Students can register for this module without taking any assessment.
Assessment group D4
Weighting Study time Eligible for self-certification
Vibration computational and analysis assignment 30% Yes (extension)
Matlab code submitted on Matlab Grader (10% of module credit) and a brief computational report (1000 words, 20% of module credit)
Online Examination 70% No
QMP test (2x 1h)
~Platforms - AEP,QMP
• Online examination: No Answerbook required
• Students may use a calculator
• Engineering Data Book 8th Edition
Feedback on assessment
• Feedback during laboratory sessions
• Feedback on assignments.
• Model solutions to exam type questions.
• Support through advice and feedback hours.
• Cohort level feedback on examinations
To take this module, you must have passed:
This module is Core for:
• Year 3 of UESA-H310 BEng Mechanical Engineering
• Year 3 of UESA-H315 BEng Mechanical Engineering
• Year 4 of UESA-H314 BEng Mechanical Engineering with Intercalated Year
• Year 3 of UESA-HH35 BEng Systems Engineering
• Year 3 of UESA-HH36 BEng Systems Engineering
• Year 4 of UESA-HH34 BEng Systems Engineering with Intercalated Year
• Year 3 of UESA-H311 MEng Mechanical Engineering
• UESA-H316 MEng Mechanical Engineering
□ Year 3 of H315 Mechanical Engineering BEng
□ Year 3 of H316 Mechanical Engineering MEng
• Year 4 of UESA-H317 MEng Mechanical Engineering with Intercalated Year
• UESA-HH31 MEng Systems Engineering
□ Year 3 of HH31 Systems Engineering
□ Year 3 of HH35 Systems Engineering
• Year 4 of UESA-HH32 MEng Systems Engineering with Intercalated Year
This module is Core optional for:
• Year 3 of UESA-H115 MEng Engineering with Intercalated Year
• UESA-H317 MEng Mechanical Engineering with Intercalated Year
□ Year 3 of H317 Mechanical Engineering with Intercalated Year
□ Year 4 of H317 Mechanical Engineering with Intercalated Year
• Year 4 of UESA-HH32 MEng Systems Engineering with Intercalated Year
• Year 3 of UESA-H11L Undergradaute Engineering (with Intercalated Year)
This module is Optional for:
• Year 3 of UESA-H113 BEng Engineering
• Year 3 of UESA-H114 MEng Engineering
• Year 4 of UESA-H115 MEng Engineering with Intercalated Year
• UESA-H11L Undergradaute Engineering (with Intercalated Year)
□ Year 3 of H11L Engineering (with Intercalated Year)
□ Year 4 of H11L Engineering (with Intercalated Year)
This module is Option list A for:
• Year 4 of UESA-H111 BEng Engineering with Intercalated Year
• Year 3 of UESA-H112 BSc Engineering | {"url":"https://courses.warwick.ac.uk/modules/2023/ES386-15","timestamp":"2024-11-13T04:59:15Z","content_type":"text/html","content_length":"23825","record_id":"<urn:uuid:242ce00f-dd7f-4379-8431-50c2d44972b4>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00415.warc.gz"} |
Stock y has a beta of 13 and an expected return of 185, Financial Management
Stock Y has a beta of 1.3 and an expected return of 18.5%. Stock Z has a beta of 0.70 and an expected return of 12.1%. If the risk-free rate is 8% and the market risk premium is 7.5%, are these
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Frontiers | The value of generalized linear mixed models for data analysis in the plant sciences
• ^1Department of Plant Pathology, The Ohio State University, Wooster, OH, United States
• ^2Center for Integrated Fungal Research, Department of Entomology and Plant Pathology, North Carolina State University, Raleigh, NC, United States
Modern data analysis typically involves the fitting of a statistical model to data, which includes estimating the model parameters and their precision (standard errors) and testing hypotheses based
on the parameter estimates. Linear mixed models (LMMs) fitted through likelihood methods have been the foundation for data analysis for well over a quarter of a century. These models allow the
researcher to simultaneously consider fixed (e.g., treatment) and random (e.g., block and location) effects on the response variables and account for the correlation of observations, when it is
assumed that the response variable has a normal distribution. Analysis of variance (ANOVA), which was developed about a century ago, can be considered a special case of the use of an LMM. A wide
diversity of experimental and treatment designs, as well as correlations of the response variable, can be handled using these types of models. Many response variables are not normally distributed, of
course, such as discrete variables that may or may not be expressed as a percentage (e.g., counts of insects or diseased plants) and continuous variables with asymmetrical distributions (e.g.,
survival time). As expansions of LMMs, generalized linear mixed models (GLMMs) can be used to analyze the data arising from several non-normal statistical distributions, including the discrete
binomial, Poisson, and negative binomial, as well as the continuous gamma and beta. A GLMM allows the data analyst to better match the model to the data rather than to force the data to match a
specific model. The increase in computer memory and processing speed, together with the development of user-friendly software and the progress in statistical theory and methodology, has made it
practical for non-statisticians to use GLMMs since the late 2000s. The switch from LMMs to GLMMs is deceptive, however, as there are several major issues that must be thought about or judged when
using a GLMM, which are mostly resolved for routine analyses with LMMs. These include the consideration of conditional versus marginal distributions and means, overdispersion (for discrete data), the
model-fitting method [e.g., maximum likelihood (integral approximation), restricted pseudo-likelihood, and quasi-likelihood], and the choice of link function to relate the mean to the fixed and
random effects. The issues are explained conceptually with different model formulations and subsequently with an example involving the percentage of diseased plants in a field study with wheat, as
well as with simulated data, starting with a LMM and transitioning to a GLMM. A brief synopsis of the published GLMM-based analyses in the plant agricultural literature is presented to give readers a
sense of the range of applications of this approach to data analysis.
1 Introduction
Whether one is conducting a planned experiment with replication, blocking, and randomization or collecting observational data such as from a survey, data analysis needs to be carried out in order to
reach conclusions (Schabenberger and Pierce, 2002). We take it as axiomatic that appropriate statistical methods are required to interpret the collected data from any study. Of course, “appropriate”
can mean many different things, depending on the situation and context of interest. Generalized linear mixed models (GLMMs) are becoming quite popular for data analysis in agriculture and other
disciplines (Gbur et al., 2012; Ruíz et al., 2023); however, there is a lot of uncertainty about which GLMM to use, how to fit GLMMs to data, and how to interpret the obtained results. Our objectives
here were to provide some background on the statistical modeling of data, particularly for data collected in the agricultural or the plant sciences, to explain GLMMs with examples, and to discuss
several important issues when using GLMMs that may be initially not appreciated by the data analyst.
Several other references are valuable for learning more about GLMMs and for conducting analyses of a wide range of datasets from different experimental designs (Littell et al., 2006; Bolker et al.,
2009; Zuur et al., 2009; Gbur et al., 2012; Stroup, 2013; Brown and Prescott, 2015; Stroup et al., 2018; Gianinetti, 2020; Li et al., 2023; Ruíz et al., 2015, 2023). For those who wish to learn more
theory, as well as applications, Stroup (2013) is an indispensable reference. Molenberghs and Verbeke (2010) and McCulloch and Searle (2001) provided considerably more theory about mixed models, but
may be of less value to the reader of this article. This article is heavily influenced by Stroup (2013, 2015) and the material in Chapters 11–13 in Stroup et al. (2018). We focus on an example
involving the percentage of plants infected by a particular disease in a field study, although the methods can be applied to any discrete data where percentages can be calculated. Gianinetti (2020)
and Li et al. (2023) are other excellent references for the GLMM-based analysis of data expressed as percentages. Readers should refer to Gbur et al. (2012); Stroup (2013), and Ruíz et al. (2023) for
details on the analysis of continuous data using GLMMs. For those interested in broader issues related to statistical analysis in horticulture, including commonly made errors, we recommend Kramer
et al. (2016).
Below, we start with some historical background on data analysis, followed by a discussion on non-normal statistical distributions and then presentations on models for data with normal and non-normal
distributions. Model expansions and alternatives are given for select experimental designs. We place an emphasis on the different methods of model fitting and the interpretation of the estimated
parameters for GLMMs, demonstrated with an example dataset. Some major challenges in the use of GLMMs are presented. Before the conclusions, a list of cases in the literature where GLMMs were used is
2 Background: from ANOVA to GLMMs
Analysis of variance (ANOVA), including the special case of t-tests, and linear regression have been the foundations for data analysis in agriculture and other disciplines for over a century (Fisher,
1918, 1935; Cochran and Cox, 1957; Steel and Torrie, 1960). The pioneering statistical works of Fisher, Yates, Cochran, and Snedecor, among others, coupled with the advances in statistical software
and the increased speed and memory of computers, have given researchers a large toolbox of methods to describe data, predict outcomes, and make inferences about hypotheses of interest (Schabenberger
and Pierce, 2002; Gbur et al., 2012).
ANOVA can be considered a special case of multiple linear regression (Speed, 2010), where several predictor or explanatory variables (X[1], X[2], etc.) are binary (0, 1). This allows data analysis to
be couched in terms of linear modeling of the response variables as functions of predictor variables, an approach that still dominates today. An explanatory variable can be discrete, generally known
as a classification or class variable, or simply a factor, and consists of two or more distinct levels (e.g., cultivar 1 and cultivar 2 or treatment A and treatment B). Otherwise, the explanatory
variable can be continuous (e.g., temperature), often called a covariate or a covariable. Continuous variables can be treated as discrete variables in a statistical model, depending on the objective
and the manner in which the experiment was conducted.
Models consist of fixed-effects and/or random-effects variables (Schabenberger and Pierce, 2002). For the fixed-effects variables, the levels (categories or groups) in the study represent all
possible levels of the factor or all the levels of interest by the investigator (i.e., they were selected for study because they are of specific interest). Examples would be fungicide treatment,
biocontrol treatment, pathogen inoculum dose, temperature, cultivar, or the nitrogen level in a fertilizer. For the random-effects variables, the levels in the study represent only a random sample of
a larger set of levels (i.e., a sample from a distribution of effects). Examples could include the location (environment), a block, or a plot in a field study. A given variable could be considered
fixed or random, depending on the circumstances. For instance, plant genotype could be considered a random-effects variable if a sample of a population of genotypes was randomly selected for study,
with the goal of characterizing the mean (expected value) and the variability of the response variable (e.g., yield); on the other hand, genotype could be considered as a fixed-effects variable if
the investigator was strictly interested in the responses of those particular genotypes. Similar arguments can be made about location–year (“environment”) as being fixed or random (see Chapter 6 in
Littell et al., 2006).
The term mixed model is used when there are both fixed- and random-effects variables in the model. More specifically, these are known as linear mixed models (LMMs) when the response variable (e.g.,
yield, biomass, or disease severity) is considered to be normally distributed and the mean (expected value) is modeled directly as a function of the fixed- and random-effects terms (Stroup, 2013).
This is explained in the next section, which includes a more technical description of LMMs. A special case of a LMM is the linear model (LM), in which there are no random effects except for the
residual; another special case is the random-effects model, in which there are no fixed-effects variables. The concept of random effects goes back to Fisher (1918, 1935), possibly earlier (but less
formally), with subsequent important early contributions by Yates (1940) and Eisenhart (1947), as well as many others.
For many years, the standard approach to handling random effects in standard software (or by brute force work on a calculator in the very early days) was to fit a LM to the data using ordinary least
squares as if all effects were fixed and then perform post-model-fitting calculations to estimate the mean squares and variances for the random-effects variables (Steel and Torrie, 1960; Littell
et al., 2006). The latter are used (automatically) to estimate the standard errors (SEs) of the least squares means and the mean differences for the fixed-effects terms and also to test for factor
effects. This approach works fine for many special cases (e.g., balanced randomized complete block designs); however, even for a design such as the popular split plot or split plot with blocks,
certain SEs cannot be calculated correctly (Littell et al., 2006). For incomplete block designs (i.e., where each block does not contain all the treatments), recovering all of the available
information in a study (e.g., treatment effects within and between blocks) requires some tedious post-model-fitting calculations when all factors (including blocks) are considered fixed in the model
(Yates, 1940). Many other situations often cannot be analyzed satisfactorily with the mean square, post-model-fitting adjustment approach, at least not without major approximations. Examples include
repeated measures in time and space or any situation when complex correlations of data need to be specified or estimated.
The methodology of treating random effects fully as random in the model-fitting process was developed by Henderson in a series of papers (e.g., Henderson, 1950, 1953, 1984), with extensive theory by
Harville (1977) and others, especially Laird and Ware (1982). The model-fitting approach is likelihood-based (rather than least squares/mean square-based), with an assumed normal distribution for the
response variable conditional on the random effects (see below). Except for special (simple) cases, model fitting is iterative, facilitated by fast computers with large memories to carry out the
multiple iterations in a rapid manner. It was not until the mid-1990s that (relatively) easy-to-use software was developed to fit LMMs to data based on the likelihood principle. The MIXED procedure
in SAS is especially important here (Littell et al., 2006). Presently, several R packages, such as “lme4” and “nlme,” can be used (Galecki and Burzykowski, 2013; Bolker et al., 2022). GenStat (VSN
International, Hemel Hempstead, UK) has been able to fit LMMs using the likelihood methodology for many years. These programs have opened up a great diversity of applications in data analysis across
all disciplines that go far beyond what was possible with the traditional ANOVA approach in the older software (Brown and Prescott, 2015). Nevertheless, the older approach is still used extensively.
It has been well understood for decades that the distributions of many response variables are not normal. Many are discrete (such as the counts of fruit or the percentages of diseased plants), while
others are continuous but not symmetrical (possibly including the severity of disease symptoms or the time to an event, such as time to seed germination). The gamma and beta distributions are
possible alternatives to the normal distribution for asymmetrical continuous distributions (Gbur et al., 2012; Ruíz et al., 2023). Counts with no (definable) upper bound (e.g., the number of insects
on plants or the number of spores in a spore trap) may be described using the Poisson distribution, while counts with an upper bound [e.g., the number of plants with disease symptoms out of n plants
observed (disease incidence) or the number of seeds germinating out of n seeds observed] may be described using a binomial distribution (Madden and Hughes, 1995; Madden et al., 2007; Gianinetti, 2020
). In the binomial case, proportions can be defined as y/n, where y is the count. In the Poisson case, proportions do not exist as n is not defined. In practice, there is always a finite upper bound
n for a count in biology; however, if n is very much larger than y, then it is reasonable to assume that there is no upper bound. An alternative for Poisson is the negative binomial (NB)
distribution, while an alternative for the binomial is the beta-binomial distribution (Madden and Hughes, 1995). Both of these alternatives represent situations with higher variances than defined by
the Poisson and the binomial.
A typical property of non-normal distributions is that the variance is a function of the mean (Stroup, 2013). However, the standard assumption of a (normality-based) LM or LMM is that the (residual)
variance is constant across all of the factor levels (i.e., the variance is independent of the mean). Linear or linear mixed modeling approaches can be (and routinely have been) used for non-normal
data, as an approximation, by transforming the response variable so that the residual variance is roughly constant across the different means (e.g., for different treatments, cultivars, etc.).
Examples include using the angular transformation (arcsine square root transformation) for proportions when the response variable has a binomial distribution and the square root transformation for
unbounded counts when the response variable has a Poisson distribution (Piepho, 2003, 2009). Although this has been a popular and practical approach for decades, there are some disadvantages, as
explained by Stroup (2015) and Gbur et al. (2012), among others. Essentially, this entails forcing the data to agree with a model (linear or LMM) rather than choosing a model that matches the
stochastic process that generates the data (i.e., choosing a model that corresponds to the distribution of the data). Among other things, the estimated treatment “means” of the transformed values, or
their back transformation to the original data scale, do not necessarily mean what users might think they mean. Additional issues are discussed further below.
The analysis of non-normal data has been possible for decades using methods that are actually based on non-normal distributions, especially for data with binary, binomial, and Poisson distributions (
McCullagh and Nelder, 1989; Schabenberger and Pierce, 2002). Typical analyses include probit or logistic modeling of bioassay (binary dead or alive) data or count data. This was primarily for models
with only fixed effects, with the models known as generalized linear models (GLMs), usually when the distribution belonged to the so-called exponential family of distributions. The addition of random
effects to these GLMs to form GLMMs has posed some considerable statistical challenges, but research was in full force by the early 1990s (Breslow and Clayton, 1993; Wolfinger and O’Connell, 1993),
although it would take another decade or more before relatively easy-to-use software became broadly available. The GLIMMIX procedure of SAS is especially important here, as is the lme4 package in R.
GenStat also has methods for fitting GLMMs. One can consider LMs, LMMs, and GLMs as special cases of GLMMs. Readers should consult Littell et al. (2006); Gbur et al. (2012); Stroup (2013), and Ruíz
et al. (2023) for more details on GLMMs, as well as on LMMs and GLMs.
There are many reasons to use GLMMs in the agricultural and plant sciences (Gbur et al., 2012) because one can account for both random and fixed effects with several different realistic statistical
distributions for the data. Nevertheless, there are also many issues that investigators need to be aware of when using GLMMs. These are described below after a more formal introduction of LMMs and
GLMMs in the context of an example dataset.
3 Non-normal distributions
3.1 Example
We consider a simple example to demonstrate several concepts and methods for the analysis of non-normal data. This is a subset of a much larger dataset analyzed in Paul et al. (2019) regarding, in
part, the effect of fungicide treatments on Fusarium head blight in wheat (caused by the fungus Fusarium graminearum), mycotoxin contamination of the grain, and crop yield. This example is from one
location–year (environment), a subset of the full dataset consisting of 29 environments (conducted by different researchers) with two factors evaluated at each environment, fungicide treatment, and
cultivar resistance. We only consider the susceptible plant cultivar here, so that we are left with just one fixed-effects factor in a randomized complete block design (RCBD). If all the treatments
were not in all of the blocks, we would have an incomplete block design. We are using this subset merely for demonstration purposes, not as a way of analyzing the full multi-environment dataset with
two factors.
The example experiment in this environment was laid out in four blocks (j = 1, ..., 4), with the six treatments (i = 1, ..., 6) randomized within each block. There were n = 100 wheat spikes randomly
chosen and visually assessed for disease symptoms in each experimental unit (plot: a block–treatment combination identified by the ij index), and each spike was categorized as either diseased or
healthy, giving a total of y[ij] diseased spikes per plot, with a proportion given by prop[ij] = y[ij]/n. This proportion is known as disease incidence (Madden et al., 2007), a measure of the
fraction of individuals that are infected. With this design, there is a single response for each experimental unit (y[ij] or prop[ij]), even though this is the sum of several (n) binary observations.
There is no requirement that n be constant as it could vary with plot (n[ij]). Here, we consider block to be a random effect and treatment to be a fixed effect. Note that, in Paul et al. (2019), more
emphasis was placed on the analysis of the severity of disease on spikes (known as an “index” in the head blight literature) instead of the disease incidence. Severity is a continuous response
variable representing the area of spikes with symptoms, expressed on a proportion or a percentage scale.
The research questions are: Does treatment affect (mean) disease incidence? If so, which treatments are different from each other? Once we consider GLMMs for these data, we can ask better questions;
for example, does treatment affect the probability of a wheat spike being infected (π) and which treatments differ from others in terms of π? To save space, the only difference we show is between the
last and first treatments.
3.2 Distribution for disease incidence
Since disease incidence is discrete with an upper bound of n, it is reasonable to consider the y in each plot to have a binomial distribution (Madden and Hughes, 1995), at least as a starting point.
Generically, without reference to a particular treatment or block (i.e., no subscripts), we can write this as y ~ Bin(π,n), where π is a location parameter representing the probability of a trait or
characteristic. For instance, π is the probability that a plant is infected or diseased; a moment estimate of π is the mean of the proportions for a given plot. The mean y for the binomial
distribution is nπ and its variance is nπ(1 − π). Converting to proportions, the mean prop is π and its variance is π(1 − π)/n for the binomial. It is well known that the binomial can be well
approximated using a normal distribution if n is large enough (Schabenberger and Pierce, 2002). To exemplify, following Stroup (2013, 2015), we simulated 100,000 observations from a binomial
distribution with π = 0.1, with three different values of n (Figure 1). With n = 10, the binomial distribution is fairly skewed and poorly approximated by the normal (smooth curve); with n = 30, the
binomial is much less skewed; and with n = 100, the binomial is very close to a normal distribution. With values of π closer to 0.5, the binomial is much closer to being symmetric (and is exactly
symmetrical at π = 0.5), and approximation to normality is achieved with a smaller n. At very small or large π (i.e., very close to 0 or 1, respectively), a very large n may be needed to approximate
Figure 1
Figure 1 Frequency distribution of 100,000 observations of the response variable y generated from a binomial distribution with n = 10, 30, 100 individuals per observation. Proportions were calculated
as y/n for each observation.
LMMs are not too sensitive to some departure from normality (Littell et al., 2006), and there are some LMM-fitting methods that do not require normality, e.g., MIVQUE0 (Rao, 1972), although
calculations of confidence intervals and inference generally assume normality. However, even when it is reasonable to assume normality as an approximation for the binomial, the problem is that the
variance of y (or of prop) will vary with the mean. The well-known angular (i.e., arcsine square root) transformation of prop does approximately stabilize variances, where prop* = sin^−1(√prop) (
Schabenberger and Pierce, 2002). Other variance-stabilizing transformations may be more successful when π is very close to 0 or 1 (Piepho, 2003). These may work well with LMMs (i.e., for assumed
normal data), although it becomes non-trivial to interpret the exact meaning of the estimated angular means for each treatment relative to the underlying distribution of the counts (or corresponding
proportions). This is especially true when there are random effects (see details and example below). There are also other challenges with the transformation-based LMM analysis. For instance, one
transformation may be appropriate to stabilize variances, but a different transformation may be needed to obtain a linear (straight line) relation between y and a continuous explanatory variable (for
a regression problem). Another transformation may be needed to obtain a symmetrical distribution. Thus, there are some big advantages to moving away from transformation-based analyses.
The binomial is a member of the exponential family of distributions (McCullagh and Nelder, 1989; Gbur et al., 2012; Ruíz et al., 2023). There are several other distributions of the exponential
family, or have statistical properties that are very similar to those of the members of the family. These include the Poisson, NB, gamma, and beta distributions (Stroup, 2013). The binomial, Poisson,
and NB are for discrete data, whereas the gamma and beta are for continuous data. GLMs and GLMMs were developed to fit data from these distributions. The normal distribution, also known as the
Gaussian, is a (symmetric) member of the exponential family, so LMM can be thought of as a special case of a GLMM. We focus on the normal and binomial distributions in this paper, which are applied
to discrete data. Readers should consult Stroup (2013); Ruíz et al. (2023), and Gbur et al. (2012) for details on the other distributions.
4 Models and analysis
4.1 Linear mixed model
Generically, we use y in the models as the response variable, with subscripts depending on the experimental and treatment design. For specific cases, we can use another symbol for the response. The
classic LMM for a RCBD with one observation per experimental unit (ij combination) is:
$\begin{array}{ll}{y}_{ij}=\text{θ}+{\tau }_{i}+{b}_{j}+{e}_{ij}& \left(1\right)\end{array}$
where θ is a constant (intercept), τ[i] is the effect of the i-th treatment (fixed), b[j] is the effect of the j-th block (random), and e[ij] is the residual (random), the latter representing
variability in the response variable not accounted for by the other terms in the model. The residual essentially gives the unique effect of each experimental unit on the response variable (after
accounting for the treatment and block main effects) and is equivalent to an interaction of the treatment and block effects (when there is one observation for each ij combination, as here). The
random effects for this RCBD model are assumed to be normally distributed, with means of 0 and with variances ${\sigma }_{b}^{2}$ and ${\sigma }_{e}^{2}$:
$\begin{array}{ll} {b}_{j}~N\left(0,{\sigma }_{b}^{2}\right),\text{\hspace{0.17em}}{e}_{ij}~N\left(0,{\sigma }_{e}^{2}\right)& \left(2\right)\end{array}$
Thus, Equations 1 and 2 are needed to jointly define the model for a RCBD with a random block effect. The (total) variance of y[ij] is ${\sigma }_{b}^{2}+{\sigma }_{e}^{2}$. In some circumstances,
block could be considered as a fixed effect (Dixon, 2016), but we do not consider this here.
Equation 1 (with the distributions in Equation 2) is defined as a mixed model because beyond the fixed effect(s), it has at least one random effect in addition to the residual (Stroup et al., 2018).
It is linear (in terms of the parameters and response variable y) because the terms on the right-hand side of Equation 1 are actually shorthand expressions for a sum of parameters (factor effects)
multiplied by binary indicator variables that identify treatments and blocks (Milliken and Johnson, 2009) and because the left-hand side does not involve any transformation of y. Note that the fixed
effects are constants (to be estimated) and the random effects are (latent) random variables to be predicted (Stroup, 2013). Many statistical programs may refer to the random effect predictions as
estimates and not predictions in the output. The non-residual random effects such as blocks do not need to be normally distributed (Lee and Nelder, 1996), but this is, by far, the most common
assumption. Statistical research has shown that the results often are not very sensitive to this normality assumption for the random effects (McCulloch and Neuhaus, 2011; Schielzeth et al., 2020). It
is a standard assumption of LMMs that the residuals have a normal distribution. The concept of a residual substantially changes when we transition to GLMMs.
For a segue to GLMMs, Equation 1 can be rewritten in terms of expected values (means). That is, determining the expected value, E(·), of the left- and right-hand sides of Equation 1 conditional on
the block random effect (more generally, conditional on all random effects other than the residual) leads to:
$\begin{array}{ll}{\mu }_{ij}=\text{θ}+{\tau }_{i}+{b}_{j}& \left(3\right)\end{array}$
where μ[ij] is the so-called conditional expected value or the mean of the response for treatment i and block j. It is “conditional” because its value is for the specific level(s) of the random
effect (the j-th block for this simple RCBD). From Equation 3 and the normality assumptions, the distribution of the response variable y[ij] conditional on the random effect, y[ij]|b[j], is defined
$\begin{array}{ll}{y}_{ij}|{b}_{j}~N\left({\mu }_{ij},{\sigma }_{e}^{2}\right)& \left(4\right)\end{array}$
Note that the mean for this conditional distribution of y[ij] is μ[ij] and that the variance of this conditional distribution for a given block (or a given level of the random effects) is ${\sigma }_
{e}^{2}$ (which was given as the residual variance in Equation 2). In other words, after accounting for the fixed and random effects, the only other variability of y is captured by the variance of
the conditional distribution (i.e., the unexplained variability). Readers should note that the μ[ij] in Equations 3 and 4 may be written as μ[ij]|b[j] in order to make the conditioning explicit (we
avoid this additional notation here). Putting it all together, the alternative to Equations 1 and 2 is:
$\begin{array}{ll}\begin{array}{l}{\mu }_{ij}=\text{θ}+{\tau }_{i}+{b}_{j}\\ {y}_{ij}|{b}_{j} ~N\left({\mu }_{ij},{\sigma }_{e}^{2}\right)\\ {b}_{j} ~N\left(0,{\sigma }_{b}^{2}\right)\end{array}& \
In this manner of writing the LMM, the residual term “disappears”; ${\sigma }_{e}^{2}$ is now seen as the variance of the conditional normal distribution for y (after accounting for other effects).
Equation 5 can actually be generalized further in a way that will help to understand GLMMs for non-normal response variables. The right-hand side of Equation 3 for the conditional mean is known as
the linear predictor (LP), which is typically written with the η symbol (η[ij] for the RCBD here). The LP specifies all the fixed and random effects in a controlled experiment or with observational
data that can affect either the conditional mean of a response variable (or a function of the conditional mean). For a LMM (i.e., normal conditional distribution), the mean simply equals the LP; that
is, we link the LP to the mean with ${\mu }_{ij}={\eta }_{ij}$. We can then rewrite the set of equations for a RCBD as:
$\begin{array}{ll}\begin{array}{l} {\eta }_{ij}=\text{θ}+{\tau }_{i}+{b}_{j}\\ {\mu }_{ij}= {\eta }_{ij}\\ {y}_{ij}|{b}_{j} ~N\left({\mu }_{ij},{\sigma }_{e}^{2}\right)\\ {b}_{j} ~N\left(0,{\sigma }_
{b}^{2}\right)\end{array}& \left(6\right)\end{array}$
The LMM is rarely seen written this way, and the inclusion of the explicit LP component is excessive for this LM with conditional normal distribution; however, it does nicely allow for a transition
to the non-normal conditional distributions of GLMMs. This formulation is also very useful for Bayesian model fitting because it defines the two statistical distributions in the model: the
distribution of the block random effect and the conditional distribution of the response variable.
Equations 1 and 2, or Equation 6 (with the four sub-equations), can be expanded for any number of fixed and random effects, as well as the interactions of random and fixed effects (Littell et al.,
2006; Stroup, 2013). Both formulations are conditional models, with the same results when fitted to data; it is just a matter of preference in how one writes the LMM. From Equations 5 or 6, the
expected mean for the i-th level of the fixed effect (the i-th treatment) can be determined. The random effects are integrated out to obtain μ[i] (instead of μ[ij]). Because the expected value of the
normally distributed b[j] is 0, the result for a LMM is simply:
$\begin{array}{ll}{\mu }_{i}=\text{θ}+{\tau }_{i}& \left(7\right)\end{array}$
This is generally of most interest to the investigator. The estimates of μ[i] and the differences of μ[i] between treatments (μ[i] − μ[i][′], where i and i′ are any two treatments), as well as their
SEs (a function of the variances in Equation 6), are derived from statistical theory and are automatically calculated with LMM software (with the right statements or options). Details are in
mixed-model textbooks (e.g., Littell et al., 2006; Galecki and Burzykowski, 2013; Stroup et al., 2018).
There is yet another way to write a LMM. The two random-effects terms (including the residual) in Equation 1 can be taken and combined into one term as: h[ij] = b[j] + e[ij]. The LMM can then be
written as:
$\begin{array}{ll}{y}_{ij}=\text{θ}+{\tau }_{i}+{h}_{ij}& \left(8\right)\end{array}$
This LMM is known as a marginal model for a RCBD experiment (Stroup, 2013). Both block and residual effects are still present, just expressed differently. The variability component(s) now has to be
written with more complexity to express both random sources. The approach is to define a vector of random effects for each level of the random effect (the j-th block here), designated h[j]. In order
to save space, we assume here that there are just three treatments, so we can write:
$\begin{array}{ll}{\mathbf{\text{h}}}_{\mathbf{\text{j}}}=\left(\begin{array}{c}{h}_{1j}\\ {h}_{2j}\\ {h}_{3j}\end{array}\right)=\left(\begin{array}{c}{b}_{j}+{e}_{1j}\\ {b}_{j}+{e}_{2j}\\ {b}_{j}+
{e}_{3j}\end{array}\right)& \left(9\right)\end{array}$
where each element of the vector corresponds to a different treatment. This leads to the derivation of the variance–covariance matrix (P) for y[ij]:
$\begin{array}{ll}P=\left(\begin{array}{ccc}{\sigma }_{\text{b}}^{2}+{\sigma }_{\text{e}}^{2}& {\sigma }_{\text{b}}^{2}& {\sigma }_{\text{b}}^{2}\\ {\sigma }_{\text{b}}^{2}& {\sigma }_{\text{b}}^{2}+
{\sigma }_{\text{e}}^{2}& {\sigma }_{\text{b}}^{2}\\ {\sigma }_{\text{b}}^{2}& {\sigma }_{\text{b}}^{2}& {\sigma }_{\text{b}}^{2}+{\sigma }_{\text{e}}^{2}\end{array}\right)& \left(10\right)\end
The diagonal elements of the matrix are the (total) variances of y[ij] (${\sigma }_{b}^{2}+{\sigma }_{e}^{2}$; the same for each treatment), while the off-diagonal elements (${\sigma }_{b}^{2}$) are
the covariances of the response variables. This formulation helps to make clear that, with a random block, observations that are from the same block are correlated. The correlations within a block
can be determined from the covariances and variances (Littell et al., 2006). This structure is known as compound symmetry (CS). The conditional model (Equations 1 and 2, or Equations 5 or 6) and the
marginal model (Equations 8 and 10) are equivalent for LMMs. Fitting either version will give the same result when fitted to data if ${\sigma }_{b}^{2}$ is 0 or positive. Although the conditional
model in Equations 5 or 6 does not explicitly show a covariance, the covariances between treatments are still there, derivable from the terms of the conditional model. For the RCBD, the correlation
of y[ij] within the same block (say, j = 1), known as the intra-class correlation, is:
$\begin{array}{ll}\rho ={\sigma }_{b}^{2}/\left({\sigma }_{b}^{2}+{\sigma }_{e}^{2}\right)& \left(11\right)\end{array}$
The reader can find more details in Stroup (2013) and Gbur et al. (2012).
The most common method to fit LMMs is through the use of restricted maximum likelihood (REML), also known as residual maximum likelihood. This can be considered the gold standard. REML produces
unbiased parameter estimates (such as with a RCBD) or less biased estimates than those produced with maximum likelihood (ML). Bayesian methods can also be used (Wolfinger and Kass, 2000; Stroup, 2021
), as well as some alternative frequentist approaches (Littell et al., 2006). It is important to note that, whether the data analyst uses the conditional (Equation 6) or the marginal (Equations 8-10)
form of the LMM, the actual (observed) data correspond to the marginal distribution (Gbur et al., 2012). In essence, the conditional and marginal models are just two different ways of generating data
from a marginal distribution (Stroup, 2013). The random effects in a conditional LMM are random latent variables that are not directly measured or observed; rather, they are predicted through the
model-fitting procedure. What one observes (measures) and then analyzes are data arising from the marginal distribution, no matter how that marginal distribution is derived or generated.
4.2 Example data analyzed with LMMs
Using the GLIMMIX procedure in SAS with REML estimation, the conditional LMM in Equation 6 (equivalent to Equations 1 and 2) was fitted to the example data with the proportion of diseased wheat
spikes as the response variable (prop[ij] for y[ij]). This is not normally done as the variance is not independent of the mean for binomial data (or discrete data, in general). Estimates of the
treatment means (e.g., $\stackrel{^}{\theta }+{\stackrel{^}{\tau }}_{1}={\stackrel{^}{\mu }}_{1}$) and the SEs are given in Table 1, together with one pairwise difference of means. The exact same
results are obtained if the marginal model (Equations 7–10) is fitted for this LMM (results not shown). The SEs are functions of the block and residual variances (details in Gbur et al., 2012;
Stroup, 2013). Interestingly, despite the problems in directly analyzing proportions, the estimated mean proportions from the fitted LMM are unbiased estimates of the true proportions for each
treatment across all the blocks (i.e., across all the levels of the random effects; the marginal means) (Stroup, 2013, 2015). However, the estimated SEs across all treatments (0.0590) are meaningless
because the LMM assumes a constant variance (${\sigma }_{e}^{2}$). Since the estimates of variability are wrong, the estimates of confidence intervals and the tests of significance are thus wrong.
Table 1
Table 1 Estimated means and standard errors (SEs) when fitting a linear mixed model (LMM; Equation 6) to the proportion of diseased wheat spikes^a with an incorrect assumption that the residual
variance is constant (i.e., independent of the mean proportion) and when fitting the LMM to the angular transformation of the proportions where it is reasonable to assume the transformed values have
constant residual variance, together with the back-transformation of the estimated angular means to obtain proportions (where the SE of the back-transformation is based on the delta method).
The LMM of Equation 6 can be generalized to allow for a separate residual for each level of the fixed effect (${\sigma }_{ei}^{2}$). This is easily done using REML algorithms such as in the MIXED
procedure in SAS. Fitting such a model can be more challenging, especially with the small sample sizes typically found in agricultural field studies. With the number of blocks in this example being
four, attempting to estimate each treatment-specific residual variance is thus not a good strategy.
The results of fitting the LMM to the transformed angular data are also shown in Table 1. The estimated SEs were constant across the treatments (0.0651), which is expected since one assumes that the
angular-transformed data have variances independent of the mean. Comparisons of the treatments and tests of significance can be done on these mean transformed values, and back-transformation of the
means can be performed to obtain a type of “mean proportion” for each treatment, as shown in the table. However, it is not intuitive what these back-transformed means represent exactly. They are not
estimates of the means of the distributions of the actual proportions, but are related to those means.
In symbols and suppressing subscripts here, using g(y) as a transformation (function) of y [such as a log and angular (arcsine square root), among others], the expected value (i.e., mean) of g(y), E(
g(y)), is not equal to the transformation of the expected value (mean) of y, g(E(y)) (Schabenberger and Pierce, 2002). These two types of means are related, but are not the same. The situation
becomes even more complicated with multiple random effects. For the example (Table 1), the back-transformed values give a type of “central” (but not truly central) value of the marginal distribution
of the data. Nevertheless, analytical approaches based on transformations still have value, as long as the investigators understand the limitations.
5 Generalized linear mixed models
5.1 Conditional non-normal distributions
With GLMMs, investigators have the opportunity to better match the model used for data analysis with a plausible model for stochastic generation of the data (Stroup, 2013, 2015). Consider the disease
incidence data in the example, where it is reasonable to assume a binomial distribution for y in a given plot (experimental unit), at least as a starting point. Suppressing subscripts temporarily, we
can write generically, y|plot ~ Bin(π,n), for the conditional distribution of the number of diseased wheat spikes (this could also be for the number of germinated seeds in a pot in a greenhouse,
etc.). The probability of disease (or the probability of some trait), π, could be influenced by the pathogen inoculum density, the favorability of the environment for infection, and so on. It is
quite plausible that π represents a parameter that has direct mechanistic interpretation. We continue with our hypothetical example from Figure 1 and consider π = 0.1 (following closely, once again,
the method in Stroup, 2013).
Consider the impact of random effects, such as blocks. If there is only one treatment (e.g., control), then the plot and the block are synonymous. With random effects, π is randomly perturbed by the
block, so that π is higher than 0.1 in some blocks (plots) and is lower than 0.1 in other blocks. The response y in each block (for any single treatment) would depend on the realized value of π for
that block. On average, the perturbation is 0, at least on one possible scale. Ultimately, we want to specify the marginal distribution of y [across the population of plots (blocks)] based on the
data-generating process in each plot (block) and the distribution of perturbations across the plots. A linear additive scale for the perturbation of the random block effects (i.e., adding b[j]
directly to π) would not work in a realistic model, in general, because a random permutation could give a probability nonsensically above 1 or below 0 in a given plot when π is relatively close to 1
or 0. A linear perturbation could work on a function of π, if the right function was used, so that π remains bounded by 0 and 1. This is the approach taken in the usual GLMM.
Being more explicit and referring back to the conditional distribution form of a LMM (Equation 6), we can write the LP for a RCBD as $\text{ }{\eta }_{ij}=\theta +{\tau }_{i}+{b}_{j}$. This is
exactly the same for a normal or a non-normal conditional distribution. Likewise, the random block effect can be assumed to have a normal distribution, as with LMMs. The question is how to link the η
[ij] LP to the location parameter. This is done through the link function, g(π[ij]), or, more generally, as g(μ[ij]), where a link function is a transformation of a parameter (not a transformation of
the observations). Thus, we can write g(π[ij]) = η[ij]. The location parameter is obtained by the inverse link function, written as π[ij] = g^−1(η[ij]) = h(η[ij]). For conditional binomial data, the
logit is the most common link function; that is, g(π[ij]) = logit(π[ij]) = ln(π[ij]/(1 − π[ij])). There are other possibilities for the link function based on specific applications or theory (
Collett, 2003). The term “identity link” or “identity link function” is used when there is no transformation of μ or π.
For a RCBD (one fixed-effects factor and random block effect factor), we can now write the conditional model form of the GLMM generically as:
$\begin{array}{ll}\begin{array}{l} {\eta }_{ij}=\text{θ}+{\tau }_{i}+{b}_{j}\\ g\left({\mu }_{ij}\right)= {\eta }_{ij}\\ {y}_{ij}|{b}_{j} ~\text{Dist}\left({\mu }_{ij},n\right)\\ {b}_{j} ~N\left(0,{\
sigma }_{b}^{2}\right)\end{array}& \left(12\right)\end{array}$
where “Dist” is generic for some conditional distribution. This equation is almost the same as that for the LMM (see Equation 6). For conditional binomial data (with π = μ), in particular, we write:
$\begin{array}{ll}\begin{array}{l}\text{ }{\eta }_{ij}=\text{θ}+{\tau }_{i}+{b}_{j}\\ \text{logit}\left({\pi }_{ij}\right)=\text{ }{\eta }_{ij}\\ {y}_{ij}|{b}_{j} ~Bin\left({\pi }_{ij},n\right)\\ {b}
_{j} ~N\left(0,{\sigma }_{b}^{2}\right)\end{array}& \left(13\right)\end{array}$
Recall that the mean of the binomial is nπ for y; the mean of y/n is π. Thus, the location parameter π with the binomial plays the role of μ in the general distribution (Equation 12). Unlike with a
LMM, it is important to note that there is no variance to estimate for the conditional binomial distribution (i.e., no unknown residual variance as the variance of the binomial is fully defined based
on n and π). The variance of the conditional binomial is nπ(1 − π). See GLMM expansions and extensions below (Section 6) for the implications of this. One can reduce Equation 13 to three components
by directly writing $\text{ logit}\left({\pi }_{ij}\right)=\text{θ}+{\tau }_{i}+{b}_{j}$ as the first sub-equation. It is important to note that, although the g(·) link function is a transformation,
a GLMM is not a model for a function or transformation of data (i.e., not a model for the transformed response variable). Put another way, a GLMM is not a model for the mean of transformed
observations; rather, it is a model for a function (transformation) of the mean (μ, π, etc.) of the actual (non-transformed) observations. This subtle difference is extremely important. The LP of the
GLMM in Equation 13 estimates a function of the probability of disease for a particular treatment and block. The inverse link to obtain π for a logit link function is given by:
$\begin{array}{ll}{\pi }_{ij}=h\left({\eta }_{\text{i}j}\right)=1/\left(1+\mathrm{exp}\left(-\text{logit}\left({\eta }_{ij}\right)\right)\right)& \left(14\right)\end{array}$
5.2 Marginal distribution
Integrating over the random effects of the conditional LMM (Equation 6) results in the marginal distribution for y (Littell et al., 2006; Stroup, 2013; Brown and Prescott, 2015). With normality for
the random effects and the conditional distribution in a LMM, the marginal distribution is determined analytically. The expected values of the fixed effects of this marginal distribution (e.g., $\
text{ }{\mu }_{ij}=\text{θ}+{\text{τ}}_{\text{i}}$) are the same as those obtained simply by inserting 0 (the expected value of the normally distributed random effects; see Equation 2) for the random
effects in the conditional LMM (Equation 3).
The situation is different with GLMMs. The marginal distribution cannot be determined analytically, requiring numerical integration; for the binomial conditional distribution, as an example, $\text{
logit}\left({\pi }_{i}\right)=\text{θ}+{\tau }_{i}$ is not the logit of the mean proportion of diseased individuals over all the blocks (i.e., over all the levels of the random effects). This is
because the inverse link is a nonlinear equation (Equation 14) and the conditional distribution is not normal. This is an extremely important concept because, as discussed above, one observes the
marginal distribution (more technically, a sample from the marginal distribution), not the conditional distribution. Numerical integration over the random effects can give an estimate of the marginal
distribution (required for model fitting), including estimates of the marginal mean (the marginal mean proportion of diseased plants here over all the blocks) and its variability. This is explained
by Stroup (2013) on pages 102–107 for a similar situation. Sometimes, this marginal distribution, which is made up of the mixture of the binomial and normal distributions, is called the
logistic–normal–binomial (LNB) distribution when used to characterize the spatial variability in fields (Hughes et al., 1998).
The marginal distribution from Equation 13 can also be obtained by simulation for known parameters for the conditional distribution and the random-effects distributions. Following Stroup (2013)
closely, we estimated the marginal distribution of the diseased proportion (y/n) across levels of the random effects by simulating 100,000 observations from Equation 13, with π = 0.1 and n = 30 for
the conditional binomial distribution, and three levels of the block effect standard deviation [${\sigma }_{b}$ = 0.5, 1.0, and 1.5 (${\sigma }_{b}^{2}$ = 0.25, 1.0, and 2.25)]. With simple random
sampling (i.e., no random effect distributions), the binomial distribution is not very skewed with n = 30 (Figure 1). However, in a mixed-model setting with symmetric (normal) random effects, the
marginal distribution of the proportion can be very skewed depending on the value of the block variance. As seen in the left-hand column of Figure 2, the marginal distribution becomes more and more
skewed with increasing block variance. A general rule of thumb is that marginal distributions are not symmetrical for GLMMs, with the skewness dependent on the skewness of the conditional
distribution (e.g., binomial) and the magnitude of the variances of all the random effects in a LP.
Figure 2
Figure 2 Marginal distributions determined by generating 100,000 observations of the response variable y with a conditional binomial distribution (with π = 0.1 and n = 30 individuals per
observation), plus a random effect for each observation (which had, for most graphs, a normal distribution with variance of ${\sigma }_{b}^{2}$or standard deviation of ${\sigma }_{b}$). Inverse link
is used to obtain y from logit(π). Proportion is calculated as y/n. Left-hand column: (A–C) Data generated with Equation 13 (without τ because there is only one treatment, with b[j] now as a plot
effect), with standard deviations of b[j] effect of 0.5 (A), 1.0 (B), and 1.5 (C). Right-hand column: (D) Data generated as in (B) (standard deviation of 1), but the proportions transformed with
angular transformation. (E) Data generated as in (B) (standard deviation of 1), but the proportions transformed with Piepho (2003) function (with δ = 5). (F) Unlike other graphs, y is generated with
beta-binomial distribution with overdispersion parameter of $\varrho$ = 0.1 and no b[j] effect, approximately giving the same variance of y/n as in (B) (with a standard deviation of 1).
In the simulation example in Figure 2, the mean proportions of the marginal distributions are estimated as 0.109, 0.134, 0.169 for block variances of 0.25, 1.0, and 2.25, respectively, all with π =
0.1 (displayed in Figures 2A–C, respectively). That is, the mean proportions increase with increasing random effect variances even though the underlying location parameter for the conditional
distribution (i.e., for the generation of observations in each experimental unit) is fixed at π. Broadly, the mean proportion will not correspond to the inverse link of θ + τ[i] for treatment i (or
just θ when there is just one treatment). When π is less than 0.5, the marginal mean proportion will be greater than π, with the degree of difference being dependent on the random effect variability.
When π is greater than 0.5, the marginal mean proportion will be less than π. The variances of the marginal distributions also increase in the same manner. Using the angular transformation of the
observations in a LMM does not lead to a symmetric marginal distribution, as seen in Figure 2D for an example block variance of 1.0. The new variance-stabilizing transformation of Piepho (2003),
with a stabilizing parameter of δ = 5 also does not lead to a symmetric distribution when the block variance is 1.0 in this example (Figure 2E). We estimated δ using the ML methodology in Piepho
(2003) (results not shown). Other tested values of δ did not result in a symmetric distribution. As discussed by Piepho (2003), the new variance-stabilizing transformation to be used in LMMs is very
effective in stabilizing variances (among treatments), but is less effective in producing a symmetrical distribution of the transformed data (especially when the original distribution of y is very
There are other ways to derive a marginal distribution than the typical method here of mixing a normal random effect distribution with conditional binomial on the link scale. For example, it is well
known that the beta-binomial provides an excellent representation of the distribution of disease incidence when there is overdispersion (i.e., higher variability than for the binomial alone,
sometimes called extra-binomial variation) (Madden and Hughes, 1995; Turechek and Madden, 1999; Madden et al., 2007). In general, the beta-binomial is utilized to characterize the spatial variability
(clustering) of incidence within fields, where each sampling location in a field is a separate observation of disease incidence (based on n observed units for each sample) (Turechek and Madden, 1999
), but the concept is easily extended to variability among blocks in planned experiments. In the context of a RCBD, the beta-binomial is derived by, once again, assuming that π is randomly perturbed
in each plot (block, etc.). However, the random block effect is not additive on the link scale of the binomial [e.g., not adding b[j] to the equation for logit(π[j])] as it is for the derived GLMM of
Equation 13. Instead, the random block effect is assumed to have a beta distribution and is multiplicative with π (i.e., working directly on the scale of π) (Madden and Hughes, 1995). The plot effect
is characterized by a $\varrho$ correlation parameter (instead of ${\sigma }_{b}^{2}$). An example of simulated observations from a beta-binomial distribution is shown in Figure 2F, where $\varrho$
was chosen to roughly have the same effect as ${\sigma }_{b}^{2}$ = 1 with the normal random effect (Figure 2B). As with the other scenarios in Figure 2, the marginal distribution is very skewed
There are many uses of the beta-binomial and the related binary power law (Hughes and Madden, 1995; Madden et al., 2018), but the beta-binomial is not commonly used for routine data analysis with
GLMMs (e.g., determining treatment effects). Since the random effect is considered non-normal, the beta-binomial is a special case of what is often called doubly (or double) generalized linear mixed
models (DGLMMs) (Lee et al., 2006). We consider this in some more detail below.
The key question in a GLMM is the measure of the “location” (and associated SE) one wants to estimate. In agreement with Stroup (2013, 2015) and Stroup et al. (2018), we argue that for conditional
binomial data, one wants to estimate logit(π[i]) = θ + τ[i] for treatment i and the conditional probability π[i] using the inverse link function (location parameter of the conditional binomial) (
Equation 14). That is, one is interested in the parameter from the conditional distribution, the probability of disease when the random effects are exactly at their means (0). This is considered a
conditional mean. This location parameter is not affected by the distribution of the random effects [although the random effect variances do affect parameter estimation and the precision (SE) of
parameter estimates]. It turns out that the π[i] from the conditional binomial (for treatment i) is the median of the marginal distribution of proportions for this treatment (Stroup, 2013). Most
importantly, when the GLMM of Equation 13 is fitted to RCBD data, estimates of logit(π[i]) for each treatment (θ + τ[i]), and therefore π[i] through the inverse link (Equation 14), can be directly
5.3 Example
The estimates of the logit of the probability of disease and the corresponding conditional probabilities (inverse links), together with their estimated SEs, from the fit of Equation 13 are given in
Table 2 (listed as model A). This is based on restricted pseudo-likelihood (RSPL) model fitting, discussed below. As expected, the SEs are not equal for the different treatments, being a function of
the variance of the conditional binomial and the block variance. The block variance estimate was small in this example (${\stackrel{^}{\sigma }}_{b}^{2}$ = 0.073), and the proportions are not very
close to 0 or 1, meaning that the ${\stackrel{^}{\pi }}_{i}$ of the conditional binomial (based on the inverse link) is close to the mean proportion of the marginal distribution obtained with the fit
of the LMM to the proportion data (Table 1). However, the SEs vary greatly between the LMM and GLMM approaches [e.g., 0.042 for treatment 1 with the GLMM (model A in Table 2) versus 0.059 for the
same treatment with the LMM (Table 1)], reflecting that the GLMM properly accounted for the variance of a binomial distribution and estimated a conditional distribution parameter.
Table 2
Table 2 Estimated means and standard errors (SEs, in parentheses) both on the linear predictor scale [logit(mean)] and the data scale (inverse link; with SEs based on the delta method) when fitting
three generalized linear mixed models (GLMMs) using restricted pseudo-likelihood (models A and B) or restricted pseudo-likelihood with a quasi-binomial conditional likelihood (model C) to the number
of diseased wheat spikes^a: naive Equation 13 (model A); Equation 16, which includes an additive random effect for the individual experiment unit (plot; model B); and Equation 17, which includes a
multiplicative overdispersion parameter instead of an additive experimental unit effect (model C).
6 GLMM expansions and alternatives
6.1 Link functions
Although it is very common to use the logit link with a conditional binomial distribution in a GLMM, there are other choices, such as the probit (inverse standard normal function) and complementary
log–log (CLL) [ln(−ln(1 − π))]. There may be theoretical reasons to choose a different link for some applications (Collett, 2003), or selection may be based on the goodness of fit of the model (Malik
et al., 2020). For instance, Kriss et al. (2012) used the CLL link in a hierarchical GLMM for surveys of clustered plant disease incidence data, where the choice of link was based on previous
theoretical and empirical work on the relationship between disease incidence and severity (a continuous variable) and the relationship between incidence at two (or more) scales in a spatial hierarchy
(Turechek and Madden, 2003; Hughes et al., 2004; Paul et al., 2005). Many other possible link functions have been proposed, and Malik et al. (2020) described several of them, with a proposed approach
for deciding on an appropriate link to use.
The reason for the logit being the default link has to do with the form of the binomial distribution. One method of writing the log of the binomial distribution (derived after a little algebraic
manipulation of the function typically given in introductory statistics courses) is:
$\begin{array}{ll}\text{ln}\left[f\left(y\right)\right]=y \text{ln}\left(\frac{\pi }{1-\pi }\right)+n\mathrm{ln}\left(1-\pi \right)+ln\left(\begin{array}{c}n\\ y\end{array}\right)& \left(15\right)\
Equation 15 is in the form for a member of the exponential family of distributions (Collett, 2003; Gbur et al., 2012). A key property of this family of distributions is that there is a term that
involves the product of the response variable (y) and a so-called canonical parameter, which is either the mean (expected value, μ) or the location parameter, or a function of the mean for the
distribution. Here, the canonical parameter is the logit of π, ln(π/(1 − π)), the typical link function for the binomial. For the Poisson, for instance, the canonical parameter is ln(μ), which is the
typical default link function for count data. The work by Gbur et al. (2012) and many of the major reference books for GLMMs give the ln[f(y)] forms for other distributions or density functions in
the exponential family. All the relevant properties of these distributions (such as the variance, among others) follow directly from the form of the ln[f(y)]. When the default canonical link is not
selected, the main software programs, such as GLIMMIX in SAS, handle the necessary extra calculations in the background to fit models and calculate statistics.
6.2 Realistic modifications of the GLMM
The GLMM with a binomial conditional distribution (Equation 13) is the obvious natural extension of the LMM (Equation 6) for a RCBD; however, there is a very good chance that the use of the simple
GLMM with agricultural (or other) experiments will give incorrect results! In particular, the SEs of the estimated location parameters (μ and π) for treatments (on the link scale of the model or for
the inverse link scale of the data) may be too small. Likewise, tests of significance are likely to be wrong or misleading, where the test statistics are too large and the p-values for significance
too small (Stroup, 2015). This is an important warning, as elaborated below. To understand this, note that the conditional normal distribution in Equation 6 has a variance (${\sigma }_{e}^{2}$) that
is estimated, but there is no variance to estimate with the conditional binomial. The variance of y for a conditional binomial distribution is fully defined by π and n; that is, var(y[ij]|b[j]) = nπ
[ij](1 − π[ij]) for the RCBD with random block effect. However, as discussed elsewhere, count data are very commonly overdispersed (Madden et al., 2007, 2018); that is, y has a higher variance at the
experimental unit (plot) level than nπ[ij](1 − π[ij]). Similarly, for counts without an upper bound, y has a higher variance at the plot level than the theoretical variance for a Poisson distribution
(μ) (Madden and Hughes, 1995; Madden et al., 2018). As an aside, the concept of overdispersion as used here does not exist for the (conditional) normal distribution because the residual variance is
independent of the mean; ${\sigma }_{e}^{2}$ can take on any value to account for the variability not accounted for by the other terms in the model (in other words, the residual variance is estimated
from the data). For plant diseases, there are numerous causes for overdispersion at the experimental unit (plot in this case) level, especially the clustering of diseased individuals in plots or
fields (Hughes and Madden, 1995; Madden et al., 2007). More broadly, any non-uniformity within the experimental units or nonrandom sampling of the units, or misspecification of the LP (right-hand
side of the equation for η), or misspecification of the conditional distribution in the model can result in overdispersion in a given data analysis. Put another way, overdispersion can occur “when
the model fails to adequately account for all the sources of variability” (Stroup et al., 2018, p. 391).
There are multiple ways to deal with overdispersion when analyzing discrete data. The primary approaches for our purposes in this article are: 1) to expand the LP to include an additional
random-effects term (or terms); 2) to use a quasi-likelihood for the conditional distribution; or 3) to use a different conditional distribution that allows for higher variability than for the
binomial (or Poisson for unbounded counts) (Stroup, 2013). There are important implications for all of these model expansions in terms of model fitting (estimation). We defer the discussion on
estimation until later.
6.3 Adding terms to the linear predictor
The first of these approaches is the expansion of the GLMM. For the RCBD conditional model of Equation 13, we add a term for the interaction of block and treatment, (bτ)[ij], which we can write as v
[ij], to the right-hand side of the LP:
$\text{ }{\eta }_{ij}=\text{θ}+{\tau }_{i}+{b}_{j}+{v}_{ij}$
The distribution of v[ij] is assumed to be normal, ${v}_{ij}~N\left(0,{\sigma }_{v}^{2}\right)$. Now, the conditional distribution of y given the random effects is written as:
${y}_{ij}|{b}_{j},{v}_{ij} ~Bin\left({\pi }_{ij},n\right)$
Conceptually, we assume that π is randomly perturbed not only by the block but also by the individual plot. Recall that the residual term in Equation 1 for a normality-based LMM is equivalent to the
interaction of block and treatment, where the residual term accounts for the variability not accounted for by the other terms in the model. Therefore, the expansion of the LP for the conditionally
binomial data is in the same spirit, taking into account the experimental design [for experiments without blocks, the b[j] term would not be present, but the v[ij] term would still be used here for
the random effect of the experimental unit (plot)]. For the RCBD with conditional binomial data, Equation 13 is now written as:
$\begin{array}{ll}\begin{array}{l}\text{ }{\eta }_{ij}=\text{θ}+{\tau }_{i}+{b}_{j}+{v}_{ij}\\ \text{logit}\left({\pi }_{ij}\right)=\text{ }{\eta }_{ij}\\ {y}_{ij}|{b}_{j},{v}_{ij} ~Bin\left({\pi }_
{ij},n\right)\\ {b}_{j} ~N\left(0,{\sigma }_{b}^{2}\right)\\ {v}_{ij} ~N\left(0,{\sigma }_{v}^{2}\right)\end{array}& \left(16\right)\end{array}$
The same general approach is used for any conditional GLMM that is based on a conditional binomial or a conditional Poisson distribution (with any number of fixed and random effects), i.e., to add a
residual-like random effect to the LP equation of the model. This may be a three-way or a higher-order interaction, generally a crossing of all the terms in the LP. Equation 16 is considered a
conditional model, and the π[ij] represents the probabilities of disease for the individual plots (combination of treatment and block), the same as with Equation 13. Interest still remains on the
estimation of logit(π[i]) (= $\text{θ}+{\tau }_{i}$) and, hence, π[i].
6.3.1 Example
Fitting Equation 16 using RSPL (see below for a discussion on model fitting), which we label as model B, resulted in estimates of ${\stackrel{^}{\sigma }}_{b}^{2}$ = 0.031 for block (compared with
0.073 for model A) and ${\stackrel{^}{\sigma }}_{v}^{2}$ = 0.248 for the block × treatment interaction [plot random effect (“residual”)] (Table 2). Although the block variance is smaller than that of
the naive GLMM above (no v[ij] term), there is now a nonzero plot variance that was not previously considered. The estimates of the LP (logit) and the inverse link for each treatment are shown in
Table 2. Here, the estimated SEs are larger than that for the fit of the simpler Equation 13 (e.g., 0.283 for treatment 1 logit for Equation 16 versus 0.168 for Equation 13). This, together with the
nonzero estimated variance for v[ij], is an indicator of the overdispersion that needed to be accounted for. The F statistic for treatment significance is now 5.24, smaller than that for the simpler
(and inadequate) Equation 13 that did not account for the extra variability (F = 27.9). The means (on the link or data scale) are similar to the results found using the simpler model. However, these
means will, in general, not be the same.
6.4 Quasi-likelihood
When there is greater variability than is possible with a binomial distribution (i.e., with overdispersion or extra-binomial variation), instead of adding a random effect term as the LP, the
conditional variance can be defined as a constant times the binomial variance. For a RCBD, this is var(y[ij]|b[j]) = ϕnπ[ij](1 − π[ij]), where ϕ is an overdispersion scale parameter. It plays the
role of the index of dispersion (D) in spatial pattern analysis (Madden et al., 2018). By allowing ϕ to take on any positive value, the conditional variance can always be expressed as a multiple of
the binomial. For unbounded counts, one would write var(y[ij]|b[j]) = ϕμ[ij], where μ[ij] is the variance (and the mean) of the Poisson distribution. With ϕ not equal to 1, the conditional
distribution is no longer binomial; in fact, there is no longer an actual true conditional statistical distribution that could stochastically generate the data. Thus, one has a so-called
quasi-likelihood, and a method suitable for quasi-likelihood is used to fit the model (Gbur et al., 2012; Stroup, 2013). See below for a discussion on estimation.
The quasi-likelihood-based model for the RCBD is given as a slight modification of Equation 13:
$\begin{array}{ll}\begin{array}{l} {\eta }_{ij}=\text{θ}+{\tau }_{i}+{b}_{j}\\ \text{logit}\left({\pi }_{ij}\right)=\text{ }{\eta }_{ij}\\ {y}_{ij}|{b}_{j} ~ quasi.Bin\left({\pi }_{ij},n;\varphi \
right)\\ {b}_{j} ~N\left(0,{\sigma }_{b}^{2}\right)\end{array}& \left(17\right)\end{array}$
where $quasi.Bin\left({\pi }_{ij},n;\varphi \right)$ is an arbitrary expression that represents a conditional quasi-likelihood (which is not for the actual distribution of the data). Note that the LP
(η[ij]) is the same as the naive GLMM in Equation 13 (i.e., no addition of a v[ij] term). It would be incorrect to add the v[ij] term (block × treatment interaction) in a model where the ϕ is also
added as this would be overparameterization; that is, two terms (${\sigma }_{v}^{2}$ and $\varphi$) would be “competing” to explain the overdispersion.
6.4.1 Example
Table 2 displays the estimated means and SEs (model-scale logits and data-scale inverse link means) for the fit of Equation 17 based on the conditional quasi-binomial likelihood (model C). This was
done with RSPL (see below). The estimated means are extremely close to those obtained for the fit of naive Equation 13 (model A), but the SEs are much larger, reflecting an estimated scale parameter
of $\stackrel{^}{\varphi }$ = 5.77 (i.e., the conditional variance is nearly six times larger than that obtainable with the binomial). The SEs are all larger than those for the fit of the naive
Equation 13, but more similar to the SEs for the expanded LP model of Equation 16 (model B). The test of treatment effect is now F = 4.80, similar to that obtained with Equation 16 and much smaller
than the F for the naive Equation 13 (model A).
The use of quasi-likelihood for model fitting is very straightforward and can be applied to many modeling situations. It is often considered a simple fix for overdispersion in GLMs or GLMMs (
McCullagh and Nelder, 1989). Nevertheless, (Stroup 2013, pp. 347–348) was somewhat critical of the approach because an actual likelihood function is not used in the estimation as there is no true
conditional distribution. That is, there is no model for the direct stochastic generation of the data. There are alternative approaches, such as the use of Equation 16 (model B), which may be
consistent with the experimental design (i.e., random block and plot effects in this example). Model B (Equation 16) can be thought of as characterizing one possible data generation process for the
given experimental design (e.g., the random effects of blocks and experimental units such as plots) and model C (Equation 17) as characterizing a consequence of the random effects affecting π (or μ
in general) that are not in the model. Piepho (1999) and Madden et al. (2002) compared these and other GLMMs for the analysis of plant disease incidence data. Piepho (1999) should also be consulted
for a more thorough discussion of data generation processes that would lead to model B or C.
6.5 Different distributions with overdispersion
For unbounded count data, Poisson is commonly replaced by the NB to account for overdispersion at the experimental unit level (or for the clustering of individual observations, in general) (Madden
et al., 2007). The conditional Poisson would be replaced with the conditional NB in Equation 13. Because the NB has its own overdispersion parameter k that is estimated, then a v[ij] term would not
be added and a ϕ not specified for a quasi-distribution. This substitution of the NB for the Poisson is easy to do with the GLIMMIX procedure in SAS. For overdispersed binomial-type data (when there
is an upper bound of n for y), such as in our example, the beta-binomial distribution (“Betabin”) can replace the binomial. This distribution is not in the exponential family and is much less used
for mixed-model analysis, although it has had more use with GLMs with only fixed effects (Hughes and Madden, 1995; Morel and Neerchal, 2010) and for quantifying aspects of the spatial heterogeneity
of plant diseases (Madden et al., 2007, 2018). To use this model, the ${y}_{ij}|{b}_{j} ~\mathit{\text{Bin}}\left({\pi }_{ij},n\right)$ in Equation 13 would be replaced with ${y}_{ij}|{b}_{j} ~\
mathit{\text{Betabin}}\left({\pi }_{ij},n, \varrho \right)$, where $\varrho$ is one formulation of the overdispersion parameter for this distribution (at $\varrho$ = 0, the Betabin reduces to the
binomial; as $\varrho$ increases, overdispersion increases). The LP would contain only the treatment and block effect (as in Equation 13). This can be considered one type of DGLMM (Lee et al., 2006).
The beta-binomial distribution is not available in the GLIMMIX procedure, but can be fitted using ML (not the REML) with the NLMIXED procedure when there are random effects (such as block effects).
Considerably more coding is required to achieve this for the beta-binomial. An alternative to fitting the beta-binomial is to utilize the h-likelihood (see below).
The results for the example with the beta-binomial as the conditional distribution (model D) are given in Table 3 based on ML estimation. Although the means are similar, the SEs are larger than those
for model A (Equation 13) as the extra-binomial variability is taken into account, but smaller than those for models B and C (which take into account the extra-binomial variability in different
ways). The smaller SEs for the beta-binomial are due, in part, to ML rather than REML being used in the estimation.
Table 3
Table 3 Estimated means and standard errors (SEs, in parentheses) both on the linear predictor scale [logit(mean)] and the data scale (inverse link; with SEs based on the delta method) when
generalized linear mixed models (GLMMs) were fitted to the number of diseased wheat spikes^a using maximum likelihood with the quadrature method for integral approximation for Equation 16 (model
B-quad), for Equation 13 but with the beta-binomial for the conditional distribution of y instead of the binomial (model D), and when using Bayesian estimation with Equation 16 (model B-Bayes).
7 Fitting GLMMs to data
Unlike with normality-based LMMs, the method for fitting GLMMs to data (i.e., parameter estimation and random effects prediction) is controversial (Stroup, 2013; Stroup and Claassen, 2020). Even the
labels for the model-fitting methods are confusing and are used differently by different researchers, with the labels even evolving over time. For LMMs, REML is the generally preferred (and
noncontroversial) method of model fitting, partly because it produces unbiased estimates (fixed effects and their SEs) or less biased estimates than ML (McCulloch and Searle, 2001; Galecki and
Burzykowski, 2013; Stroup et al., 2018). For simple cases, REML duplicates the results for the mean square (and moment)-based methods that predate the contemporary likelihood-based methods for LMMs.
Regular ML can also be used for LMMs. However, it is well known that this produces biased estimates; in particular, the variance estimates and the SEs will be too small for ML estimation, resulting
in the test statistics being too large. The latter is an issue when the number of independent observations is small. REML and ML are both iterative methods, although simple situations can result in
convergence in a single iteration.
Recall that, for any experiment or survey, the observed data are a manifestation of the marginal distribution. Thus, the estimation requires integration over all the random effects in a conditional
model to approximate the marginal distribution. However, there is no analytical solution (i.e., mathematical expression) for this integration with GLMMs; only approximations are possible, which is
one of the complexities of working with this class of models. In fact, practical use of GLMMs only became possible with the availability of fast computers with large memory (and excellent
programming). Despite the many labels in the literature, two broad frequentist methods can be used: model approximation (linearization) and integral approximation. There are also some more
specialized methods.
7.1 Linearization
Linearization involves a first-order Taylor series expansion of the inverse link function of the LP equation [π[ij] = g^−1(η[ij]) = h(η[ij])], generally centered on the current (or initial) estimates
of the fixed and random effects (Breslow and Clayton, 1993; Wolfinger and O’Connell, 1993). A so-called pseudo-variable (y*) is formed based on this expansion, which is a function of the current
parameter estimates (and random effect predictions) and the first derivative of the inverse link with respect to the LP. The expected value of y* is modeled as a function of the fixed and random
effects (the right-hand side of the LP). The variance of y* conditional on the random effects is a complex function of the first derivatives and the variance function for the conditional distribution
[e.g., nπ(1 − π) for binomial and μ for Poisson].
Wolfinger and O’Connell (1993) developed a pseudo-likelihood (PL) method based on linearization. The PL method makes the (approximating) assumption that the conditional distribution of y* (but not y
), given the random effects, is normal with a complicated variance (Stroup, 2013; Xie and Madden, 2014). As a consequence, the marginal distribution is also normal. Thus, one can use the machinery of
normality-based LMMs to fit GLMMs and then (automatically) recover the relevant statistics for the actual distribution of y after convergence of the PL algorithm. The PL fitting algorithm is doubly
iterative in that the fitting of the y* pseudo-variable is iterative (as is any LMM), and then the y* is updated at the end of each LMM fit based on the LMM results, with the LMM fitting being
repeated with the updated y*. The double iterations continue until convergence (defined in different ways). With PL, one ultimately is basing the analysis on the actual distributions specified in the
model for y (the normality assumptions for y* is only for the intermediate computational work). Restricted PL is the default in the GLIMMIX procedure in SAS (called RSPL in SAS), which uses
REML-based model fitting for the pseudo-variable. As an option, the ML version of PL (ML-based PL; known as MSPL in GLIMMIX) can be performed. An overdispersion term, ϕ, can also be added as an
option (see Equation 17) with the PL method (either RSPL or MSPL), which then becomes a quasi-likelihood approach (no longer a true statistical distribution for the conditional distribution of y). We
are not aware of any R packages for PL estimation.
Breslow and Clayton (1993) took a quasi-likelihood approach, which only assumes that the objective function to maximize in the iterative estimation process has the general form of a member of the
exponential family of conditional distributions (defined through a mean and variance, but with a multiplicative overdispersion parameter). This approach is often labeled as penalized quasi-likelihood
(PQL) or marginal quasi-likelihood (MQL). Implementation in R can be done with the “glmmPQL” function in the MASS package. In this program, the overdispersion parameter ϕ is always estimated (cannot
be restricted to 1); therefore, the results do not correspond to a (conditional) distribution for y, but to a quasi-likelihood. The PL approach taken to fit model C in the example (Table 2) has
analogy with the PQL method of Breslow and Clayton (1993).
The PL method is extremely flexible, allowing not only true distributions but also quasi-likelihoods and can easily handle correlated observations, such as in temporal or spatial repeated measures (
Stroup, 2013). The RSPL method was therefore used to fit models A, B, and C to the example data discussed so far.
7.2 Integral approximation
Instead of approximating the model to obtain a marginal distribution of a pseudo-variable, the integration over the random effects (of the original model) can be approximated to obtain an estimated
marginal distribution of y. Generally, the best method to use is the Gauss–Hermit quadrature (quadrature for short), although it is computationally demanding and can be extremely (or painfully) slow
for moderate to large data problems, sometimes requiring many hours if there are several factors and interactions. The Laplace approximation, a special case of quadrature, is another integral
approximation method that works well for many datasets and often gives estimates very similar to those obtained with the more accurate quadrature (Joe, 2008; Stroup, 2013; Ruíz et al., 2023). Laplace
can be very fast and works when quadrature is not possible or practical. Both Laplace and quadrature are available as options in GLIMMIX of SAS, and quadrature with one quadrature point (which
reduces to one way of expressing the Laplace approximation) is the default in the lme4 package of R when fitting GLMMs. Linearization methods are not done with the “lme4” package.
There are two important points worth noting with this approach: 1) Integral approximations are ML-based; that is, there is no restricted/residual (REML-like) version that reduces bias. The problems
that come with the use of ML instead of REML with normality-based models carry over to GLMMs (bias of parameter estimates and SEs, test statistics, etc.). 2) Integral approximations require a true
likelihood; therefore, quasi-likelihood-based models (e.g., ϕ > 1 with quasi-binomial likelihood) cannot be fitted. In particular, model C for overdispersion with the binomial data cannot be fitted
using these approaches.
7.2.1 Example with integral approximation
The results from model B (Equation 16, with a random v[ij] for plot effects) using quadrature are given in Table 3 (model B-quad). The results for the means are similar in this example to those
obtained with RSPL (see Table 2), although the SEs are slightly smaller with the integral approximation methods (e.g., SEs of 0.245 versus 0.283 for the logit mean of treatment 1). The variance
estimates are a little smaller here (0.175 versus 0.248 for ${\stackrel{^}{\sigma }}_{v}^{2}$), and the F statistic is a little larger for quadrature compared with that of RSPL. The smaller variances
and SEs are a consequence of the use of ML- rather than REML-based methods. If the MSPL method in GLIMMIX was used, the results would be more similar between PL and integral approximations (both
being ML-based). The linearization and integral approximation (quadrature) methods, however, will never give identical results.
7.3 Comparison of linearization and integral approximations
Early on after the linearization methods were proposed, it was accepted as common knowledge that integral approximations are more accurate (less biased) than linearization when fitting GLMMs to data
(Stroup, 2013). However, this conclusion was mostly based on assessments under extreme conditions [such as when there was only one individual (diseased or healthy) in an experimental unit]. The
“common knowledge” has not held up based on more recent assessments, at least not as a generality (Couton and Stroup, 2013; Claassen, 2014; Piepho et al., 2018; Stroup and Claassen, 2020). Estimation
performance has been recently assessed in detail by Stroup and Claassen (2020) (see also their online supplements) with extensive simulations, and the overall results showed that there is no overall
best method for fitting GLMMs.
The best method depends, in part, on how well the conditional distribution (such as binomial or Poisson) can be approximated by a normal distribution (see Figure 1). For conditional binomial data,
this partly depends on n (the number of individual observations that gives a proportion y/n) for an individual experimental unit, such as a plot (block × treatment combination for a RCBD). When n is
much less than 30, the conditional binomial may be quite skewed (especially when π is close to 0 or 1), and the linearization methods may give biased results. A small n per experimental unit coupled
with a large number of experimental units (e.g., blocks in a RCBD) would be a situation where the likelihood approximation works best as the conditional distribution need not be normal-like. Integral
approximations may also be desirable when π is very close to 0 or 1 at moderate values of n. On the other hand, the linearization methods, particularly the RSPL, are very accurate when n is 40 or
higher in each experimental unit and may produce considerably less biased estimates of the means and SEs than the integral approximations, especially with the small number of experimental units
(blocks) that are common in agricultural sciences. There are circumstances when all estimation methods are less than satisfactory (e.g., small n combined with a small number of replicates).
Since integral approximations are strictly ML-based and not REML-based, they suffer from small-sample bias (small number of replicates, such as blocks in the RCBD case). Thus, with the typical number
of four to six blocks in field experiments, integral approximations will typically lead to estimates of variances that are too small, leading to the SEs for means being too small and the test
statistics for the significance of factor effects (overall or individual contrasts) being too large. These all result in a large type I error rate (rejecting the null hypothesis when the null
hypothesis is true). The RSPL linearization method (with the same model) performs much better in these situations. The exception is when the n is very small, in which no method may be acceptable.
However, the MSPL linearization method (or quasi-likelihood) would also suffer from the small-sample bias because it is ML- and not REML-based. A much more detailed investigation is given in Stroup
and Claassen (2020).
There are other advantages of the linearization methods. We have found when analyzing multiple datasets that the linearization methods more readily converge under a wide range of situations. Since
these methods use normal distributions (as intermediate approximations), some methods strictly for LMMs are available. One is the use of the Kenward–Roger method to obtain more appropriate estimates
of the SEs of the parameter estimates and adjustments for the denominator degrees of freedom for small samples (Stroup, 2013). Another advantage is that the algorithm does not “blow up” when one or
more variance estimates are 0. Negative variances can even be obtained with the linearization methods as long as the variance of the marginal distribution is positive (Stroup et al., 2018). This
allows for better control of type 1 errors in some circumstances. In contrast, with integral approximations, an estimate of 0 for a variance leads to nonsensical SEs (and other results). For the
latter, the terms in the GLMM with a 0 variance must be removed and the model refitted.
Based on Claassen (2014) and Stroup and Claassen (2020), one can make the following general recommendations for fitting GLMMs to conditional binomial data:
● For large n and small number of replicates (e.g., blocks): Use RSPL.
● For large n and large number of replicates: Use either approach as both have similar performance.
● For small n and small number of replicates: No method has shown great performance. RSPL may converge more easily.
● For small n and large number of replicates: Use integral approximation approaches.
● When using integral approximation, quadrature may be theoretically better than the Laplace approximation, but the Laplace approach gives similar results under many circumstances. Quadrature may not
be computationally feasible in some situations, especially with several variances in the GLMM.
● If one wants to simply inflate the variance of the conditional distribution (ϕ > 1) to account for the overdispersion (extra-binomial variation due to not explicitly accounting for some sources of
variation in the model), the binomial can be replaced with a quasi-likelihood (analogous to PL with the ϕ allowed to take any value). With a small number of replicates (blocks), this should be done
with restricted PL (REML-based, such as RSPL in GLIMMIX).
Similar recommendations can be made for conditional Poisson data (Stroup and Claassen, 2020). That is, if the number of replicates is small (as in many field studies), RSPL linearization outperforms,
or strongly outperforms, the quadrature and Laplace integral approximation methods. The superior performance of RSPL may be less pronounced when the conditional distribution is highly skewed
(typically when μ is very close to 0), but it still performs well. Integral approximations perform well primarily when there are many replicates.
7.4 Goodness of fit
Even when linearization may be advantageous based on the bias of the parameter and variance estimates, among others, there are some other reasons to use integral approximations. In particular, the
goodness of fit of models, including evaluation of the model assumptions, cannot easily or directly be done using linearization methods. This is because one is dealing with the (restricted)
log-likelihood for the pseudo-variable (with RSPL or MSPL), not the (restricted) log-likelihood for the original observations. The goodness of fit of the model for the pseudo-variable, although done,
is not a reliable metric to assess the GLMM (Littell et al., 2006). For instance, with the example in Table 2, adding the v[ij] block × treatment interaction to the GLMM (Equation 16) resulted in a
point estimate of ${\sigma }_{v}^{2}$ greater than 0, with larger SEs for the means and smaller F statistic for significance compared with those of the model without this term (Equation 13). This
suggests that the added random effect term was necessary. The results between the two models would be about the same (e.g., no increase in SEs) if the v[ij] term was not needed. However, this
conclusion is only informal.
The v[ij] term (or any random-effects terms) can be formally tested if the (restricted) log-likelihood for the data is available, which is obtained with the integral approximation. A likelihood-based
confidence interval can be constructed for ${\sigma }_{v}^{2}$, and a chi-squared test of significance (null hypothesis of ${\sigma }_{v}^{2}$ = 0) is straightforward with the GLIMMIX procedure. A
simpler approach with integral approximation approaches is to compare the AIC statistics for the model fits with and without the v[ij] term. With the example above, the AIC is 194.8 with the v[ij]
term (model B-quad), while it is 229.7 with the simpler model [model A-quad (results not shown for A-quad)]. The lower AIC is an indication that there is a random plot effect that needs to be
accounted for.
Likelihood-based methods are especially appropriate for studying the magnitude of estimated variances when the focus is on random effects rather than on fixed effects. For instance, Kriss et al.
(2012) used a hierarchical GLMM to investigate the variability of the incidence of Fusarium head blight among counties, fields within counties, and sampling sites within fields. Using a so-called
small-area sampling approach, the authors demonstrated the use of likelihood-based confidence intervals for the variances in which the spatial scales had the greatest (and lowest) heterogeneity of
disease. Presently, there are new computational methods for incorporating survey weights in the random effects of GLMMs to account for unequal sampling probabilities when analyzing survey data (
Diaz-Ramirez et al., 2020).
Recent research by Piepho (2019, 2023) has shown how to estimate a coefficient of determination (R^2) for the fit of LMMs and GLMMs. An advantage of this new approach is that linearization or
integral approximation methods can be used to fit the GLMM. Interested readers should consult these papers for a discussion of other proposals for R^2 calculations with mixed models. Using the
algorithm in Piepho (2023), R^2 = 0.507 for the fit of Equation 16 (model B) to the example data. In our view, the calculation of relative measures of goodness of fit such as R^2 is most desirable
when fitting LMMs or GLMMs with continuous explanatory variables, when trying to compare mixed models with different explanatory variables or number of explanatory variables, or when analyzing
observational data rather than data from randomized trials. An example would be a random coefficient model for crop yield in relation to disease intensity measurement, with location–year as a random
effect (Madden and Paul, 2009).
In principle, graphical methods can be used to assess the goodness of fit, as well as select the most reasonable model, as they commonly are for normality-based LMMs (Littell et al., 2006; Stroup,
2013; Stroup et al., 2018). This includes assessment of different types of residual plots. There are many possible choices in viewing residuals for GLMMs, such as those determined on the model (i.e.,
linear predictor) or the data (i.e., inverse link) scale, or the type of scaling (standardization) to use for the residuals (e.g., raw, Pearson, or studentized). A useful presentation on this can be
found in Stroup et al. (2018) and Stroup (2013). We agree with the authors of the former that more research is needed to fully evaluate the fit of GLMMs, especially when trying to decide on the most
appropriate model to use.
7.5 The h-likelihood
As an alternative to the linearization and integral methods described above, the so-called h-likelihood method is also possible (Lee and Nelder, 1996; Lee et al., 2006). Instead of estimating a
marginal likelihood (either on a pseudo-variable scale or the original data scale), a joint likelihood is defined for both fixed and random effects. This approach is especially useful when one wants
to specify that the random effects have non-normal distributions. An example would be a beta-distribution for the random block effect. Thus, the h-likelihood method is well suited for DGLMMs, such as
when the beta-binomial distribution is used; however, in principle, h-likelihood can be applied to a wide range of problems. Although there is an R package (“hglm”), the methodology has still not
been broadly adopted in statistics for routine data analysis with GLMMs, and some view the approach with skepticism (Meng, 2009). For normally distributed random effects, we have found that the h
-likelihood estimation performs very similarly to RSPL in terms of fixed-effects parameter estimates, SEs, and other statistics when the data have a conditional binomial distribution (Piepho et al.,
7.6 Bayesian estimation
Although we emphasize frequentist methods in this paper, a Bayesian approach can be taken to fit a GLMM if one has an assumed true conditional distribution (binomial or Poisson, for instance) (Piepho
and Madden, 2022). Thus, the Bayesian approach follows directly from the integral approximation methods in the sense that the model is not approximated and true distributions are used. Bayesian
analysis is a different philosophy, but is becoming quite popular in many fields. Prior distributions (indicating the degree of certainty or uncertainty before the experiment) on all the parameters
(e.g., τ[i]), random effects (e.g., b[j]), and the variance and covariance parameters (e.g., ${\sigma }_{b}^{2}$) need to be specified in order to ultimately estimate the marginal posterior
distributions and credible intervals for the parameters and the differences of parameters (such as the means and the differences of means). Most Bayesian approaches (but not all) require a
considerable amount of coding to implement, and is computationally expensive, where sampling of distributions is done, usually with the Markov chain Monte Carlo (MCMC) algorithm, until convergence is
reached (although it is not always reached). The approach is much more challenging for a non-statistician. There are a range of R packages for Bayesian analysis (rstan), and the release of the BGLIMM
procedure in SAS greatly facilitates Bayesian analysis with GLMMs as the syntax is very close to that of GLIMMIX. For those interested, Section 2.3 of Brown and Prescott (2015) is a good place to
start, at least in terms of mixed models. The online tutorial by Stroup (2021) is extremely informative for the fitting of GLMMs using the BGLIMM procedure. Piepho and Madden (2022) further
demonstrated the use of this Bayesian procedure for conducting a meta-analysis of conditional binomial data.
7.6.1 Example
When fitting Equation 16 to the example data (model B-Bayes) with non-informative prior distributions, the treatment means were similar to those found with the frequentist analysis (Table 3). As is
typical, the means were not the same, however. The standard deviations of the posterior distributions (analogous to the SEs in the frequentist analysis) were larger than those for the fit of the same
model with quadrature (compare models B, B-quad, and B-Bayes). This is expected because Bayesian analysis takes into account the uncertainty of the variance estimates.
8 Overview of some additional GLMMs and general guidance for analysis
8.1 Marginal model
Although one would normally be interested in the conditional means when fitting a GLMM to discrete data (Stroup, 2013), such as π[i], there may be situations when marginal means are desired. This
type of GLMM can be fitted by expanding on the concepts explained above for marginal LMMs (Equations 7-11). The approach uses the quasi-likelihood and expands on the common method used with GLMs (no
random effects) for repeated measures (Liang and Zeger, 1986), generally called generalized estimating equations (GEEs). As with LMMs, the LP for a RCBD consists only of the fixed effects, $\text{
logit}\left({\pi }_{ij}\right)={\eta }_{ij}=\theta +{\tau }_{i}$. The effects of block and plot are accounted for through a so-called working correlation or covariance matrix for the vector of the
response variable for the j-th block, y[j] (the response vector would have six elements with six treatments). The working covariance matrix is not a true covariance matrix, but behaves like one.
Details are given in Stroup (2015) and Stroup (2013).
Fitting a marginal GLMM (with RSPL) to the example disease incidence data would produce the same mean proportions (after applying the inverse link function to the estimated logits), as found in the
first column of results in Table 1 for a normality-based LMM fitted directly to the proportion data (Stroup, 2013). This is not unexpected because the marginal means and conditional means are the
same with LMMs, as discussed above. The SEs of the estimated means with marginal GLMM do vary with the mean, as required, due to the dependency of variances on the means for non-normal data. Although
similar here, the means from a marginal GLMM generally will not agree with the means from a conditional GLMM. Since we feel that researchers are mostly interested in conditional means (see above
discussion), we do not give any more details on the marginal approach.
8.2 GLMMs for two expansions of the RCBD
It is instructive to see how the GLMM for a RCBD is expanded for two different scenarios.
8.2.1 Sampling
With field studies, it is common to collect multiple samples within each experimental unit. For instance, Madden et al. (2002) analyzed several datasets with a GLMM for the effects of fungicide
treatment on the incidence of Phomopsis leaf blight in strawberry. Within each experimental unit (plot), there were either three or five samples, each consisting of n = 15 leaflets. Although the
values from the clusters could be pooled to obtain a single y and n for each plot, sampling within plots can also be explicitly accounted for. The LMM linear predictor would be:
$\begin{array}{ll} {\eta }_{ijk}=\text{θ}+{\tau }_{i}+{b}_{j}+{v}_{ij}& \text{(18A)}\end{array}$
where μ[ijk] = η[ijk], in which the ijk subscript represents the i-th treatment, the j-th block, and the k-th sampling unit within the plot (block × treatment combination), respectively, and v[ij] is
the random plot effect (equivalent to the block × treatment interaction). The residual for the LMM (i.e., the variance of the conditional normal distribution, ${\stackrel{^}{\sigma }}_{e}^{2}$)
represents the effect of the k-th sampling unit within the ij-th plot (block × treatment × sampling unit interaction).
For a GLMM with a conditional binomial distribution, the simple use of Equation 18A, where logit(π[ijk]) = η[ijk], would fail to account for the effects of the sampling units within plots. This
approach would be a generalization of model A (Equation 13). A quasi-likelihood approach could be taken using Equation 18A with the overdispersion parameter ϕ, producing a generalization of model C (
Equation 17; quasi-conditional binomial likelihood). Alternatively, the conditional model could be expanded to:
$\begin{array}{ll}\text{ }{\eta }_{ijk}=\text{θ}+{\tau }_{i}+{b}_{j}+{v}_{ij}+{w}_{ijk}& \text{(18B)}\end{array}$
where w[ijk] is the random effect of the k-th sampling unit within the ij-th plot (analogous to the residual in a LMM, now a three-way interaction of block, treatment, and sampling unit). Now, one
has a conditional binomial for y[ijk]:
Both v[ij] and w[ijk] are assumed to have normal distributions with variances ${\sigma }_{v}^{2}$ and ${\sigma }_{w}^{2}$, respectively, when using Equation 18B. This would be an expansion of model B
(Equation 16). Based on previous work by Piepho (1999), Madden et al. (2002) compared these and some other more complex GLMMs for the analysis of five strawberry datasets for disease incidence.
Additional levels in the sampling hierarchy for disease incidence were analyzed for another crop/disease system by Piepho (1999) using data initially analyzed by Hughes and Madden (1995) using GLMs
(all fixed effects). Only the RSPL (without or with a ϕ overdispersion term) was used by Piepho (1999) and Madden et al. (2002). Recall from above that the use of the ϕ parameter with PL is actually
a form of quasi-likelihood.
8.2.2 Split plots
A split plot with blocking is a very common design for field experiments in agriculture, involving two or more fixed-effects factors (Steel and Torrie, 1960; Schabenberger and Pierce, 2002). Here,
one has three sizes of experimental units: blocks, whole plots (one of the fixed-effects factors, where the factor levels are randomized within each block), and sub-plots (another fixed-effects
factor, where the factor levels are randomized within each whole plot). For instance, Moraes et al. (2022) studied the effect of cultivar resistance level and fungicide treatment on disease incidence
for Fusarium head blight, as well as for other response variables assumed to have distributions in the exponential family. For a single year, cultivar was the whole plot (large experimental units),
randomized within blocks, and fungicide treatment was the sub-plot (small experimental units), randomized within each whole plot factor level within blocks. With random blocks and a split-plot
design, the LP for a normality-based LMM is:
$\begin{array}{ll}\text{ }{\eta }_{ijk}=\text{θ}+{\gamma }_{i}+{\tau }_{j}+{\left(\gamma \tau \right)}_{ij}+{b}_{k}+{v}_{ik}& \text{(19A)}\end{array}$
where μ[ijk] = η[ijk], in which γ[i] is the effect of the i-th whole-plot factor, τ[j] is the effect of the j-th sub-plot factor, (γτ)[ij] is the interaction of the whole-plot and sub-plot factors, b
[k] is the random effect of the k-th block, and v[ik] is the random effect of the ik-th whole plot (equivalent to the interaction of block and the whole-plot effect). The variance of the conditional
normal distribution represents the variability among sub-plots within whole plots (equivalent to the interaction of block, whole-plot, and sub-plot effects).
For a GLMM for disease incidence (conditional binomial), with logit(π[ijk]) = η[ijk], direct use of Equation 19A [an expansion of model A (Equation 13) for RCBD] would not account for the variability
of the sub-plots. The extra-binomial variability due to a sub-plot effect could be accounted for by using quasi-likelihood for the binomial, i.e., using Equation 19B, but with a quasi-binomial
likelihood with a ϕ parameter [expansion of model C (Equation 17)]. Alternatively, reflecting the random effects of sub-plots within whole plots and blocks, the LP can be expanded to:
$\begin{array}{ll} {\eta }_{ijk}=\text{θ}+{\gamma }_{i}+{\tau }_{j}+{\left(\gamma \tau \right)}_{ij}+{b}_{k}+{v}_{ik}+{w}_{ijk}& \text{(19B)}\end{array}$
where w[ijk] is the random effect of the j-th sub-plot within the ik-th whole plot (equivalent to the interaction of block, whole plot, and sub-plot). This would be an expansion of model B (Equation
16) for a conditional binomial. Both v[ij] and w[ijk] are assumed to have normal distributions with variances ${\sigma }_{v}^{2}$ and ${\sigma }_{w}^{2}$, respectively. Equation 19B is the basis for
the model used in Moraes et al. (2022), although the authors expanded it to account for an analysis of the split plot over years (where year and interactions with year were also random effects in the
8.3 Overall guidance for GLMMs with different designs
GLMMs can be expanded in numerous ways to account for different treatment and experimental designs. Readers should consult Gbur et al. (2012); Stroup (2013); Stroup et al. (2018), and Ruíz et al.
(2023) for examples. For those who are used to analyzing data with LMMs, here is the basic rule of thumb:
● Start with the LP equation that would be appropriate for a normality-based LMM analysis (there are many textbooks with this).
● For GLMMs involving either the binomial or Poisson conditional distribution, either:
o Add a random-effects term to the LP that would be analogous to the residual term of a LMM, which can then be fitted with a linearization or an integral approximation method (see above for guidance)
o Keep the LP of the LMM and use a quasi-likelihood (with the ϕ parameter) instead of a true conditional distribution.
As described above, there are other approaches as well, such as the use of marginal models or the switch to other conditional distributions for overdispersed binomial-type data (e.g., beta-binomial).
For repeated measures, or more generally when there is a variance–covariance matrix, the use of GLMM methods is more challenging, but can be done with a working covariance matrix (a GEE-type
quasi-likelihood approach). See Stroup et al. (2018) and Stroup (2013) for discussion and examples.
When the conditional distribution is not Poisson or binomial, such as the gamma or beta for (assumed) non-normal continuous random variables, there is actually a variance or scale parameter for the
conditional distribution that is estimated from the data (Gbur et al., 2012). Thus, one might not need to consider adding a random effect term to the LP or use a quasi-likelihood approach. There may
still be additional variability to account for, however, and readers should refer to Gbur et al. (2012).
8.4 The overall guidance applied to the example
Several GLMMs and model-fitting methods were useful in analyzing the example dataset that exhibited the typical situation of higher variability than can be represented by the naive model A, the model
that transfers directly from an LMM-based analysis. The use of model A would give a false sense of treatment effects with artificially low SEs. The addition of an experimental unit random-effects
term (analogous to the residual in an LMM) was consistent with the experimental design and led to more realistic SEs and tests of significance. Due to the small number of blocks (typical for field
studies in the plant sciences) and the relatively large number of observations within plots, the RSPL (model B), a linearization approach, was preferable to the quadrature (model B-quad), an integral
approximation. Quasi-likelihood methods could also be used to account for the overdispersion exhibited by the data (model C), although one no longer has a true conditional statistical distribution in
the model. A more complicated approach of changing the conditional distribution from binomial to beta-binomial was also effective in accounting for the overdispersion (model D), although this method
will probably not be for routine analysis by non-statisticians. Except for model A, the differences in the results (i.e., the means and SEs) were not large, reflecting the fact that the estimated π
values for the different treatments were not too close to 0 or 1.
9 Challenges
9.1 All zeros or all ones
Although it remains popular to fit LMMs to transformed proportion data (for conditional binomial), care must be taken regarding the selected transformation. Transformations of observations such as
logs or logits fail (i.e., are undefined) when the proportion is 0 or 1, and the log transformation fails when the proportion is 0 (Piepho, 2003) (the angular transformation is defined for these
proportions). A major advantage of GLMMs for conditional binomial data is that the observed individual zeros or ones for proportions do not pose any difficulty; no ad hoc methods (adjustments for
zeros and ones) are needed to accommodate the extreme values of the response variable. Nevertheless, there remains a potential problem with GLMMs fitted to data with a conditional binomial (or
Poisson) distribution. For a RCBD and conditional binomial data, as an example, model fitting fails if all the y observations for a given treatment are equal to 0 or n (so that the proportions are 0
and 1 for all observations) (Gianinetti, 2020). Statistically, the problem is known as quasi-complete separation (Albert and Anderson, 1984; Clark et al., 2023). Since this is typically found with a
very effective treatment for controlling disease, so that all y values are 0, this is sometimes called the all-zero problem, but it all applies to the situation where all the y values are equal to n.
When one tries to use a likelihood-based model-fitting method with these data, the iterative procedure will never converge, or apparent convergence may be obtained, but for nonsensical or meaningless
estimates of some means and absurdly large (and meaningless) values of some SEs (Albert and Anderson, 1984). Essentially, this is because with quasi-complete separation, mathematically, the ML
estimates of one or more parameters are infinite (Heinze and Schemper, 2002). That is, some parameter estimates do not exist as real numbers when fitting a model using ML. Gianinetti (2020) and
Claassen (2014) discussed the problem in more detail. One solution is to remove the treatments with all zeros (or all ns) and then refit the model. The problem is that the treatment resulting in all
zeros often is of most interest, such as a new or a very effective pesticide. The treatment with all ns may be the control treatment that the investigator wishes to compare with others. The ad hoc
approach is to add a small constant, c, to y (y′ = y + c) and add 2c to n (n′ = n + 2c). This gives values of prop′ (=y′/n′) that may be very close to 0 and 1, but never equal to these limits. The y′
data can then be analyzed. Of course, the choice of c will affect the results. A common choice for c is 0.5. Still, there are questions. In particular, should one modify all observations in this
manner (all treatments and blocks) or just for the “problem” treatments? In fact, the modification is only needed for a single observation, and not all replicates, to avoid the problem (one of the
blocks for the problem treatment).
For GLMs (i.e., all fixed effects), models can be fitted to data of this type using a penalized likelihood method (Firth, 1993; Heinze and Schemper, 2002). For a RCBD, this would mean treating block
as a fixed effect. This is straightforward with the LOGISTIC procedure in SAS and with some R packages. Although this may be reasonable for this balanced simple case, expansions to split plots and
other designs (such as repeated measures and cluster sampling, among others) are not possible as random effects are required to account for aspects of the experimental designs. It turns out that
Firth’s penalized likelihood method is related to the ad hoc approach of fitting a model to the modified y′ data (Firth, 1993). Firth (1993) also showed the link between his penalized likelihood
approach and Bayesian analysis with certain types of prior distributions. Claassen (2014) has made important contributions by expanding on the approach in Firth (1993) for GLMMs for some situations.
As far as we are aware, this generalization is not available in standard GLMM software.
Overall, there is no best approach to recommend at this point for data with a conditional binomial (or Poisson) distribution, and a lot more research is needed. For the practicing data analyst, the
ad hoc y′/n′ approach will be the easiest to implement in the GLMM context, by far. Alternatively, analyzing the proportion data (suitably transformed) using a normality-based LMM might be the most
practical when there are multiple treatments with all zeros or all ns.
The situation is different for continuous non-normal variables in the exponential family. For instance, with the commonly used gamma distribution (Gbur et al., 2012), the response variable is a
non-zero positive value (0 is not allowed). With the beta distribution for continuous proportions, the response variable is between 0 and 1 (where 0 and 1 are not allowed). When using these as the
conditional distribution in a GLMM, any individual observations outside the permitted range become missing values (just like any 0 becomes a missing value when logs are used in a LMM). An adjusted y
(y′) would need to be used to keep the response variable within the required range, although the form of the adjustment will affect the results. There are also generalizations of the gamma and beta
distributions that allow for zeros (or zeros and ones) (e.g., Basak and Balakrishnan, 2012), but these are outside the usual realm of GLMMs with exponential family distributions. More research is
needed to deal with this problem when using GLMMs.
9.2 Too many zeros
Misspecification of the conditional distribution is one of the causes of overdispersion for discrete data. Sometimes, there too many zeros in the dataset compared with what is possible to represent
with a binomial distribution (or a beta-binomial). There could also be too many observations with all n individuals with disease (or whatever the trait of interest). For plant disease in the field,
suppose, as a simple example, that there is no pathogen inoculum present in some plots (e.g., certain sections of a field), but there is inoculum in the other plots (e.g., fungal spores in soil); the
presence or absence of inoculum is unknown to the investigator. This can be considered a consequence of the spatial heterogeneity of the unknown inoculum levels. Where inoculum is present, the
conditional binomial may be appropriate for any given treatment and block (where y could be from 0 to n), but without the inoculum, there can be no infection (where y must be 0).
For the conditional distribution, one then has a mixture of two distributions or two processes. One distribution would be binomial and the other would be a degenerate discrete distribution, where
Prob(y = 0) = 1. This is known as the zero-inflated process, which is usually manifested by a relatively large frequency of zeros in a histogram compared with the rest of the frequency distribution
at a given π (Cohen, 1960; Lambert, 1992). In Equation 13, as an example, all sub-equations are the same, except that we write ${y}_{ij}|{b}_{j} ~ \text{Mixture}\left({\pi }_{ij},n;\text{ω}\right)$,
where “Mixture” is an arbitrary notation for the mixture distribution and ω is a so-called mixture probability parameter. The estimate of ω represents the proportion of the mixture that is of the
binomial type, and 1 − ω represents the proportion that is the type with all zeros. Detection of zero inflation will typically be done only when there are many observations, such as when a dataset
consists of multiple cluster samples within each experimental unit (plot). Failure to account for this situation will lead to biased estimates of the π[i] probabilities for treatments. There would
typically be insufficient data in a simple RCBD, with one observation per combination of treatment and block, to identify and model zero inflation as a mixture distribution.
There are excellent methods for fitting zero-inflated models with GLMs (all fixed effects), such as in GENMOD and FMM in SAS. It is much more challenging to fit zero-inflated GLMMs. In SAS, one would
need to utilize nonlinear mixed-model software and write the code in NLMIXED. “glmmTMB” is an R package for zero-inflated discrete data, while “NBZIMM” is a mixed-model package for zero-inflated NB
data (recall that NB is a generalization of the Poisson for overdispersed count data). Bayesian approaches may be very beneficial (Zuur et al., 2012). Readers should be on the lookout for new
procedures and packages for fitting these types of GLMMs.
10 Example GLMMs in the plant sciences in agriculture
We have assembled a list of papers where GLMMs for non-normal data (or data assumed to be non-normal) have been used in plant pathology, agronomy, and horticulture (Table 4). We also included papers
when normal (Gaussian) data were analyzed together with non-normal data (for different response variables). This list is based on a search in Google Scholar and Web of Science. The list is only
intended to give a sense of the range of applications of GLMMs and related data analyses and is not meant to be an exhaustive compilation. The response variables analyzed in these experiments can be
placed in the following categories: i) continuous symmetric or asymmetric data (e.g., amount of mycotoxins, insect body length, seed weight, sucrose concentration, yield, disease severity, leaf
damage, or beetle survival time); ii) discrete proportion data (e.g., number of leaves infected, germinated seeds, pigmented flowers, or hatched eggs out of a total of n individuals); and iii) count
(discrete) data without a definable upper limit (e.g., number of insects, aphids, female eggs, or tubers). These studies were conducted to assess either the efficacy of fungicides or insecticides on
disease and insect control or the effect of environmental or host factors on disease severity, mycotoxin accumulation, plant density, seed germination, seed weight, etc. A handful of studies are
focused on a theoretical evaluation of the different elements of GLMMs using existing datasets (e.g., Piepho, 2019).
Table 4
Table 4 Summary of some applications of generalized linear mixed models in the analysis of data from designed experiments in agricultural and horticultural systems.
Binomial, NB, gamma, and Poisson were the most commonly used conditional distributions in these studies, while beta (Bilka et al., 2021; Susi and Laine, 2021) and Bernoulli (a special case of
binomial with n = 1) (De Silva et al., 2014) were less frequently used. The link functions specified in the models in these studies were logit, log, probit, complementary log–log, and identity (i.e.,
no transformation of the mean). While several papers provided details on how model fitting was conducted and specified the estimation method [ML (quadrature or Laplace), PL, or quasi-likelihood)]
used, other papers did not provide details of the estimation method used. We made no attempt to establish whether the model, the link function, or the estimation method specified in these studies was
appropriately specified by the authors to match the data and the aims of individual studies. Furthermore, we did not evaluate whether the authors interpreted the results appropriately. This summary
shows that the application of GLMMs in the above disciplines is diverse and is used to address hypotheses in a wide range of experimental conditions. We expect this to grow as researchers appreciate
the clear argument of matching the model to the data (for a given experimental and treatment design) when using GLMMs for analysis.
11 Summary and conclusions
There are several reasons to use GLMMs for the analysis of non-normal data, some of which were discussed in this paper, with many more details in several references (Littell et al., 2006; Bolker
et al., 2009; Zuur et al., 2009; Gbur et al., 2012; Stroup, 2013; Brown and Prescott, 2015; Stroup et al., 2018; Gianinetti, 2020; Li et al., 2023; Ruíz et al., 2015, 2023). Perhaps the biggest
argument is that one is better off matching the model to the data with a GLMM rather than changing (i.e., transforming) the data to match the model, as with a LMM (Gbur et al., 2012; Stroup, 2013).
With some contemporary software, it is deceptively easy now to switch from normality-based LMMs to non-normality-based GLMMs. For instance, this simple model statement in the GLIMMIX procedure of SAS
can be used to fit a LMM to the response variable y for a RCBD (as in our example):
$\text{model} y=\text{ trt};$
(Random effects, such as blocks, are given in separate statements, and the identification of explanatory variables as factors is done in a separate statement). Using the same procedure, a
conditional–binomial GLMM with a logit link function can be fitted to the same data with only one minor change to the model statement:
$\text{model\hspace{0.17em}}y/n=\text{ trt};$
with all other statements (not shown here) being the same. The procedure automatically identifies this as a conditional binomial with logit link when the left-hand side of the equation is given as
the ratio of a response variable to the number of individuals (i.e., y/n). Of course, the distribution (dist=) and link (link=) can be explicitly specified as options with this procedure (Gbur
et al., 2012; Brown and Prescott, 2015; Ruiz et al., 2023), allowing for many different conditional distributions and links. However, this simple switch would be fitting the naive model A (Equation
13) to the RCBD data (assuming the block random effect was also specified). This is because, with an LMM, a residual is always estimated to account for the plot variability (in a field study) after
accounting for other sources of variability, but is not estimated with a conditional binomial (or conditional Poisson). The variance is always nπ(1 − π) for a conditional binomial and always μ for
the conditional Poisson. In separate statements for random effects, the block × treatment random term (v[ij]) would need to be added to fit model B (Equation 16), or the conditional distribution to
be changed from binomial to a quasi-binomial likelihood with the ϕ overdispersion parameter to account for overdispersion (model C; Equation 17). Alternatively, a more complex conditional
distribution could be used, such as the beta-binomial (but not with the GLIMMIX procedure).
The important point here is that one should never lose sight of the experimental design when analyzing data (Piepho et al., 2003; Bello et al., 2004, 2016), as well as how to account for the relevant
aspects of the design (fixed effects, random effects, and correlations) in the model (e.g., right-hand side of the equation for η, plus distributions for random effects) for any selected conditional
distribution, whether using LMMs or GLMMs. Readers should consult Stroup (2013, 2015) for a more thorough discussion on this topic. Adding terms to a GLMM (or modifying conditional distributions) may
be needed to account for the sources of variability that are automatically accounted for with LMMs.
In this paper, we emphasized the switch from LMM to GLMM, with the assumption that the reader is more familiar with LMMs. We mostly restricted our attention to one type of response variable [discrete
observations (counts) that can be represented as proportions or percentages] and one experimental design, namely, a RCBD. This approach was chosen to show the similarities and differences between
LMMs and GLMMs, focusing on one small dataset. The differences in the results between LMMs and GLMMs will be greater in other datasets with larger random effect variances and when some of the π[i]
values are closer to 0 or 1 for some treatments (Stroup, 2013). The differences between LMMs and GLMMs carry through to more complex experimental and treatment designs (Gbur et al., 2012; Ruíz
et al., 2023). Issues in data analysis that are mostly well resolved for LMMs remain debatable with GLMMs. For instance, appropriate model fitting (estimation) can be controversial for GLMMs (Stroup
and Claassen, 2020). In fact, some models, such as those involving a quasi-likelihood (e.g., models with an overdispersion parameter, ϕ), can only be fitted using certain estimation methods (either
pseudo-likelihood or quasi-likelihood). For other models, investigators have choices for model fitting, and the best choice depends on the circumstances, as presented above. Despite earlier
criticisms, linearization methods, particularly the restricted versions such as RSPL (analogous to REML for LMMs), have been shown recently to often be the best choice for model fitting, especially
for discrete data and the small sample sizes typically used in field studies in agriculture (Stroup and Claassen, 2020). Integral approximations of the marginal likelihood (such as quadrature and
Laplace) do have advantages when there are reasonably large numbers of replicates, especially when one is interested in comparing the goodness of fit of different models through differences in the
log-likelihood or have specific interest in the variances and covariances.
A key distinction of LMMs and GLMMs is the notion of conditional versus marginal means (Stroup, 2013). Conditional and marginal means are the same for normality-based LMMs, but are different, in
general, for GLMMs. The magnitude of the random effect variances has a large influence on the magnitude of the difference between the conditional and marginal means. We agree with Stroup (2013, 2015)
and Stroup et al. (2018) that most researchers would find the conditional means as being more meaningful and of primary interest, thus emphasizing the importance of conditional models in a GLMM-based
analysis. The use of a conditional model, however, still allows for broad or narrow inference when testing hypotheses or calculating the confidence intervals for parameters or parameter differences.
Inference space (broad vs. narrow) has to do with whether inference is over the entire population of random effects (say, all possible blocks, locations, etc.) or only for the specific random effect
levels in the study. Littell et al. (2006) and Stroup (2013) should be consulted for more detail on this topic.
Author contributions
LM: Conceptualization, Formal analysis, Writing – original draft, Writing – review & editing. PO: Conceptualization, Data curation, Writing – review & editing.
The author(s) declare that no financial support was received for the research, authorship, and/or publication of this article.
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.
The author(s) declared that they were an editorial board member of Frontiers, at the time of submission. This had no impact on the peer review process and the final decision.
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Supplementary material
The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fhort.2024.1423462/full#supplementary-material
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Keywords: binomial, conditional distribution, fixed effects, integral approximation, marginal distribution, pseudo-likelihood, random effects, quasi-likelihood
Citation: Madden LV and Ojiambo PS (2024) The value of generalized linear mixed models for data analysis in the plant sciences. Front. Hortic. 3:1423462. doi: 10.3389/fhort.2024.1423462
Received: 25 April 2024; Accepted: 04 June 2024;
Published: 25 June 2024.
Edited by:
Xiangming Xu
, National Institute of Agricultural Botany (NIAB), United Kingdom
Copyright © 2024 Madden and Ojiambo. 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: Laurence V. Madden, madden.1@osu.edu | {"url":"https://www.frontiersin.org/journals/horticulture/articles/10.3389/fhort.2024.1423462/full","timestamp":"2024-11-14T08:04:42Z","content_type":"text/html","content_length":"963000","record_id":"<urn:uuid:794c4976-a064-448f-838e-7fd852c784a1>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00827.warc.gz"} |
How To Do Long Division? - Steps and Methods - Chimpvine
How To Do Long Division? – Steps and Methods
Understanding Division
Division is a fundamental operation in mathematics that involves the distribution of a quantity into equal parts. It is the process of splitting a number into equal groups or determining how many
times one number is contained within another. In this article, we will explore the concept of division, division properties, and the method of long division.
The Division Equation
The division equation is represented as a ÷ b = c, where ‘a’ is the dividend, ‘b’ is the divisor, and ‘c’ is the quotient. The dividend is the number being divided, the divisor is the number by
which the dividend is divided, and the quotient is the result of the division. When the number isn’t excatly divisor, there may be a remainder as well. It is the number that cannit be further
How to Do Long Division
Division is one of the four basic mathematical operations which includes addition, subtraction, and multiplication. Long division is a method used to divide large numbers or polynomials by hand. The
process involves several steps, including dividing, multiplying, subtracting, and bringing down digits. It is a systematic approach to division that allows for the accurate calculation of the
quotient and remainder.
Steps of Doing Long Division
Step 1: Set up the division problem.
Step 2: Divide the first digit of the dividend by the divisor and write the answer on the top which will be the quotient.
Step 3: Subtract the result from the number and write down the difference,
Step 4: Bring down the next digit of the dividend
Step 5: Repeat the same process until it can’t be furter divided.
Let’s look at some examples.
Example 1: Long Division without remainder
Dividing 650 by 5
Step 1: Setting up the division problem
Step 2: Divide the first digit 6, by 5 . As 6 cannot be divided by 5, we will look for a number less than 6 that 5 can divide, which is 5 itself. So, 5 ÷ 5 = 1, we will write 1 in the quotient.
Step 3: Subtract 6 – 5 = 1.
Step 4: We bring down the next digit of the dividen wihch is 5. The new dividend will be 15, we can divide 15 by 5 with quotient 3, as 5 * 3 = 15.
Step 5: Subtract 15 – 15 =0, we will bring the next digit down, which is 0.
Step 6: As 5 ÷ 0 = 0, we will now divide the dividend of 0 by 0, which will give us 0, so our answer will be 130.
Example 2: Long Division with Remainder
Divide 263 by 3
Step 1: Setting up the division problem
Step 2: Divide the first digit 2, by 3 . As 3 is greater than 2, we will put 0 and bring the next digit down.
Step 3: Then, 26 will be the dividend. We can’t divide 26 by 3, however we know that 3*8 = 24, so we will use 8 as the quotient.
Step 4: Subtract 26 – 24 = 2. Bring down 3, so the next dividend will be 23.
Step 5: We can’t divide 23 by 3, but 7 * 3 = 21, so we will use 7 as the quotient.
Step 6: Subtract 23 – 21 = 2, there is no othher digit, so 2 is our remainder and we cannot go further.
Learn division with interactive activities on our site ChimpVine.
1. Long Division for Two Digits
Tip: If the divisor is a two digit number, we will look for the first two digit of the dividend and carry out the division.
2. Dividing with 0
Tip: If 0 is divided by any number, the answer will be 0. For example, 0 ÷55 = 0
3. Dividing with 1
Tip: If any number is divided by 1, the quotient will be the same number. For example, 40 ÷1=40
4. Division Formula
Tip: We can use the division formula to verify quotient and remainder. The formula is Dividend = (Divisor * Quotient) + Remainder.
Story: “The Long Division Adventures of Sam and Emma”
Sam and Emma were two curious explorers who encountered real-life situations where the concept of long division played a crucial role in solving problems and making decisions.
Challenge 1: The Recipe Dilemma
Sam and Emma were passionate about cooking and decided to bake a batch of cookies. The recipe called for 2 cups of flour, and they needed to divide the flour equally into 4 portions. By using the
division equation, they determined that each portion should contain 0.5 cups of flour, ensuring the perfect consistency for their cookies.
Challenge 2: The Budgeting Puzzle
As they planned a road trip, Sam and Emma needed to divide their total budget of $500 over 5 days of travel. By applying the division equation, they calculated that they could spend $100 per day,
allowing them to manage their expenses effectively and enjoy their journey without overspending.
Challenge 3: The Party Planning Adventure
Sam and Emma had to divide 60 party favors into 6 equal groups for their friend’s birthday party. Using the concept of division, they determined that each group would receive 10 party favors,
ensuring that all the guests had an equal share of the fun.
The properties of division includes division by 1 Property, division by itself property, division by 0 property, and division of 0 by any number property.
Division and multiplication are inverse operations, meaning they undo each other. When dividing a number by another, it is equivalent to multiplying the first number by the reciprocal of the second
number. This relationship allows for the conversion of division problems into multiplication problems and vice versa.
Long division is significant as it provides a systematic method for dividing large numbers or polynomials. It allows for the accurate calculation of the quotient and remainder, making it a valuable
tool for solving complex division problems by hand.
Yes, division is commonly used to solve real-life problems related to sharing, distribution, measurement, and resource allocation. It helps in determining equal portions, calculating rates, and
managing quantities, making it an essential skill for everyday tasks and decision-making.
Division performs operations with fractions and decimals, such as finding the quotient of two fractions or dividing a decimal by a whole number. It allows for the comparison, simplification, and
conversion of fractions and decimals, making it a fundamental aspect of working with rational numbers.
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The Science Behind Relationship PredictorsThe Science Behind Relationship Predictors
The Science Behind Relationship Predictors
Relationship Predictor Survey
There are now several tools that predict which genealogical relationships are most likely for given amounts of shared DNA. With the recent introduction of the cM Explainer^TM at MyHeritage, Blaine
Bettinger and I launched a citizen science initiative to evaluate them. Volunteers have been contributing match data for known relationships then running each match through each of the tools in the
study. The goal is to determine which predictor, if any, is best. You can read more about the project here.
To date, the survey has more than 6,000 entries for 50 unique relationships ranging from parent–child to 8th cousins twice removed. We’re putting the finishing touches on the analyses and are
excited to share them with the genealogy community soon. In the meantime, it helps to understand how relationship predictors work.
What Could Go Wrong?
Biology likes to throw curveballs, so even the best DNA-based predictions will be off sometimes. On average, 1st cousins (1Cs) share roughly 875 cM, but a 1st cousin once removed (1C1R) can share
that much on the rare occasion. If you happen to be that rare 1C1R, all of the known predictive tools will peg you as a 1C over a 1C1R. That doesn’t mean the tools are faulty—nor that you are!—just
that this is a particularly tough case.
Of course, not all predictive tools are equal. They are based on computer models, and those models make assumptions. If the assumptions are wrong, the predictions can be off as well.
The Assumptions
By understanding those assumptions, we can get a better sense of how much credence to give a predictive tool. Let’s consider some of them.
Genome Size
For simplicity, in this post I will assume that a parent–child match shares 3500 cM and that the amount is halved each generation.
Sex-specific Inheritance Patterns
We inherit exactly 50% (3500 cM) of our autosomal DNA from each parent but not exactly 25% (1750 cM) from each grandparent. That’s because a process called “crossing over” in a parent’s body divvies
up the grandparents’ DNA before passing it on, and that division is rarely equitable. Although the average is ≈1750 cM, some grandparent–grandchild matches will share more and some will share less.
Distributions of atDNA shared with maternal (pink) and paternal (blue) grandparents. The higher the bar, the more likely that centimorgan amount is for the relationship.
It turns out that egg production involves more crossover events than sperm production. Just as you’re more likely to get a 50-50 split of heads versus tails if you flip a coin 20 times than 10
times, you’re more likely to share close to the average of 1750 cM with your maternal grandparents than your paternal ones; the crossover events are analogous to coin flips. That means relationship
prediction for your maternal relatives should be slightly more accurate, because those matches will be closer to the average, on average.
Family Structure
How likely a DNA match is to be any given relationship depends, in part, on how many such relatives you have. For example, a match of 1750 cM could be a grandparent, grandchild, aunt or uncle, niece
or nephew, or half sibling, and which is more likely will depend on how many of each you have.
On paper, I have four grandparents, one aunt, one uncle, and one half sister. Based on those numbers, a 1750-cM match to me has a 57% chance of being a grandparent, a 29% chance of being an aunt or
uncle, and a 14% chance of being a half sibling. A younger cousin of mine has four grandparents, six aunts/uncles, and two half siblings, so for her the probabilities are 33%, 50%, and 17%,
respectively. Those probabilities are quite different.
Any tool that tries to differentiate those relationships must make assumptions about the family structure, usually based on what’s typical for the population. If your family structure doesn’t fit
those assumptions, the predictions can be off for you yet be quite good for someone else.
Population Growth
Population growth also plays a key role here. Consider an unknown match who shares somewhere between the averages for 1C and 1C1R. Which of those two relationships is more likely depends, in part,
on how many 1Cs and 1C1Rs you have. If each of your 1Cs had one child each, then you should have an equal number of 1Cs and 1C1Rs. (Let’s ignore our parents’ first cousins for now, just to get the
point across.) The distributions will look like this:
The break-even point, where a match is equally likely to be either relationship, is around 625 cM.
However, if each of those family members had two kids each, you have twice as many 1C1Rs as 1Cs, and the distributions would look like this:
In this case, the break-even point is closer to 650 cM and a 625-cM match is more likely to be a 1C1R. If your family averages four kids per couple, the probabilities are shifted even more.
The most popular predictive tools are based on proprietary data from either AncestryDNA or MyHeritage, neither of which has publicly stated the population growth factor they use. However, it’s safe
to assume it’s around 2.5, which is a fairly standard fertility rate for developed countries over the past century. Again, if your family doesn’t align well with that assumption, the predictors may
not work as well for you.
Perhaps the biggest challenge for relationship predictors is endogamy, the practice of marrying within a cultural or geographic group. Endogamy is common around the world and causes people to be
related in more than one way. Those “extra” connections can increase the amount of shared DNA and throw off predictor tools that assume each match is related only once. Thus, those of us from
endogamous populations will get less reliable relationship predictions.
The Best Tool
The ideal relationship predictor would be tailored to your particular family structure, your population’s growth rate, your level of endogamy, and so on. Such a tool does not exist yet. (But stay
tuned!) In its absence, we would like to know which of the available tools gives most people the correct relationships most of the time.
The next post in this series will reveal how the tools did!
Learn More!
Whether you love math or hate it, you’ll get more out of genetic genealogy if you understand some basic concepts and how they apply to our DNA. Join me for my upcoming class “No One Told Me There
Would Be Math! DNA Numbers Made Easy.” It’s meant to be accessible to everyone, even if you haven’t done math since high school. The same lecture is offered on two dates/times, so sign up for the
one that bet fits your schedule.
Posts in This Series
8 thoughts on “The Science Behind Relationship Predictors”
1. Thank you for the elegant explanation. Family structure would seem to be the major factor making us misfits in the current models. I would love to contribute to research in that direction.
2. You say “your parent’s 1Cs had one child each” would count in the total number of 1C1R’s. However, the children of one of my parent’s 1st cousins are the same number of generations removed from
my grandparents and are therefore my 2nd cousins, correct? The only way to get a 1C1R is the children of your 1st cousins, although technically, if we were to treat 1st cousins the same way we do
other cousins, we could say we could refer to their parents as 1C1R as well, but we more commonly refer to them as aunts and uncles. 😉
1. Good catch! And a reminder that relationship prediction is often more complicated than a simplified description can capture. The children of your 1Cs are 1C1Rs, but so are your parent’s 1Cs.
I’ll edit the text to be a bit more clear.
3. > The most popular predictive tools are based on proprietary data from either AncestryDNA or MyHeritage, neither of which has publicly stated the population growth factor they use. However, it’s
safe to assume it’s around 2.5, which is a fairly standard fertility rate for developed countries over the past century.
You should be able to calculate this from public trees, given estimated birthdates for currently-living people, and maybe given family, regional / cultural, and generational variations.
4. A small pedantic correction: I think you are more likely to get a(n exact) 50-50 split from 10 coin tosses than 20 (~17.6% against ~12.5%, if I’ve done the sums right). With 2 coin tosses its 50%
of course. But we know what you mean: the distribution is “tighter” around 50-50 with increasing n.
1. Point taken!
5. Differentiating between the 3 bundled relationship categories within that 2nd (25%) degree of relatedness is what needs to be done to figure out how the testers are related. It can obviously be
done successfully the way you describe either with advanced calculation methods or with one of the good 3rd party tools available. Another way to differentiate the relationship categories that
belong to the 25% degree is by matching the generational separation between the two testers to the generational separation that the relationship categories describe. Grandparent Grandchild is a 2
generation separation category, Uncle-Aunt/Niece-Nephew is a 1 generation separation category and 1/2 sibling is a 0 generation separation category. In each degree of relatedness there won’t be
more than one category per number of generations separated. The calculators rank probability based on the number of people related in that category, the more people the more likely the testers
are to be related in that category which is a liklihood based on statistical probability. The alternate approach requires knowledge or an estimated guess at generational separation by the user to
select the category that aligns to their situation with the other tester. I think the latter approach has a better chance of turning out to be correct on the first try just because it involves
the user and makes them look at their specific situation in a way that no tool maker can anticipate. However as assisted reproduction becomes more and more prevalent (there goes your volume) our
perception of generational separation based on age alone are less and less likely to be correct. I think at the moment the user has a better chance of picking the correct category by knowing or
estimating how many generations separate the two testers than by looking at the chances based on the number of people possible in the category but our ability to guess at how many generations
separate testers based on their ages is going to keep declining until, at some point in the future statistical probability based on an estimated head count overtakes the simplicity of eliminating
the grandparent/grandchild category completely for two 20 year old testers and rolling the dice that the two 20 year old testers are 0 generation rather than 1 generation apart (because of course
both are possible but 0 generation of separation would be more likely if we go back to estimating by a % of the population method). A hybrid of the two methodologies might work well now and in
the future. The one thing that troubled me about the analogy of counting my own relatives in those categories to get a percentage of liklihood is that it does not take the fact that the other
tester calculating from their side might have completely different percentages of liklihood which means counting known relatives is the wrong thing to be counting. The known relatives need to be
excluded from the equation completely so that the focus is only on the unkown relationship vs those population factors you mention and of course the maximum path count (number of ways two people
can be related in those categories). Anyway there are a lot of ways to go about the same problem. Thanks for explaining how most of the tools work!
1. If you have a priori reasons to gauge which generation(s) the matches are in, that’s a great way of narrowing the field. That can be hard to tell, though. For example, my grandfather was one
of 14 children, with a spread of 25 years between the youngest and oldest. The spread in the next generation down is even bigger. An alternate way of going about it is to use multiple DNA
matches to start to hone in on a generation. There’s no one-size approach. We need to use all the tools in our toolbox!
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# programming mathematical methods for machine learning
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Combine VLOOKUP with SUMIF
- Written by Puneet
Yes, you can combine VLOOKUP and SUMIF. In SUMIF, there is a criteria argument in which you can use VLOOKUP to create a dynamic value. This will allow you to change the criteria by changing the
lookup value in the VLOOKUP. The use of this combo formula is unique.
In this tutorial, we will learn to combine SUMIF and VLOOKUP to create a formula.
Combine SUMIF and VLOOKUP
1. First, in a cell enter “=SUMIF(“, for the range argument, refer to the product ID range that you have in table1.
2. After that, in the second augment, you need to use the VLOOKUP function to lookup for the product ID by using the product name from the cell above.
3. Next, in the third argument of the SUMIF, refer to the quantity column to use as a sum_range.
4. In the end, enter the closing parentheses and hit enter to get the result.
How this Formula Works
Let’s break this formula into three parts: In the first part, you have specified the range where you have the product ID.
In the second part, you have the VLOOKUP that takes the product name from cell B15 and lookup it in the table2. For Headphones, we have the Product ID OT-356.
In the third part, we have the quantity column as sum_range.
In short, VLOOKUP helps you find the Product ID with the Product Name and then SUMIF takes that Product ID and looks it up in the Product ID column, and then sums values from the Quantity column.
As I said, when you use SUMIF and VLOOKUP, this makes your formula a dynamic formula. When you change the product name in the cells it changes the result.
SUMIF and VLOOKUP in Multiple Sheets
You can use this combination even when you have both tables in multiple sheets differently.
In the above example, you have the product name table on a different sheet.
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